Shaftesbury Road, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA
U N SA C O M R PL R E EC PA T E G D ES
477 Williamstown Road, Port Melbourne, VIC 3207, Australia
314β321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi β 110025, India 103 Penang Road, #05β06/07, Visioncrest Commercial, Singapore 238467
Cambridge University Press is part of Cambridge University Press & Assessment, a department of the University of Cambridge. We share the Universityβs mission to contribute to society through the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org
Β© David Greenwood, Bryn Humberstone, Justin Robinson, Jenny Goodman and Jennifer Vaughan 2016, 2019, 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. First published 2016
Second Edition 2019 Third Edition 2024
20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Cover designed by Denise Lane (Sardine) Typeset by diacriTech
Printed in China by C & C Offset Printing Co., Ltd.
A catalogue record for this book is available from the National Library of Australia at www.nla.gov.au
ISBN 978-1-009-48064-2
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 β10%βof 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 12, 66 Goulburn Street Sydney NSW 2000 Telephone: (02) 9394 7600 Facsimile: (02) 9394 7601 Email: memberservices@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 & Assessment 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 & Assessment does not guarantee the accuracy of such information thereafter.
Cambridge University Press & Assessment acknowledges the Aboriginal and Torres Strait Islander peoples of this nation. We acknowledge the traditional custodians of the lands on which our company is located and where we conduct our business. We pay our respects to ancestors and Elders, past and present. Cambridge University Press & Assessment is committed to honouring Aboriginal and Torres Strait Islander peoplesβ unique cultural and spiritual relationships to the land, waters and seas and their rich contribution to society.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
iii
Contents ix x xii xiii xiv xviii xx
Computation with positive integers
2
U N SA C O M R PL R E EC PA T E G D ES
About the authors Acknowledgements Introduction Guide to the working programs in exercises Guide to this resource Working with unfamiliar problems: Part 1 Working with unfamiliar problems: Part 2
1
1A 1B 1C 1D 1E 1F 1G
1H 1I
2
Place value in ancient number systems Place value in HinduβArabic numbers Adding and subtracting positive integers Algorithms for adding and subtracting Multiplying small positive integers Progress quiz Multiplying large positive integers Dividing positive integers Applications and problem-solving Estimating and rounding positive integers Order of operations with positive integers Modelling Technology and computational thinking Investigation Problems and challenges Chapter summary Chapter checklist with success criteria Chapter review
4 9 15 20 25 30 31 35 40 CONSOLIDATING 42 47 52 53 55 56 57 58 59
Number properties and patterns 2A 2B
2C 2D 2E 2F
2G 2H
Factors and multiples CONSOLIDATING Highest common factor and lowest common multipleβ Divisibility tests EXTENDING Prime numbers Using indices Prime decomposition Progress quiz Squares and square roots Number patterns CONSOLIDATING
62 64
Strands: Number
Algebra
Strands: Number
Algebra
69 74 79 82 88 92 93 99
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Contents
2I 2J 2K
Applications and problem-solving Spatial patterns Tables and rules The Cartesian plane and graphs Modelling Technology and computational thinking Investigation Problems and challenges Chapter summary Chapter checklist with success criteria Chapter review
105 107 115 121 128 129 131 132 133 135 137
U N SA C O M R PL R E EC PA T E G D ES
iv
3
Fractions and percentages 3A 3B 3C 3D 3E 3F 3G
3H 3I 3J 3K 3L
4
Introduction to fractions CONSOLIDATING 144 Equivalent fractions and simplified fractions 150 Mixed numerals and improper fractions CONSOLIDATING 157 Ordering fractions 163 Adding fractions 169 Subtracting fractions 175 Multiplying fractions 180 187 Progress quiz Dividing fractions 188 Fractions and percentages 194 Finding a percentage of a number 199 204 Applications and problem-solving Introduction to ratios 206 Solving problems with ratios 213 Modelling 218 Technology and computational thinking 219 Investigation 221 Problems and challenges 222 Chapter summary 223 Chapter checklist with success criteria 224 Chapter review 226
Algebraic techniques 4A 4B 4C 4D
142
Introduction to algebra Substituting numbers into algebraic expressions Equivalent algebraic expressions Like terms
230
Strands: Number
Measurement
Strand: Algebra
232 238 243 248
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Contents
4E
253 254 259 261 267 272 273 275 276 277 278 279
U N SA C O M R PL R E EC PA T E G D ES
4F 4G
Progress quiz Multiplying and dividing expressions Applications and problem-solving Expanding brackets EXTENDING Applying algebra Modelling Technology and computational thinking Investigation Problems and challenges Chapter summary Chapter checklist with success criteria Chapter review
v
5
Decimals 5A 5B 5C 5D 5E 5F 5G
5H 5I
6
282
Place value and decimals CONSOLIDATINGβ 284 Rounding decimals 289 Adding and subtracting decimals 294 Multiplying and dividing decimals by 10, 100, 1000 etc. 299 Multiplying decimals 305 Progress quiz 310 Dividing decimals 311 Connecting decimals and fractions 317 322 Applications and problem-solving Connecting decimals and percentages 324 Expressing proportions using decimals, fractions and percentages 329 Modelling 335 336 Technology and computational thinking Investigation 338 Problems and challenges 339 Chapter summary 340 Chapter checklist with success criteria 341 Chapter review 343
Semester review 1
346
Negative numbers
354
6A 6B 6C 6D
Strand: Number
Introduction to negative integers Adding and subtracting positive integers Adding and subtracting negative integers Multiplying and dividing integers EXTENDING
356 361 366 371
Strands: Number
Algebra
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Contents
6E 6F 6G
Progress quiz Order of operations with integers EXTENDING Applications and problem-solving Substituting integers EXTENDING The Cartesian plane Modelling Technology and computational thinking Investigation Problems and challenges Chapter summary Chapter checklist with success criteria Chapter review
376 377 381 383 386 391 393 395 396 397 398 399
U N SA C O M R PL R E EC PA T E G D ES
vi
7
Geometry 7A 7B 7C 7D 7E 7F
7G 7H 7I 7J 7K 7L
8
Points, lines, intervals and angles CONSOLIDATING Adjacent and vertically opposite angles Transversal and parallel lines Solving compound problems with parallel lines EXTENDING Classifying and constructing triangles Classifying quadrilaterals and other polygons Progress quiz Angle sum of a triangle Symmetry β Reflection and rotationβ Applications and problem-solving Translationβ Drawing solids Nets of solidsβ Modelling Technology and computational thinking Investigation Problems and challenges Chapter summary Chapter checklist with success criteria Chapter review
Statistics and probability 8A 8B 8C 8D
Collecting and classifying data Summarising data numerically Column graphs and dot plots Line graphs
402
404 414 421
Strands: Space Measurement
430 436 443 451 453 460 465 474 476 482 487 493 495 497 498 499 501 505
512 514 520 526 535
Strands: Statistics
Probability Algebra
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Contents
8E 8F
542 550 556 558 564 566
U N SA C O M R PL R E EC PA T E G D ES
8G
Stem-and-leaf plots Sector graphs and divided bar graphs EXTENDING Progress quiz Describing probability CONSOLIDATING Applications and problem-solving Theoretical probability in single-step experiments Experimental probability in single-step experimentsβ Modelling Technology and computational thinking Investigation Problems and challenges Chapter summary Chapter checklist with success criteria Chapter review
vii
8H 8I
9
Equations 9A 9B 9C 9D 9E 9F
9G 9H
10
Introduction to equations Solving equations by inspectionβ Equivalent equations Solving equations algebraically Progress quiz Equations with fractions EXTENDING Equations with brackets EXTENDING Applications and problem-solving Using formulas Using equations to solve problems Modelling Technology and computational thinking Investigation Problems and challenges Chapter summary Chapter checklist with success criteria Chapter review
Measurement 10A 10B 10C 10D 10E
Metric units of length CONSOLIDATING Perimeter CONSOLIDATING Circles, βΟβ and circumference Arc length and perimeter of sectors and composite shapes EXTENDING Units of area and area of rectangles
573 578 580 582 583 584 585 587
592
Strand: Algebra
594 599 604 609 616 617 622 627 629 634 638 640 642 643 644 645 646
650 652 659 665
Strands: Measurement Algebra
672 680
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Contents
10F 10G 10H 10I
Area of parallelograms Area of trianglesβ Progress quiz Area of composite shapes EXTENDING Volume of rectangular prisms Applications and problem-solving Volume of triangular prismsβ Capacity CONSOLIDATING Mass and temperature CONSOLIDATING Modelling Technology and computational thinking Investigation Problems and challenges Chapter summary Chapter checklist with success criteria Chapter review
687 693 698 700 707 713 715 723 728 733 734 736 737 738 739 741
U N SA C O M R PL R E EC PA T E G D ES
viii
10J 10K 10L
Semester review 2
11
746
Algorithmic thinking (online only)
Introduction Activity 1: Algorithms for working with data Activity 2: Adding consecutive numbers Activity 3: Turtle graphics
Index Answers
755 758
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
ix
About the authors David Greenwood is the Head of Mathematics at Trinity Grammar School in
U N SA C O M R PL R E EC PA T E G D ES
Melbourne and has 30+ yearsβ experience mathematics from Year 7 to 12. He is the lead author for the Cambridge Essential series and has authored more than 80 titles for the Australian Curriculum and for the syllabuses of the states and territories. He specialises in analysing curriculum and the sequencing of course content for school mathematics courses. He also has an interest in the use of technology for the teaching of mathematics.
Bryn Humberstone graduated from the University of Melbourne with an
Honours degree in Pure Mathematics, and has 20+ yearsβ experience teaching secondary school mathematics. He has been a Head of Mathematics since 2014 at two independent schools in Victoria. Bryn is passionate about applying the science of learning to teaching and curriculum design, to maximise the chances of student success.
Justin Robinson is co-founder of The Wellbeing Distillery, an organisation
committed to energising, equipping and empowering teachers. He consults with schools globally on the fields of student and educator wellbeing. Prior to this, Justin spent 25 years teaching mathematics, covering all levels of secondary education including teaching VCE, IB and A-Levels. His driving passion was engaging and challenging students within a safe learning environment. Justin is an Honorary Fellow of the University of Melbourneβs Graduate School of Education.
Jenny Goodman has taught in schools for over 28 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.
Jennifer Vaughan has taught secondary mathematics for over 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; hence, facilitating student confidence, achievement and an enjoyment of maths.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Acknowledgements
Stuart Palmer was born and educated in NSW. He is a fully qualified high
school mathematics teacher with more than 25 yearsβ experience teaching students from all walks of life in a variety of schools. He has been Head of Mathematics in two schools. He is very 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.
U N SA C O M R PL R E EC PA T E G D ES
x
Acknowledgements
The author and publisher wish to thank the following sources for permission to reproduce material: Cover: Β© Getty Images / Eloku
Images: Β© Getty Images / Daniel Rocal - PHOTOGRAPHY, p.xx / matt_scherf / Getty Images, p.xxi / Bloom image / Getty Images, p.xxii / DEV IMAGES, p.xxiii / Chapter 1 Opener / deberarr / whitemay, p.4 / Β© ClassicStock / Alamy, p.5 / clu, p.7 / PBNJ Productions / Getty Images, p.8 / peterfz30 / Getty Images, p.14 / AROON PHUKEED, p.15 / Mikael Vaisanen, p.18 / D4Fish, p.20 / Grant Faint, p.23(1) / Nenov, p.23(2) / Blend Images - Peathegee Inc, p.25 / gilaxia, p.28 / Daniela Dβanna / EyeEm / Getty Images, p.31 / PhotoAlles / Getty Images, p.32 / Carlos. E. Serrano, p.33 / Westend61, p.35 / Ron Levine, p.40(1) / Image Source, p.40(2) / Marc Dozier, p.41 / Kathrin Ziegler, p.42 / PETER PARKS / AFP / Gety Images, p.42 / Jason Edwards, p.46 / kali9, p.48 / Dennis Fischer Photography, p.50 / Stanislaw Pytel, p.52 / imagenavi, p.54 / Vertigo3d, p.56 / majorosl / Getty Images, p.60 / apomares, p.61 / Chapter 2 Opener / TonyFeder / AlexRaths, p.64 / blue jean images, p.67 / Cyndi Monaghan / Gty, p.69 / SolStock, p.72 / Flavio Coelho, p.73 / Paul Bradbury, p.74 / Traceydee Photography, p.77 / kali9, p.79 / DieterMeyrl, p.87 / Baran azdemir E+, p.91 / Paul Bradbury, p.97 / Monty Rakusen, p.99(1) / kasto80, p.99(2) / KTSDesign/SCIENCEPHOTOLIBRARY, p.103 / Kanok Sulaiman, p.104 / Mint Images, p.105(1) / Isabelle Rozenbaum, p.105(2) / Tatyana Tomsickova Photography, p.106 / Riou, p.144 / JohnnyGreig, p.107 / Naeblys, p.115 / Peshkova, p.121 / primeimages, p.127 / Qi Yang, p.130 / Chapter 3 Opener / undefined undefined / agrobacter, p.144 / Halfpoint Images, p.148 / Halfdark, p.149 / Peter Dazeley, p.150 / Anadolu Agency / Contributor, p.155 / Bill Varie, p.156 / Martin Poole, p.161 / BRETT STEVENS, p.162 / fotokostic, p.163 / Joff Lee, p.167 / Gregory Van Gansen, p.173 / Alys Tomlinson, p.174 / Petri Oeschger, p.185(1) / 10β000 Hours, p.185(2) / izusek, p.188 / kuritafsheen, p.192 / ArtistGNDphotography, p.193 / Paul Bradbury, p.194 / Yulia Naumenko, p.197 / John Lamb, p.198 / filo, p.198 / fotostorm, p.199 / Hiroshi Watanabe, p.202(1) / Tom Werner, p.202(2) / The Good Brigade, p.203 / gece33, p.204(1) / Marko Geber, p.204(2) / Willie B. Thomas, p.205 / Martin Poole, p.206(2) / MoMo Productions, p.209 / Luke Stanton, p.211 / Elyse Spinner, p.215 / ozgurdonmaz, p.216(1) / Arctic-Images, p.216(2) / Jonas Pattyn, p.217(1) / Westend61, p.218 / Callista Images, p.219 / Jackyenjoyphotography, p.229 / Chapter 4 Opener / Pencho Chukov / Danny Lehman, p.263 / Maskot, p.237 / Monty Rakusen, p.239 / Andrew Merry, p.243 / XiXinXing, p.248 / Hinterhaus Productions, p.251 / Imgorthand, p.252 / andresr, p.255 / adventtr, p.259(1) / mrs, p.259(2) / Davidf, p.260 / irina88w, p.261 / Jonathan Kirn, p.266 / JasonDoiy, p.267 / Teera Konakan, p.272 / Yuichiro Chino, p.273 / Comezora, p.274 / Tao Xu, p.275 / Maskot, p.280 / Frank Rothe, p.281 / Chapter 5 Opener / moodboard / Andrew Brookes, p.284 / Malkovstock, p.289 / simonkr, p.292 / South_agency, p.293(1) / Goodboy Picture Company, p.293(2) / Ariel Skelley, p.294 / Andrew Fox, p.297 / simoncarter, p.298 / Monty Rakusen, p.299 / florintt, p.303 / photovideostock, p.305 / microgen, p.305 / sestovic, p.311 / Mitchell Gunn/Getty Images, p.311 / wihteorchid, p.315(1) / VioletaStoimenova, p.315(2) / South_agency, p.316 / Chris Sattlberger, p.317 / sankai, p.320 / PM Images, p.321 / Sadeugra, p.322(1) / ANDRZEJ WOJCICKI, p.322(2) / Stanislaw Pytel, p.323 / kajakki, p.324 / PeopleImages, p.327(1) / MarianneBlais, p.327(2) / brizmaker / Getty Images, p.329 / fstop123, p.333 / fotog, p.334 / Candy Pop Images, p.338 / Chapter 6 Opener / kavram / mrcmos, p.356 / Stephen Frink, p.360 / JazzIRT, p.361 / Icy Macload, p.364 / matejmo, p.365 / Bruce Yuanyue Bi, p.366 / Fajrul Islam, p.376 / dstephens, p.377 / Photo taken by Kami (Kuo, Jia-Wei), p.379 / George Lepp, p.381 / ultramarinfoto, p.382 / je33a, p.383 / deepblue4you, p.385 / PASIEKA, p.390 / Tim Platt, p.391 / supersizer, p.392 / xavierarnau, p.393 / damircudic, p.395 / guvendemir, p.396 / Ashley Cooper, p.401 / Chapter 7 Opener / 35007 / hatman12, p.404 / M_Arnold, p.406 / MaxCab, p.420 / Xinzheng, p.421 / Image Source, p.429 / Dan Reynolds Photography, p.430 / DLMcK, p.436 / imageBROKER/Helmut Meyer zur Capellen, p.442 / BrianScantlebury, p.443 / Tempura, p.453 / photovideostock, p.460(1) / Natchaphol chaiyawet, p.460(2) / Jessica Peterson, p.463 / Oscar Wong, p.475 / DigtialStorm, p.475 / Jung Getty, p.481 / hh5800, p.482 / Hero Images, p.494 / Chapter 8 Opener / Steve Bell / VCG, p.515 / Stringer / simpson33, p.516 / Artie Photography (Artie Ng), p.519 / Nitat Termmee, p.520 / Vertigo3d, p.521 / swilmor, p.522 / skynesher, p.523 / vm, p.526 / Phil Fisk, p.527 / simonkr, p.537 / Daly and Newton, p.548 / mixetto, p.549 / Nick Rains, p.550 / Walter Geiersperger, p.551 / Justin Paget, p.558 / Oliver Strewe, p.559 / Kaori Ando, p.654 / Matteo Colombo, p.566(1) / Aaron Foster, p.566(2) / ferrantraite, p.567 / Kutay Tanir, p.569 / Tetra Images, p.573 / shekhardino, p.575 (1) / Monty Rakusen, p.575(2) / Sandra Clegg, p.579 / Federico Campolattano / 500px, p.580 / Tobias Titz, p.581 / Sally Anscombe, p.582 / Pete Saloutos, p.583 /
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Acknowledgements
U N SA C O M R PL R E EC PA T E G D ES
Zero Creatives, p.584 / Svetlana Repnitskaya, p.592 / Thomas Barwick, p.593 / Chapter 9 Opener / TommL / SCIEPRO, p.596 / Juan Silva, p.601 / d3sign, p.605(1) / Ekaterina Goncharova, p.605(2) / Enrico Calderoni/Aflo, p.612 / Feverpitched, p.619 / Luis Alvarez, p.622 / Vithun Khamsong, p.629(1) / Maskot, p.629(2) / Vladimir Godnik, p.630 / PeopleImages, p.631 / Katrin Ray Shumakov, p.634 / James Braund, p.635(1) / Cameron Spencer, p.635(2) / OrnRin, p.636 / Michael Schmitt, p.640 / Clerkenwell, p.641 / William Fawcett, p.642 / Hero Images, p.643 / Trina Dopp Photography, p.644 / d3sign, p.650 / Chapter 10 Opener / RugliG / -Oxford-, p.652 / Darryl Peroni, p.656(1) / LUCKY gilhare / 500px, p.656(4) / Cyndi Monaghan / Gty, p.656(5) / AlexLMX, p.656(6) / Xiaodong Qiu, p.657 / Feng Wei Photography, p.658 / DavidCallan, p.659 / majorosl / Getty Images, p.664 / Elena Popova, p.670 / tzahiV, p.680 / valentyn semenov / 500px, p.685(1) / BanksPhotos, p.685(2) / Seb Oliver, p.687(1) / John C Magee, 687(2) / Allan Baxter, p.692 / rabbit75_ist, p.693 / Roy Rochlin / Contributor, p.706 / Rost-9D, p.707 / Richard Drury, p.713(1) / Jackyenjoyphotography, p.713(2) / wellsie82, p.71 / Matthias Kulka, p.722 / Matt Anderson Photography, p.723 / xbrchx, p.728(1) / Danita Delimont, p.728(2) / Jonathan Pow, p.732 / PASIEKA, p.734 / Wendy Cooper, p.736 / Jupiterimages, p.745; / TED 43 / Creative Commons Attribution 3.0 Unported license, p.98.
xi
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.
Β© Extracts from the VCE F-10 study design (2023-2027) reproduced by permission; Β© VCAA. VCE is a registered trademark of the VCAA. The VCAA does not endorse or make any warranties regarding this study resource. Current VCE Study Designs and related content can be accessed directly at www.vcaa.vic.edu.au. Readers are also advised to check for updates and amendments to VCE Study Designs on the VCAA website and via the VCAA Bulletin and the VCAA Notices to Schools.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
xii
Introduction
U N SA C O M R PL R E EC PA T E G D ES
The third edition of Essential Mathematics for the Victorian Curriculum has been significantly revised and updated to suit the teaching and learning of Version 2.0 of the Victorian Curriculum. Many of the established features of the series have been retained, but there have been some substantial revisions, improvements and new elements introduced for this edition across the print, digital and teacher resources.
New content and some restructuring New topics have come in at all year levels. In Year 7 there are new lessons on ratios, volume of triangular prisms, and measurement of circles, and all geometry topics are now contained in a single chapter (Chapter 4). In Year 8, there are new lessons on 3D-coordinates and techniques for collecting data. For Year 9, error in measurement is new in Chapter 5, and sampling and proportion is introduced in Chapter 9. In Year 10, four lessons each on networks and combinatorics have been added, and there are new lessons on logarithmic scales, rates of change, two-way tables and cumulative frequency curves and percentiles. The Year 10 book also covers all the 10A topics from Version 2.0 of the curriculum. This content can be left out for students intending to study General Mathematics, or prioritised for students intending to study Mathematical Methods or Specialist Mathematics. Version 2.0 places increased emphasis on investigations and modelling, and this is covered with revised Investigations and Modelling activities at the end of chapters. There are also many new elaborations covering First Nations Peoplesβ perspectives on mathematics, ranging across all six content strands of the curriculum. These are covered in a suite of specialised investigations provided in the Online Teaching Suite. Other new features β’ Technology and computational thinking activities have been added to the end of every chapter to address the curriculumβs increased focus on the use of technology and the understanding and application of algorithms. β’ Targeted Skillsheets β downloadable and printable β have been written for every lesson in the series, with the intention of providing additional practice for students who need support at the basic skills covered in the lesson, with questions linked to worked examples in the book. β’ Editable PowerPoint lesson summaries are also provided for each lesson in the series, with the intention of saving the time of teachers who were previously creating these themselves.
Diagnostic Assessment tool Also new for this edition is a flexible, comprehensive Diagnostic Assessment tool, available through the Online Teaching Suite. This tool, featuring around 10,000 new questions, allows teachers to set diagnostic tests that are closely aligned with the textbook content, view student performance and growth via a range of reports, set follow-up work with a view to helping students improve, and export data as needed.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
xiii
Guide to the working programs in exercises
U N SA C O M R PL R E EC PA T E G D ES
The suggested working programs in the exercises in this book provide three pathways to allow differentiation for Growth, Standard and Advanced students (schools will likely have their own names for these levels). Each exercise is structured in subsections that match the mathematical proficiencies of Fluency, Problem-solving and Reasoning, as well as Enrichment (Challenge). (Note that Understanding is covered by βBuilding understandingβ in each lesson.) In the exercises, the questions suggested for each pathway are listed in three columns at the top of each subsection: β’ β’ β’
Growth FLUENCY
Standard
Advanced
1, 2, 3(Β½), 4
2, 3(Β½), 4, 5(Β½)
3(Β½), 4, 5(Β½)
PROBLEM-SOLVING 6
6, 7
7, 8
9, 10
10, 11
REASONING 9
ENRICHMENT: Adjusting concentration
β β 12 The left column (lightest shaded colour) is the Growth The working program for Exercise 3A in Year 7. pathway The questions recommended for a Growth The middle column (medium shaded colour) is the Standard student are: 1, 2, 3(1/2), 4, 6 and 9. See note below. pathway The right column (darkest shaded colour) is the Advanced pathway.
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 β where there is an increasing level of mathematical sophistication required from Fluency to Problem-solving to Reasoning and Enrichment β but also within each proficiency; 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 Growth pathway should use the left tab, which includes all but the hardest Fluency questions as well as the easiest Problem-solving and Reasoning questions. An Advanced student can use the right tab, proceed through the Fluency questions (often half of each question), and have their main focus be on the Problem-solving and Reasoning questions, as well as the Enrichment questions. A Standard student would do a mix of everything using the middle tab.
Choosing a pathway There are a variety of ways to determine the appropriate pathway for students through Note: The nomenclature used to list questions is as follows: the course. Schools and individual teachers should follow the method that works for β’ β3β, 4β β: complete all parts of them. If required, the prior-knowledge pre-tests (now found online) can be used as a questions 3 and 4 tool for helping students select a pathway. The following are recommended guidelines: β’ β1β-β4:β complete all parts of β’ β’ β’
A student who gets 40% or lower should complete the Growth questions A student who gets above 40% and below 85% should complete the Standard questions A student who gets 85% or higher should complete the Advanced questions.
β’ β’ β’ β’
questions 1, 2, 3 and 4 ββ10β(ββΒ½β)ββββ: complete half of the parts from question 10 (a, c, e, ... or b, d, f, ...) β2β-ββ4β(ββΒ½β)ββββ: complete half of the parts of questions 2, 3 and 4 ββ4β(ββΒ½β)ββ, 5ββ: complete half of the parts of question 4 and all parts of question 5 β: do not complete any of the questions in this section.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
xiv
Guide to this resource PRINT TEXTBOOK FEATURES NEW New lessons: authoritative coverage of new topics in the Victorian Curriculum 2.0 in the form of
U N SA C O M R PL R E EC PA T E G D ES
1
new, road-tested lessons throughout each book.
2 Victorian Curriculum 2.0: content strands and content descriptions are listed at the beginning of the chapter (see the teaching program for more detailed curriculum documents) 3 In this chapter: an overview of the chapter contents
4 Working with unfamiliar problems: a set of problem-solving questions not tied to a specific topic
5 Chapter introduction: sets context for students about how the topic connects with the real world and the history of mathematics 6 Learning intentions: sets out what a student will be expected to learn in the lesson 7 Lesson starter: an activity, which can often be done in groups, to start the lesson 8 Key ideas: summarises the knowledge and skills for the lesson
9 Building understanding: a small set of discussion questions to consolidate understanding of the Key ideas (replaces Understanding questions formerly inside the exercises) 10 Worked examples: solutions and explanations of each line of working, along with a description that clearly describes the mathematics covered by the example 6
170
8
Chapter 2 Geometry and networks
2K Introduction to networks
2K Introduction to networks
KEY IDEAS
β A network or graph is a diagram connecting points using lines.
LEARNING INTENTIONS β’ To know what is meant by a network graph β’ To know the key features of a network graph β’ To be able to find the degree of a vertex and the sum of degrees for a graph β’ To be able to describe simple walks through a network using the vertex labels
5
7
171
β’ β’
The points are called vertices (or nodes). Vertex is singular, vertices is plural. The lines are called edges.
β The degree of a vertex is the number of edges connected to it.
β’ β’
A network is a collection of points (vertices or nodes) which can be connected by lines (edges). Networks are used to help solve a range of real-world problems including travel and distance problems, intelligence and crime problems, computer network problems and even metabolic network problems associated with the human body. In Mathematics, a network diagram can be referred to as a graph, not to be confused with the graph of a function like y = x 2 + 3.
A vertex is odd if the number of edges connected to it is odd. A vertex is even if the number of edges connected to it is even.
β The sum of degrees is calculated by adding up the degrees of all
the vertices in a graph. β’ It is also equal to twice the number of edges.
A
B
β A walk is any type of route through a network.
D
β’ β’
Example: A-B-C-A-D.
Lesson starter: The Konigsberg bridge problem
BUILDING UNDERSTANDING
The seven bridges of Konigsberg is a well-known historical problem solved by Leonhard Euler who laid the foundations of graph theory. It involves two islands at the centre of an old German city connected by seven bridges over a river as shown in these diagrams.
1 Here is a graph representing roads connecting three towns A, B and C. a How many different roads (edges) does this graph have? b How many different vertices (nodes) does this graph have? c If no road (edge) is used more than once and no town
B
A
C
D
The problem states: Is it possible to start at one point and pass over every bridge exactly once and return to your starting point? β’
crosses all bridges exactly once. Try starting at different places.
C
(vertex/node) is visited more than once, how many different walks are there if travelling from: i A to C? ii A to B? iii B to C? d How many roads connect to: i town A? ii town B? iii town C?
2 This graph uses four edges to connect four vertices. a How many edges connect to vertex A? b What is the degree of vertex A? c State the total number of edges on the graph. d of degrees for the graph.
A
C
B
9
A
B C
D
e What do you notice about the total number of edges and the sum of degrees for this graph?
β’ β’
Investigate if there might be a solution to this problem if one of the bridges is removed. Investigate if there might be a solution to this problem if one bridge is added.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Guide to this resource
11 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
442
Chapter 5
10
Quadratic expressions and equations
Example 13 Simplifying algebraic fractions involving quadratic expressions 2
2
25x β 10x + 1 . 4x β 9 Γ _____________ Simplify _____________ 10x 2 + 13x β 3 10x 2 β 17x + 3 SOLUTION
11
12 Gentle start to exercises: the exercise begins at Fluency, with the first question always linked to the first worked example in the lesson
xv
E X P L A NAT I O N 25x 2 β 10x + 1
4x 2 β 9
_____________ Γ _____________
10x 2 + 13x β 3 10x 2 β 17x + 3 (5x β 1)1(5x β 1)1 (2x + 3)1(2x β 3)1 _______________ = _______________ Γ (2x + 3)1(5x β 1)1 (2x β 3)1(5x β 1)1 =1
Cancel to simplify.
U N SA C O M R PL R E EC PA T E G D ES
12
First, use the range of factorising techniques to factorise all quadratics.
Now you try
2
2
16x β 24x + 9 . 9x β 4 Γ _____________ Simplify _____________ 12x 2 β 17x + 6 12x 2 β x β 6
Exercise 5E
13 Working programs: differentiated question sets for three ability levels in exercises
FLUENCY
14
Example 12a
Example 12b
Factorise the following. a 3x 2 + 10x + 3
2 Factorise the following. a 2x 2 β 11x + 5 e 7x 2 + 2x β 5 i 2x 2 β 9x β 5 m 6x 2 + x β 12 q 8x 2 β 14x + 5
1, 2(1/2)
b 2x 2 + 3x + 1 b f j n r
5x 2 + 2x β 3 2x 2 β 9x + 7 13x 2 β 7x β 6 10x 2 + 11x β 6 8x 2 β 26x + 15
3 Factorise the following. a 18x 2 + 27x + 10 d 30x 2 + 13x β 10 g 24x 2 β 38x + 15
13
14 Example references: show where a question links to a relevant worked example β the first question is always linked to the first worked example in a lesson
1
1β3(1/3)
d 3x 2 β 5x + 2
3x 2 β 11x β 4 3x 2 + 2x β 8 5x 2 β 22x + 8 6x 2 + 13x + 6 6x 2 β 13x + 6
c g k o s
b 20x 2 + 39x + 18 e 40x 2 β x β 6 h 45x 2 β 46x + 8
PROBLEMβSOLVING
1β3(1/4)
c 3x 2 + 8x + 4
3x 2 β 2x β 1 2x 2 + 5x β 12 8x 2 β 14x + 5 4x 2 β 5x + 1 9x 2 + 9x β 10
d h l p t
c 21x 2 + 22x β 8 f 28x 2 β 13x β 6 i 25x 2 β 50x + 16
4(1/2), 5
4(1/3), 5, 6(1/3)
4(1/3), 5, 6(1/3)
4
a 6x 2 + 38x + 40 d 32x 2 β 88x + 60 g β 50x 2 β 115x β 60
b 6x 2 β 15x β 36 e 16x 2 β 24x + 8 h 12x 2 β 36x + 27
c 48x 2 β 18x β 3 f 90x 2 + 90x β 100 i 20x 2 β 25x + 5
15 Problems and challenges: in each chapter provides practice with solving problems connected with the topic 16 Chapter checklist with success criteria: a checklist of the learning intentions for the chapter, with example questions
740
Chapter 10
Measurement
Chapter checklist
Chapter checklist and success criteria
17 Applications and problem-solving: a set of three extended- 16 response questions across two pages that give practice at applying the mathematics of the chapter to real-life contexts
A printable version of this checklist is available in the Interactive Textbook
10E
β
10. I can find the area of squares and other rectangles. e.g. Find the area of this rectangle.
4 mm
10 mm
10G
12. I can find the area of triangles. e.g. Find the area of this triangle.
9 cm
10 cm
10H
13. I can find the area of composite shapes. e.g. Find the area of this composite shape.
Ext
4 cm
6 cm
10I
14. I can find the volume of rectangular prisms (cuboids). e.g. Find the volume of this cuboid.
10J
15. I can find the volume of triangular prisms. e.g. Find the volume of this triangular prism.
10K
16. I can convert between units of capacity. e.g. Convert 500 mL to litres.
10K
17. I can convert between volumes and capacities. e.g. A container has a volume of 2000 cubic centimetres. Find its capacity in litres.
10L
18. I can convert between units of mass. e.g. Convert 2.47 kg to grams.
4 cm
8 cm
18 NEW Technology and computational thinking activity in each chapter addresses the curriculum's increased focus on the use of different forms of technology, and the understanding and implementation of algorithms
3 cm
2 cm
5 cm
4 cm
19 Modelling activities: an activity in each chapter gives students the opportunity to learn and apply the mathematical modelling process to solve realistic problems
Modelling
18
733
Estimating park lake area
Chapter 1
Reviewing number and financial mathematics
Technology and computational thinking
Start
Key technology: Spreadsheets
People will often use an online tax calculator to get an idea of how much income tax they will have to pay each year. Such calculators will use an in-built algorithm to help determine the tax rates depending on the given income. An added factor is the Medicare levy which is added on to most tax liabilities.
Input I
Yes
Is I β©½ 18200?
T=0
We will use the following income tax rates for this investigation. Taxable income
Tax on this income
0β$18 200
Nil
$18 201β$45 000
$45 001β$120
Is I β©½ 45000?
19 cents for each $1 over $18 200
$5092 plus 32.5 cents for each $1 over $45 000
$120 001β$180 000
$29 467 plus 37 cents for each $1 over $120 000
over
$51 667 plus 45 cents for each $1 over $180 000
$180 001
No
Yes
T = 0.19 Γ (I β 18200)
No
Yes
The Medicare levy is 2% of taxable income.
1 Getting started
T = 5092 + 0.325 Γ (I β 45000)
No
a Find the amount of tax paid, not including the Medicare levy, for the following taxable incomes. i $15 000 ii $32 000 iii $95 000 iv $150 000 v $225 000 b Calculate the total tax payable on the following taxable incomes, including the Medicare levy, which is 2% of your taxable income. i $60 000 ii $140 000
Present a report for the following tasks and ensure that you show clear mathematical workings and explanations where appropriate.
Technology and computational thinking
Technology and computational thinking
Coding tax
A local shire wants to estimate the volume of water in its local lake. The birdsβ eye view of the lake is shown here and each grid square represents 10 metres by 10 metres. The average depth of water in the lake is 3 metres.
85
Yes
Preliminary task
a Estimate the area of the shaded region in the following grids. Each grid square represents an area of one square metre. ii i
19
b A body of water has a surface area of 2000 m2. The volume of water can be found using the formula: Volume = Surface Area Γ Depth. If the average depth of the water is 3 i the volume of water in cubic metres ii the volume of water in litres. Use the fact that 1 m3 = 1000 L.
No
2 Applying an algorithm
Modelling task
T = T + 0.02 Γ I
a b Explain what the formula T = T + 0.02I c
Formulate
1 b above.
Output T
3 Using technology
Solve
End
nested IF statements.
4 Extension
a A different tax system might use the following table. Find the values of a, b and c.
a Try entering the formulas into a spreadsheet. Be careful to put all the brackets and commas in the correct positions. b Test your spreadsheet calculator by entering different taxable incomes studied earlier in this investigation. c i $67 000 ii $149 000 iii $310 000
Taxable income
Tax on this income
0β$25 000
Nil
$25 001β$60 000
20 cents for each $1 over $25 000
$60 001β$150 000
$a plus 30 cents for each $1 over $60 000
$150 001β$250 000
$b plus 40 cents for each $1 over $150 000
$250 001 and over
$c plus 50 cents for each $1 over $250 000
The Medicare levy is 2% of taxable income. b Try adjusting your tax calculator for this tax system and test a range of taxable incomes to ensure it is working correctly.
Modelling
84
Evaluate and verify
Communicate
a The problem is to estimate the volume of water in the shire lake. Write down all the relevant information that will help solve the problem. b Outline your method for estimating the surface area of the lake shown above. c Estimate the area of the lake in square metres. Explain your method, showing any calculations.
d Estimate the volume of water in the lake. Give your answer in both m3 and in litres. e Compare your results with others in your class and state the range of answers provided. f g h could use to obtain a better estimate of the volume.
Extension questions a b Compare how your estimates and methods would change if the lakeβs depth is no longer assumed to be a constant 3 metres.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Guide to this resource
20 Chapter reviews: with short-answer, multiple-choice and extended-response questions; questions that are extension are clearly signposted 21 Solving unfamiliar problems poster: at the back of the book, outlines a strategy for solving any unfamiliar problem
U N SA C O M R PL R E EC PA T E G D ES
xvi
INTERACTIVE TEXTBOOK FEATURES
22 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
23 Workspaces: almost every textbook question β including all working-out β can be completed inside the Interactive Textbook by using either a stylus, a keyboard and symbol palette, or uploading an image of the work 24 Self-assessment: students can then self-assess their own work and send alerts to the teacher. See the Introduction on page xii for more information.
25 Interactive working programs can be clicked on so that only questions included in that working program are shown on the screen
26 HOTmaths resources: a huge catered library of widgets, HOTsheets and walkthroughs seamlessly blended with the digital textbook 27 A revised set of differentiated auto-marked practice quizzes per lesson with saved scores
28
30
27
28 Scorcher: the popular competitive game
29 Worked example videos: every worked example is linked to a high-quality video demonstration, supporting both in-class learning and the flipped classroom
26
30 Desmos graphing calculator, scientific calculator and geometry tool are always available to open within every lesson
29
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Guide to this resource
xvii
31 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
U N SA C O M R PL R E EC PA T E G D ES
32 Auto-marked prior knowledge pre-test for testing the knowledge that students will need before starting the chapter 33 Auto-marked progress quizzes and chapter review multiple-choice questions in the chapter reviews can now be completed online
DOWNLOADABLE PDF TEXTBOOK
34 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.
34
ONLINE TEACHING SUITE
35 NEW 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
36 NEW PowerPoint lesson summaries contain the main elements of each lesson in a form that can be annotated and projected in front of class
37 Learning Management System with class and student analytics, including reports and communication tools
38 Teacher view of studentβs work and self-assessment allows the teacher to see their classβs workout, how students in the class assessed their own work, and any βred flagsβ that the class has submitted to the teacher 39 Powerful test generator with a huge bank of levelled questions as well as ready-made tests
40 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 41 Worksheets and four differentiated chapter tests in every chapter, provided in editable Word documents
42 More printable resources: all Pre-tests, Progress quizzes and Applications and problem-solving tasks are provided in printable worksheet versions Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
xviii
The questions on the next four pages are designed to provide practice in solving unfamiliar problems. Use the βWorking with unfamiliar problemsβ poster at the back of this book to help you if you get stuck.
For questions 1β3, try starting with smaller numbers and look for a pattern.
U N SA C O M R PL R E EC PA T E G D ES
Part 1
Working with unfamiliar problems: Part 1
In Part 1, apply the suggested strategy to solve these problems, which are in no particular order. Clearly communicate your solution and final answer. 1
How many diagonals exist for a β7β-sided regular polygon? How many diagonals can be drawn from one vertex of a β30β-sided regular polygon?
2 Find the value of β11111111βsquared.
3 Find the sum of the first β25βodd numbers.
For questions 4 and 5, try making a list or table.
4 Five students have entered a race. How many different arrangements are there for first, second and third place, given that nobody ties? 5 Arrange all the digits 1, 2, 3, 4 and 5 into the form Γ so that the β3β-digit number multiplied by the β2β-digit number gives the largest possible answer. 6 A tree surgeon charges β$15βto cut a log into β4βpieces. How much would he charge, at the same rate, to cut a log into β99βpieces?
For questions 6β8, draw a labelled diagram to help you visualise the problem.
7 How many β2β-digit numbers can be written using only the digits 0, 1, 2, 3 or 4 with no repetition?
8 An β8β-sided star is formed by drawing an equilateral triangle on each side of a square. Find the obtuse angle formed by adjacent sides of the star. 9 Approximately how many planes are needed to carry β76819β people if each plane holds β289βpeople? Give your answer to the nearest β10β planes.
10 Approximately how many times does the word βtheβ appear in this book? 11 Approximately how far would you have walked, in km, after walking β10 000β steps?
For question 9, try estimating by rounding the values in the question.
For question 10 and 11, try working with a smaller sample first.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Working with unfamiliar problems: Part 1
U N SA C O M R PL R E EC PA T E G D ES
Part 1
12 Insert operation signs between the digits β1 2 3 4 5 6 7 8 9βto make an answer of β100β. The digits must be kept in ascending order. a Use any of the four operations and keep the digits separate. b Use only addition and subtraction. Also digits may be combined to form a β2β- or β3β-digit number.
xix
For questions 12 and 13, try using a formula or rule to find a shortcut to the answer.
For question 14, try using algebra as a tool: define the pronumerals, form an equation and then solve it.
13 A glass fish tank is a rectangular prism of width β40 cmβand length β1 mβ. A scale is to be marked on the side of the tank showing β10β-litre increases in volume. How far apart should the scale markings be? If the tank is to hold β280βlitres of water and β 5 cmβheight is allowed above the water level, what is the height of the fish tank?
14 Divide β$410βbetween Bob, Zara and Ahmed so that Bob gets β$40βmore than Zara and Zara has β$20βmore than Ahmed.
15 A sailor has a cat, a mouse and a chunk of cheese that he needs to get across the lake in his boat. The boat is very small and it can only hold the sailor and one passenger, or the cheese, at a time. The other problem the sailor faces is that if he leaves the mouse and the cat alone, the cat will eat the mouse and if he leaves the cheese with the mouse alone, the cheese will get eaten. How many trips are needed for the sailor, the cat, the mouse and the cheese to arrive safely on the other side of the lake?
For question 15, try using concrete, everyday materials to represent the problem.
16 Ethan takes β6βdays to paint a house, Jack takes β8βdays to paint a house and Noah takes β12βdays to paint a house. Exactly how many days would it take to paint a house if all three of them worked together?
For question 16, try applying one or more mathematical procedures, such as a rule for adding fractions.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
xx
For the questions in Part 2, again use the βWorking with unfamiliar problemsβ poster at the back of this book, but this time choose your own strategy (or strategies) to solve each problem. Clearly communicate your solution and final answer.
U N SA C O M R PL R E EC PA T E G D ES
Part 2
Working with unfamiliar problems: Part 2
1
Maddie remembered her friendβs house number has two digits and that they added to β8β. In ascending order, list the possible house numbers that fit this description.
2 Find the smaller angle between the big hand and little hand of a clock at 2 p.m. and also at 7:10 a.m. 3 Find each β2β-digit number, from β12βto β40β, that has the sum of its factors greater than double the number.
4 Using grid paper, draw all the possible different arrangements of β5βequally sized connecting squares. Rotations and reflections donβt count as separate arrangements. Which of your arrangements would fold up to make an open box? Mark the base square for these arrangements. 5 How many prime numbers are less than β100β?
6 At the end of a soccer match, each member of the two teams of β11βplayers wishes to shake hands with everyone else who played. How many handshakes are needed?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Working with unfamiliar problems: Part 2
xxi
7 What is the smallest number that has each of the numbers β1βto β9βas a factor?
U N SA C O M R PL R E EC PA T E G D ES
9 What is the 2018th digit after the decimal point in the decimal expansion of _ ββ4 ββ? 7
Part 2
8 A game involves rolling two β8β-sided dice. The numbers shown on both dice are β1βto β 8βinclusive. How many different totals are possible in this game?
10 Approximately how many β20βcent coins are needed so that when placed next to each other they cover a β1β-metre square? Give your answer to the nearest β100β. What is their value in dollars? 11 A triangle has one angle double the smallest angle and the other angle β25Β°βless than double the smallest angle. Find the size of the smallest angle in this triangle. 12 What is the last digit in the number β3203β?
13 Find the interior angle sum of a β42β-sided regular polygon.
14 How many palindromic numbers are there that are greater than β100βand less than β 1000β? (A palindrome is the same when written forwards and backwards.)
15 In a message that is written in secret code, what letter of the alphabet would likely be represented by the most common symbol in the message? 16 How many squares of any size are on a chess board?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
U N SA C O M R PL R E EC PA T E G D ES
1 Computation with positive integers
Maths in context: Whole numbers in the ancient world and now Various number symbols and systems have been used over thousands of years. The Mayan people lived in Central America from 1500 BCE to the 16th century CE. They counted in 20s rather than 10s and their simplest number system used a shell for zero, a pebble for one unit, and a stick for ο¬ve units. For example: Number 20s units
20 29 103 (1 Γ 20 + 0) (1 Γ 20 + 9) (5 Γ 20 + 3)
Here are some everyday situations where whole numbers are used now.
β’ Number skills are needed in algebra, which is used in numerous occupations, including coding. β’ At a parkrun each runner is allocated two whole numbers: their ID bar code and a ο¬nishing place bar code. These are linked to the placement times (whole number minutes and seconds) and results uploaded onto the parkrun website. β’ Builders calculate the number of pavers required around a swimming pool. β’ The number of votes in an election for class representatives are tallied.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter contents Place value in ancient number systems Place value in HinduβArabic numbers Adding and subtracting positive integers Algorithms for adding and subtracting Multiplying small positive integers Multiplying large positive integers Dividing positive integers Estimating and rounding positive integers
U N SA C O M R PL R E EC PA T E G D ES
1A 1B 1C 1D 1E 1F 1G 1H
(CONSOLIDATING)
1I
Order of operations with positive integers
Victorian Curriculum 2.0
This chapter covers the following content descriptors in the Victorian Curriculum 2.0: NUMBER
VC2M7N02, VC2M7N08, VC2M7N10 ALGEBRA
VC2M7A02
Please refer to the curriculum support documentation in the teacher resources for a full and comprehensive mapping of this chapter to the related curriculum content descriptors.
Β© VCAA
Online resources
A host of additional online resources are included as part of your Interactive Textbook, including HOTmaths content, video demonstrations of all worked examples, auto-marked quizzes and much more.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 1
Computation with positive integers
1A Place value in ancient number systems LEARNING INTENTIONS β’ To understand that there are different number systems that have been used historically in different cultures β’ To be able to write numbers in the Egyptian number system β’ To be able to write numbers in the Babylonian number system β’ To be able to write numbers in the Roman number system
U N SA C O M R PL R E EC PA T E G D ES
4
Throughout the ages and in different countries, number systems were developed and used to help people count and communicate with numbers. From the ancient Egyptians to the modern day, different systems have used pictures and symbols to represent whole numbers. In Australia, the most commonly used number system is the HinduβArabic system (using the digits 0β9), often called the decimal system.
Lesson starter: Count like a Roman
Here are the letters used in the Roman number system for some numbers that you know.
β’
β’
Number
1
2
3
4
5
6
Roman numerals
I
II
III
IV
V
VI
Number
7
8
9
10
50
100
Roman numerals
VII
VIII
IX
X
L
C
The Roman numerals on this old milestone in Norfolk, England, show the distances in miles to nearby villages.
What numbers do you think XVII and XIX represent? Can you write the numbers 261 and 139 using Roman numerals?
KEY IDEAS
β Egyptian number system
β’ β’ β’ β’
Records show that this number system was used from about 3000 bce. Hieroglyphics were used to represent numbers. From about 1600 bce, hieroglyphics were used to represent groups of 10, 100, 1000 etc. Symbols of the same type were grouped in twos or threes and arranged vertically. Number
1
10
100
1000
10 000
100 000
1 000 000
Stick or staff
Arch or heel bone
Coil of rope
Lotus ο¬ower
Bent ο¬nger or reed
Tadpole or frog
Genie
Hieroglyphic
Description
β’
Examples: 3 5 21
342
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1A Place value in ancient number systems
β’ β’
5
Note that the hieroglyphics with the larger value are written in front (i.e. on the left). There was no symbol for the number zero.
β Babylonian number system
From about 1750 bce the ancient Babylonians used a relatively sophisticated number system and its origins have been traced to about 3000 bce. Symbols called cuneiform (wedge shapes) were used to represent numbers. The symbols were written into clay tablets, The Hanging Gardens of Babylon, built for which were then allowed to dry in the sun. his wife by King Nebuchadnezzar II around The number system is based on the number 600 BCE, were one of the seven wonders of 60, but a different wedge shape was used to the ancient world. represent groups of 10. The system is positional Number 1 10 60 in that the position of each Symbol wedge shape helps determine Description Upright Sideways Upright its value. So means 2 but wedge shape wedge wedge shape means 62. To represent zero, they used a blank space or sometimes a small slanted wedge shape for zeros inside a number. Examples: 5 11 72 121
U N SA C O M R PL R E EC PA T E G D ES
β’
β’ β’ β’
β’
β’ β’
β Roman number system
β’ β’
β’
β’
The Roman number system uses capital letters to indicate numbers. These are called Roman numerals. The Roman number system was developed in about the third century bce and remained the dominant system in many parts of the world until about the Middle Ages. It is still used today in many situations. A smaller letter value to the left of a larger letter value indicates subtraction. For example, IV means 5 β 1 = 4 and XC means 100 β 10 = 90. Only one letter can be placed to the left for subtraction. I, X and C are the numerals that can be used to reduce the next two larger numerals. So X, for example, can be used to reduce L and C but not D. Number
1
5
10
50
100
500
1000
Symbol
I
V
X
L
C
D
M
Examples: 2 4 21 59 II IV XXI LIX
90 XC
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 1
Computation with positive integers
BUILDING UNDERSTANDING 1 Which number system uses these symbols? a cuneiform (wedge shapes); e.g. b capital letters; e.g. V and L c hieroglyphics (pictures); e.g. and
U N SA C O M R PL R E EC PA T E G D ES
6
2 Write these numbers in the Egyptian number system. a β1β b β10β c β100β
d β1000β
3 Write these numbers in the Babylonian number system. a β1β b β10β c β60β 4 Write these numbers in the Roman number system. a β1β b β5β c β10β
d β50β
e β100β
5 In the Roman system, IV does not mean β1 + 5βto give β6β. What do you think it means?
Example 1
Using ancient number systems
Write each of the numbers β3β, β15βand β144βusing the given number systems. a Egyptian b Babylonian c Roman SOLUTION
EXPLANATION
a
3β β
means β1β
β15β
means β10β
1β 44β
means β100β
3β β
means β1β
1β 5β
means β10β
β144β
means β60β
b
c
β3β
III
I means β1β
β15β
XV
V means β5β X means β10β
β144β
CXLIV
C means β100β XL means β40β IV means β4β
Now you try
Write each of the numbers 4, 23 and 142 using the given number systems. a Egyptian b Babylonian c Roman
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1A Place value in ancient number systems
7
Exercise 1A FLUENCY
1
1β6
4β6
Write the numbers 14 and 131 using the Egyptian number system.
U N SA C O M R PL R E EC PA T E G D ES
Example 1a
1β5
Example 1b
2 Write the numbers 14 and 131 using the Babylonian number system.
Example 1c
3 Write the numbers 14 and 131 using the Roman number system.
Example 1
4 Write these numbers using the given number systems. a Egyptian i β3β ii β21β b Babylonian i β4β ii β32β c Roman i β2β ii β9β 5 What number do these groups of symbols represent? a Egyptian i iii b Babylonian i
iii β114β
iv β352β
iii β61β
iv β132β
iii β24β
iv β156β
ii iv ii
iii
iv
c Roman i IV iii XVI
ii VIII iv XL
6 Work out the answer to each of these problems. Write your answer using the same number system that is given in the question. a XIV β+β XXII b β c
β
PROBLEM-SOLVING
7 In ancient Babylon, a person adds
d DCLXIX β+β IX
7, 8
goats to another group of
7β9
8β10
goats.
How many goats are there in total? Write your answer using the Babylonian number system.
8 An ancient Roman counts the number of people in three queues. The first queue has XI, the second has LXII and the third has CXV. How many people are there in total? Write your answer using the Roman number system.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 1
Computation with positive integers
9 One Egyptian house is made from stones and a second house is made from stones. How many more stones does the first house have? Write your answer using the Egyptian number system. 10 Which number system (Egyptian, Babylonian or Roman) uses the least number of symbols to represent these numbers? a 55 b 60 c 3104
U N SA C O M R PL R E EC PA T E G D ES
8
REASONING
11(1/2)
11(1/2), 12
11(1/2), 12, 13
11 In the Roman system, Is, Xs and Cs are used to reduce either of the next two larger numerals. So 9 is IX, not VIIII; and 49 is XLIX, not IL. Also, only one numeral can be used to reduce another number. So 8 is VIII, not IIX. Use this to write these numbers using Roman numerals. a 4 b 9 c 14 e 29 f 41 g 49 i 99 j 449 k 922
d 19 h 89 l 3401
12 The Egyptian system generally uses more symbols to represent a number compared to other number systems. Can you explain why? In the Egyptian system, how many symbols are used for the number 999?
13 In the Babylonian system, stands for 1, but because they did not use a symbol for zero at the end of a number, it also represents 60. People would know what it meant, depending on the situation it was used. Here is how it worked for large numbers. The dots represent empty spaces. 1
60 3600 ..... ..... ..... a Write these numbers using the Babylonian system. i 12 ii 72 iii 120 iv 191 b Can you explain why . . . . . . . . . . represents 3600? c What would . . . . . . . . . . . . . . . represent?
ENRICHMENT: Other number systems
β
v 3661
β
vi 7224
14
14 Other well-known number systems include: a Mayan b modern Chinese c ancient Greek. Look up these number systems on the internet or in other books. Write a brief sentence covering the points below. i When and where the number systems were used. ii What symbols were used. iii Examples of numbers using these symbols.
An ancient Mayan carving.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1B Place value in HinduβArabic numbers
9
1B Place value in HinduβArabic numbers
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand how place value works in the HinduβArabic (decimal) number system β’ To be able to identify the place value of digits in different numbers β’ To be able to convert between basic numerals and their expanded form β’ To be able to compare two positive integers by considering the digits and their place value
The commonly used number system today, called the decimal system or base 10, is also called the HinduβArabic number system. Like the Babylonian system, the value of the digit depends on its place in the number, but only one digit is used in each position. A digit for zero is also used. The decimal system originated in ancient India about 3000 bce and spread throughout Europe through Arabic texts over the next 4000 years.
The famous βHistorie de la Mathematiqueβ, a French document showing the history of the HinduβArabic number system over thousands of years.
Lesson starter: Largest and smallest
Without using decimal points, repeated digits or a zero (0) at the start of a number, see if you can use all the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to write down: β’ β’
the largest possible number the smallest possible number.
Can you explain why your numbers are, in fact, the largest or smallest possible?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 1
Computation with positive integers
KEY IDEAS β The HinduβArabic or decimal system uses base β10β. This means powers of β10β (β1β, β10β, β100β, β1000β,
...) are used to determine the place value of a digit in a number. β Indices can be used to write powers of β10β.
U N SA C O M R PL R E EC PA T E G D ES
10
β10 = 10β β β1β β100 = 10β β 2β β β1000 = 10β β 3β β etc.
β The symbols β0, 1, 2, 3, 4, 5, 6, 7, 8βand β9βare called digits.
β Whole numbers greater than zero are called positive integers: β1, 2, 3, 4, β¦β
β The value of each digit depends on its place in the number. The place value of the digit β2βin the
number β126β, for example, is β20β.
β The basic numeral 3254 can be written in expanded form as β3 Γ 1000 + 2 Γ 100 + 5 Γ 10 +β β
4 Γ 1β.
thousands hundreds
tens
ones
3 2 5 4 = 3 Γ 1000 + 2 Γ 100 + 5 Γ 10 + 4 Γ 1 expanded form
β Numbers can be written in expanded form using indices:
β 2β β+ 5 Γ 10β β 1β β+ 4 Γ 1β β3254 = 3 Γ 10β β 3β β+ 2 Γ 10β
β Symbols used to compare numbers include the following:
β=β(is equal to)
β1 + 3 = 4β
or
β10 β 7 = 3β
β β(is not equal to) β
β1 + 3 β 5β
or
β11 + 38 β 50β
> β β(is greater than) β©Ύ β β(is greater than or equal to)
5β > 4β β5 β©Ύ 4β
or or
1β 00 > 37β β4 β©Ύ 4β
< β β(is less than) β©½ β β(is less than or equal to)
4β < 5β β4 β©½ 5β
or or
1β 3 < 26β β4 β©½ 4β
or
β8997 ββββ9000β
β β βor ββββ(is approximately equal to) β4.02 β 4β Λ
Λ
BUILDING UNDERSTANDING
1 Choose one of the words βonesβ, βtensβ, βhundredsβ or βthousandsβ to describe the 1 in each number. a β100β
b β1000β
c β10β
d β1β
2 Which number using digits (next to the capital letters) matches the given numbers written in words? A β10 001β E β7421β
B 2β 63β F β3615β
C 3β 6 015β G β2036β
D 7β 040 201β H β100 001β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1B Place value in HinduβArabic numbers
a b c d
11
two hundred and sixty-three seven thousand four hundred and twenty-one thirty-six thousand and fifteen one hundred thousand and one
3 Which symbol (next to the capital letters) matches the given words? A β=β
b is less than
B ββ β
U N SA C O M R PL R E EC PA T E G D ES a is not equal to
c is greater than or equal to
C β>β
d is equal to
D ββ©Ύβ
e is greater than
E β<β
f
is less than or equal to
F ββ©½β
g is approximately equal to
G βββ
4 State whether each of these statements is true or false. a β5 > 4β b β6 = 10β c β9 β 99β d β1 < 12β e β22 β©½ 11β f β126 β©½ 126β g β19 β©Ύ 20β h β138 > 137β
Example 2
Finding place value
Write down the place value of the digit 4 in these numbers. a β437β b β543 910β SOLUTION
EXPLANATION
a β4 Γ 100 = 400β
4β βis worth β4 Γ 100β β3βis worth β3 Γ 10β β7βis worth β7 Γ 1β
b β4 Γ 10 000 = 40 000β
5β βis worth β5 Γ 100 000β β4βis worth β4 Γ 10 000β β3βis worth β3 Γ 1000β β9βis worth β9 Γ 100β β1βis worth β1 Γ 10β
Now you try
Write down the place value of the digit 7 in these numbers. a β72β b β87 159β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 1
Computation with positive integers
Example 3
Writing numbers in expanded form
a Write β517βin expanded form.
b Write β25 030βin expanded form.
SOLUTION
EXPLANATION
a 5β 17 = 5 Γ 100 + 1 Γ 10 + 7 Γ 1β
Write each digit separately and multiply by the appropriate power of β10β.
b 25 030 = 2 Γ 10 000 + 5 Γ 1000 + 0 Γ 100 + 3 Γ 10 + 0 Γ 1 β―β―β―β―β― β β β β β = 2 Γ 10 000 + 5 Γ 1000 + 3 Γ 10
Write each digit separately and multiply by the appropriate power of 10. Remove any term where zero is multiplied.
U N SA C O M R PL R E EC PA T E G D ES
12
Now you try
a Write β2715βin expanded form.
Example 4
b Write β40 320βin expanded form.
Writing numbers in expanded form with index notation
Write the following numbers in expanded form with index notation. a 7β 050β b β32 007β SOLUTION
EXPLANATION
a 7050 = 7 Γ 10 + 5 Γ 10 3
1
7β 050βincludes β7β thousands β 3β β)ββ, β0βhundreds, β5β tens (β β7 Γ 1000 = 7 Γ 10β
β(β5 Γ 10 = 5 Γ 10β β β1β)βand β0β ones.
b 32007 = 3 Γ 104 + 2 Γ 103 + 7 Γ 1
The place value of the β3β is 30 000 = 3 Γ 10 000 ββ―β― β β β β = 3 Γ 10β β 4β β
The place value of the β2β is 2000 = 2 Γ 1000 β β β β β = 2 Γ 10β β 3β β
The place value of the β7β is β7 = 7 Γ 1β.
Now you try
Write the following numbers in expanded form with index notation. a 3β 70β b β20 056β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1B Place value in HinduβArabic numbers
13
Exercise 1B FLUENCY
1
2β7β(β 1β/2β)β
2β8β(β 1β/2β)β
Write down the place value of the digit β4βin these numbers. a β943β b β7450β
U N SA C O M R PL R E EC PA T E G D ES
Example 2
1, 2β6ββ(1β/2β)β
Example 2
Example 2
Example 3
Example 3
Example 4a
Example 4b
2 Write down the place value of the digit β7βin these numbers. a β37β b β71β c β379β e 1β 712β f β7001β g β45 720β
d 7β 04β h β170 966β
3 Write down the place value of the digit β2βin these numbers. a 1β 26β b β2143β c β91 214β
d β1 268 804β
4 Write these numbers in expanded form. a β17β b β281β
c β935β
d β20β
5 Write these numbers in expanded form. a β4491β b β2003β
c β10 001β
d β55 555β
6 Write the following numbers in expanded form with index notation. a β3080β b β450β c β90 030β
d 4β 7 500β
7 Write the following numbers in expanded form with index notation. a 4β 2 009β b β3604β c β245β
d 7β 00 306β
8 Write these numbers, given in expanded form, as a basic numeral. a 3β Γ 100 + 4 Γ 10 + 7 Γ 1β b β9 Γ 1000 + 4 Γ 100 + 1 Γ 10 + 6 Γ 1β c β7 Γ 1000 + 2 Γ 10β d β6 Γ 100 000 + 3 Γ 1β e β4 Γ 1 000 000 + 3 Γ 10 000 + 7 Γ 100β f β9 Γ 10 000 000 + 3 Γ 1000 + 2 Γ 10β PROBLEM-SOLVING
9
9ββ(1β/2β)β,β 10
10, 11
9 Arrange these numbers from smallest to largest. a β55, 45, 54, 44β b β729, 29, 92, 927, 279β c 2β 3, 951, 136, 4β d β435, 453, 534, 345, 543, 354β e 1β 2 345, 54 321, 34 512, 31 254β f 1β 010, 1001, 10 001, 1100, 10 100β
10 How many numbers can be made using the given digits? Digits are not allowed to be used more than once and all digits must be used. a β2, 8βand β9β b β1, 6βand β7β c β2, 5, 6βand β7β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 1
Computation with positive integers
11 You are given three different non-zero digits, for example: 2, 5 and 8. How many three-digit numbers can be formed from your three given digits if digits can be used more than once? (For example, 522 and 825 are both possible if the original digits were 2, 5 and 8.) REASONING
12
12
12, 13
12 The letters used here represent the digits of a number. Write each one in expanded form. For example, 7A2 means 7 Γ 100 + A Γ 10 + 2 Γ 1. a AB b ABCD c A0000A
U N SA C O M R PL R E EC PA T E G D ES
14
13 By considering some of the other number systems (Egyptian, Babylonian or Roman) explained in the previous section, describe the main advantages of the HinduβArabic (decimal) system. ENRICHMENT: Very large numbers
β
β
14
14 It is convenient to write very large numbers in expanded form with index notation. Here is an example. 50 000 000 = 5 Γ 10 000 000 = 5 Γ 107 a Explain why it is convenient to write large numbers in this type of expanded form. b 3200 can also be written in the form 32 Γ 102. All the non-zero digits are written down and then multiplied by a power of 10. Similarly, write each of these numbers in the same way. i 4100 ii 370 000 iii 21 770 000 c Write each of these numbers as basic numerals. ii 7204 Γ 103 iii 1028 Γ 106 i 381 Γ 102 d Write these numbers in expanded form, just as you did in the examples above. Research them if you do not know what they are. i 1 million ii 1 billion iii 1 trillion iv 1 googol v 1 googolplex
In 2008 in Zimbabwe, bank notes were issued in trillions of dollars, but soon became worthless due to inο¬ation.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1C Adding and subtracting positive integers
15
1C Adding and subtracting positive integers
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand the commutative and associative laws for addition β’ To be able to use the mental strategies partitioning, compensating, doubling/halving to calculate a sum or difference of positive integers mentally
The process of finding the total value of two or more numbers is called addition. The words βplusβ, βaddβ and βsumβ are also used to describe addition. The process for finding the difference between two numbers is called subtraction. The words βminusβ, βsubtractβ and βtake awayβ are also used to describe subtraction.
Lesson starter: Your mental strategy
Many problems that involve addition and subtraction can be solved mentally without the use of a calculator or complicated written working. Consider
A welder who is to join several lengths from a steel bar uses addition to calculate the total length, and uses subtraction to ο¬nd the length of the steel that will be left over.
98 + 22 β 31 + 29
How would you work this out? What are the different ways it could be done mentally? Explain your method.
KEY IDEAS
β The symbol + is used to show addition or find a sum.
+3
e.g. 4 + 3 = 7
3 4 5 6 7 e.g. 4 + 3 = 3 + 4 This is the commutative law for addition, meaning that the order does not matter.
β a+b=b+a
β’
8
β a + (b + c) = (a + b) + c
β’
e.g. 4 + (11 + 3) = (4 + 11) + 3 This is called the associative law for addition, meaning that it does not matter which pair is added first.
β The symbol β is used to show subtraction or find a difference.
β2
e.g. 7 β 2 = 5
β a β b β b β a (in general)
e.g. 4 β 3 β 3 β 4
β a β (b β c) β (a β b) β c (in general)
4
5
6
7
8
e.g. 8 β (4 β 2) β (8 β 4) β 2
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 1
Computation with positive integers
β Mental addition and subtraction can be done using different strategies.
β’
β’
Partitioning (grouping digits in the same position) 171 + 23 = 100 + (β 70 + 20)β+ (β 1 + 3)β β―β―β― β β β β β = 194 Compensating (making a β10, 100βetc. and then adjusting or compensating by adding or subtracting) 46 + 9 = 46 + 10 β 1 β ββ―β― β β β = 55 Doubling or halving (making a double or half and then adjusting with addition or subtraction) 75 + 78 = 75 + 75 + 3 124 β 61 = 124 β 62 + 1 ββ―β― β =β 62 + 1β β β―β― β =β 150 + 3β β β β = 63 β = 153
U N SA C O M R PL R E EC PA T E G D ES
16
β’
BUILDING UNDERSTANDING
1 a Give three other words that can mean the same as addition. b Give three other words that can mean the same as subtraction.
2 State the number which is: a β3βmore than β7β c β7βless than β19β
b 5β 8βmore than β11β d β137βless than β157β
3 a State the sum of β19βand β8β. b State the difference between β29βand β13β.
4 State whether each of these statements is true or false. a β4 + 3 > 6β b β11 + 19 β©Ύ 30β d β26 β 15 β©½ 10β e β1 + 7 β 4 β©Ύ 4β 5 Give the result for each of the following. a β7βplus β11β c the sum of β11βand β21β e β36βtake away β15β
Example 5
c 1β 3 β 9 < 8β f β50 β 21 + 6 < 35β
b 2β 2βminus β3β d β128βadd β12β f the difference between β13βand β4β
Using mental strategies for addition and subtraction
Use the suggested strategy to mentally work out the answer. a 1β 32 + 156β (partitioning) b β25 + 19β (compensating) c β56 β 18β (compensating) d β35 + 36β(doubling or halving) SOLUTION
EXPLANATION
a 1β 32 + 156 = 288β
100 + 30 + 2 β―β― β____________ β β100 + 50 + 6 β β―β― 200 + 80 + 8
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1C Adding and subtracting positive integers
b β25 + 19 = 44β
U N SA C O M R PL R E EC PA T E G D ES
c 5β 6 β 18 = 38β
25 + 19 = 25 + 20 β 1 β ββ―β― β =β 45 β 1β β = 44 56 β 18 = 56 β 20 + 2 β ββ―β― β =β 36 + 2β β = 38 35 + 36 = 35 + 35 + 1 ββ―β― β β =β 70 + 1β β = 71
17
d β35 + 36 = 71β
Now you try
Use the suggested strategy to mentally work out the answer. a β512 + 284β (partitioning) b β76 + 98β (compensating) c 4β 2 β 19β (compensating) d β75 + 73β(doubling or halving)
Exercise 1C FLUENCY
Example 5a
1
1β5ββ(β1/2β)β,β 6
1β5β(β β1/2β)β,β 7
2β5ββ(β1/2β)β,β 7ββ(β1/2β)β
Mentally find the answers to these sums. (Hint: Use the partitioning strategy.) a β23 + 41β b β71 + 26β c β138 + 441β d β246 + 502β e β937 + 11β f β1304 + 4293β
2 Mentally find the answers to these differences. (Hint: Use the partitioning strategy.) a β29 β 18β b β57 β 21β c β249 β 137β d β1045 β 1041β e β4396 β 1285β f β10 101 β 100β
Example 5b
Example 5c
Example 5d
Example 5
3 Mentally find the answers to these sums. (Hint: Use the compensating strategy.) a β15 + 9β b β64 + 11β c β19 + 76β d β18 + 115β e β31 + 136β f β245 + 52β
4 Mentally find the answers to these differences. (Hint: Use the compensating strategy.) a β35 β 11β b β45 β 19β c β156 β 48β d β244 β 22β e β376 β 59β f β5216 β 199β
5 Mentally find the answers to these sums and differences. (Hint: Use the doubling or halving strategy.) a β25 + 26β b β65 + 63β c β121 + 123β d β240 β 121β e β482 β 240β f β1006 β 504β 6 Use the suggested strategy to mentally work out the answer. a β123 + 145β (partitioning) b β36 + 29β (compensating) c 4β 7 β 28β (compensating) d β55 + 56β(doubling or halving) 7 Mentally find the answers to these mixed problems. a β11 + 18 β 17β b β37 β 19 + 9β d β136 + 12 β 15β e β28 β 10 β 9 + 5β g β1010 β 11 + 21 β 1β h β5 β 7 + 2β
c 1β 01 β 15 + 21β f β39 + 71 β 10 β 10β i β10 β 25 + 18β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 1
Computation with positive integers
PROBLEM-SOLVING
8β10
9β12
11β13
8 Gary worked β7βhours on Monday, β5βhours on Tuesday, β13βhours on Wednesday, β11βhours on Thursday and β2β hours on Friday. What is the total number of hours that Gary worked during the week?
U N SA C O M R PL R E EC PA T E G D ES
18
9 In a batting innings, Phil hit β126βruns and Mario hit β19β runs. How many more runs did Phil hit compared to Mario? 10 A farmer reduced his cattle numbers from β86βto β54β. How many cows were taken away?
11 Bag A has β18βmarbles and bag B has β7βfewer marbles than bag A. What is the total number of marbles?
12 Matt has β36βcards and Andy has β35βmore cards than Matt. If they combine their cards, how many do they have in total? 13 Each side on a magic triangle adds up to the same number, as shown in this example with a sum of β12βon each side. a Place each of the digits from β1βto β6βin a magic triangle with three digits along each side so that each side adds up to the given number. i β9β ii β10β
4
12
12
3
2
5
1
6
12
b Place each of the digits from β1βto β9βin a magic triangle with four digits along each side so that each side adds up to the given number. i β20β ii β23β
REASONING
14
14, 15
15, 16
14 a The mental strategy of partitioning is easy to apply for β23 + 54βbut harder for β23 + 59β. Explain why. b The mental strategy of partitioning is easy to apply for β158 β 46βbut harder for β151 β 46β. Explain why. 15 Complete these number sentences if the letters βa, bβand βcβrepresent numbers. a aβ + b = c so c β β ββββββ= aβ b βa + c = b so b β a = ββ βββββ
16 This magic triangle uses the digits β1βto β6β, and has each side adding to the same total. This example shows a side total of β9β. a How many different side totals are possible using the same digits? b Explain your method.
1
6
2
5
4
3
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1C Adding and subtracting positive integers
ENRICHMENT: Magic squares
β
β
19
17, 18
17 A magic square has every row, column and main diagonal adding to the same number, called the magic sum. For example, this magic square has a magic sum of 15.
9
2
15
3
5
7
15
U N SA C O M R PL R E EC PA T E G D ES
Find the magic sums for these squares, then fill in the missing numbers.
4 8
1
6
15
15 15 15 15 15
a
b
6 7 2
5
c
10
11
15 20
13
14
12
19
d
1
15
4
6
9
11
13
2
16
18 The sum of two numbers is 87 and their difference is 29. What are the two numbers?
This magic square was known in ancient China as a βLo Shuβ square and uses only the numbers 1 to 9. It is shown in the middle of this ancient design as symbols on a turtle shell, surrounded by the animals which represent the traditional Chinese names for the years.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 1
Computation with positive integers
1D Algorithms for adding and subtracting LEARNING INTENTIONS β’ To understand that algorithms can be used to add or subtract numbers in cases where mental strategies are ineffective β’ To be able to apply the addition algorithm to ο¬nd the sum of positive integers β’ To be able to apply the subtraction algorithm to ο¬nd the difference of positive integers
U N SA C O M R PL R E EC PA T E G D ES
20
It is not always practical to solve problems involving addition and subtraction mentally. For more complex problems a procedure involving a number of steps can be used and this helps to give the answer. Such a procedure is called an algorithm.
For the addition algorithm, if two digits add to more than 9, then the higher place value digit in the sum can be carried to the next column. For example, 10 ones can be renamed as 1 ten and carried into the tens column.
For the subtraction algorithm, if a larger digit is to be subtracted from a smaller digit in the ones column, 1 of the tens from the tens column can be renamed to form an extra 10 ones. This can be repeated where necessary in other place value columns.
A householdβs monthly power usage is found by subtracting consecutive end-of-month kWh (kilowatt-hour) readings, as displayed on its electricity meter. Adding these monthly amounts gives the power used per year.
Lesson starter: The missing digits
Discuss what numbers could go in the empty boxes. Give reasons for your answers.
1
4
+ 9 5
1
2
5
β1
4
9
5
4
KEY IDEAS
β An algorithm is a procedure involving a number of steps that eventually leads to the answer to
a problem.
β Addition algorithm
β’ β’ β’
Arrange the numbers vertically so that the digits with similar place value are in the same column. Add digits in the same column, starting on the right. If the digits add to more than 9, carry the β10β to the next column.
1 234 4 + 2 = 6 + 192 3 + 9 = 12 426 1 + 2 + 1 = 4
β Subtraction algorithm
β’
Arrange the numbers vertically so that the digits with similar place value are in the same column.
12 159
β1 82 77
9β2=7 15 β 8 = 7 1β1=0
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1D Algorithms for adding and subtracting
β’ β’
21
Subtract digits in the same column top-down and starting on the right. If a larger digit is to be subtracted from a smaller digit in the ones column, β1βof the tens from the tens column can be renamed to form an extra β10βones. This can be repeated where necessary in other place value columns.
U N SA C O M R PL R E EC PA T E G D ES
β Calculators could be used to check your answers.
BUILDING UNDERSTANDING
1 Mentally find the results to these simple sums. a β87 + 14β b β99 + 11β
c β998 + 7β
d β52 + 1053β
2 Mentally find the results to these simple differences. a β36 β 9β b β100 β 16β c β37 β 22β
d β1001 β 22β
3 What is the missing number in these problems? a b 3 6 4 6
d
+1 5 5
Example 6
+
1 1
c
4
6 7 β4 8
0
9
1 4 β 6 8
2 0
Using the addition algorithm
Give the result for each of these sums. 26 a β ββ + 66 ____
439 b β β + 172 _
SOLUTION
EXPLANATION
a
Add the digits vertically. β6 + 6 = 12β, so rename β10βof the β12βones as β1βten and carry this into the tens column.
126
+66 92
b
2 3 9
14139
+1 72 6 11
9β + 2 = 11β, so rename β10βof the β11βones as β1βten and carry this into the tens column. β1 + 3 + 7 = 11β, so rename β10βof these β11β tens as β1βhundred and carry this extra β1β hundred into the hundreds column.
Now you try
Give the result for each of these sums. 38 a β β + 54 _
276 b β β + 459 _
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
22
Chapter 1
Computation with positive integers
Example 7
Using the subtraction algorithm
b β 3240β β 2721 _
U N SA C O M R PL R E EC PA T E G D ES
Give the result for each of these differences. a β 74β β 15 _ SOLUTION
EXPLANATION
a
6 7 14
Take β1βof the tens from β7βtens and rename it as β10βones to make β 14 β 5 = 9β. Then subtract β1βten from β6βtens (not β7β tens).
2 3 12 34 10
Take β1βof the tens from β4βtens and rename it as β10βones to make β 10 β 1 = 9β. Subtract β2βfrom β3β(not β4β). Take β1βof the thousands from β3βthousands and rename it as β10βhundreds to make β12β hundreds. β12 hundreds β 7 hundreds = 5 hundreds. β Note that β2 thousands β 2 thousands = 0 thousands βand you do not need to show a β0βbefore the β5β.
β1 5 59
b
β2 7 2 1 5 1 9
Now you try
Give the result for each of these differences. a β 85β β 39 _
b β 5470β β 3492 _
Exercise 1D FLUENCY
Example 6
1
1, 2β5ββ(β1/2β)β
284 b β β + 178 _
2 Give the answer to each of these sums. Check your answer with a calculator. 17 74 36 a β β c ββ+ 24ββ b β+ 25β β+ 51β _ _ _ 129 e β β + 97 _
Example 6
3β6β(β β1/2β)β
Give the result for each of these sums. 17 a β β + 34 _
Example 6
2β6β(β β1/2β)β
f
β 458β + 287 _
1041 β g β + 882 _
47 d β β + 39 _
3092 h β β + 1988 _
3 Show your working to find the result for each of these sums. a β85 + 76β
b 1β 31 + 94β
c β1732 + 497β
d β988 + 987β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1D Algorithms for adding and subtracting
23
4 Give the result for each of these sums. 17 a β 26β + 34 _
126 b β 47β + 19 _
152 c β 247β + 19 _
2197 d β 1204β + 807 _
e β946 + 241 + 27 + 9β
f
U N SA C O M R PL R E EC PA T E G D ES
β1052 + 839 + 7 + 84β
Example 7
5 Find the answers to these differences. Check your answer with a calculator. a β 54β c β 46β b β 85β β 23 β 27 β 65 _ _ _ e β 125β β 89 _
Example 7
f
β 241β β 129 _
g β 358β β 279 _
h β 491β β 419 _
6 Show your working to find the answer to each of these differences. a β32 β 16β b β124 β 77β c β613 β 128β PROBLEM-SOLVING
7, 8
d β 94β β 36 _
d β1004 β 838β
8β10
9β11
7 Farmer Green owns β287β sheep, Farmer Brown owns β526βsheep and Farmer Grey owns β1041βsheep. How many sheep are there in total?
8 A carβs odometer shows β12 138β kilometres at the start of a journey and β12 714βkilometres at the end of the journey. How far was the journey?
9 Two different schools have β871βand β 950βstudents enrolled. a How many students are there in total? b Find the difference in the number of students between the schools.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 1
Computation with positive integers
10 Find the missing numbers in these sums. a
b
3
1 +
+5 3
c
4
+
7
1
9
1
4
7
9 1
4
U N SA C O M R PL R E EC PA T E G D ES
24
11 Find the missing numbers in these differences. a
b
6
β2 8
2
β
4
c
5
3
8 8
β
1
REASONING
12
2
9
2
1
6
12, 13
5
12, 13
12 a Work out the answer to these simple problems. i β28 + 18 β 17β ii β36 β 19 + 20β b For part a i, is it possible to work out β18 β 17βand then add this total to β28β? c For part a ii, is it possible to work out β19 + 20βand then subtract this total from β36β? d Can you suggest a good mental strategy for part a ii above that gives the correct answer? 13 a What are the missing digits in this sum? b Explain why there is more than one possible set of missing numbers in the sum given opposite. Give some examples. ENRICHMENT: More magic squares
14 Complete these magic squares. a 62 67 60 65
β
b
β
101
2
3
+
4
2
1
14β16
114
106
109
110
113 103 102 116
15 The sum of two numbers is β978βand their difference is β74β. What are the two numbers? 16 Make up some of your own problems like Question 15 and test them on a friend.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1E Multiplying small positive integers
25
1E Multiplying small positive integers
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand the commutative and associative laws for multiplication β’ To be able to use mental strategies to ο¬nd products β’ To be able to apply the multiplication algorithm to ο¬nd the product of a single digit number by a positive integer
The multiplication of two numbers represents a repeated addition.
For example, 4 Γ 2 could be thought of as 4 groups of 2 or 2 + 2 + 2 + 2.
Similarly, 4 Γ 2 could be thought of as 2 groups of 4 or 2 Γ 4 or 4 + 4.
4Γ2
To estimate a patientβs heart rate in beats per minute, a nurse counts the number of heart beats in 15 seconds and multiplies that number by 4.
2Γ4
Lesson starter: Are these mental strategies correct? Three students explain their method for finding the answer to 124 Γ 8. β’ β’ β’
Billy says that you can do 124 Γ 10 to get 1240, then subtract 2 to get 1238. Lea says that you halve 124 and 8 twice each to give 31 Γ 2 = 62. Surai says that you multiply 8 by 4 to give 32, 8 by 2 to give 16 and 8 by 1 to give 8. She says the total is therefore 32 + 16 + 8 = 56.
Are any of the students correct and can you explain any errors in their thinking?
KEY IDEAS
β Finding the product of two numbers involves multiplication. We say βthe product of 2 and 3 is 6β. β aΓb=bΓa
β’
e.g. 2 Γ 3 = 3 Γ 2 This is the commutative law for multiplication, meaning that the order does not matter.
β (a Γ b) Γ c = a Γ (b Γ c)
β’
e.g. (3 Γ 5) Γ 4 = 3 Γ (5 Γ 4) This is the associative law for multiplication, meaning it does not matter which pair is multiplied first.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 1
Computation with positive integers
β The multiplication algorithm for multiplying by a single digit
involves: β’ Multiplying the single digit by each digit in the other number, starting from the right. β’ Carrying and adding any digits with a higher place value to the total in the next column.
β1β23 β β β―β― β_ βΓ β4β 4β Γ 3 = 12β β92 4 Γ 20 + 10 = 90
U N SA C O M R PL R E EC PA T E G D ES
26
β Mental strategies for multiplication include:
β’ β’
β’
β’
β’
β’
Knowing your multiplication tables off by heart. β9 Γ 7 = 63 12 Γ 4 = 48β Using the commutative law by changing the order. For example, β43 Γ 2βmight be thought of more easily as β2β groups of β43βor β2 Γ 43β. Using the commutative and associative law by altering the 5 Γ 11 Γ 2 = 5 Γ 2 Γ 11 β β―β― β =β 10 Γ 11β β order if more than one number is being multiplied. β = 110 Using the distributive law by making a β10, 100βetc. and then adjusting by adding or subtracting. The distributive law is: βa Γ (b + c) = (a Γ b) + (a Γ c) or a Γ (b β c) = (a Γ b) β (b Γ c)β. This law will be used more extensively when algebra is covered. 7 Γ 18 = (7 Γ 20) β (7 Γ 2) 6 Γ 21 = (6 Γ 20) + (6 Γ 1) β β ββ―β― ββ―β― β β β β =β 140 β 14β = 120 + 6 β = 126 β = 126
15 Γ 18 = 30 Γ 9 β = 270 β―β― β Γ 16ββ―β― 11 =β 11 Γβ 8βΓ 2β β = 88 Γ 2 β = 176
Using the doubling and halving strategy by doubling one number and halving the other. Using factors to split a number.
BUILDING UNDERSTANDING
1 State the next three numbers in these patterns. a β4, 8, 12, 16, _ _β b β11, 22, 33, _ _β
2 Are these statements true or false? a β4 Γ 3 = 3 Γ 4β c β11 Γ 5 = 10 Γ 6β e β5 Γ 18 = 10 Γ 9β g β19 Γ 7 = 20 Γ 7 β 19β
b d f h
3 What is the missing digit in these products? a b 2 1 3 6 Γ
3
6
Γ
1 8
5
c β17, 34, 51, _ _β
2β Γ 5 Γ 6 = 6 Γ 5 Γ 2β β3 Γ 32 = 3 Γ 30 + 3 Γ 2β β21 Γ 4 = 2 Γ 42β β64 Γ 4 = 128 Γ 8β
c
Γ
7 6 2
1
2
d
Γ
1
4 02 3 06
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1E Multiplying small positive integers
Example 8
27
Using mental strategies for multiplication c β4 Γ 29β
U N SA C O M R PL R E EC PA T E G D ES
Use a mental strategy to find the answer to each of these products. a β7 Γ 6β b 3β Γ 13β d β5 Γ 24β e β7 Γ 14β SOLUTION
EXPLANATION
a β7 Γ 6 = 42β
β7 Γ 6βor 6β Γ 7βshould be memorised (from multiplication tables).
b β3 Γ 13 = 39β
3β Γ 13 = (3 Γ 10) + (3 Γ 3) = 30 + 9 = 39β (The distributive law is being used.)
c 4β Γ 29 = 116β
4β Γ 29 = (4 Γ 30) β (4 Γ 1) = 120 β 4 = 116β (The distributive law is being used.)
d β5 Γ 24 = 120β
β5 Γ 24 = 10 Γ 12 = 120β (The doubling and halving strategy is being used.)
e 7β Γ 14 = 98β
β7 Γ 14 = 7 Γ 7 Γ 2 = 49 Γ 2 = 98β (Factors of β14βare used.)
Now you try
Use a mental strategy to find the answer to each of these products. a β8 Γ 4β b 6β Γ 21β d β5 Γ 42β e β12 Γ 25β
Example 9
Using the multiplication algorithm
Give the result for each of these products. a β31 Γ 4β SOLUTION
EXPLANATION
a
4Γ1 = 4 4 Γ 3 = 12 β―β― β β βββ β 4 Γ 30 = 120 4 + 120 = 124
31 βΓ _4β 124
b β 6β β1β β 4β β9 7 β_ βΓ 7β 13 79
c β4 Γ 19β
b β197 Γ 7β
β7 Γ 7 = 49β (β7βtimes β9βtens plus the carried β4βtens makes β67βtens. Regroup β60βtens as β6βhundreds and carry β6βinto the hundreds column) β7 Γ 9 + 4 = 67β (β7βhundreds plus the carried β6βhundreds makes β13β hundreds) β7 Γ 1 + 6 = 13β
Now you try
Give the result for each of these products. a β42 Γ 7β
b β372 Γ 8β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
28
Chapter 1
Computation with positive integers
Exercise 1E FLUENCY
1
2β6β(β 1β/2β)β
3β6β(β 1β/2β)β
Using your knowledge of multiplication tables, give the answer to these products. a 3β Γ 5β b 8β Γ 4β c β6 Γ 6β d β9 Γ 3β e β7 Γ 4β f β4 Γ 9β g β8 Γ 7β h β5 Γ 8β
U N SA C O M R PL R E EC PA T E G D ES
Example 8a
1, 2β5ββ(1β/2β)β
Example 8b,c
Example 8d,e
Example 8
Example 9
2 Find the results to these products mentally. (Hint: Use the distributive law strategy (addition) for a to d and the distributive law strategy (subtraction) for e to h.) a 5β Γ 21β b 4β Γ 31β c β6 Γ 42β d β53 Γ 3β e β3 Γ 19β f β6 Γ 29β g β4 Γ 28β h β38 Γ 7β
3 Find the answer to these products mentally. (Hint: Use the double and halve strategy or split a number using its factors.) a 4β Γ 24β b 3β Γ 18β c β6 Γ 16β d β24 Γ 3β 4 Use a suitable mental strategy to find the answer to each of these products. a 8β Γ 7β b 4β Γ 13β c β3 Γ 29β d β4 Γ 52β
5 Give the result of each of these products, using the multiplication algorithm. Check your results using a calculator. a β 33β b β 43β c β 72β d β 55ββ Γ 2 Γ 3 Γ 6 Γ 3 _ _ _ _ 129 e β β Γ 2 _
Example 9
e β9 Γ 14β
f
β 407β Γ 7 _
h β 3509β Γ β9 _
g β 526β Γ 5 _
6 Find the answer to these products, showing your working. a 4β 7 Γ 5β b 1β 391 Γ 3β c β9 Γ 425β PROBLEM-SOLVING
7, 8
d β7 Γ 4170β
8β10
9β11
7 Eight tickets costing β$33βeach are purchased for a concert. What is the total cost of the tickets?
8 A circular race track is β240βmetres long and Rory runs seven laps. How far does Rory run in total? 9 Reggie and Angelo combine their packs of cards. Reggie has five sets of β13βcards and Angelo has three sets of β17βcards. How many cards are there in total? 10 Sala purchases some goods for a party at an outlet store and has β$100βto spend. She selects eight bottles of drink for β$2β each, β13βfood packs at β$6βeach and β18βparty hats at β50βcents each. Does she have enough money to pay for all the items?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1E Multiplying small positive integers
11 Find the missing digits in these products. a b 2 5 3 9 Γ 7 Γ 3
2
d
7 9 Γ
Γ
5
3 7
f
1 3 2
g
10
6
h
U N SA C O M R PL R E EC PA T E G D ES
e
1 2
c
29
2
Γ
Γ 3 5
7
8 9
2 3
9 1
REASONING
Γ
5
1
6 0
12
4
Γ
1
12, 13
9
8
13, 14
12 The commutative and associative laws for multiplication mean that numbers can be multiplied in any order. So β(a Γ b) Γ c = (b Γ a) Γ c = b Γ (a Γ c) = _____β, where the brackets show which numbers are multiplied first. Two ways of calculating β2 Γ 3 Γ 5βare (β β2 Γ 3β)βΓ 5 = 6 Γ 5β and β3 Γ (β β5 Γ 2β)β= 3 Γ 10β. Including these two ways, how many ways can β2 Γ 3 Γ 5βbe calculated?
13 The distributive law can help to work out products mentally. For example, β7 Γ 31 = (7 Γ 30) + (7 Γ 1) = 210 + 7 = 217β
Write each of the following as single products. Do not find the answer. a 3β Γ 20 + 3 Γ 1β c 7β Γ 30 + 7 Γ 2β e aβ Γ 40 β a Γ 2β
b 9β Γ 50 + 9 Γ 2β d β5 Γ 100 β 5 Γ 3β f βa Γ 200 + a Γ 3β
14 How many different ways can the two spaces be filled in this problem? Explain your reasoning. 2
3 4
8
2
Γ
ENRICHMENT: Missing digits
15 Find all the missing digits in these products. a 1 Γ
7
5 1
β
β
b
15, 16
2 9
Γ
3
8
16 The product of two numbers is β132βand their sum is β28β. What are the two numbers?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
30
Chapter 1
1B
1
Write the number β134βusing the given number systems. a Egyptian b Roman c Babylonian
2 Write the number β50862βin expanded form.
U N SA C O M R PL R E EC PA T E G D ES
Progress quiz
1A
Computation with positive integers
1C
1D
1E
3 Use the suggested strategy to mentally work out the answer. a 1β 43 + 232β (partitioning) b β35 + 29β (compensating) c β74 β 17β (compensating) d β35 + 36β (doubling) 4 Give the result for each of these problems. 18 a β β + 44 _
5 Using your knowledge of multiplication tables, give the answer to these products. a 7β Γ 4β c β12 Γ 9β
1E
1E
b β 124β β 46 _
b β9 Γ 8β d β5 Γ 9β
6 Use the distributive law strategy to find the answer to each of these products. Show your working. a 6β Γ 14β b β5 Γ 39β 7 Give the result of each of these products, using the multiplication algorithm. Show your working. a β 84β Γ 3 _ b β 237β Γ 4 _
c β2146 Γ 7β
1D
1C
8 Two different schools have β948βand β1025βstudents enrolled. a How many students are there in total? b Find the difference in the number of students between the schools.
9 Decide if the following statements are always true (T), always false (F) or sometimes true/ sometimes false (S), if a, b and c represent three different numbers. a aβ + b = b + aβ b βa Γ b = b Γ aβ c βa β b = b β aβ d βa Γ (b + c) = a Γ b + a Γ cβ e β(a + b) + c = a + (b + c)β f β(a β b) β c = a β (b β c)β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1F Multiplying large positive integers
31
1F Multiplying large positive integers
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To be able to multiply by a power of ten by adding zeros to the end of a number β’ To be able to apply the multiplication algorithm to ο¬nd the product of any two positive integers
There are many situations that require the multiplication of large numbers β for example, the total revenue from selling 40 000 tickets at $23 each, or the area of a rectangular park with length and width dimensions of 65 metres by 122 metres. To complete such calculations by hand requires the use of a suitable algorithm.
How much revenue came from selling tickets to this game?
Lesson starter: Spot the errors
There are three types of errors in the working shown for this problem. Find the errors and describe them.
271 Γ 13 _ 613 271 _ 1273
KEY IDEAS
β When multiplying by 10, 100, 1000, 10 000 etc. each digit appears to move to the left by the
number of zeros. For example, 45 Γ 1000 = 45 000.
β A strategy for multiplying by multiples of 10, 100 etc. is to first multiply by the number without
the zeros then insert the zeros at the end of the product. For example, 21 Γ 3000 = 21 Γ 3 Γ 1000 = 63 Γ 1000 = 63 000
β The algorithm for multiplying large numbers involves separating the
problem into smaller products and then adding the totals.
143 Γ 14 _ 1572 β143 Γ 4 1430 _ β143 Γ 10 2002 β1430 + 572
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 1
Computation with positive integers
BUILDING UNDERSTANDING 1 What is the missing digit in these products? a 72 Γ 10 = 7 0 b 13 Γ 100 = 130
U N SA C O M R PL R E EC PA T E G D ES
32
c 49 Γ 100 = 49
0
d 924 Γ 10 = 92
0
2 What is the missing number in these products? a 15 Γ __ = 1500 b 329 Γ __ = 3290 c 92 Γ __ = 920 000
How could you estimate the number of pieces of fruit and vegetables on this stall without counting them all?
3 State if the following calculations are correct. If they are incorrect, find the correct answer. a 26 b 39 Γ 4 Γ 14 _ _ 84 156 39 _ 195
c
92 Γ 24 _ 368 1840 _ 2208
d
102 Γ 24 _ 408 240 _ 648
Example 10 Multiplying large numbers Give the result for each of these products. a 37 Γ 100 b 45 Γ 70
c 614 Γ 14
SOLUTION
EXPLANATION
a 37 Γ 100 = 3700
Move the 3 and the 7 two places to the left, so the 3 moves into the thousands place and the 7 moves into the hundreds place. Insert two zeros at the end of the number (in the tens and ones places).
b 45 Γ 70 = 45 Γ 7 Γ 10 = 315 Γ 10 = 3150
First multiply by 7 then multiply by 10 later. 45 Γ 7 _ 315
c
First multiply 614 Γ 4. Then multiply 614 Γ 10. Add the totals to give the answer.
614 Γ 14 _ 2456 6140 _ 8596
Now you try
Give the result for each of these products. a 23 Γ 1000 b 73 Γ 40
c 752 Γ 23
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1F Multiplying large positive integers
33
Exercise 1F FLUENCY
1
1β3β(β 1β/2β)β,β 4
2β3β(β 1β/2β)β, 4, 5
Give the result for each of these products. a β4 Γ 100β b β29 Γ 10β e 5β 0 Γ 1000β f β630 Γ 100β
c 1β 83 Γ 10β g β1441 Γ 10β
d 4β 6 Γ 100β h β2910 Γ 10 000β
2 Give the result for each of these products. a β17 Γ 20β b β36 Γ 40β e 1β 38 Γ 300β f β92 Γ 5000β
c 9β 2 Γ 70β g β317 Γ 200β
d β45 Γ 500β h β1043 Γ 9000β
U N SA C O M R PL R E EC PA T E G D ES
Example 10a
1, 2β3ββ(1β/2β)β,β 4
Example 10b
Example 10c
3 Use the multiplication algorithm to find these products. b β 72β a β 37β c β 126β Γ 19 Γ 11 Γ 15 _ _ _
d β 428β Γ 22 _
g β 380β Γ 49 _
h β 1026β Γ 33 _
396 e β β Γ 46 _
Example 10a,b,c
f
β 416β Γ 98 _
4 Give the result for each of these products. a i β43 Γ 10β b i β71 Γ 20β c i β124 Γ 12β
ii 7β 2 Γ 1000β ii β26 Γ 300β ii β382 Γ 15β
5 First estimate the answers to these products, then use a calculator to see how close you were. a β19 Γ 11β b β26 Γ 21β c β37 Γ 15β d β121 Γ 18β PROBLEM-SOLVING
6, 7
7β9
8β10
6 A pool area includes β68βsquare metres of paving at β$32βper square metre. Find the value of β68 Γ 32β to state the total cost of the paving. 7 Waldo buys β215βmetres of pipe at β$28βper metre. What is the total cost of the piping?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 1
Computation with positive integers
8 How many seconds are there in one 24-hour day? 9 Find the missing digits in these products. a Γ
2 1
b 7
1
1
2
0
1 Γ
3
c
d Γ
1
3
2 Γ
7
2
U N SA C O M R PL R E EC PA T E G D ES
34
2
1
1
3 4
9
3
4
3
1 5
2 2
6
6
5
10 There are β360βdegrees in a full turn. How many degrees does the minute hand on a clock turn in one week? REASONING
11
11, 12ββ(1β/2β)β
12β(β 1β/2β)β,β 13
11 The product of two positive integers is less than their sum. What must be true about one of the numbers?
12 If both numbers in a multiplication problem have at least three digits, then the algorithm needs to be expanded. Use the algorithm to find these products. 294 1013 3947 b β β c β β d β 47126β a β β Γ 916 Γ 1204 Γ 3107 Γ 136 _ _ _ _
13 Can you work out these computations using an effective mental strategy? Look to see if you can first simplify each question. a 9β 8 Γ 16 + 2 Γ 16β b β33 Γ 26 β 3 Γ 26β c β19 Γ 15 + 34 Γ 17 β 4 Γ 17 + 1 Γ 15β d β22 Γ 19 β 3 Γ 17 + 51 Γ 9 β 1 Γ 9 + 13 Γ 17 β 2 Γ 19β ENRICHMENT: Multiplication puzzle
β
β
14 a What is the largest number you can make by choosing five different digits from the list β1, 2, 3, 4, 5, 6, 7, 8, 9βand placing them into the product shown at right? b What is the smallest number you can make by choosing five different digits from the list β1, 2, 3, 4, 5, 6, 7, 8, 9βand placing them into the product shown at right?
14, 15
Γ
15 The product of two whole numbers is β14391βand their difference is β6β. What are the two numbers?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1G Dividing positive integers
35
1G Dividing positive integers
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To know that a division of two numbers can result in a quotient and a remainder, and the result can be written as a mixed numeral if there is a remainder β’ To be able to use mental strategies to ο¬nd quotients β’ To be able to apply the short division algorithm to divide positive integers
Division involves finding the number of equal groups into which a particular number can be divided. This can be achieved both with and without a remainder or βleft overβ. Dividing 20 apples among five people and dividing $10 000 between three bank accounts are examples of when division can be used.
Multiplication and division are reverse operations, and this is shown in this simple example: 7 Γ 3 = 21
So, 21 Γ· 3 = 7
and
21 Γ· 7 = 3
To calculate the number of rafters needed to support a roof, a carpenter ο¬rst divides the roof span (its length) by the required space between each rafter.
Lesson starter: Arranging counters
A total of 24 counters sit on a table. Using whole numbers, in how many ways can the counters be divided into equal-sized groups with no counters remaining? β’ β’
Is it also possible to divide the counters into equal-sized groups but with two counters remaining? If five counters are to remain, how many equal-sized groups can be formed and why?
β’ β’ β’ β’ β’ β’ β’ β’ β’ β’ β’ β’ β’ β’ β’ β’ β’ β’ β’ β’ β’ β’ β’ β’
KEY IDEAS
β The number of equal-sized groups formed from the division operation is called the quotient. β The total being divided is called the dividend and the size of the equal groups is called the
divisor.
β Any amount remaining after division into
equal-sized groups is called the remainder. 7 Γ· 3 = 2 and 1 remainder means 7=2Γ3+1 37 Γ· 5 = 7 and 2 remainder means 37 = 7 Γ 5 + 2
7 Γ· 3 = 2 and 1 remainder = 2
total being divided (dividend)
size of equal groups (divisor)
quotient
1 3
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 1
Computation with positive integers
β βa Γ· bβand βb Γ· aβare not generally equal.
β’
The commutative law does not hold for division, e.g.ββ8 Γ· 2 β 2 Γ· 8β
β (β a Γ· b)βΓ· cβand βa Γ· (β b Γ· c)βare not generally equal.
β’
The associative law does not hold for division, e.g. ββ(β 8 Γ· 4)βΓ· 2 β 8 Γ· (β 4 Γ· 2)β
U N SA C O M R PL R E EC PA T E G D ES
36
β The short division algorithm involves first
11 Γ· 3 = 3 and dividing into the digit with the highest place value 4 Γ· 3 = 1 and 2 rem. 1 rem. and then carrying any remainder to the next digit, 23 Γ· 3 = 7 and working from left to right. 2 rem. 13 7 413 Γ· 3 = 137 and 2 remainder 3 41 3 β ββ―β― β β 2β β = 137β_ 3 β Mental division can be done using different strategies. β’ Knowing your multiplication tables off by heart. β63 Γ· 9 = ?βis the same as asking β9 Γ ? = 63β. β’ Making a convenient multiple of the divisor and then adjusting by adding or subtracting. Below is an application of the distributive law. 84 Γ· 3 = (90 β 6) Γ· 3 84 Γ· 3 = (60 + 24) Γ· 3 =β (90 Γ· 3) β = (60 Γ· 3) + (24 Γ· 3) β βββ―β― β―β― β β (6β Γ· 3)β β β β―β― ββ―β― β β β β = 20 + 8 β = 30 β 2 β = 28 β = 28
)
β’
70 β Γ· 14β =β 35 Γ· 7β β= 5
Halving both numbers. If both numbers in the division are even, then halve both numbers.
BUILDING UNDERSTANDING
1 State the number that is missing in these statements. a β8 Γ· 2 = 4 is the same as 4 Γ ? = 8.β b β36 Γ· 12 = 3 is the same as ? Γ 12 = 36.β c β42 Γ· ? = 6 is the same as 6 Γ 7 = 42.β d β72 Γ· 6 = ? is the same as 12 Γ 6 = 72.β
2 What is the remainder when: a β2βis divided into β7β? b β5βis divided into β37β? c β42βis divided by β8β? d β50βis divided by β9β?
3 State the missing digit in each of these divisions. a
7 3β 5 1
b
2 7β 8 4
c
2 5β1 2 5
d
1 9β1 3 5
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1G Dividing positive integers
37
Example 11 Using mental strategies for division Use a mental strategy to find the quotient. a β84 Γ· 7β b 9β 3 Γ· 3β
U N SA C O M R PL R E EC PA T E G D ES
c β128 Γ· 8β
SOLUTION
EXPLANATION
a β84 Γ· 7 = 12β
7β Γ ? = 84β (Use your knowledge from multiplication tables.)
b β93 Γ· 3 = 31β
9β 3 Γ· 3 = (90 Γ· 3) + (3 Γ· 3) = 30 + 1β (This uses the distributive law.)
c β128 Γ· 8 = 16β
1β 28 Γ· 8 = 64 Γ· 4 = 32 Γ· 2 = 16β (Halve both numbers repeatedly.)
Now you try
Use a mental strategy to find the quotient. a β36 Γ· 9β b 4β 84 Γ· 4β
c β520 Γ· 8β
Example 12 Using the short division algorithm
Use the short division algorithm to find the quotient and remainder. _ _ a β3ββ 37β b β7ββ 195β SOLUTION
EXPLANATION
a
3β Γ· 3 = 1βwith no remainder. β7 Γ· 3 = 2βwith β1β remainder.
12 β _β β 3ββ37β
β37 Γ· 3 = 12βand β1β remainder. 1β β= 12β_ 3 b 27 β _ ββ 7βββ19ββ5β5β
β195 Γ· 7 = 27βand β6β remainder. 6 ββ β= 27β_ 7
7β βdoes not divide into β1β. β19 Γ· 7 = 2βwith β5β remainder. β55 Γ· 7 = 7βwith β6β remainder.
Now you try
Use the short division algorithm to find the quotient and remainder. _ _ a β4ββ 93β b β6ββ 435β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
38
Chapter 1
Computation with positive integers
Exercise 1G FLUENCY
1
1β3ββ(1β/2β)β, 4, 5β7ββ(1β/2β)β
Use your knowledge of multiplication tables to find the quotient. a 2β 8 Γ· 7β b 3β 6 Γ· 12β c β48 Γ· 8β e β42 Γ· 6β f β63 Γ· 7β g β40 Γ· 5β
4, 5, 6β7ββ(1β/2β)β
d β45 Γ· 9β h β44 Γ· 4β
U N SA C O M R PL R E EC PA T E G D ES
Example 11a
1β3ββ(1β/2β)β, 4, 6ββ(1β/2β)β
Example 11b
Example 11c
Example 11
Example 12
Example 12
2 Find the answer to these using a mental strategy. (Hint: Use the distributive law strategy.) a 6β 3 Γ· 3β b 8β 8 Γ· 4β c β96 Γ· 3β d β515 Γ· 5β e β287 Γ· 7β f β189 Γ· 9β g β906 Γ· 3β h β305 Γ· 5β
3 Find the answers to these using a mental strategy. (Hint: Use the halving strategy by halving both numbers.) a 8β 8 Γ· 4β b 1β 24 Γ· 4β c β136 Γ· 8β d β112 Γ· 16β 4
Use a suitable mental strategy to find the quotient.
a 4β 8 Γ· 4β
b β81 Γ· 9β
c β63 Γ· 3β
d β105 Γ· 5β
e β120 Γ· 4β
f
β256 Γ· 16β
5 Write the answers to these divisions, which involve β0sβand β1sβ. a β26 Γ· 1β b 1β 094 Γ· 1β c 0β Γ· 7β
d β0 Γ· 458β
6 Use the short division algorithm to find the quotient and remainder. _ _ _ a β3ββ71β b 7β ββ 92β c β2ββ 139β _ _ _ e β4ββ 2173β f β3ββ 61 001β g β5ββ4093β
_ d β6ββ 247β _ h β9ββ 90 009β
7 Use the short division algorithm to find the quotient and remainder. a β525 Γ· 4β b 1β 691 Γ· 7β c β2345 Γ· 6β
d β92 337 Γ· 8β
PROBLEM-SOLVING
8, 9
9β12
12β14
8 If β36βfood packs are divided equally among nine families, how many packs does each family receive?
9 Spring Fresh Company sells mineral water in packs of six bottles. How many packs are there in a truck containing β642β bottles? 10 A bricklayer earns β$1215βin a week. a How much does he earn per day if he works Monday to Friday? b How much does he earn per hour if he works β9βhours per day?
11 A straight fence has two end posts as well as other posts that are divided evenly along the fence β4β metres apart. If the fence is to be β264βmetres long, how many posts are needed, including the end posts? 12 Friendly Taxis can take up to four passengers each. What is the minimum number of taxis required to transport β59β people? 13 A truck can carry up to β7βtonnes of rock in one trip. What is the minimum number of trips needed to transport β130 βtonnes of rock?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1G Dividing positive integers
14 All the rows, columns and main diagonals in the magic square multiply to give β216β. Can you find the missing numbers?
9
39
12 1
REASONING
15β17
18, 19, 20ββ(β1/2β)β
U N SA C O M R PL R E EC PA T E G D ES
15, 16
15 Write down the missing numbers. a β37 Γ· 3 = 12β and b β96 Γ· 7 =β
remainder means β37 =β
βΓ 3 + 1β
and β5βremainder means β96 = 13 Γβ
c β104 Γ· 20 = 5β and
β+ 5β
remainder means β104 =β
βΓ 20 + 4β
16 Pies are purchased wholesale at β9βfor β$4β. How much will it cost to purchase β153β pies? 17 Give the results to these problems, if βaβrepresents any positive integer. a βa Γ· aβ b β0 Γ· aβ c βa Γ· 1β
18 A number less than β30βleaves a remainder of β3βwhen divided by β5βand a remainder of β2βwhen divided by β3β. What two numbers meet the given conditions?
19 As you know βa Γ· bβis not generally equal to βb Γ· aβ. However, can you find a situation where βa Γ· bβand β b Γ· aβare equal? Try to find as many as possible. 20 The short division algorithm can also be used to divide by numbers with more than one digit. e.g. 215 Γ· 12 = 17 and 11 remainder. 17 β β―β―β― β―β―β― 21 Γ· 12β =β 1 and 9 remainder.β β 12 21 5 95 Γ· 12 = 7 and 11 remainder.
)
Use the short division algorithm to find the quotient and remainder. a β371 Γ· 11β b β926 Γ· 17β d β1621 Γ· 15β e β2109 Γ· 23β
ENRICHMENT: Long, short division
β
c 4β 04 Γ· 13β f β6913 Γ· 56β β
21, 22ββ(β1/2β)β
21 The magic product for this square is β6720β. Find the missing numbers.
1 6 22 Instead of carrying out a complex division algorithm, you could convert 40 the divisor into a smaller pair of factors and complete two simpler division questions to arrive at the correct answer. 14 For example: 1458 Γ· 18 = β(1458 Γ· 2)βΓ· 9 β β ββ―β― =β 729 Γ· 9β β = 81 Use factors to help you calculate the following. a β555 Γ· 15β b β860 Γ· 20β c β3600 Γ· 48β d β1456 Γ· 16β e β6006 Γ· 42β f β2024 Γ· 22β
56
2
3
10
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 1
Computation with positive integers
The following problems will investigate practical situations drawing upon knowledge and skills developed throughout the chapter. In attempting to solve these problems, aim to identify the key information, use diagrams, formulate ideas, apply strategies, make calculations and check and communicate your solutions.
Vet visits
U N SA C O M R PL R E EC PA T E G D ES
Applications and problem-solving
40
1
The following table shows the number of cats and dogs that are seen each year at Quindara Veterinary Clinic. Year
Cats
Dogs
β2018β
β124β
1β 11β
β2019β
β132β
β130β
β2020β
β118β
β122β
β2021β
β141β
1β 26β
β2022β
β128β
β122β
2β 023β
β113β
β121β
The vet owners are interested in how the number of cats and dogs compare, as well as the differences in total numbers from year to year. a How many more cats than dogs were seen in 2021 at Quindara Veterinary Clinic? b In which year was the number of cats and dogs closest? c In which year was the greatest number of cats and dogs seen? d Overall, were there more cats or dogs seen in this 6-year period? e What is the difference in number between the total number of cats and dogs seen over this period? f What was the total number of cats and dogs seen at Quindara Veterinary Clinic? g The vet owners hoped to see a total of 1600 cats and dogs over the 6-year period. How close did they get to their target?
Saving for a new bat
2 Zac is keen to save up to buy a β$229β New Balance cricket bat. He currently has β$25β and he hopes to buy the bat in β12βweeksβ time, giving him enough time to knock his bat in and prepare it for the start of the new cricket season. Zac generally earns his money from walking his neighbourβs dog, where he gets β$6β for taking the dog on a half-hour walk. He also gets β$10βpocket money per week from his parents.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Applications and problem-solving
U N SA C O M R PL R E EC PA T E G D ES
Applications and problem-solving
Zac wishes to explore his income and expenses to calculate the time that it will take to save for the new cricket bat. a If Zac takes the neighbourβs dog for three half-hour walks per week, how much can he earn from dog walking in β12β weeks? b How much pocket money will Zac receive over the β12β weeks? c Given Zac already has β$25βin his wallet, he walks the dog three times per week and he receives his weekly pocket money, how much money can Zac have in β12βweeksβ time?
41
Zac generally spends his money on food, drinks and bus fares and normally spends approximately β$15βper week. d How much money does Zac normally spend over β12β weeks? e Given Zacβs earnings and spending, how much is he likely to have saved at the end of the β12β weeks? f If Zac does not wish to reduce his spending, how many extra dog walks would Zac have to make over the β12βweeks to have enough money to buy his chosen cricket bat? g If Zac cannot do any more dog walks, how much less money per week would Zac need to spend if he wishes to buy his chosen bat in β12βweeksβ time?
Anayaβs steps
3 Anaya lives in Alice Springs and decides to keep a record of how many steps she walks each day. She discovers she walks β12 000βsteps each day. Anaya is interested in using the number of steps that she walks each day to calculate how long it will take to cover various distances. a How many steps will she walk in a fortnight? b If Anayaβs general step length is β1 mβ, how many kilometres does Anaya walk in a fortnight? c If Anaya continues to walk an average of β12 000βsteps each day, how far would she walk in one year?
Alice Springs lies in the heart of Australia, and locals joke that is the only town in Australia which is the closest to every beach in Australia. The actual closest beach to Alice Springs is in Darwin and is approximately β1500 kmβ away. d Walking at her normal rate of β12 000βsteps per day, how many days would it take Anaya to reach her closest beach in Darwin? e Investigate your own average step length and explore how many steps you take on an average day and therefore consider how far you walk in one year.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 1
Computation with positive integers
1H Estimating and rounding positive integers CONSOLIDATING LEARNING INTENTIONS β’ To understand that in some practical situations, an estimate or approximation is accurate enough β’ To be able to round numbers to a degree of accuracy (e.g. to the nearest 100) β’ To be able to estimate numerical answers to arithmetic questions by rounding each number in the question
U N SA C O M R PL R E EC PA T E G D ES
42
Many theoretical and practical problems do not need precise or exact answers. In such situations reasonable estimations can provide enough information to solve the problem.
The total revenue from the Australian Open tennis tournament depends on crowd numbers. Estimates would be used before the tournament begins to predict these numbers. An estimate for the total revenue might be $8 million.
An important ο¬nal check before take-off is the pilotβs estimate of the planeβs maximum ο¬ight distance. With fuel for 4 hours and ο¬ying at 205 km/h, the maximum ο¬ight distance is 800 km, rounded to the nearest hundred km.
Lesson starter: The tennis crowd
Here is a photo of a crowd at a tennis match. Describe how you might estimate the number of people in the photo. What is your answer? How does your answer differ from those of others in your class?
How can you estimate the number of spectators?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1H Estimating and rounding positive integers
43
KEY IDEAS β Estimates or approximations to the answers of problems can be found by rounding numbers to
the nearest multiple of β10, 100, 1000β etc.
U N SA C O M R PL R E EC PA T E G D ES
β If the next digit is β0, 1, 2, 3 or 4β, then round down. β If the next digit is β5, 6, 7, 8βor β9β, then round up.
β Leading digit approximation rounds the first digit up or down, resulting in a multiple of
β10βor β100βor β1000β etc. e.g. For β932βuse β900β,
For β968βuse β1000β, For β3715βuse β4000β
β The symbol βββmeans βapproximately equal toβ. The symbol β can also be used.
BUILDING UNDERSTANDING
1 State whether these numbers have been rounded up or down. a β59 β 60β b β14 β 10β d β255 β 260β e β924 β 900β
c 1β 37 β 140β f β1413 β 1000β
2 For the given estimates, decide if the approximate answer is going to give a larger or smaller result compared to the true answer. a β58 + 97 β 60 + 100β c β130 β 79 β 130 β 80β
b 2β 4 Γ 31 β 20 Γ 30β d β267 β 110 β 270 β 110β
Example 13 Rounding whole numbers Round these numbers as indicated. a β86β(to the nearest β10β)
b β4142β(to the nearest β100β)
SOLUTION
EXPLANATION
a β86 β 90β
The digit after the β8βis greater than or equal to β 5β, so round up.
b β4142 β 4100β
The digit after the β1βis less than or equal to β4β, so round down.
Now you try
Round these numbers as indicated. a β73β(to the nearest β10β)
b β6354β(to the nearest β100β)
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 1
Computation with positive integers
Example 14 Using leading digit approximation Estimate the answers to these problems by rounding each number to the leading digit. a 4β 2 Γ 7β b β95 Γ 326β
U N SA C O M R PL R E EC PA T E G D ES
44
SOLUTION
EXPLANATION
a 42 Γ 7 β 40 Γ 7 β β β β β = 280
The leading digit in β42βis the β4βin the βtensβ column. The nearest βtenβ to β95βis β100β, and the leading digit in β326βis in the βhundredsβ column.
b 95 ββ 100 Γ 300β β Γ 326ββ―β― β = 30 000
Now you try
Estimate the answers to these problems by rounding each number to the leading digit. a 6β 3 Γ 8β b β185 Γ 37β
Example 15 Estimating with operations
Estimate the answers to these problems by rounding both numbers as indicated. a 1β 15 Γ 92β(to the nearest β100β) b β2266 Γ· 9β(to the nearest β10β) SOLUTION
EXPLANATION
a 115 ββ 100 Γ 100β β Γ 92ββ―β― β = 10 000
β115βrounds to β100βand β92βrounds to β100β.
b 2266 Γ· 9β―β― β β ββ 2270 Γ· 10β β = 227
β2266βrounds to β2270βand β9βrounds to β10β.
Now you try
Estimate the answers to these problems by rounding both numbers as indicated. a 3β 71 Γ 915β(to the nearest β100β) b β841 Γ· 18β(to the nearest β10β)
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1H Estimating and rounding positive integers
45
Exercise 1H FLUENCY
1
1, 2β6ββ(1β/2β)β
2β6β(β 1β/2β)β
3β6β(β 1β/2β)β
Round these numbers as indicated. a i β72β(nearest β10β)
ii β39β(nearest β10β)
iii β153β(nearest β10β)
Example 13b
b i
ii β718β(nearest β100β)
iii β1849β(nearest β100β)
U N SA C O M R PL R E EC PA T E G D ES Example 13a
Example 13
Example 14
Example 15
Example 15
Example 15
β362β(nearest β100)β
2 Round these numbers as indicated. a β59β(nearest β10β) b 3β 2β(nearest β10β) d β185β(nearest β10β) e β231β(nearest β100β) g β96β(nearest β10β) h β584β(nearest β100β)
c 1β 24β(nearest β10β) f β894β(nearest β100β) i β1512β(nearest β1000β)
3 Round these numbers using leading digit approximation; e.g. β385βrounds to β400βand 52 rounds to β50β. a β21β b β29β c β136β d β857β e 5β 600β f β92 104β g β9999β h β14β 4 Estimate the answers to these problems by first rounding both numbers as indicated. a β72 + 59β(nearest β10β) b β138 β 61β(nearest β10β) c 2β 75 β 134β(nearest β10β) d β841 + 99β(nearest β10β) e 2β 03 β 104β(nearest β100β) f β815 + 183β(nearest β100β) g β990 + 125β(nearest β100β) h β96 + 2473β(nearest β100β) i β1555 β 555β(nearest β1000β) j β44 200 β 36 700β(nearest β1000β) 5 Use leading digit approximation to estimate the answer. a β29 Γ 4β b β124 + 58β c β232 β 106β e 3β 94 Γ· 10β f β97 Γ 21β g β1390 + 3244β
d β61 Γ· 5β h β999 β 888β
6 Estimate the answers to these problems by rounding both numbers as indicated. a β29 Γ 41β(nearest β10β) b β92 Γ 67β(nearest β10β) c 1β 24 Γ 173β(nearest β100β) d β2402 Γ 3817β(nearest β1000β) e 4β 8 Γ· 11β(nearest β10β) f β159 Γ· 12β(nearest β10β) g β104 Γ· 11β(nearest β10β) h β2493 Γ· 103β(nearest β100β) PROBLEM-SOLVING
7, 8
8β10
8β11
7 A digger can dig β29βscoops per hour and work β7β hours per day. Approximately how many scoops can be dug over 10 days?
8 Many examples of Aboriginal art include dot paintings. Here is one example. Estimate the number of dots it contains.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 1
Computation with positive integers
9 Most of the pens at a stockyard are full of sheep. There are β55βpens and one of the pens has β22βsheep. Give an estimate for the total number of sheep at the stockyard. 10 A whole year group of β159βstudents is roughly divided into β19βgroups. Estimate the number in each group.
U N SA C O M R PL R E EC PA T E G D ES
46
11 It is sensible sometimes to round one number up if the other number is going to be rounded down. Use leading digit approximation to estimate the answers to these problems. a 1β 1 Γ 19β c β25 Γ 36β REASONING
b β129 Γ 954β d β1500 Γ 2500β
12ββ(β1/2β)β
12
12
12 The letters βaβand βbβrepresent numbers, which could be the same as each other or could be different. Use the word βsmallerβ or βlargerβ to complete these sentences. a If βaβand βbβare both rounded up, then compared to the true answer the approximate answer to: i βa + bβwill be _______. ii βa Γ bβwill be _______. b If only βaβis rounded up, but βbβis left as it is, then compared to the true answer the approximate answer to: i βa β bβwill be _______. ii βa Γ· bβwill be _______. c If only βbβis rounded up, but βaβis left as it is, then compared to the true answer the approximate answer to: i βa β bβwill be _______. ii βa Γ· bβwill be _______. d If only βbβis rounded down, but βaβis left as it is, then compared to the true answer the approximate answer to: i βa β bβwill be _______. ii βa Γ· bβwill be _______. ENRICHMENT: Maximum error
β
β
13
13 When rounding numbers before a calculation is completed, it is most likely that there will be an error. This error can be large or small, depending on the type of rounding involved. For example, when rounding to the nearest β10β, β71 Γ 11 β 70 Γ 10 = 700β.
But β71 Γ 11 = 781β, so the error is β81β. a Calculate the error if these numbers are rounded to the nearest β10βbefore the multiplication is calculated. i β23 Γ 17β ii β23 Γ 24β iii β65 Γ 54β iv β67 Γ 56β b Explain why the error in parts i and iii is much less than the error in parts ii and iv. c Calculate the error if these numbers are rounded to the nearest β10βbefore the division is calculated. i β261 Γ· 9β ii β323 Γ· 17β iii β99 Γ· 11β iv β396 Γ· 22β d Explain why the approximate answers in parts i and ii are less than the correct answer, and why the approximate answers in parts iii and iv are more than the correct answer.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1I Order of operations with positive integers
47
1I Order of operations with positive integers
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To know the convention for determining order of operations in an expression involving more than one operation β’ To be able to evaluate arithmetic expressions involving more than one operation
When combining the operations of addition, subtraction, multiplication and division, a particular order needs to be followed. Multiplication and division sit higher in the order than addition and subtraction, and this relates to how we might logically interpret simple mathematical problems put into words. Consider these two statements. β’ β’
2 groups of 3 chairs plus 5 chairs. 5 chairs plus 2 groups of 3 chairs.
In both cases, there are 2 Γ 3 + 5 = 11 chairs. This means that 2 Γ 3 + 5 = 5 + 2 Γ 3. This also suggests that for 5 + 2 Γ 3 the multiplication should be done first.
Lesson starter: Minimum brackets β’
How might you use brackets to make this statement true? 2+3Γ5β3Γ·6+1=2
β’
What is the minimum number of pairs of brackets needed to make it true?
KEY IDEAS
β When working with more than one operation:
β’ β’ β’
Deal with brackets (also known as parentheses) first. Do multiplication and division next, working from left to right. Do addition and subtraction last, working from left to right.
β Recall (a + b) + c = a + (b + c) but (a β b) β c β a β (b β c)
(a Γ b) Γ c = a Γ (b Γ c) but (a Γ· b) Γ· c β a Γ· (b Γ· c)
4 Γ (2 + 3) β 12 Γ· 6 1st 5 2nd 3rd 20 2 last 18
β Brackets can sit inside other brackets.
β’ β’
Square brackets can also be used. For example, [2 Γ (3 + 4) β1] Γ 3. Always deal with the inner brackets first.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 1
Computation with positive integers
BUILDING UNDERSTANDING 1 Which operation (addition, subtraction, multiplication or division) is done first in the following? a 2+5β3 d 5Γ2+3 g (8 Γ· 4) β 1 j 10 β 2 + 3
b e h k
5Γ·5Γ2 7Γ·7β1 4+7Γ2 6+2Γ3β1
c f i l
2Γ3Γ·6 (6 + 2) Γ 3 8 β 10 Γ· 5 5 Γ (2 + 3 Γ· 3) β 1
U N SA C O M R PL R E EC PA T E G D ES
48
2 Classify these statements as true or false. a 5 Γ 2 + 1 = (5 Γ 2) + 1 c 21 β 7 Γ· 7 = (21 β 7) Γ· 7
b 10 Γ (3 + 4) = 10 Γ 3 + 4 d 9 β 3 Γ 2 = 9 β (3 Γ 2)
A house painter uses order of operations to calculate the area to be painted on each wall and its cost. Cost = ($/m2) Γ [L Γ W (wall) β L Γ W (window) β L Γ W (door)]
Example 16 Using order of operations Use order of operations to answer the following. a 5 + 10 Γ· 2
b 18 β 2 Γ (4 + 6) Γ· 5
SOLUTION
EXPLANATION
a 5 + 10 Γ· 2 = 5 + 5 = 10
Do the division before the addition.
b 18 β 2 Γ (4 + 6) Γ· 5 = 18 β 2 Γ 10 Γ· 5 = 18 β 20 Γ· 5 = 18 β 4 = 14
Deal with brackets first. Do the multiplication and division next, working from left to right. Do the subtraction last.
Now you try Use order of operations to answer the following. a 13 + 3 Γ 7
b 100 β (4 + 8) Γ· 2 Γ 10
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1I Order of operations with positive integers
49
Example 17 Using order of operations in worded problems
U N SA C O M R PL R E EC PA T E G D ES
Find the answer to these worded problems by first writing the sentence using numbers and symbols. a Double the sum of β4βand β3β. b The difference between β76βand β43βis tripled, and then the quotient of β35βand β7βis subtracted. SOLUTION
EXPLANATION
a 2 Γ β(4 + 3)β = 2 Γ 7 β β β β β = 14
First, write the problem using symbols and words. Brackets are used to ensure the sum of β4βand β3β is found first.
b 3 Γ β(β76 β 43β)ββ 35 Γ· 7 = 3 Γ 33 β 35 Γ· 7 β β ββ―β― =β 99 β 5β β = 94
First, write the problem using symbols and numbers. Use brackets for the difference since this operation is to be completed first.
Now you try
Find the answer to these worded problems by first writing the sentence using numbers and symbols. a The difference of β10βand β6βis tripled. b The sum of β12βand β36βis doubled and then the quotient of β40βand β2βis subtracted.
Exercise 1I
Example 16a
Example 16a
Example 16a
Example 16b
Example 16a
FLUENCY
1β4, 7
Use order of operations to answer the following. a β3 + 2 Γ 4β b β5 + 3 Γ 4β
c β10 β 2 Γ 3β
d β12 β 6 Γ 2β
2 Use order of operations to answer the following. a β2 Γ 3 + 1 Γ 2β b β7 Γ 2 + 1 Γ 3β
c β6 Γ 3 β 2 Γ 2β
d β4 Γ 5 β 3 Γ 3β
3 Use order of operations to answer the following. a β20 Γ· 2 + 2β b β20 + 2 Γ· 2β
c β20 β 2 Γ· 2β
d β20 Γ· 2 β 2β
4 Use order of operations to answer the following. a β4 Γ (3 + 2)β b β3 Γ (4 + 2)β
c β(4 + 3) Γ 2β
d β(4 β 2) Γ 3β
1
5 Use order of operations to find the answers to the following. a β2 + 3 Γ 7β b β5 + 8 Γ 2β d β22 β 16 Γ· 4β e β6 Γ 3 + 2 Γ 7β g β18 Γ· 9 + 60 Γ· 3β h β2 + 3 Γ 7 β 1β j β63 Γ· 3 Γ 7 + 2 Γ 3β k 7β 8 β 14 Γ 4 + 6β
1β5ββ(β1/2β)β,β 7
c f i l
5β8β(β β1/2β)β
β10 β 20 Γ· 2β β1 Γ 8 β 2 Γ 3β β40 β 25 Γ· 5 + 3β β300 β 100 Γ 4 Γ· 4β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
50
Chapter 1
6 Use order of operations to find the answer to the following problems. a 2β Γ β(3 + 2)β b β18 Γ· β(10 β 4)β d β(100 + 5)βΓ· 5 + 1β e β2 Γ β(9 β 4)βΓ· 5β g β16 β 2 Γ (β 7 β 5)β+ 6β h β(7 + 2)βΓ· (β 53 β 50)β ( ) ( ) j β 20 β 10 βΓ β 5 + 7 β+ 1β k β3 Γ β(72 Γ· 12 + 1)ββ 1β
c f i l
β(19 β 9)βΓ· 5β β50 Γ· β(13 β 3)β+ 4β β14 β β(7 Γ· 7 + 1)βΓ 2β β48 Γ· β(4 + 4)βΓ· (β 3 Γ 2)β
7 Find the answer to these worded problems by first writing the sentence using numbers and symbols. a Triple the sum of β3βand β6β. b Double the quotient of β20βand β4β. c The quotient of β44βand β11βplus β4β. d 5 more than the product of β6βand β12β. e The quotient of β60βand β12βis subtracted from the product of β5βand β7β. f 15 less than the difference of β48βand β12β. g The product of β9βand β12βis subtracted from double the product of β10βand 15.
U N SA C O M R PL R E EC PA T E G D ES
Example 16b
Computation with positive integers
Example 17
8 These computations involve brackets within brackets. Ensure you work with the inner brackets first. a β2 Γ [(2 + 3) Γ 5 β 1]β b β[10 Γ· (2 + 3) + 1 ] Γ 6β c β26 Γ· [10 β (17 β 9) ]β d β[6 β (5 β 3) ] Γ 7β e β2 + [103 β (21 + 52) ] β (9 + 11) Γ 6 Γ· 12β PROBLEM-SOLVING
9, 10, 11ββ(β1/2β)β
10, 11β(β β1/2β)β
11, 12
9 A delivery of β15βboxes of books arrives, each box containing eight books. The bookstore owner removes three books from each box. How many books still remain in total? 10 In a class, eight students have three televisions at home, four have two televisions, β13βhave one television and two students have no television. How many televisions are there in total? 11 Insert brackets into these statements to make them true. a 4β + 2 Γ 3 = 18β b β9 Γ· 12 β 9 = 3β c β2 Γ 3 + 4 β 5 = 9β d β3 + 2 Γ 7 β 3 = 20β e β10 β 7 Γ· 21 β 18 = 1β f β4 + 10 Γ· 21 Γ· 3 = 2β g β20 β 31 β 19 Γ 2 = 16β h β50 Γ· 2 Γ 5 β 4 = 1β i β25 β 19 Γ 3 + 7 Γ· 12 + 1 = 6β
12 An amount of β$100βis divided into two first prizes of equal value and three second prizes of equal value. Each prize is a whole number of dollars and first prize is at least four times the value of second prize. If second prize is more than β$6β, find the amount of each prize.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
1I Order of operations with positive integers
REASONING
13ββ(β1/2β)β
13β(β β1/2β)β,β 14
51
13β(β β1/2β)β, 14, 15
U N SA C O M R PL R E EC PA T E G D ES
13 Decide if the brackets given in each statement are actually necessary; that is, do they make any difference to the problem? a β2 + (β 3 Γ 6)β= 20β b β(2 + 3)βΓ 6 = 30β c β(20 Γ 2)βΓ 3 = 120β ( ) ( ) d β10 β β 5 + 2 β= 3β e β22 β β 11 β 7 β= 18β f β19 β β(10 Γ· 2)β= 14β g (β 40 Γ· 10)βΓ· 4 = 1β h β100 Γ· β(20 Γ· 5)β= 25β i β2 Γ β(3 + 2)βΓ· 5 = 2β
14 The letters βaβ, βbβand βcβrepresent numbers. Decide if the brackets are necessary in these expressions. a βa + β(b + c)β b βa β β(b β c)β c βa Γ β(b Γ c)β d βa Γ· β(b Γ· c)β 15 Simplify the following. Assume βa β 0βand βb β 0β. a βa + b β aβ b β(a β a)βΓ bβ ENRICHMENT: Operation in rules
c βa + b Γ· bβ
d βa Γ b Γ· aβ
β
β
16
16 Using whole numbers and any of the four operations β(+ , β , Γ, Γ·)β, describe how you would obtain the βFinishβ number from the βStartβ number in each of these tables. Your rule must work for every pair of numbers in its table. a
Start
Finish
1 2
b
Start
Finish
3
1
5
2
3
7
4
9
c
Start
Finish
0
3
10
3
4
17
3
6
5
26
4
9
6
37
Make up your own table with a secret rule and test it on a friend.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
52
Chapter 1
Computation with positive integers
Modelling
Cans and chips At a school fundraising fair, Mike has been given $29 to spend on soft drink and chips for himself and his friends. Cans of soft drink cost $2 each and cups of chips cost $3 each.
U N SA C O M R PL R E EC PA T E G D ES
Present a report for the following tasks and ensure that you show clear mathematical working and explanations where appropriate.
Preliminary task
a Find the total cost of buying: i 5 cans of soft drink ii 7 cups of chips iii 4 cans of soft drink and 3 cups of chips iv 6 cans of soft drink and 5 cups of chips. b If 6 cups of chips and 3 cans are purchased, find the change from $29. c Determine the maximum number of cans of soft drink that can be purchased for $29 if: i 3 cups of chips are purchased ii 4 cups of chips are purchased.
Modelling task
Formulate Solve
Evaluate and verify
Communicate
a The problem is to find the maximum number of cans and cups of chips that Mike can purchase with the money he was given. Write down all the relevant information that will help solve this problem. b Choose at least two combinations for the number of cans and number of cups of chips so that the total cost is less than $29. Show your calculations to explain why your combinations are affordable. c Choose at least two different combinations for the number of cans and chips so that the total cost is equal to $29. d Determine possible combinations which would mean that Mike: i maximises the number of cans purchased ii maximises the number of cups of chips purchased. e Determine the maximum number of items (cans and/or cups of chips) that can be purchased for $29 or less: i if at least 5 cups of chips must be purchased ii if there are no restrictions. f There are two ways of achieving the maximum number of items for part e ii. Explain why there are two combinations, and compare them. g Summarise your results and describe any key findings.
Extension questions
a If Mike had $67, investigate the maximum number of items (cans of soft drink and/or cups of chips) that can be purchased if at least 12 cups of chips must be included. b If the total amount is still $67 and the cost of chips changes to $2.50, describe how the answer to part a above would change. You can assume that the cost of the cans stays the same.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Technology and computational thinking
53
Making β100β dollars
U N SA C O M R PL R E EC PA T E G D ES
You can imagine that there are many ways in which you can make β$100βfrom β$10βand β$20β notes. Assuming you have enough of each type, you could make β$100βusing five β$20βand no β$10βnotes or perhaps three β$20βnotes and four β$10βnotes, for example.
Technology and computational thinking
Key technology: Spreadsheets
1 Getting started
a Without using one of the combinations mentioned in the introduction, list three different possible combinations of β$20βand β$10βnotes that make up a total of β$100β. b Can you find a way to make up β$100βby combining β$10βand β$20βnotes so that there is a total of seven notes being used? If so, describe it. c Can you find a way to make up β$100βby combining β$10βand β$20βnotes so that there is a total of eleven notes being used? Explain your answer.
2 Using technology
We will use a spreadsheet to explore the number of ways you can choose β$10βand β$20βnotes to form β$100β. a Create the following spreadsheet using the given information. β’ The number of β$20β notes can vary and is in cell βB3β. β’ The information in cell βA7βcreates the number of β$10β notes selected. β’ The information in cell βB6βcreates the total value of the β$10βand β$20βnotes selected. β’ The information in cell βC6βcreates the total number of notes selected. b Fill down at cells βA7β, βB6βand βC6β. Fill down until ten β$10βnotes are used. Note that the β$βsign in cells βB6βand βC6βensures that the number of β$20βnotes in cell βB3βis used for every calculation. c See if your spreadsheet is working properly by altering the number of β$20βnotes in cell βB3β. All the values should change when this cell is updated. d Choose two β$20βnotes by changing the cell βB3βto equal β2β. Read off your spreadsheet to answer the following. i How many β$10βnotes are required to make the total of β $100β? ii How many notes are selected in total if you make β$100β? iii Describe what happens if the number of β$20βnotes in cell βB3β is changed to β7β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
54
Chapter 1
Computation with positive integers
a Use your spreadsheet to systematically run through and count all the ways you can make β$100β using β$10βand β$20βnotes. Apply these steps. β’ Step β1β: Choose the smallest number of β$20βnotes possible and enter this number into cell βB3β. β’ Step β2β: Read off the number of β$10βnotes needed to make β$100β. β’ Step β3β: Increase the number of β$20βused by one and repeat Step β2β. β’ Step β4β: Continue increasing the number of β$20βnotes used until you reach a maximum possible number. b Describe the combination of notes where a total of β$100βis achieved but there are eight notes in total. c Is it possible to achieve a total of β$100βusing only six notes in total? If so, describe how. d How many ways were there of forming β$100βfrom β$10βand β$20β notes?
U N SA C O M R PL R E EC PA T E G D ES
Technology and computational thinking
3 Applying an algorithm
4 Extension
a Alter your spreadsheet so that a β$100βtotal is required but this time using β$5βand β$20βnotes. Decide if it possible to achieve a β$100βtotal with a total of: i β17β notes ii β10β notes. b Alter your spreadsheet so that a β$100βtotal is required but this time using a combination of three notes, e.g. β$10β, β$20βand β$50β. What is the total number of ways this can be achieved?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Investigation
55
The abacus
Investigation
U N SA C O M R PL R E EC PA T E G D ES
Counting boards called abacuses (or abaci) date back to 500 BCE. The abacus is a counting device that has been used for thousands of years. Abacuses were used extensively by merchants, traders, tax collectors and clerks before modern-day numeral systems were developed. These were wood or stone tablets with grooves, which would hold beans or pebbles.
There are 5 beads on one side of a modern abacus worth 1 each and 2 beads on the opposite side worth 5 each. β’ Each wire represents a different unit, e.g. ones, tens, hundreds etc. β’ Beads are counted only when they are pushed towards the centre.
th o hu usa te ndr nds n on s eds es
The modern abacus is said to have originated in China in about the thirteenth century and includes beads on wires held in a wooden frame.
A German woodcut from 1508 showing an abacus in use by the gentleman on the right, while a mathematician (at left) writes algorithms.
Here is a diagram showing the number 5716.
a What numbers are showing on the abacus diagrams below? Only the first six wires are showing. i ii iii iv
b Draw abacus diagrams showing these numbers. i 57 ii 392 iii 6804 iv 290 316 c Imagine adding two numbers using an abacus by sliding beads along their wires. Clearly explain the steps taken to add these numbers. i 11 + 7 ii 2394 + 536 d Imagine subtracting two numbers using an abacus by sliding beads along their wires. Clearly explain the steps taken to subtract these numbers. i 23 β 14 ii 329 β 243 e Multiplication is calculated as a repeated addition, e.g. 3 Γ 21 = 21 + 21 + 21. Clearly explain the steps involved when using an abacus to multiply these numbers. i 3 Γ 42 ii 5 Γ 156 f Division is calculated as a repeated subtraction, e.g. 63 Γ· 21 = 3, since 63 β 21 β 21 β 21 = 0. Clearly explain the steps involved when using an abacus to divide these numbers. i 28 Γ· 7 ii 405 Γ· 135 g See if you can find a real abacus or computer abacus with which to work. Use the abacus to show how you can do the problems in parts c to f above.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
56
Chapter 1
The extra dollar? Up for a challenge? If you get stuck The cost of dinner for two people is β$45β and on a question, check out the βWorking with unfamiliar problemsβ poster at they both give the waiter β$25βeach. Of the the end of the book to help you. extra β$5βthe waiter is allowed to keep β$3β as a tip and returns β$1βto each person. So the two people paid β$24βeach, making a total of β$48β, and the waiter has β$3β. The total is therefore β$48 + $3 = $51β. Where did the extra β$1βcome from?
U N SA C O M R PL R E EC PA T E G D ES
Problems and challenges
1
Computation with positive integers
2 The sum along each line is β15β. Can you place each of the digits β1, 2, 3, 4, 5, 6, 7, 8β and β9βto make this true?
3 Ethan starts at β2637βand counts backwards by eights. He stops counting when he reaches a number less than β10β. What is this final number? 4 Make the total of β100βout of all the numbers β2, 3, 4, 7βand β11β, using each number only once. You can use any of the operations β(+ , β, Γ, Γ· )β, as well as brackets. 5 A leaking tap loses 1 drop of water per second. If β40β of these drops of water make a volume of β10 mLβ, how many litres of water are wasted from this tap in mL: a in 1 day? (round answer to the nearest unit) b in 1 year? (round answer to the nearest β100β).
6 When this shape is folded to make a cube, three of the sides will meet at every vertex (corner) of the cube. The numbers on these three sides can be multiplied together. Find the smallest and largest of these products. 7
1
3
9
11
5
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter summary
2 Γ 100 + 7 Γ 10 + 3 Γ 1 is the expanded form of 273. This can also be written with indices as 2 Γ 102 + 7 Γ 101 + 3 Γ 1
U N SA C O M R PL R E EC PA T E G D ES
Place value
Chapter summary
The place value of 3 in 1327 is 300.
57
Ancient number systems
Addition and subtraction
Roman LXXVI is 76 XCIV is 94
Algorithms
1
Babylonian is 23 is 71
1
8
371 +_____ 843
937 β 643 _____
1214
294
Mental strategies
172 + 216 = 300 + 80 + 8 = 388 98 β 19 = 98 β 20 + 1 = 79
Egyptian
is 21
is 143
Order of operations
Brackets first, then Γ and Γ·, then + and β from left to right. 2 + 3 Γ 4 Γ· (9 Γ· 3) = 2 + 12 Γ· 3 =2+4 =6
Algorithms
2
29 Γ____ 13 87 290 ____ 377
Estimation
955 to the nearest 10 is 960 950 to the nearest 100 is 1000
Multiplication and division
Positive integers
Leading digit approximation 39 Γ 326 β 40 Γ 300 = 12 000
68
3 2025
with 1 remainder
Mental strategies 7 Γ 31 = 7 Γ 30 + 7 Γ 1 = 217 5 Γ 14 = 5 Γ 2 Γ 7 = 70 64 Γ· 8 = 32 Γ· 4 = 16 Γ· 2 = 8 156 Γ· 4 = 160 Γ· 4 β 4 Γ· 4 = 40 β 1 = 39
Multiplying by 10, 100, β¦ 38 Γ 100 = 3800 38 Γ 700 = 38 Γ 7 Γ 100 = 26 600
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
58
Chapter 1
Computation with positive integers
A printable version of this checklist is available in the Interactive Textbook 1A
β
1. I can write numbers in the Egyptian, Babylonian and Roman number systems. e.g. Write β144βin the Egyptian, Babylonian and Roman number systems.
U N SA C O M R PL R E EC PA T E G D ES
Chapter checklist
Chapter checklist with success criteria
1B
2. I can write down the place value of digits within a number. e.g. Write the place value of the digit β4βin the number β543 910β.
1B
3. I can write positive integers in expanded form with and without index notation. e.g. Write 517 in expanded form. Rewrite 517 in expanded form with index notation.
1C
4. I can use mental addition and subtraction techniques effectively. e.g. Mentally find β132 + 156β(with partitioning) and β56 β 18β(with compensating).
1D
5. I can use the addition algorithm. e.g. Find β439 + 172βby first aligning the numbers vertically.
1D
6. I can use the subtraction algorithm. e.g. Find β3240 β 2721βby first aligning the numbers vertically.
1E
7. I can use mental multiplication techniques effectively. e.g. Mentally find β4 Γ 29β.
1E
8. I can use the multiplication algorithm for single-digit products. e.g. Find β31 Γ 4βand β197 Γ 7βusing the multiplication algorithm.
1F
9. I can multiply by powers of ten. e.g. State the value of β37 Γ 100β.
1F
10. I can multiply large numbers using the multiplication algorithm. e.g. Find β614 Γ 14βusing the multiplication algorithm.
1G
11. I can use mental strategies to divide positive integers. e.g. Mentally find β93 Γ· 3β.
1G
12. I can use the short division algorithm. e.g. Find the quotient and remainder when β195βis divided by β7β.
1H
13. I can round positive integers to a power of ten. e.g. Round β4142βto the nearest β100β.
1H
14. I can estimate answers using leading digit approximation. e.g. Estimate β95 Γ 326βby rounding each number to the leading digit.
1H
15. I can estimate answers by rounding numbers. e.g. Estimate β2266 Γ· 9βby rounding both numbers to the nearest β10β.
1I
16. I can use order of operations. e.g. Find the value of β18 β 2 Γ (β 4 + 6)βΓ· 5β.
1I
17. I can use order of operations in worded problems. e.g. Find the difference between β76βand β43β, triple this result and, finally, subtract the quotient of β 35β and β7β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
59
Short-answer questions Write these numbers using the given number systems. a Egyptian i β3β ii β31β iii β326β b Babylonian i β12β ii β60β iii β132β c Roman i β14β ii β40β iii β146β
U N SA C O M R PL R E EC PA T E G D ES
1
Chapter review
1A
1B
2 Write down the place value of the digit β5βin these numbers. a β357β b β5249β c β356 612β
1C
3 Use a mental strategy to find these sums and differences. a β124 + 335β b β687 β 324β c β59 + 36β
1D
1E/G
1F/G
1D/F/G
d β256 β 39β
4 Use an algorithm and show your working for these sums and differences. c 329 a 76 b 1528 β ββ β ββ β β ββ 138β + β 52β + β_ 796β _ _
5 Use a mental strategy to answer the following. a β5 Γ 19β b β22 Γ 6β d β123 Γ· 3β e β264 Γ· 8β g β29 Γ 1000β h β36 Γ 300β
d
Find the missing digits in the following. a 2 3
b
+7 3 9 6
2
1
2
3
Γ 2
0
_ d β4ββ30 162β
β4
c
d
β
1
6
0
3
1
3
5
6
4
1
1
5
7
3
1
with no remainder
1H
8 Round these numbers as indicated. a β72β(nearest β10β) b β3268β(nearest β100β)
1H
9 Use leading digit approximation to estimate the answers to the following. a β289 + 532β b β22 Γ 19β c β452 Γ 11β
1I
2109 ββ ββ 1814β _
c 5β Γ 44β f β96 Γ· 4β i β14 678 Γ· 1β
6 Use an algorithm and show your working for the following. _ c β7ββ327β b 27 a 157 β ββ β β Γ β_ 13β βΓ _9β
7
β
10 Use order of operations to find the answers to the following. a β3 Γ (β 2 + 6)β b β6 β 8 Γ· 4β d β(5 + 2)βΓ 3 β (β 8 β 7)β e β0 Γ β(β988 234 Γ· 3β)β
c β951β(nearest β100β) d β99 Γ· 11β
c 2β Γ 8 β 12 Γ· 6β f β1 Γ β(3 + 2 Γ 5)β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
60
Chapter 1
Computation with positive integers
1A
1B
1
The correct Roman numerals for the number β24β are: A XXIII B XXIV C XXXLIV
D IVXX
E IXXV
2 3β Γ 1000 + 9 Γ 10 + 2 Γ 1βis the expanded form of: A 3β 920β B β392β C β3092β
D β3902β
E β329β
U N SA C O M R PL R E EC PA T E G D ES
Chapter review
Multiple-choice questions
1C/E
3 Which of the following is not true? A 2β + 3 = 3 + 2β B 2 β Γ 3 = 3 Γ 2β D β5 Γ· 2 β 2 Γ· 5β E β7 β 2 = 2 β 7β
C (β 2 Γ 3)βΓ 4 = 2 Γ β(3 Γ 4)β
1C
4 The sum of β198βand β103β is: A 3β 01β B β304β
C β299β
D β199β
E β95β
1C
5 The difference between β378βand β81β is: A 4β 59β B β297β C β303β
D β317β
E β299β
1E
6 The product of β7βand β21β is: A 1β 47β B β141β
D β140β
E β207β
1G
7 Which of the following is the missing digit in this division? 1 1 8 C β9β
D β8β
E β7β
1G
8 The remainder when β317βis divided by β9β is: A 7β β B β5β C β2β
D β1β
E β0β
1G
9 4β 58βrounded to the nearest β100β is: A 4β 00β B β500β
C β460β
D β450β
E β1000β
1I
10 The answer to β[2 + 3 Γ (7 β 4) ] Γ· 11β is: A β1β B 5β β C β11β
D β121β
E β0β
β
7
A 6β β
C β21β
1 5
2 6
B β1β
Extended-response questions 1
A city tower construction uses β4520βtonnes of concrete trucked from a factory that is β7β kilometres from the construction site. Each concrete mixer can carry β7βtonnes of concrete, and the concrete costs β$85β per truck load for the first β30βloads and β$55βper load after that. a How many loads of concrete are needed? Add a full load for any remainder. b Find the total distance travelled by the concrete trucks to deliver all loads, assuming they need to return to the factory after each load. c Find the total cost of concrete needed for the tower construction. d A different concrete supplier offers a price of β$65βper β8β-tonne truck, no matter how many loads are needed. Find the difference in the cost of concrete for the tower by this supplier compared to the original supplier.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
U N SA C O M R PL R E EC PA T E G D ES
Chapter review
2 One night Ricky and her brother Micky decide to have some fun at their fatherβs lolly shop. In the shop they find β7βtins of jelly beans each containing 135 beans, β9βpackets of β121β choc buds, β12βjars of β70βsmarties and β32βpackets of β5βliquorice sticks. a Find the total number of lollies that Ricky and Micky find that night. b Find the difference between the number of choc buds and the number of smarties. c Ricky and Micky decide to divide each type of lolly into groups of β7βand then eat any remainder. Which type of lolly will they eat the most of and how many? d After eating the remainders, they round the total of each lolly using leading digit approximation. If they round down they put the spare lollies in their pockets. If they round up, they borrow any spare lollies from their pockets. Any left over in their pockets, they can eat. Do Ricky and Micky get to eat any more lollies?
61
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
U N SA C O M R PL R E EC PA T E G D ES
2 Number properties and patterns
Maths in context: Number patterns around us
Mathematicians are always looking for patterns in number sequences. When a pattern is known, rules can be found, and rules can be used to predict future values. To create a number pattern with a common difference, add or subtract an equal amount. For example:
β’ When ο¬lling a swimming pool from a tap and hose, the total volume of water increases by a constant volume per minute. β’ When a plane ο¬ies at a constant speed and altitude, the volume of remaining fuel decreases by an equal amount every minute.
and decrease in value when multiplied by a number less than 1. For example:
β’ A ο¬nancial investment that increases each January by 3% (interest) would have a list of its January values forming a sequence of increasing numbers, with a common ratio of 1.03. β’ Threatened or endangered animal species have small population numbers that are declining. Their annual population totals form a sequence of decreasing numbers, with a common ratio less than 1, e.g. 0.9. About 300 of Australiaβs animal species are endangered, including various possums and koalas.
Number patterns with a common ratio increase in value when multiplied by a number greater than 1 Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter contents
U N SA C O M R PL R E EC PA T E G D ES
2A Factors and multiples (CONSOLIDATING) 2B Highest common factor and lowest common multiple 2C Divisibility tests (EXTENDING) 2D Prime numbers 2E Using indices 2F Prime decomposition 2G Squares and square roots 2H Number patterns (CONSOLIDATING) 2I Spatial patterns 2J Tables and rules 2K The Cartesian plane and graphs
Victorian Curriculum 2.0
This chapter covers the following content descriptors in the Victorian Curriculum 2.0: NUMBER
VC2M7N01, VC2M7N02 ALGEBRA
VC2M7A04, VC2M7A05
Please refer to the curriculum support documentation in the teacher resources for a full and comprehensive mapping of this chapter to the related curriculum content descriptors.
Β© VCAA
Online resources
A host of additional online resources are included as part of your Interactive Textbook, including HOTmaths content, video demonstrations of all worked examples, auto-marked quizzes and much more.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
2A Factors and multiples CONSOLIDATING LEARNING INTENTIONS β’ To know what factors and multiples are β’ To understand that each number has inο¬nitely many multiples β’ To be able to ο¬nd factors of a number β’ To be able to ο¬nd multiples of a number
U N SA C O M R PL R E EC PA T E G D ES
64
Many famous mathematicians have studied number patterns in an attempt to better understand our world and to assist with new scientific discoveries. Around 600 bce, the Greeks built on the early work of the Egyptians and Babylonians. Thales of Miletus, the βfather of Greek mathematicsβ, is credited for significant advances in Number Theory. One of his students, Pythagoras of Samos, went on to become one of the most well-known mathematicians to have lived. Pythagoras was primarily a religious leader, but he believed that the understanding of the world could be enhanced through the understanding of numbers. We start this chapter on Number Properties and Patterns by explaining the concepts of factors and multiples, which are key building blocks for Number Theory.
Factors of 24 are used by nurses who dispense prescription medicines in hospitals and nursing homes. For example: four times/day is every 6 hours (4 Γ 6 = 24); three times/ day is every 8 hours (3 Γ 8 = 24).
Imagine one dozen doughnuts packed into bags with 3 rows of 4 doughnuts each. Since 3 Γ 4 = 12, we can say that 3 and 4 are factors of 12. Purchasing βmultipleβ packs of one dozen doughnuts could result in buying 24, 36, 48 or 60 doughnuts, depending on the number of packs. These numbers are known as multiples of 12.
Lesson starter: The most factors, the most multiples Which number that is less than 100 has the most factors?
Which number that is less than 100 has the most multiples less than 100?
KEY IDEAS
β Factors of a particular number are numbers that divide exactly into that number.
β’
β’
For example: The factors of 20 can be found by considering pairs of numbers that multiply to give 20 which are 1 Γ 20, 2 Γ 10 and 4 Γ 5. Therefore, written in ascending order, the factors of 20 are 1, 2, 4, 5, 10, 20. Every whole number is a factor of itself and also 1 is a factor of every whole number.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2A Factors and multiples
65
β Multiples of a number are found by multiplying the given number by any whole number.
β’
For example: The multiples of β20βare β20, 40, 60, 80, 100, 120, β¦β Other numbers that are multiples of β20βinclude β480, 2000βand β68 600β. Every positive integer has infinitely many multiples.
U N SA C O M R PL R E EC PA T E G D ES
β Given the statements above, it follows that factors are less than or equal to the particular number
being considered and multiples are greater than or equal to the number being considered.
BUILDING UNDERSTANDING
1 For each of the following numbers, state whether it is a factor of β60β (ββ F)β, a multiple of β60β (ββ M)ββor neither β(N)ββ. a β120β e β6β i β22β
b 1β 4β f β5β j β600β
c 1β 5β g β240β k β70β
d 4β 0β h β2β l β1β
2 For each of the following numbers, state whether it is a factor of β26β (ββ F)β, a multiple of β26β (ββ M)ββor neither β(N)ββ. a β2β e β210β i β1β
Example 1
b 5β 4β f β27β j β26 000β
c 5β 2β g β3β k β13β
d 4β β h β182β l β39β
Finding factors
Find the complete set of factors for each of these numbers. a β15β b β40β SOLUTION
EXPLANATION
a Factors of β15βare β1, 3, 5, 15β.
β1 Γ 15 = 15,
b Factors of β40βare: β1, 2, 4, 5, 8, 10, 20, 40β.
1β Γ 40 = 40, 2 Γ 20 = 40β β4 Γ 10 = 40, 5 Γ 8 = 40β The last number you need to check is β7β, but β 40 Γ· 7 = 5βrem. β5β, so β7βis not a factor.
3 Γ 5 = 15β
Now you try
Find the complete set of factors for each of these numbers. a β21β b β30β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
66
Chapter 2
Number properties and patterns
Example 2
Listing multiples
U N SA C O M R PL R E EC PA T E G D ES
Write down the first six multiples for each of these numbers. a 1β 1β b β35β SOLUTION
EXPLANATION
a 1β 1, 22, 33, 44, 55, 66β
The first multiple is always the given number. Add on the given number to find the next multiple. Repeat this process to get more multiples.
b β35, 70, 105, 140, 175, 210β
Start at β35β, the given number, and repeatedly add β 35βto continue producing multiples.
Now you try
Write down the first six multiples for each of these numbers. a 4β β b β26β
Example 3
Finding factor pairs
Express β195βas a product of two factors, both of which are greater than β10β. SOLUTION
EXPLANATION
β195 = 13 Γ 15β
Systematically divide β195βby numbers greater than 10 in an attempt to find a large factor. The number β13βgoes into β195βa total of 015 β15βtimes with no remainder 13 1965 .
)
Now you try
Express β165βas a product of two factors, both of which are greater than β10β.
Exercise 2A FLUENCY
Example 1
Example 1
1
1, 2β4ββ(1β/2β)β
2β5β(β 1β/2β)β
2β5β(β 1β/2β)β
Find the complete set of factors for each of these numbers. a 1β 2β b β48β
2 List the complete set of factors for each of the following numbers. a 1β 0β b β24β d β36β e β60β g β80β h β29β
c β17β f β42β i β28β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2A Factors and multiples
Example 2
3 Write down the first six multiples for each of the following numbers. a β5β b β8β d β7β e β20β g β15β h β100β
67
c 1β 2β f β75β i β37β
U N SA C O M R PL R E EC PA T E G D ES
4 Fill in the gaps to complete the set of factors for each of the following numbers. a 1β 8βββ1, 2, __, 6, 9, __β ββ b β25βββ1, __, 25β ββ c 7β 2βββ _ββ_ , 2, 3, _ _ββ_ , _ _ , 8, _ _ , _ _ , 18, _ _ , 36, 72β ββ d β120ββ1, 2, __, __, __, 6, __, 10, __, __, 20, __, 30, __, 60, __β
5 Which number is the incorrect multiple for each of the following sequences? a β3, 6, 9, 12, 15, 18, 22, 24, 27, 30β b β43, 86, 129, 162, 215, 258, 301, 344β c 1β 1, 21, 33, 44, 55, 66, 77, 88, 99, 110β d β17, 34, 51, 68, 85, 102, 117, 136, 153, 170β PROBLEM-SOLVING
6, 7ββ(1β/2β)β
6β8
7β9
6 Consider the set of whole numbers from β1βto β25β inclusive. a Which number has the most factors? b Which number has the fewest factors? c Which numbers have an odd number of factors?
Example 3
7 Express each of the following numbers as a product of two factors, both of which are greater than β10β. a β192β b 3β 15β c β180β d β121β e β336β f 4β 94β
8 Zane and Matt are both keen runners. Zane takes β4βminutes to jog around a running track and Matt takes β5βminutes. They start at the same time and keep running until they both cross the finish line at the same time. a How long do they run for? b How many laps did Zane run? c How many laps did Matt run? 9 Anson has invited β12βfriends to his birthday party. He is making each of them a βlolly bagβ to take home after the party. To be fair, he wants to make sure that each friend has the same number of lollies. Anson has a total of β 300βlollies to share among the lolly bags. a How many lollies does Anson put in each of his friendsβ lolly bags? b How many lollies does Anson have left over to eat himself? Anson then decides that he wants a lolly bag for himself also. c How many lollies will now go into each of the β13βlolly bags?
After much pleading from his siblings, Anson prepares lolly bags for them also. His sister Monique notices that the total number of lolly bags is now a factor of the total number of lollies. d What are the different possible number of sibling(s) that Anson could have? e How many siblings do you expect Anson has?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
REASONING
10
10β12
12β14
10 Are the following statements true or false? a A multiple of a particular number is always smaller than that number. b β2βis a factor of every even number. c β3βis a factor of every odd number. d A factor is always greater than or equal to the given number. e When considering a particular number, that number is both a factor and a multiple of itself.
U N SA C O M R PL R E EC PA T E G D ES
68
11 6β 0βis a number with many factors. It has a total of β12βfactors and, interestingly, it has each of the numbers β1, 2, 3, 4, 5, 6βas a factor. a What is the smallest number that has β1, 2, 3, 4, 5, 6, 7βand β8βas factors? b What is the smallest number that has β1, 2, 3, 4, 5, 6, 7, 8, 9βand β10βas factors?
12 a What numbers can claim that the number β100βis a multiple of them? (For example, 5 could claim that 100 is a multiple of it.) b What are the factors of β100β? 13 All Australian AM radio stations have frequencies that are multiples of β9β. For example, a particular radio station has a frequency of β774β(kilohertz or kHz). Find three other βAMβradio stations and show their frequencies are, indeed, multiples of β9β.
14 A particular number is a multiple of β6βand also a multiple of β15β. This tells you that β6βand β15βare factors of the number. List all the other numbers that must be factors. ENRICHMENT: Factors and multiples with spreadsheets
β
β
β15β
15 a Design a spreadsheet that will enable a user to enter any number between β1βand β100βand it will automatically list the first 30 multiples of that number. b Design a spreadsheet that will enable a user to enter any particular number between β1βand β100β and it will automatically list the numberβs factors. c Improve your factor program so that it finds the sum of the factors and also states the total number of factors for the particular number. d Use your spreadsheet program to help you find a pair of amicable numbers. A pair of numbers is said to be amicable if the sum of the factors for each number, excluding the number itself, is equal to the other number. Both numbers in the first pair of amicable numbers fall between β200βand β300β. An example of a non-amicable pair of numbers: β12 : factor sum = 1 + 2 + 3 + 4 + 6 = 16β 1β 6 : factor sum = 1 + 2 + 4 + 8 = 15β The factor sum for 16 would need to be 12 for the pair to be amicable numbers.
Helpful Microsoft Excel formulas INT(number) β Rounds a number down to the nearest integer (whole number). MOD(number, divisor) β Returns the remainder after a number is divided by the divisor. IF(logical test, value if true, value if false) β Checks whether a condition is met and returns one value if true and another value if false. COUNTIF(range, criteria) β Counts the number of cells within a range that meet the given condition.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2B Highest common factor and lowest common multiple
69
2B Highest common factor and lowest common multiple
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To know the meaning of the terms highest common factor (HCF) and lowest common multiple (LCM) β’ To be able to ο¬nd the highest common factor of two numbers β’ To be able to ο¬nd the lowest common multiple of two numbers
In the previous section, factors and multiples of a number were explained. Remember that factors are less than or equal to a given number and that multiples are greater than or equal to a given number. given number e.g. 12
factors β€ 12 e.g. 1, 2, 3, 4, 6, 12
multiples β₯ 12 e.g. 12, 24, 36, 48, . . .
There are many applications in Mathematics for which the highest common factor (HCF) of two or more numbers must be determined. In particular, the skill of finding the HCF is required for the future topic of factorisation, which is an important aspect of Algebra.
Similarly, there are many occasions for which the lowest common multiple (LCM) of two or more numbers must be determined. Adding and subtracting fractions with different denominators requires the skill of finding the LCM.
People use HCF when calculating equivalent application rates, such as: pool owners simplifying a manufacturerβs chlorine application rates; farmers calculating equivalent fertiliser rates; and nurses and chemists evaluating equivalent medication rates.
Lesson starter: You provide the starting numbers!
For each of the following answers, you must determine possible starting numbers. On all occasions, the numbers involved are less than 100. 1 2 3 4 5 6
The HCF of two numbers is 12. The HCF of three numbers is 11. The LCM of two numbers is 30. The LCM of three numbers is 75. The HCF of four numbers is 1. The LCM of four numbers is 24.
Suggest two possible starting numbers. Suggest three possible starting numbers. Suggest two possible starting numbers. Suggest three possible starting numbers. Suggest four possible numbers. Suggest four possible numbers.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
KEY IDEAS β HCF stands for highest common factor. It is the highest (i.e. largest) factor that is common to
the numbers provided. β’ For example, to find the HCF of β24βand β40β: Factors of β24βare β1, 2, 3, 4, 6, 8, 12βand β24β. Factors of β40βare β1, 2, 4, 5, 8, 10, 20βand β40β. Therefore, common factors of β24βand β40βare β1, 2, 4βand β8β. Therefore, the highest common factor of β24βand β40βis β8β.
U N SA C O M R PL R E EC PA T E G D ES
70
β LCM stands for lowest common multiple. It is the lowest (i.e. smallest) multiple that is
common to the numbers provided. β’ For example, to find the LCM of β20βand β12β: Multiples of β20βare β20, 40, 60, 80, 100, 120, 140, β¦β Multiples of β12βare β12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, β¦β Therefore, common multiples of β20βand β12βare β60, 120, 180, β¦β Therefore, the lowest common multiple of β20βand β12βis β60β.
β The LCM of two numbers can always be found by multiplying the two numbers together and
dividing by their HCF. β’ For example, to find the LCM of β20βand β12β. The HCF of β20βand β12βis β4β. Therefore, the LCM of β20βand β12βis β20 Γ 12 Γ· 4 = 60β.
BUILDING UNDERSTANDING
1 The factors of β12βare β1, 2, 3, 4, 6βand β12β, and the factors of β16βare β1, 2, 4, 8βand β16β. a What are the common factors of β12βand β16β? b What is the HCF of β12βand β16β?
2 State the missing numbers to find out the HCF of β18βand β30β. Factors of β18βare β1, __, 3, __, __βand β18β. Factors of β__βare β1, __, __, 5, __, 10, __βand β30β. Therefore, the HCF of β18βand β30βis β__β.
3 The first β10βmultiples of β8βare β8, 16, 24, 32, 40, 48, 56, 64, 72βand β80β. The first β10βmultiples of β6βare β6, 12, 18, 24, 30, 36, 42, 48, 54βand β60β. a What are two common multiples of β8βand β6β? b What is the LCM of β8βand β6β?
4 State the missing numbers to find out the LCM of β9βand β15β. Multiples of β9βare β9, 18, __, 36, __, __, __, __, 81βand β__β. Multiples of β15βare β__, 30, __, 60, 75, __, __βand β120β. Therefore, the LCM of β9βand β15βis β__β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2B Highest common factor and lowest common multiple
Example 4
71
Finding the highest common factor (HCF)
Find the highest common factor (HCF) of β36βand β48β. EXPLANATION
Factors of β36β are: β1, 2, 3, 4, 6, 9, 12, 18βand β36β.
1 Γ 36 = 36, β2 Γ 18 = 36, β3 Γ 12 = 36, ββ―β―β―β―β 4 Γ 9 = 36, β6 Γ 6 = 36
Factors of β48β are: β1, 2, 3, 4, 6, 8, 12, 16, 24βand β48β.
1 Γ 48 = 48, 2 Γ 24 = 48, 3 Γ 16 = 48, ββ―β―β―β― β β β 4 Γ 12 = 48, 6 Γ 8 = 48
The HCF of β36βand β48βis β12β.
Common factors are β1, 2, 3, 4, 6βand β12β, of which β12βis the highest.
U N SA C O M R PL R E EC PA T E G D ES
SOLUTION
Now you try
Find the highest common factor (HCF) of β30βand β48β.
Example 5
Finding the lowest common multiple (LCM)
Find the lowest common multiple (LCM) of the following pairs of numbers. a β5βand β11β b β6βand β10β SOLUTION
EXPLANATION
a The LCM of β5βand β11βis β55β.
Note that the HCF of β5βand β11βis β1β. β5 Γ 11 Γ· 1 = 55β Alternatively, list multiples of 5 and 11 and then choose the lowest number in both lists.
b The LCM of β6βand β10βis β30β.
Note that the HCF of β6βand β10βis β2β. Alternatively, list multiples of both numbers. The LCM of β6βand β10βis β6 Γ 10 Γ· 2 = 30β. Multiples of β6βare β6, 12, 18, 24, 30, 36, β¦β Multiples of β10βare β10, 20, 30, 40, β¦β
Now you try
Find the lowest common multiple (LCM) of the following pairs of numbers. a β4βand β7β b β10βand β15β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
72
Chapter 2
Number properties and patterns
Exercise 2B FLUENCY
1
Find the HCF of the following pairs of numbers. a 1β 5βand β25β
b β20βand β30β
2 Find the HCF of the following pairs of numbers. a 4β βand β5β b 8β βand β13β e β16βand β20β f β15βand β60β i β80βand β120β j β75βand β125β
c 2β βand β12β g β50βand β150β k β42βand β63β
2β5ββ(1β/2β)β
3β6β(β 1β/2β)β
U N SA C O M R PL R E EC PA T E G D ES
Example 4
1, 2ββ(1β/2β)β,β 4ββ(1β/2β)β
Example 4
Example 4
Example 5
Example 5
d β3βand β15β h β48βand β72β l β28βand β42β
3 Find the HCF of the following groups of numbers. That is, find the highest number which is a factor of all the provided numbers. a 2β 0, 40, 50β b β6, 15, 42β c β50, 100, 81β d β18, 13, 21β e β24, 72, 16β f β120, 84, 144β
4 Find the LCM of the following pairs of numbers. a β4βand β9β b β3βand β7β d β10βand β11β e β4βand β6β g β12βand β18β h β6βand β9β j β12βand β16β k β44βand β12β
c f i l
β12βand β5β β5βand β10β β20βand β30β β21βand β35β
5 Find the LCM of the following groups of numbers. That is, find the lowest number which is a multiple of all the provided numbers. a β2, 3, 5β b β3, 4, 7β c β2, 3, 4β d β3, 5, 9β e β4, 5, 8, 10β f β6, 12, 18, 3β 6 Find the HCF of the following pairs of numbers and then use this information to help calculate the LCM of the same pair of numbers. a β15βand β20β b 1β 2βand β24β c 1β 4βand β21β d β45βand β27β PROBLEM-SOLVING
7, 8
8, 9
8β10
7 Find the LCM of β13βand β24β. 8 Find the HCF of β45βand β72β.
9 Find the LCM and HCF of β260βand β390β.
10 Andrew runs laps of a circuit in β4βminutes. Bryan runs laps of the same circuit in β3βminutes. Chris can run laps of the same circuit in β6β minutes. They all start together on the starting line and run a race that goes for β36β minutes. a What is the first time, after the start, that they will all cross over the starting line together? b How many laps will each boy complete in the race? c How many times does Bryan overtake Andrew during this race?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2B Highest common factor and lowest common multiple
REASONING
11
11, 12
73
12, 13
11 Given that the HCF of a pair of different numbers is β8β, find the two numbers: a if both numbers are less than β20β b when one number is in the β20sβand the other in the β30s.β
U N SA C O M R PL R E EC PA T E G D ES
12 Given that the LCM of a pair of different numbers is β20β, find the seven possible pairs of numbers.
xΓy 13 The rule for finding the LCM of two numbers βxβand βyβis _ β β. Is the rule for the LCM of three HCF (β x, y)β xΓyΓz numbers βxβ, βyβand βzβ ββ___________β?βExplain your answer. HCF (β x, y, z)β ENRICHMENT: LCM of large groups of numbers
14 a b c d
β
β
14
Find the LCM of these single-digit numbers: β1, 2, 3, 4, 5, 6, 7, 8, 9β. Find the LCM of these first β10βnatural numbers: β1, 2, 3, 4, 5, 6, 7, 8, 9, 10β. Compare your answers to parts a and b. What do you notice? Explain. Find the LCM of the first β11βnatural numbers.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
2C Divisibility tests EXTENDING LEARNING INTENTIONS β’ To know the meaning of the terms divisible, divisor, dividend, quotient and remainder β’ To understand that divisibility tests can be used to check if a number is divisible by another number without performing the division β’ To be able to test for divisibility by 2, 3, 4, 5, 6, 8, 9 and 10
U N SA C O M R PL R E EC PA T E G D ES
74
It is useful to know whether a large number is exactly divisible by another number. Although we can always carry out the division algorithm, this can be a difficult and tedious process for large numbers. There are simple divisibility tests for each of the single-digit numbers, with the exception of 7. These divisibility tests determine whether or not the number is divisible by the chosen divisor.
When a barcode is scanned, an algorithm ο¬rst checks whether the code is valid. The digits are combined using a formula, then a divisibility rule is applied to test if it is a genuine barcode.
Lesson starter: Five questions in 5 minutes In small groups, attempt to solve the following five questions in 5 minutes. 1 2 3 4 5
Some numbers greater than 1 are only divisible by 1 and themselves. What are these numbers called? Is 21 541 837 divisible by 3? What two-digit number is the βmost divisibleβ (i.e. has the most factors)? Find the smallest number that is divisible by 1, 2, 3, 4, 5 and 6. Find a number that is divisible by 1, 2, 3, 4, 5, 6, 7 and 8.
KEY IDEAS
β A number is said to be divisible by another number if there is no remainder after the division
has occurred. For example, 20 is divisible by 4 because 20 Γ· 4 = 5 with no remainder.
β If the divisor divides into the dividend exactly, then the divisor is said to be a factor of that
number.
β Division notation
remainder
The first number is called the dividend and the second number is called the divisor. Example: 27 Γ· 4 = 6 remainder 3 Another way of representing this information is 27 = 4 Γ 6 + 3.
dividend divisor
27 = 6 rem. 3 = 6 3 4 4 quotient
β Key terms
β’
Dividend Divisor Quotient
β’
Remainder
β’ β’
The starting number; the total; the amount you have The number doing the dividing; the number of groups The number of times the divisor went into the dividend, also known as βthe answerβ to a division calculation The number left over; the number remaining (sometimes written as βrem.β)
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2C Divisibility tests
75
β Divisibility tests
U N SA C O M R PL R E EC PA T E G D ES
1 All numbers are divisible by β1β. 2 All even numbers are divisible by β2β. Last digit must be a β0, 2, 4, 6βor β8β. 3 The sum of the digits must be divisible by β3β. 4 The number formed from the last two digits must be divisible by β4β. 5 The last digit must be a β0βor β5β. 6 Must pass the divisibility tests for β2βand β3β. 7 (There is no easy test for divisibility by β7β.) 8 The number formed from the last three digits must be divisible by β8β. 9 The sum of the digits must be divisible by β9β. 10 The last digit must be β0β.
BUILDING UNDERSTANDING
1 Give a reason why: a β8631βis not divisible by β2β c β426βis not divisible by β4β e β87 548βis not divisible by β6β g β3 333 333βis not divisible by β9β
b d f h
2 Give the remainder when: a β326βis divided by β3β c β72βis divided into six groups
b 2β 1 154βis divided into groups of four d β45 675βis shared into five groups
3β 1 313βis not divisible by β3β β5044βis not divisible by β5β β214 125βis not divisible by β8β β56 405βis not divisible by β10β
3 Which three divisibility tests involve calculating the sum of the digits?
4 If you saw only the last digit of a β10β-digit number, which three divisibility tests (apart from β1β) could you still apply?
Example 6
Applying divisibility tests
Determine whether or not the following calculations are possible without leaving a remainder. a β54 327 Γ· 3β b β765 146 Γ· 8β SOLUTION
EXPLANATION
a Digit sum β= 21β Yes, β54 327βis divisible by β3β.
5β + 4 + 3 + 2 + 7 = 21β 2β 1βis divisible by β3β, therefore β54327β must be divisible by β3β.
b
Check whether the last three digits are divisible by β8β. They are not, therefore the original number is not divisible by β8β.
18 rem. β2β 6 8 146
)
No, β765 146βis not divisible by β8β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
76
Chapter 2
Number properties and patterns
Now you try
U N SA C O M R PL R E EC PA T E G D ES
Determine whether or not the following calculations are possible without leaving a remainder. Give a reason why or why not. a β52 962 Γ· 3β b β309 482 Γ· 4β
Example 7
Testing for divisibility
Carry out divisibility tests on the given number and fill in the table with ticks or crosses. Number
Divisible by 2
Divisible by 3
Divisible by 4
Divisible by 5
Divisible by 6
Divisible by 8
Divisible by 9
Divisible by 10
Divisible by 2
Divisible by 3
Divisible by 4
Divisible by 5
Divisible by 6
Divisible by 8
Divisible by 9
Divisible by 10
β
β
β
β48 569 412β
SOLUTION Number
β48 569 412β
β
β
β
β
β
EXPLANATION
4β 8 569 412βis an even number and therefore is divisible by β2β. β48 569 412βhas a digit sum of β39βand therefore is divisible by β3β, but not by β9β. β48 569 412βis divisible by β2βand β3β; therefore it is divisible by β6β. The last two digits are β12β, which is divisible by β4β. The last three digits are β412β, which is not divisible by β8β. The last digit is a β2βand therefore β48 569 412βis not divisible by β5βor β10β.
Now you try
Carry out divisibility tests on the number β726 750βto decide which of the following numbers it is divisible by: β2, 3, 5, 6, 8, 9βand β10β.
Exercise 2C FLUENCY
Example 6
Example 6
1
1β5
1β6ββ(β1/2β)β
6, 7
Use the divisibility test for β3βto determine whether the following numbers are divisible by β3β. a 5β 72β b 8β 7β c β6012β d β7301β
2 Use the divisibility tests for β2βand β5βto determine whether the following calculations are possible without leaving a remainder. a 4β 308 Γ· 2β b 5β 22 Γ· 5β c β7020 Γ· 5β d β636 Γ· 2β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2C Divisibility tests
Example 6
3 Use the divisibility test for β9βto determine whether the following calculations are possible without leaving a remainder. a β363 Γ· 9β b β702 Γ· 9β c β2143 Γ· 9β d β8712 Γ· 9β 4 Use the divisibility test for β6βto determine whether the following numbers are divisible by β6β. a β332β b β402β c β735β d β594β
U N SA C O M R PL R E EC PA T E G D ES
Example 6
77
Example 6
Example 6
Example 7
5 Use the divisibility tests for β4βand β8βto determine whether the following calculations are possible without leaving a remainder. a β528 Γ· 4β b β714 Γ· 4β c β9200 Γ· 8β d β5100 Γ· 8β
6 Determine whether the following calculations are possible without leaving a remainder. a β23 562 Γ· 3β b β39 245 678 Γ· 4β c β1 295 676 Γ· 9β d β213 456 Γ· 8β e 3β 193 457 Γ· 6β f β2 000 340 Γ· 10β g β51 345 678 Γ· 5β h β215 364 Γ· 6β i β9543 Γ· 6β j β25 756 Γ· 2β k β56 789 Γ· 9β l β324 534 565 Γ· 5β 7 Carry out divisibility tests on the given numbers and fill in the table with ticks or crosses. Number
Divisible by 2
Divisible by 3
Divisible by 4
Divisible by 5
Divisible by 6
Divisible by 8
Divisible by 9
Divisible by 10
β243 567β
β28 080β
β189 000β
β1 308 150β
β1 062 347β
PROBLEM-SOLVING
8, 9ββ(β1/2β)β
8, 9β(β β1/2β)β,β 10
11β14
8 a Can Julie share β$113βequally among her three children? b Julie finds one more dollar on the floor and realises that she can now share the money equally among her three children. How much do they each receive? 9 Write down five two-digit numbers that are divisible by: a β5β b 3β β c 2β β d 6β β e 8β β f 9β β g β10β h 4β β
10 The game of βclustersβ involves a group getting into smaller-sized groups as quickly as possible once a particular group size has been called out. If a year level consists of β88βstudents, which group sizes would ensure no students are left out of a group? 11 How many of the whole numbers between β1βand β250βinclusive are not divisible by β5β? 12 How many two-digit numbers are divisible by both β2βand β3β?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
13 Blake is older than β20βbut younger than β50β. His age is divisible by β2, 3, 6βand β9β. How old is he? 14 Find the largest three-digit number that is divisible by both β6βand β7β. REASONING
15
15, 16
16, 17
15 A number is divisible by β15βif it is divisible by β3βand by β5β. a Determine if β3225βis divisible by β15β, explaining why or why not. b Determine if β70 285βis divisible by β15β, explaining why or why not. c The number β2β β47β is divisible by β15β. Each box stands for a single missing digit and the two missing digits could be the same as each other or different digits. What could this number be? Try to list as many as possible.
U N SA C O M R PL R E EC PA T E G D ES
78
16 a Is the number β968 362 396 392 139 963 359βdivisible by β3β? b Many of the digits in the number above can actually be ignored when calculating the digit sum. Which digits can be ignored and why? c To determine if the number above is divisible by β3β, only five of the β21βdigits actually need to be added together. Find this βreducedβ digit sum.
17 The divisibility test for the numeral β4βis to consider whether the number formed by the last two digits is a multiple of β4β. Complete the following sentences to make a more detailed divisibility rule. a If the second-last digit is even, the last digit must be either a ____, ____ or ____. b If the second-last digit is odd, the last digit must be either a ____ or ____. ENRICHMENT: Divisible by β11β?
18 a b c d
β
β
18
Write down the first nine multiples of β11β. What is the difference between the two digits for each of these multiples? Write down some three-digit multiples of β11β. What do you notice about the sum of the first digit and the last digit?
The following four-digit numbers are all divisible by β11β: β1606β, β2717β, β6457β, β9251β, β9306β
e Find the sum of the odd-placed digits and the sum of the even-placed digits. Then subtract the smaller sum from the larger. What do you notice? f Write down a divisibility rule for the number β11β. g Which of the following numbers are divisible by β11β? i β2 594 669β ii β45 384 559β iii β488 220β iv β14 641β v β1 358 024 679β vi β123 456 789 987 654 321β
An alternative method is to alternate adding and subtracting each of the digits. For example: β4 134 509 742βis divisible by β11β. Alternately adding and subtracting the digits will give the following result: β4 β 1 + 3 β 4 + 5 β 0 + 9 β 7 + 4 β 2 = 11β h Try this technique on some of your earlier numbers.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2D Prime numbers
79
2D Prime numbers
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To know what prime numbers and composite numbers are β’ To be able to determine whether a number is prime by considering its factors β’ To be able to ο¬nd the prime factors of a given number
A prime number is defined as a positive whole number that has exactly two distinct factors: 1 and itself. It is believed that prime numbers (i.e. positive whole numbers with only two factors) were first studied by the ancient Greeks. More recently, the introduction of computers has allowed for huge developments in this field. Computers have allowed mathematicians to determine which large numbers are primes. Programs have also been written to automatically generate huge prime numbers that could not be calculated previously by hand. There are some interesting prime numbers that have patterns in their digits; for example, 12 345 678 901 234 567 891. This is known as an ascending prime.
Cryptography makes online transactions secure by secretly selecting two large prime numbers and ο¬nding their product. It would take years for a powerful computer to ο¬nd the prime factors of this product, so the data is safe.
You can also get palindromic primes, such as 111 191 111 and 123 494 321.
Below is a palindromic prime number that reads the same upside down or when viewed in a mirror. |88808|80888|
Lesson starter: How many primes? How many numbers from 1 to 100 are prime?
You and a classmate have 4 minutes to come up with your answer.
KEY IDEAS
β A prime number is a positive whole number that has exactly two distinct factors: 1 and itself. β The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17 and 19.
β A number that has more than two factors is called a composite number. β The numbers 0 and 1 are neither prime nor composite numbers. β The number 2 is prime. It is the only even prime number.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
BUILDING UNDERSTANDING 1 The factors of β12βare β1, 2, 3, 4, 6βand β12β. Is β12βa prime number? 2 The factors of β13βare β1βand β13β. Is β13βa prime number? 3 State the first β10βprime numbers.
U N SA C O M R PL R E EC PA T E G D ES
80
4 State the first β10βcomposite numbers.
5 What is the first prime number greater than β100β?
6 Explain why β201βis not the first prime number greater than β200β. (Hint: find another factor besides β1βand β201β.)
Example 8
Determining whether a number is a prime or composite
State whether each of these numbers is a prime or composite: β22, 35, 17, 11, 9, 5β. SOLUTION
EXPLANATION
Prime: β5, 11, 17β
5β , 11, 17βhave only two factors (β1β and themselves).
Composite: β9, 22, 35β
β9, 22βand β35βeach have more than two factors. β9 = 3 Γ 3, 22 = 2 Γ 11, 35 = 5 Γ 7β
Now you try
State whether each of these numbers is prime or composite: β19, 32, 13, 79, 57, 95β.
Example 9
Finding prime factors
Find the prime factors of β30β. SOLUTION
EXPLANATION
Factors of β30β are:β1, 2, 3, 5, 6, 10, 15, 30β
Find the entire set of factors first.
Prime numbers from this list of factors are β 2, 3βand β5β.
Include only the prime numbers in your list.
Now you try Find the prime factors of β84β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2D Prime numbers
81
Exercise 2D FLUENCY
1
2β3β(β 1β/2β)β,β 4
2β3β(β 1β/3β)β,β 4
State whether each of the following is a prime (β P)βor composite (β C)ββnumber. a β7β b β24β c β29β
U N SA C O M R PL R E EC PA T E G D ES
Example 8
1, 2β3ββ(1β/2β)β
Example 8
Example 9
2 State whether each of the following is a prime β(P)βor composite (β C)ββnumber. a β14β b 2β 3β c β70β d β37β e β51β g β19β h 3β β i β8β j β49β k β99β m 2β β n 3β 1β o β39β p β89β q β71β
f l r
2β 7β β59β 1β 03β
3 Find the prime factors of: a β42β b 3β 9β
f
3β 6β
c 6β 0β
d β25β
4 List the composite numbers between, but not including: a 3β 0βand β50β b β50βand β70β PROBLEM-SOLVING
e 2β 8β
c 8β 0βand β100β
5
5ββ(1β/2β)β, 6, 7
6β8
5 The following are not prime numbers, yet they are the product (β Γ)βof two primes. Find the two primes for each of the following numbers. a β55β b 9β 1β c β143β d β187β e β365β f 1β 33β 6 Which one of these composite numbers has the fewest prime factors? β12, 14, 16, 18, 20β
7 Twin primes are pairs of primes that are separated from each other by only one even number; for example, β3βand β5βare twin primes. Find three more pairs of twin primes.
8 1β 3βand β31βare known as a pair of βreverse numbersβ. They are also both prime numbers. Find any other two-digit pairs of prime reverse numbers. REASONING
9
9, 10
10, 11
9 Many mathematicians believe that every even number greater than β2βis the sum of two prime numbers. For example, β28βis β11 + 17β. Show how each of the even numbers between β30βand β50βcan be written as the sum of two primes. 10 Give two examples of a pair of primes that add to a prime number. Explain why all possible pairs of primes that add to a prime must include the number β2β. 11 Find three different prime numbers that are less than β100βand which sum to a fourth different prime number. There are more than β800βpossible answers. See how many you can find. ENRICHMENT: Prime or not prime?
β
β
12
12 Design a spreadsheet that will check whether or not any number entered between β1βand β1000βis a prime number. If your spreadsheet is successful, someone should be able to enter the number β773βand very quickly be informed whether or not this is a prime number. You may choose to adapt your factor program (Enrichment activity Exercise 2A, Question 15).
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
2E Using indices LEARNING INTENTIONS β’ To know the meaning of the terms powers, index form, basic numeral, base number and index number β’ β’ β’
To understand what a b means when a and b are whole numbers To be able to write a product in index form if there are repeated factors To be able to evaluate numeric expressions involving powers using multiplication
U N SA C O M R PL R E EC PA T E G D ES
82
When repeated multiplication of a number occurs, the expression can be simplified using powers. This involves writing the repeated factor as the base number and then including an index number to indicate how many times this factor must be multiplied by itself. This is also known as writing a number in index form. For example,
2 Γ 2 Γ 2 can be written as 23. The base number is 2 and the index number is 3.
Powers are also used to represent very large and very small numbers. For example, 400 000 000 000 000 would be written as 4 Γ 1014. This way of writing a number is called standard form or scientific notation, as there are some very large numbers involved in science.
Lesson starter: A better way β¦ β’ β’
103 seconds β 17 minutes; 106 seconds β 12 days; 109 seconds β 32 years. 1012 seconds ago β 31710 years ago, before any written language, before the pyramids of Egypt, and when prehistoric artists painted the walls of Chauvet Cave, France.
What is a better way of writing the calculation 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 (that is not the answer, 20)? What is a better way of writing the calculation 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2 (that is not the answer, 1024)?
You may need to access the internet to find out some of the following answers.
Computers have the capacity to store a lot of information. As you most likely know, computer memory is given in bytes. β’ β’ β’ β’ β’ β’
How many bytes (B) are in a kilobyte (kB)? How many kilobytes are in a megabyte (MB)? How many megabytes are in a gigabyte (GB)? How many gigabytes are in a terabyte (TB)? How many bytes are in a gigabyte? (Hint: It is over 1 billion and it is far easier to write this number as a power!) Why do computers frequently use base 2 (binary) numbers?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2E Using indices
83
KEY IDEAS β Powers are used to help write expressions involving repeated multiplication in a simplified
form using indices.
U N SA C O M R PL R E EC PA T E G D ES
For example: β8 Γ 8 Γ 8 Γ 8 Γ 8βcan be written as 8β β 5β β β When writing a basic numeral as a power, you need a base number and an index number. This
is also known as writing an expression in index form.
index number
85 = 32 768
base number
basic numeral
β aβ β βbβreads as ββaβto the power of βbββ. In expanded form it would look like:
βξ½ξ ξ ξ ξΎ β―β―β― a Γβ―β― a Γ a Γ a Γ aβ¦. . . Γβ aβ the factor a appears b times
β Powers take priority over multiplication and division in the order of operations.
For example : 3 + 2 Γ 4β β 2β β = 3 + 2 Γ 16 β ββ―β― ββ β =β 3 + 32β β ββ―β― β β = 35
β Note: 2β β 3β ββ 2 Γ 3β, that is, 2β β 3β ββ 6β.
Instead: 2β β β3β= 2 Γ 2 Γ 2 = 8β.
BUILDING UNDERSTANDING
1 Select the correct answer from the following alternatives. 3β β 7β ββmeans: A β3 Γ 7β D β3 Γ 7 Γ 3 Γ 7 Γ 3 Γ 7 Γ 3β
B 3β Γ 3 Γ 3β E β3 Γ 3 Γ 3 Γ 3 Γ 3 Γ 3 Γ 3β
C 7β Γ 7 Γ 7β F β37β
2 Select the correct answer from the following alternatives. β9 Γ 9 Γ 9 Γ 9 Γ 9βcan be simplified to: A β9 Γ 5β B β5 Γ 9β 5 D 9β βββ E β99 999β
C 5β β 9β β F β95β
3 State the missing values in the table. Index form
Base number
Index number
Basic numeral
β 2β β β3β
2β β
β3β
β8β
β 5β β 2β β
10β β 4β β ββ 2β β 7β β
β 1β β β12β 12β β β1β ββ 0β β β5β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
Example 10 Converting to index form Simplify the following expressions by writing them in index form. a 5β Γ 5 Γ 5 Γ 5 Γ 5 Γ 5β b β3 Γ 3 Γ 2 Γ 3 Γ 2 Γ 3β
U N SA C O M R PL R E EC PA T E G D ES
84
SOLUTION a
5β Γ 5 Γ 5 Γ 5 Γ 5 Γ 5 = 5β β 6β β
b 3Γ3Γ2Γ3Γ2Γ3 = 2 Γ 2 Γ 3 Γ 3 Γ 3 Γβ 3ββ β―β― ββ―β― = 2β β 2β βΓ 3β β 4β β
EXPLANATION
The number β5βis the repeated factor and it appears six times.
First, write the factors in increasing order (all the 2s before the 3s). The number β2βis written two times, and the number β3βis written four times.
Now you try
Simplify the following expressions by writing them in index form. a 7β Γ 7 Γ 7 Γ 7 Γ 7β b β5 Γ 5 Γ 11 Γ 5 Γ 11 Γ 11 Γ 5β
Example 11 Expanding a power Expand and evaluate the following terms. a 2β β 4β β
b 2β β 3β βΓ 5β β 2β β
SOLUTION
EXPLANATION
=β 2 Γ 2 Γ 2 Γ 2β a 2β ββ 4β βββ―β― β = 16
Write β2βdown four times and multiply.
β 2β β = 2 Γ 2 Γ 2 Γ 5 Γ 5 b 2β β 3β βΓ 5β β =β 8 Γ 25β β β―β― β β = 200
Write the number β2βthree times, and the number β5β, two times.
Now you try
Expand and evaluate the following terms. a 5β β 3β β
b 2β β 3β βΓ 10β β 2β β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2E Using indices
85
Example 12 Evaluating expression with powers b β2 Γ 3β β 3β β+ 10β β 2β β+ 1β β 7β β
U N SA C O M R PL R E EC PA T E G D ES
Evaluate: a 7β β 2β ββ 6β β 2β β SOLUTION a
β 2β β = 7 Γ 7 β 6 Γ 6 7β β 2β ββ 6β β
ββ―β― β =β 49 β 36β β = 13
b 2 Γ 3β β β3β+ 10β β 2β β+ 1β β 7β β β = 2 Γ 3 Γ 3 Γ 3 + 10 Γ 10 + 1 Γ 1 Γ 1 β β―β―β―β― β―β― β ββ―β― β β Γ 1 Γ 1 Γ 1 Γβ 1β β = 54 + 100 + 1 β = 155
EXPLANATION
Write in expanded form (optional step). Powers are evaluated before the subtraction occurs.
Write in expanded form (optional step). Follow order of operation rules. Carry out the multiplication first, then carry out the addition.
Now you try Evaluate: β 2β β a 5β β 2β ββ 3β
b β5 Γ 2β β 3β β+ 4β β 2β ββ 3β β 2β β
Exercise 2E FLUENCY
Example 10a
Example 10a
Example 10b
Example 10b
Example 11a
1
1, 2βββ3ββ(β1/2β)β,β 5βββ7ββ(β1/2β)β
Simplify the following expressions by writing them in index form. a β3 Γ 3 Γ 3β b β2 Γ 2 Γ 2 Γ 2 Γ 2β d β10 Γ 10 Γ 10 Γ 10β e β6 Γ 6β
2β3β(β β1/2β)β,β 5βββ8ββ(β1/2β)β
2βββ3β(β β1/3β)β, 4, 5βββ8ββ(β1/3β)β
c 1β 5 Γ 15 Γ 15 Γ 15β f β1 Γ 1 Γ 1 Γ 1 Γ 1 Γ 1β
2 Simplify the following expressions by writing them in index form. a 4β Γ 4 Γ 5 Γ 5 Γ 5β b β3 Γ 3 Γ 3 Γ 3 Γ 7 Γ 7β
c β2 Γ 2 Γ 2 Γ 5 Γ 5β
3 Simplify the following expressions by writing them as powers. a β3 Γ 3 Γ 5 Γ 5β b β7 Γ 7 Γ 2 Γ 2 Γ 7β d β8 Γ 8 Γ 5 Γ 5 Γ 5β e β6 Γ 3 Γ 6 Γ 3 Γ 6 Γ 3β g β4 Γ 13 Γ 4 Γ 4 Γ 7β h β10 Γ 9 Γ 10 Γ 9 Γ 9β
c 1β 2 Γ 9 Γ 9 Γ 12β f β13 Γ 7 Γ 13 Γ 7 Γ 7 Γ 7β i β2 Γ 3 Γ 5 Γ 5 Γ 3 Γ 2 Γ 2β
4 Simplify by writing using powers. β2 Γ 3 Γ 5 Γ 5 Γ 3 Γ 3 Γ 2 Γ 2 Γ 2 Γ 5 Γ 3 Γ 2 Γ 2 Γ 5 Γ 3β 5 Expand these terms. (Do not evaluate.) a 2β β β4β b 17β β 2β β e 14β β 4β β f 8β β 8β β
c 9β β 3β β g 10β β 5β β
d 3β β 7β β h 54β β 3β β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
86
Chapter 2
Example 11b
6 Expand these terms. (Do not evaluate.) a 3β β 5β βΓ 2β β 3β β b 4β β 3β βΓ 3β β 4β β e β5 Γ 7β β 4β β f 2β β 2β βΓ 3β β 3β βΓ 4β β β1β
c 7β β 2β βΓ 5β β 3β β g 11β β 5β βΓ 9β β β2β
d 4β β 6β βΓ 9β β 3β β h 20β β 3β βΓ 30β β β2β
7 Evaluate: a 2β β 5β β e 10β β 4β β
c 10β β 3β β g 1β β 6β βΓ 2β β 6β β
d 3β β 2β βΓ 2β β 3β β h 11β β 2β βΓ 1β β 8β β
b 8β β 2β β f 2β β 3β βΓ 5β β 3β β
U N SA C O M R PL R E EC PA T E G D ES
Example 12
Number properties and patterns
Example 12
8 Evaluate: a 3β β 2β β+ 4β β 2β β d (ββ9 β 5)3β β g 1β β 4β β+ 2β β 3β β+ 3β β 2β β+ 4β β 1β β
c 8β β β2ββ 2 Γ 3β β 3β β 7 f 2β β β ββ 1 Γ 2 Γ 3 Γ 4 Γ 5β i β(1β β 27 β β+ 1β β 23 β )β βΓ 2β β 2β β
b 2β Γ 5β β β2ββ 7β β 2β β 3 4 e 2β β β βΓ 2β βββ 3 h 10β β β ββ 10β β 2β β
9βββ10ββ(1β/2β)β
PROBLEM-SOLVING
9βββ10β(β 1β/2β)β,β 11
9 Determine the index number for the following basic numerals. a 1β 6 = 2β β ?β β b 1β 6 = 4β β ?β β c β64 = 4β β ?β β e β27 = 3β β ?β β f β100 = 10β β ?β β g β49 = 7β β ?β β
9βββ10ββ(1β/4β)β, 11, 12
d β64 = 2β β ?β β h β625 = 5β β ?β β
10 Write one of the symbols β< , =βor β>βin the box to make the following statements true. a 2β β 6β ββ 2β β 9β β b 8β β 3β ββ 8β β 2β β c 2β β 4β ββ 4β β 2β β d 3β β 2β ββ e 6β β 4β ββ
5β β 3β β
f
12β β 2β ββ
3β β 4β β
g 11β β 2β ββ
4β β 2β β
h 1β β 8β ββ
2β β 7β β
2β β 3β β
11 A text message is sent to five friends. Each of the five friends then forwards it to five other friends and each of these people also sends it to five other friends. How many people does the text message reach, not including those who forwarded the message?
12 Jane writes a chain email and sends it to five friends. Assume each person who receives the email reads it and, within β5βminutes of the email arriving, sends it to five other people who have not yet received it. a Show, with calculations, why at least β156βpeople (including Jane) will have read the email within β15βminutes of Jane sending it. b If the email always goes to a new person, and assuming every person in Australia has an email address and access to email, how long would it take until everyone in Australia has read the message? (Australian population is approx. β26βmillion people.) c How many people will read the email within β1β hour? d Using the same assumptions as above, how long would it take until everyone in the world has read the message? (World population is approx. β8βbillion people.) e Approximately how many people will have read the email within β2β hours? REASONING
13
13, 14
13ββ(1β/2)β β, 14, 15
13 Write the correct operation β(+ , β , Γ, Γ· )βin the box to make the following equations true. a 3β β 2β ββ 4β β 2β β= 5β β 2β β b 2β β β4ββ 4β β 2β β= 4β β 4β β c 2β β β7ββ β5β β 3β β= 3β β 1β β d 9β β 2β ββ
3β β 4β β= 1β β 20 β β
e 10β β β2ββ
10β β 2β β= 10β β 4β β
f
10β β β2ββ
8β β 2β β= 6β β 2β β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2E Using indices
U N SA C O M R PL R E EC PA T E G D ES
14 The sequence 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 gives the powers of 2 less than 1000. a Write the sequence of powers of 3 less than 1000 (starting 1, 3, 9, β¦) b Write the sequence of powers of 5 less than 1000 (starting 1, 5, 25, β¦) c i For the original sequence (of powers of 2), list the difference between each term and the next term (for example: 2 β 1 = 1, 4 β 2 = 2, β¦) ii What do you notice about this sequence of differences? d i List the difference between consecutive terms in the powers of 3 sequence. ii What do you notice about this sequence of differences?
87
15 Find a value for a and for b such that a β b and a b = ba. ENRICHMENT: Investigating factorials
β
β
16
16 In mathematics, the exclamation mark (!) is the symbol for factorials. 4! = 4 Γ 3 Γ 2 Γ 1 = 24 n! = n Γ (n β 1) Γ (n β 2) Γ (n β 3) Γ (n β 4) Γ β¦ Γ 6 Γ 5 Γ 4 Γ 3 Γ 2 Γ 1 a Evaluate 1!, 2!, 3!, 4!, 5! and 6! Factorials can be written in prime factor form, which involves powers. For example: 6! = 6 Γ 5 Γ 4 Γ 3 Γ 2 Γ 1 = (2 Γ 3) Γ 5 Γ (2 Γ 2) Γ 3 Γ 2 Γ 1 = 24 Γ 32 Γ 5 b Write these numbers in prime factor form. i 7! ii 8! iii 9! iv 10! c Write down the last digit of 12! d Write down the last digit of 99! e Find a method of working out how many consecutive zeros would occur on the right-hand end of each of the following factorials if they were evaluated. (Hint: Consider prime factor form.) i 5! ii 6! iii 15! iv 25! f 10! = 3! Γ 5! Γ 7! is an example of one factorial equal to the product of three factorials. Express 24! as the product of two or more factorials.
If you could ο¬y at the speed of light, covering 9.5 Γ 1012 km per year (ten trillion km/year), it would take you 106 years (one million years) to cross our Milky Way galaxy.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
2F Prime decomposition LEARNING INTENTIONS β’ To understand that composite numbers can be broken down into a product of prime factors β’ To be able to use a factor tree to ο¬nd the prime factors of a number (including repeated factors) β’ To be able to express a prime decomposition using powers of prime numbers
U N SA C O M R PL R E EC PA T E G D ES
88
All composite numbers can be broken down (i.e. decomposed) into a unique set of prime factors. A common way of performing the decomposition into prime factors is using a factor tree. Starting with the given number, βbranchesβ come down in pairs, representing a pair of factors that multiply to give the number above it. This process continues until prime factors are reached.
Lesson starter: Composition of numbers from prime factors
βComposeβ composite numbers by multiplying the following sets of prime factors. The first one has been done for you. a 2 Γ 3 Γ 5 = 30 e 13 Γ 17 Γ 2
c 32 Γ 23 g 25 Γ 34 Γ 7
b 2Γ3Γ7Γ3Γ2 f 22 Γ 52 Γ 72
d 5 Γ 11 Γ 22 h 11 Γ 13 Γ 17
The reverse of this process is called decomposition, where a number like 30 is broken down into the prime factors 2 Γ 3 Γ 5.
KEY IDEAS
β Every composite number can be expressed as a product of its prime factors. β A factor tree can be used to show the prime factors of a composite number. β Each βbranchβ of a factor tree eventually terminates in a prime factor.
β Powers are often used to efficiently represent composite numbers in prime factor form.
For example:
a pair of βbranchesβ
2
βbranchesβ terminate on prime factors
starting composite number
48
4
12
2
4
2
3
2
β΄ 48 = 2 Γ 2 Γ 2 Γ 2 Γ 3 = 24 Γ 3 expressed with powers
β It does not matter which pair of factors you choose to start a factor tree. The final result of
prime factors will always be the same. β It is conventional to write the prime factors in ascending (i.e. increasing) order.
For example: 600 = 23 Γ 3 Γ 52 Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2F Prime decomposition
89
BUILDING UNDERSTANDING 1 Sort the following list of numbers into two groups: composite numbers and prime numbers. β15, 13, 7, 5, 8, 9, 27, 23, 11, 4, 12, 2β
U N SA C O M R PL R E EC PA T E G D ES
2 State the missing numbers in the empty boxes to complete the following factor trees. a b c 40 30 100 8
5
10
2
2
2
3
5
2
3 State the numbers on the next set of branches for each of the following factor trees. a b c 56 220 90 9
10
4
14
4 State the following prime factors, using powers. a β2 Γ 3 Γ 3 Γ 2 Γ 2β c β7 Γ 2 Γ 3 Γ 7 Γ 2β
55
4
b 5β Γ 3 Γ 3 Γ 3 Γ 3 Γ 5β d β3 Γ 3 Γ 2 Γ 11 Γ 11 Γ 2β
Example 13 Expressing composite numbers in prime factor form Express the number β60βin prime factor form. SOLUTION 60
5
12
3
4
2 2 β΄ 60 = 2 Γ 2 Γ 3 Γ 5 60 = 22 Γ 3 Γ 5
EXPLANATION
A pair of factors for β60βis β5 Γ 12β. The β5βbranch terminates since β5βis a prime factor. A pair of factors for β12βis β3 Γ 4β. The β3βbranch terminates since β3βis a prime factor. A pair of factors for β4βis β2 Γ 2β. Both these branches are now terminated. Hence, the composite number, β60β, can be written as a product of each terminating branch. The factors are listed in increasing order.
Now you try Express the number β180βin prime factor form.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
90
Chapter 2
Number properties and patterns
Exercise 2F FLUENCY
1
2βββ3ββ(1β/2β)β
2βββ3ββ(1β/4β)β
Express the following numbers in prime factor form. a 3β 6β b β100β
U N SA C O M R PL R E EC PA T E G D ES
Example 13
1, 2ββ(1β/2β)β
Example 13
Example 13
2 Express the following numbers in prime factor form. a 7β 2β b β24β d β44β e β124β g β96β h β16β j β111β k β64β
c f i l
β38β β80β β75β β56β
3 Express these larger numbers in prime factor form. Note that it can be helpful to start your factor tree with a power of β10β(e.g. β100β) on one branch. a β600β b β800β c β5000β d β2400β e β1 000 000β f β45 000β g β820β h β690β PROBLEM-SOLVING
4, 5
4βββ6
5βββ7
4 Match the correct composite number (a to d) to its set of prime factors (A to D). a 1β 20β A 2β β 4β βΓ 3β β 2β β b β144β B β2 Γ 3 Γ 5β β 2β β 2 2 c β180β C 2β β β βΓ 3β β β βΓ 5β d β150β D β2 Γ 3 Γ 2 Γ 5 Γ 2β
5 Find the smallest composite number that has the five smallest prime numbers as factors.
6 a Express β144βand β96βin prime factor form. b By considering the prime factor form, determine the highest common factor (HCF) of β144βand β96β.
7 a Express β25 200βand β77 000βin prime factor form. b By considering the prime factor form, determine the highest common factor (HCF) of β25 200βand β 77 000β. REASONING
8
8βββ10
9βββ12
8 The numbers β15βand β21βcan be written in prime factor form as β3 Γ 5βand β3 Γ 7β. This means the lowest common multiple (LCM) of β15βand β21βis β3 Γ 5 Γ 7β(which is β105β). a Use the prime factor forms to find the LCM of β14βand β21β. b Use the prime factor forms to find the LCM of β15βand β33β. c β60 = 2β β 2β βΓ 3 Γ 5βand β210 = 2 Γ 3 Γ 5 Γ 7β. Use this to find the LCM of β60βand β210β.
9 Represent the number β24βwith four different factor trees, each resulting in the same set of prime factors. Note that simply swapping the order of a pair of factors does not qualify it as a different form of the factor tree.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2F Prime decomposition
U N SA C O M R PL R E EC PA T E G D ES
10 Only one of the following is the correct set of prime factors for 424. A 22 Γ 32 Γ 5 B 2 Γ 32 Γ 52 C 53 Γ 8 D 23 Γ 53 a Justify why you can eliminate alternatives A and B straight away. b Why can option C be discarded as an option? c Show that option D is the correct answer.
91
11 a State the error in each of the following prime factor trees and/or prime decompositions. i ii iii 60 60 60 5
10
2 5 60 = 2 Γ 52
2
2
30
30
2
6 5 60 = 2 Γ 5 Γ 6
15
3 5 60 = 2 Γ 3 Γ 5
b What is the correct way to express 60 in prime factor form?
12 Write 15 different (i.e. distinct) factor trees for the number 72. ENRICHMENT: Four distinct prime factors
β
β
13β15
13 There are 16 composite numbers that are smaller than 1000 which have four distinct prime factors. For example: 546 = 2 Γ 3 Γ 7 Γ 13. By considering the prime factor possibilities, find all 16 composite numbers and express each of them in prime factor form.
14 A conjecture is a statement that may appear to be true but has not been proved conclusively. Goldbachβs conjecture states that βEvery even number greater than 2 is the sum of two prime numbers.β For example, 53 = 47 + 5. Challenge: Try this for every even number from 4 to 50. 15 Use the internet to find the largest known prime number.
Supercomputers like this have been used to search for prime numbers with millions of digits. Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
92
Chapter 2
2A
1
Find the complete set of factors for each of these numbers. a 1β 6β b β70β
2 Write down the first four multiples for each of these numbers. a 7β β b β20β
U N SA C O M R PL R E EC PA T E G D ES
Progress quiz
2A
Number properties and patterns
2B
2B
2C
Ext
2C
Ext
2D
2D
2E
2E
2F
3 Find the HCF of the following groups of numbers. a 1β 5βand β10β b β36β, β54βand β72β
4 Find the LCM of the following groups of numbers. a 8β βand β12β b β3β, β5βand β9β
5 Use divisibility rules to determine whether the following calculations are possible without leaving a remainder. Give a reason for each answer. a 3β 4 481 Γ· 4β b β40 827 Γ· 3β c β824 730 Γ· 6β d β5 247 621 Γ· 9β
6 The game of βclustersβ involves a group getting into smaller-sized groups as quickly as possible once a particular group size has been called out. If a year level consists of β120βstudents, which group sizes (of more than one person) would ensure no students are left out of a group?
7 State whether each of the following is a prime (P) or composite (C) number or neither (N). Give reasons. a β60β b β1β c β13β d β0β 8 Find the prime numbers that are factors of these numbers. a β35β b β36β
9 Simplify the following expressions by writing them as powers. a β5 Γ 5 Γ 5 Γ 5β b β7 Γ 3 Γ 7 Γ 7 Γ 3 Γ 7 Γ 7β 10 Expand and evaluate the following terms. a 3β β 4β β b 1β β 4β βΓ 3β β 2β β 1 c 5β β β βΓ 10β β 4β β d (ββ12 β 8)2β β e 9β β β2ββ 3β β 3β βΓ 2β
11 Express the following numbers in prime factor form, writing factors in ascending order. a β24β b β180β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2G Squares and square roots
93
2G Squares and square roots
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand what it means to square a number or to take its square root β’ To know what a perfect square is β’ To be able to ο¬nd the square of a number β’ To be able to ο¬nd the square root of a perfect square β’ To be able to locate square roots between two consecutive whole numbers
A square number can be illustrated by considering the area of a square with a whole number as its side length. For example: Area of square = 4 cm Γ 4 cm = 16 cm2
4 cm
4 cm
Therefore, 16 is a square number.
Another way of representing square numbers is through a square array of dots. For example: Number of dots = 3 rows of 3 dots = 3 Γ 3 dots = 32 dots = 9 dots Therefore, 9 is a square number.
To produce a square number you multiply the number by itself. All square numbers written in index form will have a power of 2. Finding a square root of a number is the opposite of squaring a number. _ For example: 42 = 16 and therefore β16 = 4.
To find square roots we use our knowledge of square numbers. A calculator is also frequently used to find square roots. Geometrically, the square root of a number is the side length of a square whose area is that number.
Lesson starter: Speed squaring tests
In pairs, test one anotherβs knowledge of square numbers. β’ β’ β’
Ask 10 quick questions, such as β3 squaredβ, β5 squaredβ etc. Have two turns each. Time how long it takes each of you to answer the 10 questions. Aim to be quicker on your second attempt.
Write down the first 10 square numbers. β’ β’ β’
Begin to memorise these important numbers. Time how quickly you can recall the first 10 square numbers without looking at a list of numbers. Can you go under 5 seconds?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
KEY IDEAS β Any whole number multiplied by itself produces a square number.
For example: 5β β 2β β= 5 Γ 5 = 25β. Therefore, β25βis a square number. β’ Square numbers are also known as perfect squares. β’ The first β12βsquare numbers (not including 0, which is counted as the βzeroβth square numberβ) are:
U N SA C O M R PL R E EC PA T E G D ES
94
β’
β’
Index form
1β β 2β β
2β β 2β β
3β β 2β β
4β β 2β β
5β β 2β β
6β β 2β β
7β β 2β β
8β β 2β β
9β β 2β β
10β β 2β β
11β β 2β β
12β β 2β β
Basic numeral
β1β
β4β
β9β
1β 6β
β25β
β36β
β49β
β64β
β81β
β100β β121β 1β 44β
All non-zero square numbers have an odd number of factors. The symbol for squaring is (βββ)2β β. The brackets are optional, but can be useful when simplifying more difficult expressions.
β The square root of a given number is the βnon-negativeβ number that, when multiplied by itself,
produces the given number. _ β’ The symbol for square rooting is β β 9 ββ. β’ Finding a square root of a number is the opposite of squaring a number. _ β 16 β= 4β β 2β β= 16β; hence, β For example: 4β We read this as: ββ4βsquared equals β16β, therefore, the square root of β16βequals β4.ββ β’ Squaring and square rooting are βoppositeβ operations. _
_ 2
β’
β’
ββ (ββ7)2β β = 7β For example: (βββ β 7 )β β β= 7ββalsoββ A list of common square roots is: Square root form β β 1β
_
ββ4 β
ββ9 β
ββ16 β ββ25 β ββ36 β ββ49 β ββ64 β ββ81 β ββ100 β β β 121 β β β 144 β
_
_
_
_
_
_
_
_
_
Basic numeral
β2β
3β β
β4β
β5β
6β β
β7β
β8β
9β β
β10β
1β 1β
β12β
β1β
_
_
We can locate any square root between two positive integers by considering its square. For _ β 2β β= 25βand 6β β 2β β= 36β. example, β β 31 βmust be between β5βand β6βbecause β31βis between 5β
BUILDING UNDERSTANDING
1 Consider a square of side length β6 cmβ. What would be the area of this shape? What special type of number is your answer?
2 State the first β15βsquare numbers in index form and as basic numerals.
3 We can confirm that β9βis a square number by drawing the diagram shown at right. a Explain using dots, why β6βis not a square number. b Explain using dots, why β16βis a square number. 4 Find: a 6β β 2β β d β10βto the power of β2β
b 5β β squared e 7β β 2β β
c (ββ11)2β β f β12 Γ 12β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2G Squares and square roots
95
U N SA C O M R PL R E EC PA T E G D ES
5 Find:_ a ββ 25 β b the square root of β16β _ c ββ100 β d the side length of a square that has an area of β49 cmβ β 2β β
Example 14 Evaluating squares and square roots Evaluate: a 6β β 2β β
b β β 64 β
SOLUTION
EXPLANATION
a 6β β 2β β= 36β
β6β β β2β= 6 Γ 6β
b β β 64 β= 8β
8 Γ 8 = 64 β _β β β β΄β β 64 β = 8
_
_
c ββ1600 β= 40β
_
_
c β β 1600 β
40 _ Γ 40 = 1600 ββ―β― β β β β΄β β 1600 β = 40
Now you try Evaluate: a 9β β 2β β
_
b β β 36 β
_
c β β 4900 β
Example 15 Locating square roots between positive integers State_ which consecutive whole numbers are either side of:_ a ββ43 β b ββ130 β SOLUTION
EXPLANATION
a β6βand β7β
β 2β β= 49β 6β β 2β β= 36βand 7β _ β43βlies between the square numbers β36βand β49β, so β β 43 βlies between β 6βand β7β.
b β11βand β12β
β 2β β= 144β 11β β β2β= 121βand 12β _ β130βlies between the square numbers β121βand β144β, so β β 130 ββlies between β11βand β12β.
Now you try
State_ which consecutive whole numbers are either side of:_ a β β 11 β b ββ109 β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
96
Chapter 2
Number properties and patterns
Example 16 Evaluating expressions involving squares and square roots Evaluate: _ a 3β β 2β ββ β β 9 β+ 1β β 2β β
_
U N SA C O M R PL R E EC PA T E G D ES
β 2β β b βββ8β2β β+ 6β
SOLUTION
EXPLANATION
_ β 2β β = 9 β 3 + 1 3β β 2β ββ ββ9 β+ 1β
a β
_
_
3β β β2β= 3 Γ 3, β β 9 β= 3, 1β β 2β β= 1 Γ 1β
β β β β= 7 _
β_ b βββ8β2β β+ 6β β 2β β = β 64 + 36 β ββ =β ββ100 ββ β β
β β 8β β β2β= 8 Γ 8, 6β β 2β β= 6 Γ 6 ββ―β― β _ β βββ β 100 β= 10
β = 10
Now you try
Evaluate: _ a 4β β 2β ββ β β 25 β+ 7β β 2β β
_
β 2β β b βββ5β2β ββ 4β
Exercise 2G FLUENCY
Example 14a
Example 14a
Example 14b
Example 14c
Example 15
Example 16
1
1, 2β5ββ(β1/2β)β
2β6β(β β1/2β)β
2β6β(β β1/3β)β
Evaluate: a 4β β 2β β
b 7β β 2β β
c 3β β 2β β
d 10β β 2β β
2 Evaluate: a 8β β 2β β e 6β β 2β β i 11β β 2β β
b 2β β 2β β f 15β β 2β β j 100β β 2β β
c 1β β 2β β g 5β β 2β β k 20β β 2β β
d 12β β 2β β h 0β β 2β β l 50β β 2β β
3 Evaluate: _ a β β 25 β _ e β β 0β _ i β β 4β
b ββ9 β _ f ββ81 β _ j ββ144 β
4 Evaluate: _ a ββ2500 β
b ββ6400 β
_
_
_
_
c ββ1 β _ g ββ49 β _ k ββ400 β
d ββ_ 121 β h ββ_ 16 β l ββ169 β
_
_
c ββ8100 β
d ββ729 β
5 State_ which consecutive whole numbers are either side of: _ _ 29 β b ββ40 β c ββ71 β a ββ_ _ _ e β β 85 β f ββ103 β g ββ7 β 6 Evaluate: _ a 3β β β2β+ 5β β 2β ββ ββ16 β d 1β β 2β βΓ 2β β 2β βΓ 3β β 2β β 2 2 g 6β β β βΓ· 2β β β βΓ 3β β 2β β
b e h
4β _ Γ 4β β β2β 2β β β_ βββ5β2β ββ 3β _ _ ββ9 βΓ β β 64 βΓ· β β 36 β
_
d ββ50 β _ h ββ3 β
c f i
8β β β2ββ 0β β 2β β+ 1β β 2β β _ ββ_ 81 ββ 3β β 2β β β 2β β+ 5β β 2β β ββ12β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2G Squares and square roots
PROBLEM-SOLVING
7, 8
7β9
97
8β10
U N SA C O M R PL R E EC PA T E G D ES
7 A square ο¬oor is covered with 64 square tiles. a Explain why the ο¬oor length must be 8 tile lengths. b Hence, find the perimeter of this ο¬oor in tile lengths. c If a square ο¬oor is covered with 81 square tiles, what would its perimeter be? _
_
8 The value of β19 as a decimal starts with 4 because β19 is between 4 and 5. Find the units digit for the following. _ _ _ _ a β72 b β83 c β26 d β34
9 List all the square numbers between 101 and 200 (Hint: There are only four.) 10 a Find two square numbers that add to 85. b Find two square numbers that have a difference of 85.
Squares and square roots are used by people who work with triangles, such as engineers, architects, builders, carpenters and designers. Electricians use _ a square root to calculate voltage: V = βPR , where P is power and R is resistance.
REASONING
11
11, 12
12, 13
11 The value of 742 can be thought of as the area of a square with width 74 units. This square can be broken into smaller areas as shown. 70 4 So 742 = 702 + 2 Γ 70 Γ 4 + 42 = 4900 + 560 + 16 = 5476 70 70 Γ 4 702 Use a diagram like the one on the right to calculate these values. a 342 b 812 c 1032 4 42 70 Γ 4
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
12 a Evaluate 32 Γ 42. b Evaluate 122. c The rule a 2 Γ b2 = (a Γ b)2 can be used to link 32 Γ 42 and 122. What are the values of a and b if 32 Γ 42 = 122? d Check this formula using other numbers. 13 a Evaluate 112 and 1112. b Predict an answer for 11112. c Evaluate 11112 and test your prediction.
U N SA C O M R PL R E EC PA T E G D ES
98
ENRICHMENT: Exploring square patterns
β
β
14
14 a List the square numbers from 12 to 202 in a table like the one below. n
1
2
3
β¦
20
n2
1
4
9
β¦
400
b Add a new row to your table which gives the difference between each value and the previous one. n
1
2
3
β¦
20
n2
1
4
9
β¦
400
difference
β
3
5
β¦
c What patterns do you notice in the differences between consecutive square numbers? d Try to explain why these patterns occur. You could use a visual representation like dots arranged in square as below to help illustrate.
The number pattern shown here is known as Pascalβs triangle, named after the French mathematician Blaise Pascal. Can you explain how the digits in each row of Pascalβs triangle form the powers of 11?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2H Number patterns
99
2H Number patterns CONSOLIDATING
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand what a number pattern (or number sequence) is β’ To be able to describe a pattern where there is a common difference (added or subtracted) β’ To be able to describe a pattern where there is a common ratio (multiplied or divided) β’ To be able to ο¬nd terms in a number pattern with a common difference or ratio
Mathematicians commonly look at lists of numbers in an attempt to discover a pattern. They also aim to find a rule that describes the number pattern to allow them to predict future numbers in the sequence. Here is a list of professional careers that all involve a high degree of mathematics and, in particular, involve looking at data so that comments can be made about past, current or future trends:
Statistician, economist, accountant, market researcher, financial analyst, cost estimator, actuary, stock broker, data scientist, research scientist, financial advisor, medical scientist, budget analyst, insurance underwriter and mathematics teacher!
People who study statistics ο¬nd trends in population and employment numbers by analysing a sequence of percentage changes over previous years.
Financial analysts use sequences of previous percentage increases or decreases to forecast future earnings for various shares in the stock market.
Lesson starter: Whatβs next?
A number sequence consisting of five terms is placed on the board. Four gaps are placed after the last number. 64, 32, 16, 8, 4, ___, ___, ___, ___ β’ β’
Can you work out and describe the number pattern? Make up your own number pattern and test it on a class member.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
KEY IDEAS β Number patterns are also known as sequences, and each number in a sequence is called a term.
β’
Each number pattern has a particular starting number and terms are generated by following a particular rule.
U N SA C O M R PL R E EC PA T E G D ES
100
β Strategies to determine the pattern involved in a number sequence include:
β’
β’
β’
β’
β’
Looking for a common difference Are terms increasing or decreasing by a constant amount? For example: β2, 6, 10, 14, 18, β¦βEach term is increasing by β4β. Looking for a common ratio Is each term being multiplied or divided by a constant amount? For example: β2, 4, 8, 16, 32, β¦βEach term is being multiplied by β2β. Looking for an increasing/decreasing difference Is there a pattern in the difference between pairs of terms? For example: β1, 3, 6, 10, 15, β¦βThe difference increases by β1βeach term. Looking for two interlinked patterns Is there a pattern in the odd-numbered terms, and another pattern in the even-numbered terms? For example: β2, 8, 4, 7, 6, 6, β¦βThe odd-numbered terms increase by β2β, the even-numbered terms decrease by β1β. Looking for a special type of pattern Could it be a list of square numbers, prime numbers, Fibonacci numbers β (1, 2, 3, 5, 8, 13, 21, β¦ )β etc.? ββ 3β ββ, 3β ββ 3β β, β¦β For example: β1, 8, 27, 64, 125, β¦βThis is the pattern of cube numbers: 1β β 3β ββ, 2β
BUILDING UNDERSTANDING
1 State the first five terms of the following number patterns. a starting number of β8β, common difference of adding β3β b starting number of β32β, common difference of subtracting β1β c starting number of β52β, common difference of subtracting β4β d starting number of β123β, common difference of adding β7β
2 State the first five terms of the following number patterns. a starting number of β3β, common ratio of β2β(multiply by β2βeach time) b starting number of β5β, common ratio of β4β
c starting number of β240β, common ratio of _ β1 β(divide by β2βeach time) 2
d
starting number of β625β, common ratio of _ β1 β 5
3 State whether the following number patterns have a common difference (β+βor βββ), a common ratio (βΓβor βΓ·β) or neither. a β4, 12, 36, 108, 324,β ββ¦β c β212, 223, 234, 245, 256,β ββ¦β e β64, 32, 16, 8, 4,β ββ¦β g β2, 3, 5, 7, 11,β ββ¦β
b d f h
1β 9, 17, 15, 13, 11,β ββ¦β β8, 10, 13, 17, 22,β ββ¦β β5, 15, 5, 15, 5,β ββ¦β β75, 72, 69, 66, 63,β ββ¦β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2H Number patterns
101
Example 17 Identifying patterns with a common difference
U N SA C O M R PL R E EC PA T E G D ES
Find the next three terms for these number patterns that have a common difference. a β6, 18, 30, 42,β___, ___, ___ b β99, 92, 85, 78β, ___, ___, ___ SOLUTION
EXPLANATION
a β54, 66, 78β
The common difference is β12β. Continue adding β12βto generate the next three terms.
b β71, 64, 57β
The pattern indicates the common difference is β7β. Continue subtracting β7βto generate the next three terms.
Now you try
Find the next three terms for these number patterns that have a common difference. a β7β, β10β, β13β, β16β, ____, ____, ____ b β187β, β173β, β159β, β145β, ____, ____, _____
Example 18 Identifying patterns with a common ratio
Find the next three terms for the following number patterns that have a common ratio. a β2, 6, 18, 54,β___, ___, ___ b β256, 128, 64, 32,β___, ___, ___ SOLUTION
EXPLANATION
a β162, 486, 1458β
The common ratio is β3β. Continue multiplying by β3βto generate the next three terms.
b β16, 8, 4β
The common ratio is _ β1 β. Continue dividing by β2β 2 to generate the next three terms.
Now you try
Find the next three terms for the following number patterns that have a common ratio. a β5β, β15β, β45β, ___, ____, ____ b β640β, β320β, β160β, ____, ___, ____
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
102
Chapter 2
Number properties and patterns
Exercise 2H FLUENCY
1
2β6β(β 1β/2β)β
2β6ββ(1β/4β)β
Find the next three terms for these number patterns that have a common difference. a i β5, 12, 19, ___, ___, ___β ii β32, 41, 50, ___, ___, ___β
U N SA C O M R PL R E EC PA T E G D ES
Example 17a
1, 2β5ββ(1β/2β)β
Example 17b Example 17
Example 18
b i
β37, 32, 27, ___, ___, ___β
ii β91, 79, 67, ___, ___, ___β
2 Find the next three terms for the following number patterns that have a common difference. a 3β , 8, 13, 18, ___, ___, ___β b β4, 14, 24, 34, ___, ___, ___β c β26, 23, 20, 17, ___, ___, ___β d β106, 108, 110, 112, ___, ___, ___β e β63, 54, 45, 36, ___, ___, ___β f β9, 8, 7, 6, ___, ___, ___β g β101, 202, 303, 404, ___, ___, ___β h β75, 69, 63, 57, ___, ___, ___β
3 Find the next three terms for the following number patterns that have a common ratio. a 2β , 4, 8, 16, ___, ___, ___β b β5, 10, 20, 40, ___, ___, ___β c β96, 48, 24, ___, ___, ___β d β1215, 405, 135, ___, ___, ___β e β11, 22, 44, 88, ___, ___, ___β f β7, 70, 700, 7000, ___, ___, ___β g β256, 128, 64, 32, ___, ___, ___β h β1216, 608, 304, 152, ___, ___, ___β
4 Find the missing numbers in each of the following number patterns. a 6β 2, 56, ___, 44, 38, ___, ___β b β15, ___, 35, ___, ___, 65, 75β c β4, 8, 16, ___, ___, 128, ___β d β3, 6, ___, 12, ___, 18, ___β e β88, 77, 66, ___, ___, ___, 22β f β2997, 999, ___, ___, 37β g β14, 42, ___, ___, 126, ___, 182β h β14, 42, ___, ___, 1134, ___, 10 206β
5 Write the next three terms in each of the following sequences. a β3, 5, 8, 12, ___, ___, ___β b β1, 2, 4, 7, 11, ___, ___, ___β c β1, 4, 9, 16, 25, ___, ___, ___β d β27, 27, 26, 24, 21, ___, ___, ___β e β2, 3, 5, 7, 11, 13, ___, ___, ___β f β2, 5, 11, 23, ___, ___, ___β g β2, 10, 3, 9, 4, 8, ___, ___, ___β h β14, 100, 20, 80, 26, 60, ___, ___, ___β
6 Generate the next three terms for the following number sequences and give an appropriate name to the sequence. a 1β , 4, 9, 16, 25, 36, ___, ___, ___β b β1, 1, 2, 3, 5, 8, 13, ___, ___, ___β c β1, 8, 27, 64, 125, ___, ___, ___β d β2, 3, 5, 7, 11, 13, 17, ___, ___, ___β e β4, 6, 8, 9, 10, 12, 14, 15, ___, ___, ___β f β121, 131, 141, 151, ___, ___, ___β PROBLEM-SOLVING
7, 8
7β9
7β10
7 Complete the next three terms for the following challenging number patterns. a 1β 01, 103, 106, 110, ___, ___, ___β b β162, 54, 108, 36, 72, ___, ___, ___β c β3, 2, 6, 5, 15, 14, ___, ___, ___β d β0, 3, 0, 4, 1, 6, 3, ___, ___, ___β
8 When making human pyramids, there is one less person on each row above, and it is complete when there is a row of only one person on the top. Write down a number pattern for a human pyramid with β10βstudents on the bottom row. How many people are needed to make this pyramid?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2H Number patterns
103
9 The table below represents a seating plan with specific seat numbers for a section of a grandstand at a soccer ground. It continues upwards for another 20 rows. 25
26
27
28
29
30
31
32
17
18
19
20
21
22
23
24
Row 2
9
10
11
12
13
14
15
16
Row 1
1
2
3
4
5
6
7
8
U N SA C O M R PL R E EC PA T E G D ES
Row 4 Row 3
a What is the number of the seat directly above seat number 31? b What is the number of the seat on the left-hand edge of row 8? c What is the third seat from the right in row 14? d How many seats are in the grandstand?
10 Find the next five numbers in the following number pattern. 1, 4, 9, 1, 6, 2, 5, 3, 6, 4, 9, 6, 4, 8, 1, ___, ___, ___, ___, ___ REASONING
11
11, 12
12, 13
11 Jemima writes down the following number sequence: 7, 7, 7, 7, 7, 7, 7, β¦ Her friend Peta declares that this is not really a number pattern. Jemima defends her number pattern, stating that it is most definitely a number pattern as it has a common difference and also has a common ratio. What are the common difference and the common ratio for the number sequence above? Do you agree with Jemima or Peta?
12 Find the sum for each of the following number sequences. a 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 b 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 c 1 + 2 + 3 + 4 + 5 + β¦ + 67 + 68 + 69 + 70 d 5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32 + 35 + 38
13 The great handshake problem There are a certain number of people in a room and they must all shake one anotherβs hand. How many handshakes will there be if there are: a 3 people in the room? b 5 people in the room? c 10 people in the room? d 24 people in a classroom? e n people in the room?
A
B
E
C
D
Diagram showing the possible handshakes with ο¬ve people.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
ENRICHMENT: What number am I?
β
β
14
14 Read the following clues to work out the mystery number. a I have three digits. I am divisible by β5β. I am odd. The product of my digits is β15β. The sum of my digits is less than β10β. I am less than β12 Γ 12β. b I have three digits. The sum of my digits is β12β. My digits are all even. My digits are all different. I am divisible by β4β. The sum of my units and tens digits equals my hundreds digit. c I have three digits. I am odd and divisible by β5βand β9β. The product of my digits is β180β. The sum of my digits is less than β20β. I am greater than 30β β 2β ββ. d Make up two of your own mystery number puzzles and submit your clues to your teacher.
U N SA C O M R PL R E EC PA T E G D ES
104
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Applications and problem-solving
U N SA C O M R PL R E EC PA T E G D ES
Prime number landscaping
Applications and problem-solving
The following problems will investigate practical situations drawing upon knowledge and skills developed throughout the chapter. In attempting to solve these problems, aim to identify the key information, use diagrams, formulate ideas, apply strategies, make calculations and check and communicate your solutions.
105
1
The team at Prime Landscaping only like to plant trees, lay sleepers, or group large rocks in clumps where the number of items in the group is a prime number. The landscaping team are interested in the prime number options available to them and the total number of objects that they might need to purchase for particular jobs. a Prime Landscaping want to arrange a pile of large rocks with fewer than β15βin the pile. What size groups are possible? b Would Prime Landscaping ever lay a group of β27βsleepers? Give a reason for your answer.
One of Prime Landscapingβs clients said they only wanted trees or shrubs that were planted in even numbers as they disliked odd numbers of plants (such as a group of β3βtrees or a group of β7β shrubs). Prime Landscaping felt confident they could meet the needs of their client and still stay true to their policy of planting only in groups of prime numbers. c What size group of plants did Prime Landscaping plant for their client?
A new client purchased β59βscreening plants and wanted them to be planted in three different, but similar sized groups. Prime Landscaping said that this was possible and proceeded to plant the plants. d What size groups were the screening plants planted in?
Blinking rates
2 Diviesh tends to blink his eyes every four seconds, while Jordan blinks his eyes only every seven seconds. Blinking provides moisture to the eyes and prevents irritation and damage. Diviesh and Jordan are interested in the number of times they will blink their eyes during various time intervals and how often they will blink at the same time depending on their blinking rates. a If Diviesh and Jordan both blink together, how long will it be until they blink together again? b How many times will Diviesh and Jordan blink together in one minute, and at what times will this occur?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
Concentrated staring at a computer monitor can significantly reduce the eye blinking rate, and can result in what is known as βgamer eye syndromeβ. Diviesh and Jordan were filmed playing a one-hour gaming session. Divieshβs blinking rate reduced to β 3βblinks per minute, and Jordanβs blinking rate reduced to β4βblinks per minute. c At the start of a particular three-minute time period in their computer gaming session, Diviesh and Jordan both blinked together. If this time was referred to as β0βseconds, write down the time (in seconds) of Divieshβs next β10βblinks and Jordanβs next β10β blinks. d How frequently do Diviesh and Jordan blink together during a one-hour computer gaming session?
U N SA C O M R PL R E EC PA T E G D ES
Applications and problem-solving
106
Excessive blinking can indicate a disorder of the nervous system. A doctor examined a particular patient and initially observed that their blink rate was β30βblinks per minute. She then decided to monitor the patient for a period of β5βminutes and was relieved to observe the patient only blinked β110β times. e What was the average blinking rate (blinks per minute) for the patient over the β5βminute period?
A reluctant reader
3 A keen student mathematician, but reluctant reader, was encouraged by his Maths teacher to read just two pages at night on the first night and then simply add another two pages each night β so on the second night the student would have to read four pages, and then six pages on the third night and so on. The student agreed to do this. The student is interested in how many pages he will be reading on particular nights and how long it will take to read a book of a certain size. a How many pages would the student be reading on the 5th night? b On what night was the student reading 16 pages? c How many nights did it take the student to read a 182 page book? d How big a book can the student read in a 30-day month? e If a student wishes to read a 275 page book in 10 days, describe a different rule with a different pattern that would allow this to occur. List the number of pages the student would need to read each night.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2I Spatial patterns
107
2I Spatial patterns
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To know that a spatial pattern begins with a starting design and has a repeating design β’ To understand how spatial patterns are related to number patterns β’ To be able to continue a spatial pattern given the ο¬rst few shapes β’ To be able to describe and use a rule relating the number of shapes and the number of objects required to make them
Patterns can also be found in geometric shapes. Mathematicians examine patterns carefully to determine how the next term in the sequence is created. Ideally, a rule is formed that shows the relationship between the geometric shape and the number of objects (e.g. tiles, sticks, counters) required to make such a shape. Once a rule is established it can be used to make predictions about future terms in the sequence.
Lesson starter: Stick patterns
Materials required: One box of toothpicks or a bundle of matchsticks per student. β’ β’
β’ β’
An architect uses spatial pattern analysis when designing open-plan ofο¬ces, laboratories, museums, and seating arrangements in lecture rooms and concert halls.
Generate a spatial pattern using your sticks. You must be able to make at least three terms in your pattern. For example:
Ask your partner how many sticks would be required to make the next term in the pattern. Repeat the process with a different spatial design.
KEY IDEAS
β A spatial pattern is a sequence of geometrical shapes that can be described by a number pattern.
For example:
spatial pattern
number pattern
4
12
8
β A spatial pattern starts with a simple geometric design. Future terms are created by adding on
repeated shapes of the same design. If designs connect with an edge, the repetitive shape will be a subset of the original design, as the connecting edge does not need to be repeated. For example: starting design
repeating design
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
β To help describe a spatial pattern, it is generally converted to a number pattern and a common
difference is observed. β The common difference is the number of objects (e.g. sticks) that need to be added on to create
the next term. β A table of values shows the number of shapes and the number of sticks.
U N SA C O M R PL R E EC PA T E G D ES
108
Number of squares
β1β
2β β
β3β
β4β
β5β
Number of sticks
β4β
β8β
β12β
1β 6β
β20β
β A pattern rule tells how many sticks are needed for a certain number of shapes.
For example: Number of sticks β= β4βΓβnumber of shapes
β Rules can be found that connect the number of objects (e.g. sticks) required to produce the
number of designs. For example: hexagon design
βRule: Number of sticks used = 6 Γ number of hexagons formedβ
BUILDING UNDERSTANDING
1 Describe or draw the next two terms for each of these spatial patterns. a
b
2 For each of the following spatial patterns, describe or draw the starting geometrical design and also the geometrical design added on repetitively to create new terms. (For some patterns the repetitive design is the same as the starting design.)
a
b c
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2I Spatial patterns
109
Example 19 Drawing and describing spatial patterns
U N SA C O M R PL R E EC PA T E G D ES
a Draw the next two shapes in the spatial pattern shown.
b Write the spatial pattern above as a number pattern relating the number of sticks required to make each shape. c Describe the pattern by stating how many sticks are required to make the first term, and how many sticks are required to make the next term in the pattern.
SOLUTION
EXPLANATION
a
Follow the pattern, by adding three sticks each time to the right-hand side.
b β5, 8, 11, 14, 17β
Count the number of sticks in each term. Look for a pattern.
c 5β βsticks are required to start the pattern, and an additional β3βsticks are required to make the next term in the pattern.
Now you try
a Draw the next two shapes in the spatial pattern shown.
b Write the spatial pattern as a number pattern relating the number of sticks required to make each shape. c Describe the pattern by stating how many sticks are required to make the first term, and how many sticks are required to make the next term in the pattern.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
Example 20 Finding a general rule for a spatial pattern a Draw the next two shapes in this spatial pattern.
U N SA C O M R PL R E EC PA T E G D ES
110
b Copy and complete the table. Number of triangles
β1β
Number of sticks required
β3β
2β β
β3β
β4β
β5β
c Describe a rule connecting the number of sticks required to the number of triangles. d Use your rule to predict how many sticks would be required to make β20β triangles.
SOLUTION
EXPLANATION
a
Follow the pattern by adding one triangle each time.
b
Number of triangles
β1β
2β β
β3β
β4β
β5β
Number of sticks required
β3β
β6β
β9β
1β 2β
β15β
An extra β3βsticks are required to make each new triangle.
c βNumber of sticks = 3 Γ number of trianglesβ
β3βsticks are required per triangle.
d Number of sticksβ =β 3 Γ 20ββ β β = 60 ββ΄ 60 sticks are required.β
number of triangles = 20 ββ―β― β β β β so number of sticks = 3 Γ 20
Now you try
a Draw the next two shapes in this spatial pattern.
b Copy and complete the table. Number of shapes
β1β
Number of sticks required
β5β
2β β
β3β
β4β
β5β
c Describe a rule connecting the number of sticks required to the number of shapes. d Use your rule to predict how many sticks would be required to make β20β shapes.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2I Spatial patterns
111
Exercise 2I FLUENCY
1
2ββ(1β/2β)β, 3, 4
2β(β 1β/2β)β,β 4
a Draw the next two shapes in the spatial pattern shown.
U N SA C O M R PL R E EC PA T E G D ES
Example 19
1β3
b Write the spatial pattern above as a number pattern in regard to the number of sticks required to make each shape. c Describe the pattern by stating how many sticks are required to make the first term, and how many sticks are required to make the next term in the pattern.
Example 19
2 For each of the spatial patterns below: i Draw the next two shapes. ii Write the spatial pattern as a number pattern. iii Describe the pattern by stating how many sticks are required to make the first term and how many more sticks are required to make the next term in the pattern. a
b
c
d
e
f
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
112
Chapter 2
Example 20
Number properties and patterns
3 a Draw the next two shapes in this spatial pattern.
U N SA C O M R PL R E EC PA T E G D ES
b Copy and complete the table. Number of crosses
β1β
β2β
3β β
β4β
5β β
Number of sticks required
c Describe a rule connecting the number of sticks required to the number of crosses produced. d Use your rule to predict how many sticks would be required to make β20β crosses.
Example 20
4 a Draw the next two shapes in this spatial pattern.
b Copy and complete the table. Planks are vertical and horizontal. Number of fence sections
β1β
2β β
β3β
β4β
β5β
Number of planks required
c Describe a rule connecting the number of planks required to the number of fence sections produced. d Use your rule to predict how many planks would be required to make β20βfence sections.
PROBLEM-SOLVING
5, 6
6, 7
6β8
5 At North Park Primary School, the classrooms have trapezium-shaped tables. Mrs Greene arranges her classroomβs tables in straight lines, as shown.
a Draw a table of results showing the relationship between the number of tables in a row and the number of students that can sit at the tables. Include results for up to five tables in a row. b Describe a rule that connects the number of tables placed in a straight row to the number of students that can sit around the tables. c The room allows seven tables to be arranged in a straight line. How many students can sit around the tables? d There are β65βstudents in Grade β6βat North Park Primary School. Mrs Greene would like to arrange the tables in one straight line for an outside picnic lunch. How many tables will she need?
6 The number of tiles required to pave around a spa is related to the size of the spa. The approach is to use large tiles that are the same size as that of the smallest spa.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2I Spatial patterns
U N SA C O M R PL R E EC PA T E G D ES
A spa of length β1βunit requires β8βtiles to pave around its perimeter, whereas a spa of length β4β units requires β14βtiles to pave around its perimeter. a Complete a table of values relating length of spa and number of tiles required, for values up to and including a spa of length β6β units. b Describe a rule that connects the number of tiles required for the length of the spa. c The largest spa manufactured is β15βunits long. How many tiles would be required to pave around its perimeter? d A paving company has only β30βtiles left. What is the largest spa they would be able to tile around?
113
7 Which rule correctly describes this spatial pattern of βhatsβ?
A B C D
Number of sticks β= 7 Γβnumber of hats Number of sticks β= 7 Γβnumber of hats β+ 1β Number of sticks β= 6 Γβnumber of hats β+ 2β Number of sticks β= 6 Γβnumber of hats
8 Which rule correctly describes this spatial pattern?
A B C D
Number of sticks β= 5 Γβnumber of houses β+ 1β Number of sticks β= 6 Γβnumber of houses β+ 1β Number of sticks β= 6 Γβnumber of houses Number of sticks β= 5 Γβnumber of houses
REASONING
9β(β β1/2β)β
9β(β β1/2β)β,β 10
10, 11
9 Design a spatial pattern to fit each of the following number patterns. a β4, 7, 10, 13, β¦β b β4, 8, 12, 16, β¦β c 3β , 5, 7, 9, β¦β d β3, 6, 9, 12, β¦β e β5, 8, 11, 14, β¦β f β6, 11, 16, 21, β¦β
10 A rule to describe a special window spatial pattern is written as βy = 4 Γ x + 1β, where βyβrepresents the number of βsticksβ required and βxβis the number of windows created. a How many sticks are required to make one window? b How many sticks are required to make β10β windows? c How many sticks are required to make βgβ windows? d How many windows can be made from β65β sticks?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
11 A rule to describe a special fence spatial pattern is written as βy = m Γ x + nβ, where βyβrepresents the number of pieces of timber required and βxβrepresents the number of fencing panels created. a How many pieces of timber are required to make the first panel? b What does βmβ represent? c Draw the first three terms of the fence spatial pattern for βm = 4βand βn = 1β.
U N SA C O M R PL R E EC PA T E G D ES
114
ENRICHMENT: Cutting up a circle
β
β
12
12 What is the greatest number of sections into which you can divide a circle, using only a particular number of straight line cuts? a Explore the problem above. (Hint: For a given number of lines, you may need to draw several circles until you are sure that you have found the maximum number of sections.) Note: The greatest number of sections is required and, hence, only one of the two diagrams below is correct for three straight line cuts. 5 4
3
6
1 2
Incorrect. Not the maximum number of sections for 3 lines.
3
2 4
1
7
Correct. The maximum number of sections for 3 lines.
5
6
Copy and complete this table of values. Number of straight cuts
Maximum number of sections
β1β
β2β
β3β
4β β
β5β
β6β
β7β
β7β
b Can you discover a pattern for the maximum number of sections created? What is the maximum number of sections that could be created with β10βstraight line cuts? c The formula for determining the maximum number of cuts is quite complex: βsections = _ β1 βΓ cutsβ β 2β β+ _ β1 βΓ cuts + 1β 2 2 Verify that this formula works for the values you listed in the table above. Using the formula, how many sections could be created with β20βstraight cuts?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2J Tables and rules
115
2J Tables and rules
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand that a rule connects two varying quantities β’ To be able to complete an input-output table given a rule β’ To be able to ο¬nd a rule in the form output = ? Γ input + ? for an input-output table
A rule describes a relationship between two values, often called input and output.
For example, the rule output = input + 3 describes a relationship between input and output. If the input value is 5, then the output value will be 8. This is sometimes shown as a function machine: 5
input
+3
output
8
From tables of human heights and bone lengths, rules are found, e.g. height = 2.3 Γ femur length + 65.5 cm . By measuring bones from the Petra tombs, archaeologists calculated the height of ancient humans.
If a different input is put in the function machine above, the output will change.
We can use a table to list a selection of input and output combinations, as below: input
output
5
8
9
12
51
54
100
103
Another example of a rule is output = 4 Γ input, which we could use to create a table of values. input
output
1
4
2
8
3
12
...
...
This can be drawn as a function machine too: input
Γ4
output
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
Lesson starter: Guess the output A table of values is drawn on the board with three completed rows of data. β’ β’
Additional values are placed in the input column. What output values should be in the output column? After adding output values, decide which rule fits (models) the values in the table and check that it works for each input and output pair. Four sample tables are listed below.
U N SA C O M R PL R E EC PA T E G D ES
116
input
output
input
output
input
output
input
output
β2β
β6β
β12β
β36β
β2β
β3β
β6β
β1β
β5β
β9β
β5β
β15β
β 3β
β5β
2β 0β
β8β
β6β
β10β
β8β
β24β
β 9β
β17β
1β 2β
β4β
β1β
?
β0β
?
7β ββ
?
4β 2β
?
β8β
?
β23β
?
β12β
?
β4β
?
KEY IDEAS
β A rule shows the relation between two varying quantities.
For example: βoutput = input + 3βis a rule connecting the two quantities input and output. The values of the input and the output can vary, but we know from the rule that the value of the output will always be β3βmore than the value of the input. Rules can be thought of as function machines: input output +3
β A table of values can be created from any given rule. To complete a table of values, we start
with the first input number and use the rule to calculate the corresponding output number. we do this for each of the input numbers in the table. For example: For the rule output = input + 3, If input = 4, then β β β―β― β―β― β β β β output = 4 + 3 β= 7
Replacing the input with a number is known as substitution.
β Often, a rule can be determined from a table of values. On close inspection of the values, a
relationship may be observed. Each of the four operations should be considered when looking for a connection. input
β1β
2β β
β3β
β4β
output
β6β
7β β
β8β
β9β β10β 1β 1β
β5β
6β β
β By inspection, it can be observed that every output value is β5βmore than the corresponding input
value. The rule can be written as: βoutput = input + 5β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2J Tables and rules
117
BUILDING UNDERSTANDING
U N SA C O M R PL R E EC PA T E G D ES
1 State whether each of the following statements is true or false. a If βoutput = input Γ 2β, then when βinput = 7β, βoutput = 14β. b If βoutput = input β 2β, then when βinput = 5β, βoutput = 7β. c If βoutput = input + 2β, then when βinput = 0β, βoutput = 2β. d If βoutput = input Γ· 2β, then when βinput = 20β, βoutput = 10β.
2 Choose the rule (A to D) for each table of values (a to d). Rule A: βoutput = input β 5β
Rule B: βoutput = input + 1β
Rule C: βoutput = 4 Γ inputβ
Rule D: βoutput = 5 + inputβ
a c
input
β20β
β14β
6β β
output
β15β
β9β
1β β
input
β4β
β5β
β6β
output
β5β
β6β
β7β
b d
input
β8β
β10β
1β 2β
output
β13β
β15β
1β 7β
input
β4β
β3β
2β β
output
β16β
β12β
8β β
Example 21 Completing a table of values Copy and complete each table for the given rule. a βoutput = input β 2β input
β3β
β5β
β7β
1β 2β
β20β
output
b βoutput = β(β3 Γ inputβ)β+ 1β input
β4β
β2β
β9β
β12β
0β β
output
SOLUTION
EXPLANATION
a βoutput = input β 2β
Replace each input value in turn into the rule. e.g. When input is β3β: βoutput = 3 β 2 = 1β
input
β3β
β5β
7β β
β12β
β20β
output
β1β
β3β
5β β
β10β
β18β
b βoutput = β(3 Γ input)β+ 1β input
β4β
β2β
9β β
β12β
β0β
output
β13β
β7β
2β 8β
β37β
β1β
Replace each input value in turn into the rule. e.g. When input is β4β: βoutput = β(3 Γ 4)β+ 1 = 13β
Now you try
Copy and complete each table for the given rule. a βoutput = input + 4β input
output
β3β
β5β
β7β
1β 2β
β20β
b βoutput = β(8 Γ input)ββ 5β input
β1β
β2β
β5β
β7β
β10β
output
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
118
Chapter 2
Number properties and patterns
Example 22 Finding a rule from a tables of values Find the rule for each of these tables of values. input
β3β
output
β12β β13β β14β β15β β16β
β4β
β5β
β6β
b
β7β
input
β1β
β2β
output
β7β
β14β β21β β28β β35β
β3β
β4β
β5β
U N SA C O M R PL R E EC PA T E G D ES
a
SOLUTION
EXPLANATION
a oβ utput = input + 9β
Each output value is β9βmore than the input value.
b βoutput = input Γ 7βor βoutput = 7 Γ inputβ
By inspection, it can be observed that each output value is β7βtimes bigger than the input value.
Now you try
Find the rule for each of these tables of values. a
input
β10β β11β β12β β13β β14β
output
β5β
β6β
β7β
β8β
b
β9β
input
β2β
output
β12β β18β β24β β30β β36β
β3β
β4β
β5β
β6β
Exercise 2J FLUENCY
Example 21a
1
1β5
Copy and complete each table for the given rule. a oβ utput = input + 5β input
β1β
β3β
5β β
β10β
β20β
output
Example 21a
β4β
β5β
6β β
β7β
β10β
output
β11β
β18β
β44β
β100β
β1β
β2β
3β β
β4β
β5β
output
output
β9β
β1β
1β 0β
β6β
β3β
9β β
β0β
β55β
0β β
β100β
1β 2β
β14β
7β β
β50β
b βoutput = input Γ 4β input
β5β
β1β
input
β5β
β15β
b βoutput = (β input Γ· 2)β+ 4β input
β6β
β8β
β10β
output
c βoutput = (β 3 Γ input)β+ 1β input
β3β
output
3 Copy and complete each table for the given rule. a oβ utput = (β 10 Γ input)ββ 3β input
input
d βoutput = input Γ· 5β
9β β
output
Example 21b
b βoutput = 2 Γ inputβ
output
c βoutput = input β 8β input
3β5
output
2 Copy and complete each table for the given rule. a oβ utput = input + 3β input
2β5
β5β
β12β
2β β
d βoutput = (β 2 Γ input)ββ 4β
β9β
β14β
input
β3β
β10β
β11β
output
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2J Tables and rules
Example 22
4 State the rule for each of these tables of values. a input β4β β5β 6β β β7β β8β output
β6β
7β β
β8β
β9β
5 State the rule for each of these tables of values. a input β10β β8β 3β β β1β β14β
b
input
β1β
β2β
β3β
4β β
β5β
output
β4β
8β β
β12β
β16β
β20β
input
β6β
β18β
3β 0β
β24β
β66β
output
β1β
β3β
β5β
β4β
β11β
U N SA C O M R PL R E EC PA T E G D ES
Example 22
β5β
b
119
output
β21β
β19β
1β 4β
β12β
β25β
PROBLEM-SOLVING
6, 7
6, 7
7, 8
6 A rule relating the radius and diameter of a circle is βdiameter = 2 Γ radiusβ. In the following table, radius is the input and diameter is the output. a Write the rule above using input and output, rather than radius and diameter. b Copy and complete the missing values in the table. input
β1β
β2β
β3β
7β β
output
β20β
β30β
1β 00β
β162β
7 Find a simple rule for the following tables and then copy and complete the tables. a
input
β4β
β10β
output
b
β13β
6β β
β39β
input
β12β
output
β3β
β93β
β14β
5β β
β24β
1β 7β β8β
9β β
β11β
β15β
β2β
6β β
β10β
β12β
1β β
β34β
0β β
β200β
8 Copy and complete each table for the given rule. a βoutput = input Γ input β 2β input
β3β
β6β
β8β
1β 2β
β2β
3β β
β8β
9β β
β0β
output
b βoutput = β(24 Γ· input)β+ 1β input
β6β
β12β
β1β
output
c oβ utput = inputβββ2β+ inputβ input
β5β
β12β
β2β
output
d βoutput = 2 Γ input Γ input β inputβ input
β3β
β10β
β11β
7β β
β50β
output
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
REASONING
9, 10
9β(β β1/2β)β
9ββ(β1/2β)β, 10, 11
9 Another way of writing βoutput = input + 3βis to state that βy = x + 3β. Write the rule using βyβand βxβ for the following. a
βxβ
β0β
β1β
2β β
β3β
4β β
βyβ
6β β
β7β
β8β
β9β
1β 0β
βxβ
β5β
β2β
1β β
β7β
0β β
βyβ
β15β
β6β
β3β
2β 1β
β0β
βxβ
β5β
β2β
1β 1β
β6β
3β β
βyβ
3β β
β0β
β9β
β4β
β1β
βxβ
β0β
β3β
1β 0β
β6β
4β β
βyβ
1β β
β10β
β101β
3β 7β
β17β
U N SA C O M R PL R E EC PA T E G D ES
120
b c
d
10 It is known that for an input value of β3β the output value is β7β. a State two different rules that work for these values. b How many different rules are possible? Explain.
11 Many function machines can be βreversedβ, allowing you to get from an output back to an input. For instance, the reverse of ββ+ 3ββ is βββ 3ββ because if you put a value through both machines you get this value back again, as shown: value
β3
+3
value
The reverse rule for βoutput = input + 3βis βoutput = input β 3β. Write the reverse rule for the following. a βoutput = input + 7β b βoutput = input β 4β c βoutput = 4 Γ inputβ d βoutput = 2 Γ input + 1β e βoutput = input Γ· 2 + 4β
ENRICHMENT: Finding harder rules
β
β
12
12 a The following rules all involve two operations. Find the rule for each of these tables of values. i
iii v
input
β4β
β5β
β6β
7β β
β8β
output
β5β
7β β
β9β
β11β
β13β
input
β10β
β8β
β3β
1β β
β14β
output
β49β
3β 9β
β14β
β4β
β69β
input
β4β
β5β
β6β
7β β
β8β
output
β43β
5β 3β
β63β
β73β
β83β
ii
iv vi
input
β1β
β2β
β3β
β4β
β5β
output
β5β
β9β
1β 3β
β17β
β21β
input
β6β
β18β
β30β
β24β
6β 6β
output
β3β
β5β
7β β
β6β
β13β
input
β1β
β2β
β3β
β4β
β5β
output
β0β
β4β
8β β
1β 2β
β16β
b Write three of your own two-operation rules and produce a table of values for each rule. c Swap your tables of values with those of a classmate and attempt to find one anotherβs rules.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2K The Cartesian plane and graphs
121
2K The Cartesian plane and graphs
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To know the meaning of the terms number plane (or Cartesian plane), origin and coordinates β’ To be able to interpret the location of a point described by its coordinates, e.g. (2, 4) β’ To be able to plot one or more points given their coordinates
We are already familiar with number lines. A number line is used to locate a position in one dimension (i.e. along the line).
y
What is the position of this point on the number plane?
5 4 3 2 1
A number plane is used to locate a position in two dimensions (i.e. within the plane). A number plane uses two number lines to form a grid system, so that points can be located precisely. A rule can then be illustrated visually using a number plane by forming a graph.
O
1 2 3 4 5
x
The horizontal number line is referred to as the x-axis. The vertical number line is referred to as the y-axis. The point where the two axes intersect is called the origin and can be labelled O.
Lesson starter: Estimate your location Consider the door as βthe originβ of your classroom. β’ β’
β’
Describe the position you are sitting in within the classroom in reference to the door. Can you think of different ways of describing your position? Which is the best way? Submit a copy of your location description to your teacher. Can you locate a classmate correctly when location descriptions are read out by your teacher?
A Cartesian coordinate plane is used on the touch screen of smartphones and devices to precisely locate where a touch is detected.
KEY IDEAS
β A number plane is used to represent position in two dimensions, therefore it requires two
coordinates.
β In mathematics, a number plane is generally referred to as a Cartesian plane, named after the
famous French mathematician, RenΓ© Descartes (1596β1650).
β A number plane consists of two straight perpendicular number lines, called axes.
β’ β’
The horizontal number line is known as the x-axis. The vertical number line is known as the y-axis.
β The point at which the two axes intersect is called the origin, and is often labelled O.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
β The position of a point on a number plane is given
y as a pair of numbers, known as the coordinates of 5 the point. Coordinates are always written in brackets This dot is 4 and the numbers are separated by a comma. represented by 3 the coordinates For example: (β 2, 4)ββ. the vertical 2 (2, 4) β’ The βxβ-coordinate (input) is always written first. y-axis 1 The βxβ-coordinate indicates how far to go from x the origin in the horizontal direction. O 1 2 3 4 5 β’ The βyβ-coordinate (output) is always written the origin the horizontal second. The βyβ-coordinate indicates how far to go x-axis from the origin in the vertical direction.
U N SA C O M R PL R E EC PA T E G D ES
122
BUILDING UNDERSTANDING
1 Which of the following is the correct way to describe point βAβ? A β2, 1β B β1, 2β D β(x2, y1)β
E β(1, 2)β
C β(2, 1)β
y
3 2 1
A 2 Copy and complete the following sentences. a The horizontal axis is known as the __________. O 1 2 3 b The _____________ is the vertical axis. c The point at which the axes intersect is called the ____________. d The βxβ-coordinate is always written __________. e The second coordinate is always the ___________. f ________ comes before ________ in the dictionary, and the ________ coordinate comes
x
before the ________ coordinate on the Cartesian plane.
Example 23 Plotting points on a number plane
Plot these points on a number plane: βA(β 2, 5)β Bβ(4, 3)β Cβ(0, 2)β SOLUTION
EXPLANATION
y
5 4 3 2 1
A
B
C
O 1 2 3 4 5
x
Draw a Cartesian plane, with both axes labelled from β0βto β5β. The first coordinate is the βxβ-coordinate. The second coordinate is the βyβ-coordinate. To plot point βAβ, go along the horizontal axis to the number β2β, then move vertically up β5βunits. Place a dot at this point, which is the intersection of the line passing through the point β2βon the horizontal axis and the line passing through the point β5βon the vertical axis.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2K The Cartesian plane and graphs
123
Now you try Plot these points on a number plane: βA(β 1, 4)βββββB(β 3, 2)βββββC(β 4, 0)β
U N SA C O M R PL R E EC PA T E G D ES
Example 24 Drawing graphs For the given rule βoutput = input + 1β:
a Copy and complete the given table of values. b Plot each pair of values in the table to form a graph. c Do these points lie in a straight line?
input (β x)β
output β(y)β
β0β
β1β
β1β β2β β3β
SOLUTION a
EXPLANATION
input (β x)β
output β(y)β
β0β
1β β
β1β
2β β
β2β
3β β
β3β
4β β
b
Plot each (β x, y)ββpair. The pairs are β(0, 1)ββ, ββ(1, 2)ββ, ββ(2, 3)βand (β 3, 4)ββ.
output
y
4 3 2 1
O
Use the given rule to find each output value for each input value. The rule is: βoutput = input + 1β, so add β1βto each input value.
1 2 3 input
x
c Yes
The points lie in a straight line (you could check by using a ruler to see that it is possible to draw a straight line through all the points.)
Now you try
For the given rule βoutput = input + 4β:
a Copy and complete the given table of values. b Plot each pair of values in the table to form a graph. c Do these points lie in a straight line?
input β(x)β
output (β y)β
β0β
4β β
β1β β2β β3β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
124
Chapter 2
Number properties and patterns
Exercise 2K 1, 3β7
Plot these points on a number plane. a A β (β 2, 4)β b B β β(3, 1)β
c βCβ(0, 1)β
d βDβ(2, 0)β
2 Plot the following points on a number plane. a βAβ(4, 2)β b B β β(1, 1)β e βE(β 3, 1)β f βFβ(5, 4)β
c C β β(5, 3)β g βGβ(5, 0)β
d βDβ(0, 2)β h βHβ(0, 0)β
1
2ββ(1β/2β)β,β 3β9
2β(β 1β/2β)β,β 5β10
U N SA C O M R PL R E EC PA T E G D ES
Example 23
FLUENCY
Example 23
3 Write down the coordinates of each of these labelled points. Your answers should be written with the point name then its coordinates, like βA(β 1, 4)ββ. a b y y 6 D 5 4 A 3 G 2 1 B
F
H
C
E
O 1 2 3 4
Example 24
6 T 5 4 3 M 2 1
5 6
x
O
S
Q
U
N
P
R
1 2 3 4 5 6
4 For the given rule βoutput = input + 2β: a Copy and complete the given table of values. input β(x)β
output β(y)β
0β β
β2β
β1β β2β
5 4 3 2 1
O 1 2 3 4 input
β3β
b Plot each pair of values in the table to form a graph. c Do these points lie in a straight line?
5 For the given rule βoutput = input β 1β: a Copy and complete the given table of values. input β(x)β
output β(y)β
1β β β2β β3β β4β
x
y
output
Example 24
x
y
output
Example 23
3 2 1
O
1 2 3 4 input
x
b Plot each pair of values in the table to form a graph. c Do these points lie in a straight line?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2K The Cartesian plane and graphs
6 For the given rule βoutput = input Γ 2β: a Copy and complete the given table of values. input β(x)β
y
output β(y)β
output
Example 24
β0β β1β
125
U N SA C O M R PL R E EC PA T E G D ES
β2β
6 5 4 3 2 1
β3β
O 1 2 3 input
b Plot each pair of values in the table to form a graph. c Do these points lie in a straight line?
Example 24
7 For the given rule βoutput = input Γ inputβ: a Copy and complete the given table of values. input β(x)β
output β(y)β
β0β
β0β
β1β
y
10 9 8 7 6 5 4 3 2 1
output
β2β β3β
b Plot each pair of values in the table to form a graph. c Do these points lie in a straight line?
O
8 For the given rule βoutput = 6 Γ· input : β input β(x)β
output β(y)β
β1β β2β β3β β6β
b Plot each pair of values in the table to form a graph. c Do these points lie in a straight line?
1
2 input
x
3
y
a Copy and complete the given table of values.
output
Example 24
x
6 5 4 3 2 1
O 1 2 3 4 5 6 input
x
9 Draw a Cartesian plane from β0βto β5βon both axes. Place a cross on each pair of coordinates that have the same βxβ- and βyβ-value.
10 Draw a Cartesian plane from β0βto β8βon both axes. Plot the following points on the grid and join them in the order they are given. β(2, 7), (6, 7), (5, 5), (7, 5), (6, 2), (5, 2), (4, 1), (3, 2), (2, 2), (1, 5), (3, 5), (2, 7)β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
PROBLEM-SOLVING
11, 12
12, 13
12β14
11 a Plot the following points on a Cartesian plane and join the points in the order given, to draw the basic shape of a house. (β 1, 5)ββ, (ββ 0, 5)ββ, (ββ 5, 10)ββ, (ββ 10, 5)ββ, (ββ 1, 5)ββ, (ββ 1, 0)ββ, (ββ 9, 0)ββ, (ββ 9, 5)β b Describe a set of four points to draw a door. c Describe two sets of four points to draw two windows. d Describe a set of four points to draw a chimney.
U N SA C O M R PL R E EC PA T E G D ES
126
12 The perimeter of a square is its length multiplied by β4β. We can write this as a rule: βperimeter = 4 Γ length.β a Copy and complete the given table of values (all values are in centimetres). length β(x)β
perimeter (β y)β
1β β
β4β
Length
2β β 3β β 4β β
b Plot each pair of values to form a graph with length on the βxβ-axis and perimeter on the βyβ-axis. c Do these points lie in a straight line?
13 A grid system can be used to make secret messages. Jake decides to arrange the letters of the alphabet on a Cartesian plane in the following manner. a Decode Jakeβs following message: β(3, 2), (5, 1), (2, 3), (1, 4)β b Code the word SECRET. c To increase the difficulty of the code, Jake does not include brackets or commas and he uses the origin to indicate the end of a word. What do the following numbers mean? β13515500154341513400145354001423114354β. d Code the phrase: BE HERE AT SEVEN.
y
U
V
W
X
Y
P
Q
R
S
T
K
L
M
N
O
F
G
H
I
J
1
A
B
C
D
E
O
1
2
3
4
5
5 4 3 2
x
14 βABCDβis a rectangle. The coordinates of βAβ, βBβand βCβare given below. Draw each rectangle on a Cartesian plane and state the coordinates of the missing corner, βDβ. a b c d
βA(β 0, 5)βββB(β 0, 3)βββC(β 4, 3)βββD(β ? , ?)β βAβ(4, 4)βββB(β 1, 4)βββC(β 1, 1)βββD(β ? , ?)β βAβ(0, 2)βββB(β 3, 2)βββC(β 3, 0)βββD(β ? , ?)β βAβ(4, 1)βββB(β 8, 4)βββC(β 5, 8)βββD(β ? , ?)β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2K The Cartesian plane and graphs
REASONING
15
15ββ(β1/2β)β
127
15, 16
15 Write a rule (e.g. βoutput = input Γ 2β) that would give these graphs. b
y
10 8 6 4 2
y 3 2 1
O 1 2 3 4 input
output
output
U N SA C O M R PL R E EC PA T E G D ES
6 5 4 3 2 1
c
y
output
a
x
O
1 2 3 input
O 1 2 3 4 5 6 input
x
x
16 βA(β 1, 0)βand βB(β 5, 0)βare the base points of an isosceles triangle. a Find the coordinate of a possible third vertex. b Show on a Cartesian plane that there are infinite number of answers for this third vertex. c The area of the isosceles triangle is β10βsquare units. State the coordinates of the third vertex. ENRICHMENT: Locating midpoints
β
β
17
17 a Plot the points βA(β 1, 4)βand βB(β 5, 0)βon a Cartesian plane. Draw the line segment βABβ. Find the coordinates of βMβ, the midpoint of βABβ, and mark it on the grid. (It needs to be halfway along the line segment.) b Find the midpoint, βMβ, of the line segment βABβ, which has coordinates βA(β 2, 4)βand βB(β 0, 0)ββ. c Determine a method for locating the midpoint of a line segment without having to draw the points on a Cartesian plane. d Find the midpoint, βMβ, of the line segment βABβ, which has coordinates βAβ(6, 3)βand βB(β 2, 1)ββ. e Find the midpoint, βMβ, of the line segment βABβ, which has coordinates βAβ(1, 4)βand βB(β 4, 3)ββ. f Find the midpoint, βMβ, of the line segment βABβ, which has coordinates βAβ(β 3, 2)βand βB(β 2, β 3)ββ. g βMβ(3, 4)βis the midpoint of βABβand the coordinates of βAβare (β 1, 5)β. What are the coordinates of βBβ?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
128
Chapter 2
Number properties and patterns
Modelling
Plunge pool tiling
2m
U N SA C O M R PL R E EC PA T E G D ES
The Australian Plunge Pool Company offers a free tiled path around each plunge pool constructed. The standard width of each plunge pool is β2βmetres but the length can vary (it must be a whole number of metres). The tiling is completed using β1βmetre square tiles around all sides of the pool. This diagram shows the tiling for a pool of length β3β metres.
3m
Present a report for the following tasks and ensure that you show clear mathematical workings and explanations where appropriate.
Preliminary task a b c d
Use the diagram above to count the number of tiles required for a β2βmetre by β3βmetre plunge pool. Draw an accurate diagram of a rectangle which is β2βmetres by β6βmetres. If possible, use grid paper. On your diagram, accurately draw β1β-metre square tiles around the outside of the rectangle. Use your diagram to find how many tiles would be required for a β2βmetre by β6βmetre plunge pool.
Modelling task
Formulate
Solve
a The problem is to find a rule relating the number of tiles required to the length of the pool. Write down all the relevant information that will help solve this problem, and use diagrams to illustrate the following four pools. i β2 mβby β2 mβ ii β2 mβby β3 mβ iii β2 mβby β4 mβ iv 2β mβby β5 mβ b Use a table like the one shown to summarise the number of tiles required for varying pool lengths. Pool length β(m)β
β2β
β3β
β4β
5β β
β6β
Number of tiles
Evaluate and verify
Communicate
c d e f g
If the pool length increases by β1βmetre, describe the effect on the number of tiles required. If a pool could have a length of only β1βmetre, how many tiles would need to be used? Determine the rule linking the number of tiles and the length of the pool. Explain why your rule correctly calculates the number of tiles required. Give a geometric reason. Using your rule from part e above, find the number of tiles required for a plunge pool of length: i β8β metres ii β15β metres. h Using your rule from part e above, find the length of a plunge pool if the number of tiles used is: i β22β ii β44β. i Use your rule to explain why there will always be an even number of tiles required. j Summarise your results and describe any key findings.
Extension questions
a If luxury plunge pools can have a width of β3βmetres instead of β2βmetres, determine the rule linking the number of tiles and the length of these pools. Compare it to the rule found when the width is β2β metres. b The luxury plunge pools of width β3βmetres now require the path to be two tiles wide. Investigate how this requirement affects the rule for the number of tiles required. Use diagrams to help communicate your solution.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Technology and computational thinking
129
Making patterns
U N SA C O M R PL R E EC PA T E G D ES
Number patterns might seem like an easy thing to generate but describing relationships within patterns is at the foundation of algorithms and complex mathematical functions. Such algorithms can help us to write computer programs and find rules to help estimate solutions to complex problems. Two key words used in this investigation are βsequenceβ and βtermβ. The pattern β3, 5, 7, 9, 11β, for example, is a sequence of five terms and could be described as the list of odd numbers from β5β up to β11β. We will use the letter βnβto describe which term we are talking about. So, βn = 2βrefers to the β2ndβterm, which in this example is β5β.
Technology and computational thinking
(Key technology: Spreadsheets)
1 Getting started
a This table shows nine terms of the even number sequence starting at β2β. βnβ
β1β
2β β
3β β
β4β
β5β
β6β
β7β
β8β
β9β
Term
β2β
β4β
β6β
8β β
β10β
β12β
β14β
β16β
β18β
i What is the β4thβterm in the sequence of even numbers? ii Which term is equal to β16β? b Method 1: One way to generate the sequence of even numbers is to add β2βto the previous term. So, in this sequence, the β6thβterm is β12βand so the β7thβterm is β12 + 2 = 14β. i Use this idea to find the β10thβterm in the sequence. ii What will be the β13thβterm in the sequence? c Method 2: Another way to find terms is to use a rule and the value of βnβcorresponding to the term you are finding. For example: The β5thβterm is β2 Γ 5 = 10βand the β7thβterm is β2 Γ 7 = 14βand so, in general, the βnthβterm is given by β2 Γ nβ. i Use this idea to find the β11thβterm in the sequence. ii What will be the β17thβterm in the sequence? Start iii What will be the β29thβterm in the sequence?
2 Applying an algorithm
Letβs look at generating the list of odd numbers starting at β1β using Method βπβ. Look at this flow chart which generates the sequence. a Work through the flow chart and write the outputs into this table. βnβ
β1β
Term, βtβ
β1β
β2β
n = 1, t = 1 Output n, t
3β β
n = n + 1, t = t + 2
b Why does this algorithm only deliver β5βvalues for βnβand βtβ?
c Another way of generating the list of odd numbers is to use the rule: β Termβ(t)β= 2 Γ n β 1β. This is Method βπβas described in part βπβ. i Check this rule by confirming the terms in the table above. ii Use this rule to find the β8thβ term. iii Draw a flow chart that outputs the first β8βterms of the odd number sequence using this rule.
Is n = 6?
No
Yes End
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
130
Chapter 2
Number properties and patterns
We will use a spreadsheet to generate the odd number sequence using both methods described above. a Create a spreadsheet as follows. The βnβvalue is in column A and the sequence is generated in columns B and C using the two different methods.
U N SA C O M R PL R E EC PA T E G D ES
Technology and computational thinking
3 Using technology
b Fill down at cells βA5β, βB5βand βD4βto generate β8β terms. c Describe the advantages and disadvantages of both methods. d Now construct a spreadsheet to generate the following sequences using both method β1βand β2β. i A sequence starting at β1βand increasing by β3βeach time. ii A sequence starting at β20βand decreasing by β2βeach time.
4 Extension
Use a flow chart and a spreadsheet to generate the following number sequences. a The square numbers β1, 4, 9, β¦β b The triangular numbers β1, 3, 6, 10, β¦β c A number pattern of your choice.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Investigation
131
Fibonacci sequences
U N SA C O M R PL R E EC PA T E G D ES
Fibonacciβs rabbits
Investigation
Leonardo Fibonacci was a famous thirteenth century mathematician who discovered some very interesting patterns of numbers in nature.
These rules determine how fast rabbits can breed in ideal circumstances. β’ Generation β1β: One pair of newborn rabbits is in a paddock. A pair is one female and one male. β’ Generation β2β: After it is β2βmonths old (in the third month), the female produces another pair of rabbits. β’ Generation β3β: After it is β3βmonths old (in the fourth month), this same female produces another pair of rabbits. β’ Every female rabbit produces one new pair every month from age β2β months. a Using the βrabbit breeding rulesβ, complete a drawing of the first five generations of rabbit pairs. Use it to complete the table below. b Write down the numbers of pairs of rabbits at the end of each month for β12βmonths. This is the Fibonacci sequence. c How many rabbits will there be after β1β year? d Explain the rule for the Fibonacci sequence. Month
β1β
β2β
Number of rabbits
β2β
2
Number of pairs
1
3β β
β4β
5β β
Fibonacci sequence in plants
a Count the clockwise and anticlockwise spiralling βlumpsβ of some pineapples and show how these numbers relate to the Fibonacci sequence. b Count the clockwise and anticlockwise spiralling scales of some pine cones and show how these numbers relate to the Fibonacci sequence.
Fibonacci sequence and the golden ratio
a Copy this table and extend it to show the next β10βterms of the Fibonacci sequence: β1, 1, 2, 3, 5, β¦β Fibonacci sequence Ratio
1
1
2
3
5
β¦
1
2
1.5
β¦
β¦
1Γ·1
2Γ·1
3Γ·2
b Write down a new set of numbers that is one Fibonacci number divided by its previous Fibonacci number. Copy and complete this table. c What do you notice about the new sequence (ratio)? d Research the golden ratio and explain how it links to your new sequence.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
132
Chapter 2
Sticks are arranged by a student such that the first three diagrams in the pattern are:
Up for a challenge? If you get stuck on a question, check out the βworking with unfamiliar problemsβ poster at the end of the book to help.
U N SA C O M R PL R E EC PA T E G D ES
Problems and challenges
1
Number properties and patterns
How many sticks would there be in the β50βth diagram of the pattern? 2 A number is said to be a βperfect numberβ if the sum of its factors equals the number. For this exercise, we must exclude the number itself as one of the factors. The number β6βis the first perfect number. Factors of β6β(excluding the numeral β6β) are β1, 2βand β3β. The sum of these three factors is β1 + 2 + 3 = 6β. Hence, we have a perfect number. a Find the next perfect number. (Hint: It is less than β50β.) b The third perfect number is β496β. Find all the factors for this number and show that it is a perfect number.
3 Anya is a florist who is making up bunches of tulips with every bunch having the same number of tulips. Anya uses only one colour in each bunch. She has 1β 26βred tulips, 1β 08βpink tulips and 1β 44β yellow tulips. Anya wants to use all the tulips. a What is the largest number of tulips Anya can put in each bunch? b How many bunches of each colour would Anya make with this number in each bunch? 4 Mr and Mrs Adams have two teenage children. If the teenagersβ ages multiply together to give β252β, find the sum of their ages. 5 Complete this sequence. β 2β β+ 3ββββββββββ3β β β2β= 2β β β2β+ 5ββββββββββ4β β 2β β= 3β β β2β+ 7ββββββββββ5β β β2β=β _______ββββββββββ6β β β2β=β _______ 2β β 2β β= 1β
6 Use the digits β1, 9, 7βand β2β, in any order, and any operations and brackets you like, to make as your answers the whole numbers β0βto β10β. For example: β1 Γ 9 β 7 β 2 = 0β (β 9 β 7)βΓ· 2 β 1 = 0β
7 The first three shapes in a pattern made with sticks are:
How many sticks make up the β 100βth shape?
8 Two numbers have a highest common factor of β1β. If one of the numbers is β36βand the other number is a positive integer less than β36β, find all possible values for the number that is less than β36β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter summary
Factor trees
Multiples are β₯ number 20: 20, 40, 60 , 80, ... 15: 15, 30, 45, 60 , 75, ... Lowest common multiple LCM = 60
90 9
10
U N SA C O M R PL R E EC PA T E G D ES
A composite number has more than two factors 10: factors 1, 2, 5, 10 62: factors 1, 2, 31, 62 Composite numbers are not prime.
Multiples
Chapter summary
Composite numbers
133
3
3
2
5
90 = 2 Γ 3 Γ 3 Γ 5 90 = 2 Γ 32 Γ 5
Factors are β€ number 20: 1, 2, 4, 5 , 10, 20 15: 1, 3, 5 , 15 Highest common factor HCF = 5
Number properties
243 = 3 Γ 3 Γ 3 Γ 3 Γ 3 = 3 5
basic numeral
expanded form
Square numbers 16 = 4 Γ 4 = 42 25 = 5 Γ 5 = 52
Prime numbers
A prime number only has two factors: 1 and itself. 5: factors 1 and 5 17: factors 1 and 17
index index number form
First 10 square numbers
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
Square roots β16 = 4 β25 = 5 β22 is between 4 and 5
Divisibility tests (Ext)
Division
1 is not a prime number.
base number
Indices
Factors
dividendβ 52 1 divisorβ 3 = 17 3
quotient
remainder
remainder
52 = 3 Γ 17 + 1
divisor quotient
2: last digit even (0, 2, 4, 6, 8) 3: sum digits Γ· 3 4: number from last two digits Γ· 4 5: last digit 0 or 5 6: Γ· by 2 and 3 8: number from last 3 digits Γ· 8 9: sum digits Γ· 9 10: last digit 0
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
Sequence 2, 4, 6, 8, ... ββββ terms
Common difference of decreasing by 3: 100, 97, 94, ... Common ratio of 2: 3, 6, 12, 24, ...
U N SA C O M R PL R E EC PA T E G D ES
Chapter summary
134
Number patterns
Rules, graphs and tables show a relationship between two quantities that can vary.
Common difference: increase of 4 Spatial pattern:
...
Number pattern: 4
8
12
...
Rule: Number of sticks = 4 Γ number of rhombuses
input output
20 10 37 17
5 7
Graph showing a relationship y
Sam is 5 years older than his sister Mikaela Mikaelaβs age Samβs age
0 5
3 7 13 8 12 18
Samβs age (years)
output = 2 Γ input β 3
18 16 14 12 10 8 6 4 2
O
2 4 6 8 10 12 14 Mikaelaβs age (years)
x
y
the vertical y-axis
5 4 3 2 1
O
the origin
This dot represents a point with the coordinates (2, 4)
1 2 3 4 5
x
the horizontal x-axis
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter checklist
135
Chapter checklist with success criteria 1. I can list the factors of a number. e.g. Find the complete set of factors for the number β40β.
U N SA C O M R PL R E EC PA T E G D ES
2A
β
Chapter checklist
A printable version of this checklist is available in the Interactive Textbook
2A
2. I can list the multiples of a number (up to a certain limit). e.g. Write down the first six multiples of the number β11β.
2A
3. I can list pairs of factors of a number. e.g. Express 132 as a product of two factors both of which are greater than β10β.
2B
4. I can find the highest common factor (HCF) of two numbers. e.g. Find the highest common factor (HCF) of β36βand β48β.
2B
5. I can find the lowest common multiple of two numbers. e.g. Find the lowest common multiple of β6βand β10β.
2C
6. I can determine if a number is divisible by β2, 3, 4, 5, 6, 8, 9βand/or β10β. e.g. Consider the number β48 569 412β. State which of the following numbers are factors of it: β2, 3, 4, 5, 6, 8, 9, 10β.
2D
7. I can determine whether a number is prime by considering its factors. e.g. Explain why β17βis prime but β35βis not prime.
2D
8. I can find the prime factors of a number. e.g. Find the prime factors of β30β.
2E
9. I can convert an expression to index form. e.g. Write β3 Γ 3 Γ 2 Γ 3 Γ 2 Γ 3βin index form.
2E
10. I can evaluate expressions involving powers using the order of operations. β 2β ββ. β 2β ββ 6β e.g. Evaluate 7β
2F
11. I can express a composite number in prime factor form. e.g. Express the number β60βin prime factor form.
2G
12. I can find the square of a number. e.g. Evaluate 6β β 2β ββ.
2G
13. I can find the square root of a number which is a perfect square. e.g. Find the square root of β1600β.
2G
14. I can locate square roots between consecutive whole numbers. _ e.g. State which consecutive whole numbers are either side of β β 43 ββ.
2H
15. I can find the common difference or common ratio in a sequence. e.g. Find the common ratio in the number pattern β256, 128, 64, 32, β¦β
2H
16. I can find the next terms in a number pattern. e.g. Find the next three terms in the number pattern β6, 18, 30, 42, β¦β
Ext
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
β 2I
17. I can describe a spatial pattern in terms of the number of objects required for the first shape, and for later shapes being added. e.g. Describe the pattern by stating how many sticks are required to make the first term, and how many are required to make the next term in the pattern.
U N SA C O M R PL R E EC PA T E G D ES
Chapter checklist
136
2I
18. I can describe and use a rule for a spatial patterns. e.g. Describe a rule connecting the number of sticks required to the number of triangles, and use this to find the number of sticks for β20β triangles.
2J
19. I can complete a table of values for a given rule. e.g. Fill out the table for the rule βoutput = (β 3 Γ input)β+ 1β. input
β4β
β2β
β9β
β12β
0β β
output
2J
2K 2K
20. I can find a rule for a table of values. e.g. Find the rule for the table of values shown. input
β1β
β2β
3β β
β4β
5β β
output
β7β
β14β
β21β
2β 8β
β35β
21. I can plot points on a number plane. e.g. Plot the points βA(β 2, 5)β, Bβ(4, 3)βand βC(β 0, 2)ββ.
22. I can draw a graph of a rule by completing a table of values. e.g. For the rule βoutputβ(y)β= inputβ(x)β+ 1β, construct a table of values using input values of β0, 1, 2β and β3β. Use the table of coordinates to plot a graph.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
137
Short-answer questions 1
a Find the complete set of factors of β120βand circle those that are composite numbers. b Determine three numbers between β1000βand β2000βthat each have factors β1, 2, 3, 4, 5β and itself.
U N SA C O M R PL R E EC PA T E G D ES
2 a Write down the first β12βmultiples for each of β8βand β7βand circle the odd numbers. b Which two prime numbers less than β20βhave multiples that include both β1365βand β1274β?
Chapter review
2A/D
2A/D
2B
2D
2D/F
3 a Find the HCF of the following pairs of numbers. i β15βand β40β ii β18βand β26β b Find the LCM of the following pairs of numbers. i β5βand β13β ii β6βand β9β
iii β72βand β96β iii β44βand β8β
4 a State whether each of these numbers is a prime or composite number. β21, 30, 11, 16, 7, 3, 2β b How many prime multiples are there of β13β?
5 a State the prime factors of β770β. b Determine three composite numbers less than β100β, each with only three factors that are all prime numbers less than β10β.
2E
6 Simplify these expressions by writing them in index form. a 6β Γ 6 Γ 6 Γ 6 Γ 6 Γ 6 Γ 6 Γ 6β b β5 Γ 5 Γ 5 Γ 5 Γ 2 Γ 2 Γ 2 Γ 2 Γ 2β
2F
7 Write these numbers as a product of prime numbers. Use a factor tree and then index form. a 3β 2β b β200β c β225β
2F
8 Determine which number to the power of β5βequals each of the following. a 1β 00 000β b 2β 43β c β1024β
2G
9 Evaluate each of the following. a 5β β 2β ββ 3β β 2β β b β2 Γ 4β β 2β ββ 5β β 2β β
2C
10 Determine whether the following calculations are possible without leaving a remainder. a β32 766 Γ· 4β b 1β 136 Γ· 8β c β2417 Γ· 3β
c β5 Γ 3β β 4β ββ 3β β 2β β+ 1β β 6β β
d 12β β 2β ββ β(7β β 2β ββ 6β β 2β β)β
Ext
2C
11 a Carry out divisibility tests on the given number and fill in the table with ticks or crosses. State the explanation for each result.
Ext
Number
Divisible Divisible Divisible Divisible Divisible Divisible Divisible Divisible by β2β by β3β by β4β by β5β by β6β by β8β by β9β by β10β
8β 4 539 424β
b Use divisibility rules to determine a β10β-digit number that is divisible by β3, 5, 6βand β9β. c Determine a six-digit number that is divisible by β2, 3, 5, 6, 9βand β10β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
138
Chapter 2
12 a Evaluate: _ _ i β β 25 β ii β β 2500 β _ _ _ _ v ββ16 Γ 49 βΓ· β β 4β iv 4β β 2β ββ β β 25 β+ β β 7β β 2β β b State_ which consecutive whole numbers are either side of: _ β β i β 18 β ii β 74 β
_
β 2β β+ 1_ 2β β 2β β iii β β 5β β 3β β 2β β+ 4β β 2β β vi β10β β 2β βΓ· β _
iii β β 51 β
U N SA C O M R PL R E EC PA T E G D ES
Chapter review
2G
Number properties and patterns
2H
2H
2H
2I/J
13 Find the next three terms for the following number patterns that have a common difference. a β27, 30, 33, β¦β b β67, 59, 51, β¦β c 2β 38, 196, 154, β¦β 14 Find the next three terms for the following number patterns that have a common ratio. a 3β 5, 70, 140, β¦β b β24 300, 8100, 2700, β¦β c 6β 4, 160, 400, β¦β 15 Find the next six terms for each of these number patterns. a β21, 66, 42, 61, 84, 56, β¦β b β22, 41, 79, 136, β¦β
16 a Draw the next two shapes in this spatial pattern of sticks.
b Copy and complete this table. Number of rhombuses
β1β
β2β
3β β
β4β
5β β
Number of sticks required
c Describe the pattern by stating how many sticks are required to make the first rhombus and how many sticks must be added to make the next rhombus in the pattern.
2I
2J
17 A rule to describe a special window spatial pattern is: Number of sticks β= 3 Γβnumber of windows β+ 2β a How many sticks are required to make β1β window? b How many sticks are required to make β10β windows? c How many sticks are required to make βgβ windows? d How many windows can be made from β65β sticks? 18 Copy and complete each table for the given rule. a βoutput = input + 5β input
β3β
β5β
7β β
β12β
β20β
β9β
1β 2β
β0β
output
b βoutput = 2 Γ input + 7β input
β4β
β2β
output
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
2J
β3β
4β β
β5β
β6β
β7β
output
β12β
1β 3β
1β 4β
β15β
β16β
input
β1β
2β β
β3β
4β β
β5β
output
β20β
β32β
β44β
β56β
β68β
input
β0β
1β β
β2β
3β β
β4β
output
β1β
β3β
5β β
β7β
β9β
input
β3β
4β β
β5β
6β β
β7β
output
β7β
β6β
5β β
β4β
β3β
U N SA C O M R PL R E EC PA T E G D ES
b
input
Chapter review
19 Find the rule for each of these tables of values. a
139
c
d
2K
20 a State the coordinates of each point plotted on this number plane. b State the coordinates on this grid of a point βCβso that βABCDβis a square. c State the coordinates on this grid of a point βEβon the βxβ-axis so that β ABEDβis a trapezium (i.e. has only one pair of parallel sides).
y
5 4 3 2 1
B
A
D
O 1 2 3 4 5
2K
x
21 Determine the next three terms in each of these sequences and explain how each is generated. a β1, 4, 9, 16, 25, β¦β b β1, 8, 27, 64, β¦β c (β 1, 3)β, (β 2, 4)β, (β 3, 5)β, β¦β d β31, 29, 31, 30, 31, 30, β¦β _ _ e β1, β β 2 β, β β 3 β, 2, β¦β f β1, 1, 2, 2, 3, 4, 4, 8, 5, 16, 6, 32, β¦β
Multiple-choice questions
2A
2A
2C
1
Which number is the incorrect multiple for the following sequence? β3, 6, 9, 12, 15, 18, 22, 24, 27, 30β A 1β 8β B 2β 2β C β30β D β6β
E β3β
2 Which group of numbers contains every factor of β60β? A 2β , 3, 4, 5, 10, 12, 15, 60β B β2, 3, 4, 5, 10, 12, 15, 20, 30β C 1β , 2, 3, 4, 5, 10, 12, 15, 20, 30β D β2, 3, 4, 5, 10, 15, 20, 30, 60β E β1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60β
3 Which of the following numbers is not divisible only by prime numbers, itself and 1? A β21β B β77β C β110β D β221β E β65β
Ext
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 2
Number properties and patterns
2D
4 Which of the following groups of numbers include one prime and two composite numbers? A 2β , 10, 7β B β54, 7, 11β C β9, 32, 44β D β5, 17, 23β E β18, 3, 12β
2E
5 7β Γ 7 Γ 7 Γ 7 Γ 7βcan be written as: A 5β β 7β β B 7β β 5β β
2G
6
_ β 2β β+ 4β β 2β ββ. Evaluate β β 3β
C β7 Γ 5β
D β75β
E β77 777β
U N SA C O M R PL R E EC PA T E G D ES
Chapter review
140
A β7β
B β5β
C β14β
D β25β
E β6β
2B
7 The HCF and LCM of β12βand β18β are: A β6βand β18β B β3βand β12β
C β2βand β54β
D β6βand β36β
E β3βand β18β
2F
8 The prime factor form of β48β is: A 2β β β4βΓ 3β B 2β β 2β βΓ 3β β 2β β
C β2 Γ 3β β 3β β
D β3 Γ 4β β 2β β
E 2β β 3β βΓ 6β
2E
9 Evaluate 4β β 3β ββ 3 Γ (β2β β 4β ββ 3β β 2β β)ββ. A β427β B β18β
C β43β
D β320β
E β68β
2A
2C
Ext
2K
10 Factors of β189β are: A β3, 7, 9, 18, 21, 27β D β3, 7, 9, 17, 21β
B 3β , 9, 18, 21β E β3, 7, 9, 21, 27, 63β
11 Which number is not divisible by β3β? A β25 697 403β B 3β 1 975β D β28 650 180β E β38 629 634 073β
C β3, 9, 18β
C β7 297 008β
12 Which set of points is in a horizontal line? A (β 5, 5)ββ, (ββ 6, 6)ββ, (ββ 7, 7)β B β(3, 2)ββ, (ββ 3, 4)ββ, (ββ 3, 11)β C β(2, 4)ββ, (ββ 3, 6)ββ, (ββ 4, 8)β D β(5, 4)ββ, (ββ 6, 4)ββ, (ββ 8, 4)ββ, (ββ 12, 4)β E β(1, 5)ββ, (ββ 5, 1)ββ, (ββ 1, 1)ββ, (ββ 5, 5)β
Extended-response questions 1
For the following questions, write the answers in index notation (i.e. bβ β βxβ) and simplify where possible. a A rectangle has width/breadth β27 cmβand length β125 cmβ. i Write the width and the length as powers. ii Write power expressions for the perimeter and for the area. b A squareβs side length is equal to 4β β 3β ββ. i Write the side length as a power in a different way. ii Write power expressions in three different ways for the perimeter and for the area. c β5 Γ 5 Γ 5 Γ 5 Γ 7 Γ 7 Γ 7β d 4β β 3β β+ 4β β 3β β+ 4β β 3β β+ 4β β 3β β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
two desks
three desks
four desks
five desks
U N SA C O M R PL R E EC PA T E G D ES
one desk
Chapter review
2 A class arranges its square desks so that the space between their desks creates rhombuses of identical size, as shown in this diagram.
141
a How many rhombuses are contained between: i four desks that are in two rows (as shown in the diagram above)? ii six desks in two rows? b Draw β12βdesks in three rows arranged this way. c Rule up a table with columns for the number of: β’ rows β’ desks per row β’ total number of desks β’ total number of rhombuses. If there are four desks per row, complete your table for up to β24β desks. d If there are four desks per row, write a rule for the number of rhombuses in βnβrows of square desks. e Using a computer spreadsheet, complete several more tables, varying the number of desks per row. f Explain how the rule for the number of rhombuses changes when the number of desks, βdβ, per row varies and also the number of rows, βnβ, varies. g If the number of rows of desks equals the number of desks per row, how many desks would be required to make β10 000β rhombuses?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
U N SA C O M R PL R E EC PA T E G D ES
3 Fractions and percentages
Maths in context: Percentages and your money Our economy relies on the exchange of money, making the proο¬ts and incomes we need for the costs of living. Most money calculations use rates, and these are always written as percentages. βPer centβ and β%β means βper 100β. Because one dollar contains 100 cents, a given percentage tells us the number of cents per dollar. Stores regularly have sales and offer percentage discounts. For example, a 30% discount means 30 cents (= $(30/100) = $0.30) taken off each dollar of the original price. By using percentages to calculate discounted prices, the best deals can be found.
A pay rise is always welcomed by workers. A pay rise rate of 5% means 5 cents (= $(5/100) = $0.05) extra is added for every dollar of the employeeβs current pay. By using percentages, the new income can be calculated.
When you have completed your training and earn a weekly wage, you will likely want to live independently. Then it is important to prepare a budget (i.e. a saving and spending plan) and have money put aside for cost-of-living bills and personal use. To compare the proportions of income spent on each type of expense, you can calculate each as a percentage of your annual pay, totalling to 100%.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter contents Introduction to fractions (CONSOLIDATING) Equivalent fractions and simpliο¬ed fractions Mixed numerals and improper fractions (CONSOLIDATING) Ordering fractions Adding fractions Subtracting fractions Multiplying fractions Dividing fractions Fractions and percentages Finding a percentage of a number Introduction to ratios Solving problems with ratios
U N SA C O M R PL R E EC PA T E G D ES
3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L
Victorian Curriculum 2.0
This chapter covers the following content descriptors in the Victorian Curriculum 2.0: NUMBER
VC2M7N03, VC2M7N05, VC2M7N06, VC2M7N07, VC2M7N09, VC2M7N10 MEASUREMENT VC2M7M06
Please refer to the curriculum support documentation in the teacher resources for a full and comprehensive mapping of this chapter to the related curriculum content descriptors.
Β© VCAA
Online resources
A host of additional online resources are included as part of your Interactive Textbook, including HOTmaths content, video demonstrations of all worked examples, auto-marked quizzes and much more.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
3A Introduction to fractions CONSOLIDATING LEARNING INTENTIONS β’ To know what the numerator and denominator of a fraction represent in different situations β’ To be able to represent fractions on a number line β’ To be able to represent a fraction of a shape by dividing it into several regions and shading some of the regions
U N SA C O M R PL R E EC PA T E G D ES
144
The word βfractionβ comes from the Latin word βfrangereβ, which means βto break into piecesβ.
Although the following sentences are not directly related to the mathematical use of fractions, they all contain words that are related to the original Latin term βfrangereβ and they help us gain an understanding of exactly what a fraction is. β’
The fragile vase smashed into a hundred pieces when it landed on the ground. β’ After the window was broken, several fragments were found on the ο¬oor. β’ She fractured her leg in two places. β’ The computer was running slowly and needed to be defragmented. β’ The elderly gentleman was becoming very frail in his old age. Can you think of any other related words?
Spoken time can use fractions of an hour. The digital time 6:30 can be spoken as βhalf-past 6β 30 = _ 1 ; the digital time of because _ 60 2 3:15 is the same as βa quarter past 15 = _ 1. 3β because _ 60 4
Brainstorm specific uses of fractions in everyday life. The list could include cooking, shopping, sporting, building examples and more.
Lesson starter: What strength do you like your cordial?
1 strength cordial, Imagine preparing several jugs of different strength cordial. Samples could include _ 4 1 strength cordial, _ 1 strength cordial, _ 1 strength cordial. _ 8 6 5 β’
β’
In each case, describe how much water and how much cordial is needed to make a 1 litre mixture. Note: 1 litre (L) = 1000 millilitres (mL). On the label of a Cotteeβs cordial container, it suggests βTo make up by glass or jug: add four parts water to one part Cotteeβs Fruit Juice Cordial, according to tasteβ. What fraction of cordial do Cotteeβs suggest is the best?
KEY IDEAS
β A fraction is made up of a numerator (up) and a denominator (down).
For example: 3 5
numerator denominator
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3A Introduction to fractions
β’ β’ β’
145
The denominator tells you how many parts the whole is divided up into. The numerator tells you how many of the divided parts you have selected. The horizontal line separating the numerator and the denominator is called the vinculum.
β A proper fraction or common fraction is less than a whole, and therefore the numerator must
U N SA C O M R PL R E EC PA T E G D ES
be smaller than the denominator. β2 βis a proper fraction. For example: _ 7
β An improper fraction is greater than a whole, and therefore the numerator must be larger than
the denominator. Some mathematicians also consider fractions equal to a whole, like _ β3 β, to be 3 improper fractions. 5 βis an improper fraction. For example: β_ 3
β We can represent fractions on a number line.
This number line shows the whole numbers β0, 1βand β2β. Each unit has then been divided equally into four segments, therefore creating βquartersβ. 0
1 4
0
2 4
3 4
4 4
5 4
1
1 2
6 4
7 4
8 4
9 4
10 4
2
1
12
1
22
β Whole numbers can be represented as fractions.
On the number line above we see that β1βis the same as _ β4 βand β2βis the same as _ β8 ββ. 4 4
β We can represent fractions using area. If a shape is divided into regions of equal areas, then
shading a certain number of these regions will create a fraction of the whole shape. βFraction shaded = _ β3 β 4
For example:
BUILDING UNDERSTANDING
1 a State the denominator of this proper fraction: _ β2 ββ.
9 b State the numerator of this improper fraction: _ β7 ββ. 5
2 Group the following list of fractions into proper fractions, improper fractions and whole numbers.
b β_2 β
a _ β7 β
7
6
c _ β50 β 7
d _ β3 β 3
e _ β3 β 4
f
5β β_ 11
3 Find the whole numbers amongst the following list of fractions. (Hint: There are three whole numbers to find.)
a _ β15 β 4
b _ β12 β 5
c _ β30 β 15
d _ β30 β 12
e _ β12 β 12
f
_ β28 ββ
7
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
Example 1
Understanding the numerator and denominator
A pizza is shown in the diagram. a Into how many pieces has the whole pizza been divided? b How many pieces have been selected (shaded)? c In representing the shaded fraction of the pizza: i what must the denominator equal? ii what must the numerator equal? iii write the amount of pizza selected (shaded) as a fraction.
U N SA C O M R PL R E EC PA T E G D ES
146
SOLUTION
EXPLANATION
a 8β β
The pizza is cut into β8βequal pieces.
b β3β
β3βof the β8βpieces are shaded in blue.
c i
The denominator shows the number of parts the whole has been divided into.
β8β
ii β3β
The numerator tells how many of the divided parts you have selected.
3β iii β_ 8
The shaded fraction is the numerator over the denominator; i.e. β3βout of β8β divided pieces.
Now you try
a Into how many pieces has the whole pizza been divided? b How many pieces have been selected (shaded)? c In representing the shaded fraction of the pizza: i what must the denominator equal? ii what must the numerator equal? iii write the amount of pizza selected (shaded) as a fraction.
Example 2
Representing fractions on a number line
Represent the fractions _ β3 βand _ β9 βon a number line. 5 5 SOLUTION 0
3 5
EXPLANATION
1
92 5
Draw a number line starting at β0βand mark on it the whole numbers β0, 1βand β2β. Divide each whole unit into five segments of equal length. Each of these segments has a length of one-fifth.
Now you try
Represent the fractions _ β2 βand _ β7 βon a number line. 3 3
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3A Introduction to fractions
Example 3
147
Shading areas
Represent the fraction _ β3 βin three different ways, using a square divided into four equal regions. 4 EXPLANATION
U N SA C O M R PL R E EC PA T E G D ES
SOLUTION
Ensure division of the square creates four equal areas. Shade in three of the four regions. Other answers are possible.
Now you try
Represent the fraction _ β1 βin three different ways, using a rectangle divided into six equal regions. 3
Exercise 3A FLUENCY
Example 1
Example 1
1
1, 2, 3ββ(β1/2β)β,β 4
2, 3ββ(β1/2β)β, 4, 5ββ(β1/2β)β
3ββ(β1/2β)β, 4, 5ββ(β1/2β)β
A pizza is shown to the right. a Into how many pieces has the whole pizza been divided? b How many pieces have been selected (shaded)? c In representing the shaded fraction of the pizza: i what must the denominator equal? ii what must the numerator equal? iii write the amount of pizza selected (shaded) as a fraction.
2 Answer the following questions for each of the pizzas (A to D) drawn below. A B C
D
a Into how many pieces has the whole pizza been divided? b How many pieces have been selected (shaded)? c In representing the shaded fraction of the pizza: i what must the denominator equal? ii what must the numerator equal? iii write the amount of pizza selected (shaded) as a fraction.
Example 2
3 Represent the following fractions on a number line. b _ β2 βand _ β5 β 3 βand _ a β_ β6 β 3 3 7 7 11 2 βand _ β8 β e β_βand _ d β_ β11 β 5 5 4 4
1 βand _ β5 β c β_ 6 6 f
5 ,β _ β_ β9 βand _ β3 β 4 4 2
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
148
Chapter 3
Example 3
Fractions and percentages
4 Represent each of these fractions in three different ways, using a rectangle divided into equal regions. 2β 3β a _ β1 β c β_ b β_ 4 6 8
U N SA C O M R PL R E EC PA T E G D ES
5 Write the next three fractions for each of the following fraction sequences. 3 β, _ c _ β1 β, _ β2 β, _ β3 ,β _ β4 β, ___, ___, ___β a β_ β4 β, _ β5 β, _ β6 β, ___, ___, ___β b _ β5 β, _ β6 β, _ β7 β, _ β8 β, ___, ___, ___β 3 3 3 3 8 8 8 8 5 5 5 5 f _ β3 β, _ β8 β, _ β13 ,β _ β18 β, ___, ___, ___β d _ β11 β, _ β8 β, ___, ___, ___β e _ β13 β, _ β7 β, ___, ___, ___β β10 β, _ β9 β, _ β11 β, _ β9 β, _ 7 7 7 7 4 4 4 4 2 2 2 2 PROBLEM-SOLVING
6
6, 7
7, 8
6 What fractions correspond to each of the red dots positioned on these number lines? a b 0 1 2 3 4 5 6 7 0 1 2 c
0
1
2
3
4
d
0
1
2
3
4
7 What operation (i.e. β+β, βββ, βΓβor βΓ·β) does the vinculum relate to?
8 For each of the following, state what fraction of the diagram is shaded. a
b
c
d
e
f
REASONING
9
9, 10
10, 11
9 For each of the following, write the fraction that is describing part of the total. a After one day of a β43β-kilometre hike, the walkers had completed β12β kilometres. b Out of β15βstarters, β13βrunners went on and finished the race. c Rainfall was below average for β11βmonths of the year. d One egg is broken in a carton that contains a dozen eggs. e Two players in the soccer team consisting of β11βplayers scored a goal. f The lunch stop was β144βkilometres into the β475β-kilometre trip. g Seven members in the class of β20βhave visited Australia Zoo. h One of the four car tyres is worn and needs replacing. i It rained three days this week.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3A Introduction to fractions
149
10 a Explain, using diagrams, why the fractions _ β4 βand _ β6 βboth have the value of β1β. 4 6 b Give some examples of fractions which would have the value of β2β. C
D
U N SA C O M R PL R E EC PA T E G D ES
11 Which diagram has one-quarter shaded? A B
ENRICHMENT: Adjusting concentration
β
β
12
12 A β250β-millilitre glass of cordial is made by mixing four parts water to one part syrup. a What fraction of the cordial is syrup? b What amount of syrup is required?
Fairuz drinks β50βmillilitres of the cordial but finds it too strong. So he fills the glass back up with β50βmillilitres of water. c How much syrup is in the cordial now? d What fraction of the cordial is syrup?
Fairuz drinks β50βmillilitres of the cordial but still finds it too strong. So, once again, he fills the glass back up with β50βmillilitres of water. e How much syrup is in the glass now? f What fraction of the cordial is syrup?
Lynn prefers her cordial much stronger compared with Fairuz. When she is given a glass of the cordial that is mixed at four parts to one, she drinks β50βmillilitres and decides it is too weak. So she fills the glass back up with β50βmillilitres of syrup. g How much syrup is in Lynnβs glass now? h What fraction of the cordial is syrup?
Lynn decides to repeat the process to make her cordial even stronger. So, once again, she drinks β50βmillilitres and then tops the glass back up with β50βmillilitres of syrup. i How much syrup is in Lynnβs glass now? j What fraction of the cordial is syrup? k If Fairuz continues diluting his cordial concentration in this manner and Lynn continues strengthening her cordial concentration in this manner, will either of them ever reach pure water or pure syrup? Explain your reasoning.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
3B Equivalent fractions and simplified fractions LEARNING INTENTIONS β’ To understand what it means for two fractions to be equivalent β’ To be able to simplify fractions by dividing by their highest common factor
U N SA C O M R PL R E EC PA T E G D ES
150
Often fractions may look very different when in fact they have the same value.
For example, in an AFL football match, βhalf-timeβ is the same as βthe end of the second quarterβ. We can say 1 and _ 2 are equivalent fractions. In both situations, that _ 2 4 the equivalent fraction of the game has been completed. Consider a group of friends eating homemade pizzas. Each person cuts up their pizza as they like.
Trevor cuts his pizza into only two pieces, Jackie cuts hers into four pieces, Tahlia into six pieces and Jared into eight pieces. The shaded pieces are the amount that they have eaten before it is time to start the second movie.
Radiologists diagnose medical conditions by using body imaging techniques such as X-rays, CT scans, MRI scans and ultrasound. They analyse the energy wave scatter fractions and how these fractions change with an illness.
By looking at the pizzas, it is clear to see that Trevor, Jackie, Tahlia and Jared have all eaten the same amount of pizza. We can therefore 3 and _ 1, _ 4 are equivalent fractions. 2, _ conclude that _ 2 4 6 8
Trevor
Jackie
Tahlia
Jared
3=_ 1=_ 2=_ 4. This is written as _ 2 4 6 8
Lesson starter: Fraction clumps
Prepare a class set of fraction cards. (Two example sets are provided below.) β’ β’ β’
Hand out one fraction card to each student. Students then arrange themselves into groups of equivalent fractions. Set an appropriate time goal by which this task must be completed.
Repeat the process with a second set of equivalent fraction cards. Sample sets of fraction cards Class set 1
3,_ 3,_ 10 , _ 8,_ 5,_ 3, _ 1000 , _ 100 , _ 10 , _ 10 , _ 13 , _ 5,_ 3, _ 7,_ 7,_ 1, _ 1, _ 1, _ 1, _ 1, _ 2,_ 2, _ 2,_ 4,_ 2 _ 2 12 24 80 3 40 5 6 8 40 9 4 4000 200 50 16 30 39 10 14 6 28 10 20 8
Class set 2 9,_ 30 , _ 5,_ 20 , _ 6,_ 3,_ 3, _ 3, _ 10 , _ 2000 , _ 300 , _ 6, _ 6,_ 2, _ 21 , _ 14 , _ 4,_ 2,_ 24 , _ 11 , _ 4, _ 12 , _ 1, _ 2, _ 22 _ 3 14 18 10 12 64 66 6 7 70 32 8 15 30 6 5000 49 800 9 21 5 35 30 16 55
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3B Equivalent fractions and simplified fractions
151
KEY IDEAS β Equivalent fractions are fractions that represent the same numerical value.
β1 βand _ For example: _ β2 βare equivalent fractions. 2 4
U N SA C O M R PL R E EC PA T E G D ES
β Equivalent fractions are produced by multiplying the numerator and denominator by the same
number.
β Equivalent fractions can also be produced by dividing the numerator and denominator by the
same number.
β Simplifying fractions involves writing a fraction in its βsimplest formβ in which the numerator
and denominator have no common factors other than β1β. To do this, the numerator and the denominator must be divided by their highest common factor (HCF).
β It is a mathematical convention to write all answers involving fractions in their simplest form.
β A fraction wall can be helpful when comparing fractions. For example, the widths demonstrate
that four-sixths and two-thirds are equivalent.
1
one whole
1 2
halves
1 3
thirds
1 4
quarters fifths
sixths
1 5 1 6
1 8 1 ninths 9 1 tenths 10 1 twelfths 12
eighths
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
BUILDING UNDERSTANDING 1 Which of the following fractions are equivalent to _ β1 ββ? 2
3 β, _ β_ β3 β, _ β 3 β, _ β11 β, _ β 5 β, _ β 8 β, _ β2 ,β _ β 7 β, _ β2 ,β _ β6 β 5 6 10 4 22 15 12 1 10 10
U N SA C O M R PL R E EC PA T E G D ES
152
8 ββ? 2 Which of the following fractions are equivalent to β_ 20
4 β, _ β 6 β, _ β 4 β, _ β1 β, _ β 8 β, _ β16 β, _ β2 β, _ β12 β, _ β 80 β, _ β1 β β_ 10 5 20 10 40 5 12 40 200 4
3 Use a fraction wall or number line to give an example of three fractions that are equivalent to _ β2 .ββ 3
4 In the following lists of equivalent fractions, choose the fraction that is in its simplest form. a _ β 3 β, _ β10 β, _ β 2 β, _ β1 β 15 50 10 5
Example 4
b _ β100 ,β _ β7 β β 3 β, _ β1 ,β _
c _ β4 ,β _ β2 β, _ β16 ,β _ β20 β
600 18 6 42
6 3 24 30
Producing equivalent fractions with a given numerator or denominator
Fill in the missing number for the following equivalent fractions. a 2= 3 12
b 12 = 20 10
c
4 12 = 5
SOLUTION
EXPLANATION
a
To get from β3βto β12βusing multiplication or division only, multiply by β4β. The numerator must also be multiplied by β 4βto make the fraction equivalent.
Γ4
2= 8 3 12 Γ4
b
Γ·2
12 6 = 20 10
To get from β20βto β10β, divide by β2β. The numerator must also be divided by β2βto make the fraction equivalent.
Γ·2
c
Γ3
4 = 12 5 15
Multiply by β3βto get from β4βto β12β. The denominator must also be multiplied by β3βto make the fraction equivalent.
Γ3
Now you try
Fill in the missing number for the following equivalent fractions. a 3= 5 10
b 18 = 24 12
5 20 c 6=
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3B Equivalent fractions and simplified fractions
Example 5
153
Converting fractions to simplest form
7β b β_ 42
U N SA C O M R PL R E EC PA T E G D ES
Write these fractions in simplest form. 12 β a β_ 20 SOLUTION
EXPLANATION
a
The HCF of β12βand β20βis β4β. Both the numerator and the denominator are divided by the HCF, β4β.
Γ·4
12 3 = 20 5 Γ·4
b
The HCF of β7βand β42βis β7β. The β7βis βcancelledβ from the numerator and the denominator.
Γ·7
1 7 = 42 6 Γ·7
Now you try
Write these fractions in simplest form. 16 β a β_ 20
Example 6
b _ β14 β 56
Deciding if fractions are equivalent
Write either β=βor ββ βbetween the fractions to state whether the following are equivalent or not equivalent. 8 ββ _ a β_ β20 β β3 β b _ β2 ββ _ 3 12 10 25 SOLUTION a
Γ·2
EXPLANATION
Γ·5
8 4 20 4 = and = 10 5 25 5 Γ·2
Write each fraction in simplest form. The numerators (β 4)βand denominators (β 5)βare equal, so they are equivalent.
Γ·5
so _ β 8 β= _ β20 β 10 25
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
154
Chapter 3
Fractions and percentages
SOLUTION b
Γ·2
EXPLANATION Write each fraction in simplest form. If they are equivalent their simplest form will have the same numerators and denominators. If not, they are not equivalent. The denominators are different, so they are not β 4 ββto β2 β= _ equivalent. An alternative method is to write _ 6 12 show that when the denominators are the same (β 12)ββ, the numerators are different.
Γ·3
2 1 1 3 and = = 6 3 12 4 Γ·3
U N SA C O M R PL R E EC PA T E G D ES
Γ·2
so _ β2 ββ _ β3 β 6 12
Now you try
By writing β=βor ββ βbetween the fractions, state whether the following pairs of fractions are equivalent or not equivalent. a _ β2 ββ _ β8 β β16 β b _ β 4 ββ _ 10 30 5 20
Exercise 3B FLUENCY
Example 4a
1
4β6β(β β1/2β)β
3 = 4 8
b
1 = 2 6
c
2 = 5 10
d 3= 5 20
2 Fill in the missing number for the following equivalent fractions. a
Example 4c
1β6β(β β1/2β)β
Fill in the missing number for the following equivalent fractions. a
Example 4b
1β3, 5β6ββ(β1/2β)β
10 = 20 2
b
6 = 10 5
c
15 = 20 4
3 Fill in the missing number for the following equivalent fractions. a 2= 6 b 3= 6 c 5 = 20 10 5 4
Example 4
4 Find the unknown value to make the equation true.
Example 5
5 β= _ a _ β3 β= _ β? β β? β b β_ 4 12 8 80 β14 β f β_? β= _ e _ β3 β= _ β15 β 7 1 ? 40 10 19 2 _ _ _ i β β= β β j β β= _ β190 β 7 ? 20 ? 5 Write the following fractions in simplest form. 15 β a β_ 20 e _ β14 β 35 _ i β35 β 45
12 β b β_ 18 2β f β_ 22 36 β j β_ 96
d
8 = 12 6
d 6 = 30 8
6 β= _ c β_ β18 β 11 ? ? 24 _ _ g β β= β β 10 20 _ β55 β k β11 β= _ 21 ?
2 β= _ β16 β d β_ 7 ? 13 _ β? β h β β= _ 14 42 l _ β11 β= _ β44 β ? 8
10 β c β_ 30 8β g β_ 56 120 β k β_ 144
8β d β_ 22 9β h β_ 27 700 β l β_ 140
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3B Equivalent fractions and simplified fractions
6 Write either β=βor ββ βbetween the fractions, to state whether the following pairs of fractions are equivalent or not equivalent. 4 ββ _ 3 ββ _ b β_ β2 β β30 β c β_ a _ β1 ββ _ β5 β 7 70 8 4 2 8 11 ββ _ 1 ββ _ β33 β β402 β e β_ f β_ d _ β5 ββ _ β15 β 2 804 9 18 15 45 18 ββ _ 6 ββ _ g _ β12 ββ _ β1 β h β_ β21 β β11 β i β_ 36 3 24 28 18 33
U N SA C O M R PL R E EC PA T E G D ES
Example 6
155
PROBLEM-SOLVING
7
7, 8
8, 9
7 These lists of fractions are meant to contain only fractions in their simplest form; however, there is one mistake in each list. Find the fraction that is not in simplest form and rewrite in simplest form. β3 β, _ β5 β, _ β7 β a _ β1 ,β _ 3 8 9 14
2 ,β _ b β_ β12 β, _ β16 β, _ β13 β 5 16 9 37
7 ,β _ d β_ β 9 β, _ β11 β, _ β13 β 63 62 81 72 1 βa tank of free petrol. 8 Four people win a competition that allows them to receive β_ 2 Find how many litres of petrol the drivers of these cars receive. a Ford Territory with a β70β-litre tank b Nissan Patrol with a β90β-litre tank c Holden Commodore with a β60β-litre tank d Mazda β323βwith a β48β-litre tank c _ β12 ,β _ β 4 β, _ β 5 β, _ β6 β 19 42 24 61
9 A family block of chocolate consists of β12βrows of β6βindividual squares. Tania eats β16β individual squares. What fraction of the block, in simplest terms, has Tania eaten? REASONING
10
10, 11
11, 12
10 Justin, Joanna and Jack are sharing a large pizza for dinner. The pizza has been cut into β12β equal pieces. Justin would like _ β1 βof the pizza, Joanna would like _ β1 ββof the pizza and Jack will eat whatever is 3 4 remaining. By considering equivalent fractions, determine how many slices each person gets served.
11 J.K. Rowlingβs first book, Harry Potter and the Philosopherβs Stone, is β225β pages long. Sam plans to read the book in three days, reading the same number of pages each day. a How many pages should Sam read each day? 75 βof the book is b The fraction β_ 225 equivalent to what fraction in simplest form? c By the end of the second day, Sam is
on track and has read _ β2 βof the book. 3 How many pages of the book does _ β2 ββrepresent? 3
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
12 A fraction when simplified is written as _ β3 ββ. 5 a List some possibilities for what the original fraction was. b Explain why the sum of the original fractionβs numerator and denominator must be even. ENRICHMENT: Equivalent bars of music
β
β
13
13 Each piece of music has a time signature. A common time signature is called _ β4 βtime, and is actually 4 referred to as common time! 4 βtime, means that there are four βquarter notesβ (or crotchets) in each bar. Common time, or β_ 4 Listed below are the five most commonly used musical notes. whole note (fills the whole bar) β semibreve
U N SA C O M R PL R E EC PA T E G D ES
156
half note (fills half the bar) β minim
quarter note (four of these to a bar) β crotchet eighth note (eight to a bar) β quaver
sixteenth note (sixteen to a bar) β semiquaver
When two or more quavers or shorter notes appear together, their βflagsβ are joined together to form a horizontal bar. a Write six different βbarsβ of music in _ β4 ββtime. 4 Carry out some research on other types of musical time signatures. 12 ββmeans? b Do you know what the time signature β_ 8 c Write three different bars of music for a _ β12 βtime signature. 8 d What are the musical symbols for different length rests? e How does a dot (or dots) written after a note affect the length of the note?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3C Mixed numerals and improper fractions
157
3C Mixed numerals and improper fractions CONSOLIDATING
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To know the meaning of the terms improper fractions and mixed numerals β’ To be able to convert from a mixed numerals to an improper fraction β’ To be able to convert from an improper fraction to a mixed numeral
As we have seen in this chapter, a fraction is a common way of representing part of a whole number. For 2 of a tank of petrol. example, a particular car trip may require _ 3
On many occasions, you may need whole numbers plus a part of a whole number. For example, a long 1 tanks of petrol. Note that 2_ 1 is a way of writing 2 + _ 1. interstate car trip may require 2_ 4 4 4
When you have a combination of a whole number and a proper fraction, this number is known as a mixed numeral.
Nurses use many formulas requiring accurate arithmetic calculations. The time for an intravenous drip equals the total volume divided by the drip rate. The result is often a mixed 23 = 5_ 3 hours. numerals, such as _ 4 4
Lesson starter: The sponge cake
With a partner, attempt to solve the following sponge cake problem. There is more than one answer.
At Peteβs cake shop, small cakes are cut into four equal slices, medium cakes are cut into six equal slices and large cakes are cut into eight equal slices. For a 70th birthday party, 13 cakes were ordered all of which were eaten. After the last slice was eaten, a total of 82 slices of cake had been eaten. How many cakes of each size were ordered?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
KEY IDEAS β A number is said to be a mixed numeral when it is a mix of a whole number plus a proper
fraction. 2
3 is a mixed numeral 5
U N SA C O M R PL R E EC PA T E G D ES
158
whole number
proper fraction
β Improper fractions (fractions greater than a whole, where the numerator is greater than the
denominator) can be converted to mixed numerals or whole numbers. 15 3 =3 4 4
improper fraction
16 =4 4
improper fraction
mixed numeral
whole number
β Mixed numerals can be converted to improper fractions.
β When improper fractions are written as mixed numerals, the fraction part should be written in
simplest form.
β A number line helps show the different types of fractions.
improper fractions
0
1 4
1 2
3 4
4 4
5 4
6 4
7 4
8 4
9 4
10 4
11 4
12 4
1
11
11
13
2
21
21
23
3
4
2
4
4
2
4
mixed numerals
proper fractions
BUILDING UNDERSTANDING
1 Between which two whole numbers do the following mixed numerals lie? a β2_ β1 β b β11β_1 β c β36β_8 β 2
7
9
2 Work out the total number of pieces in each of these situations. a four pizzas cut into six pieces each b β10βLego trucks, where each truck is made from β36βLego pieces c five jigsaw puzzles with β12βpieces in each puzzle d three cakes cut into eight pieces each
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3C Mixed numerals and improper fractions
159
3 The mixed numeral β2_ β3 βcan be represented in βwindow shapesβ as 4
+
3 β=β 2β β_ 4
+
U N SA C O M R PL R E EC PA T E G D ES
Draw or describe the following mixed numerals using window shapes.
b 1β _ β3 β
a 1β _ β1 β 4
d 5β _ β2 β
c 3β _ β2 β
4
4
4
4 A window shape consists of four panes of glass. How many panes of glass are there in the following number of window shapes? a β2β b β3β c 7β β 3 1 _ _ e β4β β g β2_ β2 β f β1β β 4 4 4
d 1β 1β h β5_ β4 β 4
5 What mixed numerals correspond to the red dots on each number line? a c
7
8
22
23
Example 7
b
9 10 11 12 24
25
0 1 2 3 4 5
26
Converting mixed numerals to improper fractions
Convert β3_ β1 βto an improper fraction. 5 SOLUTION
1β = _ 3β_ β3 Γ 5 + 1 β 5 β β 5 ββ―β― β 16 _ β= β β 5
EXPLANATION
3
1 1 of a whole = 3 wholes + 5 5 =
+
+
+
=
+
+
+
Multiply the whole number by the denominator and then add the numerator. β3 Γ 5 + 1 = 16β
Now you try
Convert β2_ β5 βto an improper fraction. 6
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
Example 8
Converting improper fractions to mixed numerals
Convert _ β11 βto a mixed numeral. 4 SOLUTION
EXPLANATION
Method 1 3β 8 β+ _ _ β11 β= _ β3 β= 2 + _ β3 β= 2β_ β8 + 3 β= β_ 4 4 4 4 4 4
11 = 11 quarters 4 = + +
U N SA C O M R PL R E EC PA T E G D ES
160
Method 2
+
2 rem 3 4 11
)
4
= 2
+
+
=
3β So _ β11 β= 2β_ 4
+
+
+
+
+
+
+
3 4
Now you try
Convert _ β23 βto a mixed numeral. 7
Example 9
Writing mixed numerals in simplest form
Convert _ β20 βto a mixed numeral in simplest form. 6 SOLUTION
EXPLANATION
1β 2 β= 3β_ _ β20 β= 3β_ 6
6
3
Method 1: Convert to mixed numeral and then simplify the fraction part. Γ·2 2 1 Note 6 = 3 Γ·2
or
Γ·2
1 20 10 = =3 3 6 3
Method 2: Simplify the improper fraction first and then convert to a mixed numeral.
Γ·2
Now you try Convert _ β18 βto a mixed numeral in simplest form. 4 Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3C Mixed numerals and improper fractions
161
Exercise 3C FLUENCY
1
2β4β(β 1β/3β)β
Convert the following mixed numerals to improper fractions. a β1_ β2 β b β2_ β1 β c β3_ β1 β 3 2 5
2β4ββ(1β/4β)β
d 1β _ β3 β 4
U N SA C O M R PL R E EC PA T E G D ES
Example 7
1, 2β4ββ(1β/2β)β
Example 7
Example 8
Example 9
2 Convert these mixed numerals to improper fractions. 1β a β2_ β1 β b β3β_ 3 5 2β d β2_ β1 β e β4β_ 2 3 2β g β6_ β1 β h β5β_ 9 8 15 β j β4_ β5 β k 5β β_ 12 20
1β c 4β β_ 7 2β f β8β_ 5 11 β i β1β_ 12 3β l β64β_ 10
3 Convert these improper fractions to mixed numerals. 7β 5β a β_ b β_ 5 3 12 β e β_ d _ β16 β 7 7 48 35 _ _ h β β g β β 7 8 93 231 _ _ k β β j β β 100 10
11 β c β_ 3 20 _ f β β 3 37 _ i β β 12 _ l β135 β 11
4 Convert these improper fractions to mixed numerals in simplest form. 10 β a β_ b _ β28 β c _ β16 β 4 10 12 18 30 _ _ _ e β β f β β g β40 β 9 16 15 PROBLEM-SOLVING
5
d _ β8 β 6 _ h β60 β 25
5, 6
6, 7
5 Draw a number line from β0βto β5βand mark the following fractions and mixed numerals on it. 3 β, _ 1β 1 β, 3β_ 4 β, _ 1 β, _ 2 β, 2, _ 1β b β_ β12 β, 2β_ β14 ,β 3β_ β10 β, _ c β_ β19 β a β_ β5 β, 3β_ 4 4 4 2 3 3 3 5 5 5 5 5
1 βis located between β4βand β5βon a number line, which means _ β25 βis also located between β 6 The number β4β_ 6 6 4βand β5β(as it has the same value). Convert the following to mixed numerals to find the two whole numbers located either side. 17 β 29 β 50 β a β_ b β_ c β_ 3 4 8 7 Four friends order three large pizzas for their dinner. Each pizza is cut into eight equal slices. Simone has three slices, Izabella has four slices, Mark has five slices and Alex has three slices. a How many pizza slices do they eat in total? b How many pizzas do they eat in total? Give your answer as a mixed numeral. c How many pizza slices are left uneaten? d How many pizzas are left uneaten? Give your answer as a mixed numeral.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
REASONING
8
8, 9
9, 10
8 Explain, using diagrams, why _ β7 βand β1_ β3 βare equal. You could use shapes split into quarters or a 4 4 number line. 9 a Patricia has three sandwiches that are cut into quarters and she eats all but one-quarter. How many quarters does she eat? b Phillip has five sandwiches that are cut into halves and he eats all but one-half. How many halves does he eat? c Crystal has βxβsandwiches that are cut into quarters and she eats all but one-quarter. How many quarters does she eat? d Byron has βyβsandwiches that are cut into thirds and he eats all but one-third. How many thirds does he eat? e Felicity has βmβsandwiches that are cut into βnβ pieces and she eats them all. How many pieces does she eat?
U N SA C O M R PL R E EC PA T E G D ES
162
10 A fully simplified mixed numeral between β2βand β3β is written as an improper fraction with a numerator of β17β. List all the possible mixed numerals matching this description. ENRICHMENT: Mixed numeral swap meet
β
β
11
11 Using the digits β1β, β2βand β3βonly once, three different mixed numerals can be written. a i Write down the three possible mixed numerals. ii Find the difference between the smallest and highest mixed numerals. b Repeat part a using the digits β2, 3βand β4β. c Repeat part a using the digits β3, 4βand β5β. d Predict the difference between the largest and smallest mixed numerals when using only the digits β 4, 5βand β6β. Check to see if your prediction is correct. e Write down a rule for the difference between the largest and smallest mixed numerals when using any three consecutive integers. f Extend your investigation to allow mixed numerals where the fraction part is an improper fraction. g Extend your investigation to produce mixed numerals from four consecutive digits.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3D Ordering fractions
163
3D Ordering fractions
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To know the meaning of the terms ascending and descending β’ To be able to compare two fractions and decide which one is larger β’ To be able to order a list of fractions in ascending or descending order
You already know how to order a set of whole numbers.
For example: 3, 7, 15, 6, 2, 10 are a set of six different whole numbers that you could place in ascending or descending order. In ascending order, the correct order is: 2, 3, 6, 7, 10, 15. In descending order, the correct order is: 15, 10, 7, 6, 3, 2.
In this section, you will learn how to write different fractions in ascending and descending order. To be able to do this we need to compare different fractions and we do this through our knowledge of equivalent fractions (see Section 3B).
A farm mechanic maintains agricultural machinery such as harvesters and irrigation pumps. Fraction skills are essential as many components 1 are manufactured with fractional measurements, e.g. a cover plate 6_ 4 3 inches wide. inches long by 4_ 8
Remember that a number is greater than another number if it lies to the right of it on a number line. 3>_ 1 _ 4
2
0
1 2
3 4
1
Lesson starter: The order of ο¬ve β’ β’ β’
β’
As a warm-up activity, ask five volunteer students to arrange themselves in alphabetical order, then in height order and, finally, in birthday order. Each of the five students receives a large fraction card and displays it to the class. The rest of the class must then attempt to order the students in ascending order, according to their fraction card. It is a group decision and none of the five students should move until the class agrees on a decision. Repeat the activity with a set of more challenging fraction cards.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
KEY IDEAS β To order (or arrange) fractions we must know how to compare different fractions. This is often
done by considering equivalent fractions. β If the numerators are the same, the smallest fraction is the one with the biggest denominator, as
U N SA C O M R PL R E EC PA T E G D ES
164
it has been divided up into the most pieces. For example: _ β1 β< _ β1 β 7 2
β If the denominators are the same, the smallest fraction is the one with the smallest numerator.
For example: _ β 3 β< _ β7 β 10 10
β To order two fractions with different numerators and denominators, we can use our knowledge
of equivalent fractions to produce fractions with a common denominator and then compare the numerators.
β The lowest common denominator (LCD) is the lowest common multiple of the different
denominators.
β Ascending order is when numbers are ordered going up, from smallest to largest. On a number
line, this means listing the numbers from left to right.
β Descending order is when numbers are ordered going down, from largest to smallest. On a
number line, this means listing the numbers from right to left.
BUILDING UNDERSTANDING
1 State the largest fraction in each of the following lists. 1β a _ β3 ,β _ β2 β, _ β5 β, β_ 7 7 7 7
c _ β 5 β, _ β 9 β, _ β 3 β, _ β4 β
b _ β4 ,β _ β2 β, _ β7 β, _ β5 β 3 3 3 3
d _ β8 ,β _ β4 ,β _ β6 β, _ β7 β
11 11 11 11
5 5 5 5
2 State the lowest common multiple of the following sets of numbers. a β2, 5β b β5, 4β c β3, 6β e β2, 3, 5β f β3, 4, 6β g β3, 8, 4β
d 4β , 6β h β2, 6, 5β
3 State the lowest common denominator of the following sets of fractions. a _ β1 ,β _ β3 β
b _ β2 ,β _ β3 β
3 5 _ e β4 β, _ β3 β 6 8
f
c _ β4 ,β _ β2 β
d _ β 2 β, _ β1 β
7 3 _ g β1 β, _ β2 β, _ β3 β 2 3 4
4 5 _ β 5 β, _ β2 β 12 5
10 5
h
_ β4 β, _ β3 β
3 4
4 Find the missing numbers to produce equivalent fractions. a 2= 5
d 3= 7
15
14
b 2= 3
e 3= 8
12
40
c 1= 4
f
16
5 = 18 6
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3D Ordering fractions
165
Example 10 Comparing fractions
U N SA C O M R PL R E EC PA T E G D ES
Place the correct mathematical symbol <, = or >, in between the following pairs of fractions to make a true mathematical statement. 2 ββ _ _ a β_ β4 β b _ β1 ββ β1 β 3 5 5 5 3 3 2 _ _ c _ β ββ β β β16 β d β2_ β ββ 7 7 3 5 SOLUTION
EXPLANATION
2 ββ < _ a β_ β4 β 5 5
Denominators are the same, therefore compare numerators.
b _ β1 ββ > _ β1 β 3 5
Numerators are the same. Smallest fraction has the biggest denominator.
Γ5
c
Γ3
2 10 3 9 = and = 3 15 5 15 Γ5
Γ3
_ β10 ββ > _ β 9 β. Hence, _ β3 .ββ β2 ββ > _
15
15
3 ββ d 2β β_ 7
_ β16 β
3
5
7
_ β16 β. Hence, β2_ β16 .ββ β3 ββ > _ β17 ββ > _
7
7
LCD of β3βand β5βis β15β, so write equivalent fractions with β15βas denominator. Denominators now the same, therefore compare numerators.
7
7
First, convert mixed numerals to improper fractions. Now compare the two improper fractions to decide which is bigger.
Now you try
Place the correct mathematical symbol <, = or >, in between the following pairs of fractions to make a true mathematical statement. a _ β4 ββ 7 c _ β10 ββ 16
_ β2 β
7
_ β15 β 24
b _ β1 ββ 9 d β2_ β4 ββ 9
_ β1 β
6
25 β β_ 9
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
166
Chapter 3
Fractions and percentages
Example 11 Ordering fractions and mixed numerals
1 β, _ b β1_ β3 β, _ β7 β, _ β3 ,β 2β_ β11 β 5 4 2 4 5
U N SA C O M R PL R E EC PA T E G D ES
Place the following fractions in ascending order. 3 β, _ a β_ β4 β, _ β2 β 4 5 3 SOLUTION Γ 15
a
EXPLANATION
Γ 12
LCD of β3, 4βand β5βis β60β. Produce equivalent fractions with denominator of β60β.
Γ 20
3 45 , 4 48 , 2 40 = = = 4 60 5 60 3 60 Γ 15
Γ 12
Γ 20
40 β, _ β_ β45 β, _ β48 β
Order fractions in ascending order.
2 β, _ β3 β, _ β4 β β_ 3 4 5
Rewrite fractions back in original form.
60 60 60
8 ,β _ β7 β, _ β3 β, _ β9 β, _ β11 β b β_ 5 4 2 4 5
Express all mixed numerals as improper fractions.
_ β35 β, _ β30 β, _ β45 β, _ β44 β β32 β, _
LCD of β2, 4βand β5βis β20β. Produce equivalent fractions with a denominator of β20β.
30 β, _ β_ β32 β, _ β35 β, _ β44 β, _ β45 β 20 20 20 20 20
Order fractions in ascending order.
3 β, _ 3 β, 1β_ 1β β7 β, _ β11 β, 2β_ β_ 2 5 4 5 4
Rewrite fractions back in original form.
20 20 20 20 20
Now you try
Place the following fractions in ascending order. 3 β, _ a β_ β2 β, _ β1 β 4 6 2
1 ,β _ β5 β, 2β_ β3 β, _ β9 β, _ β23 β b β1_ β7 β, _ 8 4 2 4 4 8
Exercise 3D FLUENCY
Example 10a
1
1β7ββ(β1/2β)β
4β8β(β β1/2β)β
Place the correct mathematical symbol <, = or >, in between the following pairs of fractions to make a true mathematical statement. a _ β3 ββ 7
Example 10b
1β3, 5
_ β2 β
7
b _ β4 ββ 9
_ β7 β
9
c _ β 4 ββ 12
_ β3 β
12
d _ β5 ββ 3
_ β6 β
3
2 Place the correct mathematical symbol <, = or >, in between the following pairs of fractions to make a true mathematical statement. a _ β1 ββ 5
_ β1 β
7
b _ β1 ββ 9
_ β1 β
7
c _ β3 ββ 6
_ β3 β
10
d _ β 4 ββ 11
_ β4 β
11
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3D Ordering fractions
Example 10c
3 Place the correct mathematical symbol <, = or >, in between the following pairs of fractions to make a true mathematical statement. 2 ββ a β_ 3
_ β1 β
6
b _ β4 ββ 5
_ β9 β
10
c _ β3 ββ 4
_ β7 β
d _ β2 ββ 5
10
_ β3 β
7
4 Place the correct mathematical symbol <, = or >, in between the following pairs of fractions to make a true mathematical statement. 3 ββ _ _ _ _ c _ β1 ββ β1 β d _ β 1 ββ β1 β a β_ β1 β b _ β2 ββ β3 β 4 3 10 20 2 3 5 5 4 4 3 9 15 5 5 4 _ _ _ _ _ _ _ _ e β ββ β β f β ββ β β β β β β g β ββ h β ββ 7 7 5 4 5 10 21 6 3 2 1 7 15 3 12 _ _ _ _ _ _ _ j β1β ββ β1β β i β ββ β β β2_ β3 β l β ββ β β k β3β ββ 3 2 7 8 11 5 4 5
U N SA C O M R PL R E EC PA T E G D ES
Example 10
167
Example 11a
Example 11b
5 Place the following fractions in ascending order. a _ β3 β, _ β8 β, _ β7 β 5 5 5 c _ β2 β, _ β3 β, _ β4 β 5 4 5
5 ,β _ β1 β, _ β2 β b β_ 9 3 9 5 β, _ β3 β, _ β2 β d β_ 6 5 3
6 Place the following fractions and mixed numerals in ascending order. 1β a β2_ β1 β, _ β11 β, _ β5 β, 3β_ 4 4 2 3
7 β, _ β11 β, _ β5 β b 1β β_ β7 β, _ 8 6 4 3
3β 1 β, 2β_ c 2β _ β 7 ,β _ β11 β, 2β_ β9 β, _ 10 4 5 2 5
10 β, 4β_ 4 β, _ 1β 2 ,β 4β_ d 4β β_ β15 β, 4β_ 9 3 27 3 6
7 Place the following fractions in descending order. 3 β, _ 1 β, _ a β_ β3 β, _ β 3 ,β _ β 7 β, _ β4 β b β_ β1 ,β _ β5 β, _ β3 β, _ β1 β 2 5 10 10 5 8 2 8 4 4
1 ,β _ c β_ β 5 β, _ β1 ,β _ β2 β β5 β, _ 3 12 6 2 3
8 Place the following fractions in descending order, without finding common denominators. 3 ,β _ a β_ β3 β, _ β3 β, _ β3 β 5 7 6 8
b _ β 1 ,β _ β 1 β, _ β1 β β 1 β, _ 15 10 50 100
3 β, 5β_ 4 ,β 10β_ 2β c 7β _ β 1 β, 8β_ 3 11 5 9
1 β, 2β_ 1 β, 2β_ 1 β, 2β_ 1β d 2β β_ 3 9 6 5
PROBLEM-SOLVING
9
9, 10
10, 11
9 Place the following cake fractions in decreasing order of size.
A sponge cake shared equally by four people β= _ β1 ββcake 4 B chocolate cake shared equally by eleven people β= _ β 1 ββcake 11 C carrot and walnut cake shared equally by eight people β= _ β1 ββcake 8
10 Four friends, Dean, David, Andrea and Rob, all competed in the Great Ocean Road marathon. Their respective finishing times were β3_ β1 βhours, β 3 5 βhours, β3_ 3β_ β 4 βhours. Write down the correct finishing β1 βhours and β3_ 4 12 15 order of the four friends.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
11 Rewrite the fractions in each set with their lowest common denominator and then write the next two fractions that would continue the pattern. 2 β, _ a β_ β1 β, _ β4 β, _____, _____β β5 β, 2, _____, _____β b _ β1 β, _ 9 3 9 2 4 c _ β11 β, _ β7 β, _____, _____β β3 β, _ d _ β1 β, _ β4 β, _ β 9 β, _____, _____β 2 7 14 6 2 6
U N SA C O M R PL R E EC PA T E G D ES
168
REASONING
12ββ(1β/2β)β
12β(β 1β/2β)β,β 13
13, 14
12 For each of the following pairs of fractions, write a fraction that lies between them. Hint: convert to a common denominator first, then consider the values in between. 1 β, _ 2 β, _ a _ β3 β, _ β3 β b β_ β1 β c β_ β1 β 5 4 7 6 4 2 1β 1 β, 2β_ 3β 7 β, 8β_ e β2β_ d _ β17 β, _ β7 β f β8β_ 3 5 20 10 10 4 a βis a fraction less than β1β, you can always find a fraction strictly between _ 13 If β_ βa βand β1β. For example, b b 3 7 _ _ between β βand β1βyou can find the fraction β .ββ 4 8 3 β, _ a Explain, using a common denominator of β8β, why β_ β7 ,β 1βis in ascending order. 4 8 b Find a fraction with a denominator of β16βthat lies between _ β7 βand β1β. 8 c Explain why you could keep going forever, finding a fraction less than β1βbut greater than the second largest value. 14 Write all the whole number values that β?β can take so that _ β? βlies strictly between: 3 1β a β2βand β3β b β5βand β5β_ 2 ENRICHMENT: Shady designs
β
β
15
15 a For each of the diagrams shown, work out what fraction of the rectangle is coloured purple. Explain how you arrived at each of your answers. i ii
iii
iv
b Design and shade two more rectangle designs.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3E Adding fractions
169
3E Adding fractions
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand that adding fractions requires a common denominator β’ To be able to add two fractions by considering their lowest common denominator β’ To be able to add two mixed numerals
Fractions with the same denominator can be easily added.
+
3 8
=
2 8
+
5 8
=
Fractions with different denominators cannot be added so easily.
+
1 3
Note: 1 1 1 β + 7 4 3 2 1 1 β + 7 4 3
=
1 4
+
=
?
But with a common denominator it is possible.
+
1 3 4 12
+
+
=
1 4 3 12
=
?
=
7 12
Lesson starter: βLikeβ addition
Pair up with a classmate and discuss the following. 1
Which of the following pairs of numbers can be simply added without having to carry out any form of conversion? a 6 goals, 2 goals b 11 goals, 5 behinds c 56 runs, 3 wickets d 6 hours, 5 minutes e 21 seconds, 15 seconds f 47 minutes, 13 seconds g 15 cm, 3 m h 2.2 km, 4.1 km i 5 kg, 1680 g 3 1 1 5 , 1_ 2 1 _ _ _ _ k , j , l 2_ 7 7 4 2 12 3 Does it become clear that we can only add pairs of numbers that have the same unit? In terms of fractions, we need to have the same ______________. 2 By choosing your preferred unit (when necessary), work out the answer to each of the problems above.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
KEY IDEAS β Fractions can be simplified easily using addition if they are βlikeβ fractions; that is, they have the
same denominator. This means they have been divided up into the same number of pieces. Same denominators β If two or more fractions have the same denominator, to add them simply add the numerators and keep the denominator. This allows you to find the total number of divided pieces.
U N SA C O M R PL R E EC PA T E G D ES
170
Different denominators β If the denominators are different, we use our knowledge of equivalent fractions to convert them to fractions with the same lowest common denominator (LCD). β To do this, carry out these steps.
1 Find the LCD (often, but not always, found by multiplying denominators). 2 Convert fractions to their equivalent fractions with the LCD. 3 Add the numerators and write this total above the LCD.
β After adding fractions, always look to see if your answer needs to be simplified.
BUILDING UNDERSTANDING
1 State the missing terms in the following sentences. a To add two fractions, they must have the same ______________. b When adding fractions, if they have the same ______________, you simply add the ______________.
c When adding two or more fractions where the ______________ are different, you must find the ____________ ____________ ____________.
d After carrying out the addition of fractions, you should always ______________ your answer to see if it can be ______________.
2 State the LCD for the following pairs of βincompleteβ fractions. a _ β ββ+ _ β ββ b _ β β β+ _ β ββ c _ β β β+ _ β ββ 5
3
6
3
12
d _ β β β+ _ βββ
8
12
16
3 Which of the following are correct? a _ β1 β+ _ β4 β= _ β5 β 9
9
9
b _ β1 β+ _ β1 β= _ β2 β 3
4
7
2β c _ β3 β+ _ β4 β= 1β_ 5
5
5
d _ β1 β+ _ β2 β= _ β3 β 2
5
7
Example 12 Adding βlikeβ fractions Add the following fractions. a _ β1 β+ _ β3 β 5 5
3 β+ _ b β_ β 5 β+ _ β6 β 11 11 11
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3E Adding fractions
EXPLANATION
1 β+ _ 4β a β_ β3 β= β_ 5 5 5
The denominators are the same; i.e. βlikeβ, therefore simply add the numerators.
ββ 3 β+ _ β14 β β 5 β+ _ β6 β = _ b _ ββ11 11 11 β β 11 β β 3β β = 1β_ 11
Denominators are the same, so add numerators.
U N SA C O M R PL R E EC PA T E G D ES
SOLUTION
171
Write answer as a mixed numeral.
Now you try
Add the following fractions. 4 β+ _ a β_ β1 β 7 7
β2 β+ _ β5 β b _ β4 β+ _ 9 9 9
Example 13 Adding βunlikeβ fractions Add the following fractions. a _ β1 β+ _ β1 β 5 2
3 β+ _ b β_ β5 β 4 6
SOLUTION
EXPLANATION
1 β+ _ a β_ β1 β = _ β 2 β+ _ β5 β 2 10 10 β 5 β β β 7 _ β= β β 10
LCD is β10β. Write equivalent fractions with the LCD. Γ2
Γ5
1 2 = 5 10
1 5 = 2 10
Γ2
Γ5
Denominators are the same, so add numerators.
b _ β5 β = _ β 9 β+ _ β10 β β3 β+ _ 4 6 12 12 β β β =ββ _ β19 β 12 7β β = 1β_ 12
LCD is β12β. Write equivalent fractions with the LCD. Γ3
Γ2
3 9 = 4 12
5 10 = 6 12
Γ3
Γ2
Denominators are the same, so add numerators. Write answer as a mixed numeral.
Now you try
Add the following fractions. a _ β1 β+ _ β1 β 3 5
3 β+ _ β7 β b β_ 4 10
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
172
Chapter 3
Fractions and percentages
Example 14 Adding mixed numerals
3β b 2β _ β5 β+ 3β_ 4 6
U N SA C O M R PL R E EC PA T E G D ES
Simplify: 2β a β3_ β2 β+ 4β_ 3 3 SOLUTION
a Method 1 β2 β = 7 + _ β4 β 3+4+_ β2 β+ _ 3 3 3β β β β 1β β = 8β_ 3 Method 2 ββ_ 11 β+ _ β14 β = _ β25 β 3β β 3β ββ 3 1β β = 8β_ 3
b Method 1 5 β+ _ β3 β = 5 + _ β10 β+ _ 2 + 3 + β_ β9 β 12 12 6 4 β β―β― β =ββ 5 + _ β β19 β 12 7β β = 6β_ 12 Method 2 17 β+ _ β15 β = _ β34 β+ _ β45 β β_ 4 12 12 6 β β β =ββ _ β79 β 12 7β β = 6β_ 12
EXPLANATION
Add the whole number parts. Add the fraction parts. 1 β, simplify the answer. Noting that _ β4 β= 1β_ 3 3
Convert mixed numerals to improper fractions. Have the same denominators, so add numerators. Convert improper fraction back to a mixed numerals. Add the whole number parts. LCD of β6βand β4βis β12β. Write equivalent fractions with LCD. Add the fraction parts. 7 β, simplify the answer. Noting that _ β19 β= 1β_ 12 12
Convert mixed numerals to improper fractions. Write equivalent fractions with LCD. Add the numerators. Simplify answer back to a mixed numeral.
Now you try Simplify: 3β a β1_ β3 β+ 2β_ 5 5
3β b 2β _ β 1 β+ 1β_ 4 10
Exercise 3E
Example 12a
Example 12b
1
FLUENCY
1β5ββ(1β/2β)β
Add the following fractions. 1 β+ _ a _ β2 β+ _ β5 β b β_ β6 β 9 9 12 12
3 β+ _ c β_ β4 β 15 15
1β4ββ(1β/2β)β,β 6β7ββ(1β/2β)β
2 Add the following fractions, writing your final answers as a mixed numeral. 6 β+ _ 7 β+ _ 2 β+ _ β3 β+ _ β4 β c β_ a β_ β3 β b β_ β6 β 7 7 10 10 5 5 5
3β4ββ(1β/2β)β,β 6β7ββ(1β/2β)β
3 β+ _ β2 β d β_ 9 9
12 β+ _ d β_ β 3 β+ _ β8 β 19 19 19
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3E Adding fractions
Example 13a
1 β+ _ β_ β3 β 3 5 1 β+ _ β3 β β_ 5 4
1 β+ _ c β_ β1 β 2 6 2 β+ _ g β_ β1 β 7 3
1 β+ _ d β_ β1 β 4 3 3 β+ _ β1 β h β_ 8 5
4 Add the following fractions, writing your final answers as a mixed numeral. 8 β+ _ 3 β+ _ 4 β+ _ b β_ β3 β c β_ β2 β a β_ β5 β 7 4 11 3 5 6
2 β+ _ d β_ β3 β 3 4
5 Simplify: 3β 1 β+ 2β_ a β1β_ 5 5 2β e β5_ β2 β+ 4β_ 3 3
2 β+ 4β_ 1β b 3β β_ 7 7 3 β+ 12β_ 4β f β8β_ 6 6
1 β+ 1β_ 2β c 1β 1β_ 4 4 7 β+ 9β_ 7β g β9β_ 11 11
3 β+ 4β_ 2β d β1β_ 9 9 3 β+ 7β_ 4β h β4β_ 5 5
6 Simplify: 3β 2 β+ 1β_ a β2β_ 3 4 3β e β8_ β1 β+ 6β_ 2 5
5β 2 β+ 1β_ b 5β β_ 6 5 2 β+ 6β_ 4β f β12β_ 3 9
1 β+ 8β_ 2β c 3β β_ 2 3 8 β+ 7β_ 3β g β17β_ 4 11
3β 4 β+ 7β_ d β5β_ 7 4 5β 7 β+ 5β_ h β9β_ 8 12
7 Simplify: 1 β+ _ a β_ β1 β 4 4 e β1_ β2 β+ _ β1 β 3 6
2 β+ _ b β_ β1 β 9 3 10 β+ _ f β_ β3 β 11 4
7 β+ _ c β_ β1 β 8 2 1 β+ 5β_ 1β g β2β_ 2 5
4 β+ 1β_ 1β d β_ 4 5 5β 3 β+ 2β_ h β_ 8 8
U N SA C O M R PL R E EC PA T E G D ES
Example 13b
3 Add the following fractions. a _ β1 β+ _ β1 β b 2 4 2 β+ _ e β_ β1 β f 5 4
173
Example 14a
Example 14b
PROBLEM-SOLVING
8, 9
9, 10
10, 11
8 Myles, Liza and Camillus work at a busy cinema complex. For a particular movie, Myles sells _ β3 βof all 5 1 _ the tickets and Liza sells β .ββ 3 a What fraction of movie tickets are sold by Myles and Liza, together? b If all of the movie tickets are sold, what is the fraction sold by Camillus?
9 Martine loves to run and play. Yesterday, she ran for β2_ β1 βkilometres, walked for β5_ β2 βkilometres and 4 5 1 _ skipped for β βa kilometre. What was the total distance that Martine ran, walked and skipped? 2 10 Jackson is working on a β1000β-piece jigsaw puzzle. After β1βweek, he has completed _ β 1 ββof the puzzle. 10 2 βof the puzzle. In the third week, Jackson completed After β2βweeks he has completed another β_ 5 1 _ another β βof the puzzle. 4 a By the end of the third week, what fraction of the puzzle has Jackson completed? b How many pieces of the puzzle does Jackson place in the second week? c What fraction of the puzzle is still unfinished by the end of the third week? How many pieces is this?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
11 A survey of Year β7βstudentsβ favourite sport is carried out. A total of β180βstudents participate in the survey. One-fifth of students reply that netball is their favourite, one-quarter reply rugby and one-third reply soccer. The remainder of students leave the question unanswered. a What fraction of the Year β7βstudents answered the survey question? b What fraction of the Year β7βstudents left the question unanswered? c How many students did not answer the survey question?
U N SA C O M R PL R E EC PA T E G D ES
174
REASONING
12
12, 13
13
12 Four students each read the same English novel over two nights, for homework. The table shows what fraction of the book was read on each of the two nights.
*Vesna woke up early on the third morning and read another _ β1 βof the novel before leaving for school. 6 Place the students in order, from least to most, according to what fraction of the book they had read by their next English lesson. Student
First night
Second night
Mikhail
_ β2 β 5
1β β_
Jim
1β β_ 2
1β β_ 10
Vesna*
_ β1 β
_ β1 β
Juliet
7β β_ 12
_ β1 β
4
4
5
20
13 Fill in the empty boxes to make the following fraction sums correct. 7 7 1 1 1 1 1 a b + = + + = 8 10
ENRICHMENT: Raise it to the max
β
c
3
β
+
4
=
17 20
14
14 a Using the numbers β1, 2, 3, 4, 5βand β6βonly once, arrange them in the boxes on the right to first produce the maximum possible answer, and then the + + minimum possible answer. Work out the maximum and minimum possible answers. b Repeat the process for four fractions using the digits β1βto β8βonly once each. Again, state the maximum and minimum possible answers. c Investigate maximum and minimum fraction statements for other sets of numbers and explain your findings. d Explain how you would arrange the numbers β1βto β100βfor β50βdifferent fractions if you were trying to achieve the maximum or minimum sum.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3F Subtracting fractions
175
3F Subtracting fractions
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand that subtracting fractions requires a common denominator β’ To be able to subtract two fractions by considering their lowest common denominator β’ To be able to subtract two mixed numerals
Subtracting fractions is very similar to adding fractions. You can establish the lowest common denominator (LCD) if one does not exist and this is done through producing equivalent fractions. Then, instead of adding numerators at the final step, you simply carry out the correct subtraction.
Complications can arise when subtracting mixed numerals and Example 16 shows the available methods that can be used to overcome such problems.
Plumbers use fraction skills. Copper pipes for hot water have diameters in fractions of an inch, so a 3β_ 1 _ 1 of an 1 =_ pipe wall thickness can be _ 2 ( 8 4 ) 16 inch. Plumbers also use wall thickness to calculate safe water pressures.
Lesson starter: Alphabet subtraction 0
β’ β’
1 12
2 12
3 12
4 12
5 12
6 12
7 12
8 12
9 12
10 12
Copy into your workbook the number line above. Place the following letters in the correct position on the number line. 2 1 11 1 1 5 A=_ C=_ D=_ E=_ F=_ B=_ 3 2 12 12 4 12 7 1 12 1 3 5 I=_ H=_ K=_ M=_ L=_ J=_ 12 3 12 6 4 6
11 12
1
0 G=_ 12
β’
Complete the following alphabet subtractions, giving your answer as a fraction and also the corresponding alphabet letter. a JβF b AβG c DβFβM d CβB e KβC f LβHβE g KβJβE h LβIβM
β’
What does A + B + C + D + E + F + G + H + I β J β K β L β M equal?
KEY IDEAS
β Fractions can be subtracted easily if they are βlikeβ fractions.
β The process for subtracting fractions involves finding the lowest common denominator before
subtracting the numerator. At the final step, you should also check if the result can be simplified 3β_ 1=_ 2 , which simplifies to _ 1 ). further (for example, _ 3 6 6 6
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
β When subtracting mixed numerals, you must have a fraction part that is large enough to allow
the other proper fraction to be subtracted from it. If this is not the case at the start of the problem, you may choose to regroup one of the whole number parts of the first mixed numeral. For example: 3 βββββββββββ_ β7_ β1 ββ 2β_ β1 βis not big enough to have _ β3 βsubtracted from it. 2 4 2 4 3 3 _ _ β6β ββ 2β ββββββββββTherefore, replace β7βby β6 + 1 = 6 + _ β2 ββ. 2 4 2
U N SA C O M R PL R E EC PA T E G D ES
176
β A fail-safe method for subtracting mixed numerals is to convert to improper fractions right from
the start.
3 β= _ For example:βββββββββββ7_ β1 ββ 2β_ β15 ββ _ β11 β 2 4 2 4
BUILDING UNDERSTANDING
1 State the missing terms in these sentences. a To subtract one fraction from another, you must have a common ______________. b One fail-safe method of producing a common denominator is to simply ______________
the two denominators. c The problem with finding a common denominator that is not the lowest common denominator is that you have to deal with larger numbers and you also need to ___________ your answer at the final step. d To find the LCD, you can ______________ the denominators and then divide by the HCF of the denominators.
2 State the LCD for the following pairs of βincompleteβ fractions. a _ β βββ _ β ββ b _ β β ββ _ β ββ c _ β βββ _ βββ 4
6
6
9
3 Which of the following are correct? a _ β 8 ββ _ β 5 β= _ β3 β 10 10 10 _ c β 8 ββ _ β 8 β= _ β0 β= 0β 11 10 1
8
d _ β β ββ _ βββ 9
12
21
b _ β 5 ββ _ β 5 β= _ β5 β d
12
10
2
20
20
20
_ β 3 ββ _ β 2 β= _ β1 β
Example 15 Subtracting βlikeβ and βunlikeβ fractions Simplify:
a _ β7 ββ _ β2 β 9 9
β1 β b _ β5 ββ _ 6 4
c _ β 7 ββ _ β8 β 10 15
SOLUTION
EXPLANATION
5β 7 ββ _ a β_ β2 β= β_ 9 9 9
Denominators are the same, therefore we are ready to subtract the second numerator from the first.
b ββ_ 5 ββ _ β1 β = _ β10 ββ _ β3 β ββ6 4 β β 12 12 β β7 β β= _ 12
Need to find the LCD, which is β12β. Write equivalent fractions with the LCD. We have the same denominators now, so subtract second numerator from the first.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3F Subtracting fractions
c _ β21 ββ _ β8 β = _ β16 β β 7 ββ _ 10 15 30 30 β β =ββ _ β β5 β 30 β1 β β= _ 6
177
U N SA C O M R PL R E EC PA T E G D ES
Find the LCD of β10βand β15β, which is β30β. Write both fractions with a denominator of β30β. Subtract the numerators. Simplify the result ( β_ β 5 β)βby dividing β5βand β30βby their 30 highest common factor.
Now you try Simplify: 8 ββ _ a β_ β3 β 11 11
b _ β 9 ββ _ β3 β 10 4
c _ β 5 ββ _ β1 β 12 4
Example 16 Subtracting mixed numerals Simplify: 1β a β5_ β2 ββ 3β_ 3 4 SOLUTION
3β 1 ββ 4β_ b 8β β_ 4 5
EXPLANATION
Method 1: Converting to an improper fraction 2 ββ 3β_ 1β = _ a 5β_ β17 ββ _ β13 β 3 4 3 4 β= _ β68 ββ _ β39 β β β 12 β 12 β β β29 β β= _ 12 5β β = 2β_ 12
Convert mixed numerals to improper fractions. Need to find the LCD, which is β12β. Write equivalent fractions with the LCD.
3β = _ 1 ββ 4β_ b 8β_ β41 ββ _ β19 β 4 4 5 5 164 _ β = β ββ _ β95 β 20 20β β ββ―β― β β β 69 _ β= β β 20 9β β = 3β_ 20
Convert mixed numerals to improper fractions. Need to find the LCD, which is β20β. Write equivalent fractions with the LCD. We have the same denominators now, so subtract second numerator from the first and convert back to improper fraction.
We have the same denominators now, so subtract second numerator from the first and convert back to improper fraction.
Method 2: Deal with whole numbers first, regrouping where necessary 2 ββ 3β_ 1β = β 5 + _ a 5β_ β2 β)ββ ( β 3+_ β1 β)β ( 3 4 3 4 2 1 _ _ β = (β 5 β 3)β+ ( β β ββ β ) ββ 3 β4 β β ββ―β― β β β―β― β―β― β= 2 + ( β_ β 8 ββ _ β3 β β 12 12 ) 5β β = 2β_ 12
Understand that a mixed numeral is the addition of a whole number and a proper fraction. Group whole numbers and group proper fractions. Simplify whole numbers; simplify proper fractions. Regrouping was not required.
Continued on next page
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
178
Chapter 3
Fractions and percentages
3 βcannot be taken away from _ β1 ββeasily. β_ 4 5 Therefore, we must borrow a whole. Group whole numbers and group proper fractions.
U N SA C O M R PL R E EC PA T E G D ES
3β = β 8 + _ 1 ββ 4β_ b 8β_ β1 ) β ββ ( β 4+_ β3 β)β ( 4 4 5 5 6 _ _ β = β(7 + β ) β ββ β(4 + β3 β)β 4 5 6 3 _ _ ββ―β― β―β― β―β― β =β β(7 β 4)β+ β(β ββ β ) βββ 5 4 β= 3 + ( β_ β24 ββ _ β15 β β 20 20 ) 9β β = 3β_ 20
Simplify whole numbers; simplify proper fractions. Borrowing a whole was required.
Now you try Simplify: 1β a 3β _ β2 ββ 2β_ 3 4
3β b β3_ β2 ββ 1β_ 4 5
Exercise 3F FLUENCY
Example 15a
Example 15a
Example 15b
Example 15c
Example 16a
Example 16b
1
1β(β β1/2β)β,β 3ββ(β1/2β)β,β 5ββ(β1/2β)β
2β6ββ(β1/2β)β
3β6β(β β1/2β)β
Simplify: a _ β5 ββ _ β3 β 7 7
4 ββ _ b β_ β1 β 11 11
12 ββ _ β5 β c β_ 18 18
2 ββ _ β1 β d β_ 3 3
2 Simplify: 6 ββ _ a β_ β2 β 9 9
5 ββ _ b β_ β2 β 19 19
3 ββ _ β3 β c β_ 5 5
17 ββ _ d β_ β9 β 23 23
3 Simplify: 2 ββ _ a β_ β1 β 3 4
3 ββ _ β1 β b β_ 5 2
3 ββ _ c β_ β1 β 5 6
4 ββ _ d β_ β1 β 7 4
4 Simplify the following, ensuring your final answer is in simplest form. 3 ββ _ 7 ββ _ 5 ββ _ b β_ β3 β β1 β c β_ a β_ β1 β 4 4 10 10 6 6 9 ββ _ 5 ββ _ e _ β3 ββ _ β1 β β2 β f β_ g β_ β1 β 10 5 5 10 12 4 5 Simplify: 1β 4 ββ 2β_ 9β 5 ββ 15β_ 11 ββ 7β_ 2β a β3β_ c β8β_ b 2β 3β_ 7 7 5 5 14 14 1β 3 ββ 2β_ 5 ββ 5β_ 1β 4β e β6_ β2 ββ 4β_ f β5β_ g β9β_ 7 3 4 4 9 6 6 Simplify: 1 ββ 2β_ 2β a β5β_ 3 3 3β e β8_ β 5 ββ 3β_ 4 12
PROBLEM-SOLVING
4β b β8_ β2 ββ 3β_ 5 5 3 ββ _ f β1β_ β7 β 5 9
7 ββ _ d β_ β5 β 8 8 11 ββ _ h β_ β3 β 20 10
5 ββ _ β3 β d β3β_ 9 9 3 ββ 7β_ 7β h β14β_ 4 10 2 ββ 7β_ 1β d β12β_ 9 3 3 ββ 3β_ 2β h β6β_ 3 20
5β 1 ββ 8β_ c β13β_ 2 6 1 ββ 1β_ 1β g β11β_ 4 11 7, 8
8, 10
9β11
7 Tiffany poured herself a large glass of cordial. She noticed that the cordial jug has _ β3 βof a litre in it 4 1 βof a litre in it after she filled her glass. How much cordial did before she poured her glass and only β_ 5 Tiffany pour into her glass? Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3F Subtracting fractions
179
8 A family block of chocolate is made up of β60βsmall squares of chocolate. Marcia eats β10βblocks, Jon eats β9βblocks and Holly eats β5βblocks. What fraction of the block of chocolate is left? 9 Three friends split a restaurant bill. One pays _ β1 βof the bill and one pays _ β1 βof the bill. 2 3 What fraction of the bill must the third friend pay?
U N SA C O M R PL R E EC PA T E G D ES
1 βdollars, but owes her parents β15β_ 1 βdollars. How much money does Patty have left after 10 Patty has β23β_ 4 2 she pays back her parents? Repeat this question using decimals and dollars and cents. Do you get the same answer? 11 Three cakes were served at a birthday party: an ice-cream cake, a chocolate cake and a sponge cake. Three-quarters of the ice-cream cake was eaten. The chocolate cake was cut into β12βequal pieces, of which β9βwere eaten. The sponge cake was divided into β8βequal pieces, with only β1βpiece remaining. a What fraction of each cake was eaten? b What fraction of each cake was left over? c What was the total amount of cake eaten during the party? d What was the total amount of cake left over after the party? REASONING
12
12, 13
12ββ(1β/2β)β, 13, 14
12 Fill in the empty boxes to make the following fraction sums correct. 1 1 1 a b β = 1 12 β = 2 10 5 c 2
3
β1
3
=
2 3
d 8 1 β6
4
=1
1 2
13 Today Davidβs age is one-seventh of Felicityβs age. Felicity is a teenager. a In β1βyearβs time David will be one-fifth of Felicityβs age. What fraction of her age will he be in β2βyearsβ time? b How many years must pass until David is one-third of Felicityβs age? c How many years must pass until David is half Felicityβs age?
14 a Example 16 shows two possible methods for subtracting mixed numerals: βBorrowing a whole numberβ and βConverting to an improper fractionβ. Simplify the following two expressions and discuss which method is the most appropriate for each question. 2β 5 ββ 23β_ 4β i β2_ β1 ββ 1β_ ii β27β_ 3 5 11 5 b If you have an appropriate calculator, work out how to enter fractions and check your answers to parts i and ii above. ENRICHMENT: Letter to an absent friend
β
β
15
15 Imagine that a friend in your class is absent for this lesson on the subtraction of fractions. They were present yesterday and understood the process involved when adding fractions. Your task is to write a letter to your friend, explaining how to subtract mixed numerals. Include some examples, discuss both possible methods but also justify your favourite method. Finish off with three questions for your friend to attempt and include the answers to these questions on the back of the letter.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
3G Multiplying fractions LEARNING INTENTIONS β’ To understand that multiplying fractions is easier if you ο¬rst cancel common factors from numerators and denominators in each fraction β’ To know that a whole number can be written as a fraction with a denominator of 1 β’ To be able to multiply fractions, mixed numerals and/or whole numbers, giving an answer in simplest form
U N SA C O M R PL R E EC PA T E G D ES
180
What does it mean to multiply two fractions?
Do you end up with a smaller amount or a larger amount when you multiply two proper fractions? 1Γ_ 2 equal? What does _ 3 3 β’
βStripβ method Imagine you have a strip of paper. 2 of the strip. You are told to shade _ 3
2 strip. 1 of your _ You are now told to shade in a darker colour _ 3 3 The final amount shaded is your answer.
β’
βNumber lineβ method Consider the number line from 0 to 1 (shown opposite). It is divided into ninths. 2. Locate _ 3 Divide this position into three equal pieces (shown as 1Γ_ 2 you have only one of the three pieces. To locate _ 3 3 The final location is your answer (shown as
β’
β’
2 9
1 3
0 0
1 9
2 9
3 9
2 3
4 9
5 9
6 9
1
7 9
8 9
1
).
2. ); i.e. _ 9
βShadingβ method 1 of a square multiplied by _ 2 of a Consider _ 3 3 square.
βThe ruleβ method When multiplying fractions, multiply the numerators and multiply the denominators.
Γ
= 2 9
=
1Γ2=_ 1Γ_ 2=_ 2 _ 3
3
3Γ3
9
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3G Multiplying fractions
181
Lesson starter: βClock faceβ multiplication Explain and discuss the concept of fractions of an hour on the clock face.
U N SA C O M R PL R E EC PA T E G D ES
In pairs, students match up the following β10ββclock faceβ multiplication questions with their correct answer. You may like to place a time limit of β5βminutes on the activity. Discuss answers at the end of the activity. Questions
Questions
Answers
Answers
1
_ β1 βof β4β hours
A
β25 minutesβ
6
_ β1 βof _ β3 ββhour
F
β2βhours β40 minutesβ
2
_ β1 βof β2β hours
B
1β _ β1 ββhours 2
7
_ β 1 βof _ β1 ββhour
G
_ β 1 ββhour
3
_ β1 βof β6β hours
C
β5 minutesβ
8
_ β1 ββhour β1 βof _
H 4β 0 minutesβ
4
_ β1 βof _ β1 ββhour
D
_ β1 ββhour
9
_ β2 βof β4β hours
i
_ β 1 ββhour
5
_ β1 ββhour β1 βof _
E
β2β hours
β1 ββhour 10 _ β5 βof _ 6 2
J
β3 minutesβ
2 3 4 3
4
4
3
4
3
10 5
4
2
2
3
12
10
KEY IDEAS
β Fractions do not need to have the same denominator to be multiplied.
β To multiply fractions, multiply the numerators and multiply the denominators.
In symbols: _ βa βΓ _ βc β= _ βa Γ c β b d bΓd β If possible, βsimplifyβ, βdivideβ or βcancelβ fractions before multiplying. β’ Cancelling can be done vertically or diagonally. β’ Cancelling can never be done horizontally. β’
1 _ β3 βΓ _ ββ4β β ββ
5
8ββ 2β β
1
_ βββ β3ββΓ _ β4 β
5
Never do this!
6ββ 2β β
1
2
5
7
_ βββ β3ββΓ _ ββ6β β ββ
cancelling vertically
β
cancelling diagonally
β
cancelling horizontally
β
β Any whole number can be written as a fraction with a denominator of β1β, e.g. β4 = _ β4 ββ.
1
β βofβ, ββΓββ, βtimesβ, βlots ofβ and βproductβ all refer to the same mathematical operation of
multiplying.
β Mixed numerals must be changed to improper fractions before multiplying. β Final answers should be written in simplest form.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
BUILDING UNDERSTANDING 1 State the missing terms in these sentences. a A proper fraction has a value that is between ______ and ______. b An improper fraction is always greater than ______. c A mixed numeral consists of two parts, a _______ _______ part and a _______ _______ part.
U N SA C O M R PL R E EC PA T E G D ES
182
2 When multiplying a whole number by a proper fraction, do you get a smaller or larger answer when compared with the whole number? Explain your answer.
3 Describe how many items you would have in these situations and use drawings to show the answer to these problems. a _ β1 βof β12β lollies 3
b _ β2 βof β18β doughnuts 3
c _ β3 βof β32β dots
8 4 One of the following four methods is the correct solution to the problem _ β1 βΓ _ β1 β. Find the correct 2 5 solution and copy it into your workbook. C β β β_1 βΓ β_1 β β B β β β_1 βΓ _ D β β β_1 βΓ _ A β β β_1 βΓ _ β1 β β1 β β1 β 2 5 2 5 2 5 2 5 ββ _ β =β ββ _ ββ _ β =β β =β β1 Γ 1 ββ β1 Γ 1 ββ β 5 βΓ _ ββ2 β β ββ _ β =β β1 + 1 ββ 10 10 2Γ5 2Γ5 2+5 7 2 _ _ _ 2 β= β β β β= β β β = β1 β β= _ β β 20 10 10 7
Example 17 Multiplying proper fractions Find: a _ β2 βΓ _ β1 β 3 5
b _ β3 βΓ _ β8 β 4 9
c _ β4 βof _ β3 β 8 6
SOLUTION
EXPLANATION
a _ β2 βΓ _ β1 β = _ β2 Γ 1 β 3 5 β β β 3 Γ 5β β= _ β2 β 15
Multiply the numerators. Multiply the denominators. The answer is in simplest form.
1 2 b _ β1 Γ 2 β ββ8β β ββ = _ βββ β3ββΓ _ β1β4β β9β3β 1 Γ 3 β β β β β= _ β2 β 3 1 1 c _ β4 βof _ β3 β = _ βββ β4ββΓ _ β3ββ β ββ 8 6 β2β8β β6β2β β ββ =β _ β1 Γ 1 ββ β 2Γ2 _ β = β1 β 4
Cancel first. Then multiply numerators and denominators.
Change βofβ to multiplication sign. Cancel and then multiply the numerators and the denominators. The answer is in simplest form.
Now you try Find: a _ β1 βΓ _ β2 β 5 7
β15 β b _ β4 βΓ _ 5 16
β5 β c _ β3 βof _ 4 9
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3G Multiplying fractions
183
Example 18 Multiplying proper fractions by positive integers Find: b _ β2 βof 32β 5
U N SA C O M R PL R E EC PA T E G D ES
1 βΓ 21β a β_ 3 SOLUTION
EXPLANATION
7 a β_ 1 βΓ 21 = _ β1 βΓ _ ββ2β 1ββ ββ 1ββ β3β 3 1 β β β β_ β β = β7 β 1 β= 7
Rewrite β21βas a fraction with a denominator equal to β1β. Cancel and then multiply numerators and denominators. β7 Γ· 1 = 7β
b _ β2 βΓ _ β32 β β2 βof 32 = _ 1 5 5 64 _ β =ββ β β β β 5 4β β = 12β_ 5
Rewrite βofβ as a multiplication sign. Write β32βas a fraction. Multiply numerators and denominators. Convert answer to a mixed numeral.
Now you try Find:
2 βΓ 35β a β_ 5
5 βof 16β b β_ 6
Example 19 Multiplying improper fractions Find:
5 βΓ _ a β_ β7 β 3 2
b _ β8 βΓ _ β15 β 4 5
SOLUTION
EXPLANATION
a β_ 5 βΓ _ β7 β = _ β5 Γ 7 β 3 2 β β β 3 Γ 2β β 5β β= _ β35 β= 5β_ 6 6
Multiply the numerators. Multiply the denominators. Convert the answer to a mixed numeral.
3β β 2 b β_ 8 βΓ _ β15 β = _ β βββ1β8βΓ β1β 5β 1 4 β β ββ β5βΓ β4β β β β β5 β= _ β6 β= 6 1
Cancel first. Multiply βcancelledβ numerators and βcancelledβ denominators. Write the answer in simplest form.
Now you try Find: 4 βΓ _ a β_ β10 β 7 3
β49 β b _ β30 βΓ _ 7 10
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
184
Chapter 3
Fractions and percentages
Example 20 Multiplying mixed numerals Find: 2β a 2β _ β1 βΓ 1β_ 3 5
U N SA C O M R PL R E EC PA T E G D ES
2β b β6_ β1 βΓ 2β_ 4 5
SOLUTION
EXPLANATION
1 βΓ 1β_ 2β = _ a 2β_ β7 βΓ _ β7 β 3 3 5 5 β β =ββ _ β49 β β 15 4β β = 3β_ 15
Convert mixed numerals to improper fractions. Multiply numerators. Multiply denominators.
5 3β β 2β = _ 1 βΓ 2β_ βββ1β25ββΓ _ β1ββ 2β β b 6β_ 1 4 5 ββ β4β β5β β β ββ―β― β 15 β β β β= _ β β 1 β = 15
Convert to improper fractions. Simplify fractions by cancelling. Multiply numerators and denominators. Write the answer in simplest form.
Write the answers in the simplest form.
Now you try Find: 1β a 2β _ β1 βΓ 4β_ 2 5
3β b β3_ β1 βΓ 3β_ 4 5
Exercise 3G FLUENCY
Example 17a
Example 17b
Example 17
Example 18
1
Find: a _ β1 βΓ _ β1 β 2 3 e _ β3 βΓ _ β1 β 4 5
1, 2β5ββ(β1/2β)β
b _ β1 βΓ _ β1 β 3 5 f _ β2 βΓ _ β1 β 7 3
β5 β c _ β2 βΓ _ 3 7 1 βΓ _ β5 β g β_ 4 6
2β6β(β β1/2β)β
2β6β(β β1/2β)β
2 βΓ _ d β_ β3 β 5 7 2 βΓ _ h β_ β2 β 9 5
2 Multiply the following, remembering to cancel common factors in the numerator and denominators. a _ β4 βΓ _ β1 β 7 4
b _ β 9 βΓ _ β10 β 10 27
20 βΓ _ c β_ β 80 β 40 100
7 βΓ _ β30 β d β_ 15 49
3 Find: β3 β a _ β2 βΓ _ 3 5 e _ β2 βof _ β3 β 7 5
b _ β3 βΓ _ β5 β 6 11 3 βof _ f β_ β2 β 4 5
c _ β 8 βΓ _ β3 β 11 4 5 βof _ g β_ β4 β 10 7
2 βΓ _ d β_ β10 β 5 11 6 βof _ h β_ β3 β 9 12
4 Find: 1 βΓ 18β a β_ 3 e _ β2 βof 42β 7
b _ β1 βΓ 45β 5 _ f β1 βof 16β 4
c _ β2 βΓ 24β 3 _ g β4 βof 100β 5
3 βΓ 25β d β_ 5 3 βof 77β h β_ 7
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3G Multiplying fractions
Example 19
6 βΓ _ b β_ β11 β 7 5 f _ β21 βΓ _ β8 β 4 6
6 Find: 1β a β1_ β3 βΓ 2β_ 3 5
2β b 1β _ β1 βΓ 1β_ 7 9
9 βΓ _ β13 β d β_ 4 6 14 βΓ _ h β_ β15 β 7 9
β11 β c _ β6 βΓ _ 4 5 10 _ g β βΓ _ β21 β 7 5
U N SA C O M R PL R E EC PA T E G D ES
Example 20
5 Find: a _ β5 βΓ _ β7 β 2 3 8 βΓ _ β10 β e β_ 3 5
185
PROBLEM-SOLVING
2β c β3_ β1 βΓ 2β_ 4 5 7, 8
1β d β4_ β2 βΓ 5β_ 7 3
8, 9
8β10
7 At a particular school, the middle consists of Year β7βand Year β8βstudents only. _ β2 βof the middle school 5 students are in Year β7β. a What fraction of the middle school students are in Year β8β? b If there are β120βmiddle school students, how many are in Year β7βand how many are in Year β8β?
1 βlitresβof paint 8 To paint one classroom, β2β_ 3 are required. How many litres of paint are
required to paint five identical classrooms?
9 A scone recipe requires β1_ β3 βcups of self-raising flour and _ β3 βof a cup of cream. James is catering for a 4 4 large group and needs to quadruple the recipe. How much self-raising flour and how much cream will he need?
10 Julie has finished an injury-plagued netball season during which she was able to play 2 βof the matches. The season consisted only β_ 3 of β21βmatches. How many games did Julie miss as a result of injury?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
REASONING
11
11, 12
12, 13
11 Not all of the following fraction equations are correct. Copy them into your workbook, then place a tick beside those that are correct and a cross beside those that are wrong. Provide the correct solution for those you marked as incorrect. 1 β+ _ 1 β= _ 1 βΓ β_ a _ β1 β+ _ β1 β= _ β1 β b β_ β1 β= _ β1 β β2 β c β_ 3 4 7 3 4 12 3 4 7 1 ββ _ 1 ββ _ d _ β1 βΓ _ β1 β= _ β1 β β1 β= _ β1 β e β_ β1 β= _ β0 β f β_ 3 4 12 3 4 12 3 4 β1
U N SA C O M R PL R E EC PA T E G D ES
186
12 Circle the correct alternative for the following statement and justify your answer. Using an example, explain why the other alternatives are incorrect. When multiplying a proper fraction by another proper fraction the answer is: A a whole number. B a mixed numeral. C an improper fraction. D a proper fraction. 13 Write two fractions that: a multiply to _ β3 β 5 _ b multiply to β3 β 4 _ c multiply to β1 β 7
ENRICHMENT: Who are we?
β
β
14
14 a Using the clues provided, work out which two fractions are being discussed. β’ We are two proper fractions. β’ Altogether we consist of four different digits. β’ When added together our answer will still be a proper fraction. β’ When multiplied together you could carry out some cancelling. β’ The result of our product, when simplified, contains no new digits from our original four. β’ Three of our digits are prime numbers and the fourth digit is a cube number. b Design your own similar question and develop a set of appropriate clues. Have a classmate try to solve your question. c Design the ultimate challenging βWho are we?β question. Make sure there is only one possible answer.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Progress quiz
1
Progress quiz
Consider the fraction _ β3 .ββ 4 a Represent this fraction on a diagram. b State the denominator of this fraction. c State the numerator of this fraction. d Represent this fraction on a number line. e Is this a proper fraction, an improper fraction or a mixed numeral?
U N SA C O M R PL R E EC PA T E G D ES
3A
187
3A
2 What fraction is represented on the number line shown? Write it as an improper fraction and as a mixed numeral. 0
3B 3B
1
2
3 Write three equivalent fractions for _ β2 ββ. 5
4 Write these fractions in simplest form. 4β a β_ b _ β15 β 10 30
3C
5 Convert β1_ β3 βto an improper fraction. 5
3C
6 Convert _ β13 βto a mixed numeral. 4
3D
3D 3E
3F
3G
3
c _ β14 β 6
d _ β24 β 8
7 Place the correct mathematical symbol <, = or > between the following pairs of fractions to make true mathematical statements. 2 ββ _ _ _ _ b _ β4 ββ β24 β β12 β c β1_ β1 ββ a β_ β5 β β18 β d _ β5 ββ 30 10 5 5 3 9 9 20 8 Write the following fractions in ascending order: _ β1 β, _ β2 β, _ β9 β, _ β4 .ββ 2 3 4 9
9 Add the following fractions. 4 β+ _ a β_ β2 β β3 β b _ β2 β+ _ 7 7 5 10
β2 β c _ β3 β+ _ 4 5
10 Simplify: 5 ββ _ a β_ β1 β 12 3
β1 β b _ β5 ββ _ 6 4
1β c 5β _ β1 ββ 3β_ 2 5
11 Find: 3 βof $560β a β_ 5
b _ β2 βΓ _ β7 β 3 11
2 βΓ _ β9 β c β_ 3 10
1β d 1β _ β3 β+ 3β_ 4 2
β9 β d 1β _ β1 βΓ _ 3 16
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
3H Dividing fractions LEARNING INTENTIONS β’ To be able to ο¬nd the reciprocal of a fraction or a mixed numeral β’ To understand that dividing fractions can be done by multiplying by a reciprocal β’ To be able to divide fractions, mixed numerals and/or whole numbers, giving an answer in simplest form
U N SA C O M R PL R E EC PA T E G D ES
188
Remember that division used to be referred to as βhow manyβ.
Thinking of division as βhow manyβ helps us to understand dividing fractions. 1Γ·_ 1 , think of _ 1 how many _ 1 s? For example, to find _ 2 4 2 4
or
1 s are in a _ 1? how many _ 4 2
Consider this strip of paper that is divided into four equal sections. 1Γ·_ 1 , we have only _ 1 a strip, so we will shade In our example of _ 2 4 2 in half the strip. By thinking of the Γ· sign as βhow manyβ, the question is asking how many quarters are in half the strip. From our diagram, we 1Γ·_ 1 = 2. can see that the answer is 2. Therefore, _ 2 4
In a game of football, when it is half-time, you have played two 1Γ·_ 1 = 2. quarters. This is another way of confirming that _ 2 4
Lesson starter: βDivvy upβ the lolly bag
To βdivvy upβ means to divide up, or divide out, or share equally. Consider a lolly bag containing 24 lollies. In pairs, students answer the following questions. β’ β’ β’
β’
β’ β’
β’
β’
Aircraft engineers assemble new planes and aircraft mechanics maintain, service and repair planes. Fraction skills are essential as most aircraft parts use imperial measurements, usually in fractions of an inch.
How many lollies would each person get if you divvy up the lollies between three people? 1 of the lollies in the bag, how many did you get? If you got _ 3 1 ? Therefore, Γ·3 is the same as Γ _ 1. Can you see that βdivvying upβ by 3 is the same as getting _ 3 3 How many lollies would each person get if you divvy up the lollies between eight people? 1 of the lollies in the bag, how many did you get? If you got _ 8 1 ? Therefore, Γ·8 is the same as Γ _ 1. Can you see that βdivvying upβ by 8 is the same as getting _ 8 8 What do you think is the same as dividing by n? a? What do you think is the same as dividing by _ b
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3H Dividing fractions
189
KEY IDEAS β To find the reciprocal of a fraction, you must invert the fraction. This is done by swapping the
U N SA C O M R PL R E EC PA T E G D ES
numerator and the denominator. βInvertingβ is sometimes known as turning the fraction upside down, or flipping the fraction. β’ The reciprocal of _ βa βis _ βb ββ. b a For example: The reciprocal of _ β3 βis _ β5 ββ. 5 3
β Dividing by a number is the same as multiplying by its reciprocal.
For example: β15 Γ· 3 = 5βand β15 Γ _ β1 β= 5β. 3 β’
Dividing by β2βis the same as multiplying by _ β1 .ββ 2
β When asked to divide by a fraction, instead choose to multiply by the fractionβs reciprocal.
Therefore, to divide by _ βa βwe multiply by _ βab ββ. b
β When dividing, mixed numerals must be changed to improper fractions.
BUILDING UNDERSTANDING
1 Which of the following is the correct first step for finding _ β3 βΓ· _ β4 ββ? A
_ β3 βΓ _ β7 β
B
4
5
3
7
5
_ β5 βΓ _ β4 β
C _ β5 βΓ _ β7 β
7
3
4
2 State the correct first step for each of these division questions. (Do not find the final answer.) a _ β 5 βΓ· _ β3 β 11
5
c _ β 7 βΓ· _ β12 β
b _ β1 βΓ· _ β1 β 3
10
5
5
d _ β8 βΓ· 3β 3
3 When dividing mixed numerals, the first step is to convert to improper fractions and the second step is to multiply by the reciprocal of the divisor. State the correct first and second steps for each of the following mixed numeral division questions. (Do not find the final answer.) 1β a 2β _ β1 βΓ· 1β_ 2
3
b β24 Γ· 3β_1 β 5
1β c 4β _ β 3 βΓ· 5β_ 11
4
3β d _ β8 βΓ· 11β_ 3
7
4 Make each sentence correct, by choosing the word more or less in the gap. a β10 Γ· 2βgives an answer that is __________ than β10β
b β10 Γ· _ β1 βgives an answer that is __________ than β10β 2
c
_ β3 βΓ· _ β2 βgives an answer that is __________ than _ β3 β
d
_ β3 βΓ _ β3 βgives an answer that is __________ than _ β3 β
e
_ β5 βΓ· _ β8 βgives an answer that is __________ than _ β5 β
f
4
4
3
2
4
4
7 5 7 5 5 _ β βΓ _ β βgives an answer that is __________ than _ β5 β 7 8 7
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
Example 21 Finding reciprocals State the reciprocal of the following. a _ β2 β 3
c β1_ β3 β 7
b 5β β
U N SA C O M R PL R E EC PA T E G D ES
190
SOLUTION
EXPLANATION
a Reciprocal of _ β2 βis _ β3 ββ. 3 2 b Reciprocal of β5βis _ β1 ββ. 5 β 7 ββ. c Reciprocal of β1_ β3 βis _ 7 10
The numerator and denominator are swapped.
5 βand then invert. Think of β5βas β_ 1 3 βto an improper fraction, i.e. _ β10 β, and then invert. Convert β1β_ 7 7
Now you try
State the reciprocal of the following. a _ β6 β 7
c β3_ β2 β 3
b 3β β
Example 22 Dividing a fraction by a whole number Find:
a _ β5 βΓ· 3β 8
b β2_ β 3 βΓ· 5β 11
SOLUTION
EXPLANATION
β5 βΓ· 3 = _ β5 βΓ _ β1 β a _ 8 8 3β β β β 5 _ β= β β 24 3 βΓ· 5 = _ b 2β_ β25 βΓ· _ β5 β 11 11 1 5 β =ββ _ β βββ β25ββΓ _ β1β β 11 β5β 1β β β5 β β= _ 11
β3 β)ββ. Change the βΓ·βsign to a βΓβsign and invert the β3 ( β βor _ 1 Multiply the numerators and denominators. Convert the mixed numeral to an improper fraction. Write β5βas an improper fraction. Change the βΓ·βsign to a βΓβsign and invert the divisor. Simplify by cancelling. Multiply numerators and denominators.
Now you try Find:
a _ β3 βΓ· 5β 4
b β3_ β3 βΓ· 2β 5
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3H Dividing fractions
191
Example 23 Dividing a whole number by a fraction b β24 Γ· _ β3 β 4
U N SA C O M R PL R E EC PA T E G D ES
Find: a β6 Γ· _ β1 β 3 SOLUTION
EXPLANATION
a 6Γ·_ β1 β = _ β3 β β6 βΓ _ 3 1 1β β β β β 18 _ β = β β= 18 1
β3 ββ. Instead of βΓ· _ β1 β, change to βΓ _ 3 1 Simplify.
8 3β = _ b 24 Γ· β_ βββ β24ββΓ _ β4 β β 4β β 1 β3β 1β ββ β = 32
β4 ββ. Instead of βΓ· _ β3 β, change to βΓ _ 4 3 Cancel and simplify.
Now you try Find:
a β8 Γ· _ β1 β 5
b β20 Γ· _ β4 β 9
Example 24 Dividing fractions by fractions Find: 3 βΓ· _ a β_ β3 β 5 8
3β b 2β _ β2 βΓ· 1β_ 5 5
SOLUTION
EXPLANATION
a β_ 3 βΓ· _ β3 βΓ _ β8 β β3 β = _ 8 3 5 5 β β β ββ 8 3β _ β = β β= 1β_ 5 5
Change the βΓ·βsign to a βΓβsign and invert the Βdivisor. (Note: The divisor is the second fraction.) Cancel and simplify.
3β = _ 2 βΓ· 1β_ b 2β_ β12 βΓ· _ β8 β 5 5 5 5 1 3 β β =ββ _ βββ β112ββΓ _ βββββ52 ββ β5β β β β8β β β 3 1β _ β = β β= 1β_ 2 2
Convert mixed numeral to improper fractions. Change the βΓ·βsign to a βΓβsign and invert the divisor. Cancel, multiply and simplify.
Now you try Find:
2 βΓ· _ a β_ β5 β 3 8
3β b 4β _ β1 βΓ· 2β_ 10 5
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
192
Chapter 3
Fractions and percentages
Exercise 3H FLUENCY
1β7ββ(1β/2β)β
4β8β(β 1β/2β)β
1
State the reciprocal of the following proper fractions. 3β a _ β2 β c β_ b _ β5 β 5 5 7 2 State the reciprocal of the following whole numbers. a 2β β b 6β β c β10β
4β d β_ 9
U N SA C O M R PL R E EC PA T E G D ES
Example 21a
1β4ββ(1β/2β)β,β 7ββ(1β/2β)β
Example 21b
Example 21c
Example 22a
Example 22b
Example 23
Example 24a
Example 24b
Example 24
d β4β
3 State the reciprocal of the following mixed numerals. Remember to convert them to improper fractions first. b 2β _ β2 β c β3_ β1 β d β8_ β2 β a 1β _ β5 β 3 2 3 6
4 Find: 3 βΓ· 2β a β_ 4
b _ β 5 βΓ· 3β 11
c _ β8 βΓ· 4β 5
15 βΓ· 3β d β_ 7
5 Find: a 2β _ β1 βΓ· 3β 4
b 5β _ β1 βΓ· 4β 3
4 βΓ· 8β c β12β_ 5
d β1_ β13 βΓ· 9β 14
a 5β Γ· _ β1 β 4 e β12 Γ· _ β2 β 5
b 7β Γ· _ β1 β 3 f β15 Γ· _ β3 β 8
c β10 Γ· _ β1 β 10 _ g β14 Γ· β7 β 2
d β24 Γ· _ β1 β 5 _ h β10 Γ· β3 β 2
7 Find: 2 βΓ· _ a β_ β2 β 7 5 1β e β2_ β1 βΓ· 1β_ 4 3
b _ β1 βΓ· _ β1 β 5 4 3β f β4_ β1 βΓ· 3β_ 10 5
β6 β c _ β3 βΓ· _ 7 11 3β 1 βΓ· 3β_ g β12β_ 2 4
2 βΓ· _ β8 β d β_ 3 9 4β h β9_ β3 βΓ· 12β_ 7 7
8 Find: 3 βΓ· 5β a β_ 8 e β7 Γ· _ β1 β 4
b 2β 2 Γ· _ β11 β 15 6 _ f β2β βΓ· 9β 15
3β c 2β _ β2 βΓ· 1β_ 4 5 1β g β7_ β2 βΓ· 1β_ 3 6
3 βΓ· _ d β_ β9 β 4 4 3 βΓ· _ β2 β h β_ 5 7
6 Find:
PROBLEM-SOLVING
9, 10
10β12
11β13
9 If β2_ β1 βleftover pizzas are to be shared between 4 three friends, what fraction of pizza will each friend receive?
10 A property developer plans to subdivide β7_ β1 ββ 2 acres of land into blocks of at least _ β3 βof an 5 acre. Through some of the land runs a creek, where a protected species of frog lives. How many complete blocks can the developer sell if two blocks must be reserved for the creek and its surroundings?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3H Dividing fractions
193
11 Miriam cuts a β10β-millimetre thick sisal rope into 3 βmetres long four equal pieces. If the rope is β3β_ 5 before it is cut, how long is each piece?
U N SA C O M R PL R E EC PA T E G D ES
3 βof an hour to make a chair. 12 A carpenter takes β_ 4 How many chairs can he make in β6β hours?
13 Justin is a keen runner and runs at a pace of 1 βminutes per kilometre. Justin finished a β3β_ 2 Sunday morning run in β77βminutes. How far did he run? REASONING
14
14, 15
15, 16
14 Use diagrams of circles split into equal sectors to explain the following. 3 βΓ· 2 = _ a β_ β3 β 4 8
1 β= 6β b β3 Γ· β_ 2
15 Pair up the equivalent expressions and state the simplified answer. 1 βof 8β β12 Γ· 4β β10 Γ· 2β β_ 1β 0 Γ _ β1 β 2 2 _ β3 Γ 2β β12 Γ _ β1 β β1 βΓ· _ β1 β β3 Γ· _ β1 β 2 4 2 8
16 a A car travels β180βkilometres in β1_ β1 βhours. How far will it travel in β2βhours if it travels at the same 2 speed? b A different car took β2_ β1 βhours to travel β180βkilometres. How far did it travel in β2βhours, if it 4 maintained the same speed? ENRICHMENT: You provide the question
β
β
17ββ(1β/2β)β
17 Listed below are six different answers. You are required to make up six questions that will result in the following answers. All questions must involve a division sign. Your questions should increase in order of difficulty by adding extra operation signs and extra fractions. a _ β3 β 5
b 2β _ β1 β 3
c _ β7 β 1
d β0β
e _ β1 β 100
f
β4_ β4 β 5
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
3I Fractions and percentages LEARNING INTENTIONS β’ To understand that a percentage can be thought of as a fraction with a denominator of 100 β’ To be able to convert a percentage to a fraction in simplest form β’ To be able to convert a fraction to a percentage
U N SA C O M R PL R E EC PA T E G D ES
194
We come across percentages in many everyday situations. Interest rates, discounts, test results and statistics are just some of the common ways in which we deal with percentages. Percentages are closely related to fractions. A percentage is another way of writing a fraction with a denominator of 100. Therefore, 87% means that if something is divided into 100 pieces you would have 87 of them.
Lesson starter: Student ranking
Five students completed five different Mathematics tests. Each of the tests was out of a different number of marks. The results are shown below. Your task is to rank the five students in descending order, according to their test result. β’ β’ β’ β’ β’
A project manager calculates the fraction of a constructionβs total cost for wages, building materials, tax, etc. To easily compare these fractions, they are changed to percentages of the total cost.
Matthew scored 15 out of a possible 20 marks. Mengna scored 36 out of a possible 50 marks. Maria scored 33 out of a possible 40 marks. Marcus scored 7 out of a possible 10 marks. Melissa scored 64 out of a possible 80 marks.
Change these test results to equivalent scores out of 100, and therefore state the percentage test score for each student.
KEY IDEAS
β The symbol % means βper centβ. This comes from the Latin per centum, which means out of
100. Therefore, 75% means 75 out of 100.
β We can write percentages as fractions by changing the % sign to a denominator of 100 (meaning
out of 100).
37 For example: 37% = _ 100
β We can convert fractions to percentages through our knowledge of equivalent fractions.
25 = 25% 1=_ For example: _ 4 100
β To convert any fraction to a percentage, multiply by 100.
3 Γ 100 = _ 3Γ_ 100 = _ 75 = 37_ 3 = 37_ 1 . Therefore _ 1% For example: _ 8 8 1 2 2 8 2 Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3I Fractions and percentages
195
β Common percentages and their equivalent fractions are shown in the table below. It is useful to
know these. Fraction
_ β1 β
1β β_ 3
1β β_ 4
_ β1 β
1β β_ 8
_ β2 β
3β β_ 4
Percentage
β50%β
1 β%β β33β_ 3
β25%β
β20%β
1 β%β 1β 2β_ 2
2 β%β β66β_ 3
β75%β
5
3
U N SA C O M R PL R E EC PA T E G D ES
2
BUILDING UNDERSTANDING
1 Change these test results to equivalent scores out of β100β, and therefore state the percentage. a β7βout of β10 =β______ out of β100 = _____%β b β12βout of β20 =β______ out of β100 = _____%β c β80βout of β200 =β______ out of β100 = _____%β
2 Write these fraction sequences into your workbook and write beside each fraction the equivalent percentage value. a _ β1 β, _ β2 β, _ β3 β, _ β4 β 4 4 4 4
b _ β1 ,β _ β2 β, _ β3 β, _ β4 β, _ β5 β 5 5 5 5 5
c _ β1 ,β _ β2 ,β _ β3 β 3 3 3
3 If β14%βof students in Year β7βare absent due to illness, what percentage of Year β7βstudents are at school?
Example 25 Converting percentages to fractions
Express these percentages as fractions or mixed numerals in their simplest form. a β17%β b 3β 6%β c β140%β SOLUTION
EXPLANATION
a β17% = _ β 17 β 100
Change β%βsign to a denominator of β100β.
b
Change the β%βsign to a denominator of β100β then divide the numerator and the denominator by the HCF of β4β. The answer is now in simplest form.
Γ·4
36% =
36 9 = 100 25 Γ·4
c
Γ· 20
140% =
140 7 = 100 5 Γ· 20
or 1
2 5
Change the β%βsign to a denominator of β100β then divide the numerator and the denominator by the HCF of β20β. You can convert to a mixed numeral.
Now you try
Express these percentages as fractions or mixed numerals in their simplest form. a β39%β b 4β 2%β c β220%β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
Example 26 Converting to percentages through equivalent fractions Convert the following fractions to percentages. a _ β5 β 100
11 β b β_ 25
U N SA C O M R PL R E EC PA T E G D ES
196
SOLUTION
EXPLANATION
a _ β 5 β= 5%β 100
Denominator is already β100β, therefore simply write number as a percentage
b
Require denominator to be β100β. Therefore, multiply numerator and denominator by β4βto get an equivalent fraction.
Γ4
44 11 = 100 25
Γ4 = 44%
Now you try
Convert the following fractions to percentages. a _ β9 β 100
11 β b β_ 20
Example 27 Converting to percentages by multiplying by β1 00β Convert the following fractions to percentages. a _ β3 β 8
b β3_ β3 β 5
SOLUTION
EXPLANATION
25 a _ β23 βΓ _ ββ1β 00ββ ββ β3 βΓ 100 = _ 8 1 β β8β
Multiply by β100β. Simplify by cancelling HCF. Write your answer as a mixed numeral.
β
β
1 ββ β β―β― β―β― ββ =β _ β75 β= 37β_ 2 2 1 β% β΄_ β3 β = 37β_ 8 2
20 3 βΓ 100 = _ β118 βΓ _ β1ββ 00ββ ββ b 3β_ 1 5 β ββ5β β ββ―β― β β =β 360β
Convert mixed numeral to improper fraction. Cancel and simplify.
3 β = 360% β΄ 3β_ 5
Now you try
Convert the following fractions to percentages. a _ β7 β 40
b β2_ β1 β 4
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3I Fractions and percentages
197
Exercise 3I FLUENCY
1
1, 2β4ββ(1β/2β)β
2β5β(β 1β/2β)β
2β5ββ(1β/4β)β
Example 25b
b i
U N SA C O M R PL R E EC PA T E G D ES
Example 25a
Express these percentages as fractions or mixed numerals in their simplest form. a i β19%β ii β29%β iii β57%β
Example 25a,b
Example 25c
Example 26
Example 27
β70%β
ii β45%β
iii β48%β
2 Express these percentages as fractions in their simplest form. a β11%β b β71%β c β43%β e 2β 5%β f β30%β g β15%β
d 4β 9%β h β88%β
3 Express these percentages as mixed numerals in their simplest form. a β120%β b β180%β c β237%β e 1β 75%β f β110%β g β316%β
d 4β 01%β h β840%β
4 Convert these fractions to percentages, using equivalent fractions. a _ β8 β c _ β 97 β b _ β 15 β 100 100 100 e _ β7 β f _ β8 β g _ β43 β 20 25 50 j _ β27 β k _ β20 β i _ β56 β 20 5 50
d _ β 50 β 100 h _ β18 β 20 l _ β16 β 10
5 Convert these fractions to percentages by multiplying by β100%β. a _ β1 β b _ β1 β c _ β4 β 8 3 15 f β4_ β1 β e 1β _ β3 β g β2_ β36 β 5 20 40
d _ β10 β 12 h _ β13 β 40
PROBLEM-SOLVING
6, 7
7, 8
7β9
6 A bottle of lemonade is only β25%β full. a What fraction of the lemonade has been consumed? b What percentage of the lemonade has been consumed? c What fraction of the lemonade is left? d What percentage of the lemonade is left? 7 A lemon tart is cut into eight equal pieces. What percentage of the tart does each piece represent?
8 Petrina scores β28βout of β40βon her Fractions test. What is her score as a percentage?
9 The Heathmont Hornets basketball team have won β14βout of β18βgames. They still have two games to play. What is the smallest and the largest percentage of games the Hornets could win for the season?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
REASONING
10
10, 11
11β13
10 Lee won his tennis match with the score β 6β4, 6β2, 6β1β. Therefore, he won β18β games and lost β7β. a What fraction of games did he win? b What percentage of games did he win?
U N SA C O M R PL R E EC PA T E G D ES
198
11 How many percentages between β1%βand β99%β can be written as a simplified fraction with a β 3 ,ββ denominator of β10β? (For example, β30% = _ 10 which would count as one possible answer.)
12 Scott and Penny have just taken out a home β1 β%β. Write this interest rate as a simplified fraction. loan, with an interest rate of β5_ 2
13 Give an example of a fraction that would be placed between β8%βand β9%βon a number line. ENRICHMENT: Lottery research
β
β
14
14 Conduct research on a major lottery competition to answer the following, if possible. a Find out, on average, how many tickets are sold each week. b Find out, on average, how many tickets win a prize each week. c Determine the percentage chance of winning a prize. d Determine the percentage chance of winning the various divisions. e Work out the average profit the lottery competition makes each week.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3J Finding a percentage of a number
199
3J Finding a percentage of a number
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand that ο¬nding percentages of a number can be found by multiplying by a fraction β’ To be able to ο¬nd a percentage of a number β’ To be able to apply percentages to worded problems
A common application of percentages is to find a certain percentage of a given number. Throughout life, you will come across many examples where you need to calculate percentages of a quantity. Examples include retail discounts, interest rates, personal improvements, salary increases, commission rates and more. In this exercise, we will focus on the mental calculation of percentages.
Customers, retail assistants, accountants and managers all use percentage calculations when ο¬nding a discounted price. This is the original price reduced by a percentage of its value
Lesson starter: Percentages in your head
It is a useful skill to be able to quickly calculate percentages mentally.
Calculating 10% or 1% is often a good starting point. You can then multiply or divide these values to arrive at other percentage values. β’
β’ β’ β’
In pairs, using mental arithmetic only, calculate these 12 percentages. a 10% of $120 b 10% of $35 c 20% of $160 d 20% of $90 e 30% of $300 f 30% of $40 g 5% of $80 h 5% of $420 i 2% of $1400 j 2% of $550 k 12% of $200 l 15% of $60 Check your answers with a classmate or your teacher. Design a quick set of 12 questions for a classmate. Discuss helpful mental arithmetic skills to increase your speed at calculating percentages.
KEY IDEAS
β To find the percentage of a number we:
1 2 3 4
Express the required percentage as a fraction. Change the word βofβ to a multiplication sign. Express the number as a fraction. Follow the rules for multiplication of fractions or look for a mental strategy to complete the multiplication. percentage 100
β Percentage of a number = _ Γ number
25 Γ _ 60 25% of 60 = _ 100 1 = 15
1 of 60 25% of 60 = _ 4 = 60 Γ· 4 = 60 Γ· 2 Γ· 2 = 15
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
β Useful mental strategies
β’ β’ β’ β’
To find β50%β, divide by 2 To find β10%β, divide by 10 To find β25%β, divide by 4 To find β1%β, divide by 100 Swap the two parts of the problem around if the second number is easier to work with, e.g. find β28%βof β50β, find β50%βof β28β
U N SA C O M R PL R E EC PA T E G D ES
200
β’
BUILDING UNDERSTANDING
1 State the missing number in the following sentences. a Finding β10%βof a quantity is the same as dividing the quantity by ______ . b Finding β1%βof a quantity is the same as dividing the quantity by ______ . c Finding β50%βof a quantity is the same as dividing the quantity by ______ . d Finding β100%βof a quantity is the same as dividing the quantity by ______ . e Finding β20%βof a quantity is the same as dividing the quantity by ______ . f Finding β25%βof a quantity is the same as dividing the quantity by ______ .
2 Without calculating the exact values, determine which alternative (i or ii) has the highest value. a i β20%βof β$400β ii β25%βof β$500β b i β15%βof β$3335β ii β20%βof β$4345β c i β3%βof β$10 000β ii β2%βof β$900β d i β88%βof β$45β ii β87%βof β$35β
Example 28 Finding the percentage of a number Find: a 3β 0%βof β50β
b β15%βof β400β
SOLUTION
EXPLANATION
β β β β30 βΓ _ a 30% of 50 = _ 1 β―β― β β β2β100ββ β β 30 _ β = β β= 15 2 Mental arithmetic: β10% of 50 = 5β Hence, β30% of 50 = 15β.
Write β%βas a fraction. Cancel and simplify.
4 β4ββ 00ββ ββ β β15 βΓ _ b 15% of 400 = _ 1 β1β100ββ β β 15 Γ 4 _ ββ―β― β β β= β β= 60 β 1 Mental arithmetic: β10% of 400 = 40, 5% of 400 = 20β Hence, β15% of 400 = 60β.
Write β%βas a fraction. Cancel and simplify.
5ββ 0β1β β
Multiply by β3βto get β30%β.
Halve to get β5%β. Multiply by β3βto get β15%β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3J Finding a percentage of a number
201
Now you try Find: a 4β 0%βof β90β
U N SA C O M R PL R E EC PA T E G D ES
b β12%βof β500β
Example 29 Solving a worded percentage problem
Jacqueline has saved up β$50βto purchase a new pair of jeans. She tries on many different pairs but only likes two styles, Evie and Next. The Evie jeans are normally β$70βand are on sale with a β 25%βdiscount. The Next jeans retail for β$80βand have a β40%βdiscount for the next β24βhours. Can Jacqueline afford either pair of jeans? SOLUTION
EXPLANATION
Evie jeans
Discount = 25% of $70 β= _ β 25 βΓ _ β70 β= $17.50 1 β β β β―β―β― β―β― ββ―β― β 100 β Sale price = $70 β $17.50 β = $52.50 Next jeans
Calculate the discount on the Evie jeans.
Discount = 40% of $80 β= _ β 40 βΓ _ β80 β= $32 1 ββ β―β― β ββ―β―β― ββ―β― β 100 Sale price = $80 β $32 β = $48 Jacqueline can afford the Next jeans but not the Evie jeans.
Calculate the discount on the Next jeans.
Find β25%βof β$70β.
Find the sale price by subtracting the discount.
Find β40%βof β$80β.
Find the sale price by subtracting the discount.
Now you try
Two shops sell the same jumper. In Shop A the retail price is β$80βbut there is currently a β30%β off sale. In Shop B the retail price is β$90βbut there is currently a β40%βoff sale. In which shop is the jumper cheaper and by how much?
Exercise 3J FLUENCY
Example 28
Example 28
1
Find: a β50%βof β20β
2 Find: a β50%βof β140β e 2β 5%βof β40β i β5%βof β80β m 1β 1%βof β200β
1, 2β3ββ(1β/2β)β,β 4
2β3ββ(1β/2β)β, 4, 5ββ(1β/2β)β
2β3β(β 1β/4β)β,β 5ββ(1β/3β)β
b β50%βof β40β
c β10%βof β60β
d β20%βof β80β
b f j n
c g k o
d h l p
1β 0%βof β360β β25%βof β28β β4%βof β1200β β21%βof β400β
2β 0%βof β50β β75%βof β200β β5%βof β880β β12%βof β300β
3β 0%βof β90β β80%βof β250β β2%βof β9500β β9%βof β700β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
202
Chapter 3
Fractions and percentages
3 Find the following values involving percentages greater than β100%β. a 1β 20%βof β80β b 1β 50%βof β400β c β110%βof β60β e β125%βof β12β f β225%βof β32β g β146%βof β50β
d β400%βof β25β h β3000%βof β20β
4 Match the questions with their correct answer. Answers
β10%βof β$200β
β$8β
β20%βof β$120β
β$16β
β10%βof β$80β
β$20β
β50%βof β$60β
β$24β
β20%βof β$200β
β$25β
β5%βof β$500β
β$30β
β30%βof β$310β
β$40β
β10%βof β$160β
β$44β
β1%βof β$6000β
β$60β
β50%βof β$88β
β$93β
U N SA C O M R PL R E EC PA T E G D ES
Questions
5 Find: a 3β 0%βof β$140β d β2%βof β4500β tonnes g β5%βof β30β grams PROBLEM-SOLVING
b 1β 0%βof β240β millimetres e β20%βof β40β minutes h β25%βof β12β hectares 6, 7
c β15%βof β60β kilograms f β80%βof β500β centimetres i β120%βof β120β seconds 7β9
9β11
6 Harry scored β70%βon his Percentages test. If the test is out of β50βmarks, how many marks did Harry score?
Example 29
7 Grace wants to purchase a new top and has β$40βto spend. She really likes a red top that was originally priced at β$75βand has a β40%βdiscount ticket on it. At another shop, she also likes a striped hoody, which costs β$55β. There is β20%βoff all items in the store on this day. Can Grace afford either of the tops? 8 In a student survey, β80%βof students said they received too much homework. If β300βstudents were surveyed, how many students felt they get too much homework?
9 2β 5%βof teenagers say their favourite fruit is watermelon. In a survey of β48βteenagers, how many students would you expect to write watermelon as their favourite fruit?
10 At Gladesbrook College, β10%βof students walk to school, β35%βof students catch public transport and the remainder of students are driven to school. If there are β1200βstudents at the school, find how many students: a walk to school b catch public transport c are driven to school.
11 Anthea has just received a β4%βsalary increase. Her wage before the increase was β$2000βper week. a How much extra money does Anthea receive due to her salary rise? b What is Antheaβs new salary per week? c How much extra money does Anthea receive per year? Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3J Finding a percentage of a number
REASONING
12, 13
12β14
203
14β17
U N SA C O M R PL R E EC PA T E G D ES
12 Sam has β2βhours of βfree timeβ before dinner is ready. He spends β25%βof that time playing computer games, β20%βplaying his drums, β40%β playing outside and β10%βreading a book. a How long does Sam spend doing each of the four different activities? b What percentage of time does Sam have remaining at the end of his four activities? c Sam must set the table for dinner, which takes β5 minutesβ. Does he still have time to get this done?
13 Gavin mows β60%βof the lawn in β48 minutesβ. How long will it take him to mow the entire lawn if he mows at a constant rate? 14 Find: a 2β 0%βof (β50%βof β200β) c 5β %βof (β5%βof β8000β)
b β10%βof (β30%βof β3000β) d β80%βof (β20%βof β400β)
15 Which is larger: β60%βof β80βor β80%βof β60β? Explain your answer.
16 Tom did the following calculation: β120 Γ· 4 Γ· 2 Γ 3β. What percentage of β120βdid Tom find? 17 a If β5%βof an amount is β$7β, what is β100%βof the amount? 1 β%βof the amount? b If β25%βof an amount is β$3β, what is β12β_ 2 ENRICHMENT: Waning interest
β
β
18
18 When someone loses interest or motivation in a task, they can be described as having a βwaning interestβ. Jill and Louise are enthusiastic puzzle makers, but they gradually lose interest when tackling very large puzzles. a Jill is attempting to complete a β5000β-piece jigsaw puzzle in β5βweeks. Her interest drops off, completing β100βfewer pieces each week. i How many pieces must Jill complete in the first week to ensure that she finishes the puzzle in the β5β-week period? ii What percentage of the puzzle does Jill complete during each of the β5β weeks? iii What is the percentage that Jillβs interest wanes each week? b Louise is attempting to complete an β8000β-piece jigsaw puzzle in β5βweeks. Her interest drops off at a constant rate of β5%βof the total jigsaw size per week. i What percentage of the puzzle must Louise complete in the first week to ensure she finishes the puzzle in the β5β-week period? ii Record how many pieces of the puzzle Louise completes each week and the corresponding percentage of the puzzle. iii Produce a table showing the cumulative number of pieces completed and the cumulative percentage of the puzzle completed over the β5β-week period.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
The following problems will investigate practical situations drawing upon knowledge and skills developed throughout the chapter. In attempting to solve these problems, aim to identify the key information, use diagrams, formulate ideas, apply strategies, make calculations and check and communicate your solutions.
Completing a new tunnel
U N SA C O M R PL R E EC PA T E G D ES
Applications and problem-solving
204
1
A major new tunnel is to be built in a capital city to improve traffic congestion. The tunnel will be able to take three lanes of traffic in each direction. The project manager has established a detailed plan for the job which is scheduled to take β30βmonths to complete. The overall distance of the tunnel is β5 kmβ. One year into the job, β2 kmβof the tunnel is complete. To estimate if they are on schedule to finish the project on time, the project manager wants to calculate the fraction of time the job has been going, and the fraction of the length of the tunnel completed. a What fraction of the time of the job has passed? b What fraction of the distance of the tunnel has been completed? c At the one year mark, is the job on track, in front or behind schedule? d At the two year mark, what distance of the tunnel needs to have been completed for the project to be on track? e With two months to go, how much of the tunnel should have been completed for the project to be on track? f What is the average rate (in metres per month) that needs to be completed for the tunnel project to remain on schedule? Give your answer as a fraction and as a number to the nearest metre per month.
Cutting down on screen time
2 Wasim is a student who spends a quite a bit a time in the day looking at a screen. Wasim is concerned about the increasing fraction of his day that he is on βscreen timeβ and wants to compare the amount of screen time against the time spent on other activities. a His first rule for himself is that he must spend less than _ β1 βof an entire day watching 8 a screen. Following this rule, what is the maximum time Wasim can watch a screen?
His parents suggest that Wasim should only take into account his waking hours. Wasim knows he needs to sleep on average β9β hours/day. b If Wasim still wants to spend the same total amount of time watching a screen, what fraction of his waking day is now dedicated to screen time?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Applications and problem-solving
U N SA C O M R PL R E EC PA T E G D ES
Wasim thinks that his leisure time should be spent evenly between: β’ screen time β’ outside time (which includes walking to and from school) β’ family time (which includes meals and chores), and β’ commitments time (which includes homework, music practice, sports practice on some nights).
Applications and problem-solving
Wasim decides that he should not have to count screen time for any of the β7βhours he is at school, and chooses to look at the fraction of his leisure time he devotes to watching a screen. c Given Wasim still wants to spend β3βhours/day, outside of school, watching a screen, what fraction of Wasimβs leisure time is screen time?
205
With this new guideline: d what fraction of Wasimβs leisure time is screen time? e what fraction of Wasimβs waking time is screen time? f what fraction of Wasimβs entire day is screen time? g how much time per day can Wasim spend, outside of school, watching a screen? h how many hours per year can Wasim spend watching a screen?
Islaβs test scores
3 Isla has set herself the goal of achieving a score of at least β75%βon each of her classroom tests for the upcoming term. Isla in interested in what raw scores need to be achieved on future tests to achieve her goal, as well as what scores need to be obtained so that a particular average can be achieved for the term. Determine what minimum score Isla needs to achieve on each of the following tests to reach her goal. a Maths test with β100β marks b French test with β80β marks c Science test with β50β marks d Geography test with β20β marks The actual marks Isla achieved for the above four tests were: Maths β76βmarks, French β52β marks, Science β42βmarks and Geography β12β marks. e In which tests did Isla achieve her goal? Isla still has one test remaining for the term and has now changed her goal to averaging a score of β75%βacross all her tests for the term. The remaining test is out of β30β marks. f Given that each test is equally important, what score does Isla need on her remaining test to achieve her new goal? g Given that each test is not equally important, but each mark is equally important, what score does Isla need on her remaining test to achieve her new goal?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
3K Introduction to ratios LEARNING INTENTIONS β’ To understand that a ratio compares two or more related quantities in a given order β’ To be able to write a ratio from a description β’ To be able to use a common factor to simplify a ratio
U N SA C O M R PL R E EC PA T E G D ES
206
Ratios are used to show the relationship between two or more related quantities. They compare quantities of the same type using the same units. The numbers in the ratio are therefore written without units. When reading or writing ratios, the order is very important. In a sport, a teamβs win-loss ratio during the season could be written as 5 : 2 meaning that a team has won 5 games and only lost 2 games. This is very different from a win-loss ratio of 2 : 5 which would mean that 2 games were won and 5 games were lost.
A sporting teamβs win-loss performance could be expressed as a ratio.
Lesson starter: Impossible cupcakes
Liam is baking cupcakes for a group of friends and plans to make banana cupcakes and chocolate cupcakes in the ratio 2 to 3. This means that for every 2 banana cupcakes there are 3 chocolate cupcakes.
a How many chocolate cupcakes would Liam make if he makes the following number of banana cupcakes? i 4 ii 6 iii 8 b How many banana cupcakes would Liam make if he makes the following number of chocolate cupcakes? i 9 ii 15 iii 18 c How many cupcakes will Liam make in total if he makes: i 2 banana cupcakes? ii 6 chocolate cupcakes? d Given Liamβs cupcake ratio, is it possible that he could make 7 cupcakes? Explain why or why not.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3K Introduction to ratios
207
KEY IDEAS β A ratio compares two or more related quantities.
β’ β’
U N SA C O M R PL R E EC PA T E G D ES
β’
Ratios are expressed using whole numbers in a given order. A colon is used to separate the parts of a ratio. Ratios are expressed without units.
β A ratio can be simplified in a similar way to a fraction.
β’ β’ β’ β’
The fraction _ β4 βcan be simplified to _ β2 β.β 3 6 The ratio β4 : 6βcan be simplified to β2 : 3.β A ratio can be simplified if the numbers that make up the ratio have a common factor. Dividing all numbers in a ratio by their highest common factor will lead to the simplest form.
BUILDING UNDERSTANDING
1 Of a group of β5βcounters, β2βof them are red
and β3βof them are blue. a Which ratio would describe the number of red counters compared to the number of blue counters? A β2 : 5β B β2 : 3β C β3 : 2β D β3 : 5β E β1 : 2β b Which ratio would describe the number of blue counters compared to the number of red counters? A β2 : 5β C β3 : 2β E β1 : 2β
B 2β : 3β D β3 : 5β
c True or false? The fraction of red counters out of the total is _ β2 .ββ
3 d True or false? The fraction of blue counters out of the total is _ β3 .ββ 5
2 This rectangle is divided up into β12βsquares, β4βof which are shaded. a How many squares are unshaded? b A ratio which describes the ratio of shaded to unshaded
squares could be: A β3 : 9β B β4 : 7β C β4 : 8β D β5 : 7β E β4 : 6β c True or false? The correct ratio from part b above could also be written as β8 : 4β. d True or false? The correct ratio from part b above could also be written as β1 : 2β. e True or false? The correct ratio from part b above could also be written as β2 : 1β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
Example 30 Expressing as a ratio a A box contains β3βwhite and β4βdark chocolates. Write each of the following as a ratio. i the ratio of white chocolates to dark chocolates ii the ratio of dark chocolates to white chocolates b A shop shelf contains β2βcakes, β5βslices and β9βpies. Write each of the following as a ratio. i the ratio of cakes to slices ii the ratio of cakes to slices to pies iii the ratio of pies to cakes
U N SA C O M R PL R E EC PA T E G D ES
208
SOLUTION
EXPLANATION
β3 : 4β
The order is white to dark and there are β3βwhite and β4βdark chocolates.
ii β4 : 3β
The order is dark to white and there are β4βdark and β3βwhite chocolates.
β2 : 5β
We are only concerned with the ratio of cakes to slices and the order is cakes then slices.
a i
b i
ii β2 : 5 : 9β
Three quantities can also be compared using ratios. The order is important.
iii β9 : 2β
We are only concerned with pies and cakes and the order is pies then cakes.
Now you try
a A container has β2βclear and β5βgrey marbles. Write each of the following as a ratio. i the ratio of clear to grey marbles ii the ratio of grey marbles to clear marbles. b A computer shop has β3βlaptops, β4βtablets and β7βphones on a display table. Write each of the following as a ratio. i the ratio of laptops to phones ii the ratio of laptops to tablets to phones iii the ratio of tablets to laptops.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3K Introduction to ratios
209
Example 31 Simplifying a ratio b 3β : 12 : 15β
c β15βpoints to β10β points
U N SA C O M R PL R E EC PA T E G D ES
Simplify the following ratios. a β4 : 10β SOLUTION 4 : 10 a Γ·2
2:5
EXPLANATION
Γ·2
The highest common factor of β4βand β10βis β2βso dividing both numbers by β2βwill give the ratio in simplest form.
b Γ· 3 3 : 12 : 15 Γ· 3 1:4 :5
The highest common factor of β3β, β12βand β15βis β3βso dividing all numbers by β3βwill give the ratio in simplest form.
c Γ· 5 15 : 10 Γ· 5 3:2
First use the numbers to write the information as a ratio without units. Then divide both numbers by the highest common factor β5β.
Now you try
Simplify the following ratios. a 1β 5 : 5β
b β4 : 8 : 16β
c β18βgoals to β6β goals
Exercise 3K FLUENCY
Example 30
1, 2, 4β6ββ(1β/2β)β
2, 3, 4β6β(β 1β/2β)β
3, 4β6β(β 1β/3β)β
1
a A box contains β5βwhite and β3βdark chocolates. Write each of the following as a ratio. i the ratio of white chocolates to dark chocolates ii the ratio of dark chocolates to white chocolates b A toy shop window contains β3βcars, β7βtrucks and β4βmotorcycles. Write each of the following as a ratio. i the ratio of cars to trucks ii the ratio of cars to trucks to motorcycles iii the ratio of motorcycles to cars 2 a A pencil case has β8βpencils and β5β pens. Write each of the following as a ratio. i the ratio of pencils to pens ii the ratio of pens to pencils b A coffee shop produces β6βlattes, β7β cappuccinos and β5βflat white coffees in one hour. Write the following as a ratio. i the ratio of cappuccinos to flat whites ii the ratio of lattes to cappuccinos to flat whites iii the ratio of flat whites to lattes
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
210
Chapter 3
Fractions and percentages
U N SA C O M R PL R E EC PA T E G D ES
3 Each hour, a new radio station plans to play β3βsongs from the 70s, β4βsongs from the 80s, β7βsongs from the 90s and just one song from the 60s. Write each of the following as a ratio. a 80s songs to 70s songs b 70s songs to 90s songs c 60s songs to 80s songs d 70s songs to 60s songs e 90s songs to 80s songs to 70s songs f 60s songs to 70s songs to 80s songs to 90s songs
Example 31a
Example 31b
Example 31c
4 Simplify the following ratios. a 3β : 6β b 4β : 16β e β20 : 15β f 3β 5 : 45β i β120 : 140β j β72 : 56β
c 1β 0 : 5β g β24 : 18β k β99 : 22β
d 9β : 3β h β110 : 90β l β132 : 72β
5 Simplify the following ratios. a β2 : 4 : 10β b β5 : 10 : 30β e β36 : 48 : 12β f 2β : 90 : 16β
c β12 : 6 : 18β g β75 : 45 : 30β
d 2β 8 : 14 : 21β h β54 : 27 : 36β
6 Simplify the following ratios. a β6βpoints to β3β points c β20βmetres to β15β metres e β130βmm to β260β mm g β18βkg to β12βkg to β30β kg
b d f h
PROBLEMβSOLVING
β4βgoals to β16β goals β16βcm to β10β cm β48βlitres to β60β litres β24βseconds to β48βseconds to β96β seconds
7β8ββ(1β/2β)β
7β9β(β 1β/2β)β
8β10β(β 1β/2β)β
7 Write a ratio of shaded squares to white squares in these diagrams. Simplify your ratio if possible. a b c
d
8 Find the missing number. a _β _ : 4 = 6 : 12β c β3 : 2 = __ : 14β e β26 : 39 = 2 : __β
e
f
b _β _ : 8 = 5 : 4β d β9 : 4 = __ : 8β f β120 : __ = 2 : 5β
9 Simplify the following ratios by firstly expressing the fractions using a common denominator if necessary. a _ β3 β : _ β4 β d _ β7 β : _ β13 β c _ β 9 β: _ β4 β β3 β b _ β5 β : _ 3 3 5 5 11 11 7 7 e _ β1 β : _ β1 β f _ β1 β : _ β1 β g _ β1 β : _ β1 β h _ β1 β : _ β1 β 3 2 7 3 4 8 5 3 i _ β2 β : _ β1 β β2 β j _ β1 β : _ β4 β l _ β7 β : _ β3 β k _ β3 β : _ 9 5 5 10 4 5 8 2
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3K Introduction to ratios
10 a Show how β1_ β2 β : 2βcan be simplified to β5 : 6β. 3 b Simplify these ratios that also include mixed numerals. 1β 1β ii β4 : 1β_ iii β1_ β2 β : 2β_ i β1_ β1 β : 3β 2 2 3 2 REASONING
11, 12
211
2β iv 3β _ β4 β : 2β_ 3 5 11β13
12β14
U N SA C O M R PL R E EC PA T E G D ES
11 George simplifies β4 : 2βto β1 : _ β1 β. Explain his error and write the correct ratio. 2
12 A shade of orange is created by mixing yellow and red in the ratio β2 : 3β. a Explain why this is the same as mixing β4βlitres of yellow paint with β6βlitres of red paint. b Decide if the following ratios of yellow paint to red paint will make the colour more yellow or more red compared to the given ratio above: i β1 : 2β ii β2 : 1β iii β3 : 4β
13 The following compare quantities with different units. Write them using a ratio in simplest form by first writing them using the same units. a β50 cmβto β2β metres b β30 mmβto β5 cmβ c β$4βto β250β cents d β800βgrams to β1 kgβ e 4β 5βminutes to β2β hours f 1β βday to β18β hours
14 A friend says that the ratio of a number βnβto the same number squared βnβ2βis equal to β1 : 2β. Explain why the friend is incorrect unless βn = 2β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
ENRICHMENT: Paint mixing
β
β
15
15 The three primary colours are red, blue and yellow. Mixing them together in different ways forms a variety of colours. The primary, secondary and tertiary colours are shown in this chart. YELLOW primary
U N SA C O M R PL R E EC PA T E G D ES
212
YELLOW ORANGE tertiary
YELLOW GREEN tertiary
ORANGE secondary
GREEN secondary
RED ORANGE tertiary
BLUE GREEN tertiary
RED primary
BLUE primary
BLUE VIOLET tertiary
VIOLET secondary
RED VIOLET tertiary
a The secondary colours are formed by mixing the primary colours. What colour do you think will be formed if the following pairs of primary colours are mixed into the ratio β1 : 1β? i yellow and red ii yellow and blue iii red and blue b The tertiary colours are formed by mixing a primary colour and secondary colour. What colour do you think will be formed if the following pairs of colours are mixed into the ratio β1 : 1β? i yellow and orange ii blue and violet iii green and yellow c If secondary colours are formed by mixing primary colours in the ratio β1 : 1β, and the tertiary colours are formed by mixing primary and secondary colours in the ratio β1 : 1β, give the following ratios. i the ratio of yellow to red in the colour red orange. ii the ratio of yellow to blue in the colour yellow green. d In a special colour mix red is mixed with blue in the ratio β2 : 1β. This new mix is then mixed with more blue in the ratio β3 : 2β. What is the ratio of red to blue in the final mix?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3L Solving problems with ratios
213
3L Solving problems with ratios
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand the connection between ratios, fractions and proportions β’ To be able to solve proportion problems using ratios and fractions β’ To be able to divide a given quantity in a given ratio
When considering the make up of a material like concrete, it is important to correctly balance the proportions of sand, water, cement and gravel. These ingredients could be thought of as parts of the whole, fractions or the whole or as a ratio. You might say, for example, that a particular cement mix has sand, water, cement and gravel in the ratio 3 : 2 : 1 : 4. This means that out of a total of 10 parts, 3 of them are sand, 2 are water, 1 is cement and 4 are gravel. Similarly, you could say that the proportion or fraction of 3 and so if a load of concrete is 40 tonnes, then the proportion of sand is _ 3 Γ 40 = 12 sand in the mix is _ 10 10 tonnes.
Lesson starter: Making bread
The two main ingredients in bread are flour and water. A commonly used ratio of flour to water is 5 : 3. β’ β’ β’ β’ β’
How would you describe the proportion of flour in a bread mix? How would you describe the fraction of water in a bread mix? If the ratio of flour to water is 5 : 3, does that mean that all bread loaves that are made must have a total of 8 cups of ingredients? Explain. If a bread mix consisted of a total of 16 cups of bread and water, describe a method for working out the total number of cups of flour or water required. Is it possible to make a loaf of bread using a total of 12 cups of flour and water? Describe how you would calculate the number of cups of flour or water needed.
KEY IDEAS
β Ratios can be considered as a collection of parts.
The ratio of vinegar to water for example might be 3 : 4, which means that for every 3 parts of vinegar there are 4 parts of water.
β The total number of parts can be calculated by adding up all the numbers in the ratio.
The total number of parts in the example above is 3 + 4 = 7.
β Ratio problems can be solved by considering the proportion of one quantity compared to
the whole.
3 and so a 210 mL bottle contains _ 3 Γ 210 = 90 mL of vinegar. The proportion of vinegar is _ 7 7
β To divide a quantity into a given ratio:
β’ β’ β’
calculate the total number of parts (e.g. 3 + 4 = 7)
3) calculate the proportion of the desired quantity (e.g. _ 7 3 Γ 210 = 90) use fractions to calculate the amount (e.g. _ 7
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
BUILDING UNDERSTANDING 1 A cordial mix is made up of β1βpart syrup to β9βparts water. a Write the cordial to water mix as a ratio. b What is the total number of parts in the ratio? c What proportion of the cordial mix is syrup? Write your answer as a fraction. d What proportion of the cordial mix is water? Write your answer as a fraction. e If β400βmillilitres of cordial mix is made up, how much of it is: i syrup? ii water?
U N SA C O M R PL R E EC PA T E G D ES
214
2 $β 150βis to be divided into two parts using the ratio β2 : 3βfor Jacob and Sally, in that order. a What is the total number of parts in the ratio? b What fraction of the money would Jacob receive? c What fraction of the money would Sally receive? d Calculate the amount of money received by: i Jacob ii Sally.
Example 32 Finding amounts given a ratio
Dry mix cement is combined with water in a given ratio to produce concrete. a If the dry mix to water ratio is β3 : 5β, write the proportion of the following as a fraction. i dry mix ii water 3 _ b If β βof the concrete is dry mix, write the ratio of dry mix to water. 7 c If β8βlitres of dry mix is used using a dry mix to water ratio of β2 : 3β, what amount of water is required? SOLUTION a i
EXPLANATION
_ β3 β
The total number of parts is β8βand the number of parts relating to dry mix is β3β.
8
ii _ β5 β 8 b _ β3 β : _ β4 β= 3 : 4β 7 7
The number of parts relating to water is β5βout of a total of β8β.
c dry mix water 2:3 Γ4
Γ4
If _ β3 βof the concrete is dry mix, then _ β4 βis water. 7 7 Multiply both fraction by the common denominator to produce the simplified ratio. Multiply both numbers in the ratio by β4βto see what amount of water corresponds to β8βlitres of dry mix.
8 : 12 ββ΄ 12βlitres of water required.
Now you try
Juice concentrate is to be combined with water in a given ratio to produce a drink. a If the concentrate to water ratio is β2 : 7β, write the proportion of the following as a fraction. i concentrate ii water 1 _ b If β βof the drink is concentrate, write the ratio of concentrate to water. 5 c If β100βmL of concentrate is mixed using a concentrate to water ratio of β2 : 9β, what amount of water is required?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3L Solving problems with ratios
215
Example 33 Dividing an amount into a given ratio Divide β15βkg into the ratio β2 : 1β. EXPLANATION
Total number of parts β= 2 + 1 = 3β β1 β Fractions are _ β2 βand _ 3 3
First find the total number of parts in the ratio. Calculate the proportions for each number in the ratio. Multiply each fraction by the amount given to find the amounts corresponding to each portion. Answer the question with the appropriate units.
U N SA C O M R PL R E EC PA T E G D ES
SOLUTION
2 βΓ 15 = 10βand _ β1 βΓ 15 = 5β Amounts are β_ 3 3 β10 kg and 5 kgβ
Now you try
Divide β60βseconds into the ratio β3 : 1β.
Exercise 3L FLUENCY
Example 32
Example 32
Example 32
1
1β5, 6ββ(β1/2β)β
2β5, 6β(β β1/2β)β
2, 3, 5, 6β(β β1/3β)β
Powder is to be combined with water in a given weight ratio to produce glue. a If the powder to water ratio is β2 : 3β, write the proportion of the following as a fraction. i powder ii water b If the powder to water ratio is β1 : 5β, write the proportion of the following as a fraction. i powder ii water c If the powder to water ratio is β3 : 4β, write the proportion of the following as a fraction. i powder ii water
2 Flour is to be combined with water in the ratio β4 : 3βto produce damper. a Write the proportion of the following as a fraction. i flour ii water 2 _ b If β βof the damper is flour, write the ratio of flour to water. 3 c If β60βgrams of flour is mixed using a flour to water ratio of β3 : 2β, what amount of water is required?
3 A number of hockey teams have various win-loss ratios for a season. a The Cats have a win-loss ratio of β5 : 3β. What fraction of the games in the season did the Cats win? b The Jets won _ β3 βof their matches. Write down their win-loss ratio. 4 c The Eagles lost 5 of their matches with a win-loss ratio of β2 : 1β. How many matches did they win? d The Lions have a win-loss ratio of β2 : 7β. What fraction of the games in the season did the Lions lose? e The Dozers lost _ β2 βof their matches. Write down their win-loss ratio. 5 f The Sharks won β6βof their matches with a win-loss ratio of β3 : 2β. How many matches did they lose?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
216
Chapter 3
Fractions and percentages
Example 33
4 Divide β24βkg into the ratio β1 : 3β.
Example 33
5 Divide β35βlitres into the ratio β3 : 4β. 6 Divide the following into the given ratios. a β$40βin the ratio β2 : 3β c β32βkg in the ratio β3 : 5β e β54βlitres in the ratio β5 : 4β g β96βmetres in the ratio β7 : 5β
b d f h
$β 100βin the ratio β3 : 7β β72βkg in the ratio β5 : 1β β66βlitres in the ratio β2 : 9β β150βmetres in the ratio β8 : 7β
U N SA C O M R PL R E EC PA T E G D ES
Example 33
PROBLEMβSOLVING
7, 8
8, 9ββ(β1/2β)β,β 10
9β(β β1/2β)β, 10, 11
7 Micky is making up a spray for his fruit orchard and mixes chemical with water using a ratio of β2 : 13β. a How much chemical should he use if making a spray with the following total number of litres? i β15β litres ii β45β litres iii β120β litres b If he uses β20βlitres of chemical, how much spray will he make? c If he uses β39βlitres of water, how much chemical does he use? d If he makes β60βlitres of spray, how much chemical did he use?
8 Mohammad has β$72βto divide amongst three children in the ratio β1 : 2 : 3β. a How much money is given to the first child listed? b How much money is given to the third child listed? 9 Divide the following into the given ratios. a β$180βinto the ratio of β2 : 2 : 5β c β165βlitres into the ratio β2 : 10 : 3β
b β750βgrams into the ratio of β5 : 3 : 7β d β840βkg into the ratio β2 : 7 : 3β
10 The ratio of cattle to sheep on a farm is β2 : 7β. a How many sheep are there if there are β50β cattle? b How many cattle are there if there are β84β sheep? c How many sheep are there if there are β198βcattle and sheep in total? d How many cattle are there if there are β990βcattle and sheep in total? 11 The three angles in a triangle are in the ratio β1 : 2 : 3β. What is the size of the largest angle?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3L Solving problems with ratios
REASONING
12, 13
12β14
217
13β15
12 Sisi divides β30βbeads into a ratio of β1 : 3βto make two piles. How many beads make up the remainder?
U N SA C O M R PL R E EC PA T E G D ES
13 Mal tips out his collection of β134βone dollar coins and divides them using the ratio β2 : 5β. Will Mal be able to complete the division without leaving a remainder? Give a reason. 14 A sporting squad is to be selected for a training camp with attacking, midfielder and defence players in the ratio β3 : 5 : 2β. Currently, the coach has selected β4βattacking players, β5βmidfielders and 3 defence players. Find the minimum number of extra players the coach needs to select to have the correct ratio.
15 There are initially β14βlions in an enclosure at a zoo and the ratio of male to female lions is β4 : 3β. Some of the male lions escaped and the ratio of male to female lions is now β2 : 3β. How many male lions escaped?
ENRICHMENT: Slopes as ratios
β
β
16
16 The steepness of a slope can be regarded as the ratio of the vertical distance compared to the horizontal distance. So a slope with ratio β1 : 4βwould mean for every β1βmetre vertically you travel β4βmetres horizontally. a Describe the following slopes as ratios of vertical distance to horizontal distance. i vertical β20βmetres and horizontal β80β metres ii vertical β120βmetres and horizontal β40β metres b A slope has a ratio of β2 : 9β. i Find the horizontal distance travelled if the vertical distance is β12β metres. ii Find the vertical distance travelled if the horizontal distance is β36β km. 1 βmetres is climbed c What would be the slope ratio if β20β_ 2 1 βis climbed horizontally? vertically and β5β_ 2
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
218
Chapter 3
Fractions and percentages
Modelling
Pricing plants Andrew owns a nursery and normally sells flower pots for β$15βeach and on average he sells β100βof these pots per day. The cost to produce each flower pot is β$5β, so the profit per pot under normal conditions is β $10βper pot.
U N SA C O M R PL R E EC PA T E G D ES
To attract more customers to the nursery, he considers lowering the price with one of the following deals. β’ Deal β1β: Lowering the selling price to β$12β, where he predicts he will sell β1_ β1 βtimes as many pots 2 per day. β’ Deal β2β: Lowering the selling price to β$11β, where he predicts selling β1_ β4 βtimes as many pots per day. 5
Present a report for the following tasks and ensure that you show clear mathematical workings and explanations where appropriate.
Preliminary task
a Under normal conditions, what is the total profit for Andrewβs shop after selling β100βflower pots? b Using Deal β1β, find the following amounts. i the profit made on each pot sold ii the expected number of pots to be sold iii the expected total profit for Andrewβs nursery for one day c Compared to normal conditions, by how much does Andrewβs profit increase if he uses Deal β1β? d Write the profit increase as a percentage of the original profit (found in part a).
Modelling task
Formulate Solve
Evaluate and verify
Communicate
a The problem is to create a deal for Andrew that will cause the profit to increase by at least 10%. Write down all the relevant information that will help solve this problem. b Repeat Preliminary task part b, but use Deal β2βinstead of Deal 1. c Compared to normal conditions, by how much does Andrewβs profit increase if he uses Deal β2β? Express your answer as a percentage of the original profit with no deals. d Andrew considers a third deal where the selling price is reduced to β$8β. This will cause the number of pots sold to triple. Compare this deal with the other two, considering the total profit for a day. e By considering the deals above, compare the profits for one dayβs trade for Andrewβs nursery. Sort the deals from most profitable to least profitable for Andrew. f Construct your own scenario for Andrewβs nursery which will cause the overall profit for one day to be at least 10% more than that of the normal conditions. i Describe the scenario in terms of the price sold and the number of flower pots sold. ii Show that the profit is increased by at least 10% by finding the expected total profit. g Summarise your results and describe any key findings.
Extension questions
a Is it possible to find a deal that produces exactly a β20%βincrease in profit for Andrewβs nursery? If so, find it and explain whether or not it is a realistic deal to offer. b Suppose Andrew keeps the regular price at β$15βper pot, but allows people to buy β5βpots for the price of β4β. Explain how the results would compare with charging β$12βper flower pot. Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Technology and computational thinking
219
Adding to infinity
U N SA C O M R PL R E EC PA T E G D ES
It would be natural to think that a total sum would increase to infinity if we continually added on numbers to the total, however, this is not always the case. Just like in this image, we could continually zoom in adding more and more spirals but we know that the total size of the image never gets larger.
Technology and computational thinking
(Key technology: Spreadsheets)
1 Getting started
a Consider a sequence of numbers where each number is double the previous number in the sequence. Then the total sum is obtained by adding each new number to the previous total. i Copy and complete this table using the above description. Sequence
1
2
4
Sum
1
3
7
8
16
32
ii What do you notice about the total sum? Is it increasing more or less rapidly as more numbers are being added? Give a reason. b Now consider finding a total sum from a sequence of numbers where each number is one half of the previous number. i Copy and complete this table using the above description. Sequence
1
Sum
1
_ β1 β
2 _ β3 β 2
1β β_ 4 7β β_ 4
1β β_ 8
_ β1 β
16
ii What do you notice about the total sum? Is it increasing more or less rapidly as more numbers are being added? Give a reason. Start iii What number does your sum appear to be approaching? Input n
2 Applying an algorithm
Here is an algorithm which finds a total sum of a sequence of βnβ numbers starting at β1β. Note that βSβstands for the total sum, βtβstands for the term in the sequence and βiβcounts the number of terms produced. a Copy and complete this table for βn = 6β, writing down the values for βtβand βSβfrom beginning to end. βtβ
1
βsβ
1
_ β1 β
2 _ β3 β 2
_ β1 β
4
b Describe this sequence and sum compared to the sequence studied in part 1.
i = 1, t = 1, S = 1 Output t, S
Is i = n?
Yes
No
t 2 S=S+t i=i+1 t=
End
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 3
Fractions and percentages
c Describe the parts of the algorithm that make the following happen. i Each term is being halved each time. ii The sum is increased each time by adding on the new term. iii The algorithm knows when to end. d Write an algorithm that adds up all the numbers in a sequence but this time: β’ the sequence starts at β2β β’ the sum is obtained by adding one third of the previous number each time.
U N SA C O M R PL R E EC PA T E G D ES
Technology and computational thinking
220
3 Using technology
We will use a spreadsheet to generate the sums studied above. a Create a spreadsheet as shown. b Fill down at cells A5, B5 and C5 to generate 10 terms. Check that the numbers you produce match the numbers you found in parts 1b and 2a. c Fill down further. What do you notice about the total sum? d Experiment with your spreadsheet by changing the rule in B5. Divide B4 by different numbers and also try multiplying B4 by a number. e Find a rule for cell B4 where the total sum moves closer and closer to _ β3 βas βnβ increases. 2
4 Extension
a Find a rule for cell B4 where the total sum moves closer and closer to β3βas βnβ increases. b Describe the circumstances and the way in which the terms βtβare generated that make βSβ approach closer and closer to a fixed sum rather than increasing to infinity.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Investigation
221
Egyptian fractions
Investigation
The fractions in the ancient Egyptian Eye of Horus were used for dividing up food and land, as well as portions of medicine. They are called unitary fractions because all the numerators are β1β.
U N SA C O M R PL R E EC PA T E G D ES
Clearly, the ancient Egyptians had no calculators or precise measuring instruments; nevertheless, by repeatedly dividing a quantity in half, the fractions _ β1 β, _ β 1 βwere combined to β1 ,β _ β1 ,β _ β 1 βor _ 2 4 8 16 32 estimate other fractions.
Using ancient Egyptian fractions, how could three loaves be divided equally between five people? β1 ββ( ) a loaf. The remaining half loaf can be cut First, cut the loaves in half and give each customer _ 2 1 1 1 _ _ _ ) of a loaf. There is a small portion left β into eight parts and each person is given β βof β β= β ββ( 8 2 16
(
3 portions of _ β 1 β β, so these portions can be divided in half and each customer given _ β1 βof _ β 1 β= _ β 1 ββ 2 16 32 16 ) ( ) of a loaf. loaf 1 customer 1 customer 2 1 2
1 2
loaf 2 customer 3 customer 4 1 2
loaf 3 customer 5
shared between customers
1 2
1 left over 32
Each customer has an equal share _ β1 β+ _ β 1 β+ _ β 1 ββ( 2 16 32 1 ββ( β_ ) of a loaf left over. 32
1 16
1 32
) of the loaf and the baker will have the small
If each loaf is divided exactly into five parts, the three loaves would have β15βequal parts altogether and each customer could have three parts of the β15β; _ ββ 3 β= _ β1 βof the total or _ β3 βof one loaf. _ β3 β= 0.6β 15 5 5 5 1 β+ _ and β_ β 1 β+ _ β 1 β= 0.59375 β 0.6β (βββmeans approximately equal). 2 16 32
So even without calculators or sophisticated measuring instruments, the ancient Egyptian method of repeated halving gives quite close approximations to the exact answers. Task
Using diagrams, explain how the following portions can be divided equally using only the ancient Egyptian unitary fractions of _ β1 β, _ β1 β, _ β1 ,β _ β 1 βand _ β 1 ββ. 2 4 8 16 32 a three loaves of bread shared between eight people b one loaf of bread shared between five people c two loaves of bread shared between three people
Include the Egyptian Eye of Horus symbols for each answer, and determine the difference between the exact answer and the approximate answer found using the ancient Egyptian method.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
222
Chapter 3
Find the sum of all fractions in the Up for a challenge? If you get stuck form _ βa β, where the numerator βaβ is on a question, check out the βWorking b with unfamiliar problemsβ poster at less than the denominator βbβ, and βbβ the end of the book to help you. is an integer from β2βto β10β inclusive. Fractions that are equivalent, e.g. _ β1 ββ 2 2 _ and β β, should each be counted towards the sum. 4
U N SA C O M R PL R E EC PA T E G D ES
Problems and challenges
1
Fractions and percentages
2 At the end of each practice session, Coach Andy rewards his swim team by distributing β30βpieces of chocolate according to effort. Each swimmer receives a different number of whole pieces of chocolate. Suggest possible numbers (all different) of chocolate pieces for each swimmer attending practice when the chocolate is shared between: a 4β β swimmers b β5β swimmers c β6β swimmers d β7β swimmers.
3 In this magic square, the sum of the fractions in each row, column and diagonal is the same. Find the value of each letter in this magic square. 2 5
A
4 5
B
C
D
E
1 2
1
4 You are given four fractions: _ β1 ,β _ β1 ,β _ β1 βand _ β1 β. Using any three of these fractions, complete each number 3 4 5 6 sentence below. a _β _+ __Γ __= _ β7 β 30 b β__Γ·__β __= _ β7 β 6
c β__+ __β __= _ β13 β 60
5 When a β$50βitem is increased by β20%β, the final price is β$60β. Yet when a β$60βitem is reduced by β20%β, the final price is not β$50β. Explain. 6 β$500βis divided in the ratio β13 : 7 : 5β. How much money is the largest portion?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter summary
improper fraction
2 parts selected
10 7
2 numerator 5 denominator
Equivalent fractions 50 8 = 30 = 21 = 16 = 12 100 60 42
3
simplest form
mixed numeral 5 4
5 parts in the whole
U N SA C O M R PL R E EC PA T E G D ES
Percentages
Chapter summary
proper fraction 34
223
mixed
improper
2 35 = 10 + 35 = 13 5 5
improper
mixed
40 = 2 10 = 22 15 15 3
Fractions
40 = 5 Γ 8 = 83 = 2 23 15 5Γ3
or
50 50% = 100 = 12
10 1 10% = 100 = 10
75 75% = 100 = 34
20 20% = 100 = 15
80 80% = 100 = 45
25 25% = 100 = 14
100% = 100 =1 100
Recall
Expressing as a proportion $12 out of $50 = 12 50 = 6 25 = 24%
Simplify
7 =7Γ·7=1 7
or
HCF of 42 and 63 is 21.
Ratios Simplest form uses whole numbers. 10 : 8 = 5 : 4 3 red and 2 blue Ratio of red to blue is 3 : 2. 3 are red, 2 are blue 5
1 1% = 100
42 = 77 ΓΓ 69 = 69 = 33 ΓΓ 23 = 23 63 Γ2 42 = 21 = 23 21 Γ 3 63
Comparing fractions
5
5 8
?
3 4
5 8
<
6 8
8 is the lowest common denominator (LCD) which is the lowest common multiple (LCM) of 4 and 8.
Operating with fractions
Adding fractions 4 + 23 5
Multiplying fractions
= 12 + 10 15
15
7 = 22 = 1 15 15
2
= 2 54 β 1 23
=1+( 7 = 1 12
1
2
2
5
9 = 16 Γ 1 4 5
= 15
4
Dividing fractions
= 36 = 7 15 5
1
Percentage of a quantity
5 = 13 β3 4
5 β 23 4
15 8 β 12 12
)
)
20 = 39 β 12 12
=
39 β 20 12 19
25% of $40 120% of 50 minutes = 1 Γ 40 4
= $10
7
= 12 = 1 12
1
46 Γ· 19
2 1 34 β 13
34 β 13
= (2 β 1) + (
3 15 Γ 2 14
5 20 Γ = 10 50
Subtracting fractions
1
5 of 20 50 10
50 Γ = 120 100 1 2
1
= 60 minutes = 1 hour
= 25 Γ· 10 6
9
5 93 = 25 Γ 6 2 10 2 = 15 4 = 33 4
Reciprocal of
10 9 is 10 9
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
224
Chapter 3
Fractions and percentages
A printable version of this checklist is available in the Interactive Textbook
3A
β
1. I can identify the numerator and denominator of a fraction. e.g. A pizza has been cut into eight pieces, with three of them selected. Write the selection as a fraction and state the numerator and denominator.
U N SA C O M R PL R E EC PA T E G D ES
Chapter checklist
Chapter checklist with success criteria
3A
3A
3B
3B
3B
3C
3C
3D
2. I can represent fractions on a number line.
β9 βon a number line. e.g. Represent the fractions _ β3 βand _ 5 5
3. I can represent a fraction by shading areas.
e.g. Represent the fraction _ β3 βin three different ways, using a square divided into four equal 4 regions.
4. I can produce equivalent fractions with a given numerator or denominator. β ? βand _ β12 ββ. β4 β= _ e.g. Fill in the missing numbers for _ β2 β= _ ? 5 3 12
5. I can write fractions in simplest form.
e.g. Write the fraction _ β12 βin simplest form. 20
6. I can decide if two fractions are equivalent.
β20 βequivalent? Explain your answer. e.g. Are _ β4 βand _ 5 25
7. I can convert mixed numerals to improper fractions. e.g. Convert β3_ β1 βto an improper fraction. 5
8. I can convert from an improper fraction to a mixed numeral, simplifying if required. β20 βto mixed numerals in simplest form. e.g. Convert _ β11 βand _ 4 6
9. I can compare two fractions to decide which is bigger.
e.g. Decide which of _ β2 βand _ β3 βis bigger, and write a statement involving β>βor β<βto summarise 5 3 your answer.
3D
3E
3E
3F
10. I can order fractions in ascending or descending order.
β4 ,β _ β2 ββ. β3 ,β _ e.g. Place the following fractions in ascending order: _ 4 5 3
11. I can add two fractions by converting to a common denominator if required. β5 ββ. e.g. Find the value of _ β3 β+ _ 4 6
12. I can add two mixed numerals. e.g. Find the sum of β3_ β2 βand β4_ β2 ββ. 3 3
13. I can subtract fractions by converting to a common denominator if required. β1 ββ. e.g. Find the value of _ β5 ββ _ 6 4
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter checklist
14. I can subtract mixed numerals. 1 .ββ e.g. Find the value of β5_ β2 ββ 3β_ 3 4
U N SA C O M R PL R E EC PA T E G D ES
3F
Chapter checklist
β
225
3G
3G
3H
3H
3H
3I
3I
15. I can multiply fractions (proper and/or improper fractions). β8 ββ. β3 βΓ _ e.g. Find _ 4 9
16. I can multiply mixed numerals and fractions by whole numbers 2 βand _ β1 βΓ 21β. e.g. Find β6_ β1 βΓ 2β_ 5 3 4
17. I can find the reciprocal of a fraction, whole number or mixed numeral. e.g. Find the reciprocal of β1_ β3 .ββ 7
18. I can divide fractions (proper and/or improper fractions). β3 ββ. e.g. Find _ β3 βΓ· _ 5 8
19. I can divide mixed numerals. 3 ββ. e.g. Find β2_ β2 βΓ· 1β_ 5 5
20. I can convert a percentage to a fraction or mixed numeral. e.g. Express β36%βas a fraction in simplest form. 21. I can convert a fraction to a percentage. β3 βto percentages. e.g. Convert _ β11 βand _ 8 25
3J
22. I can find the percentage of a number. e.g. Find β15%βof β400β.
3K
23. I can express related quantities as a ratio. e.g. A box contains 3 white and 4 dark chocolates. Write the ratio of dark chocolates to white chocolates.
3K
24. I can simplify a ratio. e.g. Simplify β4 : 10β
3L
25. I can find amount given a ratio. e.g. If the ratio of dry mix cement to water is β2 : 3β, what amount of water is mixed with 8 litres of dry mix cement?
3L
26. I can divide an amount into a given ratio. e.g. Divide β15 kgβinto the ratio β2 : 1β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
226
Chapter 3
Fractions and percentages
3A
1
Write the following diagrams as fractions (e.g. _ β1 β) and list them in ascending order. 6
U N SA C O M R PL R E EC PA T E G D ES
Chapter review
Short-answer questions
3B 3B
3C
3D
3D
3D
3D
3D
3E/F
2 Write four fractions equivalent to _ β3 βand write a sentence to explain why they are equal in value. 5
3 Write the following fractions in simplest form. 18 β a β_ b _ β8 β 30 28
35 β c β_ 49
4 Convert each of the following to a mixed numeral in simplest form. b _ β63 β a _ β15 β c _ β45 β 36 10 27
56 β d β_ 16
5 Place the correct mathematical symbol β< , = or >β, in between the following pairs of fractions to make true mathematical statements. _ _ β4 β a _ β2 ββ β1 β b _ β3 ββ 7 7 8 8 3 2 1 _ _ _ _ c β1β ββ β1β β β29 β d β3β ββ 3 5 9 9 6 State the largest fraction in each list. a _ β3 β, _ β2 ,β _ β5 β, _ β1 β 7 7 7 7
3 ,β _ b β_ β2 β, _ β5 β, _ β1 β 8 8 8 8
7 State the lowest common multiple for each pair of numbers. a 2β , 5β b β3, 7β
c β8, 12β
8 State the lowest common denominator for each set of fractions. β3 β b _ β2 β, _ β3 β a _ β1 β, _ 2 5 3 7
β5 β c _ β3 ,β _ 8 12
9 Rearrange each set of fractions in descending order. 1β a 1β _ β3 ,β _ β9 ,β 2β_ 5 5 5 b _ β14 ,β _ β9 β, _ β5 β β11 ,β _ 8 6 4 3 7 β, 5β_ 2 ,β _ 1β 1 ,β 5β_ c β5β_ β47 ,β 5β_ 3 9 18 9 3
10 Determine the simplest answer for each of the following. 3 β+ _ β1 β a β_ 8 8 3β d β2_ β 7 β+ 3β_ 10 15 g _ β3 ββ _ β2 β+ _ β7 β 4 5 8
b _ β1 β+ _ β1 β 3 2 7 ββ _ β3 β e β_ 8 8 7 β+ 2β_ 1β h β8_ β 7 ββ 4β_ 9 3 12
β5 β c _ β3 β+ _ 8 6 3β f β5_ β1 ββ 2β_ 4 4 3β 7 ββ 6β_ 1 β+ 5β_ i β13β_ 2 10 5
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
3G
c _ β3 βof β16β 4 2β f β3_ β1 βΓ 2β_ 8 5
b _ β4 βof β100β 5 _ e β2 βof _ β1 β 3 4
12 Determine the reciprocal of each of the following. 3β b _ β7 β c 2β _ β3 β a β_ 12 4 4
U N SA C O M R PL R E EC PA T E G D ES
d 5β _ β1 β 3
Chapter review
3H
11 Find: 1 βΓ 21β a β_ 3 d _ β 8 βΓ _ β25 β 10 4
227
3H
3I
13 Perform these divisions. 6 βΓ· 3β a β_ 10 1β b β64 Γ· 3β_ 5 6β c 6β _ β2 βΓ· 1β_ 10 5 1 βΓ· 1β_ 1β d _ β3 βΓ· 1β_ 8 4 2
14 Copy the table into your workbook and complete. Percentage form
β36%β
3K
β18%β
_ β5 β
1β 2β β_ 5
Fraction
3J
β140%β
11 β β_ 25
100
15 Determine which alternative (i or ii) is the larger discount. a i β25%βof β$200β ii β20%βof β$260β b i β5%βof β$1200β ii β3%βof β$1900β
16 A shop shelf displays β3βcakes, β5βslices and β4βpies. Write the following as ratios. a cakes to slices b slices to pies c pies to cakes d slices to pies to cakes
3K
17 Express these ratios in simplest form. a β21 : 7β b β3.2 : 0.6β
3L
18 Divide β$80βinto the following ratios. a β7 : 1β
c β$2.30 : 50 centsβ
b β2 : 3β
Multiple-choice questions
3A
1
Which set of fractions corresponds to each of the red dots positioned on the number line? 0
3 β, _ 3 β, _ β6 ,β 1β_ β12 β A β_ 8 8 8 8 5β C _ β1 β, _ β3 ,β _ β9 β, 1β_ 2 4 8 8 1 β, _ E _ β3 β, _ β3 ,β 1β_ β14 β 8 4 2 8
1
2
1 β, _ B _ β3 ,β _ β3 β, 1β_ β12 β 8 4 4 8 3 β, 1β_ 1β D _ β2 β, _ β3 β, 1β_ 8 4 8 2
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
228
Chapter 3
3C
2 Which of the following statements is not true? 6 β= _ A _ β3 β= _ β9 β B β_ β18 β 4 12 11 33 2 β= _ D _ β13 β= _ β16 β β39 β E β_ 7 56 14 42
3 β= _ C β_ β15 β 10 40
3 Which set of mixed numerals corresponds to the red dots on the number line?
U N SA C O M R PL R E EC PA T E G D ES
Chapter review
3B
Fractions and percentages
0
1
3 ,β 2β_ 2 β, 3β_ 1β A 1β _ β1 ,β 1β_ 5 5 5 5 2 ,β 2β_ 2 β, 3β_ 2β C β1_ β1 ,β 1β_ 5 5 5 5 3 ,β 2β_ 3 β, 3β_ 1β E β1_ β1 ,β 1β_ 5 5 5 5
3D
3D
3E
3F
3D/I
3C
3G
2
3
3 β, 2β_ 3 β, 3β_ 1β B β1_ β2 β, 1β_ 5 5 5 5 4 β, 2β_ 2 β, 3β_ 2β D β1_ β2 β, 1β_ 5 5 5 5
β13 β β11 β, _ 4 Which is the lowest common denominator for this set of fractions? _ β 7 β, _ 12 15 18 A 6β 0β B β120β C β180β D β3240β E β90β
5 Which of the following fraction groups is in correct descending order? 1 β, _ β1 ,β _ β2 β A β_ β3 β, _ β3 β, _ β3 β B _ β3 β, _ 5 3 2 4 5 8 7 D _ β 1 β, _ β1 β β 1 β, _ β 1 β, _ C _ β5 β, _ β4 ,β _ β3 β, _ β2 β 10 20 50 100 8 5 8 3 8 β, 2β_ 3β 2 ,β 2β_ E β2_ β1 ,β 2β_ 5 15 3 4
6 Which problem has an incorrect answer? 1 β+ _ β3 β= _ β4 β A β_ β 5 β= _ B _ β3 β+ _ β5 β 6 6 6 4 12 16 5β 1 β= 2β_ C _ β3 βΓ _ β 5 β= _ β5 β D β5_ β2 ββ 3β_ 4 12 16 3 4 12 E _ β3 βΓ _ β4 β= _ β3 β 4 5 5 7 Three friends share a pizza. Kate eats _ β1 βof the pizza, Archie eats _ β1 βof the pizza and Luke eats 3 5 the rest. What fraction of the pizza does Luke eat? 4β D _ β7 β C _ β14 β B _ β2 β A β_ E _ β8 β 15 3 12 15 15
8 Which list is in correct ascending order? A 0β .68, _ β3 ,β 0.76, 77%, _ β1 ,β 40β 4 3 7 12 _ _ B β β, 82%, 0.87, β ,β 88%β 8 15 C β21%, 0.02, 0.2, 0.22, _ β22 β 10 14 _ D β ,β 0.36, 0.3666, 37%, _ β 93 β 40 250 3 13 _ _ E β0.76, 72%, β β, 0.68, β β 4 40
60 βcan be written as: 9 β_ 14 2β B β2_ β4 β C β4_ β2 β E β5_ β1 β D β7_ β4 β A 4β β_ 7 7 7 7 14 17 _ 10 β βof a metre of material is needed for a school project. How many centimetres is this? 25 A 6β 5 cmβ B β70 cmβ C β68 cmβ D β60 cmβ E β75 cmβ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
229
Extended-response questions 3 ββ _ 1 β+ 4β_ b β5 Γ· 3β_ β5 β 3 8 12 3 ββ _ 1 βΓ· β 3β_ d β3_ β5 β+ 6β_ β3 β β 7 4 ( 8 4)
U N SA C O M R PL R E EC PA T E G D ES
Evaluate each of the following. 3 βΓ 2β_ 1β a 3β _ β1 β+ 1β_ 4 4 2 3β 1 βΓ· 2β_ 4 βΓ 3β_ c β7_ β2 β+ 2β_ 4 10 5 5
Chapter review
1
β 5 βof the 2 The length of one side of a triangle is _ β 5 βof the perimeter and a second side has length _ 12 28 perimeter. a If these two sides have a total length of β75βcentimetres, determine the triangleβs perimeter. b Find the lengths of each of the three sides.
3 a A sale on digital cameras offers β20%βdiscount. Determine the sale price of a camera that was originally priced at β$220β. b The sale price of a DVD is β$18β. This is β25%βless than the original marked price. Determine the original price of this DVD. 4 Perform the following calculations. a Increase β$440βby β25%β. (Hint: first find 25% of $440.) b Decrease β300βlitres by β12%β. c Increase β$100βby β10%βand then decrease that amount by β10%β. Explain the reason for the answer. d When β$Aβis increased by β20%β, the result is β$300β. Calculate the result if β$Aβis decreased by β20%β. 5 When a Ripstick is sold for β$200β, the shop makes β25%βprofit on the price paid for it. a If this β$200βRipstick is now sold at a discount of β10%β, what is the percentage profit of the price at which the shop bought the Ripstick? b At what price should the Ripstick be sold to make β30%β profit? 6 At Sunshine School there are β640βprimary school students and β860βsecondary students.
For their Christmas family holiday, β70%βof primary school students go to the beach and β45%β of secondary students go to the beach. Determine the overall percentage of students in the whole school that has a beach holiday for Christmas. Write this percentage as a mixed numeral.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
U N SA C O M R PL R E EC PA T E G D ES
4 Algebraic techniques
Maths in context: Apps for agriculture
Algebra is important in many occupations, including for gaming and app developers. Australian farmers now have some very useful apps to help improve food production, efο¬ciency, and proο¬ts. This technology has been developed by the scientists and programmers at CSIRO (Commonwealth Scientiο¬c and Industrial Research Organisation).
Soilmapp is an app based on CSIROβs soil data base. Australian soils are critical to our food production and farmers need to know how well their soil holds water, and its clay content and acidity.
Yield Prophet is an app that gives grain farmers a probability for higher or lower yields based on virtual choices of nitrogen fertilisers, irrigation and grain crop varieties. Using this app, farmers can adjust their plans and save large amounts of money.
Wheatcast is a website that uses technology to forecast the harvest volumes for Australiaβs 30 000 grain farmers. By relating current BOM weather data to CSIRO soil moisture maps, fortnightly forecasts of harvest volumes are provided for most locations.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter contents
U N SA C O M R PL R E EC PA T E G D ES
4A Introduction to algebra 4B Substituting numbers into algebraic expressions 4C Equivalent algebraic expressions 4D Like terms 4E Multiplying and dividing expressions 4F Expanding brackets (EXTENDING) 4G Applying algebra
Victorian Curriculum 2.0
This chapter covers the following content descriptors in the Victorian Curriculum 2.0: ALGEBRA
VC2M7A01, VC2M7A02, VC2M7A06
Please refer to the curriculum support documentation in the teacher resources for a full and comprehensive mapping of this chapter to the related curriculum content descriptors.
Β© VCAA
Online resources
A host of additional online resources are included as part of your Interactive Textbook, including HOTmaths content, video demonstrations of all worked examples, auto-marked quizzes and much more.
The EweWatch app keeps track of the mother sheep (the ewe) before, during and after giving birth to a lamb. The ewe wears a device to track her movements. This information helps farmers to locate any ewes that have birthing problems and need help.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 4
Algebraic techniques
4A Introduction to algebra LEARNING INTENTIONS β’ To know the basic terminology of algebra β’ To be able to identify coefο¬cients, terms and constant terms within expressions β’ To be able to write expressions from word descriptions
U N SA C O M R PL R E EC PA T E G D ES
232
A pronumeral is a letter that can represent a number. The choice of letter used is not significant mathematically, but can be used as an aid to memory. For instance, s might stand for the number of goals scored and w kg might stand for someoneβs weight. Remember that each pronumeral stands in place of a number (e.g. the number of goals, the number of kilograms).
The table shows the salary Petra earns for various hours of work if she is paid $12 an hour. Numbers of hours
Salary earned ($)
1
12 Γ 1 = 12
2
12 Γ 2 = 24
3
12 Γ 3 = 36
n
12 Γ n = 12n
Using pronumerals we can work out a total salary for any number of hours of work.
Rather than writing 12 Γ n, we write 12n because multiplying a pronumeral by a number is common and 18 . this notation saves space. We can also write 18 Γ· n as _ n
Lesson starter: Pronumeral stories
Ahmed has a jar with b biscuits. He eats 3 biscuits and then shares the rest equally among 8 friends. Each b β 3 biscuits. This is a short story for the expression _ b β 3 . (Note that b does not stand friend receives _ 8 8 for βbiscuitsβ. It stands in place of βthe number of biscuitsβ.) β’ β’
b β 3 , and share it with others in the class. Try to create another story for _ 8 Can you construct a story for 2t + 12? What about 4(k + 6)?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
4A Introduction to algebra
233
KEY IDEAS β In algebra, letters can be used to stand for numbers. A pronumeral is a letter that stands for a
number. If a pronumeral could represent any number rather than just a single number, it is also called a variable.
U N SA C O M R PL R E EC PA T E G D ES
β βa Γ bβis written as βabβand βa Γ· bβis written as _ βa ββ.
b
β A term consists of numbers and variables/pronumerals combined with multiplication or
4xyz division. For example, β5βis a term, βxβis a term, β9aβis a term, βabcβis a term, _ β βis a term. 3
β A term that does not contain any variables/pronumerals is called a constant term. All numbers
by themselves are constant terms.
β An (algebraic) expression consists of numbers and variables/pronumerals combined with any
mathematical operations. For example, β3x + 2yzβis an expression and β8 Γ· (β 3a β 2b)β+ 41βis also an expression.
β A coefficient is the number in front of a variables/pronumerals. For example, the coefficient of β
yβin the expression β8x + 2y + zβis β2β. If there is no number in front, then the coefficient is β1β, since β 1zβand βzβare always equal, regardless of the value of βzβ.
BUILDING UNDERSTANDING
1 The expression β4x + 3y + 24z + 7βhas four terms. a State all the terms. b What is the constant term? c What is the coefficient of βxβ? d Which letter has a coefficient of β24β?
2 Match each of the word descriptions on the left with the correct mathematical expression on the right. a the sum of βxβand β4β b β4βless than βxβ
c d e f
the product of β4βand βxβ one-quarter of βxβ
the result from subtracting βxβfrom β4β β4βdivided by βxβ
A B C D
xβ β 4β _ βx β 4 β4 β xβ β4xβ E _ βx4 β F βx + 4β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 4
Algebraic techniques
Example 1
Using the terminology of algebra
a State the individual terms in the expression β3a + b + 13cβ. b State the constant term in the expression β4a + 3b + 5β. c State the coefficient of each pronumeral in the expression β3a + b + 13cβ.
U N SA C O M R PL R E EC PA T E G D ES
234
SOLUTION
EXPLANATION
a There are three terms: β3a, bβand β13cβ.
Each part of an expression is a term. Terms are added (or subtracted) to make an expression.
b β5β
The constant term is β5βbecause it has no pronumerals (whereas β4aβand β3bβboth have pronumerals).
c The coefficient of βaβis β3β, the coefficient of βbβ is β1βand the coefficient of βcβis β13β.
The coefficient is the number in front of a pronumeral. For βbβ, the coefficient is β1β because βbβis the same as β1 Γ bβ.
Now you try
a List the individual terms in the expression β4q + 10r + s + 2tβ. b State the constant term in the expression β4q + 7r + 6β c State the coefficient of each pronumeral in β4q + 10r + s + 2tβ.
Example 2
Writing expressions from word descriptions
Write an expression for each of the following. a 5β βmore than βkβ b 3β βless than βmβ d double the value of βxβ e the product of βcβand βdβ
c the sum of βaβand βbβ
SOLUTION
EXPLANATION
a kβ + 5β
β5βmust be added to βkβto get β5βmore than βkβ.
b βm β 3β
3β βis subtracted from βmβ.
c βa + bβ
βaβand βbβare added to obtain their sum.
d β2 Γ xβor β2xβ
βxβis multiplied by β2β. The multiplication sign is optional.
e βc Γ dβor βcdβ
βcβand βdβare multiplied to obtain their product.
Now you try
Write an expression for each of the following. a 1β 0βmore than βpβ b the product of βaβand βbβ d half the value of βzβ e β4βless than βbβ
c triple the value of βkβ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
4A Introduction to algebra
Example 3
235
Writing expressions involving more than one operation
U N SA C O M R PL R E EC PA T E G D ES
Write an expression for each of the following without using the βΓβor βΓ·β symbols. a βpβis halved, then β4βis added. b The sum of βxβand βyβis taken and then divided by β7.β c xβ βis added to one-seventh of βy.β d β5βis subtracted from βkβand the result is tripled. SOLUTION p a β_β+ 4β 2
EXPLANATION
x+y β b β(x + y)βΓ· 7 = _ β 7
βxβand βyβare added. This whole expression is divided by β7β. By writing the result as a fraction, the brackets are no longer needed.
y β1 yβ β c xβ + β_βor βx + _ 7 7
y βxβis added to one-seventh of βyβ, which is _ β ββ. 7 β5βsubtracted from βkβgives the expression βk β 5β. Brackets must be used to multiply the whole expression by β3β.
d β(k β 5)βΓ 3 = 3β(k β 5)β
βpβis divided by β2β, then β4βis added.
Now you try
Write an expression for each of the following without using the βΓβor βΓ·β symbols. a βmβis tripled and then β5βis added. b The sum of βaβand βbβis halved. c 3β βis subtracted from the product of βpβand βq.β d β5βless than βmβis multiplied by β7.β
Exercise 4A FLUENCY
Example 1a
Example 1b
Example 1c
1
1β2ββ(1β/2β)β, 3, 4β 5ββ(1β/2β)β
3β6ββ(1β/2β)β
List the individual terms in the following expressions. a β3x + 2yβ b β4a + 2b + cβ c β5a + 3b + 2β
2 State the constant term in the following expressions. a β7a + 5β b β2x + 3y + 6β c β8 + 3x + 2yβ
4β7β(β 1β/2β)β
d β2x + 5β
d β5a + 1 + 3bβ
3 a State the coefficient of each pronumeral in the expression β2a + 3b + cβ. b State the coefficient of each pronumeral in the expression β4a + b + 6c + 2dβ.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
236
Chapter 4
Example 1
Algebraic techniques
4 For each of the following expressions, state: i the number of terms ii the coefficient of βnβ. a 1β 7n + 24β
d β15a β 32b + 2n + _ β4 βxyβ 3 d _ f β5bn β 12 + β β+ 12nβ 5
U N SA C O M R PL R E EC PA T E G D ES
c β15nw + 21n + 15β
b β31 β 27a + 15nβ
e βn + 51β
Example 2
Example 3
5 Write an expression for each of the following without using the βΓβor βΓ·β symbols. a 1β βmore than βxβ b the sum of βkβand β5β c double the value of βuβ d β4βlots of βyβ e half of βpβ f one-third of βqβ g β12βless than βrβ h the product of βnβand β9β i βtβis subtracted from β10β j βyβis divided by β8β 6 Write an expression for each of the following without using the βΓβor βΓ·β symbols. a 5β βis added to βxβ, then the result is doubled. b βaβis tripled, then β4βis added. c βkβis multiplied by β8β, then β3βis subtracted. d β3βis subtracted from βkβ, then the result is multiplied by β8β. e The sum of βxβand βyβis multiplied by β6β. f βxβis multiplied by β7βand the result is halved. g βpβis halved and then β2βis added. h The product of βxβand βyβis subtracted from β12β. 7 Describe each of these expressions in words. a 7β xβ b aβ + bβ PROBLEM-SOLVING
c (β x + 4)βΓ 2β 8, 9
d β5 β 3aβ
9β11
8β12
8 Nicholas buys β10βbags of lollies from a supermarket. a If there are β7βlollies in each bag, how many lollies does he buy in total? b If there are βnβlollies in each bag, how many lollies does he buy in total? (Hint: Write an expression involving βnβ.)
9 Mikayla is paid β$xβper hour at her job. Write an expression for each of the following amounts (in β$β). a How much does Mikayla earn if she works β8β hours? b If Mikayla gets a pay rise of β$3βper hour, what is her new hourly wage? c If Mikayla works for β8βhours at the increased hourly rate, how much does she earn?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
4A Introduction to algebra
237
10 Recall that there are β100βcentimetres in β1βmetre and β1000βmetres in β1βkilometre. Write expressions for each of the following. a How many metres are there in βxβ km? b How many centimetres are there in βxβ metres? c How many centimetres are there in βxβ km?
U N SA C O M R PL R E EC PA T E G D ES
11 A group of people go out to a restaurant, and the total amount they must pay is β$Aβ. They decide to split the bill equally. Write expressions to answer the following questions. a If there are β4βpeople in the group, how much do they each pay? b If there are βnβpeople in the group, how much do they each pay? c One of the βnβpeople has a voucher that reduces the total bill by β$20β. How much does each person pay now?
12 There are many different ways of describing the expression _ βa + b βin words. One way is: 4 βThe sum of βaβand βbβis divided by β4β.β What is another way? REASONING
13
13, 15
14, 15
13 If βxβis a whole number between β10βand β99β, classify each of these statements as true or false. a βxβmust be smaller than β2 Γ xβ. b βxβmust be smaller than βx + 2β. c xβ β 3βmust be greater than β10β. d β4 Γ xβmust be an even number. e 3β Γ xβmust be an odd number. 14 If βbβis an even number greater than β3β, classify each of these statements as true or false. a βb + 1βmust be even. b βb + 2βcould be odd. c 5β + bβcould be greater than β10β. d β5bβmust be greater than βbβ.
15 If βcβis a number between β10βand β99β, sort the following in ascending order (i.e. smallest to largest):β 3c, 2c, c β 4, c Γ· 2, 3c + 5, 4c β 2, c + 1, c Γ cβ. ENRICHMENT: Many words compressed
β
β
16
16 One advantage of writing expressions in symbols rather than words is that it takes up less space. For instance, βtwice the value of the sum of βxβand β5ββ uses eight words and can be written as β2(β x + 5)ββ. Give an example of a worded expression that uses more than β10βwords and then write it as a mathematical expression.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 4
Algebraic techniques
4B Substituting numbers into algebraic expressions LEARNING INTENTIONS β’ To understand that variables/pronumerals can be replaced with numbers β’ To be able to substitute numbers for variables/pronumerals β’ To be able to evaluate an expression using order of operations once all variable/pronumeral values are known
U N SA C O M R PL R E EC PA T E G D ES
238
Evaluation of expressions involves replacing pronumerals (like x and y) with numbers and obtaining a single number as a result. For example, we can evaluate 4 + x when x is 11, to get 15.
Lesson starter: Sum to 10
The pronumerals x and y could stand for any number. β’ β’
What numbers could x and y stand for if you know that x + y must equal 10? Try to list as many pairs as possible. If x + y must equal 10, what values could 3x + y equal? Find the largest and smallest values.
KEY IDEAS
β To evaluate an expression or to substitute values means to replace each variable/pronumeral
in an expression with a number to obtain a final value. For example, if x = 3 and y = 8, then x + 2y evaluated gives 3 + 2 Γ 8 = 19.
β A term like 4a means 4 Γ a. When substituting a number we must consider the multiplication
operation, since two numbers written as 42 is very different from the product 4 Γ 2.
β Once an expression contains no pronumerals, evaluate using the normal order of operations
seen in Chapter 1: β’ brackets β’ multiplication and division from left to right β’ addition and subtraction from left to right.
For example: (4 + 3) Γ 2 β 20 Γ· 4 + 2 = 7 Γ 2 β 20 Γ· 4 + 2 = 14 β 5 + 2 = 9+2 = 11
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
4B Substituting numbers into algebraic expressions
239
BUILDING UNDERSTANDING 1 Use the correct order of operations to
U N SA C O M R PL R E EC PA T E G D ES
evaluate the following. a 4+2Γ5 b 7β3Γ2 c 3Γ6β2Γ4 d (7 β 3) Γ 2
2 What number would you get if you
replaced b with 5 in the expression 8 + b?
3 What number is obtained when x = 3 is substituted into the expression 5 Γ x?
4 What is the result of evaluating 10 β u if u is 7?
Engineers, welders and metal workers regularly substitute values into formulas. A circular steel plate, used when constructing a large airduct, can have an area of: A = Οr 2 = Ο Γ 0.72 = 1.5m2 (using one decimal place).
5 Calculate the value of 12 + b if: a b=5 b b=8 c b = 60 d b=0
Example 4
Substituting a variable/ pronumeral
Given that t = 5, evaluate: a t+7
b 8t
10 + 4 β t c _ t
SOLUTION
EXPLANATION
a t+7 = 5+7 = 12
Replace t with 5 and then evaluate the expression, which now contains no pronumerals.
b 8t = 8 Γ t = 8Γ5 = 40
Insert Γ where it was previously implied, then substitute 5 for t. If the multiplication sign is not included, we might get a completely incorrect answer of 85.
c _ 10 + 4 β t = _ 10 + 4 β 5 t 5 = 2+4β5 = 1
Replace all occurrences of t with 5 before evaluating. Note that the division (10 Γ· 5) is calculated before the addition and subtraction.
Now you try
Given that u = 3, evaluate: a 4u
b 8βu
12 β 3 c 2u + _ u
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 4
Algebraic techniques
Example 5
Substituting multiple variables/pronumerals
Substitute βx = 4βand βy = 7βto evaluate these expressions. a 5β x + y + 8β b β80 β β(2xy + y)β
U N SA C O M R PL R E EC PA T E G D ES
240
SOLUTION
EXPLANATION
a 5x + y + 8 = 5 Γ x + y + 8 ββ =β 5 Γ 4 + 7 +β 8ββ β β―β― β―β― β―β― β = 20 + 7 + 8 β = 35
Insert the implied multiplication sign between β5β and βxβbefore substituting the values for βxβand βyβ.
b 80 β β(2xy + y)β = 80 β β(2 Γ x Γ y + y)β β = 80 β (β 2 Γ 4 Γ 7 + 7)β ββ β―β― β―β― βββ―β― β =β 80 β β(56β + 7)ββ β = 80 β 63 β = 17
Insert the multiplication signs, and remember the order in which to evaluate. Note that both occurrences of βyβare replaced with β7β.
Now you try
Substitute βp = 3βand βq = 10βto evaluate these expressions. a 4β p β q + 2β b βpq + 3 Γ β(q β 2p)β
Example 6
Substituting with powers and roots
If βp = 4βand βt = 5β, find the value of: a 3β pβ β β2β b tβββ2β+ pβ β β3β SOLUTION a
EXPLANATION
3pβ β β2β = 3 Γ p Γ p β
_
β β2β+ 3β β 2β β c ββ pβ
ββ =β 3 Γ 4 Γ 4β β = 48
Note that β3pβ β β2βmeans β3 Γ p Γ pβnot (ββ3 Γ p)2β ββ.
b tβββ2β+ pβ β β3β = 5β β 2β β+ 4β β 3β β βββ―β― =β 5 Γ 5 +β β4 Γ 4 Γ 4β β β = 25 + 64 β = 89
βtβis replaced with β5β, and βpβis replaced with β4β. Remember that 4β β 3β βmeans β4 Γ 4 Γ 4β.
β β2β+ 3β β 2β β = β β 4β β 2β β+ 3β β 2β β c β β pβ _ ββ =β β β β β 25 ββ
Recall that the square root of β25βis β5βbecause β5 Γ 5 = 25β.
_
_
β= 5
Now you try
If βx = 3βand βy = 8β, find the value of: a 4β xβ β β2β b xβ β β3β+ yβ β β2β
_
β 2β ββ yβ β β2β c ββ10β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
4B Substituting numbers into algebraic expressions
241
Exercise 4B FLUENCY
1β7ββ(1β/2β)β
4β8β(β 1β/2β)β
Given βt = 5β, evaluate: a βt + 2β
b βt β 1β
c βt Γ 2β
d βt Γ 3β
2 Given βx = 7β, evaluate: a βx β 3β
b βx + 5β
c β2xβ
d β10xβ
3 If βk = 4β, evaluate: a β3k + 1β
b β2k β 3β
c β10 β kβ
d β12 β 2kβ
1
U N SA C O M R PL R E EC PA T E G D ES
Example 4a,b
1β3ββ(1β/2β)β,β 6β7ββ(1β/2β)β
Example 4a,b
Example 4
Example 4
4 If βx = 5β, evaluate each of the following. a β2β(x + 2)β+ xβ c _ β20 x β+ 3β e _ βx + 7 β 4 g β7x + 3β(x β 1)β
f
βx + xβ(x + 1)β
j
k β100 β 4β(3 + 4x)β
l
i
Example 5
Example 5
b β30 β β(4x + 1)β β10 d (β x + 5)βΓ _ xβ 10 β x β β_ x
h β40 β 3x β xβ
30 β+ 2xβ(x + 3)β β_ x (3x β 8)β 6β β_ β x+2
5 Substitute βa = 2βand βb = 3βinto each of these expressions and evaluate. a βa + bβ b β3a + bβ c 5β a β 2bβ
d β7ab + bβ
e aβ b β 4 + bβ
f
g _ βa8 ββ _ β12 β b
6 _ h _ β12 a β+ βb β
6 Evaluate the expression β5x + 2yβ when: a βx = 3βand βy = 6β c xβ = 7βand βy = 3β e xβ = 2βand βy = 0β
7 Copy and complete each of these tables. a βnβ β1β β2β 3β β β4β β5β β6β nβ + 4β
c
5β β
βbβ
β2(β b β π)β
b xβ = 4βand βy = 1β d βx = 0βand βy = 4β f βx = 10βand βy = 10β b
β2β
β3β
βxβ
β1β
β2β
β12 β xβ
β8β
1β β
β2 Γ β(3a + 2b)β
4β β
β5β
β6β
d
βqβ
β3β
β4β
5β β
β6β
β4β
β5β
β6β
β9β
β1β
β2β
3β β
1β 0q β qβ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
242
Chapter 4
Example 6
Algebraic techniques
8 Evaluate each of the following, given βa = 9, b = 3βand βc = 5β. a 3β cβ β β2β
b β5bβ β β2β d β2bβ β β2β+ _ βa ββ 2cβ 3 _ 2 β β β+ 4β β 2β β f ββbβ
c aβ β β2ββ 3β β 3β β
_
_
e ββa β+ β β 3ab β β β3ββ g β24 + _ β2bβ 6
U N SA C O M R PL R E EC PA T E G D ES
β β2β h (ββ2c)2β ββ aβ
PROBLEM-SOLVING
9
9, 11
10, 11
9 A number is substituted for βbβin the expression β7 + bβand gives the result β12β. What is the value of βbβ?
10 A whole number is substituted for βxβin the expression β3x β 1β. If the result is a two-digit number, what value might βxβhave? Try to describe all the possible answers. 11 Copy and complete the table. βxβ
β5β
βx + 6β
β11β
β4xβ
β20β
9β β
β12β
7β β
2β 4β
β28β
REASONING
12
12, 13
13, 14
12 Assume βxβand βyβare two numbers, where βxy = 24β. a What values could βxβand βyβequal if they are whole numbers? Try to list as many as possible. b What values could βxβand βyβequal if they can be decimals, fractions or whole numbers?
13 Dugald substitutes different whole numbers into the expression β5 Γ (β a + a)β. He notices that the result always ends in the digit β0β. Try a few values and explain why this pattern occurs.
14 Values of βxβand βyβare substituted into the expression βx + 2yβand the result is β100β. a Find a possible value for x and y to make βx + 2yβequal 100. b If the same values of βxβand βyβare substituted into the expression β3x + 6yβ, what is the result? c Explain why your answer to part b will always be the same, regardless of which pair you chose in part a. ENRICHMENT: Missing numbers
β
β
15
15 a Copy and complete the following table. Note that βxβand βyβare whole numbers (β 0, 1, 2, 3, β¦)βfor this table. βxβ
β5β
β10β
yβ β
β3β
β4β
βxyβ
β5β
β9β
xβ + yβ
βx β yβ
β7β
β2β
4β 0β
1β 4β
7β β 3β β
β8β
1β 0β
β0β
b If βxβand βyβare two numbers where βx + yβand βx Γ yβare equal, what values might βxβand βyβ have? Try to find at least three (they do not have to be whole numbers).
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
4C Equivalent algebraic expressions
243
4C Equivalent algebraic expressions
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To know what it means for two expressions to be equivalent β’ To be able to determine whether two expressions are equivalent using substitution β’ To be able to generalise number facts using algebra
Like using words from a language, there are often many ways in algebra to express the same thing. For example, we can write βthe sum of x and 4β as x + 4 or 4 + x, or even x + 1 + 1 + 1 + 1. No matter what number x is, x + 4 and 4 + x will always be equal. We say that the expressions x + 4 and 4 + x are equivalent because of this.
By substituting different numbers for the pronumerals it is possible to see whether two expressions are equivalent. Consider the four expressions in this table.
a=0
When identical houses are built, a window supplier can multiply the number of windows, w, ordered for one house by the number of houses. This is equivalent to adding all the windows. For example, for four houses, 4w = w + w + w + w.
3a + 5
2a + 6
7a + 5 β 4a
a+a+6
5
6
5
6
a=1
8
8
8
8
a=2
11
10
11
10
a=3
14
12
14
12
a=4
17
14
17
14
From this table it becomes apparent that 3a + 5 and 7a + 5 β 4a are equivalent, and that 2a + 6 and a + a + 6 are equivalent.
Lesson starter: Equivalent expressions Consider the expression 2a + 4. β’ β’ β’
Write as many different expressions as possible that are equivalent to 2a + 4. How many equivalent expressions are there? Try to give a logical explanation for why 2a + 4 is equivalent to 4 + a Γ 2.
This collection of pronumerals and numbers can be arranged into many different equivalent expressions.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 4
Algebraic techniques
KEY IDEAS β Two expressions are called equivalent when they are always equal, regardless of what numbers
are substituted for the pronumerals. For example: β’ βx + 12βis equivalent to β12 + xβ, because the order in which numbers are added is not important. β’ β3kβis equivalent to βk + k + kβ, because multiplying by a whole number is the same as adding repeatedly.
U N SA C O M R PL R E EC PA T E G D ES
244
β The rules of algebra are used to prove that two expressions are equivalent, but a table of values
can be helpful to test whether expressions are likely to be equivalent.
BUILDING UNDERSTANDING
1 a State the missing numbers in this table. b State the missing word: β2x + 2β and
βx = 1β
βx = 2β
βx = 3β
βx = 0β
βx = 1β
βx = 2β
βx = 3β
β(x + 1) Γ 2β
(β x + 1)βΓ 2βare __________ expressions.
2 a State the missing numbers in this table. b Are β5x + 3βand β6x + 3β equivalent
5β x + 3β
expressions?
Example 7
βx = 0β
β2x + 2β
6β x + 3β
Identifying equivalent expressions
Which two of these expressions are equivalent: β3x + 4, 8 β x, 2x + 4 + xβ? SOLUTION
EXPLANATION
3β x + 4βand β2x + 4 + xβ are equivalent.
By drawing a table of values, we can see straight away that β3x + 4β and β8 β xβare not equivalent, since they differ for βx = 2β. βx = 1β
βx = 2β
βx = 3β
β3x + 4β
7β β
β10β
β13β
β8 β xβ
7β β
β 6β
β 5β
β2x + 4 + xβ
7β β
β10β
β13β
β3x + 4βand β2x + 4 + xβare equal for all values, so they are equivalent.
Now you try
Which two of these expressions are equivalent: β2a + 6, 6a + 2, (β βa + 3β)βΓ 2, 8aβ?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
4C Equivalent algebraic expressions
Example 8
245
Generalising number facts with algebra
U N SA C O M R PL R E EC PA T E G D ES
Generalise the following number facts to find two equivalent expressions. a 5β + 5 = 2 Γ 5, 7 + 7 = 2 Γ 7, 10 + 10 = 2 Γ 10β(Use βxβto stand for the number.) b β2 + 7 = 7 + 2, 4 + 5 = 5 + 4, 12 + 14 = 14 + 12β(Use βaβand βbβto stand for the numbers.) SOLUTION
EXPLANATION
a xβ + x = 2xβ
Each of the statements is in the form β+β β= 2 Γβ , where the value of is the same. Use βxβto stand for the number in , and recall that β2 Γ xβis written β2xβ.
b βa + b = b + aβ
Here there are two numbers which vary, but the pattern is always β a + b = b + aβ. For example, if βa = 2βand βb = 7,βwe get the fact β2 + 7 = 7 + 2β.
Now you try
Generalise the following number facts to find two equivalent expressions. a 1β 0 Γ 5 Γ 10 = 5 Γ 100, 10 Γ 7 Γ 10 = 7 Γ 100, 10 Γ 3 Γ 10 = 3 Γ 100β(Use βxβto stand for the number.) b β4 Γ 7 = 7 Γ 4, 3 Γ 9 = 9 Γ 3, 12 Γ 4 = 4 Γ 12β(Use βaβand βbβto stand for the numbers.)
Exercise 4C FLUENCY
Example 7
Example 7
Example 8a
1
1, 2ββ(β1/2β)β, 3, 5
2β5β(β β1/2β)β,β 6
3β5β(β β1/2β)β,β 6
Decide if the following pairs of expression are equivalent (E) or not equivalent (N). a β3x, 2x + 4β b β5 + a, a + 5β c 2β d, d + dβ
2 Pick the two expressions that are equivalent. a β2x + 3, 2x β 3, x + 3 + xβ c 4β x, 2x + 4, x + 4 + xβ e 2β k + 2, 3 + 2k, 2β(k + 1)β
b 5β x β 2, 3x β 2 + 2x, 2x + 3x + 2β d β5a, 4a + a, 3 + aβ f βb + b, 3b, 4b β 2bβ
3 Generalise the following number facts to find two equivalent expressions. In each case, use βxβto stand for the number. a β4 + 4 + 4 = 3 Γ 4, β7 + 7 + 7 = 3 Γ 7, β10 + 10 + 10 = 3 Γ 10β b β2 Γ 7 Γ 2 = 4 Γ 7, β β β 2 Γ 12 Γ 2 = 4 Γ 12, β β β 2 Γ 15 Γ 2 = 4 Γ 15β c β5 + 99 = 5 β 1 + 100, β7 + 99 = 7 β 1 + 100, 12 + 99 = 12 β 1 + 100β d β2 Γ (10 + 1) = 20 + 2 Γ 1, β2 Γ (10 + 4) = 20 + 2 Γ 4, β2 Γ (10 + 7) = 20 + 2 Γ 7β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
246
Chapter 4
Example 8b
Algebraic techniques
4 Generalise the following number facts to find two equivalent expressions. In each case, use βaβand βbβ to stand for the numbers. a Fact 1: 10 Γ (1 + 4) = 10 Γ 1 + 10 Γ 4 Fact 2: 10 Γ (3 + 9) = 10 Γ 3 + 10 Γ 9β β Fact 3: 10 Γ (7 + 2) = 10 Γ 7 + 10 Γ 2ββ β
U N SA C O M R PL R E EC PA T E G D ES
b Fact 1: 5 + 100 + 3 = 3 + 100 + 5 β β―β―β― Fact 2: 9 + 100 + 8 = 8 + 100 + 9β β Fact 3: 17 + 100 + 2 = 2 + 100 + 17 β 2β β c Fact 1: (3 + 1) (3 β 1) = 3β β 2β ββ 1β β β―β―β― Fact 2: (5 + 2) (5 β 2) = 5β β 2β ββ 2β β β2β Fact 3: (9 + 4) (9 β 4) = 9β β 2β ββ 4β β 2β β
d Fact 1: 3 Γ 4 + 4 Γ 3 = 2 Γ 3 Γ 4 β β―β―β― Fact 2: 7 Γ 8 + 8 Γ 7 = 2 Γ 7 Γ 8β β Fact 3: 12 Γ 5 + 5 Γ 12 = 2 Γ 12 Γ 5
5 Match up the equivalent expressions below. a 3β x + 2xβ
A β6 β 3xβ
b 4β β 3x + 2β
B β2x + 4x + xβ
c β2x + 5 + xβ
C β5xβ
d xβ + x β 5 + xβ
D β4 β xβ
e β7xβ
E β3x + 5β
f
β4 β 3x + 2xβ
F β3x β 5β
6 Demonstrate that β6x + 5βand β4x + 5 + 2xβare likely to be equivalent by completing the table. β6x + 5β β4x + 5 + 2xβ
xβ = 1β βx = 2β βx = 3β βx = 4β
PROBLEM-SOLVING
7
7, 8
8, 9
7 Write two different expressions that are equivalent to β4x + 2β.
w
8 The rectangle shown below has a perimeter given by βw + l + w + lβ. Write an equivalent expression for the perimeter.
l
9 There are many expressions that are equivalent to β3a + 5b + 2a β b + 4aβ. Write an equivalent expression with as few terms as possible. REASONING
10
11, 12
l
w
11β13
10 Prove that no pair of these four expressions is equivalent: β4 + x, 4x, x β 4, x Γ· 4β. (Hint: substitute a value for βxβ, like βx = 8β.)
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
4C Equivalent algebraic expressions
U N SA C O M R PL R E EC PA T E G D ES
11 Generalise each of the following patterns in numbers to give two equivalent expressions. In each case, choose pronumerals starting at βaβ (as many as required to handle the observation provided). The first one has been done for you. a Observation: 3 + 5 = 5 + 3 and 2 + 7 = 7 + 2 and 4 + 11 = 11 + 4. Generalised: The two expressions a + b and b + a are equivalent. b Observation: 2 Γ 5 = 5 Γ 2 and 11 Γ 5 = 5 Γ 11 and 3 Γ 12 = 12 Γ 3. c Observation: 4 Γ (10 + 3) = 4 Γ 10 + 4 Γ 3 and 8 Γ (100 + 5) = 8 Γ 100 + 8 Γ 5. d Observation: 100 β (4 + 6) = 100 β 4 β 6 and 70 β (10 + 5) = 70 β 10 β 5. e Observation: 20 β (4 β 2) = 20 β 4 + 2 and 15 β (10 β 3) = 15 β 10 + 3. f Observation: 100 Γ· 5 Γ· 10 = 100 Γ· (5 Γ 10) and 30 Γ· 2 Γ· 3 = 30 Γ· (2 Γ 3).
247
12 From two equivalent expressions, you can substitute values for the pronumerals to get particular number facts. For example, because x + 4 and 4 + x are equivalent, we can get number facts like 2 + 4 = 4 + 2 and 103 + 4 = 4 + 103 (and others). Use the following pairs of equivalent expressions to give three particular number facts. a x + 12 and 12 + x b 2(q + 1) and 2 + 2q c (a + b)2 and a 2 + 2ab + b2 13 a Show that the expression 4 Γ (a + 2) is equivalent to 8 + 4a, using a table of values for a between 1 and 4. b Write an expression using brackets that is equivalent to 10 + 5a. c Write an expression without brackets that is equivalent to 6 Γ (4 + a). ENRICHMENT: Thinking about equivalence
β
β
14, 15
14 3a + 5b is an expression containing two terms. List two expressions containing three terms that are equivalent to 3a + 5b.
15 Three expressions are given: expression A, expression B and expression C. a If expressions A and B are equivalent, and expressions B and C are equivalent, does this mean that expressions A and C are equivalent? Try to prove your answer. b If expressions A and B are not equivalent, and expressions B and C are not equivalent, does this mean that expressions A and C are not equivalent? Try to prove your answer.
Each shape above is made from three identically-sized tiles of width, w, and length, l. They all have the same area. Do any of the shapes have the same perimeter?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 4
Algebraic techniques
4D Like terms LEARNING INTENTIONS β’ To know what like terms are β’ To be able to identify like terms within an expression β’ To be able to simplify expressions by combining like terms
U N SA C O M R PL R E EC PA T E G D ES
248
Whenever we have terms with exactly the same pronumeral parts, they are called βlike termsβ and can be collected. For example, 3x + 5x can be simplified to 8x. If the two terms do not have exactly the same pronumerals, they must be kept separate; for example, 3x + 5y cannot be simplified β it must be left as it is. Recall from arithmetic that numbers can be multiplied in any order (e.g. 5 Γ 3 = 3 Γ 5). This means pronumerals can appear in a different order within a term and give equivalent expressions (e.g. ab and ba are equivalent).
A vet has 3 rows of animal cages with 4 cages per row. Each cage is L cm by H cm. The total length of the cages in centimetres is L + L + L + L = 4L and total height in centimetres is H + H + H = 3H. If another row is added, the height is 3H + H = 4H.
Lesson starter: Simplifying expressions β’
Try to find a simpler expression that is equivalent to
1a + 2b + 3a + 4b + 5a + 6b + β¦+19a + 20b
β’ β’
What is the longest possible expression that is equivalent to 10a + 20b + 30c? Assume that all coefficients must be whole numbers greater than zero. Compare your expressions to see who has the longest one.
KEY IDEAS
β Like terms are terms containing exactly the same pronumerals, although not necessarily in the
same order. Like
Not like
3x and 5x
3x and 5y
β12a and 7a
11d and 4c
5ab and 6ba
β8ab and 5a
4x 2 and 3x 2
x 2 and x
β Like terms can be combined within an expression to create a simpler expression that is
equivalent. For example, 5ab + 3ab can be simplified to 8ab. β If two terms are not like terms (such as 4x and 5y), they can still be added to get an expression
like 4x + 5y, but this expression cannot be simplified further. Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
4D Like terms
249
BUILDING UNDERSTANDING 1 For each of the following terms, state all the pronumerals that occur in it. a β4xyβ b β3abcβ c β2kβ
d βpqβ
U N SA C O M R PL R E EC PA T E G D ES
2 State the missing words or expressions to make the sentences true. More than one answer might be possible. a β3xβand β5xβare _______________ terms. b β4xβand β3yβare not ____________ ____________. c β4xyβand β4yxβare like ____________. d β4aβand ____________ are like terms. e βx + x + 7βand β2x + 7βare ____________ expressions. f β3x + 2x + 4βcan be written in an equivalent way as ____________.
Example 9
Identifying like terms
Classify the following pairs as like terms or not like terms. a 3β xβand β2xβ b 3β aβand β3bβ d β4kβand βkβ e β2aβand β4abβ
c 2β abβand β5baβ f β7abβand β9abaβ
SOLUTION
EXPLANATION
a 3β xβand β2xβare like terms.
The pronumerals are the same.
b β3aβand β3bβare not like terms.
The pronumerals are different.
c β2abβand β5baβare like terms.
The pronumerals are the same, even though they are written in a different order (one βaβand one βbβ).
d β4kβand βkβare like terms.
The pronumerals are the same.
e β2aβand β4abβare not like terms.
The pronumerals are not exactly the same (the first term contains only βaβand the second term has βaβand βbβ).
f
The pronumerals are not exactly the same (the first term contains one βaβand one βbβ, but the second term contains two copies of βaβand one βbβ).
β7abβand β9abaβare not like terms.
Now you try
Classify the following pairs as like terms or not like terms. a 4β xβand β3yβ b 7β aβand β9aβ d β4jβand βjβ e β8bcβand β9cbβ
c 3β xyβand β4xyxβ f β9pqβand β10pβ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 4
Algebraic techniques
Example 10 Simplifying using like terms Simplify the following by collecting like terms. a β7b + 2 + 3bβ b β12d β 4d + dβ c β5 + 12a + 4b β 2 β 3aβ d β13a + 8b + 2a β 5b β 4aβ e β12uv + 7v β 3vu + 3vβ
U N SA C O M R PL R E EC PA T E G D ES
250
SOLUTION
EXPLANATION
a β7b + 2 + 3b = 10b + 2β
7β bβand β3bβare like terms, so they are added. They cannot be combined with the term β2β because it is not βlikeβ β7bβor β3bβ.
b β12d β 4d + d = 9dβ
All the terms here are like terms. Remember that βdβmeans β1dβwhen combining them.
c
5 + 12a + 4b β 2 β 3a β―β― β 12a β 3a + 4b + 5 β 2β = = 9a + 4b + 3
1β 2aβand β3aβare like terms, so we first bring them to the front of the expression. We subtract β3aβbecause it has a minus sign in front of it. We can also combine the β5βand the β 2βbecause they are like terms.
d
13a + 8b + 2a β 5b β 4a β―β―β― β= 13a + 2a β 4a + 8b β 5bβ = 11a + 3b
First, rearrange to bring like terms together. Combine like terms, remembering to subtract any term that has a minus sign in front of it.
e
12uv + 7v β 3vu + 3v β= 12uv β 3vu + 7v + 3vβ β―β― = 9uv + 10v
Combine like terms. Remember that β12uvβ and β 3vuβare like terms (i.e. they have the same pronumerals), so β12uv β 3uv = 9uvβ.
Now you try
Simplify the following by collecting like terms. a β3a + 4 + 12aβ b β10q + 3q β 9qβ c β4x + 13y + 2x + 3 + 5yβ d β9a + 4b β 3a β 3b + aβ e β10uv + 3u β 2vu + 7u + 2vβ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
4D Like terms
251
Exercise 4D FLUENCY
1
2β5β(β 1β/2β)β
Classify the following pairs as like terms (L) or not like terms (N). a β6xβand β3xβ b β5aβand β7bβ c β3dβand β9dβ
2ββ(1β/2β)β,β 4β5ββ(1β/2β)β
d β2xβand β2yβ
U N SA C O M R PL R E EC PA T E G D ES
Example 9
1, 2β4ββ(1β/2β)β
Example 9
Example 10
Example 10
Example 10
2 Classify the following pairs as like terms (L) or not like terms (N). a β7aβand β4bβ b β3aβand β10aβ c β18xβand β32xβ e 7β βand β10bβ f βxβand β4xβ g β5xβand β5β i β7cdβand β12cdβ j β3abcβand β12abcβ k β3abβand β2baβ
d 4β aβand β4bβ h β12abβand β4abβ l β4cdβand β3dceβ
3 Simplify the following by collecting like terms. a βa + aβ b β3x + 2xβ e 1β 5u β 3uβ f β14ab β 2abβ
d 1β 2d β 4dβ h β4xy β 3xyβ
c 4β b + 3bβ g β8ab + 3abβ
4 Simplify the following by collecting like terms. a β2a + a + 4b + bβ b β5a + 2a + b + 8bβ d β4a + 2 + 3aβ e β7 + 2b + 5bβ g β7f + 4 β 2f + 8β h β4a β 4 + 5b + bβ j β10a + 3 + 4b β 2aβ k 4β + 10h β 3hβ m 1β 0 + 7y β 3x + 5x + 2yβ n β11a + 4 β 3a + 9β
c f i l o
5 Simplify the following by collecting like terms. a β7ab + 4 + 2abβ b β9xy + 2x β 3xy + 3xβ d β5uv + 12v + 4uv β 5vβ e β7pq + 2p + 4qp β qβ
c 2β cd + 5dc β 3d + 2cβ f β7ab + 32 β ab + 4β
PROBLEM-SOLVING
6, 7
β3x β 2x + 2y + 4yβ β3k β 2 + 3kβ β3x + 7x + 3y β 4x + yβ β10x + 4x + 31y β yβ β3b + 4b + c + 5b β cβ
8, 9ββ(β1/2β)β
7, 8, 9β(β β1/2β)β
6 Josh and Caitlin each work for βnβhours per week. Josh earns β$27βper hour and Caitlin earns β $31βper hour. a Write an expression for the amount Josh earns in one week (in dollars). b Write an expression for the amount Caitlin earns in one week (in dollars). c Write a simplified expression for the total amount Josh and Caitlin earn in one week (in dollars). 7 The length of the line segment shown could be expressed as βa + a + 3 + a + 1β. a
a
3
a
1
a Write the length in the simplest form. b What is the length of the segment if βaβis equal to β5β?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 4
Algebraic techniques
8 Let βxβrepresent the number of marbles in a standard-sized bag. Xavier bought β4β bags and Cameron bought β7βbags. Write simplified expressions for: a the number of marbles Xavier has b the number of marbles Cameron has c the total number of marbles that Xavier and Cameron have d the number of extra marbles that Cameron has compared to Xavier.
U N SA C O M R PL R E EC PA T E G D ES
252
9 Simplify the following expressions as much as possible. a 3β xy + 4xy + 5xyβ b β4ab + 5 + 2abβ d β10xy β 4yx + 3β e β10 β 3xy + 8xy + 4β g β4 + x + 4xy + 2xy + 5xβ h β12ab + 7 β 3ab + 2β REASONING
c β5ab + 3ba + 2abβ f β3cde + 5ecd + 2cedβ i β3xy β 2y + 4yxβ
10
10, 11
11, 12
10 a Demonstrate, using a table of values, that β3x + 2xβis equivalent to β5xβ. b Prove that β3x + 2yβis not equivalent to β5xyβ. 11 a Demonstrate that β5x + 4 β 2xβis equivalent to β3x + 4β. b Prove that β5x + 4 β 2xβis not equivalent to β7x + 4β. c Prove that β5x + 4 β 2xβis not equivalent to β7x β 4β.
12 a Generalise the following number facts, using βxβto stand for the variable number. Fact 1: 18 Γ 10 β 15 Γ 10 = 3 Γ 10 β β―β―β― β―β―β― Fact 2: 18 Γ 7 β 15 Γ 7 = 3 Γ 7β β Fact 3: 18 Γ 11 β 15 Γ 11 = 3 Γ 11
b Explain how to find β18 Γ 17 β 15 Γ 17βin the least number of calculations. You should also explain why your method will work. c State the value of β18 Γ 17 β 15 Γ 17β.
ENRICHMENT: How many rearrangements?
β
β
13
13 The expression βa + 3b + 2aβis equivalent to β3a + 3bβ. a List two other expressions with three terms that are equivalent to β3a + 3bβ. b How many expressions, consisting of exactly three terms added together, are equivalent to β3a + 3bβ? All coefficients must be whole numbers greater than β0β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Progress quiz
4A
For the expression β7a + 4b + c + 9β, answer the following. a State the number of terms. b List the individual terms. c State the coefficient of βbβ. d What is the constant term? b the sum of βaβand βkβ d β4βless than βwβ
U N SA C O M R PL R E EC PA T E G D ES
2 Write an expression for each of the following. a the product of βmβand βpβ c 8β βmore than βtβ
Progress quiz
4A
1
253
4A
4B
4B
4C
4C
3 Write an expression for each of the following without using the βΓ·βor βΓβ symbols. a βmβis halved, then β7βis added. b β7βis added to βmβand then the result is halved. c The sum of βaβand βkβis taken and then divided by β3β. d βaβis added to one-third of βkβ. e 1β 2βis subtracted from βdβand the result is tripled. f βdβis tripled and β12βis subtracted from the result. 4 If βx = 4β, evaluate each of the following. a β3x + 7β b _ β20 x β+ 2 β xβ
c β18 β (β 2x + 1)β
5 Substitute βa = 5βand βb = 2βinto each of these expressions and evaluate. a β3a + b + 7β
b β20 β β(a + 2b)β
d β5 + aβ β β2ββ 2bβ β β2β
e
_ β β2ββ 16 β ββaβ
35 β c β_ a+b
6 Which two of the following expressions are equivalent? β3a + 4, 4a + 3, 4 + 3a, 7aβ
7 Generalise the following number facts to find two equivalent expressions. a Use βxβto stand for the number. Fact 1: 2 + 10 = 10 + 2 β β―β― Fact 2: 5 + 10 = 10 + 5β β Fact 3: 11 + 10 = 10 + 11 b Use βaβand βbβto stand for the numbers. Fact 1: 2 Γ 3 + 2 Γ 5 = 2 Γ (3 + 5) β β―β―β― Fact 2: 2 Γ 7 + 2 Γ 9 = 2 Γ (7 + 9)ββ β Fact 3: 2 Γ 11 + 2 Γ 4 = 2 Γ (11 + 4)
4D
8 Classify the following pairs as like terms (L) or not like terms (N). a β3aβand β8aβ b β3xβand β3xyβ c β6βand β6aβ
4D
9 Simplify the following by collecting like terms. a β7a + 2b + 5 + a + 3bβ
4C
d β4mpβand β5pmβ
b β2cd + 4c + 8d + 5dc β c + 4β
10 Archie has two part-time jobs each paying β$8βper hour. He works βxβhours at one job and βyβ hours at the other. Write two equivalent expressions for the total amount of money, in dollars, that he earns.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 4
Algebraic techniques
4E Multiplying and dividing expressions LEARNING INTENTIONS β’ To know the different ways multiplication and division can be written β’ To be able to simplify expressions with multiplication β’ To be able to simplify expressions with division
U N SA C O M R PL R E EC PA T E G D ES
254
When multiplying a number by a pronumeral, we have already seen we can write them next to each other. For example, 7a means 7 Γ a, and 5abc means 5 Γ a Γ b Γ c. The order in which numbers or pronumerals are multiplied is unimportant, so 5 Γ a Γ b Γ c = a Γ 5 Γ c Γ b = c Γ a Γ 5 Γ b. When writing a product without Γ signs, the numbers are written first. 7xy We write _ as shorthand for (7xy) Γ· (3xz). 3xz 5Γ2=_ 10 by dividing by common factors such as _ 10 = _ 2. We can simplify fractions like _ 15 15 5 Γ 3 3 7xy 7xy 7y Similarly, common pronumerals can be cancelled in a division like _, giving _ = _. 3xz 3xz 3z
Lesson starter: Rearranging terms
5abc is equivalent to 5bac because the order of multiplication does not matter. In what other ways could 5abc be written?
5ΓaΓbΓc=?
KEY IDEAS
β a Γ b is written as ab.
a. β a Γ· b is written as _ b
β a Γ a is written as a 2.
β Because of the commutative property of multiplication (e.g. 2 Γ 7 = 7 Γ 2), the order in which
values are multiplied is not important. So 3 Γ a and a Γ 3 are equivalent.
β Because of the associative property of multiplication (e.g. 3 Γ (5 Γ 2) and (3 Γ 5) Γ 2
are equal), brackets are not required when only multiplication is used. So 3 Γ (a Γ b) and (3 Γ a) Γ b are both written as 3ab.
β Numbers should be written first in a term and pronumerals are generally written in alphabetical
order. For example, b Γ 2 Γ a is written as 2ab.
β When dividing, any common factor in the numerator and denominator can be cancelled. 24a 1b 2a For example: _ =_ c 1 21bc
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
4E Multiplying and dividing expressions
255
BUILDING UNDERSTANDING
U N SA C O M R PL R E EC PA T E G D ES
1 Chen claims that 7 Γ d is equivalent to d Γ 7. a If d = 3, find the values of 7 Γ d and d Γ 7. b If d = 5, find the values of 7 Γ d and d Γ 7. c If d = 8, find the values of 7 Γ d and d Γ 7. d Is Chen correct in his claim?
2 Classify each of the following statements as true or false. a 4 Γ n can be written as 4n. b n Γ 3 can be written as 3n. c 4 Γ b can be written as b + 4. d a Γ b can be written as ab. e a Γ 5 can be written as 50a. f a Γ a can be written as 2a.
2 Γ 6 .) 12 . (Note: This is the same as _ 3 a Simplify the fraction _
18 3Γ6 2 Γ 1000 .) 2000 b Simplify the fraction _. (Note: This is the same as _ 3000 3 Γ 1000 2 Γ a .) 2a . (Note: This is the same as _ c Simplify _ 3a 3Γa
4 Match up these expressions with the correct way to write them. a 2Γu A 3u b 7Γu B _5 u
c 5Γ·u d uΓ3
C 2u
e uΓ·5
E 7u
D _u 5
To tile a shopping mall ο¬oor area, a tiler is using rectangular tiles each l cm by w cm. There are 60 rows of tiles with 40 tiles per row. The tiler needs enough tile glue for area: A = 60w Γ 40l = 2400wl square centimetres.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 4
Algebraic techniques
Example 11 Simplifying expressions with multiplication a Write β4 Γ a Γ b Γ cβwithout multiplication signs. b Simplify β4a Γ 2b Γ 3cβ, giving your final answer without multiplication signs. c Simplify β3w Γ 4wβ.
U N SA C O M R PL R E EC PA T E G D ES
256
SOLUTION
EXPLANATION
a 4β Γ a Γ b Γ c = 4abcβ
When pronumerals are written next to each other they are being multiplied.
b 4a Γ 2b Γ 3c = 4 Γ a Γ 2 Γ b Γ 3 Γ c ββ―β― β =β 4 Γ 2 Γ 3 Γ a Γ b Γ cβ β = 24abc
First, insert the missing multiplication signs. Rearrange to bring the numbers to the front. β 4 Γ 2 Γ 3 = 24βand βa Γ b Γ c = abcβ, giving the final answer.
c 3w Γ 4w = 3 Γ w Γ 4 Γ w β β―β― β =ββ 3 Γ 4 Γ w Γ wββ β = 12βwβ2β
First, insert the missing multiplication signs. Rearrange to bring the numbers to the front. β3 Γ 4 = 12βand βw Γ wβis written as βwβ2ββ.
Now you try
a Write βd Γ 5 Γ e Γ fβwithout multiplication signs. b Simplify β3x Γ 5y Γ 2zβ, giving your final answer without multiplication signs. c Simplify β2a Γ 5aβ
Example 12 Simplifying expressions with division a Write (β 3x + 1)βΓ· 5βwithout a division sign.
b Simplify the expression _ β8ab ββ. 12b
SOLUTION
EXPLANATION
a (β 3x + 1)βΓ· 5 = _ β3x + 1 β 5
The brackets are no longer required as it becomes clear that all of β3x + 1βis being divided by β5β.
b β_ 8ab β = _ β8 Γ a Γ b β 12b 12 Γ b 2 ____________ β―β― β ββ―β― =ββ β―β― β Γ β4βΓ a βΓ βbββ 3 Γ β4βΓ βbβ 2a β= _ β β 3
Insert multiplication signs to help spot common factors. β8βand β12βhave a common factor of β4β. Cancel out the common factors of β4βand βbβ.
Now you try
a Write (β 3x + 2)βΓ· (β 2x + 1)βwithout a division sign.
12pqr b Simplify the expression β_ββ. 15pr
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
4E Multiplying and dividing expressions
257
Exercise 4E 1β5ββ(1β/2β)β
Write the following without multiplication signs. a β2 Γ a Γ bβ b β5 Γ x Γ y Γ zβ e 2β Γ xβ f β5 Γ pβ i β7 Γ 4 Γ fβ j β5 Γ 2 Γ a Γ bβ
c 8β Γ q Γ rβ g β8 Γ a Γ bβ k β2 Γ b Γ 5β
d 1β 2 Γ s Γ t Γ u Γ vβ h β3 Γ 2 Γ aβ l βx Γ 7 Γ z Γ 4β
2 Simplify these expressions. a β3a Γ 12β b 7β d Γ 9β e 4β a Γ 3bβ f β7e Γ 9gβ i βa Γ 3b Γ 4cβ j β2a Γ 4b Γ cβ
c 2β Γ 4eβ g β8a Γ bcβ k β4d Γ 3e Γ 5fgβ
d 3β Γ 5aβ h β4d Γ 7afβ l β2cb Γ 3a Γ 4dβ
3 Simplify these expressions. a βw Γ wβ b aβ Γ aβ e pβ Γ 7pβ f βq Γ 3qβ
c 3β d Γ dβ g β6x Γ 2xβ
d 2β k Γ kβ h β9r Γ 4rβ
1
2β5β(β 1β/2β)β
3β5β(β 1β/2β)β
U N SA C O M R PL R E EC PA T E G D ES
Example 11a
FLUENCY
Example 11b
Example 11c
Example 12a
Example 12b
4 Write each expression without a division sign. a βx Γ· 5β b βz Γ· 2β d βb Γ· 5β e β2 Γ· xβ g βx Γ· yβ h βa Γ· bβ j β(2x + y)βΓ· 5β k β(2 + x)βΓ· β(1 + y)β
c f i l
aβ Γ· 12β β5 Γ· dβ β(4x + 1)βΓ· 5β β(x β 5)βΓ· β(3 + b)β
a β= aβ. 5 Simplify the following expressions by dividing by any common factors. Remember that β_ 1 2x β a β_ d _ β2ab β b _ β5a β c _ β9ab β 5x 9a 5a 4b 2x 9x 30y 10a _ _ _ e β β g β β f β β h _ β β 4 12 15a 40y 4xy 9x β i _ β4a β j _ β21x β l β_ k _ β β 7x 2 3xy 2x PROBLEM-SOLVING
6, 7
7, 9
7β9
6 Write a simplified expression for the area of the following rectangles. Recall that for rectangles, β Area = width Γ lengthβ. a b c 6 3x k 3
x
4y
7 The weight of a single muesli bar is βxβ grams. a What is the weight of β4βbars? Write an expression. b If Jamila buys βnβbars, what is the total weight of her purchase? c Jamilaβs cousin Roland buys twice as many bars as Jamila. What is the total weight of Rolandβs purchase? 8 Five friends go to a restaurant. They split the bill evenly, so each spends the same amount. a If the total cost is β$100β, how much do they each spend? b If the total cost is β$Cβ, how much do they each spend? Write an expression.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 4
Algebraic techniques
9 Replace the question marks with algebraic terms to make these equivalence statements true. a 4β c Γβ β?ββ Γ b = 12abcβ b β2a Γ 2b Γ ?ββ = 28abcβ c _ β14ab β= 2aβ ? 12xy _ β= xβ d β ? 50x βΓ y = 5yβ e β_ ?
U N SA C O M R PL R E EC PA T E G D ES
258
REASONING
10
10, 11
11β13
10 The expression β3 Γ 2pβis the same as the expression 2p + 2p + 2p . (1) (2) (3)
a What is a simpler expression for β2p + 2p + 2pβ? (Hint: Combine like terms.) b β3 Γ 2pβis shorthand for β3 Γ 2 Γ pβ. How does this relate to your answer in part a?
11 The area of the rectangle shown is β3aβ. The length and width of this rectangle are now doubled. a Draw the new rectangle, showing its dimensions. b Write a simplified expression for the area of the new rectangle. c Divide the area of the new rectangle by the area of the old rectangle. What do a you notice? d What happens to the area of the original rectangle if you triple both the length 3 and the width? 12 Consider the expression β5a Γ 4b Γ 5cβ. a Simplify the expression. b If βa = 3, b = 7βand βc = 2βfind the value of β5a Γ 4b Γ 5cβwithout a calculator. c Explain why it is easier to evaluate the simplified expression.
13 We can combine like terms after simplifying any multiplication and division. For example, β 3a Γ 8b β 2a Γ 7bβsimplifies to β24ab β 14abβ, which simplifies to β10abβ. Use this strategy to simplify the following as much as possible. a β7a Γ 4b β 6a Γ 2bβ b β10a Γ 5b β 4b Γ 9aβ c β5a Γ 12b β 6a Γ 10bβ ENRICHMENT: Managing powers
β
β
14
14 The expression βa Γ aβcan be written as βaβ2βand the expression βa Γ a Γ aβcan be written as βaβ3ββ. a What is β3a βbβ2βwhen written in full with multiplication signs? b Write β7 Γ x Γ x Γ y Γ y Γ yβwithout any multiplication signs. c Simplify β2a Γ 3b Γ 4c Γ 5a Γ b Γ 10c Γ aβ. d Simplify β4βaβ2βΓ 3a bβ β2βΓ 2βcβ2ββ.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Applications and problem-solving
U N SA C O M R PL R E EC PA T E G D ES
Comparing printing costs
Applications and problem-solving
The following problems will investigate practical situations drawing upon knowledge and skills developed throughout the chapter. In attempting to solve these problems, aim to identify the key information, use diagrams, formulate ideas, apply strategies, make calculations and check and communicate your solutions.
259
1
Glenn doesnβt currently own a home printer and looks into the costs of purchasing a printer, toner cartridges and paper. Glenn finds out the following information: β’ Black and white laser printer (β23βpages per minute) β= $98β β’ Toner cartridge (lasts for β1000βpages) β= $72β β’ Reams of paper (β500βpages in a ream) β= $5β
Another option for Glenn is to print at school, but the school charges β10βcents per βA4βpage of black and white printing. Glenn is interested in comparing the cost of printing on his home printer compared to the cost of printing on one of the schoolβs printers. a What is the cost of Glenn printing β2000βpages at school over the course of the year? b If Glenn wishes to print β2000βpages at home over the course of the year, how many toner cartridges and how many reams of paper will he need to purchase? c What will it cost Glenn to buy a printer and print β2000βpages at home? d What is the total cost of Glennβs printing at home per page including the cost of the printer? e What is the total cost of Glennβs printing at home per page excluding the cost of the printer? f Do you recommend Glenn prints at home or at school? g What is the cost of Glenn printing βnβpages at school? h What is the cost of Glenn printing βnβpages at home, excluding the cost of the printer? i Find a value of βnβfor which it is cheaper to print: i at home, including the cost of the printer ii at school, including the cost of the printer.
Calculating walking distance
2 Carmen determines her average stride length by walking a distance of β100 mβand taking β125βsteps to do so. Carmen is curious about how far she walks in various time intervals including a whole year. a What is Carmenβs average stride length? Give your answer in metres and in centimetres. b Write an expression for the distance in kilometres that Carmen walks in βxβ steps.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 4
Algebraic techniques
Carmen is determined to walk at least β10 000βsteps every day. c How far will Carmen walk on a day when she takes β10 000βsteps? Give your answer in metres and in kilometres. d If Carmen averages β10 000βsteps per day over an entire year (non-leap year), how far will Carmen walk in the year? Give your answer in kilometres. e How many steps per day would Carmen have to average over the year if she wanted to be able to say she had walked the βlength of Australiaβ, a distance equivalent to walking from Melbourne to Darwin (β 3740 km)ββ? f How many steps per day would Carmen have to average over the year if she wanted to be able to say she had walked the βwidth of Australiaβ, a distance equivalent to walking from Perth to Sydney (β 3932 km)ββ? g How many steps per day would a student have to average over a year if they have a stride length of β sβmetres and want to walk βnβ kilometres?
U N SA C O M R PL R E EC PA T E G D ES
Applications and problem-solving
260
Earning money from dog-walking
3 Eskander is keen to earn some money from offering to take their neighboursβ dogs for a walk. Eskander is interested in the total amount of money that can be earned depending on the size of the fee charge per walk, the duration of the walks and the total number of walks.
Eskander initially thinks they will simply charge β$5βper dog walk and writes down the rule βM = 5 Γ w.β a What do you think βMβand βwβstand for? b How much will Eskander earn from β20βdog walks? Eskander then thinks that it makes more sense to charge a different rate for 20-minute walks and 4β 0β-minute walks. They decide to charge β$4βfor β20β-minute walks and β$8βfor β40β-minute walks. c Using βaβto represent the number of β20β-minute walks, and βbβto represent the number of β40β-minute walks, write a rule for the money Eskander will earn. d How much will Eskander earn from βtenβ20-minute walks and βtenβ40-minute walks?
Eskander realises that they are really charging the same rate per minute for the β20β- and β40β-minute walks. e How much does Eskander earn per minute of dog walking? Give your answer in dollars and also in cents. f Write an equation for the amount of money Eskander will earn per βtβminutes of walking. g Using this new equation, how much will Eskander earn from β600βminutes of walking? Check if this is the same amount as your answer to part d.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
4F Expanding brackets
261
4F Expanding brackets EXTENDING
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand what it means to expand brackets β’ To be able to expand brackets using repeated terms or rectangle areas β’ To be able to expand brackets using the distributive law
We have already seen that there are different ways of writing two equivalent expressions. For example, 4a + 2a is equivalent to 2 Γ 3a, even though they look different.
Note that 3(7 + a) = 3 Γ (7 + a), which is equivalent to 3 lots of 7 + a.
So, 3(7 + a) = 7 + a + 7 + a + 7 + a = 21 + 3a It is sometimes useful to have an expression that is written with brackets, like 3 Γ (7 + a), and sometimes it is useful to have an expression that is written without brackets, like 21 + 3a.
A builder designs a low-set house, 15 m long by 10 m wide. Customers can select the option of adding a deck of variable width, x m. The total ο¬oor area is: A = 15(10 + x) = 150 + 15x (house)
(deck)
Lesson starter: Total area
7
What is the total area of the rectangle shown at right? Try to write two expressions, only one of which includes brackets.
a
3
KEY IDEAS
β Expanding (or eliminating) brackets involves writing an equivalent expression without brackets.
This can be done by writing the bracketed portion a number of times or by multiplying each term. 2(a + b) = a + b + a + b = 2a + 2b
or 2(a + b) = 2 Γ a + 2 Γ b = 2a + 2b
β To eliminate brackets, you can use the distributive law, which states that:
a(b + c) = ab + ac
and
a (b β c) = ab β ac
β The distributive law is used in arithmetic.
For example:
5 Γ 27 = 5(20 + 7) = 5 Γ 20 + 5 Γ 7 = 100 + 35 = 135
β The process of removing brackets using the distributive law is called expansion. β When expanding, every term inside the brackets must be multiplied by the term outside the brackets.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 4
Algebraic techniques
BUILDING UNDERSTANDING 1 The expression β3(β a + 2)βcan be written as β(a + 2) + (a + 2) + (a + 2)β. a Simplify this expression by collecting like terms. b State β2(β x + y)βin full without brackets and simplify the result. c State β4(β p + 1)βin full without brackets and simplify the result. d State β3(β 4a + 2b)βin full without brackets and simplify the result.
U N SA C O M R PL R E EC PA T E G D ES
262
2 The area of the rectangle shown can be written as β4(β x + 3)ββ. a What is the area of the green rectangle? b What is the area of the red rectangle? c State the total area as an expression, without using brackets.
x
3
4
3 State the missing parts in the following calculations which use the distributive law. a 3 Γ 21 = 3 Γ (20 + 1)
= 3 Γ 20 + 3 Γ 1 = + =
b 7 Γ 34 = 7 Γ (30 + 4)
=7Γ +7Γ = + =
c 5 Γ 19 = 5 Γ (20 β 1) =5Γ = β =
β5Γ
4 a State the missing results in the following table. Remember to follow the rules for correct order of operations. βx = 1β
β4(x + 3)β
β4x + 12β
β = 4(1 + 3) β =β 4(4)β ββ β = 16
β = 4(1) + 12 β =β 4 + 12β β β = 16
xβ = 2β
βx = 3β βx = 4β
b State the missing word. The expressions β4(β x + 3)βand β4x + 12βare __________.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
4F Expanding brackets
263
Example 13 Expanding brackets using rectangle areas
U N SA C O M R PL R E EC PA T E G D ES
Write two equivalent expressions for the area of each rectangle shown, only one of which includes brackets. a b 12 5 x a
2
3
SOLUTION
EXPLANATION
a Using brackets: β2(β 5 + x)β Without brackets: β10 + 2xβ
The whole rectangle has width β2βand length β5 + xβ. The smaller rectangles have area β2 Γ 5 = 10βand β 2 Γ x = 2xβ, so they are added.
b Using brackets: β12β(a + 3)β
The dimensions of the whole rectangle are β12βand βa + 3β. Note that, by convention, we do not write (β a + 3)β12β. The smaller rectangles have area β12 Γ a = 12aβand β 12 Γ 3 = 36β.
Without brackets: β12a + 36β
Now you try
Write two equivalent expressions for the area of each rectangle shown, only one of which includes brackets. a b r
6
7
x
5
7
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 4
Algebraic techniques
Example 14 Expanding using the distributive law Expand the following expressions. a 5β β(x + 3)β b 8β β(a β 4)β
c β3β(5a + 2)β
d β5aβ(3p β 7q)β
U N SA C O M R PL R E EC PA T E G D ES
264
SOLUTION
EXPLANATION
a 5β(x + 3)β = 5 Γ x + 5 Γ 3 β β β β―β― β β = 5x + 15
Use the distributive law: 5(x + 3) = 5x + 5 Γ 3 Simplify the result. Alternative method x +3 5 5x +15 so 5(x + 3) = 5x + 15
( ) = 8Γaβ8Γ4 b 8β β β β a β 4 βββ―β― β = 8a β 32
Use the distributive law with subtraction:
8(a β 4) = 8a β 8 Γ 4 Simplify the result.
Alternative method a β4 8 8a β32 so 8(a β 4) = 8a β 32
c 3β(5a + 2)β = 3 Γ 5a + 3 Γ 2 β β β―β― β β β = 15a + 6
Use the distributive law:
3(5a + 2) = 3 Γ 5a + 3 Γ 2
Simplify the result, remembering that β3 Γ 5a = 15aβ. Alternative method 5a +2 3 15a +6 so 3(5a + 2) = 15a + 6
d 5aβ(3p β 7q)β = 5a Γ 3p β 5a Γ 7q β―β― β β β β β = 15ap β 35aq
Expanding: 5a(3p β 7q) = 5a Γ 3p β 5a Γ 7q
Simplify the result, remembering that β5a Γ 3p = 15apβ and β5a Γ 7q = 35aqβ. Alternative method 3p β7q
5a 15ap β35aq so 5a(3p β 7q) = 15ap β 35aq.
Now you try
Expand the following expressions. a 4β β(b + 7)β b 9β β(k β 5)β
c 5β β(2p + 5)β
d β9xβ(3z β 2)β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
4F Expanding brackets
265
Exercise 4F FLUENCY
1
2, 3β5β(β 1β/2β)β
3β5β(β 1β/3β)β
Write two equivalent expressions for the area of each rectangle shown, only one of which includes brackets. a b 4 x 2
U N SA C O M R PL R E EC PA T E G D ES
Example 13
1, 2, 3β4ββ(1β/2β)β
2
a
1
c
x
d
4
8
z
12
9
Example 14a
Example 14b
Example 14c
Example 14d
2 Use the distributive law to expand the following. a β6(y + 8)β b β7β(l + 4)β ( ) d β4β 2 + a β e β7β(x + 5)β
c 8β β(s + 7)β f β3β(6 + a)β
3 a 4β β(x β 3)β d β8(β e β 7)β
c 8β β(y β 8)β f β10β(8 β y)β
b 5β β(βj β 4β)β e β6β(3 β e)β
4 Use the distributive law to expand the following. a β10β(6g β 7)β b β5β(3e + 8)β ( ) d β5β 2u + 5 β e β7β(8x β 2)β g β7(β q β 7)β h β4β(5c β v)β ( ) j β6β 8l + 8 β k 5β β(k β 10)β
c f i l
5β β(7w + 10)β β3β(9v β 4)β β2β(2u + 6)β β9β(o + 7)β
5 Use the distributive law to expand the following. a β6iβ(t β v)β b β2dβ(v + m)β d β6eβ(s + p)β e βdβ(x + 9s)β g β5jβ(r + 7p)β h βiβ(n + 4w)β j βfβ(2u + v)β k 7β kβ(2v + 5y)β
c f i l
5β cβ(2w β t)β β5aβ(2x + 3v)β β8dβ(s β 3t)β β4eβ(m + 10y)β
PROBLEM-SOLVING
6, 7
7, 8
7β9
6 Write an expression for each of the following and then expand it. a A number, βxβ, has β3βadded to it and the result is multiplied by β5β. b A number, βbβ, has β6βadded to it and the result is doubled. c A number, βzβ, has β4βsubtracted from it and the result is multiplied by β3β. d A number, βyβ, is subtracted from β10βand the result is multiplied by β7β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 4
Algebraic techniques
7 At a school assembly there are a number of teachers and a number of students. Let β tβstand for the number of teachers and βsβ stand for the number of students. a Write an expression for the total number of people at the assembly (assume there are just teachers and students in attendance). b If all people at the assembly are wearing two socks, write two equivalent expressions for the total number of socks being worn (one expression with brackets and one without).
U N SA C O M R PL R E EC PA T E G D ES
266
8 When expanded, β4(β 3x + 6y)βgives β12x + 24yβ. Find two other expressions that expand to β12x + 24yβ.
9 The distance around a rectangle is given by the expression β2(β l + w)β, where βlβis the length and βwβis the width. What is an equivalent expression for this distance? REASONING
10
10, 11
11, 12
10 a Use a diagram of a rectangle like that in Example 13 to demonstrate that β5(β x + 3)β= 5x + 15β. b Use a diagram of a rectangle to prove that β(a + 2)β(b + 3)β= ab + 2b + 3a + 6β. c Use a diagram to find an expanded form for β(βa + 3β)β(βb + 5β)ββ.
11 To multiply a number by β99β, you can multiply it by β100βand then subtract the original number. For example, β17 Γ 99 = 1700 β 17 = 1683β. Explain, using algebra, why this method always works (not just for the number β17β).
12 When expanded, β5β(2x + 4y)βgives β10x + 20yβ. a How many different ways can the missing numbers be filled with whole numbers for the equivalence ( βx+β βyβ) β= 10x + 20yβ? b How many different expressions expand to give β10x + 20yβif fractions or decimals are included? ENRICHMENT: Expanding sentences
β
β
13
13 Using words, people do a form of expansion. Consider these two statements: Statement A: βJohn likes tennis and football.β
Statement B: βJohn likes tennis and John likes football.β
Statement B is an βexpanded formβ of statement A, which is equivalent in its meaning but more clearly shows that two facts are being communicated. Write an βexpanded formβ of the following sentences. a Rosemary likes Maths and English. b Priscilla eats fruit and vegetables. c Bailey and Lucia like the opera. d Frank and Igor play video games. e Pyodir and Astrid like fruit and vegetables. (Note: There are four facts being communicated in part e.)
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
4G Applying algebra
267
4G Applying algebra
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To know that algebra can model a variety of situations β’ To be able to apply an expression in a modelling situation β’ To be able to construct an expression from a problem description
Algebraic expressions can be used to describe problems relating to many different areas, including costs, speeds and sporting results. Much of modern science relies on the application of algebraic rules and formulas. It is important to be able to convert word descriptions of problems to mathematical expressions in order to solve these problems mathematically.
A standard Uber fare is calculated by adding a base fare to a cost for t minutes and a cost for d km. For example, a fare in dollars could be 2.55 + 0.38t + 1.15d.
Lesson starter: Garden bed area
The garden shown below has an area of 34 m2, but the width and length are unknown. l=?
w=?
area = 34 m2
2m
3m
β’ β’
What are some possible values that w and l could equal? Try to find the dimensions of the garden that make the fencing around the outside as small as possible.
KEY IDEAS
β Many different situations can be modelled with algebraic expressions.
For example, an algebraic expression for perimeter is 2l + 2w
β To apply an expression, the pronumerals should be defined clearly. Then known values should
be substituted for the pronumerals. For example: If l = 5 and w = 10, then perimeter = 2 Γ 5 + 2 Γ 10 = 30
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 4
Algebraic techniques
BUILDING UNDERSTANDING 1 The area of a rectangle is given by the expression βw Γ lβ, where βwβis its width and βlβis its length in metres. a Find the area in square metres if βw = 5βand βl = 7β. b Find the area in square metres if βw = 2βand βl = 10β.
U N SA C O M R PL R E EC PA T E G D ES
268
2 The perimeter of a square with width βwβcm is given by the expression β4wβ cm. a Find the perimeter of a square with width β6 cmβ(i.e. βw = 6β). b Find the perimeter of a square with width β10 mβ(i.e. βw = 10β). 3 Consider the equilateral triangle shown. a State an expression that gives the perimeter of this triangle. b Use your expression to find the perimeter if βx = 12β.
x
x
x
Example 15 Applying an expression
The perimeter of a rectangle is given by the expression β2w + 2lβ, where βwβis the width and βlβis the length. a Find the perimeter of a rectangle if βw = 5βand βl = 7β. b Find the perimeter of a rectangle with width β3 cmβand length β8 cmβ. SOLUTION
EXPLANATION
a 2w + 2l = 2β(5)β+ 2β(7)β β β―β― β =β 10 + 14β β β = 24
To apply the rule, we substitute βw = 5βand β l = 7βinto the expression. Evaluate using the normal rules of arithmetic (i.e. multiplication before addition).
b 2w + 2l = 2β(3)β+ 2β(8)β β =β 6 + 16β β―β― β β β = 22 cm
Substitute βw = 3βand βl = 8βinto the expression. Evaluate using the normal rules of arithmetic, remembering to include appropriate units (cm) in the answer.
Now you try
The area of a triangle is given by the expression _ βbh βwhere βbβis the 2 base width and βhβis the height. a Find the area of a triangle if βb = 7βand βh = 10β. b Find the area of a triangle with base β20 cmβand height β12 cmβ.
h
b
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
4G Applying algebra
269
Example 16 Constructing expressions from problem descriptions
U N SA C O M R PL R E EC PA T E G D ES
Write expressions for each of the following. a The total cost, in dollars, of β10βbottles, if each bottle costs β$xβ. b The total cost, in dollars, of hiring a plumber for βnβhours. The plumber charges a β$30β call-out fee plus β$60βper hour. c A plumber charges a β$60βcall-out fee plus β$50βper hour. Use an expression to find how much an β 8β-hour job would cost. SOLUTION
EXPLANATION
a β10xβ
Each of the β10βbottles costs β$xβ, so the total cost is β10 Γ x = 10xβ.
b β30 + 60nβ
For each hour, the plumber charges β$60β, so must pay β60 Γ n = 60nβ. The β$30βcall-out fee is added to the total bill.
c Expression for cost: β60 + 50nβ If βn = 8β, then cost is β60 + 50 Γ 8 = $460β
Substitute βn = 8βto find the cost for an β8β-hour job. Cost will be β$460β.
Now you try
Write expressions for each of the following. a The total volume, in litres, of soft drink in βnβbottles, if each bottle contains β2β litres. b The total time, in minutes, it takes to paint βkβsquare metres of fencing, given that it takes β20βminutes to set up the painting equipment and then β5βminutes per square metre. c Hiring a car costs β$80βhiring fee plus β$110βper day. Use an expression to find the cost of hiring the car for seven days.
Exercise 4G FLUENCY
Example 15
Example 15
1
1β6
2β7
3β7
The perimeter of a rectangle is given by the expression β2w + 2lβwhere βwβis the width and βlβ is the length. a Find the perimeter of a rectangle if βw = 4βand βl = 10β. b Find the perimeter of a rectangle with width β7 cmβand length β15 cmβ.
2 The perimeter of an isosceles triangle is given by the expression βx + 2yβwhere βxβ is the base length and βyβis the length of the other sides. a Find the perimeter if βx = 3βand βy = 5β. b Find the perimeter of an isosceles triangle if the base length is β4 mβand the other sides are each β7 mβ.
y
y
x
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
270
Chapter 4
Example 16a
3 If pens cost β$2βeach, write an expression for the cost, in dollars, of βnβ pens. 4 If pencils cost β$xβeach, write an expression for the cost, in dollars, of: a 1β 0β pencils b β3βpackets of pencils, if each packet contains β5β pencils c βkβ pencils
U N SA C O M R PL R E EC PA T E G D ES
Example 16
Algebraic techniques
5 A car travels at β60 km / hβ, so in βnβhours it has travelled β60nβ kilometres. a How far does the car travel in β3βhours (i.e. βn = 3β)? b How far does the car travel in β30β minutes? c Write an expression for the total distance (in βkmβ) travelled in βnβhours for a motorbike with speed β 70 km / hβ. 6 A carpenter charges a β$40βcall-out fee and then β$80βper hour. This means the total cost, in dollars, for βxβhours of work is β40 + 80xβ. a How much would it cost for a β2β-hour job (i.e. βx = 2β)? b How much would it cost for a job that takes β8β hours? c The call-out fee is increased to β$50β. What is the new expression for the total cost, in dollars, of βxβ hours?
Example 16b
7 Match up the word problems with the expressions (A to E) below. a The area of a rectangle with length β5βand width βxβ. b The perimeter of a rectangle with length β5βand width βxβ. c The total cost, in dollars, of hiring a DVD for βxβdays if the price is β$1βper day. d The total cost, in dollars, of hiring a builder for β5βhours if the builder charges a β $10βcall-out fee and then β$xβper hour. e The total cost, in dollars, of buying a β$5βmagazine and a book that costs β$xβ. PROBLEM-SOLVING
8, 9
9β11
A B C D
β10 + 2xβ β5xβ β5 + xβ xβ β
E β10 + 5xβ 10β12
8 A plumber charges a β$50βcall-out fee and β$100βper hour. a Copy and complete the table below. Number of hours
β1β
β2β
3β β
β4β
β5β
Total costs (β $)β
b Find the total cost if the plumber works for βtβhours. Give an expression. c Substitute βt = 30βinto your expression to find how much it will cost for the plumber to work β30β hours.
9 To hire a tennis court, you must pay a β$5βbooking fee plus β$10βper hour. a What is the cost of booking a court for β2β hours? b What is the cost of booking a court for βxβhours? Write an expression. c A tennis coach hires a court for β7βhours. Substitute βx = 7βinto your expression to find the total cost. 10 Adrianβs mobile phone costs β30βcents to make a connection, plus β60βcents per minute of talking. This means that a βtβ-minute call costs β30 + 60tβ cents. a What is the cost of a β1β-minute call? b What is the cost of a β10β-minute call? Give your answer in dollars. c Write an expression for the cost of a βtβ-minute call in dollars.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
4G Applying algebra
U N SA C O M R PL R E EC PA T E G D ES
11 In Australian Rules football a goal is worth β6βpoints and a βbehindβ is worth β1βpoint. This means the total score for a team is β6g + bβ, if βgβgoals and βbβbehinds are scored. a What is the score for a team that has scored β5βgoals and β3β behinds? b What are the values of βgβand βbβfor a team that has scored β8βgoals and β5β behinds? c If a team has a score of β20β, this could be because βg = 2βand βb = 8β. What are the other possible values of βgβand βbβ?
271
12 In a closing-down sale, a shop sells all CDs for β$cβeach, books cost β$bβeach and DVDs cost β$dβ each. Claudia buys β5βbooks, β2βCDs and β6β DVDs. a What is the cost, in dollars, of Claudiaβs order? Give your answer as an expression involving βb, cβ and βdβ. b Write an expression for the cost, in dollars, of Claudiaβs order if CDs doubled in price and DVDs halved in price. c As it happens, the total price Claudia ends up paying is the same in both situations. Given that CDs cost β$12βand books cost β$20β(so βc = 12βand βb = 20β), how much do DVDs cost? REASONING
13
13
13, 14
13 A shop charges β$cβfor a box of tissues. a Write an expression for the total cost, in dollars, of buying βnβboxes of tissues. b If the original price is tripled, write an expression for the total cost, in dollars, of buying βnβ boxes of tissues. c If the original price is tripled and twice as many boxes are bought, write an expression for the total cost in dollars. 14 Hiring a basketball court costs β$10βfor a booking fee, plus β$30βper hour. a Write an expression for the total cost in dollars to hire the court for βxβ hours. b For the cost of β$40β, you could hire the court for β1βhour. How long could you hire the court for the cost of β$80β? Assume that the time does not need to be rounded to the nearest hour. c Explain why it is not the case that hiring the court for twice as long costs twice as much. d Find the average cost per hour if the court is hired for a β5βhour basketball tournament. e Describe what would happen to the average cost per hour if the court is hired for many hours (e.g. more than β50β hours). ENRICHMENT: Mobile phone mayhem
β
β
15
15 Rochelle and Emma are on different mobile phone plans, as shown below. Connection
Cost per minute
Rochelle
β20β cents
β60β cents
Emma
β80β cents
β40β cents
a b c d e
Write an expression for the cost of making a βtβ-minute call using Rochelleβs phone. Write an expression for the cost of making a βtβ-minute call using Emmaβs phone. Whose phone plan would be cheaper for a β7β-minute call? What is the length of call for which it would cost exactly the same for both phones? Investigate current mobile phone plans and describe how they compare to those of Rochelleβs and Emmaβs plans.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
272
Chapter 4
Algebraic techniques
Modelling
Internet service provider
U N SA C O M R PL R E EC PA T E G D ES
A small internet service provider is exploring a number of pricing structures for its customers. Three structures that it is considering are listed below. β’ Structure 1: No initial connection fee plus β$40βper month β’ Structure 2: β$80βinitial connection plus β$30βper month β’ Structure 3: β$150βinitial connection plus β$25βper month
Present a report for the following tasks and ensure that you show clear mathematical workings and explanations where appropriate.
Preliminary task
a Find the total cost of purchasing β12βmonthsβ service using: i pricing structure β1β ii pricing structure β2β iii pricing structure β3β. b How many monthsβ service will β$380βprovide using pricing structure β2β? c Which pricing structure is cheapest for β6βmonths of service? Give reasons.
Modelling task
Formulate
Solve
Evaluate and verify
Communicate
a The problem is to find the pricing structure that provides the best and worst value for money for customers, depending on how long they use the service. Write down all the relevant information that will help solve this problem. b If a customer receives βnβmonths of service, write expressions for the total cost in dollars using: i pricing structure β1β ii pricing structure β2β iii pricing structure β3β.
c Use your expressions to find the total cost of β3, 6, 9, 12βand β15βmonths of service using the three different pricing structures. Summarise your results using a table like the one shown.
Months
β3β
β6β
9β β
β12β
β15β
Structure 1 cost (β $)β
Structure 2 cost β($)β Structure 3 cost β($)β
d Determine the pricing structure that produces most money for the internet provider for: i β9βmonths of service ii β15βmonths of service. e For how many months will the overall cost of the following pricing structures be the same? i β1βand β2β ii β2βand β3β iii β1βand β3β f Describe the pricing structure(s) which deliver(s) the cheapest service for customers. If it depends on the number of months of service, explain how this affects the answers. g Summarise your results and describe any key findings.
Extension questions
a Construct a possible pricing structure which has a higher connection fee, but is cheaper than the above pricing structures for β12βmonths of service. Justify your choice. b Construct a possible pricing structure with a β$100βconnection fee that has a total cost of β$520βfor β 12βmonths of service. Justify your answer with calculations. Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Technology and computational thinking
273
Making microchips
U N SA C O M R PL R E EC PA T E G D ES
One of the key components of computers is the microchip processor. It is the nerve centre of the system which can hold a large amount of information and perform mathematical operations. The cost of making microchips involves significant set up costs and an ongoing cost per chip. These costs are passed on to the computer companies that buy and install the chips into their machines.
Technology and computational thinking
Key technology: Spreadsheets and graphing
1 Getting started
Firstly, consider two computer chip making companies, Mybot and iChip. Their cost structures are given as follows. β’ Mybot: Set-up costs: β$100 000β, Cost per item: β$32β β’ iChip: Set-up costs: β$50 000β, Cost per item: β$55β a Find the total cost of producing β2000βchips from: i Mybot ii iChip. b Find the total cost of producing β3000βchips from: i Mybot ii iChip. c Which company has the least overall cost for making: i β2000β chips? ii β3000β chips? d If βxβis the number of chips produced, write an expression for the total cost of producing chips for: i Mybot ii iChip.
2 Using technology
a Create a spreadsheet as shown to help find the overall costs for each company. b Fill down at cells A5, B4 and C4 to show the production costs of making 1 through to β10β chips. c Which company has the least costs for producing somewhere between β1β and β10β chips? d Alter the starting number in cell A4 to β 1500βand compare the costs of producing β 1500βto β1509βchips. Which company has the least overall cost of producing chips in this range? e Alter the starting number in cell A4 to β 2500βand compare the costs of producing β 2500βto β2509βchips. Which company has the least overall cost of producing chips in this range?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
274
Chapter 4
Algebraic techniques
a Apply the following algorithm to help search for the number of chips produced where the costs for both companies are very close. β’ Step 1: Alter the starting number in cell A4 to β2000βand compare the costs of producing β2000β chips. β’ Step 2: Note which company has the least costs. β’ Step 3: Increase or decrease the number in cell A4 until your answer to the question in step β2β changes. β’ Step 4: Repeat from step β3βuntil you have found the number of chips where the cost of production for the companies is the closest. b Add a graph by inserting a chart into your spreadsheet showing where the costs of production for the companies intersect. Investigate how changing the value of βxβin cell A4 changes the graph.
U N SA C O M R PL R E EC PA T E G D ES
Technology and computational thinking
3 Applying an algorithm
4 Extension
a Upgrade your spreadsheet to include editable upfront costs and per chip costs for each company as shown. Note that the $ sign is used to ensure that the rules in cell B8 and C8 always refer to the upfront and per chip cost information while you fill down. b Vary these costs, and in each case, change the number in A8 to help find the number of chips where the overall costs for both companies are very similar.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Investigation
275
Fencing paddocks
U N SA C O M R PL R E EC PA T E G D ES
n=2
Investigation
A farmer is interested in fencing off a large number of β n=1 1 m Γ 1 mβforaging regions for his chickens. Consider the pattern shown at right. a For βn = 2β, the outside perimeter is β8 mβ, the area is β4 βm2β ββ and the total length of fencing required is β12 mβ. Copy and complete the following table. βnβ
β1β
β2β
Outside perimeter (β m)β
β8β
Area β(βmπβ β)β
4β β
Fencing required β(m)β
β12β
3β β
β4β
n=3
β5β
β6β
b Write an expression for: i the total outside perimeter of the fenced section ii the total area of the fenced section. c The farmer knows that the expression for the total amount of fencing is one of the following. Which one is correct? Prove to the farmer that the others are incorrect. iii βn Γ 2 Γ β(n + 1)β i β6nβ ii β(n + 1)2β β d Use the correct formula to work out the total amount of fencing required if the farmer wants to have a total area of β100 βm2β βfenced off. In a spreadsheet application, these calculations can be made automatically. Set up a spreadsheet as follows.
Drag down the cells until you have filled all the rows from βn = 0βto βn = 30β. e Find the amount of fencing needed if the farmer wants the total area to be at least: i β25 βm2β β ii β121 βm2β β iii β400 βm2β β iv β500 βm2β ββ. f If the farmer has β144 mβof fencing, what is the maximum area his grid could have? g For each of the following lengths of fencing, give the maximum area, in βm2β ββ, that the farmer could contain in the grid. i β50 mβ ii β200 mβ iii β1 kmβ iv β40 kmβ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
276
Chapter 4
If βx + y = 8βand βy + m = 17βfind the value of β x + 2y + mβ.
2 A square is cut in half and the two identical rectangles are joined to form a rectangle as shown in this diagram.
Up for a challenge? If you get stuck on a question, check out the βWorking with unfamiliar problemsβ poster at the end of the book to help you.
U N SA C O M R PL R E EC PA T E G D ES
Problems and challenges
1
Algebraic techniques
Find an expression for the perimeter of the rectangle if the square has a side length of: a 2β mβ b β4β(x + 3)β c βw + yβ
3 These two identical βLβ shapes are to be joined along identical (matching) sides without any overlap. Find a simplified algebraic expression for the largest and smallest possible perimeters of the joined shapes and also for the difference between these two perimeters. Calculate this difference when βx = 10β. The diagrams are not drawn to scale. x+3
x+3
2(x + 5)
2(x + 5)
4 In a list of five consecutive integers, the middle integer is β3a + 2β. Find two equivalent expressions for the sum of these five integers: one expanded and simplified, and one factorised. (Note that consecutive integers are whole numbers that follow each other. For example, β3, 4, 5βand β6βare four consecutive integers.) 5 Find the values of the pronumerals below in the following sum/product tables. a b Sum
Sum
βaβ
βbβ
βcβ
dβ β
β24β
3β 2β
β12β
βeβ
β48β
Product
Product
βaβ
bβ β
β18β
2β β
βcβ
dβ β
β12β
βeβ
β180β
6 What is the coefficient of βxβonce the expression βx + 2β(x + 1)β+ 3β(x + 2)β+ 4β(x + 3)β+ ... + 100β(x + 99)ββ is simplified completely?
7 Think of any number and then perform the following operations. Add β5β, then double the result, then subtract β12β, then subtract the original number, then add β2β. Use algebra to explain why you now have the original number again. Then design a puzzle like this yourself and try it on friends.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter summary
Creating expressions
U N SA C O M R PL R E EC PA T E G D ES
2 Γ a Γ 3 Γ a Γ b = 6a2b 6xy 6xy Γ· 12x = 12x y = 2
Pronumerals are letters used to represent numbers g : number of grapes in a bunch d : distance travelled (in metres)
Chapter summary
6 more than k: k + 6 Product of 4 and x : 4x 10 less than b: b β 10 q Half of q: 2 The sum of a and b is tripled: 3(a + b)
Multiplying and dividing expressions
277
Terms are pronumerals and numbers combined with Γ, Γ· e.g. 4x, 10y, 3a , 12 3a means 3 Γ a b means b Γ· 10 10
Algebraic expressions
Like terms have exactly the same pronumerals. 5a and 3a 2ab and 12ba 7ab and 2a
Combination of numbers, pronumerals and operations, e.g. 2xy + 3yz, 12 β3 x
Equivalent expressions
Always equal when pronumerals are substituted. e.g. 2x + 3 and 3 + 2x are equivalent. 4 Γ 3a and 12a are equivalent.
Algebra
To simplify an expression, find a simpler expression that is equivalent.
Applications
Expanding brackets (Ext)
3(a + 4) = 3a + 12
a +4 3 3a +12
5k(10 β 2j ) = 50k β 10kj
10 β2j 5k 50k β10kj
Using the distributive law gives an equivalent expression.
Substitution Replacing pronumerals with values. e.g. 5x + 2y when x = 10 with and y = 3 becomes 5 Γ 10 + 2 Γ 3 = 50 + 6 = 56 e.g. q 2 when q = 7 becomes 72 = 49
Combining like terms gives a way to simplify. e.g. 4a + 2 + 3a = 7a + 2 3b + 4c + 2b β c = 5b + 4c 12xy + 3x β 5yx = 7xy + 3x
Commonly used expressions A=lΓw P = 2l + 2w w l
Cost is 50 + 90x
call-out fee
hourly rate
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
278
Chapter 4
Algebraic techniques
A printable version of this checklist is available in the Interactive Textbook 4A
β
1. I can state the coefficients of pronumerals within an expressions. e.g. What is the coefficient of βcβin β3a + b + 13cβ?
U N SA C O M R PL R E EC PA T E G D ES
Chapter checklist
Chapter checklist with success criteria
4A
2. I can list terms within expressions, and identify constant terms. e.g. List the terms in β3a + b + 13βand circle the constant term.
4A
3. I can write algebraic expressions from word descriptions. e.g. Write an expression for βthe sum of βaβand βbββ.
4B
4. I can substitute a number for a pronumeral and evaluate. e.g. Given that βt = 5β, evaluate β8tβ.
4B
5. I can substitute multiple numbers for multiple pronumerals and evaluate. e.g. Substitute βx = 4βand βy = 7βto evaluate β80 β (2xy + y)β.
4B
6. I can substitute into expressions involving powers and roots. e.g. If βp = 4βfind the value of β3βpβ2ββ.
4C
7. I can decide whether two expressions are equivalent. e.g. Indicate the two expressions which are equivalent: β3x + 4, 8 β x, 2x + 4 + xβ.
4C
8. I can generalise number facts using algebra. e.g. Write an equation involving βxβto generalise the following: β1 + 1 = 2 Γ 1, β7 + 7 = 2 Γ 7, β15 + 15 = 2 Γ 15β
4D
9. I can decide whether two terms are like terms. e.g. Decide whether β2abβand β5baβare like terms, giving reasons.
4D
10. I can simplify using like terms. e.g. Simplify β5 + 12a + 4b β 2 β 3aβby collecting like terms.
4E
11. I can simplify expressions involving multiplication. e.g. Simplify β4a Γ 2b Γ 3cβ, giving your final answer without multiplication signs.
4E
12. I can simplify expressions involving division. e.g. Simplify the expression _ β8ab ββ. 12b
4F
13. I can expand brackets. e.g. Expand β3(β a + 2b)ββ.
4G
14. I can apply an expression in a modelling problem. e.g. Given the perimeter of a rectangle is β2w + 2lβ, find the perimeter of a rectangle with length β 8 cmβand width β3 cmβ.
4G
15. I can construct an expression from a problem description. e.g. Write an expression for the total cost in dollars of hiring a plumber for βnβhours, if they charge β$30βcallout fee plus β$60βper hour.
Ext
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
279
Short-answer questions 1
Chapter review
a List the four individual terms in the expression β5a + 3b + 7c + 12β. b What is the constant term in the expression above? c What is the coefficient of βcβin the expression above?
2 Write an expression for each of the following. a β7βis added to βuβ b βkβis tripled c 7β βis added to half of βrβ d β10βis subtracted from βhβ e the product of βxβand βyβ f βxβis subtracted from β12β
U N SA C O M R PL R E EC PA T E G D ES
4A
4A
4B
3 If βu = 12β, find the value of: a βu + 3β b β2uβ
c _ β24 uβ
d β3u β 4β
4B
4 If βp = 3βand βq = 5β, find the value of: a βpqβ b βp + qβ
c 2β β(q β p)β
d β4p + 3qβ
4B
5 If βt = 4βand βu = 10β, find the value of: a βtβ2β b β2βuβ2β
c β3 + ββt β
4C
4D
4D
4E
4E
_
d ββ10tu β
6 For each of the following pairs of expressions, state whether they are equivalent (E) or not equivalent (N). a β5xβand β2x + 3xβ b β7a + 2bβand β9abβ c 3β c β cβand β2cβ d β3(β x + 2y)βand β3x + 2yβ
7 Classify the following pairs as like terms (L) or not like terms (N). a β2xβand β5xβ b β7abβand β2aβ c β3pβand βpβ e 4β abβand β4abaβ f β8tβand β2tβ g β3pβand β3β 8 Simplify the following by collecting like terms. a β2x + 3 + 5xβ b β12p β 3p + 2pβ d β12mn + 3m + 2n + 5nmβ e β1 + 2c + 4h β 3o + 5cβ
9 Simplify the following expressions involving products. a β3a Γ 4bβ b β2xy Γ 3zβ c β12f Γ g Γ 3hβ
d 9β xyβand β2yxβ h β12kβand β120kβ
c 1β 2b + 4a + 2b + 3a + 4β f β7u + 3v + 2uv β 3uβ d β8k Γ 2 Γ 4lmβ
10 Simplify the following expressions involving quotients. 3u β a β_ 2u
4F
_
12y b _ β β 20y
c _ β2ab β 6b
11 Expand the following expressions using the distributive law. a β3β(x + 2)β b β4β(p β 3)β c β7β(2a + 3)β
12xy d _ β β 9yz
d β12β(2k + 3l)β
Ext
4F
12 Give two examples of expressions that expand to give β12b + 18cβ.
Ext
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
280
Chapter 4
4G
13 If a single tin of paint weighs β9 kgβ, write an expression for the weight in βkgβof βtβtins of paint. 14 If there are βcβcats and βdβdogs at a vet, write an expression for the total number of animals at the vet (assume this vet only has cats and dogs present).
U N SA C O M R PL R E EC PA T E G D ES
Chapter review
4G
Algebraic techniques
4G
15 Write an expression for the total number of books that Analena owns if she has βxβfiction books and twice as many non-fiction books.
Multiple-choice questions
4A
1
In the expression β3x + 2y + 4xy + 7yzβ, the coefficient of βyβ is: A 3β β B β2β C β4β D β7β
E β16β
4B
2 If βt = 5βand βu = 7β, then β2t + uβis equal to: A 1β 7β B β32β C β24β
D β257β
E β70β
4B
3 If βx = 2β, then β3βxβ2βis equal to: A 3β 2β B β34β
D β25β
E β36β
4C
4D
4D
4E
4E
4F
C β12β
4 Which one of the following equivalences best generalises the number facts β 2 Γ 7 + 7 = 3 Γ 7, 2 Γ 12 + 12 = 3 Γ 12, 2 Γ 51 + 51 = 3 Γ 51 ?β A 7β x + 7 = 3xβ B βx + x = 2xβ C βx Γ x Γ x = 3xβ D β2x + x = 3xβ E β2x + 7 = 3xβ 5 Which of the following pairs does not consist of two like terms? A 3β xβand β5xβ B β3yβand β12yβ D β3cdβand β5cβ E β3xyβand βyxβ
C β3abβand β2abβ
6 A fully simplified expression equivalent to β2a + 4 + 3b + 5aβ is: A 4β β B β5a + 5b + 4β D β7a + 3b + 4β E β11abβ
C β10ab + 4β
7 The simplified form of β4x Γ 3yzβ is: A 4β 3xyzβ B β12xyβ
C β12xyzβ
D β12yzβ
E β4x3yzβ
8 The simplified form of _ β21ab ββis: 3ac 7b 7ab _ _ A βc β B β ac β
C _ β21b β 3c
D β7β
E _ βb β 7c
D β24xβ
E β8x + 12yβ
9 When brackets are expanded, β4(β 2x + 3y)ββbecomes: A β8x + 3yβ B β2x + 12yβ C β8x + 8yβ
Ext
4A
10 A number is doubled and then 5 is added. The result is then tripled. If the number is represented by βkβ, then an expression for this description is: A β3β(2k + 5)β B β6β(k + 5)β C β2k + 5β D β2k + 15β E β30kβ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
281
Extended-response questions
U N SA C O M R PL R E EC PA T E G D ES
A taxi driver charges β$3.50βto pick up passengers and then β$2.10β per kilometre travelled. a State the total cost if the trip length is: i β10 kmβ ii β20 kmβ iii β100 km.β b Write an expression for the total cost, in dollars, of travelling a distance of βdβ kilometres. c Use your expression to find the total cost of travelling β40 kmβ. d Prove that your expression is not equivalent to β2.1 + 3.5dβby substituting in a value for βdβ. e Another taxi driver charges β$6βto pick up passengers and then β$1.20βper kilometre. Write an expression for the total cost (in dollars) of travelling βdβkilometres in this taxi.
Chapter review
1
2 An architect has designed a room, shown below, for which βxβand βyβare unknown. (All measurements are in metres.)
x+5
x
x+y
x+2
y
3
a Find the perimeter of this room if βx = 3βand βy = 2β. b It costs β$3βper metre to install skirting boards around the perimeter of the room. Find the total cost of installing skirting boards if βx = 3βand βy = 2β. c Write an expression for the perimeter (in metres) of the room and simplify it completely. d Write an expanded expression for the total cost, in dollars, of installing skirting boards along the roomβs perimeter. e Write an expression for the total floor area in βm2β ββ.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
U N SA C O M R PL R E EC PA T E G D ES
5 Decimals
Maths in context: Decimal times and sport Decimal numbers and sport go together. From recording the times to reporting the statistics, decimals can be responsible for sponsorship funding and sporting fame. Olympic medals and sporting champions are often decided by tenths or hundredths of a second.
The challenging 400 m freestyle womenβs swimming event has had its world record (WR) times topped recently. In 2016, Katie Ledecky of USA made a WR time of 3:56.46 s. In 2022, Australiaβs Ariarne Titmus beat that time by 0.06 s, swimming a WR time of 3:56.40 s. Early 2023, Canadian Summer McIntosh swam 0.32 s faster again, making a new WR time of 3:56.08 s.
Major athletic track events now use an individual electronic starter beep, replacing the pistol sound that took 0.15 s to the farthest athlete. A camera scans the ο¬nish line 2000 times per second, signalling the digital timer the instant athletes cross the line. Exact times are recorded in milliseconds, then rounded to hundredths.
The Gold Coast 500 records times to four decimal places of a second. The previous lap time record for the S5000 race was 1:10.0499 s, set in 2013. However, in October 2022, Australiaβs Nathan Herne broke this record with a qualifying lap time of 1:09.6681 s, exactly 0.3818 s faster. Herne went on to win his ο¬rst ever S5000 race. Then, at Fukuoka 2023 World Aquatics Championships, Ariarne Titmus beat her 400 m rivals in an awesome new WR time of 3:55.38 s.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter contents
U N SA C O M R PL R E EC PA T E G D ES
5A Place value and decimals (CONSOLIDATING) 5B Rounding decimals 5C Adding and subtracting decimals 5D Multiplying and dividing decimals by 10, 100, 1000 etc. 5E Multiplying decimals 5F Dividing decimals 5G Connecting decimals and fractions 5H Connecting decimals and percentages 5I Expressing proportions using decimals, fractions and percentages
Victorian Curriculum 2.0
This chapter covers the following content descriptors in the Victorian Curriculum 2.0: NUMBER
VC2M7N03, VC2M7N04, VC2M7N05, VC2M7N06, VC2M7N07, VC2M7N10
Please refer to the curriculum support documentation in the teacher resources for a full and comprehensive mapping of this chapter to the related curriculum content descriptors.
Β© VCAA
Online resources
A host of additional online resources are included as part of your Interactive Textbook, including HOTmaths content, video demonstrations of all worked examples, auto-marked quizzes and much more.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 5
Decimals
5A Place value and decimals CONSOLIDATING LEARNING INTENTIONS β’ To understand the meaning of the decimal point β’ To know the place value of digits after a decimal point β’ To be able to decide if one decimal is larger or smaller than another decimal β’ To be able to convert proper fractions and mixed numerals to decimals, when their denominators are powers of ten
U N SA C O M R PL R E EC PA T E G D ES
284
Some quantities change by whole number amounts, such as the number of people in a room, but there are many quantities that increase or decrease continuously, such as your height, weight and age. We often talk about age as a whole number (e.g. Mike is 12 years old) but, in reality, our age is an ever-increasing (continuous) quantity. For example, if Mike is 12 years, 4 months, 2 weeks, 3 days, 5 hours, 6 minutes and 33 seconds old, then Mike is actually 12.380 621 47 years old! There are many numbers in todayβs society that are not whole numbers. For example, it is unusual to buy an item in a supermarket that is simply a whole number of dollars. The price of almost all shopping items involves both dollars and cents. A chocolate bar may cost $1.95, which is an example of a decimal number.
Nurses calculate exact volumes to be injected for prescribed medications. To allow accurate measurements, syringe scales show tenths or hundredths of a millilitre.
Lesson starter: Split-second timing
Organise students into pairs and use a digital stopwatch. Many studentsβ watches will have a suitable stopwatch function. β’ β’
Try to stop the stopwatch on exactly 10 seconds. Have two attempts each. Were you able to stop it exactly on 10.00 seconds? What was the closest time? Try these additional challenges with your partner. a Stop the watch exactly on: 7 seconds 1 seconds ii 8.37 seconds iii 9_ i 12_ 10 2 b How quickly can you start and stop the stopwatch? c How accurately can you time 1 minute without looking at the stopwatch?
iv 14.25 seconds
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5A Place value and decimals
285
KEY IDEAS β A decimal point is used to separate the whole number from the decimal or fraction part. β When dealing with decimal numbers, the place value table must be extended to involve tenths,
U N SA C O M R PL R E EC PA T E G D ES
hundredths, thousandths etc. The number β428.357β means:
Decimal point
Hundreds
Tens
Ones
.
Tenths
4
2
8
.
3
Hundredths Thousandths 5
7
4 Γ 100
2 Γ 10
8Γ1
.
1 3Γ_ 10
1 5Γ_ 100
1 7Γ_ 1000
400
20
8
.
3 _
5 _
7 _
10
100
1000
BUILDING UNDERSTANDING
1 For the number β58.237β, give the value of the following digits. a β2β b β3β
c β7β
2 A stopwatch is stopped at β36.57β seconds. a What is the digit displayed in the tenths column? b What is the digit displayed in the ones column? c What is the digit displayed in the hundredths column? d Is this number closer to β36βor β37β seconds?
3 Write the following number phrases as decimals. a seven and six-tenths b twelve and nine-tenths c thirty-three and four-hundredths d twenty-six and fifteen-hundredths e eight and forty-two hundredths f ninety-nine and twelve-thousandths
Example 1
Understanding decimal place value
What is the value of the digit β8βin the following numbers? a β12.85β b β6.1287β SOLUTION
EXPLANATION
8β a β_ 10
The β8βis in the first column after the decimal point, which is the tenths column.
b _ β 8 β 1000
The β8βis in the third column after the decimal point, which is the thousandths column.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 5
Decimals
Now you try What is the value of the digit β3βin the following numbers? a 1β 5.239β b β9.3508β
U N SA C O M R PL R E EC PA T E G D ES
286
Example 2
Converting simple fractions (with denominator 10, 100 etc) to decimals
Express each of the following proper fractions and mixed numerals as decimals. a _ β7 β c β3_ β 17 β b _ β5 β 10 100 100 SOLUTION
EXPLANATION
a _ β 7 β= 0.7β 10
7 βmeans seven-tenths, so put the β7βin the tenths column. β_ 10
b _ β 5 β= 0.05β 100 c β3_ β 17 β= 3.17β 100
5 βmeans five-hundredths, so put the β5βin the hundredths column. β_ 100 β3 _ β 17 βmeans β3βones and β17βone-hundredths. β17βhundredths is one-tenth and 100 seven-hundredths.
Now you try
Express each of the following proper fractions and mixed numerals as decimals. b _ β3 β c β4 _ β 307 β a _ β9 β 100 1000 10
Example 3
Comparing and sorting decimals
Place the correct mathematical symbol β< or >β, in between the following pairs of decimals to make a true mathematical statement. a 3β .5β β3.7β b 4β .28β β4.263β c β0.00 102β β0.01 001β SOLUTION
EXPLANATION
a 3β .5 < 3.7β
The first digit from left-to-right where the decimals are different is in the tenths position. β5βis less than β7β, so β3.5βis less than β3.7β.
b β4.28 > 4.263β
The ones digits β(4)βand tenths digits (β 2)βare equal, so look at the hundredths digit. β8 > 6β, so β4.28 > 4.263β Note: even though the thousandths digits β(β0 and 3β)βare in the opposite order, the hundredths digit is more important, so no further comparison is needed.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5A Place value and decimals
c 0β .00102 < 0.01001β
287
The first digit where the decimals differ is the hundredths digit. The number 0.00102 has 0 and the number 0.01001 has 1, which is greater.
β
β
hundredth
U N SA C O M R PL R E EC PA T E G D ES
hundredth
Now you try
Place the correct mathematical symbol < or > between the following pairs of decimals to make a true mathematical statement. a β8.7β
β8.44β
b 1β 0.184β
β10.19β
c β0.0401β
β0.00928β
Exercise 5A FLUENCY
Example 1
Example 1
Example 2a,b
Example 2c
Example 3
1
1, 2, 3β5ββ(1β/2β)β
2β5β(β 1β/2β)β
What is the value of the digit β3βin the following numbers? a β1.37β b β4.213β
c β10.037β
2 What is the value of the digit β6βin the following numbers? a 2β 3.612β b β17.46β d β0.693β e β16.4β g β2.3641β h β11.926β
c 8β 0.016β f β8.568 13β
3 Express each of the following proper fractions as a decimal. 8β a _ β3 β b β_ 10 10 d _ β 23 β 100
9β e β_ 10
g _ β 121 β 1000
74 β h β_ 1000
15 β c β_ 100 f
4 Express each of the following mixed numerals as a decimal. 7β b β5β_ a β6_ β4 β 10 10 d 1β _ β 16 β 100
83 β e β14β_ 100
g 5β _ β7 β 100
612 β h β18β_ 1000
2β6β(β 1β/3β)β
2 β β_ 100
3β c β212β_ 10 f
51 β β7β_ 100
5 Place the correct mathematical symbol β< or >βbetween the following pairs of decimals to make a true mathematical statement. β4.2β b β3.65β β3.68β c β1.05β β1.03β d β5.32β β5.4β a β4.7β e 6β .17β
β6.2β
f
β8.25β
β8.19β
g β2.34β
β2.3β
h β5.002β
β5.01β
β6.03β
β6.00β
j
β8.21β
β7.3β
k β9.204β
β8.402β
l
β4.11β
β4.111β
i
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 5
Decimals
6 State whether each of the following statements is true or false. a 7β .24 < 7.18β b 2β 1.32 < 20.89β c β4.61 > 4.57β
d β8.09 > 8.41β
e β25.8 β©½ 28.5β
f
2β .1118 β©½ 2.8001β
g β7.93 β©Ύ 8.42β
h β11.11 β©Ύ 11.109β
_ β 3 β= _ β 30 β
j
7 β= _ β_ β 70 β 10 100
5 ββ 5β k β_ 10
l
i
10
100
PROBLEM-SOLVING
_ β 2 ββ _ β 20 β
10
8, 9
7β(β β1/2β)β,β 8
100
8β10
U N SA C O M R PL R E EC PA T E G D ES
288
7 Find the difference between each of the following decimal numbers and their nearest whole number. a 6β .9β b 7β .03β c β18.98β d β16.5β e β17.999β f β4.99β g β0.85β h β99.11β 8 Arrange these groups of numbers in ascending order (i.e. smallest to largest). a 3β .52, 3.05, 3.25, 3.55β b β30.6, 3.06, 3.6, 30.3β c β17.81, 1.718, 1.871, 11.87β d β26.92, 29.26, 29.62, 22.96, 22.69β
9 The batting averages for five retired Australian Cricket test captains are: Adam Gilchrist β47.60β, Steve Waugh β51.06β, Mark Taylor β43.49β, Allan Border β50.56βand Kim Hughes β37.41β. a List the five players in descending order of batting averages (i.e. largest to smallest). b Ricky Pontingβs test batting average is β56.72β. Where does this rank him in terms of the retired Australian test captains listed above? 10 The depth of a river at β9 : 00βam on six consecutive days was: Day 1: β1.53 mβββ Day 2: β1.58 mβββ Day 3: β1.49 mβ Day 4: β1.47 mβββ Day 5: β1.52 mβββ Day 6: β1.61 mβ a On which day was the river level highest? b On which day was the river level lowest? c On which days was the river level higher than the previous day? REASONING
11
11, 12
12, 13
11 You can insert the digit β0β in some parts of the number β12.34βwithout changing the value (e.g. β12.340, 012.34β) but not others (e.g. β102.34, 12.304β). Explain why this is the case.
12 βAβ, βBβand βCβare digits and βA > B > Cβin decimal numbers. Write these numbers from smallest to largest. Note that the dot represents the decimal point. a βA.B, B.C, A.C, C.C, C.A, B.Aβ b βA.BC, B.CA, B.BB, C.AB, C.BC, BA.CA, AB.AB, A.AA, A.CAβ 13 Write the following as decimals, if βAβis a digit. a _ βA β b _ βA β 10 100
ENRICHMENT: Different decimal combinations
c _ βA β+ _ βA β 10 100 β
d βA + _ βA β+ _ β A β 10 1000
β
14
14 a Write as many different decimal numbers as you can and place them in ascending order using: i the digits β0, 1βand a decimal point. Each digit can be used only once. ii the digits β0, 1, 2βand a decimal point. Each digit can be used only once. iii the digits β0, 1, 2, 3βand a decimal point. Each digit can be used only once. b Calculate the number of different decimal numbers that could be produced using the digits β 0, 1, 2, 3, 4βand a decimal point.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5B Rounding decimals
289
5B Rounding decimals
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand that rounding involves approximating a decimal number to fewer decimal places β’ To be able to round decimals to a given number of decimal places
Decimal numbers sometimes contain more decimal places than we need. It is important that we are able to round decimal numbers when working with money, measuring quantities, including time and distance, or writing answers to some division calculations. For example, the distance around the school oval might be 0.39647 km, which rounded to 1 decimal place is 0.4 km. The rounded figure, although not precise, is accurate enough for most applications.
A carβs engine capacity is calculated by ο¬nding the total volume of its cylinders. This is given in litres, rounded to the nearest tenth. A 3.6 L engine is generally more powerful than a 2.5 L engine.
Running events are electronically measured in seconds and rounded to 2 decimal places. Usain Bolt has repeatedly broken his own world records. In August 2009, he set a new world record of 9.58 seconds over 100 m at the World Championships in Germany, which was 11-hundredths (0.11) of a second faster than his Beijing Olympic Games (August 2008) record of 9.69 seconds.
Lesson starter: Rounding brainstorm 1
In small groups, brainstorm occasions when it may be useful to round or estimate decimal numbers. Aim to get more than 10 common applications. 2 In pairs one person states a decimal number and the partner needs to state another decimal number that would allow the two numbers to add up to a whole number. Use mental arithmetic only. Start with 1 decimal place and try to build up to 3 or 4 decimal places.
KEY IDEAS
β Rounding involves approximating a decimal number to fewer decimal places. β To round a decimal:
β’ β’
Cut the number after the required decimal place; e.g. round to 2 decimal places. To determine whether you should round your answer up or down, consider only the digit immediately to the right of the specified place. For rounding purposes, this can be referred to as the critical digit. βcutβ
|
15.63 27
2 is the critical digit in this example
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 5
β’ β’
Decimals
If the critical digit is less than β5β(i.e. β0, 1, 2, 3βor β4β), then you round down. This means write the original number to the place required, leaving off all other digits. If the critical digit is 5 or more (i.e. β5, 6, 7, 8βor β9β), then you round up. This means write the original number to the place required, but increase this digit by β1β, carrying if necessary. Leave off all other digits.
U N SA C O M R PL R E EC PA T E G D ES
290
BUILDING UNDERSTANDING
1 For each of the following, select the closer alternative. a Is β5.79βcloser to β5.7βor β5.8β? b Is β6.777βcloser to β6.77βor β6.78β?
2 To round correctly to a specified number of places, you must know which digit is the critical digit. a State the critical digit in each of the following numbers. i β25.8174βrounded to β1βdecimal place. ββββCritical digit =_______ ii β25.8174βrounded to β2βdecimal places. β Critical digit =_______ iii β25.8174βrounded to β3βdecimal places. β Critical digit =_______ iv β25.8174βrounded to the nearest whole number. β Critical digit =_______ b State the correct rounded numbers for the numbers in parts i to iv above.
Example 4
Rounding decimals to 1 decimal place
Round both of the following to β1βdecimal place. a 2β 5.682β
b β13.5458β
SOLUTION
EXPLANATION
a 2β 5.7β
The critical digit is β8βand therefore the tenths column must be rounded up from a β6βto a β7β.
b β13.5β
The critical digit is β4βand therefore the tenths column remains the same, in effect rounding the original number down to β13.5β.
Now you try
Round both of the following to β1βdecimal place. a 5β 3.741β
b β4.1837β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5B Rounding decimals
Example 5
291
Rounding decimals to different decimal places
U N SA C O M R PL R E EC PA T E G D ES
Round each of the following to the specified number of decimal places. a Round β18.34728βto β3βdecimal places. b Round β0.43917βto β2βdecimal places. c Round β7.59967βto β3βdecimal places. SOLUTION
EXPLANATION
a β18.347β
The critical digit is β2β, therefore round down.
b β0.44β
The critical digit is β9β, therefore round up.
c 7β .600β
The critical digit is β6β, therefore round up. Rounding up has resulted in digits being carried over, since rounding up 7.599 in the thousandths digit means adding 0.001, giving the result of 7.600, correct to 3 decimal places.
Now you try
Round each of the following to the specified number of decimal places. a Round β18.62109βto β2βdecimal places. b Round β3.14159βto β4βdecimal places. c Round β0.1397βto β3βdecimal places.
Exercise 5B FLUENCY
1
1, 2β7ββ(β1/2β)β
Example 4a
Round each of the following to β1βdecimal place. a i β32.481β
ii β57.352β
Example 4b
b i
ii β46.148β
Example 4
Example 5
Example 5a,b
β19.7431β
2 Round each of the following to β1βdecimal place. a β14.82β b β7.38β e 6β .85β f β9.94β
c 1β 5.62β g β55.55β
3 Write each of the following correct to β2βdecimal places. a β3.7823β b β11.8627β c 5β .9156β e 1β 23.456β f β300.0549β g β3.1250β i β56.2893β j β7.121 999β k β29.9913β
2β7β(β β1/2β)β
3β7ββ(β1/4β)β
d 0β .87β h β7.98β
d 0β .93225β h β9.849β l β0.8971β
4 Round each of the following to the specified number of decimal places, given as the number in the brackets. a β15.913 (β 1)β b β7.8923 (β 2)β c β235.62 (β 0)β d β0.5111 (β 0)β e 2β 31.86 (β 1)β f β9.3951 (β 1)β g β9.3951 (β 2)β h β34.712 89 (β 3)β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
292
Chapter 5
Example 5c
Decimals
5 Round each of the following to the specified number of decimal places. a 2β 3.983 (β 1)β b 1β 4.8992 (β 2)β c β6.95432 (β 0)β
d β29.999731 (β 3)β
U N SA C O M R PL R E EC PA T E G D ES
6 Round each of the following to the nearest whole number; that is, round them to β0βdecimal places. a β27.612β b 9β .458β c β12.299β d β123.72β e β22.26β f β117.555β g β2.6132β h β10.7532β 7 Round each of the following amounts to the nearest dollar. a $β 12.85β b $β 30.50β c β$7.10β e β$120.45β f β$9.55β g β$1.39β PROBLEM-SOLVING
8
d β$1566.80β h β$36.19β
8, 9
8, 9
8 Some wise shoppers have the habit of rounding all items to the nearest dollar as they place them in their shopping basket. They can then keep a running total and have a close approximation as to how much their final bill will cost. Use this technique to estimate the cost of the following. a Jeanette purchases β10β items: β$3.25, $0.85, $4.65, $8.99, $12.30, $7.10, $2.90, $1.95, $4.85, $3.99β b Adam purchases β12βitems: β $0.55, $3.00, $5.40, $8.90, $6.90, $2.19, $3.20, $5.10, $3.15, $0.30, $4.95, $1.11β c Jeanetteβs actual shopping total is β$50.83βand Adamβs is β$44.75β. How accurate were Jeanetteβs and Adamβs estimations? 9 Electronic timing pads are standard in National Swimming competitions. In a recent β100 mβ freestyle race, Edwina receives a rounded time of β52.83βseconds and Jasmine a time of β53.17β seconds. a If the timing pads could only calculate times to the nearest second, what would be the time difference between the two swimmers? b If the timing pads could only calculate times to the nearest tenth of a second, what would be the time difference between the two swimmers? c What is the time difference between the two swimmers, correct to β2βdecimal places? d If the timing pads could measure in seconds to β3βdecimal places, what would be the quickest time in which Edwina might have swum the race?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5B Rounding decimals
REASONING
10
10, 11
293
10β12
U N SA C O M R PL R E EC PA T E G D ES
10 Henry and Jake each go to a petrol station to refuel their cars. a Henry buys β31.2βlitres of petrol at a cost of β158.9βcents per litre. Given that β31.2 Γ 158.9 = 4957.68β, round β$49.5768βto two decimal places to state the total price for the petrol. b If the price per litre had first been rounded to β$1.59β, find the total price for β31.2β litres. c Jake buys β52.8βlitres of premium petrol at a cost of β171.6βcents per litre. Find the total price he pays. (Remember to round your final answer to the nearest cent.) d How much extra would Jake have to pay if the cost per litre were rounded to the nearest cent before Jake bought β52.8β litres?
11 Without using a calculator, evaluate β15.735629 Γ· 7β, correct to β2βdecimal places. What is the least number of decimal places you need to find in the quotient to ensure that you have rounded correctly to β 2βdecimal places? 12 Samara believes β0.449999βshould be rounded up to β0.5β, but Cassandra believes it should be rounded down to β0.4β. Make an argument to support each of their statements, but then show the flaw in one girlβs logic and clearly indicate which girl you think is correct. ENRICHMENT: Rounding with technology
β
β
13, 14
13 Most calculators are able to round numbers correct to a specified number of places. Find out how to do this on your calculator and check your answers to Questions 3 and 4.
14 Spreadsheet software packages can also round numbers correct to a specified number of places. Find out the correct syntax for rounding cells in a spreadsheet program, such as Microsoft Excel, and then check your answers to Questions 5 and 6.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 5
Decimals
5C Adding and subtracting decimals LEARNING INTENTIONS β’ To understand that to add or subtract decimals, additional zeros might need to be entered for some decimal places β’ To be able to add decimals β’ To be able to subtract decimals
U N SA C O M R PL R E EC PA T E G D ES
294
Addition and subtraction of decimals follows the same procedures as those for whole numbers. To add or subtract whole numbers you must line up the ones, tens, hundreds and so on, and then you add or subtract each column. When dealing with the addition or subtraction of decimals the routine is the same. Consider how similar the following two sums are:
5 11 4 2 7 2 1 0 6 8 9 2
5 11 . 4 2 7.2 1 0.6 8 9.2
Lesson starter: Whatβs the total?
Each student thinks of three coins (gold or silver) and writes their total value on a sheet of paper. Each student in the class then estimates the total value of the amounts written down in the classroom. Record each studentβs estimated total. β’ β’
β’
Each student then writes the value of the three coins they thought of on the board (e.g. $2.70, $0.80 etc.). Students copy down the values into their workbooks and add the decimal numbers to determine the total value of the coins in the classroom. Which student has the closest estimated total?
Accountants, ο¬nancial planners and auditors are some of the ο¬nance professionals for whom adding and subtracting with decimals is an everyday calculation.
KEY IDEAS
β When adding or subtracting decimals, the decimal points and each of the decimal places must
be aligned under one another.
β The location of the decimal point in the answer is directly in line with the location of each of
the decimal points in the question.
β Once the numbers are correctly aligned, proceed as if completing whole number addition or
subtraction.
β If the numbers of decimal places in the numbers being added or subtracted are different, it can
be helpful to place additional zeros in the number(s) with fewer digits to the right of the decimal point to prevent calculation errors.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5C Adding and subtracting decimals
295
BUILDING UNDERSTANDING 1 β7.12, 8.5βand β13.032βare added together. Which of the following is the best way to prepare these numbers ready for addition?
B
7.12 8.5 + 13.032
7.12 8.5 + 13.032
C
7.120 8.500 + 13.032
D
7.12 8.5 + 13.032
U N SA C O M R PL R E EC PA T E G D ES
A
2 Which of the following is the correct way to present and solve the subtraction problem 7β 7.81 β 6.3β? A
B
77.81 β 6.3 84.11
Example 6
77.81 β 6.30 71.51
C
77.81 β 6.3 14.81
D
77.81 β 6.3 77.18
Adding decimals
Find the value of: a β8.31 + 5.93β
b β64.8 + 3.012 + 5.94β
SOLUTION
EXPLANATION
a
18.31 + 5.93 14.24
Make sure all decimal points and decimal places are correctly aligned directly under one another.
b
1614.800
Align decimal points directly under one another. Fill in missing decimal places with additional zeros. Carry out addition, following the same procedure as that for addition of whole numbers.
3.012 + 5.940 73.752
Now you try
Find: a β12.52 + 23.85β
b β4.28 + 5.9 + 2.152β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
296
Chapter 5
Decimals
Example 7
Subtracting decimals
Find the value of: a 5β .83 β 3.12β
U N SA C O M R PL R E EC PA T E G D ES
b β146.35 β 79.5β
SOLUTION
EXPLANATION
a
5.83 β 3.12 2.71
Make sure all decimal points and decimal places are correctly aligned directly under one another.
13 15 1
Align decimal points directly under one another and fill in missing decimal places with additional zeros. Carry out subtraction, following the same procedure as that for subtraction of whole numbers.
b
1 4 6 .35 β 7 9.50 6 6.85
Now you try
Find: a 9β .48 β 4.16β
b β12.294 β 5.41β
Exercise 5C FLUENCY
Example 6a
Example 6
Example 7
1
1, 2β4ββ(β1/2β)β
2β4β(β β1/2β)β
2β(β β1/2β)β,β 4ββ(β1/2β)β
Find the value of the following. a
2.14 + 3.61
b
3.40 + 1.35
c
10.053 + 24.124
d
5.813 + 2.105
e
3.12 + 4.29
f
2.38 + 4.71
g
8.90 + 3.84
h
2.179 + 8.352
b d f h
β5.37 + 13.81 + 2.15β β2.35 + 7.5β β1.567 + 3.4 + 32.6β β323.71 + 3.4506 + 12.9β
c
23.94 β 17.61
2 Find each of the following. a 1β 2.45 + 3.61β c β3.6 + 4.11β e β312.5 + 31.25β g β5.882 + 3.01 + 12.7β 3 Find: a
17.2 β 5.1
b
128.63 β 14.50
d
158.32 β 87.53
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5C Adding and subtracting decimals
4 Find: a β14.8 β 2.5β d β31.657 β 18.2β
b 2β 34.6 β 103.2β e β412.1 β 368.83β
PROBLEM-SOLVING
c 2β 5.9 β 3.67β f β5312.271 β 364.93β
5β7
5 Find the missing numbers in the following sums. a 3 .
5, 7, 9
5, 8, 10
U N SA C O M R PL R E EC PA T E G D ES
Example 7
297
b
+4 . 6
8 .
+
. 3
c
d
1
. 3
. 1 1
1
. 7 5
4 . 4
1 .
+
9
+
1 . 1
6
2 .
4
1 . 8
9
1 . 3
9
3
5
6 How much greater is β262.5βthan β76.31β?
7 Stuart wants to raise β$100βfor the Rainbow Club charity. He already has three donations of β$30.20β, β $10.50βand β$5.00β. How much does Stuart still need to raise? 8 Daily rainfalls for β4βdays over Easter were β12.5 mmβ, β3.25 mmβ, β 0.6 mmβand β32.76 mmβ. What was the total rainfall over the β4β-day Easter holiday?
9 Copy and complete the addition table below. β+β
0.01
0.05
0.38
β1.72β
0.3
β1.13β
0.75 1.20 1.61
1.42
β1.21β
β1.58β
β3.03β
10 Michelle earned β$3758.65βworking part-time over a β1β-year period. However, she was required to pay her parents β$20βper week for board for β52βweeks. Michelle also spent β$425.65βon clothing and β$256.90β on presents for her family and friends during the year. She placed the rest of her money in the bank. How much did Michelle bank for the year?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 5
Decimals
REASONING
11
11, 12
12, 13
11 An addition fact can be converted into two subtraction facts. For example, because β2.1 + 4.7 = 6.8β we know that β6.8 β 2.1 = 4.7βand β6.8 β 4.7 = 2.1βmust be true. Write two subtraction facts for each of the following. a 3β .2 + 4.95 = 8.15β b 2β .8 + 3.7 = 6.5β c β8.3 + 1.8 = 10.1β d β2.51 + 3.8 = 6.31β
U N SA C O M R PL R E EC PA T E G D ES
298
12 a Is it possible to add two decimal numbers together and get a whole number? If so, give an example, if not, explain why not. b Is it possible to subtract a decimal number from a different decimal number and get a whole number? Explain why or why not.
13 a Write down three numbers between β1βand β10β, each with β2βdecimal places, that would add to β11.16β. b Can you find a solution to part a that uses each digit from β1βto β9βexactly once each? ENRICHMENT: Money, money, money β¦
β
β
14
14 Investigate the following procedures and share your findings with a friend. a Choose an amount of money that is less than β$10.00β(e.g. β$3.25β). b Reverse the order of the digits and subtract the smaller number from the larger number (e.g. β$5.23 β $3.25 = $1.98β). c Reverse the order of the digits in your new answer and now add this number to your most recent total (e.g. β$1.98 + $8.91 = $10.89β). Did you also get β$10.89β? Repeat the procedure using different starting values. Try to discover a pattern or a rule. Justify your findings.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5D Multiplying and dividing decimals by 10, 100, 1000 etc.
299
5D Multiplying and dividing decimals by 10, 100, 1000 etc.
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand that a power of ten is a number like 10, 100, 1000 etc. β’ To be able to multiply a decimal by a power of ten β’ To be able to divide a decimal by a power of ten
Powers of 10 include 101, 102, 103, 104, β¦, which correspond to the numbers 10, 100, 1000, 10 000, β¦ Note that the number of zeros in the number is the same as the power of 10 for that number. For example, 104 = 10 000, the number ten thousand has four zeros and it is equal to ten to the power of four.
Lesson starter: Dynamic leap frog
A set of large number cards, enough for one card per student in the class, is required. The set of cards should include the following digits, numbers and symbols:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ., Γ,Γ·,1, 10, 100, 1000, 10 000, 1 00 000, 1 000 000 The decimal place card is vital! Cards should be big enough to be read from the back of the classroom. Any of the digits can be doubled up to increase the total number of cards. Each student receives one card. β’ β’
β’ β’
β’
Four students with one of the 0 to 9 digit cards stand up at the front and make a 4-digit number. The student with the decimal place card then positions themselves somewhere within this number or on either end. Now a student with the Γ or Γ· operation comes up the front. Finally, a student with a power of 10 card comes up and performs the actual calculation by gently moving the decimal place! Repeat a number of times with students swapping cards on several occasions.
Engineers, scientists and trade workers frequently multiply and divide by powers of 10. Medical researchers use microscopes to study viruses, bacteria and cells such as 8 mm = 0.008 mm in diameter. human red blood cells, _ 1000
KEY IDEAS
β Every number can be written so it contains a decimal point but it is usually not shown in
integers. For example: 345 and 345.0 are equivalent.
β Extra zeros can be added in the column to the right of the decimal point without changing the
value of the decimal. For example: 12.5 = 12.50 = 12.500 = 12.5000 etc. Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 5
Decimals
β When multiplying by powers of β10β, follow these rules.
β’
Move the decimal point to the right the same number of places as there are zeros in the multiplier. For example, if multiplying by β1000β, move the decimal point β3βplaces to the right. 5.7839 Γ 1000 = 5783.9
U N SA C O M R PL R E EC PA T E G D ES
300
(Note: The decimal point actually stays still and all the digits move three places to the left, but this is harder to visualise.)
β When dividing by powers of β10β, follow these rules.
β’
Move the decimal point to the left the same number of places as there are zeros in the multiplier. For example, if dividing by β100β, move the decimal point β2βplaces to the left.
2975.6 Γ· 100 = 29.756
(Note: The decimal point actually stays still and all the digits move two places to the right, but this is harder to visualise.)
BUILDING UNDERSTANDING
1 State the missing power of β10, (β 10, 100, 1000, etc.)β a β56.321 Γβ β= 5632.1β b β0.03572 Γβ
β= 3.572β
2 State the missing power of β10, (β 10, 100, 1000, etc.)β a β2345.1 Γ·β β= 2.3451β b β0.00367 Γ·β
β= 0.000367β
3 a How many places and in what direction does the decimal point in the number appear to move if the following operations occur? i βΓ 100β ii βΓ 1 000 000β
iii βΓ·1000β
iv βΓ·10 000 000β
b If all of the operations above had taken place on a number, one after the other, what would be the final position of the decimal place relative to its starting position?
Example 8
Multiplying by 10,100,1000 etc.
Calculate: a 3β 6.532 Γ 100β
b β4.31 Γ 10 000β
SOLUTION
EXPLANATION
a 3β 6.532 Γ 100 = 3653.2β
β100βhas two zeros, therefore decimal point appears to move β2β places to the right.ββ 36.532
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5D Multiplying and dividing decimals by 10, 100, 1000 etc.
b β4.31 Γ 10 000 = 43 100β
301
Decimal point appears to move β4βplaces to the right and additional zeros are inserted as necessary.ββ 4.3100
U N SA C O M R PL R E EC PA T E G D ES
Now you try Calculate: a β51.0942 Γ 1000β
Example 9
b β3.9 Γ 100β
Dividing by 10,100,1000 etc.
Calculate: a β268.15 Γ· 10β
b β7.82 Γ· 1000β
SOLUTION
EXPLANATION
a β268.15 Γ· 10 = 26.815β
β10βhas one zero, therefore decimal point is moved β1βplace to the left.ββ 268.15
b β7.82 Γ· 1000 = 0.00782β
Decimal point is moved β3βplaces to the left and additional zeros are inserted as necessary.ββ .00782
Now you try
Calculate: a β392.807 Γ· 10β
b β8.12 Γ· 100β
Example 10 Working with the βmissingβ decimal point Calculate: a β567 Γ 10 000β
b β23 Γ· 1000β
SOLUTION
EXPLANATION
a β567 Γ 10 000 = 5 670 000β
If no decimal point is shown in the question, it must be at the very end of the number. Four additional zeros must be inserted to move the invisible decimal point β4βplaces to the right. ββ 5670000.
b β23 Γ· 1000 = 0.023β
Decimal point is moved β3βplaces to the left.ββ 0.023
Now you try Calculate: a β289 Γ 1000β
b β78 Γ· 10 000β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
302
Chapter 5
Decimals
Example 11 Evaluating using order of operations Evaluate this expression, using the order of operations: β426 Γ· 100 + 10β(0.43 Γ 10 β 1.6)β EXPLANATION
426 Γ· 100 + 10β(β β0.43 Γ 10 β 1.6β)β β = 4.26 + 10β(β β4.3 β 1.6β)β β―β― β―β―β― β―β― β β β ββ―β― β =β 4.26 + 10βΓ 2.7β β = 4.26 + 27 β = 31.26
First, we must calculate the brackets. The division by β100β can also be done in the first step. β10β(4.3 β 2.6)βmeans 10 Γ (β 4.3 β 2.6)ββ.
U N SA C O M R PL R E EC PA T E G D ES
SOLUTION
Now you try
Evaluate β32.5 β 47 Γ· 10 + (β 3.1 + 5.4)βΓ 100β
Exercise 5D FLUENCY
Example 8
Example 8
Example 9
Example 9
Example 10
1
Calculate: a 2β 4.327 Γ 100β
1, 2β5ββ(1β/2β)β
b β43.61 Γ 10β
2β5β(β 1β/2β)β
2β5β(β 1β/3β)β
c β1.84 Γ 1000β
2 Calculate: a β4.87 Γ 10β d β14.304 Γ 100β g β12.7 Γ 1000β j β213.2 Γ 10β
b e h k
3β 5.283 Γ 10β β5.69923 Γ 1000β β154.23 Γ 1000β β867.1 Γ 100 000β
c f i l
β422.27 Γ 10β β1.25963 Γ 100β β0.34 Γ 10 000β β0.00516 Γ 100 000 000β
3 Calculate: a β42.7 Γ· 10β d β5689.3 Γ· 100β g β2.9 Γ· 100β j β36.7 Γ· 100β
b e h k
β353.1 Γ· 10β 1β 2 135.18 Γ· 1000β β13.62 Γ· 10 000β β0.02 Γ· 10 000β
c f i l
β24.422 Γ· 10β 9β 3 261.1 Γ· 10 000β β0.54 Γ· 1000β β1000.04 Γ· 100 000β
4 Calculate: a β22.913 Γ 100β d β22.2 Γ· 100β
b 0β .031 67 Γ 1000β e β6348.9 Γ 10 000β
c 4β .9 Γ· 10β f β1.0032 Γ· 1000β
5 Calculate: a 1β 56 Γ 100β d β16 Γ· 1000β g β7 Γ· 1000β
b 4β 3 Γ 1000β e β2134 Γ 100β h β99 Γ 100 000β
c β2251 Γ· 10β f β2134 Γ· 100β i β34 Γ· 10 000β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5D Multiplying and dividing decimals by 10, 100, 1000 etc.
PROBLEM-SOLVING
6 Evaluate the following, using the order of operations. a β1.56 Γ 100 + 24 Γ· 10β b c 3β + 10β(24 Γ· 100 + 8)β d e 3β 5.4 + 4.2 Γ 10 β 63.4 Γ· 10β f g β14 Γ· 100 + 1897 Γ· 1000β h
6β(β β1/2β)β, 8, 9
6ββ(β1/4β)β,β 8β10
1β 6 Γ· 100 + 32 Γ· 10β β10β(6.734 Γ 100 + 32)β β4.7 β 24 Γ· 10 + 0.52 Γ 10β β78.1 β 10β(64 Γ· 100 + 5)β
U N SA C O M R PL R E EC PA T E G D ES
Example 11
6β(β β1/2β)β,β 7
303
7 A service station charges β$1.87 per litreβof petrol. How much will it cost Tanisha to fill her car with 100 litres of petrol? 8 A large bee farm produces 1200 litres of honey per day. a If there are β1000 millilitresβin β1 litreβ, how many millilitres of honey can the farmβs bees produce in one day? b The farmβs honey is sold in β 100 millilitreβjars. How many jars of honey can the farmβs bees fill in one day?
9 Wendy is on a mobile phone plan that charges her β3βcents per text message. On average, Wendy sends β10βtext messages per day. What will it cost Wendy for β100βdays of sending text messages at this rate? Give your answer in cents and then convert your answer to dollars.
10 Darren wishes to purchase β10 000βshares at β$2.12βper share. Given that there is also an additional β$200β brokerage fee, how much will it cost Darren to purchase the shares? REASONING
11
11, 12
12, 13
11 The weight of a matchstick is β0.00015 kgβ. Find the weight of β10 000βboxes of matches, with each box containing β100βmatches. The weight of one empty match box is β0.0075 kgβ. 12 Complete the table below, listing at least one possible combination of operations that would produce the stated answer from the given starting number. Starting number
Answer
Possible two-step operations
β12.357β
β1235.7β
Γ β 1000β, βΓ·10β
3β 4.0045β
β0.0340045β
0β .003601β
β360.1β
B β AC.DFGβ
βBA.CDFGβ
D β .SWKKβ
βDSWKKβ
Fβ WYβ
βF.WYβ
βΓ·100β, βΓ 10β
13 The number β12 345.6789βundergoes a series of multiplication and division operations by different powers of β10β. The first four operations are: βΓ·1000, Γ 100, Γ 10 000βand βΓ·10β. What is the fifth and final operation if the final number is β1.234 567 89β?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 5
Decimals
ENRICHMENT: Scientific notation
β
β
14
14 Extremely large numbers and extremely small numbers are often written in a more practical way, known as standard form or scientific notation. For example, the distance from Earth to the Sun is β150 000 000 kilometresβ! The distance of β 150 million kilometresβcan be written in standard form as β1.5 Γ β10β8β βkilometresβ.
U N SA C O M R PL R E EC PA T E G D ES
304
On a calculator, β150 000 000βcan be represented as β1.5E8β.
1β .5 Γ β10β8β βand β1.5E8βrepresent the same large number and indicate that the decimal place needs to be moved β8βplaces to the right. 1.5E8 = 1.5 Γ 108 = 1.5 Γ 100 000 000 = 150000000
a Represent these numbers in standard form. i β50 000 000 000 000β ii β42 000 000β iii β12 300 000 000 000 000β b Use a calculator to evaluate the following. i β40 000 000 000 Γ 500 000 000β ii β9 000 000 Γ 120 000 000 000 000β c The distance from Earth to the Sun is stated above as β150 million kilometresβ. The more precise figure is β149 597 892 kilometresβ. Research how astronomers can calculate the distance so accurately. (Hint: It is linked to the speed of light.) d Carry out further research on very large numbers. Create a list of ten very large numbers (e.g. distance from Earth to Pluto, the number of grains in β1 kgβof sand, the number of stars in the galaxy, the number of memory bytes in a terabyteββ¦β). List your ten large numbers in ascending order. e How are very small numbers, such as β0.000000000035β, represented in standard form? You can research this question if you do not know. f Represent the following numbers in standard form. i β0.000001β ii β0.0000000009β iii β0.000000000007653β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5E Multiplying decimals
305
5E Multiplying decimals
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To be able to multiply decimals β’ To understand that it is helpful to estimate to check the position of the decimal point in the ο¬nal answer
There are countless real-life applications that involve the multiplication of decimal numbers. For example, finding the area of a block of land that is 34.5 m long and 5.2 m wide, or pricing a 4.5-hour job at a rate of $21.75 per hour. In general, the procedure for multiplying decimal numbers is the same as multiplying whole numbers. There is, however, one extra final step, which involves placing the decimal point in the correct position in the answer.
Lesson starter: Multiplication musings Consider the following questions within your group. β’ β’ β’ β’ β’
To score a synchronised dive, the sum of the execution and synchronisation scores is multiplied by the degree of difο¬culty and then by 0.6, to match the scoring of individual dives. For example: 39.5 Γ 3.2 Γ 0.6 = 75.84.
What happens when you multiply by a number that is less than 1? Consider the product of 15 Γ 0.75. Will the answer be more or less than 15? Why? Estimate an answer to 15 Γ 0.75. What is the total number of decimal places in the numbers 15 and 0.75? Calculate 15 Γ 0.75. How many decimal places are there in the answer?
KEY IDEAS
β When multiplying decimals, start by ignoring any decimal points and perform the multiplication
as you would normally. On arriving at your answer, place the decimal point in the correct position.
β The correct position of the decimal point in the answer is found by following the rule that
the total number of decimal places in the numbers in the question must equal the number of decimal places in the answer. 534 For example: Γ 12 decimal points 3 decimal places in the question 1068 ignored here 5340 5.34 Γ 1.2 = 6.408 6408 3 decimal places in the answer
β It is always worthwhile estimating your answer. This allows you to check that your decimal
point is in the correct place and that your answer makes sense.
β Answers can be estimated by first rounding each value to the nearest whole number.
e.g. 2.93 Γ 5.17 β 3 Γ 5 = 15
(β means approximately equal to)
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 5
Decimals
β When multiplying by multiples of β10β, initially ignore the zeros in the multiplier and any
decimal points and perform routine multiplication. On arriving at your answer, position your decimal point, remembering to move your decimal point according to the rules of multiplying by powers of β10β.
U N SA C O M R PL R E EC PA T E G D ES
306
BUILDING UNDERSTANDING
1 State out the total number of decimal places in each of the following product statements. a β4 Γ 6.3β b β3.52 Γ 76β c β42 Γ 5.123β d β8.71 Γ 11.2β e β5.283 Γ 6.02β f β0.00103 Γ 0.0045β
2 State where the decimal point should be inserted into each of the following answers so that the multiplication is true. a β6.4 Γ 3 = 192β
b β6.4 Γ 0.3 = 192β
c β0.64 Γ 0.3 = 192β
3 Why is it worthwhile to estimate an answer to a multiplication question involving decimals? 4 a What is the difference between a decimal point and a decimal place? b How many decimal points and how many decimal places are in the number β423.1567β?
5 State the missing words for the rule for multiplying decimal numbers (see the Key ideas in this section). The total number of decimal places _____________________ must equal the number of ________________ in the answer.
Example 12 Estimating decimal products
Estimate the following by first rounding each decimal to the nearest whole number. a 4β .13 Γ 7.62β b β2.8 Γ 12.071β SOLUTION
EXPLANATION
a 4.13 Γ 7.62β ββ 4 Γ 8β β β = 32
Round each value to the nearest whole number (β0βdecimal places). β4.13 β 4β, β7.62 β 8β Multiply the rounded values to obtain an estimate.
b 2.8 Γ 12.071 β 3 Γ 12 β β β β β = 36
2β .8 β 3βand β12.071 β 12β Multiply the rounded values to obtain an estimate.
Now you try
Estimate the following by first rounding each decimal to the nearest whole number. a 2β .9 Γ 4.18β b β7.05 Γ 10.9β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5E Multiplying decimals
307
Example 13 Multiplying decimals b β3.63 Γ 6.9β
U N SA C O M R PL R E EC PA T E G D ES
Calculate the following and check your answer. a β12.31 Γ 7β SOLUTION
EXPLANATION
a
Perform multiplication, ignoring the decimal point. There are β2βdecimal places in the numbers in the question, so there will be β2βdecimal places in the answer.
Β΄
b
1231 7 8617
β12.31 Γ 7 = 86.17β
Check that this value is close to the estimated value β (β 12 Γ 7 = 84)β, which it is.
363 69 3267 21780 25047
Ignore both decimal points. Perform routine multiplication. Total of β3βdecimal places in the numbers in the question, so there must be β3βdecimal places in the answer.
β3.63 Γ 6.9 = 25.047β
Check that this value is close to the estimated value β (β 4 Γ 7 = 28)β, which it is.
Β΄
Now you try Calculate: a β32.15 Γ 9β
b β2.6 Γ 1.78β
Example 14 Multiplying decimals by multiples of β1 0β Calculate: a β2.65 Γ 40 000β
b β0.032 Γ 600β
SOLUTION
EXPLANATION
a β2.65 Γ 40 000 = 106 000β
Ignore the decimal point and zeros. Multiply β265 Γ 4β.
Β΄
265 4 1060
β΄ 10.60 Γ 10 000 = 106000. = 106000
Position the decimal point in your answer. There are β 2βdecimal places in β2.65,βso there must be β2β decimal places in the answer. Move the decimal point β4βplaces to the right, to multiply β10.60βby β10 000β.
Continued on next page
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
308
Chapter 5
Decimals
SOLUTION
EXPLANATION
b β0.032 Γ 600 = 19.2β
Ignore the decimal point and zeros. Multiply β32 Γ 6β.
32 Β΄ 6 192
Position the decimal point in the answer.
U N SA C O M R PL R E EC PA T E G D ES
Shift the decimal point β2βplaces to the right to multiply β 0.192βby β100β.
β΄ 0.192 Γ 100 = 19.2
Now you try
Calculate: a 5β 2.9 Γ 3000β
b β0.0061 Γ 700β
Exercise 5E FLUENCY
Example 12
Example 13a
Example 13
Example 14
1
1, 2β4ββ(β1/2β)β
1, 3β5β(β β1/2β)β
1ββ(β1/2β)β,β 3β5ββ(β1/3β)β
Estimate the following by first rounding each decimal to the nearest whole number. a 3β .6 Γ 2.1β b 4β .9 Γ 8.31β c β7.2 Γ 10.3β d β2.95 Γ 7.832β
2 Calculate: a 1β 3.51 Γ 7β
b 2β .84 Γ 6β
c β5.13 Γ 6β
d β7.215 Γ 3β
3 Calculate: a 5β .21 Γ 4β e β3 Γ 72.82β i β0.34 Γ 16β
b 3β .8 Γ 7β f β1.293 Γ 12β j β43.21 Γ 7.2β
c 2β 2.93 Γ 8β g β3.4 Γ 6.8β k β0.023 Γ 0.042β
d β14 Γ 7.2β h β5.4 Γ 2.3β l β18.61 Γ 0.071β
4 Calculate: a 2β .52 Γ 40β d β1.4 Γ 7000β g β0.0034 Γ 200β
b 6β .9 Γ 70β e β3000 Γ 4.8β h β0.0053 Γ 70 000β
5 Calculate and then round your answer to the nearest dollar. a 5β Γ $6.30β b β3 Γ $7.55β d β$1.45 Γ 12β e β$30.25 Γ 4.8β g β34.2 Γ $2.60β h β0.063 Γ $70.00β PROBLEM-SOLVING
6
c β31.75 Γ 800β f β7.291 Γ 50 000β i β3.004 Γ 30β
c β4 Γ $18.70β f β7.2 Γ $5200β i β0.085 Γ $212.50β 6β8
7β9
6 Use decimal multiplication to answer the following. a Anita requires β4.21 mβof material for each dress she is making. She is planning to make a total of seven dresses. How much material does she need? b The net weight of a can of spaghetti is β0.445 kgβ. Find the net weight of eight cans of spaghetti. c Jimbo ran β5.35 kmβeach day for the month of March. How many kilometres did he run for the month? 7 Bernard is making a cubby house for his children. He needs β32βlengths of timber, each β2.1 mβ long. a What is the total length of timber needed to build the cubby house? b What is the cost of the timber if the price is β$2.95βper metre?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5E Multiplying decimals
309
8 A lawyer charges β$125.00βper hour to assist her client. How much does the lawyer charge the client if she works on the job for β12.25β hours?
U N SA C O M R PL R E EC PA T E G D ES
9 According to its manufacturer, a particular car can travel β14.2 kmβon β1βlitre of petrol. a How far could the car travel on β52βlitres of petrol? b The car has β23.4βlitres of fuel in the tank and must complete a journey of β310 kmβ. Will it make the journey without refuelling? c If the car does make the journey, how much petrol is left in the tank at the end of the trip? If the car does not make the journey, how many extra litres of fuel are needed? REASONING
10
10, 11
11β13
10 When estimating a decimal product, the actual answer is usually higher or lower than the estimated value. For example, β3.8 Γ 2.1 = 7.98β, which is lower than the estimate of β4 Γ 2 = 8β. a Without calculating the actual value, explain why β3.92 Γ 6.85βwill be lower than the estimated value of β4 Γ 7 = 28β. b Explain why β4.21 Γ 7.302βis higher than the estimated product. c Give an example of a situation where one value rounds up, and the other rounds down but the actual product is higher than the estimate. 11 Write down two numbers, each with β2βdecimal places, that when multiplied by β1.83βwill give an Βanswer between β0.4βand β0.5β.
12 Write down one number with β4βdecimal places that when multiplied by β345.62βwill give an answer between β1βand β2β. 13 a b c d
If β68 Γ 57 = 3876β, what is the answer to β6.8 Γ 5.7β? Why? If β23 Γ 32 = 736β, what is the answer to β2.3 Γ 32β? Why? If β250 Γ 300 = 75 000β, what is the answer to β2.5 Γ 0.3β? Why? What is β7 Γ 6β? What is the answer to β0.7 Γ 0.6β? Why?
ENRICHMENT: Creating a simple cash register
β
β
14
14 Using a spreadsheet (e.g. Microsoft Excel, Google sheets, Apple numbers), design a user-friendly cash register interface. You must be able to enter up to β10βdifferent items into your spreadsheet. You will need a quantity column and a cost per item column. Using appropriate formulas, the total cost of the bill should be displayed, and there should then be room to enter the amount of money paid and, if necessary, what change should be given. When your spreadsheet is set up, enter the following items. β4βchocolate bars β@ $1.85β each toothpaste β@ $4.95β β3βloaves of bread β@ $3.19β each
β2 kgβsausages β@ $5.99 per kgβ
newspaper β@ $1.40β
tomato sauce β@ $3.20β
β2 Γ 2βlitres of milk β@ $3.70β each
washing powder β@ $8.95β
β2βpackets of Tim Tams β@ $3.55β each
β5 Γ 1.25βlitres of soft drink β@ $0.99β each
βMoney paid = $80.00β
If your program is working correctly, the amount of change given should be β$13.10β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
310
Chapter 5
5A
1
What is the place value of the digit β6βin the following numbers? a 3β .5678β b β126.872β
2 Express each of the following fractions as a decimal. a _ β9 β b _ β 19 β 10 1000
c 3β _ β1 β 4
3 Place the correct mathematical symbol β< or >βin between the following pairs of decimals to make a true mathematical statement. β4.1β b β3.719β β3.71β a 4β .02β
U N SA C O M R PL R E EC PA T E G D ES
Progress quiz
5A
Decimals
5A
c β8.2β
5B
β8.19β
d β14.803β
β14.8005β
4 Round each of the following to the specified number of decimal places. a β16.8765βto β2βdecimal places b β2.34999βto β3βdecimal places c β0.66667βto β1βdecimal place 5 Find: a β0.9 + 4.5β c β12.89 β 9.37β
b β12.56 + 3.671 + 0.8β d β8.06 β 2.28β
5D
6 Evaluate: a 3β .45 Γ 1000β
b β65.345 Γ· 100β
5E
7 Estimate the following products by first rounding each value to the nearest whole number. a 3β .28 Γ 4.7β b β9.12 Γ 2.04β c β3.81 Γ 5.12β
5C
5E
5C/E
5C/E
8 Calculate: a 4β 5 Γ 2000β c β4.78 Γ 0.4β
b β23.8 Γ 5β d β4.56 Γ 30 000β
9 Insert the decimal point in the answer so that each mathematical sentence is true. a β12 β 3.989 = 8011β b β1.234 Γ 0.08 Γ 2000 = 19 744β 10 It costs β$59.85βfor a large bottle of dog shampoo. Find: a the change from paying with one β$50βnote and one β $20β note b the cost of buying three bottles.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5F Dividing decimals
311
5F Dividing decimals
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To be able to divide decimals by whole numbers β’ To be able to divide decimals by other decimals
Similar to multiplication of decimal numbers, there are countless real-life applications that involve the division of decimal numbers. However, unlike multiplying decimal numbers, where we basically ignore the decimal points until the very end of the calculation, with division we try to manipulate the numbers in the question in such a way as to prevent dividing by a decimal number.
Division using decimal places is required in many calculations of an average. One example is calculating a batting average in cricket, where Australiaβs highest Test cricket batting average is 99.94, held by Sir Donald Bradman.
Lesson starter: Division decisions
Consider the following questions within your group. β’ β’ β’ β’ β’
What happens when you divide by a number that is less than 1 but greater than 0? Consider the answer of 10 Γ· 0.2. Will the answer be more or less than 10? Why? Estimate an answer to 10 Γ· 0.2. Calculate the answer of 100 Γ· 2. How does this compare to the answer of 10 Γ· 0.2? Can you think of an easier way to calculate 21.464 Γ· 0.02?
KEY IDEAS
β Division of decimal numbers by whole numbers
β’ β’
Complete as you would normally with any other division question. The decimal point in the quotient (answer) goes directly above the decimal point in the dividend. For example: 60.524 Γ· 4 1 5.131
β2
1
4 6 0.524
β Division of decimal numbers by other decimals
β’ β’
β’
Change the divisor into a whole number. Whatever change is made to the divisor must also be made to the dividend. For example: 24.524 Γ· 0.02
24.562 Γ· 0.02 = 2456.2 Γ· 2 When dividing by multiples of 10, initially ignore the zeros in the divisor and perform routine division. On arriving at your answer, you must then re-position your decimal point according to the rules of dividing by powers of 10. For each zero in the divisor that you ignored initially, the decimal point must move 1 place to the left.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 5
Decimals
BUILDING UNDERSTANDING 1 For the question β36.52 Γ· 0.4 = 91.3β, which of the following options uses the correct terminology? A β36.52βis the divisor, β0.4βis the dividend and β91.3βis the quotient. B β36.52βis the dividend, β0.4βis the divisor and β91.3βis the quotient. C β36.52βis the quotient, β0.4βis the dividend and β91.3βis the divisor. D β36.52βis the divisor, β0.4βis the quotient and β91.3βis the dividend.
U N SA C O M R PL R E EC PA T E G D ES
312
2 Explain where you place the decimal point in the quotient (i.e. answer), when dividing a decimal by a whole number.
3 Calculate: a β1200 Γ· 20β b β120 Γ· 2β c β12 Γ· 0.2β e Explain why these questions all give the same answer.
d β1.2 Γ· 0.02β
4 For each of the following pairs of numbers, move the decimal points the same number of places so that the second number becomes a whole number. a β3.2456, 0.3β b β120.432, 0.12β c 0β .00345, 0.0001β
d β1234.12, 0.004β
Example 15 Dividing decimals by whole numbers Calculate: a 4β 2.837 Γ· 3β
b β0.0234 Γ· 4β
SOLUTION
EXPLANATION
a 1β 4.279β
Carry out division, remembering that the Βdecimal point in the answer is placed directly above the decimal point in the dividend.
1 4.2 7 9
β1 22
3 4 2.8 3 7
b β0.00585β
0.00 5 8 5 2
3 2
4 0.02 3 4 0
Remember to place zeros in the answer every time the divisor βdoesnβt goβ. Again, align the decimal point in the answer Βdirectly above the decimal point in the question. An additional zero is required at the end of the dividend to terminate the decimal answer.
Now you try Calculate: a 1β 5.184 Γ· 4β
b β0.413 Γ· 5β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5F Dividing decimals
313
Example 16 Dividing decimals by decimals b β0.03152 Γ· 0.002β
U N SA C O M R PL R E EC PA T E G D ES
Calculate: a β62.316 Γ· 0.03β SOLUTION
EXPLANATION
a β62.316 Γ· 0.03β β―β― β = 6231.6 Γ· 3 = 2077.2
Need to divide by a whole number.
20 7 7.2
β
2 2
3 62 3 1.6
b 0.031 52 Γ· 0.002β β ββ―β― = 31.52 Γ· 2 = 15.76 1 5.7 6
β
2 1 1
2 3 1.5 2
62.316 Γ· 0.03
Move each decimal point β2βplaces to the right. Carry out the division question β6231.6 Γ· 3β. Multiply the divisor and dividend by β1000β. 0.03152 Γ· 0.002
Move each decimal point β3βplaces to the right. Carry out the division question β31.52 Γ· 2β.
Now you try
Calculate: a β47.188 Γ· 0.04β
b β0.2514 Γ· 0.003β
Example 17 Dividing decimals by multiples of β1 0β Calculate β67.04 Γ· 8000.β SOLUTION
EXPLANATION
08.38 8 67.04
Ignore the three zeros in the β8000β. Divide β67.04βby β8β.
8.38 Γ· 1000 = 0.00838 67.04 Γ· 8000 = 0.00838
Now divide by β1000β, resulting in moving the decimal point β3βplaces to the left.
β
Now you try
Calculate β149.8 Γ· 7000β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
314
Chapter 5
Decimals
Example 18 Evaluating using order of operations
U N SA C O M R PL R E EC PA T E G D ES
Calculate using the order of operations: β3.8 β 1.6 Γ 0.45 + 5 Γ· 0.4β SOLUTION
EXPLANATION
3.8 β 1.6 Γ 0.45 + 5 Γ· 0.4 βββ―β― =β 3.8 β 0.72 +β 12.5β β β β―β― β―β― β = 3.08 + 12.5 β = 15.58
First carry out βΓβand βΓ·β, working from left to right. Then carry out β+βand βββ, working from left to right.
Now you try
Calculate using the order of operations: β9.3 + 1.3 Γ 4.05 β 1.2 Γ· 0.25β
Exercise 5F FLUENCY
Example 15a
Example 15
Example 16a
Example 16b
Example 17
1
1, 2β5ββ(β1/2β)β
1, 2β3ββ(β1/3β)β,β 4β6ββ(β1/2β)β
2β4β(β β1/4β)β,β 5β6ββ(β1/3β)β
Calculate: a 7β .02 Γ· 2β
b 1β 5.57 Γ· 3β
c β17.48 Γ· 4β
d β45.565 Γ· 5β
2 Calculate: a 8β .4 Γ· 2β e β4.713 Γ· 3β i β0.0144 Γ· 6β
b 3β 0.5 Γ· 5β f β2.156 Γ· 7β j β234.21 Γ· 2β
c 6β 4.02 Γ· 3β g β38.786 Γ· 11β k β3.417 Γ· 5β
d β2.822 Γ· 4β h β1491.6 Γ· 12β l β0.01025 Γ· 4β
3 Calculate: a 6β .14 Γ· 0.2β e β45.171 Γ· 0.07β
b 2β 3.25 Γ· 0.3β f β10.78 Γ· 0.011β
c 2β .144 Γ· 0.08β g β4.003 Γ· 0.005β
d β5.1 Γ· 0.6β h β432.2 Γ· 0.0002β
4 Calculate: a 0β .3996 Γ· 0.009β
b 0β .0032 Γ· 0.04β
c β0.04034 Γ· 0.8β
d β0.948 Γ· 1.2β
5 Calculate: a 2β 36.14 Γ· 200β d β0.846 Γ· 200β
b 4β 13.35 Γ· 50β e β482.435 Γ· 5000β
6 Calculate the following, rounding your answers to β2βdecimal places. a β35.5 kg Γ· 3β b β$213.25 Γ· 7β d β287 g Γ· 1.2β e β482.523 L Γ· 0.5β
c β3.71244 Γ· 300β f β0.0313 Γ· 40β c β182.6 m Γ· 0.6β f β$5235.50 Γ· 9β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5F Dividing decimals
PROBLEM-SOLVING
8, 10β12
7β(β β1/2β)β,β 8β10
7 Calculate the following, using the order of operations. a β3.68 Γ· 2 + 5.7 Γ· 0.3β b 6β (β 3.7 Γ 2.8 + 5.2)β ( ) c 1β 7.83 β 1.2β 8.1 β 2.35 β d β9.81 Γ· 0.9 + 75.9 Γ· 10β e (β 56.7 β 2.4)βΓ· (β 0.85 Γ· 2 + 0.375)β f β34.5 Γ 2.3 + 15.8 Γ· β(0.96 β 0.76)β
U N SA C O M R PL R E EC PA T E G D ES
Example 18
7β(β β1/2β)β,β 8
315
8 Find the missing digits in these division questions. a b 0. 6 4 0. 3β2. 6
7
β
3 1.
2
c
2.
β10. 7
5
d
2. 1 4
β15.
2 9
9 Charlie paid β$12.72βto fill his ride-on lawnmower with β8βL of fuel. What was the price per litre of the fuel that he purchased?
10 Dibden is a picture framer and has recently purchased β214.6 mβof timber. The average-sized picture frame requires β90 cmβ (ββ 0.9 m)βof timber. How many average picture frames could Dibden make with his new timber?
11 A water bottle can hold β600 mLβof water. How many water bottles can be filled from a large drink container that can hold β16 Lβ?
12 Six friends go out for dinner. At the end of the evening, the restaurantβs bill is β$398.10β. a As the bill is split equally among the six friends, how much does each person pay? b Given that they are happy with the food and the service, they decide to round the amount they each pay to β$70β. What is the waiterβs tip?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 5
Decimals
REASONING
13
13, 14
14, 15
13 Clara purchases β1.2 kgβof apples for β$3.90β. Her friend Sophia buys β900 gβof bananas for β$2.79βat the same shop. Find the cost per kilogram of each fruit. Which type of fruit is the best value in terms of price per kilogram? 14 A police radar gun measures a car to be β231.5 mβaway. Exactly β0.6βseconds later, the radar gun measures the same car to be β216.8 mβ away. a Determine the speed of the car in metres per second (m/s). b Multiply your answer to part a by β3.6βto convert your answer to km/h. c The car is travelling along an β80 βkmβ/βhβstretch of road. Is the car speeding?
U N SA C O M R PL R E EC PA T E G D ES
316
15 Given that β24.53 Γ 1.97 = 48.3241β, write down the value of each of the following divisions, without using a calculator. a β48.3241 Γ· 1.97β b β48.3241 Γ· 2.453β c β4832.41 Γ· 1.97β d β483.241 Γ· 245.3β e β0.483241 Γ· 0.197β f β483 241 Γ· 2453β ENRICHMENT: What number am I?
β
β
16
16 In each of the following, I am thinking of a number. Given the following clues for each, find each number. a When I add β4.5βto my number and then multiply by β6β, the answer is β30β. b When I divide my number by β3βand then add β2.9β, the answer is β3β. c When I multiply my number by β100βand then add β9β, the answer is β10β. d When I multiply my number by β5βand then add a half, the answer is β6β. e When I subtract β0.8βfrom my number, then divide by β0.2βand then divide by β0.1β, the answer is β200β. f Make up three of your own number puzzles to share with the class.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5G Connecting decimals and fractions
317
5G Connecting decimals and fractions
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To be able to convert decimals to fractions β’ To be able to convert fractions to decimals β’ To understand the symbols to indicate recurring decimals
Decimals and fractions are both commonly used to represent numbers that are not simply whole numbers. It is important that we know how to convert a decimal number to a fraction, and how to convert a fraction to a decimal number.
Lesson starter: Match my call β’
β’ β’
In pairs, nominate one student to be βFraction kidβ and the other to be βDecimal expertβ. βFraction A civil engineer can design a tunnel for an underground kidβ starts naming some common fractions and railway having an upward slope with a gradient of 0.03. By βDecimal expertβ tries to give the equivalent converting 0.03 to 3/100 the engineer then knows that for every horizontal 100 m, the tunnel must rise by 3 m. decimal value. Start with easy questions and build up to harder ones. After 10 turns, swap around. This time βDecimal expertβ will name some decimal numbers and βFraction kidβ will attempt to call out the equivalent fraction. Discuss the following question in pairs: Which is easier, converting fractions to decimals or decimals to fractions?
KEY IDEAS
β Converting decimals to fractions
β’
Using your knowledge of place value, express the decimal places as a fraction whose denominator is 10, 100, 1000 etc. Remember to simplify the fraction whenever possible.
25 = _ 1 e.g. 0.25 = _ 100 4
β Converting fractions to decimals
β’
β’
β’
When the denominator is 10, 100, 1000 etc. we can simply change the fraction to a decimal through knowledge of place value. When the denominator is not 10, 100, 1000 etc. try to find an equivalent fraction whose denominator is 10, 100, 1000 etc. and then convert to a decimal. A method that will always work for converting fractions to decimals is to divide the numerator by the denominator.
37 = 0.37 e.g. _ 100
2=_ 4 = 0.4 e.g. _ 5 10
0. 6 2 5 5 5. 50 20 40 = 0.625 = 8 e.g. 8
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 5
Decimals
β Recurring decimals are decimals with a repeated pattern.
β’
A dot, dots or a bar above a number or numbers indicates a repeated pattern. βΎβ βe.g. _ β1 β= 0.33333 β¦ = 0.β3Λ ββββββββββββββββββββββ_ β13 β= 1.181818 β¦ = 1.β1Λ β8Λ βor 1.β18β 3 11 12 β= 1.714285714285 β¦ = 1.β7Λ β1428β5Λ βor 1.β714285β βΎβ β_ 7
U N SA C O M R PL R E EC PA T E G D ES
318
BUILDING UNDERSTANDING
1 State the missing numbers in each of these statements, which convert common fractions to decimals.
a 1= 2
c 3= 4
10
100
b 4=
25
2
=
1
= 0.5 = 0.
d
5
= 0.25
4 = 0. 10
2 State the missing numbers in each of these statements, which convert decimals to fractions, in simplest form.
a 0.2 = c 0.8 =
10 8
=
=
1 5
b 0.15 =
100
=
d 0.64 = 64 =
5
100
3
25
3 State whether each of the following is true or false. Use the examples in the Key ideas to help. a β0.333β¦= 0.3β b β0.1111β¦= 0.β1Λ β Λ c β3.2222β¦= 3.β2β d β1.7272β¦= 1.7β2Λ β e β3.161616β¦= 3.β1Λ β6Λ β f β4.216216β¦= 4.ββΎ 216ββ
Example 19 Converting decimals to fractions
Convert the following decimals to fractions or mixed numerals in their simplest form. a 0β .239β b β10.35β SOLUTION
EXPLANATION
a _ β 239 β 1000
β0.239 = 239β thousandths Note _ β 239 βcannot be simplified further. 1000 β0.35 = 35βhundredths, which can be simplified further by dividing the numerator and denominator by the highest common factor of β5β.
35 β= 10β_ 7β b β10β_ 100 20
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5G Connecting decimals and fractions
319
Now you try
U N SA C O M R PL R E EC PA T E G D ES
Convert the following decimals to fractions in their simplest form. a β0.407β b β7.45β
Example 20 Converting fractions to decimals Convert the following fractions to decimals. 17 β a β_ 100
c _ β7 β 12
b 5β _ β3 β 5
SOLUTION
EXPLANATION
17 β= 0.17β a β_ 100 6 β= 5.6β b β5_ β3 β= 5β_ 10 5
1β 7β hundredths
_ β 6 βis an equivalent fraction of _ β3 ββ, whose
10 5 denominator is a power of β10β.
7 β= 0.58333 β¦ or 0.58β3Λ β c β_ 12
0.5 8 3 3 3...
β 7 10 4 4 4
12 7.0 0 0 0 0
Now you try
Convert the following fractions to decimals. a _ β 123 β 1000
3β b 1β 9β_ 4
c _ β5 β 6
Exercise 5G FLUENCY
Example 19a
Example 19
Example 19
Example 20a
1
1, 2β6ββ(1β/2β)β
2β6β(β 1β/2β)β,β 7
2β6β(β 1β/3β)β,β 7
Convert the following decimals to fractions in their simplest form. a β0.3β b β0.41β c β0.201β
d β0.07β
2 Convert the following decimals to fractions in their simplest form. a β0.4β b β0.22β c β0.15β
d β0.75β
3 Convert the following decimals to fractions or mixed numerals in their simplest form. a β0.5β b β6.4β c β10.15β d β18.12β e 3β .25β f β0.05β g β9.075β h β5.192β 4 Convert each of these fractions to decimals. 7β a β_ b _ β9 β 10 10 e _ β121 β 100
f
3β _ β 29 β 100
31 β c β_ 100
79 β d β_ 100
g _ β 123 β 1000
3 β h β_ 100
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
320
Chapter 5
5 Convert the following fractions to decimals, by first changing the fraction to an equivalent fraction whose denominator is a power of β10β. 23 β a _ β4 β c _ β7 β b _ β1 β d β_ 5 20 2 50 3β 1β 5β f β3β_ h β_ e _ β19 β g β_ 4 8 20 2
U N SA C O M R PL R E EC PA T E G D ES
Example 20b
Decimals
Example 20c
6 Convert the following fractions to decimals, by dividing the numerator by the denominator. 1β 2β a β_ d β_ c _ β3 β b _ β3 β 2 5 4 6 e _ β1 β 3
_ β3 β
f
3β h β_ 7
5β g β_ 12
8
7 Copy and complete the following fraction/decimal tables. The quarters table (part c) has already been done for you. It is worth trying to memorise these fractions and their equivalent decimal values. a halves b thirds Fraction
_ β0 β
2
1β β_ 2
2β β_ 2
Fraction
Decimal
Decimal
3
1β β_ 3
2β β_ 3
3β β_ 3
Decimal
c quarters
Fraction
_ β0 β
d fifths
0β β_ 4 β0β
_ β1 β
4 β0.25β
_ β2 β
4 β0.5β
_ β3 β
4 β0.75β
_ β4 β
Fraction
4 β1β
PROBLEM-SOLVING
5β 0 β β_ 3 β β_ 1 β β_ 4 β β_ 2 β β_ β_ 5 5 5 5 5 5
Decimal
8, 9
8, 9
9, 10, 11
8 Arrange the following from smallest to largest. a _ β1 ,β 0.75, _ β5 ,β 0.4, 0.99, _ β1β 2 8 4
β1 β, 0.58, 0.84, _ β4 β b _ β3 ,β 0.13, _ 7 9 5
9 Tan and Lillian are trying to work out who is the better chess player. They have both been playing chess games against their computers. Tan has played β37βgames and beaten the computer β11β times. Lillian has played only β21βgames and has beaten the computer β6β times. a Using a calculator and converting the appropriate fractions to decimals, determine who is the better chess player. b Lillian has time to play another four games of chess against her computer. To be classified as a better player than Tan, how many of these four games must she win?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5G Connecting decimals and fractions
U N SA C O M R PL R E EC PA T E G D ES
10 To estimate the thickness of one sheet of A4 paper, Christopher measures a ream of paper, which consists of β500β sheets of A4 paper. He determines that the pile is β55 mmβthick. How thick is one sheet of A4 paper? Express your answer as a decimal number and also as a fraction.
321
11 When _ β4 βis expressed in decimal 7 form, find the digit in the β23rdβ decimal place. Give a reason for your answer.
REASONING
12
12, 14
13β15
12 a Copy and complete the following fraction/decimal table. Fraction
1β β_ 2
1β β_ 3
1β β_ 4
1β β_ 5
1β β_ 6
1β β_ 7
1β β_ 9
1β β_ 8
_ β1 β
10
Decimal
b Comment on the trend in the decimal values as the denominator increases. c Try to explain why this makes sense.
13 a Copy and complete the following decimal/fraction table. Decimal
β0.1β
0β .2β
β0.25β
β0.4β
0β .5β
0β .6β
β0.75β
β0.8β
β0.9β
Fraction
b Comment on the trend in the fractions as the decimal value increases. c Try to explain why this makes sense.
14 Write three different mixed numerals with different denominators that are between the decimal values of β2.4βand β2.5β. 15 A decimal that is not a recurring decimal is called a terminating decimal. βΎβand β2.β4Λ βare recurring decimals, but β12.432βand β9.07βare terminating decimals. For example, β3.β71β a Can all fractions be converted to terminating decimals? Explain why or why not. b Can all terminating decimals be converted to fractions? Explain why or why not.
ENRICHMENT: Design a decimal game for the class
β
β
16
16 Using the skill of converting decimals to fractions and vice versa, design an appropriate game that students in your class could play. Ideas may include variations of Bingo, Memory, Dominoes etc. Try creating a challenging set of question cards.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 5
Decimals
The following problems will investigate practical situations drawing upon knowledge and skills developed throughout the chapter. In attempting to solve these problems, aim to identify the key information, use diagrams, formulate ideas, apply strategies, make calculations and check and communicate your solutions.
Average height of a group of friends
U N SA C O M R PL R E EC PA T E G D ES
Applications and problem-solving
322
1
The following table shows the name and height of six friends. Name
Height
Sam
β1.45 mβ
Ellie
β1.53 mβ
Leo
β1.71 mβ
Cammo
β1.54 mβ
Sabina
β1.44 mβ
Felix
β1.87 mβ
The friends are interested in comparing their heights, calculating the average height and looking at how the average might be impacted by the addition of another friend to the group. a Place the friends in ascending height order. b How much taller is Felix than Sam? c What is the total height of the six friends? d What is the average height of the six friends? e When a seventh friend joins the group, the new average height becomes β1.62 mβ. What is the height of the new friend?
A drive to the moon?
2 Linh learns that the approximate distance from Earth to the Moon is β384 400 kmβand that an average car cruising speed is β100 km / hβ. Linh is curious about how long it would take her and her family to drive to the Moon. While she of course knows this is not possible, she does wonder how long it would take if it was. a Given the above information, how many hours would it take Linh and her family to drive to the Moon? b Convert your answer to days, giving your answer: i correct to two decimal places ii in exact form using decimals iii in exact form using fractions iv in exact form using days and hours. c Investigate how long would it take Linh and her family to drive to the Sun.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Applications and problem-solving
323
Changing a poolβs dimensions
U N SA C O M R PL R E EC PA T E G D ES
Applications and problem-solving
3 A local swimming pool is a rectangular prism and has the following dimensions: βlength = 15.2 m, width = 4.8 m, depth = 1.2 mβ. The council are interested in changing the size of the pool by a certain factor and want to be clear about the impact of such changes on its depth and perimeter. a Aiko works for the local council and has put in a submission for increasing each of the dimensions of the pool by a factor of β1.25β. What are the dimensions of the pool in Aikoβs submission? b Kaito wishes to make a scale model of Aikoβs new pool, where the dimensions are _ β 1 βthβof the real 10 size. What would be the dimensions of Kaitoβs model pool? c The local council decide to increase the size of the pool, but they are concerned about the pool being too deep and only want to increase the depth of the pool by a factor of β1.1β. They are happy to increase the width by a factor of β1.25β, but because of the available room they are actually keen to increase the length of the pool by a factor of β1.75β. What are the dimensions of the new council approved pool? d A different local councillor argued that the original perimeter of the pool must remain the same. If Aiko still wishes to increase the length of the pool by a factor of β1.25β, what would the reduced width of the new pool need to be?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 5
Decimals
5H Connecting decimals and percentages LEARNING INTENTIONS β’ To understand the meaning of βper centβ (%). β’ To be able to convert between percentages and decimals
U N SA C O M R PL R E EC PA T E G D ES
324
Percentages are commonly used in todayβs society. Per cent is derived from the Latin words per centum, meaning βout of 100β.
Percentages give an idea of proportion. For example, if a newspaper states that 2000 people want a council swimming pool constructed, then we know how many want a pool but we do not know what proportion of the community that is. However, if there are 2500 people in this community, the newspaper can state that 80% want a swimming pool. This informs us that a majority of the community (i.e. 80 out of every 100 people) want a swimming pool constructed.
A restaurant uses only parts of some vegetables such as broccoli, capsicum and cabbage. The vegetable order can be increased by 15%, so 1.15 kg is purchased for each 1 kg of vegetables that are cooked.
Lesson starter: Creative shading β’
β’ β’
Draw a square of side length 10 cm and shade exactly 20% or 0.2 of this figure. Draw a square of side length 5 cm and shade exactly 60% or 0.6 of this figure. Draw another square of side length 10 cm and creatively shade an exact percentage of the figure. Ask your partner to work out the percentage you shaded.
What percentage is shaded?
KEY IDEAS
β The symbol, %, means per cent. It comes from the Latin words per centum, which translates to
βout of 100β.
23 = 0.23 For example: 23% means 23 out of 100 which equals _ 100
β To convert a percentage to a decimal, divide by 100. This is done by moving the decimal point 2
places to the left.
For example: 42% = 42 Γ· 100 = 0.42.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5H Connecting decimals and percentages
325
β To convert a decimal to a percentage, multiply by β100β. This is done by moving the decimal
point β2βplaces to the right. For example: 0.654 Γ 100 = 65.4. Therefore 0.654 = 65.4%
U N SA C O M R PL R E EC PA T E G D ES
(Note: As in Section 5D, it is not actually the decimal point that moves; rather, it is the digits that move around the stationary decimal point.)
BUILDING UNDERSTANDING
1 7β 2.5%βis equivalent to which of the following decimals? A β72.5β B β7.25β C β0.725β
D β725.0β
2 1β 452%βis equivalent to which of the following decimals? A β0.1452β B β14.52β C β145 200β
D β145.20β
3 0β .39βis equivalent to which of the following percentages? A β39%β B β3.9%β C β0.39%β
D β0.0039%β
4 Prue answered half the questions correctly for a test marked out of β100β. a What score did Prue get on the test? b What percentage did Prue get on the test? c What score would you expect Prue to get if the test was out of: i β10β ii β200β iii β40β d What percentage would you expect Prue to get if the test was out of: i β10β ii β200β iii β40β
iv β2β iv β2β
5 State the missing numbers or symbols in these numbers sentences. a 58% = 58 out of b 35% =
= 58
out of 100 = 35 Γ·
100 =
=
58
100
=0
=
58
.35
Example 21 Converting percentages to decimals Express the following percentages as decimals. a β30%β b β3%β c β240%β
d β12.5%β
e 0β .4%β
SOLUTION
EXPLANATION
a 30% = 0.3
β30 Γ· 100 = 0.3β
b 3% = 0.03
3β Γ· 100 = 0.03β Note that 3 can be written as 003. before moving the decimal point: 003.
Continued on next page Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
326
Chapter 5
Decimals
β240 Γ· 100 = 2.4β
d 12.5% = 0.125
Decimal point moves β2βplaces to the left.
e 0.4% = 0.004
Decimal point moves β2βplaces to the left.
U N SA C O M R PL R E EC PA T E G D ES
c 240% = 2.4
Now you try
Express the following percentages as decimals. a 9β 2%β b β4%β c β150%β
d β7.6%β
e β0.05%β
Example 22 Converting decimals to percentages Express the following decimals as percentages. a 0β .045β
b β7.2β
SOLUTION
EXPLANATION
a 0β .045 = 4.5%β
Multiplying by β100βmoves the decimal point β2βplaces to the right.
b β7.2 = 720%β
Multiply β7.2βby β100β.
Now you try
Express the following decimals as percentages. a 0β .703β
b β23.1β
Exercise 5H FLUENCY
Example 21a
Example 21b
1
1β3ββ(1β/2β)β,β 5ββ(1β/2β)β
1β6ββ(1β/2β)β
3β6β(β 1β/2β)β
Express the following percentages as decimals. a 7β 0%β b 4β 0%β e β48%β f β32%β
c 5β 0%β g β71%β
d β20%β h β19%β
2 Express the following percentages as decimals. a 8β %β b 2β %β
c β3%β
d β5%β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5H Connecting decimals and percentages
Example 21aβc
c 6β 8%β g β100%β k β75%β
d 5β 4%β h β1%β l β199%β
4 Express the following percentages as decimals. a β22.5%β b β17.5%β e 1β 12.35%β f β188.8%β i β0.79%β j β0.025%β
c 3β 3.33%β g β150%β k β1.04%β
d 8β .25%β h β520%β l β0.95%β
5 Express the following decimals as percentages. a β0.8β b β0.3β
c β0.45β
d β0.71β
6 Express the following decimals as percentages. a β0.416β b β0.375β e 0β .025β f β0.0014β
c 2β .5β g β12.7β
d 2β .314β h β1.004β
U N SA C O M R PL R E EC PA T E G D ES
Example 21d,e
3 Express the following percentages as decimals. a β32%β b β27%β e 6β %β f β9%β i β218%β j β142%β
327
Example 21a
Example 22
PROBLEM-SOLVING
7, 8
7β9
8β10
7 Place the following values in order from highest to lowest. (Hint: Convert all percentages to decimals first.) a 86%, 0.5%, 0.6, 0.125, 22%, 75%, 2%, 0.78 b 124%, 2.45, 1.99%, 0.02%, 1.8, 55%, 7.2, 50 8 At a hockey match, β65%βof the crowd supports the home team. What percentage of the crowd does not support the home team?
9 Last Saturday, Phil spent the β24βhours of the day in the following way: β0.42βof the time was spent sleeping, β0.22βwas spent playing sport and β0.11β was spent eating. The only other activity Phil did for the day was watch TV. a What percentage of the day did Phil spend watching TV? b What percentage of the day did Phil spend not playing sport?
10 Sugarloaf Reservoir has a capacity of β96βgigalitres. However, as a result of the drought it is only β25%β full. How many gigalitres of water are in the reservoir?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 5
Decimals
REASONING
11
11, 12
12, 13
11 A β β, βBβ, βCβand βDβare digits. Write the following decimal numbers as percentages. a 0β .ABCDβ b βA.ACβ c βAB.DCβ d β0.0DDβ e βC.DBAβ f β0.CCCDDDβ 12 βAβ, βBβ, βCβand βDβare digits. Write the following percentages as decimal numbers. a βA.B%β b βBCD%β c βAC%β d β0.DA%β e βABBB%β f βDD.D%β
U N SA C O M R PL R E EC PA T E G D ES
328
13 Trudy says that it is impossible to have more than β100%β. She supports her statement by saying that if you get every question correct in a test, then you get β100%βand you cannot get any more. a Do you agree with Trudyβs statement? b Provide four examples of when it makes sense that you cannot get more than β100%β. c Provide four examples of when it is perfectly logical to have more than β100%β. ENRICHMENT: AFL ladder
β
β
14
14 The Australian Rules football ladder has the following column headings. Pos
Team
P
W
L
D
F
6
A
%
Pts
Brisbane Lions
22
13
8
1
7
Carlton
22
13
9
0
2017
1890
106.72
54
2270
2055
110.46
8
Essendon
22
10
11
52
1
2080
2127
97.79
9
Hawthorn
22
9
42
13
0
1962
2120
92.55
36
10
Port Adelaide
22
9
13
0
1990
2244
88.68
36
a Using a calculator, can you determine how the percentage column is calculated? b What do you think the βFβ and the βAβ column stand for? c In their next match, Essendon scores β123βpoints for their team and has β76βpoints scored against them. What will be their new percentage? d By how much do Hawthorn need to win their next game to have a percentage of β100β? e If Port Adelaide plays Hawthorn in the next round and the final score is Port Adelaide β124β beats Hawthorn β71β, will Port Adelaideβs percentage become higher than Hawthornβs?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5I Expressing proportions using decimals, fractions and percentages
329
5I Expressing proportions using decimals, fractions and percentages
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand that finding one number as a proportion of a total generally involves division β’ To be able to express a proportion as a fraction, decimal or percentage β’ To be able to use fractions, decimals and percentages to compare proportions.
Sometimes we want to know the proportion of a certain quantity compared to a given total or another quantity. This may be done using a fraction, percentage or ratio. Earthβs surface, for example, is about 70% ocean. So, the proportion of land could be 3 (as a fraction) written as 30% (as a percentage) or _ 10 or 0.3 (as a decimal). For different situations, it could make more sense represent a proportion as a percentage, a fraction or as a decimal.
Lesson starter: Municipal parkland
A bricklayer can mix 4 parts of sand to 1 part of cement to make a dry mortar mix. These proportions can also be expressed as fractions and percentages: 4 = 80%, cement _ 1 = 20%. i.e. sand _ 5 5
A municipal area is set aside in a new suburb and is to be divided into three main parts. The total area is 10 000 m2 and the three parts are to be divided as follows. Parkland: 6000 m2 Shops: 2500 m2 Playground and skate park: 1500 m2 β’ Express each area as a proportion of the total, first as a fraction and then as a percentage and finally as a decimal. 3 of the parkland is to be a lake, what percentage of the total area is the lake? β’ If _ 5
KEY IDEAS
β To express one quantity as a proportion of the other, start with the fraction:
red parts _ amount e.g. _ fraction = _ =2 total ( total parts 5 ) 2=_ 4 = 0.4 β This fraction can be converted to a decimal e.g. _ ( ) 5 10 β The decimal or fraction can be converted to a percentage by multiplying by 100, 2 Γ 100 = 40 so _ 2 = 40% e.g. 0.4 Γ 100 = 40 so 0.4 = 40%, _ 5 5
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 5
Decimals
β An object or quantity divided into parts can be analysed using fractions, decimals or percentages.
Shaded proportion as a fraction: _ β2 β 5 Shaded proportion as a decimal: β0.4β Shaded proportion as a percentage: β40%β
U N SA C O M R PL R E EC PA T E G D ES
330
β Compare proportions using the same type of number (e.g., by converting all the proportions to
percentages). e.g. Use percentages to compare the scores β17βout of β20β, β0.65βand β82%β _ β17 β= _ β 85 β= 85% and 0.65 = _ β 65 β= 65%β 20 100 100
BUILDING UNDERSTANDING
1 This square shows some coloured triangles and some white triangles. a How many triangles are coloured? b How many triangles are white? c What fraction of the total is coloured? d What percentage of the total is coloured? e What fraction of the total is white? f What percentage of the total is white?
2 A farmerβs pen has β2βblack sheep and β8βwhite sheep. a How many sheep are there in total? b What fraction of the sheep are black? c What fraction of the sheep are white? d What percentage of the sheep are black? e What percentage of the sheep are white? 3 a What fraction of the diagram is shaded? b convert your answer to part a into a decimal. c What percentage of the diagram is shaded?
Example 23 Expressing proportions as fractions, decimals and percentages Express the proportion β$40βout of a total of β$200β: a as a fraction b as a decimal SOLUTION
c as a percentage
EXPLANATION
Γ· 40
40 1 a Fraction: 200 = 5 Γ· 40
Write the given amount and divide by the total. Then simplify the fraction by dividing the numerator and denominator by the common factor (40).
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5I Expressing proportions using decimals, fractions and percentages
331
Γ2 b Decimal:
1= 2 5 10 Γ2
Convert the simplified fraction to a decimal by making a β10β, β100β, β1000β, etc.
U N SA C O M R PL R E EC PA T E G D ES
β= 0.2β
c Percentage: _ β 40 βΓ _ β100 β = 20 200 1 β β β β β β―β― 40 _ therefore β β = 20% 200
Multiply the fraction by β100βto convert to a percentage, or convert the decimal to a percentage. Γ·2
40 20 Alternative method: 200 = 100 = 20% Γ·2
Now you try
Express the proportion β$30βout of a total of β$120β: a as a fraction b as a decimal
c as a percentage
Example 24 Finding fractions and percentages from parts A glass of cordial is β3βparts syrup to β7βparts water. a Express the amount of syrup as a fraction of the total. b Express the proportion of syrup as a decimal. c Express the amount of water as a percentage of the total.
10 9 8 7 6 5 4 3 2 1
water
syrup
SOLUTION
EXPLANATION
a βFraction = _ β3 β 10 3 β= 0.3β b β_ 10
There is a total of β10βparts, including β3βparts syrup.
β100% β c Percentage = _ β 7 βΓ _ ββ―β― β 10 β 1 β β = 70%
There is a total β7βparts water in a total of β10β parts.
Convert the fraction to a decimal.
Now you try
A mixture of cordial is β2βparts syrup to β3βparts water. a Express the amount of syrup as a fraction of the total. b Express the proportion of water as a decimal. c Express the amount of water as a percentage of the total.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
332
Chapter 5
Decimals
Example 25 Comparing proportions
U N SA C O M R PL R E EC PA T E G D ES
On a test, Jenny scored _ β16 β, Brendan managed to get β0.6βof the questions correct and Raj received a 25 score of β61%β. Use percentages to compare the three test scores and decide who scored the highest. SOLUTION
EXPLANATION
Jenny: _ β16 β= _ β 64 β= 64%β 25 100 Brendan: β0.6 = _ β 60 β= 60%β 100 Raj: β61%β So Jenny scored the highest
Convert Jennyβs fraction to a percentage and Brendanβs decimal to a percentage. Then compare the three percentages.
Now you try
Millie, Adam and Yoe are all completing the same hike. Millie has completed _ β 7 βof the hike, Adam 20 has completed β0.4βof the hike and Yoe has completed β39%βof the hike. Use percentages to compare the three proportions and decide who has completed the largest proportion of the hike.
Exercise 5I FLUENCY
Example 23a
Example 23b
Example 23c
Example 23
Example 24
1
1β5β(β β1/2β)β, 6, 8
4, 6β9
Express the following proportions as a fraction of the total. Remember to simplify all fractions. a 4β βout of β10β b β5βout of β20β c β$30βout of β$300β d β15βsheep out of a total of β25β sheep
2 Express the following proportions as a decimal. a 3β βout of β10β c β29βout of β100β 3
1β5ββ(β1/2β)β,β 6β9
Express the following proportions as a percentage. a 2β 5βout of β100β c β23βout of β50β
b β7βout of β10β d β5βout of β100β b β7βout of β10β d β13βout of β20β
4 Express the following proportions as a fraction, a decimal and a percentage of the total. a 3β 0βout of a total of β100β b β3βout of a total of β5β c β$10βout of a total of β$50β d β$60βout of a total of β$80β e β2 kgβout of a total of β40 kgβ f β30 mLβout of a total of β200 mLβ 5 Write the shaded proportions of each shapeβs total area as a decimal and as a percentage. a b c
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
5I Expressing proportions using decimals, fractions and percentages
d
e
333
f
6 A jug of lemonade is made up of β2βparts lemon juice to β18βparts water. Express the proportion of the total that is lemon juice as: a a fraction b a decimal c a percentage.
U N SA C O M R PL R E EC PA T E G D ES Example 24
Example 24
Example 25
7 A mix of concrete is made up of β1βpart cement to β4βparts sand. a Express the amount of cement as a fraction of the total. b Express the amount of cement as a percentage of the total. c Express the proportion of the total that is sand as a decimal.
7 β, Mel managed to get β0.67βof the questions correct and Wendy received a score 8 On a test, Val scored β_ 10 of β69%β. Use percentages to compare the three test scores and decide who scored the highest.
9 Three siblings walk home from school. By 4:00 pm Wally has completed _ β29 βof the walk, Su has 50 completed β0.55βof the walk and Drew has completed β53%βof the walk. Use percentages to compare the three proportions and decide who has walked the most. PROBLEM-SOLVING
10, 11
10 In a new subdivision involving β20 000 βm2β ββ, specific areas are set aside for the following purposes. β’ Dwellings: β12 000 βm2β β β’ Shops: β1000 βm2β β β’ Roads/Paths: β3000 βm2β β β’ Park: β2500 βm2β β β’ Factories: Remainder a What area is set aside for factories? b Express the area of the following as both a fraction and a percentage of the total area. i shops ii dwellings c What fraction of the total area is either dwellings, shops or factories? d What percentage of the total area is park or roads/paths?
10β12
11β13
iii park
11 Gillian pays β$80βtax out of her income of β$1600β. What percentage of her income does she keep? Also state this proportion as a decimal. 12 Express the following proportions as a fraction, percentage and decimals of the total. Remember to ensure the units are the same first. a β20βcents of β$5β b β14βdays out of β5β weeks c 1β 5βcentimetres removed from a total length of β3β metres d β3βseconds taken from a world record time of β5β minutes e 1β 80βgrams of a total of β9β kilograms f 1β 500βcentimetres from a total of β0.6β kilometres
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 5
Decimals
13 Of β20βstudents, β10βplay sport and β12βplay a musical instrument, with some of these students playing both sport and music. Two students do not play any sport or musical instrument. a What fraction of the students play both sport and a musical instrument? b What percentage of the students play a musical instrument but not a sport?
music ?
sport ?
? ?
U N SA C O M R PL R E EC PA T E G D ES
334
REASONING
14, 15
14β16
15β17
14 For recent class tests, Ross scored β45βout of β60βand Maleisha scored β72βout of β100β. Use percentages or decimals to show that Ross got a higher proportion of questions correct. 15 The prices of two cars are reduced for sale. A hatch priced at β$20 000βis now reduced by β$3000βand a 4WD priced at β$80 000βis now reduced by β$12 800β. Determine which car has the largest percentage reduction, giving reasons.
16 A yellow sports drink has β50 gβof sugar dissolved in fluid that weighs β250 gβ, including the weight of the sugar. A blue sports drink has β57 gβof sugar dissolved in fluid that weighs β300 gβ, including the weight of the sugar. Which sports drink has the least percentage of sugar? Give reasons.
17 A mixture of dough has βaβparts flour to βbβparts water. a Write an expression for the fraction of flour within the mixture. b Write an expression for the percentage of water within the mixture. ENRICHMENT: Transport turmoil
β
β
18
18 A class survey of β30βstudents reveals that the students use three modes of transport to get to school: bike, public transport and car. All of the students used at least one of these three modes of transport in the past week. Twelve students used a car to get to school and did not use any of the other modes of transport. One student used all three modes of transport and one student used only a bike for the week. There were no students who used both a bike and a car but no public transport. Five students used both a car and public transport but not a bike. Eight students used only public transport.
Use this diagram to help answer the following. bike a How many students used both a bike and public transport but not a car? b What fraction of the students used all three modes of transport? c What fraction of the students used at least one mode of transport, with one of their modes being a bike? car public d What fraction of the students used at least one mode of transport, transport with one of their modes being public transport? e What percentage of students used public transport and a car during the week? f What percentage of students used either public transport or a car or both during the week?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Modelling
335
ASX share market game
U N SA C O M R PL R E EC PA T E G D ES
They consider three possible investment scenarios for the year.
Modelling
Seven friends join together in a share market game and buy a total of β10 000βshares at β$1.40βeach. In the game they pay a brokerage fee of β2%βof the value of the shares whenever they buy or sell shares. They plan to sell the shares at the end of the year.
Scenario β1: 10%βshare price growth. Scenario β2: 25%βshare price growth. Scenario β3: 5%βshare price fall.
Present a report for the following tasks and ensure that you show clear mathematical workings and explanations where appropriate.
Preliminary task a b c d
State the total cost of buying the β10 000βshares not including the brokerage fee. Find the value of the brokerage fee for buying the β10 000β shares. If the share price of β$1.40βincreased by β10%β, what would be the new value of a single share? If the share price of β$1.40βdecreased by β5%β, what would be the new value of a single share?
Modelling task
Formulate Solve
Evaluate and verify
Communicate
a The problem is to decide if the friends will earn a profit of at least β$500βeach at the end of the year, after all fees are considered. Write down all the relevant information to help solve this problem. b Assume the shares increase in value by β10%βas in scenario β1β. i What would be the value of the shares at the end of the year? ii Find the value of the brokerage fee for selling the β10 000βshares at the end of the year. iii If the friends sell the shares after a year, what is the total profit after all the buy/sell fees are paid? iv How much profit does each of the seven friends receive? (Assume they share the profits equally.) c Assume the shares increase in value by β25%βas in scenario β2β. Find the resulting profit per person. d If scenario β3βoccurs and the shares decrease in value by β5%β, find the resulting loss per person. e By comparing your results from above, decide if the friends earn more than β$500βeach from any of the three scenarios. f By increasing the expected growth rate, construct a scenario so that the profit per person is greater than β$500βafter one year. Justify your choice by showing workings. g Summarise your results and describe any key findings.
Extension questions
a Construct a scenario that delivers exactly β$500βprofit per person after one year, when rounded to the nearest dollar. b Investigate the effect on the profit per person in the following cases: i if the fee paid on each purchase or sale of shares is reduced ii if the number of friends involved is changed from seven.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
336
Chapter 5
Decimals
_
Key technology: Spreadsheets We know that decimals can be used to write numbers which are of equal value to certain fractions and β8 ββ, β2.β3Λ βis equal to _ percentages. β1.6βfor example is equal to _ β7 βand β0.57βis equal to β57%β. Such decimals are 3 5 either terminating or infinitely recurring. The are other numbers, however, that cannot be described using a terminating or infinitely recurring decimal. βΟβis one such number and is introduced in the Measurement
U N SA C O M R PL R E EC PA T E G D ES
Technology and computational thinking
Approximating β β 2β
_
3_
chapter and you will have already seen numbers like β β 2 βand ββ5 βwhich also fit in this category. The square root of β2βhas been studied since the ancient times and is the diagonal distance across a square of side _ length one unit. A Babylonian clay tablet dated approximately β1700βBC shows an approximation of β β 2 ββ which is correct to β6βdecimal places.
_
β2
1
1
1 Getting started
_
One way to approximate numbers which have infinitely recurring decimals uses continued fractions. β β 2 ββas 1 ________________ a continued fraction is shown here. β1 + ββ―β―β 1 2 + _____________ ββ―β― β 1 _ 2+β β 2+_ β 1 β 2+β± _
a Find an approximation of ββ2 βby evaluating the following. Write your answer as a decimal each time. 1β i β1 + β_ 2 ii β1 + _ β 1 β 2+_ β1 β 2
1 β iii β1 + _ β 2+_ β 1 β 2+_ β1 β 2
1 iv β1 + β____________ β β―β― 1 β 2+_ β 2+_ β 1 β 2+_ β1 β 2
b Describe how the next continued fraction is obtained from the previous one.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Technology and computational thinking
2 Applying an algorithm
337
_
Technology and computational thinking
Here is an algorithm which finds an approximation to β2 using a continued fraction with n calculations. a Complete this table for n = 5 writing down the values for i and f from beginning to end.
Start
U N SA C O M R PL R E EC PA T E G D ES
Input n
i
0
1
2
f
2
2.5 2.4
3
b What is the value of d outputted near the end of the _ program? Check that this value is close to the value of β2 obtained using a calculator. c Describe the parts of the algorithm that make the following happen. i The f value is updated each time to a new f value. ii The algorithm knows when to end. d Why do you think the rule d = f β 1 is used at the end to deliver the final result? Hint: Look at the continued fraction _ for β2 .
i = 0, f = 2
i = i + 1, f = 2 +
Is i = n?
1 f
No
Yes
d=fβ1
3 Using technology
Letβs now use a spreadsheet to apply the above algorithm.
Output d
End
_
a Construct a spreadsheet to calculate the continued fraction for β2 using the information provided here. b Fill down at cells A5, B5 and C4 for about 10 rows. _ c What do you notice about the approximation of β2 as you go down the list? _ d Use your result to write β2 correct to: i 3 decimal places ii 7 decimal places.
4 Extension
a Use the above technique to find an approximation for: _ _ i β5 ii β3 _ b You can also use dynamic geometry to estimate the value of β2 by constructing a special triangle and measuring the length of the diagonal as shown. Try this technique. Then try constructing _ _ another number like β3 or β5 .
1.42
1
1
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
338
Chapter 5
Decimals
The concept of a βbest buyβ relates to purchasing a product that is the best value for money. To determine the βbest buyβ you can compare the prices of similar products for the same weight.
U N SA C O M R PL R E EC PA T E G D ES
Investigation
Best buy
Converting units
a Convert the following to a price per kg. i β2 kgβof apples for β$3.40β ii β5 kgβof sugar for β$6.00β iii β1.5 kgβof cereal for β$4.50β iv β500 gβof butter for β$3.25β b Convert the following to a price per β100 gβ. i β300 gβof grapes for β$2.10β ii β1 kgβof cheese for β$9.60β iii β700 gβof yoghurt for β$7.49β iv β160 gβof dip for β$3.20β
STRAWBERRY JAM jar β375 gβ β$3.95β β$10.53 per kgβ
STRAWBERRY JAM jar β250 gβ β$2.95β β$11.80 per kgβ
Finding βbest buysβ
a By converting to a price per kg, determine which is the best buy. i β2 kgβof sauce A for β$5.20βor β1 kgβof sauce B for β$2.90β ii β4 kgβof pumpkin A for β$3.20βor β3 kgβof pumpkin B for β$2.70β iii β500 gβof honey A for β$5.15βor β2 kgβof honey B for β$19.90β iv β300 gβof milk A for β$0.88βor β1.5 kgβof milk B for β$4.00β b By converting to a price per β100 gβ, determine which is the best buy. i β500 gβof paper A for β$3.26βor β200 gβof paper B for β$1.25β ii β250 gβof salami A for β$4.50βor β150 gβof salami B for β$3.10β iii 7β 20 gβof powder A for β$3.29βor β350 gβof powder B for β$1.90β iv 1β .1 kgβof shampoo A for β$12.36βor β570 gβof shampoo B for β$6.85β
Applications of βbest buysβ
a Star Washing Liquid is priced at β$3.85βfor β600 gβ, while Best Wash Liquid is priced at β$5.20β for β1 kgβ. Find the difference in the price per β100 gβ, correct to the nearest cent. b Budget apples cost β$6.20βper β5 kgβbag. How much would a β500 gβbag of Sunny apples have to be if it was the same price per β100 gβ? c β1.5 kgβof cheddar cheese costs β$11.55β, and β800 gβof feta cheese costs β$7.25β. Sally works out the best value cheese, then buys β$5βworth of it. How much and what type of cheese did Sally buy?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Problems and challenges
Problems and challenges
a When _ β3 βis written as a decimal, state the Up for a challenge? If you get stuck on 7 a question check out the βWorking with digit in the β20thβdecimal place. unfamiliar problemsβ poster at the end b Given that βA Γ 2.75 = 10.56β, write an of the book to help you. expression for β1.056 Γ· 0.00275β. c Find this product: β(1 β 0.5)(β 1 β 0.β3Λ )β (β 1 β 0.25)(β 1 β 0.2)β d Write the recurring decimal β1.4β51β βΎβas an improper fraction. e Callum pays β$4.10βfor his coffee using eight coins. They are made up of β$1β coins and β50βand β20βcent pieces. How many of each coin did Callum use?
U N SA C O M R PL R E EC PA T E G D ES
1
339
2 Consider the ladder in the diagram. The heights of each rung on the ladder are separated by an equally sized gap. Determine the heights for each rung of the ladder.
1.44 m high
0.54 m high
3 Find the digits represented by the letters in these decimal problems. a
b
A. 2B β_β + 9. C5ββ 11. 12
0 .757 d β _ββ Aββ2 .2B1β
c 3β .A Γ B.4 = 8.16β _
_
4 We know that β β 9 βis β3βsince β32β β= 9β, but what about β β 2 ββ? a Find the value of: i β1. β4β2β ii β1. β5β2β _ b Now try to find the value of β β 3 βcorrect to: i two decimal places
5 Find a fraction equal to the following decimals. a β0.β4Λ β b β3.β2Λ β
2A. 43 β_β β 9. B4ββ C7. 8D
iii β1. 4β 5ββ2β
ii three decimal places.
c β0.β17β βΎβ
d β4.ββΎ 728ββ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 5
Decimals
Changing decimals to fractions 16 4 0.16 = 100 = 25 8 1 = 2 125 2.008 = 2 1000
Comparing decimals 12.3 > 12.1 6.72 < 6.78 0.15 β 0.105 284.7 β€ 284.7
Changing fractions to decimals 2 4 = 10 = 0.4 5 7 35 = 100 = 0.35 20
Rounding
U N SA C O M R PL R E EC PA T E G D ES
Chapter summary
340
Decimals as fractions
The critical digit is circled.
1 8 4 0.184 = 10 + 100 + 1000 184 = 1000
2.34 | 2 2.34 2.34 | 5 2.35 2.34 | 6 2.35 5.89 | 9 5.90 5.99 | 7 6.00 If critical digit is β₯ 5 round up. If critical digit is < 5 round down.
Place value of digits 0.184 1 tenth 8 hundredths 4 thousandths
Decimals
Subtraction 1
1
216.94 β 31.53 185.41 Align decimal points.
Addition 1
9.807 + 26.350 36.157 1
Align decimal points.
Division by 10, 100, 1000 etc.
Division
2.76 Γ· 10 000 = 0.000 276 Decimal point moves left.
8.547 Γ· 0.03 = 854.7 Γ· 3 284.9 2 1 2
3 854.7
Multiplication 278 Γ 34 1112 8340 9452 2.78 Γ 34 = 94.52 2.78 Γ 3.4 = 9.452 0.278 Γ 3.4 = 0.9452 0.278 Γ 0.34 = 0.09452
Number of decimal places in the question equals number of decimal places in the answer.
Multiplication by 10, 100, 1000 etc.
2.76 Γ 10 000
= 27 600.0 Decimal point moves right.
Proportions
13 out of 20 = 13 (Fraction) 20
Decimals and percentage 0.63 = 63 Γ· 100 = 63% 8% = 8 Γ· 100 = 0.08 240% = 240 Γ· 100 = 2.4
13 20
65 = 100 = 0.65 (Decimal) 65
0.65 = 100 = 65% (Percentage)
Fractions to decimals
13 = 0.13 100
3 = 0.375 8
2 = 0.2222... β’ 9
= 0.2
0.375 8 3.000 0.222... 9 2.000...
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter checklist
341
Chapter checklist with success criteria 1. I can state the value of a digit after the decimal point in a number. e.g. What is the value of the digit β8βin the number β6.1287β?
U N SA C O M R PL R E EC PA T E G D ES
5A
β
Chapter checklist
A printable version of this checklist is available in the Interactive Textbook
5A
2. I can convert fractions to decimals (if the denominator is 10, 100, 1000 etc.). β317 βto a decimal. e.g. Convert _ 100
5A
3. I can compare two or more numbers written as decimals. e.g. Decide which is smallest out of β2.37, 2.72, 3.27β.
5B
4. I can determine the critical digit in a rounding operation. e.g. The number β23.5398βis to be rounded to two decimal places. Circle the critical digit.
5B
5. I can round decimals to a given number of decimal places. e.g. Round β0.43917βto β2βdecimal places.
5C
6. I can add decimals. e.g. Find β64.8 + 3.012 + 5.94β.
5C
7. I can subtract decimals. e.g. Find β146.35 β 79.5β.
5D
8. I can multiply a decimal by a power of β10β. e.g. Evaluate β4.31 Γ 10 000β.
5D
9. I can divide a decimal by a power of 10. e.g. Evaluate β7.82 Γ· 1000β.
5D
10. I can multiply and divide a whole number by a power of 10, introducing a decimal place if required. e.g. Evaluate β23 Γ· 1000β.
5E
11. I can estimate decimal products using whole number approximations e.g. Estimate β4.13 Γ 7.62βby first rounding each decimal to the nearest whole number.
5E
12. I can multiply decimals. e.g. Calculate β3.63 Γ 6.9βand β0.032 Γ 600β.
5F
13. I can divide decimals. e.g. Calculate β2.3 Γ· 1.1 and 67.04 Γ· 8000β.
5G
14. I can convert decimals to fractions. e.g. Convert β10.35βto a mixed numeral in simplest form.
5G
5G
15. I can convert fractions to decimals. e.g. Convert _ β53 βto a decimal. 5
16. I can convert fractions to recurring decimals. e.g. Convert _ β 7 βto a decimal. 12
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 5
Decimals
β 5H
17. I can convert a percentage to a decimal. e.g. Convert β12.5%βto a decimal.
U N SA C O M R PL R E EC PA T E G D ES
Chapter checklist
342
5H 5I
5I
18. I can convert a decimal to a percentage. e.g. Convert β0.045βto a percentage.
19. I can write a value as a proportion of a total, giving the proportion as a fraction, decimal or percentage. e.g. Write the proportion β$40βout of β$200βas a fraction, a decimal and a percentage. 20. I can compare proportions by converting to percentages.
e.g. By first writing the proportions as percentages, compare the scores _ β 8 β, 0.78 and 82% and 10 decide which is largest.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
343
Short-answer questions Arrange each group in descending order, from largest to smallest. a β0.4, 0.04, 0.44β b β2.16, 2.016, 2.026β c 0β .932, 0.98, 0.895β
U N SA C O M R PL R E EC PA T E G D ES
1
Chapter review
5A
5G
2 Write each fraction as a decimal. 81 β a β_ b _ β 81 β 10 100
c _ β801 β 100
d _ β 801 β 1000
5A
3 What is the place value of the digit β3βin the following numbers? a β12.835β b β6.1237β
c β13.5104β
5A
4 State whether each of the following is true or false. a β8.34 < 8.28β b β4.668 > 4.67β
c β8.2 > 8.182β
d β3.08 β©½ _ β308 β 100
5B
5B
5D
_ β 7 β= _ β 70 β
10
100
6 Round each of the following to the specified number of decimal places (which is given in Βbrackets). b β15.8892 (β 2)β c β7.254 32 (β β1)β β a β423.46 (β 1)β
g _ β 5 β(β 3)β 11
5A/B/D
f
5 List all possible numbers with β3βdecimal places that, when rounded to β2βdecimal places, result in β45.27β.
d β69.999 531 (β β3β)β
5C
e _ β 62 ββ©Ύ 6.20β 100
3 β(β 1)β 2 β(β 2)β f β_ e 2β β_ 3 4 h _ β 1 β(β 44)ββ(Hint: Look for a shortcut!) 81
7 Evaluate: a β13.85 β 4.32β b β19.12 β 14.983β c 2β 7.6 + 15.75β d β204.708 37 + 35.7902β e β472.427 β 388.93β f β210.8 β (β 26.3 β 20.72)β
8 State whether each of the following is true or false. a β10.34 Γ· 100 = 0.1034β b β3.125 Γ 0.1 = 31.25β c 1β 15.23 Γ· 10 = 1.1523 Γ 1000β d β115.23βhas β3βdecimal places e 2β 4.673 = 24.7βwhen rounded to β1βdecimal place
9 Solve each of the following, using the order of operations. a β1.37 Γ 100β b β0.79 Γ 1000β c 2β 25.1 Γ· 10β d β96.208 Γ· 1000β e 7β 5.68 + 6.276 Γ 100 β 63.24 Γ· 10β f β3.56 Γ 100 + 45 Γ· 10β ( ) g β100 Γ β 56.34 Γ 100 + 0.893 β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
344
Chapter 5
5E/F
10 Estimate the following products by first rounding each decimal to the nearest whole number. a 3β .8 Γ 7.1β b β5.3 Γ 9.2β c β7.24 Γ 1.94β d β9.97 Γ 3.17β e 4β .94 Γ 8.7β f β3.14 Γ 2.72β 11 Calculate the following. a 2β .4 Γ 8β d β3.8 Γ· 4β g β4 Γ· 0.25β
b 9 β Γ 7.11β e 1β 2.16 Γ· 8β h β1.2 Γ· 0.4β
c 2β .3 Γ 8.4β f β3 Γ· 0.5β i β3.42 Γ· 1.1β
U N SA C O M R PL R E EC PA T E G D ES
Chapter review
5E
Decimals
5F/G/H
12 Copy and complete this table, stating fractions both with the denominator β100βand in their simplest form. Decimal
Fraction
Percentage
β0.45β
_ β ? β= _ β7 β
100
10
β32%β
β0.06β
_ β 79 β
100
β1.05β
? β= _ β_ β7 β 100 20
β65%β
_ β1 β β ? β= _ 1000 8
5I
5I
13 Write each of the following proportions as a fraction, a decimal and a percentage. a $β 3βout of β$10β b β12βlitres out of β15β litres c β6 kmβout of β8 kmβ d β180 mLβout of β1 litreβ
14 On a test, Dillon scored _ β15 β, Judy managed to get β0.8βof the questions correct and Jon received a 20 score of β78%β. Use percentages to compare the test scores and decide who scored the highest.
Multiple-choice questions
5A
1
The next number in the pattern β0.023, 0.025, 0.027, 0.029β is: A 0β .0003β B β0.030β C β0.0031β D β0.031β
E β0.033β
5G
2 β0.05βis equivalent to: 5β A β_ B _ β5 β 10 100
5A
3 The smallest number out of β0.012, 10.2, 0.102, 0.0012βand β1.02β is: A 0β .012β B β0.102β C β0.0012β D β1.02β
E β10.2β
5F
4 0β .36 Γ· 1000βis equal to: A 3β .6β B β360β
E β0.000β36β
C _ β 5 β 1000
C β0.036β
5 β D β_ 500
D β0.0036β
E β5β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
C β0.124β
D 1β 24β
E β0.0124β
5E
6 What is the answer to β0.08 Γ 0.6β? A β0.48β B β4.8β
C β0.0048β
D 0β .048β
E β48β
5B
7 When rounded to β1βdecimal place, β84.553β becomes: A β80β B β84β C β84.5β
D 8β 4.6β
E β84.55β
5G
8 As a decimal, _ β23 βis equal to: 90 A β0.2β B β0.2β5Λ β
C β0.26β
D 0β .β2Λ β5Λ β
E β0.25β6Λ β
5B/C
9 7β + 0.7 + 0.07 + 0.007β, to β2βdecimal places, is: A 7β .78β B 7β .77β C β7β
D β7.7β
E β7.777β
U N SA C O M R PL R E EC PA T E G D ES
5 6β .2 Γ 0.2βis equal to: A β1.24β B β12.4β
Chapter review
5E
345
5G
10 β5.β624β βΎβ means: A β5.62444β¦β D β5.6246464β¦β
B 6β .6242424β¦β E β5.62456245624β¦β
C β5.624624624β¦β
Extended-response questions 1
Find the answer in these practical situations. a Jessica is paid β$125.70βfor β10βhours of work and Jaczinda is paid β$79.86βfor β6βhours of work. Who receives the higher rate of pay per hour, and by how much? b Petrol is sold for β124.9βcents per litre. Jacob buys β30 Lβof petrol for his car. Find the total price he pays, to the nearest β5β cents. c The Green family are preparing to go to the Great Barrier Reef for a holiday. For each of the four family members, they purchase a goggles and snorkel set at β$37.39βeach, fins at β$18.99β each and rash tops at β$58.48βeach. How much change is there from β$500β? d For her school, a physical education teacher buys β5βeach of basketballs, rugby union and soccer balls. The total bill is β$711.65β. If the rugby balls cost β$38.50βeach and the basketballs cost β$55.49βeach, what is the price of a soccer ball?
2 A car can use β25%βless fuel per km when travelling at β90 km / hβthan it would when travelling at β 110 km / hβ. Janelleβs car uses β7.8βlitres of fuel per β100 kmβwhen travelling at β110 km / hβ, and fuel costs β155.9βcents per litre. a How much money could Janelle save on a β1000-kmβtrip from Sydney to Brisbane if she travels at a constant speed of β90 km / hβinstead of β110 km / hβ? b During a β24β-hour period, β2000βcars travel the β1000-kmβtrip between Sydney and Brisbane. How much money could be saved if β30%βof these cars travel at β90 km / hβinstead of β110 km / hβ? 3 Siobhan is on a β6β-week holiday in the United Kingdom, and is using her phone to keep in contact with her friends and family in Australia. The phone charge for voice calls is β$0.40β βflagfallβ and β$0.65βper β45βseconds; text messages are β$0.38β each. During her holiday, Siobhan makes β27βvoice calls and sends β165βtext messages to Australia. If her phone bill is β$832.30β, determine the average length of Siobhanβs voice calls.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
346
Semester review 1
Short-answer questions 1
Write the number seventy-four in: a Babylonian numerals b Roman numerals
c Egyptian numerals.
U N SA C O M R PL R E EC PA T E G D ES
Semester review 1
Computation with positive integers
2 Write the numeral for: a 6β Γ 10000 + 7 Γ 1000 + 8 Γ 100 + 4 Γ 10 + 9 Γ 1β b β7 Γ 100 000 + 8 Γ 100 + 5 Γ 10β 3 Calculate: a β96 481 + 2760 + 82β d β980 Γ 200β
b 1β 0 963 β 4096β e 4β 932 Γ· 3β
c 1β 47 Γ 3β f β9177 Γ· 12β
4 State whether each of the following is true or false. a 1β 8 < 20 β 2 Γ 3β b β9 Γ 6 > 45β
c β23 = 40 Γ· 2 + 3β
5 How much more than β17 Γ 18βis β18 Γ 19β? 6 Calculate: a 7β Γ 6 β 4 Γ 3β d β16 Γ [14 β (6 β 2) ]β
b 8 β Γ 8 β 16 Γ· 2β e 2β 4 Γ· 6 Γ 4β
c 1β 2 Γ (β 6 β 2)β f β56 β (β 7 β 5)βΓ 7β
7 State whether each of the following is true or false. a β4 Γ 25 Γ 0 = 1000β b β0 Γ· 10 = 0β d β8 Γ 7 = 7 Γ 8β e 2β 0 Γ· 4 = 20 Γ· 2 Γ· 2β
c 8β Γ· 1 = 1β f β8 + 5 + 4 = 8 + 9β
8 Insert brackets to make β18 Γ 7 + 3 = 18 Γ 7 + 18 Γ 3β true.
9 How many times can β15βbe subtracted from β135βbefore an answer of zero occurs? 10 Write β3 859 643βcorrect to the nearest: a β10β b thousand
c million.
Multiple-choice questions 1
Using numerals, thirty-five thousand, two hundred and six is: A β350 260β B 3β 5 260β C β35 000 206β D β3526β
2 The place value of β8βin β2 581 093β is: A β8β thousand B 8β 0β thousand D β8β tens E β8β ones
E β35 206β
C β8β hundred
3 The remainder when β23 650βis divided by β4β is: A β0β B 4β β C β1β
D β2β
E β3β
4 1β 8 β 3 Γ 4 + 5βsimplifies to: A β65β B 1β 35β
C β11β
D β1β
E β20β
5 8β 00 Γ· 5 Γ 4βis the same as: A β160 Γ 4β B 8β 00 Γ· 20β
C β800 Γ· 4 Γ 5β
D β40β
E β4 Γ 5 Γ· 800β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Semester review 1
U N SA C O M R PL R E EC PA T E G D ES
a During a particular week, Tom works 7:00 a.m. to 2:00 p.m. Monday to Thursday. How many hours does he work that week? b How much does Tom earn for this work? c If Tom works β5βhours on Saturday in the same week, what is his total income for the week? d How many more hours on a Friday must Tom work to earn the same amount as working β5β hours on a Saturday?
Semester review 1
Extended response question Tom works as a labourer, earning β$25βan hour on weekdays and β$60βan hour on weekends.
347
Number properties and patterns Short-answer questions 1
List the factors of: a β15β
2 List the first five multiples of: a β3β
b β30β
c β100β
b β7β
c β11β
3 List all factors common to β30βand β36β.
4 What is the highest factor common to β36βand β40β? 5 Find the value of: a 11β β 2β β
c β33 β 2β β 3β β
b 6β β 2β βΓ 2β β β2β
6 What is the square root of β14 400β? _
β β2β+ 4β β β2β = 3 + 4βtrue or false? 7 Is the equation β β 3β
Ext
8 Find the smallest number that must be added to β36 791βso that it becomes divisible by: a β2β b 3β β c β4β 9 A pattern is shown using matchsticks. term 1
term 2
term 3
a How many matchsticks are needed to build the β12thβterm in this pattern? b Which term in this pattern uses exactly β86β matchsticks?
10 Find the missing values in the table. input
β4β
β5β
output
β19β
β23β
β6β
β39β
4β 7β
β403β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
348
Semester review 1
y E
4 3 2 1
D C
U N SA C O M R PL R E EC PA T E G D ES
Semester review 1
11 Write down the coordinates of each point.
B
A
O
1 2 3 4
x
Multiple-choice questions 1
The first prime number after β20β is: A 2β 1β B β22β
C β23β
D β24β
E β25β
2 The highest common factor (HCF) of β12βand β18β is: A 6β β B β12β C β4β
D β2β
E β9β
3 2β Γ 2 Γ 2 Γ 3βis the same as: A 6β Γ 3β B 2β β 3β βΓ 3β
C 8β β 3β β
D 6β β 3β β
E 4β β 3β β
C β4β
D β17β
E β7β
D 2β β β4βΓ 3β
E 2β β 3β βΓ 3β
_
4 Evaluating 3β β 2β ββ ββ25 β+ 3β gives: A β8β B β5β
5 The number β48βin prime factor form is: A 2β β β4βΓ 5β B β2 Γ 3 Γ 5β C 2β β 3β βΓ 3β β β2β
Extended-response question
term 1
term 2
term 3
The diagrams above show the tile pattern being used around the border of an inground swimming pool. a b c d
Draw the fourth term in the pattern. How many coloured tiles are used in term β4βof the pattern? Which term uses β41βcoloured tiles in its construction? If each coloured tile costs β$1βand each white tile costs β50βcents, what is the cost of completing the pattern using β41βcoloured tiles?
Fractions and percentages Short-answer questions 1
Arrange _ β1 β, _ β1 β, _ β2 βand _ β 3 ββin ascending order. 2 3 5 10
2 βas an improper fraction. 2 Express β5β_ 3
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Semester review 1
3β c 2β _ β1 β+ 3β_ 2 4 β5 β f β1_ β1 βΓ _ 5 12
1β b β4 β 1β_ 3 e _ β2 βΓ· _ β1 β 3 6
Semester review 1
3 Find each of the following. 2 β+ _ β1 β a β_ 3 4 d _ β2 βΓ _ β1 β 5 2
349
U N SA C O M R PL R E EC PA T E G D ES
4 Write β15%βas a simple fraction. 5 Find β25%βof β$480.β
1 β%βof β$480.β 6 Find β12β_ 2
7 State whether each of the following is true or false. a β25%βof βx = x Γ· 4β b 1β 0%βof βw = _ βw β 10 c 2β 0%βof β50 = 50%βof β20β d β1%βof βx = 100xβ 8 Simplify the following ratios. a β7 : 21β
b 4β : 12 : 16β
9 Divide β$60βin the given ratios. a β5 : 1β
c β24βpoints to β10β points
b β2 : 3β
Multiple-choice questions
Which of the following is equivalent to _ β12 ββ? 7 24 5 _ A β β C β1_ β5 β B β1_ β β 7 7 12 1 1 _ _ 2 β β+ β βis equal to: 2 3 2β A β_ B _ β2 β C _ β5 β 6 5 6 350 _ 3 β βin simplest form is: 450 35 β B _ β4 β C _ β3 β A β_ 5 4 45 1
D _ β112 β 17
E _ β7 β 12
D _ β1 β 5
7β E β_ 6
D _ β3.5 β 4.5
E _ β7 β 9
C _ β5 β 1
D _ β1 β 5
E _ β1 β 40
C _ β2 βΓ· _ β3 β 5 4
D _ β2 βΓ _ β4 β 5 3
β3 β E _ β3 βΓ _ 2 4
4 What fraction of β$2βis β40β cents? 1β A β_ B _ β20 β 20 1 3 1 5 β2β_βΓ· β_βis the same as: 2 4 5 βΓ _ A β_ β4 β B _ β5 βΓ _ β3 β 2 3 2 4
Extended-response question Calebβs cold and flu prescription states: βTake two pills three times a day with food.β The bottle contains β54β pills. a b c d
How many pills does Caleb take each day? What fraction of the bottle remains after Day β1β? How many days will it take for the pills to run out? If Caleb takes his first dose on Friday night before going to bed, on what day will he take his last dose?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
350
Semester review 1
Semester review 1
Algebraic techniques Short-answer questions Consider the expression β5x + 7y + 3x + 9β. a How many terms are in this expression? b Can the expression be simplified? c What is the value of the constant term? d What is the coefficient of βyβ?
U N SA C O M R PL R E EC PA T E G D ES
1
2 Write an algebraic expression for each of the following. a the sum of βxβand β3β b the product of βaβand β12β c the sum of double βxβand triple βyβ d βwβdivided by β6β e double βxβtaken from βyβ 3 Find how many: a cents in β$mβ c millimetres in βpβ kilometres
b hours in βxβ days d days in yβ β hours.
4 If βm = 6β, find the value of each of the following. a m β + 7β b β2m β 1β m + 6β d β2(β m β 3)β e β_ 2 5 Evaluate the expression β3(β 2x + y)βwhen βx = 5βand βy = 2β.
6 Simplify each of the following. a β6a + 4aβ d βm + m β mβ
b 7β x β 3xβ e 6β + 2a + 3aβ
c 6β m + 3β m β+ 4m β 3β f β_ 2
c 9β a + 2a + aβ f βx + y + 3x + yβ
7 a Write an expression for the perimeter of rectangle βABCDβ. b Write an expression for the area of rectangle βABCDβ. A
B
3
D
Ext
C
x+4
8 Find the missing term. a 3β a Γ __ = 18abcβ
b β10ab Γ· __ = 2aβ
c β2p + 2p + 2p = 6__β
9 Expand: a 2β β(a + 3)β
b β12β(a β b)β
c β8β(3m + 4)β
10 Write the simplest expression for the perimeter of this figure. 2xy
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Semester review 1
351
Multiple-choice questions 1β 2 β xβ means: A β12βless than βxβ D βxβis less than β12β
B xβ βless than β12β E βxβis more than β12β
C βxβhas the value of β12β
C βx + 2yβ
D β(βx + y)β2β
E βx + y + 2β
3 Half the product of βaβand βbβ is: a + bβ A β2abβ B β_ 2
ab β C β_ 2
β1 βbβ D _ β1 βa + _ 2 2
E _ βa β+ bβ 2
U N SA C O M R PL R E EC PA T E G D ES
2 Double the sum of βxβand βyβ is: A β2β(x + y)β B β2x + yβ
Semester review 1
1
4 4β a + 3b + c + 5b β cβis the same as: A β32abβ B β4a + 8b + 2cβ D β64abcβ E β4a + 8bβ 5 If βa = 3βand βb = 7β, then β3aβ β β2β+ 2bβis equal to: A β66β B β95β C β23β
C β8a + 4bβ
D β41β
E β20β
Extended-response question
A bottle of soft drink costs β$3βand a pie costs β$2β. a Find the cost of: i β2βbottles of soft drink and β3β pies ii βxβbottles of soft drink and β3β pies iii βxβbottles of soft drink and βyβ pies. b If Anh has β$50β, find his change if he buys βxβbottles of soft drink and βyβ pies.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
352
Semester review 1
Short-answer questions 1
Write each of the following as a decimal. a two-tenths b _ β 13 β 100
c _ β17 β 10
U N SA C O M R PL R E EC PA T E G D ES
Semester review 1
Decimals
2 In the decimal β136.094β: a what is the value of the β6β? b what is the value of the β4β? c what is the decimal, correct to the nearest tenth? 3 Round β18.398741βcorrect to: a the nearest whole number b β1βdecimal place c β2βdecimal places. 4 Evaluate: a 1β 5 β 10.93β d β0.6 Γ 0.4β
b β19.7 + 240.6 + 9.03β e
(ββ0.3)2β β
c 2β 0 β 0.99β 12 β f β_ 0.2
5 Find: a 1β .24 β 0.407β b β1.2 + 0.6 Γ 3β c β1.8 Γ 0.2 Γ· 0.01β
6 If β369 Γ 123 = 45 387β, write down the value of: a β3.69 Γ 1.23β b β0.369 Γ 0.123β c β45.387 Γ· 36.9β 7 Find: a β36.49 Γ 1000β b β1.8 Γ· 100β c β19.43 Γ 200β
8 For each of the following, state the larger of each pair. a _ β4 β, 0.79β 5 b β1.1, 11%β c _ β2 β, 0.6β 3 9 State if each of the following is true or false. a 0β .5 = 50%β b β0.15 = _ β2 β 20 4 _ d β126% = 1.26β e β β= 0.08β 5
c β38% = 0.19β f
1β _ β3 β= 1.75β 4
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Semester review 1
353
Multiple-choice questions 8β 0 + _ β 6 β+ _ β 7 βis the same as: 10 1000 A β8067β B β867β
D 8β 0.067β
B β0.770 = _ β 77 β 100 E β0.7 Γ 10 = 7β
E β80.607β
C β0.07 Γ 0.7 = 0.49β
U N SA C O M R PL R E EC PA T E G D ES
2 Select the incorrect statement. A β0.707 > 0.7β
C β80.67β
Semester review 1
1
D β0.7 Γ _ β 1 β= 0.07β 10
3 The best estimate for β23.4 Γ 0.96β is: A 2β 34β B 2β 30β
C β0.234β
D β23β
E β20β
4 _ β3 βis the same as: 8 A β0.375β
C β0.38β
D 2β .β6Λ β
E β38%β
C β17β
D _ β4 β 68
E β7 Γ· 0.05β
B β3.8β
5 6β .8 Γ· 0.04βis the same as: A β68 Γ· 4β B 6β 80 Γ· 4β
Extended-response question
The cost of petrol is β116.5βcents per litre. a Find the cost of β55 Lβof petrol, correct to the nearest cent. b Mahir pays cash for his β55 Lβof petrol. What is the amount that he pays, correct to the nearest β5β cents? c If the price of petrol per litre is rounded to the nearest cent before the cost is calculated, how much would β55 Lβof petrol cost now? d By how much is Mahir better off if the total cost is rounded rather than rounding the price per litre of the petrol? e If the price drops to β116.2βcents per litre, is the comparison between rounding at the end versus rounding at the beginning the same as it was above?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
U N SA C O M R PL R E EC PA T E G D ES
6 Negative numbers
Maths in context: Positive and negative measurements The Antarctic has the coldest climate on Earth. Near the coast, the temperature can range from +10Β°C in summer to β60Β°C in winter, making winter up to 70Β°C colder than summer. High in the mountains, temperatures average β30Β°C in summers and β80Β°C in winters, 50Β°C colder than summer.
When measured from its base, on the ocean ο¬oor, the Hawaiian mountain of Mauna Kea is the highest mountain in the world. The top is at +4205 m and the base at β6000 m making a total height of over 10 200 m from the base. That is much higher than Mt Everest at 8848 m above sea level.
Antarctic icebergs start out as huge broken ice shelves. Measuring from sea level, the top of an iceberg could be at +50 m and the bottom at β350 m, making a total height of 400 m. These gigantic icebergs are many kilometres across and weigh millions of tonnes.
There are several places on the Earthβs continents that are below sea level. Examples include Australiaβs Lake Eyre at β15m, Californiaβs Death Valley at β86m, Israelβs Sea of Galilee with its shores at β212m and the very salty Dead Sea (pictured) with its shores at β423m, the lowest dry land on Earth.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter contents 6A 6B 6C 6D
Introduction to negative integers Adding and subtracting positive integers Adding and subtracting negative integers Multiplying and dividing integers (EXTENDING)
6E Order of operations with integers
U N SA C O M R PL R E EC PA T E G D ES
(EXTENDING)
6F Substituting integers (EXTENDING) 6G The Cartesian plane
Victorian Curriculum 2.0
This chapter covers the following content descriptors in the Victorian Curriculum 2.0: NUMBER
VC2M7N03, VC2M7N08, VC2M7N10 ALGEBRA
VC2M7A05
Please refer to the curriculum support documentation in the teacher resources for a full and comprehensive mapping of this chapter to the related curriculum content descriptors.
Β© VCAA
Online resources
A host of additional online resources are included as part of your Interactive Textbook, including HOTmaths content, video demonstrations of all worked examples, auto-marked quizzes and much more.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 6
Negative numbers
6A Introduction to negative integers LEARNING INTENTIONS β’ To understand that integers can be negative, positive or zero β’ To be able to represent integers on a number line β’ To be able to compare two integers and decide which is greater
U N SA C O M R PL R E EC PA T E G D ES
356
The numbers 1, 2, 3, β¦ are considered to be positive because they are greater than zero (0). Negative numbers extend the number system to include numbers less than zero. Integers can be positive, zero or negative.
The winning golf score above is β7, shown in the left column. Golf scores are positive and negative integers giving the number of strokes above or below the par, the total strokes of an expert golfer.
The use of negative numbers dates back to 100 bce when the Chinese used black rods for positive numbers and red rods for negative numbers in their rod number system. These coloured rods were used for commercial and tax calculations. Later, a great Indian mathematician named Brahmagupta (598β670) set out the rules for the use of negative numbers, using the word fortune for positive and debt for negative. Negative numbers were used to represent loss in a financial situation. An English mathematician named John Wallis (1616β1703) invented the number line and the idea that numbers have a direction. This helped define our number system as an infinite set of numbers extending in both the positive and negative directions. Today, negative numbers are used in all sorts of mathematical calculations and are considered to be an essential part of our number system.
Lesson starter: Simple applications of negative numbers β’ β’
Try to name as many situations as possible in which negative numbers are used. Give examples of the numbers in each case.
KEY IDEAS
β Negative numbers are numbers less than zero.
β Integers can be negative, zero or positive and include
β’ β’
β¦, β 4, β3, β2, β1, 0, 1, 2, 3, 4, β¦ The number β3 is read as βnegative 3β. The number 3 is sometimes written as +3 and is sometimes read as 'positive 3'.
β A number line shows:
β’ β’
positive numbers to the right of zero. negative numbers to the left of zero.
negative
positive
β4 β3 β2 β1 0 1 2 3 4
β Each negative number has a positive opposite and each positive number has a negative
opposite. 3 and β3 are examples of opposite numbers.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
6A Introduction to negative integers
357
β Numbers can be thought of as having a sign (β+βor βββ) and a magnitude, so numbers like β3βand β
β 3βboth have a magnitude of β3βbut have opposite signs. The β+βsign is rarely used, as β+ 3β is generally written as β3β.
β Negative fractions that are not integers can also be located on a
U N SA C O M R PL R E EC PA T E G D ES
number line.
β 32 or β1 21
β A number is greater than another number if it occurs to the right
β2
β1 β 2 3
0
of it on a number line. For example, ββ 2βis greater than ββ 4β. This can be written as ββ 2 > β 4 or β 4 < β 2.β
BUILDING UNDERSTANDING
1 What are the missing numbers on these number lines? a b β3
β1 0 1
3
β10 β9 β8
β6
2 ββ 5βis the opposite number of β5β, and β5βis the opposite number of ββ 5β. State the opposite of these numbers. a β2β
b ββ 7β
c β21β
d ββ 1071β
3 Select from the words greater or less to complete these sentences. a β5βis __________ than β0β b ββ 3βis __________ than β0β c β0βis __________ than ββ 6β d β0βis __________ than β1β
Example 1
Drawing a number line including integers
Draw a number line, showing all integers from ββ 4βto β2β. SOLUTION
β4 β3 β2 β1 0 1 2
EXPLANATION
Use equally spaced markings and put ββ 4βon the left and β2βon the right.
Now you try
Draw a number line, showing all integers from ββ 6βto β1β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
358
Chapter 6
Negative numbers
Example 2
Comparing integers using inequality symbols
Insert the symbol β<β(less than) or β>β(greater than) into these statements to make them true. b ββ 1β
β3β
ββ 6β
U N SA C O M R PL R E EC PA T E G D ES
a β β 2β
SOLUTION
EXPLANATION
a β β 2 < 3β
ββ 2βis to the left of β3βon a number line, so it is less than β3β. β2 β1 0 1 2 3
β 1βis to the right of ββ 6βon a number line, so it β is greater than ββ 6β.
b ββ 1 > β 6β
β6 β5 β4 β3 β2 β1 0
Now you try
Insert the symbol β<β(less than) or β>β(greater than) into these statements to make them true. a 5β β
ββ 7β
b ββ 4β
ββ 1β
Exercise 6A FLUENCY
Example 1 Example 1
1
1, 3β6ββ(β1/2β)β
3β6β(β β1/2β)β
Draw a number line, showing all integers from ββ 2βto β3β.
2 Draw a number line for each description, showing all the given integers. a from ββ 2βto β2β b from ββ 10βto ββ 6β 3 List all the integers that fit the description. a from ββ 2βup to β4β c greater than ββ 5βand less than β1β
Example 2
2, 3β6β(β β1/2β)β
b greater than ββ 3βand less than β2β d less than β β 3βand greater than ββ 10β
4 Insert the symbol β<β(less than) or β>β(greater than) into these statements to make them true. a β7β
β9β
b β3β
c β0β
ββ 2β
d ββ 4β
β0β
e ββ 1β
β5β
f
ββ 7β
ββ 6β
g ββ 11β
ββ 2β
h ββ 9β
ββ 13β
β3β
j
β3β
l
ββ 2β
i
ββ 3β
k β β 130β
β1β
β2β
ββ 3β ββ 147β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
6A Introduction to negative integers
5 State the temperature indicated on these thermometers. a b c Β°C Β°C
d
5
10
0
Β°C 20
0
0
β10
0
U N SA C O M R PL R E EC PA T E G D ES
Β°C
359
β10
β5
β20
β20 β40
6 Write down the negative fraction or negative mixed numeral illustrated by the dot on these number lines. a b β2 β1 0 β2 β1 0 c e
β5
β12
d
β4
β11
β10
PROBLEM-SOLVING
f
β10
β9
β20
7, 8
β19
8ββ(β1/2β)β,β 9
8β(β β1/2β)β,β 9
7 Arrange these numbers in ascending order. Each number should be less than the next number in the list. a ββ 3, β 6, 0, 2, β 10, 4, β 1β b ββ 304, 126, β 142, β 2, 1, 71, 0β 8 Write the next three numbers in these simple patterns. a 3β , 2, 1, ___, ___, ___β b ββ 8, β 6, β 4, _ _ββ_ , _ _ββ_ , _ _ _β c 1β 0, 5, 0, ___, ___, ___β d ββ 38, β 40, β 42, ___, ___, ___β e ββ 91, β 87, β 83, _ _ββ_ , _ _ββ_ , _ _ _β f β199, 99, β 1, ___, ___, ___β
9 These lists of numbers show deposits (positive numbers) and withdrawals (negative numbers) for a month of bank transactions. Find the balance at the end of the month. a Start balance
Final balance
β$200β
b Start balance
β$0β
ββ $10β
β$50β
ββ $130β
ββ $60β
β$25β
ββ $100β
ββ $100β
β$200β
β_ $20β
ββ $100β _
β_ββ
Final balance
β_ββ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 6
Negative numbers
REASONING
10
10, 11ββ(β1/2β)β
11β(β β1/2β)β,β 12
10 If the height above sea level for a plane is a positive number, then the height for a submarine could be written as a negative number. What is the height relative to sea level for a submarine at these depths? a 5β 0 mβ b β212.5 mβ c β0 mβ
U N SA C O M R PL R E EC PA T E G D ES
360
11 The difference between two numbers could be thought of as the distance between the numbers on a number line. For example, the difference between ββ 2βand β1βis β3β.
3
β3 β2 β1 0 1 2 3
By considering them on a number line, find the difference between these pairs of numbers. a β β 1 and 1β b ββ 2 and 2β c ββ 3 and 1β d ββ 4 and 3β e ββ 3 and 0β f ββ 4 and β 1β g ββ 10 and β 4β h ββ 30 and 14β
12 The opposite of any number other than β0βhas the same magnitude but a different sign. (For example, β5βand ββ 5βare opposites.) a Draw a number line to show β4βand its opposite, and use this to find the difference between the two numbers. b State a number that has a difference of β9βwith its opposite. c If numbers are listed in ascending order, explain by using a number line why their opposites would be in descending order. (For example, ββ 4, 2, 7βare ascending so β4, β 2, β 7βare descending.) ENRICHMENT: The final position
β
13 For these sets of additions and subtractions, an addition means to move to the right and a subtraction means to move left. Start at zero each time and find the final position. a β β 1, 4, β 5β b β3, β 5, β 1, 4β c ββ 5, β 1, 3, 1, β 2, β 1, 4β d ββ 10, 20, β 7, β 14, 8, β 4β e ββ 250, 300, β 49, β 7, 36, β 81β f β β 7001, 6214, β 132, 1493, β 217β
β
negative
13
positive
β3 β2 β1 0 1 2 3
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
6B Adding and subtracting positive integers
361
6B Adding and subtracting positive integers
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand the number line model for adding and subtracting positive integers β’ To be able to use a number line to add a positive integer to any integer β’ To be able to use a number line to subtract a positive integer from any integer
Adding and subtracting a positive integer can give both positive and negative answers. For example, when a newly installed fridge at 20Β°C is switched on, the temperature inside the freezer might fall by 25Β°C. The final temperature is β5Β°C; i.e. 20 β 25 = β5. If a temperature of β10Β°C rises by 5Β°C, the final temperature is β5Β°C; i.e. β10 + 5 = β5.
When food goes into a fridge or freezer its temperature begins to change, sometimes reaching temperatures below zero.
Lesson starter: Positive and negative possibilities
Decide if it is possible to find an example of the following. If so, give a specific example. β’ β’ β’ β’ β’ β’ β’ β’
A positive number added to a positive number gives a positive number. A positive number added to a positive number gives a negative number. A positive number added to a negative number gives a positive number. A positive number added to a negative number gives a negative number. A positive number subtracted from a positive number gives a positive number. A positive number subtracted from a positive number gives a negative number. A positive number subtracted from a negative number gives a positive number. A positive number subtracted from a negative number gives a negative number.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 6
Negative numbers
KEY IDEAS β If a positive number is added to any number, you move right on a number line.
β2 + 3 = 5β
Start at β2βand move right by β3β.
+3 1
2
3
4
5
6
U N SA C O M R PL R E EC PA T E G D ES
362
ββ 5 + 2 = β 3β
Start at ββ 5βand move right by β2β.
+2
β6 β5 β4 β3 β2
β If a positive number is subtracted from any number, you move left on a number line.
β2 β 3 = β 1β
β3
Start at β2βand move left by β3β.
β2 β1 β
ββ 4 β 2 = β 6β
0
Start at ββ 4βand move left by β2β.
1
2
3
β2
β7 β6 β5 β4 β3
BUILDING UNDERSTANDING
1 In which direction (i.e. right or left) on a number line do you move for the following calculations? a β2βis added to ββ 5β c β4βis subtracted from β2β
b 6β βis added to ββ 4β d β3βis subtracted from ββ 4β
2 Match up the problems a to d with the number lines A to D. a β5 β 6 = β 1β
A
b ββ 2 + 4 = 2β
B
β3 β2 β1
c ββ 1 β 3 = β 4β
C
d ββ 6 + 3 = β 3β
D
0
1
2
β5 β4 β3 β2 β1
0
3
β7 β6 β5 β4 β3 β2 β2 β1
0
1
2
3
4
5
6
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
6B Adding and subtracting positive integers
Example 3
363
Adding and subtracting positive integers
Calculate the answer to these additions and subtractions. a ββ 2 + 3β b ββ 8 + 1β c β5 β 7β
U N SA C O M R PL R E EC PA T E G D ES
d ββ 3 β 3β
SOLUTION
EXPLANATION
a ββ 2 + 3 = 1β
+3
β3 β2 β1
b ββ 8 + 1 = β 7β
0
1
2
+1
β9 β8 β7 β6 β5
c 5β β 7 = β 2β
β7
β3 β2 β1
d ββ 3 β 3 = β 6β
0
1
2
3
4
5
6
β3
β7 β6 β5 β4 β3 β2 β1
Now you try
Calculate the answer to these additions and subtractions. a ββ 4 + 7β b ββ 8 + 3β c β4 β 8β
d ββ 6 β 4β
Exercise 6B FLUENCY
Example 3a,b
Example 3a,b
Example 3c,d
1
Calculate the answers to these additions. a ββ 2 + 5β b ββ 6 + 2β
1, 2β4ββ(1β/2β)β
c ββ 4 + 3β
2β5β(β 1β/2β)β
2β5ββ(1β/4β)β
d ββ 3 + 8β
2 Calculate the answers to these additions. a ββ 1 + 2β b ββ 1 + 4β e β β 4 + 3β f ββ 5 + 2β i ββ 4 + 0β j ββ 8 + 0β m β β 130 + 132β n ββ 181 + 172β
c g k o
β 3 + 5β β ββ 11 + 9β ββ 30 + 29β ββ 57 + 63β
d h l p
β 10 + 11β β ββ 20 + 18β ββ 39 + 41β ββ 99 + 68β
3 Calculate the answers to these subtractions. a β4 β 6β b β7 β 8β e β β 3 β 1β f ββ 5 β 5β i ββ 37 β 4β j β39 β 51β m β β 100 β 200β n β100 β 200β
c g k o
3β β 11β ββ 2 β 13β β62 β 84β β328 β 421β
d h l p
1β β 20β ββ 7 β 0β ββ 21 β 26β ββ 496 β 138β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 6
Negative numbers
4 Find the missing number. a 2β +β
β= 7β
b ββ 2 +β
β= 7β
c ββ 2 +β
β= 3β
d ββ 4 +β
β= β 2β
e β5 ββ
β= 0β
f
β3 ββ
β= β 4β
g β β 9ββ
β= β 12β
h ββ 20 ββ
β= β 30β
β= β 1β
j
ββ 8 ββ
β= β 24β
k
β+ 1 = β 3β
l
β+ 7 = 2β
ββ 4 = β 10β
n
ββ 7 = β 20β
o
β+ 6 = β 24β
p
ββ 100 = β 213β
i
ββ 6 +β
U N SA C O M R PL R E EC PA T E G D ES
364
m
5 Evaluate the following. Remember to work from left to right. a β3 β 4 + 6β b 2β β 7 β 4β d ββ 5 β 7 β 1β e ββ 3 + 2 β 7 + 9β g β0 β 9 + 7 β 30β h β β 15 β 20 + 32 β 1β PROBLEM-SOLVING
6, 7
c β β 1 β 4 + 6β f ββ 6 + 1 β 20 + 3β 7, 8
7β9
6 Determine how much debt remains in these financial situations. a owes β$300βand pays back β$155β b owes β$20βand borrows another β$35β c owes β$21 500βand pays back β$16 250β 7 a The reading on a thermometer measuring temperature rises β 18Β° Cβfrom ββ 15Β° Cβ. What is the final temperature? b The reading on a thermometer measuring temperature falls β 7Β° Cβfrom β4Β° Cβ. What is the final temperature? c The reading on a thermometer measuring temperature falls β 32Β° Cβfrom ββ 14Β° Cβ. What is the final temperature?
8 For an experiment, a chemical solution starts at a temperature of β25Β° Cβ, falls to ββ 3Β° Cβ, rises to β15Β° Cβ and then falls again to ββ 8Β° Cβ. What is the total change in temperature? Add all the changes together for each rise and fall. 9 An ocean sensor is raised and lowered to different depths in the sea. Note that ββ 100 mβmeans β100 mβ below sea level. a If the sensor is initially at ββ 100 mβand then raised to ββ 41 mβ, how far does the sensor rise? b If the sensor is initially at ββ 37 mβand then lowered to ββ 93 mβhow far is the sensor lowered?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
6B Adding and subtracting positive integers
REASONING
10
10
365
10, 11
U N SA C O M R PL R E EC PA T E G D ES
10 Give an example that suits the description. a A positive number subtract a positive number equals a negative number. b A negative number subtract a positive number equals a negative number. c A negative number add a positive number equals a positive number. d A negative number add a positive number equals a negative number.
11 a a is a positive integer, b is a positive integer and a > b. For each of the following, decide if the result will be positive, negative or zero. i a+b ii a β b iii b β a iv a β a b a is a negative integer and b is a positive integer. Decide if each of the following is always true. i a + b is positive ii a β b is negative ENRICHMENT: + or β combinations
β
β
12
12 Insert + or β signs into these statements to make them true. a 3
4
5=4
b 1
7
9
4 = β5
c β4
2
d β20
10
e βa
b
a
b=0
βa
a
3a
b
f
1
4=0
3
7
36
1
18 = β4
b = a β 2b
Positive and negative numbers are used to show the changes in international money exchange rates.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 6
Negative numbers
6C Adding and subtracting negative integers LEARNING INTENTIONS β’ To understand that adding a negative integer is the same as subtracting a positive integer β’ To understand that subtracting a negative integer is the same as adding a positive integer β’ To be able to use a number line to add and subtract negative integers
U N SA C O M R PL R E EC PA T E G D ES
366
By observing patterns in number calculations, we can see the effect of adding and subtracting negative integers. Addition
2+3=5 2+2=4 2+1=3 2+0=2 2 + (β1) = 1 2 + (β2) = 0 2 + (β3) = β1
Subtraction
β1 β1 β1 β1 β1 β1
2 β 3 = β1 2β2=0 2β1=1 2β0=2 2 β (β1) = 3 2 β (β2) = 4 2 β (β3) = 5
+1 +1 +1 +1 +1 +1
There are places where negative temperatures can occur daily in winter. Temperature differences are found by subtracting negative numbers. Finding an average temperature requires adding negative numbers.
So adding β3 is equivalent to subtracting 3, and subtracting β3 is equivalent to adding 3.
Lesson starter: Dealing with debt
Let β$10 represent $10 of debt. Write a statement (e.g. 5 + (β10) = β5) to represent the following financial situations. β’ β’ β’
$10 of debt is added to a balance of $5. $10 of debt is added to a balance of β$5. $10 of debt is removed from a balance of β$15.
KEY IDEAS
β Adding a negative number is equivalent to subtracting its opposite.
a + (βb) = a β b 2 + (β3) = 2 β 3 = β1
β4 + (β2) = β4 β 2 = β6
β3
β2 β1 0
1
2
3
β2
β7 β6 β5 β4 β3
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
6C Adding and subtracting negative integers
367
β Subtracting a negative number is equivalent to adding its opposite.
+2 4
5
6
7
8
+3
U N SA C O M R PL R E EC PA T E G D ES
βa β β(β b)β= a + bβ β5 β β(β 2)β= 5 + 2 = 7β
ββ 2 β β(β 3)β= β 2 + 3 = 1β
β3 β2 β1
0
1
2
β On a number line, the effect of adding or subtracting a negative number is to reverse the
direction of the operation.
BUILDING UNDERSTANDING
1 State the missing numbers in these sentences. The first one has been done for you. a β2 + 5βmeans that 5 is added to 2 . b ββ 3 + 6βmeans that is added to . ( ) c β1 + β β 3 βmeans that is added to . ( ) d ββ 7 + β β 11 βmeans that is added to . e β5 β 3βmeans that is subtracted from . f ββ 2 β 6βmeans that is subtracted from . ( ) g β7 β β β 3 βmeans that is subtracted from . h ββ 7 β (β β 11)βmeans that is subtracted from .
2 State the missing number or phrase to complete these sentences. a Adding ββ 4βis equivalent to subtracting . b Adding ββ 6βis equivalent to _______________ β6β. c Adding 5 is equivalent to subtracting . d Adding ββ 11βis equivalent to _______________ β11β. e Subtracting ββ 2βis equivalent to adding . f Subtracting ββ 7βis equivalent to _____________ β7β. 3 State whether each of the following is true or false. a β2 + β(β 3)β= 5β b c ββ 5 + (β β 3)β= β 8β d e β5 β β(β 1)β= 4β f g β2 β β(β 3)β= 1β h
1β 0 + β(β 1)β= 9β ββ 6 + (β β 2)β= β 4β β3 β (β β 9)β= 12β ββ 11 β (β β 12)β= β 1β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
368
Chapter 6
Negative numbers
Example 4
Adding and subtracting negative integers
Calculate the answer to these additions and subtractions. a 7β + (β β 2)β b β β 2 + β(β 3)β c β1 β (β β 3)β
U N SA C O M R PL R E EC PA T E G D ES
d ββ 6 β β(β 2)β
SOLUTION
EXPLANATION
Adding ββ 2βis equivalent to subtracting β2β.
7 + (β β 2)β = 7 β 2
a β
β β β β= 5 ( ) b β β 2 + β β 3 ββ =β β 2 β 3β β = β5 c 1β β (β β 3)ββ =β 1 + 3β β= 4 (β 2)β = β 6 + 2 d β 6 β β β β β β β = β4
Adding ββ 3βis equivalent to subtracting β3β.
Subtracting ββ 3βis equivalent to adding β3β.
Subtracting ββ 2βis equivalent to adding β2β.
Now you try
Calculate the answer to these additions and subtractions. b β β 4 + β(β 5)β c β3 β (β β 7)β a 9β + (β β 3)β
d ββ 11 β β(β 6)β
Exercise 6C FLUENCY
Example 4a
Example 4a,b
Example 4c,d
1
1, 2β4ββ(1β/2β)β
Calculate the answer to these additions. a 5β + (β β 3)β b 7β + β(β 4)β
2β5β(β 1β/3β)β
2β5ββ(1β/4β)β
c β3 + β(β 5)β
d β5 + β(β 9)β
2 Calculate the answer to these additions. a 3β + β(β 2)β b 8β + β(β 3)β e β1 + (β β 4)β f β6 + β(β 11)β ( ) i ββ 2 + β β 1 β j ββ 7 + (β β 15)β m β β 7 + (β β 3)β n β β 20 + (β β 9)β
c g k o
1β 2 + β(β 6)β β20 + β(β 22)β ββ 5 + (β β 30)β ββ 31 + (β β 19)β
d h l p
β9 + β(β 7)β β0 + β(β 4)β ββ 28 + (β β 52)β ββ 103 + (β β 9)β
3 Calculate the answer to these subtractions. a β2 β (β β 3)β b 5β β β(β 6)β e ββ 5 β (β β 1)β f ββ 7 β (β β 4)β i ββ 4 β (β β 6)β j ββ 9 β (β β 10)β ( ) m 5β β β β 23 β n 2β 8 β β(β 6)β
c g k o
2β 0 β β(β 30)β ββ 11 β (β β 6)β ββ 20 β (β β 20)β ββ 31 β β(β 19)β
d h l p
β29 β β(β 61)β ββ 41 β (β β 7)β ββ 96 β (β β 104)β ββ 104 β (β β 28)β
4 Find the missing number. a β2 +β β= β 1β d
g β5 ββ j m 5β ββ p
β+ (β 3) = 1β β= 6β
b β3 +β e
h β2 ββ
β= β 7β
c ββ 2 +β
β+ (β 10) = β 11β
f
β= 7β
i
β= β 6β
β+ (β 4) = 0β
ββ 1 ββ
ββ (β 3) = 7β
k
ββ (β 10) = 12β
l
β= 11β
n
ββ (β 2) = β 3β
o ββ 2 ββ
β= 3β
ββ (β 4) = β 20β β= β 4β
β+ (β 5) = β 1β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
6C Adding and subtracting negative integers
5 Calculate the answer, working from left to right. a β3 + (β β 2)β+ (β β 1)β b β2 + β(β 1)β+ β(β 6)β d β10 β (β β 6)β+ (β β 4)β e ββ 7 β (β β 1)β+ (β β 3)β g ββ 9 β (β β 19)β+ (β β 16)β h ββ 15 β β(β 20)β+ (β β 96)β ( ) ( ) j ββ 2 β β β 3 ββ β β 5 β k β β 18 β β(β 16)ββ (β β 19)β
369
β3 β β(β 1)ββ β(β 4)β ββ 20 β β(β 10)ββ (β β 15)β ββ 13 β β(β 19)β+ (β β 21)β β5 + β(β 20)ββ (β β 26)β
U N SA C O M R PL R E EC PA T E G D ES
c f i l
PROBLEM-SOLVING
6, 7
7β9
8β10
6 An ocean sensor is initially at ββ 90 mβ; i.e. β90 mβbelow sea level. The sensor is raised β50 mβ, lowered β 138 mβand then raised again by β35 mβ. What is the probeβs final position?
7 A small business has a bank balance of ββ $50 000β. An amount of β$20 000βof extra debt is added to the balance and, later, β$35 000βis paid back. What is the final balance? 8 $β 100βof debt is added to an existing balance of β$50βof debt. Later, β$120βof debt is removed from the balance. What is the final balance? 9 Here is a profit graph showing the profit for each month of the first half of the year for a bakery shop. a What is the profit for: i February? ii April? b What is the overall profit for the 6 months?
10 8 6 4 2
Month
O J F M A M J β2 β4 β6 β8 β10
10 Complete these magic squares, using addition. The sum of each row, column and diagonal should be the same. a
ββ 2β
β5β
b
ββ 3β
1β β
β4β
REASONING
11
β 6β β β β 17β β β 7β
11, 12
12β14
11 Write the following as mathematical expressions, e.g. β2 + (β β 3)ββ. a The sum of β3βand β4β b The sum of ββ 2βand ββ 9β c The difference between β5βand ββ 2β d The difference between ββ 2βand β1β e The sum of βaβand the opposite of βbβ f The difference between βaβand the opposite of βbβ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 6
Negative numbers
12 An addition number fact like 3 + 4 = 7 can be turned into two subtraction number facts (7 β 3 = 4, 7 β 4 = 3). a Use the number fact 7 + (β2) = 5 to state two subtraction number facts. b Use the number fact β3 + 7 = 4 to state two subtraction number facts. c Find the value of β2 + (β3) and hence state two subtraction number facts. d If you wanted to convince somebody that 1 β (β3) = 4, how could you do this using an addition number fact?
U N SA C O M R PL R E EC PA T E G D ES
370
13 Simplify these numbers. (Hint: In part a, β (β4) is the same as 0 β (β4).) a β (β4) b β (β (β1)) c β (β (β (β (β3))))
14 a If a is a positive number and b is a negative number, decide if each of the following statements is always true. i a + b is negative ii a β b is positive b If a and b are both negative numbers, decide if each of the following statements is always true. i a + b is negative ii a β b is positive c If a and b are both negative numbers and b < a, is a β b always positive? Give reasons. ENRICHMENT: Negative fractions
β
β
15(1/2)
15 Negative decimals and fractions can be added and subtracted using the same rules as those for integers. Calculate the answer to these sums and differences of fractions. 1 a 2 + (β _ 2)
4 b 5 + (β _ 3)
3 1 + β_ c β_ 2 ( 2)
10 2 + β_ d β_ 3 ( 3)
1 e 5 β (β _ 3)
f
3 10 β (β _ 2)
5 β β_ 3 g β_ 4 ( 4)
4 β β_ 1 h β_ 7 ( 2)
9 + β_ 9 _ 2 ( 3)
j
9 β β_ 9 _ 2 ( 3)
2 + β1_ 1 k 4_ 3 ( 2)
l
7 β β_ 2 n β_ 4 ( 5)
2 β β1_ 1 o 3_ 7 ( 2)
1 β β3_ 2 p β5_ 6 ( 5)
i
3 + β_ 1 m β_ 2 ( 3)
5 + β4_ 4 5_ 7 ( 5)
The temperature of the Moonβs sunlit side is around 125Β°C and the dark side temperature is around β175Β°C. The sunlit side is 125 β (β175) = 300Β°C warmer than the dark side.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
6D Multiplying and dividing integers
371
6D Multiplying and dividing integers EXTENDING
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To know that the sign of a product or quotient of two negative numbers is positive β’ To know that the sign of a product or quotient of a negative and positive number is negative β’ To be able to multiply and divide integers
The rules for multiplication and division of integers can be developed by considering repeated addition. For example: 4 groups of β3 is β3 + (β3) + (β3) + (β3) = β12. So, 4 Γ (β3) = β12. Also, β3 Γ 4 = β12 since a Γ b = b Γ a.
We also know that if 5 Γ 7 = 35, then 35 Γ· 7 = 5, so if 4 Γ (β3) = β12 then β12 Γ· (β3) = 4. This is saying there are 4 groups of β3 in β12, which we know from the repeated addition above. Also, β12 Γ· 4 = β3.
These examples give rise to the rules governing the multiplication and division of negative numbers.
Lesson starter: Patterns in tables
Complete this table of values for multiplication by noticing the patterns. What does the table of values tell you about the rules for multiplying negative integers?
Β΄ β3 β2 β1 0
1
2
3
0
0
0
0
1
0
1
2
0
2
3
0
β3
0
β2
0
β1
0
0
0
0
0
4
KEY IDEAS
β The product or quotient of two numbers of the same sign (i.e. two positive numbers or two
negative numbers) is a positive number. So positive Γ positive = positive e.g. 3 Γ 4 = 12 e.g.
positive Γ· positive = positive 12 Γ· 4 = 3
and
negative Γ negative = positive β3 Γ (β4) = 12
and
negative Γ· negative = positive β12 Γ· (β4) = 3
β The product or quotient of two numbers of the opposite sign (i.e. a positive number and a
negative number) is a negative number. So negative Γ positive = negative e.g. β3 Γ 4 = β12 negative Γ· positive = negative e.g. β12 Γ· 3 = β4
and
and
positive Γ negative = negative 3 Γ (β4) = β12 positive Γ· negative = negative 12 Γ· (β3) = β4
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 6
Negative numbers
β To find the product or quotient of two numbers, use their magnitudes to find the magnitude of
the answer, and use their signs to decide the sign of the answer. (For example, to find ββ 3 Γ 5β, the magnitude of ββ 3βis β3βand the magnitude of β5βis β5β, so the product has magnitude β15β. The sign is negative since the two numbers are of opposite sign.)
U N SA C O M R PL R E EC PA T E G D ES
372
BUILDING UNDERSTANDING 1 State the missing number.
a β2 Γ (β β 3)β= β 6β, so ββ 6 Γ· (β β 3)β=β
b β2 Γ (β β 3)β= β 6β, so ββ 6 Γ· 2 =β
c ββ 16 Γ· 4 = β 4β, so
d β16 Γ· (β β 4)β= β 4β, so
βΓ 4 = β 16β
βΓ β(β 4)β= 16β
2 Insert the missing word positive or negative for each sentence. a The product (β Γ)βof two positive numbers is _____________. b The product (β Γ)βof two negative numbers is _______________. c The product (β Γ)βof two numbers with opposite signs is ___________________. d The quotient (β Γ·)βof two positive numbers is _________________. e The quotient (β Γ·)βof two negative numbers is ____________________. f The quotient (β Γ·)βof two numbers with opposite signs is __________________.
Example 5
Multiplying and dividing integers
Calculate these products and quotients. a β5 Γ (β β 6)β b β β 3 Γ β(β 7)β
c ββ 36 Γ· (β β 4)β
SOLUTION
EXPLANATION
a β5 Γ β(β 6)β= β 30β
The two numbers are of opposite sign, so the answer is negative.
b ββ 3 Γ (β β 7)β= 21β
The two numbers are of the same sign, so the answer is positive.
c ββ 36 Γ· (β β 4)β= 9β
Both numbers are negative, so the answer is positive.
d ββ 18 Γ· 9 = β 2β
The two numbers are of opposite sign, so the answer is negative.
d ββ 18 Γ· 9β
Now you try
Calculate these products and quotients. a ββ 4 Γ 8β b β β 9 Γ (β β 5)β
c β21 Γ· (β β 3)β
d ββ 50 Γ· β(β 10)β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
6D Multiplying and dividing integers
Example 6
373
Working with multiple operations
Work from left to right to find the answer to β7 Γ 4 Γ· (β2). EXPLANATION
β7 Γ 4 Γ· (β2) = β28 Γ· (β2) = 14
First, calculate β7 Γ 4. Then calculate β28 Γ· (β2).
U N SA C O M R PL R E EC PA T E G D ES
SOLUTION
Now you try
Work from left to right to find the answer to β12 Γ· 2 Γ (β4).
Accountants work with credit (positive amounts) and debit; i.e. debt (negative amounts). When calculating monthly repayments, a debt (e.g. β$50 000) is divided by the number of months of the loan.
Exercise 6D FLUENCY
Example 5a
Example 5a,b
1
State the value of the following. a 2 Γ (β3) d β4 Γ 2
1, 2β6(1/2)
b β2 Γ 3 e 4 Γ (β2)
2 Calculate the answer to these products. a 3 Γ (β5) b 1 Γ (β10) ( ) e β8 Γ β4 f β2 Γ (β14) i β13 Γ 3 j 7 Γ (β12) m β6 Γ (β11) n 5 Γ (β9)
2β7(1/3)
2β7(1/4)
c β2 Γ (β3) f β4 Γ (β2)
c g k o
β3 Γ 2 β12 Γ (β12) β19 Γ (β2) β21 Γ (β3)
d h l p
β9 Γ 6 β11 Γ 9 β36 Γ 3 β36 Γ (β2)
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
374
Chapter 6
3 Calculate the answer to these quotients. a 1β 4 Γ· (β β 7)β b β36 Γ· (β β 3)β e ββ 9 Γ· (β β 3)β f β β 19 Γ· (β β 19)β i β84 Γ· (β β 12)β j ββ 108 Γ· 9β ( ) m ββ 132 Γ· β β 11 β n β β 39 Γ· (β β 3)β
ββ 40 Γ· 20β β 25 Γ· 5β β ββ 136 Γ· 2β β78 Γ· β(β 6)β
c g k o
d h l p
ββ 100 Γ· 25β 3β 8 Γ· (β β 2)β ββ 1000 Γ· (β β 125)β ββ 156 Γ· (β β 12)β
4 Work from left to right to find the answer. Check your answer using a calculator. a 2β Γ β(β 3)βΓ β(β 4)β b β β 1 Γ 5 Γ β(β 3)β c ββ 10 Γ· 5 Γ 2β d ββ 15 Γ· (β β 3)βΓ 1β e β β 2 Γ 7 Γ· β(β 14)β f β100 Γ· β(β 20)βΓ 2β g β48 Γ· (β β 2)βΓ (β β 3)β h β β 36 Γ 2 Γ· β(β 4)β i ββ 125 Γ· 25 Γ· β(β 5)β j ββ 8 Γ· (β β 8)βΓ· (β β 1)β k 4β 6 Γ· (β β 2)βΓ (β β 3)βΓ (β β 1)β l ββ 108 Γ· (β β 12)βΓ· (β β 3)β
U N SA C O M R PL R E EC PA T E G D ES
Example 5c,d
Negative numbers
Example 6
5 Write down the missing number in these calculations. a β5 Γβ
β= β 35β
b
βΓ β(β 2)β= β 8β
c β16 Γ·β
β= β 4β
e
βΓ· (β 3) = β 9β
f
d ββ 32 Γ·β
g ββ 5000 Γβ j
β50 Γ·β
β= β 10 000β
β= β 50β
β= β 4β
βΓ· 7 = β 20β
h ββ 87 Γβ
β= 261β
i
β243 Γ·β
k β β 92 Γβ
β= 184β
l
ββ 800 Γ·β
β= β 81β
β= β 20β
6 Remember that _ β9 βmeans β9 Γ· 3β. Use this knowledge to find the following values. 3 β 12 21 β a β_ββ b β_ c _ ββ 40 β β7 4 β5 β 124 β β 15 _ _ d β β f β 100 β β e _ β β4 β 20 β5 β 900 20 000 _ _ h β β g β β β 200 30 7 Remember that 3β β β2β= 3 Γ 3 = 9β, and (βββ 3)2β β= β 3 Γ (β β 3)β= 9β. Use this knowledge to find the following values. a β(ββ 2)2β β b (βββ 1)2β β c (βββ 9)2β β d (βββ 10)2β β e (βββ 6)2β β f (βββ 8)2β β g (βββ 3)2β β h (βββ 1.5)2β β PROBLEM-SOLVING
8
8, 9
8 List the different pairs of integers that multiply to give these numbers. a 6β β b β16β c ββ 5β
9, 10
d ββ 24β
9 Insert a multiplication or division sign between the numbers to make a true statement. a β2β
β(β 3)ββ
β(β 6)β= 1β
b ββ 25β
β(β 5)ββ
β3 = 15β
c ββ 36β
β2β
β(β 3)β= 216β
d ββ 19β
β(β 19)ββ
β15 = 15β
10 a There are two distinct pairs of numbers whose product is ββ 8βand difference is β6β. What are the two numbers? b The quotient of two numbers is ββ 11βand their difference is β36β. What are the two numbers? There are two distinct pairs to find.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
6D Multiplying and dividing integers
REASONING
11
11, 12
375
12β14
U N SA C O M R PL R E EC PA T E G D ES
11 Recall that the opposite of a number has the same magnitude but a different sign (e.g. ββ 7βand β7β). a What number can you multiply by to go from a number to its opposite? b How could you get from a number to its opposite using division? c Use one of the operations from parts a and b to explain why you get back to the original number if you take the opposite of an opposite. 12 2β β 4β βmeans β2 Γ 2 Γ 2 Γ 2β, and (βββ 2)4β β= β 2 Γ (β β 2)βΓ β(β 2)βΓ β(β 2)ββ. a Calculate: i (βββ 2)3β β ii β(ββ 2)6β β iii β(ββ 3)3β β b Which questions from part a give positive answers and why? c Which questions from part a give negative answers and why?
iv β(ββ 3)4β β
13 βa Γ bβis equivalent to βabβ, and β2 Γ (β 3)βis equivalent to ββ (2 Γ 3)β. Use this information to simplify these expressions as much as possible. a βa Γ (β b)β b ββ a Γ bβ c ββ a Γ (β b)β 14 (β1) + (β2) + (β3) + (β4) = β10 and (β1) Γ (β2) Γ (β3) Γ (β4) = 24. Therefore, it is possible to use the numbers β1, β2, β3 and β4 to achieve a βresultβ of β10 and 24. What other βresultsβ can you find using those four numbers and any mathematical operations? For example: What is (β1) Γ (β2) + (β3) Γ (β4)? Can you find expressions for every integer result from β20 to 20? ENRICHMENT: βΓβand βΓ·βwith negative fractions
β
β
15ββ(β1/2β)β
15 Calculate the answer to these problems containing fractions. Simplify where possible. β3 β c ββ _ β5 βΓ _ a _ β1 βΓ β(β _ β1 β)β b _ β3 βΓ β(β _ β2 β)β 7 5 2 2 4 3 1 βΓ· β β _ β1 β f ββ _ β5 βΓ· _ d ββ _ β3 βΓ β(β _ β4 ) ββ e β_ β1 β β 8 2 4 3 4 ( 4) a βΓ β β _ g ββ _ β 6 βΓ· β(β _ β12 ) h ββ _ β3 βΓ· ( β β_ β1 β)β ββ βb β β i β_ 2 4 11 11 b ( a) βa β k β β _ βa βΓ· _ j ββ _ βab βΓ β(β _ βa ) ββ l ββ _ βab βΓ· ( β β_ βab ) ββ b b b
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
376
Chapter 6
6A
1
Draw a number line from ββ 2βto β3β, showing all the given integers.
2 Copy and insert the symbol β<β(less than) or β>β(greater than) into these statements to make them true. a β β 2β
β4β
b ββ 9β
ββ 12β
c β4β
ββ 5β
3 Arrange these numbers in ascending order: ββ 6, 8, β 4, 0, 7β.
U N SA C O M R PL R E EC PA T E G D ES
Progress quiz
6A
Negative numbers
6A 6B
4 Calculate the answer to these additions and subtractions. a ββ 10 + 12β b β β 4 β 5β c β26 β 34β
d β β 5 β 8 + 9 β 22β
6C
5 Calculate the answer to these additions and subtractions. a β9 + (β 4)β b β β 8 + (β 7)β c β0 + (β 3)β
d β12 β (β 8)β
6D
6 Calculate the answer, working from left to right. a β β 20 β (β 10) β (β 15)β b β10 β (β 6) + (β 4)β
6D
7 Calculate these products and quotients. a β4 Γ (β 3)β b β β 5 Γ (β 12)β
c β β 56 Γ· 8β
8 Work from left to right to find the answer. a β5 Γ (β 2) Γ (β 4)β c β64 Γ· (β 8) Γ (β 2)β
b β25 Γ· (β 5) Γ 6β d ββ 40 Γ· (β 4) Γ· (β 5)β
9 Simplify each of the following. a (β β 5)β2β β
b (β β 2)β3β β
6D
Ext
6D
Ext
6C
Ext
c _ ββ 72 β β6
d ββ 20 Γ· (β 5)β
d _ ββ 1260 β 4
10 Ethan has a debt of β$120βon his credit card. He buys another item using his credit card, which adds an extra debt of β$90β. At the end of the month Ethan paid β$140βoff his credit card debt. What is the final balance on Ethanβs credit card?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
6E Order of operations with integers
377
6E Order of operations with integers EXTENDING
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To know the convention for determining order of operations in an expression involving more than one operation β’ To be able to evaluate arithmetic expressions involving more than one operation with positive and/or negative numbers
We have learnt from our study of positive integers that there is a particular order to follow when dealing with mixed operations and brackets. This order also applies when dealing with negative numbers. For example: β2 + 3 Γ (β4) is different from (β2 + 3) Γ (β4).
Lesson starter: Brackets or not? During a classroom debate about truth of the statement 3 Γ (β4) β 8 Γ· (β2) = β8: β’ β’ β’
Submarine operators and scuba divers apply the order of operations to calculate a negative height, h metres, relative to sea level. If a diver or submarine at h = β5 m descends for 3 minutes at β20 m/min, h = β5 + 3 Γ (β20) = β65, so the depth is β65 m.
Lil says that the statement needs to have more brackets to make it true. Max says that even with brackets it is impossible to make it true. Riley says that it is correct as it is and there is no need for more brackets.
Who is correct and why?
KEY IDEAS
β When working with more than one operation and with positive and/or negative numbers, follow
these rules. β’ Deal with brackets first. β’ Do multiplication and division next, working from left to right. β’ Do addition and subtraction last, working from left to right.
β2 Γ 3 β (10 + (β2)) Γ· 4 1st = β2 Γ 3 β 8 Γ· 4 2nd 3rd = β6 β 2 last = β8
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 6
Negative numbers
BUILDING UNDERSTANDING 1 Which operation (i.e. brackets, addition, subtraction, multiplication or division) is done first in each of the following problems? a ββ 2 Γ· 2 + 1β c β7 β (β β 8)βΓ· 4β e β(2 + 3 Γ (β β 2)β)β+ 1β
b β β 3 + 2 Γ (β β 6)β d ββ 6 Γ· (β 4 β β(β 2)β)β f ββ 11 Γ· (β 7 β 2 Γ β(β 2)β)β
U N SA C O M R PL R E EC PA T E G D ES
378
2 Classify each of the following statements as true or false. a ββ 4 + 2 Γ 3 = β 4 + (β 2 Γ 3)β b ββ 4 + 2 Γ 3 = (β β 4 + 2)βΓ 3β c β8 Γ β(2 β β(β 2)β)β= 8 Γ 4β d β8 Γ β(2 β β(β 2)β)β= 8 Γ 0β ( ) ( ) ( ) e ββ 40 β 20 Γ· β β 5 β= β β 40 β 20 βΓ· β β 5 β f ββ 40 β 20 Γ· (β β 5)β= β 40 β (β 20 Γ· (β β 5))β β
Example 7
Using order of operations
Use order of operations to evaluate the following. a 5β + 2 Γ (β β 3)β
b ββ 6 Γ 2 β 10 Γ· β(β 5)β
SOLUTION
EXPLANATION
a 5β + 2 Γ (β β 3)ββ =β 5 + β(β 6)ββ β = β1
Do the multiplication before the addition.
b β 6 Γ 2 β 10 Γ· (β β 5)β = β 12 β (β β 2)β ββ―β― β =β β 12 + 2β β β = β 10
Do the multiplication and division first. When subtracting ββ 2β, add its opposite.
Now you try
Use order of operations to evaluate the following. a β10 + β(β 20)βΓ· 5β
Example 8
b β5 Γ β(β 7)β+ β(β 24)βΓ· 4β
Using order of operations with brackets
Use order of operations to evaluate the following. a (β β 2 β 1)βΓ 8β
b β5 Γ· β(β 10 + 5)β+ 5β
SOLUTION
EXPLANATION
(β β 2 β 1)βΓ 8 = β 3 Γ 8
a β
β β β = β 24
β
b 5 Γ· (β β 10 + 5)β+ 5 = 5 Γ· (β β 5)β+ 5 β β ββ―β― =β β 1 + 5β β= 4
Deal with brackets first.
Deal with brackets first. Then do the division before the addition.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
6E Order of operations with integers
379
Now you try Use order of operations to evaluate the following. a β(β 10 + 2)βΓ· β(3 β 5)β
U N SA C O M R PL R E EC PA T E G D ES
b β(36 Γ· (β β 4)β+ 2)βΓ β(1 β 3)β
Exercise 6E FLUENCY
Example 7a
Example 7a
Example 7
Example 8
1
1, 2, 3β4ββ(1β/3β)β
1β4β(β 1β/2β)β
Use order of operations to evaluate the following. a β3 + 2 Γ β(β 4)β c 2β + (β 3) Γ 4β
b 5β + 8 Γ β(β 2)β d β5 + (β 2) Γ 3β
2 Use order of operations to evaluate the following. a β10 + 4 Γ· (β 2)β c 2β 4 Γ· (β 4) Γ 6β
b 1β 2 + 8 Γ· (β 4)β d β40 Γ· 5 Γ (β 2)β
3 Use order of operations to evaluate the following. a β2 + 3 Γ (β β 3)β b β9 + 10 Γ· (β β 5)β d β18 + (β β 9)βΓ 1β e β10 β 2 Γ (β β 3)β ( ) g ββ 8 β β β 7 βΓ 2β h ββ 2 Γ 4 + 8 Γ (β β 3)β j β12 Γ· (β β 6)β+ 4 Γ· (β β 2)β k β β 30 Γ· 5 β 6 Γ 2β m 8β Γ (β β 2)ββ (β β 3)βΓ 2β n ββ 1 Γ 0 β (β β 4)βΓ 1β
4 Use order of operations to evaluate the following. a β(3 + 2)βΓ (β β 2)β c β β 3 Γ (β β 2 + 4)β e 1β 0 Γ· (β 4 β (β β 1))β β g (β 24 β 12)βΓ· (β 16 + (β β 4)β)β i ββ 2 Γ (β 8 β 4)β+ (β β 6)β k β0 + (β β 2)βΓ· (β 1 β 2)β m (β β 3 + (β β 5))β βΓ (β β 2 β (β β 1))β β o ββ 5 β (β 8 + (β β 2)β)β+ 9 Γ· (β β 9)β PROBLEM-SOLVING
b d f h j l n
c f i l o
3β4ββ(1β/4β)β
2β 0 + (β β 4)βΓ· 4β β10 β 1 Γ (β β 4)β ββ 3 Γ (β β 1)β+ 4 Γ (β β 2)β ββ 2 Γ 3 β 4 Γ· (β β 2)β β0 Γ β(β 3)ββ (β β 4)βΓ 0 + 0β
β(8 β 4)βΓ· (β β 2)β ββ 1 Γ (β 7 β 8)β (β 2 + (β β 3))β βΓ (β β 9)β β(3 β 7)βΓ· (β β 1 + 0)β ββ 2 β 3 Γ (β β 1 + 7)β β1 β 2 Γ (β β 3)βΓ· β(β 3 β (β β 2)β)β ββ 3 Γ· β(β 1 + 4)βΓ 6β
5, 6
6, 7ββ(β1/2β)β
7β(β β1/2β)β,β 8
5 A shop owner had bought socks at β$5βa pair but, during an economic downturn, changed the price to sell them for β$3βa pair. In a particular week, β124βpairs are sold and there are other costs of β$280β. What is the shop ownerβs overall loss for the week?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 6
Negative numbers
6 A debt of β$550βis doubled and then β$350βof debt is removed each month for β3βmonths. What is the final balance? 7 Insert brackets to make each statement true. a β β 2 + 3 Γ 8 = 8β b ββ 10 Γ· 4 + 1 = β 2β d ββ 5 β 1 Γ· (β β 6)β= 1β e β3 β 8 Γ· 5 + 1 = 0β g ββ 2 Γ 3 β (β β 7)ββ 1 = β 21β h ββ 3 + 9 Γ· (β β 7)β+ 5 = β 3β
c β β 1 + 7 Γ 2 β 15 = β 3β f β50 Γ 7 β 8 Γ β(β 1)β= 50β i β32 β β(β 8)βΓ· (β β 3)β+ 7 = 10β
U N SA C O M R PL R E EC PA T E G D ES
380
8 By inserting only one pair of brackets, how many different answers are possible for this calculation? Also include the answers for which brackets are not required. ββ 2 + 8 Γ (β β 4)ββ (β β 3)β REASONING
9
10, 11
9β10ββ(β1/2β)β
9 If brackets are removed from these problems, does the answer change? a (β 2 Γ 3)ββ (β β 4)β b (β 8 Γ· (β β 2)β)ββ 1β c (β β 2 + 3)βΓ 4β d β9 Γ· (β β 4 + 1)β e (β 9 β (β β 3)βΓ 2)β+ 1β f (β β 1 + 8 Γ· (β β 2)β)βΓ 2β
10 State if each of the following is always true or false. a (β β 3 + 1)ββ + (β 7) = β 3 + (1 + (β 7) )β b β(β 3 + 1) β (β 7) = β 3 + (1 β (β 7) )β ( ) ( ) c β a + b β+ c = a + β b + c β d β(a + b)ββ c = a + (β b β c)β e (β a β b)β+ c = a β (β b + c)β f β(a β b)ββ c = a β (β b β c)β
11 a Is the answer to each of the following positive or negative? i ββ 6 Γ (β β 4)βΓ β(β 8)βΓ β(β 108)βΓ β(β 96)β ii ββ 100 Γ· (β β 2)βΓ· 2 Γ· (β β 5)β iii (βββ 3)3β β iv ββ 1 Γ β(ββ 2)3β β ( )βΓ 4 Γ 7 Γ (β β 3)β ( )2 ( ) ___________ v _____________________ β―β―β― ββ 6 Γ β β 3 β―β― β vi β―β― ββββ 1 β βΓ β β 1 ββ (βββ 2)2β β (βββ 1)3β βΓ (β β 1)β b Explain the strategy you used to answer the questions in part a.
ENRICHMENT: Powers and negative numbers
β
β
12, 13
12 First, note that: β’ 2β β 4β β= 2 Γ 2 Γ 2 Γ 2 = 16β β’ (βββ 2)4β β= β 2 Γ (β β 2)βΓ (β β 2)βΓ (β β 2)β= 16β β’ ββ2β β 4β β= β (β 2 Γ 2 Γ 2 Γ 2)β= β 16β When evaluating expressions with powers, the power is dealt with first in the order of operations. For example: (β (β β 2)β3β ββ 1β)βΓ· (β 3) = (β 8 β 1) Γ· (β 3) = β 9 Γ· (β 3) = 3β Evaluate each of the following. a 2β β 2β β b (β β 2)β2β β d (β β 2)β5β β e ββ2β β 5β β 3 g (β (β β 3)ββ ββ 1)βΓ· (β 14)β h β30 Γ· (β 1 β 4β β 2β )β β
c ββ2β β 2β β f (β3β β β2ββ 1)βΓ 4β i ββ 10 000 Γ· (β β 10)β4β β
13 Kevin wants to raise ββ 3βto the power of β4β. He types ββ3β β 4β βinto a calculator and gets ββ 81β. Explain what Kevin has done wrong.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Applications and problem-solving
U N SA C O M R PL R E EC PA T E G D ES
Sunnyβs monthly budget
Applications and problem-solving
The following problems will investigate practical situations drawing upon knowledge and skills developed throughout the chapter. In attempting to solve these problems, aim to identify the key information, use diagrams, formulate ideas, apply strategies, make calculations and check and communicate your solutions.
381
1
Sunny wanted to keep a close track of all the money she earned and spent and decided to keep her own balance sheet, just like at a bank. Below is a copy of Sunnyβs transaction history for the month of October. Sunny is able to have a negative balance, because her bank is her parentsβ bank and they are okay if Sunny borrows money from them, as long as she pays them back at a later date. Date
Transaction Details
Debit
Credit
Balance
β4 / 10β
Earned β$25βfrom babysitting
β$25β
ββ $30β
β5 / 10β
Spent β$90βon a new pair of shoes
β11 / 10β
Received β$15βfrom Grandma
1β 8 / 10β
Spent β$12βgoing to the movies
β$12β
ββ $117β
2β 2 / 10β
Donated β$5βto βOdd Socksβ day
β$5β
ββ $122β
2β 3 / 10β
Earned β$60βfrom babysitting
β27 / 10β
Spent β$25βon Hollyβs birthday present
β30 / 10β
Paid β$40βfor magazine subscription
β$90β
β β $120β
β$15β
ββ $105β
Sunny is interested in managing her money and analysing her earnings and expenditure during the month of October. a What was Sunnyβs balance on β1β October? b How much did Sunny owe her parents on β1β October? c Complete the last three rows of Sunnyβs balance sheet. d How much did Sunny spend in October? e How much did Sunny earn in October? f How much does Sunny owe her parents at the end of October?
Body temperature of a polar bear
2 Polar bears are the worldβs largest land predators and can weigh up to β600 kgβ. They live in the Arctic where temperatures are regularly below β0Β° Cβ. The average winter temperatures in the Arctic (month of January) range from ββ 34Β° Cβto ββ 5Β° Cβ. The average summer temperatures in the Arctic (month of July) range from ββ 12Β° Cβto β10Β° Cβ. A polar bear must maintain its body temperature at β37Β° Cβand has two coats of fur and a thick layer of fat to help with this task. You are interested in the various differences between minimum and maximum air temperatures and the body temperature of a polar bear. a What is the maximum difference in winter between the outside temperature and a polar bearβs body temperature? b What is the minimum difference in summer between the outside temperature and a polar bearβs body temperature?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 6
Negative numbers
c What would the outside temperature be if there was β100Β° Cβdifference between the outside temperature and a polar bearβs body temperature? d The coldest temperature recorded in the Arctic was ββ 69.6Β° Cβin Greenland in β1991β. What was the difference between the outside temperature and a polar bearβs body temperature at this time? e Write an expression for the difference between the outside temperature, βTΒ° Cβ, and a polar bearβs body temperature.
U N SA C O M R PL R E EC PA T E G D ES
Applications and problem-solving
382
Depth of a scuba dive
3 Alessandro is an experienced scuba diving instructor who teaches groups of adults how to safely scuba dive. During a class, Alessandro travels up and down in the water moving between the different members of his group and constantly checking in on them. Alessandro wears a depth gauge watch and below is a table showing his depths recorded at β2β-minute intervals during his most recent 1-hour class. Time (min)
Depth (m)
Time (min)
Depth (m)
Time (min)
Depth (m)
Time (min)
Depth (m)
β0β
β0β
β16β
β β 10β
β32β
ββ 15β
β48β
β β 31β
β2β
β β 1β
1β 8β
ββ 8β
β34β
ββ 21β
β50β
β β 34β
β4β
β β 3β
β20β
ββ 8β
β36β
ββ 24β
β52β
β β 30β
β6β
β β 8β
β22β
ββ 6β
β38β
ββ 28β
β54β
β β 26β
β8β
β β 4β
β24β
β β 12β
β40β
ββ 29β
β56β
β β 20β
β10β
β β 9β
2β 6β
β β 15β
β42β
ββ 25β
β58β
β β 12β
β12β
β β 4β
2β 8β
β β 11β
β44β
ββ 22β
β60β
β0β
β14β
β β 5β
3β 0β
ββ 9β
β46β
ββ 29β
Alessandro is interested in using positive and negative numbers to analyse the change in depth and rate of change of depth over the course of a dive. a Using the data provided from the depth gauge watch, what was the total vertical distance Alessandro travelled during the β1β-hour class? b Scuba divers need to be careful of ascending too quickly as it can cause decompression sickness. It is commonly understood that divers should not exceed an ascent rate of β9βm/minute. From the depth gauge watch data, what was Alessandroβs maximum ascent rate and when did this occur? c What was Alessandroβs average vertical speed over the β1β-hour class in m/min? travelledβ ______________ βSpeed = β―β― βDistance β―β― Time taken
d A scuba diver instructor assisting a group of beginner scuba divers ended up diving to a depth of βdβmetres and resurfacing on βfiveβ occasions, and also diving to a depth of β2dβ metres and resurfacing on βtwoβoccasions, all in the one βtβminute class. Write an expression for the instructorβs average vertical speed in m/min during the class.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
6F Substituting integers
383
6F Substituting integers EXTENDING
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand that substitution involves replacing a pronumeral (letter) with a number β’ To be able to substitute positive or negative integers into an expression and evaluate
The process known as substitution involves replacing a pronumeral (sometimes called a variable) with a number. A carβs speed after t seconds could be given by the expression 10 + 4t. So, after 3 seconds we can calculate the carβs speed by substituting t = 3 into 10 + 4t.
So 10 + 4t = 10 + 4 Γ 3 = 22 metres per second.
Lesson starter: Order matters
Two students substitute the values a = β2, b = 5 and c = β7 into the expression ac β bc. Some of the different answers received are 21, β49, β21 and 49. β’
When a vehicle or plane brakes to a stop, its speed is changing at a negative rate, e.g. β5 m/s per second. Aircraft engineers substitute negative numbers into formulas to calculate the stopping distances of various planes.
Which answer is correct and what errors were made in the calculation of the three incorrect answers?
KEY IDEAS
β Substitute into an expression by replacing pronumerals (letters
representing numbers) with numbers.
If a = β3 then 3 β 7a = 3 β 7 Γ (β3) = 3 β (β21) = 3 + 21 = 24
β Brackets can be used around negative numbers to avoid confusion with other symbols.
BUILDING UNDERSTANDING
1 Which of the following shows the correct substitution of a = β2 into the expression a β 5? A 2β5 B β2 + 5 C β2 β 5 D 2+5
2 Which of the following shows the correct substitution of x = β3 into the expression 2 β x? A β2 β (β3) B 2 β (β3) C β2 + 3 D β3 + 2 3 Rafe substitutes c = β10 into 10 β c and gets 0. Is he correct? If not, what is the correct answer?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
384
Chapter 6
Negative numbers
Example 9
Substituting integers c βb Γ· 5 β aβ
U N SA C O M R PL R E EC PA T E G D ES
Evaluate the following expressions using βa = 3βand βb = β 5β. a 2β + 4aβ b 7β β 4bβ SOLUTION
EXPLANATION
a 2 + 4a = 2 + 4 Γ 3 β =β 2 + 12β β β―β― β β = 14
Replace βaβwith β3βand evaluate the multiplication first.
b 7 β 4b = 7 β 4 Γ (β β 5)β ββ =β 7 β (β ββ20)ββ β β β―β― β―β― β = 7 + 20 β = 27
Replace the βbβwith ββ 5βand evaluate the multiplication before the subtraction.
c b Γ· 5 β a = β5 Γ· 5 β 3 β ββ―β― =β β 1 β 3β β β = β4
Replace βbβwith ββ 5βand βaβwith β3β, and then evaluate.
Now you try
Evaluate the following expressions using βx = β 7βand βy = 2β. a β4 β 5yβ b β2x + 5β
c β5(β y β x)β
Exercise 6F FLUENCY
1
Example 9a
1, 2β4ββ(β1/2β)β
b i
β4 β 2bβ
ii β10 β 7bβ
Example 9c
c i
βb Γ· 3 β aβ
ii βb Γ· 1 β aβ
Example 9c
Example 9
2β4β(β β1/3β)β,β 5
Evaluate the following expressions using βa = 4βand βb = β 3β. a i β1 + 2aβ ii β5 + 3aβ
Example 9b
Example 9a,b
2β5β(β β1/2β)β
2 Evaluate the following expressions using βa = 6βand βb = β 2β. a 5β + 2aβ b β β 7 + 5aβ c βb β 6β e β4 β bβ f β7 β 2bβ g β3b β 1β i β5 β 12 Γ· aβ j β1 β 60 Γ· aβ k β10 Γ· b β 4β
d βb + 10β h ββ 2b + 2β l β3 β 6 Γ· bβ
3 Evaluate the following expressions using βa = β 5βand βb = β 3β. a βa + bβ b aβ β bβ c βb β aβ e β5b + 2aβ f β6b β 7aβ g ββ 7a + b + 4β
d β2a + bβ h ββ 3b β 2a β 1β
4 Evaluate these expressions for the given pronumeral values. a 2β 6 β 4x (β x = β 3)β b ββ 2 β 7k (β k = β 1)β ( ) c β10 Γ· n + 6 β n = β 5 β d ββ 3x + 2y (β x = 3, y = β 2)β e β18 Γ· y β x (β x = β 2, y = β 3)β f ββ 36 Γ· a β ab (β a = β 18, b = β 1)β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
6F Substituting integers
385
5 These expressions contain brackets. Evaluate them for the given pronumeral values. (Remember that ab means a Γ b.) a 2 Γ (a + b) (a = β1, b = 6) b 10 Γ· (a β b) + 1 (a = β6, b = β1) c ab Γ (b β 1) (a = β4, b = 3) d (a β b) Γ bc (a = 1, b = β1, c = 3) PROBLEM-SOLVING
6, 7
7, 8
U N SA C O M R PL R E EC PA T E G D ES
6, 7
6 The area of a triangle for a fixed base of 4 metres is given by the rule Area = 2h m2, where h metres is the height of the triangle. Find the area of such a triangle with these heights. a 3m b 8m
7 For a period of time a motorcycleβs speed, in metres per second, is modelled by the expression 20 + 3t, where t is the number of seconds after the motorcycle passes a particular building. The rule applies from 6 seconds before the motorcyclist reached the building. a Find the motorcycleβs speed after 4 seconds. b Find the motorcycleβs speed at t = β2 seconds (i.e. 2 seconds before passing the t = 0 point). c Find the motorcycleβs speed at t = β6 seconds. 8 The formula for the perimeter, P, of a rectangle is P = 2l + 2w, where l and w are the length and the width, respectively. a Use the given formula to find the perimeter of a rectangle with: i l = 3 and w = 5 ii l = 7 and w = β8 b What problems are there with part a ii above? REASONING
9
9, 10
10, 11
9 Write two different expressions involving x that give an answer of β10 if x = β5.
10 Write an expression involving the pronumeral a combined with other integers, so if a = β4 the expression would equal these answers. a β3 b 0 c 10
11 If a and b are non-zero integers, explain why these expressions will always give the result of zero. (a β a) a ab β a a aβb+bβa b _ c _ d _ a β1 b b ENRICHMENT: Celsius/Fahrenheit
β
β
12
12 The Fahrenheit temperature scale (Β°F) is still used today in some countries, but most countries use the Celsius scale (Β°C). 32Β°F is the freezing point for water (0Β°C). 212Β°F is the boiling point for water (100Β°C). 5 Γ (Β°F β 32). The formula for converting Β°F to Β°C is Β°C = _ 9
a Convert these temperatures from Β°F to Β°C. i 41Β°F ii 5Β°F iii β13Β°F b Can you work out the formula that converts from Β°C to Β°F? c Use your rule from part b to check your answers to part a.
The temperature of boiling water is 100Β°C or 212Β°F.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 6
Negative numbers
6G The Cartesian plane LEARNING INTENTIONS β’ To understand that the Cartesian plane can be extended to include negative numbers on both axes β’ To understand what a coordinate pair means if one or both numbers is negative β’ To be able to plot a point at a location expressed as x- and y-coordinates
U N SA C O M R PL R E EC PA T E G D ES
386
During the seventeenth century, two well-known mathematicians, RenΓ© Descartes and Pierre de Fermat, independently developed the idea of a number plane. The precise positions of points are illustrated using coordinates, and these points can be plotted using the axes as measuring guides. This invention revolutionised the study of mathematics and provided a vital link between geometry and algebra. The number plane or coordinate plane, is also called the Cartesian plane (named after Descartes). It uses two axes at right angles that extend in both the positive and negative directions.
Lesson starter: North, south, east and west
Mathematician and philosopher RenΓ© Descartes.
The units for this grid are in metres.
N
RenΓ© starts at position O and moves: β’ β’
3 m east 4 m south
β’ β’
Pierre starts at position O and moves: β’ β’
1 m west 4 m east
β’ β’
3 2 1
2 m south 5 m north.
3 m south 5 m north.
W
O β3 β2 β1 β1
E
1 2 3
β2 β3
Using the number plane, how would you describe RenΓ© and Pierreβs final positions?
S
KEY IDEAS
β The Cartesian plane is also called the number plane.
β The diagrams show the two axes (the x-axis and the y-axis) and
the four quadrants.
2nd quadrant
1st quadrant
3rd quadrant
4th quadrant
y
(β3, 3) 4 3 2 1
(2, 3)
O β4 β3 β2 β1 β1 1 2 3 4 (β1, β2) β2 β3 (4, β3) β4
x
β A point plotted on the plane has an x-coordinate and y-coordinate, which are written as (x, y). β The point (0, 0) is called the origin and labelled O.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
6G The Cartesian plane
387
β To plot points, always start at the origin.
β’ β’ β’
U N SA C O M R PL R E EC PA T E G D ES
β’
For (β β2, 3β)βmove β2βright and β3β up. For (4, β3) move β4βright and β3β down. For (β3, 3) move β3βleft and β3β up. For (β1, β2) move β1βleft and β2β down.
BUILDING UNDERSTANDING
1 Match the points βAβ, βBβ, βCβ, βDβ, βEβ, βFβ, βGβand βHβwith the given coordinates. y a β(β 1, 3)β b β(2, β 3)β E D 3 c β(2, 1)β 2 1 d β(β 2, β 2)β H A e β(3, 3)β β3 β2 β1 O 1 2 3 Gβ1 f β(β 3, 1)β β2 F C g β(1, β 2)β β3 B h β(β 1, β 1)β
2 Count the number of points, shown as dots, on this plane that have: a both βxβ- and βyβ-coordinates as positive numbers b an βxβ-coordinate as a positive number c a βyβ-coordinate as a positive number d an βxβ-coordinate as a negative number e a βyβ-coordinate as a negative number f both βxβ- and βyβ-coordinates as negative numbers g neither βxβnor βyβas positive or negative numbers.
x
y
3 2 1
β1 O β1 β2
1 2 3
x
Example 10 Finding coordinates
y
For the Cartesian plane shown, write down the coordinates of the points labelled βAβ, βBβ, βCβand βDβ.
4 3 2 1
D
O β4 β3 β2 β1 β1
C
β2 β3 β4
A
1 2 3 4 B
x
Continued on next page
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
388
Chapter 6
Negative numbers
EXPLANATION
A = (β 1, 1)β B = (β 3, β 2)β β―β― β β β β β β C = (β β 2, β 4)β D = (β β 3, 3)β
For each point, write the βxβ-coordinate first (from the horizontal axis) followed by the βyβ-coordinate (from the vertical axis).
U N SA C O M R PL R E EC PA T E G D ES
SOLUTION
Now you try
y
For the Cartesian plane shown, write down the coordinates of the points labelled βAβ, βBβ, βCβand βDβ.
B
4 3 2 1
A
β4 β3 β2 β1 O β1 D β2 β3 β4
1 2 3 4 C
x
Exercise 6G FLUENCY
Example 10
1
1, 3β5
2β5
3β6
For the Cartesian plane shown, write down the coordinates of the points labelled βAβ, βBβ, βCβand βDβ.
y
4 3 2 1
D
A
β4 β3 β2 β1 O β1 C β2 β3 β4
Example 10
2 For the Cartesian plane given, write down the coordinates of the points labelled βAβ, βBβ, βCβ, βDβ, βEβ, βFβ, βGβand βHβ.
x
1 2 3 4
B
y
H
D
4 3 2 1
O β4 β3 β2 β1 β1 G β2 β3 Cβ4
E
A
1 2 3 4
x
B F
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
6G The Cartesian plane
3 a Draw a set of axes using β1 cmβspacings. Use ββ 4βto β4βon both axes. b Now plot these points. i (β β 3, 2)β ii β(1, 4)β iii β(2, β 1)β v (β 2, 2)β vi β(β 1, 4)β vii β(β 3, β 1)β
389
iv β(β 2, β 4)β viii (β 1, β 2)β
4 For the number plane given, write down the coordinates of the points labelled βAβ, βBβ, βCβ, βDβ, βEβ, βFβ, βGβand βHβ.
U N SA C O M R PL R E EC PA T E G D ES
y
H
C
4 D 3 2 E 1
O β4 β3 β2 β1 β1
F
A
1 2 3 4
x
β2 B β3 β4 G
5 Seven points have the following βxβ- and βyβ-coordinates. βxβ
ββ 3β
ββ 2β
ββ 1β
0β β
β1β
β2β
β3β
βyβ
ββ 2β
ββ 1β
0β β
β1β
β2β
3β β
β4β
a Plot the seven points on a Cartesian plane. Use ββ 3βto β3βon the βxβ-axis and ββ 2βto β4βon the βyβ-axis. b What do you notice about these seven points on the Cartesian plane?
6 Seven points have the following βxβ- and βyβ-coordinates. βxβ
ββ 3β
ββ 2β
ββ 1β
0β β
β1β
β2β
β3β
βyβ
β5β
β3β
β1β
ββ 1β
ββ 3β
β β 5β
ββ 7β
a Plot the seven points on a Cartesian plane. Use ββ 3βto β3βon the βxβ-axis and ββ 7βto β5βon the βyβ-axis. b What do you notice about these seven points on the number plane?
PROBLEM-SOLVING
7, 8
8β10
9β11
7 When plotted on the Cartesian plane, what shape does each set of points form? a βA(β β 2, 0)β, Bβ(0, 3)β, Cβ(2, 0)β b βA(β 3, β 1), B(β 3, 2), C(1, 2), D(1, β 1)β c A β (β β 4, β 2)β, Bβ(3, β 2)β, Cβ(1, 2)β, Dβ(β 1, 2)β d βAβ(β 3, 1)β, Bβ(β 1, 3)β, Cβ(4, 1)β, Dβ(β 1, β 1)β
8 Using the origin as one corner, the point βAβ(3, 2)βas the opposite corner and the axes as two of the sides, a rectangle can be positioned on a set of axes, as shown opposite. Its area is β6βsquare units. Find the area of the rectangle if the point βAβ is: b β(β 3, 2)β a (β 2, 2)β d (β 3, β 5)β c β(β 1, β 4)β
y
3 2 1
A
O
1 2 3
x
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 6
Negative numbers
9 Karenβs bushwalk starts at a point (2, 2) on a grid map. Each square on the map represents 1 km. If Karen walks to the points (2, β7), then (β4, β7), then (β4, 0) and then (2, 0), how far has she walked in total? 10 The points A(β2, 0), B(β1, ?) and C(0, 4) all lie on a straight line. Find the y-coordinate of point B. 11 The points A(β4, 8), B(β1, ?) and C(2, β2) all lie on a straight line. Find the y-coordinate of point B.
U N SA C O M R PL R E EC PA T E G D ES
390
REASONING
12
12, 13
12, 13
12 Consider the points A(β2, 2), B(0, 2) and C(3, β2). a Which point is closest to (0, 0)? b Which point is farthest from (0, 0)? c List the given points in order from closest to farthest from the origin, O.
13 A point (a, b) sits on the number plane in one of the four regions 1, 2, 3 or 4, as shown. These regions are called quadrants. a Name the quadrant or quadrants that include the points where: i a>0 ii a > 0 and b < 0 iii b < 0 iv a < 0 and b < 0 b Shade the region that includes all points for which b > a. ENRICHMENT: Rules and graphs
β
β
y
2
1
x
3
4
14
14 Consider the rule y = 2x β 1. a Substitute each given x-coordinate into the rule to find the y-coordinate. Then complete this table. x
β3
β2
β1
0
1
2
3
y
b Draw a Cartesian plane, using β3 to 3 on the x-axis and β7 to 5 on the y-axis. c Plot each pair of coordinates (x, y) onto your Cartesian plane. d What do you notice about the set of seven points?
The Cartesian plane was the starting point for the development of computer-generated graphics and design.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Modelling
391
Mathematical orienteering
U N SA C O M R PL R E EC PA T E G D ES
For example, if starting at the origin, then the route defined by [β β 3, β 2]βfollowed by [β 4, 5]βtakes the person β3βunits left and β2βunits down from the origin to the point (β β 3, β 2)βthen β4βunits right and β5βunits up to point (β 1, 3)βas shown. The instruction [β β 1, β 3]βwould then take the person back to the origin as shown. Units are in kilometres and north is up on this map.
Modelling
Mathematical orienteering involves hiking in directions defined by two numbers in square brackets giving the horizontal (east/west) and vertical (north/south) distances to travel.
y
4 3 2 1
O β4 β3 β2 β1 β1
1 2 3 4
x
β2 β3 β4
When a course is designed it must meet the following criteria. β’ The course must start and finish at the origin (β 0, 0)ββ. β’ The total north/south movement must not be more than β20β km. β’ The total east/west movement must not be more than β20β km.
Present a report for the following tasks and ensure that you show clear mathematical workings and explanations where appropriate.
Preliminary task
a For the three-component course described in the introduction, find: i the total north/south movement (this includes β2 kmβsouth, β5 kmβnorth and then β3 kmβ south) ii the total east/west movement. b A three-component course starting at (β 0, 0)βis defined by the instructions β[4, β 2]β, followed by β [β 6, β 1]β, followed by [β 2, 3]ββ. i Draw a diagram of this course. ii Find the total north/south movement. iii Find the total east/west movement. c Explain why the course from part b satisfies all the criteria for a mathematical orienteering course.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
392
Chapter 6
Negative numbers
Modelling task Formulate
U N SA C O M R PL R E EC PA T E G D ES
Solve
a The problem is to design a mathematical orienteering course where the total north/south and east/ west movements are equal to β20 kmβ. Write down all the relevant information that will help solve this problem with the aid of a diagram. b A mathematical orienteering course with four components is defined by the instructions: [β 2, 5]ββ, [ββ β 3, β 1]ββ, [ββ β 1, 6]βand [β 2, 2]βand the starting location of (β 0, 0)ββ. i Draw a number plane and use arrows to illustrate this course. ii Explain whether this course satisfies the criteria for a mathematical orienteering course. c Another four-component course is defined by the instructions: [β β 4, 1]ββ, [ββ 3, 5]ββ, [ββ 3, β 7]βand [β β? , ? ]β ββ. Determine the instruction [β β? , ? ]β βto ensure it satisfies all the criteria. d Construct your own four-component course which starts and ends at the origin and with the north/south and east/west movements each equal to exactly β20βkm. Illustrate your course using a number plane. e For your course from part d, write down the four instructions and explain why the total north/south and east/west movements are each equal to exactly β20β km. f Is it possible to design a four-component course which meets the criteria where one of the instructions is [β β 11, 2]β? Explain why or why not. g Summarise your results and describe any key findings.
Evaluate and verify
Communicate
Extension questions
a Design a mathematical orienteering course using six components, with the north/south and east/ west movements each exactly β20 kmβ. Try to use each of the four quadrants of a number plane. b Rather than describing the total north/south distance and the total east/west distance, it is possible to write the conditions for a mathematical orienteering course using total distance north and total distance east. Explain how to do this.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Technology and computational thinking
393
Tracking lift movements
U N SA C O M R PL R E EC PA T E G D ES
Negative numbers can be used to describe the position and direction of a lift in a building. Assume the lift starts at level β8β. The addition of β β 3βwould mean that the lift travels down three levels. Additionally, a lift stopping at level ββ 2βfor example would mean it travels to the floor which is two levels below ground sometimes referred to as Basement β 2βor βB2β.
Technology and computational thinking
Key technology: Spreadsheets
1 Getting started
Imagine a building where a lift has β4βbasement levels, a ground level and β12βfloors above ground. Assume that the lift starts at ground level which is level β0β. Each movement from one level to a different level will be called an operation. The following table shows the position of the lift after β5β operations. Operation
Initial
β1β
β2β
3β β
β4β
β5β
Position
β0β
β5β
ββ 2β
β3β
β11β
ββ 1β
a At what floor is the lift positioned after the following operations? i β2β ii β3β iii β5β b What number is added to the previous position to obtain the new position after the following operations? i β1β ii β2β iii β5β c The number of floors travelled by the lift in the β2ndβoperation is β7βas it travels from level β5βdown to level ββ 2β. What is the total number of floors travelled by the lift after the β5β operations?
2 Applying an algorithm
In a scenario using a single lift, we will consider the following type of lift sequence. β’ The lift starts at ground level which is level β0β. β’ A person presses the call button at a random floor level and the lift moves to that level. β’ The person enters the lift and presses a desired floor level. The lift moves to that level. β’ The person exits the lift and the lift waits for the next call. a The following table summarises the position of the lift for a scenario with β5βpeople and β5β operations. Note the following: β’ The lift starts at ground level, level β0β. β’ The Entry level is the level from which the person calls the lift. β’ The Exit level is the level at which the person exits the lift. β’ Rise/fall β Exit to entry is the number added to the previous Exit level to travel to the new Entry level. β’ Rise/fall β Entry to exit is the number added to the Entry level to travel to the new Exit level. Person
Entry level
Exit level
β1β
β3β
β8β
Rise/fall β Exit to entry Rise/fall β Entry to exit β3β(starting at β0β)
β5β
β2β
β11β
β4β
β3β
ββ 7β
3β β
β0β
β8β
β4β
β9β
β1β
ββ 12β
β5β
β2β
β5β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 6
Negative numbers
a Follow the given algorithm to find the missing numbers in the table above. b Draw your own table representing a scenario of β7βpeople and using numbers between ββ 6βand β15β to represent a lift in a building with β6βbasement levels and β15βabove ground levels. Enter random numbers of your choice in the Entry level and Exit level columns then calculate the values in both Rise/fall columns.
U N SA C O M R PL R E EC PA T E G D ES
Technology and computational thinking
394
3 Using technology
We will use a spreadsheet to construct a lift scenario using a special random number function to automatically select our Entry level and Exit level column values. a Create a spreadsheet to represent a scenario of β10βpeople using random numbers between ββ 4βand β 12βto represent a lift in a building with β4βbasement levels and β12βabove ground levels. Use the following as a guide. Note: β’ The RANDBETWEEN function gives a random integer between the two given values inclusive, in this case ββ 4βand β12β. β’ The rules in columns C and D calculate the column values as per the tables in part 2 above.
b Fill down at cells βA5β, βB4β, βC4β, βD5βand βE4βto create a scenario for β10β people. c Look at your results. Did your lift ever rise or fall by more than β10β floors? d Press Shift βF9βto create a new set of numbers and results.
4 Extension
a Make changes to your spreadsheet so that the scenario of β20βpeople using random numbers between ββ 7βand β18βto represent a lift in a building with β7βbasement levels and β18βabove ground levels. Looking at your spreadsheet, is there a time where your lift rises or falls by more than β22βfloors? Use Shift βF9β to create a new set of numbers. b Use the following rules to find the total number of floors that the lift travelled. β’ For total Exit to entry movement, use =SUM(ABS(D4:D23)) β’ For total Entry to exit movement, use =SUM(ABS(E4:E23)) c Use Shift βF9βand see if you can create a scenario where the total Exit to entry movement plus the total Entry to exit movement is: i greater than β250β ii less than β100.β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Investigation
U N SA C O M R PL R E EC PA T E G D ES
If you have money saved in a bank account, your account balance should be positive. If you take out or spend too much money, your account balance may become negative. a Set up a spreadsheet to record and calculate a bank balance. Enter the given information describing one week of deposits and withdrawals, as shown.
Investigation
Account balance with spreadsheets
395
b i For the given spreadsheet, what is the balance at the end of May β1βst? ii On which day does the balance become negative? c Enter this formula into cell E5: = E4 +ββC5- D5 Fill down to reveal the balance after each day. d Enter another week of deposits and withdrawals so that the balance shows both positive and negative amounts. e Now alter your opening balance. What opening balance is needed so that the balance never becomes negative? Is there more than one value? What is the least amount? f Investigate how positive and negative numbers are used on credit card accounts. Give a brief explanation.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
396
Chapter 6
Find the next three numbers in these Up for a challenge? If you get stuck patterns. on a question, check out the βworking with unfamiliar problemsβ poster at a 3β , β 9, 27β, _____, _____, _____ the end of the book to help you. b ββ 32, 16, β 8β, _____, _____, _____ c β0, β 1, β 3, β 6β, _____, _____, _____ d ββ 1, β 1, β 2, β 3, β 5β, _____, _____, _____
U N SA C O M R PL R E EC PA T E G D ES
Problems and challenges
1
Negative numbers
2 Evaluate the following. a ββ 100 + (β β 98)β+ (β β 96)β+ β¦ + 98 + 100β b (β 50 β 53)β+ (β 49 β 52)β+ (β 48 β 51)β+ β¦ + (β 0 β 3)β c β2 β 3 + 4 β 5 + 6 β 7+ β¦ β 199 + 200β
3 Insert brackets and symbols (β+β, βββ, βΓβ, βΓ·β) into these number sentences to make them true. a ββ 3β
β4β
β(β 2)β= β 6β
b ββ 2β
β5β
β(β 1)ββ
β11 = 21β
c β1β
β30β
(β β 6)ββ
(β β 2)β= β 3β
4 a The difference between two numbers is β14βand their sum is β8β. What are the two numbers? b The difference between two numbers is β31βand their sum is β11β. What are the two numbers?
5 If βxβand βyβare integers less than β10βand greater than ββ 10β, how many different integer pairs (β x, y)ββmake the equation βx + 2y = 10β true?
6 In the sequence of numbers ββ¦ , e, d, c, b, a, 0, 1, 1, 2, 3, 5, 8, 13, β¦βeach number is the sum of its two preceding numbers, e.g. β13 = 5 + 8β. What are the values of βa, b, c, dβand βeβ? 7 Given the rule βx ββ2β= _ β 12 β, evaluate ββ(βββ 5)β2 β ββ. xβ β
8 If βp > q > 0βand βt < 0β, insert β>βor β<βto make each of these a true statement for all values of βp, qβand βtβ. a pβ + tβ
βq + tβ
b βt β pβ
βt β qβ
c βptβ
βqtβ
9 Describe the set of all possible numbers for which the square of the number is greater than the cube of the number. 10 A formula linking an objectβs velocity βv m / sβwith its initial velocity βu m / sβ, acceleration βa m /sβ β 2β βand its β β2β+ 2asβ. Find: change of distance βs mβis given by vβ β β2β= uβ a the velocity if the initial velocity is β0 m / sβ, the acceleration is β4 m /sβ β 2β ββ and change of distance is β8 mβ. b the acceleration if the velocity is β70 m / sβ, the initial velocity is β 10 m / sβand the change of distance is β100 mβ.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter summary
β3 + 5 = 2 β4 + 3 = β1 5 β 7 = β2 β1 β10 = β11
Number line negative
U N SA C O M R PL R E EC PA T E G D ES
positive
Adding and subtracting negative integers
Chapter summary
Adding and subtracting positive integers
397
β3 β2 β1 0 β3 < 2
1
2 + (β3) = 2 β 3 = β1 β5 + (β4) = β5 β 4 = β9 4 β (β3) = 4 + 3 = 7 β10 β (β6) = β10 + 6 = β4
2 3 1 > β1
Cartesian plane y
(β3, 0)
O β3 β2 β1 β1
1 2 3
β2 β3
(1, β2)
(β2, β2)
Multiplication (Ext)
3 2 (0, 2) (3, 1) 1
x
Integers β¦, β3, β2, β1, 0, 1, 2, 3, β¦
2Γ3=6 2 Γ (β3) = β6 β2 Γ 3 = β6 β2 Γ (β3) = 6
Substitution (Ext)
Division (Ext)
If a = β2 and b = 4, then: b β a = 4 β (β2) = 6 ab + 2a = β2 Γ 4 + 2 Γ (β2) = β8 + (β4) = β12
10 Γ· 5 = 2 10 Γ· (β5) = β2 β10 Γ· 5 = β2 β10 Γ· (β5) = 2
Order of operations (Ext)
First brackets, then Γ or Γ· then + or β, from left to right. 3 Γ (5 β (β2)) + 8 Γ· (β4) = 3 Γ 7 + (β2) = 21 + (β2) = 19
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
398
Chapter 6
Negative numbers
A printable version of this checklist is available in the Interactive Textbook 6A
β
1. I can represent integers on a number line. e.g. Draw a number line showing all integers from ββ 4βto β2β.
U N SA C O M R PL R E EC PA T E G D ES
Chapter checklist
Chapter checklist with success criteria
6A
2. I can compare two integers. e.g. Insert the symbol β<β(less than) or β>β(greater than) into the statement ββ 1β make it true.
6B
3. I can add a positive integer to another integer. e.g. Evaluate ββ 2 + 3β.
6B
4. I can subtract a positive integer from another integer. e.g. Evaluate ββ 3 β 3β.
6C
5. I can add a negative integer to another integer. e.g. Evaluate ββ 2 + (β β 3)ββ.
6C
6. I can subtract a negative integer from another integer. e.g. Evaluate β1 β (β β 3)ββ.
6D
7. I can multiply two integers. e.g. Evaluate ββ 3 Γ (β β 7)ββ.
Ext
6D
8. I can divide two integers. e.g. Evaluate ββ 18 Γ· 9β.
Ext
6D
9. I can evaluate expressions involving integers and both multiplication and division. e.g. Evaluate β24 Γ· (β3) Γ (β 1)β
Ext
6E
10. I can use order of operations to evaluate expressions involving negative numbers. e.g. Evaluate ββ 6 Γ 2 β 10 Γ· (β β 5)ββ.
Ext
6F
11. I can substitute positive or negative integers into an expression and evaluate. e.g. Evaluate βb Γ· 5 β aβif βa = 3βand βb = β 5β.
Ext
6G
12. I can state the coordinates of a point labelled on a number plane. e.g. State the coordinates of the points labelled βAβ, βBβ, βCβand βDβ.
ββ 6β to
y
4 3 2 1
D
O β4 β3 β2 β1 β1
C
6G
β2 β3 β4
A
1 2 3 4 B
x
13. I can plot a point from a given pair of coordinates. e.g. Draw a set of axes from ββ 4βto β4βon both axes, and plot the points (β β 3, 2)βand β (β 2, β 4)ββ.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
399
Short-answer questions 1
Insert the symbol β<β(less than) or β>β(greater than) into each of these statements to make it true. a β0β
2 Evaluate: a β2 β 7β d ββ 36 + 37β g β10 β (β β 2)β j ββ 3 + 7 β (β β 1)β
b e h k
ββ 7β
c β3β
β4β
β 4 + 2β β β5 + (β β 7)β ββ 21 β (β β 3)β β0 + (β β 1)ββ 10β
d ββ 11β c f i l
ββ 6β
0β β 15β ββ 1 + β(β 4)β β1 β 5 + (β β 2)β ββ 2 β (β β 3)ββ (β β 4)β
U N SA C O M R PL R E EC PA T E G D ES
6B/C
b ββ 1β
β7β
Chapter review
6A
6C
3 Find the missing number for each of the following. a ββ 2 +β c 5β ββ
6D
Ext
6D
Ext
6E
Ext
6F
β= β 3β
β= 6β
e β β 1 ββ
β= 20β
g β7 +β
β= β 80β
b ββ 1 +β
β= β 10β
d ββ 2 ββ
β= β 4β
ββ 15 ββ
β = β 13β
h ββ 15 +β
β= 15β
f
4 Evaluate: a β5 Γ β(β 2)β d β10 Γ· (β β 2)β g ββ 3 Γ 2 Γ· (β β 6)β
b β β 3 Γ 7β e ββ 36 Γ· 12β h β β 38 Γ· (β β 19)βΓ β(β 4)β
c β β 2 Γ β(β 15)β f ββ 100 Γ· (β β 25)β
5 Find the missing number. a β4 Γβ
β= β 8β
b
βΓ· β 5 = 10β
c
βΓ· 9 = β 4β
6 Use order of operations to find the answers to these expressions. a ββ 2 + 5 Γ (β β 7)β b β β 1 β 18 Γ· (β β 2)β ( ( ) ( ) d β5 β 4 Γ β β 3 βΓ· β β 3 β e β β 2 β 5)βΓ (β 8 Γ· (β β 1))β β
7 Evaluate the following expressions if βa = 7β, βb = β 3βand βc = β 1β. a βa β bβ b β2b β 5aβ c βab + cβ
d ββ 1 Γβ
β= 1β
c β β 15 Γ· (β 1 + 4)β f ββ 7 Γ (β (β β 4)ββ 7)β+ 3β d βbc β 2aβ
Ext
6G
8 For the Cartesian plane shown, write down the coordinates of the points labelled βAβ, βBβ, βCβ, βDβ, βEβ and βFβ. y
4 3 C 2 1
B
A x O β4 β3 β2 β1 1 2 3 4 β1 β2 D β3 E F β4
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
400
Chapter 6
Negative numbers
6A
1
When the numbers ββ 4, 0, β 1, 7βand ββ 6βare arranged from lowest to highest, the correct sequence is: A β0β, ββ 1β, ββ 4β, ββ 6β, β7β B β0β, ββ 4β, ββ 6β, ββ 1β, β7β C ββ 6β, ββ 4β, ββ 1β, β0β, β7β D ββ 1β, ββ 4β, ββ 6β, β0β, β7β E ββ 6β, ββ 1β, β0β, ββ 4β, β7β
U N SA C O M R PL R E EC PA T E G D ES
Chapter review
Multiple-choice questions
6B
2 The difference between ββ 19βand β8β is: A β152β B β β 11β
C ββ 27β
D β11β
E β27β
6C
3 The missing number in β2 ββ A 1β β B ββ 1β
C β5β
D ββ 5β
E β2β
6C
4 5β β (β β 2)β+ β(β 7)βis equal to: A ββ 4β B β10β
C β7β
D β0β
E β14β
6A
6D
Ext
6D
β= 3β is:
5 The temperature inside a mountain hut is initially ββ 5Β° Cβ. After burning a fire for β2βhours the temperature rises to β17Β° Cβ. What is the rise in temperature? A ββ 12Β° Cβ B β12Β° Cβ C β22Β° Cβ D ββ 85Β° Cβ E ββ 22Β° Cβ 6 The product or quotient of two negative numbers is: A positive B negative D added E different
7 β β 2 Γ (β β 5)βΓ· β(β 10)βis equal to: A ββ 5β B β10β
C ββ 20β
C zero
D β1β
E ββ 1β
Ext
6E
Ext
6F
8 Which operation (i.e. addition, subtraction, multiplication or division) is completed second in the calculation of (β β 2 + 5)βΓ β(β 2)β+ 1β? A addition B subtraction C multiplication D division E brackets 9 If βa = β 2βand βb = 5β, then βab β aβis equal to: A β β 12β B ββ 8β C β8β
D β12β
E β9β
Ext
6G
10 The points βA(β β 2, 3)ββ, βB(β β 3, β 1)ββ, βC(β 1, β 1)βand βD(β 0, 3)βare joined on a number plane. What shape do they make? A triangle B square C trapezium D kite E parallelogram
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
401
Extended-response questions
U N SA C O M R PL R E EC PA T E G D ES
A scientist, who is camped on the ice in Greenland, records the following details in his notepad regarding the temperature over five days. Note that βmin.β stands for minimum and βmax.β stands for maximum. β’ Monday: βmin . = β 18Β° Cβ, βmax . = β 2Β° Cβ. β’ Decreased β29Β° Cβfrom Mondayβs max. to give Tuesdayβs min. β’ Wednesdayβs min. was ββ 23Β° Cβ. β’ Max. was only ββ 8Β° Cβ on Thursday. β’ Fridayβs min. is β19Β° Cβcolder than Thursdayβs max. a What is the overall temperature increase on Monday? b What is Tuesdayβs minimum temperature? c What is the difference between the minimum temperatures for Tuesday and Wednesday? d What is the overall temperature drop from Thursdayβs maximum to Fridayβs minimum? e By how much will the temperature need to rise on Friday if its maximum is β0Β° Cβ?
Chapter review
1
2 When joined, these points form a picture on the number plane. What is the picture? βA(β β0, 5β)β, Bβ(β β1, 3β)β, Cβ(β β1, 1β)β, Dβ(β β2, 0β)β, Eββ(β1, 0β)β, Fβ(β β1, β 2β)β, Gβ(β β3, β 5β)β, Hβ(β ββ3, β 5β)β, Iββ(ββ1, β 2β)β, β βJ(β ββ1, 0β)β, Kβ(β ββ2, 0β)β, L(β1, 1) , Mβ(β ββ1, 3β)β, Nβ(β β0, 5β)β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
U N SA C O M R PL R E EC PA T E G D ES
7 Geometry
Maths in context: Cable stayed bridges Skills in geometry are essential for all the practical occupations including animators, architects, astronomers, bricklayers, boilermaker, builders, carpenters, concreters, construction workers, designers, electricians, engineers, jewellers, navigators, plumbers, ship builders and surveyors.
Cable stayed bridges use pylons with attached cables that fan out to support the bridge deck, such as the Langkawi Sky Bridge in Malaysia. Here we can see the application of angles at a point, angles formed by parallel lines and angles in a triangle. The bridge is a 125 m curved pedestrian bridge with steel and
concrete panels forming the bridge deck, supported by inverted triangular steel trusses. It is about 100 m above ground and suspended by eight cables from single pylon. The pylon is 81.5 m high and must bear the total load of the bridge via the cables. Engineers had to solve the critical problem of how to balance the immense weight of the bridge from a single point at the top of the pylon. Cable stayed bridges in Australia include the Anzac Bridge in Sydney, the Eleanor Schonell Bridge and the Kurilpa Bridge in Brisbane, the Seafarers Bridge in Melbourne and the Batman Bridge in Tasmania.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter contents Points, lines, intervals and angles (CONSOLIDATING) Adjacent and vertically opposite angles Transversal and parallel lines Solving compound problems with parallel lines (EXTENDING) Classifying and constructing triangles Classifying quadrilaterals and other polygons Angle sum of a triangle Symmetry Reο¬ection and rotation Translation Drawing solids Nets of solids
U N SA C O M R PL R E EC PA T E G D ES
7A 7B 7C 7D 7E 7F 7G 7H 7I 7J 7K 7L
Victorian Curriculum 2.0
This chapter covers the following content descriptors in the Victorian Curriculum 2.0: SPACE
VC2M7SP01, VC2M7SP02, VC2M7SP03, VC2M7SP04 MEASUREMENT
VC2M7M04, VC2M7M05
Please refer to the curriculum support documentation in the teacher resources for a full and comprehensive mapping of this chapter to the related curriculum content descriptors.
Β© VCAA
Online resources
A host of additional online resources are included as part of your Interactive Textbook, including HOTmaths content, video demonstrations of all worked examples, auto-marked quizzes and much more.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
7A Points, lines, intervals and angles CONSOLIDATING LEARNING INTENTIONS β’ To know the meaning of the terms point, vertex, intersection, line, ray, segment and plane β’ To know the meaning of the terms acute, right, obtuse, straight, reο¬ex and revolution β’ To understand what collinear points and concurrent lines are β’ To be able to name lines, segments, rays and angles in terms of labelled points in diagrams β’ To be able to measure angles using protractors β’ To be able to draw angles of a given size
U N SA C O M R PL R E EC PA T E G D ES
404
The fundamental building blocks of geometry are the point, line and plane. They are the basic objects used to construct angles, triangles and other more complex shapes and objects. Theoretically, points and lines do not occupy any area, but we can represent them on a page using drawing instruments.
Angles are usually described using the unit of measurement called the degree, where 360 degrees (360Β°) describes one full turn. The idea to divide a circle into 360Β° dates back to the Babylonians, who used a sexagesimal number system based on the number 60. Because both 60 and 360 have a large number of factors, many fractions of these numbers are very easy to calculate.
Engineers calculate the angles between the straight structural supports used on bridges such as the angles seen here on the Story Bridge, Brisbane.
Lesson starter: Estimating angles How good are you at estimating the size of angles? Estimate the size of these angles and then check with a protractor.
Alternatively, construct an angle using interactive geometry software. Estimate and then check your angle using the angle-measuring tool.
KEY IDEAS
β A point is usually labelled with a capital letter.
P
β A line passing through two points, A and B, can be called line AB or
line BA and extends indefinitely in both directions. β’ upper-case letters are usually used to label points.
B
A
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7A Points, lines, intervals and angles
405
U N SA C O M R PL R E EC PA T E G D ES
β A plane is a flat surface and extends indefinitely.
β Points that all lie on a single line are collinear.
A
C
B
β If two lines meet, an intersection point is formed.
β Three or more lines that intersect at the same point are concurrent.
β A line segment (or interval) is part of a line with a fixed length
B
and end points. If the end points are βAβand βBβthen it would be named segment βABβor segment βBAβ(or interval βABβor interval βBAβ).
A
β A ray βABβis a part of a line with one end point βAβand passing through
B
point βBβ. It extends indefinitely in one direction.
A
β When two rays (or lines) meet, an angle is formed at the intersection
arm
point called the vertex. The two rays are called arms of the angle.
vertex
arm
β An angle is named using three points, with the vertex as
the middle point. A common type of notation is ββ ABCβ or ββ CBAβ. The measure of the angle is βaΒ°β. β’ lower-case letters are often used to represent the size of an angle.
or
A
B
aΒ°
C
β Part of a circle called an arc is used to mark an angle.
B
A
β These two lines are parallel. This is written βAB β₯ DCβ.
C
D
β These two lines are perpendicular. This is written βAB β₯ CDβ.
C
B
A
β The markings on this diagram show that βAB = CD, AD = BC,β
D
A
ββ BAD = β BCD and β ABC = β ADCβ.
D
B
Γ Γ C
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
β Angles are classified according to their size.
Angle type
Size
acute
between 0Β° and 90Β°
Examples
U N SA C O M R PL R E EC PA T E G D ES
406
right
90Β°
obtuse
between 90Β° and 180Β°
straight
180Β°
reο¬ex
between 180Β° and 360Β°
revolution
360Β°
β A protractor can be used to measure angles to within an accuracy of about half a degree. Some
protractors have increasing scales marked both clockwise and anticlockwise from zero. To use a protractor: 1 Place the centre of the protractor on the vertex of the angle. 2 Align the base line of the protractor along one arm of the angle. 3 Measure the angle using the other arm and the scale on the protractor. 4 A reflex angle can be measured by subtracting a measured angle from 360Β°.
BUILDING UNDERSTANDING
1 Describe or draw the following objects. a a point P c an angle β ABC e a plane
b a line AN d a ray ST f three collinear points A, B and C
2 Match the words line, segment, ray,
collinear or concurrent to the correct description. a Starts from a point and extends indefinitely in one direction. b Extends indefinitely in both directions, passing through two points. c Starts and ends at two points. d Three points in a straight line. e Three lines intersecting at the same point.
Why would we use the geometric term rays to describe the sunlight showing through the trees?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7A Points, lines, intervals and angles
407
3 Without using a protractor, draw or describe an example of the following types of angles. a acute b right c obtuse d straight e reflex f revolution
U N SA C O M R PL R E EC PA T E G D ES
4 What is the size of the angle measured with these protractors? a b
c
Example 1
d
Naming objects
Name this line segment and angle. a A B
b
P
Q
R
SOLUTION
EXPLANATION
a segment βABβ
Segment βBAβ, interval βABβor interval βBAβare also acceptable.
b ββ PQRβ
Point βQβis the vertex so the letter βQβsits in between βPβand βRβ. ββ RQPβis also correct.
Now you try
Name this ray and angle. a A
b B
B
A
C
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
Example 2
Classifying and measuring angles
For the angles shown, state the type of angle and measure its size. a b G A
c
D
O
U N SA C O M R PL R E EC PA T E G D ES
408
O
B
SOLUTION
E
E
F
EXPLANATION
a acute ββ AOB = 60Β°β
A
B
O
The angle is an acute angle so read from the inner scale, starting at zero.
b obtuse ββ EFG = 125Β°β
G
E
F
The angle is an obtuse angle so read from the outer scale, starting at zero.
c reflex
obtuse β DOE = 130Β° βreflex β 130Β°β β―β― β DOEββ―β― =β 360Β° β β = 230Β°
D
O
E
First measure the obtuse angle before subtracting from 3β 60Β°βto obtain the reflex angle.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7A Points, lines, intervals and angles
409
Now you try c O
D
U N SA C O M R PL R E EC PA T E G D ES
For the angles shown, state the type of angle and measure its size. a b E B A
F
O
E
G
Example 3
Drawing angles
Use a protractor to draw each of the following angles. a ββ AOB = 65Β°β b β β WXY = 130Β°β SOLUTION
EXPLANATION
a
B
Y
X
c
Step 1: Draw a base line βOBβ. Step 2: Align the protractor along the base line with the centre at point βOβ. Step 3: Measure β65Β°βand mark a point, βAβ. Step 4: Draw the arm βOAβ.
A
O
b
c ββ MNO = 260Β°β
W
Step 1: Draw a base line βXWβ. Step 2: Align the protractor along the base line with the centre at point βXβ. Step 3: Measure β130Β°βand mark a point, βYβ. Step 4: Draw the arm βXYβ. Step 1: Draw an angle of β360Β° β 260Β° = 100Β°β. Step 2: Mark the reflex angle on the opposite side to the obtuse angle of β100Β°β. Alternatively, draw a β180Β°βangle and measure an β80Β°βangle to add to the β180Β°β angle.
O
N
M
Now you try
Use a protractor to draw each of the following angles. a ββ AOB = 30Β°β b β β WXY = 170Β°β
c ββ MNO = 320Β°β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
410
Chapter 7
Geometry
Exercise 7A FLUENCY
Name these line segments and angles. a i Q
2, 3, 4β6β(β 1β/2β)β, 7, 8
ii X
2, 3β6β(β 1β/2β)β, 7, 8
U N SA C O M R PL R E EC PA T E G D ES
1
1β3, 4β6ββ(1β/2β)β, 7, 8
Example 1a
Y
P
Example 1b
Example 1
b i
ii
A
T
S
B O 2 Name the following objects. a T
U
b
D
C
c
d
B
A
C
e
f
Q
S
P
Example 1b
T
3 Name the angle marked by the arc in these diagrams. a b A
B
B
C
A
C
D
D
O
c
d
B
D
A
O
E
C
E
A
B
C
D
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7A Points, lines, intervals and angles
4 For the angles shown, state the type of angle and measure its size. a b
c
U N SA C O M R PL R E EC PA T E G D ES
Example 2
411
Example 3
d
e
f
g
h
i
5 Use a protractor to draw each of the following angles. a β40Β°β b β75Β°β c β90Β°β f β205Β°β g β260Β°β h β270Β°β
d 1β 35Β°β i β295Β°β
e 1β 75Β°β j β352Β°β
6 For each of the angles marked in the situations shown, measure: a the angle that this ramp makes with the ground
b the angle the Sunβs rays make with the ground
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
c the angle or pitch of this roof
d the angle between this laptop screen and the keyboard.
U N SA C O M R PL R E EC PA T E G D ES
412
7 Name the set of three labelled points that are collinear in these diagrams. b B a D
C
B
D
C
A
A
8 State whether the following sets of lines are concurrent. a b
PROBLEM-SOLVING
9, 10
9β11
11, 12
9 Count the number of angles formed inside these shapes. Count all angles, including ones that may be the same size and those angles that are divided by another segment. a b
10 A clock face is numbered β1βto β12β. Find the angle the minute hand turns in: a 3β 0 minutesβ b β45 minutesβ c β5 minutesβ e β1 minuteβ f β9 minutesβ g β10.5 minutesβ
d 2β 0 minutesβ h β21.5 minutesβ
11 A clock face is numbered β1βto β12β. Find the angle between the hour hand and the minute hand at: a β6 : 00 p.m.β b β3 : 00 p.m.β c β4 : 00 p.m.β d β11 : 00 a.m.β 12 Find the angle between the hour hand and the minute hand of a clock at these times. a β10 : 10 a.m.β b β4 : 45 a.m.β c β11 : 10 p.m.β d β2 : 25 a.m.β e β7 : 16 p.m.β f β9 : 17 p.m.β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7A Points, lines, intervals and angles
REASONING
13
13, 14
413
14, 15
U N SA C O M R PL R E EC PA T E G D ES
13 a If points βAβ, βBβand βCβare collinear and points βAβ, βBβand βDβare collinear, does this mean that points βBβ, β Cβand βDβare also collinear? Use a diagram to check. b If points βAβ, βBβand βCβare collinear and points βCβ, βDβand βEβare collinear, does this mean that points βBβ, β Cβand βDβare also collinear? Use a diagram to check. 14 An acute angle ββ AOBβis equal to β60Β°β. Why is it unnecessary to use a protractor to work out the size of the reflex angle ββ AOBβ?
A
?
O
60Β°
B
15 The arrow on this dial starts in an upright position. It then turns by a given number of degrees clockwise or anticlockwise. Answer with an acute or obtuse angle. a Find the angle between the arrow in its final position with the arrow in its original position, as shown in the diagram opposite, which illustrates part i. Answer with an acute or obtuse angle. i β290Β°β clockwise ii β290Β°β anticlockwise iii β450Β°β clockwise iv 4β 50Β°β anticlockwise v 1β 000Β°β clockwise vi 1β 000Β°β anticlockwise b Did it matter to the answer if the dial was turning clockwise or anticlockwise? c Explain how you calculated your answer for turns larger than β360Β°β. ENRICHMENT: How many segments?
β
β
?
290Β°
16
16 A line contains a certain number of labelled points. For example, this line has three points. a Copy and complete this table by counting the total number of segments for the given number of labelled points. C
B
A
Number of points
β1β
β2β
β3β
β4β
β5β
β6β
Number of segments
b Explain any patterns you see in the table. Is there a quick way of finding the next number in the table? c If βnβis the number of points on the line, can you find a rule (in terms of βnβ) for the number of segments? Test your rule to see if it works for at least three cases, and try to explain why the rule works in general.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
7B Adjacent and vertically opposite angles LEARNING INTENTIONS β’ To know the meaning of the terms adjacent, complementary, supplementary, vertically opposite and perpendicular β’ To be able to work with vertically opposite angles and perpendicular lines β’ To be able to ο¬nd angles at a point using angle sums of 90Β°,180Β° and 360Β°
U N SA C O M R PL R E EC PA T E G D ES
414
Not all angles in a diagram or construction need to be measured directly. Special relationships exist between pairs of angles at a point and this allows some angles to be calculated exactly without measurement, even if diagrams are not drawn to scale.
Lesson starter: Special pair of angles
By making a drawing or using interactive geometry software, construct the diagrams below. Measure the two marked angles. What do you notice about the two marked angles? A
A
B
aΒ°
O
People who calculate angles formed by intersecting lines include: glass cutters who design and construct stained-glass windows; jewellers who cut gemstones at precise angles; and quilt makers who design and sew geometric shapes.
bΒ°
B
A
aΒ°
O
bΒ°
C
eΒ° O fΒ°
C
B
D
C
KEY IDEAS
β Adjacent angles are side by side and share a vertex and an arm.
A
β AOB and β BOC in this diagram are adjacent angles.
β AOB
β BOC
B
C
O
β Two adjacent angles in a right angle are complementary.
They add to 90Β°. β’ If the value of a is 30, then the value of b is 60 because 30Β° + 60Β° = 90Β°. We say that 30Β° is the complement of 60Β°.
bΒ°
aΒ° a + b = 90
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7B Adjacent and vertically opposite angles
415
β Two adjacent angles on a straight line are supplementary.
They add to β180Β°β. If the value of b is 50, then the value of a is 130 because 50Β° + 130Β° = 180Β°. We say that 130Β° is the supplement of 50Β°.
U N SA C O M R PL R E EC PA T E G D ES
aΒ° bΒ° a + b = 180
β Angles in a revolution have a sum of β360Β°β.
aΒ°
bΒ° a + b = 360
β Vertically opposite angles are formed when two lines intersect.
bΒ°
The opposite angles are equal. The name comes from the fact that the pair of angles has a common vertex and they sit in opposite positions across the vertex.
aΒ°
aΒ°
bΒ°
β Perpendicular lines meet at a right angle (β 90Β°)β. We write βAB β₯ CDβ.
D
B
A
C
BUILDING UNDERSTANDING
aΒ°
1 a Measure the angles βaΒ°βand βbΒ°βin this diagram. b Calculate aβ + bβ. Is your answer β90β? If not, check your measurements. c State the missing word: βaΒ°βand βbΒ°βare ____________ angles.
2 a Measure the angles βaΒ°βand βbΒ°βin this diagram. b Calculate aβ + bβ. Is your answer β180β? If not,
aΒ°
bΒ°
bΒ°
check your measurements.
c State the missing word: βaΒ°βand βbΒ°β are ____________ angles.
3 a Measure the angles βaΒ°β, βbΒ°β, βcΒ°βand βdΒ°βin this diagram. b What do you notice about the sum of the four angles? c State the missing words: βbΒ°βand βdΒ°β are _____________ angles.
dΒ°
aΒ°
bΒ°
cΒ°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
4 a Name the angle that is complementary to ββ AOBβin this diagram.
A
B C
O
b Name the two angles that are supplementary to ββ AOBβin this diagram.
D
U N SA C O M R PL R E EC PA T E G D ES
416
O
C
A
B
c Name the angle that is vertically opposite to ββ AOBβin this diagram.
B
C
O
A
Example 4
D
Finding angles at a point using complementary and supplementary angles
Without using a protractor, find the value of the pronumeral βaβ. a b
55Β°
aΒ°
aΒ°
35Β°
SOLUTION
EXPLANATION
a aβ β =β 90 β 35β β = 55
Angles in a right angle add to β90Β°β. βa + 35 = 90β
b βaβ =β 180 β 55β β = 125
Angles on a straight line add to β180Β°β. βa + 55 = 180β
Now you try
Without using a protractor, find the value of βaβfor the following. a b A
65Β°
aΒ°
aΒ°
130Β°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7B Adjacent and vertically opposite angles
Example 5
417
Finding angles at a point using other properties
U N SA C O M R PL R E EC PA T E G D ES
Without using a protractor, find the value of the pronumeral βaβ. a b aΒ° 47Β° 120Β° aΒ°
SOLUTION
EXPLANATION
a βa = 47β
Vertically opposite angles are equal.
b a = 360 β (90 + 120) β ββ―β― β β―β― =β 360 β 210β β = 150
The sum of angles in a revolution is β360Β°β. βa + 90 + 120 = 360β βaβis the difference between β210Β°βand β360Β°β.
Now you try
Without using a protractor, find the value of βaβfor the following. a b aΒ°
116Β°
aΒ°
50Β°
Exercise 7B FLUENCY
Example 4a
1
Without using a protractor, find the value of the pronumeral a. a b aΒ°
Example 4
1, 2β5ββ(β1/2β)β
1β5β(β β1/2β)β
c
aΒ° 60Β°
20Β°
2β6β(β β1/2β)β
aΒ°
15Β°
2 Without using a protractor, find the value of the pronumeral βaβ. (The diagrams shown may not be drawn to scale.) a b c 40Β° 75Β° aΒ° aΒ° 30Β° aΒ° d
e
110Β° aΒ°
f
aΒ° 120Β°
aΒ° 45Β°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
418
Chapter 7
Geometry
g
h
i
aΒ° 50Β°
3 Without using a protractor, find the value of the pronumeral βaβ. (The diagrams shown may not be drawn to scale.) a b c aΒ° aΒ° 77Β° 115Β° aΒ°
U N SA C O M R PL R E EC PA T E G D ES
Example 5
60Β°
aΒ° 49Β°
aΒ°
37Β°
d
e
aΒ°
aΒ°
120Β°
f
220Β°
aΒ°
g
h
i
160Β°
135Β°
140Β°
aΒ°
100Β°
aΒ°
aΒ°
4 For each of the given pairs of angles, write βCβif they are complementary, βSβif they are supplementary or βNβif they are neither. a 2β 1Β° , 79Β°β b β130Β° , 60Β°β c β98Β° , 82Β°β d β180Β° , 90Β°β e β17Β° , 73Β°β f β31Β° , 59Β°β g β68Β° , 22Β°β h β93Β° , 87Β°β 5 Write a statement like βAB β₯ CDβfor these pairs of perpendicular line segments. a b S c W U H T E F
Y
V
X
G
6 Without using a protractor, find the value of βaβin these diagrams. a b 30Β° aΒ° 30Β°
c
40Β°
aΒ°
aΒ°
65Β°
100Β°
110Β°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7B Adjacent and vertically opposite angles
d
e aΒ°
419
f 40Β°
aΒ° 45Β°
135Β°
U N SA C O M R PL R E EC PA T E G D ES
aΒ° PROBLEM-SOLVING
7
8, 9
7-8β(β 1β/2β)β
7 Decide whether the given angle measurements are possible in the diagrams below. Give reasons. a b c 60Β° 25Β°
40Β°
140Β°
50Β°
310Β°
d
e
f
42Β°
35Β°
138Β°
80Β°
250Β°
35Β°
8 Find the value of βaβin these diagrams. a b aΒ°
c
aΒ°
aΒ° aΒ°
(3a)Β°
(2a)Β°
(2a)Β°
aΒ°
d
e
f
(2a)Β°
(a + 10)Β°
(a β 10)Β°
(a β 60)Β°
(3a)Β°
(a + 60)Β°
9 A pizza is divided between four people. Bella is to get twice as much as Bobo, who gets twice as much as Rick, who gets twice as much as Marie. Assuming the pizza is cut into slices from the centre outwards, find the angle at the centre of the pizza for Marieβs piece.
REASONING
10
10, 11
11, 12
10 a Is it possible for two acute angles to be supplementary? Explain why or why not. b Is it possible for two acute angles to be complementary? Explain why or why not.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
11 Write down a rule connecting the letters in these diagrams, e.g. a + b = 180. a b c aΒ° bΒ° aΒ° bΒ° bΒ°
cΒ°
aΒ°
U N SA C O M R PL R E EC PA T E G D ES
420
12 What is the minimum number of angles you would need to measure in this diagram if you wanted to know all the angles? Explain your answer.
ENRICHMENT: Pentagon turns
β
β
13
13 Consider walking around a path represented by this regular pentagon. All sides have the same length and all internal angles are equal. At each corner (vertex) you turn an angle of a, as marked.
aΒ°
aΒ°
bΒ°
bΒ°
bΒ°
aΒ°
aΒ°
bΒ°
bΒ°
aΒ°
a How many degrees would you turn in total after walking around the entire shape? Assume that you face the same direction at the end as you did at the start. b Find the value of a. c Find the value of b. d Explore the outside and inside angles of other regular polygons using the same idea. Complete this table to summarise your results.
Regular shape
a
b
Triangle Square
Pentagon Hexagon
Heptagon Octagon
Each of the identical shapes that make up this quilt design is called a rhombus. Four line segments form the sides of each rhombus. How many lines intersect at each vertex? How many angles meet at each vertex? Can you determine the size of the angles in each pattern piece?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7C Transversal and parallel lines
421
7C Transversal and parallel lines
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To know the meaning of the terms transversal, corresponding, alternate, cointerior and parallel β’ To be able to identify angles that are in a given relation to another angle (for example, identifying an angle cointerior to a given angle) β’ To be able to ο¬nd the size of angles when a transversal crosses parallel lines β’ To be able to determine whether two lines are parallel using angles involving a transversal
When a line, called a transversal, cuts two or more other lines a number of angles are formed. Pairs of these angles are corresponding, alternate or cointerior angles, depending on their relative position. If the transversal cuts parallel lines, then there is a relationship between the sizes of the special pairs of angles that are formed.
Surveyors use parallel line geometry to accurately measure the angles and mark the parallel lines for angle parking.
Lesson starter: Whatβs formed by a transversal? Draw a pair of parallel lines using either: β’ β’
two sides of a ruler; or interactive geometry software (parallel line tool).
Then cross the two lines with a third line (transversal) at any angles.
Measure each of the eight angles formed and discuss what you find. If interactive geometry software is used, drag the transversal to see if your observations apply to all the cases that you observe.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
KEY IDEAS β A transversal is a line passing through two or more
other lines that are usually, but not necessarily, parallel.
U N SA C O M R PL R E EC PA T E G D ES
422
transversal
transversal
β A transversal crossing two lines will form special pairs
of angles. These are: β’ corresponding (in corresponding positions)
β’
β’
Γ
alternate (on opposite sides of the transversal and inside the other two lines)
Γ
cointerior (on the same side of the transversal and inside the other two lines).
B
β Parallel lines are marked with the same arrow set.
β’
If βABβis parallel to βCDβ, then we write βAB | |CDβ.
Γ
Γ
A
D
C
β If a transversal crosses two parallel lines, then:
β’ β’ β’
corresponding angles are equal alternate angles are equal cointerior angles are supplementary (i.e. sum to β180Β°β). corresponding alternate aΒ°
bΒ°
a=b
aΒ°
aΒ°
a=b
bΒ°
bΒ°
aΒ° bΒ°
cointerior
aΒ° bΒ°
a + b = 180
a=b
aΒ°
bΒ°
a + b = 180
a=b
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7C Transversal and parallel lines
423
BUILDING UNDERSTANDING 1 Use a protractor to measure each of the eight angles in
U N SA C O M R PL R E EC PA T E G D ES
this diagram. a How many different angle measurements did you find? b Do you think that the two lines cut by the transversal are parallel?
2 Use a protractor to measure each of the eight
angles in this diagram. a How many different angle measurements did you find? b Do you think that the two lines cut by the transversal are parallel?
3 Choose the word equal or supplementary to complete these sentences. If a transversal cuts two parallel lines, then: a alternate angles are _____________. b cointerior angles are _____________. c corresponding angles are ________. d vertically opposite angles are ______.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
Example 6
Naming pairs of angles A
Name the angle that is: a corresponding to ββ ABFβ b alternate to ββ ABFβ c cointerior to ββ ABFβ d vertically opposite to ββ ABF.β
H G
B
F
U N SA C O M R PL R E EC PA T E G D ES
424
C
D
SOLUTION
E
EXPLANATION
a β β HFGβ(or ββ GFHβ)
H
F
b ββ EFBβ(or ββ BFEββ )
F
These two angles are in corresponding positions, both G above and on the right of the intersection.
These two angles are on opposite G sides of the transversal and inside the other two lines.
E
c ββ HFBβ(or ββ BFHββ )
H
Γ
B
F
d ββ CBDβ(or ββ DBCβ)
C
B
These two angles are on the same side of the transversal and inside the other two lines.
These two angles sit in opposite positions across the vertex B.
D
Now you try
Using the same diagram as in the example above, name the angle that is: a corresponding to ββ EFBβ b alternate to ββ DBFβ c cointerior to ββ EFBβ d vertically opposite to ββ GFE.β
A
H
G
B
F
C
D
E
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7C Transversal and parallel lines
Example 7
425
Finding angles in parallel lines
Find the value of βaβin these diagrams and give a reason for each answer. a b c aΒ°
U N SA C O M R PL R E EC PA T E G D ES
115Β°
aΒ°
55Β°
aΒ°
110Β°
SOLUTION
EXPLANATION
a βa = 115β alternate angles in parallel lines
Alternate angles in parallel lines cut by a transversal are equal.
b aβ = 55β corresponding angles in parallel lines
Corresponding angles in parallel lines are equal.
c aβ β =β 180 β 110β β―β― β = 70
Cointerior angles in parallel lines sum to β180Β°β.
cointerior angles in parallel lines
Now you try
Find the value of βaβin these diagrams and give a reason for each answer. a b c 50Β°
aΒ°
Example 8
120Β°
aΒ° 65Β°
aΒ°
Determining whether two lines are parallel
Giving reasons, state whether the two lines cut by the transversal are parallel. a b 75Β° 78Β°
122Β°
58Β°
SOLUTION
EXPLANATION
a not parallel Alternate angles are not equal.
Parallel lines cut by a transversal have equal alternate angles.
b parallel The cointerior angles sum to β180Β°β.
1β 22Β° + 58Β° = 180Β°β Cointerior angles inside parallel lines are supplementary (i.e. sum to β180Β°β).
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
426
Chapter 7
Geometry
Now you try Giving reasons, state whether the two lines cut by the transversal are parallel. a b 60Β°
109Β°
U N SA C O M R PL R E EC PA T E G D ES
71Β°
55Β°
Exercise 7C FLUENCY
Example 6
1
Name the angle that is: a corresponding to ββ BGAβ b alternate to ββ FGHβ c cointerior to ββ CHGβ d vertically opposite to ββ FGA.β
1, 2, 4β6ββ(β1/2β)β
2, 4β6β(β β1/2β)β
3, 4β6β(β β1/2β)β
A
B
F
G
E
C
H
D
Example 6
Example 6
2 Name the angle that is: a corresponding to ββ ABEβ b alternate to ββ ABEβ c cointerior to ββ ABEβ d vertically opposite to ββ ABE.β
3 Name the angle that is: a corresponding to ββ EBHβ b alternate to ββ EBHβ c cointerior to ββ EBHβ d vertically opposite to ββ EBH.β
C
F
E
H
G
B
A
D
C
D
E
B
F
A
H
G
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7C Transversal and parallel lines
Example 7
4 Find the value of βaβin these diagrams, giving a reason. a b
427
c 110Β°
130Β°
aΒ°
aΒ°
aΒ°
U N SA C O M R PL R E EC PA T E G D ES
70Β° d
e
aΒ°
aΒ°
f
67Β°
130Β°
aΒ°
120Β°
g
h
115Β° aΒ°
j
aΒ°
aΒ° 62Β°
k
117Β°
Example 7
i
100Β°
l
116Β°
64Β° aΒ°
aΒ°
5 Find the value of each pronumeral in the following diagrams. a b aΒ° 120Β° 70Β° bΒ° cΒ° bΒ° cΒ°
aΒ°
c
dΒ°
aΒ°
aΒ°
d
82Β°
aΒ°
e
f
bΒ° cΒ°
cΒ°
Example 8
bΒ° cΒ°
85Β° aΒ° bΒ°
119Β°
aΒ° bΒ°
6 Giving reasons, state whether the two lines cut by the transversal are parallel. a b c 59Β°
d
132Β°
132Β°
112Β° 68Β°
81Β° 81Β°
58Β°
e
f
79Β° 78Β°
60Β°
100Β°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
PROBLEM-SOLVING
7β8ββ(β1/2β)β
7 Find the value of βaβin these diagrams. a b 35Β° aΒ°
7β8β(β β1/2β)β,β 9
8β(β β1/2β)β, 9, 10
c aΒ°
aΒ° 41Β°
U N SA C O M R PL R E EC PA T E G D ES
428
70Β°
d
e
f
60Β°
aΒ°
141Β°
aΒ°
150Β°
aΒ°
8 Find the value of βaβin these diagrams. a b
c
80Β°
aΒ°
aΒ°
115Β°
aΒ°
62Β°
d
e
f
aΒ°
aΒ°
57Β°
42Β°
aΒ°
67Β°
g
aΒ°
80Β°
h
aΒ°
i
121Β°
130Β°
aΒ°
9 A transversal cuts a set of three parallel lines. a How many angles are formed? b How many angles of different sizes are formed if the transversal is not perpendicular to the three lines? 10 Two roads merge into a freeway at the same angle, as shown. Find the size of the obtuse angle, βaΒ°β, between the parallel roads and the freeway.
aΒ°
60Β°
freeway
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7C Transversal and parallel lines
REASONING
11
11, 12
429
11β13
11 a This diagram includes two triangles with two sides that are parallel. Give a reason why: i βa = 20β ii βb = 45.β
20Β°
bΒ°
45Β°
U N SA C O M R PL R E EC PA T E G D ES
aΒ°
b Now find the values of βaβand βbβin the diagrams below. i ii 25Β° bΒ° bΒ° aΒ°
iii
aΒ°
35Β°
aΒ° 50Β°
bΒ°
25Β°
41Β°
35Β°
12 This shape is a parallelogram with two pairs of parallel sides. a Use the β60Β°βangle to find the value of βaβand βbβ. b Find the value of βcβ. c What do you notice about the angles inside a parallelogram?
13 Explain why these diagrams do not contain a pair of parallel lines. a b
cΒ°
aΒ°
bΒ°
60Β°
c
130Β°
130Β°
150Β°
40Β°
140Β°
300Β°
ENRICHMENT: Adding parallel lines
β
β
14 Consider this triangle and parallel lines. a Giving a reason for your answer, name an angle equal to: i ββ ABDβ ii ββ CBE.β
b What do you know about the three angles ββ ABDβ, ββ DBEβand ββ CBEβ? c What do these results tell you about the three inside angles of the triangle βBDEβ? Is this true for any triangle? Try a new diagram to check.
14, 15
A
D
B
C
E
15 Use the ideas explored in Question 14 to show that the angles inside a quadrilateral (i.e. a four-sided shape) must sum to β360Β°β. Use this diagram to help.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
7D Solving compound problems with parallel lines EXTENDING LEARNING INTENTION β’ To be able to combine facts involving parallel lines and other geometric properties to ο¬nd missing angles in a diagram
U N SA C O M R PL R E EC PA T E G D ES
430
Parallel lines are at the foundation of construction in all its forms. Imagine the sorts of problems engineers and builders would face if drawings and constructions could not accurately use and apply parallel lines. Angles formed by intersecting beams would be difficult to calculate and could not be transferred to other parts of the building.
A builder makes sure that the roof rafters are all parallel, the ceiling joists are horizontal and parallel, and the wall studs are perpendicular and parallel.
Lesson starter: Not so obvious
Some geometrical problems require a combination of two or more ideas before a solution can be found. This diagram includes an angle of size aΒ°. β’ β’
aΒ°
Discuss if it is possible to find the value of a. Describe the steps you would take to find the value of a. Discuss your reasons for each step.
65Β°
KEY IDEAS
β Some geometrical problems involve more than one step.
Step 1: β ABC = 75Β° (corresponding angles on parallel lines) Step 2: a = 360 β 75 (angles in a revolution sum to 360Β°) = 285
A
E
B
75Β°
C
aΒ°
D
BUILDING UNDERSTANDING
1 In these diagrams, first find the value of a and then find the value of b. a b c 65Β°
aΒ°
bΒ°
aΒ°
bΒ°
74Β°
125Β° aΒ° bΒ°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7D Solving compound problems with parallel lines
431
2 Name the angle in these diagrams (e.g. ββ ABCβ) that you would need to find first before finding the value of βaβ. Then find the value of βaβ.
a
b
E B
60Β°
c
C
A 70Β°
U N SA C O M R PL R E EC PA T E G D ES
A aΒ° B
aΒ°
A
F
D
C
D 60Β° aΒ°
E
70Β°
D
G
B
F
C
Example 9
Finding angles with two steps
Find the value of βaβin these diagrams. a A B 170Β° F aΒ° D C
b
D
A
60Β°
E
aΒ°
70Β°
C
B
SOLUTION
EXPLANATION
a β BDE = 360Β° β 90Β° β 170Β° ββ―β― β β =β 100Β°β β΄ a = 100
Angles in a revolution add to β360Β°β. ββ ABCβcorresponds with ββ BDEβ, and β BCβand βDEβare parallel.
b β ABC = 180Β° β 70Β° β =β 110Β°β β ββ―β― β β β΄ a = 110 β 60 β = 50
β ABCβand ββ BCDβare cointerior β angles, with βABβand βDCβ parallel. ββ ABC = 110Β°βand βaΒ° + 60Β°ββ = 110Β°β
Now you try
Find the value of βaβin these diagrams. a A G C
60Β°
B
aΒ°
D
F
b
C
aΒ°
A
D
F
H
50Β°
B
E
G
105Β°
J
I
E
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
432
Chapter 7
Geometry
Exercise 7D FLUENCY
1
Find the value of βaβin these diagrams. a E
2β(β 1β/2β)β,β 4ββ(1β/2β)β
b
U N SA C O M R PL R E EC PA T E G D ES
Example 9a
2, 4
1, 2ββ(1β/2β)β, 3, 4ββ(1β/2β)β
C
65Β°
aΒ°
D
F
165Β°
aΒ°
B
A
Example 9a
2 Find the value of βaβin these diagrams. a b A 300Β° E B
c
B
C
D
C
D
C
F
B
F
E
aΒ°
A
aΒ°
150Β°
aΒ°
A
E
65Β° D
d
D
aΒ°
e
E
A
F
f
G
B
130Β°
C
107Β°
E
F
B
G
C
aΒ°
aΒ°
H
I
D
57Β°
A
H
Example 9b
3 Find the value of βaβin these diagrams. a
b
B
C
30Β°
55Β°
aΒ°
100Β°
60Β°
A
aΒ°
D
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7D Solving compound problems with parallel lines
4 a
b
C
c
A
A
aΒ°
D
D 62Β° 38Β° B
30Β°
45Β°
A
85Β° D aΒ°
aΒ°
B
75Β°
U N SA C O M R PL R E EC PA T E G D ES
Example 9b
433
B
C
C
d
A
B
aΒ°
e
E
D
C
f
E
A
aΒ° 80Β°
45Β°
C
D
A
E
40Β°
aΒ°
D
B
35Β°
C
B
PROBLEM-SOLVING
5
5
5 Find the size of ββ ABCβin these diagrams. a b A
c
A
60Β° B
5ββ(1β/2)β β,β 6
C
A
110Β°
70Β°
B
130Β°
C
B
75Β°
130Β°
C
d
A
B 50Β°
e
B
25Β°
60Β°
B
C
30Β°
C
f
35Β°
40Β°
C
A
A
6 Find the value of βxβin each of these diagrams. a b
c
140Β°
xΒ°
110Β°
130Β°
xΒ°
100Β°
xΒ°
60Β° 280Β°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
REASONING
7
8, 9
7, 8ββ(β1/2β)β
7 What is the minimum number of angles you need to know to find all the angles marked in these diagrams? a b c dΒ° dΒ° dΒ° cΒ° eΒ° eΒ° cΒ° f Β° bΒ° f Β° aΒ° gΒ° bΒ° bΒ° aΒ° hΒ° cΒ° aΒ° eΒ°
U N SA C O M R PL R E EC PA T E G D ES
434
8 In these diagrams, the pronumeral βxβrepresents a number and β2xβmeans β2 Γ xβ. Find the value of βxβ in each case. a b 120Β°
(2x)Β°
60Β°
(2x)Β°
c
d
(3x)Β°
(x + 20)Β°
50Β°
60Β°
e
f
70Β°
(x β 10)Β°
9 Find the value of βaβin these diagrams. a (2a)Β°
60Β° (4x)Β°
b
(3a)Β°
aΒ°
80Β°
aΒ°
c
(5a)Β°
150Β°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7D Solving compound problems with parallel lines
ENRICHMENT: Adding parallel lines
β
β
435
10ββ(β1/2β)β
10 Adding extra parallel lines can help to solve more complex geometry problems.
U N SA C O M R PL R E EC PA T E G D ES
You can see in this problem that the value of aΒ° is the sum of two alternate angles after adding the extra (dashed) parallel line. 40Β°
40Β°
aΒ°
40Β° 70Β°
70Β°
70Β°
Find the value of βaβin these diagrams. a
b
c
50Β°
50Β°
aΒ°
80Β° aΒ°
120Β°
aΒ°
50Β°
60Β°
d
e
f
aΒ°
300Β°
aΒ°
30Β°
aΒ°
20Β°
280Β°
140Β°
70Β°
260Β°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
7E Classifying and constructing triangles LEARNING INTENTIONS β’ To understand that triangles can be classiο¬ed by their side lengths (scalene, isosceles, equilateral) or by their interior angles (acute, right, obtuse) β’ To know that in an isosceles triangle, the angles opposite the apex are equal and the two sides (legs) adjacent to the apex are of equal length β’ To be able to classify triangles based in side lengths or angles β’ To be able to construct triangles using a protractor and ruler
U N SA C O M R PL R E EC PA T E G D ES
436
The word βtriangleβ (literally meaning βthree anglesβ) describes a shape with three sides. The triangle is an important building block in mathematical geometry. Similarly, it is important in the practical world of building and construction owing to the rigidity of its shape.
In this Melbourne sports stadium, engineers have used triangular structural supports, including equilateral and isosceles triangles. These triangles are symmetrical and evenly distribute the weight of the structure.
Lesson starter: Stable shapes
Consider these constructions, which are made from straight pieces of steel and bolts.
Assume that the bolts are not tightened and that there is some looseness at the points where they are joined. β’ β’ β’
Which shape(s) do you think could lose their shape if a vertex is pushed? Which shape(s) will not lose their shape when pushed? Why? For the construction(s) that might lose their shape, what could be done to make them rigid?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7E Classifying and constructing triangles
437
KEY IDEAS β Triangles can be named using the vertex labels.
C
triangle βABCβor βΞABCβ β Triangles are classified by their side lengths.
B
U N SA C O M R PL R E EC PA T E G D ES
A
scalene
equilateral
isosceles
60Β°
3 different sides 3 different angles
60Β°
2 equal sides 2 equal angles
60Β°
3 equal sides 3 equal angles (60Β°)
β Triangles are also classified by the size of their interior angles.
acute
right
obtuse
all acute angles
one right angle
one obtuse angle
β The parts of an isosceles triangle are named as shown opposite. The
apex
base angles are equal and two sides (called the legs) are of equal length. The two sides of equal length are opposite the equal angles.
β Sides of equal length are indicated by matching markings.
β Rulers, protractors and arcs drawn using a pair of compasses
can help to construct triangles accurately. right triangles
isosceles triangles
legs
base angles
base
equilateral triangles
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
Three side lengths (e.g. constructing Two sides and the angle a triangle with side lengths 3 cm, between them (e.g. constructing 5 cm, 6 cm) a triangle with side lengths 4 cm and 5 cm with a 40Β° angle between them)
Two angles and a side (e.g. constructing a triangle with angles 35Β° and 70Β° and a side length of 5 cm)
U N SA C O M R PL R E EC PA T E G D ES
438
C
C
C
3 cm
5 cm
4 cm
A
A
6 cm
B A
40Β°
5 cm
B
70Β° 35Β° 5 cm
B
BUILDING UNDERSTANDING
1 Describe or draw an example of each of the triangles given below. Refer back to the Key ideas in this section to check that the features of each triangle are correct. a scalene b isosceles c equilateral d acute e right f obtuse
2 Answer these questions, using the point labels βA, Bβand βCβfor the given isosceles triangle. a Which point is the apex? b Which segment is the base? c Which two segments are of equal length? d Which two angles are the base angles?
A
C
B
Example 10 Classifying triangles
These triangles are drawn to scale. Classify them by: i their side lengths (i.e. scalene, isosceles or equilateral) ii their angles (i.e. acute, right or obtuse). a b
SOLUTION
EXPLANATION
a i isosceles ii acute
Has β2βsides of equal length. All angles are acute.
b i scalene ii obtuse
Has β3βdifferent side lengths. Has β1βobtuse angle.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7E Classifying and constructing triangles
439
Now you try
U N SA C O M R PL R E EC PA T E G D ES
These triangles are drawn to scale. Classify them by: i their side lengths ii their angles. a b
Example 11 Constructing triangles using a protractor and ruler Construct a triangle βABCβwith βAB = 5 cm, β ABC = 30Β°βand ββ BAC = 45Β°β. SOLUTION
A
EXPLANATION
30Β°
5 cm
First, measure and draw segment βABβ. Then use a protractor to form the angle β30Β°β at point βBβ.
B
Then use a protractor to form the angle β45Β°β at point βAβ. Mark point βCβand join with βAβand with βBβ.
C
A
45Β°
30Β°
5 cm
B
Now you try
Construct a triangle βABCβwith βAB = 4 cm, β ABC = 45Β°βand ββ BAC = 60Β°β.
Example 12 Constructing a triangle using a pair of compasses and ruler Construct a triangle with side lengths β6 cm, 4 cmβand β5 cmβ. SOLUTION 6 cm
EXPLANATION
Use a ruler to draw a segment β6 cmβin length.
Continued on next page
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
440
Chapter 7
Geometry
Construct two arcs with radius β4 cmβand β5 cmβ, using each end of the segment as the centres.
5 cm
4 cm
U N SA C O M R PL R E EC PA T E G D ES
6 cm
5 cm
4 cm
Mark the intersection point of the arcs and draw the two remaining segments.
6 cm
Now you try
Construct a triangle with side lengths β5 cm, 6 cmβand β7 cmβ.
Exercise 7E FLUENCY
Example 10i
Example 10ii
Example 10
1
1β8
3β8
3β5, 7, 8
Classify each of these triangles according to their side lengths (i.e. scalene, isosceles or equilateral). a b c
2 These triangles are drawn to scale. Classify them according to their angles (i.e. acute, right or obtuse). a b c
3 These triangles are drawn to scale. Classify them by: i their side lengths (i.e. scalene, isosceles or equilateral) ii their angles (i.e. acute, right or obtuse). b a
c
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7E Classifying and constructing triangles
Example 11
4 Use a protractor and ruler to construct the following triangles. a triangle βABCβwith βAB = 5 cmβ, ββ ABC = 40Β°βand ββ BAC = 30Β°β b triangle βDEFβwith βDE = 6 cmβ, ββ DEF = 50Β°βand ββ EDF = 25Β°β c triangle βABCβwith βAB = 5 cmβ, ββ ABC = 35Β°βand βBC = 4 cmβ 5 Construct a triangle with the given side lengths. a β7 cm, 3 cmβand β5 cmβ.
b β8 cm, 5 cm and 6 cmβ.
U N SA C O M R PL R E EC PA T E G D ES
Example 12
441
6 Construct an isosceles triangle by following these steps. a Draw a base segment of about β4 cmβin length. b Use a pair of compasses to construct two arcs of equal radius. (Try about β5 cmβ but there is no need to be exact.) c Join the intersection point of the arcs (apex) with each end of the base. d Measure the length of the legs to check they are equal. e Measure the two base angles to check they are equal.
apex
base
7 Construct an equilateral triangle by following these steps. a Draw a segment of about β4 cmβin length. b Use a pair of compasses to construct two arcs of equal radius. Important: Ensure the arc radius is exactly the same as the length of the segment in part a. c Join the intersection point of the arcs with the segment at both ends. d Measure the length of the three sides to check they are equal. e Measure the three angles to check they are all equal and β60Β°β. 8 Construct a right triangle by following these steps. a Draw a segment, βABβ, of about β4 cmβin length. b Extend the segment βABβto form the ray βADβ. Make βADβ about β2 cmβin length. c Construct a circle with centre βAβand radius βADβ. Also mark point βEβ. d Draw two arcs with centres at βDβand βEβ, as shown in the diagram. Any radius will do as long as they are equal for both arcs. e Mark point βCβand join with βAβand βBβ. PROBLEM-SOLVING
9
C
D
A
9, 10
E
B
10, 11
9 Is it possible to draw any of the following? If yes, give an example. a an acute triangle that is also scalene b a right triangle that is also isosceles c an equilateral triangle that is also obtuse d a scalene triangle that is also right angled
10 Without using a protractor, accurately construct these triangles. Rulers can be used to set the pair of compasses. a triangle βABCβwith βAB = 5.5 cmβ, βBC = 4.5 cmβand βAC = 3.5 cmβ b an isosceles triangle with base length β4 cmβand legs β5 cmβ c an equilateral triangle with side length β3.5 cmβ d a right triangle with one side β4 cmβand hypotenuse β5 cmβ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
11 Copy and complete the following table, making the height of each cell large enough to draw a triangle in each cell. Draw an example of a triangle that fits the triangle type in both the row and column. Are there any cells in the table for which it is impossible to draw a triangle? Triangles
Scalene
Isosceles
Equilateral
Acute Right
U N SA C O M R PL R E EC PA T E G D ES
442
Obtuse
REASONING
12
12, 13
13, 14
12 a Is it possible to divide every triangle into two right triangles using one line segment? Explore with diagrams. b Which type of triangle can always be divided into two identical right triangles? 13 Try drawing a triangle with side lengths β4 cm, 5 cmβand β10 cmβ. Explain why this is impossible. 14 a Is the side opposite the largest angle in a triangle always the longest? b Can you draw a triangle with two obtuse angles? Explain why or why not. ENRICHMENT: Gothic arches
β
β
15
15 a The Gothic, or equilateral arch, is based on the equilateral triangle. Try to construct one, using this diagram to help.
b The trefoil uses the midpoints of the sides of an equilateral triangle. Try to construct one, using this diagram to help.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7F
Classifying quadrilaterals and other polygons
443
7F Classifying quadrilaterals and other polygons
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To be able to determine if a polygon is convex or non-convex β’ To be able to determine if a polygon is regular or irregular β’ To be able to classify a polygon by the number of sides it has β’ To know what a quadrilateral is β’ To be able to classify a quadrilateral as a parallelogram, rectangle, rhombus, square, trapezium or kite based on a diagram or a description
Polygons are closed plane shapes with straight sides. Each side is a segment and joins with two other sides at points called vertices. The number of sides, angles and vertices are the same for each type of polygon, and this number determines the name of the polygon. The word polygon comes from the Greek words poly meaning βmanyβ and gonia meaning βangleβ. Quadrilaterals are polygons with four sides. There are special types of quadrilaterals and these are identified by the number of equal side lengths and the number of pairs of parallel lines.
Architects designed these giant domes for plant greenhouses at the Eden Project, England. Steel hexagons, pentagons and triangles support plastic βpillowsβ full of air that insulate plants from the cold.
Lesson starter: Quadrilaterals that you know
You may already know the names and properties of some of the special quadrilaterals. Which ones do you think have: β’ β’ β’
2 pairs of parallel sides? all sides of equal length? 2 pairs of sides of equal length?
Are there any types of quadrilaterals that you know which you have not yet listed?
KEY IDEAS
β Polygons are closed plane figures with straight sides.
β A vertex is the point at which two sides of a shape meet.
side
convex
vertex
(Vertices is the plural form of vertex.)
β Convex polygons have all vertices pointing outward and all interior
(inside) angles smaller than 180Β°.
β Non-convex (or concave) polygons have at least one vertex pointing inward
nonconvex
and at least one interior angle larger than 180Β°.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
β Polygons are classified by the number of sides
they have.
Number of sides
Type
β3β
Triangle or trigon
β Regular polygons have sides of equal length and
β4β
Quadrilateral or tetragon
angles of equal size. β’ In a diagram, sides of equal length are shown using markings (or dashes).
β5β
Pentagon
β6β
Hexagon
β7β
Heptagon or septagon
β8β
Octagon
β9β
Nonagon
β10β
Decagon
β11β
Undecagon
β12β
Dodecagon
U N SA C O M R PL R E EC PA T E G D ES
444
regular pentagon
irregular pentagon
D
β Polygons are usually named with capital letters for each
quadrilateral ABCD C
A
vertex and in succession, clockwise or anticlockwise.
B
β A diagonal is a segment that joins two vertices, dividing a
diagonals
shape into two parts.
β Special quadrilaterals
Rectangle
Square
Rhombus
Parallelogram
Kite
Trapezium
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7F
Classifying quadrilaterals and other polygons
445
β Quadrilaterals with parallel sides contain two pairs of cointerior angles.
aΒ°
dΒ°
cΒ°
c + d = 180
U N SA C O M R PL R E EC PA T E G D ES
bΒ° a + b = 180
BUILDING UNDERSTANDING
1 Consider these three polygons. i
a b c d
ii
iii
The three shapes are examples of what type of polygon? Which shape(s) are convex and why? Which shape(s) are non-convex and why? State the missing words in this sentence. The third shape is called a __________ __________.
2 Draw an example of each of the quadrilaterals listed. Mark any sides of equal length with single or double dashes, mark parallel lines with single or double arrows and mark equal angles using the letters βaΒ°βand βbΒ°β. (Refer back to the Key ideas in this section should you need help.) a square b rectangle c rhombus d parallelogram e trapezium f kite
3 a Draw two examples of a non-convex quadrilateral. b For each of your drawings, state how many interior angles are greater than β180Β°β.
Example 13 Classifying polygons
a State the type of polygon and whether it is convex or non-convex. b Is the polygon regular or irregular?
Continued on next page
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
SOLUTION
EXPLANATION
a convex pentagon
The polygon has β5βsides and all the vertices are pointing outward.
b irregular
The sides are not of equal length and the angles are not equal.
U N SA C O M R PL R E EC PA T E G D ES
446
Now you try
a State the type of polygon and whether it is convex or non-convex. b Is the polygon regular or irregular?
Example 14 Classifying quadrilaterals State the type of each quadrilateral given below. a
b
SOLUTION
EXPLANATION
a non-convex quadrilateral
One interior angle is greater than β180Β°β.
b trapezium
There is one pair of parallel sides.
Now you try
State the type of each quadrilateral given. a
b
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7F
Classifying quadrilaterals and other polygons
447
Exercise 7F FLUENCY
1
2β8
2β6, 8
a State the type of polygon shown below and whether it is convex or non-convex. b Is the polygon shape regular or irregular?
U N SA C O M R PL R E EC PA T E G D ES
Example 13
1β6
2 How many sides do each of these polygon have? a pentagon c decagon e undecagon g nonagon i octagon
Example 13
triangle heptagon quadrilateral hexagon dodecagon
3 a Which of the given shapes are convex? b State the type of polygon by considering its number of sides. i ii
iv
Example 14a
b d f h j
iii
v
4 Classify each of these quadrilaterals as either convex or non-convex. a b
vi
c
175Β°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
448
Chapter 7
Example 14b
Geometry
5 State the type of special quadrilateral given below. a b
e
f
U N SA C O M R PL R E EC PA T E G D ES
d
c
Example 14b
6 List all the types of quadrilateral that have: a 2β βdifferent pairs of sides of equal length b β2βdifferent pairs of opposite angles that are equal in size c β2βdifferent pairs of parallel lines d only β1βpair of parallel lines e only β1βpair of opposite angles that are equal in size.
7 State whether the following are polygons (P) or not polygons (N). a circle b square c rectangle e cylinder f cube g line
d oval h segment
8 Use your knowledge of the properties of quadrilaterals to find the unknown angles and lengths in each of these diagrams. a b bm c 10 cm 5m aΒ° 100Β° aΒ° bΒ° aΒ° 130Β° b cm
50Β°
PROBLEM-SOLVING
9
9, 10
10, 11
9 Draw line segments to show how you would divide the given shapes into the shapes listed below. a b c
two triangles
one rectangle and two triangles
d
e
four triangles and one square
three triangles
f
two quadrilaterals
one pentagon and one heptagon
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7F
Classifying quadrilaterals and other polygons
449
U N SA C O M R PL R E EC PA T E G D ES
10 A diagonal between two vertices divides a polygon into two parts.
a What is the maximum (i.e. largest) number of diagonals that can be drawn for the following shapes if the diagonals are not allowed to cross? i convex pentagon ii convex decagon b What is the maximum number of diagonals that can be drawn for the following shapes if the diagonals are allowed to cross? i convex pentagon ii convex decagon
11 Using the given measurements, accurately draw this equilateral triangle onto a piece of paper and cut it into β4βpieces, as shown. Can you form a square with the four pieces? 3 cm
6 cm
6 cm
6 cm
3 cm
6 cm
REASONING
6 cm 12
12, 13
13, 14
12 State whether each of the following statements is true or false. a A regular polygon will have equal interior (i.e. inside) angles. b The sum of the angles inside a pentagon is the same as the sum of the angles inside a decagon. c An irregular polygon must always be non-convex. d Convex polygons are not always regular. 13 The diagonals of a quadrilateral are segments that join opposite vertices. a List the quadrilaterals that have diagonals of equal length. b List the quadrilaterals that have diagonals intersecting at β90Β°β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
14 a Are squares a type of rectangle or are rectangles a type of square? Give an explanation. b Are rhombuses a type of parallelogram? Explain. c Is it possible to draw a non-convex trapezium? ENRICHMENT: Construction challenge
β
β
15
15 Use a pair of compasses and a ruler to construct these figures. Use the diagrams as a guide, then measure to check the properties of your construction. a a rhombus with side length 5 cm
U N SA C O M R PL R E EC PA T E G D ES
450
b a line parallel to segment AB and passing through point P P
A
P
B
A
B
Each trapezium-shaped tabletop is a quadrilateral and two together forms a hexagon. Could these two trapezium-shaped tables be joined to make a parallelogram?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Progress quiz
1
E
A
F
B
K
G
I
Progress quiz
Consider the diagram below and answer the following. a Name the point where the line βEHβ intersects βKFβ. b Name an angle which has its vertex at βGβ. c Name an angle adjacent to ββ FGHβ. d Name a set of three concurrent lines. e Name an obtuse angle with its vertex at B and use your protractor to measure the size of this angle.
U N SA C O M R PL R E EC PA T E G D ES
7A
451
7B
C
J
D
H
2 Find the value of each pronumeral below and give a reason for each answer. a b xΒ°
xΒ°
105Β°
62Β°
c
d
xΒ°
64Β°
xΒ°
157Β°
e
f
xΒ°
g
7C
h
60Β°
xΒ°
7C
300Β°
47Β°
xΒ°
140Β°
xΒ° 75Β°
xΒ°
3 Name the angle that is: a corresponding to ββ EGBβ b alternate to ββ AGHβ c vertically opposite to ββ GHDβ d cointerior to ββ CHG.β
F
D
H
C
A
B
G
E
4 Find the value of βaβin these diagrams and give a reason for each answer. b c a aΒ°
116Β°
128Β° aΒ°
76Β° aΒ°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
452
Chapter 7
Progress quiz
7C
Geometry
5 Giving reasons, state whether the two lines cut by the transversal are parallel. a b 57Β° 111Β° 69Β°
U N SA C O M R PL R E EC PA T E G D ES
56 Β°
7D
6 Find the value of βaβin these diagrams. a
b
Ext
54Β°
aΒ°
aΒ°
145Β°
7E
7E
7F
47Β°
7 Classify this triangle by: a its side lengths b its angles.
8 Construct an equilateral triangle with each side having a length of β6 cmβ. 9 State the special type of quadrilateral given in each diagram below. a b
c
d
Γ
Γ
7F
10 Name the type of shape stating whether it is concave or convex, regular or irregular. a b
c
d
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7G Angle sum of a triangle
453
7G Angle sum of a triangle
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To know that the sum of interior angles in a triangle is 180Β° β’ To know what an exterior angle is β’ To be able to ο¬nd exterior angles in triangles using supplementary angles β’ To understand that the angle sum of a polygon can be determined by decomposing into triangles
The three interior angles of a triangle have a very important property. No matter the shape of the triangle, the three angles always add to the same total.
The Millau Viaduct in France is the tallest bridge structure in the world; its tallest pylon is 343 m. The different triangles formed by the bridgeβs cables all have the same angle sum.
Lesson starter: A visual perspective on the angle sum
Use a ruler to draw any triangle. Cut out the triangle and tear off the three corners. Then place the three corners together. aΒ°
bΒ°
cΒ°
aΒ°
bΒ°
cΒ°
What do you notice and what does this tell you about the three angles in the triangle? Compare your results with those of others. Does this work for other triangles?
KEY IDEAS
β The angle sum of the interior angles of a triangle is 180Β°.
bΒ°
aΒ°
cΒ°
a + b + c = 180
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
A
β If one side of a triangle is extended, an exterior angle is
formed. In the diagram shown opposite, ββ DBCβis the exterior angle. The angle ββ DBCβis supplementary to ββ ABCβ(i.e. adds to β180Β°β).
B D
aΒ° C
(180 β a)Β°
U N SA C O M R PL R E EC PA T E G D ES
454
β The angle sum βSβof a polygon with βnβsides can be determined by decomposing into triangles.
Quadrilateral (β n = 4)β
Pentagon β(n = 5)β
Angle sum = β3 Γ 180Β°ββ = 540Β°β
Angle sum = β2 Γ 180Β°ββ = 360Β°β
BUILDING UNDERSTANDING
1 a Use a protractor to measure the three angles in this triangle.
b Add your three angles. What do you notice?
2 For the triangle opposite, give reasons why: a βaβmust equal β20β b βbβmust equal β60.β
aΒ°
160Β°
bΒ°
100Β°
3 What is the size of each angle in an equilateral triangle? 4 For the isosceles triangle opposite, give a reason why: a βa = 70β b βb = 40.β
bΒ°
70Β° aΒ°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7G Angle sum of a triangle
455
Example 15 Finding an angle in a triangle Find the value of βaβin these triangles. a aΒ° 60Β°
b
95Β°
U N SA C O M R PL R E EC PA T E G D ES
aΒ°
70Β°
SOLUTION
EXPLANATION
a a = 180 β β(60 + 95)β β β β―β― ββ―β― =β 180 β 155β β = 25
The sum of angles in a triangle is β180β. Add the two known angles. Find the difference between β180βand β155β.
b a = 180 β β(70 + 70)β β β β―β― ββ―β― =β 180 β 140β β = 40
The two angles opposite the sides of equal length (i.e. the base angles) in an isosceles triangle are equal in size. Add the two equal angles. Find the difference between β140βand β180β.
Now you try
Find the value of βaβin these triangles. a
b
aΒ°
20Β°
20Β°
135Β°
aΒ°
Example 16 Finding an exterior angle
Find the size of the exterior angle (β xΒ°)βin this diagram.
aΒ°
xΒ°
62Β°
Continued on next page
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
456
Chapter 7
Geometry
EXPLANATION
a = 180 β (β 90 + 62)β ββ―β― β β β―β― =β 180 β 152β β = 28
The angle sum for a triangle is β180Β°β. Add the two known angles. βaβis the difference between β180βand β152β.
xβ β =β 180 β 28β β = 152
Angles of size βxΒ°βand βaΒ°βare supplementary (i.e. they add to β180Β°β). βxβis the difference between β180βand β28β.
U N SA C O M R PL R E EC PA T E G D ES
SOLUTION
β βThe size of the exterior angle is β152Β°β. β΄
Now you try
Find the size of the exterior angle (β xΒ°)βin this diagram.
20Β°
132Β°
xΒ°
aΒ°
Exercise 7G FLUENCY
Example 15a
1
1, 2ββ(1β/2β)β, 3, 4β5ββ(1β/2β)β
Find the value of βaβin these triangles. a aΒ° 30Β°
40Β°
110Β°
2 Find the value of βaβin each of these triangles. a b aΒ° 20Β° 40Β°
d
130Β°
80Β°
e
aΒ°
25Β°
2ββ(1β/2β)β,β 4β5ββ(1β/2β)β
b
110Β°
Example 15a
2β5ββ(1β/2β)β
35Β°
aΒ°
c
aΒ°
35Β°
aΒ°
f
15Β°
120Β°
aΒ°
aΒ°
20Β°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7G Angle sum of a triangle
Example 15b
3 Find the value of βaβin each of these isosceles triangles. a b 30Β° aΒ°
457
aΒ° 72Β°
4 Find the value of βaβin each of these isosceles triangles. a b 80Β° aΒ°
U N SA C O M R PL R E EC PA T E G D ES Example 15b
c
74Β°
aΒ°
65Β°
aΒ°
d
e
f
aΒ°
30Β°
70Β°
aΒ°
Example 15
110Β°
aΒ°
5 The triangles below have exterior angles. Find the value of βxβ. For parts b to f, you will need to first calculate the value of βaβ. a b c xΒ°
60Β°
150Β°
xΒ°
80Β°
d
aΒ°
aΒ°
xΒ°
150Β°
e
f
100Β°
xΒ°
60Β°
xΒ°
aΒ°
aΒ°
xΒ°
82Β°
aΒ°
70Β°
40Β°
60Β°
PROBLEM-SOLVING
6β(β β1/2β)β
6 Find the value of βaβin each of these triangles. a
6β7β(β β1/2β)β
6β7β(β β1/2β)β,β 8
b
aΒ°
110Β°
aΒ°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
c
d 100Β°
42Β°
aΒ°
35Β°
U N SA C O M R PL R E EC PA T E G D ES
458
aΒ°
e
f
aΒ°
(2a)Β°
56Β°
aΒ°
40Β°
7 Each of these diagrams has parallel lines. Find the value of βaβ. a b aΒ° 40Β°
80Β°
c
70Β°
35Β°
aΒ°
50Β°
aΒ°
d
e
aΒ°
35Β°
f
15Β°
20Β°
aΒ°
aΒ°
100Β°
30Β°
8 A plane flies horizontally β200 mβabove the ground. It detects two beacons on the ground. Some angles are known, and these are shown in the diagram. Find the angle marked βaΒ°β between the line of sight to the two beacons.
120Β°
aΒ°
200 m
140Β° beacons
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7G Angle sum of a triangle
REASONING
9
9, 10
459
10, 11
U N SA C O M R PL R E EC PA T E G D ES
9 In the Key Ideas, we can see how polygons can be decomposed into triangles. For example: A quadrilateral can be decomposed into two triangles without any intersecting diagonals so the angle sum of a quadrilateral equals β2 Γ 180Β°ββ = 360Β°β. a By decomposing into triangles, without any intersecting diagonals, how many triangles are formed in these polygons? i pentagon ii hexagon iii heptagon b Using your results from part a, determine the angle sum of the following polygons. i pentagon ii hexagon iii heptagon
10 Determine the angle sum of the following polygons. a octagon b nonagon
c decagon
11 If βSβis the angle sum of a polygon with βnβsides, find a rule linking βSβwith βnβ. ENRICHMENT: Exploring triangle theorems
12 a b c d
β
β
Find the sum β75Β°ββ + 80Β°β. Find the value of βaβin the diagram opposite. What do you notice about the answers to parts a and b? Do you think this would be true for other triangles with different angles? Explore.
12β14
75Β°
aΒ°
80Β°
13 This diagram includes two parallel lines. aΒ° bΒ° cΒ° a The angles marked βaΒ°βare always equal. From the list (corresponding, alternate, cointerior, vertically opposite), give a bΒ° aΒ° reason why. b Give a reason why the angles marked βbΒ°βare always equal. c At the top of the diagram, angles βaΒ°β, βbΒ°βand βcΒ°βlie on a straight line. What does this tell you about the three angles βaΒ°β, βbΒ°βand βcΒ°βin the triangle? 14 Complete these proofs. Give reasons for each step where brackets are shown. a The angle sum in a triangle is β180Β°β. β DCA = aΒ°β(Alternate to ββ BACβand βDEβis parallel to βABβ.) β ββ ECB = β_______ (_____________________) ββ DCA + β ACB + β ECB = β_______ (_____________________) ββ΄ a + b + c = _______________β b The exterior angle outside a triangle is equal to the sum of the two interior opposite angles.
D
C
E
cΒ°
aΒ°
A
bΒ°
B
B
bΒ°
aΒ° A
cΒ° C
D
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
7H Symmetry LEARNING INTENTIONS β’ To understand what a line of symmetry is β’ To be able to determine the order of line symmetry for a shape β’ To understand what rotational symmetry is β’ To be able to determine the order of rotational symmetry for a shape
U N SA C O M R PL R E EC PA T E G D ES
460
You see many symmetrical geometrical shapes in nature. The starfish and sunflower are two examples. Shapes such as these may have two types of symmetry: line and rotational.
Lesson starter: Working with symmetry On a piece of paper, draw a square (with side lengths of about 10 cm) and a rectangle (with length of about 15 cm and width of about 10 cm), then cut them out. β’
β’
Starο¬sh and sunο¬owers are both symmetrical, but in different ways.
How many ways can you fold each shape in half so that the two halves match exactly? The number of creases formed will be the number of lines of symmetry. Now locate the centre of each shape and place a sharp pencil on this point. Rotate the shape 360Β°. How many times does the shape make an exact copy of itself in its original position? This number describes the rotational symmetry.
KEY IDEAS
β An axis or line of symmetry divides a shape into two equal
parts. It acts as a mirror line, with each half of the shape being a reflection of the other. β’ An isosceles triangle has one line (axis) of symmetry. β’ A rectangle has two lines (axes) of symmetry.
β The order of rotation is the number of times a shape makes an exact copy
2
of itself (in its original position) after rotating 360Β°. β’ We say that there is no rotational symmetry if the order of rotational symmetry is equal to 1.
1
3
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7H Symmetry
461
BUILDING UNDERSTANDING 1 How many ways could you fold each of these shapes in half so that the two halves match
U N SA C O M R PL R E EC PA T E G D ES
exactly? (To help you solve this problem, try cutting out the shapes and folding them.) a square b rectangle c equilateral triangle d isosceles triangle e rhombus f parallelogram
2 For the shapes listed in Question 1 , imagine rotating them β360Β°βabout their centre.
1
How many times do you make an exact copy of the shape in its original position?
2
Example 17 Finding the symmetry of shapes
Give the number of lines of symmetry and the order of rotational symmetry for each of these shapes. a rectangle
SOLUTION
b regular pentagon
EXPLANATION
a 2 lines of symmetry
rotational symmetry: order β2β
b 5 lines of symmetry
rotational symmetry: order β5β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
462
Chapter 7
Geometry
Now you try Give the number of lines of symmetry and the order of rotational symmetry for each of these shapes. a isosceles trapezium Γ
U N SA C O M R PL R E EC PA T E G D ES
Γ
b equilateral triangle
Exercise 7H FLUENCY
Example 17
Example 17
1
1β6
2β7
2β4, 6, 7
Give the number of lines of symmetry and the order of rotational symmetry for this regular hexagon.
2 Give the number of lines of symmetry and the order of rotational symmetry for each shape. a b
c
d
e
f
3 Name a type of triangle that has the following properties. a 3 lines of symmetry and order of rotational symmetry β3β b 1 line of symmetry and no rotational symmetry c no line or rotational symmetry
4 List the special quadrilaterals that have these properties. a lines of symmetry: i β1β ii β2β iii β3β b rotational symmetry of order: i β1β ii β2β iii β3β
iv β4β iv β4β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7H Symmetry
463
U N SA C O M R PL R E EC PA T E G D ES
5 State the number of lines of symmetry and the order of rotational symmetry for each of the following. a b
c
d
6 Of the capital letters of the alphabet shown in the font here, state which have: a 1 line of symmetry A B C D E F G H I J K L M b 2 lines of symmetry N O P Q R S T U V W X Y Z c rotational symmetry of order 2. 7 Complete the other half of these shapes for the given axis of symmetry. b c a
PROBLEM-SOLVING
8
8, 9
8, 9
8 Draw the following shapes, if possible. a a quadrilateral with no lines of symmetry b a hexagon with one line of symmetry c a shape with line symmetry of order 7 and rotational symmetry of order 7 d a diagram with no line of symmetry but rotational symmetry of order 3 e a diagram with line of symmetry of order 1 and no rotational symmetry
9 These diagrams are made up of more than one shape. State the order of line symmetry and of rotational symmetry. a b c
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
REASONING
10
10
10, 11
10 Many people think a rectangle has four lines of symmetry, including the diagonals.
U N SA C O M R PL R E EC PA T E G D ES
464
a Complete the other half of this diagram to show that this is not true. b Using the same method as that used in part a, show that the diagonals of a parallelogram are not lines of symmetry.
try
me
line
m f sy
o
11 A trapezium has one pair of parallel lines. a State whether trapeziums always have: i line symmetry ii rotational symmetry. b What type of trapezium will have one line of symmetry? ENRICHMENT: Symmetry in βthreeβ dimensions
β
β
12
12 Some solid objects also have symmetry. Rather than line symmetry, they have plane symmetry. This cube shows one plane of symmetry, but there are more that could be drawn.
State the number of planes of symmetry for each of these solids. a cube b rectangular prism
c right square pyramid
d right triangular prism
f
e cylinder
sphere
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7I
Reflection and rotation
465
7I Reflection and rotation
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand that a shape can be reο¬ected or rotated to give an image β’ To be able to draw the result of a point or shape being reο¬ected in a mirror line β’ To be able to draw the result of a point or shape being rotated about a point
Reο¬ection and rotation are two types of transformations that involve a change in position of the points on an object. If a shape is reflected in a mirror line or rotated about a point, the size of the shape is unchanged. Hence, the transformations reflection and rotation are said to be isometric.
Lesson starter: Draw the image Here is a shape on a grid. β’ β’ β’ β’ β’
When designing βThe Mirrored Staircaseβ in this picture, the architect has drawn one staircase and then reο¬ected its design across the vertical line that is the axis of symmetry.
Draw the image (result) after reflecting the shape in the mirror line A. Draw the image (result) after reflecting the shape in the mirror line B. Draw the image after rotating the shape about point O by 180Β°. Draw the image after rotating the shape about point O by 90Β° clockwise. Draw the image after rotating the shape about point O by 90Β° anticlockwise.
mirror line A
mirror line B
O
Discuss what method you used to draw each image and the relationship between the position of the shape and its image after each transformation.
KEY IDEAS
β Reflection and rotation are isometric transformations that give an
image of an object or shape without changing its shape and size.
β The image of point A is denoted Aβ².
β A reflection involves a mirror line, as shown in the diagram
opposite.
A B F E D
C
Bβ² Aβ² Fβ² Eβ² image Cβ² Dβ²
mirror line
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
β A rotation involves a centre point of rotation (β C)βand an angle of
A
rotation, as shown. β’ A pair of compasses can be used to draw each circle, to help find the position of image points.
C
D
B
90Β°
U N SA C O M R PL R E EC PA T E G D ES
466
Aβ²
Dβ² image
Bβ² rotation 90Β° clockwise about C
BUILDING UNDERSTANDING
1 Use the grid to reflect each shape in the given mirror line. a b
c
d
e
f
2 Give the coordinates of the image point βAβ²βafter the point βA(β 2, 0)ββis rotated about point βCβ(0, 0)βby the following angles. a β180Β°β clockwise b β180Β°β anticlockwise c β90Β°β clockwise d β90Β°β anticlockwise
y
2 1
β2 β1 β1
C A 1 2
x
β2
3 a Are the size and shape of an object changed after a reflection? b Are the size and shape of an object changed after a rotation?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7I
Reflection and rotation
467
Example 18 Drawing reflections y
Draw the reflected image of this shape and give the coordinates of βAβ²β, βBβ²β, βCβ²βand βDβ²ββ. The βyβ-axis is the mirror line.
4 3 2 1
U N SA C O M R PL R E EC PA T E G D ES
mirror line ( y-axis)
β4 β3 β2 β1 O β1 D A β2 B β3 C β4
SOLUTION
1 2 3 4
x
EXPLANATION
Reflect each vertex βAβ, βBβ, βCβand β Dβabout the mirror line. The line segment from each point to its image should be at β90Β°βto the mirror line.
y
4 3 2 1
mirror line ( y-axis)
O 1 2 3 4 β4 β3 β2 β1 β1 Aβ² D A image Dβ² β2 Bβ² B β3 C Cβ² β4
x
βAβ²β= (1, β 1) , βBβ²β= (1, β 2) , βCβ²β= (3, β 3) , βDβ²β= (3, β 1)β
Now you try
Draw the reflected image of this shape and give the coordinates of βAβ²β, βBβ²β, βCβ²βand βDβ²ββ. The βyβ-axis is the mirror line.
y
B 4 3 2 1 D A β4 β3 β2 ββ 1 1O β2 β3 β4
C
1 2 3 4
x
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
Example 19 Drawing rotations Draw the image of this shape and give the coordinates of βAβ²β, βBβ²βand βDβ²βafter carrying out the following rotations. a 9β 0Β°βanticlockwise about βCβ b β180Β°βabout βCβ
y 4 3 D 2 1 B C A 2 β3 β2 β1 1 3 β1
U N SA C O M R PL R E EC PA T E G D ES
468
x
β2 β3 β4
SOLUTION
EXPLANATION
a
Rotate each point on a circular arc around point βCβby β90Β°β anticlockwise. Join the three image points (ββAβ²β, βBβ²βand βDβ²ββ) with line segments to form the result.
y
4 3 2 Bβ² D 1 Aβ² Dβ² B C A β3 β2 β1 β1 1 2 3
x
β2 β3 β4
βAβ²β= (0, 1), βBβ²β= (0, 2), βDβ²β= (β2, 1)β
b
Rotate each point on a circular arc around point βCβby β180Β°βin either direction. Join the three image points (ββAβ²β, βBβ²βand βDβ²ββ) with line segments to form the result.
y
4 3 D 2 1 Bβ² Aβ² C A B β3 β2 β1 β1 1 2 3
x
β2 Dβ²β3 β4
βAβ²β= (β1, 0), βBβ²β= (β2, 0), βDβ²β= (β1, β2)β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7I
Reflection and rotation
469
Now you try y
Draw the image of this shape and give the coordinates of βAβ²β, βBβ²βand βDβ²βafter carrying out the following rotations. a β90Β°βanticlockwise about βCβ b β180Β°βabout βCβ
D
U N SA C O M R PL R E EC PA T E G D ES
4 3 2 1 C
β5 β4 β3 β2 ββ 11
A
B
1 2 3 4
x
β2 β3 β4
Exercise 7I FLUENCY
Example 18
1
1, 2ββ(1β/2β)β, 3, 4
2β3β(β 1β/2β)β, 4, 5
2β3β(β 1β/2β)β, 4, 5
Draw the reflected image of this shape and give the coordinates of βAβ²β, βBβ²β, βCβ²βand βDβ²ββ. The βyβ-axis is the mirror line. y
5 4 A D 3 2 1 C B β5 β4 β3 β2 ββ 1 1O β2 β3 β4 β5
Example 18
1 2 3 4 5
x
mirror line (y-axis)
2 Draw the image of each shape in the mirror line and give the coordinates of βAβ²β, βBβ²β, βCβ²βand βDβ².ββ Note that the βyβ-axis is the mirror line for parts a to c, whereas the βxβ-axis is the mirror line for parts d to f. a b y y B4
C
4 3 2 1
3 2 1
A D O β4 β3 β2 β1 β1 β2 β3 β4
1 2 3 4
mirror line
x
O β4 β3 β2 β1 β1 β2 β3 β4
A
D
B C 1 2 3 4
x
mirror line
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
470
Chapter 7
Geometry
c
d
y 4 3 2 1
y B
4 3 2 1
mirror line
C
A D O β4 β3 β2 β1 β1 1 2 3 4 β2 β3 mirror line β4
x
x
U N SA C O M R PL R E EC PA T E G D ES O β4 β3 β2 β1 β1 β2 β3 β4
e
1 2 3 4 A D B
C
f
y
4 3 2 1
O β4 β3 β2 β1 β1 A Dβ2 β3 B β4 C
y
A
D
mirror line
1 2 3 4
x
4 3 2 1
C O β4 β3 β2 β1 β1
3 Give the new coordinates of the image point βAβ²βafter point βAβ has been rotated around point βC(β 0, 0)ββby: a 1β 80Β°β clockwise b β90Β°β clockwise c β90Β°β anticlockwise d β270Β°β clockwise e β360Β°β anticlockwise f β180Β°β anticlockwise.
Example 19
B
4 Draw the image of this shape and give the coordinates of βAβ²β, βBβ²ββ and βDβ²βafter the following rotations. a 9β 0Β°βanticlockwise about βCβ b β180Β°βabout βCβ c β90Β°βclockwise about βCβ
mirror line
1 2 3 4
x
β2 β3 β4
y
4 3 2 1
A
C
β4 β3 β2 β1 β1
1 2 3 4
x
β2 β3 β4
y
4 3 2 1
A D C β4 β3 β2 β1 β1 1 2 3 4
x
β2 B β3 β4
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7I
5 Draw the image of this shape and give the coordinates of βAβ²β, βBβ²βand βDβ²ββ after the following rotations. a β90Β°βanticlockwise about βCβ b β180Β°βabout βCβ c 9β 0Β°βclockwise about βCβ
471
y 4 B 3 2 A 1 C
D x
U N SA C O M R PL R E EC PA T E G D ES
Example 19
Reflection and rotation
β3 β2 β1 β1
1 2 3
β2 β3 β4
PROBLEM-SOLVING
6, 7
6β8
6, 8, 9
6 The mirror lines on these grids are at a β45Β°βangle. Draw the reflected image. a b c
d
e
f
7 On the Cartesian plane, the point βA(β β 2, 5)βis reflected in the βxβ-axis and this image point is then reflected in the βyβ-axis. What are the coordinates of the final image? 8 A point, βB(β 2, 3)β, is rotated about the point βC(β 1, 1)β. State the coordinates of the image point βBβ²βfor the following rotations. a β180Β°β b β90Β°β clockwise c 9β 0Β°β anticlockwise
y
3 2 1
β3 β2 β1 O β1 β2 β3
B
C
1 2 3
x
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
9 For each shape given, by how many degrees has it been rotated and in which direction? a b c C
C C
U N SA C O M R PL R E EC PA T E G D ES
472
REASONING
10
10, 11
11, 12
10 a By repeatedly reflecting a shape over a moving mirror line, patterns can be formed. This right-angled triangle for example is reflected β4βtimes by placing the mirror line vertically and on the right side each time. Create a pattern using these starting shapes by repeatedly placing the vertical mirror line on the right side. i ii
iii Create your own using reflection. b By repeatedly rotating a shape about a point, patterns can be formed. This diagram shows a semicircle rotated by β90Β°βthree times about the given point. Create a pattern using three β90Β°βrotations about the given point. i ii
iii Create your own using rotation. c See if you can combine reflections and rotations to create more complex patterns.
11 Write the missing number in these sentences. a Rotating a point β90Β°βclockwise is the same as rotating a point ________ anticlockwise. b Rotating a point β38Β°βanticlockwise is the same as rotating a point ________ clockwise. c A point is rotated β370Β°βclockwise. This is the same as rotating the point ________ clockwise.
12 A point βSβhas coordinates (β β 2, 5)ββ. a Find the coordinates of the image point βSβ²βafter a rotation β180Β°βabout βCβ(0, 0)ββ. b Find the coordinates of the image point βSβ²βafter a reflection in the βxβ-axis followed by a reflection in the βyβ-axis. c What do you notice about the image points in parts a and b? d Test your observation on the point βTβ(β 4, β 1)βby repeating parts a and b.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7I
ENRICHMENT: Interactive geometry software exploration
β
Reflection and rotation
β
473
13, 14
U N SA C O M R PL R E EC PA T E G D ES
13 Explore reflecting shapes dynamically, using interactive geometry software. a On a grid, create any shape using the polygon tool. b Construct a mirror line. c Use the reflection tool to create the reflected image about your mirror line. d Drag the vertices of your original shape and observe the changes in the image. Also try dragging the mirror line.
14 Explore rotating shapes dynamically, using interactive geometry software. a On a grid, create any shape using the polygon tool. b Construct a centre of rotation point and a rotating angle (or number). c Use the rotation tool to create the rotated image that has your nominated centre of rotation and angle. d Drag the vertices of your original shape and observe the changes in the image. Also try changing the angle of rotation.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
The following problems will investigate practical situations drawing upon knowledge and skills developed throughout the chapter. In attempting to solve these problems, aim to identify the key information, use diagrams, formulate ideas, apply strategies, make calculations and check and communicate your solutions.
Roof trusses
U N SA C O M R PL R E EC PA T E G D ES
Applications and problem-solving
474
1
When building a house, the frame to hold the roof up is constructed of roof trusses. Roof trusses come in different designs and define the pitch, or angle, of the external roof and the internal ceiling. The image here shows a roof with W trusses. The standard W truss provides an external pitch for the roof and a flat internal ceiling for the plaster.
Standard W truss
A builder is interested in how the lengths and angles work together for a standard W roof truss and the overall height of the truss above the ceiling which depends on the pitch angle. a Why do you think this design of truss is called a W truss? b How many segments of timber are required to construct one W truss? c Using the guidelines below, construct an accurate scale of a W truss in your workbook with a roof pitch of β30Β°β. i Draw a horizontal base beam of β12 cm.β ii Divide the base beam into five equal segments. iii Measure β30Β°βangles and draw the two sloping roof beams. iv Divide each sloping roof beam into three equal segments. v Draw the internal support beams by connecting lines between the equal segment markings on the roof and base beams. d On your accurate diagram, measure and label each of the internal angles formed by the support beams. e From your answers in part d, label any parallel support beams. f From your accurate diagram, what is the vertical height from the top of the roof to the ceiling (base beam)? g Investigate the angle (pitch) of either a roof at school or a roof at home. If possible, take a photo of the roof trusses and measure the relevant angles.
The perfect path for a hole-in-one
2 Visualising angles is a key to successfully playing mini-golf. When a golf ball bounces off a straight wall we can assume that the angle at which it hits the wall (incoming angle) is the same as the angle at which it leaves the wall (outgoing angle).
wall
incoming angle
centre angle
outgoing angle
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Applications and problem-solving
U N SA C O M R PL R E EC PA T E G D ES
Applications and problem-solving
You are interested in drawing a path where incoming angles equal outgoing angles so that you can score a hole-in-one. A sample mini-golf hole is shown here. A two-bounce pathway hole a Your first try. You may need to copy the diagram onto barrier paper first. Select a point on the starting line where you choose to tee off from and choose an angle to hit the ball at. Trace this path, ensuring each outgoing angle equals each incoming angle and see how close barrier you end up to going in the hole. b Label each of your angles along your path. c Can you draw an accurate path for a hole-in-one? A three-bounce pathway d Can you design a three-bounce path for a hole-inone? Label and measure your angles. start line (tee) You can place the ball Design your own anywhere along this line. e Design your own mini-golf hole that is possible for someone to score a hole-in-one through bouncing off walls or barriers. Show the correct pathway. (Hint: You may wish to draw the correct pathway first and then place in the barriers to make it challenging for players.)
475
Tessellating bathroom tiles
3 A tessellation is defined as an arrangement of shapes closely fitted together in a repeated pattern without gaps or overlapping. Elise is interested in what simple shapes tesselate and how to tile a bathroom using regular and irregular shapes. a Name the only three regular polygons that tessellate by themselves. b Why canβt a regular pentagon or a regular octagon tessellate by themselves? c Create your own tessellation using a combination of regular triangles, squares and hexagons. Elise is designing her new bathroom and wishes to have a floor tile pattern involving just pentagons. Eliseβs builder says that it is impossible to tessellate the bathroom floor with regular pentagons, but he is confident that he can tessellate the floor with irregular pentagons if he can find the right tiles. d Suggest what the internal angles of an irregular pentagon tile should be to ensure they tessellate with one another. e Draw an irregular pentagon that would tessellate with itself. f Create a bathroom floor pattern to show Elise how irregular pentagon tiles can tessellate.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
7J Translation LEARNING INTENTIONS β’ To understand that a shape can be translated left, right, up or down β’ To be able to draw the result of a point or shape being translated in a given direction β’ To be able to describe a translation given an original point/shape and an image point/shape
U N SA C O M R PL R E EC PA T E G D ES
476
Translation involves a shift in an object left, right, up or down. The orientation of a shape is unchanged. Translation is another isometric transformation because the size and shape of the image is unchanged.
Lesson starter: Describing a translation Consider this shape ABCD and its image Aβ²Bβ²Cβ²Dβ².
A
β’ β’ β’
Aβ²
Bβ²
Dβ²
Cβ²
D
B
In a dragster race along 300 m of straight track, the main body of the car is translated down the track in a single direction.
C
Use the words left, right, up or down to describe how the shape ABCD, shown opposite, could be translated (shifted) to its image. Can you think of a second combination of translations that give the same image? How would you describe the reverse translation?
KEY IDEAS
β Translation is an isometric transformation involving a shift left, right, up or down.
β Describing a translation involves saying how many units a shape is shifted left, right, up and/or
down.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7J
Translation
477
BUILDING UNDERSTANDING y
1 Point βAβhas coordinates (β 3, 2)β. State the coordinates of the 5 4 3 2 1
U N SA C O M R PL R E EC PA T E G D ES
image point βAβ²βwhen point βAβis translated in each of the following ways. a β1βunit right b β2βunits left c β3βunits up d β1βunit down e β1βunit left and β2βunits up f β3βunits left and β1βunit down g β2βunits right and β1βunit down h β0βunits left and β2βunits down
O
A
1 2 3 4 5
x
2 A point is translated to its image. State the missing word (i.e. left, right, up or down) for each sentence. a (β 1, 1)βis translated _______ to the point (β 1, 3)ββ. b (β 5, 4)βis translated _______ to the point (β 1, 4)ββ. c (β 7, 2)βis translated _______ to the point (β 7, 0)ββ. d (β 3, 0)βis translated _______ to the point (β 3, 1)ββ. e (β 5, 1)βis translated _______ to the point (β 4, 1)ββ. f (β 2, 3)βis translated _______ to the point (β 1, 3)ββ. g (β 0, 2)βis translated _______ to the point (β 5, 2)ββ. h (β 7, 6)βis translated _______ to the point (β 11, 6)ββ.
3 The point (β 7, 4)βis translated to the point (β 0, 1)ββ. a How far left has the point been translated? b How far down has the point been translated? c If the point (β 0, 1)βis translated to (β 7, 4)ββ: i How far right has the point been translated? ii How far up has the point been translated?
Example 20 Translating shapes
Draw the image of the triangle βABCβafter a translation β2βunits to the right and β 3βunits down.
A
B
C
Continued on next page
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
SOLUTION
EXPLANATION
A
Shift every vertex β2βunits to the right and β3βunits down. Then join the vertices to form the image.
B
C Aβ²
U N SA C O M R PL R E EC PA T E G D ES
478
Bβ²
Cβ²
Now you try
Draw the image of the triangle βABCβafter a translation β 1βunit left and β3βunits down.
B
A
C
Example 21 Describing translations
A point βB(β 5, β 2)βis translated to βBβ²β(β 1, 2)β. Describe the translation. SOLUTION
EXPLANATION
Translation is β6βunits left and β4βunits up.
y
3 Bβ² 2 1
O β2 β1 β1
1 2 3 4 5
β2 β3
x
B
Now you try
A point βB(β β 2, 5)βis translated to βBβ²(β β4, β 3β)β. Describe the translation.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7J
Translation
479
Exercise 7J FLUENCY
1
2β4β(β 1β/2β)β
2ββ(1β/2β)β,β 3β4ββ(1β/3β)β
Draw the image of the triangle βABCβafter a translation β3βunits to the left and β 2βunits down.
A
C
U N SA C O M R PL R E EC PA T E G D ES
Example 20
1, 2, 3β4ββ(1β/2β)β
B
Example 20
2 Draw the image of these shapes after each translation. a β3βunits left and β1βunit up b β1βunit right and β2βunits up
c 3β βunits right and β2βunits down
d β4βunits left and β2βunits down
3 Point βAβhas coordinates (β β 2, 3)β. Write the coordinates of the image point βAβ²βwhen point βAβis translated in each of the following ways. a β3βunits right b β2βunits left c 2β βunits down d β5βunits down e 2β βunits up f β10βunits right g β3βunits right and β1βunit up h β4βunits right and β2βunits down i β5βunits right and β6βunits down j β1βunit left and β2βunits down k β3βunits left and β1βunit up l β2βunits left and β5βunits down
y
5 4 A 3 2 1
β5 β4 β3 β2 ββ 1 1O
1 2 3 4 5
x
β2 β3 β4 β5
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
480
Chapter 7
4 Describe the translation when each point is translated to its image. Give your answer similar to these examples: ββ4βunits rightβ or ββ2βunits left and β3βunits upβ. a A β (β 1, 3)βis translated to βAβ²(β β1, 6β)ββ. b βBβ(4, 7)βis translated to βBβ²(β β4, 0β)ββ. c βC(β β 1, 3)βis translated to βCβ²β(β 1, β 1)β. d βD(β β 2, 8)βis translated to βDβ²(β β 2, 10)β. e βE(β 4, 3)βis translated to βEβ²(β β 1, 3)β. f βFβ(2, β 4)βis translated to βFβ²β(4, β 4)β. g βG(β 0, 0)βis translated to βGβ²β(β 1, 4)β. h βHβ(β 1, β 1)βis translated to βHβ²(β β2, 5β)ββ. i βI(β β 3, 8)βis translated to βIβ²(β β0, 4β)ββ. j βJβ(2, β 5)βis translated to βJβ²β(β 1, 6)β. k K β (β β 10, 2)βis translated to βKβ²β(2, β 1)β. l βL(β 6, 10)βis translated to βLβ²β(β 4, β 3)β.
U N SA C O M R PL R E EC PA T E G D ES
Example 21
Geometry
PROBLEM-SOLVING
5
5, 6
6, 7
5 A point, βAβ, is translated to its image, βAβ².β Describe the translation that takes βAβ²βto βAβ(i.e. the reverse translation). a A β (β 2, 3)βand βAβ²(β β4, 1β)β b βBβ(0, 4)βand βBβ²(β β4, 0β)β c βCβ(0, β 3)βand βCβ²(β β 1, 2)β d βD(β 4, 6)βand βDβ²β(β 2, 8)β
6 If only horizontal or vertical translations of distance β1βare allowed, how many different paths are there from points βAβto βBβon each grid below? No point can be visited more than once. a b B B A
A
7 Starting at (β 0, 0)βon the Cartesian plane, how many different points can you move to if a maximum of β3βunits in total can be translated in any of the four directions of left, right, up or down with all translations being whole numbers? Do not count the point (β 0, 0)ββ. REASONING
8
8
8, 9
8 A shape is translated to its image. Explain why the shapeβs size and orientation is unchanged.
9 A combination of translations can be replaced with one single translation. For example, if (β 1, 1)ββis translated β3βunits right and β2βunits down, followed by a translation of β6βunits left and β5βunits up, then the final image point β(β 2, 4)βcould be obtained with the single translation β3βunits left and β3βunits up. Describe the single translation that replaces these combinations of translations. a (β 1, 1)βis translated β2βunits left and β1βunit up, followed by a translation of β5βunits right and β2β units down. b β(6, β 2)βis translated β3βunits right and β3βunits up, followed by a translation of β2βunits left and β1β unit down. c (β β 1, 4)βis translated β4βunits right and β6βunits down, followed by a translation of β6βunits left and β2βunits up. d (β β 3, 4)βis translated β4βunits left and β4βunits down, followed by a translation of β10βunits right and β11βunits up.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7J
ENRICHMENT: Combined transformations
β
β
Translation
481
10
U N SA C O M R PL R E EC PA T E G D ES
10 Write the coordinates of the image point after each sequence of transformations. For each part, apply the next transformation to the image of the previous transformation. a (2, 3) β’ reflection in the x-axis β’ reflection in the y-axis β’ translation 2 units left and 2 units up b (β1, 6) β’ translation 5 units right and 3 units down β’ reflection in the y-axis β’ reflection in the x-axis c (β4, 2) β’ rotation 180Β° about (0, 0) β’ reflection in the y-axis β’ translation 3 units left and 4 units up d (β3, β7) β’ rotation 90Β° clockwise about (0, 0) β’ reflection in the x-axis β’ translation 6 units left and 2 units down e (β4, 8) β’ rotation 90Β° anticlockwise about (0, 0) β’ translation 4 units right and 6 units up β’ reflection in the x- and the y-axis
Construction material moved onto a building by a modern crane has undergone a combination of transformations.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
7K Drawing solids LEARNING INTENTIONS β’ To be able to draw pyramids, cylinders and cones β’ To be able to use square or isometric dot paper to accurately draw solids
U N SA C O M R PL R E EC PA T E G D ES
482
Three-dimensional solids can be represented as a drawing on a two-dimensional surface (e.g. paper or computer screen), provided some basic rules are followed.
Architects create 3D models of building plans by hand or with computer software.
Lesson starter: Can you draw a cube? Try to draw a cube. Here are two bad examples. β’ β’
What is wrong with these drawings? What basic rules do you need to follow when drawing a cube?
KEY IDEAS
β Draw cubes and rectangular prisms by keeping:
β’ β’
parallel sides pointing in the same direction parallel sides the same length.
β Draw pyramids by joining the apex with the vertices on the base.
triangular pyramid (tetrahedron) apex
triangular base
square pyramid apex
square base
β Draw cylinders and cones by starting with an oval shape.
cylinder
cone
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7K Drawing solids
483
β Square and isometric dot paper can help to accurately draw solids. Drawings made on
isometric dot paper clearly show the cubes that make up the solid. isometric dot paper
U N SA C O M R PL R E EC PA T E G D ES
square dot paper
BUILDING UNDERSTANDING
1 Copy these diagrams and add lines to complete the solid. Use dashed line for invisible sides. a cube b cylinder c square pyramid
2 Cubes are stacked to form these solids. How many cubes are there in each solid? a b c
Example 22 Drawing solids Draw these solids. a a cone on plain paper b this solid on isometric dot paper SOLUTION
EXPLANATION
a
Draw an oval shape for the base and the apex point. Dot any line or curve which may be invisible on the solid. Join the apex to the sides of the base.
b
Rotate the solid slightly and draw each cube starting at the front and working back.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
484
Chapter 7
Geometry
Now you try
U N SA C O M R PL R E EC PA T E G D ES
Draw these solids. a a square-based pyramid on plain paper b this solid on isometric dot paper
Exercise 7K FLUENCY
Example 22a Example 22a
Example 22b
Example 22b
1
1β4
2β4, 5ββ(1β/2β)β
2, 4, 5β(β 1β/2β)β
Draw a square-based prism on plain paper.
2 On plain paper, draw an example of these common solids. a cube b tetrahedron c cylinder d cone e square-based pyramid f rectangular prism 3 Copy these solids onto square dot paper. a
b
4 Draw these solids onto isometric dot paper. a b
c
5 Here is a cylinder with its top view (circle) and side view (rectangle). top
top
side
side
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7K Drawing solids
c cone
U N SA C O M R PL R E EC PA T E G D ES
Draw the shapes which are the top view and side view of these solids. a cube b square prism
485
d square pyramid
g square pyramid on cube
e octahedron
h hemisphere β(_ β1 βsphere)ββon 2 square prism
PROBLEM-SOLVING
6
f
sphere
i
cone on hemisphere
6, 7
7, 8
6 Here is the top (plan or birdβs eye) view of a stack of β5βcubes. How many different stacks of β5βcubes could this represent?
7 Here is the top (plan) view of a stack of β7βcubes. How many different stacks of β7βcubes could this represent?
8 Draw these solids, making sure that: i each vertex can be seen clearly ii dashed lines are used for invisible sides. a tetrahedron (solid with β4β faces) b octahedron (solid with β8β faces) c pentagonal pyramid (pyramid with pentagonal base)
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
REASONING
9
9
9, 10
9 Andrea draws two solids as shown. Aiden says that they are drawings of exactly the same solid. Is Aiden correct? Give reasons. and
U N SA C O M R PL R E EC PA T E G D ES
486
10 Match the solids a, b, c and d with an identical solid chosen from A, B, C and D. a
b
c
d
A
B
C
D
ENRICHMENT: Three viewpoints
β
β
11
11 This diagram shows the front and left sides of a solid.
left
a Draw the front, left and top views of these solids. i
b Draw a solid that has these views. i front
ii front
front
ii
left
top
left
top
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7L
Nets of solids
487
7L Nets of solids
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand that a net is a two-dimensional representation of a solidβs faces β’ To know what a polyhedron is β’ To know what the five Platonic solids are β’ To be able to draw a net of simple solids
The ancient Greek philosophers studied the properties of polyhedra and how these could be used to explain the natural world. Plato (427β347 bce) reasoned that the building blocks of all three-dimensional objects were regular polyhedra which have faces that are identical in size and shape. There are five regular polyhedra, called the Platonic solids after Plato, which were thought to represent fire, earth, air, water and the universe or cosmos.
Lesson starter: Net of a cube
Platonic solids form excellent dice; their symmetry and identical faces provide fair and random results. Platonic dice have 4, 6, 8, 12 or 20 faces and are used in role-playing games such as βDungeons and Dragonsβ.
Here is one Platonic solid, the regular hexahedron or cube, and its net.
If the faces of the solid are unfolded to form a net, you can clearly see the 6 faces.
Can you draw a different net of a cube? How do you know it will fold to form a cube? Compare this with other nets in your class.
KEY IDEAS
β A net of a solid is an unfolded two-dimensional
square pyramid
cylinder
representation of all the faces. Here are two examples.
β A polyhedron (plural: polyhedra) is a solid with flat faces.
β’
They can be named by their number of faces, e.g. tetrahedron (4 faces), hexahedron (6 faces).
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
β The five Platonic solids are regular polyhedra each with identical regular faces and the same
number of faces meeting at each vertex. β’ regular tetrahedron (β4βequilateral triangular faces)
U N SA C O M R PL R E EC PA T E G D ES
488
β’
regular hexahedron or cube (β6βsquare faces)
β’
regular octahedron (β8βequilateral triangular faces)
β’
regular dodecahedron (β12βregular pentagonal faces)
β’
regular icosahedron (β20βequilateral triangular faces)
BUILDING UNDERSTANDING
1 State the missing words in these sentences. a A regular polygon will have _________________ length sides. b All the faces on regular polyhedra are __________________ polygons. c The ________________ solids is the name given to the β5βregular polyhedra.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7L Nets of solids
C
U N SA C O M R PL R E EC PA T E G D ES
2 Which of the following nets would not fold up to form a cube? A B
489
3 Name the type of shapes that form the faces of these Platonic solids. a tetrahedron b hexahedron c octahedron d dodecahedron e icosahedron 4 Name the solids that have the following nets. a b
c
Example 23 Drawing nets Draw a net for these solids. a rectangular prism
b regular tetrahedron
SOLUTION
EXPLANATION
a
This is one possible net for the rectangular prism, but others are possible.
Continued on next page
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
490
Chapter 7
Geometry
Each triangle is equilateral. Each outer triangle folds up to meet centrally above the centre triangle.
U N SA C O M R PL R E EC PA T E G D ES
b
Now you try
Draw a net for these solids. a triangular prism
b square-based pyramid
Exercise 7L FLUENCY
Example 23a
1
Example 23b
2 Draw a net for this pyramid.
Example 23
1β5
2β6
3ββ(β1/2β)β,β 4β6
Draw a net for this rectangular prism.
3 Draw one possible net for these solids. a b
c
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7L Nets of solids
e
f
U N SA C O M R PL R E EC PA T E G D ES
d
491
4 Which Platonic solid(s) fit these descriptions? There may be more than one. a Its faces are equilateral triangles. b It has β20β faces. c It has β6β vertices. d It is a pyramid. e It has β12β sides. f It has sides which meet at right angles (not necessarily all sides).
5 Here are nets for the five Platonic solids. Name the Platonic solid that matches each one. a b
c
d
e
6 How many faces meet at each vertex for these Platonic solids? a tetrahedron b hexahedron d dodecahedron e icosahedron
c octahedron
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
PROBLEM-SOLVING
7
7
7, 8
7 Try drawing a net for a cone. Check by drawing a net and cutting it out to see if it works. Here are two cones to try. a b
U N SA C O M R PL R E EC PA T E G D ES
492
8 How many different nets are there for these solids? Do not count rotations or reflections of the same net. a regular tetrahedron b cube REASONING
9
9
9, 10
9 Imagine gluing two tetrahedrons together by joining two faces as shown, to form a new solid. a How many faces are there on this new solid? b Are all the faces identical? c Why do you think this new solid is not a Platonic solid. (Hint: Look at the number of faces meeting at each vertex.) 10 Decide if it is possible to draw a net for a sphere. ENRICHMENT: Number of cubes
β
β
11
11 Consider a number of β1 cmβcubes stacked together to form a larger cube. This one, for example, contains β3 Γ 3 Γ 3 = 27β cubes. a For the solid shown: i how many β1 cmβcubes are completely inside the solid with no faces on the outside? ii how many β1 cmβcubes have at least one face on the outside? b Copy and complete this table. βnβ(side length)
β1β
β2β
nβ β β3β(number of 1 cm cubes)
β1β
8β β
Number of inside cubes
β0β
Number of outside cubes
β1β
3β β
β4β
β5β
c For a cube stack of side length βn cmβ, βn β©Ύ 2β, find the rule for: i the number of cubes in total ii the number of inside cubes iii the number of outside cubes.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Modelling
493
BMX ramp
U N SA C O M R PL R E EC PA T E G D ES
Modelling
Marion is designing a BMX ramp and wishes to use three equal length pieces of steel for each side of the ramp. Her design is shown in this diagram with the three pieces of equal length steel shown as the line segments βAB, BC and CDβ, shown in green.
C
A
B
D
The points βAβ, βBβand βDβare on a straight line and represent the base of the ramp. The line segment βACβ represents the ramp slope. Present a report for the following tasks and ensure that you show clear mathematical working and explanations where appropriate.
Preliminary task
a Copy the diagram above, putting dashes on βABβ, βBCβand βCDβto indicate they are the same length. b What type of triangle is βΞABCβ? Give a reason. c If ββ BAC = 15Β°β, use your diagram to find the size of: i ββ ACBβ ii ββ ABCβ. d If ββ ABC = 140Β°β, determine the size of all the other angles you can find in the diagram.
Modelling task
Formulate
Solve
Evaluate and verify
Communicate
a The problem is to find the steepest ramp that Marion can build using the three equal length pieces of steel. Write down all the relevant information that will help solve this problem, including any diagrams as appropriate. b Starting with ββ BAC = 25Β°β, determine the following angles giving reasons for each calculation. Illustrate with a diagram. i ββ ACBβ ii ββ ABCβ iii ββ CBDβ iv β β BDCβ v β β BCDβ c Determine the angle ββ BDCβ for: i ββ BAC = 30Β°β ii ββ BAC = 40Β°β. d Describe the problem that occurs when using the steel to make a ramp with ββ BAC = 45Β°β and illustrate with a diagram. e Is it possible for Marion to use an angle ββ BACβgreater than β45Β°β? Explain why or why not. f If the angle ββ BACβmust be a whole number of degrees, determine the slope angle for the steepest ramp possible. g Summarise your results and describe any key findings.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
494
Chapter 7
Geometry
Now Marion has five pieces of equal length steel (AB, BC, CD, DE and EF) and she uses them to make a ramp, AE, in the following way. E
U N SA C O M R PL R E EC PA T E G D ES
Modelling
Extension questions
C
D
A
F
B
a Copy out the diagram and find all the angles if β BAC = 15Β°. b Determine the steepest possible slope angle if all angles in the diagram must be a whole number of degrees. c Compare your answer to the angle found in the case when she used three pieces of equal length steel.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Technology and computational thinking
495
Classifying shapes
Technology and computational thinking
Key technology: Dynamic geometry
U N SA C O M R PL R E EC PA T E G D ES
Classification systems are used all around the world to help sort information and make it easier for us to find what we are looking for. Similarly, shapes are also classified into subgroups according to their properties, including the number of parallel sides, side lengths and angles.
1 Getting started
Letβs start by classifying triangles by using this flowchart. Start
Are there 3 equal sides?
Yes
Output equilateral
Output right isosceles
Output obtuse isosceles
No
Are there 2 equal sides?
Yes
Yes
Is there a right angle?
Yes
No
Is there an obtuse angle?
No
No
Output acute isosceles
Is there a right angle?
Yes
Output right scalene
No
Is there an obtuse angle?
Yes
Output obtuse scalene
No
Output acute scalene
End
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
Work through the flowchart for each of these triangles and check that the algorithm classifies each triangle correctly. b c a
U N SA C O M R PL R E EC PA T E G D ES
Technology and computational thinking
496
d
e
2 Applying an algorithm
The special quadrilaterals that we consider here are: parallelogram, rectangle, rhombus, square, trapezium and kite. a Use the definitions in this chapter to think about what shared properties they have. Note the following definitions: β’ Parallelogram: A quadrilateral with two pairs of parallel sides β’ Rectangle: A parallelogram with all angles 90 degrees β’ Rhombus: A parallelogram with all sides equal β’ Square: A rhombus with all angles 90 degrees OR a Rectangle with all sides equal β’ Trapezium: A quadrilateral with one pair of parallel sides β’ Kite: A quadrilateral with two pairs of adjacent equal sides b Draw a flowchart similar to the one in part 1 for triangles, that helps to classify quadrilaterals. Test your algorithm using a range of special quadrilaterals.
3 Using technology
Construct these shapes using dynamic geometry. The construction for an isosceles triangle is shown in the diagram. a isosceles triangle b equilateral triangle c right-angled triangle
4 Extension
a Construct as many of the special quadrilaterals as you can using dynamic geometry. The construction for a rectangle is shown here. Note that the perpendicular line tool is used in this construction to save having to construct multiple perpendicular lines using circles. b Test that your construction is correct by dragging one of the initial points. When dragging, the properties of the shape should be retained.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Investigation
497
The perfect billiard ball path
Investigation
U N SA C O M R PL R E EC PA T E G D ES
When a billiard ball bounces off a straight wall (with no side spin), we can assume that the angle at which it hits the wall (incoming angle) is the same as the angle at which it leaves the wall (outgoing angle). This is similar to how light reflects off a mirror.
Single bounce
Use a ruler and protractor to draw a diagram for each part and then answer the questions. a Find the outgoing angle if: i the incoming angle is 30Β° ii the centre angle is 104Β°. b What geometrical reason did you use to calculate the answer to part a ii above?
wall
incoming angle
centre angle
outgoing angle
Two bounces
Two bounces of a billiard ball on a rectangular table are shown here. aΒ° 30Β° a Find the values of angles a, b, c, d and e, in that order. Give a cΒ° bΒ° reason for each. eΒ° dΒ° b What can be said about the incoming angle on the first bounce and the outgoing angle on the second bounce? Give reasons for your answer. c Accurately draw the path of two bounces using: i an initial incoming bounce of 20Β° ii an initial incoming bounce of 55Β°.
More than two bounces
a Draw paths of billiard balls for more than two bounces starting at the midpoint of one side of a rectangular shape, using the starting incoming angles below. i 45Β° ii 30Β° b Repeat part a but use different starting positions. Show accurate diagrams, using the same starting incoming angle but different starting positions. c Summarise your findings of this investigation in a report that clearly explains what you have found. Show clear diagrams for each part of your report.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
498
Chapter 7
Rearrange six matchsticks to make up four equilateral triangles.
Up for a challenge? If you get stuck on a question, check out the βWorking with unfamiliar problemsβ poster at the end of the book to help you.
2 How many equilateral triangles of any size are in this diagram?
U N SA C O M R PL R E EC PA T E G D ES
Problems and challenges
1
Geometry
3 What is the angle between the hour hand and minute hand of a clock at β9 : 35 a.m.β?
4 Two circles are the same size. The shaded circle rolls around the other circle. How many degrees will it turn before returning to its starting position? 5 A polygonβs vertices are joined by diagonals. How many diagonals can be drawn in each of these polygons? a decagon (β10β sides) b β50β-gon
6 This solid is made by stacking β1 cmβcubes. How many cubes are used?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter summary
U N SA C O M R PL R E EC PA T E G D ES
Angles at a point
Chapter summary
Angles acute 0Β°β90Β° right 90Β° obtuse 90Β°β180Β° straight 180Β° Β° 360Β° reflex 180Β°β revolution 360Β°
Measuring angles
499
bΒ° aΒ°
cΒ° dΒ°
Complementary a + b = 90 Supplementary c + d = 180 Vertically opposite a=c Revolution a + b + 90 + c + d = 360
Geometrical objects A
E
C
D
B
β ABC F ray BD line EF segment AB collinear points B, C, D vertex B
Angles and parallel lines
Parallel lines
a Β° cΒ°
dΒ°
tran
bΒ°
sve
rsa
l
a = b (corresponding) a = d (alternate) a + c = 180 (cointerior)
If a = 120, then b = 120, d = 120 and c = 60.
Constructions Two sides and the Three side lengths angle between them C C 4 cm 3 cm 5 cm 40Β° B A B A 6 cm 5 cm Two angles and a side C
Compound problems with parallel lines (Ext) A
C
A
35Β° 70Β° B 5 cm
30Β° B
60Β°
β ABC = 30Β° + 60Β°= 90Β°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
500
Chapter 7
Geometry
irregular regular convex non-convex pentagon octagon
scalene
isosceles
acute
right
aΒ°
equilateral
110Β°
60Β°
30Β°
obtuse
a = 180 β (110 + 30) = 180 β 140 = 40
U N SA C O M R PL R E EC PA T E G D ES
Chapter summary
Angle sum
Type
Polygons
cylinder
Solids rectangular prism
Exterior angle
Triangles
aΒ° 70Β°
cΒ°
bΒ°
If a = 70 b = 180 β (70 + 70) = 40 c = 180 β 40 = 140
Platonic solids
Regular polyhedron tretrahedron (4) hexahedron (6) octahedron (8) dodecahedron (12) icosahedron (20)
Polygons, solids and transformations
Quadrilaterals
Nets rectangular cylinder prism
Symmetry
Special quadrilaterals parallelogram β rectangle β rhombus β square trapezium kite
5 lines of symmetry rotational symmetry of order 5
regular pentagon
Transformations
Rotation
A (β2, 3)
y
3 2 1
O β3 β2 β1 β1 mirror β2 line β3 (y-axis)
Aβ² (2, 3)
1 2 3
Translation
y
A
180Β° 3 rotation 2 1
x
O β3 β2 β1 β1 90Β° β2 clockwise rotation β3
D
1 2 3
triangle
x
B
C Aβ²
Dβ²
Bβ²
Cβ²
2 units right and 3 units down
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter checklist
501
Chapter checklist with success criteria 1. I can name lines, rays and segments. e.g. Name this line segment.
U N SA C O M R PL R E EC PA T E G D ES
7A
β
Chapter checklist
A printable version of this checklist is available in the Interactive Textbook
A
7A
B
2. I can name angles. e.g. Name the marked angle.
P
Q
R
7A
3. I can classify an angle based on its size. e.g. Classify β134Β°βas an acute angle, a right angle, an obtuse angle, a straight angle, a reflex angle or a revolution.
7A
4. I can measure the size of angles with a protractor. e.g. Use a protractor to measure the angle ββ EFGβ.
G
E
F
7A
5. I can draw angles of a given size using a protractor. e.g. Use a protractor to draw an angle of size β260Β°β.
7B
6. I can find angles as a point using complementary or supplementary angles. e.g. Find the value of βaβin these diagrams.
a
b
aΒ°
7B
65Β° aΒ°
130Β°
7. I can find the size of angles without a protractor using other angles at a point. e.g. Find the value of βaβwithout a protractor.
aΒ°
120Β°
7C
8. I can name angles in relation to other angles involving a transversal. e.g. Name the angle that is (a) alternate to ββ ABFβ, and (b) cointerior to ββ ABFβ.
A
H
G
B
F
C D
E
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
β 7C
9. I can find the size of unknown angles in parallel lines. e.g. Find the value of βaβ, giving a reason for your answer.
U N SA C O M R PL R E EC PA T E G D ES
Chapter checklist
502
aΒ°
110Β°
7C
10. I can determine whether two lines are parallel given a transversal. e.g. State whether the two lines cut by this transversal are parallel.
122Β°
7D
58Β°
11. I can solve problems involving parallel lines and angles at a point. e.g. Find the value of βaβin this diagram.
Ext
D
A
60Β°
aΒ°
70Β°
C
B
7E
12. I can classify a triangle as scalene, isosceles or equilateral. e.g. Classify this triangle based on the side lengths.
7E
13. I can classify a triangle as acute, right or obtuse. e.g. Classify this triangle based on the angles.
7E
14. I can construct a triangle with given lengths and angles. e.g. Construct a triangle βABCβwith βAB = 5cm, β ABC = 30Β°βand ββ BAC = 45Β°β.
7E
15. I can construct a triangle using a ruler and pair of compasses. e.g. Construct a triangle with side lengths β6 cm, 4 cmβand β5 cmβ.
7F
16. I can name polygons based on the number of sides. e.g. State the name for a polygon with five sides.
7F
17. I can classify polygons as convex/non-convex and regular/irregular. e.g. State whether this shape is convex or non-convex, and whether it is regular or irregular.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter checklist
18. I can classify quadrilaterals. e.g. Determine whether the quadrilateral shown is convex or non-convex and what type(s) of special quadrilateral it is.
U N SA C O M R PL R E EC PA T E G D ES
7F
Chapter checklist
β
503
7G
19. I can use the angle sum of a triangle to find an unknown angle. e.g. Find the value of βaβin this triangle.
120Β°
20Β°
7G
aΒ°
20. I can find an unknown angle within an isosceles triangle. e.g. Find the value of βaβin this diagram.
aΒ°
70Β°
7G
21. I can find exterior angles for a triangle. e.g. Find the value of βxβin this diagram.
62Β°
7H 7I
aΒ°
xΒ°
22. I can determine the line and rotational symmetry of a shape. e.g. Give the order of line symmetry and of rotational symmetry for a rectangle.
23. I can find the result of a reflection of a point or shape in the coordinate plane. e.g. The shape βABCDβis reflected in the βyβ-axis. State the coordinates of βAβ²,ββ ββBβ²ββ, ββCβ²βand βDβ²β and connect them to draw the image.
y
4 3 2 1
β4 β3 β2 β1 O β1 D A β2 B β3 C
mirror line (y-axis)
1 2 3 4
x
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 7
Geometry
β 7I
24. I can find the result of a rotation of a point or shape in the coordinate plane. e.g. The triangle βABDβis rotated β90Β°βanticlockwise about βCβ. State the coordinates of βAβ²β, βBβ²βand β Dβ²βand hence draw the image.
U N SA C O M R PL R E EC PA T E G D ES
Chapter checklist
504
y
4 3 D 2 1 B C A β3 β2 β1 1 2 3 β1
x
β2 β3 β4
7J
25. I can draw the result of a translation. e.g. Draw the image of the triangle βABCβafter a translation β2βunits to the right and β3βunits down.
A
B
C
7J
26. I can describe a translation. e.g. A point βB(β 5, β 2)βis translated to βBβ²β(β 1, 2)β. Describe the translation.
7K
27. I can draw simple solids. e.g. Draw a cone.
7K
28. I can draw solids on isometric dot paper. e.g. Draw this solid on isometric dot paper.
7L
29. I can draw a net for a solid. e.g. Draw a net for a rectangular prism and for a regular tetrahedron.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
505
Short-answer questions Name each of these objects. a D
b
c
A
P
U N SA C O M R PL R E EC PA T E G D ES
1
Chapter review
7A
C
B
O
d
e
f
T
C
A
7A
S
2 For the angles shown, state the type of angle and measure its size using a protractor. a b
c
7A
3 Find the angle between the hour and minute hands on a clock at the following times. Answer with an acute or obtuse angle. a 6:00 a.m. b 9:00 p.m. c 3:00 p.m. d 5:00 a.m.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
506
Chapter 7
Chapter review
7B
Geometry
4 Without using a protractor, find the value of βaβin these diagrams. a b
c aΒ°
70Β° aΒ°
130Β°
145Β°
U N SA C O M R PL R E EC PA T E G D ES
aΒ°
d
e
aΒ°
f
aΒ° 75Β°
41Β°
g
aΒ°
h
aΒ°
52Β°
i
(a + 30)Β°
(2a)Β°
aΒ°
aΒ°
(2a)Β°
7C
7C
5 Using the pronumerals βaβ, βbβ, βcβor βdβgiven in the diagram, write down a pair of angles that are: a vertically opposite b cointerior c alternate d corresponding e supplementary but not cointerior.
aΒ°
dΒ°
cΒ°
6 For each of the following, state whether the two lines cut by the transversal are parallel. Give reasons for each answer. a b c 65Β° 65Β°
7D
bΒ°
60Β°
92Β°
89Β°
7 Find the value of βaβin these diagrams. a b
130Β°
c
Ext
80Β°
85Β° aΒ°
aΒ°
59Β°
70Β°
aΒ°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
d
e
507
f 70Β° aΒ° 32Β° 140Β°
aΒ°
U N SA C O M R PL R E EC PA T E G D ES
150Β°
Chapter review
aΒ°
7E
7E
7F
7F
7F
7G
8 Use a protractor and ruler to construct these triangles. a triangle βABCβwith βAB = 4 cmβ, ββ CAB = 25Β°βand ββ ABC = 45Β°β b triangle βABCβwith βAB = 5 cmβ, ββ BAC = 50Β°βand βAC = 5 cmβ
9 Use a protractor, pair of compasses and a ruler to construct these triangles. a triangle βABCβwith βAB = 5 cmβ, βBC = 6 cmβand βAC = 3 cmβ b triangle βABCβwith βAB = 6 cmβ, βBC = 4 cmβand βAC = 5 cmβ 10 How many sides do these polygons have? a pentagon b heptagon
c undecagon
11 A diagonal inside a polygon joins two vertices. Find how many diagonals can be drawn inside a quadrilateral if the shape is: a convex b non-convex. 12 Name each of these quadrilaterals. a b
c
13 Find the value of βaβin each of these shapes. a b 42Β° 80Β° aΒ° aΒ° 70Β° d
e
65Β°
aΒ°
f
aΒ°
aΒ°
c
40Β°
20Β°
aΒ°
g
h
i
aΒ°
110Β°
aΒ°
75Β° 25Β°
15Β° aΒ°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
508
Chapter 7
14 Name the order of line and rotational symmetry for each of these diagrams. a b c
U N SA C O M R PL R E EC PA T E G D ES
Chapter review
7H
Geometry
7I
15 Write the coordinates of βAβ²β, βBβ²βand βCβ²βwhen this shape is reflected in the following mirror lines. a the βyβ-axis b the βxβ-axis
y
4 3 2 1
O β4 β3 β2 β1 β1 β2 β3 β4
7I
16 Points βA(β 0, 4)ββ, βB(β 2, 0)βand βD(β 3, 3)βare shown here. Write down the coordinates of the image points βAβ²β, βBβ²βand βDβ²β after each of the following rotations. a 1β 80Β°βabout βCβ(0, 0)β b β90Β°βclockwise about βC(β 0, 0)β c β90Β°βanticlockwise about βCβ(0, 0)β
1 2 3 4 C A
x
B
y
4 A 3 2 1 C
β4 β3 β2 β1 β1
D
B 1 2 3 4
x
β2 β3 β4
7J
17 Write the coordinates of the vertices βAβ²β, βBβ²βand βCβ²β after each of these translations. a 4β βunits right and β2βunits up b β1βunit left and β4βunits up
y
4 3 2 1
O 1 2 3 4 β4 β3 β2 β1 C β1 β2 β3 B A β4
7K
x
18 Draw a side view, top view and net for each of these solids. a b
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
509
Multiple-choice questions Three points are collinear if: A they are at right angles. B they form a β60Β°β angle. C they all lie in a straight line. D they are all at the same point. E they form an arc on a circle.
U N SA C O M R PL R E EC PA T E G D ES
1
Chapter review
7A
7B
7A
7A
R
B sum to β270Β°β E sum to β45Β°β
C sum to β360Β°β
4 A reflex angle is: A β90Β°β D between β0Β°βand β90Β°β
B β180Β°β E between β90Β°βand β180Β°β
C between β180Β°βand β360Β°β
6 The angle a minute hand on a clock turns in β20β minutes is: A β72Β°β B β36Β°β C β18Β°β
80 90 100 11 70 0 60 110 100 90 80 70 120 0 60 13 0 2 5 01 50 0 13
D 1β 44Β°β
0 180 60 17 0 1 10 0 15 20 30
5 What is the size of the angle measured by the protractor? A β15Β°β C β105Β°β E β195Β°β B β30Β°β D β165Β°β
40
7E
Q
3 Complementary angles: A sum to β180Β°β D sum to β90Β°β
0
7C
P
E ββ PQPβ
14
7A
2 The angle shown here can be named: A ββ QRPβ C ββ QPRβ B ββ PQRβ D ββ QRRβ
0 10 20 180 170 1 60 30 150 40 14 0
7A
E β120Β°β
7 If a transversal cuts two parallel lines, then: A cointerior angles are equal. B alternate angles are supplementary. C corresponding angles are equal. D vertically opposite angles are supplementary. E supplementary angles add to β90Β°β.
8 The three types of triangles all classified by their interior angles are: A acute, isosceles and scalene. B acute, right and obtuse. C scalene, isosceles and equilateral. D right, obtuse and scalene. E acute, equilateral and right.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
510
Chapter 7
9 A non-convex polygon has: A all interior angles of 90Β°. B six sides. C all interior angles less than 180Β°. D all interior angles greater than 180Β°. E at least one interior angle greater than 180Β°.
U N SA C O M R PL R E EC PA T E G D ES
Chapter review
7F
Geometry
7F
10 The quadrilateral that has β2βpairs of sides of equal length and β1βpair of angles of equal size is called a: A kite B trapezium C rhombus D triangle E square
7H
11 A rhombus has line symmetry of order: A 0β β B β1β C β2β
7I
12 The point βT(β β 3, 4)βis reflected in the βxβ-axis; hence, the image point βTβ²βhas coordinates: A (β 3, 4)β B β(β 3, 4)β C β(0, 4)β D β(3, β 4)β E β(β 3, β 4)β
7J
D β3β
E β4β
13 The translation that takes βA(β 2, β 3)βto βAβ²β(β 1, 1)βcould be described as: A 3β βunits left. B 4β βunits up. C β3βunits left and β4βunits up. D β1βunit right and β2βunits down. E β1βunit left and β2βunits down.
Extended-response questions 1
A factory roof is made up of three sloping sections. The E F sloping sections are all parallel and the upright supports are at β90Β°βto the horizontal, as shown. Each roof section makes A B C a β32Β°βangle (or pitch) with the horizontal. factory a State the size of each of these angles. i β β EABβ ii β β GCDβ iii ββ ABFβ iv β β EBFβ b Complete these sentences. i β β BAEβis ____________________________ to ββ CBFβ. ii β β FBCβis ____________________________ to ββ GCBβ. iii ββ BCGβis ____________________________ to ββ GCDβ. c Solar panels are to be placed on the sloping roofs and it is decided that the angle to the horizontal is to be reduced by β11Β°β. Find the size of these new angles. i β β FBCβ ii β β FBAβ iii ββ FCGβ
G
D
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
D
Chapter review
aΒ°
U N SA C O M R PL R E EC PA T E G D ES
2 Two cables support a vertical tower, as shown in the diagram opposite, and the angle marked βaΒ°βis the angle between the two cables. a Find ββ BDCβ. b Find ββ ADCβ. c Find the value of βaβ. d If ββ DABβis changed to β30Β°βand ββ DBCβis changed to β65Β°β, will the value of βaβstay the same? If not, what will be the new value of βaβ?
511
3 Shown is a drawing of a simple house on a Cartesian plane. Draw the image of the house after these transformations. a translation β5βunits left and β4βunits down b reflection in the βxβ-axis c rotation β90Β°βanticlockwise about βC(β 0, 0)β
60Β°
25Β°
A
B
C
y
4 3 2 1
O β4 β3 β2 β1 β1
1 2 3 4
x
β2 β3 β4
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
U N SA C O M R PL R E EC PA T E G D ES
8 Statistics and probability
Maths in context: Collecting all types of data Statistical calculations, tables, charts and graphs are essential for interpreting all types of data and are widely used, including by governments, hospitals, medical professions, sports clubs, farmers, insurance companies and many businesses.
There are more than one hundred sports played in Australia. Each has a national and state organisation that keeps a record of results, calculates statistical measures such as the mean, median and mode, and provides progress graphs and charts. Australia has sport websites for athletics, basketball, canoeing and kayaking, cricket, cycling, diving, football, hockey, netball, Paralympics, rowing, rugby, sailing, skiing, snowboarding, swimming, volleyball and water polo. The following Australian government organisations obtain and record data and provide statistical measures and displays.
β’ ABS (Australian Bureau of Statistics): collects data from samples and also runs the Australian Census every ο¬ve years. ABS provides the government with population numbers and data on peopleβs health, education levels, employment, and housing. β’ BOM (Bureau of Meteorology): records weather data and forecasts the probabilities of rain, storms, wind strengths, temperatures, ο¬res, ο¬oods and droughts. This is important information for planning by emergency services, air trafο¬c control, farmers, and even holiday-makers!
β’ CSIRO (Commonwealth Scientiο¬c and Industrial Research Organisation): provides agricultural data, such as water availability, soil types, land use and crop forecasts.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter contents 8A 8B 8C 8D 8E 8F
Collecting and classifying data Summarising data numerically Column graphs and dot plots Line graphs Stem-and-leaf plots Sector graphs and divided bar graphs
U N SA C O M R PL R E EC PA T E G D ES
(EXTENDING)
8G Describing probability (CONSOLIDATING) 8H Theoretical probability in single-step experiments 8I Experimental probability in single-step experiments
Victorian Curriculum 2.0
This chapter covers the following content descriptors in the Victorian Curriculum 2.0: STATISTICS
VC2M7ST01, VC2M7ST02, VC2M7ST03 PROBABILITY
VC2M7P01, VC2M7P02 ALGEBRA
VC2M7A04
Please refer to the curriculum support documentation in the teacher resources for a full and comprehensive mapping of this chapter to the related curriculum content descriptors.
Β© VCAA
Online resources
A host of additional online resources are included as part of your Interactive Textbook, including HOTmaths content, video demonstrations of all worked examples, auto-marked quizzes and much more.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
8A Collecting and classifying data LEARNING INTENTIONS β’ To know the meaning of the terms primary source, secondary source, census, sample and observation β’ To be able to classify variables as numerical (discrete or continuous) or categorical β’ To understand that different methods are suitable for collecting different types of data, based on the size and nature of the data
U N SA C O M R PL R E EC PA T E G D ES
514
People collect or use data almost every day. Athletes and sports teams look at performance data, customers compare prices at different stores, investors look at daily interest rates, and students compare marks with other students in their class. Companies often collect and analyse data to help produce and promote their products to customers and to make predictions about the future.
Lesson starter: Collecting data
Consider, as a class, the following questions and discuss their implications. β’ β’ β’ β’
A doctor records a patientβs medical data to track their recovery. Examples include temperature, which is continuous numerical data, and number of heart beats recorded during a ο¬xed time period, which is discrete numerical data.
Have you or your family ever been surveyed by a telemarketer at home? What did they want? What time did they call? Do you think that telemarketers get accurate data? Why or why not? Why do you think companies collect data this way? If you wanted information about the most popular colour of car sold in Victoria over the course of a year, how could you find out this information?
KEY IDEAS
β In statistics, a variable is something measurable or observable that is expected to change over
time or between individual observations. It can be numerical or categorical. β’ Numerical (quantitative) data can be discrete or continuous: β Discrete numerical β data that can only be particular numerical values, e.g. the number of TV sets in a house (could be 0, 1, 2, 3 but not values in between such as 1.3125). β Continuous numerical β data that can take any value in a range. Variables such as heights, weights and temperatures are all continuous. For instance, someone could have a height of 172 cm, 172.4 cm or 172.215 cm (if it can be measured accurately). β’ Categorical β data that are not numerical such as colours, gender, brands of cars are all examples of categorical data. In a survey, categorical data come from answers which are given as words (e.g. βyellowβ or βfemaleβ) or ratings (e.g. 1 = dislike, 2 = neutral, 3 = like).
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8A Collecting and classifying data
515
β Data can be collected from primary or secondary sources.
β’
U N SA C O M R PL R E EC PA T E G D ES
β’
Data from a primary source are firsthand information collected from the original source by the person or organisation needing the data, e.g. a survey an individual student conducts to answer a question that interests them. Data from a secondary source have been collected, published and possibly summarised by someone else before we use it. Data collected from newspaper articles, textbooks or internet blogs represent secondary source data.
β Samples and populations
β’ β’ β’
When an entire population (e.g. a maths class, all the cars in a parking lot, a company, or a whole country) is surveyed, it is called a census. When a subset of the population is surveyed, it is called a sample. Samples should be randomly selected and large enough to represent the views of the overall population. When we cannot choose which members of the population to survey, and can record only those visible to us (e.g. people posting their political views on a news website), this is called an observation.
BUILDING UNDERSTANDING
1 Match each word on the left to its meaning on the right. a sample A only takes on particular numbers within a range b categorical B a complete set of data c discrete numerical C a smaller group taken from the population d primary source D data grouped in categories like βblueβ, βbrownβ, βgreenβ e continuous numerical E data collected firsthand f population F can take on any number in a range
2 Give an example of: a discrete numerical data b continuous numerical data c categorical data.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
Example 1
Classifying variables
Classify the following variables as categorical, discrete numerical or continuous numerical. a the Australian state or territory in which a baby is born b the length of a newborn baby
U N SA C O M R PL R E EC PA T E G D ES
516
SOLUTION
EXPLANATION
a categorical
As the answer is the name of an Australian state or territory (a word, not a number) the data are categorical.
b continuous numerical
Length is a continuous measurement, so all numbers are theoretically possible.
Now you try
Classify the following variables as categorical, discrete numerical or continuous numerical. a the number of children a person has b the brand of shoes someone wears
Example 2
Collecting data from primary and secondary sources
Decide whether a primary source or a secondary source is suitable for collection of data on each of the following and suggest a method for its collection. a a coffee shop wants to know the average number of customers it has each day b a detergent manufacturer wants to know the favourite washing powder or liquid for households in Australia SOLUTION
EXPLANATION
a primary source by recording daily customer numbers
This information is not likely to be available from other sources, so the business will need to collect the data itself, making this a primary source
b secondary data source using the results from a market research agency
A market research agency might collect these results using a random phone survey. Obtaining a primary source would involve conducting the survey yourself but it is unlikely that the sample will be large enough to be suitable.
Now you try
Decide whether a primary source or secondary source is suitable for collection of data on each of the following and suggest a method for its collection. a the maximum temperature each year in Australia for the past β100β years b the number of pets owned by everyone in a class at school
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8A Collecting and classifying data
517
Exercise 8A FLUENCY
1
2β3β(β 1β/2β)β,β 4
2β3β(β 1β/2β)β,β 4
Classify the following as categorical or numerical. a the eye colour of each student in your class b the date of the month each student was born, e.g. the β9thβof a month c the weight of each student when they were born d the brands of airplanes landing at Melbourneβs international airport e the temperature of each classroom f the number of students in each classroom period one on Tuesday
U N SA C O M R PL R E EC PA T E G D ES
Example 1
1, 2β3ββ(1β/2β)β,β 4
Example 1
2 Classify the following variables as categorical, continuous numerical or discrete numerical data. a the number of cars in each household b the weights of packages sent by Australia Post on Wednesday β20 βDecember 2023 c the highest temperature of the ocean each day d the favourite brand of chocolate of the teachers at your school e the colours of the cars in the school car park f the brands of cars in the school car park g the number of letters in different words on a page h the number of advertisements in a time period over each of the free-to-air channels i the length of time spent doing this exercise j the arrival times of planes at JFK airport k the daily pollution levels in the Burnley Tunnel on the City Link Freeway l the number of text messages sent by an individual yesterday m the times for the β100 mβfreestyle event at the world championships over the last 10 years n the number of Blu-ray discs someone owns o the brands of cereals available at the supermarket p marks awarded on a maths test q the star rating on a hotel or motel r the censorship rating on a movie showing at the cinema 3 Is observation or a sample or a census the most appropriate way for a student to collect data on each of the following? a the arrival times of trains at Southern Cross Station during a day b the arrival times of trains at Southern Cross Station over the year c the heights of students in your class d the heights of all Year β7βstudents in the school e the heights of all Year β7βstudents in Victoria f the number of plastic water bottles sold in a year g the religions of Australian families h the number of people living in each household in your class i the number of people living in each household in your school j the number of people living in each household in Australia
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
518
Chapter 8
k l m n
the number of native Australian birds found in a suburb the number of cars travelling past a school between 8 a.m. and 9 a.m. on a school day the money spent at the canteen by students during a week the ratings of TV shows
4 Identify whether a primary or secondary source is suitable for the collection of data on the following. a the number of soft drinks bought by the average Australian family in a week b the age of school leavers in far North Queensland c the number of soft drinks consumed by school age students in a day d the highest level of education by the adults in Australian households e the reading level of students in Year β7β in Australia
U N SA C O M R PL R E EC PA T E G D ES
Example 2
Statistics and probability
PROBLEM-SOLVING
5, 6
5, 7β9
8β10
5 Give a reason why someone might have trouble obtaining reliable and representative data using a primary source to find the following. a the temperature of the Indian Ocean over the course of a year b the religions of Australian families c the average income of someone in India d drug use by teenagers within a school e the level of education of different cultural communities within Victoria
6 Secondary sources are already published data that are then used by another party in their own research. Why is the use of this type of data not always reliable? 7 When obtaining primary source data you can survey the population or a sample. a Explain the difference between a βpopulationβ and a βsampleβ when collecting data. b Give an example situation where you should survey a population rather than a sample. c Give an example situation where you should survey a sample rather than a population.
8 A Likert-type scale is for categorical data where items are assigned a number; for example, the answer to a question could be β1 = dislike, 2 = neutral, 3 = likeβ. a Explain why the data collected are categorical even though the answers are given as numbers. b Give examples of a Likert-type scale for the following categorical data. You might need to reorder some of the options. i strongly disagree, somewhat disagree, somewhat agree, strongly agree ii excellent, satisfactory, poor, strong iii never, always, rarely, usually, sometimes iv strongly disagree, neutral, strongly agree, disagree, agree 9 A sample should be representative of the population it reports on. For the following surveys, describe who might be left out and how this might introduce a bias. a a telephone poll with numbers selected from a phone book b a postal questionnaire c door-to-door interviews during the weekdays d a Dolly magazine poll conducted online via social media e a Facebook survey
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8A Collecting and classifying data
U N SA C O M R PL R E EC PA T E G D ES
10 Another way to collect primary source data is by direct observation. For example, the colour of cars travelling through an intersection (categorical data) is best obtained in this way rather than through a questionnaire. a Give another example of a variable for which data could be collected by observation. b Explain how you could estimate the proportion of black cars parked at a large shopping centre car park without counting every single one.
519
REASONING
11
12, 13
12, 14
11 Television ratings are determined by surveying a sample of the population. a Explain why a sample is taken rather than conducting a census. b What would be a limitation of the survey results if the sample included β50βpeople nationwide? c If a class census was taken on which (if any) television program students watched from 7.30β8.30 last night, why might the results be different to the official ratings? d Research how many people are sampled by Nielsen Television Audience Measurement in order to get an accurate idea of viewing habits and stick within practical limitations.
12 Australiaβs census surveys the entire population every five years. a Why might Australia not conduct a census every year? b Over β40%βof all Australians were born overseas or had at least one of their parents born overseas. How does this impact the need to be culturally sensitive when designing and undertaking a census? c The census can be filled out on a paper form or using the internet. Given that the data must be collated in a computer eventually, why does the government still allow paper forms to be used? d Why might a country like India or China conduct their national census every β10β years? 13 When conducting research on Indigenous Australians, the elders of the community are often involved. Explain why the elders are usually involved in the research process. 14 Write a sentence explaining why two different samples taken from the same population can produce different results. How can this problem be minimised? ENRICHMENT: Collecting a sample
β
β
15
15 a Use a random number generator on your calculator or computer to record the number of times the number 1 to 5 appears (you could even use a die by re-rolling whenever you get a β6β) out of β50β trials. Record these data. i Tabulate your results. ii Compare the results of the individuals in the class. iii Explain why differences between different students might occur. b Choose a page at random from a novel or an internet page and count how many times each vowel β(βA, E, I, O, Uβ)βoccurs. Assign each vowel the following value βA = 1, E = 2, I = 3, O = 4, U = 5β and tabulate your results. i Why are the results different from those in part a? ii How might the results for the vowels vary depending on the webpage or novel chosen?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
8B Summarising data numerically LEARNING INTENTIONS β’ To understand that numerical data can be summarised as a single number by ο¬nding its range, mean, median or mode β’ To be able to ο¬nd the range of a set of numerical data β’ To be able to ο¬nd the mean, median and mode of a set of numerical data
U N SA C O M R PL R E EC PA T E G D ES
520
Although sometimes it is important to see a complete set of data, either as a list of numbers or as a graph, it is often useful to summarise the data with a few numbers.
For example, instead of listing the height of every Year 7 student in a school, you could summarise this by stating the median height and the difference (in cm) between the tallest and shortest people.
Lesson starter: Class summary
For each student in the class, find their height (in cm), their age (in years), and how many siblings they have. β’ β’ β’
Which of these three sets of data would you expect to have the largest range? Which of these three sets of data would you expect to have the smallest range? What do you think is the mean height of students in the class? Can you calculate it?
The median (i.e. middle) height of school students, of relevant ages, is used to determine suitable dimensions for classroom chairs and desks.
KEY IDEAS
β The range of a set of data is given by:
Range = highest number β lowest number. 1 6 7
1 5
range = 7 β 1 = 6
lowest highest
β Mean, median and mode are three different measures that can be used to summarise a set of
data. The word average is used to refer to the mean. β’ These are also called measures of centre or measures of central tendency.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8B Summarising data numerically
521
β The mean of a set of data is given by:
Mean = (sum of all the values) Γ· (total number of values) 1 + 6 + 7 + 1 + 5 = 20
mean = 20 Γ· 5 = 4
U N SA C O M R PL R E EC PA T E G D ES
β The median is the middle value when the values are sorted from lowest to highest. If there are
two middle values, then add them together and divide by 2.
1 1 β― 5 6 7
middle
1 3 4 9 10 12
median = 5
middle
median =
1 (4 + 9) = 6.5 2
β The mode is the most common value. It is the value that occurs most frequently. We also say
that it is the value with the highest frequency. There can be more than one mode. β― 1 β― 1 5 6 7
mode = 1
BUILDING UNDERSTANDING
1 Consider the set of numbers 1, 5, 2, 10, 3. a State the largest number. b State the smallest number. c What is the range?
2 State the range of the following sets of numbers. a 2, 10, 1, 3, 9 b 6, 8, 13, 7, 1 c 0, 6, 3, 9, 1 d 3, 10, 7, 5, 10 3 For the set of numbers 1, 5, 7, 7, 10, find the: a total of the numbers when added c median
b mean d mode.
These people are lined up in order of height. Whose heights are used to calculate: the range? the median? the mean?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
Example 3
Finding the range, mean and mode
Consider the ages (in years) of seven people who are surveyed in a shop: β15, 31, 12, 47, 21, 65, 12.β a Find the range of values. b Find the mean of this set of data. c Find the mode of this set of data.
U N SA C O M R PL R E EC PA T E G D ES
522
SOLUTION
EXPLANATION
a range = 65 β 12 β β β β β = 53
βHighest number = 65, lowest number = 12β The range is the difference.
b mean β =β 203 Γ· 7β β β = 29
S β um of values = 15 + 31 + 12 + 47 + 21 + 65 + 12 = 203β β Number of values = 7β
c βmode = 12β
The most common value is β12β.
Now you try
Consider the test scores of β5βpeople: β25, 19, 32, 25, 29β. a Find the range of values. b Find the mean test score. c Find the mode test score.
Example 4
Finding the median
Find the median of the following: a 7β , 2, 8, 10, 9, 7, 13β
b β12, 9, 15, 1, 23, 7β
SOLUTION
EXPLANATION
a Values: 2, 7, 7, 8, 9, 10, 13
Place the numbers in ascending order. (There is just one middle value, because there is an odd number of values.) The median is β8β, because it is the middle value of the sorted list.
βmedian = 8β
b Values: 1, 7, 9, 12, 15, 23 median = _ β9 + 12 β 2 21 _ β =ββ β β β β 2 β = 10.5
β
β βor 10β β)β ( 2 1 _
Place the numbers in ascending order and circle the two middle values. (There are two middle values because there is an even numbers of values.) The median is formed by adding the two middle values and dividing by β2β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8B Summarising data numerically
523
Now you try b β10, 2, 5, 7, 3, 3, 1, 8β
U N SA C O M R PL R E EC PA T E G D ES
Find the median of the following. a β7, 2, 9, 3, 5, 1, 8β
Exercise 8B
Example 3a
Example 3b
Example 3
Example 3
Example 4a
Example 4b
Example 4
1
FLUENCY
1β6ββ(β1/2β)β
Find the range of the following. a β3, 6, 7, 10β b β2, 8, 11, 12, 15β
c β4, 1, 8, 2, 9β
d β3, 12, 20, 2, 4, 11β
c β2, 2, 7, 1, 4, 2β
d β9, 2, 10β
2 Find the mean of the following. a β4, 3, 2, 5, 6β b β2, 10, 5, 7β
3β6β(β β1/2β)β,β 7
4β6β(β β1/2β)β, 7, 8
3 Consider the ages (in years) of nine people who are surveyed at a train station: β 18, 37, 61, 24, 7, 74, 51, 28, 24.β a Find the range of values. b Find the mean of this set of data. c Find the mode of this set of data. 4 For each of the following sets of data, calculate the: i range ii mean iii mode. a β1, 7, 1, 2, 4β c 3β , 11, 11, 14, 21β e 1β , 22, 10, 20, 33, 10β g β114, 84, 83, 81, 39, 12, 84β
b d f h
5 Find the median of: a β2, 5, 10, 12, 15β c 3β , 1, 5, 2, 9β e 1β 2, 18, 31, 15, 19, 10, 12β
b 1β , 7, 8, 10, 11β d β12, 5, 7, 10, 2β f β17, 63, 4, 13, 97, 82, 56β
6 Find the median of: a β3, 8, 10, 14, 16, 19β c 1β , 5, 2, 9, 13, 17β e 3β , 2, 3, 1, 8, 7, 6, 9β
b 2β , 7, 8, 10, 13, 18β d β5, 2, 3, 11, 7, 15β f β4, 9, 2, 7, 8, 1, 5, 6β
2β , 2, 10, 8, 13β β25, 25, 20, 37, 25, 24β β55, 24, 55, 19, 15, 36β β97, 31, 18, 54, 18, 63, 6β
7 The median for the data set 5 7 7 10 12 13 17 is 10. What would be the new median if the following number is added to the data set? a β9β b β12β c β20β d β2β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
8 The number of aces that a tennis player serves per match is recorded over eight matches. Match
β1β
β2β
β3β
β4β
β5β
6β β
β7β
β8β
Number of aces
β11β
β18β
β11β
β17β
β19β
β22β
β23β
β12β
a What is the mean number of aces the player serves per match? Round your answer to β1β decimal place. b What is the median number of aces the player serves per match? c What is the range of this set of data?
U N SA C O M R PL R E EC PA T E G D ES
524
PROBLEM-SOLVING
9, 10
10, 11
10β12
9 Brent and Ali organise their test marks for a number of topics in Maths, in a table. Test β1β
a b c d
Test β2β
Test β3β
Test β4β
Test β5β
Test β6β
Test β7β
Test β8β
Test β9β
Test β10β
Brent
β58β
β91β
β91β
7β 5β
β96β
β60β
9β 4β
β100β
β96β
β89β
Ali
β90β
β84β
β82β
β50β
β76β
β67β
β68β
β71β
β85β
5β 7β
Which student has the higher mean? Which student has the higher median? Which student has the smaller range? Which student do you think is better at tests? Explain why.
10 Alyshaβs tennis coach records how many double faults Alysha has served per match over a number of matches. Her coach presents the results in a table. Number of double faults
β0β
β1β
β2β
3β β
β4β
Number of matches with this many double faults
β2β
β3β
β1β
4β β
β2β
a b c d e f
In how many matches does Alysha have no double faults? In how many matches does Alysha have β3βdouble faults? How many matches are included in the coachβs study? What is the total number of double faults scored over the study period? Calculate the mean of this set of data, correct to β1βdecimal place. What is the range of the data?
11 A soccer goalkeeper recorded the number of saves he makes per game during a season. He presents his records in a table. Number of saves
β0β
β1β
β2β
β3β
β4β
5β β
Number of games
β4β
β3β
β0β
β1β
β2β
2β β
a How many games did he play that season? b What is the mean number of saves this goalkeeper made per game? Hint: first find the total number of saves made for the season. c What is the most common number of saves that the keeper had to make during a game?
12 The set β1, 2, 5, 5, 5, 8, 10, 12βhas a mode of β5βand a mean of β6β. a If a set of data has a mode of β5β(and no other modes) and a mean of β6β, what is the smallest number of values the set could have? Give an example. b Is it possible to make a data set for which the mode is β5β, the mean is β6βand the range is β20β? Explain your answer.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8B Summarising data numerically
REASONING
13
13, 14
525
14β16
13 Evie surveys all the students in her class to find the distance from their homes to school. One of the students is on exchange from Canada and reports a distance of 16 658 km. Would this very large value have a greater effect on the mean or median distance? Explain your answer.
U N SA C O M R PL R E EC PA T E G D ES
14 Consider the set of values 1, 3, 5, 10, 10, 13. a Find the mean, median, mode and range. b If each number is increased by 5, state the effect this has on the: i mean ii median iii mode iv range c If each of the original numbers is doubled, state the effect this has on the: i mean ii median iii mode iv range d Is it possible to include extra numbers and keep the same mean, median, mode and range? Try to expand this set to at least 10 numbers, but keep the same values for the mean, median, mode and range.
15 a Two whole numbers are chosen with a mean of 10 and a range of 6. What are the numbers? b Three whole numbers are chosen with a mean of 10 and a range of 2. What are the numbers? c Three whole numbers are chosen with a mean of 10 and a range of 4. Can you determine the numbers? Try to find more than one possibility. 16 Prove that for three consecutive numbers, the mean will equal the median. ENRICHMENT: Mean challenges
β
β
17, 18
17 a Give an example of a set of numbers with the following properties. i mean = median = mode ii mean > median > mode iii mode > median > mean iv median < mode < mean b If the range of a set of data is 1, is it still possible to find data sets for each of parts i to iv above? 1, _ 1, _ 1, _ 1. 18 Find the mean and median of the fractions _ 2 3 4 5
An important aspect of scientiο¬c investigation is collecting data and summarising it numerically.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
8C Column graphs and dot plots LEARNING INTENTIONS β’ To understand that graphs should include titles and labelled axes, with any numerical values drawn to scale β’ To know what an outlier is β’ To be able to construct and interpret a column graph β’ To be able to construct and interpret a dot plot
U N SA C O M R PL R E EC PA T E G D ES
526
Discrete numerical data can be counted and presented as a dot plot, with the number of dots representing the frequency.
Favourite colour
red
Colour
Categorical data can be counted and presented as a column graph. Each columnβs length indicates the frequency of that category. Column graphs can also be useful for labelled continuous data (e.g. height of people).
green
yellow blue
pink
0
Consider a survey of students who are asked to choose their favourite colour from five possibilities, as well as state how many people live in their household. The colours could be shown as a column graph, and the number of people shown in a dot plot.
5 10 15 20 Number of students As a column graph (horizontal)
Favourite colour
Number of students
20 15 10 5
0
1
2 3 4 Household size
5
nk
pi
ue
bl
w
ye llo
ee n
gr
re d
0
Colour As a column graph (vertical)
Lesson starter: Favourite colours
Survey the class to determine the number of people in each studentβs family and each studentβs favourite colour from the possibilities red, green, yellow, blue and pink. β’ β’
Each student should draw a column graph and a dot plot to represent the results. What are some different ways that the results could be presented into a column graph? (There are more than 200 ways.)
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8C Column graphs and dot plots
527
KEY IDEAS β A dot plot can be used to display discrete numerical data, where each dot represents one datum. β A column graph is a way to show data in different categories, and is useful when more than a
U N SA C O M R PL R E EC PA T E G D ES
few items of data are present.
β Column graphs can be drawn vertically or horizontally. Horizontal column graphs are
sometimes also called bar graphs.
β Graphs should have the following features:
A title explaining what the graph is about
Favourite colour
20 15
10
A label on each axis
Colour
nk
pi
ue
bl
llo
n
ee
gr
re
d
0
w
5
ye
Number of students
An even scale for the numerical axis
Category labels for any non-numerical data
β Any numerical axis must be drawn to scale.
β An outlier is a value that is noticeably distinct from the main cluster of points.
main cluster
an outlier
0 1 2 3 4 5 6 7 8 9 10
β Data represented as a dot plot could be described as symmetrical or skewed (or neither).
Negatively skewed
Symmetrical
Positively skewed
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8
0 1 2 3 4 5 6 7 8
mean β median
median < mean
mean < median
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
BUILDING UNDERSTANDING 1 The graph opposite shows the height of four boys. Answer true or false to each of the following statements. a Mick is β80 cmβ tall. b Vince is taller than Tranh. c Peter is the shortest of the four boys. d Tranh is β100 cmβ tall. e Mick is the tallest of the four boys.
110 100 90 80 70 60 50 40 30 20 10 0
Height chart
Tr an h V in ce Pe te r
M
ic k
Height (cm)
U N SA C O M R PL R E EC PA T E G D ES
528
Child
After-school activities
2 The favourite after-school activity of a number of Year β7βstudents is
8 6 4
social networking
television
sport
2
video games
Enrolments
recorded in the column graph shown opposite. a How many students have chosen television as their favourite activity? b How many students have chosen social networking as their favourite activity? c What is the most popular after-school activity for this group of students? d How many students participated in the survey?
Favourite activity
Example 5
Interpreting a dot plot
The dot plot on the right represents the results of a survey that asked some children how many pets they have at home. a Use the graph to state how many children have β2β pets. b How many children participated in the survey? c What is the range of values? d What is the median number of pets? e What is the outlier? f What is the mode?
Pets at home survey
0 1 2 3 4 5 6 7 8 Number of pets Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8C Column graphs and dot plots
EXPLANATION
a β4β children
There are β4βdots in the ββ2βpetsβ category, so 4 children have β2β pets.
b β22β children
The total number of dots is β22β.
U N SA C O M R PL R E EC PA T E G D ES
SOLUTION
529
c 8β β 0 = 8β
βRange = highest β lowestβ In this case, βhighest = 8, lowest = 0β.
d β1β pet
As there are β22βchildren, the median is the average of the β11thβand β12thβvalue. In this case, the β11thβand β12thβvalues are both β1β.
e the child with β8β pets
The main cluster of children has between β0βand β 3βpets, but the person with β8βpets is significantly outside this cluster.
f
The most common number of pets is β1β.
β1β pet
Now you try
The dot plot shows the number of visits a sample of pet owners have made to the vet in the past year. a How many people were surveyed? b How many people made more than three visits to the vet in the past year? c What is the range? d What is the median? e What is the outlier? f What is the mode?
Example 6
Visiting the vet
0 1 2 3 4 5 6 7 Number of visits
Constructing a column graph
Draw a column graph to represent the following peopleβs heights. Name
Tim
Phil
Jess
Don
Nyree
Height β(cm)β
β150β
1β 20β
β140β
1β 00β
β130β
Continued on next page
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
530
Chapter 8
Statistics and probability
SOLUTION
EXPLANATION First decide which scale goes on the vertical axis.
160 140 120 100 80 60 40 20 0
β aximum height = 150 cmβ, so axis goes from β M 0 cm to 160 cmβ(to allow a bit above the highest value).
U N SA C O M R PL R E EC PA T E G D ES
Height (cm)
Height chart
Remember to include all the features required, including axes labels and a graph title.
Tim Phil Jess Don Nyree Name
Now you try
Draw a column graph to represent the following peopleβs arm spans. Name
Matt
Pat
Carly
Kim
Tristan
Arm span (β cm)β
β180β
β190β
β150β
β160β
β170β
Exercise 8C FLUENCY
Example 5
Example 5
1
1β6
The dot plot on the right represents the results of a survey that asked some people how many times they had flown overseas. a Use the graph to state how many people have flown overseas once. b How many people participated in the survey? c What is the range of values? d What is the median number of times flown overseas? e What is the outlier? f What is the mode?
2 In a Year β4βclass, the results of a spelling quiz are presented as a dot plot. a What is the most common score in the class? b How many students participated in the quiz? c What is the range of scores achieved? d What is the median score? e Identify the outlier.
2β7
2, 3, 5β7
Overseas flights
0 1 2 3 4 5 6 7 Number of times o/s
Spelling quiz results
0 1 2 3 4 5 6 7 8 9 10 Score out of 10 Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8C Column graphs and dot plots
Favourite colours in Year 7B
Favourite colour
pink blue yellow
U N SA C O M R PL R E EC PA T E G D ES
3 From a choice of pink, blue, yellow, green or red, each student of Year β7Bβchose their favourite colour. The results are graphed on the right. a How many students chose yellow? b How many students chose blue? c What is the most popular colour? d How many students participated in the class survey? e Represent these results as a dot plot.
531
green red
0
2 4 6 8 Number of students
10
Height (cm)
4 Joan has graphed her height at each of her past five birthdays.
a b c d
Example 6
Joanβs height at different birthdays 180 160 140 120 100 80 60 40 20 0 8 9 10 11 12 Joanβs age ( years)
How tall was Joan on her β9thβ birthday? How much did she grow between her β8thβbirthday and β9thβ birthday? How much did Joan grow between her β8thβand β12thβ birthdays? How old was Joan when she had her biggest growth spurt?
5 Draw a column graph to represent each of these studentsβ heights at their birthdays. a Mitchell b Fatu Age (years)
Height (β cm)β
Age (years)
Height (β cm)β
β 8β
β120β
β 8β
β125β
β9β
β125β
β9β
β132β
β10β
β135β
β10β
β140β
β11β
β140β
β11β
1β 47β
β12β
β145β
β12β
β150β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
6 Every five years, a company in the city conducts a transport survey of peopleβs preferred method of getting to work in the mornings. The results are graphed below. Number of employees
Transport methods 70 60 50 40 30 20 10 0
public transport car walk/bicycle
U N SA C O M R PL R E EC PA T E G D ES
532
1990
1995
2000 2005 Year of survey
2010
2015
a Copy the following table into your workbook and complete it, using the graph. 1990
b c d e
Use public transport
β30β
Drive a car
β60β
Walk or cycle
β10β
1995
2000
2005
2010
2015
In which year(s) is public transport the most popular option? In which year(s) are more people walking or cycling to work than driving? Give a reason why the number of people driving to work has decreased. What is one other trend that you can see from looking at this graph?
7 a Draw a column graph to show the results of the following survey of the number of male and female puppies sold by a commercial dog breeder. Put time (years) on the horizontal axis. 2011
2012
2013
2014
2015
Number of male puppies born
β40β
β42β
5β 8β
β45β
β30β
β42β
Number of female puppies born
β50β
4β 0β
β53β
β41β
β26β
3β 5β
During which year(s) were there more female puppies sold than male puppies? Which year had the fewest number of puppies sold? Which year had the greatest number of puppies sold? During the entire period of the survey, were there more male or female puppies sold?
PROBLEM-SOLVING
8 The average (mean) income of adults in a particular town is graphed over a β6β-year period. a Describe in one sentence what has happened to the income over this period of time. b Estimate what the average income in this town might have been in β2012β. c Estimate what the average income might be in β2028β if this trend continues.
8
8, 9
Income (Γ $10 000)
b c d e
2010
7 6 5 4 3 2 1 0
9, 10
Average income in a town
2013 2014 2015 2016 2017 2018 Year
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8C Column graphs and dot plots
10 Mr Martin and Mrs Stevensson are the two Year β3β teachers at a school. For the latest arithmetic quiz, they have plotted their studentsβ scores on a special dot plot called a parallel dot plot, shown opposite. a What is the median score for class β3Mβ? b What is the median score for class β3Sβ? c State the range of scores for each class. d Based on this test, which class has a greater spread of arithmetic abilities? e If the two classes competed in an arithmetic competition, where each class is allowed only one representative, which class is more likely to win? Justify your answer.
En gl ish H ist or y Sc ie nc e
at hs
M
A
rt
U N SA C O M R PL R E EC PA T E G D ES
9 A survey is conducted of studentsβ favourite subjects 30 from a choice of Art, Maths, English, History and Science. Someone has attempted to depict the results in a column graph. 25 a What is wrong with the scale on the vertical axis? 20 b Give at least two other problems with this graph. 5 c Redraw the graph with an even scale and appropriate 0 labels. d The original graph makes Maths look twice as popular as Art, based on the column size. According to the survey, how many times more popular is Maths? e The original graph makes English look three times as popular as Maths. According to the survey, how much more popular is English? f Assume that Music is now added to the surveyβs choice of subjects. Five students who had previously chosen History now choose Music, and β16βstudents who had previously chosen English now choose Music. What is the most popular subject now?
533
Arithmetic quiz scores Class 3M
0 1 2 3 4 5 6 7 8 9 10
Score
Class 3S
REASONING
11
11, 12
12, 13
No. of passengers
11 At a central city train station, three types of services Train passengers at Urbanville Station 2500 run: local, country and interstate. The average 2000 number of passenger departures during each week is 1500 shown in the stacked column graph. 1000 a Approximately how many passenger departures 500 per week were there in β2013β? 0 2013 2014 2015 2016 2017 2018 b Approximately how many passenger departures Year were there in total during β2018β? interstate country local c Does this graph suggest that the total number of passenger departures has increased or decreased during the period β2013β2018β? d Approximately how many passengers departed from this station in the period β2013β2018β? Explain your method clearly and try to get your answer within β10 000βof the actual number.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
12 The mean value of data represented in a dot plot can be found by adding the value represented by each dot, then dividing by the number of dots. a Use this method to find the mean number of children per household, shown in this dot plot of β10β households. b A faster method is to calculate β(2 Γ 1) + (3 Γ 2) + (4 Γ 3) + (1 Γ 4)β and divide by β10β. Explain why this also works to find the mean.
0 1 2 3 4 5 Number of children
U N SA C O M R PL R E EC PA T E G D ES
534
13 Classify the following dot plots as representing symmetrical, positively skewed or negatively skewed data. Find the mean number in each case and compare it to the median. a b
0 1 2 3 4 5 6 7 8
0 1 2 3 4 5 6
c
d
0 1 2 3 4 5
0 1 2 3 4 5 6 7 8 9
ENRICHMENT: How many ways?
β
β
14
Animal sightings
Animal sightings
2 4 6 Number of sightings
8
ar be
an
te lo
o ng
ar ug co
ha nt el
pe
Animal sightings
8 6 4 2 0
ep
How many different column graphs could be used to represent the results of this survey? (Assume that you can only change the order of the columns, and the horizontal or vertical layout.) Try to list the options systematically to help with your count.
Number of sightings
Animal
di
ha nt
0
ep
o
dingo cougar antelope bear elephant
el
ng
di
ar
ar
ug
co
te
an
be
pe
Animal
8 6 4 2 0
lo
Number of sightings
14 As well as being able to draw a graph horizontally or vertically, the order of the categories can be changed. For instance, the following three graphs all represent the same data.
Animal
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8D Line graphs
535
8D Line graphs
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand that a line graph can be used to display continuous numerical data β’ To be able to draw a line graph β’ To be able to use a line graph to estimate values β’ To be able to interpret a travel graph
A line graph is a connected set of points joined with straight line segments. The variables on both axes should be continuous numerical data. It is often used when a measurement varies over time, in which case time is conventionally listed on the horizontal axis. One advantage of a line graph over a series of disconnected points is that it can often be used to estimate values that are unknown.
A business can use line graphs to display data such as expenses, sales and proο¬ts, versus time. A line graph makes it easy to visualise trends, and predictions can then be made.
Lesson starter: Room temperature
30 25 20 15 10 5
0
β’ β’ β’
Room A
Temperature ( Β°C )
Temperature (Β°C)
As an experiment, the temperature in two rooms is measured hourly over a period of time. The results are graphed below.
1 2 3 Time (hours)
4
35 30 25 20 15 10 5
0
Room B
3 1 2 Time (hours)
4
Each room has a heater and an air conditioner to control the temperature. At what point do you think these were switched on and off in each room? For each room, what is the approximate temperature 90 minutes after the start of the experiment? What is the proportion of time that room A is hotter than room B?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
KEY IDEAS segments. β’
The variables on both axes should be continuous numerical data.
Weight
β A line graph consists of a series of points joined by straight line
U N SA C O M R PL R E EC PA T E G D ES
536
β’
Time is often shown on the horizontal axis.
Time
graph. β’ Time is shown on the horizontal axis. β’ Distance is shown on the vertical axis. β’ The slope of the line indicates the rate at which the distance is changing over time This is called speed.
Distance (km)
β A common type of line graph is a travel
60 50 40 30 20 10
0
at rest
30 km in 3 hours or 10 km/h
30 km in 1 hour or 30 km/h
1 2 3 4 5 6 Time (hours)
BUILDING UNDERSTANDING
Catβs weight over time
It is weighed at the start of each month. State the catβs weight at the start of: a January b February c March d April
6 5 4 3 2 1 0 Jan
Feb Mar Apr Month
Lillianβs height
Height (cm)
2 The graph shows Lillianβs height over a β10β-year period from when she was born. a What was Lillianβs height when she was born? b What was Lillianβs height at the age of β7β years? c At what age did she first reach β130 cmβ tall? d How much did Lillian grow in the year when she was 7 years old? e Use the graph to estimate her height at the age of β9_ β1 ββyears. 2
Weight (kg)
1 The line graph shows the weight of a cat over a β3β-month period.
160 140 120 100 80 60 40 20 0
1 2 3 4 5 6 7 8 9 10 Age (years)
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8D Line graphs
Example 7
537
Drawing a line graph
The temperature in a room is noted at hourly intervals. Time
10 a.m.
11 a.m.
12 p.m.
1 p.m.
β10β
1β 5β
β20β
β23β
β18β
U N SA C O M R PL R E EC PA T E G D ES
Temperature (Β°C)
9 a.m.
a Present the results as a line graph. b Use your graph to estimate the room temperature at 12:30 p.m.
SOLUTION
EXPLANATION
a
β’ The vertical axis is from β0 to 25β. The scale is even (i.e. increasing by β5βeach time). β’ Dots are placed for each measurement and joined with straight line segments.
.
m
p.
.
m
p.
12
1
a.m
.
11
10
a.m
a.m
9
.
25 20 15 10 5 0
.
Temperature (Β°C )
Room temperature
Time
b About β20Β° Cβ
By looking at the graph halfway between 12 p.m. and β1βp.m., an estimate is formed.
Now you try
The temperature outside is noted at hourly intervals. Time
Temperature (Β°C)
4 p.m.
5 p.m.
6 p.m.
7 p.m.
8 p.m.
β25β
3β 0β
β20β
β18β
β15β
a Present the results as a line graph. b Use your graph to estimate the outside temperature at 5:30 p.m.
Interpreting a travel graph
This travel graph shows the distance travelled by a cyclist over 5 hours. a How far did the cyclist travel in total? b How far did the cyclist travel in the first hour? c What is happening in the second hour? d When is the cyclist travelling the fastest? e In the fifth hour, how far does the cyclist travel?
Distance cycled over 5 hours
Distance (km)
Example 8
30 25 20 15 10 5 0
1 2 3 4 5 Time (hours) Continued on next page
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
538
Chapter 8
Statistics and probability
EXPLANATION
a 3β 0 kmβ
The right end point of the graph is at (β 5, 30)ββ.
b β15 km β
At time equals β1βhour, the distance covered is β15 kmβ.
c at rest
The distance travelled does not increase in the second hour.
d in the first hour
This is the steepest part of the graph.
e β5 kmβ
In the last β3βhours, the distance travelled is β15 kmβ, so in β1βhour, β 5 kmβis travelled.
U N SA C O M R PL R E EC PA T E G D ES
SOLUTION
Now you try
This travel graph shows the distance travelled by a hiker over 4 hours. a How far did the hiker travel in total? b How far did the hiker travel in the first hour? c When is the hiker stationary? d When is the hiker travelling the fastest? e How far was travelled in the fourth hour?
Distance (km)
Distance hiked over 4 hours 10 9 8 7 6 5 4 3 2 1 0
1 2 3 4 5 Time (hours)
Exercise 8D FLUENCY
Example 7
1
1β4
2β5
2, 4, 5
A dog is weighed at the beginning of each month for five months as shown. a Draw a line graph of its weight. Month
Jan
Feb
March
April
May
Weight (kg)
β5β
β6β
8β β
β7β
β6β
b Use your graph to estimate the weight of the dog mid April.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8D Line graphs
Example 7
539
2 Oliver measures his pet dogβs weight over the course of a year, by weighing it at the start of each month. He obtains the following results. Weight (kg)
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
β7β
7β .5β
β8.5β
9β β
β9.5β
9β β
β9.2β
β7.8β
7β .8β
7β .5β
β8.3β
β8.5β
U N SA C O M R PL R E EC PA T E G D ES
a Draw a line graph showing this information, making sure the vertical axis has an equal scale from β 0 kg to 10 kgβ. b Describe any trends or patterns that you see. c Oliver put his dog on a weight loss diet for a period of 3β βmonths. When do you think the dog started the diet? Justify your answer.
m
id
ni gh 2 t a.m 4 . a. 6 m. a. 8 m. a 10 .m . a m .m id . d 2 ay p. 4 m. p. 6 m. p. 8 m. p 1 .m m 0p . id .m ni . gh t
Temperature ( Β°C)
3 Consider the following graph, which shows the outside temperature over a β24β-hour period that starts at midnight. a What was the temperature at midday? Temperature during a day b When was the hottest time of the day? c When was the coolest time of the day? 30 d Use the graph to estimate the temperature 25 at these times of the day: 20 i 4:00 a.m. 15 ii 9:00 a.m. 10 iii 1:00 p.m. 5 iv 3:15 p.m. 0
Time
4 This travel graph shows the distance travelled by a van over β6β hours. a How far did the van travel in total? Distance travelled over 6 hours b How far did the van travel in the first hour? 200 c What is happening in the fourth hour? (i.e. from 160 βt = 3βto βt = 4β) 120 d When is the van travelling the fastest? 80 e Estimate how far the van travels in the sixth hour? 40 Distance (km)
Example 8
0
5 This travel graph shows the distance travelled by a bushwalker over β5β hours. a For how long was the bushwalker at rest? b How far did the bushwalker walk in the second hour? c During which hour did the bushwalker walk the fastest?
Distance walked over 5 hours
Distance (km)
Example 8
1 2 3 4 5 6 Time (hours)
20 15 10 5 0
1 2 3 4 5 Time (hours)
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
PROBLEM-SOLVING
6 The water storage levels for a given city are graphed based on the percentage of water available on the first day of each month. For this question, assume that the amount of water used does not change from month to month. a During which month did it rain the most in this city? b At what time(s) in the year is the water storage below β40%β? c From August 1 to September 1, if a total of β20β megalitres of water went into storage, how much water was used during this period?
6
6
6, 7
Water storage levels 50 45 40 35 30 25 20 15 10 5 0
Ja n Fe Mb a Ar pr M a Ju y n Ju e ly A ug Se p O ct N ov D ec
Percentage
U N SA C O M R PL R E EC PA T E G D ES
540
6 a. 8 m. a 10 .m. m a.m id . d 2 ay p. 4 m. p. 6 m. p. 8 m. p 1 .m m 0 p. . id m ni . gh t
Temperature ( Β°C)
Month 7 The temperature in a living room is measured frequently throughout a particular day. The results Temperature in a living room are presented in a line graph, as shown below. The individual points are not indicated on this graph to 30 reduce clutter. a Twice during the day the heating was switched on. 20 At what times do you think this happened? Explain your reasoning. 10 b When was the heating switched off? Explain your reasoning. 0 c The house has a single occupant, who works during the day. Describe when you think that person is: i waking up Time ii going to work iii coming home iv going to bed. d These temperatures were recorded during a cold winter month. Draw a graph that shows what the lounge room temperature might look like during a hot summer month. Assume that the room has an air conditioner, which the person is happy to use when at home. REASONING
8 Draw travel graphs to illustrate the following journeys. a A car travels: b β’ β120 kmβin the first β2β hours β’ β0 kmβin the third hour β’ β60 kmβin the fourth hour β’ β120 kmβin the fifth hour.
8
8, 9
8β10
A jogger runs: β’ β12 kmβin the first hour β’ β6 kmβin the second hour β’ β0 kmβin the third hour β’ at a rate of β6 kmβper hour for β2β hours.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8D Line graphs
541
9 Explain how the steepness of sections of a travel graph can be used to describe how fast someone or something is travelling.
U N SA C O M R PL R E EC PA T E G D ES
10 A person records the length (β cm)βand weight (grams) of newborn babies at a hospital over the course of a year. a Explain what would be wrong with drawing a line graph with length on the horizontal axis and weight on the vertical axis. b Give an example of a possible use of line graphs when dealing with newborn baby length or weight data. ENRICHMENT: Which hemisphere?
β
β
11
11 The following line graph shows the maximum temperature in a city for the first day of each month.
40 35 30 25 20 15 10 5 0
Ja n Fe b M ar A p M r ay Ju n Ju e ly A ug Se p O ct N ov D ec
Temperature (Β°C)
Temperature in a year
Month
Is this city in the Northern or Southern hemisphere? Explain why. Is this city close to the equator or far from the equator? Explain why. Redraw the graph to start the β12β-month period at July and finish in June. Describe how the new graphβs appearance is different from the one shown above. In another city, somebody graphs the Temperature in a year maximum temperature over a β12β-month period, as shown opposite. 30 In which hemisphere is this city likely to be? 25 Explain your answer. 20 15 10 5 0 J F M A M J J A S O N D Month Average temperature ( Β°C)
a b c d e
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
8E Stem-and-leaf plots LEARNING INTENTIONS β’ To be able to interpret a stem-and-leaf plot β’ To be able to represent data in a stem-and-leaf plot
U N SA C O M R PL R E EC PA T E G D ES
542
A stem-and-leaf plot is a useful way of presenting numerical data in a way that allows trends to be spotted easily. Each number is split into a stem (the first digit or digits) and a leaf (the last digit). Stem Leaf
53 is 78 is 125 is
5 3 7 8
1 89
12 5
By convention, leaves are shown in increasing order as you work away from the stem, and stems are shown in increasing order going down the page.
The advantage of presenting data like this comes when multiple numbers have the same stem. For example, the list 122, 123, 124, 124, 127, 129 can be represented as shown below. Stem Leaf
12 2 3 4 4 7 9
Lesson starter: Test score analysis
In a class, studentsβ most recent test results out of 50 are recorded. Test 1 results 43, 47, 50, 26, 38, 20, 25, 20, 50, 44, 33, 47, 47, 50, 37, 28, 28, 22, 21, 29
Test 2 results
2 3357 3 04
Stem-and-leaf plots are used to display data such as in government, scientiο¬c and business reports. This plot lists daily temperatures (e.g. 2|5 means 25Β°C). The range = 34 β 18 = 16Β°C. Can you ο¬nd the median and the mode?
Stem Leaf 1 8
2 7 8
3 2 2 4 5 5 7 9
4 0 1 2 3 3 6 8 8 5 0 0
4|3 means 43 out of 50 marks
β’
β’ β’
For each test, try to find how many students: β achieved a perfect score (i.e. 50) β failed the test (i.e. less than 25) β achieved a mark in the 40s. If there are 100 test results that you wish to analyse, would you prefer a list or a stem-and-leaf plot? What is it that makes a stem-and-leaf plot easier to work with? Discuss.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8E Stem-and-leaf plots
543
KEY IDEAS β A stem-and-leaf plot is a way to display numerical data. β Each number is usually split into a stem (the first digit or digits) and a leaf (the last digit).
U N SA C O M R PL R E EC PA T E G D ES
β For example:
Stem Leaf
The number 7 is The number 31 is The number 152 is
β0β β7β β3β β1β
β15β β2β
β3|ββ1βmeans β31β
β A key is usually connected to each stem-and-leaf plot to show the value of the stem and leaf.
For example: β4 | 8βmeans β48βor β4 | 8βmeans β4.8 cmβ.
β Leaves should be aligned vertically, listed in ascending order as you move away from the stem. β Any outliers can be identified by looking at the lowest value or highest value to see if they are
far away from all the other numbers.
β The shape of a distribution can be seen from a stem-and-leaf plot:
Symmetrical
Positively Skewed
Negatively Skewed
Stem Leaf
Stem Leaf
Stem Leaf
1 36
1 126
2 9
2 139
2 4788
3 6
3 0125
3 259
4 3
4 023
4 38
5 18
5 45
5 4
6 2457
6 8
7 35
7 0
BUILDING UNDERSTANDING
1 The number β52βis entered into a stem-and-leaf plot. a What digit is the stem? b What digit is the leaf?
2 If 8|5 represents 85, what number is represented by the following combinations? a β3β|β9β b β2β|β7β c β13β|β4β 3 In this stem-and-leaf plot, the smallest number is β35β. What is the largest number?
Stem Leaf
β3β β5 7 7 9β β4β β2 8β
5β β β1 7β
β3|ββ5βmeans β35β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
Example 9
Interpreting a stem-and-leaf plot
Average daily temperatures are shown for some different countries. Stem Leaf β1β β3 6 6β
U N SA C O M R PL R E EC PA T E G D ES
544
2β β β0 0 1 2 5 5 6 8 9β β3β β0 2β
β2|ββ5βmeans β25Β° Cβ
a b c d e
Write out the temperatures as a list. How many countriesβ temperatures are represented? What are the minimum and maximum temperatures? What is the range of temperatures recorded? What is the median temperature recorded?
SOLUTION
EXPLANATION
a 13, 16, 16, 20, 20, 21, 22, 25, 25, β β―β―β― β 26, 28, 29, 30, 32
Each number is converted from a stem and a leaf to a single number. For example, β1|ββ3βis converted to β13β.
b β14β
The easiest way is to count the number of leaves β each leaf corresponds to one country.
c ββ―β― minimum = 13Β° Cβ maximum = 32Β° C
The first stem and leaf is β1|ββ3βand the last stem and leaf is β3β|β2ββ.
d βrange = 19Β° Cβ
βRange = maximum β minimum = 32 β 13 = 19β.
e βmedian = 23.5Β° Cβ
The middle value is halfway between the numbers β2|ββ2ββ β1 β(22 + 25)β= 23.5β. and β2β|β5β, so βmedian = _ 2
Now you try
The age of a number of people is shown in a stem-and-leaf plot. a Write out the ages as a list. b How many peopleβs ages are represented? c What are the youngest and oldest peopleβs ages? d Give the range of ages. e What is the median age?
Stem Leaf β1β β8β
2β β β1 1 4β
β3β β0 2 8 9β 4β β β3β
β3|ββ2βmeans β32βyears old
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8E Stem-and-leaf plots
545
Example 10 Creating a stem-and-leaf plot
U N SA C O M R PL R E EC PA T E G D ES
Represent this set of data as a stem-and-leaf plot: β23, 10, 36, 25, 31, 34, 34, 27, 36, 37, 16, 33β SOLUTION
EXPLANATION
Sorted: 10, 16, 23, 25, 27, 31, 33, 34, 34, ββ―β―β―β―β 36, 36, 37
Sort the list in increasing order so that it can be put directly into a stem-and-leaf plot.
β2β 3β 5 7β
Split each number into a stem and a leaf. Stems are listed in increasing order and leaves are aligned vertically, listed in increasing order down the page.
3β β β1 3 4 4 6 6 7β
Add a key to show the value of the stem and leaf.
Stem Leaf β1β 0β 6β
2β |ββ5βmeans β25β
Now you try
Represent this set of data as a stem-and-leaf plot: β36, 42, 38, 24, 32, 25, 29, 41, 48, 35β.
Exercise 8E FLUENCY
Example 9
Example 9a-c
1
1β3, 4ββ(1β/2β)β
2, 3, 4β5β(β 1β/2β)β
The number of buttons on some calculators are shown in this stem and leaf plot. a Write out all the numbers in a list. b How many calculators are represented? c What are the minimum and maximum number of buttons? d What is the range of the number of buttons recorded? e What is the median number of buttons recorded?
2 This stem-and-leaf plot shows the ages of people in a group. a Write out the ages as a list. b How many ages are shown? c Answer true or false to each of the following. i The youngest person is aged β10β. ii Someone in the group is β17βyears old. iii Nobody listed is aged β20β. iv The oldest person is aged β4β.
3, 4β5β(β 1β/2β)β
Stem Leaf β1β β3 7β
2β β β0 2 4 8 9β β3β β1 3 5β
2β | 4βmeans β24β
Stem Leaf β0β β8 9β
1β β β0 1 3 5 7 8β 2β β β1 4β
1β |ββ5βmeans β15βyears old
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
546
Chapter 8
3 For each of the stem-and-leaf plots below, state the range and the median. c a b Stem Leaf Stem Leaf
Stem Leaf
β0β β9β
β1β β1 4 8β
β3β 1β 1 2 3 4 4 8 8 9β
1β β β3 5 6 7 7 8 9β
2β β β1 2 4 4 6 8β
β4β β0 1 1 2 3 5 7 8β
β2β β0 1 9β
β3β β0 3 4 7 9β
β5β β0 0 0β
U N SA C O M R PL R E EC PA T E G D ES
Example 9d,e
Statistics and probability
1β |ββ8βmeans β18β
4β β β2β
4β |ββ2βmeans β42β
3β |ββ7βmeans β37β
Example 10
Example 10
4 Represent each of the following sets of data as a stem-and-leaf plot. a 1β 1, 12, 13, 14, 14, 15, 17, 20, 24, 28, 29, 31, 32, 33, 35β b β20, 22, 39, 45, 47, 49, 49, 51, 52, 52, 53, 55, 56, 58, 58β c β21, 35, 24, 31, 16, 28, 48, 18, 49, 41, 50, 33, 29, 16, 32β d β32, 27, 38, 60, 29, 78, 87, 60, 37, 81, 38, 11, 73, 12, 14β
5 Represent each of the following data sets as a stem-and-leaf plot. (Remember: β101βis represented as β10 | 1β.) a 8β 0, 84, 85, 86, 90, 96, 101, 104, 105, 110, 113, 114, 114, 115, 119β b β120, 81, 106, 115, 96, 98, 94, 115, 113, 86, 102, 117, 108, 91, 95β c 1β 92, 174, 155, 196, 185, 178, 162, 157, 173, 181, 158, 193, 167, 192, 184, 187, 193, 165, 199, 184ββ d β401, 420, 406, 415, 416, 406, 412, 402, 409, 418, 404, 405, 391, 411, 413, 408, 395, 396, 417ββ
PROBLEM-SOLVING
6, 7
6 This back-to-back stem-and-leaf plot shows the ages of all the people in two shops. The youngest person in Shop β1βis β15β(not β51β).
For each statement below, state whether it is true in Shop β1βonly (β 1)ββ, Shop β2βonly β(2)β, both shops (β B)βor neither shop (β N)ββ. a b c d e f g h i
This shop has a β31β-year-old person in it. This shop has six people in it. This shop has a β42β-year-old person in it. This shop has a β25β-year-old person in it. This shop has two people with the same age. This shop has a β52β-year-old person in it. This shop has a β24β-year-old person in it. This shopβs oldest customer is an outlier. This shopβs youngest customer is an outlier.
7, 8
8β10
Shop β1β Stem
Shop β2β
β5β
β1β
β6 7 β
β7 7 5 3β
β2β
β4 5β
β3β
β1β
4β β
β5β
β2β
1β |ββ5βmeans β15βyears old
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8E Stem-and-leaf plots
7 A company recorded the duration (in seconds) that visitors spent on its websiteβs home page. a How many visitors spent less than β20βseconds on the home page? b How many visitors spent more than half a minute? c How many visitors spent between β10βand β30β seconds? d What is the outlier for this stem-and-leaf plot? e The company wishes to summarise its results with a single number. βVisitors spend approximately ____ on our home page.β What number could it use?
547
Stem Leaf β0β β2 4 6 8 9β β1β β0 0 1 2 8β β2β β2 7 9β β3β
U N SA C O M R PL R E EC PA T E G D ES
β4β
8 Two radio stations poll their audience to determine their ages. a Find the age difference between the oldest and youngest listener polled for: i station β1β ii station β2.β b One of the radio stations plays contemporary music that is designed to appeal to teenagers and the other plays classical music and broadcasts the news. Which radio station is most likely to be the one that plays classical music and news? c Advertisers wish to know the age of the stationsβ audiences so that they can target their advertisements more effectively (e.g. to β38βto β57βyear olds; note that β38-57β is a 20-year age range, not 19 years, because it starts at 38 years 0 days and extends to 57 years 364.999... days.). Give a β20β-year age range for the audience majority who listen to: i station β1β ii station β2.β
β5β β8β
β2|ββ7βmeans β27β seconds
Station β1β
Stem
Station β2β
0β β
β1β
2β 3 3 4 5 6 8 9β
β8 7β
2β β
β0 0 1 2 4 5 8 8β
β9 7 5 4 3 3β
β3β
1β 1 2β
β7 6 5 5 4 4 1β
4β β
β8β
β9 3 2 0β
β5β
9 A group of students in Year β6βand Year β7βhave their heights recorded in a back-to-back stem-and-leaf plot, shown here. a State the range of heights for: i Year β7β ii Year β6.β b Which year level has a bigger range? c State the median height for: i Year β7β ii Year β6.β d Which year level has the larger median height? e Describe how you might expect this back-to-back stem-and-leaf plot to change if it recorded the heights of Year 1 students on the left, and Year 9 students on the right.
β3|ββ4βmeans β34βyears old
Year β6β
Stem
Year β7β
β10β β11β
β6 3 1β
β12β
8β 4 3 2β
1β 3β
β8β
7β 6 5 4 0β
β14β
3β 4 7 9β
6β 4 1β
1β 5β
β0 1 2 4 6β
2β 0β
β16β
2β 3 6 8β
β3β
β17β
β14β|β6βmeans β146 cmβ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
10 A teacher has compiled her studentsβ recent test scores out of β50βas a stem-and-leaf plot. However, some values are missing, as represented by the letters βa, b, cβand βdβ.
Stem Leaf β1β β5β β2β β4 5 a 6 7 9β
a How many students took the test? b How many students passed the test (i.e. achieved a mark of β25β or higher)? c State the possible values for each of the missing digits βaβto βdβ.
3β β βb 0 1 5β β4β β2 8 cβ
U N SA C O M R PL R E EC PA T E G D ES
548
REASONING
11
5β β βdβ
β3|ββ1βmeans β31β
11, 12
12, 13
11 a Explain why it is important that leaves are aligned vertically. (Hint: Consider how the overall appearance could be helpful with a large data set.) b Why might it be important that data values are sorted in stem-and-leaf plots?
12 a Give an example of positively skewed data in a stem-and-leaf plot, and investigate how the mean and median compare. b Give an example of negatively skewed data in a stem-and-leaf plot, and investigate how the mean and median compare. c Give an example of symmetrical data in a stem-and-leaf plot, and investigate how the mean and median compare. 13 A stem-and-leaf plot is constructed showing the ages of all the people who attended a local farmersβ market at a certain time of the day. However, the plotβs leaves cannot be read.
a For each of the following, either determine the exact answer or give a range of values the answer could take. i How many people were at the market? ii How many people aged in their β30sβwere at the market? iii How old is the youngest person? iv What is the age difference between the youngest and oldest person? v How many people aged β40βor over were at the market? vi How many people aged β35βor over were at the market? b Classify each of the following as true or false. i The majority of people at the market were aged in their β 30sβor β40sβ. ii There were five teenagers present. iii Exactly two people were aged β29βyears or under. iv Two people in their β40sβmust have had the same age. v Two people in their β30sβmust have had the same age. vi Two people in their β20sβcould have had the same age.
Stem Leaf β1β β? β β2β β? β
β3β β? ? ? ? ? ? ? ? ? ?β
β4β β? ? ? ? ? ? ? ? ? ? ? ? ? ? ?β β5β β? ? ?β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8E Stem-and-leaf plots
549
c Explain why it is possible to determine how many people were aged β40βor over, but not the number of people who are aged β40βor under. d It is discovered that the person under β20βyears of age is an outlier for this market. What does that tell you about how old the next oldest person is? β
β
14
U N SA C O M R PL R E EC PA T E G D ES
ENRICHMENT: Negative stem-and-leaf plots
14 Negative numbers can also be displayed in stem-and-leaf plots. This Stem Leaf stem-and-leaf plot gives the average winter temperatures in β15β different ββ 2β β9 4 4β cities. β β 1β β7 5 3 2β a What are the minimum and maximum temperatures listed? ββ 0β β8 5β b Find how many cities had average temperatures: β0β β3 4 6β i between ββ 10Β° Cβand β10Β° Cβ β1β β5 8β ii between ββ 25Β° Cβand β5Β° Cβ 2β β β3β iii below β5Β° Cβ. ββ 1β|β5βmeans ββ 15Β° Cβ c Why is there a β0βrow and a ββ 0βrow, even though β0βand ββ 0βare the same number? d What is the average (or mean) of all the listed temperatures in the β15βcities? Give your answer correct to β1βdecimal place. e What is the median of all the listed temperatures? Compare this to the mean found in part d.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
8F Sector graphs and divided bar graphs EXTENDING LEARNING INTENTIONS β’ To understand that pie charts and divided bar graphs are used to represent proportions of a total β’ To be able to draw and interpret a pie chart β’ To be able to draw and interpret a divided bar graph
U N SA C O M R PL R E EC PA T E G D ES
550
A pie chart (also called a sector graph) consists of a circle divided into different sectors or βslices of pieβ, where the size of each sector indicates the proportion occupied by any given category. A divided bar graph is a rectangle divided into different rectangles or βbarsβ, where the size of each rectangle indicates the proportion of each category. If a student is asked to describe how much time they spend each evening doing different activities, they could present their results as either type of graph: internet
TV
sport
homework internet
sport
homework
0%
TV
20%
sport
TV
40%
60%
homework
80%
100%
internet
From both graphs, it is easy to see that most of the studentβs time is spent playing sport and the least amount of time is spent using the internet.
Lesson starter: Student hobbies
Rania, Kristina and Ralph are asked to record how they spend their time after school. They draw the following graphs. Rania
Kristina
Ralph
homework
internet
TV
homework
homework
internet
β’ β’
TV
sport
sport
Based on these graphs alone, describe each student in a few sentences. Justify your descriptions based on the graphs.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8F Sector graphs and divided bar graphs
551
KEY IDEAS β To calculate the size of each section of the graph, divide the value in a given category by the
sum of all category values. This gives the categoryβs proportion or fraction.
U N SA C O M R PL R E EC PA T E G D ES
β To draw a sector graph (also called a pie chart), multiply each categoryβs proportion or
fraction by β360Β°βand draw a sector of that size.
β To draw a divided bar graph, multiply each categoryβs proportion or fraction by the total width
of the rectangle and draw a rectangle of that size.
BUILDING UNDERSTANDING
1 Jasna graphs a sector graph of how she spends her leisure time. TV a What does Jasna spend the most time doing? b What does Jasna spend the least time doing? homework c Does she spend more or less than half of her time playing sport?
sport
internet
2 Thirty students are surveyed to find out their favourite sport and their results are graphed below. rugby (12)
a b c d
soccer (6)
basketball (4)
AFL (8)
What is the most popular sport for this group of students? What is the least popular sport for this group of students? What fraction of the students has chosen soccer as their favourite sport? What fraction of the students has chosen either rugby or AFL?
Example 11 Drawing a sector graph and a divided bar graph
On a particular Saturday, Sanjay measures the number of hours he spends on different activities. Television
Internet
Sport
Homework
β1β hour
β2β hours
β4β hours
β3β hours
Represent the table as: a a sector graph
b a divided bar graph. Continued on next page
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
552
Chapter 8
Statistics and probability
SOLUTION
EXPLANATION
a
The total amount of time is β 1 + 2 + 4 + 3 = 10 hoursβ. Then we can calculate the proportions and sector sizes:
TV internet
Category
Proportion
Television
_ β1 β 10
Sector size (Β°)
Internet
_ β 2 β= _ β1 β
_ β1 βΓ 360 = 72β
Sport
4 β= _ β_ β2 β 10 5
_ β2 βΓ 360 = 144β
3β β_ 10
_ β 3 βΓ 360 = 108β
U N SA C O M R PL R E EC PA T E G D ES
homework
sport
10
Homework
b
TV internet
sport
homework
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
_ β 1 βΓ 360 = 36β
10
5
5
5
10
Using the same proportions calculated above, make sure that each rectangle takes up the correct amount of space. For example, if the total width is β15 cmβ, then sport β2 βΓ 15 = 6 cmβ. occupies _ 5
Now you try
Consider the following time allocation: Entertainment
Sport
Homework
β5β hours
β2β hours
β3β hours
Represent this table as: a a sector graph
b a divided bar graph.
Exercise 8F FLUENCY
Example 11
1
1β3
2β3
3
Consider the following results of a study on supermarket shopping habits. Items
Food
Drinks
Household items
Other
Proportion of money spent
β50%β
β25%β
β20%β
β5%β
a Represent this information in a divided bar graph. b Graph this information as a sector graph.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8F Sector graphs and divided bar graphs
Example 11a
553
2 A group of passengers arriving at an airport is surveyed to establish which countries they have come from. The results are presented below. Country No. of passengers
China
United Kingdom
USA
France
β6β
β5β
β7β
β2β
U N SA C O M R PL R E EC PA T E G D ES
a What is the total number of passengers who participated in the survey? b What proportion of the passengers surveyed have come from the following countries? Express your answer as a fraction. i China ii United Kingdom iii USA iv France c On a sector graph, determine the angle size of the sector representing: i China ii United Kingdom iii USA iv France d Draw a pie chart showing the information calculated in part c.
Example 11b
3 A group of students in Years β7βand β8βis polled on their favourite colour, and the results are shown at right.
Colour
Year β7β votes
Year β8β votes
Red
β20β
β10β
a Draw a sector graph (pie chart) to represent the Green β10β Year β7βcolour preferences. Yellow β5β b Draw a different pie chart to represent the Year β8β Blue β10β colour preferences. Pink β15β c Describe two differences between the charts. d Construct a divided bar graph that shows the popularity of each colour across the total number of Years β7βand β8βstudents combined
PROBLEM-SOLVING
4
4, 5
β4β
β12β β6β β8β
5, 6
4 A group of Year β7βstudents was polled on their favourite foods, and the results are shown in this sector graph. a If β40βstudents participated in the survey, find how many of them chips chose: i chocolate ii chips chocolate pies iii fruit iv pies. b Health experts are worried about what these results mean. They fruit would like fruit to appear more prominently in the sector graph, and to not have the chocolate sector next to the chips. Redraw the sector graph so this is the case. c Another β20βstudents were surveyed. Ten of these students chose chocolate and the other β10β chose chips. Their results are to be included in the sector graph. Of the four sectors in the graph, state which sector will: i increase in size ii decrease in size iii stay the same size.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
5 Yakob has asked his friends what is their favourite school subject, and he has created the following divided bar graph from the information.
English
Science
0%
Maths
History
50%
100%
U N SA C O M R PL R E EC PA T E G D ES
554
a If Yakob surveyed 30 friends, state how many of them like: i Maths best ii History best iii either English or Science best. b Redraw these results as a sector graph.
6 Friends Krishna and Nikolas have each graphed their leisure habits, as shown below. a Which of the two friends spends more of their time playing sport? b Which of the two friends does more intellectual activities in their leisure time? c Krishna has only 2 hours of leisure time each day because he spends the rest of his time doing homework. Nikolas has 8 hours of leisure time each day. How does this affect your answers to parts a and b above? Krishnaβs leisure time
TV
internet
Nikolasβ leisure time board games
reading
sport
sport
playing piano
When creating a personal budget, people often include a sector graph. The sector sizes show the proportion of income allocated for items such as bills, rent, food, clothing, car expenses and insurance.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8F Sector graphs and divided bar graphs
REASONING
7
7
555
7, 8
7 In two surveys, people were asked what is their favourite pet animal. a If β16βpeople participated in survey β1β, how many chose dog? Survey 1
U N SA C O M R PL R E EC PA T E G D ES
Survey 2
cat
dog
cat
bird
bird
b c d e
dog
If β30βpeople participated in survey β2β, how many chose bird? Jason claims that β20βpeople participated in survey β1β. Explain clearly why this cannot be true. Jaimee claims that β40βpeople participated in survey β2β. Explain clearly why this cannot be true. In fact, the same number of people participated for each survey. Given that fewer than β100β people participated, how many participants were there? Give all the possible answers.
8 a In a sector graph representing three different categories (red, blue and green), it is known that the proportion of red is _ β1 βand the proportion of blue is _ β1 β. What is the proportion of green? Answer as a 3 4 fraction and the number of degrees of this sector. b Explain why it is impossible to create a sector graph where the proportion of red is _ β1 ,ββ the 2 1 1 _ _ proportion of green is β βand the proportion of blue is β ββ. 3 4 ENRICHMENT: Rearranging graphs
β
β
9
9 Consider the divided bar graph shown below. a Show how this graph will look if the segments are placed in the order βC, D, A, Bβ(from left to right).
A
B
C
D
b In how many different ways could this divided bar graph be drawn (counting βABCDβand βCDABβ as two different ways)? c If this bar graph is redrawn as a sector graph, how many ways could the segments be arranged? Try to list them systematically. Do not consider two sector graphs to be different if one is just a rotation of another.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
556
Chapter 8
1
Classify the following variables as categorical, discrete numerical or continuous numerical. a eye colour b animal weight (β kg)β c number of siblings d time to run β100βmetres β(sec)β
2 For each of the following sets of data, calculate the: i range ii mean iii median iv mode. a 5β , 12, 3, 8, 2, 9, 3β b β15, 24, 22, 28, 16, 15β
U N SA C O M R PL R E EC PA T E G D ES
Progress quiz
8A
Statistics and probability
8B
8C
8D
3 This dot plot represents the number of children in each family of some Year β7β students. a What is the most common family size in this class? (Family size means the number of children.) b How many families are shown by this graph, assuming there are no siblings within the class? c What is the range of family sizes? d What is the median family size? e Identify the outlier.
Family size
1 2 3 4 5 6 7 8 Number of children in this family
4 The temperature outside a classroom was recorded four times during one school day. The following results were obtained. Time
9 a.m.
11 a.m.
1 p.m.
3 p.m.
Temperature
β15Β° Cβ
β20Β° Cβ
β28Β° Cβ
2β 5Β° Cβ
a Draw a line graph showing this information. b Use your graph to estimate the temperature at noon.
5 This travel graph shows the distance travelled by a cyclist over β 5β hours. a How far did the cyclist travel in total? b How far had the cyclist travelled after β3β hours? c What is happening in the fourth hour? d In the fifth hour, how far did the cyclist travel? e During what hour was the cyclist travelling the fastest?
Distance cycled
Distance (km)
8D
20 15 10 5
0
1 2 3 4 5 Time (hours)
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Progress quiz
β1β β4 7 8β β2β β0 2 4 5 7 8 9β 3β β β0 3 7β β2|ββ7βmeans β27Β° Cβ
U N SA C O M R PL R E EC PA T E G D ES
a How many towns have their temperatures listed in this stem-and-leaf table? b What is the maximum and minimum noon temperature recorded? c What is the range of temperatures recorded? d What is the median temperature recorded?
Stem Leaf
Progress quiz
6 This stem-and-leaf table shows the noon temperatures (in βΒ° Cβ) of different towns around Australia on one particular day.
8E
557
7 Represent this set of data as a stem-and-leaf plot: β10, 21, 16, 18, 7, 19, 18, 9, 20, 12β
8E
8 Some Year β7βstudents were asked how they travelled to school. The results are shown in this table.
8F
Ext
Public transport
Bicycle
Car
β14β
β2β
β4β
a Represent the data as a sector graph. b Represent the data as a divided bar graph of total length β15 cmβ.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
8G Describing probability CONSOLIDATING LEARNING INTENTIONS β’ To understand that we can describe the likelihood of events using phrases such as βeven chanceβ, βunlikelyβ and βcertainβ β’ To be able to describe the likelihood of an event
U N SA C O M R PL R E EC PA T E G D ES
558
Often, there are times when you may wish to describe how likely it is that an event will occur. For example, you may want to know how likely it is that it will rain tomorrow, or how likely your sporting team will win this yearβs premiership, or how likely it is that you will win a lottery. Probability is the study of chance.
Lesson starter: Likely or unlikely?
Try to rank these events from least likely to most likely. Compare your answers with other students in the class and discuss any differences. β’ β’ β’ β’ β’ β’
The Bureau of Meteorology (BOM) uses percentages to describe the chance of rain. For example: very high (90%) chance; high (70%) chance; medium (50%) chance; slight (20%) chance; near zero chance.
It will rain tomorrow. The Matildas will win the soccer World Cup. Tails landing uppermost when a 20-cent coin is tossed. The Sun will rise tomorrow. The king of spades is at the top of a shuffled deck of 52 playing cards. A diamond card is at the bottom of a shuffled deck of 52 playing cards.
KEY IDEAS
β When using the English language to describe chance, there are a number of phrases that can
be used.
100%
certain likely
50%
even chance unlikely
0%
more likely less likely
impossible
β If two events have the same chance of occurring, then we say that it is equally likely they will
occur.
β A fair coin or die is one for which all outcomes are equally likely. For example, a fair coin has
an equal chance of landing heads or tails up and is not biased by being heavier on one side than the other.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8G Describing probability
559
BUILDING UNDERSTANDING
U N SA C O M R PL R E EC PA T E G D ES
1 Match each of the events a to d with a description of how likely they are to occur (A to D). a A tossed coin landing heads up. A unlikely B likely b Selecting an ace first try from a fair deck C impossible of β52βplaying cards. D even chance c Obtaining a number other than β6βif a fair β6β-sided die is rolled.
d Obtaining a number greater than β8βif a fair β 6β-sided die is rolled.
2 State the missing words in these sentences. a If an event is guaranteed to occur, we say it is __________. b An event that is equally likely to occur or not occur has an ________ ____________. c A rare event is considered __________. d An event that will never occur is called __________.
Example 12 Describing chance
Classify each of the following statements as either true or false. a It is likely that children will go to school next year. b It is an even chance for a fair coin to display tails. c Rolling a β3βon a β6β-sided die and getting heads on a coin are equally likely. d It is certain that two randomly chosen odd numbers will add to an even number. SOLUTION
EXPLANATION
a true
Although there is perhaps a small chance that the laws might change, it is (very) likely that children will go to school next year.
b true
There is a β50β-β50β, or an even chance, of a fair coin displaying tails. It will happen, on average, half of the time.
c false
These events are not equally likely. It is more likely to flip heads on a coin than to roll a β3βon a β6β-sided die.
d true
No matter what odd numbers are chosen, they will always add to an even number.
Now you try
Classify each of the following statements as true or false. a It is an even chance that it will rain every day this year. b It is likely that it will rain some time in the next year. c Rolling an even number and rolling an odd number on a fair β6β-sided die are equally likely. d It is certain that heads will be flipped at least once on a fair coin if it is flipped β10β times.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
560
Chapter 8
Statistics and probability
Exercise 8G FLUENCY
1
2β4
3, 4
Classify each of the following statement as either true or false. a It is likely that some people will use public transport next week. b Getting a tail on a coin and rolling a β6βon a β6β-sided die are equally likely. c There is a β1βin β5βchance of randomly selecting the letter A from the word MARCH. d It is an even chance for a β6β-sided die to show a number less than β3β.
U N SA C O M R PL R E EC PA T E G D ES
Example 12
1β3
Example 12
2 Consider a fair β6β-sided die with the numbers β1βto β6βon it. Answer true or false to each of the following. a Rolling a β3βis unlikely. b Rolling a β5βis likely. c Rolling a β4βand rolling a β5βare equally likely events. d Rolling an even number is likely. e There is an even chance of rolling an odd number. f There is an even chance of rolling a multiple of β3β. 3 Match up each of the events a to d with an equally likely event A to D. a Rolling a β2βon a β6β-sided die A b Selecting a heart card from a fair deck of β 52βplaying cards B c Flipping a coin and tails landing face up C d Rolling a β1βor a β5βon a β6β-sided die D
Example 12
Selecting a black card from a fair deck of β52βplaying cards Rolling a number bigger than β4βon a β6β-sided die Selecting a diamond card from a fair deck of β52βplaying cards Rolling a β6βon a β6β-sided die
4 Consider the spinner shown, which is spun and could land with the arrow pointing to any of the three colours. (If it lands on a boundary, it is spun again until it lands on a colour.) a State whether each of the following is true or false. i There is an even chance that the spinner will point to green. ii It is likely that the spinner will point to red. iii It is certain that the spinner will point to purple. iv It is equally likely that the spinner will point to red or blue. v Green is twice as likely to occur as blue. b Use the spinner to give an example of: i an impossible event ii a likely event iii a certain event iv two events that are equally likely.
blue
green
red
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8G Describing probability
PROBLEM-SOLVING
5
5, 6
561
6, 7
5 Three spinners are shown below. Match each spinner with the description.
blue
green
red
U N SA C O M R PL R E EC PA T E G D ES
red
red
blue
green
blue
red
Spinner 2
Spinner 1
Spinner 3
a Has an even chance of red, but blue is unlikely. b Blue and green are equally likely, but red is unlikely. c Has an even chance of blue, and green is impossible.
6 Draw spinners to match each of the following descriptions, using blue, red and green as the possible colours. a Blue is likely, red is unlikely and green is impossible. b Red is certain. c Blue has an even chance, red and green are equally likely. d Blue, red and green are all equally likely. e Blue is twice as likely as red, but red and green are equally likely. f Red and green are equally likely and blue is impossible. g Blue, red and green are all unlikely, but no two colours are equally likely. h Blue is three times as likely as green, but red is impossible.
7 For each of the following spinners, give a description of the chances involved so that someone could determine which spinner is being described. Use the colour names and the language of chance (i.e. βlikelyβ, βimpossibleβ etc.) in your descriptions. a
b
red
blue
green
d
c
red
green
green
red
e
blue
green
red
blue
f
red
blue
blue
red
red
blue
green blue
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
REASONING
8
8, 9
8, 9
8 A bag contains 4 blue marbles, 4 red marbles and 2 yellow marbles. A single marble is chosen at random. a Describe two equally likely events for this situation. b Once a marble has been chosen, it is not put back into the bag. i If the first marble chosen was blue, what is the most likely outcome for the next marbleβs colour? Explain your answer. ii If the first marble chosen was red, what is the most likely outcome for the next marbleβs colour? Explain your answer. iii If a yellow marble was chosen first, explain why there are two equally likely outcomes for the next marble.
U N SA C O M R PL R E EC PA T E G D ES
562
9 A coin consists of two sides that are equally likely to occur when tossed. It is matched up with a spinner that has exactly the same chances, as shown below.
red
heads
Tossing the coin with heads landing uppermost is equally likely to spinning red on the spinner. Tossing the coin with tails landing uppermost is equally likely to spinning blue on the spinner. Hence, we say that the coin and the spinner are equivalent. a Draw a spinner that is equivalent to a fair 6-sided die. (Hint: The spinner should have six sections of different colours.) b How can you tell from the spinner you drew that it is equivalent to a fair die? c A die is βweightedβ so that there is an even chance of rolling a 6, but rolling the numbers 1 to 5 are still equally likely. Draw a spinner that is equivalent to such a die. d How could you make a die equivalent to the spinner shown in the diagram? e Describe a spinner that is equivalent to selecting a card from a fair deck of 52 playing cards.
blue
1
6
2
4
3
The game of βTwisterβ uses a spinner to determine the positions to be attempted.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8G Describing probability
ENRICHMENT: Spinner proportions
β
β
563
10
U N SA C O M R PL R E EC PA T E G D ES
10 The language of chance is a bit vague. For example, for each of the following spinners it is βunlikelyβ that you will spin red, but in each case the chance of spinning red is different.
red
green
blue
green
blue
blue
Spinner 1
red
red
Spinner 2
Spinner 3
Rather than describing this in words, we could give the fraction (or decimal or percentage) of the spinner occupied by a colour. a For each of the spinners above, give the fraction of the spinner occupied by red. b What fraction of the spinner would be red if it had an even chance? c Draw spinners for which the red portion occupies: i β100%βof the spinner ii β0%βof the spinner. d For the sentences below, fill in the gaps with appropriate fraction or percentage values. i An event has an even chance of occurring if that portion of the spinner occupies _________ of the total area. ii An event that is impossible occupies _________ of the total area. iii An event is unlikely to occur if it occupies more than _________ but less than _________ of the total area. iv An event is likely if it occupies more than _________ of the total area. e How can the fractions help determine if two events are equally likely? f Explain why all the fractions occupied by a colour must be between β0βand β1β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
The following problems will investigate practical situations drawing upon knowledge and skills developed throughout the chapter. In attempting to solve these problems, aim to identify the key information, use diagrams, formulate ideas, apply strategies, make calculations and check and communicate your solutions.
Using Australian Census data
U N SA C O M R PL R E EC PA T E G D ES
Applications and problem-solving
564
1
The Australian Census occurs every five years, and the Australian Bureau of Statistics (ABS) website shows key statistics, called QuickStats, which provide Census data available for most areas, from small areas to state, territory and national level. The QuickStats search button can be found after clicking the βCensusβ heading on the ABS website. You are interested in using this resource to find various statistics about the people in the area in which you live and about the people in particular regions of Australia. a Using QuickStats on the ABS website, for your local suburb: i find the total number of people who live in your suburb ii find the median age of the people who live in your suburb iii find the average number of people per household in your suburb. b Complete the above questions from part a for the state or territory you live in. c Research the median age for each of Australiaβs six states and two territories and represent this data in an appropriate graph. Which state has the oldest median age? Which state has the youngest median age? d Research a topic of interest, for example: education, language, ancestry, religion, marriage, employment, income, size of family, size of home etc. Compare your suburb, region or state for your topic of interest and present your findings using suitable graphs and make appropriate summary comments.
Head-to-head
2 Many media enterprises like to compare elite sport players via the analysis of particular statistics. You are interested in making a fair comparison between two elite players using statistical data to calculate measures of centre and construct graphs. a Lionel Messi and Cristiano Ronaldo are two exceptional soccer players, but which one is better? The following data compares key statistics for these two players at a particular point in time. Statistic
Messi
Ronaldo
Matches
β784β
β936β
Goals
β637β
β670β
Goal assists
β264β
β215β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Applications and problem-solving
U N SA C O M R PL R E EC PA T E G D ES
For example: Emma Kearney vs Daisy Pearce in AFLW, or Serena Williams vs Steffi Graf in tennis i Determine which statistics are most important and carry out research on your two individual players. ii Present your findings in appropriate graphs, including column, line or pie graphs. iii Include numerical values for mean, median, mode and/or range for key statistics. iv Conclude with your personal opinion, based on the statistics, on the most valuable player.
Applications and problem-solving
i Represent this data in appropriate graphs. ii Calculate Messiβs and Ronaldoβs average number of goals per match and show this information in an appropriate graph. iii Which player do you think is better? Use your graph to justify your answer. b Carry out your own research to compare two exceptional players for your chosen sport.
565
Spin for a spin class
3 Chloe loves to exercise, but she quickly becomes bored with the same exercise routine. To help add variety and fun to her exercise routine she wants to design a spinner that she can spin each day to determine which activity she will do.
Chloe would like to design a spinner that makes it equally likely that she will do a yoga, bike or swim session, but twice as likely that she will do a running session compared to each of the other exercises. a Design a spinner for Chloeβs four possible exercise options.
Chloe is loving the variety and surprise of the exercise spinner and decides to introduce a second spinner related to the duration of her fitness session. She would like a β15βminute, β30βminute and β60β minute session to be equally likely to occur, her favourite β45βminute session to be β1.5βtimes more likely to occur and her longest and hardest β90βminute session to be half as likely to occur. b Design a spinner for Chloeβs five exercise duration times c When Chloe uses both spinners, what are the details of her most likely exercise session? d If Chloe uses the spinners for β100βdays, how many β90βminute sessions might she expect to complete? e If Chloe uses the spinners for βnβdays, how many yoga sessions might she expect to complete?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
8H Theoretical probability in single-step experiments LEARNING INTENTIONS β’ To know the meaning of the terms experiment, trial, outcome, event and sample space β’ To understand that the probability of an event is a number between 0 and 1 inclusive representing the chance it will occur β’ To be able to calculate probabilities of simple events
U N SA C O M R PL R E EC PA T E G D ES
566
The probability of an event occurring is a number between 0 and 1 inclusive. This number states precisely how likely it is for an event to occur. It is often written as a fraction and can indicate how frequently the event would occur over a large number of trials. For example, if you toss a fair coin many times, you would expect heads to come up half the time, so the 1 . If you roll a fair 6-sided die many times, you probability is _ 2 should roll a 4 about one-sixth of the time, so the probability 1. is _ 6
To be more precise, we should list the possible outcomes of rolling the die: 1, 2, 3, 4, 5, 6. Doing this shows us that there is a 1 out of 6 chance that you will roll a 4 and there is a 0 out of 6 (= 0) chance of rolling a 9.
.
.
The study of genetics applies theoretical probability calculations to predict the chance of various traits in offspring. The above eye colour probabilities are one example.
Lesson starter: Spinner probabilities Consider the three spinners shown below.
green
red
β’ β’
blue
green
red
blue
red
β’
red
blue
What is the probability of spinning blue for each of these spinners? What is the probability of spinning red for each of these spinners?
4 and the probability of spinning Try to design a spinner for which the probability of spinning green is _ 7 blue is 0.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8H Theoretical probability in single-step experiments
567
KEY IDEAS β A random experiment is a chance activity which produces varying results. β A trial is a single component of an experiment such as tossing a coin, rolling a die or spinning
U N SA C O M R PL R E EC PA T E G D ES
a spinner. β An outcome is a possible result of the experiment, like rolling a 5 or a coin showing tails. β The sample space is the set of all possible outcomes of an experiment.
β An event is either a single outcome (e.g. rolling a 3) or a collection of outcomes (e.g. rolling a
3, 4 or 5).
β When considering an event, a favourable outcome is one of the outcomes we are interested in. (e.g.
in considering the event of rolling an even number on a die, the favourable outcomes are 2, 4 and 6.)
β The probability of an event is a number between 0 and 1 inclusive, to represent the chance of the
event occurring. The probability of an event occurring, if all the outcomes are equally likely, is: number of outcomes where the event occurs __________________________________ total number of outcomes
β Probability is often written as a fraction, but it can be written as a decimal or as a percentage. β We write Pr(green) to mean βthe probability that the outcome is greenβ.
more likely 1 2
probability:
0
word description:
impossible
1
even chance
certain
BUILDING UNDERSTANDING
1 Match up each experiment a to d with the list of possible outcomes A to D. a tossing a coin A 1, 2, 3, 4, 5, 6 b rolling a die B red, white, blue c selecting a suit from a fair C heads, tails deck of 52 playing cards D hearts, diamonds, d choosing a colour on the clubs, spades
The French ο¬ag is divided into three sections of different colours and equal size.
French flag
2 State the missing words in the following sentences. a The _________ _________ is the set of possible outcomes. b An impossible event has a probability of _________. c If an event has a probability of 1, then it is _________. d The higher its probability, the _________ likely the event will occur. e An event with a probability of _1 has an _________ of occurring. 2
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
Example 13 Listing sample spaces and favourable outcomes A spinner with numbers β1βto β7βis spun. The event being considered is βspinning an odd numberβ. a List the sample space. b List the favourable outcomes.
U N SA C O M R PL R E EC PA T E G D ES
568
SOLUTION
EXPLANATION
a sample space β= {β 1, 2, 3, 4, 5, 6, 7}β
For the sample space, we list all the possible outcomes. Technically, the sample space is {spin βa 1β, spin βa 2β, spin βa 3βetc.} but we do not usually include the additional words.
b favourable outcomes β= β{1, 3, 5, 7}β
Because we are considering the event βspinning an odd numberβ, we list the outcomes from the sample space that are odd.
Now you try
A spinner with numbers β1βto β5βis spun. The event being considered is βspinning an even numberβ. a List the sample space. b List the favourable outcomes.
Example 14 Calculating probabilities
A fair β6β-sided die is rolled. a Find the probability of rolling a β3β, giving your answer as a fraction. b Find the probability of rolling an even number, giving your answer as a decimal. c Find the probability of rolling a number greater than β4β. Give your answer as a percentage correct to one decimal place. SOLUTION
EXPLANATION
a sample space β= {β 1, 2, 3, 4, 5, 6}β favourable outcomes β= β{3}β βPrβ(3)β= _ β1 β 6
The number of possible outcomes is 6. The number of favourable outcomes is β1β. We divide to find the probability.
b sample space β= {β 1, 2, 3, 4, 5, 6}β favourable outcomes β= β{2, 4, 6}β β3 β Prβ(even)β = _ 6 β ββ β β_ β = β1 β 2 β = 0.5
Find the probability as a fraction first, then convert to a decimal.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8H Theoretical probability in single-step experiments
Find the probability as a fraction first, then convert to a percentage and round.
U N SA C O M R PL R E EC PA T E G D ES
c sample space β= {β 1, 2, 3, 4, 5, 6}β favourable outcomes β= β{5, 6}β β2 β Pr (β βgreater than 4β)β = _ 6 ββ =β _ β β β1 ββ 3 β = 33.3% (1 decimal place)
569
Now you try
A spinner has the numbers β1, 2, 3, 4βand β5βon it, each with an equal area. It is spun once. a Find the probability of spinning a β3β, giving your answer as a fraction. b Find the probability of spinning an odd number, giving your answer as a decimal. c Find the probability of spinning an even number, giving your answer as a percentage.
Exercise 8H FLUENCY
Example 13a
Example 13b
Example 13
Example 14
1
1β5, 6ββ(β1/2β)β
2β7
2, 3, 6β8
List the sample space for the following experiments. a A β6β-sided die is rolled. b A spinner with numbers β1βto β4βis spun c A spinner with colours red, green and blue is spun. d A letter is chosen at random from the word LUCK.
2 A β6β-sided die is rolled. List the favourable outcomes for the following events. a An odd number is rolled. b A number less than β4βis rolled. c A number other than β6βis rolled. d A number other than β3βor β5βis rolled.
3 A letter from the word TRACE is chosen at random. a List the sample space. b The event being considered is choosing a vowel (remember that vowels are A, E, I, O, U). List the favourable outcomes. c The event being considered is choosing a consonant (remember that a consonant is any letter that is not a vowel). List the favourable outcomes. d The event being considered is choosing a letter that is also in the word MATHS. List the favourable outcomes. 4 A fair β4β-sided die (numbered β1, 2, 3βand β4β) is rolled. a List the sample space. b Find the probability of rolling a β3β, given your answer as a fraction. c Find the probability of rolling an odd number, given your answer as a decimal. d Find the probability of rolling a number more than β1β, giving your answer as a percentage.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
570
Chapter 8
5 Consider the spinner shown. a How many outcomes are there? List them. b Find βPrβ(red)β; i.e. find the probability of the spinner pointing to red. c Find βPrβ(red or green)ββ. d Find βPrβ(not red)ββ. e Find βPrβ(yellow)ββ.
green
blue
U N SA C O M R PL R E EC PA T E G D ES
Example 14
Statistics and probability
Example 14
6 A spinner with the numbers β1βto β7βis spun. The numbers are evenly spaced. a List the sample space. b Find βPr (β 6)ββ. c Find βPr (β 8)ββ. d Find βPr (β 2 or 4)ββ. e Find βPr (β even)ββ. f Find βPr (β odd)ββ. g Give an example of an event having the probability of β1β.
red
3
2
1
4
7
5
6
7 The letters in the word MATHS are written on β5βcards and then one is drawn from a hat. a List the sample space. b Find βPr (β T)β, giving your answer as a decimal. c Find βPr (β consonant is chosen)β, giving your answer as a decimal. d Find the probability that the letter drawn is also in the word TAME, giving your answer as a percentage.
8 The letters in the word PROBABILITY are written on β11βcards and then one is drawn from a hat. a Find βPr (β P)ββ. b Find βPr (β P or L)ββ. c Find βPr (β letter chosen is in the word BIT)ββ. d Find βPr (β not a B)ββ. e Find βPr (β a vowel is chosen)ββ. f Give an example of an event with the probability of _ β 3 ββ. 11 PROBLEM-SOLVING
9
9, 10
10, 11
9 A bag of marbles contains β3βred marbles, β2βgreen marbles and β5βblue marbles. They are all equal in size and weight. A marble is chosen at random. a What is the probability that a red marble is chosen? (Hint: It is not _ β1 βbecause the colours are not all 3 equally likely.) Give your answer as a percentage. b What is the probability that a blue marble is chosen? Give your answer as a percentage. c What is the probability that a green marble is not chosen? Give your answer as a percentage.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8H Theoretical probability in single-step experiments
9
2 3 4
U N SA C O M R PL R E EC PA T E G D ES
10 Consider the spinner opposite, numbered β2βto β9β. a List the sample space. b Find the probability that a prime number will be spun, giving 8 your answer as a decimal. (Remember that β2βis a prime number.) c Giving your answers as decimals, state the probability of 7 getting a prime number if each number in the spinner opposite is: i increased by β1β ii increased by β2β iii doubled (Hint: It will help if you draw the new spinner.) d Design a new spinner with at least three sectors for which βPrβ(prime)β= 1β.
571
6
5
11 A bag contains various coloured marbles β some are red, some are blue, some are yellow and some are green. You are told that βPrβ (red)β= _ β1 ββ, βPrβ(blue)β= _ β1 βand βPrβ(yellow)β= _ β1 ββ. 2 4 6 You are not told the probability of selecting a green marble. a If there are β24β marbles: i find how many there are of each colour. ii what is the probability of getting a green marble? b If there are β36β marbles: i find how many there are of each colour. ii what is the probability of getting a green marble? c What is the minimum number of marbles in the bag? d Does the probability of getting a green marble depend on the actual number of marbles in the bag? Justify your answer. REASONING
12
12, 13
13, 14
12 Consider an experiment where a spinner has numbers 1β9 which are each equally likely to be spun. a Explain why the events βspinning an even numberβ and βspinning an odd numberβ are not equally likely, by first listing the favourable outcomes in each case. b Find the probability of spinning a number less than β4βand, hence, give an example of an equally likely event. c Find the probability of spinning a number less than β10β. d Give an example of an event that has a probability of _ β2 .ββ 3
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
13 a State the values of the pronumerals in the following table. Event
β rβ(event occurs)β P
β rβ(event does not occurs)β P
Sum of two numbers
Rolling a die, get a β3β
_ β1 β
_ β5 β
βaβ
Tossing a coin, get βHβ
_ β1 β
βbβ
βcβ
Rolling a die, get β2βor β5β
dβ β
_ β2 β
βeβ
Selecting a letter from βHEARTβ, getting a vowel
βfβ
βgβ
βhβ
6
6
2
U N SA C O M R PL R E EC PA T E G D ES
572
3
b If the probability of selecting a vowel in a particular word is _ β 3 β, what is the probability of selecting 13 a consonant? 4 β, what is the probability of spinning c If the probability of spinning blue with a particular spinner is β_ 7 a colour other than blue?
14 A box contains different coloured counters of identical size and shape. A counter is drawn from the box, with βPrβ(purple)β= 10%β, βPrβ(yellow)β= _ β2 βand βPrβ(orange)β= _ β1 ββ. 7 3 a Is it possible to obtain a colour other than purple, yellow or orange? If so, state the probability. b What is the minimum number of counters in the box? c If the box cannot fit more than β1000βcounters, what is the maximum number of counters in the box? ENRICHMENT: Designing spinners
β
β
15
15 a For each of the following, design a spinner using only red, green and blue sectors to obtain the desired probabilities. If it cannot be done, then explain why. i βPrβ(red)β= _ β1 ,β Prβ(green)β= _ β1 β, Prβ(blue)β= _ β1 β 2 4 4 β1 ,β Prβ(green)β= _ β1 β, Prβ(blue)β= _ β1 β ii βPrβ(red)β= _ 2 2 2 β1 ,β Prβ(green)β= _ β1 β, Prβ(blue)β= _ β1 β iii βPrβ(red)β= _ 4 4 4
iv βPrβ(red)β= 0.1, Prβ(green)β= 0.6, Prβ(blue)β= 0.3β
b If βPrβ(red)β= xβand βPrβ(green)β= yβ, write a formula using βxβand βyβto determine what βPrβ(blue)ββmust equal.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8I
Experimental probability in single-step experiments
573
8I Experimental probability in single-step experiments
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand what experimental probability is and how it is related to the (theoretical) probability of an event for a large number of trials β’ To be able to find the expected number of occurrences of an event β’ To be able to find the experimental probability of an event given experiment results
Although the probability of an event tells us how often an event should happen in theory, we will rarely find this being exactly right in practice. For instance, if you flip a coin 100 times, it might come up heads 53 times out of 100, which is 1 of the times you flipped it. Sometimes we will not exactly _ 2 not be able to find the exact probability of an event, but we can carry out an experiment to estimate it.
Lesson starter: Flipping coins
Flipping a fair coin 100 times does not necessarily mean it will come up heads 50 times.
For this experiment, each class member needs a fair coin that they can flip. β’ β’ β’
Each student should flip the coin 20 times and count how many times heads occurs. Tally the total number of heads obtained by the class.
1? How close is this total number to the number you would expect that is based on the probability of _ 2 Discuss what this means.
KEY IDEAS
β The experimental probability of an event occurring based on a particular experiment is
defined as:
number of times the event occurs ______________________________ total number of trials in the experiment
β The expected number of occurrences = probability Γ number of trials.
β If the number of trials is large, then the experimental probability is likely to be close to the
actual probability of an event.
BUILDING UNDERSTANDING
1 A 6-sided die is rolled 10 times and the following numbers come up: 2, 4, 6, 4, 5, 1, 6, 4, 4, 3. a What is the experimental probability of getting a 3? b What is the experimental probability of getting a 4? c What is the experimental probability of getting an odd number?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
574
Chapter 8
Statistics and probability
U N SA C O M R PL R E EC PA T E G D ES
2 When a coin is flipped β100βtimes, the results are β53βheads and β47β tails. a What is the experimental probability of getting a head? b What is the experimental probability of getting a tail? c What is the theoretical probability of getting a tail if the coin is fair?
Example 15 Working with experimental probability
When playing with a spinner with the numbers β1βto β4βon it, the following numbers come up: β1, 4, 1, 3, 3, 1, 4, 3, 2, 3β. a What is the experimental probability of getting a β3β? b What is the experimental probability of getting an even number? c Based on this experiment, how many times would you expect to get a β3βif you spin β1000β times? SOLUTION
EXPLANATION
a _ β2 βor β0.4βor β40%β 5
number of 3s β= _ ______________ ββ―β― β―β― β2 β β 4 β= _
3 βor 0.3 or 30% b β_ 10
times with even result β= β_ 3β ____________________________ β―β―β― β―β― βnumber of 10 number of trials
c β400β times
2 βΓ 1000 = 400β βprobability Γ number trials = β_ 5
number of trials
10
5
Now you try
When rolling a die, the following numbers come up: β1, 4, 3, 6, 6, 2, 4, 3β. a What is the experimental probability of rolling a β3β? b What is the experimental probability of rolling an even number? c Based on this experiment, how many times would you expect to get a β3βif you rolled the die β80β times?
Exercise 8I FLUENCY
Example 15
1
1β6
1β7
3β7
A spinner with the numbers β1βto β5βon it is spun and the following numbers come up: β 2, 5, 5, 3, 4, 4, 1, 2, 3, 5, 3, 3β a What is the experimental probability of getting a β5β? b What is the experimental probability of getting an odd number? c Based on this experiment, how many times would you expect to get an odd number if you spin β150β times?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8I
Experimental probability in single-step experiments
575
2 A survey is conducted on peopleβs television viewing habits. Number of hours per week
β0β4β
β5β9β
β10β19β
β20β29β
β30+β
Number of people
β20β
1β 0β
β15β
β5β
β0β
a b c d
U N SA C O M R PL R E EC PA T E G D ES
How many people participated in the survey? What is the probability that a randomly selected participant watches less than β5βhours of television? What is the probability that a randomly selected participant watches β20βββ29βhours of television? What is the probability that a randomly selected participant watches between β5βand β29βhours of television? e Based on this survey, the experimental probability of watching β30+βhours of television is β0β. Does this mean that watching β30+βhours is impossible?
Example 15
Example 15
3 A fair coin is flipped. a How many times would you expect it to show tails in β 1000β trials? b How many times would you expect it to show heads in β 3500β trials? c Initially, you flip the coin β10βtimes to find the probability of the coin showing tails. i Explain how you could get an experimental probability of β0.7β. ii If you flip the coin β100βtimes, are you more or less likely to get an experimental probability close to β0.5β?
4 A fair β6β-sided die is rolled. a How many times would you expect to get a β3βin β600β trials? b How many times would you expect to get an even number in β600β trials? c If you roll the die β600βtimes, is it possible that you will get an even number β400β times? d Are you more likely to obtain an experimental probability of β100%βfrom two throws or to obtain an experimental probability of β100%βfrom β10β throws? 5 Each time a basketball player takes a free throw there is a β4βin β6βchance that the shot will go in. This can be simulated by rolling a β6β-sided die and using numbers β1βto β4βto represent βshot goes inβ and numbers β5βand β6βto represent βshot missesβ. a Use a β6β-sided die over β10βtrials to find the experimental probability that the shot goes in. b Use a β6β-sided die over β50βtrials to find the experimental probability that the shot goes in. c Working with a group, use a β6β-sided die over β100βtrials to find the experimental probability that the shot goes in. d Use a β6β-sided die over just one trial to find the experimental probability that the shot goes in. (Your answer should be either β0βor β1β.) e Which of the answers to parts a to d above is closest to the theoretical probability of β66.67%β? Justify your answer.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
6 The colour of the cars in a school car park is recorded. Colour
Red
Black
White
Blue
Number of cars
β21β
β24β
β25β
β20β
Purple Green β3β
β7β
Based on this sample: a What is the probability that a randomly chosen car is white? b What is the probability that a randomly chosen car is purple? c What is the probability that a randomly chosen car is green or black? d How many purple cars would you expect to see in a shopping centre car park with β2000β cars?
U N SA C O M R PL R E EC PA T E G D ES
576
7 The number of children in some families Number of children β0β 1β β β2β β3β β4β is recorded in the table shown. Number of families β5β 2β 0β β32β β10β β3β a How many families have no children? b How many families have an even number of children? c How many families participated in the survey? d Based on this experiment, what is the probability that a randomly selected family has β1βor β2β children? e Based on this experiment, what is the probability that a randomly selected family has an even number of children? f What is the total number of children considered in this survey? PROBLEM-SOLVING
8
8, 9
8, 9
8 A handful of β10βmarbles of different colours is placed into a bag. A marble is selected at random, its colour recorded and then returned to the bag. The results are shown below. Red marble chosen
Green marble chosen
Blue marble chosen
β21β
β32β
β47β
a Based on this experiment, how many marbles of each colour do you think there are? Justify your answer in a sentence. b For each of the following, state whether or not they are possible colours for the β10β marbles. i β3βred, β3βgreen, β4β blue ii β2βred, β4βgreen, β4β blue iii β1βred, β3βgreen, β6β blue v β2βred, β0βgreen, β8β blue
iv 2β βred, β3βgreen, β4βblue, β1β purple
9 Match each of the experiment results a to d with the most likely spinner that was used (A to D). A B blue green red red
C
red
blue
green
Red
Green
Blue
a
β18β
β52β
β30β
b
β27β
β23β
0β β
c
β20β
β23β
β27β
d
β47β
β0β
β53β
blue
D
red
green
green red
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8I
REASONING
Experimental probability in single-step experiments
10
10
577
10, 11
U N SA C O M R PL R E EC PA T E G D ES
10 Thomas and his dog Loui have a trick where Thomas throws a ball and Loui catches it with his mouth. Assume that there is a β50%β chance that the trick will be successful on any occasion, and they will try four times to see how many catches are successful. This can be simulated with four coin flips (use heads to represent a successful catch). a Count the number of successful catches in each group of four coin flips and write your β20βresults down. b Copy and complete a table like the one below to summarise your results. Number of catches
β0β 1β β β2β β3β 4β β
Frequency
Total β20β
c Based on your simulation, what is the experimental probability that there will be just one successful catch? d Based on your simulation, what is the experimental probability that there will be four catches? e Explain why you might need to use simulations and experimental probabilities to find the answer to parts c and d above. f If you had repeated the experiment only β5βtimes instead of β20βtimes, how might the accuracy of your probabilities be affected? g If you had repeated the experiment β500βtimes instead of β20βtimes, how might the accuracy of your probabilities be affected?
11 Classify the following statements as true or false. Justify each answer in a sentence. a If the probability of an event is _ β1 ,β then it must have an experimental probability of _ β1 ββ. 2 2 1 _ β1 .ββ b If the experimental probability of an event is β β, then its theoretical probability must be _ 2 2 c If the experimental probability of an event is β0β, then the theoretical probability is β0β. d If the probability of an event is β0β, then the experimental probability is also β0β. e If the experimental probability is β1β, then the theoretical probability is β1β. f If the probability of an event is β1β, then the experimental probability is β1β. ENRICHMENT: Improving estimates
β
β
12
12 A spinner is spun β500βtimes. The table opposite Red Green Blue shows the tally for every β100β trials. First set of β100β trials β22β β41β β37β a Give the best possible estimate for βPrβ(red)ββ, β Second set of β100β trials β21β β41β β38β Prβ(green)βand βPrβ(blue)βbased on these trials. Third set of β100β trials β27β β39β β34β b If your estimate is based on just one set of Fourth set of β100β trials β25β 4β 6β β29β trials, which one would cause you to have the Fifth set of β100β trials β30β 4β 4β 2β 6β most inaccurate results? c Design a spinner that could give results similar to those in the table. Assume you can use up to β10βsectors of equal size. d Design a spinner that could give results similar to those in the table if you are allowed to use sectors of different sizes.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
578
Chapter 8
Statistics and probability
Modelling
Water restrictions A dam near the town of Waterville has received limited flows from its catchment area in recent times. As a percentage of its full capacity, the volume of water in the dam at the start of each of the last β12βmonths is given in the following table. Jan
Feb
Mar
Apr
May
Jun
July
Aug
Sep
Oct
Nov
Dec
Water volume for month (%)
β75β
7β 2β
β68β
6β 5β
6β 3β
β60β
β59β
β56β
β52β
β49β
β47β
4β 4β
U N SA C O M R PL R E EC PA T E G D ES
Month
The water restrictions policy for Waterville include the following statements. β’ Once the volume of water decreases to β60%βof full capacity, level β1βrestrictions apply. β’ Once the volume of water decreases to β50%βof full capacity, level β2βrestrictions apply. β’ Once the volume of water decreases to β35%βof full capacity, level β3βrestrictions apply.
Present a report for the following tasks and ensure that you show clear mathematical workings and explanations where appropriate.
Preliminary task
a By considering the water volume for the Waterville dam for each of the given β12βmonths, find: i the mean (round to one decimal place) ii the median iii the range. b Use the given data to construct a line graph for the water volume of the dam near Waterville. Use the following axes on your graph. β’ Time (in months) on the horizontal axis β’ Water volume (β %)βon the vertical axis
Modelling task
Formulate Solve
a The problem is to estimate how many months it will be until level β3βrestrictions apply, based on the current trends. Write down all the relevant information that will help solve this problem. b Describe the general trend/pattern that you see in the data. c Find the volume decrease as a percentage of the full capacity from: i the month of January to the month of July ii the month of May to the month of December. d Consider the decrease in water volume from one month to the next. i State the β11βdifferent decreases across the β12β months. ii Find the mean percentage decrease using your data from part d i above. Round your answer to one decimal place. e If the trend continued, estimate the water volume (as a percentage of the full capacity) in: i January of the following year ii December of the following year.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Modelling
Evaluate and verify
f
U N SA C O M R PL R E EC PA T E G D ES
Using the data from above, determine the month when the following were applied: i level β1β restrictions ii level β2β restrictions. g Estimate when level β3βwater restrictions will apply. Justify your answer with the use of the mean percentage decrease from part d and your graph. h Summarise your results and describe any key findings.
Modelling
Communicate
579
Extension questions
a Determine the mean monthly percentage increase required after December of the given year for the dam to be at β50%βcapacity in β12βmonthsβ time. b Determine the mean monthly percentage increase required after December of the given year for the dam to be back at β100%βcapacity in β12βmonthsβ time. c Describe some reasons why it might be unreasonable to expect the volume of water in the dam to increase or decrease by a fixed amount each month.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
580
Chapter 8
Statistics and probability
Key technology: Spreadsheets Unfortunately for many sporting clubs, matches are sometimes cancelled due to rain. Frances, a keen soccer player, is interested in the probability of a certain number of matches being cancelled across a β10β-match season. To be eligible for state-wide selection, she needs to have played in at least β7βnon-cancelled matches, meaning that no more than β3βmatches can be cancelled.
U N SA C O M R PL R E EC PA T E G D ES
Technology and computational thinking
Soccer in the rain
1 Getting started
Information provided by Francesβs sporting club shows that from the past β100βplanned matches, β20β of them were cancelled due to rain. a Based on the data provided by the club, what is the experimental probability that any particular match will be cancelled? b Out of a β10β-match season, what is the expected number of matches that will be cancelled due to rain? c Do you think that in her next β10β-match season, Frances has a good chance of playing at least β7βmatches? Give a reason.
2 Using technology
We will use Excel to simulate the outcomes of Francesβs β10β-match soccer season. a First, use the RANDBETWEEN function as shown to generate a set of random integers between β1βand β10β inclusive. b Given that the probability of a wet day is β0.2β, we will assign the following: β’ β1βor β2βmeans that it is wet and the match is cancelled β’ β3βto β10βmeans that it is dry and the match is not cancelled.
Press the βF9βkey to recalculate your random number. Keep doing this until you record at least two wet days (β1βor β2β).
c Upgrade your spreadsheet so it simulates a full β10β-match season in one go as shown. Column B will confirm if the day is wet or dry and row β14βcounts the total number of wet days in the simulated season. d Fill down at cells βA4βand βB4βto row β13β. This should produce results for a β10β-match season.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Technology and computational thinking
581
3 Applying an algorithm
U N SA C O M R PL R E EC PA T E G D ES
Technology and computational thinking
a Press the βF9βkey to recalculate your random numbers in your β10β-match simulation spreadsheet. b Keep a record of how many wet days there are for each β10β-match season. Use the following table to record your results. If, for example, your next simulation results in β3βwet days, then add a dash in the tally under ββ3ββ. Repeat for a total for β20β simulations. Number of wet days
β0β
β1β
β2β
β3β
β4β
β5β
β6β
β7β
β8β
β9β
β10β
Tally
Frequency
c Out of your β20βsimulations, what is the frequency corresponding to: i β0βwet days? ii β1βwet day? iii β2βwet days? iv β3βwet days? v no more than β3βwet days? d Based on these results, what do you think Francesβs chances are of playing in at least 7 matches? Give reasons.
4 Extension
a To improve your simulation results, increase the total number of simulations completed using your spreadsheet. Update the tally/frequency table to include a total of one hundred β10β-match season results. b Using your new results, what is the frequency of no more than β3βwet days? c Use your result to calculate the probability that Frances will play in at least β7β matches. d Compare your result from part c with the theoretical result of β0.88β, correct to two decimal places.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
582
Chapter 8
Statistics and probability
In the game of Monopoly, two β6β-sided dice are rolled to work out how far a player should go forward. For this investigation, you will need two β6β-sided dice or a random number simulator that simulates numbers between β1βand β6β. a Roll the two dice and note what they add up to. Repeat this β100βtimes and complete this table.
U N SA C O M R PL R E EC PA T E G D ES
Investigation
Monopoly risk
Dice sum Tally
β2β
β3β
β4β
β5β
β6β
β7β
8β β
β9β
1β 0β
1β 1β
β12β
Total
β100β
b Represent the results in a column graph. Describe the shape of the graph. Do you notice any patterns? c Use the results of your experiment to give the experimental probability of two dice adding to: i β3β ii β6β iii β8β iv β12β v β15β. d What is the most likely sum for the dice to add to, based on your experiment? Is this the mean, median or mode that you are describing? e If the average Monopoly game involves β180βrolls, find the expected number of times, based on your experiment, that the dice will add to: i β3β ii β6β iii β8β iv β12β v β15β. f Why do you think that certain sums happen more often than others? Explain why this might happen by comparing the number of times the dice add to β2βand the number of times they add to β8β. g What is the mean dice sum of the β100βtrials you conducted above?
To conduct many experiments, a spreadsheet can be used. For example, the spreadsheet below can be used to simulate rolling three β6β-sided dice. Drag down the cells from the second row to row β1000β to run the experiment β1000β times.
h Investigate what the most likely dice sums are when you roll more than two dice. You should use a spreadsheet like the one above to find the most likely values. (Note: Instead of using the MODE Βfunction to help you, you can also use the AVERAGE function.)
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Problems and challenges
Problems and challenges
Six numbers are listed in ascending Up for a challenge? If you get stuck order and some are removed. The on a question, check out the βWorking with unfamiliar problemsβ poster at mean and median are both β6β, the the end of the book to help you. mode is β2βand the range is β10β. Fill in the missing numbers. β? , ? , 5, ? , ? , ?β
U N SA C O M R PL R E EC PA T E G D ES
1
583
2 A survey is conducted at a school and the results are presented as a pie chart. Find the minimum number of people who participated in the survey if the smallest sector has an angle of: a 9β 0Β°β b β36Β°β c β92Β°β
d β35Β°β
3 In a class of β20βstudents, a poll was taken of the number of cars owned by each family. The median number of cars owned is β1.5βand the mean number is β1.4βcars. Complete the table of the results, shown below. Number of cars
β0β
Number of students
β4β
β1β
β2β
β3β
4 Each of the β8βletters of a word is written on a separate card. Given the following probabilities, what is the word? βPr (letter P) = Pr (letter R) = 12.5%, Pr (letter B) = _ β1 β, Pr (vowel ) = 0.375β 4
5 Frank the fisherman enjoys beach fishing on Fraser Island, Qld. flathead, 12 One year he kept a count of the fish types that he caught and displayed these numbers as a pie chart in his fish shop. Calculate tailor, 75 the answers to these questions, showing all steps. dart, 160 golden a What angle did Frank use to represent his whiting catch? trevally, 3 whiting, b Find the difference between the smallest and largest angles in 50 this pie chart. c What is the probability of Frank catching a tailor? d Find the probability that Frank does not catch a golden trevally. e What is the probability of Frank catching a flathead or golden trevally? 6 A circular spinner is made using the colours red, green, purple and yellow in four sectors with two sectors being equal in size. The spinner is spun β120βtimes and the results obtained are shown in the table below. Sector colour
Red
Green
Purple
Yellow
Frequency
β40β
β32β
β19β
β29β
Design a spinner that is likely to give these results, labelling the sector colours and angles. Explain the mathematical reason for your answer and show relevant calculations.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
Data types β’ Numerical e.g. number of people (discrete), height of trees (continuous) β’ Categorical e.g. colours, drink sizes (S, M, L)
Chapter summary
584
U N SA C O M R PL R E EC PA T E G D ES
Data collection β’ Primary source (firsthand) or secondary source (collected by someone else) β’ Census (whole population) or sample (selection)
Graphical representations
Height (cm)
Column graphs Height chart
title
Line graphs
Dot plots
Child
Time is often displayed on the horizontal axis.
axis labels
Pie charts (Ext)
proportion =
Stem-and-leaf plots
Divided bar graphs (Ext)
width of bar = proportion Γ total width
number total
basketball
tennis squash
Stem
hockey
Leaf
2 3 6 7
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
β 23, 26, 27
angle = 360Β° Γ proportion
Statistics and probability
Summarising data numerically
range = 10 β 1 = 9 1, 2, 2, 3, 4, 4, 4, 5, 5, 6, 7, 8, 8, 9, 10
Experimental probability
Use an experiment or survey or simulation to estimate probability. e.g. Spinner lands on blue 47 times out of 120 47 Experimental probability = 120
median = 5 mode = 4 (most common value) (middle value) sum of values mean = number of values 78 = = 5.2 15
Outcome: possible result of an experiment Event: either a single outcome or a collection of outcomes
Experiment/trial: e.g. roll a fair die Sample space: {1, 2, 3, 4, 5, 6} Pr(roll a 5) = 16 Pr(roll odd number) = 36 = 12
Probability: how likely an event is unlikely 1 likely 0 1 2
impossible even chance more likely
certain
Expected number is Pr(event) Γ number of trials e.g. Flip coin 100 times, expected number of heads
Sample space: e.g. red {red, green, blue} green 1 blue Pr(spin red) = 3 Pr(donβt spin blue) = 23
= 12 Γ 100 = 50
e.g. Roll die 36 times, expected number of 5s 1
= 6 Γ 36 = 6
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter checklist
585
Chapter checklist with success criteria 1. I can classify variables. e.g. Classify the length of a newborn baby as categorical, discrete numerical or continuous numerical.
U N SA C O M R PL R E EC PA T E G D ES
8A
β
Chapter checklist
A printable version of this checklist is available in the Interactive Textbook
8A
2. I can choose a suitable method for collecting data. e.g. Decide whether a primary or secondary source is suitable for an economist collecting data on the average income of Australian households, and suggest a method for its collection.
8B
3. I can find the range for a set of data. e.g. Find the range of β15, 31, 12, 47, 21, 65, 12β.
8B
4. I can find the mean for a set of data. e.g. Find the mean of β15, 31, 12, 47, 21, 65, 12β.
8B
5. I can find the median for a set of data (including an even number of values). e.g. Find the median of β15, 31, 12, 47, 21, 65, 12, 29β.
8B
6. I can find the mode for a set of data. e.g. Find the mode of β15, 31, 12, 47, 21, 65, 12β.
8C
7. I can interpret a column graph and a dot plot. e.g. Use the dot plot to state how many children have β2βpets, and identify any outliers.
Pets at home survey
0 1 2 3 4 5 6 7 8 Number of pets
8C
8D
8. I can construct a column graph and a dot plot from a set of data. e.g. Construct a column graph to represent the heights of five people: Tim (β 150 cm)ββ, Phil β(120 cm)β, Jess (β 140 cm)β, Don (β 100 cm)β, Nyree (β 130 cm)ββ.
9. I can draw a line graph. e.g. Draw a line graph to represent the temperature in a room at the following times: 9 a.m. (β β10Β° Cβ)β, 10 a.m. (β β15Β° Cβ)β, 11 a.m. (β β20Β° Cβ)β, 12 p.m. (β β23Β° Cβ)β, 1 p.m. (β β18Β° Cβ)ββ.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 8
Statistics and probability
β 8D
10. I can interpret a travel graph. e.g. Describe what is happening in the travel graph shown, including stating when the cyclist is travelling fastest and when they are stationary.
Distance cycled over 5 hours
Distance (km)
U N SA C O M R PL R E EC PA T E G D ES
Chapter checklist
586
30 25 20 15 10 5 0
8E
11. I can interpret a stem-and-leaf plot. e.g. Average daily temperatures (in Β°C) are shown in the stem-and-leaf plot. Write the temperatures as a list and state the median.
1 2 3 4 5 Time (hours)
Stem Leaf
β1β β3 6 6β
β2β β0 0 1 2 5 5 6 8 9β 3β β β0 2β
β2 | 5βmeans 25Β°C
8E
12. I can create a stem-and-leaf plot. e.g. Represent this set of data as a stem-and-leaf plot: β 23, 10, 36, 25, 31, 34, 34, 27, 36, 37, 16, 33β.
8F
13. I can draw a sector graph (pie chart) with correct sector sizes. e.g. Draw a pie chart to represent the following allocation of a 10-hour day: Television (β1βhour), Internet (β2βhours), Sport (β4βhours), Homework (β3β hours).
8F
14. I can draw a divided bar graph with correct rectangle widths. e.g. Draw a divided bar graph to represent the following allocation of a β10β-hour day: Television (β1βhour), Internet (β2βhours), Sport (β4βhours), Homework (β3β hours).
8G
15. I can describe the chance of events using standard English phrases. e.g. Decide whether rolling a β3βon a β6β-sided die and getting heads on a coin are equally likely.
8H
8H
8I
8I
Ext
Ext
16. I can list the sample space of an experiment and the favourable outcomes of an event. e.g. If a fair β6β-sided die is rolled, list the sample space. List the favourable outcomes for the event βroll an odd numberβ.
17. I can calculate the probability of a simple event, giving an answer as a fraction, decimal or percentage. e.g. If a fair β6β-sided die is rolled, find the probability of getting a number less than β3β. Answer as a percentage rounded to one decimal place.
18. I can find the expected number of occurrences of an event. e.g. If an event has probability β0.4β, find the expected number of times the event would occur in β1000β trials.
19. I can calculate the experimental probability of an event. e.g. When playing with a spinner the following numbers come up: β1, 4, 1, 3, 3, 1, 4, 3, 2, 3β. Find the experimental probability of getting an even number.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
587
Short-answer questions Draw a column graph to represent the following peopleβs ages. Name
Sven
Dane
Kelly
Hugo
Frankie
Age (years)
β20β
β12β
β15β
2β 2β
β25β
U N SA C O M R PL R E EC PA T E G D ES
1
Chapter review
8C
8D
8F
Ext
2 A Year β7βgroup was asked how many hours of television they watch in a week. The results are given in the table. a How many students participated in the survey? b What is the total number of hours of television watched? c Find the mean number of hours of television watched. d Show this information in a column graph.
TV watched (hours)
No. of students
β8β
β5β
β9β
β8β
β10β
β14β
β11β
β8β
β12β
β5β
3 The number of students in the library is recorded hourly, as displayed in the graph. a How many students entered the library when it first opened? b How many students were in the library at β8β hours after opening? c If the library opens at β9βa.m, at what time are there the most number of students in the library? d How many students were in the library at β4β p.m? 4 1β 20βpeople were asked to nominate their favourite take-away food from the list: chicken, pizza, hamburgers, Chinese. The results are given in the table.
a If you want to show the data in a pie chart, state the angle needed to represent Chinese food. b What percentage of people prefer hamburgers? c Represent the results in a pie chart.
Number of students
8C
14 12 10 8 6 4 2 0
Visiting in the library
0 1 2 3 4 5 6 7 8 Hours after library opening
Food
Frequency
Chicken
β15β
Pizza
β40β
Hamburgers
β30β
Chinese
β35β
8B
5 Consider the data β1, 4, 2, 7, 3, 2, 9, 12β. State the: a range b mean c median
d mode.
8B
6 Consider the data β0, 4, 2, 9, 3, 7, 3, 12β. State the: a range b mean c median
d mode.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
588
Chapter 8
7 The stem-and-leaf plot shows the price of various items in a shop. a How many items cost more than β$20β? b What is the cost of the most expensive item? c State the range of the prices. d If a customer had β$20βto spend, how much money would they have left after buying the most expensive item they can afford?
Stem Leaf β0β β1 1 4 5 8β β1β β0 2 2 2 3 4 7β 2β β β4 5 5β 3β β β1β
U N SA C O M R PL R E EC PA T E G D ES
Chapter review
8E
Statistics and probability
8G
8H
8H
8H
8I
8 For each of the following descriptions, choose the probability from the set β0, _ β1 β, _ β3 β, 1, _ β19 βthat matches best. 8 4 20 a certain b highly unlikely c highly likely d likely e impossible
9 List the sample space for each of the following experiments. a A fair β6β-sided die is rolled. b A fair coin is tossed. c A letter is chosen from the word DESIGN. d Spinning the spinner shown opposite.
10 Vin spins a spinner with nine equal sectors, which are numbered β 1βto β9β. a How many outcomes are there? b Find the probability of spinning: i an odd number ii a multiple of β3β iii a number greater than β10β iv a prime number less than β6β v a factor of β8β vi a factor of β100.β
β1 | 4βmeans β$14β
blue
yellow
green
11 One card is chosen at random from a standard deck of β52βplaying cards. Find the probability of drawing: a a red king b a king or queen c a jack of diamonds d a picture card (i.e. king, queen or jack). 12 A coin is flipped β100βtimes, resulting in β42βheads and β58β tails. a What is the experimental probability of getting heads? Give your answer as a percentage. b What is the actual probability of getting heads if the coin is fair? Give your answer as a percentage.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
red green
Chapter review
13 Consider the spinner shown. a State the probability that the spinner lands in the green section. b State the probability that the spinner lands in the blue section. c Tanya spins the spinner β100βtimes. What is the expected number of times it would land in the red section? d She spins the spinner β500βtimes. What is the expected Βnumber of times it would land in the green section?
U N SA C O M R PL R E EC PA T E G D ES
8H
589
blue
Multiple-choice questions
8C
1
In the column graph shown, the highest income is earned by: A Michael B Alice C Dan D Laura
E Victoria
ic
to
ria
s
le
V
Ch
ar
an
D
e
a
lic
A
ur
La
M
ic
ha
el
Income ( Γ $1000)
Annual income
90 80 70 60 50 40 30 20 10 0
Name
Questions 2 and 3 relate to the following information. The results of a survey are shown below.
8F
Ext
8I
Instrument learned
piano
violin
drums
guitar
Number of students
β10β
β2β
β5β
β3β
2 If the results above are presented as a sector graph, then the angle occupied by the drums sector is: A β360Β°β B β180Β°β C β120Β°β D 9β 0Β°β E β45Β°β
3 Based on the survey, the experimental probability that a randomly selected person learns the guitar is: 1β C β3β A β_ B _ β1 β D _ β3 β E _ β3 β 4 2 20 5
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
590
Chapter 8
4 Which one of the following variables is continuous numerical? A the Australian State or Territory in which babies are born B the number of babies born in a given year C the number of hairs on a babyβs head D the weight (in kg) of newborn babies E the length (in number of letters) of a babyβs first name
U N SA C O M R PL R E EC PA T E G D ES
Chapter review
8A
Statistics and probability
Questions 5 and 6 relate to the following information.
In a class of β20βstudents, the number of days each student was absent over a β10β-week period is recorded. β1, 0, 1, 2, 2, 3, 2, 4, 3, 0, 1, 1, 2, 3, 3, 3, 2, 2, 2, 2β
8B
5 The mode is: A 0β β
8B
8B
8H
8H
8I
B β1β
C β2β
D β3β
E β4β
6 The mean number of days a student was absent is: A 1β β B β2β C β1.95β
D β3β
E β39β
7 The range of the numbers β1, 5, 3, 9, 12, 41, 12β is: A 4β 0β B β41β C β12β
D β3β
E β1β
8 Which of the following events has the same probability as rolling an odd number on a fair β6β-sided die? A rolling a number greater than 4 on a fair β6β-sided die B choosing a vowel from the word CAT C tossing a fair coin and getting heads D choosing the letter T from the word TOE E spinning an odd number on a spinner numbered β1βto β7β
9 Each letter of the word APPLE is written separately on five cards. One card is then chosen at random. βPr(letter P)β is: A β0β B 0β .2β C β0.4β D β0.5β E β1β
10 A fair β6β-sided die is rolled β600βtimes. The expected number of times that the number rolled is either a β1βor a β2β is: A β100β B 2β 00β C β300β D β400β E β500β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
591
Extended-response questions The number of rainy days experienced throughout a year in a certain town is displayed below. Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
No. of rainy days
β10β
β11β
β3β
β7β
β2β
β0β
β1β
β5β
6β β
β9β
β7β
β5β
U N SA C O M R PL R E EC PA T E G D ES
a Show this information in a column graph. b For how many days of the year did it rain in this town? c What is the probability that it will rain in any day during winter (i.e. during June, July and August)? d What type of variable (e.g. continuous numerical) is the number of rainy days? e What type of variable is the month? f What other type of graph could be used to present this data?
Chapter review
1
2 At a school camp, a survey was conducted to establish each studentβs favourite dessert. Ice-cream
Yoghurt
Danish pastry
Jelly
Pudding
Cheesecake
β10β
β5β
2β β
β7β
4β β
β12β
a How many students participated in the survey? b What is the most popular dessert selected? c What is the probability that a randomly selected student chooses jelly as their favourite dessert? d For each of the following methods listed below, state whether it would be a reasonable way of presenting the surveyβs results. i column graph ii line graph iii sector graph (pie chart) iv divided bar graph e If the campers attend a school with β800βstudents, how many students from the entire school would you expect to choose pudding as their preferred dessert?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
U N SA C O M R PL R E EC PA T E G D ES
9 Equations
Maths in context: Equations are solved every day in many occupations
Solving equations is an essential skill used in solving problems in a vast number of occupations, including business, agriculture, construction, engineering, ο¬nancial services, food production, manufacturing, technology and the trades. It is important for the managers of both the large stores and small independent retailers, such as cafes, fashion boutiques, electronics shops and shoe shops, to keep track of cash ο¬ow and ensure a proο¬t. Financial calculations involve using formulas and solving equations to calculate measures such as the number of sales required to break-even (no proο¬t and no loss), and overall proο¬t as a percentage of sales.
Construction workers, such as engineers, builders, concreters, plumbers and roofers, solve equations to ο¬nd the time a job will take, the cost of materials and possible proο¬t. Service businesses who have clients, such as dog groomers, ο¬tness and dance instructors, hairdressers, hotel owners, optometrists and physiotherapists must all manage their ο¬nances. Equations are solved to ο¬nd the number of clients needed to make a certain proο¬t per week.
Electronic engineers and electricians require skills in algebra to apply formulas and solve equations related to the power supply in buildings, vehicles, electronic components, appliances and various devices.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter contents Introduction to equations Solving equations by inspection Equivalent equations Solving equations algebraically Equations with fractions (EXTENDING) Equations with brackets (EXTENDING) Using formulas Using equations to solve problems
U N SA C O M R PL R E EC PA T E G D ES
9A 9B 9C 9D 9E 9F 9G 9H
Victorian Curriculum 2.0
This chapter covers the following content descriptors in the Victorian Curriculum 2.0: ALGEBRA
VC2M7A01, VC2M7A03, VC2M7A06
Please refer to the curriculum support documentation in the teacher resources for a full and comprehensive mapping of this chapter to the related curriculum content descriptors.
Β© VCAA
Online resources
A host of additional online resources are included as part of your Interactive Textbook, including HOTmaths content, video demonstrations of all worked examples, auto-marked quizzes and much more.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 9
Equations
9A Introduction to equations LEARNING INTENTIONS β’ To understand the difference between an equation and an expression β’ To be able to determine if an equation is true or false, substituting for pronumerals if required β’ To be able to write an equation given an English description
U N SA C O M R PL R E EC PA T E G D ES
594
An equation is a mathematical statement stating that two expressions have the same value. It consists of two expressions that are separated by an equals sign (=).
Sample equations include:
3+3 = 6 30 = 2 Γ 15 100 β 30 = 60 + 10
which are all true equations.
An equation does not have to be true. For instance, 2 + 2 = 17 and 5 = 3 β 1 and 10 + 15 = 12 + 3 are all false equations.
Many mathematical equations need to be solved to build and launch space stations into orbit.
If an equation contains pronumerals, one cannot generally tell whether the equation is true or false until values are substituted for the pronumerals. For example, 5 + x = 7 could be true (if x is 2) or it could be false (if x is 15).
Lesson starter: Equations: True or false?
Rearrange the following five symbols to make as many different equations as possible. 5, 2, 3, +, =
β’ β’
Which of them are true? Which are false? Is it always possible to rearrange a set of numbers and operations to make true equations?
KEY IDEAS
β An expression is a collection of pronumerals, numbers and operators without an equals sign
(e.g. 2x + 3).
β An equation is a mathematical statement stating that two expressions are equal
(e.g. 2x + 3 = 4y β 2).
β Equations have a left-hand side (LHS), a right-hand side (RHS) and an equals sign in between.
2x + 3 = 4y β 2 β LHS β RHS
β’
The equals sign indicates that the LHS and RHS have the same numerical value.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9A Introduction to equations
595
β Equations are mathematical statements that can be true (e.g. β2 + 3 = 5β) or false
(e.g. β2 + 3 = 7β).
β If a pronumeral is included in an equation, you need to know the value to substitute before
U N SA C O M R PL R E EC PA T E G D ES
deciding whether the equation is true. For example, β3x = 12βwould be true if β4βis substituted for βxβ, but it would be false if β10βis substituted.
BUILDING UNDERSTANDING
1 Classify each of these equations as true or false. a β2 + 3 = 5β b β3 + 2 = 6β c β5 β 1 = 6β
2 If βx = 2β, is β10 + x = 12βtrue or false?
3 Consider the equation β4 + 3x = 2x + 9β. a If βx = 5β, state the value of the left-hand side (LHS). b If βx = 5β, state the value of the right-hand side (RHS). c Is the equation β4 + 3x = 2x + 9βtrue or false when βx = 5β?
Example 1
Identifying equations
Which of the following are equations? a β3 + 5 = 8β b β7 + 7 = 18β
c β2 + 12β
d β4 = 12 β xβ
e 3β + uβ
SOLUTION
EXPLANATION
a β3 + 5 = 8βis an equation.
There are two expressions (i.e. β3 + 5βand β8β) separated by an equals sign.
b β7 + 7 = 18βis an equation.
There are two expressions separated by an equals sign. Although this equation is false, it is still an equation.
c 2β + 12βis not an equation.
This is just a single expression. There is no equals sign.
d β4 = 12 β xβis an equation.
There are two expressions separated by an equals sign.
e 3β + uβis not an equation.
There is no equals sign, so this is not an equation.
Now you try
Decide whether the following are equations (β E)βor not (β N)ββ. a β3 + 7β b β12 = 4 + 8β c β10 + 3 = 20β
d β2 β 3xβ
e 5β + q = 12β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 9
Equations
Example 2
Classifying equations
For each of the following equations, state whether it is true or false. a 7β + 5 = 12β b β5 + 3 = 2 Γ 4β ( ) c β12 Γ β 2 β 1 β= 14 + 5β d β3 + 9x = 60 + 6β, if βx = 7β e β10 + b = 3b + 1β, if βb = 4β f β3 + 2x = 21 β yβ, if βx = 5βand βy = 8β
U N SA C O M R PL R E EC PA T E G D ES
596
SOLUTION
EXPLANATION
a true
The left-hand side (LHS) and right-hand side (RHS) are both equal to β12β, so the equation is true.
b true
βLHS = 5 + 3 = 8βand βRHS = 2 Γ 4 = 8β, so both sides are equal.
c false
βLHS = 12βand βRHS = 19β, so the equation is false.
d true
If βxβis β7β, then: β HS = 3 + 9 Γ 7 = 66, RHS = 60 + 6 = 66β L
e false
If βbβis β4β, then: βLHS = 10 + 4 = 14, RHS = 3β(4)β+ 1 = 13β
f
If βx = 5βand βy = 8β, then: βLHS = 3 + 2β(5)β= 13, RHS = 21 β 8 = 13β
true
Now you try
For each of the following equations, state whether it is true or false. a 4β + 9 = 15β b β12 β 3 = 5 + 4β c β4 Γ β(3 + 2)β= 10 + 10β d β10 + 2x = 16β, if βx = 6β e β16 β a = 7aβ, if βa = 2β f β3a + b = 2b β aβ, if βa = 10βand βb = 2β
Example 3
Writing equations from a description
Write equations for each of the following scenarios. a The sum of βxβand β5βis β22β. b The number of cards in a deck is βxβ. In β7βdecks there are β91β cards. c Priyaβs age is currently βjβyears. In β5βyearsβ time her age will equal β17β. d Corey earns β$wβper year. He spends _ β 1 βon sport and _ β 2 βon food. The total amount Corey spends 12 13 on sport and food is β$15 000β. SOLUTION
EXPLANATION
a xβ + 5 = 22β
The sum of βxβand β5βis written βx + 5β.
b β7x = 91β
7β xβmeans β7 Γ xβand this number must equal the β91β cards.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9A Introduction to equations
c jβ + 5 = 17β
In β5βyearsβ time Priyaβs age will be β5βmore than her current age, so βj + 5βmust be β17β.
d _ β 1 βΓ w + _ β 2 βΓ w = 15 000β 12 13
_ β 1 βof Coreyβs wage is _ β 1 βΓ wβand _ β 2 βof his wage
12
13
U N SA C O M R PL R E EC PA T E G D ES
12 2 βΓ wβ. is β_ 13
597
Now you try
Write equations for each of the following. a The product of β3βand βaβis β21β. b The sum of βpβand βqβis β10β. c Tristanβs age is currently βtβyears. Three years ago he was β24β. d John has βaβmarbles and Michael has βbβmarbles. When John combines half of his marble collection with a quarter of Michaelβs collection, the total is β150β marbles.
Exercise 9A FLUENCY
Example 1
Example 1
Example 2aβc
Example 2d
Example 2e
Example 2f
1
1β5ββ(1β/2β)β,β 7
State whether the following are equations (Yes or No). a β2 + 6 = 8β b β1 + 1 = 3β c β9 β 3β
2β7β(β 1β/2β)β
2ββ(1β/2β)β,β 5β7ββ(1β/2β)β
d β3 = 9 + xβ
e 5β β yβ
2 Classify each of the following as an equation (β E)βor not an equation (β N)ββ. a β7 + x = 9β b β2 + 2β c 2β Γ 5 = tβ d β10 = 5 + xβ e β2 = 2β f β7 Γ uβ g β10 Γ· 4 = 3pβ h β3 = e + 2β i βx + 5β
3 For each of the following equations, state whether it is true or false. a 1β 0 Γ 2 = 20β b β12 Γ 11 = 144β d β100 β 90 = 2 Γ 5β e β30 Γ 2 = 32β g β2(β 3 β 1)β= 4β h β5 β β(2 + 1)β= 7 β 4β j β2 = 17 β 14 β 1β k 1β 0 + 2 = 12 β 4β
c f i l
4 If βx = 3β, state whether each of these equations is true or false. a β5 + x = 7β b βx + 1 = 4β c β13 β x = 10 + xβ
3β Γ 2 = 5 + 1β β12 β 4 = 4β β3 = 3β β1 Γ 2 Γ 3 = 1 + 2 + 3β d β6 = 2xβ
5 If βb = 4β, state whether each of the following equations is true or false. a β5b + 2 = 22β b β10 Γ β(b β 3)β= b + b + 2β c 1β 2 β 3b = 5 β bβ d βb Γ (β b + 1)β= 20β
6 If βa = 10βand βb = 7β, state whether each of these equations is true or false. a βa + b = 17β b βa Γ b = 3β c aβ Γ β(a β b)β= 30β d βb Γ b = 59 β aβ e β3a = 5b β 5β f βb Γ β(a β b)β= 20β g β21 β a = bβ h β10 β a = 7 β bβ i β1 + a β b = 2b β aβ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
598
Chapter 9
7 Write equations for each of the following. a The sum of β3βand βxβis equal to β10β. c The sum of βaβand βbβis β22β. e The product of β8βand βxβis β56β. g One-quarter of βtβis β12β.
b d f h
When βkβis multiplied by β5β, the result is β1005β. When βdβis doubled, the result is β78β. When βpβis tripled, the result is β21β. The sum of βqβand βpβis equal to the product of β qβand βpβ.
U N SA C O M R PL R E EC PA T E G D ES
Example 3a
Equations
PROBLEM-SOLVING
Example 3bβd
8, 9
8, 10
8, 10, 11
8 Write true equations for each of these problems. You do not need to solve them. a Chairs cost β$cβat a store. The cost of β6βchairs is β$546β. 1 βhours in total. b Patrick works for βxβhours each day. In a β5β-day working week, he works β37β_ 2 c Pens cost β$aβeach and pencils cost β$bβ. Twelve pens and three pencils cost β$28βin total. d Amy is βfβyears old. In β10βyearsβ time her age will be β27β. e Andrewβs age is βjβand Haileyβs age is βmβ. In β10βyearsβ time their combined age will be β80β.
9 Find a value of βmβthat would make this equation true: β10 = m + 7β.
10 Find two possible values of βkβthat would make this equation true: βk Γ β(8 β k)β= 12β. (Hint: Try whole numbers between β1βand β10β.) 11 If the equation βx + y = 6βis true, and βxβand βyβare both whole numbers between β1βand β5β, what values could they have? REASONING
12, 13
14
13, 14ββ(β1/2β)β
12 Explain why the equation βa + 3 = 3 + aβis always true, regardless of the value of βaβ. 13 Explain why the equation βb = b + 10βis never true, regardless of the value of βbβ. 14 Equations involving pronumerals can be split into three groups: A: Always true, no matter what values are substituted. N: Never true, no matter what values are substituted. S: Sometimes true but sometimes false, depending on the values substituted. Classify each of these equations as βA, Nβor βSβ. a xβ + 5 = 11β b 1β 2 β x = xβ c βa = aβ e βa = a + 7β f β5 + b = b β 5β g β0 Γ b = 0β i β2x + x = 3xβ j β2x + x = 4xβ k β2x + x = 3x + 1β ENRICHMENT: Equation permutations
β
β
d β5 + b = b + 5β h βa Γ a = 100β l βa Γ a + 100 = 0β 15, 16
15 For each of the following, rearrange the symbols to make a true equation. a 6β , 2, 3, Γ,ββ =β b β1, 4, 5, β , =β c β2, 2, 7, 10, β , Γ·,ββ =β d β2, 4, 5, 10, β , Γ·,ββ =β
16 a How many different equations can be produced using the symbols β2, 3, 5, + ,ββ =β? b How many of these equations are true? List them. c Is it possible to change just one of the numbers above and still produce true equations by rearranging the symbols? Explain your answer. d Is it possible to change just the operation above (i.e. β+β) and still produce true equations? Explain your answer.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9B Solving equations by inspection
599
9B Solving equations by inspection
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand that a solution to an equation is the value of the pronumeral that makes the equation true β’ To be able to use inspection (also called trial and error) to ο¬nd a solution
Solving an equation is the process of finding the values that pronumerals must take in order to make the equation true. Pronumerals are also called βunknownsβ when solving equations. For simple equations, it is possible to find a solution by trying a few values for the pronumeral until the equation is true. This method does not guarantee that we have found all the solutions (if there is more than one) and it will not help if there are no solutions, but it can be a useful and quick method for simple equations. A house painter solves an equation to calculate the number of litres of paint needed. If undercoat paint covers 12m2/L and the total wall area is 240m2, the painter solves: 12x = 240, giving x = 20, so 20 litres are required.
Lesson starter: Finding the missing value β’
Find the missing values to make the following equations true. 10 Γ β17 = 13 27 = 15 + 3 Γ 2Γ
β’
+ 4 = 17
Can you always find a value to put in the place of
in any equation?
KEY IDEAS
β Solving an equation means finding the values of any pronumerals that make the equation true.
These values are called solutions to the equation.
β An unknown in an equation is a pronumeral whose value needs to be found in order to make
the equation true.
β One method of solving equations is by inspection (also called trial and error), which involves
inspecting (or trying) different values and seeing which ones make the equation true.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 9
Equations
BUILDING UNDERSTANDING 1 If the missing number is β5β, classify each of the following equations as true or false. a β+ 3 = 8β b β10 Γβ β+ 2 = 46β c β10 ββ
d β12 = 6 +β
β= 5β
βΓ 2β
U N SA C O M R PL R E EC PA T E G D ES
600
2 For the equation
β+ 7 = 13β:
a State the value of the LHS (left-hand side) if b State the value of the LHS if
β= 10β.
c State the value of the LHS if
β= 6β.
d What value of
β= 5β.
would make the LHS equal to β13β?
3 State the unknown pronumeral in each of the following equations. Note that you do not need to find its value. a β4 + x = 12β
Example 4
b β50 β c = 3β
c β4b + 2 = 35β
d β5 β 10d = 2β
Finding the missing number
For each of these equations, find the value of the missing number that would make it true. a βΓ 7 = 35β b β20 ββ β= 14β SOLUTION
EXPLANATION
a 5β β
5β Γ 7 = 35βis a true equation.
b β6β
2β 0 β 6 = 14βis a true equation.
Now you try
Find the value of the missing number that would make the following equations true. a βΓ· 3 = 7β b β12 +β β= 27β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9B Solving equations by inspection
Example 5
601
Solving equations by inspection
Solve each of the following equations by inspection. a βc + 12 = 30β b 5β b = 20β
U N SA C O M R PL R E EC PA T E G D ES
c β2x + 13 = 21β
SOLUTION
EXPLANATION
a c + 12 = 30 β β β β c = 18
The unknown variable here is βcβ. β18 + 12 = 30βis a true equation. An alternative method is to βundoβ the operation + 12 c + 12 30
c 18
β 12
b β 5b = 20β b=4
The unknown variable here is βbβ. Recall that β5bβmeans β5 Γ bβ, so if βb = 4βthen β5b = 5 Γ 4 = 20β. An alternative method is to βundoβ the operation. Γ5 b 4
5b 20
Γ·5
c 2x + 13 = 21 β β β β x= 4
The unknown variable here is βxβ. Trying a few values: βx = 10βmakes βLHS = 20 + 13 = 33β, which is too large. βx = 3βmakes βLHS = 6 + 13 = 19β, which is too small. βx = 4βmakes βLHS = 21β. An alternative method is to βundoβ the operation. + 13
Γ2
2x + 13 21
2x 8
x 4
Γ·2
β 13
Now you try
Solve the following equations by inspection. a βq β 5 = 13β b 2β x = 16β
c β3a + 5 = 35β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
602
Chapter 9
Equations
Exercise 9B FLUENCY
1
2β5β(β 1β/2β)β
3β5β(β 1β/3β)β
For each of these equations, find the value of the missing number that would make it true. a βΓ 4 = 24β b βΓ· 5 = 10β c β10 ββ β= 2β d β9 +β
β= 26β
U N SA C O M R PL R E EC PA T E G D ES
Example 4
1, 2β4ββ(1β/2β)β
Example 4
2 Find the value of the missing numbers. a 4β +β β= 7β b 2β Γβ β= 12β e β42 =β
Example 5a,b
Example 5c
Example 5c
βΓ 7β
f
β100 ββ
β= 30β
c β13 =β
g
β+ 3β
d β10 = 6 +β
βΓ 4 = 80β
h
β+ 12 = 31β
3 Solve the following equations by inspection. a 8β Γ y = 64β b 6β Γ· l = 3β e βl + 2 = 14β f βa β 2 = 4β i β12 = e + 4β j βr Γ· 10 = 1β
c lβ Γ 3 = 18β g βs + 7 = 19β k β13 = 5 + sβ
d β4 β d = 2β h βx Γ· 8 = 1β l β0 = 3 β zβ
4 Solve the following equations by inspection. a 2β p β 1 = 5β b 3β p + 2 = 14β e β2b β 1 = 1β f β5u + 1 = 21β ( ) i β45 = 5β d + 5 β j β3d β 5 = 13β
c 4β q β 4 = 8β g β5g + 5 = 20β k β8 = 3m β 4β
d β4v + 4 = 24β h β4(β e β 2)β= 4β l β8 = 3o β 1β
5 Solve the following equations by inspection. (All solutions are whole numbers between β1βand β10β.) a β4 Γ β(x + 1)ββ 5 = 11β b β7 + x = 2 Γ xβ c β(3x + 1)βΓ· 2 = 8β d β10 β x = x + 2β e β2 Γ β(x + 3)β+ 4 = 12β f β15 β 2x = xβ PROBLEM-SOLVING
6, 7
6β8
7β11
6 Find the value of the number in each of these examples. a A number is doubled and the result is β22β. b β3βless than a number is β9β. c Half of a number is β8β. d β7βmore than a number is β40β. e A number is divided by β10β, giving a result of β3β. f β10βis divided by a number, giving a result of β5β.
7 Ezekielβs current age is unknown. In β5βyearsβ time he will be β18βyears old. a If βxβis Ezekielβs current age, which of the following equations can be used to describe this situation? A xβ + 18 = 5β B βx + 5 = 18β C β5x = 18β D βx β 5 = 18β b Solve the equation by inspection to find the value of βxβ. c How old is Ezekiel?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9B Solving equations by inspection
603
8 Arabella is paid β$10βan hour for βxβhours. During a particular week, she earns β$180β. a Write an equation involving βxβto describe this situation. b Solve the equation by inspection to find the value of βxβ.
U N SA C O M R PL R E EC PA T E G D ES
9 One bag of sand weighs βw kgβand another bag which is twice as heavy, weighs β70 kgβ. a Write an equation involving βwβto describe this situation. b Solve the equation by inspection to find the value of βwβ. 10 Chloe buys βxβkg of apples at β$4.50βper kg. She spends a total of β$13.50β. a Write an equation involving βxβto describe this situation. b Solve the equation by inspection to find βxβ. 11 Yanniβs current age is βyβyears. In β12β yearsβ time he will be three times as old. a Write an equation involving βyβto describe this situation. b Solve the equation by inspection to find βyβ. REASONING
12
12
12, 13
12 a Solve the equation βx + (β x + 1)β= 19βby inspection. b The expression βx + β(x + 1)βcan be simplified to β2x + 1β. Use this observation to solve βx + β(x + 1)β= 181βby inspection.
13 There are three consecutive positive integers that add to β45β. a Solve the equation βx + (x + 1) + (x + 2) = 45βby inspection to find the three numbers. b An equation of the form βx + (x + 1) + (x + 2) = ?βhas a positive integer solution only if the right-hand side is a multiple of β3β. Explain why this is the case. (Hint: Simplify the LHS.) ENRICHMENT: Multiple pronumerals
β
β
14
14 When multiple pronumerals are involved, inspection can still be used to find a solution. For each of the following equations find, by inspection, one pair of values for βxβand βyβthat make them true. a βx + y = 8β b βx β y = 2β c β3 = 2x + yβ d βx Γ y = 6β e β12 = 2 + x + yβ f βx + y = x Γ yβ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 9
Equations
9C Equivalent equations LEARNING INTENTIONS β’ To understand what it means for two equations to be equivalent β’ To be able to apply an operation to both sides of an equation to form an equivalent equation β’ To be able to determine that two equations are equivalent by ο¬nding an operation that has been applied to both sides
U N SA C O M R PL R E EC PA T E G D ES
604
Sometimes, two equations essentially express the same thing. For example, the equations x + 5 = 14, x + 6 = 15 and x + 7 = 16 are all made true by the same value of x. Each time, we have added one to both sides of the equation. The following diagram shows that if we do the same to both sides of an equation such as x + 5 = 14, we can make equivalent equations, x + 2 = 11, x + 6 = 15 and 2x + 10 = 28. x + 2 = 11
1 1 x
11
subtract 3 from both sides
1 1 1 1 1 x
initial equation
1 1 1 1 1 1 x
x + 5 = 14
14
add 1 to both sides
x + 6 = 15
1 14
2x + 10 = 28
double both sides
1 1 1 1 1 x
1 1 1 1 1 x
14
14
A true equation stays true if we βdo the same thing to both sidesβ, such as adding a number or multiplying by a number. The exception to this rule is that multiplying both sides of any equation by zero will always make the equation true, and dividing both sides of any equation by zero is not permitted because nothing can be divided by zero. Other than this exception, if we do the same thing to both sides we will have an equivalent equation.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9C Equivalent equations
605
Lesson starter: Equations as scales The scales in the diagram show β2 + 3x = 8β. β’
1 1
1 1 1 1
x x x
1 1 1 1
U N SA C O M R PL R E EC PA T E G D ES
β’
What would the scales look like if two ββ1 kgββ blocks were removed from both sides? What would the scales look like if the two ββ1 kgββ blocks were removed just from the left-hand side? (Try to show whether they would be level.) Use scales to illustrate why β4x + 3 = 4βand β4x = 1β are equivalent equations.
β’
KEY IDEAS
β Two equations are equivalent if substituting pronumerals always makes both equations true or
both equations false. For example, ββx + 3 = 10ββ and ββ10 = x + 3ββ are always either both true (if βx is 7β) or both false (if βx is not equal to 7β).
β To show that two equations are equivalent you can repeatedly:
β’ β’ β’ β’ β’
add the same number to both sides subtract the same number from both sides multiply both sides by the same number (other than zero) divide both sides by the same number (other than zero) swap the left-hand side with the right-hand side of the equation
BUILDING UNDERSTANDING
1 Give an equation that results from adding β10βto both sides of each of these equations. a β10d + 5 = 20β b β7e = 31β c β2a = 12β d βx = 12β
2 Match up each of these equations (a to e) with its equivalent equation (i.e. A to E), where β3β has been added to both sides. a
β10 + x = 14β
A
β12x + 3 = 123β
b
βx + 1 = 13β
B
βx + 13 = 11x + 3β
c
1β 2 = x + 5β
C
β13 + x = 17β
d
βx + 10 = 11xβ
D
xβ + 4 = 16β
e
β12x = 120β
E
β15 = x + 8β
Example 6
Applying an operation
For each equation, find the result of applying the given operation to both sides and then simplify. a β2 + x = 5 [+ 4]β b β7x = 10 [Γ 2]β c 3β 0 = 20b [Γ· 10]β d β7q β 4 = 10 [+ 4]β
Continued on next page
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 9
Equations
SOLUTION a
b
EXPLANATION
2+x=5 ββ―β― 2 +β―β― x + 4 = 5β+ 4β x+6=9
The equation is written out, and β4βis added to both sides.
7x = 10 β 7x Γ 2 = 10βΓ 2β 14x = 20
The equation is written out, and both sides are multiplied by β2β.
30 = 20b
The equation is written out, and both sides are divided by β10β.
10 10 3 = 2b
Simplify the expressions on each side.
Simplify the expressions on each side.
U N SA C O M R PL R E EC PA T E G D ES
606
c
Simplify the expressions on each side.
30 β= _ β β_ β20bββ
d
7q β 4 = 10 7q β β 4 + 4ββ =ββ 10β+ 4ββ β―β― 7q = 14
The equation is written out, and β4βis added to both sides. Simplify the expressions on each side.
Now you try
Find the result of applying the given operation to both sides and then simplifying. a qβ + 5 = 8 [+ 11]β b β7k = 10 [Γ 3]β c β15 = 25b [Γ· 5]β d β9 + 3b = 8 + 2a [β 8]β
Example 7
Showing that equations are equivalent
Show that these pairs of equations are equivalent by stating the operation used. a 2β x + 10 = 15βand β2x = 5β b β5 = 7 β xβand β10 = 2β(7 β x)β c β10β(b + 3)β= 20βand βb + 3 = 2β SOLUTION
EXPLANATION
a Both sides have had β10β subtracted.
2β x + 10 β 10βsimplifies to β2xβ, so we get the second equation by subtracting β10β.
2x + 10 = 15
β 10
β 10
2x = 5
b Both sides have been multiplied by β2β. 5=7βx
Γ2
2β (β 7 β x)βrepresents the RHS; i.e. β7 β xβ, being multiplied by β2β.
Γ2
10 = 2(7 β x)
c Both sides have been divided by β10β. 10(b + 3) = 20
Γ· 10
Γ· 10
Remember β10β(b + 3)βmeans β10 Γ (β b + 3)ββ. If we have β10β(b + 3)β, we get βb + 3β when dividing by β10β.
b+3=2
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9C Equivalent equations
607
Now you try
U N SA C O M R PL R E EC PA T E G D ES
Show that these pairs of equations are equivalent by showing the operation used. a β3a β 4 = 10βand β3a = 14β b β3r = 5 β 4rβand β6r = 2β(5 β 4r)β c xβ β 7 = 9 β 3xβand βx β 16 = β 3xβ
Exercise 9C FLUENCY
Example 6
Example 6
Example 7
1
1, 2β3ββ(β1/2β)β
2β4β(β β1/2β)β
2β4β(β β1/3β)β
For each equation, find the result of applying the given operation to both sides and then simplify. a β3 + x = 7 [+ 3]β b β4x = 7 [Γ 3]β c β15 = 5b [Γ· 5]β d 5β q + 2 = 6 [β 2]β
2 For each equation, show the result of applying the listed operations to both sides. a β5 + x = 10 [+ 1]β b 3β x = 7 [Γ 2]β c β12 = 8q [Γ· 4]β d β9 + a = 13 [β 3]β e β7 + b = 10 [+ 5]β f β5 = 3b + 7 [β 5]β g β2 = 5 + a [+ 2]β h β12x β 3 = 3 [+ 5]β i β7p β 2 = 10 [+ 2]β
3 Show that these pairs of equations are equivalent by stating the operation used. a 4β x + 2 = 10βand 4β x = 8β b β7 + 3b = 12βand β9 + 3b = 14β c 2β 0a = 10βand β2a = 1β d β4 = 12 β xβand β8 = 2β(12 β x)β e 1β 8 = 3xβand β6 = xβ f β12 + x = 3βand β15 + x = 6β g β4β(10 + b)β= 80βand β10 + b = 20β h β12x = 5βand β12x + 4 = 9β
4 For each of the following equations, find the equivalent equation that is the result of adding β4βto both sides and then multiplying both sides by β3β. a βx = 5β b β2 = a + 1β c βd β 4 = 2β d β7 + a = 8β e β3y β 2 = 7β f β2x = 6β PROBLEM-SOLVING
5
5, 6
6, 7
5 Match up each of these equations (a to e) with its equivalent equation (i.e. A to E), stating the operation used. a
βm + 10 = 12β
A
7β β m = 6β
b
β3 β m = 2β
B
β5m = 18β
c
1β 2m = 30β
C
β6m = 10β
d
β5m + 2 = 20β
D
6β m = 15β
e
β3m = 5β
E
βm + 12 = 14β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 9
Equations
6 For each of the following pairs of equations, show they are equivalent by listing the two steps required to transform the first equation to the second. a xβ = 5βand β3x + 2 = 17β b βm = 2βand β10m β 3 = 17β c β5β(2 + x)β= 15βand βx = 1β d β10 = 3x + 10βand β0 = xβ 7 For each of the following equations, write an equivalent equation that you can get in one operation. Your equation should be simpler (i.e. smaller) than the original. a 2β q + 7 = 9β b 1β 0x + 3 = 10β c β2β(3 + x)β= 40β d βx Γ· 12 = 5β
U N SA C O M R PL R E EC PA T E G D ES
608
REASONING
8
8, 9
8β10
8 Sometimes two equations that look quite different can be equivalent. a Show that β3x + 2 = 14βand β10x + 1 = 41βare equivalent by copying and completing the following. 3x + 2 = 14
β2
β2
3x = 12
Γ·3
Γ·3
___ = ___
Γ 10
Γ 10
___ = ___
+1
+1
10x + 1 = 41
b Show that β5x β 3 = 32βand βx + 2 = 9βare equivalent. (Hint: Try to go via the equation βx = 7β.) c Show that (β x Γ· 2)β+ 4 = 9βand (β x + 8)βΓ· 2 = 9βare equivalent.
9 As stated in the rules for equivalence listed in the Key ideas, multiplying both sides by zero is not permitted. a Write the result of multiplying both sides of the following equations by zero. i β3 + x = 5β ii β2 + 2 = 4β iii β2 + 2 = 5β b Explain in a sentence why multiplying by zero does not give a useful equivalent equation.
10 Substituting pronumerals into expressions can be done by finding equivalent equations. Show how you can start with the equation βx = 3βand find an equivalent equation with: a β7x + 2βon the LHS b β8 + 2xβon the LHS ENRICHMENT: Equivalence relations
β
β
11
11 Classify each of the following statements as true or false, justifying your answer. a Every equation is equivalent to itself. b If equation β1βand equation β2βare equivalent, then equation β2βand equation β1βare equivalent. c If equation β1βand equation β2βare equivalent, and equation β2βand equation β3βare equivalent, then equation β1βand equation β3βare equivalent. d If equation β1βand equation β2β are not equivalent, and equation β2βand equation β3β are not equivalent, then equation β1β is not equivalent to equation β3β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9D Solving equations algebraically
609
9D Solving equations algebraically
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand that we can ο¬nd simpler equations by using opposite operations (e.g. dividing by 3 when the pronumeral was multiplied by 3) β’ To be able to solve one-step equations algebraically (using equivalent equations) β’ To be able to solve two-step equations algebraically (using equivalent equations) β’ To understand that solutions can be checked by substitution into both sides of an equation
A soccer player preparing for a game will put on shin pads, then socks and, finally, soccer boots. When the game is over, these items are removed in reverse order: first the boots, then the socks and, finally, the shin pads. Nobody takes their socks off before their shoes. A similar reversal of procedures occurs with equivalent equations. Here are three equivalent equations.
x=3
Γ2
2x = 6
+4
Γ2
+4
2x + 4 = 10
We can undo the operations around x by doing the opposite operation in the reverse order. β4
2x + 4 = 10
β4
2x = 6
Γ·2
Γ·2
x=3
Because these equations are equivalent, this means that the solution to 2x + 4 = 10 is x = 3. An advantage with this method is that solving equations by inspection can be very difficult if the solution is not just a small whole number.
Lesson starter: Attempting solutions
Georgia, Kartik and Lucas try to solve the equation 4x + 8 = 40. They present their attempted solutions below. Georgia
Γ·4
4x + 8 = 40
Kartik
Γ·4
β8
x + 8 = 10
β8
β’ β’ β’
x=2
4x + 8 = 40
Lucas
+8
β8
Γ·4
Γ·4
x = 12
β8
4x = 32
4x = 48
β8
4x + 8 = 40
Γ·4
Γ·4
x=8
Which of the students has the correct solution to the equation? Justify your answer by substituting each studentβs final answer. For each of the two students with the incorrect answer, explain the mistake they have made in their attempt to have equivalent equations. What operations would you do to both sides if the original equation was 7x β 10 = 11?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 9
Equations
KEY IDEAS β Sometimes it is difficult to solve an equation by inspection, so a systematic approach is
required. β To solve an equation, find a simpler equation that is equivalent. Repeat this until the solution is
U N SA C O M R PL R E EC PA T E G D ES
610
found.
β A simpler equation can be found by applying the opposite operations in reverse order. e.g. for
5x + 2 = 17, we have:
Γ5
x
5x
+2
5x + 2
So we solve the equation by βundoingβ them in reverse order. 5x + 2
This gives the solution:
β2
β2
Γ·5
5x
5x + 2 = 17
x
β2
5x = 15
Γ·5
Γ·5
x=3
β A solution can be checked by substituting the value to see if the equation is true.
LHS = 5x + 2 RHS = 17 = 5Γ3+2 = 17
BUILDING UNDERSTANDING
1 State whether each of the following equations is true or false. a x + 4 = 7, if x = 3 b b β 2 = 7, if b = 5 c 7(d β 6) = d, if d = 7 d g + 5 = 3g, if g = 2 e f Γ 4 = 20, if f = 3
2 Consider the equation 7x = 42. a State the missing number in the
diagram opposite. b What is the solution to the equation 7x = 42?
7x = 42
Γ·7
Γ·7
x = __
The order in which things are done matters in both sports and maths.
3 The equations g = 2 and 12(g + 3) = 60 are equivalent. What is the solution to the equation 12(g + 3) = 60?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9D Solving equations algebraically
611
4 State the operation applied in each of the following. Remember that the same operation must be used for both sides.
a
b
5 + a = 30
b = 7.2
U N SA C O M R PL R E EC PA T E G D ES
a = 25
10b = 72
c
12 =
d
c 4
8 = c β 12 20 = c
48 = c
Example 8
Solving one-step equations
Solve each of the following equations algebraically. a β5x = 30β b 1β 7 = y β 21β
q c β10 = _ β β 3
SOLUTION
EXPLANATION
a
The opposite of βΓ 5βis βΓ· 5β. By dividing both sides by β5β, we get an equivalent equation. Recall that β5x Γ· 5βsimplifies to βxβ.
5x = 30
Γ·5
Γ·5
x=6
So the solution is βx = 6β.
b
The opposite of ββ 21βis β+ 21β.
17 = y β 21
+ 21
+ 21
38 = y
So the solution is βy = 38β.
c
Γ3
10 =
q 3
Γ3
Write the pronumeral on the LHS.
Multiplying both sides by β3βgives an equivalent equation that q is simpler. Note that _ β βΓ 3 = qβ. 3
30 = q
So the solution is βq = 30β.
Write the pronumeral on the LHS.
Now you try
Solve each of the following equations algebraically. a β10a = 30β b 2β 3 = 19 + xβ
y c β7 = _ ββ 4
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 9
Equations
Example 9
Solving two-step equations
Solve each of the following equations algebraically and check the solution. a 7β + 4a = 23β c β12 = 2β(e + 1)β b _ βd ββ 2 = 4β 3
U N SA C O M R PL R E EC PA T E G D ES
612
SOLUTION
EXPLANATION
a
At each step, try to make the equation simpler by applying an operation to both sides. Choose the opposite operations based on β7 + 4aβ: Γ4 +7 a 4a 7 + 4a
7 + 4a = 23
β7
β7
4a = 16
Γ·4
Γ·4
a=4
Check: LHS = 7 + 4a βRHS = 23 β β= 7 + 4 Γ 4 β β ββ β―β― β―β― β β β β = 7 + 16 β = 23
b
d β2=4 3 d =6 3
+2
Γ3
+2
At each step, try to make the equation simpler by applying an operation to both sides.
Γ3
The opposite of ββ 2βis β+ 2βand the opposite of βΓ· 3βis βΓ 3β.
d = 18
Check:
RHS = 4 β 3 =β _ β18 ββ β2β ββ―β― ββ 3 β= 6 β 2 β= 4
Check that our equation is true by substituting β d = 18βback into the LHS and RHS. Both sides are equal, so βd = 18βis a solution.
12 = 2(e + 1)
At each step, try to make the equation simpler by applying an operation to both sides. The opposite of βΓ 2βis βΓ· 2βand the opposite of β+ 1βis ββ 1β.
βd ββ 2 LHS = _ β
Opposite operations: ββ 7β, then βΓ· 4β. Check that our equation is true by substituting β a = 4βback into the LHS and RHS. Both sides are equal, so βa = 4βis a solution.
c
Γ·2
Γ·2
6=e+1
β1
β1
5=e
So the solution is βe = 5β. Check: LHS = 12 RHS = 2(e + 1) = 2(5 + 1) = 2Γ6 = 12 β
Write the solution on the LHS. Check that our equation is true by substituting β e = 5βback into the equation.
Now you try
Solve each of the following equations algebraically and check the solution. a 9β + 3x = 15β b _ βm ββ 3 = 2β c β4(β p β 2)β= 24β 4 Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9D Solving equations algebraically
613
Exercise 9D FLUENCY
1
Solve each of the following equations algebraically. a i β4x = 20β
2ββ(1β/2β)β, 3, 4, 5β7ββ(1β/2β)β
2β(β 1β/4β)β, 4, 5β7ββ(1β/3β)β
ii β7x = 56β
U N SA C O M R PL R E EC PA T E G D ES
Example 8a
1, 2ββ(1β/2β)β, 3, 4, 5ββ(1β/2β)β
Example 8b
b i
Example 8c
c i
Example 8
β5 = y β 2β q β20 = _ β β 2
ii β38 = y β 26β q ii β7 = _ β β 3
2 Solve the following equations algebraically. a β6m = 54β b βg β 9 = 2β e 7β + t = 9β f β8 + q = 11β i β24 = j Γ 6β j β12 = l + 8β m kβ Γ· 5 = 1β n β2 = y β 7β q βb Γ 10 = 120β r βp β 2 = 9β
c g k o s
sβ Γ 9 = 81β β4y = 48β β1 = v Γ· 2β β8z = 56β β5 + a = 13β
d h l p t
3 Copy and complete the following to solve the given equations algebraically. a b 7a + 3 = 38 4b β 10 = 14 + 10
c
Γ·2
7a = 35
__ = __
__ = __
__ = __
2(q + 6) = 20 q + 6 = __
d
Γ·2
β3
x +3 10 x __ = 10
5=
__ = __
iβ β 9 = 1β β7 + s = 19β β19 = 7 + yβ β13 = 3 + tβ βn β 2 = 1β
+ 10
β3
__ = __
4 For each of these equations, state the first operation you would apply to both sides to solve it. a β2x + 3 = 9β b β4x β 7 = 33β c 5β β(a + 3)β= 50β d β22 = 2β(b β 17)β
Example 9
5 Solve each of the following equations algebraically. Check the solution by substituting the value into the LHS and RHS. a β6f β 2 = 64β c β5x β 4 = 41β d β3β(a β 8)β= 3β b β_k β+ 9 = 10β 4 e 5β k β 9 = 31β f _ βa β+ 6 = 8β g β2n β 8 = 14β h _ βn β+ 6 = 8β 3 4 i β1 = 2g β 7β j β30 = 3q β 3β k β3z β 4 = 26β l β17 = 9 + 8pβ m 1β 0d + 7 = 47β n β38 = 6t β 10β o β9u + 2 = 47β p β7 = 10c β 3β q β10 + 8q = 98β r β80 = 4β(y + 8)β s β4β(q + 8)β= 40β t β7 + 6u = 67β 6 Solve the following equations, giving your solution as fractions. a β4x + 5 = 8β b β3 + 5k = 27β d β10 = 3 Γ β(2 + x)β e β3 = β(8x + 1)βΓ· 2β
c 2β 2 = β(3w + 7)βΓ 2β f β3β(x + 2)β= 7β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 9
Equations
7 Solve the following equations algebraically. (Note: The solutions for these equations are negative numbers.) a 4β r + 30 = 2β b β2x + 12 = 6β c β10 + β_t β= 2β 2 y e ββ 3x = 15β f β4 = 2k + 22β d _ β β+ 10 = 4β 4 g β2x = β 12β h β5x + 20 = 0β i β0 = 2x + 3β
U N SA C O M R PL R E EC PA T E G D ES
614
PROBLEM-SOLVING
8, 9
9β11
10β12
8 For each of the following, write an equation and solve it algebraically. a The sum of βxβand β5βis β12β. b The product of β2βand βyβis β10β. c When βbβis doubled and then β6βis added, the result is β44β. d β7βis subtracted from βkβ. This result is tripled, giving β18β. e β3βis added to one-quarter of βbβ, giving a result of β6β. f β10βis subtracted from half of βkβ, giving a result of β1β.
9 Freddy gets paid β$12βper hour, plus a bonus of β$50βfor each week. In one week he earned β$410β. a Write an equation to describe this, using βnβfor the number of hours worked. b Solve the equation algebraically and state the number of hours worked. 10 Eliana buys β12βpencils and β5βpens for the new school year. The pencils cost β$1.00β each. a If pens cost β$xβeach, write an expression for the total cost. b The total cost was β$14.50β. Write an equation to describe this. c Solve the equation algebraically, to find the total cost of each pen. d Check your solution by substituting your value of βxβinto β12 + 5xβ.
11 Write equations and solve them algebraically to find the unknown value in each of the following diagrams. a b 4 w 3
c
area = 15
10
x
perimeter = 28
x
area = 12
w
perimeter = 28
d
12 Solve the following equations algebraically. a 7β β(3 + 5x)ββ 21 = 210β b β(100x + 13)βΓ· 3 = 271β
c β3β(12 + 2x)ββ 4 = 62β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9D Solving equations algebraically
REASONING
13
13, 14
14, 15
14 a Show that β2x + 5 = 13βand β5x = 20βare equivalent by filling in the missing steps. b Show that β10 + 2x = 20βand β2(β x β 3)β= 4βare equivalent. c Two equations are written down and they each have a single solution. If the solution is the same for both equations, does this guarantee the equations are equivalent? Justify your answer. d If two equations have different solutions, does this guarantee they are not equivalent? Justify your answer.
2x + 5 = 13
615
13 Write five different equations that give a solution of βx = 2β. β5
β5
U N SA C O M R PL R E EC PA T E G D ES
__ = __ x=4
5x = 20
15 Nicola has attempted to solve four equations. Describe the mistake she has made in each case. a b 4x + 2 = 36 3x + 10 = 43 Γ·4 Γ·4 β 10 β 10 x+2=9
β2
3x = 33
β2
β3
β3
x=7
c
β5
2a + 5 = 11
x = 30
d
β5
β 12
Γ·2
β7
2a = 16
Γ·2
7 + 12a = 43
β 12
7 + a = 31
a=8
β7
a = 24
ENRICHMENT: Pronumerals on both sides
β
β
16ββ(1β/2β)β
16 If an equation has a pronumeral on both sides, you can subtract it from one side and then use the same method as before. For example:
5x + 4 = 3x + 10
β 3x
β4
2x + 4 = 10
β 3x
β4
2x = 6
Γ·2
Γ·2
x=3
Solve the following equations using this method. a β5x + 2 = 3x + 10β c 5β + 12l = 20 + 7lβ e 1β 2s + 4 = 9 + 11sβ g β5j + 4 = 10 + 2jβ
b d f h
8β x β 1 = 4x + 3β β2 + 5t = 4t + 3β β9b β 10 = 8b + 9β β3 + 5d = 6 + 2dβ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
616
Chapter 9
1
Classify each of the following as an equation (β E)βor not an equation (β N)ββ. a β4 + w = 9β b βx + 12β c β3 + 10 = 17β d β20 + 6β
2 If βx = 4β, state whether each of these equations is true β(T)βor false (β F)ββ. a β3x β 4 = 8β b β5x + 6 = 25β c β9 β x = x + 1β d β2x = 5x β 8β
U N SA C O M R PL R E EC PA T E G D ES
Progress quiz
9A
Equations
9A
9A
9B
9C
9C
9D
9D
9D
3 Write an equation for each of the following. (You do not need to solve the equations you write.) a The sum of β5βand βmβis equal to β12β. b When βdβis doubled, the result is β24β. c The product of β9βand βxβis β72β. d Canaries cost β$cβeach and budgies cost β$bβeach. Three canaries and four budgies cost β$190β. 4 Solve the following equations by inspection. a βa β 5 = 4β b β36 = x Γ 9β c βm Γ· 5 = 6β d β3a + 1 = 7β
5 For each equation, find the result of applying the given operation to both sides. You do not need to solve the equation. a β4x = 7β(multiply by β5β) b β2a + 5 = 21β(subtract β5β) 6 State the operation applied to get from the first to the second equation in each of the following. a b c 4x β 8 = 12 18 = 6x 3a = 5 4x = 20 3=x 6a = 10 7 Solve each of the following equations algebraically. a aβ + 7 = 19β b βw β 4 = 27β c βk Γ· 9 = 7β d β5m = 40β
8 Solve each of the following equations algebraically and check your solution. a 1β 2x = 132β b β8 + 3a = 29β c β42 = 8m β 6β d β100 = 5β(y + 12)β
9 For each of the following, write an equation and solve it algebraically. a When βxβis doubled then β2βis added, the result is β12β. b β5βis subtracted from βkβ. The result is tripled giving β21β. c β1βless than the product of β5βand βyβis β19β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9E Equations with fractions
617
9E Equations with fractions EXTENDING
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand that to solve an equation with a fraction, it is helpful to multiply both sides by its denominator β’ To be able to solve equations involving fractions
Solving equations that involve fractions is straightforward once a means a Γ· b. This means that if we we recall that, in algebra, _ b have a fraction with b on the denominator, we can multiply both sides by b to get a simpler, equivalent equation.
Lesson starter: Fractional differences
Fraction equations are solved when a business or a person calculates annual expenses. If rent is $500/week and the x = 500. Solving annual rent is $x, then _ 52 this equation gives x = 26 000, so rent is $26 000 p.a.
Consider these three equations. 2x + 3 = 7 a _ 5
β’ β’ β’
c 2(_x ) + 3 = 7 5
2x + 3 = 7 b _ 5
Solve each of them (by inspection or algebraically). Compare your solutions with those of your classmates. Why do two of the equations have the same solution?
KEY IDEAS
a means a Γ· b. β Recall that _ b
β To solve an equation that has a fraction on one side, multiply both sides by the denominator.
Γ5
x =4 5
Γ5
x = 20
β If neither side of an equation is a fraction, do not multiply by the denominator.
Γ3
x +5=8 3 . . .
β5
Do not do this because the LHS is not yet a fraction by itself.
Γ3
Γ3
x +5=8 3 x =3 3
x=9
β5
Do this
Γ3
x + 2 are different, as demonstrated in these ο¬ow charts. x + 2 and _ β The expressions _ 3
x
Γ·3
3
x 3
+2
x+2 3
vs
x
+2
x+2
Γ·3
x+2 3
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 9
Equations
BUILDING UNDERSTANDING 1 Classify each of the following as true or false. a _ βa βmeans βa Γ· 5β.
q 12 d _ β4 + a βmeans β4 + β(a Γ· 3)ββ. 3
b _ β βmeans β12 Γ· qβ.
5 c _ β4 + a βmeans β(4 + a)βΓ· 3β. 3 12 + 3q e _ β βmeans (β 12 + 3q)βΓ· 4β. 4
U N SA C O M R PL R E EC PA T E G D ES
618
f
β2 + _ βx βmeans (β 2 + x)βΓ· 5β. 5
2 a If βx = 10β, state the value of _ βx + 4 ββ. 2
b
If βx = 10β, state the value of _ βx β+ 4β. 2
x + 4 βand β_x β+ 4βare equivalent expressions. c State whether the following is true or false: β_ 2
2
3 State the missing parts to solve each of these equations. a b b Γ4
4
= 11
Γ5
Γ4
b = __
c
d
h =7 4
__ = __
d =3 5 __ = __
Γ5
p =2 13
__ = __
4 For each of the following equations (a to d), choose the appropriate first step (i.e. A to D) needed to solve it. a _ βx β= 10β 3 _ b βx β+ 2 = 5β 3 c _ βx β 3 β= 1β 2 d _ βx ββ 3 = 5β 2
A Multiply both sides by β2β. B Multiply both sides by β3β.
C Subtract β2βfrom both sides.
D Add β3βto both sides.
Example 10 Solving equations with fractions Solve each of the following equations. a _ βa β= 3β 7
5y b _ β β= 10β 3
3x β+ 7 = 13β c β_ 4
2x β 3 β= 3β d β_ 5
SOLUTION
EXPLANATION
a
Multiplying both sides by β7βremoves the denominator of β7β.
Γ7
a =3 7
Γ7
a = 21
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9E Equations with fractions
b
5y = 10 3
Γ3
5y = 30
Γ·5
Multiplying both sides by β3βremoves the denominator of β3β. Γ3
The equation β5y = 30βcan be solved normally.
Γ·5
U N SA C O M R PL R E EC PA T E G D ES
y=6
619
c
3x + 7 = 13 4
β7
3x =6 4
Γ4
Γ·3
d
Γ5
+3
Γ·2
3x = 24 x=8
2x β 3 =3 5
2x β 3 = 15 2x = 18 x=9
β7
Γ4
First, we subtract β7βbecause we do not have a fraction by itself on the LHS. Once there is a fraction by itself, multiply by its denominator (in this case, β4β) and solve the equation β3x = 24βas you would normally.
Γ·3
First, multiply both sides by β5βto remove the denominator.
Γ5
+3
Then solve the equation β2x β 3 = 15βas you would normally.
Γ·2
Now you try
Solve each of the following equations. p 2m β= 4β a β_β= 5β b β_ 10 5
3u β+ 8 = 11β c β_ 5
2x + 3 β= 3β d β_ 7
Exercise 9E FLUENCY
Example 10a
Example 10a,b
1
Solve each of the following equations. a β= 2β a β_ b _ β a β= 3β 4 10
2 Solve the following equations algebraically. m β= 2β a β_ b β_c β= 2β 9 6 2y e _ β3u β= 12β f _ β β= 4β 5 9 3j 4h _ i β β= 8β j _ β β= 9β 5 5
1, 2β3ββ(1β/2)β β
2β4β(β 1β/2)β β
2β4ββ(1β/4)β β
c _ βx β= 7β 5
d β_x β= 100β 2
c β_s β= 2β 8 5x β= 10β g β_ 2
d _ βr β= 2β 5 3a β= 6β h β_ 8
k _ β5v β= 5β 9
l
3q β_β= 6β 4
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
620
Chapter 9
3 Solve the following equations algebraically. Check your solutions using substitution. j+8 b β 2 β= 1β y+5 d β_ a _ βh + 15 β= 2β c β_β= 1β b β_β= 1β 2 12 11 11 7u β 12 4t 4r β 13 β w + 5 _ _ e β β= 1β f β14 + β β= 18β h β1 = β_ g β1 = β_β 9 9 3 11 2z 2q s + 2 3l k β_β+ 9 = 21β β= 1β j _ β l β12 = _ β β+ 10β i _ β β+ 2 = 4β 2 5 7 9 v β 4 f β 2 7 + 4d β 5x m 1β = _ β β o β9 = 4 + β_β p β3 = β_ n β_β= 1β 7 2 9 7 7n 7m + 7 7p q _ β β+ 14 = 21β t _ β4a β 6 β= 6β r β_β= 21β s β3 = β_ββ 11β 5 4 5 4
U N SA C O M R PL R E EC PA T E G D ES
Example 10c,d
Equations
Example 10
4 Solve the following equations algebraically. (Note: The solutions to these equations are negative numbers.) a β+ 2 = 1β y+4 2x β+ 10 = 6β d β_x β+ 12 = 0β b β_ c β_ a β_β= 1β 4 10 5 3 e β0 = 12 + _ β2u β 5
PROBLEM-SOLVING
f
3y _ β β+ 8 = 2β 5
g β1 = _ ββ 2u β 3 β 5 5
h ββ 2 = _ β4d β+ 2β 5
5, 6
6, 7
5 In each of the following cases, write an equation and solve it to find the number. a A number is halved and the result is β9β. b One-third of βqβis β14β. c A number, βrβ, is doubled and then divided by β5β. The result is β6β. d β4βis subtracted from βqβand this is halved, giving a result of β3β. e β3βis added to βxβand the result is divided by β4β, giving a result of β2β. f A number, βyβ, is divided by β4βand then β3βis added, giving a result of β5β.
6 A group of five people go out for dinner and then split the bill evenly. They each pay β$31.50β. a If βbβrepresents the total cost of the bill, in dollars, write an equation to describe this situation. b Solve this equation algebraically. c What is the total cost of the bill? 7 Lee and Theo hired a tennis court for a cost of β$xβ, which they split evenly. Out of his own pocket, Lee also bought some tennis balls for β$5β. a Write an expression for the total amount of money that Lee paid. b Given that Lee paid β$11βin total, write an equation and solve it to find the total cost of hiring the court. c State how much money Theo paid for his share of hiring the tennis court.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9E Equations with fractions
REASONING
8
8, 9
621
8β10
U N SA C O M R PL R E EC PA T E G D ES
2x β+ 3β. 8 a Explain, in one sentence, the difference between _ β2x + 3 βand β_ 5 5 2x + 3 β= 7β? b What is the first operation you would apply to both sides to solve β_ 5 2x β+ 3 = 7β? c What is the first operation you would apply to both sides to solve β_ 5 2x + 3 2x _ _ d Are there any values of βxβfor which β βand β β+ 3βare equal to each other? 5 5
9 Sometimes an equationβs solution will be a fraction. For example, β2x = 1βhas the solution βx = _ β1 ββ. 2 1 _ a Give another equation that has βx = β βas its solution. 2 5 .ββ b Find an equation that has the solution βx = β_ 7 c Could an equation have the solution βx = β _ β1 β? Justify your answer. 2
1 βhave the same effect. 10 Dividing by β2βand multiplying by β_ 2 For example, β6 Γ· 2 = 3βand β6 Γ _ β1 β= 3β. 2 a Show how each of these equations can be solved algebraically. 1 βΓ x = 5β i β_x β= 5β ii β_ 2 2 1 (β x + 4)β= 10βalgebraically, showing the steps you would b Solve the two equations _ βx + 4 β= 10βand β_ 3 3 use at each stage clearly. c How does rewriting divisions as multiplications change the first step when solving equations?
ENRICHMENT: Fractional answers
β
β
11
11 Solve each of the following equations, giving your answers as a fraction. 3x β 4 β= β_ 3β 2x + 5 β= 3β b β_ a β_ 4 6 4 1 β= _ d β_ β3x β 1 β c ( β_ β7 + 2x β)βΓ 3 = 10β 2 5 4 5x β 3 β= 6β. The solution is βx = 9β. Change one number or one operator 12 Consider the equation β_ 7 (i.e. βΓβ, βββor Γ· ) in the equation so that the solution will be βx = 12β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 9
Equations
9F Equations with brackets EXTENDING LEARNING INTENTIONS β’ To be able to expand brackets and collect like terms in algebraic expressions β’ To understand that expanding brackets and collecting like terms is a useful technique when solving equations involving brackets
U N SA C O M R PL R E EC PA T E G D ES
622
Recall from Chapter 4 that expressions with brackets can be expanded using the picture shown to the right about rectanglesβ areas. So 3(x + 5) is equivalent to 3x + 15.
5
x
3 Γ 5 = 15
3 3 Γ x = 3x
When solving 3(x + 5) = 21, we could first divide both sides by 3 or we could first expand the brackets, giving 3x + 15 = 21, and then subtract 15. For some equations, the brackets must be expanded first.
x+5
Lesson starter: Removing brackets
a Draw two rectangles, with areas 4(x + 3) and 5(x + 2). b Use these to show that 4(x + 3) + 5(x + 2) is equivalent to 9x + 22. c Can you solve the equation 4(x + 3) + 5(x + 2) = 130?
KEY IDEAS
β To expand brackets, use the distributive law, which states that:
a(b + c) = ab + ac
e.g. 3(x + 4) = 3x + 12
x +4 3 3x +12
a(b β c) = ab β ac
e.g. 4(b β 2) = 4b β 8
b β2 4 4b β8
β Like terms are terms that contain exactly the same pronumerals and can be collected to
simplify expressions. For example, 3x + 4 + 2x can be simplified to 5x + 4.
β Equations involving brackets can be solved by first expanding brackets and collecting like terms.
BUILDING UNDERSTANDING
1 Which of the following is the correct expansion of 5(x + 2)? A 5Γx+2 B 5x + 2 C 5x + 10
2 State the missing numbers. a 3(x + 2) = 3x+ c 2(b + 1) =
b+2
D 10x
b 4(3a + 1) = 12a+ d 6(2c + 3) =
c + 18
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9F Equations with brackets
b 2β a + 4bβcan be simplified to β6abβ. d β7a + 3 + 2aβcan be simplified to β9a + 3β. f β20x β 12x + 3yβcan be simplified to β 32x + 3yβ.
U N SA C O M R PL R E EC PA T E G D ES
3 Answer true or false to each of the following. a β4x + 3xβcan be simplified to β7xβ. c β6p β 4pβcan be simplified to β2pβ. e β2b + 3βcan be simplified to β5bβ.
623
Example 11 Expanding brackets Expand each of the following. a β4β(x + 3)β
b 6β (β q β 4)β
c β5(β 3a + 4)β
SOLUTION
EXPLANATION
a β4β(x + 3)β= 4x + 12β
Using the distributive law:
b β6β(q β 4)β= 6q β 24β
Using the distributive law:
c 5β β(3a + 4)β= 15a + 20β
Using the distributive law:
x +3 4(x + 3) = 4x + 12ββ4 4x +12
q β4 6(q β 4) = 6q β 24ββ6 6q β24
3a +4 5(3a + 4) = 5 Γ 3a + 20ββ5 15a +20
Now you try
Expand each of the following. a β3(β x + 7)β
b 8β (β m β 6)β
c β5(β 8y + 7)β
Example 12 Simplifying expressions with like terms Simplify each of these expressions. a β2x + 5 + xβ
b β3a + 8a + 2 β 2a + 5β
SOLUTION
EXPLANATION
a β2x + 5 + x = 3x + 5β
Like terms are β2xβand βxβ. These are combined to give β3xβ.
b β3a + 8a + 2 β 2a + 5 = 9a + 7β
Like terms are combined: β3a + 8a β 2a = 9aβand β2 + 5 = 7β
Now you try
Simplify each of these expressions. a β4a + 7 + 2aβ
b β4a + 9a + 3 β 4a β 6β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
624
Chapter 9
Equations
Example 13 Solving equations by expanding brackets c β3(β b + 5)β+ 4b = 29β
U N SA C O M R PL R E EC PA T E G D ES
Solve each of these equations by expanding brackets first. a 3β β(x + 2)β= 18β b 7β = 7β(4q β 3)β SOLUTION
EXPLANATION
a
x +2 Expand brackets. 3 3x +6
3(x + 2) = 18 3x + 6 = 18 β6 β6 3x = 12 Γ·3 Γ·3 x=4
b
7 = 7(4q β 3) 7 = 28q β 21
+ 21
Γ· 28
28 = 28q 1=q
+ 21
Γ· 28
Solve the equation by performing the same operations to both sides. 4q β3 Expand brackets. 7 28q β21
Solve the equation by performing the same operations to both sides.
so q = 1 is the solution.
b +5 Expand brackets. 3 3b +15
c 3β (β b + 5)β+ 4b = 29β β3b + 15 + 4b = 29β
7b + 15 = 29
β 15
Γ·7
7b = 14 b=2
β 15
Γ·7
Collect like terms to simplify the expression. Solve the equation by performing the same operations to both sides.
Now you try
Solve each of these equations by expanding brackets first. a 5β β(m + 4)β= 30β b 5β 0 = 10β(3x β 1)β
c β5(β q β 1)β+ 2q = 23β
Exercise 9F FLUENCY
1
Example 11a
Expand each of the following. a i β3β(x + 2)β
1, 2β6ββ(β1/2β)β
2β7β(β β1/2β)β
2β5ββ(β1/4β)β,β 6β7ββ(β1/2β)β
ii β5β(x + 7)β
Example 11b
b i
β5β(q β 2)β
ii β11β(q β 4)β
Example 11c
c i
β3β(2a + 7)β
ii β6β(4a β 1)β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9F Equations with brackets
2 Expand each of the following. a β2β(x + 1)β c 2β (β 3a β 4)β e 4β β(3x + 4)β g β12β(4a + 3)β
b d f h
3 Simplify these expressions by collecting like terms. a β3a + a + 2β c 2β b β 4 + bβ e 5β x + 3 + xβ g β7 + 2b β 1β
b d f h
5β β(2b + 3)β β5β(7a + 1)β β3β(8 β 3y)β β2β(u β 4)β
U N SA C O M R PL R E EC PA T E G D ES
Example 11
625
Example 12
Example 13a
Example 13b
Example 13c
Example 13c
5β + 2x + xβ β5a + 12 β 2aβ β3k + 6 β 2kβ β6k β k + 1β
4 Solve the following equations by expanding the brackets first. Check your solutions by substituting them in. a β2β(10 + s)β= 32β b β2β(5 + l)β= 12β c 3β (β p β 7)β= 6β d β8β(y + 9)β= 72β e 8β (β 4 + q)β= 40β f β7β(p + 7)β= 133β ( ) g β8β m + 7 β= 96β h β22 = 2β(b + 5)β i β25 = 5β(2 + p)β j β63 = 7β(p + 2)β k β9(β y β 6)β= 27β l β2β(r + 8)β= 32β
5 Solve these equations by expanding the brackets first. a β6β(3 + 2d)β= 54β b c 3β (β 2x β 4)β= 18β d ( ) e 4β 4 = 4β 3a + 8 β f g β10 = 5β(9u β 7)β h
8β β(7x β 7)β= 56β β27 = 3β(3 + 6e)β β30 = 6β(5r β 10)β β3β(2q β 9)β= 39β
6 Solve the following equations by first expanding the brackets. You will need to simplify the expanded expressions by collecting like terms. a β5β(4s + 4)β+ 4s = 44β b β5i + 5β(2 + 2i)β= 25β c β3β(4c β 5)β+ c = 50β d β3β(4 + 3v)ββ 4v = 52β e β5β(4k + 2)β+ k = 31β f β4q + 6β(4q β 4)β= 60β g β40 = 4y + 6β(2y β 4)β h β44 = 4f + 4β(2f + 2)β i β40 = 5t + 6β(4t β 3)β
7 Solve the following equations algebraically. (Note: The answers to these equations are negative numbers.) a β3(β u + 7)β= 6β b β2β(k + 3)β= 0β c β6β(p β 2)β= β 18β d β16 = 8β(q + 4)β e β5β(2u + 3)β= 5β f β3 = 2β(x + 4)β+ 1β ( ) g β4(β p β 3)β+ p = β 32β h β3β r + 4 β+ 2r + 40 = 2β i β2β(5 + x)ββ 3x = 15β PROBLEM-SOLVING
8
8
8, 9
8 For each of the following problems: i write an equation. ii solve your equation by first expanding any brackets. a β5βis added to βxβand then this is doubled, giving a result of β14β. b β3βis subtracted from βqβand the result is tripled, giving a final result of β30β. c A number, βxβ, is doubled and then β3βis added. This number is doubled again to get a result of β46β. d β4βis added to βyβand this is doubled. Then the original number, βyβ, is subtracted, giving a result of β17β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 9
Equations
9 Solve the following equations. a β β 3β(x β 2)β= 3β
b ββ 5β(4 β 3x)β= β 35β
REASONING
10
c β1 β 2β(x β 1)β= β 7β 10, 11
11, 12ββ(β1/2β)β
10 For each of the following equations, explain why there are no solutions by first simplifying the LHS. a 2β β(x + 5)ββ 2x = 7β b β3β(2x + 1)β+ 6β(2 β x)β= 4β c β4β(2x + 1)ββ 10x + 2β(x + 1)β= 12β
U N SA C O M R PL R E EC PA T E G D ES
626
11 Consider the equation β2(β 3x + 4)ββ 6x + 1 = 9β. a Show that this equation is true if βx = 0β. b Show that this equation is true if βx = 3β. c Explain why this equation is always true. d Give an example of another equation involving brackets that is always true, where one side contains a variable but the other side is just a number.
12 For equations like β4β(3x + 2)β= 44β, you have been expanding the brackets first. Since β4(β 3x + 2)β= 44β is the same as β4 Γ β(3x + 2)β= 44β, you can just start by dividing both sides by β4β. Without expanding brackets, solve the equations in Question 4 by dividing first. ENRICHMENT: Expanding multiple brackets
β
β
13ββ(1β/2β)β
13 Solve each of the following equations by expanding all sets of brackets. a 6β (β 2j β 4)β+ 4β(4j β 3)β= 20β b β3β(4a + 5)β+ 5β(1 + 3a)β= 47β c β2β(5a + 3)β+ 3β(2a + 3)β= 63β d β222 = 3β(4a β 3)β+ 5β(3a + 3)β e β77 = 2β(3c β 5)β+ 3β(4c + 5)β f β240 = 4β(3d + 3)β+ 6β(3d β 2)β g β2β(x + 3)β+ 4β(x + 5)β= 32β h β4β(x + 5)β+ 4β(x β 5)β= 24β i β2β(3x + 4)β+ 5β(6x + 7)β+ 8β(9x + 10)β= 123β j β2β(βx + 1β)β+ 3ββ(βx + 1β)β+ 4ββ(βx + 1β)β= 36β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Applications and problem-solving
U N SA C O M R PL R E EC PA T E G D ES
Zanaβs earning goal
Applications and problem-solving
The following problems will investigate practical situations drawing upon knowledge and skills developed throughout the chapter. In attempting to solve these problems, aim to identify the key information, use diagrams, formulate ideas, apply strategies, make calculations and check and communicate your solutions.
627
1
Zana helps her uncle at the local garden nursery and gets paid β$12βper hour. Zana is interested in the total amount of money she can earn for a certain number of hours worked, as well as the time taken for her to reach an earning goal, including and excluding bonuses. a Write an equation for Zanaβs situation using the pronumerals βPβto represent Zanaβs pay (in dollars) and βhβto represent the number of hours Zana works in a month. b If Zana works for a total of β15βhours for the month of May, how much pay will she receive?
If Zana works more than β20βhours in a month, she also receives a monthly bonus of β$30βfrom her uncle to thank her for her hard work. c Write an equation for Zanaβs pay when she works more than β20βhours in a month. d If Zana works β24βhours in June, what would be her total pay for this month? e If Zana wishes to earn β$4500βover the course of the year, how many hours will she need to work on average each month? Use your equation from part c and round your answer to the nearest hour.
Local cafeβs rent
2 The local cafe is forecasting the profit they will make each day. On average, the cafe owners are confident they can make a profit of β$10βfor every breakfast ordered, a β$15βprofit for every lunch ordered and a β$3βprofit on every drink ordered. The cafe wishes to determine if they will be able to pay the rent for their current premises or whether they will need to move to cheaper premises. a Using appropriate pronumerals, write an equation for the total profit the cafe makes per day. The cafe kept a record of orders for the week and these are shown below. Breakfasts
Lunches
Drinks
Monday
Day
β36β
β42β
β90β
Tuesday
β30β
β54β
β75β
Wednesday
β28β
β57β
β96β
Thursday
β42β
β51β
β99β
Friday
β35β
β70β
β120β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 9
Equations
b Using your equation from part a, calculate the profit the cafe made on each of the above days. c What was the average daily profit for this week? d Using this average daily profit, forecast what profit the cafe can make over the course of the year, assuming they are open β5βdays per week for β52βweeks of the year. e The current rent for the premises is β$42 000βper annum. What percentage of the cafeβs profit goes towards rent? Give your answer to the nearest per cent. f What other expenses do you think the cafe needs to pay out of their profit?
U N SA C O M R PL R E EC PA T E G D ES
Applications and problem-solving
628
Human heart beat
3 Zachary has a resting heart rate of β70βbeats per minute, and a general light activity heart rate of β90βbeats per minute. Zachary is interested in the number of times his heart beats over the course of various time intervals including a whole year. a If Zachary is at rest on average β14βhours a day, and in a light activity state on average β10βhours a day, how many times will Zacharyβs heart beat in: i β1β day? ii β1βyear? (assume a non-leap year) iii βtβyears? (assume a leap year every four years) b Write an expression for the number of times a heart beats in βtβyears, if the average heart rate is βnβbeats per minute. c If an average human heart can beat β4βbillion times (β 4 000 000 000)β, and Tamara has an average heart rate of β90βbeats per minute, how many years can Tamaraβs heart beat for? d Investigate how many times your heart beats in a year. Take the following things into account: β’ your average resting heart rate, your average sitting heart rate, your average walking heart rate, your average exercise heart rate β’ the average number of hours you sleep, sit, walk, exercise per day. e Set up a spreadsheet that would allow a user to input their different state heart rates and the average number of hours they spend at each state in a day. Using appropriate equations in the spreadsheet, calculate the number of heart beats in a year.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9G Using formulas
629
9G Using formulas
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand that formulas and rules are types of equation β’ To be able to apply a formula by substituting and evaluating β’ To be able to apply a formula by substituting and solving an equation
Often, two or more pronumerals (or variables) represent related quantities. For example, the speed (s) at which a car travels and the time (t) it takes to arrive at its destination are related variable quantities. A formula is an equation that contains two or more pronumerals and shows how they are related.
The volume, V, of a rugby ball is calculated using the formula: ΟL W 2 Volume = _ 6 where L is the length and W is the width.
Lesson starter: Fahrenheit and Celsius
In Australia, we measure temperature in degrees Celsius, whereas in the USA it is measured in degrees 9C + 32. Fahrenheit. A formula to convert between them is F = _ 5 β’
β’ β’
At what temperature in degrees Fahrenheit does water freeze? At what temperature in degrees Fahrenheit does water boil? What temperature is 100Β° Fahrenheit in Celsius? What is significant about this temperature?
KEY IDEAS
β A variable is a pronumeral that represents more than one value.
β A formula is a rule or equation that shows the relationship between two or more variables.
For example:
A=lw
area of rectangle length
width
β In the example above there are three variables, whereby A = 6, l = 2 and w = 3 is one solution
for this formula because 6 = 2 Γ 3. β’ A = lw has three variables: A, l and w. If two variables are known, then the formula can be used to find the unknown variable. β’ The subject of an equation is a pronumeral that occurs by itself on the left-hand side. For example: T is the subject of T = 4x + 1. β’ To use a formula, first substitute all the known values into the formula, then solve the resulting equation, if possible.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 9
Equations
BUILDING UNDERSTANDING 1 State whether each of the following equations is a rule (β R)βor not a rule (β N)ββ. a β2x + 5 = 10β b βy = 3x + 5β c βF = maβ d β5 β q = 3β e βw = 12 β vβ f βP = I + k β 3β
U N SA C O M R PL R E EC PA T E G D ES
630
2 Substitute: a βx = 3βinto the expression β5xβ
b βx = 7βinto the expression β4(x + 2)β y+4 7
c βy = 3βinto the expression β20 β 4yβ
d βy = 10βinto the expression _ β β
Example 14 Applying a formula
Consider the rule βk = 3b + 2β. Find the value of: a kβ βif βb = 5β b kβ βif βb = 10β
c βbβif βk = 23.β
SOLUTION
EXPLANATION
a kβ β―β― β =β 3 Γ 5 + 2β β = 17 b k = 3 Γ 10 + 2 β β ββ―β― β β = 32
Substitute βb = 5βinto the equation.
c
Substitute βk = 23βinto the equation. Now solve the equation to find the value of βbβ.
23 = 3b + 2
β2
Γ·3
21 = 3b 7=b
Therefore, βb = 7β.
Substitute βb = 10βinto the equation.
β2
Γ·3
Write the final solution as an equation with the variable on the left-hand side.
Now you try
Using the formula βB = 5A + 7β, find the value of: a B β βif βA = 2β b B β βif βA = 12β
c βAβif βB = 52.β
Example 15 Applying a formula involving three pronumerals Consider the rule βQ = w(4 + t)β. Find the value of: a Q β βif βw = 10βand βt = 3β
b βtβif βQ = 42βand βw = 6β
SOLUTION
EXPLANATION
a Q = 10(4 + 3) β β―β― β =β 10 Γ 7β β β = 70
Substitute βw = 10βand βt = 3βto evaluate.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9G Using formulas
42 = 6(4 + t) 42 = 24 + 6t
b β 24
3=t
Substitute βQ = 42βand βw = 6β. Expand the brackets and then solve the equation.
β 24 Γ·6
U N SA C O M R PL R E EC PA T E G D ES
Γ·6
18 = 6t
631
Therefore βt = 3β.
Write the final solution as an equation with the variable on the left-hand side.
Now you try
Consider the rule βF = mg β 5β. Find the value of: a βFβif βm = 7βand βg = 10β
b βmβif βF = 93βand βg = 9.8.β
Exercise 9G FLUENCY
1
Example 14a,b Example 14c Example 14
Example 14
Example 15
Example 15
1β4
Consider the rule βk = 5b + 3β. Find the value of: a i βkβif βb = 2β b i
βbβif βk = 23β
2β5
2, 4, 5
ii βkβif βb = 7β
ii βbβif βk = 58β
2 Consider the rule βh = 2m + 1β. Find: a βhβif βm = 3β b βhβif βm = 4β c m β βif βh = 17β. Set up an equation and solve it algebraically d βmβif βh = 21β. Set up an equation and solve it algebraically. 3 Consider the formula βy = 5 + 3xβ. Find: a βyβif βx = 6β b βxβif βy = 17β
c βxβif βy = 26β
4 Consider the rule βA = q + t.β Find: a βAβif βq = 3βand βt = 4β b βqβif βA = 5βand βt = 1β
c βtβif βA = 3βand βq = 3β
5 Consider the formula βG = 7x + 2yβ. Find: a βGβif βx = 3βand βy = 3β b βxβif βy = 2βand βG = 11β
c βyβif βG = 31βand βx = 3β
PROBLEM-SOLVING
6, 7
6β8
7β9
6 The formula for the area of a rectangle is βA = l Γ wβ, where βlβis the rectangleβs length and βwβis its width. A
w
l
a Set up and solve an equation to find the width of a rectangle with βA = 20βand βl = 5β. b A rectangle is drawn for which βA = 25βand βw = 5β. i Set up and solve an equation to find l. ii What type of rectangle is this? Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 9
Equations
7 The perimeter for a rectangle is given by βP = 2(w + l)β. Find the: a perimeter when βw = 3βand βl = 5β b value of βlβwhen βP = 10βand βw = 2β c area of a rectangle if its perimeter is β20βand width is β5.β 8 A formula relating density (β d)β, mass (β m)βand volume (β v)βis βd = _ βm v β.β a Find the value of βdβif βm = 12βand βv = 4β. b Find the density of an object with a mass of 12 grams, and a volume of β4 cβ mβ3β β, giving an answer
U N SA C O M R PL R E EC PA T E G D ES
632
with the units βg / βcmβ3β ββ.
c Another object has a density of β10 g/ βcmβ3β βand a volume of β5 βcmβ3β β. Write an equation and solve it to find its mass, m.
9 To convert between temperatures in Celsius 9C β+ 32β. and Fahrenheit the rule is βF = β_ 5 a Find βFβif βC = 20β. b Find the value of βCβif βF = 50β. c Find the temperature in Celsius if it is β 53.6Β°β Fahrenheit. d Marieko claims that the temperature in her city varies between β68Β°βFahrenheit and β 95Β°βFahrenheit. What is the difference, in Celsius, between these two temperatures? REASONING
10
10, 11
11, 12
10 There are two different formulas that can be used to calculate a personβs maximum heart rate β(H)β from their age β(a)β. Formula β1β: βH = 220 β aβ Formula β2β: βH = 206.9 β 0.67aβ a Find the two values of βHβthat the formulas give for a β10β-year-old person by substituting βa = 10.β b Find the two values of βHβthat the formulas give for a β100β-year-old person. c Demonstrate that both formulas give a similar maximum heart rate for a β40β-year-old person. d Use formula β1βto find the age of someone with a maximum heart rate of β190β. e Use formula β2βto find the age of someone with a maximum heart rate of β190β. Round to the nearest whole number. 11 Rearranging a formula involves finding an equivalent equation that has a different variable on one side by itself. For example, the formula βS = 6g + bβ can be rearranged to make βgβby itself. Now we have a formula that can be used to find βgβonce βSβand βbβare known. a Rearrange βS = 5d + 3bβto make a rule where βdβis by itself. b
Rearrange the formula βF = _ β9C β+ 32βto make βCβby itself.
5 c Rearrange the formula βQ = 3(x + 12) + xβto make βxβby itself. (Hint: You will need to expand the brackets first.)
βb Γ·6
S = 6g + b S β b = 6g
βb
Γ·6
Sβb =g 6
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9G Using formulas
633
12 A taxi company charges different amounts of money based on how far the taxi travels and how long the passenger is in the car. Although the company has not revealed the formula it uses, some sample costs are shown below. Time (t) in minutes
Cost (C) in dollars
β10β
β20β
3β 0β
β20β
β30β
5β 0β
U N SA C O M R PL R E EC PA T E G D ES
Distance (D) in km
a Show that the rule βC = D + tβis consistent with the values above. b Show that the rule βC = 3Dβis not consistent with the values above. c Show that the rule βC = 2D + 10β is consistent with the values above. d Try to find at least two other formulas that the taxi company could be using, based on the values shown.
ENRICHMENT: AFL equations
β
β
13
13 In Australian Rules Football (AFL), the score, βSβ, is given by βS = 6g + bβ, where βgβis the number of goals scored and βbβis the number of βbehindsβ (i.e. near misses). a Which team is winning if the Abbotsford Apes have scored β11βgoals (βg = 11β) and β9βbehinds (βb = 9β), whereas the Brunswick Baboons have scored β12βgoals and β2β behinds? b The Camberwell Chimpanzees have scored β7βbehinds and their current score is βS = 55β. Solve an equation algebraically to find how many goals the team has scored. c In some AFL competitions, a team can score a βsupergoalβ, which is worth β9βpoints. If βqβis the number of supergoals that a team kicks, write a new formula for the teamβs score. d For some rare combinations of goals and behinds, the score equals the product of βgβand βbβ. For example, β4βgoals and β8βbehinds gives a score of β4 Γ 6 + 8 = 32β, and β4 Γ 8 = 32β. Find all the other values of βgβand βbβthat make the equation β6g + b = gbβ true.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 9
Equations
9H Using equations to solve problems LEARNING INTENTIONS β’ To know that equations can be applied to real-world situations β’ To be able to solve problems using equations
U N SA C O M R PL R E EC PA T E G D ES
634
Our methods for solving equations can be applied to many situations in which equations occur.
Digital advertisers calculate cost per click, $c. If an ad that cost $2000 had 10 000 clicks, then 10 000 c = 2000. Solving this equation gives c = 0.2, i.e. 20 cents per click.
Lesson starter: Stationery shopping
Sylvia bought 10 pencils and 2 erasers for $20.40. Edward bought 5 pencils and 3 erasers for $12.60. β’ β’ β’
Use the information above to work out how much Karl will pay for 6 pencils and 5 erasers. Describe how you got your answer. Is there more than one possible solution?
KEY IDEAS
β To solve a problem, follow these steps.
Step 1: Use a pronumeral to stand in for the unknown.
Let $p = the cost of a pencil.
Step 2: Write an equation to describe the problem.
10p + 2 Γ 3.5 = 25.
Step 3: Solve the equation.
This can be done by inspection or algebraically.
Step 4: Make sure that you answer the original question and the solution seems reasonable and realistic.
Donβt forget to include the correct units (e.g. dollars, years, cm).
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9H Using equations to solve problems
635
BUILDING UNDERSTANDING
U N SA C O M R PL R E EC PA T E G D ES
1 For each of the following problems, choose the best pronumeral definition. a Problem: Moniqueβs age next year is β12β. How old is she now? A Let βm =βMoniqueβs current age. B Let βm = Moniqueβ. C Let βm = 12β. D Let βm =βMoniqueβs age next year. E Let βm = this yearβ. b Problem: Callan has β15βboxes, which weigh a total of β300 kgβ. How much does each box
weigh? A Let βw = 15β. B Let βw = 300β. C Let βw = the weight in kilograms of one boxβ. D Let βw = the number of boxesβ. E Let βw = the total weight in kilogramsβ. c Problem: Jaredβs family has a farm with cows and sheep. The total number of animals is β200β and there are β71βcows. How many sheep are there? A Let βx = the size of a sheepβ. B Let βx = the total number of animalsβ. C Let βx = the number of sheepβ. D Let βx = the number of cowsβ. E Let βx =βJaredβs age.
Example 16 Solving a problem using equations
The sum of Kateβs current age in years and her age next year is β19β. How old is Kate? SOLUTION
EXPLANATION
Let βk =βKateβs current age in years.
Define a pronumeral to stand for the unknown number.
βk + (k + 1) = 19β
Write an equation to describe the situation. Note that β k + 1βis Kateβs age next year.
β1
Γ·2
2k + 1 = 19 2k = 18 k=9
β1
Simplify the LHS and then solve the equation algebraically.
Γ·2
Kate is currently β9βyears old.
Answer the original question.
Now you try
Hiring a cabin costs β$80βper day, plus you must pay a β$30βcleaning fee. Given that the total cost is β $350β, how many days was the cabin hired?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
636
Chapter 9
Equations
Exercise 9H FLUENCY
1
2β5
3β5
The sum of Jerryβs current age and his age next year is β43β. How old is Jerry?
U N SA C O M R PL R E EC PA T E G D ES
Example 16
1β4
Example 16
Example 16
Example 16
Example 16
2 Launz buys a car and a trailer for a combined cost of β$40 000β. The trailer costs β$2000β. a Define a pronumeral for the carβs cost. b Write an equation to describe the problem. c Solve the equation algebraically. d Hence, state the cost of the car. 3 Valentina buys β12βpens for a total cost of β$15.60β. a Define a pronumeral for the cost of one pen. c Solve the equation algebraically.
b Write an equation to describe the problem. d Hence, state the cost of one pen.
4 Isabella is paid β$17βper hour and gets paid a bonus of β$65βeach week. One particular week she earned β$643β. a Define a pronumeral for the number of hours Isabella worked. b Write an equation to describe the problem. c Solve the equation algebraically. d How many hours did Isabella work in that week? 5 This rectangular paddock has an area of β720 βm2β ββ. a Write an equation to describe the problem, using βwβfor the paddockβs width in metres. b Solve the equation algebraically. c How wide is the paddock? d What is the paddockβs perimeter? PROBLEM-SOLVING
6, 7
wm
24 m
7β10
10β13
6 A number is doubled, then β3βis added and the result is doubled again. This gives a final result of β34β. Set up and solve an equation to find the original number, showing all the steps clearly. 7 The perimeter of the shape shown is β30β. Find the value of βxβ.
6
x
2x
12
8 Alexa watches some television on Monday, then twice as many hours on Tuesday, then twice as many 1 βhours from Monday to Wednesday, how much hours again on Wednesday. If she watches a total of β10β_ 2 television did Alexa watch on Monday?
9 Marcus and Saraβs combined age is β30β. Given that Sara is β2βyears older than Marcus, write an equation and find Marcusβ age. 10 An isosceles triangle is shown at right. Write an equation and solve it to find βxΒ°β, the unknown angle. (Remember: The sum of angles in a triangle is β180Β°β.)
154Β° xΒ°
xΒ°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9H Using equations to solve problems
637
11 Find the value of βyβin the triangle shown here, by first writing an equation. yΒ° (2y)Β°
U N SA C O M R PL R E EC PA T E G D ES
12 A rectangle has width βwβand height βhβ. The perimeter and area of the rectangle are equal. Write an equation and solve it by inspection to find some possible values for βwβand βhβ. (Note: There are many solutions to this equation. Try to find a few.) 13 Find the values of βxβand βyβin the rectangle shown.
2x + 3
3y
6
10
REASONING
14
14, 15
15, 16
14 In some situations an equation makes sense, but in other situations it does not make sense. In this question you will consider the equation β3c + 10 = 4β. a Theodore set up an equation to be used to find the number of chairs in a room (β c)β. The equation he wrote was β3c + 10 = 4β. Explain why he must have made a mistake. b Benedict set up an equation to be used to find the temperature outside in βΒ° Cβ. He also wrote the equation β3c + 10 = 4β. Explain why this equation could be used for temperature even though it could not be used for the number of chairs. 15 If photocopying costs β35βcents a page and βpβis the number of pages photocopied, which of the following equations have possible solutions? Justify your answers. (Note: Fraction answers are not possible because you must still pay β35βcents even if you photocopy only part of a page.) a β0.35p = 4.20β b β0.35p = 2.90β c β0.35p = 2.80β
16 Assume that an isosceles triangle is drawn so that each of its three angles is a whole number of degrees. Prove that the angle βaβmust be an even number of degrees.
aΒ°
bΒ°
ENRICHMENT: Strange triangles
β
β
17 Recall that the sum of angles in a triangle is β180Β°β. a David proposes the following triangle, which is not drawn (x + 100)Β° to scale. i Find the value of βxβ. (x β70)Β° ii Explain what makes this triangle impossible. b Helena proposes the following triangle, which is also not drawn to scale. (60 β x)Β° i Explain why the information in the diagram is not enough to find βxβ. (70 + x)Β° ii What are the possible values that βxβcould take? c Design a geometric puzzle, like the one in part a, for which there can be no solution.
bΒ°
17
30Β°
50Β°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
638
Chapter 9
Equations
Modelling
β100 mβGift handicap
U N SA C O M R PL R E EC PA T E G D ES
Each year the town of Walpole runs a special race called the Walpole Gift where each runner is assigned a handicap advantage. A runnerβs handicap advantage is the number of metres provided as a head-start in front of the start line. This means a runner with a β10βmetre handicap advantage only has to run the remaining β90βmetres. Their handicap advantage is determined by their previous performance. Three runners, Peter, Jethro and Leonard, compare their given handicap for the upcoming Walpole Gift. They also estimate their average speed for the race in m/s (metres per second). Runner
Peter
Jethro
Leonard
Handicap advantage
β10 mβ
4β mβ
β0 mβ
Average speed estimate
β7 m / sβ
β8 m / sβ
β9 m / sβ
Present a report for the following tasks and ensure that you show clear mathematical workings and explanations where appropriate.
Preliminary task
a Complete the following table for Peter where time is in seconds and the distance is measured from the start line. The first two distances are completed for you. Time β(s)β
β0β
Distance from start line β(m)β
β10β β17β
β1β
β2β
β3β
β4β
β5β
β6β
β7β
β8β
β9β
1β 0β
b Approximately how long does it take for Peter to complete the race? Round to the nearest second. c How long does it take for Peter to reach the point β6βmetres before the finish line?
Modelling task
Formulate
a The problem is to determine who wins the Walpole Gift and how this is affected by the handicaps. Write down all the relevant information that will help solve this problem. b Using βdβmetres as the distance from the starting line and βtβfor the time in seconds, write rules for the distance covered for: i Peter using his estimated speed of β7 m / sβand handicap of β10 mβ ii Jethro using his estimated speed of β8 m / sβand handicap of β4 mβ iii Leonard using his estimated speed of β9 m / sβwith no handicap. c Use your rules to find the distance from the start line of: i Peter after β5β seconds ii Jethro after β8β seconds iii Leonard after β6β seconds. d Use your rules to find when: i Peter reaches the β52 mβ mark ii Jethro reaches the β28 mβ mark iii Leonard reaches the β90 mβ mark.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Modelling
Solve
U N SA C O M R PL R E EC PA T E G D ES
Modelling
Evaluate and verify
e After how many seconds does: i Leonard catch Peter? ii Jethro catch Peter? iii Leonard catch Jethro? f Determine the placings for the three men. g Compare the performance of each of the men in the race. Include a description of when and where on the β100 mβtrack anyone got overtaken. h Assume that the three men kept the same speed in next yearβs race, but their handicaps could change. Give an examples of handicaps that would result in: i Peter beating Leonard ii Jethro beating Leonard. i Summarise your results and describe any key findings.
639
Communicate
Extension questions
a Assume the original handicap advantage stayed the same next year, but that Peter and Jethro could run faster. i At what speed would Peter need to travel so that he beats Leonard? ii At what speed would Jethro need to travel so that he beats Leonard? b Assume the original handicap advantage stayed the same next year and all three runners had the same speed as this year, but the track could be longer or shorter than β100βmetres. Investigate how this would affect the handicaps required by Peter and Jethro.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
640
Chapter 9
Equations
Key technology: Spreadsheets The size and shape of the things we make depend on the amount of material that we have and any restrictions that we might have regarding the shapeβs dimensions. For example, making a rectangular pen with β 100βmetres of fencing can result in many different shaped rectangles. Depending on the relationship between the length and the width, we can determine the dimensions of the rectangle using equations.
U N SA C O M R PL R E EC PA T E G D ES
Technology and computational thinking
Making a pen with β100β metres
1 Getting started
Working with β100βmetres of fencing, letβs consider making a rectangular pen where the length is β10β metres more than the width. Let βxβmetres be the width of the rectangle. a Write an expression for the following. i length of the rectangle ii perimeter of the rectangle b Using the fact that there is a total of β100βm of fencing available, write an equation. c Simplify and solve your equation to find the width of the pen. d Repeat parts aβc above if the length of the pen is β5βmetres more than the width.
2 Using technology
We will set up a spreadsheet to assist in the calculation of the dimensions of the pen for varying lengths and widths. a Construct a spreadsheet as shown including an adjustable amount which is the difference between the length and width of the pen. b Fill down at cells βA7β, βB6βand βC6βuntil you find the width and length that give the correct perimeter of β100 mβ. c Now change the length and width difference in cell βC3βto β20βmetres. What does your spreadsheet say about the dimensions required to produce a rectangular pen using β100 mβof fencing?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Technology and computational thinking
641
3 Applying an algorithm
U N SA C O M R PL R E EC PA T E G D ES
Technology and computational thinking
a Now change the length and width difference to β5βmetres. What do you notice about the results in your spreadsheet? b Make an adjustment to the formula in cell β A7βso that your spreadsheet will show the required width and length to create a pen using β100 mβof fencing. Shown here is the formula in cell βA7βwhere the increment has been adjusted to 0.5. c Now try adjusting the difference in cell βC3βto be β25βand write down the dimensions of the pen where the perimeter is β100 mβ. d Describe the shape of the pen if the value entered into cell βC3βis β0β. e Further adjustment of the formula in cell βA7βmay be needed for non-integer values in cell βC3β. Use your spreadsheet to find the dimensions of the pen if the difference in the width and length is the following. i β12.5β metres ii β24.2β metres iii 1β 8.7β metres
4 Extension
a Using the pronumeral βdβfor the difference between the length and the width of the pen, write expressions for the following. Remember that βxβmetres is the width of the pen. i length of the rectangle ii perimeter of the rectangle b Using the fact that there is a total of β100 mβof fencing available, write an equation using the variables βxβand βdβ. c Simplify and solve your equation to find the width of the pen. Your answer should be an expression in terms of βdβ. d Use your result from part c to check some of your conclusions from part 3 above. For example, if β d = 12.5β, find the value of βxβthat creates a pen using β100 mβof fencing.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
642
Chapter 9
Equations
There are thousands of theme parks all over the world which offer a vast array of rides that are built to thrill. By searching the internet, you can discover the longest, tallest, fastest and scariest rides. Although prices are kept competitive, theme parks need to make a profit so that they can maintain safety standards and continue to build new and more exciting rides.
U N SA C O M R PL R E EC PA T E G D ES
Investigation
Theme parks
Thrill World and Extreme Park are two theme parks. Both charge different prices for entry and for each ride. Their prices are: β’ Thrill World: β$20βentry and β$5βper ride β’ Extreme Park: β$60βentry and β$3βper ride. a Copy and complete the table below for each theme park. The total cost for the day includes the entry cost and cost of the rides. β...β
Number of rides (n)
β1β
Thrill World total cost $T
β$25β
.β..β
Extreme Park total cost $E
β$63β
.β..β
β2β β3β β4β β5β β6β β7β β8β
2β 0β β21β β22β β23β β24β β25β
b Write a formula for: i βTβ, the total cost, in dollars, for βnβrides at Thrill World ii βEβ, the total cost, in dollars, for βnβrides at Extreme Park. c For each of these thrill seekers, use an equation to calculate how many rides they went on. i Amanda, who spent β$105β at Thrill World ii George, who spent β$117βat Extreme Park d Refer to your completed table to determine the number of rides that will make the total cost for the day the same at each theme park. e A third theme park, Fun World, decides to charge no entry fee but to charge β$10βper ride. Find the minimum number of rides that you could go on at Fun World before it becomes cheaper at: i Thrill World ii Extreme Park.
Investigate how much Fun World should charge to attract customers while still making profits that are similar to those of Thrill World and Extreme Park. Provide some mathematical calculations to support your conclusions.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Problems and challenges
Problems and challenges
Find the unknown number in the following Up for a challenge? If you get stuck problems. on a question, check out the βWorking with unfamiliar problemsβ poster at a A number is added to half of itself and the the end of the book to help you. result is β39β. b A number is doubled, then tripled, then quadrupled. The result is β696β. c One-quarter of a number is subtracted from β100βand the result is β8β. d Half of a number is added to β47β, and the result is the same as the original number doubled. e A number is increased by β4β, the result is doubled and then β4βis added again to give an answer of β84β.
U N SA C O M R PL R E EC PA T E G D ES
1
643
2 A triangle has sides β2x + 7, 3x + 4βand β4(β x β 4)βmeasured in cm. If the perimeter of the triangle is β220 cmβ, find the difference between the longest and shortest sides.
(3x + 4) cm
4(x β 4) cm
P = 220 cm
(2x + 7) cm
3 For this rectangle, find the values of βxβand βyβand determine the perimeter. The sides are measured in cm.
2x + 13
y
2x β 5
27
4 A rectangular property has a perimeter of β378 mβ. It is divided up into five identical rectangular house blocks as shown in this diagram. Determine the area of each house block.
5 Find the values of βaβ, βbβand βcβ, given the clues: β5(β a + 2)β+ 3 = 38βand β2(β b + 6)ββ 2 = 14βand β3a + 2b + c = 31β
6 Find the values of βaβand βbβfor each of these geometric figures. a b (2b)Β°
bΒ°
(3a)Β°
(5a)Β°
32Β°
(7a + 4)Β°
(3(b + 2) β 1)Β°
7 Jamesβs nanna keeps goats and chickens in her backyard. She says to James, βThere are β41βanimals in this yardβ. James says to his nanna, βThere are β134βanimal legs in this yardβ. How many goats and how many chickens are in the paddock?
8 Five consecutive numbers have the middle number β3x β 1β. If the sum of these five numbers is β670β, find the value of βxβand list the five numbers.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
644
Chapter 9
Equations
Equivalent equations
U N SA C O M R PL R E EC PA T E G D ES
Chapter summary
Solving an equation Finding pronumeral values to make equation true e.g. 15 + x = 20 solution: x = 5
Equivalent equations are obtained by performing the same operation to both sides
Formulas
Formulas or rules are types of equations. e.g. F = ma, A = l Γ w, P = 4x Solving equations
e.g.
e.g.
e.g.
β 5 3x + 5 = 38 β 5 3x = 33 Γ· 3 x = 11 Γ· 3 Γ4
x =7 4
x = 28
Γ4
Equations with fractions are solved by multiplying by the denominator when fraction is by itself.
β2
Γ7
e.g.
+8
2 + x7 = 4 x =2 7
x = 14
β2
Γ7
20 + 3x = 13
+8
To solve algebraically, use equivalent equations and opposite operations. Main expression
First step
3x + 5 12x 5x β 2
β5 Γ·12 +2 Γ4
x 4
Equations with brackets (Ext)
Brackets should be expanded and like items collected. Use the distributive law to expand. e.g.
4(x + 2) = 4x + 8
5(x + 3) + 2x = 29 5x + 15 + 2x = 29 β 15 7x + 15 = 29 β 15 7x = 14 Γ·7 x = 2 Γ·7
e.g.
3x + 9 =6 4
Γ4 3x + 9 = 24 β9 β9 3x = 15 Γ·3 x = 5 Γ·3
Γ4
12 + 3x = 5
Equations
+ 2 5x β 2 = 33 + 2 Γ· 5 5x = 35 Γ· 5 x=7
Equations with fractions (Ext)
e.g.
e.g.
Applications
1 Use a pronumeral for the unknown. 2 Write an equation. 3 Solve the equation. 4 Answer the original question.
e.g. Johnβs age in 3 years will be 15. How old is he now? 1 Let j = Johnβs current age in years 2 j + 3 = 15 3 j + 3 = 15 β3 β3 j = 12 4 John is currently 12 years old.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter checklist
645
Chapter checklist with success criteria 1. I can identify an equation. e.g. Which of the following are equations? β7 + 7 = 18, 2 + 12, 4 = 12 β x, 3 + uβ.
U N SA C O M R PL R E EC PA T E G D ES
9A
β
Chapter checklist
A printable version of this checklist is available in the Interactive Textbook
9A
2. I can classify an equation as true or false, given the value of any pronumerals. e.g. Decide whether β10 + b = 3b + 1βis true when βb = 4β.
9A
3. I can write an equation from an English description. e.g. Write an equation for βthe sum of βxβand β5βis β22β.β
9B
4. I can solve equations by inspection. e.g. Solve the equation β5b = 20βby inspection.
9C
5. I can apply an operation to both sides of an equation. e.g. State the result of adding β4βto both sides of the equation β7q β 4 = 10β.
9C
6. I can show that two equations are equivalent. e.g. Show that β5 = 7 β xβand β10 = 2β(7 β x)βare equivalent by stating the operation used.
9D
7. I can solve one-step equations using equivalent equations. e.g. Solve β5x = 30β algebraically.
9D
8. I can solve two-step equations using equivalent equations. e.g. Solve β7 + 4a = 23β algebraically.
9D
9. I can check the solution to an equation using substitution. e.g. Check that the solution to β12 = 2β(e + 1)βis βe = 5β.
9E
10. I can solve equations involving fractions. e.g. Solve _ β3x β+ 7 = 13β. 4
Ext
9F
11. I can expand brackets. e.g. Expand β5(β 3a + 4)ββ.
Ext
9F
12. I can collect like terms. e.g. Simplify β3a + 8a + 2 β 2a + 5β.
Ext
9F
13. I can solve equations involving brackets. e.g. Solve β3(β b + 5)β+ 4b = 29β.
Ext
9G
14. I can apply formulas when the unknown is by itself. e.g. For the rule βk = 3b + 2β, find the value of βkβif βb = 10β.
9G
15. I can apply formulas when an equation must be solved. e.g. For the rule βk = 3b + 2β, find the value of βbβif βk = 23β.
9H
16. I can solve problems using equations. e.g. The sum of Kateβs current age and her age next year is β19β. Use an equation to determine how old Kate is.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
646
Chapter 9
Equations
9A
1
Classify each of the following equations as true or false. a 4β + 2 = 10 β 2β b β2β(3 + 5)β= 4β(1 + 3)β c β5w + 1 = 11,βif βw = 2β d β2x + 5 = 12,βif βx = 4β e βy = 3y β 2,βif βy = 1β f β4 = z + 2,βif βz = 3β
U N SA C O M R PL R E EC PA T E G D ES
Chapter review
Short-answer questions
9A
9B
9C
9D
9E
Ext
9F
Ext
2 Write an equation for each of the following situations. You do not need to solve the equations. a The sum of β2βand βuβis β22β. b The product of βkβand β5βis β41β. c When βzβis tripled the result is β36β. d The sum of βaβand βbβis β15β. 3 Solve the following equations by inspection. a βx + 1 = 4β b βx + 8 = 14β d βy β 7 = 2β e 5β a = 10β
c 9β + y = 10β f _ βa β= 2β 5
4 For each equation, find the result of applying the given operation to both sides and then simplify. a 2β x + 5 = 13 β[β 5]β b β7a + 4 = 32 β[β 4]β c β12 = 3r β 3 β[+ 3]β d β15 = 8p β 1 β[+ 1]β 5 Solve each of the following equations algebraically. a 5β x = 15β b βr + 25 = 70β c β12p + 17 = 125β d β12 = 4b β 12β e β5 = 2x β 13β f β13 = 2r + 5β g β10 = 4q + 2β h β8u + 2 = 66β 6 Solve the following equations algebraically. 8p a _ β3u β= 6β b _ β β= 8β 4 3 2y + 20 5y _ e β4 = _ β β d β β+ 10 = 30β 7 2
c β3 = _ β2x + 1 β 3 4x β+ 4 = 24β f β_ 3
7 Expand the brackets in each of the following expressions. a 2β β(3 + 2p)β b β4β(3x + 12)β c β7β(a + 5)β d β9β(2x + 1)β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
9F
U N SA C O M R PL R E EC PA T E G D ES
Chapter review
Ext
8 Solve each of these equations by expanding the brackets first. a β2(β x β 3)β= 10β b β27 = 3β(x + 1)β c 4β 8 = 8β(x β 1)β d β60 = 3y + 2β(y + 5)β e 7β β(2z + 1)β+ 3 = 80β f β2β(5 + 3q)β+ 4q = 40β
647
9F
Ext
9F
Ext
9G
9G
9H
9H
9 Consider the equation β4(β x + 3)β+ 7x β 9 = 10β. a Is βx = 2βa solution? b Show that the solution to this equation is not a whole number.
10 a Does β3β(2x + 2)ββ 6x + 4 = 15βhave a solution? Justify your answer. b State whether the following are solutions to β5(β x + 3)ββ 3β(x + 2)β= 2x + 9β. i xβ = 2β ii xβ = 3β
1 βhβ(a + b)β, where βhβis the height of the 11 The formula for the area of a trapezium is βA = β_ 2 trapezium, and βaβand βbβrepresent the parallel sides. a Set up and solve an equation to find the area of a trapezium with height β20 cmβand parallel sides of 15 cm and 30 cm. b Find the height of a trapezium whose area is β55 βcmβ2β βand has parallel sides β6 cmβand β5 cmβ, respectively. 12 Consider the rule βF = 3a + 2bβ. Find: a F β βif βa = 10βand βb = 3β b βbβif βF = 27βand βa = 5β c aβ βif βF = 25βand βb = 8β
13 For each of the following problems, write an equation and solve it to find the unknown value. a A number is added to three times itself and the result is β20β. What is the number? b The product of β5βand a number is β30β. What is the number? c Juanitaβs mother is twice as old as Juanita. The sum of their ages is β60β. How old is Juanita? d A rectangle has a width of β21 cmβand a perimeter of β54 cmβ. What is its length? 14 Find the value of βyβfor each of these figures. a (y + 3)Β° yΒ°
(y + 2)Β°
(2y)Β°
b
(2y)Β°
(4y + 20)Β°
(2y + 10)Β°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
648
Chapter 9
Equations
9A
1
If βx = 3β, which one of the following equations is true? A 4β x = 21β B 2β x + 4 = 12β C β9 β x = 6β D β2 = x + 1β E βx β 3 = 4β
U N SA C O M R PL R E EC PA T E G D ES
Chapter review
Multiple-choice questions
9A
9B
9C
9D
9D
9E
Ext
9E
Ext
2 When β11βis added to the product of β3βand βxβ, the result is β53β. This can be written as: A 3β x + 11 = 53β B 3β β(x + 11)β= 53β C β_x β+ 11 = 53β 3 _ D βx + 11 β= 53β 3 E β3x β 11 = 53β
3 Which of the following values of βxβmake the equation β2β(x + 4)β= 3xβ true? A β2β B β4β C β6β D β8β E β10β
4 The equivalent equation that results from subtracting β3βfrom both sides of β12x β 3 = 27β is: A 1β 2x = 24β B 1β 2x β 6 = 24β C β12x β 6 = 30β D β9x β 3 = 24β E β12x = 30β 5 To solve β3a + 5 = 17β, the first step to apply to both sides is to: A add β5β B divide by β3β C subtract β17β D divide by β5β E subtract β5β 6 The solution to β2t β 4 = 6β is: A tβ = 1β D βt = 7β
B tβ = 3β E βt = 9β
C βt = 5β
7 The solution of _ β2x β= 10β is: 7 A βx = 35β D βx = 30β
B xβ = 70β E βx = 5β
C βx = 20β
3p + 5 8 The solution to the equation β10 = _ β ββis: 2 A pβ = 5β B βp = 20β D βp = 7β E βp = 1β
C βp = 15β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
9F
10 A formula relating βAβ, βpβand βtβis βA = 3p β tβ. If βA = 24βand βt = 6β, then βpβ equals: A β18β B β4β C β30β D β2β E β10β
U N SA C O M R PL R E EC PA T E G D ES
9G
C βu = 11β
Chapter review
Ext
9 The solution of β3(β u + 1)β= 15β is: A βu = 5β B uβ = 4β D βu = 6β E βu = 3β
649
Extended-response questions 1
Udhavβs satellite phone plan charges a connection fee of β15βcents and then β2βcents per second for every call. a How much does a β30β-second call cost? b Write a rule for the total cost, βCβ, in cents, for a call that lasts βtβ seconds. c Use your rule to find the cost of a call that lasts β80β seconds. d If a call cost β39βcents, how long did it last? Solve an equation to find βtβ. e If a call cost β$1.77β, how long did it last? f On a particular day, Udhav makes two calls β the second one lasting twice as long as the first, with a total cost of β$3.30β. What was the total amount of time he spent on the phone?
2 Gemma is paid β$xβper hour from Monday to Friday, but earns an extra β$2βper hour during weekends. During a particular week, she worked β30βhours during the week and then β10β hours on the weekend. a If βx = 12β, calculate the total wages Gemma was paid that week. b Explain why her weekly wage is given by the rule βW = 30x + 10(x + 2)β. c Use the rule to find Gemmaβs weekly wage if βx = 16β. d If Gemma earns β$620βin one week, find the value of βxβ. e If Gemma earns β$860βin one week, how much did she earn from Monday to Friday?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
U N SA C O M R PL R E EC PA T E G D ES
10 Measurement
Maths in context: Measurement is important everywhere Measurement skills are essential for all practical occupations, including: β’ all engineers, architects, auto mechanics, bakers, boiler makers, bricklayers, builders, carpenters and chefs β’ construction workers, electricians, farmers, hairdressers, house painters, mechanics, miners and nurses β’ pharmacists, plumbers, roboticists, seamstresses, sheet metal workers, surveyors, tilers, vets and welders. Practical workers use a variety of measurement units, illustrated by these facts. β’ The Eiffel Tower in France is painted with 50 tonnes of paint every 7 years. β’ The Sydney Harbour Bridge is repainted every 5 years by two robots and 100 painters, using 30 000 litres of paint weighing 140 tonnes.
β’ The Sydney Harbour Bridgeβs surface area is 485 000 m2 (around 60 football ο¬elds). β’ The volume of water in Sydney Harbour is about 500 gigalitres or ο¬ve hundred thousand million litres or 0.5 km3. β’ The Great Wall of China is more than 6000 km long. β’ Australiaβs unbroken dingo-proof fence is 5614 km long and runs from South Australia to Queensland. β’ The Great Pyramid of Giza, built in 2500 BCE, is made of 2.5 million stone blocks that each weigh about 2300 kg (2.3 tonnes), heavier than a Tesla Model 3 with a mass of 1800 kg or 1.8 tonnes! β’ The maximum daytime temperature on Mars is about 20Β°C, but a freezing β65Β°C at night.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter contents
U N SA C O M R PL R E EC PA T E G D ES
10A Metric units of length (CONSOLIDATING) 10B Perimeter (CONSOLIDATING) 10C Circles, Ο and circumference 10D Arc length and perimeter of sectors and composite shapes (EXTENDING) 10E Units of area and area of rectangles 10F Area of parallelograms 10G Area of triangles 10H Area of composite shapes (EXTENDING) 10I Volume of rectangular prisms 10J Volume of triangular prisms 10K Capacity (CONSOLIDATING) 10L Mass and temperature (CONSOLIDATING)
Victorian Curriculum 2.0
This chapter covers the following content descriptors in the Victorian Curriculum 2.0: MEASUREMENT
VC2M7M01, VC2M7M02, VC2M7M03, VC2M7M06 ALGEBRA
VC2M7A01, VC2M7A03, VC2M7A06
Please refer to the curriculum support documentation in the teacher resources for a full and comprehensive mapping of this chapter to the related curriculum content descriptors.
Β© VCAA
Online resources
A host of additional online resources are included as part of your Interactive Textbook, including HOTmaths content, video demonstrations of all worked examples, auto-marked quizzes and much more.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10
Measurement
10A Metric units of length CONSOLIDATING LEARNING INTENTIONS β’ To be able to choose a suitable unit for a length within the metric system β’ To be able to convert between metric lengths (km, m, cm and mm) β’ To be able to read a length shown on a ruler or tape measure
U N SA C O M R PL R E EC PA T E G D ES
652
The metric system was developed in France in the 1790s and is the universally accepted system today. The word metric comes from the Greek word metron, meaning βmeasureβ. It is a decimal system where length measures are based on the unit called the metre. The definition of the metre has changed over time.
Originally, it was proposed to be the length of a pendulum that beats at a rate of one beat per second. It was later defined as 1/10 000 000 of the distance from the North Pole to the equator on a line on Earthβs surface passing through Paris. In 1960, a metre became 1 650 763.73 wavelengths of the spectrum of the krypton-86 atom in a vacuum. In 1983, the metre was defined as the distance that light travels in 1/299 792 458 seconds inside a vacuum. To avoid the use of very large and very small numbers, an appropriate unit is often chosen to measure a length or distance. It may also be necessary to convert units of length. For example, 150 pieces of timber, each measured in centimetres, may need to be communicated as a total length using metres. Another example might be that 5 millimetres is to be cut from a length of timber 1.4 metres long because it is too wide to fit a door opening that is 139.5 centimetres wide.
A carpenter may need to measure lengths in metres, centimetres and millimetres. Making accurate measurements and converting units are essential skills for construction workers.
Lesson starter: How good is your estimate?
Without measuring, guess the length of your desk, in centimetres. β’ β’ β’
Now use a ruler to find the actual length in centimetres. Convert your answer to millimetres and metres. If you lined up all the class desks end to end, how many desks would be needed to reach 1 kilometre? Explain how you got your answer.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10A Metric units of length
653
KEY IDEAS β Metric system
β’ β’
km
Γ 100
Γ 10 cm
m
mm
U N SA C O M R PL R E EC PA T E G D ES
β’
Γ 1000
1β centimetre (cm) = 10 millimetres (mm)β β1 metre (m) = 100 centimetres (cm)β β1 kilometre (km) = 1000 metres (m)β
Γ· 1000
β Conversion
Γ· 100
Γ· 10
β’
When converting to a smaller unit, multiply by β10βor β100β or β1000β. The decimal point appears to move to the right. For example: 28 cm = (28 Γ 10) mm 2.3 m = (2.3 Γ 100) cm ββ―β― β β β ββ―β― β β β β = 230 cm β = 280 mm
β’
When converting to a larger unit, divide by a power of β10β(i.e. β10, 100, 1000β). The decimal point appears to move to the left. For example: 47 mm = (47 Γ· 10) cm 4600 m = (4600 Γ· 1000) km ββ―β― β β β β ββ―β― β β β = 4.7 cm β = 4.6 km
β When reading scales, be sure about what units are showing
on the scale. This scale shows β36 mmβ.
mm cm
1
2
3
4
BUILDING UNDERSTANDING
1 Use the metric system to state how many: a millimetres in β1β centimetre b centimetres in β1β metre c metres in β1β kilometre d millimetres in β1β metre e centimetres in β1β kilometre f millimetres in β1β kilometre.
2 State the missing number or word in these sentences. a When converting from metres to centimetres, you multiply by _______. b When converting from metres to kilometres, you divide by _______. c When converting from centimetres to metres, you _______ by β100.β d When converting from kilometres to metres, you ________ by β1000β. 3 Calculate each of the following. a β100 Γ 10β c β100 Γ 1000β
b 1β 0 Γ 100β d β10 Γ 100 Γ 1000β
4 a When multiplying by a power of β10β, in which direction does the decimal point move β left or right? b When dividing by a power of β10β, in which direction does the decimal point move β left or right?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10 Measurement
Example 1
Choosing metric lengths
Which metric unit would be the most appropriate for measuring these lengths? a dimensions of a large room b thickness of glass in a window
U N SA C O M R PL R E EC PA T E G D ES
654
SOLUTION
EXPLANATION
a metres (β m)β
Using βmmβor βcmβwould give a very large number, and using km would give a number that is very small.
b millimetres β(mm)β
The thickness of glass is likely to be around β 5 mmβ.
Now you try
Which metric unit would be the most appropriate for measuring these lengths? a distance between the centre of the city and a nearby suburb b height of a book on a bookshelf
Example 2
Converting metric units of length
Convert to the units given in brackets. a 3β m (cm)β b β25 600 cm (km)β SOLUTION
EXPLANATION
a 3 m = 3 Γ 100 cm β β β ββ―β― β = 300 cm
1β m = 100 cmβ Multiply since you are converting to a smaller unit.
b 25 β =β 25 600 Γ· 100 000β β 600 cmβ―β― β = 0.256 km
There are β100 cmβin β1 mβand β1000 mβin β1 kmβ and β100 Γ 1000 = 100 000β.
Now you try
Convert to the units given in brackets. a 2β km (m)β b β3400 mm (m)β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10A Metric units of length
Example 3
655
Reading length scales
U N SA C O M R PL R E EC PA T E G D ES
Read the scales on these rulers to measure the marked length. a 0 cm 1 cm 2 cm 3 cm
b
4m
5m
6m
7m
SOLUTION
EXPLANATION
a 2β 5 mmβ b β70 cmβ
β2.5 cmβis also accurate. Each division is _ β 1 βof a metre, which is β10 cmβ. 10
Now you try
Read the scales on these rules to measure the marked length. a Answer in millimetres.
0 cm 1 cm 2 cm 3 cm
b Answer in centimetres.
0m
1m
2m
3m
Exercise 10A FLUENCY
Example 1
1
1β3, 4β7ββ(β1/2β)β
2, 4β7β(β β1/2β)β,β 8
2, 4β7β(β β1/2β)β, 8, 9
Which metric unit would be the most appropriate for measuring the following? a the distance between two towns b the thickness of a nail c height of a flag pole d length of a garden hose e dimensions of a small desk f distance across a city
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
656
Chapter 10 Measurement
2 Choose which metric unit would be the most suitable for measuring the real-life length indicated in each of these photos. a b
U N SA C O M R PL R E EC PA T E G D ES
Example 1
Example 2a
Example 2a
c
d
e
f
3 Convert to the units given in the brackets. a 4β m (cm)β b 6β km (m)β
c β30 mm (cm)β
4 Convert these measurements to the units shown in brackets. a 5β cm (mm)β b β2 m (cm)β c β3.5 km (m)β e β40 mm (cm)β f β500 cm (m)β g β4200 m (km)β i β6.84 m (cm)β j β0.02 km (m)β k β9261 mm (cm)β
d β700 cm (m)β
d β26.1 m (cm)β h β472 mm (cm)β l β4230 m (km)β
5 Add these lengths together and give the result in the units shown in brackets. a β2 cmβand β5 mm (cm)β b 8β cmβand β2 mm (mm)β c 2β mβand β50 cm (m)β d β7 mβand β30 cm (cm)β e 6β kmβand β200 m (m)β f β25 kmβand β732 m (km)β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10A Metric units of length
Example 2b
c 2β .4 m (mm)β f β913 000 mm (m)β i β0.7 km (mm)β
7 These rulers show centimetres with millimetre divisions. Read the scale to measure the marked length. i in centimetres ii in millimetres a b
U N SA C O M R PL R E EC PA T E G D ES
Example 3
6 Convert to the units shown in the brackets. a β3 m (mm)β b β6 km (cm)β d β0.04 km (cm)β e β47 000 cm (km)β g β216 000 mm (km)β h β0.5 mm (m)β
657
0
1
2
c
0
1
2
3
4
5
0
1
2
3
0
1
2
3
4
5
0
1
2
3
4
d
0
1
2
e
f
0
1
2
3
4
g
h
0
1
2
3
8 Read the scale on these tape measures. Be careful with the units given! a b 0 m
1
2
11 km
3
12
13
9 Use subtraction to find the difference between the measurements, and give your answer with the units shown in brackets. a β9 km, 500 m (km)β b β3.5 m, 40 cm (cm)β c β0.2 m, 10 mm (cm)β PROBLEM-SOLVING
10β12
12β15
14β17
10 Arrange these measurements from smallest to largest. a β38 cm, 540 mm, 0.5 mβ b 0β .02 km, 25 m, 160 cm, 2100 mmβ c 0β .003 km, 20 cm, 3.1 m, 142 mmβ d β0.001 km, 0.1 m, 1000 cm, 10 mmβ 11 Joe widens a β1.2 mβdoorway by β50 mmβ. What is the new width of the doorway, in centimetres?
12 Three construction engineers individually have plans to build the worldβs next tallest tower. The Titan tower is to be β 1.12 kmβtall, the Gigan tower is to be β109 500 cmβtall and the Bigan tower is to be β1210 mβtall. Which tower will be the tallest?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10 Measurement
13 Steel chain costs β$8.20βper metre. How much does it cost to buy chain of the following lengths? a 1β kmβ b β80 cmβ c β50 mmβ 14 A house is β25βmetres from a cliff above the sea. The cliff is eroding at a rate of β40 mmβper year. How many years will pass before the house starts to fall into the sea? 15 Mount Everest is moving with the Indo-Australian plate at a rate of about β10 cmβper year. How many years will it take to move β5 kmβ?
U N SA C O M R PL R E EC PA T E G D ES
658
16 A ream of β500βsheets of paper is β4 cmβthick. How thick is β1βsheet of paper, in millimetres? 17 A snail slides β2 mmβevery β5βseconds. How long will it take to slide β1 mβ? REASONING
18
18
18, 19
18 Copy this chart and fill in the missing information. km m
Γ· 100
cm
Γ 10
mm
19 Many tradespeople measure and communicate with millimetres, even for long measurements like timber beams or pipes. Can you explain why this might be the case? ENRICHMENT: Very small and large units
β
β
20
20 When β1βmetre is divided into β1βmillion parts, each part is called a micrometre (βΞΌmβ). At the other end of the spectrum, a light year is used to describe large distances in space. a State how many micrometres there are in: i β1 mβ ii β1 cmβ iii β1 mmβ iv 1β kmβ b A virus is β0.000312 mmβwide. How many micrometres is this? c Research the length called the light year. Explain what it is and give example of distance using light years, such as to the nearest star other than the Sun.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10B Perimeter
659
10B Perimeter CONSOLIDATING
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand that perimeter is the distance around the outside of a two-dimensional shape β’ To understand that marks can indicate two (or more) sides are of equal length β’ To be able to ο¬nd the perimeter of a shape when the measurements are given
The distance around the outside of a two-dimensional shape is called the perimeter. The word perimeter comes from the Greek words peri, meaning βaroundβ, and metron, meaning βmeasureβ. We associate perimeter with the outside of all sorts of regions and objects, like the length of fencing surrounding a block of land or the length of timber required to frame a picture.
This fence marks the perimeter (i.e. the distance around the outside) of a paddock. Farmers use the perimeter length to calculate the number of posts needed for the fence.
Lesson starter: Is there enough information?
This diagram includes only 90Β° angles and only one side length is marked. Discuss if there is enough information given in the diagram to find the perimeter of the shape. What additional information, if any, is required?
10 cm
KEY IDEAS
β Perimeter, sometimes denoted as P, is the distance around the
1.6 cm
outside of a two-dimensional shape.
β Sides with the same markings are of equal length.
β The unknown lengths of some sides can sometimes be determined
by considering the given lengths of other sides.
2.8 cm
4.1 cm
P = 1.6 + 1.6 + 2.8 + 4.1 = 10.1 cm
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10 Measurement
BUILDING UNDERSTANDING 1 These shapes are drawn on β1 cmβgrids. Give the perimeter of each. a b
U N SA C O M R PL R E EC PA T E G D ES
660
2 Use a ruler to measure the lengths of the sides of these shapes, and then find the perimeter. a b
c
d
Example 4
Finding the perimeter
Find the perimeter of each of these shapes. a 3 cm
b
3m
6m
5m
5 cm
2m
SOLUTION
EXPLANATION
a perimeter = 2 Γ 5 + 3 β―β― β β β β β = 13 cm
There are two equal lengths of β5 cmβ and one length of β3 cmβ.
b perimeter = 2 Γ 6 + 2 Γ 8 β―β― β β β β β = 28 m
3m
6m
6β2=4m
5m
2m
3+5=8m
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10B Perimeter
661
Now you try Find the perimeter of each of these shapes. a 6 cm
b 4m
U N SA C O M R PL R E EC PA T E G D ES
3 cm
6m
Exercise 10B
Example 4a
1
FLUENCY
1β4
Find the perimeter of each of the shapes. a 4 cm
b
2, 4, 5
7m
11 m
6 cm
Example 4a
2β5
2 Find the perimeter of these shapes. (Diagrams are not drawn to scale.) a b 5 cm 3 cm 6m
8m
8m
7 cm
5m
10 m
c
d
1m
0.2 m
10 km
5 km
e
f
10 cm
2.5 cm
6 cm
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
662
Chapter 10 Measurement
3 Find the perimeter of each of these shapes. a 2m 5m 3m 2m
b
4 cm
3 cm
4 cm
U N SA C O M R PL R E EC PA T E G D ES
Example 4b
2 cm
Example 4b
4 Find the perimeter of these shapes. All corner angles are β90Β°β. a b 4 cm 2m
5m
8m
7 cm
10 cm
5 cm
c
4m
d
6 km
5 mm
3 km
1 mm
3 mm
9 km
5 a A square has a side length of β2.1 cmβ. Find its perimeter. b A rectangle has a length of β4.8 mβand a width of β2.2 mβ. Find its perimeter. c An equilateral triangle has all sides the same length. If each side is β15.5 mmβ, find its perimeter. PROBLEM-SOLVING
6, 7
7β9
9β12
6 A grazing paddock is to be fenced on all sides. It is rectangular in shape, with a length of β242 mβand a width of β186 mβ. If fencing costs β$25βper metre, find the cost of fencing required. 7 The lines on a grass tennis court are marked with chalk. All the measurements are shown in the diagram and given in feet. 39 feet
21 feet
27 feet
36 feet
a Find the total number of feet of chalk required to do all the lines of the given tennis court. b There are β0.305βmetres in β1βfoot. Convert your answer to part a to metres.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10B Perimeter
663
8 Only some side lengths are shown for these shapes. Find the perimeter. (Note: All corner angles are 9β 0Β°β.) a b 4 cm 20 mm 10 cm
U N SA C O M R PL R E EC PA T E G D ES
15 mm 18 cm
9 Find the perimeter of each of these shapes. Give your answers in centimetres. a b c 271 mm
430
mm
168 mm
1.04 m
0.38 m
7.1 cm
10 A square paddock has β100βequally-spaced posts that are β4βmetres apart, including one in each corner. What is the perimeter of the paddock? 11 The perimeter of each shape is given. Find the missing side length for each shape. a b c 4 cm 2 cm
?
?
? P = 11 cm
12 km P = 38 km
P = 20 m
12 A rectangle has a perimeter of β16 cmβ. Using only whole numbers for the length and width, how many different rectangles can be drawn? Do not count rotations of the same rectangle as different. REASONING
13
13, 14
14, 15
13 Write an algebraic rule (e.g. βP = 2a + bβ) to describe the perimeter of each shape. a b c a
b
a
b
a
d
e
a
f
b
a
a
b
c
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10
Measurement
14 Write an algebraic rule for the perimeter of each given shape. a b a
b
a c
c
b
U N SA C O M R PL R E EC PA T E G D ES
664
15 a A square has perimeter P. Write an expression for its side length. b A rectangle has perimeter P and width a. Write an expression for its length. ENRICHMENT: Picture frames
β
β
16
16 The amount of timber used to frame a picture depends on the outside lengths of the overall frame. Each side is first cut as a rectangular piece, and two corners are then cut at 45Β° to make the frame. a A square painting of side length 30 cm is to be framed with 30 cm 5 cm timber of width 5 cm. Find the total length of timber required for the job. 30 cm b A rectangular photo with dimensions 50 cm by 30 cm is framed with timber of width 7 cm. Find the total length of timber required to complete the job. c Kimberley uses 2 m of timber of width 5 cm to complete a square picture frame. What is the side length of the picture? d A square piece of embroidery has side length a cm and is framed by timber of width 4 cm. Write an expression for the total length of timber used in cm.
The perimeter of a house is found by adding the lengths of all the outside walls. Scaffolding is required around the perimeter for construction work above ground level.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10C
Circles, Ο and circumference
665
10C Circles, Ο and circumference
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To know the features of a circle including the radius, diameter and circumference β’ To know that Ο is the ratio of the circumference of a circle to its diameter β’ To be able to calculate a circleβs circumference, diameter or radius if given one of the other two measurements
The special number Ο, pronounced pi, has fascinated mathematicians, scientists, designers and engineers since the ancient times. This mathematical constant, which connects a circleβs circumference to its diameter, is critical in any calculation that involves circular shapes. It also appears in many other areas of 22 to help make calculations mathematics. Evidence suggests that the Egyptians used the approximation _ 7 25 _ and that the Babylonians used the number = 3.125. Around 250 ad, the Greek mathematician 8 Archimedes used an algorithm to prove that Ο was between 3.1410 and 3.1429 and in a similar manner around 265 ad, a Chinese mathematician, Liu Hui, proved that Ο β 3.1416. Today we can use computers and complex mathematics to approximate Ο correct to millions of decimal places.
Lesson starter: How close can you get?
For this activity, you will need some string, scissors, a ruler and a pair of compasses. We know that Ο is the C . In this activity, we will estimate the ratio of a circleβs circumference, C, to its diameter, d, and so Ο = _ d value of Ο by drawing a circle and measuring both its diameter and circumference. β’ β’ β’ β’
β’
Use a pair of compasses to construct a circle of any size and draw in a diameter. Use a piece of string to trace around the circle and cut to size. Measure the length of the string (the circumference) and the circleβs diameter. C. Use your measurements to approximate the value of Ο using Ο = _ d Calculate a class average to see how this approximation compares to an accurate value of Ο.
KEY IDEAS
β Features of a circle
β’ β’ β’ β’ β’ β’
Diameter (d) is the distance across the centre of a circle. Radius (r) is the distance from the centre to the circle. Note: d = 2r. Chord: A line interval connecting two points on a circle. Tangent: A line that touches the circle at a point. β A tangent to a circle is at right angles to the radius. Sector: A portion of a circle enclosed by two radii and a portion of a circle (arc). Segment: An area of a circle βcut offβ by a chord.
cumference cir
eter
diam
radius
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10 Measurement
β Circumference (β βC)β βis the distance around a circle.
β’ β’
t en m β = 2Οrβor βC = Οdβ C g se hord c Pi β(Ο)βis a constant numerical value and is an irrational number, sector meaning that it cannot be expressed as a fraction. β As a decimal, the digits have no pattern and continue forever. The ratio of the circumference to the diameter of any circle is nt nge a t C β ββ. equal to pi β(Ο)β; i.e. βΟ = _ d βΟ = 3.14159β(correct to β5βdecimal places) β Common approximations include β3β, β3.14βand _ β22 .ββ 7 β A more precise estimate for pi can be found on most calculators or on the internet.
U N SA C O M R PL R E EC PA T E G D ES
666
β’
β’
BUILDING UNDERSTANDING
1 State whether the following statements relating features of a circle are true or false. a βr = 2dβ b βC = 2Οrβ c βd = ΟCβ C C _ _ f βd = 2rβ d βΟ = β β e βΟ = βr β d
2 Name the features of the circles shown.
f
d
e
a
g
b
c
3 The following relate to a circle's radius and diameter. a Find a circle's diameter if its radius is β15β mm. b Find a circle's radius if its diameter is β8β cm.
Example 5
Approximating βΟβ
A common approximation of βΟβis the fraction _ β22 ββ. 7 22 _ a Write β βas a decimal rounded to: 7 i two decimal places ii three decimal places. b Decide if _ β22 βis a better approximation to βΟβthan the number β3.14β. 7
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10C
SOLUTION a i
Circles, Ο and circumference
667
EXPLANATION
_ β22 β= 3.14 β(β2 d.p.β)β
7 22 β= 3.143 (β β3 d.p.β)β ii β_ 7
_ β22 ββ Ο = 0.0013β(β β4 d.p.β)βand Ο β β 3.14 = 0.0016β(β β4 d.p.β)β
U N SA C O M R PL R E EC PA T E G D ES
b _ β22 β= 3.14285β... and βΟ = 3.14159β... 7 So, _ β22 βis a better approximation 7 compared to β3.14β.
22 β= 3.142β... so round down for β2βdecimal places β_ 7 _ β22 β= 3.1428β... so round up for β3βdecimal places 7 7
22 βis closer to βΟβ. So, β_ 7
Now you try
A possible approximation of βΟβis the fraction _ β25 ββ. 8 25 _ a Write β βas a decimal rounded to: 8 i two decimal places ii three decimal places. b Decide if _ β25 βis a better approximation to βΟβthan the number β3.14β. 8
Example 6
Calculating the circumference
Calculate the circumference of these circles using a calculator and rounding your answer correct to two decimal places. a b
m
7c
2.3 mm
SOLUTION
EXPLANATION
a C = Οd β β β =ββ Ο(7)β β β = 21.99 cm
Since you are given the diameter, use the β C = Οdβformula. Substitute βd = 7βand use a calculator for the value of βΟβ. Round as required.
Continued on next page
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
668
Chapter 10
Measurement
b C = 2Οr = 2Ο(2.3) = 14.45 mm
U N SA C O M R PL R E EC PA T E G D ES
Since you are given the diameter, use the C = 2Οr formula. Substitute r = 2.3 and use a calculator for the value of Ο. Round as required.
Now you try
Calculate the circumference of these circles using a calculator and rounding your answer correct to two decimal places. a b 4.7 cm
11 mm
Exercise 10C FLUENCY
Example 5
Example 5
1
1, 2, 3β5(1/2)
2, 3β5(1/2)
2, 3β5(1/4)
28 . A reasonable approximation of Ο is the fraction _ 9 28 _ a Write as a decimal rounded to: 9 i two decimal places ii three decimal places. 28 is a better approximation to Ο than b Decide if _ 9 the number 3.14.
355 . 2 A good approximation of Ο is the fraction _ 113 355 as a decimal rounded to: a Write _ 113 i three decimal places ii four decimal places.
355 is a better approximation to Ο than b Decide if _ 113 the number 3.142.
The Ancient Greek mathematician and astronomer Archimedes was able to prove that Ο was between 223 and _ 22 . _ 7 71 Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10C
Example 6a
Circles, Ο and circumference
669
3 Calculate the circumference of these circles using βC = Οdβand round your answer correct to two decimal places. a b
13
U N SA C O M R PL R E EC PA T E G D ES
m
5c
c
d
9.8 m
4.6 m
Example 6b
cm
4 Calculate the circumference of these circles using βC = 2Οrβand round your answer correct to two decimal places. a b 4 mm
21 cm
c
d
16.2 m
8.3 mm
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10 Measurement
5 Estimate the circumference of these circles using the given approximation of βΟβ. a b 2 cm
m
U N SA C O M R PL R E EC PA T E G D ES
670
5m
p =3
c
p = 3.14
d
113
14 m
p=
22 7
PROBLEM SOLVING
p=
6, 7
cm
355 113
6β8
7β9
6 A log in a wood yard has a diameter of β40 cmβ. Find its circumference, correct to one decimal place.
7 We know that the rule βC = Οdβcan be rearranged to give βd = _ βC Ο β. Use this rearranged rule to calculate the following, correct to two decimal places. a a circleβs diameter if the circumference is β10 cmβ b a circleβs diameter if the circumference is β12.5 mβ c a circleβs radius if the circumference is β7 mmβ d a circleβs radius if the circumference is β37.4 mβ 8 A circular pond has radius β1.4βmetres. Find its circumference, correct to one decimal place. 9 A bicycle wheel completes β1024β revolutions while on a Sunday ride. Find the distance travelled by the bicycle if the wheelβs diameter is β85 cmβ. Round to the nearest metre.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10C
REASONING
10
Circles, Ο and circumference
10, 11
671
11, 12
U N SA C O M R PL R E EC PA T E G D ES
10 The following are four fractions which ancient cultures used to approximate βΟβ. β’ Egyptian _ β22 β 7 β’ Babylonians _ β25 β 8 β’ Indian _ β339 β 108
Chinese _ β142 β 45 a Which one of these ancient historical approximations is closest to the true value of βΟβ? Note that βΟβ written to five decimal places is β3.14159β. b In more recent times, _ β22 βhas been a very popular approximation throughout many countries. 7 Explain why you think this is the case. c Find your own simple fraction which is a reasonable approximation of βΟβ. Write your fraction as a decimal and compare it with βΟβwritten in decimal form with at least five decimal places. β’
11 If βC = Οdβ, find a rule for the radius of a circle, βrβ, in terms of its circumference, βCβ, and use it to find the radius of a circle if its circumference is β25 cmβ. Round to two decimal places. 12 A professor expresses the circumference of a circle with diameter β5 cmβusing the exact value, β5Ο cmβ. Use an exact value to give the circumference of a circle with the given diameter or radius. a Diameter β7 cmβ b Radius β3 mmβ ENRICHMENT: Planetary circumnavigation
β
β
13
13 While not being perfect spheres, the circumference of the planets in our solar system can be approximated by using a circle. Here are some planetary facts with some missing details. Calculate the missing numbers, rounding to the nearest integer. Part
Planet
Diameter (βkmβ approx.)
a
Earth
β12 750β
Circumference (βkmβ approx.)
b
Mars
β 6 780β
c
Jupiter
β439 260β
d
Mercury
β 15 330β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10
Measurement
10D Arc length and perimeter of sectors and composite shapes EXTENDING LEARNING INTENTIONS β’ To know that an arc is a portion of a circleβs circumference β’ To understand how the length of an arc relates to the angle at the centre of the circle β’ To be able to calculate the length of an arc β’ To be able to calculate the perimeter of a sector
U N SA C O M R PL R E EC PA T E G D ES
672
Whenever a portion of a circleβs circumference is used in a diagram or construction, an arc is formed. To determine the arcβs length, the particular fraction of the circle is calculated by considering the angle at the centre of the circle that defines the arc.
Lesson starter: The rule for ο¬nding an arc length Complete this table to develop the rule for finding an arc length (l). Angle
Fraction of circle
Arc length
180Β°
180 = _ 1 _
1 Γ Οd l=_
90 = ________ _
l = ________ Γ ________
90Β°
360
360
Diagram
2
2
180Β°
90Β°
45Β° 30Β°
150Β° π
KEY IDEAS
β A circular arc is a portion of the circumference of a circle.
In the diagram: r = radius of circle aΒ° = number of degrees in the angle at the centre of a circle l = arc length Formula for arc length: aΒ° Γ 2Οr or l = _ aΒ° Γ Οd l=_ 360 360
β The sector also has two straight edges, each with length of r.
Formula for perimeter of sector: aΒ° Γ 2Οr + 2r or P = _ aΒ° Γ Οd + d P=_ 360 360
l
r
aΒ°
r
r
aΒ°
r
l
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10D Arc length and perimeter of sectors and composite shapes
673
β Common circle portions
quadrant
semicircle
U N SA C O M R PL R E EC PA T E G D ES
r r
P=
1 Γ 2Οr + 2r 4
P=
1 Γ 2Οr + 2r 2
β A composite figure is made up of more than one basic shape.
BUILDING UNDERSTANDING
1 What fraction of a circle is shown in these diagrams? Name each shape. a b
2 What fraction of a circle is shown in these sectors? Simplify your fraction. a b c
60Β°
d
e
f
315Β°
120Β°
3 Name the two basic shapes that make up these composite figures. a b
c
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10 Measurement
Example 7
Finding an arc length
Find the length of each of these arcs for the given angles, correct to two decimal places. a b 2 mm
U N SA C O M R PL R E EC PA T E G D ES
674
50Β°
230Β°
10 cm
SOLUTION
EXPLANATION
a l = _ β 50 βΓ 2Ο Γ 10 βββ―β― β 360 β β = 8.73 cm
The fraction of the full circumference is _ β 50 ββ 360 and the full circumference is β2Οrβ, where βr = 10β.
230 βΓ 2Ο Γ 2 b l = β_ βββ―β― β 360 β β = 8.03 mm
The fraction of the full circumference is _ β230 ββ 360 and the full circumference is β2Οrβ, where βr = 2β.
Now you try
Find the length of each of these arcs for the given angles, correct to two decimal places. a b 7 cm
112Β°
5m
Example 8
290Β°
Finding the perimeter of a sector
Find the perimeter of each of these sectors, correct to one decimal place. a b
3m
5 km
SOLUTION
EXPLANATION
a P =_ β1 βΓ 2Ο Γ 3 + 2 Γ 3 β ββ―β― β β4 β = 10.7 m
The arc length is one-quarter of the circumference and included are two radii, each of β3 mβ.
b P =_ β1 βΓ Ο Γ 5 + 5 β ββ―β― β β2 β = 12.9 km
A semicircleβs perimeter consists of half the circumference of a circle plus a full diameter. In this case, the whole circleβs circumference is found using βΟ Γ dβsince we have the diameter.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10D Arc length and perimeter of sectors and composite shapes
675
Now you try Find the perimeter of each of these sectors, correct to one decimal place. a b
U N SA C O M R PL R E EC PA T E G D ES
9 mm 1.6 cm
Example 9
Finding the perimeter of a composite shape
Find the perimeter of the following composite shape, correct to one decimal place. 10 cm
5 cm
SOLUTION
EXPLANATION
P = 10 + 5 + 10 + 5 + _ β1 βΓ 2Ο Γ 5 β β β ββ―β―β― 4 β = 37.9 cm
There are two straight sides of β10 cmβand β5 cmβ shown in the diagram. The radius of the circle is β5 cmβ, so the straight edge at the base of the diagram is β15 cmβlong. The arc is a quarter circle.
Now you try
Find the perimeter of the following composite shape, correct to one decimal place.
3 cm
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
676
Chapter 10 Measurement
Exercise 10D FLUENCY
1
1β4β(β 1β/2β)β
1ββ(1β/3β)β,β 2β4ββ(1β/2β)β
Find the length of each of the following arcs for the given angles, correct to two decimal places. a b
U N SA C O M R PL R E EC PA T E G D ES
Example 7
1,ββ(1β/2β)β,β 2β4
60Β° 8 cm
80Β°
c
4m
d
225Β°
100Β°
7 mm
0.5 m
e
f
300Β°
0.2 km
330Β°
26 cm
Example 8a
2
Find the perimeter of each of these quadrants, correct to one decimal place. a b 4 cm
10 m
c
d
2.6 m
8 cm
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10D Arc length and perimeter of sectors and composite shapes
Example 8b
3
677
Find the perimeter of these semicircles, correct to one decimal place. a b 14 km mm 0 2
d
U N SA C O M R PL R E EC PA T E G D ES
c
4.7 m
17 mm
4 Find the perimeter of these sectors, correct to one decimal place. Include the two radii in each case. a b
60Β° 4.3 m
115Β°
3.5 cm
c
d
6 cm
250Β°
270Β°
7m
PROBLEM-SOLVING
Example 9
5, 6
5β7
5β8
5 Find the perimeter of each of these composite shapes, correct to one decimal place. a b c 2 cm 30 cm
3m
12 cm
4m
d
e
f
6 km
10 cm
10 km
10 mm 8 cm Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10 Measurement
6 A window consists of a rectangular part of height β2 mβand width β1 mβ, with a semicircular top having a diameter of β1 mβ. Find its perimeter, correct to the nearest β1 cmβ.
U N SA C O M R PL R E EC PA T E G D ES
678
7 For these sectors, find only the length of the arc, correct to two decimal places. a b c 140Β°
85Β°
1.3 m
100 m
7m
330Β°
8 Calculate the perimeter of each of these shapes, correct to two decimal places. a b c 4 cm
5m
9m
10 m
REASONING
9
9, 10
9β11
9 Give reasons why the circumference of this composite shape can be found by simply using the rule β P = 2Οr + 4rβ. r
10 Explain why the perimeter of this shape is given by βP = 2Οrβ.
r Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10D Arc length and perimeter of sectors and composite shapes
679
45
km
11 Find the radius of each of these sectors for the given arc lengths, correct to one decimal place. a b c 4m 60 m 30Β° r r
r
U N SA C O M R PL R E EC PA T E G D ES
100Β°
ENRICHMENT: Exact values and perimeters
β
β
12, 13
12 The working to find the exact perimeter of this composite shape is given by: P = 2Γ8+4+_ β1 Ο β Γ4 2 β ββ―β― β β β = 20 + 8Ο cm
4 cm
8 cm
Find the exact perimeter of each of the following composite shapes. a b 6 cm
2 cm
5 cm
13 Find the exact answers for Question 8 in terms of βΟβ.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10
Measurement
10E Units of area and area of rectangles LEARNING INTENTIONS β’ To understand what the area of a two-dimensional shape is β’ To be able to convert between metric areas (square millimetres, square centimetres, square metres, square kilometres, hectares) β’ To be able to ο¬nd the area of squares and other rectangles
U N SA C O M R PL R E EC PA T E G D ES
680
Area is measured in square units. It is often referred to as the amount of space contained inside a flat (i.e. plane) shape; however, curved three-dimensional (3D) solids also have surface areas. The amount of paint needed to paint a house and the amount of chemical needed to spray a paddock are examples of when area would be considered.
Agricultural pilots ο¬y small planes that spray crops with fertiliser or pesticide. The plane is ο¬own up and down the paddock many times, until all the area has been sprayed.
Lesson starter: The 12 cm2 rectangle
A rectangle has an area of 12 square centimetres (12 cm2). β’ β’ β’
Draw examples of rectangles that have this area, showing the length and width measurements. How many different rectangles with whole number dimensions are possible? How many different rectangles are possible if there is no restriction on the type of numbers allowed to be used for length and width?
KEY IDEAS
β The metric units of area include:
β’
1 square millimetre (1 mm2)
β’
1 square centimetre (1 cm2) 1 cm2 = 100 mm2
β’
1 square metre (1 m2)
1 mm 1 mm
1 cm
1 cm
1 m2 = 10 000 cm2
1 m (Not drawn to scale.)
1m
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10E
β’
1β βsquare kilometre (β 1 βkmβ2β )β β β1 βkmβ2β β= 1 000 000 βm2β β
Units of area and area of rectangles
1 km
681
(Not drawn to scale.)
U N SA C O M R PL R E EC PA T E G D ES
1 km β’
1β βhectare (β 1 ha)β 1β ha = 10 000 βm2β β
100 m (Not drawn to scale.)
100 m
β The dimensions of a rectangle are called length (β βlβ)βand width (β βw)β ββ. β The area of a rectangle is given by the number of rows multiplied
by the number of columns. Written as a formula, this looks like: A = l Γ wβ. This also works for numbers that are not integers.
β The area of a square is given by: βA = l Γ l = βlβ2β
A=lΓw
w
β
l
A = l2
l
BUILDING UNDERSTANDING
1 For this rectangle drawn on a β1 cmβgrid, find each of the following. a the number of single β1 cmβ squares b the length and the width c length βΓβ width
2 For this square drawn on a centimetre grid, find the following. a the number of single β1 cmβ squares b the length and the width c length βΓβ width
3 Count the number of squares to find the area in square units of these shapes.
a
b
c
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10 Measurement
4 Which unit of area (ββmmβ2β β, βcmβ2β β, βmβ2β, haβor βkmβ2β β) would you choose to measure these areas? Note that β1 βkmβ2β βis much larger than β1 haβ. a area of an βA4βpiece of paper c area of a small farm e area of a large football oval
b area of a wall of a house d area of a large desert f area of a nail head
U N SA C O M R PL R E EC PA T E G D ES
682
Example 10 Counting areas
Count the number of squares to find the area of the shape drawn on this centimetre grid.
SOLUTION
EXPLANATION
β6 βcmβ2β β
There are β5βfull squares and half of β2βsquares in the triangle, giving β1β more.
1 of 2 = 1 2
Now you try
Count the number of squares to find the area of the shape drawn on this centimetre grid.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10E
Units of area and area of rectangles
683
Example 11 Finding areas of rectangles and squares Find the area of this rectangle and square. a
b
U N SA C O M R PL R E EC PA T E G D ES
4 mm
2.5 cm
10 mm
SOLUTION
EXPLANATION
a Area = l Γ w β =ββ 10 Γ β4β β β β = 40 βmmβ2β β
The area of a rectangle is the product of the length and width.
b Area = βlβ2β β β =β 2.β5β 2β β
The width is the same as the length, so
β
A = l Γ l = βlβ2β β β ββ―β― β(2.5)β2β β= 2.5 Γ 2.5
β = 6.25 βcmβ2β β
Now you try
Find the area of this rectangle and square. a
b
3 cm
1.5 km
8 cm
Exercise 10E FLUENCY
Example 10
1
1, 2, 3ββ(1β/2β)β,β 4β6
2, 3β(β 1β/2β)β, 4, 5, 7
2β3β(β 1β/2β)β, 4, 5, 7, 8
Count the number of squares to find the area of these shapes on centimetre grids. a b
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
684
Chapter 10 Measurement
2 Count the number of squares to find the area of these shapes on centimetre grids. a b
U N SA C O M R PL R E EC PA T E G D ES
Example 10
c
Example 11
3
d
Find the area of these rectangles and squares. Diagrams are not drawn to scale. a b c
2 cm
10 cm
11 mm
20 cm
3.5 cm
d
e
2 mm
f
5m
2.5 mm
1.2 mm
g
h
i
0.8 m
1.7 m
17.6 km
0.9 cm
10.2 km
4 Find the side length of a square with each of these areas. Use trial and error if you are unsure. a 4β βcmβ2β β b β25 βm2β β c β144 kβ mβ2β β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10E
Units of area and area of rectangles
685
5 There are β10 000 βm2β βin one hectare (β ha)β. Convert these measurements to hectares. a β20 000 m β 2β β b β100 000 m β 2β β c β5000 βm2β β
U N SA C O M R PL R E EC PA T E G D ES
6 A rectangular soccer field is to be laid with new grass. The field is β100 mβlong and β50 mβ wide. Find the area of grass to be laid. 7 Glass is to be cut for a square window of side length β50 cmβ. Find the area of glass required for the window.
8 Two hundred square tiles, each measuring β10 cmβ by β10 cmβ, are used to tile an open floor area. Find the area of flooring that is tiled. PROBLEM-SOLVING
9, 10
9β11
9, 11, 12
9 a A square has a perimeter of β20 cmβ. Find its area. b A square has an area of β9 βcmβ2β β. Find its perimeter. c A squareβs area and perimeter are the same number. How many units is the side length?
10 The carpet chosen for a room costs β$70β per square metre. The room is rectangular and is β6 mβ long by β5 mβwide. What is the cost of carpeting the room?
11 Troy wishes to paint a garden wall that is β11 mβ long and 3β mβhigh. Two coats of paint are needed. The paint suitable to do the job can be purchased only in whole numbers of litres and covers an area of β15 βm2β βper litre. How many litres of paint will Troy need to purchase?
12 A rectangular area of land measures β200 mβby β400 mβ. Find its area in hectares. REASONING
13
13 a Find the missing length for each of these rectangles. i ii A = 50 cm2 5 cm
13, 14
A = 22.5 mm2
14β16
2.5 mm
? ? b Explain the method that you used for finding the missing lengths of the rectangles above.
14 Explain why the area shaded here is exactly β2 βcmβ2β ββ.
1 cm
4 cm
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10
Measurement
15 A square has perimeter P cm. a If P = 44, find the area of the square. b If P is unknown, write an expression for the area of the square, using P. P = 44 16 A square has all its side lengths doubled. How does this change the area? Investigate and justify your answer.
U N SA C O M R PL R E EC PA T E G D ES
686
ENRICHMENT: Area conversions
β
17 a Use this diagram or similar to help answer the following. i How many mm2 in 1 cm2? ii How many cm2 in 1 m2? iii How many m2 in 1 km2? b Complete the diagram below. Γ ____ Γ ____
km2
m2
17
10 mm
10 mm
A = 1 cm2
Γ ____
mm2
cm2
Γ· ____ Γ· ____
β
Γ· ____
c Convert these units to the units shown in brackets. i 2 cm2 (mm2) ii 10 m2 (cm2)
iii 3.5 km2 (m2)
iv 300 mm2 (cm2)
v 21 600 cm2 (m2)
vi 4 200 000 m2 (km2) vii 0.005 m2 (mm2)
viii 1 km2 (ha) ix 40 000 000 cm2 (ha)
Working with different units of area is important in the division of land.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10F
Area of parallelograms
687
10F Area of parallelograms
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand that the area of a parallelogram is related to the area of a rectangle β’ To be able to ο¬nd the area of a parallelogram given its base and height
Recall that a parallelogram is a quadrilateral with two pairs of parallel sides. Opposite sides are of the same length and opposite angles are equal.
aΒ°
bΒ°
bΒ°
aΒ°
Like a triangle, the area of a parallelogram is found by using the length of one side (called the base) and the height (which is perpendicular to the base.)
Whenever pairs of parallel straight lines meet a parallelogram is formed, including an area which can be calculated using a base length and a perpendicular height.
Lesson starter: Developing the rule
Start this activity by drawing a large parallelogram on a loose piece of paper. Ensure the opposite sides are parallel and then use scissors to cut it out. Label one side as the base and label the height, as shown in the diagram. β’ β’ β’
height
Cut along the dotted line. Now shift the triangle to the other end of the parallelogram to make a rectangle. Now explain how to find the area of a parallelogram.
base
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10 Measurement
KEY IDEAS β A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
β’ β’ β’
For parallelograms, the dimensions are called base (β βbβ)βand perpendicular height (β βhβ)ββ. The base can be any side of the parallelogram. The height is perpendicular to the base.
U N SA C O M R PL R E EC PA T E G D ES
688
h
h
b
b
β’
The area of a parallelogram is base βΓβperpendicular height.
β The formula for the area of a parallelogram is:
βA = b Γ hβ
β β= area A bβ β= length of the base βhβ= perpendicular height
BUILDING UNDERSTANDING
1 State the missing numbers, using the given values of βbβand βhβ. a b = 5, h = 7
c b = 8, h = 2.5
b b = 20, h = 3
A =β bhβ β―β― β ββ―β― β β β = ___ Γ ___ β = 35
A = ___ β―β― ββ β β β β β = 8 Γ ___ β = ___
A = ___ β β ββ―β― β β β = 20 Γ ___ β = ___
2 For each of these parallelograms, state the side length of the base and the height that might be used to find the area.
a
b
c
2 cm
4m
6 cm
7m
10 m
5m
d
e
f
5 cm
1.3 m
1.8 m
1.5 cm
2 cm
6.1 cm
0.9 m
5.8 cm
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10F
Area of parallelograms
689
Example 12 Finding the area of a parallelogram Find the area of these parallelograms. a
b
3 cm
U N SA C O M R PL R E EC PA T E G D ES
5m
12 m
2 cm
SOLUTION
EXPLANATION
a A = bh β β =ββ 12βΓ 5ββ β = 60 βm2β β
Choose the given side as the base β(12 m)β and note the perpendicular height is β5 mβ.
b A = bh β β =ββ 2 Γβ 3β β β = 6 βcmβ2β β
Use the given side as the base β(2 cm)β, noting that the height is β3 cmβ.
Now you try
Find the area of these parallelograms. a
b
6m
2.5 m
6 cm
3 cm
Exercise 10F FLUENCY
Example 12a
1
2, 4, 5
1, 2ββ(β1/2β)β, 3, 4ββ(β1/2β)β,β 5
Find the area of these parallelograms. a 2m
5m
2β(β β1/2β)β,β 4ββ(β1/2β)β, 5, 6
b
8 cm
10 cm
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
690
Chapter 10 Measurement
Example 12a
2 Find the area of these parallelograms. a 4m
b 7m
U N SA C O M R PL R E EC PA T E G D ES
10 m 4m
c
d
7m
12 km
2.5 m
3 km
Example 12b
3 Find the area of these parallelograms. a 7 cm
b
12 mm
4 cm
9 mm
Example 12b
4 Find the area of these parallelograms. Note that you will need to consider which dimensions are the base and the height. a b 5 cm 4 cm 3 cm 10 cm
4 cm
2.1 cm
c
2m
d
2.1 cm
15 m
1.8 cm
1 cm
3m 5 These parallelograms are on β1 cmβgrids (not to scale). Find their area. a b c
d
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10F
Area of parallelograms
691
6 The floor of an office space is in the shape of a parallelogram. The longest sides are β9 mβand the perpendicular distance between them is β6 mβ. Find the area of the office floor. PROBLEM-SOLVING
7, 8
8, 9
8β10
U N SA C O M R PL R E EC PA T E G D ES
7 Find the height of a parallelogram when its: a aβ rea = 10 βm2β βand βbase = 5 mβ b βarea = 28 βcmβ2β βand βbase = 4 cmβ c βarea = 2.5 βmmβ2β βand βbase = 5 mmβ 8 Find the base of a parallelogram when its: a βarea = 40 βcmβ2β βand βheight = 4 cmβ b βarea = 150 βm2β βand βheight = 30 mβ c βarea = 2.4 βkmβ2β βand βheight = 1.2 kmβ
9 A large wall in the shape of a parallelogram is to be painted with a special red paint, which costs β$20βper litre. Each litre of paint covers β5 βm2β β. The wall has a base length of β30 mβand a height of β10 mβ. Find the cost of painting the wall. 10 A proposed rectangular flag for a new country is yellow with a red stripe in the shape of a parallelogram, as shown. a Find the area of the red stripe. b Find the yellow area.
60 cm
30 cm
REASONING
11
70 cm
11, 12
12, 13
11 Explain why this parallelogramβs area will be less than the given rectangleβs area. 5 cm
5 cm
10 cm
10 cm
12 A parallelogram includes a green triangular area, as shown. What fraction of the total area is the green area? Give reasons for your answer.
13 The area of a parallelogram can be thought of as twice the area of a triangle. Use this idea to complete this proof of the rule for the area of a parallelogram. Area = twice triangle area β β―β― ββ―β― =β 2 Γ ________β β β = ___________
h
b
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10 Measurement
ENRICHMENT: Glass façade
β
β
14
14 The Puerta de Europa (Gate of Europe) towers are twin office buildings in Madrid, Spain. They look like normal rectangular glass-covered skyscrapers but they lean towards each other at an angle of β15Β°β to the vertical. Two sides are parallelograms and two sides are rectangles. Each tower has a vertical height of β120 mβ, a slant height of β130 mβand a square base of side β50 mβ.
U N SA C O M R PL R E EC PA T E G D ES
692
All four sides are covered with glass. If the glass costs β$180βper square metre, find the cost of covering one of the towers with glass. (Assume the glass covers the entire surface, ignoring the beams.)
130 m
120 m
50 m
50 m
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10G
Area of triangles
693
10G Area of triangles
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand that a triangle can be considered as half of a parallelogram β’ To be able to identify the base and the (perpendicular) height of a triangle β’ To be able to ο¬nd the area of a triangle
In terms of area, a triangle can be considered to be half a parallelogram, which is why the formula for the area of a triangle is the same as for a 1 . One of the sides of a triangle parallelogram but with the added factor of _ 2 is called the base (b), and the height (h) is the distance between the base and the opposite vertex. This is illustrated using a line that is perpendicular (i.e. at 90Β°) to the base. Any shape with all straight sides (i.e. polygons) can be divided up into a combination of rectangles (or squares) and triangles. This can help to find areas of such shapes.
Lesson starter: Half a parallelogram
Consider a triangle which is duplicated, rotated and joined to the original triangle as shown. h
h
b
β’ β’ β’
The Flatiron Building, New York City, was built in 1902 on a triangular block of land. Each of its 22 ο¬oors is an isosceles triangle, like the triangular shape of an iron.
b
What type of shape is the one to the right? What is the rule for the area of the shape to the right? What does this tell you about the rule for the original triangle?
KEY IDEAS
β A triangleβs dimensions are called the base (b) and height (h).
β’
β’
The base can be any side of the triangle. The height is perpendicular to the base.
h
h
b
h
b
b
β The area of a triangle is given by the formula:
1 bh = _ 1 Γ base Γ height A=_ 2 2 bh or b Γ h Γ· 2. 1 bh is equivalent to _ Note that _ 2 2 β’ The area of triangle is half the area of a parallelogram. Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10 Measurement
BUILDING UNDERSTANDING 1 For each of these triangles, what length would be used as the base? a b 7m 8 cm
2.1 m
U N SA C O M R PL R E EC PA T E G D ES
694
20 cm
c
d
2m
5 cm
3m
6.3 cm
2 For each of these triangles, what length would be used as the height? a b 11 m
11 mm
6m
15 mm
c
4.1 m
d
1.9 m
3.2 mm
3 State the value of βAβin βA = _ β1 βbhβ if: 2 a βb = 5βand βh = 4β b βb = 7βand βh = 16β
4.7 mm
c βb = 2.5βand βh = 10β
Example 13 Finding areas of triangles Find the area of each given triangle. a
b
6m
9 cm
10 cm
9m
SOLUTION
EXPLANATION
β1 βbh a area = _ 2 β ββ―β― β Γ 9β β _ β = β1 βΓ 10 2 β = 45 βcmβ2β β 1 βbh b area = β_ 2 β ββ―β― β =β _ β1 βΓ β6 Γ 9β 2 β = 27 βm2β β
Use the formula and substitute the values for base length and height.
The length measure of β9 mβis marked at β90Β°β to the side marked β6 mβ. So β6 mβis the length of the base and β9 mβis the perpendicular height.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10G
Area of triangles
695
Now you try Find the area of each given triangle. a
b
U N SA C O M R PL R E EC PA T E G D ES
6m
7 cm
8m
12 cm
Exercise 10G FLUENCY
Example 13a
1
1, 2ββ(1β/2β)β, 3, 4ββ(1β/2β)β,β 5
Find the area of each triangle given. a
2ββ(1β/2β)β,β 4ββ(1β/2β)β, 5, 6
b
6m
2ββ(1β/2β)β,β 4β5ββ(1β/2β)β,β 7
8 cm
3 cm
10 m
Example 13a
2 Find the area of each triangle given. a b
c
12 cm
5 mm
4m
5 cm
10 mm
8m
d
e
f
2.4 m
16 m
2.5 cm
1 cm
20 m
1m
Example 13b
3 Find the area of each triangle given. a
b
10 mm
4 cm
12 mm
7 cm
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
696
Chapter 10 Measurement
Example 13b
4 Find the area of each of the following triangles. a
b
7 km
2 mm
7 km
U N SA C O M R PL R E EC PA T E G D ES
3 mm
c
d
1.3 cm
4 cm
6 cm
e
10 m
2 cm
f
4m
1.7 m
5m
5 Find the area of these triangles, which have been drawn on β1 cmβgrids. Give your answer in βcmβ2β ββ. a b
c
d
6 A rectangular block of land measuring β40 mβlong by β24 mβwide is cut in half along a diagonal. Find the area of each triangular block of land.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10G
7 A square pyramid has a base length of β120 mβand a triangular face of height β80 mβ. Find the area of one triangular face of the pyramid.
Area of triangles
697
80 m
U N SA C O M R PL R E EC PA T E G D ES
120 m
PROBLEM-SOLVING
8, 9
9, 10
9β11
8 Each face of a β4β-sided die is triangular, with a base of β2 cmβand a height of β1.7 cmβ. Find the total area of all β4βfaces of the die. 9 A farmer uses fencing to divide a triangular piece of land into two smaller triangles, as shown. What is the difference in the two areas? 10 A yacht must have two of its sails replaced as they have been damaged by a recent storm. One sail has a base length of β2.5 mβ and a height of β8 mβand the bigger sail has a base length of β4 mβand a height of β16 mβ. If the cost of sail material is β$150βper square metre, find the total cost to replace the yachtβs damaged sails.
18 m
26 m
40 m
11 a The area of a triangle is β10 βcmβ2β βand its base length is β4 cmβ. Find its height. b The area of a triangle is β44 βmmβ2β βand its height is β20 mmβ. Find its base length. REASONING
12
12, 13
13, 14
12 The midpoint, βMβ, of the base of a triangle joins the opposite vertex. Is the triangle area split in half exactly? Give reasons for your answer.
M
h
13 If the vertex βCβfor this triangle moves parallel to the base βABβ, will the area of the triangle change? Justify your answer.
C
14 The area of a triangle can be found using the formula βA = _ β1 βbhβ. Write down 2 the formula to find the base, βbβ, if you are given the area, βAβ, and height, βhβ. ENRICHMENT: Estimating areas with curves
β
A
β
B
15
15 This diagram shows a shaded region that is _ β1 βof β3 βcmβ2β β= 1.5 βcmβ2β ββ. 2 Using triangles like the one shown here, and by counting whole squares also, estimate the areas of these shapes below. a b c
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
698
Chapter 10 Measurement
10A
1
Which metric unit would be most appropriate for measuring the lengths of: a the height of the classroom? b the width of your thumb c the distance from Sydney to Canberra?
2 Convert to the units given in brackets. a 4β m (β cm)β b β2 m (β mm)β d β3 km (β m)β e 1β .45 km (β m)β
c 3β .5 cm (β mm)β f β23 000 m (β km)β
U N SA C O M R PL R E EC PA T E G D ES
Progress quiz
10A
10B
3 Find the perimeter of each of these shapes. a b 5 cm 46 mm 4m 5 cm
c
70 cm
32 cm
25 cm
1.2 m
52 cm
10C
10C
4 A good approximation of βΟβis the fraction _ β69 ββ. 22 69 βas a decimal, rounded to three decimal places. a Write β_ 22 b Decide if _ β69 βis a better approximation to βΟβthan the number β3.14β. 22
5 Calculate the circumference of these circles using a calculator and rounding your answer correct to two decimal places. a
b
9 mm
10D
6 cm
6 Find the perimeter of these shapes, correct to one decimal place. a b
74Λ 5 cm
12 cm
c
6 cm
10 cm
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Progress quiz
10E
699
7 Find the area of this rectangle and square. b 45 cm
7 cm
U N SA C O M R PL R E EC PA T E G D ES
2m
Progress quiz
a
10F
8 Find the area of these parallelograms. a b 5 cm
c
9m
12 mm
mm
8.1 m
16
11 cm
10G
9. Find the area of these triangles. a 4m 3m
20 mm
b
7m
5m
9m
c
4 4.
m
d
8 mm
5 mm
10B/E
10G
10 The perimeter of a square made of thin wire is β60 cmβ. Find: a the area of this square b the area of a rectangle made with this wire, if its length is twice its width c the number of smaller squares that can be made from this wire if each square has an area of β 4βsquare centimetres. 11 What fraction of the entire shape is the shaded region?
3 cm
6 cm
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10
Measurement
10H Area of composite shapes
EXTENDING
LEARNING INTENTIONS β’ To understand what a composite shape is β’ To be able to identify simple shapes that make up a composite shape β’ To be able to ο¬nd the area of a composite shape
U N SA C O M R PL R E EC PA T E G D ES
700
The areas of more complicated shapes can be found by dividing them up into more simple shapes, such as the rectangle and triangle. We can see this in an aerial view of any Australian city. Such a view will show that many city streets are not parallel or at right angles to each other. As a result, this causes city blocks to form interesting shapes, many of which are composite shapes made up of rectangles and triangles.
Streets, parks and buildings form complex shapes that can be made up of triangles and rectangles.
Lesson starter: Dividing land to ο¬nd its area
Working out the area of this piece of land could be done by dividing it into three rectangles, as shown. β’ β’ β’
Can you work out the area using this method? What is another way of dividing the land to find its area? Can you use triangles? What is the easiest method to find the area? Is there a way that uses subtraction instead of addition?
20 m
30 m
30 m
10 m
20 m
KEY IDEAS
β Composite shapes are made up of more than one simple shape.
β The area of composite shapes can be found by adding or subtracting the areas of simple shapes.
A square plus a triangle
A rectangle minus a triangle
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10H Area of composite shapes
701
BUILDING UNDERSTANDING 1 Describe where you would draw a dotted line to divide these shapes into two more simple shapes.
b
c
U N SA C O M R PL R E EC PA T E G D ES
a
2 To find the area of each of the following shapes, decide if the easiest method would involve the addition of two shapes or the subtraction of one shape from another.
a
b
c
3 State the missing numbers to complete the solutions for the areas of these shapes. a b 5m 1 cm 4m
2 cm
4m
1 cm
13 m
3 cm
A = lw β ____
A = lββ2β+ ____ β = 1β 2β β+ 3 Γ ____ β ββ β ββ―β― β―β― β―β― β β = ____ + ____ β = ____ βcmβ2β β
β = ____ Γ 8 β _ β1 βΓ ____ Γ ____ β β ββ―β― β β―β―β― β―β― β 2β β = ____ β ____ β = ____ βm2β β
Example 14 Finding the area of composite shapes using addition
Find the area of each of these composite shapes. a 2 cm
b
12 m
5m
3 cm
2 cm
9m
5 cm
SOLUTION
EXPLANATION
a A = l Γ w + βlβ2β =β 5 Γ 2 +β β22βββββ β―β― β βββ―β― β = 10 + 4 β = 14 cβ mβ2β β
The shape is made up of a rectangle of length β 5 cmβand width β2 cmβand a square with side length β2 cm.β
Continued on next page
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10 Measurement
b A = lΓw+_ β1 bβ h 2 1 _ β =β 9 Γ 5 + β2β βΓ β3 Γ 5β β―β― β β―β― β = 45 + 7.5 β = 52.5 βm2β β
Divide the shape into a rectangle and triangle and find the missing lengths. 9m 5m
3m 5m
U N SA C O M R PL R E EC PA T E G D ES
702
9m
Now you try
Find the area of these composite shapes. a 3m
b
4 cm
3 cm
6 cm
8m
Example 15 Finding the area of composite shapes using subtraction
Find the area of each of these composite shapes. a 4 cm
b
4 cm
3 cm
6 cm
7 cm
7 cm
SOLUTION
EXPLANATION
a A = βlβ2ββ _ β1 βbh 2 2 β =β β7β ββ _ β1 βΓβ 4 Γ 3β β β―β― 2 β = 49 β 6 β = 43 cβ mββ2β
β1 βΓ 4 Γ 3)βfrom the Subtract the triangle ( β_ 2 square β(7 Γ 7)β. 4 cm
3 cm
7 cm
7 cm
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10H Area of composite shapes
β1 βΓ 6 Γ 4)βat the top of Subtract the triangle ( β_ 2 the shape from the larger square β(6 Γ 6)β. 6 cm 4 cm
U N SA C O M R PL R E EC PA T E G D ES
b A = βlβ2ββ _ β1 βbh 2 2 =β β6β ββ _ β1 βΓβ β 6 Γ 4β β ββ―β― 2 β = 36 β 12 β = 24 cβ mβ2β β
703
6 cm
6 cm
Now you try
Find the area of each of these composite shapes. a 7 cm 8 cm
10 cm
b
5 cm
2 cm
3 cm
2 cm
Exercise 10H FLUENCY
Example 14
1
1, 2ββ(1β/2β)β, 3, 4ββ(1β/2β)β
2ββ(1β/2β)β,β 4ββ(1β/2β)β
2ββ(1β/2β)β,β 4ββ(1β/2β)β
Find the area of these composite shapes. Assume that all angles that look like right angles are right angles. a 11 cm 4 cm
7 cm
4 cm
b
10 m
4m
6m
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
704
Chapter 10 Measurement
Example 14
2 Find the area of these composite shapes by adding together the area of simpler shapes. Assume that all angles that look like right angles are right angles. a b 3m 10 mm 10 mm
U N SA C O M R PL R E EC PA T E G D ES
4m 40 mm
6m
c
d
2m
5m
3m
8m
9m
3m
e
f
21 cm
Example 15
6 km
3 km
9 cm
10 km 17 cm 3 Find the area of these composite shapes. Assume that all angles that look like right angles are right angles. a b 6 cm 10 cm 5 cm
2 cm
9 cm
5 cm
Example 15
4 Use subtraction to find the area of these composite shapes. Assume that all angles that look like right angles are right angles. a b 3m 2m
10 cm
6m
3 cm
3 cm
7m
c
d
2m
12 m
5m
5m
7m
6m
e
f
1m
20 cm
2.5 m
4m
14 cm Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10H Area of composite shapes
PROBLEM-SOLVING
5, 6
5β7
705
6β8
5 Find the areas of these composite shapes. Assume that all angles that look like right angles are right angles. a b c 4 cm
U N SA C O M R PL R E EC PA T E G D ES
8 cm
5 cm
15 cm
2m
4 cm
20 cm
6 By finding the missing lengths first, calculate the area of these composite shapes. a b
2m
3.5 m
4m
12 cm
c
d
5m
2 cm
3.5 cm
1 cm
2m
7 A wall has three square holes cut into it to allow for windows, as shown. Find the remaining area of the wall.
6 cm
2.4 m
1.2 m
6m
8 A factory floor, with dimensions shown opposite, is to be covered with linoleum. Including underlay and installation, the linoleum will cost β$25βper square metre. The budget for the job is β$3000β. Is there enough money in the budget to cover the cost?
10 m
4m
12 m
5m
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10
Measurement
REASONING
9
10
9, 10
9 Explain why using subtraction is sometimes quicker than using addition to find the area of a composite shape. Refer to the diagram as an example. 10 The 4-sided shape called the trapezium has one pair of parallel sides. a For the trapezium shown opposite, is it possible to find the base length of each triangle on the sides? Justify your answer. b Can you come up with a method for finding the area of this trapezium using the rectangle and triangles shown in the diagram? Use diagrams to explain your method.
U N SA C O M R PL R E EC PA T E G D ES
706
ENRICHMENT: Adding to infinity
β
6m
5m
10 m
β
11
11 The square given opposite, which has an area of 1 unit, is divided to show 1, _ 1, _ 1, β¦ the areas of _ 2 4 8 a Similar to the one shown opposite, draw your own square, showing as many fractions as you can. Try to follow the spiral pattern shown. (Note: The bigger the square you start with, the more squares you will be able to show.) b i Write the next 10 numbers in this number pattern. 1, _ 1, _ 1, β¦ _ 2 4 8
1 4
1 8
1 32
1 2
1
1 16
1
ii Will the pattern ever stop? c What is the total area of the starting square? d What do your answers to parts b ii and c tell you about the answer to the sum below? 1+_ 1+_ 1+_ 1 + β¦ (continues forever) _ 2 4 8 16
Architects and engineers calculate the areas of many geometric shapes when designing and constructing modern multistorey towers, such as the 182 m high Hearst Tower in New York City.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10I
Volume of rectangular prisms
707
10I Volume of rectangular prisms
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand what the volume of a three-dimensional object is β’ To know that common metric units for volume include cubic millimetres, cubic centimetres, cubic metres and cubic kilometres β’ To know what a rectangular prism (or cuboid) is β’ To be able to ο¬nd the volume of a cube and other rectangular prisms
The amount of space inside a three-dimensional (3D) object is called volume. Volume is measured in cubic units such as the cubic centimetre, which is 1 cm long, 1 cm wide and 1 cm high.
Just like the topics of length and area, different units can be selected, depending on the size of the volume being measured. For example, the volume of water in the sea could be measured in cubic kilometres and the volume of concrete poured from a cement mixing truck could be measured in cubic metres.
The Paciο¬c Ocean contains around 700 million cubic kilometres of water.
Lesson starter: Volume
We all know that there are 100 cm in 1 m, but do you know how many cubic centimetres are in 1 cubic metre? β’ β’
Try to visualise 1 cubic metre: 1 metre long, 1 metre wide and 1 metre high. Guess how many cubic centimetres would fit into this space. Describe a method for working out the exact answer. Explain how your method works.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10 Measurement
KEY IDEAS β Volume is measured in cubic units. β The common metric units for volume include:
β’
cubic millimetres (mm3)
1 mm 1 mm 1 mm
U N SA C O M R PL R E EC PA T E G D ES
708
(Not drawn to scale.)
β’
cubic centimetres (cm3)
1 cm
1 cm
1 cm
β’
cubic metres (m3)
1m
(Not drawn to scale.)
1m
1m
β’
cubic kilometres (km3)
1 km
(Not drawn to scale.)
1 km
1 km
β The volume of a rectangular prism is given by the
formula:
l
V = length Γ width Γ height β β β ββ―β―β― β = lwh β’
w
h V = lwh
A rectangular prism is also called a cuboid.
β The volume of a cube is given by:
V = lΓlΓl ββ β 3 β β = βlβ β
l
l
V = l3
l
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10I
Volume of rectangular prisms
709
BUILDING UNDERSTANDING
U N SA C O M R PL R E EC PA T E G D ES
1 For each of these solids, count the number of cubic units to find its volume. a b
c
d
e
f
2 State the missing numbers to complete the working shown for each of these solids. a
3 cm
4 cm
b
c
2 km
2 cm
6m
2 km
1m
V = lwh β β―β― β =β 4 Γ β__Γ __β β = __βcmβ3β β
3m
V = lwh =ββ 1 Γ β__Γ __ββ β ββ―β― β = __ __
2 km
V = lwh β =β 2 Γ β__Γ __β β β―β― β = __ βkmβ3β β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
710
Chapter 10 Measurement
Example 16 Finding the volume of a cuboid Find the volume of this rectangular prism.
4 cm 3 cm
U N SA C O M R PL R E EC PA T E G D ES
8 cm
SOLUTION
EXPLANATION
V = lwh =ββ 8 Γ β3 Γ 4ββ β ββ―β― β = 96 cβ mβ3β β
Use the formula for the volume of a rectangular prism, then substitute the three lengths into the formula.
Now you try
2 cm
Find the volume of this rectangular prism.
4 cm
6 cm
Exercise 10I FLUENCY
Example 16
1
Find the volume of these rectangular prisms. a 3m 7m
Example 16
1, 2ββ(1β/2)β β,β 3β5
2β(β 1β/2)β β,β 3β6
b
4m
2β(β 1β/2)β β,β 4β6
1 cm
2 cm
5 cm
2 Find the volume of these rectangular prisms. a b
c
7 cm
7 cm
d
3 cm
2 cm
e
10 km
1m
10 m
f
1.5 km 5 km
1.4 mm
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10I
Volume of rectangular prisms
711
3 A fruit box is β40 cmβlong, β30 cmβwide and β20 cmβhigh. Find its volume. 4 A shipping container is β3 mβwide, β4 mβhigh and β8 mβlong. Find its volume. 5 A short rectangular ruler is β150 mmβlong, β40 mmβwide and β2 mmβthick. Find its volume.
U N SA C O M R PL R E EC PA T E G D ES
6 There is enough ice on Earth to fill a cube of side length β300 kmβ. Find the approximate volume of ice on Earth. PROBLEM-SOLVING
7β9
7β(β β1/2β)β,β 8
7ββ(β1/2β)β, 9, 10
7 These solids are made up of more than one rectangular prism. Use addition or subtraction to find the volume of the composite solid. a b 4m 3m
4 cm
2 cm
3 cm
10 m
2 cm
5 cm
c
d
8 cm
4m
4 cm
2m
5m
9 cm
4m
10 m
2 cm
8 A box measuring β30 cmβlong, β20 cmβhigh and β30 cmβwide is packed with matchboxes, each measuring β 5 cmβlong, β2 cmβhigh and β3 cmβwide. How many matchboxes will fit in the box? 9 The outside dimensions of a closed wooden box are β20 cmβby β20 cmβby β20 cmβ. If the box is made of wood that is β2 cmβthick, find the volume of air inside the box. 10 a The area of one face of a cube is β25 βcmβ2β β. Find the cubeβs volume. b The perimeter of one face of a cube is β36 mβ. Find the cubeβs volume. REASONING
11
11
11, 12
11 We can find the area of this shaded triangle by thinking of it as half a rectangle. Use the same idea to find the volume of each of these solids. a
b
c
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10 Measurement
12 a Find the height of these rectangular prisms with the given volumes. Use trial and error, or set up and solve an equation. i ii hm 4m V = 16 m3
2m
h cm
U N SA C O M R PL R E EC PA T E G D ES
712
3 cm V = 45 cm3
iii
iv
hm
h cm
4 cm
2m
9m V = 54 m3
2 cm V = 56 cm3
b Can you explain a method that always works for finding the height of a rectangular prism? c Use V β , lβand βwβto write a rule for βhβ.
ENRICHMENT: Cubic conversions
β
β
13
13 a The diagram shows a β1 βcmβ3β βblock that is divided into cubic millimetres. i
How many βmmββ3βare there along one side of the cube?
1 cm
ii How many βmmβ3β βare there in one layer of the cube? (Hint: How many cubes sit on the base?)
iii How many layers of βmmβ3β βare there in the cube? iv Use your answers from parts i to iii above to now calculate how
many βmmββ3βthere are in β1 βcmβ3β ββ. b Use a similar method to calculate the number of: i βcmββ3βin β1 βm3β β ii βmβ3βin β1 βkmβ3β β c Complete the diagram shown. Γ ____ Γ ____
km3
m3
1 cm
Γ ____
cm3
Γ· ____ Γ· ____
1 cm
mm3
Γ· ____
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Applications and problem-solving
U N SA C O M R PL R E EC PA T E G D ES
Painting a bedroom
Applications and problem-solving
The following problems will investigate practical situations drawing upon knowledge and skills developed throughout the chapter. In attempting to solve these problems, aim to identify the key information, use diagrams, formulate ideas, apply strategies, make calculations and check and communicate your solutions.
713
1
Braxton wishes to redesign his bedroom and wants to start by painting the walls and the ceiling. Braxton measures his room to be β4.5 m Γ 3.0 mβ, and the height of his ceiling to be β2.4 mβ. Braxton wants to calculate the total wall and ceiling area of his room in order to determine the number of litres of paint to purchase. a What is the area of Braxtonβs ceiling? b What is the total area of Braxtonβs walls (assuming there are no doors or windows)? c Braxton does have three doors (one entry door and two wardrobe doors) each measuring β800 mm Γ 2.1 mβ, and one window measuring β1500 mm Γ 1.8 mβin his bedroom. What is the total area of Braxtonβs doors and window? d Using your answers to parts b and c, what is the actual total area of Braxtonβs walls? e Braxton decides to paint the walls and ceiling with two coats of paint. What is the total area Braxton will be painting? f Braxton decides to use the same colour paint for both the walls and the ceiling. He has chosen his colour and the paint tin says that the coverage for the paint is β12 βm2β βper litre of paint. How many litres of paint does Braxton need to buy?
Building a sponge pit
2 The Year β7βStudent Representative Council (SRC) at Ashwelling Secondary College wish to build a βsponge pitβ as a fun activity where students can connect and interact. They would like the sponge pit to be big enough to have four students sit comfortably in there to ask one another questions. They decide to make the sponge pit in the shape of a cube with side length β1.5 mβ. Materials: β’ β45 mm Γ 90 mmβpine for the sponge pit frame at β$3.89βper linear metre β’ β18 mmβMarine plywood β (1500 mm Γ 750 mm)βfor the bottom and sides of the sponge pit at β$58.95βper sheet β’ β5 cmβlength foam sponge cubes at β $7.95βfor a pack of β100β The SRC need to determine the quantity and cost of the materials needed for the schoolβs maintenance team to build the sponge pit.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10 Measurement
a How many metres of pine is required for the sponge pit frame? b How many sheets of marine plywood β(1500 mm Γ 750 mm)βare required to line the bottom and the sides of the sponge pit? c What is the volume of the sponge pit? Give your answer in β3 βcmβ3β ββ. d For the βballsβ, the SRC actually decide to use blue cubes of a dense sponge material. The side length of the cube is β5 cmβ. What is the volume of one blue sponge cube? e While there always will be some space between cubes in the sponge pit, assuming that there are no gaps, what is the maximum number of sponge cubes needed to fill the empty pit? f Given that students need to sit in the sponge pit, and that there will be some gaps, a decision is made to order only β75%βof the maximum number of sponge cubes. How many packs of β100β sponge cubes need to be ordered? g What is the total cost of materials for the sponge pit? Round your answer to the nearest dollar. h If the school decides to build a sponge pit, but they only have β$1000βfor the total budget for materials, what would be the maximum side length of the pit that could be built?
U N SA C O M R PL R E EC PA T E G D ES
Applications and problem-solving
714
Planting pine trees on a farm
3 A farmer has a vacant rectangular paddock which is β50 mβlong and β30 mβwide. She would like to plant a pine plantation on the paddock to be harvested every β15βyears. She decides all pine trees must be planted at least β2 mβaway from the paddock fence, so the trees do not overhang the paddock boundary or damage her fences. β’ Pine trees planted β4 mβapart will generally reach their maximum growth height. β’ Pine trees planted β3 mβapart will generally reach only β70%βof their maximum growth height. β’ Pine trees planted β2 mβapart will generally reach only β35%βof their maximum growth height. The farmer is interested in how many trees to plant depending on how far apart they should be and how to maximise the harvest which depends on the planting arrangements. a How many trees can the farmer plant in their vacant paddock if they decide to plant them β4 mβ apart? b How many trees can the farmer plant in their vacant paddock if they decide to plant them β3 mβ apart? c How many trees can the farmer plant in their vacant paddock if they decide to plant them β2 mβ apart? d Given the different growth heights that will be achieved, how far apart should the farmer plant the pine trees to maximise their harvest?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10J
715
Volume of triangular prisms
10J Volume of triangular prisms
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand that the volume of a prism can be calculated using the area of the cross-section and the perpendicular height β’ To know what a triangular prism is β’ To be able to ο¬nd the volume of a triangular prism
We know that the rule for the volume of a rectangular prism, V = l Γ w Γ h, includes the product l Γ w which represents the area of the base. So, V = l Γ w Γ h could be rewritten as V = A Γ h where A is the area of the base. The area of the base also tells us how many cubes fit on one layer inside the solid. Multiplying by the height gives the total number of cubes in the solid and hence the volume. This rule V = A Γ h can therefore be applied to other prisms, including a triangular prism provided that the area of the base (cross-section) can be calculated.
h
A
w
l
Lesson starter: A trio of triangular prisms
We know that the volume of a prism is given by V = A Γ h and for the triangular prisms shown below, that area A corresponds to the cross-section which is a triangle. a
b
2 cm
c
10 cm
6 cm
6 cm
18 cm
4 cm
3 cm
3 cm
β’ β’ β’ β’
4 cm
Draw a two-dimensional representation of the triangular cross-sections of the three prisms labelling the base and the height. Calculate the area, A of the triangular cross sections. Use V = A Γ h to find the volume of each prism by firstly deciding what measurement represents the height. Discuss the shape and orientation of the triangular cross-sections of the three prisms and the direction of the height (h) used in the formula V = A Γ h.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10 Measurement
KEY IDEAS β A right triangular prism has:
β’ β’
a uniform (constant) triangular cross-section (base) three rectangular faces a height measured perpendicular (at β90Β°β) to the base.
U N SA C O M R PL R E EC PA T E G D ES
716
β’
A
h
β The volume of a triangular prism is given
A by the formula βV = A Γ hβ, where: β’ βVβis the volume β’ βAβis the area of the triangular cross-section (base) β’ βhβis the height of the prism which is perpendicular (at β90Β°β) to the base β’ the area of the cross-section βAβcan be found using the β1 βbh.β formula for the area of a triangle βA = _ 2
h
h
b
BUILDING UNDERSTANDING
1 For each triangular prism decide which measurement would be used as the height, βhβ, in the formula βV = A Γ hβ.
a
b
4m
3m
5m
4m
7m
8m
c
2 cm
1.8 cm
2.5 cm
2 For the triangular prisms in question β1βabove, find the area of the triangular cross-section using β A=_ β1 βbhβ. 2
3 For the triangular prisms in question β1βabove, find the volume using βV = A Γ hβ. Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10J Volume of triangular prisms
717
Example 17 Finding the volume of a triangular prism b 5 cm
U N SA C O M R PL R E EC PA T E G D ES
Find the volume of these triangular prisms. a
7 cm
5m
6m
10 cm
8m
SOLUTION
EXPLANATION
a A =_ β1 βbh 2 _ β = β1 βΓ 8 Γ 5 2 β βββ―β― β = 20 βmβ2β ββ β β V = AΓh β = 20 Γ 6 β = 120 βm3β β b A =_ β1 βbh 2 _ β = β1 βΓ 10 Γ 5 2 β β2β βββ β β β ββ―β― β = 25 βcmβ V = AΓh β = 25 Γ 7 β = 175 βcmβ3β β
First find the area of the triangular cross-section.
The height used for the prism is β6 cmβand is perpendicular to the cross-section.
First find the area of the triangular cross-section with base β10 cmβand height β5 cmβ.
Use the rule for the volume of a prism noting that the height is β7 cm.β
Now you try
Find the volume of these triangular prisms. a
b
3m
3 cm
6 cm
6m
8m
5 cm
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
718
Chapter 10 Measurement
Example 18 Finding the volume of a triangular prism with an obtuse angled triangular cross-section
U N SA C O M R PL R E EC PA T E G D ES
Find the volume of this triangular prism.
6 mm
9 mm
5 mm
SOLUTION
EXPLANATION
A =_ β1 βbh 2 _ β = β1 βΓ 5 Γ 6 2 β βββ―β― β β2β ββ β =β 15 m β βmβ V = AΓh β = 15 Γ 9 β = 135 m β mβ3β β
First find the area of the triangular cross-section.
The height used for the prism is β9 mmβand is perpendicular to the cross-section.
Now you try
Find the volume of this triangular prism.
10 m
4m
7m
Exercise 10J
Example 17
1
FLUENCY
1β4
Find the volume of these triangular prisms. a
b 4 cm
2β5
2β(β 1β/3β)β,β 4ββ(1β/3β)β,β 6
2 mm
4 mm
6 cm
5 mm 7 cm
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10J Volume of triangular prisms
Example 17a,b
2 Find the volume of these triangular prisms. a 2m
719
b
3m 5 cm
U N SA C O M R PL R E EC PA T E G D ES
7 cm
10 cm
3m
c
d
8 mm
4 mm
5 mm
12 mm
9 mm
5 mm
e 6m
f
9 mm
5 mm
10 mm
13 m
15 m
Example 18
3
Find the volume of these triangular prisms. a
b
4m
6m
4m
3 cm
3 cm
2 cm
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
720
Chapter 10 Measurement
Example 18
4
Find the volume of these triangular prisms. a
b
3m 2m
3m
U N SA C O M R PL R E EC PA T E G D ES
5m
3m
4m
c
1.1 cm
1.5 cm
1.2 cm
5
A simple tent is in the shape of a triangular prism as shown. Find the tentβs volume.
1.4 m
3m
2m
6 A concrete ramp is in the shape of a triangular prism as shown. Find the volume of concrete in βcmβ3β ββ.
80 cm
100 cm
90 cm
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10J Volume of triangular prisms
PROBLEM-SOLVING
7, 8
7, 8ββ(β1/2β)β
721
8, 9
7 A chocolate bar is in the shape of a triangular prism. The triangular cross-section has base β3 cmβ and height β2 cmβ. Its total length is β12 cmβ. Find the volume of the chocolate in the bar.
U N SA C O M R PL R E EC PA T E G D ES
8 The following composite solids combine rectangular and triangular prisms. Find the volume. a b
3 cm
5m
4m
4 cm
2m
2m
5 cm
4 cm
c
d
7 cm
4 cm
3 cm
2 cm
4 cm
8 cm
3 cm
5 cm
9 A box in the shape of a cube has side length β20 cmβ. It contains β400βsmall metal wedges, each of which is a triangular prism with cross-sectional area β2.8 βcmβ2β βand length β3.5 cmβ. Find the volume of air space remaining in the box. REASONING
10
10, 11
10ββ(β1/2β)β, 11, 12
10 The following refer to triangular prisms with given volumes and unknown cross-sectional areas or height measurements. a If the volume is β40 βcmβ3β βand the cross-sectional area is β10 βcmβ2β β, find its height.
b If the volume is β8.2 βm3β βand the cross-sectional area is β1.6 βm2β β, find its height.
c If the volume is β125 βmmβ3β βand the height is β20 mmβ, find its cross-sectional area.
d If the volume is β4000 βcmβ3β βand the height is β400 cmβ, find its cross-sectional area.
11 A triangular prism has volume β24 βcmβ3β βand height β8 cmβ. The height of the triangular cross-section is β 2 cmβ. Find the base length of the triangular cross-section.
12 A triangular prism has volume β100 βmmβ3β βand height β20 mmβ. The base of the triangular cross-section is β 4 mmβ. Find the height of the triangular cross-section.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10 Measurement
ENRICHMENT: Reversing the rules
β
13 The given triangular prism has volume βVβ, height βh2β ββ and the triangular cross-section has base βbβand height βh1β ββ. a Write a rule for the volume βVβin terms of βb, βh1β β, and βh2β ββ. b Write a rule for the volume βh2β βin terms of βV, bβand βh1β ββ. c Write a rule for the volume βh1β βin terms of βV, bβand βh2β ββ. d Write a rule for the volume βbβin terms of βV, βh1β βand βh2β ββ. e Use one of your rules above to find: i βh2β βif βV = 12, b = 4βand βh1β β= 4β. ii βh1β βif βV = 20, b = 5βand βh2β β= 8β. iii βbβif βV = 80, βh1β β= 20βand βh2β β= 10β.
β
13
h1
U N SA C O M R PL R E EC PA T E G D ES
722
h2
b
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10K
Capacity
723
10K Capacity CONSOLIDATING
U N SA C O M R PL R E EC PA T E G D ES
LEARNING INTENTIONS β’ To understand that capacity is the volume of ο¬uid or gas that an object can hold β’ To know that common metric units include millilitres, litres, kilolitres and megalitres β’ To be able to convert between common units for capacity β’ To be able to convert between the volume and capacity of a container
Capacity relates to the volume of ο¬uid or gas that a container can hold. For example, the capacity of a water tank may be 5000 litres, or a farmerβs water allocation might be 300 megalitres (meaning 300 million litres). The basic unit is the litre, which contains 1000 cm3 of space. Other common metric units for capacity include the millilitre, kilolitre and megalitre. There is a clear link between capacity and volume, as they both relate to the space occupied by a three-dimensional object.
Lesson starter: Water containers
The space that a surfboard occupies is given in litres, as it is measured by the volume of water it displaces when submerged in a bath. Boards with a higher litre capacity ο¬oat higher in the water.
Will, Tony and Ethan each bring a container to collect some water from a fountain. β’
Will says his container holds 2 litres.
β’
Tony says his container holds 2000 cm3. Ethan says his container holds 2000 millilitres.
β’
Who can collect the most water? Give reasons for your answer.
KEY IDEAS
β Capacity is the volume of ο¬uid or gas that an object can hold. β Common metric units include:
β’ β’ β’
1 litre (L ) = 1000 millilitres (mL) 1 kilolitre (kL ) = 1000 litres (L) 1 megalitre (ML ) = 1000 kilolitres (kL)
Γ 1000
ML
Γ 1000
kL
Γ· 1000
Γ 1000
L
Γ· 1000
mL
Γ· 1000
β Relating volume and capacity
β’ β’
1 cm3 = 1 mL 1 m3 = 1000 L = 1 kL
M I L K
M I L K
1L (1000 mL)
1000 cm3
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10 Measurement
BUILDING UNDERSTANDING 1 State the missing number. a β1 mLβcontains the volume of _____________ βcmβ3β ββ. b β1 Lβcontains _______________ βmLβ. c β1 Lβcontains _________________ βcmβ3β ββ. d β1 kLβcontains ___________________ βLβ. e β1 MLβcontains __________________ βkLβ.
U N SA C O M R PL R E EC PA T E G D ES
724
2 State which volumes are the same. a β1 L, 10 kL, 1000 mL, 1 βm3β β, 1000 βcmβ3β β b β1 βm3β β, 100 L, 1000 L, 1000 ML, 1 kLβ
3 From options A to F, choose the capacity that best matches the given container. a teaspoon A β18 Lβ b cup B β250 mLβ c bottle C β10 kLβ d kitchen sink D β20 mLβ e water tank E β45 MLβ f water in a lake F β0.8 Lβ
Example 19 Converting units for capacity
Convert to the units shown in brackets. a 5β 00 mL (L) β
b β4 kL (L)β
SOLUTION
EXPLANATION
a 500 =β 500 Γ· 1000 Lβ β mLββ―β― β = 0.5 L
When converting to a larger unit, divide. There are β 1000 mLβin β1 Lβ.
b 4β kLβ―β― β =β 4 Γ 1000 Lβ β = 4000 L
When converting to a smaller unit, multiply. There are β1000 Lβin β1 kL.β
Now you try
Convert to the units shown in brackets. a β7200 mL (L)β
b β7 ML (kL)β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10K Capacity
725
Example 20 Converting units for capacity using multiple steps Convert to the units shown in the brackets. a β8 400 000 mL (kL)β
U N SA C O M R PL R E EC PA T E G D ES
b β3 ML (L) β
SOLUTION
EXPLANATION
a 8 400 000 mL = 8 400 000 Γ· 1000 L ββ―β― β―β― β =β 8400 Γ· 1000 kLβ β β = 8.4 kL
There are β1000 mLβin β1 Lβand β1000 Lβin β1 kL.β
b 3 ML = 3 Γ 1000 kL ββ―β― β β β―β― =β 3 Γ 1000 Γ 1000 Lβ β = 3 000 000 L
There are β1000 kLβin β1 MLβand β1000 Lβin β1 kLβ. So, β1 MLβis β1 million litresβ.
Now you try
Convert to the units shown in the brackets. a β67 000 000 L (ML)β
b β9 kL (mL)β
Example 21 Converting βcmβ3β β to litres
Find the capacity of this container, in litres.
10 cm
SOLUTION
EXPLANATION
V = 20 Γ 10 Γ 10 β = 2000 βcmβ3β β β ββ―β― β―β― =β 2000 mLββ β β β = 2000 Γ· 1000 L β= 2L
V β = lwhβ 1β βcmβ3β β= 1 mLβ There are β1000 mLβin β1 litreβ.
20 cm
Now you try
Find the capacity of this container, in litres.
10 cm
30 cm
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
726
Chapter 10 Measurement
Exercise 10K FLUENCY
1
2ββ(1β/2β)β, 3β5, 7ββ(1β/2β)β
2β(β 1β/4β)β, 4β6, 7ββ(1β/2β)β
Convert to the units shown in the brackets. a 7β 00 mL (L) β b β8000 mL (L)β
c β60 mL (L)β
d β2 kL (L)β
f
U N SA C O M R PL R E EC PA T E G D ES
Example 19a
1, 2ββ(1β/2β)β, 3, 5, 7ββ(1β/2β)β
Example 19b Example 19
e β5 kL (L)β
2 Convert to the units shown in brackets. a β2 L (mL)β b β0.1 L (mL)β e β2000 L (kL)β f 3β 500 mL (L)β i β0.257 L (mL)β j β9320 mL (L)β m 0β .5 kL (L)β n 9β 1 000 kL (ML)β
c g k o
β7.5 kL (L)β
6β ML (kL)β β70 000 mL (L)β β3.847 ML (kL)β β0.42 L (mL)β
β24 kL (L)β β2500 kL (ML)β β47 000 L (kL)β β170 L (kL)β
d h l p
3 Read these scales to determine the amount of water in each of the containers. a b c 25 4 20 3 15 2 10 1 5 m3 mL
1L 750 500 250 mL
4 A cup of β200 mLβof water is added to a jug already containing β1 Lβof water. Find the total volume in: a βmLβ b βL.β
Example 20a
Example 20b
5 Convert to the units shown in brackets. a 4β 700 000 mL (kL)β b 2β 60 000 mL (kL)β e β9 ML (L)β
f
β0.4 ML (L)β
c β5 000 000 L (ML)β
d β3 840 000 L (ML)β
g β0.2 kL (mL)β
h β0.005 kL (mL)β
6 A farmer purchases β3.3 MLβof water for her apple orchard. How many litres is this?
Example 21
7 Find the capacity of each of these rectangular prism containers, in litres. a b
10 cm
15 cm
8 cm
10 cm
12 cm
5 cm
c
d
6 cm
3 cm
21 cm
9 cm
32 cm
e
13 cm
f
10 cm 50 cm 32 cm Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10K Capacity
PROBLEM-SOLVING
8, 9
9, 10
727
10β12
8 A swimming pool in the shape of a rectangular prism has length β50 mβ, width β25 mβand depth β2 mβ. Find the swimming poolβs: a volume, in βm3β β b capacity, in βL.β
U N SA C O M R PL R E EC PA T E G D ES
9 A dripping tap leaks about β10 mLβevery minute. a If there are β50βdrips per minute, find the volume of one drip. b Find the approximate volume of water, in litres, that has leaked from a tap after the following time periods. i β100 minutesβ ii β1 hourβ iii β1 dayβ iv 1β yearβ
10 A dose of 1β 2 mLβof medicine is to be taken twice each day from a 0β .36 Lβbottle. How many days will it take to finish the medicine? 11 A gas bottle contains β50 Lβof liquid gas. If the liquid gas is used at a rate of β20 mLβper minute, how many hours will the gas bottle last?
12 A cityβs dams have β2βmillion megalitres of water and the average daily consumption of the cityβs people is β400 Lβper day per person. If the cityβs population is β5βmillion people, how long will the dam supply last without further water catchment? REASONING
13
13
13, 14
13 If βxβis any number, then βxβlitres is the same as β1000 Γ x = 1000xβmillilitres because there are β1000 mLβ in β1 Lβ. Write expressions for βx Lβin the following units. a βcmβ3β β b βmβ3β c βkLβ d βMLβ
14 a A rectangular prism has length βl cmβ, width βw cmβand height βh cmβ. Write an expression for the capacity of the container measured in: i βcmβ3β β ii βmLβ iii βLβ iv βkLβ b A rectangular prism has length βl mβ, width βw mβand height βh mβ. Write an expression for the capacity of the container measured in: i βm3β β ii βLβ iii βkLβ iv βMLβ ENRICHMENT: Added depth
β
β
15 A container is β10 cmβlong, β5 cmβwide and β8 cmβ high. a Find the depth of water when the following amounts of water are poured in. (Remember: β1 mL = 1 βcmβ3β ββ.) i β400 mLβ ii β200 mLβ
iii β160 mLβ
15
8 cm
10 cm
5 cm
b After adding β200 mLβ, a further β30 mLβis added. What is the increase in depth? c A β1β-litre container of milk has a base area of β8 cmβby β7 cmβ. After β250 mLβof milk is poured out, what is the depth of the milk remaining in the container? Give your answer to the nearest βmmβ.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10
Measurement
10L Mass and temperature CONSOLIDATING LEARNING INTENTIONS β’ To know that common metric units for mass are milligrams, grams, kilograms and tonnes β’ To know that temperatures are commonly measured in degrees Celsius, where 0Β°C is the freezing point of water and 100Β°C is the boiling point of water β’ To be able to convert between common units of mass
U N SA C O M R PL R E EC PA T E G D ES
728
The scales for both mass and temperature are based on the properties of water. In France in 1795, the gram was defined
as being the weight of 1 cm3 of water at 0Β°C. Later it was redefined to be the weight at 4Β°C, as this is considered to be the temperature at which water is the most dense. So, 1 litre of water is very close to 1 kilogram, which is the basic unit for mass. Up until 2018, the kilogram was defined as the weight of a platinum-based ingot stored in Paris but recently the kilogram has been redefined in terms of an electric current. Other units for mass include the tonne, gram and milligram. A small car has a mass of about 1 tonne and a 20-cent coin has a mass of about 11 grams.
A small car has a mass of about 1 tonne. A vehicle ferry operator must keep the ship stable by balancing the mass of vehicles as the ferry is loaded and unloaded.
Temperature tells us how hot or cold something is. Anders Celsius (1701β1744), a Swedish scientist, worked to define a scale for temperature. After his death, temperature was officially defined by: β’ β’
0Β°C (0 degrees Celsius) β the freezing point of water. 100Β°C (100 degrees Celsius) β the boiling point of water (at one standard unit of pressure).
This is still the common understanding of degrees Celsius. As mentioned in Chapter 9, Fahrenheit is another scale used for temperature. This is investigated further in the Enrichment questions.
Lesson starter: Choose a unit of mass
Choosing objects for given masses Name five objects of which their mass would commonly be measured in: β’ β’ β’ β’
tonnes kilograms grams milligrams.
Choosing places for given temperatures Is it possible for the temperature to drop below 0Β°C? How is this measured and can you give examples of places or situations where this might be the case?
Ice melts at 0Β°C.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10L Mass and temperature
729
KEY IDEAS β The basic unit for mass is the kilogram (β kg)ββ. β1βlitre of water has a mass that is very close to
β1β kilogram. β Metric units for mass include:
Γ 1000
Γ 1000
U N SA C O M R PL R E EC PA T E G D ES
Γ 1000
β’ β’ β’
1β gram (g) = 1000 milligrams (mg)β β1 kilogram (kg) = 1000 grams (g)β β1 tonne (t) = 1000 kilograms (kg)β
t
kg
Γ· 1000
g
Γ· 1000
mg
Γ· 1000
β The common unit for temperature is degrees Celsius (β Β° C)ββ.
β’ β’
0β Β° Cβis the freezing point of water. β100Β° Cβis the boiling point of water.
BUILDING UNDERSTANDING
1 Choose the pair of equal mass measurements. a β1 kg, 100 g, 1000 g, 10 tβ
b β1000 mg, 10 kg, 1 g, 1000 tβ
2 From options A to F, choose the mass that best matches the given object. a human hair A β300 gβ b β10β-cent coin B β40 kgβ c bottle C β100 mgβ d large book D β1.5 kgβ e large bag of sand E β13 tβ f truck F β5 gβ
3 From options A to D, choose the temperature that best matches the description. a temperature of coffee A β15Β° Cβ b temperature of tap water B β50Β° Cβ c temperature of oven C ββ 20Β° Cβ d temperature in Antarctica D β250Β° Cβ
Example 22 Converting units of mass Convert to the units shown in brackets. a β2.47 kg (g)β
b β170 000 kg (t)β
SOLUTION
EXPLANATION
a 2.47 kg = 2.47 Γ 1000 g ββ―β― β β β β = 2470 g
1β kg = 1000 gβ Multiply because you are changing to a smaller unit.
Continued on next page
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
730
Chapter 10 Measurement
1β t = 1000 kgβ Divide because you are changing to a larger unit.
b 170 000 kg = 170 000 Γ· 1000 t β―β― β β β β β = 170 t
U N SA C O M R PL R E EC PA T E G D ES
Now you try
Convert to the units shown in brackets. a 3β 2.5 kg (g)β
b β9500 kg (t)β
Exercise 10L FLUENCY
1
Example 22a
Example 22b Example 22
1, 2ββ(β1/2β)β,β 3β7
Convert to the units shown in the brackets. a i β4.29 kg (g)β b i
2β(β β1/2β)β, 3, 4ββ(β1/2β)β, 5, 7, 8
2ββ(β1/2β)β,β 6β9
ii β7500 g (kg)β
β620 000 kg (t)β
ii β5.1 t (kg)β
2 Convert to the units shown in brackets. a 2β t (kg)β c β2.4 g (mg)β e β4620 mg (g)β g β0.47 t (kg)β
b d f h
i
β27 mg (g)β
j
k
_ β1 βkg (g)β
l
8
m 2β 10 000 kg (t)β o 5β 92 000 mg (g)β
7β 0 kg (g)β β2300 mg (g)β β21 600 kg (t)β β312 g (kg)β _ β3 βt (kg)β 4 β10.5 g (kg)β
n 0β .47 t (kg)β p β0.08 kg (g)β
3 What mass is indicated on these scales? a
b
c
26
t
1014 g
20 16
kg 1 2 3 45 6 7
120
40
80
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10L Mass and temperature
4 What temperature is indicated on these scales? a b Β°C Β°C 40
10
38
5
36
Β°C 20
U N SA C O M R PL R E EC PA T E G D ES
15
c
731
15
34
100
150
e
f
1
200
27
250
0
Β°C
9 1
50
2
36
45
8
d
10
Β°C
300
Β°C
5 A small truck delivers β0.06 tβof stones for a garden. Write the mass of stones using these units. a βkgβ b βgβ c βmgβ
6 A box contains β20βblocks of cheese, each weighing β150 gβ. What is the approximate mass of the box in the following units? a βgβ b βkgβ 7 The temperature of water in a cup is initially β95Β° Cβ. After half an hour, the temperature is β62Β° Cβ. What is the drop in temperature? 8 An oven is initially at a room temperature of β25Β° Cβ. The oven dial is turned to β172Β° Cβ. What is the expected increase in temperature? 9 Add all the mass measurements and give the result in kg. a β3 kg, 4000 g, 0.001 tβ b β2.7 kg, 430 g, 930 000 mg, 0.0041 tβ PROBLEM-SOLVING
10,11
11,12
12β14
10 Arrange these mass measurements from smallest to largest. a β2.5 kg, 370 g, 0.1 t, 400 mgβ b β0.00032 t, 0.41 kg, 710 g, 290 000 mgβ 11 The highest and lowest temperatures recorded over a β7β-day period are as follows. Day
β1β
β2β
3β β
β4β
5β β
β6β
β7β
Lowest temperature (β Β° C)β
β8β
β6β
β10β
β9β
7β β
β8β
β10β
Highest temperature (β Β° C)β 2β 4β
β27β
β31β
β32β
β21β
β19β
β29β
a Which day had the largest temperature range? b What is the largest temperature drop from the highest temperature on one day to the lowest temperature on the next day? c What would have been the final temperature on Day 7 if the temperature increased from the minimum by β16Β° Cβ?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10
Measurement
12 A 10 kg bag of ο¬our is used at a rate of 200 g per day. How many days will the bag of ο¬our last? 13 A boat has a weight limit of 3.5 t carrying capacity. Loaded onto the boat are 1500 tins of coffee at 500 g each, 36 bags of grain at 20 kg each, 190 boxes of mangoes at 5.5 kg each and 15 people, averaging 80 kg each. Is the load too much for the weight limit of the boat? 14 A truck tare mass (i.e. mass with no load) is 13.2 t. The truckβs gross mass is 58.5 t. This is the total maximum mass allowed, including the load. a What is the maximum load the truck can carry? b The truck is loaded with 120 timber beams at 400 kg each. Will it exceed its gross weight limit?
U N SA C O M R PL R E EC PA T E G D ES
732
REASONING
15
15,16
16,17
15 Water weighs 1 kg per litre. What is the mass of these volumes of water? a 1 mL b 1 kL c 1 ML
16 The containers shown below are in the shape of rectangular prisms and are filled with water. Calculate the mass of water in each container, in kg. b c a 15 cm 2m 15 m 12 m 40 cm 20 cm 1m 17 The kelvin (K) is a temperature unit used by many scientists, where 273 K is approximately 0Β°C. (The kelvin used to be called the βdegree kelvinβ or Β°K.) An increase in 1 K is the same as an increase in 1Β°C. a Write the following temperatures in Β°C. i 283 K ii 300 K iii 1000 K b Write the following temperatures in kelvins. i 0Β°C ii 40Β°C iii β273Β°C ENRICHMENT: Fahrenheit
β
β
18
18 Daniel Fahrenheit (1686β1736) proposed the Fahrenheit temperature scale in 1724. It was commonly used in Australia up until the mid-twentieth century, and is still used today in the United States. 32Β°F is the freezing point of water. 212Β°F is the boiling point of water. a What is the difference between the temperature, in Fahrenheit, for the boiling point of water and the freezing point of water? Liquid nitrogen freezes at β210Β°C and boils at b An increase of temperature by 1Β°F is the same as an β196Β°C. increase by what amount in Β°C? c An increase of temperature by 1Β°C is the same as an increase by what amount in Β°F? 5 . Convert these Fahrenheit d To convert from Β°F to Β°C, we use the formula C = (F β 32) Γ _ 9 temperatures to Β°C. i 32Β°F ii 68Β°F iii 140Β°F iv 221Β°F e Find the rule to convert from Β°C to Β°F. Test your rule to see if it works and write it down.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Modelling
733
Estimating park lake area
U N SA C O M R PL R E EC PA T E G D ES
Modelling
A local shire wants to estimate the volume of water in its local lake. The birdβs-eye view of the lake is shown here and each grid square represents 10 metres by 10 metres. The average depth of water in the lake is 3 metres. Present a report for the following tasks and ensure that you show clear mathematical workings and explanations where appropriate.
Preliminary task
a Estimate the area of the shaded region in the following grids. Each grid square represents an area of one square metre. i ii
b A body of water has a surface area of 2000 m2. The volume of water can be found using the formula: Volume = Surface Area Γ Depth. If the average depth of the water is 3 metres, find: i the volume of water in cubic metres ii the volume of water in litres. Use the fact that 1 m3 = 1000 L.
Modelling task
Formulate
Solve
Evaluate and verify
Communicate
a The problem is to estimate the volume of water in the shire lake. Write down all the relevant information that will help solve the problem. b Outline your method for estimating the surface area of the lake shown above. c Estimate the area of the lake in square metres. Explain your method, showing any calculations.
d e f g h
Estimate the volume of water in the lake. Give your answer in both m3 and in litres. Compare your results with others in your class and state the range of answers provided. Review your method for estimating the surface area of the lake and refine your calculation. Explain how you improved your method to find the surface area of the lake. Summarise your results and describe any key findings. You should also describe any methods you could use to obtain a better estimate of the volume.
Extension questions
a Use the internet to find a map or satellite view of a lake in your local area, then estimate its area. b Compare how your estimates and methods would change if the lakeβs depth is no longer assumed to be a constant 3 metres.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
734
Chapter 10
Measurement
(Key technology: Dynamic geometry) In many industries like architecture, design and construction, it is common to calculate areas of shapes. To do this we use formulas including two or more variables. An understanding of how these formulas work helps to make calculations and solve more complex problems involving composite shapes and three-dimensional solids.
U N SA C O M R PL R E EC PA T E G D ES
Technology and computational thinking
Dynamic area formulas
1 Getting started
1 bh. Consider the formula for the area of a triangle, A = _ 2 a Use the formula to find the area of these triangles. i
ii
3m
6 cm
4m
10 cm
iii
6 mm
7 mm
b By redrawing the above triangles and adding other line segments, show that it is possible to view all of the above three triangles as half of a parallelogram. c Explain how the area formula for a triangle is connected to the area of a parallelogram.
2 Applying an algorithm
a Open a dynamic geometry program or website and follow these steps to construct a triangle with fixed height. D β’ Construct a line AB. β’ Construct a line parallel to AB and passing through D. 2.54 β’ Place a point C on the parallel line. β’ Construct a line perpendicular to 2.4 A B AB passing through C and construct point E. β’ Construct the triangle ABC by constructing AB, BC and AC. β’ Construct the segment CE. β’ Construct the triangle ABC using the polygon tool. β’ Measure the lengths of the segments AB and CE and the area of triangle ABC.
C
2.12
E
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Technology and computational thinking
U N SA C O M R PL R E EC PA T E G D ES
3 Using technology
Technology and computational thinking
b Drag the point C on the constructed parallel line. What do you notice about the three measurements? Do your observations remain the same for an acute, a right and an obtuse triangle? c Adjust one of the points A or B to change the base length, then repeat part b above. What do you notice?
735
a Explore the formula for the area of a parallelogram using a dynamic D geometry, by following these steps. C β’ Construct a line AB and segment AB. 1.5 3.28 1.32 β’ Construct a circle with centre A and F radius AE with E on line AB. E B 2.48 A β’ Construct point D on the circle. β’ Construct a line parallel to AB passing through D. β’ Construct the line segment AD. β’ Construct a line parallel to AD passing through B. β’ Construct point C. β’ Construct a line perpendicular to AB passing through C. β’ Construct the point F and line segment CF. β’ Construct the quadrilateral ABCD using the polygon tool. β’ Measure the lengths AB, CF and AD and the area of ABCD. b By dragging point D on the circle, the length AD does not change. Drag point D to change the area of the parallelogram. What do you notice? c As you drag point D, what do you notice about the length CF? When the length CF is close to zero, what is the area of the parallelogram? d Drag the point E or B to change the size of the parallelogram. Then repeat parts b and c above. e What do your observations tell you about the formula for the area of a parallelogram?
4 Extension
a Use dynamic geometry to construct a circle. Calculate the circumference and diameter. Show that regardless of the size of the circle, the circumference divided by the diameter is equal to Ο. b Construct a quadrilateral of your choice using dynamic geometry and calculate its area. Explore how the area depends on the lengths corresponding to the variables in the area formula.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
736
Chapter 10 Measurement
Greg, Sally and Alston apply for a mining licence to look for opals at Coober Pedy in South Australia. They are required to choose an area and mark it out with special orange tape so that others will know which areas are already taken. Their length of tape is β200 mβ.
U N SA C O M R PL R E EC PA T E G D ES
Investigation
Opal mining
Square mining areas
They first decide to mark out an area as a square. a Make a drawing of their square area. b Calculate the side length and area and show this on your diagram. Also show any working.
Rectangular mining areas
They then change the mining area and experiment with different side lengths. a Show three possible lengths and areas for rectangular mining sites. b Complete a table similar to the one below. Fill in the missing numbers for the side lengths given, then add your own rectangle measurements from part a above. Length
Width
Perimeter
β10β
β200β
2β 0β
β200β
3β 5β
β200β
Area
β200β β200β β200β
c Are there any rectangles that give a larger area than the square mining area from above?
Circular mining areas
They now decide to try to arrange the tape to form a circle. For this section, you will need the rule to calculate the distance around a circle (circumference). The circumference βCβis given by βC = 2 Γ Ο Γ rβwhere βrβis the length of the radius and β r C Ο β 3.14β. a Calculate the radius of the circle correct to one decimal place. Use a trial and error (guess and check) technique and remember that the circumference will be β 200 mβ. Explain and show your method using a table of values. b Calculate the area of the circular mining area correct to the nearest square metre. Use the special rule for the area of a circle βAβwhich is given by βA = Ο Γ βrβ2ββ.
The largest area
Compare the areas marked out with the β200 mβtape by Greg, Sally and Alston. Comment on any differences. Which shape gives the largest area for the given perimeter? Would your answer be the same if any shape were allowed to be used? Explain.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Problems and challenges
1
B
OR
U N SA C O M R PL R E EC PA T E G D ES
A
Up for a challenge? If you get stuck on a question, check out the βWorking with unfamiliar problemsβ poster at the end of the book to help you.
Problems and challenges
Without measuring, state which line looks longer: βAβor βBβ? Then measure to check your answer.
737
2 You have two sticks of length β3 mβand β5 mβ, both with no scales. How might you mark a length of β1 mβ? 3 Count squares to estimate the area of these circles if each grid square is 1 cm across. a b
4 A house roof has β500 βm2β βof area. If there is β1 mmβof rainfall, how much water, in litres, can be collected from the roof? 5 Work out the volume of this rectangular prism with the given face areas.
12 m2
8 m2
6 m2
6 Find the area of the shaded region in the diagram shown at right.
7 cm
3 cm
7 A cube of side length β20 cmβcontains β5 Lβof water. a What is the depth of water in the cube? b What is the increase in depth if β1.5 Lβis added to the cube of water? 8 These two rectangles overlap, as shown. Find the total area of the shaded region.
8 cm
5 cm
3 cm 2 cm
4 cm 6 cm Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10
Measurement
mm2
cm2 m2 ha km2 1 ha = 10 000 m2
A = bh =5Γ2 = 10 cm2
2 cm 5 cm
Square
Units of area
Parallelogram
A = s2 = 92 = 81 m2
9m
U N SA C O M R PL R E EC PA T E G D ES
Chapter summary
738
Triangle
2 cm
3m A = 12 bh
4 cm
A = lw lb =4Γ2 = 8 cm2
= 12 Γ 3 Γ 2 = 3 m2
Volume
mfe r e n c
e
ci
rcu
= 18 Γ 10 + 12 Γ 10 Γ 9
Area
c d = 2r p = β, d
eter
A = l Γ b + 12 bh = 225 m2
Circles radius
diam
Composite shapes (Ext) 18 m 10 m 19 m 8m
Rectangle
2m
Measurement
Units of length
10 mm = 1 cm 100 cm = 1 m 1000 m = 1 km
Rectangular prism
Length
Temperature
Perimeter
Β°C (Celsius)
2.1 cm
Mass
1 g = 1000 mg 1 kg = 1000 g 1 t = 1000 kg
3 cm
1.5 cm
Units of capacity
1 L = 1000 mL 1 kL = 1000 L 1 ML = 1000 kL and 1 mL = 1 cm3 1 m3 = 1000 L
2 cm 4 cm
5 cm V = lbh = 5 Γ 4 Γ 2 = 40 cm3 Units of volume mm3 cm3 m3 km3
P = 2 Γ 2.1 + 1.5 + 3 = 8.7 cm
Circumference 8 cm
C=pΓd =pΓ8 = 25.13 cm
Sectors (Ext)
40 Γ2ΓpΓ3+2Γ3 P = 360 = 8.1 cm 40Β° 3 cm
Triangular prism
4 cm
8 cm
10 cm
22 1 bh= =12 12Γ Γ8 Γ8 Γ4 4= =1616cmcm AA== bh 2
VV==Ah Ah==16 16ΓΓ10 10==160 160cm cm3 3
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter checklist
739
Chapter checklist with success criteria 1. I can choose a suitable metric unit for measuring a length. e.g. Which metric unit would be most appropriate for measuring the width of a large room?
U N SA C O M R PL R E EC PA T E G D ES
10A
β
Chapter checklist
A printable version of this checklist is available in the Interactive Textbook
10A
2. I can convert metric units of length. e.g. Convert β25 600 cmβto βkmβ.
10A
3. I can read a length scale. e.g. Measure the marked length on the scale.
4m
10B
4. I can find the perimeter of a shape. e.g. Find the perimeter of this shape.
5m
6m
7m
3m
6m
5m
2m
10C
5. I can approximate Ο using a fraction. e.g. Round _ β25 βusing three decimal places and decide if _ β25 βis a better approximation to βΟβ 8 8 than the number β3.14β.
10C
6. I can calculate the circumference of a circle. e.g. Calculate the circumference of this circle, rounding your answer correct to two decimal places.
10D
7. I can find the perimeter of a sector. e.g. Find the perimeter of this sector, correct to two decimal places.
8 cm
Ext
100Β°
4m
10D
8. I can find the perimeter of a composite shape. e.g. Find the perimeter of this composite shape, correct to two decimal places.
Ext
5m
8m
10E
9. I can find the area of a shape by considering squares on a grid. e.g. Count the number of squares to find the area of the shape drawn on this centimetre grid.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter 10 Measurement
β 10E
10. I can find the area of squares and other rectangles. e.g. Find the area of this rectangle.
U N SA C O M R PL R E EC PA T E G D ES
Chapter checklist
740
4 mm
10 mm
10F
11. I can find the area of parallelograms. e.g. Find the area of this parallelogram.
5m
12 m
10G
12. I can find the area of triangles. e.g. Find the area of this triangle.
9 cm
10 cm
10H
13. I can find the area of composite shapes. e.g. Find the area of this composite shape.
Ext
4 cm
6 cm
10I
14. I can find the volume of rectangular prisms (cuboids). e.g. Find the volume of this cuboid.
4 cm
8 cm
10J
3 cm
15. I can find the volume of triangular prisms. e.g. Find the volume of this triangular prism.
2 cm
5 cm
4 cm
10K
16. I can convert between units of capacity. e.g. Convert β500 mLβto litres.
10K
17. I can convert between volumes and capacities. e.g. A container has a volume of β2000βcubic centimetres. Find its capacity in litres.
10L
18. I can convert between units of mass. e.g. Convert β2.47 kgβto grams.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
741
Short-answer questions Using the metric system, state how many: a millimetres in β1 cmβ b metres in β1 kmβ
2 Convert to the units shown in brackets. a β5 cm (mm)β b β200 cm (m)β e 7β .1 kg (g)β f β24 900 mg (g)β i β4000 mL (L)β j β29 903 L (kL)β m 1β day (min) β n β3600 s (β min)β
c g k o
3 Read these scales. a
b
c centimetres in β1 km.β β3.7 km (m)β 2β 8 490 kg (t)β β0.4 ML (kL)β β84 h (β days)β
d h l p
4β 21 000 cm (km)β 0β .009 t (g)β β0.001 kL (mL)β β2.5 h (β s)β
U N SA C O M R PL R E EC PA T E G D ES
10A
1
Chapter review
10A
10A
2
1
cm
3
cm
c
d
3
2
1
kg
10B
1
2
3
10 8 6 4 2 L
4
5
4 Find the perimeter of these shapes. Assume that all angles that look like right angles are right angles. a b 4m
7.1 cm
3.2 cm
c
5m
d
8 km
4 km
9m
e
7 km
9 km
f
6m
0.4 mm
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
742
Chapter 10 Measurement
5 A common approximation to βΟβis _ β22 ββ. 7 22 βrounded to three decimal places. a Write β_ 7 b Decide if _ β22 βis a better approximation to βΟβthan the number β3.14β. 7 6 Calculate the circumference of these circles using a calculator and rounding your answer correct to two decimal places. a b 5m 11 cm
U N SA C O M R PL R E EC PA T E G D ES
Chapter review
10C
10C
10D
7 Find the perimeter of these shapes, correct to one decimal place. a b
Ext
6 cm
105Β°
4 cm
6 cm
10F/G
8
Find the area of each of the following shapes. Assume that all angles that look like right angles are right angles. a b 2 km
7 km
4.9 cm
c
9m
d
6 cm
4 cm
15 m
e
8m
f
1 cm
1 cm
3.5 m
1 cm
3 cm
g
h
2m
1.5 km
7m
0.6 km
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
10H
a
b
7m
U N SA C O M R PL R E EC PA T E G D ES
4m
Chapter review
Ext
9 Find the area of these composite shapes. Assume that all angles that look like right angles are right angles.
743
4m
2 mm 5 mm
c
d
9 cm
6m
15 cm
9 cm
2m
4m
22 cm
10I
12 m
10 Find the volume contained in each of these rectangular prisms. a b
c
2.5 cm
1 cm cubes
3 cm
1 cm
10J
11 Find the volume of these triangular prisms. a
4 mm
b
4m
3 cm
6 cm
5 cm
8m
7m
c
6 mm
10 mm
7 mm
10K
10L
12 A rectangular fish tank is of length β60 cmβ, width β40 cmβand height β30 cmβ. Give the tankβs capacity in: a βcmβ3β β b βmLβ c βL.β
13 Arrange these measurements from smallest to largest. a β3 t, 4700 kg, 290 000 g, 45 mgβ b β50 000 mL, 1 ML, 51 L, 0.5 kLβ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
744
Chapter 10 Measurement
Chapter review
Multiple-choice questions 10A
How many millimetres are there in β1 mβ? A β1β B 1β 0β C β100β
D β1000β
E β10 000β
2 Shonali buys β300 cmβof wire that costs β$2βper metre. How much does she pay for the wire? A β$150β B $β 600β C β$1.50β D β$3β E β$6β
U N SA C O M R PL R E EC PA T E G D ES
10A
1
10B
10C
3 The triangle given has a perimeter of β20 cmβ. What is the missing base length? A 6β cmβ B 8β cmβ C β4 cmβ D β16 cmβ E β12 cmβ
4 Which of the following is the best approximation of βΟβ? B _ β41 β C _ β20 β A _ β61 β 7 12 20
8 cm
?
D
_ β10 β
E
3
_ β16 β
5
10D
5 The length of the arc only on a sector with a centre angle of 160Β° and radius β7 cmβis closest to: A β19.0 cmβ B β20.1 cmβ C β19.5 cmβ D β9.8 cmβ E β33.5 cmβ
10E
6 The area of a rectangle with length β2 mβand width β5 mβ is: B β5 βm2β β C β5 mβ A β10 βm2β β
10G
7 A triangle has base length β3.2 cmβand height β4 cmβ. What is its area? A β25.6 βcmβ2β β B β12.8 cmβ C β12.8 βcmβ2β β D β6 cmβ
Ext
10H
Ext
10I
10J
10K
10L
10G
D β5 βm3β β
B 4β 5.5 βkmβ2β β E β24.5 βkmβ2β β
E β6.4 βcmβ2β β
3 km
8 The total area of this composite shape is: A 5β 6 βkmβ2β β D β10.5 βkmβ2β β
E β10 mβ
C β35 βkmβ2β β
5 km
9 A cube has a side length of β3 cmβ. Its volume is: A β27 βcmβ3β β B β9 βcmβ2β β C β3 cmβ
7 km
D β9 βcmβ3β β
E β36 βcmβ3β β
10 A triangular prism has cross-sectional area βA βcmβ2β β, height β4 cmβand volume β 36 cβ mβ3β β. The value of βAβ is: A 3β β D β4.5β
11 2β 000 βcmβ3β βis the same as: A 2β βm3β β B β2 Lβ
B 1β 8β E β27β
C β9β
C β2 kLβ
D β2 mLβ
4 cm
E β2 tβ
12 β9βtonnes of iron ore is being loaded onto a ship at a rate of β20 kgβper second. How many minutes will it take to load all of the β9βtonnes of ore? A β0.75β min B β45β min C β7.3β min D β450β min E β7.5β min 13 The base length of a parallelogram is β10 cmβand its area is β30 βcmβ2β ββ. The parallelogramβs height is: A β10 cmβ B 3β cmβ C β30 cmβ D β3 βcmβ2β β E β10 βm2β β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Chapter review
745
Extended-response questions
U N SA C O M R PL R E EC PA T E G D ES
A truck carries a large rectangular container of dimensions β6 mβlong, β3 mβwide and β2 mβ high. The truck weighs β15.4βtonnes without its load. a Find the area of the base of the container in: i m β 2β β ii βcmβ2β ββ. b Find the volume of the container in βm3β ββ. c How many litres of water could the truck hold in the container? d Since β1 Lβof water weighs β1 kgβ, give the weight of the truck if the container was filled with water. Give your answer in tonnes. e The truck completes three journeys, which take, on average, β1βhour, β17βminutes and β38βseconds per trip. What is the total time for the three journeys?
Chapter review
1
Ext
2 Lachlan builds a race track around the 36.3 m outside of his family house block. The block combines rectangular and 15 m house triangular areas, as shown in the 10 m diagram. a How far is one complete circuit of the track? 18 m b Lachlan can jog β10βlaps at about β33βseconds each. What is the total time, in minutes and seconds, that it takes him to complete the β10β laps? c What is the total area of the block? d The house occupies β100 βm2β βand the rest of the block is to have instant turf, costing β$12β per square metre. What will be the cost for the instant turf?
e The house sits on a concrete slab that is β50 cmβdeep. What is the volume of concrete, in βm3β ββ?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
746
Semester review 2
Short-answer questions 1
For each of the following, insert β=, >βor β<β. a ββ 3__3β b ββ 10 Γ· 2__5β
β 40 β c ββ 20 Γ (β β 1)β__β_ β2
U N SA C O M R PL R E EC PA T E G D ES
Semester review 2
Negative numbers
2 Calculate: a ββ 5 + (β β 8)β d ββ 4 β 8 β 9β
Ext
Ext
Ext
Ext
b 1 β 2 β 96β e β β 12 + 96β
c β β 12 β 96β f ββ 7 β (β β 7)β
3 Find: a ββ 6 Γ 4β b ββ 9 Γ 8 Γ (β β 1)β c (β β 12)β2β ( ) β 7 β β 9 Γ β f ββ 10 + 7 Γ (β β 3)β e ββ 150 Γ· (β β 2 β 3)β d _ β β 3 4 State whether the answer to each of the following is positive or negative. a ββ 3 Γ (β β 3)βΓ (β β 3)β b ββ 109 Γ 142 Γ (β β 83)β c ββ 2 Γ (β β 1 β (β β 3)β)β
5 Copy and complete. a β__ + 9 = β 6β
b β__ Γ β(β 3)β= β 6 Γ (β β 4)β
c β16 Γ __ = β 64β
6 If βa = 6βand βb = β 4β, find the value of: a β β a + bβ b βa β bβ d ββ abββ2β e βaβ2β+ βbβ2β
c 2β β(b β a)β f β24 Γ· β(ab)β
Multiple-choice questions 1
Ext
Ext
Which of the following statements is incorrect? A ββ 2 > β 4β B 0β < 5β C β0 < β 10β
D ββ 9 < β 8β
E ββ 5 < 3β
2 1β 2 + (β β 9)ββ (β β 3)βis the same as: A β12 + 9 + 3β B β12 β 9 + 3β
C β12 β 9 β 3β
D β12 β 12β
E β12β
3 If βa = β 3β, the value of βaβ2βΓ β 2β is: A β36β B ββ 36β
C β18β
D ββ 18β
E β12β
4 The point that is β3βunits below (β 3, 1)βhas coordinates: A β(0, 1)β B β(0, β 2)β C β(0, β 1)β
D β(3, 4)β
E β(3, β 2)β
5 1β 2 Γ β(β 4 + (β β 8)βΓ· 2)ββequals: A ββ 96β B β72β
D β60β
E β96β
C ββ 72β
Extended-response question Refer to the given Cartesian plane when answering these questions.
a Name any point that lies in the first quadrant. b Name any point(s) with a βyβ-value of zero. Where does each point lie? c Which point has coordinates (β β 1, β 2)ββ? d Find the distance between points: i βAβand βBβ ii βDβand βE.β e What shape is formed by joining the points βIDAGβ?
y
G
3 I 2 C 1
β3 β2 β1β1O Fβ2 H β3
D
A
B
1 2 3 4 5
x
E
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Semester review 2
U N SA C O M R PL R E EC PA T E G D ES
Geometry
Semester review 2
f What is the area of βIDAGβ? g βABXDβare the vertices of a square. What are the coordinates of βXβ? h Decode: (β 2, 2)β, β(2, β 3)β, β(0, 2)β, β(β 1, 2)β, β(2, 2)β, β(2, β 3)β
747
Short-answer questions 1
a Name two pairs of parallel lines. b Name a pair of perpendicular lines. c List any three lines that are concurrent. At what point do they cross? d Name two points that are collinear with point βCβ. e Name the point at which line βBEβand line βFDβ intersect.
2 Measure these angles. a
A
B
F
C
E
D
b
c
3 What angle is complementary to β65Β°β?
4 What angle is supplementary to β102Β°β?
5 Find the value of βaβin each of the following angles. a b 40Β°
d
e aΒ°
f aΒ°
25Β°
aΒ° 120Β° 100Β°
aΒ°
40Β°
aΒ°
c
62Β°
aΒ° 56Β°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Semester review 2
6 Find the value of each angle formed when these two parallel lines are crossed by the transversal, as shown.
aΒ° 80Β° bΒ° cΒ° gΒ° dΒ° f Β° eΒ°
U N SA C O M R PL R E EC PA T E G D ES
Semester review 2
748
7 Name each of the following shapes. a b
d
c
e
8 Find the value of βaβin these diagrams. a b aΒ°
40Β°
f
c
50Β°
aΒ°
aΒ°
85Β°
80Β°
20Β°
9 For each of the following, find the value of every pronumeral. a b
c
aΒ°
80Β°
65Β°
d
f
60Β°
bΒ°
aΒ°
65Β°
aΒ°
e
110Β°
70Β°
aΒ° 100Β°
aΒ° 55Β°
10 State the coordinates of the vertices of the image triangle β (βAβ², Bβ² and Cβ²β)βif triangle βABCβis transformed in the following ways. a reflected in the βxβ-axis b reflected in the βyβ-axis c rotated about βOβ(0, 0)βby β90Β°β clockwise d rotated about βOβ(0, 0)βby β90Β°β anticlockwise e rotated about βOβ(0, 0)βby β180Β°β f translated β4βunits to the left and β1βunit up g translated β3βunits to the left and β2βunits down
67Β°
dΒ°
cΒ°
aΒ°
bΒ°
48Β°
y
3 2 1
β3 β2 β1β1O β2 β3
B
A
C
1 2 3 4
x
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Semester review 2
749
Multiple-choice questions Which statement is correct? A Line βmβis perpendicular to line βlβ. B Line βmβbisects line βlβ. C Line βmβis parallel to line βlβ. D Line βmβis shorter than line βlβ. E Line βmβis longer than line βlβ.
m
U N SA C O M R PL R E EC PA T E G D ES
l
Semester review 2
1
2 An angle of β181Β°βis classified as: A acute B reflex
C straight
D obtuse
3 Which two angles represent alternate angles? A βaΒ°βand βeΒ°β B βdΒ°βand βf Β°β C βaΒ°βand βf Β°β D βgΒ°βand βbΒ°β E βcΒ°βand βf Β°β
E sharp
aΒ° bΒ° dΒ° cΒ°
eΒ° f Β°
hΒ° gΒ°
4 Which of the following shows a pair of supplementary angles? Assume that lines which look parallel are parallel. A B C Γ Γ Β° Β° Β° Γ D
E
Β°
Γ
Β°
Γ
5 The order of rotation of a parallelogram is: A β2β B β4β C β3β
D 0β β
E β1β
Extended-response question a b c d e
Draw any triangle in your workbook and extend each side, as shown below. Measure the angles βa, bβand βcβ. What should they add to? Find the values of βx, yβand βzβ(i.e. the exterior angles of the triangle). What is the total of angles βx + y + zβ? Repeat for any quadrilateral, as shown below. What is the value of βw + x + y + zβ? yΒ°
yΒ°
cΒ°
aΒ° zΒ°
dΒ°
zΒ°
bΒ°
xΒ°
aΒ°
cΒ°
bΒ° xΒ°
wΒ°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
750
Semester review 2
Short-answer questions 1
Consider the set of numbers β1, 2, 5, 5, 8, 9, 10, 5, 3, 8β. a List them in ascending order. b How many numbers are in the set? c Calculate the: i mean ii mode iii median iv range. d If each number in the set is doubled, what is the new mean?
U N SA C O M R PL R E EC PA T E G D ES
Semester review 2
Statistics and probability
2 A spinner is designed with different numbers in each sector. Consider spinners A to D shown below. A B 1 2 1 2 5 4
3
C
3
1
2
1
D
1
2
6
2
5
3
1
4
3
a Which spinner has the lowest probability of landing on the number β1βin a single spin? b Which spinner has a β50%βprobability of landing on the number β1βin a single spin? c List the spinners in order, from the most likely to land on the number β1βto the least likely.
3 One card is randomly selected from a standard deck of β52βplaying cards. Find the probability that the selected card is: a red b black c a heart d an ace e a king f a red β7.β 4 This stem-and-leaf plot shows the ages of a group of people in a room. a How many people are in the room? b What are the ages of the youngest and oldest people? c For the data presented in this plot, find the: i range ii median iii mode. 5 A set of six scores is given as β17, 24, 19, 36, 22β and
.
a Find the value of
if the mean of the scores is β23β.
b Find the value of
if the median of the scores is β22β.
Stem
Leaf
β0β β3β
5β β
β1β β1β
7β β
β9β
β2β β0β
2β β
β2β
β3β β6β
β9β
β4β β3β
β7β
β3β
β2β|β3βmeans β23βyears old
c If the value of is β14β, and the score of β17βis removed from the set, calculate the new mean and median of the group.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Semester review 2
751
Multiple-choice questions
Pink
Green
Purple
Black
β12β
β20β
β6β
β12β
β10β
U N SA C O M R PL R E EC PA T E G D ES
Blue
Semester review 2
The following information is relevant for Questions 1 and 2. A survey asked β60βparticipants their favourite colour. The results are shown below.
Ext
1
Ext
2 If the results are shown as a sector graph, the size of the smallest sector would be: A 3β 6Β°β B 6β Β°β C β10Β°β D β60Β°β E β30Β°β
If a divided bar graph β10 cmβlong is constructed to display the data, the length required for purple would be: A β12 cmβ B 2β cmβ C β1 cmβ D β5 cmβ E β3 cmβ
3 For the set of numbers β1, 5, 2, 10, 1, 6β, the difference between the mode and median is: A 5β .5β B 2β .5β C β3.5β D β3.2β E β2β 4 For the set of numbers β3, 2, 1, 5, 7, 3, 1, 3β, the mean is: A 3β β B 5β β C β6β
E β3.125β
D β22.375β
5 Which of the following sets of numbers has a mode of β3βand a mean of β2β? A 1β , 1, 2, 2, 3β B 2β , 2, 2, 4, 5β C β1, 2, 3, 4, 5β D β1, 3, 3, 3, 0β
E β1, 2, 3, 3, 3β
Extended-response question
A pack of playing cards includes β13βcards for each suit: hearts, diamonds, clubs and spades. Each suit has an ace, king, queen, jack, β2, 3, 4, 5, 6, 7, 8, 9βand β10β. One card is drawn at random from the pack. Find the following probabilities. a Pr(heart) b Pr(club) c Pr(diamond or spade) d Pr(ace of hearts) e Pr(number less than β4βand not an ace) f Pr(king) g Pr(ace or heart) h Pr(queen or club)
Equations
Short-answer questions 1
Solve: a βx + 9 = 12β
2 Solve: a β3x + 3 = 9β
b _ βx β= 12β 9 Ext
b β_x β+ 6 = 12β 2
c βx β 9 = 12β Ext
d β9x = 12β
c β3β(m β 1)β= 18β
3 If βy = mx + bβ, find: a βyβwhen βm = 3, x = 4βand βb = 8β c m β βwhen βy = 36, x = 3βand βb = 12β
b βbβwhen βy = 20, m = 4βand βx = 4β
4 If βP = S β Cβ, find: a βPβwhen βS = 190βand βC = 87β c C β βwhen βP = 384βand βS = 709.β
b βSβwhen βP = 47.9βand βC = 13.1β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Semester review 2
5 Use your knowledge of geometry and shapes to find the value of βxβin each of the following. a b c (2x + 5) cm 36Β° 2xΒ° (x + 2) cm
Semester review 2
752
17 cm
U N SA C O M R PL R E EC PA T E G D ES
P = 28 cm
6 The perimeter of this triangle is β85 cmβ. Write an equation and then solve it to find the value of βxβ.
2x cm
25 cm
Multiple-choice questions
The solution to the equation βx β 3 = 7β is: A β4β B 1β 0β C β9β
D β11β
E β3β
2 The solution to the equation β2(β x + 3)β= 12β is: A 4β .5β B β2β C β7β
D β6β
E β3β
1
Ext
3 m β = 4βis a solution to: A 3β m + 12 = 0β D βm + 4 = 0β
4 The solution to β2p β 3 = 7β is: A pβ = 4β B βp = 5β
m β= 16β B β_ 4 E β3m β 6 = 2β
C βp = 2β
C β10 β 2m = 2β
D βp = 10β
E βp = 3β
5 Ying thinks of a number. If he adds β4βto his number and then multiplies the sum by β5β, the result is β35β. What equation represents this information? A yβ + 9 = 35β B β5y β 4 = 35β C β5y + 4 = 35β D β5(β y + 4)β= 35β E βy + 20 = 35β
Extended-response question
The cost of hiring a hall for an event is β$200βplus β$40βper hour. a What is the cost of hiring the hall for β3β hours? b What is the cost of hiring the hall for β5β hours? c What is the cost of hiring the hall for βnβ hours? d If the cost of hiring the hall totals β$460β, for how many hours was it hired?
Measurement
Short-answer questions 1
Complete these conversions. a 5β m = __ cmβ d β1.7 cm = __ mβ
b 6 β km = __ mβ e 1β 80 cm = __ mβ
c β1800 mm = __ mβ f β5_ β1 βL = __ mLβ 2
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Semester review 2
c
85 cm
1.3 m
68 cm
1.5 m
4.2 m e
60 c
m
40
f
cm
7m
U N SA C O M R PL R E EC PA T E G D ES
d
Semester review 2
2 Find the perimeter of each of the following. a b
753
1.2 m
12 m
55 cm
20 m
3 Find the area of each of the following. a b 1.3 m
c
8m
3m
8m
15 m
d
e
4.5 m
7m
12 m
c
7m
12 m
7m
8m
12 m
4 Calculate the volume of these solids. a 9 cm 9 cm
f
b
3m
5m
8m
9 cm
10 m
12 m
Ext
d
4m
4m
4 cm
8m
9 cm
10 cm
6m
6m
5 Convert: a β5 L = __ mLβ c _ β1 βL = __ mLβ 4 e 8β βm3β β= __ kLβ
b β7 βcmβ3β β= __ mLβ d β3 kL = __ Lβ
f
β25 000 mL = __ Lβ
6 Find the capacity of a fish tank in litres if it is a rectangular prism β50 cmβlong, β40 cmβwide and β 30 cmβ high.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
754
Semester review 2
8 A collection of objects have the following individual masses: β220 kg, 0.05 tβand β67000 gβ. What is their total combined mass in βkgβ? 9 A factory freezer is installed in a warehouse and is at an initial temperature of β28Β° Cβ. After it is switched on, the temperature drops by β31Β° Cβ. What is the new temperature?
U N SA C O M R PL R E EC PA T E G D ES
Semester review 2
7 Find the circumference of a circle with radius β3 cmβ. Round to two decimal places.
Multiple-choice questions 1
1β 7 mmβis the same as: A 0β .17 mβ B β0.17 cmβ
C β0.017 mβ
D β170 cmβ
E β1.7 mβ
2 0β .006 Lβis the same as: A 6β mLβ B 3β 6 mLβ
C β10 mLβ
D β60 mLβ
E β600 mLβ
D
E
3 Which of the following shapes has the largest perimeter? A B C
4 The perimeters of the two shapes shown here are equal. The area of the square is: B β7.5 βm2β β C β56.25 βm2β β A 3β 0 βm2β β 13 m 5m 2 2 D β120 βmβ β E β60 βmβ β 12 m
5 Which of the following rectangular prisms holds exactly β1β litre? A B 5 cm 20 cm 2 cm 20 cm 5 cm
10 cm
10 cm
5 cm
10 cm
D
C
E
10 cm
20 cm
2 cm
10 cm
20 cm
10 cm
Extended-response question
Robert has been given β36 mβof fencing with which to build the largest rectangular enclosure that he can, using whole number side lengths. a Draw three possible enclosures and calculate the area of each one. b What are the dimensions of the rectangle that gives the largest possible area? c If Robert chooses the dimensions in part b and puts a post on each corner, and then posts every metre along the boundary, how many posts will he need? d If it takes β15βminutes to dig each hole for each post, how many hours will Robert spend digging?
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
755
Index common factors 254, 256 common fractions 145 common ratios 100β1 commutative law 15, 25β6, 36, 254 comparison 10, 151, 163β5, 213, 286β7, 329, 332, 358, 514, 564 compasses 439, 466, 665 compensating 16β17 complementary angles 414β16 composite numbers 79β80, 88β9 composite shapes 673, 675, 700β3 concurrent lines 405 cones 482 constant terms 233 constants 665β6 continuous numerical data 514β16, 526, 535β6 continuous quantities 284 conventions 88, 151, 263, 542 conversion 84, 108, 153, 158β60, 165, 170, 172, 176β7, 183β4, 189β90, 194β6, 267, 286, 317β19, 325β6, 329, 331β2, 544, 568β9, 629, 652β4, 724β5, 729 convex polygons 443, 445β6 coordinates 121β2, 386β8, 467β8 corresponding angles 421β2, 424β5, 430β1 critical digits 289β91 cubes/cuboids 482, 487, 708, 710, 713, 715 cubic units 707β8 cuneiform 5 cylinders 482
discrete numerical data 514, 526 distance 322, 536, 652, 666 distributive law 26, 36β7, 261, 264, 622 divided bar graphs 551β2 dividend 35, 74 divisibility tests 74β6 division 35, 47β8, 64, 66, 70, 100β1, 114, 145β6, 151β3, 181, 188β90, 209, 215, 233, 238, 254, 256, 299β302, 311β13, 317, 371β2, 617, 724, 729 division algorithm 36β7, 74 divisors 35β6, 74, 311, 313 dot plots 526β9 dots 93, 318, 537 doubling 16β17, 26
U N SA C O M R PL R E EC PA T E G D ES
acute angles 406, 437 addition 15β17, 20β1, 25, 31, 36, 47β8, 85, 169β72, 238, 244, 294β5, 361β3, 366, 368, 377, 701β2 adjacent angles 414 algebra/algebraic expressions 69, 233, 238, 244β5, 267, 386, 612, 634 algorithm 20β1, 31, 665 alignment (decimals/places) 294β6 alphabetical order 254 alternate angles 421β2 alternate method 264, 331 amicable numbers 68 angle sum 415β17, 422, 425, 453β5 angles 404β9, 417, 421, 424β5, 431, 437β9, 455, 474β5, 672, 687 approximation 42β4, 289, 665β7, 671 arcs 405, 437, 440, 665, 672, 674 area 93, 145, 261, 263, 267, 329, 680β3, 687β8, 693β4, 700β3, 713 arms 405, 414 arrows 422 ascending order 64, 88β9, 163β4, 166, 522, 543 associative law 15, 25β6, 36, 254 averaging 260, 322 axes 121β2, 386, 460, 527, 530 bar graphs 527 base, length, height 687β9, 693β4, 715β18 base numbers/numerals 10, 82β3, 94 brackets 47, 122, 235, 238, 254, 261, 263, 266, 302, 377β8, 383, 622, 626 βbranchesβ (of factor trees) 88β9 cancelling 181β4, 190β1, 195β6, 200, 254, 256 capacity 723β5 Cartesian planes 121β2, 386 categorical data 514β16, 526 Celsius (degree) 385, 629, 728β9 census 515, 564 central tendency, measures of 520 chance 558β9, 566 chord 665 circles 114, 404β5, 665, 668, 672 circumference 665β7, 671, 674 closed plane figures 443 coefficients 233β4, 248 cointerior angles 421β2, 425, 431, 445 collinear points 405 colon (:) 207 colour 527 column graphs 526β7, 529β30 commas 122 common denominators 169, 214 common difference 100β1, 108
equal signs 594β5 equal to 65, 69, 158 equally likely 558 equation sides 604β6 equations 594β7, 617β19, 622, 627, 629, 634β5 equilateral triangles 437, 490 equivalent equations 604β6, 609β11 equivalent expressions 243β5, 247β8, 261, 263 equivalent fractions 150β3, 163β4, 166, 171, 175β7, 194β6 error 294, 383 estimation 42, 44, 305β7, 404, 535, 537, 573, 652 even numbers 36, 75, 79, 100, 559, 568β9 events occurrence 558β9, 566β7 expanded form 10, 12 data collection/analysis 514, 516, 520, expanding brackets 84, 261, 263, 266, 526, 545, 564 622β3, 626, 631 decimal places/points 285, 289β91, 294β6, experimental probability 573β4 299β301, 305β8, 311, 313, 324β6, 568, expressions 84β5, 94, 96, 150β3, 653, 666β7, 674β5 163β4, 166, 171, 175β7, 199, 208, decimal system 4, 9, 652 233β5, 254, 261, 264, 268β9, 330, 594, decimals 284β7, 289, 294β6, 299, 305β7, 628 311β13, 317β19, 325β6, 329, exterior angles 454β5 331, 567 decomposition 88, 454 factor sets 65β6 degrees 404, 406, 414β15, 431, 453, 672 factor trees 88 denominators 69, 144β6, 151β4, 159, factorisation 69 164β6, 169β71, 175β7, 180β4, factors 26, 64β6, 69β71, 74, 79β80, 88, 94, 189β90, 194β6, 214, 254, 317, 319, 207, 254, 256, 323, 404 330, 617β19 Fahrenheit (degrees) 629, 728 descending order 163β4, 194 fair (coin/die) 558β9, 566 description 109, 234, 269, 476, 478, favourable outcomes 567β8 558β9, 596β7, 634 formulas 629β30 diameter 665, 668 fractions 144β5, 147, 150β4, 157β9, digits 9β10, 20, 31, 74, 289β91, 299, 325, 169β72, 175β7, 188, 191, 194β5, 542β3 199β200, 213β14, 254, 317β19, 329, dimensions 121, 323 357, 404, 566, 568, 617β19, 666 direction 367 function machine 115β16
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Index
generalising 245 geometric shapes 107 gram 729 graphs 121β3, 526, 564β5 greater than 65, 69, 158, 163, 357β8 groups/grouping 4β5, 16, 25β7, 35, 176β8, 254
magnitude 357, 372 main cluster (points) 527, 529 marks/markings 437, 440, 444, 659 mass 728β9 mathematical statements 165, 594β5 mean 520β2, 527 median 520β2, 527, 544, 564 halving 16β17, 26, 36β7, 200 metric system 652β4 height 322, 687β9, 693β4, 715β18 metric units 708, 723 hieroglyphics 4 mirror line 465, 467 highest common factor (HCF) 69β71, 151, mixed numerals 157β60, 165β6, 171β2, 195β6, 207, 209, 318 175β7, 181, 183β4, 189, 195β6 horizontal axis 535β6 mode 520β2, 527 βhow manyβ 188 models/modelling 267 multiples 36, 43, 64β6, 69β71, 164, 240, image 465, 468 306β7, 542 improper fractions 145, 157β60, 165β6, multiplication 25, 31β2, 47β8, 70, 85, 172, 176β7, 181, 183β4 100β1, 152, 159, 180, 182β4, 196, increasing order 545 214β15, 232β4, 238, 244, 254, 256, index form/notation 12, 82β4, 93 299β301, 305, 307, 326, 371β2, 377, indices 10, 82β3 617, 654, 715, 724, 729 input 115β16, 122 multiplication algorithms 26β7, 31 inspection 599, 601 multipliers 300 integers 299, 356β8, 371β2, 377, 383β4 interior angles 437, 443, 453 naming 407, 424, 437 interpretation 47, 528β9, 537β8, 544 negative integers 356, 361, 366, 368, 371, intersection points 405, 424, 440 377, 382β3, 386 intervals 404β5 nets 487, 489β90 inversion 189β91 no remainder 37, 74 irrational numbers 666 non-convex polygons 443, 445β6 irregular polygons 446 non-like terms 248 isometric dot paper 483 notation 74, 82, 232, 304, 405 isometric transformations 465β6, 476 number facts 245 isosceles triangles 437, 460, 693 number lines 121, 145β6, 151, 158, 163, 175, 180, 357, 367 keys 545 number pairs 122 kilogram 729 number patterns 64, 99, 107 kites 444 number separation 285 number systems 4β6, 356, 404 labels/labelling 404, 413, 526β7, Number Theory 64 530, 687 numbers 32, 42, 49, 199β200, 233, 254, leading digit approximation 43β4 311, 356, 520 leaf 542β3 numerators 144β6, 151β4, 159, 164β5, left to right (working) convention 47β8, 171, 180β2, 184, 189β90, 254, 238, 314, 373, 377 317, 330 left-hand sides (LHS) 594, 596, 601, 605, numerical (quantitative) data 514, 610β12, 619, 630β1, 635 542β3 length 437, 652, 654β5, 659β60, 672, 681, 683, 687β9, 693β4, 715β18 observation 515 less than 65, 69, 358 obtuse angles 406, 408, 437 letters 5, 404 odd numbers 100, 559, 568β9 βlikeβ fractions 170β1, 175β7 βofβ 182β3, 199 like terms 248β50, 622β4 one-step equations 611 line graphs 535β7 operations 35, 44, 47β9, 83, 85, 94, 116, line of symmetry 460β2 163β4, 166, 233, 235, 238, 314, lines 404, 413, 665 367, 373, 377β8, 605β6, 609β10, lower case letters 405 612, 624 lowest common denominator (LCD) opposite angles 415, 424 164β6, 170β2, 175β7 opposite operations 94, 366β7, lowest common multiple (LCM) 612 69β71 opposite sign 371β2
order/ordering 15, 25β6, 47β9, 64, 83, 85, 163β4, 166, 194, 206β8, 238, 240, 249, 254, 302, 314, 377β8, 383, 542, 545, 609β10 origin 121β2, 386β7 outcomes 558, 566β8 outliers 527, 543 output 115β16, 122
U N SA C O M R PL R E EC PA T E G D ES
756
parallel lines 405, 421β2, 424β5, 430 parallel sides 445β6, 687 parallelograms 444, 687β9 partitioning 16 parts 157, 172, 176, 213, 215, 329, 331, 460 patterns 64, 79, 98β101, 107, 109β10, 245, 371 percentages 194β6, 199β201, 324β6, 329, 331, 567 perfect squares 94 perimeters 268, 659β60, 672, 674 perpendicular lines 405, 415, 693 pi 665β7 place value 4β6, 9, 11, 20, 36, 284β5, 317 planes 122, 405 Platonic solids 487 plots/plotting 122β3, 386β8, 526β7 points 121β2, 386, 404, 413, 416β17, 468, 527, 535 polygons 443β4, 453, 475 polyhedrons 487β8 populations 515 position 121, 286, 305, 307, 311, 360, 386, 466 positive integers 10, 15, 25, 31, 35, 42, 47, 79, 95, 183, 238, 356, 361β3, 371, 377, 382 powers 10, 82β5, 88, 93, 240, 299β300, 306, 311, 319, 653 prediction 99, 107, 514, 535 primary (data) sources 515β16 prime factors and form 80, 88β9 prime numbers 79β80 prisms 707β8, 715 probability 558β9, 566β9, 573β4 problem solving 201, 213, 259, 267, 381β2, 430, 474β5, 564β5, 609β11, 617β19, 627β8, 634β5, 713β14 products 25, 31, 66, 306, 371β2, 683 pronumerals 232β3, 238β40, 243, 248β9, 254, 256, 267, 383, 416, 594β5, 611, 622, 627, 629β30, 634β5 proper fractions 145, 157β8, 177, 180, 182β3 proportions 213, 215, 329, 332, 550β2 protractors 404, 406, 409, 437, 439 pyramids 482 quadrants 386, 673 quadrilaterals 443β6, 454, 687
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Index
simplifying (to simplest form) 151β4, 158, 160, 170, 172, 175, 177β8, 181β2, 184, 190β1, 195β6, 200, 208β9, 214, radius 665, 672, 674 248, 250, 254, 256, 264, 317β18, 330, ranges 514, 520, 522, 529, 544 605β6, 609β10, 612, 617, rates 382, 536 622, 624 ratios 100β1, 206β9, 213β15, 217, 329, single-step experiments 566β7, 573β4 666 skewed data 527, 543 rays 405, 407 slope 217, 536 reciprocals 189β90 solids 482β3, 487 rectangle areas 263, 680, 683, 693, space 567, 707 700β2 spatial patterns 107β8, 110 rectangles 268, 444, 460 speed 322, 382, 536 rectangular prisms 707β8, 710 square areas 683 recurring decimals 318 square brackets 47 reflections 465β7 square numbers 93β4 reflex angles 406, 408 square roots 93β6, 240 regular polygons 444, 446 square units 680 remainder (rem.) 35β7, 74β5 squares 93, 95β6, 444, 483, 682, 693, repeated addition 371, 605 702β3 repeated pattern 318 stem 542β3 revolution 406, 415, 417, 431 stem-and-leaf plots 542, 544β5 rhombus 444 straight angles 406 right angles 386, 406, 437 strategy (mental) 16β17, 25β7, 31, 36β7, right triangular prisms 716 100, 199β200 right-hand sides (RHS) 594, 596, 605β6, βstripβ method 180, 188 610, 612 subject 629 roots 240 substitution 116, 238β40, 243β4, 267, rotation 460β1, 465β6, 468 383β4, 594β5, 610, 612, 629β30, 694, rounding (up/down) 43, 289β91, 305β6, 710 666β8, 671 subtraction 5, 15β17, 20β2, 36, 47β8, 85, rule(s) 100, 107β8, 110, 115β18, 121, 123, 175β7, 234, 238, 250, 264, 294, 296, 180, 244, 267β8, 300, 306, 377, 629, 361β3, 366β8, 377, 702β3 672, 717 subtraction algorithm 22 supplementary angles 415β16, 422, 425, samples 515, 519, 567β8 454, 456 scalene triangles 437 surface area 680 scales 530, 653, 655, 680, 728 symbols/signs 4β5, 10, 15, 43, 49, 94, 165, scientific notation (standard form) 181, 188, 190β1, 194, 199, 235, 254, 82, 304 256, 286β7, 299, 357β8, 365, 371β2, secondary (data) sources 515β16 383, 594 sectors 550β2, 665, 672, 674 symmetrical data 527, 543 segments 405, 407, 443, 467β8, 535, symmetry 460β2 537, 665 sequences 99β100, 107 tables 115β17, 244, 285 βshadingβ method 180, 324 tables (multiplication) 26, 36β7 shapes 404, 436, 443, 460β1, 475β7, 543, tangents 665 659β60, 672β3, 682 terminology 233β4 short division algorithm 36β7 terms 100, 107, 238, 248, 250, 254, 261
theoretical probability 566β7 three-dimensional (3D) solids 680 time 536 to scale 414, 527, 680β1, 708 tonne 729 transformation 465β6, 476 translation 476β8 transversals 421β2, 424β5 trapezium 444, 446 travel graph 536β7 trial and error 599 trials 567, 573β4 triangles 98, 404, 436β9, 453, 455, 693β4, 700β2 triangular prisms 715, 717 true/false equations 594β6, 599, 604, 610, 612 two-step equations 611
U N SA C O M R PL R E EC PA T E G D ES
quantitative data 514 quotient 35, 37, 49, 74, 311, 371β2
757
units 206, 209, 215, 268, 634, 652β4, 680, 707, 724β5, 728β9 unknowns (pronumerals/variables) 599β601, 629, 634β5 βunlikeβ fractions 171, 176β7 upper case letters 404
value(s) 5, 10, 115β18, 238, 244, 267, 285, 294, 299, 417, 431, 455, 514, 521β2, 543, 545, 594, 599 variables 233, 239β40, 514β16, 535β6, 629 vertex 405β6, 414, 424, 443, 467, 478, 693 vertically opposite angles 414β15, 417 vinculum 145 volume 707β8, 710, 715, 717β18, 723β5 whole numbers 10, 35, 43, 64β5, 79, 93β5, 145, 151, 157β8, 172, 176β8, 181, 183, 190β1, 284, 305, 311β13 width 681, 683 x-axis/y-axis 121β2, 386β8, 467
zero 31, 248, 294, 296, 299, 301, 307β8, 312β13, 356, 408, 604
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
758
Answers
Chapter 1 1A
1 β14, 27β 2 123456787654321 3 β625β 4 β60β 5 β431 Γ 52β 6 β$490β 7 β16β 8 β150Β°β 9 β270β 10 Discuss: e.g. count how many times βtheβ appears on one page and multiply by the number of written pages in the book. 11 Varies: e.g. β7.5βββkmβ for an average step of β75βββcmβ 12 a β1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 Γ 9 = 100β b Many solutions e.g. β123 β 45 β 67 + 89 = 100β 13 β2.5βββcm, 75βββcmβ(answers vary if thickness of glass considered) 14 β$110, $130, $170β 15 β7β trips 16 β2_ β 2 ββ days 3
Building understanding 1 a Babylonian 2 a βIβ
b Roman b
c Egyptian d
c
U N SA C O M R PL R E EC PA T E G D ES
Answers
Working with unfamiliar problems: Part 1
W ο»Ώ WUP1
Working with unfamiliar problems: Part 2 1 2 3 4
β17, 26, 35, 44, 53, 62, 71, 80β β60Β° , 155Β°β β12, 18, 20, 24, 30, 36βand β40β β12βarrangements possible, β8βopen box nets
3 a 4 a βIβ 5 β5 β 1 = 4β
b
b βVβ
c
c βXβ
e βCβ
iii X β XIVβ iii 2β 13β iii 7β 1β iii 1β 6β
iv βCLVIβ iv β241β iv β205β iv β40β
Now you try Example 1
b 4 is 23 is
a 4 is
23 is
142 is
142 is c 4 is IV 23 is XXIII 142 is CXLII
Exercise 1A 1
,
2
, 3 βXIV, CXXXIβ 4 a i iii
ii iv
b i
ii
iii
c 5 a b c 6 a
iv
i βIIβ i β33β i β12β i β4β βXXXVIβ
ii ii ii ii
βIXβ β111β β24β β8β
b
d βDCLXXVIIIβ
c
5 β25β 6 β231β 7 β2520β 8 β15β 9 β7β 10 β1200β approximately β$240β 11 β41Β°β 12 β7β 13 β7200Β°β 14 β99β 15 e 16 β204β
d βLβ
(ββ 21)ββ
7
8 βCLXXXVIIIβββββ(ββ188β)ββββ
9
(74)
10 a Roman b Babylonian c Roman 11 a βIVβ b βIXβ c βXIVβ d βXIXβ e βXXIXβ f βXLIβ g βXLIXβ h βLXXXIXβ i βXCIXβ j βCDXLIXβ k βCMXXIIβ l βMMMCDIβ 12 A separate picture is to be used for each β1, 10, 100βetc. The number β999β uses β27β pictures. 13 a i ii iii
......
v
iv vi
......
b βthirdβββposition = 60 Γ 60 = 3600β c β216βββ000β 14 Answers may vary. Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
1B
Hundreds Tens βBβ βBβ βCβ True False
b b f b f
βEβ βEβ βFβ False True
b d c c g c g
Thousands Ones βCβ βDβ βGβ True False
d H β β d A β β
U N SA C O M R PL R E EC PA T E G D ES
1 a c 2 a 3 a e 4 a e
Answers
Building understanding
13 Position gives the place value and only one digit is needed for each place. There is also a digit for zero. 14 a You do not need to write the zeros. b i β41 Γβ β10ββ2ββ ii β37 Γβ β10ββ4ββ iii 2β 177 Γβ β10ββ4ββ c i β38βββ100β ii β7βββ204βββ000β iii 1β βββ028βββ000βββ000β d i β1 Γβ β10ββ6ββ ii β1 Γβ β10ββ9ββ iii 1β Γβ β10ββ12ββ iv β1 Γβ β10ββ100ββ v β1 Γβ β10ββgoogolββ
759
d True h True
Now you try
Example 2 a β7 Γ 10 = 70β
b β7 Γ 1000 = 7000β
Example 3 a β2715 = 2 Γ 1000 + 7 Γ 100 + 1 Γ 10 + 5 Γ 1β b 4β 0βββ320 = 4 Γ 10βββ000 + 3 Γ 100 + 2 Γ 10β Example 4 a β370 = 3 Γβ β10ββ2ββ + 7 Γβ β10ββ1ββ b 2β 0βββ056 = 2 Γ 1β04ββ ββ + 5 Γ 1β0ββ1ββ + 6 Γ 1β
Exercise 1B
Building understanding 1 a b 2 a 3 a 4 a d 5 a d
Add, plus, sum Minus, take away, difference 1β 0β b β69β c 1β 2β 2β 7β b 1β 6β True b True False e True 1β 8β b β19β 1β 40β e β21β
d β20β
c f c f
True False β32β β9β
Now you try Example 5 a 7β 96β
1B
1 a β40β b β400β 2 a 7β β b β70β c β70β d 7β 00β e β700β f β7000β g β700β h 7β 0βββ000β 3 a β20β b β2000β c β200β d 2β 00βββ000β 4 a β1 Γ 10 + 7 Γ 1β b 2β Γ 100 + 8 Γ 10 + 1 Γ 1β c β9 Γ 100 + 3 Γ 10 + 5 Γ 1β d 2β Γ 10β 5 a β4 Γ 1000 + 4 Γ 100 + 9 Γ 10 + 1 Γ 1β b 2β Γ 1000 + 3 Γ 1β c β1 Γ 10βββ000 + 1 Γ 1β d 5β Γ 10βββ000 + 5 Γ 1000 + 5 Γ 100 + 5 Γ 10 + 5 Γ 1β 6 a β3 Γβ β10ββ3ββ + 8 Γβ β10ββ1ββ b 4β Γβ β10ββ2ββ + 5 Γβ β10ββ1ββ c β9 Γβ β10ββ4ββ + 3 Γβ β10ββ1ββ d 4β Γβ β10ββ4ββ + 7 Γβ β10ββ3ββ + 5 Γβ β10ββ2ββ 7 a β4 Γβ β10ββ4ββ + 2 Γβ β10ββ3ββ + 9 Γ 1β b 3β Γβ β10ββ3ββ + 6 Γβ β10ββ2ββ + 4 Γ 1β c β2 Γβ β10ββ2ββ + 4 Γβ β10ββ1ββ + 5 Γ 1β d 7β Γβ β10ββ5ββ + 3 Γβ β10ββ2ββ + 6 Γ 1β 8 a β347β b 9β 416β c β7020β d 6β 00βββ003β e β4βββ030βββ700β f β90βββ003βββ020β 9 a β44, 45, 54, 55β b 2β 9, 92, 279, 729, 927β c β4, 23, 136, 951β d 3β 45, 354, 435, 453, 534, 543β e β12βββ345, 31βββ254, 34βββ512, 54βββ321β f β1001, 1010, 1100, 10βββ001, 10βββ100β 10 a β6β b 6β β c β24β 11 β27β 12 a βA Γ 10 + B Γ 1β b A β Γ 1000 + B Γ 100 + C Γ 10 + D Γ 1β c βA Γ 100βββ000 + A Γ 1β
1C
b β174β
c 2β 3β
d β148β
Exercise 1C
1 a β64β b β97β d 7β 48β e β948β 2 a 1β 1β b β36β d 4β β e β3111β 3 a 2β 4β b β75β d 1β 33β e β167β 4 a 2β 4β b β26β d 2β 22β e β317β 5 a 5β 1β b β128β d 1β 19β e β242β 6 a 2β 68β b 6β 5β 7 a 1β 2β b β27β d 1β 33β e β14β g 1β 019β h 0β β 8 β38β hours 9 β107β runs 10 3β 2β cows 11 β29β marbles 12 1β 07β cards 13 a i 1 6 2
β579β β5597β β112β β10βββ001β β95β β297β β108β β5017β β244β β502β d β111β c β107β f β90β i β3β c f c f c f c f c f
c 1β 9β
ii
5
5 4
2 3
3
4 6
1
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
760
Answers
ii
6 9
2
1 4 14 a b 15 a 16 a b 17 a
9 1
7 8
3
2
6 5
7
4 3
5
8
Because β3 + 9βis more than β10β, so you have to carry. Because β8 β 6βis easy, but β1 β 6βmeans you have to carry. βc β b = aβ b βb β a = cβ Four ways (totals are β9, 10, 11β and β12β) Answers may vary.
6 a 1β 6β b 4β 7β 7 1β 854β sheep 8 β576β kilometres 9 a 1β 821β students b 10 a 3 8 +5 3 9 1
c β485β
d 1β 66β
b β79β students c 6 7 + 8 4 7 9 1 4
1 1 4 + 7 7 1 9 1
U N SA C O M R PL R E EC PA T E G D ES
Answers
b i
b
c
1D
d
β6β
β1β
β8β
β7β
β5β
β3β
β2β
β9β
β4β
β10β
β15β
ββ 8β
ββ 9β
β11β
β13β
β14β
ββ 7β
β12β
β15β
β20β
β13β
β14β
β16β
β18β
β19β
β12β
β17β
11 a
12 a b c d 13 a b 14 a
b
b
6 2 β2 8 3 4
β62β
β67β
β60β
β61β
β63β
6β 5β
β66β
β59β
6β 4β
β101β
β115β
β114β
β104β
β106β
β107β
β109β
ββ 1β
β15β
β14β
ββ 4β
β12β
ββ 6β
ββ 7β
ββ 9β
β108β
β110β
β111β
β105β
β103β
β102β
β116β
ββ 8β
β10β
β11β
ββ 5β
β113β
β13β
ββ 3β
ββ 2β
β16β
15 β452β and β526β 16 Answers may vary.
2 1 6 5
Building understanding
Building understanding b β110β b 8β 4β b 6β β
c 1β 005β c 1β 5β c 1β β
d 1β 105β d 9β 79β d 3β β
Now you try
1 a β20, 24, 28β 2 a True e True 3 a β3β
b β44, 55, 66β b True c False f True g False b β0β c β5β
c β68, 85, 102β d True h False d β2β
Now you try
Example 6 a β92β
b β735β
Example 7 a β46β
b β1978β
Example 8 a β32β d β210β
b β99β f β745β b 2β 25β b 1β 92β e β1223β b β20β f β112β
b c g c
β462β β41β β1923β β2229β
c 1β 9β g 7β 9β
b β126β e 3β 00β
Example 9 a β294β
Exercise 1D 1 a β51β 2 a β87β e β226β 3 a β161β 4 a β77β d β4208β 5 a β31β e β36β
30 9 2 9 2 7
1E
1D
1 a β101β 2 a β27β 3 a β1β
β
i β29β ii β37β Yes No The balance of ββ 19 + 20βis β+ 1β, so add β1β to β36β. Answers may vary. Different combinations in the middle column can be used to create the sum.
β112β
18 β29β and β58β
c
2 6 5 β 1 8 4 8 1
c β76β
b 2β 976β
Exercise 1E
d β86β h β5080β d β1975β c 4β 18β f β1982β d β58β h β72β
1 a e 2 a e 3 a 4 a d
β15β β28β β105β β57β β96β β56β β108β
b f b f b
β32β β36β β124β β174β β54β b β52β e 1β 26β
c g c g c
β36β 5β 6β β252β 1β 12β β96β
d h d h d c β87β
β27β β40β β159β β266β β72β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
b β129β f β2849β b β4173β
c 4β 32β g β2630β c β3825β
d 1β 65β h 3β 1βββ581β d 2β 9βββ190β
Now you try Example 10 a 2β 3βββ000β
b β2920β
c β17βββ296β
Exercise 1F 25 Γ 5 12 5
d
13 2 Γ 8
1 a 4β 00β b d 4β 600β e g 1β 4βββ410β h 2 a 3β 40β b d 2β 2βββ500β e g 6β 3βββ400β h 3 a 4β 07β b d 9β 416β e g 1β 8βββ620β h 4 a i β430β b i β1420β c i β1488β 5 a 2β 09β b β546β 6 β$2176β 7 $β 6020β 8 β86βββ400β seconds 9 a 2 3
β290β β50βββ000β β29βββ100βββ000β β1440β β41βββ400β β9βββ387βββ000β β1368β β18βββ216β β33βββ858β
c 1β 830β f 6β 3βββ000β
U N SA C O M R PL R E EC PA T E G D ES
b
Answers
5 a β66β e β258β 6 a β235β 7 β$264β 8 β1680β metres 9 β116β cards 10 No 11 a 3 9 Γ 7 27 3
761
c
79 Γ 3
10 5 6
2 37
e
g
2 7 Γ 7 1 8 9
f
2 3 2 Γ 5 11 6 0
h
3 9 Γ 9 35 1
Γ
3 1 4 7
2 1 9 8
Γ
b Answers may vary; e.g. 2 9 5 Γ 3 88 5
Progress quiz 1 a 2 3 4 5 6 7 8 9
b C β XXXIVβ
c
β5 Γ 10 000 + 8 Γ 100 + 6 Γ 10 + 2 Γ 1β a β375β b β64β c β57β d 7β 1β a β62β b β78β a β28β b β72β c β108β d 4β 5β a β84β b β195β a β252β b 9β 48β c β15βββ022β a β1973β students b β77β students a T b T c F d T e T f S
1F
Building understanding
1 a β2β b β0β 2 a 1β 00β b 1β 0β 3 a Incorrect, β104β c Correct
b
Γ
d β2178β
14 3 1 3
4 2 9 1 4 3 0
3 9 1
c
16 β6, 22β
ii β72βββ000β ii β7800β ii β5730β c 5β 55β
Γ 1 7 1 6 1 2 3 0
7 Γ 1 51 9
c 1β 890β f 4β 0βββ768β
Ch1 Progress quiz
12 β12β ways 13 a β3 Γ 21β b 9β Γ 52β c β7 Γ 32β d 5β Γ 97β e βa Γ 38β f βa Γ 203β 14 Three ways: (ββ 0, 1)β,β β(1, 5)β,β β(2, 9)ββ. You cannot carry a number to the hundreds column. 15 a Answers may vary; e.g. 2 1 7
c 6β 440β f 4β 60βββ000β
1 8 5 9
d
4 9 3 7
Γ
1 2 6 2 1
3 4 3 1 4 7 0
1 2 6 2 5 2 0
1 8 1 3
2 6 4 6
10 6β 0βββ480β degrees 11 One number is a β1β. 12 a 3β 9βββ984β c β4βββ752βββ188β 13 a 1β 600β b β780β 14 a 8β 4βββ000β 15 1β 23, 117β
b d c b
9β 27βββ908β 1β 46βββ420βββ482β 8β 10β d β1000β 3β 185β
1G
Building understanding 1 a 2β β 2 a 1β β 3 a 1β β
b 3β β b 2β β b 1β β
c 7β β c 2β β c 5β β
d 1β 2β d 5β β d 5β β
Now you try
c β0β
d 4β β c β10βββ000β b Incorrect, β546β d Incorrect, β2448β
Example 11 a 4β β
b β121β
Example 12 a 2β 3β and β1β remainder
c β65β b 7β 2β and β3β remainder
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
762
Answers
β3β c β6β d β5β β9β g β8β h β11β β22β c β32β d β103β β21β g β301β h β61β β31β c β17β d β7β b β9β c β21β d β21β e 3β 0β f β16β b β1094β c β0β d β0β 2 ββ 1 ββ 6 a β23β rem. β2 = 23β_ b β13β rem. β1 = 13β_ 7 3 1 ββ 1 ββ c β69β rem. β1 = 69β_ d β41β rem. β1 = 41β_ 2 6 1 ββ 2 ββ e β543β rem. β1 = 543β_ f 2β 0βββ 333β rem. β2 = 20βββ333β_ 4 3 3 ββ h β10βββ 001β rem. β0 = 10001β g β818β rem. β3 = 818β_ 5 1 ββ 4 ββ 7 a β131β rem. β1 = 131β_ b β241β rem. β4 = 241β_ 7 4 5 ββ 1 ββ c β390β rem. β5 = 390β_ d β11βββ 542β rem. β1 = 11542β_ 6 8 8 4β β packs 9 1β 07β packs 10 a β$243β b β$27β 11 6β 7β posts 12 β15β taxis 13 β19βtrips; any remainder needs β1βmore trip 14 ββ 2β β9β β12β 1 a e 2 a e 3 a 4 a 5 a
β4β β7β β21β β41β β22β β12β β26β
b f b f b
Example 14 a 4β 80β
b β8000β
Example 15 a 3β 60βββ000β
b β42β
Exercise 1H
1H
U N SA C O M R PL R E EC PA T E G D ES
Answers
Exercise 1G
3β 6β
β6β
ββ 1β
ββ 3β
β4β
β18β
15 a β1, 12β 16 β$68β 17 a β1β 18 β8β or β23β 19 βa = bβor βa = β bβ
b 1β 3, 7β
c β4, 5β
b 0β β
c a
8 ββ 20 a β33β rem. β8 = 33β_ 11 1 ββ c β31β rem. β1 = 31β_ 13 16 ββ e β91β rem. β16 = 91β_ 23 21 ββ 1β ββ 6β β20β β56β 4β 0β
β28β
ββ 2β
ββ 3β
1β 4β
ββ 5β
β24β
ββ 4β
1β 2β
ββ 8β
ββ 7β
β10β
22 a β37β d β91β
8 ββ b 5β 4β rem. β8 = 54β_ 17
1 ββ d 1β 08β rem. β1 = 108β_ 15 25 ββ f 1β 23β rem. β25 = 123β_ 56
1I
b 4β 3β e β143β
c 7β 5β f β92β
1H
Building understanding 1 a d 2 a c
Up Up Larger Smaller
b Down e Down
c Up f Down
b Smaller d Larger
Now you try Example 13 a β70β
1 a i β70β ii β40β iii β150β b i β400β ii β700β iii β1800β 2 a β60β b β30β c β120β d β190β e β200β f β900β g β100β h β600β i β2000β 3 a β20β b β30β c β100β d β900β e β6000β f β90βββ000β g β10βββ000β h β10β 4 a β130β b β80β c β150β d β940β e β100β f β1000β g 1β 100β h β2600β i β1000β j β7000β 5 a β120β b β160β c β100β d β12β e β40β f β2000β g 4β 000β h β100β 6 a β1200β b β6300β c β20βββ000β d β8βββ000βββ000β e β5β f β16β g β10β h β25β 7 ββ 2100βscoops 8 Answers may vary. 9 ββ 1200βsheep 10 ββ 8βpeople 11 a β200β b 1β 00βββ000β c β800β d 3β βββ000βββ000β or β4βββ000βββ000β 12 a i Larger ii Larger b i Larger ii Larger c i Smaller ii Smaller d i Larger ii Larger 13 a i β9β ii β152β iii 1β 0β iv β448β b One number is rounded up and the other is rounded down. c i β3β ii β3β iii 1β β iv β2β d If the numerator is decreased, then the approximation will be smaller. If the denominator is increased then the approximation will also be smaller. If the opposite occurs the approximation will be larger.
Building understanding 1 a d g j 2 a
Addition b Division Multiplication e Division Division h Multiplication Subtraction k Multiplication True b False c False
c f i l
Multiplication Addition Division Division d True
Now you try Example 16 a β34β
b 4β 0β
Example 17 a β12β
b 7β 6β
b β6400β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
Exercise 1I
U N SA C O M R PL R E EC PA T E G D ES
Answers
a β11β b β17β c 4β β d 0β β a β8β b β17β c 1β 4β d 1β 1β a β12β b β21β c 1β 9β d 8β β a β20β b β18β c 1β 4β d 6β β a β23β b β21β c 0β β d 1β 8β e β32β f β2β g β22β h 2β 2β i β38β j β153β k β28β l β200β 6 a β10β b β3β c 2β β d 2β 2β e β2β f β9β g β18β h 3β β i β10β j β121β k β20β l β1β 7 a β27β b β10β c 8β β d 7β 7β e β30β f β21β g β192β 8 a β48β b β18β c β13β d 2β 8β e β22β 9 7β 5β books 10 4β 5β TV sets 11 a β(4 + 2) Γ 3 = 18β b 9β Γ· (12 β 9) = 3β c β2 Γ (3 + 4) β 5 = 9β d (β 3 + 2) Γ (7 β 3) = 20β e β(10 β 7) Γ· (21 β 18) = 1β f β(4 + 10) Γ· (21 Γ· 3) = 2β g [β 20β(31 β 19) ] Γ 2 = 16β h 5β 0 Γ· (2 Γ 5) β 4 = 1β i β(25 β 19) Γ (3 + 7) Γ· 12 + 1 = 6β 12 First prize β$38β, second prize β$8β 13 a No b Yes c No d Yes e Yes f No g No h Yes i Yes 14 a No b Yes c No d Yes 15 a βbβ b 0β β c βa + 1β d βbβ 16 a Multiply by β2βand add β1β. b Multiply by β3βand subtract β3β, or subtract β1βand then multiply by β3β. c Multiply by itself and add β1β. 1 2 3 4 5
5 β611β 6 β519β 7 β116β 8 β124β; β1379β 9 β3700β 10 β8596β 11 β31β 12 quotient β27β, remainder β6β 13 β4100β 14 β30βββ000β 15 β200β 16 β14β 17 β94β
763
Short-answer questions 1 a i iii
ii
b i
ii
c a a a a d g 6 a c 7 a 2 3 4 5
6
3 4 5 6
5
1
8
β5β One way is β(2 + 7) Γ 11 + 4 β 3β a β22βββL / dayβ b β7900βββL / yearβ β21, 495β
;
2 2 3 +7 3 8 9 6 1
; CXLIV
2 4β 0βββ000β 3 β5 Γ 100 + 1 Γ 10 + 7 Γ 1; 5 Γβ β10ββ2ββ + 1 Γβ β10ββ1ββ + 7 Γ 1β 4 β288β; β38β
ii βXLβ b β5000β b β363β c b β2324β c b β132β e β33β h β10βββ800β b d b
1
3β 51β 7β 540β rem. β2β
7 2 9 β4 7 3 2 5 6
1 8 3
5
3 7 1
iii C β XLVIβ c 5β 0βββ000β d β217β d β295β c β220β f 2β 4β i 1β 4βββ678β
β95β β191β
d
5 3 Γ 2 7
4
9 11 5
0 6 0
1 4 3 1
8 a 7β 0β 9 a 8β 00β 10 a 2β 4β
3
Chapter checklist with success criteria 1
i βXIVβ 5β 0β β459β β128β β95β β41β β29βββ000β β1413β β46β rem. β5β
c
1 The two people pay β$24βeach, which is β$48βin total. Of that β $48βthe waiter has β$3β, leaving a balance of β$45βfor the bill. 2 2 9 4 7
......
iii
b 3β 300β b β400β b β4β c 1β 4β
c 5β 000β d 2β 0β
c β1000β d β10β e β0β f β13β
4 A β β 9 B β β
5 B β β 10 βAβ
Multiple-choice questions 1 B β β 6 βAβ
2 C β β 7 D β β
3 Eβ β 8 C β β
Extended-response questions 1 a c 2 a c
β646β loads β$36βββ430β β3034β sweets Liquorice sticks, β6β
b d b d
9β 044β kilometres $β 295β 2β 49β Yes (ββ 124)ββ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Ch1 Problems and challenges
Problems and challenges
Chapter review
764
Answers
2A Building understanding 1 a e i 2 a e i
βMβ βFβ βNβ βFβ βNβ βFβ
b f j b f j
βNβ Fβ β β β M N β β N β β β β M
c g k c g k
βFβ M β β βNβ βMβ N β β βFβ
d h l d h l
βNβ βFβ βFβ βNβ βMβ βNβ
9 a 2β 5β b β0β c β23β d β2, 7, 12, 17, 37, 47, 62, 87, 137, 287β e β2β 10 a False b True c False d False e True 11 a 8β 40β b 2β 520β 12 a 1β , 2, 4, 5, 10, 20, 25, 50, 100β b β1, 2, 4, 5, 10, 20, 25, 50, 100β 13 Answers may vary, but they should be multiples of β9β. 14 1, 2, 3, 5, 10, 30 must also be factors. 15 Check the output each time.
U N SA C O M R PL R E EC PA T E G D ES
Answers
Chapter 2
Now you try Example 1 a β1, 3, 7, 21β
b β1, 2, 3, 5, 6, 10, 15, 30β
Example 2 a β4, 8, 12, 16, 20, 24β
b β26, 52, 78, 104, 130, 156β
Example 3 β165 = 11 Γ 15β
2A
Exercise 2A 1 a b 2 a b c d e f g h i 3 a b c d e f g h i 4 a b c d 5 a 6 a 7 a b c d e f 8 a
β1, 2, 3, 4, 6, 12β β1, 2, 3, 4, 6, 8, 12, 16, 24, 48β β1, 2, 5, 10β β1, 2, 3, 4, 6, 8, 12, 24β β1, 17β β1, 2, 3, 4, 6, 9, 12, 18, 36β β1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60β β1, 2, 3, 6, 7, 14, 21, 42β β1, 2, 4, 5, 8, 10, 16, 20, 40, 80β β1, 29β β1, 2, 4, 7, 14, 28β β5, 10, 15, 20, 25, 30β β8, 16, 24, 32, 40, 48β β12, 24, 36, 48, 60, 72β β7, 14, 21, 28, 35, 42β β20, 40, 60, 80, 100, 120β β75, 150, 225, 300, 375, 450β β15, 30, 45, 60, 75, 90β β100, 200, 300, 400, 500, 600β β37, 74, 111, 148, 185, 222β β3, 18β β5β β1, 4, 6, 9, 12, 24β β3, 4, 5, 8, 12, 15, 24, 40, 120β β22β b β162β c β21β β24β b 1β β β12 Γ 16β β21 Γ 15β β12 Γ 15β β11 Γ 11β β12 Γ 28βor β24 Γ 14βor β21 Γ 16β β19 Γ 26βor β38 Γ 13β β20β min b 5β β laps
2B
Building understanding
1 a β1, 2, 4β b 4β β 2 Factors of β18βare β1, 2, 3, 6, 9β and β18β. Factors of β30βare β1, 2, 3, 5, 6, 10, 15β and β30β. Therefore, the HCF of β18βand β30βis β6β. 3 a β24, 48β b 2β 4β 4 Multiples of β9βare β9, 18, 27, 36, 45, 54, 63, 72, 81β and β90β. Multiples of β15βare β15, 30, 45, 60, 75, 90, 105β and β120β. Therefore, the LCM of β9βand β15βis β45β.
Now you try Example 4 β6β Example 5 a 2β 8β
b 3β 0β
Exercise 2B
d β117β c 1β , 4, 9, 16, 25β
c 4β β laps
1 a β5β b 1β 0β 2 a 1β β b β1β c β2β d β3β e β4β f 1β 5β g β50β h β24β i β40β j 2β 5β k β21β l β14β 3 a β10β b β3β c β1β d β1β e 8β β f β12β 4 a β36β b β21β c β60β d β110β e β12β f 1β 0β g β36β h β18β i β60β j 4β 8β k β132β l β105β 5 a β30β b β84β c β12β d β45β e 4β 0β f β36β 6 a βHCF = 5, LCM = 60β b H β CF = 12, LCM = 24β c βHCF = 7, LCM = 42β d H β CF = 9, LCM = 135β 7 3β 12β 8 β9β 9 βLCM = 780, HCF = 130β 10 a β12β min b Andrew β9βlaps, Bryan β12βlaps, Chris β6βlaps c β3βtimes (including the finish) 11 a β8, 16β b 2β 4, 32β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
9 a b c d e f g h
10, 15, 20, 25, 30 12, 15, 18, 21, 24 10, 12, 14, 16, 18 12, 18, 24, 30, 36 16, 24, 32, 40, 48 18, 27, 36, 45, 54 10, 20, 30, 40, 50 12, 16, 20, 24, 28
}
(other answers possible)
2C
10 2β , 4, 8, 11, 22, 44β 11 β200β 12 1β 5β 13 3β 6β 14 9β 66β 15 a Yes, because it is divisible by β3βand β5β. b No, because it is not divisible by β3β. c As many as possible from:β 22βββ470, 25βββ470, 28βββ470, 20βββ475, 23βββ475, 26βββ475, 29βββ475β. 16 a Yes b Multiples of β3β; adding a multiple of β3βdoes not change the result of the divisibility test for β3β. c β18β 17 a β0, 4, 8β b 2β , 6β 18 a β11, 22, 33, 44, 55, 66, 77, 88, 99β b 0β β c β110, 121, 132, 143, 154, β¦β d Equals the centre digit or β11βplus the centre digit e Difference is β0βor β11β f Sum the odd- and even-placed digits. If the difference between these two sums is β0βor is divisible by β11β, then the number is divisible by β11β. g i Yes ii Yes iii No iv Yes v Yes vi Yes h Answers may vary.
Building understanding
1 a Not even b Digits do not sum to a multiple of β3β c β26βis not divisible by β4β d Last digit is not β0βor β5β e Not divisible by β3β(sum of digits is not divisible by β3β) f β125βis not divisible by β8βand it is not even g Sum of digits is not divisible by β9β h Last digit is not β0β 2 a β2β b 2β β c β0β d β0β 3 β3, 6βand β9β 4 β2, 5βand β10β
Now you try
Example 6 a Yes, digit sum is β24β, which is divisible by β3β. b No, β82βis not divisible by β4β. Example 7 β2, 3, 5, 6, 9βand β10β(all except β4βand β8β).
Exercise 2C 1 2 3 4 5 6
a a a a a a e i
2D
No Yes No No Yes Yes No No
b b b b b b f j
Yes No Yes Yes No No Yes Yes
c c c c c c g k
Yes Yes No No Yes Yes No No
d d d d d d h l
Number
Divisible by β2β
Divisible by β3β
Divisible by β4β
Divisible by β5β
Divisible by β6β
Divisible by β8β
Divisible by β9β
Divisible by β10β
β β β β β
β β β β β
β β β β β
β β β β β
β β β β β
β β β β β
β β β β β
β β β β β
7
β243βββ567β
β28βββ080β
β189βββ000β
β1βββ308βββ150β β1βββ062βββ347β
No Yes Yes Yes No Yes Yes Yes
Building understanding 1 2 3 4 5 6
No Yes β2, 3, 5, 7, 11, 13, 17, 19, 23, 29β β4, 6, 8, 9, 10, 12, 14, 15, 16, 18β β101β β201βis divisible by β3.β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2C
U N SA C O M R PL R E EC PA T E G D ES
d 2β 7βββ720ββββ(β23ββ ββ Γ 5 Γ 7 Γ 9 Γ 11)ββ
b β$38β
8 a No
Answers
12 β1β and β20; 2βand β20; 4βand β20; 5βand β20; 10βand β20; 4βand β5; 4β and β10β. 13 No 14 a β2520β b 2β 520β c Identical answers; β2520βis already divisible by β10β, so adding β10βto list does not alter LCM.
765
766
Answers
Example 8 Prime: β13, 19, 79β Composite: β32, 57, 95β Example 9 β2, 3, 7β
Example 11 a 1β 25β
b 8β 00β
Example 12 a 1β 6β
b 4β 7β
Exercise 2E 1 a ββ33ββ ββ b ββ2ββ5ββ c ββ15ββ4ββ 4 2 d 1ββ 0ββ ββ e ββ6ββ ββ f ββ1ββ6ββ 3 2 4 2 2 a ββ4ββ ββ Γβ β5ββ ββ b ββ3ββ ββ Γβ β7ββ ββ c ββ2ββ3ββ Γβ β5ββ2ββ 3 a ββ32ββ ββ Γβ β5ββ2ββ b ββ2ββ2ββ Γβ β7ββ3ββ c ββ9ββ2ββ Γβ β12ββ2ββ d 5ββ 3ββ ββ Γβ β8ββ2ββ e ββ3ββ3ββ Γβ β6ββ3ββ f ββ7ββ4ββ Γβ β13ββ2ββ 3 3 1 1 2 g 4ββ ββ ββ Γβ β7ββ ββ Γβ β13ββ ββ h ββ9ββ ββ Γβ β10ββ ββ i ββ2ββ3ββ Γβ β3ββ2ββ Γβ β5ββ2ββ 6 5 4 4 ββ2ββ ββ Γβ β3ββ ββ Γβ β5ββ ββ 5 a β2 Γ 2 Γ 2 Γ 2β b 1β 7 Γ 17β c β9 Γ 9 Γ 9β d 3β Γ 3 Γ 3 Γ 3 Γ 3 Γ 3 Γ 3β e β14 Γ 14 Γ 14 Γ 14β f 8β Γ 8 Γ 8 Γ 8 Γ 8 Γ 8 Γ 8 Γ 8β g 1β 0 Γ 10 Γ 10 Γ 10 Γ 10β h 5β 4 Γ 54 Γ 54β 6 a β3 Γ 3 Γ 3 Γ 3 Γ 3 Γ 2 Γ 2 Γ 2β b 4β Γ 4 Γ 4 Γ 3 Γ 3 Γ 3 Γ 3β c β7 Γ 7 Γ 5 Γ 5 Γ 5β d 4β Γ 4 Γ 4 Γ 4 Γ 4 Γ 4 Γ 9 Γ 9 Γ 9β e β5 Γ 7 Γ 7 Γ 7 Γ 7β f 2β Γ 2 Γ 3 Γ 3 Γ 3 Γ 4β g 1β 1 Γ 11 Γ 11 Γ 11 Γ 11 Γ 9 Γ 9β h 2β 0 Γ 20 Γ 20 Γ 30 Γ 30β 7 a β32β b β64β c β1000β d β72β e β10βββ000β f β1000β g β64β h β121β 8 a β25β b 1β β c β10β d 6β 4β e β128β f β8β g 2β 2β h β900β i β8β 9 a β4β b β2β c β3β d β6β e β3β f β2β g β2β h β4β 10 a β<β b β>β c β=β d β<β e β>β f β>β g β<β h β<β 11 β125β 12 a β1 + 5 +β β5ββ2ββ +β β5ββ3ββ = 156β b 5β 5βββminsβ c β305βββ175βββ781βββpeopleβ d 7β 5βββminsβ e βApprox. 75βββ000βββ000βββ000βββ000βββ000β 13 a β+β b βΓβ c βββ d Γ· ββ e βΓβ f βββ 14 a β1, 3, 9, 27, 81, 243, 729β b 1β , 5, 25, 125, 625β c i β1, 2, 4, 8, 16, 32, 64, 128, 256β ii It is the powers of β2.β d i β2, 6, 18, 54, 162, 486β ii They are being multiplied by β3βeach time. This sequence is double the powers of β3β. 15 aβ = 2, b = 4β 16 a β1, 2, 6, 24, 120, 720β b i ββ24ββ ββ Γβ β3ββ2ββ Γ 5 Γ 7β ii ββ2ββ7ββ Γβ β3ββ2ββ Γ 5 Γ 7β 7 4 iii 2ββ ββ ββ Γβ β3ββ ββ Γ 5 Γ 7β iv ββ2ββ8ββ Γβ β3ββ4ββ Γβ β5ββ2ββ Γ 7β
U N SA C O M R PL R E EC PA T E G D ES
Answers
Now you try
2E
Exercise 2D
1 a βPβ b C β β c βPβ 2 a βCβ b P β β c C β β d βPβ e βCβ f βCβ g βPβ h P β β i βCβ j βCβ k βCβ l βPβ m βPβ n P β β o βCβ p βPβ q βPβ r βPβ 3 a β2, 3, 7β b 3β , 13β c β2, 3, 5β d β5β e 2β , 7β f 2β , 3β 4 a β32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49β b β51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69β c 8β 1, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99β 5 a β5, 11β b β7, 13β c β11, 13β d β11, 17β e 5β , 73β f 7β , 19β 6 β16β 7 β5β and β7, 11βand β13, 17βand β19β, as well as other pairs 8 β17βand β71β, as well as other pairs such as β37βand β73, 79βand β 97, 11β and β11β. 9 30 = 13 + 17, 32 = 29 + 3, 34 = 29 + 5, 36 = 29 + 7, 38 = 31 + 7, 40 = 37 + 3, 42 = 31 + 11, 44 = 41 + 3, β ββ β―β―β―β― β―β―β―β―β― βββ 46 = 41 + 5, 48 = 41 + 7, 50 = 19 + 31 (other answers are possible) 10 β2 + 3 = 5βand β2 + 5 = 7β. All primes other than β2βare odd and two odds sum to give an even number that is not a prime. So, any pair that sums to a prime must contain an even prime, which is β2β. 11 ββ(5, 7, 11)ββ, as well as other groups 12 Check your spreadsheet using smaller primes.
2E
Building understanding 1 βEβ 3
2 βDβ
Value
Base number
Index number
ββ2ββ3ββ
β 2β
β3β
β8β
ββ5ββ2ββ
β 5β
β2β
β 25β
ββ10ββ4ββ
1β 0β
β4β
β 10000β
ββ2ββ7ββ
β2β
β7β
β 128β
ββ 1ββ12ββ
1β β
1β 2β
β1β
ββ12ββ1ββ
1β 2β
β1β
β12β
ββ0ββ5ββ
0β β
β5β
β0β
Now you try Example 10 a ββ75ββ ββ
b 5ββ ββ4ββ Γβ β11ββ3ββ
Basic numeral
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
10 a 4β 24βcannot have a factor of β5β. b 8β βis not a prime number. c 424 iv 6β β 4
2F
2 2
53
U N SA C O M R PL R E EC PA T E G D ES
2
106
Answers
c 0β β d 0β β e It is the index number on the base β5β. i β1β ii 1β β iii β3β f e.g. β23 !ββ Γ 4 !β
767
Building understanding 1 2 3 4
Composite: β15, 8, 9, 27, 4, 12β; Prime: β13, 7, 5, 23, 11, 2β a β5, 4β b β6, 2β c β10, 2, 5β a β3, 3, 2, 5β b β2, 2, 2, 7β c β5, 11, 2, 2β a ββ23ββ ββ Γβ β3ββ2ββ b ββ3ββ4ββ Γβ β5ββ2ββ c ββ22ββ ββ Γ 3 Γβ β7ββ2ββ d ββ2ββ2ββ Γβ β3ββ2ββ Γβ β11ββ2ββ
β 424 =β β2ββ3ββ Γ 53β β΄ 11 a i β5 Γ 10 β 60β ii β6βis not a prime number. iii A β2βhas been left off the prime factor form. b 6β 0 =β β2ββ2ββ Γ 3 Γ 5β 12 72 72 2
Now you try
36
2
Example 13 β180 =β β22ββ ββ Γβ β32ββ ββ Γ 5β
18
2
Exercise 2F
ββ2ββ2ββ Γβ β5ββ2ββ
9
4
6
2
2 2
2
36
2
2
3
6
2
3
4
2
2
3
2
36
9
2 3
3
6
4
72
36
2
12
3
72
2
3
36
12
6
3
2
6
3 2
3
72
24
3
3
8
24
6
4
24
4
2
2
72
2
24
3
6
72
3
12
3
3
72
2
18
2F
1 a b 2 a 2ββ 3ββ ββ Γβ β3ββ2ββ b 2ββ 3ββ ββ Γ 3β c 2β Γ 19β d ββ22ββ ββ Γ 11β e ββ2ββ2ββ Γ 31β f 2ββ 4ββ ββ Γ 5β g ββ25ββ ββ Γ 3β h ββ24ββ ββ i β3 Γβ β5ββ2ββ j 3β Γ 37β k ββ26ββ ββ l 2ββ 3ββ ββ Γ 7β 3 5 2 2 3 a 2ββ ββ ββ Γ 3 Γβ β5ββ ββ b ββ2ββ ββ Γβ β5ββ ββ c ββ23ββ ββ Γβ β5ββ4ββ d ββ2ββ5ββ Γ 3 Γβ β5ββ2ββ e ββ26ββ ββ Γβ β5ββ6ββ f ββ2ββ3ββ Γβ β3ββ2ββ Γβ β5ββ4ββ g 2ββ 2ββ ββ Γ 5 Γ 41β h β2 Γ 3 Γ 5 Γ 23β 4 a βDβ b A β β c βCβ d βBβ 5 β2310β 6 a β144 =β β2ββ4ββ Γβ β3ββ2β, 96 =β β2ββ5ββ Γ 3β b H β CF =β β24ββ ββ Γ 3 = 48β 7 a β25βββ200 =β β2ββ4ββ Γβ β3ββ2ββ Γβ β5ββ2ββ Γ 7, 77βββ000 =β β2ββ3ββ Γβ β5ββ3ββ Γ 7 Γ 11β b H β CF =β β23ββ ββ Γβ β52ββ ββ Γ 7 = 1400β 8 a β2 Γ 3 Γ 7 = 42β b 3β Γ 5 Γ 11 = 165β c ββ22ββ ββ Γ 3 Γ 5 Γ 7 = 420β 9 24 24
36
2
3
ββ22ββ ββ Γβ β3ββ2ββ
2
2
12
3
3
4
2
8
4
2 2
2
2
2
3 2
2
2
2
2
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
768
Answers
10 a b c d e
72
3
24
3
2
24
12 6
2
12
2
4
β3 Γ 3 Γ 3 Γ 3 = 81β 1β Γ 1 Γ 1 Γ 1 Γ 3 Γ 3 = 9β β5 Γ 10 Γ 10 Γ 10 Γ 10 = 50βββ000β 4β Γ 4 = 16β 9 Γ 9 β 3 Γ 3 Γ 3 Γ 2 = 81 β 54 ββ― ββ― βββ ββ β = 27
11 a 24 = 2 Γ 2 Γ 2 Γ 3 ββ ββ―β―β― ββ― βββ β = β23ββ ββ Γ 3 b β180 =β β22ββ ββ Γβ β32ββ ββ Γ 5β
3
U N SA C O M R PL R E EC PA T E G D ES
Answers
72
2
2
3
72
2
72
2G
4
2
18
2 9 3
4
2
2
2 6 2
3
72
6
Ch2 Progress quiz
2
Building understanding
3
3 6
6
2
2
3
72
12
3 4 2
1 3β 6ββββcmββ2ββ, a square number 2 β1ββ2ββ = 1,β β2ββ2ββ = 4,β β3ββ2ββ = 9,β β4ββ2ββ = 16,β β5ββ2ββ = 25,β β6ββ2ββ = 36,β β7ββ2ββ = 49, 2ββ ββ = 64,β β9ββ2ββ = 81,β β10ββ2ββ = 100,β β11ββ2ββ = 121,β β12ββ2ββ = 144,β β―β―β―β―β― βββ βββ8
3
72
12
2
18
8
3
2
4
2
9
2 3
2
13 2β Γ 3 Γ 5 Γ 7 = 210β
β2 Γ 3 Γ 7 Γ 11 = 462β
β2 Γ 3 Γ 5 Γ 11 = 330β
β2 Γ 3 Γ 7 Γ 13 = 546β
β2 Γ 3 Γ 5 Γ 13 = 390β
β2 Γ 3 Γ 7 Γ 17 = 714β
β2 Γ 3 Γ 5 Γ 17 = 510β
β2 Γ 3 Γ 7 Γ 19 = 798β
β2 Γ 3 Γ 5 Γ 19 = 570β
β2 Γ 3 Γ 7 Γ 23 = 966β
β2 Γ 3 Γ 5 Γ 23 = 690β
β2 Γ 5 Γ 7 Γ 11 = 770β
β2 Γ 3 Γ 5 Γ 29 = 870β
β2 Γ 5 Γ 7 Γ 13 = 910β
β2 Γ 3 Γ 5 Γ 31 = 930β
β2 Γ 3 Γ 11 Γ 13 = 858β
14 Check with your teacher. 15 Answers may vary.
Progress quiz
1 a β16 : 1, 2, 4, 8, 16β b β70 : 1, 2, 5, 7, 10, 14, 35, 70β 2 a β7, 14, 21, 28β b β20, 40, 60, 80β 3 a β5β b β18β 4 a β24β b β45β 5 a No, Last β2βdigits not βΓ·β β4β b Yes, Sum of β21 Γ· 3β c Yes, Sum of β24 Γ· 3β, and even βΓ·β β2β d Yes, Sum of β27 Γ· 9β 6 β2, 3, 4, 5, 6, 10, 12, 20, 24, 30, 40, 60β 7 a βC :βlarge number of factors b βN :βonly one factor c βP :βonly two factors, β1βand itself d βNβ 8 a β5, 7β b β2, 3β 9 a ββ54ββ ββ b ββ3ββ2ββ Γβ β7ββ5ββ
3
1β32ββ ββ = 169, 1β42ββ ββ = 196, 1β52ββ ββ = 225 3 a A square is not possible. b Draw a β4βby β4βsquare. 4 a β36β b 2β 5β d β100β e 4β 9β 5 a β5β b β4β c β10β
c 1β 21β f 1β 44β d β7βββcmβ
Now you try Example 14 a 8β 1β
b 6β β
c β70β
Example 15 a 3β β and β4β
b 1β 0β and β11β
Example 16 a 6β 0β
b 3β β
Exercise 2G
1 a β16β b β49β c β9β 2 a 6β 4β b β4β c β1β e β36β f 2β 25β g β25β i β121β j 1β 0βββ000β k β400β 3 a β5β b β3β c β1β e β0β f 9β β g β7β i β2β j 1β 2β k β20β 4 a β50β b β80β c β90β 5 a β5β and β6β b 6β β and β7β d β7β and β8β e 9β β and β10β g β2β and β3β h 1β β and β2β 6 a β30β b 6β 4β d β36β e 4β β g β81β h 4β β _ 7 a Because β ββ 64β―ββ = 8β b β32β tile lengths c β36β tile lengths 8 a β8β b β9β c β5β 9 β121, 144, 169, 196β 10 a 4 and 81, or 36 and 49 b 3β 6βand β121β; other answers possible 11 a β1156β b β6561β
d β100β d β144β h β0β l β2500β d β11β h β4β l β13β d β27β c β8β and β9β f β10β and β11β c 6β 5β f 0β β i β13β
d β5β
c β10βββ609β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
β144β 1β 44β βa = 3, b = 4β Answers may vary; e.g. ββ4ββ2ββ Γβ β5ββ2ββ and ββ20ββ2ββ β121β and β12βββ 321β β1βββ234βββ321β 1β βββ234βββ321β
β17, 23, 30β 1β 6, 22, 29β β36, 49, 64β 1β 7, 12, 6β β17, 19, 23β 4β 7, 95, 191β 5β , 7, 6β 3β 2, 40, 38β
14 a β8β b
βnβ
β1β
β2β
β3β
β4β
β5β
β6β
β7β
β8β
β9β
β10β
β11β
β12β
β13β
β14β
β15β
β16β
β17β
β18β
β19β
β20β
ββnβββ― 2ββ
β1β
β4β
β9β
β16β
β25β
β36β
β49β
β64β
β81β
β100β
β121β
β144β
β169β
β196β
β225β
β256β
β289β
β324β
β361β
β400β
Difference
β
β3β
β5β
β7β
β9β
β11β
β13β
β15β
β17β
β19β
β21β
β23β
β25β
β27β
β29β
β31β
β33β
β35β
β37β
β39β
c The difference between consecutive square numbers increases by β2βeach time. d Answers may vary.
2H
Building understanding
β8, 11, 14, 17, 20β β52, 48, 44, 40, 36β β3, 6, 12, 24, 48β β240, 120, 60, 30, 15β Ratio of β3β Difference adding β11β e Ratio of _ ββ1 ββ 2 g Neither
1 a c 2 a c 3 a c
b d b d b d
β32, 31, 30, 29, 28β β123, 130, 137, 144, 151β β5, 20, 80, 320, 1280β β625, 125, 25, 5, 1β Difference subtracting β2β Neither
f Neither h Difference subtracting β3β
Now you try
6 a β49, 64, 81β; square numbers b 2β 1, 34, 55β; Fibonacci numbers c β216, 343, 512β; cubes (i.e. powers of β3β) d 1β 9, 23, 29β; primes e β16, 18, 20β; composite numbers f 1β 61, 171, 181β; palindromes (or increasing by 10) 7 a β115, 121, 128β b 2β 4, 48, 16β c β42, 41, 123β d 9β , 6, 13β 8 β1, 2, 3, 4, 5, 6, 7, 8, 9, 10ββββ(total = 55)ββ 9 a β39β b β57β c β110β d β192β 10 β1, 0, 0, 1, 2β 11 Difference is β0β, ratio is β1β. 12 a β55β b β100β c β2485β d β258β 13 a β3β b β10β c β45β d 2β 76β e βn Γβ β(n β 1)ββ Γ· 2β 14 a β135β b β624β c β945β
2I
Example 17 a β19, 22, 25β
b β131, 117, 103β
Example 18 a β135, 405, 1215β
b β80, 40, 20β
Building understanding 1 a
Exercise 2H 1 a b 2 a c e g 3 a b c d e f g h 4 a c e g
i β26, 33, 40β i β22, 17, 12β β23, 28, 33β β14, 11, 8β β27, 18, 9β 5β 05, 606, 707β β32, 64, 128β 8β 0, 160, 320β β12, 6, 3β 4β 5, 15, 5β β176, 352, 704β β70βββ000, 700βββ000, 7βββ000βββ000β 1β 6, 8, 4β 7β 6, 38, 19β β50, 32, 26β β32, 64, 256β β55, 44, 33β 7β 0, 98, 154β
b d f h
ii β59, 68, 77β ii β55, 43, 31β 4β 4, 54, 64β β114, 116, 118β β5, 4, 3β β51, 45, 39β
b
b d f h
β25, 45, 55β β9, 15, 21β β333, 111β β126, 378, 3402β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2H
U N SA C O M R PL R E EC PA T E G D ES
5 a b c d e f g h
Answers
12 a b c d 13 a b c
769
770
Answers
ii 4β , 7, 10, 13, 16β iii 4β βsticks are required to start, then β3βare added. b i
Answers
2 a and b and
U N SA C O M R PL R E EC PA T E G D ES
ii 3β , 5, 7, 9, 11β iii 3β βsticks are required at the start, then β2βare added.
c
and
c i
Now you try Example 19 a
b β4, 7, 10, 13, 16β c 4β βmatchsticks are required to start the pattern, and an additional β3βmatchsticks are required to make the next term in the pattern.
ii 6β , 11, 16, 21, 26β iii 6β βsticks are required to start, then β5βare added.
d i
2I
Example 20 a
ii 7β , 12, 17, 22, 27β iii 7β βsticks are required to start, then β5βare added.
e i
b
Number of shapes
β1β
2β β
β3β
β4β
β5β
Number of sticks required
β5β
9β β
β13β
β17β
2β 1β
c βNumberβββofβββsticks = 1 + 4 Γ numberβββofβββshapesβ
ii 6β , 11, 16, 21, 26β iii 6β βsticks are required to start, then β5βare added.
d β81β
f i
Exercise 2I 1 a
ii 4β , 7, 10, 13, 16β iii 4β βsticks are required to start, then β3βare added.
3 a
b β3, 5, 7, 9, 11β c 3β βsticks are required to start, then β2βare added. 2 a i
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
b
β1β
β2β
β3β
β4β
5β β
No. of sticks required
β4β
β8β
β12β
β16β
2β 0β
f
Answers
No. of crosses
771
c N β umberβββofβββsticks = 4 Γ numberβββofβββcrossesβ d 8β 0β sticks
U N SA C O M R PL R E EC PA T E G D ES
4 a
b
No. of fence sections
β1β
β2β
β3β
β4β
5β β
No. of planks required
β4β
β7β
β10β
β13β
1β 6β
c N β umberβββofβββplanks = 3 Γ numberβββofβββfenceβββsections + 1β d 6β 1β planks 5 a No. of tables β1β β2β β3β β4β β5β No. of students
β14β
β17β
b N β umberβββofβββstudents = 3 Γ numberβββofβββtables + 2β c β23β students d 2β 1β tables 6 a Spa length β1β β2β β3β β4β β5β
β6β
No. of tiles
β5β
β8β
β10β
β8β
β12β
b
β14β
β16β
β18β
12 a
Number of straight cuts
β1β
β2β
β3β
β4β
β5β
β6β
β7β
Maximum number of sections
β2β
β4β
β7β
β11β
β16β
β22β
β29β
b 5β 6β c β211β
2J
Building understanding 1 a True 2 a A β β
c
d β16β
b False b βDβ
c True c βBβ
d True d C β β
Now you try Example 21
d
a
e
b
input
β3β
5β β
β7β
β12β
β20β
output
β7β
β9β
1β 1β
β16β
β24β
input
β1β
β2β
β5β
β7β
1β 0β
output
β3β
β11β
β35β
β51β
7β 5β
Example 22 a oβ utput = input β 5β b oβ utput = input Γ 6 (β orβββoutput = 6 Γ input)ββ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2J
b N β umberβββofβββtiles = 2 Γ spaβββlength + 6β c β36β tiles d 1β 2β units 7 βAβ 8 A β β 9 Answers may vary, examples below. a
β11β
10 a 5β β b β41β c β4g + 1β 11 a m β + nβ b Number of pieces for each new panel c
772
Answers
1 a
b
8 a
input
β1β
β3β
β5β
β10β
β20β
output
β6β
β8β
β10β
β15β
β25β
b
input
β3β
β6β
β8β
β12β
β2β
output
β7β
β34β
β62β
β142β
β2β
input
β6β
β12β
β1β
β3β
β8β
output
β5β
β3β
β25β
β9β
β4β
input
β3β
β9β
β1β
β10β
β6β
output
β6β
β18β
β2β
β20β
β12β
input
β4β
β5β
β6β
β7β
β10β
output
β7β
β8β
β9β
β10β
β13β
input
β5β
β1β
β3β
β9β
β0β
output
β20β
β4β
β12β
β36β
β0β
input
β11β
β18β
β9β
β44β
β100β
output
β3β
β10β
β1β
β36β
β92β
input
β5β
β15β
β55β
β0β
β100β
output
β1β
β3β
β11β
β0β
β20β
input
β1β
β2β
β3β
β4β
β5β
output
β7β
β17β
β27β
β37β
β47β
input
β6β
β8β
β10β
β12β
β14β
output
β7β
β8β
β9β
β10β
β11β
input
β5β
β12β
β2β
β9β
β14β
output
β16β
β37β
β7β
β28β
β43β
input
β3β
β10β
β11β
β7β
β50β
Building understanding
output
β2β
β16β
β18β
β10β
β96β
1 C β β 2 a βxβ-axis d first
U N SA C O M R PL R E EC PA T E G D ES
Answers
Exercise 2J
2 a
b
c
d
2K
3 a
b
c
d
4 a b 5 a b 6 a b
c
d
input
β5β
β12β
β2β
β9β
β0β
output
β30β
β156β
β6β
β90β
β0β
input
β3β
β10β
β11β
β7β
β50β
output
β15β
β190β
β231β
β91β
β4950β
yβ = x + 6β yβ = 3xβ yβ = x β 2β yβ =β βxβββ― 2ββ + 1β
9 a b c d
βoutput = 2 Γ input + 1 ; output = 3 Γ input β 2β Infinitely many βoutput = input β 7β βoutput = input + 4β βoutput = input Γ· 4β βoutput = (input β 1) Γ· 2β βoutput = (input β 4) Γ 2β i βoutput = 2 Γ input β 3β ii βoutput = 4 Γ input + 1β iii βoutput = 5 Γ input β 1β iv βoutput = input Γ· 6 + 2β v βoutput = 10 Γ input + 3β vi βoutput = 4 Γ input β 4β b Answers may vary.
10 a b 11 a b c d e 12 a
2K
βoutput = input + 1β βoutput = 4 Γ inputβ βoutput = input + 11β βoutput = input Γ· 6β βoutput = 2 Γ inputβ
b yβ β-axis e yβ -β coordinate
c origin f βx, y, x, yβ
Now you try Example 23
input
β1β
β2β
β3β
β7β
β10β
β15β
β50β
β81β
output
β2β
β4β
β6β
β14β
β20β
β30β
β100β
β162β
y
5
7 a βoutput = 3 Γ inputβ
4
input
β4β
β10β
β13β
β6β
β8β
β3β
β5β
β11β
β2β
3
output
β12β
β30β
β39β
β18β
β24β
β9β
β15β
β33β
6β β
2
A
B
1
b βoutput = input β 9β
O input
β12β
β93β
β14β
β17β
β21β
β10β
β43β
β9β
β209β
output
β3β
β84β
β5β
β8β
β12β
β1β
β34β
β0β
β200β
C
1
2
3
4
5
x
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
b
y
output (y)
β1β
β5β
β2β
β6β
β3β
β7β
5 4 3 2 1
output
β4β
x
U N SA C O M R PL R E EC PA T E G D ES
β0β
Answers
Example 24 a input (x)
773
O
b
1 2 3 4 5 input
y
c Yes 5 a
7 6 5 4 3 2 1
x
O
β2β
β1β
β3β
β2β
β4β
3
5 4 3 2 1
aβd y
O
1 2 3 4 5 input
A
4
2
C
input (x)
output (y)
β0β
β0β
β1β
β2β
β2β
β4β
β3β
β6β
B
D
O
1
2
3
x
4
aβh y
b
y
F C
D
H
A
B
E
G
1 2 3 4 5
output
5 4 3 2 1
x
6 5 4 3 2 1
3 a A(1, 4), B(2, 1), C(5, 3), D(2, 6), E(4, 0), F(6, 5), G(0, 3), H(4, 4)
b M(1, 2), N(3, 2), P(5, 1), Q(2, 5), R(2, 0), S(6, 6), T(0, 6), U(5, 4)
4 a
input (x)
output (y)
β0β
β2β
β1β
β3β
β2β
β4β
β3β
β5β
x
O
c Yes 7 a
1 2 3 4 5 input
x
input (x)
output (y)
β0β
β0β
β1β
β1β
β2β
β4β
β3β
β9β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
2K
c Yes 6 a
3
2
β0β
y
output
Exercise 2K
1
output (y)
β1β
b
1 2 3
c Yes
1
input (x)
774
Answers
b βPoints at (0, 0), (1, 1), (2, 4), (3, 9)β
11 a
Answers
y
y 10 9 8 7 6 5 4 3 2 1
output
U N SA C O M R PL R E EC PA T E G D ES
10 9 8 7 6 5 4 3 2 1
O
Ch2 Problems and challenges
c No 8 a
O
x
1 2 3 input
input (x)
output (y)
β1β
6β β
β1β
β4β
3β β
β2β
β8β
β3β
2β β
β3β
β12β
β6β
1β β
β4β
β16β
output
y
6 5 4 3 2 1
1 2 3 4 5 6 input
x
c No
9
y
5 4 3 2 1
O
10
1 2 3 4 5
b c 13 a b c d 14 a b c d 15 a b c 16 a b c
βPoints at (1, 4), (2, 8), (3,12), (4, 16)β Yes HELP ββ(4, 4)β,β β(5, 1)β,β β(3, 1)β,β β(3, 4)β,β β(5, 1)β,β β(5, 4)ββ KEY UNDER POT PLANT β21510032513451001154004451255143β βDβ(4, 5)ββ βDβ(4, 1)ββ βDβ(0, 0)ββ βDβ(1, 5)ββ βoutput = input + 2β βoutput = input Γ 3β βoutput = input Γ· 2β ββ(3, 6)ββor any with βx = 3β Any point on a vertical line with βx = 3β ββ(3, 5)ββ
17 a M β β(3, 2)ββ b ββ(1, 2)ββ c Find the average of the βxβ-values, then the average of the βyβ-values. d ββ(4, 2)ββ e ββ(2.5, 3.5)ββ f ββ(β 0.5, β 0.5)ββ g (ββ 5, 3)ββ
x
y
8 7 6 5 4 3 2 1
O
b ββ(4, 0)β,β β(4, 3)β,β β(6, 3)β,β β(6, 0)ββ, but answers may vary. c Answers may vary. d Answers may vary. 12 a length (x) perimeter (y)
β2β
b βPoints at (1, 6), (2, 3), (3, 2), (6, 1)β
O
x
1 2 3 4 5 6 7 8 9 10
Problems and challenges
x 1 2 3 4 5 6 7 8
1 1β 01β 2 a β28ββββββββ(ββ1, 2, 4, 7, 14β)ββββ b β1, 2, 4, 8, 16, 31, 62, 124, 248β 3 a β18βtulips per bunch b β7βred, β6βpink and β8βyellow bunches 4 β14 + 18 = 32β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
5 5ββ 2ββ ββ =β β4ββ2ββ + 9,β β6ββ2ββ =β β5ββ2ββ + 11β 6 Answers may vary. Check that the answer is the whole number that you are looking for. 7 β602β 8 β1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35β
775
Chapter review 1 a
Answers
Short-answer questions 1, 2, 3, 4 , 5, 6 , 8 , 10 , 12 , 15 ,
20 , 24 , 30 , 40 , 60 , 120 b For example, β1200, 1440, 1800β 2 a 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96
U N SA C O M R PL R E EC PA T E G D ES
Chapter checklist with success criteria
output
13
7
28
37
1
20 βoutput = input Γ 7β 21 y
A
B
C
O 1 2 3 4 5
22
input (x)
output (y)
0
1
1
2
2
3
3
4
x
y
output
5 4 3 2 1
4 3 2 1
O
1 2 3 input
x
7 , 14, 21 , 28, 35 , 42, 49 , 56, 63 , 70, 77 , 84
b 7β , 13β
3 a i β5β ii β2β iii 2β 4β b i β65β ii β18β iii 8β 8β 4 a Composite β21, 30, 16β; prime β11, 7, 3, 2β b Only one, β13β 5 a β2, 5, 7, 11β b 3β 0, 42, 70β 6 a ββ68ββ ββ b 5ββ 4ββ ββ Γβ β2ββ5ββ 7 a ββ25ββ ββ b 2ββ 3ββ ββ Γβ β5ββ2ββ c ββ32ββ ββ Γβ β5ββ2ββ 8 a β10β b 3β β c β4β 9 a β16β b 7β β c β397β d 1β 31β 10 a No b Yes c No 11 a Divisible Divisible Divisible Number by β2β by β3β by β4β
Divisible by β5β
8β 4βββ539βββ424β
β
β
β
β
Number
Divisible by β6β
Divisible by β8β
Divisible by β9β
Divisible by β10β
β84βββ539βββ424β
β
β
β
β
Explanation: β84βββ539βββ424βis an even number, therefore is divisible by β2β. β84βββ539βββ424βhas a digit sum of β39β, therefore is divisible by β3β, but not by β9β. β84βββ539βββ424βis divisible by β2βand β3β, therefore is divisible by β6β. The last two digits are β24β, which is divisible by β4β. The last three digits are β424β, which is divisible by β8β. The last digit is a β4β, therefore not divisible by β5βor β10β. b Answers may vary. c Any six-digit number ending in β0βwith sum of digits which is divisible by β9β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Ch2 Chapter checklist with success criteria
β1, 2, 4, 5, 8, 10, 20, 40β β11, 22, 33, 44, 55, 66β β12 Γ 11β β12β β30β β2, 3, 4βand β6β β17βhas exactly two factors (β1βand β17β), but β35βhas more than two factors (β1, 5, 7, 35β) 8 β2, 3, 5β 9 ββ22ββ ββ Γβ β34ββ ββ 10 β13β 11 β60 =β β2ββ2ββ Γ 3 Γ 5β 12 β36β 13 β40β 14 β6β and β7β 15 _ ββ 1 ββ 2 16 β54, 66, 78β 17 Start with β5βsticks, add three sticks each time. 18 number of sticks = β3 Γβnumber of triangles; β60βsticks required for β20β triangles 19 input 4 2 9 12 0 1 2 3 4 5 6 7
Answers
12 a i β5β ii β50β iv β18β v β14β b i 4 and 5 ii 8 and 9 13 a β36, 39, 42β b β43, 35, 27β c β112, 70, 28β 14 a β280, 560, 1120β b β900, 300, 100β c β1000, 2500, 6250β 15 a β168, 51, 336, 46, 672, 41β b β212, 307, 421, 554, 706, 877β 16 a
iii β13β vi β20β iii 7 and 8
Extended-response questions 1 a i βWidthβββββ33ββ ββββcm,β βββlengthβββββ53ββ ββββcmβ ii βArea =β β3ββ3ββ Γβ β5ββ3βββββcmββ2β, perimeter = 2 Γβ β33ββ ββ + 2 Γβ β5ββ3ββββcmβ b i ββ2ββ6ββ ii βArea =β β16ββ3β or ββ46ββ β or ββ212 ββ β, perimeter =β β16ββ2β or ββ44ββ β or ββ28ββ ββ 3 4 c ββ5ββ ββ Γβ β7ββ ββ d 1ββ 6ββ2ββ or ββ44ββ ββ or ββ28ββ ββ 2 a i β1β ii β2β b
U N SA C O M R PL R E EC PA T E G D ES
Answers
776
c
Ch2 Chapter review
b
No. of rhombuses
β1β
β2β
β3β
β4β
β5β
No. of sticks required
β4β
β7β
1β 0β
β13β
β16β
c Four sticks are required to start the pattern, and an additional three sticks are required to make each next term in the pattern. 17 a β5β b β32β c β3g + 2β d β21β windows 18 a input β3β β5β β7β β12β β20β output
b
19 a b c d 20 a b c 21 a b c d e f
β8β
β10β
β12β
β17β
β25β
input
β4β
β2β
β9β
β12β
β0β
output
β15β
β11β
β25β
β31β
β7β
βoutput = input + 9β βoutput = 12 Γ input + 8β βoutput = 2 Γ input + 1β βoutput = 10 β inputβ βA(1, 3), B(3, 4), D(2, 1)β βC(4, 2)β βE(5, 0)β β36, 49, 64β; square the counting numbers β125, 216, 343β; cube of the counting numbers (ββ 4, 6)β,β β(5, 7)β,β β(6, 8)ββ; each βyβ-value is β2βmore than the βxβ-value β3_ 1, 31, 30β; days of the month in a leap year _ _ βββ5 β,β ββ6 β,β ββ7 ββ; square root of the counting numbers β7, 64, 8β; two sequences: β1, 2, 3, 4, β¦β and β1, 2, 4, 8, 16, β¦β
Multiple-choice questions 1 βBβ 7 βDβ
2 Eβ β 8 A β β
3 C β β 9 C β β
4 βEβ 10 βEβ
5 βBβ 11 βBβ
6 B β β 12 βDβ
Rows
β1β
β2β
Desks per row
β4β
β4β
β4β
β4β
β4β
β4β
Total no. of desks
β4β
β8β
β12β
β16β
β20β
β24β
Total no. of rhombuses
β0β
β3β
β6β
β9β
β12β
β15β
β3β
β4β
β5β
β6β
d Numberβββofβββrhombuses = 3 Γ (numberβββofβββrows β 1) ββ β―β―β― ββ― ββ― βββ β = 3(n β 1)
e
Rows
β1β
Desks per row
β5β
β5β
β5β
β5β
β5β
β5β
β5β
β5β
Total desks
β5β
β10β
β15β
β20β
β25β
β30β
β35β
β40β
Rhombuses
β0β
β4β
β8β
β12β
β16β
β20β
β24β
β28β
Rows
β1β
β2β
β3β
β4β
β5β
β6β
β7β
β8β
Desks per row
β6β
β6β
β6β
β6β
β6β
β6β
β6β
β6β
Total desks
β6β
β12β
β18β
β24β
β30β
β36β
β42β
β48β
Rhombuses
β0β
β5β
β10β
β15β
β20β
β25β
β30β
β35β
Rows
β1β
β2β
β3β
β4β
β5β
β6β
β7β
β8β
Desks per row
β7β
β7β
β7β
β7β
β7β
β7β
β7β
β7β
Total desks
β7β
β14β
β21β
β28β
β35β
β42β
β49β
β56β
Rhombuses
β0β
β6β
β12β
β18β
β24β
β30β
β36β
β42β
Rows
β1β
β2β
β3β
β4β
β5β
β6β
β7β
β8β
Desks per row
β8β
β8β
β8β
β8β
β8β
β8β
β8β
β8β
Total desks
β8β
β16β
β24β
β32β
β40β
β48β
β56β
β64β
Rhombuses
β0β
β7β
β14β
β21β
β28β
β35β
β42β
β49β
β2β
β3β
β4β
β5β
β6β
β7β
β8β
f No.βββofβββrhombuses = (no.βββofβββdesks β 1) Γ (no.βββofβββrows β 1) ββ ββ― ββ― βββ β―β―β―β― β = (d β 1) (n β 1) g β10βββ201β desks
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
b
3A
c
Building understanding 1 a β9β 2 Proper: βb, e, fβ Improper: βa, cβ Whole numbers: βdβ 3 βc, e, fβ
d
b β7β
1 3
2 3
1
4 3
5 3
2
0
1 6
2 6
3 6
4 6
5 6
1
0
1 4
2 4
3 4
1
5 4
6 4
2
7 4
9 4
10 4
11 4
3
U N SA C O M R PL R E EC PA T E G D ES
e
0
Answers
Chapter 3
777
f
0
1 5
2 5
3 5
4 5
0
1 4
2 4
3 4
1
1
6 5
5 4
7 5
6 4
8 5
7 4
2
9 5
2
9 4
11 5
10 4
12 5
11 4
3
4 Answers may vary. a
Now you try Example 1 a β10β c i β10β
b β7β
iii _ ββ 7 ββ 10
ii 7β β
Example 2 2 3
7 3
0
1
2
3
Example 3 (Answers may vary.)
b
3A
Exercise 3A
c
1 a β5β c i
β5β
iii _ ββ 2 ββ 5
ii 2β β
2 A a 4β β
b β1β
c i β4β
b β7β
c i β8β
iii _ ββ 7 ββ 8
ii β7β
C a 3β β
b β2β
c i β3β
iii _ ββ 2 ββ 3
ii β2β
D a 1β 2β
b β5β
c i β12β 0
iii _ ββ 1 ββ 4
ii β1β
B a 8β β
3 a
b β2β
1 7
iii _ ββ 5 ββ 12
ii β5β 2 7
3 7
4 7
5 7
6 7
1
8 ,ββ β_ 9 ββ 5 a _ ββ 7 ,ββ β_ 5 5 5 6 ,ββ β_ 7 ββ c _ ββ 5 ,ββ β_ 3 3 3 3 ,ββ β_ 1 ββ e _ ββ 5 ,ββ β_ 2 2 2 3 ;β 3β_ 10 ββ 1 ,ββ β_ 7 β; 5,β β_ 6 a 1β _ β 1 ,ββ β_ 2 2 2 2 2 3 16 4 11 2 _ _ _ _ _ c ββ ;β 1β ,ββ β ;β 2β β,β β ββ 7 7 7 7 7
9 ,ββ β_ 10 ,ββ β_ 11 ββ b ββ_ 8 8 8 6 ,ββ β_ 5 ββ 7 ,ββ β_ d ββ_ 7 7 7 23 β,β β_ 28 ,ββ β_ 33 ββ f ββ_ 4 4 4 4 β; 2β_ 1 β,β β_ 11 ββ b _ ββ 1 ;ββ β_ 5 5 5 5 10 ;β 3β_ 11 ;β 4β_ 14 ββ 2 β,β β_ 2 ,ββ β_ d 3β _ β 1 ,ββ β_ 3 3 3 3 3 3
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
7 Division 6 ββ 8 a ββ_ 11 d _ ββ 5 ββ 6 _ 9 a ββ 12 ββ 43 144 ββ f ββ_ 475 10 a
b _ ββ 13 ββ 15 g _ ββ 7 ββ 20
c _ ββ 7 ββ 12 f _ ββ 5 ββ 9 1 _ d ββ ββ 12 i _ ββ 3 ββ 7
1 ββ 4 ββ =β β_ b ββ_ 8 2 3 ββ =β β_ 1 ββ e ββ_ 12 4 c _ ββ 11 ββ 12 h _ ββ 1 ββ 4
e _ ββ 2 ββ 11
5 a _ ββ 3 ββ 4 _ e ββ 2 ββ 5 i _ ββ 7 ββ 9
2 ββ b ββ_ 3 1 ββ f ββ_ 11 3 ββ j ββ_ 8 b β=β g β=β
6 a ββ β f = β β
1 ββ c ββ_ 3 1 ββ g ββ_ 7 5 ββ k ββ_ 6 c β=β h = β β
d ββ β i β=β
4 ββ d ββ_ 11 1 ββ h ββ_ 3 5 ββ = 5β l ββ_ 1 e β=β
3B
U N SA C O M R PL R E EC PA T E G D ES
Answers
778
Four quarters makes up one whole, so _ ββ4 ββ = 1β. A similar 4 diagram can be used for sixths. (This could also be shown on a number line.) 10 ,ββ β_ 40 ββ(other answers possible). _ b ββ 6 β,β β_ 3 5 20 11 βCβ 1 ββ b β50βββmLβ 12 a ββ_ 5 c β40βββmLβ d _ ββ 4 ββ 25 _ e β32βββmLβ f ββ 16 ββ 125 g 9β 0βββmLβ h _ ββ 9 ββ 25 i β122βββmLβ j _ ββ 61 ββ 125 k Approximately, yes they will.
1 ββ 3 ββ 2 ββ 7 a _ ββ 7 ββ =β β_ c _ ββ 4 ββ =β β_ d b _ ββ 12 ββ =β β_ 14 2 42 21 16 4 8 a β35β b 4β 5β c β30β d 2 _ 9 ββ ββ 9 10 Justin β4β, Joanna β3β, Jack β5β 11 a β75β c β150β b _ ββ 1 ββ 3 6 30 36 _ _ _ 12 a ββ β,β β β,β β ββ(other answers possible) 10 50 60
1 ββ _ ββ 7 ββ =β β_
63 β24β
9
b 3β βand β5βare both multiplied by the same value, so the new sum of numerator and denominator will be β8 Γβthat value, which is even (because β8βis even). 13 Answers may vary. a
b β12βquavers (eighth notes) to a bar c
3B
Building understanding 3 β,β β_ 5 ββ 2 ,ββ β_ 11 β,β β_ 1 ββ_ 6 4 22 10 80 ββ 16 ,ββ β_ 2 β,β β_ 2 _ ββ 4 β,β β_ 10 40 5 200 6 ,ββ β_ 8 ββ 3 _ ββ 4 β,β β_ 6 9 12 4 a _ ββ 1 ββ 5
d
Exercise 3B 6 ββ 1 a ββ_ 8 2 a _ ββ 1 ββ 2 _ 3 a ββ 6 ββ 15 4 a β9β e β8β i β35β
British name / American name
Rest
Breve / Double whole note
Semibreve / Whole note
1 ββ b ββ_ 6
2 ββ c ββ_ 3
Now you try
Example 4 3 ββ =β β_ 6 ββ a ββ_ 5 10 Example 5 4 ββ a ββ_ 5 Example 6 a β=β
Note
Minim /Half note
20 ββ c _ ββ 5 ββ =β β_ 6 24
9 ββ b _ ββ 18 ββ =β β_ 24 12
1 ββ b ββ_ 4
4 ββ c ββ_ 10 3 ββ c ββ_ 4 20 ββ c ββ_ 40 c β33β g β12β k β105β
Or
Quaver / Eighth note For notes of this length and shorter, the note has the same number of flags (or hooks) as the rest has branches.
b ββ β
3 ββ b ββ_ 6 3 ββ b ββ_ 5 6 ββ b ββ_ 8 b β50β f β2β j 2β 00β
Crotchet / Quarter note
12 ββ d ββ_ 20 4 ββ d ββ_ 6 30 ββ d ββ_ 40 d 5β 6β h β39β l β2β
Semiquaver / Sixteenth note
e Each dot increases the length of the note by another β50%β. e.g. β©β’ = _ ββ― 3β―βnoteββββ(value = 3βββbeats)β 4 ββ βββ β―β―β― β ββ―β―β― ββ― β = halfβββnote + 50% β = halfβββnote + quarterβββnote
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
3C b 1β 1β and β12β b β360β c β60β
c β36β and β37β d 2β 4β
9 a 1β 1β d 3β y β 1β
b 9β β e βmnβ
c 9β β
c β6β and β7β 1 ββ d β1β_ 8 c β4x β 1β
3 ,β 2β_ 5 ββ 10 2β _ β 1 ,β 2β_ 8 7 6 5 ββ 1 β, 3β_ 1 ββ 11 a i β1_ β 2 ,β 2β_ ii β1β_ 6 3 3 2 11 ββ 2 β, 4β_ 2 ββ ii β1β_ b i β2_ β 3 ,β 3β_ 12 4 4 3 3 β, 5β_ 3 ββ 19 ββ ii β1β_ c i β3_ β 4 ,β 4β_ 5 5 4 20 d β1_ β 29 ββ 30 e A mixed numeral with a whole number part equal to 1. Denominator equal to product of two largest numbers, numerator is one less than denominator. f, g Answers may vary.
U N SA C O M R PL R E EC PA T E G D ES
1 a β2β and β3β 2 a β24β 3 a
b β7β and β8β
b 1β _ β 7 ββ 8 8 Answers may vary.
Answers
Building understanding
6 a β5β and β6β 7 a 1β 5β
779
b c
d
4 a β8β b β12β e 1β 7β f β7β 1 ββ 5 a β7_ β 1 β, 10β_ 2 2 1 β, 4β_ 2 ββ b 1β _ β 2 β, 2β_ 3 3 3 2 β, 25β_ 4 β, 26β_ 1 ββ 1 β, 24β_ c 2β 3β_ 5 5 5 5
c 2β 8β g β10β
d 4β 4β h 2β 4β
3D
Building understanding
Now you try
1 a _ ββ 5 ββ 7
Example 7
6 Example 8 β3_ β 2 ββ 7 Example 9 β4_ β 1 ββ 2 5 ββ a ββ_
2 a e i
3 a e i
4 a e
3 _ ββ 11 ββ 5 _ ββ 14 ββ 3 _ ββ 23 ββ 12 β1_ β 2 ββ 5 _ β1β 5 ββ 7 β3_ β 1 ββ 12 β2_ β 1 ββ 2 β1_ β 1 ββ 8
5 a
7 ββ d ββ_
b
c
d
b f j
b f
5 10 ββ ββ_ 3 42 ββ ββ_ 5 53 ββ ββ_ 12 2 ββ β1β_ 3 2 ββ β6β_ 3 3 ββ β9β_ 10 4 ββ β2β_ 5 1 ββ β3β_ 3
5 3
1
3 4
0
1 4 5
0
7 ββ c ββ_
j
0
c
11 ββ b ββ_
f
2 3
b
b f b f
b β8β
β20β β12β β20β β60β c β4β
8 ββ d ββ_ 5 d β12β h β30β d β10β h β12β e β15β f 1β 5β
Now you try
Exercise 3C 1
β10β β30β β15β β24β β6β
9 ββ c ββ_ 11 c β6β g 2β 4β c 2β 1β g 1β 2β d 6β β
1
2
g k
c
g k
c
g
2 29 ββ ββ_ 7 55 ββ ββ_ 9 115 ββ ββ_ 20 2 ββ β3β_ 3 3 ββ β4β_ 8 31 ββ β2β_ 100 1 ββ β1β_ 3 2 ββ β2β_ 3
1
33
2
3
1 24
12 1 32 4
2
3
10 5
14 1 35 5
2
3
4
4 19 5
4
h l
d h l
d h
4 5 ββ ββ_ 2 42 ββ ββ_ 8 643 ββ ββ_ 10 2 ββ β2β_ 7 6 ββ β6β_ 7 3 ββ β12β_ 11 1 ββ β1β_ 3 2 ββ β2β_ 5
Example 10 a > β β
b β<β
c = β β
d β<β
Example 11 3 ββ 1 ,ββ β_ a _ ββ 2 ,ββ β_ 6 2 4
5 β, 1β_ 9 β, 2β_ 23 ββ 7 β,β β_ 1 ,ββ β_ b _ ββ 3 ,ββ β_ 4 4 8 4 2 8
Exercise 3D 1 2 3 4
a a a a e i
β β > β>β β>β β>β β<β > β β
b b b b f j
β β < β<β β<β β=β β>β β>β
8 ββ 7 ,ββ β_ 5 a _ ββ 3 ,ββ β_ 5 5 5 3 ,ββ β_ 4 ββ c _ ββ 2 ,ββ β_ 5 4 5 5 ,ββ β_ 11 β, 3β_ 1 ββ 6 a 2β _ β 1 ,ββ β_ 4 2 4 3 3 β, 2β_ 9 β, 2β_ 1 β, 2β_ 7 ββ c _ ββ 11 ,ββ β_ 5 4 2 5 10 3 ββ 3 ,ββ β_ 7 β,β β_ 1 ,ββ β_ 7 a _ ββ 4 ,ββ β_ 5 10 5 2 10 5 β,β β_ 2 ,ββ β_ 1 β,β β_ 1 ββ c _ ββ 5 ,ββ β_ 6 3 2 12 3
c c c c g k
> β β > β β > β β < β β < β β β<β
d d d d h l
β β < β=β β<β β>β β=β β>β
5 ββ 2 ,ββ β_ 1 ,ββ β_ b ββ_ 9 3 9 3 ,ββ β_ 5 ββ 2 ,ββ β_ d ββ_ 5 3 6 7 ,ββ β_ 11 β, 1β_ 7 ββ b _ ββ 5 ,ββ β_ 3 4 6 8 10 ,β 4β_ 15 ββ 2 ,ββ β_ 4 ,β 4β_ d 4β _ β 1 ,β 4β_ 6 27 9 3 3 5 ,ββ β_ 3 ,ββ β_ 1 ,ββ β_ 1 ββ b _ ββ 3 ,ββ β_ 4 8 2 8 4
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3C
2 a e 3 a e 4 a
17 ββ ββ_
7 ββ b ββ_ 3
Answers 3 β,β β_ 3 ,ββ β_ 3 β,β β_ 3 ββ 8 a ββ_ 5 6 7 8 3 ,β 7β_ 2 β, 8β_ 1 β, 5β_ 4 ββ c β10β_ 3 5 11 9 1 β,β β_ 1 ,ββ β_ 1 ββ 9 ββ_ 4 8 11 10 Andrea, Rob, Dean, David 6 ββ 14 ββ b _ ββ 11 β,β β_ 11 a _ ββ 5 β,β β_ 4 4 9 9
1 β,β β_ 1 β,β β_ 1 ββ b _ ββ 1 β,β β_ 10 15 50 100 1 β, 2β_ 1 ,β 2β_ 1 ββ d β2_ β 1 ,β 2β_ 3 5 6 9
3 ββ c _ ββ 5 ,ββ β_ 6 6
11 ββ d _ ββ 10 ,ββ β_ 14 14
9 ββ 5 ββ 7 ββ 1 ββ d β13β_ b β7β_ c β12β_ 6 a 4β β_ 6 12 30 28 5 ββ 1 ββ 21 ββ 1 ββ e β15β_ g β25β_ f β19β_ h β15β_ 9 24 10 44 5 ββ 3 ββ 1 ββ 7 a _ ββ 1 ββ b ββ_ d β2β_ c β1β_ 9 2 20 8 7 ββ 29 ββ h β3β g β7β_ e β1_ β 5 ββ f β1β_ 10 6 44 14 ββ 1 ββ 8 a ββ_ b ββ_ 15 15 9 β8_ β 3 ββββkmβ 20 1 ββ, β250β pieces 3 ββ b β400β c ββ_ 10 a ββ_ 4 4 13 ββ c 3β 9β 11 a _ ββ 47 ββ b ββ_ 60 60 ββ― 36β―β)ββ, Vesna ( ββ _ ββ― 37β―β)ββ, Juliet ( ββ _ ββ― 38β―β)ββ, Mikhail ( ββ _ ββ― 39β―β)ββ 12 Jim ββ(_ 60 60 60 60
U N SA C O M R PL R E EC PA T E G D ES
Answers
780
12 Answers may vary. 5 ββ 13 ββ 1 ββ c ββ_ b ββ_ a ββ_ 3 21 20 1 ββ 29 ββ e β2β_ d _ ββ 3 ββ f β8β_ 4 4 40 8 ,ββwhich is in ascending order. 7 ,ββ 1 can be written _ 7 ,ββ β_ ββ6 β,β β_ 13 a _ ββ 3 β,β β_ 4 8 8 8 8 b _ ββ 15 ββ 16 c If you keep doubling the numerator and denominator, then add β1βto the numerator you will generate a new βsecond 15 β,β β_ 31 β,β β_ 63 β,ββββ¦β largestβ value each time, βe.g.ββββ_ 16 32 64 14 a β7, 8β b 1β 6β 3 1 ββ 1 ββ 1 _ _ iii ββ_ iv ββ_ 15 a i ββ ββ ii ββ ββ 4 2 3 8 b Answers may vary.
3E
3E Building understanding 1 a b c d 2 a 3 a
Denominator Denominator, numerators Denominators, lowest common denominator Check, simplified β15β b β6β c β24β β b β c β
d β48β d β
Now you try Example 12 5 ββ a ββ_ 7 Example 13 a _ ββ 8 ββ 15 Example 14 a β4_ β 1 ββ 5
2 ββ b 1β β_ 9
1 a Denominator c Simplify 2 a β12β b β18β 3 a β b β
9 ββ b 1β β_ 20
7 ββ c ββ_ 15 4 ββ c β1β_ 5 2 ββ c ββ_ 3 13 ββ g ββ_ 21 13 ββ c β1β_ 33 3 ββ c β12β_ 4 3 ββ g 1β 9β_ 11
b Multiply d Multiply c β24β c β
d β63β d β
Now you try
17 ββ b 3β β_ 20
7 ββ b ββ_ 12 3 ββ b β1β_ 10 14 ββ b ββ_ 15 19 ββ f ββ_ 20 9 ββ b β1β_ 28 3 ββ b β7β_ 7 1 ββ f 2β 1β_ 6
6 ββ +β β_ 5 ββ +β β_ 5 ββ 4 ββ = 9β_ 14 a Maximum: ββ_ 1 2 3 6 3 ββ = 1β_ 3 ββ 2 ββ +β β_ 1 ββ +β β_ Minimum: ββ_ 4 5 6 20 6 ββ +β β_ 5 ββ = 14β_ 3 ββ 7 ββ +β β_ b Maximum: _ ββ 8 ββ +β β_ 1 2 3 4 4 3 ββ +β β_ 97 ββ 2 ββ +β β_ 4 ββ = 1β_ Minimum: _ ββ 1 ββ +β β_ 5 6 7 8 210 c Answers may vary. d Maximum: Largest numbers as numerators, smallest numbers as denominators; combine largest numerator available with smallest denominator available to produce the fractions. Minimum: Smallest numbers as numerators, largest numbers as denominators; combine smallest numerator available with smallest denominator available to produce the fractions.
3F Building understanding
Exercise 3E 7 ββ 1 a ββ_ 9 2 a β1_ β 2 ββ 7 3 a _ ββ 3 ββ 4 _ e ββ 13 ββ 20 4 a β1_ β 13 ββ 30 5 a β3_ β 4 ββ 5 1 ββ e β10β_ 3
13 a 5β , 2β b 2β , 4, 8βor β2, 3, 24β c β5, 1βor β30, 3β
5 ββ d ββ_ 9
4 ββ d β1β_ 19 7 ββ d ββ_ 12 23 ββ h ββ_ 40 5 ββ d β1β_ 12 5 ββ d β5β_ 9 2 ββ h β12β_ 5
Example 15 a _ ββ 5 ββ 11 Example 16 a β1_ β 5 ββ 12
3 ββ b ββ_ 20
1 ββ c ββ_ 6
13 ββ b 1β β_ 20
Exercise 3F 1 a _ ββ 2 ββ 7 _ 2 a ββ 4 ββ 9 3 a _ ββ 5 ββ 12 4 a _ ββ 2 ββ 3 e _ ββ 1 ββ 2
3 ββ b ββ_ 11 3 ββ b ββ_ 19 1 ββ b ββ_ 10 2 ββ b ββ_ 5 1 ββ f ββ_ 2
7 ββ c ββ_ 18 c β0β
13 ββ c ββ_ 30 1 ββ c ββ_ 2 1 ββ g ββ_ 6
1 ββ d ββ_ 3 8 ββ d ββ_ 23 9 ββ d ββ_ 28 1 ββ d ββ_ 4 1 ββ h ββ_ 4
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers 1 c 1_ 7 7 g 4_ 18 2 c 4_ 3 37 g 9_ 44
2 d 3_ 9 1 h 7_ 20 8 d 4_ 9 29 h 2_ 60
Now you try Example 17 2 a _ 35 Example 18
5 c _ 12
3 b _ 4 1 b 13_ 3
a 14
U N SA C O M R PL R E EC PA T E G D ES
3 b 8_ 7 5 f 3_ 28 3 b 4_ 5 37 f _ 45
Answers
3 5 a 1_ 5 5 e 2_ 12 2 6 a 2_ 3 2 e 4_ 3 11 L 7 _ 20 3 8 _ 5 1 9 _ 6 3 , $7.75 10 $7_ 4
781
Example 19 19 a 1_ 21 Example 20 9 a 9_ 10
14 a i
b 4 years
c 10 years
8 : βConverting to an improper fractionβ is quick and _
3G
Building understanding
1 a 0, 1 b 1 c Whole number, proper fraction 2 You get a smaller answer because you are multiplying by a number that is less than 1. 3 a
c
1 1 a _ 6 3 e _ 20 1 2 a _ 7 2 3 a _ 5 6 e _ 35 4 a 6 e 12
1 b _ 15 2 f _ 21 1 b _ 3 5 b _ 22 3 f _ 10 b 9 f 4
10 c _ 21 5 g _ 24 2 c _ 5 6 c _ 11 2 g _ 7 c 16 g 80
5 31 3 5 a 5_ b 1_ c 3_ 6 35 10 1 f 7 g 6 e 5_ 3 11 25 4 6 a 3_ c 7_ b 1_ 5 15 63 3 7 a _ 5 b 48 in Year 7, 72 in Year 8. 2L 8 11_ 3 9 7 cups of self-raising ο¬our, 3 cups of cream 10 7 games 7 11 a Γ _ 12 d
β
7 b Γ_ 12 e
β
6 d _ 35 4 h _ 45 2 d _ 7 4 d _ 11 1 h _ 6 d 15 h 33
7 d 4_ 8 1 h 3_ 3 d 24
1 c Γ_ 12 1 f Γ_ 12
3=_ 6 . The two numerators will always multiply 2Γ_ 12 D; e.g. _ 7 5 35 to give a smaller number than the two larger denominators. Hence, the product of two proper fractions will always be a proper fraction. 13 Answers may vary. 3 3Γ_ 5Γ_ 3 2Γ_ 2 a _ c _ b _ 5 2 7 6 4 5 3 2, _ 14 a _ 7 8
Progress quiz
1 a A square with 4 equal parts, 3 of which are shaded; answers may vary.
4 D
b 4 c 3
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3G
15 efficient for this question. 36 : βBorrowing a whole numberβ keeps the numbers ii 3_ 55 smaller and easier to deal with for this question. 15 Answers may vary.
b
b 12
Exercise 3G
3 , sponge _ 3 , chocolate _ 7 11 a Ice-cream _ 4 4 8 1 , chocolate _ 1 , sponge _ 1 b Ice-cream _ 4 4 8 5 3 d _ c 2_ 8 8 12 a 3, 4. Other answers possible. b 3, 1. Other answers possible. c 1, 2. Other answers possible. d 4, 3; 1, 6; or 2, 4. 1 13 a _ 4
b 21
782
Answers
β2 β1
0
1
2
3
e Proper fraction 2 1β _ β 2 ββ or _ ββ 5 ββ or β1_ β 4 ββ or _ ββ 10 ββ 3 3 6 6 6 ,ββ β_ 20 β,β β_ 10 β,β β_ 8 βββββ¦β(Answers may vary.) 4 ,ββ β_ 3 _ ββ 2 ββ =β β_ 5 10 15 50 25 20 1 ββ 7 ββ = 2β_ 1 ββ b ββ_ c ββ_ 4 a _ ββ 2 ββ 5 2 3 3 8 ββ 5 ββ_ 5 6 β3_ β 1 ββ 4 7 a β>β b β=β c β=β
d 3β β
6 a β20β
b β21β
c β100β
e β30β
f β40β
g β4β
7 a _ ββ 5 ββ 7 e β1_ β 11 ββ 16 3 _ 8 a ββ ββ 40
4 ββ b ββ_ 5
11 ββ c ββ_ 14 1 ββ g β3β_ 3 13 ββ c β1β_ 35 4 ββ g β6β_ 7
3 ββ f 1β β_ 11 b β30β
d β120β h β6_ β 2 ββ 3 3 ββ d ββ_ 4 3 ββ h ββ_ 4 1 ββ d ββ_ 3 1 ββ h β2β_ 10
U N SA C O M R PL R E EC PA T E G D ES
Answers
d
9 ββ 4 β,β β_ 1 ,ββ β_ 2 β,β β_ 8 ββ_ 9 2 3 4 6 ββ 9 a ββ_ 7 10 a _ ββ 1 ββ 12 11 a β$336β
7 ββ b ββ_ 10
3 ββ c β1β_ 20
7 ββ b ββ_ 12 b _ ββ 14 ββ 33
3 ββ c ββ_ 5
e β28β
9 _ ββ 3 ββ 4
d β<β
10 β9β
1 ββ d 5β β_ 4 3 ββ c β2β_ 10 3 ββ d ββ_ 4
12 8β β
11 _ ββ 9 ββββmetresββββ(orβββ90βββcm)ββ 10
13 2β 2βββkmβ 14 a
=
3H
6 ββ, so dividing the shaded region in half gives three So _ ββ 3 ββ =β β_ 4 8 ββ3 ββ eighths, i.e. _ 8
Building understanding
3H
1 βAβ
5 ββ Γβ β_ 5 ββ 5 ββ 2 a ββ_ b _ ββ 1 ββ Γβ β_ 3 1 11 3 5 ββ Γ·β β_ 5 ββ Γβ β_ 3 ββ 4 β,β β_ 3 a ββ_ 2 3 2 4 21 β,β β_ 47 ββ Γβ β_ 4 ββ c _ ββ 47 ββ Γ·β β_ 11 4 11 21 4 a less d more
5 ββ 1 ββ c _ ββ 7 ββ Γβ β_ d _ ββ 8 ββ Γβ β_ 10 12 3 3 16 β,β β_ 5 ββ 24 ββ Γβ β_ b β24 Γ·β β_ 5 1 16 80 β,β β_ 8 ββ Γβ β_ 7 ββ d _ ββ 8 ββ Γ·β β_ 3 7 3 80
b more e less
c more f less
Now you try Example 21 7 ββ a ββ_ 6 Example 22 a _ ββ 3 ββ 20 Example 23 a β40β
1 ββ b ββ_ 3
Building understanding 1 a β70, 70β
b β45β
5 ββ c ββ_ 3 1 ββ c ββ_ 10 2 ββ c ββ_ 7 2 ββ c ββ_ 5 3 ββ c β1β_ 5
b β60, 60β
c β40, 40β
3 ββ = 75% ,ββ_ 2 ββ = 50% ,ββ_ 4 ββ = 100%β 2 a _ ββ 1 ββ = 25% ,ββ_ 4 4 4 4 3 ββ = 60% ,ββ_ 5 ββ = 100%β 2 ββ = 40% ,ββ_ 4 ββ = 80% ,ββ_ 1 ββ = 20% ,ββ_ b ββ_ 5 5 5 5 5 3 ββ = 100%β 1 ββ % ,ββ_ 2 ββ = 66β_ 2 ββ % ,ββ_ c _ ββ 1 ββ = 33β_ 3 3 3 3 3
b 1β _ β 19 ββ 23
7 ββ b ββ_ 5 1 ββ b ββ_ 6 3 ββ b ββ_ 8 5 ββ b ββ_ 33 1 ββ b β1β_ 3
1 ββ = 6β. You can fit six halves into three wholes, so β3 Γ·β β_ 2 1 ββ = 4, 12 Γ· 4βand β12 Γβ β_ 1 ββ = 3,β β10 Γβ β_ 1 ββ and β ββ 1 ββ Γ·β β_ 15 _ ββ 1 ββ of β8β and _ 2 2 8 4 2 1 ββ and β3 Γ 2 = 6β 10 Γ· 2 = 5β, β3 Γ·β β_ 2 16 a β240βββkmβ b β160βββkmβ
3I
Exercise 3H 5 ββ 1 a ββ_ 2 2 a _ ββ 1 ββ 2 _ 3 a ββ 6 ββ 11 4 a _ ββ 3 ββ 8 5 a _ ββ 3 ββ 4
b
17 Answers may vary.
3 ββ c ββ_ 11
4 ββ b 1β β_ 5
Example 24 a β1_ β 1 ββ 15
f _ ββ 4 ββ 15
9 ββ d ββ_ 4 1 ββ d ββ_ 4 3 ββ d ββ_ 26 5 ββ d ββ_ 7 3 ββ d ββ_ 14
3 β86%β
Now you try Example 25 a _ ββ 39 ββ 100
21 ββ b ββ_ 50
1 ββ c 2β β_ 5
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
Exercise 3J
b β55%β
Example 27 1 β%β a β17β_ 2
b β225%β
1 a i
_ ββ 19 ββ
29 ββ ii ββ_ 100 9 ββ ii ββ_ 20
100 i _ ββ 7 ββ 10 11 _ ββ ββ 100 1 _ ββ ββ 4 β1_ β 1 ββ 5 _ β1β 3 ββ 4 β8%β β35%β β112%β
71 ββ b ββ_ 100 3 ββ f ββ_ 10 4 ββ b β1β_ 5 1 ββ f β1β_ 10 b β15%β f β32%β j β135%β
43 ββ c ββ_ 100 3 ββ g ββ_ 20 37 ββ c β2β_ 100 4 ββ g β3β_ 25 c 9β 7%β g β86%β k β400%β
1 β%β 5 a β12β_ 2 e β115%β
1 β%β b β33β_ 3 f β420%β
2 β%β c β26β_ 3 g β290%β
6 a _ ββ 3 ββ 4 1 _ 7 β12β β%β 2
b β75%β
1 ββ c ββ_ 4
b
2 a e
3 a e
4 a e i
57 ββ iii ββ_ 100 12 ββ iii ββ_ 25 49 ββ d ββ_ 100 22 ββ h ββ_ 25 1 ββ d 4β β_ 100 2 ββ h 8β β_ 5
d 5β 0%β h 9β 0%β l 1β 60%β
1% d β83β_ β β 3 1% h 3β 2β_ β β 2 d β25%β
8 7β 0%β 9 7β 0%β, β80%β
18 ββ b β72%β 10 a ββ_ 25 11 β4β (they are β10%β, β30%β, β70%β, β90%β) 55 ββ =β β_ 11 ββ 12 ββ_ 1000 200
17 ββ. Other answers ββ 85 ββ =β β_ 13 For example, β8_ β1 β%β, which is _ 2 1000 200 possible. 14 Answers may vary.
3J
Building understanding 1 a β10β d 1β β 2 a ii
b 1β 00β e β5β
b ii
c β2β f 4β β
c i
Now you try Example 28 a β36β
b β60β
Example 29 In Shop B the jumper is cheaper by β$2β.
d i
1% 16 3β 7β_ β β 2 17 a β$140β b $β 1.50β 18 a i β1200β ii β24%,β βββ22%,β βββ20%, 18%, 16%β iii β2%β b i β30%β ii Week β1β: β30%β, β2400β pieces; week β2β: β25%β, β2000β pieces; week β3β: β20%β, β1600β pieces; week β4β: β15%β, β1200β pieces; week β5β: β10%β, β800β pieces iii Pieces Week
Cumulative β%β
completed
Week β1β
3β 0%β
β2400β
Week β2β
5β 5%β
β4400β
Week β3β
7β 5%β
β6000β
Week β4β
9β 0%β
7β 200β
Week β5β
β100%β
8β 000β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
3J
U N SA C O M R PL R E EC PA T E G D ES
Exercise 3I
1 a β10β b β20β c 6β β d β16β 2 a 7β 0β b β36β c 1β 0β d β27β e β10β f β7β g 1β 50β h β200β i 4β β j β48β k β44β l β190β m 2β 2β n β84β o β36β p β63β 3 a β96β b β600β c 6β 6β d β100β e β15β f β72β g 7β 3β h β600β 4 β10%βof β$200 = $20β β20%βof β$120 = $24β β10%βof β$80 = $8β β50%βof β$60 = $30β β20%βof β$200 = $40β β5%βof β$500 = $25β β30%βof β$310 = $93β β10%βof β$160 = $16β β1%βof β$6000 = $60β β50%βof β$88 = $44β 5 a β$42β b β24βββmmβ c β9βββkgβ d 9β 0β tonnes e β8βββminβ f β400βββcmβ g 1β .5βββgβ h β3β hectares i β144β seconds 6 β35β 7 No (red β$45β, striped β$44β) 8 β240β 9 β12β 10 a β120β b β420β c β660β 11 a β$80 / weekβ b β$2080β c β$4160β 12 a Computer games β30βββminβ, drums β24βββminβ, outside β48βββminβ, reading β12βββminβ b 5β %βtime remaining c Yes, with β1βββminβto spare 13 8β 0βββminβ 14 a β20β b β90β c 2β 0β d β64β 15 They are the same
Answers
Example 26 a β9%β
783
Answers
3K
3L
Building understanding
Building understanding
1 a c 2 a d
βBβ False β8β True
1 a β1 : 9β
b C β β d True b C β β e False
e i
c False
c _ ββ 1 ββ 10
b β10β
β40βββmLβ
2 a β5β
9 ββ d ββ_ 10
ii β360βββmLβ b _ ββ 2 ββ 5
3 ββ c ββ_ 5
U N SA C O M R PL R E EC PA T E G D ES
Answers
784
d i
Now you try
ii 3β : 4 : 7β
Example 31 a β3 : 1β
ii β5 : 2β
b β1 : 2 : 4β
Example 32 2 ββ a i ββ_ 9 b β1 : 4β c β450βββmLβ
iii β4 : 3β
c β3 : 1β
3K
ii 3β : 7 : 4β
b f j
b f
b f j
ii β6 : 7 : 5β b 3β : 7β e β7 : 4 : 3β β1 : 4β c β7 : 9β g 9β : 7β k b 1β : 2 : 6β e β3 : 4 : 1β h β6 : 3 : 4β β1 : 4β c β4 : 5β g b 1β : 1β e β3 : 2β b 1β 0β e β3β β5 : 3β c β3 : 5β g 5β : 18β k
ii β3 : 5β
iii β4 : 3β
ii β5 : 8β
c f
β2 : 1β β4 : 3β β9 : 2β
β4 : 3β β3 : 2 : 5β
β9 : 4β β3 : 7β β15 : 16β
7 ββ ii ββ_ 9
Example 33 β45 : 15β
Exercise 3K i β5 : 3β i β3 : 7β i β8 : 5β i 7β : 5β β4 : 3β 3β : 1β β1 : 2β β4 : 3β β6 : 7β β1 : 2 : 5β 4β : 2 : 3β 5β : 3 : 2β β2 : 1β β1 : 2β β1 : 2β 1β : 1β β2β 1β 8β β3 : 4β β2 : 3β β4 : 1β
ii β$90β
Now you try
Example 30 a i β2 : 5β b i β3 : 7β
1 a b 2 a b 3 a d 4 a e i 5 a d g 6 a e 7 a d 8 a d 9 a e i
β$60β
c f
c f c f
iii β5 : 6β β1 : 4β β1 : 3 : 4 : 7β d 3β : 1β h β11 : 9β l β11 : 6β β2 : 1 : 3β β1 : 45 : 8β d h β1 : 2β β5 : 7β β21β β300β d h l
β8 : 5β β1 : 2 : 4β
Exercise 3L 1 a i
b i c i
2 a i
3 ββ ii ββ_ 5 5 ββ ii ββ_ 6 4 ββ ii ββ_ 7 3 ββ ii ββ_ 7
2 ββ ββ_ 5 _ ββ 1 ββ 6 _ ββ 3 ββ 7 _ ββ 4 ββ 7
b 2β : 1β c 4β 0βββgβ
3 a _ ββ 5 ββ 8 d _ ββ 7 ββ 9 4 β6βββkgβ, β18βββkgβ
b β3 : 1β
c 1β 0β
e β3 : 2β
f β4β
5 β15βββL,β β20βββLβ
β7 : 13β β2 : 1β β7 : 12β
5 ββ :ββ_ 6 ββ = 5 : 6β 10 a β1_ β 2 ββ : 2 =β β_ 3 3 3 b i β1 : 2β ii β8 : 3β iii β2 : 3β iv 5β 7 : 40β 11 β2 : 1β. His answer should use whole numbers only. 12 a β4βlitres to β6βlitres is the ratio β4 : 6 = 2 : 3β b i More red ii More yellow iii More yellow 13 a β1 : 4β b β3 : 5β c β8 : 5β d 4β : 5β e β3 : 8β f β4 : 3β 14 βn :ββnβββ― 2ββ = n : n Γ n = 1 : nβ(by dividing both sides by βnβ) So, βn :ββnβββ― 2ββ is not equal to β1 : 2β, except in the case where βn = 2βwhere βn :ββnβββ― 2ββ = 2 : 4 = 1 : 2β. 15 a i Orange ii Green iii Violet b i Yellow orange ii Blue violet iii Yellow green c i β1 : 3β ii β3 : 1β d 2β : 3β
6 a c e g
β$16β, β$24β β12βββkgβ, β20βββkgβ β30βββL,β β24βββLβ β56βββm,β β40βββmβ
7 a i β2βββLβ b β150βββLβ
b d f h
β$30β, β$70β β60βββkgβ, β12βββkgβ β12βββL,β β54βββLβ β80βββm,β β70βββmβ
ii 6β βββLβ c β6βββLβ
8 a β$12β 9 a $β 40, $40, $100β c β22βββL, 110βββL, 33βββLβ 10 a β175β b β24β 11 β90Β°β 12 β2β 13 No, β134 Γ· 7 = 19βremainder β1β 14 β8β 15 β4β 16 a i β1 : 4β b i β54βββmβ c β41 : 11β
iii β16βββLβ d 8β βββLβ
b b d c
β$36β β250βββg, 150βββg, 350βββgβ β140βββkg, 490βββkg, 210βββkgβ β154β d β220β
ii 3β : 1β ii 8β βββkmβ
Problems and challenges 1 2β 2.5β 2 Answers will vary, but possible answers are: a 6β , 7, 8, 9β b β4, 5, 6, 7, 8β c β2, 3, 4, 6, 7, 8β d β1, 2, 3, 4, 5, 6, 9βor β1, 2, 3, 4, 5, 7, 8β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
3
5
9 ββ βA =β β_ 10
_ ββ 4 ββ
11 ββ βB =β β_ 10 3 ββ βE =β β_ 5
7 ββ βC =β β_ 10 1 ββ ββ_ 2
3 ββ D β =β β_ 10
3 ββ > _ b ββ_ ββ 1 ββ 8 8
5 a _ ββ 2 ββ < _ ββ 4 ββ 7 7
5
c 1β _ β 2 ββ > β1_ β 3 ββ 5 3
β1β
b 2β 1β b 2β 1β
U N SA C O M R PL R E EC PA T E G D ES
1 ββ Γβ β_ 1 ββ 1 ββ ββ β_ 1 ββ 1 ββ ββ β_ 1 ββ b _ ββ 1 ββ Γ·β β_ c _ ββ 1 ββ +β β_ 4 a _ ββ 1 ββ +β β_ 6 5 3 3 4 6 6 4 5 5 Because the final β20%βreduction is off the original β$50β plus also off the β$10βincrease, the β20%βreduction will be β$12β overall. 6 β$260β
6 a _ ββ 5 ββ 7 7 a 1β 0β 8 a β10β
1 ββ < _ d 3β β_ ββ 29 ββ 9 9 b _ ββ 5 ββ 8 c β24β c β24β
Answers
_ ββ 2 ββ
785
Chapter checklist with success criteria
16 ββ 7 ββ_
b 8β 0β
c β12β
d 5β β
1 ββ e ββ_
f 7β _ β 1 ββ 2
14
b β20β
β220%β
β140%β 4β 4%β
1 ββ 19 _ ββ 3 ββ = 1β_
β18%β
2
15 a 16 a 17 a 18 a
21 4β 4%β; β37.5%β 22 6β 0β 23 4β : 3β 24 2β : 5β 25 1β 2β litres 26 1β 0kgβ and β5kgβ
ii 3β : 5β 3β : 1β β$70, $10β
b β5 : 4β b β16 : 3β
1 B β β 6 βBβ
Short answer questions
Extended-response questions
1 β,β β_ 1 ,ββ β_ 1 β,β β_ 1 β,β β_ 1 ββ 1 β,β β_ 1 β,β β_ 1 _ ββ 1 ,ββ β_ 14 10 8 6 5 4 3 2 6 ββ =β β_ 9 ββ =β β_ 15 ββ etc 12 ββ =β β_ 2 _ ββ 3 ββ =β β_ 5 10 15 20 25 Multiplying by the same number in the numerator and denominator is equivalent to multiplying by β1β, and so does not change the value of the fraction.
1 a 7β _ β 5 ββ 8 2 a 1β 26βββcmβ
b _ ββ 2 ββ 7
3 ββ b 1β β_ 4
2 ββ c β1β_ 3
c _ ββ 5 ββ 7
b i c 4β : 3β
d β5 : 4 : 3β c β23 : 5β b $β 32, $48β
Multiple-choice questions
Chapter review
3 a _ ββ 3 ββ 5 1 ββ 4 a β1β_ 2
3 ββ d ββ_ 16 1 ββ d ββ_ 5
Fraction _ ββ 9 ββ 25 β2_ β 1 ββ 5 5 _ ββ ββ 100 β1_ β 2 ββ 5 _ ββ 11 ββ 25 _ ββ 9 ββ 50
β36%β
3 ββ 8 ββ = 1β_ 18 ββ_ 5 5
4 ββ c ββ_ 11
c β4β
5β %β
20 _ ββ 9 ββ 25
15 _ ββ 2 ββ 3
12 ββ b ββ_ 7
17 _ ββ 7 ββ 10
2
6
Percentage form
16 β15β; β7β
5 3 ββ; β3_ β 1 ββ 8 β2β_ 4 3 3 ββ ββ 2 ββ >β β_ 9 _ ββ 2 ββ is bigger; _ 3 3 5 3 β,β β_ 4 ββ 10 _ ββ 2 β,β β_ 3 4 5 19 ββ = 1β_ 7 ββ 11 ββ_ 12 12 1 ββ 12 _ ββ 25 ββ = 8β_ 3 3 7 ββ 13 ββ_ 12 5 ββ 29 ββ = 2β_ 14 ββ_ 12 12
c 1β _ β 5 ββ 24 _ f 2β β 1 ββ 2 3 ββ i 1β 2β_ 5
11 a β7β
12 a _ ββ 4 ββ 3 _ 13 a ββ 1 ββ 5
4 β8β; β15β 5 _ ββ 3 ββ 5 6 Yes, multiplying numerator and denominator by β5βshows they are equivalent.
9 ,ββ β_ 5 ββ 11 β,β β_ 14 β,β β_ b ββ_ 4 6 8 3
1 ββ d β3β_ 2
2 C β β 7 D β β
3 A β β 8 D β β
1% 5 a β12β_ β β 2 2% 6 β55β_ β β 3
5 βDβ 10 βCβ
11 ββ b 5β β_ 24
17 ββ c 1β 0β_ d β6_ β 2 ββ 80 21 1 ββββcm, 51βββcmββ 1 ββββcm, 22β_ b β52β_ 2 2 b $β 24β
b 2β 64βββLβ
c β$99β
3 a β$176β 4 a β$550β
4 C β β 9 A β β
d $β 200β
b β$208β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Ch3 Chapter checklist with success criteria
3 ββ; numerator = β3β; denominator = β8β 1 ββ_ 8 2 3 1 92 0 5 5 3
9 ,β 1β_ 3 ββ 1 ,ββ β_ 9 a β2β_ 5 5 5 7 β, 5β_ 47 β, 5β_ 1 ,ββ β_ 1 ββ c β5_ β 2 ,β 5β_ 3 18 3 9 9 10 a _ ββ 1 ββ b _ ββ 5 ββ 6 2 23 _ _ e ββ 1 ββ d 5β β ββ 2 30 9 _ h 6β _ β 5 ββ g 1β β ββ 36 40
786
Answers
13 a d 14 a c
4A Building understanding 1 a c 2 a d
β4x, 3y, 24z, 7β β4β βFβ βBβ
True True False True
b True e False
c False b False d True
15 βc Γ· 2, c β 4, c + 1, 2c, 3c, 3c + 5, 4c β 2, c Γ cβ
b 7β β d zβ β
16 βThe sum of 2 and twice the value of βxβis taken from β3ββ becomes β3 β (2x + 2)β. (Answers may vary.)
U N SA C O M R PL R E EC PA T E G D ES
Answers
Chapter 4
b βAβ e C β β
c D β β f Eβ β
Now you try
Building understanding
Example 1 a β4q, 10r, s, 2tβ b β6β c The coefficient of βqβis β4β, the coefficient of βrβ is β10β, the coefficient of βsβis β1βand the coefficient of βtβ is β2β.
1 2 3 4 5
Example 2 a βp + 10β
Now you try
z 1 βzβ or _ d ββ_ ββ ββ 2 2 Example 3
a β3m + 5β c βpq β 3β
b βabβ
c β3kβ
e βb β 4β
a + b ββ or ββ_ 1 β(a + b)β b ββ_ 2 2 d β7(m β 5)β
β3x, 2yβ b β4a, 2b, cβ β5a, 3b, 2β d β2x, 5β β5β b β6β c β8β d β1β The coefficient of βaβis β2β, the coefficient of βbβis β3βand the coefficient of βcβ is β1β. b The coefficient of βaβis 4, the coefficient of βbβis 1, the coefficient of βcβis 6 and the coefficient of βdβis 2. 4 a i β2β ii β17β b i β3β ii β15β c i β3β ii β21β d i β4β ii β2β e i β2β ii β1β f i β4β ii β12β 5 a βx + 1β b 5β + kβ c β2uβ d β4yβ p q g βr β 12β h β9nβ e ββ_ββ f ββ_ββ 2 3 y _ i β10 β tβ j ββ ββ 8 6 a β2(x + 5)β b β3a + 4β c β8k β 3β d β8(k β 3)β p _ 7x _ e β6(x + y)β h β12 β xyβ g ββ ββ + 2β f ββ ββ 2 2 7 a The product of β7βand βx.β b The sum of βaβand βb.β c The sum of βxβand β4βis doubled. d βaβis tripled and subtracted from β5β. 8 a β70β b β10nβ 9 a β8xβ b βx + 3β c β8(x + 3)β 10 a β1000βββxβ b β100βββxβ c β100βββ000βββxβ $A $A $A β 20 _ _ 11 a ββ ββ b ββ n ββ c _ ββ n ββ 4 12 βOne-quarter of the sum of βaβand βbβ.β (Answers may vary.) 1 a c 2 a 3 a
a β14β β13β β15β β3β a β17β
b β1β
c β10β
d β8β
b β20β
c β72β
d β12β
Example 4 a 1β 2β
Exercise 4A
4A
4B
b 5β β
c β7β
Example 5 a 4β β
b 4β 2β
Example 6 a 3β 6β
b 9β 1β
c β6β
Exercise 4B 1 2 3 4
a a a a e i 5 a e 6 a d 7 a
b
c
d
β7β β4β β13β β19β β3β β35β β5β β5β β27β β8β
b b b b f j b f
β4β β12β β5β β9β 1β β 8β 6β β9β 2β 4β b 2β 2β e 1β 0β
β10β β14β β6β β7β β47β β8β β4β β0β
d d d d h l d h c β41β f β70β
β15β β70β β4β 2β 0β β20β β6β β45β β8β
βnβ
β1β
β2β
β3β
4β β
β5β
6β β
βn + 4β
β5β
6β β
β7β
β8β
9β β
β10β
c c c c g k c g
βxβ
β1β
2β β
β3β
4β β
β5β
β6β
β12 β xβ
β11β
β10β
9β β
β8β
7β β
β6β
βbβ
β1β
2β β
β3β
4β β
β5β
β6β
β2(β b β 1)ββ
0β β
2β β
β4β
6β β
β8β
β10β
βqβ
β1β
β2β
3β β
β4β
β5β
6β β
β10q β qβ
β9β
1β 8β
β27β
3β 6β
β45β
β54β
8 a β75β b β45β c β54β e β12β f 5β β g 3β 3β 9 β5β 10 xβ βis between β4βand β33βinclusive.
d 1β 1β h β19β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
11
βxβ
β5β
9β β
β12β
β1β
6β β
β7β
β11β
β15β
β18β
7β β
β12β
β13β
β20β
β36β
β48β
4β β
2β 4β
β28β
c Eβ β f βDβ
β6x + 5β
β4x + 5 + 2xβ
βx = 1β
β11β
β11β
xβ = 2β
β17β
β17β
xβ = 3β
β23β
β23β
xβ = 4β
β29β
β29β
15 a
βxβ
β5β
β10β
β7β
β9β
β5β
β8β
yβ β
3β β
β4β
β2β
β5β
β2β
β0β
xβ + yβ
β8β
β14β
β9β
β14β
β7β
β8β
xβ β yβ
β2β
6β β
β5β
β4β
β3β
β8β
βxyβ
β15β
β40β
β14β
β45β
β10β
β0β
b 2β β and β2, 3βand β1_ β1 ,β 6βand β1_ β1 ββ; answers may vary. 5 2
4C
Building understanding 1 a
βx = 0β
xβ = 1β
βx = 2β
xβ = 3β
β2x + 2β
β2β
4β β
β6β
β8β
(ββ x + 1)ββ Γ 2β
β2β
4β β
β6β
β8β
b equivalent 2 a
They are equivalent because they are always equal. 7 2β x + 2 + 2x, 2(2x + 1)β; answers may vary. 8 β2(w + l)β; answers may vary. 9 β9a + 4bβ; answers may vary (must have β2βterms). 10 If βx = 8β, all four expressions have different values. 11 b βabβand βbaβ c βa Γ (b + c)βand βa Γ b + a Γ cβ d aβ β (b + c)βand βa β b β cβ e βa β (b β c)βand βa β b + cβ f aβ Γ· b Γ· cβand βa Γ·β β(b Γ c)ββ 12 a β3 + 12 = 12 + 3, 17 + 12 = 12 + 17, 58 + 12 = 12 + 58β (other answers possible) b 2(8 + 1) = 2 + 2 Γ 8, 2(100 + 1) = 2 + 2 Γ 100, 2(3 + 1) β―β―β―β― ββ β ββ― ββ― ββ β= 2 + 2 Γ 3 (other answers possible) c (β 3 + 8)ββ2ββ =β β3ββ2ββ + 2 Γ 3 Γ 8 +β β8ββ2β,β β(5 + 9)ββ2β ββ β―β―β―β― ββ β =β β5ββ2ββ + 2 Γ 5 Γ 9 +β β9ββ2β,β β(7 + 7)ββ2ββ =β β7ββ2ββ + 2 Γ 7 Γ 7 +β β7ββ2β, (other answers possible) 13 a 8β + 4aβ β4 Γβ β(a + 2)ββ βx = 1β
β12β
β12β
βx = 2β
β16β
β16β
βx = 0β
βx = 1β
βx = 2β
βx = 3β
βx = 3β
β20β
β20β
5β x + 3β
β3β
8β β
β13β
β18β
βx = 4β
β24β
β24β
β6x + 3β
β3β
9β β
β15β
β21β
b No
Now you try
Example 7 β2a + 6βand β(a + 3) Γ 2βare equivalent Example 8 a β10 Γ x Γ 10 = 100xβ b βab = baβ
4D
Building understanding
Exercise 4C
1 a βNβ b βEβ 2 a β2x + 3, x + 3 + xβ c β2x + 4βand βx + 4 + xβ e β2k + 2βand β2(k + 1)β 3 a βx + x + x = 3xβ c βx + 99 = x β 1 + 100β 4 a β10(a + b) = 10a + 10bβ b aβ + 100 + b = b + 100 + aβ
b 5β (2 + a)β c β24 + 6aβ 14 2β a + a + 5b, 3a + 12b β 7bβ; answers may vary. 15 a Yes; for any value of βxβ, expressions βAβand βBβare equal, and expression βBβand βCβare equal, so expressions βAβand βCβare equal. b No; e.g. if expression βAβis β7xβ, expression βBβ is β3xβ and expression βCβ is β5x + 2xβ.
b d f b d
c βEβ β5x β 2, 3x β 2 + 2xβ β5aβ and β4a + aβ βb + bβ and β4b β 2bβ β2x Γ 2 = 4xβ β2(10 + x ) = 20 + 2xβ
1 a c 2 a b c d e f
xβ βand βyβ βkβ like like terms terms 1β 5aβ(Answers may vary.) equivalent 5β x + 4β(Answers may vary.)
b aβ , bβand βcβ d pβ βand βqβ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
4C
U N SA C O M R PL R E EC PA T E G D ES
12 a β1β and β24, 2βand β12, 3βand β8, 4βand β6β 1 ,β y = 120β. b Infinitely many answers; e.g. βx =β β_ 5 13 Because β5 Γ (a + a)β is β5 Γ 2 Γ aβ, which is β10aβ. Every multiple of β10β ends with β0β. 14 a xβ = 10, y = 45β(other answers possible) b 3β 00β c Because if βx + 2y = 100β, then multiplying both sides by β3β results in β3x + 6y = 300β.
(β a + b)(a β b) =β βaβββ― 2ββ ββ βbβββ― 2ββ βab + ba = 2abβ C β β b βAβ Fβ β e βBβ
Answers
βx + 6β β4xβ
c d 5 a d 6
787
788
Answers
Example 9 a Not like terms d Like terms
b Like terms e Like terms
c Not like terms f Not like terms
Example 10 a β15a + 4β d β7a + bβ
b β4qβ e 8β uv + 10u + 2vβ
c β6x + 18y + 3β
2 a m β Γ pβor βmpβ b βa + kβ c βt + 8β d w β β 4β m _ m + 7 _ 3 a βββ― β―ββ + 7β c _ ββ a + k ββ b ββ ββ 2 3 2 e β3(d β 12)β f 3β d β 12β d βa +β β_k ββ 3 4 a β19β b 3β β c β9β 5 a β24β b 1β 1β c β5β d β22β e 3β β 6 β3a + 4βand β4 + 3aβ 7 a βx + 10 = 10 + xβ b β2(a + b) = 2a + 2bβ 8 a βLβ b βNβ c βNβ d βLβ 9 a β8a + 5b + 5β b β7cd + 3c + 8d + 4β 10 8β x + 8y, 8(x + y)β(Answers may vary.)
U N SA C O M R PL R E EC PA T E G D ES
Answers
Now you try
Ch4 Progress quiz
Exercise 4D
1 a βLβ b βNβ c βLβ d βNβ 2 a N β β b Lβ β c βLβ d βNβ e βNβ f Lβ β g N β β h βLβ i βLβ j βLβ k βLβ l βNβ 3 a β2aβ b 5β xβ c β7bβ d β8dβ e β12uβ f 1β 2abβ g 1β 1abβ h βxyβ 4 a β3a + 5bβ b 7β a + 9bβ c xβ + 6yβ d β7a + 2β e β7 + 7bβ f β6k β 2β g β5f + 12β h β4a + 6b β 4β i β6x + 4yβ j β8a + 4b + 3β k β7h + 4β l β14x + 30yβ m β2x + 9y + 10β n β8a + 13β o β12bβ 5 a β9ab + 4β b β6xy + 5xβ c β7cd β 3d + 2cβ d β9uv + 7vβ e β11pq + 2p β qβ f 6β ab + 36β 6 a β27nβ b β31nβ c β58nβ 7 a β3a + 4β b β19β 8 a β4xβ b β7xβ c β11xβ d β3xβ 9 a β12xyβ b 6β ab + 5β c 1β 0abβ d β6xy + 3β e β5xy + 14β f β10cdeβ g β6xy + 6x + 4β h β9ab + 9β i β7xy β 2yβ 10 a β3x + 2xβ β5xβ xβ = 1β
β5β
β5β
xβ = 2β
β10β
β10β
xβ = 3β
β15β
β15β
4E
Building understanding 1 a c 2 a d
3 a _ ββ 2 ββ 3 4 a βCβ d βAβ
xβ = 1β
β7β
β7β
xβ = 2β
β10β
β10β
xβ = 3β
β13β
β13β
For example, if βx = 10, 5x + 4 β 2x = 34βbut β7x + 4 = 74β. For example, if βx = 1, 5x + 4 β 2x = 7βbut β7x β 4 = 3β. β18x β 15x = 3xβ Calculate β3 Γ 17βinstead, because that will equal β 18 Γ 17 β 15 Γ 17.β c β3 Γ 17 = 51.β 13 a β2a + a + 3b, b + 3a + 2bβ(Answers may vary.) b 1β 2β b c 12 a b
Example 11 a 5β defβ
1 a e i 2 a d g j 3 a e 4 a e i
e b 7β a, 4b, c, 9β d 9β β
b _ ββ 2 ββ 3 b Eβ β e D β β
c _ ββ 2 ββ 3 c βBβ
b 3β 0xyzβ
c β10βaβββ― 2ββ
4q b _ ββ ββ 5
Exercise 4E
5 a
Progress quiz
b True e False
b Both are β35β d Yes c False f False
Now you try
Example 12 a _ ββ 3x + 2 ββ 2x + 1
b For example, if βx = 5βand βy = 10β, then β3x + 2y = 35βbut β 5xy = 250β. 11 a β5x + 4 β 2xβ β3x + 4β
1 a β4β c 4β β
Both are β21β Both are β56β True True
i
β2abβ β2xβ β28fβ β36βββaβ β15βββaβ β8βββabcβ β8βββabcβ ββwβββ― 2ββ β7βpβββ― 2ββ _ βββ― xβ―ββ 5 _ ββ x2 ββ _ ββ 4x + 1 ββ 5 2 _ ββ ββ 5 _ βββ― xβ―ββ 2 β2aβ
b 5β xyzβ c f 5β pβ g j 1β 0abβ k b β63βββdβ e 1β 2βββabβ h 2β 8βββadfβ k β60βββdefgβ b ββaβββ― 2ββ c f β3βqβββ― 2ββ g z b βββ―_β―ββ c 2 5 ββ g f ββ_ d 2x + y k j ββ_ββ 5 5 ββ b ββ_ c 9 3x ββ g f ββ_ 4 j 3β β k
β8qrβ β8abβ β10βββbβ
c f i l
β3βdβββ― 2ββ β12βxβββ― 2ββ a β―ββ βββ―_ 12 x βββ―_ y ββ
2 + x ββ ββ_ 1+y 9a ββ ββ_ 4 _ ββ 2 ββ 3 β2yβ
d β12stuvβ h β6aβ l β28βββxzβ β8βββeβ β63βββegβ β12βββabcβ β24βββabcdβ d β2βkβββ― 2ββ h β36βrβββ― 2ββ d _ ββ b ββ 5 a ββ h βββ―_ b l _ ββ x β 5 ββ 3+b _ d ββ 2b ββ 5 3 ββ h ββ_ 4 l _ ββ y3 ββ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
6 a β3kβ 7 a β4xβ g
b 6β xβ b nβ xβ g
c 1β 2βββxyβ c β2nxβ g
β3aβ b 7β cβ c 7β bβ 1β 2βββyβ e β10βββxβ β6pβ 3β Γ 2pβalso simplifies to β6pβ, so they are equivalent. 2a
1 a β2(x + 4)β and β2x + 8β b 2β (a + 1)β and β2a + 2β c β12(x + 4)β and β12x + 48β d 8β (z + 9)β and β8z + 72β 2 a β6y + 48β b β7l + 28β c β8s + 56β d 8β + 4aβ e β7x + 35β f β18 + 3aβ 3 a β4x β 12β b β5j β 20β c β8y β 64β d 8β e β 56β e β18 β 6eβ f β80 β 10yβ 4 a β60g β 70β b β15e + 40β c β35w + 50β d 1β 0u + 25β e β56x β 14β f β27v β 12β g 7β q β 49β h β20c β 4vβ i β4u + 12β j 4β 8l + 48β k 5β k β 50β l β9o + 63β 5 a β6it β 6ivβ b β2dv + 2dmβ c β10cw β 5ctβ d 6β es + 6epβ e βdx + 9dsβ f β10ax + 15avβ g 5β jr + 35jpβ h βin + 4iwβ i β8ds β 24dtβ j 2β fu + fvβ k 1β 4kv + 35kyβ l β4em + 40eyβ 6 a β5(x + 3) = 5x + 15β b 2β (b + 6) = 2b + 12β c β3(z β 4) = 3z β 12β d 7β (10 β y) = 70 β 7yβ 7 a βt + sβ b 2β (t + s) = 2t + 2sβ 8 β3(4x + 8y)β and β2(6x + 12y)β(Answers may vary.) 9 β2l + 2wβ 10 a b x 3 a 2
U N SA C O M R PL R E EC PA T E G D ES
9 a d 10 a b 11 a
Exercise 4F
Answers
$C b ββ_ββ 5
8 a β$20β
789
6
b 1β 2βββaβ
c _ ββ 12a ββsimplifies to β4β. It has four times the area. 3a
Area is multiplied by β9β. 1β 00β βabcβ 4β 200β Multiplying a number by 100 is simpler than multiplying by 4 or 5. 13 a 1β 6βββabβ b β14βββabβ c β0β 14 a 3β 8 Γ a Γ b Γ bβ b β7βxβββ― 2ββyβββ― 3ββ d 12 a b c
c β1200βββββaβββ― 3ββbβββ― 2ββcβββ― 2ββ
d β24βββββaβββ― 3ββbβββ― 2ββcβββ― 2ββ
5
5x
15
4F
3
3a
6
4β (β x + 3)ββ
β4x + 12β
11 βa Γ 99βis the same as βa Γ (100 β 1)β, which expands to β 100a β aβ. 12 a β4β ways, including β1(10x + 20y)β b Infinitely many ways 13 a Rosemary likes Maths and Rosemary likes English. b Priscilla eats fruit and Priscilla eats vegetables. c Bailey likes the opera and Lucia likes the opera. d Frank plays video games and Igor plays video games. e Pyodir likes fruit and Pyodir likes vegetables and Astrid likes fruit and Astrid likes vegetables.
xβ = 1β
β16β
β16β
4G
βx = 2β
β20β
β20β
βx = 3β
β24β
β24β
Building understanding
βx = 4β
β28β
β28β
b equivalent
1 a β35β 2 a 2β 4βββcmβ 3 a β3xβ
Now you try
Now you try
Example 13 a Using brackets: β6(7 + x)β, without brackets: β42 + 6xβ b Using brackets: β7(r + 5)β, without brackets: β7r + 35β
Example 15 a 3β 5β
Example 14 a β4b + 28β c β10p + 25β
2b
b 9β k β 45β d 2β 7xz β 18xβ
b 2β 0β b 4β 0βββmβ b 3β 6β
b 1β 20ββββcmββ2ββ
Example 16 a 2β nβ b β20 + 5kβ c 8β 0 + 110nβfor βnβdays, so β$850β total.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
4F
β3a + 6β xβ + y + x + y = 2x + 2yβ βp + 1 + p + 1 + p + 1 + p + 1 = 4p + 4β 4β a + 2b + 4a + 2b + 4a + 2b = 12a + 6bβ β4xβ b β12β c β4x + 12β β60, 3, 63β 3β 0, 4, 210, 28, 238β β20, 1, 100, 5, 95β
4 a
ab
c βab + 3b + 5a + 15β
Building understanding 1 a b c d 2 a 3 a b c
b
790
Answers
1 2 3 4 5 6 7 8
a β28β a β13β 2β nβ a β10xβ a β180βββkmβ a β$200β a βBβ a Hours
b 4β 4βββcmβ b β18βββmβ b β15xβ b β30βββkmβ b β$680β c βDβ
c kβ xβ c β70nβ c β50 + 80xβ d βEβ e βCβ
9 β2abβ and β5baβare like terms; the pronumerals are the same, even though they are written in a different order (one βaβ and one βbβ). 10 β9a + 4b + 3β 11 β24abcβ 2a ββ 12 ββ_ 3 13 3β a + 6bβ 14 2β 2cmβ 15 3β 0 + 60nβ (dollars)
U N SA C O M R PL R E EC PA T E G D ES
Answers
Exercise 4G
b βAβ
Ch4 Problems and challenges
Total cost (ββ $)ββ
b c 9 a 10 a 11 a b c 12 a c 13 a 14 a b c d e 15 a c e
β1β
β2β
β3β
β4β
β5β
β150β
β250β
β350β
β450β
β550β
β100t + 50β β$3050β β$25β b β$(10x + 5)β c β$75β β90β cents b β$6.30β c β0.3 + 0.6tβ β33β βg = 8, b = 5β βg = 3βand βb = 2, g = 1βand βb = 14, g = 0βand βb = 20β β$(5b + 2c + 6d)β b β$(5b + 4c + 3d)β β$8β βcnβ b β3cnβ c β6cnβ β10 + 30xβ β2βββhβββ20βββminβ Because you donβt pay the booking fee twice. β$32β It will get closer to β$30β. β$(0.2 + 0.6t)β b β$(0.8 + 0.4t)β Emmaβs d β3βββminβ Answers may vary.
Problems and challenges
1 β25β 2 a βP = 10mβ b βP = 20(x + 3)β c βP = 5(w + y)β 3 Largest β18x + 70β, smallest β12x + 44β, difference β 6x + 26 = 86βββcmβ 4 β15a + 10, 5(3a + 2)β 5 a βa = 4, b = 12, c = 16, d = 8, e = 36β b βa = 6, b = 3, c = 5, d = 10, e = 15β 6 β5050β 7 Because β2(x + 5) β 12 β x + 2βsimplifies to βxβ.
Chapter checklist with success criteria 1 2 3 4 5 6 7 8
β13β β3aβ, βbβ, β13β; there is no constant term. βa + bβ β40β β17β β48β β3x + 4βand β2x + 4 + xβare equivalent. βx + x = 2xβ
Chapter review
Short-answer questions 1 a β5a, 3b, 7c, 12β 2 a uβ + 7β d a a a a a e 8 a c e
3 4 5 6 7
b 1β 2β b 3β kβ
hβ β 10β e xβ yβ β15β b β24β β15β b β8β β16β b β200β βEβ b βNβ βLβ b βNβ βNβ f Lβ β β7x + 3β β7a + 14b + 4β β1 + 7c + 4h β 3oβ
9 a β12abβ
b β6xyzβ
10 a _ ββ 3 ββ
3 ββ b ββ_
c c c c c g b d f
c 7β β c 7β +β ββ―_rβ―ββ 2 f β12 β xβ d β32β d β27β d β20β d N β β d βLβ h βLβ
β2β β4β β5β βEβ Lβ β βNβ β11pβ 3β m + 17mn + 2nβ β4u + 3v + 2uvβ
c β36βββfghβ aβ―ββ c βββ―_ 3
4x ββ d ββ_ 3z
4 D β β 9 Eβ β
5 βDβ 10 βAβ
5 2 11 a β3x + 6β b β4p β 12β c β14a + 21β d 2β 4k + 36lβ 12 β2(6b + 9c), 6(2b + 3c)β(Answers may vary.) 13 β9tβ 14 βc + dβ 15 β3xβ
d β64klmβ
Multiple-choice questions 1 B β β 6 D β β
2 A β β 7 C β β
3 C β β 8 A β β
Extended-response questions 1 a b c d e 2 a c e
i β$24.50β ii β$45.50β iii β$213.50β 3β .5 + 2.1dβ β$87.50β If βd = 40, 2.1 + 3.5d = 142.10β, not β87.50β. β6 + 1.2dβ β26β b $β 78β β4x + 2y + 10β d 1β 2x + 6y + 30β βx(x + 5) + 3yβ(Answers may vary.)
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
Chapter 5 5A Building understanding b _ β3 β 100 b β6β
c β7β
c _ β 7 β 1000 d β37β c 3β 3.04β f β99.012β
U N SA C O M R PL R E EC PA T E G D ES
2β 1 a β_ 10 2 a β5β 3 a 7β .6β d 2β 6.15β
β .C, C.A, B.C, B.A, A.C, A.Bβ C βC.BC, C.AB, B.CA, B.BB, A.CA, A.BC, A.AA, BA.CA, AB.ABβ β0.Aβ b β0.0Aβ c β0.AAβ d βA.A0Aβ i β0.1, 1.0β (β2β ways) ii β0.12, 0.21, 1.02, 1.20, 2.01, 2.10, 10.2, 12.0, 20.1, 21.0β (β10β ways) iii 0.123, 0.132, 0.213, 0.231, 0.312, 0.321, 1.023, 1.032, 1.203, 1.230, 1.302, 1.320, 2.013, 2.031, β β β―β―β―β― 2.103, 2.130, 2.301, 2.310, 3.012, 3.021, 3.102, β β―β―β―β― β 3.120, 3.201, 3.210, 10.23, 10.32, 12.03, 12.30, ββ―β―β―β― 13.02, 13.20, 20.13, 20.31, 21.03, 21.30, 23.01,β 23.10, 30.12, 30.21, 31.02, 31.20, 32.01, 32.10, β―β―β―β― β 102.3, 103.2, 120.3, 123.0, 130.2, 132.0, 201.3,ββ 203.1, 210.3, 213.0, 230.1, 231.0, 301.2, 302.1, β―β―β―β― β β 310.2, 312.0, 320.1, 321.0 (60 ways)
Answers
12 a b 13 a 14 a
791
b 1β 2.9β e β8.42β
Now you try
Example 1 3 β a β_ 100 Example 2 a β0.9β
b _ β3 β 10
b β408β ways
b β0.03β
c β4.307β
Example 3 a β8.7 > 8.4β b 1β 0.184 < 10.19β c β0.0401 > 0.00928β
Exercise 5A 3β 1 a β_ 10 2 a _ β6 β 10
Building understanding 1 a β5.8β 2 a i β1β b i β25.8β iii 2β 5.817β
ii β7β
b 6β .78β iii β4β ii β25.82β iv β26β
iv β8β
Now you try Example 4 a 5β 3.7β Example 5 a 1β 8.62β
5A
b _ β 3 β c _ β3 β 1000 100 b _ β6 β c _ β 6 β d _ β6 β 100 1000 10 e β6β f _ β6 β g _ β6 β h _ β 6 β 100 100 1000 3 a 0β .3β b β0.8β c β0.15β d β0.23β e β0.9β f 0β .02β g 0β .121β h β0.074β 4 a β6.4β b 5β .7β c β212.3β d β1.16β e β14.83β f 7β .51β g 5β .07β h β18.612β 5 a β>β b < β β c β>β d β<β e β<β f > β β g > β β h β<β i β>β j β>β k β>β l β<β 6 a False b False c True d False e True f True g False h True i True j True k True l False 7 a β0.1β b 0β .03β c β0.02β d β0.5β e β0.001β f 0β .01β g 0β .15β h β0.11β 8 a β3.05, 3.25, 3.52, 3.55β b β3.06, 3.6, 30.3, 30.6β c β1.718, 1.871, 11.87, 17.81β d β22.69, 22.96, 26.92, 29.26, 29.62β 9 a Waugh, Border, Gilchrist, Taylor, Hughes b First 10 a Day β6β b Day β4β c Days β2, 5βand β6β 11 When you insert it at the start or end you do not change the place value of any other digits, but when you insert the β0β in the middle somewhere, you do change the place value. For example, in β12.34β, the value of β4βis _ β 4 β, but in β12.304βit is 100 now _ β 4 β. Similarly, β102.34βchanges the value of the digit β1β 1000 from β10βto β100β.
5B
b 4β .2β
b 3β .1416β
c β0.140β
Exercise 5B
i β32.5β ii β57.4β i β19.7β ii β46.1β β14.8β b β7.4β c β15.6β d β0.9β β6.9β f 9β .9β g β55.6β h β8.0β β3.78β b β11.86β c β5.92β d β0.93β β123.46β f 3β 00.05β g β3.13β h β9.85β β56.29β j 7β .12β k β29.99β l β0.90β β15.9β b β7.89β c β236β d β1β β231.9β f 9β .4β g β9.40β h β34.713β β24.0β b β14.90β c β7β d β30.000β β28β b β9β c β12β d β124β β22β f 1β 18β g β3β h β11β β$13β b β$31β c β$7β d β$1567β β$120β f $β 10β g β$1β h β$36β β$51β b $β 44β c Very accurate β0 sβ b β0.4 sβ c β0.34 sβ d β52.825 sβ β$49.58β β$49.61β β$90.60β β22 cents βmore, as β52.8 Γ 1.72 = 90.82βto two decimal places. 11 β2.25, 3βdecimal places
1 a b 2 a e 3 a e i 4 a e 5 a 6 a e 7 a e 8 a 9 a 10 a b c d
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
12 Samara: Round to β2βdecimal places β= 0.45β, then round this to β1βdecimal β= 0.5β. Cassandra: Rounding to β1βdecimal place, critical digit is the second β4β, which is less than β5β, therefore rounded to β1βdecimal place β= 0.4β. Samara has a flaw of rounding an already rounded number. Cassandra is correct. 13 Depends on your calculator. 14 Depends on your software package.
5D
5C
Now you try
Building understanding
Example 8 a 5β 1 094.2β
b β390β
Example 9 a 3β 9.2807β
b β0.0812β
Example 10 a 2β 89 000β
b β0.0078β
Building understanding 1 a β100β 2 a 1β 000β 3 a i Right β2β places iii Left β3β places b Left β2β places
b 1β 00β b 1β 0β ii Right β6β places iv Left β7β places
U N SA C O M R PL R E EC PA T E G D ES
Answers
792
1 βCβ
2 βBβ
Now you try Example 6 a β36.37β
b β12.332β
Example 7 a β5.32β
b β6.884β
5C
Exercise 5C
1 a β5.75β b β4.75β c β34.177β d β7.918β e β7.41β f 7β .09β g 1β 2.74β h β10.531β 2 a β16.06β b 2β 1.33β c 7β .71β d β9.85β e β343.75β f β37.567β g β21.592β h β340.0606β 3 a β12.1β b β114.13β c β6.33β d β70.79β 4 a β12.3β b 1β 31.4β c 2β 2.23β d β13.457β e β43.27β f β4947.341β 5 a β7, 8β b β6, 5, 1, 4β c β0, 1, 0, 2β d β7, 5, 1, 6, 1β 6 β186.19β 7 β$54.30β 8 β49.11 mmβ 9 β+β 0β .01β 0β .05β 0β .38β 1β .42β β0.3β 0β .31β 0β .35β 0β .68β 1β .72β
0β .75β 0β .76β 0β .80β 1β .13β 2β .17β
1β .20β 1β .21β 1β .25β 1β .58β 2β .62β
1β .61β 1β .62β 1β .66β 1β .99β 3β .03β
10 β$2036.10β 11 a β8.15 β 3.2 = 4.95, 8.15 β 4.95 = 3.2β b β6.5 β 3.7 = 2.8, 6.5 β 2.8 = 3.7β c β10.1 β 8.3 = 1.8, 10.1 β 1.8 = 8.3β d β6.31 β 3.8 = 2.51, 6.31 β 2.51 = 3.8β 12 a Yes, e.g. β2.3 + 1.7 = 4, 3.81 + 1.19 = 5β b Yes, e.g. β4.8 β 1.8 = 3, 7.92 β 2.92 = 5β 13 a Answers may vary; e.g. β3.57 + 4.15 + 3.44β b Answers may vary; e.g. β1.35 + 2.87 + 6.94 = 11.16β 14 Always end up with β$10.89βunless the starting value has the same first digit, in which case we end up with zero. The β8β forms as it comes from adding two 9s. This produces a β1β to be carried over, which results in the answer being a β$10β answer, rather than a β$9β answer.
Example 11 β877.8β
Exercise 5D
1 a β2432.7β b 4β 36.1β c 2 a 4β 8.7β b 3β 52.83β c d β1430.4β e 5β 699.23β f g β12 700β h β154 230β i j β2132β k β86 710 000β l 3 a β4.27β b β35.31β c d β56.893β e 1β 2.13518β f g β0.029β h 0β .001362β i j β0.367β k β0.000002β l 4 a β2291.3β b 3β 1.67β c d β0.222β e 6β 3 489 000β f 5 a β15 600β b β43 000β c d β0.016β e 2β 13 400β f g β0.007β h 9β 900 000β i 6 a β158.4β b β3.36β c β85.4β e β71.06β f 7β .5β g β2.037β 7 $β 187β 8 a β1 200 000 mLβ b β12 000β 9 β3000βcents, β$30β 10 β$21 400β 11 β225 kgβ 12 Answers may vary.
1β 840β β4222.7β β125.963β β3400β β516 000β β2.4422β β9.32611β β0.00054β β0.0100004β β0.49β 0β .0010032β β225.1β β21.34β β0.0034β d β7054β h β21.7β
Starting number
Answer
Possible twostep operations
β12.357β
1β 235.7β
βΓ 1000, Γ·10β
β34.004 5β
0β .034004 5β
βΓ·100, Γ·10β
β0.003601β
3β 60.1β
βΓ 100, Γ 1000β
βBAC.DFGβ
βBA.CDFGβ
βΓ·100, Γ 10β
βD.SWKKβ
βDSWKKβ
βΓ 100 000, Γ·10β
βFWYβ
βF.WYβ
βΓ·1000, Γ 10β
13 Γ· β 1 000 000β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
ii β4.2 Γ β10ββ7β ii β1.08 Γ β10ββ21β
11 Answers may vary; β0.25, 0.26β 12 Answers may vary; β0.0043β 13 a 3β 8.76β b β73.6β
c β0.75β
d β42, 0.42β
Progress quiz 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 10 a
6β β hundredths β0.9β β<β b β16.88β b β5.4β b β3450β b β15β b β90 000β b β8.011β b β$10.15β b
b 6β β ones b β0.019β
c β3.25β d β>β
U N SA C O M R PL R E EC PA T E G D ES
ii β9 Γ β10βββ10β
Answers
14 a i β5 Γ β10β13 β β iii β1.23 Γ β10β16 β β b i β2 Γ β10β19 β β c, d Answers may vary. e β3.5 Γ β10ββ11 β β f i β1 Γ 1β 0ββ6 β β iii β7.653 Γ β10ββ12 β β
793
5E
Building understanding
1 a β1β b 2β β c β3β d β3β e β5β f β9β 2 a β19.2β b β1.92β c β0.192β 3 It helps you check the position of the decimal point in the answer. 4 a The decimal point is the actual βdotβ; decimal places are the numbers after the decimal point. b 1β βdecimal point, β4βdecimal places 5 in the question; decimal places
β β > β2.350β β17.031β β0.65345β β18β 1β 19β β197.44β β$179.55β
c > β β c β0.7β c β3.52β
d β5.78β
c 2β 0β c β1.912β
d 1β 36 800β
5F
Building understanding
b β77β
Example 13 a β289.35β
b β4.628β
Example 14 a β158 700β
b β4.27β
Now you try
Exercise 5E
1 a β8β b 4β 0β c β70β d β24β 2 a β94.57β b β17.04β c β30.78β d β21.645β 3 a 2β 0.84β b β26.6β c β183.44β d 1β 00.8β e β218.46β f 1β 5.516β g 2β 3.12β h β12.42β i β5.44β j β311.112β k β0.000966β l β1.32131β 4 a β100.8β b β483β c β25 400β d 9β 800β e β14 400β f β364 550β g 0β .68β h β371β i β90.12β 5 a β$31.50, $32β b β$22.65, $23β c β$74.80, $75β d β$17.40, $17β e β$145.20, $145β f β$37 440, $37 440β g $β 88.92, $89β h β$4.41, $4β i β$18.0625, $18β 6 a β29.47 mβ b β3.56 kgβ c β165.85 kmβ 7 a β67.2 mβ b β$198.24β 8 β$1531.25β 9 a β738.4 kmβ b Yes c β1.57 Lβleft in the tank 10 a β3.92βis less than β4β, β6.85βis less than β7β, so β3.92 Γ 6.85β is less than β4 Γ 7β. b β4.21βis greater than β4β, β7.302βis greater than β7β, so β 4.21 Γ 7.302βis greater than β4 Γ 7.β c Answer may vary, e.g. β3.9 Γ 7.4 = 28.86βis greater than β 4 Γ 7 = 28β.
5E
Example 12 a β12β
1 B ββ 2 Directly above the decimal point in the dividend 3 a 6β 0β b β60β c β60β d β60β e An identical change has occurred in both the dividend and the divisor. 4 a β32.456, 3β b β12 043.2, 12β c β34.5, 1β d 1β 234 120, 4β
Now you try
Example 15 a 3β .796β
b 0β .0826β
Example 16 a 1β 179.7β
b 8β 3.8β
Example 17 β0.0214β Example 18 β9.765β
Exercise 5F
1 a β3.51β 2 a 4β .2β d 0β .7055β g 3β .526β j 1β 17.105β 3 a β30.7β d 8β .5β g 8β 00.6β 4 a β44.4β c β0.050425β 5 a β1.1807β d 0β .00423β 6 a β11.83 kgβ d 2β 39.17 gβ 7 a β20.84β d 1β 8.49β
b β5.19β c β4.37β b 6β .1β e β1.571β h β124.3β k 0β .6834β b β77.5β e β645.3β h β2 161 000β b β0.08β d 0β .79β b β8.267β e β0.096487β b $β 30.46β e β965.05 Lβ b 9β 3.36β e β67.875β
c f i l c f
d β9.113β 2β 1.34β β0.308β β0.0024β β0.0025625β β26.8β β980β
c f c f c f
β0.0123748β β0.0007825β β304.33 mβ $β 581.72β β10.93β β158.35β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
8 a β8, 9β b β9β c β1, 5, 5β d β7, 7, 0β 9 β$1.59 / Lβ 10 β238β frames 11 β26.67, 26βcan be filled 12 a β$66.35β b β$21.90β 13 Apples β$3.25 / kgβ; bananas β$3.10 / kgβ; hence, bananas are better value. 14 a β24.5 m / sβ b β88.2 km / hβ c Yes 15 a β24.53β b β19.7β c β2453β d β1.97β e β2.453β f β197β 16 a β0.5β b β0.3β c β0.01β d β1.1β e β4.8β f Answers may vary.
10 0β .11 mm, _ β 11 βmmβ 100 4 _ 571428ββ, repeating pattern 11 β2, β β= 0.ββΎ 7 12 a β0.5, 0.β3Λ β, 0.25, 0.2, 0.1β6Λ β, 0.ββΎ 142857β, 0.125, 0.β1Λ β, 0.1β b, c Answers may vary. 1 β, _ 13 a β_ β1 β, _ β2 β, _ β1 β, _ β3 ,β _ β3 ,β _ β4 β, _ β9 β β1 β, _ 10 5 4 5 2 5 4 5 10 b, c Answers may vary. 45 β, 2β_ 3β 14 Answers may vary; β2_ β4 ,β 2β_ 9 100 7 15 a No, for example, _ β1 βis β0.β3Λ β, which is recurring. 3 b Yes, by writing the decimal places as a numerator with a suitable power of β10β, e.g. β0.3701βis _ β 3701 ββ. 10000 16 Answers may vary.
U N SA C O M R PL R E EC PA T E G D ES
Answers
794
5G Building understanding 1 a β5β 2 a 2β β 3 a False d False
b 1β 00β b 1β 5, 20β b True e True
c 7β 5, 7β c β10, 4β
d β5, 4β d β16β c True f True
Now you try Example 19
5G
407 β a β_ 1000 Example 20 a β0.123β c β0.83333 . . .βor β0.8β3Λ β
b 7β _ β9 β 20
b β19.75β
5H
Building understanding ββ C βBβ βAβ a β50β c i β5β d i β50%β 5 a β100, Γ·, 100, .β
1 2 3 4
ii 1β 00β ii β50%β
b 5β 0%β iii β20β iv β1β iii β50%β iv 5β 0%β b β35, 100, 35, 0β
Now you try
Exercise 5G 3β 1 a β_ 10 2 a _ β2 β 5 _ 3 a β1 β 2 e β3_ β1 β 4
b _ β 41 β 100 _ b β11 β 50 2β b β6β_ 5 1β f β_ 20
c _ β 201 β 1000 _ c β3 β 20 3β c β10β_ 20 g β9_ β3 β 40
d _ β7 β 100 _ d β3 β 4 3β d 1β 8β_ 25 h 5β _ β 24 β 125
4 a 0β .7β e β1.21β
b 0β .9β f 3β .29β
c 0β .31β g 0β .123β
d β0.79β h β0.03β
b _ β 5 β= 0.5β 5 a _ β 8 β= 0.8β 10 10 46 _ _ d β β= 0.46β e β 95 β= 0.95β 100 100 25 β= 2.5β g β_ h _ β 375 β= 0.375β 10 1000 6 a β0.5β b β0.5β d β0.4β e β0.β3Λ β Λ g β0.41β6β h β0.ββΎ 428571ββ 7 a 0β , 0.5, 1β b β0, 0.β3Λ β, 0.β6Λ β, 0.β9Λ (β 0.9999999β¦= 1)β c β0, 0.25, 0.5, 0.75, 1β d β0, 0.2, 0.4, 0.6, 0.8, 1.0β
c _ β 35 β= 0.35β 100 f 3β _ β 25 β= 3.25β 100
c β0.75β f β0.375β
1 β, 0.4, _ 8 a β_ β1 β, _ β5 β, 0.75, 0.99β b _ β1 ,β 0.13, _ β3 β, 0.58, _ β4 ,β 0.84β 4 2 8 7 5 9 11 β= 0.ββΎ 9 a Tan: β_ 297ββ; Lillian: _ β 6 β= 0.ββΎ 285714ββ; hence, Tan is the 37 21 better chess player. b Two or more
Example 21 a 0β .92β d β0.076β Example 22 a 7β 0.3%β
b 0β .04β e 0β .0005β
c β1.5β
b 2β 310%β
Exercise 5H
1 a β0.7β b β0.4β c β0.5β d β0.2β e β0.48β f 0β .32β g β0.71β h β0.19β 2 a β0.08β b β0.02β c β0.03β d β0.05β 3 a β0.32β b β0.27β c β0.68β d β0.54β e β0.06β f 0β .09β g β1β h 0β .01β i β2.18β j 1β .42β k β0.75β l β1.99β 4 a β0.225β b β0.175β c β0.3333β d β0.0825β e β1.1235β f 1β .888β g β1.50β h β5.20β i β0.0079β j 0β .00025β k β0.0104β l β0.0095β 5 a β80%β b β30%β c β45%β d β71%β 6 a β41.6%β b β37.5%β c β250%β d β231.4%β e β2.5%β f 0β .14%β g β1270%β h β100.4%β 7 a β86% , 0.78, 75% , 0.6, 22% , 0.125, 2% , 0.5%β b β50, 7.2, 2.45, 1.8, 124% , 55% , 1.99% , 0.02%β 8 β35%β 9 a β25%β b β78%β 10 β24β gigalitres 11 a βAB.CD%β b βAAC%β c A β BDC%β d βD.D%β e C β DB.A%β f βCC.CDDD%β 12 a β0.0ABβ b B β .CDβ c β0.ACβ d β0.00DAβ e A β B.BBβ f β0.DDDβ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers 1β b β0.1β c β10%β 6 a β_ 10 1 _ b β20%β c β0.8β 7 a β β 5 8 Val (β 70%)β, Mel (β 67%)β, Wendy (β 69%)βso Val scored the highest.
U N SA C O M R PL R E EC PA T E G D ES
9 Wally β(58%)β, Su β(55%)β, Drew (β 53%)βso Wally has walked the most.
Answers
13 a No b Answers may vary. Examples include: percentage score on a Maths test, percentage of damaged fruit in a crate, percentage of spectators wearing a hat, percentage of the day spent sleeping. c Answers may vary. Examples include: percentage profit, percentage increase in prices, percentage increase in the price of a house, percentage score on a Maths test with a bonus question. 14 a βF Γ· A Γ 100β b Fβ :βpoints scored for the team; A: points scored against the team c β100%β d 1β 58β points e Yes; Hawthorn β90.60%β, Port Adelaide β91.32%β
795
5I Building understanding 1 a β4β
d 5β 0%β
2 a 1β 0β
d 2β 0%β 3β 3 a β_ 10
ii _ β3 β, 60%β 5
iii _ β1 β, 12.5%β 8
d β27.5%β
11 β95% , 0.95β 1 β, 4% , 0.04β 12 a β_ 25 b _ β2 ,β 40% , 0.4β 5
b 4β β 1β e β_ 2 b _ β1 β 5 e β80%β
1β c β_ 2 f 5β 0%β
c _ β 1 β, 5% , 0.05β 20
c _ β4 β 5
d _ β 1 β, 1% , 0.01β 100
b β0.3β
c β30%β
1 β, 2% , 0.02β e β_ 50
f _ β 1 β, 2.5% , 0.025β 40
Now you try
Exercise 5I 2β 1 a β_ 5 2 a β0.3β
1β b β_ 4 b 0β .7β
1β c β_ 10 c 0β .29β
d _ β3 β 5 d 0β .05β
3 a β25%β
b β70%β
c β46%β
d β65%β
4 a _ β 3 ,β 0.3, 30%β 10 3 ,β 0.6, 60%β b β_ 5 c _ β1 ,β 0.2, 20%β 5
13 a _ β1 β 5
5I
Example 23 b β0.25β c β25%β a _ β1 β 4 Example 24 b β0.6β c β60%β a _ β2 β 5 Example 25 Millie β(35%)β, Adam β(40%)β, Yoe β(39%)βso Adam has completed the most.
b β40%β
14 Ross β75%β, Maleisha β72%βor Ross β0.75β, Maleisha β0.72β 15 Hatch β15%β, β4WD 16%β; hence, β4WDβhas larger price reduction. 16 Yellow β20%β, blue β19%β; hence, blue has least percentage of sugar. 17 a _ β a β b _ β100b β a+b a+b 1β 18 a β3β b β_ c _ β1 β 6 30 2 β%β e β20%β d _ β17 β f β96β_ 3 30
Problems and challenges
1 a 2β β b 1β 00Aβ c β0.2β d _ β479 βor _ β1437 β 330 990 e 2β Γ $1βcoins, β3 Γ 50β-cent pieces and β3 Γ 20β-cent pieces 2 Height of rungs: 0.18 m, 0.36 m, 0.54 m, 0.72 m, 0.9 m, β β―β―β―β―β― β 1.08 m, 1.26 m, 1.44 m, 1.62 m, 1.8 m
d _ β3 ,β 0.75, 75%β 4
1 ,β 0.05, 5%β e β_ 20
3 β, 0.15, 15%β f β_ 20 5 a β0.6, 60%β c β0.25, 25%β e β0.75, 75%β
10 a β1500 βm2β β b i _ β 1 β, 5%β 20 29 β c β_ 40
b β0.5, 50%β d β0.4, 40%β f β0.8, 80%β
β = 1, B = 7, C = 8β A βA = 7, B = 5, C = 1, D = 9β βA = 4, B = 2β A β = 3, B = 7β i β1.96β ii 2β .25β i β1.73β ii 1β .732β 4 2 17 β _ _ 5 a β β b β3 β β c β_ 9 9 99 3 a b c d 4 a b
iii 2β .1025β d 4β _ β728 β 999
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
796
Answers
12
8 β β_
1 1000 2 3β .17β 3 2β .37β 4 23.53β¨8 5 0β .44β 6 7β 3.752β 7 β66.85β 8 43100 9 0β .00782β 10 β0.023β 11 3β 2β 12 β25.047β; β19.2β 13 β2.ββΎ 09βand 0.00838β 7β 14 β10β_ 20 15 β10.6β _ 16 β0.58β3Λ β(or β0.58β3ββ) 17 β0.125β 18 β4.5%β 1 ββ; β0.2β; β20%β 19 β_ 5 20 As percentages they are β80%β, β78%βand β82%β, so the largest is β82%β.
Decimal
Fraction
β0.45β
_ β9 β β 45 β= _ 100 20 _ β 70 β= _ β7 β 100 10 _ β 32 β= _ β8 β 100 25 _ β 6 β= _ β3 β 100 50 _ β 79 β 100 _ β105 β= _ β21 β 100 20 _ β 35 β= _ β7 β 100 20 _ β 65 β= _ β13 β 100 20 _ β1 β β 125 β= _ 1000 8
β0.7β β0.32β
Percentage 4β 5%β β70%β β32%β
Ch5 Chapter checklist with success criteria
U N SA C O M R PL R E EC PA T E G D ES
Answers
Chapter checklist with success criteria
Chapter review
β0.44, 0.4, 0.04β β2.16, 2.026, 2.016β β0.98, 0.932, 0.895β β8.1β b β0.81β
β0.79β
β1.05β
β0.35β
β0.65β
β0.125β
β6%β
7β 9%β
1β 05%β 3β 5%β
6β 5%β
β12.5%β
3 ,β 0.3, 30%β 13 a β_ 10 4 ,β 0.8, 80%β b β_ 5
c _ β3 β, 0.75, 75%β 4
d _ β 9 β, 0.18, 18%β 50
Short-answer questions 1 a b c 2 a
β0.06β
14 Dillon β(75%)β, Judy (β 80%)β, Jon (β 78%)βso Judy scored the highest.
c β8.01β
d β0.801β
3 a β3 hundredths = _ β3 β b β3 thousandths = _ β 3 β 100 1000 c β3 ones = 3β 4 a False b False c True d True e False f True 5 45.265, 45.266, 45.267, 45.268, 45.269, 45.270, 45.271, 45.272, 45.273, 45.274 6 a β423.5β b β15.89β c β7.3β d β70.000β e β2.8β f 0β .67β g 0β .455β h β0.01234567901234567901234567901234567901234568β 7 a β9.53β b 4β .137β c 4β 3.35β d β240.49857β e β83.497β f β205.22β 8 a True b False c False d False e True 9 a 1β 37β b β790β c 2β 2.51β d β0.096208β e β696.956β f β360.5β g β563 489.3β 10 a β28β b 4β 5β c 1β 4β d β30β e β45β f β9β 11 a β19.2β b β63.99β c β19.32β d β0.95β e β1.52β f β6β g β16β h β3β i β3.1β0Λ β9Λ β
Multiple-choice questions 1 βDβ 2 βBβ 3 βCβ 4 βEβ 5 βAβ 6 βDβ 7 D ββ 8 βBβ 9 βAβ 10 βCβ
Extended-response questions
1 a Jessica β$12.57β; Jaczinda β$13.31β; hence, Jaczinda earns higher pay rate by β74βcents per hour. b β$37.47, $37.45βto the nearest β5β cents c β$40.56β d β$48.34β 2 a β$30.40β b β$18 240.30β 3 β32βminutes and β26βseconds, on average
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
Semester review 1
10
b LXXIV
β6β
β9β
β11β
β100β
output
β19β
β23β
2β 7β
β39β
β47ββββ
β403ββββ
c
Multiple-choice questions
b 7β 00βββ850β b β6867β d β196βββ000β 3 ββ f β764β_ 4
U N SA C O M R PL R E EC PA T E G D ES
2 a β67βββ849β 3 a β99βββ323β c β441β e β1644β
5
11 βA(1, 0), B(4, 1), C(3, 2), D(1, 3), E(3, 4)β
Short-answer questions 1 a
β4β
Answers
Computation with positive integers
input
797
4 a False b True c True 5 β36β 6 a β30β c β48β e β16β 7 a False b True c False d True e True f True 8 β18 Γ (7 + 3)β 9 β9β times 10 a β3βββ859βββ640β b 3β βββ860βββ000β c β4βββ000βββ000β
1 C
4 E
5 D
4 D
5 A
a
b 1β 8β c β9thβ d $β 61β
b β56β d 1β 60β f 4β 2β
Fractions and percentages Short-answer questions
2 B
11 ββ 3 a ββ_ 12 2 ββ b β2β_ 3 1 ββ c β6β_ 4
3 D
b β$700β
Semester review 1
2 ,ββ β_ 1 ββ 1 ,ββ β_ 1 _ ββ 3 β,β β_ 10 3 5 2 2 _ ββ 17 ββ 3
4 C
5 A
Extended-response question a β28β
3 B
Extended-response question
Multiple-choice questions 1 E
2 A
c β$1000β
d β12βββhβ
Number properties and patterns
e 4β β f _ ββ 1 ββ 2
3 ββ 4 ββ_ 20 5 β$120β 6 β$60β
Short-answer questions
1 a β1, 3, 5, 15β b 1β , 2, 3, 5, 6, 10, 15, 30β c β1, 2, 4, 5, 10, 20, 25, 50, 100β 2 a β3, 6, 9, 12, 15β b 7β , 14, 21, 28, 35β c β11, 22, 33, 44, 55β 3 β1, 2, 3βand β6β 4 β4β 5 a β121β b β144β 6 β120β 7 False 8 a β1β b β1β 9 a β61β b β17β
1 ββ d ββ_ 5
7 a b c d 8 a b c 9 a b
c β25β
True True True False β1 : 3β β1 : 3 : 4β β12 : 5β β$50, $10β β$24, $36β
Multiple-choice questions
c β1β
1 B
2 C
3 E
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
Extended-response question
Extended-response question
a β6β b _ ββ 8 ββ 9 c β9β
a i β$12β ii β$(β 3x + 6)ββ iii $β β(3x + 2y)ββ b β$β(50 β 3x β 2y)ββ
d Last dose on Sunday week
Decimals
U N SA C O M R PL R E EC PA T E G D ES
Answers
798
Algebraic techniques
Short-answer questions 1 a b c d
β4β Yes β9β 7β β
βx + 3β 1β 2aβ β2x + 3yβ _ ββ w ββ 6 e βy β 2xβ
Semester review 1
2 a b c d
3 a β100mβ b β24xβ c 1β βββ000βββ000pβ y d _ ββ ββ 24
4 a b c d e f
β13β β11β β39β β6β β6β β24β
5 3β 6β 6 a β10aβ b β4xβ c β12aβ d βmβ e β6 + 5aβ f β4x + 2yβ 7 a β2x + 14β b β3β(x + 4)ββ = 3x + 12β 8 a β6bcβ b β5bβ c βpβ 9 a β2a + 6β b β12a β 12bβ c β24m + 32β 10 β12xyβ
Short-answer questions β0.2β β0.13β β1.7β β6β units b _ ββ 4 ββ 1000
1 a b c 2 a
c β136.1β
3 a b c 4 a b c d e f 5 a 6 a 7 a b c 8 a
9 a b c d e f
β18β 1β 8.4β β18.40β β4.07β 2β 69.33β β19.01β 0β .24β β0.09β 6β 0β β0.833β β4.5387β β36βββ490β 0β .018β β3886β 4 ββ ββ_ 5 True False False True False True
b β3β b β0.045βββ387β
c 3β 6β c 1β .23β
b β1.1β
2 ββ c ββ_ 3
Multiple-choice questions 1 2 3 4 5
E C D A B
Extended-response question
Multiple-choice questions 1 2 3 4 5
B A C E D
a b c d e
$β 64.08β β$64.10β β$64.35β β25cβ It becomes β$63.90βrather than β$63.80β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
6A
6B
Building understanding 1 a ββ 2, 2β 2 a β β 2β 3 a greater
c β β 1β f β357β
b 1β β e ββ 51β
b ββ 7, β 5β c ββ 21β c greater
Building understanding d 1β 071β d less
1 a Right 2 a D β β
b Right b βAβ
c Left c B β β
d Left d βCβ
U N SA C O M R PL R E EC PA T E G D ES
b β7β b less
Answers
13 a ββ 2β d β β 7β
Chapter 6
799
Now you try
Now you try
Example 1
β6 β5 β4 β3 β2 β1
Example 2 a β5 > β 7β
0
Example 3 a 3β β
1
β2 β1
2 a
b
1
2
3
β2 β1 0 1 2
β10 β9 β8 β7 β6
β 2, β 1, 0, 1, 2, 3, 4β β β β 2, β 1, 0, 1β ββ 4, β 3, β 2, β 1, 0β β β 9, β 8, β 7, β 6, β 5, β 4β β<β b β>β β<β f β<β β<β j β>β β4Β° Cβ b ββ 1Β° Cβ
c g k c
β>β β<β β<β ββ 7Β° Cβ
d ββ 10β
d h l d
< β β > β β > β β β β 25Β° Cβ
1 ββ 2 ββ 3 ββ 6 a ββ 1β_ b ββββ_ c ββ 4β_ 2 3 5 1 ββ 1 ββ 4 ββ d β β 9β_ e ββ 11β_ f ββ 19β_ 7 4 4 7 a β β 10, β 6, β 3, β 1, 0, 2, 4β b ββ 304, β 142, β 2, 0, 1, 71, 126β 8 a 0β , β 1, β 2β b ββ 2, 0, 2β c ββ 5, β 10, β 15β d ββ 44, β 46, β 48β e ββ 79, β 75, β 71β f ββ 101, β 201, β 301β 9 a β$5β b ββ $10β 10 a ββ 50βββmβ b β β 212.5βββmβ c β0βββmβ 11 a β2β b 4β β c β4β d 7β β e β3β f β3β g 6β β h 4β 4β 12 a Γ Γ β4 0 4
difference β= 8β 1β―ββ ββ b 4β _ β 1 ββ βββ(ββor β 4ββ―_ 2 2) c The opposites are reflected to the other side of β0βon a number line, so if the originals are listed left-to-right (ascending) then their opposites will be right-to-left (descending).
1 a 3β β b ββ 4β 2 a 1β β b β3β e ββ 1β f ββ 3β i β β 4β j ββ 8β m 2β β n ββ 9β 3 a β β 2β b ββ 1β e ββ 4β f ββ 10β i β β 41β j ββ 12β m β β 300β n ββ 100β 4 a 5β β b β9β e β5β f β7β i 5β β j β16β m β β 6β n ββ 13β 5 a 5β β b ββ 9β e β1β f ββ 22β 6 a $β 145β b β$55β 7 a 3β Β° Cβ b ββ 3Β° Cβ 8 β69Β° Cβ 9 a 5β 9βββmβ 10 Answers may vary. 11 a i Positive iii Negative b i No 12 Answers may vary. a β β ββ,β β+β c β+ββ,β β+ββ,β ββββ,β β+β e β+ββ,β β+ββ,β βββor ββββ,β β+ββ,β β+β
c c g k o c g k o c g k o c g
β β 1β 2β β β β 2β ββ 1β β6β β β 8β β β 15β ββ 22β ββ 93β 5β β 3β β ββ 4β ββ 30β 1β β β β 32β
d β5β d β1β h ββ 2β l β2β p ββ 31β d ββ 19β h ββ 7β l ββ 47β p ββ 634β d β2β h β10β l ββ 5β p ββ 113β d ββ 13β h ββ 4β c β$5250β c ββ 46Β° Cβ
b 5β 6βββmβ
ii Positive iv Zero ii Yes
b + β ββ,β ββββ,β βββ d β β ,ββ β β+ββ,β β+ββ,β β+ββ,β βββ f β β ,ββ β β+ββ,β ββββ,β βββ
6C
Building understanding 1 b f 2 a d 3 a d g
6β , β 3β c ββ 3, 1β d β β 11, β 7β 6β , β 2β g β β 3, 7β h β β 11, β 7β β4β b Subtracting c Subtracting e β2β f False b True c False e False f False h False
e β3, 5β
β β 5β Adding True True
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
6A
3 a b c d 4 a e i 5 a
0
c β β 4β
Exercise 6B
b ββ 4 < β 1β
Exercise 6A 1
b ββ 5β
800
Answers
6D
Example 4 a β6β
b ββ 9β
d β β 5β
c 1β 0β
Exercise 6C 1 a β2β 2 a 1β β e ββ 3β i ββ 3β m β β 10β 3 a β5β e ββ 4β i β2β m 2β 8β 4 a ββ 3β e ββ 1β i ββ 4β m β β 6β 5 a β0β e ββ 9β i ββ 15β 6 ββ 143βββmβ 7 ββ $35βββ000β 8 ββ $30β 9 a i $β 8000β b $β 2000β 10 a
b b f j n b f j n b f j n b f j
β3β β5β ββ 5β ββ 22β ββ 29β β11β ββ 3β β1β β34β ββ 10β β4β β4β ββ 5β ββ 5β β5β β6β
c c g k o c g k o c g k o c g k
β β 2β 6β β β β 2β ββ 35β ββ 50β 5β 0β β β 5β β0β ββ 12β β β 4β β β 1β β2β β2β 8β β β β 6β β17β
d d h l p d h l p d h l p d h l
β β 4β 2β β β β 4β ββ 80β β β 112β 9β 0β β β 34β β8β β β 76β 4β β β β 5β ββ 24β 4β β 1β 2β β β 91β β11β
Building understanding 1 a β2β 2 a Positive d Positive
b β β 3β c ββ 4β b Positive e Positive
d ββ 4β c Negative f Negative
6C
U N SA C O M R PL R E EC PA T E G D ES
Answers
Now you try
ii β β $6000β
b
ββ 2β
β0β
β5β
ββ 13β
ββ 11β
ββ 6β
8β β
β1β
ββ 6β
ββ 3β
ββ 10β
ββ 17β
β β 3β
2β β
β4β
β β 14β
ββ 9β
ββ 7β
11 a 3β + 4β c β5 β (β 2)β e βa + (β b)β
b ββ 2 + (β 9)β d (ββ β 2)ββ β 1β f βa β (β b)β
12 a 5 β 7 = β 2 ββ β ββ 5 β (β 2) = 7 b 4 β 7 = β3 ββ β ββ 4 β (β 3) = 7 c β 2 + (β 3) = β 5, ββ so βββββββββββββ―β― β―β― β 5 β (β 2)ββ―ββ― =ββ―ββ― β 3,ββββ β 5 β (β 3) = β 2
d Either from ββ 3 + 4 = 1βor β4 + (β 3) = 1β, you can rearrange to get β1 β (β 3) = 4β. 13 a 4β β b β β 1β c ββ 3β 14 a i No ii Yes b i Yes ii No c Yes, if βb < aβ, then subtracting βbβtakes the result to a number bigger than zero. 15 a _ ββ 3 ββ 2
e _ ββ 16 ββ 3 i _ ββ 3 ββ 2 11 ββ m β β ββ_ 6
11 ββ b ββ_ 3
c β β 2β
d β β 4β
23 ββ f ββ_ 2 15 ββ j ββ_ 2 27 ββ n β β ββ_ 20
1 ββ g β β ββ_ 2 19 ββ k ββ_ 6 o _ ββ 67 ββ 14
1 ββ h β β ββ_ 14 32 ββ l ββ_ 35 53 ββ p β β ββ_ 30
Now you try Example 5 a ββ 32β
b 4β 5β
c ββ 7β
d β5β
Example 6 β24β
Exercise 6D
1 a ββ 6β b ββ 6β c β6β d ββ 8β e ββ 8β f 8β β 2 a ββ 15β b β β 10β c ββ 6β d ββ 54β e β32β f 2β 8β g β144β h β β 99β i β β 39β j ββ 84β k β38β l ββ 108β m β66β n β β 45β o β63β p β72β 3 a ββ 2β b β β 12β c ββ 2β d ββ 4β e β3β f 1β β g ββ 5β h ββ 19β i β β 7β j ββ 12β k ββ 68β l β8β m β12β n 1β 3β o ββ 13β p β13β 4 a β24β b 1β 5β c ββ 4β d 5β β e β1β f β β 10β g β72β h β18β i 1β β j ββ 1β k ββ 69β l ββ 3β 5 a ββ 7β b 4β β c ββ 4β d β8β e β27β f β β 140β g β2β h ββ 3β i β β 3β j ββ 1β k ββ 2β l β40β 6 a ββ 3β b β β 3β c β8β d β31β e β3β f 5β β g ββ 30β h ββ 100β 7 a β4β b 1β β c β81β d β100β e β36β f 6β 4β g β9β h β2.25β 8 a β(1, 6), (2, 3), (β 1, β 6), (β 2, β 3)β b (1, 16), (2, 8), (4, 4), (β 1, β16), βββ―β―β―βββ (β 2, β8), (β 4, β4) c β(β 1, 5), (β 5, 1)β d (β 1, 24), (β 24, 1), (β 2, 12), (β 12, 2), βββ―β―β―βββ (β 3, 8), (β 8, 3), (β 4, 6), (β 6, 4) β , Γ·β Γ b Γ· β , Γβ c βΓ, Γβ d βΓ·βββ, Γβ β(4, β 2), (β 4, 2)β b β(33, β 3), (β 33, 3)β ββ 1β Divide by (ββ β 1)ββ If the original number is βxβthen its opposite is βx Γ (β1)β, and the opposite of that is βx Γ (β1) Γ (β1)β. But β (β1) Γ (β1) = 1β, so you end up with βx Γ 1βwhich is the original number, βxβ. 12 a i ββ 8β ii β64β iii β β 27β iv β81β b Parts ii and iv, even number of negative factors c Parts i and iii, odd number of negative factors
9 a 10 a 11 a b c
13 a ββ abβ
b ββ abβ
c βabβ
14 Answers will vary.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
1 ββ 15 a β β ββ_ 4 e ββ 1β i ββ 1β
3 ββ c β β ββ_ 7 g _ ββ 1 ββ 2 k ββ 1β
d 1β β h β6β l 1β β
U N SA C O M R PL R E EC PA T E G D ES
Progress quiz
h β(β 3 + 9) Γ· (β 7 + 5) = β 3β i β(32 β (β 8) ) Γ·(β 3 + 7) = 10β 8 Three answers β(β 10, β 21, β 31)β 9 a No b No c Yes d Yes e No f Yes 10 a True b True c True d True e False f False 11 a i Negative ii Negative iii Negative iv Positive v Negative vi Negative b If there is an even number of negative factors the result will be positive, if odd then negative 12 a 4β β b β4β c ββ 4β d β β 32β e ββ 32β f β32β g 2β β h ββ 2β i ββ 1β 13 Kevin should have typed (βββ β 3)4βββ ββto raise ββ 3βto the power of β4β. ββββ34ββ ββ is ββ 1 Γβ β34ββ ββ.
Answers
1 ββ b β β ββ_ 2 5 ββ f ββββ_ 4 j β1β
801
1
β2 β1
0
1
2
3
2 a ββ 2 < 1β b β β 9 > β 12β 3 ββ 6, β 4, 0, 7, 8β 4 a β2β b ββ 9β c ββ 8β 5 a β5β b ββ 15β c ββ 3β 6 a β5β b β12β 7 a ββ 12β b β60β c ββ 7β 8 a β40β b ββ 30β c β16β 9 a β25β b ββ 8β c β12β 10 Debt of β$70βor ββ $70β
c β4 > β 5β
d β β 26β d 2β 0β
d 4β β d β β 2β d β β 315β
6F
Building understanding
6E
1 C β β
Building understanding Division Subtraction True False
b e b e
Multiplication Multiplication False False
c f c f
Division Multiplication True True
3 No, β20β
b ββ 9β
c β45β
Now you try Example 9 a β β 6β
Now you try
Exercise 6F
Example 7 a β6β
b ββ 41β
Example 8 a β4β
b β14β
1 a i β9β ii β17β b i β10β ii β31β c i ββ 5β ii ββ 7β 2 a 1β 7β b β23β c β β 8β d β8β e β6β f β11β g β β 7β h β6β i 3β β j ββ 9β k ββ 9β l β6β 3 a β β 8β b ββ 2β c 2β β d ββ 13β e ββ 25β f β17β g 3β 6β h β18β 4 a 3β 8β b β5β c 4β β d β β 13β e ββ 4β f β β 16β 5 a 1β 0β b ββ 1β c β β 24β d ββ 6β 6 a 6β ββββm2ββ ββ b 1β 6ββββm2ββ ββ 7 a β32βmetres per second b β14βmetres per second c β2βmetres per second 8 a i β16β ii ββ 2β b A negative dimension is not possible. 9 Answers may vary. 10 Answers may vary. 11 a βa β a = 0βand ββ b + b = 0β a b βββ―_ a ββ = 1β c βa β a = 0β ab ββcancels to simply give βa.β d ββ_ b
Exercise 6E
1 a ββ 5β b ββ 11β 2 a 8β β b β10β 3 a β β 7β b β7β e β16β f β14β i ββ 5β j ββ 4β m ββ 10β n β4β 4 a ββ 10β b ββ 2β e β2β f β9β i ββ 14β j ββ 20β m β8β n ββ 6β 5 β$528β 6 ββ $50β 7 a β(β 2 + 3) Γ 8 = 8β b β β 10 Γ· (4 + 1) = β 2β c β(β 1 + 7) Γ 2 β 15 = β 3β d (ββ β 5 β 1)ββ Γ·β β(β 6)ββ = 1β e β(3 β 8) Γ· 5 + 1 = 0β f β50 Γ (7 β 8) Γ (β 1) = 50β g β β 2 Γ (3 β (β 7)ββ) β 1 = β 21β
c c c g k o c g k o
β 10β β ββ 36β β19β β6β ββ 18β β0β ββ 6β β1β β2β ββ 12β
d d d h l
β β 1β β β 16β 9β β β β 32β β β 4β
d 1β β h 4β β l β β 5β
12 a i
β5Β° Cβ
ii ββ 15Β° Cβ
iii β β 25Β° Cβ
9C ββ + 32β b βF =β β_ 5
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Ch6 Progress quiz
1 a d 2 a d
2 βBβ
802
Answers
6 a
Building understanding 1 a e 2 a e
βDβ βEβ β9β 10 β β
b f b f
ββ B βHβ 18 β β 2β β
c g c g
ββ A βFβ 15 β β 1β β
d h d
β β C βGβ 6β β
y 5 4 3 2 1 x
U N SA C O M R PL R E EC PA T E G D ES
Answers
6G
β4 β3 β2 β1β1O β2 β3 β4 β5 β6 β7
Now you try
Example 10 βA = (1, 3)β βB = (β 3, 4)β C β = (4, β 1)β D β = (β 1, β 2)β
Exercise 6G
1 βA(2, 2), B(2, β 3), C(β 2, β 1), D(β 4, 3)β
2 A(2, 1), B(3, β 2), C(β 1, β 4), D(β 2, 2),β βββE(4, 3), β―β―β―β― ββF(2, β 3),β βββG β 3, β 1β , H β 4, 4β βββ (ββ ββ ββ )ββββ (ββ ββ ββ )ββββ
3 a, b
y
4 3 2 1
6G
1 2 3 4
β4 β3 β2 β1β1O
x
1 2 3 4
β2 β3 β4
Rectangle Kite β6β square units β15β square units
c βB, A, Cβ
ii β4β iv β3β
1 2 3 4
x
β2 β3 β4
14 a yβ =β β{β 7, β 5, β 3, β 1, 1, 3, 5}ββ b, c y
y
4 3 2 1
β4 β3 β2 β1β1O
b d b d
4 3 2 1
β4 β3 β2 β1β1O
4 A(3, 0),β ββββββB(0, β 2),β βββC(β 1, 0),β βββD(0, 4),β βββ βββ―β―β―βββ E(0, 2),β βββF(1, 0),β βββG(0, β 4), H(β 3, 0) 5 a
b They lie in a straight line. 7 a Triangle (isosceles) c Trapezium 8 a 4β β square units c β4β square units 9 β28βββkmβ 10 βy = 2β 11 βy = 3β 12 a B ββ b βCβ 13 a i β1β and β4β iii 3β β and β4β b y
1 2 3 4
β2 β3 β4
b They lie in a straight line.
x
5 4 3 2 1
β4 β3 β2 β1β1O
1 2 3 4
x
β2 β3 β4 β5 β6 β7
d They are in a straight line.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
Chapter review
β 81, 243, β 729β β 4β , β 2, 1β ββ 10, β 15, β 21β β β 8, β 13, β 21β β0β β β 153β β101β ββ 3 Γ (4 + (β 2)ββ) = β 6β ββ 2 Γ 5 Γ (β 1) + 11 = 21β or ββ 2 Γ 5 Γ· (β 1) + 11 = 21β c 1β Γ 30 Γ· (β 6) β (β 2) = β 3β 4 a β11β and ββ 3β b 2β 1β and ββ 10β 5 9β β pairs 6 aβ = 1, b = β 1, c = 2, d = β 3, e = 5β
Short-answer questions
1 a b c d 2 a b c 3 a b
1 ββ 7 β β ββ_ 25 8 a βp + t > q + tβ b βt β p < t β qβ c βpt < qtβ 9 All numbers less than β1βexcept β0β 10 a β8βββm / sβ b β24βββm /ββs2ββ ββ
Chapter checklist with success criteria 1
Multiple-choice questions
β4 β3 β2 β1 0 1 2
2 ββ 1 > β 6β 3 β1β 4 ββ6β 5 ββ5β 6 β4β 7 β21β 8 ββ2β 9 β8β 10 ββ10β 11 ββ4β 12 βA =β β(1, 1)ββ; βB =β β(3, β 2)ββ; βC =β β(β 2, β 4)ββ; βD =β β(β 3, 3)ββ
13
y
Extended-response questions 1 a 1β 6Β° Cβ d 1β 9Β° Cβ
4 3 2 1
β4 β3 β2 β1 O β1 β2 β3 β4
1 C β β 2 βEβ 3 βBβ 4 βDβ 5 βCβ 6 βAβ 7 Eβ β 8 βCβ 9 βBβ 10 C β β
b β β 31Β° Cβ e β27Β° Cβ
c β8Β° Cβ
2 Rocket
1 2 3 4
x
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Ch6 Problems and challenges
U N SA C O M R PL R E EC PA T E G D ES
1 a < β β b < β β c β>β d < β β 2 a β β 5β b β β 2β c ββ 15β d 1β β e ββ 2β f β β 5β g 1β 2β h β β 18β i β β 6β j 5β β k β β 11β l 5β β 3 a β β 1β b β β 9β c ββ 1β d 2β β e ββ 21β f β β 2β g β β 87β h 3β 0β 4 a β β 10β b β β 21β c β30β d β β 5β e ββ 3β f 4β β g 1β β h β β 8β 5 a β β 2β b β β 50β c ββ 36β d β β 1β 6 a β β 37β b 8β β c ββ 3β d 1β β e β56β f 8β 0β 7 a 1β 0β b β β 41β c ββ 22β d β β 11β 8 βA(3, 0), B(2, 3), C(β 1, 2), D(β 4, β 2), E(0, β 3), F(4, β 4)β
Answers
Problems and challenges
803
804
Answers
7A Building understanding 1 a
b
P
N
A
3 a c 4 a d g
β BOCβββorββββ COBβ β b ββ BACβββorββββ CABβ ββ BEAβββorββββ AEBβ d β β AOCβββorββββ COAβ Acute, β40Β°β b Acute, β55Β°β c Right, β90Β°β Obtuse, β125Β°β e Obtuse, β165Β°β f Straight, β180Β°β Reflex, β230Β°β h Reflex, β270Β°β i Reflex, β325Β°β
5 a
b
U N SA C O M R PL R E EC PA T E G D ES
Answers
Chapter 7
c
d
A
40Β°
T
75Β°
S
c
C
B
e
d
135Β°
f
C
B
90Β°
e
A
f
205Β°
175Β°
2 a Ray b Line d Collinear e Concurrent 3 Answers may vary. 4 a β50Β°β b β145Β°β c β90Β°β
c Segment
g 260Β°
h 270Β°
i 295Β°
j 352Β°
d β250Β°β
Now you try
7A
Example 1 a Ray βBAβ
b ββ BCAβββorββββ ACBβ
Example 2 a Acute, ββ AOB = 50Β°β b Straight, ββ EFG = 180Β°β c Reflex ββ EOD = 250Β°β Example 3 a
A
30Β°
B
O
b W
170Β° X
c
Y
320Β°
M
N
O
Exercise 7A 1 a b 2 a c e
i Segment βPQβ i ββ AOBβββorββββ BOAβ Point βTβ ββ BACβββorββββ CABβ Ray βPQβ
6 a β29Β°β b β55Β°β c β35Β°β d β130Β°β 7 a βC,βββBβand βDβ b βA,βββCβand βDβ 8 a No b Yes 9 a β8β b β14β 10 a β180Β°β b β270Β°β c β30Β°β d β120Β°β e β6Β°β f 5β 4Β°β g β6β3Β°ββ ββ h 1β 29Β°β 11 a β180Β°β b β90Β°ββββor 270Β°β c β120Β°ββββor 240Β°β d 3β 0Β°ββββor 330Β°β 12 a β115Β°β b β127.5Β°β c β85Β°β d β77.5Β°β e β122Β°β f 1β 76.5Β°β 13 a Yes b No 14 Use the revolution to get β360Β° βββ60Β° = 300Β°β. 15 a i β70Β°β ii β70Β°β iii β90Β°β iv β90Β°β v 8β 0Β°β vi β80Β°β b No c Subtract β360Β°βuntil you have a number that is less than β 180Β°β, then change the sign if it is negative. 16 a Missing numbers are β0, 1, 3, 6, 10, 15β. b For β5βpoints, add β4βto the previous total; for β6βpoints, add β5β to the previous total, and so on. n (ββ n β 1)ββ c βNumberβββofβββsegments =β β_ 2
ii Segment βXYβ ii ββ STUβββorββββ UTSβ b Line βCDβ d Plane f Segment βSTβ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
Building understanding
Building understanding
1 a, b Angles should add to β90Β°β c Complementary 2 a, b Angles should add to β180Β°β c Supplementary 3 a, b Angles should add to β360Β°β c Vertically opposite angles 4 a ββ BOCβββorββββ COBβ b β β AODβββorββββ DOAβand ββ BOCβββorββββ COBβ c ββ CODβββorββββ DOCβ
1 a β4β 2 a 2β β 3 a Equal c Equal
b b b d
No Yes Supplementary Equal
U N SA C O M R PL R E EC PA T E G D ES
7C
Answers
7B
805
Now you try
Example 6 a β β DBCβ(or ββ CBDβ) c ββ DBFβ(or ββ FBDβ)
Now you try Example 4 a βa = 25β
b βa = 50β
Example 5 a βa = 116β
b βa = 220β
b β β BFHβ(or ββ HFBβ) d β β HFBβ(or ββ BFHβ)
Example 7 a aβ = 120β, corresponding angles in parallel lines b aβ = 50β, alternate angles in parallel lines c βa = 115Β°β, cointerior angles in parallel lines
Example 8 a Parallel, because the cointerior angles sum to β180Β°β b Not parallel, because the corresponding angles are not equal
Exercise 7B
Exercise 7C
Triangle
β120β
Square
β90β
β90β
Pentagon
β72β
β108β
β60β
Hexagon
β60β
β120β
Heptagon
ββ 51β
ββ 129β
Octagon
β45β
β135β
1 a ββ CHGβββorββββ GHCβ b β β CHGβββorββββ GHCβ c ββ BGHβββorββββ HGBβ d β β BGHβββorββββ HGBβ 2 a ββ DEHβββorββββ HEDβ b β β BEFβββorββββ FEBβ c ββ DEBβββorββββ BEDβ d β β CBGβββorββββ GBCβ 3 a ββ FEGβββorββββ GEFβ b β β DEBβββorββββ BEDβ c ββ GEBβββorββββ BEGβ d β β ABCβββorββββ CBAβ 4 a β130β, corresponding b 7β 0β, corresponding c β110β, alternate d 1β 20β, alternate e β130β, vertically opposite f β67β, vertically opposite g 6β 5β, cointerior h 1β 18β, cointerior i 1β 00β, corresponding j β117β, vertically opposite k 1β 16β, cointerior l β116β, alternate 5 a βa = 70, b = 70, c = 110β b aβ = 120, b = 120, c = 60β c βa = 98, b = 82, c = 82, d = 82β d aβ = 90, b = 90, c = 90β e βa = 95, b = 85, c = 95β f aβ = 61, b = 119β 6 a No, corresponding angles should be equal. b Yes, alternate angles are equal. c Yes, cointerior angles are supplementary. d Yes, corresponding angles are equal. e No, alternate angles should be equal. f No, cointerior angles should be supplementary. 7 a β35β b β41β c β110β d 3β 0β e β60β f β141β 8 a β65β b β100β c β62β d 6β 7β e β42β f β57β g 1β 00β h β130β i β59β 9 a β12β angles b 2 angles 10 1β 20β 11 a i The angle marked βaΒ°βis alternate to the β20Β°β angle. ii The angle marked βbΒ°βis alternate to the β45Β°β angle. b i βa = 25, b = 50β ii βa = 35, b = 41β iii aβ = 35, b = 25β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7B
1 a β70β b β30β c β75β 2 a β50β b β60β c β15β d 7β 0β e β60β f 1β 35β g 4β 0β h β30β i β41β 3 a β115β b β37β c β77β d 2β 40β e β140β f 2β 70β g 1β 10β h β130β i β125β 4 a N b N c S d N e C f C g C h S 5 a βEFββββ₯βββGHβ b βSTββββ₯βββUVβ c βWYββββ₯βββXYβ 6 a β30β b β75β c β60β d β135β e β45β f 1β 30β 7 a No, should add to β90Β°β. b Yes, they add to β180Β°β. c Yes, they add to β360Β°β. d Yes, they are equal. e No, they should be equal. f No, should add to β360Β°β. 8 a β30β b 6β 0β c β60β d β45β e β180β f 3β 6β 9 β24Β°β 10 a No, if both are less than β90Β°βthey cannot add to β180Β°β b Yes, e.g. β20Β°β and β70Β°β. 11 a βa + b = 90β b βa + b + c = 180β c βa + b = 270β 12 Only one angle β the others are either supplementary or vertically opposite. 13 a β360Β°β b β72β c β108β d Regular shape a b
Answers
βa = 120, b = 120β β60β Opposite angles are equal. The two angles do not add to β180Β°β. The cointerior angles do not add to β180Β°β. Alternate angles are not equal. i ββ BDEβ, alternate ii ββ BEDβ, alternate Add to β180Β°β Three inside angles of a triangle add to β180Β°β, which is always true. 15 The angles of each triangle add to β180Β°β, so the total is β360Β°β. 12 a b c 13 a b c 14 a b c
c
d
e
f
U N SA C O M R PL R E EC PA T E G D ES
Answers
806
7D
b B β Cβ d β β ACBβand ββ ABCβ
Now you try
Building understanding 1 a c 2 a c
2 a A ββ c βACβand βABβ
βa = 65, b = 115β βa = 55, b = 55β ββ BEDβ, βa = 30β ββ ADCβββorββββ ABDβ, βa = 50β
b βa = 106, b = 106β
Example 10 a i Isosceles b i Equilateral
b ββ EBDβ, βa = 70β
Example 11
ii Right ii Acute
C
Now you try Example 9 a βa = 30β
7D
b b f b b f b f b b b f b b f
β115β β120β 7β 3β β65β β60β 5β 5β β120Β°β 7β 5Β°β β150β 2β β 6β 0β 1β 0β β45β 2β 50β 3β 00β
B
Example 12
c β115β
d β123β
c β65β
d β45β
c β55Β°β
d β75Β°β
c 6β 0β c 2β β c 4β 0β
d β30β
c 1β 2β c 4β 0β
7E
Building understanding 1 a
4 cm
A
Exercise 7D 1 a β105β 2 a β60β e β50β 3 a β50β 4 a β80β e β60β 5 a β130Β°β e β90Β°β 6 a β50β 7 a β1β 8 a β30β e β120β 9 a β60β 10 a β110β e β40β
45Β°
60Β°
b βa = 55β
b
5 cm
6 cm
7 cm
Exercise 7E
d β110β
1 a Equilateral b Isosceles c Scalene 2 a Right b Obtuse c Acute 3 a i Scalene ii Right b i Equilateral ii Acute c i Isosceles ii Obtuse 4 Check measurements with a ruler and protractor. 5 Check your answer by measuring the lengths of the sides. 6 Check by doing parts d and e. 7 Check by doing parts d and e. 8 Check that ββ CAB = 90Β°β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
9 a Yes
b Yes
e
f
aΒ°
3 a
U N SA C O M R PL R E EC PA T E G D ES
d Yes
Answers
aΒ°
c No
807
10 a Start with one side and the intersection of two arcs to give the other two lengths. b Start with the base and use two arcs of equal radius. c Start with the base and use two arcs with the same radius as the base. d Start with the β4βββcmβsegment, construct a right angle (see Q7), then the hypotenuse. 11 Triangles Scalene Isosceles Equilateral
Acute
12 a Yes b Isosceles 13 The two shorter sides together must be longer than the longest side. 14 a Yes b No, the three sides will not join. 15 a Check that the dashed lines form an equilateral triangle. b Use the diagram to assist.
7F
Building understanding
c
Pentagon i and iii; all interior angles are less than β180Β°β ii; there is one interior angle greater than β180Β°β Regular pentagon b aΒ° aΒ° aΒ°
bΒ°
aΒ°
Example 13 a Hexagon, non-convex
b Irregular
Example 14 a Convex quadrilateral
b Convex kite
aΒ°
aΒ°
aΒ°
bΒ°
d
aΒ°
aΒ°
c
d
e
f
aΒ°
bΒ°
aΒ°
1 a Octagon, non-convex b Irregular 2 a 5β β b β3β c β10β d β7β e β11β f β4β g β9β h β6β i 8β β j β12β 3 a i, iv and vi b i Quadrilateral ii Pentagon iii Hexagon iv Octagon v Octagon vi Triangle 4 a Convex b Non-convex c Non-convex 5 a Square b Trapezium c Kite d Rhombus e Rectangle f Parallelogram 6 a Rectangle, kite, parallelogram b Rhombus, parallelogram c Square, rectangle, rhombus, parallelogram d Trapezium e Kite 7 a βNβ b βPβ c βPβ d βNβ e βNβ f βNβ g βNβ h βNβ 8 a βa = 90β, βb = 10β b βa = 100β, βb = 5β c βa = 50β, βb = 130β 9 Answers may vary. Some possibilities are given. a b
bΒ°
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7F
Obtuse
aΒ°
Now you try
Exercise 7F
Right
1 a b c d 2 a
b 1β β
Answers
10 a i β2β ii β7β b i β5β ii β35β 11 Forming a square is possible. 12 a True b False c False d True 13 a Square, rectangle b Square, rhombus, kite 14 a A square is a type of rectangle because a rectangle can be constrained to form a square. b Yes, a parallelogram can be constrained to be a rhombus. c No 15 a Use given diagram and check that all sides are the same length. b Use given diagram and check that the distance between the lines are always equal.
7G Building understanding 1 The three angles should add to β180Β°β. 2 a The β160Β°βangle and βaβare on a straight line, which should add to β180Β°β. b βa + b + 100 = 180β; so if βa = 20β, βbβ must be β60β. 3 β60Β°β 4 a The two base angles in an isosceles triangle are equal. b The sum must be β180Β°β (i.e. β70Β° + 70Β° + 40Β° = 180Β°β).
U N SA C O M R PL R E EC PA T E G D ES
Answers
808
Ch7 Progress quiz
Example 15 a aβ = 25β
b aβ = 140β
Example 16 βx = 152β
Progress quiz 1 a b c d e 2 a b c d e f g h 3 a c 4 a b c 5 a b 6 a b 7 a b 8
Now you try
G Angle βEGFβor βFGHβor βEGKβor βKGHβand other names Angle βEGFβor βKGHβand other names βADβ, βIJβand βEHβ Angle βABFβor angle βKBDβ measures β125Β°β βx = 28β(angles in a right angle add to β90Β°β) βx = 75β(angles on a straight line add to β180Β°β) βx = 64β(vertically opposite) βx = 23β(angles in a straight line add to β180Β°β) βx = 43β(angles in a right angle add to β90Β°β) βx = 60β(revolution) βx = 150β(revolution) βx = 65β(vertically opposite) ββ GHDβ b ββ DHGβ ββ FHCβ d ββ AGHβ βa = 76β(alternate angles in parallel lines) βa = 116β(corresponding angles in parallel lines) βa = 52β(cointerior angles in parallel lines) No, corresponding angles are not equal. Yes, cointerior angles add to β180Β°β. β125β β79β Isosceles triangle Right-angled triangle
Exercise 7G
1 a β40β b 3β 0β 2 a 6β 0β b 3β 0β c β55β d β65β e 2β 5β f β145β 3 a β120β b 3β 6β 4 a β65β b 2β 0β c β32β d β75β e 5β 5β f β35β 5 a β30β b 1β 40β c β60β d β60β e 1β 0β f β142β 6 a β40β b 1β 20β c β45β d β132β e 1β 6β f β30β 7 a β60β b 6β 0β c β55β d β55β e 1β 45β f β50β 8 β20Β°β 9 a i β3β ii β4β iii β5β b i β540Β°β ii β720Β°β iii β900Β°β 10 a β1080Β°β b 1β 260Β°β c β1440Β°β 11 βS = 180ββ(ββn β 2β)ββββ 12 a β155Β°β b 1β 55β c They are the same. d Yes, always true 13 a Alternate angles in parallel lines b Alternate angles in parallel lines c They must add to β180Β°β. 14 a βββ DCAββ― =ββ― a (ββ ββAlternate to β BAC and DE is parallel to AB.β)βββββ β ββ ECBββ― =ββ― b ββ(ββAlternate to β ABC and DE is parallel to AB.β)βββββ
6 cm
6 cm
6 cm
9 a c 10 a b c d
Square b Trapezium Rhombus d Parallelogram Concave irregular pentagon Convex regular hexagon Convex irregular rectangle Non-convex irregular decagon
β β DCA + β ACB + β ECB = 180Β°β(Angles on a line add to β180Β°β.) ββ΄ a + b + c = 180β b aβ + b + c = 180ββββββββββββββββββββββ(Anglesβββinβββaβββtriangleβββsumβββtoβββ180Β° .)ββ βa + b = 180 β cβββββ(1)ββ βAlsoββββ ACB + β BCD = 180Β°β(Angles in a straight line sum to β180Β°β.) βc + β BCD = 180Β°β ββ BCD =β β(180 β c)βΒ°ββββ(2)ββ From (1) and (2) we have ββ BAC = aΒ° + bΒ°β.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
7H Building understanding β4β ways 1β β way β4β 1β β
b e b e
β2β ways β2β ways β2β β2β
c f c f
β3β ways 0β β ways β3β 2β β
11 a b 12 a d
b
i No Isosceles trapezium β9β b 3β β 1β β e Infinite
ii No c 4β β f Infinite
U N SA C O M R PL R E EC PA T E G D ES
1 a d 2 a d
b 1β β and β1β
Answers
9 a 4β β and β4β c 1β β and β1β 10 a
809
Now you try
Example 17 a Line symmetry: order β1βand rotational symmetry: order β1β b Line symmetry: order β3βand rotational symmetry: order β3β
7I
Building understanding 1 a
b
c
d
e
f
Exercise 7H
2 a (β β 2, 0)β c β(0, β 2)β 3 a No
b (β β 2, 0)β d (β 0, 2)β b No
c
Now you try Example 18
8 a
y
b
B
B' 4 C' 3 image 2 1 A D A' D' x O β4 β3 β2 β1 β1 1 2 3 4 C
c
d
β2 β3 β4
e
Aβ² = (1, 1) Bβ² = (1, 4) βββ ββ ββ― ββ― Cβ² = (4, 3) Dβ² = (3, 1) Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
7H
1 6β β and β6β 2 a β4β and β4β b β2β and β2β c β2β and β2β d 1β β and β1β e β1β and β1β f 0β β and β2β 3 a Equilateral b Isosceles c Scalene 4 a i Kite ii Rectangle, rhombus iii None iv Square b i Trapezium, kite ii Rectangle, rhombus, parallelogram iii None iv Square 5 a 5, β5β b 1, β1β c 1, β1β d 4, β4β 6 a βA, B, C, D, E, M, T, U, V, W, Y.βK is almost symmetric but not quite. b H β , I, O, Xβ c βH, I, N, O, S, X, Zβ 7 a b
810
Answers
Answers
Example 19 a
c
d
y B' image
D
4 3 2
A
B
β5 β4 β3 β2 β1 β1
1
2 3 4
U N SA C O M R PL R E EC PA T E G D ES
1 A' C
D'
x
e
f
β2 β3 β4
b
βAβ²= (β 1, 1), Bβ²= (β 1, 3), Dβ²= (β 4, 1)β y
7 (β 2, β 5)β 8 a (β 0, β 1)β b (β 3, 0)β c β(β 1, 2)β 9 a β180Β°β anticlockwise b 9β 0Β°β anticlockwise c β90Β°β clockwise 10 a i
D
4
3 2
1 C
β5 β4 β3 β2 β1 B' image β1 A' β2
B
A
1
2
3 4
x
β3
ii
7I
β4 D'
βAβ²= (β 1, β 1), Bβ²= (β 3, β 1), Dβ²= (β 1, β 4)β
Exercise 7I
1 βAβ²(1, 3), Bβ²(1, 1), Cβ²(2, 1), Dβ²(3, 3)β 2 a βAβ²(1, 1), Bβ²(1, 4), Cβ²(2, 2), Dβ²(3, 1)β b βAβ²(β 3, 4), Bβ²(β 3, 1), Cβ²(β 2, 1), Dβ²(β 1, 2)β c βAβ²(β 1, β 2), Bβ²(β 2, β 4), Cβ²(β 4, β 4), Dβ²(β 4, β 3)β d βAβ²(2, β 1), Bβ²(2, β 4), Cβ²(4, β 2), Dβ²(4, β 1)β e βAβ²(β 3, 2), Bβ²(β 3, 3), Cβ²(β 1, 4), Dβ²(β 1, 1)β f βAβ²(β 3, β 4), Bβ²(β 1, β 4), Cβ²(β 1, β 1), Dβ²(β 2, β 3)β 3 a β(β 3, β 3)β b β(3, β 3)β c β(β 3, 3)β d β(β 3, 3)β e β(3, 3)β f β(β 3, β 3)β 4 a βAβ²(0, β 1), Bβ²(2, 0), Dβ²(0, β 3)β b βAβ²(1, 0), Bβ²(0, 2), Dβ²(3, 0)β c βAβ²(0, 1), Bβ²(β 2, 0), Dβ²(0, 3)β 5 a βAβ²(β 1, 0), Bβ²(β 3, 0), Dβ²(β 1, 2)β b βAβ²(0, β 1), Bβ²(0, β 3), Dβ²(β 2, β 1)β c βAβ²(1, 0), Bβ²(3, 0), Dβ²(1, β 2)β 6 a b
iii Answers may vary. b i
ii
iii Answers may vary. c Answers may vary.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
c β10Β°β b (β 2, β 5)β d ββ(4, 1)ββ for both
c f i l
β(β 2, 1)β β(8, 3)β β(3, β 3)β β(β 4, β 2)β
U N SA C O M R PL R E EC PA T E G D ES
7J
3 a β(1, 3)β b β(β 4, 3)β d (β β 2, β 2)β e β(β 2, 5)β g (β 1, 4)β h β(2, 1)β j (β β 3, 1)β k (β β 5, 4)β 4 a β3β units up b 7β β units down c β4β units down d 2β β units up e β5β units left f 2β β units right g 1β βunit left and β4βunits up h 3β βunits right and β6βunits up i 3β βunits right and β4βunits down j 3β βunits left and β11βunits up k 1β 2βunits right and β3βunits down l 1β 0βunits left and β13βunits down 5 a β2βunits left and β2βunits up b 4β βunits left and β4βunits up c β1βunit right and β5βunits down d 6β βunits right and β2βunits down 6 a β4β b 7 2β 4β points 8 It is neither rotated nor enlarged. 9 a β3β right and β1β down b c β2β left and β4β down d 10 a β(β 4, β 1)β b c β(β 7, 2)β d e β(4, β 2)β
Answers
11 a β270Β°β b β322Β°β 12 a β(2, β 5)β c The same point 13 Check with your teacher. 14 Check with your teacher.
811
Building understanding 1 a e 2 a e 3 a b c
ββ(4, 2)ββ ββ(2, 4)ββ up left β7β units β3β units i β7β units
b f b f
(βββ ββ1, 2β)ββββ (ββ 0, 1)ββ left left
c g c g
(βββ ββ3, 5β)ββββ (ββ 5, 1)ββ down right
ii β3β units
Now you try Example 20
B
A
C'
(ββ 3, 1)ββ (ββ 3, 0)ββ up right
1β 2β
1β β right and β2β up 6β β right and β7β up (β β 4, β 3)β (β β 13, β 5)β
7J
A'
C
B'
d h d h
7K
Example 21 Translation is β6βunits right and β8βunits down.
Building understanding 1 a
b
Exercise 7J 1
A
A'
C
C'
c
B
B'
2 a
b
2 a 2β β c 6β β
b 4β β
Now you try Example 22 a
c
b
d
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
812
Answers
f
(2 circles)
1
, g
(square, triangle on square) ,
2 a
b
h
(circle in square, semicircle on rectangle)
U N SA C O M R PL R E EC PA T E G D ES
Answers
Exercise 7K
,
i
c
d
e
f
(circle, triangle on semicircle)
,
6 6β β 7 2β 0β 8 a
(4-faced pyramid)
b
3 See given diagrams. 4 a
b
(8-faced double pyramid)
7K
c
c
(pyramid with 5-sided base)
9 Yes, one can be rotated to match the other.
5 a
(two squares)
10 a C 11 a i
,
b
b A
front
c B
left
d D
top
(square and rectangle with the same top side length)
,
c
(circle, isoceles triangle)
,
d
ii
front
left
top
(square, isoceles triangle)
,
e
(square, rhombus)
,
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
b i
ii
2
Answers
7L 3 Answers may vary. a
U N SA C O M R PL R E EC PA T E G D ES
Building understanding
813
1 a Equal c Platonic 2 A and C 3 a Equilateral triangle c Equilateral triangle e Equilateral triangle 4 a Cube c Triangular pyramid
b
b Regular
b Square d Regular pentagon b Cylinder
c
d
e
f
Now you try Example 23 a
7L
b
Exercise 7L
4 a b c d e f 5 a c e 6 a d 7 a
Tetrahedron, octahedron, icosahedron Icosahedron Octahedron Tetrahedron Hexahedron, octahedron Hexahedron, octahedron Octahedron b Hexahedron Dodecahedron d Icosahedron Tetrahedron β3β b β3β c β4β 3β β e β5β b
1 Answers may vary.
8 a β2β b 1β 1β 9 a 6β β b Yes c There is not the same number of faces meeting at each vertex. 10 No 11 a i β1β ii β26β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
814
Answers
n (side length)
1
2
3
4
5
n3(number of 1 cm cubes)
1
8
27
64
125
Number of inside cubes
0
0
1
8
27
15
5 cm
4 cm 6 cm
16 Pentagon 17 Convex; irregular 18 Convex; trapezium 19 β40Β°β 20 βa = 40β 21 βx = 152β 22 Line symmetry order β= 2β; rotational symmetry order β= 2β 23 ββAβ² =β β(ββ1,ββββ 1β)ββ,βββBβ² =β β(ββ1,ββββ 2β)ββ,βββCβ² =β β(ββ3,ββββ 3β)ββ,βββDβ² =β β(ββ3,ββββ 1β)ββββ
U N SA C O M R PL R E EC PA T E G D ES
Answers
b
Number of outside cubes
1
8
26
56
98
c i ββnβββ― 3ββ ii (ββ n β 2)ββ3ββ iii ββnβββ― 3ββ ββ β(n β 2)ββ3ββ
Problems and challenges
y
Ch7 Problems and challenges
1 Tetrahedron
4 3 2 1
β27β β77.5Β°β β720Β°β a β35β b β1175β 6 β161β 2 3 4 5
mirror line ( y-axis)
O 1 2 3 4 β4 β3 β2 β1 β1 Aβ² D A image Dβ² β2 Bβ² B β3 C Cβ² β4
x
Chapter checklist with success criteria 1 2 3 4 5
Segment AB ββ PQRβ(or ββ RQPβ) Obtuse β125Β°β
24 βββAβ²β―ββ =β β(0,βββ1)β,ββββBβ²β―ββ =β β(0,βββ2)β,ββββDβ²β―ββ =β β(βββ β 2,βββ1)β ββ
y
260Β°
6 a β50β b β25β 7 β150Β°β 8 (a) ββ EFBβ(or ββ BFEβ); (b) ββ HFBβ(or ββ BFHβ) 9 βa = 70β(cointerior angles are supplementary) 10 Yes, the two lines are parallel sinceββββ122 + 58 = 180β, cointerior angles are supplementary in parallel lines. 11 βa = 50β 12 Isosceles 13 Obtuse 14
C
A
45Β°
x
β2 β3 β4
25
A
B
C
Aβ²
Bβ² 30Β°
5 cm
4 3 2 Bβ² D 1 Aβ² Dβ² B C A β3 β2 β1 1 3 2 β1
Cβ²
B
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
β β²(β 1, β 2), Bβ²(β 3, β 3), Cβ²(β 3, β 1)β A βAβ²(1, 2), Bβ²(3, 3), Cβ²(3, 1)β ββ Aβ²(0, β 4), Bβ²(β 2, 0), Dβ²(β 3, β 3)β βAβ²(4, 0), Bβ²(0, β 2), Dβ²(3, β 3)β βAβ²(β 4, 0), Bβ²(0, 2), Dβ²(β 3, 3)β βAβ²(1, β 1), Bβ²(4, β 1), Cβ²(3, 1)β βAβ²(β 4, 1), Bβ²(β 1, 1), Cβ²(β 2, 3)β
Answers
15 a b 16 a b c 17 a b 18 a
U N SA C O M R PL R E EC PA T E G D ES
26 Translation is β6βunits left and β4βunits up. 27
815
28
29
b
Chapter review
Short-answer questions
1 a Segment βCDβ b ββ AOBβββorββββ BOAβ c Point βPβ d Plane e Ray βACβ f Line βSTβ 2 a Acute, β35Β°β b Obtuse, β115Β°β c Reflex, β305Β°β 3 a β180Β°β b 9β 0Β°β c β90Β°β d β150Β°β 4 a β20β b β230β c β35β d β41β e β15β f 3β 8β g β60β h β120β i β30β 5 a βaβand βbβ b βaβand βdβ c βaβand βcβ d βbβand βcβ e βcβand βdβor βbβand βdβ 6 a Yes, corresponding angles are equal. b No, alternate angles should be equal. c No, cointerior angles should be supplementary. 7 a β100β b β95β c β51β d 3β 0β e β130β f 7β 8β 8 Check lengths and angles with a ruler and pair of compasses. 9 Check lengths and angles with a ruler and pair of compasses. 10 a β5β b β7β c β11β 11 a β2β b β1β 12 a Trapezium b Rhombus c Kite 13 a β30β b β48β c β50β d 6β 0β e β80β f 1β 30β g 4β 0β h β80β i β105β 14 a β2, 2β b β1, 1β c β0, 2β
1 C 5 D 9 E 13 C
2 B 6 E 10 A
3 D 7 C 11 C
4 C 8 B 12 E
Extended-response questions
1 a i β32Β°β ii β32Β°β b i Corresponding iii Supplementary c i β21Β°β ii 1β 59Β°β 2 a β30Β°β b c β35β d 3 y c
iv β58Β°β
iii 6β 9Β°β 6β 5Β°β It stays the same.
4 3 2 1
β4 β3 β2 β1β1O a
iii β148Β°β ii Cointerior
β2 β3 β4
x
1 2 3 4
b
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Ch7 Chapter review
Multiple-choice questions
816
Answers
8A Building understanding 1 a C d E 2 Answers may vary.
b D e F
c A f B
c Extremely large population makes primary data difficult to collect. d Sensitive topic might make students less keen to give honest and reliable answers. e Cultural issues and the different cultural groups that exist in the community make collection difficult. 6 The data is often collected by a market research company. It is not always possible to know how the data is collected, the areas it is collected from and whether there was a bias introduced in the surveys. 7 a Population is the entire group of people but a sample is a selection from within it. b If the population is small enough (e.g. a class) or there is enough time/money to survey the entire population (e.g. national census). c When it is too expensive or difficult to survey the whole population, e.g. television viewing habits of all of Victoria 8 a The answers stand for different categories and are not treated as numbers. They could have been AβE rather than 1β5. b i β1 =βstrongly disagree, β2 =βsomewhat disagree, β3 =βsomewhat agree, β4 =βstrongly agree. ii β1 =βpoor, β2 =βsatisfactory, β3 =βstrong, β4 =β excellent. iii 1β =βnever, β2 =βrarely, β3 =βsometimes, β4 =β usually, β5 =β always. iv β1 =βstrongly disagree, β2 =βdisagree, β3 =β neutral, β4 =βagree, β5 =βstrongly agree. 9 a Excludes people who have only mobile numbers or who are out when phone is rung; could bias towards people who have more free time. b Excludes people who do not respond to these types of mail outs; bias towards people who have more free time. c Excludes working parents; bias towards shift workers or unemployed. d Excludes anyone who does not read this magazine; bias towards girls; bias towards those who use social media. e Excludes people who do not use Facebook; bias towards specific age groups or people with access to technology. 10 a For example, number of babies at a local playground. Other answers possible. b Count a sample, e.g. just one floor of one car park. 11 a Too expensive and difficult to measure television viewing in millions of households. b Not enough people β results can be misleading. c Programs targeted at youth are more likely to be watched by the students. d Research required. 12 a Too expensive and people might refuse to respond if it came too often. b English as a second language can impact the collection of data (simple, unambiguous English is required). Some people from particular cultures may not be keen to share information about themselves. c Some people cannot access digital technologies and they would be excluded from the results. d Larger populations and a greater proportion of people in poverty can make census data harder to obtain.
U N SA C O M R PL R E EC PA T E G D ES
Answers
Chapter 8
Now you try
Example 1 a Discrete numerical b Categorical
Example 2 a Secondary, e.g. find the results published in a magazine or journal. b Primary, e.g. run a survey of everyone in the class.
8ο»ΏA
Exercise 8A
1 a Categorical b Numerical c Numerical d Categorical e Numerical f Numerical 2 a Discrete numerical b Continuous numerical c Continuous numerical d Categorical e Categorical f Categorical g Discrete numerical h Discrete numerical i Continuous numerical j Continuous numerical k Continuous numerical l Discrete numerical m Continuous numerical n Discrete numerical o Categorical p Discrete numerical q Discrete numerical r Categorical 3 a Observation b Sample of days using observation or secondary source records within each day c Census of the class d Sample e Sample f Sample using secondary source data g Census (this question appears on the population census) h Census of the class i Sample j Results from the population census k Observation l Observation m Sample n Sample 4 a Secondary β a market research company b Secondary β department of education data c Primary data collection via a sample d Secondary source using results from the census e Secondary source using NAPLAN results or similar 5 a Proximity to the Indian Ocean makes first hand collection of the data difficult and size of the ocean makes it hard to get representative data. b Too many people to ask and a sensitive topic means that using the census results as your source would be better.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
U N SA C O M R PL R E EC PA T E G D ES
14 a Mean: β7β, median: β7.5β, mode: β10β, range: β12β b i Increases by β5β ii Increases by β5β iii Increases by β5β iv No effect c i Doubles ii Doubles iii Doubles iv Doubles d Yes; e.g. if the set were duplicated: β1, 1, 3, 3, 5, 5, 10, 10,β β 10, 10, 13, 13β. 15 a β7βand β13β b 9β , 10, 11β c They must be β8, 10βand β12β. 16 Numbers are: βx, x + 1, x + 2β;
Answers
13 It gives ownership and establishes trust where there may not have been any. It also ensures a deeper understanding of the process and need for honesty in the collection and use of any data. 14 Different people are chosen in the samples. Larger, randomly selected samples give more accurate guides. 15 a i Answers may vary. ii Answers may vary. iii Random processes give different results. b i Different vowels have different frequencies of occurring. ii If a high frequency word has an unusual range of vowels, e.g. a web page about Mississippi.
817
8B
Building understanding
b β1β b 1β 2β b β6β
c β9β
c 9β β c β7β
d 7β β d β7β
Now you try Example 3 a β13β Example 4 a β5β
8C
b β26β
c β25β
b β4β
b 1β 3β c β8β d β18β b β6β c β3β d β7β b 3β 6β c β24β ii 3β β iii β1β ii 7β β iii β2β ii 1β 2β iii β11β ii 2β 6β iii β25β ii 1β 6β iii β10β ii 3β 4β iii β55β ii 7β 1β iii β84β ii 4β 1β iii β18β b β8β c β3β e β15β f 5β 6β b β9β c β7β 1β 1β e β4.5 or 4β_ f 5β .5 or 5β_ 2 2 7 a β9.5β b 1β 1β c β11β d β8.5β 8 a β16.6β b β17.5β c β12β 9 a Brent b Brent c Ali d Brent 10 a β2β b β4β c β12β d 2β 5β e β2.1β f 4β β 11 a β12β b β2β c β0β saves 12 a β3βvalues; e.g. {β5, 5, 8}β b Yes, e.g. β{0, 0, 5, 5, 5, 7, 20}β 13 The mean distance is increased by a large amount (the median is basically unaffected). a a a a b c d e f g h 5 a d 6 a d
β7β β4β 6β 7β i β6β i β11β i β18β i β17β i β32β i β40β i β102β i β91β 1β 0β 7β β β12β 6β β
Building understanding 1 a True d True 2 a β5β
b False e False
b β3β
c True
c Sport
d β20β
Now you try Example 5 a 1β 5β d 2β β
b 5β β e β7β
c 6β β f β1β
Example 6 200 180 160 140 120 100 80 60 40 20 0
Arm span chart
Matt
Pat Carly Kim Tristan Student
Exercise 8C 1 a c e 2 a
4β β β7β Person with β7β trips β8β b β24β
b 1β 7β d 2β β f β2β trips c β8β d β7β
e β2β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8B
Exercise 8B 1 2 3 4
18 βmean = _ β 77 β, median = _ β7 β 240 24
Arm span (cm)
1 a β10β 2 a β9β 3 a β30β
(β 3x + 3)β βmean = _ β β= x + 1 = medianβ 3 17 a i β4, 4, 4β; answers may vary. ii β1, 4, 3, 9, 1β; answers may vary. iii 2β , 3, 8, 10, 10β; answers may vary. iv β1, 2, 3, 4, 4, 16β; answers may vary. b It is possible. (Hint: Divide every number by the current range.)
818
Answers
c Red
d β27β
7 a 60 50 40 30 20 10 0
Dogs sold
U N SA C O M R PL R E EC PA T E G D ES
Number of sales
Answers
3 a β2β b β7β e Favourite colour in 7B
2010 2011 2012 2013 2014 2015 Year
pink blue yellow green red
male
Colour
4 a β120 cmβ c β60 cmβ
Mitchellβs height
160 140 120 100 80 60 40 20 0
Fatuβs height
8
9
10 11 Age (years)
e
or
nc
ie
Sc
ish
ist
H
gl
hs
Subject
12
6 a
1β 990β 1β 995β 2β 000β 2β 005β 2β 010β 2β 015β
b c d e
En
12
at
10 11 Age (years)
rt
9
A
8
y
Number of votes
160 140 120 100 80 60 40 20 0
b 2β 0 cmβ d 1β 1βyears old
M
b
Height (cm)
8C
Height (cm)
5 a
female
2β 010β c 2β 014β d β2012β e Male It has increased steadily. Approx. β$38 000β Approx. β$110 000 β $130 000β It is unequal. The axes have no labels and the graph does not have a title. c Favourite subjects 35 30 25 20 15 10 5 0
b 8 a b c 9 a b
Using public transport
β30β
β25β
β40β
β50β
6β 0β
β55β
Driving a car
β60β
β65β
β50β
4β 0β
β20β
β20β
Walking or cycling
β10β
β20β
1β 5β
1β 5β
β25β
β60β
2β 005βand β2010β 2β 010βand β2015β Environmental concerns; answers may vary. Public transport usage is increasing; answers may vary.
Four times as popular One and a half times as popular Music β7β β7β β3M : 8, 3S : 4β β3Mβ β3Mβbecause best student got β10β. β1500β β104 000β Increased Approx. β590 000β passengers β2.4β Because β1 + 1 + 2 + 2 + 2 + 3 + 3 + 3 + 3 + 4βcan be thought of as (β 2 Γ 1)β+ (β 3 Γ 2)β+ (β 4 Γ 3)β+ (β 1 Γ 4)ββto calculate the total number of children. 13 a Symmetric, with a mean of β4.5β, equal to the median of β4.5β b Negatively showed, with a mean of β4.25β, less than the median of β4.5.β c Positively showed, with a mean of β4.85β, greater than the median of β4.5.β d Symmetric, with a mean of 3, equal to median of 3. 14 β240β d e f 10 a b c d e 11 a b c d 12 a b
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
8D
b β4 kgβ c β5 kgβ b β110 cmβ e β140 cmβ
d 4β .5 kgβ c β8β years
3
Now you try
4
Example 7 a
Outside temperature
5 6
Temperature (Β°C)
35 30
7
25 20 15 0
4 pm
5 pm
6 pm Time
7 pm
8 pm
8
b 2β 5Β° Cβ
Exercise 8D 1 a
O
Distance (km)
Dogβs weight over time
10 8 6 4
Ja n Fe b M ar A p Mr ay Ju n Ju l A ug Se p O ct N ov D ec
2 0
Month
1 2 3 4 5 Time (hours)
Distance jogged
30 24 18 12 6
O
Ja n Fe b M a Ar pr M ay
Weight (kg)
10 8 6 4 2 0
Month
Weight (kg)
300 240 180 120 60
b
Dogβs weight over time
b 6β .5 kgβ 2 a
Distance (km)
Example 8 a β10 kmβ b β5 kmβ c In the third hour (from β2βto β3β hours) d In the first hour e 3β kmβ
1 2 3 4 5 Time (hours)
9 The steeper sections of graph correspond to the person/object travelling faster. Horizontal section correspond to stationary periods. 10 a Time should be on the horizontal axis, not length; in practice, this would be a meaningless graph with a number of dots and lines going in each direction. b Length of new born babies over time, or weight over time. Possibly you could calculate the average (mean) length or weight month by month so it is easier to see on the graph if these are trends. 11 a The city is in the Southern hemisphere because it is hot in JanuaryβDecember. b The city is quite close to the equator because the winter temperatures are reasonably high.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
8D
U N SA C O M R PL R E EC PA T E G D ES
1 a β3 kgβ 2 a β45 cmβ d 1β 5 cmβ
Answers
Building understanding
b Weight increases from January until July, then goes down suddenly. c July a β23Β° Cβ b 2β : 00β pm c β12 : 00β am d i β10Β° Cβ ii β18Β° Cβ iii 2β 4Β° Cβ iv β24.5Β° Cβ a β200 kmβ b β80 kmβ c At rest d In the first hour e β40 kmβ a β2β hours b 5β kmβ c Fifth hour a July b Jan to July and Oct to Dec c β20βmegalitres because the level stayed the same a At β7 amβand β8 pmβ b At β8 amβand β11 pmβ c i Around β7 amβ(heater goes on) ii Around β8 amβ(turns heater off) iii Around β8 pmβ(heater put back on) iv Around β11 pmβ(heater turned off) d Answers may vary. a Distance travelled
819
820
Answers
c Average temperature (Β°C)
Stem Leaf β1β 1β 2 3 4 4 5 7β β2β β0
4
8
9β
β3β β1
2
3
5β
β2 | 8 means 28β
U N SA C O M R PL R E EC PA T E G D ES
Answers
4 a
Temperature in a year 40 35 30 25 20 15 10 5
b
Stem Leaf β2β β0
Ju l A ug Se p O ct N ov D ec Ja n Fe b M ar A p M r ay Ju n
0
3β β β9β
Month
4β β β5
d Maximum occurs in the middle. e The city is in the Northern hemisphere because it is hot in JuneβJuly.
5β β β1
c
8E
Building understanding 1 a β5β 2 a β39β 3 β57β
b β2β
b β27β
c β134β
2β 7
9
9β
2 2 4β | 7 means 47β
3
5
6
8
8β
Stem Leaf β1β β6
6
8β
2β β β1
4
8
9β
3β β β1
2
3
5β
4β β β1
8
9β
5β β β0β
3β | 5 means 35β
8E
Now you try
Example 9 a β18, 21, 21, 24, 30, 32, 38, 39, 43β b β9β c Youngest is β18βyears old. Oldest is β43βyears old. d β25β years e β30βyears old
d
β1β β1
2
2β β β7
9β
3β β β2
7
4β 8
8β
5
6β
4β β
Example 10
β5β
Stem Leaf β2β 4β
5
9β
β3β 2β
5
6
β4β 1β
2
8β
8β
5 a
Exercise 8E
β13, 17, 20, 22, 24, 28, 29, 31, 33, 35β β10β β13, 35β β22β β26β β8, 9, 10, 11, 13, 15, 17, 18, 21, 24β β10β i False ii True iii True βRange = 20, median = 17β βRange = 31, median = 26β R β ange = 19, median = 40.5β
β6β β0
0β
7β β β3
8β
8β β β1
7β
6β | 0 means 60β
β3 | 2 means 32β
1 a b c d e 2 a b c 3 a b c
Stem Leaf
Stem Leaf β8β β0
4
9β β β0
6β
1β 0β β1
4
5β
1β 1β 0β 3 4 4 5 9β
β10 | 4 means 104β
iv False
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
b
Stem Leaf 6β 4
5
β10β 2β
6
8β
β11β 3β
5
5
6
8β
b c
7β
β12β 0β β
U N SA C O M R PL R E EC PA T E G D ES
d 14 a b c d e
Answers
β8β 1β β9β 1β
iv Between β31βand β49β years v β18β vi Between β18βand β28β i True ii False iii True iv True v False vi False Cannot determine how many people are exactly aged β40β years. Close to β30β years Minimum: ββ 29Β° Cβ, maximum: β23Β° Cβ i β5β ii 1β 0β iii 1β 1β Because ββ 05βand β05βare different numbers ββ 5.2Β° Cβ ββ 8Β° Cβ; it is β2.8Β° Cβlower than the mean.
821
β11 | 5 means 115β Stem Leaf β15β 5β
7
8β
β16β 2β
5
7β
β17β 3β
4
8β
β18β 1β
4
4
5
7β
β19β 2β
2
3
3
6
8F
Building understanding
9β
1 a Playing sport c More 2 a Rugby c _ β1 β 5
β16 | 2 means 162β
d
Stem Leaf β39β 1β
5
6β
β40β 1β
2
4
5
6
6
8
β41β 1β
β2
β3
β5
β6
β7
β8β
9β
β42β 0β β
Now you try Example 11 a Sport
10 a b c 11 a b 12 a b c 13 a
k
9 a b c d e
β2β b B c β1β d B e β1β N g β2β h 1β β i N 1β 0β b β1β c β8β 5β 8β e β10β seconds i β49β years ii β36β years Radio station β1β i β33βto β52β years ii β12βto β31βyears, or 13 to 32 years i β30β ii β52β Year β6β i β151.5 cmβ ii β144.5 cmβ Year β7β The difference between the two plots would increase, with the median on the left going down and median on the right going up. 1β 5β 1β 3β aβ βis β5βor β6β, βbβis β0β, βcβis β8βor β9β, βdβis β0β. Easier to compare sizes of different stems visually. Helps in noticing trends and calculating the median. Answers may vary, but the mean should be greater than the median. Answers may vary, but the mean should be less than the median. Answers may vary but the mean and median should be approximately equal. i β30β ii 1β 0β iii Between β10βand β19βyears old
r wo me
Ho
6 a f 7 a d 8 a b c
b Basketball d _ β2 β 3
8F
β41 | 3 means 413β
b Watching television
Entertainment
c
b
Entertainment
Sport
Homework
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Exercise 8F 1 a
food
drinks
b
household items
other
other
household items
food
drinks
2 a β20β b i
_ β3 β
10 iii _ β7 β 20 c i β108Β°β iii 1β 26Β°β
ii _ β1 β 4 iv _ β1 β 10 ii β90Β°β iv β36Β°β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
822
Answers
d
Krishna Nikolas It means Nikolas spends more time playing sport. β6β β10β c Bird was chosen by _ β1 β, which would be β2.5β people. 8
6 a b c 7 a b
Answers
France
China USA United Kingdom
U N SA C O M R PL R E EC PA T E G D ES
β1 βof β40βis not a whole number. d Each portion is _ β1 ,β but _ 3 3
e β24, 48, 72βor β96βpeople participated in each survey.
3 a
5 ,β 15β0Β°β β 8 a β_ 12
pink
1 β, which is greater than β100%β. β1 β+ _ β1 β= 1β_ b Because _ β1 β+ _ 2 3 4 12
red
9 a
C
blue
green
yellow
b
pink
A
D
B
b 2β 4β c β6β
red
blue
green
yellow
c Higher proportion of Year β7βs like red; higher proportion of Year β8βs likes yellow. d
red
4 a i b
green
β20β
yellow
ii β10β
blue
iii β6β
1 a b c d 2 a b 3 a d 4 a
iv β3β iv β15β
25
pink
iv β4β
fruit
chips
Categorical Continuous numerical Discrete numerical Continuous numerical i β10β ii β6β iii β5β i β13β ii β20β iii β19β β2β children b 1β 7β families c β7β β3β children e 8β β children Outside temperature 30
Temperature (Β°C)
Ch8 Progress quiz
Progress quiz
chocolate
20 15 10 5 0
9 am
pies
c i Chips iii Chocolate 5 a i β9β b History
ii Fruits and pies
ii β6β
iii β15β
English
Maths Science
1 pm 11 am Time
b 5 a d 6 a b c d
Around β24Β° Cβ β20 kmβ b β10 kmβ β10 kmβ e 5th hour β13β towns Maximum β37Β° Cβ, minimum β14Β° Cβ β23Β° Cβ β25Β° Cβ
7
Stem Leaf β0β β7
9β
1β β β0
2
2β β β0
1β
6
8
8
3 pm
c At rest
9β
1|6 means 16
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
Car 9 a Public transport
2 3
1
U N SA C O M R PL R E EC PA T E G D ES
Bicycle
Answers
ii Blue, because there are still β4βblue marbles (and only β3β red). iii Red and blue are equally likely because there are still β4β of each colour present.
8 a
823
4
6
5
b
b All sectors have the same size; i.e. β60Β°β. c
Methods of transport
Public transport
Bicycle
Car
6
1
2
8G
Building understanding 1 a D 2 a Certain c Unlikely
b A
c B d C b Even chance d Impossible
3
4
d By replacing the β5βwith a β6β(so that there are two faces with β6β). e With β52βequal segments 10 a Spinner 1: _ β1 ,β spinner 2: _ β1 β, spinner 3: _ β1 β 4 3 9 b _ β1 β 2
Now you try
c i
b True
c True
8G
Example 12 a False
5
d False
Exercise 8G
1 a True b False c True d False 2 a True b False c True d False e True f False 3 a D b C c A d B 4 a i True ii False iii False iv True v True b i Spinner landing on yellow (other answers possible) ii Spinner not landing on red iii Spinner landing on green, blue or red iv Spinner landing on blue or on red 5 a Spinner β3β b Spinner β2β c Spinner β1β 6 Answers may vary. 7 a Blue, red and green are equally likely. b Red and green both have an even chance. c Green and blue are equally likely; red and blue are not equally likely. d Blue is certain. e Blue, red and green are all possible, but no two colours are equally likely. f Red and blue both have an even chance. 8 a Choosing a red marble and choosing a blue marble b i Red, because there are still β4βred marbles (and only β3β blue).
red
ii
blue
Other answers possible
green
d i β50%β ii β0%β iii 0β % , 50%β iv β50%β e If the two fractions are equal, the two events are equally likely. f The proportion of the spinnerβs area cannot exceed β100%β (or β1β) and must be greater than or equal to β0%β.
8H
Building understanding
1 a C b A c D 2 a Sample space b Zero d More e Even chance
d B c Certain
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
824
Answers
Example 13 a β{1, 2, 3, 4, 5}β Example 14 a _ β1 β 5
c 1β β d Spinning a number less than β7β. Other answers are possible.
b β{β2, 4β}β b β0.6β
c β40%β
Exercise 8H
13 a βa = 1β, βb = _ β1 ββ, βc = 1β, βd = _ β1 ββ, βe = 1β, βf = _ β2 ,ββ βg = _ β3 ,ββ βh = 1β 5 5 2 3 10 β b β_ 13 _ c β3 β 7 19 β b 2β 10β c β840β 14 a Yes, β_ 210 15 a i
U N SA C O M R PL R E EC PA T E G D ES
Answers
Now you try
1 a c 2 a c 3 a c
β{1, 2, 3, 4, 5, 6}β β red, green, blue}β { {β1, 3, 5}β {β1, 2, 3, 4, 5}β {βT, R, A, C, E}β {βT, R, C}β
β{1, 2, 3, 4}β β{L, U, C, K}β β{1, 2, 3}β β{1, 2, 4, 6}β β{A, E}β β{T, A}β
c β0.5β
5 a β3β: red, green, blue
c _ β2 β 3 e β0β 6 a β{1, 2, 3, 4, 5, 6, 7}β c β0β
green
1β b β_ 7 2β d β_ 7 f _ β4 β 7
2
5
31
β12βred, β6βblue, β4βyellow, β2β green ii _ β1 β 12 b i β18βred, β9βblue, β6βyellow, β3β green 1β ii β_ 12 c 12 1 ββ _ d No, because it is always β1 β β_ β1 ββ _ β1 ββ. 2 4 6 β4 ββ, βodd = {β1, 3, 5, 7, 9}ββ 12 a βEven = {ββ2, 4, 6, 8β}βwith probability _ 9 with probability _ β5 β 9 b _ β1 β, spinning a number greater than β6β(other answers 3 possible)
blue
ii Cannot be done because adds to more than β1β. iii Cannot be done because adds to less than β1β. iv
1β b β_ 3 d _ β2 β 3
e _ β3 β 7 g Number chosen is less than β10β; answers may vary. 7 a β{M, A, T, H, S}β b β0.2β c β0.8β d β60%β 1β 8 a β_ b _ β2 β c _ β5 β e _ β4 β d _ β9 β 11 11 11 11 11 f Choosing a letter in the word ROPE; answers may vary. 9 a 3β 0%β b β50%β c 8β 0%β 10 a {β2, 3, 4, 5, 6, 7, 8, 9}β b β0.5β c i β0.375β ii β0.375β iii β0β d Possible spinner:
11 a i
red
1β b β_ 4 d β75%β
4 a {β1, 2, 3, 4}β
8I
b d b d b d
red
green
blue
b 1β β x β yβ. Also βxβ, βyβand β1 β x β yβmust all be between β0βand β1β.
8I
Building understanding 1β 1 a β_ 10 2 a _ β 53 β 100
b _ β2 β 5 _ b β 47 β 100
c _ β3 β 10 c _ β1 β 2
Now you try
Example 15 a _ β1 βor β0.25βor β25%β 4 c β20β
b _ β5 βor β0.625βor β62.5%β 8
Exercise 8I
1 βor β0.25βor β25%β 1 a β_ 4 b _ β2 β 3 c 1β 00β 2 a β50β c
e 3 a b c
b _ β2 β 5 1 _ _ β β d β3 β 5 10 No, just that nobody did it within the group surveyed. β500β 1β 750β i β7β tails ii More
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
yellow
red
U N SA C O M R PL R E EC PA T E G D ES
90Β° 120Β° 60Β° 90Β°
Answers
4 a β100β b 3β 00β c Yes (but this is very unlikely) d From β2β throws 5 Answers may vary. 1β 3 β d β60β 6 a β_ b β_ c _ β 31 β 4 100 100 7 a β5β b β40β c β70β f 1β 26β e _ β4 β d _ β26 β 7 35 8 a 2β βred, β3βgreen and β5β blue b i Yes ii Yes iii Yes iv Yes v No 9 a C b D c B d A 10 a Answers may vary. b Answers may vary. c Answers may vary. d Answers may vary. e No technique for finding theoretical probability has been taught yet. f Less accurate g More accurate 11 a False; there is no guarantee it will occur exactly half of the time. b False; e.g. in two rolls, a die might land β3βone time. The β1 ,ββ though. theoretical probability is not _ 2 c False; perhaps the event did not happen yet but it could. d True; if it is theoretically impossible it cannot happen in an experiment. e False; experiment might have been lucky. f True; if it is certain, then it must happen in an experiment. 12 a Red: β25%β, green: β42.2%β, blue: β32.8%β b Fifth set is furthest from the final estimate. c Have β4βgreen sectors, β3βblue and β3βred sectors d Red: β90Β°β, green: β150Β°β, blue: β120Β°β
825
Chapter checklist with success criteria
Continuous numerical Secondary source by looking at the census data β53β β29β β25β β12β 4β βchildren have β2βpets; β8βpets is an outlier.
Height chart
160 140 120 100 80 60 40 20 0
Tim Phil Jess Don Nyree Name Room temperature
4 PROBABLE (or PEBBLIER) 5 a β60Β°β b β192 β 3.6 = 188.4Β°β c _ β1 βor β25%β 4 d _ β 99 βor β99%β 100 _ e β 1 βor β5%β 20 6 A spinner with these sector angles has actual probabilities close to the experimental probabilities. Sector colour
Red
Green
Purple
Yellow
Sector angle
1β 20Β°β
9β 0Β°β
6β 0Β°β
9β 0Β°β
Actual probability
_ β 40 β
_ β 30 β
_ β 20 β
_ β 30 β
Experimental probability
40 β β_ 120
_ β 32 β
_ β 19 β
_ β 29 β
120
120 120
120 120
120 120
. a.m . 12 p. m . 1 p. m .
β2β
a.m
β8β
.
β6β
10
No. of students
25 20 15 10 5 0
a.m
β3β
Temperature (Β°C)
β2β
d β72β
9
β1β
c β90β
11
Height (cm)
1 2 3 4 5 6 7 8
9
1 2β , 2, 5, 7, 8, 12β 2 a β4β b 1β 0β 3 No. of cars 0β β β4β
green
Time
10 The cyclist travels β15 kmβin the first hour, is at rest during the second hour and travels β15 kmβbetween the second and fifth hour. The cyclist is travelling fastest during the first hour. 11 β13, 16, 16, 20, 20, 21, 22, 25, 25, 26, 28, 29, 30, 32β; β median = 23.5β(Β° C)β 12
Stem Leaf 1 06
2 357 3 1344667 β2 | 5βmeans β25β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Ch8 Problems and challenges
Problems and challenges
purple
826
Answers
5 a β11β 6 a 1β 2β 7 a β4β
13
Answers
TV
homework internet
b 5β β b β5β b β$31β
b _ β1 β c _ β19 β 8 20 β{1, 2, 3, 4, 5, 6}β b β{D, E, S, I, G, N}β d β9β 5β 1β i β_ ii β_ 9 3 1β 4β iv β_ v β_ 3 9 2β _ β1 β b β_ c 26 13 β42%β b
8 a β1β 9 a c 10 a
d 2β β d 3β β d $β 3β
e β0β d _ β3 β 4 {βheads, tails}β β blue, yellow, green}β {
U N SA C O M R PL R E EC PA T E G D ES
sport
c 3β .5β c β3.5β c β$30β
b
14
TV Internet
Sport
Homework
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
11 a
Multiple-choice questions
Chapter review
Extended-response questions
Short-answer questions
1 a
1
1 B 6 C
2 D 7 A
1β β_ 52 β50%β
4β vi β_ 9
c β25β
3 E 8 C
4 D 9 C
3β d β_ 13
d β250β
5 C 10 B
Rainy days during a year
12 10 8 6 4 2 0
Ja n Fe b M ar A pr M ay Ju n Ju l A ug Se p O ct N ov D ec
No. of rainy days
Age (years)
1β b β_ 4
13 a _ β1 β 2
Studentsβ ages
30 25 20 15 10 5 0
Month
Sven Dane Kelly Hugo Frankie Name
2 a β40β d No. of students
Ch8 Chapter review
12 a
15 These events are not equally likely. It is more likely to flip heads on a coin than to roll a β3βon a β6β-sided die. 16 Sample space {ββ1, 2, 3, 4, 5, 6β}β; favourable outcomes {ββ1, 3, 5β}β 17 β33.3%β 18 β400β 3 β(or β0.3β, or β30%β) 19 β_ 10
iii β0β
b 4β 00β
b β66β
c 1β 0β hours
Studentsβ TV watching
18 15 12 9 6 3 0
c _ β 6 β= _ β3 β 92 46
Discrete numerical Categorical Line graph β40β Cheesecake c _ β7 β 40 d i Yes ii No e β80β
d e f 2 a b
8
9 10 11 TV watched (hours)
3 a 4β β students c β1β pm 4 a β105Β°β c
12
iii Yes
iv Yes
b 2β β students d β6β students b β25%β
Chicken
Chinese
Pizza Hamburgers
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
Chapter 9 9A Building understanding b False
c False
9B Building understanding
U N SA C O M R PL R E EC PA T E G D ES
1 a True 2 True 3 a β19β
1β 2β β4β, β(e.g. 2 + 3 = 5,β βββ3 + 2 = 5,β βββ5 = 2 + 3,β βββ5 = 3 + 2)β Yes; if β5βis changed to β1β, then β1 + 2 = 3βis a true equation. Yes; if β+βis changed to βββ, then β5 β 3 = 2βis a true equation.
Answers
16 a b c d
827
b 1β 9β
c True
1 a True 2 a 1β 2β 3 a xβ β
Now you try Example 1 a βNβ
b βEβ
Example 2 a False d False
Example 3 a β3a = 21β
c βt β 3 = 24β
c βEβ
b True e True
d βNβ
e βEβ
c True f False
b βp + q = 10β
1 ββ Γ b = 150β d _ ββ 1 ββ Γ a +β β_ 2 4
Exercise 9A
c True c 1β 3β c bβ β
d False d β6β d βdβ
Now you try Example 4 a 2β 1β
Example 5 a qβ = 18β
b 1β 5β
b βx = 8β
c βa = 10β
Exercise 9B
1 a β6β b β50β c 8β β d β17β 2 a 3β β b β6β c 1β 0β d β4β e β6β f β70β g 2β 0β h β19β 3 a βy = 8β b βl = 2β c lβ = 6β d βd = 2β e βl = 12β f βa = 6β g sβ = 12β h βx = 8β i eβ = 8β j βr = 10β k βs = 8β l βz = 3β 4 a βp = 3β b βp = 4β c qβ = 3β d βv = 5β e βb = 1β f βu = 4β g gβ = 3β h βe = 3β i dβ = 4β j βd = 6β k βm = 4β l βo = 3β 5 a βx = 3β b βx = 7β c βx = 5β d xβ = 4β e βx = 1β f βx = 5β 6 a β11β b β12β c β16β d 3β 3β e β30β f β2β 7 a βBβ b xβ = 13β c β13β years old 8 a β10x = 180β b xβ = 18β 9 a β2w = 70β b w β = 35β 10 a β4.5x = 13.5β b xβ = 3β 11 a βy + 12 = 3yβ b yβ = 6β 12 a βx = 9β b 2β x + 1 = 181βso βx = 90β 13 a βx = 14β, so β14, 15βand β16βare the numbers. b LHS is β3x + 3βor β3(β x + 1)ββ, which will always be a multiple of β3β. 14 a βx = 2βand βy = 6β; answers may vary. b xβ = 12βand βy = 10β; answers may vary. c βx = 1βand βy = 1β; answers may vary. d xβ = 12βand βy = 0.5β; answers may vary. e βx = 10βand βy = 0β; answers may vary. f xβ = 2βand βy = 2β; answers may vary.
9C
Building understanding 1 a 1β 0d + 15 = 30β c 2β a + 10 = 22β 2 a C β β b βDβ
b 7β e + 10 = 41β d xβ + 10 = 22β c Eβ β d βBβ
e βAβ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9A
1 a Yes b Yes c No d Yes e No 2 a Eβ β b N β β c Eβ β d Eβ β e βEβ f βNβ g βEβ h Eβ β i βNβ 3 a True b False c True d True e False f False g True h False i True j True k False l True 4 a False b True c False d True 5 a True b True c False d True 6 a True b False c True d True e True f False g False h True i True 7 a β3 + x = 10β b 5β k = 1005β c βa + b = 22β d 2β d = 78β e β8x = 56β f β3p = 21β g _ βββ― t β―ββ = 12β h qβ + p = q Γ pβ 4 8 a β6c = 546β b β5x = 37.5β c β12a + 3b = 28β d βf + 10 = 27β e βj + 10 + m + 10 = 80β 9 βm = 3β 10 βk = 2, k = 6β 11 βx = 1βand βy = 5, x = 2βand βy = 4, x = 3βand βy = 3, x = 4βand β y = 2, x = 5βand βy = 1β 12 Because it does not make a difference what order you add, so the value of βa + 3βwill always equal the value of β3 + aβ. 13 Because for any value of b, the two sides will differ by β10β(and therefore never be equal.) 14 a βSβ b βSβ c βAβ d A β β e βNβ f βNβ g βAβ h S β β i βAβ j βSβ k βNβ l N β β 15 a β6 = 2 Γ 3β; answers may vary. b 5β β 4 = 1β; answers may vary. c β10 Γ· 2 = 7 β 2β; answers may vary. d 4β β 2 = 10 Γ· 5β; answers may vary.
b False b β17β b βcβ
828
Answers
10 a
Example 6 a βq + 16 = 19β c β3 = 5bβ
b 2β 1k = 30β d 1β + 3b = 2aβ
Example 7 a β4βis added to both sides b Both sides are doubled c β9βis subtracted from both sides
+2
b
x=3
Γ7
Γ2
Γ7
7x = 21
+8
+2
7x + 2 = 23
x=3 2x = 6 8 + 2x = 14
Γ2 +8
11 a True; you can β+ 3βto both sides and then ββ 3βto get the original equation again. b True; simply perform the opposite operations in the reverse order, so β+ 4βbecomes ββ 4β. c True; use the operations that take equation 1 to equation 2 and then the operations that take equation 2 to equation 3. d False; e.g. equation β1β: βx = 4β, equation β2β: βx = 5β, equation β3β: β 2x = 8β.
U N SA C O M R PL R E EC PA T E G D ES
Answers
Now you try
9C
Exercise 9C 1 a c 2 a d g 3 a c e g 4 a c e 5 a d 6 a c 7 a c 8 a
β6 + x = 10β b β12x = 21β β3 = bβ d β5q = 4β β6 + x = 11β b β6x = 14β c 3β = 2qβ 6β + a = 10β e β12 + b = 15β f β0 = 3b + 2β 4β = 7 + aβ h β12x + 2 = 8β i β7p = 12β Subtracting β2β b Adding β2β Dividing by β10β d Multiplying by β2β Dividing by β3β f Adding β3β Dividing by β4β h Adding β4β β3(β x + 4)ββ = 27β b β18 = 3β(a + 5)ββ β3d = 18β d β3(β 11 + a)ββ = 36β β3(β 3y + 2)ββ = 33β f 3β (β 2x + 4)ββ = 30β βE β(+ 2)ββ b βA β(+ 4)ββ c βD (Γ·2)β B β (β 2)β e βC (Γ 2)β βΓ 3βthen β+ 2β b βΓ 10βthen ββ 3β βΓ·5βthen ββ 2β d ββ 10βthen βΓ·3β β2q = 2β b 1β 0x = 7β β3 + x = 20β d βx = 60β 3x + 2 = 14 β2 β2 3x = 12 Γ·3 Γ·3 x=4 Γ 10
10x = 40
Γ 10
+1
+1
10x + 1 = 41
b
+3
Γ·5
+2
c
β4
Γ2
+8
Γ·2
5x β 3 = 32
+3
5x = 35 x=7
x+2=9
Γ·5
9D
Building understanding 1 2 3 4
a True a β6β βg = 2β a ββ 5β
x = 10
x + 8 = 18
(x + 8) Γ· 2 = 9
c βy = 28β
b βm = 20β
c βp = 8β
Exercise 9D
1 a i βx = 5β b i βy = 7β c i βq = 40β 2 a βm = 9β b βg = 11β e βt = 2β f βq = 3β i βj = 4β j βl = 4β m βk = 5β n βy = 9β q βb = 12β r pβ = 11β 3 a 7a + 3 = 38 β3 β3 7a = 35 Γ·7 Γ·7 a=5
c
+ 10
Γ·2
β6
+8
9 a i β0 = 0β ii β0 = 0β iii β0 = 0β b Regardless of original equation, will always result in β0 = 0β.
d β+ 12β
b βx = 4β
Γ2
Γ·2
c βΓ 4β
Example 9 a xβ = 2β
Γ·4
β4
b βΓ·10β
e False
Example 8 a aβ = 3β
b
xΓ·2=5
c True d False b xβ = 6β
Now you try
+2
(x Γ· 2) + 4 = 9
b False
d
β3 Γ 10
4b β 10 = 14
c g k o s
ii βx = 8β ii βy = 64β ii βq = 21β βs = 9β βy = 12β βv = 2β βz = 7β βa = 8β
d h l p t
iβ = 10β βs = 12β yβ = 12β βt = 10β nβ = 3β
+ 10
4b = 24
Γ·4
b=6
2(q + 6) = 20
Γ·2
q + 6 = 10 q=4
5=
x +3 10
2=
x 10
β6
β3 Γ 10
20 = x
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
βk = 4β βa = 6β βq = 11β βt = 8β βy = 12β
Add β7β Divide by β2β βx = 9β βn = 11β βz = 10β βu = 5β βq = 2β
d h l p t
aβ = 9β nβ = 8β pβ = 1β cβ = 1β uβ = 10β
4 5 6 7 8 9
a a a a a a
aβ = 9β 2β 0x = 35β + β 8β aβ = 12β xβ = 11β xβ = 5β
b xβ = 4β
c m β = 30β d βa = 2β b β2a = 16β b βΓ· 6β c Γ β 2β b w β = 31β c βk = 63β d βm = 8β b aβ = 7β c βm = 6β d yβ = 8β b βk = 12β c βy = 4β
U N SA C O M R PL R E EC PA T E G D ES
b f j n r
b d c g k o s
Answers
4 a Subtract β3β c Divide by β5β 5 a βf = 11β e βk = 8β i βg = 4β m βd = 4β q qβ = 11β
829
3 ββ 6 a βx =β β_ 4 4 ββ d xβ =β β_ 3 7 a βr = β 7β d yβ = β 24β
24 ββ b kβ =β β_ 5 5 _ e βx =β β ββ 8 b βx = β 3β e βx = β 5β
g xβ = β 6β
h xβ = β 4β
4 ββ c w β =β β_ 3 1 ββ f βx =β β_ 3 c βt = β 16β f βk = β 9β 3 ββ i βx = βββ_ 2 β2y = 10 β y = 5β β3(β k β 7)ββ = 18 β k = 13β ββ_k ββ β 10 = 1 β k = 22β 2 βn = 30βββhβ β12 + 5x = 14.5β
Γ5
b
β 10
Γ·2
β3
Γ2
c d 15 a b c d 16 a e
5x = 20
Γ5
10 + 2x = 20
β 10
2x = 10 x=5
xβ3=2
2(x β 3) = 4
Γ·2
β3
Γ2
Yes Yes First step, should have subtracted β2βfirst. Second step, LHS divided by β3β, RHS has β3βsubtracted. First step, RHS has β5βadded not subtracted. First step, LHS has β11aβ subtracted, not β12β. βx = 4β b βx = 1β c βl = 3β d tβ = 1β βs = 5β f βb = 19β g βj = 2β h dβ = 1β
Progress quiz
1 a βEβ b βNβ 2 a βTβ b βFβ 3 a β5 + m = 12β c β9x = 72β
c c b d
βEβ d N β β Tβ β d Fβ β β2d = 24β β3c + 4b = 190β
Building understanding 1 a d 2 a 3 a c 4 a
True b False False e True β7β b β9β βb = 44β βΓ 4, h = 28β βBβ b βCβ
c True f False c False
b dβ = 15β d Γ β 13, p = 26β c A β β d βDβ
Now you try Example 10 a pβ = 50β
b βm = 10β
c uβ = 5β
d βx = 9β
Exercise 9E
1 a βa = 8β b aβ = 30β c βx = 35β d xβ = 200β 2 a βm = 12β b βc = 18β c sβ = 16β d βr = 10β e βu = 20β f βy = 18β g xβ = 4β h aβ = 16β i hβ = 10β j βj = 15β k βv = 9β l βq = 8β 3 a βh = 9β b βy = 6β c jβ = 3β d βb = 4β e βu = 3β f βt = 9β g w β = 6β h βr = 4β i qβ = 9β j βs = 3β k βl = 8β l βz = 7β m vβ = 11β n βf = 9β o βx = 2β p βd = 5β q nβ = 5β r βm = 11β s βp = 8β t βa = 9β 4 a βy = β 1β b βa = β 10β c xβ = β 10β d βx = β 48β e βu = β 30β f βy = β 10β g uβ = β 4β h βd = β 5β q 5 a _ βββ― t β―ββ = 9 β t = 18β b ββ_ββ = 14 β q = 42β 2 3 2r qβ4 _ _ c ββ ββ = 6 β r = 15β d ββ ββ = 3 β q = 10β 5 2 y f ββ_ββ + 3 = 5 β y = 8β e _ ββ x + 3 ββ = 2 β x = 5β 4 4 b βb = 157.5β c β$157.50β 6 a _ ββ b ββ = 31.50β 5 7 a βββ―_xβ―ββ + 5β 2 b _ ββ x ββ + 5 = 11 β x = $12β 2 c $β 6β 8 a The different order in which β3βis added and the result is multiplied by β5β. b Multiply by β5β c Subtract β3β d No, the difference between them is always β2.4βfor any value of βxβ. 9 a β4x = 2β(other solutions possible) b 7β x = 5β c Yes, e.g. β2x + 1 = 0β
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Ch9 Progress quiz
8 a xβ + 5 = 12 β x = 7β b c β2b + 6 = 44 β b = 19β d b e _ ββ ββ + 3 = 6 β b = 12β f 4 9 a β12n + 50 = 410β b 10 a β12 + 5xβ b c βx = 0.5β, so pens cost β50βcents. 11 a β3w = 15 β w = 5β b β4x = 12 β x = 3β c β2(β 10 + x)ββ = 28 β x = 4β d β4w = 28 β w = 7β 12 a βx = 6β b xβ = 8β c βx = 5β 13 Examples include: β x 4 _ x + 1 = 3, 7x = 14, 21 β x = 19,β β_ x ββ = x,β β2 ββ = 1β. 14 a 2x + 5 = 13 β5 β5 2x = 8 Γ·2 Γ·2 x=4
9E
830
Answers ii Divide by _ ββ1 ββ 2
Multiply by β2β
b xβ = 26βfor both of them. c Makes the first step a division (by a fraction) rather than multiplication. 7 ββ 7 ββ 17 ββ 19 ββ 11 a βx =β β_ d βx =β β_ b βx =β β_ c βx =β β_ 2 6 6 6 12 Answers will vary. Substitute βx = 12βinto your equation to ensure it is a valid solution.
12 a e i 13 a d g j
βs = 6β qβ = 1β βp = 3β βj = 2β βa = 8β βx = 1β βx = 3β
b lβ = 1β c βp = 9β d βy = 0β f βp = 12β g m β = 5β h βb = 6β j βp = 7β k βy = 9β l rβ = 8β b βa = 1β c aβ = 3β e βc = 4β f βd = 8β h βx = 3β i βx = 0β
U N SA C O M R PL R E EC PA T E G D ES
Answers
10 a i
9F
Building understanding
Building understanding 1 C β β 2 a β6β 3 a True d True
b β4β
c β2β
b False e False
d β12β c True f False
b β8m β 48β
Example 12 a β6a + 7β Example 13 a βm = 2β
c 4β 0y + 35β
b β9a β 3β
b βx = 2β
9F
b R β β e R β β b β36β
c R β β f R β β
c β8β
d β2β
Now you try
Example 15 a F β = 65β
b βB = 67β
c βA = 9β
b βm = 10β
Exercise 9G
c qβ = 4β
Exercise 9F 1 a b c 2 a d g 3 a e 4 a e i 5 a e 6 a d g 7 a d g 8 a b c d 9 a 10 a b c 11 a b c d
1 a N β β d N β β 2 a 1β 5β
Example 14 a B β = 17β
Now you try Example 11 a β3x + 21β
9G
i β3x + 6β ii β5x + 35β i β5q β 10β ii β11q β 44β i β6a + 21β ii β24a β 6β β2x + 2β b β10b + 15β c 6β a β 8β 3β 5a + 5β e β12x + 16β f β24 β 9yβ 4β 8a + 36β h β2u β 8β β4a + 2β b β5 + 3xβ c β3b β 4β d β3a + 12β β6x + 3β f βk + 6β g β2b + 6β h β5k + 1β βs = 6β b lβ = 1β c βp = 9β d βy = 0β βq = 1β f βp = 12β g βm = 5β h βb = 6β βp = 3β j pβ = 7β k βy = 9β l βr = 8β βd = 3β b xβ = 2β c βx = 5β d eβ = 1β βa = 1β f βr = 3β g βu = 1β h βq = 11β βs = 1β b βi = 1β c cβ = 5β vβ = 8β e βk = 1β f βq = 3β yβ = 4β h βf = 3β i βt = 2β βu = β 5β b βk = β 3β c pβ = β 1β qβ = β 2β e βu = β 1β f βx = β 3β pβ = β 4β h βr = β 10β i βx = β 5β i β2(β 5 + x)ββ = 14β ii βx = 2β i β3β(q β 3)ββ = 30β ii βq = 13β i β2β(2x + 3)ββ = 46β ii βx = 10β i β2β(y + 4)ββ β y = 17β ii βy = 9β βx = 1β b βx = β 1β c xβ = 5β LHS simplifies to β10β, but β10 = 7βis never true. LHS simplifies to β15β, not β4β. LHS simplifies to β6β, not β12β. βLHS = 9, RHS = 9β, therefore true. βLHS = 9, RHS = 9β, therefore true. LHS simplifies to β9β. For example β2(β x + 5)ββ β 3 β 2x = 7β. (Answers may vary.)
i βk = 13β ii βk = 38β i βb = 4β ii βb = 11β βh = 7β b βh = 9β c βm = 8β d βm = 10β βy = 23β b βx = 4β c βx = 7β βA = 7β b βq = 4β c βt = 0β βG = 27β b βx = 1β c βy = 5β β20 = w Γ 5, w = 4β i β25 = 5l, l = 5β ii Square βP = 16β b βl = 3β β25β units squared βd = 3β b β3βββg /ββcmββ3ββ m _ c 10 = ββ― β―β ββ ββ― ββ― 5 β ββ m = 50βββgrams
1 a b 2 a 3 a 4 a 5 a 6 a b 7 a c 8 a
9 a 10 a b c d e
11 a c
12 a b c d
13 a b c d
β = 68β F b βC = 10β c β12Β° Cβ d β15Β° Cβ βH = 210, H = 200.2β βH = 120, H = 139.9β The formulas give β180βand β180.1βrespectively, which are close. β30β years old β25β years old S β 3b ββ 5β(F β 32)βββ βd =β β_ b βC =β β_ 5 9 Q β 36 βx =β β_ββ 4 Check by substituting values back into equation. If βD = 20, Cβshould equal β60βnot β50β, as in row β2β. Check by substituting values back into equation. Dt ββ + 20β. (Answers may For example, βC = 2t β 10, C =β β_ 20 vary.) Abbotsford Apes β8β goals βS = 9q + 6g + bβ β0βgoals and β0βbehinds, β2βgoals and β12βbehinds, β3βgoals and β 9βbehinds, β7βgoals and β7βbehinds.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
9H Building understanding
6 Both sides have been multiplied by β2β.
5=7βx
b βCβ
Γ2
c βCβ
Γ2 10 = 2(7 β x)
Now you try
7 xβ = 6β 8 βa = 4β 9 Yes, the solution is βe = 5βbecause β2(β 5 + 1)ββ equals β12β. 10 βx = 8β 11 β15a + 20β 12 β9a + 7β 13 βb = 2β 14 βk = 32β 15 βb = 7β 16 βk + k + 1 = 19β; Kate is currently β9βyears old
U N SA C O M R PL R E EC PA T E G D ES
Example 16 β4β days
Answers
1 a βAβ
831
Exercise 9H
Problems and challenges 1 a β26β
2 3 4 5 6 7 8
b β29β
c β368β
1 ββ d 3β 1β_ 3
2β 7βββcmβ βx = 7, y = 9, P = 72βββcmβ β1458ββββmββ2ββ βa = 5, b = 2, c = 12β a βa = 22.5, b = 37.5β b βa = 10, b = 23β β26β goats, β15β chickens βx = 45; 132, 133, 134, 135, 136β
Chapter checklist with success criteria 1 2 3 4 5
β7 + 7 = 18βand β4 = 12 β xβare both equations. It is false because the LHS is β14βand the RHS is β13β. βx + 5 = 22β βb = 4β β7q = 14β
e β36β
Chapter review
Short-answer questions
False b True c True False e True f False β2 + u = 22β b 5β k = 41β β3z = 36β d aβ + b = 15β βx = 3β b βx = 6β c βy = 1β yβ = 9β e βa = 2β f βa = 10β β2x = 8β b β7a = 28β c β15 = 3rβ d β16 = 8pβ βx = 3β b rβ = 45β c βp = 9β d βb = 6β βx = 9β f βr = 4β g βq = 2β h βu = 8β βu = 8β b pβ = 3β c βx = 4β yβ = 8β e βy = 4β f βx = 15β β6 + 4pβ b 1β 2x + 48β β7a + 35β d 1β 8x + 9β βx = 8β b βx = 8β c βx = 7β yβ = 10β e βz = 5β f βq = 3β No 7 ββ, which is not whole. b Solution is βx =β β_ 11 10 a No; LHS simplifies to β10β. b i Is a solution. ii Is a solution.
1 a d 2 a c 3 a d 4 a 5 a e 6 a d 7 a c 8 a d 9 a
11 a 12 a 13 a 14 a
A β rea = 450ββββcmββ2ββ β36β b β6β β5β b β6β βy = 35β
b H β eight = 10βββcmβ c β3β c β20β d β6βββcmβ b βy = 30β
Multiple-choice questions 1 C β β 6 βCβ
2 βAβ 7 βAβ
3 D β β 8 A β β
4 βBβ 9 βBβ
5 βEβ 10 βEβ
Extended-response questions 1 a c e f
2 a b c d e
β75β cents b C β = 15 + 2tβ β$1.75β d 1β 2β seconds β81β seconds β2.5βmin in total (β50βseconds for the first call, then β100β seconds) β$500β β30β hours at β$x / hourβ, and β10βββ hoursβ at β$(x + 2) / hourβ. β$660β βx = 15β βx = 21β, so Gemma earned β$630βfrom Monday to Friday.
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
9H
1 2β 1β years 2 a Let βc =βcarβs cost b βc + 2000 = 40βββ000ββ c βc = 38βββ000β d β$38βββ000β 3 a Let βp =βcost of one pen b β12p = 15.6β c βp = 1.3β d β$1.30β 4 a Let βh =βnumber of hours worked b β17h + 65 = 643β d 3β 4β hours c βh = 34β 5 a β24w = 720β b βw = 30β c β30βββmβ d β108βββmβ 6 β2(β 2x + 3)ββ = 34 β x = 7β 7 βx = 4β 8 β1.5βββhβ 9 β14β years old 10 β2x + 154 = 180 β x = 13β 11 β3y = 90 β y = 30β 12 Examples include: βh = 3, w = 6βor βh = 12, w = 2.4β. (Answers may vary.) 13 βx = 3.5, y = 2β 14 a Solution is βc = β 2βbut you cannot have a negative number of chairs. b Solution of βc = β 2βmeans the temperature is ββ 2Β° Cβ, which is possible although unpleasant. 15 a Possible (βp = 12β) b Not possible because solution is not a whole number. c Possible (ββ p = 8)ββ 16 βa + 2b = 180β, so βa = 180 β 2b = 2β(90 β b)ββis always even. 17 a i βx = 60β ii One angle is ββ 10Β°β, which is impossible. b i β60 β x + 70 + x + 50βis always β180β, regardless of the value of βxβ. ii Any value less than β60βand greater than ββ 70β. c Answers may vary.
832
Answers
10A Building understanding 1 a d 2 a c 3 a c 4 a
β10β β1000β β100β divide β1000β β100βββ000β Right
b β100β e 1β 00βββ000β
c 1β 000β f 1β βββ000βββ000β
13 a β$8200β 14 β625β years 15 β50 000β years 16 β0.08βββmmβ 17 β2500βββsβ 18
b $β 6.56β
c β41βββcβ
km Γ·1000
Γ1000
U N SA C O M R PL R E EC PA T E G D ES
Answers
Chapter 10
b d b d b
β1000β multiply β1000β β1βββ000βββ000β Left
Now you try
Example 1 a Kilometres (ββ km)ββ
b Centimetres ββ(cm)ββ
Example 2 a β2000βββmβ
b 3β .4βββmβ
Example 3 a β32βββmmβ
b 2β 40βββcmβ
10A
Exercise 10A
1 a Kilometres b Millimetres c Metres d Metres e Centimetres f Kilometres 2 a Metres b Millimetres c Kilometres d Kilometres e Centimetres f Centimetres 3 a β400βββcmβ b β6000βββmβ c β3βββcmβ d β7βββmβ 4 a β50βββmmβ b β200βββcmβ c β3500βββmβ d β2610βββcmβ e β4βββcmβ f β5βββmβ g β4.2βββkmβ h β47.2βββcmβ i β684βββcmβ j β20βββmβ k β926.1βββcmβ l β4.23βββkmβ 5 a β2.5βββcmβ b β82βββmmβ c β2.5βββmβ d β730βββcmβ e β6200βββmβ f β25.732βββkmβ 6 a β3000βββmmβ b β600βββ000βββcmβ c β2400βββmmβ d β4000βββcmβ e β0.47βββkmβ f β913βββmβ g β0.216βββkmβ h β0.0005βββmβ i β700βββ000βββmmβ 7 a i β2βββcmβ ii β20βββmmβ b i β5βββcmβ ii β50βββmmβ c i β1.5βββcmβ ii β15βββmmβ d i β3.2βββcmβ ii β32βββmmβ e i β3βββcmβ ii β30βββmmβ f i β3βββcmβ ii β30βββmmβ g i β1.2βββcmβ ii β12βββmmβ h i β2.8βββcmβ ii β28βββmmβ 8 a β2.7βββmβ b β0.4βββkmβ 9 a β8.5βββkmβ b β310βββcmβ c β19βββcmβ 10 a β38βββcm, 0.5βββm, 540βββmmβ b β160βββcm, 2100βββmm, 0.02βββkm, 25βββmβ c β142βββmm, 20βββcm, 0.003βββkm, 3.1βββmβ d β10βββmm, 0.1βββm, 0.001βββkm, 1000βββcmβ 11 β125βββcmβ 12 Bigan tower
m
Γ·100 000
Γ·100
Γ·1 000 000
Γ·1000
Γ100 000
Γ100
cm
Γ·10
Γ 1 000 000
Γ1000
Γ10
mm
19 So that only one unit is used, and mm deliver a high degree of accuracy. 20 a i β1β million ii β10βββ000β iii β1000β iv β1β billion (ββ 1βββ000βββ000βββ000)ββ b β0.312βββΞΌmβ c The distance you travel in β1βyear at the speed of light
10B
Building understanding 1 a β10βββcmβ 2 a 6β .3βββcmβ c β5.2βββcmβ
b 1β 2βββcmβ b 1β 2βββcmβ d 6β .4βββcmβ
Now you try Example 4 a 2β 1βββcmβ
b 3β 6βββmβ
Exercise 10B
1 a β16βββcmβ b 2β 5βββmβ 2 a 1β 5βββcmβ b 3β 7βββmβ c β30βββkmβ d β2.4βββmβ e 2β 6βββcmβ f β10βββcmβ 3 a β20βββmβ b 2β 2βββcmβ 4 a β42βββcmβ b β34βββmβ c β36βββkmβ d β18βββmmβ 5 a β8.4βββcmβ b β14βββmβ c β46.5βββmmβ 6 β$21βββ400β 7 a β516βββftβ b β157.38βββmβ 8 a β70βββmmβ b β72βββcmβ 9 a β40.7βββcmβ b β130.2βββcmβ c β294βββcmβ 10 4β 00βββmβ 11 a β5βββcmβ b β5βββmβ c β7βββkmβ 12 4β β, including a square 13 a βP = a + 2bβ b βP = 4aβ c βP = 2a + 2bβ d P β = 4aβ e βP = 2a + 2bβ f βP = 2a + b + cβ 14 a βP = 2a + 2b + 2c or 2ββ(ββa + b + cβ)ββββ b βP = 2a + 2b + 2c or 2ββ(ββa + b + cβ)ββββ P ββ ( ) 15 a ββ_ b _ ββ β P β 2a βββ 4 2 16 a β160βββcmβ b 2β 16βββcmβ c β40βββcmβ d 4β a + 32βββorβββ4(β a + 8)ββ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
Now you try
Building understanding
Example 7 a 9β .77βββmβ
b 3β 5.43βββcmβ
Example 8 a 3β 2.1βββmmβ
b 4β .1βββcmβ
βFalseβ Tβ rueβ Diameter Circumference Chord Tangent β30βββmmβ
b Tβ rueβ e Fβ alseβ
c Fβ alseβ f Tβ rueβ b Radius d Segment f Sector
Example 9 β13.7βββcmβ
U N SA C O M R PL R E EC PA T E G D ES
1 a d 2 a c e g 3 a
Answers
10C
833
b β4βββcmβ
Now you try
Example 5 a i β3.13β b 3β .14βis closer
ii β3.125β
Example 6 a β34.56βββmmβ
b β29.53βββcmβ
Exercise 10C
1 a β8.38βββcmβ b β5.59βββmβ c 1β 2.22βββmmβ d β1.96βββmβ e β136.14βββcmβ f 1β .15βββkmβ 2 a β14.3βββcmβ b 3β 5.7βββmβ c β28.6βββcmβ d 9β .3βββmβ 3 a β51.4βββmmβ b 3β 6.0βββkmβ c β43.7βββmmβ d 1β 2.1βββmβ 4 a β13.1βββmβ b 1β 4.0βββcmβ c β40.3βββcmβ d 4β 4.5βββmβ 5 a β18.7βββmβ b β11.1βββcmβ c 1β 01.1βββcmβ d β35.4βββkmβ e β67.1βββmmβ f 4β 5.1βββcmβ 6 β657βββcmβ 7 a β26.88βββmβ b β52.36βββmβ c β6.24βββmβ 8 a β25.13βββcmβ b β56.55βββmβ c β35.71βββmβ 9 The four arcs make one full circle (ββ 2Οr)ββand the four radii make β4rβ. 10 The perimeter includes one large semicircle ( ββ _ ββ― 1β―ββ Γ 2Οr = Οr)ββ 2 and the two smaller semicircles. Which make one full smaller circle (ββ Ο Γ r)ββ. So the total is β2Οrβ. 11 a β7.6βββmβ b β34.4βββmβ c 9β .5βββkmβ 12 a β24 + 3Οβββcmβ b 9β + 2.5Οβββcmβ 13 a β8Οβββcmβ b β18Οβββmβ c β5Ο + 20βββcmβ
10E
Building understanding 1 a c 2 a c 3 a c 4 a d
β8β 8β ββββcmββ2ββ β9β β9ββββcmββ2ββ β5β square units β96β square units ββcmββ2ββ ββkmββ2ββ
Example 10 β12ββββcmββ2ββ
Building understanding
Example 11 a 2β 4ββββcmββ2ββ
_ ββ 1 ββ
4
_ ββ 3 ββ
4
b _ ββ 1 ββ semicircle 2 c _ ββ 1 ββ 6 f _ ββ 7 ββ 8 b quadrant, rectangle
b 3β βββcmβ and β3βββcmβ b 8β β square units
b m ββ 2ββ ββ e hβ aβ
c hβ aβ f m ββ mββ2ββ
Now you try
10D
1 ββ quadrant 1 a ββ_ 4 1 ββ 2 a ββ_ b 2 d _ ββ 1 ββ e 3 3 a square, semicircle c sector, triangle
b 4β βββcmβ and β2βββcmβ
b 2β .25ββββkmββ2ββ
Exercise 10E 1 a β6ββββcmββ2ββ 2 a 2β ββββcmββ2ββ 3 a β200ββββcmββ2ββ d β25ββββm2ββ ββ g β1.36ββββm2ββ ββ
b
β5ββββcmββ2ββ
b 3β ββββcmββ2ββ c 4β .5ββββcmββ2ββ
b 2β 2ββββmmββ2ββ e 1β .44ββββmmββ2ββ h 0β .81ββββcmββ2ββ
d 9β ββββcmββ2ββ c 7β ββββcmββ2ββ f 6β .25ββββmmββ2ββ i 1β 79.52ββββkmββ2ββ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
10C
1 a i β3.11β ii β3.111β b 3β .14βis closer 2 a i β3.142β ii β3.1416β b _ ββ 355 ββ is closer 113 b β40.84βββcmβ 3 a β15.71βββcmβ c β14.45βββmβ d β30.79βββmβ 4 a β25.13βββmmβ b β131.95βββcmβ c β52.15βββmmβ d β101.79βββmβ 5 a β12βββcmβ b β31.4βββmmβ c β44βββmβ d β355βββcmβ 6 β125.7βββcmβ 7 a β3.18βββcmβ b β3.98βββmβ c β1.11βββmmβ d β5.95βββmβ 8 β8.8βββmβ 9 β2734βββmβ 22 ββ 10 a ββ_ 7 b It is a relatively simple fraction and also quite accurate. c Answers will vary. C β, 3.98βββcmβ 11 βr =β β_ 2Ο 12 a 7β Οβββcmβ b β6Οβββmmβ 13 a 4β 0βββ055βββkmβ b β21βββ300βββkmβ c β139βββ821βββkmβ d β4880βββkmβ
Exercise 10D
Answers
4 a β2βββcmβ b β5βββmβ c 5 a β2βββhaβ b β10βββhaβ c 6 β5000ββββm2ββ ββ 7 β2500ββββcmββ2ββ 8 2β 0βββ000βββββcmββ2ββ 9 a β25ββββcmββ2ββ b β12βββcmβ c 10 β$2100β 11 β5βββLβ 12 β8βββhaβ 13 a i β10βββcmβ ii β9βββmmβ b Divide the area by the given length. 14 Half of a rectangle with area β4ββββcmββ2ββ 2 15 a 1β 21ββββcmββ2ββ b βββ(_ ββ― Pβ―β)βββ ββ 4 16 Area is quadrupled ββ(Γβββ4)ββ. 17 a i β100β ii β10βββ000β b Γ1 000 000 Γ10 000 Γ100
1β 2βββkmβ β0.5βββhaβ
β4β units
10G Building understanding 1 a β20βββcmβ 2 a 6β βββmβ 3 a β10β
b 7β βββmβ b β11βββmmβ b β56β
c 2β βββmβ c 1β .9βββmβ
d 6β .3βββcmβ d β3.2βββmmβ c β12.5β
U N SA C O M R PL R E EC PA T E G D ES
Answers
834
km2
m2
Γ·1 000 000
cm2
Γ·10 000
β200ββββmmββ2ββ
Now you try Example 13 a 4β 2ββββcmββ2ββ
Exercise 10G
iii β1βββ000βββ000β
mm2
Γ·100
ii β100βββ000ββββcmββ2ββ iv β3ββββcmββ2ββ vi β4.2ββββkmββ2ββ viii β100βββhaβ
c i iii β3βββ500βββ000ββββm2ββ ββ v β2.16ββββm2ββ ββ vii β5000ββββmmββ2ββ ix β0.4βββhaβ
10F
10F
Building understanding 1 a A = bh ββ ββ―β =ββ― 5 Γβ 7βββ β = 35
b A = bh ββ ββ―β =ββ― 20βΓ 3βββ β = 60
2 a βb = 6βββcm, h = 2βββcmβ c βb = 5βββm, h = 7βββmβ e βb = 5βββcm, h = 1.5βββcmβ
c A = bh ββ ββ―β =ββ― 8 Γβ 2.5βββ β = 20
b bβ = 10βββm, h = 4βββmβ d βb = 5.8βββcm, h = 6.1βββcmβ f bβ = 1.8βββm, h = 0.9βββmβ
Now you try Example 12 a β15ββββm2ββ ββ
b β24ββββm2ββ ββ
b β18ββββcmββ2ββ
Exercise 10F
a β10ββββmββ2ββ b β80ββββcmββ2ββ a β40ββββm2ββ ββ b β28ββββm2ββ ββ c β36ββββkmββ2ββ d β17.5ββββm2ββ ββ 2 2 a 2β 8ββββcmββ ββ b β108ββββmmββ ββ a β40ββββcmββ2ββ b β6.3ββββcmββ2ββ c β30ββββmββ2ββ d β1.8ββββcmββ2ββ 5 a β6ββββcmββ2ββ b β4ββββcmββ2ββ c β15ββββcmββ2ββ d β8ββββcmββ2ββ 2 6 5β 4ββββmββ ββ 7 a β2βββmβ b β7βββcmβ c β0.5βββmmβ 8 a β10βββcmβ b β5βββmβ c β2βββkmβ 9 β$1200β 10 a β1800ββββcmββ2ββ b β4200ββββcmββ2ββ 11 Because height must be less than β5βββcmβ. 1 2 3 4
1 ββββbhβ 12 βHalf;βββareaβββ(parallelogram) = bhβ and βareaβββ (triangle ) =β β_ 2 13 Area = twice triangle area 1β―ββββbhβ ββ ββ―ββ―β― βββ =ββ― 2 Γβ ββ―_ 2 β = bh 14 β$4βββ500βββ000β
1 a β30ββββm2ββ ββ b β12ββββcmββ2ββ 2 2 2 a 1β 6ββββmββ ββ b β30ββββcmββ ββ c 2β 5ββββmmββ2ββ d β160ββββmββ2ββ e β1.2ββββmββ2ββ f β1.25ββββcmββ2ββ 3 a β60ββββmmββ2ββ b β14ββββcmββ2ββ 4 a β3ββββmmββ2ββ b β24.5ββββkmββ2ββ c 1β 2ββββcmββ2ββ 2 2 d β1.3ββββcmββ ββ e β20ββββmββ ββ f β4.25ββββmββ2ββ 5 a β2ββββcmββ2ββ b β6ββββcmββ2ββ c β3ββββcmββ2ββ d 3β ββββcmββ2ββ 6 β480ββββm2ββ ββ 7 4β 800ββββm2ββ ββ 8 β6.8ββββcmββ2ββ 9 β160ββββmββ2ββ 10 β$6300β 11 a β5βββcmβ b β4.4βββmmβ 12 Yes, the base and height for each triangle are equal. 13 No, the base and height are always the same. 2A ββ 14 βb =β β_ h 15 a β7ββββcmββ2ββ
b β8ββββcmββ2ββ
c 1β 1ββββcmββ2ββ
Progress quiz
1 a Metres 2 a 4β 00βββcmβ d β3000βββmβ 3 a 1β 4.6βββcmβ 4 a β3.136β 5 a β28.27βββmmβ 6 a β16.5βββcmβ 7 a β0.9ββββmββ2ββ 8 a β55ββββcmββ2ββ 9 a β6ββββmββ2ββ 10 a β225ββββcmββ2ββ 1 ββ 11 ββ_ 3
Millimetres c 2β 000βββmmβ c β1450βββmβ f 1β 0.4βββmβ c b No b 3β 7.70βββcmβ b 3β 0.8βββcmβ c b β49ββββcmββ2ββ b β72.9ββββmββ2ββ c b β31.5ββββm2ββ ββ c β9.68ββββm2ββ ββ b β200ββββcmββ2ββ c b b e b
Kilometres 3β 5βββmmβ 2β 3βββkmβ β354βββcmβ
β41.4βββcmβ
1β 92ββββmmββ2ββ d 2β 0ββββmmββ2ββ 7β β
10H
Building understanding 1 a
b
c 2 a Addition b Subtraction c Subtraction
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
3 a A = βlβββ― 2ββ + lw
835
Now you try Example 16 β48ββββcmββ3ββ
β = 4ββββcmββ2β
1β―β bh b A = lw ββ ββ―_ 2 1β―ββ Γ 5 Γ 4 β =ββ― 13 Γ 8 ββ ββ―β_ β―β― βββ ββ ββ―β―β― 2β β = 104 β 10
Exercise 10I
U N SA C O M R PL R E EC PA T E G D ES
1 a β84ββββm3ββ ββ b 1β 0ββββcmββ3ββ 3 3 2 a 9β 8ββββcmββ ββ b β27ββββcmββ ββ c β1000ββββkmββ3ββ d 2β .744ββββmmββ3ββ e β100ββββmββ3ββ f β11.25ββββkmββ3ββ 3 β24βββ000ββββcmββ3ββ 4 β96ββββmββ3ββ 5 β12βββ000ββββmmββ3ββ 6 β27βββ000βββ000ββββkmββ3ββ 7 a β28ββββcmββ3ββ b β252ββββm3ββ ββ c β104ββββcmββ3ββ d β120ββββm3ββ ββ 8 β600β 9 β4096ββββcmββ3ββ 10 a β125ββββcmββ3ββ b 7β 29ββββmββ3ββ 11 a 6β β cubic units b β4β cubic units c β14β cubic units 12 a i β2βββmβ ii 5β βββcmβ iii β7βββcmβ iv β3βββmβ b Use: Volume βΓ·βarea of base c hβ = V Γ·β β(l Γ w)ββ 13 a i β10β ii β100β iii β10β iv β1000β b i β1βββ000βββ000β ii β1βββ000βββ000βββ000β c Γ1 000 000 000 Γ1 000 000 Γ1000
Answers
βββ― =ββ― β12ββ ββ + 3β Γ β 1βββ ββ β―β― β= 1 + 3
β = 94βββββm2ββ β
Now you try Example 14 a β33ββββm2ββ ββ
b β30ββββcmββ2ββ
Example 15 a 6β 8ββββcmββ2ββ
b β10ββββcmββ2ββ
Exercise 10H
1 β,β β_ 1 β,β β_ 1 β,β β_ 1 β,β β_ 1 β,β β_ 1 β,β β_ 1 β,β β_ 1 β,β β_ 1 β,β β_ 1 ββ ββ_ 16 32 64 128 256 512 1024 2048 4096 8192 ii No c β1β square unit d Total must equal β1β.
b i
10I
β = 24ββββcmββ3β
b V = lwh β =ββ―ββ― 1 Γ β3 Γ 6βββ ββ ββ―β―β― β = 18ββββm3ββ β
c V = βlβββ― 3β ββ―β =ββ― 2 βΓ 2 Γ 2βββ ββ β―β― β = 8ββββkmββ3β
b 2β 4β e 5β 6β
m3
cm3
Γ·1 000 000 000 Γ·1 000 000
mm3
Γ·1000
10J
Building understanding 1 a β5βββmβ 2 a 6β ββββm2ββ ββ 3 a 3β 0ββββm3ββ ββ
b 7β βββmβ b β16ββββmββ2ββ b β112ββββmββ3ββ
c 2β βββcmβ c β2.25ββββcmββ2ββ c β4.5ββββcmββ3ββ
Now you try Example 17 a 4β 5ββββcmββ3ββ
b 7β 2ββββmββ3ββ
Example 18 β140ββββm3ββ ββ
Exercise 10J
Building understanding 1 a β6β d 1β 44β 2 a V = lwh ββ―β =ββ―ββ― 4 Γ β2 Γ 3βββ ββ
km3
10I
1 a β60βββββcmββ2ββ b β32ββββmββ2ββ 2 a 3β 3ββββm2ββ ββ b β600ββββmmββ2ββ c β25ββββmββ2ββ 2 2 d 2β 1ββββmββ ββ e β171ββββcmββ ββ f 4β 5ββββkmββ2ββ 2 2 3 a 4β 5ββββcmββ ββ b β54ββββcmββ ββ 4 a β39ββββm2ββ ββ b β95.5ββββcmββ2ββ c β26ββββmββ2ββ d 5β 4ββββm2ββ ββ e β260ββββcmββ2ββ f 4β ββββmββ2ββ 2 2 5 a 5β 3βββββcmββ ββ b β16ββββmββ ββ c β252ββββcmββ2ββ 2 2 6 a 8β 0ββββcmββ ββ b β7ββββmββ ββ c β14.5ββββmββ2ββ d β11.75ββββcmββ2ββ 7 1β 0.08ββββm2ββ ββ 8 Yes, with β$100βto spare. 9 Subtraction may involve only two simple shapes. 10 a No; bases could vary depending on the position of the top side. b Yes, β40ββββm2ββ ββ; take out the rectangle and join the triangles to give a base of β10 β 6 = 4β. 11 a See given diagram in Chapter 10, Exercise 10G, Question 11.
c 2β 4β f 1β 3β
1 a β20ββββmmββ3ββ 2 a 9β ββββm3ββ ββ d 2β 70ββββmmββ3ββ 3 a β9ββββcmββ3ββ 4 a 3β 0βββββm3ββ ββ 5 β4.2ββββm3ββ ββ 6 β360 000ββββcmββ3ββ 7 3β 6ββββcmββ3ββ 8 a β110ββββcmββ3ββ
9 β4080ββββcmββ3ββ 10 a β4βββcmβ c β6.25ββββmmββ2ββ
b 8β 4ββββcmββ3ββ
β175ββββcmββ3ββ
b e β585ββββmββ3ββ b β9ββββmββ3ββ
b β60ββββm3ββ ββ
b 4β 8ββββmββ3ββ
c 8β 0ββββmmββ3ββ f β225ββββmmββ3ββ c β0.99ββββcmββ3ββ
c β160ββββcmββ3ββ
d β140ββββcmββ3ββ
b 5β .125βββmβ d 1β 0ββββcmββ3ββ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
11 β3βββcmβ 12 β2.5βββmmβ
15 a i β8βββcmβ b β0.6βββcmβ
bββh1ββ― βββh2ββ― ββ _ 13 a βV =β β_ ββ =β β1 βbββh1ββ― βββh2ββ― βββ 2 2
2V ββ b hββ 2ββ― βββ =β β_ bββh1ββ― ββ
2V ββ c ββh1ββ― βββ =β β_ e i
βh1ββ― βββh2ββ― ββ
ii β1β
c β13.4βββcmβ
iii β3.2βββcmβ
10L
2V ββ d βb =β β_
bββh2ββ― ββ β1.5β
ii β4βββcmβ
iii β0.8β
Building understanding 1 a β1kg, 1000βββgβ 2 a C β β b βFβ c A β β 3 a B β β b βAβ
b 1β 000βββmg, 1βββgβ d D β β e βBβ f Eβ β c βDβ d βCβ
U N SA C O M R PL R E EC PA T E G D ES
Answers
836
10K
Building understanding 1 a d 2 a 3 a
β1β b β1000β β1000β e β1000β β1βββL, 1000βββmL, 1000ββββcmββ3ββ βDβ b βBβ c βFβ
c β1000β
b 1β ββββmββ3β, 1000βββL, 1βββkLβ d A β β e C β β f βEβ
Example 19 a β7.2βββLβ
b β7000βββkLβ
Example 20 a β67βββMLβ
b β9βββ000βββ000βββmLβ
Example 21 β3βββLβ
10K
Exercise 10K
b f j n
b β8βββLβ e β5000βββLβ β100βββmLβ c 3β .5βββLβ g β9.32βββLβ k β91βββMLβ o b 2β .5ββββm3ββ ββ b b d f h b β0.48βββLβ e 2β 5βββLβ
c 0β .06βββLβ f 7β 500βββLβ β6000βββkLβ d β24βββ000βββLβ 7β 0βββLβ h β2.5βββMLβ β3847βββkLβ l β47βββkLβ β420βββmLβ p β0.17βββkLβ c 8β 75βββmLβ 1β .2βββLβ 0β .26βββkLβ β3.84βββMLβ 4β 00βββ000βββLβ β5000βββmLβ
c β0.162βββLβ f β32.768βββLβ b β2βββ500βββ000βββLβ ii β0.6βββLβ iv β5256βββLβ
13 a β1000xββββcmββ3ββ c 0β .001xβββkLβ 14 a i βlwhβ lwh ββ iii ββ_ 1000
b 0β .001xββββmββ3ββ d 0β .000βββ001βββxβββMLβ ii βlwhβ iv _ ββ lwh ββ 1βββ000βββ000
βlwhββββm3ββ ββ
ii β1000βββlwhβββL β lwh ββββMLβ iv ββ_ 1000
b i
iii βlwhβββkLβ
Example 22 a 3β 2βββ500βββgβ
b 9β .5βββtβ
Exercise 10L
Now you try
1 a β0.7βββLβ d β2000βββLβ 2 a 2β 000βββmLβ e β2βββkLβ i β257βββmLβ m β500βββLβ 3 a β12βββmLβ 4 a 1β 200βββmLβ 5 a 4β .7βββkLβ c β5βββMLβ e β9βββ000βββ000βββLβ g β200βββ000βββmLβ 6 β3βββ300βββ000βββLβ 7 a β1.5βββLβ d β8.736βββLβ 8 a β2500ββββm3ββ ββ 9 a β0.2βββmLβ b i β1βββLβ iii β14.4βββLβ 10 β15β days 11 β41βββhβββ40βββminβ 12 β1000β days
Now you try
1 a i β4290βββgβ ii β7.5βββkgβ b i β620βββtβ ii β5100βββkgβ 2 a β2000βββkgβ b 7β 0000βββgβ c β2400βββmgβ d β2.3βββgβ e 4β .620βββgβ f β21.6βββtβ g β470βββkgβ h 0β .312βββkgβ i β0.027βββgβ j β750βββkgβ k β125βββgβ l β0.0105βββkgβ m β210βββtβ n 4β 70βββkgβ o β592βββgβ p β80βββgβ 3 a β4βββkgβ b 1β 2βββgβ c β65βββtβ 4 a β12Β° Cβ b 3β 7Β° Cβ c β17Β° Cβ d β225Β° Cβ e 1β .7Β° Cβ f β31.5Β° Cβ 5 a β60βββkgβ b 6β 0βββ000βββgβ c 6β 0βββ000βββ000βββmgβ 6 a β3000βββgβ b 3β βββkgβ 7 3β 3Β° Cβ 8 β147Β° Cβ 9 a β8βββkgβ b 8β .16βββkgβ 10 a β400βββmg, 370βββg, 2.5βββkg, 0.1βββtβ b β290βββ000βββmg, 0.00032βββt, 0.41βββkg, 710βββgβ 11 a β4βth day b 2β 5Β° Cβ c β26Β° Cβ 12 β50β days 13 Yes, by β215βββkgβ 14 a β45.3βββtβ b Yes, by β2.7βββtβ 15 a β1βββgβ b 1β βββtβ c β1000βββtβ 16 a β12βββkgβ b 1β 000βββkgβ c β360βββ000βββkgβ 17 a i β10Β° Cβ ii β27Β° Cβ iii β727Β° Cβ b i β273βββKβ ii β313βββKβ iii β0βββKβ 5 4 βΒ° Fβ _ _ 18 a β180Β° Fβ b ββ βΒ° Cβ c ββ 9 βΒ° F or 1β_ 9 5 5 d i β0Β° Cβ ii β20Β° Cβ iii 6β 0Β° Cβ iv β105Β° Cβ 9C _ e βF =β β ββ + 32β 5
Problems and challenges
1 Both line segments are the same length. 2 Mark a length of β5βββmβ, then use the β3βββmβstick to reduce this to β2βββmβ. Place the β3βββmβstick on the β2βββmβlength to show a remainder of β1βββm.β 3 a β3ββββcmββ2ββ b 2β 0ββββcmββ2ββ 4 β500βββLβ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
β24ββββm3ββ ββ β10.5ββββcmββ2ββ a β12.5βββcmβ β48ββββcmββ2ββ
b β3.75βββcmβ
1β 6βββmβ b β23βββmβ d β3.2βββmmβ f β3.143β b β34.56βββcmβ b β23.0βββcmβ b β24.01ββββcmββ2ββ 1β 4ββββkmββ2ββ β67.5ββββmββ2ββ 1β 2ββββcmββ2ββ β14ββββmββ2ββ 5β ββββcmββ2ββ 1β 4ββββmββ2ββ 0β .9ββββkmββ2ββ β22ββββmββ2ββ 2β 1ββββmmββ2ββ β291ββββcmββ2ββ 5β 2ββββmββ2ββ β18ββββcmββ3ββ 7β .5ββββcmββ3ββ β64ββββmmββ3ββ β45ββββcmββ3ββ 1β 12ββββmββ3ββ β210ββββmmββ3ββ β72βββ000ββββcmββ3ββ 7β 2βββ000βββmLβ β72βββLβ β45βββmg, 290βββ000βββg, 3βββt, 4700βββkgβ 5β 0βββ000βββmL, 51βββL, 0.5βββkL, 1βββMLβ
β20.6βββcmβ 3β 4βββkmβ β24βββmβ Yes 3β 1.42βββmβ 2β 6.3βββcmβ
Metres (m) β0.256βββkmβ β0.7βββmβββ(orβββ70βββcm)β β28βββmβ β3.125β; it is a less accurate approximation than β3.14β(both are less than βΟβbut β3.14βis closer). 6 β25.13βββcmβ 7 β14.980βββmβ 8 β28.85βββmβ 9 β6βββββcmββ2ββ 10 β40βββββmmββ2ββ 11 β60βββββm2ββ ββ 12 β45βββββcmββ2ββ 13 β24βββββcmββ2ββ 14 β96βββββcmββ3ββ 15 β20βββββcmββ3ββ 16 β0.5βββLβ 17 2β βββLβ 18 β2470βββgβ 1 2 3 4 5
Chapter review
Short-answer questions 1 a β10β b 1β 000β c β100βββ000β 2 a β50βββmmβ b 2β βββmβ c β3700βββmβ d 4β .21βββkmβ e β7100βββgβ f β24.9βββgβ g 2β 8.49βββtβ h 9β 000βββgβ i β4βββLβ j β29.903βββkLβ k β400βββkLβ l β1000βββmLβ m β1440βββminβ n 6β 0βββminβ o β3.5βββdaysβ p 9β 000βββsβ 3 a β2.5βββcmβ b 2β .3βββcmβ c β4.25βββkgβ d 5β βββLβ
Multiple-choice questions 1 D β β 2 βEβ 3 βCβ 4 βEβ 5 βCβ 6 βAβ 7 Eβ β 8 βBβ 9 βAβ 10 C β β 11 βBβ 12 Eβ β 13 B β β
Extended-response questions 1 a b d 2 a d
i β18ββββm2ββ ββ β36ββββm3ββ ββ 5β 1.4βββtβ β97.3βββmβ $β 2040β
ii β180βββ000ββββcmββ2ββ c β36βββ000βββLβ e β3βββhβββ52βββminβββ54βββsβ b β5βββminβββ30βββsβ c β270ββββmββ2ββ e β50ββββmββ3ββ
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Ch10 Chapter checklist with success criteria
U N SA C O M R PL R E EC PA T E G D ES
Chapter checklist with success criteria
4 a c e 5 a 6 a 7 a 8 a b c d e f g h 9 a b c d 10 a b c 11 a b c 12 a b c 13 a b
Answers
5 6 7 8
837
838
Answers
Negative numbers Short-answer questions 1 a β<β 2 a ββ 13β d ββ 21β 3 a ββ 24β d β21β 4 a Negative 5 a ββ 15β 6 a ββ 10β d ββ 96β
β<β ββ 84β β84β β72β β30β Positive ββ 8β β10β β52β
b b e b e b b b e
c c f c f c c c f
β β = ββ 108β β0β β144β ββ 31β Negative ββ 4β ββ 20β ββ 1β
10 a b c d e f g
ββAβ²β―β(2, 0),β βBβ²β―β(2, β 3),β βCβ²β―β(4, 0)β ββAβ²β―β(β 2, 0),β βBβ²β―β(β 2, 3),β βCβ²β―β(β 4, 0)β ββAβ²β―β(0, β 2),β βBβ²β―β(3, β 2),β βCβ²β―β(0, β 4)β ββAβ²β―β(0, 2),β βBβ²β―β(β 3, 2),β βCβ²β―β(0, 4)β ββAβ²β―β(β 2, 0),β βBβ²β―β(β 2, β 3),β βCβ²β―β(β 4, 0)β ββAβ²β―β(β 2, 1),β βBβ²β―β(β 2, 4),β βCβ²β―β(0, 1)β ββAβ²β―β(β 1, β 2),β βBβ²β―β(β 1, 1),β βCβ²β―β(1, β 2)β
U N SA C O M R PL R E EC PA T E G D ES
Answers
Semester review 2
Multiple-choice questions 1 C
2 B
3 D
4 E
5 A
Semester review 2
Extended-response question a b c d e f g h
βDβ βAβ, βBβ, βOβand βGβ; all lie on the βxβ-axis. βFβ i β2β units ii β5β units Trapezium β8β square units βX(β 4, 2)ββ DECIDE
Geometry
Short-answer questions
1 a βAC || FDββββ or βEB || DCβ b βBFβand βBDβ c βCD, ED, BDβat βDβor any three of βAC, BD, BEβor βBFβat βBβ d βAβand βBβ e βEβ 2 a β30Β°β b β80Β°β c β150Β°β 3 β25Β°β 4 β78Β°β 5 a βa = 140β b βa = 50β c βa = 140β d βa =βββ65β e βa =βββ62β f βa = 56β 6 βa = 100, b = 80, c = 100, d = 80, e = 100, f = 80, g = 100β 7 a Regular pentagon b Regular octagon c Isosceles right triangle d Rhombus e Trapezium f Kite 8 a β130β b β45β c β80β 9 a βa = 60β b βa = 65β c βa = 115β d βa = 90, b = 90β e βa = 65β f βa = 132, b = 48, c = 48, d = 65β
Multiple-choice questions 1 A
2 B
3 B
4 B
5 A
Extended-response question a b c d e
Answers may vary. βa + b + c = 180Β°β Answers may vary. βx + y + z = 360Β°β β360Β°β
Statistics and probability Short-answer questions 1 a b c d 2 a
β1, 2, 3, 5, 5, 5, 8, 8, 9, 10β β10β i β5.6β ii β5β 1β 1.2β A b B
iii β5β
iv β9β
c βB, C, D, Aβ
3 a _ ββ 1 ββ b _ ββ 1 ββ 2 2 d _ ββ 1 ββ e _ ββ 1 ββ 13 13 4 a β13β b β3, 47β c i β44β ii β22β 5 a β20β b β22β c Mean is β23β, median is β22β
c _ ββ 1 ββ 4 f _ ββ 1 ββ 26
iii β22β
Multiple-choice questions 1 B
2 A
3 B
4 E
5 D
Extended-response question a _ ββ 1 ββ 4 d _ ββ 1 ββ 52 g _ ββ 4 ββ 13
b _ ββ 1 ββ 4 e _ ββ 2 ββ 13 h _ ββ 4 ββ 13
c _ ββ 1 ββ 2 f _ ββ 1 ββ 13
Equations
Short-answer questions 1 a βx = 3β
b βx = 108β
c βx = 21β
4 ββ d βx =β β_ 3
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400
Answers
a βx = 2β b a βy = 20β b a βP = 103β b a βx = 5β b 4β x + 25 = 85; x = 15β
xβ = 12β βb = 4β βS = 61β βx = 6β
c c c c
β = 7β m βm = 8β βC = 325β βx = 18β
b 2β 4ββββmββ2ββ e 1β 14ββββmββ2ββ
b 7β β e β8β
c 6β 0ββββmββ2ββ f β171ββββmββ2ββ
b 1β 20ββββmββ3ββ d 1β 80ββββcmββ3ββ c 2β 50β f β25β
U N SA C O M R PL R E EC PA T E G D ES
Multiple-choice questions
3 a β1.69ββββmββ2ββ d 7β 5ββββm2ββ ββ 4 a β729ββββcmββ3ββ c β160ββββmββ3ββ 5 a 5β 000β d 3β 000β 6 β60βββLβ 7 1β 8.85βββcmβ 8 β337βββkgβ 9 ββ 3Β° Cβ
Answers
2 3 4 5 6
839
1 B
2 E
3 C
4 B
5 D
Extended-response question a β$320β
b β$400β
c β$(β 200 + 40n)ββ
1 βhβ d 6β β_ 2
Multiple-choice questions 1 C
2 A
3 E
Measurement
Extended-response question
Short-answer questions
a b c d
1 a d 2 a d
β500β 0β .017β β272βββcmβ 2β 20βββcmβ
b e b e
β6000β β1.8β β11βββmβ β3.4βββmβ
c f c f
β1.8β 5β 500β β3.2βββmβ 9β 2βββmβ
4 C
5 A
Answers may vary; e.g. β8βββm Γ 10βββm, 4βββm Γ 14βββm, 15βββm Γ 3βββmβ 9β βββmβ by β9βββmβ (area β= 81ββββm2ββ ββ) β36β posts 9β βββhβ
Semester review 2
Uncorrected 3rd sample pages β’ Cambridge University Press Β© Greenwood, et al 2024 β’ 978-1-009-48064-2 β’ Ph 03 8671 1400