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JACARANDA MATHS QUEST

AUSTRALIAN CURRICULUM | THIRD EDITION

JACARANDA MATHS QUEST

AUSTRALIAN CURRICULUM | THIRD EDITION

MARK BARNES | JOANNE BRADLEY | LYN ELMS | ROBERT CAHN

CONTRIBUTING AUTHORS

Roger Blackman | Coral Connor | Catherine Hughes | Anita Cann

Elena Iampolsky | Irene Kiroff | Carol Patterson | Lee Roland

Robert Rowland | Nilgun Safak | Douglas Scott | Robyn Williams

Third edition published 2018 by John Wiley & Sons Australia, Ltd

42 McDougall Street, Milton, Qld 4064

First edition published 2011

Second edition published 2015

Typeset in 11/14 pt Times LT Std

The moral rights of the authors have been asserted.

ISBN: 978-0-7303-4632-6

Reproduction and communication for educational purposes

The Australian Copyright Act 1968 (the Act) allows a maximum of one chapter or 10% of the pages of this work, 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).

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 book 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.

Trademarks

Jacaranda, the JacPLUS logo, the learnON, assessON and studyON logos, Wiley and the Wiley logo, and any related trade dress are trademarks or registered trademarks of John Wiley & Sons Inc. and/or its affiliates in the United States, Australia and in other countries, and may not be used without written permission. All other trademarks are the property of their respective owners.

Front cover image: Subbotina Anna/Shutterstock

Cartography by Spatial Vision, Melbourne and MAPgraphics Pty Ltd, Brisbane

Illustrated by various artists, diacriTech and the Wiley Art Studio.

Typeset in India by diacriTech

Printed in Singapore by Markono Print Media Pte Ltd

10 9 8 7 6 5 4 3

Topic 4 Linear equations

CONTENTS v CONTENTS About this resource �� x Acknowledgements xii Topic 1 Numeracy 1 1 1.1 Overview 1 1.2 Set A 2 1.3 Set B 4 Answers 8 Topic 2 Number skills 9 2.1 Overview 9 2.2 Rational numbers 10 2.3 Surds �� 14 2.4 Real numbers �� 24 2.5 Scientific notation �� 27 2.6 Review 32 Answers 37 Topic 3 Algebra 42 3.1 Overview 42 3.2 Using pronumerals 43 3.3 Algebra in worded problems 49 3.4 Simplifying algebraic expressions 53 3.5 Expanding brackets 58 3.6 Expansion patterns 65 3.7 Further expansions 69 3.8 The highest common factor��71 3.9 The highest common binomial factor �� 75 3.10 Applications �� 78 3.11 Review 85 Answers 91

101 4.1 Overview 101 4.2 Solving linear equations 102 4.3 Solving linear equations with brackets 109 4.4 Solving linear equations with pronumerals on both sides 111 4.5 Solving problems with linear equations 115 4.6 Rearranging formulas 119 4.7 Review 122 Answers �� 128

Topic 5 Congruence and similarity

Topic 6 Pythagoras and trigonometry

Topic

Topic 8 Proportion and rates

vi CONTENTS

132

Overview 132 5.2 Ratio and scale 133 5.3 Congruent figures 140 5.4 Similar figures 147 5.5 Area and volume of similar figures 157 5.6 Review 162 Answers �� 169

5.1

174

Overview 174 6.2 Pythagoras’ theorem 175 6.3 Applications of Pythagoras’ theorem 183 6.4 What is trigonometry? 189 6.5 Calculating unknown side lengths 196 6.6 Calculating unknown angles 203 6.7 Angles of elevation and depression ��209 6.8 Review 215 Answers 222

229

6.1

Project: Learning

earning?

graphs 231 7.1 Overview 231 7.2 Plotting linear graphs 232 7.3 The equation of a straight line 236 7.4 Sketching linear graphs 245 7.5 Technology and linear graphs 250 7.6 Determining linear rules 253 7.7 Practical applications of linear graphs 260 7.8 Midpoint of a line segment and distance between two points 264 7.9 Non-linear relations (parabolas, hyperbolas, circles)�� 269 7.10 Review �� 276 Answers �� 282

7 Linear and non-linear

297

Overview 297 8.2 Direct proportion 298 8.3 Direct proportion and ratio 302 8.4 Inverse proportion 306 8.5 Introduction to rates 310 8.6 Constant and variable rates ��312 8.7 Rates of change ��318

Review 323 Answers 329

8.1

8.8

Probability

CONTENTS vii

2 336 9.1 Overview 336 9.2 Set C 337 9.3 Set D 340 Answers 344 Topic 10

345 10.1 Overview �� 345 10.2 Review of index laws �� 346 10.3 Raising a power to another power �� 352 10.4 Negative indices 356 10.5 Square roots and cube roots 360 10.6 Review 363 Answers 368 Topic 11 Financial mathematics 373 11.1 Overview 373 11.2 Salaries and wages ��374 11.3 Special rates 377 11.4 Piecework 382 11.5 Commission and royalties 385 11.6 Loadings and bonuses �� 388 11.7 Taxation and net earnings �� 392 11.8 Simple interest �� 396 11.9 Compound interest 401 11.10 Review 406 Answers 411 Topic

416 12.1 Overview 416 12.2 Measurement 417 12.3 Area 426 12.4 Area and perimeter of a sector 436 12.5 Surface area of rectangular and triangular prisms 442 12.6 Surface area of a cylinder 447 12.7 Volume of prisms and cylinders��451 12.8 Review �� 461 Answers �� 469

475 13.1 Overview 475 13.2 Theoretical probability 476 13.3 Experimental probability 482 13.4 Venn diagrams and two-way tables 488

Topic 9 Numeracy

Indices

12 Measurement

Topic 13

Topic 14 Statistics

Topic 15 Numeracy 3

Project: Backyard flood?

Topic 16 Quadratic algebra

Topic 17 Quadratic functions

viii CONTENTS 13.5 Two-step experiments 498 13.6 Mutually exclusive and independent events ��507 13.7 Conditional probability 516 13.8 Review 521 Answers 527

537 14.1 Overview 537 14.2 Sampling 538 14.3 Collecting data ��548 14.4 Displaying data �� 557 14.5 Measures of central tendency �� 568 14.6 Measures of spread �� 577 14.7 Review 586 Answers 592

604 15.1 Overview 604 15.2 Set E 605 15.3 Set F 609 Answers 613

614

617 16.1 Overview 617 16.2 Factorisation patterns 618 16.3 Factorising monic quadratics 622 16.4 Factorising non-monic quadratics 625 16.5 Simplifying algebraic fractions 628 16.6 Quadratic equations 634 16.7 The Null Factor Law 637 16.8 Solving the quadratic equation ax2 + bx + c = 0 640 16.9 Solving quadratic equations with two terms �� 644 16.10 Applications �� 647 16.11 Review �� 650 Answers 656

663 17.1 Overview 663 17.2 Graphs of quadratic functions 664 17.3 Plotting points to graph quadratic functions 669 17.4 Sketching parabolas of the form y = ax2 675 17.5 Sketching parabolas of the form y = ax2 + c 679

Topic 18 STEM extension: Programming

18.1

18.6

18.7

CONTENTS ix 17.6 Sketching parabolas of the form y = (x – h)2 683 17.7 Sketching parabolas of the form y = (x – h)2 + k 687 17.8 Sketching parabolas of the form y = (x + a) (x + b) 691 17.9 Applications 696 17.10 Review 699 Answers 706

Overview 18.2 Programs 18.3 Arrays �� 18.4 Loops ��

Set structures��

18.5

Sorting algorithms

The Monte Carlo method

Review Answers Glossary 724 Index 732

18.8

ABOUT THIS RESOURCE

Jacaranda Maths Quest 9 Australian Curriculum Third Edition has been completely revised to help teachers and students navigate the Australian Curriculum Mathematics syllabus. The suite of resources in the Maths Quest series is designed to enrich the learning experience and improve learning outcomes for all students. Maths Quest is designed to cater for students of all abilities: no student is left behind and none is held back. Maths Quest is written with the specific purpose of helping students deeply understand mathematical concepts. The content is organised around a number of features, in both print and online through Jacaranda’s learnON platform, to allow for seamless sequencing through material to scaffold every student’s learning.

Topic introductions put the topic into a real-world context.

The learning sequence at a glance

Fully worked examples throughout the text demonstrate key concepts.

Your FREE online learnON resources contain hundreds of videos, interactivities and traditional WorkSHEETs and SkillSHEETs to support and enhance learning.

Carefully graded questions cater for all abilities. Question types are classified according to strands of the Australian Curriculum.

Fully worked solutions to every question are provided online, and answers are provided at the end of each print

An extensive glossary of mathematical terms in print, and as a hoverover feature in your learnON title

Visit your learnON title to watch videos which tell the story of mathematics.

Each topic concludes with comprehensive Review questions, in

x ABOUT THIS RESOURCE

TOPIC 3 Algebra 61 c03Algebra.indd Page 61 11/10/17 10:17 AM 2. The areas of the four small rectangles can be added together. A ac + ad + bc + bd So (a + b)(c + d) ac + ad + bc + bd • There are several methods that can be helpful in remembering how to expand binomial factors. One commonly used method is the FOIL method. This method is given the name FOIL because the letters stand for: First — multiply the first term in each bracket. Outer — multiply the 2 outer terms of each bracket. Inner — multiply the 2 inner terms of each bracket. Last — multiply the last term of each bracket. (a + b)(c + d) II F ‘FOIL’– First Outer I Last OO LL (a + b)(c + d) nosenose eyebrow eyebroweyebrow mouthmouth eyebrow eyebrow nose and mouth (a + b)(c + d) ac + ad + bc + bd (a + b)(c + d) ac + bd + bc + ad OR WORKED EXAMPLE 13 TI CASIO Expand and simplify each of the following expressions. a (x 5)(x + 3) b (x + 2)(x + 3) c (2x + 2)(2x + 3) THINK WRITE a1 Expand using the FOIL method. a (x 5)(x + 3) 2 Simplify the expression by collecting like terms. x2 + 3x 5x 15 x2 2x 15 b1 Expand the brackets using FOIL. b (x + 2)(x + 3) 2 Simplify the expression by collecting like terms. = x2 + 3x + 2x + 6 x2 + 5x + 6 c1 Expand the brackets using FOIL. c (2x + 2)(2x + 3) 2 Simplify the expression by collecting like terms. 4x2 + 6x + 4x + 6 4x2 + 10x + 6 Complete this digital doc: SkillSHEET: Expanding brackets (doc-10819) Complete this digital doc: SkillSHEET: Expanding a pair of brackets (doc-10820) RESOURCES — ONLINE ONLY 42 Jacaranda Maths Quest 9 c03Algebra.indd Page 42 11/10/17 10:15 AM NUMBER AND ALGEBRA TOPIC 3 Algebra 3.1 Overview Numerous videos and interactivities are embedded just where you need them, at the point of learning, in your learnON title at www.jacplus.com.au. They will help you to learn the concepts covered in this topic. 3.1.1 Why learn this? Algebra relates to you and the world around you. It is part of everyday life and you will be using it without knowing it. If you want to be a scientist or engineer, you will certainly need algebra. If you want to do well in Maths, you will need to study algebra. When you learn algebra you learn a different way of thinking. It helps with problem solving, decision making, reasoning and creative thinking. 3.1.2 What do you know? 1. THINK Use a thinking tool such as a concept map to list what you know about algebra. 2. PAIR Share what you know with a partner and then with a small group. 3. SHARE As a class, create a thinking tool such as a concept map to show your class’s knowledge of algebra. LEARNING SEQUENCE 3.1 Overview 3.2 Using pronumerals 3.3 Algebra in worded problems 3.4 Simplifying algebraic expressions 3.5 Expanding brackets 3.6 Expansion patterns 3.7 Further expansions 3.8 The highest common factor 3.9 The highest common binomial factor 3.10 Applications 3.11 Review Watch this eLesson: The story of mathematics: One small step for man . . . (eles-1690) RESOURCES — ONLINE ONLY

110 Jacaranda Maths Quest 9 c04LinearEquations.indd Page 110 11/10/17 12:23 PM WORKED EXAMPLE 6 TI CASIO Solve each of the following linear equations. a 7 (x 5) 28 b 6 (x + 3) 7 THINK WRITE a1 7 is a factor of 28 so divide both sides by 7 a 7(x 5) 28 7 (x 5) 7 28 7 2 Add 5 to both sides. x 5 4 3 Write the value of x x = 9 b1 6 is not a factor of 7, so it will be easier to expand the brackets first. b 6(x + 3) 7 6x + 18 = 7 2 Subtract 18 from both sides. 6x + 18 7 18 6x 11 3 Divide both sides by 6 x 11 6 (or 1 5 6) Exercise 4.3 Solving linear equations with brackets Individual pathways U PRACTISE Questions: 1a–f, 2a–h, 3a–f, 4a–f, 5, 6, 8, 10 U CONSOLIDATE Questions: 1d–i, 2d–i, 3d–i, 4d–i, 5, 7–11 U MASTER Questions: 1g–l, 2g–l, 3g–l, 4g–l, 5, 7–12 U U U Individual pathway interactivity: int-4490 ONLINE ONLY To answer questions online and to receive immediate feedback and sample responses for every question, go to your learnON title at www.jacplus.com.au. Note: Question numbers may vary slightly. Fluency 1. WE6 Solve each of the following linear equations. a. 5(x 2) 20 b. 4(x + 5) 8 c. 6(x + 3) 18 d. 5(x 41) 75 e. 8(x + 2) 24 f. 3(x + 5) 15 g. 5(x + 4) 15 h. 3(x 2) 12 i. 7( 6) 0 j. 6( 2) 12 k. 4( + 2) 4.8 l. 16( 3) 48 2. WE6 Solve each of the following equations. a. 6(b 1) = 1 b. 2(m 3) = 3 c. 2(a + 5) = 7 d. 3(m + 2) = 2 e. 5(p 2) 7 f. 6( 4) 8 g. 10( + 1) 5 h. 12(p 2) 6 i. 9(a 3) 3 j. 2(m + 3) 1 k. 3(2a + 1) 2 l. 4(3m + 2) 5 Complete this digital doc: SkillSHEET: Expanding brackets (doc-10827) RESOURCES — ONLINE ONLY Individual pathway interactivities in each sub-topic ensure consolidation of learning for every skill level. 126 Jacaranda Maths Quest 9 c04LinearEquations.indd Page 126 11/10/17 12:25 PM Investigation 1 Rich Task Forensic science Studies have been conducted on the relationship between the height of a human and measurements of a variety of body parts. One such study relates the height of a person to the length of the upper arm bone (humerus). The relationships are different for (1) males and females and (2) for different races. Let us consider the relationships for white adult Australians. For males, h 3.08 + 70.45 and for females, h 3.36 + 57.97 where h represents the body height in centimetres and the length of the humerus in centimetres. Imagine the following situation. A decomposed body was found in the bushland. A team of forensic scientists suspects that the body could be the remains of either Alice Brown or James King; they have been missing for several years. From the description provided by their Missing Persons file, Alice is 162 cm tall and James’ file indicates that he is 172 cm tall. The forensic scientists hope to identify the body based on the length of the body’s humerus. 1. Complete the following tables for both males and females, using the equations on the previous page. Calculate the body height to the nearest centimetre. Table for males Length of humerus (cm) 20 25 30 35 40 Body height h (cm) Table for females Length of humerus (cm) 20 25 30 35 40 Body height h (cm) 2. On a piece of graph paper, draw the first quadrant of a Cartesian plane. Since the length of the humerus is the independent variable, place it on the -axis. Place the dependent variable, body height, on the y-axis. 3. Plot the points from the two tables representing both male and female bodies from question 1 onto the set of axes drawn in question 2 Join the points with straight lines, using different colours to represent males and females. Engaging Investigations at the end of each topic to deepen conceptual understanding Jacaranda Maths Quest 9 c14Statistics.indd Page 586 11/10/17 11:25 AM 14.7 Review 14.7.1 Review questions Fluency 1. Which one of the following is an example of numerical discrete data? a. Your favourite weeknight television show b. The speed of a car recorded on a speed camera C. The number of home runs by a baseball player d. Placing in a field of eight swimmers of a 100 m freestyle race The following information relates to questions 2 and 3 Number of bedrooms 123456 Number of homes 49323672 2. For the data summarised, the most likely type of distribution is: a. a negatively skewed distribution b. a positively skewed distribution C. skewed to the left d. a symmetric distribution. 3. The mode for the data collected is: a. 36 b. 5 C. 4 d. 3 4. The results of a Science test marked out of 60 are represented by the stem plot shown below. Key: 2 3 23 StemLeaf 0 8 1 8 2 3 5 3 0 3 6 4 2 4 5 5 5 8 5 4 7 8 8 9 6 0 0 0 0 The median, mean and mode, respectively, are: a. 60, 42, 52 b. 40, 45, 60 C. 45, 60, 45 d. 45, 44, 60 5. The following is a grouped frequency table. The best estimate of the mean score is: ScoreFrequency 0– <50 12 50–<100 24 100–<150 23 150–<200 4 a. 63 b. 90 C. 92 d. 100 6. The range and IQR for the following set of data is: 46, 46, 49, 53, 61, 63, 67, 72, 81, 84, 93 a. 46 and 34 b. 46 and 65 C. 47 and 63 d. 47 and 32 7. The sample standard deviation for the following set of data is (to 1 decimal place): 25, 27, 33, 42, 47, 54, 58, 59, 65, 66 a. 15.3 b. 14.5 C. 47.6 d. 26.0

both print and online.

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Maths Quest contains a free activation code for learnON (please see instructions on the inside front cover), so students and teachers can take advantage of the benefits of both print and digital, and see how learnON enhances their digital learning and teaching journey.

includes:

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• Hundreds of videos and interactivities to bring concepts to life

• Fully worked solutions to every question

• Immediate feedback for students

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• Comprehensive summaries for each topic

• Code puzzles and dynamic interactivities help students engage with and work through challenging concepts

• Formative and summative assessments

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ABOUT THIS RESOURCE xi

ACKNOWLEDGEMENTS

The authors and publisher would like to thank the following copyright holders, organisations and individuals for their assistance and for permission to reproduce copyright material in this book.

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Every effort has been made to trace the ownership of copyright material. Information that will enable the publisher to rectify any error or omission in subsequent reprints will be welcome. In such cases, please contact the Permissions Section of John Wiley & Sons Australia, Ltd.

xii ACKNOWLEDGEMENTS

TOPIC 1 Numeracy 1

1.1 Overvi ew

Numerous videos and interactivities are embedded just where you need them, at the point of learning, in your learnON title at www.jacplus.com.au. They will help you to learn the concepts covered in this topic.

1.1.1 Why learn this?

Our lives are interwoven with mathematics. Counting, measuring and pattern-making are all part of everyday life. We use numbers to mark significant events (such as birthdays) and for identification (such as passports and credit cards). We use numbers to describe ourselves (for example our height and weight). Shopping involves understanding numbers, and tallying scores in sports requires a comparison of numbers. You may not realise just how much you rely on numbers.

1.1.2 What do you know?

1. THINK List what you know about numeracy. Use a thinking tool such as a concept map to show your list.

2. PAIR Share what you know with a partner, then with a small group.

3. SHARE As a class, create a thinking tool such as a large concept map that shows your class’s knowledge of numeracy.

LEARNING SEQUENCE

TOPIC 1 Numeracy 1 1 NUMERACY

Watch this eLesson: NAPLAN: Strategies and tips (eles-1688) RESOURCES — ONLINE ONLY

Set A

Set B

1.1 Overview 1.2

1.3

These questions provide the opportunity for students to use their numeracy skills in everyday situations.

1.2 Set A

1.2.1 Calculator allowed

1. Which of the following is the longest time?

a. 95 000 seconds b. 22 hours

C. 1800 minutes d. 1 day

2. Anne bought 20 pens and Jo bought 30 pens. Each pen costs the same amount, and the pens cost $75 altogether. How much did Jo pay for her pens?

3. When a ball is dropped it bounces on the ground. The maximum height reached after each bounce is half the distance the ball drops. If a ball is dropped from a ladder 6 m above the ground, what is the total distance travelled by the time the ball hits the ground the third time?

a. 9 m b. 18 m

C. 10.5 m d. 15 m

4. Last year at Fern Hill High School, there were 985 students. The student population has increased by 20% this year. How many students are at Fern Hill High School this year?

a. 1000 b. 1182

C. 1218 d. 1320

5. The length of a rectangle is 9 cm and its perimeter is 30 cm. What is the area of the rectangle?

6. The graph shows the number of juices sold each day of a particular week at a school. Calculate the mean daily number of juices sold at the school that week.

7. (102 + 12 ÷ 3) ÷ ( 8) = ?

a. 13 b. 14

C. 13 d. 14

8. 3 8 expressed as a percentage is:

a. 30.8% b. 13%

C. 37.5% d. 0.375%

9. Lena wants to make 2 chocolate cakes. The recipe for a chocolate cake is:

1 1 2 cups self-raising flour

1 2 cup sugar

2 tablespoons cocoa

1 egg 125 grams butter. How much of each ingredient will she need to use?

___ cups self-raising flour

___ cup sugar

___ tablespoons cocoa ___ eggs ___ grams butter

10. In the diagram shown, what is the value of x? a. 110°

11. Express in binary form. 113ten

105°

of juices sold

120°

100°

12. Jen bought a car on sale for 15% off the original price, which was $12 500. How much did she pay for the car?

a. $1875 b. $8505 C. $10 625 d. $10 510

2 Jacaranda Maths Quest 9

20 10 40 30 Mon. Tues.

Wed. Thurs. Fri. 160° 160° x° x°

Day of week Number

13. Connor climbs to the summit of a mountain 4250 m above sea level. Inga is scuba diving at the bottom of the ocean. The ocean floor is 27 metres below sea level. What is the vertical distance between Connor and Inga?

14. The ratio of cordial to water in Shinji’s favourite drink is 2 : 5. How much water does he need to add to 80 mL of cordial to make his favourite drink?

a. 100 mL b. 200 mL

300 mL

15. Which of the following represents the right side view of the 3-dimensional figure shown? a.

400 mL

16. A rectangular paddock is surrounded by a fence. The perimeter of the paddock is 870 m. If the length of the paddock is 65 m longer than its width, calculate the dimensions of the paddock.

17. Calculate the area of material required to make the tent shown, including the floor.

a. 12 m2

C. 12.45 m2

b. 12.5 m2

d. 24.45 m2

18. Write the rule for the following pattern made from sticks.

Let n = number of sticks.

Let s = number of squares.

a. n = 2s + 2 b. n = 3s + 1 C. n = 4s

n = s + 3

19. If 1 teaspoon = 5 mL and 1 tablespoon = 20 mL, how many teaspoons are there in a tablespoon? a. 3 b. 4 C. 5 d. 2

20. The total surface area (TSA) of a cylinder can be found using the formula TSA = 2πr (r + h) where r is the radius and h is the height of the cylinder. Find h when r = 4 cm and the TSA is 377 cm2

21. A car travelled at 60 kilometres per hour for 2 hours and 15 minutes. How far did it travel? a. 165 km b. 120 km C. 75 km

135 km

h = ?

TOPIC 1 Numeracy 1 3

C. d.

1.8 m 1.5 m 1.5 m 2 m r

4 cm

22. The table below shows the times for 4 students to run 100 m

What is the average (mean) time the 4 students ran 100 m?

23. Which of the following is the smallest number? a. √ 3 6 b. 0.18

24. Complete the following to make this equation true. 13 × (15 1) = 180 + ?

25. The diagram shows a bag of tokens. Aneko chooses a token from the bag without looking. What is the chance of Aneko choosing a star token? a. 3 10 b. 1 3

1 4

26. Which of the following diagrams shows an equilateral triangle? a.

1 2

27. What is the volume of this rectangular prism?

28. What is the longest pencil that can fit inside the box shown? (Round to 1 decimal place.)

a. 26.9 mm b. 25 mm C. 22.4 mm d. 24.6 mm

29. Which coordinates will the graph of y =−2x + 1 pass through? a. (5, 11) b. (5, 9) C. (1, 3) d. (1, 3)

30. If a =−3, then 3a3 equals: a. 27

1.3 Set B

1.3.1 Non-calculator

1. How many faces does a rectangular prism have? a. 4 b. 6 C. 8 d. 12

2. Oranges sell for $2.50 per kilogram. What is the total cost of 3 kg of oranges?

a. $2.50 b. $5.00 C. $4.50 d. $7.50

4 Jacaranda Maths Quest 9

Student MitchJoshBenLiam Time (s) 14.2 15.4 14.8 14.4

d. 3 8

0.52

b. C. d.

d. 81

12 mm 6 mm 5 mm 20 mm 10 mm 15 mm

3. A fair, six-sided die is rolled. What is the chance of it landing on a 2?

1 6

4. 7 + (2 × 5) 9 = ?

a. 36

5. Evaluate 1.3 × 0.6.

1 3

6. What is the equation of the graph shown?

a. y = x + 3

b. y = x 3

C. y = 3x + 3

d. y = 3x 3

7. Draw a linear graph using the table of values below.

1 2

8. Sivan recorded the colour of cars travelling along her street between 9 am and 10 am. Colour BlueWhiteRedSilverBlackGold Number 1 6 2 5 3 3

What was the percentage of white cars?

9. What type of angle is this?

a. Acute

b. Obtuse

C. Reflex

d. Straight

10. What is the inequation that represents the interval shown on the number line?

11. A store is having a 25% off sale. What would be the reduced price of an item that was originally $120?

a. $40

b. $30

C. $100

d. $90

12. Two friends share 24 lollies. One friend gets twice as many as the other. How many lollies do they each receive?

a. 10 and 14

C. 6 and 18

8 and 16

4 and 20

TOPIC 1 Numeracy 1 5

d. 7

x 3 2 1 0 1 2 3 y 2 1 0 1 2 3 4

0123456 a. 1 ≤ x ≤ 6 b. 1 ≤ x < 6 C. 1 ≥ x ≥ 6 d. 1 < x ≤ 6

y x –1 1 2 3 –2 –3 –4 4 0 123 4 –1 –2 –3 –4 A B C

13. Calculate the product of 3 4 and 5 6 .

a. 1 7 12 b. 9 10

19 12 d. 5 8

14. The sum of 3 consecutive odd numbers is 21. Which equation best represents this situation?

a. x + 3 = 21 b. 3x = 21

15. Find the mean and the median of the data below.

2, 1, 5, 5, 4, 0, 2, 5, 2, 6

a. Mean = 3, median = 2

C. Mean = 3, median = 3

C. 3x + 6 = 21 d. 6x + 3 = 21

b. Mean = 3.2, median = 4

d. Mean = 3.2, median = 3

16. The Venn diagram shows the number of students in Year 9 who play basketball, tennis and hockey. Find n(tennis ∩ hockey)′

a. 61 b. 78

C. 70 d. 64

17. Evaluate 32 + 23 + 33

a. 40 b. 44

C. 36 d. 38

18. What is the missing number?

5000 + 700 + ? + 6 = 5746

19. A piece of string 30 cm long forms a rectangle. If the length of the rectangle formed is 10 cm, what is its width?

a. 5 cm b. 10 cm

C. 15 cm d. 20 cm

20. Which of the following is the top view of the 3-dimensional figure shown? a. b.

21. This ruler is used to measure the height of a teacher. How tall is the teacher?

a. 1.6 m

b. 1.75 m

C. 1.075 m

d. 175 m

22. A roof makes an angle of 40° with the ceiling. What is the value of x?

a. 50°

b. 40°

C. 45°

d. 90°

23. The stem-and-leaf plot shows the number of goals scored in each netball match over a season. The range is:

a. 48 b. 50

C. 14 d. 42

24. A circular spinner with 5 equal-sized sectors is labelled from 1 to 5. The spinner is spun. What is the probability of it landing on an odd number?

a. 3 b. 1 2

C. 3 5 d. 1 3

6 Jacaranda Maths Quest 9

3 ξ 11 BasketballTennis Hockey 16 114 21 14 9 m 2.5 2 1.5 1 x 40° 40° StemLeaf 1 4 9 2 5 7 9 3 1 2 4 8 4 0 4 5 4 7 6 2

25. Write an expression to represent the following: p is added to the square of q and then the sum is divided by y.

26. If one hour of parking costs $4.50, how much will it cost to park from 9 am until 2 pm?

a. $45

b. $22.50

C. $20 d. $16.50

27. A map has a scale showing that 1 cm on the map represents an actual distance of 2 km. Write the appropriate scale ratio for this map.

a. 1 : 2

b. 1 : 200

C. 1 : 2000 d. 1 : 200 000

28. The shape shown is rotated 90° in a clockwise direction around the black dot. Which of the following choices represents the result?

29. ABC is an isosceles triangle. What is the value of y?

a. 5 b. 7

C. 9 d. 11

30. Which of the following is the net of a cube?

TOPIC 1 Numeracy 1 7

a. b. C. d.

a. b. C. d. 2y + 5 A C B 3y – 2

Answers

Topic 1 Numeracy 1

Exercise 1.2 Set A 1.2.1 Calculator allowed

Exercise 1.3 Set B 1.3.1 Non-calculator

8 Jacaranda Maths Quest 9

1. C 2. $45 3. D 4. B 5. 54 cm2 6. 21 7. C 8. C 9. 3, 1, 4, 2, 250 10. A 11. 1 110 0012 12. C 13. 4277 m 14. B 15. C 16. l = 250 m w = 185 m 17. C 18. B 19. B 20. h ≈ 11 cm 21. D 22. 14.7 s 23. B 24. 2 25. C 26. D 27. 360 mm3 28. A 29. B 30. D

1. B 2. D 3. A 4. C 5. 0.78 6. C 7. y x –1 1 2 3 –2 –3 –4 0 123 –1 –2 –3 8. 30% 9. B 10. D 11. D 12. B 13. D 14. C 15. D 16. D 17. B 18. 40 19. A 20. C 21. B 22. A 23. A 24. C 25. q2 + p y 26. B 27. D 28. D 29. B 30. D

TOPIC 2 Number skills

2.1 Overview

2.1.1 Why learn this?

Mathematics has been a part of every civilisation throughout history. From the most primitive times people have needed numbers for counting and calculating. Our modern world is linked by computers, which rely heavily on numbers to store, find and track information.

2.1.2 What do you know?

1. THINK List what you know about numbers. Use a thinking tool such as a concept map to show your list.

2. PAIR Share what you know with a partner, then with a small group.

3. SHARE As a class, create a thinking tool such as a large concept map to show your class’s knowledge of real numbers.

LEARNING SEQUENCE

2.1 Overview

2.2 Rational numbers

2.3 Surds

2.4 Real numbers

2.5

2.6 Review

TOPIC 2 Number skills 9 NUMBER AND ALGEBRA

Scientific notation

• Note: The quotient 9 ÷ 0 has no answer; therefore, numbers such as 9 0 do not exist. They are said to be ‘undefined’. Because they ‘do not exist’, they are not rational numbers.

WORKED EXAMPLE 1

By writing each of the following in fraction form, show that the numbers are rational. a 7

THINK

Each number must be written as a fraction using integers.

a The number 7 has to be written in fraction form. To write a number as a fraction, it must be written with a numerator and denominator. For a whole number, the denominator is 1.

b The number 11 has to be written in fraction form. To write a number as a fraction, it must be written with a numerator and denominator. For a whole number, the denominator is 1

c The number 0 has to be written in fraction form. To write a number as a fraction, it must be written with a numerator and denominator. For a whole number, the denominator is 1.

d Change 4 2 5 to an improper fraction.

e The number 1.2 can be expressed as 1 + 0.2. This can then be expressed as 1 + 2 10 . Write this as an improper fraction.

WRITE

a 7 = 7 1 is rational

b 11 = 11 1 is rational

c 0 = 0 1 is rational

d 4 2 5 = 22 5 is rational.

e 1.2 = 12 10 is rational.

TOPIC 2 Number skills 11

b −11 c 0 d −4 2 5 e 1.2

2.2.4 Rational numbers written as decimals

• When a rational number is written as a decimal there are two possibilities.

1. The decimal terminates, e.g. 5 4 = 1.25

2. The decimal repeats, or recurs, e.g. 7 6 = 1.1666 . . .

For 1.1666 . . . the 6 is a repeating digit. This number, 7 6 , is called a recurring decimal, and it can also be written as 1.16.

1.615 or 1.615 means 1.615 615 615 . . . (The digits 615 in the decimal repeat.)

1.615 or 1.615 means 1.615 151 515 . . . (The digits 15 in the decimal repeat.)

1.615 . means 1.615 555 (Only the digit 5 in the decimal repeats.)

WORKED EXAMPLE 2

Write the first 8 digits of each of the following recurring decimals. a 3.02

a In 3.0 . 2 . , 02 recurs.

3.020 202 0 b In 47.1 . , 1 recurs.

47.111 111

c In 11.54 . 9 . , 49 recurs. c 11.549 494

WORKED EXAMPLE 3

Write each fraction as a recurring decimal using dot notation. (Use a calculator for the division.) a 5 6

a 5 ÷ 6 = 0.833 333 3

3 recurs — the dot goes above the 3

b 57 ÷ 99 = 0.575 757 575 7 57 recurs — dots go above the 5 and the 7.

c 25 ÷ 11 = 2.272 727 27 27 recurs — dots go above the 2 and the 7.

d 4 ÷ 7 = 0.571 428 571 It looks as though 571 428 will recur — dots go above the 5 and the 8. d 4 7 = 0.571428

2.2.5 Is every decimal a rational number?

• 1.237 is rational, because it is a terminating decimal. It is easy to show that 1.237 = 1237 1000

• 0.86, or 0.868 686 . . . , is rational because it is a recurring decimal. It can be shown that 0.86 = 86 99 .

• Decimals that do not terminate and also do not recur are not rational. They cannot be written as integer fractions or ratios, and are called irrational numbers.

12 Jacaranda Maths Quest 9

WRITE

b 47.1

11.549 THINK

b 57 99 c 25 11 d 4 7 THINK WRITE

5 6 = 0.83 .

b 57 99 = 0.5

7 .

25 11 = 2.2 . 7 .

2.2.6 The number system

Real numbers R

Irrational numbers I (surds; non-terminating, non-recurring decimals; π, e)

Negative integers Z ‒

Integers Z

Zero (neither positive nor negative)

Rational numbers Q

Non-integer rationals (terminating and recurring decimals)

Positive integers Z + (natural numbers N )

Note: A real number is any number that lies on the number line. Further explanation can be found in the next section.

RESOURCES — ONLINE ONLY

Complete this digital doc: SkillSHEET: Operations with directed numbers (doc-6100)

Exercise 2.2 Rational numbers

Individual pathways

U PRACTISE

Questions: 1a−l, 2a−f, 3a−e, 4−7

U CONSOLIDATE

Questions: 1d−l, 2c−j, 3c−g, 4−11

U MASTER

Questions: 1e−l, 2f−j, 3e−j, 4−11

U U U Individual pathway interactivity: int-4476 ONLINE ONLY

To answer questions online and to receive immediate feedback and sample responses for every question, go to your learnON title at www.jacplus.com.au. Note: Question numbers may vary slightly.

Fluency

1. WE1 Show that the following numbers are rational by writing each of them in fraction form.

TOPIC 2 Number skills 13

c. 22 3 d. 51 8 e.

f. 7

10 g. 0.002 h. 87.2 i.

j. 1.56 k. 3.612 l. 0.08

√16

2. WE2 Write the first 8 digits of each of the following recurring decimals.

a. 0.5 .

c. 0.51

e. 5.1 . 83 .

g. 8.9 . 124 .

i. 5.1234

b. 0.5 . 1 .

d. 6.031

f. 7.024 .

h. 5.123 . 4 .

j. 3.002

3. WE3 Write the following fractions as recurring decimals using dot notation. (Use a calculator.)

a. 5 9

c. 2 2 11

e. 173 99

g. 35 6

i. 46 99

Understanding

4. From the following list of numbers:

3, 3 7 , 0, 2.3, 2.3, 2 3 5 , 15

a. write down the natural numbers

b. write down the integers

c. write down the rational numbers.

5. Write these numbers in order from smallest to largest. 2.1, 2.12, 2.121, 2.121, 2.12

Reasoning

6. Explain why all integers are rational numbers.

b. 3 11

d. 3 7

f. 73 990

h. 7 15

j. 46 990

7. Are all fractions in which both the numerator and denominator are integers rational? Explain why or why not.

Problem solving

8. a. Using the fraction a b , where a and b are natural numbers, write 3 recurring decimals in fractional form using the smallest natural numbers possible.

b. What was the largest natural number you used?

9. Write 2 fractions that have the following number of repeating digits in their decimal forms.

a. 1 repeating digit

c. 3 repeating digits

b. 2 repeating digits

d. 4 repeating digits

10. If a $2 coin weighs 6 g, a $1 coin weighs 9 g, a 50c coin weighs 15 g, a 20c coin weighs 12 g, a 10c coin weighs 5 g and a 5c coin weighs 3 g, explain what the maximum value of the coins would be if a bundle of them weighed exactly 10 kg.

11. A Year 9 student consumes 2 bottles of water every day, except on every 5th day when she only has 1. Calculate her annual bottled water consumption.

Reflection

How many recurring digits will there be for 1 13 ?

2.3 Surds

2.3.1 Rational and irrational numbers

• Ancient mathematicians were shocked to find that rational numbers could not be used to label every point on the number line. In other words they discovered lengths that could not be expressed as fractions. These numbers are called irrational numbers.

14 Jacaranda Maths Quest 9

• A number is irrational if it is not rational; that is, if it cannot be written as a fraction, nor as a terminating or recurring decimal.

• Irrational numbers are denoted by the letter I

2.3.2 Roots

•The nth root of any positive number can be found. That is, √ n b = a or an = b. For example,

√25 = 5 because 25 = 5 × 5

Note: The 2 is usually not written in square roots, for example √25 =

2 25 √ 3 64 = 4 because 64 = 4 × 4 × 4.

WORKED

EXAMPLE 4

Evaluate each of the following.

a √16 b √ 3 27

√ 5 32 THINK WRITE

a Determine the number that when multiplied by itself gives 16 : 16 = 4 × 4

b Determine the number that when multiplied by itself three times gives 27 : 27 = 3 × 3 × 3.

c Determine the number that when multiplied by itself five times gives 32 : 32 = 2 × 2 × 2 × 2 × 2.

√16 = 4

TOPIC 2 Number skills 15

√

TI | CASIO

√ 3 27 = 3

√ 5 32 = 2

2.3.3 Surds

• When the square root of a number is an irrational number, it is called a surd. For example, √10 cannot be written as a fraction, nor as a recurring or terminating decimal. It is therefore irrational and is called a surd.

√10 ≈ 3.162 277 660 17 . . .

• The value of a surd can be approximated using a number line. For example, √21 will lie between 4 and 5, because √16 = 4 and √25 = 5. This can be shown on a number line.

WORKED EXAMPLE 5

Place √34 on a number line. THINK WRITE

1 The square number that is smaller than 34 is 25 The square number that is larger than 34 is 36. Write the numbers just below and just above √34. √34 lies between √25 and √36.

2 Draw a number line to show the approximate position of √34.

• Note: Negative numbers do not have square roots within the set of real numbers. However, in the 18th century mathematicians described the square root of a negative number as an ‘imaginary number’. Imaginary numbers and real numbers together form the set of complex numbers. Imaginary numbers have many uses in higher level mathematics used in science and engineering, but they are beyond the scope of this current course.

WORKED EXAMPLE 6

Which of the following are surds?

a √0 = 0. This is a rational number and therefore not a surd. a √0 = 0, which is not a surd.

b √20 ≈ 4.472 135 955 . . . It is an irrational number and therefore a surd. b √20 is a surd.

c √9 =−3. This is a rational number and therefore not a surd. c √9 =−3, which is not a surd.

√ 3 6 ≈ 1.817 120 592 83 . . . It is an irrational number and therefore a surd. d √ 3 6 is a surd.

16 Jacaranda Maths Quest 9

4567 25 36 34

b √20 c √9 d √ 3 6

√0

THINK WRITE

34567 16

25 21 (approximately)

2.3.4 Multiplying and dividing surds

• Consider that 3 = √9, 2 = √4, and 6 = √36.

The multiplication 3 × 2 = 6 could be written as √9 × √4 = √36

• In general, √a × √b = √ab. For example, √7 × √3 = √21.

•Similarly, √a ÷ √b = √ a b

For example, √18 ÷ √3 = √6.

WORKED EXAMPLE 7

Evaluate the following, leaving your answer in surd form. a √7 × √2

THINK WRITE

a Apply the rule √a × √

b Only √3 is a surd. It is multiplied by 5, which is not a surd.

c Apply the rule √

d Multiply the whole numbers by each other. Multiply the surds by each other.

WORKED EXAMPLE 8

Evaluate the following, leaving your answer in surd form.

THINK

a Apply the rule

b Simplify the fraction first.

c Simplify the whole numbers. Then apply the rule

d Apply the rule

e Rewrite the numerator as the product of two surds and then simplify.

TOPIC 2 Number skills 17

b 5 × √3 c √5 × √5 d 2√3 × 4√5

a √7 × √2 = √14

b = √ab

b 5 × √3 = 5√3

a × √b

√ab

c √5 × √5 = √25 = 5

d 2√3 × 4√5 =−2 × 4 × √3 × √5 =−8 × √15 =−8√15

a √10 √5 b √ 10 5 c 6√8 4√4 d √20 √5 e 5 √5

WRITE

√a ÷ √b = √ a b a √10 √5 = √ 10 5 = √2

b √ 10 5 = √2

√a ÷ √b = √ a b . c 6√8 4√4 = 3√2 2

√ a b d √20 √5 = √4 = 2

√a ÷ √b =

e 5 √5 = √5 × √5 √5 = √5