Sandra Tisdell–Clifford DEVELOPMENTAL MATHEMATICS BOOK...

DEVELOPMENTAL MATHEMATICS BOOK 3

Sandra Tisdell–Clifford

Founding authors Allan Thompson • Effie Wrightson

Series editor Robert Yen

Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States

iiiISBN 9780170351027

Preface/About the author vi

Features of this book vii

Curriculum grids viii

Series overview x

CHAPTER 1 PYTHAGORAS’ THEOREM1–01 Squares, square roots and surds 2 1–02 Pythagoras’ theorem 41–03 Finding the hypotenuse 61–04 Finding a shorter side 81–05 Mixed problems 101–06 Testing for right-angled triangles 121–07 Pythagorean triads 141–08 Pythagoras’ theorem problems 16Practice Test 1 19

CHAPTER 2WHOLE NUMBERS AND DECIMALS2–01 Mental addition 222–02 Mental subtraction 242–03 Rounding decimals and money 262–04 Adding and subtracting decimals 282–05 Mental multiplication 302–06 Multiplying decimals 322–07 Mental division 342–08 Dividing decimals 362–09 Terminating and recurring decimals 38Practice Test 2 41

CHAPTER 3INTEGERS AND FRACTIONS3–01 Adding and subtracting integers 443–02 Multiplying and dividing integers 463–03 Order of operations 483–04 Fractions 503–05 Fraction of a quantity 523–06 Ordering fractions 543–07 Adding and subtracting fractions 563–08 Multiplying and dividing fractions 58Practice Test 3 61

CHAPTER 4ALGEBRA4–01 From words to algebraic expressions 644–02 Substitution 664–03 Adding and subtracting terms 684–04 Multiplying terms 704–05 Dividing terms 724–06 Expanding expressions 744–07 Factorising algebraic terms 764–08 Factorising expressions 784–09 Factorising with negative terms 80Practice Test 4 83

CONTENTS

ivDevelopmental Mathematics Book 3 ISBN 9780170351027

CHAPTER 5TRIGONOMETRY5–01 The sides of a right-angled triangle 865–02 The trigonometric ratios 885–03 Finding the tangent ratio 905–04 Trigonometry on a calculator 925–05 Using tan to find an unknown side 945–06 Using tan to find an unknown angle 965–07 Finding an unknown side 985–08 Finding more unknown sides 1005–09 Finding an unknown angle 102Practice Test 5 105

CHAPTER 6 PERCENTAGES6–01 Percentages, fractions and decimals 1086–02 Percentage of a quantity 1106–03 Expressing quantities as fractions and

percentages 1126–04 Percentage increase and decrease 1146–05 The unitary method 1166–06 Profit, loss and discounts 1186–07 Simple interest 120Practice Test 6 123

CHAPTER 7INDICES7–01 Multiplying terms with the same base 1267–02 Dividing terms with the same base 1287–03 Power of a power 1307–04 Zero and negative indices 1327–05 Index laws review 1347–06 Significant figures# 1367–07 Scientific notation for large numbers 1387–08 Scientific notation for small numbers 1407–09 Scientific notation on a calculator 142Practice Test 7 145

CHAPTER 8GEOMETRY8–01 Types of angles 1488–02 Angle geometry 1518–03 Angles on parallel lines 1548–04 Types of triangles 1578–05 Triangle geometry 1608–06 Types of quadrilaterals 1638–07 Quadrilateral geometry 1668–08 Congruent figures 1698–09 Tests for congruent triangles 172Practice Test 8 176

CHAPTER 9EQUATIONS9–01 One-step equations 1809–02 Two-step equations 1829–03 Equations with variables on both sides 1849–04 Equations with brackets 1869–05 Simple quadratic equations x2 = c# 1889–06 Equation problems 190Practice Test 9 193

CHAPTER 10EARNING MONEY#

10–01 Wages and salaries# 19610–02 Overtime pay# 19810–03 Commission and piecework# 20010–04 Annual leave loading# 20210–05 Income tax# 20410–06 PAYG tax and net pay# 206Practice Test 10 209

# = NSW additional content

vISBN 9780170351027

CHAPTER 11INVESTIGATING DATA11–01 Types of data 21211–02 Sample vs. census 21411–03 The mean and mode 21611–04 The median and range 21811–05 Dot plots and stem-and-leaf plots 22011–06 Frequency tables 22311–07 Frequency histograms and polygons 22611–08 The shape of a distribution 229Practice Test 11 233

CHAPTER 12LENGTH AND TIME12–01 The metric system 23612–02 Limits of accuracy of measuring

instruments 23912–03 Perimeter 24212–04 Circumference of a circle 24412–05 Time 24712–06 Timetables 25012–07 International time zones# 253Practice Test 12 257

CHAPTER 13AREA AND VOLUME13–01 Area 26013–02 Area of a circle 26313–03 Areas of trapeziums, kites and

rhombuses 26613–04 Surface area of a rectangular prism 26913–05 Surface area of a triangular prism 27213–06 Surface area of a cylinder* 27413–07 Volume of a prism 27613–08 Volume of a cylinder 279Practice Test 13 282

CHAPTER 14GRAPHING LINES14–01 Tables of values 28614–02 The number plane 28814–03 Graphing tables of values 29014–04 Graphing linear equations 29214–05 Testing if a point lies on a line 29414–06 Horizontal and vertical lines 29614–07 Solving linear equations graphically 298Practice Test 14 301

CHAPTER 15PROBABILITY15–01 Probability 30415–02 Complementary events 30715–03 Relative frequency 30915–04 Venn diagrams 31115–05 Two-way tables 31515–06 Probability problems 318Practice Test 15 322

CHAPTER 16RATIOS AND RATES16–01 Ratios 32616–02 Ratio problems 32816–03 Scale maps and diagrams 33016–04 Dividing a quantity in a given ratio 33416–05 Rates 33616–06 Rate problems 33816–07 Speed 340Practice Test 16 343

Answers 345

Index 381

* = NSW STAGE 5.2, # = NSW additional content

viDevelopmental Mathematics Book 3 ISBN 9780170351027

PrEFACEIn schools for over four decades, Developmental Mathematics has been a unique, well-known and trusted Years 7–10 mathematics series aimed at developing key numeracy and literacy skills. This 5th edition of the series has been revised for the new Australian curriculum as well as the NSW syllabus Stages 4 and 5.1. The four books of the series contain short chapters with worked examples, definitions of key words, graded exercises, a language activity and a practice test. Each chapter covers a topic that should require about two weeks of teaching time.

Developmental Mathematics supports students with mathematics learning, encouraging them to experience more confidence and success in the subject. This series presents examples and exercises in clear and concise language to help students master the basics and improve their understanding. We have endeavoured to equip students with the essential knowledge required for success in junior high school mathematics, with a focus on basic skills and numeracy.

Developmental Mathematics Book 3 is written for students in Years 9–10, covering the Australian curriculum (mostly Year 9 content) and NSW syllabus (see the curriculum grids on the following pages and the teaching program on the NelsonNet teacher website). This book presents concise and highly-structured examples and exercises, with each new concept or skill on a double-page spread for convenient reading and referencing.

Students learning mathematics need to be taught by dynamic teachers who use a variety of resources. Our intention is that teachers and students use this book as their primary source or handbook, and supplement it with additional worksheets and resources, including those found on the NelsonNet teacher website (access conditions apply). We hope that teachers can use this book effectively to help students achieve success in secondary mathematics. Good luck!

AbouT ThE AuThorSandra Tisdell-Clifford teaches at Newcastle Grammar School and was the Mathematics coordinator at Our Lady of Mercy College (OLMC) in Parramatta for 10 years. Sandra is best known for updating Developmental Mathematics for the 21st century (4th edition, 2003) and writing its blackline masters books. She also co-wrote Nelson Senior Maths 11 General for the Australian curriculum, teaching resources for the NSW senior series Maths in Focus and the Years 7–8 homework sheets for New Century Maths/NelsonNet.

Sandra expresses her thanks and appreciation to the Headmaster and staff of Newcastle Grammar School and dedicates this book to her husband, Ray Clifford, for his support and encouragement. She also thanks series editor Robert Yen and editor Anna Pang at Cengage Learning for their leadership on this project.

Original authors Allan Thompson and Effie Wrightson wrote the first three editions of Developmental Mathematics (published 1974, 1981 and 1988) and taught at Smith’s Hill High School in Wollongong. Sandra thanks them for their innovative pioneering work, which has paved the way for this new edition for the Australian curriculum.

vii

FEATurES oF ThIS book• Eachchapterbeginswithatableofcontentsandlistofchapteroutcomes• Eachteachingsectionofachapterispresentedclearlyonadouble-pagespread

ISBN 9780170351027 Chapter 1 Pythagoras’ theorem1

1Pythagoras’ theorem

In thIs Chapter you will:

calculate the square and square root of a number

understand what a surd is

understand and write Pythagoras’ theorem for right-angled triangles

use Pythagoras’ theorem to find the length of the hypotenuse or shorter side in a right-angled triangle, giving the answer as a surd or a rounded decimal

use Pythagoras’ theorem to test whether a triangle is right-angled

investigate Pythagorean triads

solve problems involving Pythagoras’ theorem

what’s in Chapter 1?1–01 squares, square roots and surds

1–02 Pythagoras’ theorem

1–03 Finding the hypotenuse

1–04 Finding a shorter side

1–05 Mixed problems

1–06 testing for right-angled triangles

1–07 Pythagorean triads

1–08 Pythagoras’ theorem problems

shutterstock.com/isaravut

BK-CLA-DEV_MATH_BK_3-140196-Chp01.indd 1 17/12/14 2:06 PM

5 cm c

12 cm7 m p

4 m

8.25 m4.22 m

d

Developmental Mathematics Book 3 ISBN 97801703510276

WorDBanKexact form When an answer is written as an exact number, such as a whole number, decimal

or a surd, and not rounded.

example 6

Find the length of the hypotenuse in each triangle below, writing your answer in exact form.

a b

solutIon

a c2 = a2 + b2 b c2 = a2 + b2

= 52 + 122 p2 = 42 + 72 p is the hypotenuse. = 169 = 65 c 169= p 65= m This is in exact surd form. = 13 cm This is in exact form.

example 7

Find d correct to one decimal place.

solutIon

c2 = a2 + b2

d2 = 4.222 + 8.252

= 85.8709d = d 85.8709=

= 9.266655276 ≈ 9.3 m Rounded to one

decimal place.

to find the length of the hypotenuse in a right-angled triangle: write down Pythagoras’ theorem in the form c2 = a2 + b2 where c is the length of the hypotenuse solve the equation check that your answer is the longest side

Finding the hypotenuse 1–03

From the diagram, a hypotenuse of length 13 cm looks reasonable. It is also the longest side.

= 65p does not simplify to an exact decimal

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1BK-CLA-DEV_MATH_BK_3-140196-Chp01.indd 6 17/12/14 2:06 PM

c6 m

8 m

24 cm

10 cm

m

a

12 cm

9 cm8 m

z 15 m

30 m16 m

b

y

1 cm 2 cm

1 cm 1 cm

x

7.8 cm

9.6 cm 12.4 m

5.2 m

22.6 m

6.3 m

ISBN 9780170351027 Chapter 1 Pythagoras’ theorem7

1 Draw a right-angled triangle and mark the hypotenuse.

2 Copy and complete for the triangle shownc2 = a2 + b2

c2 = 62 + 2

= c = __

=

3 Find the length of the hypotenuse in each triangle below. Answer in exact form.

a b c

d e f

4 Round your answers to question 3 e and f to one decimal place.

5 Find the length of the hypotenuse in each triangle below. Answer correct to one decimal place.a b c

exercise 1–03

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• Theleftpagecontainsexplanations,workedexamples,andifappropriate,aWordbankof mathematical terminology and a fact box

• Therightpagecontainsanexerciseset,includingmultiple-choicequestions,scaffoldedsolutions and realistic applications of mathematics

• EachchapterconcludeswithaLanguage activity (puzzle) that reinforces mathematical terminology in a fun way, and a Practice test containing non-calculator questions on general topics and specific topic questions grouped by chapter subheading

Language activity

18Developmental Mathematics Book 3 ISBN 9780170351027

FInd-A-woRd PuzzleMake a copy of this page, then find all the words listed below in this grid of letters.

A O J S R O O T W Y P R T G B K J E Z G

W A B B A W J M X E B X M D A I H S S T

F P H V Q R Q M T X D J A F Y U N U V A

C B D R U E O C O N V E R S E H X N D G

A F Z U N M X G B W G Q Y H D S V E G L

K F F I Q C B U A S V L G R E U F T I V

A E L R D Y Y D X H M A S Y Q G L O U I

W T P V Q Q P L A K T Y A D E Q O P O S

N G W B B E K E N I D Y R Q V N V Y R T

W C M U S Z L Q J E R G P H O I X H P Y

K Q Q L O G S B A D A T Z J R Z U T L F

C C Q E N Y Q A E S W F Y T P U B Y P S

O R S A U V Q M T G O K D X H E K S R Z

F G I D Y U O M S M R L J N D E W K O J

B R V D K F Q Q V F Q D G I F V O I B U

T Q Z C Z S U E L G N A S A G W L R L Y

O I H B J A Q I C E P L Z U T J D G E C

A E L D R G P Y H C K Z L E R Q W E M M

D Q O E F C X J B U L C A N K D D N P V

B G A W L E I W U I U G U A R C A I B I

anGlE ConVERsE DRaw hyPotEnusE PRoBlEM PRoVE PythaGoRas RootsiDE sQuaRE suRD thEoREMtRiaD tRianGlE


12 m

10 m

4 m

h

kj

6 cm

c

5 cm 9 m13 m

r

Practice test 1

19ISBN 9780170351027 Chapter 1 Pythagoras’ theorem

Part a General topicsCalculators are not allowed.

1 Write 2108 in 12-hour time.

2 Complete: 60.8 cm = mm

3 Evaluate 44 × 5.

4 Find the perimeter of this shape.

5 Evaluate 27

314

− .

6 Expand –2(x – 4).

7 Find 23

of $36.

8 Find the mean of 1, 2, 8, 3, 6.

9 How many faces has a triangular prism?

10 Kathy pays a grocery bill of $83.45 with a $100 note. Calculate the change.

Part B Pythagoras’ theoremCalculators are allowed.

1–01 squares, square roots and surds

11 Evaluate each expression, correct to two decimal places.

a 34 b 6 32 2+ c 7.6 2.42 2−

12 Select all the surds from this list of square roots.

16 324 82 169 28 225 144 75 289 68

1–02 Pythagoras’ theorem

13 Name the hypotenuse and write Pythagoras’ theorem for this triangle.

1–03 Finding the hypotenuse

14 Find the value of each pronumeral, giving your answer as a surd.

a b


• AdditionalteachingresourcescanbedownloadedfromtheNelsonNetteacherwebsiteat www.nelsonnet.com.au: worksheets, puzzle sheets, skillsheets, video tutorials, technology worksheets, teaching program, curriculum grids, chapter PDFs of this book

• Note: Complimentary access to NelsonNet is only available to teachers who use Developmental Mathematics as a core educational resource in their classroom. Contact your sales representative for information about access codes and conditions.

ISBN 9780170351027 Features of this book

Developmental Mathematics Book 3 ISBN 9780170351027viii

Strand and SubStrand

developmental mathematicS BOOk 1chapter


number and alGebra

Number and place value 1 Integers and the number plane3 Working with numbers4 Factors and primes5 Powers and decimals6 Multiplying and dividing

decimals9 Algebra and equations

1 Working with numbers2 Primes and powers4 Integers

real numbers 5 Powers and decimals6 Multiplying and dividing

decimals7 Fractions8 Multiplying and dividing fractions17 Percentages and ratios

5 Decimals11 Fractions12 Percentages16 Ratios and rates

Money and financial mathematics 6 Multiplying and dividing decimals 12 Percentages

Patterns and algebra 9 Algebra and equations 6 Algebra15 Further algebra

Linear and non-linear relationships 1 Integers and the number plane9 Algebra and equations

15 Further algebra17 Graphing lines

meaSurement and GeometrY

using units of measurement 12 Length and time13 Area and volume

9 Length and time10 Area and volume

Shape 10 Shapes and symmetry

Location and transformation 10 Shapes and symmetry 7 Angles and symmetry

Geometric reasoning 2 Angles11 Geometry

7 Angles and symmetry8 Triangles and quadrilaterals

Pythagoras and trigonometry 3 Pythagoras’ theorem

StatiSticS and probabilitY

Chance 16 Probability 14 Probability

Data representation and interpretation

14 Statistical graphs15 Analysing data

13 Investigating data

curriculum Grid AUSTRALIAN CURRICULUM

ISBN 9780170351027ix

curriculum Grid AUSTRALIAN CURRICULUM

Strand and SubStrand



number and alGebra

real numbers 2 Whole numbers and decimals3 Integers and fractions6 Percentages7 Indices16 Ratios and rates

1 Working with numbers2 Percentages7 Ratios and rates

Money and financial mathematics

6 Percentages 3 Earning and saving money

Patterns and algebra 4 Algebra7 Indices

4 Algebra10 Indices

Linear and non-linear relationships

9 Equations14 Graphing lines

13 Equations and inequalities15 Coordinate geometry16 Graphing lines and curves

meaSurement and GeometrY

using units of measurement 12 Length and time13 Area and volume

9 Length and time11 Area and volume

Geometric reasoning 8 Geometry 8 Congruent and similar figures

Pythagoras and trigonometry 1 Pythagoras’ theorem5 Trigonometry

5 Pythagoras’ theorem6 Trigonometry

StatiSticS and probabilitY

Chance 15 Probability 14 Probability

Data representation and interpretation

11 Investigating data 12 Investigating data

Developmental Mathematics Book 3 ISBN 9780170351027x

booK 1

1 Integers and the number plane2 Angles3 Working with numbers4 Factors and primes5 Powers and decimals6 Multiplying and dividing decimals7 Fractions8 Multiplying and dividing fractions9 Algebra and equations10 Shapes and symmetry11 Geometry12 Length and time13 Area and volume14 Statistical graphs15 Analysing data16 Probability17 Percentages and ratios

booK 2

1 Working with numbers2 Primes and powers3 Pythagoras’ theorem4 Integers5 Decimals6 Algebra7 Angles and symmetry8 Triangles and quadrilaterals9 Length and time10 Area and volume11 Fractions12 Percentages 13 Investigating data14 Probability15 Further algebra16 Ratios and rates17 Graphing lines

booK 3

1 Pythagoras’ theorem2 Whole numbers and decimals3 Integers and fractions4 Algebra5 Trigonometry 6 Percentages7 Indices8 Geometry9 Equations10 Earning money11 Investigating data 12 Length and time13 Area and volume 14 Graphing lines15 Probability 16 Ratios and rates

booK 4

1 Working with numbers2 Percentages3 Earning and saving money4 Algebra5 Pythagoras’ theorem6 Trigonometry7 Ratios and rates8 Congruent and similar figures9 Length and time10 Indices11 Area and volume 12 Investigating data13 Equations and inequalities14 Probability15 Coordinate geometry16 Graphing lines and curves

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Sandra Tisdell–Clifford DEVELOPMENTAL MATHEMATICS BOOK...

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