Pearson Mathematics 9 Teacher Companion - Content

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ContentsPearson Mathematics writing and development team iii

Series features vi

Using Pearson Mathematics Teacher Companion viii

Pearson Mathematics 9 Curriculum Correlation ix

Chapter 1 Financial mathematics 2Recall 1 4

Exploration Task: Get rich quick? 4

1.1 Percentages review 5

Problem solving: Age-old dilemmas 13

1.2 Buying and selling 14

Problem solving: Patrick’s prized painting 21

Gamespace: ShopBay 22

1.3 Earning an income 24

Problem solving: Careless Carly 30

Investigation: Working out the roster 31

1.4 Tax and other pay deductions 33

Game: Master of the market 39

Maths 4 Real: A fair wage 40

Half-time 1 42

1.5 Simple interest 43

Exploration Spreadsheet: Using iteration to investigate savings plans 50

1.6 Payment methods 52

1.7 The cost of resources 59

1.8 Communication costs 67

Puzzle: Sudoku 73

Challenge 1 74

Chapter review 1 75

Numeracy practice 1 79

Chapter 2 Pythagoras’ theorem 80Recall 2 82

Exploration Task: Pythagorean triples 82

2.1 Pythagoras’ theorem and right-angled triangles 83

Problem solving: Olives and pots 89

2.2 Finding the length of the hypotenuse 90

Game: Perfect square root bingo 99

2.3 Finding the length of a shorter side 100

Puzzle: Roots but no plants 104

Half-time 2 105

Maths 4 Real: Pythagoras and plasma screens 106

Investigation: Estimating with Pythagoras 108

2.4 Applications of Pythagoras’ theorem 109

Puzzle: Pythagoras and the missing fig 114

2.5 Pythagorean triples 115

Problem solving: How old was Pythagoras? 119

Gamespace: The Pythagoreans 120

Challenge 2 122

Chapter review 2 123


Mixed review A 127

Chapter 3 Algebra 130Recall 3 132

Exploration Task: How many different areas of the same rectangle? 132

3.1 Introducing index laws using variables 133

Problem solving: Square cubes 141

Maths 4 Real: The digital world comes in byte-sized bits 142

3.2 More index laws and index properties 144

Puzzle: What is a ‘googolplex’? 155

3.3 Scientific notation and significant figures 156

Problem solving: Stairway to the Moon 165

Gamespace: Super powers 166

3.4 Rearranging formulas 168

Puzzle: In the prime of his life 172

Half-time 3 173

3.5 Expanding the brackets 174

Problem solving: Divisibility dilemma? 181

3.6 Expanding special products 182

Problem solving: Oranges and apples 185

3.7 Factorising using common factors 186

Game: Matching expressions 189

Investigation: Mr Gershwin’s piano 190

3.8 Factorising by grouping in pairs 192

Puzzle: Trouble in variable paradise 197

Challenge 3 198



Exploration STEM: Help find the best loan! 202

Exploration STEM: Grow your own Pythagoras tree 202

Exploration STEM: How healthy is your garden? 203

Exploration Coding: The bubble sort 203

Pearson Maths 2e 9 TC-00 PT1.indd 4 22/01/2017 3:00 pm

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Chapter 4 Measurement 204Recall 4 206

Exploration Task: Seeing double 206

4.1 Perimeter 207

4.2 Area 215

Maths 4 Real: Veterinary maths 224

Half-time 4 226

4.3 Surface area 227

Problem solving: Icosahedron 234

4.4 Volume and capacity 235

Problem solving: Gold, set, match 241

Exploration CAS: The properties of cylinders 242

Gamespace: Origami sports shirt 244

Investigation: The humble milk carton 246

Challenge 4 247



Mixed review B 253

Chapter 5 Linear relationships 256Recall 5 258

Exploration Task: Straight lines 258

5.1 Solving linear equations 259

Problem solving: Burning down 267

Gamespace: A day out 268

5.2 Solving problems using linear equations 270

Problem solving: Fractions in fractions in fractions 272

5.3 Coordinate geometry 273

Puzzle: Hitori 277

5.4 Plotting linear graphs 278

Problem solving: How many edges? 285

Half-time 5 286

5.5 Gradient 287

Problem solving: Lost in the desert 296

Exploration CAS: Linear relationships 297

5.6 Sketching linear graphs using the gradient and y-intercept 300

Game: Equation battleships 311

Maths 4 Real: Engineering and linear relationships 312

5.7 Sketching linear graphs using intercepts 314

5.8 Vertical and horizontal graphs 318

Problem solving: Area of a rectangle 321

Investigation: A fun park dilemma 322

Challenge 5 324



Appendices: Templates and BLMs A1-1

Answers for appendices A1-20

For Chapters 6–9, see Teacher Companion Part Two.


Student BookThe Second Edition Student Book includes updated questions, activities and design, with full coverage of the Australian Curriculum: Mathematics as well as the Victorian Curriculum: Mathematics.

It incorporates the latest research as well as feedback from teachers and learners across Australia.

Content caters for students of all abilities, with improved differentiation of all exercise questions and more questions for students consolidating their skills.

Homework ProgramThe Homework Program provides a collection of tear-out worksheets for students to practise and revise mathematical concepts.

Teacher CompanionThe Teacher Companion makes lesson preparation easy by combining full-colour Student Book pages with teacher support including improved contextual teaching suggestions and strategies, class activities, extra questions, worked solutions and answers for every question in the Student Book.

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Teacher Companion 1 Teacher Companion 2

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Pearson Lightbook StarterLightbook Starter is an innovative digital resource powered by Pearson’s award-winning Lightbook technology. It has been developed to help students learn key mathematical concepts, evaluate their understanding and track their progress. ‘Before you begin’ sections assess learner readiness before each chapter topic, while ‘Check-in’ questions can be used to evaluate learner understanding and practice after every chapter section.

Auto-correcting questions are linked to the Progress Tracker dashboard for easy analysis and viewing of results, which are mapped to progression through the Student Book as well as to Australian Curriculum: Mathematics and Victorian Curriculum: Mathematics content descriptions.

Pearson eBookMuch more than just pages on a screen, Pearson eBook is an online or offline version of your Student Book linked to interactive content, rich media resources and other useful content specifically developed for Mathematics. It supports you with appropriate online resources and tools for every section of the Student Book, including videos, eWorked Examples, interactive lessons, worksheets and more. Teacher resources include chapter tests, full teaching programs and curriculum mapping for the Australian Curriculum: Mathematics and for the Victorian Curriculum: Mathematics.

Pearson Places is the gateway to digital learning material for teachers and students across Australia. Access your content at www.pearsonplaces.com.au.

Professional Learning, Training and DevelopmentDid you know that Pearson also offers teachers a diverse range of training and development product-linked learning programs? We are dedicated to supporting your implementation of Pearson Mathematics, but it doesn’t stop there.

Our courses align closely with Pearson Mathematics Second Edition and offer an in-depth learning experience, combining both practical and theoretical elements, enabling you to implement the resource effectively in your classroom.

Find out more about our product-linked learning, workshops, courses and conferences at Pearson Academy www.pearsonacademy.com.au.

We believe in learning.All kinds of learning for all kinds of people,delivered in a personal style.Because wherever learning flourishes, so do people.

mathematics


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Support for the whole department!The Pearson Mathematics 9 Teacher Companion has been designed to provide support for all mathematics teachers at your school, from least to most experienced.

Active participation and inquiry

Class activities• suggested games and activities that teachers

might use to introduce, reinforce or revise mathematical concepts and skills

• useful BLMs provided

Copyright © 2017 Pearson Australia (a division of Pearson Australia Group Pty Ltd)

Spot the hypotenuseFind as many hypotenuses as possible in the diagram below. Highlight them once you have checked that the right-angled triangle in question has an angle that is 90°. It is a race to the finish!

Target:

Excellent: greater than 7

Very good: greater than 5

Good: greater than 3

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Spot the hypotenuseEquipment required: protractor (optional), coloured pencils, 1 copy of Appendix 2C per student

This 5-minute activity can be used when students’ concentration starts to wane, as a quick break, or as a concluding activity. The aim is for students to practise identifying the hypotenuse of a right-angled triangle. The diagram in Appendix 2C shows many right-angled triangles. How many can your students find? Have students trace over any hypotenuse they identify. You could turn this activity into a game where the winner is the student who can correctly identify the most. Encourage your students to use a protractor to check their work or if they aren’t sure whether or not an angle is a right angle (the diagram is drawn to scale).

Recap• quick questions for

the beginning or end of class

• encouraging a calm, ordered beginning or end to the lesson

Resource summaries• a list at the beginning

of each section of all the digital and print resources available, including videos, interactives, tutorials and more

Comprehensive teaching support

Teaching strategies• tips of the trade you

would tell a new teacher if you had time

• common student misconceptions

• help for students experiencing difficulties

• suggestions for students who finish a task quickly

Suggested examples• examples not in the

Student Book that help model the working of questions in each section

Answers and worked solutions• answers and solutions

showing the working required for every Student Book question and feature

Question Answer

1 Simplify: 4a2b × -2a3b5

-8a5b6

2 Simplify: 24x5y4 ÷ 8x2y

3x3y3

3 Simplify: 4a

4 Simplify: (p3)5 × (p2)2

p19

5 Evaluate 4k0. 4

Recap

2a2 4b×2ab

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ResourceseWorked examples• Finding a percentage of an amount• Increasing and decreasing by a given

percentage• Writing one amount as a percentage

of another

Lessons• Percentages (1)• Percentages (2)

Lightbook Starter• Check-in 1.1

Appendices• 1A Fraction, decimal and percentage

equivalence cards• 1B Ordering fractions and

percentages cards

Which short side is a and which short side is b?The side labels a and b can be interchanged. Emphasise this point to students so that they are confident when labelling all three sides of a right-angled triangle. Show students that they will obtain the same answer to a problem regardless of which side is labelled a and which side is labelled b. Take this discussion further. What alternative pronumerals could be used to replace a and b? What about c? What should one do to solve a problem if a and b are the same size? (See suggested example 4.)

Suggested examples1 Jeanne pays simple interest on her

loan of $3790 at a rate of 2% p.a. for 3 years. How much interest does she pay?

Answer:I = PrTI = 3790 × 0.02 × 3I = $227.40Jeanne paid back $227.40 in simple interest.

2 Marcus borrowed $3400 to buy a boat for 18 months at a rate of 19% p.a. simple interest. How much will he repay in total?

Answer:I = PrTI = $3400 × 0.19 × 1.5I = $969

Total payment = 3400 + 969 = $4369Marcus will pay $4369 for the boat.

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Teacher Support

Suggested Examples

1 Shade of each of the following:

(a) (b) (c)

Answer:Divide each shape or collectio

n

into 8 equal parts and shade one part.

(a) (b) (c)

2 What fraction of the following shapes and collections is shaded?

(a) (b) (c)

Answer:Count the total number of parts: this is the denominator. Count the number of shaded parts this is the numerator.

(a) (b) (c)

3 Place the following fractions on a number line: ,

, , , , 1

Answer:To place the fractions with a denominator of 2, 3 and 6, we need to partition each whole into 6 equal parts.

Note: You could use this example to discuss how whole numbers can be represented as fractions.

4 Write the number 3 as an improper fraction with a denominator of 4.

Answer:

3

=

3

×

4

=

12

3

=

18---

27--- 1

6--- 3

13-----

12--- 5

6--- 3

3--- 3

2--- 12

6----- 2

3---

0 21

12

56

33

32

23

1261

?4---

124-----

Resources

Interactives

• Fractions of a whole drag and drop

Tutorials and quizzes

• One quantity as a fraction of another

eWorked Examples

• eWorked Example 1• eWorked Example 2• eWorked Example 3• eWorked Example 4

Question Answer

1

45

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9 5

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7

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2 3.5 or 3 or

3 remainder 1

3

of 24 6

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of 100 20

5

Write in ascending order:

1 , 0, , 1, 0, , , 1, 1

3.1

Recap

12---

14---

15---

12--- 3

4--- 1

4--- 1

4--- 3

4--- 1

2---

Resources

Worksheets

• R3.1—Fractions of shapes: the importance of equal parts

• R3.2—Fractions of shapes and expressing fractions in words

• R3.3—Ordering fractions• R3.4—Addition and subtraction of

fractions with like denominators• R3.5—Multiples and factors

RECALL 3

1 (a)

,

(b)

,

(e)

2 (a) (b) (c)

2 or

(d) (e) (f)

(g) (h)

3 (a)

2, 1 , 1, , 0

(b)

0, , , , 1, ,

4 (a) (b) (c)

or 1

5 (a)

8, 16, 24, 32, 40

(b)

12, 24, 36, 48, 60

(c)

24

(d)

1, 2, 3, 4, 6, 8, 12, 24

(e)

1, 2, 3, 4, 6, 9, 12, 18, 36

(f)

12

Worksheets

Answers

15--- 3

8--- 2

5--- 12

5------

110------ 5

8--- 2

3---

125

------ 716------

14--- 3

4---

18--- 3

8--- 7

8--- 9

8--- 11

8------

57--- 6

11------ 4

3--- 1

3---

AA Pearson Maths 7 TC-03-02.fm Page 107 Monday, August 22, 2011 11:06 AM

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Teacher Companionmathematics


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Pearson Mathematics 9 Curriculum CorrelationAustralian Curriculum: Mathematics correlationThis maps the Australian Curriculum: Mathematics syllabus to Pearson Mathematics 9.

For further details and for correlations to the Victorian Curriculum, see the Teacher Resources available to download from the eBook, or from the ProductLink page on the Pearson Places website.

Number and Algebra Pearson Mathematics 9 Real numbers Chapter 3 Algebra

Chapter 9 Non-linear relationships and proportion

Solve problems involving direct proportion. Explore the relationship between graphs and equations corresponding to simple rate problems (ACMNA208)

• identifying direct proportion in real-life contexts

9.6 Direct proportion9.7 Inverse proportion

Apply index laws to numerical expressions with integer indices (ACMNA209)

• simplifying and evaluating numerical expressions, using involving both positive and negative integer indices

3.1 Introducing index laws using variables3.2 More index laws and index properties

Express numbers in scientific notation (ACMNA210)

• representing extremely large and small numbers in scientific notation, and numbers expressed in scientific notation as whole numbers or decimals

3.3 Scientific notation and significant figures

Money and financial mathematics Chapter 1 Financial mathematics

Solve problems involving simple interest (ACMNA211)

• understanding that financial decisions can be assisted by mathematical calculations

1.1 Percentages review1.2 Buying and selling1.3 Earning an income1.4 Tax and other pay deductions1.5 Simple interest1.6 Payment methods1.7 The cost of resources1.8 Communication costs

Patterns and algebra Chapter 3 Algebra

Extend and apply the index laws to variables, using positive integer indices and the zero index (ACMNA212)

• understanding that index laws apply to variables as well as numbers

3.1 Introducing index laws using variables3.2 More index laws and index properties

Apply the distributive law to the expansion of algebraic expressions, including binomials, and collect like terms where appropriate (ACMNA213)

• understanding that the distributive law can be applied to algebraic expressions as well as numbers

• understanding the relationship between expansion and factorisation and identifying algebraic factors in algebraic expressions

3.4 Rearranging formulas3.5 Expanding the brackets3.6 Expanding special products3.7 Factorising using common factors3.8 Factorising by grouping in pairs


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Number and Algebra Pearson Mathematics 9 Linear and non-linear relationships Chapter 5 Linear relationships

Chapter 9 Non-linear relationships and proportion

Find the distance between two points located on the Cartesian plane using a range of strategies, including graphing software (ACMNA214)

• investigating graphical and algebraic techniques for finding distance between two points

• using Pythagoras’ theorem to calculate distance between two points

5.3 Coordinate geometry

Sketch linear graphs using the coordinates of two points and solve linear equations (ACMNA215)

• determining linear rules from suitable diagrams, tables of values and graphs and describing them using both words and algebra

5.1 Solving linear equations5.2 Solving problems using linear equations5.4 Plotting linear graphs5.6 Sketching linear graphs using the gradient and y-intercept5.7 Sketching linear graphs using the x- and y-intercepts5.8 Vertical and horizontal graphs

Find the midpoint and gradient of a line segment (interval) on the Cartesian plane using a range of strategies, including graphing software (ACMNA294)

• investigating graphical and algebraic techniques for finding midpoint and gradient

• recognising that the gradient of a line is the same as the gradient of any line segment on that line

5.3 Coordinate geometry5.5 Gradient5.6 Sketching linear graphs using the gradient and y-intercept

Graph simple non-linear relations with and without the use of digital technologies and solve simple related equations (ACMNA296)

• graphing parabolas, and circles connecting x-intercepts of a graph to a related equation

9.1 Quadratic relationships9.2 Solving quadratic equations9.3 Sketching parabolas using transformations9.4 Circle relationships9.5 Exponentials and hyperbolas

Measurement and Geometry Pearson Mathematics 9 Using units of measurement Chapter 3 Algebra

Chapter 4 MeasurementChapter 6 Geometric reasoning

Calculate areas of composite shapes (ACMMG216)

• understanding that partitioning composite shapes into rectangles and triangles is a strategy for solving problems involving area

4.1 Perimeter4.2 Area

Calculate the surface area and volume of cylinders and solve related problems (ACMMG217)

• analysing nets of cylinders to establish formulas for surface area

• connecting the volume and capacity of a cylinder to solve authentic problems

4.3 Surface area4.4 Volume and capacity6.7 Solids and nets

Solve problems involving the surface area and volume of right prisms (ACMMG218)

• solving practical problems involving surface area and volume of right prisms

4.3 Surface area4.4 Volume and capacity

Investigate very small and very large time scales and intervals (ACMMG219)

• investigating the usefulness of scientific notation in representing very large and very small numbers

3.3 Scientific notation and significant figures


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Measurement and Geometry Pearson Mathematics 9 Geometric reasoning Chapter 6 Geometric reasoning

Use the enlargement transformation to explain similarity and develop the conditions for triangles to be similar (ACMMG220)

• establishing the conditions for similarity of two triangles and comparing this to the conditions for congruence

• using the properties of similarity and ratio, and correct mathematical notation and language, to solve problems involving enlargement (for example, scale diagrams)

• using the enlargement transformation to establish similarity, understanding that similarity and congruence help describe relationships between geometrical shapes and are important elements of reasoning and proof

6.1 Angle review6.2 Triangles and congruency6.3 Special quadrilaterals and their properties6.5 Similarity and similar triangles6.6 Solving problems using similar triangles

Solve problems using ratio and scale factors in similar figures (ACMMG221)

• establishing the relationship between areas of similar figures and the ratio of corresponding sides (scale factor)

6.4 Enlarging and reducing6.5 Similarity and similar triangles6.6 Solving problems using similar triangles

Pythagoras and trigonometry Chapter 2 Pythagoras’ theoremChapter 7 Trigonometry

Investigate Pythagoras’ Theorem and its application to solving simple problems involving right angled triangles (ACMMG222)

• understanding that Pythagoras’ Theorem is a useful tool in determining unknown lengths in right-angled triangles and has widespread applications

• recognising that right-angled triangle calculations may generate results that can be integers, fractions or irrational numbers

2.1 Pythagoras’ Theorem and right-angled triangles2.2 Finding the length of the hypotenuse2.3 Finding the length of a shorter side2.4 Applications of Pythagoras’ Theorem2.5 Pythagorean triples

Use similarity to investigate the constancy of the sine, cosine and tangent ratios for a given angle in right-angled triangles (ACMMG223)

• developing understanding of the relationship between the corresponding sides of similar right-angled triangles

7.1 Introduction to trigonometry7.2 Trigonometric ratios

Apply trigonometry to solve right-angled triangle problems (ACMMG224)

• understanding the terms ‘adjacent’ and ‘opposite’ sides in a right-angled triangle

• selecting and accurately using the correct trigonometric ratio to find unknown sides (adjacent, opposite and hypotenuse) and angles in right-angled triangles

7.1 Introduction to trigonometry7.2 Trigonometric ratios7.3 Using trigonometry to find side lengths7.4 Using trigonometry to find angles7.5 Applications of trigonometry


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Statistics and Probability Pearson Mathematics 9 Chance Chapter 8 Statistics and probability

List all outcomes for two-step chance experiments, both with and without replacement using tree diagrams or arrays. Assign probabilities to outcomes and determine probabilities for events (ACMSP225)

• conducting two-step chance experiments

• using systematic methods to list outcomes of experiments and to list outcomes favourable to an event

• comparing experiments which differ only by being undertaken with replacement or without replacement

8.6 Probability events8.7 Representing probability

Calculate relative frequencies from given or collected data to estimate probabilities of events involving ‘and’ or ‘or’ (ACMSP226)

• using Venn diagrams or two-way tables to calculate relative frequencies of events involving ‘and’, ‘or’ questions

• using relative frequencies to find an estimate of probabilities of ‘and’, ‘or’ events

8.5 Understanding probability8.6 Probability events8.7 Representing probability

Investigate reports of surveys in digital media and elsewhere for information on how data were obtained to estimate population means and medians (ACMSP227)

• investigating a range of data and its sources, for example the age of residents in Australia, Cambodia and Tonga; the number of subjects studied at school in a year by 14-year-old students in Australia, Japan and Timor-Leste

8.1 Investigating data8.2 Interpreting data8.4 Comparing data sets

Data representation and interpretation Chapter 8 Statistics and probability

Identify everyday questions and issues involving at least one numerical and at least one categorical variable, and collect data directly and from secondary sources (ACMSP228)

• comparing the annual rainfall in various parts of Australia, Pakistan, New Guinea and Malaysia

8.1 Investigating data8.2 Interpreting data8.4 Comparing data sets

Construct back-to-back stem-and-leaf plots and histograms and describe data, using terms including ‘skewed’, ‘symmetric’ and ‘bi modal’ (ACMSP282)

• using stem-and-leaf plots to compare two like sets of data such as the heights of girls and the heights of boys in a class

• describing the shape of the distribution of data using terms such as ‘positive skew’, ‘negative skew’ and ‘symmetric’ and ‘bi-modal’

8.3 Statistics from grouped data8.4 Comparing data sets

Compare data displays using mean, median and range to describe and interpret numerical data sets in terms of location (centre) and spread (ACMSP283)

• comparing means, medians and ranges of two sets of numerical data which have been displayed using histograms, dot plots, or stem and leaf plots

8.2 Interpreting data8.3 Statistics from grouped data8.4 Comparing data sets


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