Pearson Mathematics 8 Teacher Companion - Contents

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ContentsPearson Mathematics writing and development team iii

Series features vi

Using Pearson Mathematics Teacher Companion viii

Pearson Mathematics 8 Curriculum Correlation ix

Chapter 1 Integers and indices 2Recall 1 4

Exploration Task: Comparing powers 4

1.1 Integers review 5

Problem solving: Lab maths 12

Investigation: Walking the plank 13

1.2 Integer multiplication 14

Puzzle: The 1 dilemma 18

1.3 Integer division 19

Game: 3 in a row 23

Gamespace: Cryogenic crisis 24

Half-time 1 26

1.4 Combined operations with integers 27

Puzzle: Animal speed challenge 31

Maths 4 Real: The ultimate cool 32

1.5 Multiplying and dividing numbers in index form 34

Game: Closest to 500 43

Investigation: Cyclic powers 44

1.6 Powers of powers, products and quotients 45

Challenge 1 50

Chapter review 1 51

Numeracy practice 1 55

Chapter 2 Fractions, decimals and percentages 56Recall 2 58

Exploration Task: Is this right? 58

2.1 Working with fractions and decimals 59

Puzzle: Sudoku 66

2.2 Types of decimals 67

Puzzle: Hitori 74

Investigation: Terminating and recurring decimals 75

2.3 Negative fractions and decimals 76

2.4 Estimating percentages 84

Game: Best estimate 88

2.5 Writing fractions and decimals as percentages 89

Puzzle: KenKen 97

Half-time 2 98

2.6 Writing percentages as fractions and decimals 99

Game: First to change 105

2.7 Writing one amount as a percentage of another 106

Problem solving: League tables 112

2.8 Finding a percentage of an amount 113

Problem solving: Cordial contents 117

Gamespace: Alice in Numberland 118

2.9 Increasing or decreasing by a given percentage 120

Puzzle: Break the code 123

Investigation: Supermarket specials 124

2.10 Financial applications of percentages 126

Problem solving: The plummeting price 135

Exploration Spreadsheet: Sam’s Skate Shop 136

Maths 4 Real: Relative wealth 138

Challenge 2 140

Chapter review 2 141


Mixed review A 146

Chapter 3 Algebra 148Recall 3 150

Exploration Task: Which is larger? 150

3.1 Variables and expressions 151

Problem solving: Tricky algebra 157

3.2 Substitution for variables 158

Game: 4 in a row 162

3.3 Using formulas 163

Puzzle: How old am I? 167

Investigation: Take your medicine 168

Maths 4 Real: Astronomical algebra 170

3.4 Simplifying expressions 172

Half-time 3 177

Gamespace: Magic algebra 178

Pearson Maths 2e 8 TC-00 PT1.indd 4 18/01/2017 11:14 am

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3.5 Multiplying and dividing algebraic terms 180

Problem solving: Make a profit 183

3.6 Expanding brackets 184

Puzzle: The rat race 189

Exploration Spreadsheet: Farmer Jones’ dilemma 190

3.7 Factorising 192

Challenge 3 197



Exploration STEM: Planes, trains, boats and automobiles 202

Exploration STEM: The scale of the universe 202

Exploration STEM: Fishing, angling, netting 203

Exploration Coding: Quick division 203

Chapter 4 Ratio and rate 204Recall 4 206

Exploration Task: Ratios 206

4.1 Writing ratios 207

Game: The game of Nim 212

4.2 Simplifying ratios 213

Gamespace: Human ratios 218

4.3 Unit ratios and scale factors 220

Problem solving: Cartoon capers 228

4.4 Using ratios to find amounts 229

Puzzle: Sudoku 235

Investigation: Bicycle gears 236

Maths 4 Real: The golden ratio – fact or fiction? 238

Half-time 4 240

4.5 Scale drawings 241

4.6 Sharing an amount in a given ratio 248

Problem solving: Catch 22 251

4.7 Rates 252

Problem solving: Numberrrrrrrrrs 260

Challenge 4 261



Mixed review B 266

Chapter 5 Measurement 268Recall 5 270

Exploration Task: Can squares and rectangles be equal? 270

5.1 Perimeter 271

Game: That formula is mine 277

5.2 Circle relationships 278

Exploration CAS: The relationship between the circumference and diameter of a circle 281

5.3 Circumference 283

Problem solving: The icing on the cake 288

Investigation: Staggered starts 289

5.4 Area 291

Problem solving: Tiling a floor 302

Half-time 5 303

5.5 Area of a circle 304

Problem solving: Victa the goat 310

Exploration CAS: Area of equilateral triangles 311

5.6 Finding the area of composite shapes 314

Problem solving: Doubling a square 321

Gamespace: Are we there yet? 322

5.7 Volume and capacity 324

Problem solving: Baffling box 333

5.8 Time 334

Puzzle: Time gone cuckoo 344

Challenge 5 345

Maths 4 Real: Today’s date is … 346



Appendices: Templates and BLMs A1-1

Answers for appendices A1-16

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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PEARSON mathematics

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CHAPTERS 1–5PART ONE

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PEARSON mathematics

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Student Book

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Teacher Companion 1 Teacher Companion 2

Lightbook Starter

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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 8 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)

Paper sizesYou are probably familiar with A4, which is the most common standard paper size in Australia. A5 is half the size of A4, A6 is half the size of A5 and so on, proportionally as shown:

Procedure1 To begin, measure the width and length of a blank sheet of A4 paper to the nearest millimetre. For consistency,

always take the longer of the two measurements to be the length.

2 Fold the A4 piece of paper in half. Use scissors to cut along the fold to make an A5 piece of paper. Measure the A5 paper’s length and width.

3 Fill in the information for A4 and A5 in the following table. Calculate the ratio of the paper’s length and width.

4 Fold the A5 piece of paper in half. Use scissors and cut along the fold to make an A6 piece of paper. Measure the A6 paper’s length and width and fill in the information for A6.

5 Repeat the above process until you have folded, cut and measured the length and width of the A7, A8, A9 and A10 paper sizes.

6 Do you notice any relationship between the ratios of each paper size?

Paper size Length (mm) Width (mm) Ratio of length : width

A4

A5

A6

A7

A8

A9

A10

A5A4

A6

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Equivalent fractions and decimal Snap/Memory/Go FishStudents are to prepare 20 pairs of decimal and fraction cards that are equivalent. For example, one pair is

and 0.5. They may also have a pair that

is and 0.5 as this would enable more

snaps. There is a limit of four of the same decimal, and it is best that students prepare this for their game. Once they have their cards, students may play any of the three games using their decimal fraction cards.To modify the cards, some students may also have cards with diagrams on them.

12---

36---

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 If v = u + at, find v where u = 10, a = 32, t = 2.

74

2 If F = ma, find F where m = 8 and a = 12.

96

3 Which of the following are like terms to 4a?

3ab, 2a, -a, 4a2, -

2a, -a, -

Recap

a7---

a7---

ResourceseWorked examples• Ordering negative fractions and

decimals• Adding and subtracting negative

decimals• Adding and subtracting negative

fractions

Lessons• Operations on fractions 1• Operations with fractions and

decimals

Homework Program• Skills/Practice 2A

Lightbook Starter• Check-in 2.3

Magnitude and sizeTest that your students understand that -19 is less than -2, even though the magnitudes of 19 and 2 imply the opposite. You could plot the numbers on a number line and point out that the further right on a number line a number is, the larger it is. Alternatively, for an advanced class this could lead into a discussion about ‘absolute value’ (absolute value is the magnitude of an integer, and is enclosed in straight lines: ||). Thus, although -19 < -2,|-19| > |-2|.

Suggested examples1 Place the following fractions on a

number line: - , , - , , , -1

Answer:To place the fractions with a denominator of 2, you need to partition each whole into 2. To place the fractions with a denominator of 3, you need to partition each whole number into 3. To place the fractions with a denominator of 6, you need to partition each whole number into 6.

2 Calculate: −

Answer:

= +

= + =

22--- 3

3--- 5

6--- 3

2--- 12

6----- 2

3---

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126- -

23--- - 1

6---

23--- 1

6---

46--- 1

6--- 5

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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

=

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27--- 1

6--- 3

13-----

12--- 5

6--- 3

3--- 3

2--- 12

6----- 2

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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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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---

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Teacher Companionmathematics


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Pearson Mathematics 8 Curriculum CorrelationAustralian Curriculum: Mathematics correlationThis maps the Australian Curriculum: Mathematics syllabus to Pearson Mathematics 8.

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 8 Number and place value Chapter 1 Integers and indices

Chapter 2 Fractions, decimals and percentages

Use index notation with numbers to establish the index laws with positive integral indices and the zero index (ACMNA182)• evaluating numbers expressed as powers of positive integers

1.5 Multiplying and dividing numbers in index form1.6 Powers of powers, products and quotients

Carry out the four operations with rational numbers and integers, using efficient mental and written strategies and appropriate digital technologies (ACMNA183)• using patterns to assist in finding rules for the multiplication

and division of integers• using the number line to develop strategies for adding and

subtracting rational numbers

1.1 Integers review1.2 Integer multiplication1.3 Integer division1.4 Combined operations with integers2.1 Working with fractions and decimals2.3 Negative fractions and decimals2.5 Writing fractions and decimals as percentages2.6 Writing percentages as fractions and decimals

Real numbers Chapter 2 Fractions, decimals and percentagesChapter 4 Ratio and rate

Investigate terminating and recurring decimals (ACMNA184)• recognising terminating, recurring and non‑terminating

decimals and choosing their appropriate representations

2.1 Working with fractions and decimals2.2 Types of decimals2.3 Negative fractions and decimals

Investigate the concept of irrational numbers, including π (ACMNA186)• understanding that the real number system includes irrational

numbers

2.2 Types of decimals

Solve problems involving the use of percentages, including percentage increases and decreases, with and without digital technologies (ACMNA187)• using percentages to solve problems, including those

involving mark‑ups, discounts, and GST• using percentages to calculate population increases and

decreases

2.4 Estimating percentages2.5 Writing fractions and decimals as percentages2.6 Writing percentages as fractions and decimals2.7 Writing one amount as a percentage of another2.8 Finding a percentage of an amount2.9 Increasing or decreasing by a given percentage2.10 Financial applications of percentages

Solve a range of problems involving rates and ratios, with and without digital technologies (ACMNA188)• understanding that rate and ratio problems can be solved

using fractions or percentages and choosing the most efficient form to solve a particular problem

• calculating population growth rates in Australia and Asia and explaining their difference

4.1 Writing ratios4.2 Simplifying ratios4.3 Unit ratios and scale factors4.4 Using ratios to find amounts4.5 Scale drawings4.6 Sharing an amount in a given ratio4.7 Rates

Money and financial mathematics Chapter 2 Fractions, decimals and percentages

Solve problems involving profit and loss, with and without digital technologies (ACMNA189)• expressing profit and loss as a percentage of cost or selling

price, comparing the difference• investigating the methods used in retail stores to express

discounts

2.10 Financial applications of percentages

Patterns and algebra Chapter 3 Algebra

Extend and apply the distributive law to the expansion of algebraic expressions (ACMNA190)• applying the distributive law to the expansion of algebraic

expressions using strategies such as the area model

3.6 Expanding brackets


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Number and Algebra Pearson Mathematics 8 Factorise algebraic expressions by identifying numerical factors (ACMNA191)• recognising the relationship between factorising and

expanding• identifying the greatest common divisor (highest common

factor) of numeric and algebraic expressions and using a range of strategies to factorise algebraic expressions

3.5 Multiplying and dividing algebraic terms3.7 Factorising

Simplify algebraic expressions involving the four operations (ACMNA192)• understanding that the laws used with numbers can also be

used with algebra

3.1 Variables and expressions3.2 Substitution for variables3.3 Using formulas3.4 Simplifying expressions3.5 Multiplying and dividing algebraic terms

Linear and non-linear relationships Chapter 6 Linear graphsChapter 7 Linear equations

Plot linear relationships on the Cartesian plane with and without the use of digital technologies (ACMNA193)• completing a table of values, plotting the resulting points and

determining whether the relationship is linear• finding the rule for a linear relationship

6.1 Interpreting line graphs6.2 Linear relationships6.3 Finding the rule6.4 Using linear relationships

Solve linear equations using algebraic and graphical techniques. Verify solutions by substitution (ACMNA194)• solving real life problems by using variables to represent

unknowns

6.4 Using linear relationships7.1 The language of equations7.2 Solving linear equations7.3 Solving more complex equations7.4 Solving equations with the unknown on both sides7.5 Solving problems using equations

Measurement and Geometry Pearson Mathematics 8 Using units of measurement Chapter 5 Measurement

Choose appropriate units of measurement for area and volume and convert from one unit to another (ACMMG195)• choosing units for area including mm2, cm2, m2, hectares, km2,

and units for volume including mm3, cm3, m3

• recognising that the conversion factors for area units are the squares of those for the corresponding linear units

• recognising that the conversion factors for volume units are the cubes of those for the corresponding linear units

5.4 Area5.5 Area of a circle5.6 Finding the area of composite shapes5.7 Volume and capacity

Find perimeters and areas of parallelograms, trapeziums, rhombuses and kites (ACMMG196)• establishing and using formulas for areas such as trapeziums,

rhombuses and kites

5.1 Perimeter5.4 Area5.6 Finding the area of composite shapes

Investigate the relationship between features of circles such as circumference, area, radius and diameter. Use formulas to solve problems involving determining circumference and area (ACMMG197)• investigating the circumference and area of circles with

materials or by measuring, to establish an understanding of formulas

• investigating the area of circles using a square grid or by rearranging a circle divided into sectors

5.2 Circle relationships5.3 Circumference5.5 Area of a circle5.6 Finding the area of composite shapes

Develop formulas for volumes of rectangular and triangular prisms and prisms in general. Use formulas to solve problems involving volume (ACMMG198)• investigating the relationship between volumes of rectangular

and triangular prisms

5.7 Volume and capacity

Solve problems involving duration, including using 12‑ and 24‑hour time within a single time zone (ACMMG199)• identifying regions in Australia and countries in Asia that

are in the same time zone

5.8 Time


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Measurement and Geometry Pearson Mathematics 8 Geometric reasoning Chapter 8 Geometry

Define congruence of plane shapes using transformations (ACMMG200)• understanding the properties that determine congruence

of triangles and recognising which transformations create congruent figures

• establishing that two figures are congruent if one shape lies exactly on top of the other after one or more transformations (translation, reflection, rotation), and recognising that the matching sides and the matching angles are equal

8.3 Congruence and transformation

Develop the conditions for congruence of triangles (ACMMG201)• investigating the minimal conditions needed for the unique

construction of triangles, leading to the establishment of the conditions for congruence (SSS, SAS, ASA and RHS)

• solving problems using the properties of congruent figures• constructing triangles using the conditions for congruence

8.3 Congruence and transformation8.4 Congruent triangles

Establish properties of quadrilaterals using congruent triangles and angle properties, and solve related numerical problems using reasoning (ACMMG202)• establishing the properties of squares, rectangles,

parallelograms, rhombuses, trapeziums and kites• identifying properties related to side lengths, parallel sides,

angles, diagonals and symmetry

8.1 Angles review8.2 Shapes review8.4 Congruent triangles8.5 Congruence and quadrilaterals

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

Identify complementary events and use the sum of probabilities to solve problems (ACMSP204)• identifying the complement of familiar events• understanding that probabilities range between 0 to 1

and that calculating the probability of an event allows the probability of its complement to be found

9.5 Understanding probability9.6 Theoretical probability for single‑step experiments9.7 Venn diagrams and two‑way tables

Describe events using language of ‘at least’, exclusive ‘or’ (A or B but not both), inclusive ‘or’ (A or B or both) and ‘and’ (ACMSP205)• posing ‘and’, ‘or’ and ‘not’ probability questions about objects

or people

9.6 Theoretical probability for single‑step experiments9.7 Venn diagrams and two‑way tables

Represent events in two‑way tables and Venn diagrams and solve related problems (ACMSP292)• using Venn diagrams and two‑way tables to calculate

probabilities for events, satisfying ‘and’, ‘or’ and ‘not’ conditions• understanding that representing data in Venn diagrams or

two‑way tables facilitates the calculation of probabilities• collecting data to answer the questions using Venn diagrams

or two‑way tables

9.7 Venn diagrams and two‑way tables

Data representation and interpretation Chapter 9 Statistics and probability

Explore the practicalities and implications of obtaining data through sampling using a variety of investigative processes (ACMSP206)• investigating the uses of random sampling to collect data

9.1 Population sampling9.4 Statistics from grouped data

Investigate the effect of individual data values, including outliers, on the mean and median (ACMSP207)• using displays of data to explore and investigate effects

9.2 Using sample measures of centre and spread9.3 Frequency tables and graphs9.4 Statistics from grouped data

Investigate techniques for collecting data, including census, sampling and observation (ACMSP284)• identifying situations where data can be collected by census

and those where a sample is appropriate

9.1 Population sampling9.4 Statistics from grouped data

Explore the variation of means and proportions of random samples drawn from the same population (ACMSP293)• using sample properties to predict characteristics

of the population

9.2 Using sample measures of centre and spread9.4 Statistics from grouped data


Pearson Mathematics 8 Teacher Companion - Contents

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