E s s E n t i a l MatheMatics 8 - Cambridge University...

E s s E n t i a lMatheMaticsfor thE australian CurriCulum

YEar8DaviD GrEEnwooD franCa frank

JEnnY GooDman BrYn humBErstonE

Justin roBinson JEnnifEr vauGhan

Cambridge University Press978-0-521-17864-8 - Essential Mathematics: For the Australian Curriculum: Year 8David Greenwood, Franca Frank, Jenny Goodman, Bryn Humberstone, Justin Robinson and Jennifer VaughanFrontmatterMore information

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iii

Table of Contents

About the authors viiiIntroduction and how to use this book ix

Integers 2

Pre-test 4 1.1 Whole number addition and subtraction 5 1.2 Whole number multiplication and division 9 1.3 Number properties 13 1.4 Divisibility and prime factorisation 18 1.5 Negative numbers 23 1.6 Addition and subtraction of negative integers 27 1.7 Multiplication and division of integers 31 1.8 Order of operations and substitution 35 Investigation 39

Puzzles and challenges 41Review: Chapter summary 42 Multiple-choice questions 43 Short-answer questions 43 Extended-response questions 45

Lines, shapes and solids 46

Pre-test 48 2.1 Angles at a point 49 2.2 Parallel lines 55 2.3 Triangles 62 2.4 Unique triangles 69 2.5 Quadrilaterals 74 2.6 Polygons 80 2.7 Solids and Euler’s rule 85 Investigation 91

Puzzles and challenges 93Review: Chapter summary 94 Multiple-choice questions 95

1 Number and Algebra

Number and place value

Strand and content description

2 Measurement and Geometry

Geometric reasoning






iv

Short-answer questions 96 Extended-response questions 98

Fractions, decimals and percentages 100

Pre-test 102 3.1 Equivalent fractions 103 3.2 Operations with fractions 109 3.3 Operations with negative fractions 118 3.4 Understanding decimals 124 3.5 Operations with decimals 130 3.6 Terminating, recurring and rounding decimals 137 3.7 Percentages 143 3.8 Percentages: expressing and fi nding 150 3.9 Decreasing and increasing by a percentage 156 3.10 Calculating percentage change 161 3.11 Percentages and the unitary method 166

Investigation 170Puzzles and challenges 172Review: Chapter summary 173 Multiple-choice questions 175 Short-answer questions 175 Extended-response questions 177

Measurement and introduction to Pythagoras’ theorem 178

Pre-test 180 4.1 Length and perimeter 181 4.2 Circumference of a circle 187 4.3 Area 192 4.4 Area of special quadrilaterals 199 4.5 Area of a circle 204 4.6 Sectors and composite shapes 209 4.7 Surface area of a prism 215 4.8 Volume and capacity 220 4.9 Volume of prisms and cylinders 225 4.10 Time 230 4.11 Introduction to Pythagoras’ theorem 238 4.12 Using Pythagoras’ theorem 243 4.13 Finding the length of a shorter side 248 Investigation 253

Puzzles and challenges 255Review: Chapter summary 256

4Measurement and Geometry

Using units of measurement

Number and Algebra

Real numbers


Real numbers

Money and fi nancial mathematics






v

Multiple-choice questions 257 Short-answer questions 258 Extended-response questions 260

Algebra 262

Pre-test 264 5.1 The language of algebra 265 5.2 Substitution and equivalence 270 5.3 Adding and subtracting terms 274 5.4 Multiplying and dividing terms 279 5.5 Adding and subtracting algebraic fractions 283 5.6 Multiplying and dividing algebraic fractions 288 5.7 Expanding brackets 293 5.8 Factorising expressions 298 5.9 Applying algebra 302 5.10 Index laws: multiplying and dividing powers 306 5.11 Index laws: raising powers 311 Investigation 315


Semester Review 1 322

Ratios and rates 332

Pre-test 334 6.1 Introducing ratios 335 6.2 Simplifying ratios 340 6.3 Dividing a quantity in a given ratio 345 6.4 Scale drawings 351 6.5 Introducing rates 358 6.6 Ratios and rates and the unitary method 363 6.7 Solving rate problems 368 6.8 Speed 373 Investigation 379



Real numbers

Algebra 2625 Number and Algebra

Number and place value

Patterns and algebra






vi

Equations and inequalities 386

Pre-test 388 7.1 Equations review 389 7.2 Equivalent equations 394 7.3 Equations with fractions 399 7.4 Variables on both sides of equations 404 7.5 Equations with brackets 409 7.6 Formulas and relationships 413 7.7 Applications 417 7.8 Inequalities 422 7.9 Solving inequalities 426 Investigation 431


Probability and statistics 440

Pre-test 442 8.1 Interpreting graphs and tables 443 8.2 Frequency tables and tallies 450 8.3 Histograms 456 8.4 Measures of centre 464 8.5 Measures of spread 470 8.6 Surveying and sampling 475 8.7 Probability 480 8.8 Multiple events 485 8.9 Tree diagrams 490 8.10 Venn diagrams and two-way tables 495 8.11 Experimental probability 502 Investigation 508


Straight line graphs 518

Pre-test 520 9.1 The number plane 521 9.2 Rules, tables and graphs 525

8Chance

Data representation and interpretation

Statistics and Probability


Linear and non-linear relationships


Linear and non-linear relationships






vii

9.3 Finding the rule using tables 529 9.4 Gradient 535 9.5 Gradient-intercept form 542 9.6 The x-intercept 550 9.7 Applying linear graphs 555 9.8 Non-linear graphs 561 Investigation 567


Transformation and congruence 576

Pre-test 578 10.1 Refl ection 579 10.2 Translation 585 10.3 Rotation 589 10.4 Congruent fi gures 595 10.5 Congruent triangles 600 10.6 Similar fi gures 606 10.7 Similar triangles 612 10.8 Congruence and quadrilaterals 619 Investigation 624


Semester Review 2 632

Answers 639

10 Measurement and Geometry

Geometric reasoning






viii

About the Authors

David Greenwood is head of mathematics at trinity Grammar school in melbourne and

has 17 years experience teaching maths from Years 7 to 12. he has run numerous

workshops in australia and overseas in the use of technology for the teaching of

maths and has a keen interest in the development of the australian Curriculum with

particular interest in the sequencing of content and the profi ciency strands.

Bryn Humberstone graduated from university of melbourne with an honours degree

in Pure mathematics, and is currently teaching both junior and senior mathematics

in victoria. Bryn is particularly passionate about writing engaging mathematical

investigations and effective assessment tasks for students with a variety of

backgrounds and ability levels.

Justin Robinson is head of Positive Education and mathematics teacher at Geelong

Grammar school. Prior to this, he spent 20 years teaching mathematics and was a

key writer of in-house maths material. he has a keen interest in engaging all

students through a wide variety of effective teaching methods and materials.

Jenny Goodman has worked in comprehensive state and selective high schools in

nsw and has a keen interest in teaching students of differing abilities, from the

gifted and talented students to those with learning diffi culties.

Jennifer Vaughan has taught secondary mathematics for 30 years in nsw, wa, QlD and

new Zealand, and has tutored and lectured mathematics at Queensland university of

technology. Jennifer currently teaches at ormiston College, Queensland.

Franca Frank has been teaching for over 30 years. she has taught all levels of mathematics including

Extension 2 in nsw, and has been involved in writing programs for new courses.






ix

Development of this series started in 2008 with an analysis of state curricula, in parallel with the work of

aCara. the manuscripts were drafted to this analysis, aCara documents and feedback provided by

the states. revisions were made to each curriculum release including version 1.0 published in December

2010. future versions will be covered in (1) printable supplements published on the companion website,

(2) online electronic versions revised immediately and (3) regular reprints of the books.

as the australian Curriculum specifi es minimum content to be covered, the series includes some topics

which are necessary pre-requisites for specifi ed content and which have always been covered by individual

state curricula. the series also includes logical extensions in a range of topics. Both these categories are

indicated in the teaching program, which also maps the content to version 1.0. to help schools manage

the transition, curriculum grids are also available mapping the content to existing state syllabuses and

curricula.

through these approaches the series is founded on a reliable teaching structure and sequence

which can confi dently be used to manage differentiated learning in the classroom with the minimum of

preparation. for more information see the companion website.

Guide to this book

Features:

australian curriculum: strands and

content descriptions for chapter (also

available in a grid)

What you will learn: an

overview of chapter contents

chapter introduction: use

to set a context for students

3

Chapter

What you will learn

Public key encryption

Whole number addition and subtractionWhole number multiplication and divisionNumber propertiesPrime factorisation and divisibilityNegative numbersAddition and subtraction of negative integersMultiplication and division of integersOrder of operations and substitution

1.11.21.31.41.51.61.71.8

Today, most of the world’s electronic commercial transactions are encrypted so that important information does not get into the wrong hands. Public key encryption uses the RSA algorithm named after its inventors Rivest, Shamir and Ademan, who invented the mathematical procedure in 1977. The algorithm creates public and private number keys that are used to encrypt and decrypt information. These keys are generated using products of prime numbers. Because prime factors of large numbers are very difficult to find (even for today’s powerful computers) it is virtually impossible to decrypt the code without a private key. The algorithm uses prime numbers, division and remainders, equations and the 2300-year-old Euclidean division algorithm to complete the task. If it wasn’t for Euclid (about 300 bce) and the prime numbers, today’s electronic transactions would not be secure.

Australian curriculumN u m b e r a N d a l g e b r a

N u m b e r a n d p l a c e v a l u e

Carry out the four operations with integers, using efficient mental and written strategies and appropriate digital technologies

IntegersIntegers111111Chapter1Chapter11111111Chapter1Chapter

•

CUAU110-01.indd 2-3 12/22/10 4:00:15 AM

Pre-test: establishes prior knowledge

(also available as a printable worksheet)

Chapter 1 Whole numbers4 p

For each of the following, match the word with the symbol.

a add A −b subtract B ÷c multiply C +d divide D ×

Write each of the following as numbers.

a fi fty-seven b one hundred and sixteen

c two thousand and forty-four d eleven thousand and two

Answer which number is:

a 2 more than 11 b 5 less than 42

c 1 less than 100 d 3 more than 7997

e double 13 f half of 56f

Complete these patterns, showing the next seven numbers.

a 7, 14, 21, 28, 35, __, __, __, __, __, __, __.

b 9, 18, 27, 36, 45, __, __, __, __, __, __, __.

c 11, 22, 33, 44, 55, __, __, __, __, __, __, __.

1

2

3

4

Introduction and how to use this book






x

Guide to this book (continued)topic introduction: use to relate the

topic to mathematics in the wider world

hOtmaths icons: links to interactive

online content via the topic number,

1.3 in this case (see next page for more)

Let’s start: an activity (which can

often be done in groups) to start the lesson

Key ideas: summarises the knowledge and

skills for the lesson (digital version also

available for use with iwB)

examples: solutions with explanations and

descriptive titles to aid searches (digital

versions also available for use with iwB)

exercise questions categorised by

the four profi ciency strands

and enrichment

Questions are linked to examples

investigations: inquiry-based

activities Puzzles and challenges

chapter summary: mind map of

key concepts & interconnections 2 semester reviews per book

Chapter reviews with multiple-choice, short-answer and extended-response questions

Chapter 1 Whole numbers14

tion and subtractionThe process of fi nding the total value of two or more numbers is called

addition. The words ‘plus’, ‘add’ and ‘sum’ are also used to describe

addition.

The process for fi nding the difference between two numbers is

called subtraction. The words ‘minus’, ‘subtract’ and ‘take away’ are

also used to describe subtraction.

Let’s start: Your mental strategyMany problems that involve addition and subtraction can be solved

mentally without the use of a calculator or complicated written

working.

Consider 98 + 22 − 31 + 29

How would you work this out? What are the different ways it could be

done mentally? Explain your method.

Addit1.3

What’s the difference in our heights?

■ The symbol + is used to show addition or fi nd a sum.

e.g. 4 + 3 = 7

a■ + b = b + a e.g. 4 + 3 = 3 + 4

– This is the commutative law for addition, meaning that

the order does not matter.

The symbol■ − is used to show subtraction or fi nd a difference.

e.g. 7 − 2 = 5

a■ − b ≠ b − a e.g. 4 − 3 ≠ 3 − 4

Mental addition and subtraction can be done using different strategies.■

Partitioning– (Grouping digits in the same position)

171 + 23 = 100 + (70 + 20) + (1 + 3)

= 194

Compensating– (Making a 10, 100 etc. and then adjusting or compensating by adding or

subtracting)

46 + 9 = 46 + 10 − 1

= 55

Doubling or halving– (Making a double or half and then adjusting with addition or

subtraction)

75 + 78 = 75 + 75 + 3 124 − 61 = 124 − 62 + 1

= 150 + 3 = 62 + 1

= 153 = 63

765

+3

4 83

76

−2

5 84

Key

idea

s

CUAU093-c01.indd 14 9/12/10 9:38:56 AM


tion and subtractionThe process of fi nding the total value of two or more numbers is called

addition. The words ‘plus’, ‘add’ and ‘sum’ are also used to describe

addition.

The process for fi nding the difference between two numbers is

called subtraction. The words ‘minus’, ‘subtract’ and ‘take away’ are

also used to describe subtraction.

Let’s start: Your mental strategyMany problems that involve addition and subtraction can be solved

mentally without the use of a calculator or complicated written

working.

Consider 98 + 22 − 31 + 29

How would you work this out? What are the different ways it could be

done mentally? Explain your method.

Addit1.3

What’s the difference in our heights?

■ The symbol + is used to show addition or fi nd a sum.

e.g. 4 + 3 = 7

a■ + b = b + a e.g. 4 + 3 = 3 + 4

– This is the commutative law for addition, meaning that

the order does not matter.

The symbol■ − is used to show subtraction or fi nd a difference.

e.g. 7 − 2 = 5

a■ − b ≠ b − a e.g. 4 − 3 ≠ 3 − 4

Mental addition and subtraction can be done using different strategies.■

Partitioning– (Grouping digits in the same position)

171 + 23 = 100 + (70 + 20) + (1 + 3)

= 194

Compensating– (Making a 10, 100 etc. and then adjusting or compensating by adding or

subtracting)

46 + 9 = 46 + 10 − 1

= 55

Doubling or halving– (Making a double or half and then adjusting with addition or

subtraction)

75 + 78 = 75 + 75 + 3 124 − 61 = 124 − 62 + 1

= 150 + 3 = 62 + 1

= 153 = 63

765

+3

4 83

76

−2

5 84

Key

idea

s

CUAU093-c01.indd 14 9/12/10 9:38:56 AM

Number and Algebra 15

Example 4 Mental addition and subtraction

Use the suggested strategy to mentally work out the answer.

a 132 + 156 (partitioning) b 25 + 19 (compensating)

c 56 – 18 (compensating) d 35 + 36 (doubling or halving)

SOLUTION EXPLANATION

a 132 + 156 = 288 100 30 2100 50 6

200 80 8

+ +30+ +50

+ +80

b 25 + 19 = 44 25 + 19 = 25 + 20 – 1

= 45 – 1

= 44

c 56 – 18 = 38 56 – 18 = 56 – 20 + 2

= 36 + 2

= 38

d 35 + 36 = 71 35 + 36 = 35 + 35 + 1

= 70 + 1

= 71

1 a List three words that mean addition.

b List three words that mean subtraction.

2 Write the number which is:

a 3 more than 7 b 58 more than 11

c 7 less than 19 d 137 less than 157

3 a Find the sum of these pairs of numbers.

i 2 and 6 ii 19 and 8 iii 62 and 70

b Find the difference between these pairs of numbers.


4 State whether each of these statements is true or false.

a 4 + 3 > 6 b 11 + 19 ≥ 30 c 13 − 9 < 8

d 26 − 15 ≤ 10 e 1 + 7 − 4 ≥ 4 f 50 − 21 + 6 < 35

5 Give the result for each of these problems.

a 7 plus 11 b 22 minus 3 c the sum of 11 and 21

d 128 add 12 e 36 take away 15

f the difference between 13 and 4

Exercise 1C

Unde

rsta

ndin

g

CUAU093-c01.indd 15 9/12/10 9:39:21 AM


Example 4 Mental addition and subtraction

Use the suggested strategy to mentally work out the answer.

a 132 + 156 (partitioning) b 25 + 19 (compensating)

c 56 – 18 (compensating) d 35 + 36 (doubling or halving)

SOLUTION EXPLANATION

a 132 + 156 = 288 100 30 2100 50 6

200 80 8

+ +30+ +50

+ +80

b 25 + 19 = 44 25 + 19 = 25 + 20 – 1

= 45 – 1

= 44

c 56 – 18 = 38 56 – 18 = 56 – 20 + 2

= 36 + 2

= 38

d 35 + 36 = 71 35 + 36 = 35 + 35 + 1

= 70 + 1

= 71

1 a List three words that mean addition.

b List three words that mean subtraction.

2 Write the number which is:

a 3 more than 7 b 58 more than 11

c 7 less than 19 d 137 less than 157

3 a Find the sum of these pairs of numbers.


b Find the difference between these pairs of numbers.


4 State whether each of these statements is true or false.

a 4 + 3 > 6 b 11 + 19 ≥ 30 c 13 − 9 < 8

d 26 − 15 ≤ 10 e 1 + 7 − 4 ≥ 4 f 50 − 21 + 6 < 35

5 Give the result for each of these problems.

a 7 plus 11 b 22 minus 3 c the sum of 11 and 21

d 128 add 12 e 36 take away 15

f the difference between 13 and 4

Exercise 1C

Unde

rsta

ndin

g

CUAU093-c01.indd 15 9/12/10 9:39:21 AM


6 Mentally fi nd the answers to these sums. Hint: Use the partitioning strategy.

a 23 + 41 b 71 + 26 c 138 + 441

d 246 + 502 e 937 + 11 f 1304 + 4293

g 140 273 + 238 410 h 390 447 + 201 132 i 100 001 + 101 010

7 Mentally fi nd the answers to these differences. Hint: Use the partitioning strategy.

a 29 − 18 b 57 − 21 c 249 − 137

d 1045 − 1041 e 4396 − 1285 f 10 101 − 100

8 Mentally fi nd the answers to these sums. Hint: Use the compensating strategy.

a 15 + 9 b 64 + 11 c 19 + 76

d 18 + 115 e 31 + 136 f 245 + 52

9 Mentally fi nd the answers to these differences. Hint: Use the compensating strategy.

a 35 − 11 b 45 − 19 c 156 − 48

d 244 − 22 e 376 − 59 f 5216 − 199

10 Mentally fi nd the answers to these sums and differences. Hint: Use the doubling or

halving strategy.

a 25 + 26 b 65 + 63 c 121 + 123

d 240 − 121 e 482 − 240 f 1006 − 504

11 Mentally fi nd the answers to these mixed problems.

a 11 + 18 − 17 b 37 − 19 + 9 c 101 − 15 + 21

d 136 + 12 − 15 e 28 − 10 − 9 + 5 f 39 + 71 − 10 − 10

g 1010 − 11 + 21 − 1 h 5 − 7 + 2 i 10 − 25 + 18

12 Gary worked 7 hours on Monday, 5 hours on Tuesday, 13 hours on Wednesday, 11 hours on

Thursday and 2 hours on Friday. What is the total number of hours that Gary worked during the

week?

13 In a batting innings, Phil hit 126

runs and Mario hit 19 runs. How

many more runs did Phil hit

compared to Mario?

Example 4a

Example 4b

Example 4c

Example 4d

Flue

ncy

Prob

lem

-sol

ving

CUAU093-c01.indd 16 9/12/10 9:39:31 AM


6 Mentally fi nd the answers to these sums. Hint: Use the partitioning strategy.

a 23 + 41 b 71 + 26 c 138 + 441

d 246 + 502 e 937 + 11 f 1304 + 4293

g 140 273 + 238 410 h 390 447 + 201 132 i 100 001 + 101 010

7 Mentally fi nd the answers to these differences. Hint: Use the partitioning strategy.

a 29 − 18 b 57 − 21 c 249 − 137

d 1045 − 1041 e 4396 − 1285 f 10 101 − 100

8 Mentally fi nd the answers to these sums. Hint: Use the compensating strategy.

a 15 + 9 b 64 + 11 c 19 + 76

d 18 + 115 e 31 + 136 f 245 + 52

9 Mentally fi nd the answers to these differences. Hint: Use the compensating strategy.

a 35 − 11 b 45 − 19 c 156 − 48

d 244 − 22 e 376 − 59 f 5216 − 199

10 Mentally fi nd the answers to these sums and differences. Hint: Use the doubling or

halving strategy.

a 25 + 26 b 65 + 63 c 121 + 123

d 240 − 121 e 482 − 240 f 1006 − 504

11 Mentally fi nd the answers to these mixed problems.

a 11 + 18 − 17 b 37 − 19 + 9 c 101 − 15 + 21

d 136 + 12 − 15 e 28 − 10 − 9 + 5 f 39 + 71 − 10 − 10

g 1010 − 11 + 21 − 1 h 5 − 7 + 2 i 10 − 25 + 18

12 Gary worked 7 hours on Monday, 5 hours on Tuesday, 13 hours on Wednesday, 11 hours on

Thursday and 2 hours on Friday. What is the total number of hours that Gary worked during the

week?

13 In a batting innings, Phil hit 126

runs and Mario hit 19 runs. How

many more runs did Phil hit

compared to Mario?

Example 4a

Example 4b

Example 4c

Example 4dFl

uenc

yPr

oble

m-s

olvi

ng

CUAU093-c01.indd 16 9/12/10 9:39:31 AM


19 Complete these number sentences if the letters a, b and c represent numbers.

a a + b = c so c − __ = a b a + c = b so b − a = __

20 This magic triangle uses the digits 1 to 6, and has each side adding to the same

total. This example shows a side total of 9.

a How many other different side totals are possible using the same digits?

b Explain your method.

Enrichment: Magic squares

21 A magic square has every row, column and main diagonal adding

to the same number, called the magic sum. For example, this magic

square has a magic sum of 15.

Find the magic sums for these squares, then fi ll in the

missing numbers.

a6666666666666666666666666

77777777777777777777777777777 5555555555555555555555555

22222222222222222222222222222222

b10111111100000000000001111101110000000000000

11111111111111111111111111111111111111 13111111111133333333333333111113333333

12111111111111122222222222222222222112222

c15111111111555555555555111111511155555555555 20222222222222200000000000022222222222222000000000000

14111411111111144444444444444411111144444444

191111119111111111911199999999999999999999999999

d1111111111111111 15111115515515555551111115515551555555 4444444444444444444444444

6666666666666666666666 9999999999999999999999

111111111111111111111111111111111111

1311113113113111131113113333333333333111333 2222222222222222222222222222 161161116611166666666116116111666666661116666

22 The sum of two numbers is 87 and their difference is 29. What are the two numbers?

6 5

2 4 3

1

4444444444444444444 9999999999999999999999 2222222222222222222222 15111111555555551111115555555

3333333333333333333 5555555555555555555 7777777777777777777 15115511111155555555111155555

8888888888888888888 1111111111111 6666666666666666666 15111155555111111555555551155

15111155155515551111551551555 15111111155515551555155515551555 15111111555555555111111555555555 15111111111111111555555555555555555 15111111555555511111155555555

Reas

onin

g

This magic square was known in ancient China as a ‘Lo Shu’ square and uses only the numbers 1 to 9. It is shown in the middle of this ancient design as symbols on a turtle shell, surrounded by the animals which represent the traditionalchinese names for the years.

CUAU093-c01.indd 18 9/12/10 9:39:56 AM


19 Complete these number sentences if the letters a, b and c represent numbers.

a a + b = c so c − __ = a b a + c = b so b − a = __

20 This magic triangle uses the digits 1 to 6, and has each side adding to the same

total. This example shows a side total of 9.

a How many other different side totals are possible using the same digits?

b Explain your method.

Enrichment: Magic squares

21 A magic square has every row, column and main diagonal adding

to the same number, called the magic sum. For example, this magic

square has a magic sum of 15.

Find the magic sums for these squares, then fi ll in the

missing numbers.

a6666666666666666666666666

77777777777777777777777777777 5555555555555555555555555

22222222222222222222222222222222

b10111111100000000000001111101110000000000000

11111111111111111111111111111111111111 13111111111133333333333333111113333333

12111111111111122222222222222222222112222

c15111111111555555555555111111511155555555555 20222222222222200000000000022222222222222000000000000

14111411111111144444444444444411111144444444

191111119111111111911199999999999999999999999999

d1111111111111111 15111115515515555551111115515551555555 4444444444444444444444444

6666666666666666666666 9999999999999999999999

111111111111111111111111111111111111

1311113113113111131113113333333333333111333 2222222222222222222222222222 161161116611166666666116116111666666661116666

22 The sum of two numbers is 87 and their difference is 29. What are the two numbers?

6 5

2 4 3

1

4444444444444444444 9999999999999999999999 2222222222222222222222 15111111555555551111115555555

3333333333333333333 5555555555555555555 7777777777777777777 15115511111155555555111155555

8888888888888888888 1111111111111 6666666666666666666 15111155555111111555555551155

15111155155515551111551551555 15111111155515551555155515551555 15111111555555555111111555555555 15111111111111111555555555555555555 15111111555555511111155555555

Reas

onin

g

This magic square was known in ancient China as a ‘Lo Shu’ square and uses only the numbers 1 to 9. It is shown in the middle of this ancient design as symbols on a turtle shell, surrounded by the animals which represent the traditionalchinese names for the years.

CUAU093-c01.indd 18 9/12/10 9:39:56 AM


The abacuse abacus is a counting device that has been used for thousands

years. They were used extensively by merchants, traders, tax

llectors and clerks before modern-day numerals systems were

veloped. Counting boards called Abax date back to 500 BCE.

ese were wood or stone tablets with grooves, which would hold

ans or pebbles.

The modern abacus is said to have originated in China in about

e 13th century and includes beads on wires held in a wooden

me.

A modern abacus with thirteen wires

There are 5 beads on one side of a modern abacus

worth 1 each and 2 beads on the opposite side worth

5 each.

Each wire represents a different unit; e.g. ones,

tens, hundreds etc.

Beads are counted only when they are pushed

toward the centre.

Here is a diagram showing the number 5716.

ones

tens

hund

reds

thou

sand

s

ThThe

of y

col

dev

The

bea

the

fram

erman woodcut from 1508 showing an abacus in use by gentleman on right, A GGele a mathematician (at left) writes algorithms.whiil

CUAU093-c01.indd 44 9/12/10 9:48:19 AM


11 The extra dollar. The cost of dinner for two people is $45 and they both give the waiter $25 each.

Of the extra $5 the waiter is allowed to keep $3 as a tip and returns $1 to each person.

So the two people paid $24 each, making a total of $48, and the waiter has $3. The total is therefore

$48 + $3 = $51. Where did the extra $1 come from?

2 The sum along each line is 15. Can you place each of the digits 1, 2, 3, 4, 5,

6, 7, 8 and 9 to make this true?

3 The sum along each side of this triangle is 17. Can you place each

of the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 to make this true?

4 Make the total of 100 out of all the numbers 2, 3, 4, 7 and 11, using each number only once. You

can use any of the operations (+, –, ×, ÷), as well as brackets.

5 Sudoku is a popular logic number puzzle made up of a 9 by 9 square, where each column and row

can use the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 only once. Also, each digit is to be used only once in

each 3 by 3 square. Solve these puzzles.

7 6 9 3

4 1 8 7

8 2 9 1

3 1

2 8 5 3

5 6 9 2

3 9 5

6 8 4

5 9 7

4 2 8 7

2 8 7 9 1

6 3 5

3 7 2 8

6 5 4 7 8 2

2 6

7 5

8 3 9 2 7

CUAU093-c01.indd 46 9/12/10 9:48:53 AM


Order of OperationsBrackets first, then ×and ÷, then + and –

from left to right.2 + 3 × 4 ÷ (9 ÷ 3)

= 2 + 12 ÷ 3= 2 + 4= 6

Place value

Addition andSubtraction

AncientNumber Systems

Estimation

Roman

371+ 843_____1214

29× 13____

87290____377

937– 643_____

294

68

3 205

with 1remainder

Mental strategies172 + 216 = 300 + 80 + 8

= 38898 – 19 = 98 – 20 + 1

= 79

Mental strategies7 × 31 = 7 × 30 + 7 × 1 = 217

5 × 14 = 10 × 7 = 70 64 ÷ 8 = 32 ÷ 4 = 16 ÷ 2 = 8156 ÷ 4 = 160 ÷ 4 – 4 ÷ 4

The place value of 3 in1327 is 300.

2 × 100 + 7 × 10 + 3 × 1is the expanded form of 273.

Algorithms

Algorithms

Multiplicationand DivisionWhole numbers

Egyptian

Babylonian

is 71is 23

11

2

22

8

LXXVI is 76 XCIV is 94

is 143

is 21

156 ÷ 4 = 160 ÷ 4 4 ÷ 4= 40 – 1= 39

955 to the nearest 10 is 960 Leading digit approximation955 to the nearest 10 is 96606950 to the nearest 100 is 10000000

LLLeLL ading g g g g digit apprppp oximation3939393939 × 322222666 66 ≈ 40 × 3300 = 12222 000 0000000 Multiplying by 10, 100, ……

38 × ×× ××× 1010101000 0 = === 333833 003838388838838 ×× ××× 707070700 = 38388388888 × × ×× 777 77 × 100 = = 2 6 606060606000

CUAU093-c01.indd 47 9/12/10 9:49:17 AM


Multiple-choice questions 1 The correct Roman numerals for the number 24 is:

A XXIII B XXIV C XXXLIV

D IVXX E IXXV

2 3 × 1000 + 9 × 10 + 2 × 1 is the expanded form of:

A 3920 B 392 C 3092

D 3902 E 329

3 Which of the following is not true?

A 2 + 3 = 3 + 2 B 2 × 3 = 3 × 2 C (2 × 3) × 4 = 2 × (3 × 4)

D 5 ÷ 2 ≠ 2 ÷ 5 E 7 − 2 = 2 − 7

4 The sum of 198 and 103 is:

A 301 B 304 C 299

D 199 E 95

5 The difference between 378 and 81 is:

A 459 B 297 C 303

D 317 E 299

6 The product of 7 and 21 is:

A 147 B 141 C 21

D 140 E 207

7 The missing digit in this division is:

A 6 B 1 C 9

D 8 E 7

8 The remainder when 317 is divided by 9 is:

A 7 B 5 C 2

D 1 E 0

9 458 rounded to the nearest 100 is:

A 400 B 500 C 460

D 450 E 1000

10 The answer to [2 + 3 × (7 – 4)] ÷ 11 is:

A 1 B 5 C 11

D 121 E 0

) 27) 61 1 8

1 52

CUAU093-c01.indd 48 9/12/10 9:50:55 AM

Semester review 1

350

Sem

este

r rev

iew

1 Whole numbersMultiple-choice questions

1 Using numerals, thirty-fi ve thousand, two hundred and six is:

A 350 260 B 35 260 C 35 000 206 D 3526 E 35 206

2 The place value of 8 in 2 581 093 is:

A 8 thousand B 80 thousand C 8 hundred D 8 tens E 8 ones

3 The remainder when 23 650 is divided by 4 is:

A 0 B 4 C 1 D 2 E 3

4 18 − 3 × 4 + 5 simplifi es to:

A 65 B 135 C 11 D 1 E 20

5 800 ÷ 5 × 4 is the same as:

A 160 × 4 B 800 ÷ 20 C 800 ÷ 4 × 5 D 40 E 4 × 5 ÷ 800

Short-answer questions

1 Write the number seventy-four in:

a Babylonian numerals

b Roman numerals

c Egyptian numerals

2 Write the numeral for:

a 6 × 10 000 + 7 × 1000 + 8 × 100 + 4 × 10 + 9 × 1

b 7 × 100 000 + 8 × 100 + 5 × 10

3 Calculate:

a 96 481 + 2760 + 82 b 10 963 − 4096 c 147 × 3

d 980 × 200 e 4932 ÷ 3 f 9177f ÷ 12

4 State whether each of the following is true or false.

a 18 < 20 − 2 × 3 b 9 × 6 > 45 c 23 = 40 ÷ 2 + 3

5 How much more than 17 × 18 is 18 × 19?

6 Calculate:

a 7 × 6 − 4 × 3 b 8 × 8 − 16 ÷ 2 c 12 × (6 − 2)

d 16 × [14 − (6 − 2)] e 24 ÷ 6 × 4 f 56f − (7 − 5) × 7

7 State whether each of the following is true or false.

a 4 × 25 × 0 = 1000 b 0 ÷ 10 = 0 c 8 ÷ 0 = 0

d 8 × 7 = 7 × 8 e 20 ÷ 4 = 20 ÷ 2 ÷ 2 f 8 f + 5 + 4 = 8 + 9

8 Insert brackets to make 18 × 7 + 3 = 18 × 7 + 18 × 3 true.

9 How many times can 15 be subtracted from 135 before an answer of zero occurs?

CUAU093-SR-1.indd 350 9/12/10 12:07:22 PM

textbooks also include:■ Complete answers■ Glossary■ Using technology

activities






xi

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