Connections Maths 8

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_in

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id

e

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connections

8 m a t h s

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C O N T E N T S

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5 Properties of geometrical

figures 165

Angles and lines 167

Speed skills 5.1: Geometry revision 168

Parallel lines 172

Tests for parallel lines 172

Triangles 177

Triangle notation 177

Types of triangles 177 The angle sum of a triangle 179 The exterior angle of a triangle 179

Constructions 186

Bisecting an angle 186 Bisecting an interval 186 Constructing parallel lines 187 Constructing a perpendicular 187 Constructing a triangle given two angles

and one side 188

Using technology: Bisecting an angle using Wingeom 188

Symmetry 192

Quadrilaterals 196

The angle sum of a quadrilateral 197 Special quadrilaterals 198 Problem solving 5 207 Literacy skills 5 209 Chapter review 5 210 6 Percentages 213 Percentages 215 Percentage conversions 215

Speed skills 6.1: Percentage conversions 216

Percentage problems 217 Expressing quantities as percentages 219 Finding percentages of quantities 222

Speed skills 6.2: Finding percentages of quantities 223

Percentage composition 227 Increasing and decreasing by a given

percentage 229

Discount 233

Profit and loss 237

Using technology: Profit and loss percentages 241

Finding the whole amount using the

unitary method 241

Problem solving 6 243

Literacy skills 6 244

Chapter review 6 245

7 Substitution and linear

relationships 247

Substitution 249

Speed skills 7.1: Substitution 249

Indices and substitution 252

Using technology: Spreadsheets and substitutions 254

Formulas 254

Using technology: Formulas in spreadsheets 260

Completing a table of values 260

The number plane 262

Quadrants in the number plane 264

Speed skills 7.2: Quadrants in the number plane 264

Coordinate geometry 265

Graphing lines 269

Linear equations 270

The gradient of a line 270

Using technology: Graphing lines using a spreadsheet 275

The intersection of two straight lines 277

Deriving a rule 279

8 Perimeter and area 287

Metric conversions 289

Length 289

Mass 289

Capacity 289

Limits of accuracy 290

Speed skills 8.1: Metric measurements 291

Perimeter 293

Perimeter by measurement 293 Perimeter by calculation 293 Perimeter formulas 294

Area 299

Units of measurement for area 299

Area conversions 300

Area formulas 304

Areas of special quadrilaterals 309 The area of a parallelogram 309

Using technology: Maximum area 313

The parts of a circle 314

Speed skills 8.2: The parts of a circle 315

The circumference of a circle 316 The number (pi) 317 Formulas for the circumference of a circle 317 Giving an exact answer in terms of 317

Using technology: Finding the value for 322

The area of a circle 323

9 Data analysis and probability 333

Data collection 336 Census 336 Sample 336 Types of data 337 Quantitative data 337 Categorical data 337 c c c c

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Using technology: Using a graphics calculator to

draw a histogram 344

More methods of displaying data 346

Dot plots 346

Stem-and-leaf-plots 347

Analysing data 351

Measures of spread 351 Measures of location 351

Using technology: Finding measures of location 357

Statistical notation 358

Using technology: Calculating the mean 363

Using a calculator to find the mean 364

Probability 368

The range of probability 369 Complementary events 370

10 Equations and inequalities 381

Equations 383

Speed skills 10.1: Checking solutions of equations 384

Solving equations by inspection 385 Solving equations by backtracking 386 The formal solution of equations 390 Collecting like terms in equations 393 Backtracking with two steps 394 Solving equations with variables

on both sides 396

Removing grouping symbols 397

Speed skills 10.2: Removing grouping symbols 397

Solving equations with grouping symbols 398 Solving equations with fractions 399 Solving equations by substitution 401 Translating worded problems into symbols 404

Speed skills 10.3: Translating worded problems

into symbols 404

Using equations to solve problems 405 Solving equations in geometry 409

Inequalities 414

Solving inequalities 414

11 Surface area and volume 421

Solids 422

Prisms 423

Pyramids 423

Other solids 424

Intersecting, parallel and skew edges 424 Lines in the environment 425

Nets of solids 425

Surface area 429

Surface areas of triangular prisms 434

Cross-sections 437

Speed skills 11.1: Solids and cross-sections 438

Building and drawing solids 441

Volume 443

Volume conversions 444 Volumes of right prisms 447 Volumes of rectangular prisms, triangular

prisms and composite solids 450

Using technology: Investigating volumes 453

Volumes of right cylinders 453

Using technology: Maximum volumes 456

Volume and capacity conversions 457

12 Ratios and rates 465

Ratios 468

Equivalent ratios 472

Ratios involving fractions, decimals,

algebraic expressions and different units 475

Speed skills 12.2: Simplifying ratios 477

The unitary method 478

Ratio problems 479

Dividing a quantity in a given ratio 483

Scale drawing 485

Rates 489

Speed 491

Using technology: Calculating average speed 493

Solving rate problems 493

13 Congruent and similar figures 501

Transformations and congruent figures 502

Transformations 502

Congruent figures 503

Problems involving congruent figures 508 Congruent triangle constructions 510

Similar figures 512

Enlarging and reducing figures 517 Identifying similar figures 523 Constructing similar figures 525 Finding unknown angles and sides of

similar figures 527

Problems involving similar figures 531

Problem solving 13 536 Literacy skills 13 537 Chapter review 13 538 Diagnostic test 540 Answers 562 Index 609 c c c c c

vii

C O N T E N T S

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Introduction

Preface to the teacher

This book is the second of two books written for the Mathematics Stage 4 syllabus in New South Wales. It is written for Year 8 students who have achieved most of the Stage 3 learning outcomes and are achieving Stage 4 learning outcomes. This book, together with Connections Maths 7, covers the entire new Stage 4 course.

This book is a fresh, interesting presentation of this important first stage of high school mathematics. Each topic has numerous worked examples and each exercise has graded questions.

The text clearly explains the mathematical concepts used in language that is appropriate for students beginning high school.

The sequence of the work has been planned to maximise student interest and enjoyment while assisting teachers to identify the learning outcomes achieved. Students are encouraged to develop their mathematical skills to achieve the learning outcomes of the course.

Most chapters have these features:

p Working mathematically Students are presented with interesting information and are asked to apply their mathematical knowledge to explore possible outcomes and extensions. This may be done individually or in small groups. Students have the opportunity to develop new skills and concepts and to focus on different learning processes.

p Speed skills Students use their mental arithmetic skills to complete short exercises in a set time.

p Using technology Students are encouraged to use technology to explore and extend mathematical concepts. Calculators, spreadsheets and geometry software are used to enhance students’ learning.

p Problem solving Students can explore different ways of finding solutions to original problems.

p Literacy skills Students can review the language used in the chapter and test their understanding of the terms they have learnt.

p Chapter review Students can complete a comprehensive review of the work treated in the chapter using this sequence of questions, which help to determine student learning outcomes.

The teaching and assessment book for this text gives additional teaching information and a sample teaching program that is linked to student learning outcomes.

This book is intended to interest and motivate students and provide them with a firm basis for future mathematical studies. We wish them well in their study of mathematics.

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What you will find in the student text and CD-ROM

This textbook and CD-ROM package have been produced to suit the new Mathematics Stage 4 syllabus for New South Wales. Features of this book include:

p outcomes at the start of every chapter

p a dynamic full-colour design that clearly distinguishes theory, examples, exercises and special features

p carefully graded exercises with worked examples and solutions linked to each

p cartoons offering helpful hints

p Working mathematically strands that are fully integrated, many of which feature challenging sections designed as extension material and also contain interesting historical and present-day explorations of mathematics

p a Chapter review to revise and consolidate learning in each chapter

p Speed skills sections to revise and develop mental arithmetic skills

p Problem solving sections requiring application of strategies, communication and reasoning through an inquiry approach

p a comprehensive Diagnostic test providing a cumulative review of learning in all chapters, cross-referenced to relevant exercises

p integrated technology activities

p Literacy skills activities to develop language skills relevant to each chapter

p fully linked icons to the accompanying CD-ROM

p a comprehensive index

The student CD-ROM accompanying this textbook can be used at school or at home for further explanation and learning. Each CD-ROM contains:

p animated worked examples

p movies related to selected topics, offering explanations for visual learners. These feature bright, energetic young presenters in appealing locations.

p technology files featuring geometry demonstrations and formatted spreadsheets

p the entire textbook, with hyperlinks to the above features

How to use the CD-ROM

Insert the CD-ROM into your CD-ROM drive. The main menu contains four folders:

p Worked examples Click on this to access a selection of animated worked examples from

every chapter in the book.

p Using technology Click on this to access geometry demonstrations and formatted

spreadsheets.

p View movies Click on this to access all the movies.

p Textbook Click on this to access the entire textbook in PDF format, with hotlinks to all the

above features.

For further details and troubleshooting, read the Read-Me file contained in the CD-ROM.

ix

I N T R O D U C T I O N CD-ROM CD-ROM

c

R e com mended tim e

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Acknowledgements

The authors and publishers would like to thank the following for granting permission to reproduce copyright material:

Photographic material: Brand X Pictures, Comstock, Corel, Corbis, Creativ Collection, Eyewire, Goodshoot, John Foxx, Microsoft®Excel screenshots reproduced with permission from Microsoft Corporation, PhotoAlto, Photo Essentials, Photodisc, Royal Australian Mint, Stockbyte.

Every effort has been made trace the ownership of copyright material. However, should any infringement have occurred, the publishers would be pleased to be contacted by the copyright owners.

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Working

mathematically,

problem solving

and number

S y l l a b u s o u t c o m e s

NS4.1 Recognises the properties of special groups of whole numbers and applies a range of strategies to aid computation

NS4.2 Compares, orders and calculates with integers

NS4.3 Operates with fractions and decimals

MS4.3 Performs calculations of time that involve mixed units

WMS4.1 Asks questions that could be explored using mathematics in relation to Stage 4 content

WMS4.2 Analyses a mathematical or real-life situation, solving problems using technology where appropriate

WMS4.3 Uses mathematical terminology and notation, algebraic symbols, diagrams, text and tables to communicate mathematical ideas

WMS4.4 Identifies relationships and the strengths and weaknesses of different strategies and solutions, giving reasons

WMS4.5Links mathematical ideas and makes connections with, and generalisations about, existing knowledge and understanding in relation to Stage 4 content

I n t h i s c h a p t e r y o u w i l l l e a r n t o :

n

use problem-solving steps and strategies to solve unfamiliar problems

n

round off whole numbers and decimals

n

use mental strategies to aid computation

n

use the correct order of operations (with or without a calculator)

n

use spreadsheets to solve problems

n

perform operations with whole numbers, directed numbers, fractions

and decimals

n

perform operations involving time units

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W O R K I N G M A T H E M A T I C A L L Y

Understanding ‘Working mathematically’

The ‘Working mathematically’ sections in this book provide opportunities for you to apply your knowledge and to develop new skills that will help you solve familiar and unfamiliar problems within and beyond mathematics. They allow you to explore different concepts and problem-solving techniques.

‘Working mathematically’ involves 5 interrelated processes: 1 Questioning

p Ask questions about mathematical situations and problems—to help understand the problem and to help choose the best strategy to use to solve the problem.

p Ask ‘what if’ questions—to extend the problem, to raise different possibilities and to make predictions.

2 Applying strategies

p Develop and use a range of problem-solving strategies (see below).

p Use appropriate technology to investigate and solve problems.

3 Communicating

p Develop and use appropriate mathematical language and symbols.

p Explain mathematical ideas in writing, orally and by using diagrams.

4 Reasoning

p Develop and use strategies to investigate problems, check solutions and support conclusions.

p Develop and use logical reasoning to prove and explain the solution to a problem. 5 Reflecting

p Reflect on (consider, think about) your experiences to see the connections/relationships between different parts of mathematics.

p Reflect on where and how mathematical ideas and concepts can be used to investigate and explain situations in the real world.

Problem-solving steps and strategies

Problem-solving skills are developed over time.

Problems can be solved in different ways, but the general steps are:

1 Understand the problem:

p Read the problem carefully.

p State the problem in your own words.

p Note important information.

p Decide what has to be found or done.

Problem-solving strategies:

p Look for patterns.

p Make a table or list.

p Draw a diagram.

p Solve a simpler similar problem.

p Work backwards.

p Act it out.

p Guess, check and improve.

p Use a combination of these.

These processes complement each

other. That is, they combine to make a complete method. Interrelated

means ‘connected’.

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2 Solve the problem:

p Determine a strategy that might help to solve the problem.

p Discuss the strategy with other students. Is there another way to solve the problem?

p Describe the process you use. If possible, check and improve your solution. 3 Record your solution:

p Record your solution to the problem and discuss it with others.

p Is your answer sensible? Have you used all the necessary information?

p Check that you have answered the problem.

p Are there alternative solutions?

p Can you make some general conclusion(s) about your solution? Are there exceptions to the rule?

p Can the problem be extended further?

Examples

●

1 Ten people meet at a conference. Everyone shakes hands once with each other person. How many handshakes are there altogether?

p Understand the problem:

As each person shakes the hand of every other person once only, it is important not to count more than 1 handshake for each

pair of people.

p Solve the problem:

A diagram may help, but 10 people is a lot, so try some smaller numbers.

Note the pattern of triangular numbers. Put these in a table:

Following the pattern shows that for 10 people the number of handshakes is:

9 8 7 6 5 4 3 2 1 45 handshakes 1 2 people 1 handshake 3 people 2 1 3 handshakes 4 people 3 2 1 6 handshakes 5 people 4 3 2 1 10 handshakes 2 1 2 3 3 4 3 4 5 1 2 1 2

3

C H A P T E R 1 W O R K I N G M A T H E M A T I C A L LY, P R O B L E M S O LV I N G A N D N U M B E R Number of people 2 3 4 5 6 7 8 9 10 Handshakes 1 3 6 10 ? ? ? ? ?

~|

2

|_~|

3

|_~|

4

|_~|

5

|_

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p Record your solution:

The first person shakes hands with 9 people, the second person shakes hands with 8 people and so on. So there are 45 handshakes altogether. Could you illustrate this problem by representing the 10 people as the vertices of a decagon or as points around a circle? Show how.

●

2 Lollies come in large and small packets. When an order was delivered to a shop, 19 large and 3 small packets had split open, spilling all their contents of 224 lollies. How many lollies were in each large and each small packet?

p Understand the problem:

The contents of the 19 large and 3 small packets was a total of 224 lollies. So we require:

(19 …) (3 …) 224

Remember that there must be more lollies in each large packet than in each small packet.

p Solve the problem:

Method 1: Try using ‘guess, check and improve’.

Guess 1 Large packet contains 8 lollies; small packet contains 4 lollies.

Check Since there are 19 large packets and 3 small packets:

Number of lollies 19 8 3 4

152 12 164 lollies

Improve We require 224 lollies, so there must be more lollies in the packets.

Guess 2 Large packet contains 10 lollies; small packet contains 4 lollies. Check Number of lollies 19 10 3 4

190 12 202

Improve We require 224 lollies, so there must be more in the packets.

Note that 19 12 228, which would be impossible, since there is a

total of 224 lollies. So each large packet can have at most 11 lollies.

This method is sometimes called trial and error.

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Guess 3 Large packet contains 11 lollies; small packet contains 5 lollies.

Check Number of lollies 19 11 3 5 209 15

224

Method 2: Now try solving the problem using a table (or a spreadsheet).

First, note that 19 12 228, which would be impossible. Hence each

large packet can have at most 11 lollies.

Step 1 Construct a table (or set up a spreadsheet) showing all the

possibilities with each large packet containing from 1 through to 11 lollies. You can use a calculator to complete the table.

Step 2 Analyse the table of values.

Remember that each large packet must contain more lollies than each small packet.

Using a spreadsheet:

5

C H A P T E R 1 W O R K I N G M A T H E M A T I C A L LY, P R O B L E M S O LV I N G A N D N U M B E R

Number of lollies Large packets Remainder in small packet Number of lollies

in packet Total (224 total) (remainder 3)

1 19 205 68.33333333 2 38 186 62 3 57 167 55.66666667 4 76 148 49.33333333 5 95 129 43 6 114 110 36.66666667 7 133 91 30.33333333 8 152 72 24 9 171 53 17.66666667 10 190 34 11.33333333 11 209 15 5 Large packets Number of lollies

Number of Remainder in small packet

lollies in packet Total (224 _{total) (remainder 3)}

1 19 1 … 224 … … … 3 … 2 19 2 … 224 … …

. . .

plarge packet 2 lollies small packet 62 lollies 19 2 3 62 224 lollies

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p Record your solution:

Using ‘guess, check and improve’ or using a table (or a spreadsheet), we have found that there are 11 lollies in each large packet and 5 lollies in each small packet.

Note: From the table, we can see that this is the only possible solution.

P R O B L E M - S O L V I N G S T E P S A N D S T R A T E G I E S

Solve these problems, showing all working out. Look for alternative answers and try different strategies.

■

1 How many ways can a team of 2 players be selected from:

a 3 players? b 5 players? c 7 players?

■

2 Ten friends sent each other Christmas cards. How many cards were sent?

■

3 Place or signs between the numbers to make this calculation correct:

6 3 2 12 3 5 7

■

4 What 5 consecutive numbers add up to 140?

■

5 A wire frame is made with 9 intersections labelled A to I.

An ant is to walk from A to I but can pass through an intersection no more than once and walk over a section of wire no more than once. How many ways can the ant walk from A to I ?

■

6 There are 12 teams in a netball competition. How many games need to be played if each team is to play each other team twice?

■

7 What 3 numbers, all less than 10, have their sum equal to their product?

■

8 The difference between 2 numbers is 54. The digits of the numbers are in reverse order. (For example, 21 and 12 have digits in reverse order.)

a What are the numbers? b Have you found all possible answers?

■

9 Bacteria in a sample double their number every 2 hours. In 12 hours the sample is completely full. How many hours does it take to half fill the sample with bacteria?

■

10 How many times between 1:30 pm and 6:15 pm do the hands on a clock cross?

■

11 Three equal-sized gears mesh together.

a If A rotates in a clockwise direction, in which direction will C rotate?

b Suppose gear B has 48 teeth.How many times does it rotate for 3 rotations of gear A?

E x e r c i s e

1A

A

B C D E F G H

I

A B C

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■

12 A basketball goal can be worth 3 points (goal outside the keyhole), 2 points (goal inside the keyhole) or 1 point (penalty shot). How many different ways can a player score 10 points?

■

13 Ken and Min were given the same amount of money. Ken bought 2 oranges and had 70c left. Min bought 4 oranges and had 20c left. Each orange cost the same amount. How much did each person have to start with?

■

14 A group of people attended a meeting. Before it began, each person shook hands with every other person once. There were 253 handshakes. How many people were there?

■

15 Find the number that each letter in MATHS stands for so that the sum of each row is 36.

■

16 1₃and 3₄are changed to the same lowest common denominator. It is then possible to write at least 4 fractions that are between 1₃and 3₄. What are they?

■

17 The New Age Pet Shop had a sale. The Wright family bought 100 animals for $100.

How many baby dinosaurs did they buy? Use a table to solve this.

■

18 The area of a 10 cm by 8 cm rectangular sheet of paper is reduced by 25%. What might the length and breadth of the new sheet be? Have you found all the answers?

■

19 A number pattern starts with 2 and ends with 74, and has 5 numbers in between these. Each number after the first is the sum of the 2 preceding terms. What is the full pattern?

■

20

Four clowns are waiting to practise a routine. Three of them are sitting in a line one behind the other. The fourth clown is behind a curtain, so cannot be seen by the others. Each clown is wearing either a white or a black hat.

A B C D

7

C H A P T E R 1 W O R K I N G M A T H E M A T I C A L LY, P R O B L E M S O LV I N G A N D N U M B E R M S 3 6 A 5 4 H T 7

Baby dinosaurs 8 for $1

Stegosaurs $1 each

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The clowns know that between them there are 2 black and 2 white hats, but no clown knows what colour he is wearing. B is facing the curtain; C is facing B; D is facing both C and B; A cannot see any of the others. All must remain this way, without turning around or leaving their seats or using tricks (such as a mirror).

They are not allowed to talk to or signal each other and have only 10 minutes to work their hat colours.

After a short time, one of them calls out, indicating that he knows his hat colour.

a Which clown is this?

b Why is he 100% sure of the colour of his own hat?

Rounding off

Some problems only need approximate answers.

The process of working out an approximate answer is called rounding off.

Example

Round off 4375 to the nearest 100.

4375 is closer to 4400 than to 4300.

4375 rounded to the nearest 100 is 4400.

4300 4375 4400

The number line shows that 4375 is closer to 4400 than

to 4300.

To round off a number to a given number of places, look at the digit after the specified place.

p If it is 5 or more, round up.

p If it is less than 5, round down.

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9 Examples

●

1 Round 642.8 to the nearest unit.

As the next digit is more than 5, 642.8 to the nearest unit is 643.

●

2 Round 25.615 correct to:

a 1 decimal place b 2 decimal places

a b

Using the FIXfeature, you can set your calculator to round off the display to a given number of decimal places. Check your calculator manual for details.

As the next digit is 5,

25.615 correct to 2 decimal places is 25.62.

As the next digit is less than 5, 25.615 correct to 1 decimal place is 25.6.

2nd decimal place

¶

25.615

˝

next digit 3rd decimal place 1st decimal place

¶

25.615

˝

next digit 2nd decimal place

R e co mmended_ti m e

S P E E D S K I L L S

1 . 1

Rounding off

1 Round off to the nearest 100:

a 765 b 4940 c 15 379 d 15 867 e 4692.8

2 Round off to the nearest 10:

a 464 b 1597 c 155.8 d 9.31 e 2147.51

3 Round off to the nearest unit:

a 41.93 b 181.49 c 2.1396 d 0.761 e 1275.51

4 Round off to 1 decimal place:

a 4.317 b 150.049 c 18.961 d 1.2916 e 25.555

f 6.049 g 37.999 h 0.0761 i 507.289 j 8.987

5 Write correct to 2 decimal places:

a 3.2166 b 18.4092 c 1.5555 d 310.0107 e 6.546

f 82.057 g 6.1188 h 173.009 i 8.849 j 72.956

6 Write correct to the nearest hundredth:

a 14.013 b 11.299 c 12.8196 d 1.4549 e 25.0017

7 Write correct to the nearest tenth:

a 8.1157 b 0.256 c 27.778 d 721.45 e 0.0843

✗

units

¶

642.8

˝

next digit 1st decimal place

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Operations with whole numbers

You can use mental computation strategies to

perform operations with whole numbers. Mental computation is also useful when you are finding an estimate, or when checking if a calculator answer is sensible.

Examples

Find:

●

1 67 45

●

2 103 57 67 40 107 103 50 53 107 5 112 53 7 46 67 45 112 103 57 46

●

3 19 8

●

4 2700 30 19 8 (20 1) 8 27 3 9 20 8 1 8 So 2700 30 90 160 8 152

●

5 617 6

Mental method Written method

617 6 600 6 10 6 7 6 3600 60 42 3600 102 3702 617 6 3702

●

6 345 29

●

7 245 6

345 Mental method Written method

29 240 6 40 40 remainder 5 _______ 3 105 5 6 5₆ 6245 6 900 _______ 10 005 245 6 405₆ _______ 345 29 10 005 1 6417 6 –––––– 3702 –––––– Using mental computation strategies is sometimes quicker than using a calculator.

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11

●

8 281 23 Method 1 Method 2 12 remainder 5 23 10 230 23281 23 2 46 23¶ So 23 12 23 10 23 2 ___ 51 230 46 46 276 ___ 5 So 281 23 12 5 281 23 12₂5₃ O P E R A T I O N S W I T H W H O L E N U M B E R S

Complete all questions in this exercise without using a calculator.

■

1 Find: a 38 7 b 54 6 c 8 7 d 36 9 e 23 8 f 107 9 g 80 25 h 75 32 i 5 60 j 200 4 k 47 50 l 98 60 m 640 8 n 87 45 o 49 52 p 30 60 q 91 36 r 139 53 s 27 8 t 9 14

■

2 Find: a 5 17 2 b 35 17 5 c 20 8 7 5 d 13 19 e 37 3 37 97 f 18 53 82 g 19 103 19 3 h 399 4

■

3 Find: a 317 5 b 523 6 c 153 18 d 49 36 e 73 42 f 227 34 g 39 5 h 324 6 i 459 8 j 632 9 k 165 7 l 232 6

■

4 Complete the following:

183₄ 384₇

a If 475 then 75 18

b If 7270 then 270

7

24₁5₈ 231₄1₂

c If 18437 then 437

d If 42977 then 977

■

5 Find (writing your answers as fractions in simplest form):

a 67 22 b 178 17 c 57 18 d 735 23 e 715 26

■

6 A jug contains 750 mL of water. 287 mL of water is poured from the jug. How much water remains in the jug?

■

7 How many ice-blocks can you buy for $10 if they cost $1.20 each?

■

8 How many 55-cent stamps could you buy for $20?

E x e r c i s e

1B

Method 1 is often called long division.

✗

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Order of operations and the calculator

When a question involves more than one operation, follow these steps:

Example

Evaluate 3 2 4 (4 7).

3 2 4 (4 7) 3 2 4 11 (Do operations inside the brackets.)

6 44 (Do any multiplications and divisions.)

50 (Do any additions and subtractions.)

When you are using a calculator, the correct order of operations will be performed automatically. For this question, press:

3 2 4 ( 4 7 ) 5 0 Modern calculators follow the correct order of operations.

Step 1 Perform operations inside the grouping symbols. Step 2 Starting from the left, do all multiplications and

divisions as you come to them.

Step 3 Starting from the left again, do all additions and

subtractions as you come to them.

S P E E D S K I L L S

1 . 2

Order of operations without a calculator

Evaluate (without using a calculator):

1 10 (6 4) 2 7 (4 2) 3 5 (18 10) 4 (45 5) 5 5 (28 8) 4 6 32 (15 7) 7 3 2 0 8 24 6 2 9 6 7 2 10 30 5 3 11 24 8 3 12 24 (8 3) 13 36 4 3 14 23 (10 7) 15 40 (5 4) 16 40 5 4 17 (27 21) (3 3) 18 42 7 5 19 [3 (5 1)] 22 20 8 [19 (9 1)] 21 2 5 2 22 10 4 5 23 6 3 1 24 5 (3) 2 25 64 (16 8) 26 3 (2 5) 27 9 (16 4) 28 20 5 7 29 27 3 2 6 30 15 6 4 31 15 8 10 32 30 10 20 33 10 7 11 34 (7 6) 8 35 6 4 5 16 36 6 4 4 2 37 10 70 7 (20) 38 5₆4₃ 39 6₉3₃ 40 ₃4₀3₆2₄

✗

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13 Examples

Use a calculator to evaluate, correct to 2 decimal places where necessary:

●

1 58 32 15 97 Press 58 32 15 97 58 32 15 97 441

●

2 14₃.₄9₈3_.3₆.8 Press 14.9 33.8 34 8.6

14 3 . 4 9 8 3 . 3 6 .8

_{0.17 (correct to 2 decimal places)}

O R D E R O F O P E R A T I O N S A N D T H E C A L C U L A T O R

■

1 Use a calculator to evaluate, correct to 2 decimal places where necessary:

a 35 17 25 b 250 20 17 c 3 42 36 5 d 20 7 3 e (37.4 8.6) 23 f 8.3 7.2 6.1 35 g ₁₆37₄ h 17₅_..9₇6 i 7₅4₂ j ₂6₀₀12₅₆ k 30 6.2 9.7 10 l 7 5 3 m 7 (5 3) n 62 (3 7 4) o 62 3 + 7 4 p (84 53) (27 19) q 57 26 3 8.2 r ₁₅2.2₁₃ s 14.3₂₂_.22.8 t ₈74₂0₁ u 2 5 (18) v 12 25 3 9 w 11.7 (12.2 16.9) x 205 10 5 5 y 351₉7 z 9₉8₆9

■

2 If one Australian dollar is worth 55 American cents, what are the values of the following Australian dollar amounts, in American dollars?

a $600 b $460 c $7 145 000

■

3 12 people equally share a prize of $415 000. How much does each person receive? How much is left over?

■

4 Change 1000 days to weeks and days.

■

5 Clive saved $130 in 5 months. At this rate, how much would he save in a year?

■

6 How much change would there be from $2800 after buying 65 calculators at $25.65 each and 450 grid books at $2.40 each?

■

7 A new car was advertised for $25 990. A buyer decided to buy this car but wanted alloy wheels, which cost an extra $750. On-road costs were also additional.

E x e r c i s e

1C

0 . I 6 6 5 5 2 6 6 7 ) ( ) (

4 4 I Only round to 2 decimal_{places if the calculator}

answer has more than 2 decimal places.

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a What were the on-road costs if the total cost was $27 980?

b The buyer borrowed money to finance the car. She paid back 48 monthly payments of $730. How much would she have saved if she could have paid cash for the car?

■

8 Seven people went to a restaurant and their bill was $273. They decided to share the bill equally, but one person only had $21. The others said they would each pay more to cover the bill. How much extra did each person pay?

■

9 Tickets to The Big Night Out were being sold for $95.50 each. Michael bought tickets for himself and his friends. He paid $764. How many friends did he buy tickets for?

■

10 This table shows the populations of some Australian states in 1901 and 2001. Between 1901 and 2001, what was:

a the increase in the population of New South Wales?

b the total increase in the combined population of New South Wales, Victoria and Queensland?

■

11 Competition diving was first included in the Olympics in 1904 at St Louis, USA. Women’s springboard events were first introduced in 1920, and at the 2000 Sydney Olympics synchronised diving was included for the first time. In springboard events, men do 11 dives and women do 10 dives from a 3 m platform.

To determine a competitor’s score for a single dive:

p A panel of 7 judges score the dive out of 10.

p The lowest and highest scores are eliminated.

p The sum of the remaining 5 scores is multiplied by the degree of difficulty (between 1.2 and 3.8).

p This total is multiplied by 0.6; the answer is the final score. The table below shows the 7 judges’ scores for each

competitor after the first round of a diving competition.

a Calculate the score for each competitor.

b Rank the divers (from first to last) at the conclusion of the first round of the competition. State Population 1901 2001 NSW 1 354 000 6 448 000 Vic. 1 201 000 4 753 000 Qld 498 000 3 549 000

Degree of difficulty Judges’ scores

Julia 2.0 6.5 6.5 7 7 7 7.5 8 Jody 1.2 7.5 8 8 8.5 7 9 8 Kate 1.8 8 8 8.5 9 8 8.5 8.5 Kim 2.2 7.5 7 7.5 8 7 7.5 7 Laura 1.9 6.5 7.5 8.5 8 8 8 8.5 Leanne 2.5 6.5 7.5 9 8 8.5 7 7

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Spreadsheets

Computer spreadsheets are very useful when solving mathematical problems. They can be used to investigate data and can be programmed to perform calculations. We will use Microsoft Excel spreadsheets in examples, but most other spreadsheets perform the same basic operations.

Terms used in spreadsheets

p cell A cell is a box on a spreadsheet and is identified by a letter and a number. In the following example, the word ‘square’ is in cell B1. (This could also be called b1.)

p formula A formula can be used for calculation in a spreadsheet just as it can be used to solve a mathematical problem. Every formula in a spreadsheet begins with an sign. For example, the formula =B2*B3 placed in cell B4 multiplies the number in B2 by the number in B3 and displays the answer in B4.

p Fill Down The Fill Down instruction copies cell formulas down the column. This

instruction is in the Edit menu. The formula in each cell is changed as it is copied down the column.

Example

Use a formula to write the numbers 1 to 15, then find the square of each number and the sum of the squares.

p Open the spreadsheet program (Microsoft Excel or another

spreadsheet program). In the spreadsheet program, open a new spreadsheet.

p Copy the headings ‘Number’ and ‘Square’ as

shown. Enter 1 in cell A2.

p Click on cell A3.

p In cell A3 write the formula =A2+1, then press the

Enter key.

1 will be added to the 1 in A2 and the answer of 2 will appear in A3.

p Click on cell A3 and hold the mouse button down.

Drag the mouse pointer down until the cells A3 to A16 are highlighted. Let the mouse button up now.

p Go to the Edit menu and click on the Fill Down

command. All the numbers from 2 to 15 should appear.

15

9 squared 92 9 9 81

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p In cell B2 write the formula =A2*A2 and press the Enter key.

The content of cell A2 (1) will be multiplied by itself to give an answer of 1 in cell B2.

p Click on cell B2 and drag down to cell B16. Go to the Edit

menu and click on the Fill Down command to copy the formula. This should give you the squares of the numbers from 1 to 15.

p Write the word ‘Total’ in cell A17.

p In cell B17 write the formula =SUM(B2:B16) and press

the Enter key.

This will give you the sum of the squares of the numbers from 1 to 15, which is 1240.

You can move around a spreadsheet using

the arrow keys or the mouse.

Spreadsheet exercises

■

1 For this spreadsheet, what formula is written in:

a cell D3? b cell D5? c cell C6?

■

2 For the spreadsheet in question 1:

a Write a formula in D6 that will give the total of the 9 numbers that are not blue.

b What is the total?

c Change the number in A3 to 35. What else changes in the spreadsheet?

■

3 Copy the numbers that are not shaded in the

given spreadsheet.

a Write a formula in each of the shaded cells to give the total of the row or column.

b Rearrange the numbers 1 to 9 so that all the totals in the shaded cells are 15.

CD-ROM

The formula in cell D1 is =A1+B1+C1.

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17 W O R K I N G M A T H E M A T I C A L L Y

Write a spreadsheet to answer each problem.

■

4 Find the total of:

a all the counting numbers from 1 to 1000

b all the squares of the counting numbers from 1 to 100

c all the cubes of the counting numbers from 1 to 100

d all the even numbers from 2 to 100

e all the odd numbers from 1 to 99

f all the numbers from100 to 100

■

5 Find 2 numbers between 1 and 100 whose sum is 85 and whose product is 1566.

■

6 Find 4 consecutive numbers between 1 and 100:

a that have a sum of 150

b that have a product of 32 760

c that have a sum of 110 and a product of 570 024

■

7 Is it possible to find 4 consecutive even numbers that have a sum of:

a 50? b 100? c 150? d 300?

A short history of spreadsheets

Accountants and other business people have used spreadsheets in their work for over 150 years, but using them manually was simply too slow, mechanical and awkward to be very effective.

In 1961 Professor Richard Mattessich devised computer spreadsheets for use in mainframe programs. They were very simple and limited in their capacity. Then, in 1978, Dan Bricklin developed a software program in which users could enter and manipulate 5 columns and 20 rows of data. This was soon refined, extended and marketed with faster speeds, scrolling and greater memory.

As the personal computer market grew rapidly, so did the need for more powerful

spreadsheet software. Integrated charting, plotting and database features were added, and later graphical interfaces with pull down menus and point-and-click features (using a mouse) were added.

Now programmers are looking to the future with the development of a spreadsheet solver add-in that can be used for both equation solving (called goalseeking) and for linear and non-linear programming.

p List some examples of where spreadsheets can be used.

p List some advantages of using spreadsheets.

Use the Help menu on your spreadsheet if you

need extra information.

21, 22, 23 and 24 are 4 consecutive

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Directed numbers

Directed numbers include all positive and negative whole numbers, fractions and decimals. Integers are all the positive and negative whole numbers, including zero.

To add and subtract directed numbers, use the number line.

Examples

Find:

●

1 3 5 3 5 2

●

2 2 3 2 3 5

To multiply or divide a pair of directed numbers, use this rule:

If a number has no sign, it is positive. Therefore, 7 is7 (but we don’t usually write the sign).

Examples

Find:

●

1 3 6

●

2 2 7

●

3 12₃ 3 6 18 2 7 14 12₃_{12 (3)} 4 –5 –4 –3 –2 –1 0 1 2 –3 Start –2 –1 0 1 2 3 4 direction

¶

3 5

˝

start move 3 and 5 have the same signs. Both are negative.

If the signs are the same, the answer is positive. If the signs are different, the answer is negative.

3 6 could also be written as 3 (6). CD-ROM

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On a calculator there are usually 2 negative keys:

This is the key used to change a number from positive to negative. This is the key used to subtract one number from another.

Try these calculations to find 3 4:

p Press 3 4

Which method is correct for your calculator?

Example

Find 5 (3).

5 (3) 5 3 On a calculator, press:

8 ₅ ₃

Class discussion

■

1 What happens when you add a negative number, for example … (4)?

■

2 What happens when you subtract a negative number, for example … (4)? () () () ()

19

C H A P T E R 1 W O R K I N G M A T H E M A T I C A L LY, P R O B L E M S O LV I N G A N D N U M B E R Does 4 () 3 1? (3) is really 1 (3), which gives3. R e co mmended_ti m e

S P E E D S K I L L S

1 . 3

The 4 operations with directed numbers

Find: 1 13 7 2 6 8 3 15 6 4 27 34 5 8 12 6 5 40 7 4 16 8 19 10 9 0 9 10 12 18 11 5 8 12 7 6 13 8 3 14 6 9 15 4 5 16 6 3 17 6 3 18 (4)2 19 (7)2 20 1₄2 21 35 7 22 48 6 23 24 (4) 24 100(20) 25 0 8 26 20 (7) 27 8 (13) 28 30 (20) 29 0 (15) 30 36₆ 31 35 (35) 32 2 8 7 33 6 9 12 34 30 (1₂) 35 50 1₂ 36 ₁1₀ 400 37 81₂ 101₂ 38 41₂ 41₂ 39 1 51₂ 40 4₉5 41 8 7 42 0 60 43 5.6 1.6 44 121₂ 8 45 57 (57) 46 9.2 9.2 47 0 3.7 48 11 (2) 49 6.2 10 50 1₁3₃

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Examples

●

1 Insert or to make each statement true:

a 5

3 b 2

3 a 5 3 b 2 3

●

2 Find: a 3 2 4 b 2 (3) 5 a 3 2 4 5 b 2 (3) 5 1 D I R E C T E D N U M B E R S

■

1 This thermometer shows2°C. Give the new reading if the temperature:

a rose by 6°C

b dropped by 5°C

c rose by 4°C and then dropped by 7°C

d dropped by 1°C and then dropped by another 2°C

■

2 Graph the following numbers on a number line: 3, 1, 0, 21₂, 3

■

3 Graph the integers between4 and 5 on a number line.

■

a 8

1 b 2

0 c 3

5 d 5

1

e 10

5 f 3

2 g 5

6 h 4

3

■

5 Arrange each group of numbers in ascending order (from smallest to largest):

a 5, 0,4, 3, 2 b 12, 61₂, 6, 31₂,10 c 4,2, 3, 1, 0, 5, 5

■

6 Arrange each group of numbers in descending order:

a 6, 4, 7,2 b 3.4,2.9, 7.6, 3, 0 c 9,3, 7, 16, 2, 5

■

7 This table shows the minimum temperatures recorded in these cities on a certain day:

What was the difference between the temperatures in:

a Calgary and London? b London and Melbourne?

c Bangkok and Calgary? d Bangkok and London?

°C 15 10 5 0 5 10 15

E x e r c i s e

1D

City Temperature (°C) Bangkok 26 Calgary 15 London ₂ Melbourne 14 greater than less than

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W O R K I N G M A T H E M A T I C A L L Y

■

8 Find: a 3 2 1 b 5 (2) (1) c 3 5 7 d 2 7 16 e 2 9 50 f 3 13 g 36 (12) 8 h 12₈0 13 i 8 (40 6) j 85₅ 7₂ k 16 (3) 51 l 6₈0 (4 17) m 3 (2) 4 n 2581₃ 1 o 125 (3 12) p 6 18 (16)

■

9 What 2 numbers give a product of15? Is there more than one answer?

■

10 Write this spreadsheet to perform operations with directed numbers.

In B4 write =B1+B2. In B5 write =B1-B2. In B6 write =B1*B2. In B7 write =B1/B2.

Enter the following numbers and check that the computer answers are what you expect. The a number goes in B1 and the b number in B2.

a a 10, b 5 b a 10, b 5 c a 10, b 5

d a 10, b 5 e a 12, b 4 f a 12, b 4

Repeating sequences

When 1₇is changed into a decimal, it gives the repeating sequence 0.1 42857 . Using this repeating sequence, 14 28 57 99 and 142 857 999!

p See if this also applies to the decimal for ₁1₃.

Look carefully at these products using the number 15 873:

15 873 7 111 111 p What do you notice about all the multipliers?

14 222 222 p Why does the pattern stop here? What happens now? 21 333 333 p What do you get if you multiply 15 873 by 70? 77? 84? 28 444 444 91? 98 and more? beyond 133?

35 555 555 p What is 15 873 9? How is this connected with 1₇? 42 666 666 49 777 777 56 888 888 63 999 999

21

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Marathon

In 1896, in a re-enactment of the Greek legend of the marathon was run over a distance of 40 km (the distance between the ancient towns of Marathon and Athens in Greece). However, the distance for the modern marathon was set at the 1908 London Olympic Games. The distance of the marathon was increased to 42.195 km to allow the start of the race to be seen by the Royal family on the lawns of Windsor Castle! Although the Italian athlete Dorando Pietri finished first, he was not awarded the medal because he was helped over the finishing line after almost collapsing near the end of the race.

p Convert 42.195 km to metres.

p State in metres the increase in distance of the marathon when it was increased from 40 km to 42.195 km.

p Estimate a place 42.195 km from your school.

p Find out the names (and countries of origin) of the current Olympic gold medal holders for the men’s and women’s marathons.

Fractions

A proper fraction is a fraction that has a numerator of less value than the denominator. Some examples are 1₄, ₅2and ₁9₀.

An improper fraction is a fraction that has a numerator of greater value than the denominator or a numerator that is equal to the denominator. Some examples are 5₅, 7₅and 4₁9₀.

A mixed number is a mixture of a whole number and a proper fraction. Some examples are 13₄, 2₅4and 5₁1₀.

A fraction is in its simplest form when the numerator and denominator cannot be divided evenly by the same number (other than 1). For example:

1₂5₀1₂5

3

0

4

(dividing the numerator and denominator by 5)

3₄

3₄is the simplest form.

d n en u o m m er in a a to to r r

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Basic fraction rules

Addition and subtraction

For example:

2₅₁1₀ The LCM of 5 and 10 is 10. 3₅1₄ The LCM of 5 and 4 is 20. ₁4₀₁1₀ 1₂2₀₂5₀ ₁5₀ ₂7₀ 1₂

Multiplication

For example: 3₄5₉ ₁7₀5₈ ₄3 1 5₉ 3 (cancelling) 1 7 0 2 8 51 (cancelling) ₁5₂ ₁7₆

Division

For example: 2 3 1 4 The reciprocal of 1 4is 4 1. 7 81 7 6 2₃4₁ 7₈ 1 1 1 7 1 6 2 8₃ 2₁ 22₃ ₂

23

C H A P T E R 1 W O R K I N G M A T H E M A T I C A L LY, P R O B L E M S O LV I N G A N D N U M B E R Dividing by a fraction is the same as multiplying

by its reciprocal.

R

T

R

p Change the fractions so that they have the same denominator.

(The denominator will be the lowest common multiple of the denominators.)

p Add or subtract the numerators—the denominator stays the same.

p Cancel any common factors.

p Multiply the numerators, then multiply the denominators.

p The first fraction stays the same.

p Change the sign to a sign.

p Write down the reciprocal of the second fraction; that is, turn the second fraction upside-down.

p Proceed as with multiplication.

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Examples

●

1 Express 53₄as an improper fraction.

●

2 Express 1₅9as a mixed number.

M

1₅9 19 5 53₄ 5 3₄

m

3 remainder 4 5 4₄ 3 3 4₅ 2₄3

●

3 Reduce 1₃2₀to a fraction in its simplest form. 1₃2₀1₃2

2

0 5

(Divide the numerator and denominator by 6.)

2₅

●

4 What is the reciprocal of 21₂?

M

21₂ 2 1₂

m

4₂1

5₂ _{the reciprocal of 2}1₂(or 5₂) is 2₅.

S P E E D S K I L L S

1 . 4

Fractions

1 Write the fraction (in simplest form) that is shaded in each diagram:

2 State whether the fraction is:

A less than 1 B equal to 1 C greater than 1

a 7₇5₃ b ₁13₄ c 1₁9₉ d 1₁5₆₁0 e 8₇9₀ 3 Complete:

a 1₃₁₈ b ₄3₁₂ c 5₆₂₄ d ₁7₀₈₀ e 7₈₄₀ 4 Express as a fraction with a denominator of 10:

a 1₂ b 1₅ c 3₅ d 5₂ e 3 a b c d

✗

CD-ROM

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25 Examples

●

1 Find 3₄2₃. 3₄ 2₃₁9₂₁8₂ (The LCM of 4 and 3 is 12.) 1₁7₂ 1₁5₂

●

2 Simplify 21₂1₄.

●

3 Simplify 13₄2₃. 21₂1₄5₂ 1₄ 13₄₃27₄2₃ 5₈ 7₄3₂ 2₈1₂5₈

5 Express as a fraction with a denominator of 100:

a 1₄ b 1₅ c ₂1₅ d ₂3₀ e 1₅3₀ 6 Insert, or to make each statement true:

a 1₂7₀

₂3₀ b 1₂

1₂1₀ c 1₃

1₄ d 2₃

6₉ e 3₄

4₅ 7 Express as an improper fraction:

a 11₅ b 12₃ c 2₁7₀ d 32₅ e 8₁7₀

8 Write as a mixed number:

a 1₁3₀ b 9₈ c 1₃3 d 2₅8 e 3₇9 9 Write each fraction in its simplest form:

a 1₁0₅ b ₂6₁ c ₁8₀ d 2₆4₄ e 4₆5₀ 10 Find the reciprocal of:

a 1₈ b 3₇ c 12₃ d 71₂ e 47₈ f 10

11 Complete the following:

a Dividing by 1₄is the same as multiplying by

.

b Dividing by 2 is the same as multiplying by

.

c Dividing by 2₃is the same as multiplying by

.

d Dividing by 11₄is the same as multiplying by

.

e Dividing by 3₅is the same as multiplying by

.

a 51₂

5 3 b 12 4

121₂ c 91₃

9 2

Connections Maths 8

C

D

-R

OM

in

s

id

e

8

8

8

8

connections

8

m a t h s

Contents

v

vii

Introduction

Preface to the teacher

What you will find in the student text and CD-ROM

How to use the CD-ROM

ix

c

Acknowledgements

Working

mathematically,

problem solving

and number

S y l l a b u s o u t c o m e s

I n t h i s c h a p t e r y o u w i l l l e a r n t o :

n

n

n

n

n

n

n

W O R K I N G M A T H E M A T I C A L L Y

Understanding ‘Working mathematically’

Problem-solving steps and strategies

Examples

●

3

~|

|_~|

|_~|

|_~|

|_

●

5

■

■

■

■

■

■

■

■

■

■

■

E x e r c i s e

1A

A

I

■

■

■

■

■

■

■

■

■

7

Rounding off

Example

9

_in