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66MATHEMATICS

Haese & Harris Publications

Stan PulgiesRobert HaeseSandra Haese

second edition

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MATHEMATICS FOR YEAR 6SECOND EDITION

This book is copyright

Copying for educational purposes

Acknowledgement

Stan Pulgies M.Ed., B.Ed., Grad.Dip.T.Robert Haese B.Sc.Sandra Haese B.Sc.

Haese & Harris Publications3 Frank Collopy Court, Adelaide Airport SA 5950Telephone: (08) 8355 9444, Fax: (08) 8355 9471email:web:

National Library of Australia Card Number & ISBN 1 876543 62 0

© Haese & Harris Publications 2003

Published by Raksar Nominees Pty Ltd, 3 Frank Collopy Court, Adelaide Airport SA 5950

First Edition 2000Second Edition 2003

Cartoon artwork by John Martin and Chris Meadows.Artwork by Piotr Poturaj, Joanna Poturaj and David PurtonCover design by Piotr Poturaj.Cover photograph: © Nicholas Birks,Computer software by David Purton and Eli Sieradzki

Typeset in Australia by Susan Haese (Raksar Nominees). Typeset in Times Roman 10 /11

. Except as permitted by the Copyright Act (any fair dealing for thepurposes of private study, research, criticism or review), no part of this publication may bereproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic,mechanical, photocopying, recording or otherwise, without the prior permission of the publisher.Enquiries to be made to Haese & Harris Publications.

: Where copies of part or the whole of the book are madeunder Part VB of the Copyright Act, the law requires that the educational institution or the bodythat administers it has given a remuneration notice to Copyright Agency Limited (CAL). Forinformation, contact the Copyright Agency Limited.

: the descriptors listed at the beginning of each chapter are taken from thepublished by the Department of Education and

Children’s Services.

Western Pygmy Possum Wildflight Australia Photography

R-7 SACSA Mathematics Teaching Resource

\Qw_ \Qw_

[email protected]


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FOREWORD

[email protected]

www.haeseandharris.com.au

Mathematics for Year 6 (second edition) offers a comprehensive and rigorous course of study atYear 6 level. Through its worked examples, exercises, activities and answers, and the support of theinteractive Student CD, the book provides students with the structure and content to work efficientlyat their own rate.

The book and CD package is designed to supplement classroom practice and give teachers time toexplore other creative strategies, depending on the needs of their students. It is not the Year 6Mathematics Curriculum, nor does it proclaim to provide the most effective teaching program.

The knowledge, skills and understandings listed at the beginning of each chapter incorporate thedescriptors used in the R-7 SACSA Mathematics Teaching Resource for Middle Years. We have listedthe descriptors in the hope that this will be a helpful guide for Year 6 teachers who may wish to usethe book to support their teaching practice.

About this second edition: the second edition is a general revision and updating of the original text,with some reorganisation of chapters. Changes include:

new sections on Speed and Temperature in the Time chapter

a new section on Mass in the Solids chapter

a new chapter Data Collection and Analysis, which combines Reading Graphs and Charts, andData Analysis, and includes a new section Using Technology to Graph Data

an extended chapter on Measurement, including perimeter, area, volume and capacity.

the inclusion of an interactive Student CD

In the table of contents, page numbers for corresponding sections in the first edition are given inbrackets. This is intended as a guide for teachers who may wish to use the first and second editionswithin the same classroom. A glance at the contents pages will show how both books correlate (theabsence of a page number in brackets denotes the introduction of a new section).

About the interactive Student CD: the CD contains the text of the book. Students can leave thetextbook at school and keep the CD at home, to save carrying a heavy textbook to and from schooleach day. But more than that, by clicking on the ‘active icons’ within the text, students can access arange of interactive features: graphing and geometry software, spreadsheets, video clips, computerdemonstrations and simulations, and worksheets.

The CD is ideal for independent study. Students can revisit concepts taught in class and explore newideas for themselves. It is fantastic for teachers to use for demonstrations and simulations in theclassroom. In summary, the book offers structure and rigour, and the CD makes maths come alive. Wehave endeavoured to provide as broad a base of activity and learning styles as we can, but we alsocaution that no single book should be the sole resource for any classroom teacher.

We welcome your feedback.

Email:

Web:

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4 TABLE OF CONTENTS

TABLE OF CONTENTS Numbers in brackets denote pages in the first edition.

They have been included to assist teachers using the

first and second editions in the same classroom.

1 WHOLE NUMBERS 9

2 POINTS, LINES AND CIRCLES 43

3 NUMBER FACTS 73

(9)

(41)

(67)

A Different number systems 10

B How many? The language of number 14

C A numbered world 15

D Our number system 16

E Zero 19

F Rounding numbers 21

G Estimation and approximation 24

H Place value 32

I Number line 38

J Number puzzles (Extension) 40

Review Set A (Chapter 1) 41

Review Set B (Chapter 1) 41

Review Set C (Chapter 1) 42

A Points and lines 44

B Polygons 48

C Angles 52

D Perpendicular lines 58

E Triangles and quadrilaterals 59

F Angles in triangles 63

G Angles in quadrilaterals 65

H Circles 67




A Addition and subtraction 74

B Multiplication and division by powers of 10 78

C Calculator use 81

D Multiplication 84

E Division 86

F Using number operations (problem solving) 88

G Zero and one 90

H Factors of whole numbers 91

(10)

(13)

(15)

(15)

(18)

(19)

(22)

(30)

(35)

(37)

(42)

(47)

(51)

(57)

(58)

(60)

(62)

(68)

(70)

(73)

(76)

(76)

(81)

(86)


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TABLE OF CONTENTS 5

I Divisibility 93

J Multiples of whole numbers 94

K One operation after another 96




Extension (A different base system) 100

A Fractions 102

B Representation of fractions 103

C Fractions of regular shapes 104

D Fractions of quantities 105

E Finding the whole from a fraction 109

F Ordering of fractions 110

G Equivalent fractions 111

H Fractions to lowest terms 115

I Mixed numbers & improper fractions 115

J Adding fractions 118

K Subtraction of fractions 123

L Miscellaneous problem solving 125

M Ratio 126




A Representing decimals 134

B Using a number line 135

C Showing place value with blocks 137

D Place value 138

E Value of money 143

F Adding decimal numbers 146

G Subtracting decimal numbers 149

H Multiplying & dividing by 10, 100 and 1000 151

I Converting fractions to decimals 154

J Converting decimals to fractions 156

K Percentage 156

L Multiplication with whole numbers 158

M Division by whole numbers 162

N Operating with money 164

O Problem solving 165

(89)

(91)

(84)

(124)

(125)

(126)

(128)

(131)

(133)

(134)

(140)

(147)

(151)

(154)

(142)

(156)

(157)

(159)

(161)

(165)

(168)

(170)

(172)

(175)

(177)

(179)

(183)

4 FRACTIONS 101

5 DECIMALS 133

(123)

(155)


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6 TABLE OF CONTENTS

P Rounding decimal numbers 166




A Units of measure 172B Converting length units 174

C Perimeter 176

D Perimeters of special figures 180

E Problem solving with perimeters 182

F Comparing area units 183

G Area 185

H Metric area units 186

I Area of a rectangle 188

J Areas of composite shapes 192

K Problem solving with areas 193

L Volume 194

M Capacity 197

N Calculating volumes and capacities 200




A Scales 209

B Grids 210

C Maps 212

D Direction 216

E Plans 221

F Coordinates 224

G Locus 227



A Polyhedra 234

B Drawing solids 235

C Making solids from nets 240

D Different views of objects 244

E Intersection of solids and planes 245

F Mass 248


(96)

(100)

(106)

(109)

(111)

(115)

(117)

(189)

(190)

(191)

(195)

(198)

(200)

(212)

(214)

(220)

(223)

(224)

6 MEASUREMENT 171

7 LOCATION AND POSITION 207

8 SOLIDS AND MASS 231

(187)


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TABLE OF CONTENTS 7



A Sampling 257

B Organising data 257

C Displaying data 262

D Using technology to graph data 272

E Interpreting data 275

F Measuring the 'middle' of a data set 277




A Time lines 285

B Units of time 287

C A date with a calendar 291

D Reading clocks and watches 294

E Clockwise direction 302

F Timetables 303

G Speed 306

H Temperature 308

I Maximum, minimum and average temperatures 310

Review Set A (Chater 10) 312



A Number patterns (sequences) 316

B Dot patterns 317

C Matchstick patterns 319

D Rules and problem solving 321

E Graphing patterns 323

F Using word formulae 323

G Converting words to symbols 326

H Using a spreadsheet 328

I Algebraic expressions 329

J Graphing from a rule 332

K Graphs of real world situations 335



(275)

(275)

(279)

(282)

(284)

(251)

(252)

(257)

(259)

(266)

(268)

(292)

(293)

(295)

(296)

(298)

(299)

(301)

(303)

(304)

9 DATA COLLECTION AND REPRESENTATION 255

10 TIME AND TEMPERATURE 283

11 PATTERNS AND ALGEBRA 315

(249)

(291)


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8 TABLE OF CONTENTS

12 TRANSFORMATIONS 339

13 CHANCE AND PROBABILITY 363

ANSWERS 381

INDEX 415

(311)

(341)

A The language of transformations 340

B Tessellations 343

C Line symmetry 346

D Rotations 349

E Enlargements and reductions 356




A Describing chance 364

B All possible results 367

C Probability 372

D Most likely to least likely order 373

E Statistics and probability 374

F Life insurance tables (Extension) 376




(312)

(332)

(316)

(321)

(327)

(342)

(344)

(349)

(350)

(351)


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Wholenumbers

Wholenumbers

11Chapter

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

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recognise the existence of different numbersystems (e.g. Greek, Roman, Hindu-Arabic)

provide examples of the use of number ineveryday life

read, write and record numbers to onemillion, using numerals and words

explain place value of digits in numbers to000 0001� �

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write numbers to 000 000 in expandedform

round to the nearest 10 100 1000 10 000and 100 000

place numbers in descending andascending order

compare numbers and use symbols(e.g., and

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In this chapter you will learn how to

Knowledge, skills and understandings


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Y:\...\SA_06-2\SA06_01\009SA601.CDRMon Sep 29 13:45:19 2003


10 WHOLE NUMBERS (CHAPTER 1)

The number system we use is called the Hindu-Arabic System.

Archaeologists and anthropologists study ancient civilizations. They have helped us to

understand how people long ago counted and recorded numbers.

Before the Hindu-Arabic system many ancient number systems were used to tally.

For example, items they wanted to show could be represented or matched by:

scratches on a

cave wall (show

new moons since

the buffalo herd

came through)

knots on the rope

(show rows of

corn planted)

pebbles on sand

(show traps set

for fish)

notches cut on

the branch (show

new lambs born).

A tally of jjjjj eventually was replaced by jjjj©© and jjjj©© jjjj©© was much easier for tallying

than jjjjjjjjjj :

The Ancient Egyptians developed

the symbol to show jjjjjjjjjj

and to represent ten lots of

or 100.

The Mayans used a

to represent ,

to represent

and to represent 100.

DIFFERENT NUMBER SYSTEMSA

...............

To ,aus now this would not seem to

be very efficient way for record-ing larger numbers.

ANCIENT GREEK OR ATTIC SYSTEM

Like the Mayans, the Ancient Greeks could see the need to include a symbol for Someexamples of Ancient Greek numbers are

5.


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1000 6000

1324 6781

300 700

20 80

+ 4 + 1

This system depends on addition and multiplication.

Change the following Ancient Greek numerals into a Hindu-Arabic number:

a b

a b

1 Change the following Ancient Greek numerals into Hindu-Arabic numbers:

a b c

d e f

2 Write the following Hindu-Arabic numbers as Ancient Greek numerals:

a 14 b 31 c 99 d 555 e 4082 f 5601

EXERCISE 1A

1

20

700 1000 5000

2

30

3

50

4

60

5

100

6

400

7

500

8 9 10 Can you see whatthe smaller , ,

and X symbols do tothe symbol whenthey are joined?

��

Example 1

Like the Ancient Greeks, the Romans also

had a system which used a combination of

symbols. The first four numbers could be

represented by the fingers on one hand.

ROMAN NUMERALS

WHOLE NUMBERS (CHAPTER 1) 11

I II III IIIIbut later IV

1 2 3 4


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The V formed by the thumb and forefinger of

an open hand represented 5.

Two V’s joined together became two lots

of 5, i.e., ten or X.

C represented one hundred, and half of ,

i.e., L became 50.

One thousand was represented by an .

With a little imagination you should see that

an split in half and turned 90o would

look like a , so D became half a thou-

sand or 500.

1 2 3 4 5 6 7 8 9 10I II III IV V VI VII VIII IX X

20 30 40 50 60 70 80 90 100 500 1000XX XXX XL L LX LXX LXXX XC C D M

For example, IV stands for 1 before 5, i.e., 4

whereas VI stands for 1 after 5, i.e., 6

and XC stands for 10 before 100, i.e., 90

whereas CX stands for 10 after 100, i.e., 110.

5000 10 000 50 000 100 000 500 000 1 000 000V X L C D M

Look for Roman numerals onclocks and watches, at the end of

movies when the credits arebeing shown, on plaques and on

the top of buildings, and aschapter numbers to novels.

Unlike some other ancient systems, the Roman system had to be written in order and thevalue would change if the order changed.

There were also rules for the order in which symbols could be used. The could only appearbefore or X, the could only appear before or C, and could only appear before oran M. For example, the number could be written as MCMXCIX, but could be writ-ten as MIM.

Larger numerals were formed by placing stroke above the symbol which made the numbertimes as large:

IV X L C a D

a

1999

1000

not

EXERCISE 1A (continued)

3 What numbers are represented by the following symbols:

a XVI b XXXI c LXXXIII

d CXXIV e MCCLVII f D L VDCC?

4 Write the following numbers in Roman numerals:

a 18 b 34 c 279 d 902 e 1046 f 2551


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RESEARCH OTHER WAYS OF COUNTING

5 a Which Roman numeral less than one hundred has the greatest number of symbols?

b What is the highest Roman numeral between M and MM which has the least number

of symbols?

c Write the year 1999 using Roman symbols.

6 Use Roman numerals to answer the following

questions.

a

b

c In one week beginning on Monday and ending on Saturday, Marcus baked LXVIII,

LVI, XLIV, XLIX, XCVIII and CLIV loaves.

i How many loaves did he bake for the week?

ii What was his profit if he made III denarii profit on each loaf?

d Julius and his road building crew were expected to build MDC metres of road

before the end of spring. If they completed CCCLX metres in summer, CDLXXX

in autumn, and CCCXV in winter, how much more did they need to complete in

the last season?

Find out

1 How the Ancient Egyptians and Mayans represented numbers larger than 1000.

2 Whether the Egyptians used a symbol for zero.

3 How to write the symbols 1 to 10 using

Chinese or Japanese characters.

4 What larger Braille numbers feel like.

5 How deaf people ‘sign’ numbers.

6 What the Roman numerals were for

the two Australian Olympiads.

Denarii is the unitof currency usedby the Romans.

1

6

2

7

3

8

4

9

5

0

INVESTIGATION BIRTHDAYS IN ROMAN

What to do:

Use a calendar to help you.

1 In Roman numerals write:

a your date of birth, for example,

XXI-XI-MCMXLVI

b what the date will be when you are

i XV ii L iii XXI iv C

VI

XIII

XX

XXVII

VII

XIV

XXI

XXVIII

VIII

I

XV

XXII

XXIX

IX

II

XVI

XXIII

X

III

XVII

XXIV

XI

IV

XVIII

XXV

XII

V

XIX

XXVI

Each week Octavius sharpened CCCLIV swordsfor the General. How many would he need tosharpen if the General doubled his order?

What would it cost Claudius to finish his court-yard if he needed to pay for CL pavers at VIIIdenarii each and labour costs of XCIV denarii?


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ACTIVITY WORDS FOR NUMBERS

What to do:

Some people think that mathematics is just working with numbers. Mathematics is more than

just working with numbers. Mathematics has symbols and language besides numbers.

Numbers also have a language of their own.

The following are examples of how number has developed its own language to tell us “how

many”:

a couple of people a pair of sox

a brace of ducks twins

Three people playing musical instruments or

singing together are a trio. In cricket, a

bowler getting three batsmen out in three suc-

cessive balls is said to have bowled a hat

trick. Writing an order in triplicate means

that three copies of the order form are made.

Usually each form is a different colour. Some

other words meaning three include: thrice,

triad, trilogy, triple, and trifecta.

1

HOW MANY? THE LANGUAGE OF NUMBERB

a dueta dyad

a doubletwo apples


2 Ignoring any leap year what will be

a XIV days after your XVI birthday b LX days after your XIX birthday?

the date:

In groups, brainstorm list of as many words as you can which can meanor Check your list for the correct spelling and meanings using

dictionary or thesaurus.

a. afour five


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2 a

b

3

Use dictionaries and thesauri (more than one thesaurus) to find what number isrepresented by the following words:

.

rite sentences using these words to illustrate numbers.

octave, dozen, duck, decade, score, naught, century zilch, gross, ton, hexagon,zero, quadruplet, quartet, hexahedron, pentathlon, nonet, quintet, quinella, hepta-gon, pentahedron, grand, sextet

,

W

Use the ideas from to write half dozen questions to challenge yourclass mates.

Exercise 1A a

1 How much did each twin get when they shared equally a seven hundred and fifty dollar

prize?

2 Each member of the string quartet played a solo for a minute and a half at the concert.

What amount of time was played by soloists?

3 The trio travelling around the country each paid $125:50 for petrol and motel accommo-

dation. What was the total amount paid?

4 The septuagenarian (person in their 70’s) celebrated her birthday with a cake which had

a candle for each decade she had lived. How many candles were on her cake?

5

6 The quintet each trebled their $400 investment.

What was their new total amount?

Besides the numbers that we use at school, numbers

have many other uses.

For example, you will find an ISBN number near the

front of this book. This 10 digit International Standard

Book Number is a record of the language in this book.

This number also tells the place where it was written

and who the publisher was.

Your birth certificate will have a number on it. You

have an enrolment number for the school you are at-

tending. Your address, telephone, bus route and shoe

size all have numbers.

A NUMBERED WORLDC

715 896 772 323 260 379 104 694 030

417 099 915 341 993 944 641 232 819

500 973 801 564 442 385 558 485 023

687 594 667 185 788 236 727 251 127

803 222 418 803 192 246 004 309 526

582 751 955 660 681 390 204 600 990

123 324 924 871 916 500 864 791 408

437 256 005 081 072 775 276 510 718

234 240 758 247 081 714 797 128 389

020 284 647 317 371 074 422 745 994

256 477 391 628 302 672 851 185 657

906 854 522 688 389 396 421 819 891

961 367 677 716 644 657 086 398 874

346 024 253 141 456 775 211 744 603

140 278 283 653 127 988 721 310 309

997 794 681 582 081 040 199 242 257� ��

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CREDIT CARD

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Sir Donald Bradman scored tons and double centuries during his first classcricket career What was the smallest totalnumber of runs he may have scored justin tons and double centuries?

78 36.

EXERCISE 1B


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ACTIVITY USING NUMBERS

What to do:

1 In groups, brainstorm as many uses of number as you can for the follow-

ing categories:

a money b travel c information d records.

2 Find out how a bar code works and how it is used.

From our brief look at the History of Number, you

would remember that the method of writing num-

bers is called a number system. The system we use

was developed in India, 2000 years ago, and was in-

troduced to European nations by Arab traders about

1000 years ago. We therefore call our system the

Hindu-Arabic System.

The marks we use to represent numbers are called

numerals. They are made up by using the symbols

1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. These symbols are

called digits.

The digits3 and 8 can be used to form the numeral ‘38’ for number ‘thirty-eight’ and numeral

‘83’ for number ‘eighty-three’.

In Latin “numerus” means number.

1 2 3 4 5 6 7 8 9

oneordinal number

Hindu-Arabic

Numeral

our numeral

two three four five six seven eight nine

Most of the time weare not worried

about the differencebetween ‘numeral’

and ‘number’.We use the word

most of thetime.

number

There are three kinds ofpeople in the world.

Those who can countand those who cannot.

OUR NUMBER SYSTEMD

The numbers we use for countingare called Thepossible combination of naturalnumbers is There is nolargest number say that theset of all natural numbers is

Natural numbers are alsocalled

If we include in our set of num-bers the number zero, then ourset now has new name, the setof

natural numbers

endless

infi-

nite

counting numbers

whole numbers

.

.. We

..

,a

.

0


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There are three features which make the Hindu-Arabic system useful and more efficient than

most ancient systems:

² It uses only 10 digits to make all the natural numbers.

² It uses the digit 0 or zero to show an empty space.

² It has a place value system where digits represent different

numbers when placed in different value columns.

Each digit in a number has a place value. For example: in 567 942

1 What number is represented by the digit 7 in the following?

a 27 b 74 c 567 d 758e 1971 f 7635 g 3751 h 27 906i 80 007 j 72 024 k 87 894 l 478 864

2 What is the place value of the digit 5 in the following?

a 385 b 4548 c 32 756 d 577 908

3 Write down the place value of the 2, the 4 and the 8 in each of the following:

a 1824 b 32 804 c 80 402 d 248 935

4 a Use the digits 9, 5 and 7 once only to make the smallest number you can.

b Write the largest number you can, using the digits 3, 2, 0, 9 and 8 once only.

c What is the largest 6 digit number you can write using each of the digits 1, 4 and

7 twice?

d What are the different numbers you can write using the digits 6, 7 and 8 once only?

5 Place the following numbers in order beginning with the smallest (ascending order):

a 62, 26, 20, 16, 60

b 67, 18, 85, 26, 64, 29

c 770, 70, 700, 7, 707

d 2808, two thousand and eight, 2080, two thousand eight hundred

What number is represented by the digit 9 in the numeral 1972?

nine hundred or 900

Example 2

5 6 7 9 4 2

units

tens

hundre

ds

thousa

nds

ten

thousa

nds

hundre

dth

ousa

nds

EXERCISE 1D


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6 Place the following numbers in order beginning with the largest (descending order):

a 17, 21, 20, 16, 32

b 77, 28, 95, 36, 64, 49

c 880, 800, 80, 808, 8

d 2606, two thousand and six, 2060, two thousand six hundred

7 Often the symbols we use in mathematics are used instead of a group of words (called

a phrase). Examples are:

= means is equal to 6= means is not equal to

> means is greater than < means is less than

Rewrite the following with the symbol or symbols =, 6=, > or < to replace the ¤ to

make a correct statement:

a 7 ¤ 9 b 9 ¤ 7 c 2 + 2 ¤ 4

d 3 ¡ 1 ¤ 9 ¡ 7 e 6 + 1 ¤ 5 f 7 ¡ 3 ¤ 9 ¡ 5

g 16 ¤ 5 h 5 ¤ 16 ¡ 9 i 12 ¤ 24 ¥ 2

j 11 £ 2 ¤ 44 ¥ 2 k 15 ¡ 9 ¤ 2 £ 3 l 7 + 13 ¤ 5 £ 4

m 118 ¡ 17 ¤ 98 + 3 n 7900 ¤ 9700 o 7900 ¤ 7090

p 25 £ 4 ¤ 99 q 99 ¤ 25 £ 4 r 345 678 ¤ 345 687

8 Express the following in simplest form:

a 4 £ 10 + 9

b 7 £ 100 + 4

c 3 £ 100 + 8 £ 10 + 6

d 2 £ 1000 + 6 £ 100 + 3 £ 10 + 4

e 6 £ 10 000 + 5 £ 100 + 8 £ 10 + 3

f 9 £ 10 000 + 3 £ 1000 + 8

g 3 £ 100 + 4 £ 10 000 + 7 £ 10 + 6 £ 1000 + 5

h 2 £ 10 + 9 £ 100 000 + 8 £ 1000 + 3

i 2 £ 100 000 + 3 £ 100 + 7 £ 10 000 + 8

9 Write in expanded form:

a 486 b 340 c 2438 d 4083e 24 569 f 39 804 g 400 308 h 254 372

a Express 5 £ 10 000 + 7 £ 1000 + 9 £ 100 + 4 in simplest form.

b Write 3908 in expanded form.

a 5 £ 10 000 + 7 £ 1000 + 9 £ 100 + 4 = 57 904

b 3908 = 3 £ 1000 + 9 £ 100 + 8

Example 3

DEMO


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10 Write the following in numeral form:

a thirty six b seventy

c thirty d eighteen

e nine hundred f nine thousand

g five hundred and twenty h five hundred and two

i six thousand and fourteen j six thousand four hundred and forty

k fourteen thousand and four l forty thousand and forty

m fifteen thousand eight hundred and sixty nine

n ninety five thousand three hundred and eleven

o seven hundred and eight thousand one hundred and ninety eight.

11 Write the following numbers in words:

a 66 b 660 c 715 d 888e 4389 f 6010 g 90 000 h 38 700i 15 040 j 44 444 k 408 804 l 246 357m 50 500 n 505 000 o 500 500 p 50 050

12 Write the following operations and their answers in numerical form:

a four more than forty b six greater than eleven

c three less than two hundred d eight fewer than eighty

e eighteen fewer than six thousand

f three thousand reduced by two hundred

g an additional fifty to eleven thousand

h 38 more than five hundred and nine thousand

a Write “two thousand seven hundred and four” in numeral form.

b Write the numeral 36 098 in words.

a 2704

b thirty six thousand and ninety eight

Example 4

Neither the Egyptians nor the Romans had a

symbol to represent nothing.

The symbol 0 was called zephirum in Arabic.

Our word zero comes from this.

In the Hindu-Arabic System, the digit for zero

is used as a place holder in numerals.

ZEROE


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For example, in 580 the 0 is a place holder for units to show that the 8 means 8 tens and

there are no single units. Also, in the number 6032 the 0 shows that there are no hundreds.

However, because of the place that the zero takes, the digit to the left of it takes on the value

of ‘thousands’.

In the number 789 place the zero digit

between 7 and 8.

a Write the new number.

b Write the new number in words.

a 7089

b seven thousand and eighty nine

Example 5

With whole numbers thezero is never placed

before any other digit,unless there is a very

special reason.

1 With the number 543:

a i place a zero between the 4 and 3 ii write the new number in words

b i place two zeros between the 5 and 4 ii write the new number in words

c i place a zero between the 5 and 4 and two zeros between the 4 and 3ii write the new number in words

d i place two zeros after the 3 ii write the new number in words

e i place two zeros between the 5 and the 4 and three zeros after the 3ii write the new number in words

f

The rules for operating with zero are:

² Any number + 0 = the same number Example: 3 + 0 = 3

² Any number ¡ 0 = the same number Example: 8 ¡ 0 = 8

² Any number £ 0 = 0 Example: 7 £ 0 = 0

² 0 ¥ any number = 0 Example: 0 ¥ 9 = 0

² Any number ¥ 0 has no answer Example: 5 ¥ 0 has no answer.

EXERCISE 1E

Place four zeros in the number write the five highestnumbers you can make. Start with the highest.

, and by rearranging the digits,


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Often we are not really interested in the exact value

of a number, but rather we want a reasonable esti-

mate of it.

For example, there may be 48 students in the library

or 315 competitors at the athletics carnival or 38 948spectators at the football match. If we are only in-

terested in an approximate number, then 50 students,

300 competitors and 40 000 spectators would be a

good approximation in each of the above examples.

We may round off numbers by making them into, for example, the nearest number of tens.

368 is roughly 37 tens or 370

363 is roughly 36 tens or 360

We say 368 is rounded up to 370 and 363 is rounded down to 360.

ROUNDING NUMBERSF

Rules for rounding off are:

² If the digit after the one being rounded off is less than 5 (i.e., 0, 1, 2, 3 or 4)

we round down.

² If the digit after the one being rounded off is 5 or more (i.e., 5, 6, 7, 8, 9)

we round up.

1 Round off to the nearest 10:

a 23 b 65 c 68 d 97

e 347 f 561 g 409 h 598

i 3015 j 2856 k 3094 l 8885

m 2895 n 9995 o 30 905 p 49 895

EXERCISE 1F

Round off the following to the nearest 10:

a 38 b 483 c 8605

a 38 is approximately 40 fRound up, as 8 is greater than 5g

b 483 is approximately 480 fRound down as 3 is less than 5g

c 8605 is approximately 8610 fRound up, halfway is rounded upg

Example 6DEMO


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a 81 b 671 c 617 d 850

e 349 f 982 g 2111 h 3949

i 999 j 13 484 k 99 199 l 10 074

Example 7


a 89 b 152 c 19 439

a 89 is approximately 100 fRound up as 8 is greater than 5g

b 152 is approximately 200 fRound up for 5 or moreg

c 19 439 is approximately 19 400 fRound down, as 3 is less than 5g

Go first to the digit afterthe one being rounded

off. That is, the first oneto the right.

DEMO


a 834 b 695 c 1089 d 5485

e 7800 f 6500 g 9990 h 9399

i 13 095 j 7543 k 246 088 l 499 359


a 932 b 4500 c 44 482

a 932 is approximately 1000 fRound up as 9 is greater than 5g

b 4500 is approximately 5000 fRound up for 5 or moreg

c 44 482 is approximately 44 000 fRound down, as 4 is less than 5g

Example 8

Round off the following to the nearest 10 000:

a 42 635 b 60 895 c 99 981

a 42 145 is approximately 40 000 fRound down as 2 is less than 5g

b 60 895 is approximately 60 000 fRound down for 0g

c 99 981 is approximately 100 000 fRound up for 5 or moreg

Example 9

DEMO


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4 Round off to the nearest 10 000:

a 18 124 b 47 600 c 54 500 d 75 850

e 89 888 f 52 749 g 90 555 h 99 776

Example 10

Round off the following to the nearest 100 000:

a 124 365 b 350 984 c 547 690

a 124 365 is approximately 100 000 fRound down as 2 is less than 5g

b 350 984 is approximately 400 000 fRound up for 5 or moreg

c 547 690 is approximately 500 000 fRound down as 4 is less than 5g

5 Round off to the nearest 100 000:

a 181 000 b 342 000 c 654 000 d 709 850

e 139 888 f 450 749 g 290 555 h 89 512

6 Round off to the accuracy given:

a $187:45 (to nearest $10)

b $18 745 (to nearest $1000)

c 375 km (to nearest 10 km)

d $785 (to nearest $100)

e the population of a town is 29 295(to nearest one thousand)

f 995 cm (to nearest metre)

g 8945 litres (to nearest kilolitre)

h the cost of a house was $274 950 (to nearest $10 000)

i the number of sheep on a farm is 491 560 (nearest 100 000)

One kilolitre is onethousand litres.

DEMO

RESEARCH ROUNDING AROUND YOU

Check

What to do:

around your home, class and school to find the following and then round off

to the accuracy asked for.

1 To the nearest 10, find the number of

a pieces of cutlery in your home

b i pencils ii exercise books in your classroom

c adults in the school

d vehicles in the school carpark.


a students in the school

b items sold in the canteen each day

c books in the school library

d bricks in a house closest to your school.

A few pages furtheron in this chapter

you will findexamples of how toestimate a number.


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a kilometres your family car has travelled

b kilolitres of water your household used last year

c pages in the Yellow Pages

d digits in all the phone numbers on a typical page in the White Pages.

4 To the nearest 100 000, find

a the size of the crowd of last year’s biggest outdoor event

b the cost of the dearest house in Sunday’s Real Estate pages of the newspaper.

Calculators and computers are part of everyday life. They save lots of time, energy and

money by the speed and accuracy with which they complete different operations.

However, the people operating the

computers and calculators can, and do,

make mistakes when keying in the in-

formation.

It is very important that when we use

calculators we have a strategy for mak-

ing an estimate of what the answer

should be. An estimate is not a guess.

It is a quick and easy approximation

to the correct answer.

By making an estimate we can tell if

our calculated or computed answer is

reasonable.

ESTIMATION AND APPROXIMATIONG

Not likely!

Three doublebeef burgersand fries...

$175 thanks!

ROUNDING TO THE NEAREST 5 CENTS

Because we no longer use 1 cent and 2 cent coins, amounts of money to be paid in cash must

be rounded to the nearest 5 cents. For example, a supermarket bill and the bill for fuel at a

service station must be rounded to the nearest 5 cents.

If the number of cents ends in

² 0 or 5, the amount remains unchanged.

² 1 or 2, the amount is rounded down to 0.

² 3 or 4, the amount is rounded up to 5.

² 6 or 7, the amount is rounded down to 5.

² 8 or 9, the amount is rounded up to 10.


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Round the following amounts to the nearest 5 cents:

a $1:42 b $12:63 c $3:16 d $24:99

a $1:42 would be rounded down to $1:40:

2 is rounded down.

b $12:63 would be rounded up to $12:65.

3 is rounded up to 5.

c $3:16 would be rounded down to $3:15.

6 is rounded down to 5.

d $24:99 would be rounded up to $25:00:

9 is rounded up to 10, so 99 becomes 100 and $24:99 becomes $25:00

1 Round the following amounts to the nearest 5 cents:

a 99 cents b $2:74 c $1:87 d $1:84

e $34:00 f $25:05 g $16:77 h $4:98

i $13:01 j $102:23 k $430:84 l $93:92

2 a Rachel paid cash for her supermarket bill of $84:72. How much did she pay?

b Jason filled his car with petrol and the amount shown at the petrol pump was $31:66.

How much did he pay in cash?

c Nicolas used the special dry-cleaning offer of ‘3 items for $9:99’. How much money

did he pay?

EXERCISE 1G

Example 11

DEMO

For the purposes of estimation, money is rounded to the nearest whole dollar. Amounts

between $1:00 and $1:49 are rounded to one dollar and amounts $1:50 and up to $1:99 are

rounded to $2:00.

3 For the purpose of estimation, round the following to the nearest whole dollar:

a $3:87 b $9:28 c $4:39 d $11:05 e $7:55

f $19:45 g $19:55 h $39:45 i $39:50 j $61:19

Approximate a $4:37 b $16:85 to the nearest dollar.

a $4:37 is rounded down to $4:00 f37 cents is less than 50 centsg

b $16:85 is rounded up to $17:00 f85 cents is greater than 50 centsg

Example 12


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When estimating sums, products, quotients and differ-

ences we usually round the first digit (from the left) and

put zeros in other places.

For example 68 would round to 70

374 would round to 400

5396 would round to 5000

and 43 875 would round to 40 000

4

Estimate the cost of 28 chocolates at $1:95 each.

28 £ 1:95 is approximately 30 £ $2

is approximately $60

Example 13

Rounding to the firstdigit means the same asrounding to one figure.

Icecream $2.10

Crisps $1.05

Cheese snacks $1.30Ice block $0.85

Licorice rope $0.75

Health bar $1.95

Pineapple lumps $1.80

Chocolate bar $1.30

Jubes $1.20

Honeycomb bar $0.95

300mL drink $1.15

Estimate the total cost (by rounding the prices to the nearest dollar) of

a one icecream, a packet of crisps, a health bar and a drink

b 5 licorice ropes, 4 icecreams, 2 honeycomb bars and 4 drinks

c 3 ice blocks, 2 pkts pineapple lumps, 4 chocolate bars and 3 cheese snacks

d 10 health bars, 4 icecreams, 6 jubes and 3 licorice ropes

e 19 ice blocks, 11 drinks, 12 pkts cheese snacks and 9 pkts pineapple lumps

f 21 pkts crisps, 18 choc bars, 28 health bars and 45 drinks

g 4 dozen drinks, half a dozen packets of pineapple lumps and a dozen health bars

h 192 honeycomb bars, 115 icecreams, 189 pkts crisps and 237 drinks

i 225 licorice ropes, 269 drinks, 324 honeycomb bars and 209 ice blocks.


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ACTIVITY SHOPPING AROUND

What to do:

1 Estimate how many of each of the items in the table you can buy for

$20 000.

Use� only the cheapest prices and brand new items.

You� cannot buy a fraction of an item.

2 Write your estimate in the chart.

3 Check your estimate through advertisements in newspapers and catalogues.

4 Complete the following chart.

A B C D E F

Item

Estimated Price from Price Correct number Balance

number catalogue rounded of items bought. or amount

you could or to one Divide $20 000 left from

buy newspaper figure by D $20 000

lap topcomputer

colour printer

109 cm screen

TV

return air faresto Disneyland

wheel size 26mountain bikes

microwaveoven

basketball

5 Write down the names of the items in order from most to least expensive.

6 If you bought one of each of the above items, how much change would you have

from $20 000?

PRINTABLE

TEMPLATE

Estimate the sum 594 + 317 + 83

Round off to the first digit then put zeros in the other places:

594 + 317 + 83 is approximately 600 + 300 + 80which is approximately 980

Example 14


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EXERCISE 1G (continued)

5 Estimate the following:

a 78 + 42 b 478 + 242 c 196 + 324

d 83 + 61 + 59 e 834 + 615 + 592 f 815 ¡ 392

g 3189 + 4901 h 6497 ¡ 2981 i 34 614 ¡ 19 047

j 89 139 ¡ 31 988 k 59 104 + 20 949 l 1489 + 2347 + 6618

Go back over the above exercises and compare your estimates with the exact answers.

6 For 5 a, c, e, g, i and k, show the difference between the estimate and the exact answer.

7 Estimate the following products:

a 19 £ 8 b 31 £ 7 c 28 £ 4 d 52 £ 6

e 87 £ 5 f 92 £ 3 g 39 £ 9 h 88 £ 8

i 54 £ 7 j 36 £ 9 k 94 £ 5 l 67 £ 3

Estimate the difference 2164 ¡ 897

Round off to the first digit then put zeros in the other places:

2164 ¡ 897 is approximately 2000 ¡ 900which is approximately 1100

Example 15

Estimate the product a 39 £ 7 b 891 £ 4

a Round off to the first digit then put

zeros in the other places

39 £ 7 is approximately 40 £ 7which is approximately 280

b Round off to the first figure then

put zeros in other places

891 £ 4 is approximately 900 £ 4which is approximately 3600

Example 16

8 Multiply the following. Use estimation to check that your answers are reasonable:

a 41£ 9

b 78£ 7

c 53£ 4

d 92£ 9

e 27£ 6

f 69£ 8

g 82£ 5

h 38£ 3


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9 Estimate the products:

a 484 £ 3 b 197 £ 9 c 521 £ 6 d 238 £ 8

e 729 £ 8 f 381 £ 4 g 2158 £ 7 h 3948 £ 5

10 Multiply the following. Use estimation to check that your answers are reasonable:

a 214£ 9

b 694£ 3

c 808£ 7

d 376£ 8

e 497£ 6

f 941£ 4

g 522£ 5

h 658£ 7

i 374£ 4

j 783£ 5

k 413£ 9

l 863£ 7

11 Estimate the following products using 1 figure approximations:

a 49 £ 32 b 83 £ 57 c 58 £ 43 d 389 £ 21

e 519 £ 38 f 88 £ 307 g 728 £ 65 h 921 £ 78

i 58 975 £ 8 j 31 942 £ 6 k 6412 £ 37 l 29 £ 7142

Estimate the product 427 £ 89

Round off the first digit then put zeros in

the other places:

427 £ 89 is approximately 400 £ 90 f5 digits in the questiong

is approximately 36 000 f5 digits in the answerg

The estimate tells us that the correct answer should

have 5 digits in it.

Example 17The sum of the

number of zeros isthe number of zeros

which shouldappear in the

product, unless theproduct of the twodigits ends in zero.

If two factors in a product are both“halfway” numbers, a closer approximationis obtained by rounding the smaller number

up and the larger number down.


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By rounding each number off to 1 digit, estimate the following:

a 45 £ 35 b 650 £ 25

a If both are rounded up: 45 £ 35 is approximately 50 £ 40 fRound up for 5gwhich is approximately 2000

The correct answer is 1575.

Notice that 45 £ 35 is approximately 40 £ 40 fBoth end in 5 so

which is approximately 1600 round one up and

which is closer to 1575 than 2000. the other downg

) a closer approximation is found by rounding the smaller one up and the

larger one down.

b 650 £ 25 is approximately 600 £ 30 fBoth end in 5 so

which is approximately 18 000 round one up and

the other downg

12 Estimate the products

a 45 £ 15 b 65 £ 25 c 75 £ 85

d 550 £ 35 e 95 £ 95 f 750 £ 15

g 950 £ 45 h 9500 £ 45 i 2500 £ 85

Find the difference in the estimates in f and h , when both factors are rounded up, and

when one is rounded up and the other rounded down.

Find the approximate value of the quotient of 5968 ¥ 51

5968 ¥ 51 is approximately 6000 ¥ 50

which is approximately 600 ¥ 5

which is approximately 120

13 Use estimation to find which of these calculator answers is

reasonable:

a 126 £ 9 1134 14310 11 344

b 93 £ 28 2804 26 404 2604

c 685 £ 72 49 320 43 920 4392

d 897 ¥ 3 2999 209 299

e 79 £ 196 16 684 15 484 160 484

f 3945 £ 32 120 400 12 040 126 240

g 8151 ¥ 19 3209 429 329

Example 18

Example 19

4

205

divisor

quotient

dividend

When multiplying, thetotal number of digits in

the questions oftenshows how many digitswill be in the answer.


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14 In the following questions, round the given data to one figure to find the approximate

value asked for:

a A large supermarket has 12 rows

of cars in its carpark. If each row

has approximately 50 cars, esti-

mate the total number of cars in

the park.

b A school canteen has 11 shelves

in its fridge. Estimate the num-

ber of drinks in the fridge if there

are approximately 21 drinks on

each shelf.

c Scott reads 19 pages in one hour. At this rate, estimate how long it will take him

to read a 413 page novel.

d Each student is expected to raise approximately $28 in a school’s spellathon. If 397students take part, estimate the amount the school could expect to raise.

e A school trip needs one adult

helper for every 5 students. Ap-

proximately how many adults are

needed if 95 students are going

on the trip?

f Estimate the number of students

in a school if there are 21 classes

with approximately 28 students

in each class.

1 Divide the paper into

equal parts as shown:

2 Count the number of paper clips in one part.

3 Multiply the paper clips in one part by the total number of parts.

Number of paper clips in 1 part £ number of parts = 7 £ 8= 56 paper clips

Estimate: 56 paper clips are lying on the sheet of paper.

Example 20

Estimate the number ofpaper clips on the sheetof paper:



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15 Using the method outlined in Example 20, estimate the number in each of the following:

a b

c d

e f

Over the next few pages there are a number of activities designed to help you understand

place value.

An understanding of values up to the hundred thousand place will make the understanding of

very large whole numbers and very small decimal numbers much easier to understand.

Tiles Gears

Buttons

Wordspresently; but towards noon the raft had been found lodged against the Missouri shore some five or sixmiles below the village and then hope perished; they must be drowned else hunger would have driventhem home by nightfall if not sooner It was believed that the search for the bodies had been a fruitlesseffort merely because the drowning must have occurred in mid-channel since the boys being goodswimmer would otherwise hive escaped to shore This was Wednesday night If the bodies contemnedmissing until Sunday all hope would be given over; and the funerals would be preached on that morningTom shuddered Mrs Harare gave a sobbing goodnight and turned to go Then with a mutual impulse thetwo bereaved women flung themselves into each other’s arms and had a good consoling cry and thenparted Aunt Pole was tender far beyond her wont in her goodnight to Sid and Mary Sid snuffed a bit andMary went off crying with all her heart Aunt Pole knelt down and prayed for Tom so touchingly sokingly and with such measureless love in her words and her trembling that he was weltering in tea againlong she was through He had to keep still long after she went to bed for she kept making broken-heartedfrom time to time tossing and turning over But at last she was still only moaning a little in her sleep Nowthe boy stole out rose gradually by the bedside shaded the candlelight with his hand and stood regardingher His heart was full of pity for her He took out his sycamore scroll and placed it by the candle Butsomething occurred to him and he lingered considering His face lighted with a happy solution of histhought; he put the bark hastily in his pocket then he bent over and kissed the faded lips and straightwaymade his stealthy exit latching the door behind him He threaded his way back to the ferry landing foundnobody at large there and walked boldly on board the boat for he knew she was tenantless except thatthere was a watch man who always turned in and sae t like a graven image He untied the skiffat the sternslip Into it and was soon rowing cautiously up 8 stream When he had pulled a the village startedquartering across and bent himself stoutly to his work He hit the landing on the other side neatly for thiswas a familiar bit of work to him He was moved to capture the skiffarguing that’ it I might be considered

Arrows

PLACE VALUEH

Spheres

PRINTABLE

TEMPLATE


Multi Attribute blocks (MA blocks) are one practical way of showing the value in basesystem.

This diagram of MA blocks represents the number three thousand five hundred and forty nine):

a

(

10

3549

DEMO


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In some older number systems, the order in which

the symbols were written did not change the

value of the number. However, in the Hindu-

Arabic system the order of the digits and the

place in which they are put is very impor-

tant.

Consider what would happen in the above exam-

ple if the digits 3 and 9 changed positions. The

PV blocks would need to be changed as below:

The value of the 9 which represented 9 units has now changed to represent 9 thousand. Each

time a digit moves one place to the left its value increases ten times. Conversely, each time

a digit moves one place to the right its value decreases ten times.

1 What number is represented by the following?

a b

2 Draw representations of the following:

a 3094 b 4186

ACTIVITY NOTATION CARDS

What to do:

1 To make a notation card

draw up a piece of card like

the one given. Name all

the places but leave off the

numbers.

The place or a position ofa digit in a number

determines its value.

EXERCISE 1H

1 cm

1 cm

2 cm

6 cm 6 cm

Thousands Units

Hundreds HundredsTens TensUnits Units


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2 Cut up another piece of card into ten 4-square-cm

squares and write the digits from 0 to 9 on one side.

Shuffle the squares then write the numbers 0 to 9 on the

other side of the squares. This gives you 2 lots of digits

from 0 to 9.

3

2 cm

2 cm

Thousands Units

Hundreds HundredsTens TensUnits Units

4 1 9 2 5 7

Grouping or chunking a smallnumber of digits makes them

easier to say. We often“group” phone numbers.

Ascending meansgoing up.

Place the digits at random on theNotation Card. Practice saying thenumbers by starting from the leftand reading the numbers in groupsof three.

For example, group the numbergiven like this:

(thousand group) (unitgroup).419 257

3 Write in numerals and words the a largest b smallest six digit number you can

make with the digits 0 to 9 (not repeated).

4 In numerals and words, what is the difference between the largest and smallest number

in 3?

5 What is the sum of the largest and smallest numbers in 3?

6 Starting with the smallest number that can

be made with all the digits 0 to 3 using

them once only, list in ascending order all

the numbers that can be formed.

EXERCISE 1H (continued)


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Y:\HAESE\SA_06\SA06_01\034SA601.CDRMon Sep 01 12:55:20 2003



Ten thousands Thousands Hundreds Tens Units

ten thousandcents

one thousandcents

two thousandcents

five thousandcents

one hundredcents

two hundredcents

five hundredcents

ten cents

twenty cents

fifty cents

one cent

two cents

five cents

7 Write the place values for the sum of the following amounts:

a b

c d

e f

On the chart above, if one cent represents the unit, then ten cents represents the tens, dollarrepresents the hundreds, ten dollars represents the thousands and one hundred dollars repre-sents the ten thousands place.

a


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8 What would be the monetary values of

a 8765 cents b 24 075 cents c 56 908 cents

d fifteen thousand four hundred and forty cents

e ninety eight thousand three hundred and seven cents

f the sum of all the money 7 a , b and c?

9

a A B

b A B

c A B

10 Using the abacus

a b

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VIDEO

DEMOIn this simple abacus each column represents place value. Using wholenumbers, the unit is the far right column. The beads in the first examplerepresent which we say as thousand What numbers dothe beads in and represent?

a

.251463 251 463�

a b

ACTIVITY CARDS AND PLACE VALUE

What to do:

This is a card game for 2 to

6 players. Each player has

a place value chart in which

each space is large enough to

comfortably place a standard

sized playing card. The chart

should have the place value

names written on it.

A 8

8

6

6

9

9

7

7

4

4

Hundredthousands

Tenthousands

Thousands Hundreds Tens Units

In the place-value card game for the highest number which hand of each pair of hands“wins”?

,

In words, write the sum of difference between each pair of hands

Find the sum of each column ( and ).

i ii

A B


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From a full pack, remove all the picture cards and the 10’s, leaving the aces as ones. There

are thirty six cards from ones to nines.

The cards are then shuffled and placed face down in a pack.

Taking it in turns each player must first nominate the place-value of the card they are about

to pick up from the pack.

In the example above, the player first nominated the ten thousand space and then picked

up a 9.

The same player’s second turn was to nominate the hundreds place and she picked up an

ace.

Her third pick was for the hundred thousand place and she picked up a 6.

Each turn she had to nominate a place value that she had not used before she picked up a

card.

In this example she finished the game with 697 148.

You could play this game to see who could get either the highest or lowest possible number

with the six cards chosen.

11 Using a calculator, key in the numbers as shown:

a Now subtract 5678. Say the number and write down the

digit which appears in the thousands place. Repeat this

subtraction process 5 more times, that is, say the number

and write the digit.

b What number is left after you have subtracted 6 times?

c What is the highest number you can make with the digits

you have written?

d What is the lowest number you can make with the digits

you have written?

e In the highest number, what digit appears in the thousands

place?

12 a Key in the number 23, then multiply it by 3 and write down your answer. Multiply

your new answer by 3, say the number then write it down. Keep on repeating this

pattern until you have a 4 digit in the hundred thousands place.

b How many times did you multiply by 3?

c What is the number?

d In your answers, how many times did the 7 digit appear in the hundreds place?

EXERCISE 1H (continued)

123609


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A line on which equally spaced points are marked is called a number line.

not a number line correct number line

A number line allows the order and relative positions of numbers to be shown.

The arrow head shows that the line can continue indefinitely.

Many number lines like rulers, tape measures, scales and speedometers have positive integers.

They start from zero.

Some number lines like weather and fridge

thermometers and devices for measuring

depth in submarines and charges in batter-

ies, have positive and negative integers.

Show the numbers 9, 15, 3 and 6 with dots on a number line.

We rearrange 9, 15, 3 and 6 in ascending order i.e., 3, 6, 9, 15:15 ¡ 3 = 12 fcalculate the range of equal spaces needed, highest to lowestg

7 8 9 10 11 12 13 7 8 9 10 11 12 13

order and positions not relative

21 26 23 22 25 24 27

order and relative positions

120 130 140 150 160 170 180

Example 21

0 5

3 6 9 15

10 15 20

This example shows agraph of the set of

numbers 3, 6, 9 and 15.

NUMBER LINEI

KM/H

0

20

40

6080

100120

140

160

180

200

E

F

FUEL

0

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3

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6

7

8

910

1112

1314

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Number lines can also be used to show the four basic operations of adding, subtracting,

multiplying and dividing, with number.

Perform the following operations on a number line:

a 3 + 8 ¡ 6 b 4 £ 3 + 2 c 23 ¥ 5

a 3 + 8 ¡ 6 = 5

b 4 £ 3 + 2 = 14

c 23 ¥ 5

Choose a suitable scale f¥ is opposite of £g ) start from right side.

23 ¥ 5 = 4 with a remainder of 3.

1 Use dots to show the following numbers on a number line:

a 9, 4, 8, 2, 7

b 14, 19, 16, 18, 13

c

multiples of 4 below 40d

70, 30, 60, 90, 40

e 250, 75, 200, 25, 125

f 4000, 3000, 500, 2500, 1500

2 What operations do the following number lines show? Give a final answer.

a b

c d

e f

3 Draw a number line and show the following operations. Give a final answer.

a 9 + 8 ¡ 6 b 2 + 4 + 8 ¡ 2 c 40 + 70 + 90 ¡ 50

d 55 + 60 + 75 ¡ 40 e 3 £ 9 ¡ 8 f 4 £ 6 ¥ 5

EXERCISE 1I

Example 22

0 5 10 15

0 5 10 15

0 3 10 15 20 25

0 5 10 15 20

0 5 10 15 20

0 5 10 15 20

0 100 200 300 400 500 600 700

0 10 20 30 40 50 60 70

0 10 20 30 40 50

DEMO


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Y:\...\SA_06-2\SA06_01\039SA601.CDRFri Sep 19 17:29:33 2003



Number lines can also show order and relative positions for fractions.

For example

and decimals

1

2

3

4

EXERCISE 1J

0 Qt_ Wt_ Et_ Rt_ 1 1\Qt_ 1\Wt_ 1\Et_ 1\Rt_ 2 2\Qt_

1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7

NUMBER PUZZLES (EXTENSION)J

Draw triangles like the one shown. Using eachnumber once only place the numbers to in thesquares so that each side of the triangle adds up to

three

, 2 7

12 13 14a b c

In the eleven squares write the numbers from to sothat every set of three numbers in straight line adds up to

all 1 1118a .

Draw three triangles like the one shown. Using each numberonce only place the numbers to in the triangles so that

each side of the triangle adds up to

, 11 1957 59 63a b c

Draw three shapes as shown. Using each numberonce only place the numbers to in the circlesso that each line leading to the centre adds up to

, 1 10

19 21 25a b c

PRINTABLE

TEMPLATE


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REVIEW SET B CHAPTER 1

REVIEW SET A CHAPTER 1

5 Copy the grid below and then complete these cross numbers.

There is only one way in which all the numbers will fit.

2 digits

eighty six, ninety, twenty five, seventy eight,

forty five, forty one, seventy five, forty two,

forty three, seventy two, eighty five

3 digits

739, 246, 208, 267, 846, 540

4 digits

9306, 9346, 4098, 8914, 2672, 1984, 2635,

89615 digits

fifty six thousand three hundred and eighty four, 53 804, forty four thousand nine

hundred and sixty seven, 36 495.

1 Give the numbers represented by the Roman symbols: a VIII b LIV

2 Write the following numbers in Roman symbols: a 23 b 110

3 Give the number represented by the digit 2 in a 253 b 12 467

4 Express 3 £ 10 000 + 4 £ 100 + 5 £ 10 + 9 in simplest form.

5 Write the smallest whole number you can make with the digits 6, 3, 1, 1, 2.

6 Write fifty three thousand and seventy two in numerical form.

7 Write the operations and answer in numerical form: three hundred and six more

than four thousand and eleven.

8 Round the following:

a 64 762 to the nearest 10 000 b 1976 grams to the nearest kilogram

c $1:98 to the nearest 5 cents

9 Estimate the cost of 31 calculators at $37:85 each.

10 Estimate the difference between 2061 and 477.

11 Write $1620 as cents.

12 Show the first six even numbers as dots on a number line.

13 What operations does the number line show? Give a final answer.

1 Give the numbers represented by the Greek symbols: a b

6

0 5 10 15 20

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REVIEW SET C CHAPTER 1


2 Write the following numbers in Greek symbols: a 23 b 1000

3 Give the number represented by the digit 6 in a 45 362 b 63 549


5 Write 2469 in expanded form.

6 Write the numeral 51 602 in words.

7 Write the operations and answer in numerical form: twenty seven less than two thousand

and three.


a 52 794 to the nearest 1000 b 375 cm to the nearest metre

c $4:92 to the nearest 5 cents

9 Estimate the sum of 69 753 and 4690.

10 Estimate the product of 671 and 49 using 1 figure approximations.

11 Write 6005 cents as dollars.

12 Show the multiples of 3 less than 20 as dots on a number line.

13 Show the operations 4 £ 3 ¡ 8 on a number line. Give a final answer.

1 Give the numbers represented by the Roman symbols: a XIX b XXXV

2 Write the following numbers using Roman symbols: a 11 b 43

3 Give the number represented by the digit 9 in 59 632.


5 a Write the largest whole number possible with the digits 0 2 3 7 9, , , , .

b Write 37 029 in expanded form.

6 Write fifty thousand six hundred and ten in numerical form.

7 Write the operations and answer in numerical form: increase 863 by 794.


a 5607 to the nearest 10 b $634:27 to the nearest 5 cents

9 Estimate the following:

a 6493 + 2172 ¡ 3698 b 450 £ 65 c 23 £ $49:20

10 Show the first three multiples of 5 on a number line.

11 What operations does the number line show? Give a final answer.

12 Show the operations 5 + 5 £ 2 on a number line. Give a final answer.

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Y:\HAESE\SA_06\SA06_01\042SA601.CDRTue Aug 26 11:59:27 2003


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