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MATHEMATICS
for year
6
second edition
Stan Pulgies
Robert Haese
Sandra Haese
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Haese & Harris Publications
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MATHEMATICS FOR YEAR 6
SECOND EDITION
Stan Pulgies
Robert Haese
Sandra Haese
M.Ed., B.Ed., Grad.Dip.T.
B.Sc.
B.Sc.
Haese & Harris Publications
3 Frank Collopy Court, Adelaide Airport SA 5950
Telephone: (08) 8355 9444, Fax: (08) 8355 9471
email: [email protected]
web:
www.haeseandharris.com.au
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
Second Edition
2000
2003
Cartoon artwork by John Martin and Chris Meadows.
Artwork by Piotr Poturaj, Joanna Poturaj and David Purton
Cover design by Piotr Poturaj.
Cover photograph: Western Pygmy Possum © Nicholas Birks, Wildflight Australia Photography
Computer software by David Purton and Eli Sieradzki
Typeset in Australia by Susan Haese (Raksar Nominees). Typeset in Times Roman 10\Qw_ /11\Qw_
This book is copyright. Except as permitted by the Copyright Act (any fair dealing for the
purposes of private study, research, criticism or review), no part of this publication may be
reproduced, 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.
Copying for educational purposes: Where copies of part or the whole of the book are made
under Part VB of the Copyright Act, the law requires that the educational institution or the body
that administers it has given a remuneration notice to Copyright Agency Limited (CAL). For
information, contact the Copyright Agency Limited.
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Acknowledgement: the descriptors listed at the beginning of each chapter are taken from the
R-7 SACSA Mathematics Teaching Resource published by the Department of Education and
Children’s Services.
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FOREWORD
Mathematics for Year 6 (second edition) offers a comprehensive and rigorous course of study at
Year¡¡6 level. Through its worked examples, exercises, activities and answers, and the support of the
interactive Student CD, the book provides students with the structure and content to work efficiently
at their own rate.
The book and CD package is designed to supplement classroom practice and give teachers time to
explore other creative strategies, depending on the needs of their students. It is not the Year 6
Mathematics 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 the
descriptors used in the R-7 SACSA Mathematics Teaching Resource for Middle Years. We have listed
the descriptors in the hope that this will be a helpful guide for Year 6 teachers who may wish to use
the 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, and
Data 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 in
brackets. This is intended as a guide for teachers who may wish to use the first and second editions
within the same classroom. A glance at the contents pages will show how both books correlate (the
absence 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 the
textbook at school and keep the CD at home, to save carrying a heavy textbook to and from school
each day. But more than that, by clicking on the ‘active icons’ within the text, students can access a
range of interactive features: graphing and geometry software, spreadsheets, video clips, computer
demonstrations and simulations, and worksheets.
The CD is ideal for independent study. Students can revisit concepts taught in class and explore new
ideas for themselves. It is fantastic for teachers to use for demonstrations and simulations in the
classroom. In summary, the book offers structure and rigour, and the CD makes maths come alive. We
have endeavoured to provide as broad a base of activity and learning styles as we can, but we also
caution that no single book should be the sole resource for any classroom teacher.
We welcome your feedback.
Email: [email protected]
Web: www.haeseandharris.com.au
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TABLE OF CONTENTS
TABLE OF CONTENTS
1
WHOLE NUMBERS
9
(9)
10
14
15
16
19
21
24
32
38
40
41
41
42
(10)
POINTS, LINES AND CIRCLES
43
(41)
A Points and lines
B Polygons
C Angles
D Perpendicular lines
E Triangles and quadrilaterals
F Angles in triangles
G Angles in quadrilaterals
H Circles
Review Set A (Chapter 2)
Review Set B (Chapter 2)
Review Set C (Chapter 2)
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52
58
59
63
65
67
70
71
72
(42)
73
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90
91
(68)
A
B
C
D
E
F
G
H
I
J
2
NUMBER FACTS
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Addition and subtraction
Multiplication and division by powers of 10
Calculator use
Multiplication
Division
Using number operations (problem solving)
Zero and one
Factors of whole numbers
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B
C
D
E
F
G
H
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Different number systems
How many? The language of number
A numbered world
Our number system
Zero
Rounding numbers
Estimation and approximation
Place value
Number line
Number puzzles (Extension)
Review Set A (Chapter 1)
Review Set B (Chapter 1)
Review Set C (Chapter 1)
25
3
5
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.
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TABLE OF CONTENTS
I Divisibility
J Multiples of whole numbers
K One operation after another
Review Set A (Chapter 3)
Review Set B (Chapter 3)
Review Set C (Chapter 3)
Extension (A different base system)
4
FRACTIONS
A
B
C
D
E
F
G
H
I
J
K
L
M
DECIMALS
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Representing decimals
Using a number line
Showing place value with blocks
Place value
Value of money
Adding decimal numbers
Subtracting decimal numbers
Multiplying & dividing by 10, 100 and 1000
Converting fractions to decimals
Converting decimals to fractions
Percentage
Multiplication with whole numbers
Division by whole numbers
Operating with money
Problem solving
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B
C
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F
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J
K
L
M
N
O
100
5
Fractions
Representation of fractions
Fractions of regular shapes
Fractions of quantities
Finding the whole from a fraction
Ordering of fractions
Equivalent fractions
Fractions to lowest terms
Mixed numbers & improper fractions
Adding fractions
Subtraction of fractions
Miscellaneous problem solving
Ratio
Review Set A (Chapter 4)
Review Set B (Chapter 4)
Review Set C (Chapter 4)
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100
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101
(123)
102
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109
110
111
115
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118
123
125
126
129
130
131
(124)
133
(155)
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165
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TABLE OF CONTENTS
P Rounding decimal numbers
Review Set A (Chapter 5)
Review Set B (Chapter 5)
Review Set C (Chapter 5)
6
MEASUREMENT
A
B
C
D
E
F
G
H
I
J
K
L
M
N
7
Scales
Grids
Maps
Direction
Plans
Coordinates
Locus
Review Set A (Chapter 7)
Review Set B (Chapter 7)
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(106)
(109)
(111)
(115)
(117)
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Polyhedra
Drawing solids
Making solids from nets
Different views of objects
Intersection of solids and planes
Mass
Review Set A (Chapter 8)
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SOLIDS AND MASS
A
B
C
D
E
F
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Units of measure
Converting length units
Perimeter
Perimeters of special figures
Problem solving with perimeters
Comparing area units
Area
Metric area units
Area of a rectangle
Areas of composite shapes
Problem solving with areas
Volume
Capacity
Calculating volumes and capacities
Review Set A (Chapter 6)
Review Set B (Chapter 6)
Review Set C (Chapter 6)
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LOCATION AND POSITION
A
B
C
D
E
F
G
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TABLE OF CONTENTS
Review Set B (Chapter 8)
Review Set C (Chapter 8)
A
B
C
D
E
F
TIME AND TEMPERATURE
A
B
C
D
E
F
G
H
I
PATTERNS AND ALGEBRA
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Number patterns (sequences)
Dot patterns
Matchstick patterns
Rules and problem solving
Graphing patterns
Using word formulae
Converting words to symbols
Using a spreadsheet
Algebraic expressions
Graphing from a rule
Graphs of real world situations
Review Set A (Chapter 11)
Review Set B (Chapter 11)
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Time lines
Units of time
A date with a calendar
Reading clocks and watches
Clockwise direction
Timetables
Speed
Temperature
Maximum, minimum and average temperatures
Review Set A (Chater 10)
Review Set B (Chapter 10)
Review Set C (Chapter 10)
5
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Sampling
Organising data
Displaying data
Using technology to graph data
Interpreting data
Measuring the 'middle' of a data set
Review Set A (Chapter 9)
Review Set B (Chapter 9)
Review Set C (Chapter 9)
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DATA COLLECTION AND REPRESENTATION
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7
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12
TRANSFORMATIONS
A
B
C
D
E
13
CHANCE AND PROBABILITY
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360
361
(312)
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380
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INDEX
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Describing chance
All possible results
Probability
Most likely to least likely order
Statistics and probability
Life insurance tables (Extension)
Review Set A (Chapter 13)
Review Set B (Chapter 13)
Review Set C (Chapter 13)
(311)
ANSWERS
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F
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The language of transformations
Tessellations
Line symmetry
Rotations
Enlargements and reductions
Review Set A (Chapter 12)
Review Set B (Chapter 12)
Review Set C (Chapter 12)
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Chapter
1
Whole
numbers
····································
Knowledge, skills and understandings
In this chapter you will learn how to
!
!
!
!
!
recognise the existence of different number
systems (e.g. Greek, Roman, HinduArabic)
provide examples of the use of number in
everyday life
read, write and record numbers to one
million, using numerals and words
explain place value of digits in numbers to
1¡000¡000
!
!
!
write numbers to 1¡000¡000 in expanded
form
round to the nearest 10, 100, 1000, 10¡000
and 100¡000
place numbers in descending and
ascending order
compare numbers and use symbols
(e.g., = , =
6 , < and >)
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10
WHOLE NUMBERS (CHAPTER 1)
A
DIFFERENT NUMBER SYSTEMS
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).
© and ©
© ©
© was much easier for tallying
A tally of jjjjj eventually was replaced by ©
jjjj
jjjj
jjjj
than jjjjjjjjjj:
To us now, this would not seem to
be a very efficient way for recording larger numbers.
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.
ANCIENT GREEK OR ATTIC SYSTEM
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Like the Mayans, the Ancient Greeks could see the need to include a symbol for 5. Some
examples of Ancient Greek numbers are
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WHOLE NUMBERS (CHAPTER 1)
1
2
3
4
20
30
700
1000
5
50
6
7
8
60
100
9
10
400
500
11
Can you see what
the smaller D, H,
and X symbols do to
the symbol when
they are joined?
5000
This system depends on addition and multiplication.
Example 1
Change the following Ancient Greek numerals into a Hindu-Arabic number:
a
b
a
b
1000
6000
300
700
20
80
+
4
1324
+
1
6781
EXERCISE 1A
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
ROMAN NUMERALS
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.
I
II
III
IIII
but later IV
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12
WHOLE NUMBERS (CHAPTER 1)
The V formed by the thumb and forefinger of
Look for Roman numerals on
clocks and watches, at the end of
movies when the credits are
being shown, on plaques and on
the top of buildings, and as
chapter numbers to novels.
represented 5.
an open hand
Two V’s joined together
of 5, i.e., ten or X.
became two lots
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 thousand or 500.
1
I
2
II
3
III
20
XX
30
XXX
4
IV
40
XL
50
L
5
V
6
VI
7
VII
8
VIII
60
LX
70
LXX
80
LXXX
9
IX
90
XC
10
X
100
C
500
D
1000
M
Unlike some other ancient systems, the Roman system had to be written in order and the
value would change if the order changed.
For example,
whereas
and
whereas
IV stands for 1 before 5, i.e., 4
VI stands for 1 after 5, i.e., 6
XC stands for 10 before 100, i.e., 90
CX stands for 10 after 100, i.e., 110.
There were also rules for the order in which symbols could be used. The I could only appear
before V or X, the X could only appear before L or C, and C could only appear before a D or
an M. For example, the number 1999 could be written as MCMXCIX, but could not be written as MIM.
Larger numerals were formed by placing a stroke above the symbol which made the number
1000 times as large:
5000
V
10 000
X
50 000
L
100 000
C
500 000
D
1 000 000
M
EXERCISE 1A (continued)
3 What numbers are represented by the following symbols:
a XVI
b XXXI
d CXXIV
e MCCLVII
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4 Write the following numbers in Roman numerals:
a 18
b 34
c 279
d 902
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LXXXIII
D L VDCC?
e
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WHOLE NUMBERS (CHAPTER 1)
a Which Roman numeral less than one hundred has the greatest number of symbols?
5
b What is the highest Roman numeral between M and MM which has the least number
of symbols?
Denarii is the unit
c Write the year 1999 using Roman symbols.
of currency used
by the Romans.
6 Use Roman numerals to answer the following
questions.
a Each week Octavius sharpened CCCLIV swords
for the General. How many would he need to
sharpen if the General doubled his order?
b What would it cost Claudius to finish his courtyard if he needed to pay for CL pavers at VIII
denarii each and labour costs of XCIV denarii?
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?
RESEARCH
OTHER WAYS OF COUNTING
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.
1
2
3
4
5
6
7
8
9
0
INVESTIGATION
BIRTHDAYS IN ROMAN
What to do:
Use a calendar to help you.
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1 In Roman numerals write:
a your date of birth, for example,
XXI-XI-MCMXLVI
b what the date will be when you are
iii XXI
i XV
ii L
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IX
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2 Ignoring any leap year what will be the date:
a XIV days after your XVI birthday
b LX days after your XIX birthday?
B HOW MANY? THE LANGUAGE OF NUMBER
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”:
two apples
a couple of people
a brace of ducks
a dyad
a pair of sox
twins
a double
a duet
Three people playing musical instruments or
singing together are a trio. In cricket, a
bowler getting three batsmen out in three successive 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.
ACTIVITY
WORDS FOR NUMBERS
What to do:
cyan
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1 In groups, brainstorm a list of as many words as you can which can mean
four or five. Check your list for the correct spelling and meanings using a
dictionary or thesaurus.
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15
a Use dictionaries and thesauri (more than one thesaurus) to find what number is
represented by the following words:
octave, dozen, duck, decade, score, naught, century, zilch, gross, ton, hexagon,
zero, quadruplet, quartet, hexahedron, pentathlon, nonet, quintet, quinella, heptagon, pentahedron, grand, sextet.
b Write sentences using these words to illustrate numbers.
2
3 Use the ideas from Exercise 1A to write half a dozen questions to challenge your
class mates.
EXERCISE 1B
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 accommodation. 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 Sir Donald Bradman scored 78 tons and 36 double centuries during his first class
cricket career. What was the smallest total
number of runs he may have scored just
in tons and double centuries?
6 The quintet each trebled their $400 investment.
What was their new total amount?
C
A NUMBERED WORLD
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.
cyan
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Your birth certificate will have a number on it. You
have an enrolment number for the school you are attending. Your address, telephone, bus route and shoe
size all have numbers.
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715 896 772 323 260 379 104 694
417 099 915 341 993 944 641 232
CRED
IT C
ARD 558 485
500 973 801 564 442
385
687 594 667 185 788 236 727 251
803 222 418 80354192
37 87246 004 309
55 35
JOH681
55 1600
N CI390
582 751 955 660
441
TIZEN204
/02
123 324 924 871 916 500 86407791
437 256 005 081 072 775 276 510
234 240 758 247 081 714 797 128
020 284 647 317 371 074 422 745
ISBN 1-876543-00-0
256 477 391 628 302 672 851 185
906 854 522 688 389 396 421 819
961 367 677 716 644 657 086 398
346 024 253 141 456 775 211 744
140
278 283 653 127 988 721 310
9 781876 543006 >
997 794 681 582 081 040 199 242
030
819
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127
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408
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994
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ACTIVITY
USING NUMBERS
What to do:
1 In groups, brainstorm as many uses of number as you can for the following categories:
a
b
money
c
travel
information
d
records.
2 Find out how a bar code works and how it is used.
D
OUR NUMBER SYSTEM
From our brief look at the History of Number, you
would remember that the method of writing numbers is called a number system. The system we use
was developed in India, 2000 years ago, and was introduced 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.
ordinal number
one
two
three
four
five
six
seven
eight
nine
1
2
3
4
5
6
7
8
9
Hindu-Arabic
Numeral
our numeral
The digits 3 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.
The numbers we use for counting
are called natural numbers. The
possible combination of natural
numbers is endless. There is no
largest number. We say that the
set of all natural numbers is infinite. Natural numbers are also
called counting numbers.
There are three kinds of
people in the world.
Those who can count
and those who cannot.
Most of the time we
are not worried
about the difference
between ‘numeral’
and ‘number’.
We use the word
number most of the
time.
cyan
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If we include in our set of numbers the number zero, 0, then our
set now has a new name, the set
of whole numbers.
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17
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.
un
5
2
6
7
9
4
its
hu
thondre
us d
ten ands
th o
us
an
ds
th o
us
an
ds
hu
nd
red
s
ten
s
Each digit in a number has a place value. For example: in 567 942
Example 2
What number is represented by the digit 9 in the numeral 1972?
nine hundred or 900
EXERCISE 1D
1 What number is represented by the digit 7 in the following?
a 27
b 74
c 567
e 1971
f 7635
g 3751
i 80 007
j 72 024
k 87 894
d
h
l
758
27 906
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
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?
4
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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
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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:
=
>
6=
<
means is equal to
means is greater than
means is not equal to
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
Example 3
a
b
Express 5 £ 10 000 + 7 £ 1000 + 9 £ 100 + 4 in simplest form.
Write 3908 in expanded form.
a
5 £ 10 000 + 7 £ 1000 + 9 £ 100 + 4 = 57 904
b
3908 = 3 £ 1000 + 9 £ 100 + 8
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
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9 Write in expanded form:
a 486
b
e 24 569
f
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19
Example 4
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
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
a
e
i
m
the following numbers in words:
66
b 660
4389
f 6010
15 040
j 44 444
50 500
n 505 000
c
g
k
o
715
90 000
408 804
500 500
d
h
l
p
888
38 700
246 357
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
E
ZERO
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.
cyan
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In the Hindu-Arabic System, the digit for zero
is used as a place holder in numerals.
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WHOLE NUMBERS (CHAPTER 1)
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’.
Example 5
In the number 789 place the zero digit
between 7 and 8.
a Write the new number.
b Write the new number in words.
With whole numbers the
zero is never placed
before any other digit,
unless there is a very
special reason.
a 7089
b seven thousand and eighty nine
EXERCISE 1E
1 With the number 543:
a i place a zero between the 4 and 3
ii
write the new number in words
ii
write the new number in words
b
i
place two zeros between the 5 and 4
c
i
ii
place a zero between the 5 and 4 and two zeros between the 4 and 3
write the new number in words
d
i
place two zeros after the 3
e
i
ii
place two zeros between the 5 and the 4 and three zeros after the 3
write the new number in words
f
ii
write the new number in words
Place four zeros in the number, and by rearranging the digits, write the five highest
numbers you can make. Start with the highest.
cyan
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Example:
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² 0 ¥ any number = 0
0
7£0=0
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Example:
95
² Any number £ 0 = 0
100
8¡0=8
50
Example:
75
² Any number ¡ 0 = the same number
25
3+0=3
0
Example:
5
95
² Any number + 0 = the same number
100
50
75
25
0
5
The rules for operating with zero are:
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F
21
ROUNDING NUMBERS
Often we are not really interested in the exact value
of a number, but rather we want a reasonable estimate of it.
For example, there may be 48 students in the library
or 315 competitors at the athletics carnival or 38 948
spectators at the football match. If we are only interested 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.
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.
Example 6
DEMO
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
EXERCISE 1F
cyan
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1 Round off to the nearest 10:
a
b
23
e 347
f
i
3015
j
m 2895
n
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3094
30 905
d
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598
8885
49 895
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Example 7
DEMO
Round off the following to the nearest 100:
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 after
the one being rounded
off. That is, the first one
to the right.
2 Round off to the nearest 100:
a 81
b 671
e 349
f 982
i 999
j 13 484
c
g
k
617
2111
99 199
d
h
l
850
3949
10 074
Example 8
DEMO
Round off the following to the nearest 1000:
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
3 Round off to the nearest 1000:
a 834
b 695
e 7800
f 6500
i 13 095
j 7543
c
g
k
1089
9990
246 088
d
h
l
5485
9399
499 359
Example 9
cyan
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fRound up for 5 or moreg
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99 981 is approximately 100 000
0
c
5
fRound down for 0g
95
60 895 is approximately 60 000
100
b
50
fRound down as 2 is less than 5g
75
42 145 is approximately 40 000
25
0
a
5
95
100
50
75
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5
Round off the following to the nearest 10 000:
a 42 635
b 60 895
c 99 981
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4 Round off to the nearest 10 000:
18 124
89 888
a
e
47 600
52 749
b
f
c
g
54 500
90 555
Example 10
d
h
75 850
99 776
DEMO
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
e 139 888
f 450 749
c
g
654 000
290 555
d
h
709 850
89 512
6 Round off to the accuracy given:
a $187:45 (to nearest $10)
One kilolitre is one
thousand litres.
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)
RESEARCH
ROUNDING AROUND YOU
What to do:
Check 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 few pages further
on in this chapter
you will find
examples of how to
estimate a number.
cyan
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2 To the nearest 100, find the number of
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.
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3 To the nearest 1000, find the number of
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.
G
ESTIMATION AND APPROXIMATION
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.
Three double
beef burgers
and fries...
$175 thanks!
However, the people operating the
computers and calculators can, and do,
make mistakes when keying in the information.
It is very important that when we use
calculators we have a strategy for making 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.
Not likely!
By making an estimate we can tell if
our calculated or computed answer is
reasonable.
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.
cyan
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amount remains unchanged.
amount is rounded down to 0.
amount is rounded up to 5.
amount is rounded down to 5.
amount is rounded up to 10.
5
the
the
the
the
the
95
5,
2,
4,
7,
9,
100
or
or
or
or
or
75
0
1
3
6
8
0
95
100
50
75
25
0
5
95
100
50
75
25
0
5
²
²
²
²
²
5
If the number of cents ends in
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25
Example 11
Round the following amounts to the nearest 5 cents:
a $1:42
b $12:63
c $3:16
a
d $24:99
$1:42 would be rounded down to $1:40:
2 is rounded down.
b
$12:63 would be rounded up to $12:65.
DEMO
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
EXERCISE 1G
1 Round the following amounts to the nearest 5 cents:
a 99 cents
b $2:74
c $1:87
e $34:00
f $25:05
g $16:77
i $13:01
j $102:23
k $430:84
2
d
h
l
$1:84
$4:98
$93:92
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?
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.
Example 12
a $4:37
Approximate
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
cyan
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Tue Aug 26 11:50:03 2003
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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
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26
WHOLE NUMBERS (CHAPTER 1)
When estimating sums, products, quotients and differences we usually round the first digit (from the left) and
put zeros in other places.
For example
Rounding to the first
digit means the same as
rounding to one figure.
68 would round to 70
374 would round to 400
5396 would round to 5000
and
43 875 would round to 40 000
Example 13
Estimate the cost of 28 chocolates at $1:95 each.
28 £ 1:95 is approximately 30 £ $2
is approximately $60
4
Ice block $0.85
Cheese snacks $1.30
300mL drink $1.15
Crisps $1.05
Pineapple lumps $1.80
Licorice rope $0.75
Icecream $2.10
Jubes $1.20
Honeycomb bar $0.95
Health bar $1.95
Chocolate bar $1.30
cyan
Y:\...\SA_06-2\SA06_01\026SA601.CDR
Mon Sep 29 13:52:30 2003
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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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WHOLE NUMBERS (CHAPTER 1)
ACTIVITY
27
SHOPPING AROUND
What to do:
1
Estimate how many of each of the items in the table you can buy for
$20 000.
PRINTABLE
· Use only the cheapest prices and brand new items.
TEMPLATE
· You cannot buy a fraction of an item.
Write your estimate in the chart.
Check your estimate through advertisements in newspapers and catalogues.
Complete the following chart.
2
3
4
A
B
Estimated
number
you could
buy
Item
C
Price from
catalogue
or
newspaper
D
Price
rounded
to one
figure
E
Correct number
of items bought.
Divide $20 000
by D
F
Balance
or amount
left from
$20 000
lap top
computer
colour printer
109 cm screen
TV
return air fares
to Disneyland
wheel size 26
mountain bikes
microwave
oven
basketball
5
6
Write down the names of the items in order from most to least expensive.
If you bought one of each of the above items, how much change would you have
from $20 000?
Example 14
Estimate the sum 594 + 317 + 83
cyan
Y:\HAESE\SA_06\SA06_01\027SA601.CDR
Mon Aug 25 11:26:21 2003
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Round off to the first digit then put zeros in the other places:
594 + 317 + 83 is approximately 600 + 300 + 80
which is approximately 980
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WHOLE NUMBERS (CHAPTER 1)
Example 15
Estimate the difference 2164 ¡ 897
Round off to the first digit then put zeros in the other places:
2164 ¡ 897 is approximately 2000 ¡ 900
which is approximately 1100
EXERCISE 1G (continued)
5 Estimate the following:
a 78 + 42
d 83 + 61 + 59
g 3189 + 4901
j 89 139 ¡ 31 988
478 + 242
834 + 615 + 592
6497 ¡ 2981
59 104 + 20 949
b
e
h
k
c
f
i
l
196 + 324
815 ¡ 392
34 614 ¡ 19 047
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.
Example 16
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 £ 7
which is approximately 280
b
Round off to the first figure then
put zeros in other places
891 £ 4 is approximately
which
is approximately
7 Estimate the following products:
a 19 £ 8
b 31 £ 7
e 87 £ 5
f 92 £ 3
i 54 £ 7
j 36 £ 9
c
g
k
900 £ 4
3600
28 £ 4
39 £ 9
94 £ 5
d
h
l
52 £ 6
88 £ 8
67 £ 3
cyan
Y:\HAESE\SA_06\SA06_01\028SA601.CDR
Fri Aug 29 16:25:42 2003
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£ 3
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h
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82
£ 5
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g
50
69
£ 8
75
f
25
27
£ 6
0
e
5
92
£ 9
95
d
100
53
£ 4
50
c
75
78
£ 7
25
b
0
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£ 9
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8 Multiply the following. Use estimation to check that your answers are reasonable:
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WHOLE NUMBERS (CHAPTER 1)
9 Estimate the products:
a 484 £ 3
e 729 £ 8
197 £ 9
381 £ 4
b
f
c
g
521 £ 6
2158 £ 7
d
h
29
238 £ 8
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
Example 17
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.
The sum of the
number of zeros is
the number of zeros
which should
appear in the
product, unless the
product of the two
digits ends in zero.
11 Estimate the following products using 1 figure approximations:
a 49 £ 32
b 83 £ 57
c 58 £ 43
e 519 £ 38
f 88 £ 307
g 728 £ 65
i 58 975 £ 8
j 31 942 £ 6
k 6412 £ 37
d
h
l
389 £ 21
921 £ 78
29 £ 7142
cyan
Y:\HAESE\SA_06\SA06_01\029SA601.CDR
Mon Aug 25 09:19:53 2003
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If two factors in a product are both
“halfway” numbers, a closer approximation
is obtained by rounding the smaller number
up and the larger number down.
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30
WHOLE NUMBERS (CHAPTER 1)
Example 18
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 5g
which 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.
fBoth end in 5 so
round one up and
the other downg
650 £ 25 is approximately 600 £ 30
which is approximately 18 000
b
12 Estimate the products
a 45 £ 15
d 550 £ 35
g 950 £ 45
65 £ 25
95 £ 95
9500 £ 45
b
e
h
75 £ 85
750 £ 15
2500 £ 85
c
f
i
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.
Example 19
Find the approximate value of the quotient of 5968 ¥ 51
4
20
5
5968 ¥ 51 is approximately 6000 ¥ 50
which is approximately 600 ¥ 5
which is approximately 120
quotient
dividend
divisor
79 £ 196
16 684
15 484
160 484
f
3945 £ 32
120 400
12 040
126 240
g
8151 ¥ 19
3209
429
329
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Fri Sep 19 17:28:03 2003
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897 ¥ 3
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d
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4392
95
43 920
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49 320
50
685 £ 72
75
c
25
2604
0
26 404
5
2804
95
93 £ 28
100
b
50
11 344
75
14310
25
1134
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126 £ 9
5
95
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13 Use estimation to find which of these calculator answers is
reasonable:
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WHOLE NUMBERS (CHAPTER 1)
31
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, estimate the total number of cars in
the park.
b A school canteen has 11 shelves
in its fridge. Estimate the number 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 397
students take part, estimate the amount the school could expect to raise.
e A school trip needs one adult
helper for every 5 students. Approximately 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.
Example 20
Estimate the number of
paper clips on the sheet
of paper:
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.
cyan
Y:\...\SA_06-2\SA06_01\031SA601.CDR
Wed Sep 17 16:36:17 2003
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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.
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WHOLE NUMBERS (CHAPTER 1)
15 Using the method outlined in Example 20, estimate the number in each of the following:
a Buttons
b Arrows
Words
c
d
Spheres
f
Gears
presently; but towards noon the raft had been found lodged against the Missouri shore some five or six
miles below the village and then hope perished; they must be drowned else hunger would have driven
them home by nightfall if not sooner It was believed that the search for the bodies had been a fruitless
effort merely because the drowning must have occurred in mid-channel since the boys being good
swimmer would otherwise hive escaped to shore This was Wednesday night If the bodies contemned
missing until Sunday all hope would be given over; and the funerals would be preached on that morning
Tom shuddered Mrs Harare gave a sobbing goodnight and turned to go Then with a mutual impulse the
two bereaved women flung themselves into each other’s arms and had a good consoling cry and then
parted Aunt Pole was tender far beyond her wont in her goodnight to Sid and Mary Sid snuffed a bit and
Mary went off crying with all her heart Aunt Pole knelt down and prayed for Tom so touchingly so
kingly and with such measureless love in her words and her trembling that he was weltering in tea again
long she was through He had to keep still long after she went to bed for she kept making broken-hearted
from time to time tossing and turning over But at last she was still only moaning a little in her sleep Now
the boy stole out rose gradually by the bedside shaded the candlelight with his hand and stood regarding
her His heart was full of pity for her He took out his sycamore scroll and placed it by the candle But
something occurred to him and he lingered considering His face lighted with a happy solution of his
thought; he put the bark hastily in his pocket then he bent over and kissed the faded lips and straightway
made his stealthy exit latching the door behind him He threaded his way back to the ferry landing found
nobody at large there and walked boldly on board the boat for he knew she was tenantless except that
there was a watch man who always turned in and sae t like a graven image He untied the skiffat the stern
slip Into it and was soon rowing cautiously up 8 stream When he had pulled a the village started
quartering across and bent himself stoutly to his work He hit the landing on the other side neatly for this
was a familiar bit of work to him He was moved to capture the skiffarguing that’ it I might be considered
e
Tiles
DEMO
PRINTABLE
TEMPLATE
H
PLACE VALUE
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.
Multi Attribute blocks (MA blocks) are one practical way of showing the value in a base 10
system.
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Mon Sep 29 12:20:24 2003
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This diagram of MA blocks represents the number three thousand five hundred and forty nine
(3549):
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WHOLE NUMBERS (CHAPTER 1)
In some older number systems, the order in which
the symbols were written did not change the
value of the number. However, in the HinduArabic system the order of the digits and the
place in which they are put is very important.
33
The place or a position of
a digit in a number
determines its value.
Consider what would happen in the above example 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.
EXERCISE 1H
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 cm
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.
1 cm
Thousands
Hundreds
Tens
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Y:\HAESE\SA_06\SA06_01\033SA601.CDR
Fri Aug 29 16:27:30 2003
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Units
Hundreds
Tens
Units
2 cm
6 cm
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34
WHOLE NUMBERS (CHAPTER 1)
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 Place the digits at random on the
Notation Card. Practice saying the
numbers by starting from the left
and reading the numbers in groups
of three.
2 cm
2 cm
Thousands
Hundreds
Tens
Units
Units
Hundreds
Tens
Units
4 1 9 2 5 7
For example, group the number
given like this:
419 (thousand group) 257 (unit
group).
Grouping or chunking a small
number of digits makes them
easier to say. We often
“group” phone numbers.
EXERCISE 1H (continued)
3 Write in numerals and words the a largest
make with the digits 0 to 9 (not repeated).
b smallest six digit number you can
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?
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Ascending means
going up.
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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.
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WHOLE NUMBERS (CHAPTER 1)
Ten thousands
Thousands
Hundreds
Tens
Units
ten thousand
cents
one thousand
cents
one hundred
cents
ten cents
one cent
two thousand
cents
two hundred
cents
twenty cents
two cents
five thousand
cents
five hundred
cents
fifty cents
five cents
35
On the chart above, if one cent represents the unit, then ten cents represents the tens, a dollar
represents the hundreds, ten dollars represents the thousands and one hundred dollars represents the ten thousands place.
cyan
Y:\...\SA_06-2\SA06_01\035SA601.CDR
Wed Sep 10 09:19:50 2003
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7 Write the place values for the sum of the following amounts:
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36
WHOLE NUMBERS (CHAPTER 1)
8 What would be the monetary values of
a 8765 cents
b 24 075 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 ?
56 908 cents
c
9 In the place-value card game for the highest number, which hand of each pair of hands
“wins”?
6
ªª
ªªª
ªª
6
9
§ §
§ §
§ §
6
8
9
A
9
A
6
6
i sum of
In words, write the
B
7
ªª
ªª
ªª
9
§ §
§ §
§§§
§ §
§ §
§
§ §
§ §
§
§
A
9
9
7
6
8
9
7
2
§ §
§
§ §
§ §
9
7
7
ªª
ªªª
ªª
A
9
9ª ª
B
A
4
6
§ §
§ §
§§§
§ §
§ §
§
§ §
7
9
7
7
5
7
7
8
§§§
§ §
§ §
9
9
6
§
8
A
A
2
c
ª
9§ §
4
5
A
ªª
ª
ªª
4
A
B
5
ª
9
b
2
8
ªª
ªªª
ªª
8
9ª ª
4
5
A
2
a
ii difference between each pair of hands
Find the sum of each column (A and B).
10 Using the abacus
VIDEO
DEMO
In this simple abacus each column represents a place value. Using whole
numbers, the unit is the far right column. The beads in the first example
represent 251¡463 which we say as 251 thousand 463. What numbers do
the beads in a and b represent?
a
b
ACTIVITY
CARDS AND PLACE VALUE
What to do:
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Ten
Hundred
Thousands Hundreds
thousands thousands
25
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0
A
9
5
7
4
8
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Wed Sep 17 13:50:59 2003
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Tens
8
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.
Units
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WHOLE NUMBERS (CHAPTER 1)
37
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.
EXERCISE 1H (continued)
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?
cyan
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Fri Aug 29 16:28:08 2003
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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?
25
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123609
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WHOLE NUMBERS (CHAPTER 1)
I
NUMBER LINE
A line on which equally spaced points are marked is called a number line.
7 8
9 10
11 12
13
7
8
not a number line
9
10
11
12
13
correct number line
A number line allows the order and relative positions of numbers to be shown.
21
26
23
22
25
24
27
120 130 140 150 160 170 180
order and positions not relative
order and relative positions
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.
F
0
1
2
3
4
5
6
E
7
8
9
10
60
FUEL
11
12
40
20
13
14
80 100 120
140
160
KM/H
180
0
15
200
16
Some number lines like weather and fridge
thermometers and devices for measuring
depth in submarines and charges in batteries, have positive and negative integers.
This example shows a
graph of the set of
numbers 3, 6, 9 and 15.
Example 21
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
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Wed Sep 17 16:47:00 2003
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WHOLE NUMBERS (CHAPTER 1)
39
Number lines can also be used to show the four basic operations of adding, subtracting,
multiplying and dividing, with number.
Example 22
DEMO
Perform the following operations on a number line:
a
b 4£3+2
3+8¡6
3+8¡6 =5
a
4 £ 3 + 2 = 14
b
c
0
5
10
15
0
5
10
15
23 ¥ 5
Choose a suitable scale f¥ is opposite of £g
c
0
3
23 ¥ 5
10
) start from right side.
15
20
25
23 ¥ 5 = 4 with a remainder of 3.
EXERCISE 1I
1 Use
a
b
c
d
e
f
dots to show the following numbers on a number line:
9, 4, 8, 2, 7
14, 19, 16, 18, 13
70, 30, 60, 90, 40
multiples of 4 below 40
250, 75, 200, 25, 125
4000, 3000, 500, 2500, 1500
2 What operations do the following number lines show? Give a final answer.
a
b
0
5
10
15
20
0
c
5
10
15
20
d
0
10
20
30
40
50
0
e
100
200
300
400
500
600
700
f
0
10
20
30
40
50
60
70
0
5
10
15
20
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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
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WHOLE NUMBERS (CHAPTER 1)
Number lines can also show order and relative positions for fractions.
For example
0
Qt_
Wt_
Et_
Rt_
1
1\Qt_
1\Wt_
1\Et_
1\Rt_
2
2.5
2.6
2\Qt_
and decimals
1.7
1.8
1.9
J
2.0
2.1
2.2
2.3
2.4
2.7
NUMBER PUZZLES (EXTENSION)
EXERCISE 1J
1 In the eleven squares write all the numbers from 1 to 11 so
that every set of three numbers in a straight line adds up to 18.
2
Draw three triangles like the one shown. Using each
number once only, place the numbers 2 to 7 in the
squares so that each side of the triangle adds up to
12
a
b
13
c 14
3 Draw three triangles like the one shown. Using each number
once only, place the numbers 11 to 19 in the triangles so that
each side of the triangle adds up to a 57
b 59 c 63
4
Draw three shapes as shown. Using each number
once only, place the numbers 1 to 10 in the circles
so that each line leading to the centre adds up to
19
a
b 21
c 25
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TEMPLATE
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WHOLE NUMBERS (CHAPTER 1)
41
PRINTABLE
TEMPLATE
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
6
3 digits
739, 246, 208, 267, 846, 540
4 digits
9306, 9346, 4098, 8914, 2672, 1984, 2635,
8961
5 digits
fifty six thousand three hundred and eighty four, 53 804, forty four thousand nine
hundred and sixty seven, 36 495.
REVIEW SET A
CHAPTER 1
1 Give the numbers represented by the Roman symbols:
a 23
2 Write the following numbers in Roman symbols:
3 Give the number represented by the digit 2 in
a VIII
a 253
b LIV
b 110
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
c $1:98 to the nearest 5 cents
b
1976 grams to the nearest kilogram
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.
0
5
10
15
REVIEW SET B
CHAPTER 1
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1 Give the numbers represented by the Greek symbols:
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WHOLE NUMBERS (CHAPTER 1)
a 23
2 Write the following numbers in Greek symbols:
3 Give the number represented by the digit 6 in
b 1000
a 45 362
b 63 549
4 Express 5 £ 10 000 + 2 £ 1000 + 3 £ 10 + 7 in simplest form.
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.
8 Round the following:
a 52 794 to the nearest 1000
c $4:92 to the nearest 5 cents
b
375 cm to the nearest metre
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.
REVIEW SET C
CHAPTER 1
1 Give the numbers represented by the Roman symbols:
a XIX
a 11
2 Write the following numbers using Roman symbols:
b XXXV
b 43
3 Give the number represented by the digit 9 in 59 632.
4 Express 9 £ 10 000 + 5 £ 1000 + 4 £ 100 + 6 in simplest form.
a Write the largest whole number possible with the digits 0, 2, 3, 7, 9.
b Write 37 029 in expanded form.
5
6 Write fifty thousand six hundred and ten in numerical form.
7 Write the operations and answer in numerical form: increase 863 by 794.
8 Round the following:
a 5607 to the nearest 10
b
9 Estimate the following:
a 6493 + 2172 ¡ 3698
$634:27 to the nearest 5 cents
450 £ 65
b
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.
0
5
10
15
20
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12 Show the operations 5 + 5 £ 2 on a number line. Give a final answer.
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