Mathematics with Understanding

253 Pages·1972·9.163 MB·English

by  Harold Fletcher, Arnold A. Howell and C. Plumpton (Auth.)

MATHEMATICS
WITH
UNDERSTANDING
BOOK 2
HAROLD FLETCHER AND ARNOLD A. HOWELL
PERGAMON PRESS
Oxford · New York · Toronto · Sydney
Braunschweig
Pergamon Press Ltd., Headington Hill Hall, Oxford
Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford,
New York 10523
Pergamon of Canada Ltd., 207 Queen's Quay West, Toronto 1
Pergamon Press (Aust.) Pty. Ltd., 19a Boundary Street,
Rushcutters Bay, N.S.W. 2011, Australia
Vieweg & Sohn GmbH, Burgplatz 1, Braunschweig
Copyright © 1972 H. Fletcher and A. A. Howell
All Rights Reserved. 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
Pergamon Press Ltd.
First edition 1972
Library of Congress Catalog Card No. 78-111361
Printed in Hungary
This book is sold subject to the condition
that it shall not, by way of trade, be lent,
resold, hired out, or otherwise disposed
of without the publisher's consent,
in any form of binding or cover
other than that in which
it is published.
08 016745 4 (hard cover)
PREFACE
IN RECENT years many changes have taken place in the content of and
the approach to the teaching of mathematics in primary schools. As a
result of this it was felt necessary to provide a guide to the new ideas
in primary mathematics for students in colleges of education. It is
hoped that qualified teachers will also find it helpful and thought-
provoking. Thus whenever the words teacher or student are used, note
that they refer to teachers of mathematics in primary schools or
students in training to be such teachers. For ease of presentation the
subject-matter has been divided into two books. It is essential that
students and teachers integrate the appropriate sections of both books.
We have attempted to present the contents in such a way that will
encourage careful analysis of known techniques and constructive
discussion between tutor and student or teacher and child as well as
increasing the reader's knowledge of the new developments in mathe­
matics.
Set language is used throughout the books. This is not simply to be
modern but because of a sincere belief that this is a natural language
for children and that it helps them and teachers alike to a deeper
understanding of the structure behind many mathematical ideas and
processes.
It was very sad that Mr. Fletcher died suddenly shortly after this
second book had gone to press. He had made a profound influence
on the teaching of mathematics and will be greatly missed.
CHAPTER 1
NUMBER SYSTEMS
Introduction
In Book 1 the cardinal number of a set was considered in detail.
It will be remembered that cardinal number is an abstract property of
a set. It is possible to handle three cups, three beads, three coins but
it is not possible to handle three. All sets which contain three objects
can be matched with each other as in Fig. 1.1, and thus the sets have
a property in common :
FIG. 1.1.
In order to communicate this property we often wish to make some
record. In this case the numeral 3 is recorded to signify that the
sets contain three members. Various ways of recording were also
dealt with in detail in Book 1. Sets which contain respectively one,
two, three, four, etc., objects can be put in order as in Fig. 1.2. It will
be noticed that as the sets are now matched in each case there
1
2 MATHEMATICS WITH UNDERSTANDING
is one member not matched from the previous set. The numerals
associated with these sets are, of course, 1, 2, 3, 4. The idea of the
ordinal aspect of number is thus shown. When sets are matched in
FIG. 1.2.
this way the set of numbers obtained is called the set of natural
numbers. The set of natural numbers is an infinite set, i.e. there is not
a greatest member. This fact can be recorded as
{1,2,3,4,5,6,7,8,9,10,11, ...}.
The dots are important. They indicate that not all the members have
been recorded.
In mathematics it is often helpful to illustrate ideas in either arith­
metic or algebra by means of a geometrical picture. The set of natural
numbers can be illustrated in this way by points on a number line
as in Fig. 1.3.
5 12 3 4 5 6
FIG. 1.3.
The numeral 1 has been placed one step from the starting point;
the numeral 2 two steps from the start and so on. The starting point
has been labelled S rather than 0 at this stage since zero is not a
natural number.
NUMBER SYSTEMS 3
Binary Operations
When a set containing three toy cars is united with a set of four
toy boats there are seven toys in the union set. We say that the numbers
3 and 4 have been added together. Addition is called a binary opera­
tion. The word binary is used because two sets have been united or
two numbers added together and the word operation is used to denote
that something has been done to the sets or to the numbers.
Figure 1.4 illustrates the operation.
FIG. 1.4.
Note that in this example it is not possible for a car to be a boat
and so the sets A and B have no members in common. They are
disjoint sets. When two disjoint sets are united the cardinal number of
the union is the sum of the cardinal numbers of the sets. This is written
as
n(A)+n(B) = n(AÖB).
For the sets in Fig. 1.4 we can record in several ways, thus:
(3,4)-^U 7
4 MATHEMATICS WITH UNDERSTANDING
(3,4)
or
3+4 = 7.
or
In words the first two would probably be spoken as, "three and four
under addition maps onto seven", whilst the third could be spoken
as "three plus four is a way of writing seven". This latter emphasizes
that 3, 4, and 7 strictly are not numbers but numerals, i.e. ways of
recording numbers. At a later stage 3 + 4 = 7 would be read as three
plus four equals seven. The fact could also be illustrated on a number
line as shown in Fig. 1.5.
J_
3 4 5
FIG. 1.5.
It is clear that 4 + 3 is also a way of writing 7, and this can be illustrated
as in Fig. 1.6.
Clearly Figs. 1.5 and 1.6 are different illustrations showing that 3 + 4
and 4 + 3 are different situations. Both, however, are ways of writing 7.
Tn general, if a and b are natural numbers it is always true that
a+b = b + a.
This is an important property in the structure of the natural numbers.
It is called the COMMUTATIVE PROPERTY OF ADDITION. We say that the
binary operation addition is commutative.
Q. 1. Is subtraction commutative?
A. 1. No. A single example will serve to prove this, for 5 - 3 is not a way of writing
3-5, i.e. 5-3 7± 3-5. (The sign ^ means "is not equal to.")
NUMBER SYSTEMS 5
Q. 2. Are the operations of multiplication and division commutative?
A. 2. Multiplication is commutative but division is not commutative
(e.g. 3X4 = 12 = 4x3,8^2 * 2^-8)
Note. Only a single example is needed to show that division is not commutative.
The single example 3x4 = 4x3 = 12, however, does not prove that multiplication
is commutative. It merely illustrates it.
One other important aspect of a binary operation is that the result
must be unique. The ordered pair (a, b) under a given operation must
map onto only one image.
Closure
Another important idea concerning the structure of the natural
numbers is that of closure.
If a and b are members of the set of natural numbers, is [a 4- b]
always a member?
The answer is clearly yes.
This property is called "closure". We may look at this in a slightly
different way.
Suppose we have the open sentence
a+b = D
where the universal set is the set of natural numbers and a and b are
both natural numbers. Is it always true that the truth set contains
exactly one member? Here again the answer to this question is
clearly in the affirmative. The following example illustrates.
Example
Universal set: The set of natural numbers.
Open sentence: 5 + 6 = D.
Truth set: {11}.
6 MATHEMATICS WITH UNDERSTANDING
Clearly, no matter what natural numbers a and b represent, the
truth set will always contain just one member. Briefly, closure simply
means that a solution can be found within the universal set. The
following examples will help to reinforce the idea of closure.
Q. 1. The universal set is the set of odd natural numbers. The binary operation is
addition. Is closure satisfied?
2. The universal set is the set of even natural numbers and the binary opera­
tion is addition. Is closure satisfied?
3. The universal set is the set of odd natural numbers and the binary operation
is multiplication. Is closure satisfied?
4. The universal set is the set of even natural numbers and the operation is
multiplication. Is closure satisfied?
5. The universal set is {1,4,7,10,13, ...}, i.e. is the set of all the numbers which
are one greater than a multiple of three and the operation is multiplication.
Is closure satisfied?
6. Make up some examples where closure is satisfied.
A. 1. No. The sum of two odd numbers is always even.
2. Yes. The sum of two even numbers is always even.
3. Yes. An odd number multiplied by an odd number is always odd.
4. Yes. The product of two even numbers is always even.
5. The universal set is {1,4, 7,10,13,16, ...}. Consider, first, two members of
this set, e.g. 7 and 13. When 91 is divided by 3 the remainder is 1 and also
7x13 = 91. Hence 91 is a member of the set. However, this one example
does not prove that closure is satisfied. Even if we test with many different
examples and find that for each of these cases the result is a member of the
set we have not proved that the result always holds. A proof could be devised
as follows.
A general member of the set can be expressed in the form 3«+1, where
n is a natural number. Another member is of the form 3m+1.
Now by simple algebra
(3/f+l)(3m+l) = 9wm+3>i-f 3/n+l
= 3(3/im+/i+m)+l
= 3/7+1,
where/? is a natural number since clearly 3nm+n+m is a natural number.
Hence closure is satisfied.
6. Many more examples of the same type as question 5 can be devised. One
such example might be that the set of all the natura lnumbers which are one
greater than multiplies of five are closed under multiplication [i.e., the set
{1, 6,11,16, 21, ...} is closed under multiplication]. The proof is similar to
that given in question 5.
NUMBER SYSTEMS 7
The Whole Numbers and the Integers
Consider now the open sentence
a+ D = b, where a and b are any natural numbers.
The universal set is the set of natural numbers. Is it still true that the
truth set always contains exactly one member no matter which natural
numbers a and b represent? A few examples show that it is not true.
Consider the open sentences
5+D = 5 or 3+D = 3.
When each of the set of natural numbers {1, 2, 3, 4, 5, ...} are tried
it is found that none of them are members of the truth set. In each of
these cases the truth set is the empty set denoted by { } or 0.
In order that the truth set for these open sentences should not be
empty it is necessary to extend the universal set of natural numbers to
include the number zero. We call this new extended set the set of
WHOLE NUMBERS.
Thus the whole numbers are {0, 1, 2, 3, 4, 5, 6 ,...}.
Note. Many teachers refer to 0 as just a place holder. This is not a
good practice. In a sense, in the Hindu-Arabic system of recording
which we use, all numerals are place holders. In, for example, the
recording of one hundred and twenty-three by 123, the "3" holds
the "2" in the tens column and the "2" and "3" together hold the
"1" in the hundreds column. It should be remembered that "0" is
a number in its own right. It is the cardinal number of the empty
set. It is also better to refer to "0" as zero rather than nought. Hence
the set of whole numbers starts with zero and is an infinite set.
Another point which must be borne in mind about the open sen­
tence α+Π = b is that it implies the open sentence b—a— D.
In symbols this is written as
a+ D = b =>b — a = □.
An example will illustrate this.
A bar of chocolate costs 7p and I give a lOp coin. How much