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Overview
Sawyer addresses this difficulty, speaking both for those who like recreational mathematics and for those who teach, suggesting a variety of methods used by many effective teachers — techniques of visualizing, dramatizing, and analyzing numbers that attract and retain the attention and understanding of students. His topics, ranging from basic multiplication and division to algebra, encompass word problems, graphs, negative numbers, fractions, and many other practical applications of elementary mathematics.
A valuable resource for parents and teachers, this book will captivate any reader seeking an improved understanding of mathematics.
Product Details
| ISBN-13: | 9780486143620 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 08/29/2012 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 352 |
| File size: | 14 MB |
| Note: | This product may take a few minutes to download. |
Read an Excerpt
Vision in Elementary Mathematics
By W. W. Sawyer
Dover Publications, Inc.
Copyright © 1964 W. W. SawyerAll rights reserved.
ISBN: 978-0-486-14362-0
CHAPTER 1
Even and Odd
SOME results in arithmetic can be grasped by a single act of mental vision. For example, suppose someone has played dice sufficiently to realize that the pattern [ILLUSTRATION OMITTED] represents 6. It is immediately apparent that 6 consists of 2 rows of 3. In this sense, the result 6 = 2 × 3 is one that we can see directly.
Unfortunately, this kind of direct vision leaves off almost as soon as it has begun. In the lines above, the words immediately, unfortunately, arithmetic have occurred. How many letters are there in each of these words? Few people can answer without counting or breaking these words into smaller groups. One might perhaps see arithmetic as arith metic and realize that it contained 2 groups of 5 letters, making 10 in all. Even so, we have gone beyond the bounds of direct vision. We have a fairly clear picture of the smallest numbers. Beyond them, we see only a blur.
The average man is often too modest. He sees only a blur. He may blame himself for this; cleverer people, he may think, see clearly. But this is not so. The blur into which all numbers dissolve soon after 4 or 5 is the common experience of us all. If I am shown the answer to a multiplication question, such as 127 × 419, I do not know at a glance whether the answer is correct or not. I can sympathize with children who are helpless before the question 'What is 7 x 8?' I know, of course, that the answer is 56, because that was well drilled into me in my childhood, but it is not something that I can see directly. So it is understandable that children have no idea at all of how to cope with this question, and give all kinds of answers at random.
How to organize the chaos that lies beyond the smallest numbers is therefore a problem that confronts the entire human race.
While we cannot see the correct answer to 127 × 419, there are certain tests we can apply. We would, for example, reject immediately the answer 23 as being much too small, or 1,000,000,000 as being much too large. We would also reject the answer 53,312 although it is about the right size. For 127 and 419 are both odd numbers. It is impossible that their product should be an even number, 53,312.
The classification into even and odd brings us once more within the scope of vision. If we look at this number:
[ILLUSTRATION OMITTED]
we can probably not tell without counting what number it is, but we can see at a glance that it is even. It has the characteristic shape of an even number. It can be divided into two equal parts, like this:
[ILLUSTRATION OMITTED]
Also, it can be broken up into pairs, like this:
[ILLUSTRATION OMITTED]
At a dance, if there are several couples on the floor, and no eccentric individuals dancing alone or in threes or other groupings, we can be sure that the number of people dancing is even. Evenness is thus a quality we can recognize in collections which we have not counted.
An odd number, on the other hand, has a shape such as:
[ILLUSTRATION OMITTED]
You cannot break it up into couples. One dot is left without a partner. Of course, we could place this shape the other way up:
[ILLUSTRATION OMITTED]
It would still represent an odd number.
The addition properties of even and odd are now apparent. Addition may be pictured as 'putting together'. If we put two even numbers together:
[ILLUSTRATION OMITTED]
they form an even number, as we can see from the shape above. But if we put together an odd number and an even number, we obtain an odd number, as is shown by these shapes.
[ILLUSTRATION OMITTED]
Finally, if we join together two odd numbers they dovetail to form the shape of an even number.
[ILLUSTRATION OMITTED]
For multiplication, we might picture five sixes like this:
[ILLUSTRATION OMITTED]
This has the shape of an even number. Clearly any number of sixes will give an even shape:
[ILLUSTRATION OMITTED]
There is nothing special about six; the same illustration will serve for any even number. An even number multiplied any number of times gives an even number.
If we take an odd number and repeat it we obtain the following pictures.
[ILLUSTRATION OMITTED]
Here it is fairly evident that odd and even shapes occur alternately. If we repeat an odd number an even number of times, the total is even. If we repeat an odd number an odd number of times, the total is odd. So 'even times odd is even, odd times odd is odd'.
PICTURES OF MULTIPLICATION
If you are seeking to understand, or to remember, or to teach mathematics through pictures, you should never allow yourself to be tied to one particular image of a mathematical idea. A picture that is suitable for one purpose may be most awkward for another.
In attempting to illustrate the result 'odd times odd is odd', I had to consider different ways of representing multiplication and see which was most suitable. I am by no means sure I chose the best.
Here are a few ways of picturing multiplication. We might, for example, picture 3 fours and 4 threes like this:
[ILLUSTRATION OMITTED]
In some connexion it might be desirable to picture 3 as [ILLUSTRATION OMITTED] and 4 [ILLUSTRATION OMITTED] as In that case, we might use these pictures:
Method B
[ILLUSTRATION OMITTED]
In both of these, the special patterns for 3 and 4 are apparent.
None of the above brings out the fact that 3 fours and 4 threes are the same number, 12. This can be shown by using a rectangle, as in Method C:
[ILLUSTRATION OMITTED]
The rectangle is a very useful way of showing multiplication, and it will be used frequently in this book.
When we use the rectangle to illustrate multiplication, the numbers involved have to be represented by dots arranged in a straight line. For example, in the illustration of 4 x 3 above, the 4 is shown as 4 dots in a row, while the 3 is shown by 3 dots in a column. This arrangement does not emphasize the fact that 4 is even and 3 odd.
Method C is generally the most convenient way of picturing multiplication. However, it does not agree with the shapes we had earlier for even and odd. This means that there is a conflict when we want to portray simultaneously multiplication and the property of being even or odd. If we want to use Method C to indicate multiplication, we must find some new way of indicating even and odd. If we want to use our shapes for even and odd, we cannot use Method C to picture multiplication. Because of conflicts such as this, the pictorial representation of mathematical results calls for ingenuity and judgement. To devise such pictures, you need sustained meditation on the matter in hand, and this helps to fix the ideas in the memory. Learners should always be encouraged to make their own pictures. Even if the pictures are not good, the effort of making them will leave lasting traces in the mind, and cause the work to be remembered.
If we have 7 dots in line, and we want to emphasize that 7 is an odd number, we can do this by pairing the dots off, like this:
[ILLUSTRATION OMITTED]
In the picture of 5 × 7 as a rectangle we may first stress the oddness of 7, like this:
[ILLUSTRATION OMITTED]
Unpaired dots now remain only in the last column, which contains, of course, 5 dots. Pairing these to emphasize the oddness of 5, we obtain the following picture of 5 x 7:
[ILLUSTRATION OMITTED]
The solitary dot shows that 5 x 7 is in fact odd.
Here we have held to Method C for showing multiplication. If instead we decided it was more important to keep the shapes: [ILLUSTRATION OMITTED] and [ILLUSTRATION OMITTED] for 7 and 5, we might represent 5 × 7 in this way:
[ILLUSTRATION OMITTED]
It might then occur to us that the oddness of this arrangement could be emphasized by turning the blocks like this:
[ILLUSTRATION OMITTED]
Again, a solitary unpaired dot stands out.
EVEN AND ODD IN ALGEBRA
If we place 2 fives side by side, we obtain the shape for an even number, 10.
[ILLUSTRATION OMITTED]
There is nothing special about 5. If we took 2 sevens, or 2 fours, or 2 thirteens we should in each case obtain an even number. An even number results when we have 2 of any number you like to mention. We may picture this as here:
[ILLUSTRATION OMITTED]
A cloud has come in front of the dots, so that I cannot see how many there are. The letter n is short for 'any number you like to mention'. As soon as you mention a number the clouds roll away and I see that number in the top row, and also in the bottom row. Whatever number you choose, the number of dots I see when the clouds lift will be even. I suppose this picture has a defect. If you were to choose 'zero' or 'one', my picture fails because I already have two dots in each row. I ask you to forgive this weakness. For any other number, the picture does what it is meant to. You may have doubts about what would happen if you chose 'two'. If you did that, the clouds would part and reveal a blank space.
Our picture thus helps us to imagine 'any even number'. In algebra the symbol corresponding to this picture is 2n, which means 'twice any number you like to choose'.
The picture for an odd number will be:
[ILLUSTRATION OMITTED]
Here we see the number you chose twice, and in addition a single dot. Using n as short for 'the number you chose', this picture contains n twice and 1 extra. The symbol for it in algebra is 2n + 1.
We might make a little table to show the effect of choosing different values.
Every even number will appear, sooner or later, in the column under 2n. Every odd number will appear, sooner or later, in the column under 2n + 1.
GENERALIZATION
An even number is a number that can be split into two equal parts. But two is no better than any other number. We might equally well consider numbers that could be split into 3 equal parts, or for that matter into 5 or 11 or 17 equal parts. A similar theory could be developed for each.
If we are interested in splitting into 3 parts, we must recognize 3 types of numbers. There are numbers such as 15 which break exactly into 3 parts, and have the shape:
[ILLUSTRATION OMITTED]
There are numbers such as 19 which leave a surplus of 1 when we try to split them into 3 equal pieces. These have the shape:
[ILLUSTRATION OMITTED]
And finally there are numbers like 14 which leave a surplus of 2 and have the shape:
[ILLUSTRATION OMITTED]
We have the words 'even' and 'odd' to describe how numbers stand in relation to 2. We have no words to describe how numbers stand in relation to 3. Rather than invent words, which would be a burden on the reader's memory, it seems wise to use the language of algebra, which gives us easily-remembered labels for these 3 types of number, as follows:
[ILLUSTRATION OMITTED]
In the same way, a number would be placed in type 7n+5 if, when you tried to split it into 7 equal parts, you found you had 5 dots left over. A number would be of type 10n+3 if when you tried to split it into 10 equal parts you had 3 left over.
When we were dealing with even and odd numbers, we saw that we could predict the result of an addition or a multiplication if we knew the types to which the numbers belonged. For example, we could predict that an odd number multiplied by an even number would give an even number. We did not need to be told which odd number and which even number were involved in the multiplication.
This suggests a subject for investigation: what happens when we do not divide numbers into the types odd and even, based on 2, but into types based on other numbers. For example, if I ask you to write down any number that leaves remainder 3 on division by 5, and also any number that leaves remainder 4 on division by 5, and multiply together the numbers you have written, can I make any prediction about the answer you obtain? (I assume you carry out the multiplication correctly.)
In terms of our symbolism, I have asked you to multiply a number of type 5n+3 by a number of type 5n+4. Can we predict the type of your answer?
Someone with a working knowledge of algebra can deal with this question immediately. This does not mean that it is a trivial question. Many who have learnt algebra at school cannot do it. But for the moment we are not supposed to know anything about algebra. The question has been posed as one that might arise in the teaching of arithmetic. Two lines of attack are open. We might try to picture the multiplication. On page 13 we drew a picture to show that 5 × 7 must be odd. It is possible to adapt this idea and make it fit the present problem. Some readers may like this idea: others may not. It depends on their ingenuity and skill in devising diagrams. The other line of attack is to collect evidence by testing particular cases. This is straightforward, and could be done by children in an arithmetic class. They might choose 8 as a number of the type 5n+3 and 9 as a number of the type 5n+4. Then 8 × 9 = 72 and when 72 is divided by 5, the remainder is 2. So 72 is of type 5n+2. Then they would select other numbers, say 13 and 4. Then 13 × 4 = 52, which again is of type 5n+2. The suspicion begins to grow that any number of type 5n+3 multiplied by one of type 5n+4 gives an answer of type 5n+ 2.
Addition is much easier to handle than multiplication. From the pictures we had earlier, it is immediately evident that when you add together numbers of types 3n+1 and 3n+2 you obtain a number of type 3n.
[ILLUSTRATION OMITTED]
Any problem that involves only the addition of number types can be settled by drawing pictures. For example, it would not be hard to solve the question: if you add a number of type 7n+4 to one of type 7n +6, of what type will the answer be? As has been mentioned, it is possible to devise pictures to answer the question: 'What type of number do you get if you multiply a number of type 7n+4 by one of type 7n +6?' But it is simpler to deal with this by the methods of algebra, and we shall return to this question later on.
Exercises
1. Arrange 23 dots so as to show that 23 is of the type 3n+2.
2. Draw a picture to illustrate a number of the type 4n+1. Draw another picture to show a number of type 4n+2. If the two pictures are joined together, what type of number do they represent?
3. We call a number of type 10n+3 if on division by 10 it leaves remainder 3. Mention 5 different numbers of this type. If they are written, what do you notice about them?
4. Choose any number of the type 5n+2 and any number of type 5n +3. Multiply them together. What remainder is left when your answer is divided by 5? Repeat this work several times, with different numbers. Does the same remainder come each time?
Even and Odd
5. I multiply two numbers together. The first number is of the type 6n+2. The second is of the type 6n+3. What remainder does my answer leave when divided by 6?
6. If you multiply a number of type 3n+1 by another number of the same type, what remainder do you get when you divide the answer by 3? To what type does the answer belong?
7. Choose two numbers, each of type 4n+1. Multiply them together. Divide the result by 4. What remainder is there?
8. Choose two numbers, each of type 5n+1. Multiply them together. Divide the result by 5. What remainder is there?
9. Questions (6), (7), (8) have something in common. Examine these questions and your answers to them. Do they suggest anything to you?
10. (An application of even and odd.) A football league contains 5 teams, whose names are given in the table below. The teams played only within the league. Do you find anything in the table below to suggest that an error has crept into the records?
(Continues...)
Excerpted from Vision in Elementary Mathematics by W. W. Sawyer. Copyright © 1964 W. W. Sawyer. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
Table of Contents
Introduction1. Even and Odd
2. Divisibility
3. An Unorthodox Point of Entry
4. "Tricks, Bags, and Machines"
5. "Words, Signs, and Pictures"
6. Sudden Appearance of a Practical Result
7. A Miniature Problem in Design
8. Investigations
9. The Routines of Algebra I
10. The Routines of Algebra II
11. Graphs
12. Negative Numbers
13. Fractions
Answers