What does it mean
To divide by
a fraction
you turn it
upside down
and multiply
This is fairly clear, and is most easily shown by an
example such as s+_ We are dividing by , so the rule
tells us to turn this upside down, giving 7, and we now
multiply to get 5x7 735
which is the correct answer.
More generally we shall have
a.c -a xd ad
6d -bc bc
Why does it work
To explain this we have to go a long way back into the
nature of fractions and the meaning of multiplication
and division. Fractions first arise in a natural way when
something has to be divided into equal parts.
Figure 1
As in the previous article in this series we seek to
find out (a) what the rule means, (b) why it works
and (c) is it misleading, and if so is there a better
way of wording it.
Figure 2
I
22
In each diagram of Fig. 1 the shape has been split up
into three equal parts, and in each case one of the three
Figure 3
parts has been shaded. We say that one-third has been
shaded, and write this as -. All this is fairly trivial, but
what are we to make of 3? In that the "1" has been
replaced by a "2" we can interpret this as "Take two
identical objects and divide them up into three equal
parts, 3 represents one of these parts". See Fig. 2.
is a third of two (- of 2).
But it is fairly clear that in each case the shaded area
for 2 is twice the shaded area for 3, i.e. . is two-thirds
or 3+3, and in that multiplication can be considered as
repeated addition we can write 1+1 = 2x- and we have
that 1 of 2
We can generalize this to have the fraction 6 interpreted
as 1/b of a a/ba
Multiplying a fraction by a natural number
Multiplication first arises as repeated addition. 3x4
means "write down the number 4 three times and add
them together".
3x4 = 4+4+4 = 12.
On the other hand 4x3 = 3+3+3+3, which is not the
same process, does give the same answer, i.e. 12.
3x4 = 12 = 4x3.
The commutative property for multiplication of the
natural numbers is useful, and we use it when we come
to multiplication of a fraction by a natural number.
If we want to evaluate 4xm, we can easily interpret
this as repeated addition
4x+ = }+1+3+3 = (
We have written down + four times and added.
But how do we interpret 'x4. We cannot write down
"4" a third number of times, it is meaningless. We want
to give it a meaning so that the answer is 3, for in this
way we shall still have multiplication being commutative. The obvious extension is to define 3x4 to be
the same as 4 of 4 for (i) by the above reasoning i of 4
has the value 4 and (ii) it means we can extend the idea
of multiplication to the product of two fractions. With
this extension it quickly follows that ax1
= -a
b =
b1xb*
Multiplying two fractions (See Fig. 3)
Suppose we wish to evaluate Ix-. We can now identify
this as being the same as finding ( of 3. Taking a rectangle we first shade out 2 of it. We now wish to shade
out three quarters of this area, and this we do as shown.
The resulting cross-hatched area represents 4xi or 4 of 4,
and clearly we have shaded in 2, A. On the
other hand 5x4 would be represented by the cross-
hatched region of the next figure, and again it is a
region of A. So we have 2x = fx = A. It is not difficult
to extend this idea to show that
a c c a _ ac
bX-+ X +
b d d b bd
With all this as background, and with the idea of
equivalent fractions we can move on to the problem of
division.
We can now consider division as repeated subtraction
and interpret this as meaning "How many sets of size 9
thirds can I take from a set of size 22 thirds?" This is
the same as saying "How many sets of size 9 can I take
from a set of size 22?", which is 9 sets. 2"-3 = = (iii) We can use the properties of multiplication and
division by 1, and the idea of equivalent fractions to
write 9+3
Cong as before.
(iv) Division is the inverse of multiplication. This means
Division of a fraction by a whole number
Consider the problem of finding 9+3. Then we can
approach this in several ways.
(i) Dividing an object into 3 equal parts is the same as
finding a third of that object so
9+3 = + of 9 = jx ] = 2
(ii) Using equivalent fractions we can write 3 as ] and
9+3 becomes 2+ .
that if we divide by 3 and then multiply by 3 our
original number is unchanged. This gives us
(-2+3)x3
We now use the fact that the inverse of 3 is +, i.e.
3x- = 1. If we multiply by 3 and then multiply by +
our original number will be unchanged. Multiplying
both sides of the equation by A gives
3
so Cong as before.
23
This same method is also expressed more algebraically
by saying that to find O such that
9+3 = O
is the same as finding the number 0 such that
0x3 = ~ [Compare 21+3 = 7 and 7x3 = 21]
now multiply both sides by the inverse of 3, i.e. 3, to
get Dx3x' = 22x1
S3 X3before.
9 ~i~ tlr~
A= -2as
Division of a fraction by a fraction
We can adapt any of the last three of the four ways
outlined in the above paragraph. Consider the problem
of 22+3
(i) Using the idea that division is repeated subtraction
we get.2 22X5
"How many sets of size 9 fifteenths can I take from a
set of size 110 fifteenths" is the same as "How many
sets of size 9 can I take from a set of size 110".
This gives the answer le sets, which came from 2X.
(ii) Using equivalent fractions and unity we get
22 22 5 22 5
=n
_= 3 3 2X=
229
- .3
3 3 s 3~5 3 33 9
3 X3 1
as before.
(iii) Using the idea of inverses, and noting that the
inverse of 2 is since +x = 1, we have
22
3) X3 - 22
3 5 ' /\5 3
so multiplying both sides by the inverse of 3, i.e.5
(22 -3) , 5 = 22 5
3 * 5 X5X3 3 X3
22. 3Xjx 2= 22
3 5 3 "A
22
22
5
3 ( -5 3 X3
Here we see most clearly that dividing by I has the
same effect as multiplying by -.
Note that this method relies on the fact that ba is the
inverse of in that if we multiply by and then by
we are multiplying by 1 and our original number is
unchanged.
As in the case of division by a whole number this last
method can also be represented more algebraically. We
are looking for a number 0 such that
.-5= 0
522.1=
A
Critical
Review
Why a critical review? Ten years after the beginning of
the revolution in school mathematics, can we find any
overall patterns and any future trends? British teachers
are proud of a system which has allowed so much
experiment and change, yet their pride is tinged with
bewilderment when they examine the present situation
in mathematics teaching in secondary schools.
They are bewildered by the number of choices of
books, each with a different syllabus mixture and
claiming to cover different examinations. They are
bewildered by the foolish dichotomy which persists
between "traditional" and "modern" mathematics-
indeed one may be led to believe that they are different
subjects. At the same time they are bewildered by a
school system with the age of transfer being 11, 12, 13,
14 and sometimes 16. What can we say to reassure
them and can we say anything to all teachers of mathematics of the age range 11-16?
These questions may be answered as the project
develops. It is certain however that there is a present
demand that something should be said and done and
we outline some of the needs and hopes that are being
expressed.
Firstly there is a need to review syllabus, books,
workcards and other teaching aids. Some teachers are
and this is the same as finding a number E such that
1 22
Ox5 - 3
asking that the project should help to break a monopoly
now multiplying both sides by the inverse of d i.e. 5 we
the project should make two or three recommendations
get Ox x- = 9x
which leads to Dxl = O = 2x = -Qas before.
is the rule misleading?
that may be established by one syllabus. Others have
deplored the diversification of syllabuses and ask that
as the best buy (a "Which" report). It would not be
right for the project to go as far as either of these
extremes. We intend to review possible contents and
methods of presentation against the background of
existing books and syllabuses. Some recommendations
will emerge, some omissions will occur, and teachers
On the whole the rule satisfies the two conditions for
will have to draw their own final conclusions.
such a rule of thumb. It is easy to remember and fairly
Secondly there is a need to throw light on school
mathematics and on mathematical language. Teachers
will recognize some difficulties. Children change
foolproof in application. There are, however, two
places where pupils make errors when trying to use the
rule as quoted. One is to turn the wrong fraction upside
down. The other is the problem of applying the rule
when dividing by a whole number. This latter case leads
to problems as to the meaning of "turn it upside
down". What is 6 upside down? Is it 9? A number such
as 6 is not always seen as 9, and the rule fails. Both
these errors reveal a flaw in the rule as stated, it does
not convey the reason for its validity. Could we reword
it to the more generally correct
To divide by a number multiply by its inverse
or, if this would not be readily understood, perhaps the
wording
To divide by a fraction, invert it and multiply
may help to remind pupils of the derivation of the rule,
and foster understanding.
24
schools, teachers change schools, schools are reorganized
and questions are asked. Are you doing modern maths?
Transformation Geometry? Functions? Even the wellinformed teacher is inclined to reply, "what do you
mean by. . . ?" We hope to provide mathematical back-
ground articles to help teachers to clarify their language,
to understand the relative importance of new ideas and
the later development of the mathematics.
What is a group? Another name for a Latin Square!
Why do the vector books stop at the Angle in a Semicircle Theorem? What is a matrix? A representation of
a linear transformation! Why are mappings important?
Many courses do not answer these questions, and books
that do provide the answers are likely to be remote
from the teacher. The project aims to produce material