Part 5. Fractions:
Math or Arithmetic texts I have used seldom gave much space to
fractions, and usually
it was only fractions of solid objects such as cakes, pies, and pieces
of paper.
The emphasis seemed to be on providing a visual image of particular
fractions.
Fractions are not such a simple subject. They cause a lot of trouble to
students in later
years so whatever can be done with them in the early years will be
appreciated later.
A fraction is a special kind of number that we use to talk about a
quantity
that is less than one, that is, where something is a portion of a whole.
There are two basic fraction situations, one where a fraction is
used to
name a part of a whole one, two, or three dimensional object, and the
other where a fraction is used to name a part of one group of objects.
Common fractions have two numerals each, one on top of the other, as 7
/ 8.
When I dealt with fractions in the primary grades I taught the children
that the number
on the bottom, 8, counted the number of equal parts the whole was
divided into.
The number on the top, 7, counted the same parts as the bottom number,
but only those being talked about.
The bottom number described the size of the parts, the top number
counted them.
I did mention that the top number was called the numerator,
and the bottom number
was the denominator. I tried to relate these name to the jobs
the numbers did.
Most of the time we did not use these names. We talked instead of
the counting
number, and the describing number. These were more familiar words.
By telling us how many parts there was in one whole the bottom
number
let us know what the parts would be like. It let us picture them.
This was the number we had to focus on first.
We talked about a party where there was just one cake. How big a piece
of that cake
would you get if you invited one person to share it, or 4, or 10, or
100, or 1000?
What if you invited no one.
We did the same with a plate of 8 cookies. How many would you get if
you shared
them with 1 other person, or 3 others, or 7 others? What if you didn't
share?
Changing the bottom number, the describing number, changed the amount 1
portion
would be. As the describing number got larger, each portion got
smaller. The smallest
number, 1, would mean the portion was the whole thing.
23.
The top number was there to tell us how many of the bottom number's
parts we were using the fraction to talk about. It counted just as we
were
used to counting, only it counted special equal parts of a whole.
It counted the kind of parts the bottom number told us about.
When this top number, or counting number, got bigger it meant there
were more parts.
What if a mother put a cake on the table for her child to eat some of
after school, and
she had cut it into 8 pieces. If her child ate 1 / 8 of the cake there
would be a lot left.
But what if her child ate 2 / 8, or 5 / 8, or 7 / 8, or 8 / 8?
By discussing such things, and by cutting up real objects and pictured
objects, the
children got a sense of what a fraction was about.
A fraction is a single number in one sense, but to understand it we
have
to consider both numbers used to form it.
There may be emphasis in texts placed on knowing what a quarter, a
third, and a half
are like. There may be work done on relating a quarter to a half, or to
a whole.
What I think was lacking in the texts and curriculum I had to use was
an effort to
promote a real understanding of why 1 / 4 was the size it was, and why
1 / 3 was larger
than 1 /4.
For this reason I concentrated on the meaning of the two numbers, on
what they were
telling us, rather than just picturing how big an amount the fraction
was as a single
number.
I wanted to look at the three and the four in 3 / 4,
not just the three-quarters.
In a sense this was like looking at the value of the four and the eight
in 48, rather than
simply the forty-eight.
Using squares and circles on the chalkboard, and sets of real objects,
my class
worked on understanding just what fractions meant.
I might put up a square, and the fraction 3 / 4. I would ask how many
equal parts I had
to divide the square into. Then I would ask how many of those parts had
to be made
different in some way.
I might put out 8 blocks, and the fraction 3 / 4. I would ask someone
to divide the blocks
into the correct number of equal groups. Then I would ask how many of
those groups
was the fraction counting.
Working like this we were able to compare fractions. The square divided
into 4 could
be divided into 8. We could note that each of the former parts became
two smaller
parts. If the 3 / 4 was marked out, we could see it also had 6 / 8 so
we knew that these
two fractions talked about the same amount, but in a different way.
24.
This idea could be extended so we could see there were many fractions
that could
mean the same amount of a whole, yet look very different from one
another.
These were equal fractions. 6 / 8 = 3 / 4 .
It wasn't hard to see that there was the same kind of relationship
between the counting
numbers in these fractions as there was between the describing numbers.
This was especially true if we did 1 / 2, 2 / 4, and 4 / 8. When we
made the describing
number twice as big, the counting number became twice as big.
We could also see that 7 / 8 was larger than 3 / 4, and 5 / 8 was
smaller, but it was
easier to see this if we changed the 3 / 4 to 6 / 8. Then we had 5 / 8,
6 / 8, and 7 / 8.
For children who knew a little about multiplication and division it
wasn't difficult to learn
how to change fractions so they could be easily compared as to size,
that is changed
to a common denominator.
This need to convert fractions to a common denominator is one area
where children
have difficulty later on in school when they begin to add and subtract
fractions.
I have taught this long ago while teaching grades 4 to 7, and I can
remember it was
very difficult work for some children.
There are things we can do in the primary grades to make this work
easier.
We can start to teach fractions in Kindergarten. I don't mean just the
names, as
one-half, and one-quarter. We can start to teach the meaning of the
parts of fractions.
We can create a situation where there is sharing to be done. It can be
real or the
children can pretend it is real.
Two children have a big cookie to share. How many pieces should the
cookie be cut
into? What should we be careful to do as we cut the cookie? Is each
child going to
feel their share is fair?
Each child has a piece of the cookie. That piece is one piece out of
two pieces in the
whole cookie. Both pieces are the same size.
We say each child has half a cookie. That is written with numbers as 1
/ 2. The bottom
number tells how many pieces the cookie was cut into. The top number
tells how
many of those pieces each child got.
This focus on the meaning of the two numbers in a fraction can be
started
at an early age.
Over the next three years the world of fractions can be developed.
I'm not trying to say that fractions become a big topic, like the one
to follow this section,
but they can be brought in here and there throughout every year.
25.
Asking questions such as,
"How many children are in our class?",
"How many boys are there?",
"How many girls are there?",
"What fraction of the children in our class are boys?", and
"What fraction are girls?",
can provide opportunities to explore the meaning of the two numbers in
a fraction.
Say the answer to the first question is 30. This becomes the bottom
number since it
tells how many parts are in the whole.
If the second answer is 17, then the fraction that are boys is 17 / 30
since the 17 on the
top tells us how many of the 30 in the class we are talking about with
this fraction.
If the third answer is 13, then the fraction that are girls is 13 / 30
since the 13 on the top
tells us how many of the 30 in the class we are talking about with this
fraction.
We can notice that the two top numbers in these fractions add up to the
number in the
whole class.
We can ask questions about fractions relating to many things to do with
children: hair,
shoes, brothers, sisters, pets, ...
There are many objects in a classroom that provide the same
opportunities: windows,
desks, walls, light fixtures, ....
When children begin to multiply and divide there are other
opportunities to start
working with fractions.
I have had second grade children learn to do questions such as 1 / 4 =
__ / 8 after
they have started multiplication and division.
They can think of 4 x __ = 8, and then 1 x 2 = 2 to see that 1 / 4 = 2
/ 8.
Fractions can exist on their own for us to work with but when they are
really used they
are related to something else.
I have had them find fractions of a group of objects. This required
them to divide first,
and then multiply.
I would have asked them to find 3 / 4 of 8. First they had to divide 8
by 4 to find that
there were 2 in each part. Then they had to multiply that 2 by 3
because the counting
part of the fraction said there was to be 3 of those groups of 2. They
would find that
3 / 4 of 8 was 6.
Now to talk about how work done in the primary grades can be of help
later on.
In my Simple Math booklet, starting on p.19, I discuss the use of
factors in
multiplication and division. Then, starting on p.28, I discuss the use
of factors in
working with fractions, particularly the addition and subtraction of
fractions where one
must find a common denominator, hopefully the LCD or lowest common
denominator.
26.
A common denominator is simply a way of saying that two or more
fractions are talking about exactly the same sized parts or groups. The
lowest common denominator would be one where the parts or groups are
as large as possible, yet the fractions will have parts of the same
size.
This may be where some children have difficulty.
When adding or subtracting fractions, what they really add or
subtract
are the counting numbers on the tops of the fractions, the numerators,
but those numbers must be counting the same types of things.
It wouldn't make sense to say that we were adding 3 apples and 4
oranges to get 7
apples. We have to find a common name. It would make sense to say we
are adding
3 pieces of fruit and 4 pieces of fruit to get 7 pieces of fruit.
Unfortunately finding a common name for fractions is not so simple.
As we change the part of the fraction which names through description,
the
denominator, we also change the counting number, the numerator.
As well, those numbers can only be changed through multiplication and
division to a
unique set of numbers if they are to be useful to us.
This is where a working knowledge of factors, and multiples, can help,
and very young
children can learn to use them.
Factor, as a name, is a relationship word. It tells us that a
number will divide evenly
into another, using whole numbers. 3 is a factor of 6, but not of 7.
Multiple is the name related to 'factor'. It tells us a number can
be created by
multiplying a whole number factor by itself or another whole number
factor.
6 is a multiple of 3, but 7 is not.
Factors and multiples are directly tied to working with fractions.
They
determine what the describing part of the fraction can be changed to.
If 3 / 8 and 4 /12 are to be added, we must find a number of parts that
both bottom
numbers can be changed to before we can add the same kinds of things.
A knowledge of factors and multiples would tell us that the smallest
number both could
be changed into is 24.
8's multiples are 8, 16, 24, ... , or 8 is a factor of 8, 16, 24, ...
12's multiples are 12, 24, 36, ... , or 12 is a factor of 12, 24, 36,
...
Each of the 8 parts in 3 / 8 must be divided into 3 parts to give us 24
in a whole.
8 x 3 = 24 does this for us.
Each of the 12 parts in 4 / 12 must be divided into 2 parts to give us
24 in a whole.
12 x 2 = 24 does this.
27.
But, when we change the bottom number we also change the top number
because it
counts the parts in the bottom number. What we do to the bottom we do
to the top.
So, if we've done 8 x 3 = 24, we must also do 3 x 3 = 9. 3 / 8 becomes
9 / 24.
If we've done 12 x 2 = 24, we must also do 4 x 2 = 8. 4 / 12 becomes 8
/ 24.
These can be added by combining the counting numbers, 9 and 8.
The answer is 17 / 24.
(We don't add the bottom numbers because they are just like names.)
You have probably noticed I used 4 / 12, a fraction not in its simplest
form. This is
another problem area in later grades. By simplest form we mean that
the
denominator has been made as small in number as it can be, or
the parts it
names are as large as they can be, and still have the counting part
of the
fraction, the numerator, be a whole number.
Changing a fraction to its simplest form requires factors and multiples
again.
12 can be made from 1, 2, 3, 4, 6, and 12.
4 can be made from 1, 2, and 4.
4 is the biggest common factor, and it lets us make the smallest common
denominator.
If we make the 12 into groups of 4 we will have 3 groups. The 4 in 4 /
12 can become
1 group and still count the same amount it did before.
4 / 12 = 1 / 3.
If you add the 3 / 8 to 1 /3 you will still get the same answer, 17 /
24.
In my Simple Math booklet, on p.28, I also talk about prime and
composite
numbers, and the way prime factors can be used to help find
the least common
denominator (LCD) of larger denominators. I won't go into that here.
Prime and composite numbers were also a part of my work with
multiplication and
division, a part that can be valuable in later grades.
There are other fraction topics that can be started in the primary
grades.
We talked about and did a little work related to proper fractions and
improper fractions.
Proper fractions are those that really do talk about a part of
something. The
counting number on the top in these is always going to be less than the
describing
number on the bottom. All the fractions I have used so far have been
proper fractions.
Improper fractions are those that are formed like a proper
fraction with a counting
number and a describing number, but the counting number is equal to, or
larger than,
the describing number. 4 / 4 and 5 / 3 are examples of improper
fractions.
Improper fractions can become whole numbers plus fractions, or mixed
numbers.
4 / 4 is a fraction that really tells us we have a whole object or
group.
5 / 3 is a fraction that tells us we have enough parts to make one
whole object or
group, plus some more parts. 5 / 3 = 1 and 2 / 3.
28.
One last topic related to fractions is this.
Talking about a fraction of something can mean that we really do have
something that
is already divided into parts, or that it really will be divided into
parts.
It can also mean that we only pretend to divide something in parts. We
use our minds
to make a division so that we can talk about something that we may not
wish to divide,
or cannot divide.
The children always liked to think of a fraction of their own or
someone else's body.
Sometimes the topics were gruesome. "3 / 4 of my body is covered with
spots", is a
tame example.
I'm sure there's more to say about fractions but I've gone on long
enough.
I do believe that they are something young children can and should
learn.
It is most important that they understand that a fraction is a special
kind of number, and
how a fraction gets its value through the relationship between the two
numbers in it.
With this knowledge they can understand how changes to one number
affect the other,
and can do so without changing the value of the fraction, only the form.
29a.