Q&A #1591

Working with fractions

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From: Gail (for Teacher2Teacher Service)
Date: May 22, 1999 at 22:44:35
Subject: Re: Working with fractions

It looks to me that these students do not really have a firm understanding of
what fractions are. How much experience do they have with using models and
manipulatives? Having trouble regrouping fractions when subtracting could be
a result of not really understading equivalent fractions. It could also be
that they are not clear on what is happening when one has to regroup during
subtraction. (Are they able to apply this skill when subtracting wholes?)
Using models (cuisenaire rods, pattern blocks, fraction squares or circles)
could help them make the connections between regrouping of whles and

How do you view reducing and simplifying fractions? I am thinking they are
the same thing. If your students are having difficulty there, it might be
that they are not comfortable with equivalent fractions. My students had to
play around a lot with fraction models and fraciton charts before they were
comfortable with the idea that the same amount can be represented so many
different ways.

As for the operations on fractions (multiplying, for example), when I
introduced my students (fifth grade) to this skill, I reminded them what we
are doing when we multiply two whole numbers (eg. 5 X 3 is five groups of 3,
2 X 7 is two groups of 7). I tried to help them understand that multiplying
is actually being able to see how the English sentence fits with the number
sentence (eg. 2 X 1/3 is two groups of 1/3, 1/4 X 8 is one fourth of a group
of 8, and 1/2 X 1/3 is half of a third).

Do your students have any trouble determining the equivalent fraction for a
mixed number? If not, that is the key to multiplying mixed numbers and
fractions. Once they can represent the amount 1 and 1/5 as 6/5, they are on
their way to solving multiplication problems involving fractions.

When we divided fractions by fractions, we can still relate that to division
with whole numbers. Think about what happens when we divide. We are looking
for the number of groups of a certain size, or the size of a certain number
of groups (eg. 28 divided by 4 means how many groups of four are in
28, or if you make four groups from 28, how many will be in each group.)

If you think about fraction division that way, 1/4 divided by 1/8 means "how
many groups of 1/8 can you get from 1/4 (and the answer is two...), or you
could think of it this way. 1/8 of a group is worth 1/4, so the group must be
worth 8 of the fourths and that is 2. Also, 1/3 divided by 1/2 means how many
halves can you get from a group of 1/3 (think of a model of a circle. If you
have only a third of a circle, then you do not have enough for a half, do
you?). You have 2/6, but you need 3/6 to make one half, so you have two of
the three pieces you need to make the half, right. Therefore, you have 2/3
(of a half) or you could say 1/2 of a group is worth 1/3, so the entire group
must be worth 2/3.

Dividing fractions by whole numbers and mixed numbers uses the same idea. 3/4
divided by 2 means, if you make two groups from the 3/4, how much would be in
each group. 3/8 would be in each group, because making two equal groups means
finding half, right? Or, you could think of it this way. 3/4 divided by 2
means how many groups of 2 can you get from a group of 3/4? There is not
enough to make one full group, right? Not even enough to make half of a full
group, so you know the quotient must be smaller that 1/2. In fact, to make a
group of 2 you actually need 8 fourths, and you have only 3 fourths, so you
have only 3 of the 8 you need, which is 3/8.

3/4 divided by 1+1/2 is like saying 3/4 divided by 3/2. You need to figure
out how many groups of 3/2 you can find in a group of 3/4, and you know the
answer will be less than one, right? Since 3/2 is the same as saying 6/4, you
could use that fraction instead, and notice that while you have only 3
fourths, you actually need 6 fourths, so you have 3 of the 6, or 1/2 of what
you need.

Or you could say, you have 1 and a half groups, and all together that is
worth 3/4. So, 2/4 and 1/4 is 1 and a half groups worth. See where this is

Whatever you decide to do, be sure to allow lots of time for working with
manipulatives, models, and illustrations. Students who play with these sorts
of things while working on number sense topics such as these seem to have an
easier time making the generalizations they need to work with fractions and
decimals. Hope this starts you off in a direction that will help you help
your students.

 -Gail, for the Teacher2Teacher service

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