Q&A #19482

Multiplying fractions

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From: Gail (for Teacher2Teacher Service)
Date: Apr 03, 2008 at 18:33:55
Subject: Re: Multiplying fractions

Hi Linda,

I think I see one way you could help them past this hurdle. You say they are
considering multiplication as a short cut for addition.  Well, I agree, up to
a point.  But when they consider the numbers being multiplied, they need to
remember that one of the numbers names the amount in each group, and the
other names the number of groups. So they can use that same thought when
looking at fractions.  You could guide them with a table, and help them look
for patterns to make a generalization, like this:

 2  x 4 = 8       2  x 1/2 = 2/2 or 1
 1  x 4 = 4       1  x 1/2 = 1/2
1/2 x 4 = 2      1/2 x 1/2 = 1/4
1/4 x 4 = 1      1/4 x 1/2 = 1/8

Since you said they understand the idea of "half of", maybe they will see
that each step uses half of the factor before it (2, 1, 1/2, 1,4), and the
answers (products) are also one half of the preceding product.

The first set just convinces them that there is a pattern, that they can
predict. The second set establishes that the patterns involve both fractions
and wholes.

Now that they are certain it "works", perhaps you could return to the idea
that 2 x 3 means 2 groups of 3, and that 1/2 x 1/3 means half of 1/3.  When
you find 2 x 3 you are really finding 2/1 x 3/1, but we just don't think
about the wholes as fractions.  We multiply those denominators. They just
don't make a difference to the way the answer looks because of properties.

I hope this gives you something to begin with. The patterns usually convince
my students...   :-)

 -Gail, for the T2T service

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