Picturing Dividing Fractions

Date: 04/02/2001 at 20:12:28
From: Katherine Stagnitti
Subject: Dividing fractions

Dr. Math,

My fifth graders can divide fractions without a problem. They use the 
reciprocal of a given fraction without any trouble. Our problem is 
that we can't draw the division problems out to prove our answers.  

For instance, please draw 2/3 divided by 1/2. We know the answer is 
1 1/3. We're working with fraction pieces from circles. First we took 
2/3 and taped them together. Next we place a 1/2 fraction piece on top 
of the 2/3 section. We could see that it took 1 piece that was 1/2 in 
size to cover up part of the 2/3. However, the part of the 2/3 piece 
that was not covered up by the 1/2 section did not equal 1/3. Help!

Thanks,
Mrs. S.

Date: 04/02/2001 at 23:36:38
From: Doctor Peterson
Subject: Re: Dividing fractions

Hi, Katherine.

I'll use rectangles rather than circles, since that's easier to draw 
in text form. Here's 2/3:

    +---------+---------+---------+
    |XXXXXXXXX|XXXXXXXXX|         |
    +---------+---------+---------+

Now I'll lay several 1/2's next to it, so we can find out how many 
1/2's it takes to make 2/3:

    0                  2/3        1
    +---------+---------+---------+
    |XXXXXXXXX|XXXXXXXXX|         |
    +---------+---------+---------+
    +--------------+--------------+
    |11111111111111|22222222222222|
    +--------------+--------------+
     <------1-----> <-?->

Now, that extra piece after the first half is 1/6 of the original bar 
(or circle in your case). But the question is not what number is left, 
but HOW MANY HALVES IS IT?

Since 1/6 is 1/3 of 1/2, that extra piece is 1/3 of a half, and the 
whole 2/3 is 1 1/3 halves. That's the answer you're looking for.

It's a little tricky, isn't it? When we divide, we're looking for how 
many of the things we're dividing by it takes to make the thing we're 
dividing; but when we work with fractions it's easy to get mixed up 
and count units rather than divisors. To demonstrate it, you'll want 
to cover the second 1/2 with three 1/6's, and explain that they are 
thirds of the 1/2.

It might help to work up to this problem with an intermediate one, 
where we get a fractional answer, but are dividing whole numbers. Try 
dividing 3 by 2:

    0         1         2         3
    +---------+---------+---------+
    |XXXXXXXXX|XXXXXXXXX|XXXXXXXXX|
    +---------+---------+---------+
    +-------------------+-------------------+
    |1111111111111111111|2222222222222222222|
    +-------------------+-------------------+
     <--------1--------> <---?--->

This time we have 1 1/2 2's: one 2, and a 1 left over, which is 1/2 of 
a 2. You can probably figure out a better way to say that.

Here's an answer to a similar question with a different example:

  Fraction Division Diagrams
  http://mathforum.org/dr.math/problems/mitchell.04.21.99.html   

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/