Picturing Dividing FractionsDate: 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/ |
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