OR they change a mixed number into an improper fraction and seem to subconsciously think that since they did something to that mixed number, the flipping had already occurred...and then they don't flip anything.
Why does this happen? I'm going to say that it happens because they don't see the sense in it - it doesn't mean anything. Yes, you CAN PROVE to them why multiplying by the reciprocal works, but at the age they learn this process, the proof still doesn't really seem to mean much.
So, I have another way to teach fraction division - perhaps you've heard of it, or you use it. I never learned it this way as a child, but I like it and it makes more sense to some students. I learned this method when I had a student teacher a few years back. She was teaching the fraction unit, and when her supervisor came in to observe and discuss, she asked if I had ever taught fraction division using common denominators. Having only learned (and then taught) to multiply by the reciprocal, of course, I said no.
The next time she visited, she brought me a page from a textbook that explained dividing fractions using common denominators. These are the steps:
Step 1: Find common denominators, just as when adding and subtracting and then make equivalent fractions (students are already used to doing this - hopefully).
Step 2: The answer is the first numerator over the second numerator.
Done (unless you need to reduce)!
I was shocked - it seemed SO simple!
Check out this example - it's a simple one, for starters:
5/6 divided by 2/3.
1) Find the common denominator of 6. This gives you 5/6 divided by 4/6.
2) The first numerator (5) becomes the numerator in the answer. The second numerator (4) becomes the denominator. Then reduce.
Let's look at another one, with mixed numbers:
1 and 4/7 divided by 1 and 3/4.
1) Convert the mixed numbers to improper fractions, which gives you 11/7 divided by 7/4.
2) Find the common denominator of 28 and make equivalent fractions. This gives you 44/28 divided by 49/28.
2) The first numerator (44) becomes the numerator in the answer. The second numerator (49) becomes the denominator. No reducing, in this case.
I've shown both methods to my sixth-graders. Some really like it. Others stick to the flipping method - but I don't know if this is because they like it better or because it was the first way they learned it.....most of them had been taught something about fraction division in 5th grade.
What do you think? Do you see any advantages or disadvantages to teaching fraction division using common denominators?