I had about 20 minutes after a test today, and RIGHT before the test, a high school student in precalculus was confused about how to divide 2 fractions, and she couldn't remember the method: "do you multiply the top and bottom?" were words that came out of her mouth.
So. I thought I'd give the following short lesson on why the "flip and multiply" that they seem to spout works.
First I asked them the following questions:
Simplify: (7/3) / (4/5)
Then after they did that and checked, I asked them to discuss WHY it worked, and "my teacher said so" or some such thing wasn't an option.
Then we walked through the following 4 scenarios:
So to recap, it basically boils down to seeing how many groups of the denominator fit into 1 unit (that's where the "flip" comes from), and then how many 1's fit into the numerator, and then you combine the 2 numbers and MATH.