“When you divide by a fraction, you multiply by the reciprocal.”This is a mathematical truth, but when I asked my students to explain why I was amazed by all the tricks.Answers ranged from “you kiss and flip” to “you can’t divide fractions.”The latter bothered my mathematical core greatly.
After hearing these “math tricks”, I decided to write about my interpretation on why you multiply by the reciprocal and how I explained it to my students.
I started with a simple question, “8 pies were baked and cut into thirds, how many pieces are there?”My students answered 24 quickly.We then had a brief discussion on why when you divide a number greater than 1 by a number between 0 and 1, the answer increases.
We then looked at the thought process of the above question. “8 divided by 1/3, is actually the equivalent of multiplying 8 by 3”.I still, however, was not happy about the explanation.Lucy then shouted out “each pie will create 3 pieces, and since there are 8 pies we can multiply 3 by 8”.I smiled.
Now the tough question, “If you made 8 pies, and everyone you invited over ate 2/3 of a pie, how many people could you invite such that all the pies were eaten?”Students realized that the work was 8 divided by 2/3, but were having troubles explaining how to do this in words.After 5 minutes of struggling, the light bulbs started to come on.
Paraphrased here is the explanation:
First we need to figure out how many thirds are in 8 pies.We multiply 8 by 3 (using the same logic from above) and the product is 24. However this time, each person needs 2 pieces, therefore we now divide 24 by 2, and the answer is 12.
In the end, 8 divided by 2/3 is the same as 8 times 3, divided by 2.
Did my students know this already? Yes, but none of them could explain why.
Through my own experiences, I have realized the mathematical shortcuts, tricks and magic need to stop, while rationalizations, explanations, and justifications need to begin.