Justifications first, shortcuts second.

“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.