The Sharp Mathematics Contest #2 -- Winners

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In The Sharp Mathematics Contest #2, the following challenge was given:-

The Challenge

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Comments on Participation

The challenge was meant to encourage readers to read and apply information from my article Visualising the Division of Fractions and its follow-up article Visual Division of Non-Unit Fractions. The purpose of the articles is to help teachers, young children, parents and all human beings in general to really understand meanings and make sense in mathematics, and not just know how to use formulas.

As in contest 1, only two people submitted entries, and there are 3 prizes. So the third prize (2 SBD) is left unclaimed.

Part (a) was quite easy.

Actually both participants put in a lot of effort with the entries, and at first I had trouble deciding which the better one is.

@beachhermit's entry uses software graphics. For part (b) she explains division of 6 by two-thirds and verifies that the result 9 is the same as multiplying 6 by 3/2. (c) is done by comparing the result of dividing by 3/2 with multiplication by 2/3.

@orbital753's entry uses hand-drawn diagrams. He seems to be following some algorithm for doing part (b) and (c). For example, when dividing 6 by 2/3, he seems to be doing as . Although the answer is correct, if I were a young kid learning it this way, I could be quite confused because I cannot directly see where the "2/3" is until later.

For part (b), instead of verifying LHS and RHS are both equal to 9 (which we all know) I think @beachhermit could directly show that they are equal by saying that every 3 of the new units came from two of the old units (shown in yellow and purple), so the number of new units is multiplied from the number of old units via 3/2.

Similarly for part (c), we can directly see that LHS = RHS, since every 2 of the new units came from 3 of the old units (shown in green, blue and pink). So the number of new units is 2/3 of the number of old units.