Fractions Are Instructions to Divide! Sir!

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Just how do you go about solving a question like: What is 3/12 of 36 penguins? There seems to be an almost endless list of strategies that don't work well. However, there is one strategy that works VERY well.

In my class, I have the kids repeat, military style, "Fractions are instructions to divide by the denominator and then multiply by the numerator! Sir!"

Almost all kids can remember this"direct order". Many do not, however, know what needs dividing up, or into how many groups. In the problem above, the 36 penguins need to be divided into 12 even groups, and that puts 3 penguins in each group. 3 would be a fine answer to the question, "What is 1/12 of 36 penguins?", but is not a good answer for 3/12. This is where the "and multiply by the numerator" comes into play. We need to take into account 3 of the 12 equal groups of penguins, or 3 X 3 = 9 penguins.

The work also shows an array of 36 "penguins", and it would be quite correct to say that 3/12 could be understood as 3 out of every 12. This is a great model for small numbers, but I would not want to split up 3600 penguins into an array!

So, "Fractions are an instruction to divide by the denominator and then multiply the answer by the numerator! Sir!"