Notes on Fractions – Deferred Computation

What would happen if we thought of a fraction as a division that hadn’t been done yet and that we weren’t necessarily interested in doing yet?  I’ve suggested in earlier posts (e.g. here and here)  that kids often don’t see the connection between fractions and division, that they often seem to work hard to make sure they don’t confuse the two.  I venture to guess that depending on your own background you are likely to respond either “of course fractions are just like division” or “of course, fractions and divisions are very different.”

Kids in school certainly have plenty of reasons to suspect that fractions are different from division.  In no particular order: one, fractions are written as \frac{17}{5} where division is written as 17 ÷ 5 or 17 / 5; two, in a division you can sometimes give an answer as a whole and a remainder (e.g. 17 ÷ 5 is 3, remainder 2) and you would never do this for a fraction; three, a division has a dividend and a divisor, a fraction has a numerator and a denominator; four, fractions are shown in the book as slices of a single pizza, where division is shown as 159 candies divided over 13 kids; five, a division problem you can “do”, where a fraction problem you can only “reduce”.

Conversely, if you think that fractions are just like divisions, you would have to answer why we’d ever bother to write \frac{3}{4} instead of .75, or why we’d ever write \frac{1}{3} instead of .3 with the usual overbar over the 3 to show that the 3 repeats indefinitely.  You might conclude that even if doing the division fully (that means sometimes getting repeating decimals) is possible, doing the division each time must at one point in history have been considered very impractical to do.

This leads me to suggest we try out the notion that a fraction might be usefully thought of as a way of writing a division we haven’t done yet and that we don’t necessarily have any urgent plans to do.  We defer the computation till a later date, hoping that something will intervene that saves us from having to do the division, a division that would result in a (perhaps repeating) decimal.  To put it crudely, if we have to divide 17 by 5, we may decide to punt, and leave “the answer” in the form \frac{17}{5} to suggest we can’t be bothered to convert the whole thing into a (perhaps repeating) decimal right now.

In later posts, I intend to play with this notion some more and see what it leads to.

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