The Background
It's confession time: I was a compulsive fraction simplifier. Not in lowest terms? Simplify it. Improper fraction? No question. Change it to a mixed number. No rhyme or reason, except that the simplest form or mixed number was the only correct answer most of the time... and an improper fraction just felt wrong.

As a student I learned the tricks for converting improper fractions to mixed numbers and vice versa as isolated procedures, and as a 5th and 6th teacher B.C.C. (Before Common Core) I taught it that way. There was very little concept or context... as

Phil Daro would say, it was answer-getting, not math. Something like this:

If I was lucky, students put all the digits from their division problem into the correct spot in the mixed number. Spoiler alert: I wasn't always that lucky. Why do we convert improper fractions to mixed numbers anyway? (wait time) No, really, why?

Turns out, there's no mathematical reason why we MUST convert an improper fraction t…

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