Fractions are Hard!


The truth is that fractions are hard! And the reason why is that we are never told what a fraction actually is.

First, fractions are “calls to action” – identify a third of the kittens, take a fourth of the pie – and certainly are not quantities we can add or multiply. (How do you add kittens and pie?) But then we draw fractions on the number line to imply that they are numbers after all. Okay then. We can add and subtract numbers on the number line, but how do we multiply and divide them? For these latter two operations we might just be told that “of means multiply” and to divide fractions just “multiply by the reciprocal.” (What’s the reciprocal of a portion of kittens?)

Our first eight years of learning fractions is a confusing mash-up of ideas drifting in and out of grade-levels in an apparently random way. (Grade 4/5: Don’t draw portions of pie any more – fractions are numbers in their own right. Grade 6/7: To multiply fractions, think portions of pie. Grade 7/8: To divide fractions – just follow this mechanical rule.)

Here’s my approach to fractions to help young adults (and not so young adults) ready to make some personal sense of the fraction story. Part 1 of these notes goes through the various, confusing stories we typically see in grade and middle school and illustrates why we are confused. Part 2 of these notes takes a more solid approach, providing one consistent model that seems to give good solid intuition for it all. But we’re being mathematically honest here, noting that we are really just playing the game of identifying the key beliefs that seem to make all the fraction models we ever seem to talk about “click.” This is the university-level approach to explaining what fractions are, and I explain it here too. (And this shows why we can’t give the true story to grade-schoolers and middle-schoolers: it is just not pedagogically appropriate.) Part 3 fills in all the extra bits you might want to do with fractions, and part 4 offers some cool mathematics extensions.

These notes are not classroom notes – just a detailed outline of the ideas of how a set of curriculum notes might go. I have teams of teachers looking into bringing these materials to classroom life (adjusting the pacing, adding worksheets and lesson plans, and producing – joyous, not sour – assessment pieces). Work through these notes for your own personal edification. I hope they help.


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