Fractions are Hard!
Overview
Many a student—and adult!—is scared of fractions. And their reasons why are natural and absolutely appropriate: fractions are nuanced and subtle and hard.
In our early years we learn that fractions are “parts of a whole,” portions of pie, points on a number line between the whole numbers, and the like. That is, we are given a series of models for what a fraction is. But it is never made clear which of these models speaks the final, definitive truth. We march from one model to the next and are left to believe that, somehow, that covers it all.
The fact is that each model only speaks partial truth. No one model speaks the entire truth.
For example, the portions of pie model “explains” the addition of fractions: to add half a pie and a third of a pie, just bring those two portions of pie physically together and figure out what portion of a whole pie that is. But the pie model is useless for making sense of the multiplication of fractions: What on earth is half a pie times a third of a pie? (“One sixth pie squared”?)
The number line model suggests that fractions are numbers (there they are: on the number line!), but to explain their multiplication we go back to saying “of means multiply,” harking back to the parts of a whole model.
It’s a spaghetti bowl of ideas and justifications!
But the work of these early grades is important and right. It develops our intuition and insight as to how a number system of “fractions,” whatever they are, should behave. High school students and adults are ready for the next final step of the fraction story: to step back from the story presented in the early grades and reflect on what it is all trying to say, and create a comprehensive overview of what is really going on in that pasta bowl. I’ve never seen this done in a high school curriculum. So let’s do it now!
These notes, Arithmetic Chapter 5, from a new text on Arithmetic, are for all those who realise that they never really “got” fractions, but want to get them now.
Enjoy!
Chapter 5: Fractions
32. Too Many Models for Intuition
33. One (still not perfect) Model
Pies and Students
Going Backwards
Some Apparent Properties of Fractions
ASIDE: Something Quirky
More Apparent Properties of Fractions
A Logical Consequence of our Four Properties
Comparing Fractions
Zero as a Denominator
Checking Values of Fractions
Mixed Numbers
Dividing Fractions
Multiplying Fractions
A Happy Coincidence: The word “of”
Division and “of”
Adding Fractions
Subtracting Fractions
Mixed Numbers Again
Multiplying and Dividing by Positive Numbers Bigger and Smaller than One
34. The Mathematical Truth about Fractions
35. All the Rules of Arithmetic in One Place
36. So … What is a Fraction?
If you are interested in decimals too, and the play of fractions and decimal, let me know and I’ll send you chapter six!
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