Fractions are Hard!

1.1 The first story of fractions

When we were very young we were introduced to fractions as PARTS OF A WHOLE and we were taught to be flexible as to what the “whole” might be. We answered questions such as the following.

 

Example: In this picture circle a third of the six kittens.

F1

Here the “whole” is the group of six kittens. Here’s a third of this whole:

F1

Example: Circle half the stars.

F1

Here the “whole” is now four stars and we’d circle two of them.

 

Notice here that a fraction isn’t a number in and of itself. A fraction is simply a call to action. Why would you think a fraction is a number? [Actually, if a fraction is a number, then we should be able to add fractions. But we can’t! What is a third of the kittens plus half of the stars? Addition makes no sense in this mode of thinking. This proves that fractions can’t be numbers.]

 

Pretty soon pies enter the picture.

 

Example: Draw a picture of half a pie, a third of a pie, and a quarter of a pie.

 

Here the “whole” is a pie.

F1

It’s actually easier, and psychologically easier, to work with square pies (or pans of brownies).

F1

With pies, we start associating pictures with the results of sharing problems: if one pie is shared equally between two people, then each person gets this much pie: F1.

 

We develop familiarity with fractions of the basic form \(\frac{1}{N}\) (that is, \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\), and so on), but still in the sense of being a “call to action” (circle half the houses, draw a fifth of a pie).

 

A complex fraction, \(\dfrac{a}{N}\), is interpreted as \(a\) copies of \(\frac{1}{N}\).

 

Example: Two thirds of six kittens is two copies of one third of six kittens.

F1

Three-sevenths is three groups of one-seventh of the whole, and so on.

 

UPSHOT: In this story, our first exposure, fractions aren’t numbers. They are simply “calls to action” to identify parts of a whole.

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