## 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**: *I**n this picture circle a third of the six kittens.*

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

**Example: ***Circle half the stars.*

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.

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

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: .

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. *

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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