Fractions are Hard!

1.2 The next story of fractions

At some point in our grade-school experience we construct a geometric model of the number system: the number line.

Here’s a picture of 5 unit segments:

It seems natural to arrange these along a line:

We can mark, with whole numbers, the counts of segments that line up to reach locations along the line. The line segments are all placed to the right of a special point labeled “0.”

In this way, we learn to think of these numbers as specifying actual points on the line. The point labeled 5, for example, is the point on the line five units to the right of the point labeled 0. The line with points labeled this way is called the number line.

UPSHOT: We learn to associate numbers with points on the line, and – surreptitiously – the reverse too: all points on the line likely correspond to numbers.

We are next asked to consider parts of line segments, that is, to work with a unit line segment as though it is a pie. We develop familiarity with parts of segments.

Placing these parts of line segments on the number line then, again surreptitiously, suggests that there are points on the number line that deserve to have fraction labels.

UPSHOT: It is implied that fractions are numbers: they are drawn in a number line after all.

Following the work of the very early grades, a quantity such as $$\frac{7}{3}$$ is read as “seven thirds” (as is done in everyday language) and is interpreted as the final location of seven copies of a third stacked together:

$$\dfrac{7}{3} =$$ “seven thirds”  $$=$$  seven of these $$\dfrac{1}{3}$$

$$= \dfrac{1}{3} + \dfrac{1}{3} + \dfrac{1}{3} + \dfrac{1}{3} + \dfrac{1}{3} + \dfrac{1}{3} + \dfrac{1}{3}$$

$$=$$  seven groups of  $$\dfrac{1}{3}$$

$$= 7 \times \dfrac{1}{3}$$.

At the same time we start to learn that $$\dfrac{7}{3}$$, for example, is “7 divided by 3.” That is, it is the result of taking a segment 7 units long, dividing it into three equal parts, and selecting one of those parts.  This is fine, but is it at all obvious that $$7 \div 3$$ lands you at the same place on the number line as $$7 \times \dfrac{1}{3}$$?

CHALLENGE: Can you personally explain why dividing a section of length seven units long into three equals parts

is sure to land you it the same location as seven copies of $$\dfrac{1}{3}$$? (It is not “obvious”.)

THE START OF SOME ARITHMETIC FOR FRACTIONS

The notion of addition and subtraction is natural in the number line model: just stack and remove sections of length.

So adding $$\dfrac{7}{3}$$ and $$\dfrac{10}{3}$$, for instance, is naturally seen as 7 thirds and 10 thirds stacked together making 17 thirds.

$$\dfrac{7}{3} + \dfrac{10}{3} = \dfrac{17}{3}$$.

If all fractions are indeed in the same units, such as thirds, then adding and subtracting fractions is just as easy as adding and subtracting apples. A trouble comes if one wants to add or subtract fractions expressed in different units. For example, how does one handle $$\dfrac{1}{2} + \dfrac{2}{3}$$? In this number line model, we do see that it has an answer, but it is hard to figure out what the answer is.

MORE ARITHMETIC?

If fractions are now allegedly numbers and we can perform some arithmetic with them – namely addition and subtraction – surely then we must be able to multiply and divide fractions too? How?

UPSHOT:  We first have the impression that fractions are not numbers and that we certainly cannot add and subtract them (after all, adding a third of a group of kittens and a half of a group of stars is meaningless). Then we have the impression that fractions are numbers, they appear on the number line, and we can add them after all. (So what happened to kittens and stars?) Since fractions are now supposedly numbers, we should be able to multiply and divide them too – but it is not at all obvious from the number line how this would work. (And if it does work, this means we can divide kittens and stars too?)

This is philosophically all very strange!

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