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So far, we have been thinking about a fraction as the answer to a division problem.  For example, $\frac 2 3$ is the result of sharing two pies among three children.

Of course, pies do not have to be round. We can have square pies, or triangular pies or squiggly pies or any shape you please.

This “Pies Per Child Model” has served us perfectly well in thinking about the meaning of fractions, equivalent fractions, and even adding and subtracting fractions.

However, there is no way to use this model to make sense of multiplying fractions! What would this mean?

So what are fractions, if we are asked to multiply them? We are forced to switch models and think about fractions in a new way.

This switch is fundamentally perturbing. Think about students learning this for the first time. We keep switching concepts and models, and speak of fractions in each case as though all is naturally linked and obvious. None of this is obvious, it is all absolutely confusing. This is just one of the reasons that fractions can be such a difficult concept to teach and to learn in elementary school!

Think / Pair / Share (What’s wrong here?)

For each of the following visual representations of fractions, there is a corresponding incorrect symbolic expression.

• Why is the symbolic representation incorrect?
• What might elementary students find confusing in these visual representations?

$\frac 1 3$	$\frac 2 3 > \frac 3 4$
$\frac 1 3 \neq 1 3$

Units and unitizing

In thinking about fractions, it is important to remember that there are always units attached to a fraction, even if the units are hidden. If you see the number $\frac 1 2$ in a problem, you should ask yourself “half of what?” The answer to that question is your unit, the amount that equals 1.

So far, our units have been consistent: the “whole” (or unit) was a whole pie, and fractions were represented by pies cut into equal-sized pieces. But this is just a model, and we can take anything, cut it into equal-sized pieces, and talk about fractions of that whole.

One thing that can make fraction problems so difficult is that the fractions in the problem may be given in different units (they may be “parts” of different “wholes”).

Ordering Fractions

If we think about fractions as “portions of a segment,” then we can talk about their locations on a number line. We can start to treat fractions like numbers. In the back of our minds, we should remember that fractions are always relative to some unit. But on a number line, the unit is clear: it is the distance between 0 and 1.

This measurement model makes it much easier to tackle questions about the relative size of fractions based on where they appear on the number line. We can mark off different fractions as parts of the unit segment. Just as with whole numbers, fractions that appear farther to the right are larger.

You probably came up with benchmarks and intuitive methods to think about the relative sizes of fractions. Here are some of these methods. (Did you come up with others?)

Arithmetic Sequences

Consider the patterns below.

Pattern 1: $\displaystyle 5, \quad 8, \quad 11, \quad 14, \quad 17, \quad 20, \quad 23, \quad 26, \dots$

Pattern 2: $\displaystyle2, \quad 9, \quad 16, \quad 23, \quad 30, \quad 37, \quad 44, \quad 51, \dots$

Pattern 3: $\displaystyle\frac15, \quad \frac35, \quad 1, \quad \frac75, \quad \frac95, \quad \frac{11}{5}, \quad \frac{13}{5}, \quad 3, \dots$

The patterns above are called arithmetic sequences: a sequence of numbers where the difference between consecutive terms is a constant. Here are some other examples:

Pattern A: $\displaystyle\underbrace{1, \quad }_{+1}\underbrace{2, \quad }_{+1}\underbrace{3, \quad }_{+1}\underbrace{4, \quad }_{+1}5, \dots$

Pattern B: $\displaystyle \underbrace{2, \quad }_{+2}\underbrace{4, \quad }_{+2}\underbrace{6, \quad }_{+2}\underbrace{8, \quad }_{+2}10, \dots$

Pattern C: $\displaystyle \underbrace{\frac13, \quad }_{+\frac{2}{3}}\underbrace{1, \phantom{\frac 12} \ }_{+\frac23}\underbrace{\frac53, \quad }_{+\frac23}\underbrace{\frac73, \quad }_{+\frac23}3, \dots$

Here are several more number patterns:

Pattern 4: $\displaystyle 1, \quad 2, \quad 4, \quad 8, \quad 16, \quad 32, \quad 64, \quad 128,\dots$

Pattern 5: $\displaystyle 1, \quad 3, \quad 6, \quad 10, \quad 15, \quad 21, \quad 28, \quad 36, \dots$

Pattern 6: $\displaystyle \frac25, \quad \frac{7}{10}, \quad 1, \quad \frac{13}{10}, \quad \frac85, \quad \frac{19}{10}, \quad \frac{11}{5}, \quad \frac52,\dots$

Pattern 7: $\displaystyle \frac35, \quad \frac65, \quad \frac{12}{5}, \quad \frac{24}{5}, \quad \frac{48}{5}, \quad \frac{96}{5}, \dots$