Well, all good things must come to an end. In our previous three blog posts, we discussed some affordances of using diagrams to understand fraction division. In this post, we will talk about why it is important for students to go beyond diagrams.

###### The limits of diagrams for solving fraction division problems

In our earlier posts, we argued that diagrams can help students see the structure of a problem and understand why it can be represented by division. However, diagrams are rarely efficient for carrying out the resulting fraction division. For example:

Mateo filled a 1 pint measuring cup with water until it was $\frac{7}{16}$ full. If a recipe calls for $\frac23$ pints of water, what fraction of the recipe can Mateo make with the water in the measuring cup?

Drawing a diagram for this problem is not the most efficient method (try it!). A student who has learned to see this as $\frac{7}{16}\div \frac23$ (through working with diagrams) would most efficiently calculate that value using the invert-and-multiply rule without worrying about a diagram for that particular calculation.

###### Explaining the fraction division rule using algebra

Last time we argued that the “How much in one group” interpretation with the right kind of diagram can help us see why dividing by a fraction is the same as multiplying by its reciprocal.

For example, a diagram that represents a situation where $\frac25$ of a number is $1\frac34$ can show that we can multiply $1\frac34$ by $\frac52$ to find that number.

What if we were to think about this from a completely algebraic perspective? By the definition of division,$$1\frac34 \div \frac25 = x$$ means that: $$\frac25 x = 1 \frac34.$$

To solve an equation like this, we simply multiply both sides of the equation by the multiplicative inverse of $\frac25$:

$$\frac52 \cdot \frac25 x = \frac52 \cdot 1 \frac34.$$

In other words: $$ x =1 \frac34 \cdot \frac52.$$

We are not claiming that students need to be able to make a formal argument like this in order to justify the general rule for dividing fractions! But they do, eventually, need to be able to solve specific equations of the form $$\frac25 x = 1 \frac34.$$

Students who can solve equations flexibly might find the solution by rewriting an unknown factor problem as a division problem: $$x = 1\frac34 \div \frac25,$$ or by multiplying both sides of the equation by the reciprocal of $\frac25$: $$\frac52 \cdot \frac25 x = \frac52 \cdot 1 \frac34.$$

Both methods were implicit in many of the fraction division problems students have been conceptualizing with the help of diagrams, although there may not have been an explicit equation in those problems. Using equations formalizes, makes explicit, and encapsulates the implicit understandings. So students who investigate fraction division with diagrams should have the opportunity to make connections to algebraic approaches as well.

###### Final thoughts

Fraction division is a topic that students encounter at a key time in their transition from their work in elementary school arithmetic to their study of algebra as generalized arithmetic in middle school and beyond. Appropriate use of diagrams can help them understand how fraction division relates to their earlier study of division of whole numbers and when a problem can be represented by fraction division. Diagrams can also mediate students’ transition to a more structural, abstract understanding of fraction division that is represented using numeric and algebraic expressions and equations. In general, diagrams can play a key role in helping students make the transition from arithmetic to algebra, as we have illustrated in the particular case of fraction division.

Another important perspective that shows coherence with whole number division is this:

Starting with a naked problem,

3/4➗2/5

Notice that we have 3 of one unit divided by 2 of another. We know how to use equivalent fractions to convert to common units; we have done it to add, subtract and compare fractions.

Compare is relevant here. How much of 3 is 2? Or how much smaller or bigger is 2 than 3? Why 2 is 2/3 of 3.

Now back to 3/4➗2/5

Change both fractions to equivalents with common unit fractions in the usual way:

3/4 = 15/20

2/5 = 8/20

Now we have 15 of 1/20 divided by 8 of 1/20. Since the units are the same, this is equivalent to 15 of 1 unit divided by 8 of 1 unit, which is 15/8.

You could also write (15 x 1/20)/(8 x 1/20) and cancel (1/20)/(1/20).