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.

In our last post, we asked people if they could come up with a division story problem for $1\frac34 \div \frac12$. Interestingly, almost all of the responses used the “how many groups?” interpretation of division. When interpreting multiplication in terms of groups, the two factors play different roles, and so there is another interpretation of division worth exploring.

If we say that $a \times b$ means $a$ groups of $b$, then

• a division situation where $b$ and $a\times b$ are known but $a$ is unknown is called a “how many groups?” division problem
• a division situation where $a$ and $a\times b$ are known but $b$ is unknown is called a “how many (or how much) in each (or in one) group?” division problem.

[Pause here and see if you can come up with a “how much in one group?” story problem for $1 \frac34 \div \frac12$.]

How do these two interpretations of division come into play as students learn about fraction division? In grade 5, students solve problems like $6\div \frac12$ and $\frac12 \div 6$. What’s nice about problems involving a whole number divided by a unit fraction or a unit fraction divided by a whole number is that we can think of them using the same structure that we thought about division of whole numbers.

1. Kiki has 6 kg of chocolate chips. How many 2 kg packets of chocolate chips can she make?
2. Kiki has 6 kg of chocolate chips. How many $\frac12$ kg packets of chocolate chips can she make?

Notice that these are both a “how many groups?” division problem, and because there is always a whole number of unit fractions in 1, the solution will be a whole number of groups (so students do not have to worry about fractions of a group). If we write equations to represent these problems, that can also help us see the structure:

$$? \times 2 = 6$$

$$? \times \frac12 = 6$$

3. Nero had 6 cupcakes and 3 friends he wanted to share them with equally. How many cupcakes does each friend get?
4. Nero had $\frac12$ of a cupcake and 3 friends he wanted to share them with equally. How many cupcakes does each friend get?

Notice that these are both a “how many in each (or how much in one) group?” division problem, and students don’t have to worry about fractional groups because the whole number in the problem is the number of groups.

Again, with equations:

$$3 \times ? = 6$$

$$3 \times ? = \frac12$$

So in grade 5, students can build on their understanding of whole number division without having to grapple with fractional groups, so long as they understand both of these interpretations of division.

In grade 5, students also learn about fraction multiplication, so they do encounter fractions of a group, but they are not required to put these two understandings together until grade 6 when they extend their understanding of division to all fractions. This provides some scaffolding for students on their way to understanding division of fractions in general.

Let’s look at these two interpretations of division for $1\frac34 \div \frac12$.

• “How many groups?” : I need $1\frac34$ cups flour, but I only have a $\frac12$ cup measure. How many times do I have to fill the $\frac12$ measure to get $1\frac34$ cups flour? (A version of this was suggested by two different people on our last post.)
• “How much in one group?” : I have a container with $1\frac34$ cups flour. The container is $\frac12$ full. How much flour does the container have when it is full?

Here are two possible diagrams to represent these two interpretations of division:

Next time: how the different interpretations of division and diagrams can be used to understand the “invert and multiply rule” and other approaches to understanding this procedure.

In her book Knowing and Teaching Elementary Mathematics, Liping Ma wrote about this question and how teachers responded to it:

Write a story problem for $1 ¾ \div ½$.

[Pause here and think about the answer yourself.]

Many people find it hard to come up with a story problem that represents fraction division (including many math teachers, engineers, and mathematicians). Why is this hard to do? For many people, their schema for dividing fractions consists almost entirely of the “invert and multiply” rule. But there is much more to thinking about fraction division than that. So much in fact, that we can’t say it all in a single blog post. This is the first of several musings about fraction division.

The trouble with English

Consider this problem:

If you have 12 liters of tea and a container holds 2 liters, how many containers can you fill?

You probably know instantly that this is a division problem and that the answer is 6, because you know your times tables, and specifically you know that $2 \times 6 = 12$. If we say that $a \times b$ means $a$ equal groups of $b$ things in group, then a division problem where $b$ and $a\times b$ are known but $a$ is unknown is called a “how many groups?” problem. Here are some other questions that ask “how many groups?”

• If you have 1 ½ liters of tea and a container holds ¼ liter, how many containers can you fill?
• If you have 1 ¼ liters of tea and a container holds ¾ liters, how many containers can you fill?
• If you have ¾ liter of tea and a container holds 1 ¼ liters, how many containers can you fill?

Some people think that the last one feels like a trick question because you can’t even fill one completely. Because we know the answer is less than one, we could also ask it this way:

• If you have ¾ liter of tea and a pitcher holds 1 ⅓ liters, how much of a container can you fill?

So a division problem that asks “how many groups?” is structurally the same as a division problem that asks about “how much of a group?”, but because of the way we speak about quantities greater than 1 and quantities less than 1, the language makes the structure harder to see.

What other ways might we see the parallel structure?

Diagrams:

Equations: $$? \times2 = 12, \quad ? \times \frac14 = 1\frac12, \quad ? \times \frac34 = 1\frac14, \quad ? \times 1\frac14 = \frac34.$$  The diagrams don’t have the language problem. In all cases the upper and lower braces show the relation between the size of a container and the amount you have.  Whether a whole number of containers can be filled (diagrams 1 and 2), a container plus a part of a container can be filled (diagram 3), or only a part of a container can be filled (diagram 4), the underlying story is the same.

Many people think of diagrams primarily as tools to solve problems. But sometimes diagrams can help students see structure or reveal other important aspects of the mathematics. This is an example of looking for and making use of structure (MP7).

The equations have an even clearer structure, but more abstract. They all have the structure $$\mbox{(quantity of containers)}\times\mbox{(size of a container)} = \mbox{(how much you have)}.$$

The intertwining of the abstraction of the equations and the concreteness of the diagrams is a good example of MP2 (reason abstractly and quantitatively).

Coming up next week: what else are diagrams good for?