The following was sent along to me by a dear friend and mentor. It is an excerpt from a conversation she was having with a seventh-grade teacher she works with:
What are your thoughts on teaching students to divide fractions with common denominators?
[With respect to the invert-and-multiply algorithm] I know that I have walked students through this process with fraction circles. Given the right practice problems using the reciprocal makes a lot of sense. It also gives us an opportunity to talk about reciprocals!
I just wonder if, since students appear to struggle immensely with fractions in general, that having a common process for the 4 operations would make it easier to find more success. Students who easily see patterns, with a good dose of number sense, would then discover the reciprocal piece on their own.
The first thing that struck me on reading this was meaning. As math teachers, we pay very careful attention to abstract representations and we sometimes allow the meaning to slip away.
Notice that the problem (learning to divide fractions) and the proposed solution (an algorithm) are both phrased in abstract terms. Neither mentions the meanings of the division process. A quick tutorial on meanings of division, then I’ll apply to fraction division.
Write a story problem that leads to whole-number division. Most adults given that task will produce something equivalent to:
I have 35 apples to share equally among 7 people. How many apples does each person get?
In this problem, we have some number of objects (35) that we are dividing up into a known number of groups (7). We need to figure out how many are in each group. This first meaning for division is variously known as sharing, partitive and how many groups? division. These terms all mean the same thing.
But there is another meaning for division. Consider the following problem.
I have 35 apples to package in bags. I can fit 7 apples in each bog. How many bags can I fill?
In this case, we know how many are in each group, but we don’t know how many groups we can make. This meaning for division is referred to as measuring, measurement, quotative and how many in each group? division. These terms all mean the same thing.
Application to fraction division
While the sharing interpretation of division is the iconic one for most people (adults included), it is harder to conceptualize in the world of fraction division than the measuring interpretation is. Consider the following two problems:
- I have 3/4 lb. of cookie dough. I am making large cookies using 1/4 lb. of dough. How many cookies can I make?
- I have 3/4 lb. of cookie dough. I am making large cookies and that 3/4 lb. is enough to make 1/4 of a cookie. How much do I need to make a whole cookie?
The first problem is the measuring problem. How many 1/4’s are in 3/4? Each “group” is of size 1/4 lb. I need to know how many groups I can make.
The second problem is the sharing problem. What is 3/4 one-fourth of? If 3/4 is 1/4 of a group, how much is in one whole group?
With enough exploration of these kinds of contexts, the common denominator algorithm for dividing fractions can emerge. If we are asking how many 2/3 are in 7/9, we can notice that it would be easier to think of 2/3 as 6/9. Then we can see that we have one group of 6/9 in 7/9, with 1/9 left over. That 1/9 is 1/6 of a group, so 7/9 ÷ 6/9 = 1 and 1/6.
And ultimately, 7/9 ÷ 6/9 just refers to 7 things and we are making groups of 6 things. The common denominators ensure that the things are the same as each other-they are each ninths of the same whole. So 7/9 ÷ 6/9 is equivalent to 7÷6.
The invert-and-multiply algorithm is more commonly associated with sharing situations.
There have been several good articles on fraction division in the journal Mathematics Teaching in the Middle School in recent years. The best of them (in my opinion) is titled “Measurement and Fair-Sharing Models for Dividing Fractions” by Gregg and Gregg. But a quick search on the MTMS website will turn up several more.
So, what do I think of teaching students the common denominator algorithm for dividing fractions? I want to start with meaningful situations for doing the division. If we do that, then the common denominator algorithm is the one that stands a chance of making sense.
If, by contrast, we are working on a purely abstract level, then I don’t much care which algorithm we use. Both are equally efficient (less canceling in the quotient with common denominators, but more time spent looking for common denominators). Both extend to algebraic fractions, which is what mathematicians are usually concerned with. Indeed, I can teach (and have taught!) an entire calculus course in which the only algorithm for division I use is common denominators.
But I would be wary of the surface similarities implicit in wanting to “hav[e] a common process for the 4 operations”. The process isn’t really the same. Notice that in the common denominator algorithm for dividing fractions, the common denominator disappears in the quotient, while it does not do so when we add or subtract. And using common denominators for multiplying fractions seems like a lot of wasted effort, since we’ll need to cancel out those extra factors at the end of the computation.
I say go for meaning. Then the commonalities will be within each operation-division always means sharing or measuring whether we are using whole numbers or fractions-rather than across the operations on an abstract level.