Questions from middle school teachers: Division of fractions

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.

sharing

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.

measuring

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:

  1. 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?
  2. 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.

Summary

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.

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14 responses to “Questions from middle school teachers: Division of fractions

  1. Addendum

    For more on meaning in the operations, see also:
    Cardinal and ordinal numbers
    Inverse operations and
    The end of word problems redux.

  2. Follow up

    I got an email this morning from the friend who asked the question in the first place:

    Yes, exactly what we both needed to think about.
    A very mundane thought…teaching division by fractions would be so much easier if students
    a) understood division and
    b) understood fractions
    I think this helps me appreciate more so the culture of learning and teaching in countries like Singapore and China where so much more emphasis is placed on the development of conceptual understanding in the early years. It also makes me appreciate the work my friends in England did to develop their curriculum.

  3. I love the quote, “teaching divisions by fractions would be so much easier if students understood division and understand fractions.”

  4. Well written post, Christopher. I agree with you completely, that students need to understand the math they are learning, or there is very little point in teaching it to them. I always find the division by fractions operation really challenging to teach, since students, as you point out, often don’t understand either fractions or division. The “invert and multiply” method is so simple and yet so complicated for novices to understand that I avoid it almost completely.

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  10. Iztchel. This is so great. I always have a hard time teaching fraction division with common denominators even though my students always want to come with common denominators as their first strategy when developing algorithms for fraction division (Bits & Pieces II). Last time, I taught it, I completely forgot to bring it up in my conclusion and then it came out in the getting ready activity in Bits and Pieces III. It felt a bit force when I brought it then. But your well thought chosen examples makes it a lot easier to bring it up againg even if I forgot it the first time. Thanks!!

  11. Michael Paul Goldenberg

    Am I missing something here, or has no one mentioned that if you divide fractions with common denominators that the denominators no longer matter and that hence you wind up “merely” with the ratio of the original numerators? That is to say that a/b / c/b = a/c, and it seems almost superfluous to multiply the compound numerator and denominator by the reciprocal of the denominator (“invert and multiply”).

    I would suggest that the ratio of “a pieces of size b” to “c pieces of size b” should be exactly what it is: a/b or a:b.

    To my way of thinking, this isn’t an algorithm but an insight into division and fractions and ratios.

    Now, it’s trivial to look at the case of common numerators with different denominators ( a/b / a/c ) and conclude via algebra that this must equal c/b. But it’s not quite so obvious what this means and is definitely worth considering. Perhaps everyone here already knows all of this, in which case my apologies.

  12. I have responded to MPG‘s comment on a more recent post on the common numerator algorithm.

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  14. Great post! I also agree that conceptual understanding of division and fractions would be the biggest help. I must confess, though, that I’m not actually familiar with the common denominator algorithm – I’ve only ever seen and taught the invert & multiply method! Interesting to read the discussion, and it gives me another method to use with my students. Thanks!

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