I just finished helping facilitate a week-long workshop for most of the people teaching 5th - 9th grade math in two of our school districts. It was very interesting to be a part of, and to see what kinds of content and pedagogical knowledge teachers do (and don't) have. I'm not going to write much more about it, except I did want to post about common denominator division (CDD).

I brought this technique up to see if any teachers use it, and none had even heard of it. I've been teaching it to my students, as part of the scaffolding toward understanding the "multiply by the reciprocal" algorithm. Though not as efficient (typically), I think the CDD method can really help students see what is going on in fraction division, in terms of "the number of times the divisor fits into the dividend". It also fits nicely with multiplication, as "divide across" works just like "multiply across". The only problem is that "divide across" only comes out nicely when the denominator of the divisor is a factor of the denominator of the dividend, which is why (I assume) it is not the standard algorithm that is taught.

Once students know both methods, I give them a set of fraction division problems, and ask them to solve them with both methods. They then discuss which types of problems are better solved with which method, and why.

Here are some pictures that illustrate what is happening:

In this last one, you can see that the bottom shaded area fits into the top shaded area 1.5 times. This is the type of problem that I think really shows the benefit of CDD.

Here is a short proof that "divide across" works. Note: when the denominators are the same, dividing across yields 1 on the bottom, which I didn't write above.

Math Teachers at Play #78 via 1001 Math Problems

8 hours ago

## 5 comments:

It actually has some points going for it. Students already know how to find a common denominator (in theory) because they did it for addition and subtraction, and it's consistent with multiplication of fractions.

There's a good argument to be made that it's smart to teach this way, and then go to "invert and multiply" later.

Why not teach them the how and the why of multiplying by the reciproal?

Today we are going to talk about identity. If you multiply or divide any number by one, then what do you have? And you are off to the races...

I've always wondered why fraction division is not taught using CDD approach - it is such a natural follow-through from fraction multiplication --> simply the inverse!

Granted, when the denominators are the same (or when the divisor is a factor of the dividend), it is a lot easier to do. The challenge becomes what happens when the denominators (or the numerators) are not divisible.

One thought I had is that you could multiply one (or both) of the fractions by a form of one (as you would with adding or subtracting fractions) and then divide.

For example 1/2 divided by 1/3. If I multiply the first fraction by 3/3, I get 3/6 divided by 1/3. Dividing across now, I get 3/2, which is the same result as if I had inverted and multiplied, but without that magic trick.

I fully support CDD!!!

CDD was how I learned it when I was in 6th grade (1960-1961). The text used was "Arithmetic We Need" by Buswell, Brownell and Sauble, copyright 1955. After learning that, we went to invert and multiply and then did both for a while before abandoning CDD. Invert and multiply was introduced as a consequence of seeing the "pattern" of invert and multiply in other problems. Cutting 8 oranges into halves yields 16 halves, or 8 divided by 1/2 = 8 x 2. 8 divided by 2 is the same as 8 x 1/2. Dividing a half into two yields two quarters so 1/2 divided by 2 = 1/2 x 1/2. With the pattern established they ask the students to make the leap.

It looks to me like "invert and multiply" is really the same thing as CDD using the product of the denominators as the common denominator.

(a/b) / (c/d) = (ad/bd) / (bc/bd) = ad/bc

I suppose the advantage of "invert and multiply" is that it introduces the idea that division is the same as multiplying by the inverse.

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