“Okay, listen up! Today’s lesson will be on adding fractions. Let’s start with an easy one like 1/3 + 1/6. The first step is to find a common numerator, which, in this example, we already have. This becomes the numerator of the sum so let’s write a 1 up there. The denominator is, of course, itself a fraction whose numerator is the product of the denominators and whose denominator is the sum of the denominators. This gives us 1/(18/9), or 1/2.
Let’s kick it up a notch and try 2/3 + 1/4. Remember, the first step is to find the lowest common numerator, or LCN. You guys look a little puzzled. You remember learning this in grade 7, right? Since the LCN is 2, we have 2/3 + 2/8. Write a 2 up top. To determine the denominator, simply multiply and add to get 24/11. We have 2/(24/11). This is a tricky one since 24/11 doesn’t reduce nicely. Multiplying the common numerator by the denominator of the denominator gives us 22/24. One more thing… if you don’t reduce to lowest terms, I’ll have to deduct half a mark. 22/24 should be written as 11/12. I’ve typed up some notes. Take one sheet and pass the rest back.”
Christopher Danielson over at OMT shared the method above with me earlier this year. Recently, I presented it to a group of secondary math teachers. Christopher’s algorithm brilliantly initiates conversation about what is important in teaching and learning mathematics. For example, one teacher said “It works. I can prove that it works. But, it doesn’t make sense.” Another asked “It’s quick and easy, but does that matter?”
I think Christopher (@Trianglemancsd) plays it straight when he shows his algorithm to pre-service teachers. I couldn’t pull this off – more of a tongue-in-cheek thing for me. This elicited some (nervous?) laughter as teachers put themselves in the role of their students learning about LCD’s.
This segued to activities that do build conceptual understanding of fraction operations. We looked at:
- using an area model to represent multiplication,
- using pattern blocks to explore quotative division, and
- using a common denominator to divide fractions.
These last two are connected… more on this later.
Chris Hunter 
You do know, of course, that you can divide fractions using common numerators, too. Surely you remember that from grade 7.
Get a common numerator, put the denominator of the divisor in the numerator of the quotient and the denominator of the dividend in the denominator of the quotient.
So 2/3÷1/4 becomes 2/3÷2/8=8/3
There will be a quiz.
The whole story of the development of the common numerator algorithm is in a published article in Mathematics Teaching in the Middle School.
So happy to know it was useful for you.
This is so dumb, algorithms show no understanding, just memorization