A student in a developmental algebra course was struggling with problems involving least common multiples. The lesson she was working on involved finding the least common multiple of numbers first, and then using that process as an analogy for finding least common multiples for variable expressions. Surprisingly, she felt confident with the variable expressions and was struggling with the numbers.
When I sat down with her, Karen (not her real name) could not understand why the technique she was using for variables was not working with numbers. She was able to correctly find the prime factors of numbers, so she wrote:
Then, drawing on her experience with fractions, she cancelled the common factors and used what was left:
She ended up with only a 2 and a 3 remaining, which is six and she knew this was not correct. Six is not even a multiple of these numbers, never mind the least common multiple.
I restated our textbook’s approach, to wit: We want to use each factor the same number of times as it appears in the number in which it appears most often. There really is no simple way to state this and I worked a couple of examples for her.
But I was intrigued by her ‘cancelling’ approach. In addition I had offered a strategy that, while supported by our textbook, bore very little relation to her way of thinking about the problem. This is not a recipe for success. We need to help our students refine their ways of thinking, not give them yet another rule to remember. So I explored her idea of cancelling and suggested this:
When we use each factor the greatest number of times as it appears most often, we can think of this as gathering all of the prime factors, then getting rid of the ones that come from numbers where they appear less often. When we have:
We want to “cancel” the 2′s in 36 and the 3 in 24-we don’t need those. So Karen’s Cancelling Algorithm-perhaps new to the world, and perhaps new only to her and to me is this:
Cancel the common factors only in one of the two numbers:
We cancel two of the 2′s in 24 because they match up with the two 2′s in 36. And we cancel the one 3 in 24 because it matches up with one of the 3′s in 36. We could have cancelled the two 2′s in 36 instead-that’s not important. What is important is that we cancel them only once.
Karen loved this algorithm and was very, very pleased to have her thinking changed into an algorithm that works.
The interaction was partly satisfying for me and partly disturbing. Karen began very frustrated and ended feeling successful and bright. That is always satisfying. But what had she really learned? She is not studying least common multiples for their interesting mathematical properties. Instead she is studying them in order to be able to add, subtract, simplify and solve rational expressions. Without questioning the larger goal of the whole enterprise of developmental college mathematics, it is still reasonable to ask how important least common multiple is for operating on rational expressions.
The only argument for finding least common multiple in this context is that it gives us a simpler form of the resulting rational expression than any other multiple will. If I am working with numbers, I can use the least common multiple to add:
But I can use any common multiple:
The least common multiple results in a simpler fraction, but it’s the same answer either way. Indeed, I can always use the common multiple that results from multiplying the two denominators:
And the same is true of the rational expressions Karen will be working with shortly. So why do we induce this stress in our students? If the only reason to find least common multiple is to work with rational expressions, and if at the same time any common multiple will do, why do put this artificial barrier in front of our students? And why do we, as teachers, allow ourselves to work as though these barriers were real?