We had a great time last week working with cogs and I think the mathematical thinking my students did is the kind I want them doing all year. Unfortunately, this lesson is not one of them.
My colleagues and I discussed whether we should even teach these topics. They are definitely not sixth grade topics, but we need somewhere to start and once we start simplifying fractions, we need some common language. I start out by reminding students how they usually find the greatest common factor, by listing out all common factors. I’m always hoping that someone will have a better method, using prime factors, maybe? I then show students the ladder method we used for prime factorization and how it can help us here.
Once again we ask ourselves the same question, “What is the smallest prime that goes into…?”
Using the same method we used for prime factorization we divide out prime common factors. What we are left with is the prime factorization down the left side of the chart, the LCM can be found by multiplying all the outside factors together and there are several other patterns in the chart as well. The bottom factor on the right multiplied by the number on the top left also equal the LCM. Here is the same page with a few completed examples.
There are obvious implications here for finding common denominators. Students can quickly find common denominators and the chart will give the factors that the denominators need to be multiplied by to get to that common denominator. There are a lot of patterns for students to see here and I think it gives a real understanding of how prime numbers can help us work with much larger numbers.