## LCM, GCF, and Fractions.

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…?”

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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.

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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.

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## Cogs

I was really looking forward to this problem. I couldn’t wait to see what my classes would do with it and it did not disappoint. I mixed up my classes (assigned seats) and gave them very little direction. I wanted to see what kind of mathematical thinking they would do on their own.

nrich.maths.org

The Counting Cogs is very specific and Nrich has even included step-by-step group directions so students can easily manage the problem in a group. Students need to discover which pairs of cogs will allow a colored tooth on one cog to go into every gap on the other cog. Students cut out the cogs, colored one tooth, and started spinning cogs. Nrich even has an interactive that helped me give some classes an idea of what they should be doing. Thank you Nrich for a great problem.

I was thrilled with the thinking. Some students quickly realized they needed some way to keep track of pairs and created great tables and charts. Some students wanted to immediately begin making conjectures. We talked about proving what we believe to be true, and they began to see that one example is not enough to be a proof. Some students even started creating other cogs than the ones supplied to see if their conjecture really worked. I heard a lot of conjectures! Tomorrow when we begin discussing this problem students will quickly discover the cogs that are relatively prime to each other work and the others do not. I did hear this conjecture today, but students just didn’t have enough time to really prove it.

This problem let students extend their thinking about factors, primes, and relatively prime numbers. They were actually applying these concepts rather than completing another worksheet. I really like the way multiple concepts are intertwined in this problem.