## 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.

## Primes

I spent a few summers working in the PROMYS program for teachers at BU a few years ago. You spend the six weeks working with other teachers on number theory topics. Somehow the beauty of number theory had escaped me until this time. Ever since that summer I have really enjoyed teaching prime numbers.

Here are my thoughts on the next week. My sixth graders have seen prime numbers before. I always start teaching it as if they’ve never seen them before. I’m going to have them start by completing a frayer model with four vocabulary words, prime, multiple, factor, and composite. I’m taking the foldable frayer model from here, but we’re not going to fold it.  My thought is once their memories are jogged most students already know these concepts. Why am I reteaching them?

Next up, a little DIVISIBLE Lab. Students once again have seen the divisibility rules in the fifth grade. It drives the faster students crazy when I make them sit and outline all the rules again for the students that have forgotten them. This lab allows students to get up and move around the room. They cannot leave a station until they have the teacher sign off that they are correct. Each letter in DIVISIBLE is a station and students have a lab sheet with questions they complete at each station. I will give students a sheet with all rules written out to keep in their binders after the activity.

Now, the Sieve of Eratosthenes. I used to think it was a waste of time, but I think it is a good activity for students to really see the power of primes. Students go through and cross out all the multiples of two, all the multiples of three, etc. In the end they are left with only the primes. Students then see that there really is no pattern to the primes, that not all odd numbers are primes, all the common misconceptions.

After this, we’ll start prime factorization. I used to teach factor trees, but I had students missing factors when they went to write out the factorization. Last year I tried the box method. Students only have to ask themselves one question, What is the smallest prime that goes in to _________? It looks something like this:

 2 24 2 12 2 6 3 3 1

The prime factorization is easy for kids to pull out down the left side. They know when they are done because they get one in the bottom. This makes everyone’s life easier.

Tagged , ,