First Week

I've only taught three days at my new (and first!) school, and it feels like I've been there at least a month already. The first week went well enough, and I enjoyed the math I did with the students in those first few days. Though I probably didn't implement them to their full potential, these problems really got students thinking and gave me some insight into their abilities in math.

In Geometry, we did the Square-ness problem from Balanced Assessments. Students are given 9 rectangles and asked to put them in order of "square-ness." Many had no problem with this, but struggled with the follow-up questions asking why they arranged them like that, whether they could find a rectangle with an intermediate value of square-ness if given two other rectangles, and if they could come up with a formula for measuring square-ness. I didn't expect students to have all of the answers, just to be challenged and to start thinking geometrically. I let them work for most of one day on it, and we went over it two days later. I didn't spend very much time coming up with an accurate formula for measuring square-ness, but we came up with something that was on its way to being accurate. Many students realized you could subtract the side lengths to determine how square something is. Then in my last hour of geometry, when we were talking about finding the difference between the side lengths, one student piped up that it wouldn't quite work. He asked what would happen if one rectangle was much larger than another -- then our measure of square-ness wouldn't work. He came up with the idea of dividing the side lengths. Others thought maybe we could scale one rectangle to have an equivalent side length to the other rectangle, and then subtract the sides. That's much harder to put into a formula, but the idea works. Many students conflated the concept of area with square-ness while working on this activity, and now I think I see why. First of all, when asked to come up with a formula, many came up with l * w because that's a formula they already know related to rectangles. They didn't think about whether it works or not. But I think there's a deeper intuition here, as well, that some students had. While area doesn't affect square-ness -- a square is a square no matter how large it is -- you have to think about area when coming up with a measure of square-ness to avoid accidentally allowing area to mess up your measure. That's what the student was getting at in my last hour of geometry. I was surprised and delighted with his insight. I thought most students were barely holding on to the argument, and here he was ahead of me. One of the signs of a good task is a low entry point and a high ceiling, and I think this task hits the mark on both of those.

In Math 8, we did the Handshake (or Supreme Court Welcome) problem from Illuminations. I also showed this video of Sandra Day O'Connor talking about shaking hands with the other justices during meetings.

 I had students make guesses at the beginning of class to get them invested. Most came up with 81 (9*9), which was more of a quick calculation than a guess. I saw some good representations on paper as I watched them work. In my smaller class of 13, we had time to let students present their solutions to the class. I wanted them to see that there are different ways of representing math using diagrams. No one came up with the network solutions, so I showed them that, and they were excited by the pattern it makes. In the larger class (35 students), we didn't have time to go over it, which I regret. But I hope it still got them thinking and gave them a good first experience with 8th grade math.

Now that the first few days are over and we are heading into the curriculum, I feel like there's less room to do activities like this. My goal is to find ways of integrating problem solving tasks like these into my curriculum, but I'm not even near that yet. One step at a time.