This year I have just 6 kids in Calculus. This is my third year teaching these kids so I feel like I know them pretty well. So I don’t feel like I need to spend much time on the first day of school going over rules, procedures, etc. I just wanted to jump right into doing math. So I gave my students some of my kids’ toys and had them build something with them.
If you’re not sure what this is, it’s a collection of plastic track pieces that can be put together lots of different ways. Then you put a marble at the top and let it roll to the bottom. After the kids had built a simple track, I asked them how fast the marble rolled down the track. One kid immediately said that the speed was different at each point on the track and that we could find the marble’s average speed by finding the velocity at every point along the track and then averaging them all together. After a little discussion we decided that the ole distance/time equation might be easier. So they set to work on that. Then I asked them to find out how fast the marble was moving at a particular point on the track. They didn’t every hesitate, just immediately knew that they needed to find the average speed on an interval near that point. (I thought they might have to think about it for a little bit at least.) Before they started, I asked them to guess what it would be. After we had talked about the difference between average and instantaneous velocity (something that we will revisit more tomorrow), I asked them to graph the marbles height vs. time and the velocity vs. time. What I hoped they took away from this was a beginning understanding of using limits to find rates of change and some practice in graphing motion. The graphs that they came up with were not totally accurate but they generated some great discussion about the motion of the marble that I think will be useful later on in the year. Overall I was happy with the way the lesson turned out. Tomorrow we will be going into a little more detail about the limit process and looking at examples of numerical data as well as equations. But this activity gave me something that I can refer back to as we move into more abstract stuff.