I’m sure that several hundred precal teachers through the years have done this before, but it was new and exciting for me.
This year, I’m working very hard to make the students move around during the day. I shadowed some of our students recently and got so, so bored and tired sitting still all day. Yes, they’re juniors and seniors, and yes, they’ll need to be able to sit through lecture in college, but I was bored, and I don’t want them to be. I’m also trying to make the definitions of conic sections a little more tangible.
For the double-napped cone definition, I brought play-dough, which the kids made into cones and cut up with dental floss to see the shapes. It mostly worked, they had fun with the play-dough, and they could visualize it way better than with my two-dimensional drawings on the board. (This idea was completely stolen from Jonathan at http://infinitesums.com/commentary/2011/7/1/conic-play-doh.html) I liked it a lot because we could talk about where to cut it and then they could immediately cut it and see. It made it a lot more tangible, and they could even tell me how cutting at different angles will change the shape, or how the size of your cone will change the shape. I could have pushed more there, but it was so much better than past years, when I mostly just threw a picture on the board and moved on with life.
For the locus-of-points definition for an ellipse, I typically bring a piece of string, tape two ends to the board, and draw an ellipse. The kids more-or-less get the point of the sum of the distances, and we move on to the equations and graphing things. My beef with that is that it focuses on memorizing procedures and graphing things without really exploring where it comes from.
So, today, I brought a much larger piece of string. We left the classroom (always makes class more fun!) and went out to the atrium where we’d have room. I recruited one of our front desk people to come be a focus with me. We held the ends of the string, and I had a student grab the end and walk around. The students could really see how, while the distance from a point to each focus was the same, the sum of the distances was fixed. Then I had each of them grab the string and stand at a different point on the ellipse so we made a human ellipse. This turned out to be especially great later in the class period when we were proving that , because I could refer to points by who was standing where: “So, J was this point at the end of the minor axis. How far was she from me? From the other focus? A was this point, the vertex. How far was she from me? From the other focus?” It made for a lot less vocabulary on the first day of a topic.
After they all had formed the ellipse and could really see it, we went back to one person holding the string and walking around in the shape of the ellipse. I moved closer to the other focus and asked how that would change the shape. Where would we need to be for a circle? How would it change if we had less string? More string?
What I especially liked about this was that they could immediately verify their conjectures by watching the person walk while holding the string. It got them to really think about the shapes in a way that they hadn’t been able to in the past because it was a physical object that they could see and touch and move around in, not just a drawing on the board.
Oh, and this whole thing took about five minutes. We were a little short on time at the end, but that had more to do with our discussion of April Fool’s Day then it did with this. Will definitely be doing this again next year.