Grouping Activity

A long while back I read a post from Frank Noschese called subversive lab grouping. In a nutshell, you give students cards which tell them what groups they're in. But it's not as easy as it sounds- there is overlap between the words on the cards. For example, I've given students cards with letters on them. They start by trying to put all the vowels together, or maybe all the capital letters. But it doesn't form the right number of groups (4 groups with three students each), so they have to discard their model and start over. The key turns out to be the number of straight lines used to form each letter. X is two, as are T and L. So that's one group. W, M, and E are similarly grouped. Frank and his followers have thrown down a bunch of other ideas for groups, some of which I've adopted, but I've also made up my own.

My students love this activity and wanted me to write about it. Since this is the first request I've ever received for a blog entry, I figured I ought to honor it! I usually use it in calculus- I'm not sure why, but I haven't implemented it with my other classes yet. Maybe I will... one limitation is that you have to specify the number of groups and their sizes. If students are unexpectedly absent, it can throw a wrench in the works.

If anyone is interested in the groups I use, just say the word and I'll be happy to put them up.

Hands-on calculus

I started this post back in December, but never finished it. Here's the final product.

We've been working on related rates in calculus. One of the "classic" calculus problems involves a ladder in motion. Its typically moves away from a wall at a constant rate, and the students are asked to determine how fast the top of the ladder is falling.

At least the problem has context, even if the constant rate bit is a stretch. Last year I tried to turn this problem into reality, and it didn't quite work out. This year we did much better. Key things:

-put wheels on the top of the ladder so it rolls smoothly down the wall
-rest the bottom of the ladder on a constant velocity buggy (borrow one from the physics teacher)
-use a good tripod

Here's a snapshot of our setup (screenshot from LoggerPro):




I scaled the video, set up a coordinate system, and tracked the bottom of the meterstick (using the rearmost wheel on the vehicle). This produced the graph of the horizontal position of the bottom of the meterstick below.



I added a trendline so we could get velocity, and then I asked the students to use the length of the meterstick/wheel contraption along with this velocity to predict how rapidly the top of the meterstick would be falling when the base was 32 cm away from the wall (hence the cluster of points near that position).

This is where related rates came in, and the students ended up with an answer. We confirmed it by analyzing the same video and tracking the top of the meterstick.

We got a really nice parabolic shape, and the instantaneous velocity of the top of the meterstick matched their prediction. It was pretty successful- as one student put it, "it's so nice to use math to analyze something complicated that happens in the real world!"