A really fun physics/calculus lesson

This was the last week before winter break, and my honors calculus classes spent it working on The Great Calculus Challenge. In a nutshell, I built a wooden ramp and I told them we'd be putting the ramp on the roof of the school and letting a metal cube (I used density cubes - brass in one class, copper in another) slide off of it. They had 3 days to figure out where on the ground it would land, and as a class they'd get to place one blue plastic cup on the ground. The goal, of course, was for the block to land in the cup. I made a big show of the fact that the entire class would get only one try to get the block in, so they all had to agree on their answer.

This was a sort of "culmination" of our first semester of calculus. We've spent a lot of time talking about derivatives and antiderivatives in the context of motion - position, velocity, and acceleration. My students had done tons of problems about motorcycles screeching to a halt, potatoes being projected off of cliffs, etc. The new mathematical element here was that students had to calculate the velocity of the block leaving the ramp, which required them to take into account acceleration other than that due to gravity (like friction).

So what happened? In the first class, as my students were feverishly perfecting their calculation, my boyfriend (who is finishing his Ph.D. in math, and who I'm trying to convince to become a high school physics teacher - hence dragging him to school for the day) did his own calculation in about 10 minutes. (He actually wrote a little Python code to help him.) When we went outside, the class put down their cup and my boyfriend put down his (it was just short of theirs ... very "Price is Right" of him!) and, lo and behold, the block landed in his cup! I would say that the excitement this caused was a very close second to what would have happened had the block landed in their cup. In the second class, we did the same and this time everyone agreed on where the block should land. However, it fell about 2 inches short of the cup. We talked about why this might be, and I blame it on the shoddy craftsmanship (and therefore variable initial conditions) of the ramp (for which I am completely responsible). In any case, it was a fun way to spend the week before break:

Some students chose to solve the problem
by experimenting from different heights...

Others took a pencil and paper approach...

A good time was had by all...

Especially by my colleague Kyle, who got to
scale the building and drop the block for us!

The anticipation was INTENSE...

And in the end, the block fell just a smidge short.

For me, this turned out to be an experiment in what happens when you tell a group of 25-ish bright, motivated students that they have three days to come up with one answer to an open-ended problem. The two classes approached the task completely differently - one class relied heavily on a couple of "leaders" and many students were quiet or worked mostly independently, while the other class naturally split into a few truly collaborative groups. It was fun for me to be a bystander, observing the classroom dynamic and occasionally giving a cryptic nod of my head or raise of my eyebrows to indicate whether or not they were on the right track.

I really liked this activity for one main reason: when a couple of students asked how they would be graded, I got to be really dramatic and say something like "Graded??? This is SO much more than a grade! This isn't me versus you, it's you versus the laws of physics!" and that kind of silenced that conversation. I would love to have more of these activities in my back pocket for next semester, where there's a high level of intrinsic motivation, especially because I'll have second-semester seniors ... any thoughts?