## Saturday, December 18, 2010

### 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?

1. That's amazing, I love it!

2. Hey Allison,
Great activity, and as an observer I thought your enthusiasm for the concepts and importance of it all really helped push the students forward.
Just a thought, but what about using the definite integral in some sort of building project? Students get a set function, certain parameters and must build a model with the given area, or something of that nature, could be used the other way around as well... given object find area

3. Sounds fantastic, but I have a question about the math involved. If I were to do this, I'd do several trials with the ramp at different heights to record time of landing and horizontal dist of landing. That'd allow me to estimate initial velocity in both x and y directions (and I guess I can average across the different trials). Then I can make the prediction for new airborne time, and therefore new horizontal distance traveled.

BUT, it sounds like your kids didn't do that and instead calculated frictional acceleration? Is that necessary? I'd think that as long as we know the takeoff speed in x and y as the cube leaves the end of the ramp, it shouldn't matter what acceleration had gotten it there.

(I'm also curious as to how you built such an accurately forward ramp. It's kind of amazing that it didn't land either left or right of the cup!)

Thanks!
Mimi

4. Hi Mimi - We did not calculate frictional acceleration. Instead, we just assumed constant acceleration as the block slid down the ramp. They did lots of trials (though probably not enough) to figure out how long it took the block to slide down the ramp. Call that time t* - you can then use t* along with the length of the ramp to figure out the velocity of the block (in the direction of the ramp - no need to decompose vectors here) as it leaves the ramp. From there, it's just a projectile motion problem (although we always work backwards from acceleration in the x- and y-directions). Also, I definitely would not call this ramp "accurate" in any sense :-) However, I did build it with the two pieces of wood in a v-shape to funnel the block in the right direction. Hope that helps! -Allison

5. This sounds awesome! I love the "you versus the laws of physics" thing. As a calculus TA, I feel like most of my students have been trained to read story problems as anything but stories---to abandon all their common sense and knowledge about how stuff works the moment they step into math class. It's very refreshing to see a problem where "the right answer" is given by experience, rather than a solution manual.

p.s. Are you, by any chance, a WCYDWT fan? :)