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?

I can prove it, but I don't believe it!

There's a problem I really like whose result always surprises me. First, imagine you have a piece of string that’s long enough to stretch all the way around a basketball (the circumference of a basketball is 30 inches). Then you realize you have an extra 24 inches of string in your pocket, which you want to add to the string. So, you cut the circle of string somewhere, add exactly 24 inches, and then smooth it out until it makes a circle all around the ball (kind of like a ring orbiting a planet). The question is to figure out how high is the string off the basketball? It's a simple geometry calculation, and you wind up getting around 3.8 inches.

Got it? Now, try the same problem, except instead of a basketball imagine that you start by wrapping a string around the equator of the earth. Then, just as before, you find an extra 24 inches of string in your pocket, which you add on to the string, and then smooth out the resulting string until it makes a circle around the earth. How high is the string off the earth?

My intuition always tells me that the gap should be minuscule - after all, what is a mere 24 inches compared with a 25,000 mile equator? But of course, every time, the answer comes out to be ... around 3.8 inches.This definitely falls under the category of "I can prove it, but I don't believe it!"

It occurred to me that there just HAD to be some calculus in this problem (is there a problem for which that couldn't be said?), and lo and behold I found it. For their weekly challenge set, I gave my students the basketball/earth problems, and also this last one: Express the radius of a sphere as a function of its circumference, and then find the derivative of this function. Why does this make sense in light of your previous answers?

The answer is that r(c)=c/(2*pi), so r'(c) is just the constant 1/(2*pi)! So, regardless of the starting circumference, a constant change in circumference will result in a constant change in the radius. Neat, huh?

So I get it. I can prove it with calculus and without calculus. Yet somehow, I still don't really understand how this can be true. And I have to admit, after years doing math calculations that "give" the answer, it's always refreshing to come across one that doesn't...

Order of Operations?

One of my calculus classes just finished a long introduction to the derivative -- we learned it (or rather, I taught it) once, at which point I realized that the students were still completely confused about the concept of the derivative even though they were making progress in actually computing derivatives. So, we spent another week reviewing both the concept and the skill (as I attempted to explain, possibly in vain, why both of those things are important ... I told them that if they left my class without being able to explain the derivative to someone, I would have failed at my job). After another week of review, I am confident that they are ready to move on to the next topic.

This of course brings up the question of what the "next" topic is. Calculus is not linear, and at a school with absolutely no prescribed curriculum the thought of moving on often propels me into a panic-ridden tizzy. What's the best next topic - the one that will tap into the students' current understanding, keep their interest, draw on their mathematical strengths, and work on their weaknesses? (Am I asking too much of a single topic?) My two thoughts are:

1. Having them discover the power rule. They're primed to appreciate it now that they've spent weeks calculating derivatives using limits. Also, my other classes have known about the power rule for awhile now and have already tried to ruin the surprise (I have to admit, part of me loves that there even is a math rumor mill!), so I'd feel a little slimy hiding it any longer.
2. Having them practice graphing functions and their derivatives. This will get back to the concept of the derivative as the "rate of change," which is something I really want to hammer home. I would have them start out simply by making observations using this super-rad calculus grapher, which I just read about on Sam Shah's blog. Gosh I love the internet.

I teach three calculus classes - two honors and one regular. Teaching this non-honors class in particular is really forcing me to hone in on the essence of calculus. Somehow I'd be missing the point if I tried to push the "standard" calculus curriculum on these students, because they're still getting comfortable with so much of the algebra that successful calc students take for granted. I'm sure that we could eventually get to the point where they could apply the quotient rule and the chain rule to complicated functions, but it would be at the cost of a greater conceptual understanding. This reminds me of something my office-mate Kyle said the other day: "A year from now, I'd rather have students say that they understand what derivatives are and they used to be able to calculate them, than that they know the derivative of x^2 is 2x and they used to actually understand why."

This brings up the juicy debate over how and why calculus is taught in high school, but that's for another day...