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...

The Agony of Math Review Games

My calculus classes took a quiz on the power rule and "reverse power rule" before Thanksgiving break. (The "reverse power rule" is my name for the elementary anti-derivatives they've been taking ... it's always funny to make up terminology on the spot, and then all of a sudden realize that the "reverse power rule" and the "freeloader rule" are now recorded in 50 calculus notebooks). The results of the quiz suggested that my kids needed more time to digest the material, so that's what we've been doing for the past couple of days. They have a re-quiz tomorrow, so I thought that a good way to practice in a relaxed, positive atmosphere would be to play a review game. Turns out, I was wrong.

I stole the game from one of my colleagues Ted. Students work in pairs to solve a problem on the mini white-boards. For the first problem, Partner A is the only one who can write on the board; Partner B can help and offer advice, but cannot touch the marker. The roles switch for each problem. When they are done, they hold up their board. The first team to hold up the correct answer gets to throw a hacky sack at a target I've drawn on the board, and depending on where the sack lands they either get 1, 2, or 3 points. The throwing of the hacky sack clearly has no purpose other than getting the students up and excited, which kind of reminds me of Solve Crumple Toss, except that they don't have to throw away their work when they finish a problem.

I'm not sure what it was -- maybe I made the problems too hard, maybe I should have given them time in between rounds to finish up their solution even if they weren't the winning team, or maybe I should've figured out a way to get rid of the time pressure. In any case, a couple of teams wound up dominating the game and many students expressed frustration after class that they weren't able to get any of the problems right and they didn't like the time pressure. I tend to agree about the time pressure - after all, it's not about how fast you can do a problem, but how well - but if there's one calculus skill that lends itself to this kind of a game, it has to be taking derivatives using the power rule. It's like the multiplication tables, but for calculus.

I'm really curious if anyone out there has developed the perfect math review game -- one that is fun and competitive, but encourages (or better yet, forces) everyone to participate and doesn't automatically favor the quickest kids in the in the class.

"When will we ever use this?"

At my school, there's a huge emphasis on "adult-world connection." A question that teachers (or at least I) often hear when introducing a new concept is "How is this used in the real world?" (As an aside: How much do teachers at other schools encounter this?) Last year, I taught 11th grade math: an ill-defined combination of pre-calculus, algebra II, statistics, and random topics that don't really fit into any standards. I tried to create projects that touched on several different applications of math (such as linear programming to solve a land use problem, and statistical analysis of a survey the students created). Granted, I'm sure I could have done a better job creating and executing projects, but based on my limited experience I came to the following conclusion: real real-world math is really messy ... mostly too messy to survive the attention span of the average high schooler. So the real-world math we did was really fake real-world math. And guess what? The students totally didn't fall for it. It felt like a completely contrived textbook word problem that was extended into a month-long project.

Lately, I've been feeling that the question "How is this used in the real world?" is a bit misguided. Sometimes there is a really good answer that fits into a 2-minute sound bite without completely disrupting the flow of your class. Quadratics are the classic example of this: How will you ever know where your rocket/football/rubber chicken will land if you don't understand projectile motion??? However, once you step into the realm of calculus it's difficult to encapsulate the usefulness of something like the derivative in 5 sentences or less. (Or not ... please let me know if I'm totally off-base here!) Most often, I find myself wanting to give one of these two responses to that dreaded question (although I never would):

1. Asking how [insert complicated math concept here] is used in the real world is like asking how multiplication is used in the real world. It's everywhere, but could you sum up how it's useful in 2 sentences to a fourth grader who was just learning his times tables?
2. The real-world connection is that you're going to have to be able to [insert tedious technical skill here] in order to pass your intro-level calculus/physics/engineering classes in college so that you can move on to bigger and better things.
3. Does it matter??? Can't we just do it because it's interesting in its own right? A little abstract thinking never hurt anyone! (Shockingly, this argument doesn't actually seem to make concepts more interesting.)

My point is that its connection to the real world is only one (small) reason why I love math. What I really love is the feeling of success when I finally figure out something that I never thought I'd understand. I love discovering beautiful patterns. And yes, I sometimes even love the rules and the stability that they offer in the midst complete chaos. I'm also convinced that (most) high school students don't really want to learn about real-world math, except for maybe the occasional pretty picture of a fractal or 2-second snapshot of some grotesque computation that is used to solve a "real" problem. Instead, they (like all of us) just need a hook. Usually that hook is sinking their teeth into a problem that captures their attention, not doing a "realistic" mathematical analysis.

Don't get me wrong -- I do want my students to understand why mankind has spent hundreds of years studying the derivative. I want them to understand the equations of motion, and optimization, and differential equations -- but it's going to take an entire year of calculus to really answer that question, and I'd rather answer it well in a year than answer it superficially in 5 minutes. In the mean time, I can try my best to make the concepts engaging in their own right so that my students want to learn them, even if they don't instantly see the "real-world connection."

I swore I'd never do this...

Before I started teaching a year ago, I didn't understand the blogosphere. What made people think that there was an audience out there for their every musing? I swore that I would never be one of "those people" ... and then I started teaching. It took me about five minutes to realize that I had absolutely no idea what I was doing and I sought solace in - where else? - the internet! I quickly found an online community of math teachers whose stories made me laugh, whose struggles I could identify with, and whose ideas I started stealing (er, "borrowing") by the handful. I survived my first year with my head (barely) above water, and now I'm a seasoned (ha) second-year teacher. As my profile indicates, I teach math and computer science at a public charter school in San Diego, CA.

I'm blogging now because I have a lot to say and, to be quite honest, my boyfriend is sick of hearing me blather on about teaching. I am now one of those people who knows there's an audience out there for my every musing. :) Seriously, though, I feel that if I can contribute in some small way to this amazing online collaboration that's going on, then it's worth doing.

I'll keep my first post short. I know that many have said not to let a catchy name stand in your way of creating a blog, but I just couldn't do it until I had the right name. And then, a couple of months ago, one of my students inspired me with a comment he made in class. My calculus class was starting limits by figuring out the area of a circle by taking the limit of the areas of inscribed polygons with more and more sides. At the end of our discussion, one students made a comment that was so absolutely perfect that I couldn't believe that in all of my years of studying math it had never occurred to me: "So a circle is really an infinigon?" I got super-excited as I tend to do when my students astonish me, and I told him that I would include that in my book one day ... for now, a blog will have to suffice.

What makes a "good" challenge problem?

I've been giving my calculus students challenge sets every couple of weeks or so (8-10 total for the semester). I'm requiring that they complete at least 5 of them, and their challenge set grade will count for 10% of their final grade. Many students work on all of them (for some combination of enjoyment and extra credit, I think) and I've learned that before each set is due there's a group of them who loiter at a local chain establishment and argue for hours about the problems, probably spending a grand total of $1 on a bottle of water. I'm not going to lie -- this brings back fond memories of math study groups as an undergraduate, and it makes my heart smile to know that they've started their own little geek-fest.

It turns out that some of my students are pretty demanding and tell me when I've given them a "good" problem and when I haven't. They didn't like their last set so much - it basically asked them to investigate the limits of a function similar to f(x)=sin(1/x). I have lots of thoughts as to why they didn't enjoy this one as much, and it's primarily because they didn't really understand what I was asking for and started it too late to ask me any questions. However, it makes me incredibly happy that they even have an opinion one way or another about the problems: passion in the classroom = a math teacher's dream.

I fear that I've ventured far, far away from the course content with these challenge sets. On one hand I feel like I should be able to come up with some honest-to-goodness calculus challenge problems; on the other hand, it's actually more important to me to generate some excitement about math, period. It's probably also not a bad idea to give the kids who aren't loving calculus another point of entry into math.

I really, really like this week's challenge problem:

Alice and Bob need your help! They have been captured by pirates and will only be released if they can accomplish the following task:

A pirate will deal Bob five random cards out of a standard deck of 52 playing cards (no jokers). He gets to choose one card to put aside as the “mystery card.” He must use the other four cards to communicate the identity of the mystery card to Alice. They may not talk or look at each other’s faces once the cards are dealt. However, they can communicate and agree on a strategy beforehand.

Alice gets one guess – if she’s right (about the number and the suit of the card), the couple will get released. If she’s wrong, it’s bad news for Alice and Bob.

What should their strategy be?

Note: Your strategy shouldn’t contain anything shady like “If Bob points his index finger a certain way, the card is a heart.” There is at least one purely logical/mathematical strategy.

I'm trying to figure out what makes a "good" challenge problem - one that's rich with mathematical ideas but also approachable and engaging. So far, I've got:

• Anything having to do with infinity (like this back-handed approach to the harmonic series);
• Something they can "play around with"/brute force until they figure out a more "elegant" solution

Finally, I also tapped into current grad students in my old math graduate program for help. Every grad student has their favorite puzzles and a desire to do anything other than their research, so not surprisingly I got a ton of great problems from them. Here they are, for your perusing pleasure.