WCYDWT: Loop-the-loop, big possibilities

I'm having trouble following the physics in these, but it seems like they'd make some great problems for a more extended investigation. Check out a bike and a car, both going through a loop-the-loop where they are completely upside-down in the middle.

I wonder if I could design a calculus unit around these...

WCYDWT: A Scene from Holes

I'm reading Holes to my son. Am I obsessing, or would this make a good problem?


"... X-Ray had his own special shovel, which no one else was allowed to use. X-Ray claimed it was shorter than the others, but if it was, it was only by a fraction of an inch.

The shovels were five feet long, from the tip of the steel blade to the end of the wooden shaft. Stanley's hole would have to be as deep as his shovel, and he'd have to be able to lay the shovel flat across the bottom in any direction."

On the next page, after his first successful shovelful of dirt, Stanley thinks "only ten million more to go."

Really? Is it anything like ten million, I wondered...

And I decided it would be a good problem for my calculus II students. I don't know much about video clips, so I asked for help, Dan Meyer came through, and here's a clip I was able to use in class...

On Obviousness
I'm embarrassed to say this, because it seems so obvious now that I see it. For years I've been intrigued by the fact that, when you take the derivative of a volume formula, you always get the object's surface area. Suddenly, thinking about the problem I had in mind for the Holes scene above, it was obvious to me. (Painfully obvious, considering how many times I pointed that 'cool' fact out to my students, and wondered aloud why it was so.) Don't worry if it's not obvious to you, you probably haven't been teaching calculus for the past 20 years.

Derivatives measure rate of change. The volume changes in a small bit of time by adding or subtracting at the boundary. The surface area determines how much space there is for the change to happen in. Does that make any sense? No? Maybe it'll help if I get more specific.

Stanley's hole has to be a certain diameter and height, and X-Ray gets to dig a hole with a slightly smaller diameter and height (the difference is called delta x in calculus). If you imagine a thin layer all around the edge of X-Ray's hole that Stanley will still have to dig out, you see that the surface of the hole is (sort of) the difference between their holes. The difference in shovel lengths is the change in x (delta x), and the difference in amount of dirt they have to dig is change in y (delta y). Change in y over change in x is rate of change (aka slope, aka derivative).

If it's still not obvious, either you'll want to play with rate of change ideas more before trying to understand, or I'm not explaining well.

What Can You Do With This: Estimating Coin Value

Adding the blog of Albany Area Math Circle to my Google Reader has added delight to my mornings. I'm seeing so many fun problems on there! Yesterday she linked to one here with a picture that begs you (well, me at least) to start computing.

Guess the total dollar value of the change in this box, and win a galley copy of Chad Orzel's soon-to-be-published book, How to Teach Physics to Your Dog.

~~~

Edited on 8-17-09:
He's posted the answer. I was way low. As was the average answer, dubbed "the wisdom of the crowd". Orzel asks why. I'm thinking we tend to estimate low on money.

What would help us estimate better?

Mathsemantics, by Edward MacNeal, addresses this in one chapter. (I've assigned it as reading to many of my classes.) He talks about having a semantic web in your head that includes a few important numbers, like:

• population of the earth
• population of the U.S.
• population of your state
• radius of the earth
  

And then he recommends estimating often, committing to your estimate somehow, and then finding out the real value of what you estimated. For example, estimate your arrival time when you're in the car, tell the person next to you, and notice the time when you do arrive at your destination.

There are also books that give lots of examples of estimation problems that involve thinking step by step. I've skimmed through Guesstimation: solving the world's problems on the back of a cocktail napkin, by Lawrence Weinstein and John Adam, and Geekspeak: How Life + Mathematics = Happiness, by Graham Tattersall. (I liked Guesstimation better. Neither book was compelling reading, but Guesstimation will work well as a reference, whereas Geekspeak doesn't live up to its title at all.)

One classic of this type of problem is "How many piano tuners are there in New York City?" I've worked through this with many classes: What is the population of New York City? What proportion of households have pianos? How many more pianos are in the city? How long does it take to tune a piano? How often do the pianos get tuned? Put it all together, and if you estimated the pieces decently, you get an answer that's between half and double the true value, which is pretty good. [What does true value mean? How do we count people who work part-time, or who have another profession and tune pianos once in a while. Perhaps there is no exact right answer...]

I've done all this, and I'm still not that great at estimating. (I thought the visible layer of coins in that box was worth about $10, and that there were probably 6 to 10 layers like that in the box.)

Anyone have other ideas about how to help others, and myself, learn to estimate better? Anything you'd add to this if you used it in a class?