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...
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.