"... 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.
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Actually the WCYDWT that first came to my mind was - wow, that's a lot of dirt. How big is this old lake bed they're digging up?
ReplyDeleteHoles: it's not just for Calculus students.
Your interpretation sounds spot on though. Another analogy I use to explain the same thing is everlasting gobstoppers which are spheres you digest one surface layer at a time.
I love this way of thinking about it. Lovely, just lovely.
ReplyDeletei never saw this clearly until
ReplyDeleteabout the third time i did calc i.
"visual" thinkers probably get it in one;
this is the *very* kind of thing that
made me an algebra major!
how does one make it clearer?
lower the dimension maybe.
the rate of change in the area
A = \pi * r^2
of a growing *circle* is, hmm.
A' = 2\pi*r*r'
whattaya know, the length
(times, alas, the "time"
derivative of the radius;
make it change a meter per second
or something if it suits...).
oh, but look... "where"
is all the "change"
taking place?
(this must be why economic
applications like to call
derivatives "marginal"
functions....)
sort of like a 2-D everlasting gobstopper....
(terminological quibble:
the gobstopper is a *ball*...
a "sphere" is itself one of the
"surface layers"...
if i were being this careful
in my own work i might have
referred to the *interior*
of a circle... the thing
having an area... as the "disk".
this distinction is much less
commonly made, though.)
@Alison:
ReplyDelete>How big is this old lake bed they're digging up?
Big, but they're not gonna dig up the whole thing. They each have to dig a hole a day, to 'build character'. It's a juvenile detention center.
>Holes: it's not just for Calculus students.
Well, I figured students could figure this out, approximately anyway, without calculus. My question being how much digging does X-Ray save himself by using that slightly shorter shovel?
I just had to write about the calculus because it was a bit of an epiphany for me. (My son was getting frustrated because I kept stopping my reading out loud, so I could think about the math.)
I put it this way: To infinitesimally increase the volume of a solid, think about putting on a coat of paint
ReplyDeleteYep. I think my feel for the rate of change meaning (as opposed to slope) was weak. Coat of paint is easier than telling this story from Holes.
ReplyDeleteBut since the Holes story is why I got it, I'll probably start with that when talking about this connection, and then we can think coat of paint.
Thanks.
Wow. We ran through this problem in a workshop I facilitated in North Carolina last week and it played like gangbusters. Not having the book in front of me, we assumed two inches shorter and calculated the difference in cubic inches. We discussed methods for making that number more relevant to learners, going so far as to call a local composting company to determine that, over a year, Shia's character would dig thirteen dumptrucks more dirt than X-Ray's. Great times. Thanks for the inspiration.
ReplyDeleteI'm so excited that you tried this out! Who were you working with? High school teachers?
ReplyDeleteSo that's about six and a half truckloads a year for each inch. The dialogue I quoted above said it couldn't have been more than a fraction of an inch shorter, but that would still be a few truckloads.
(Am I right to think of this linearly? I don't think it gets an exact right answer, but I think it comes close. Double the difference, double the thickness of the 'skin'.)