Sometimes this stuff is just sitting there for the taking, like a gold brick in the middle of the road, and all it takes is the right kind of eye to see it. David Petro e-mailed me a link to an episode of the long-running radio series, Car Talk, which does all the heavy lifting for us. I just shot a couple of supplementary photos.
Start with discussion, brainstorming, estimation.
The fuel gauge [on my 18-wheeler] is completely useless. It's completely unreliable. So I have gone with a very low-tech method. I have gotten a wooden dowel which I use as a dipstick. Now the tanks on my 18-wheeler are cylinder-shaped and they're on their sides. Now when I stick the dipstick in and I mark the level when the tank is completely full, it happens to be 20 inches. I mark ten inches at the half. My question is how I do I accurately mark one quarter?
I mean, seriously? Christmas already?!
I'd crop this clip pretty tightly. Rich says, "It obviously isn't five inches," but we'd rather put that to our students, asking them, "Just gimme a guess: how many inches do you think?" Some students — the impatient ones, maybe even a few advanced students — will snap to "five inches." Have them sketch out their solution. Ask them if they still like it. Ask for a number they know is too large, too small.
Important questions your students may have to reckon with.
- What information do we care about here? What information don't we care about here?
- Why does the trucker care about the level of a quarter tank anyway?
- If you try out your guess, is there more than or less than or exactly a quarter of a tank left?
- What kind of shape does the gas form in the bottom of the tank?
- How is that shape formed from other shapes and how could we find its area?
These questions and their answers will vary significantly with the particular math course you're teaching.
We finally came up with the answer, that in order for his dipstick to work he's gotta be about 5.96 inches off the bottom. That's a quarter of a tank. And it seemed like it should've been a lot more than that.
Voila: a zipped archive. Thanks, David.
Extra Credit Assignment
Someone cook up a dynamic Geogebra applet for this scenario and pass me a link.
BTW: Woo! Check out Tim's.
BTW: Updated to fix an error in my math. Thanks, RM.