Wednesday, July 27, 2011

Act 2's a Killer

I need a little help.  I think I've nailed the question:

Barbecue Q2 from David Cox on Vimeo.

But I can't figure out what to give students to help them through Act 2.

Here's the raw footage and the current conversation and Greg's run at the data.  (Thanks to @maxmathforum for archiving)

Any help would be appreciated.


Becca Schutzengel said...

Hmmm, my first thought was to use Newton's Law of Cooling: dT/dt = k(A-T), where A is the ambient temperature (so in the grill, in this case). But just from glancing over the data, it doesn't seem like it's working that way. As I think about it more, the Law of Cooling applies to conduction, so it probably only works well for modeling the temperature of the outside of the meat. But the thermometer is telling you the temperature in the middle, so you'd want to account for convection from the outside to the inside? I think you can use the heat equation for that (d^2T/dt^2 = k*d^2T/dx^2). So if I were doing this I'd try to model the temperature with those two equations combined. There's three constants in there that Act 1 doesn't give values for (two proportionality constants and the ambient temperature), so I'd want more temperature vs. time data, and then I'd try to adjust the constants to fit the data. And then predict when it'll be done cooking?

Oh, just thought of this: it's probably not good that I just formulated that all in terms of calculus, since I'm assuming you want to do this with your middle school students. Ummm, I've dealt with the heat equation in lower level classes by only considering steady state, in which case temperature is just linear in space. But this is clearly not steady state. I don't know: maybe I'm just making everything way too complicated, and we should just be assuming that the temperature is linear in time. My gut is telling me that's oversimplifying it, but maybe I just have a tendency to over-complicate things? The data does look relatively linear (but I haven't plotted it).

Sorry that this comment is so long, but still doesn't have any useful answers. :(

David Cox said...

Yeah, I'd be using this with middle school kids. The models they would have at their disposal are linear and, maybe, quadratic. I keep looking at this problem like a math teacher who assumes there's a right model to use. But that's not the way my students, or yours probably, would look at it. Our students will hopefully have a question and do their darndest to answer it.

For some kids that may mean plotting some points and making their best guess. For others, that may mean doing a regression of some sort. And calculus kids will realize that they don't have enough information because there are some constants that aren't identified. Funny thing about that is that the 7th grader may have an easier time getting at a solid guess than the calculus student.

I don't think you're complicating things at all. I know the data is relatively linear. So that model could be useful.

At the end of the day, I'd like my students to recognize that they have some tools (whether it be graphing, modeling with equations or doing a regression) at their disposal that can be useful to them.

Anonymous said...

My only question was "what cut of meat is that"? The first time through the video it looked like chicken legs, but then you used the "beef" setting, which confused me. I had to watch the video again.

I never got around to thinking about what math you were trying to do. Showing the video around here would probably result in massive debates between the carnivores and the vegetarians, particularly if done around lunch time. I can't see it triggering any math interest from middle-schoolers without a lot of prodding. After all, they mostly don't cook, but just come to meals when called, so how long something takes to cook would not occur to most of them as an interesting question.