I have to get this down before I forget.

A discussion of what it means to be exponential (which colloquially is beyond misunderstood and bordering on abused) led to a discussion of what it means to grow because you’re growing.

Does gravity work this way? They asked. Thankfully not. You don’t fall faster because you’re getting faster (that’d be some *free* energy). You certainly do gain speed, but it’s not exponentially gaining.

“So what does?” I asked. “Money!” They all say, because that’s the only example anyone can ever think of.

Until that awesome kid in the back of the room says, “Chicken Wings!”

What?

Oh, right. Because the more wings you eat, the slower you go. Like:

[latex size=3]\frac{dw}{dt} = -w [/latex]

[latex size=3]\frac{dw}{w} = -dt [/latex]

[latex size=3]\ln{w} = -t + C [/latex]

[latex size=3]w = e^{-t + C} [/latex]

[latex size=3]w = C e^{-t} [/latex]

We added the coefficient (*b*) on the *-t* later, because at first it’s not motivated, unless someone brings up the fact that [wings/minute] can’t be equal to [wings], but, hey, don’t be that guy–at first.

This of course quickly devolved into thinking of other foods that have this same dying exponential behavior.

We postulated that Ghost Chiles would behave the same way, which of course ended up with Ghost Chile salsa making an appearance the next day.^{1}

Finally, this ends up exactly where you’d expect it to end up:

# MAN v. FOOD

The most ~~ridiculous~~ ~~crapulent~~ ~~fall-of-the-empire-esque~~ differential-equation-y show on the TV!

The students pulled a bunch of episodes off Youtube, because Travel Channel’s MvF website really sucks,^{2} and we got to work analyzing which challenges behaved according to what exponential behavior.

The best one we found was Memphis’ Sasquatch Burger:

Have fun with that.

The students graphed his completion as a function of time. They then fit a natural exponential function to it and worked back for the parameters on the diff eq.

We talked heavily about the units on the coefficients in the equations, which was awesome, because it turns out that students rarely work with anything but super-science-y units. ([burger] and [Hz] don’t often go together)

What’s most valuable to me is that the students created math where once there was none.

They’re getting practice being analytical in situations where they shouldn’t need to, but can, elucidating things that weren’t known before. That’s what math is, it’s not about getting 15 year-olds to struggle through the quadratic formula by sitting them in front ever-increasingly trite computer animations and “prompts” in order to get even the least analytical to memorize an algorithm for 20 minutes; its about finding out Adam Richman’s critical burger mass from a Diff Eq.

I’VENEVERSEENAMANEATSOMANYCHICKENWINGS

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