# How I Teach Calculus: A Comedy (Sound Waves and the Chain Rule)

The fundamental rift between what Math is and what math education looks like tears me apart regularly. In a very blue-collar way, I have to prepare these calculus students to do well in college Calc or Calc II. They have to know the strange tricks and unwritten rules of passing math exams and ignoring that part of their brain that asks, “Ok, but why this material now?”

What I really love, and the students claim to also enjoy, is contextualization of math by finding patterns in things that appeared, at first, unassailable.

Sadly, contextualization is not graded beyond my classroom in any consistent way.

Let me be very specific:

I am at least two orders of magnitude more interested in a student being able to identify when to use the chain rule (or create some new math/pattern) than I am with them being able to do the “tricky problems.” Although the latter was pretty much the only experience I had in college.

So, here’s a little project that attempts to suture the two competing goals of math ed together.

# Sound Wave Shenanigans:

I showed the students the real-time output of a program like AudioXplorer. I’m sure there are other, better programs, so please leave those link in the comments.

Here’s a whistle:

Here’s an “ah” at the same pitch:

We then played with several GarageBand effects modules. The question that came up (that I was hoping for) was, “What math is GarageBand doing?”

Students chose a pet sound wave that they could recreate.

• A gong
• Whistles (two-finger, tongue, etc…)
• Steven Tyler
• Guitar
• Trumpet
• Clapping
• Pop can opening

They captured them, and then attempted to model them in Grapher (or Desmos, or whatever you use). This naturally lead to a discussion of Fourier and the spectrum graph, which until now they had ignored.

A spectrum, for the uninitiated, is a bunch of spikes of telling you which frequency of waves are present in a more complex wave. The students love this, because it gave direction to the graph model, but it wasn’t just plug-and-chug.

Check out the spectrum from the “ah” above (upper left window).

Which led students to model something like this:

[latex size=3]f(x)=325\sin{800x}+100\sin{1000x}+100\sin{500x}[/latex]

They then took that wave and attempted to create delay effects (with phase changes), they were perplexed by chorus and other effects, and that part of the project kind of fell off.

Here’s a slide put together by some students looking at our wind band’s gong.

# Oh, where’s the chain rule?

We reasoned that some of the effects were working off knowledge of the speed of the wave, which demanded that we use derivatives.

The students initially posited something like this:

[latex size=3]\frac{d}{dx}\sin{(40x)} = \cos{(40x)}[/latex]

This, for most students, just seems natural, which is a poor motivation for truth. We had a long discussion about speed vs. position. Mostly centering around the idea that you’ll have to go way faster to travel back-and-forth from town-to-town if you’re going to do so 100 times each day instead of just once.

So, even though your physical position isn’t really that different, if you’re oscillating more often, you’re speedometer is going to way wilder. That motivates this:

[latex size=3]\frac{d}{dx}\sin{(40x)} = 40\cos{(40x)}[/latex]

Now you can get all formal with the chain rule, if you want.