How I Teach Calculus: A Comedy (Drubbles)

Dry Ice! Bubbles! Drubbles!

There’s really nothing better than a spontaneously created investigation. We were bumbling through the idea of the derivative operator (implicit differentiation and all that). When the question got asked:

Hey, what if the variable in the bottom (z in d/dz) isn’t in the equation at all?

I don’t think they knew how crazy the question really was, but it didn’t take long for them to grab onto the idea of shrinking and growing bubbles.

So, like, the radius grows and the volume grows, and that means time is a variable, but you can’t see it in:

[latex]V=\frac{4}{3}\pi r^3[/latex]

Which means all the variables become little implicit chain-rule problems:

[latex]\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}[/latex]

At this point, we run the very real and very fatal risk of this becoming yet another “process.” So many math teachers are obsessed with this. Just follow the effing directions and no one gets hurt, said every math teacher, ever.

Well, the directions are cute, but this isn’t IKEA, and most students literally couldn’t care less; honestly, you’d have to use Planck Lengths to measure their interest level about the speed at which this sphere’s volume grows when the blah blah blah I can’t even type it out it bores me so much.

So instead, we thought of something that actually does this, that was fun, and not boring at all:


Imagine an entire 90 minutes, 5 pounds of dry ice (grocery stores have dry ice), some displacement measuring vessels, and some soapy water. Freaking awesome math is what you should be imagining.


Their job is to make related rates out of this however they want.

Here’s the #anyqs that came up:

  • What happens when they become hemispheres?
  • Do they shrink or grow as the gas heats up?
  • What if a bubble gets trapped in another bubble, the heat transfer?
  • Do they form spheres or ellipsoids?
  • The one that has the wobbly wave, does that exchange width and height evenly in time?
  • What if you grow a bunch of bubbles in the vessel, do they change size when they grow on each other?