How I Teach Calculus: A Comedy (Glow Sticks)

The traditional pedagogy of calculus is almost entirely bereft of any rates that anyone actually cares about. Seeing as how calculus is the mathematics of rates, this is probably not such a good state of affairs.

Thus, doing my due diligence means looking for rates of change that students (and myself) might actually find interesting enough to study for a few days. Re-read that. Seriously. Have you ever asked yourself if what you’re learning about in class is actually worth studying? Not in a crack-the-books sense, but in a live-in-the-rainforest-for-five-years sense.

Anyhoo, I think glow sticks are where it’s at. They have rates that matter, they’re affected by about a jillibillion fun and graph-able variables, they won’t necessary kill you if you cut one open, and kids can learn some ancillary standards about chemical engineering (which is where most boots-on-the-ground calculus is done, anyways.)

I haven’t done this with students yet, but I have cut my fair share of glow sticks open–which by the way is done very similarly to circling the foil of a wine bottle but with a utility knife–and toyed with the proportions of reactants while measuring the rate of decrease in luminosity.

Yes that's a LEGO Stormtrooper alarm clock. No, there is not shame in merchandizing.

Why? Because I wanted to know if the reaction would work in the presence of atmospheric oxygen levels (it does).

How? With a homemade light sensor!

Light Sensing:

I anticipate about 10 minutes of Socratic dialogue getting us to the point where the students buy into the idea of needing light sensors (otherwise I think we’d get a lot of “kinda bright,” kinda bright,” “dim sot of,” “dimmingish” data; otherwise known as “useless”)

Here’s the circuit diagram:

Try not to sprain anything. The left thingy is an ohmmeter. The right thing is an LDR.

LDR = Light Dependent Resistor (or CdS). I prefer these, because the circuit is super simple, and my students have almost zero electronics skills. There are way better ways to do this that TieAndJeans can probably let us in on.

This is an LDR. This image would be about 3.5 cm wide in real life.


Aside: I got into using LDRs when I built this theremin. No, I don’t remember who I stole the diagram from, but let me know if it was you, please. Yes, you should build it.

Chemiluminescence, in Math!

Here’s the best part. The reaction is just right for high schoolers. It stretches their chemistry and their understanding of equilibrium. This reaction also introduces quantum mechanics in a friendly non-cat-murdering way.

So, there are three essential parts to making glow-y awesomeness:

  1. Hydrogen Peroxide: the chemical knife that will release the stored energy in…
  2. Diphenyl Oxalate! Two carbon rings with a whole boatload of energy stored in some C-O, C-H, and C-C bonds.

    Thanks, Wikipedia!
  3. A fluorescent dye that doesn’t react. It just absorbs the energy from the breaking apart DPhOx and then releases that energy as visible light.

Turns out that it sucks to try and buy Diphenyl Oxalate, I tried. I even tried some stuff recommended by the guys at MAKE Magazine, but I don’t really want to go through the work of synthesizing DPhOx in my non-lab classroom (yea, I still don’t have a real science room yet, ugh).

So, it’s buying glow sticks in bulk for me! These are identical to the ones Coleman sells, except for about 1/3 the price.

Contrary to the Wikipedia, the dye is in the glass tube and the Hydrogen Peroxide is in the plastic-glass space.

Warning: the stuff in the glow stick will ruin clothes, hair, eyes, dignity, hands, and whatever else you can think of that shouldn’t smell like pee and glow pink. Wear gloves, don’t be a dumb4ss.

The Actual Calculus:

If you’re wondering what principal and ancillary standards apply here; I think this is the list:

  • Differentiate exponential functions
  • Differential equations
  • Student can design and conduct a valid investigation
  • Reaction rates (I will be doing explicit direct instruction on this)
  • Limiting Reactants
  • Various photonics standards

The experiments I envision students wanting to do range from:

  • Testing different amount of reactants (hyrdogen peroxide, diphenyl oxalate) and measuring intensity over time.
  • Testing for photonic saturation by adding hydrogen peroxide periodically and measuring intensity.
  • The reaction is pH dependent (there’s a gaggle of experiments there, it prefers alkalinity)
  • The reaction is severely temperature dependent.
  • The third ingredient (dye) is ultraviolet sensitive, and can be recharged without the reaction of the first two ingredients. Go!
  • Your students will think of something so awesome that there’s no way any of us can predict how cool it will be. Please post when it happens.

The ultimate goal being a time-dependent model that takes into account all of the variables listed above. It will probably come out something like this:

Luminosity (whatever) vs. time (hours)

[latex size=3]L(t) = L_0e^{(kt-t_{1/2})}[/latex]

Where L is luminosity, t is time, L-zero is the initial luminosity (which reaching is also worth studying with a high speed camera, btw), k is that lovely constant with units that are fun to discuss, and t-half is some sort of weird half life idea; I think.

The derivative, just for giggles is:

[latex size=3]\frac{dL}{dt}=kL_0e^{(kt-t_{1/2})}[/latex]

The integral, which may or may not be related to total photons released (eeck! science!) is:

[latex size=3]\int^6_0 L(t) dt= \frac{L_0}{k}e^{(kt-t_{1/2})}[/latex]

* “jillibillion” is really enjoyable to type, and it happens to be equal to ten multiplied by the highest number that Sam Shah can think of…

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