# Harmonic Behavior Part 1

### The Abstract

Around October we start digging into trigonometry in Pre Cal. The way curriculum is structured here, this is the first time they really play with sin/cos/tan outside of a brief exposure in Geometry. Once upon a time trigonometry was a foundation of Algebra II, but not anymore. It starts with the unit circle and moves into graphs. I present an image similar to this:

We discuss what we see. Are there any trends? I show them another bill from about 9 months later. Does this appear to be predictable? Almost always someone will point out that the summer months are much higher because you're running the air conditioner. Then we make some jokes about the weather in Texas (weather jokes, they're great!). Next, I have them peak at their unit circle and plot the points associated with sin/cos for various angles around the circle. We compare the result to the power bills we were looking at. At this point we discuss amplitude and period and I assign them a graphing task.

The assignment is to draw at least two periods of their given function, identifying the amplitude and period. This year's version of the taks doesn't look much different than last year's. My intent was to add a more in depth writing component and to require them to overlay their new function on top of the parent function. However, a few things conspired against this project and it got rushed. My initial day for conducting the poster portion got pushed and I wasn't at school for a couple days immediately after, so the result suffered a bit. Now, the results were still good but there's room for improvement. So that inspired what came next.

### The Relevant

So graphing sin/cos waves (and as a supplement, tangent waves) is great and all. But something gets lost when you're marking an x-axis with numbers in terms of pi. What does that mean? Nobody thinks in terms of pi. In our discussion of these functions, I always mention the high-voltage power lines that run near the school. In the right weather conditions you can hear them hum. I ask if any of them have noticed. This leads to a discussion about how electricity works, AC/DC, why their phone has this huge power brick, etc. Then we talk about sound. I pull up a tone generator (only seems to like Firefox).

If I mess with the volume, what aspect am I changing? If I adjust the frequency, what aspect am I changing? What's the highest frequency you can hear? Why can you hear 18kHz, but I can't? What's the lowest frequency we can hear? So, that's how dog whistles work? It was an interesting discussion. I tune it to 60Hz so they can hear how electricity "sounds." We make some connections to physics, as these words get thrown around in there too. One or two of them might realize that there might be something to this science stuff.

Continued in Part 2: Signals, Phase Shifting, and High Level Thinking

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