One of my classes has spent the last month or so on basic trigonometry: right triangle trig, the unit circle, and graphs of trig functions. We have a school-wide spring exhibition coming up in exactly one month, and since trig seems to be one of the few high school math topics that has actual real-world applications that are both authentic and accessible to teenagers (a balance that's difficult to strike) I've decided to do a project surrounding trigonometry and sound (specifically music).
I've spent the last couple of days just fiddling around with Audacity, a great free audio editor that you can download here. I've experimented with importing entire songs and just with playing single notes or chords on the piano. You can also record yourself, which gets really fun for obvious reasons. In any case, any sound will generate a neat image which, if you zoom in enough, is just a complicated sound wave:
If you keep zooming in it looks more like a combination of the sine waves that we all know and love, and if you go to Analyze --> Frequency Analysis, you get a nifty frequency distribution of all the different notes in your clip. Even niftier for those of us who dig this sort of thing is that Audacity is really performing a Fourier transform on the sound wave! Gosh I wish someone had showed me this when I took Linear Algebra. Orthonormal, schmorthonormal.
What's cool is that even if you play a single note, like an "A", you don't just get a single peak at 440 Hz (a perfect A). Instead, you get a bunch of overtones that contribute to what you hear on different instruments.Once I had the basics (and I mean, the very basics. I know literally nothing about music) of Audacity down, it was just so much fun to play around. You can generate tones, superpose waves on each other, and create all sorts of nifty fade and filter effects. I wonder about having my kids try to re-create sound waves in GeoGebra by superposing simple waves until they get close to the original complex waveform. What about looking at what happens to a sound wave when it has to travel through a glass of water? Or, recording their voice and a friend's voice and trying to harmonize them by changing the pitch - a little experiment in intervals.
A colleague and I spent a good hour this afternoon recording ourselves saying simple words or phrases, and then playing them back in reverse and trying to repeat the reversed word and then play that back in reverse to see if it sounded anything like the original word - which was a jolly good time. It turns out that the word "potato" is incredibly difficult to do this with because of the little "p" at the beginning which is just a puff of air that is somehow really difficult to capture in reverse. But I digress.
... or maybe I don't. Maybe this is actually the point. I still can't think of a "project". But what I really, truly want is just for my kids to play around like I did, generate some interesting questions or "what if?"s or ideas, and try to tackle them. From experience, my fear is that when I ask my kids to simply play around and write down some observations, they freak out a little. They want you to ask a specific question so they can come up with a specific answer. It strikes me as a little odd that research - for some, that coveted culmination of one's academic career - is almost exclusively about asking good questions, yet K-12 education (and even college, in many cases) is all about getting the right answer to questions that are given to you.
So my conundrum: I want my kids to experience the "wow" factor of playing around with some neat software and exploring the connection between the mathematics of sound and why we hear things the way we do. I don't want to give them a laundry list of questions to answer because that immediately decreases the coolness by a factor of at least 100. But at the same time, I want them to have enough direction that they don't immediately flounder. Any words of wisdom out there?