Listening to Math

Finding ways to engage students sensory and creative faculties in the cause of teaching math is, in my opinion, essential to helping increase their interaction and long term retention of otherwise abstract concepts. Although much of the material on the Internet focuses on visualizing mathematics, developing techniques to relate audio to the topics they are learning seems equally important.

The applet I have been using most often to achieve this is the one found here:

Demonstrating Audio Beat Frequencies

This tool allows you to control the amplitude and frequency of two sinusoids being added together, and to listen to the resulting output. It is immediately obvious that this can be used to teach the relationship of frequency and amplitude to the resulting sound generated, which is important for topics in the Math Level IIC SAT, as well as pre-calculus and physics. It is interesting to incorporate a phase shift in the sinusoids, and see how that affects both the resulting sound as well as the resulting waveform.

One of my favorite uses of this tool, however, is to demonstrate the difference between linear functions and non-linear functions through sound. Typically the argument of a sinusoid is LINEAR in time, such as sin(10t) or cos(3t). Consequently, there is a character to the sound that is even and repeating. The frequency stays the same over the length of the tone, making the sound uniform. If, however, we change the argument in the sin wave to one that is NON-LINEAR in time, such as sin(10t^2), the sound changes noticeably. No longer is the frequency the same over time, and the sound takes on a very psychedelic nature. Try it yourself and see what it sounds like!

A sample lesson for this applet might be something like:


(1) Using only one sine wave, ask students to guess how changing the number within the sine's arguments (i.e. the frequency) and the coefficient in front of the sine (i.e. the amplitude) should affect the resultant sound. Pick a few different amplitudes and frequencies, and have them guess which one is which from just the sound. What is the effect of a negative amplitude or frequency?

(2) Try applying a phase shift to the wave. Does this affect the resulting sound? Why or why not?

(3) Instead of using a linear function of time in the argument, try using sin (10t^2) or sin (10t^3 - 10t). How does this affect the tone? Use this as a way to affirm the notion that something that is linear has the same general character over its entire domain, but something that is non-linear does not.

(4) Using two sine waves, what relationships of the frequencies will produce a beat? Help students to see that when a beat forms, there are TWO frequencies that are occuring: one is the reference frequency of the underlying tone, and the other is the frequency of the "beat", i.e. the pulsing of the tone's amplitude. One should be at the sum of the two frequencies, and the other at the difference of the two frequencies. Have the students use the trigonometric identities for angle addition and angle subtraction to generate particular beat patterns or tones.