Using Technology and Music to Talk About Sinusoids Mike Thayer Summit High School, Summit, NJ 25 th Annual “Good Ideas in Teaching Precalculus And…” Conference Rutgers University, New Brunswick, NJ March 18, 2011

What do we talk about when we talk about sinusoids? Period. AMPLITUDE. Phase. Sinusoidal axis. ▫Making graphs…lots and lots of graphs… Formulas: ▫Sums & differences ▫Product-to-sum and sum-to-product ▫Double angle, half angle ▫Other identities (Pythagorean, reciprocal, etc.)

So what do we DO with all of this information? Eventually, some smart student says, “Why are we learning this?”

Why sinusoids? In the style of Milne… Because they’re Important! ▫(they must be, because we devote several chapters to them!) Because learning to Verify their Identities is Good Practice! ▫(practice for what, I wonder?) Because they’re Needed for Calculus! ▫(actually, in calculus, they’re usually simply treated as another example of a function that can be integrated, differentiated, or power-series’d) Because they’re Interesting! ▫(this is actually true….!)

Interesting? Sinusoids? Really? We know that anything that happens in the world in a repeating sort of way can be modeled sinusoidally, or as sums of sinusoids… We also know lots of phenomena that repeat in exactly (well, kind of) this fashion ▫The tides ▫The seasons ▫Light (if you’re into physics) ▫Sound (ditto)

Why music, then? Music is the single most important example in our students’ lives of a periodic, sinusoidally- based function It’s a topic that for almost all students immediately grabs their interest (even if they’re not “musical” themselves) It actually is something we teachers can use to do DEMOS (demos?) with.

Okay, I’m still not convinced. Fine…here you go! (Extremely boring sound clip here: ) Fair point, but what if we tell our students it “looks like” THIS….

What does sound LOOK like? Do you see a periodic function? Can you estimate its period? (it’s about sec; frequency about 170 Hz)

Technology, Music and Sinusoids The goal: to use technology that is readily available to demonstrate interesting connections between sinusoids and sound. ▫We will talk about 3 programs today:  Audacity (a free music recording, mixing and editing program)  GarageBand (Apple’s music generating and recording software)  Mathematica (a computational software program – with lots of applications written for teachers!)

Audacity (audacity.sourceforge.net) From the website: “Audacity is a free, easy-to-use and multilingual audio editor and recorder for Windows, Mac OS X, GNU/Linux and other operating systems. You can use Audacity to: ▫Record live audio. ▫Convert tapes and records into digital recordings or CDs ▫Edit Ogg Vorbis, MP3, WAV or AIFF sound files. ▫Cut, copy, splice, or mix sounds together. ▫Change the speed or pitch of a recording. And more!”

So now, let’s play with Audacity a bit!play

Summary of Audacity Easy to create simple waveforms Can import samples of musical instruments to hear AND see the differences in the waveforms ( Can do simple spectral analysis Can apply pitch changes, effects to existing sounds, can “see” what they do to the waves What if we want other sounds?

GarageBandGarageBand (Mac OS X only) A software program by Apple that allows users to create music or podcasts (Wikipedia entry) Ships with all new Macs Fairly easy to use – can use software instruments as well as real instruments Part of the iLife suite for Macintosh Possible PC equivalent: Mixcraft 5 (see for more info)

Mathematica ( A computational software program that is immensely powerful. Lots of possible ways for mathematics teachers to use it – including lots of demos! How does it work? Let’s run it and see.

Conclusion There are many ways that we can bring technology into the classroom – but bringing music and technology in together can give our students a deeper understanding of sinusoids, their utility, and what all of these “different properties” actually mean. There is much more out there than I’ve had time to discuss today – if you find something good, let me know! Thank you very much for your time!