Harmonic Motion in the Real World

Best Curriculum Practice

Kathleen Smith

It is my opinion that students remember best those concepts which they have the opportunity to experience in action. Toward that end I have designed a three day workshop on harmonic motion that I use to introduce sinusoidal application problems in my trigonometry class. The NCTM Standards state that the study of trigonometry should allow students to "explore periodic real-world phenomena using the sine and cosine functions" and "use circular functions to model periodic real-world phenomena" By using the graphing calculators and the CBL the students have the chance to actually see raw data turn into a sinusoid before their eyes. The effect is dramatic and I find they easily transfer the experience to the application problems.

Because I think the use of a sinusoid to model real world situations is an important topic, I am willing to spend three days on this workshop. You could chose only one of the experiments, limit the discussion, and it would not even take a class period. You could also expand on the experiments and spend an entire week.
Prior to this workshop the students have studied the sine and cosine waves and understand amplitude, frequency, period and vertical and horizontal shifts. They are expected to describe, discuss and analyze the graphs we generate using these parameters.

I will give information on the reference materials I use for the workshop as they come up. If you have difficulty finding any of them you can e-mail me at
smithka@knight.cmi.k12.il.us and I will help you get up and running.

THE HARMONIC MOTION WORKSHOP

Day 1

I start with a discussion of what types of real world applications might be modeled with a sine or cosine wave. The students already know that sound and light move in waves. Depending on the year and the students, the discussion might be a lot of fun and full of insight, or it might be like pulling teeth......at any rate I generally get them to the point where we decide we could try to model sound and see if it graphs as a wave.

I will have set up the CBL linked to a TI-82 with the sound probe attached. We use the generic "Sound" program that comes with the CBL. I will have on hand a set of tuning forks which I get from our physics teacher, but you may also try the music teacher. I do the first run through and we talk about the results. I generally do not get a good looking curve first time out, so then the kids come up and try to show me how to do it. When we get a good looking wave we downlink the lists to the whole class and I then walk the students though how to find an equation that fits the data well. We talk about how hard it is to take good data, how important it is to use the window settings as guidelines, etc., etc., etc....

To end the class we take a second reading of good looking data from a different tuning fork, downlink it and the students have to find an equation for homework.

Day 2

We begin class by having several students who think they got a good fit with their equations come up and type them (rounded to two decimal places) into the overhead unit. We then have a class discussion about what seems to remain fixed (amplitude, horizontal axes, period, etc.) and what varies (sine,cosine, phase shift, etc.) This may go quickly, or may take some time. I generally allow 15 minutes.

Fifteen minutes into the class the music teacher will enter the room with his trombone. He says he hears we are looking at harmonic motion and asks if he might try his horn at producing a nice wave. Of course we acquiesce and he sits down in front of the sound probe. The waves he produces will vary depending on how long he has the horn extended, what note he plays, and how the wind passes through the instrument. Our particular music teacher is great at talking math since he has a degree in engineering as well as music. He goes into a whole discussion of how the waves are produced, how we are actually seeing the sum of waves produced by the same note bouncing around at different wave lengths.....

I will not even begin to explain what it all means, but I can assure you it is fascinating. I learn something new every time he comes in. More importantly, the students see that a "real person" understands this stuff and views it as practical useful knowledge. I allow the students to discuss waves with the music teacher for as long as interest stays active.

For homework I give some application problems from the book and ask the students to determine the amplitude, frequency and shifts (if any) for the situations. I do not ask them to work the problem out, just look for the parameters.

Day 3

We start day three with a pendulum problem. I have taken this lesson from Laura Clarke's article Getting into the "Swing" of Functions in the February 1997 Mathematics Teacher (page 102). It is a very neat experiment where the light probe for the CBL is outfitted with a heavy ball to make it swing like a pendulum. This unit is then suspended over a light source (I use an overhead with the projection arm folded down) that has been covered in an acetate screen with graduated shading. As the pendulum swings back and forth across the shaded light source it produces a very nice wave. There is enough material in the article to last several days and cover all aspects of functions, not just sinusoids, but I only use a limited part. (I have loaded the programs from page 106 on my calculator and I could try to send them to you if you have Mathlink. No guaranties....)

When the students enter I will have the overhead set up with the light probe pendulum on the overhead using screen #1 on page 107. I will asked them what they think I intend to do. After they have figured it out I asked why they think I need the acetate sheet and what type of effect it will have......they generally say they don't have a clue, but guess it will be a sinusoid.

So we talk a little about how waves can model situations that do not look like waves at all, the pendulum just goes back and forth, but looks like a wave when graphed. We discuss input and output, domain, range, horizontal and vertical axes, etc....until I think they are convinced that the data will in fact look like a wave. Then we try swinging the pendulum to see if we get is what we thought we should. We take several data sets and compare and contrast them. I put an acetate sheet over the LCD panel and trace each graph with an overhead pen. We can then put three or four of them on the screen together after we remove the LCD panel.

The discussion that ensues is always lively and full of questions, conjectures (some correct, some not) and insights. If time permits I will bring out shaded sheet #4 from page 107 and see if they can anticipate what the data might look like jumping from black to white rather than having a gradual transition as with the first sheet.

To end class I ask if there were any problems with the previous night's homework. Generally the fact that some of the problems did not involve a motion that the students perceived as a "wave" gave some of them trouble when looking for amplitude and shifts. Hopefully by opening with the pendulum problem they will realize when they go to ask their question that they can now answer it themselves. I love to see the students experience the power of gaining knowledge and knowing it!
For homework I have them finish the problem they had started the night before.

After the three day workshop I can refer back whenever needed to one of the three experiments to make a point, help clarify a question, focus the discussion or stimulate ideas. Some teachers may feel that the three days are a waste of direct instruction time, but I think what the students gain in three days is worth a month of my lectures, extra help sessions and one-on-one experiments. Yes the room is noisy, and yes some students will not be as attentive as we might want, but they all go away with a much better sense of what harmonic motion is, why it is such a good modeling tool and why it might just be worth studying.