I. Rides: Think of a smaller circle whose center fixed on a point on the larger circle. Now think about rotating both circles about their centers and study the relationship between height and time for a point on the large circle and then for a point on smaller circle. You can build such constructions using Geometers' Sketchpad or GeoGebra. If you want, you can introduce this problem in the context of fair rides. A vertically ride example would be Ferris Wheels. Lateral thrill rides such as tea cups ride can be fun to model as well instead. For the vertical ride, you can then ask students trace the changes in height with time, which is generated by combination of sinusoids. It would be great if you keep them adding smaller and smaller circular rides, and observe the changes in combined sinusoids. The rate of change in height can be associated with the amount of thrill. You can ask students predict most thrilling moments of those rides, in larger, and smaller circles.
II. Sound: Another idea I would suggest is to incorporate sound. You can use GeoGebra Playsound[f(x), xmin, xmax] command to produce the corresponding the sound for a function. You can ask them alter pitch and frequency by introducing a*f(b*x) or a*f(x/b) and changing the constants a and b. You can start from creating simple tones using single sinusoids, by finding out the frequency required for Do, Re, Mi,... You can later extend into building chords by combining corresponding functions. Depending on their readiness, you may also directly start by asking students to create a beat by combining sine waves with slightly different periods. Examine how their graphs look like, they may need to work on their windows setting to better make sense of the behavior. Next you can ask students how to create sound of an heart beat, tap tap tap sound, or rain drops, or an emergency siren.
III. Rotational systems that convert circular motion to linear and drill square holes: What if we rotate smaller circle inside the bigger one. We can now examine further the relationship when the radius of smaller circle is the half, the third, and the fourth of the size of the larger circle. We can ask students explore the behavior of the center of the smaller circle as the circle rotates inside. This can help one to discover a rotational system converting circular motion into linear. We can further discuss a mathematical pattern that can be used as principal in building build a sewing machine. Parametric representations can work well to express the behavior of a point on the rotating circle. A great challenge problem is to examine a rotational system that can drill a square hole using a circular rotation. For this, we need to reconceive the former smaller circle inside as another circle like contour composed by three 60 degree arcs with the same radius as the large circle. This circle/triangle like contour is known as Reuleaux Triangle!
I hope you enjoy at least reading these activities. I had fun writing them to share with you.
On Sat, Oct 12, 2013 at 4:34 PM, Nghiem <firstname.lastname@example.org>wrote:
> I teach Pre-Calculus and I am in need of ideas for a project on Sinusoids. > I have done plotting temperatures in the past, but I am looking for a new > idea. My goals for my students are: > - - understand the connection between the unit circle and sinusoidal > functions. > - - connect sinusoids to real-world data. > - - transform sin(x) or cos(x) to model the real-world data. > > thanks. > >