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Bob Hasson, College of San Mateo, requested ideas about problems to use in a Math Club, as their existing pool of intriguing problems was growing old. Here is Bob's description of the context:
The book I have been using for Math Club is Thinking Mathematically by Mason, Burton, and Stacey. It’s an old book. In the first Math Club meeting this year we (myself and the other faculty advisor) asked the students what they were interested in. This led to lectures by the teachers (one on RSA cryptography, another on the CORDIC algorithm used for computations in calculators) -- that clearly bored the students.
So I brought in some logical puzzles from one of Raymond Smullyan’s books. (Among other things Smullyan is the inventor of Knight-Knave puzzles — Knights always tell the truth, Knaves always lie, if two people said X and Y, then what must the people be?) The students did the problems as best they could with much discussion back and forth and clearly enjoyed them. This taught the teachers that the students want to do problems within their reach and socialize both. And they don’t want to be talked at. It’s really as much of a social club as a Math Club.
So logical puzzles are good. (I found some good non-Smullyan logic puzzles at http://www.skonline.org/website/mathonline/logic/logicpreface.htm.) Problems on digits and simple number theory are good. Mathematical history is also good. One student did a presentation on Ramanujan, the Indian mathematician.
Here are some of the ideas and Bob's responses - more ideas welcome!
Christine Will, Laney College, writes:
Tom Carey, San Diego State University, adds another possible site:
www.mathprojects.com links to a book from the "Math Projects Journal". It looks like most of these activities for schools could be adopted for Dev Math - not sure if that is part of the territory you cover in the Math Club.
Bob Hasson, College of San Mateo:
A couple of ideas:
1. Help the students develop a set of parametric equations with a time parameter to simulate a sun-earth-moon-mars system so that all objects are is at the right distances from the sun and go around with the right relative speeds. (Assume circular orbits to greatly simplify things.)
2. What if cars had square wheels? Would rides necessarily be bumpy? No, not if the road surfaces have the right shape. Help students explore what the shape should be, again using parametric equations. (This is not originally my idea. I first saw it, but not a solution, in a Geometer's Sketchpad presentation twenty years ago.)
Bob Hasson, College of San Mateo
I tried the sun-earth-moon-mars system in Math Club this week. It went quite well.
First I did a very quick lesson for the students on parametric graphs of circles (like <5cos t + 4, 5sin t + 3>, with 0 <= t <= 2pi). Some of the students but not all knew this stuff already.
Then I asked the students how to represent the sun as a circle of radius 1 centered at (0,0). Of course they knew that immediately.
Next I asked them how to parametrically represent the earth's orbit, if the orbit is 5 units out from the center of the sun. I told them to assume that the orbit is circular for simplicity (it almost is). Another no-brainer.
After that I asked them how to show the earth as a circle of radius 1 with parametric equations containing a time parameter a, such that, as a changes, the center of the earth circle goes around the sun circle. (<cos t + 5 cos a, sin t + 5 sin a>, with 0 <= t <= 2pi.) They needed a bit of help with that question, but not very much. To see it we used a homemade, dynamic, pretty buggy graphing applet I wrote a number of years ago. I'm sure there is better, maybe free software out there for this purpose. Perhaps someone can chime in about one of these.
Then I asked them to add the moon going around the earth 12 times for each time the earth goes around the sun. To my surprise they got that one pretty quickly, too. The students enjoyed watching their handywork: the earth circling the sun on the projected computer screen, and the moon chugging its way around the earth as the earth circled.
Finally I asked them to add Mars. I told them that Mars is half again farther out from the sun, and the relationship between radius of orbit to period of orbit is radius^2 = k period^3 (Kepler's third law). The students used the case of the earth to figure out k, and then computed the martian period. Then they added parametric equations with the time parameter a to get Mars to circle the sun with the right distance and period. This completed the functioning sun-earth-moon-mars system
All of this took about an hour and a half, our usual Math Club meeting time.
Bob Hasson College of San Mateo
In Math Club we tried the square wheels idea. Only one student was present, so I can't tell how it would have gone with a larger group. But I thought that it was a challenge to make the idea work for students.
My approach looks at the motion of two points (corners of the square wheel) on a rolling circle around the wheel, the vector between these points, and how to make this vector the tangent vector of the curve of an equation for the roadway. And of course as this happens the center of the square (and of the circle) must make no vertical change.
It's a challenge to make this idea accessible to students to an extent that they can do much of it themselves. Don't think we got there this time.
A note: people have built short roadways that are friendly to square wheeled vehicles. I'm told there is one at San Francisco's Exploratorium, with a vehicle to ride on it. That would be quite a math club project.
Some of our math club students are interested in getting a comet into the sun-earth-moon-mars system discussed before. I have figured out how to do this, and I've made a model in Geogebra, but I will have to solve the same kind of student accessibility problem. Food for thought.