Geography

Geometry and Geography

Tom Davis

tomrdavis@earthlink.net http://www.geometer.org/mathcircles March 12, 2011

1 Pedagogical Advice

I have been leading mathematical circles using this topic for many years, and it seems to work well for both young and old students. The youngest group I’ve tried it on is third graders, and it works equally well for high school students. Of course the selection of questions and the presentation varies, depending on the level of the students. I usually start the circle with a general plan, but depending on the direction of the discussion, I omit some topics or go off on a tangent if that seems right. There is plenty of material here for two hours, so in most cases you will never cover all the topics. A good mathematician must be good at visualization. Almost all problems have a geometric interpretation, and being able to visualize and manipulate geometric images in your head can vastly improve your ability to solve mathematical problems. Visualizing things in two dimensions is difficult, and three-dimensional visualization is very difficult. One of the main goals for presenting this circle is to improve the students’ three-dimensional visualization ability. It’s probably easier for students to visualize the Earth as a ball in space than to visualize a mathematical sphere. They also know what the moon, sun and stars look like. Many have seen models of the solar system with planets orbiting the sun and moons orbiting some planets. For all these reasons, geography is more concrete and easier to visualize than spheres or circles or planes in 3-space. Most students have looked at models of the Earth and know about the equator and lines of longitude (which are great circles or parts of them) have seen that most globes are mounted on a tilted axis, and that there are the perhaps somewhat mysterious arctic and antarctic circles together with the tropics of Cancer and Capricorn. Many know that when it is summer in the United States, it’s...