Using technology in the classroom can often present old problems in new ways, and in some cases that can lead to a chance to extend classical and traditional topics in new directions. The midpoint polygon problem is a good example of this phenomenon, and in this article, we attempt to explore several different ideas that arise in this way.
At one level, we want to illustrate how an Internetbased course can foster interaction of students with instructors and with other students by encouraging class members to investigate complicated problems in ways that would be difficult in a general discussion or in individual homework exercises. We do this by presenting a series of responses from students in an upper level course in geometry open to those who have completed the calculus sequence and one semester of linear algebra. The course was elected by students in mathematics, physics, and computer science, and it was specifically recommended for those interested in secondary school teaching.
We also want to introduce some natural extensions of standard approaches in the geometry of polygons to admit wider classes of geometrically interesting objects. For example many presentations of the geometry of polygons treat almost exclusively polygons that are convex. Some will deal with nonconvex polygons, but most avoid any treatment of polygons that intersect themselves. Introducing interactive geometric demonstrations changes all that. If the student is allowed to move a particular vertex in a polygon with more than three vertices, then it is a difficult programming problem to keep him or her from moving that vertex so as to create something nonconvex, let alone selfintersecting. A curious student will ask whether or not certain formulas and relationships still hold even when the polygon is not convex or when it selfintersects. Although such investigations fall outside traditional treatments, they arise very naturally when computer programs are used.
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Math 104
Original Discussion
Oriented Area Part 1
Oriented Area Part 2
Vector Notation