Visualization Talk proposal

Nathaniel Miller

In this talk, I will discuss several ways in which I have used Geometer’s Sketchpad in activities developing students’ visualization skills in a Modern Geometry class consisting mainly of pre-service elementary and secondary teachers.  In particular, I will discuss a lab in which students are asked to construct a model of a pool table, showing all of the paths that a cue ball can take before hitting another ball after bouncing off of a given number of rails.  They do this by looking at all possible repeated reflections of the target ball over sides of the table.  This leads naturally to a kind of covering space construction and a proof that there are 4n paths in which the ball hits exactly n rails.  Later in the course, the students are asked to write an open-ended paper in which they explore aspects of geometry on the cone on their own, in groups.   I will show how some students have used the ideas developed in the pool table lab to construct 2D covering space models of the cone in Sketchpad; have printed and cut these out in order to make physical models of the cone; and have used them to answer the question of how many straight paths connect two points on a particular cone.