  # Using Hyperbolic Tools in the Classroom

Here are some very brief ideas on ways that I have used Geometer's Sketchpad scripts for hyperbolic constructions in the classroom. For more teaching ideas, you might want to look at the article "The Hyperbolic Toolbox" that I wrote for the Journal of Online Mathematics and its Applications.

In my geometry classes at the University of Wisconsin Oshkosh, I use the tools in three essentially different ways:

1. Students develop the scripts themselves. In my mathematics classes, I try to avoid the "black box". Whenever possible, I have my students develop their own techniques, shortcuts and tools, thereby providing them with some sense of "ownership". The construction tools in hyperbolic geometry offer a great opportunity for students to do this, as the different constructions span a wide range of complexity. For example, in the Klein model, constructing segments and lines is trivial, dropping and raising perpendiculars is fairly straightforward, measuring angles can provide a nice application of hyperbolic trigonometry, and drawing circles is hard. Developing tools in any of the three models can serve as a great class project.
2. Use the tools to illustrate hyperbolic theorems. A difficulty with non-Euclidean geometry is visualization. How can we picture a world in which Euclid's fifth axiom fails? Even worse are some of the counterintuitive theorems of non-Euclidean geometry. The models provide us with examples of hyperbolic worlds where students can see the theorems of hyperbolic geometry illustrated. For example:

 Theorem: In hyperbolic geometry, if m and m' are parallel lines for which there exists a pair of points A and B on m equidistant from m', then m and m' have a common perpendicular that is also the shortest segment between m and m'. Exercise: Give the students a copy of the Geometer's Sketchpad Klein model with two Klein lines m m' constructed. Ask them to find two points on m which are equidistant from m'. This forces students to use their "measuring length" tool as well as to think about dropping perpendiculars to get the segment they want to measure. (Note that both of these tools can be created by students - see #1 above.) The followup is to have the students find the shortest segment joining a point on m to a point on m'. If students have not seen the theorem above, they can actually discover it themselves, while if they have seen the theorem, this activity serves to illustrate it.

3. Use the tools for more extensive constructions in the models. There are several ways to do this. On a basic level, it is a fun and education exercise for students to try to construct standard geometric figures in each model. Ask them to formally construct an equilateral triangle, a rectangle (or square), a Lambert quadrilateral, and a Saccheri quadrilateral.

A second, slightly more advanced exercise is, given two points P and Q in a model, to have students construct and measure the angle of parallelism corresponding to segment PQ. This again involves perpendiculars and requires an understanding of limiting parallel rays.