The Hyperbolic Toolbox:
Non-Euclidean Constructions in Geometer's Sketchpad
6. Teaching Examples
There are several ways in which the hyperbolic construction scripts can
be used in classroom applications. It should be emphasized, however,
that the scripts are tools and not ends in themselves. They can be
adapted by an instructor to produce applications suitable for classroom
demonstrations. Perhaps more importantly, though, they can be used
as building blocks for students to develop their own constructions and
gain a deeper understanding of hyperbolic geometry. The examples
below provide a sample of how the scripts have been used in teaching geometry,
but the list is not exhaustive by any means. Creative instructors
will be able to find many other uses.
- Have students develop some of the tools themselves. The difficulty
of writing the Sketchpad scripts for the basic "hyperbolic straightedge
and compass" constructions varies greatly. In the Klein model, for example,
constructing lines and segments are trivial, calculating lengths, dropping
and raising perpendiculars is relatively straightforward, while calculating
angle measure, constructing midpoints, angle bisectors, and circles is quite
Interestingly, the corresponding constructions in the other models
do not necessarily have corresponding difficulty levels. For example,
writing a tool that constructs a hyperbolic line in the Klein model is merely
a matter of drawing a chord passing through two points interior to a circle.
Constructing a half-circle passing through two points in the Poincaré
half-plane is slightly more challenging, though students who are able to circumscribe
a circle about a triangle should be able to complete this task. Constructing
a hyperbolic line in the Poincaré disk requires several steps, including
the ability to find "inverse points" (see Greenberg
p.243-247). On the other hand, calculating angle measure in the Poincaré
models is relatively straightforward since these models are conformal, while
in the Klein model, this action requires the use of hyperbolic trigonometry.
In my classroom, I have had students develop the tools for constructing lines
and segments in the three models as an introduction to the models and to writing
scripts in Geometer's Sketchpad after reading the appropriate section
of Greenberg for the Poincaré disk
construction. In addition, I have the students discover on their own
and without a computer how to "raise perpendiculars" in the Poincaré
models. For more advanced classes, the more challenging constructions
could be assigned as individual or group projects.
- Use the scripts as a toolbox for more extensive hyperbolic constructions.
The scripts can serve as building blocks for Geometer's Sketchpad activities
that lead to a more complete conceptualization of hyperbolic geometry.
The power of the tools becomes most apparent here: students are able
to investigate the subject interactively as they develop their own constructions.
Moreover, constructions require justification. After completing a construction,
the natural question to ask is, "Is it correct?" Here are some samples
that illustrate these principles:
- Use the scripts to either motivate or help students visualize major
theorems in hyperbolic geometry. Many of the theorems of hyperbolic
geometry are counterintuitive for students who have until now considered only
Euclid's geometry. For these students, the hyperbolic models can be
a welcome relief. Abstract theorems can give way to illuminating examples,
illustrating many of the properties of non-Euclidean geometry.
Although not truly applications in themselves, the hyperbolic tools can be
used (by either the instructor or students) to develop applets illustrating