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The Hyperbolic Toolbox - The Role of Technology

Author(s): 
Stephen Szydlik

"Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning."

- The Technology Principle, the Principles and Standards for School Mathematics (p. 24)

Technology in the form of a dynamic geometry package such as Geometer's Sketchpad offers a powerful means of studying geometry.  As the value of this technology has become recognized, instructors have integrated it into their geometry courses (see, for example, the text by Baragar).

Dynamic geometry software offers access to hyperbolic geometry in at least three fundamental ways. First, Geometer's Sketchpad allows the user to perform basic constructions in the hyperbolic models. Indeed, though not conceptually difficult, many of these constructions are quite challenging from a practical standpoint. Consider, for example, the problem of constructing a hyperbolic line in the Poincaré disk model. That is, given two points A and B interior to the disk, construct with straightedge and compass, the arc of a circle orthogonal to the boundary of the disk that passes through A and B. Theoretically, this is not a particularly difficult construction. If the center of the Poincaré disk is labeled O, then the steps are as follows:
 

Constructing a Poincare line by hand using a straightedge and compass offers a challenge in careful record-keeping. (Click for an enlarged view).

 

  1. Construct ray OA.
  2. Construct the line through A perpendicular to ray OA. Label the points where this line intersects the boundary of the disk as C and D.
  3. Construct the lines that are tangent to the boundary of the disk at C and D. Assuming A and B are not on a diameter of the circle, let A' be the point of intersection of these tangents.
  4. Construct the Euclidean circle through A, B, and A'. The arc of this circle
  5. is interior to the disk is the desired Poincaré line.


 
 
 

This construction is not conceptually difficult. However, it requires quite a competent student to carry out the steps by hand using a straightedge, compass, and pencil without getting lost in the details of the construction. The capability of the Geometer's Sketchpad software to hide and show parts of a construction on demand helps to eliminate this difficulty.

Second, Geometer's Sketchpad allows users to use basic constructions to build more complex structures.  Though understanding how to construct a line in the Poincaré disk is a worthwhile and important skill,  one does not want to go through each of the above steps when a hyperbolic line is called for.  If one wants to discuss dropping a perpendicular from a point to a line, for example, one does not want to spend too much time on the construction of the given line!  As with other technologies, Sketchpad extends the range of problems accessible to students.

Third, through its dynamic nature, Geometer's Sketchpad presents an opportunity for students to explore, experiment, and make conjectures.  With its "click-and-drag" capabilities, students are able to change a defining object in a sketch and watch the resulting effect on the rest of the sketch.  For example, a typical activity in the Poincaré models is to construct a triangle and observe that its angle sum is less than 180 degrees.  However, with dynamic geometry software, one can change the defining vertices of the triangle and observe how the angle sum depends on the triangle, specifically that the larger the triangle, the smaller its angle sum.  This investigation can provide a context for an introduction to Gauss's theorem on the proportionality of a hyperbolic triangle's area to its defect.  See the Teaching Examples section (Page 8) for more examples.

Stephen Szydlik, "The Hyperbolic Toolbox - The Role of Technology," Loci (January 2005)

JOMA

Journal of Online Mathematics and its Applications

Dummy View - NOT TO BE DELETED