Title: MAA MathFest 2015 Logo - Description: Left-click to go to MAA MathFest 2015 web site.

Show Me Geometry

Geometry Software and Tablet Demonstrations


Wednesday, August 5, 2015, 1 PM – 3 PM

Marriott Wardman Park, Virginia C


Organizer:  Sarah Mabrouk, Framingham State University



This session invites presenters to share demonstrations, using geometry software or tablet applications, which help students to understand aspects of undergraduate geometry. These demonstrations should be suitable for Euclidean and non-Euclidean geometry courses as well as for courses frequently referred to as “modern” or “higher” geometry but not those related to differential geometry or (low-level) graduate courses. Presenters must perform the full demonstration (or a key portion of it) and discuss the aspects of the demonstration that help students to understand an associated theorem. Information regarding prerequisite topics and related areas with which students have difficulty should be discussed as should problems, if any, experienced in using the software or tablet application. Presenters are invited to discuss how they have modified the demonstration over time as well as to share information about software or tablet explorations performed with students that have helped students understand the associated theorem. Abstracts should include the name of the software or application, the platform (computer or tablet), and the associated theorem as well as a brief description of the demonstration. Presenters must provide their own laptop or tablet.


1 PM -  1:15 PM

Investigation of Geometric Theorems Using Geometer’s Sketchpad

Nora Strasser, Friends University


Many students studying Modern Geometry have difficulty understanding and visualizing the basic theorems in Euclidean Geometry. To assist students in the visualization of these theorems, Geometer’s sketchpad software has been used to create activities for the students. These activities allow students to better understand the theorems and to investigate further. These are interactive activities that require a written response by the student. After the students are familiar with the software, they are invited to create their own demonstrations. I will demonstrate the Postulate of Pasch activity that has been used by students. I will include other examples if time permits.


1:20 PM -  1:35 PM

Active Exploration of Desargues' Theorem and Projective Geometry

Michael Hvidsten, Gustavus Adolphus College


Students often have difficulty in conceptualizing properties of Projective Geometry. Intuition developed from perspective drawing is sometimes used to motivate lines and points at infinity, but for many there is still a conceptual hurdle to working in projective space. This is unfortunate, as Projective Geometry is one of the most beautiful and elegant ideas in mathematics. This talk will demonstrate a project that is used in the presenter's geometry class where students investigate Desargues' Theorem in models of Euclidean, Hyperbolic, and Elliptic geometries. Investigations in these models reveal the universality of Desargues' Theorem and also makes apparent how the idea of points and lines at infinity arise naturally from these models.


1:40 PM -  1:55 PM

The Poincaré Disk Model in GeoGebra

Martha Byrne, Earlham College


The free geometry and algebra software, GeoGebra, can be used for graphing functions and Euclidean constructions, but it can also be used to illustrate the Poincaré disk model for hyperbolic geometry. Given the non-intuitive nature of hyperbolic geometry, hands-on models are important for student comprehension.


This talk will present the construction of the model in GeoGebra, and the creation of simple that instructors can share with students. These tools will allow instructors and students to draw the line connecting two arbitrary points, see limiting parallel rays, find common perpendiculars, and measure distances.


2 PM -  2:15 PM

GeoGebra and Hyperbolic Geometry

Violeta Vasilevska, Utah Valley University


Understanding non-Euclidian Geometries has been a huge struggle for my geometry students. I have been trying for a while to find ways to overcome this struggle. Recently, I assigned projects that asked students to use GeoGebra software and perform geometric constructions in the Poincare´ disc model of the hyperbolic plane. Use of GeoGebra was motivated by two factors: first it is free and second, most of our math education students are familiar with it. Students were asked to construct the model and then explore some of the properties of the hyperbolic plane. They were expected to demonstrate some hyperbolic results. The idea was to construct the model from “scratch,” which meant that they would need to understand the model well. Students who constructed the model (and did not use a template) showed deeper understanding of the model and the discussed results. In this talk I will demonstrate what the students were asked to do, some of the results that they were asked to demonstrate, what worked/did not work in terms of understanding the model, and their survey feedback about the project.


2:20 PM -  2:35 PM

Math on a Sphere: an Interactive Programming System for Spherical Geometry

Michael Eisenberg, University of Colorado

Hilary Peddicord, National Oceanic and Atmospheric Administration

Sherry Hsi, Lawrence Hall of Science, Berkeley


This demonstration will present Math on a Sphere (MoS), an interactive programming system for exploring and understanding spherical geometry. The basic idea behind MoS is that it permits users to create geometric patterns on a representation of a sphere in much the same fashion as the interactive "turtle" in the well-known Logo and Scratch languages. In MoS, the spherical turtle-a programmable pen-can be given commands to move forward or back along the arc of a great circle and to turn right or left, in addition to many other graphical commands (e.g., for changing the color or the width of drawn lines). MoS is freely available over the Web (see the site mathsphere.org) and its website is accompanied by introductory materials for understanding the language and exploring basic preliminary ideas of spherical geometry. To date, we have used the system in several museum workshops (at the Lawrence Hall of Science, Berkeley CA) and to introduce spherical geometry to middle school students in Boulder, CO.


This demonstration will illustrate the rich mathematical (and aesthetic) capabilities of MoS. We will demonstrate how the system can be used to explore spherical triangles: the dependence of their interior angle total on area, and the (spherical) Law of Sines. We will present simple programs to generate the projection of the five Platonic solids on the surface of the sphere; and will use the system to explore predator-prey curves and spirals when realized on the sphere.


2:40 PM -  2:55 PM

Using a Dynamic Software Program to Develop Geometric Constructions

Laura Singletary, Lee University


To help students engage with geometrical concepts in meaningful ways, researchers and educators have recommended the use of dynamic software programs to teach undergraduate geometry. The use of a dynamic software program in a geometry course provides students with a digital environment to test conjectures and to develop the informal understandings necessary for students to derive general conclusions and rigorous proofs. During this interactive presentation, I will share a problem I use with my geometry students, asking them to construct a net for a truncated tetrahedron. In my class, I use construction problems such as this one to help my students build on and extend their understanding of important geometric concepts, theorems, and constructions. Using these problems with dynamic software, students are able to construct accurate and dynamic diagrams. Students are then able to interact with these dynamic diagrams, providing a means for students to deduce general properties and relationships and prove their constructions are viable. Research suggests that students’ uses of such programs have the potential to improve their understandings of geometric concepts in a meaningful way.



This page was created and is maintained by S. L. Mabrouk, Framingham State University.

This page was last modified on Thursday, June 03, 2015.