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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 |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |