Park City Mathematics Institute
Geometrical Concepts from Constructions, Models, and Investigations
Project Abstracts

Drafts of Project Files (password required)

Incorporating Sketchpad into a Mathematics Curriculum
Points of Concurrency

by Melissa Cragg, Gary Kovacs, and Jerry Neidenbach
The high school geometry concepts explored in this project focus on the points of concurrency of a triangle. Students are provided with problem situations and investigate them using Geometer's Sketchpad. Discoveries from initial explorations are reinforced through investigations from several sources. The purpose of this project is to demonstrate how the dynamic software, Geometer's Sketchpad, can be integrated into a high school geometry curriculum.
Hand Signs - an exploration of transformations in analytic geometry
James Stallworth and Judy Lauper
Students will use a graphing calculator to explore the connections between geometric transformations, such as reflection and rotation, and ordered pairs in the coordinate plane. The students will create a connected scatterplot in the first quadrant of their calculator and observe the coordinates as the preimage is reflected and rotated to other quadrants in the plane. Discussions can lead to seeing rotation as a composition of reflection over intersecting lines and what happens to the orientation of points of the preimage as it undergoes various transformations.
Measure by Measure
Cal Armstrong
Distance is typically calculated by using the distance formula developed using Pythagoras. In fact, there are a multitude of distance formulae, or metrics, available to us depending on the context. For example, the taxi cab metric restricts us to moving horizontally & vertically while the Paris Metro (or Tim Horton) metric requires us to visit points along the way. By reading & working their way through the article, teachers & students will be able to develop some understanding of how which metric you choose changes the way you perceive your world.

Skunk Metric or Bully Metric
"It's Plane to Me"
A Comparison of Geometries for High School Students

Nancy Wilson

Plane geometry as originally outlined in Euclid's Elements is the traditional geometry taught in high school math courses. Other geometries that are considered "non-Euclidean" geometries can be included as well. These other geometries include spherical geometry and hyperbolic geometry. The uniqueness of Euclidean geometry is based on the "Parallel Postulate," given a line and a point not on that line, there is exactly one line parallel to the given line.

In the nature of mathematics, there are other possibilities to propose. Suppose that there are no lines that pass through a given point and are parallel to the given line. Suppose there are an infinite number of lines parallel to the given line. These other two suppositions give rise to spherical geometry and hyperbolic geometry.

The purpose of this project is to introduce students to spherical and hyperbolic geometries using models for each plane, and to examine the basic postulates of Euclidean geometry.

Hands-On Hyperbolic
Peg Cagle
Hyperbolic geometry is typically taught as an advanced geometry topic at the university level, using the upper half plane model or the Poincare disk. Both of these models depend on projections and do not give students a visceral understanding of the hyperbolic plane. Having a better understanding of the hyperbolic plane, allows students to compare and contrast space with positive curvature, zero curvature and negative curvature. By providing a model that students can touch, measure and manipulate, even high school geometry students can begin to develop an understanding of the hyperbolic plane. This project focuses on the creation of Bill Thurston's annular model, and an examination of the issues of constant curvature, hyperbolic trigonometry and tilings on the hyperbolic plane.
Explorations in Hyperbolic Geometry
Charles Schwartz

In this lesson, students compare constructions and measurements in the hyperbolic plane with the corresponding constructions and measurements in the Euclidean plane. Students review the construction of the circumcircle of a triangle in the Euclidean plane, and make the same construction in the hyperbolic plane. Students also make measurements to discover that, if you have two parallel lines in the hyperbolic plane, then the distance from a point on one line to the other line varies, depending upon the choice of the point. This contrasts with the situation in the Euclidean plane. Finally, students discover that the usual formula for the circumference of a circle is not valid on the hyperbolic plane, and as a consequence, the protractor is not a useful tool to measure angles in the hyperbolic plane.

This lesson presumes that the students have each constructed a copy of Thurston's model of the hyperbolic plane (which they are prepared to draw on). This lesson is designed as an Inquiry Lesson for honors geometry students in high school or for students enrolled in a modern geometry class in college.

Speaking Geometrically: An Introduction to Quadrilaterals
Jo-Ann Zisa

Students will use The Geometer's Sketchpad, a dynamic software tool, in a sequence of brief Math Vocabulary-Building exercises. As the year progresses, students continue to build a comprehensive bank of mathematical vocabulary fostering a deeper understanding of concepts, as they are introduced. Armed with the tools, students will be able to communicate their mathematical ideas, conjectures, responses, and questions in a clear, effective manner.

Once students have demonstrated an understanding of key geometric terms related to polygons and angles, they will explore the properties of quadrilaterals, develop generalizations in regard to classifying these shapes according to their properties, formulate and defend conjectures, and communicate their results verbally, through the use of graphic organizers, and in their math journals. Students will engage in meaningful activities from "Exploring Geometry with the Geometer's Sketchpad" as well as a variety of hands-on experiences using math manipulatives and concept/vocabulary games.

Lessons will be designed with the intention of guiding students through the transition from empirical based, inductive reasoning to the deductive use of rules and abstract thinking necessary for more advanced geometry.

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IAS/Park City Mathematics Institute is an outreach program of the School of Mathematics
at the Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540

Send questions or comments to: Suzanne Alejandre and Jim King

This material is based upon work supported by the National Science Foundation under Grant No. 0314808.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.