Park City Mathematics Institute Investigating Geometry Project Abstracts Drafts of Project Files (password required)
A Budget of Hyperbolic Constructions
Philip Mallinson

These files constitute a collection of instructions for basic constructions for exploring the geometry of the upper half-plane model of hyperbolic geometry. All files are written in GeometerŐs Sketchpad (GSP).

Robert Upton

These labs were written to help guide Geometry students to explore concepts within Geometry using the Geometer's Sketchpad software. The labs are step by step scripts to lead the student through specific tasks in the Sketchpad program. The student will learn how to use the Sketchpad software through these labs, and they will also discover many topics within the study of Geometry. Each lab contains questions that students are expected to answer, either within the lab or in a separate document. Access to a computer with the Geometer's Sketchpad software is required to complete each of these labs.

Center Pivot Irrigation Problem
Kim Wollard

This project is designed to use Geometer's Sketchpad to explore area of squares, circles, and irregular shapes to determine the solution to a problem. The exploration is presented in the form of an irrigation problem in which students will use Sketchpad to determine a solution. There is an introductory piece that asks the students to come up with an initial design. Then they will be presented with the problem and asked to come up with a solution. The project is intended for users with Sketchpad experience.

Sprockets in the Mind of a Mad Man
M. Rich Salaz

Students will work with sprockets on bicycles in order to help understand gears. They will explore and develop ratios for radii, work with angular motion, and view sprockets in motion through Geometer's Sketchpad. Students should gain more understanding of gears upon completion.

Transformations
Reflecting on Reflections - Line and Circle Inversions
Chris Bolognese, William Schultheis

Transformations normally studied in a standard high school geometry course include rigid motions in the plane (reflections, rotations, translations) and dilations. Using GSP, this project allows students to explore the reflections in a line and extends the idea of reflection into circles. A custom toolset allows students to investigate the inversion of objects such as points, segments, angles, and circles into and out of a circle. A tool manual ("help" file) has been included that discusses how to use each tool and the various constructions (and their pros/cons) employed by the designers to accomplish inversions in a circle.

Slides, Flips and Turns with Geometer's Sketchpad
Deb Guthrie

This project has students create core squares and designs using geometric transformations. They locate and mark lines of symmetry, and make conjectures about the relationship between the lines of symmetry in the core square and the transformation designs. Connections are made between the information vocabulary of "slides, flips, and turns" and the formal vocabulary of "translations, reflections, and rotations."

Exploration of Transformations - a guided activity with Geometer Sketchpad for High School Students
Ralph Polley

This is the skeleton of a script for a teacher guided exploration of transformations. Students will experiment with transformations, rotations, reflections, and combinations thereof, in a variety of ways designed to create experiential familiarity with the properties of transformations. The script, as it stands, is not designed to be followed rigidly but rather serves as a guide for the teacher of the important concepts to be developed.

Transformations on Grids
L. Jill Ehlen

Geometer's Sketchpad is a software program that will facilitate students in exploration of mathematics learning in geometry. This program is increasingly being used in secondary settings so it is useful to begin at an earlier age. Transformations are one topic that lends itself to using this program.

Topics in Geometric Measurement
Measurement is a fundamental concept in geometry. These lessons use Cabri 3D and Geometer's Sketchpad (GSP) software to demonstrate and/or provide practice in applying measurement to explore geometric concepts. The lessons are:
• Using Cabri 3D to break apart prisms in to pyramids.
• Using basic trigonometry functions to analyze area and component dimensions of regular polygons.
• Classroom activities that explore centers of balance for triangles and irregular polygons using geometric and algebraic methods.
Centers of Balance: A Weighty Issue
Carla Parker

Although the centers of balance for triangles and regular polygons are covered in a typical Geometry course, exploration of the centers of balance for irregular polygons offers an excellent opportunity for spatial and logical reasoning. These lessons examine the properties of the triangle's centroid and examine Ceva's Theorem. Students are guided to arrive at geometric methods for finding centroid of irregular quadrilaterals, pentagon and hexagons. Finally there is instruction for calculated the coordinates for the centroids of irregular polygons.

SOHCAHTOA and Regular Polygons using Geometer's Sketchpad
Mario Shaunette

Year 2 of the Interactive Mathematics Program unit "Do Bees Build it Best" introduces sine, cosine and tangent as tools to help determine the area of regular polygons of a given perimeter. This lesson uses Geometer's Sketchpad to reinforce student understanding of and facility with the use of basic trigonometric functions to determine lengths of line segments in geometric figures.

Volume of Solids: A 3D Approach
Josué Martinez* and Juan Esparza

The visualization of formulas is often a good approach for the understanding of where some formulas come from instead of just memorizing. For example, taking two (2) congruent right triangles and putting them together in such a way that it makes a rectangle. We use Cabri 3D v2 to make several dynamic models of prism and their pyramids. The software has also helped us make nets for hands models to build.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. 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.