- Given images, the student will be able to correctly identify all forms
of symmetry.

- The student will recognize the relationship between types of symmetries.

- At the challenge level, the student will learn a new way of classifying
symmetry groups.

In order to use this lesson most effectively, it is suggested that the instructor read all materials given and choose those that best suit the needs of the class. One suggestion use is to have students begin with the definition page. Following each definition, there is a link to an investigation usingThe Geometer's Sketchpad.

If

The Geometer's Sketchpadis not available, you may download a demo copy atThe Geometer's SketchpadAfter that, students can do more exploring using Java applications and Kali. For the Java applications, your browser needs to be Netscape version 3.0 or later. Do not try the Java applications on versions before 3.0. Kali can be downloaded from The Geometry Center A Web guide to using Kali on a Macintosh is also available. If your browser supports JavaScript, answers to the questions on the Kali Web pages can be checked interactively.

Paper and pencil exercises are also available to reinforce concepts explored in the computer activities. Feel free to print any of them and duplicate them for classroom use.

- Paper and Pencil Worksheets (no software required)
- Worksheets for Geometer's Sketchpad Activities
- Worksheets for Kali Activities
- Getting Started

If the Web page for using Kali is too long, print out this sheet. - Rosette Patterns
[
*Answers*] - Wallpaper Patterns
[
*Answers*]

- Getting Started
- Worksheet for Letterforms and Illusions

- Image gallery

All of the images used in this document and more can be found in the image gallery. You can use these images however you want. - Java activities

Interactive Web applications with Java can be used to enhance students' learning. Be careful, however, as they do not run on all Web browsers.

- The Geometer's Sketchpad:
Presentation Sketches-Glide Reflect Tiling
- The Math Forum: Steve's Dump
*- General Mathematics Resources* - The Math Forum: Steve's Dump
*- General Geometry Resources* - Geometry and the Imagination
*- Conway, Doyle, Gilman, Thurstan* - Introduction to Symmetry
*- Math 5337, The Geometry Center* - Introduction to Wallpaper Patterns
*- Math 5337, The Geometry Center* - Symmetry and the Shape of Space
*- Chaim Goodman-Strauss* - Classification of Patterns
*- 1996 summer teacher project* - Java Kali
- Kali worksheets
- How
to use Kali on a Macintosh
- Symmetry and Group Theory
*- The Geometry Junkyard* - Bodypaint
*- Interactive surface painting* - Symmetry in Tesselations
- Symmetry Studio: Computer Aided Surface Design
*- Artware Incorporated* - NC Computer Skills Curriculum
*- Identify and make figures with line symmetry* - Symmetry in the Plane
*-17 different ways a plane pattern can be symmetrical.* - Wallpaper Groups
*- Appropriate for college-level readers.* - The Geometer's Sketchpad:
Presentation Sketches-Glide Refelect Tiling

Student artwork courtesy of The Minnesota Center for Arts Education

Special Thanks to:

Naomi Hancock '92 Sarah Johnson '98 Katy Celeste Petersen '94 Sarah Wojiechowski '94 John Vang '98 Holly Vrieze '92

*National Council of Teachers of Mathematics. Curriculum and Evaluation
Standards for School Mathematics*. Reston,VA, NCTM, 1989.

Serra, Michael. *Discovering Geometry-An Inductive Approach.* Berkeley, CA.
Key Curriculum Press, 1993.

Clemens, Stanley R., et al. *Geometry*.
Menlo Park, CA. Addison Wesley, 1990

Brown, Richard G., *Transformational Geometry*, USA, Silver,
Burdett & Ginn Inc., 1973.

Washburn, Dorothy. K. and Crowe, Donald. W.,
*Symmetries of Culture*,
Washington Press, 1987.

Harter, Jim,
*Animals: 1419 Copyright-free Illustrations of Mammals, Birds, Fish,
Insects, etc.*,
Dover Publications, 1979.

*Java(tm) Developers Kit version 1.0.2*, Sun
Microsystems, Inc., Mountain View, California, 1995

2 Ibid., p129.

3 Ibid., p162.