Up: An Introduction to Symmetry

Teacher Resources

Objectives of Unit

This unit is designed to recognize the NCTM Curriculum and Evaluation Standards for School Mathematics concerning the need to give increased attention to "Coordinate and Transfomation Approaches" 1 in Geometry at the secondary level. The lesson also demonstrates increased attention to: "The active involvement of students in constructing and applying mathematical ideas; the use of a variety of instructional formats; and the use of computers for learning and doing mathematics."2 "Transformations serve as a powerful problem-solving tool and permit students to develop a broad concept of congruence and similarity that applies to all figures."3
  1. Given images, the student will be able to correctly identify all forms of symmetry.
  2. The student will recognize the relationship between types of symmetries.
  3. At the challenge level, the student will learn a new way of classifying symmetry groups.

Using This Unit

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 using The Geometer's Sketchpad.

If The Geometer's Sketchpad is not available, you may download a demo copy at
The Geometer's Sketchpad

After 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.


Other Resources Available Here.

Additional Resources


Nature photographs courtesy of Chuck Dorsey

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


1 National Council of Teachers of Mathematics, Curriculum and Evaluation Standards for School Mathematics, p126.
2 Ibid., p129.
3 Ibid., p162.

Up: An Introduction to Symmetry