Illuminations: Symmetries I

# Symmetries I

This is the first i-Math in a four-part series of i-Maths entitled Symmetries and Their Properties. In this first i-Math you will investigate rotational symmetry. Fix a center, turn, and you have a rotation. Many objects in nature—flowers, starfish, and crystals—and objects we use every day, such as wheels, CDs, and drinking glasses, have rotational symmetry. Here, you will learn about the mathematical properties of rotations and have an opportunity to make your own designs.

### Math Content

Students will learn about the mathematical properties of rotations and have an opportunity to make their own designs.

### Individual Lessons

Lesson 1 - Describing Rotations

Fix a center, turn, and you have a rotation. Many objects in nature—flowers, starfish, and crystals—and objects we use every day, such as wheels, CDs, and drinking glasses, have rotational symmetry. The first thing to think about is how to describe a rotation mathematically.

Lesson 2 - Finding What Doesn't Change

Now that you have found out how to describe rotations of a figure, you can predict the effect of a rotation through a given angle and even the effect of two or more rotations performed one after the other. You will also be able to find angles that leave a figure unchanged.

Lesson 3 - Relating Rotations to Symmetry

In this part, you will investigate the relationship between rotations and the symmetry you recognize in a figure or a design. This is the secret behind how mathematicians describe symmetry. The figure rotates, but the figure does not appear to have moved. Mathematicians understand change by investigating what doesn't change. You will use this idea to find the rotations underlying various shapes and designs. We start with the rotations that describe cyclic symmetry in particular figures and designs. Look for what doesn't change!

Lesson 4 - Conclusions

We finish this 4-part i-Math Investigation on rotational symmetry by putting things together. You will have the opportunity to think about the ideas you have discovered in this i-Math as a whole by answering some review questions.

### NCTM Resources

 More and Better Mathematics for All Students
 © 2000 National Council of Teachers of Mathematics Use of this Web site constitutes acceptance of the Terms of Use The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring mathematics learning of the highest quality for all students. The views expressed or implied, unless otherwise noted, should not be interpreted as official positions of the Council.