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# Symmetry

## Investigate symmetry, one of the most important ideas in mathematics. Explore geometric notions of symmetry by creating designs and examining their properties. Investigate line symmetry and rotation symmetry; then learn about frieze patterns.

### In This Session

Part A: Line Symmetry
Part B: Rotation Symmetry
Part C: Translation Symmetry and Frieze Patterns
Homework

Symmetry is one of the most important ideas in mathematics. There can be symmetry in an algebraic calculation, in a proof, or in a geometric design. It’s such a powerful idea that when it’s used in solving a problem, we say that we exploit the symmetry of the situation. In this session, you will explore geometric versions of symmetry by creating designs and examining their properties.

For information on required and/or optional materials for this session, see Note 1.

### Learning Objectives

In this session, you will do the following:

• Explore line or reflection symmetry
• Explore rotation symmetry
• Explore translation symmetry and frieze patterns

### Previously Introduced

Coordinates: Points are geometric objects that have only location. To describe their location, we use coordinates. We begin with a standard reference frame (typically the x- and y-axes). The coordinates of a point describe where it is located with respect to this reference frame. They are given in the form (x,y) where the x represents how far the point is from 0 along the x-axis, and the y represents how far it is from 0 along the y-axis. The form (x,y) is a standard convention that allows everyone to mean the same thing when they reference any point.

Reflection: Reflection is a rigid motion, meaning an object changes its position but not its size or shape. In a reflection, you create a mirror image of the object. There is a particular line that acts like the mirror. In reflection, the object changes its orientation (top and bottom, left and right). Depending on the location of the mirror line, the object may also change location.

Rotation: Rotation is a rigid motion, meaning an object changes its position but not its size or shape. In a rotation, an object is turned about a “center” point, through a particular angle. (Note that the “center” of rotation is not necessarily the “center” of the object or even a point on the object.) In a rotation, the object changes its orientation (top and bottom). Depending on the location of the center of rotation, the object may also change location.

Translation: Translation is a rigid motion, meaning an object changes its position but not its size or shape. In a translation, an object is moved in a given direction for a particular distance. A translation is therefore usually described by a vector, pointing in the direction of movement and with the appropriate length. In translation, the object changes its location, but not its orientation (top and bottom, left and right).

### New in This Session

Frieze Pattern: A frieze pattern is an infinite strip containing a symmetric pattern.

Glide Reflection: A glide reflection is a combination of two transformations: a reflection over a line followed by a translation in the same direction as the line.

Reflection or Line Symmetry: A polygon has line symmetry, or reflection symmetry, if you can fold it in half along a line so that the two halves match exactly. The folding line is called the line of symmetry.

Rotation Symmetry: A figure has rotation symmetry if you can rotate (or turn) that figure around a center point  by fewer than 360° and the figure appears unchanged.

Symmetry: A design has symmetry if you can move the entire design by either rotation, reflection, or translation, and the design appears unchanged.

Translation Symmetry: Translation symmetry can be found only on an infinite strip. For translation symmetry, you can slide the whole strip some distance, and the pattern will land back on itself.

Vector: A vector can be used to describe a translation. It is drawn as an arrow. The arrowhead points in the direction of the translation, and the length of the vector tells you the length of the translation.

### Notes

Note 1

Materials Needed:

• Mira (a transparent image reflector)
• tracing paper or patty paper and fasteners
• large paper or poster board (optional)
• scissors
• markers

Mira
You can purchase a Mira from the following source:

ETA/Cuisenaire
500 Greenview Court
Vernon Hills, IL 60061
800-445-5985/847-816-5050
800-875-9643/847-816-5066 (fax)
http://www.etacuisenaire.com/