## Recognizing Transformations

This lesson introduces students to the world of symmetry and rotation in figures and patterns. Students learn how to recognize and classify symmetry in decorative figures and frieze patterns, and get the chance to create and classify their own figures and patterns using Java Sketchpad applets.

To begin the lesson, ask the class what they know about symmetry. Who can give examples? Who can develop a definition?

A figure isOffer the following examples:symmetricunder a certain operation when that operation doesn't change the position of the shape.

vertical line symmetry | horizontal line symmetry | horizontal and vertical symmetry (and rotational symmetry) |

- How many lines of symmetry does the triangle have?
[There are 3 lines of symmetry.] - How many ways can you rotate the triangle so that it ends up in the same position?
[There are 3 rotations possible,

at 120° increments.] - What objects in your life or in nature can you think of that have symmetries?

[Answers will vary. Sample solutions: face, butterfly, snowflake, etc.]

### Types of Symmetry for 2-Dimensional Figures and Frieze Patterns

Two-dimensional figures and frieze patterns can exhibit *reflection symmetry*, which is symmetry that occurs across a line of reflection; or *rotational symmetry*,
which is symmetry that occurs about a point of rotation. Students may
already be familiar with these types of symmetry. In the elementary
grades, students may have used the terms "flip" and "turn" to refer to
them.

**Reflection Symmetry **

Give students the first page of the Recognizing Symmetry Activity Sheet.

Activity Sheet: Recognizing Symmetry

Go over how figures and frieze patterns can have various lines of symmetry. In addition to the examples on the page, it would be helpful to have other examples ready so students can hone their recognition skills. (See the links to Dihedral Figures and Frieze Patterns below.)Some points to emphasize when going through the Activity Sheet:

- When a frieze pattern has vertical reflection symmetry, that means we can draw at least 1 vertical line so that 1 side of the pattern is a mirror image of the other side. There is often more than 1 possible vertical line.
- When a frieze pattern has horizontal symmetry, the only possible horizontal line is the line through the center of the pattern.

**Rotational Symmetry**

Give students the second page from the Recognizing Rotational Symmetry – Notes Activity Sheet and review how figures and frieze patterns can have various rotational symmetries.

Recognizing Rotational Symmetry – Notes

In addition to the examples on the page, it would be helpful to have other examples ready so students can hone their recognition skills. (See the links to Cyclic Figures and Frieze Patterns below.)

### Demonstrate the Use of the Java Sketchpad Applets

Go to the Cyclic Figures activity and show students how to create figures of their own with rotational symmetry. The figures and their transformations will change as the red vertices are dragged. Also, you should demonstrate how to use the Print Screen command to copy the figure from the site (by holding down the Ctrl key and the Print Screen keys at the same time), how to paste it into Word, and then how to crop it so that only the figure is showing.

Go to the Dihedral Figures activity and show students how to create figures of their own with both reflection and rotational symmetry. Once again, show students how to use the Print Screen command to copy the figure from the site, how to paste it into Word, and then how to crop it so that only the figure is showing.

Go to the Frieze Patterns activity and show students how to create strip patterns of their own with various types of symmetry. Once again, be sure students remember to paste their images into Word.

### References

- Armstrong, M.A. Groups and Symmetry. New York: Springer-Verlag, 1988.
- Christie, Archibold. Pattern Design. New York: Dover, 1969.
- Crowe, Donald. Symmetry, Rigid Motions, and Patterns. Arlington, MA: COMAP, 1986.
- Gallian, Joseph. Contemporary Abstract Algebra. Lexington, MA: D.C. Heath, 1986.
- Jacobs, Harold. Geometry. New York: W.H. Freeman, 1987.
- Martin, George. Transformation Geometry. New York: Springer-Verlag, 1982.
- Weyl, Hermann. Symmetry. Princeton, NJ: Princeton University Press, 1952.

- Recognizing Symmetry Activity Sheet
- Paper cut-out of and equilateral triangle
- Protractor and straight edge
- Computer with Internet access

**Assessment Option**

Create a design that uses at least 2 symmetries. Explain the types of symmetries you used, which may include lines of reflection and degrees of rotation.

**Extensions**

- Several corporate logos are figures that contain both reflection and rotation symmetry. Search the Internet or dihedral. Copy and paste these figures into a word processing document and classify them.
- Just as letters can have reflection and rotation symmetry, words can also have these symmetries. For example, 'SOS' has 180° rotational symmetry and 'OX' has horizontal symmetry. Come up with real words (the longer the better!) that have each type of symmetry.
- Move on to the last lesson,
*Classifying Transformations*.

**Questions for Students**

- What geometric figures will always have both reflection and rotational symmetries?
[Answer: Regular polygons (equilateral and equiangular polygons)]

8 lines of symmetry 8 rotations, at 45° increments - Is it possible for a figure to have reflection symmetries, but not have rotational symmetries? If so, draw one.
[Answer: No.

If a figure has 1 line of symmetry, it also has one rotational symmetry: 360°.

If a figure has two lines of symmetry, then it will have two rotational symmetries: 180° and 360° and so forth.

The reason for this is that 2reflection lines that intersect is equivalent to 1 rotation about that point of intersection.]↔ 2 reflections **is equivalent to**1 rotation

3. Tell what type(s) of rotational symmetry are possible for a frieze pattern. Explain your reasoning.

[Answer: Only 180° rotation (looking at the figure upside down) will put the figure along the same orientation as before. For example, if you rotated a horizontal strip 90°, it would become a vertical strip, which will not look the same as the original horizontal strip.]

**Teacher Reflection**

- Describe students’ reactions to exploring the possibilities for reflection and rotational symmetries using the Java Sketchpad website.
- How did students demonstrate understanding of the materials presented? Did your students demonstrate their understanding of the materials in ways other than you expected? If so, describe what they did.
- Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments, and were they effective

### Transformations and Frieze Patterns

Introduce students to the world of symmetry and rotation in figures and patterns.

### Classifying Transformations

### Learning Objectives

By the end of this lesson, students will:

- Recognize both reflection and rotational symmetry in 2-dimensional figures and frieze patterns.
- Create cyclic and dihedral figures and frieze patterns using JavaSketchpad applets.
- Classify 2-dimensional figures by class (cyclic or dihedral) and by number.

### NCTM Standards and Expectations

- Understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices.

- Use various representations to help understand the effects of simple transformations and their compositions.

- Draw and construct representations of two- and three-dimensional geometric objects using a variety of tools.