## Classifying Transformations

• Lesson
9-12
1

Students will identify and classify reflections and symmetries in figures and patterns. They will also create frieze patterns from each of the seven classes using the supplemental activity sheets.

Start class by reviewing the Recognizing Symmetry – Practice Activity Sheet from the first lesson. Tell students to hold on to their creations from Question #5 (generating patterns), since they will be sharing them with a partner later on in class. However, you could choose to have some students show off their creations to the rest of the class.

Cyclic and Dihedral Notation

Classifying Cyclic Figures

On the web site Creating Cyclic Figures, how would you describe the figures you created? What kind of symmetry do they have?

Answer: Cyclic figures have only rotation symmetry

Pass out the Cyclic Figures – Notes page from the following activity sheet.

Go over each of the examples.

1. To determine the classification, determine the number of repeating patterns.
2. To determine the angles of rotation, have students draw lines to the center from corresponding points on consecutive repeating patterns and use a protractor to measure (see below). This angle is the base angle; all the other angles will be multiples of this angle, up to 360°.
 If a figure is Cyclic n, then the formula is:

Classifying Dihedral Figures

To introduce the next section, ask students, On the website Creating Dihedral Figures, what type of figures did you create?

Answer: Dihedral figures have both reflection and rotation symmetries.

Distribute the Dihedral Figures – Notes page from the Classifying Symmetry Activity Sheet and go over each example.

Points to emphasize when going over this page of the activity sheet:

1. The rotation symmetries are determined the same way as in classifying cyclic figures.
2. The lines of reflection can always be drawn through the center of each repeating pattern, and also between each repeating pattern.
3. It is very easy for students to count each reflection line twice after they have drawn the lines. However, the number of reflection lines will always equal the number of rotation symmetries.

Rigid Motions of the Plane (Basic Transformations)

Ideally, you should teach this lesson after students have discussed transformations of the plane. If that is not possible, review the four basic rigid motions of the plane.

• Rigid motions are motions that preserve the size and shape of the figure.
• The 4 rigid motions are translations, reflections, rotations and glide reflections.

There are many good resources that demonstrate the fundamental transformations, including this complete unit from NCTM.

Frieze Patterns

A frieze is a decorative horizontal strip that is made up by a repeating pattern.

To explore the 7 classes of frieze patterns, distribute the Seven Classes of Frieze Patterns – Notes page of the activity sheet and go over each pair of examples.

1. All frieze patterns have translation symmetry
2. 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.
3. When a frieze pattern has horizontal symmetry, the only possible horizontal line is the line through the center of the pattern.
4. The best way to determine if a frieze pattern has rotation symmetry is to look at it upside down and see if it looks the same.
5. The best way to recognize glide reflection symmetry is to picture a set of footprints in the sand.

Notes:
Different books and websites use different notations for the 7 classes of frieze patterns. The use of Class I, Class II, etc. is more for ease of use than for any other reason.

Now distribute Classifying Frieze Patterns – Notes page off the activity sheets and have students work in pairs or small groups to fill in the blanks. Then, give the groups an opportunity to share their answers agree. Make sure students explain their reasoning. Compare student responses with the suggested answers to see if there is agreement. Is it possible that different transformation sets will produce the same image? Use the flow-chart to classify the 2 examples on the page.

Have students exchange the results from their work at the end of the previous day's lesson Recognizing Symmetry - Practice from Recognizing Symmetry - Practice activity sheet from last class, where they used JavaSketchpad applets to create the two cyclic figures, two dihedral figures, and two frieze patterns.

1. Activity Sheet: Classifying Symmetries
2. Computer with internet access
3. Protractor

Assessments

1. Assign the Classifying Symmetry - Practice from Classifying Symmetry Activity Sheet. Note that for Question 6, student swaps their creations for Question 5 from the Recognizing Symmetry-Practice assignment since they will using them for question 6. Assign the Classifying Symmetry – Practice page of the Classifying Symmetry Activity Sheet. Note that for Question 6, students exchange their results for Question 5 on the Recognizing Symmetry Activity Sheet from the previous lesson, where they used Java Sketchpad applets to create the 2 cyclic figures, 2 dihedral figures, and 2 frieze patterns.

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### Transformations and Frieze Patterns

9-12

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

### Recognizing Transformations

9-12
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.

### Learning Objectives

By the end of this lesson, students will be able to:

• Recognize the symmetries present in frieze patterns.
• Classify frieze patterns into their appropriate class.