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Recognizing Transformations

  • Lesson
9-12
1
Geometry
James Reeder
Location: unknown

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?

Symmetry: A figure is symmetric under a certain operation when that operation doesn't change the position of the shape.
Offer the following examples:
RecognizingTransformations IMAGE LetterMSymmetry2591 the b 2591 the h 
vertical line
symmetry
horizontal line
symmetry
horizontal and
vertical symmetry
(and rotational symmetry)
Distribute a paper equilateral triangle to each student. Ask them to think about the following questions, using the triangle to determine their answers.
  1. How many lines of symmetry does the triangle have?
    2591 triangle symm 
    [There are 3 lines of symmetry.]
  2. How many ways can you rotate the triangle so that it ends up in the same position?
    2591 triangle rots 
    [There are 3 rotations possible,
    at 120° increments.]
  3. 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.

pdficonActivity 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.

pdficonRecognizing 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.

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

Assessments 

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 

  1. Go to http://www.scottkim.com/inversions/ to see some works by Scott Kim that use reflection and rotatioin symmetry.
  2. 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.
  3. 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.
 

Questions for Students 

  1. What geometric figures will always have both reflection and rotational symmetries?
    [Answer: Regular polygons (equilateral and equiangular polygons)]
    2591 octo rots 2591 octo symm 
    8 lines of symmetry 8 rotations, at 45° increments
     
  2. 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.]
    2591 color tri 1  2591 color tri 2 
    2 reflectionsis 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 JavaSketchpad 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
 
2626icon2
Geometry

Classifying Transformations

9-12
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.

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.