Illuminations: Symmetries I

Symmetries I

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.

Learning Objectives
 

Students will

  • predict the effect of a rotation through a given angle and even the effect of two or more rotations performed one after the other
  • find angles that leave a figure unchanged

Materials
 
  • Computer and Internet connection

Instructional Plan

How Does a Rotated Figure Look?

Given below is a green flag and a center of rotation C.

7. Sketch on a piece of paper how you think the flag will appear after it has been rotated counterclockwise through an angle of 90°. Then move point B until angle AOB measures 90° and click on the "Show Rotated Figure" button to see how the flag should appear after a rotation of 90°. Does its position match your drawing?

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8. Click on the "Start Over" button, choose another angle, and sketch how the rotated image should look. Move point B until angle AOB is the same as your chosen angle and then click on the "Show Rotated Figure" button again to show the rotated image. Compare the light blue figure with the one you have sketched.

Do some experimentation by dragging point C to change the rotocenter or dragging the points on the flag to change its shape. Repeat exercise 8.

9. Where can you place the rotocenter so that there is a point on the figure which will never move, no matter what the angle of rotation.

10. For a given center of rotation, what one point will always be left fixed by every rotation?


Composition of Rotations

If we perform one rotation followed immediately by another rotation, we call the combination a composition of rotations.

11. What happens when an object is rotated twice about the same center?

12. What is a rotation of 90° followed by another rotation of 90°?

13. What is a rotation of 40° followed by a rotation of 60°?

14. What is a rotation of 240° followed by a rotation of 240°?

15. How can you convert an angle greater than 360° to one less than 360°?


Figures Unchanged by a Rotation

The center of rotation will always be left fixed by the rotation, but sometimes the whole figure will appear unchanged once it has been rotated.

16. What angle of rotation will always leave every figure unchanged?

17. For each of the following figures, give all angle(s) of rotation about point C that rotate the red figure counterclockwise so that it lands exactly on top of the green figure.

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18. What pattern do you see in the measure of these angles?

19. What would happen if a different center of rotation were used?

20. What center of rotation and angles of rotation will leave a regular hexagon (six-sided figure) apparently unchanged? Can you generalize this observation?

You are now in a position to understand the secret behind how mathematicians describe symmetry in a figure. This will be revealed in the next part of this i-Math: Relating Rotations to Symmetry.

NCTM Standards and Expectations
 
Geometry 9-12
  1. Use various representations to help understand the effects of simple transformations and their compositions.
  2. Understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices.
  3. Draw and construct representations of two- and three-dimensional geometric objects using a variety of tools.

References
 
  • For All Practical Purposes: Introduction to Contemporary Mathematics, W.H. Freeman and Company, New York, 1997

  • Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures: Visualizing Transformations, NCTM Principles and Standards for School Mathematics: E-example 6.4. http://standards.nctm.org/document/eexamples/chap6/6.4/index.htm

  • Java Applets were created using the Geometer’s Sketchpad™ and JavaSketchpad™.
  
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NCTM Resources

Principles and Standards for School Mathematics

Navigating through Geometry in Grades 9-12

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