Illuminations: Symmetries I

Symmetries I

Describing Rotations

Fix a center, turn, and you have a rotation. Many objects in nature—flowers, starfish, and crystals—and objects we use every day, such as wheels, CDs, and drinking glasses, have rotational symmetry. The first thing to think about is how to describe a rotation mathematically.

Learning Objectives
 

Students will

  • think about how to describe rotations of a figure and then have the opportunity to investigate the effect of rotations through different angles and on different shapes

Materials
 
  • Computer and Internet connection

Instructional Plan

In this part of your investigation of rotational symmetry, you will think about how to describe rotations of a figure and then have the opportunity to investigate the effect of rotations through different angles and on different shapes.

Think About...Rotations

1. How can a rotation be described?

2. Draw or describe some shapes that look the same after they have been rotated.

3. What is the net result when you rotate something twice by two different angles?


Describing Rotations Applet

Given below is a green figure for you to rotate. Drag point B to change the angle of rotation and drag point C to change the center of rotation or rotocenter. The light blue figure shows the initial position of the green figure before it was rotated.

Sorry, this page requires a Java-compatible web browser.

4. What information must you know to determine exactly what rotation is being made?

5. What happens if you rotate the figure 360°? What smaller rotation is this the same as?

6. What happens if you rotate the figure more than 360°?

In this investigation, you will always use angles of rotation that are less than 360° and are counterclockwise in direction; if you want to denote a clockwise rotation, give a negative angle.


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

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