Illuminations: Symmetries II

Symmetries II

Describing Reflections

This investigation will help you to understand how reflections work and what happens when two or more reflections are applied one after the other.

Learning Objectives
 

Students will

  • understand how reflections work and what happens when two or more reflections are applied one after the other

Materials
 
  • Computer and Internet connection

Instructional Plan

Think About...Reflections

When we look into a mirror, we see our reflection. Similarly, in geometry, a shape in the plane can be reflected across a line in the plane to give the reflected image of the shape. In this activity, you'll be learning the mathematical properties these reflections have.

1. What information must be given to describe a reflection?

2. How can you describe the process of reflecting an object or shape?

3. What shapes look the same after they have been reflected?

4. What is the net result when we reflect something twice across the same mirror line?

5. What is the net result when we reflect something across two different mirror lines?

 

Reflection of a Point

The first step in understanding reflections is to understand how a point is reflected. In the following diagram, point A is reflected across mirror line PQ and has image point A'. You can drag point A to change its position or you can drag points P and Q to change the mirror line.

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6. How is the distance from point A to the mirror line related to the distance from the mirror line to the image point A' ?

7. Will this distance relationship be true for every reflection? Why or why not?

8. What can you say about the measures of angle PRA and the measure of angle PRA' ?

9. Will this angle relationship be true for every reflection? Why or why not?

Describing Reflections

In the diagram below, the green figure has been reflected across the blue line to give the red figure. You can drag points P and Q to change the location of the mirror line and you can drag any of the points on the green figure to to change its shape.

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10. When does the reflected shape overlap the original shape?

11. When is the reflected shape near the original shape?

12. When is the reflected shape far away from the original shape?

Click the "Hide Reflected Image" button to hide the reflected image. Drag any of the red points to change the original figure or the mirror line.

13. Draw how you think your new shape should appear after it has been reflected across the mirror line. Then click on the "Show Reflected Image" button to see how the shape will appear after reflection. Does its position match your drawing?

You may repeat exercise 13 until you are sure that you know how images will appear after they have been reflected.

14. Click on the "Hide Reflected Image" button. Change the green figure until you think it will look exactly like its mirror image. Click on the "Show Reflected Image" button to compare your figure to its image.

15. Click on the "Hide Reflected Image" button. Change the green figure and the mirror line until you think the reflected image will land exactly on top of the original image. Click on the "Show Reflected Image" button to see if you are correct.

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. Use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture.

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

Web Sites
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