Illuminations: Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures

Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures

Identifying Unknown Transformations

In this lesson, the user is challenged to identify the transformation that has been used.

Learning Objectives
 

Students will

  • identify the transformation that has been used

Materials
 
  • Computer and Internet connection

Instructional Plan

Task

In this task you must determine what transformation has been applied to a shape by comparing it to its image and using what you know about transformations. Consider the red circle in the interactive figure below. Drag it and observe the behavior of its image, shown as a red outline. The image is the result of a single transformation (either a reflection, a rotation, or a translation/slide) on the original shape. The symmetry of the circle may make it difficult to tell exactly what is happening in the transformations. A different shape may provide more-useful information. You may also test or develop conjectures by accessing the three transformations with the small icons at the lower left. The images for these transformations are shown as blue outlines.

Choose a different shape and observe the behavior of its image. Change the shape of the red square or red triangle by dragging it by an edge or a vertex while pressing the Control key. Change the orientation of a shape by dragging it by a vertex. What is the transformation used in challenge 1? How can you decide? Describe the position and orientation of the resulting image in relation to the original shape. Now try another challenge.

[How to Use the Interactive Figure]

[Stand-alone applet]

Discussion

Using dynamic geometry software, teachers can pose additional challenges for middle-grades students to develop their understanding of transformations and congruence. In each challenge above, a red shape and its image under an unknown transformation are shown. Students can learn about the nature of the unknown transformation by investigating the dynamic behavior of a shape and its image under the transformation and analyzing the relationships that remain constant between the original shape and its image. Using dynamic geometry software, students can identify an unknown transformation in several ways: by comparing the orientation of the shapes, by analyzing the trace of the image and of the original shape or of points on them, or by finding the locus of invariant points. Students can use the software to check their conjecture by constructing the image of the original shape under the transformation they identified.

Teacher Reflection
 
  • What new insights into transformations can students gain as they work on activities like this?

  • What are some specific ways in which middle-grades students can identify the unknown transformation used in each of the challenges in this example?

  • What can teachers do to help their students better describe transformations?

NCTM Standards and Expectations
 
Geometry 6-8
  1. Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.
  2. Examine the congruence, similarity, and line or rotational symmetry of objects using transformations.
  
1 period   

NCTM Resources

Principles and Standards for School Mathematics
National Council of Teachers of Mathematics Thinkfinity
More and Better Mathematics for All Students
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