Table of Contents Appendix E-examples Resources

Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures: Composing Transformations Visualizing Transformations Identifying Unknown Transformations Composing Reflections Composing Transformations

Rotations; translations, or slides; and reflections, or flips, are geometric transformations that change an object's position or orientation but not its shape or size. The interactive figures in this four-part example allow a user to manipulate a shape and observe its behavior under a particular transformation or composition of transformations. In the first part, Visualizing Transformations, one can choose a transformation and apply it to a shape to observe the resulting image. In the next part, Identifying Unknown Transformations, the user is challenged to identify the transformation that has been used. In Composing Reflections, users can examine the result of reflecting a shape successively through two different lines. And in this part, Composing Transformations, the users are challenged to compose equivalent transformations in two different ways. Activities like these allow students to deepen their understanding of congruence, similarity, and reflection, and they also contribute to the study of transformations, as described in the Geometry Standard.

Task

Use the icons from the upper left panel to choose up to three transformations to be successively applied to the red shape. The black shape shows the resulting image. Use the icons from the lower left panel similarly. The blue shape shows the results from successively transforming the red shape using the lower left panel. Consider the red shape in the interactive figure below. Drag it and observe the behavior of its image after one or more successive transformations are applied using the top left panel. Choose a different shape and observe the behavior of its image under the same transformation or composition. Change the shape of the red square or red triangle by dragging it from an edge or vertex while pressing the Control key. Change their orientation by dragging them from a corner. Which single transformation or composition of transformations, if any, would have the same effect on the original figure? Use the lower left panel to test your conjecture.

[How to Use the Interactive Figure]

[Stand-alone applet]

Discussion

Using dynamic geometry software, teachers can ask students to consider what happens when transformations are composed. Teachers can then ask students to make conjectures about which single transformation, if any, would have the same effect on the original figure as the composition does. A challenging assessment of students' understanding of transformations can be given if two congruent shapes are given and students are asked to specify a transformation or composition of transformations that will map one to the other.

Take Time to Reflect

What new insights into transformations can students gain as they work on activities like this? What are specific ways in which middle-grades students can identify each of the transformations or composition of transformations that would have the same effect on the original figure as the composition does? What are some ways in which teachers can assess students' understanding of transformations?

Visualizing Transformations	Identifying Unknown Transformations	Composing Reflections	Composing Transformations

Copyright © 2000-2004 by the National Council of Teachers of Mathematics. Terms of Use.