Table of Contents Appendix E-examples Resources

Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures: Composing Reflections 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 this part, Composing Reflections, users can examine the result of reflecting a shape successively through two different lines. And in the fourth 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

Each of the compositions in the interactive figure below shows the results of successive reflections over two different lines. In composition 1 the lines of reflection are perpendicular, in composition 2 they are parallel, and in composition 3 they intersect but are not necessarily perpendicular. Your task is to explore each of these compositions and then determine what single transformation, if any, would produce the same effect. First, consider the red triangle in the interactive figure below. Drag it and observe the behavior of its image after two successive reflections when the lines of reflection are perpendicular. Now 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 by dragging it by a vertex. Which single transformation, if any, would have the same effect on the original figure as the double reflection has? Now try answering the same question using another composition.

[How to Use the Interactive Figure]

[Stand-alone applet]

Discussion

Using dynamic geometry software, students can consider what happens when reflections 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 has. The tools made available by the software allow students to test their conjectures. In these activities, the final image that results from reflecting a figure using one line, then reflecting the image using a second line, will be either a translation of the original figure (if the lines are parallel) or a rotation (if the lines intersect). A challenging test of students' understanding of transformations is to give them two congruent shapes and ask them to specify a transformation or a composition of transformations that will map one to the other.

Visualizing Transformations	Identifying Unknown Transformations	Composing Reflections	Composing Transformations

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