Table of Contents Appendix E-examples Resources

Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures: Identifying Unknown 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 this 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 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

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.

Visualizing Transformations	Identifying Unknown Transformations	Composing Reflections	Composing Transformations

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