Table of Contents Appendix E-examples Resources

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

The goal of this task is to explore the effects of applying various transformations to a shape. Eventually you should be able to predict how each transformation will change the shape's image. Consider the red shape in the interactive figure below. Drag it and observe the behavior of its image, shown as a black outline. Choose a different shape, and using the same transformation, observe the behavior of its image. Change the shape of the red square or the red triangle by dragging it by an edge or vertex while pressing the "Control" key. Change the orientation by dragging the shape by a vertex. Describe the position and orientation of the resulting image in relation to the original shape. What is the relationship between the side lengths and angle measures of the original shape and those of the resulting image? Now consider the same tasks using other transformations.

[How to Use the Interactive Figure]

[Stand-alone applet]

Discussion

Dynamic geometry software allows students to visualize a transformation by manipulating a shape and observing the effect of each manipulation on its image. By focusing on the positions, side lengths, and angle measures of the original and resulting figures, middle-grades students can gain new insights into congruence. Transformations can become an object of study in their own right. Teachers can ask students to visualize and describe the relationship among lines of reflection, centers of rotation, and positions of preimages and images. Using the interactive figure, students might see that the result of a reflection is the same distance from the line of reflection as the original shape. In a rotation, students might note that the corresponding vertices in the preimage and image are the same distance from the center of rotation and all the angles formed by connecting the center to the corresponding vertices are congruent in the image and the preimage.

Visualizing Transformations	Identifying Unknown Transformations	Composing Reflections	Composing Transformations

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