**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.

**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.

**Take Time to Reflect**

- What new insights into congruence can students gain as they work on activities like this?
- What relationships between the original shape and its image are important for students to note in a translation?
- What relationships between the original shape and its image are important for students to note in a reflection?
- What relationships between the original shape and its image are important for students to note in a rotation?