To help them form images of shapes through different transformations, students can use physical objects, figures traced on tissue paper, mirrors or other reflective surfaces, figures drawn on graph paper, and dynamic geometry software. They should explore the characteristics of flips, turns, and slides and should investigate relationships among compositions of transformations. These experiences should help students develop a strong understanding of line and rotational symmetry, scaling, and properties of polygons. By exploring the positions, side lengths, and angle measures of the original and resulting figures, middle-grades students can gain new insights into congruence. Thus congruence does not depend on position and orientation.
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 the positions of preimages and images. Using dynamic geometry software, students might see that each point in a reflection is the same distance from the line of reflection as the corresponding point in the preimage, as shown in figure 6.19a. 6.19b, students might note that the corresponding vertices in the preimage and image are the same distance from the center of rotation and that the angles formed by connecting the center of rotation to corresponding pairs of vertices are congruent. Teachers can pose additional challenges to develop students' understanding of transformations and congruence. For example, given the three pairs of congruent shapes in figure 6.20, students might be asked to identify a transformation applied to transform one shape into the other.