
Composing Reflections (applet)
Examine the result of
reflecting a shape successively through two different lines. 
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
Each of the compositions in the interactive figure shows the results of successive reflections of a shape over two different lines. In composition 1 the reflection lines are perpendicular, in composition 2 they are parallel, and in composition 3 they intersect but are not necessarily perpendicular. Drag the red shape to observe the behavior of its image shown in black. To select a shape click on the shape from the icons at the top. To select a different composition click on the icons on the left. Change the shape of the red square or red triangle by dragging from an edge or vertex while pressing the Control key. Change their orientation by dragging from a vertex. Resize the circle by dragging from any point on the circumference.
 Composition 1
To change the slope of the first reflection line, drag the line. To change the position of the reflection lines, drag the point at the intersection of the lines. To change the angle of rotation, drag the angle from its vertex.
 Composition 2
To change the slope of the reflection lines, drag any of the lines. To change the position of the reflection lines, drag the point on the lines. To change the translation, drag the arrow from any of its endpoints.
 Composition 3
To change the slope of any of the reflection lines, drag any of the lines. Drag the point at the intersection of the reflection lines to change their position. To change the angle of rotation, drag the angle from its vertex.
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.
Take Time to Reflect
 What new insights into transformations can students gain as they work on activities like this?
 What are some specific ways in which middlegrades students can identify the transformation that would
have the same effect on the original figure as the composition has?
 What are some ways in which teachers can assess students' understanding of transformations?
Also see:
 6.4 Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures