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.
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 middle-grades 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?