
Composing Transformations (applet)
Users are challenged
to compose equivalent transformations in two different ways. 
Task
Use the icons from the upper left panel to choose up to three transformations to be successively applied to the red shape. The black shape shows the resulting image. Use the icons from the lower left panel similarly. The blue shape shows the results from successively transforming the red shape using the lower left panel. Consider the red shape in the interactive figure below. Drag it and observe the behavior of its image after one or more successive transformations are applied using the top left panel. Choose a different shape and observe the behavior of its image under the same transformation or composition. Change the shape of the red square or red triangle by dragging it from an edge or vertex while pressing the Control key. Change their orientation by dragging them from a corner. Which single transformation or composition of transformations, if any, would have the same effect on the original figure? Use the lower left panel to test your conjecture.
How to Use the Interactive Figure
The results of composing transformations in two different ways can be compared using this interactive figure. Use icons from the upper left panel to choose up to three transformations to be successively applied to the red shape. The black shape, labeled Transformation 1, shows the resulting image. Use buttons from the lower left panel to generate a blue shape labeled Transformation 2. The goal is to have the blue and black shapes overlap for all possible initial figures, positions, and orientations. This would suggest that the results of these two sets of transformations are congruent.
Drag the red shape to observe the behavior of its image. To select a shape click on a shape from the icons at the top. Click on the icons on the left to select a different unknown transformation. Change the shape of the red square or red triangle by dragging it from an edge or vertex while pressing the Control key. Change the orientation of the shapes by dragging them from a corner. Resize the circle by dragging it from a point on the circumference.
Discussion
Using dynamic geometry software, teachers can ask students to consider what happens when transformations 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 does. A challenging assessment of students' understanding of transformations can be given if two congruent shapes are given and students are asked to specify a transformation or 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
specific ways in which middlegrades students can identify each of the
transformations or composition of transformations that would have the same
effect on the original figure as the composition does?
 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