Composition of Transformations Lab

By Craig Russell, University of Illinois Laboratory High School

Purpose: Investigate the result of two different geometric transformations when combined. Determine whether (or under what conditions) the order of transformations may be reversed, and which compositions may be written as a single transformation.

NCTM Standards:

· Understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices.

· Use various representations to help understand the effects of simple transformations and their compositions.

The lab consists of seven “experiments,” arranged with about two experiments per tab in a Geometer’s Sketchpad file

• http://www.uni.uiuc.edu/~hcrussel/TransformationCompositionLab.gsp

Experiment 1: Students dilate and rotate through a common center (commutative).

Experiment 2: Dilate and rotate through different centers. Students discover that dilation and rotation appear to be commutative only when both transformations have a common center.

Experiment 3: Dilate and reflect. Students discover that these transformations are commutative when the center of dilation lies in the line of reflection.

Experiment 4: Rotate and reflect—commutative? These aren’t commutative in general; if the center of rotation lies on the reflection line, and the rotation is 180°, then order may be reversed. Students are asked to try placing the center of rotation on the line of reflection, then reversing the rotation; they then discover that Rotation(q) ° Reflection = Reflection ° Rotation(-q) when the center of rotation is on the line of reflection; they are asked to explain why it was necessary to reverse the rotation.

Experiment 5: Students compose two rotations with different centers, and note that the result may be written as a single rotation.

Experiment 6: Rotate and reflect—single transformation? Perceptive students will note that this composition results in a glide reflection, in which the “mirror” for the glide reflection is rotated from the original line of reflection by half the rotation angle.

Experiment 7: Students compose two dilations with different centers, and note that the result may be written as a single dilation.