In this unit students apply transformations to entire shapes such as angles or triangles by restricting the domain of the independent point to the pre-image shape and tracing the dependent point to form the image, in the process internalizing the idea that they can map an entire set of pre-image points to a transformed set of image points. They solve problems and play games that require them to apply each of the transformations to entire shapes and, given two shapes, develop the ability to identify the transformation family used and locate the required parameters (mirror, center point, angle of rotation, etc.) that define the specific function.

Once students are comfortable with transforming shapes and figuring out the transformation that relates two shapes, they use shapes to investigate the composition of two transformations. Students investigate how the composition of two reflections can be equivalent either to a translation or to a rotation, and they investigate how other compositions turn out to be equivalent to a single transformation. Students solve problems and play games in which they use the composition of several transformations to transform a given pre-image shape to a given image shape, in the process realizing that any such problem has multiple solutions.

(Though this unit does not formally describe the isometry group mathematically, student do learn that the isometries closed under composition: that is, the composition of any two isometries is equivalent to a single isometry.)

This unit also contains activities in which students apply isometry transformations to angles, triangles, and other shapes to discover and verify standard geometry theorems such as the triangle sum theorem.

Triangle Sum Theorem: In this activity you’ll move transformed images of a triangle around to dicover and verify some interesting and useful relationships among the angles.

Mirror, Mirror: In this activity you’ll figure out how multiple reflections can be used to perform other transformations.

Angles in Triangles: In this activity you’ll move transformed images of a triangle around to dicover and verify some interesting and useful relationships among the angles.

Transforming Angles: In this activity you’ll transform the angles in several geometric constructions in order to understand the constructions better.

Congruence Puzzles: This activity has puzzles in which two objects might be the same shape and size, or might be pretending. You have to find the imposters.

Similar Triangles: In this activity you’ll stretch and shrink objects, while keeping them the same shape.

Segment Congruence: In this activity you’ll demonstrate congruence by finding multiple ways to transform one segment to put its image on top of another segment.

Segment Congruence by Reflection: In this activity you'll demonstrate congruence using only reflection as you transform one segment to put its image on top of another segment.

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Update History:

28 October 2016: Created this page.
13 December 2016: Updated the description.
To do: Move the Segment Congruence Task to a new Congruence, Similarity, and Inverse unit.