The goal of this activity is to give students an opportunity to practice focusing on individual points when they write and draw a sequence of rotations and translations. This will help them in the next unit as they work on proving triangles are congruent. 

For the second obstacle, students will need to rotate the figure, perform a sequence of translations, and then rotate the figure back. This idea of transforming a problem, working with the transformed problem, and then transforming back is a powerful one throughout mathematics. For example, with coordinates, if we know how to dilate using the origin as the center but not from another point, we can translate the center to the origin, dilate the image, and translate back.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Invite a few students to share their responses with the class. Display each intermediate image in the sequence, and ask other students to explain what that single translation or rotation helped to accomplish. The purpose of this discussion is to highlight the benefits of thinking about one point at a time when setting up a sequence of transformations to take one figure to another.  

The goal of this activity is for students to start to develop a systematic, point-by-point sequence of transformations that will work to take any pair of congruent polygons onto one another. 

Monitor for students who:

• Look for a single transformation that takes quadrilateral onto quadrilateral all at once.
• Systematically line up pairs of corresponding points, then use transformations with fixed points to line up edges. For example, translation, then rotation, then reflection, or reflections across perpendicular bisectors and then edges.

Plan to have students present in this order to support the transition toward a point-by-point strategy that can be applied in more challenging problems where a single transformation is difficult to see.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Invite previously selected students to share their transformations Sequence the discussion of the approaches by the order listed in the Activity Narrative. If possible, record and display their work for all to see.

Connect the different responses to the learning goals by asking questions such as:

• “What are the benefits of each approach?”
• “How can you check the accuracy of each solution?”

Tell students that while systematically lining up one point at a time may seem tedious, it will be necessary to prove things about triangles in general in an upcoming unit.
 

If students are stuck on the idea of translating the second figure and unsure what to do, ask them why translating won't work (the obstacles are in the way) and what other transformation might help (rotating so that it fits between the obstacles).