The purpose of this activity is to allow students to explore special cases of rotating a line segment . In general, rotating a segment produces a parallel segment the same length as the original. This activity also addresses two special cases:

• When the center of rotation is the midpoint, the rotated segment is the same segment as the original, except the vertices are switched.
• When the center of rotation is an endpoint, the segment together with its image form a segment twice as long as the original.

Students experiment with a particular line segment, then make conclusions about what happens when a line segment is rotated 180° in order to generalize for any line segment (MP8).

Watch for how students explain that the rotation of segment in the second part of the question is parallel to . Some students may say that they “look parallel” while others might try to reason using the structure of the grid. Students will continue to investigate this in a future lesson, so formal language is not necessary at this time.

Ask students why it is not necessary to specify the direction of a 180-degree rotation (because a 180-degree clockwise rotation around point has the same effect as a 180-degree counterclockwise rotation around ). Invite groups to share their responses. Ask the class if they agree or disagree with each response. When there is a disagreement, have students discuss possible reasons for the differences.

Three important ideas that emerge in the discussion are:

• Rotating a segment 180-degrees around a point that is not on the original line segment produces a parallel segment the same length as the original.
• When the center of rotation is the midpoint, the rotated segment is the same segment as the original, except the vertices are switched.
• When the center of rotation is an endpoint, the segment together with its image form a segment twice as long.

If any of the ideas above are not brought up by the students during the class discussion, be sure to make them known.

In this activity, students use rotations to build a pattern of triangles that form an interesting pattern.

Triangle can be mapped to each of the three other triangles in the pattern with a single rotation. As students work on the first three questions, watch for any students who see that a single rotation can take triangle to . The center for the rotation is not drawn in the diagram (it is the intersection of segment and segment ). For students who finish early, prompt them to look for a single transformation taking to each of the other triangles.

This is the first time Math Language Routine 1: Stronger and Clearer Each Time is suggested in this course. In this routine, students are given a thought-provoking question or prompt and asked to create a first draft response in writing. It is not necessary that students finish this draft before moving to the structured partner meetings step. Students then meet with 2–3 partners to share and refine their response through conversation. While meeting, listeners ask questions such as, “What did you mean by . . .?” and “Can you say that another way?” Finally, students write a second draft of their response reflecting ideas from partners, and improvements on their initial ideas. Students should be encouraged to incorporate any good ideas and words they got from their partners to make their second draft stronger and clearer.

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to “Do segments , , , and  all have the same length? Explain your reasoning.” In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.

If time allows, display these prompts for feedback: 

• “_____ makes sense, but what do you mean when you say . . . ?”
• “Can you describe that another way?”
• “How do you know . . . ? What else do you know is true?”

Close the partner conversations and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer.

As time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.

Listen for student thinking that incorporates the following ideas and invite them to share:

• The hypotenuse of each triangle is a corresponding side to the hypotenuse in all of the other triangles.
• Rotations are a type of rigid transformation, which means that lengths of corresponding sides will stay the same.
• Quadrilateral must be a rhombus because its side lengths are all the same.

If no students suggest that quadrilateral must be a rhombus, ask students what type of quadrilateral must be. If any students claim that must be a square, leave this as an open question for now, as this will be revisited in a later lesson.