The goal of this activity is for students to generalize the transformations that they have used in specific cases and to be able to assert that a certain sequence of rigid motions will always work to take a triangle onto a congruent triangle, no matter how the triangles are initially arranged.

If students complete the cards in order, they will build from one to three transformations required. For more of a challenge, consider shuffling the cards.

Monitor for groups that are trying different sequences of rigid motions each time and for groups that have a clear routine.

This is the first time Math Language Routine 2: Collect and Display is suggested in this course. In this routine, the teacher circulates and listens to student talk while jotting down words, phrases, drawings, or writing students use. The language collected is displayed visually for the whole class to use throughout the lesson and unit. The purpose of this routine is to capture a variety of students’ words and phrases—including especially everyday or social language and non-English—in a display that students can refer to, build on, or make connections with during future discussions, and to increase students’ awareness of language used in mathematics conversations.

If students are asking their partner, “Is it a rotation? Is it a reflection?” instead, encourage them to use the sentence frames. Can they think of a transformation that will definitely line up at least one pair of corresponding points or segments?

If students are not clarifying whether they want to transform points, segments, or the whole triangle, direct them to the sentence frames: “What’s the goal?” (to take one triangle onto the other) “So what goes in that first blank?” (a triangle)

The goal is to make sure students have a set of transformations they feel confident using to take one triangle onto another congruent triangle, regardless of starting orientation. 

Ask students to share what was difficult about the game. 

Direct students’ attention to the reference created using Collect and Display. Ask students to share their method for taking one triangle to the other. Invite students to borrow language from the display as needed, and update the reference to include additional phrases as they respond.

If many groups struggled with the game, invite students to work together to develop a systematic way to line up the vertices. Then play an additional round, as a class or in groups.

The goal of the activity is to develop informal reasoning about which properties of rigid transformations and which items of given information are useful in writing a triangle congruence proof. This activity presents a partially completed proof that a sequence of a translation, reflection, and rotation will always work to take one triangle to exactly coincide with another if all the pairs of corresponding parts are congruent. 

If students struggle with the task, encourage them to find the triangle card from “Invisible Triangles” that requires a translation, rotation, and reflection and work through Noah and Priya’s steps. As they perform the transformations, why do points and rays coincide?

The goal of this discussion is to make sure all students understand why we can be sure that points and rays coincide after translating, rotating, and reflecting. Invite students to share their reasoning for each part.

“Why does Noah say ‘so that rays and line up’ rather than ‘so that segments and line up’?” (There are two steps to this part of the proof. First, use the angle to get the rays to coincide. Then, use the side length to get the points to coincide. We can‘t say the segments coincide until we have given both reasons.)