Lesson 3

This warm-up both reviews previous work on corresponding parts of congruent figures and previews the subsequent activity in which students will use rigid transformations to take one figure onto another.

Students start by drawing conclusions based on a congruence statement, and then are given a specific figure to reason about. This gives the opportunity to talk about how, often in geometry, the static picture is just one example of many different possible images that match the given information. 

Ask students to share what they wrote down before they saw the image. Next, invite a student to mark congruent sides and angles of the image for all to see.

If students made claims like segment \(AC\) can’t possibly be congruent to segment \(B’C’\), ask for a counterexample. If you have access to the digital version of the curriculum, drag points on the applet to demonstrate the case in which triangle \(ABC\) is isosceles with segment \(AC\) congruent to segment \(BC\).

The goal of this activity is for students to generalize the transformations that they have used in specific cases, 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 1 to 3 transformations required. For more 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.

Arrange students in groups of 2. Distribute one transformer card and one set of three playing cards to each group. If feasible, have students use folders or books so that students can’t see each other’s desktops. You may choose to demonstrate one round of the game for the students, with you holding the triangle card, and the students working together to be the transformer.

For demonstration, you can either use this image or use the applet here, ggbm.at/cwtjcwvv, but do not show it to students until after the round is over.

Display the transformer sentence stems from the transformer card for students.

Tell students: \(\overline {JK} \cong \overline {PQ}, \overline {JL} \cong \overline {PR}, \overline {KL} \cong \overline {QR}, \angle J \cong \angle P, \angle K \cong \angle Q, \angle L \cong \angle R\).

Do not display the triangles in the figure. Instead, as students give you transformation instructions, use tracing paper to perform the requested transformations. Record each step as you go. After each transformation, tell students how many vertices coincide.

Once all three vertices coincide (or after a few minutes, if the struggle is unproductive), display the triangles for the students, with the intermediate transformations recorded.

As students play the game, support the transformers to use the language on the cards to give precise statements, and support the students with triangle cards to ask questions, accurately carry out their partner’s instructions, and record each step as they go.

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. Invite a group that initially struggled but ultimately developed a systematic way to line up the vertices, to share their method and process. If no group had that experience, invite a group that was confident in their sequence of transformations from the beginning to share their strategy. 

If many groups struggled with the game, have a group with more success teach their strategy to the class, and either play an additional round, or have the class play you.

\(\overline{AB} \cong \overline{DE}, \overline{AC} \cong \overline{DF}, \overline{BC} \cong \overline{EF}, \angle A \cong \angle D, \angle B \cong \angle E, \angle C \cong \angle F\)

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. 

Reading, Listening, Conversing: MLR6 Three Reads. Use this routine to support reading comprehension of this word problem. Use the first read to orient students to the situation. Ask students to describe what the situation is about (Noah and Priya were playing Invisible Triangles with Card 3.). Use the second read to identify geometric relationships. Listen for and amplify important relationships such as \(\overline{AB} \cong \overline{DE}\), \(\angle A \cong \angle D\), and \(\overline{AC} \cong \overline{DF}\). After the third read, ask students to brainstorm possible strategies to answer the first question, “Why do points \(B’’\) and \(E\) have to be in the exact same place?”. This routine will help students make sense of the language in the word problem and the reasoning needed to solve the problem.
Design Principle(s): Support sense-making

If students struggle with the task, encourage them to find the triangle card from Invisible Triangles that required 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 \(A''B''\) and \(DE\) line up’ rather than ‘so that segments \(A''B''\) and \(DE\) 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. You can‘t say the segments coincide until you‘ve given both reasons.)