Lesson 18

The purpose of this warm-up is for students to understand the idea behind the Obstacle Course activity in this lesson. Students imagine all rigid motions are being done physically in the plane, so they are not allowed to do a translation or rotation where the physical motion would take the figure through a solid obstacle. While students may notice and wonder many things about this diagram, the idea that two translations were necessary to avoid the obstacle is the most important discussion point. This warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1).

Display the diagram for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the diagram. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If the question asking why figure \(ABC\) wasn't translated directly to \(DEF\) does not come up during the conversation, ask students to discuss this idea.

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).

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).

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. This is especially important because when transformations are used in triangle congruence proofs in a subsequent lesson, students will be justifying how they know that a certain transformation will take one triangle onto another. Finding a single transformation that takes each vertex of one triangle onto each corresponding vertex of another cannot be generalized because it is specific to one pair of congruent triangles. In a proof, students must define sequences of rigid transformations that take a given triangle to any triangle with congruent corresponding parts, and justify why that sequence is guaranteed to take each vertex to its corresponding vertex.

Monitor for students who:

• Find a single, separate transformation that takes \(A\) to \(A’\) and \(B\) to \(B’\), so that when \(B\) goes to \(B’\), \(A\) no longer remains on \(A’\).
• Look for a single transformation that takes quadrilateral \(ABCD\) onto quadrilateral \(A’B’C’D’\) all at once.
• Line up one or two points and then get stuck.
• Systematically line up pairs of corresponding points, then use transformations with fixed points to line up edges. You might find students who use translation, then rotation, then reflection, as well as students who use reflections across perpendicular bisectors and then edges.

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

Select previously identified students to share in this order:

• Look for a single transformation that moves all the points at once.
• Line up one or two points and then get stuck.
• Systematically line up pairs of corresponding points, then use transformations with fixed points to line up edges.

For each strategy shared, focus on what question that strategy helps to answer or what is unique and valuable about that strategy.

• If a student who shares looked for a single transformation, focus on the elegance of finding a single rigid motion, as well as inviting the student to share what was hard about their strategy.
• If a student who shares got stuck, emphasize the strategic choice to try to line up just one or two points at a time, and why that might be easier than finding one or two more complicated transformations.
• If a student who shares systematically lined up one point at a time using transformations with fixed points to ensure that previously lined-up points did not move, emphasize why this strategy could be used for lots of different starting conditions.