Lesson 17

The purpose of this warm-up is to elicit strategies and understandings students have for using rigid motions on a point-by-point basis without a grid. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to put together rigid transformations that take one polygon to another again without making reference to a grid. While participating in this activity, students need to be precise in their word choice and use of language (MP6) because the rigid motions can only be defined in reference to labeled points.

Display the diagram and one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Some students may think that \(C\) can be reflected over line \(AB\) onto \(D\). Ask them what we would need to know for that to work.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”

• “Did anyone have the same strategy but would explain it differently?”

• “Did anyone solve the problem in a different way?”

• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”

• “Do you agree or disagree? Why?”

If no student brings it up, ask students how a rotation may be used to take \(A\) to \(B\).

The purpose of this activity is to activate students’ prior knowledge about rigid and non-rigid transformations. Asking students to choose their own categories invites them to determine what characteristics might be important to notice. Students also begin to practice the skill of defining a rigid transformation that takes one figure onto the other without using a grid to estimate or define centers, angles, lines of reflection, or directed line segments.

Students may struggle to describe the location of the line of reflection. There is no need to address this during this activity as it will be covered in a subsequent activity.

Monitor for students who first translate to line up one pair of corresponding vertices, and then rotate for the card with triangle \(ABC\).

Arrange students in groups of 2. Distribute pre-cut cards. Give students 3 minutes to sort and then pause for discussion. Ask a few groups to explain how they categorized their cards.

Invite students to write the transformations and follow with a whole-class discussion.

One purpose of discussion is to re-emphasize that translations, rotations, and reflections are rigid transformations, which maintain the size and shape of polygons while other transformations do not. Remind students that figures are called congruent when there is a sequence of rigid transformations that takes one figure onto another.

The second purpose of discussion is to help students identify what is hard about defining rigid transformations off the grid. If students struggle to define rigid motions precisely, there is an optional lesson on point-by-point transformations to use in addition to this discussion.

Consider the card with triangles \(ABC\) and \(A'B'C'\). If any student first translated to line up one pair of corresponding vertices, and then rotated, call on that student to share their method. If not, discuss the difficulty of estimating where to place the center of rotation so that a single rotation will definitely line up all three points. Demonstrate for students how you can translate by the directed line segment from \(C\) to \(C'\), and then rotate by the angle formed at vertex \(C\).

If no students used a point-by-point method, encourage them to take another look at the card with \(VWXYZ\) and try to use a point-by-point method to define rigid motions that take the original figure onto the image without estimating the locations of any new lines or points. 

Wait on a detailed discussion of the transformation of the card with \(RSTU\), as it will be the focus of the next activity.

In this activity, students determine whether a reflection can take one figure to another and how to find the precise line of reflection for disjoint figures.

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

The purpose of this discussion is to reinforce using defintions to justify a response. Invite a few students to share their responses with the class. Highlight any improvements over previous justifications such as good use of a well-labeled diagram.