Lesson 6

The purpose of this Math Talk is to elicit strategies and understandings students have for rigid transformations. These understandings help students develop fluency and will be helpful later in this unit when students will need to be able to define transformations rigorously and use transformations in proofs. While participating in this activity, students need to be precise in their word choice and use of language (MP6). Students will continue developing transformation vocabulary throughout the unit, it is not necessary for students to use phrases such as directed line segment at this point. It is okay if there is not enough time to discuss all 4 problems.

Display 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.

The goal of this discussion is to identify parallel lines. 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 students do not mention parallel lines, ask, “Why don't we need to use a rotation for this pair?“ (The lines are parallel.) 

The purpose of this activity is to extend what students know about constructing a perpendicular line through a point on the given line to a new situation in which the constructed perpendicular line goes through a point not on the given line.

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

Some students may struggle more than is productive. Ask these students to draw a line segment and construct the perpendicular bisector of it. In that construction, the perpendicular bisector will go through an intersection point of two circles. Ask, “What happens if you create a circle centered at that intersection point that goes through an endpoint of the segment? Why does that happen? How can you use this idea in this new activity?”

Focus on the process of using a previous construction to generate new constructions. Here are some questions for discussion:

• “How was this construction different from perpendicular line constructions you have done before? How did thinking about the differences help you plan what to do?” (I didn't have a segment to bisect, because the point wasn't on the line. I realized I could still make a segment using the new point.)
• “How does knowing some constructions help you do other, more complicated constructions?” (I can use the same moves, but just change them a little bit.)

The purpose of this activity is for students to think strategically about how to apply previous constructions to a new construction.

It is not expected that students will use any method other than two consecutive perpendicular lines, but if students come up with other methods, consider discussing those methods during the lesson synthesis.

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

Some students may struggle more than is productive. Ask these students to consider what they just learned to construct starting from the point and line. (A perpendicular line.) Invite them to consider the relationship between the line they could construct and the parallel line they want to construct. (Those are also perpendicular.)

The purpose of the discussion is to focus on the process of using previous constructions to generate new constructions. Ask students, “How does knowing some constructions help you do other, more complicated constructions?” (In this construction, I repeated a construction I already knew twice.)

Add an additional item to the display of constructions students already know:

• A parallel line through a point not on the given line