The purpose of this activity is for students to explore why straightedge and compass constructions can be used to communicate geometric information precisely and consistently.

The goal of this discussion is to make sure students understand the straightedge and compass moves that will be allowed during activities that involve constructions. 

Ask students, “What makes this construction more precise than the sketch you made in the Warm-up?” (The compass makes exact circles. The straightedge makes straight lines. The compass keeps the right length for the radius.)

Make one class display that incorporates all valid moves. This display should be posted in the classroom for the remaining lessons within this unit. It should include: 

• Mark a starting point.
• Mark intersection points.
• Mark a point on a circle.
• Mark a point on a line or line segment.
• Create a line or line segment between two marked points.
• Create a circle centered at a marked point going through another marked point.
• Create a circle centered at a marked point with a certain radius after setting the compass to the length between two marked points.

Tell students that using these moves guarantees a precise construction. Conversely, eyeballing where a point or segment should go means that there is no guarantee someone will be able to reproduce it accurately.

The purpose of this activity is to let students determine how to use straightedge and compass moves to construct a regular hexagon precisely. Students should play with construction moves until they reach their goal rather than follow an explicit demonstration of construction steps. While the term “regular” appears in the task, it is not important for students to know the precise definition of “regular polygon” at this time.

This is the first time Math Language Routine 1: Stronger and Clearer Each Time is suggested in this course. In this routine, students are given a thought-provoking question or prompt and asked to create a first draft response in writing. It is not necessary that students finish this draft before moving to the structured pair meetings step. Students then meet with 2–3 partners to share and refine their response through conversation. While meeting, listeners ask clarifying questions such as, “What did you mean by . . . ?” and “Can you say that another way?” Finally, students write a second draft of their response reflecting ideas from partners and improvements on their initial ideas. Students should be encouraged to incorporate any good ideas and words they got from their partners to make their second draft stronger and clearer.

The purpose of this discussion is to build toward the concept of a proof by asking students to informally explain why a fact about a geometric object must be true. Ask previously identified students to share their responses to “How do you know each of the sides of the shape are the same length?”

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to “How do you know each of the sides of the shape are the same length?” In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.

Consider displaying these prompts for feedback:

• “_____ makes sense, but what do you mean when you say . . . ?”
• “Can you show me using your diagram?”
• “How do you know . . . ? What else do you know is true?”

Close the partner conversations and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer.

Here is an example of a second draft: We were given segment , which is the radius of the circle around . Then I drew segment , which is another radius of the same circle, so it’s the same length. I kept drawing more circles with the same radius, so the segments are all the same length.

If time allows, have students compare their first and second drafts.

After Stronger and Clearer Each Time, ask students what makes a good explanation. (Use math vocabulary, such as “radius.” Don’t say “it.” Label the diagram.)

If students spend more than a few minutes without significant progress, tell them the segment given in the figure is one of the six sides of the hexagon. Invite students to compare the given hexagon to the start of the construction. Then ask if they can draw another segment to make an adjacent side of the hexagon.