The purpose of this activity is for students to develop intuition that the set of points equidistant to two given points forms a perpendicular bisector by asking them to play the role of the points. Students will formalize this conjecture in the Lesson Synthesis and prove it in a subsequent lesson.

The purpose of this discussion is to establish the conjecture that the perpendicular bisector of a segment is the set of points that are the same distance to each endpoint.

Tell students that in mathematics, things that people wonder or hypothesize about are often referred to as “conjectures.” A conjecture is a reasonable guess that we make about something we are wondering about. Ask students to make a conjecture about the collection of all the points whose distance from is the same as their distance from . Invite several students to share their conjectures.

The goal of this activity is to use what students know about points equidistant to two given points to develop the construction for the perpendicular bisector using a straightedge and compass.

Monitor for students who use these methods:

• Paper folding
• Freehand drawing
• Compass and straightedge construction

Sharing paper folding and freehand drawing before the construction ensures that all students have access to the idea before they have to understand the construction.

The routine of Anticipate, Monitor, Select, Sequence, Connect (5 practices) requires a balance of planning and flexibility. The anticipated approaches might not surface in every class, and there may be reason to change the order in which strategies are presented. While monitoring, keep in mind the learning goal and adjust the order to ensure all students have access to the first idea presented (whether that be a common misconception or a different approach).

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

If students struggle to get started, direct them to their diagram from the Warm-up. How does that diagram relate to this activity?

The purpose of this discussion is to compare different methods for drawing a perpendicular bisector, highlighting the pros and cons of each.

Invite previously selected students to share their method. Sequence the discussion of the approaches by the order listed in the Activity Narrative. If possible, record and display their work for all to see.
Connect the different responses to the learning goals by asking questions, such as:

• “How can we check a perpendicular bisector for accuracy?” (Measure the angle and the segments created. Measure the distance from and to points on the line.)

If no student uses a compass, encourage the class to consider how to use that tool. Display the image from the Warm-up again, and invite students to explain how to use that construction to find a perpendicular bisector.

Emphasize that both paper folding and construction with a compass and straightedge are valid, accurate methods, but freehanding only works for a sketch. Choosing which one to use will depend on the problem and tools available.