The purpose of this Warm-up is to apply the precise definition of a circle to explore points that are equidistant from two points.

The purpose of discussion is to emphasize the definition of a circle. Here are some questions to consider:

• “Why do all the points create a circle?” (A circle is the set of points that are the same distance away from the center.)
• “What do you notice about the points that are the same distance, , from both and at the same time?” (The points that are distance away from both and are the two intersection points of the circles.)
• “Could there be 3 points that are all distance from and ?” (No. The points on the circle centered at that are inside the other circle are closer to than , and the rest of the points are farther from than .)

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 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).