This lesson is algebra-based and I don't want to assume that my students have all of the necessary algebra skills at the ready. This section gives them an opportunity to warm up the skills.
First we sketch the perpendicular bisector of a segment to make sure that students know what the perpendicular bisector is and what it does.
Next we practice writing linear equations (a) through a point with a given slope, and (b) through a point perpendicular to a given line.
Finally we practice squaring binomials (without having to distribute) and finding distance in the coordinate plane.
Some of my students will be able to work through this activity quickly on their own. Others will need to have things demonstrated for them. Typically, what I'll do is give the class enough time to work on #1, for example, then I'll demonstrate #1 for anyone who needs to see it. Then I continue in this fashion until I've demonstrated all of the problems.
Students work with their A-B partners to complete Perpendicular Bisector Theorem. As I'm walking around checking on students, I provide just-in-time feedback. After teaching this lesson several times, I know the following are likely Points of Intervention:
Once students have had enough time to complete the handout, I call their attention to the front of the class so that I can recap the major concepts and procedures from the lesson. First, I remind students that when we want to write the equation of a line, we usually need a point and a slope. In this case, with the line being the perpendicular bisector, our point is the midpoint and our slope is the opposite reciprocal of the segment's slope.
Next I discuss how we chose specific points on the perpendicular bisector to see if they were equidistant from A an B, the endpoints of the segments. I explain that even if we had chosen a million specific points on the perpendicular bisector and they all were equidistant from the endpoints of the segment, that would not constitute proof of the perpendicular bisector theorem. Next I ask students to discuss with their partners the purpose of introducing the generic point P. Then I explain that P represents ALL points on the perpendicular bisector (infinitely many of them) and how that which is true for P must be true for all points on the perpendicular bisector. Therefore if we prove that P is equidistant from the endpoints of the segment then all points on the perpendicular bisector must be equidistant from the endpoints of the segment.
Finally, I show the algebra required to show that P is equidistant from A and B. That work can be seen on the Perpendicular Bisector Theorem_model response.
Before students leave, I ask them to respond to the following prompt:
What were we attempting to show in this lesson? Explain in detail how we went about showing it. What were our findings?
When I read these responses, it will let me know how well students have seen the coherence in the lesson and how well they have understood the reasons for the individual steps we took.