Prove Circles Conjectures

To get class started, I facilitate a whole-class discussion in which we brainstorm all the possible cases of inscribed angles in circles (the center is on one side of the angle, the center is inside the angle, and the center is outside the angle).  As a whole-class, we prove the case where the center is on the inscribed angle.

After this proof, groups of students split into two pairs, where one pair proves the second case, and the other pair proves the last case.  My expectation is that each group produces one high quality proof for each case and that they be prepared to share out.  I have at least one proof for each of the two cases displayed under the document camera for questions and feedback (MP3).

During today's classwork, I ask students to prove the following circles conjectures in their groups:

• All inscribed angles intercepting the same arc are congruent
• The opposite angles of an inscribed quadrilateral are supplementary
• Any parallelogram inscribed in a circle is a rectangle
• Parallel lines intercept congruent arcs

Before I have students work on these proofs, I ask them to share their strategies for thinking about circles.  Inevitably, students will share out the idea of "adding a line" (radius or chord) to try to see inscribed angles or central angles more easily, which is essentially them looking for and making use of structure (MP7). Additionally, I encourage students to use their notes as they work through these proofs so that they have several of resources at hand.   

As I circulate the room, I initially walk around to make sure every student starts with a well-labeled sketch to represent the situation since starting these kinds of proofs is often the most difficult part.

During the debrief of these proofs, I ask student volunteers to share their work with the class.  Students may choose to project their work under the document camera or use the whiteboard to sketch a well-labeled representative diagram for the class as they talk through their ideas.  

I tell the audience that I expect at least one question and one comment about each of the proofs presented--students can use "I agree…" and "I disagree…" as sentence frames to help them get started.  I set these expectations because:

1. These proofs are often challenging, so we need a rich discussion
2. I need to validate the risks students take when they publicly share their work, and
3. Getting students to comment on each other's work creates a natural way for them to talk to each other instead of through the teacher.

When the discussion works well, there is a lot of opportunity for students to engage in MP3, as well as other important mathematical practices.