Arcs and Angles: Central and Inscribed Angles

I begin today's lesson with this Tangents and Arc Measure Warm-Up to assess students' understanding of tangent properties, arcs, and angles.  I give students about five minutes to work individually before comparing their work with their peers.

As students work I circulate the room to listen in on students' conversations and to encourage students to justify their reasoning.  During this time, I also try to select a student to present his/her thinking with the class.  

I like to facilitate a quick whole-class discussion (about 5 minutes) where we discover the relationship between inscribed and central angles that intercept the arc.  I pass out tracing paper during this time so students can convince themselves that the inscribed angle is half the measure of the central angle that intercepts the same arc.

After this discussion, I do a Four Corners activity, in which one person from each group goes to a corner of the room to become an "expert" on one of the big ideas about inscribed angles (MP1). Each expert group conducts experiments to eventually come up with a conjecture along with at least two examples that will help them convince their home group of their findings. Here are the topics for the groups to explore: 

1. Inscribed angles intercepting same arc
2. Angles in a semicircle
3. Cyclic quadrilaterals (quadrilaterals whose four vertices are located anywhere on the circle)
4. Arcs intersected by parallel lines

After the expert groups finish their investigations, all students return to their home group to share out about their findings.  I tell students that I expect presenters to share their conjectures and accompanying evidence and that I expect listeners to ask questions and try to make sense of what they are hearing.

During the final segment of today's lesson I give students a group quiz. I use group assessments to encourage group interdependency and positive group work behaviors.  For this particular quiz, I want students to focus on showing high quality work and writing good justifications, which requires them to use precise academic and geometric vocabulary (MP6). I will raise this focus area with them before I handout the assessment.