To begin the lesson, I'll give each student a copy of APK_Proving Circle Segment Relationships. This resource consists of four items that involve concepts and/or skills students will need to have when they begin to write proofs later in the lesson.
I give the students 5 minutes to work on completing the four items and then I spend five minutes going over the correct answers, reasoning, and processes.
In this section, I'll be getting students prepared to write Proof #1 on Prove Segment Relations in Circles. The prompt instructs students to:
Prove that when two chords intersect, the product of the segments of one chord equals the product of the segments of the other chord.
I'll start by having students read this prompt closely. I'll ask, for example, "Where in the diagram do we have intersecting chords?" and "Does it mean when they say 'the product of the segments of one chord?" All of this is moving us toward stating what we're trying to prove: (AE)(ED) = (CE)(EB). Once we've established this, I'll move on to modeling for students how we develop a plan for a proof like this. See the following screencast to get an idea of how that goes.
Once students have the overall plan for the proof, I have them practice re-voicing the plan with their seat partners. They should be saying something like, "We'll prove that two triangles are similar then use the fact that corresponding sides in similar triangles are proportional to set up a proportion. Finally we'll use that proportion and the means-extremes property to reach our final conclusion.
Then it's time for students to complete the proof. I'll walk around the room engaging students as needed. As I walk around, I'm looking for exemplars that I show under the document camera for other students to learn from.
In this section students will be writing proof #2 on Prove Segment Relations in Circles. As in previous sections, I'll start by having students spend some time reading the prompt closely in order to state what we are proving and have it make sense in terms of the diagram.
Once we get to that point, I give students a couple of minutes to think independently. I want them to have a chance to apply what they learned from the last proof to this new situation. There are similarities in the approaches to the two proofs. For example they both require us to draw auxiliary segments and make use of similar triangles and the means-extreme property. That said, this proof is not as straightforward as proof #1. For that reason, I give students some extra scaffolding to help them as they attempt to write the proof. This extra support comes in the form of the Scaffold for Secant Segment Product Proof. I give each student a copy.
Students will still have significant work to do to complete the proof. I'll give them 15-20 minutes to work on it and see how they do. I encourage them to collaborate because some students will progress further than others and I want my students either to benefit from explaining to someone or from having things explained to them. As students are finishing up, I will bring the attention to the front of the classroom as I show and explain the steps of the proof.
Later on a quiz and/or unit test, students will need to be able to write the proof on their own. Because of that, I want to make sure that I model it well so that all students walk away from class with the correct information and that they have it explained to them thoroughly so that they can understand it.