On the first page of Triangle Proportionality Theorem and Converse, I spend some time helping students to figure out what we are trying to prove and how we will go about proving it. That first page reads as a narrative so we read together as a class. I select students randomly to read a bit of text. There are several places in the narrative where students are asked to pause and think or answer a question. At these places, I give students a minute or so to think privately, then I have them share their thoughts/answers with their A-B partners. I also clarify or elaborate on what is stated in the text if students start to look glazed over as if it's not sinking in.
At the end of the first side, students are asked to make predictions as to what the plan for the proof will be. I take a few minutes to hear multiple predictions from students. I really try to emphasize the importance of having a big picture plan for proofs as this is really the most important aspect of proof writing.
In this section, I'll be showing students how to write the proof of the triangle proportionality theorem. Although I am employing direct instruction, I want to make sure that students are engaged and actively thinking throughout the process. I don't want them just passively taking in information. With proofs, my main goal is to make students believe that they could come up with the proof on their own. It's usually the case that students can understand a proof after it is explained to them, but they are still left wondering how they would have ever come up with the proof on their own.
So I try to involve students at every step of the proof, giving them opportunities to predict what will happen next and asking them to explain how one step connects to previous steps and next steps. In this way, students are participating in the writing of the proof, even though there is significant scaffolding.
Before I begin, I explain to students that they will be having a quiz in which they will simply be asked to prove the Triangle Proportionality Theorem and its converse. I do this so that they have motivation to work to understand and start to assimilate the information I'm about to give.
On page 2 of Triangle Proportionality Theorem and Converse, students have a partially completed proof of the triangle proportionality theorem. For each blank that students need to fill in, I pause to give them private time to think, then I let them discuss their ideas with their partner. Finally, I show what goes in the blanks and show/explain very explicitly all of the thinking and decision making that went into the particular statement and reason. See Triangle Proportionality Theorem and Converse_Key for the completed proof.
On page 3 of Triangle Proportionality Theorem and Converse, students are given a partially completed proof of the converse of the Triangle Proportionality Theorem. Since I have just led them through a related proof with many similar features, I give them more time on this proof to try completing the proof on their own. I allow them to collaborate with their neighbors as this is good way to get ideas circulating around the room.
Even though the theorem we are proving in this section is the converse of the theorem we proved in the previous section, writing the proof is not as simple as reversing the steps from the last section. Therefore, I still want to make sure to explain the proof from start to finish after students have had a chance to try it on their own. See Triangle Proportionality Theorem and Converse_Key to see the completed proof that I write for students.
As an aside, one key change with the CCSS is the de-emphasizing of the rigor of formal of proofs and renewed focus on understanding the major concepts and key steps involved in a particular proof. For some students, the sheer number of steps involved in the two proofs in this lesson may be too much. For that reason, I am also including Proving Theorems_Paragraph Proofs, which students can study, understand and memorize in lieu of remembering all of the steps demonstrated in this lesson.
Proving Theorems_Paragraph Proofs also includes a completed proof of the Triangle Angle Bisector Theorem. I use this proof as an enrichment activity for students who are advanced and need an additional challenge. At first I just give them the given and what is to be proved. Usually, I will also give a hint on where to draw the auxiliary line. If a student is stuck and appears to be going nowhere, I'll show them the completed proof and let them explain why it makes sense.
Students know beforehand that the assessment for this lesson will be a quiz in which they have to write both proofs that were demonstrated in the lesson. I explain that the letters on the quiz diagram may be different or switched around to make sure they aren't just memorizing text.
I grade this particular quiz on a pass now/pass later basis. Students must keep re-taking the quiz until they have passed both sections.