I created this lesson as an alternate approach to proving triangle angle sums. I find that many students are already familiar with the activity in which they cut off the angles and line them up to form a straight line. So instead of repeating the experiment, I start by reviewing some homework problems and then I show the class this clear although somewhat bizarre video:
After the video I start a discussion around the concept of proof, "Could we use this experiment to show that this works for every type of triangle? Would that be possible?"
Here I want students to deduce that we need a better way to approach the problem. With a proof that looks at specific cases, we will never be able to prove that it always works for every case of a triangle, since we can always make a new case slightly different from the one before. I conclude by announcing, "Today you are going to prove that all triangles have 180 degrees. And you are going to do it without cutting up any triangles."
I have created a Geogebra Module for my students that represents a basic variation of the triangle sum proof:
My goal for my students is that they can build on our work from yesterday's class to create a proof of the Triangle Angle Sum theorem. The demonstration in the module by encourages students to use what they already know about alternate interior and supplementary angles.
I give them Module 6 Guide as a source of prompts to guide their exploration. A key to supporting the success of this lesson is to allow students to find the words that describe the angle relationship and support the writing of a proof.
In this part of the lesson, we ask several students to share their versions of their Triangle Angle Sum proof. I will show show the Powerpoint slides (Transversals) that include the Geogebra module again to spark conversation. I will ask students to volunteer to present their findings. I like to quote students when they have written particularly effective arguments. I will often put several proofs on the board during this lesson.
At this point many students still use specific number cases to make their argument, but I challenge them to think about the different between using specific angle measurements and variable angle measurements.
We finish today with about 10 minutes on these extension problems:
Before departing, we review students' answers and discuss the meaning of having similar or congruent triangles (these words appear in the last problem).