I begin this climactic lesson by projecting Slide 1 of the Two Slide Intro. As students view the first slide I ask them to recall the geometric proof of the Pythagorean theorem demonstrated in previous lessons. After a few moments I'll then say, "Have you ever wondered if the theorem holds true if we used other shapes on the sides of the triangle?" And then I reveal Slide 2.
When they see Slide 2 some students will instinctively say, "Yes, of course I can see it." Others will shake (or scratch) their heads. I'll say:
Look at this slide for an example, then take a few minutes to explore this idea. Can you prove that the Pythagorean Theorem works for shapes other than a square on each side of a right triangle?
My plan is to give students as little information as possible. "The formula for the area of a circle?" I'll say, "Better ask your neighbor." Ultimately all of my students will see that the Pythagorean Theorem works for other shapes, such as a semicircle. I hope to have a volunteer go up to the board and show the class his or her work. Once he or she has finished, I'll make a point of explaining to students that there are many proofs of the Pythagorean Theorem, including one by a US president (Chester Arthur) and one by Napolean Bonaparte. To conclude this Launch and give students a sense of how some people may find beauty in proof I'll show the following video of six visual proofs of the Theorem:
Pythagorean Theorem, Six Proofs by Beau Janzen
Before beginning this activity I make sure to create homogeneous groups of students who will work well together as they travel from station to station. From the 7 Pythagorean Problems resource I place copies of today's explorations at each station, one problem per station. I want my students to have the opportunity to get up and move throughout the lesson. Students can complete the problems in any order that they choose. I tell the class that each student should do their own work, but I will collect one paper per group at the end of the lesson.
In groups, students will visit each of the seven stations. I ask them to complete (successfully) any four of the problems. Of the four, at least two of the chosen problems must involve a 3-D figure.
When students begin working I will keep an eye out for students who I know have been struggling this unit. I may just ask that they skip (or choose) certain problems in order to make sure that they practice problem solving, and complete the four assigned problems. I generally check-in with groups at each station, or ask them to check in with me, before they move on.
I forget where I picked up this strategy, but I use it in this lesson with good results. As we came to the end of the lesson I asked my students to write a response completing each of the following prompts:
"I used to think............."
"But now I think............"
Before we depart for the day I call on various students to share their responses. It is is a great way of gaining insight into how students' ideas about the Pythagorean Theorem may have changed during this unit.