The Warm-Up prompt checks to see if students understand the Pythagorean Theorem. I am on the lookout for students who assume that they can simply add the lengths of the legs to get the length of the hypotenuse. I ask students to follow our Team Warm-up routine, which involves sharing their responses with the other members of their cooperative learning team. I choose students at random to write the team's chosen response on the board.
I use the team answers to highlight misconceptions and focus attention on the correct meaning of the Pythagorean Theorem. I display a slide in the slide show and use cold calling to check for understanding. (This is also an opportunity to see if students recall how to find the side of a square given its area.)
Students will no doubt recall that the Pythagorean Theorem is usually used to find the missing side of a triangle. I agree. The squares on the hypotenuse of the triangle in the warm-up problem and the check for understanding both have areas that are perfect squares. I ask students to find the lengths of their sides.
I display the Agenda and Learning Goals Slide for the lesson and briefly review it with the class.
In this section, students work in teams to re-create a transformation proof of the Pythagorean Theorem. The activity uses a Team Jigsaw format. I distribute the handout for the activity and make sure students know where they can obtain scissors and glue. I give the instructions for the activity with the help of a slide in the slide show.
As students get to work, I display a digital timer. Students will probably enjoy this part of the lesson, but it is important to keep it to a time limit. The focus should be on using reason to analyze the proof and evaluate its claims for rigor in the next section. If a team is stumped, I invite them to send out a 'spy' to see what other teams are doing. After 5 minutes or so, I use a document camera to display student work showing the solution to the first re-assembling task.
Things to Be On the LookOut for (BOLOs):
In this section, students work in pairs to evaluate the rigor of the proof. The activity uses a Rally Coach format. I begin by reviewing the previous activity with the whole class to ensure that students understand why the demonstration proves the Pythagorean Theorem. I may invite teams to share some of their learnings or insights. I use a slide in the slide show, which contains a hyperlink to a website with an animated demonstration.
The animation shows that the transformation works for any right triangles, but--I ask students to be super-skeptical for a minute--how do we know that the left-over regions are really squares? (MP3) (What is a square? How do we know these sides are all the same length? How do we know these acute angles form a right angle?) I invite discussion. (One way I may do this is by using a game called Math Ball.)
If the major points come out in a whole-class discussion, the activity can just be used to summarize and check for understanding, if there is time remaining.
Things to Be On the LookOut for (BOLOs):
I ask students to draw squares on the sides of the triangles, because I want to reinforce the meaning of the theorem. This turns out to be good differentiation, and I will often see students drawing squares on right triangles weeks later.
The lesson close follows our Individual Size-Up Routine. The prompt asks students to recap the argument they used to show that the acute angles of a right triangle can be put together to form a right anle into their Learning Journals.
For homework, I assign problems #4-6 of Homework Set 1 for this unit. Problems #4-5 ask students to practice applying the Pythagorean Theorem to find the missing side of a triangle. Problem #6 asks students to re-call one of the arguements they used to justify the claims in the proof of the Pythagorean Theorem they analyzed in class. The second part of this problem is a challenge for students who are ready to use algebra in a related proof of the Pythagorean Theorem.