In many of my lessons involving an important theorem, I like to have the students “stumble upon” the theorem through a well-designed lesson plan and scaffolding activities. However, in today’s lesson, it is my intention to roll the theorem out to the students, and ask them WHY it works and where it comes from. I have chosen to do this for three reasons:
1) They really have already learned the theorem and it is just a matter of applying our knowledge of trigonometry to extend it.
2) Explanation will be needed when it comes to the notation.
3) I would prefer to focus on the benefits of using the theorem, rather than deriving it.
I begin the class period by writing the Pythagorean Identity on the board. Next, I tell the students that this equation is always satisfied for any angle; and today we are going to study where it comes from and why it is helpful. Following this unorthodox roll out, I provide the students with the Pythagorean Identity Notes and we begin working through it as a class.
For the first section of the notes, I like to have the students try a couple of angles to see if they can show that the theorem works for them. The students will likely initially struggle with and ask about what the sine squared and cosine squared symbols mean, but I am confident that with a little educated guessing they can figure it out. After having the students satisfy the theorem for 2-3 angles, I ask the class if anyone found any angles that didn’t work. Typically this allows us to highlight a computational error in the student’s work. Even though they were wrong, I commend them in front of the class for allowing us to learn from their mistakes. I commonly use the phrase “I LOVE wrong answers!”
Once we have shown the students that the identity appears to work, I ask them if all of the angles our class tried (and worked) are sufficient information to prove that the identity is true. In response, most students will typically say, YES. However, I quickly warn the students that no number of trials would be sufficient to prove that a theorem is true. Proof by example is not a sufficient mathematical approach, but proof by counterexample is! If we would have found a single angle that did not satisfy the Pythagorean Identity, then we can say that the identity is not valid. Having this conversation with your emphasizes the importance of proof in mathematics. For a fun twist on the discussion, ask the students for real life examples when “proof by example” is not a sufficient approach.
Through the Unit Circle Diagram on the class notes, students will be able to see that the x andy coordinates of the points represent the sine and cosine of the angle... something that we have seen before! After revisiting this feature of the unit circle, I challenge my students to guess where the "PYTHAGOREAN" Identity came from... with a big emphasis on the PYTHAGOREAN part. Now we are only a whiteboard diagram away from showing the students that the triangle formed from the angle and the point is a right triangle. Therefore, the Pythagorean Identity is nothing more than an application of the Pythagorean Theorem.
At this point it might be appropriate to begin to ask the students to think about when signs might be significant in working with the identity. I remind the class that all of our sine and cosine answers will be squared, and thus always produce a value that is greater than zero. Questions* to toss out to the class might be: How do we account for angles in different quadrants? What problems might you anticipate?
*This level of questioning might not be appropriate for all classes, but it is nice to have this discussion as an "Ace in the Hole" if the lesson takes you in this direction. If the students are not ready, or you do not wish to engage in a whole class conversation, they will be exposed to this situation directly in Example 2 of the notes handout.
At this time I circulate the Pythagorean Identity Homework and allow the students to begin working. Most days, I encourage professional collaboration strategies in my classroom and allow my students to work together as long as each person is contributing and everyone is completing his or her own worksheet. Today is no exception to this method.
When the students finish, they should begin working on the unit review from yesterday's lesson.