Here's Proof

For this lesson I changed things a bit from what I’d been doing with this trigonometry unit and returned to a more traditional format because I know that is what they are most likely to experience in college.  I begin this lesson with the following equation and question on the board:  sin²Θ + cos²Θ = 1 and “What familiar equation does this resemble?”   Many of my students will begin discussing this before class begins, so if you want it to be an independent activity, be sure to state that on the board and/or wait to post the question and equation until after your class begins.  As the bell rings I ask my students to think-pair-share about the post on the front board and give them 2-3 minutes to complete the task.  This question is really about helping my students learn to apply prior knowledge when introduced to a new concept, in this case recognizing the similarity between the structure of the posted equation and the Pythagorean Theorem (MP7).    I walk around during this time making note of which students/pairs are on track and which ones seem lost.  When I call time, I either share comments I’ve heard or ask students to share their comments with the class.  As they share, I also prompt them to explain how they’ve arrived at their conclusions. (MP2)  If this is the first time you’ve asked your students to explain their thinking, you may have to be more specific in your questions.  For example, if a student responds with “I thought it was like the Pythagorean theorem because they look the same” you might follow up with “In what way(s) does this look like the Pythagorean?”   When I’ve gotten responses from all the groups, I put the Pythagorean equation under my original equation, lining up the respective components.  You can see a picture of my whiteboard entitled “Pythagorean Identity” in my resources.  Then I tell my students that the first equation is called a “trigonometric identity” and that today we are going to walk through a proof of the equation, using our understanding of the Pythagorean Theorem. 

PROOF

• Beginning with the 2 equations on the board, I walk my students through a proof of the trig identity, with the requirement that they copy the steps as we go through.  I stop periodically to ask or answer questions.  You can see a video of a portion of this proof in my resources entitled here's proof narrative and the entire written portion as an educreations lesson entitled "Trig Identity Proof".  I didn’t video the entire exercise because it takes over ten minutes with my questions to the students, their responses, and their questions of me.  I ask questions while I demonstrate the verification on the board.  For a question like, “What part of the Pythagorean Theorem does the sin² Θ correspond to?” I expect a simple response, but for more in-depth questions like "What happens to the denominator in the sin² Θ + cos² Θ expression?” I expect a more detailed explanation of why we can cross these denominators out (MP1).   
• For the second part of this section I tell my students that they will each be working out a proof for two additional Pythagorean trigonometric identities, and that they need to complete these independently (MP2).  I also ask them to write out an explanation of the steps they take for each proof and why, to help them build their ability to communicate mathematically (MP3).  I post the two identities on the board then walk around as my students work on these, offering support, redirection and encouragement as needed.  The most common stumbling block for my students is being unsure of where to begin.  Since this is a challenge for them in many mathematical situations, I don’t generally tell them what to do, but instead try to help them choose by asking them questions like “What do you know about the tan Θ?” or “How are the cotangent and cosecant related?”  Some students still want more direction, but for the most part I try to refrain from staying with any one student for more than a minute (which is actually a long time in terms of total class time!). 
• As most students finish this activity, I have them pass these papers in so I can look them over.  I don’t usually collect this kind of work and instead use it to initiate class discussion, but I feel that these papers give me insight into how well my students are making the connections between what they already know – Pythagorean – and what we’re learning – trigonometry.  I then ask if anyone can see any reason to understand or use these identities.  This question is usually met with blank stares or a response about needing to understand it for a later class or college.  I then work an example on the board showing how an understanding of the identities can help them solve for trig values.  The example I use is: “given sin Θ = 4/5 for π/2 < Θ < π, find the values of the other five trigonometric functions of Θ”.  I require my students to copy this problem as I work it on the board so they have an example in front of them for their homework. 

When I’ve completed the example and answered any questions from my students I tell them that for their homework, they will be practicing applying what they’ve just learned.  I assign problems that specifically work with the Pythagorean Identities and finding values.  I tell students that I need to see their mathematics, but that I also need them to write about the problems to explain why they chose to solve each one as they did. (MP2)  A copy of the problems assigned is in my resources as Homework and the second page is an answer key.