Here's Proof, Too

Because we’ve already spent time verifying the Pythagorean trig identities, I choose to focus this lesson more on using those and other fundamental trig identities to simplify and find exact values for trigonometric expressions. I begin this lesson with the identities we’ve already discussed on the board along with the co-function and negative angle identities.  I also post the question “How are these equations related?”  There is a photo of this board entitled “Identities” in my resources for this lesson.  To begin class I ask my students to think-pair-share with their left shoulder partner about the information on the board.  This allows them to build their MP7skills as they look for similarities and differences between the equations.  It also reinforces their general mathematical communication skills as they talk with their partner.  While they’re discussing, I walk around and make note of any particularly strong observations as well as general trends in the discussion.  As the discussions wind down, I call the class back to a full-class discussion and share what I’ve heard, without naming specific students or teams.  For example, I might say “I heard one group talking about how all the equations use trig functions without any values given so I don’t see how we can use them to solve anything” or “Another group asked if any of these equations was over a specific domain for θ.”  I then call on students randomly to give their response to the statements, an opportunity to work on MP3 skills.  If none of these comments were actually made, I either lead my class to them using guided questioning, or I state them as having been said and move on from there.  I optimistically anticipate my students will be able to reach a consensus about the equations and recognize that they are all different ways of organizing the relationships between the trigonometric functions.  That leads us to our main activity of the day, working with these identities.

• Student work and Presentations: You will want to have copies of the "Examples" handout ready for this section.  The first section of this activity is based on some textbook work, but with a twist.  Instead of simply having my students read through the examples in their book or watch me work through them on the board, I assign teams of students to present each example to the class.  We’ve done this before, so my students are familiar with the process.  To summarize for you, I tell my students they will be working in teams of three and will have approximately five minutes to prepare their presentation, which must include all team members in some capacity.  Some students are very shy about speaking in front of the class, but can usually contribute by writing on the board or “playing Vanna” and pointing as someone else speaks.  While the teams are working on this, I walk around to answer questions and clarify for those teams that need it.  At the end of five minutes I randomly call teams to the board to present their example to the class and to answer questions if their class mates have any.  By requiring both a presentation and the responsibility of answering questions, I help my students gain expertise in MP3 and MP6 since I expect accurate answers.  Each presentation takes two to three minutes, with additional time to get to/from the board and to answer questions, so I allow approximately 30 minutes for this part of the activity, including the five minute preparation time at the beginning. 

• Whole Class Practice: You may want to review "Parametric Example" in my resources before beginning this section. I also have a video narrative for this section explaining how I question my students.  When all the teams are done presenting, I ask if there are any additional questions, then tell my students that they will now get to help me put what they’ve seen into practice.  I post a final example on the board that incorporates parametric equations and ask for suggestions on how to solve it using trig identities.  This is a tough problem for my students because they don’t easily make the connection between parametrics and trigonometry.  I’ve found that their previous mathematical experience has been very compartmentalized.  They “learn” a concept one day, apply it as homework and then move on.  They may next see that concept in the review and exam and then not see it again all year!  To make this connection easier, I offer hints, with the condition that for each hint I give, they owe me two good responses.  The first hint I make is to ask if they remember any other problems that had variables we had to rearrange or get rid of somehow.  This often leads to someone recalling working with systems of equations or working with logarithms.  In either case, that may be enough for a student to make the suggestion that we use the Pythagorean identity to eliminate the parameter “t”.   If an additional hint is needed, I ask if any of the equations can be rewritten to look more like our trig identities.  I also try to tie the questions back to what the teams presented earlier, even to the point of using similar phrases or descriptions.  For example, one of my teams used the expression “because it’s the simplest thing to do next” several times during their presentation, so I use that as a question; “What’s the simplest thing to do next?”  I continue questioning until we’ve worked through the entire example, with me serving mostly as a scribe. The example and solution are in my resources as “Parametric example” and I’ve added a video explaining how I question my students as we go through this problem.The questioning and discussion really encompass several skills but I particularly like the focus on MP2 as we move from the information given to a model and back to the original questions.  I also like the way this example lets me give some guided practice at both modeling MP4 and using appropriate tools MP5 as my students decide what the equations are modeling and which trig identities will help them find a solution.  I’m always a little surprised at how well this lesson goes, because I know there are a lot of connections that need to be made.  I think giving my students most of the responsibility for teasing out those connections rather than simply telling or showing them what they are makes it a stronger lesson.

You will need copies of the “Homework” handout for this section.  I close this lesson by assigning specific problems from the text that I’ve selected for their relevance and rigor.  I am not a fan of simply assigning the evens with the hope that extensive repetition will make the content understandable, so I try to choose problems carefully and always expect my students to show their thinking both mathematically and in writing.  The first three problems allow my students an opportunity to practice simplifying expressions using the double and half angle formulas without having to worry about going beyond that step, but they also each incorporate more than just a one-step simplification and reinforce an understanding of the relationships between the trig functions.  The next three problems give my students a different perspective on manipulating quantities and also on understanding limits to the domain of the function.  The final four problems take my students to the next level by asking them to verify trig formulas and to use those formulas to simplify and solve a real-world problem.  Verifying may seem like a make-work assignment, but I appreciate the ways my students find to accomplish this task and the depth of understanding it helps them build for themselves.