I've included a video narrative of some of the pedagogy for this section in my resources. I begin class with this question on the board, “How would you find an exact value for cos(π/12)?” This leads to a discussion of what it means to have an “exact value”, a good example of MP6. My students reach consensus that they must use known values for the tangent function, which results in several of them pulling out their unit circles to look at all the values. They are quick to recognize that they don’t have an exact value for π/12, so I ask for other ways to find what we’re looking for. Usually someone suggests that we can subtract π/3 – π/4 (MP2) but they’re unsure of how to deal with the cosine part of the expression. I encourage them to use their graphing calculators to find an approximate value and see if their ideas for subtracting work. As students are trying different combinations, I walk around observing and making note of their comments. This may seem like a waste of time, but I’ve found that providing this time to explore options makes my students more ready to accept the sum and difference formulas when I present them. I think they need to convince themselves that there’s no “easier” way. As the class comes to the realization that they cannot get the answer with their current knowledge, I offer the sum and difference formulas as an option. I refer them to the list of formulas in their textbook and suggest that they select one that will work with this problem. It’s a short step to the solution, and now my students see the value of using these formulas.
To finish this lesson, I ask my students to choose one of the problems to write a detailed explanation for. They have done this before so they know that I expect the solution written out mathematically and then a written explanation of each step including what was done and why they chose that action. This assignment is due at the beginning of class tomorrow if they don’t finish them before the end of class