Pluses and Minuses

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.

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  Solving: You need to have copies of the Challenge Problems handout for this section of the lesson.  I begin this section of the lesson by telling my students they will be working in teams of three and assign them by “randomly” drawing popsicle sticks.  Check out my strategies folder for more information about grouping.  I explain that each team will be responsible for sharing their solutions of the challenge problems at the end of class.  I then give each student a copy of the problems and advise them to budget their time so that they can develop a strategy and solve each problem(MP1).  As the teams are working I walk around offering assistance, encouragement, and occasionally redirecting. Some teams will struggle with identifying which formula to use, because they don’t really understand the structure of the problem.  To help them I ask questions like “What part of this problem do these variables represent?” and “What kind of answer are we looking for?”  Other teams have difficulty with the algebra parts of the problems, because they still confuse what the sin, cos, tan, etc represent and want to separate sin(θ + π/2) into sinθ +sin(π/2).  For those students I try to use simpler examples like using the square root sign.  I remind them that √(4+9) is not the same as √4 + √9 and that usually helps.  When there are about 20 minutes left of class, I advise the teams to wrap up their work and be ready to share their solutions in about five minutes.  This gives my students time to figure out who will speak about which problem and what they’ll say.

• Sharing: To begin the sharing portion of this lesson, I ask for a show of hands from teams that feel ready to share for all four problems.  If all the teams indicate that they’re ready, I randomly select one group to share their results for the problem of their choice.  I generally have them share this kind of work using the document camera and projector, but you can also use the whiteboard or just have them share verbally.  As the first team finishes, they “tag” the next team to share that problem, and so on until everyone has shared on the first problem.  For this type of sharing, I don’t have a long question/answer period after each team, and instead allow discussion after each problem.  My students continue to share until all teams have spoken about all four problems.  If more than one team indicates that they are not ready to share I ask for volunteers, starting with whichever problem gave them the most difficulty according to my observations while they were working.  I allow all teams that want to discuss that problem the opportunity, then move on to the next most difficult.  My expectation is that every team will discuss at least two problems and will record step-by-step solutions for the ones they don’t discuss. 

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