My students have already learned the value of the sum and difference formulas in working with trigonometric expressions and equations in a previous lesson. To begin today’s lesson I put the following equation on the board:
cos(θ + π/6) – 1 = cos(θ - π/6) for 0 ≤ θ ≤ 2π
I intentionally don’t include any directions so that my students begin discussing what can be done with the equation. I hear them talking about simplifying and solving and which formulas to use demonstrating to me that most of them are becoming much more comfortable with trigonometry. (MP1 and MP2) That’s one of the reasons I use an equation that they already have tools to solve, rather than introducing something new immediately. I’ve found that reinforcing previous instruction with a problem that my students can solve without my assistance makes that instruction more meaningful for them and also makes them more willing to learn new material. After allowing discussion for a few minutes, I ask for volunteers to come up and work through the problem on the board. I allow up to three students to give it a try encouraging them to ask their classmates for assistance as needed. Generally it only takes a few minutes for the volunteers to complete their task. I ask my students to review the solutions posted and then ask for any questions or additional discussion. It’s nice to have two or three different interpretations of how to solve the problem, since it reinforces my frequent comment that there is rarely “one right way” to do mathematics. I then clear the board and post a new equation:
2sin(θ/2) +1 = 0
and tell my students to think-pair-share a possible approach to solving this equation. After a minute or two (or when the talk dies down) I ask if there are any volunteers to solve this equation. I rarely get any takers on this because most of my students recognize that they need some new tools for this problem, and those who don’t see that are not going to volunteer! I tell them that today’s lesson will increase their trigonometry toolbox.
As we approach the last 5 minutes or so of class I advise my students to complete as much as possible the problem they are working on. I then collect the work from all my students to review what they’ve completed, assuring them that their grade for this assignment will be based on the quality of their work rather than the quantity. I choose to collect this work to get a better feel for how well my students are individually progressing in their understanding of this lesson.