Double or Half
Objective
Big Idea
Set the Stage
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.
Put into Action
- Whole Class Practice: I begin this section by saying that the new tools are called double-angle and half-angle formulas and post two examples on the board, one for sin(2θ) and one for sin(θ/2). My students are usually pretty quick to pick up on the “double” and “half” part of the formula. When I ask for volunteers to solve the problem {2sin(θ/2) +1 = 0} now, I generally get at least a couple. (MP2) Again, I select two or three students to work through the problem on the board, then ask for questions and/or discussion.
- Student Reflection: I then have them look up the rest of the double and half angle formulas in their textbook and ask if they have any questions. The first thing I’m usually asked is whether or not they have to memorize all these new formulas. My response is the same as I’ve given for the earlier trig formulas; I expect them to make a good reference page for themselves using either one of my organizers (you can see these in my strategies folder) or a system of their own. The next question is often how to tell which formula to use when. I understand this question and the confusion it implies because there are quite a few formulas to use with trigonometric functions! If this question is not asked by a student, I ask it of the class at large. I tell them to reflect individually for a moment on what they could do to most easily determine how to make this decision. (MP1, MP6) I don’t ask them to share their reflections, but suggest that they apply what they’ve just decided to the problem set I give them.
- Individual Practice: You will want to have copies of the Problem Set handout ready for this section of the lesson. There is also a video narrative supplement in my resources that explains why I have my students work independently for this section. I hand out the problem set and tell my students that for today they will be working independently. I also remind them to include written explanations as well as their mathematical manipulations. (MP2) As my students are working through the problem set, I walk around the room observing and offering encouragement and suggestions as needed. After about 15 minutes of work, I ask my students to stop their work for a brief discussion. I have already ascertained how they are progressing with problem set during my observations, but I want to get a sense of where my students think they are in the work. I ask for my students to give me a fist-to-five to show where they are in understanding the double and half angle formulas, with zero fingers being “still clueless” and five being “I’ve got it nailed”. I thank them for their responses and encourage them that they have 15 or 20 minutes remaining to work. I note which students show a fist or single finger and make a point of giving them additional instruction and support as needed.
Wrap Up
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.
