No Angle Left Behind: using trig functions for all angles

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Objective

SWBAT apply understanding of trig functions and the unit circle to acute and obtuse angles.

Big Idea

What about all the non-right triangles? Expand on the trig functions using the unit circle.

Set the Stage

7 minutes

Begin class with the first slide of the PowerPoint projected on your screen or whiteboard.  I allow students to talk about what they think it means for a few moments without additional direction, then bring the class together for a class discussion. (MP1) Usually someone will say they think we’re going to look at non-right triangles or obtuse angles.  I don’t edit or comment, but just let those ideas be out there for everyone to think about.

I assign students to pairs (as described in my strategies folder – if you have an odd number, I suggest teaming up three of your better students or allowing one strong student to work alone.) and ask them to discuss what they remember and understand about the unit circle and trig ratios. 

I put up slide 2 and allow 2-3 minutes for discussion, then have one member of each pair share a summary with the class. (MP6) If I’ve been doing my job and we haven’t just come back from vacation, the students should be able to identify values for at least sine and cosine using X1 and Y1.  I tell my students that before we can figure out how to find trig values for all angles, we need to review one more concept, and that both partners need to be active participants in the next activity.

Put it into Action

40 minutes

Kinesthetic Activity (10 min): This first activity is really not an Algebra II content piece, but I've found that it sets my students up for success for the rest of the lesson by reviewing in a new and different way.  They become comfortable again with the signs for x and y for each quadrant so that they have fewer obstacles to success in the next section of the lesson...it's well worth the time it takes!  I put up slide 3 and tell students that they will be figuring out the signs for X and Y in each quadrant.  I say that they will be getting a handout after the activity to record the signs, but that during the activity I just want them to figure it out and be sure they understand what the signs are in each quadrant and why.  The directions for the activity are on slide 3, but you will probably have to model the activity, or you can have a volunteer come up and demonstrate for the class. I have also included a video explaining the activity with this lesson entitled Trig Quadrant video.  I have each pair of students find a space where they can stand and stretch their arms straight out to the sides and review the directions on the slide.  Give students time to play with it, reminding them to switch positions after about 3 minutes. (MP6) I call time after about 6 minutes or when teams start to get off track. You will need to keep a careful eye on some of your teams during this activity to ensure that they are actually getting the correct answers. 

Teamwork (25 min): You will need copies of the Signs of Trig Functions handout and the Trig Functions handout for this section of the lesson.  Slide 4 is the same as the first part of the handout entitled “Signs of Trig Function Values” which I now give the students to complete individually. This should only take a few minutes during which time I move around making sure that everyone has the correct signs. 

I put up slide 5 which is the bottom of their handout and have students work in pairs again to complete this chart. (MP1, MP7) Some students will need prompting to connect X and Y values with the trig values, but I find that a gentle reminder about trig ratios usually puts them on the right track.  When all students have successfully completed their charts, I tell them that we are ready to help all those poor angles being left behind.  I begin this section with slide 6 as a further reminder of what we started the class with and have already studied regarding trigonometric functions.  I move quickly to slide 7 and ask for suggestions for finding the trig values for that angle.  Students may volunteer that we can find the measure of the acute angle, but if not, I draw a right triangle using the reference angle and students can usually take it from there.  I also take this time to introduce the term “reference angle” and either give additional examples or have students draw additional examples on the board, depending on time.

I ask students to work with their partner again and give each team a copy of the Trig Functions Challenge handout.  I tell the students they may use any notes or reference materials we’ve created in class and encourage them to be creative with their problem solving strategies. (MP1, MP2, MP5) I allow 8-10 minutes to complete their work on the challenge then have each pair team up with another pair to critique each other’s work.  (MP3) I remind them to focus on what the team did well and to ask questions about anything they don’t understand.  This method of students critiquing each other’s work is one I use often, but I still find that reminding students of appropriate courtesy helps keep things going smoothly.

When all teams have finished sharing, I ask for any volunteers to share the solution they found most interesting and/or elegant.  I keep the emphasis on the process and find that students generally like to show off new and different ways of getting a solution. (MP4)

Wrap Up

5 minutes

I close this lesson by putting up the last slide and asking students to think about one circumstance where they might use trig functions to find measurements like the examples we looked at today.  I give each student a notecard and ask them to write at least one example for using trig functions in real life. (MP4)  This is much more difficult for some students than it sounds, because they aren’t accustomed to projecting mathematics onto their world outside the classroom.  I encourage students to think outside the box and to consider times they might encounter circles or circular motion.