Today’s lesson is the first time students will be verifying trigonometric identities. I expect that all of my students will be comfortable with simplifying trig expressions using the fundamental identities. Today, we build on this foundation by asking students to verify identities. By the conclusion of today’s lesson students should be able to verify a variety of trigonometric identities (HSF-TF.C).
To get things started, students will work independently on the Warm-Up Clicker Questions on Page 2 of Verifying Trig Identities (Day 1 of 2). During the Warm-Up I will be looking to see if any of my students are still struggling with simplifying trig expressions. I need to be prepared to assist them later in the lesson. I have prepared an additional resource that I can provide to students who I am concerned about later in the lesson.
I hope to be hands-off during the beginning of today’s lesson. I will encourage students to use the Identities books we’ve been making in class to help with problems. It is important that students learn to make sense of the problem, and, to make use of available resources (MP1, MP5).
Preparation Note: For today’s lesson I will have a half sheet of colored paper for every student. Half the class will have one color and the other half will have a different color. We will use this paper in the last section of the lesson.
At the conclusion of the Warm-Up, I will transition to a period of direct instruction using a PowerPoint Presentation on Trigonometric Identities. My students often have a hard time with this lesson, so I feel that my students are in need of some hints and strategies before they begin verifying complex identities.
The bulk of my work during this section is modeling the process of verifying trigonometric identities. I will work from the PowerPoint. This resource was modified from a document that I downloaded from www.mathxtc.com. The initial presentation was prepared by Shawna Haider. Since we have already been studying identities, the definitions on Slide 1 should not be new to my students. I will encourage my students to use their notes to verify that these definitions are correct. This is a great time for students who were absent or who work slowly to make sure they got all the identities they may have missed last week. For others, it is a time to study.
While making my presentation, I will make clear to students that we are shifting gears. Previously, we had been simplifying one expression, writing it in the most concise way. Today, our goal is to verify an identity. Be sure to explicitly point out the students how these two situations are different. Yes, we will often use our simplification skills. However, the ultimate goal is to verify whether both sides of the equation would always be equal for all defined values of the variable.
There are often many ways to verify an identity. In my view, it is essential to only work on one side of the equal sign to rework the expression so that it is congruent to the other expression (MP7). The problems we will be presenting to students are being verified from one side. If you have longer class periods or have an honors section, I think it would be great to see if teams could verify the same identities, but in a different way. See differentiation reflection for more information on this.
For a closure to today’s lesson, students will complete a Commit and Toss. If you have never seen a Commit and Toss, watch the video for a quick intro. (Note: In the video I said that both sides simplify to 1. This is no longer true as the problem was changed after teaching the lesson to make verifying the identity more student-friendly. However, everything else in the video still applies!)
Once students Commit and Toss their simplification, peers will randomly select a paper and verify the work. Then, the papers are shared with the owner, including a brief discussion of any errors found. If it will be productive, I can follow up this activity with a discussion about whether or not we have verified that this problem is an identity. I would start the discussion by simply asking the question “Is this an identity?” The students who had side A will probably say yes since when they simplified the expression they most likely re-wrote it to be like side B. However, students who had side B should understand that they did not verify the identity because they didn’t re-write their side to be like side A so they cannot say for sure. (Note: I intentionally gave the struggling students side B? This was also to help solidify for these students the difference between simplifying and verifying!)
Following the Commit and Toss, I will present page 4 of the Flipchart. Now, the goal is for students to verify that this is an identity. So, I have students switch which side they are working on and verify the identity from that side. So students who had simplified Expression A will now be working on the right hand side of the equation to re-write it to be like the left side. This process is more challenging and should be given to students who need this challenge. While students who were assigned Expression B to simplify will now be working to re-write the left hand side of the equation to be like the right hand side. Since the students will have already seen some of the simplification work for this side, the verification process should be more accessible to them.
Once again, when students have committed to their verification, have them toss it to someone with a different color. And have them verify their peers work. Hopefully, this activity will help my students better understand the difference between simplifying an expression and verifying an identity. It will also reinforce the idea of working on one side of the equal sign when verifying trig identities.
Tonight I will assign students Trig Equations Homework 4.