Problem Solving Identities
Lesson 7 of 14
Objective: SWBAT use fundamental identities to verify other identities.
Today's bell work gives students a chance to see how Pythagorean Identities can be used to quickly verify other identities. The key is to use one's knowledge and imagination to rearrange the Pythagorean Identities productively. Many students do not see the possibilities at first. This lesson attempts to foster their sensibility with respect to replacing one trigonometric expression with an equivalent expression.
Page 2 of the bellwork shows how many of my students work today's warmup problem. This student replaced sec^2(x) with tan^2(x) +1 and later changed 1 to cos^2(x)+sin^2(x). The student did not recognize the opportunity replace sec^2 (x)-1 with tan^2 (x). The student's work is correct, which is part of the point of today's lesson. Today we focus on reasoning, as a way of making things easier.
As we discuss the bellwork problem, I expect that some students will ask if they can replace sec^2x-1 with tan^2 (x). Discussing different methods, raises ideas in students minds that they can apply later themselves. Again, my goal today is to encourage students to help each other reason in different ways and increase their efficiency when verifying identities.
Some students do get upset when they see another way that is shorter to verify the identity. My standard response to this is "use the method which is clearest for you." It is quite important that students can explain their reasoning to someone else. If that results in more steps, then that is fine.
Today students need time to work on verifying identities with me being available to answer questions. Most students are not confident in what they are doing and giving time to work where they can ask questions will help students gain confidence in their ability to reason mathematically.
I do assign a few more problems for students to verify. These problems are the most difficult we have done (p. 385 # 30, 32, 46 from Larson, "Precalculus with Limits")
Many of the students have similar struggles some include:
- When a problem has a fraction with a single trigonometric function students are not sure what to do. I remind students how to make any expression a fraction by dividing by 1 (i.e. sin x as sin x/1).
- Factoring is also a common confusion. I write the problem in terms of a single variable such as 1-s^2 where s=sin x. Once the expression is written without the trigonometric function students are able to factor. After factoring the student replaces the variable with the trigonometric function so the students can finish the problem.
- Student make errors when multiplying expressions such as (1-sin x)(1-sin x). Students always forget to distribute the problem correctly. I again rewrite the factors as an algebra problem and we review how to multiply the expression.
- Finally students are confused with how to determine which identity should be used when a function is equivalent to more than one expression. For example, tangent can be written as sine/cosine or 1/cot x, many students are not able to determine which one to use. I will tell students if all else fails change everything to sine and cosine. I try not to give this advice to all the students since I want the students to develop their own method. Some students get overly frustrated when there is not a routine to the problems, these are the students I suggest changing everything to sine and cosine. Students who use this method usually gain confidence and then develop other strategies to use.
As the students are working, I am moving around the room answering questions, checking student work and determining which students are needing more instruction. I will also identify problems that students struggle with. These are the ones we will discuss as a class. I also like to see unique arguments. I like to share the unique arguments and compare the argument with what most students do with the class.
Today is a review lesson, so students worked on problems for the entire class. Towards the end I remind students that we will have a quiz tomorrow over verifying identities. I ask if students have developed any new procedures today as they have worked. I expect there to be several hands, so I will ask volunteers to explain the strategies they are using when verifying identities. I hope to hear explanations like, "I make everything into sine or cosine, looking to see if I have a property that can be used, if one term is a fraction make all terms fractions." This is important time for all of the students. Some have their work acknowledged. Others benefit from seeing what their peers do, which helps students think about what might be done.