Simplifying Expressions

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Objective

SWBAT use the fundamental identities to simplify expressions.

Big Idea

Through the use of fundamental trigonometric identities expressions can be rewritten.

Bell Work

5 minutes

The first two lessons in this unit had students record fundamental identities on a reference sheet and then use those identities to find trigonometric values when information was known about other trigonometric functions. Today we will move to using the identities discussed to rewrite trigonometric expressions.

Students are given bell work that asks them to rewrite a trigonometric expression as a single trigonometric function. The class has been introduced to this type of work, but the students have not worked on this skill individually. I move around as students work looking for different processes used by the students. These will eventually be put on the board (page 2 of Bell Work) for students to review.

After the students share their work, I will refer to these examples to discuss how to write out work so that it is easy to follow (MP6). My students tend to skip steps as they record their work, a habit that makes it difficult for them to identify and correct missteps. Several also need assistance with organizing the written work to make it more readable by others, again important to finding and fixing errors. I will take the time to write out a division process, so students see a model of representing a process.

For many students being precise in communicating their work is a struggle. By discussing modeling and practicing students do improve. It is easy for teachers to go "easy" on this but this will hurt students as they progress to higher levels of mathematics.

Being clear with your process

10 minutes

After discussing students' work I share an example from the book.  I have taken out all the words. I cover up the process to begin the example and show one step at a time asking questions as we move through the problem?

  • What was done to get the first step? Students should have their reference sheets out at this point. Some will struggle with determining the process.
  • What would help you understand this process? Students will say I need to see a reason or explanation for what was done.
  • Have you ever done a problem and been asked to show someone how to do the problem and you can't remember what you did?
  • What would help you remember what to do? I want students to think about annotating the work (writing out a reason for the step.)

After talking about how not having reasons can make understanding the process hard, I show the second slide. Does having the reasons beside the steps help you understand? How could you do this with your work?

I discuss writing the work downward instead of across.  I suggest folding a piece of paper "hot dog" style and putting the work on one side with the reason on the other side to make the process clear. As this is discussed students begin to shake their heads as if to say yes. This is a very good technique for struggling students. It gives students a method to remember what they did so they can explain their reasoning to others.

 As a class we review each step and clarify any explanations that are confusing. 

Simplifying Expressions

5 minutes

Students work  in groups on another example.  Some students are a little timid about what they are doing while others understanding completely.  As I move around the room I find a student that usually struggles to put the problem on the board. I want to validate the students that are unsure so I make sure the student's process is correct.  Students know which students struggle in the class by putting the struggling student on the board, other students will realize they can do these problems. 

As students review the work questions such as "can I write (1/cos x) /(1/sin x) as a division problem?" will be asked.  I respond with "Does it mean the same thing as what is written?" Once students understand it is the same thing I write the the division on the board so students can see what the students are talking about.

On this problem some students will say they know that sec x is the reciprocal of cos x. Since sec x is in the numerator they put cos x in the denominator. They do the same reasoning for csc x to get sin x/cos x which is tan x.  I can tell by the discussion that these students are understanding the process and approve this as a correct process.

Rearranging Pythagorean Identities

15 minutes

We have discussed using algebra techniques as we simplify expressions.  I now put up a Pythagorean Identity. "Using only algebra rewrite this identity in 2 different ways."

Students share their results.  I then say: If I have 1-sin^2 (x) what can I replace it with?

On page 2 I put one of the rearranged expressions up and have the students factor the expression. I pick a student to show the factoring. Students need to see how they can rearrange the identities to use for substitutions. 

After working with the first Pythagorean Identity I put another identity on the board for the students to re-write. After rewriting I ask the students to factor one of the rewritten forms.

Students get confused when the Pythagorean Identities are not in the original form. By manipulating the identities students begin to see different tools that can be used when simplifying.

Closure

5 minutes

With 5 minutes left in class I put  3 problems on the board for student to simplify. These problems work with the Pythagorean identities as well as the other (reciprocal identities and quotient identities) discussed so far this week.

I let the students know that we will discuss these problems the next class period. I also tell students not to worry if they get stuck. The students need to keep track of the different attempts so that we can discuss why the method did not work. I tell students that every time you do something and it does not work you have learned something. Namely that this process does not work so don't do it again.

I also tell students that what I think will be easy may be hard for you.  For these questions the first problem will probably be problem many students will not have correct. This will lead for a great discussion tomorrow.