Solving Equations

Learning Targes for solving equations

I begin this unit on solving trigonometric equations with a review of solving algebraic equations more generally. I do this because many equations involving trigonometric functions can be classified into a few categories based on their structure:

• Linear form- isolate the function than use inverses to remove the function
• Quadratic form- factor of use quadratic equation to isolate the variable
• Function=Function- both sides of equation have the same function so set the insides equal to each other and solve.
• Rational form- Solve like a ratio or make denominators equal and set numerators equal.

In the days that follow this lesson we will use substitution to help us learn to solve trig equations. When students are fooled by the structure of the equation, we will learn to rewrite the equation using substitution to help students solve.

Students proficiency with solving one or more types of algebraic equation will also benefit from this quick review. I have in mind some of the common errors that my students are prone to as we begin this lesson. For example, I know that I will work with several students who always want to start by isolating a variable. I have included problems like this one: 3x - 4=x^2 - 2x + 4

The lesson starts with four problems to solve.  The first two have no solution, which will confuse some students. Students generally assume that all equations have solutions. To help students understand what is happening in these two problems, I will suggest that they graph the left and right sides of the equation. Depending on the level of confusion we may do this as a class. I want to make sure that my students are correctly interpreting the meaning of a situation in which two graphs do not intersect. In addition, methods that can be used to solve Question 2 give a possible solution. But, the proposed solution does not fit in the domain of the original problem, so it must be excluded from the solution set. This fact usually emits groans. Students immediately think that if they find a possible solution, it is the answer. So, we'll discuss how some techniques, such as squaring both sides of an equation, can result in extraneous solutions. The important message is that algebra is powerful, but not perfect.

Questions 3 and 4 ask students to solve a quadratic equation. There are a range of missteps and misconceptions that may arise. With these problems, I hope that my students will help each other to overcome many of these mistakes. I am ready to remind students about some of the techniques that they have learned over their time in high school. 

Once students have had time to complete these four problems, I will use Popsicle Sticks to randomly call on a student to share what they have on their paper for each problem. I often use this simple strategy when we are reviewing content. It helps students to recognize that they are responsible for working to remember the concepts we have studied and apply them when needed.

We will continue today's review  of algebraic solving techniques with a set of set of equations to solve. To foster independent recall, but provide collaborative support, I will ask my students to use a Think-Pair-Share strategy as they attempt these problems.

• Think - I have students work on a problem for a minute or two on their own.
• Pair -  I give students 2 minutes to discuss ideas for how to solve the problem with another student. 
• Share  - I randomly pick one student to do the problem on the board while others attempt the problem at their seats. Then, we discuss the problem.

With this protocol, we will focus on one question at a time. Giving one problem allows students to focus on the techniques for that problem without worrying about doing all the problems. As students finish, I will encourage them to compare their approach to the one being taken by our volunteer at the board. 

Consistent with the goals for the lesson, the problem structures in this set are ones that may prove useful when we are solving trigonometric equations in upcoming lessons. We may or may not get through all of the problems. We may also jump around in the list, based on which problems I think will most help my students. With each problem that we discuss, we will discuss the domain of the original equation. This step in the process is important as we prepare to learn to solve trigonometric equations.