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:
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.
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.
During class we spent time reviewing techniques for solving. Of course students will need to use the techniques when solving word problems. As class comes to an end I give students four application problems to solve. These problems allow students to not only practice their solving strategies but also help students continue to work on reading mathematical text. In my class, I will assign:
p. A61, #156, 158, 161, 162 from Larson "Precalculus with Limits"
At the end of today's lesson I will ask students to complete an Exit Slip that will give me a sense of how they are feeling about solving algebraic equations:
1. Was their a particular misconception that you recognized in your work today when we went over the solutions to the equations?
2. Is there a particular type of equation that you need more practice with in order to become proficient?