Yesterday we built some conceptual understanding about parametric equations. Today's goal is to be given parametric equations and to think about the corresponding equation in rectangular form.
I start by giving students this worksheet and have them work through the first page on their own with their table groups. They should have enough information from yesterday to get through most of it.
Making the graph for question #4 shouldn't be too difficult, but finding the rectangular form may be difficult. Since we just finished our unit on conic sections, many students will just look at the graph and find the equation with techniques they already know. As I am walking around I will encourage students to start with the parametric equations and see if they can convert it to rectangular form.
After students have given a shot at converting parametric equations to rectangular form, we will discuss it as a class. I will start by selecting a student who just looked at the graph and found the equation that way and have them explain their method.
Then, if a student figured out how to eliminate the parameter with substitution, I will select them to share their work with the class. If no student used that method, I will ask the class how we can take two equations with three variables and end up with one equation with two variables. I might make use a really easy set of parametric equations if they need some scaffolding.
Once students realize how to use substitution, I will have them do #5 with their table. Then I will select a student to share their work on the document camera.
To finish the lesson, I will explain how to graph these parametric equations on a graphing calculator. I will also explain the process on Desmos. Here is a YouTube video that explains the process on a Texas Instruments calculator.
In the video below, I explain the process using Desmos.
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Finally, I will assign some problems from our textbook to summarize the important concepts of this section and to give more practice with parametric equations.