Polar Equations of Conics - Day 2 of 2

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Objective

SWBAT graph and write equations of conic sections in polar form.

Big Idea

Apply the formula we derived for conic sections in polar form.

Launch and Explore

20 minutes

Yesterday we spent a considerable amount of time developing a general formula for the polar form of a conic section. I start today's class by seeing if students can recap the formula and explain what the different variables stand for. Students will usually give me one "version" of the formula and I will ask what it means if there is cosine in the formula instead of sine, and what the plus or minus determines. In the video below I talk about how I used Desmos to review these aspects.

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In this two-day lesson, we spent yesterday in a more conceptual place by deriving formulas and thinking about how to describe conic sections using a directrix. Today's purpose will be to solidify this knowledge by applying it to examples of specific conic sections. I give my students this worksheet and have them work on it in their table groups. Usually 15 minutes is enough for them to get through most of them.

Share and Summarize

20 minutes

After students have worked on these tasks with their table groups, I will select students to share their work with the class. I will select students to share each problem on the document camera. For each question there will be a specific aspect that I want to address, so I will make sure the student I choose has that in their work.

  • Problem 1: I make sure that the student has the algebraic work of dividing the fraction by 2 to get a 1 in the denominator. I also make sure the student made a table to show the important values of r and Ɵ.When writing the equation of the hyperbola in conic form, I usually show two ways: just using the graph the find the center, a, and b values and writing it in standard form and using the polar to rectangular conversions.
  • Problem 2: I choose a student who has important values marked on their graph. The focus should be labeled at the origin. The common hangup is that students will forget that the focus must be at the origin when a conic is written in polar form.
  • Problem 3: Students must know that this is an ellipse since the eccentricity is 0.8. I make sure the student I choose to share has a neat graph and who showed the substitution of the points into the equation since the location of the directrix is unknown.

After going through these problems, students are usually in good shape to start the homework. I will assign problems from the textbook to give some reinforcement to polar form of conics.