I start class today by spending a few minutes going over the homework from yesterday's lesson on polar form of conic sections. There will inevitably be a few questions on this topic so I want to clear up any misconceptions before we start reviewing for our assessment.
Usually the questions will center around a few concepts that can be elusive to some students. Here is a list of things that usually come up after the assignment:
In an attempt to solidify the knowledge of conic sections, I assign question #1 of the review worksheet and give student about 10 minutes to work on it. I want to do this question together before I let them loose on the review for the rest of the hour since this is a good summary problem that connects what we worked on in this polar unit with our work with conic sections in the previous unit. Students will work on this question with their table groups. If they finish early, I instruct them that they can continue working on the rest of the review since it will be due tomorrow.
Question #1 is difficult for students because students must move fluidly between rectangular and polar coordinate systems. Students can usually graph it pretty easily, but it is difficult for them to find the rectangular equation of the ellipse. I think students can get "stuck" and think that if the equation is in polar form then they cannot consider any properties of rectangular form. For example, for #1 students know that (1.5, 90°) and (3, 270°) are the endpoints of the major axis of the ellipse, but may not know how to find the center of the ellipse. It is a conceptual jump to switch these points to (0, 1.5) and (0, -3) on the rectangular coordinate plane and to just find the midpoint in order to know the center.
There are two main methods that students may use to come up with the rectangular equation of the ellipse. I will try to choose students to show both ways. If one method does not come up, you can introduce it to students and finish it up as a class.
Another way to find the rectangular equation is to just start with the polar form and use the conversions in order to get an equation with x and y. Some clever algebra may be needed but one way is to cross multiply so you have 6r + 2rsinθ = 12. Now r can be written as sqrt(x^2 + y^2) and rsinθ = y. The form that it is written in may not be recognizable as an ellipse, so method 1 may be more useful.
After going through this problem, it is time to keep working on the rest of the unit review and finish it for homework. This will be a good time for students to recap all of the important concepts that we worked on in this unit.