Yesterday we focused on converting one point from polar to rectangular form or vice versa. Today we are going to be converting entire graphs instead of a single point. To begin the lesson, I give students this worksheet and have them recap the conversions that we used to convert single points. After students fill these in we will share out and I will ask them where these equations came from just to drive home the point that we are merely using right triangle trigonometry.
In the section titled "additional conversions," I will just remind students that our trigonometric ratios still hold true. For example, we know that sin(θ) = y/r and sec(θ) = r/x. Since this feels like a new context for many students, it is good to remind them that these ratios we know and love are still valuable to us.
After this quick review, I have students get through as much of questions #2 and #3 from the worksheet as they can with their table groups. Usually 15 minutes is sufficient for them to get through most of the problems. Here are a few things I look for as they work:
After students have had sufficient time to work, we will share out as a class. I noticed that #2c, #2d, and #3a from the worksheet were particularly interesting, so I choose to focus on these. Here are some notes on what I observe and how we approach it as a class.
#2c - Students often have difficulty visualizing this as a line, even though they had no problems knowing #2b was a line. I start with a student who took the inverse tangent of 4 and found the angle measure of 76 degrees to know it was a line. Then we talk about how the slope must be 4 since it is y/x. I find a student who noticed that one of the conversions is tan(θ) = y/x, so we could substitute into the equation to get y/x = 4, and then rewrite to have y = 4x.
#2d - This was another tough one to visualize. Students often cannot just look at the equation and know what the graph looks like. Again, they had to rely on the substitutions and know that they could replace csc(θ) with r/y. I have a student present their algebraic steps to show that this is equivalent to y = 3.
#3a - This one was really interesting. Many students know that it was a circle with a radius of 4 so they think the polar equation should be r = 4. I discuss this problem in the video below.
After these examples, I give students this homework assignment to give them some more practice with converting equations of graphs.