To start class I will lead a discussion about the homework problems that students had to complete. For questions #4-8 from this worksheet, they have to compare two more polar graphs to the rectangular counterpart. I will choose students to present their work on the document camera so that we can have a discussion about it.
Again, I will use the method of looking at individual intervals of the rectangular graph to sketch the polar graph. I explain more in the video below.
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I also encourage students to highlight portions of the graph to show the corresponding features (like in this image). This seems to really drive home the visual connection between the two graphs.
This homework assignment is a good summation of all of the work we have done with graphing polar equations. At this point students should feel comfortable with using the rectangular counterpart to guide the sketch of the polar graph.
I give this to students in class so they have a lot of time to work with their table groups and bounce ideas and questions off of each other. Here are a few aspects that I will focus on as they work:
To end the lesson today, I want to come to some conclusions about how we can predict how many petals the graph has by looking at the equation.
Students usually notice that r = sin(2θ) has four petals while r = cos(3θ) + 2 has three petals. A student may hypothesize that if it is a sine equation then the number of petals is double the coefficient. Or a student may think that if it is an even coefficient there twice as many petals as the coefficient and if the coefficient is odd it is equal to the number of petals. If these conjectures do not come up I usually present them to the class and have them discuss.
Usually students will come to the correct conclusion by looking at other examples from the worksheet or playing around on their graphing calculator. One of my students made a very insightful observation when he said that r = cos(3θ) does have 6 petals, but they perfectly overlap on what is already present.