When we looked at rotated conic sections in an earlier lesson, my students realized that the point (1,1) could also be thought of as a rotation of 45 degrees and a distance of sqrt(2) from the origin. I remind students of this on this worksheet and use this as an introduction to polar coordinates. I outline my teaching strategies in the video below.
Once students understand the premise behind polar coordinates, I will have them complete the table on the worksheet and brainstorm generalizations about the conversions from polar coordinates to rectangular and vice versa.
I encourage students to sketch out graphs for each one and to make right triangles. That will establish the strong connection to trigonometry. As students are working I will circulate and see how they are doing. Here is a list of things I will look for:
Once students have had time to brainstorm, I will have students share their thoughts about these conversions with the class. I find that they usually have an easier time converting from polar to rectangular, than from rectangular to polar. Here are the conversions we are looking for:
It is important that students realize that the tan(θ) = y/x conversion is not perfect; they will have to think about the quadrant that the point is in and adjust the point accordingly.
The remainder of the worksheet provides an opportunity for students to graph a few points on the polar axis and to predict what some simple polar equations will look like. I encourage students to list points that satisfy each equation if they do not know what it will look like. That is usually a helpful strategy for them to envision the graph. We will share out our thoughts after they have a chance to think about it.
Finally, I will assign some problems from our textbook with polar coordinates. I will stick to problems that focus on single points or very simple graphs. We will work on more complicated equations as we progress in this unit.