Over the next few lessons I will be working with an idea is not specifically discussed in the Common Core standards. By teaching the polar coordinate system and its equations I am able to apply many of the previous topics we learned in new and insightful ways. For example, we will review trigonometric concepts, such as trigonometric identities and real valued functions with points on the coordinate plane, when learning the polar coordinate system.
I begin today's lesson by reviewing how to graph points when of real-valued functions. Polar graphing has a very similar structure, the only aspect that is different is that the ordered pair is (r,theta) instead of (x,y)
To begin, I have one student in the class give an example by graphing the point. I will then review with the students any idea they may have forgotten since the real-valued unit last semester.
I now share the definition provided in the Larson textbook "Precalculus with Limits, 2nd ed." We analyze together the definition that written in the book. My goal is for students to see that you can relate a rectangular coordinate plane with a polar coordinate system. I give every student a copy of a polar coordinate (These are shared at the end of this section.)
Some questions I ask students about the polar graph are:
After examining a plane graph, I have the students practice by graphing some points. I spend time on points that have negative r values since this is usually one of the most challenging parts of a polar coordinate system. I review the concept of how the negative value of an x or y coordinate reveals the direction the point is from the origin. A negative value on the r coordinate also gives the direction of the point.
Once the students see how to graph a point, I put a point in polar form on the board and ask the students to find another way to label the point. I let the students consider this question and discuss what a possible solution. I put the students' ideas on the board and ask each student to explain how they arrived at their answer. If a student does not use a -r, then I will ask what would the label be if the r is negative.
Lastly, I ask "How many ways can we label a specific point in polar coordinates?" Inevitably, I want students to realize that there are infinite many ways. It is important to reinforce that the value is not just the angle to 2pi.
Overall, students usually find graphing a point straightforward, allowing me to move forward to the topic of converting points between polar and rectangular form. I put a point that is in polar form on the board and label the known facts about the point. I also remind students that this process will look similar to what we did when we worked with the points on the unit circle.
Using the points, my students will consider making a right triangle. Once they do, I ask them how we will find the x and y coordinates. This is just right triangle trigonometry and the students should be able to find the x and y coordinates easily.
Once the students have done a problem with numbers I generalize the process and develop the formulas x=r cos (theta) and y= r sin (theta). Understanding these two formulas will help when the students convert equations between rectangular and polar forms. When we do this conversion and find the formulas for x and y, I make a comment about how these formulas look like parametric equations. I will say something like, "notice how x and y are dependent on the angle and r. If I know the value of r then I can write a parametric equation. Or if I know the value of theta then I can write a parametric equation. This is interesting."
After rewriting ordered pairs from polar to rectangular I ask the students how to convert in the opposite direction, rewriting rectangular to polar forms. I give the students the point (-1,1). I usually let the students have a minute or so to think about how this can be done. Students will usually plot the point. Next, my students will again make a triangle and find the parameter r by using the Pythagorean Theorem. We discuss how to find the angle. I use the point (-1,1) because most students know this will have a reference angle of pi/4.
Again we generalize this concept and find the formulas for determining the parameter r and theta.
This entire process is a great review of the unit on real-valued trigonometric functions, without actually reviewing. The students are problem solving and deepening their knowledge with concepts that are strongly connected. In this section of the lesson I have connected the parametric equations to polar and rectangular which is interesting idea for students. Seeing this connection is important for topics in physics and calculus.
The students now have a process to rewrite points in polar and rectangular form. These formulas can also be used to rewrite equations.
I give the students a rectangular form equation for a circle to convert to polar form. To help guide my students, I also rewrite the formulas we just developed in the previous section of this lesson so that they are clearly displayed on the board. Since we practiced converting the parametric equation in the last lesson, students can think about how we can use this previous concept to convert the polar equation.
To practice, we replace the x and y with their equations. It is always interesting to show the students how to simplify the equation.
The class then works to convert a polar equation to rectangular form.
Most students find this process straightforward since it involves substitution, equivalent forms of the variables or functions, and then simplifying the expression. The problems students generally have with these practice problems are algebra errors.
The students are given time to work on a few problems from this section. I assign page 781 #6, 8, 10, 20, 24, 26, 38, 46, 50, 66, 74, 75, 86, 96, 100 from Larson, "Precalculus with Limits"
These problems practice graphing and converting equations between polar and rectangular form.