Katharine Sparks WILLIAM CHRISMAN HIGH, INDEPENDENCE, MO
12th Grade Math : Unit #12 - Parametric and Polar graphs and equations : Lesson #4

What is the connection between trigonometric graphs and polar graphs? (Day 1 of 2)

Objective: SWBAT use a trigonometric graph on the rectangular coordinate plane to graph a polar graph.
Standards: HSF-IF.C.7 HSF-IF.C.9 MP1 MP4 MP6 MP7
Subject(s): Math
60 minutes
1 Teacher note on lesson development - 1 minutes

My instruction on polar graphing was developed after reading an article 

As I read this article, I realized that this process was exactly what I needed to help my students connect the concepts from the unit Graphing Trigonometric Functions and polar graphing.  

Textbooks usually have students graph by making tables, finding zeros, and looking for symmetry. However, the method used in this lesson has students use prior knowledge to find the same concepts. 

2 Bell work - 5 minutes

The lesson begins with students graphing a function as done in the unit Graphing Trigonometric Functions.  I give the students a few minutes to make a graph and then ask a student to put the graph on the board.

I quickly review the the key concepts of translations, amplitude, period, and reflection by asking:

  • How would the equation change the graph was shifted up 2 units?
  • How would the equation changed if the graph was shifted pi/3 to the left?
  • How would the graph change if the graph reflected across the x-axis?
  • How would the graph change if the graph had 3 periods from 0 to 2 pi?

This review helps students remember graph transformations, allowing me to review any unfamiliar concepts.

3 Connecting trigonometric graphing with polar graphing - 15 minutes

I am now ready to help students see the connections between trigonometric graphing and polar graphing.

I begin by adjusting the equation a little for them. I explain that y=r and x=theta.  I have now changed the graph from a rectangular equation to a polar equation. This is not a conversion like we did in yesterday's lesson, but this will help us with graphing.

"If I change the axises so that what is usually y is now r and x is now theta, will this change how the graph looks?". Such a question reinforces to students that the labels do not change the graph. 

I re-draw the graph and label the vertical axis with r and the horizontal axis with theta.  I now explain how polar graphing is similar to taking the horizontal axis and rotating it around the origin.  We can look at the graph to determine how to polar graph. (See graph)

Using the graph, we examine the points for theta equaling 0, pi/2, pi, 3pi/2 and 2pi. As we move along the graph I ask the students if the graph is positive of negative. We connect the points so that the graph begins at polar coordinate (0,0) and goes through (3, pi/2). I ask "is the graph linear? How can you tell? Should we make a line or a curve between these 2 points?".

I move from (3, pi/2) to (0, pi) asking the same questions from above.  Students will see that when we move to 3pi/2 we have a negative r. "What does a negative r tell us? How will we graph it?". I point out how the graph is below the theta axis so that all the points between (0,pi) and (-3, 3pi/2) are negative.

See the  example for how I work through this example. I use different colors so so that the students can see how the graph is moving. I ask the students why the graph is actually complete after a half period?" Students think about symmetry and how the points with a negative 4 are in the same place as the points with a positive r.  This is a great example of how symmetry with the origin will result in this type of graph.  I remind students that sine is an odd function and has symmetry with the origin which causes this to happen.

I now ask the students what would the graph of r=3cos(theta) look like.  I know that some students will realize that the graph will show the circle's diameter on the polar axis. We also discuss what happens if the r is negative.

4 Using technology to verify the result. - 10 minutes

Towards the end of this lesson, I want my students to verify their results and predictions by using technology. I explain to my class how to put the calculator into polar mode. On the TI-84 you go to "mode" and change to polar. We look at the window on the calculator and see what is different. I do make sure the students have their calculators in radians. We then graph r=3cos(theta). As the students graph we review the window settings. 

I will ask "Do we need to graph from [0,2pi]?". In this case when we look at the rectangular graph the students see that we have all the points on the polar graph from [0,pi].

 

5 Closure - 5 minutes

As the class ends, students can experiment with their calculators.  I give them a chance to practice graphing while displaying their results on a Desmos graph to share with everyone. 

The students also begin working on the Polar Graphing packet.  Each problem has a place for students to graph in both rectangular and polar. I have printed off several different types of polar graph paper (in the resource section of bell work) that can be used. However, students can use any paper that works for them.

As the students prepare to leave I ask them how to determine the line of symmetry by looking at the equation.  Students will need to think about whether the function is written in terms of sine or cosine.  When it is written in terms of sine, there will be symmetry along the pi/2 axis. If it is written in terms of cosine, there will be symmetry along the polar axis.

Example of Student graph
Joy

Here is a student example of a polar graph depicting what she called a "spider". The student was very excited to see this graph produced.  After displaying her graph to the class, another student added that he wanted to add a rose on the back of the spider. This student added the equation and together they both adjusted the equation until they agreed on the look of the spider and rose.

Other students really liked adjusting the equation for a rose. One student commented that he had made the sun and then showed others his calculator. The students really liked being able to create very different graphs as they worked.

Creating a design would be a great extension to this unit when time allows.  There are several activities like this on the web: site 1 and site 2.