Limaçons and Roses - Day 1 of 2
Lesson 6 of 12
Objective: SWBAT identify characteristics of limaçons and roses and sketch their graphs.
Yesterday we spent some time converting equations from rectangular to polar form or vice versa on a purely algebraic level. Today we will be looking at graphs and thinking about how a rectangular trig graph can relate to its polar counterpart.
I start class by going over the Homework from yesterday. I plan to pay particularly close attention to Questions #1 and #2. One hurdle that will probably present itself is that students will think that the rectangular form of r = cos θ is y = cos x. The best proof that this is not true is that the graph of r = cos θ is a circle and y = cos x is a sine wave. If the graphs look different, then they can't be equivalent formulas. However, these graphs are related, so I want to leverage that fact while still maintaining that they are not equivalent equations. I will elicit responses for #1 and see how the class reacts to them until we are all in agreement.
Using Question #2, I want to drive home the point that the angle measure is the input of a polar equation and the r value is the output. I want my students to realize that we can simply plug in values to get information about the graph. Again, I will choose a student to share his/her table with the class.
After our conversation about yesterday's homework, I will give students this worksheet and explain that we are going to be spending some time looking at how a polar graph and its related rectangular graph are similar (or different). The first three questions on the worksheet let students explore r = 1 + cos θ and y = 1 + cos x.
Teacher note: I do not want my students to use calculators at all during this activity because I really want them to focus on how characteristics of a rectangular graph are present in the polar graph. I give them about 10-15 minutes to work on questions #1-3.
Here are a list of things I will be checking for as students work:
- The rectangular equation of the graph of r = 1 + cos θ is really complicated if we use our conversions from the last few days - it doesn't give us much information at all.
- Hopefully students will recall how to graph y = 1 + cos x. If they get stuck I ask them about the transformation that is present and what they remember about the regular cosine graph.
- Students should notice that the table for r = 1 + cos θ is identical to y = 1 + cos x; the ordered pairs will be the same but they are in two different coordinate planes.
- Their graphs should be connected with smooth curves and should not be jagged. They may have to add in more points to the table.
When students sketch the graph of r = 1 + cos θ, I will choose a student to share his/her work with the class. I will go over questions that other students have to make sure that everyone is on the same page. Again, I will stress that the rectangular form of r = 1 + cos θ is not y = 1 + cos x, but rather that complicated equation they found in Question #1. Next, I will introduce this graph as a special type of limaçon called a cardioid and relate its name to the word "heart."
After this first example, students are ready to match other rectangular and polar pairs of graphs. I will give them this set of 10 cards and have them work with their tables to match the polar graph with the related rectangular graph. The intent of this matching activity is to get students to recognize that the ordered pairs are identical in both forms, and for them to recognize that the characteristics of the rectangular graph affect the shape of the polar graph. I explain more in this video.
I give students about 10 minutes to complete the matching. I encourage them to make notes on the cards if they notice equivalent points or characteristics that are important. After this time, we will share out our pairs and give reasons for the matches.
One strategy during the sharing is to highlight the corresponding parts of the graphs on my interactive whiteboard for students to see. For example, if a student notices that the negative portion of the rectangular graph causes the inner loop of the polar graph, I will highlight both parts in the same color (like in this image) to show that they correspond. This is a great visual way for students to see the connection between the two graphs.
To bring this lesson to a productive close, I stress to my students that this approach of graphing the rectangular graph is an alternative to memorizing a litany of rules that dictate what the polar graph should look like. I believe that this approach empowers students by enabling them to reason through the process of graphing any polar equation.
For a homework assignment, students will finish up Questions #4-8 on the notes worksheet we started off class with. This will give students some more practice with the process of graphing polar equations and will allow them to compare the graph to a rectangular equation.