Experience First:

Today students continue yesterday’s exploration of polar graphs by looking specifically at limacons. They first graph equations with cosine in them and later equations with sine in them. They notice patterns about when the graphs make dimples, heart shapes, and loops, though their vocabulary to describe these might be different.

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Throughout the activity, the idea of maximum distance away, or maximum radius will become important. Many students noticed that the sum of the a and b values is the furthest point away from the pole, but were not able to explain why. Similarly, students saw that the a-value represented the y-intercept for cosine graphs and the x-intercept for sine graphs, but needed more prompting to explain what this had to do with the behavior of sine and cosine at the angles of 0, π/2, π, 3π/2, and 2π. Because the experience portion of the lesson is meant to spark curiosity and help students use inductive reasoning to form conjectures, it is okay to save some of these conversations for the debrief.

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Monitoring Questions:

• What would happen if the value of a was zero? 

• What about if b was zero? 

• How can you determine the length and location of a loop from the equation? 

• Of the three types of limacons, which one never touches the pole? Why?

• What is the maximum output of r=2-3cos(theta)? Why?

• What is the minimum output of r=2-3cos(theta)? Why?

• At which angles do the minimum and maximum radius values occur?

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Formalize Later:

Today’s lesson lends itself well to a whole-class discussion. Assign a reporter in each group that will share out the main take-aways or patterns observed from their group. In the debrief we use the round-robin protocol to collect these ideas on the whiteboard or poster paper. Each group shares one thing and we keep circling back to the groups until groups have no more unique contributions. Be ready to hear some things you may not have thought about yourself! My students always surprise me with their observations! 

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Our main goal in this lesson is not that students draw perfect limacon graphs without a calculator/graphing tool but that they would be able to connect key features of a polar graph with its equation. While we do have students practice graphing limacons by hand we also give plenty of short answer questions or have students describe key features of the graph without actually drawing it.

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Try to transition students to function language during the debrief to describe the function’s behavior, instead of just referring to the equation. Symmetry, maximum distance from the pole, and intercepts are not just cool graph features, they are intrinsically linked to the key characteristics of the function. Pay particular attention to the range of a polar function that represents a limacon. Understanding this function feature requires a robust understanding of the behavior of sine and cosine.