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Equations in Polar and Cartesian Form (Lesson 5.6)

Unit 5 - Day 9

​Learning Objectives​
  • Use the conversion formulas to rewrite equations of graphs in their alternate forms.

  • Recognize that some graphs are more easily described in polar coordinates whereas others are more easily described in Cartesian coordinates

Quick Lesson Plan
Activity: Converting Equations Card Sort



Card Sort Handout

Card Sort Key


Recording Sheet

Experience First

We spent the first ten minutes of class reviewing the big ideas from yesterday’s lesson, since it was the first day they had ever seen polar coordinates. We gave students the prompt: “Suppose a classmate was absent yesterday. How would you describe to them what polar points are?” First, we gave students 5 minutes of individual writing time. Then we had them share their response to their group members and make any edits they wanted to make. We then had each group share one response back to the whole group.


For the next activity, you will need to print the card sort that contains 6 graphs, 6 equations in rectangular coordinates and 6 equations in polar coordinates. We printed each set on a different color. Each group will need one set of cards and they will work together to find matching trios. Additionally, give each student a recording sheet where they will keep track of their matches. They will then work on the final column which requires showing algebraically why the polar and Cartesian equations are equivalent. It may be helpful to have the conversion formulas up on the board or in a place that’s easily visible for students.


A Desmos version of this card sort can be found here. The answer key is embedded.

Formalize Later

On the back of their recording sheets or in a notebook have students identify and write down the major strategies they used to match the cards. An important take-away is that equations can be manipulated so that they match the conversion formula, and then can be substituted for the variable in the other form.     As time permits, allow students time to work on homework or other practice problems in their small groups, applying their new strategies.


We found that our students struggled most when converting polar equations like θ=π/3 into rectangular. Many just gave the ordered pair on the unit circle at that angle. Two main approaches can be used here. I like to have students graph the set of points where the angle is π/3, regardless of what the radius is. They recognize that the set of these points represents the graph of the equation and it is a line passing through the origin that makes a 60˚ angle with the x-axis/polar axis. We talk about how we are used to writing lines in y=mx+b form, and then we go about finding the y-intercept (0) and the slope. We talk about the fact that rise/run is simply the ratio of the opposite over the adjacent side, which can be found by taking the tangent of the angle.


Other students use the conversion formula tan θ=y/x and then plug in the angle of π/3 and solve for y. While this is of course true, we find that it obscures the idea of trying to represent the same graph in two different coordinate systems, and students tend to focus only on the procedure instead of the larger goal.

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