Maryland Math G.GPE.D.11

MathGrades 9–12EXPRESSING GEOMETRIC PROPERTIES WITH EQUATIONS

The Standard

Convert equations between polar and rectangular forms.

Maryland College and Career Ready Standards for Mathematics (2025) · Polar Equations.

Teacher's field guide

What Students Need to Do: Students use x = r cos θ, y = r sin θ, and r² = x² + y² to rewrite equations. They work in both directions and check that the equations describe the same graph.
What Mastery Looks Like: Given r = 4 cos θ, a student writes x² + y² = 4x. Given x² + y² = 9, they write r² = 9 and recognize the same circle.
Common Misconceptions: Students swap sine and cosine when replacing x and y. They may leave r cos θ unchanged, or divide by r and lose the origin from the graph.
How to Assess It: Exit ticket: Convert r = 6 cos θ to Cartesian form, then convert x² + y² = 16 to polar form. Show the substitution used for each.

Lesson moves

Plot points for r = 4 cos θ on polar graph paper, then mark their matching (x, y) coordinates on Cartesian paper.
Have partners explain on mini whiteboards why r = 4 cos θ becomes x² + y² = 4x rather than r = 4x.
Make cards labeled r = 3, r cos θ = 2, and r = 4 cos θ, then match them to Cartesian equation cards.
On a radar map, mark a 5-mile detection boundary around the station; write its polar and Cartesian equations.

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