Maryland Math G.GPE.D.13

MathGrades 9–12EXPRESSING GEOMETRIC PROPERTIES WITH EQUATIONS

The Standard

Solve systems of polar equations.

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

Teacher's field guide

What Students Need to Do: Students graph two polar curves and find the points they share. They solve the equations and check for intersections that may have different polar coordinates, including the pole.
What Mastery Looks Like: A student finds both intersections of r = 2 and r = 4 cos θ: (2, π/3) and (2, 5π/3). They check the points on a sketch and know to test the pole separately.
Common Misconceptions: Students may set the radius expressions equal and assume they have found every intersection, missing the pole. They may treat (−2, π/3) and (2, 4π/3) as different points, though they name the same point.
How to Assess It: Ask: Find every intersection of r = 2 and r = 4 cos θ for 0 ≤ θ < 2π. Give exact polar coordinates and check them on a sketch.

Lesson moves

Trace r = 3 and r = 6 sin θ on separate transparent polar grids, stack them, and mark their shared points.
Ask pairs to draw (−2, π/3) and (2, 4π/3), then write why the points land in the same place.
Give pairs cards labeled r = 2, r = 4 cos θ, and r = 2 sin θ; draw two, find intersections, and compare sketches.
Draw a radar map with range r = 2 km and a signal boundary r = 4 cos θ km; mark where their boundaries cross.

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