Precalculus Lesson

Polar Coordinates & Curves: From Points to Petals

Shared by A STEM Educator

11th GradeBTC Style / Thinking Classrooms50 min

Learning Objective

Students will convert between polar and rectangular coordinates, apply symmetry tests to polar equations, and sketch classic polar curves by identifying key features including negative radii behavior and petal count rules.

Lesson Overview

Students begin by placing single polar points on a grid, then build through symmetry tests and negative radii to efficiently sketch limaçons, cardioids, rose curves, and lemniscates — culminating in petal-count analysis and a multi-curve overlay challenge.

Materials

Whiteboard or vertical chart paper
Dry-erase markers (multiple colors)
Pencil and paper
Basic calculator (phone OK)

Scaffolded Task Progression

1Task 1

A sonar contact is logged at

(r, θ) = (3, \frac{π}{3})

. Plot this point on a polar grid. Then convert it to rectangular coordinates

(x, y)

using

x = r cos θ

and

y = r sin θ

. Round to one decimal place.

[ Interactive graph / diagram, available in the full lesson ]

2Task 2

A second contact is logged at

(r, θ) = (- 3, \frac{π}{3})

. Plot this point. How does it compare to the point in Task 1? Write a rule for what

r < 0

does geometrically.

[ Interactive graph / diagram, available in the full lesson ]

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