Navigating the Polar Plane

From Grid-Lines to Radial Horizons

Module 1: Foundations

Mission: Polar Battleship

How do we locate an object on a radar screen?

"Instead of 'go right 3, up 4', we say 'look at 53 degrees and travel 5 units out'."

Distance (r) from the origin (Pole)
Direction (\(\theta\)) from the 0° line (Polar Axis)

The Anatomy of a Polar Point

\( (r, \theta) \)

Unlike Cartesian coordinates, the radial distance \(r\) comes before the angular direction \(\theta\).

r: Directed distance from the Pole (origin).
\(\theta\): Directed angle from the Polar Axis (positive x-axis).

Pro Tip: Negative \(r\) values

If \(r\) is negative, move in the opposite direction of \(\theta\) (across the origin).

Plotting Example

\( P(3, \frac{2\pi}{3}) \)

The "Infinite Address" Problem

Every point in Cartesian space has exactly one name. In Polar space, every point has infinite names.

Add Full Rotations

Keep \(r\) the same, but spin around the circle.

\( (r, \theta + 2\pi k) \)

Switch the Sign of \(r\)

Flip \(r\) and rotate exactly half-way around.

\( (-r, \theta + \pi) \)

Find three other names for:

\( (2, \frac{\pi}{4}) \)

\( (2, \frac{9\pi}{4}) \)
\( (-2, \frac{5\pi}{4}) \)
\( (2, -\frac{7\pi}{4}) \)

Bridging Two Worlds

Linking Rectangular \((x, y)\) and Polar \((r, \theta)\)

Polar to Rectangular

\( x = \) \( r \cos \theta \)

\( y = \) \( r \sin \theta \)

Rectangular to Polar

\( r^2 = \) \( x^2 + y^2 \)

\( \tan \theta = \) \( \frac{y}{x} \)

Caution: Always check the quadrant of \((x, y)\) when finding \(\theta\)!

Polar Coordinate Systems and Curve Analysis

Navigating the Polar Plane

Graphing Basic Polar Equations

Symmetry and Roses

Limacons and Cardioids

Lemniscates and Spirals

Navigating the Polar Plane Slides

Polar Battleship Activity

Teacher Navigation Notes

Basic Polar Graphs Slides

Rectangular vs Polar Worksheet

Equation Transformation Key

Floral Geometry Slides

Rose Curve Investigation Sheet

Symmetry Tests Teacher Guide

Heart of the Loop Slides

Limacon Family Analysis Worksheet

Parameter Sliders Activity Guide

Infinite Curves Slides

Spirals and Infinity Worksheet

Polar Analysis Final Challenge

Similar Sequences

Slice Lab Fraction Division

dfredericks

Ratio Lab Intervention

dfredericks

Number Scouts Sequence

dfredericks

Statistiques Exploratoires

Hugo Pierrard

Division in Daily Life

Saiko Otake

Radical Relationships

Calculus Foundations

Transformation Surgeons

Sigma Summit

Solution Sleuths

Period Power Remediation

GCF Visual Remediation

Recursive Sequence Mastery

Angle Avengers

Slope and Symmetry