The coordinate plane

Builds on

Required prerequisites, all the way down: 5

You will learn to

• Plot and read points (x, y) with integer and rational coordinates in all four quadrants.
• Find horizontal and vertical distances between points using absolute value.
• Reflect points in the axes and describe the effect on their coordinates.

Definition

• The coordinate plane is formed by two perpendicular number lines, the x-axis and the y-axis, meeting at the origin (0, 0).
• The coordinates (a, b) of a point give its horizontal and vertical positions relative to the origin.

Theorem

• Reflecting (a, b) in the x-axis gives (a, −b), and in the y-axis gives (−a, b).
• The distance between (a, b) and (c, b) is |a − c|.
• Each point of the plane corresponds to exactly one ordered pair of real numbers.

Common misconceptions

• Swapping the coordinates, plotting (3, 2) at (2, 3).
• Counting grid lines rather than unit intervals when finding distances.

Leads to

Optional discoveries

• Linear algebra Points of the plane can be added and scaled as vectors, giving the vector space ℝ².
• Surfaces, polyhedra and the Euler characteristic In some video games, walking off the right edge of the screen brings you back on the left, and walking off the top brings you back at the bottom. If that world were a real surface, what shape would it be?Wonder Trail →