Practice Maths

Cartesian Plane: Plotting

Plotting Points

Key Terms

x-axis
the horizontal number line on the Cartesian plane
y-axis
the vertical number line on the Cartesian plane
origin
the point (0, 0) where the x-axis and y-axis cross
coordinate
a pair of numbers written as (x, y) that describes the exact position of a point on the plane
quadrant
one of the four regions of the Cartesian plane, formed when the x-axis and y-axis divide the plane

The Grid

The Cartesian plane is made of two number lines that cross at zero. We call this center point the Origin (0, 0).

  • The horizontal (left-right) line is the x-axis.
  • The vertical (up-down) line is the y-axis.
x y Quad 1 Quad 2 Quad 3 Quad 4

How to Plot Coordinates (x, y)

Coordinates are always written as a pair in brackets: (x, y).

  • First number (x): Move left (negative) or right (positive).
  • Second number (y): Move down (negative) or up (positive).

Mnemonic: You must walk along the hall (x) before you can take the elevator up or down (y).

Hot Tip: Zeroes
If a coordinate has a 0, the point sits directly on a line.
(5, 0) sits on the x-axis.
(0, 5) sits on the y-axis.

Worked Example

Plot the point (3, −2) on the Cartesian plane.

Step 1 — Start at the origin (0, 0).

Step 2 — The x-value is 3 (positive), so move 3 units to the right along the x-axis.

Step 3 — The y-value is −2 (negative), so move 2 units down from that position to reach the point (3, −2). Mark the point.

Remember: always move horizontally first (x), then vertically (y).

The Man Who Watched a Fly on the Ceiling

The Cartesian plane was invented by a French mathematician called René Descartes in the 1600s. The story goes that he was lying in bed, sick, watching a fly crawl across the ceiling. He realised he could describe exactly where the fly was at any moment by saying how far across and how far up it was from the corner of the ceiling. That simple insight created an entire branch of mathematics!

Today, every GPS system, every Google Maps location, every video game character position, and every graph you'll ever see in science uses Descartes' idea. When your phone tells you you're at a certain location, it's giving you coordinates — exactly like (x, y) on the Cartesian plane.

The "Corridor and Stairs" Rule

The most important rule in plotting: always move horizontally first, then vertically. Here's a memorable way to think about it:

Walk along the corridor (x-axis), then take the stairs up or down (y-axis).

The point (4, −2) means: walk 4 units right along the corridor, then go 2 floors down the stairs. If you go to the stairs first, you'll end up in the wrong room!

Negative Coordinates: The Four Quadrants

The x-axis and y-axis divide the plane into four regions called quadrants. Using the sign of the coordinates, you can identify which quadrant any point belongs to — without even drawing it:

  • Quadrant 1: x positive, y positive → upper right (+, +)
  • Quadrant 2: x negative, y positive → upper left (−, +)
  • Quadrant 3: x negative, y negative → lower left (−, −)
  • Quadrant 4: x positive, y negative → lower right (+, −)

If either coordinate is 0, the point sits exactly on an axis — not inside any quadrant.

Remember: (5, 0) is on the x-axis and (0, 5) is on the y-axis. A zero in a coordinate always means "on an axis." The origin (0, 0) is the only point where both axes meet.

Coordinates in Real Life

Think about any board game that uses a grid — Battleship, chess, or Snakes and Ladders. When you say "B6" in Battleship, you're giving coordinates! The letter is the x-position and the number is the y-position. The Cartesian plane formalises this idea into a powerful mathematical tool used by architects, engineers, game designers, and scientists every day.

Common Mistake: Plotting (3, 5) as if it were (5, 3). The x-coordinate always comes first. One useful memory trick: in the alphabet, x comes before y. In coordinates, x is written first. Always go across before going up.

Practice Questions

  1. Reading Coordinates Fluency

    Look at the grid below. Write the coordinates for points A, B, C, and D.

    x y 2 4 6 -2 -4 -6 2 4 6 -2 -4 -6 A B C D
    1. Point A
    2. Point B
    3. Point C
    4. Point D

  2. Identifying Quadrants Fluency

    Without drawing a grid, state which Quadrant (1, 2, 3, or 4) these points are in:

    1. (5, 2)
    2. (−3, −6)
    3. (−4, 7)
    4. (2, −5)

  3. Points on Axes (Zeroes) Fluency

    State whether these points lie on the x-axis or the y-axis:

    1. (0, 8)
    2. (−5, 0)
    3. (0, −3)

  4. The Origin Fluency

    What are the coordinates of the Origin?

  5. Halves and Decimals Fluency

    Imagine a number line. Describe where the point (2.5, −1.5) would be:

    1. Between which two whole numbers is the x-value?
    2. Between which two whole numbers is the y-value?

  6. Creating Shapes Understanding

    If you plotted the following points and connected them in order, what shape would you make?

    Points: (2, 2), (2, −2), (−2, −2), (−2, 2) and back to (2, 2).

  7. Navigation Understanding

    Start at the point (1, 1).

    1. Move 3 units to the left. What is your new x-coordinate?
    2. From there, move 4 units down. What is your new y-coordinate?
    3. What are the final coordinates?

  8. Theme Park Map Understanding

    Use the Theme Park Map below to answer the questions. Each grid line represents 1 unit.

    x y 2 4 6 -2 -4 -6 2 4 6 -2 -4 -6 R W G C

    Key: R = Rollercoaster, W = Water Slide, G = Gardens, C = Cafe

    1. What are the coordinates of the Rollercoaster (R)?
    2. What are the coordinates of the Gardens (G)?
    3. Which attraction is in Quadrant 4?
    4. You are at the Water Slide (W). You walk 11 units Right and 6 units Down. Where do you end up?

  9. Battleship Logic Problem Solving

    You are playing a game of Battleships.

    1. You hid your submarine at coordinates (−3, 4). Your opponent fires a missile at (−3, 5). Did they hit you?
    2. Your opponent has a ship that is 3 units long. One end is at (2, 2) and the other end is at (2, 5). If you fire at (2, 4), is it a HIT or MISS?

  10. Linear Pattern Plotting Problem Solving

    Look at this table of values:

    x012
    y135
    1. Write these as coordinates: (0, 1), (1, 3), ...
    2. If you plotted these on a graph and connected them, would they form a straight line or a curved line?
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