Math Vale

🏰 Geometry Kingdom · Geometry

The Coordinate Plane

Name every point on the plane with an ordered pair, read and plot points in all four quadrants, and measure along the grid lines between them.

In short

  • An ordered pair (x, y) is an address measured from the origin: across first, then up. (3, 5) and (5, 3) are different points.
  • The signs alone name the region: (+, +) is Quadrant I, (-, +) is II, (-, -) is III, (+, -) is IV, and a coordinate of 0 puts the point on an axis instead.
  • When two points share a coordinate, the distance between them is |larger - smaller| — never negative, and the two counts add when the points straddle zero.
  • Reflecting flips a sign and nothing else: across the x-axis gives (x, -y), across the y-axis gives (-x, y), and a half turn about the origin gives (-x, -y).

Two number lines, one address

A single number line gives every point one number. Lay a second number line across the first, at right angles and crossing at zero, and now every point on the whole flat surface has an address made of two numbers.

The horizontal line is the x-axis. The vertical line is the y-axis. Where they cross is the origin, written (0, 0). Everything is measured from there.

An address is written as an ordered pair: (x, y). The first number says how far to go across, the second says how far to go up. "Ordered" is the whole point — (3, 5) and (5, 3) are two different places.

Say it out loud as you move: across, then up. Crawl before you climb. That one habit prevents most of the errors on this page.

Reading and plotting a point

To read a plotted point, start at the origin and take the journey in two straight legs.

  • Leg one: move along the x-axis until you are directly below or above the point. Right is positive, left is negative. That count is x.
  • Leg two: move straight up or down to the point. Up is positive, down is negative. That count is y.

To plot the pair (-4, 2), do the same thing in reverse: begin at the origin, count 4 to the left, then 2 up, and mark the spot.

Negative coordinates are not a special case; they are the same instruction pointing the other way. The minus sign is a direction, not a difficulty.

A point with a coordinate of 0 sits on an axis. (7, 0) is on the x-axis, because it never left it. (0, -3) is on the y-axis.

The four quadrants

The two axes cut the plane into four regions called quadrants. They are numbered with Roman numerals, starting at the top right and going anticlockwise:

  • Quadrant I: x is positive, y is positive — (+, +), top right.
  • Quadrant II: x is negative, y is positive — (-, +), top left.
  • Quadrant III: both negative — (-, -), bottom left.
  • Quadrant IV: x is positive, y is negative — (+, -), bottom right.

You never have to plot a point to name its quadrant. The two signs are enough: (-6, -1) has two minus signs, so it is in Quadrant III.

The axes themselves belong to no quadrant. A point with x = 0 or y = 0 is on an axis, and the honest answer to "which quadrant?" is "none of them — it is on the y-axis".

Distance along a grid line

When two points share a coordinate, the path between them runs straight along a grid line and you can measure it by subtraction alone.

Same y-coordinate means a horizontal segment: only the x-values differ, so the length is the gap between them. Same x-coordinate means a vertical segment, measured the same way with the y-values.

Distance is how far apart, never which way round, so it is never negative. Take the larger coordinate minus the smaller one, which is exactly what the absolute value bars mean: distance = |x2 - x1|.

That matters most when the two points are on opposite sides of zero. From (-4, 3) to (9, 3) the distance is 9 - (-4) = 13, not 5. Walk in from -4 to 0 (4 units), then out from 0 to 9 (9 units), and the two counts add.

Once you can measure the sides, a rectangle drawn on the grid gives up its perimeter and area: find the width from the two x-values, the height from the two y-values, then use 2 x (width + height) or width x height.

Reflections: folding the grid

Imagine the plane is a sheet of paper and an axis is a crease. Fold along it, and every point lands on its image — the same distance from the crease, on the other side.

  • Fold along the x-axis and nothing moves left or right, so x is unchanged and y changes sign: (x, y) goes to (x, -y).
  • Fold along the y-axis and nothing moves up or down, so y is unchanged and x changes sign: (x, y) goes to (-x, y).
  • A half turn about the origin is one fold followed by the other, so both signs change: (x, y) goes to (-x, -y).

Only the sign changes. The size of each coordinate stays exactly the same, because the image is the same distance from the crease as the original.

Check every image by asking one question: is it as far from the fold line as the point I started with, and on the opposite side? If it is not, the wrong coordinate was flipped.

Worked examples

Example 1

A marker sits 3 units left of the origin and 5 units below it. Write its ordered pair, and name the quadrant it lies in.

  1. Start at the origin and travel across first. Left is the negative direction, so x = -3.
  2. Now travel up or down. Below is the negative direction, so y = -5.
  3. Across first, up second: the marker is at (-3, -5).
  4. Both coordinates are negative, which is the (-, -) corner, so the marker lies in Quadrant III.

Example 2

Two survey posts stand at A(-6, 2) and B(7, 2). How far apart are they?

  1. Both posts have y = 2, so the line between them is horizontal and only the x-values matter.
  2. Distance is never negative, so subtract the smaller x from the larger: 7 - (-6).
  3. Subtracting a negative adds: 7 + 6 = 13.
  4. The posts are 13 units apart. Check by counting: 6 units from -6 in to 0, then 7 units out to 7, and 6 + 7 = 13.

Example 3

Three corners of a rectangle with sides along the grid lines are (2, 1), (9, 1) and (9, 6). Find the fourth corner, then the perimeter of the rectangle.

  1. A rectangle with sides along the grid lines uses only two different x-values and two different y-values: here x is 2 or 9, and y is 1 or 6.
  2. The value 2 appears once among the given corners, and so does 6, so the missing corner is (2, 6).
  3. Width = 9 - 2 = 7. Height = 6 - 1 = 5.
  4. Perimeter = 2 x (7 + 5) = 24 units. (The area, for comparison, is 7 x 5 = 35 square units.)

Example 4

Reflect the point (4, -3) across the y-axis.

  1. The fold line is the y-axis, so the fold moves the point left and right, never up and down.
  2. The y-coordinate is untouched: y stays -3.
  3. The point is 4 units right of the fold, so its image is 4 units left of it: x goes from 4 to -4.
  4. The image is (-4, -3). It is the same distance from the y-axis as the original, on the opposite side, so it checks out.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

Master Pytha has pinned the marker M to the survey grid. Write the ordered pair for M.

Answer: (2, 4)

  1. From the origin, M is 2 units to the right, so x = 2.
  2. From there, M is 4 units up, so y = 4.
  3. Across first, up second: M = (2, 4).

Problem 2

Difficulty 3 of 5

In which quadrant does the point (7, 8) lie?

  1. Quadrant I
  2. Quadrant IV
  3. Quadrant III
  4. Quadrant II

Answer: A. Quadrant I

  1. x = 7: right of the y-axis.
  2. y = 8: above the x-axis.
  3. (7, 8) lies in Quadrant I.

Problem 3

Difficulty 4 of 5

Two survey posts stand on the same vertical grid line: A(-1, -10) and B(-1, 6). How many units of rope reach from A to B?

Answer: 16 units

  1. Both points have x = -1, so the rope is vertical.
  2. The y-coordinates are -10 and 6.
  3. Distance = |6 - (-10)| = 16.
  4. A and B are 16 units apart.

Common mistakes

  • Writing the pair the wrong way round — plotting (2, 7) as 7 across and 2 up. Across always comes first.
  • Dropping a minus sign when reading a point in Quadrant II, III or IV, and answering (3, 4) for a point that is really at (-3, 4).
  • Calling a point on an axis a quadrant point. (0, -5) is on the y-axis and belongs to no quadrant at all.
  • Reporting a distance as a - b instead of |a - b|, so a length comes out negative, or subtracting the sizes of two coordinates on opposite sides of zero and getting far too small an answer.
  • Flipping the wrong coordinate in a reflection: folding along the x-axis changes y, not x.

What you should be able to do

  • Plot and read ordered pairs in all four quadrants, and name the quadrant or axis a point lies on.
  • Find the distance between two points that share an x- or y-coordinate.
  • Reflect a point across an axis and describe how its coordinates change.
  • Find the perimeter or a missing vertex of a rectangle drawn on the grid.

Where this fits in the curriculum

Common Core

  • 5.G.A.1

    Grade 5 — Use a pair of perpendicular number lines to define a coordinate system, and understand how the two numbers of an ordered pair locate a point.

  • 5.G.A.2

    Grade 5 — Represent real-world and mathematical problems by graphing points in the first quadrant and interpret coordinate values in context.

  • 6.NS.C.6.C

    Grade 6 — Find and position integers and other rational numbers on a number line and pairs of them on a coordinate plane.

  • 6.NS.C.8

    Grade 6 — Solve problems by graphing points in all four quadrants, and use coordinates and absolute value to find distances between points with the same first or second coordinate.

Ontario

  • G6.E1.3

    Grade 6 — Plot and read coordinates in all four quadrants of a Cartesian plane, and describe the translations that move a point from one coordinate to another.

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