Math Core

How do you tell a friend exactly where something is on a map? You can give two numbers: how far to go across and how far to go up. The coordinate plane is a grid that lets you name any spot with just two numbers.

Two number lines

A coordinate plane is made of two number lines that cross at a right angle.

The x-axis runs across, left to right.
The y-axis runs up and down.
The point where they cross is called the origin. It sits at 0 on both number lines.

Every point on the grid gets a pair of numbers called its coordinates.

Definition

Ordered pair

An ordered pair $(x, y)$ tells you where a point is.

The first number, the x-coordinate, tells how far to move across (right) from the origin.
The second number, the y-coordinate, tells how far to move up from the origin.

The origin is $(0, 0)$ .

The order matters, which is why it's called an ordered pair. The point $(2, 5)$ is not the same place as $(5, 2)$ .

Plotting and reading points

To plot the point $(2, 3)$ , start at the origin. Move 2 units to the right along the x-axis. Then move 3 units up. Put a dot there.

To read a point, do the same steps backward. Look straight down from the point to the x-axis to find the first number. Look straight across to the y-axis to find the second number.

A is at (2, 3), B at (5, 1), C at (0, 4) and D at (6, 0).Open in grapher →

Look at points C and D. Point C has an x-coordinate of 0, so you don't move right at all. It sits on the y-axis. Point D has a y-coordinate of 0, so you don't move up. It sits on the x-axis.

Across first, then up

In $(x, y)$ , go across first, then up. A helpful memory trick: you walk into a building before you take the elevator up.

Worked example: Read the coordinates

In the graph above, what are the coordinates of point B?

Start at the origin. Point B is straight above 5 on the x-axis, so $x = 5$ . It is straight across from 1 on the y-axis, so $y = 1$ .

Point B is at $(5, 1)$ .

Distance along a grid line

When two points have the same y-coordinate, they are on the same horizontal line. Subtract the x-coordinates to find how far apart they are. When two points have the same x-coordinate, subtract the y-coordinates.

Worked example: How far apart?

How far apart are the points $(2, 6)$ and $(7, 6)$ ?

Both points have a y-coordinate of 6, so they are on the same horizontal line. Subtract the x-coordinates: $7 - 2 = 5$ .

They are 5 units apart.

Worked example: A walk around town

On a town map, each unit is one block. Nia's home is at $(1, 2)$ . The library is at $(1, 7)$ . The pool is at $(6, 7)$ . Nia walks from home to the library, then to the pool, along the streets. How many blocks does she walk?

Home to library: the x-coordinates match, so subtract the y-coordinates. $7 - 2 = 5$ blocks.

Library to pool: the y-coordinates match, so subtract the x-coordinates. $6 - 1 = 5$ blocks.

Total: $5 + 5 = 10$ blocks.

Common mistake

Don't flip the numbers. $(4, 1)$ means 4 across and 1 up. If you go up 4 and across 1, you land on $(1, 4)$ , which is a different point.

Practice

Use this graph for the first three problems.

Points P, Q, R, S and T.Open in grapher →

Practice 1

What are the coordinates of point P?

Enter a point like (2, -3)

Practice 2

What are the coordinates of point Q?

Enter a point like (2, -3)

Practice 3

Which point is on the y-axis?

Practice 4

To plot the point $(7, 2)$ , you start at the origin. What do you do next?

Practice 5

How many units apart are the points $(3, 2)$ and $(3, 9)$ ?

Type a number.

Practice 6

A bug starts at the point $(4, 3)$ . It crawls 3 units to the right and then 2 units up. What are the coordinates of the point where it stops?

Enter a point like (2, -3)

Practice 7

On a map where each unit is one block, Maya's house is at $(2, 1)$ , the park is at $(2, 6)$ and the store is at $(8, 6)$ . Maya walks from her house to the park, then from the park to the store, along the streets. How many blocks does she walk in all?

Type a number.