Ordered Pairs on the Coordinate Plane | Grade 5 Math | LevelKite

A coordinate plane is a flat surface with two number lines on it. The number lines cross at a right angle. Each one is called an axis, and the two together are called axes.

The x-axis runs across, from left to right. Its numbers get larger as you move right.
The y-axis runs up and down. Its numbers get larger as you move up.
The origin is the point where the two axes cross. It is at $0$ on both number lines.

In this lesson, grid lines are 1 unit apart, so you can count units as you move away from the origin.

To say exactly where a point is, you use an ordered pair: two numbers inside parentheses with a comma between them, such as $(4, 2)$ . The two numbers are the point's coordinates.

The first number is the x-coordinate. It goes with the x-axis.
The second number is the y-coordinate. It goes with the y-axis.

The names match the axes, and the order matches the alphabet: x comes before y. The origin is at $0$ on both axes, so its ordered pair is $(0, 0)$ .

Key concept

An ordered pair gives directions from the origin

An ordered pair tells you how to get from the origin to its point. The x-coordinate says how many units to move right, along the x-axis. The y-coordinate then says how many units to move up, in the direction of the y-axis. A $0$ means you do not move that way.

Marking a dot where these moves end is called plotting the point. To read the pair of a point that is already drawn, go right until you are directly below the point, then go up to it. Count the units each way.

These moves work for every pair in this lesson, because every coordinate here is $0$ or greater. Each number's place gives it its job, so when the two numbers are different, switching them names a different point.

(\underbrace{x}_{\text{right}},\ \underbrace{y}_{\text{up}})

Start at the origin

(0, 0)

, move right 4 units to the point labelled 4, then move up 2 units to Point A.

Worked example

Read the coordinates of Point A

What ordered pair names Point A in the figure above?

In the figure, the x-axis is the bottom edge and the y-axis is the left edge. Start at the origin, labelled (0, 0), where the axes meet. Every ordered pair counts its moves from this point.
Move across first. Go right along the x-axis until you are directly below Point A. You stop at the point labelled 4 after 4 units, so the x-coordinate is 4.
$x = 4$
Then go straight up from that spot until you reach Point A. That takes 2 units, so the y-coordinate is 2.
$y = 2$
Write the x-coordinate first and the y-coordinate second, because an ordered pair always lists them in that order.
$(x, y) = (4, 2)$

Answer

Point A is at $(4, 2)$ , which is 4 units right of the origin and 2 units up.

Worked example

Plot (3, 5) and (5, 3)

Plot the points $(3, 5)$ and $(5, 3)$ . Do the two ordered pairs name the same point?

For (3, 5), start at the origin. Move 3 units right, because the x-coordinate is 3. Then move 5 units up, because the y-coordinate is 5. Mark a dot where you stop.
$(3, 5) \rightarrow \text{right } 3,\ \text{up } 5$
For (5, 3), start at the origin again. This time the x-coordinate is 5 and the y-coordinate is 3, so move 5 units right, then 3 units up. Mark a second dot.
$(5, 3) \rightarrow \text{right } 5,\ \text{up } 3$
Compare the two dots, shown in the figure below. The dot for (3, 5) is 2 units farther left and 2 units higher than the dot for (5, 3). Both pairs use 3 and 5, but since 3 and 5 are different numbers, switching them changes both moves.
$(3, 5) \neq (5, 3)$

Answer

The pairs name different points: $(3, 5)$ is 3 units right and 5 units up, while $(5, 3)$ is 5 units right and 3 units up.

Same two numbers, different points:

(3, 5)

sits higher and farther left than

(5, 3)

.

Ordered pairs can name real places, too. A first-quadrant map is a map drawn on the first quadrant: the part of the coordinate plane above the x-axis and to the right of the y-axis. The two axes form the map's bottom and left edges.

The map below shows part of a town. The train station is at the origin, $(0, 0)$ , and each grid unit stands for 1 block. Point A is the library, Point B is the pool and Point C is the school. On this map, a place's x-coordinate tells how many blocks it is to the right of the station. Its y-coordinate tells how many blocks it is above the station. For example, the library is at $(2, 6)$ : 2 blocks to the right of the station and 6 blocks up.

Key: Point A is the library, Point B is the pool, Point C is the school, and the train station is at

(0, 0)

.

Worked example

Find places on the town map

Use the town map above. Which place is at $(6, 2)$ ? What ordered pair names the school?

Start at the station, which is the origin. The x-coordinate is 6, so move 6 blocks right. The y-coordinate is 2, so move 2 blocks up. You reach Point B, the pool.
$(6, 2) \rightarrow \text{right } 6,\ \text{up } 2$
Check the order. The library, Point A, is at (2, 6). It uses the same two numbers, but since 2 and 6 are different, switching them leads to a different place. So the library is not the answer.
$(2, 6) \neq (6, 2)$
Now read the school, Point C. Moving right from the origin, you are directly below Point C after 5 blocks. Moving up from there, you reach it after 7 blocks. Write the right move first and the up move second.
$(x, y) = (5, 7)$

Answer

The place at $(6, 2)$ is the pool, Point B. The school, Point C, is at $(5, 7)$ : 5 blocks to the right of the station and 7 blocks up.

On a first-quadrant grid or map, an ordered pair is a set of directions from the origin. The x-coordinate tells you how far to go right, and the y-coordinate tells you how far to go up. Keep the two numbers in that order, and each pair leads you to exactly one point.