Imagine you are playing an open-world video game. Your character runs around a giant map. The game always knows exactly where you are: maybe 5 steps east and 3 steps north of the center. That idea of “how far east” and “how far north” is the same idea we use in math on something called the coordinate plane. Learning to graph points and measure distances on this plane lets you read maps, track positions, and solve many real-world problems. 🎯
The coordinate plane is like a map where every point has an address made of two numbers. In this lesson, you will learn how to:
The cool part: you will be using rational numbers (fractions and decimals) and negative numbers all at once, just like a real GPS system does. 🌍
The coordinate plane is made by two number lines that cross each other at right angles.
On each axis, to the right and up are positive numbers, and to the left and down are negative numbers. [Figure 1] shows the basic layout of the coordinate plane with axes and numbers labeled.
The axes split the plane into four regions called quadrants. They are numbered using Roman numerals:
Points that lie exactly on an axis are not in any quadrant. For example, \((0, 3)\) is on the y-axis, and \((-4, 0)\) is on the x-axis.
Every point on the coordinate plane is named by an ordered pair: \((x, y)\).
A helpful way to remember the order is: “x comes before y in the alphabet,” or “go across first, then go up/down.”
Example: The point \((4, -2)\)
This lands you in Quadrant IV.
To plot any point \((x, y)\):
Let’s walk through four points, one in each quadrant. These are shown together in [Figure 2].
Coordinates do not have to be whole numbers. They can be:
To plot \((1.5, -2.5)\):
You follow the same steps as with whole numbers; your moves are just smaller or in between the tick marks.
Now we want to answer a very common question: How far apart are two points? For this lesson, we focus on a special but very important case: when the points share the same x-coordinate or the same y-coordinate.
When this happens, we can find the distance using absolute value.
The absolute value of a number is its distance from 0 on the number line, ignoring the sign. We write it with vertical bars, like \(|-3|\).
Absolute value is always non-negative because distance can never be negative.
On a number line, the distance between two points a and b is \(|a - b|\). We can use this same idea on the coordinate plane when one coordinate is the same.
Suppose we have two points: \((x, y_1)\) and \((x, y_2)\). They have the same x-coordinate, so they lie on a vertical line. The distance between them is the difference between their y-coordinates:
Distance = \(|y_2 - y_1|\).
This works even if the y-values are negative, positive, or a mix, because absolute value gives the actual distance.
[Figure 3] illustrates two points with the same x-coordinate and shows the vertical distance between them.
Now suppose we have two points: \((x_1, y)\) and \((x_2, y)\). They have the same y-coordinate, so they lie on a horizontal line. The distance between them is the difference between their x-coordinates:
Distance = \(|x_2 - x_1|\).
Again, we use absolute value to make sure the distance is not negative.
Problem: Find the distance between the points \((2, 3)\) and \((2, -1)\).
Step 1: Check which coordinate is the same.
Both points have x = 2, so they are vertically aligned.
Step 2: Identify the y-coordinates.
First point: \((2, 3)\) has y = 3.
Second point: \((2, -1)\) has y = -1.
Step 3: Use the distance formula for same x.
Distance = \(|y_2 - y_1|\).
Let \(y_1 = 3\) and \(y_2 = -1\).
Distance = \(|-1 - 3|\) = \(|-4|\) = 4.
Answer: The distance between the points is 4 units.
Notice that one point is in Quadrant I (2, 3) and the other is in Quadrant IV (2, -1), but the distance is still easy to find with absolute value.
Problem: Find the distance between the points \((-5, -2)\) and \((1, -2)\).
Step 1: Check which coordinate is the same.
Both points have y = -2, so they are horizontally aligned.
Step 2: Identify the x-coordinates.
First point: x = -5.
Second point: x = 1.
Step 3: Use the distance formula for same y.
Distance = \(|x_2 - x_1|\).
Let \(x_1 = -5\) and \(x_2 = 1\).
Distance = \(|1 - (-5)|\) = \(|1 + 5|\) = \(|6|\) = 6.
Answer: The distance between the points is 6 units.
Even though both x-values cross over 0 (from -5 to 1), absolute value helps you find the correct distance.
Problem: Find the distance between the points \((1.5, 4)\) and \((1.5, -2.5)\).
Step 1: Check which coordinate is the same.
Both points have x = 1.5, so they form a vertical line.
Step 2: Identify the y-coordinates.
First point: y = 4.
Second point: y = -2.5.
Step 3: Use the distance formula for same x.
Distance = \(|y_2 - y_1|\).
Let \(y_1 = 4\) and \(y_2 = -2.5\).
Distance = \(|-2.5 - 4|\) = \(|-6.5|\) = 6.5.
Answer: The distance between the points is 6.5 units.
This example shows that rational numbers, including decimals, work perfectly with the same ideas.
These ideas are not just for math class. They appear in many real-life situations you already know about.
a) GPS and Maps
Phones and GPS devices use numbers like coordinates to track locations. While actual GPS uses latitude and longitude, the idea is similar: each place has a pair of numbers. If two places are directly north-south of each other, you are basically finding the distance using the difference in just one type of coordinate, like we did with same x or same y.
b) Video Game Worlds 🎮
In many 2D games, your character’s position is stored as coordinates \((x, y)\). If your character moves straight up or down, only the y-coordinate changes. The game engine can quickly find how far you moved by subtracting the starting y from the ending y and taking absolute value.
c) Sports Fields and Courts
Imagine a coach mapping out player spots on a basketball court using a coordinate system. If two players stand in a vertical line under the hoop (same x-coordinate), the coach can describe how far one is from the other using the difference in their y-coordinates.
d) City Maps and Grids
Some cities are formed like grids. Streets running east-west and north-south can be numbered, like 1st Street, 2nd Street, etc. You can use coordinates to describe intersections, and if two intersections share the same east-west street, then the distance between them is based on the difference in their north-south positions.
When working on the coordinate plane, you are really doing the same kind of thinking you used on a number line, just with two directions at once.
This shows how your understanding of positive and negative numbers extends from one dimension (a line) to two dimensions (the plane).
For same x: Distance = \(|y_2 - y_1|\).
For same y: Distance = \(|x_2 - x_1|\).
Coordinate Plane and Quadrants
Ordered Pairs
Rational Number Coordinates
Absolute Value
Distance with Same Coordinate
By understanding how to plot points in all four quadrants and how to measure distances using coordinates and absolute value, you can solve many real-world and mathematical problems involving position, movement, and mapping. 🎉
