The Cartesian plane is like a map grid. Just like the map helps you find the shoe store or post office, the Cartesian plane helps you locate a pair of values.
There is a myth that the mathematician Rene Descartes was watching a fly on the ceiling when he was in bed. He figured out a way of tracking the fly by thinking of the ceiling as a grid system. This system is like a map, and we call it the Cartesian plane. Just like on a map, you can locate any point in the Cartesian plane if you know two things - how far the point is to the left/right and how far it is up/down.
Using a Map Grid
Let's start by looking at a map.
In this city map, the dog is right at the center of town. As you travel east of the center of town, the streets are named 100 East, 200 East, 300 East, and so forth. You find similar patterns as you drive west, north, or south. Main Street is really 0 East AND 0 West simultaneously, so it is easiest to just give it a name. Similarly, Center Street is really 0 North and 0 South.
You can easily find any location on the map just by knowing how far it is east/west and north/south of the center of town. The car is at about 200 East and 200 North. The shoes are at about 200 West and 100 South.
Using Numbers Instead of Street Names
The Cartesian Plane is very much like a map grid. Instead of using the compass directions (north, south, east, west), you use positive and negative numbers. East and north are positive, while west and south are negative. The picture below is exactly the same as the last map, except for replacing the compass directions with positive and negative numbers (integers).
Simple Cartesian plane
Now the car is at the crossing of the red line '2' and the blue line '2.' You could say it is at 'red 2' and 'blue 2.' That approach works and keeps the vertical and horizontal straight, but it is a little cumbersome. Instead of relying on colors or compass directions, mathematicians use a different trick: they use order to describe which number is the horizontal distance from the center of town and which is vertical. The horizontal distance always comes first.
Here is an easy way to remember which direction comes first: an airplane has to go over before it can go up. To find the shoes, go over to -2 then up to -1. Strange as it sounds, going 'up' by -1 means going down!
Since you always go horizontally first and vertically second, you just need to list the two numbers (or coordinates) in order. The car is at (2,2). You always put parentheses around the two coordinates as a clue that the numbers represent a location on the Cartesian plane. Written this way, we call (2,2) an ordered pair.
As another example, the shoes are at (-2,-1).
We still don't have an accurate Cartesian plane, though. Let's make a few more changes.
The Cartesian Plane
In the picture above, several things are different. First, we've put arrows at the ends of all the lines. This simply means that the grid goes on forever in all directions.
Next, do you see the word origin? You don't really have to write that in. It shows us that the middle point, the 'center of town,' has a special name. The point (0,0) is the origin, or the place that you start from when finding your way through the Cartesian plane.
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The next thing you need to know is that the horizontal direction is usually called the x direction. So if you want to go 2 to the right, you can just say x = 2. For two to the left, you would say x = -2. If you start at the origin and go to x = 2, you will slide your finger along the red line. On this picture, it is labeled x-axis.
Similarly, the vertical direction is usually called the y direction. If y = 1, then you will go up to the first red line above the thicker x-axis. As you travel upward to y = 1, you will slide your finger along the blue line, the y-axis.
The last thing to notice is that the car and the shoes have been replaced by small circles. Of course, there is nothing wrong with using a shoe symbol instead of a dot, but dots are easier and faster to draw. You call these dots points. The green point is at (-2,-1).
Parts of the Plane
You might also notice that the thicker red and blue lines divide the Cartesian plane into four sections. These areas are called quadrants. Quadrant 1 (the upper right-hand box) is the section where both x and y are positive. Then we rotate counter-clockwise to number the other quadrants. In Quadrant 2, x is negative, and y is positive. In Quadrant 3, they are both negative. Finally, in Quadrant 4, x is positive, and y is negative.
After you've completed this lesson on the Cartesian plane, determine your readiness to:
Specify the purpose and components of the Cartesian plane
Compare the Cartesian plane to a map grid
Write ordered pairs
Explain the relationship between x values and y values in all four quadrants of the Cartesian plane
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