The Cartesian Plane

Image Source: http://fc05.deviantart.net

In the above picture we have a pier going out at right angles from the horizontal coastline.

We could walk away from where the man and his boat are, to many different locations, by wading out into the water.

We could then locate our new position, by using the man as our starting point, and calling this point the “Origin”.

We could then see how far down the beach we have traveled, (either left or right), as well as how far out into the water we have gone by counting the number of pier pylons that we are from the shoreline.

Our location is then a unique “Point”, made up of how far left or right from the man we are, and how far vertically out from the shore we are.

Referencing our position like this is called “Coordinate Geometry”.

It involves using “Points” in a two dimensional “Cartesian Plane”.

Introduction to the Cartesian Plane.

Let’s start with a video that describes how we set up and use an X-Y Grid for locating points.

“Points” are dots which show our position on the grid.

The X-Y Grid is called the “Cartesian Plane”.

“Cartesian”, because a guy called “Rene Descartes” invented it.

“Plane”, meaning that we are working with a flat two dimensional surface, just like a flat sheet of graph paper.

[youtube http://www.youtube.com/watch?v=E2FqyJL2w8U]

The “Cartesian Plane” consists of an “X-Y” grid of squares that looks like this.

The across ways Horizontal line is called the “X-axis”.

The up and down Vertical line is called the “Y-axis”.

The center point where the lines cross over is called the “Origin”.

Points are drawn as dots at the corner points of squares; they are never placed inside the squares.

Each axis is a number line that has negative and positive values along it. The middle of each axis is zero, and is located at the center “Origin”.

Quadrants in the Cartesian Plane

The Cartesian plane is divided into four squares or “Quadrants”.

The quadrants start at one, and move around anti-clockwise to two, three, and four.

The Quadrants are assigned Roman numerals so they do not look like questions, eg. QII is Quadrant two, but Q2 in maths usually means “Question 2”.

The Blue point in the middle of all the Quadrants is at a special location called the “Origin”, which is not part of any quadrant.

Eg. The point of origin is not in any quadrant. In fact, any point located on the X or Y axis is not in a quadrant.

For a point to be in QI, QII, QIII or QIV, it must not be on an axis.

Let’s take a closer look at each of the four quadrants in detail.

The First Quadrant

The first Quadrant takes in the top right hand area of the Cartesian Plane.

Points located in Quadrant One always have (x,y) coordinates that are both positive numbers.

For example the point (5,3) shown above is in the first Quadrant.

The X-coordinate is positive 5, which means it is located five units (squares) TO THE RIGHT of the Origin.

The Y-coordinate is positive 3, which means it is located three units (squares) UP from the Origin.

Note that if either of the x or y coordinates are zero, the point will be located on one of the “Axis” lines and is not in any Quadrant.

Eg. Instead of being in a Quadrant, it is directly on the Axis.

The Second Quadrant

The second Quadrant takes in the top left hand area of the Cartesian Plane.

Points located in Quadrant Two always have (x,y) coordinates that are a negative number, followed by a positive number.

For example the point (-5,3) shown above is in the second Quadrant.

The X-coordinate is -5, which means it is located five units (squares) TO THE LEFT of the Origin.

The Y-coordinate is positive 3, which means it is located three units (squares) UP from the Origin.

Note that if either of the x or y coordinates are zero, the point will be located on one of the “Axis” lines and is not in any Quadrant. (Eg. Instead of being in a Quadrant, it is directly on the Axis).

The Third Quadrant

The third Quadrant takes in the bottom left hand area of the Cartesian Plane.

Points located in Quadrant Three always have (x,y) coordinates that are both negative numbers.

For example the point (-5,-3) shown above is in the third Quadrant.

The X-coordinate is -5, which means it is located five units (squares) TO THE LEFT of the Origin.

The Y-coordinate is -3, which means it is located three units (squares) DOWN from the Origin.

The Fourth Quadrant

The fourth Quadrant takes in the bottom right hand area of the Cartesian Plane.

Points located in Quadrant Four always have (x,y) coordinates that are a positive number, followed by a negative number.

For example the point (5,-3) shown above is in the fourth Quadrant.

The X-coordinate is 5, which means it is located five units (squares) TO THE RIGHT of the Origin.

The Y-coordinate is -3, which means it is located three units (squares) DOWN from the Origin.

Note that if either of the x or y coordinates are zero, the point will be located on one of the “Axis” lines and is not in any Quadrant. (Eg. Instead of being in a Quadrant, it is directly on the Axis).

Cartesian Plane Summary

The X-Y Grid of the Cartesian Plane is divided up into four equal areas called “Quadrants”.

These Quadrants are named in a counter-clockwise direction.

Points in each of these Quadrants have coordinates that have specific combinations of positive and negative x and y values.

If either of the x or y coordinates are zero, the point will be located on one of the “Axis” lines and is not in any of the Quadrants. (Instead of being in a Quadrant, the point is directly on the Axis).

The “Origin”, which is located in the center of the Cartesian Plane at (0,0) is also not in any of the Quadrants.

Here is a simplified summary of the Cartesian Plane Quadrants that can be copied into your maths notebook.

Image Source: http://0.tqn.com

Graphing Ordered Pairs

Points are located on the X-Y Grid using “Coordinates” which are “Ordered Pairs”.

The order is always alphabetical, and the x coordinate number always comes before the y coordinate number.

The number pairs are enclosed in brackets with a comma used to separate them.

The sign of each coordinate number tells us in what direction to leave the Origin to travel to our coordinate point.

If x is positive we go across to the right (eg. (5,?) means go five units or squares to the RIGHT.

If x is negative we go across to the Left (eg. (-5,?) means go five units or squares to the LEFT.

If y is positive we go UP (eg. (?,3) means go three units or squares UP.

If y is negative we go DOWN (eg. (?,-3) means go three units or squares DOWN.

By going horizontally Across first, and then Vertically second, using the supplied x and y values, we can reach the exact location of any point on the Cartesian Plane.

If you have already done Directed Number Integers in your maths course, then think about doing what you know first, by locating the x coordinate onto the horizontal number line.

Then from that position go up or down in the vertical direction to where the y coordinate needs to be located.

Finally place a dot at this finishing position to mark the point.

Here is a video with plenty of examples on how to plot x-y points, or “Ordered Pairs” .

[youtube http://www.youtube.com/watch?v=YlT726odQcM]

Here is another video where a mathematics teacher introduces the coordinate plane, and plots some example points onto it.

[youtube http://www.youtube.com/watch?v=ZT848fdqOzk]

Cartesian Plane Practice Grid

Here is a blank Cartesian Plane you can print out or project onto a whiteboard to practice plotting points.

(Clicking the Image should take you to an 800×494 pixels grid that can be projected or printed).

Cartesian Plane Revision

The following slideshare presentation will test how well you know the Cartesian Plane and Quadrants.

It also covers how points can combine to make Horizontal and Vertical Lines.

[slideshare id=85218&doc=the-coordinate-plane-geometry-24211]

Interactive Point Plotting Test

The web page shown above supplies some interactive activities for plotting points, and supplies answers.

It will test your point plotting skills and knowledge.

Click the picture above, or the link below to do this activity.

Interactive Point Plotting Test

Graph Mole Plotting Points Game

This is the easy version of Graph Mole, a fun little Algebra game that teaches you how to plot points on a Cartesian coordinate plane.

Play the game by clicking on the correct x-y coordinates at the bottom of the game screen that show the correct position of the mole.

On the game page there are also links to medium difficulty and challenging versions of the mole game.

Click the picture above, or the link below to play this game.

Graph Mole Plotting Points Game
That’s it for this lesson on the “Cartesian Plane”.

Related Items

Plotting Graphs from Horizontal Values Tables
Plotting a Linear Graph using a Rule Equation
Plotting Graphs from T-Tables of Values
Real World Straight Line Graphs I
Real World Straight Line Graphs II

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