The Coordinate Plane

In this lesson we go over how to graph point on the coordinate plane. The slides discuss how we move from plotting points on a number line in one dimension to using a coordinate plane for two-dimensional coordinate points as ordered pairs. We dive into how to plot a point, what quadrants are, how points look in different parts of the plane, and even how to use the Pythagorean Theorem to find the distance between points!

So without further ado, read through the slides below to get a feel for how to plot points on the coordinate plane!

Phew, that’s a lot to take in. Once you’ve gone over this and found some practice problems to cement the idea in your head what integers are, you may move on to the next lesson below!

Transformations In the Coordinate Plane

This lesson is a video generated from the fully animated PowerPoint I created that is available in my Teachers Pay Teachers store page. You may also head over to my YouTube channel to see the slideshows in video form there!

Just in case the slides aren’t your thing, here is a text outline of the main points of the lessons above!

  • The Coordinate Plane
  • Graphing Points and Finding Distance
  • Objectives
  • By the end of this lesson you should feel comfortable:
  • Recognizing parts of the coordinate plane
  • Graphing points on the coordinate plane
  • Finding distances between points
  • Dimensions
  • Until now, we have plotted numbers on a number line.
  • But the number line only goes in two directions, left and right.
  • This is one dimension, a horizontal line.
  • But what if we need to move up or down?
  • Could we add another line?
  • Yes!
  • Now we can graph in two dimensions!
  • Let’s see what this second line can do…
  • X and y axis
  • The number lines are now called axes (pronounced ax-ees).
  • The horizontal line we are used to is called the x-axis.
  • The vertical line is called the y-axis.
  • The y-axis, just like the x, is numbered.
  • Upward is considered positive.
  • Downward is considered negative.
  • The center where the axes meet is called the origin (more on that in a moment).
  • The Coordinate Plane
  • This….
  • …is called the coordinate plane!
  • A plane (no not the one you fly on) is any flat surface that extends forever in every direction.
  • An infinite 2-dimensional surface!
  • A coordinate is an ordered pair in the form (x, y).
  • So this grid is an infinite flat surface we can plot coordinate points on!
  • Coordinates
  • So exactly what are coordinates and what is an ordered pair?
  • An ordered pair is simply the form of a coordinate.
  • They look like this: (x,y)
  • An example of a coordinate would be (2, 4).
  • But how did I get there?
  • The key to plotting a point on the coordinate plane is the axes!
  • So for this point, our x coordinate is 2 and our y coordinate is 4.
  • Coordinates
  • Let’s look at a diagram for how to plot a point.
  • We know that in (2,4), 2 represents the x.
  • So on our x-axis we travel 2 from the center:
  • Then 4 represents the y, so we travel 4 in the y direction.
  • Tada! We have arrived at our point!
  • Remember:
  • x travels left or right first
  • then y travels up or down
  • Quadrants
  • Now we can learn to navigate our way around the coordinate plane.
  • Notice how the lines divide the plane into four sections?
  • Four means quad, so we call each a quadrant.
  • We start in the upper right quadrant with quadrant I, where both coordinates are positive.
  • Then we circle around counterclockwise.
  • Take note that we use Roman Numerals to count the numbers of the quadrants so as not to get confused later.
  • Signs
  • So what’s special about each quadrant?
  • Each section has a specific form of ordered pair due to the signs of the coordinates.
  • To get to quadrant I, we go right (positive in the x direction), then up (positive in the y direction):
  • To land in quadrant II, however, we move left first (negative in the x direction), then go up:
  • Quadrant III requires two negative moves:
  • Finally Quadrant IV moves right and then down:
  • On the axes
  • There are a few other special locations on the graph, though.
  • Have you noticed that we always start from the center?
  • This is why we call where the axes meet the origin, or beginning.
  • What if we had a point not in a quadrant but directly on one of the axes, though?
  • What are the coordinates of these points?
  • Hmmm, so what pattern do we notice here?
  • On the Axes
  • In our first point, we went 4 to the right:
  • Then we don’t move up or down, so the distance we move vertically (in the y direction) is 0.
  • Thus our coordinate would have to be (4,0):
  • What about (0,3)?
  • First we did not move left or right, so 0 horizontally (in the x direction).
  • Then up 3 for the y coordinate!
  • The other points would work in the same way!
  • Distances
  • Hopefully now you know your way around the coordinate plane.
  • Which of course means we should take it a step farther!
  • Let’s throw up a couple of points:
  • What if we want to know how far apart these points are?
  • On a coordinate plane we can only move horizontally and vertically, but the shortest distance between two points is a straight line:
  • So how do we find the distance this diagonal line traveled?
  • Distances
  • The Pythagorean theorem!
  • But that means we need a right triangle…
  • We know we can only move left or right, and up or down.
  • So the path we would have to take to get from the blue point to the red point would be:
  • Or:
  • Both would create the same leg distances, so we will focus on the first path, but the top would work the same!
  • Distances
  • To find how far we moved right and up, first we should know the coordinates of our points:
  • So we started at -2 on the x and moved to 4.
  • To figure out this distance, we subtract 4 – (-2):
  • For the vertical distance, we moved from -3 to 5.
  • Thus, we need to subtract our y coordinates:
  • Now we can use either the Pythagorean theorem or our Pythagorean triples to figure out the missing hypotenuse!
  • Therefore our distance, d, is 10.
  • conclusion
  • We can use a coordinate plane to graph coordinates in the form of ordered pairs in two dimensions: horizontal and vertical.
  • Ordered pairs have the form (x,y) with each coordinate representing how far the point moved along the horizontal x-axis and vertical y-axis.
  • The center of the plane where the axes meet is called the origin or (0,0).
  • To plot a point, you must move in the x direction first, then the y direction.
  • The quadrants are the four sections of the coordinate plane beginning with I in the top right and counting to IV counterclockwise. These help determine signs of ordered pairs.
  • A point may fall on the x or y axis.
  • We use the Pythagorean theorem to find the diagonal distance between two points.
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