Math Background

Graphing Integers: Overview

Graphing on the coordinate plane is a way to visualize relationships between two quantities. René Descartes, a French mathematician and philosopher, discovered this unique way to combine algebra and geometry. He recognized that by drawing two perpendicular number lines called axes, and labeling them with positive and negative numbers, he could locate any point on the plane with x- and y-coordinates. The diagram below shows the different parts of the coordinate plane.

different parts of the coordinate plane

The point where the x-axis and y-axis intersect is called the origin. Notice that x-values to the right of the y-axis are positive and x-values to the left of the y-axis are negative. Similarly, y-values above the x-axis are positive and y-values below the x-axis are negative. The axes divide the plane into four quadrants, which are numbered starting in the upper right-hand quadrant and moving counterclockwise as I, II, III, and IV.

An ordered pair of numbers, (x, y) is used to locate a point on the plane. The first number in an ordered pair is called the x-coordinate and represents a location on the x-axis. The second number is called the y-coordinate and represents a location on the y-axis. To locate a point on the coordinate plane, do the following:

(1) Start at the origin.
(2) The first number is the x-coordinate. If it is positive, move to the right the appropriate number of units. If it is negative, move to the left the appropriate number of units.
(3) The second number is the y-coordinate. If it is positive, move up the appropriate number of units. If it is negative, move down the appropriate number of units.

Look at the coordinate grid below. The ordered pair for point A is (3, -2). To locate point A, we move three units to the right and two units down. Also shown on the coordinate plane are points B (-3, 5), C (-1, -4), D (0, -3), E (2, 4), and F (4, 0).

coordinate grid

Notice that points in Quadrant I have a positive x- and a positive y-coordinate. Points in Quadrant II have a negative x-coordinate and a positive y-coordinate. Points in Quadrant III have a negative x-coordinate and a negative y-coordinate, and points in Quadrant IV have a positive x-coordinate and a negative y-coordinate.

Students can connect points on coordinate grids to form geometric shapes. Use Learning Tool 20 in the Learning Tools Folder. You can use these shapes to introduce transformations to students. A transformation is a change in the position of a plane figure. Three kinds of transformations are translation, sliding a figure a given distance in a given direction; reflection, flipping a figure across a given line; and rotation, turning a figure about a given point. See examples below.

shapes in coordinate grids
Translation Reflection Rotation

One way to represent relationships between pairs of numbers is by using an equation such as x − 2 = y. A second way to represent a relationship is by making a function table like the one below.

x x − 2 = y y(x, y)
-3 -3 − 2 = -5 -5 (-3, -5)
-2 -2 − 2 = -4 -4 (-2, -4)
-1 -1 − 2 = -3 -3 (-1, -3)
 0 0 − 2 = -2 -2 (0, -2)
 1 1 − 2 = -1 -1 (1, -1)
 2 2 − 2 = 0 0 (2, 0)
 3 3 − 2 = 1 1 (3, 1)

This table can then be used to create the graph of the relationship. By plotting the points from the table above, we can see the relationship that exists between the x- and y- values that satisfy the equation x − 2 = y.

coordinate grid

We can now see that the pattern of points lies along a straight line. By connecting the points with a solid line and indicating that it can be extended in both directions, we have created the graph of x − 2 = y. A linear function is a function with a straight-line graph.


Houghton Mifflin Math Grade 5