Teaching Models

Coordinate Graphing

A geometric transformation changes the position or size of a figure in a plane without changing the shape of the figure. All transformations studied in this chapter involve only changes in location or position; therefore figures remain congruent under these transformations.

A translation moves a figure by sliding it along a line in one direction. A rotation turns a figure around a given point. A reflection flips the figure across a line, thus producing a mirror image. At this grade level, it may be easier for students to complete translations by working on grid paper.

Coordinate Plane
A coordinate plane is composed of a horizontal number line (the x-axis) and a vertical number line (the y-axis). The point where the two axes intersect is the origin. A location, or point, on the coordinate plane is identified by an ordered pair such as (3, 2), which names the coordinates of that location or point. The first number, or the x-coordinate, tells how far to the right or left the point is located in the horizontal direction. The second number, or the y-coordinate, tells how far up or down the point is located in the vertical direction. The coordinates of the origin are (0, 0). Students must realize that, for example, the point (3, 2) is not the same as point (2, 3).

Coordinate plane showing 3,2

The coordinate axes divide the plane into four regions called quadrants. The quadrants are numbered I, II, III, and IV as shown. All points in Quadrant I have two positive coordinates. All points in Quadrant II have a negative x-coordinate and a positive y-coordinate. All points in Quadrant III have two negative coordinates. All points in Quadrant IV have a positive x-coordinate and a negative y-coordinate.

Coordinate plane showing quadrants

At this grade level, students locate and graph ordered pairs of integers on a coordinate plane. They then use coordinates to describe transformations of points and simple figures such as triangles.

A function is a rule that associates one and only one value of one variable with each value of another variable. The function y = 2x expresses y in terms of x. For each value of x there is one and only one value of y. An equation determines y as a function of x, if, for each x, the equation can be solved to give exactly one value of y. The equations 3x + 2 = 14, xy = 7, and y = x + 3 determine y as a function of x. The equation y = x2 determines y as a function of x, but the equation x = y2 does not determine y as a function of x because for some values of x, such as 4, the two values y = 2 and y = +2 for y both satisfy the equation. The graph of x = y2 is not a straight line.

In general, the graph of an equation that can be written Ax + By = C, where A, B, and C are fixed numbers, is a straight line. Each point on the line is a solution to the equation. Such an equation is called a linear equation.

To graph a linear equation, students first make a table of values for x and y. Then they graph the ordered pairs. Finally, they draw a line through the points.


Graph the equation y = x + 3.
x y
2 +1
1 +2
0 +3
+1 +4
+2 +5

These values can be interpreted as the x-coordinates and y-coordinates of points in the coordinate plane. Graphing these points means placing a dot at the points (2, +1), (1, +1), (0, +3), (+1, +4), and (+2, +5). To graph the equation, students connect the points with a straight line.

Coordinate graph showing y=x+3

Teaching Model 23.2: Integers and Functions

Houghton Mifflin Math Grade 5