Teaching Models

Algebra and Graphing

Points on a Grid
A location, or point, on a grid can be identified by an ordered pair such as (3, 2), which names the coordinates of that location or point. The first number tells how far to the right or left the point is located in the horizontal direction. The second number tells how far up or down the point is located in the vertical direction. Students must realize that, for example, that the point (3, 2) is not the same as point (2, 3). The numbers in an ordered pair are called coordinates.

Grid

Integers
The set of integers consists of the counting numbers 1, 2, 3,…, their opposites 1, 2, 3,…, and the number 0. The negative integers are needed to solve exercises such as 5 − 7 = 2 as well as to record quantities like the temperature below zero, the depth below sea level, how much has been lost, and how much is owed.

The integers can be represented on a number line that extends both to the left and to the right of zero. The positive integers are represented to the right of 0 and the negative integers are represented to the left of 0.

Number line

Point out to students that numbers such as 3 and +3 are opposite numbers.

Coordinate Plane
A coordinate plane is composed of a horizontal number line (the x-axis) and a vertical number line (the y-axis). With the extension of the number system to include negative numbers, each axis can be extended in two directions. The point where the two axes intersect is the origin. The coordinates of the origin are (0, 0). The first number in an ordered pair is the x-coordinate. The second number is the y-coordinate.

Coordinate plane

The coordinate axes divide the plane into four regions called quadrants. The quadrants are numbered I, II, III, and IV as shown on the grid below. At this grade level, students do not need to know the numbering system for the quadrants, but should recognize that the appropriate quadrants for ordered pairs can be identified according to the signs of the numbers. 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.

Quadrants

The length of a horizontal or vertical line segment on a coordinate plane can be found by counting units or by using subtraction. For horizontal line segments, the difference between the x-coordinates of the endpoints is the length. For vertical line segments, the difference between the y-coordinates of the endpoints is the length.

Functions
At this grade level, functions are graphed from tables of given values. All values are positive, thus the graphs of functions studied in this chapter are located in Quadrant I only. Formal definitions and extended work with functions are not necessary at this time.

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 graphs of such equations are lines in the plane. An equation that can be written Ax + By = C, where A, B, and C are fixed numbers, is called a linear equation.

The graph of a linear equation is a straight line. To graph a linear equation, first a table of values for x and y is completed and then the ordered pairs are graphed. Finally a line is drawn through the points.

Equation: y = x + 3

Table and line plot

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, +2), (0, +3), (+1, +4), and (+2, +5). To graph the equation, students connect the points with a straight line.


Teaching Model 24.1: Locate Points on a Grid


Houghton Mifflin Math Grade 4