Math Background

Lesson: Graphing Integers
Introducing the Concept

Students learned about linear equations earlier. Now they will review how to solve those equations and learn how to graph linear equations in x and y.

Materials: overhead transparency or poster paper

Preparation: Draw a coordinate grid on a transparency or poster paper, or use Learning Tool 20 in the Learning Tools Folder.

Prerequisite Skills and Concepts: Students should know how to graph ordered pairs. They should also be able to solve simple addition, subtraction, multiplication, and division linear equations.

Write the equation x − 3 = y on the board.

  • Ask: Who could tell me what this equation means in words?
    Students should say, “One number minus 3 is equal to another number,” or something equivalent.
  • Say: Earlier in the year we worked with similar equations, but those equations had only one unknown, an x or a y. This equation has two unknowns, x and y. If I were to let x = 2 in this equation, what would the value of y be?
    Students will calculate that y = -1. Draw a table with four columns. Label the first column x, the second column x − 3 = y, the third column y, and the fourth column (x, y). See the table below.
x x − 3 = y y (x, y)
       
       
       
       

Put 2, 2 − 3 = -1, -1, and (2, -1) in the appropriate columns of the first row, as in the table shown.

x x − 3 = y y (x, y)
2 2 − 3 = -1 -1 (2, - 1)
       
       
       
  • Say: Since y = -1 when x = 2, I put 2 in the x-column, 2 − 3 = -1 in the x − 3 = y column, -1 in the y-column, and the ordered pair (2, -1) in the (x, y) column.
    Solicit four or five other values for x and find their corresponding values for y. List them in the table accordingly.
  • Say: We now have two ways to represent the relationship between the numbers x and y. The equation describes a general relationship between the numbers; in words, the equation states that one number minus 3 is equal to another number. The table now gives us some specific values that satisfy the relationship. If you look at the table, you will see that every y-value is 3 less than its corresponding x-value.
  • Say: Now we are going to look at a third way to represent the equation x − 3 = y. This way is a visual representation, using a graph. We are going to use the values from our table and graph the ordered pairs on the graph.
  • Ask: Using this graph (the overhead or large poster graph), who can show us where the point (2, -1) is located?
    Have a student graph the point on the grid. Do the same for the other points in the table.
  • Ask: What do you notice about the points we graphed?
    Students should respond that they appear to lie in a straight line. If they don't, show them by drawing a line through all the points. Make sure you indicate that the line goes on indefinitely in both directions by drawing arrows on the ends of the lines.
  • Say: Let's try one more equation. (Write the equation 2x = y on the board.) How are the two variables in this equation related?
    Students should respond that the y-value is double the x-value.
  • Ask: If we want to graph this equation, what should we do next?
    Students should say to create a table of values for x and y. Create a table of values with at least five points in it.
  • Ask: Now what do we do?
    Graph the points on a grid. Have volunteers plot the points from the table of values on the coordinate grid for the class to see.
  • Ask: What do you notice about these values?
    Students should say that they too seem to lie in a straight line. Draw the double-arrowhead line on the grid.
Houghton Mifflin Math Grade 5