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Grade 5
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Introducing the Concept

Graphing Integers

You and your students learned about linear equations in Units I and 3 in MathSteps. Now they will review how to solve those equations and learn how to graph linear equations in for x and y.

Materials: overhead transparency or poster paper

Preparation: Draw a coordinate grid on a transparency or poster paper.

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 that "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 only had 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). Put 2, 2 – 3 = -1, -1, and (2, -1) in the appropriate columns of the first row.

  • Say: Since y = -1 when x = 2, I put 2 in the x-column, 2 – 3 = -1 in the x – 3 = y, -1 in the y-column, and the ordered pair (2, -1) in the (x, y) column.
    Solicit 4 or 5 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, which in words says that one number minus 3 is equal to another number. The table now gives us some specific values which satisfy that 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 could show us where the point (2, -1) is located?
    Have a student graph the point on the grid. Do the same for the other 4 or 5 points in the table.

  • Ask: What do you notice about the points we graphed?
    Students will 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 will say to create a table of values for x and y. So, create a table of values with at least 5 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.
       Ordered Pairs         Graphing Linear Equations

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