Math Background

Lesson: Functions and Graphing
Introducing the Concept

Your students may have graphed on a coordinate plane before, but it may not be a skill they have mastered. You will need to review how to graph points, the names of the four quadrants, and terms such as coordinate plane, ordered pair, x-coordinate, y-coordinate, and origin.

Materials: graph paper for students, two worksheets shown below, graph paper for the overhead, a ruler

Preparation: Prepare the worksheets shown below with some points graphed on one and some figures (for which students will find the areas) drawn on the other. Use Learning Tool 8 in the Learning Tools Folder.

Prerequisite Skills and Background: Students should be able to perform the four basic operations, and they should be able to find the area of a triangle and square by using formulas.

  • Say: Today we are going to work on graphing points in the coordinate plane. (Place the overhead graph paper on the overhead and draw in the axes as students watch. Label the x-axis and y-axis.) I am now drawing two perpendicular lines on my graph paper. These are called the x-axis and y-axis. Do the same on your sheet of graph paper using your ruler. Label your axes as I have done.
  • Ask: Does anyone know what we call the point where the two perpendicular lines intersect? (the origin) The origin is the ordered pair (0, 0) because it is at zero units on the x-axis and zero units on the y-axis. Points to the right of the y-axis have positive x-values, points to the left of the y-axis have negative x-values, and points on the y-axis have an x-value of zero. (Label the x-axis.) Similarly, points above the x-axis have positive y-values, points below the x-axis have negative y-values and points on the x-axis have a y-value of zero.
    (Label the y-axis.)
  • Say: The 2 in the ordered pair (2, 4) is the x-coordinate for the point and the 4 is the y-coordinate for the point. What would we label a point that is 3 units to the right of the origin and 5 units below the x-axis? (3, -5)
    Mark this point on the overhead and have students do the same on their graph paper.
  • Say: Now, on your graph paper, graph the following ordered pairs and label the points.
    (-4, 6)     (-3, -3)     (4, 0)     (0, -5)     (-3, 2.5)

    While students are graphing and labeling those points, walk around the room to check students' progress. When they have finished, have volunteers come up and do the same on the overhead.

    Distribute the worksheet shown below.

  • Say: Use the worksheet I have just given you and write the ordered pairs for the 6 points on it.
    Walk around the room to check students' progress. After students have finished, have volunteers come up and do the same on the overhead.
  • Ask: If we go from point C on the graph to point F, how many units do we move?(6) When we have two points like C and F above, we can find how far apart they are by subtracting their x-coordinates. Point C is at (-4, -4) and point F is at (2, -4). Therefore, the distance between the points is 2 − (-4), or 6 units. In general, we can find the distance between two points that lie on a line parallel to the x-axis by subtracting their x-coordinates.
  • Say: On your worksheet, plot point G at coordinates (-2, -1).
  • Ask: How do you think we could find the distance between points G and D? (We could count them on the graph, or we could subtract their y-coordinates.) Let's do both. If we subtract the y-coordinates, we get 6 − (-1), or 7, units.
    Have students count to check that the answer is 7.

    Distribute the second worksheet, which has a triangle and a rectangle drawn on it as shown below.

  • Say: Look at rectangle WXYZ and write the ordered pairs.
    Give students some time to label the points, and then ask a volunteer to read off the coordinates for each point.
  • Ask: Who can find the length of side WX without counting? (4 units) How did you figure that out? (subtracted the x-coordinates: 6 − 2 = 4) Find the length of XY the same way. (10 units)
    Have a volunteer explain how to do it.
  • Ask: How do we find the area of rectangles? (Multiply the length times the width.) Good. So we can find the area of rectangle WXYZ by multiplying: 10 x 4 = 40 square units.
    Have students check the answer by counting the square units.
  • Ask: Look at triangle ABC. What is the length of side AC? (3 units) What do we call AC? (base) What is the length of side BC? (6 units) What do we call BC? (height) Who remembers the formula for the area of a triangle?
    (A = one-halfbh)
  • Say: Using the formula, find the area of the triangle. (9 units)
    When students have finished, ask a volunteer to show the work on the board.

Houghton Mifflin Math Grade 6