Activity 3. - Determining the Slope Graphically.
Use the StraightLineEqn.xls tool, which displays a graph of
an equation of the form:
y = mx + b.
Do This:
Using the tool, first change the coefficient of x represented by “m” to .2 and the value of the constant term represented by “b” to 2. NOTE: To complete a change press the enter key after typing the new value.
Based on our discovery in Activity 2, we know that the graph
of the equation
y = 0.2x + 2 has a slope of .2
For this activity, we will use our spreadsheet tools to help us to clearly understand the concepts involved. In the grid at the top right hand corner, change the x-coordinates to 0, and 5, respectively. The tool will automatically calculate and change the corresponding y values. As you observe the graph note the changes and determine the vertical distance and the horizontal distance between the two points. The vertical change is called the rise while the horizontal change is called the run.
(a) What is the vertical change or Δy?
(b) What is the horizontal change or Δx?
(c) Now substitute the appropriate values and calculate the slope, remembering that graphically, the slope is determined in the following way:
Δy
change(y)
rise
Slope = ---- or ------------ or in words, ------
Δx change(x) run
(d) How does your answer compare with the slope that was determined by simply examining the equation? Did you get the same slope? If not, there is an error. Try again.
(e) Try the same activity by changing the x coordinates to -5 and 5.
(1) What is the vertical change or Δy?
(2) What is the horizontal change or Δx?
(3) Now calculate the slope, as done in part “c”.
(f) A regular graph spans the plane and is not cut off in small segments as we have done using the tool.
Now, inserting x coordinates of -10 and 10, try to locate a pair of points on the graph and use them to calculate the slope.
For this activity, we need to locate pairs of points at which the graph line passes through the intersection of a vertical line and a horizontal line shown on the graph.
(1) What is the vertical change or Δy?
(2) What is the horizontal change or Δx?
(3) Now calculate the slope.
(e) Give two reasons why you would expect to get the same slope with each pair of points on the graph for the given equation.
TEACHER NOTES
Just in case student continue to experience difficulty do it together.
When you inserted the x coordinates of 0 and 5, your corresponding y coordinates of 2 and 3 should have been displayed, resulting in the ordered pairs, (0, 2) and (5, 3). These are the end points of the line segment shown in the graph.
Let us do this together:
We need to remember that graphically, the slope is determined in the following way:
Δy change(y) rise
Slope = ---- or ------------ or in words, ------
Δx change(x) run
How do we get a numerical value using this method?
Let us consider the end points shown on the graph, (0. 2) and (5, 3). Using a pencil point and beginning with (0, 2) move horizontally in the direction of (5, 3) until the pencil point touches the vertical line passing through (5, 3). We have moved 5 spaces in the positive direction. This is a “run” of +5. Now moving the pencil point vertically in the direction of the second point, (5, 3 ), we move up 1 space to get to ( 5, 3 ). This is a rise of +1. Substituting the numbers obtained, we get
rise 1
Slope = ------ = ---- = 0.2
run 2
This lesson incorporates the following teaching strategies bases on the NCTM standards.
In addition the lesson meets the following technology guidelines specified by Garofolo, Drier, Harper, Timmerman, and Shockey (2000):