Activity 3. - Determining the Slope Graphically.

Use the StraightLineEqn.xls tool, which displays a graph ofan equation of the form:
y = mx + b.

Do This:

Using the tool, first change the coefficient of x representedby “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 graphof the equation
y = 0.2x + 2 has a slope of .2

For this activity, we will use our spreadsheet tools to helpus to clearly understand the concepts involved. In the grid at the top righthand corner, change the x-coordinates to 0, and 5, respectively. The tool willautomatically calculate and change the corresponding y values. As you observethe graph note the changes and determine the vertical distance and thehorizontal distance between the two points. The vertical change is called the rise while the horizontal change iscalled the run.

(a)      What is thevertical change or Δy?

(b)      What is thehorizontal change or Δx?

(c)      Nowsubstitute the appropriate values and calculate the slope, remembering thatgraphically, the slope is determined in the following way:

Δy              change(y)                               rise

Slope = ----    or       ------------      or in words,    ------

Δx               change(x)                                run

(d)      Howdoes your answer compare with the slope that was determined by simply examiningthe equation?  Did you get the sameslope?  If not, there is an error.  Try again.

(e)      Try the sameactivity 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 notcut off in small segments as we have done using the tool.

Now, inserting x coordinates of -10and 10, try to locate a pair of points on the graph and use them to calculatethe slope.

For this activity, we need to locate pairs ofpoints at which the graph line passes through the intersection of a verticalline 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 tworeasons why you would expect to get the same slope with each pair of points onthe graph for the given equation.

TEACHER NOTES

Just in case student continue to experience difficulty do ittogether.

When you inserted the x coordinates of 0 and 5, yourcorresponding y coordinates of 2 and 3 should have been displayed, resulting inthe ordered pairs, (0, 2) and (5, 3). These are the end points of the linesegment shown in the graph.

Let us do this together:

We need to remember that graphically, the slope isdetermined 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 thedirection of (5, 3) until the pencil point touches the vertical line passingthrough (5, 3). We have moved 5 spaces in the positive direction. This is a“run” of +5.  Now moving the pencilpoint vertically in the direction of the second point, (5, 3 ), we move up 1space to get to ( 5, 3 ). This is a rise of +1. Substituting the numbersobtained, we get

rise        1

Slope = ------  = ----   = 0.2

run        2

This lesson incorporates the following teaching strategiesbases on the NCTM standards.

• Technology integration
• Interactive learning
• Concept-building through discovery
• Actively building new knowledge from experience and prior knowledge
• Discourse in mathematics
• Formulate explanations
• Mathematical literacy
• Using modeling to solve problems
• Making connections by investigating various models

In addition the lesson meets the following technologyguidelines specified by Garofolo,Drier, Harper, Timmerman, and Shockey (2000):

• Applying technology to meaningful problems
• Employing technology to improve pedagogy
• Connect mathematical topics
• Use multiple representations