Understanding the Slope of a Line
Use Geoboards to Model Slope
A geoboard is a board with a grid of pegs on it. Using rubber bands on a geoboard gives us a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line.
Manipulative Mathematics
Doing the Manipulative Mathematics activity "Exploring Slope" will help you develop a better understanding of the slope of a line. (Graph paper can be used instead of a geoboard, if needed).
We'll start by stretching a rubber band between two pegs as shown in Figure 4.19.
Figure 4.19
Doesn't it look like a line?
Now we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle, as shown in Figure 4.20.
Figure 4.20
We carefully make a 90º angle around the third peg, so one of the newly formed lines is vertical and the other is horizontal.
To find the slope of the line, we measure the distance along the vertical and horizontal sides of the triangle. The vertical distance is called the rise and the horizontal distance is called the run, as shown in Figure 4.21.
Figure 4.21
If our geoboard and rubber band look just like the one shown in Figure 4.22, the rise is 2. The rubber band goes up 2 units. (Each space is one unit).
Figure 4.22 The rise on this geoboard is 2, as the rubber band goes up two units.
What is the run?
The rubber band goes across 3 units. The run is 3 (see Figure 4.22).
The slope of a line is the ratio of the rise to the run. In mathematics, it is always referred to with the letter \(m\).
Slope of a Line
The slope of a line of a line is \(m=\frac{\text { rise }}{\text { run }}\).
The rise measures the vertical change and the run measures the horizontal change between two points on the line.
What is the slope of the line on the geoboard in Figure 4.22?
\(\begin{align} \begin{array}{l} m=\frac{\text { rise }}{\operatorname{run}} \\ m=\frac{2}{3} \end{array} \end{align}\)
The line has slope \(\dfrac{2}{3}\). This means that the line rises 2 units for every 3 units of run.
When we work with geoboards, it is a good idea to get in the habit of starting at a peg on the left and connecting to a peg to the right. If the rise goes up it is positive and if it goes down it is negative. The run will go from left to right and be positive.
Example 4.25
What is the slope of the line on the geoboard shown?
Solution
Use the definition of slope: \(m=\frac{\text { rise }}{\text { run }}\).
Start at the left peg and count the spaces up and to the right to reach the second peg.
| The rise is 3. | \(m=\frac{3}{\mathrm{run}}\) |
| The run is 4. | \(m=\frac{3}{4}\) |
| The slope is \(\frac{3}{4}\). |
Table 4.29
This means that the line rises 3 units for every 4 units of run.
Try It 4.49
What is the slope of the line on the geoboard shown?
Try It 4.50
What is the slope of the line on the geoboard shown?
Example 4.26
What is the slope of the line on the geoboard shown?
Solution
Use the definition of slope: \(m=\frac{\text { rise }}{\text { run }}\).
Start at the left peg and count the units down and to the right to reach the second peg.
| The rise is −1. | \(=\frac{-1}{\mathrm{run}}\) |
| The run is 3. | \(\begin{align} \begin{array}{l} m=\frac{-1}{3} \\ m=-\frac{1}{3} \end{array} \end{align}\) |
| The slope is \(−\frac{1}{3}\). |
Table 4.30
This means that the lne drops 1 unit for every 3 units of run.
Try It 4.51
What is the slope of the line on the geoboard?
Try It 4.52
What is the slope of the line on the geoboard?
Notice that in Example 4.25 the slope is positive and in Example 4.26 the slope is negative. Do you notice any difference in the two lines shown in Figure 4.23(a) and Figure 4.23(b)?
Figure 4.23
We 'read' a line from left to right just like we read words in English. As you read from left to right, the line in Figure 4.23(a) is going up; it has positive slope. The line in Figure 4.23(b) is going down; it has negative slope.
Positive and Negative Slopes
Example 4.27
Use a geoboard to model a line with slope \(\dfrac{1}{2}\).
Solution
To model a line on a geoboard, we need the rise and the run.
| Use the slope formula. | \(m=\frac{\text { rise }}{\text { run }}\) |
| Replace \(m\) with \(\frac{1}{2}\). | \(\frac{1}{2}=\frac{\text { rise }}{\text { run }}\) |
Table 4.31
So, the rise is 1 and the run is 2.
Start at a peg in the lower left of the geoboard.
Stretch the rubber band up 1 unit, and then right 2 units.
The hypotenuse of the right triangle formed by the rubber band represents a line whose slope is \(\dfrac{1}{2}\).
Try It 4.53
Model the slope \(m=\frac{1}{3}\). Draw a picture to show your results.
Try It 4.54
Model the slope \(m=\frac{3}{2}\). Draw a picture to show your results.
Example 4.28
Use a geoboard to model a line with slope \(m=\frac{-1}{4}\).
Solution
| Use the slope formula. | \(m=\frac{\text { rise }}{\text { run }}\) |
| Replace \(m\) with \(\frac{-1}{4}\). | \(\frac{-1}{4}=\frac{\text { rise }}{\text { run }}\) |
Table 4.32
So, the rise is −1 and the run is 4.
Since the rise is negative, we choose a starting peg on the upper left that will give us room to count down.
We stretch the rubber band down 1 unit, then go to the right 4 units, as shown.
The hypotenuse of the right triangle formed by the rubber band represents a line whose slope is \(\frac{-1}{4}\).
Try It 4.55
Model the slope \(m=\frac{-2}{3}\). Draw a picture to show your results.
Try It 4.56
Model the slope \(m=\frac{-1}{3}\). Draw a picture to show your results.