In this section, we will explore the concepts of slope.
Using rubber bands on a geoboard gives 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. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill.
Doing the Manipulative Mathematics activity "Exploring Slope" will help you develop a better understanding of the slope of a line.
We’ll start by stretching a rubber band between two pegs to make a line as shown in the image below.
Slope of a line
The slope of a line is
m=runrise.
The rise measures the vertical change and the run measures the horizontal change.
What is the slope of the line on the geoboard in the image above?
example
What is the slope of the line on the geoboard shown?

Solution
Use the definition of slope.
m=runrise
Start at the left peg and make a right triangle by stretching the rubber band up and to the right to reach the second peg.
Count the rise and the run as shown.
The rise is 3units.The run is4units.m=run3m=43The slope is 43.
example
What is the slope of the line on the geoboard shown?
Answer:
Solution
Use the definition of slope.
m=runrise
Start at the left peg and make a right triangle by stretching the rubber band to the peg on the right. This time we need to stretch the rubber band down to make the vertical leg, so the rise is negative.
The rise is −1.The run is3.m=run−1m=3−1m=−31The slope is −31.
example
Use a geoboard to model a line with slope
21.
Answer:
Solution
To model a line with a specific slope on a geoboard, we need to know the rise and the run.
Use the slope formula. |
m=runrise |
Replace m with 21 . |
21=runrise |
So, the rise is
1 unit and the run is
2 units.
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 with a slope of
21.
try it
Use a geoboard to model a line with the given slope:
m=31.
Answer:
Use a geoboard to model a line with the given slope:
m=23.
Answer:
example
Use a geoboard to model a line with slope
4−1
Answer:
Solution
Use the slope formula. |
m=runrise |
Replace m with −41 . |
−41=runrise |
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 to the right
4 units.

The hypotenuse of the right triangle formed by the rubber band represents a line whose slope is
−41.
try it
Use a geoboard to model a line with the given slope:
m=2−3.
Answer:
Use a geoboard to model a line with the given slope:
m=3−1.
Answer: