4.4 Understand slope of a line  (Page 2/14)

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.

$\begin{array}{ccc}\text{The rise is 3.}\hfill & & m=\frac{3}{\text{run}}\hfill \\ \text{The run is 4.}\hfill & & m=\frac{3}{4}\hfill \\ & & \text{The slope is}\frac{3}{4}.\hfill \end{array}$

This means that the line rises 3 units for every 4 units of run.

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.

$\begin{array}{cccc}\text{The rise is}\mathrm{-1}.\hfill & & & m=\frac{\mathrm{-1}}{\text{run}}\hfill \\ \text{The run is 3.}\hfill & & & m=\frac{\mathrm{-1}}{3}\hfill \\ & & & m=-\frac{1}{3}\hfill \\ & & & \text{The slope is}-\frac{1}{3}.\hfill \end{array}$

This means that the line drops 1 unit for every 3 units of run.

Solution

To model a line on a geoboard, we need the rise and the run.

$\begin{array}{ccccc}\text{Use the slope formula.}\hfill & & \hfill m& =\hfill & \frac{\text{rise}}{\text{run}}\hfill \\ \text{Replace}m\text{with}\frac{1}{2}.\hfill & & \hfill \frac{1}{2}& =\hfill & \frac{\text{rise}}{\text{run}}\hfill \end{array}$

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 $\frac{1}{2}$ .

Solution

$\begin{array}{ccccc}\text{Use the slope formula.}\hfill & \hfill & \hfill m& =\hfill & \frac{\text{rise}}{\text{run}}\hfill \\ \text{Replace}m\text{with}\frac{\mathrm{-1}}{4}.\hfill & \hfill & \hfill \frac{\mathrm{-1}}{4}& =\hfill & \frac{\text{rise}}{\text{run}}\hfill \end{array}$

So, the rise is $\mathrm{-1}$ 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 $\frac{\mathrm{-1}}{4}$ .

Solution