Slope and Rate of Change
Define slope as rise over run and calculate it from two points or a graph. Interpret slope as a constant rate of change and distinguish positive, negative, zero, and undefined slopes in context.
A lesson of Pre-Algebra, in The Coordinate Plane & Linear Relationships, about 10 minutes.
Slope
By the end of this lesson, you will be able to calculate slope from two points or a graph, interpret slope as a rate of change, and classify lines by their slope.
Definition of Slope
Slope measures the steepness and direction of a line.
$$m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$$
where $(x_1, y_1)$ and $(x_2, y_2)$ are any two points on the line.
Rise: the vertical change ($\Delta y$). Run: the horizontal change ($\Delta x$).
Calculating Slope
Example: Find the slope through $(2, 3)$ and $(5, 9)$.
$m = \frac{9-3}{5-2} = \frac{6}{3} = 2$
For every 1 unit you move right, the line goes up 2 units.
Example: Find the slope through $(-1, 4)$ and $(3, -2)$.
$m = \frac{-2-4}{3-(-1)} = \frac{-6}{4} = -\frac{3}{2}$
Negative slope — the line goes downward from left to right.
Classifying Lines by Slope
| Slope | Line | Behavior |
|---|---|---|
| $m > 0$ | Rising | Goes up left to right |
| $m < 0$ | Falling | Goes down left to right |
| $m = 0$ | Horizontal | $y = k$ (constant) |
| Undefined | Vertical | $x = k$ |
Steeper lines have larger $|m|$. A slope of $-5$ is steeper than a slope of $-2$.
Slope as Rate of Change
In real-world contexts, slope = rate of change.
Example: A car travels at constant speed. After 2 hours it has gone 120 miles; after 5 hours, 300 miles.
$m = \frac{300-120}{5-2} = \frac{180}{3} = 60$ miles per hour
The slope is the speed — how fast the distance is increasing per hour.
Units of slope: $\frac{\Delta y \text{ units}}{\Delta x \text{ units}}$
Slope from a Graph
Count the rise and run between two lattice points:
- Choose two clear points on the line
- Count the vertical change (rise)