Rise Over Run
Rise over run is a way to describe slope: the vertical change (rise) divided by the horizontal change (run) between two points on a line.
Formula
\text{slope} = \frac{\text{rise}}{\text{run}}
Definition
Rise over run is a simple way to find slope: "rise" is how many units you move up or down between two points, and "run" is how many units you move to the right, so slope equals rise divided by run. Rise is positive going up and negative going down, while run is always measured in the positive (rightward) direction. This is the geometric interpretation of the difference quotient $(f(x + h) - f(x))/h$ as $h$ approaches $0$, which defines the derivative; for a linear function $f(x) = mx + b$, the difference quotient equals $m$ for every $h$, confirming that a line's slope is constant everywhere.
Example
Moving up $4$ units and right $2$ units gives slope $4/2 = 2$; moving down $3$ and right $1$ gives slope $-3/1 = -3$. From $(2, 1)$ to $(5, 7)$, rise $= 7 - 1 = 6$ and run $= 5 - 2 = 3$, so slope $= 6/3 = 2$, matching the formula $(y_2-y_1)/(x_2-x_1)$. For $f(x) = 3x + 1$, $(f(x+h) - f(x))/h = 3h/h = 3$, the rise over run, independent of $x$ and $h$.
Key Insight
Imagine walking up a hill: rise is how high you climb, run is how far forward you walk, and a steep hill has a big rise for a small run. You can read slope directly off a graph by picking two clear lattice points on the line and counting up (or down) and over. The difference-quotient version of this idea is exactly rise over run in the limit, and linear functions are the only ones where the ratio is constant everywhere, which is why they have one well-defined slope.