# Rise Over Run - The Slope Method Explained  
## What Is Rise Over Run?  
**Rise over run** is the method for measuring the **slope** (or [gradient](/content/math/geometry/gradient-of-a-line/index.html)) of a straight line. "Rise" is how far the line goes **up or down** – the vertical change, written $\Delta y$. "Run" is how far it goes **across** – the horizontal change, written $\Delta x$. The slope is one divided by the other:  
$$\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$$  
Where "gradient" and "slope" name the _quantity_, rise over run names the _procedure_ for getting it. You either count the rise and run as steps on a graph, or subtract coordinates from two known points.  
**Reading the sign as you count:** moving up is a positive rise, moving down is a negative rise; moving right is a positive run. So a line falling left to right gives a negative rise over a positive run - a **negative** slope. A line climbing gives a positive over positive - a **positive** slope.  
Roofers, road engineers, and stair builders all size their work with one phrase — "rise over run" - long before anyone writes $y = mx + c$.  
## Examples of Rise Over Run  
### Example 1  
**A line rises $3$ units for every $4$ units it runs to the right. What is its slope?**  
Put the rise over the run:  
$$\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{3}{4}$$  
Final answer: the slope is $\dfrac{3}{4}$.  
### Example 2  
**Find the slope of the line through $(1, 2)$ and $(4, 8)$ using rise over run.**  
Your first instinct might be to count the run first because $x$ comes first, writing run over rise: $\dfrac{4-1}{8-2} = \dfrac{3}{6} = \dfrac{1}{2}$. Let's check that against the picture.  
From $(1,2)$ to $(4,8)$ the line climbs $6$ but only moves across $3$, so it rises faster than it runs - the slope must be greater than $1$. A value of $\tfrac{1}{2}$ is too small, which flags the fraction as upside down.  
Rise goes on top:  
$$\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{8 - 2}{4 - 1} = \frac{6}{3} = 2$$  
A slope of $2$ matches the steep climb, correcting the flip.  
Final answer: the slope is $2$.  
### Example 3  
**Find the slope of the line through $(2, 7)$ and $(6, 3)$.**  
Subtract in the same order for both, second point minus first:  
$$\text{slope} = \frac{3 - 7}{6 - 2} = \frac{-4}{4} = -1$$  
The rise is negative because the line drops as it moves right.  
Final answer: the slope is $-1$, a falling line.  
### Example 4  
**On a graph, a line goes down $5$ squares and right $2$ squares between two marked points. What is its slope?**  
Down $5$ is a rise of $-5$; right $2$ is a run of $+2$:  
$$\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{-5}{2}$$  
Final answer: the slope is $-\dfrac{5}{2}$, a steep falling line.  
### Example 5  
**A wheelchair ramp rises $1$ m over a run of $12$ m. What is its slope, and why does the number matter?**  
$$\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{1}{12} \approx 0.083$$  
Final answer: the slope is $\dfrac{1}{12}$. Accessibility codes cap ramps near this value so the climb is gentle enough to use safely, which is exactly why rise over run is the language builders use.  
### Example 6  
**Does the order of the two points change the slope? Use $(1, 2)$ and $(4, 8)$ both ways.**  
Point $2$ minus point $1$:  
$$\frac{8 - 2}{4 - 1} = \frac{6}{3} = 2$$  
Now swap which point is first:  
$$\frac{2 - 8}{1 - 4} = \frac{-6}{-3} = 2$$  
Both give $2$, because flipping the order negates the top and the bottom together.  
Final answer: the slope is $2$ either way - order does not matter, as long as you subtract $x$ and $y$ in the _same_ order.  
## Why Rise Over Run Matters: "Turning Steepness Into a Countable Fraction"  
Rise over run is older than coordinate geometry - it is how builders described steepness for centuries: the **pitch of a roof, the grade of a road, the fall of a drainpipe.** The method matters because it makes steepness _countable_: you do not need the equation of a line to know how steep it is, only two points and a subtraction.  
- **You can do it straight from a picture.** On graph paper, count squares up for the rise and squares across for the run, then form the fraction. No formula required.  
- **It reveals direction as it goes.** Because a downward rise is negative, the sign falls out of the counting itself - the method tells you whether the line climbs or falls.  
- **It scales to any two points.** Given coordinates, "rise over run" becomes $\dfrac{y_2 - y_1}{x_2 - x_1}$, the same idea written with subtraction.  
Drainage is the unforgiving version of this: a pipe laid with too small a rise over run will not carry water away, and one laid too steep lets water outrun the solids and clog. Plumbers work to a standard [drainage fall](https://en.wikipedia.org/wiki/Drainage), often around a rise of $1$ over a run of $40$, computed as rise over run before a single pipe is cut. The method turns "steep enough, but not too steep" into a fraction a code can specify.  
## Common Mistakes With Rise Over Run  
### Mistake 1: Putting run over rise  
**Where it slips in:** Counting the horizontal move first and placing it on top.  
**Don't do this:** Writing $\dfrac{\text{run}}{\text{rise}}$, so a line rising $6$ over a run of $3$ gets a slope of $\tfrac{1}{2}$.  
**The correct way:** Rise goes on **top**, run on the **bottom**: $\dfrac{\text{rise}}{\text{run}} = \dfrac{6}{3} = 2$. The rusher who counts across first and forgets to check often ends up with the reciprocal; a quick check against the graph catches it.  
### Mistake 2: Subtracting the coordinates in different orders  
**Where it slips in:** Taking $y_2 - y_1$ on top but $x_1 - x_2$ on the bottom.  
**Don't do this:** Computing $\dfrac{y_2 - y_1}{x_1 - x_2}$, which flips the sign of the slope.  
**The correct way:** Keep the **same point first** in both the numerator and the denominator: $\dfrac{y_2 - y_1}{x_2 - x_1}$. The second-guesser who recomputes with the points swapped, then panics at a sign change, usually just mismatched the order on one line. Swapping _both_ is fine; swapping _one_ is the error.  
### Mistake 3: Losing the sign of a downward rise  
**Where it slips in:** Counting a line that drops but recording the rise as positive.  
**Don't do this:** For a line going down $4$ and right $2$, writing $\dfrac{4}{2} = 2$ and calling it positive.  
**The correct way:** Down is a **negative** rise: $\dfrac{-4}{2} = -2$. The sign is part of the answer - it says the line falls. The memorizer who recalls "rise over run" as a bare fraction but drops the minus sign turns a falling line into a rising one.  
## Conclusion  
- **Rise over run** is the method for finding slope: $\dfrac{\text{rise}}{\text{run}} = \dfrac{\Delta y}{\Delta x} = \dfrac{y_2 - y_1}{x_2 - x_1}$.  
- **Rise** is the vertical change (up positive, down negative); **run** is the horizontal change.  
- Rise always goes on top; putting run on top gives the reciprocal, not the slope.  
- Subtract $x$ and $y$ in the **same order** - the point order does not change the slope.  
- The sign of the rise carries the direction: a falling line has a negative slope.  
## Practise What You Have Learned  
Work through these to test your understanding: find the slope through $(0, 0)$ and $(4, 6)$ using rise over run (Answer to Question 1: $\tfrac{3}{2}$); a line drops $6$ and runs right $3$ - state its slope (Answer to Question 2: $-2$); and confirm the slope through $(5, 1)$ and $(2, 7)$ is the same taken both ways.