On a graph, the rate of change can be counted directly from the grid.
Why it works
Between two points on a line, the run is the horizontal change and the rise is the vertical change. They are exactly the change in $x$ and the change in $y$, so $$\text{slope} = \frac{\text{rise}}{\text{run}}$$
Key idea
Pick two points where the line crosses grid corners. Going from left to right, count the rise (up is positive, down is negative) and the run, then divide rise by run. Check each axis’s scale before counting squares.
Worked examples
Worked example: Up and to the right
A line passes through the grid points $(-2, -1)$ and $(2, 5)$. Find its slope.
- Run: from $x = -2$ to $x = 2$ is 4 to the right.
- Rise: from $y = -1$ to $y = 5$ is 6 up.
- Slope $= \frac{6}{4} = \frac{3}{2}$.
Answer: $\frac{3}{2}$
Worked example: A falling line
A line goes through $(-3, 4)$ and $(3, 0)$. Find its slope.
- Run: 6 to the right.
- Rise: from 4 down to 0, which is $-4$.
- Slope $= \frac{-4}{6} = -\frac{2}{3}$. A line that falls from left to right has a negative slope.
Answer: $-\frac{2}{3}$
Worked example: Mind the scale
In the graph just above, find the slope.
- The line goes 3 squares right and 2 squares up.
- But each square across is 2 units and each square up is 5 units: run $= 3 \cdot 2 = 6$, rise $= 2 \cdot 5 = 10$.
- Slope $= \frac{10}{6} = \frac{5}{3}$. Counting squares alone would give $\frac{2}{3}$.
Answer: $\frac{5}{3}$
Common mistake
Dividing run by rise, or counting grid squares without reading the scale. Slope is rise over run, the change in $y$ over the change in $x$. Read actual coordinates from the axes before you count.