On a graph, the rate of change can be counted directly from the grid.
Por qué funciona
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 $$\text{slope} = \frac{\text{rise}}{\text{run}}$$
Idea clave
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
Ejemplo resuelto: 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 $= \frac{6}{4} = \frac{3}{2}$.
Respuesta: $\frac{3}{2}$
Ejemplo resuelto: 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.
Respuesta: $-\frac{2}{3}$
Ejemplo resuelto: 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 $\frac{2}{3}$.
Respuesta: $\frac{5}{3}$
Error común
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.