Slope
Two lines both climb, but one creeps and one shoots up. What single number says how steep a line is, and why is it the same everywhere along the line?
Where you are 4 inside
What this area is about
Slope is rise over run: how many units a non-vertical line goes up for each unit it goes across, with a sign that says whether it climbs or falls. It is the rate of change of a linear relationship, and the m in y = mx + b.
How these fit together
The idea. Rate of change is the change in output divided by the change in input, with units such as dollars per hour. Slope is that rate for a straight line.
Procedures. Slope from a graph or a table counts the rise and the run between two points. The slope formula and the four kinds of slope computes \displaystyle m = \frac{y_2 - y_1}{x_2 - x_1} from coordinates alone, valid when x_2 \ne x_1, and sorts lines into positive, negative, zero and undefined slope.
A result. Slope triangles explains why any two points on a non-vertical line give the same slope: on a slanted line, the right triangles drawn along it are scaled copies of each other.
A path through it
Rate of change first, then slope from a graph or table, then the slope formula, and last slope triangles, which proves what the earlier lessons kept finding.
Next: slope-intercept form, direct variation and, much later, the derivative at a point.