Slope From Two Points or a Graph

Slope takes two points and compresses the whole tilt of a line into a single number. Pick two points (x1, y1) and (x2, y2). The rise is the change in y, y2 minus y1. The run is the change in x, x2 minus x1. Slope is rise divided by run. A steep line climbs a lot of y for a little x, so that ratio is large. A flat horizontal line has no rise at all, so the ratio is zero no matter how far you run.

On a graph, you don't need the formula if the line passes through clean lattice points. Pick two points where the line crosses grid intersections, count the boxes up or down between them (that's the rise), then count the boxes left or right (that's the run). Divide. Going down counts as negative rise; going left counts as negative run. This is the same formula, just counted by eye instead of subtracted with coordinates.

Slope is the natural next step after distance and midpoint, because all three turn two points into one descriptive number: distance gives size, midpoint gives location, slope gives direction. Slope then feeds directly into writing the equation of a line, since point-slope and slope-intercept forms both start by finding this number first. It also sets up parallel and perpendicular line questions, where equal slopes or negative reciprocal slopes are the whole test.