Slope of a Line

Geometry · Axiom Academy

Measuring how steep a line is, and which way it leans. Slope measures the rate of change of a line. Take any two points on it: the vertical change between them is the rise , the horizontal change is the run , and the slope is their ratio. Watch a point climb the line below — it drags a right triangle whose orange leg is the rise and teal leg is the run. As the triangle was built, its two legs gave the slope directly. Writing the rise and run with the two points' coordinates gives the slope formula : Rise = vertical change (up positive, down negative) · Run = horizontal change (right positive, left negative) The sign of m tells you which way a line leans, and two boundary cases sit between "up" and "down." Watch one line below pivot about a fixed point: it starts steeply rising , flattens to horizontal , tilts into falling , then stands straight up where the slope is undefined . The live readout shows m change as it turns. Each orientation the line passed through is one of the four cases: m > 0 : the line rises from left to right. m < 0 : the line falls from left to right. m = 0 : a horizontal line — the rise is 0 . A vertical line — the run is 0 , so you would divide by 0 . 3. Reading Rise and Run off a Grid On a coordinate grid you can just count the rise and the run between two points, then divide. The line below passes through (2, 1) and (8, 5) . Watch the figure step across the grid (the run) and lift up (the rise), counting squares as it goes.

This is the written version of the interactive lesson above. See the full Geometry course.