Slope

Slope is a number that describes how steep a line is and which way it tilts. A bigger number means a steeper line. A negative number means the line runs downhill as you move to the right instead of uphill.

Rise over run

The standard way to find slope is "rise over run": pick two points on the line, count how far up or down you moved between them (the rise), then count how far left or right you moved (the run), and divide the first number by the second. A line that goes up 2 for every 1 it goes across has a slope of 2, which reads as "steep and climbing." A line that goes up 1 for every 4 it goes across has a slope of 1/4, which reads as "gentle and climbing." A line that drops as it goes right has a negative slope, and a perfectly flat, horizontal line has a slope of 0, since it never rises at all.

Reading it instead of only calculating it

The formula is only half of understanding slope. The other half is being able to look at a graphed line and describe it in plain words before doing any arithmetic at all: is it climbing or falling, and is it steep or gentle? A child who can answer that from the picture has a way to sanity-check the number they calculate afterward; if the line clearly falls left to right but their answer comes out positive, something in the count went wrong.

Picture an 8th grader, call her Alicia (an illustrative example, not a real student): she started catching her own sign errors once she got in the habit of describing the line out loud first ("this one's going down, so I already know my answer has to be negative") before she ever touched the rise-over-run formula.

Slope shows up again later as the "m" in the equation y = mx + b, and everything above still applies there: it is still just rise over run, describing the same steepness and direction, only written as part of an equation instead of read off a graph.

If seeing this worked through on a whiteboard would help, watch the free Explanova demo: a 75-second recording of Coach Nova working a problem step by step (a tape-diagram example, not a slope problem, but the same walkthrough style).