Average rate of change
A quantity speeds up, slows down, even turns around. What one number says how fast it changed between two moments?
Where you are 4 inside
What this area is about
When a graph is a straight line, one rate describes all of it. Most graphs are not straight, so the honest answer to “how fast?” depends on which stretch of inputs you mean. The average rate of change over an interval is the change in output divided by the change in input, \displaystyle \frac{f(b) - f(a)}{b - a} with a \ne b: the slope of the line joining two points of the graph.
How these fit together
Two procedures for one number. Average rate of change from a table or graph reads the two outputs off rows or points, and average rate of change from a formula works them out from the rule.
One expression for every interval. The difference quotient writes the rate over the interval from x to x + h as a single expression in x and h.
A new idea. Rates at an instant shrinks the interval around one point and watches the averages settle, which is how a rate can belong to a single moment.
A path through it
Tables and graphs or formulas first, in either order. Then the difference quotient, and last the rate at an instant, which needs it.
This area feeds concavity, where the rate itself changes, and leads straight into calculus through the derivative at a point.