Slope and rate of change
The slope of a line measures how steep it is: the rate of change between any two points on the line, like rising 2 units for every 1 unit across, so a line through (1, 3) and (2, 5) has slope (5-3)/(2-1) = 2/1 = 2. Similar triangles formed along the line explain why this rate stays the same no matter which two points you pick. For a curve, this same idea (the change in output divided by the change in input) gives the average rate of change over an interval, though unlike a line's slope, it's different for every different interval you pick.
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