Slopes Beyond Graphs

A rate of change is how much one quantity shifts when another moves. Salary per year, calories per mile, temperature per hour — all rates of change. They compress the relationship between two moving quantities into a single, usable number.

You already think in rates constantly. "That restaurant is 5 miles away and takes 20 minutes to reach" combines distance and time into an average speed. "I burn about 100 calories per mile running" combines energy and distance. Making these rates explicit is the first step toward mathematical thinking.

The concept scales from the everyday to the scientific. GDP growth rate, tumor growth rate, and the rate at which glaciers retreat all use the same underlying math — change in one variable divided by change in another.