Developing math vocabulary
Rate of Change
Pronunciation: RAYT of CHAYNJ
A rate of change compares how much one quantity changes relative to a change in another quantity.
Δy/ΔxDeveloping
Plain language
What it means
A rate of change compares how much one quantity changes relative to a change in another quantity.
Formal meaning
Mathematical definition
Average rate of change over [a,b] is [f(b)-f(a)]/(b-a); instantaneous rate of change is a derivative.
Where it fits
Its place in mathematics
Rates connect slope, speed, unit rates, growth, decay, economics, science, and calculus.
Why it matters
The practical reason to learn it
They show not only that something changed, but how rapidly it changed relative to another measurement.
Worked example
Find an average rate
A distance changes from 20 km at 1 hour to 80 km at 3 hours. Find the average rate.
- Find the distance change: 80 - 20 = 60 km.
- Find the time change: 3 - 1 = 2 hours.
- Divide change in distance by change in time.
The average rate is 30 kilometers per hour.
Real-life example
Where this appears
A business owner tracks how quickly costs, sales, or inventory change over time.
Common mistake
What to watch for
Dividing final output by final input instead of dividing the changes over the stated interval.
Memory tip
Keep this in mind
Write delta output over delta input and include units.
Little-known fact
Keep curiosity alive
A rate can be negative, indicating that the output decreases as the input increases.
A profession that uses this idea
Trends are rates, not just totals
Businesses compare changes per day, customer, item, or dollar to understand growth and efficiency.
Explore Merchants & Business OwnersFollow the learning trail
Prerequisites, related ideas and next concepts
People behind the ideas
Related Math Heroes
Nicole Oresme
Nicole Oresme was mathematician, philosopher, and bishop. He represented changing qualities with geometric diagrams and studied fractional exponents and infinite series.
René Descartes
A French thinker who helped unite algebra and geometry by representing curves and shapes with coordinates and equations.
Put the idea to work