Eleventh Grade · Math · week 26

Exponential growth and decay

Model exponential growth and decay, including compound interest and half-life, and evaluate the models.

Exponential growth and decay both follow the same shape, y = a*b^x, where a is the starting amount and b is the growth factor per time period: b > 1 means growth, and 0 < b < 1 means decay. Compound interest is the most common growth example: A = P(1 + r/n)^(nt), where P is the principal, r is the annual rate, n is the number of times it compounds per year, and t is the number of years.

Half-life is the classic decay example: the amount remaining after time t is A = A0 * (1/2)^(t / h), where h is the half-life -- the exponent counts how many "half-life periods" have elapsed, not the raw time itself. Both formulas are really the same idea underneath: repeated multiplication by a fixed factor, tracked through an exponent.

📋 For the teacher
Teach it (10–15 min):
  • Walk through the compound interest formula piece by piece the first time, naming what each letter means in plain words before plugging in any numbers.
  • Have them compute r/n and nt as their own separate mini-steps before touching the main formula -- rushing this is where most errors happen.
  • For half-life, emphasize that the exponent is elapsed time DIVIDED BY the half-life, not the elapsed time itself -- a common and costly mix-up.
  • Ask "growth or decay?" for every new scenario before computing anything, just by checking whether the base is above or below 1.
Talk about it:
  • What does each letter in A=P(1+r/n)^(nt) represent?
  • Why does the half-life formula divide elapsed time by the half-life inside the exponent?
  • How can you tell growth from decay just by looking at the base of an exponential model?

Mastery looks like: They correctly set up and evaluate a compound interest or half-life model, identifying growth versus decay from the base.

Common stumbles: Using elapsed time directly as the exponent in a half-life problem instead of dividing by the half-life first; mixing up which number is the rate and which is the number of compounding periods.

See it visually

Interactive, spoken-aloud tools that show this idea in motion.

ExponentsRepeated multiplication and how fast powers grow.try →
Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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