Ninth Grade · Math · week 18

Exponential models: growth and decay

Build and evaluate exponential growth and decay models of the form y = a(1 + r)^t and y = a(1 - r)^t.

A quantity that grows by the same PERCENT every period follows y = a(1 + r)^t: a is the starting amount, r is the growth rate written as a decimal, and t counts how many periods have passed. Each period multiplies the current amount by the growth factor (1 + r) - that one multiplication, repeated, is the entire model.

Decay works the same way with a factor below 1: y = a(1 - r)^t, since losing r of the amount each period leaves (1 - r) of it behind. Investments, populations, and viral spread are classic growth; car depreciation, medicine leaving the bloodstream, and radioactive decay are classic decay. The single most important habit: r is a rate (like 0.05 for 5%), never the whole factor - the factor is 1 + r or 1 - r.

📋 For the grown-up teacher
Teach it (10–15 min):
  • Physically build one period at a time on a calculator: start with a, multiply by the growth factor, multiply again, and watch t grow the exponent - the exponent IS "how many times we multiplied."
  • Insist r stays a decimal in the model itself - 5% becomes 0.05, and the FACTOR is 1.05, two separate ideas that get blurred constantly.
  • Connect to something they already track: a video/post's view count doubling, or a phone's battery percentage draining - name the real r if you can estimate it.
  • Round only the FINAL answer, never a step in the middle - rounding early quietly ruins the later multiplications.
Talk about it:
  • Why is the growth factor 1 + r instead of just r?
  • What real-life situation would you model with decay instead of growth?
  • If two towns start at the same size but grow at different rates, does the FASTER one always stay ahead by the same amount, or does the gap change?

Mastery looks like: They can build y = a(1±r)^t from a percent-change story and evaluate it correctly at a given t, keeping r as a decimal throughout.

Common stumbles: Using r itself as the base instead of 1±r (e.g., using 0.05 instead of 1.05); rounding in the middle of a multi-step calculation.

Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
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