Ninth Grade · Math · week 7

Exponents & exponential growth

Apply product/quotient/power rules for exponents and contrast exponential vs linear growth.

Exponents are compressed multiplication: x⁵ means five x's multiplied. The rules follow from just counting factors: multiplying powers ADDS exponents (x³ · x⁴ = x⁷ — three x's then four more), dividing SUBTRACTS them, and a power of a power MULTIPLIES them ((x³)⁴ = x¹²). Anything (nonzero) to the zero power is 1 — you've multiplied no copies, leaving the multiplicative "nothing," which is 1.

Exponential growth means multiplying by the same factor each step: y = a·bˣ. It starts sleepy and then explodes past any straight line. Linear adds; exponential multiplies — that one word of difference is why doubling pennies beats a million dollars in a month.

📋 For the grown-up teacher
Teach it (10–15 min):
  • Derive each rule by writing the factors out ONCE — rules memorized without the "why" evaporate.
  • The rice-on-a-chessboard story (1 grain doubling per square) is the feel of exponential — tell it.
  • Build the two towers side by side in a table: +2 world vs ×2 world, ten rows. Let the table shock them.
  • Catch the classic error early: x³ · x⁴ is NOT x¹². Ask "how many x's are actually multiplied?"
Talk about it:
  • Why does multiplying powers add the exponents?
  • What's the difference between +5 each step and ×5 each step?
  • Where does exponential growth show up in real life?

Mastery looks like: They simplify with all three rules by counting factors, and can write y = a·bˣ from a doubling/tripling story.

Common stumbles: Multiplying exponents when they should add; treating b⁰ as 0; calling steep linear growth "exponential."

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