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Algebra 2 9-12 / Week 06 / Garden Day
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Week 06 · Exponential Functions and Geometric Series

Garden Day

Garden Day: the doubling cards
// Duckweed doubling across the trough
⏱ about 20 min

Garden Day: The Doubling Cards

Saturday the beans come home in a jar, and the kitchen table becomes the trough.

"Each card has a start and a ratio," Comet says. "You find the day 5 count, then the total of days 1 to 5."

The first card reads "start 3, ratio 2." Someone lines up beans and calls out "Forty-eight, and ninety-three in all!"

"How did you get the total so fast?" Wren asks. "Double the last pile and subtract three," comes the answer.

Nova hovers above the jar. "Would you like to check it with the formula? a times r to the n minus 1, over r minus 1."

"3 times 31 over 1. Ninety-three," Comet says. "The beans and the formula agree."

"Next card has ratio one half," Wren says. "Watch the signs."

  1. An exponential function multiplies by the same factor each step: N(d) = 6 × 2ᵈ. Build it from a table or two points.
  2. Rewrite the exponent to read a different rate: 1.1ᵈ = (1.1⁷)^(d/7), a weekly factor of about 1.95.
  3. A geometric sequence has a recursive formula, aₙ = r × aₙ₋₁, and an explicit one, aₙ = a × rⁿ⁻¹.
  4. The sum of n terms is a(rⁿ - 1)/(r - 1) when r is not 1. Derive it by subtracting S from rS.
  5. Schedules that multiply each day are geometric series, and the formula totals them.
◇ FAMILY DOUBLING CARDS
Write five cards, each with a start and a ratio, like "start 3, ratio 2" or "start 16, ratio 1/2."
Deal a card. Everyone writes the first five terms, then the day 5 term from the explicit formula.
Total the five terms two ways: by adding, and by the formula a(r⁵ - 1)/(r - 1). Check they agree.
Trade roles so everyone writes a card and everyone totals one. Use beans or buttons to check a small card.
For a grown-up
This is a paper and bean activity. Beans are counters: nothing goes in the mouth, and keep them from small children and pets.
A wrong total is a good moment: list the five terms and add them by hand, then compare with the formula.
Every start and ratio on the cards is made up for the game, not a fact about any real plant.
GARDEN DAY CHECK
  • Read the question.
  • Tap your answer.
A geometric sequence starts at 3 and multiplies by 2 each time. What is term 5?
A geometric series starts at 3 with ratio 2. What is the sum of the first 5 terms?
A card reads "start 16, ratio 1/2." What is the total of the first 5 terms?
A card reads "start 1, ratio 2." What is the total of the first 10 terms? Type the number.
WHY THIS EXERCISEThe formula turns ten additions into one calculation, and the doubling pattern makes the answer one less than 2¹⁰.
Draw two of the family cards with their first five terms, and write the total under each.
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