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Algebra 2 9-12 / Week 07 / Garden Day
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Week 07 · Logarithms

Garden Day

Garden Day: the doubling tray
// The exponent, found backwards
⏱ about 20 min

Garden Day: The Doubling Tray

Saturday the seed tray comes to the kitchen table with a bag of dried beans and a stack of index cards.

"Three beans in the first cell," Comet says. "Each round, every bean gets a partner. Double them."

"Three, six, twelve, twenty-four," someone counts. "How many rounds until ninety-six?"

Wren holds up a card with 3 × 2ʳ = 96 on it. "Before you count, write the logarithm."

Nova dims her light to a hint. "Would you like a hint? Divide by three first."

"2ʳ = 32, so r = log₂ 32 = 5," comes the answer. The beans agree, five rounds later.

"Now the backwards card," Comet says. "Start at 96 and halve. When do you get to three?"

  1. A logarithm is an exponent: log₂ 128 = 7 because 2⁷ = 128. The sentence log_b N = k means b to the k equals N.
  2. To solve a × bᵗ = d: divide by a, take logs, divide by log b. Check against a table.
  3. A quantity that shrinks by a steady factor, like the jug's gap, is read with the same logarithm.
  4. y = log₂ x is the inverse of y = 2ˣ, its mirror across y = x. It passes (1, 0) and never touches the y-axis.
  5. Shifts and stretches move the log graph the same way they move any graph.
◇ FAMILY DOUBLING TRAY
Put 3 beans in one cell of a seed tray or egg carton. Each round, double the beans into the next cell.
Before each round, someone writes the power sentence: 3 × 2ʳ. Someone else writes the matching log sentence.
Pick a target like 96 or 192. Write 3 × 2ʳ = target on a card and solve it with a logarithm before the beans get there.
Trade roles so everyone writes a card and everyone solves one. Beans are counted and put away, never eaten.
For a grown-up
This is a counting activity with dried beans. Keep beans away from small children and off the floor.
A wrong answer is a good moment: double the beans one more round and see which side does not match.
The doubling is the family's own game, not a fact about any real plant or seed.
GARDEN DAY CHECK
  • Read the question.
  • Tap your answer.
The tray starts with 3 beans and doubles each round. After how many rounds are there 96 beans?
A card reads 3 × 2ʳ = 192, so 2ʳ = 64. What is r = log₂ 64?
A card reads 3 × 2ʳ = 384. How many rounds of doubling does it take? Type the number.
WHY THIS EXERCISEEvery card is the same shape. Divide away the start, then the logarithm is the number of rounds.
Draw the tray as a row of cells. Write the bean count in each cell and the round number above it.
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