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Algebra 2 9-12 / Week 06 / Wednesday
3/6
Week 06 · Exponential Functions and Geometric Series

Wednesday

Glass House Lab: the bean tray
// Duckweed doubling across the trough
⏱ about 20 min

Wednesday: Glass House Lab, the Bean Tray

Comet tips a bag of dried beans onto the potting bench and lines up six index cards. "Lab day. We model the trough."

"Three beans on day one," she says, placing them on the first card. "Double each day."

Wren counts along. "Six. Twelve. Twenty-four." The piles grow fast. "What do you notice about the last card?"

"It is almost as big as all the others together," Comet says. "Ninety-six beans."

Nova projects a running total above the bench. "Would you like a hint? Add the cards one by one and watch the sum."

"Three, nine, twenty-one, forty-five," Comet reads. "Each total is one less than the next pile times... something."

"Hold that thought for tomorrow," Wren says. "Today, count everything and write both formulas."

What you need

  • A bag of dried beans, or buttons, or paper squares cut with scissors.
  • Six index cards numbered day 1 to day 6, and your Glass House Log.
  • A tray or a towel to keep the beans from rolling off the bench.
Safety first
Beans are counters, not snacks: nothing goes in the mouth. Keep them away from small children and pets.
Cut paper squares with scissors on cardboard. Work at a bench, not on a ladder, and lift nothing heavy alone.
Sweep up stray beans so no one slips on them.

Run the lab

  1. Put 3 beans on the day 1 card. Record it as term 1.
  2. For each next card, count the beans on the card before and place twice as many.
  3. Stop at day 6. Record each card's count in your table.
  4. Write the recursive formula and the explicit formula for your bean sequence.
  5. Add the cards one at a time and record the running total. Keep it for tomorrow.

The crew's bean table

These are the crew's own counts from Nova's log, made up for the Glass House. With the same start, yours should match.

Day nBeans aₙRunning total
133
269
31221
42445
54893
696189

Recursive: a₁ = 3 and aₙ = 2 × aₙ₋₁. Explicit: aₙ = 3 × 2ⁿ⁻¹. Day 6 is 3 × 2⁵ = 96.

All six cards together hold 189 beans. Notice that 189 is 3 less than 96 × 2 = 192. Tomorrow you see why.

The beans are a model of the duckweed, and the duckweed rule is the crew's model of the trough. Models all the way down.

COUNT THE CARDS
  • Read the question.
  • Tap your answer.
A geometric sequence starts at 3 and multiplies by 2 each time. What is term 6?
Beans start at 3 on day 1 and double each day. How many are on the card 4 days later, day 5?
How many beans are on all 6 cards together?
Using aₙ = 3 × 2ⁿ⁻¹, how many beans are on the day 5 card? Type the number.
WHY THIS EXERCISEThe explicit formula jumps straight to any card. The exponent is one less than the day number.
How many beans are on the day 3 card? Type the number.
What is the running total after day 4? Type the number.
In the recursive formula aₙ = 2 × aₙ₋₁, what is the ratio? Type the number.
StatementTrue or false?
3 × 2 × 2 × 2 × 2 × 2 = 96?
3 + 6 + 12 + 24 + 48 + 96 = 189?
The day 6 card alone holds more beans than days 1 to 5 together.?
The bean sequence 3, 6, 12, 24 is arithmetic.?
Term n of a geometric sequence uses the exponent n - 1.?
WHY THIS EXERCISEThe last pile beating all the earlier piles together is the signature of doubling.
Draw your six cards with the bean counts. Under each card, write the running total so far.

Great lab work. Tomorrow you prove the pattern in the running totals: the sum of a geometric series.

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