Week 06 · Exponential Functions and Geometric Series

Monday

The duckweed count

// Duckweed doubling across the trough

⏱ about 20 min

Monday: The Duckweed Count

A few green specks float in the water trough by the back wall. Comet counts them with a pencil tip.

"Six duckweed fronds. Nova, log it." The next morning she counts again. "Twelve." The morning after: "Twenty-four."

Wren reads the log over her shoulder. "What do you notice? Not plus six each day."

"Times two each day," Comet says. "Six, twelve, twenty-four. Tomorrow should be forty-eight."

Nova projects the counts as rising dots on the glass. "Would you like a hint? Write the count after d days as one rule."

"Six times two to the d," Comet says. "An exponential function. Our own model, from our own counts."

"Then when does it cover the trough?" Wren asks. "That is this week's question, and next week's too."

Duckweed spreading across a water trough in the Glass House while Nova projects dots that double and Comet and Wren count.

An exponential function from a table

The crew's counts are 6, 12, 24, 48. Each day's count is the day before times 2. The pattern is multiply, not add.

So the rule is N(d) = 6 × 2ᵈ: start at 6 and multiply by 2 once for each day. This is the crew's model from their own counts, not a fact about any plant.

Check it: N(3) = 6 × 2³ = 6 × 8 = 48. It matches the log.

A linear rule adds the same amount each step. An exponential rule multiplies by the same factor each step.

A solved problem to study: the rule from two counts

Two readings from the log: day 1 had 12 fronds and day 3 had 48.
From day 1 to day 3 is 2 days. The count was multiplied by 48 ÷ 12 = 4 over those 2 days.
So the daily factor r satisfies r² = 4, which gives r = 2.
Run back one day from day 1: the day 0 count is 12 ÷ 2 = 6. The rule is N(d) = 6 × 2ᵈ.

The crew's trough log

Every row is the crew's own count from Nova's log, made up for the Glass House.

Day d	Count N(d) = 6 × 2ᵈ	Times the day before
0	6	start
1	12	× 2
2	24	× 2
3	48	× 2
4	96	× 2
5	192	× 2

$The crew's duckweed model N(d) = 6 × 2ᵈ from day 0 to day 5, a curve bending upward faster each day.$

USE THE MODEL

Read the question.
Tap your answer.

The crew's model is N(d) = 6 × 2ᵈ. What is the count on day 5?

The outputs 6, 12, 24, 48 come from equal steps in x. Is the pattern linear, exponential or neither?

The outputs 6, 9, 12, 15 come from equal steps in x. Is the pattern linear, exponential or neither?

Using N(d) = 6 × 2ᵈ, how many fronds does the model give on day 4? Type the number.

WHY THIS EXERCISEAn exponential rule grows by the same factor each step. Each new day is one more multiplication.

BUILD THE RULE

Read the question.
Tap your answer.

A count is 10 on day 0 and 30 on day 1. Which exponential rule fits?

A count is 5 on day 0 and 20 on day 2. What is the daily factor?

A rule reads N(d) = 6 × 2ᵈ. What does the 6 mean?

Statement	True or false?
In N(d) = 6 × 2ᵈ, the 2 is the growth factor.	?
6 × 2 × 2 × 2 = 48	?
6 × 2 × 5 = 192	?
An exponential rule adds the same amount each step.	?
The counts 6, 12, 24, 48 come from an exponential rule.	?

WHY THIS EXERCISETelling multiply-each-step from add-each-step is the first skill of the week.

Try it

Fold a sheet of paper in half, then in half again, up to five folds. Count the layers after each fold.
Write the layer count as a rule in the number of folds. Is it linear or exponential?

Glass House Log: write what an exponential function is, in your own words, with the duckweed example.Explain how two readings from the log are enough to find the whole rule.Comet called the rule "our own model." Write why the crew does not call it a fact about duckweed.Write one thing at home that grows by multiplying and one that grows by adding.

Strong start. Tomorrow you rewrite the rule to read its rate by the day or by the week.

Grown-up checked my answers

Tuesday →