Week 07 · Exponents and Exponential Change

Monday

The strip that doubles

// Linear change adds, exponential change multiplies

⏱ about 20 min

Monday: The Strip That Doubles

Comet unrolls a long strip of paper across the Loft floor. "New week, new build. What can we make with one strip?"

She folds it in half. "Two layers." Again. "Four." Again. "Eight. Wren, what do you notice?"

"Each fold doubles the layers," Wren says. "Last week the cup tower added three cups a row. This multiplies."

"So after ten folds it is huge?" Comet asks, already folding a fifth time. The strip fights back.

Nova hovers over the fold, her light drawing a small table in the air. "Would you like a hint? Count folds and layers side by side."

"Zero folds, one layer. One fold, two. Two folds, four," Wren reads. "Powers of two."

Comet looks at you. "Our engineer, how many layers after six folds? And why can I not fold it a seventh time?"

Comet folds a long paper strip in the Loft while Wren tracks the layers and Nova watches a bouncing ball.

Adding or multiplying

A linear function adds the same amount every step. The cup tower added 3 cups a row.

An exponential function multiplies by the same factor every step. The strip multiplies its layers by 2 at every fold.

The crew's rule is L(n) = 2n: after n folds there are 2n layers. The 2 is the growth factor.

A solved example to study: L(6) = 2⁶ = 2 × 2 × 2 × 2 × 2 × 2 = 64. Six factors of 2, not 2 × 6.

Folds n	Layers L(n)	How it got there
0	1	the flat strip
1	2	1 × 2
2	4	2 × 2
3	8	4 × 2
4	16	8 × 2
5	32	16 × 2
6	64	32 × 2

$A curve for 2 to the power n from 0 to 6 beside the straight line 2n + 1 for comparison.$

Words for the week

In 2n, 2 is the base and n is the exponent. The exponent counts the factors of 2.
The growth factor is the number you multiply by each step. For the strip it is 2.
Linear change: equal differences from step to step. Exponential change: equal ratios from step to step.
A table tells you which. Subtract neighbors for the differences, divide neighbors for the ratios.

FOLD THE STRIP

Read the question.
Tap your answer.

The strip starts as 1 layer and every fold doubles it. How many layers after 5 folds?

How many layers would 8 folds make, if the paper allowed it?

The cup tower from week 5 has 3 cups in row 1 and adds 3 each row. How many cups in row 8?

ADDS OR MULTIPLIES?

Read the question.
Tap your answer.

The strip's layers at folds 0, 1, 2, 3 are 1, 2, 4, 8. Linear, exponential or neither?

The cup tower rows hold 3, 6, 9, 12 cups. Linear, exponential or neither?

The outputs 5, 10, 20, 40 come from equal steps in x. Is the pattern linear, exponential or neither?

The outputs 1, 4, 9, 16 come from equal steps in x. Is the pattern linear, exponential or neither?

Statement	True or false?
2⁶ = 64	?
2⁵ = 10	?
2 × 2 × 2 × 2 = 16	?
An exponential function multiplies by the same factor every step.	?
Doubling layers at every fold is a linear pattern.	?

WHY THIS EXERCISETelling adding from multiplying is the whole week. Equal differences mean linear; equal ratios mean exponential.

Try it

Take any strip of paper and fold it in half as many times as you can. Count the layers after each fold.
Write the folds and layers as a table. Check that each layer count is double the one before.
Keep fingers clear of the sharp crease edge, and use tape, not teeth, to hold a fold.

Draw the folds and layers as two graphs: the strip's doubling curve and the cup tower's straight line.

Loft Log: write the difference between a linear pattern and an exponential pattern in your own words.Your strip stopped folding at some point. Write how many layers it had, and why the next fold was so hard.Write one thing in your own build that adds the same amount each step, and one that could double.

Strong start. Tomorrow the rules of exponents stretch to fractions, and roots turn out to be powers in disguise.

Grown-up checked my answers

Tuesday →