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Algebra 1 9-12 / Week 07 / Loft Day
6/6
Week 07 · Exponents and Exponential Change

Loft Day

Loft Day: fold and count
// Linear change adds, exponential change multiplies
⏱ about 20 min

Loft Day: Fold and Count

Saturday the Loft fills with family and a stack of paper strips.

"Everyone takes a strip," Comet says. "Before you fold, guess how many layers you can reach."

Guesses fly. "Sixteen." "A hundred." "Ten."

"What do you notice after each fold?" Wren asks, handing out pencils for the tally.

"It doubles! One, two, four, eight, sixteen. And now it will not fold."

Nova hovers above the table, her light dimming to a hint. "Would the longest strip reach more folds?"

Someone grabs the longest strip and gets one more fold. Thirty-two layers. Cheers all around.

"Next round," Comet says, "the ball. Predict its fourth bounce before it drops."

  1. Linear change adds the same amount each step. Exponential change multiplies by the same factor each step.
  2. Equal differences mean linear. Equal ratios mean exponential.
  3. A fraction exponent is a root on the bottom and a power on the top. 8 to the 1/3 is the cube root of 8.
  4. A growth factor of 1.05 means growing 5%. A decay factor of 0.8 means losing 20%.
  5. An exponential function eventually passes any linear or quadratic one, and rewriting a rule can reveal its rate.
◇ FAMILY FOLD AND BOUNCE
Each person folds a paper strip as many times as possible and tallies the layers after every fold.
Compare tallies. Did everyone double each time? Who reached the most folds, and was their strip longer or thinner?
Then drop a ball from 100 centimeters and record four bounce heights. Find the family's decay factor together.
For a twist, ask everyone to predict bounce 6 from the model before measuring it.
For a grown-up
Paper edges can be sharp at the crease. Tape, not teeth, holds a stubborn fold.
The ball bounces in a cleared space away from windows, shelves and small children's heads.
A guess that is far off is a good moment: ask whether the pattern added or multiplied.
LOFT DAY CHECK
  • Read the question.
  • Tap your answer.
A strip starts as 1 layer and doubles with each fold. How many layers after 6 folds?
A family tally reads 2, 4, 8, 16 at folds 1 to 4. Linear, exponential or neither?
What is 16^(3/4)?
The family ball keeps 80% of its height each bounce. By what factor is the height multiplied each bounce?
A strip is folded 4 times. How many layers does it have? Type the number.
WHY THIS EXERCISECounting layers fold by fold is the exponential pattern you can hold in your hands.
Draw the family tally as a graph, folds across and layers up, and mark where the strip stopped folding.
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