A sheet of paper has an initial thickness of 0.1 millimeters. Each time the paper is folded in half, the...

1. TRANSLATE the problem information

• Given information:
  • Initial thickness: \(0.1 \text{ mm}\)
  • Each fold doubles the thickness
  • Need equation for thickness \(\mathrm{t}\) after \(\mathrm{n}\) folds

2. INFER the growth pattern

• Since each fold doubles the thickness, this creates exponential growth
• We need to track how the thickness changes with each successive fold
• The key insight: after each fold, we multiply by 2 again

3. INFER the mathematical pattern

• Let's trace a few cases:
  • Start: \(0.1 \text{ mm}\)
  • After 1 fold: \(0.1 \times 2 = 0.1(2)^1 \text{ mm}\)
  • After 2 folds: \(0.1 \times 2 \times 2 = 0.1(2)^2 \text{ mm}\)
  • After 3 folds: \(0.1 \times 2^3 \text{ mm}\)

• The pattern shows: thickness = (initial thickness) × 2^(number of folds)

4. TRANSLATE into final equation

• After \(\mathrm{n}\) folds: \(\mathrm{t} = 0.1(2)^\mathrm{n}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students recognize that something increases with each fold but confuse the type of growth pattern.

They might think "each fold adds something" instead of "each fold multiplies by 2," leading them to consider linear growth like \(\mathrm{t} = 0.1 + 2\mathrm{n}\). Or they might think the growth accelerates but choose quadratic growth \(\mathrm{t} = 0.1\mathrm{n}^2\) instead of exponential.

This may lead them to select Choice A (\(\mathrm{t} = 0.1\mathrm{n}^2\)) or Choice D (\(\mathrm{t} = 0.1 + 2\mathrm{n}\)).

Second Most Common Error:

Poor TRANSLATE execution: Students misinterpret "thickness doubles" as meaning the thickness becomes half (gets thinner when folded).

This backwards interpretation makes them think each fold reduces thickness, leading them toward the decreasing exponential pattern.

This may lead them to select Choice C (\(\mathrm{t} = 0.1(1/2)^\mathrm{n}\)).

The Bottom Line:

This problem tests whether students can recognize exponential growth patterns in real-world contexts. The key insight is understanding that "doubling with each step" means multiplying by 2 repeatedly, which creates the base-2 exponential function structure.