A certain town has an area of 4.36 square miles. What is the area, in square yards, of this town?...

1. TRANSLATE the problem information

• Given information:
  • Town area: 4.36 square miles
  • Conversion factor: 1 mile = 1,760 yards
  • Need to find: area in square yards

2. INFER the conversion approach

• Key insight: When converting area measurements, we must square the linear conversion factor
• Since \(\mathrm{1\text{ mile} = 1,760\text{ yards}}\), then \(\mathrm{1\text{ square mile} = (1,760\text{ yards})^2}\)
• Strategy: First find square yards per square mile, then multiply by 4.36

3. Calculate the area conversion factor

• 1 square mile = \(\mathrm{(1,760)^2}\) square yards
• \(\mathrm{1,760^2 = 3,097,600}\) square yards per square mile (use calculator)

4. SIMPLIFY to find the final answer

• Town area = 4.36 square miles × 3,097,600 square yards per square mile
• \(\mathrm{4.36 \times 3,097,600 = 13,505,536}\) square yards (use calculator)

Answer: D. 13,505,536

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that area conversions require squaring the linear conversion factor. They see "\(\mathrm{1\text{ mile} = 1,760\text{ yards}}\)" and think "\(\mathrm{1\text{ square mile} = 1,760\text{ square yards}}\)."

Using this incorrect reasoning: \(\mathrm{4.36 \times 1,760 = 7,673.6 \approx 7,674}\)

This may lead them to select Choice B (7,674)

Second Most Common Error:

Inadequate SIMPLIFY execution: Students understand the squaring concept but make calculation errors when computing \(\mathrm{1,760^2}\) or the final multiplication, especially if attempting these complex calculations without a calculator.

This leads to confusion and selecting an incorrect answer choice through computational mistakes.

The Bottom Line:

The key challenge is recognizing that area measurements involve two dimensions, so when you convert the linear dimension (miles to yards), the area conversion factor must be squared. This is a fundamental concept about how area units relate to linear units that many students haven't fully internalized.