A triangular field has a base of 5,280 yards and a height of 1,760 yards. What is the area of...

1. TRANSLATE the problem information

• Given information:
  • Triangular field with \(\mathrm{base = 5,280\ yards}\)
  • \(\mathrm{Height = 1,760\ yards}\)
  • Need area in square miles
  • Conversion factor: \(\mathrm{1\ mile = 1,760\ yards}\)
• What this tells us: We need to find triangle area and convert to square miles

2. INFER the solution approach

• We can convert units before or after calculating area
• Method 1: Convert lengths to miles first, then calculate area
• Method 2: Calculate area in square yards, then convert to square miles
• Method 1 involves simpler numbers, so let's start there

3. SIMPLIFY the unit conversions

• Convert base: \(\mathrm{5,280\ yards \div 1,760\ yards/mile = 3\ miles}\)
• Convert height: \(\mathrm{1,760\ yards \div 1,760\ yards/mile = 1\ mile}\)

4. SIMPLIFY the area calculation

• \(\mathrm{Area = \frac{1}{2} \times base \times height}\)
• \(\mathrm{Area = \frac{1}{2} \times 3\ miles \times 1\ mile = 1.5\ square\ miles}\)

Answer: 1.5 (Choice C)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students incorrectly convert square units by using the linear conversion factor instead of squaring it.

When using Method 2, they calculate the area as 4,646,400 square yards correctly, but then convert by dividing by 1,760 instead of \(\mathrm{1,760^2}\). This gives them \(\mathrm{4,646,400 \div 1,760 = 2,640\ square\ miles}\), which doesn't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students forget they need the answer in square miles and give their answer in square yards.

They correctly calculate \(\mathrm{\frac{1}{2} \times 5,280 \times 1,760 = 4,646,400}\), but then select this as their final answer without noticing that none of the choices are anywhere near this large number. This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests whether students can handle unit conversions systematically while keeping track of whether they're working with linear units (yards to miles) or area units (square yards to square miles). The key insight is that area conversion requires squaring the linear conversion factor.