A certain region has a population density of 2,313.83 people per square kilometer. What is the population density, in people...

1. TRANSLATE the problem information

• Given information:
  • Population density: 2,313.83 people per square kilometer
  • Linear conversion: 1 kilometer = 0.62 miles
• What we need: Population density in people per square mile

2. INFER the conversion approach

• Key insight: Population density involves area (square units), not just linear distance
• Since we have a linear conversion (kilometers to miles), we need to convert it to an area conversion (square kilometers to square miles)
• Strategy: First find the area conversion factor, then apply it to the density

3. Convert linear distance to area conversion

• Linear: 1 kilometer = 0.62 miles
• Area: 1 square kilometer = \((0.62)^2\) square miles
• Calculate: \((0.62)^2 = 0.3844\) square miles (use calculator)

4. SIMPLIFY the density conversion

• We have: 2,313.83 people per 1 square kilometer
• Since 1 square kilometer = 0.3844 square miles, we need to find how many people are in 1 square mile
• People per square mile = \(2,313.83 \div 0.3844 = 6,016.2\) (use calculator)

Answer: B (6,016.2)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Using the linear conversion factor instead of the area conversion factor

Students often think: "1 kilometer = 0.62 miles, so I'll just multiply or divide by 0.62." They fail to recognize that population density involves area measurements, which require squaring the linear conversion factor. This leads them to calculate either \(2,313.83 \times 0.62 = 1,434.6\) or \(2,313.83 \div 0.62 = 3,732.0\), neither of which matches the given choices exactly but 1,434.6 is closest to choice A.

This may lead them to select Choice A (1,439.9).

Second Most Common Error:

Poor SIMPLIFY execution: Correctly identifying the need for area conversion but making calculation errors

Students understand they need \((0.62)^2\), but make arithmetic mistakes like calculating \(0.62 \times 0.62\) incorrectly, or making errors in the final division. These calculation errors can lead to results that appear to match other answer choices.

This causes them to get stuck between multiple choices or select an incorrect answer based on their miscalculation.

The Bottom Line:

The key insight is recognizing that "per square kilometer" means you're dealing with area, not linear distance. Many students miss this connection and apply linear conversions to area-based measurements, leading to systematic errors rather than just arithmetic mistakes.