Square A has side lengths that are 246 times the side lengths of square B. The area of square A...

1. TRANSLATE the problem information

• Given information:
  • Square A has side lengths that are 246 times the side lengths of square B
  • The area of square A is k times the area of square B
  • Need to find the value of k

• What this tells us:
  • If square B has side length s, then square A has side length 246s

2. INFER the relationship between scaling and area

• Key insight: When you scale the sides of a square by some factor, the area gets scaled by the square of that factor
• This means we need to find how the areas compare when the sides are in a 246:1 ratio

3. Set up the area expressions

• Area of square B = \(\mathrm{s}^2\)
• Area of square A = \((246\mathrm{s})^2 = 246^2 \times \mathrm{s}^2\)

4. SIMPLIFY to find the scaling factor

• Since area of square A = k × area of square B:
  \(246^2 \times \mathrm{s}^2 = \mathrm{k} \times \mathrm{s}^2\)
• The \(\mathrm{s}^2\) terms cancel out:
  \(\mathrm{k} = 246^2\)
• Calculate \(246^2 = 60,516\) (use calculator)

Answer: A. 60,516

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that area scales by the square of the linear scale factor. Instead, they think that if the sides are 246 times larger, then the area is also just 246 times larger.

This leads them to conclude \(\mathrm{k} = 246\) and select Choice C (246).

Second Most Common Error:

Poor SIMPLIFY execution: Students understand the correct relationship but make arithmetic errors when computing \(246^2\). They might calculate something like \(246 \times 2 = 492\) instead of \(246^2\).

This may lead them to select Choice B (492).

The Bottom Line:

This problem tests whether students understand the fundamental relationship between linear scaling and area scaling. The key insight is recognizing that area is a two-dimensional measurement, so it scales with the square of any linear scaling factor.