The floor of a ballroom has an area of 600 square meters. An architect creates a scale model of the...

1. TRANSLATE the problem information

• Given information:
  • Actual ballroom floor area: 600 square meters
  • Scale model: each side length = \(\frac{1}{10}\) × actual side length
  • Find: area of the scale model

• This tells us we have a linear scale factor of \(\frac{1}{10}\)

2. INFER how area scaling works

• Key insight: When you scale linear dimensions by a factor k, area scales by \(\mathrm{k}^2\)
• Why? Because area involves multiplying two linear dimensions \(\mathrm{length} \times \mathrm{width}\)
• If both length and width are scaled by \(\frac{1}{10}\), then:

\(\mathrm{Area} = (\frac{1}{10} \times \mathrm{length}) \times (\frac{1}{10} \times \mathrm{width}) = (\frac{1}{10})^2 \times (\mathrm{length} \times \mathrm{width})\)

3. SIMPLIFY the area scale factor

• Linear scale factor = \(\frac{1}{10}\)
• Area scale factor = \((\frac{1}{10})^2 = \frac{1}{100}\)

4. Calculate the scale model area

• Scale model area = (area scale factor) × (actual area)
• Scale model area = \(\frac{1}{100} \times 600 = 6\) square meters

Answer: A. 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students apply the linear scale factor directly to area instead of squaring it.

They think: "If the sides are \(\frac{1}{10}\) the size, then the area is also \(\frac{1}{10}\) the size."

Scale model area = \(\frac{1}{10} \times 600 = 60\) square meters

This may lead them to select Choice C (60).

Second Most Common Error:

Conceptual confusion about scaling: Students might think they need to find the area of one side of the model rather than understanding that area scaling applies to the entire floor area.

This leads to confusion about what calculation to perform and may cause them to get stuck and guess.

The Bottom Line:

The key insight is recognizing that area is a two-dimensional quantity, so when linear dimensions scale by k, area scales by \(\mathrm{k}^2\). This is not just a formula to memorize—it's a logical consequence of how area measurement works.