A rectangular tile measures 16 inches by 4 inches. A square tile has the same area as the rectangular tile....

1. TRANSLATE the problem information

• Given information:
  • Rectangular tile: 16 inches by 4 inches
  • Square tile has same area as rectangular tile
  • Find: side length of square tile

• This tells us we need to find areas and set them equal

2. INFER the solution strategy

• Strategy: First find the rectangular area, then use that area to find the square's side length
• We'll need both area formulas: rectangle \(\mathrm{(length \times width)}\) and square \(\mathrm{(side}^2\mathrm{)}\)

3. SIMPLIFY to find the rectangular area

• Area of rectangle = \(\mathrm{16 \times 4 = 64}\) square inches

4. TRANSLATE the equal area condition

• Since areas are equal: Area of square = 64 square inches
• Using square area formula: \(\mathrm{s}^2 \mathrm{= 64}\)

5. SIMPLIFY to find the side length

• Take the square root: \(\mathrm{s = \sqrt{64} = 8}\) inches
• Since length is positive, we take the positive root

Answer: B (8)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Confusing area with perimeter

Students might think "same area" means "same perimeter" and set up \(\mathrm{4s = 16 + 16 + 4 + 4}\), leading to \(\mathrm{s = 10}\).

This may lead them to select Choice C (10)

Second Most Common Error:

Poor SIMPLIFY execution: Calculation errors in basic operations

Students might incorrectly calculate \(\mathrm{16 \times 4 = 32}\) instead of 64, then find \(\mathrm{\sqrt{32} \approx 5.7}\), leading to confusion about which answer to choose, or they might make errors finding \(\mathrm{\sqrt{64}}\).

This leads to confusion and guessing among the available choices

The Bottom Line:

This problem tests whether students can translate a verbal area relationship into mathematical equations and execute basic area calculations accurately. The key insight is recognizing that "same area" creates an equation between different area formulas.