The width of a rectangular dance floor is w feet. The length of the floor is 6 feet longer than...

1. TRANSLATE the problem information

• Given information:
  • Width = \(\mathrm{w}\) feet
  • Length is 6 feet longer than width = \(\mathrm{w + 6}\) feet
• What we need to find: Perimeter in terms of \(\mathrm{w}\)

2. INFER the approach

• Perimeter means the distance around the entire rectangle
• For a rectangle, this means adding all four sides: length + length + width + width
• We have both length and width expressions, so we can substitute them

3. SIMPLIFY by substituting and combining terms

• Perimeter = length + length + width + width
• Perimeter = \(\mathrm{(w + 6) + (w + 6) + w + w}\)
• Perimeter = \(\mathrm{w + 6 + w + 6 + w + w}\)
• Perimeter = \(\mathrm{4w + 12}\)

Answer: B. \(\mathrm{4w + 12}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students correctly identify width as \(\mathrm{w}\) but struggle with translating "6 feet longer than its width" into \(\mathrm{w + 6}\). They might think the length is just 6, leading to a perimeter calculation of \(\mathrm{w + w + 6 + 6 = 2w + 12}\).

However, the more common translation error is understanding what perimeter means - some students add only one length and one width instead of all four sides: \(\mathrm{w + (w + 6) = 2w + 6}\).

This may lead them to select Choice A (\(\mathrm{2w + 6}\)).

Second Most Common Error:

Conceptual confusion about area vs. perimeter: Students might recall that rectangles involve multiplying length and width, leading them to calculate \(\mathrm{w \times (w + 6) = w^2 + 6w}\), or they might misremember and think \(\mathrm{w^2 + 6}\) is somehow correct.

This may lead them to select Choice C (\(\mathrm{w^2 + 6}\)) or Choice D (\(\mathrm{w^2 + 6w}\)).

The Bottom Line:

This problem tests whether students can accurately translate word relationships into algebraic expressions and whether they understand that perimeter means adding ALL sides, not just two. The key insight is recognizing that a rectangle has four sides: two lengths and two widths.