The area A, in square centimeters, of a rectangular cutting board can be represented by the expression \(\mathrm{w(w + 9)}\),...

1. TRANSLATE the problem information

• Given information:
  • Area \(\mathrm{A = w(w + 9)}\) square centimeters
  • \(\mathrm{w}\) = width in centimeters
  • Need to find: expression for length in centimeters

2. INFER the relationship strategy

• Since this is a rectangle, we know \(\mathrm{Area = width \times length}\)
• We have the area expression and we know the width
• We can use algebra to find the length expression

3. SIMPLIFY to find the length

• Set up the equation: \(\mathrm{Area = width \times length}\)
• Substitute: \(\mathrm{w(w + 9) = w \times length}\)
• Divide both sides by \(\mathrm{w}\): \(\mathrm{length = (w + 9)}\)

Answer: D. (w + 9)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they need to use the fundamental rectangle area relationship \(\mathrm{Area = width \times length}\) to work backwards and find the length expression.

Instead, they might look at the given expression \(\mathrm{w(w + 9)}\) and think this directly represents the length, leading them to select Choice A \(\mathrm{(w(w + 9))}\).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret what the problem is asking for, thinking they need to find a numerical value rather than an algebraic expression for length.

This confusion about what exactly they're solving for causes them to get stuck and randomly select an answer, often gravitating toward the simpler expressions like Choice B \(\mathrm{(w)}\) or Choice C \(\mathrm{(9)}\).

The Bottom Line:

This problem requires students to work backwards from a given area expression using the fundamental rectangle formula, which many students struggle with because they're more accustomed to using formulas in the forward direction (given dimensions, find area) rather than reverse engineering one dimension from the area and the other dimension.