A rectangular garden plot has an area of 96 square feet. If the length of the garden is 4 feet...

1. TRANSLATE the problem information

• Given information:
  • Area of rectangular garden = \(96\) square feet
  • Length = width + \(4\) feet
  • Find: the length

• What this tells us: We have one unknown (width) that determines both dimensions

2. INFER the approach

• Since we know area and the relationship between length and width, we can set up one equation with one unknown
• Let width = \(\mathrm{w}\), so length = \(\mathrm{(w + 4)}\)

3. TRANSLATE into mathematical equation

• Using Area = length × width:

\(\mathrm{96 = (w + 4) \times w}\)

\(\mathrm{96 = w^2 + 4w}\)

4. SIMPLIFY to solve the quadratic

• Rearrange to standard form: \(\mathrm{w^2 + 4w - 96 = 0}\)
• Factor by finding two numbers that multiply to \(-96\) and add to \(4\)
• Those numbers are \(12\) and \(-8\): \(\mathrm{(w + 12)(w - 8) = 0}\)
• Solutions: \(\mathrm{w = -12}\) or \(\mathrm{w = 8}\)

5. APPLY CONSTRAINTS to select valid solution

• Since width represents a physical dimension, it must be positive
• Therefore: \(\mathrm{w = 8}\) feet
• Length = \(\mathrm{w + 4 = 8 + 4 = 12}\) feet

Answer: 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might set up the length relationship backwards, writing length = \(\mathrm{w - 4}\) instead of \(\mathrm{w + 4}\), or confuse which dimension is which.

This leads to the equation \(\mathrm{96 = w(w - 4) = w^2 - 4w}\), giving \(\mathrm{w^2 - 4w - 96 = 0}\). Factoring gives \(\mathrm{(w - 12)(w + 8) = 0}\), so \(\mathrm{w = 12}\) or \(\mathrm{w = -8}\). Taking the positive solution gives width = \(12\), so length = \(\mathrm{12 - 4 = 8}\). This backwards answer might cause confusion or lead to guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students might make algebraic errors when expanding \(\mathrm{w(w + 4)}\) or when factoring the quadratic equation.

For example, they might incorrectly expand to get \(\mathrm{w^2 + 4}\) instead of \(\mathrm{w^2 + 4w}\), or struggle with factoring and resort to guessing factors. This leads to confusion and random answer selection.

The Bottom Line:

This problem requires careful translation of the word relationship into algebra, systematic quadratic solving, and remembering that real-world dimensions must make sense. The key insight is recognizing that "4 feet more than" means addition, not subtraction.