Question:A rectangle has an area of 91 square units. The length of the rectangle is 6 units greater than its...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{Area = 91}\) square units
  • \(\mathrm{Length = width + 6}\) units

• What this tells us: We need to express both dimensions in terms of one variable and use the area formula.

2. TRANSLATE into mathematical expressions

• Let \(\mathrm{w = width}\) of rectangle
• Then \(\mathrm{length = w + 6}\)
• Using \(\mathrm{Area = length × width}\): \(\mathrm{w(w + 6) = 91}\)

3. SIMPLIFY the equation

• Expand: \(\mathrm{w(w + 6) = 91}\)
• \(\mathrm{w² + 6w = 91}\)
• \(\mathrm{w² + 6w - 91 = 0}\)

4. SIMPLIFY by factoring the quadratic

• Need two numbers that multiply to -91 and add to 6
• Since \(\mathrm{91 = 7 × 13}\), try -7 and 13: \(\mathrm{(-7) + 13 = 6}\) ✓ and \(\mathrm{(-7)(13) = -91}\) ✓
• Factor: \(\mathrm{(w - 7)(w + 13) = 0}\)
• Solutions: \(\mathrm{w = 7}\) or \(\mathrm{w = -13}\)

5. APPLY CONSTRAINTS to select valid solution

• Since width must be positive in real-world context: \(\mathrm{w = 7}\)
• Therefore: \(\mathrm{length = w + 6 = 7 + 6 = 13}\)

Answer: 13

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to set up the relationship between length and width algebraically.

Instead of letting \(\mathrm{w = width}\) and expressing length as \(\mathrm{w + 6}\), they might:

• Try to solve with two separate variables without connecting them
• Misinterpret "6 units greater" as multiplication rather than addition
• Set up the equation backwards (\(\mathrm{width = length + 6}\))

This leads to confusion and an inability to form the correct quadratic equation, causing them to guess randomly.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make factoring errors or don't complete the quadratic solution properly.

They might:

• Factor incorrectly and get wrong values for w
• Make sign errors when expanding \(\mathrm{w(w + 6)}\)
• Give up on factoring and guess instead of using systematic approaches

This causes them to arrive at incorrect dimensions and select a wrong answer.

The Bottom Line:

This problem requires strong algebraic translation skills to convert the word relationship into mathematical expressions, followed by solid quadratic solving techniques. Students who struggle with setting up variables or factoring will find themselves stuck early in the solution process.