The area of a rectangular piece of fabric is represented by the expression 9k^2 square units. A smaller rectangular piece...

1. TRANSLATE the problem information

• Given information:
  • Original fabric area: \(9\mathrm{k}^2\)
  • Area of piece cut out: \(4\mathrm{k}^2\)
  • Need to find: Area of remaining fabric

• What this tells us: When something is "cut from" or "removed from" something else, we subtract to find what remains.

2. TRANSLATE the approach

• "Area of remaining fabric" means: Original area - Cut out area
• Set up the subtraction: \(9\mathrm{k}^2 - 4\mathrm{k}^2\)

3. SIMPLIFY by combining like terms

• Both terms have the same variable part \(\mathrm{k}^2\), so they are "like terms"
• When subtracting like terms: subtract the coefficients, keep the variable part
• \(9\mathrm{k}^2 - 4\mathrm{k}^2\)
  \(= (9 - 4)\mathrm{k}^2\)
  \(= 5\mathrm{k}^2\)

Answer: B. \(5\mathrm{k}^2\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might interpret "cut from" as addition rather than subtraction, thinking they need to combine the areas.

This leads them to calculate \(9\mathrm{k}^2 + 4\mathrm{k}^2 = 13\mathrm{k}^2\), causing them to select Choice C (\(13\mathrm{k}^2\)).

Second Most Common Error:

Poor SIMPLIFY execution: Students might lose track of the variable part during subtraction, performing only \(9 - 4 = 5\) and forgetting about \(\mathrm{k}^2\).

This leads them to select Choice A (5).

Third Potential Error:

Conceptual confusion about operations: Students might think they need to multiply the areas instead of subtract, calculating \(9\mathrm{k}^2 \times 4\mathrm{k}^2 = 36\mathrm{k}^4\).

This may lead them to select Choice D (\(36\mathrm{k}^4\)).

The Bottom Line:

This problem tests whether students can correctly translate a word problem involving removal into subtraction, and then properly handle algebraic terms. The key insight is recognizing that "cutting out" means subtraction, and that like terms combine by operating on their coefficients.