The area of a rectangle is 57 square inches. The length of the longest side of the rectangle is 19...

1. TRANSLATE the problem information

• Given information:
  • Area of rectangle = 57 square inches
  • Longest side = 19 inches
  • Need to find: shortest side length

2. INFER the approach

• Since we know the area and one dimension, we can use the rectangle area formula to find the other dimension
• Set up an equation where the unknown shortest side is our variable

3. TRANSLATE into mathematical notation

• Let \(\mathrm{x}\) = length of shortest side
• Using \(\mathrm{Area = length \times width}\): \(\mathrm{57 = 19 \times x}\)

4. SIMPLIFY by solving the equation

• Divide both sides by 19: \(\mathrm{x = 57 \div 19 = 3}\)

Answer: 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not immediately recognize that they can use the area formula to set up an equation with the unknown dimension.

Instead, they might try to guess and check values or become confused about how to proceed when they have area and only one dimension. This leads to confusion and guessing among answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{57 = 19x}\) but make an arithmetic error when calculating \(\mathrm{57 \div 19}\).

Common mistakes include getting \(\mathrm{57 \div 19 = 4}\) or other incorrect values due to rushing through the division or not checking their work.

The Bottom Line:

This problem tests whether students can strategically apply the area formula in reverse - using known area and one dimension to find the other dimension. The key insight is recognizing that you can solve for the unknown side algebraically rather than trying other approaches.