xy = 12 Which table gives three values of x and their corresponding values of y for the given equation?...

1. TRANSLATE the problem requirements

• Given information:
  • Equation: \(\mathrm{xy = 12}\)
  • Four tables with different \(\mathrm{(x, y)}\) pairs
  • Need to find which table gives "corresponding values"

• What this tells us: The correct table will have \(\mathrm{(x, y)}\) pairs where each pair satisfies \(\mathrm{xy = 12}\)

2. INFER the verification strategy

• To check if a table is correct, substitute each \(\mathrm{(x, y)}\) pair into \(\mathrm{xy = 12}\)
• If \(\mathrm{xy = 12}\) for ALL pairs in a table, that's our answer
• If ANY pair fails to equal 12, eliminate that table

3. Check Table A systematically

• \(\mathrm{x = 1, y = 12}\): \(\mathrm{(1)(12) = 12}\) ✓
• \(\mathrm{x = 2, y = 6}\): \(\mathrm{(2)(6) = 12}\) ✓
• \(\mathrm{x = 3, y = 4}\): \(\mathrm{(3)(4) = 12}\) ✓

All three pairs work! But let's verify the others don't work.

4. Quickly eliminate other tables

• Table B: \(\mathrm{x = 1, y = 11}\) gives \(\mathrm{(1)(11) = 11 ≠ 12}\) ✗
• Table C: \(\mathrm{x = 1, y = 6}\) gives \(\mathrm{(1)(6) = 6 ≠ 12}\) ✗
• Table D: \(\mathrm{x = 1, y = 1}\) gives \(\mathrm{(1)(1) = 1 ≠ 12}\) ✗

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor TRANSLATE reasoning: Students may misunderstand what "corresponding values for the equation" means and think they need to solve \(\mathrm{xy = 12}\) for specific x-values rather than check which given pairs satisfy the equation.

They might try to create their own table by solving \(\mathrm{y = 12/x}\) for \(\mathrm{x = 1, 2, 3}\), then get confused when their calculated values don't match any of the answer choices exactly. This leads to confusion and guessing.

Second Most Common Error:

Arithmetic mistakes: Students understand the checking strategy but make multiplication errors, particularly with larger numbers.

For example, they might calculate \(\mathrm{(3)(4) = 10}\) instead of 12, leading them to incorrectly eliminate Table A. This may lead them to select Choice B, C, or D based on faulty calculations rather than correct verification.

The Bottom Line:

This problem tests whether students can verify solutions to equations rather than generate them. The key insight is recognizing that "corresponding values" means the given pairs should satisfy the equation when substituted.