A geologist needs to collect at least 67 samples of lava from a volcano. If the geologist has already collected...

1. TRANSLATE the problem information

• Given information:
  • Geologist needs at least 67 samples total
  • Already collected 63 samples
  • Need to find minimum additional samples

• What this tells us: We need to find how many more samples are required to reach the goal of at least 67

2. TRANSLATE into mathematical language

• Let \(\mathrm{x}\) = number of additional samples needed
• "At least 67" means \(\geq 67\)
• Current samples + additional samples \(\geq\) total needed
• Set up inequality: \(63 + \mathrm{x} \geq 67\)

3. SIMPLIFY by solving the inequality

• \(63 + \mathrm{x} \geq 67\)
• Subtract 63 from both sides: \(\mathrm{x} \geq 67 - 63\)
• \(\mathrm{x} \geq 4\)

4. INFER the final answer

• Since \(\mathrm{x} \geq 4\), the minimum value of x is 4
• The geologist needs at least 4 additional samples

Answer: C. 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might misinterpret what the question is asking for and think they need to find the total number of samples rather than additional samples.

This leads them to think the answer should be 67 (the total needed) or get confused about what exactly they're solving for. Since 67 isn't an option, this may lead them to select Choice A (130) by adding 63 + 67, or Choice B (63) thinking it's asking for current samples.

Second Most Common Error:

Poor INFER reasoning: Students might set up the problem correctly but fail to recognize that they need the minimum number of additional samples.

They might think that since the geologist already has 63 samples and that's "close enough" to 67, no additional samples are needed. This leads them to select Choice D (0), not realizing that \(63 \lt 67\) means more samples are definitely required.

The Bottom Line:

This problem tests whether students can translate a real-world constraint into mathematical language and understand the difference between "current amount," "total needed," and "additional amount required." The key insight is recognizing that "at least" creates an inequality, not an equation.