At a community food drive, volunteers pack canned goods into boxes at a constant rate. The function \(\mathrm{P(t) = 414...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{P(t) = 414 - 18t}\)
  • \(\mathrm{P(t)}\) = number of cans remaining to be packed after \(\mathrm{t}\) minutes
  • Packing happens at a constant rate

• What this tells us: We have a linear function where \(\mathrm{t}\) represents time and \(\mathrm{P(t)}\) represents a quantity that changes over time.

2. INFER what each part of the function represents

• In the function \(\mathrm{P(t) = 414 - 18t}\):
  • 414 represents the starting amount (when \(\mathrm{t = 0}\))
  • -18 represents how much the remaining amount changes each minute
  • Since it's negative, the remaining amount decreases by 18 each minute

• Key insight: If remaining cans decrease by 18 per minute, then 18 cans are being packed each minute.

3. INFER by checking specific values

• At \(\mathrm{t = 0}\): \(\mathrm{P(0) = 414 - 18(0) = 414}\) cans remaining initially
• At \(\mathrm{t = 1}\): \(\mathrm{P(1) = 414 - 18(1) = 396}\) cans remaining after 1 minute
• Change: \(\mathrm{414 - 396 = 18}\) cans were packed in that first minute

This confirms that 18 cans are packed per minute.

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students recognize that 18 is important but misinterpret what it represents in the context. They might think the coefficient directly tells them what remains at a specific time, rather than understanding it represents a rate of change.

This confusion about the meaning of the coefficient may lead them to select Choice C (18 cans remain after 1 minute) or cause them to guess randomly.

Second Most Common Error:

Missing conceptual knowledge of linear functions: Students don't understand that in \(\mathrm{y = mx + b}\), the coefficient \(\mathrm{m}\) represents rate of change. Without this foundation, they might focus on the number 18 without understanding its role as a rate.

This conceptual gap may lead them to select Choice B (18 initial cans) or Choice D (18 minutes duration) by associating 18 with other aspects of the problem.

The Bottom Line:

This problem tests whether students can move beyond just identifying numbers to understanding what coefficients mean in real-world contexts. The key insight is connecting the negative rate of "remaining cans" to the positive rate of "cans packed."