The table below shows the height h (in inches) of a plant measured at different times t (in weeks) during...

1. TRANSLATE the problem information

• Given information:
  • Height h measured at times t: \((1,3), (2,12), (3,27), (4,48), (5,75)\)
  • Weekly growth rate = change in height per week
  • Need to determine what type of relationship exists between time and growth rate
• What this tells us: We need to calculate how much the height changes each week, then see how those changes behave over time.

2. INFER the approach

• To find weekly growth rate, calculate consecutive differences in height
• Then analyze the pattern in these growth rates to determine the relationship type

3. Calculate the weekly growth rates

• Week 1 to 2: \(12 - 3 = 9\) inches per week
• Week 2 to 3: \(27 - 12 = 15\) inches per week
• Week 3 to 4: \(48 - 27 = 21\) inches per week
• Week 4 to 5: \(75 - 48 = 27\) inches per week
• Growth rate sequence: \(9, 15, 21, 27\) inches per week

4. INFER the pattern in growth rates

• Check differences between consecutive growth rates:
  • \(15 - 9 = 6\)
  • \(21 - 15 = 6\)
  • \(27 - 21 = 6\)
• Since the growth rate increases by the same amount (6) each week, this represents a linear relationship with positive slope.

Answer: B (increasing linear relationship)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse "weekly growth rate" with the original height data, thinking they should graph the height values directly rather than calculating the rate of change.

This leads them to analyze the original sequence \((3, 12, 27, 48, 75)\) instead of the growth rates, potentially selecting Choice D (curved relationship) since the original heights follow a quadratic-like pattern.

Second Most Common Error:

Poor INFER reasoning: Students correctly calculate the growth rates \((9, 15, 21, 27)\) but fail to recognize that constant differences indicate a linear relationship. They might see that the numbers are increasing and incorrectly assume this means a curved or exponential relationship.

This may lead them to select Choice D (increasing curved relationship) instead of recognizing the linear pattern.

The Bottom Line:

This problem requires careful distinction between the original data and the derived rate information, plus understanding that "increasing at a constant rate" describes linear behavior, not curved behavior.