Question:A model predicts that a certain plant was 47 inches tall when it was first measured and that the plant...

1. TRANSLATE the problem information

• Given information:
  • Plant was 47 inches tall when first measured
  • Plant grows 1.5 inches per week
  • Model equation: \(\mathrm{h(w) = p + qw}\)
  • \(\mathrm{h(w)}\) = height after \(\mathrm{w}\) weeks
  • Need to find: value of \(\mathrm{p}\)

2. INFER what p represents in the equation

• In the linear equation \(\mathrm{h(w) = p + qw}\):
  • \(\mathrm{p}\) is the constant term (like the \(\mathrm{b}\) in \(\mathrm{y = mx + b}\))
  • This represents the starting value when \(\mathrm{w = 0}\)
  • \(\mathrm{q}\) is the coefficient of \(\mathrm{w}\) (the growth rate)

3. INFER the key insight about initial measurement

• "When first measured" means \(\mathrm{w = 0}\) weeks have passed
• At \(\mathrm{w = 0}\): \(\mathrm{h(0) = p + q(0) = p + 0 = p}\)
• So \(\mathrm{p}\) equals the height at first measurement

4. TRANSLATE the final connection

• The plant was 47 inches tall when first measured
• Therefore: \(\mathrm{p = 47}\)

Answer: 47

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse which parameter represents what in the linear equation. They might think \(\mathrm{p}\) represents the growth rate (1.5 inches per week) instead of the initial height.

This confusion stems from not carefully reading "when it was first measured" and not connecting this phrase to \(\mathrm{w = 0}\) in the equation. They might incorrectly conclude \(\mathrm{p = 1.5}\), thinking \(\mathrm{p}\) represents the weekly growth.

The Bottom Line:

This problem tests whether students can connect real-world linear relationships to standard mathematical form. The key insight is recognizing that \(\mathrm{p}\) represents the y-intercept - the value when the input variable equals zero.