Question:A certain plant grows at a constant rate. On day 2, the plant is 14 centimeters tall, and on day...

1. TRANSLATE the problem information

• Given information:
  • Plant grows at constant rate (this means linear relationship)
  • On day 2: height = 14 cm → coordinate point \(\mathrm{(2, 14)}\)
  • On day 5: height = 23 cm → coordinate point \(\mathrm{(5, 23)}\)
  • Need equation in form \(\mathrm{h = md + b}\)

2. INFER the solution approach

• Since we have two points and need a linear equation, we must:
  • Find the slope (m) using the two points
  • Use one point to find the y-intercept (b)

3. SIMPLIFY to find the slope

• Use slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• \(\mathrm{m = \frac{23 - 14}{5 - 2}}\)
  \(\mathrm{= \frac{9}{3}}\)
  \(\mathrm{= 3}\)

4. SIMPLIFY to find the y-intercept

• Substitute slope and one point into \(\mathrm{h = md + b}\)
• Using point \(\mathrm{(2, 14)}\): \(\mathrm{14 = 3(2) + b}\)
• \(\mathrm{14 = 6 + b}\)
• \(\mathrm{b = 8}\)

5. Write the final equation and verify

• \(\mathrm{h = 3d + 8}\)
• Check with second point \(\mathrm{(5, 23)}\): \(\mathrm{h = 3(5) + 8 = 23}\) ✓

Answer: A) h = 3d + 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may incorrectly identify which variable represents which quantity, mixing up day and height in their coordinate points, or they might not recognize that "constant rate" means linear relationship.

This confusion about variable assignment or the meaning of constant rate can lead them to set up the problem incorrectly from the start, causing them to select any answer choice through guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the problem but make arithmetic errors when calculating the slope (getting 9 instead of 3, for example) or when solving for the y-intercept.

An arithmetic error in slope calculation might lead them to select Choice B (\(\mathrm{h = 9d - 4}\)) if they use the incorrect slope of 9 and then make additional errors finding the y-intercept.

The Bottom Line:

This problem tests whether students can translate a real-world linear growth scenario into mathematical language and then execute the standard procedure for finding a linear equation from two points. The key insight is recognizing that "constant rate" automatically means you're working with a linear function.