2n + 6 = 14 A tree had a height of 6 feet when it was planted. The equation above...

1. TRANSLATE the equation components

• Given equation: \(\mathrm{2n + 6 = 14}\)
• Context translation:
  • 6 feet = initial height when planted
  • 14 feet = final height reached
  • \(\mathrm{n}\) = number of years of growth
  • We need to figure out what 2 represents

2. INFER what each part means mathematically

• The equation structure is: (something × years) + initial height = final height
• This means: \(\mathrm{2n}\) must represent the total amount the tree grew
• Since \(\mathrm{2n}\) = total growth over \(\mathrm{n}\) years, then 2 = growth per year

3. TRANSLATE back to real-world meaning

• The coefficient 2 represents the average number of feet the tree grew per year
• We can verify: If the tree grows 2 feet per year for \(\mathrm{n}\) years, total growth = \(\mathrm{2n}\) feet
• Starting height (6) + total growth (\(\mathrm{2n}\)) = final height (14)

Answer: B. The average number of feet that the tree grew per year

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students focus on the number 2 in isolation rather than understanding its role as a coefficient in the context of the equation structure.

Students might think "2 means double" and select Choice A, thinking it represents doubling the height. They miss that 2 is multiplied by years (\(\mathrm{n}\)), making it a rate, not a doubling factor.

This may lead them to select Choice A (The number of years it took the tree to double its height)

Second Most Common Error:

Poor unit analysis during TRANSLATE: Students confuse what 2 represents by not carefully tracking units (feet vs. years).

They might substitute \(\mathrm{n = 1}\) and calculate \(\mathrm{2(1) + 6 = 8}\), then think the "2" represents this height calculation, leading them to confuse the coefficient with an actual height measurement.

This may lead them to select Choice C (The height, in feet, of the tree when the tree was 1 year old)

The Bottom Line:

Success requires recognizing that coefficients in linear equations often represent rates when the variable represents time, and interpreting mathematical relationships within their real-world context.