For a training program, Juan rides his bike at an average rate of 5.7 minutes per mile. Which function m...

1. TRANSLATE the problem information

• Given information:
  • Rate: 5.7 minutes per mile
  • Distance: x miles
  • Need to find: Function for total minutes

2. INFER the mathematical relationship

• To find total time, we use the fundamental rate relationship:
  \(\mathrm{Total = Rate × Quantity}\)
• In this context: \(\mathrm{Total\ minutes = (minutes\ per\ mile) × (number\ of\ miles)}\)

3. TRANSLATE into function form

• Substituting our values:
  \(\mathrm{m(x) = 5.7 × x = 5.7x}\)

Answer: D. m(x) = 5.7x

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret what "5.7 minutes per mile" means in the mathematical relationship. They might think that to find minutes, they need to divide miles by the rate, creating the function \(\mathrm{m(x) = x/5.7}\).

This reasoning stems from confusing rate problems with other division scenarios they've seen, leading them to select Choice A (\(\mathrm{x/5.7}\)).

Second Most Common Error:

Poor INFER reasoning: Students recognize they need to use both 5.7 and x but don't understand the multiplicative relationship. Instead, they might think the function involves adding or subtracting these values.

This confusion about how rates combine with quantities may lead them to select Choice B (\(\mathrm{x + 5.7}\)) or Choice C (\(\mathrm{x - 5.7}\)).

The Bottom Line:

This problem tests whether students understand that rates describe multiplicative relationships. The key insight is recognizing that "minutes per mile" tells you how to scale the number of miles to get total minutes.