The given function \(\mathrm{g(m) = -0.05m + 12.1}\) models the number of gallons of gasoline that remains from a full...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{g(m) = -0.05m + 12.1}\)
  • \(\mathrm{g(m)}\) = gallons remaining after driving m miles
  • Need to find: gallons used per mile

2. INFER the meaning of function components

• In the linear function \(\mathrm{g(m) = -0.05m + 12.1}\):
  • The coefficient \(\mathrm{-0.05}\) tells us the rate of change
  • The \(\mathrm{12.1}\) represents the starting amount (when \(\mathrm{m = 0}\))

• Key insight: Since \(\mathrm{g(m)}\) represents gallons remaining, a negative rate of change means gas is being consumed

3. INFER the consumption rate

• The coefficient \(\mathrm{-0.05}\) means:
  • For every 1 mile driven, remaining gas decreases by \(\mathrm{0.05}\) gallons
  • If remaining gas decreases by \(\mathrm{0.05}\) gallons, then \(\mathrm{0.05}\) gallons were used

Answer: A. 0.05

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students focus on the y-intercept \(\mathrm{(12.1)}\) instead of the coefficient, thinking it represents the rate of consumption rather than the total tank capacity.

They see \(\mathrm{12.1}\) as the most prominent number and assume it must be the answer without considering what each part of the function represents.

This may lead them to select Choice B (12.1).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret the relationship between "remaining gas" and "used gas," potentially getting confused about the sign or meaning of the coefficient.

This leads to confusion about which number in the function answers the question, causing them to get stuck and guess.

The Bottom Line:

This problem tests whether students understand that in a linear function modeling a decreasing quantity, the coefficient (not the constant term) represents the rate of change - and therefore the rate of consumption.