\(\mathrm{E(h) = 90 - 1.5h}\) The given function E models the error rate, in errors per 1000 units produced, at...

1. TRANSLATE the problem information

• Given function: \(\mathrm{E(h) = 90 - 1.5h}\)
• Question asks: "by how much does the error rate decrease for each additional hour of training?"
• This is asking for the rate of change in the error rate per hour

2. INFER the connection to linear function structure

• Recognize this is a linear function in form \(\mathrm{y = mx + b}\)
• In \(\mathrm{E(h) = 90 - 1.5h}\):
  • 90 is the y-intercept (starting error rate)
  • -1.5 is the coefficient/slope (rate of change)
• The coefficient tells us how much E changes when h increases by 1

3. INFER the meaning of the coefficient

• Coefficient of h is -1.5
• This means: when h increases by 1 hour, E(h) changes by -1.5
• The negative sign means the error rate decreases
• The magnitude (1.5) tells us by how much it decreases

4. State the final answer

• The error rate decreases by 1.5 errors per 1000 units produced for each additional hour

Answer: B (1.5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misunderstand what the question is asking and focus on the wrong part of the function.

Instead of recognizing the question asks for rate of change, they might think it's asking for the initial error rate or some other calculation. This confusion leads them to select Choice D (90) by incorrectly identifying the y-intercept as the answer.

Second Most Common Error:

Poor INFER execution: Students attempt unnecessary calculations instead of directly reading the coefficient.

They might think they need to perform division like \(\mathrm{90 \div 1.5 = 60}\), misunderstanding the relationship between the components of the linear function. This leads them to select Choice C (60).

The Bottom Line:

Linear function problems often seem more complex than they are. The key insight is that coefficients directly tell you rates of change - no additional calculation needed, just proper interpretation of the function's structure.