A car decelerates from a speed of 50 miles per hour to 0 miles per hour at a constant rate...

1. TRANSLATE the problem information

• Given information:
  • Initial speed: \(50\text{ mph}\)
  • Final speed: \(0\text{ mph}\)
  • Rate of change: \(10\text{ mph per second}\) (deceleration)
• What this tells us: The speed decreases by exactly \(10\text{ mph}\) every second

2. INFER the mathematical relationship

• Key insight: "Constant rate" is the crucial phrase here
• When something changes by the same amount per unit time, that's the definition of a linear relationship
• The rate is \(-10\text{ mph/s}\) (negative because it's deceleration)

3. INFER the function behavior

• Since the rate of change is negative (\(-10\text{ mph/s}\)), the function is decreasing
• Linear relationship + decreasing behavior = decreasing linear function
• The function would look like: \(\mathrm{Speed} = 50 - 10\mathrm{t}\)

Answer: B (Decreasing linear)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse "constant deceleration" with exponential decay, thinking that slowing down always means exponential behavior.

They might reason: "The car is slowing down, and things that slow down follow exponential patterns like radioactive decay." This misconception leads them to select Choice A (Decreasing exponential).

Second Most Common Error:

Poor INFER reasoning: Students correctly identify that it's linear but get confused about whether deceleration means increasing or decreasing.

They might think: "The deceleration rate is increasing the stopping, so it's increasing linear." This backward reasoning about what's actually changing leads them to select Choice D (Increasing linear).

The Bottom Line:

The key insight is recognizing that "constant rate" always signals linear behavior, regardless of the context. Students who focus on the physical situation (car slowing down) instead of the mathematical relationship (constant rate of change) often miss this connection.