A research team monitored the temperature of a cooling liquid over several hours. The data collected is displayed in the...

1. TRANSLATE the problem information

• Given information:
  • A scatterplot showing temperature (T) in °C versus time (t) in hours
  • The liquid is cooling (temperature decreasing over time)
  • Four possible linear equations in the form \(\mathrm{T = mt + b}\)
• What we need to find:
  • Which equation best represents the data

2. INFER the direction of the relationship

• Key observation: As you scan from left to right across the scatterplot, the points move downward. This means as time increases, temperature decreases.
• Strategic reasoning: In the equation \(\mathrm{T = mt + b}\):
  • If \(\mathrm{m}\) is positive, temperature increases with time
  • If \(\mathrm{m}\) is negative, temperature decreases with time
• Since the liquid is cooling, we need a negative slope.

Eliminate choices C and D (both have slope +1.8)

3. INFER the y-intercept

Now we're down to:

• Choice A: \(\mathrm{T = -1.8t - 11.9}\)
• Choice B: \(\mathrm{T = -1.8t + 11.9}\)

The difference is the y-intercept (the b value).

• What does the y-intercept mean? It's the temperature at time \(\mathrm{t = 0}\) (when timing began).
• Look at the leftmost point on the scatterplot: at about \(\mathrm{t ≈ 0.5}\) hours, \(\mathrm{T ≈ 10.8°C}\)
• Think backwards: If the temperature is dropping at 1.8°C per hour, and it's 10.8°C at \(\mathrm{t = 0.5}\), then half an hour earlier:
  • Temperature change = \(\mathrm{1.8 × 0.5 = 0.9°C}\) warmer
  • At \(\mathrm{t = 0}\): \(\mathrm{T ≈ 10.8 + 0.9 = 11.7°C}\)
• This is close to 12°C, meaning the y-intercept must be positive (around +11.9, not -11.9)

Eliminate choice A

4. VERIFY the remaining choice

Let's check if \(\mathrm{T = -1.8t + 11.9}\) fits some data points:

• At \(\mathrm{t = 1}\): \(\mathrm{T = -1.8(1) + 11.9 = 10.1°C}\) (graph shows ≈9.4°C) ✓ Close
• At \(\mathrm{t = 3}\): \(\mathrm{T = -1.8(3) + 11.9 = 6.5°C}\) (graph shows ≈5.0°C) ✓ Close
• At \(\mathrm{t = 6}\): \(\mathrm{T = -1.8(6) + 11.9 = 1.1°C}\) (graph shows ≈0.3°C) ✓ Close

The equation provides a reasonable fit for the data.

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not connecting the concept of "cooling" to the sign of the slope

Some students look at the answer choices and don't immediately recognize that a cooling liquid must have a negative rate of change. They might:

• Focus only on the numbers (1.8 and 11.9) without considering signs
• Try to calculate the exact slope from two points instead of using the pattern
• Get confused by all four choices looking plausible

This leads to confusion and potentially guessing randomly among all four choices.

Second Most Common Error:

Misunderstanding y-intercept concept: Confusing the y-intercept with one of the plotted points

Students might think the y-intercept should be one of the temperatures they can see on the graph (like 10.8 at \(\mathrm{t = 0.5}\)). They fail to recognize that:

• The y-intercept occurs at \(\mathrm{t = 0}\) (off the visible data range)
• They need to extrapolate backward to find it

Without this understanding, they might select Choice A (\(\mathrm{T = -1.8t - 11.9}\)) because they correctly identify the negative slope but then guess on the y-intercept, choosing the wrong sign.

The Bottom Line:

This problem tests whether you can connect the physical situation (cooling) to mathematical properties (negative slope) and whether you understand what a y-intercept represents in context. The key is using strategic reasoning to eliminate choices systematically rather than trying to calculate everything precisely from the graph.