A diver ascends toward the surface at a constant rate. The relationship between the diver's depth d, in meters, and...

1. TRANSLATE the equation components

• Given equation: \(\mathrm{d = 120 - 0.8t}\)
• This can be rewritten as: \(\mathrm{d = 120 + (-0.8)t}\)
• This is in the form: (dependent variable) = (initial value) + (rate of change)(independent variable)

2. INFER the meaning of each component

• In linear equations, the coefficient of the variable represents the slope
• Slope = rate of change = how much d changes for each unit change in t
• Here: \(\mathrm{-0.8}\) means depth decreases by \(\mathrm{0.8}\) meters per second
• The \(\mathrm{0.8}\) represents the magnitude (size) of this rate of change

3. TRANSLATE this back to the real-world context

• Since the diver is ascending (moving toward surface), depth should decrease over time
• The negative slope confirms this: depth decreases as time increases
• \(\mathrm{0.8}\) = the decrease in depth (in meters) for each 1-second increase in time

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse the slope with the y-intercept, thinking that because 120 is the initial depth, \(\mathrm{0.8}\) must also represent some kind of initial condition.

They don't recognize that in \(\mathrm{d = 120 - 0.8t}\), the \(\mathrm{-0.8}\) coefficient tells us how d changes as t changes. Instead, they focus on the numbers without understanding their roles in the linear relationship.

This may lead them to select Choice A (The initial depth, in meters, of the diver).

Second Most Common Error:

Conceptual confusion about slope: Students recognize that \(\mathrm{0.8}\) relates to the rate of change but miss the negative sign's significance. They think that since \(\mathrm{0.8}\) is positive, it represents an increase in depth.

They fail to understand that the equation \(\mathrm{d = 120 - 0.8t}\) has a negative slope (\(\mathrm{-0.8}\)), meaning depth decreases. The \(\mathrm{0.8}\) by itself represents the magnitude of this decrease.

This may lead them to select Choice D (The increase in the diver's depth, in meters, for each one-second increase in time).

The Bottom Line:

Success requires understanding that in linear models, the coefficient of the independent variable represents the rate of change, and interpreting both the magnitude and direction (sign) of that rate in the real-world context.