Temperature Change

Task

Each expression below gives the rate of change of temperature in degrees Celsius over a certain period of time. Interpret each expression below in terms of the temperature change.

1. $-\frac{2 \text{ degrees}}{3 \text{ hours}}$
2. $\frac{-2 \text{ degrees}}{3 \text{ hours}}$
3. $\frac{2 \text{ degrees}}{-3 \text{ hours}}$

IM Commentary

The goal of this task is to provide a context for interpreting the expressions $-\frac{p}{q}$, $\frac{-p}{q}$, and $\frac{p}{-q}$  (with $q \neq 0$): this matches the last part of the standard 7.NS.2.b, ''Interpret quotients of rational numbers by describing real-world contexts,'' though in this case the numerator and denominator are integers. Because of the context, students will also gain experience working with rates.

Each expression calculates the rate for the change in temperature over the three hour period and this means that the three expressions must be equal. The teacher may wish to discuss the fact that the task shows that $$-\frac{p}{q} = \frac{-p}{q} = \frac{p}{-q}$$ (when $p = 2$ and $q = 3$) but this is not formally requested. The teacher may wish to have students choose their own expression indicating that the temperature is falling 2 degrees every 3 hours and then have them share and consider other possible expressions: while it is unlikely any one would use $\frac{2}{-3}$, the other two expressions may both appear and would naturally lead to a good discussion.

Solution

1. In the first expression, $-\frac{2 \text{ degrees }}{3 \text{ hours}}$, the negative sign indicates that the temperature was decreasing. The expression $\frac{2 \text{ degrees}}{3 \text{ hours}}$ indicates that it was falling at a rate of 2 degrees for every 3 hours. This is the same as falling at a unit rate of $\frac{2}{3}$ of a degree per hour. 
2. The expression $\frac{-2 \text{ degrees }}{3 \text{ hours}}$ means that the rate of change of temperature is -2 degrees in 3 hours. This means that the temperature is falling at a rate of 2 degrees every 3 hours.

3. The expression $\frac{2 \text{ degrees }}{-3 \text{ hours}}$ means that the temperature is increasing by 2 degrees every -3 hours. This is an odd way of talking about a rate of change but it makes sense. For every 3 hours we go back in time, the temperature goes up by 2 degrees. Again, the temperature is decreasing 2 degrees every 3 hours (moving forward in time).

   This way of viewing the change of temperature does not feel natural because in our experience time flows forward and this expression requires looking at time in reverse.