# Temperature Change

## Task

The table below shows the temperature, $T,$ in Tucson, Arizona $t$ hours after midnight.

When does the temperature decrease the fastest: between midnight and 3 a.m. or between 3 a.m. and 4 a.m.?

$t$ (hours after midnight) | 0 | 3 | 4 |
---|---|---|---|

$T$ (temp. in $^{\circ}$F) | 85 | 76 | 70 |

## IM Commentary

This task gives an easy context to introduce the idea of average rate of change. While the biggest absolute drop in temperature occurs during the first time interval, the change per unit time is larger during the second time interval. It makes sense to normalize and divide by the length of the time interval and to get to the idea of change of temperature per unit time to make a meaningful comparison.

This problem could be done as a Think-Pair-Share activity. After posing the question, students can decide what they think and why and then discuss their answer with their neighbor. Different students are likely to make an argument for both time intervals and their reasoning will contrast the difference between absolute change and average rate of change. This will allow the teacher to point out the two different ways to measure change and introduce the usefulness of the idea of average rate of change. Drawing a possible graph of the temperature function during this discussion provides an opportunity to introduce slope as average rate of change.

Task based on a problem by Jerry Morris, Sonoma State University. Used with permission.

## Solution

Even though the temperature drops more between midnight and 3 a.m. than it does between 3 a.m. and 4 a.m., the temperature decreases fastest between 3 a.m. and 4 a.m., as the calculations below demonstrate:

Interval | $\Delta T$ | $\Delta t$ | $\displaystyle{\frac{\Delta T}{\Delta t}}$ |
---|---|---|---|

$0 \leq t \leq 3$ | $-9^{\circ}$F | 3 hours | $\displaystyle{\frac{-9}{3} = -3^{\circ}}$F per hour |

$3 \leq t \leq 4$ | $-6^{\circ}$F | 1 hour | $\displaystyle{\frac{-6}{1} = -6^{\circ}}$F per hour |

Thus, the temperature decreases at an average rate of only $3^{\circ}$F per hour between midnight and 3 a.m., but it decreases at an average rate of $6^{\circ}$F per hour between 3 and 4 a.m.

## Temperature Change

The table below shows the temperature, $T,$ in Tucson, Arizona $t$ hours after midnight.

When does the temperature decrease the fastest: between midnight and 3 a.m. or between 3 a.m. and 4 a.m.?

$t$ (hours after midnight) | 0 | 3 | 4 |
---|---|---|---|

$T$ (temp. in $^{\circ}$F) | 85 | 76 | 70 |