In this Warm-up, students compare the changes in output (temperature) over two intervals of input (time). The temperature in one interval changes by a greater amount than in the other interval, but in the latter, temperature changes more rapidly.

Thinking about what it means for temperature to drop “faster” activates the idea of rates of change and prepares for the work later in the lesson.

Some students may ask for a clarification about what is meant by “faster” in this situation. Acknowledge that thinking about its meaning in context is a great way to approach the task. Encourage these students to interpret the word by using their understanding of the given information.

Select 1–2 students from each group to explain their reasoning. As they explain, record and display their thinking for all to see. After both groups have had a chance to present, ask if anyone changed their mind because of the explanation they heard. If so, invite them to share their reasons.

It is not necessary to resolve the question at this point. Students will continue thinking about this question in the next activity.

This activity introduces students to average rate of change by building on what students know about rate of change and slope.

Students see that finding the change in the output for every unit of change in the input can be a useful way to generalize what happens between two function values, regardless of the behaviors of individual data points between them. They recognize that this number tells us how—on average—one quantity is changing relative to the other, and that it can be useful for comparing the trends in different intervals of a function.

In the activity, students find average rates of change by reasoning and using their knowledge of linear relationships. Monitor for students who use these strategies (the examples shown are for finding the average rate of change between 6 p.m. and 10 p.m.):

• Find the hourly changes in temperature, and calculate the average. The temperature fell by the first hour, then , , and in the next three hours. The average is , or per hour.
• Find the overall change in temperature, and divide it by the overall change in hours. The temperature fell in 4 hours, which means an average drop of per hour.
• Find the slope of the line connecting the two end points of the interval. The slope between and is , which means an average drop of per hour.

Plan to have students present in this order to support connecting rate of change to what students already know about slope.

In the Activity Synthesis, students generalize their reasoning and see that finding the average rate of change is indeed equivalent to finding the slope of the line connecting the two points.

This activity allows students to practice finding and interpreting average rates of change in a different context. No tables of values are given here, so students will need to estimate the coordinates on the graph to compute the average rates of change.

Students reason quantitatively and abstractly (MP2) as they extract contextual information from a graph, manipulate it symbolically, and then interpret the numerical results in context. To interpret the average rates of change, students need to pay attention to the units used to measure the input (years) and output (people in millions). This is a chance to practice attending to precision (MP6).