Using Graphs to Find Average Rate of Change

5 min

Narrative

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.

Launch

Display the task for all to see. Give students 2 minutes of quiet time to work the question, and then follow with a whole-class discussion.

Student Task

Here are the recorded temperatures at three different times on a winter evening. 

time 4 p.m. 6 p.m. 10 p.m.
temperature 25F25^\circ \text{F} 17F17^\circ \text{F} 8F8^\circ \text{F}
  • Tyler says the temperature dropped faster between 4 p.m. and 6 p.m. 
  • Mai says the temperature dropped faster between 6 p.m. and 10 p.m.

Who do you agree with? Explain your reasoning.

Sample Response

Sample responses:

  • I agree with Mai. The temperature dropped 8F8^\circ \text{F} between 4 p.m. and 6 p.m., but it fell 9F9^\circ \text{F} between 6 p.m.and 10 p.m.
  • I agree with Tyler. The temperature dropped 4F4^\circ \text{F} per hour between 4 p.m. and 6 p.m. It fell less than 4F4^\circ \text{F} per hour in the evening.
Activity Synthesis (Teacher Notes)

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.

Anticipated Misconceptions

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.

Standards
Building Toward
  • F-IF.6·Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
  • F-IF.6·Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
  • F-IF.6·Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
  • F-IF.6·Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
  • F-IF.6·Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
  • F-IF.6·Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
  • HSF-IF.B.6·Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

20 min

10 min