Average rate of change
• Calculate the average rate of change of a function over a specified interval; •Interpret the average rate of change of a function over a specified interval.
In this section, students are introduced to the notion of average rate of change over an interval. They work with expressions for average rates of change, and compute and estimate average rates of change. In grade 8, students learned that the rate of change of a linear function is the slope of its graph and that the slope can be computed from the coordinates of any two distinct points on the line. For nonlinear functions, the story is more complicated because their rates of change vary depending on the interval chosen. As for linear functions, average rates of change over an interval can be computed from the coordinates of two points on their graphs, namely those corresponding to the endpoints of the interval.
Tasks
1
Temperature Change
WHAT: Students are given a table with entries for temperature and time. They are asked: “When does the temperature decrease the fastest: between midnight and 3 a.m. or between 3 a.m. and 4 a.m.?”
WHY: This task gives an easy context to introduce the idea of average rate of change F-IF.B.6. 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.
2
The High School Gym
WHAT: This task describes how the temperature changed in a high school gym over a period of time. Students are asked questions about the relationship between two variables: $T$ (temperature of the gym in degrees Fahrenheit) and $M$ (time, in minutes, since noon) F-IF.A.1: Is $M$ a function of $T$? Is $T$ a function of $M$? In the second two parts of the task, students choose equations and expressions to represent statements that involve average rate of change F-IF.B.6.
WHY: The first two parts of this task return to the definition of function. The second two parts move students toward expressing average rate of change in function notation.
3
Mathemafish Population
WHAT: In this task, students are presented with fictional data about a fish population shown in a table and graph. Students are asked to calculate and interpret average rates of change over intervals that they select in order to support the statement that the fish population is being decimated F-IF.B.6, MP.3.
WHY: Students build on their experiences with expressions for average rate of change from The High School Gym. By interpreting key features of the graph F-IF.B.4, they use their understanding of how the fish population has changed over time in order to select intervals and to contrast rates of change over various intervals.