Recognize scenarios that depict exponential growth or decay by identifying a fixed percent change or common ratio.
Write equations of the form y=ab^x to model scenarios that grow or decay by a fixed percent or factor.
Quick Lesson Plan
For this experience, students will work through this Desmos activity: Game, Set, Flat. This will take them through the science behind the bounce of a tennis ball. Students will investigate whether certain bounces indicate if a tennis ball is good or bad. The International Tennis Federation has ruled that tennis balls must have a rebound height between 53% and 58% of its previous bounce height. This models an exponential relationship that the students will discover as they work through the activity.
For Slide 8 (#3), make sure students know to fill in the table and draw the graph for the heights corresponding to a good tennis ball according to the rules set by the Tennis Federation.
Do you think everyone in the class will have the same multiplier in their equation?
Is there truly a constant multiplier? Why or why not?
Why is the initial value in your equation not 4 if it was dropped from a height of 4 feet?
Why are we using the word “model” here?
What does the graph of the bounces reveal about how the heights of each bounce are changing?
This activity ties the previous day’s learning specifically to a modeling context. A key skill is determining if a situation exhibits exponential change, and whether that change is growth or decay. As you are monitoring and debriefing, continue to formatively assess the main goal of this unit, which is for students to be able to explain how the pattern in an exponential function is different from the pattern in a linear function. In the previous lesson, students were introduced to the exponential function with an initial value (a) and a growth/decay factor (b), so be sure to connect those parameters to the values they got from the Desmos activity. Explain that the Desmos used estimates, so the “initial height” of the tennis ball will vary from student to student, but that they should all be around 4 since that’s where the ball was dropped from.
For #2c in the Check Your Understanding, students may see “decreases by 10% every hour” and associate it with exponential decay. However, phone batteries are already expressed as percentages, so if battery decreases by 10% per hour, then it goes down by a fixed amount (starts at 100%, then 90%, then 80%, etc.). This is linear, not exponential.