Today's lesson involves two of my favorite problems that allow students to puzzle with ideas of exponential growth and decay. Depending on how quickly the class works, I might give both problems to students or I might divide the class in two and give each half one of the problems. If I divide the class in two, the discussion period of the class can be used to have students share out their problem and work with the other class. If I want to differentiate this lesson for students, I might give the groups that are still developing their ideas about the difference between linear and exponential growth the money problem, and give the rainforest problem to students who are ready to try to quantify exponential decay.
We read through One Million Dollars or One Penny and the The Disappearing Rainforest together. Again, depending on how I've decided to divide up the class, I let students ask clarifying questions and then they get to work. As I circulate, I look out for:
One Million Dollars Problem
I may encourage students to use graphing software like desmos.com rather than having them do graphs by hand depending on their level of competency with graph making.
I have different groups share out their problems and findings here. In the money problem, we are looking to show again how quickly 1 cent can grow if it continues to double. Students are often shocked by the results of this problem. Again we can compare exponential and linear growth here. We can also talk about whether or not this function is growing discretely or continuously and if it fits the criteria to be an arithmetic or geometric sequence.
For the rainforest problem, we want to focus on writing the explicit equation for exponential decay (students have not learned the standard formula yet). I'll try to elicit from students that they are keeping 93% of the rainforest so .93 becomes our common ratio. We can talk about whether or not the rainforest would ever completely disappear and what this means mathematically versus in reality.
The focus of today's discussion should really be about students sharing their work and representations and defending their answers.
To close today's class I'll ask students to respond to the following prompt on an exit ticket: