Students will be able to describe exponential decay using multiple representations based on the example of eating some fraction of what remains of a candy bar each day.

Students create visual examples of exponential decay and develop formulas to describe this. Students relate exponential decay to exponential growth.

30 minutes

After two days of investigation, some students will have fully mastered the bunny problems. This means that they have already figured out how to use equations to solve all forms of the problem. Other students still need more time to investigate them in order to reach this level of abstraction. So the purpose of this lesson is to present a new type of problem to all students. Some students will simply investigate the simple example, while other students will investigate different versions of this problem in order to generalize.

The warm-up is designed to enable all students to review the bunny problems. Some students will spend the whole 30 minutes working on the first three problems, because they are still figuring out those problems. The warm-up is also an informal formative assessment: as students work, you can circulate and figure out which students are ready for the more challenging and abstract exponential decay problems and which students need to work on the more basic exponential decay problems with the fraction of ½.

You can spend however much time you want on the warm-up, depending upon how much time students need to figure out the bunny problems. As you circulate, ask students questions about the bunny problems. The goal is to help students move towards a more abstract understanding of each problem. In this case, at the most concrete level, students will use blocks or diagrams to represent the problem. Then, students will start to organize their data into tables. As students try to complete these data tables, they will find a way to figure out the next number in the table without using a diagram. From this point, they will start to try to find an equation for this data table. Once they have found equations, they will then figure out how to use these equations to work backwards. So during the warm-up today, figure out at which level of abstraction each student is working and ask them questions to help them move towards a more abstract understanding of the problem. The point is for students to move slowly—if they just start writing equations that don’t make sense to them, they have not really learned anything. So during this time, it is important to encourage students to take their time and to remind them explicitly that the purpose of the warm-up is not completion but understanding. If you notice students writing down equations that they don’t seem to understand, ask them questions to make sure that they do understand or have clarifying conversations with them to address some of the following questions:

In an example where each pair of bunnies has 3 pairs of bunnies per month, you might ask students the following questions:

- If you know that there are a certain number of pairs of bunnies in one month, how can you find the number of bunnies the next month?
- You multiply by 3 and this gives you the new babies. What do you still need to do to find the total number of bunnies? (add the parents.)
- Is multiplying a number by 3 and then adding the number to itself the same as multiplying by 4? Why would this be true?
- If you always multiply by the same amount each time, how could you express this using a function rule?
- Once you have written a function rule, could this help you solve the problems with different missing information?

30 minutes

10 minutes