We launch the investigation almost immediately and ask the class, "can you change the world?" We follow by showing the trailer for the movie Pay it Forward. The trailer shows a clear idea of what it means to "pay it forward."
After the clip shows, I would ask students to rephrase what it means to "pay it forward" and outline the discussion on the board.
Paying It Forward:
1. Help three people
2. Those three people help three other people
I would ask students to take 2 minutes and draw a visual of this process.
"Imagine you tried this. What would you do? How much of an impact would it have. Write out three ideas you have to help others and draw a model that represents what happens if they pass it on and then the next group passes it again. How many people will you have impacted?"
After they have had a chance to reflect, I ask students to share. I want them to use their mathematical model to make a social argument.
"Would paying it forward make a difference? How do you know?"
This is a discussion around would could happen and what students think would happen if they tried to pay it forward. This is not a debate with a correct or even predictable result, it is a chance to spend about 5 minutes sharing ideas around the concept of paying it forward. The more they talk about it, the more they will be ready to work on the math surrounding the concept.
I like to give students plenty of room in this investigation, since I want them to find a way to naturally reach an exponential model. The question I ask is, "Could paying it forward reach everyone in the world?" Specifically, I will offer them two paying it forward models.
Model 1: Help 3 and have those people help 3 others (like the movie)
Model 2: Help 3 every day.
Get a group of 1000 people together who will help 3 people each day. Is this a stronger model than the one in the movie. Why?
If you have a group of people willing to help 3 others each day, how large would the group need to be to reach everyone in the world in 21 days?
After I present the models and prompts, I ask students, "What do you need to solve this problem?" Students need to know how paying it forward works (which we discussed at the start) and the current world population. They also need to have a time frame for how long it takes to complete a "good deed." This depends on what they consider acceptable as a "good deed" and what they consider to have an impact.
I provide all the tools needed to solve this problem. I give them the population number (displaying the link on the projector) and have a station with graph paper, graphing calculators, etc.
As I circulate, I will nudge students toward functions, graphs and tables, but only if they don't have another working algorithm. For example, if they really like drawing a tree to represent the growth of paying it forward, I would ask them to look at a smaller population before they approach the population of the entire world. I wouldn't discourage them from their algorithm, since the tree diagram will help them make sense of this problem in a way that is natural for them.
During their investigation, I like to record ideas and quotes from the class. I start off the summary by sharing some of the more compeling student ideas and use these to launch a quick conversation. For example, a student might say, "If everyone followed through, this wouldn't take long at all." I would ask the class if they agree and how they could know. Students would share their approaches in tables, graphs, functions, etc. We would discuss the equation y = 3^x with questions like, what does x represent? What does y represent? How does this connect to the columns in a table and the axis in a graph?
For students who graphed the function by hand, I would show their work, demo it on the graphing calculator and extend it by using Desmos and other online graphing calculators. I like to discuss the meaning of the intersection points and the reasoning as to why exponential growth is so much greater than linear growth.
I might do this by simply showing a multiplicative (exponential) vs. an additive (linear) model and comparing the slopes in the linear and exponential paying it forward models.
With the graphs, tables and functions shared, I would ask students to summarize how they can recognize an exponential relationship. They could respond in a variety of ways, but I would quote students around the following ideas:
1. Linear functions make straight lines and exponential functions make "curves"
2. Exponential functions can grow a lot faster than exponential functions.
3. Linear functions have a constant rate of change or slope. Exponential functions do not
4. Linear functions look like y = mx + b and exponential functions look like y = a^x