Environment: Students will work in pairs on this investigation and each pair of students will have an opportunity to share their results with the class and give their decision as to which fitness center they would join and why.
Before beginning their work on this investigation, students will need to work with their partner to make sense of the problem(MP1) being asked. There is a lot of information given and students will need to determine how to use the information in a way that will help them arrive at a viable solution.
As a scaffold/extension, this investigation has two versions. The first requires that students define their variables, write two rate of change equations in y=mx+b form. They can then graph both of their equations on the accompanying axes. Students will then answer a series of questions which ask them to pull meaning from the graphs. For example, when the graph of Ironman Fitness center is "above" Barbell Fitness Center that indicates that it is more expensive. There is also a break even point when both centers will cost the same amount. After that break even point, the fitness center that was more expensive to begin with actually becomes the better deal in the long run.
The second version of the investigation is similar but adds a third function which is a horizontal line. There are now three points of intersection which students can find and use in their explanation of the meaning of the graph. In all three cases, students are using a mathematical model (MP4) to represent the situaiton and determining how various points in that model can convey important information about the cost of each fitness center.
In this task, students are asked to make a decision about the fitness center that they would join and why. Each group will be given an opportunity to share their ideas, graphs, and findings with the class. Lastly, students will write a short reflection on the investigation using a version of 3-2-1. (3 things that I learned or thought that I did well, 2 questions I still have, 1 thing I like/disliked about the investigation). This ticket out the door can be used to assess both the investigation itself and student understanding of the content.