What Does Mathematical Modeling Look Like in a Precalculus Classroom? Dan Teague NC School of Science and Mathematics teague@ncssm.edu

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Real World Taken Seriously?

The Real World? When watching basketball on television one day, I observed the following incident involving Michael Jordan: As he drove for the basket, he was fouled. According to the announcer, at that point in the season, "Michael Jordan is making 78% of his free throws". He misses the first shot and makes the second. Later in the game, Michael Jordan was again fouled. This time, as he came to the free throw line to take his shot, the announcer stated that "Michael Jordan is making 76% of his free throws".

The Math is Fairly Simple

Rounding?

The GAIMME Modeling Cycle

Keys to Modeling Modeling happens when students make important decisions about what problem to solve, how to proceed, and when to turn back.

Small Problems: Students Making Decisions

How do you do mathematics?

How far will the mantid walk when it walks for food?

Keys to Modeling Modeling happens when students make important decisions about what problem to solve, how to proceed, and when to turn back. Create the simplest form of the problem that contains the essence of the problem. Use your basic solution and the iterative process to add more reality to your initial solution.

 

Create a Problem You Can Solve

Create a Specific Example of the Problem

Driving for gas Station A is on your normal route from home to school and is selling gas this week for $3.00 a gallon while Station B, which is 5 miles off your normal route, is selling gas for $2.85 a gallon. Your car gets 30 mpg, and your friend’s car gets only 10 mpg. Should either of you drive to Station B? Explain your decisions.

Create a Specific Example of the Problem

Driving for gas Station A is on your normal route from home to school and is selling gas this week for $3.00 a gallon while Station B, which is 5 miles off your normal route, is selling gas for $2.85 a gallon. Station C has the least expensive gas, but is 8 miles off your route. Your car gets 30 mpg, and your across-the-street friend’s car gets only 10 mpg. Should either of you drive to Station B or Station C for gas? Explain your decisions.

Generalize the Problem

Great Lakes Problem

Loss-Gain Equation Drug next = Drug now – Loss + Gain

Keys to Modeling Modeling happens when students make important decisions about what problem to solve, how to proceed, and when to turn back. Create the simplest form of the problem that contains the essence of the problem. Use your basic solution and the iterative process to add more reality to your initial solution. Pay close attention to errors. Try to understand why, how, and by how much they are wrong.

The Elevator Problem

Altering Student Beliefs Math is rigid. It requires remembering how. Math is open and empowering. It requires creativity and figuring out how. Mathematics isn’t just for physicists and engineers.

Simplifying Questions Suppose it is a holiday and only 5 people come to work today. Each person works on a different floor, and they all ride the same elevator. How long will it take for everyone to get to work? Now suppose that 5 people come to work and these five people do not all work on different floors. How long will it take for everyone to get to work? Why was question (2) harder to answer than question (1)? What assumptions will you need to make in order to simplify the problem?

Assumptions my students made to simplify the problem The elevator is filled to capacity (10 people) for each trip. The elevator must stop at every floor during each trip. No one takes the stairs. No one uses the elevator to go down during this time (or if they do, it does not impact the time for the elevator to complete it’s trip). The elevator doors don’t have to re-open on any floors.

Updated Questions If all elevators go to all floors, how long will it take everyone to get to work? If 80 people were late using the unrestricted elevators, approximately what time did the employees begin arriving at the ground floor? Reassign the elevators to transport the employees to their offices as quickly as possible. What arrangement produces the shortest time? If this arrangement had been used today, would everyone have arrived at their floor on time?

What strategies did we use that we can apply to future mathematical modeling adventures? We used a simple case to understand the structure of the problem. We drew a diagram to help us visualize the scenario. We thought about what made the problem hard to help us figure out simplifying assumptions. We considered the worst case scenario and solved this rather than trying to think about all of the different possibilities. We found a solution that worked, then we modified it to see if we could improve it. We had to make sure our solution was realistic. (Sometimes “mathematically optimal” is not optimal in the real world.)

What did students have to say about the Elevator Problem? “Never before, in all my math experiences, had I seen a problem as open ended and varying as this one. Working on a problem like this with no obvious answer and many different options was a wholly new experience for me. This problem helped me visualize the role math could and most likely will play in my future.”

Things I’ve learned… Think about what information you give to students and when you give them that information. Check in often. Struggle is important, but it needs to be productive. Communicate with students how they will be graded on this assignment. If this is their first experience, consider grading only on effort (or not at all). If you want to assess their ability to communicate their findings, consider giving students the chance to revise their work. You want to encourage creativity. What grade do you assign to an incorrect answer that is based on a really good idea versus a correct answer that was very straightforward?

It’s hard for us to let go of details. “I loved how realistic it was, aside from no one taking the stairs.” What else did students have to say? We see the power of collaboration. “In my pod, I felt like none of us could have solved the problem on our own but we pooled together our knowledge and we found that it was possible to solve it together.”

We struggled. And we learned that we can persist through the struggle. “It was also a very complex problem, much more complex than I had ever done before. We had to set assumptions to reduce the amount of variables and make the project manageable. Even with the assumptions the problem was daunting. We had to break it down logically instead of just trying to plug it into a memorized equation. This thought process is very common in this class, and while I found it confusing and hard, I end with deeper understanding of how to do the problems.” “The elevator problem was probably the first time in the class that I felt like I was trying to comprehend something completely beyond my intelligence, but eventually I figured out what we are doing.”

We can see the relevance of this problem and this process to our lives. “I don't know what a profession that focuses on efficiency (workplace or public) is called, but I would love to work out things like this for a living.” “In most of my other math classes, the concepts were mostly superficial; in the sense that we only learned the basics and processes of a certain idea without working on how it could be used in real life. Of course, this was often nice and easy, bit if I'm looking to work in a STEM field one day, it is crucial to understand the applications of the different things we learn.”

We liked it! “I absolutely loved the elevator problem because it was so intricate and complex. I also liked that it might actually be helpful one day and have real world application. I liked that in order to find the one of many possible final solutions, you must first solve for one tiny section, how long it takes to get to one floor, and then apply it to the whole process. I think this was also one of my favorite problems because it was a reasoning problem instead of a computation problem.

Modeling Classes are Different Discuss with Parents Discuss with Faculty Colleagues Discuss with Administration

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Dan Teague North Carolina School of Science and Mathematics teague@ncssm.edu Dan Teague North Carolina School of Science and Mathematics

Sample Question

Make the problem a bit simpler