Several years ago a friend shared a video with me that changed my perspective on teaching math forever. It was Dan Meyer’s Ted Talk, Math Class Needs a Makeover. Before I had ever heard the term modeling, the concept was lodged in my mind and consequently my practice because of the following clip. After you check it out, let’s take a look at a few tried and true classroom activities on modeling.

Meyer goes on to make five suggestions for implementing these kinds of activities. (I highly recommend watching the entire video!)

- Use multimedia
- Encourage student intuition
- Ask the shortest question you can
- Let students build the problem.
- Be less helpful.

Using these suggestions, my students have created (and enjoyed!) several mathematical models. For example, I modified Meyer’s idea of filling up a water cooler and set up a 10 gallon Gatorade cooler outside the school. I had my students fill it with water and then told them they had to drain it only using the spout at the bottom. Of course this led to grumbling about how long they were going to have to stand there, so I told them I would make them a deal. They could go back inside if they could tell me how long it would take to do it. The first student idea was to wait until the cooler had drained halfway and then double the time. Another student built on that idea and said we could drain it a quarter of the way and multiply the time by four. This led to a discussion that we could drain it to whatever fraction we wanted as long as we proportionally compensated for the time. From that, some students created a graph of direct variation that showed the amount of drained water as a function of time. Others argued that it wasn’t direct variation because the water started out with 10 gallons in it and therefore created a graph with the number of gallons remaining vs. the time the spout was held. I tended to like the second one because it gave tangible meanings to the x and y-intercepts of their graphs. When they gave me a compelling estimate as a class, I ended the activity. Knowing what I know about modeling now, I should have recorded a video beforehand of the entire cooler draining and compared their models to the actual time it took because reconciling the theoretical to the practical is an important part of the modeling process.

In one of our more comprehensive modeling activities, my students developed a model for a country’s success in the Olympics. After watching the games one summer, I started to wonder why some countries seemed to do better than others. I posed that question to my students and we brainstormed two main categories that we thought might correlate with a country’s Olympic performance: population (greater probability that gifted athletes live there) and per capita income (more opportunities for athletes to practice and/or have access to high quality facilities and equipment.) I had each student pick three to five countries, research their populations, per capita incomes, and total medal counts in the past four summer Olympics, and add their information to the class spreadsheet. Then, in groups, they created a scatterplot for their assigned factor and analyzed the data using linear regressions to see which factors more highly correlated with Olympic performance. If you want more specifics or want to see the results, then check out the award winning lesson plan, our class spreadsheet, or my reflection on the lesson. You can also find instructions for a similar project at Mathalicious.

If you’re ready to create your own modeling activities, check out this guide that highlights some of the most important things I’ve learned about mathematical modeling. If the idea of creating your own modeling activities is intimidating, then start out by using one of the many activities already out there. When Math Happens is a great site that organizes 3 act tasks into grade bands. NCTM’s Mathematical Modeling and Modeling Mathematics has several well researched modeling activities, too. I particularly liked the ones in which students determined the stopping distance of a vehicle (pg. 18), the amount of food and water needed for emergency relief after a disaster (pg. 22), and the best placement of a fire station (pg. 55). Don’t forget to search online NCTM resources as well. For example, there is a really well done MTMS article called The Footprint Problem: A Pathway to Modeling based on a task developed by Koellner-Clark and Lesh. (See below)

I hope you’ve learned something new and have been motivated to integrate more modeling activities because of this series. Remember, our students’ thinking is either supported or constrained because of the activities we select for them to explore (Lesh pg. 135). Do you have a favorite resource for modeling activities? Care to share some of your own? Let us know!

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