By Graham Fletcher, posted on April 25, 2016 – 

In case you missed my last post, I discussed what the fourth of the eight Standards for Mathematical Practice (SMP 4), modeling with mathematics, is and what it is not. I introduced the Whopper Jar Three-Act task, for which we left estimations and the information we needed to solve our question.

ACT 2—The Missing Information

So, you have asked for certain pieces of information that are critical to answering our main question: How many Whoppers are in the jar? Here’s information you have identified as the unknown variables:

• How many Whoppers come in a bag
• How many bags did he use to fill the bag?
• Did he use the whole bag?
  

At this point, you’re ready to go and so are the students. You have made an estimate, identified the variables, and put them to good use. Now it’s time to see how close your estimate was to the actual number of Whoppers in the jar.

Act 3—The Reveal

First view the video. The third and final act reveals the answer to our driving question. You’ll notice that no solutions are shared; that is because it’s left up to the teacher to orchestrate this conversation during the closing of the lesson.

Do you feel a sense of accomplishment knowing that you got the right answer? Maybe you were off a little, and now you want to backtrack to see where something went wrong. One exciting outcome of three-act tasks is that the urgency to go back and analyze mistakes along the way doesn’t happen with adults only; it happens with students as well. This is just one of one of the major benefits of using three-act tasks.

The Benefits of Including Three-Act Tasks

While engaging students in Three-Act tasks, I have found that many students are consistently asking themselves, Does this make sense? Whether it is the missing variables they have identified, or checking the reasonableness of their estimate, we’re asking students to take more ownership of their learning and engage in mathematics differently.

Having worked through this Three-Act task, you can begin to recognize the type of reasoning and problem solving that we need to expect from our students. I can honestly say that I thought I was doing a great job of engaging students in modeling until I watched Dan Meyer’s TED Talk: Math Class Needs a Makeover. Many Three-Act Tasks are composed of a fixed beginning with a particular ending in mind. It’s everything that happens in between the initial question and the third act that I’m after.

Employing the open-middle concept of Three-Act tasks has proven instrumental for me in providing authentic opportunities for our students to engage in modeling with mathematics. In my experience, we tend to engage students in problems that have a limited number of solution paths, which tends to pigeonhole a student’s thinking. As an added bonus, having a variety of student solutions makes it much easier to make connections between the models presented during the closing of the lesson.

As you begin to find ways to incorporate Three-Act tasks in your classroom, you’ll begin to see that modeling is about a mathematical process. Modeling with mathematics requires that students engage in a collection of actions, rather than a single action alone. Asking questions, making estimates, and identifying missing variables are everyday actions that students could use; but it’s when the actions are used in concert that the modeling magic occurs.

Below are the actions and processes we’ve engaged in while tackling the Whopper Jar task. They should be the same actions we should expect our students to engage in when they are modeling with mathematics. Are any of these actions in the modeling process missing from your instruction?

When we begin to ask students to engage in the modeling process, we begin to ask them to look at the world through the lens of the Standards of Mathematical Practice. We begin to ask that students do the following:

• Make sense of problems and persevere in solving them
• Reason abstractly and quantitatively
• Construct viable arguments and critique the reasoning of others
• Model with mathematics
• Use appropriate tools strategically
• Attend to precision
• Look for and make use of structure
• Look for and express regularity in repeated reasoning

After all, isn’t this what we should be teaching?

To find the Whopper Jar or other three-act tasks, visit Dan Meyer’s blog. Find more tasks from these other teachers who are fully invested in the power of mathematical modeling:

Dane Ehlert

Robert Kaplinsky

Jon Orr

Kyle Pearce

Andrew Stadel

Mike Wiernicki

Graham Fletcher has worked in education for over ten years as a classroom teacher, math coach, and district math specialist. He graduated from the University of Georgia, where he earned his specialist degree in Math Education. Fletcher’s passion for conceptual understanding through problem-based lessons has led him to present internationally and throughout the United States. He continues to be an advocate for best practice and a change agent for elementary school mathematics.