Modeling Prompt

This first modeling prompt is structured differently from the others. Instead of responding to a prompt, students evaluate a given sample response to a prompt. The purpose is for students to learn more about what a mathematical modeling prompt is and what a response to a modeling prompt might look like. Modeling with mathematics is a complex set of skills, and evaluating those skills is also complex. Students may not have been asked to create models before. Trying something new when the expectations are unclear can be a stressful experience. By evaluating a sample response to a modeling prompt themselves, students will better understand the expectations their own responses will be evaluated on, and this will help them gain confidence in their ability to succeed at modeling.

Understanding the Sample Model and Rubric

Before distributing the prompt with the sample response to evaluate, students will need some background information before they can understand what they're being asked to do in this task. They should know:

• What a modeling prompt is, and in particular how it's different from a word problem, which is likely more familiar. Modeling prompts are also often expressed in words, and involve more thought than a more straightforward exercise. But unlike word problems, modeling prompts challenge the modeler to make reasonable assumptions, decide what information is important, ask for or research for more information if needed, think creatively within constraints, and consider the implications of the model. The process of modeling is cyclical, and it does not end by producing a “correct answer.” A mathematical model expresses a simplified relationship in the real world; models can be rough but still useful. Models can often be refined to represent the real-world relationship more accurately.
• The expectations for work students produce in response to a modeling prompt. Students should have an opportunity to discuss the modeling rubric and ask clarifying questions about it before they grade the sample prompt.

Students can learn this background information in many ways, and the best method will be different for each classroom. Here is one way.

Display the sample modeling prompt for all to see. Give students quiet think time to answer these questions:

1. What do I understand about this task?
2. What would I need to know in order to answer this question myself?

Ask "What questions do you have about the prompt?" After answering any clarifying questions, ask "If you were going to respond to this prompt, what would you try?" Write down students' ideas for all to see. They don't need to find a complete strategy for answering the prompt. The goals of this discussion are to make sure students understand what the prompt is asking, and to put students in the position of the modeler trying to answer the prompt. This will prepare them to understand the sample model when they read it.

Next, show students the rubric that they will use to evaluate the sample model. After quiet think time, invite them to share with a partner something they notice and something they wonder about each of the rubric categories. Once students have shared, ask “What would you like to know that would help you use this rubric?” Students should have a good enough understanding of the rubric that they can feel comfortable trying to use it, but they may discover the answers to some of their questions themselves as they evaluate the sample model, so deep understanding of the rubric is not needed at this time.

Evaluating the Sample Response

To each student, distribute a copy of the sample prompt and response (for this prompt only, these are included in the same document), and the modeling rubric. Students should be able to write or otherwise make notes on the rubric and sample response if possible.

Responses to modeling prompts can vary widely, but they often contain certain pieces: assumptions, calculations, a mathematical model (stated with an equation or equations, with a graph, with a geometric diagram, or in words), conclusions, and generalizations. Ask students to identify these different sections of the sample model in some way (for example, by outlining them in different colors) before they evaluate it.

Students should have ample opportunity to discuss the sample response and the degree to which it meets each skill on the rubric, but if they work in groups, they needn't reach a group consensus on how to evaluate the response. Instead, students should be encouraged to justify the way they evaluated the response by writing comments in the "Notes or Comments" boxes in the rubric (or on separate paper if needed).

After students evaluate the sample response, conduct a whole-class discussion. Here are some questions for discussion:

• "What did the modeler do well?"
• "How could this response be improved?"
• "What part of the response was easiest to understand? What could have been explained better?"
• "Which skill should the modeler work on?"
• "Was there something you disagreed with your group about?"
• "If you were the modeler's teacher, what advice would you give them?"
• "When you create your own mathematical models, which of the skills on the rubric do you think will be difficult for you? Which will be easier?"

The goals of this discussion are to make sure students can interpret the rubric, and to prepare them to apply what they've learned from the process of understanding and evaluating the sample response when they respond to a modeling prompt for the first time themselves. 

Students should also understand that modeling is a cycle, and that they should evaluate their own models and then refine them. Introduce them to this process by asking them to work together to improve the response based on their evaluation of it. After sufficient work time, each group or pair should share their improved version with the class so that everyone can see a variety of ways that the response could be improved. Students could share by presenting to the class, doing a gallery walk, uploading a scan or photo of their work to a shared online space, or by any method that works best for the class.