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 is a stressful experience. By evaluating a sample response to a modeling prompt themselves, students better understand the expectations on which their own responses are evaluated, and this helps 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, provide students with the background information they need in order to perform 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, though often expressed in words, involve more thought than a more straightforward exercise. Unlike word problems, modeling prompts challenge the modeler to make reasonable assumptions, decide what information is important, ask for or research the 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; rough models are still useful. Models often can 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 learn this background information in many ways, and the best method is 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 students "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 prepares them to understand the sample model when they read it.

Next, show students the rubric to 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 rubric category. Once students have shared, ask: “What would you like to know that would help you use this rubric?” Students should have a sufficient understanding of the rubric such that they 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 a 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 the response (for this prompt only, these are included in the same document), along with the modeling rubric. Students should write or otherwise make notes on the rubric and the sample response, if possible.

Responses to modeling prompts can vary widely, but they often contain certain pieces: assumptions, calculations, a mathematical model (stated with one or more 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 begin their evaluation.

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

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 you improve this response?"
• "What part of the response was easiest to understand? What could have been explained better?"
• "Which skill should the modeler work to improve?"
• "Was there something you disagreed about with your group?"
• "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 more 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 also should 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, either as partners or in small groups, to improve the response, based on their evaluation of it. After sufficient work time, each group should share their improved version with the class so that everyone can see a variety of ways to improve the response. Students could share by presenting to the class, doing a Gallery Walk, uploading a scan or a photo of their work to a shared online space, or by any method that works best for the class.