Experience First

This lesson is a high-level task designed to get students thinking about multiple paths to finding a solution and interpreting the meaning of that solution in context and across multiple representations. Students may be surprised about the open-ended nature of the task which differs from the usual progression of questions. Encourage students to clearly demonstrate how they are thinking about this problem (using color, diagrams, etc.) and look for students using numeric/tabular, graphical, and analytical representations as you are monitoring. We strongly recommend using Margaret Smith and Mary Kay Stein’s 5 practices approach for facilitating this lesson. Express curiosity about students’ thinking and be looking for ways to connect different students’ work during the debrief.

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Formalize Later

Although this lesson features a quadratic function, the focus of this lesson is about connecting solutions across multiple representations, and less about the characteristics of parabolas. Invite at least three groups to share their method for solving this problem, intentionally selecting groups that showcase the three main solution paths. Ask students to summarize each others’ ideas and make connections between representations. For example, ask how a revenue of $0 shows up in the table and how it is visible in the graph. Then ask how information from the equation allowed the analytical group to find the same value. 

If time allows, ask students to discuss the advantages and disadvantages of each representation and when one representation or solution method might be preferable over another.

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Use the debrief or Important Ideas to review calculator keys for finding an intersection and have students state in context what it means for the two expressions (curves) to be equal