This Warm-up prompts students to compare four data sets. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.

Invite each group to share one reason why a particular set of three goes together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations, and ensure the reasons given are correct.
During the discussion, ask students to explain the meaning of any terminology they use, such as “increasing” or “decreasing.” Also, press students on unsubstantiated claims.

The purpose of this activity is to build on the work of the Warm-up as students determine which of the given functions is a better fit for the data. For this activity, students should use any language that makes sense to them to describe why a function is or is not a good fit and how they would change the function to be a better fit for the data. Since both functions offer reasonably good fits for the data, students have the opportunity to make an argument about why they think a particular function is a better fit (MP3).

Monitor for students using different explanations of what makes a function fit the data, such as by focusing on the general shape of the graph, the accuracy of individual points, or the average error for all the points.

In the digital version of the activity, students use an applet to check their initial response to the question about which function better fits the shape of the data. The applet allows students to zoom out on the graph and see the part of the rational function that is not visible in the initial graphing window.

This activity works best when each student has access to devices that can run the Desmos applet because students will benefit from seeing the relationship in a dynamic way. If students don't have individual access, projecting the applet will be helpful during the Activity Synthesis.

In this partner activity, students take turns describing transformations of a graph and sketching the transformed graph from the description. As students trade roles explaining their thinking and listening, they have opportunities to refine and use more precise language when describing transformations (MP6).