The purpose of this activity is to learn or remember a straightforward way to check if a relationship is linear or exponential (subtracting and dividing successive terms). Also, in the Activity Synthesis, there is an opportunity for students to rehearse creating a scatter plot, and to recall how to recognize that a graph of a relationship might indicate that it is linear or exponential.

In this activity, students critique a statement or response that is intentionally unclear, incorrect, or incomplete and improve it by clarifying meaning, correcting errors, and adding details (MP3).

The goal is to identify key information that differentiates linear and exponential situations.

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response to the last question by correcting errors, clarifying meaning, and adding details.

• Display this first draft:
  Linear relationships change in the same way every time, and exponential relationships do not.
  Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
• Give students 2–4 minutes to work with a partner to revise the first draft.
• Select 1–2 students or groups to slowly read aloud their draft. Record for all to see as each draft is shared. Then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

Demonstrate how to use the table of values about the farmer’s grain and technology to create a scatter plot. Then, ask students to use technology and the table of values about the car depreciation to create a scatter plot. Display these graphs side by side and observe ways in which the graphs suggest that one would be better modeled with a linear function and the other with an exponential function.

If students have access to the digital version of the materials, the “Graphing Calculator” tool under Math Tools is recommended. Add a new table, enter the values, and then adjust the graphing window (under the wrench button) so that the scatter plot is visible.

This activity is an opportunity to practice using data given in a table to decide between a linear and exponential model. It is also an opportunity to practice creating a scatter plot using technology. The data is not perfectly linear or exponential, which is addressed in the Launch. Making decisions about how best to model a data set is a part of modeling with mathematics (MP4).

The purpose of this discussion is to compare models and discuss strategies. Invite a few students to display their scatter plots and models. Because the data is a little bit messy, different students might come up with models that are slightly different from each other but still reasonable. Here are some questions for discussion:

• “How did you decide whether a linear or exponential model would be better?” (If the data was being added by about the same amount, I knew the model should be linear. If the data was being multiplied by about the same amount, I knew the model should be exponential.)
• “How did you determine the growth factor or rate of change?” (I divided a number by the previous number to calculate the growth factor. I subtracted a number from the next number to calculate the rate of change. In both cases they weren’t the same number each time, but it was close.)
• “How did you construct your equation?” (Linear equations use an initial value, or -intercept, and the rate of change. Exponential equations use an initial value and the growth factor. Both of the initial values are in the tables.)
• “How does the scatter plot and graph show that your model is a good fit for the data?” (The points on the scatter plot are close to the line of the model.)