The purpose of the discussion is to provide insight into how to model data with functions.

Display 2–3 approaches from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “Why do the different approaches lead to the same or different outcomes?”
• “Are there any benefits or drawbacks to one approach compared to another?”

Select students to share their predictions, estimates, and model functions. After each response is shared, ask if there are other possible solutions from other students. Here are some questions for discussion:

• “How did you come up with your model function?” (I drew a line that went through the middle of the data and it was close to going through the points and , so I wrote the equation of the line going through those points.)
• “On day 0 the price of the toy was \$5, does that mean your linear function should also have 5 as the vertical intercept?” (Not necessarily. It did work out in this case, but there could be another vertical intercept, especially if the point is very different from the rest of the data near zero.)
• “How did you use the average rate of change in your model function?” (I used the average rate of change for the 10 days as the slope of my function because it seemed to be a good fit.)
• “How confident are you in your prediction for the price of the toy after 12 days?” (I’m not very confident. While there does seem to be a nice upward trend from the data we have, at any point the price could drop significantly.)

The purpose of the discussion is to recognize that additional data can provide a better understanding of a situation, but understanding what is actually happening in the situation can help a lot as well. Select students to share their responses. After each response, ask if there are additional solutions. Here are some questions for discussion:

• “How does knowing additional numerical data about the average price of the toy help your prediction?” (It shows me that the trend I saw at the beginning was not expected to continue.)
• “How does knowing the actual situation about the additional shipment help your prediction?” (It let me know that the trend for the price increase over the first 10 days was likely to continue until the new shipment arrived, when the average price dropped down and began rising again.)