This Warm-up helps students get oriented to the situation they will be working with throughout this lesson. Students also make initial predictions based on very limited information. They look for and make use of the structure of the graph to solve this problem (MP7).

The purpose of the discussion is to recognize that it can be difficult to make predictions based on very limited data and information.

Poll the class on their predictions. Ask selected students to explain why they made that prediction. Record and display the responses for all to see. If possible, display and refer to the graph as students explain their reasoning.

Ask students, “What other information would help improve your predictions?” (More data or a more detailed explanation of what the toy is and what is happening would help me make a better prediction.)

In this activity students use some data to find an average rate of change and write a linear function to model data for the price of a collectible toy over several days. As students use the graph to create a linear model, they are modeling with mathematics (MP4). In the associated Algebra 1 lesson students examine battery life on a phone by modeling a graph with a function. Students are supported in this activity by being given some additional steps to think through while modeling a situation.

If possible, do not allow students to look at the next activity while they work on this activity. The next activity gives students additional information about the price that may affect their thinking for the questions here.

Monitor for students who use these different strategies to decide the rate of change for their model:

• Use the rate of change from two days that are near each other
• Use the rate of change from the first and last data points
• Use the rate of change of most of the points

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.)

In this activity students get additional information about the average price of the toy, which does not follow the same trend as in the previous activity. The additional information shows students that trends can change and that knowing additional data is helpful, but also knowing the situation can help make better predictions. Students model with mathematics by adjusting their model when given new information (MP4).

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.)