The purpose of this activity is for students to consider a situation that seems like it could be modeled with a linear function from the limited information given, but where a linear model fails with just a bit more data (MP4). While study of periodic functions is in high school, students' familiarity with the cyclical nature of the moon should be enough for them to make sense of the nonlinear nature of the situation.

The goal of this discussion is for students to understand that while we can create a linear function out of 2 data points, doing so does not always lead to accurate predictions. Here are some questions for discussion:

• “The Moon’s surface viewed from Earth is actually 100% illuminated on day 14. Does this agree with the prediction you made?” (No, I assumed a 4% increase each day which means the Moon would be 100% illuminated on day 26.)
• “Is the percentage illumination of the Moon’s surface a linear function of the day?” (No, the amount of the Moon's surface that is illuminated increases and decreases repeatedly over a year about every 28 days. That is not linear.)

Tell students that if we only use two data points, it is always possible to model a situation with a linear function. We need additional data to help us determine if a linear model is appropriate. In this case, we have observed ourselves that the moon phases go in cycles that repeat over an over again, which isn't something a linear function is meant to model.

The purpose of this activity is for students to determine if a given set of data can be modeled by a linear function. Students first view a set of pictures and data for the length of a shadow at 0, 20, and 60 minutes. Then, they make a prediction about how long a shadow will be after 95 minutes. Students then compare their estimate with the actual length of the shadow and make conclusions about the model they used to make their estimate.

Monitor for students using different strategies to make their prediction. For example, students may use different pairs of points to make their prediction, or they may try to use all three. The discussion for this activity focuses on how if we are given two input-output pairs, we can always find a linear function with these inputs and outputs, but that doesn’t mean a linear function is actually appropriate for the situation. In this case, we can see when we get more data that a linear function is not appropriate.

In the digital version of the activity, students use an applet to estimate the length of the shadow. The applet allows students to plot points quickly and accurately. This activity works best when students have access to the applet because the points are close to linear but make a distinct downward curve.

Select groups that had different strategies for making their original prediction to share their reasoning about whether or not a linear model is a good fit for predicting the length of the shadow. In particular, make sure it is pointed out how much using the rate of change determined by the first two points overpredicts the length of the shadow after 95 minutes.

If not brought up by students, ask, “What will the length of the shadow be near sunset? After sunset?” (Near sunset the shadow will be very long, and after sunset there will not be a shadow.) Emphasize that over the short term, a linear model might be useful even for nonlinear data, but if the actual situation isn’t considered, like here where the sun will eventually set, conclusions will be wrong if we try to force a linear model.

The purpose of this activity is for students to approximate different parts of a graph with an appropriate line segment. While they have seen previously that not everything can be modeled with a linear function, students see in this activity that it is possible to model pieces of a situation but that the model should not be stretched beyond the piece it is meant for. This activity prepares for work with piecewise linear functions in a future lesson.

The graph is one students encountered in a previous activity, but here students interact with it differently by sketching a linear function that models a certain part of the data. They take this model and consider its ability to predict inputs and outputs for other parts of the graph (MP4). This helps students think about subsets of data that might have different models from other parts of the data.

The purpose of this discussion is for students to understand that although they might find a good model for one part of a graph, that does not mean that model will work for other parts.

Select previously identified students to share their models. Display these for all to see throughout the entire discussion. Questions for discussion:

• “How much does your model for 1991 to 1995 overestimate the percent recycled in 1996? In 1997?”
• “If we drew in a single line to model 1997 to 2013, what would that line predict well? What would that line predict poorly?” (A single line modeling those years would reasonably predict the percent recycled from 1997 to 2010, but it wouldn’t be able to show how the percent recycled from 2011 to 2013 is decreasing.)

Conclude the discussion by telling students that there is a trade-off between the accuracy of the model and the number of years to include in the interval. We could “connect the dots” and be accurate about everything, but then our model has limited use and is complicated with so many parts. (Just imagine writing an equation for each piece!)