Lesson 9

In this warm-up, students work with data to determine if the situation represented by the data could be modeled by a linear function (MP4). Students are given 3 different data points and use what they know about linear functions and proportional relationships to estimate when the candle will burn out. Students are then asked to determine if this situation could be modeled by a linear function. The focus of the discussion should be around the last question and how students justify their reasoning.

The purpose of this discussion is for students to justify how this situation can be modeled by a linear equation. Select students who answered yes to the last question and ask:

• “Was the data exactly linear? If not, what made you decide that you could treat it as such?”
• “Which data points did you use to predict when the candle would burn out?”
• “What was the slope between the first two data points? What was the slope between the last two data points? What does it mean that their slopes are different?”

Tell students that although the data is not precisely linear, it does makes sense to model the data with a linear function because the points resemble a line when graphed. We can then use different data points to help predict when the candle would burn out. Answers might vary slightly, but it results in a close approximation.

Conclude the discussion by asking students to reconsider the range of values posted earlier for the first question and ask if they think that range is acceptable or if it needs to change (for example, students may now think the range should be smaller after considering the different slopes).

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.

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 over-predicts the length of the shadow after 95 minutes.

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, mathematicians have also used the geometry of Earth traveling around the Sun to provide a better model for the length of the shadow as a function of time that is not a linear function.

The purpose of this activity is for students to approximate different parts of a graph with an appropriate line segment. This graph is from a previous activity, but 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 input and output for other parts of the graph. This helps students think about subsets of data that might have different models from other parts of the data.

Identify students who draw in different lines for the first question to share during the Activity Synthesis.

Arrange students in groups of 2. Provide students with access to straightedges. Give students 3–5 minutes of quiet work time and then time to share their responses with their partner. Follow with a whole-class discussion.

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

Select students previously identified 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 1996? 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 in number of years to include in the interval and accuracy. 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 imaging writing an equation for each piece!)