In this activity, students draw their own linear model to fit the data in a scatter plot. In one scatter plot, the data points are nearly linear, and in another there is much more variation in the data. A discussion follows about what makes some lines a better fit than others (MP3).

In the digital version of the activity, students use an applet to fit a linear model to the data. The applet allows students to drag two points on a line around the graph to find a good linear model. Use the digital version if available to allow students to try different models easily.

Monitor for groups who use these strategies:

• Find the middle point of the data and twist the line to find a slope that fits the rest of the data.
• Make a visual estimate of an appropriate slope, and then slide the line around to go through the middle of the data.

The purpose of this discussion is to look at some strategies for drawing a line that fits the data well.

Invite previously selected groups to share how they found a good linear model. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display their work for all to see.

Connect the different responses to the learning goals by asking questions such as:

• “An important part of a line is its slope. How did you find a good slope for a line that can fit the data?” (We twisted the line to be more or less steep to find an angle that kind of goes along with the points.)
• “Another important part of a line is its position on the plane. Why do you think your line fits best where it is rather than sliding it up, down, left, or right?” (With the line where we put it, it goes through the middle of the data so that there are some points on either side of it. Moving it would not seem as balanced to us.)

If desired and time allows, demonstrate this procedure:

1. Enclose all of the points in the scatter plot with a blob.
2. Use a straightedge to draw a line “through the middle” of the blob. Some students find it helpful to think of the blob as a hot dog bun and the line as the hot dog.

If students understand what makes for a good fit from the previous activity, then this activity may be considered optional. The next activity will give students less scaffolded practice deciding if a line fits the data well.

Students have seen linear models for data in a previous lesson. In this activity, students begin to determine what makes a good model for data. They compare two different lines with the same data set to determine which model fits the data better. A formal, quantitative discussion of lines that best fit data will come in later grades. At this stage, students are only asked to informally determine whether the line fits the data well based on how close the points are to the line.

The purpose of this discussion is for students to see some strategies for evaluating the fit of a model.

Some questions for discussion:

• “Which model did a better job of fitting with the data?” (Model A)
• “What were some things that helped you determine which model was better for this data?” (The line went through the “middle” of the data. There were some points on each side of the line, so that it looks to be in the middle. The model predicts the values fairly well for most points.)
• “If a person was looking to buy a used car made in 2006 and incorrectly used Model B, approximately how much money would they be predicting to pay? If they used Model A?” (Model B predicts the cost to be about \$12,000 instead of \$8,000 from Model A, which is a \$4,000 difference, or a 50% increase in price!)

We say that Model A “fits the data” better than Model B, or that model A is a “better fit.”

This activity gives students additional practice finding linear models that match the association of the data. In the first scatter plot, students are given a linear model that has a good slope, but is shifted up from the center of the data. In the second set of data, students are given a linear model that goes through the middle of the data, but has a slope that is too steep. Students are given the opportunity to correct these issues by drawing their own linear models on the same scatter plots.

The purpose of the discussion is for students to recognize the important aspects of a linear model for a set of data.

Consider asking some of the following questions.

• "Compare the given lines to the ones you drew in terms of their slopes and vertical positions." (In the first scatter plot, they should have about the same slope, but the given line is shifted up. In the second scatter plot, they should both go through a point near the center of the data, but have different slopes.)
• "What are some ways you thought about trying to find a good line to draw?"