The mathematical purpose of this activity is for students to:

• Distinguish between linear and nonlinear relationships in bivariate, numerical data.
• Informally assess the fit of a linear model.
• Compare the slope and the vertical intercepts of different linear models.
• Describe the relationship between two variables.

Students sort different scatter plots showing a linear model during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).

The purpose of this discussion is for students to discuss the goodness of fit for linear models.

Here are some questions for discussion.

• “How are scatter plots of A and F the same? How are they different?” (They have the same slope, and the linear model for each scatter plot is equally well fit. They are different because each of them has a different vertical intercept.)
• “How do you know if a linear model is a good fit?” (We need to look at the scatter plot and the line of best fit and make a decision about whether or not the data follows a linear trend.)
• “Why is the goodness fit for the linear model in scatter plot B better than the fit for the linear model in scatter plot A?” (The data in B falls on or very close to the linear model. The data in A is scattered around the line of best fit and has roughly the same number of values below the line of best fit as it does above the line of best fit.)

It is recommended to use the digital version of this activity. The mathematical purpose of this activity is for students to use technology to compute a line of best fit for data given in a table and to understand the meaning of the slope and -intercept.

In the digital version of the activity, students use an applet to create a scatter plot from data. The applet allows students to enter data into a table and accurately create a scatter plot, then drag given points to create a linear model to fit the data, and finally use the technology to find a least-squares regression line. The digital version may be helpful for quickly creating the scatter plot and increasing the accuracy of the plot and of fitting a line to the data.

If the digital version of the activity is not available, students should be guided through using available technology to find the least-squares regression line as the line of best fit.

The purpose of this discussion is for students to make connections between the scatter plot and the equation of the line of best fit. Display each scatter plot, the line of best fit, and the equation of the line of best fit.

Here are some questions for discussion:

• “How does using technology help model the data represented by the scatter plot?” (It allows different people to come up with the same equation for the line of best fit. If the line is just drawn by hand, there can be different linear equations that seem to fit the data well, but there is only one “best” fit line.)
• “What does the -intercept represent in each scatter plot? When is it reasonable to use this interpretation?” (It represents the value of estimated by the linear model when . When the intercept is near the range of the data, it can be reasonable to use this interpretation because otherwise, the linear trend may not continue. There are also some situations in which a value of 0 for does not make sense.)
• “Why is the slope the same in scatter plot A and scatter plot F?” (It is the same because the data in scatter plot F is the same data as in scatter plot A, except that the values for have all been increased by 4.5 units.)

Tell students they should be careful when predicting values outside the range of the data, in particular, for the -intercept. Even when the data is fit well by a linear model, the behavior of the variables farther away may not be linear. It is important to remember that all predictions using the best-fit line are estimates and the reasonableness of the predictions should be considered.

Students may struggle with interpreting slope and -intercept. Remind students of how each relates to a situation. To help students interpret slope, ask them, “What does the -variable represent? What does the -variable represent? How is slope connected to the - and -variables? What happens to as increases (or decreases)?” To help students interpret the -intercept, ask them: “What does the point on the scatter plot mean? What are the coordinates of the -intercept? What do each of the coordinates mean in the situation described? What is the -value when  is 0? Which variable has a value of 0? Which variable is represented with ?”