Getting Started 
Misconception/Error The student does not compute the residuals correctly. 
Examples of Student Work at this Level When calculating the residuals, the student:
 Calculates the predicted values incorrectly or rounds unecessarily.
 Calculates the absolute values of the residuals.
 Subtracts in the wrong order, that is, subtracts the observed values of b from the predicted values of b.
 Subtracts the predicted values from the values of a.
 Adds the observed values of b and the predicted values.
 Indicates he or she does not know how to calculate residuals.

Questions Eliciting Thinking How are predicted values calculated?
How are residuals calculated?
What does a residual indicate? 
Instructional Implications If needed, review how to use the linear model to calculate a predicted value and ask the student to calculate the predicted value for each subject. Then explain that a residual is the difference between the observed values of b and the predicted values of b for each subject (residual = observed value â€“ predicted value). Explain that the residuals describe the extent to which the predicted values deviate from the observed values and are useful in evaluating how well a linear model fits a set of data. Ask the student to calculate the residuals and plot them on the graph.
Provide additional opportunities to calculate residuals in the context of linear models. 
Moving Forward 
Misconception/Error The student makes systematic errors when graphing the residuals. 
Examples of Student Work at this Level The student correctly calculates the predicted values and the residuals. When graphing the residuals, the student:
 Graphs the absolute value of the residuals.
 Is confused by the scale and plots points in the wrong location.
 Graphs some residuals against a and some against b.Â

Questions Eliciting Thinking Where should negative residual values be graphed?
I think you made an error in graphing some of your residuals. Can you check your graph again?
What are you plotting the residuals against â€“ values of a or values of b? 
Instructional Implications Remind the student that a residual plot is a scatter plot of the residuals typically graphed against the independent variable. Residual plots are used to assess the degree of fit of a linear model. Provide feedback regarding any errors in graphing and ask the student to revise his or her graph and assessment of the fit of the linear model.
Provide additional opportunities to create residual plots and assess the fit of a linear model. 
Almost There 
Misconception/Error The student does not understand how to interpret a residual plot. 
Examples of Student Work at this Level The student correctly calculates both the predicted values and the residuals and correctly graphs the residuals against values of a. When explaining what the residual plot indicates about the fit of the equation, the student says the linear model:
 Is a good fit because the residuals are close to the horizontal axis or close to the actual values.
 Is not a good fit because the residuals are not close to the horizontal axis or are not near each other.

Questions Eliciting Thinking How do you analyze a residual plot? What should it look like if the model is a good fit for the data? What can it look like when the model is not a good fit for the data?
Is the fact that some residuals are positive and others negative important? 
Instructional Implications Remind the student that residual plots are used to assess the degree of fit of a linear model. Explain that a plot that indicates good fit shows uniform scatter of the residuals about the horizontal axis. Indicate to the student that it is difficult to interpret a residual plot for small data sets, but this particular plot is reasonably good given that the data set is small.
Provide residual plots of larger data sets that indicate a good fit (by displaying even scatter of residuals about the horizontal axis). Also show the student residual plots that indicate a poor fit. For example, show the student a plot in which:
 All of the residuals are either positive or negative.
 The residuals have a paraboliclike shape either opening up or down.
 Have a funnel shape (narrow at one end and fanning out at the other).
Ask the student to relate the residual plot to the fit of the model (e.g., if all of the residuals are positive, then the model is systematically predicting smaller values of the dependent variable than the observed values).
Provide additional opportunities to create residual plots and assess the fit of a linear model. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student correctly calculates both the predicted values and the residuals:
Predicted Values 15.32 12.64 Â 11.3 Â 7.28 Â 4.6
Residuals Â Â Â Â Â Â 0.68 Â 0.64 Â 0.3 Â 0.28 Â 0.4
and correctly graphs the residuals against values of a.
When explaining what the residual plot indicates about the fit of the equation, the student says the linear model is a pretty good fit since there is scatter of residuals about the horizontal axis. The student may indicate that the data set is small so it is difficult to be certain. 
Questions Eliciting Thinking What is a residual? What does it indicate?
What features of a residual plot would indicate that a model is not a good fit for the data?
Is a line the only possible fit to the data? 
Instructional Implications Provide residual plots of larger data sets which indicate a poor fit. For example, show the student a plot in which:
 All of the residuals are either positive or negative.Â
 The residuals have a paraboliclike shape either opening up or down.
 Have a funnel shape (narrow at one end and fanning out at the other).
Ask the student to relate the residual plot to the fit of the model (e.g., if all of the residuals are positive, then the model is systematically predicting smaller values of the dependent variable than the observed values).
