Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 68884
Residuals
Students are asked to compute, graph, and interpret the residuals associated with a line of best fit.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, residuals, best fit
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_Residuals_Worksheet.docx
MFAS_Residuals_Worksheet.pdf
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
This task can be implemented individually, with small groups, or with the whole class.
1. The teacher asks the student to complete the problems on the Residuals worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
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.
page 1 of 4 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 – 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.
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 – 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.
page 2 of 4 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 parabolic-like 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 11.3
Residuals
7.28 4.6
0.68 -0.64 -0.3 -0.28 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.
The residuals have a parabolic-like shape either opening up or down.
Have a funnel shape (narrow at one end and fanning out at the other).
page 3 of 4 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).
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Residuals worksheet
SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
Description
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. ★
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given
functions or choose a function suggested by the context. Emphasize linear, and exponential models.
b. Informally assess the fit of a function by plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear association.
MAFS.912.S-ID.2.6:
Remarks/Examples:
Students take a more sophisticated look at using a linear function to model the relationship between two numerical
variables. In addition to fitting a line to data, students assess how well the model fits by analyzing residuals.
page 4 of 4