Lesson 6

Residuals

Lesson Narrative

The mathematical purpose of this lesson is to informally assess the fit of a function by plotting and analyzing residuals. The term residual is introduced as the difference between the \(y\)-value for a point in a scatter plot and the value predicted by the linear model for the associated \(x\)-value. The work of this lesson connects to previous work because students analyzed bivariate data by creating scatter plots and fitting linear functions to the data. The work of this lesson connects to upcoming work because students will use the correlation coefficient to formally assess the fit of a function.

When students take turns with a partner matching graphs of residuals to scatter plots that display linear models, students trade roles explaining their thinking and listening, providing opportunities to explain their reasoning and critique the reasoning of others (MP3).

Learning Goals

Teacher Facing

  • Calculate and plot the residuals for a given data set and use the information to determine the goodness of fit for a linear model.
  • Comprehend the connection between residuals, variability, and whether or not using a linear model is appropriate.

Student Facing

  • Let’s examine how close data is to linear models.

Required Preparation

Prepare 1 copy of the blackline master for every 2 students.

Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.) 

Learning Targets

Student Facing

  • I can plot and calculate residuals for a data set and use the information to judge whether a linear model is a good fit.

CCSS Standards

Glossary Entries

  • residual

    The difference between the \(y\)-value for a point in a scatter plot and the value predicted by a linear model. The lengths of the dashed lines in the figure are the residuals for each data point.