Reflection: Grappling with Complexity Linear Regression and Residuals  Section 2: Direct Instruction
To many students, a residual plot looks exactly like a scatter plot, but yet they are told that a pattern on a scatter plot is a good thing, while a pattern on a residual plot is a bad thing. Especially for lower level students, this can be extremely frustrating. At best, they might memorize the rules so that they can pass the assessments, never fully understanding the value in the distinctions and similarities.
A compare and contrast discourse can be helpful in making the distinction. Showing several examples of scatter plots (with regression equations superimposed) sidebyside with their residual plots can help students generalize the following:

A residual plot is an interpretation of how much residual error exists between the data and the model on the scatter plot (i.e. “Is this the best model here?”)

A Scatter Plot and its Residual Plot will always have the same horizontal label. (i.e. if the scatter plot is showing height in terms of time, then the residual plot will show residual error (for height) in terms of time)

A pattern on a residual plot says that I should have been able to make a better model. Some examples are useful here:

a pattern that shows residuals that are consistently low (or high), means I should have used a model that gave me lower (or higher) values.

residual patterns that are symmetric or curved means that I could get a better model by making those predictable changes.

The absence of a pattern on a residual plot does not necessarily mean that the model is a ‘good’ model, only that there is nothing better.
For example, show a scatter plot that has a linear regression imposed on a pattern that is clearly (or even subtley) exponential. Display the residual plot and discuss its implications. Change the model to an exponential regression and view the resulting residual plot. Also, show a scatter plot with a linear regression for data that clearly has no association. View the residual plot, noting that the lack of pattern doesn’t mean our linear model is any good… only that nothing else will be better!
Linear Regression and Residuals
Lesson 4 of 4
Objective: SWBAT create residual plots and use them to determine if a linear model is an appropriate for a given twovariable data set.
Big Idea: Examining the size and distribution of errors made by a model can help us determine if the model is appropriate.
To introduce the idea of residuals, I ask my students to calculate the prediction errors of two models. In Warm up Prediction Errors, students use models to calculate a predicted value of output and then compare the predicted output to an observed value [MP4].
When students have completed this warmup, I ask if they think the other values predicted by the model would be too high too low or just right. In this way, I start a conversation about the relationship between the distribution of residuals and the appropriateness of a given model. Specifically I hope that they will agree that a good model will sometimes overpredict and sometimes underpredict, and that the errors should be about the same size for small and large values of output.
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Direct Instruction
I begin this notetaking session by reminding my students of the three conditions for using the linear model.
 the data set includes paired values of two quantitative variables
 the scatter plot displays a linear form
 there is no pattern in the residual plot when the linear model is applied
In the previous lesson, we discussed the first two conditions and this lesson will cover the third. In the warmup, my students practiced calculating prediction errors using models that they encountered in previous units of study (exponential and polynomial). I use this work as a springboard for discussing how residuals are used to determine if the linear model is a good fit.
The final test of whether it is appropriate to use a linear model is to create a plot with the residuals on the y axis and the input values on the xaxis and examine the plot for patterns. If it appears that there is regularity to the residual plot, we can conclude that the linear model is NOT a good fit. That is, we want to see very scattered residuals before accepting the linear model as appropriate (MP4).
Students practice calculating and interpreting residuals by completing WS Calculating Residuals with their table partners. This activity sheet provides students with two data sets to work with. Their job is to use the graphing calculator to find the line of best fit and calculate residuals. Although students may do each of the calculations separately, I guide them towards the use of a spreadsheet page on their calculator for these repeated calculations (MP5).
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Close and Assignment
When students have completed the worksheet, we come together as a class to discuss the results. I ask volunteers to share answers to each part, with students coming to the board as necessary. I allow time for questions and then pass out HW Calculating Residuals, which is the homework assignment for the evening.
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