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 warm-up, 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 over-predict and sometimes under-predict, and that the errors should be about the same size for small and large values of output.
I begin this note-taking session by reminding my students of the three conditions for using the linear model.
In the previous lesson, we discussed the first two conditions and this lesson will cover the third. In the warm-up, 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 x-axis 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).
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.