Working with Residuals

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Objective

SWBAT calculate and graph residual plots to understand the strength of their data.

Big Idea

Is the graph random? Students develop an understanding of residuals and how they can indicate a linear relationship.

Opening

5 minutes

Today students learn about residuals for the first time. I like the way this task walks them through the process. I find students have trouble understanding residuals, especially with respect to reading a residual plot so we spend a lot of time on this lesson and I look for more examples of graphs for them to help solidify their understanding.  We start class by looking at Rockin' the Residuals together and working through the first few questions at the board as a whole group.

Investigation

30 minutes

As we work through the first few questions together, students get a sense of how to determine the value of a residual. Students in my class often struggle with the comparing that needs to happen between the given y value in the table (or in the data) and the y value that the equation would give them.  We practice with a bunch of points until they get this idea.  Then students need to remember that their x values will stay the same, but their y values will now represent the distance from the actual data point to where the point would be if it was on the line of best fit.

Next students work on creating the residual plot. While I don't find that students have difficulty creating the plot, they often have trouble reading it. I find that we spend so much time with students asking them to look for patterns, it's hard for them to look for a random assortment of dots.  I try to explain to students that if a pattern emerges in their residual plot, a linear fit might not be best, or the line of best fit may not have been drawn well. In my class, students seem to have trouble determining when a residual graph is random and when it shows a pattern.  Therefore, we spend a lot of time looking at the residual plots in Questions #8 through #11. I first have students take a look on their own or in small groups and then we share out in the Discussion section of the lesson.

Discussion + Closing

25 minutes

Once students know how to create a residual plot, we look the examples in Questions #8 through #11 to try and figure out what's happening there.  I have students share out their thoughts and arguments for whether or not the graphs look "random."  If they are not random, we puzzle through what might be happening in the graph.  For example, in Question #8, the graph looks pretty random at first, but then we see points getting further from the line on both sides. I ask students what that might mean the original data set looks like. I might have some students come to the board and try to sketch some rough data points and a line of best fit. I want to elicit from students that things start out ok, with points on both sides of the line, but then they steadily get further away.  This might suggest that a linear fit is good for the beginning of the data but not the end.

To close today's class, I want students to reflect on the value of creating a residual plot. Depending on their level of understanding and how well the class went, I might use either of the following prompts:

1.  What is looking at the pattern in the residual plot important?

2.  How does a residual plot give you different information about a linear relationship from the correlation coefficient? What are the advantages and disadvantages of each?

Citations

Rockin' the Residuals is licensed by © 2012 Mathematics Vision Project | MVP In partnership with the Utah State Office of Education Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license.

Source: http://www.mathematicsvisionproject.org/secondary-1-mathematics.html