Lesson 13: Improving your Model
Lesson 13: Improving Your Model
Objective:
Students will learn to describe associations that are not linear.
Materials:
 Describe the Association handout (LMR_4.16_Describe the Association)
Vocabulary:
Essential Concepts:
Essential Concepts:
If a linear model is fit to a nonlinear trend, it will not do a good job of predicting. For this reason, we need to identify nonlinear trends by looking at a scatterplot or the model needs to match the trend.
Lesson:

Remind students that they have been learning a great deal about linear associations. However, there are other types of associations, and today they will learn to describe them.

Distribute the Describe the Association handout (LMR_4.16). In teams, students will examine the trend of each plot. Their task is to write a description of the trend that they see in the data and what the trend means.

Allow students time to discuss and record their descriptions for each plot in their DS journals. Walk around the room monitoring student teamwork. Look for descriptions that are interesting to share with the whole class.

Select a team to present a description of one plot to the class. Teams will listen to each presentation, compare it to their description of the plot, and as a team they will agree or disagree. If there is disagreement, lead a discussion that guides students to reason toward the correct description.

Summarize the discussion for each plot and ask students take notes or revise their descriptions in their DS journals.

Repeat steps 4 and 5 for the rest of the plots.
Plot Descriptions for Describe the Association (LMR_4.16):

Plot A: There is no trend (perhaps some may see a very, very weak linear trend), so there is no/hardly any association. There is a great deal of scatter in the data. It means that y does not depend on x.

Plot B: There appears to be a linear trend. The association is negative and appears somewhat strong. It means that as x increases, y decreases.

Plot C: There is a linear trend. The association is positive and it is very strong. It means that the yvalue increases at approximately the same rate for every increase in x value. This is a line.

Plot D: The trend is nonlinear. There seems to be a weak association because there is scatter in the data. We cannot tell if the association is positive or negative. It has the shape of a parabola; therefore, it is quadratic. For smaller xvalues, the yvalue is decreasing and for larger x values, the y value is increasing.

Plot E: The trend is nonlinear. There seems to be a strong association because there is little scatter in the data. It is also in the shape of a parabola, so it is quadratic.


Using the Cheat Notes strategy, ask teams to write notes about how to describe associations.

Plots A, B, and C should be familiar to the students by now. However, plots D and E show a different type of trend. Although the trends are nonlinear, they can still tell us important information about the yvalues based on values of x. Ask:
 What happens if we were to fit a linear model to these nonlinear trends? Would it still make good predictions? Answer: Fitting a linear model to a nonlinear trend would not properly describe the trend of the data. Therefore no, it would not make good predictions.

To examine why they would not make good predictors, draw an approximate linear bestfit line and get students to understand that in some regions, the model would almost always overpredict, and in others would almost always underpredict. We want a model that goes, more or less, through the 'middle' of the points. Ask:
 How can we get a model that goes, more or less, through the middle of all the data points? Answer: We need to change the model.

Trends like the quadratic ones shown in plots D and E can be described as polynomial trends. Plots that follow quadratic, cubic, quartic, etc. shapes all exhibit polynomial trends. We need to adjust the model. You may show students several choices of equations (quadratic, trinomial, linear) along with their graphs and ask them which might be a good candidate.

When investigating the data for trends, the model needs to fit the data.
Class Scribes:
One team of students will give a brief talk to discuss what they think the 3 most important topics of the day were.
Homework & Next Day
Students may finish their Cheat Notes for homework, if not completed in class.