Illuminations: Least Squares Regression

Least Squares Regression


Automobile Mileage: Age vs. Mileage

In this lesson, students plot data about automobile mileage and interpret the meaning of the slope and y-intercept of the least squares regression line. By examining the graphical representation of the data, students analyze the meaning of the slope and y-intercept of the line and put those meanings in the context of the real-life application.

The activity is very similar to that in Lesson Five of this Unit Plan. However, by graphing the data in a different format, the students will produce a line with a positive slope in this activity, while the line in Lesson Five had a negative slope. Doing both lessons allows students to investigate how changing the independent variable affects the resulting graph and equation.

Learning Objectives
 

Students will

  • interpret slope as a rate of change in the context of real-life data
  • interpret the y-intercept in the context of “real-world” data
  • make predictions based on the least squares regression line
  • explain the difference between actual values and predicted values

Materials
 

Instructional Plan

This lesson should begin with a review of what the students learned in Lesson Five of this Unit Plan. Having them restate what they learned will help reinforce the concepts of that activity and give the teacher another opportunity to assess students’ understanding.

Today’s activity should begin with a general discussion of what students know about the age of an automobile and the mileage they would expect to be on the automobile.

Students should be divided into teams of two to work at the computer. If possible, the students should continue to work the same teams as those in Lesson Five. Each student should be given a copy of the handout Automobile Mileage—Age vs. Mileage (or a similar handout of student produced data). They should visit the Web site: http://illuminations.nctm.org/index_d.aspx?id=454. Working together, the partners can share the responsibility of making sure the data is plotted correctly. One student should plot the data, while the second reads out loud the data and makes sure it is plotted correctly.

Students should click on the applet and make the changes in the viewing window indicated on the handout. As the students begin to plot the data, walk from group to group, making sure the students are plotting the data correctly. Encourage students to think about the data they are plotting and the resulting graphs. Ask Guiding Questions such as the following as you monitor and facilitate the group work.

  1. Are you beginning to notice any pattern in the shape of the plot?
  2. What type of function do you think will fit this data?
  3. Do you think the slope of the line will be positive or negative?
  4. What do you think the y-intercept of the regression line might be?

Allow the students to complete the plot and answer the questions on the handout. Continue to circulate and facilitate discourse between the partners while they complete this portion of the activity.

After completing the questions on the handout, students should be given the opportunity to discuss their findings as a class. The Guiding Questions at the beginning of this lesson and the questions on the handout can be used to help guide this discussion. This will encourage students to reflect on what they have discovered about the graphical and algebraic estimations of the real data and allow them to strengthen their understanding of slope as a rate of change. The teacher should pay particular attention to the students’ understanding of the units used on the axes.

Assessment Options
 

The discussion will give both the teacher and students an opportunity to assess the students’ understanding of the lesson. At this stage of the activity it is important to know if students can:

  • Correctly plot data points on the applet
  • Determine if a least squares regression line is actually a good fit for the data being graphed
  • Interpret slope as a rate of change
  • Explain a positive rate of change
  • Interpret the meaning of the y-intercept
  • Understand the meaning of the correlation coefficient
  • Use the units on the axes as an aid in interpreting the meaning of the slope and y-intercept
  • Explain the difference between actual values and predicted values

The Guiding Questions at the beginning of this lesson and the questions on the handout help the students focus on the mathematics and aid you in determining the students’ level of understanding of the mathematical concepts in this lesson. If you began using Status of the Class to record students’ understanding of Lesson One of this Unit Plan, continue to document student understanding and progress. Documenting information about students’ understanding throughout the lesson(s) can help you focus on the needs and strengths of individual students, and thus can increase student learning opportunities.

Extensions
 

The following questions can be used to help students relate what they have done during this lesson to other topics about linear functions.

  1. For what values of x did your equation seem appropriate? Why?
  2. What values of y do you think would be appropriate for the “real-life data” used in this example? Why?
  3. What would be an appropriate domain for your graph?
  4. What would be an appropriate range for your graph?
  5. What was the independent variable in this application?
  6. What was the dependent variable in this application?

Teacher Reflection
 
  1. Which students met all the objectives of this lesson? What extension activities are appropriate for those students?
  2. Which students did not meet the objectives of this lesson? What instructional experiences do they need next? What mathematical ideas need clarification?
  3. What adjustments would you make the next time you teach this lesson?

NCTM Standards and Expectations
 
Data Analysis & Probability 9-12
  1. Understand how basic statistical techniques are used to monitor process characteristics in the workplace.
  2. Evaluate published reports that are based on data by examining the design of the study, the appropriateness of the data analysis, and the validity of conclusions
  
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NCTM Resources

Principles and Standards for School Mathematics

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