Illuminations: Least Squares Regression

Least Squares Regression

This Unit Plan consists of lessons in which students interpret the slope and y-intercept of least squares regression lines in the context of real-life data. Students use an e-example applet to plot the data and calculate the correlation coefficient and equation of the least squares regression line. These lessons develop skills in connecting, communicating, reasoning, and problem solving as well as representing fundamental ideas about data.

Math Content

Each of the lessons in this Unit Plan builds skills necessary for interpreting the slope and y-intercept of linear equations. The lessons provide students with a real-life context in which to relate both concepts. Students need multiple opportunities to explore slope as a rate of change and to view the y-intercept of a graph as having real meaning. These lessons give students the opportunity to view data in tabular, graphic, and algebraic form. Opportunities for the students to discuss and display their work are also provided.

Prerequisite Knowledge

It is strongly recommended that teachers go online and become familiar with the program before conducting any of the lessons in this Unit Plan.

Individual Lessons

Lesson 1 - Traveling Distances

In this lesson, students interpret the meaning of the slope and y-intercept of the graph of real-life data. By examining the graphical representation of the data, students relate the slope and y-intercept of the least squares regression line to the real-life data. They also interpret the correlation coefficient of the resulting least squares regression line.

Lesson 2 - Bathtub Water Levels

This lesson is similar to Lesson One: Traveling Distances; however, this lesson is designed so students examine real-life data that illustrates a negative slope. Students interpret the meaning of the negative slope and y-intercept of the graph of the real-life data. By examining the graphical representation of the data, students relate the slope and y-intercept of the least squares regression line to the real-life data. They also interpret the correlation coefficient of the least squares regression line.

Lesson 3 - My Graph Isâ€¦

This lesson is designed to allow students to select their own real-life data to plot and interpret. Interpreting the meaning of the slope and y-intercept of their least squares regression lines will help reinforce the concepts introduced in Lessons One and Two of this Unit Plan. The students are then given the opportunity to display their work.

Lesson 4 - Gallery Walk

This lesson is designed to allow students to view the work of other students in the class and to explain their own work. Some teachers may be tempted to skip this step in the Unit Plan, but it is very important that students be given the opportunity to verbalize what the mathematics means that they performed in Lesson Three.

Lesson 5 - Automobile Mileage: Year vs. Mileage

In this lesson, students plot data about automobile mileage and interpret the meaning of the slope and y-intercept in the resulting equation for 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 interpret them in the context of the real-life application. Students also make decisions about the age and mileage of automobiles based on the equation of the least squares regression line.

Lesson 6 - 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.

Lesson 7 - Automobile Mileage: Years Since 1990 vs. Mileage

This lesson is a third representation of the automobile mileage data used in Lessons Five and Six of this Unit Plan. This lesson provides another opportunity for the students to analyze how changing the independent variable in a set of data can result in a different least squares regression line. Students can then use the new equation to make some of the same predictions they made in Lessons Five and Six.

Lesson 8 - Automobile Mileage: Comparing and Contrasting

In this lesson, students compare and contrast their findings in Lessons Five, Six, and Seven of this Unit Plan. This lesson allows students the time they need to think about and discuss what they have done in the previous lessons. This lesson will provide the teacher with another opportunity to listen to student discourse and assess student understanding.

Lesson 9 - Looking Back and Moving Forward

As you review student work in this unit, it is important to remember the mathematical objectives/expectations of this Unit Plan that are stated in Principles and Standards for School Mathematics.

NCTM Resources

 More and Better Mathematics for All Students
 © 2000 National Council of Teachers of Mathematics Use of this Web site constitutes acceptance of the Terms of Use The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring mathematics learning of the highest quality for all students. The views expressed or implied, unless otherwise noted, should not be interpreted as official positions of the Council.