Illuminations: Least Squares Regression

Least Squares Regression

Unit Overview

Lesson 1

Lesson 2

Lesson 3

Lesson 4

Lesson 5

Lesson 6

Lesson 7

Lesson 8

Lesson 9

My Graph Is…

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.

Learning Objectives

Students will

select linear data to investigate
plot data points
interpret the slope of a line as a rate of change in the context of real-life data
interpret the y-intercept of a line in the context of real-life data
interpret the meaning of the correlation coefficient of the least squares regression line

Materials

Computer and Internet connection
Graphing Real-Life Data recording sheet
Status of the Class

Instructional Plan

It is important that the students be given the opportunity to graph and interpret their own sets of data. This will reinforce the skills introduced in the first two lessons of this Unit Plan. It will also give the teacher another opportunity to observe the students at work and assess their understanding of the material in the previous lessons.

Divide the class into pairs of students. Give each student another copy of the handout Graphing Real-Life Data. Each pair of students is to think of two sets of data that will be linear when graphed. These might include such items as how many pounds of ground beef are purchased vs. the total cost, the number of hours worked vs. gross pay, and the number of marbles removed from a bag vs. the weight of the bag.

Encourage the students to do one graph with a positive slope and another with a negative slope. (The students could be encouraged to find sets of data on the Internet. There are numerous sites that provide statistical data. The students might be interested in sports records, consumer consumption of products, etc.)

Once each pair of students has decided what they want to graph and collected the data set if necessary, they should go to the Web site:
http://illuminations.nctm.org/index_d.aspx?id=454 and plot their first set of data points on the applet. Students should use no more than 8 data points. This may require them to change the window settings. After plotting the data, they should have the applet draw the least squares regression line and calculate the correlation coefficient. Each student should record all this information on his or her own copy of the handout and then collaborate with his or her partner to complete the other information on the handout for the first graph.

After completing the first graph, the students should follow the same procedure to complete their second graph.

The students should then tape one copy of their a href="lessons/9-12/reallife/sheet1.pdf">Graphing Real-Life Data recording sheet to the classroom wall. These will be used during Lesson Four of this Unit Plan.

Questions for Students

What factors did you consider when selecting sets of data that might be linear when graphed?
What mathematical knowledge did you need to be able to select data sets that might be linear when graphed?
What additional knowledge and skills are required when plotting the data using the Web-based tools?
What caused you trouble? Why do you think it was a problem for you?
How would you explain to a younger student the characteristics of data sets that might be linear when graphed?

Assessment Options

At this point in the unit it is important to know if individual students can:

Select data that is linear in nature
Correctly plot data points, both by hand and on the applet
Interpret the meaning of the slope of a line as a rate of change in the context of real-life data
Interpret the meaning of the y-intercept of a line in the context of real-life data
Understand the meaning of correlation coefficients
Correctly label the axes on the graph of real-life data
Correctly scale the axes for a set of real-life data

Using the handout during the peer discussion will help students focus on the mathematics involved in this lesson. The handout provides display material for each pair of students and also provides information the students can use to help them answer questions in other parts of this Unit Plan.

The teacher should continue to collect information about the students’ understanding of the material on the Teacher Resource Sheet, Status of the Class. The assessment information you collect can help you monitor student learning, adjust instruction, and plan future lessons for the class. Data on individual students can be used to plan strategies for regrouping students, remediation, and extension activities. This information is extremely useful when discussing progress toward learning targets with students, parents, administrators, and colleagues.

Teacher Reflection

What mathematical content did the students seem to grasp?
What mathematical content did the students seem not to grasp? What do you need to reinforce during the next lesson?
Did two graphs provide enough practice for the students in your class?
What adjustments would you make the next time you teach this lesson?

NCTM Standards and Expectations

Data Analysis & Probability 9-12

Compute basic statistics and understand the distinction between a statistic and a parameter.
For univariate measurement data, be able to display the distribution, describe its shape, and select and calculate summary statistics.
Understand histograms, parallel box plots, and scatterplots and use them to display data.


		1 period
		NCTM Resources Principles and Standards for School Mathematics Web Sites Investigating Linear Relationships: The Regression Line and Correlation Coefficient