Illuminations: Least Squares Regression

Least Squares Regression


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.

Learning Objectives
 

Students will

  • explain their reasoning for selecting data
  • explain their findings about the slope and y-intercept of their own data

Materials
 
  • Graphs from the previous lesson

Instructional Plan

This lesson should begin with what is known as “Gallery Walk.” Depending on the size of the class, the students should be given a specified amount of time to view the graphs that were taped to the wall during Lesson Three of this Unit Plan. The students are to move from graph to graph during this time and, without any talking, view the work of their classmates. While on their “Gallery Walk,” the students should think about the data their classmates plotted and whether or not the information accompanying each graph seems appropriate.

After the “Gallery Walk” is completed, each pair of students should be given the opportunity to stand next to their graphs. Each student should explain one of the graphs. Their explanations should include such topics as why they thought their data would be linear in nature; how they chose the labels for their axes; what rate of change the slope of their line represents; what the y-intercept of their line means in the context of their data; what the correlation coefficient was; and, what the correlation coefficient meant in terms of the fit of their line.

Questions for Students
 
  1. When you took the Gallery Walk, what similarities did you notice among the graphs?
  2. What differences did you observe on the Gallery Walk? What caused these differences?
  3. Was the information presented on the graph represented appropriately?
  4. What rate of change does the slope of the line represent?
  5. What is the meaning of the y-intercept in the context of the data you presented?

Assessment Options
 

These presentations will give both the teacher and students an opportunity to assess the students' understanding of the material in the first four lessons of this Unit Plan. At this stage of the Unit Plan it is important to know if students can:

  • 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 as used with their data sets
  • Correctly label the axes on the graph of real-life data
  • Correctly scale the axes for a set of real-life data

It is important to determine the students’ understanding and appropriate use of the mathematical vocabulary used thus far in this Unit Plan. Are the students’ grasping the concept of slope as a rate of change? Can they correctly interpret the y-intercept of the least squares regression line? Do they understand the relationship between the correlation coefficient and the “fit” of the least squares regression line? Determine if students may need some other data sets to really understand the relationship between the correlation coefficient computed by the calculator and the calculator’s regression equation.

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
 
  1. What key words did the students use correctly?
  2. What key words did the students use incorrectly?
  3. What adjustments would I make in this lesson the next time I teach it?
  4. What mathematical content should I stress in earlier lessons in order to make this lesson more successful?

NCTM Standards and Expectations
 
Data Analysis & Probability 9-12
  1. Understand how sample statistics reflect the values of population parameters and use sampling distributions as the basis for informal inference.
  2. Identify trends in bivariate data and find functions that model the data or transform the data so that they can be modeled.
  3. 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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