Illuminations: Least Squares Regression

Least Squares Regression


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.

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-life data
  • make predictions based on the equation of the least squares regression line
  • compare and contrast two (three) interpretations of a set of real-life data

Materials
 

Instructional Plan

This lesson should begin with a brief discussion of the activities done in Lessons Five, Six, and Seven of this Unit Plan. This discussion should focus on the mathematics of interpreting the slope as a rate of change and the y-intercept as representing a specific data point on the graph. The following guiding questions will help you with your discussion.

  1. How did you determine the rate of change in each of the previous activities?
  2. What did the y-intercept tell you about the automobile and its mileage in each of the previous activities?
  3. Why didn’t the predicted values for the mileage match the actual mileage?

Students should be divided into the same teams as in Lessons Five and Six of this Unit Plan. Each student should be given a copy of the handout Automobile Mileage—Comparing and Contrasting Our Findings<.

Working together, the partners can discuss their findings from the two previous lessons and consolidate their understanding of the material. Giving the students the time to talk to each other about what they observed in Lessons Five, Six, and Seven of this Unit Plan will serve to strengthen their overall understanding of the ideas presented in the lessons.

While the students talk to each other and answer the questions on the handout, it is very important that the teacher circulates and monitors the discussions. This gives him or her a better understanding of what the students understand and what they may still need help with. Circulating also gives the teacher an opportunity to listen to the mathematical terminology the students use.

After completing the questions on the handout, students should be given the opportunity to discuss their findings as a class. The questions on the handout can be used to help guide this discussion. This process encourages students to reflect on what they have discovered about the graphical and algebraic estimates 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’ ability to interpret the slope in the context of the automobile warranties and leasing agreements. Here is where the teacher can allow the students to discuss if leasing is better. This slope is a representation that now means something real to students and they can discuss their findings.

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 Unit Plan it is important to know if students can:

  • Interpret slope as a rate of change
  • Interpret the meaning of the y-intercept
  • Make real-life decisions based on an analysis of the data given
  • Use correct mathematical vocabulary

The guiding questions may assist you in understanding your students’ level of attainment of the concepts in this lesson. They can also guide you in determining entries for the Status of the Class recording sheet.

As you reach the final lesson in this unit, it may be useful to consider how much individual students have grown, and to use this information to plan strategies for remediation and extension activities. This would also be a good time to ask individual students to talk with you about the entries in their unit portfolios.

Extensions
 

Students can contact local automobile dealerships or use the Internet to learn more about the warranties offered on new automobiles and the different types of leasing agreements that are available. They can also examine what happens to the trade-in value of an automobile as its mileage increases. Can the students determine the optimum time to trade in an automobile? These topics and others can be used as the basis of written or oral presentations.

Teacher Reflection
 
  1. What similarities did the students notice?
  2. What differences did the students notice?
  3. What other learning experiences would help the students identify similarities? Differences?
  4. Were the students able to successfully translate their understanding of slope into real-life decisions?
  5. Were students able to use correct mathematical terminology?
  6. Which students met all the objectives of this lesson? What extension activities are appropriate for those students?
  7. Which students did not meet the objectives of this lesson? What instructional experiences do they need next? What mathematical ideas need clarification?
  8. What adjustments would you make the next time you teach this lesson?

NCTM Standards and Expectations
 
Data Analysis & Probability 9-12
  1. Identify trends in bivariate data and find functions that model the data or transform the data so that they can be modeled.
  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
  3. Understand how basic statistical techniques are used to monitor process characteristics in the workplace.
  4. Understand how sample statistics reflect the values of population parameters and use sampling distributions as the basis for informal inference.
  
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NCTM Resources

Principles and Standards for School Mathematics

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