Does It Fit?

Lesson Content

• Lesson Plan Template:

  General Lesson Plan

• Learning Objectives: What should students know and be able to do as a result of this lesson?

  
  
  • Students will know the meaning of residuals and how they relate to a set of data and a linear regression.
  • The student will be able to calculate and plot residuals, both by hand and using a graphing calculator. given a set of data.
  • Students will be able to interpret a positive versus a negative residual and explain the implication of large residuals.
  • Students will be able to informally assess the fit of a function by analyzing residual plots for patterns that may indicate that a linear model is not the best fit.

• Prior Knowledge: What prior knowledge should students have for this lesson?

  
  

  The student should indicate mastery of the following eighth grade standards.

  • MAFS.8.SP.1.1 Students construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities.
  • MAFS.8.SP.1.2 Students suggest a linear association informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

  The student should be familiar with the following algebra standards.

  • MAFS.912.S-ID.3.8 Students compute (using technology) and interpret the correlation coefficient of a linear fit. The student should understand that the correlation coefficient measures the strength of the association of the two variables and how well a model fits a particular set of data.
  • MAFS 912.S-ID.2.6a Students fit a function to the data and use functions fitted to data to solve problems in the context of the data for linear, quadratic and exponential models.

• Guiding Questions: What are the guiding questions for this lesson?

  
  

  How can the appearance of a linear model be deceiving when analyzing data?

• Teaching Phase: How will the teacher present the concept or skill to students?

  
  

  Phase 1 Warm Up

  • The teacher should provide the students with the Warm Up worksheet (see attached). The teacher may have the students compare their model with those of a partner or group and then ask the class if their model was a good fit for the data.
  • The teacher should have the students locate data points that have the least and greatest difference from the model they chose. Students should explain how they selected these points. The teacher may want to ask the students how they would measure this difference and have them draw dotted vertical lines to visually represent the residuals. Students should be able to generate the basic formula for calculating the residual, i.e., the observed dependent value minus the corresponding dependent value predicted by the regression equation. The teacher should explain that the measurement of the vertical distance from the data point to the regression line is called a residual.
  • The teacher should include in this discussion the meaning of positive residuals (values greater than expected) and negative residuals (values less than expected). The student should be able to explain the meaning of a positive or negative residual in context, e.g., Bennie's height was greater than his predicted height that year, or his height was less. The teacher should lead the students into realizing that plotting the residuals would provide a better perspective on the reliability of the model that was chosen.
  • The teacher should work with the students that did not correctly complete the worksheet to clear up any misconceptions or fill any gaps the students may be having. The remaining students should, independently or in pairs, write an equation for their model and/or use a graphing calculator to find the regression equation. The following website can provide assistance in the use of a graphing calculator if needed. https://epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=100841. (If the calculator is not generating the r value, the student should choose 'Diagnostics On' from the 'Catalog' or 'Mode' menu.) The teacher can provide an approximation of the regression equation (y = 2.4x + 30.5) generated by the calculated for students that do not have time to complete the calculator portion of this activity.

  Phase 2 See Guided Practice

  Phase 3 See Independent Practice

• Guided Practice: What activities or exercises will the students complete with teacher guidance?

  
  

  Phase 2 Guided Practice

  • The teacher should provide the students with the Residual Activity 1 worksheet (see attached). (The equation used in this worksheet, and all other worksheets in this lesson, is an approximation of the actual regression found by using the graphing calculator. The change was made in order to reduce the amount of time needed for calculations made by hand. To save time finding the predicted values, the students may enter the equation into the graphing calculator and use the corresponding values in the table to complete the chart on the worksheet.)
  • It may be helpful for the teacher to emphasize the domain throughout the lesson when discussing the reasonableness of the prediction equation.
  • The teacher can have the students discuss whether or not the model is a good fit using the residual plot. The students should be able to recognize the negative and positive residuals and may need to discuss the expected level of accuracy for the differences between the observed value and predicted value. The students should understand that residuals closer to the x-axis are smaller and therefore indicate that the observed value is closer to the predicted value.
  • The teacher should explain that the significance of the size of the residual could vary in context. For example, a difference of two can be considered large if the dependent variable is shoe size but may be considered small if the dependent variable is body weight.
  • If it does not arise naturally, the teacher should introduce the possibility of the significance of the apparent pattern in the graph of the residuals. The teacher should lead the students into an understanding of the implications of a pattern in the distance from the observed values and the predicted values and what it suggests about the relationship between the data. The teacher should encourage the students to discuss their interpretation of the relationship between the data.
  • The students should begin to question the accuracy of the regression equation. The teacher should encourage this speculation by asking the students to use the regression equation to predict Bennie's height at age 18, 21 and 25 and have them discuss the accuracy of the model at each age.
  • As a whole group, the teacher should discuss possible reasons the predicted values are not realistic and encourage the students to consider different models for the data. If needed, the teacher should review the characteristics of the graphs of different functions, including linear, quadratic and exponential.
  • The teacher should lead the students into realizing that sometimes a calculated regression may not be an appropriate summary of a given set of bivariate data. The teacher should explain that if there is a pattern in the margin of error there is a problem with the regression. The teacher should lead the students to understand that for the data to be linear the residuals should be scattered around the x-axis and that the points should be unpredictable.
  • The teacher should provide the students with the Residuals Activity 2 worksheet (see attached) and monitor their responses as they are completing the task. The students can compare their responses with a partner or group. The teacher should provide the answers to the worksheet and allow the students to correct their responses.
  • Based on the students' understanding, the teacher should have the students that completed the worksheet correctly begin on the Residuals Activity 3 worksheet (see attached). The teacher should work with the students that did not correctly complete the Residuals Activity 2 worksheet to clear up any misconceptions or fill any gaps the students may have.
  • In going over the Residuals Activity 2 worksheet, it may be advantageous for the teacher to again discuss points outside of the given domain, e.g., 20, 25, 30, and have the class discuss the accuracy of the model at each age.
  • If time allows, the teacher may want to briefly discuss the use of piecewise functions when modeling multiple stages of growth.
  • The teacher should note that completion of the Residual Activity 3 worksheet is not necessary for students to achieve mastery of the standard.

• Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the lesson?

  
  

  Phase 3 Independent Practice

  • The teacher should provide the students with the Independent Practice worksheet (see attached). As the students are completing this worksheet, the teacher should monitor their responses. This activity can be collected to determine mastery of the standard for each student.
  • The teacher may decide to use the Independent Practice (Optional Scatter Plot) worksheet if he or she determines that the students would benefit from being able to visualize each residual by comparing the observed values in the data plot and the predicted values in the graph of the linear regression.
  • The teacher can group the students based on their responses to this worksheet. Group 1 - Students that need additional assistance in completing the Independent Practice worksheet, Group 2-Students that can complete the Enrichment worksheet (see attached) if provided with the regression equations (these students may need two copies of this worksheet), and Group 3 - Students that can easily use their calculator to determine the quadratic and exponential regression and graph the residual plot.
  • The following website can provide assistance in the use of the graphing calculator if needed. https://epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=100465.
  • The teacher should note that the enrichment worksheet does not need to be completed in order for the students to achieve mastery of the standard.

• Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?

  
  

  On a notecard or a separate sheet of paper, the students will complete a 3-2-1 exit slip by answering the following questions. The teacher should collect the exit slip from the students as they are leaving.

  1. List three types of regression models? (linear, quadratic, exponential)
  2. Name two characteristics of residual plots that can help us analyze regressions. (pattern and size)
  3. What is one thing you discovered about the correlation coefficient in terms of the linearity of the data? (a strong correlation does not necessarily ensure linearity of data.)

• Summative Assessment

  
  

  The teacher should have the students independently answer the following questions on a note card or on a sheet of paper at the end of class or at the beginning of the following period.

  1. What are residuals and how are they calculated? (Residuals are the measurement of the vertical distance from the data point to the regression line. Residuals are calculated by subtracting the predicted dependent value from the corresponding observed dependent value.)
  2. What does a negative residual indicate about the data point? (A negative residual indicates that the observed value is less than the predicted value.)
  3. Sketch an example of a residual plot from a linear model. (Answers will vary but should include plots with points that are equally and randomly dispersed above and below the horizontal axis.)
  4. What will be true about the residuals if the linear model is a good predictor of the data? (The residuals will be small in context of the problem and have no distinct pattern.)

• Formative Assessment

  
  

  The teacher should monitor student understanding with attached activities and questions throughout the lesson.

  Teaching Phase 1 Warm Up

  • The teacher should quickly assess prior knowledge of each student from the responses to Question 1 on the Warm Up worksheet (attached) and assist any student that is unable to complete this question while the other students are completing Questions 2 and 3 using a graphing calculator.

  Teaching Phase 2 Guided Practice

  • The teacher should monitor students' understanding of residuals and a residual plot by observing student responses to the Residuals Activity 2 worksheet. The students that responded correctly to the questions on this worksheet should begin Residual Activity 3 while the teacher assists the remaining students in their understanding.

  Teaching Phase 3 Independent Practice

  • The Independent Practice worksheet is intended to fully measure the anticipated learning goals for this lesson. Therefore, the teacher may want to not only monitor student responses but also document questions and concerns the students are still having at this point in the lesson. The teacher should address these misconceptions and gaps while the remaining students are working on the Enrichment worksheet.

• Feedback to Students

  
  

  The teacher will provide feedback throughout the lesson.

  Teaching Phase 1 Warm Up

  • The teacher will provide the students with feedback regarding prior knowledge related to the standard in this lesson during the completion of the Warm Up worksheet. An opportunity for understanding will be provided when the teacher checks the students' responses to the Warm Up worksheet.

  Teaching Phase 2 Guided Practice

  • The teacher will provide feedback on the students' understanding of residuals and residual plots during the completion of the Residuals Activity 2 worksheet. An opportunity for understanding will be provided after the teacher checks the students' responses to the Residuals Activity 2 worksheet.

  Teaching Phase 3 Independent Practice

  • The teacher will provide feedback on students' mastery of the standard during the completion of the Independent Practice worksheet. An opportunity for understanding will be provided after the teacher checks the students' responses to the Independent Practice worksheet.