TEACHER: Spaghetti Residuals & Correlation

Transcription

1 Summary In this task, students will analyze the data from Breaking Spaghetti. Develop understanding of residuals. Use technology to draw residual plot. Use residual plots to determine whether a linear function, exponential function, or neither is a good model. Language: linear model correlation residual residual plot correlation coefficient Then, find the sum of the residuals above and below the line. When you have the linear regression model, the difference between those is zero. They also calculate the correlation coefficient, r, with technology. Then they view the applet Guessing Correlations to develop understanding of correlation coefficient. Differentiation All students # 1-15 New learning. Most students # Students who need a challenge # 16 What happens when you consume caffeine every 12 hours? Grouping: Teams Formative Assessment: #1 A Catch and Release is available if students have trouble completing the table. #5 Write function with growth rate less than 1. A Catch and Release is available if students have trouble writing an equation. #7 The growth rate is (1 growth factor). Ticket Out Write an exponential function from the initial value and caffeine level one hour later. Formative This is a separate document Assessment What to bring out in the Debrief: o What is a half-life? o The growth factor for an exponential decay function, f(x) = ab x, when 0 < b < 1 it is a decay function. Algebra 1 by Southwest Washington Common Core Mathematics Consortium is licensed under a Creative Commons Attribution 4.0 International License 2/10/14 Page 1 of 8

2 Resources: o Student version of the Task o Snapshot of graphs from video. Give these to students after they have attempted to make their own sketch. You can cut these and have them paste them onto their task paper. o Video clips without solution so students may view them on a personal device. o Complete videos. Regression Equations Learning Targets Common Core Standards Practice 4. Model with mathematics. Practice 5. Use appropriate tools strategically. Practice 6. Attend to precision. S-ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit. S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. b. Informally assess the fit of a function by plotting and analyzing residuals. Regression Equations Learning Targets S-IDc I can interpret linear models. Explain that the correlation coefficient, r, must be between -1 and 1 inclusive and explain what each of these values means. Determine whether the correlation coefficient shows a weak positive, strong positive, weak negative strong negative, or no correlation. S-IC I can collect and analyze data to make inferences and justify conclusions. Use technology to create and analyze a residual plot to determine whether the function is an appropriate fit. BIG Idea: Students will be able to graph data, find a function that best fits data, find and interpret correlation coefficient, graph residual plots to determine if the function is the best model, and use the model to make predictions. Page 2 of 8

3 How Strong is Spaghetti? 1. Why all the points in this scatterplot do not all lie on the line? When we actually perform the experiment, the spaghetti does not break as predicted. The scatterplot was made and the line of best fit was drawn with 2. Use the scatterplot and the line of best fit on the last page to complete the table. Pieces of spaghetti x Observed y-value from experiment Predicted y-value from equation. y = 13.1x+7.1 observed - predicted above, on or below line? Page 3 of 8

4 below below below above above below SUM Describe in words what the difference you found in the observed predicted column represents. This shows how far off the observed value is from the predicted value. 4. Describe in words what the description in the above, on or below line represents. If the data point is above the line, then the residual is positive and it is greater than the data point is above the linear model. That difference between the observed y-value and the predicted y-value is called a residual. Each data point has one residual. 5. Use a colored pencil or highlighter to draw in the residuals on the scatterplot (connect the observed and predicted y-values), and write the length of the residual next to the line. 6. Use the table above to find the sum of the residuals. zero 7. Write one sentence comparing the sums of the residuals above the line with the sums of the residuals below the line. The sum of the residuals above the line is the same as the sum of the residuals below the line. Page 4 of 8

5 Some classes did a lab called, Barbie Bungee. In this lab they tied rubber bands to the feet of a Barbie. Then they held here at the top of a meter stick. Then they bungeed (dropped) her and recorded the distance she fell. Brianna made a scatterplot of her data, and drew what she thought was a good linear model. Then she drew and measured the residuals. Suppose Brianna asked you if this was a good linear model. What would you tell her? Support your response with information from the scatterplot. This is not a good linear model because the difference of the residuals above and below the line is not close to zero. above: = 20 below: = 10 0 Page 5 of 8

6 Use technology to find the correlation coefficient for each of the data sets we used in this assignment. TI graphing calculator internet How Strong is Spaghetti? 8. correlation coefficient: _ Manatees & Motorboats 9. correlation coefficient: _ Car Care 10. correlation coefficient: _ Pieces of spaghetti x Year Observed y-value from experiment Motorboat Registrations (thousands) Manatee Motorboat Deaths Oil Changes Per Year Cost of Repairs ($) Page 6 of 8

7 11. Play the game Guessing Correlations Use watch you learned by playing Guessing Correlations to answer the questions below. 12. Which scatterplot has a higher correlation coefficient, How Strong is Spaghetti or Manatees Killed by Motorboats? How Strong is Spaghetti What do you think it means when a scatterplot has a higher correlation coefficient? The data falls closer to a line. 13. Car Care has a negative correlation coefficient. What do you think it means when a scatterplot has a negative correlation coefficient? The slope of the linear model is negative 14. How Strong is Spaghetti and Manatees Killed by Motorboats have a positive correlation coefficient. What do you think it means when a scatterplot has a negative correlation coefficient? The slope of the linear model is positive. Page 7 of 8

8 Modeling Data Learning Targets Practice 4: Model with mathematics. Practice 5: Use appropriate tools strategically. S-IDc I can interpret linear models. F-LE I can construct and compare linear, quadratic and exponential models and solve problems. S-IDb I can summarize, represent and interpret data on two categorical and quantitative variables. How do you determine whether a line is the function that best fits the data? Choose 2 data sets: 15. one you think is best fit by a linear function (How Strong is spaghetti is in L1 an d L2) 16. another you think is best fit by an exponential function Use technology to find the correlation coefficient for each of the data sets we used in this assignment. TI graphing calculator internet Use technology to find the correlation coefficient of 17. data that has a scatterplot with a strong correlation. 18. data that has a scatterplot with a weak correlation. 19. data that has a scatterplot with a positive correlation. 20. data that has a scatterplot with a negative correlation. 21. data that has a scatterplot with no correlation. How do you determine whether a line is the function that best fits the data? Page 8 of 8