Residuals (IM Algebra 1, Unit 3, Lesson 6)
Task
Lesson Purpose The mathematical purpose of this activity is to introduce the concept of residuals and to have students plot and analyze the residuals to informally assess the fit of a function. In this activity, students also fit a function to data and use the function to solve problems. Students learn that a residual is the difference between the actual y-value of a point and the expected y-value of the point with the same associated x-value predicted by the linear model. How Tuva Enhances the lesson Illustrative Mathematics recommends using a digital applet so students can fit a precise line of best fit to data and attend to the accuracy of the linear model. Tuva supports this recommendation by allowing students to generate a least squares line and view the corresponding equation with appropriate precision, ensuring consistency across students’ models. Tuva also automatically calculates residuals for each data point and generates a residual plot. This removes the computational burden of manually calculating differences between actual and estimated values, reducing opportunities for arithmetic error. As a result, students can focus on interpreting residuals as vertical distances from the line of best fit and on using residuals to evaluate how well the linear model represents the data. By providing accurate, dynamic representations of both the linear model and the residuals, Tuva supports students in making meaningful connections between the scatter plot, the equation of the line, and the residual plot. This alignment allows teachers to emphasize conceptual understanding—such as what it means for a model to fit data well—while maintaining fidelity to the mathematical goals of the IM lesson.
Oranges Return Let’s examine how close data is to linear models. The data in this scatter plot represents the linear relationship between number oranges and their weights. A linear estimate is close to actual data. It is an approximation based on the data represented by the scatter plot.
Topics and standards: HSS-ID-B-6-C, HSS-ID-B-6-B, Algebra I, Illustrative Math & Tuva, HSN-Q.A.3