This lesson uses the data from the video of orange weights from the activity "Orange You Glad We’re Boxing Fruit." 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 -value of a point and the expected -value of the point with the same associated -value predicted by the linear model.

In the digital version of the activity, students use an applet to fit a line to data. Use the digital version if possible to allow precision in the line of best fit.

Compare student answers to the question about the point that the line estimates best to the answer for the question about the residual closest to zero.

Show a graph of the residuals.

Ask students:

• “What does it mean for the residual to be positive? Negative?” (The residual is positive when the actual data value is greater than what the model estimates for that -value and negative when the actual data value is less than the estimate.)
• “What does it mean when a residual is on or close to the horizontal axis?” (It means that the line of best fit passes through or comes close to passing through that point in the graph.)
• “Find the residual that has the furthest vertical distance from the horizontal axis. What does this mean in the context of the scatter plot and the line of best fit?” (The residual that is furthest from the horizontal axis has the same -coordinate as the point that is the greatest vertical distance away from the line of best fit for the data represented by the scatter plot.)

Students sort different graphs during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7). They should begin to recognize that a plot of the residuals for data that is fit well by a linear model shows residuals that are close to the -axis and do not show a noticeable trend.

The goal is to make sure students understand the connections between a scatter plot displaying a linear model and a graph of the residuals. A good linear model for the data will have residuals that are close to the x-axis and scattered on both sides without a clear pattern.

Much discussion takes place between partners. Invite students to share how they made the matches.

• “What were some ways you handled finding the matches for B and C? Recall that B and C used the same data but different linear models.” (The line in C was not as good of a fit as line B, so I knew the graph of the residuals would be farther away from the horizontal axis.)
• “Look at the matches for E and J. How can you tell from the graph of the residuals that the linear model is not a line of best fit?” (The first half of the residuals are positive, and the second half are negative. This lets me know that the line does not pass through the middle of the data.)
• “What do you notice about the residuals in graph K? Explain what you notice in the context of scatter plot A.” (The residuals go in a u-shaped pattern. The data represented by the scatter plot does not appear linear. It is curved, so when I plot a line through it, I would expect the residuals to show the curvature.)
• “Describe any difficulties you experienced and how you resolved them.” (It was tough figuring out how to decide where to begin when finding matches. I used my partner’s strategy of looking at the values on the -axis to help narrow down the choices.)

Students may not understand how to determine if the linear model estimates the weight of oranges well or poorly. Ask them to determine the weight that the model estimates, then ask how close that estimate is to the actual weight.