Lesson 6

The purpose of this Math Talk is to elicit strategies and understandings students have for subtracting an estimated value from an actual value. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to compute residuals from a linear model.

Display the instructions for all to see. Ask students, ”How would the answers to these two questions be different?“

• Actual value: 21 cars. Estimated value: 20 cars
• Actual value: 20 cars. Estimated value: 21 cars

(The first question has an answer of 1 car. The second question has an answer of -1 car.)

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

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 \(y\)-value for a point and the expected \(y\)-value for the point on the linear model with the same associated \(x\)-value.

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.

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 \(x\) 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 \(x\)-coordinate as the point that is the greatest vertical distance away from the line of best fit in the scatter plot.)

In this activity, students take turns with a partner, matching graphs of residuals to scatter plots that display linear models. Students trade roles, explaining their thinking and listening, providing opportunities to explain their reasoning and critique the reasoning of others (MP3). 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 \(x\)-axis and do not show a noticeable trend.

Arrange students in groups of 2. Demonstrate how to set up and find matches. Choose a student to be your partner. Mix up the cards and place them face up. Point out that the cards contain either a scatter plot with a linear model or a graph of the residuals. Select one of each style of card and then explain to your partner why you think the cards do or do not match. Demonstrate productive ways to agree or disagree—for example, by explaining your mathematical thinking or asking clarifying questions. Give each group a set of cut-up cards for matching.

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 scattered on either side of the \(x\)-axis without a clear pattern and close to the axis.

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 in the scatter plot does not appear linear and is curved, so when you plot a line through it, you 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 \(x\)-axis to help narrow down the choices.)