Residuals

5 min

Narrative

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.

Launch

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.

Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Student Task

Mentally calculate how close the estimate is to the actual value using the difference: actual value−estimated value\text{actual value} - \text{estimated value}.

  • Actual value: 24.8 grams. Estimated value: 19.6 grams.
  • Actual value: $112.11. Estimated value: $109.30.
  • Actual value: 41.5 centimeters. Estimated value: 45.90 centimeters.
  • Actual value: -1.34 degrees Celsius. Estimated value: -2.45 degrees Celsius.

Sample Response

  • 5.2 grams. Sample reasoning: There are 0.4 grams between 19.6 and 20 and another 4.8 grams between 20 and 24.8 and 0.4+4.8=5.20.4 + 4.8 = 5.2.
  • $2.81 Sample reasoning: From 109.3 to 110 is 0.7, and then from 110 to 112.11 is 2.11, so 0.7+2.11=2.810.7 + 2.11 = 2.81.
  • -4.4 centimeters. Sample reasoning: 45.9−41.5=4.445.9 - 41.5 = 4.4, but this should be in the other order, so it is negative.
  • 1.11 degrees Celsius. Sample reasoning: 2.45−1.34=1.112.45 - 1.34 = 1.11, but these are both negative and in the other order, so the sign stays the same.
Activity Synthesis (Teacher Notes)

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?”
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . . .” or “I noticed _____, so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing
Standards
Building Toward
  • HSS-ID.B.6.b·Informally assess the fit of a function by plotting and analyzing residuals.
  • S-ID.6.b·Informally assess the fit of a function by plotting and analyzing residuals.
  • S-ID.6.b·Informally assess the fit of a function by plotting and analyzing residuals.
  • S-ID.6.b·Informally assess the fit of a function by plotting and analyzing residuals.

15 min

15 min