Barbie Bungee Lesson 2

  • Barbie's Line of Best Fit

    Lesson 2 of 3
    8th grade and High School Statistics / Algebra

    45-60 minutes

    Description

    Find the line of best fit for the Barbie Bungee data, interpret the slope and y-intercept, and understand the correlation coefficient.

    Materials

    Introduce

    In Lesson 1 students collected data relating the number of rubber bands attached to Barbie and the lowest point that Barbie's head reaches. Remind students of the predictions they made in their previous work.

    Display the data from one of your groups using Residuals and Linear Regression.

    “What pattern do you notice in these data?”

    In front of the class, model the features of the applet. Add a movable line. Allow a student to move the line to where they believe it is a “best fit”.

    “The vertical distance from each point to the line is the error from the prediction made using the line of best fit (a residual). Some residuals are positive (the point is above the line), and some residuals are negative (the point is below the line). We square each residual to make all of the values positive. Now we would like to find the line that minimizes the sum of the areas of the squared residuals.

    Allow another student to move the line to try to minimize the sum of the areas of the squared residuals of the area.

    Have groups of students explore with the movable line before sharing the “Show line of best fit and residuals”.
    Click “Show line of best fit and residuals”.

    Explore

    In the same groups from Lesson 1, students will input their data into the NCTM applet or into L1 and L2 on the graphing calculator. They will find the line of best fit using the NCTM applet of their graphing calculator (STAT, CALC, LinReg(ax+b)). Students will then work through the Lesson 2 Student Activity Sheet in their groups. In this worksheet, students will think about the physical interpretation of slope and y-intercept. They will also find the correlation coefficient (r) for their data set.

    Teacher Notes

    The activity sheet is intended for students to work through in groups. The teacher is walking around and listening to the group discussions. Prompt and cue with ideas or questions to check for and deepen student understanding.
    Here are some questions you ask students as they are working in groups:

    1. “Is the slope related to Barbie or the rubber bands?" "What does the value of the slope mean in the context of the problem?" (SMP 2) If students are struggling, make sure they know which variable goes on which axis.               
      1. Extension: "What would happen to the slope if we used a taller Barbie? Thicker rubber bands?” “Longer rubber bands?” (SMP 7) (For students who struggle to visualize, maybe have a few other types of rubber bands for them to explore - thick, thin, long short.)
    2. “Is the y-intercept related to Barbie or the rubber band?" What does the value of the y-intercept mean in the context of the problem?”(SMP 2)
      1. Extension: “What would happen to the y-intercept if we used a taller Barbie? Different types of rubber bands?”
    3. “What number is your correlation coefficient, r, close to?” “How does the scatter plot relate to value of r?”

    Synthesize

    If necessary, connect slope to the definition that students know from previous learning. Most students know slope as the riseover the run or the change in y over the change in x. In this context, the slope is the predicted change in the lowest point of Barbie's head for every additional rubber band that is added.

    If necessary, connect y-intercept to the definition that students know from previous learning. Most students know the y-intercept as the y-value when the x-value is zero. In this context, the y-intercept is the predicted lowest point of Barbie's head for zero rubber bands. This corresponds to the height of Barbie.

    The correlation coefficient, r, is a measure of the strength of the linear relationship. Scatter plots with a positive linear relationship and very little scatter are close to 1. Scatter plots with a negative linear relationship and very little scatter are close to -1. Scatter plots with very little linear relationship and lots of scatter are close to 0.

    Assesment

    Teacher Reflection

    • What questions did you ask students that best helped you to assess their understanding of slope, y-intercept, and correlation?
    • What questions did you ask students that best helped them to move forward when they were stuck or to deepen their understanding?
    • What misconceptions did your students have related to the topics addressed in this lesson and how did you address those misconceptions?

    Leave your thoughts in the comments below.

    Related Material

    NCTM Classroom Resources Interactive

    From Mathematics Teaching in the Middle School

    Other Lessons in This Activity

    Lesson 1 of 3
    Students will collect and present data for dropping a Barbie (or other object) from a set height using rubber bands in order to make predictions.
    Lesson 3 of 3
    Help Barbie make a better prediction for her bungee jump by using a line of best fit and analyzing residuals.
  • Comments

  • 1 Comments

    • Avatar

      Nice worksheet. Thanks for the comments to teacher (in red).

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  • Essential Question(s)

    • How can we create a mathematical model for our Barbie Bungee data? How good is the model for making predictions?

    Standards

    CCSS, Content Standards to specific grade/standard

    • 8.SP.A.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.
    • S-ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
    • S-ID.6.a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
    • S-ID.6.b Informally assess the fit of a function by plotting and analyzing residuals.
    • S-ID.6.c Fit a linear function for a scatter plot that suggests a linear association.
    • S-ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
    • S-ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.
    • S-ID.9 Distinguish between correlation and causation.

    CCSS, Standards for Mathematical Practices

    • SMP 2 Reason abstractly and quantitatively.
    • SMP 4 Model with mathematics.

    PtA, highlighted Effective Teaching Practice and/or Guiding Principle CCSS

    • Implement tasks that promote reasoning and problem solving.
    • Use and connect mathematical representations.
    • Facilitate meaningful mathematical discourse.
    • Pose purposeful questions.
    • Elicit and use evidence of student thinking.