# 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 fi*t (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:

- “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.
- 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.)

- “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)
- Extension: “What would happen to the
*y-intercept*** **if we used a taller Barbie? Different types of rubber bands?”

- “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.