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• Activate prior experience in grade 8 with informally fitting a line to a scatter plot and informally judging its goodness of fit (8.SP.A.2).

• Model the relationship with an equation for a line and use it to make predictions (8.SP.A.3).

• Represent data on two quantitative variables on a scatter plot (S-ID.B.6).

• Describe how two variables on a scatter plot are related (S-ID.B.6).

• Interpret the slope and the intercept of a linear model in the context of the data (S-ID.C.7).

(Note: 8.SP.A.2 and 8.SP.A.3 are prerequisites, not target standards in this unit. However, they are standards involved in one of the suggested activities.)

Students are presented with bivariate data that suggest a strong linear correlation. They plot the data points by hand or with technology, informally create a line of best fit, write an equation for the line, and use the equation to make predictions and interpret them in the context. Other tasks in this section focus on the last step: using the equation of a line fitted to data to make predictions and interpret them in the context of the data. Familiarity with informal fitting a line to data and assessing that fit is a basis for work in this unit: assessing fit more precisely by analyzing residuals.

## Tasks

WHAT: Students are given a graph that shows Number of Texts Sent vs. GPA for 52 high school students. The equation of a line of best fit is given and students are asked to interpret its coefficients in the context of the data.

WHY: This task gives more opportunities to practice interpreting a linear model, probing into the referents of coefficients and points on the line (MP2). It is also an opportunity to discuss some of the subtleties of modeling mentioned in the task commentary, e.g., the slope is the predicted change, not the actual change.

WHAT: In this task, students are given a scatter plot with a line of best fit and its equation that show how the finishing time for the men’s 100-meter dash has changed since 1900. Students are asked to interpret the coefficients of the equation in the context of the data. Then, students are asked to use the model to extrapolate the finish time from an earlier Olympics, and asked to determine a realistic domain for the function. A short (under 3 minutes) video that gives an overview of times for this race is here.

WHY: Students again interpret coefficients of a linear equation in the context of data (MP2), but are also asked to make predictions (MP4) and determine a reasonable domain, again illustrating subtleties of modeling.

## External Resources

#### Description

WHAT: In the animated film Tangled, Rapunzel’s hair is long—really long! How fast must her hair grow in order for it to get that long? To address this question, students collect data from photographs of a real-life Rapunzel named Ida Pedersen, who did not cut her hair for ten years. They create a scatter plot of Pedersen’s hair length over time, fit a line to match the data (8.SP.A.2), and interpret the slope of this line as her hair’s growth rate (8.SP.A.3). Then, they analyze the realism of the portrayal of Rapunzel’s super-long locks.

WHY: This lesson is placed here because there is no question about whether one variable depends on the other, allowing students on what they remember about linear functions (8.F.B.4) without the complicating factor of evaluating how well-correlated the variables are. This task is focused on eighth grade standards so can be used to review as students need.

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