# Lines of best fit and residuals

• Use available technology to find lines of best fit (S-ID.B.6a).

• Quantify the goodness of fit by plotting and analyzing residuals (S-ID.B.6b).

Until now, students have estimated lines of best fit by eyeballing. This section shows how to use technology to plot lines of best fit and to quantify goodness (or badness) of their fit by analyzing residuals.

## Tasks

WHAT: Students are given a scenario about a laptop whose battery is partially charged together with pairs of times and battery levels that occur while the battery is charging. They are asked to draw conclusions about when the battery will be charged, the rate at which it’s charging, and how long it would take to fully charge if the battery had no charge. Students are not told how to represent the given data, in particular, there are several choices for the variable representing time. They examine change over different intervals, a preliminary to examining residuals, and use technology to obtain a line of best fit.

Note: Using the given times may pose a challenge. When setting up the axis for this graph students need to understand the referent of time 0. The teacher may need to scaffold this part. A more scaffolded version of this task is Laptop Battery Charge (designed for students in grade 8).

WHY: Students demonstrate that they can closely examine bivariate data and notice, both graphically and by comparing rate of change in different intervals, when something isn’t linear. Despite the fact that the data aren’t linear, students should be able to note that a linear model does give a fairly accurate account of the relationship between time and the laptop battery charge.