Park City Mathematics Institute Data, Statistics, and Probability

Parabolic Path to a Best Best-Fit Line
by Jeff Reinhardt and Joe Simons

 Download:     Teacher Notes: PDF format     Student Activity: PDF format     Student File: Fathom™ file WHAT IS IT? How does one determine the best best-fit line for a set of data? One approach is to find the line with the smallest overall error in prediction, or more precisely, the line that minimizes the sum of the squared deviations from the observed data to that line. This activity uses dynamic statistical software (Fathom™ 2) to engage students in an exploration of this concept. Building on intuitive guess-and-check methods to estimate a best-fit line, students experiment with the underlying quadratic relationship between slope and residuals to find the least squares regression line (LSRL). This optimization process connects students' knowledge of quadratic functions with statistics, and in the process, deepens their understanding of the LSRL. This activity would be appropriate in a course introducing concepts in regression analysis (such as Algebra 2) or at a more advanced level (such as AP Statistics) to develop and extend these ideas. GRADE LEVEL 9 - 12 STRAND Statistics, Functions, Geometry CLASS TIME 1-2 class periods MATERIALS     Student Activity Sheet and Fathom™ 2 file(one for each student)     Computer with Fathom™ 2 software
OBJECTIVES
In this activity, students use dynamic statistical software to explore important characteristics of a least squares regression line, especially the quadratic relationship between its slope and the squared residuals.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.