# Guiding students through approximating trendsAn instructor's guide to Best-fit Lines

Many exercises in introductory geoscience courses require the construction of a best-fit line (or approximating linear trends in data). Many students struggle with this, particularly because they often want to just "connect the dots". The Best-Fit Line module is designed to give students the tools to construct (approximate) best-fit lines through data points plotted on X-Y graphs.

## What should the student get out of the page?

By the time the student has worked through the page, he or she should be able to:

• recognize trends in plotted data that are appropriate for a best fit line
• use one of two procedures for approximating a best fit line
• construct a best fit line for plotted data (see plotting points for a tutorial on constructing plots from data)
• demonstrate their newly acquired procedure by solving sample problems

## Why is it hard for students?

For whatever reason, many students do not understand the concept of fitting a line to a set of data. Perhaps it is because, in many mathematics courses, the process is reversed - students construct a table of data from a line. Most students arriving in their first geoscience class will be familiar with the equation for a line (many can recite "y = mx + b" if you ask) but most of them cannot conceptualize what that means.

The most common mistake that students make is to connect the lines; however, in the geosciences, we are most often describing a natural, somewhat chaotic system and we are not interested in the absolute value of the data, we are interested in the general trend of the data. Instead of needing to know all the values in a data set, geoscientists often want to be able to describe a set of data with an equation that approximates a natural phenomenon - a concept with which many introductory geoscience students will be unfamiliar. Students are also often unfamiliar with the "messy data" of natural systems and try to make sure the line goes through all the points by connecting the dots. Try to get students to understand that the trend is more important than the actual data set - we plot the data to help us visualize the trend.

For more help on teaching about trends in data, please see Understanding Trends in the Teaching Quantitative Literacy section of SERC.

## What have we left out of this page?

This page only touches on the approximation of a linear regression and does not cover any exponential or logarithmic curves. Almost all of the exercises in introductory geoscience books are formatted for linear regression (often with the use of semi-log or log-log plots) and thus we do not cover the issue of other types of regressions.

## Instructor resources

Below are a couple of examples of exercises in which regression is needed.
• This student resource on Regression by Eye has an applet that allows the student to estimate the trend of a large set of data. The student can draw a "best-fit" line, can estimate the value of r (Pearson's correlation) and can try to minimize the mean square error.
• Science Courseware's Geology Labs Online has a unit on virtual dating that allows a student to work on regressing a line visually (it also allows you to change the intercept and shows error). An excellent resource for students to work on best-fit lines.