# Lines of Fit and Scatter Plots

Course: Algebra
Time Allotted for Lesson: 90 mins

I. Goals:  The students will explore scatter plots and lines of best fit.  The will learn the terms positive and negative correlation.  They will also learn the definition of a line of fit and of the best-fit line.  We will be plotting several points of data and determine a best-fit line.  We will write the equation of the best fit line.

II. Objectives:  The students will be exploring best-fit lines.  We will plot points and use Fathom to create data sets and then graph the linear regression lines( best fit lines).  The students will also learn to graph these lines by hand and determine the equations for the best fit lines.  They will also explore how accurate best fit lines are using a java applet from http://www.mste.uiuc.edu/activity/regression/.

III. Materials and Resources:  internet access, Fathom, and computers, graph paper or marker boards with grid lines

IV.  Attention Grabber: Fathom and internet explorations

V.  Procedure:
Today we will be talking about lines of fit on scatter plots.  There are several terms that we will need to define first.  When we talk about scatter plots we are talking about several points of data that have been plotted on a graph, and the line of best fit is the line that most closely goes through the points of data.  We can use the line of best fit to then estimate new data points, for example if we wanted to know larger values etc.

Scatter Plots:  two sets of data are plotted as ordered pairs

Positive correlation:  the data is said to have a positive correlation if the x values increase as the y values increase...for graphs with positive correlation the lines of best fit also have positive slopes.

Negative correlation:  the data is said to have a negative correlation if the x values increase as the y values decrease, the slope of the line of best fit will be negative.

Some scatter plots have no correlation.  This means that you can not draw a line of best fit.

Line of fit:  if the data points closely resemble a line the a line of best fit can be drawn.

Line of Best Fit:  the line that has the least error margin, meaning the best-fit line most closely resembles the data.

Let's explore Lines of Best Fit.  Go to the following website.
http://www.mste.uiuc.edu/activity/regression/.  This simulation allows students to place data points and find the line of best fit for the points that they have made.  The instructions are on the web.  Ask the students to create one graph that has a positive correlation, one with a negative correlation.  The students should observe that the positive correlation graph has a best-fit line with positive slope and the negative correlation with negative slope.

Now we want to explore actual data points, with your graph paper.  Plot the following.  (Taken from Glencoe Mathematics Text)

The Table shows an estimate for the number of bald eagle pairs in the US for certain years since 1985.

Plot these points and draw a line of fit to the data.  In your graph, pick two data points that the line of fit goes through.  We then are going to use those two points to determine the slope and then write the equation of the line of fit.  So, for example if your points are (3,2500) and (11,5000), then the slope would be m=2500/8=312.5, and then we would used point slope form so, y-2500=312.5(x-3), then put the equation into slope intercept form.  y=321.5x + 1562.5.

This is not a line of best fit though, in order to do this we are going to use fathom and then we are going to see how close our line of fit is to the actual best-fit line that fathom creates.  The computer program will be the most accurate approximation for the line of best fit.

Refer to your hand out on how to do this.

From Fathom we get the following.

We can see that our line of fit is fairly close to the line of best fit from Fathom.  Now we want to approximate what may happen in a few years.  So, if we want to know what the number of eagle pairs there would be 20 years after 1985,
What would be our first step?  Do you remember that we said by writing equations of lines we can determine any value of x or y that lies on that line if we are given either a x or a y.  So if we have our x=20, how would we find the y that lies on this line?  Plug in our x into the equation of the line of fit we determined and the line of best fit, let's see how far off our line is.

Line of Fit we calculated:

y= 312.5(20)+1562.5=7812.5 we would round up because cant have half a pair so there would be 7,813 pairs of eagles 20 years after 1985.

Best-Fit Line

y=309(20)+1560=7740, so there would be approximately 7,740 pairs of eagles 20 years after 1985 according to the line of best fit from Fathom.

VI.  Closure:  Today we talked about lines of best fit and scatter plots.  What were some advantages to using best fit lines?  Can you think of people that may use best fit lines everyday?  (Statisticians)  Talk about other real world examples of scatter plots and lines of best fit.

VII.  Extension: This lesson may take two days, if so take notes and do best-fit lines by hand first day and computer explorations the second day.