Note: This lesson
requires a previous nights homework to be completed before the lesson
can be started. The homework requires the students to collect
data, and is mentioned within the lesson write-up.
To learn about data collection
To introduce Fathom2 and methods to
To introduce scatter plots and all of
To visually manipulate a line of best
To understand the basic concept of line
of least squares
The goal of this lesson is to allow students to explore a data set in
Fathom2 ultimately investigating the concept of least squares.
This use of technology will make the topic of study more interesting by
survey that the students will complete allowing them to work with data
that seems pertinent to their lives. The students will ultimately
be responsible for understanding scatter plots and the basic concept of
Before the students begin this lesson,
the teacher needs to create a survey using Fathom2. The survey
should include questions that produce statistical information that is
straight-forward and easily obtainable for the students.
Begin by discussing the many different methods of collecting
data. Have the students discuss the different methods they are
aware of. Some of these methods include experiments, surveys,
random sampling and
interviews. We are going to focus on using a survey that has been
created by the teacher. In our
example, we will ask for information such as length of foot, hand, and
arm span. Below is our sample survey. If you would like to take
the survey, the user name and
password are survey.
To view the results of the survey, go to
and sign in using the
above user name and password. Click on surveys, and view the
for a description of how
to create a survey using
The homework from the previous night was to collect data pertaining to
the survey from their families and friends. It is important that
the measurements are in the same units. For our sample survey,
the students will measure the foot, hand, and arm span of family
members and friends and bring that data to class. The collection
of out of class data is to
insure a wider range of age groups and a larger data sample.
The students will begin the activity by completing the survey with
the previously collected data as well as their own. They
can do this by logging on to the Fathom website or the teachers
personal webpage and inputting their user name and password. The
way the survey program was created allows each user name to take the
survey multiple times. This data
will all be recorded into the Fathom2 program whether the survey is
completed on the Fathom Survey website or the teachers own personal
webpage. Once all of the
data has been collected, the students will import the results into the
software program. This is done by first opening the Fathom2
program. Next, the students need to log in to the Fathom Survey
website using their user name and password. They will then click
on the surveys link, and click the Results line next to the survey they
took. Below is an example of our results as viewed from the
Fathom Survey website.
Once the results have been displayed, the students will click the icon
next to the web address and drag this icon on top of their blank
document. The results have now been downloaded into the Fathom
document. The students will now use this data to explore
the concept of line of least squares.
Before the students can investigate the line of least squares,
they must first create a scatter plot using the Fathom2 program.
First, they have to create a table of the data set. With
the 'collection' icon highlighted, they will click the 'Table' icon and
a table onto the screen.
Fathom2 will automatically fill in the
table with the data.
Next the students construct a graph using two of
categories from the data set. In the following example, students
would plot arm span versus height. This is done by clicking
the 'Graph' icon and dragging a graph onto the screen. Next,
on the 'Arm_Span' column in the table and drag it onto the x-axis of
graph. Do the same to place 'Height' on the y-axis. The
students now have a scatter plot of their data.
Now that their graphs are constructed, the students will insert a
"moveable line" from the graph menu. This is done by clicking
'Graph', 'Movable Line' with the scatter plot highlighted. They
this line so that it appears to follow the trend of the data set.
The line can be manipulated by hovering the cursor over the middle of
it and then dragging the line up and down or by pivoting the line with
the cursor on either end of the line.
After they have their best guestimate, they can insert the "least
squares line" from the graph menu. The students will now compare
this to their moveable line. The equations for each line are
provided at the bottom of the graph.
The students will repeat this
process to compare the other categories of the data, to give them a
better understanding of the least squares line.
Below is the graph from Fathom2 with the movable line plotted.
Below is the graph from Fathom2 with the Least-Squares line plotted.
Below is the graph from Fathom2 with both the movable line and the
Least-Squares line plotted.
Finally have the students remove the Least-Squares line by highlighting
the graph and going to 'Graph', 'Least-Squares', so that they
only have the movable line visible on their scatter plot. With
the graph highlighted, have them go to 'Graph', 'Show Squares'.
will make a square where one corner is on the data point and the
opposite corner on the movable line.
These squares will help the students to realize that by moving the line
they are trying to minimize the distance from the data points to the
line. Hence the term, line of least squares.
As the students manipulate the movable line, they will be able to see
that the size of the squares changes. As the movable line is
further from the least squares line, the squares are larger.
The closer the movable line is to the least squares line, the smaller
the squares are. To help the students see this, have them
reinsert the least squares line by going to 'Graph',
Next they can view the squares of the least squares line as well by
going to 'Graph', 'Show Squares' while the least squares line is
highlighted. Students can manipulate the movable line and see how
the size of the squares varies as the line is closer to or further away
from the least squares line.
Now that this activity has been completed, the students should
discuss what they believe the line of
least squares represents. It is important for the teacher to
insure that the students understand that the line of least squares
represents a general equation for the data and not just a randomly
placed line within the data points.
This will conclude the first day's lesson. Day 2 will include the
procedure for finding the line of least squares.
To assess the students progress, have them print off their work or
check each students computer before he leaves the class.