Introducing the Line of Least Squares Using Fathom2
Katherine Huffman & Brooke Norman
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.
Objectives:
Technology:
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 using a 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 least squares.
Lesson:
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 http://survey.keypress.com/web/loginForm.php and sign in using the above user name and password. Click on surveys, and view the survey.
Click here for a description of how to create a survey using Fathom2.
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 Fathom2 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 drag a table onto the screen.
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, click on the 'Arm_Span' column in the table and drag it onto the x-axis of the 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 will then manipulate 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'. This 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', 'Least-Squares'. 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.
Next Days Lesson
Unit Homepage
Objectives:
To learn about data collection
To introduce Fathom2 and methods to
import data
To introduce scatter plots and all of
its aspects
To visually manipulate a line of best
fit
To understand the basic concept of line
of least squares
Technology:
Fathom2
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 using a 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 least squares.
Lesson:
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 http://survey.keypress.com/web/loginForm.php and sign in using the above user name and password. Click on surveys, and view the survey.
Click here for a description of how to create a survey using Fathom2.
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 Fathom2 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 drag 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, click on the 'Arm_Span' column in the table and drag it onto the x-axis of the 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 will then manipulate 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'. This 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', 'Least-Squares'. 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.
Next Days Lesson
Unit Homepage