Best Fit

 Date March 2006 Class Algebra 2 Teacher Janet Wineland Notes Pass the Ball activity.  This is a fun activity and the kids learned so much from it!  We went outside, practiced a few times so that we would be fairly consistent, but I also explained that data is never perfect!  I liked the way we started with three students and kept adding more students while predicting the time for the whole class until the whole class was added.  The data was good – not perfect.  For the bounce, catch, and pass activity we changed things a little.  We added a few students at a time and then did the whole class.  Next, we found our best- fit line without the actual whole class data.  We compared actual data to predicted data for the whole class.  I then introduced the concept of residual and we talked about the way the best- fit line is really found using the sum of least squares.  I think this really made sense to the students.  The last thing we did was to predict some other times for different numbers of students.  Two of our predictions were using the students over again.  The last number was three times around the circle.  The data for the last time was way off – I used this to show them that data outside of the range can make meaningless predictions. I have enjoyed all of these activities and have definite ideas on how I will change them next year.  Thanks for putting it in the grant that we had to send these in each month – it really encouraged me use the activities and not forget about them! Date March, 2006 Class Algebra2 Teacher Mary Beth Weier Notes DATA COLLECTION AND THE OLYMPICS Objectives: Students will research real-world data that can be expressed in ordered pairs.                     Students will graph the data by hand and find the equation of the best fit line.                     Students will verify their results by entering their data into the TI82 graphing                         calculator and finding the regression equation. Materials:  World Almanac                    Graph paper                    TI82 graphing calculator Procedure: 1.  In groups of two or three, students select an Olympic event where the winner is determined by time or distance and record the winning times each year the modern Olympics were held. 2.  Each student graph the data by hand; draw the line of best fit, find its slope and equation. 3.  Each student enters their data into the lists in the TI82 graphing calculator, generates a scatter plot of the data and the line of best fit.  4.  Students compare the equation of the line they calculated by hand and the one the calculator generated.  5. Students use the equation of the line to predict the results in their events for the next Olympics.  Notes:  The activities we did at the workshop helped me realize the importance of students finding their own ordered pairs of real world numbers which makes finding the equation of a line more meaningful.  I used write-ups from some of the experiments that we did as a model to design my activity.  The students were very involved and did a very good job with this project.              Students had trouble at first finding the line of best fit because they weren’t sure where they should draw the line.  They are used to connecting the dots since most of the problems in the book give the students ordered pairs that are on the same line.  It was good for them to see that in the real world not everything comes out smoothly. Date March 2006 Class Algebra 1 Teacher Johnnie Ostermeyer Notes Ages who’s older I did the Ages project with my class to have better them understand best-fit line. One thing that I would recommend doing first is change some of the names to people that your students would have heard of before.  I know that the next time I use this I will take off Annette Funicello and Tom Watson. I like this project because you do not have to use a lot of paper. Here is the set up that I used:  Groups of two with one graphing calculator and one piece of scratch paper per group. I read off a name and then give the students about 1 min to discuss with their partner how old the person was and then they would put their answer into List 1 in their calculator and write it on there scratch paper (just in case they deleted it some how).  This way I did not have to give everyone a copy of the project. Once we finished with the guesses I had them put the real ages into List 2 of their calculators.  We then graphed their results using the actual ages as our X’s and the guesses as our Y’s. I then made them find the best-fit line on their scratch paper.  Then we graphed their line.  Then we used the calculator to find the best-fit line and graphed that line.  (Some students were really close and others were way off.) We then did the rest of the steps on the project hand out. ages of people . I asked the students which group was the best guesser of ages? How do you know? They finally said with some hints that the group with the slope closest to 1 and a y-intercept closest to 0. This went really well and I think that the students liked using the calculators to do the things that we had been doing with pencil and paper. This was a good and easy activity to do and I will do it again. Date November, 2006 Class Algebra I Teacher Johnnie Ostermeyer Notes Elbow vs. Height A good review of slope, y-intercept, plotting points, Slope-Intercept form, graphing a line and best fit-line. The student really seemed to enjoy this activity also. I had them partner up with one calculator per group but each had to have their own paper.  They had to go around and get ten peoples measurements from their tip of their middle finger to the point of their elbow and then their height all in centimeters. They then just used the nice and simple activity sheet provided by the class CD. This went well and I would recommend this as a review activity or as a project during the best fit-line section of an Algebra I book. Date November 6, 2005 Class Algebra 2 (Grades 10-12) Teacher Mary Beth Weier Notes Notes:  The students enjoyed the guessing game. Entering the information in the lists was good practice on using the STAT key on the calculator.  A good discussion was held on why the sum of the absolute value of the difference is a good indicator of who was the best guesser, but why the sum of the squares of this difference is a better indicator.  The students also had a better understanding of the best-fit line by doing this activity. Date October 2005 Class Teacher Notes Forearm vs. Height.  The students gathered the class data and made a scatter plot on their graphing calculators.  Next they used a transparency of a line over the graph and found their own line of best fit.  We compared their results with the linear regression equation found by the calculator.  I then introduced the idea of residual plots and after a day on this concept, we found the residual plot for our forearm-height data.  We finished this section by writing a paragraph in which they interpreted their graphs, their correlation coefficient, and their proportion of variation.  At the end of the unit we did the activity with the M&M’s.  I varied the activity slightly by having them start with 4 M&M’s in a cup, shaking the cup and rolling out the M&M’s.  Then they counted the number of ‘m’s that were visible and added that many more to the cup for the next trial.  When they were done we experimented with the quadratic, the exponential, and the cubic graphs to see which fit the best.  Next we reversed the process by starting with all of the M&M’s in the cup, shaking the cup and rolling out the M&M’s and than removing all of the ones that showed an ‘m’, putting the rest in the cup and repeating.  This makes an interesting curve that gave them a lot to think about! Date Sept. 30, 2005 Class Teacher Stephanie Reynolds Notes Ball Drop Experiment Students were placed in groups of 3.  Needed supplies were: two different balls, a measuring tape and a graphing calculator.  In their groups, they selected 8 heights to drop the ball from.  For each ball at each height, they measured the height that the ball bounced back up.  Students then entered all of their data into lists in the graphing calculator.  They determined a best fit line using a clear plastic line, and then by using the linear regression function on the calculator.  They then answered questions on the Ball Drop Experiment worksheet given to us in class. Students found that the height the ball bounces compared to the height it is dropped is a linear function. Students enjoyed doing an “active” lesson.  I realized that in the future, I will do this later in chapter 1.  I will also give them more background information about regression functions first. Date Sept. 26, 2005 Class Algebra Teacher Johnnie Ostermeyer Notes I had the students use the calculators to find the best-fit line for data. I was not able to find the time to use a project along with this task. I just used a problem out of my Algebra book and we found the best-fit line by hand picking two points that we felt were the best and then finding the equation in slop-intercept form.  Then we plugged the points into list 1 and 2 in our calculator and plotted them.  We then used the calculator to find the Regression line for the data.  We then graphed the equation that we came up with along with the equation that the calculator figured for us to see how close we were. The students seemed to like this and enjoyed using the calculators to find the line even after we already had the equation.  I then let them use the calculator for a big data set that was in the book on their homework and they appreciated that instead of doing everything by hand. Next time I would like to collect our own data making it a project, but I still think that it went well.