I teach eighth grade math at State College Area Junior High School which is part of the State College Area School District. The lesson about which I am writing was taught in my eighth grade Algebra class. This class is an accelerated class for the district. Most of the eighth grade students are enrolled in Pre-Algebra or GeneralMath. Only ten percent of the student population takes Algebra as an eighth grader so most of these students are highly motivated and have experienced success in math. They are very interested in class and are inquisitive about math problems.

The activity that I am writing about is "the wave" problem. The Problem was to predict the number of seconds/minutes it takes to do "the wave" in a given stadium. This problem involved collecting data, creating a median fit line, and interpreting the data. Prior to this lesson the students had done lines of best fit by just eyeballing the data. They also had done two extensive problems in which they had to use median fit lines. However, in the other two problems I provided the data. Then with "the wave" problem the students collected the data. Overall this activity took three forty minute periods. I am going to divide the summary into the events that took place during those three class periods.

At the beginning of the period, I introduced the problem by telling the class some facts about "the wave". Basically I told them that "the wave" originated in Seattle, Washington during a Seattle Mariners baseball game. I also said that "the wave" is believed to behave like a linear function, and that is what we are going to explore. I told them that the problem is to predict the number of seconds/minutes it takes to do "the wave" in a given stadium. I then asked the class to brainstorm ideas of what we need to know to answer the problem. Their ideas varied from the number of people in the stadium and number of sections in a stadium to the rate of speed of "the wave".

After we discussed the student's ideas on the problem, we started to experiment to determine the amount of time it took various numbers of people to do "the wave". As a class, we determined the independent variable (number of students) and the dependent variable (time). One student commented that in our other problems, time was usually the independellt variable and wondered why it varied for this experiment. To collect the data, I had two students keeping the time. Then after a round of "the wave", we averaged the two times. This helped to eliminate some error in timing. We did "the wave" for groups of 3, 6, 10, 16, 22, 31, 37, 74, 111, 164, 246, and 410 people. I had another teacher bring her class over to do some experimenting, and we also went around numerous times to get some higher data values.

At the end of the period, the class had a little bit of time so we started to make the axes for the graph on which they would plot their data tomorrow.

Most of the students seemed to enjoy doing "the wave". They liked the idea that they were collecting the data. The students were thinking about the actual wave when we experimented because various students commented on the class's enthusiasm and starting position. The students had come to the front of the room so they were not sitting in a chair when we did "the wave", but they did crouch down to get the effect of sitting. The only drawback was that it did take awhile to collect the data, but it was helpful to have another class participating.

In the middle of the data, the time for 37 students came out less than the time for 31 students. This puzzled some of the students, but others suggested that some students may have slowed down on the trial that involved 31 students. A student commented that it would work out because the median fit line would handle the error in data collecting. I thought it was great that the students were looking ahead to the next part of the problem and fitting the parts together.

The students now used the data which they collected yesterday to create a scatter plot. From there, they graphed the median fit line and determined its eciuation, slope, and intercepts. The students were working in groups and found it interesting to compare equations, slopes, and intercepts. They understood that their answers would vary, but wanted to see how close the values were. The class also started to interpret the data. They used their model to answer some questions and thought about graphs for varying situations such as having the students turn around twice before sitting down.

The students did a good job in figuring out the median fit line. I found that they talked through the process as they did it in their group. It was great to see them sharing their knowledge and helping each other. A few students ran into problems when they calculated their intercepts and compared their calculations to the graphs. First of all, I thought it was great that they were checking their calculations with their graphs. Too often students will just perform the calculations without thinking about what they are doing. As I checked their calculations, everything looked correct, and I became puzzled as to why their graphs did not match their work. I then looked closer at the graph and realized that their scales on the x-axis were off. The students were disappointed that they had to redo the graph, but understood why the calculations and graph did not match.

Some students wanted to add (0, 0) to their data points, but I told them to just use the data we collected. Because they used our data that involved some error, most students median fit lines did not go through the origin. This bothered a lot of students, and they had trouble understanding why their data did not go through the origin. As I think back, maybe we could have included the origin as a data point. Then we could have used that point when calculating the median fit line. The result probably would have been a median fit line closer to the origin.

Today most students were at the point of giving meaning to the slope and intercepts. The class was familiar with understanding the slope and intercepts so this did not take much time. The activity which filled most of the period was calculating how long it would take to do the wave in Beaver Stadium. One student suggested to get some information from the phone book. We did this and found out that the stadium contains 39 sections as well as 11 new sections in the upper deck. The stadium's capacity is about 93,000 (including 10,000 people in the upper deck sections). This data gave the students a start, but they also wanted to know how many seats there are in a horizontal row. Some students asked about the circumference of the stadium. The class also started to think about the overlap that may occur in some people's wave. The groups ended up getting different answers which was fine as long as they showed sensible work to support their answer.

Finally "the wave" packet asked some questions about varying slopes for two different classes. This was designed to make the students think about things that may have occurred to make the data vary greatly. The final question dealt with the problem of the origin and why the graphs did not go through it. I wanted to see how the students would explain this problem.

To end class, I had each student write a journal entry expressing thoughts, problems, questions, or any feelings on "the wave" problem. I left this pretty open-ended because I just wanted the students to express their feelings or concerns on the problem.

I really liked relating the problem to Beaver Stadium. This was something most of the students were familiar with and interested in. This application made the students think about the problem. Some students started by figuring out the length of time for 93,000 people. Then when they got large answers that they knew did not make sense, they went back and thought about the problem some more. Eventually all of the groups came up with a good strategy. The only problem was that I did not have enough data for them to use. I need to gather some more data before I do this problem again. I also think it would be interesting to time an actual wave at Beaver Stadium so that my class can make comparisons of the real to their model.

The journal was a good way for some students to express some questions that were still troubling them. It also gave me a chance to see what the students thought of the problems. I only had one student express a dislike for doing "the wave". For your interest, I have included the student journals in my binder. Overall the students enjoyed the problem, and I think they learned a lot from it.

Name: Date:

Welcome to the 1996 baseball season!! (Roarrr! Roarrr!) As with most things in life, baseball has its share of excitement. Similar to all the other sports, when the fans at a baseball game get excited, they start to do something known as "the wave". Most of you probably have no idea that "the wave" originated in Seattle, Washington during a Seattle Mariners baseball game. But that historic day in Seattle when the crowd went nuts and began to do the wave was also a history day in the world of mathematics. It sent us mathematicians like myself into a frenzy. (Made us bonkers!) Since us mathematicians have no social life, when we hear of something new happening in the world, we try to relate it to mathematics.

So I get out on my journey to somehow relate "the way" to mathematics. What I found was astonishing. "The wave" behaves like a mathematical linear equation. In other words, similar to the equations that we have worked during these past few weeks. (y=mx+b) So I tested my theory on other sports and noticed the patterns of other waves. And true to form, "the wave" matched a linear equation every time.

So what we are going to do today is an experiment involving "the wave" and how it is related to mathematics.

The independent variable, x, is
The units for this variable are

The dependent variable, y, is
The units for this variable are

Now that we have ordered pairs, we can graph our linear equation. Use a sheet of graph paper to graph the ordered pairs.

After plotting your data on graph paper, find the median fit line. Circle the two points on your graph that the median fit line goes through. Write their coordinates below.

POINT #1 (,)POINT #2 (,)

Use these two points to find the equation of your line. Show all work.

FIND THE SLOPE OF YOUR LINE.

FIND THE Y-INTERCEPT OF YOUR LINE.

FIND THE X-INTERCEPT OF YOUR LINE.

WRITE THE EQUATION OF YOUR LINE.

REWRITE THE EQUATION USING THE NAME OF THE VARIABLES INSTEAD OF X AND Y.

Use the equation to answer the following questions. Show all work!!

1. How long would it take 40 students to make a wave?

2. How many students are needed for a 25 second wave? (Make sure your answers make sense for the question.)

3. How would your graph look if every student stood up and turned around twice before sitting down?

4. Describe in words what the slope represents. (Be sure to include numbers)

5. Describe in words what the x and y-intercepts represent. (Be sure to include numbers)

X-INTERCEPT:

Y-INTERCEPT:

Do the intercepts make sense for the situation? Explain.

6. How long would it take to do the wave in Beaver Stadium? Explain how you figured this out.

7. Other classes have completed this experiment. Their graphs are shown below.

A. Give a possible explanation of why the slopes are different.

B. If the experiment was done correctly, the line should go through the origin. (Zero students should produce a time of zero seconds). Give some reasons why their graphs do not go through the origin.