As his participant project, Evan Glazer presented a student project on modeling temperature data (http://www.glenbrook.k12.il.us/gbsmat/temp/temp.html). He noted that the temperature data work well with a cosine graph; however, it is possible to adapt the exercise by modeling parts of the data with a linear graph. As Evan described the project, the student must always live in a city where the temperature ranges between 65 and 85 degrees. If, at some point during the year, the temperature is outside of this range, the student must move to a new city that fits the temperature criteria. The students get the data from an Internet site (http://nimbo.wrh.noaa.gov/Reno/max.html), and analyze the data to fit a cosine function. They then find airline reservations, go sightseeing on suggested Web pages, and present their findings in a written report.
Isaac then demonstrated how to integrate MathCad into this project. In particular, he illustrated how the student can use the defined function to predict a temperature on any given day, and how to show graphically the period in which the temperature falls in the 65 to 85 degree range.
Judy noted that when you give middle school students only linear models to test, they get a false sense that all data can be modelled linearly. She suggested that perhaps students could learn how to model the data using a spreadsheet without having to fully understand all of the mathematics. Isaac suggested that while the overall model is not linear, students can use linear approximations to predict daily temperatures, as opposed to the more general monthly average temperature.
Bob Panoff mentioned that in differentiability and continuity, the concept of piecewise linear is important, and so it may be valuable to introduce these concepts early on. He also suggested that another lesson could present the idea that by introducing more data, the model becomes more exact, and at some point, too much data produces extra noise. Suzanne and Mel, middle school teachers, suggested that one valuable lesson might be graph interpretation. Evan added that this project piques the interest of non-mathematically inclined students, appealing to those who are good writers and have good organizational skills. Judy said this might be a good activity for group work, giving the teacher the chance to bring students with different resources together. Bob encouraged the participants to use any of the resources he has made available on the Internet.
Bob also said that the Internet can be useful for posting data from students' weather recordings from around the country. For example, in Maryland high school students are comparing weather data for eastern versus western Maryland. This provides an opportunity for the sciences and mathematics to work together.
After some individual project development time, Suzanne Alejandre gave a presentation of the thousand locker problem (http://forum.swarthmore.edu/alejandre/frisbie/unit7.html). She suggested that students begin in groups of two and use physical manipulatives to model the problem: when a partner counts, it helps get the problem to "click." Mel noted that by using many different techniques of presenting (physical modeling, computer, mathematics), a teacher can hope to reach many more students. By repeating the problem in different ways, there is more reinforcement and a better chance of understanding. Sarah encouraged using manipulatives in many situations from early on in a child's education - for example, grouping foods in sets during dinner. As a variant, Suzanne and Sarah also mentioned lining the students up in a row and having them step forward and backward to model opening and closing a locker.
Bobbie noted that this problem is rich in teaching about problem-solving. It teaches the students that they should begin with a simpler problem and use simulations when possible. In addition, this is a good lesson for learning about divisibility and factors. Judy mentioned two twists to the problem: keeping track of how many times a specific locker is open and closed, and finding how many lockers are opened exactly twice.