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Q&A #3790


Advanced Math projects

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From: Jim and Carolyn (for Teacher2Teacher Service)
Date: Apr 26, 2000 at 21:11:15
Subject: Re: Advanced Math projects

Hi Renae,

A project that has worked well for me has students working across the
topics of parametric equations, composition of functions, data collection
and analysis, and technology to foreshadow the study of related rates in
calculus.

I begin the unit by helping students tie together parametric equations and
composition of functions. Throughout the unit students will use graphing
calculators.

A typical problem I might provide:

Each of Frosty's parts is a spherical snowball. When the sun comes out, his 
head (volume = 6 cubic feet) begins to melt at the rate of .25 cubic feet per 
minute.  Frosty's corncob pipe falls from his mouth when the radius of his 
head is 9 inches.  How soon after the sun pops out does Frosty lose his pipe?
 
           Solution:
                The volume is a function of time.
                V = f(t) = 6 - t/4
                The radius is a function of the volume.
                R = g(V) = ((3V)/(4pi))^(1/3)
                Now compose the functions the functions.
                R = g(V) = g(f(t)) = (g o f)(t) = g(6 - t/4)
                =((3(6 - t/4))/(4pi))^(1/3)
                We now have 2 parametric equations in terms of t.

Graph these equations on the graphing calculator and trace to radius = .75
feet. Frosty's pipe will fall when time equals 16.9 minutes.

The next dimension I add in the unit is the use of real data and the 
regression capabilities of the graphing calculator.

A typical problem I might provide:

Using data from the United States National Center for Health Statistics
showing birth years and the life expectancy for a selected population such
as white males, black females or others, students are asked to enter the data 
in their calculator.  They are to discover the graph and regression formula 
and graphically estimate the life expectancy for a selected person born in the 
year 2020.  Then estimate in what year the life expectancy would be 100 years.

Most of my students are in physics classes where they have used paper and
pencil to find linear regressions.  They were surprised during this phase of
our work to find that with their graphing calculator a natural log, power, or 
exponential regression may be more appropriate.

Next I provide students with data and ask them to analyze it and then students 
themselves propose several useful problem situations.  The first time I asked 
students to design problems I was surprised to find how difficult it was for 
students to write clear solvable problems.  However, as students develop their 
problem writing skills I am always amazed at the variety and creativity they 
display in their problems.

A typical situation I might provide:

From the American Journal of Public Health, a list of several countries, the
cigarette consumption per adult per year in each country, and the coronary
heart disease mortality per 100,000 people in that country is given. Students 
are to propose and solve several practical problems involving the
information.

One cooperative group proposed that the United States embark on a campaign
to decrease the average cigarette consumption by 50 cigarettes per year with
the goal being to establish when the mortality from coronary heart disease
would be cut in half. Another group was quick to point out that even though
the data for cigarette consumption and coronary heart disease showed a high
positive correlation there could possibly have been other factors involved.
The heated discussion that followed helped students see the importance of
questioning hastily arrived at conclusions.

The foundation has now been laid. Students can work with parametrics and
composition of functions. They can use regression capabilities of graphing
calculators to analyze data and look at sets of data and suggest meaningful
ways for using the data. It is now time for students to step outside the
classroom and use their skills. I challenge my students to use these skills to 
solve real life problems. As a unit project, working in teams, students will 
scatter throughout the school and community to gather data of related events 
and design appropriate problems. Some cooperative groups select a department 
in our high school to work with and some work with businesses in our 
community. Students will prepare ideas for projects before meeting with their 
resource persons. Contacts will be made and meetings held. The resource person 
will discuss the team's ideas and perhaps describe needs they have for data 
that could be gathered. Students design a method for gathering the data and 
the data is gathered. Students write a useful problem from the data that can 
be solved employing parametrics, composition of functions and technology.

In past years, students have written problems investigating related rates
such as:

   -school attendance vs. outside temperatures
   -number of plate lunches sold in the cafeteria vs. weather conditions
   -number of books checked out of the school library vs. number of students 
        entering library
   -school baseball team wins vs. ERA over several seasons
   -pop rivet (Maurer Industrial Supplies) sales and the Dow Jones 
   -furnaces (church plumbing, heating, and air conditioning, etc) sold and 
        serviced vs. temperatures

Through class discussions students design effective modes for communicating 
results. During this discussion many students come to realize that their 
future careers will very likely entail this same need for communication. Most 
students choose to include portions of their project in their portfolio.

 -Jim and Carolyn, for the Teacher2Teacher service

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