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Proof, The
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Ideas from Teachers
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(Gr. 7-12)
This activity could be used with NOVA's "The Proof"
program. Fermat reduced great concepts to simple truths. Sometimes these truths are disturbing. The following discussion is used to expand my math students thinking about the nature of numbers we take for granted. The elegance and peculiarities of numbers can be readily observed when anaylzing our concept of 0 (zero). The usual description includes such characteristics as "place holder" and "nothing." However, what if zero were "something"—something that began to appear only when we let our minds think on a universal scale. Step 1: A Mathematical Oddity
Ask any math student what x^0 is, and the answer will be 1. How is that possible? If 0 is nothing, how can something to the nothing be something (1)? Experiment: Using a calculator, key in the following statement. 1000^(1/2)^1000 (What do you think the answer will be?) 1. So 1 is the result of reducing a number multiple times by a root. Question for investigation: What is the logical solution of taking the infinte root of an infinite number? Step 2: A Thermodynamic Math Corollary
The laws of thermodynamics are now understood to mean that the complete absence of anything is not possible. In other words, "nothing" isn't possible as we know it. However, isn't zero described a nothing? Experiment: Using a calculator with a square root function, key in 123456 and begin pressing the square root key repeatedly. What do you see happening to the answers appearing on the calculator screen? (Answers approach zero) Step 3: Making a Connection
If, therefore, the infinite root of an infinite number equals 1, and an infinite number to the zero power equals 1, then an infinite root equals zero. We can write our statement as: x^(1/infinity) = 1 Step 4: A Digression
square root (0) > 0 Experiment: Which of the following fractions is larger? 9/16 or sqrt(9/16)
1/infinity or sqrt(1/infinity) Step 5: Making Some Sense
Why then can we say that zero times anything is zero? For numbers within our ability to manipulate, what would be an infinite portion? Infinitely small, too small to have meaning. Nothing? Editor's note: Branting is in the process of writing the following sets of classroom discussions and investigations (grades 7-12). Contact him for more information. "The Zero Seminars: Thought Experiments About Nothing"
A mathematical exploration of the abstraction of using zero to denote something that doesn't exist. "Fermi Problems"
Mental exercises in which information must be estimated. A series of estimation situations, beginning with a classic problem used by Enrico Fermi (How many piano tuners live in Chicago?). Sent in by
Steven Branting
Jenifer Junior High School
Lewiston, ID
sbranting@mail.lewiston.k12.id.us 
(Gr. 8-12)
I designed a two-week lesson plan around NOVA's "The Proof"
program and Web site. It was
challenging for me since I teach 8th grade, and the topics are directed toward more
mature students. The mini-unit started with squaring a number and the opposite operation;
square root. This was reinforced by the activity "Square Pennies." I made a simple worksheet
on solving equations with squares and square roots, so the students could "algebraically" use
the Pythagorean theorem.
An introduction to the Pythagorean theorem was next. The students
explored the theorem using scissors and paper making tangrams—I'd like to find something
better next year. I also used the Pythagorean Puzzles
online activity as examples for
the class. A couple of worksheets on Pythagorean problems,
similar to Pythagorean Puzzle, brought closure to the theorem. I had the students make
up their own Pythagorean problem and design a mini poster.
We then had a class
discussion on the relationship between the Pythagorean theorem and Fermat's last
theorem. We also discussed the importance of Andrew Wiles proof. To spark the
student's interest, I printed out a large banner that stretched along the
chalkboard that read "I have discovered a truly marvelous proof, which this
margin is to narrow to contain—Pierre de Fermat." This reinforced the historical
mysticism behind the proof. In addition, I printed on a transparency the NOVA Online
Web site
that featured Andrew Wiles.
By this time the students were ready to watch the program. Since
there is a lot to absorb in the program, I broke it down into three sections (one day per section)
and designed a program worksheet for each section. I included pictures of Andrew Wiles and other
clipart from the NOVA Web site. The worksheets allowed me to stop the program and have a class
discussion on what was presented. I'm sure the students would have been overwhelmed if I
showed the program all at once. I would be happy to share these worksheets or any other
ideas, suggestions on this topic. Please e-mail me if you are interested. Sent in by
Derek J. Hogan
Pennichuck Junior High
Nashua, NH
hogand@nashua.edu
(Gr. 9-10)
Objective
To learn about Fermat's Last Theorem as a natural extension of the Pythagorean Theorem; to make connections between Wiles' proof, the Taniyama-Shimura Conjecture, Euclid's Fifth Postulate, and the non-Euclidean geometries that exist as a result of the negation of the Fifth Postulate; to learn about what it might be like to be a research mathematician and the importance of cooperation with other mathematicians; and to recognize the progression and building of mathematics throughout time.
Materials
Procedure
I show "The Proof" to my honors geometry students as a culminating activity of a unit on right triangles and the Pythagorean Theorem.
Ask students to consider the problem an + bn = cn. Students work with partners and try to find whole number solutions to the equation. After some time spent on this, students will start to question whether they will be able to find any solutions. At this point, give them a brief history of Fermat's Last Theorem and write the theorem on the board. I often incorporate pieces from other disciplines that relate to what we are doing, and have found many fun poems on the theorem and on Andrew Wiles that I have students read aloud before watching the film (these can be found on the Internet). Students are given the option of writing their own poem for fun or for extra credit. Students are given questions about the film ahead of time so that they can focus on certain pieces. We watch the film, stopping at various places for class discussion. Students answer questions and turn in their answers as a final assessment.
Assessment
Answers to questions about "The Proof." Sent in by
Michele Lender
East Haven High School
East Haven, CT
(Graduate Studies)
I've been using NOVA's "The Proof"
program as an example of consilience in a public policy class. The idea is to show the students that solutions come from many, many different ideas and places. That the truth is paramount and that preconceived notions are a hinderance to the discovery process, which is the most important part of developing public policy. Sent in by
Remzey L. Samarrai
University of Florida
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