Name: ___________________________
Period: ___________________________
Date: _____________________
Mathematics Research Paper
The field of mathematics is rich with content, containing a myriad of topics and concepts that cannot
possibly be studied over the course of a regular school year. As a student enrolled in the Precalculus
Honors class, you will be required to write a mathematics paper on a topic in mathematics that you
have not studied during your high school career.
A mathematics paper is different than most other papers. In the past, you may have written a paper by
formulating an idea or opinion and then using various sources to find evidence that defended that
opinion or supported that idea. When writing a mathematics paper, you choose a mathematical topic
and do research in order to understand the topic. After understanding the topic, the paper you write
develops the topic using logical arguments and examples. Your paper is not going to be a summary of
the research you do; for example, you are not going to write about “the history of code-breaking”.
Your paper will either develop mathematical theories and concepts that you haven’t yet learned, or it
will apply math to various situations.
In order to write a mathematical paper, you first need to choose a topic. At the end of this packet is a
list of suggestions that you can write about. You do not have to choose a topic from the list. You
should ask me about topics not on the list; some things might seem like good ideas, but might become
very involved and beyond the scope of a high school mathematics student.
Requirements
1. Students will write a 6 – 10 page research paper on a math topic of their choice (with normal
margins, spacing, and font size!)
2. Students will have a bibliography that uses at least two books or print journal articles.
3. Students may use the internet as a resource in addition to books and print articles.
4. Students must work independently
Optional (Math Fair Participants)
1. Students will rewrite their paper, turning in a second draft and then a final version of their
paper
2. Students will work on developing a presentation of their paper for the Long Island Math Fair
3. Students will work on visuals for their presentations
4. Students and their parents / guardians will complete the necessary paperwork for participation
in the Math Fair
Timeline
In order to help you develop your paper, I have set up due dates for various benchmarks:
9/29:
Topics are due:
You will submit a paper with your name, your class period, the topic you are going to be studying / writing about,
and a brief description of why this topic interests you or why you have chosen your topic
10/20
3 sources, not Wikipedia
On the same paper on which you submitted your topic, you will list three sources you intend to use for your paper.
Two of them must be non-internet sources.
10/27
Outline due
You will submit an outline of your paper. It will include your topic, and a breakdown of the flow of your paper;
what each paragraph (or groups of paragraphs) is going to discuss, examples that you might use, etc.
11/3
Introductory Paragraph
You will turn in, on a piece of paper with a proper heading, your topic and your introductory paragraph.
11/17
Papers are Due; Do you wish to participate in the Math Fair?
You must submit a copy of your paper. The paper will have a cover page that contains your name, your class
period, and the name of your topic. In addition, if you wish to participate in the Math Fair, you will add a
statement on the bottom of the cover page which says “I wish to participate in the Math Fair”.
Math Fair Timeline
The Math Fair is significantly different from your other math experiences, such as Mathletes and Math Competitions, in
many respects. You will investigate a project for 2 months, developing your own ideas about the topic. Then, you will
present them both written and orally. In math class or Mathletes, the time span for a problem is minutes, not months. The
Math Fair gives you the opportunity to develop your "math power" -the ability to investigate one topic and then to stand
before a group of judges, peers, and parents and defend your work.
Your desire to attend the Math Fair is the first criterion for actually participating. In addition to desire, you must meet all of
the deadlines mentioned above. Further, your paper will have to meet rigorous mathematical standards, and not simply be a
summary of information that you have obtained from a variety of sources. Your paper will need to be a reflection of your
own understanding of the topic which you researched, demonstrating a mastery of the material. If you meet the
requirements of mathematical understanding, punctual submissions, and personal desire, then you will be chosen to attend
the Math Fair.
Attending the Math Fair is a reward in itself, allowing you the pleasure of demonstrating your newly acquired knowledge.
For those who need some extrinsic motivation, you will have a nationally-recognized activity to put on your resume! Also,
because the extra amount of work and effort that is required to produce a high quality paper is beyond the scope of a normal
high school student’s workload, I will give you 2 points on your second quarter average, and 2 more points on your third
quarter average.
This is going to be a lot of work, but you are going to learn a lot from this experience. You will be learning how to write a
research paper and learning about how to present a project to a group of people that you don’t know. You will also be
focusing on a mathematical topic that you will become an expert on. This will be a fantastic experience that you will be
able to put on your college applications!
For those wishing to participate in the math fair, then the following additional timeline is for you.
12/1
Permission slips for Hofstra University and second draft
You will turn in a signed permission slip from your parents allowing you to attend the Math Fair on February 27 th.
You will submit a second draft of your paper, based upon the original suggestions that I have made. You may
incorporate new or further research and development of your topic, but the paper must still be within the 10-page
maximum. And don’t fix the font or the margins, or the spacing so it fits.
1/5
Visuals due, Paperwork due
Whatever visuals you plan to use at the Math Fair must be completed. This could include, for example, a trifold or
a PowerPoint presentation. Also, all paperwork related to the Math Fair must be completed and submitted to me.
1/19
Final Papers Due!
Three copies of your paper must be submitted, along with the Math Fair Cover Sheet completely filled out
For more information about the Math Fair, visit my Math Fair webpage:
http://hicksvillepublicschools.org/Page/11091
Format:
1.
Cover page:
Your name, your class period, the name of your topic (a title), all centered
If you want to participate in the math fair.
2.
Abstract:
A separate page that contains a short summary of what the reader can expect to find in your paper. It is
not an introduction. It is a summary.
3.
Introduction
The first paragraph of your paper. This orients the reader to your topic and covers basic material
connected to something the reader is already familiar with.
4.
Body
This follows directly from the Introduction, not a separate page, and is the “meat” of your paper
5.
Conclusion or “For Further Research”
The final paragraph(s) of your paper. This follows directly from the body, and either sums up the
content of your paper or discusses other topics that you may research in the future that are based upon
what you learned
6.
Citations
Your paper should include internal documentation. This means, whenever you reference something
from your research, you write the author’s name and the page number in parentheses.
You will also need a Works Cited, or Bibliography page, written in correct MLA Format.
Suggested Topics:
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
Hyperbolic Trigonometric Functions
 Sinh, cosh, tanh, sech, csch, coth
 Inverse hyperbolic trig functions
Curves defined by parametric equations
Polar Coordinates
Complex Numbers
 Polar Form
 DeMoivre’s Theorem
Vectors
 Dot and cross-products
 Vector Fields
Matrices
The Pigeonhole Principle
Conditional Probability
The Monty Hall Problem
 Extended to more than 3 doors with only 1 prize
 Extended to more than 3 doors with multiple prizes
Field Theory
 Fields
 Groups
 Rings
Solving cubic equations
 Cardano’s equation / formula
Solving quartic (4th degree polynomial) equations
Abel and the unsolvability of 5th-degree polynomial equations
The number e
Cardinality of infinite sets
 Cantor’s set
 Countable and Uncountable sets
Transfinite number arithmetic
Set Theory
 Russel’s Paradox (The Barber Paradox)
 The Banach-Tarski Paradox
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
Geometry: “Squaring the Lune” (Finding the area of a crescent)
Pi
 Historical calculations
 Methods of calculation
Archimedes area problem (area bounded by the parabola y = x² on [0,1])
Prime numbers
Fibonacci Numbers
 The golden ratio
Game Theory
 Backward Induction
 The Hangman’s Paradox
 Prisoner’s Dilemma
Special number types
 Lazy Caterer
 Amiable
 Friendly
 Abundant / Perfect / Deficient
 Vampire
Base-n number systems
 Binary
 Hexadecimal
 Modular Arithmetic
The Chinese Remainder Theorem
Fermat’s Last Theorem
 History (including Andrew Wiles successfully proving it)
 Basic mathematics of it
Non-Euclidean geometry
 Hyperbolic geometry
 Spherical (Riemann) geometry
Magic Squares
 Solutions for odd / singly even / doubly even
Sudoku Mathematics
Lego Mathematics (Combinatorics)
The Traveling Salesman problem
 The Bridges of Koenigsburg
Pick’s Theorem
Abacus mathematics
Slide-rule mathematics
Fractals and Complex Numbers
Ancient Number Systems
 Arithmetic
Ciphers and cryptography
 Vigenere ciphers
 Affine ciphers
 Keyed Vigenere Ciphers
 Application to binary operations
 Modular Ciphers
 Combination Ciphers
Four-dimensional geometry
 Investigate 4-dimensional objects
 The story of Flatland
Euler’s equation
i
o
o
e 1  0

UPC mathematics
 Modular arithmetic
 Check digits
 Credit card applications
Linear Programming
 Solving systems of inequalities graphically
 Solving systems of inequalities with a tableaux
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
The Catalan numbers
Stirling numbers
Recursion
 The Tower of Hanoi
 Combinations in Pascal’s Triangle
The Art Gallery Problem
 Extended to oddly-shaped museums
The Four-Color Theorem (Also called the Map-coloring theorem)
Napier’s bones
The Koch (Snowflake) curve
Sierpinsky’s Triangle
Euler’s Theorem (involving 1, 2, and 3-dimensional figures)
Cycloids
Knot Theory
The Paper Folding Problem
 Incidentally, solved by a junior in high school!
The Collatz Conjecture
The Paper Airplane Problem
 Which designs give the furthest distance? Greatest height? Long flight? Why?
Solving a Rubik’s Cube
 Rubik’s Cube Mathematics
 God’s number
Mathematics of creating a sundial
 Trigonometry
 Astronomy
Probability
 Expected Value
 Game Show Mathematics
 Deal or No Deal
 Wheel of Fortune
 Let’s Make a Deal
Statistics
 Applied to various hypotheses, such as effects of sleeping patterns on work habits
The “Random Number” Problem
 Asking people for a random number from 1 to n seems to generate more numbers than others.
Why?
Mathematical Applications
 Bridge Building
 Chinese Postman Problem
 Actuarial Math: How do Insurance Companies make money?
 Sabremetrics: Baseball Mathematics
 Linear and Angular Kinematics (Physics)