 First Digits  Ivars Peterson (MathTrek)
Take a look at a newspaper page listing stock market prices. You might think that each of the numbers from 1 to 9 would occur equally often among the first digits of all the listed prices. Instead, however, you're very likely to find that numbers starting
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 The Five Percent Solution  Keith Devlin (Devlin's Angle)
"Mathematical disciplines such as statistics, optimization theory, probability theory, queuing theory, control theory, game theory, and operations research are all used routinely in making difficult choices in public policy, health, business, manufacturing,
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 Five Strategies for Integrating the Internet into the Curriculum  American Library Association (ALA)
An essay organized into Strategy One: Use technology to teach information literacy; Strategy Two: Go on a webquest; Strategy Three: Teach students to use computer applications; Strategy Four: Join a telecollaborative project; and Strategy Five: Look at
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 FiveThirtyEight  Nate Silver, Editor in Chief
Quantitative features, interactives, and "datalabs" on politics, economics, sports, and other topics by Silver, Carl Bialik, and others. Science posts have included "How Statisticians Could Help Find That Missing Plane," "Toilet Seat Covers: To Use or
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 Fixed Point Theory on the Web  M. A. Khamsi
Fixed Point Theory and Applications sites; Mailing List; Conferences: previous and forthcoming; books on fixed point theory; other interesting sites on the Web; bibliography database; abstracts; papers on fixed point theory; general information.
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 Flight of the Bumblebee  Ivars Peterson (MathLand)
How did proving that a bumblebee can't fly originate? Who started the story? It apparently first surfaced in Germany in the 1930s, and the story was about a prominent Swiss aerodynamicist. One evening, the researcher happened to be talking to a biologist
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 Fluid Mechanics, Chaos and Mixing  Julio M. Ottino, Chemical Engineering Dept., Northwestern University
Research Interests of the Chaos and Mixing Group include: Granular Flows and SolidSolid Mixing; Competition Between Chaos and Order; LiquidLiquid Mixing; Mixing in 3D Flows; Mixing in Cell Culture Systems; SolidLiquid Mixing; and Geometric Aspects
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 Focus on Student Practice  Suzanne Alejandre
Article about how students worked on the Problem of the Week (PoW) "Wooden Legs" in a way that developed the Common Core State Standards (CCSS) Mathematical Practice of "making sense of problems and persevering in solving them." This article, originally
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 Fold Everything  National Geographic Society
See how paperfolding inspires the design of aircraft, stent grafts, and the packaging and transportation of telescope lenses in this overview of the history and future of origami. With downloadable instructions for folding a peacock, as well as an online
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 Folding Maps  Ivars Peterson (MathTrek)
Erik D. Demaine of the computer science department at the University of Waterloo
in Ontario and his coworkers have developed an efficient method for recognizing when a creased sheet is foldable into a flat package (results are reported in an unpublished
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 Following Benford's Law, or Looking Out for No.1  Malcolm W. Browne; The New York Times Company
An article from the New York Times on the Web: Dr. Theodore P. Hill asks his mathematics students at the Georgia Institute of Technology to go home and either flip a coin 200 times and record the results, or merely pretend to flip a coin and fake 200
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 Food Counts (MatheMUSEments!)  Ivars Peterson (Math Muse for Kids)
You tear open a package of M&M's chocolate candies. Fiftyseven little candies spill out. You notice that certain colors are more common than others. You might count 7 brown, 17 red, 18 yellow, 6 green, 5 orange, and 4 blue candies in this bag. Does
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 Forget 'Back to Basics.' It's Time for 'Forward to (the New) Basics.'  Keith Devlin (Devlin's Angle)
What is the best way to prepare our children for life in the twentyfirst century? The first thing to realize is that the nation needs only 3 or 4 percent of the population to be highly skilled in mathematics. Of the remainder, hardly any will ever need
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 Formulas for fairness  Ivars Peterson  Science News Online
Applying the math of cake cutting to conflict resolution: researchers turn to the mathematics of cake cutting and point allocation to develop improved methods for people to use to settle disputes.
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 Forum of Geometric Constructor  Yasuyuki Iijima, Aichi University of Education, Japan
A dynamic geometry environment for DOS and Windows used in Japan  see the Tsukuba Journal of Educational Study in Mathematics.
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 Foundations of Greek Geometry  Michael Tirabassi; Tufts University
A linked historical essay outlining the contributions of Thales, Pythagoras, and Plato to the study of geometry.
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 The Four Color Theorem  Robertson, Sanders, Seymour, Thomas; Georgia Tech
A brief summary of a new proof of the Four Color Theorem, with a fourcoloring algorithm found by Neil Robertson, Daniel P. Sanders, Paul Seymour and Robin Thomas, illustrated using a map of the United States. Contents: History; Why a new proof?; Outline
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 The Four Colour Theorem  MacTutor Math History Archives
Linked essay describing work on the theorem from its posing in 1852 through its solution in 1976, with two other web sites and 9 references (books/articles).
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 Four Construction Problems  Cut the Knot!, Alexander Bogomolny
Construct a triangle, given the three points in the plane that are the outer vertices of equilateral triangles constructed outward on the sides of the desired triangle. Construct a triangle, given the three points in the plane that are the centers of
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 Four Corners, Four Faces (MatheMUSEments!)  Ivars Peterson (Math Muse for Kids)
To Arthur Silverman, a sculptor in New Orleans, tetrahedrons, or triangular pyramids, are very special. He's been creating sculptures based on the tetrahedron for more than 20 years. You might see examples on display in plazas and office buildings in
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 Fractal Clouds  Robert F. Cahalan
Information on the fantastic variety of cloud forms and structures, and their implications for climate. Albedo (abstract of a short technical paper for downloading); authors of papers on fractal clouds; cloud types and associated geometry, processes,
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 Fractal Dimension  Robert L. Devaney; Dept. of Mathematics, Boston University
A section from the paper "Chaos in the Classroom," describing the concept of fractal dimension, using the Sierpinski triangle as an example.
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 Fractal Expressionism  Richard Taylor, Adam P. Micolich and David Jonas
Can science further our understanding of art? An article discussing the connection between fractals and Jackson Pollock's paintings.
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 Fractal Geometry  Introduction  Harold Brochmann
Links to a series of articles exploring: The notion of dimension; The notion of selfsimilarity; The length of a coastline; The Sierpinsky Carpet; The Mandelbrot Set  the icon of chaos; and Applications of Fractal Geometry.
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 Fractal Metaphysics Home Page  Christopher Sunami
A clearinghouse of (serious) materials on the web focused on the relationship between fractal geometry, chaos theory, and metaphysics. Essays, related sites, galleries, and recommended reading.
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 Fractal Music: Research, Publications, and Compositions by Harlan Brothers  Harlan J. Brothers
In 2003, Benoit Mandelbrot suggested to Harlan Brothers that he undertake a rigorous treatment of the subject of fractal music. Here is an overview of Brothers’ subsequent research, including papers, definitions, and examples.
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 Fractal Past, Fractal Future  Ivars Peterson  Science News Online
Fractals have invaded the popular imagination. Calendars, computer screens, and books feature vivid, phantasmagorical images of weirdly branched, wildly swirling structures. Some fractal history, and an essay about the geometric complexity of natural
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 Fractions, Cycles, and Time  Ivars Peterson (MathTrek)
In ancient times, people had to confront awkward numbers in astronomical contexts when they compared the motions of the sun and moon. The unfailing, daily passages of the sun across the sky and the corresponding movements of the stars at night represented
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 Fragments of the Past  Ivars Peterson (MathLand)
Historians of mathematics now generally agree that scholars in China, India, and the Islamic world produced remarkably sophisticated mathematics between the fifth and the fifteenth centuries. However, most would probably still argue that Europeans in
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 Frank Garvan
Frank Garvan is a mathematics professor at the University of Florida. His combinatorics research articles are available for download in PostScript and .dvi formats. A Maple package for qseries may also be downloaded. One may also find the syllabi
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 Franklin's Mathematics  Paul C. Pasles, Villanova University
From the author of Benjamin Franklin's Numbers: articles on the founding father's "littleknown role as a mathematician," and examples of Franklin's magic squares and magic circle.
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 Frank's World: Math on Cable  Ivars Peterson (MathLand)
"Hi, you're live on Math Chat!" The question from the caller sounds pretty tough: "Three salesmen visit a shop. One comes every four days, another every five days, and the third every 10 days. If all of them were at the shop today, how long would you
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 Freaky Links  Cut the Knot!, Alexander Bogomolny with William A. McWorter Jr.
Mathematics impinges on itself and the sciences in unexpected and bizarre ways. This page is a small illustration of this phenomenon, taking up recreational mathematics and the game of Chinese checkers, the strategy game called Hex, quasigroups, and block
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 Frederic Chyzak
Frederic Chyzak researches combinatorics and computer algebra, specifically holonomic functions. His thesis and other articles are available as abstracts and as PostScript files. Slides for a series of talks on holonomic functions and computer algebra
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 Fred Rickey's Home Page  Fred Rickey, Department of Mathematical Sciences, United States Military Academy
Information about courses Rickey has taught about the history of mathematics. Historical notes for calculus teachers cover Torricelli's trumpet, Clepsydra, the history of the brachistochrone, the bridge and the catenary, Cauchy's famous wrong proof, and
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 Fritz Dooley's Mancala Center  Fritz Dooley
Game theory analysis of the board game Mancala, written as a research paper for Harvard Business School.
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 From Constructivism to Active Learning  Siegfried M. Holzer
Summary of an essay defining constructivism and outlining its implications for education: active learning, authentic activities, students' views, authentic assessment, and innovative curriculum. Published in the hypermedia publication of SUCCEED: The
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 From Counting to Writing  Ivars Peterson (MathLand)
Abstract numbers are the product of a long cultural evolution. They also apparently played a crucial role in the development of writing in the Middle East. Indeed, numbers came before letters, contends archaeologist Denise SchmandtBesserat of the University
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 From Galaxies to Electrons (Math Chat)  Frank Morgan, MAA Online
What are the largest and smallest objects anyone has ever seen with the naked eye? heard? felt? (For the first part, the whole object must be seen, though not necessarily in full detail or from every side.)
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 From Lewis Carroll to Archimedes  Cut the Knot!, Alexander Bogomolny
On March 29, 1879, Vanity Fair began offering its subscribers a new weekly puzzle invented by Lewis Carroll... on March 17, 1953 Frank Gray, a research scientist at Bell Labs, filed patent no. 2632058, for the Gray code encoding
the vacuum tube. An ndigit
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 From Linear to Nonlinear Optimization: The Missing Chapter  Prof. Hossein Arsham; University of Baltimore
This presentation in the form of a research paper provides a general solution algorithm for a large class of problems with linear constraints and a continuous objective function. A hypertext companion site for Applied Management Science: Making Good Decisions.
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 From Number to Formula  Ivars Peterson (MathLand)
Given the number 1.6180339887, how can you find out whether this particular number is special in some way, as the output of a specific formula or the value of a familiar mathematical function? Do you have a favorite number: Pi, the golden ratio, e (Napier's
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 From Starting Line to Ancient Angle  Ivars Peterson (MathTrek)
The ruins of ancient Corinth include a stadium that featured several courses for foot races. In 1980, archaeologists found and excavated the curved starting line for one such racecourse, dating from about 500 B.C. Curved to allow a staggered start, the
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 Frustration Solitaire  Doyle, Grinstead, Snell
A paper on the rankderangement problem, which asks for the number of permutations of a deck of cards such that each card is replaced by a card of a different rank, and which arises when computing the probability of winning the game of 'frustration solitaire'.
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 Fun and Games in Nevada  Ivars Peterson (Mathland)
...in the hotel's casino, the sights and sounds were... the glare of neon lights, the jangle of coins erupting from slot machines, the clink of chips at blackjack tables, and the mutter of avid gamblers testing their luck. It was the middle of April,
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 The Fundamental Theorem of Algebra  MacTutor Math History Archives
Linked essay covering from Cardan in the 1500's to the Euler and Argand proofs through the 1800's, with 8 references (books/articles).
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 The Funnel Cosmology  Vladimir Trifonov; American Mathematical Society (AMS)
Haar Measure and Geodesics in Funnel Cosmology  direct computation of the geodesics in the funnel model. Cosmological shape, dark matter, accelerating expansion, etc.
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 Fun with Probability! The Probable Pen in the Cereal Box  Michael Cornell; College of Education, Univ. of Illinois at UrbanaChampaign
A cooperative classroom project for grades K9. Over a threeweek period (24 April  10 May 1996), project participants jointly calculated the expected value of a simple probability problem via experimentation. This joint effort among interested schools
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 The Fun Works  Education Development Center, Inc. (EDC)
A career exploration digital library for middle school youth (ages 1114 yrs), designed to capture imaginations and direct youth toward careers in science, technology, engineering and mathematics (STEM). It places particular emphasis on engaging youth
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 Future of Leap Seconds  Steve Allen
Allen, an astronomer, discusses leap years, leap seconds  and how to avoid their use when redefining Coordinated Universal Time (UTC): three pictures that show the situation; elapsed time (JavaScript that shows POSIX's inability to provide an interface
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