 The Three Jugs Problem  Cut the Knot!, Alexander Bogomolny
Two friends who have an eightquart jug of water wish to share it evenly. They also have two empty jars, one holding five quarts, the other three. How can they each measure exactly 4 quarts of water? ... The description of the 3 Jugs problem as a triple
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 TI89 Calculus Calculator Programs
TI89 calculator programs for sale. Enter your variables and see answers worked out step by step: a and b vectors, acceleration, area of parallelogram, component of a direction u, cos(a and b), cross product, curl, derivative, divergence of vector field,
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 Tiling with Polyominoes  Ivars Peterson (MathTrek)
"Mathematicians have proved that the general question of whether it's possible to cover the plane with identical copies of a given finite set of tiles is, in principle, computationally undecidable. In other words, there's no cookbook recipe or handbook
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 Tim's Triangular Page  Alex Sakharov
Information, definitions, and illustrations of triangles, their parts, and congruency. Includes a few min/max problems relating to triangles at the end.
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 Tips & Tricks to Gothic Geometry  Joe Chiffriller; NewYorkCarver.com
The geometry of gothic cathedrals: how to construct a trifoil design, an ogee arch, and a rose window. With links to cathedral and castle tours.
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 Totally Tessellated  Bhushan, Kay, & Williams
A comprehensive introduction to tessellations and tilings  the basic underlying mathematics and examples of tessellations in real life. The highly illustrated, printerfriendly site includes history (tessellations in mathematics and science, decorative
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 Toying with Reality: Java Applets and VRML  Paul Flavin
Virtual Reality: Modeling with Math, and a little Java. Applets to manipulate: The Pythagorean Theorem; Equation Grapher: "You type, I Graph"; Graphing Vector Calculator; Java 3d Visualizer; Virtual Rubik's Cube; Raytracing applet; Java Calculator; Play
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 Traffic Signs  Richard C. Moeur
Graphics of the most commonly used traffic signs in the United States, useful for challenging students to find symmetrical signs (in addition to polygons).
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 Traveling Salesman's Sketchpad  Selim Tezel
The Traveling Salesman's Sketchpad lets you investigate the Euclidean Traveling Salesman Problem (TSP) using an interface inspired by The Geometer's Sketchpad®. The Traveling Salesman's Sketchpad facilitates algorithm and data visualization, conjecture
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 Travel Triangles (SMILE)  Sarah Barrett, Mars Hill School
A middle and high school geometry lesson designed to teach students to review and demonstrate an understanding of the three kinds of angles (by degrees). From the Geometry and Measurement section of a collection of almost 200 single concept lessons by
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 TrianCal  Jesús Sánchez
This free dynamic calculator accepts any combination of triangle sidelengths, heights, angles, areas, or perimeters. It then solves for the remaining values and measures, further offering a GeoGebra sketch. Watch TrianCal in action, with subtitles in
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 Triangle Calculator  Peter Chrenko
Select SideSideSide (SSS), SideAngleSide (SAS), AngleSideAngle (ASA), or SideSideAngle (SSA) congruence, then enter three known sidelengths or angles, and this JavaScript returns the triangle's remaining measurements. This calculator also uniquely
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 Triangle Congruence  Math Forum, Ask Dr. Math Common Question
A selection of answers to questions about triangle congruences, such as "Why can't the SSA Theorem be used to prove congruence?" and "When should we use Congruent Parts Congruent Triangles Congruent (CPCTC), and how does it prove anything?"
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 Triangle: Proof  Michal Wojcik
A proof that the sum of the angles in a triangle is 180 degrees. This page is written in HTML with two graphical images.
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 Triangle Puzzle  Evelyn Sander, Geometry Forum Archives, The Geometry Center
Pick an arbitrary point in the interior of an equilateral triangle. Now go halfway towards one of the three vertices (you get to pick which vertex). Now you are at a new point in the triangle. Again go halfway towards any vertex. Continue this process.
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 Triangles and Simplices (The Geometry Junkyard)  David Eppstein, Theory Group, ICS, UC Irvine
An extensive annotated list of links to material on triangles and their higherdimensional generalizations (simplices), with a particular interest in partitioning regions into triangles, tetrahedra, or higher dimensional simplices, for various applications
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 Triangle with one 90degree Angle  NASA Lewis Learning Technologies Project
A triangle with one 90 degree angle is always: an equilateral triangle; an acute triangle; an obtuse triangle; or, a right triangle? A detailed answer is provided. From NASA's 9th Grade Math Proficiency Test.
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 Triangulation (SMILE)  Ann M. W. Brandon, Oak Forest High School
A lesson designed to teach students the basics of how to perform and calculate
triangulation. From the Practical and Applied Math section of a collection of almost 200 single concept lessons by the Science and Mathematics Initiative for Learning Enhancement.
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 A Tribute to Miro  Kathy Sederquist, Namaste Art Center
Students learn how contemporary artists work by doing a painting in the style of the Catalan artist Joan Miró. The basic lines in drawing, including vertical, horizontal, diagonal, and curved, are reviewed through experimentation.
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 Trigonometry  Technical Tutoring
Site provides an introduction to trigonometry, includes illustrative examples and exercises in basic trigonometry as well as a summary of the important basic trig identities and formulae.
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 Trisecting an Angle  MacTutor Math History Archives
Linked essay tracing the history of the classical Greek problem of trisecting an
arbitrary angle using for the construction only ruler and compass (which is
impossible)  but, failing that, finding some other method.
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 The Tunnel of Samos  Project MATHEMATICS!, California Institute of Technology
A videotapeandworkbook module that explores a basic topic in high school mathematics in ways that cannot be done at the chalkboard or in a textbook. This video describes a remarkable engineering work of ancient times: excavating a onekilometer tunnel
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 Turtle Tracks  Ivars Peterson (MathTrek)
One way to describe a geometric figure is in terms of the path generated by a moving point. Using the computer language LOGO, children can produce a list of commands to govern the motion of a "turtle" and trace out a geometric track on the computer screen.
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 The Tutor House: math  Adrianne Meldrum
Tutoring elementary and middle school students, Meldrum would often "hunt the Internet for tips and helps on teaching concepts to my students. I find great stuff, but I wanted somewhere central that other tutors like me could find them...." Every Wednesday
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 UMIGO  UMIGO, Inc.
Games, music videos, and "appisodes" for early math literacy and skills: "Let's All Umiball," "Scootie Doo," "Dr. Screwloose Strikes Again," "Double Trouble," "The Pizza Party," "Nobody Rides the Soakster," and "Who Woke the Mummy?" and "Who Stinks Greater?"
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 Uncle Bob's Puzzles  Robert & Claire Mead, Haverhill, NH, USA
Math and logic problems (no solutions); if you have questions, want a hint, or want to submit a solution you're invited to email the author. Asterisks indicate greater difficulty: Blind Cowboy; A Game of Catch; The Next Three; Grandpa's Birthday; Square
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 Using Sketchpad: Triangles  Mel Thornton, University of Nebraska
A sketch about relations between the areas of certain triangles, and another about how to perform translations using only a compass. Notes on Mel Thornton's presentation at the Sketchpad User's Group Meeting in Cincinatti in January 1994.
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 Vagn Lundsgaard Hansen
A personal home page with links to: a mathematical story "I am the greatest," solving and proving/explaining the isoperimetric problem for quadrilaterals; Mathematics and the Public  some experiences with the popularization of mathematics; and Mathematics
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 Vertices and Radius  Patrick Gunderson
Dynamically draws all chords of regular polygons up to 128 sides. Adjust the sliders to change the number of polygon vertices or size of the radius.
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 Virtual Polyhedra  George W. Hart
A growing collection of over 1000 virtual reality polyhedra to explore, complementing Hart's Pavilion of Polyhedreality. Includes instructions for building paper models of polyhedra including modular origami, with ideas for classroom use. Each of the
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 Visualizing An Infinite Series  Cynthia Lanius
This lesson uses a trapezoid successively divided in 4ths as a visual representation of an infinite geometric series, a mathematical concept that is often only treated symbolically. Explorations are outlined for investigating a number of series, and links
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 VisualMath  G. Fonthal, VisualPhysics
Math shareware programs that feature visual and interactive techniques under Windows 95 may be downloaded from this site: for numbers (real, complex, binary); set theory (Venn diagrams); geometry (shapes and areas, 3D Geometry); algebra (equations, polynomials,
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 WallaceBolyaiGerwien Theorem  Interactive Mathematics Miscellany and Puzzles, Alexander Bogomolny
According to the WallaceBolyaiGerwien theorem, all simple polygons of equal area are equidecomposable by dissection. The site offers a proof of the theorem and several dynamic illustrations of equidecomposition in special cases.
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 WAVE: Web Access Virtual Education  University Laboratory High School; University of Illinois at UrbanaChampaign
The Illinois WAVE Project provides Mathematica Tutorial notebooks categorized as pertaining to Algebra, Applications, Calculus, Conics, Geometry, Precalculus, Statistics, or Trigonometry, for use in the high school mathematics curriculum. From the University
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 WebGL Chorded Polygons
Dynamically draws all chords of regular polygons up to 250 sides. Adjust the sliders to change the number of polygon vertices, to zoom in, and to pan around.
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 The Web Wizard's Math Challenge  J. Mooser
An ongoing Internet contest; register (free) to submit answers and score points. Problems are designed to be readily understood but not readily solved. Speed counts, because points are awarded based on the order in which contestants submit correct answers.
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 What Is Your Answer to That Question?  Cut the Knot!, Alexander Bogomolny, with Don Greenwell
The question  Why study mathematics?  appears in a multitude of guises and a plethora of variants... Mathematics is often compared to art. Like literature, mathematics has metaphor, ambiguity, paradox, and mystery. It has history. Mathematics has contributed
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 What's My Area? (SMILE)  Karen L. Mickel, William Carter Elementary School
An elementary lesson designed to teach students to cover an area with uniform tiles and count the number of tiles needed; recognize rectangles, triangles and squares; and review basic colors. From the Geometry and Measurement section of a collection of
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 Whistler Alley Mathematics  Paul Kunkel
Kunkel's Geometer's Sketchpad lessons convey a conceptual understanding, usually without rigorous proof, with questions and suggestions for extensions: Buffon's Needle (an old probability exercise); Chinese Handcuffs (questions with applications for geometry,
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 Width, Diameter, and Geometric Inequalities (The Geometry Junkyard)  David Eppstein, Theory Group, ICS, UC Irvine
An extensive annotated list of links to material on width, diameter, and geometric inequalities.
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 Wonders of Ancient Greek Mathematics  Timothy Reluga; Tufts University
An arbitrary collection of interesting solutions to geometric problems discovered and solved by the Greeks, limited to intuitive, elegant, or beautiful solutions and some that do not meet these requirements, serving as reminders of the level to which
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 Working with Fractions Using Pattern Blocks  Margo Lynn Mankus
The objective of What's My Number? is to name the size of an object given the size of 1 unit. Materials include pattern blocks cutouts, a pattern block grid, a pattern block Java applet, and links to other pattern block activities elsewhere on the web.
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 Writing to Develop Understanding: The Math Forum @ Drexel's PoWs  Suzanne Alejandre
What can we ask students about their problem solving that will help them the most? This article, originally published in the December, 2006, issue of the CMC ComMuniCator, journal of the California Mathematics Council, addresses that question by examining
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 wrotniak.net  J. Andrzej Wrotniak
Includes shareware and freeware programs for Windows written by Wrotniak: scientific and regular calculators, a spherical geometry calculator, a logic and strategy game, a statistics graphing program, and a simple program to compute the area of a polygon.
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