 3D Drawing and Geometry  Cathi Sanders
A Math Forum Summer 1998 Institute project that uses examples of paintings, architecture, etc. to analyze different types of 3D drawings, and teaches students how to create them. Careers in 3D drawing that use these techniques, from architecture to movies, are also illustrated. Types include isometric, oblique, and perspective drawings. A drawing project for students is outlined and submissions are invited.
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 Adventures in Statistics  Scavo, Petraroja
A Web unit preprint of a paper by teachers Tom Scavo and Byron Petraroja that describes a mathematics project involving fifth grade students and the area of classrooms, including measurement, graphing, computation, data analysis, and presentation of results; to appear in "Teaching Children Mathematics".
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 Algebra  Fun with Calendars  Cynthia Lanius
Take any calendar. Tell a friend to choose 4 days that form a square. If your friend tells you only the sum of the four days, you can tell her what the four days are. How does the puzzle work? Includes a extension page for designing your own puzzle, teachers notes, and links to calendar pages on the Web. Mathematics topics: assigning variables, solving simple linear equations, factoring.
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 Algebraic Factoring  Suzanne Alejandre
A Math Forum Web Unit. Vocabulary, objectives, materials. Students use algebra tiles to explore algebraic factoring.
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 Bricks Activity  Suzanne Alejandre
A classroom activity (also called the Masonry Problem; a variation on polyominoes) aligned to the NCTM and California Standards, to be explored through manipulatives (dominoes). Students explore different possibilities of making brick walls with and without fault lines, using diagram, process, and solution in their problem solving. A teacher lesson plan is provided.
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 Chameleon Graphing: Lines and Slope  Ursula Whitcher
A Web unit for middle school and early high school students, in which Joan the Chameleon introduces and explores lines and slope, to accompany a unit for elementary and middle school students on The Coordinate Plane. See also Whitcher's unit on Plane History.
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 Classification of Patterns  BrownHerbst, Donnelly, Stratton, AndersonNielsen; The Geometry Center
Materials about symmetry and classification of repeating patterns for students in grades 710 using wallpaper patterns, to be used as either an introduction or a review: a classroomready source of information. A final project created for Math 5337, Technology in the Geometry Classroom, at the Geometry Center, University of Minnesota.
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 Crystals: Crystallography & Systems  Suzanne Alejandre
A Math Forum Web Unit. Students studying polyhedra enjoy seeing the structures as they occur in the real world. Crystalline structures can be categorized into seven crystal systems: see photographs of beautiful crystals or make paper models of crystals by printing out nets of crystals and constructing the models. CrystalMaker software gives students experience with ballandstick, spacefilling, wireframe, stick, dot surface cloud, and polyhedral models of crystals. Also Java applets that show some of the major forms for the hexoctahedralclass (symmetry 4/m3bar2/m) of crystals; and links to Web sites about crystals.
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 Designs with Circles  Suzanne Alejandre
A Math Forum Web Unit. Includes some background on circles in Islamic cultures, and an activity exploring the geometry involved in some circle designs.
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 Dominoes Activity  Suzanne Alejandre
A classroom activity (similar to Pentomino + 1 = Hexomino; a variation on polyominoes) aligned to the NCTM and California Standards, to be explored through manipulatives (paper dominoes). Students explore whether it is possible to cover a 6x5 grid with 15 dominoes, then build a variety of "brick walls" and create a brick wall catalogue. A simple freeware program (Macintosh only) by Kurt Kaufman can be used very easily to model this problem. Links to related pentomino and Fibonacci sites on the Web and a teacher lesson plan are provided.
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 Dueling Pinwheels  A Geometer's Sketchpad Activity  Cynthia Lanius
Construction directions for rotations, translation, and reflection, with questions to explore and a demonstration sketch to download.
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 EggMath  Bradlow, Sullivan, Levy, UIUC
A collection of modules (including many interactive applets) for K12 classrooms, based on the theme of eggs. Topics include: the shape of an egg (addressing symmetry and crosssection, surfaces of revolution, pin and string construction of ellipses, Cartesian and Cassini Ovals), the White/Yolk Theorem (how any two regions of the plane can be equally divided  a specific case of the BorsukUlam Theorem, with a proof of the Theorem included), spherical geometry, and embryo calculus (exponential growth and the number e). Each module offers interactive Java components and additional references. Part of the Chickscope project at the Beckman Institute. Also available at http://new.math.uiuc.edu/eggmath/.
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 Fractals  Cynthia Lanius
This lesson plan for exploring fractals is designed so 4th through 8th grade students can work independently and be assessed innovatively. It conforms to the 1989 NCTM standards, and provides links to other fractal sites. Contents: Why study fractals? Making fractals: Sierpinski Triangle, Sierpinski Meets Pascal, Jurassic Park Fractal, Koch Snowflake. Fractal Properties: Selfsimilarity, Fractional dimension, Formation by iteration. TeachertoTeacher notes; Fractals on the Web.
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 Fractals: Definition & Activities  Suzanne Alejandre
A Math Forum Web Unit. Includes definitions and descriptions of fractals, links to pages on the Sierpinski triangle, the Koch edge, the Peano curve, the Lorenz attractor, and the Dragon curve; and more links to fractal sites on the Web.
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 Games on Graphs (MegaMath)  Nancy Casey; Los Alamos National Laboratory
Students use algebraic methods to explore, model, and describe patterns and functions involving numbers, shapes, data, and graphs in problemsolving situations and communicate the reasoning used in solving these problems. Graphs, stories and games provide scenarios for games that student can play on graphs. Also Three for the Money: The Degree/Diameter Problem, an unsolved problem for students to work on, and other games that can help students increase the range of possibility for games that they can invent on graphs. Big Ideas and Key Concepts include pages on Graphs; Properties of mathematical objects; Modeling; and Abstraction.
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 Geometric Factoring  Suzanne Alejandre
A Math Forum Web Unit. Understanding factoring through geometry: students work cooperatively to display a numeral as the area of a rectangle, and make as many rectangular arrangements as possible for each numeral given.
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 The Golden Ratio  Blacker, Polanski, Schwach; The Geometry Center
Introduction to the Golden Ratio and Fibonacci Sequence. Instead of simply supplying definitions and asking the student to engage in mindless practice, students work through several activities to discover the applications of the Golden Ratio and Fibonacci Sequence. With Geometer's Sketchpad activities. A final project created for Math 5337, Technology in the Geometry Classroom, at the Geometry Center, University of Minnesota.
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 The Hand Squeeze: A Data Collection and Analysis Class Experiment  Cynthia Lanius
An experiment: measuring the amount of time that it takes for a hand squeeze to pass around a circle. Record, graph, and analyze the data, and make predictions about the time it would take for more people/greater distances.
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 Hotel Infinity (MegaMath)  Nancy Casey; Los Alamos National Laboratory
Will a hotel with an infinite number of rooms always have a vacancy? Students develop number sense, use numbers and number relationships in problemsolving situations, and communicate the reasoning used in solving these problems. Key concepts include Infinity, Transfinite arithmetic, and Onetoone correspondence.
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 The Hot Tub: Interpreting Graphs  Cynthia Lanius
Questions to answer about a graph... Notice the connection between the slope of the lines and the rate of change of the water depth. On what segments is the slope positive, and the water depth increasing? negative, and the water depth decreasing? On what segments is the slope 0, and the water depth is constant?
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 Just a Usual Day at Unusual School (MegaMath)  Nancy Casey; Los Alamos National Laboratory
Students perform a play that takes place in a school where some of the students always lie and the rest always tell the truth. Terry, the protagonist, is trying to find out which students are which, but at the beginning, there is no way of knowing whom to believe. Can you figure out whom to believe in this play? With accompanying lesson plan and discussion questions.
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 Locker Problem  Suzanne Alejandre
A classroom activity (also called 1000 Lockers) aligned to the NCTM and California Standards, to be explored through the use of manipulatives and a ClarisWorks spreadsheet. Students then look for patterns and write the answer algebraically. The problem: imagine you are at a school that still has student lockers. There are 1000 lockers, all shut and unlocked, and 1000 students. Suppose the first student goes along the row and opens every other locker. The second student then goes along and shuts every other locker beginning with number 2. The third student changes the state of every third locker beginning with number 3. (If the locker is open the student shuts it, and if the locker is closed the student opens it.) The fourth student changes the state of every fourth locker beginning with number 4. Imagine that this continues until the thousand students have followed the pattern with the thousand lockers. At the end, which lockers will be open and which will be closed? Why? A teacher lesson plan is provided.
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 Machines That Eat Your Words (MegaMath)  Nancy Casey; Los Alamos National Laboratory
An introduction to the concept of finite state machines, and how they are used to design computer systems that will recognize patterns. A finite state machine is an imaginary (or abstract) machine that is used to study and design systems that recognize and identify patterns. The idea of a finite state machine is a powerful one that has many applications in computer science. Mathematics as Problem Solving, Mathematics as Communication, Mathematics as Reasoning, and Mathematical Connections are critical items throughout the NCTM Standards. They appear at every level because they form the core of what it means to do mathematics.
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 Magic Squares: Math, History, Geography  Suzanne Alejandre
Designed primarily for middle school and older elementary school students, this Math Forum Web unit includes classroom activities for four different squares: Lo Shu, Sator, Dürer, and Franklin, three of them demonstrating increasing complexity using numbers, and one made up of letters, with number and symmetry questions and links and suggestions for history / geography / writing activities for teachers interested in interdisciplinary work. Includes directions for building magic squares, definitions, discussion of some special properties of magic squares, a Java applet, and links to other magic square Web sites.
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 Mathematics of Cartography  Cynthia Lanius
A map is a set of points, lines, and areas all defined both by position with reference to a coordinate system and by nonspatial attributes. These pages discuss how maps are used, give examples of different kinds of maps, and cover map history and math topics  lines, points, areas, coordinates, etc., in particular scale, coordinate systems, and projection. Also Problems, Resources, Careers in mapmaking, Teachers' Notes, and References.
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 The Million $ Mission  Cynthia Lanius
You have your choice of two payment options on your new job: 1. One cent on the first day, two cents on the second day, and double your salary every day thereafter for the thirty days; or 2. Exactly $1,000,000. (That's one million dollars!) What's the best choice? Includes pages on exponential growth and patterns, links to exponentials on the Web, questions, and teachers notes.
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 The Most Colorful Math of All (MegaMath)  Nancy Casey; Los Alamos National Laboratory
Coloring is a profound mathematical topic with multimilliondollar industrial applications. The problem presented here has been of interest to mathematicians for over a hundred years. With a few crayons or markers and some handdrawn maps, children can quickly find themselves grappling with the fourcolor map problem. Activities, Background Information, Evaluation.
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 Mozart's Musikalisches Würfelspiel: A Musical Dice Game for Composing a Minuet  John Chuang
During his life, Mozart wrote the measures and instructions for a musical
composition dice game that cuts and pastes prewritten measures of music together to create a minuet. Compose a minuet online in one of three ways: CGI generates all random numbers, you pick the "random" numbers, or make a fractal with Chris Seidel's Fractal Generator (The Mandelbrot Collaboration). Links to other Mozartian sites and listening to MIDI files.
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 No Matter What Shape Your Fractions Are In  Cynthia Lanius
Students explore geometric models of fractions and discover relations among them, reinforcing their knowledge of fractions by using online pattern block activities. The lessons are designed for students to work independently or with guidance from the teacher. Lessons should be printed so students can draw and color the appropriate shapes. Teachers notes are included.
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 Pascal's Triangle  Math Forum/USI
A Web unit designed to support workshops given by the Math Forum for the Urban Systemic Initiative (Philadelphia and San Diego). Read about the history of Pascal's triangle and learn to construct it; view illustrations of number patterns to be discovered; carry out interactive investigations in JavaScript or the Geometer's Sketchpad, and explore this famous triangle through lesson plans that feature questions, answers, discussion, and student worksheets.
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 PlaneMath  InfoUse, in cooperation with NASA
Materials for elementary school students about math and aeronautics, designed to stimulate and motivate students with physical disabilities in grades 47 to pursue aeronauticsrelated careers via the development and delivery of accessible math education materials on the Internet. Recognizing that math curricula for students in these grades is most often built around the manipulation of tools such as pencils, compasses, and rulers, the designers of this site have endeavored to teach the same concepts without relying on the physical acuity of the student. Activities involve finding the shortest path between two cities or how many people can board your plane, flying a herd of buffalo to the prairies, learning to fly a rescue helicopter and how planes lift, knowing when an overcast sky is really overcast, flying a kite, and planning a flight around the country. Teachers are invited to register their classes.
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 Polyhedra in the Classroom  Suzanne Alejandre
Middle School student activities to pursue with a computer in the classroom. Introduction to Polyhedra; paper nets to print out and fold; Kaleidotile; Buckyballs; Crystals (paper nets, systems); Cube coloring problems; links to polyhedra on the Web.
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 Rectangle Pattern Challenges  Cynthia Lanius
Examine different stges of rectangle patterns, and describe what you must do to get from one to the next. Observe the designs looking for patterns. Use the symmetry of the design to help you count. Organize your information into a table. On square grid paper create your own design, showing at least 3 stages. It must have at least two lines of symmetry, and it must follow a regular numerical growth pattern. On a separate sheet of paper, fill in the calculations in a table like the one shown. Teachers notes are included.
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 River Crossing (Math Exploration Quilt)  Rik Littlefield; Hanford School
You want to cross a river to reach a point exactly opposite where you are currently standing. Explore this problem stepbystep, encountering the following basic ideas: 1) Pythagorean theorem; 2) time = distance / speed; 3) distance = time * speed; 4) sums and differences of distances; and 5) the arcsine function for right triangles (which we didn't really need to solve the problem, just to get the angle expressed in a familiar way).
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 Stressed Out: Slope as Rate of Change  Cynthia Lanius
It's the night of the big game. You're in the locker room. The coach is pumping the team up. "Now, I know you people are nervous. That's okay, in fact, that's what we want. You're going to perform better on the court (stage) if you're a little nervous." Does the graph shown confirm what the coach told you? Write a statement that describes performance as stress increases... This lesson introduces basic knowledge and skills important to a study of functions in algebra and lays the groundwork for calculus.
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 Studying Polyhedra  Suzanne Alejandre
What is a polyhedron? A definition and a Java applet to help in exploring the five regular polyhedra to find how many faces and vertices each has, and what polygons make up the faces. Also links to a page of information about buckyballs, stories written by students after studying polyhedra, other sites with information about regular polyhedra, and other sites with information about Greece and Greek mathematicians.
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 TableTop Earthquakes  John C. Lahr
A demonstration of seismology for middleschool teachers and students that can be used to augment lessons in earth science (faulting, elastic rebound, plate motions), physics (forms of energy, elasticity, friction, magnetism, waves), math (graphing, logarithms, probability), social studies (hazard mitigation), and geography (global distribution of earthquakes).
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 Tangrams  Tom Scavo
Tangrams, a puzzle that helps develop spatialvisualization skills, may also be used to introduce or reinforce geometric concepts such as congruence, similarity, symmetry, etc. This unit for grades 4 through 6 uses tangrams to compute the area of polygons without formulas, introducing the terms congruent and similar. Contents: Constructing Your Own Set of Tangrams; The Area of Tangram Pieces; More Tangram Activities. Links to other tangram resources on the Web are also provided.
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 Tessellation Tutorials  Suzanne Alejandre
A series of tutorials that teach students how to tessellate (somewhat in the style of the art of M.C. Escher) using HyperCard for black and white and/or HyperStudio for color, ClarisWorks, LogoWriter, templates, or simple straightedge and compass. The tessellation lessons include units incorporating rotations and glide reflections, a section called "Where's the Math" that elaborates on some underlying geometric principles, comments contributed by others, and samples of student work. See, in particular, What Is a Tessellation?
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 Three for the Money: The Degree/Diameter Problem (MegaMath)  Nancy Casey; Los Alamos National Laboratory
Students can understand and work on an unsolved problem in mathematics. There is a good chance that this problem can be solved by someone who spends enough time experimenting with it. The only skills required to work on it are the ability to draw dots and connect them with lines, and the understanding of four ideas related to graphs: degree, diameter, planarity, and size. With ideas for discussion.
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 Traffic Jam Activity  Suzanne Alejandre
A classroom activity (also called Hop, Skip, Jump) aligned to the NCTM and California Standards, to be explored through large movement experience, manipulatives, and an interactive Java applet. Students then revisit the activity, look for patterns, and write the answer algebraically. The activity: there are seven stepping stones and six people. On the three lefthand stones, facing the center, stand three of the people. The other three people stand on the three righthand stones, also facing the center. The center stone is not occupied. Everyone must move so that the people originally standing on the righthand stepping stones are on the lefthand stones, and those originally standing on the lefthand stepping stones are on the righthand stones, with the center stone again unoccupied. A teacher lesson plan is provided.
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 The Twelve Days of Christmas and Pascal's Triangle  Judy Brown, DIMACS and The Math Forum
A lesson in which, using Pascal's triangle, students find the number of items given each day in the song, "The 12 Days of Christmas."
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 The Twelve Days of Christmas  Judy Brown, DIMACS and The Math Forum
A lesson in which students find the number of items given each day in the song, "The 12 Days of Christmas." This project is designed to be used as a warmup activity during the 12 days preceding Christmas.
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 Untangling the Mathematics of Knots (MegaMath)  Nancy Casey; Los Alamos National Laboratory
Fundamentals of knot theory for a wide range of levels  a variety of activities for exploring knots made from pieces of rope. Students can make and verify observations about knots, classify them, combine them, and find ways to determine if two knots are alike. The activities outlined here can be combined to form a single lesson about mathematical knots, or a larger investigative unit that extends over a longer period of time. Key concepts include knot theory, topology, operations, and proof.
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