 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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 Building an Icosahedron as a Class Project  Frederick J. Wicklin
A Teacher's Guide. This document describes how to build an icosahedron as a class project. The size is up to you. Stepbystep instructions are provided, as well as figures, diagrams, and templates.
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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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 Concurrency Points in a Triangle  Brakalova, Johnson, Scalzitti, Thornberry; The Geometry Center
An interactive Webbased exploration designed to facilitate the teaching of triangle concurrency, both as an integral part of a geometry curriculum and as enrichment material. These pages can be used by individual students as a computer exploration, as classroom demonstration material, or as an alternative to a more traditional investigation of triangle congruency topics. Includes notes to students and teachers, and instructions for using the Geometer's Sketchpad in the activities. 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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 Euclid's Geometry: History and Practice  Alex Pearson, The Episcopal Academy, Merion, Pennsylvania
A series of interdisciplinary lessons on Euclid's Elements, researched and written by a Classicist and hosted by the Math Forum. The material is organized into class work, short historical articles, assignments, essay questions, and a quiz.
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 Ferris Wheel  Jim King, University of Washington
Physical devices can be modeled using dynamic geometry. A vital tool for moving objects around in the model are the isometries, or distancepreserving transformations. This model of a Ferris wheel provides a good example. A stepbystep description of how to model a Ferris wheel using translations and animation in Sketchpad.
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 Folding Shapes: Are the Sides the Same?  Janna Moore
Folding Shapes gives students practice in identifying typical shapes (circles, squares, triangles, etc.), in the process of teaching about lines of symmetry. Students find symmetry in alphabet cutouts by folding them in half in various ways.
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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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 Introduction to Geometry for Primary Students  Varnelle Moore
A Web unit designed to introduce young children (K2) to beginning concepts in geometry: describing shapes, making patterns, building shapes, rotational symmetry, and line symmetry. Each lesson includes an interactive, manipulativebased project, technology, paper/pencil practice, and literature connections.
Helpful links and teacher support extension ideas are also provided.
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 Introduction to Symmetry for Primary Students  Varnelle Moore
A unit designed to give young children (K2) an introduction to symmetry, with lessons on describing tangrams, slides, turns, and flips. Each lesson includes: an interactive, manipulativebased project; an activity incorporating technology; pencil and paper practice; and literature connections. Helpful links and teacher support extension ideas are also provided.
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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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 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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 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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 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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 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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