 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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 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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 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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 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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 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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 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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 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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 Introduction to Fractions for Primary Students  Varnelle Moore
A Web unit designed to introduce young children (K2) to beginning concepts in fractions: equal parts, divide and shade, parts to whole, and writing fractions. Each lesson includes an interactive, manipulativebased project, technology, paper/pencil practice, and literature connections. Helpful links and teacher support extension ideas are also provided including alignment to NCTM Standards.
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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 Measurement for Primary Students  Varnelle Moore
A Web unit designed designed to engage young children in a series of activities where they measure and describe the attributes of integer bars: building houses, short vs. tall, designing bridges, measuring using integer bars. 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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 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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 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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 Place Value: K3  Andra McLeod
A "box full" of place value manipulatives for the classroom, with descripion, lesson ideas, and objectives listed for each of five types of manipulative.
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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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 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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 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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 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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