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17 November 1997 Vol. 2, No. 46 THE MATH FORUM INTERNET NEWS Modular Origami | Eye Opener Series | Implication & Denial THE MATHEMATICS OF PAPER-FOLDING: MODULAR ORIGAMI MOSTLY MODULAR ORIGAMI - VALERIE VANN http://people.delphi.com/vvann/index.html Modular origami is a branch of the art of paperfolding. Unlike regular origami, where a figure or sculpture such as a crane, frog, or airplane is made from a single sheet of paper without glue or cutting, modular origami figures are built up from multiple individually folded pieces, usually identical. Modular origami is often very geometric. Valerie Vann offers her own designs and developments, examples of other significant modular origami works, and favorite models by other modular origami creators. MODULAR ORIGAMI - JIM PLANK http://www.cs.utk.edu/~plank/plank/origami/origami.html Pictures of many origami polyhedra, with instructions for making the modules and the polyhedra, and putting the modules together: - polyhedrons with the penultimate module - the compound of 5 tetrahedrons - greater/lesser stellated dodecahedrons - the compound of 12 dodecahedrons ORIGAMI AND MATHEMATICS - VICTORIA BEATTY http://mahogany.lib.utexas.edu:1000/Exhibits/origami/math/ Origami and form, architecture and design: buildings, housewares, and furnishings; with a select list of related sites. \|/ For other ORIGAMI POLYHEDRA sites, see David Eppstein's GEOMETRY JUNKYARD: http://www.ics.uci.edu/~eppstein/junkyard/origami.html or use the MATH FORUM's Quick Search: http://mathforum.org/dumpgrepform.html -|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|- JAVA EYE OPENER SERIES - BOGOMOLNY http://www.cut-the-knot.com/pythagoras/tricky.html A collection of Java applets, all of which depend on a piece of Java code that controls two eyes, that illustrate and thereby help to solve or prove math problems such as 99 = 100 and: - Euclid's Proof of the Pythagorean Theorem - Construct an n-gon - The Disappearing Lines puzzle - The Sam Loyd's fifteen - The Sliders puzzle - The Lucky 7 puzzle - The Happy 8 puzzle. - The Blithe 12 puzzle. - The Binary Color Device - Analog gadgets - Wythoff's Nim - Changing Colors - Breaking Chocolate Bars - Calendar Magic - Squares and Circles - Diagonal Count - Flipping pancakes - Four Knights - Latin Squares - Marriage Problem From Alexander Bogomolny's Interactive Mathematics Miscellany and Puzzles: http://www.cut-the-knot.com/front.html -|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|- IMPLICATION AND DENIAL A conversation from MATHEDU, an unmoderated distribution list for discussing post-calculus teaching and learning of mathematics. http://mathforum.org/epigone/mathedu/zolgrendlerm Questions about student understandings of implication, ranging among causality, spreadsheets, Boolean functions, ambiguity, proof, how to communicate reasons why a result is true, how to elicit connection, mathematical language, and native modes of thinking - with sample problems, suggestions, and references for helping clarify the logic of x<y => x <=y. "One of the things that has surprised me in teaching beginning analysis... is the difficulty experienced by a number of students in accepting the statement: x<y => x <=y (x less than y implies x less than or equal to y). Working through this caused fierce arguments between some students. Some of them wanted the implication to be reversed... Has anyone had similar experiences? Is there any theoretical framework for understanding the real difficulty for students in this situation?" - David Epstein "If students in their work up to calculus have had no trouble with inequalities, it may be that they were operating symbolically and the difference was not apparent to them. ... Only when you start using explicit language does the difficulty crop up." - George Tintera "It is an annual event to find the class thinking an implication works the opposite way to the way I mean. One of the examples I habitually use is: P and Q are congruent triangles. P and Q are similar triangles. Which implies which?" - Robert P. Burn "...part of learning mathematics (at least for those who plan to become mathematicians) is learning to think in ways that are closer to mathematical language." - Brian M. Scott "...two things might help... The first is Johnson-Laird's notion of mental models (which are not necessarily images). What mental models do students have for implication? The second is that if students interpret implication as in a spreadsheet, then it's more like an injunction to act, rather than a logical statement..." - Gary E. Davis \|/ The Math Forum's Epigone software archives, threads, and searches the MATHEDU discussions: http://mathforum.org/epigone/mathedu -|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|- CHECK OUT OUR WEB SITE: The Math Forum http://mathforum.org/ Ask Dr. Math http://mathforum.org/dr.math/ Problem of the Week http://mathforum.org/geopow/ Internet Resources http://mathforum.org/~steve/ Join the Math Forum http://mathforum.org/join.forum.html Send comments to the Math Forum Internet Newsletter editors _o \o_ __| \ / |__ o _ o/ \o/ __|- __/ \__/o \o | o/ o/__/ /\ /| | \ \ / \ / \ /o\ / \ / \ / | / \ / \ |

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