The Geometry Junkyard: Polyominoes Numerous links, sorted alphabetically.Category Science Math Recreations Polyominoes Pentomino projectof-the-month from the geometry Forum. List the pentominoes;fold them to form a cube; play a pentomino game. See http://www.ics.uci.edu/~eppstein/junkyard/polyomino.html
Extractions: Polyominoes and Other Animals Connected subsets of the square lattice tiling of the plane are called polyominoes . These are often classified by their number of squares, so e.g. a tetromino has four squares and a pentomino has five; this nomenclature is by analogy to the word "domino" (a shape formed by two connected squares, but unrelated in etymology to the roots for "two" or "square"). If a polyomino or a higher-dimensional collection of cubes forms a shape topologically equivalent to a ball, it is called an animal . A famous open problem asks whether any animal in three dimensions can be transformed into a single cube by adding and removing cubes, at each step remaining an animal (it is known that removal alone does not always work). Other related figures include polyiamonds (collections of equilateral triangles), polyabolos (collections of half-squares), and polyhexes (collections of regular hexagons). Anna's pentomino page . Anna Gardberg makes pentominoes out of sculpey and agate. Arranging six squares . This Geometry Forum problem of the week asks for the number of different hexominoes, and for how many of them can be folded into a cube. Blocking polyominos . R. M. Kurchan asks, for each k, what is the smallest polyomino such that k copies can form a "blocked" configuration in which no piece can be slid free of the others, but in which any subconfiguration is not blocked.
Extractions: Ask students how they can tell if two shapes are congruent. They may be encouraged to talk about it with a partner before sharing with the entire class. Ask students to create a shapes mural based on a book read to the class or on a class activity. Students could use a theme from another subject area. G-19 Geoboards and elastics offer students the creative transition between the concrete (hexagon attribute block) and the pictorial (picture of a hexagon). Geoboards can be used for developing a wide range of mathematical concepts such as shape, congruency, fractions, symmetry, and measurement. Encourage students to be creative.
The Geometry Junkyard: All Topics This page collects in one place all the entries in the geometry junkyard. Quadrilaterals in which the sides and diagonals form more rational angles with each other than one might expect. trisection, from the geometry forum archives. Anna's pentomino page. Anna Gardberg makes pentominoes out of sculpey http://www.math.ntnu.edu.tw/~jcchuan/all.html
Extractions: All Topics This page collects in one place all the entries in the geometry junkyard. Adventitious geometry . Quadrilaterals in which the sides and diagonals form more rational angles with each other than one might expect. Dave Rusin's known math pages include another article on the same problem. Almost research-related maths pictures . A. Kepert approximates superellipsoids by polyhedra. Angle trisection , from the geometry forum archives. Anna's pentomino page . Anna Gardberg makes pentominoes out of sculpey and agate. Antipodes . Jim Propp asks whether the two farthest apart points, as measured by surface distance, on a symmetric convex body must be opposite each other on the body. Apparently this is open even for rectangular boxes. Antiprism (Archimedean) . Some self-explanatory pictures from Eric Weisstein's treasure trove of mathematics. Aperiodic space-filling tiles: John Conway describes a way of glueing two prisms together to form a shape that tiles space only aperiodically. Ludwig Danzer speaks at NYU on various aperiodic 3d tilings including Conway's biprism.
The Geometry Junkyard: All Topics This page collects in one place all the entries in the geometry junkyard. Anna Gardberg makes pentominoes out of sculpey and agate. http://www.ics.uci.edu/~eppstein/junkyard/all.html
Extractions: All Topics This page collects in one place all the entries in the geometry junkyard. Jan Abas' Islamic Patterns Page Acme Klein Bottle . A topologist's delight, handcrafted in glass. Acute square triangulation . Can one partition the square into triangles with all angles acute? How many triangles are needed, and what is the best angle bound possible? Adventitious geometry . Quadrilaterals in which the sides and diagonals form more rational angles with each other than one might expect. Dave Rusin's known math pages include another article on the same problem. Adventures among the toroids . Reference to a book on polyhedral tori by B. M. Stewart. The Aesthetics of Symmetry , essay and design tips by Jeff Chapman. 1st and 2nd Ajima-Malfatti points . How to pack three circles in a triangle so they each touch the other two and two triangle sides. This problem has a curious history, described in Wells' Penguin Dictionary of Curious and Interesting Geometry : Malfatti's original (1803) question was to carve three columns out of a prism-shaped block of marble with as little wasted stone as possible, but it wasn't until 1967 that it was shown that these three mutually tangent circles are never the right answer. See also this Cabri geometry page and the MathWorld Malfatti circles page The Albion College Menger Sponge Algorithms for coloring quadtrees Are all triangles isosceles?
The Geometry Junkyard: Unfolded Polyhedra Paper models of polyhedra. Pentomino projectof-the-month from the geometry Forum.List the pentominoes; fold them to form a cube; play a pentomino game. http://www.ics.uci.edu/~eppstein/junkyard/unfold.html
Extractions: Unfolded Polyhedra A common way of making models of polyhedra is to unfold the faces into a planar pattern, cut the pattern out of paper, and fold it back up. Is this always possible? Arranging six squares . This Geometry Forum problem of the week asks for the number of different hexominoes, and for how many of them can be folded into a cube. Build your own polyhedra , Paul Bourke. Dodecahedron calendar , generated by a postscript program. The 85 foldings of the Latin cross , E. Demaine et al. Examples, Counterexamples, and Enumeration Results for Foldings and Unfoldings between Polygons and Polytopes , Erik D. Demaine, Martin L. Demaine, Anna Lubiw, Joseph O'Rourke, cs.CG/0007019. Find all polytopes . Koichi Hirata's web software for finding all ways of gluing the edges of a polygon so that it can fold into a convex polytope. Flexagons . Folded paper polyiamonds which can be "flexed" to show different sets of faces. See also Harold McIntosh's flexagon papers , including copies of the original 1962 Conrad-Hartline papers, also mirrored on Erik Demaine's website HyperGami program for unfolding polyhedra, also described in
Mathematics: Geometry Grades 1-8 The contents include Defining Polygons, Transformation, Tessellating Shapes, ProblemSolving with pentominoes, geometry Number, Properties of Polyhedrons, 2 http://teams.lacoe.edu/documentation/support/subscriber/content/geometry.html
Extractions: MATHEMATICS: GEOMETRY The Geometry series, Shapes in Space , provides lessons for first through middle school grades and gives students rich experiences with both two- and three-dimensional shapes. The content of the programs is correlated to the California Standards and NCTM standards. Students are introduced to properties and attributes of geometric shapes, angle measures, area and perimeter, transformations, and symmetry. Pre and Post Tests are available and can be downloaded from the TEAMS Website. Teaching Geometry Concepts: Staff Development These staff development programs help teachers understand geometry and the instructional strategies for effectively teaching geometry concepts. Program design, teacher guides, and online opportunities for students, teachers, and parents are explored. Strategies and resources are presented to help students learn how to use technology as a tool for processing information, visualizing and solving problems, exploring and testing conjectures, accessing data, and verifying solutions. Teaching Geometry Concepts Staff 1 Wed 9:00-10:00 am Teaching Geometry Concepts Staff 2 Fri 9:00-10:00 am Geometry Concepts for Primary Grades: Grades 1-2 This module stresses the development of geometric ideas through activities that engage students in doing, thinking, and reflecting on geometric concepts. Activities help build self-confidence, nurture natural curiosity, and challenge students with rich problems that help them learn to appreciate and value mathematics. The content includes Sorting Geometric Shapes, Classifying Geometric Shapes, Discovering Attributes of Geometric Shapes, Constructing Shapes, Slides, Flips, and Turns and Three-Dimensional Geometry.
O One Knows Pentominoes Topic pentominoes. geometry. Student will explore concepts in geometry by manipulating pentominoes. http://www.wvpt4learning.org/lessons/pdf99/noone.pdf
Pentomino Example Problem Solving with pentominoes. The Task pentominoes are the namesof shapes created by 5 congruent squares connected together. http://teams.lacoe.edu/documentation/classrooms/amy/geometry/3-4/activities/pent
Extractions: Pentominoes are the names of shapes created by 5 congruent squares connected together. The object of the puzzle is to place all 12 pentominoes inside the square with 4 boxes of your choice remaining empty. (This puzzle requires a Java-enabled browser.) Read the instructions before clicking "Start." To begin solving the puzzle, click the " begin " button. Choose 4 squares, anywhere on the grid, to remain empty. Click on them, and they will turn black. Drag over the pentominoes, one at a time, and place them on the puzzle board. Rotate and flip the pieces if needed Fill all the remaining squares. If you change your mind, drag the pieces out of the puzzle area. If you can fill all the spaces and use all the pentominoes, you are a Champion Pentominoes Puzzle Solver To rotate a pentomino: PC - Right click and hold your mouse on it. Mac - Click the mouse, while holding down the command key. To reflect - double click.
InterMath | Investigations | Geometry Click to go to Contact Us Sitemap nav Search the Site Investigations geometry Polygons Additional Investigations Perplexing pentominoes A pentominoe is a http://www.intermath-uga.gatech.edu/topics/geometry/polygons/a10.htm
Extractions: Search the Site Investigations Geometry Polygons ... Additional Investigations A pentominoe is a shape that can be made using five different squares, with each square touching an entire side of another square. How many pentominoes are possible? Do all pentominoes have the same area? Do all pentominoes have the same perimeter? Examples of pentominoes are:
Tetris: Reflection Of Tetrominoes And Pentominoes A thread from the geometry Forum newsgroup archive Tetris Reflection of Tetrominoes and pentominoes http://mathforum.com/~sarah/HTMLthreads/articletocs/pentris.html
E-z Geometry Project Topics Dr. Math Taxicab geometry; Rice U. - School Bus geometry. Go to Top. pentominoespentominoes - Introduction; pentominoes Wood Mosiac; Rice U - Lanius pentominoes; http://www.e-zgeometry.com/links/plinks1.htm
Extractions: e-zgeometry Project Topics A to F G to M N to R S to Z Networks: Go to Top Non Euclidean Geometry: Go to Top Number Theory: Go to Top Optical Illusions: Go to Top Orgami: Go to Top Pascal's Triangle: Go to Top Pentominoes: Go to Top Pi: Go to Top Prime Numbers: Go to Top Pythagorean Theorem: Go to Top Similarity: still searching.......
Extractions: This staff development program helps teachers identify important properties of number that help build a firm foundation for understanding geometry and algebra. Program design, teacher guides, and online opportunities for students, teachers, and parents are explored. Strategies and resources are presented to help students learn how to use technology as a tool for processing information, visualizing and solving problems, exploring and testing conjectures, accessing data, and verifying solutions. The two staff development programs help teachers understand what algebraic thinking is and the importance of teaching algebraic concepts throughout the grades. Emphasis is placed on how these concepts relate to learning basic skills. Strategies for helping students develop algebraic thinking are modeled. Program design, teacher guides, and online opportunities for students, teachers, and parents are explored. Teachers discover that they can teach algebra to their students. Algebra In My World: Grades 3-4 Algebra in My World presents activities that have students thinking algebraically as they work with functions, patterns and algebraic symbols. Problem solving strategies are used to help students solve word problems. Topics include: Fun with Function, Looking at Patterns, Relationships of Numbers, Word Problems, Working with Inequalities and Looking for a Balance.
Geometry Forum Project Of The Month October 1996 pentominoes A pentomino consists of five unit squares stuck togetherso that each square shares at least one whole side with another square. http://mathforum.org/pom/toc.oct96may97.pom.html
Extractions: A pentomino consists of five unit squares stuck together so that each square shares at least one whole side with another square. There are 12 types of pentominoes. Your tasks: 1. Find the other 11 types of pentominoes. 2. Find all different rectangles with integer sides greater than 2 whose area equals 60. Show that each of these rectangles can be exactly covered (tiled) by the set of 12 pentominoes. You are allowed to rotate and reflect the pieces to tile the rectangles. November 1996 - How many cubes will be painted? Rangefinders are like binoculars. They are about 10" long with two windows on either end that face the animal. You look through the viewfinder on the back with one eye. You see two pictures of the animal. You turn something until the two images are on top of each other. The rangefinder tells you how far away the animal is. How does it work? How can it figure out how far away the animal is? January 1997 - Overlapping polygons.
Geometric Puzzles In The Classroom to work on the puzzles on the worksheet, students need to know the names of the tetrominoesand pentominoes. (You may duplicate page 54 of geometry Labs, or if http://www.picciotto.org/math-ed/puzzles/
Extractions: http://www.picciotto.org/math-ed/puzzles/ Visit Henri Picciotto's Math Education Page Send me email by Henri Picciotto This page includes some background information about my puzzle books, some articles, puzzles, and activities you can print for yourself or duplicate for your students, and a few links to relevant web sites. (Parts of this web page are adapted, with permission, from articles I wrote in 1989 for Michael Keller 's games and puzzles 'zine.) Article outline: Polyforms Polyarcs I have found geometric puzzles to be an excellent springboard for mathematics lessons, as they are interesting to both students and teachers. They lend themselves to teaching some specific concepts as well as to building students' spatial sense and problem-solving skills. Moreover they show to a wide range of learners that exploring mathematics can be rewarding, irrespective of any "practical" application. As a result, geometric puzzles have consistently found their way into my classes, my workshops, and my books. In fact they were how I first broke into print. There are many geometric puzzles that can be used in the classroom, such as for example tangrams and rep-tiles as presented in my book
Elementary Level Mathematics: Grade Five (Strand: Geometry) problem solving using some or all pentominoes to form rectangles and squares -finding perimeters and areas. A geometry Fair is an exciting, motivational http://www.sasked.gov.sk.ca/docs/elemath/g542note.html
Extractions: Have students share their thinking and the strategies they used as they experiment with various geometrical concepts. Students may still work with squares and triangles through matchstick or toothpick games. Concept attainment can be used to develop many geometrical concepts such as shapes, lines, and congruency. G-22(b)
Pentominoes -- LEGO LEGO + geometry. My first 'LEGO puzzle' (I expect there to be manymore) is a recreation of the classic pentominoes Pieces Puzzle. http://www.ericharshbarger.org/lego/pentominoes.html
Extractions: Besides my fascination with LEGO bricks, I have many other hobbies and interests. One such pasttime has been crafting puzzles, games, and brainteasers of various sorts. Often I craft the necessary pieces out of wood, but other times I make use of whatever is available (local school supply shops love me because I'm always buying tokens, dice, letter tiles, marbles and the like). It finally sunk in recently that for rectagular pieces that need to be fashioned with high precision, LEGO bricks would be a fine material with which to work. My first 'LEGO puzzle' (I expect there to be many more) is a recreation of the classic Pentominoes Pieces Puzzle. A 'pentomino' is a geometric shape formed by the edge-to-edge joining of five unit squares. There are twelve such unique pieces (not including reflected images). With twelve pieces at five squares each, this creates pieces which cover an area of sixty units (12 X 5 == 60). Thus, one challenge is to try to fit the pieces into a perfect rectangle of given dimensions; say, a 6 x 10 rectangle. Much more can be discussed about the mathematical properties of pentominoes (see below for related links). The challenge I posed for myself was to build twelve such pentominoes out of LEGO bricks and plates and tiles, and a tray into which to place them.
Grades Pre-K - 2: Geometry Specify locations and describe spatial relationships using coordinate geometry andother The task should include keeping a record of the pentominoes that are http://standards.nctm.org/document/chapter4/geom.htm
Extractions: Expectations Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships recognize, name, build, draw, compare, and sort two- and three-dimensional shapes; describe attributes and parts of two- and three-dimensional shapes; investigate and predict the results of putting together and taking apart two- and three-dimensional shapes. Specify locations and describe spatial relationships using coordinate geometry and other representational systems describe, name, and interpret relative positions in space and apply ideas about relative position; describe, name, and interpret direction and distance in navigating space and apply ideas about direction and distance; find and name locations with simple relationships such as "near to" and in coordinate systems such as maps.
Geometry In My World Problem Solving with pentominoes, 2/11/03400pm, 2/12/03-900am 2/13/03-230pm2/14/03-900am. geometry Number, 2/19/03-400pm, 2/20/03-1100am 2/21/03-230pm. http://www.sarasota.k12.fl.us/ITV/staff_programs/geometry_grades3_4.htm
Extractions: Program provider: TEAMS Distance Learning This module helps students use geometry as a means of describing the physical world. Programs include identifying, describing, comparing, and classifying geometric figures, developing spatial sense, understanding geometric properties and relationships, and exploring symmetry. This module has three components to be utilized in obtaining inservice credit Two 60-minute staff development programs, Teaching Geometry Concepts Eight 30-minute programs to be previewed and possibly used with students An interactive Electronic Classroom for accessing related resources and interacting with the studio instructor These programs for students are designed to be viewed on tape. The studio instructor initiates the lesson, shows examples, and gives directions. Then a YOUR TIME screen appears and indicates a pause the tape mode. At this point, the classroom teacher facilitates the hands-on activity. When students have completed the activity the tape is continued and the studio teacher proceeds with the lesson.
Loading L4U IPAC Summary, Students develop geometry skills and spatial sense as they use pentominoes(a shape made of five congruent, simplyconnected squares) to think http://drc.sd62.bc.ca/DT000064.HTM
Loading L4U IPAC ANALYSIS AND APPLICATIIONS (V1396) EARTH geometry (V1383) MATH BY ALL MANS geometryGRADE 2 (0266) THREE-DIMENSIONAL pentominoes (MAK0014) Tangrams (MAK0006 http://drc.sd62.bc.ca/BROWSE/31000068.HTM
