Extractions: TEAMS Distance Learning - MATHEMATICS PROGRAMS TEAMS mathematics programs complement classroom curriculum for first through eighth grades. They model effective teaching strategies and are starting points to a rich and rigorous mathematics program. Students explore, investigate and use problem solving skills as they develop mathematical thinking skills related to algebra, number and geometry concepts. Through the TEAMS Distance Learning Home Page , teachers and students can access resources that extend and support the mathematics program. The algebra series provides students with activities that build a foundation for understanding algebraic concepts. Algebra: Staff Development Teaching Algebraic Concepts (Two 60-minute programs) These staff development programs help teachers understand what algebraic thinking is and the importance of teaching algebraic concepts throughout the grades. Strategies for helping students develop algebraic thinking are modeled. Program design, teacher guides, and online opportunities for students, teachers, and parents are explored. (Six 30-minute programs) This module develops an understanding of how naturally occurring situations can be represented algebraically. Through active investigation, students model, represent, and interpret number relationships. They use algebraic symbols to create and solve problems using addition and subtraction. Equations are introduced as a way to generalize patterns and describe number relationships.
Extractions: Skip Navigation You Are Here ENC Home Curriculum Resources Search the Site More Options Don't lose access to ENC's web site! Beginning in August, goENC.com will showcase the best of ENC Online combined with useful new tools to save you time. Take action todaypurchase a school subscription through goENC.com Classroom Calendar Digital Dozen ENC Focus ... Ask ENC Explore online lesson plans, student activities, and teacher learning tools. Search Browse Resource of the Day About Curriculum Resources Read articles about inquiry, equity, and other key topics for educators and parents. Create your learning plan, read the standards, and find tips for getting grants. Grades:
The Geometry Junkyard: Polyominoes 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 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. NebulaSearch: Polyomino 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.
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. Ant on a block. 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 Algebraic surface advent calendar 2002 Algebraic surface models . Oliver Labs makes models of algebraic geometry examples using a 3d printer. Algorithmic mathematical art , Xah Lee.
InterMath | Investigations | Geometry Click to go to Recommended Investigations Polygons geometry Investigations How many pentominoes are possible? Do all pentominoes have the same area? 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:
InterMath | Investigations | Geometry Click to go to Additional Investigations Polygons geometry Investigations Perplexing pentominoes Explore with pentominoes. http://www.intermath-uga.gatech.edu/topics/geometry/polygons/add.htm
Plane Geometry To promote a greater awareness of geometry in the real world, and Problem Solvingwith pentominoes, 1992, Learning Resources, Inc. Wildsmith, Brian. http://mathcentral.uregina.ca/RR/database/RR.09.96/lockhart1.html
Extractions: Topic: Plane The students should: Demonstrate confidence, desire and an ability to solve a variety of mathematically related problems. Demonstrate knowledge and understanding of why, when and how to collect, organize and interpret numerical data: Demonstrate and understanding of numbers, patterns, counting, operations, and estimation: Demonstrate a sense of spatial awareness and familiarity with two dimensional shapes and recognize relationships between geometry and the environment. Note: Some activities may be extended if time is available. Manipulatives could be available for free time exploration. The stations would take a varied number of hours depending on such variables as number of students and their abilities, as well as the number of stations each student is expected to complete. 1. Free Exploring
PENTOMINOES pentominoes are said to have been invented by Solomon W. Golomb in 1953 at a talk The informal geometry that is ingrained in pentomino discovery is an http://www.andrews.edu/~calkins/math/pentos.htm
Extractions: November 20, 1995 Shapes that use five square blocks joined together with at least one common side are called pentominoes. There are twelve shapes in the set of unique pentominoes, named T, U, V, W, X, Y, Z, F, I, L, P, and N respectively. As a mnemonic device, one only has to remember the end of the alphabet (TUVWXYZ) and the word FILiPiNo. In order to make a set of unique pentominoes, there are only two rules which must be followed. First, if one shape can be rotated to look like another, the two shapes are not considered to be different. Second, if one shape can be flipped to look like another, the two shapes are not considered to be different. Since there are twelve distinct pentomino shapes with each covering five squares, their total area is sixty squares. There are several ways to place all twelve different pentominoes on an 8 x 8 board, with four squares always left over. By artistically specifying the positions of the four extra squares, many interesting patterns can be created. Another evident possibility is to require that the four extra squares form a 2 x 2 area (a square tetromino) in a specified position on the board. This placement leads to a remarkable theorem, which can be proved by using only three constructions: Wherever on the checkerboard a square tetromino is placed, the rest of the board can be covered with the twelve pentominoes. Another problem is to determine the least number of pentominoes which will span the checkerboard. In other words, some of the pentominoes are placed on the board in such a way that none of the rest can be added. Although there are several different configurations for solving this puzzle, the minimum number is five pentominoes. Other patterns include forming the configurations of 6 x 10, 5 x 12, 4 x 15, and 3 x 20 rectangles using all twelve pentominoes. The 3 x 20 is the most difficult to derive, and there is only one unique solution, except for the possibility of rotating the shaded central portion by 180 degrees.
NLVM Geometry Manipulatives NLVM manipulatives for geometry. pentominoes icon pentominoes Use the 12pentomino combinations to solve problems. http://nlvm.usu.edu/en/nav/topic_t_3.html
NLVM Pre-K - 2 - Geometry Manipulatives NLVM manipulatives for PreK - 2 - geometry. pentominoes icon pentominoes Use the 12 pentomino combinations to solve problems. http://nlvm.usu.edu/en/nav/category_g_1_t_3.html
Pentominoes -- LEGO LEGO + geometry. Besides my fascination with LEGO bricks, The challenge Iposed for myself was to build twelve such pentominoes out of LEGO bricks and 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.
Eric Harshbarger's Pentominoes Page: Article This page is more about wordplay rather than geometry. The fact that thetwelve pentominoes are assigned 12 letters of the alphabet for easy distinction http://www.ericharshbarger.org/pentominoes/article_05.html
Extractions: This page is more about wordplay rather than geometry. Both topics are of interest to me, so this short article should not be a great surprise. The fact that the twelve pentominoes are assigned 12 letters of the alphabet for easy distinction (as below): F I L N P T U V W X Y Z leads to a very obvious question: what is the longest word that can be formed using only "Pentomino Letters"? "Word" is a rather subjective, um, word... but let's not get too technical. I'll settle for an entry in Webster's Third New International Dictionary These types of puzzles can be facilitated greatly by using searchable Dictionaries-on-CDs and such, but I always like to at least start considering them using only my inherent vocabulary (and simply checking them against the dictionary). Having only two vowels (and the "I" and "U", at that) is not very helpful, but some words come pretty easily: ZIPPY, FUZZY, and such. Turns out FUZZY's adverbial form works as well: FUZZILY. FLUFFILY is even better ("in a fluffy manner" reads the dictionary... um, okay). That's eight letters long... now it starts getting tougher...
Math Tools Browse pentominoes, geometry in the plane (Math 3), Java Applet, Tool, 0. A virtualmanipulative of pentominoes which can be rotated and also reflected (flipped) http://mathforum.org/mathtools/cell/m3,5.7,ALL,ALL/
Extractions: Browse Catalog Discussions All Resource Types Tools Lesson Plans Stories Activities Technology PoWs Support Materials All Technology Types Calculator Texas Instruments Computer Cabri Computer Algebra System Mathematica Fathom Flash Java Applet JavaScript Presentation software PowerPoint Shockwave Sketchpad Spreadsheet Excel PDA Palm OS PocketPC You are not logged in. login register Home About ... Developers Area Submissions Resources Write a Story Newsletter Browsing: All Content in Math 3 for Geometry in the plane Browse discussions Login to Subscribe / Save Results Resource Name Topic (Course) Technology Type ... Using Kali Symmetry (Math 3) Computer Activity An exercise (in English and Spanish) in using an interactive 2D Euclidean symmetry pattern editor. This activity was originally written as one of the stations in Frisbie Middle School's Multicultural ...
Math Tool: Pentominoes , A virtual manipulative of pentominoes which can be rotated and alsoreflected Math 1, geometry in the plane. Math 2, geometry in the plane http://mathforum.org/mathtools/tool/473/g,10.13.6,ALL,ALL/
Extractions: Browse Catalog Discussions All Resource Types Tools Lesson Plans Stories Activities Technology PoWs Support Materials All Technology Types Calculator Texas Instruments Computer Cabri Computer Algebra System Mathematica Fathom Flash Java Applet JavaScript Presentation software PowerPoint Shockwave Sketchpad Spreadsheet Excel PDA Palm OS PocketPC You are not logged in. Go to: http://nlvm.usu.edu/en/nav/frames_asid_114_g_2_t_2.html (opens a new window) Description: A virtual manipulative of pentominoes which can be rotated and also reflected (flipped). Technology Type: Java Applet Guide: User's Guide (opens a new window) Author: National Library of Virtual Manipulatives (Utah State University) Language: English Cost: Does not require payment for use Average Rating: Login to rate this resource My MathTools: Login to Subscribe / Save This Reviews: be the first to review this resource Discussions: start a discussion of this resource Tool: Polyominoes Courses: Kindergarten Geometry in the plane Math 1 Geometry in the plane Math 2 Geometry in the plane Math 3 Geometry in the plane Math 4 Geometry in the plane Math 5 Similarity Math 6 Similarity Math 7 Similarity Geometry Other polygons, Tessellations
Geometric Puzzles In The Classroom Fourteen activities in my book geometry Labs are dedicated to polyominoes. pentominoes are made of five squares, and are the subject of many, http://www.picciotto.org/math-ed/puzzles/
Extractions: Some rights reserved Visit Henri Picciotto's Math Education Page Send me e-mail 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
NCTM : Illuminations Lessons : pentominoes are not considered different if the same figure can be obtained by A lesson for the middle grades on the geometry of odds and evens, http://illuminations.nctm.org/index_d.aspx?id=256
Exploring Geometry Building pentominoes is a challenge, to find all the pentominoes. To viewthe full evaluation version of Exploring geometry live in your browser go to http://www.smilemathematics.co.uk/explgeom.html
Extractions: These programs can be used flexibly in the classroom, either as lesson starters, main activities, plenary sessions or consolidation. They are particularly suited for use with data projectors and interactive whiteboards. Angle Fit consolidates the angle sum of triangles, quadrilaterals, angles on a straight line and about a point. Square Jigsaw is a challenge in spatial reasoning. Building Pentominoes is a challenge, to find all the pentominoes. Symmetry Match is an introduction to line symmetry. Enlarging Pentominoes explores enlargement. Symmetry Review consolidates line and rotational symmetry. Polygon Names consolidates properties of polygons.
EAI Education - Pentominoes, Set Of 6 pentominoes, Set of 6 Scored pentomino sets in red, blue, green, yellow, pentominoes Cooperative Informal geometry Problem Solving W/ pentominoes http://www.eaieducation.com/530188.html
GEOMETRY geometry. NOTES AND LINKS Virtual Manipulatives pentominoes pentominoes Click above and to the left of the pentominoes you want to group. http://www.sad6.k12.me.us/~dpeterson/GEOMETRY
