1. Home - Search Geometry geometry. pi geometry. pentominoes geometry. pascals triangle geometry.origami paper folding geometry. geometry. euclidean geometry. analytic http://www.algebraic.net/cgi-bin/988.cgi?q=geometry&show_page=1

2. Math Tools Support Material: Lesson Plans Teacher Information pascals triangle. Author National Library of Virtual Manipulatives (Utah State Rect coordinate geometry. geometry in the plane. geometry in space http://mathforum.com/mathtools/support.html?&new_id=424

3. Pascals Triangle Patterns found in Pascal s triangle. There are many patterns thatcan be found in Pascal s triangle. Some of the patterns that can http://www.ga.k12.pa.us/academics/us/Math/Geometry/stwk00/sloanelikemen/paterns.

Patterns found in Pascal's Triangle There are many patterns that can be found in Pascal's Triangle. Some of the patterns that can be found include: The Hockey Stick pattern, The Sum of the Rows, and The Magic 11's. Here you can learn more about each of these interesting relationships. To further understand these patterns you can refer to the completed diagram of Pascal's Triangle at the bottom of the page. The Hockey Stick pattern To form the "hockey stick", and understand the basis of this neat pattern, first draw a diagonal line downwards from one of the number 1's on the triangle. After you have selected your diagonal, you must then choose a number that is adjacent to the diagonal below it. It should then look like a hockey stick, hence the name for the pattern. What is interesting about this pattern is that when you add up the numbers in the diagonal the sum of the numbers inside the selection is equal to the number below the end of the selection that is not on the same diagonal itself. Look at the triangle with the numbers filled at the bottom of the page and try to find a "hockey stick" for your self. Here are some examples like below; the numbers that are being added up are the ones in the diagonal and the bottom part of the "hockey stick" or the blade of the stick is the number that the diagonal equals. The Sum of the Rows The sum of the numbers in any row is equal to 2 to the nth power (n is the number of the row). Here are some examples of this:

4. Pascals Triangle At the top of Blaise Pascal s triangle is the number 1. The very topnumber 1 lies in Row Zero. Row One contains two number 1 s. http://www.ga.k12.pa.us/academics/us/Math/Geometry/stwk00/sloanelikemen/backgrou

The History of Blaise Pascal and How His Triangle is Constructed Blaise Pascal was born in 1623-1662, in Clermont- Ferrand, France, but lived mostly in Paris. He was a religious philosopher, scientist, and mathematician. Besides the triangle, he also invented a calculating machine (1647), and later the barometer and the syringe. At the top of Blaise Pascal's Triangle is the number 1. The very top number 1 lies in Row Zero. Row One contains two number 1's. Each of the 1's is made by adding the two numbers above it (in this case, the two numbers are and 1, the being the number outside of the triangle). This is how you create the rest of the triangle; add the top two numbers to get the number below the two numbers. For example if you take Row Two, 1+0=1; 1+1=2; 1+0=1; And thus, Row Two's numbers are 1-2-1. So this is all you have to do to construct Pascal's Triangle. This means that there are many more rows in Pascal's Triangle than we show with our diagram (see Activity Page for diagram). There are actually an infinite amount of rows. Another way that you can find a number in the triangle is through the nCr (n Choose r) method. In this equation, "n" stands for the number of the row, and "r" is the number of spaces in that row that you want to find the number for. For example, in Row Three, the first 1 is the "zeroeth" element, the first 3 is the first element, the next three is the second element, and the last 1 is the third element.

5. Math Tools Support Material: Lesson Plans Pascal's triangle Lessons. Tools pascals triangle. Author Doctor Math Rect coordinate geometry. geometry in the plane. geometry in space http://mathforum.com/mathtools/support.html?&new_id=240

6. Math Tool: Pascals Triangle combs., Pascal s triangle, Recurrence relations. BROWSE. Site Map. COURSES. AllCourses. PreK. Kindergarten. Math1. Math2. Math3. Math4. Math5. Math6. Math7.geometry. http://mathforum.org/mathtools/tool.html?tp=9.3&id=424

7. Pascal's Triangle Interactive Pascal's triangle. Pascal's triangle interface. Fractal geometry Pascal's triangle mod 2, and mod will be like constructing a version of pascals triangle on a pyramid http://www.hverrill.net/pages~helena/pascal

URL http://shimura.math.berkeley.edu/~helena/pascal/ Pascal's Triangle and related triangles THIS PAGE IS UNDER CONSTRUCTION Contents Links Puzzles Related Triangles Clown Problem (Catalan Numbers) Tchebychev Polynomials Bessel Polynomials Stirling numbers Pascal's triangle comes up all over the place, in lots of lovely ways; there must be whole books devoted to applications of Pascal's triangle and related triangles, but I don't know them - if you do, please let me know! Links Pascal's triangle gives the coefficients of (x+1) n Probably the best place to find out all about Pascal's triangle is at Math Forum web pages on Pascal's triangle Here are some pages that will show you rows of Pascal's triangle: Math Forum's Dr Math page on Pascal's Triangle Pascal's Probability: The Triangle and the Binomial Coefficient Jay's Corner - including links to several pages on Pascal's triangle Pascal's Triangle ... Pascal's triangle interface Fractal Geometry - Pascal's triangle mod 2, and mod n, etc: Pascal's Triangle using Clock Arithmetic - Part I Cut the Knot on Sierpinski's triangle Ivars Peterson's MathLand Pascal Triangle fractal ... Program for drawing Pascal's triangle For Children: The Twelve Days of Christmas and Pascal's Triangle Pascal Triangle activities Binomial Expansion and Infinite Series More interesting links to Pascals triangle stuff: Pascal's triangle via cellular automata Binomial coefficients modulo a power The Arithmetic Properties of Binomial Coefficients I From searching on AMS MathSciNet search page About Blaise Pascal

8. Math Tool: Pascals Triangle Pascal s triangle. Fractions. Percentages. Integers. Algebra expressions. Equations functions. Linear relationships. Ratio proportion. Rect coordinate geometry. http://mathforum.org/mathtools/tool.html?co=m7&tp=9.3&id=424

9. Untitled integral 29 queries convolution 28 queries geometry 28 queries probability 27 queries lagrange queries unit vector 3 queries pascals triangle 3 queries index group 3 http://aux.planetmath.org/misc/pmsearches_200205230243.txt

10. FUCHS MIZRACHI SCHOOL Middle And Upper School Math Curriculum TOPIC CONTENT geometry Unit 1 Measurement, Capacity Weight US Customary Metric Expansion by pascals triangle Binomial Expansion by Binomial Theorem geometry TOPIC CONTENT http://www.fbsm.org/MS-US Math Curriculum.pdf

11. TI-92 BASIC Math Programs - Ticalc.org 0104-29. Analytic geometry v2.0. The most comprehensive geometry (analytic) program for TI-89 and TI-92 This stores many rows of pascals triangle to a matrix for easy http://www.ticalc.org/pub/92/basic/math

Basics Archives Community Services ... File Archives TI-92 BASIC Math Programs File Archives TI-92 BASIC MATH PROGRAMS Click a filename to download that file. Click a folder name to view files in that folder. Click for file information. indicates files with screen shots. indicates files with animated screen shots. indicates featured programs. Archive Statistics Number of files Last updated Wednesday, 2 June 2004 Total downloads Most popular file Advanced Math v1.1 with 7,325 downloads. Name Size Date Description (Parent Dir) folder Up to TI-92 BASIC Files acalcprgs.zip A+ Assorted Calculus Programs This is a collection of several programs I have recently been working on. They are all calculus-related, except for a program to solve three equations simultaneously. I have included programs to help you use Newton's Method to approximate roots for the Nth iteration, a program for finding tangent points on a curve to a given point, a program for finding tangent points on two curves (tangent to each other), programs for both the regular Mean Value Theorem and the Mean Value Theorem for integrals, and the aforementioned program for solving three equations. These should help you speed up some tedious calculations in calculus class. acst.zip

12. User:Jaredwf - Wikipedia, The Free Encyclopedia projection (cartography) Orthographic projection (geometry) Paul Bachmann pascals triangle Spline (mathematics) Timeline of algorithms Translation (geometry) http://en.wikipedia.org/wiki/User:Jaredwf

User:Jaredwf From Wikipedia, the free encyclopedia. Server will be down for maintenance on 2004-06-11 from about 18:00 to 18:30 UTC. A cultivated person's first duty is to be always prepared to rewrite the encyclopedia. Umberto Eco Table of contents 1 Local pages 2 Topics of interest 3 Current projects 4 Pages created ... edit Local pages User:Jaredwf/Math Markup edit Topics of interest Computer science List of computing topics Timeline of computing Mathematics ... Computational geometry and Computer graphics List of computer graphics and descriptive geometry topics Computability theory and Computational complexity theory List of computability and complexity topics List of complexity classes edit Current projects Computing Mathematics edit Pages created Approximation error Axis-aligned bounding box Bounding volume Buffon's needle problem ... edit Pages significantly edited Ambiguous grammar Bézier splines Cross product Finite state machine ... edit Images created DFAexample.png Graphstructuredstack jaredwf.png Leftmostderivations jaredwf.png Line Segment jaredwf.png ... edit Shortcuts Announcements Disputes Accuracy disputes NPOV disputes ... Special pages Views User page Discussion Edit History Personal tools Log in Navigation Main Page Community portal Current events Recent changes ... Donations Search Toolbox What links here Related changes User contributions Special pages This page was last modified 22:54, 28 May 2004.

13. National Library Of Virtual Manipulatives Ladybug Mazes — geometry (PreK2), geometry (3-5), Measurement (PreK-2),Measurement (3-5). pascals triangle — Numbers Operations (9-12). http://matti.usu.edu/nlvm/applets/controller/query/query.htm?qt=

14. Pearson Education Chaos and Fractal geometry. Collaborative Investigation Generalizing the AngleSum Concept. Chapter 9 Test. Using pascals triangle and the Binomial Theorem. http://www.pearsoned.co.uk/Academics/Book.asp?prodID=100000000039979&d=MM&sd=MMP

15. A Brief History work on conics and published several papers in the field of geometry. this provedthe theory of baromic pressure, he created pascals triangle (1654) which http://www.bath.ac.uk/~ma1mrw/history.html

Interesting Patterns In Pascal's Triangle Main Page A brief history Sum of the rows Hockey stick pattern ... Petal Pattern A Brief History Pascals Triangle is a well known and famous mathematical pattern. Athough this pattern was originally discovered in China it was named after the first westener to study it. According to Yuhnze He the triangle was first developed during the Song Dynasty by a mathematician named Hue Yang. Blaise Pascal was a French mathematician who was alive in the 17 th century. His mother was Antoinette Begon and his father was Etienne. His mother died when Pascal was three and his father had the responsibility of bringing up Pascal and his two sisters Gilberte and Jaqueline. In 1635 Pascal began his studies and had mastered Euclid's Elements by age 12. This won him great respect in mathematical circles. Instead of going to school Etienne took Pascal to lectures and mathematical gatherings at the "Academie Parsienne". By age 16 Pascal was playing an active role here as the principle disciple of Girard Desargues, a proffesor in geometry. Pascal began work on conics and published several papers in the field of geometry. In fact, in June 1639 Pascal had already made a significant discovery with his mysical hexagram. In 1641, a bad case of ill health delayed his research for a year.

16. Connect-ME - Weblinks various modules triangle module, pentagon moduel, sonobe module, pascals trianglemodule, that The connection with geometry is clear and yet multifaceted; a http://educ.queensu.ca/connectme/weblinks/strands.htm

Classroom Resources for Teachers Curriculum Strands Measurement Magic Squares - Math Forum http://mathforum.org/alejandre/magic.square.html Late Primary - Intermediate The Math Page - Skill In Arithmetic http://www.themathpage.com/ARITH/arithmetic.htm Multiplication: An Adventure in Number Sense http://naturalmath.com/mult/index.html This site illustrates patterns in multiplying 2 numbers up to 10 x 10. By using patterns and and 'twin facts' a 10 by 10 multiplication table is reduced to only 13 facts to be memorized. Numbers: Facts, Figures and Fiction http://richardphillips.org.uk/number/ Primary/Junior Numbers: Facts, Figures and Fiction Place Value http://www.sedl.org/scimath/compass/v03n02/place.html Primary The activities described in this lesson reinforce students' understanding of place value by using rhythm, physical action, and introspection. Simply Number Sense http://library.thinkquest.org/17888/opening.shtml

17. Pascal Lines: Steiner And Kirkman Theorems II Projective geometry lied abandoned for about two hundred years; and Pascal s result Next,have a look at the following triangle formed by three pascals http://www.cut-the-knot.org/Curriculum/Geometry/MorePascal.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Pascal Lines: Steiner and Kirkman Theorems II At the age of sixteen B. Pascal proved a remarkable theorem The three intersections of the pairs of opposite sides of a hexagon inscribed in a conic are collinear. It is said [ Bell , p. 78] that from that result (and two other lemmas) Pascal derived all Apollonius' theorems on conics and more, no fewer than 400 propositions in all. Little wonder he called it the Mystic Hexagram Hexagrammum Mysticum ). The original manuscript (that was lost) was examined and praised by Leibniz. Descartes was been stunned to learn that the work had been performed by a sixteen years old. A shorter version written a year later has survived, see [ Source , p. 326-330], but contains no derivation. The theorem is clearly of projective nature, and in the surviving manuscript Pascal leaves no doubt of his intention to imitate the methods of Projective Geometry that Desargues introduced a short time beforehand. In the 17 th century, identification with Desargues' work was certainly detrimental to any discovery. The ridicule with which his concepts and notations were met caused Desargues to give up on mathematical research. Projective Geometry lied abandoned for about two hundred years; and Pascal's result had shared its fate. However, in the 19

18. Do You Speak Mathematics? Projective geometry lay abandoned for about two hundred years; and Pascal s result Next,have a look at the following triangle formed by three pascals http://www.cut-the-knot.org/ctk/Mathematics.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Cut The Knot! An interactive column using Java applets by Alex Bogomolny Do You Speak Mathematics? November 2002 Mr Thomas Hobbes told me, that this Mr Cavendish told him, that the Greeks doe sing their Greeke. John Aubrey Brief Lives , Penguin Books, 2000, p. 74 I call on him to explain everything as clearly as if it were in Latin. Epictetus Discourses and Enchiridion , Classics Club, 1944, p. 47 Do you speak mathematics? is a very valid question assuming mathematics is a language. Many think it is. Josiah Willard Gibbs gave a speech on that account. Galileo and R. Feynman thought so. Mathematics is even judged to be a universal language, the only one suitable to initiate extraterrestrial communication [ Jacobs , p. 1]. Some object. For example, Jean-Pierre Bourguignon argues [ Basis , p. 173] that For some people, mathematics is just the language of the quantitative . This opinion is shared by some of our fellow scientists. ... We mathematicians know how wrong this opinion is, and how much effort goes into building concepts, making new links, establishing facts, and following avenues we once thought plausible but turned out to be dead ends. I generally accept the above sentiment with reservations concerning mathematics being just the language of quantitative The potency of the bond between mathematics and its language is such that many mathematicians do indeed identify the two [ Spectrum , p. 112]:

19. Enrichment - Secondary School realms of our solar system and learn estimation, geometry, and problem topics includemagic squares, topology, pascals triangle, Fibonacci numbers, polyominoes http://www.awl.ca/school/math/mr/enr/books/enrbooks.html

Titles for Enrichment Seconday Level Reading, Writing, and Doing Mathematical Proofs Daniel Solow (Grades 10-12) SS6-0-86651-148-2, 288 PP, $29.10 The little insights and ideas we all so laboriously discovered for ourselves come together in this carefully-structured, systematic book about mathematical proofs. Once students understand and analyze the structure of proofs, theyll be able to follow the more informal versions in texts and learn to create their own. two workbooks tailored to the high school curriculum short, easy-to-follow lessons followed by exercise sets with solutions Number Treasury Stanley Bezuszka and Margaret Kenney (Grades 7-12) SS6-0-86651-078-8, 208 pp., $29.10 This resource offers an extensive source of information about number patterns, classical and modern. covers prime and composite numbers, plane and figurate numbers, digital patterns, and more solutions included Math Space Mission Regional Math Network Kay Merseth, Director (Grades 7-9) SS6-0-86651-439-2, 256 pp., $33.45 Students explore the outer realms of our solar system and learn estimation, geometry, and problem solving with this four-module, spiral-bound kit. activity book includes teaching notes, reproducible student worksheets and problem cards, answers, and more

20. Patterns - Secondary Level In this book, students study history and geometry as they arithmetic, symmetry,design transformation, Latin squares, pascals triangle, repeating circles http://www.awl.ca/school/math/mr/pat/pattxt.html

Patterns Secondary Level Prices with a (*) denote Ontario PST applicable Line Design Poster Sets (All Grades) Dale Seymour Set A: SS6-0-86651-587-9, $14.95* Set B: SS6-0-86651-588-7, $14.95* The eye sees a series of graceful curves although these beautiful geometric designs are created with simple straight lines. Perfect for framing and enhancing any classroom. two sets available four posters per set each poster 11.5 x 11.5 Introduction to Line Designs (Grade 6 and Up) Dale Seymour SS6-0-86651-579-8, 292 pp., $32.00 Written especially for beginners, Line Designs helps students create string sculptures, curve stitchings and line designs. Explorations also explore line designs with personal computers, new geometry tools, and artistic creativity. complete with designs, instructions and examples The Pythagorean Theorem Poster Set and Book (Grades 8-12) Sidney J. Kolpas Book: SS6-0-86651-598-4, 48 pp., $13.50 4 posters: SS6-0-86651-597-6, 16" x 22", $24.70* Book and Posters: SS6-0-201-68657-0, $32.05* In this book, students study history and geometry as they explore eight elegant proofs of the Pythagorean Theorem from across the centuries. includes interesting facts, biography of Pythagoras and a list of concepts needed to understand the proofs

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