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Lesson 19: the Euclidean Algorithm As an Application of the Long Division Algorithm
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 19 6•2 Lesson 19: The Euclidean Algorithm as an Application of the Long Division Algorithm Student Outcomes . Students explore and discover that Euclid’s Algorithm is a more efficient means to finding the greatest common factor of larger numbers and determine that Euclid’s Algorithm is based on long division. Lesson Notes MP.7 Students look for and make use of structure, connecting long division to Euclid’s Algorithm. Students look for and express regularity in repeated calculations leading to finding the greatest common factor of a pair of numbers. These steps are contained in the Student Materials and should be reproduced, so they can be displayed throughout the lesson: Euclid’s Algorithm is used to find the greatest common factor (GCF) of two whole numbers. MP.8 1. Divide the larger of the two numbers by the smaller one. 2. If there is a remainder, divide it into the divisor. 3. Continue dividing the last divisor by the last remainder until the remainder is zero. 4. The final divisor is the GCF of the original pair of numbers. In application, the algorithm can be used to find the side length of the largest square that can be used to completely fill a rectangle so that there is no overlap or gaps. Classwork Opening (5 minutes) Lesson 18 Problem Set can be discussed before going on to this lesson. Lesson 19: The Euclidean Algorithm as an Application of the Long Division Algorithm 178 Date: 4/1/14 This work is licensed under a © 2013 Common Core, Inc. -
A MATHEMATICIAN's SURVIVAL GUIDE 1. an Algebra Teacher I
A MATHEMATICIAN’S SURVIVAL GUIDE PETER G. CASAZZA 1. An Algebra Teacher I could Understand Emmy award-winning journalist and bestselling author Cokie Roberts once said: As long as algebra is taught in school, there will be prayer in school. 1.1. An Object of Pride. Mathematician’s relationship with the general public most closely resembles “bipolar” disorder - at the same time they admire us and hate us. Almost everyone has had at least one bad experience with mathematics during some part of their education. Get into any taxi and tell the driver you are a mathematician and the response is predictable. First, there is silence while the driver relives his greatest nightmare - taking algebra. Next, you will hear the immortal words: “I was never any good at mathematics.” My response is: “I was never any good at being a taxi driver so I went into mathematics.” You can learn a lot from taxi drivers if you just don’t tell them you are a mathematician. Why get started on the wrong foot? The mathematician David Mumford put it: “I am accustomed, as a professional mathematician, to living in a sort of vacuum, surrounded by people who declare with an odd sort of pride that they are mathematically illiterate.” 1.2. A Balancing Act. The other most common response we get from the public is: “I can’t even balance my checkbook.” This reflects the fact that the public thinks that mathematics is basically just adding numbers. They have no idea what we really do. Because of the textbooks they studied, they think that all needed mathematics has already been discovered. -
The "Greatest European Mathematician of the Middle Ages"
Who was Fibonacci? The "greatest European mathematician of the middle ages", his full name was Leonardo of Pisa, or Leonardo Pisano in Italian since he was born in Pisa (Italy), the city with the famous Leaning Tower, about 1175 AD. Pisa was an important commercial town in its day and had links with many Mediterranean ports. Leonardo's father, Guglielmo Bonacci, was a kind of customs officer in the North African town of Bugia now called Bougie where wax candles were exported to France. They are still called "bougies" in French, but the town is a ruin today says D E Smith (see below). So Leonardo grew up with a North African education under the Moors and later travelled extensively around the Mediterranean coast. He would have met with many merchants and learned of their systems of doing arithmetic. He soon realised the many advantages of the "Hindu-Arabic" system over all the others. D E Smith points out that another famous Italian - St Francis of Assisi (a nearby Italian town) - was also alive at the same time as Fibonacci: St Francis was born about 1182 (after Fibonacci's around 1175) and died in 1226 (before Fibonacci's death commonly assumed to be around 1250). By the way, don't confuse Leonardo of Pisa with Leonardo da Vinci! Vinci was just a few miles from Pisa on the way to Florence, but Leonardo da Vinci was born in Vinci in 1452, about 200 years after the death of Leonardo of Pisa (Fibonacci). His names Fibonacci Leonardo of Pisa is now known as Fibonacci [pronounced fib-on-arch-ee] short for filius Bonacci. -
Reproductions Supplied by EDRS Are the Best That Can Be Made from the Original Document
DOCUMENT RESUME ED 456 844 IR 058 192 TITLE Sources of Custom-Produced Books: Braille, Audio Recordings, and Large Print. INSTITUTION Library of Congress, Washington, DC. National Library Service for the Blind and Physically Handicapped. REPORT NO DI015 ISSN ISSN-1535-1505 PUB DATE 2001-00-00 NOTE 107p. AVAILABLE FROM Reference Section, National Library Service for the Blind and Physically Handicapped, Library of Congress, Washington, DC 20542. Web site: http://www.loc.gov/nls/reference/directories.html. PUB TYPE Reference Materials Directories/Catalogs (132) EDRS PRICE MF01/PC05 Plus Postage. DESCRIPTORS Audiotape Recordings; *Books; Braille; Individual Needs; *Information Sources; Large Type Materials; Reading Materials; Talking Books; Visual Impairments; Volunteers IDENTIFIERS National Library Service for the Blind; *Transcription ABSTRACT This directory lists the names of volunteer groups, individual transcribers, and nonprofit and commercial organizations that transcribe and record books and other reading materials for persons who are blind and physically handicapped. It was compiled from information supplied by organizations and groups who perform these services. The listing is alphabetical by state. Each entry is assigned an index number and specifies such services as Braille transcription, computer-assisted transcription, print enlargements, tape recording, duplication, and binding. Entries also give such Braille code specialties as music, mathematics, and specific languages. The directory contains information in separate sections on state special education contacts and proofreaders certified by the Library of Congress. Wherever Braille groups are listed, it is understood that there is at least one transcriber or proofreader certified by the Library of Congress working with the group/organization. The introduction includes a list of other related documents on the National Library Service for the Blind and Physically Handicapped (NLS) Web site or available upon request, as well as additional resources for materials available in different formats. -
The Astronomers Tycho Brahe and Johannes Kepler
Ice Core Records – From Volcanoes to Supernovas The Astronomers Tycho Brahe and Johannes Kepler Tycho Brahe (1546-1601, shown at left) was a nobleman from Denmark who made astronomy his life's work because he was so impressed when, as a boy, he saw an eclipse of the Sun take place at exactly the time it was predicted. Tycho's life's work in astronomy consisted of measuring the positions of the stars, planets, Moon, and Sun, every night and day possible, and carefully recording these measurements, year after year. Johannes Kepler (1571-1630, below right) came from a poor German family. He did not have it easy growing Tycho Brahe up. His father was a soldier, who was killed in a war, and his mother (who was once accused of witchcraft) did not treat him well. Kepler was taken out of school when he was a boy so that he could make money for the family by working as a waiter in an inn. As a young man Kepler studied theology and science, and discovered that he liked science better. He became an accomplished mathematician and a persistent and determined calculator. He was driven to find an explanation for order in the universe. He was convinced that the order of the planets and their movement through the sky could be explained through mathematical calculation and careful thinking. Johannes Kepler Tycho wanted to study science so that he could learn how to predict eclipses. He studied mathematics and astronomy in Germany. Then, in 1571, when he was 25, Tycho built his own observatory on an island (the King of Denmark gave him the island and some additional money just for that purpose). -
Newton.Indd | Sander Pinkse Boekproductie | 16-11-12 / 14:45 | Pag
omslag Newton.indd | Sander Pinkse Boekproductie | 16-11-12 / 14:45 | Pag. 1 e Dutch Republic proved ‘A new light on several to be extremely receptive to major gures involved in the groundbreaking ideas of Newton Isaac Newton (–). the reception of Newton’s Dutch scholars such as Willem work.’ and the Netherlands Jacob ’s Gravesande and Petrus Prof. Bert Theunissen, Newton the Netherlands and van Musschenbroek played a Utrecht University crucial role in the adaption and How Isaac Newton was Fashioned dissemination of Newton’s work, ‘is book provides an in the Dutch Republic not only in the Netherlands important contribution to but also in the rest of Europe. EDITED BY ERIC JORINK In the course of the eighteenth the study of the European AND AD MAAS century, Newton’s ideas (in Enlightenment with new dierent guises and interpre- insights in the circulation tations) became a veritable hype in Dutch society. In Newton of knowledge.’ and the Netherlands Newton’s Prof. Frans van Lunteren, sudden success is analyzed in Leiden University great depth and put into a new perspective. Ad Maas is curator at the Museum Boerhaave, Leiden, the Netherlands. Eric Jorink is researcher at the Huygens Institute for Netherlands History (Royal Dutch Academy of Arts and Sciences). / www.lup.nl LUP Newton and the Netherlands.indd | Sander Pinkse Boekproductie | 16-11-12 / 16:47 | Pag. 1 Newton and the Netherlands Newton and the Netherlands.indd | Sander Pinkse Boekproductie | 16-11-12 / 16:47 | Pag. 2 Newton and the Netherlands.indd | Sander Pinkse Boekproductie | 16-11-12 / 16:47 | Pag. -
Mathematics in the Modern Age - the 18Th Century: Crossing Bridges Transcript
Mathematics in the modern age - The 18th century: Crossing bridges Transcript Date: Wednesday, 4 October 2006 - 12:00AM Location: Barnard's Inn Hall MATHEMATICS IN THE MODERN AGE - THE 18TH CENTURY: CROSSING BRIDGES Professor Robin Wilson Introduction After Isaac Newton had published his celebrated PrincipiaMathematicain 1687 – the book that explained gravitation and how the planets move – his life changed for ever, and in a way that would also affect the nature of British mathematics for the next 150 years or so. His book was highly praised in Britain, even though few people could understand it, and the reclusive Newton was now a public figure. After spending a year as an ineffective Member of Parliament for Cambridge University – apparently he spoke only once, and that was to ask an official to open a window – Newton became Warden, and later Master, of the Royal Mint, living in the Tower of London, supervising the minting of coins, and punishing counterfeiters. He had now left Cambridge for good. In 1703 Newton’s arch-rival Robert Hooke died, with repercussions for both Gresham College where he was Gresham Professor of Geometry, and the Royal Society which still held its meetings in Sir Thomas Gresham’s former house. The rebuilding of the Royal Exchange after the Great Fire of 1666 had been costly, and attendances at Gresham lectures were then sparse, so proposals were made to save money by rebuilding the College on a smaller scale. Parliament was petitioned for approval, with only Robert Hooke, now frail and the only professor resident in the College, holding out against the plans. -
Famous Mathematicians Key
Infinite Secrets AIRING SEPTEMBER 30, 2003 Learning More Dzielska, Maria. Famous Mathematicians Hypatia of Alexandria. Cambridge, MA: Harvard University The history of mathematics spans thousands of years and touches all parts of the Press, 1996. world. Likewise, the notable mathematicians of the past and present are equally Provides a biography of Hypatia based diverse.The following are brief biographies of three mathematicians who stand out on several sources,including the letters for their contributions to the fields of geometry and calculus. of Hypatia’s student Synesius. Hypatia of Alexandria Bhaskara wrote numerous papers and Biographies of Women Mathematicians books on such topics as plane and spherical (c. A.D. 370–415 ) www.agnesscott.edu/lriddle/women/ Born in Alexandria, trigonometry, algebra, and the mathematics women.htm of planetary motion. His most famous work, Contains biographical essays and Egypt, around A.D. 370, Siddhanta Siromani, was written in A.D. comments on woman mathematicians, Hypatia was the first 1150. It is divided into four parts: “Lilavati” including Hypatia. Also includes documented female (arithmetic), “Bijaganita” (algebra), resources for further study. mathematician. She was ✷ ✷ ✷ the daughter of Theon, “Goladhyaya” (celestial globe), and a mathematician who “Grahaganita” (mathematics of the planets). Patwardhan, K. S., S. A. Naimpally, taught at the school at the Alexandrine Like Archimedes, Bhaskara discovered and S. L. Singh. Library. She studied astronomy, astrology, several principles of what is now calculus Lilavati of Bhaskaracharya. centuries before it was invented. Also like Delhi, India:Motilal Banarsidass, 2001. and mathematics under the guidance of her Archimedes, Bhaskara was fascinated by the Explains the definitions, formulae, father.She became head of the Platonist concepts of infinity and square roots. -
Cavendish the Experimental Life
Cavendish The Experimental Life Revised Second Edition Max Planck Research Library for the History and Development of Knowledge Series Editors Ian T. Baldwin, Gerd Graßhoff, Jürgen Renn, Dagmar Schäfer, Robert Schlögl, Bernard F. Schutz Edition Open Access Development Team Lindy Divarci, Georg Pflanz, Klaus Thoden, Dirk Wintergrün. The Edition Open Access (EOA) platform was founded to bring together publi- cation initiatives seeking to disseminate the results of scholarly work in a format that combines traditional publications with the digital medium. It currently hosts the open-access publications of the “Max Planck Research Library for the History and Development of Knowledge” (MPRL) and “Edition Open Sources” (EOS). EOA is open to host other open access initiatives similar in conception and spirit, in accordance with the Berlin Declaration on Open Access to Knowledge in the sciences and humanities, which was launched by the Max Planck Society in 2003. By combining the advantages of traditional publications and the digital medium, the platform offers a new way of publishing research and of studying historical topics or current issues in relation to primary materials that are otherwise not easily available. The volumes are available both as printed books and as online open access publications. They are directed at scholars and students of various disciplines, and at a broader public interested in how science shapes our world. Cavendish The Experimental Life Revised Second Edition Christa Jungnickel and Russell McCormmach Studies 7 Studies 7 Communicated by Jed Z. Buchwald Editorial Team: Lindy Divarci, Georg Pflanz, Bendix Düker, Caroline Frank, Beatrice Hermann, Beatrice Hilke Image Processing: Digitization Group of the Max Planck Institute for the History of Science Cover Image: Chemical Laboratory. -
Primality Testing for Beginners
STUDENT MATHEMATICAL LIBRARY Volume 70 Primality Testing for Beginners Lasse Rempe-Gillen Rebecca Waldecker http://dx.doi.org/10.1090/stml/070 Primality Testing for Beginners STUDENT MATHEMATICAL LIBRARY Volume 70 Primality Testing for Beginners Lasse Rempe-Gillen Rebecca Waldecker American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell Gerald B. Folland (Chair) Serge Tabachnikov The cover illustration is a variant of the Sieve of Eratosthenes (Sec- tion 1.5), showing the integers from 1 to 2704 colored by the number of their prime factors, including repeats. The illustration was created us- ing MATLAB. The back cover shows a phase plot of the Riemann zeta function (see Appendix A), which appears courtesy of Elias Wegert (www.visual.wegert.com). 2010 Mathematics Subject Classification. Primary 11-01, 11-02, 11Axx, 11Y11, 11Y16. For additional information and updates on this book, visit www.ams.org/bookpages/stml-70 Library of Congress Cataloging-in-Publication Data Rempe-Gillen, Lasse, 1978– author. [Primzahltests f¨ur Einsteiger. English] Primality testing for beginners / Lasse Rempe-Gillen, Rebecca Waldecker. pages cm. — (Student mathematical library ; volume 70) Translation of: Primzahltests f¨ur Einsteiger : Zahlentheorie - Algorithmik - Kryptographie. Includes bibliographical references and index. ISBN 978-0-8218-9883-3 (alk. paper) 1. Number theory. I. Waldecker, Rebecca, 1979– author. II. Title. QA241.R45813 2014 512.72—dc23 2013032423 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. -
JBI 4.05 Newsletter
FOCUS ON JBI The Newsletter of JBI International • Spring 2007 JBI International JBI Reunites Senior and Volunteer After 40 Years (established in 1931 become a well-respected actor in the Moscow as the Jewish Braille Theater, but never saw her classmate again. Institute) provides the visually impaired, blind, Irina hurried upstairs to her apartment and physically handicapped returned with a photo album with a picture of and reading disabled of their graduating acting class. Sure enough, it was Victor! all backgrounds and ages with free books, Until that presentation, Irina did not realize how magazines and special valuable her acting skills would be for recording publications of Jewish materials for JBI’s Russian Collection . She and general interest in JBI volunteer Victor Persik reunites and reminisces with jumped at the opportunity to “do some acting Audio, Large Print and old friend Irina Gusso. again,” and we arranged for her to visit the JBI Braille. JBI, an Affiliated familiar name from the past touched off a studios. Little did Victor anticipate the Library of the United Amemory, and then a happy reunion, at JBI. wonderful surprise we had arranged for him States Library of It all began last December during a JBI Russian when he arrrived for his recording session. Congress, enables language outreach presentation at a senior There stood Irina, waiting to greet her old individuals with residence in Irvington, New Jersey, where 20 friend with arms wide open! diminished vision to seniors listened as Inna Suholutsky, our Russian understand and Liaison and Outreach Assistant, spoke about JBI JBI offers a wide variety of materials and participate in the cultural Library’s wealth of Russian Talking Books programs for the Russian community world- life of their communities. -
Extending Tangible Interactive Interfaces for Education: a System for Learning Arabic Braille Using an Interactive Braille Keypad
(IJACSA) International Journal of Advanced Computer Science and Applications, Vol. 11, No. 2, 2020 Extending Tangible Interactive Interfaces for Education: A System for Learning Arabic Braille using an Interactive Braille Keypad Hind Taleb Bintaleb1, Duaa Al Saeed2 Information Technology Department, College of Computer and Information Sciences King Saud University, Riyadh, KSA Abstract—Learning Braille for visual impairments means A basic Braille template (cell) is a tactile configuration of being able to read, write and communicate with others. There six raised\embossed dots. It is upright rectangular shapes exist several educational tools for learning Braille. Unfortunately, made of two vertical columns made of three dot positions. The for Arabic Braille, there is a lack of interactive educational tools cell is organized as a matrix of 2 × 3 dots. Those dots are and what is mostly used is the traditional learning tools, such as numerically identified by the numbers 1 to 6, see Fig. 2. There the Braille block. Replacing those tools with some more effective are different combinations of raised\embossed dots, each and interactive e-learning tools would help to improve the unique configuration represents an alphabetical letter or a learning process. This paper introduces a new educational number or a symbol. a consonant, a vowel, a number, a system with a tangible and interactive interface. This system aims diacritical mark or an abbreviated suffix [2][3]. Through the to help blind children to learn Arabic Braille letters and numbers combination of dot positions and their distribution on the two using an interactive tactile Braille keypad together with the educational website.