...The definition of the Butterfly Theorem is here below: “Let M be the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn; AD cuts PQ at X and BC cuts PQ at Y. Prove that M is also the midpoint of XY.” (Bogomolny) This is the most accurate definition currently. However, Butterfly Theorem has experienced some changes and developments. The first statement of the Butterfly Theorem appeared in the early 17th century. In 1803, a Scottish mathematician, William Wallace, posed the problem of the Butterfly Theorem in the magazine The Gentlemen’s Mathematical Companion. Here is the original problem below: “If from any two points B, E, in the circumference of a circle given in magnitude and position two right lines BCA, EDA, be drawn cutting the circle in C and D, and meeting in A; and from the point of intersection A to the centre of the circle AO be drawn, and the points E, C; B, D joined, and produced to meet an indefinite perpendicular erected at A on AO; then will FA be always equal AF. Required the demonstration?”(Bogomolny) (Figures of W Wallace’s question) Soon afterwards, there were three solutions published in 1804. And in 1805, William Herschel, a British...
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...History of Geometry Geometry was thoroughly organized in about 300 BC, when the Greek mathematician Euclid gathered what was known at the time, added original work of his own, and arranged 465 propositions into 13 books, called 'Elements'. The books covered not only plane and solid geometry but also much of what is now known as algebra, trigonometry, and advanced arithmetic. Through the ages, the propositions have been rearranged, and many of the proofs are different, but the basic idea presented in the 'Elements' has not changed. In the work facts are not just cataloged but are developed in a fashionable way. Even in 300 BC, geometry was recognized to be not just for mathematicians. Anyone can benefit from the basic learning of geometry, which is how to follow lines of reasoning, how to say precisely what is intended, and especially how to prove basic concepts by following these lines of reasoning. Taking a course in geometry is beneficial for all students, who will find that learning to reason and prove convincingly is necessary for every profession. It is true that not everyone must prove things, but everyone is exposed to proof. Politicians, advertisers, and many other people try to offer convincing arguments. Anyone who cannot tell a good proof from a bad one may easily be persuaded in the wrong direction. Geometry provides a simplified universe, where points and lines obey believable rules and where conclusions are easily verified. By first studying how to reason......
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...Kindergarten Common Core State Standards Flip Book This document is intended to show the connections to the Standards of Mathematical Practices for the content standards and to get detailed information at each level. Resources used: CCSS, Arizona DOE, Ohio DOE and North Carolina DOE. This ―flip book‖ is intended to help teachers understand what each standard means in terms of what students must know and be able to do. It provides only a sample of instructional strategies and examples. The goal of every teacher should be to guide students in understanding & making sense of the mathematics they are presented. Construction directions: Print on cardstock. Cut the tabs on each page starting with page 2. Cut the bottom off of this top cover to reveal the tabs for the subsequent pages. Staple or bind the top of all pages to complete your flip book. Compiled by Melisa Hancock (Send feedback to: melisa@ksu.edu) 1 Mathematical Practice Standards (MP) summary of each standard 1. Make sense of problems and persevere in solving them. Mathematically proficient students interpret and make meaning of the problem looking for starting points. In Kindergarten, students begin to build the understanding that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Younger students may use concrete objects or pictures to help them conceptualize and solve problems....
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...Failing in this effort, he was disgraced and established himself as a copyist of mathematical documents there still exists in Istanbul a document of the Banu Musa's version of Conics copied by him in 1024. He continued to practice the scribal art in Cairo for the remainder of his life. He did not cease to follow his scientific studies, however, and published a large number of highly original works. He produced two catalogs of his own work, which are preserved by Ibn abi Usaybia. The first of these, compiled in 1027, comprises 25 books on mathematics and 44 on physics and metaphysics, including On the Structure of the World. The second, supplementary catalog was complied in 1028. Work in Starwatching The primary interest of al-Hassan was the explanation of wonders by both mathematical and physical hypotheses. His interest in astronomy was motivated by the difference between physical and mechanistic model of the holy scopes and the mathematical model. On the Structure of the World, of which only the Latin translation has been published, describes world of four elements and the holy scopes in all their difficulty (his only change is to accept the theory that the solar apogee is fixed with respect to the fixed stars) as if they were material. He inserts a discussion of the perception of lunar...
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...Brahmagupta (598-668),who had worked on mathematics and astronomy, was the head of the astronomy observatory in Ujjain, which was at that point of time, the foremost mathematical centre in India; he and Bhaskar the second (1114-1185), who reached understanding on the number systems and solving equations, have together provided many rules for arithmetical operations with the zero. Varahamihira (505-668) who was educated in Kapitthaka and was one of the patrons of the school of mathematics in Ujjain, worked on Hindu astronomy before Aryabhata.He wrote manuals called Panchasiddhantika which refer to the addition and subtraction of zero. Vasubhandu (around 400 AD), who was born into a Hindu family but later converted to Buddhism,...
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...Some people also go as far as calling it the mathematical equivalent of Da Vinci's Mona Lisa or Michaelangelo's David. Named after Leonhard Euler, the formula establishes the deep relationship between trigonometric functions and complex exponential functions. According to the formula, for any real number x, In the above formula, e is the base of the natural logarithm, i the imaginary unit. Cos and Sin are trigonometric functions (the arguments are to be taken in radians and not degrees.). The formula applies even if x is a complex number. Particularly with x = π, or half a turn around the circle, e^iπ = cos π + i sin π Since cos π = -1 and sin π = 0, It can be deduced that e^iπ = -1 + i0 which brings us to the identity The identity successfully links five fundamental mathematical constants: 1. The number 0(the additive identity). 2. The number 1(the multiplicative identity). 3. The number pi (3.14159265…). 4. The number e (base of all natural logarithms, which occurs widely in mathematics and scientific analysis). 5. The number i (the imaginary unit of the complex numbers) The formula describes two equivalent ways to move in a circle. One of its major applications is that in the complex number theory....
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...Maths in nature "The laws of nature are but the mathematical thoughts of God" - Euclid Mathematics is everywhere in this universe. We seldom note it. We enjoy nature and are not interested in going deep about what mathematical idea is in it. Here are a very few properties of mathematics that are depicted in nature. SYMMETRY Symmetry is everywhere you look in nature . Symmetry is when a figure has two sides that are mirror images of one another. It would then be possible to draw a line through a picture of the object and along either side the image would look exactly the same. This line would be called a line of symmetry. There are two kinds of symmetry. One is bilateral symmetry in which an object has two sides that are mirror images of each other. The human body would be an excellent example of a living being that has bilateral symmetry. The other kind of symmetry is radial symmetry. This is where there is a center point and numerous lines of symmetry could be drawn. The most obvious geometric example would be a circle. Shapes Sphere: A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. The shape of the Earth is very close to that of an oblate spheroid, a sphere flattened along the axis from pole to pole such that there is a bulge around the equator....
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...These components are integral to the development of mathematical problem solving ability. Emphasis is also given to reasoning, applications, and use of technology. Advances in technology have changed the way we teach and learn mathematics. The computer and hand-held calculator, for example, offer great potential to enhance the teaching and learning of mathematics. Students will have opportunities to discover, reason and communicate mathematics. They will engage in stimulating discussions and activities where they can explore possibilities and make connections. These qualitative changes require a change in the teaching and learning approaches; incorporating activity-based and learnercentred methodologies. The syllabuses are conceptualised after extensive consultation with teachers. We hope that teachers will find the document useful and continue to provide us with valuable feedback on syllabus matters. We wish you an enjoyable and successful teaching experience. Mathematics Unit Curriculum Planning and Development Division March 2006 2 CONTENTS Preface 4 PART A INTRODUCTION 1 Rationale 2 Aims of Mathematics Education in Schools 3 Mathematics Framework 3.1 Concepts 3.2 Skills 3.3 Processes 3.4 Attitudes 3.5 Metacognition 5 5 6 PART B PRIMARY MATHEMATICS CURRICULUM 4...
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...Mathematical Conventions for the Quantitative Reasoning Measure of the GRE® revised General Test www.ets.org Overview The mathematical symbols and terminology used in the Quantitative Reasoning measure of the test are conventional at the high school level, and most of these appear in the Math Review. Whenever nonstandard or special notation or terminology is used in a test question, it is explicitly introduced in the question. However, there are some particular assumptions about numbers and geometric figures that are made throughout the test. These assumptions appear in the test at the beginning of the Quantitative Reasoning sections, and they are elaborated below. Also, some notation and terminology, while standard at the high school level in many countries, may be different from those used in other countries or from those used at higher or lower levels of mathematics. Such notation and terminology are clarified below. Because it is impossible to ascertain which notation and terminology should be clarified for an individual test taker, more material than necessary may be included. Finally, there are some guidelines for how certain information given in test questions should be interpreted and used in the context of answering the questions—information such as certain words, phrases, quantities, mathematical expressions, and displays of data. These guidelines appear at the end. Copyright © 2012 by Educational Testing Service. All rights reserved....
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...CARIBBEAN EXAMINATIONS COUNCIL Caribbean Secondary Education Certificate CSEC MATHEMATICS SYLLABUS Effective for examinations from May/June 2010 CXC 05/G/SYLL 08 Published in Jamaica © 2010, Caribbean Examinations Council All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form, or by any means electronic, photocopying, recording or otherwise without prior permission of the author or publisher. Correspondence related to the syllabus should be addressed to: The Pro-Registrar Caribbean Examinations Council Caenwood Centre 37 Arnold Road, Kingston 5, Jamaica, W.I. Telephone: (876) 630-5200 Facsimile Number: (876) 967-4972 E-mail address: cxcwzo@cxc.org Website: www.cxc.org Copyright © 2008, by Caribbean Examinations Council The Garrison, St Michael BB11158, Barbados CXC 05/OSYLL 00 Contents RATIONALE. .......................................................................................................................................... 1 AIMS. ....................................................................................................................................................... 1 ORGANISATION OF THE SYLLABUS. ............................................................................................. 2 FORMAT OF THE EXAMINATIONS ................................................................................................ 2 CERTIFICATION AND PROFILE DIMENSIONS......
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...|Skills: | | |A. 2 |Identify the givens | | |B. 3 |Circle all the values | | |C. 4 |Underline the “clue words” (mental triggers) in the word problem that tell us | | |D. 5 |what mathematical symbols to use | | |13 = 5 + 4(m-1) |+...
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...They were less concerned with developing mathematical predictive models than with developing an explanation of the reasons for the motions of the Cosmos. In his Timaeus, Plato described the universe as a spherical body divided into circles carrying the planets and governed according to harmonic intervals by a world soul. Aristotle, drawing on the mathematical model of Eudoxus, proposed that the universe was made of a complex system of concentric spheres, whose circular motions combined to carry the planets around the earth. This basic cosmological model prevailed, in various forms, until the 16th century AD. In the 3rd century BC Aristarchus of Samos was the first to suggest a heliocentric system, although only fragmentary descriptions of his idea survive. Eratosthenes, using the angles of shadows created at widely separated regions, estimated the circumference of the Earth with great accuracy. Greek geometrical astronomy developed away from the model of concentric spheres to employ more complex models in which an eccentric circle would carry around a smaller circle, called an epicycle which in turn carried around a planet. The first such model is attributed to Apollonius of Perga and further developments in it were carried out in the 2nd century BC by Hipparchus of Nicea. Hipparchus made...
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...South Carolina Math Kindergarten Standard K-4 The student will demonstrate through the mathematical processes an emerging sense of two- and three- dimensional geometric shapes and relative positions in space (Standards, 2012). Instructional Goal 1 Identify two-dimensional shapes and three-dimensional shapes. Learning Objective 1: Students will take and identify foam two-dimensional shapes square, circle, triangle, and rectangle from a mystery bag with 80% accuracy. Justification: The mystery bag is used to cover the foam shapes from view. Students will need to use their sense of touch and their knowledge of the properties of the solids to identify them. Students are able to explore different orientations, sizes, and types to recognize that each shape has distinguishable characteristics. Learning Objective 2: Students will identify and record on paper three-dimensional shapes cube, sphere, and cylinder using real-world examples of the solids with 80% accuracy. Justification: Students explore these shapes using concrete models, pictures, and real world examples to generalize connections among mathematics, the environment, and other subjects. Learning Objective 3: Students will identify two out of three attributes (color, sides, size) from two-dimensional shapes square, circle, triangle, and rectangle. Justification: Students will use memorization and recognition skills to tackle these mathematical ideas....
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...What’s In A Name? Yan Martel’s award winning novel, “Life of Pi”, is about the life of Piscine Molitor Patel, narrating many of the experiences he encounters that give light to many philosophies and ideologies about life. The protagonist in the story, Piscine Molitor, was named after a Parisian swimming pool, which was described in the story as a breathtaking area to take a good swim. However, Piscine having been fed up with the negative attention his name attracted in school, comes up with a new nickname for himself: Pi. Like many other concepts that Martel uses within the plot as a tool for readers to gain a better understanding on things, the two aforementioned meanings of his name is one of which the utilized. The two names of the protagonist in the story of “Life of Pi” is one of the many tools which Yan Martel uses to embody much more meaning than what meets the eye. The narration of how he acquired the name “Piscine Molitor” is told early on in the novel within the third chapter. This close association to a body of water foreshadows the entire second chapter where he spends in the Pacific Ocean, a much bigger body of water, as a cast-away for 227 days. Moreover, the controlled and confined features of a pool imply the concept of boundaries, (a concept which the author constantly used/ made to appear in the novel), like the boundary that Pi made between him and Richard Parker was an essential tool that kept them both alive and the boundary between fact......
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...Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.[11] Galileo Galilei...
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