This paper investigates whether it is reasonable to bestow credit to one person or group for the famed theorem that relates to the side lengths of any right-angled triangle, a theorem routinely referred to as the “Pythagorean Theorem”. The author investigates the first-documented occurrences of the theorem, along with its first proofs. In addition, proofs that stem from different branches of mathematics and science are analyzed in an effort to display that credit for the development of the theorem should be shared amongst its many contributors rather than crediting the whole of the theorem to one man and his supporters.

This paper investigates whether it is reasonable to bestow credit to one person or group for the famed theorem that relates to the side lengths of any right-angled triangle, a theorem routinely referred to as the “Pythagorean Theorem”. The author investigates the first-documented occurrences of the theorem, along with its first proofs. In addition, proofs that stem from different branches of mathematics and science are analyzed in an effort to display that credit for the development of the theorem should be shared amongst its many contributors rather than crediting the whole of the theorem to one man and his supporters.

The aim of this study is to analyze students¡¦ mathematics concepts in solving Pythagorean Theorem problems presented in two different representations (word problems and word problems with diagrams). The investigators employed the mathematics competence indicators in Grade 1-9 Integrated Curriculum in developing such problems. In analyzing data, the investigator used Schoenfeld¡¦s method in depicting their problem-solving processes, with attention to students¡¦ sequence and difference in time consumption. Four eight grade students with good competence in mathematics and expressions from a secondary school were selected as research subjects. Problems related to Pythagorean Theorem were divided into three types: Shape, Area, and Number. Data were collected using thinking aloud method and semi-structured interview, and triangulation was further applied in protocol analysis. The research results revealed 3 findings: (1) For the ¡§Shape¡¨ type problems, students¡¦ problem-solving concepts varied with different problem representation. For the ¡§Area¡¨ and ¡§Number¡¨ types of problems (without diagram), students were required to use their geometric concept when processing word problems. Students¡¨ use of problem-solving concepts would not significantly vary with problem representation types. However, students¡¦ use of problem-solving methods would affect the types and priorities of concepts used. Generally, the types of mathematics concepts could be made up by the frequency of concepts used, and more types of problem-solving concepts would be used for word problems representation than for word problems with diagrams representation. (2) In terms of the time consumed in the first three problem-solving stages of Schoenfeld, the time required to solve word problems was 1.6 times of that required to solve word problems with diagrams. In terms of the total time consumed, the time required to solve word problems was 1.25 times of that required to solve word problems with diagrams. In the problem-solving stages, students needed to explore the problem first when dealing with word problems before they could go on to solve the problem, and such repetition was more frequent when they dealt with word problems. (3) For both type of problem representations, there is a higher number of correctly-answered problems. This finding indicated that a higher frequency of problem-solving concepts and less repetition in the problem-solving stage were required; and vice versa. As to the sequence of Pythagorean Theorem concepts to be taught, the investigator suggest teachers to start with the concept of area filling in the ¡§Shape¡¨ type of problems to derive Pythagorean Theorem, and further apply the formula to - III - solving ¡§Number¡¨ problems. After students have acquired basic competency in ¡§Shape¡¨ and ¡§Number¡¨ Pythagorean Theorem problems, teachers could explain and introduce this theorem from the perspective of ¡§Area¡¨. Finally, in problem posing, teachers were also advised to apply various contexts; covering all kinds of representations of problems that enhance students¡¦ utilization of mathematics concepts; and to cater for various needs of students.

SILVA, Danniel Emanuel Bruno. O teorema de Pitágoras: abordagem no cotidiano da educação matemática e suas diversas demonstrações. 2014. 55 f. Dissertação (Mestrado Profissional em Matemática em Rede Nacional) - Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2014. / Submitted by Jessyca Silva (jessyca@mat.ufc.br) on 2017-08-21T19:55:33Z No. of bitstreams: 1 2014_dis_debsilva.pdf: 1316565 bytes, checksum: f9f30b38921af75056aba52ae1360c98 (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Bom dia, Na folha de aprovação não deve constar as assinaturas dos membros da banca. Rocilda on 2017-08-22T13:51:25Z (GMT) / Submitted by Jessyca Silva (jessyca@mat.ufc.br) on 2017-08-23T14:35:03Z No. of bitstreams: 1 2014_dis_debsilva.pdf: 1008247 bytes, checksum: e15ea5795b214cc268197d35bb86e018 (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Boa tarde, encontrei alguns pequenos erros na Dissertação de DANNIEL EMANUEL, e já envie uma cópia do email para ele avisando quais eram os erros. 1 – FOLHA DE ROSTO (centralize o nome Fortaleza e o ano da Dissertação) 2- FICHA CATALOGRÁFICA (o título da Dissertação que aparece na ficha catalográfica deve estar em letra minúscula, só a primeira letra deve ser maiúscula) 3- FOLHA DE APROVAÇÃO (está faltando seu nome na parte superior da folha de aprovação, acima do título da Dissertação) 4- EPÍGRAFE (retire o sublinha que aparece no nome EINSTEIN) 5- NUMERAÇÃO DAS PÁGINAS (coloque a numeração das páginas no CANTO SUPERIOR DIREITO) Atenciosamente, on 2017-08-23T15:50:40Z (GMT) / Submitted by Jessyca Silva (jessyca@mat.ufc.br) on 2017-08-23T19:33:51Z No. of bitstreams: 1 2014_dis_debsilva.pdf: 1008605 bytes, checksum: 92cfe1499970b1c1f1a10d77725f68b3 (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2017-08-25T11:13:23Z (GMT) No. of bitstreams: 1 2014_dis_debsilva.pdf: 1008605 bytes, checksum: 92cfe1499970b1c1f1a10d77725f68b3 (MD5) / Made available in DSpace on 2017-08-25T11:13:24Z (GMT). No. of bitstreams: 1 2014_dis_debsilva.pdf: 1008605 bytes, checksum: 92cfe1499970b1c1f1a10d77725f68b3 (MD5) Previous issue date: 2014 / This work is about the Pythagorean Theorem and its various statements, addressing its importance and as has been shown in mathematics education. We will present some of the history of Pythagoras, as well as on his theorem, and well see a search on their importance, and how students receive this information. We'll talk a bit about how the textbooks, from elementary school to the middle, deal about it, and finally we will see some statements about this theorem seeking to open a lot more range of options for students and lovers of mathematics teachers. These statements are part of the collection assembled by Loomis, who managed to submit more than three hundred statements of the theorem in a publication of 1940. / Este trabalho trata sobre o Teorema de Pitágoras e suas diversas demonstrações, abordando sua importância e como vem sendo apresentado na educação matemática. Apresentaremos um pouco da história de Pitágoras, bem como, sobre o seu teorema, e veremos uma pesquisa sobre a sua importância, e como os alunos recebem essa informação. Falaremos um pouco, sobre como os livros didáticos, tanto do ensino fundamental como no médio, tratam sobre o assunto, e por fim veremos algumas demonstrações sobre este Teorema buscando abrir muito mais o leque de opções para os alunos e professores amantes da matemática. Essas demonstrações fazem parte do acervo reunido por Loomis, que conseguiu apresentar mais de trezentas demonstrações do teorema em uma publicação de 1940.

This study concerns a teaching design with the purpose to facilitate the students’ objectification of the Pythagorean Theorem. Twelve 14-year old students (N=12) participated in the study before the theorem was introduced to them at school. The design incorporated ideas from the ‘embodied mind’ framework, history and realistic mathematics, linking ‘embodied verticality’ with ‘perpendicularity’. The qualitative analyses suggested that the participants were led to the conquest of the ‘first level of objectification’ (through numbers) of the Pythagorean Theorem, showing also evidence of appropriate ‘fore-conceptions’ of the ‘second level of objectification’ (through proof) of the theorem. The triangle the sides of which are associated with the Basic Triple (3,4,5) served as a primary instrument for the students’ objectification, mainly, by facilitating their ‘generic abstraction’ of the Pythagorean Triples.

Submitted by MARCOS EDER CUPAIOLI (marcoscupaioli@hotmail.com) on 2016-09-13T14:53:52Z No. of bitstreams: 1 Dissertação-MARCOS-EDER-CUPAIOLI-Matemática-Final Repositório.pdf: 3030917 bytes, checksum: 5fee5216541ccf70c5a9acb075b9976f (MD5) / Rejected by Felipe Augusto Arakaki (arakaki@reitoria.unesp.br), reason: Solicitamos que realize uma nova submissão seguindo as orientações abaixo: No campo “Versão a ser disponibilizada online imediatamente” foi informado que seria disponibilizado o texto completo porém no campo “Data para a disponibilização do texto completo” foi informado que o texto completo deverá ser disponibilizado apenas 6 meses após a defesa. Caso opte pela disponibilização do texto completo apenas 6 meses após a defesa selecione no campo “Versão a ser disponibilizada online imediatamente” a opção “Texto parcial”. Esta opção é utilizada caso você tenha planos de publicar seu trabalho em periódicos científicos ou em formato de livro, por exemplo e fará com que apenas as páginas pré-textuais, introdução, considerações e referências sejam disponibilizadas. Se optar por disponibilizar o texto completo de seu trabalho imediatamente selecione no campo “Data para a disponibilização do texto completo” a opção “Não se aplica (texto completo)”. Isso fará com que seu trabalho seja disponibilizado na íntegra no Repositório Institucional UNESP. Por favor, corrija esta informação realizando uma nova submissão. Agradecemos a compreensão. on 2016-09-14T21:55:41Z (GMT) / Submitted by MARCOS EDER CUPAIOLI (marcoscupaioli@hotmail.com) on 2016-09-15T01:47:57Z No. of bitstreams: 1 Dissertação-MARCOS-EDER-CUPAIOLI-Matemática-Final Repositório.pdf: 3030917 bytes, checksum: 5fee5216541ccf70c5a9acb075b9976f (MD5) / Approved for entry into archive by Juliano Benedito Ferreira (julianoferreira@reitoria.unesp.br) on 2016-09-15T18:14:38Z (GMT) No. of bitstreams: 1 cupaioli_me_me_sjrp.pdf: 3030917 bytes, checksum: 5fee5216541ccf70c5a9acb075b9976f (MD5) / Made available in DSpace on 2016-09-15T18:14:38Z (GMT). No. of bitstreams: 1 cupaioli_me_me_sjrp.pdf: 3030917 bytes, checksum: 5fee5216541ccf70c5a9acb075b9976f (MD5) Previous issue date: 2016-08-19 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Este trabalho aborda um conjunto de atividades experimentais com a finalidade de demonstrar um dos mais belos e importantes teoremas da Matemática: o Teorema de Pitágoras. São conhecidas mais de 400 demonstrações, aqui optamos por utilizar uma demonstração devido a Rudolf Wolf, por possibilitar uma abordagem geométrica lúdica através da dissecção de figuras planas. Inicialmente apresentamos o conceito geral de semelhança e áreas das figuras planas que utilizam propriedades e áreas de polígonos equidecomponíveis. Posteriormente, realizamos um breve resgate histórico sobre diversas demonstrações do Teorema e da vida de Pitágoras. Destacamos, também, uma maneira de achar algumas ternas pitagóricas, utilizando a sequência de Fibonacci. Por fim, foram propostas e desenvolvidas atividades experimentais em sala de aula com a utilização de moldes em EVA, explorando o Teorema de Pitágoras e algumas de suas aplicações. / This work contains a set of experimental activities in order to prove one of the most beautiful and important theorems in Mathematics: the Pythagorean Theorem. There are known more than 400 proofs, here we chose to use a proof due to Rudolf Wolf, by allowing a playful geometric approach by dissection of plane figures. Initially we present the general concept of similarity and areas of plane figures using properties and areas of equidecomposable polygons. Later, we do a brief historical review of some proofs of Theorem and Pythagoras's life. We also highlight a way to find some Pythagorean triples using the Fibonacci sequence. Finally, it was proposed and developed experimental activities in the classroom with the use of molds EVA, exploring the Pythagorean theorem and some of its applications.

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We give some background information about the Cauchy-Schwarz inequality including its history. We then continue by providing a number of proofs for the inequality in its classical form using various proof techniques, including proofs without words. Next we build up the theory of inner product spaces from metric and normed spaces and show applications of the Cauchy-Schwarz inequality in each content, including the triangle inequality, Minkowski's inequality and Hölder's inequality. In the final part we present a few problems with solutions, some proved by the author and some by others.

Made available in DSpace on 2015-03-26T14:00:04Z (GMT). No. of bitstreams: 1 texto completo.pdf: 3212056 bytes, checksum: 62f0b7432d4f49ef25c24f61a158a66d (MD5) Previous issue date: 2013-03-18 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This dissertation is devoted to studying the Pythagorean Theorem about various aspects. We begin by tracing a historical about this theorem, and then we present several demonstrations from him, as well as applications and workshops for your teaching. The continuation we present a proposal of activities motivational and creative in the use of this theorem in order to help teachers and arouse interest in students. We finalized this work by presenting an innovative Theorem Primer that should directing the use of labor by mathematics teachers in the classroom. / Esta dissertação é dedicada ao estudo do Teorema de Pitágoras sob vários aspectos. Começamos traçando um histórico deste teorema, e então apresentamos várias demonstrações dele, assim como aplicações e oficinas para o ensino do mesmo. A continuação apresentamos uma proposta de atividades motivacionais e criativas para a utilização deste teorema, a fim de ajudar os professores e despertar o interesse nos estudantes. Concluímos este trabalho com a apresentação de uma inovadora Cartilha do Teorema, a Cartilha Pitagórica, que deverá orientar a utilização do trabalho por professores de Matemática em sala de aula.

This work deals with a basic question of modern science and it is its indefectibility. Quality of education is reduce to an evaluation of conformity to a common known knowledge and its quantity representation. Seeds of this long process go back to an ancient academia of Gondisapur established in an Arabic world. Author proclaims that the main goal of philosophy is to show, that this is not the only way of thinking and in the same time that the main goal and power of phenomenology is to apply the transcendental epoche to overcame the truth in its regularization shape. The hardcore of modern science is located in the world of mathematics and a lot of thinkers find the Math as a land of pure sureness - the core of this work in an opposite proofs, that in fact nowadays math is all, but the correct way of thinking. The two examples are explicit: the Pythagorean Theorem and the Sum of the geometric row. This work brings a quite new view on the mathematical problem of "the point" and "the nothing" as a border of things. In the second part uses as a frame of its topic the first 18 §§ of the work "Paradoxes of the infinite" by Czech mathematician of German mother tongue Bernard Bolzano. The important idea of this study is a new ontological view on the set of prime numbers.