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Torres, Mário Régis Rebouças. "Números algébricos e transcendentes." reponame:Repositório Institucional da UFC, 2017. http://www.repositorio.ufc.br/handle/riufc/25736.
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TORRES, Máro Règis Rebouças. Números algébricos e transcendentes. 66 f. Dissertação (Mestrado Profissional em Matemática) - Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017.Submitted by Jessyca Silva (jessyca@mat.ufc.br) on 2017-09-15T05:05:08Z No. of bitstreams: 1 2017_dis_mrrtorres.pdf: 1191154 bytes, checksum: bcb31593bd1a02e84caee6bd47906dab (MD5)
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The present work deals with algebraic and transcendent numbers characterizing them under different aspects. In particular we bring some demonstrations of the irrationality of the number π and the number of Euler, base of the natural logarithm. We will also present a demonstration of the transcendence of the number and based on the script of exercises proposed by D.G. de Figueiredo, in addition to a small historical survey on π, and, algebraic and transcendent numbers.
O presente trabalho trata sobre números algébricos e transcendentes caracterizando-os sob diferentes aspectos. Em particular trazemos algumas demonstrações da irracionalidade do número π e do número de Euler, base do logaritmo natural. Também apresentaremos uma demonstração da transcendência do número e baseada no roteiro de exercícios propostos por D.G. de Figueiredo em [4], além de um pequeno apanhado histórico sobre π, e, números algébricos e transcendentes.
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Bezerra, Rafael Tavares Silva. "Frações contínuas - um estudo sobre "boas" aproximações." Universidade Federal da Paraíba, 2016. http://tede.biblioteca.ufpb.br:8080/handle/tede/9341.
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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
The study of ontinued fra tions will start with some histori al fa ts, aiming at a better understanding of the subje t. We will bring the de nition of ontinued fra tions for a number α real, with the de nition for α rational and α irrational. The dis ussion will fo us on meaning results for the al ulation of redu ed and good approximations of irrational numbers, also aimed at determining the error between the redu ed and the irrational number. We will bring a study of the periodi ontinued fra tions, with emphasis on Lagrange theorem, whi h relates a periodi ontinued fra tion and a quadrati equation. Finishing with a fo us on problem solving, as the al ulation of ontinued fra tions of irrational numbers of the form √a2 + b, as well as proof of the irrationality of e by al ulating its ontinued.
O estudo das frações ontínuas terá ini io om alguns fatos históri os, visando uma melhor ompreensão do tema. Traremos a de nição de frações ontínuas para um erto número α real, apresentando a de nição para α ra ional e para α irra ional. A dis ussão será entrada em resultados importantes para o ál ulo de reduzidas e boas aproximações de números irra ionais, visando também a determinação do erro entre a reduzida e o número irra ional. Traremos um estudo sobre as frações ontínuas periódi as, om enfase ao teorema de Langrange, que rela iona uma fração ontínua periódi a e uma equação do segundo grau. Finalizando om enfoque na resolução de problemas, omo o ál ulo de frações ontínuas de números irra ionais da forma √a2 + b, assim omo a prova da irra ionalidade de e através do ál ulo de sua fração ontínua.
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Menezes, Fernanda Martinez. "Propriedades da expansão decimal." Universidade de São Paulo, 2016. http://www.teses.usp.br/teses/disponiveis/55/55136/tde-05102016-085553/.
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Este trabalho tem como objetivo principal o estudo da expansão decimal dos números reais. Primeiramente provamos que todo número real possui ao menos uma expansão decimal. Na sequência, um método para encontrar a expansão decimal de um número entre 0 e 1 é apresentado, bem como um estudo sobre a expansão decimal de números racionais e irracionais. Em seguida, o estudo apresenta métodos que permitem encontrar aproximações racionais de números irracionais, além dos erros cometidos por essas aproximações. Na parte final, por seu turno, o foco do trabalho recai sobre a análise da regularidade (frequência) dos dígitos das expansões decimais.This work has as main objective the study of the decimal expansion of the real numbers. First we prove that every real number has at least one decimal expansion. Further, a method to find the decimal expansion of real numbers between 0 and 1 is provided as well as a the study of the decimal expansion of rational and irrational numbers. Next, the study presents methods that provide rational approximations to irrational numbers, in addition to the errors committed by these approximations. At the end, by its turn, the focus of the work is put on the analysis of the regularity (frequency) of the digits of the decimal expansion.
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Silva, Guimarães Vieira da. "Irracionalidade e transcendência: aspectos elementares." Universidade Federal do Tocantins, 2018. http://hdl.handle.net/11612/978.
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O presente trabalho tem como perspectiva a caracterização dos números Racionais e
Irracionais, e a sua devida aplicabilidade e variações no que tange o aspecto algébrico e
transcendental. Sabe-se que o Número e (de Euler), pode ser classificado como um número
transcendental, isto é, aqueles que não são raízes de nenhum polinômio que possua coeficientes
inteiros. Nesse pressuposto, o Número deve ser considerado existente e irracional. O
objetivo desta pesquisa consiste em caracterizar os fatores que abrangem os Números Racionais
e Irracionais, oferecendo a compreensão necessária referente ao Número e e a sua
ação nos Números Algébricos e Transcendentes. Como recurso metodológico, utilizou-se
uma revisão de literatura, com um crivo pautado nos fatores qualitativos e quantitativos,
a fim de se refletir sobre a temática proposta. Assim, nesta presente pesquisa, buscouse
apresentar informações dentro das melhores formas e possibilidades de favorecer a
compreensão, considerando a dificuldade em torno deste respectivo tema, devido a sua
característica abstrata, o que dificulta o entendimento por parte de muitos. Portanto,
destacam-se as iniciativas e argumentos em torno deste princípio temático, como forma
de, possivelmente, fomentar o interesse de muitos pelo mesmo, além de que, tal trabalho
possa ser relevante às necessidades de investigação de outros desejosos por este universo
de pesquisa.The present work has as its perspective the characterization of Rational and Irrational numbers, and their due applicability and variations regarding the algebraic and transcendental aspects. It is known that the number e (of Euler) can be classified as a transcendental number, that is, those that are not roots of any polynomial that has integer coefficients. In this assumption, the Number should be considered existent and irrational. The objective of this research is to characterize the factors that comprise the Rational and Irrational Numbers, offering the necessary understanding regarding Number e and its action in Algebraic and Transcendent Numbers. As a methodological resource, a literature review was used, based on qualitative and quantitative factors, in order to reflect on the proposed theme. Thus, in this present research, we sought to present information within the best ways and possibilities to favor understanding, considering the difficulty around this respective theme, due to its abstract feature, which makes it difficult for many to understand. Therefore, we highlight the initiatives and arguments around this thematic principle as a way of possibly fostering the interest of many by the same, and that such work may be relevant to the research needs of others desirous by this universe of research.
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SANTOS, Ana Cláudia Guedes dos. "Uma contribuição ao ensino de números irracionais e de incomensurabilidade para o ensino médio." Universidade Federal de Campina Grande, 2013. http://dspace.sti.ufcg.edu.br:8080/jspui/handle/riufcg/2161.
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Capes
Este trabalho tem como proposta pedagógica apresentar aos alunos o conceito de segmentos comensuráveis e de segmentos incomensuráveis, mostrando a importância desses conceitos para o estudo dos números racionais e irracionais. Veremos um processo de verificação da comensurabilidade de dois segmentos, doravante P.V.C.D.S, que é um processo geométrico de verificação de comensurabilidade de dois segmentos. A partir do P.V.C.D.S, apresentamos a demonstração clássica de que p2 é irracional, com uma abordagem geométrica, mostrando que o segmento do lado de um quadrado de medida 1 e o segmento de sua diagonal são incomensuráveis. Ainda apresentamos um estudo sobre expressões decimais, no qual será apresentado um teorema que nos permite verificar se uma fração irredutível possui representação decimal finita ou infinita e periódica. Também apresentamos outro teorema que nos permite transformar expressões decimais finitas e infinitas e periódicas na sua forma de fração. Por fim, apresentaremos algumas sugestões de atividades, que englobam todo conteúdo do presente TCC. Essas atividades foram aplicadas a uma turma de 1 ano do Ensino Médio de uma escola pública, e as respostas dos alunos estão anexadas ao trabalho.
This work have pedagogical proposed to introduce the concept of commensurable segments and incommensurable segments, showing the importance of these concepts for the study of rational and irrational numbers. We will stabelish a verification process to detect the mensurability of two segments, which is a geometric process. We present the classic demonstration that root of 2 is irrational with a geometric approach, showing that the segment of the side of a square measuring its diagonal are immeasurable. We still will present a study on decimal expressions, and prove a theorem that allows to check that an irreducible fraction has decimal representation finite or infinite and periodic. We also present another theorem that allows us to turn decimal expressions finite or infinite and periodic on its fraction form. Finally we present some suggestions for activities that include all content of the TCC. These activities have been applied to a class of 1st year of high school at a public school, and the students’ answers are attached to the work.
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Santos, Simone de Carvalho. "Uma construção geométrica dos números reais." Universidade Federal de Sergipe, 2015. https://ri.ufs.br/handle/riufs/6478.
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Coordenação de Aperfeiçoamento de Pessoal de Nível SuperiorThis study aims to present a geometric construction of real numbers characterizing them as numbers that express a measure. In this construction, each point in an oriented line represents the measure of a segment (a real number). Based on ve axioms of Euclidean geometry it was de ned an order relation, a method to add and multiply points so that it was possible to demonstrate that the line has a full ordered body of algebraic structure that we call the set of real numbers. To do so, it were presented historical elements that allow us to understand the emergence of irrational numbers as a solution to the insu ciency of rational numbers with respect to the measuring problem, the evolution of the concept of number, as well as the importance that the strict construction of real numbers had to the Foundations of Mathematics. We display a construction of rational numbers from the integernumbers as motivation for construction of numerical sets. Using the notion of measure,we show a geometric interpretation of rational numbers linking them to the points of an oriented line to demonstrate that they leave holes in the line and conclude on the need to build a set that contains the rational numbers and that ll all the points of a line. The theme is of utmost importance to the teaching of mathematics because one of the major goal of basic education is to promote understanding of numbers and operations, to develop number sense and to develop uency in the calculation. To achieve this, it is necessary to assimilate the r
O presente trabalho tem por objetivo apresentar uma construção geométrica dos números reais caracterizando-os como números que expressam uma medida. Nesta construção cada ponto de uma reta orientada representa a medida de um segmento (um número real), com base nos cinco axiomas da geometria euclidiana de niu-se uma relação de ordem, um método para somar e multiplicar pontos de tal forma que fosse possível demonstrar que a reta possui uma estrutura algébrica de corpo ordenado completo a qual chamamos de conjunto dos números reais. Para tanto, foram apresentados elementos históricos que permitem compreender o surgimento dos números irracionais como solução para a insu - ciência dos números racionais no que diz respeito ao problema de medida, a evolução do próprio conceito de número, bem como a importância que a construção rigorosa dos nú- meros reais tiveram para os Fundamentos da Matemática. Exibimos uma construção dos números racionais a partir dos números inteiros como motivação para construções de conjuntos numéricos. Usando a noção de medida mostramos uma interpretação geométrica dos números racionais associando-os aos pontos de uma reta orientada para demonstrar que eles deixam buracos na reta e concluir sobre a necessidade de construir um conjunto que contenha os números racionais e que preencham todos os pontos de uma reta. O tema é de extrema importância para o ensino da matemática, visto que um dos principais objetivos do ensino básico é promover a compreensão dos números e das operações, desenvolver o sentido de número e desenvolver a uência no cálculo, sendo necessário para tal assimilar os números reais, em especial os irracionais, os quais são tratados a partir do ensino fundamental.
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Penteado, Cristina Berndt. "Concepções do professor do ensino médio relativas à densidade do conjunto dos números reais e suas reações frente a procedimentos para a abordagem desta propriedade." Pontifícia Universidade Católica de São Paulo, 2004. https://tede2.pucsp.br/handle/handle/11180.
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Previous issue date: 2004-09-30The work approaches the subject of the density of the real numbers, here taking in the direction of the existence of infinite rational numbers and infinite irrationals between two distinct real numbers. Some research evidences difficulties of the students in the classification of rational numbers and irrationals, as well as the unfamiliarity of the property of the density of the set of the real numbers. The objective of the study is to investigate the conception and the reaction of the teachers of high-school front to the different registers of representations of the numbers, when analyzed the property of the density, as much the density of the set of the rational numbers in the set of the real numbers how much of the irrationals in reals. Is considered to investigate it the viability of two types of distinct procedures for the attainment of real numbers between two supplied: the procedure of the arithmetic mean and other inspired in the process of diagonal line of Cantor, using the representation decimal of the real numbers. For in such a way it was carried through an intervention by means of the elaboration, application and analysis of an education sequence, composed of ten activities, based in the Theory of the Registers of Representation Semiotics of Raymond Duval. The education sequence was based on principles of the Didactic Engineering of Michèle Artigue. Although to evidence envolvement of the participants, some difficulties identified in the research persist as for example, the association of the infinite representation with irrationality and the identification of a rational number as being only that one that has finite representation. Some teachers had demonstrated the intention to apply similar questions to the ones of the sequence, to its students of high-school
O trabalho aborda o tema da densidade dos números reais, aqui tomada no sentido da existência de infinitos números racionais e infinitos irracionais entre dois números reais distintos. Várias pesquisas evidenciam dificuldades dos alunos na classificação de números racionais e irracionais, bem como o desconhecimento da propriedade da densidade do conjunto dos números reais. O objetivo do estudo é investigar a concepção e a reação dos professores do Ensino Médio frente aos diferentes registros de representações dos números, quando analisada a propriedade da densidade, tanto a densidade do conjunto dos números racionais no conjunto dos números reais quanto a dos irracionais nos reais. Propõe-se a investigar a viabilidade de dois tipos de procedimentos distintos para a obtenção de números reais entre dois dados: o procedimento da média aritmética e outro inspirado no processo de diagonal de Cantor, utilizando a representação decimal dos números reais. Para tanto foi realizada uma intervenção por meio da elaboração, aplicação e análise de uma seqüência de ensino, composta de dez atividades, embasada na Teoria dos Registros de Representação Semiótica de Raymond Duval. A seqüência de ensino foi fundamentada em princípios da Engenharia Didática de Michèle Artigue. Apesar de constatar envolvimento dos participantes, algumas dificuldades identificadas nas pesquisas persistem como por exemplo, a associação da representação infinita com irracionalidade e a identificação de um número racional como sendo somente aquele que tem representação finita. Alguns professores demonstraram a intenção de aplicar questões similares às da seqüência, aos seus alunos do Ensino Médio
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Ciano, Susan A. "Architecture, dwelling, and process: between rational and irrational." Thesis, Virginia Polytechnic Institute and State University, 1986. http://hdl.handle.net/10919/53096.
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This inquiry attempts to examine different aspects of architecture and an understanding of space by exploring architectural expectations. The process described herein depicts the struggle to find a translation between two languages, one verbal and one visual. The key to my search was a constant set of ideas. The challenge was first to discern, and then to learn to use, the tools that would become my guide to the language of architecture.Master of Architecture
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Chu, Yim Tonia 1974. "Bidding, playing, or competing? : rational and irrational determinants in Internet auctioning." Thesis, Massachusetts Institute of Technology, 2000. http://hdl.handle.net/1721.1/8986.
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Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, Center for Transportation Studies, 2000.Includes bibliographical references (leaves 49-50).
Auctions, especially Internet auctions, are claimed to be efficient pricing mechanisms, assuming rational behavior and recognition of individual fundamental values. This assumption and its implication are at the heart of the thesis work. The research includes both survey studies which utilize one of the most popular sites for Internet auctions, and experiments developed in the laboratory. Both field studies and laboratory experiments paid specific attention to the psychology of bidders with main focuses on starting prices, price comparison, competition, and auction formats. Two surveys and two field studies were conducted to collect statistics in real auctions and subjective opinions from real auction participants. One field experiment was performed on a real web site to test price sensitivity. A simulation bidding system was built in the laboratory to examine auction formats and the effect of competition. Four primary results are shown. First, game-playing attitude towards auctions generally exists among bidders. Second, bidders hold strong winning aspects and suffer either "winner's curse" or regret losing. Third, bidders are price sensitive when price comparison is available and their price preferences are affected by the original starting prices. Fourth, auction formats convey different information to bidders and influence the way bidders behave. (Keywords: Auctions, Internet auctioning, Individual fundamental values, Bidders' behavior)
by Yim Tonia Chu.
S.M.
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Coward, Daniel R. "Sums of two rational cubes." Thesis, University of Oxford, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.320587.
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Ketkar, Pallavi S. (Pallavi Subhash). "Primitive Substitutive Numbers are Closed under Rational Multiplication." Thesis, University of North Texas, 1998. https://digital.library.unt.edu/ark:/67531/metadc278637/.
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Lehr (1991) proved that, if M(q, r) denotes the set of real numbers whose expansion in base-r is q-automatic i.e., is recognized by an automaton A = (Aq, Ar, ao, δ, φ) (or is the image under a letter to letter morphism of a fixed point of a substitution of constant length q) then M(q, r) is closed under addition and rational multiplication. Similarly if we let M(r) denote the set of real numbers α whose base-r digit expansion is ultimately primitive substitutive, i.e., contains a tail which is the image (under a letter to letter morphism) of a fixed point of a primitive substitution then in an attempt to generalize Lehr's result we show that the set M(r) is closed under multiplication by rational numbers. We also show that M(r) is not closed under addition.
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Brown, Bruce John Lindsay. "The initial grounding of rational numbers : an investigation." Thesis, Rhodes University, 2007. http://hdl.handle.net/10962/d1006351.
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This small scale exploratory research project investigated the grounding of rational number concepts in informal, everyday life situations. A qualitative approach was taken to allow for the identification and then in depth investigation, of issues of importance for such a grounding of rational number understanding. The methodology followed could be seen as a combination of grounded theory and developmental research. And the data was generated through in-depth and clinical interviews structured around a number of grounded tasks related to rational numbers. The research comprised three cycles of interviews that were transcribed and then analysed in detail, interspersed with periods of reading and reflection. The pilot cycle involved a single grade three teacher, the second cycle involved 2 grade three teachers and the third cycle involved 2 grade three children. The research identified a number of different perspectives that were all important for the development of a fundamental intuitive understanding that could be considered personally meaningful to the individual concerned and relevant to the development of rational number concepts. Firstly in order to motivate and engage the child on a personal level the grounding situation needed to be seen as personally significant by the child. Secondly, coordinating operations provided a means of developing a fundamental intuitive understanding, through coordination with affording structures of the situation that are relevant to rational numbers. Finally, goal directed actions that are deliberately structured to achieve explicit goals in a situation are important for the development of more explicit concepts and skills fundamental for rational number understanding. Different explicit structures give rise to different interpretations of rational numbers in grounding situations. In addition to these perspectives, it became evident that building and learning representations was important for developing a more particularly mathematical understanding, based on the fundamental understanding derived from the child's grounded experience. The conclusion drawn in this research as a result of this complexity, is that to achieve a comprehensive and meaningful grounding, children's learning of rational numbers will not follow a simple linear trajectory. Rather this process forms a web of learning, threading coordinating operations for intuitive development, interpretations for explicit grounding and representations to develop more formal mathematical conceptions.
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Spörrle, Matthias. "Irrational, rational, egal? – Empirische Untersuchungen zum Beitrag der Rational-Emotiven Verhaltenstherapie nach Albert Ellis für die psychologische Grundlagenforschung." Diss., lmu, 2006. http://nbn-resolving.de/urn:nbn:de:bvb:19-70916.
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LORIO, MARCELO NASCIMENTO. "APPROXIMATIONS OF REAL NUMBERS BY RATIONAL NUMBERS: WHY THE CONTINUED FRACTIONS CONVERGING PROVIDE THE BEST APPROXIMATIONS?" PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2014. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=23981@1.
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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIROCOORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
Frações Contínuas são representações de números reais que independem da base de numeração escolhida. Quando se trata de aproximar números reais por frações, a escolha da base dez oculta, frequentemente, aproximações mais eficientes do que as exibe. Integrar conceitos de aproximações de números reais por frações contínuas com aspectos geométricos traz ao assunto uma abordagem diferenciada e bastante esclarecedora. O algoritmo de Euclides, por exemplo, ao ganhar significado geométrico, se torna um poderoso argumento para a visualização dessas aproximações. Os teoremas de Dirichlet, de Hurwitz-Markov e de Lagrange comprovam, definitivamente, que as melhores aproximações de números reais veem das frações contínuas, estimando seus erros com elegância técnica matemática incontestável.
Continued fractions are representations of real numbers that are independent of the choice of the numerical basis. The choice of basis ten frequently hides more than shows efficient approximations of real numbers by rational ones. Integrating approximations of real numbers by continued fractions with geometrical interpretations clarify the subject. The study of geometrical aspects of Euclids algorithm, for example, is a powerful method for the visualization of continued fractions approximations. Theorems of Dirichlet, Hurwitz-Markov and Lagrange show that, definitely, the best approximations of real numbers come from continued fractions, and the errors are estimated with elegant mathematical technique.
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Carbone, Rose Elaine. "Elementary Teacher Candidates’ Understanding of Rational Numbers: An International Perspective." Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-79565.
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This paper combines data from two different international research studies that used problem posing in analyzing elementary teacher candidates’ understanding of rational numbers. In 2007, a mathematics educator from the United States and a mathematician from Northern Ireland collaborated to investigate their respective elementary teacher candidates’ understanding of addition and division of fractions. A year later, the same US mathematics educator collaborated with a mathematics educator from South Africa on a similar research project that focused solely on the addition of fractions. The results of both studies show that elementary teacher candidates from the three different continents share similar misconceptions regarding the addition of fractions. The misconceptions that emerged were analyzed and used in designing teaching strategies intended to improve elementary teacher candidates’ understanding of rational numbers. The research also suggests that problem posing may improve their understanding of addition of fractions.
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Clark, David Alan. "The Euclidean algorithm for Galois extensions of the rational numbers." Thesis, McGill University, 1992. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=39408.
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Let K be a totally real, quartic, Galois extension of $ doubq$ whose ring of integers R is a principal ideal domain. If there is a prime ideal p of R such that the unit group maps onto $(R/{ bf p} sp2$)*, then R is a Euclidean domain. This criterion is generalized to arbitrary Galois extensions.Let E be an elliptic curve over a number field F. Suppose ($F: doubq rbrack le 4$ and $F(E lbrack q rbrack ) not subseteq F$ for all primes q such that F contains a primitive $q sp{ rm th}$ root of unity, then the reduced elliptic curve $ tilde{E}(F sb{ bf p})$ is cyclic infinitely often. In general, if $ Gamma$ a subgroup of $E(F)$ with the range of $ Gamma$ sufficiently large, there are infinitely many prime ideals p of F such that the reduced curve $ tilde{E}(F sb{ bf p}) = Gamma sb{ bf p}$, where $ Gamma sb{ bf p}$ is the reduction modulo p of $ Gamma$.
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Bruyns, P. "Aspects of the group of homeomorphisms of the rational numbers." Thesis, University of Oxford, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.375224.
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Millsaps, Gayle M. "Interrelationships between teachers' content knowledge of rational number, their instructional practice, and students' emergent conceptual knowledge of rational number." Connect to resource, 2005. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1124225634.
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Thesis (Ph. D.)--Ohio State University, 2005.Title from first page of PDF file. Document formatted into pages; contains xviii, 339 p.; also includes graphics (some col.). Includes bibliographical references (p. 296-306). Available online via OhioLINK's ETD Center
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Pham, Van Anh. "Loop Numbers of Knots and Links." TopSCHOLAR®, 2017. http://digitalcommons.wku.edu/theses/1952.
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This thesis introduces a new quantity called loop number, and shows the conditions in which loop numbers become knot invariants. For a given knot diagram D, one can traverse the knot diagram and count the number of loops created by the traversal. The number of loops recorded depends on the starting point in the diagram D and on the traversal direction. Looking at the minimum or maximum number of loops over all starting points and directions, one can define two positive integers as loop numbers of the diagram D. In this thesis, the conditions under which these loop numbers become knot invariants are identified. In particular, the thesis answers the question when these numbers are invariant under flypes in the diagram D.
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Tolmie, Julie, and julie tolmie@techbc ca. "Visualisation, navigation and mathematical perception: a visual notation for rational numbers mod1." The Australian National University. School of Mathematical Sciences, 2000. http://thesis.anu.edu.au./public/adt-ANU20020313.101505.
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There are three main results in this dissertation.
The first result is the construction of an abstract visual space for rational
numbers mod1, based on the visual primitives, colour, and rational radial
direction. Mathematics is performed in this visual notation by defining
increasingly refined visual objects from these primitives. In particular,
the existence of the Farey tree enumeration of rational numbers mod1
is identified in the texture of a two-dimensional animation.
¶
The second result is a new enumeration of the rational numbers mod1,
obtained, and expressed, in abstract visual space, as the visual object
coset waves of coset fans on the torus. Its geometry is shown to encode
a countably infinite tree structure, whose branches are cosets, nZ+m,
where n, m (and k) are integers. These cosets are in geometrical 1-1
correspondence with sequences kn+m, (of denominators) of rational
numbers, and with visual subobjects of the torus called coset fans.
¶
The third result is an enumeration in time of the visual hierarchy of the
discrete buds of the Mandelbrot boundary by coset waves of coset fans.
It is constructed by embedding the circular Farey tree geometrically into
the empty internal region of the Mandelbrot set. In particular, coset fans
attached to points of the (internal) binary tree index countably infinite
sequences of buds on the (external) Mandelbrot boundary.
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Amaca, Edgar Gilbuena. "On rational functions with Golden Ratio as fixed point /." Online version of thesis, 2008. http://hdl.handle.net/1850/6212.
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Jensen, Peter Eli. "Evaluating the ABC model of rational emotive behavior therapy theory an analysis of the relationship between irrational thinking and guilt /." Click here for download, 2008. http://proquest.umi.com/pqdweb?did=1588785791&sid=1&Fmt=2&clientId=3260&RQT=309&VName=PQD.
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Shaughnessy, John F. "Finding Zeros of Rational Quadratic Forms." Scholarship @ Claremont, 2014. http://scholarship.claremont.edu/cmc_theses/849.
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In this thesis, we introduce the notion of quadratic forms and provide motivation for their study. We begin by discussing Diophantine equations, the field of p-adic numbers, and the Hasse-Minkowski Theorem that allows us to use p-adic analysis determine whether a quadratic form has a rational root. We then discuss search bounds and state Cassels' Theorem for small-height zeros of rational quadratic forms. We end with a proof of Cassels' Theorem and suggestions for further reading.
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Ng, Catherine Wai-Man. "The dilution of the law of passing-off : toward a rational basis for irrational trade mark protection." Thesis, University of Oxford, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.413527.
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SHINOHARA, Hisato, and 尚人 篠原. "小学生の対人関係ビリーフに関する研究 : 対人関係ビリーフ尺度（小学生版）の開発." 名古屋大学大学院教育発達科学研究科, 2013. http://hdl.handle.net/2237/19519.
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Hyland, Philip. "A rational emotive behaviour therapy perspective on the nature and structure of posttraumatic stress responses : the mediating and moderating effects of rational and irrational beliefs." Thesis, Ulster University, 2015. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.654102.
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Rational Emotive Behaviour Therapy (REBT: Ellis, 2001) represents the original cognitive behavioural therapy (eBT) model of psychopathology. Although there is much empirical support for the basic theory of REBT (see David, Lynn, & Ellis, 2010), the model has never been tested in the context of posttraumatic stress responses to adverse life events. The first empirical chapter of the thesis investigated the construct validity of the Attitudes and Belief Scale 2 (ABS-2: DiGiuseppe, Leaf, Exner, & Robin, 1988). This chapter employed traditional confirmatory factor analysis and confirmatory bifactor modelling to investigate the psychometric properties of the ABS-2. Results indicated that a bifactoral model conceptualisation was found to offer an adequate representation of the underlying factor structure of the scale. Based on these results, an abbreviated version of the ABS-2 with superior psychometric properties was thus constructed. In the second empirical chapter confirmatory bifactor modelling and composite reliability analysis were employed to investigate the psychometric properties of the Profile of Emotional Distress (PED: Opris & Macavei, 2007). The PED was designed to capture the qualitative distinction between dysfunctional emotions, as predicted by REBT theory. Results indicated that the PED does not capture the distinction between functional and dysfunctional negative emotions, however a bifactor model inclusive of a single general distress factor, and four method factors was found to be an acceptable fit of the data. The third empirical chapter utilised structural equation modelling to test the organisation of the irrational beliefs in the prediction of posttraumatic stress responses. A model consistent with the predictions of REBT theory was found to be a good fit of the data and explained a large percentage of variance in each symptom class of posttraumatic stress. The fourth empirical chapter provided the first piece of empirical evidence that generalised irrational beliefs impact upon posttraumatic stress symptoms via trauma-specific irrational beliefs; a frequently hypothesised relationship which had hitherto remained untested. Results of structural equation modelling offered support for this core hypothesis. Subsequently, the fifth empirical chapter investigated the impact of trauma-specific irrational beliefs in the prediction of reporting posttraumatic stress symptoms while controlling for a number of important sociodemographic factors. Binary logistic regression ~ .. ~ ... analysis was employed and found that three irrational belief process positively predicted belong to the strongly symptomatic group. Finally, the sixth empirical chapter employed sequential moderated multiple regression analysis to determine if rational beliefs could positively moderate the impact of irrational beliefs of posttraumatic stress symptoms. Rational beliefs were found to exert a negative, direct effect on posttraumatic stress symptoms, and to lessen the impact of irrational beliefs on posttraumatic stress responses.
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Tolmie, Julie. "Visualisation, navigation and mathematical perception : a visual notation for rational numbers mod 1." View thesis entry in Australian Digital Theses Program, 2000. http://thesis.anu.edu.au/public/adt-ANU20020313.101505/index.html.
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Moss, Joan. "Deepening children's understanding of rational numbers, a developmental model and two experimental studies." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape3/PQDD_0021/NQ49900.pdf.
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Lozier, Stephane. "On simultaneous approximation to a real number and its cube by rational numbers." Thesis, University of Ottawa (Canada), 2010. http://hdl.handle.net/10393/28701.
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One of the fundamental problems in Diophantine approximation is approximation to real numbers by algebraic numbers of bounded degree. In 1969, H. Davenport and W. M. Schmidt developed a new method to approach the problem. This method combines a result on simultaneous approximation to successive powers of a real number xi with geometry of numbers. For now, the only case where the estimates are optimal is the case of two consecutive powers. Davenport and Schmidt show that if a real number xi is such that 1, xi, xi² are linearly independent over Q , then the exponent of simultaneous approximation to xi and xi² by rational numbers with the same denominator is at most ( 5 - 1}/2 = 0.618..., the inverse of the Golden ratio. In this thesis, we consider the case of a number and its cube. Our main result is that if a real number xi is such that 1, xi, xi³ are linearly independent over Q , then the exponent of simultaneous approximation to xi and xi³ by rational numbers with the same denominator is at most 5/7 = 0.714.... As corollaries, we deduce a result on approximation by algebraic numbers and a version of Gel'fond's lemma for polynomials of the form a + bT + cT³.
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Tobias, Jennifer. "Preservice Elementary Teachers' Diverlopment of Rational Number Understanding Through the Social Perspective and the Relationship Among Social and Individual Environments." Doctoral diss., University of Central Florida, 2009. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/4233.
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A classroom teaching experiment was conducted in a semester-long undergraduate mathematics content course for elementary education majors. Preservice elementary teachers' development of rational number understanding was documented through the social and psychological perspectives. In addition, social and sociomathematical norms were documented as part of the classroom structure. A hypothetical learning trajectory and instructional sequence were created from a combination of previous research with children and adults. Transcripts from each class session were analyzed to determine the social and sociomathematical norms as well as the classroom mathematical practices. The social norms established included a) explaining and justifying solutions and solution processes, b) making sense of others' explanations and justifications, c) questioning others when misunderstandings occur, and d) helping others. The sociomathematical norms established included determining what constitutes a) an acceptable solution and b) a different solution. The classroom mathematical practices established included ideas related to a) defining fractions, b) defining the whole, c) partitioning, d) unitizing, e) finding equivalent fractions, f) comparing and ordering fractions, g) adding and subtracting fractions, and h) multiplying fractions. The analysis of individual students' contributions included analyzing the transcripts to determine the ways in which individuals participated in the establishment of the practices. Individuals contributed to the practices by a) introducing ideas and b) sustaining ideas. The transcripts and student work samples were analyzed to determine the ways in which the social classroom environment impacted student learning. Student learning was affected when a) ideas were rejected and b) ideas were accepted. As a result of the data analysis, the hypothetical learning trajectory was refined to include four phases of learning instead of five. In addition, the instructional sequence was refined to include more focus on ratios. Two activities, the number line and between activities, were suggested to be deleted because they did not contribute to students' development.Ph.D.
Department of Teaching and Learning Principles
Education
Education PhD
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Conley, Randolph M. "A survey of the Minkowski?(x) function." Morgantown, W. Va. : [West Virginia University Libraries], 2003. http://etd.wvu.edu/templates/showETD.cfm?recnum=3055.
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Tobias, Jennifer M. "Preservice elementary teachers' development of rational number understanding through the social perspective and the relationship among social and individual environments." Orlando, Fla. : University of Central Florida, 2009. http://purl.fcla.edu/fcla/etd/CFE0002737.
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Cruz, Junior Jorge Mageste da. "A matemática por trás de um número: razão áurea." Universidade Federal de Juiz de Fora, 2014. https://repositorio.ufjf.br/jspui/handle/ufjf/702.
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CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
O presente trabalho tem por objetivo descrever e conceituar a importância dos números áureos. Sua aplicabilidade acompanha o ser humano e frequentemente são vivenciados em situações cotidianas. Durante a elaboração deste estudo procurou-se demonstrar as diferentes aparições do número áureo, nas mais diversas áreas em que vivemos, seja na natureza, nos animais, na arquitetura e até mesmo no corpo humano. A pesquisa foi realizada através de consultas em livros escritos por autores renomados e em artigos publicados em bases de dados confiáveis. Esta pesquisa visa ampliar o conhecimento e apresentar aos alunos uma maneira diferente de ver e entender a matemática e sua aplicabilidade e influência no dia-a-dia.
The present work aims to describe and conceptualize the importance of golden numbers. Its applicability with humans and often are experienced in everyday situations. During the preparation of this study sought to demonstrate the different appearances of the Golden number, in the most diverse areas in which we live, whether in nature, animals and even in the human body. The survey was conducted through consultations in books written by renowned authors and in articles published in reliable databases. This research aims to expand the knowledge and present to students a different way to see and understand the mathematics and its applicability and influence in everyday life.
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Zangiacomo, Tassia Roberta [UNESP]. "Sobre as construções dos sistemas numéricos: N, Z, Q e R." Universidade Estadual Paulista (UNESP), 2017. http://hdl.handle.net/11449/149948.
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Este trabalho tem como objetivo construir os sistemas numéricos usuais, a saber, o conjunto dos números naturais N, o conjunto dos números inteiros Z, o conjunto dos números racionais Q e o conjunto dos números reais R. Iniciamos o trabalho tratando de noções sobre conjuntos e relações binárias. Em seguida, apresentamos o conjunto dos números naturais, definido através dos axiomas de Peano; o conjunto dos números inteiros via uma relação de equivalência com o conjunto dos números naturais; o conjunto dos números racionais, que são obtidos também via relação de equivalência, mas dessa vez com o conjunto dos números inteiros; a construção do conjunto dos números reais, feita via cortes no conjunto dos números racionais; e, para todos esses casos, mostramos a imersão do conjunto anterior no conjunto que surge na sequência. Por fim, observamos alguns materiais do ensino fundamental e médio com o intuito de investigar de que forma esses temas estão sendo apresentados para os alunos.
This work aims to construct the usual numerical systems, namely the set of natural numbers N, the set of integers Z, the set of rational numbers Q and the set of real numbers R. We begin the work dealing with notions about sets and binary relations. Next, we present the set of natural numbers, defined by Peano's axioms; the set of integers via an equivalence relation with the set of natural numbers; the set of rational numbers, which are also obtained via equivalence relation, but this time with the set of integers; the construction of the set of real numbers, made through cuts in the set of rational numbers; end for all these cases we show the immersion of the previous set in the ensemble that appears in the sequence. Finally, we observed some materials in elementary school and high school in order to investigate how these themes are being presented to the students.
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Filho, José Souto Sobrinho. "O surgimento dos números irracionais." Universidade do Estado do Rio de Janeiro, 2015. http://www.bdtd.uerj.br/tde_busca/arquivo.php?codArquivo=9522.
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Coordenação de Aperfeiçoamento de Pessoal de Nível SuperiorEste é um trabalho de pesquisa sobre um conjunto de números (irracionais) que é pouco trabalhado no ensino básico de matemática. Foi uma procura muito interessante e enriquecedora, pois encontrei matemáticos e historiadores com visões bem diferentes. Muitos deles não aceitavam este novo conjunto. Para Leopold Kronecker, só existia o conjunto dos números inteiros. Já para Cantor e Dedekind, o aparecimento dos irracionais foi extremamente importante para o desenvolvimento da matemática, abrindo novos horizontes. Menciono aqui um pouco da vida e da obra de alguns matemáticos que se envolveram com os números irracionais. Tratamos ainda da descoberta dos incomensuráveis, ou seja, como iniciou-se o problema da incomensurabilidade, e do retângulo áureo e sua importância em outras áreas. O trabalho mostra também dois grupos de números que não são mencionados quando ensinamos equações algébricas, que são os números algébricos e os números transcendentes, assim como teoremas essenciais para a prova da transcendência dos irracionais especiais e . Por fim, proponho uma aula para uma turma do 3 ano do Ensino Médio com o objetivo de mostrar a irracionalidade de alguns números, usando os teoremas pertinentes
This is a research about a set of numbers (irrationals) that is little explored in secondary school mathematics teaching. It was a very interesting and enriching search, because quite contrary facts were found. Several 19th century mathematicians did not accept this new set of numbers. To Leopold kronecker, only the set of the integers existed. To Cantor and Dedekind, the irrational numbers were extremely important for the development of mathematics, opening new horizons. I also mention the life and work of some mathematicians who were involved with the irrational numbers the discovery of the incommensurability was iniciated. The golden rectangle and its importance in other areas. The work also presents two groups of numbers that are not mentioned when algebraic equations are taught, the algebraic numbers and transcendental numbers. Essential theorems for the proof of the special irrational numbers e . Finnaly, I propose a lesson to a 3rd year high school class in order to show the irrationality of some numbers, using the relevant theorems
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Rakotoniaina, Tahina. "Explicit class field theory for rational function fields." Thesis, Link to the online version, 2008. http://hdl.handle.net/10019/1993.
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Lewis, Raynold M. Otto Albert D. "The knowledge of equivalent fractions that children in grades 1, 2, and 3 bring to formal instruction." Normal, Ill. Illinois State University, 1996. http://wwwlib.umi.com/cr/ilstu/fullcit?p9633409.
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Thesis (Ph. D.)--Illinois State University, 1996.Title from title page screen, viewed May 24, 2006. Dissertation Committee: Albert D. Otto (chair), Barbara S. Heyl, Cheryl A. Lubinski, Nancy K. Mack, Jane O. Swafford, Carol A. Thornton. Includes bibliographical references (leaves 188-198) and abstract. Also available in print.
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Santos, Edson Kretle dos. "O equilíbrio entre o elemento irracional e racional na ideia de sagrado em Rudolf Otto." Universidade Federal do Espírito Santo, 2012. http://repositorio.ufes.br/handle/10/6282.
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Previous issue date: 2012-04-26The purpose of this dissertation is to elucidate in what way the rational and the irrational elements frame the human notion of the sacred, according to the philosopher Rudolf Otto. Marked by XX Century s rule of reason, Otto reacts to the Enlightenment, which interprets the sacred solely as metaphysical, moral and evolutional formulations. After the Kantian critique, the knowledge path becomes an impossible way to the understanding of the sacred and of the religious phenomenon. Thus, the author recaptures the sentiment as the origin and the openness of the human search for the divine. Since the experience of the nouminous belongs to the sphere of the sentiment, it follows that it is understood, constituted and characterized by the irrational aspect, for the religious experience has the peculiarity of the mysterious. Given this argument, Otto responds to the Enlightenment Age stating that the mystery of the sacred shall never be encompassed by reason alone. On the other hand, a religion based exclusively on sentiment contributes to the emergence of various self-denominated miraculous, magical religious practices. The theological reflections of such religions tend to mix capitalist exchanges and divine promises of prosperity. Otto reveals himself as a supremely important philosopher in the analysis of many contemporary religious practices which emphasize the emotional aspect. In such religions the exclusion of reason is evident, which the author rejects as inconceivable, given the argument that the rational element is a fundamental component of religion. Hence, Otto s intention, as well as that of this dissertation, is to demonstrate that the religious experience is composed both by the irrational and the rational elements and that only when these two aspects are in a state of healthy balance it is possible to achieve a profound experience of the divine.
Pretende-se elucidar, a partir do pensador Rudolf Otto, de que maneira o elemento racional e irracional compõem a noção humana de sagrado. Otto, marcado pelo século XX, responde ao Iluminismo, que interpreta o sagrado apenas como formulações metafísicas, morais e evolutivas. Após a crítica kantiana, a via do conhecimento tornou-se um o caminho impossível para compreensão do sagrado e do fenômeno religioso. Por isso, o autor resgata o sentimento como origem e abertura do humano na busca pelo divino. A experiência do numinoso por está situada no âmbito do sentir passa a ser compreendida, constituída e caracterizada pelo atributo irracional uma vez que a vivência da religião possui peculiaridade do misterioso. Nesse sentido, então, Otto responde à Era do Esclarecimento afirmando que o mistério do sagrado jamais será abarcado pela razão. Em contrapartida, uma religião demasiadamente baseada apenas no sentimento fez com emergisse no contexto religioso contemporâneo muitas práticas religiosas mágicas e que se auto-intitulam milagrosas. Geralmente, as reflexões teológicas dessas religiões mesclam trocas capitalistas e prósperas bênçãos divinas. Otto torna-se um pensador de suma importância para analisar muitas práticas religiosas atuais, como dito acima, que enfatizam em excesso o aspecto emotivista na religião. Em tais posturas religiosas percebe-se a exclusão da racionalidade na religião, que para o autor em questão é algo inconcebível uma vez que ele atesta que o elemento racional é componente fundamental da religião. Portanto, o intuito de Otto e também dessa dissertação, é mostrar que a experiência religiosa é composta pelo elemento irracional e racional e somente quando ambos aspectos estão em sadio equilíbrio é que se faz uma profunda vivência do divino
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Spolaor, Silvana de Lourdes Gálio. "Números irracionais: e e." Universidade de São Paulo, 2013. http://www.teses.usp.br/teses/disponiveis/55/55136/tde-02102013-160720/.
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Nesta dissertação são apresentadas algumas propriedades de números reais. Descrevemos de maneira breve os conjuntos numéricos N, Z, Q e R e apresentamos demonstrações detalhadas da irracionalidade dos números \'pi\' e e. Também, apresentamos um texto sobre o número e, menos técnico e mais intuitivo, na tentativa de auxiliar o professor no preparo de aulas sobre o número e para alunos do Ensino Médio, bem como, alunos de cursos de Licenciatura em MatemáticaIn this thesis we present some properties of real numbers. We describe briefly the numerical sets N, Z, Q and R, and we present detailed proofs of irrationality of numbers \'pi\' and e. We also present a text about the number e less technical and more intuitive in an attempt to assist the teacher in preparing lessons about number e for High School students as well as for Teaching degree in Mathematics students
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Henry, Michael A. "Various Old and New Results in Classical Arithmetic by Special Functions." Kent State University / OhioLINK, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=kent1524583992694218.
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Persson, Frida. "Hur introducerar och arbetar lärare med bråkräkning i grundskolans tidigare år?" Thesis, Luleå tekniska universitet, Institutionen för konst, kommunikation och lärande, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:ltu:diva-75090.
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Syftet med denna studie är att ta reda på hur lärare i grundskolans tidigare år introducerar och arbetar med området bråkräkning. Utifrån detta syfte så formulerades tre stycken frågeställningar: Hur beskriver lärare att de introducerar området för sina elever? Hur beskriver lärare i grundskolans tidigare år att de arbetar med området? Samt är lärare medvetna om någon svårighet med området bråk? För att kunna besvara dessa tre frågeställningar genomfördes kvalitativa intervjuer med sju stycken lärare som arbetar runt om i Sverige. Studiens resultat visar att bråkräkning är någonting som upplevs som svårt av många elever samt att grunden till förståelse för området ligger vid en tydlig introduktion av både området i sig, men även av väsentliga begrepp. De intervjuade lärarna har även beskrivit hur de introducerar och arbetar med området bråkräkning och detta diskuteras sedan i enighet med tidigare forskning.
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Trespalacios, Jesus. "The Effects of Two Generative Activities on Learner Comprehension of Part-Whole Meaning of Rational Numbers Using Virtual Manipulatives." Diss., Virginia Tech, 2008. http://hdl.handle.net/10919/26508.
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The study investigated the effects of two generative learning activities on students’ academic achievement of the part-whole representation of rational numbers while using virtual manipulatives. Third-grade students were divided randomly in two groups to evaluate the effects of two generative learning activities: answering-questions and generating-examples while using two virtual manipulatives related to part-whole representation of rational numbers. The study employed an experimental design with pre- and post-tests. A 2x2 mixed analysis of variance (ANOVA) was used to determine any significant interaction between the two groups (answering questions and generating-examples) and between two tests (pre-test and immediate post-test). In addition, a 2x3 mixed analysis of variance (ANOVA) and a Bonferroni post-hoc analysis were used to determine the effects of the generative strategies on fostering comprehension, and to determine any significant differences between the two groups (answering-questions and generating-examples) and among the three tests (pre-test, immediate post-test, and delayed posttest).
Results showed that an answering-questions strategy had a significantly greater effect than a generating-examples strategy on an immediate comprehension posttest. In addition, no significant interaction was found between the generative strategies on a delayed comprehension tests. However a difference score analysis between the immediate posttest scores and the delayed posttest scores revealed a significant difference between the answering-questions and the generating-examples groups suggesting that students who used generating-examples strategy tended to remember relatively more information than students who used the answering-questions strategy. The findings are discussed in the context of the related literature and directions for future research are suggested.Ph. D.
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Armstrong, Barbara Ellen. "The use of rational number reasoning in area comparison tasks by elementary and junior high school students." Diss., The University of Arizona, 1989. http://hdl.handle.net/10150/184910.
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The purpose of this study was to determine whether fourth-, sixth-, and eighth-grade students used rational number reasoning to solve comparison of area tasks, and whether the tendency to use such reasoning increased with grade level. The areas to be compared were not similar and therefore, could not directly be compared in a straightforward manner. The most viable solution involved comparing the part-whole relationships inherent in the tasks. Rational numbers in the form of fractional terms could be used to express the part-whole relationships. The use of fractional terms provided a means for students to express the areas to be compared in an abstract manner and thus free themselves from the perceptual aspects of the tasks. The study examined how students solve unique problems in a familiar context where rational number knowledge could be applied. It also noted the effect of introducing fraction symbols into the tasks after students had indicated how they would solve the problems without any reference to fractions. Data were gathered through individual task-based interviews which consisted of 21 tasks, conducted with 36 elementary and junior high school students (12 students each in the fourth, sixth, and eighth grades). Each interview was video and audio taped to provide a record of the students' behavioral and verbal responses. The student responses were analyzed to determine the strategies the students used to solve the comparison of area tasks. The student responses were classified into 11 categories of strategies. There were four Part-Whole Categories, one Part-Whole/Direct Comparison Combination category and six Direct Comparison categories. The results of the study indicate that the development of rational number instruction should include: learning sequences which take students beyond the learning of a set of fraction concepts and skills, attention to the interaction of learning and the visual aspects of instructional models, and the careful inclusion of different types of fractions and other rational number task variables. This study supports the current national developments in curriculum and evaluation standards for mathematics instruction which stress the ability of students to problem solve, communicate, and reason.
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Bledsoe, Ann M. "Implementing the connected mathematics project : the interaction between student rational number understanding and classroom mathematical practices /." free to MU campus, to others for purchase, 2002. http://wwwlib.umi.com/cr/mo/fullcit?p3074374.
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Dugaich, Valéria Cristina Brumati. "Jogos como possibilidade para a melhoria do desempenho e das atitudes em relação às frações e aos decimais nos anos finais do ensino fundamental /." Bauru, 2020. http://hdl.handle.net/11449/192109.
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Orientador: Nelson Antonio PirolaResumo: Tendo em vista que o desempenho em matemática de significativo percentual de alunos do 9º ano do ensino fundamental da Rede Estadual de Ensino no Sistema de Avaliação de Rendimento Escolar do Estado de São Paulo-SARESP, é ruim, no presente estudo, investigou-se a relação entre o uso de jogos pedagógicos, as atitudes e o desempenho em matemática. Teve como objetivo geral pesquisar e criar jogos como ferramenta pedagógica com potencial para criar situações e experiências favoráveis ao ensino das diferentes representações de um número racional, podendo impactar positivamente nas atitudes dos alunos dos anos finais do ensino fundamental em relação a esses números, bem como no desempenho em tarefas relacionadas a eles. Para tanto, foi necessário investigar: o desempenho desses alunos em matemática no SARESP; suas atitudes em relação à matemática e de modo específico, às frações e aos números decimais; como o uso dos jogos pode contribuir para o ensino e a aprendizagem dos números racionais, sobretudo para o reconhecimento das diferentes representações de um número racional; construir, testar e apresentar um caderno de jogos e por fim, avaliar o possível impacto que os mesmos podem produzir sobre as atitudes e aprendizagem de conceitos e procedimentos pertinentes aos números racionais. Realizou-se, então, uma pesquisa quanti-qualitativa sendo utilizados para a coleta de dados: questionário informativo do aluno; escalas de atitudes em relação à matemática, às frações e aos números d... (Resumo completo, clicar acesso eletrônico abaixo)
Abstract: In view of the fact that the performance in mathematics of a significant percentage of students in the 9th grade of elementary school in the State Education Network in the School Performance Assessment System of the State of São Paulo-SARESP is poor, in the present study, we investigated the relationship between the use of educational games, attitudes and performance in mathematics. Its general objective was to research and create games as a pedagogical tool with the potential to create situations and experiences favorable to the teaching of different representations of a rational number, which may positively impact the attitudes of students in the final years of elementary school in relation to these numbers, as well as performance on related tasks. Therefore, it was necessary to investigate: the performance of these students in mathematics at SARESP; their attitudes towards mathematics and specifically, fractions and decimal numbers; how the use of games can contribute to the teaching and learning of rational numbers, especially to the recognition of different representations of a rational number; build, test and present a game book and, finally, evaluate the possible impact that they can have on attitudes and learning concepts and procedures relevant to rational numbers. Then, a quantitativequalitative research was carried out and used for data collection: student's questionnaire; scales of attitudes towards mathematics, fractions and decimal numbers (validated in the scop... (Complete abstract click electronic access below)
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Felix, Saulo Ferreira. "Estudo de abordagens dos números irracionais nos anos finais do ensino fundamental." Universidade Federal de Goiás, 2018. http://repositorio.bc.ufg.br/tede/handle/tede/8874.
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This work aimed to carry out an investigation in the approaches developed on the set of irrational numbers in the final years of elementary school. The methodology used is of a qualitative and technical nature of documentary analysis. Therefore, it does not need immediate interference in empirical practice and does not impose immediate interaction of the construction of theory and practice. In this way it was observed how the teaching of these numbers is exposed to these students and how this subject is approached in the textbooks of mathematics of this series. As support for the work is presented the exposition of mathematical procedures of the elements pertinent to the study and analysis of literatures in two collections of primary school mathematics. An alternative model is proposed. There was a static and repetitive approach to the teaching of irrational numbers, which is summarized in a didactic material without changes related to this approach or methodology applied to the teaching of rational and irrational numbers.
Este trabalho objetivou realizar uma investigação nas abordagens desenvolvidas sobre o conjunto dos números irracionais nos anos finais do ensino fundamental A metodologia utilizada é de natureza qualitativa e técnica de análise documental. Portanto, não necessita de interferência imediata na prática empírica e não impõe interação de imediato da construção da teoria e a prática. Deste modo foi observado como o ensino destes números é exposto a estes alunos e como este assunto é abordado nos livros didáticos de matemática destas séries. Como suporte para o trabalho é apresentada a exposição de procedimentos matemáticos dos elementos pertinentes ao estudo e à análise de literaturas em duas coleções de matemática do ensino fundamental. Um modelo alternativo é proposto. Verificou-se uma abordagem estática e repetitiva para o ensino dos números irracionais, a qual se resume em um material didático sem mudanças relacionadas a essa abordagem ou metodologia aplicada ao ensino dos números racionais e irracionais.
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Johnson, Gwendolyn Joy. "Proportionality in Middle-School Mathematics Textbooks." Scholar Commons, 2010. https://scholarcommons.usf.edu/etd/1670.
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Some scholars have criticized the treatment of proportionality in middle-school textbooks, but these criticisms seem to be based on informal knowledge of the content of textbooks rather than on a detailed curriculum analysis. Thus, a curriculum analysis related to proportionality was needed.
To investigate the treatment of proportionality in current middle-school textbooks, nine such books were analyzed. Sixth-, seventh-, and eighth-grade textbooks from three series were used: ConnectedMathematics2 (CMP), Glencoe's Math Connects, and the University of Chicago School Mathematics Project (UCSMP). Lessons with a focus on proportionality were selected from four content areas: algebra, data analysis/probability, geometry/measurement, and rational numbers. Within each lesson, tasks (activities, examples, and exercises) related to proportionality were coded along five dimensions: content area, problem type, solution strategy, presence or absence of a visual representation, and whether the task contained material regarding the characteristics of proportionality. For activities and exercises, the level of cognitive demand was also noted.
Results indicate that proportionality is more of a focus in sixth and seventh-grade textbooks than in eighth-grade textbooks. The CMP and UCSMP series focused on algebra in eighth grade rather than proportionality. In all of the sixth-grade textbooks, and some of the seventh- and eighth-grade books, proportionality was presented primarily through the rational number content area.
Two problem types described in the research literature, ratio comparison and missing value, were extensively found. However, qualitative proportional problems were virtually absent from the textbooks in this study. Other problem types (alternate form and function rule), not described in the literature, were also found.
Differences were found between the solution strategies suggested in the three textbook series. Formal proportions are used earlier and more frequently in the Math Connects series than in the other two. In the CMP series, students are more likely to use manipulatives.
The Mathematical Task Framework (Stein, Smith, Henningsen, & Silver, 2000) was used to measure the level of cognitive demand. The level of cognitive demand differed among textbook series with the CMP series having the highest level of cognitive demand and the Math Connects series having the lowest.
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Dopico, Evelyn. "The Impact of Small Group Intervention Focusing on Operations with Rational Numbers on Students' Performance in the Florida Algebra I End-of-Course Examination." Thesis, Nova Southeastern University, 2018. http://pqdtopen.proquest.com/#viewpdf?dispub=10845405.
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In Florida, passing the Algebra I end-of-course examination (EOCE) is a graduation requirement. The test measures knowledge of basic algebra. In spring 2015, the Department of Education introduced a different version of the test. For the first two administrations of the new test, the failure rate for 9th-grade students in the state was almost 50%. In contrast, the failure rate for students in the school where this study was implemented exceeded 70%. The purpose of this study was to determine the outcome of small group intervention focusing on operations with rational numbers of high school students’ performance on the Algebra I EOCE.
After analyzing several potential methods of instruction, small group instruction with the incorporation of the use of manipulatives, visuals, and guided inquiry was selected. In addition, the focus of the study was chosen to be operations with rational numbers, an area many researchers have identified as critical for student understanding of algebraic concepts. Twenty students from the target population of 600 10th and 11th grade students volunteered to participate in the study. These participants received three to six small group instruction sessions before retaking the test. In Sept 2016, all the students in the target population were administered the Algebra I EOCE again. A t-test yielded no significant difference in the learning gains of those who participated in the study and the other students in the target population. The implications of the results were that the interventions had no significant impact on student achievement. A possible reason for the lack of success could have been that six intervention sessions were not enough to produce significant results. It is recommended that future research includes a substantially larger number of interventions.
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Sehlmeyer, Peter August. "Use of learning-logs in high school pre-algebra classes to improve mastery of rational numbers and linear equations for high-risk minority students." CSUSB ScholarWorks, 1997. https://scholarworks.lib.csusb.edu/etd-project/1497.
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The purpose of this study was to determine whether there was a relationship in the use of learning-logs to traditional or current math instruction in secondary school pre-algebra classes to improve the mastery of single-variable equations by high-risk minority students.
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Brown, Bruce J. L. "Numbers: a dream or reality? A return to objects in number learning." Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-82378.
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