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Uma abordagem diferenciada dos números racionais na forma fracionária

Prochnow, Karine Zangalli Schiling
Tipo: Trabalho de Conclusão de Curso Formato: application/pdf
POR
Relevância na Pesquisa
46.26%

As diferentes “personalidades” do número racional trabalhadas através da resolução de problemas

Onuchic, Lourdes de la Rosa; Gomes Allevato, Norma Suely
Tipo: Artigo de Revista Científica Formato: 79-102
POR
Relevância na Pesquisa
46.22%
We address the different "personalities" of the rational number and the concept of proportionality, analyzing the possibilities for using the Mathematics Teaching and Learning through Problem-solving Method. This method is based on the principle that knowledge can be constructed through the use of problems that generate new concepts and new contents. The different meanings of rational number - rational point, quotient, fraction, ratio, and operator - are constructs that depend on mathematical theories in which they are imbedded and the situations that evoke them in problem-solving. Some data will be presented from continuing education courses for teachers, aiming to contribute to understanding regarding the different "personalities" of the rational number. In general, these "personalities" are not easily identified by teachers and students, which is the reason for the many difficulties encountered during problem-solving involving rational numbers. One of these "personalities", the ratio, provides the basis for the concept of proportionality, which is relevant because it is a unifying idea in mathematics.

Prática de Ensino Supervisionada no 2.º Ciclo do Ensino Básico – Matemática e Ciências da Natureza :o ensino e a aprendizagem dos números racionais e a utilização de materiais manipuláveis

Casaca, Ana
Fonte: Instituto Politécnico de Santarém Publicador: Instituto Politécnico de Santarém
Relevância na Pesquisa
46.26%

The Cognitive Predictors of Computational Skill with Whole versus Rational Numbers: An Exploratory Study

Seethaler, Pamela M.; Fuchs, Lynn S.; Star, Jon R.; Bryant, Joan
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.26%
The purpose of the present study was to explore the 3rd-grade cognitive predictors of 5th-grade computational skill with rational numbers and how those are similar to and different from the cognitive predictors of whole-number computational skill. Students (n = 688) were assessed on incoming whole-number calculation skill, language, nonverbal reasoning, concept formation, processing speed, and working memory in the fall of 3rd grade. Students were followed longitudinally and assessed on calculation skill with whole numbers and with rational numbers in the spring of 5th grade. The unique predictors of skill with whole-number computation were incoming whole-number calculation skill, nonverbal reasoning, concept formation, and working memory (numerical executive control). In addition to these cognitive abilities, language emerged as a unique predictor of rational-number computational skill.

A Compreensão da idéia do número racional e suas operações na EJA: uma forma de inclusão em sala de aula

Silva, Tácio Vitaliano da
Fonte: Universidade Federal do Rio Grande do Norte; BR; UFRN; Programa de Pós-Graduação em Ensino de Ciências Naturais e Matemática; Ensino de Ciências Naturais e Matemática Publicador: Universidade Federal do Rio Grande do Norte; BR; UFRN; Programa de Pós-Graduação em Ensino de Ciências Naturais e Matemática; Ensino de Ciências Naturais e Matemática
Tipo: Dissertação Formato: application/pdf
POR
Relevância na Pesquisa
46.29%
The awareness of the difficulty which pupils, in general have in understanding the concept and operations with Rational numbers, it made to develop this study which searches to collaborate for such understanding. Our intuition was to do with that the pupils of the Education of Young and Adults, with difficulty in understanding the Rational numbers, feel included in the learning-teaching process of mathematics. It deals with a classroom research in a qualitative approach with analysis of the activities resolved for a group of pupils in classroom of a municipal school of Natal. For us elaborate such activities we accomplished the survey difficulties and obstacles that the pupils experience, when inserted in the learning-teaching process of the Rational numbers. The results indicate that the sequence of activities applied in classroom collaborated so that the pupils to overcome some impediments in the learning of this numbers; A consciência da dificuldade que alunos, em geral, têm em compreender o conceito e operações com Números Racionais, nos fez desenvolver este estudo que busca colaborar para tal compreensão. Nosso intuito foi fazer com que os alunos da Educação de Jovens e Adultos, com dificuldade em compreender os Números Racionais...

Visualisation, navigation and mathematical perception: a visual notation for rational numbers mod1

Tolmie, Julie
Tipo: Thesis (PhD); Doctor of Philosophy (PhD)
EN_AU
Relevância na Pesquisa
66.41%
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...

Cálculo mental com números racionais não negativos: um estudo sobre as estratégias utilizadas por alunos do 4.º ano de escolaridade

Santos, Ana Catarina Granado Rebelo dos
Fonte: Instituto Politécnico de Lisboa Publicador: Instituto Politécnico de Lisboa
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Acculturation institutionnelle du chercheur, de l’enseignant et des élèves de 1re secondaire présentant des difficultés d’apprentissage dans la conception et la gestion de situations-problèmes impliquant des nombres rationnels

Lessard, Geneviève
Fonte: Université de Montréal Publicador: Université de Montréal
Tipo: Thèse ou Mémoire numérique / Electronic Thesis or Dissertation
FR
Relevância na Pesquisa
46.43%
Notre recherche s’intéresse à la transformation des rapports aux nombres rationnels d’élèves de 1re secondaire présentant des difficultés d’apprentissage. Comme le montrent plusieurs recherches, le défi majeur auquel sont confrontés les enseignants, ainsi que les chercheurs, est de ne pas s’enliser dans le cercle vicieux d’une réduction des enjeux de l’apprentissage des nombres rationnels et des possibilités d’apprentissage de l’élève en difficultés d’apprentissage, cet élève n’ayant pas ainsi la chance de mettre à l’épreuve ses connaissances, d’oser s’engager dans une démarche de construction de connaissances et d’apprécier les effets de son engagement cognitif. Afin de relever ce défi, nous avons misé sur l’intégration harmonieuse de situations problèmes. Il nous a semblé que, dans une démarche d’acculturation, l’approche écologique soit tout indiquée pour penser une «dé-transposition/re-transposition didactique» (Antibi et Brousseau, 2000) et reconstruire une mémoire porteuse d’espoirs (Brousseau et Centeno, 1998). Notre recherche vise à: 1) caractériser la progression des démarches d’acculturation institutionnelle de l’enseignant, du chercheur et des élèves et leurs effets sur les processus d’élaboration et de gestion des situations d’enseignement; 2) préciser l’évolution des connaissances...

Um estudo sobre construções dos Números Reais; A study on construction of the Real Numbers

Queiroz, Fabiana Moura de
Fonte: Universidade Federal de Goiás; Brasil; UFG; Programa de Pós-graduação em Matemática (IME); Instituto de Matemática e Estatística - IME (RG) Publicador: Universidade Federal de Goiás; Brasil; UFG; Programa de Pós-graduação em Matemática (IME); Instituto de Matemática e Estatística - IME (RG)
Tipo: Dissertação Formato: application/pdf
POR
Relevância na Pesquisa
56.3%
The main objective of this paper is to present the subtle passage of rational numbers to the real numbers, using a construction via Dedekind cuts and other by Cauchy sequences .We present a construction of rational numbers by equivalence classes, so that the reader has a foundation that serves as a support for a good understanding of proposed constructions of real numbers . We use the axiomatic method for buildings that are made on real numbers, in order to show the existence of an orderly and complete field and characterize it. It is also discussed, and a more synthesized form, the real numbers and its application to elementary and high school students.; O objetivo central deste trabalho é apresentar a sutil passagem dos números racionais aos números reais, utilizando uma construção via Cortes de Dedekind e outra por sequências de Cauchy. Apresenta-se uma construção dos números racionais por classes de equivalência, para que o leitor tenha um alicerce que sirva de apoio para um bom entendimento das construções propostas dos números reais. Utiliza-se o método axiomático para as construções que são feitas sobre números reais, com o intuito de mostrar a existência de um corpo ordenado e completo e caracterizá-lo. Discute-se ainda...

A divisão e os números racionais : uma pesquisa de intervenção psicopedagógica sobre o desenvolvimento de competências conceituais de alunos e professores; Division and rational numbers : a survey of psychopedagogical intervention on the development of conceptual competencies among students and teachers

Neves, Regina da Silva Pina
Tipo: Tese
POR
Relevância na Pesquisa
46.26%

Conhecimento do professor do 1º ciclo sobre números racionais

Perfeito, Margarida de Jesus Lucas
Fonte: Instituto Politécnico de Lisboa Publicador: Instituto Politécnico de Lisboa
Relevância na Pesquisa
46.43%
Dissertação apresentada à Escola Superior de Educação de Lisboa para obtenção do grau de mestre em Educação Matemática na Educação Pré-escolar e nos 1º e 2º Ciclos do Ensino Básico; Este estudo visa analisar o conhecimento matemático dos professores do 1º ciclo sobre números racionais, procurando responder às seguintes questões: (1) Que conhecimento revelam os professores do 1º ciclo sobre os números racionais e as suas várias representações?; (2) Como avaliam os professores o conhecimento que têm sobre os números racionais?; e (3) Que dificuldades manifestam os professores relativamente ao trabalho dos números racionais com os alunos? O estudo seguiu uma abordagem metodológica mista, reunindo componentes da investigação quantitativa e qualitativa. A recolha de dados decorreu entre 3 e 10 de janeiro de 2014 e foi feita a partir da aplicação de um questionário impresso a 18 professores do 1º ciclo de três escolas públicas, de um agrupamento situado numa zona limítrofe de Lisboa. O questionário pretendeu analisar o conhecimento matemático dos professores sobre números racionais e simultaneamente recolher informações sobre o modo como analisam o seu conhecimento e as dificuldades que sentem no seu ensino. Os dados indicam que a maioria dos professores não possui conhecimentos sólidos sobre os números racionais...

Conhecimento e formação de futuros professores dos primeiros anos – o sentido de número racional

Pinto, Hélia; Ribeiro, C. Miguel
Fonte: CIED – Centro Interdisciplinar de Estudos Educacionais/Escola Superior de Educação de Lisboa Publicador: CIED – Centro Interdisciplinar de Estudos Educacionais/Escola Superior de Educação de Lisboa
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.22%
Nos últimos anos o conhecimento do professor tem vindo a ser reconhecido como um dos aspetos nucleares no, e para o, desenvolvimento do conhecimento matemático dos alunos. Atendendo a essa centralidade, a formação deverá focar-se onde é, efetivamente, necessária, de modo a potenciar um incremento do conhecimento dos alunos, pelo conhecimento (e práticas) dos professores. Sendo os números racionais um dos tópicos problemáticos para os alunos, é fundamental identificar quais as situações matematicamente (mais) críticas para os professores de modo que, pela formação facultada, possam deixar de o ser. Neste artigo, tendo por foco o conhecimento matemático do professor e as suas especificidades, discutimos alguns aspetos desse conhecimento de futuros professores sobre números racionais, em concreto o sentido de número racional, identificando as suas componentes mais problemáticas e equacionando alguns dos porquês em que se sustentam. Terminamos com algumas considerações sobre implicações para a formação de professores e responsabilidade dos seus formadores.; Abstract: In recent years, teachers’ knowledge has come to be recognized as one of the core aspects in and for the development of students’ mathematical knowledge. This being the case...

Rational numbers with purely periodic $\beta$-expansion

Adamczewski, Boris; Frougny, Christiane; Siegel, Anne; Steiner, Wolfgang
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.29%
We study real numbers $\beta$ with the curious property that the $\beta$-expansion of all sufficiently small positive rational numbers is purely periodic. It is known that such real numbers have to be Pisot numbers which are units of the number field they generate. We complete known results due to Akiyama to characterize algebraic numbers of degree 3 that enjoy this property. This extends results previously obtained in the case of degree 2 by Schmidt, Hama and Imahashi. Let $\gamma(\beta)$ denote the supremum of the real numbers $c$ in $(0,1)$ such that all positive rational numbers less than $c$ have a purely periodic $\beta$-expansion. We prove that $\gamma(\beta)$ is irrational for a class of cubic Pisot units that contains the smallest Pisot number $\eta$. This result is motivated by the observation of Akiyama and Scheicher that $\gamma(\eta)=0.666 666 666 086 ...$ is surprisingly close to 2/3.

A negative result on algebraic specifications of the meadow of rational numbers

Bergstra, Jan A.; Bethke, Inge
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.17%
$\mathbb{Q}_0$ - the involutive meadow of the rational numbers - is the field of the rational numbers where the multiplicative inverse operation is made total by imposing $0^{-1}=0$. In this note, we prove that $\mathbb{Q}_0$ cannot be specified by the usual axioms for meadows augmented by a finite set of axioms of the form $(1+ \cdots +1+x^2)\cdot (1+ \cdots +1 +x^2)^{-1}=1$.; Comment: 5 pages, 2 tables

Representation, simplification and display of fractional powers of rational numbers in computer algebra

Rich, Albert D.; Stoutemyer, David R.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.18%
Simplification of fractional powers of positive rational numbers and of sums, products and powers of such numbers is taught in beginning algebra. Such numbers can often be expressed in many ways, as this article discusses in some detail. Since they are such a restricted subset of algebraic numbers, it might seem that good simplification of them must already be implemented in all widely used computer algebra systems. However, the algorithm taught in beginning algebra uses integer factorization, which can consume unacceptable time for the large numbers that often arise within computer algebra. Therefore some systems apparently use various ad hoc techniques that can return an incorrect result because of not simplifying to 0 the difference between two equivalent such expressions. Even systems that avoid this flaw often do not return the same result for all equivalent such input forms, or return an unnecessarily bulky result that does not have any other compensating useful property. This article identifies some of these deficiencies, then describes the advantages and disadvantages of various alternative forms and how to overcome the deficiencies without costly integer factorization.; Comment: 23 pages, 1 figure, 4 tables

Free monoids and forests of rational numbers

Nathanson, Melvyn B.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.17%
The Calkin-Wilf tree is an infinite binary tree whose vertices are the positive rational numbers. Each such number occurs in the tree exactly once and in the form $a/b$, where are $a$ and $b$ are relatively prime positive integers. This tree is associated with the matrices $L_1 = \left( \begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix} \right)$ and $R_1 = \left( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right)$, which freely generate the monoid $SL_2(\mathbf{N}_0)$ of $2 \times 2$ matrices with determinant 1 and nonnegative integral coordinates. For other pairs of matrices $L_u$ and $R_v$ that freely generate submonoids of $GL_2(\mathbf{N}_0)$, there are forests of infinitely many rooted infinite binary trees that partition the set of positive rational numbers, and possess a remarkable symmetry property.; Comment: 10 pages

Properties of proper rational numbers

Zelator, Konstantine
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.28%
This short article is aimed at educators and teachers of mathematics.Its goal is simple and direct:to explore some of the basic/elementary properties of proper rational numbers.A proper rational number is a rational which is not an integer. A proper rational r can be written in standard form: r=c/b,where c and b are relatively prime integers; and with b greater than or equal to 2. There are seven theorems, one proposition, and one lemma; Lemma1, in this paper. Lemma1 is a very well known result, commonly known as Euclid's lemma.It is used repeatedly throughout this paper, and its proof can be found in reference[1]. Theorem4(i) gives precise conditions for the sum of two proper rationals to be an integer.Theorem5(a) gives exact conditions for the product to be an integer. Theorem7 states that there exist no two proper rationals both of whose sum and product are integers.This follows from Theorem6 which states that if two rational numbers have a sum being an integer; and a product being an integer;then these two rationals must both be in fact integers.; Comment: 6 pages

Beta-expansions of rational numbers in quadratic Pisot bases

Hejda, Tomáš; Steiner, Wolfgang
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.21%
We study rational numbers with purely periodic R\'enyi $\beta$-expansions. For bases $\beta$ satisfying $\beta^2=a\beta+b$ with $b$ dividing $a$, we give a necessary and sufficient condition for $\gamma(\beta)=1$, i.e., that all rational numbers $p/q\in[0,1)$ with $\gcd(q,b)=1$ have a purely periodic $\beta$-expansion. A simple algorithm for determining the value of $\gamma(\beta)$ for all quadratic Pisot numbers $\beta$ is described.; Comment: 12 pages, 3 figures, 2 tables

Fields of quantum reference frames based on different representations of rational numbers as states of qubit strings

Benioff, Paul
Tipo: Artigo de Revista Científica