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1
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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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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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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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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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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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.<br>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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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.<br>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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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 JANEIRO<br>COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR<br>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.<br>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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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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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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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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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.<br>Ph.D.<br>Department of Teaching and Learning Principles<br>Education<br>Education PhD
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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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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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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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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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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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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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Previous issue date: 2017-02-20<br>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.<br>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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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.<br>Submitted by Jessyca Silva (jessyca@mat.ufc.br) on 2017-09-15T05:05:08Z
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Previous issue date: 2017<br>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.<br>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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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.<br>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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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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Dolma, Phuntsho. "The relationship between estimation skill and computational ability of students in years 5, 7 and 9 for whole and rational numbers." Thesis, Edith Cowan University, Research Online, Perth, Western Australia, 2002. https://ro.ecu.edu.au/theses/742.
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This study explored the relationship between estimation skill and computational ability for whole and rational numbers. The methods carried out were both quantitative as well as qualitative and data were collected from three primary schools along with their associated high school in the Perth area. The year levels chosen were 5, 7 and 9. There were two classes from each chosen primary school representing Year 5 and Year 7 and three classes of Year 9 from the high school. The total number of students involved was 91, 77 and 73 from the three respective year levels. Instruments used for collecting data were group-administered tests and interview. Two parallel tests with identical items, where one of the pair was estimation and the other written computation were administered to all the students in the chosen year levels. Interviews were conducted for the group of selected students based on the criteria: slightly above the average and slightly below the average. There were eighteen students with nine in each group. The results of the correlation shows that performance in estimation is positively correlated with written computation in all the year levels. Moreover, the t-test result reveals that there is no significant difference between the two tests expect in Year 7. Hence, the findings indicate that a child who is good in estimation skill can also perform well in written computation. As such, the importance of achieving estimation skill in a child would be very helpful in solving computation problems with understanding. On the other hand, children's performance related to the development of estimation skill and computational ability seems to be in positive direction from Year 5 to Year 7. Whereas the Year 9's performance is lower than Year 7. Among the topics, the children fared better in whole numbers compared to other topics. Performance tends to follow in a descending order from whole number to ratios. The disparities between estimation skill and computational ability are also more towards the difficult topics like division and multiplication of fractions and decimals. At the same time, the feedback from the interviewees tend to show that, the children from slightly above the average are better at choosing their own sensible strategies for solving the problems, whereas the students from slightly below average are more prone to the rote-learned algorithms. Although, male students appeared to perform better than the female students, the differences in performances are not that high. Thus, the result depicts that there are no significant gender issues in the selected year levels and topics. Further research needs to be carried out in order to determine the relationship between estimation skill and computational ability with topics other than whole and rational numbers, especially in measurement topics. Moreover, such studies can be done involving larger samples, and in other countries as well, Doing so can highlight the importance of the integration of estimation skill in teaching and learning mathematics.
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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.<br>Ph. D.
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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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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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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 Pirola<br>Resumo: 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)<br>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)<br>Mestre
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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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Previous issue date: 2016-02-26<br>Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES<br>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.<br>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.
31
Johnson, Gwendolyn Joy. "Proportionality in Middle-School Mathematics Textbooks." Scholar Commons, 2010. https://scholarcommons.usf.edu/etd/1670.
Full textAbstract:
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.
32
Silva, Guimarães Vieira da. "Irracionalidade e transcendência: aspectos elementares." Universidade Federal do Tocantins, 2018. http://hdl.handle.net/11612/978.
Full textAbstract:
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.<br>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.
33
Lopes, Ana Paula. "Desenvolvimento do sentido de número no ensino básico: um estudo no sétimo ano de escolaridade." Master's thesis, Universidade de Évora, 2010. http://hdl.handle.net/10174/20773.
Full textAbstract:
A sociedade contemporânea exige do cidadão raciocínio quantitativo e os novos paradigmas económico-sociais colocam a Matemática escolar perante um novo desafio: desenvolver a literacia matemática dos alunos. A literacia matemática contempla um vasto conjunto de conhecimentos e capacidades entre eles o sentido de número. O desenvolvimento do sentido de número dos alunos tem suscitado alguns trabalhos de pesquisa, em particular ao nível do primeiro e segundo ciclos do Ensino Básico. O objectivo desde estudo é recolher evidências sobre o sentido de número de alunos do terceiro ciclo, mais concretamente sobre o sentido de número racional dos alunos do sétimo ano de escolaridade. Para tal foi necessário considerar uma vasta e complexa rede de competências que caracterizam o sentido de número, entre elas o sentido de operação, mas também os subconstructos que definem os números racionais. Desta forma emergem deste objectivo duas questões: (i) que compreensão têm os alunos das formas equivalentes (inteiros, fracções, decimais, percentagens) dos números? (ii) como entendem os alunos o efeito das operações nos números, as propriedades e as relações entre as operações? Este estudo é uma investigação de natureza qualitativa com recurso ao design de caso. A recolha de dados empíricos foi realizada ao longo do ano lectivo 2007/2008 numa turma de sétimo ano de escolaridade onde a investigadora era a docente da disciplina de Matemática. A investigadora assumiu um papel de observadora participante e foram considerados três estudos de caso: dois alunos e a turma. A informação analisada resultou de vários métodos de recolha: a) inquéritos por questionário e entrevista com tarefas; b) observação da resolução das tarefas em ambiente de sala de aula; c) análise documental. A análise deste conjunto de contributos foi feita tendo em conta as questões de investigação e de forma individual para cada um dos casos. O estudo mostrou que os alunos atribuem pouco significado aos números, às operações e aos contextos. Revelam alguma compreensão dos números racionais escritos na forma de fracção e na forma decimal, mas têm dificuldade em compreender e manipular racionais escritos com recurso a outras formas de representação e a maioria dos alunos revela poucas competências de comparação e ordenação de números racionais. Nesta investigação ressalta também o fraco sentido de operação dos alunos. Mostram pouca compreensão do efeito das operações nos números e têm dificuldade em reconhecer, no contexto do problema, a operação que mais se adequa à situação, manifestando dificuldade quer na interpretação das situações quer na definição de estratégias apropriadas. ABSTRACT: Nowadays society requires of the citizen quantitative reasoning and the new economic-social paradigms place the school Mathematics before a new challenge: to develop the mathematical literacy of the pupils. The mathematical literacy contemplates a vast set of knowledge and capacities such as number sense. The number sense development has excited some works of research, in particular referring to grades 1-6. The aim of this study is to collect evidences about the number sense of pupils of grades 7-9, more concretely on the 7th grade pupils' sense of rational numbers. For such it was necessary to consider vast and complex net of abilities that characterize number sense, including operation sense, but also the rational numbers' sub constructs. Therefore, of this aim two questions emerge: (i) which understanding pupils have of the equivalents forms (entire, fractions, decimals, percentages) of numbers? (ii) how pupils understand the effect that operations have on numbers, the properties and relations between the operations? This study is a qualitative nature investigation that resorts to case studies. Empirical data collection was carried along the school year of 2007/2008 in a 7th grade class where the investigator was the Mathematics teacher. The investigator assumed the role of participant observer and had been considered three studies of case: two pupils and the class. The analyzed data was collected by means of: a) survey - questionnaire and interview with tasks; b) observation of the resolution of the tasks in classroom environment; c) documentary analysis. The data analysis was made according to the questions of the study and individually for each one of the cases. The study showed that pupils attribute slight meant to the numbers, to the operations and the contexts. They reveal some understanding of rational numbers represented as fractions or decimal numbers but disclose difficulty in understanding and manipulating rational numbers' other forms of representation. The majority of the pupils shows few abilities of comparison and ordinance of rational numbers. This investigation also revealed the weak sense of operation of the pupils. They show little understanding of the effect that operations have on numbers as well as difficulty in recognizing, according to the problem's context, the operation that more suits the situation. Pupils disclose difficulty in the interpretation of the situations and in the definition of appropriate strategies.
34
Lack, Brian S. "Student Participation in Mathematics Discourse in a Standards-based Middle Grades Classroom." Digital Archive @ GSU, 2010. http://digitalarchive.gsu.edu/ece_diss/11.
Full textAbstract:
The vision of K-12 standards-based mathematics reform embraces a greater emphasis on students’ ability to communicate their understandings of mathematics by utilizing adaptive reasoning (i.e., reflection, explanation, and justification of thinking) through mathematics discourse. However, recent studies suggest that many students lack the socio-cognitive capacity needed to succeed in learner-centered, discussion-intensive mathematics classrooms. A multiple case study design was used to examine the nature of participation in mathematics discourse among two low- and two high-performing sixth grade female students while solving rational number tasks in a standards-based classroom. Data collected through classroom observations, student interviews, and student work samples were analyzed via a multiple-cycle coding process that yielded several important within-case and cross-case findings. Within-case analyses revealed that (a) students’ access to participation was mediated by the degree of space they were afforded and how they attempted to utilize that space, as well as the meaning they were able to construct through providing and listening to explanations; and (b) participation was greatly influenced by peer interactional tendencies that either promoted or impeded productive contributions, as well as teacher interactions that helped to offset some of the problems related to unequal access to participation. Cross-case findings suggested that (a) students’ willingness to contribute to task discussions was related to their goal orientations as well as the degree of social risk perceived with providing incorrect solutions before their peers; and (b) differences between the kinds of peer and teacher interactions that low- and high-performers engaged in were directly related to the types of challenges they faced during discussion of these tasks. An important implication of this study’s findings is that the provision of space and meaning for students to participate equitably in rich mathematics discourse depends greatly on teacher interaction, especially in small-group instructional settings where unequal peer status often leads to unequal peer interactions. Research and practice should continue to focus on addressing ways in which students can learn how to help provide adequate space and meaning in small-group mathematics discussion contexts so that all students involved are allowed access to an optimally rich learning experience.
35
Wolffenbüttel, Reni. "Investigando números racionais com o software GeoGebra." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2015. http://hdl.handle.net/10183/133653.
Full textAbstract:
A presente pesquisa tem como foco o ensino dos números racionais no Ensino Fundamental. Seu objetivo é analisar as potencialidades e proposta de ensino que utiliza o computador, em particular do so fltiwmaitraeç dõee sg edoem uemtriaa dinâmica GeoGebra, e a metodologia de aulas de matemática investigativa. Essa proposta de ensino foi aplicada em turmas de 8º ano do Ensino Fundamental de uma escola pública, localizada na cidade de Sapucaia do Sul/RS. Esses estudantes já traziam conhecimentos sobre números racionais e, diante disso, esse campo numérico foi retomado com a intenção de verificar e contornar possíveis déficits de aprendizagem, assim como de ampliar o conhecimento sobre esses números por meio de investigações em que pudessem ser observadas algumas de suas características. Para isso, propusemos atividades que articulavam simultaneamente diferentes representações dos números racionais. A metodologia de pesquisa empregada foi a qualitativa. A proposta de ensino apresentada no final deste texto como alternativa de trabalho para o ensino de números racionais, e para professores, diante da análise realizada, poderem refletir acerca de suas vpiostueanicsi adliod asdoefstw ea rliem GitaeçoõGese.b Drao, sq ruees uflatavdooresc oebratimdo sa,o pso daelumnooss dae sctoamcaprr eoesn rseãcou rdsooss números racionais e suas regularidades, e o cenário investigativo-tecnológico, que fez com que eles se mantivessem engajados na investigação como agentes de seu aprendizado.<br>This research focuses on the teaching of rational numbers in Elementary School. It's aim is to analyse The potentialities and limitations of a teaching approach which proposes the use of the computer, particularly the dynamic geometry software GeoGebra, and the methodology of investigative math classes. This teaching approach was applied to students of the 8th year of an Elementary Education public school located at Sapucaia do Sul/RS, Brazil. These students already studied rational numbers at school in previous years. Thus this numerical field has been taken into account with the intention to check and bypass possible learning deficits, as well as increase knowledge of these numbers through investigations in which it could be observed some your characteristics. For this purpose, we proposed activities that simultaneously articulated different representations of rational numbers. The research is based in a qualitative paradigm. The teaching approach is presented at the end of this text as an alternative way for teaching rational numbers and other teachers in view of the analysis can consider its potentialities and limitations. From the results, we can highlight the visuals of the GeoGebra software, which favored students the understanding of rational numbers and their regularities and the investigative - technological scenario, which caused them to remain engaged in research as their learning agents.
36
Zakrzewski, Jennifer. "Effect of Interactive Digital Homework with an iBook on Sixth Grade Students' Mathematics Achievement and Attitudes when Learning Fractions, Decimals, and Percents." Scholar Commons, 2015. https://scholarcommons.usf.edu/etd/5611.
Full textAbstract:
Over the past decade, technology has become a prominent feature in our lives. Technology has not only been integrated into our lives, but into the classroom as well. Teachers have been provided with a tremendous amount of technology related tools to educate their students. However, many of these technologically enhanced tools have little to no research supporting their claims to enhance learning.
This study focuses on one aspect of technology, the iBook, to complete homework relating to fractions, decimals, and percents in a sixth grade classroom. An iBook is a digital textbook that allows the user to interact with the book through various features. Some of these features include galleries, videos, review quizzes, and links to websites. These interactive features have the potential to enhance comprehension through interactivity and increased motivation.
Prior to this study, two pilot iterations were conducted. During each pilot study, students in two sixth grade classrooms used the iBook to supplement learning of fractions, decimals, and percents. A comparison group was not included during either iteration, as the goal was to fine-tune the study prior to implementation. The current study was the third iteration, which included a comparison and treatment group. During this study, three research questions were considered:
1) When learning fractions, decimals, and percents, in what ways, if any, do students achieve differently on a unit test when using an interactive iBook for homework as compared to students who have access to the same homework questions in an online static PDF format?
2) What are students' perceptions of completing homework regarding fractions, decimals, and percents with an interactive iBook compared to students who complete homework in an online static PDF format?
3) In what ways does students' achievement on homework differ when completing homework related to fractions, decimals, and percents from an interactive iBook and a static PDF online assignment?
Thirty students from a small charter school in southeast Florida participated in the third iteration of this study. Fifteen students were in the comparison group and fifteen were in the treatment group. Students in both groups received comparable classroom instruction, which was determined through audio recordings and similar lesson plans. Treatment group students were provided with a copy of the iBook for homework. Comparison group students were provided with a set of questions identical to the iBook questions in a static digital PDF format. The comparison group students also had access to the textbook, but not the iBook nor the additional resources available within the iBook.
The study took place over three weeks. At the commencement of the study, all students were given a pretest to determine their prior knowledge of fractions, decimals, and percents. Students were also asked to respond to questions regarding typical homework duration, level of difficulty, overall experience, and additional resources used for support. During the study, both classes received comparable instruction, which included mini lessons, manipulative based activities, mini quizzes, and group activities. Nightly homework was assigned to each group. At the conclusion of the study, both groups were given a posttest, which was identical to the pretest. Students were asked identical questions about their homework perceptions as prior to the study, but were asked to respond in regards to the study alone. All participating students completed a questionnaire to describe their perceptions of completing homework regarding fractions, decimals, and percents with an iBook as opposed to static digital PDF homework. Lastly, six students from the comparison group participated in a focus group and six students from the treatment group participated in a separate focus group.
Data were collected from the pretest and posttest, pre and post homework responses, collected homework, mini quizzes, audio recordings, teacher journal, questionnaires, and the focus group. No difference in achievement was found between the two groups. However, both groups improved significantly from the pretest to posttest. Based on the questionnaires and focus groups, both groups of students felt they learned fractions, decimals, and percents effectively. However, the questionnaire data showed the treatment group found the iBook more convenient than the comparison group did the textbook.
Data from this study provide a baseline for future studies regarding iBooks in middle school mathematics. Although the data show no difference in achievement between the two groups, further studies should be conducted in regards to the iBook. Questionnaire and focus group data suggest, with modifications, students may be more inclined to use the resources within the iBook, which may enhance achievement with fractions, decimals, and percents.
37
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.
Full textAbstract:
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.
38
Menezes, Fernanda Martinez. "Propriedades da expansão decimal." Universidade de São Paulo, 2016. http://www.teses.usp.br/teses/disponiveis/55/55136/tde-05102016-085553/.
Full textAbstract:
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.<br>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.
39
Eriksson, Helena. "Rationella tal som tal : Algebraiska symboler och generella modeller som medierande redskap." Licentiate thesis, Stockholms universitet, Institutionen för matematikämnets och naturvetenskapsämnenas didaktik, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-129269.
Full textAbstract:
In this study the teaching of mathematics has been developed in relation to rational numbers and towards a learning activity. At the same time topic-specific mediated tools have been studied. The iterative model for learning study has been used as research approach. The purpose of the study was to explore what in an algebraic learning activity enables knowledge of rational numbers to develop. The specific questions answered by the study are how an algebraic learning activity can be formed in an otherwise arithmetic teaching tradition, what knowledge is mediated in relation to different mediated tools and what in these tools that enable this knowledge. The result of the study shows how an algebraic learning activity can be developed to support the students to understand rational numbers even in an arithmetic teaching tradition. The important details that developed the algebraic learning activity were to identify the problem to create learning tasks and the opportunity for the students to reflect that are characteristic of a learning activity. The result also shows that the mediating tools, the algebraic symbols and the general model for fractional numbers, have had significant importance for the students' possibilities to explore rational numbers. The conditions for the algebraic symbols seem to be the possibilities for these symbols to include clues to the meaning of the symbol and that the same symbol can be used in relation to several of other mediated tools. The conditions in the general model consisted of that the integer numbers and the rational numbers in the model could be distinguished and that the students could reflect on the meaning of the different parts. The general model consists of the algebraic symbols, developed in the learning activity. The algebraic symbols make the structure of the numbers visible and the general model mediates the structure of additive and multiplicative conditions that are contained in a rational number. The result of the study contributes in part to the field of mathematics education research by examining Elkonin's and Davydov's Mathematical Curriculum in a western teaching practice and in part to a development of the model of Learning study as a didactical research approach by using an activity-theoretical perspective on design and analysis.
40
LUCENA, Alexandre Marcelino de. "A metacognição no livro didático de matemática : um olhar sobre os números racionais." Universidade Federal Rural de Pernambuco, 2013. http://www.tede2.ufrpe.br:8080/tede2/handle/tede2/5414.
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Previous issue date: 2013-03-04<br>Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES<br>The present study aimed to investigate the extent to which the activities of mathematics textbooks could favor the development of students' metacognitive strategies during its resolution. We direct our focus to the rational numbers, for being a very present content in students' daily lives, and even then, be a content that generates many learning difficulties. Decided to investigate two math textbooks approved by PNLD/2011 with different perspectives regarding to a teaching methodology, one more attuned to the new conceptions of teaching (LD 1) and other more traditional (LD 2). To answer our research question, initially we selected in the evaluation form of mathematics textbooks from the Guide PNLD/2011, the activities and skills that, in our view, could favor the development of metacognition. Then we seek to categorize activities selected according to the categories proposed by Araújo (2009). Following this analysis, we found that the two books surveyed offer few activities that may favor the development of metacognitive strategies, because the LD 1 only 7.87% of the activities of the chapters related to rational numbers were
classified, while in LD 2 this number was lower, accounting for only 4.03% of the activities that may favor the development of metacognition. According to the categories of Araújo (2009), the few activities proposed by this material that favor metacognition propose reflections regarding the mathematical rules in 1st place (metacognitive strategies in order of procedure), followed by strategies that lead to reflections related to understanding the problem (strategies of the order of understanding the problem). We do not found the activities in the personal category, but on the other hand, we found problems that beckon metacognitive strategies in the sense of knowledge of knowledge itself, which did not appear in Araújo’s research (2009), and add these findings to its rating. Therefore, the results show that the two math textbooks surveyed bring in their chapters related to rational numbers, few activities that may favor the development of metacognitive strategies. However it is important to remember that the textbook is just a tool used by the teacher, then the development of metacognition in students will be dependent on the way the teacher uses this book and the activities proposed for this material.<br>A presente pesquisa teve como objetivo investigar em que medida as atividades de livros didáticos de matemática poderiam favorecer o desenvolvimento de estratégias metacognitivas dos alunos, durante a sua resolução. Direcionamos nosso foco para os números racionais, por ser um conteúdo muito presente no cotidiano dos estudantes e, mesmo assim, ser um conteúdo que gera muitas dificuldades de aprendizagem. Resolvemos investigar dois livros didáticos de matemática aprovados pelo PNLD/2011, com perspectivas distintas em relação à metodologia de ensino; um mais afinado com as novas concepções de ensino (LD 1) e outro mais tradicional (LD 2). Para responder nossa questão de pesquisa, inicialmente, selecionamos na
ficha de avaliação dos livros didáticos de matemática do Guia PNLD/2011, as atividades e habilidades que, em nossa avaliação, poderiam favorecer o desenvolvimento da metacognição. Em seguida buscamos categorizar as atividades selecionadas de acordo com as categorias propostas por Araújo (2009). Após a
referida análise, constatamos que os dois livros pesquisados disponibilizam poucas atividades que podem favorecer o desenvolvimento de estratégias metacognitivas, pois no LD 1 apenas 7,87% das atividades dos capítulos relacionados aos números racionais foram classificadas, enquanto que no LD 2 esse número foi menor, correspondendo a apenas 4,03% das atividades que podem favorecer o desenvolvimento da metacognição. De acordo com as categorias de Araújo (2009), as poucas atividades proposta por esse material que favorecem a metacognição, propõem reflexões em relação as regras matemáticas em 1º lugar (estratégias metacognitivas de ordem do procedimento), seguidas pelas estratégias que conduzem a reflexões relacionadas a compreensão do problema ( estratégias da ordem da compreensão do problema). Não encontramos atividades na categoria de ordem pessoal, mas em contrapartida, encontramos problemas que acenam para estratégias metacognitivas no sentido do conhecimento do próprio conhecimento, que não apareceram na pesquisa de Araújo (2009) e acrescentamos esses achados
a sua classificação. Portanto, os resultados mostram que os dois livros didáticos de matemática pesquisados trazem, em seus capítulos referentes aos números racionais, poucas atividades que podem favorecer o desenvolvimento de estratégias metacognitivas. No entanto é importante lembrar que o livro didático é apenas uma ferramenta utilizada pelo professor, então o desenvolvimento da metacognição nos
alunos vai estar na dependência da forma como o professor utiliza esse livro e as atividades propostas por esse material.
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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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<p> 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. </p><p> 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.</p><p>
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Smith, Scott. "An Exploratory Study of Fifth-Grade Students’ Reasoning About the Relationship Between Fractions and Decimals When Using Number Line-Based Virtual Manipulatives." DigitalCommons@USU, 2017. https://digitalcommons.usu.edu/etd/5625.
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Understanding the relationship between fractions and decimals is an important step in developing an overall understanding of rational numbers. Research has demonstrated the feasibility of technology in the form of virtual manipulatives for facilitating students’ meaningful understanding of rational number concepts. This exploratory dissertation study was conducted for the two closely related purposes: first, to investigate a sample of fifth-grade students’ reasoning regarding the relationship between fractions and decimals for fractions with terminating decimal representations while using virtual manipulative incorporating parallel number lines; second, to investigate the affordances of the virtual manipulatives for supporting the students’ reasoning about the decimal-fraction relationship.
The study employed qualitative methods in which the researcher collected and analyzed data from fifth-grade students’ verbal explanations, hand gestures, and mouse cursor motions. During the course of the study, four fifth-grade students participated in an initial clinical interview, five task-based clinical interviews while using the number line-based virtual manipulatives, and a final clinical interview. The researcher coded the data into categories that indicated the students’ synthetic models, their strategies for converting between fractions and decimals, and evidence of students’ accessing the affordances of the virtual manipulatives (e.g., students’ hand gestures, mouse cursor motions, and verbal explanations).
The study yielded results regarding the students’ conceptions of the decimal-fraction relationship. The students’ synthetic models primarily showed their recognition of the relationship between the unit fraction 1/8 and its decimal 0.125. Additionally, the students used a diversity of strategies for converting between fractions and decimals. Moreover, results indicate that the pattern of strategies students used for conversions of decimals to fractions was different from the pattern of strategies students used for conversions of fractions to decimals. The study also yielded results for the affordances of the virtual manipulatives for supporting the students’ reasoning regarding the decimal-fraction relationship. The analysis of students’ hand gestures, mouse cursor motions, and verbal explanations revealed the affordances of alignment and partition of the virtual manipulatives for supporting the students’ reasoning about the decimal-fraction relationship. Additionally, the results indicate that the students drew on the affordances of alignment and partition more frequently during decimal to fraction conversions than during fraction to decimal conversions.
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Valio, Denise Teresa de Camargo. "Frações: estratégias lúdicas no ensino da matemática." Universidade Federal de São Carlos, 2014. https://repositorio.ufscar.br/handle/ufscar/5964.
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Previous issue date: 2014-08-21<br>Financiadora de Estudos e Projetos<br>The main aim of this master´s degree dissertation is the Mathematics teaching as well as the pedagogical and teaching practices related to the topic of rational numbers, particularly, fractions. The Fractions project, main title of this paper, aims at reaching not only basic education students but also educators with the objective of proposing understandable and attractive ways of teaching the subject matter in focus.The methodology applied in learning rational numbers in their fractional form was achieved through the playful and practical exercises involving two groups of 6th graders in the Fundamental from public schools. The material handling and display of results contribute to the construction of knowledge on Mathematics and consequently its learning. The teaching activities and experiments are the hallmark and engine for the development of this dissertation project due to the fact that the use of manipulable material (graded PET bottles, funnels and water) ensures originality to the teaching/learning relationship of Mathematics.Mathematical concepts such as equivalence and comparison between fractions and even basic operations (addition and subtraction) purposely confine in the teaching sequences carried out in the educational project.<br>O objetivo desta dissertação de mestrado é o Ensino da Matemática bem como as práticas didático-pedagógicas acerca do tema números racionais , em particular, frações. O projeto Frações , título principal desse trabalho, pretende atingir estudantes da Educação Básica e também educadores com a intenção de propor meios compreensíveis e atrativos do ensino da disciplina em questão. A metodologia empregada para a aprendizagem de números racionais na forma fracionária foi o exercício prático e lúdico envolvendo alunos de duas turmas do 6º ano do Ensino Fundamental da escola pública. A manipulação de materiais e a visualização de resultados concorrem para a construção do conhecimento da Matemática e, consequentemente, de seu aprendizado. As atividades e experimentações didáticas são a marca e o propulsor do desenvolvimento deste projeto dissertativo, pois ao empregar materiais manipuláveis (garrafas PET graduadas, funis e água) isto nos garante originalidade para a relação ensino/aprendizagem da Matemática. Os conceitos matemáticos como a equivalência e comparação entre frações e ainda as operações básicas (adição e subtração) circunscrevem, propositadamente, as sequências didáticas executadas no projeto educacional.
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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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Previous issue date: 2013-08<br>Capes<br>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.<br>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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Valera, Alcir Rojas [UNESP]. "Uso social e escolar dos números racionais: representação fracionária e decimal." Universidade Estadual Paulista (UNESP), 2003. http://hdl.handle.net/11449/90210.
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valera_ar_me_mar.pdf: 594283 bytes, checksum: 7fa747413b18f73739f058ca4ea1146e (MD5)<br>Os números racionais apresentam-se como conteúdo que os alunos do Ensino Fundamental e Médio têm dificuldades para aprender. Parte dessas dificuldades decorre da diferença instituída entre o uso cotidiano dos números racionais pelo aluno e a maneira como são ensinados na escola e, também pelo desconhecimento, por parte da escola, da multiplicidade dos significados dos racionais. Enquanto o uso social centra-se na forma decimal o uso escolar recai mais sobre a forma fracionária dos números racionais. É uma separação indesejável que as práticas escolares trataram de acentuar ao longo do tempo. A partir de pesquisa bibliográfica e de estudo documental procurou-se caracterizar, nesse trabalho, a dicotomização existente entre o uso e o ensino da Matemática, que acabam sendo responsáveis por prejuízos na aprendizagem dos alunos. Isto pode ser verificado nos erros que os alunos cometeram nas provas oficiais (SARESP, SAEB...). Procurou-se analisar como essa separação vem sendo reforçada nos documentos oficiais, por meio das propostas pedagógicas e curriculares. Verificaram-se como diferentes documentos e publicações oficiais abordam os números racionais e tratam da articulação entre a perspectivas do uso escolar e a do uso cotidiano dos números racionais. Essa análise possibilitou compreender diferentes tipos de argumentações e justificativas para o ensino das frações, presentes nos currículos oficiais, bem como explicitar os conteúdos e metodologias adequadas às concepções apresentadas em tais documentos. Tudo isso possibilitou conhecer parte dos problemas que ocorrem com o ensino de frações e suas causas e por isso sugerir propostas que sinalizam para a sua superação. Embora o estabelecimento de relações entre o uso social e uso escolar ainda não ocorra de maneira efetiva, reconhece-se que aquelas orientações...
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Valera, Alcir Rojas. "Uso social e escolar dos números racionais : representação fracionária e decimal /." Marília : [s.n.], 2003. http://hdl.handle.net/11449/90210.
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Orientador: Vinício de Macedo Santos<br>Banca: Célia Maria Carolino Pires<br>Banca: José Carlos Miguel<br>Abstract: The rational numbers are shown as a subject that the students of the Elementary and High School have difficulties to learn. Some of these difficulties are due to the difference established between the daily use of the rational numbers by the student and the way it is taught at the school and, also for the ignorance, on the part of the school, of the multiplicity of their meanings. While the social use is centered in the decimal form, the school use lies more on the fractional form of the rational numbers. It is an undesirable separation that the school practices have accentuated through time. This study tried to characterize the existent dichotomization between it the use and the teaching of the Mathematics, starting from bibliographical research and of documental study that end up being responsible for damages in the students' learning.. This can be verified in the mistakes committed in the official tests (SARESP, SAEB...). It was sought to analyze how that separation has been reinforced in the official documents, by the pedagogic proposals and curricula. It was verified how the different documents and official publications deal with the rational numbers and the articulation among perspectives of the school use and the daily use of the rational numbers. That analysis made possible to understand different types of arguments and justifications for the teaching of the fractions, present in the official curricula, as well as explain the contents and the most appropriate methodologies of the conceptions presented in such documents. All this made possible to know part of the problems that happen with the teaching of fractions and their causes, and so, make suggestions on how these problems can be solved. Although the establishment of relationships between the social use and school use still doesn't happen in an effective way, it is recognized... (Complete abstract, click electronic address below)<br>Resumo: Os números racionais apresentam-se como conteúdo que os alunos do Ensino Fundamental e Médio têm dificuldades para aprender. Parte dessas dificuldades decorre da diferença instituída entre o uso cotidiano dos números racionais pelo aluno e a maneira como são ensinados na escola e, também pelo desconhecimento, por parte da escola, da multiplicidade dos significados dos racionais. Enquanto o uso social centra-se na forma decimal o uso escolar recai mais sobre a forma fracionária dos números racionais. É uma separação indesejável que as práticas escolares trataram de acentuar ao longo do tempo. A partir de pesquisa bibliográfica e de estudo documental procurou-se caracterizar, nesse trabalho, a dicotomização existente entre o uso e o ensino da Matemática, que acabam sendo responsáveis por prejuízos na aprendizagem dos alunos. Isto pode ser verificado nos erros que os alunos cometeram nas provas oficiais (SARESP, SAEB...). Procurou-se analisar como essa separação vem sendo reforçada nos documentos oficiais, por meio das propostas pedagógicas e curriculares. Verificaram-se como diferentes documentos e publicações oficiais abordam os números racionais e tratam da articulação entre a perspectivas do uso escolar e a do uso cotidiano dos números racionais. Essa análise possibilitou compreender diferentes tipos de argumentações e justificativas para o ensino das frações, presentes nos currículos oficiais, bem como explicitar os conteúdos e metodologias adequadas às concepções apresentadas em tais documentos. Tudo isso possibilitou conhecer parte dos problemas que ocorrem com o ensino de frações e suas causas e por isso sugerir propostas que sinalizam para a sua superação. Embora o estabelecimento de relações entre o uso social e uso escolar ainda não ocorra de maneira efetiva, reconhece-se que aquelas orientações... (Resumo completo, clicar acesso eletrônico abaixo)<br>Mestre
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Queiroz, Fabiana Moura de. "Um estudo sobre construções dos Números Reais." Universidade Federal de Goiás, 2015. http://repositorio.bc.ufg.br/tede/handle/tede/4555.
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Previous issue date: 2015-03-06<br>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.<br>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, e de uma forma mais sintetizada,
os números reais e a sua aplicação com alunos de ensino fundamental e médio.
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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 Superior<br>This 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<br>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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Pimentel, Thiago Trindade. "Construção dos números reais via cortes de Dedekind." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/55/55136/tde-18102018-164352/.
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O objetivo desta dissertação é apresentar a construção dos números reais a partir de cortes de Dedekind. Para isso, vamos estudar os números naturais, os números inteiros, os números racionais e as propriedades envolvidas. Então, a partir dos números racionais, iremos construir o corpo dos números reais e estabelecer suas propriedades. Um corte de Dedekind, assim nomeado em homenagem ao matemático alemão Richard Dedekind, é uma partição dos números racionais em dois conjuntos não vazios A e B em que cada elemento de A é menor do que todos os elementos de B e A não contém um elemento máximo. Se B contiver um elemento mínimo, então o corte representará este elemento mínimo, que é um número racional. Se B não contiver um elemento mínimo, então o corte definirá um único número irracional, que preenche o espaço entre A e B. Desta forma, pode-se construir o conjunto dos números reais a partir dos racionais e estabelecer suas propriedades. Esta dissertação proporcionará aos estudantes do Ensino Médio, interessados em Matemática, uma formação sólida em um de seus pilares, que é o conjunto dos números reais e suas operações algébricas e propriedades. Isso será muito importante para a formação destes alunos e sua atuação educacional.<br>The purpose of this dissertation is to present the construction of the real numbers from Dedekind cuts. For this, we study the natural numbers, the integers, the rational numbers and some properties involved. Then, based on the rational numbers, we construct the field of the real numbers and establish their properties. A Dedekind cut, named after the German mathematician Richard Dedekind, is a partition of the rational numbers into two non-empty sets A and B, such that each element of A is smaller than all elements of B and A does not contain a maximum element. If B contains a minimum element, then the cut represents this minimum element, which is a rational number. If B does not contain a minimal element, then the cut defines a single irrational number, which \"fills the gap\" between A and B. In this way, one can construct the set of real numbers from the rationals and establish their properties. This dissertation provides students who like Mathematics a solid basis in one of the pillars of Mathematics, which is the set of real numbers and their algebraic operations and properties. This text will be very important for your educational background and performance.
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Matos, Raphael Neves de. "Uma contribui??o para o ensino aprendizagem dos n?meros racionais: a rela??o entre d?zimas peri?dicas e progress?es geom?tricas." UFVJM, 2017. http://acervo.ufvjm.edu.br/jspui/handle/1/1641.
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Previous issue date: 2017<br>Este trabalho teve como objetivo principal apresentar uma contribui??o para o ensino aprendizagem dos n?meros racionais, destacando principalmente a rela??o entre d?zimas peri?dicas e progress?es geom?tricas. A metodologia utilizada permitiu a an?lise da abordagem e sequ?ncia did?tica dos t?picos D?zima peri?dica e Progress?o Geom?trica Infinita, contemplada nos livros did?ticos aprovados pelo Programa Nacional do Livro Did?tico. Nesta abordagem as fra??es e os n?meros decimais, especialmente os decimais infinitos e peri?dicos, e por consequ?ncia o c?lculo de sua fra??o geratriz, foram objetos de estudo centrais e instigadores dessa pesquisa. Realizou-se um estudo mais detalhado sobre a representa??o decimal dos n?meros racionais e analisando a compreens?o destes n?meros em n?vel fundamental e m?dio. Foi ainda proposto uma abordagem das maneiras mais usuais do c?lculo da fra??o geratriz, bem como, explorado a rela??o entre os decimais infinitos e peri?dicos e as progress?es geom?tricas. Durante o desenvolvimento deste trabalho, foi poss?vel perceber que h? mais de uma abordagem did?tica dos t?picos de ensino inerentes ao tema central analisado. O reconhecimento de que a parte decimal das d?zimas peri?dicas pode ser expressa como uma soma infinita de parcelas que, a partir de certo ponto, descreve uma progress?o geom?trica infinita de raz?o compreendida entre zero e um, ? um ponto chave na proposta de interven??o apresentada para a sala de aula. Diante desse quadro, foi verificado a ordem atualmente seguida pelos professores do 1? Ano do Ensino M?dio, o que permitiu constatar que os conte?dos D?zimas Peri?dicas e Progress?es Geom?tricas Infinitas s?o tratados sem liga??o significativa e, diante disso, foi proposta uma altera??o na ordem de abordagem desses conte?dos no Ensino M?dio. Ao final foram propostas algumas sugest?es de atividades resolvidas e outras para serem desenvolvidas em sala de aula.<br>Disserta??o (Mestrado Profissional) ? Programa de P?s-Gradua??o Matem?tica, Universidade Federal dos Vales do Jequitinhonha e Mucuri, 2017.<br>The aim of this work was to present a contribution to the teaching of rational numbers, emphasizing mainly the relation between periodic tithe and geometric progression. The methodology used allowed the analysis of the approach and didactic sequence of the topics Periodic Dizima and Infinite Geometric Progression, contemplated in textbooks approved by the National Textbook Program. In this approach fractions and decimal numbers, especially the infinite and periodic decimals, and consequently the calculation of their generative fraction, were central objects and instigators of this research. A more detailed study on the decimal representation of rational numbers was carried out and the understanding of these numbers at the fundamental and medium levels was analyzed. It was also proposed an approach of the most usual ways of calculating the generative fraction, as well as exploring the relationship between infinite and periodic decimals and geometric progressions. During the development of this work, it was possible to perceive that there is more of a didactic approach of the teaching topics inherent to the central theme analyzed. The recognition that the decimal part of the periodic tithe can be expressed as an infinite sum of plots which, from a certain point, describes an infinite geometric progression of ratio between zero and one, is a key point in the proposal of intervention presented for the classroom. In view of this situation, we verified the order currently being followed by teachers of the 1? Year of High School, which allowed to verify that the Periodic Dictionaries and Infinite Geometric Progressions are treated without significant connection and, accordingly, a change was proposed in order to approach these contents in High School. At the end, some suggestions for solved activities and others to be developed in the classroom were proposed.