Academic literature on the topic 'Mathematical division problems'
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Journal articles on the topic "Mathematical division problems"
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Greenwood, Jonathan Jay. "On The Nature of Teaching and Assessing “Mathematical Power” and “Mathematical Thinking”." Arithmetic Teacher 41, no. 3 (1993): 144–52. http://dx.doi.org/10.5951/at.41.3.0144.
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What exactly is “mathematical power” to omeone who has always identified mathematics as being the mastery of facts, such as the multiplication table, and procedures, such as the long division algorithm? What does it mean to “think mathematically” to a teacher who always struggled wit11 story problems as a student? To those teachers who fit these descriptions, and a sizable number do. assessing students mathematical power and mathematical thinking is even more bewildering.
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Setiadi, Nanang. "The Use of Realistic Mathematics Education (RME) to Help Indonesian 5th-Grade Students to Learn Multiplication and Division." Southeast Asian Mathematics Education Journal 10, no. 1 (2020): 41–53. http://dx.doi.org/10.46517/seamej.v10i1.98.
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Abstract This paper discusses the use of Realistic Mathematics Education (RME) as an alternative approach to enhance Indonesian 5th-grade students’ ability in multiplication and division. It presents the analysis of Indonesian 5th-grade students’ difficulties in applying stacking method for multiplication and division. Furthermore, it describes a mathematics teaching learning practice to stimulate students constructing their strategies, mathematical models and number sense in solving mathematical problems that involve multiplication and division. The teaching learning practice aims to apply RME for helping students develop their multiplication and division ability.Findings shows that stacking methods for multiplication and division are difficult for the students. The main students’ problem in multiplication and division stacking methods is in reapplying the steps of the methods. The steps taken to improve the learning process by implementing RME are: (1) analyze in detail the difficulties of students in multiplication and division stacking methods, (2) provide contexts of mathematical problems that can stimulate students to think mathematically, (3) hold a class mathematics congress, and (4) conduct a test to measure students’ achievement. Based on the students’ achievement, there has been several improvements. After RME, there were more students whose grades passed the Minimum Mastery Criteria. Moreover, there was a student who got 100. Then, the average test was higher. Meanwhile, there were only 3 children whose grades were 0. Thus, the application of RME has helped the 5th-grade students to improve their ability in multiplication and division.
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Moyer, Patricia Seray. "Links to Literature: A Remainder of One: Exploring Partitive Division." Teaching Children Mathematics 6, no. 8 (2000): 517–21. http://dx.doi.org/10.5951/tcm.6.8.0517.
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Children's literature can be a springboard for conversations about mathematical concepts. Austin (1998) suggests that good children's literature with a mathematical theme provides a context for both exploring and extending mathematics problems embedded in stories. In the context of discussing a story, children connect their everyday experiences with mathematics and have opportunities to make conjectures about quantities, equalities, or other mathematical ideas; negotiate their understanding of mathematical concepts; and verbalize their thinking. Children's books that prompt mathematical conversations also lead to rich, dynamic communication in the mathematics classroom and develop the use of mathematical symbols in the context of communicating. The National Council of Teachers of Mathematics (1989) emphasizes the importance of communication in helping children both construct mathematical knowledge and link their informal notions with the abstract symbols used to express mathematical ideas.
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Sidney, Pooja Gupta, and Martha Wagner Alibali. "Creating a context for learning: Activating children’s whole number knowledge prepares them to understand fraction division." Journal of Numerical Cognition 3, no. 1 (2017): 31–57. http://dx.doi.org/10.5964/jnc.v3i1.71.
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When children learn about fractions, their prior knowledge of whole numbers often interferes, resulting in a whole number bias. However, many fraction concepts are generalizations of analogous whole number concepts; for example, fraction division and whole number division share a similar conceptual structure. Drawing on past studies of analogical transfer, we hypothesize that children’s whole number division knowledge will support their understanding of fraction division when their relevant prior knowledge is activated immediately before engaging with fraction division. Children in 5th and 6th grade modeled fraction division with physical objects after modeling a series of addition, subtraction, multiplication, and division problems with whole number operands and fraction operands. In one condition, problems were blocked by operation, such that children modeled fraction problems immediately after analogous whole number problems (e.g., fraction division problems followed whole number division problems). In another condition, problems were blocked by number type, such that children modeled all four arithmetic operations with whole numbers in the first block, and then operations with fractions in the second block. Children who solved whole number division problems immediately before fraction division problems were significantly better at modeling the conceptual structure of fraction division than those who solved all of the fraction problems together. Thus, implicit analogies across shared concepts can affect children’s mathematical thinking. Moreover, specific analogies between whole number and fraction concepts can yield a positive, rather than a negative, whole number bias.
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Крюков, V. Kryukov, Зольников, Konstantin Zolnikov, Евдокимова, and S. Evdokimova. "Problems of modeling the basic elements CMOS LSI dual-purpose CAD." Modeling of systems and processes 6, no. 4 (2014): 41–44. http://dx.doi.org/10.12737/4045.
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The paper considers problems of modeling the basic elements of micro-schemes, creation of mathematical, informational and software taking into account the division of responsibilities between enterprises and modeling capabilities of ionizate-organizational and structural effects.
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Simon, Martin A. "Prospective Elementary Teachers' Knowledge of Division." Journal for Research in Mathematics Education 24, no. 3 (1993): 233–53. http://dx.doi.org/10.5951/jresematheduc.24.3.0233.
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Prospective teachers' knowledge of division was investigated through an open-response written instrument and through individual interviews. Problems were designed to focus on two aspects of understanding division: connectedness within and between procedural and conceptual knowledge and knowledge of units. Results indicated that the prospective teachers' conceptual knowledge was weak in a number of areas including the conceptual underpinnings of familiar algorithms, the relationship between partitive and quotitive division, the relationship between symbolic division and real-world problems, and identification of the units of quantities encountered in division computations. The research also characterized aspects of individual conceptual differences. The research results suggest conceptual areas of emphasis for the mathematical preparation of elementary teachers.
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Flores, Alfinio, Erin E. Turner, and Renee C. Bachman. "Research, Reflection, and Practice: Posing Problems to Develop Conceptual Understanding: Two Teachers Make Sense of Division of Fractions." Teaching Children Mathematics 12, no. 3 (2005): 117–21. http://dx.doi.org/10.5951/tcm.12.3.0117.
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Several scholars (e.g., Brown and Walter 1990; English 1997; Silver 1994) have highlighted the benefits of students posing mathematical problems (for example, students become better problem solvers). Posing mathematical problems can also help teachers develop their own mathematical knowledge and understanding. Teachers who learned mathematics mostly as “rules without reasons” now must learn how to teach for conceptual understanding. This article describes how two teachers, Elizabeth and Carolyn, posed problems to develop their own conceptual understanding of division of fractions in terms that would also be meaningful for their students. Each teacher taught a combined fourth-fifth grade in an urban school. The problems that the teachers posed and solved were collected during an initial session and from the draft of an article they wrote. Additional insights and information were obtained from interviews.
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Manger, Terje, and Ole-Johan Eikeland. "Gender Differences in Mathematical Sub-Skills." Research in Education 59, no. 1 (1998): 59–68. http://dx.doi.org/10.1177/003452379805900107.
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Gender differences in mathematical sub-skills Significant gender differences favouring boys were found among Norwegian sixth-grade students in total mathematical test score and in the sub-scores of numeracy, measurement, fractions, geometry and word problems. No significant differences were found in addition and subtraction or in multiplication and division. Items requiring an understanding of decimal numbers discriminated in favour of the boys. The study revealed the dominance of boys in the upper ranges of ability in mathematics, showing that small average gender differences can hide large differences in a highly able group of students.
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Fuson, Karen C. "Toward Computational Fluency in Multidigit Multiplication and Division." Teaching Children Mathematics 9, no. 6 (2003): 300–305. http://dx.doi.org/10.5951/tcm.9.6.0300.
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Traditionally in the United States and Canada, students have first learned how to compute with whole numbers and then have applied that kind of computation. This approach presents several problems. First, less-advanced students sometimes never reach the application phase, so their learning is greatly limited. Second, word problems usually appear at the end of each section or chapter on computation, so sensible students do not read the problems carefully: They simply perform the operation that they have just practiced on the numbers in the problem. This practice, plus the emphasis on teaching students to focus on key words in problems rather than to build a complete mental model of the problem situation, leads to poor problem solving because students never learn to read and model the problems themselves. Third, seeing problem situations only after learning the mathematical operations keeps students from linking those operations with aspects of the problem situations. This isolation limits the meaningfulness of the operations and the ability of children to use the operations in a variety of situations.
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Oshanova, N. T. "MATHEMATICAL FOUNDATIONS OF MUSIC BY AL-FARABI." BULLETIN Series of Physics & Mathematical Sciences 71, no. 3 (2020): 24–30. http://dx.doi.org/10.51889/2020-3.1728-7901.03.
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This article discusses the mathematical foundations of al - Farabi music, and suggests the main actions of arithmetic operations with Farabi relations to obtain musical intervals. Farabi drew attention to some problems for the study of musical art. This article provides an extensive understanding of how the three main problems are needed to get the ratio. Musical intervals have different values. You can divide them, multiply them, and listen to them. In music theory, you need to be familiar with the mathematical foundations for working with ratios, as well as arithmetic operations like multiplication, division, and addition. Farabi not only gives a scientific idea of the ratio of sounds, but also reveals the mathematical foundations of the emergence of harmony and musical melodies.
Dissertations / Theses on the topic "Mathematical division problems"
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Peairs, Matthew (Matthew S. ). "Machine analysis of students' mathematical representations for multiplication and division problems." Thesis, Massachusetts Institute of Technology, 2013. http://hdl.handle.net/1721.1/85467.
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Thesis: M. Eng., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2013.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (page 35).<br>This project extends Classroom Learning Partner classroom interaction software to include a semantic interpretation component. This semantic interpretation, combined with existing syntactic interpretation, enables the software to tag and group student work using knowledge of the math used in both creating and solving problems. The analysis is being prototyped using student work in grades 4 and 5, with focus on multiplication and division. First, during the authoring step, the notebook author gives each page a "page definition" that encapsulates the mathematical problem presented on that page. For a multiplication or division problem, this involves setting the three numbers connected by the product relation (e.g., 6 * 3 = 18), marking which of those numbers are given by the problem or otherwise unknown, and selecting an overall context for the problem, such as equal groups or area. Then, once students have submitted their work, the analysis component takes the raw output of the syntactic interpretation step and relates it back to the mathematical content of the page to assign each student's work a set of automatically generated tags. These tags address the correctness of a student's methods and results, as well as highlighting different problem-solving strategies that students might have used to arrive at the same answer. Finally, the teacher can sort student submissions by these various tags to quickly find noteworthy or contrasting examples to present to the class.<br>by Matthew Peairs.<br>M. Eng.
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BARCELLOS, JESSICA SILVA. "THE WORDS MAKE THIS ONE DIFFICULT: A PSYCOLINGUISTICAL INVESTIGATION ABOUT THE ROLE OF LANGUAGE IN MATHEMATICAL DIVISION PROBLEMS." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2017. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=30221@1.
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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO<br>COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR<br>PROGRAMA DE SUPORTE À PÓS-GRADUAÇÃO DE INSTS. DE ENSINO<br>Esta dissertação investiga a interface linguagem-matemática, com foco em tarefas de resolução de problemas de divisão partitiva e por quotas. Investigamos se dificuldades nesse tipo de tarefa podem estar relacionadas à complexidade linguística dos enunciados. Discute-se em que medida o padrão composicional e as estruturas linguísticas utilizadas nos enunciados podem afetar o desempenho dos alunos nesses dois tipos de problemas. Para realizar essa investigação, foram conduzidos três experimentos com alunos do segundo ano do Ensino Fundamental de uma escola da rede pública federal de ensino no Rio de Janeiro. No primeiro experimento, foram utilizados como itens experimentais os enunciados dos livros didáticos e os resultados indicam diferença significativa entre as condições, com maior número de acertos em divisão partitiva. No segundo experimento, novos enunciados foram criados, controlando-se tanto a estrutura informacional quanto a complexidade gramatical nos dois tipos de problemas. Os resultados mostram desempenho similar nas duas condições. No experimento 3, investigamos o tipo de interpretação preferida para enunciados ambíguos com sujeito composto. Verificou-se clara preferência por leituras coletivas e constatou-se que, quando estruturas ambíguas são utilizadas, o desempenho dos alunos volta a diferir entre as condições, com pior desempenho na divisão por quotas. Esta pesquisa indica que a dificuldade dos alunos em enunciados de divisão pode ser reduzida com o controle da complexidade gramatical, o que mostra o papel fundamental da observação de variáveis linguísticas na aferição de conhecimento matemático e na elaboração de materiais didáticos.<br>This work investigates the interface of language-mathematics, focusing on partitive and quotative division problem solving tasks. We investigate whether the difficulties students face when solving mathematical verbal problems can be related to linguistic complexity of the commands. We also discuss how the composition and linguistic structures that are used in the verbal problems can affect student s performance. We conducted three experiments with students of the second year of a primary school in Rio de Janeiro. In the first experiment, we used problems extracted from textbooks as experimental items; the results indicate a significant difference between the partitive and quotative conditions, resulting in a bigger number of correct answers regarding partitive division. In the second experiment, we created new commands, controlling their informational structure as well as their grammatical complexity. The results show a similar performance in both conditions. As for experiment 3, our aim was to investigate the type of interpretation students would prefer in ambiguous propositions, in which the subject of the sentence is a compound subject (coordinated structure). A preference for collective readings was observed. Also, when ambiguous structures are present, the performance of the students tends to vary depending on the conditions, declining on quotative division. Thus, the results of this research indicates that the difficulties students usually face in mathematical verbal problems can be reduced when the grammatical complexity is controlled - pointing towards the central role of linguistic variables in mathematical knowledge and in the elaboration of school materials.
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Piva, Rosalina. "Estratégias mobilizadas na resolução de problemas matemáticos de divisão por alunos da sala de articulação da 2ª fase do 2º ciclo do ensino fundamental de uma escola estadual de Várzea Grande-MT." Universidade Federal de Mato Grosso, 2014. http://ri.ufmt.br/handle/1/316.
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Previous issue date: 2014-04-22<br>CAPES<br>O presente trabalho trata de uma pesquisa de mestrado que teve como objetivo
investigar que estratégias alunos da 2ª fase do 2º ciclo (5º ano do Ensino Fundamental
de nove anos) que frequentam a sala de articulação de uma Escola Estadual no
município de Várzea Grande, em Mato Grosso, mobilizam na Resolução de Problemas
de Divisão. A estratégia é definida nesta dissertação, segundo Palhares (2004), como
um conjunto de técnicas a serem dominadas pelos solucionadores ajudam o aluno a
resolver o problema ou progredir no sentido de encontrar a sua solução. A pesquisa foi
delineada pela seguinte problemática: Que estratégias os alunos da 2ª fase do 2º ciclo
(5º ano do Ensino Fundamental de nove anos) que frequentam a sala de
articulação de uma Escola Estadual de Várzea Grande, em Mato Grosso,
mobilizam na Resolução de Problemas matemáticos de divisão? Levando em
consideração a natureza dessa pesquisa, a metodologia de investigação adotada foi uma
abordagem qualitativa e se configura em um estudo de caso, em que nos baseamos nos
autores Fiorentini; Lorenzato (2012); Yin (2010); Bogdan; Biklen (1994) e Merriam
(1998). A pesquisa foi desenvolvida em três momentos: no primeiro momento fizemos
um levantamento das pesquisas brasileiras que discutem a Resolução de Problemas de
divisão, com o objetivo de conhecer o que os autores dizem sobre essa temática. Sendo
assim, a base teórica sobre resolução de problemas é embasada por Onuchic (1999,
2011), Onuchic; Allevato (2009), e Sánchez Huete; Fernández Bravo (2006), entre
outros. Em relação à resolução de problemas, como metodologia de ensino, buscou-se
suporte em Onuchic (2011); Brasil (1997); entre outros. Quanto à definição de
estratégia na resolução de problemas, utilizamos as definições de Sánchez Huete;
Fernández Bravo (2006) e Palhares (2004). Mediante o referencial teórico citado num
segundo momento, aplicamos um estudo piloto com o objetivo de avaliar a metodologia
de coleta de dados. O terceiro momento de nossa investigação estabeleceu-se com
alunos da 2ª fase do 2º ciclo (5º ano do Ensino Fundamental) que frequentam a sala de
articulação em uma escola estadual no município de Várzea Grande-MT. As estratégias
mobilizadas pelos alunos foram identificadas por: desenho; algoritmo da divisão com
chave longa; algoritmo da divisão com chave breve; algoritmo da multiplicação;
algoritmo adição e algoritmo subtração. Como resultado, temos que os alunos da sala de
articulação têm muitas dificuldades com as operações, especialmente com a de divisão,
não conseguem identificar nos problemas as operações matemáticas, o que justifica pelo
fato de alguns alunos estarem em fase de alfabetização e, por isso, fazem tentativas,
buscando acertar qual operação deverá ser utilizada na resolução.<br>This paper deals with a Master thesis aimed to investigate which strategies students of
Stage 2nd level of the 2nd cycle (5th year of elementary school for students of nine
years) attending the resource room of a state school in the count of Várzea Grande, in
Mato Grosso, how they manage the Solve Problem Division. The strategy is defined in
this dissertation, according Palhares (2004), as a set of techniques to be mastered by
solvers helps the student to solve the problem or progress towards finding a solution.
This research was designed with the following problem: What strategies students from
Stage 2nd of the 2nd cycle (5th year of elementary school for students of nine years)
attending the resource room of a State School of Várzea Grande, Mato Grosso, how
they manage the Mathematical Problem Divison? Considering the nature of this
research, the research methodology adopted was a qualitative approach and is
configured in a case study, in which we relied on the authors Fiorentini; Lorenzato
(2012); Yin (2010); Bogdan; Biklen (1994) and Merriam (1998). The research was
developed in three stages: at first we did a survey of Brazilian research discussing the
Solve Problem Division, with the aim of knowing what the authors say about this
subject. Thus, the theoretical basis for problem solving is grounded by Onuchic (1999,
2011), Onuchic; Allevato (2009) and Sánchez Huete; Fernandez Bravo (2006), among
others. Regarding the definition of strategy in solving problems, we use the definitions
of Huete Sánchez; Fernandez Bravo (2006) and Palhares (2004). Through the
theoretical framework mentioned subsequently, applied a pilot study aiming to assess
the methodology of data collection. The third stage of our investigation it was
established with students from Stage 2 of the 2nd cycle ( 5th year of elementary school )
attending the living joint in a state school in the count of Várzea Grande- MT. The
strategies deployed by students were identified by: design; long division algorithm with
key; division algorithm with short key; algorithm of multiplication; addition and
subtraction algorithm. As a result, the students of the resource room have many
difficulties with operations, especially with the division, They fail to identify problems
in mathematical operations, which explains the fact that some students are beginning
literacy and therefore make attempts, trying to hit what operation should be used in the
resolution.
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Farlow, Brian. "Square Peg Thinking, Round Hole Problems: An Investigation of Student Thinking About and Mathematical Preparation for Vector Concepts in Cartesian and Non-Cartesian Coordinates Used in Upper-Division Physics." Diss., North Dakota State University, 2019. https://hdl.handle.net/10365/31479.
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Pinheiro, Fernanda Machado. "Explorando o jogo "Avançando com o resto" como recurso didático para o ensino e aprendizagem de alguns conteúdos matemáticos, na perspectiva da resolução de problemas /." Universidade Estadual Paulista (UNESP), 2017. http://hdl.handle.net/11449/152188.
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Previous issue date: 2017-11-01<br>Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)<br>Explorar o jogo “Avançando com o resto” sob a perspectiva da resolução de problemas constitui-se num recurso didático que favorece consolidar e ampliar o conhecimento dos alunos sobre diversos conteúdos matemáticos, em particular, o algoritmo da divisão euclidiana. Escrevemos a fundamentação teórica deste trabalho a partir de problemas que representam situações vivenciadas no jogo. Conceituamos e demonstramos os principais teoremas e resultados relacionados à divisão euclidiana, como Múltiplos e Divisores de números naturais, Paridade, Números primos e compostos, Congruência (aritmética modular), Sistema de Numeração Decimal e Critérios de divisibilidade por 2, 3, 4, 5 e 6. Este estudo nos permitiu apresentar uma sugestão de atividade direcionada às turmas do 5º ano do Ensino Fundamental, explorando os diferentes significados da divisão e o estudo reflexivo do algoritmo, planejada de acordo com as orientações curriculares presentes nos Parâmetros Curriculares Nacionais e no Currículo de Matemática do Estado de São Paulo. Considerando as dificuldades encontradas no ensino desta operação matemática e o potencial lúdico do jogo para aprendizagem, apresentamos uma possibilidade de abordagem significativa do algoritmo da divisão euclidiana, envolvendo e motivando os alunos para a aprendizagem de conceitos, competências e habilidades inerentes ao Currículo de Matemática.<br>Exploring the game "Advancing with the rest" from the perspective of problem solving is a didactic resource that favours consolidating and expanding students' knowledge about the various mathematical contents, in particular algorithm of the Euclidean division. We write the theoretical basis of this study from problems that represent situations experienced in the game. We conceptualize and demonstrate the main theorems and results related to the Euclidean division, such as Multiples and Divisors of Natural Numbers, Parity, Prime and Compound Numbers, Congruence (Modular Arithmetic), Decimal Numbering System and Divisibility criteria by 2, 3, 4, 5 e 6. This study allowed us to present a suggestion of activity directed to the 5th grade classes of Elementary School, exploring the different meanings of the division and the reflexive study of the algorithm, planned according to the curricular guidelines present in National Curricular Parameters and the Mathematics São Paulo State Curriculum. Considering the difficulties encountered in teaching this mathematical operation and the playful potential of the game for learning, we present a possibility of a significant approach to the algorithm of the Euclidean division, involving and motivating students to learn concepts, skills and abilities inherent to the Mathematics Curriculum.
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Nermyr, Linda, Elisabeth Göransson, and Jennie Andersson. "Matematisk problemlösning med förskolebarn : Lösningsstrategier på matematiska problem innehållande division/delning." Thesis, Växjö University, School of Mathematics and Systems Engineering, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:vxu:diva-1685.
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<p>Vårt syfte med det här arbetet var att undersöka hur förskolebarn agerar när de ställs inför två matematiska problem innehållande division/delning. Vilka lösningsstrategier använder de för att lösa givna problem? Vi valde att genomföra en fallstudie på den förskola där vi alla tre arbetar, och metoden vi använde oss av var deltagande observation, samt videofilmning. I studien ingick nio stycken femåringar. När vi transkriberade videoinspelningen såg vi att barnen använde sig av de olika lösningsstrategierna som vi tog upp i den teoretiska bakgrunden, i olika utsträckning. De vanligaste var pekräkning/förflyttningsräkning och fördelningsstrategin. Av resultatet framgick att de flesta barnen kombinerade flera lösningsstrategier vid samma problem. Det framkom också att barnen hade svårt att sätta ord på sina lösningsstrategier i genomförandet, och förklara hur de tänkte när de löste problemen.</p>
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Hill, Sally. "Problems related to number theory : sum-and-distance systems, reversible square matrices and divisor functions." Thesis, Cardiff University, 2018. http://orca.cf.ac.uk/111467/.
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We say that two sets $A$ and $B$, each of cardinality $m$, form an $m+m$ \emph{sum-and-distance system} $\{A,B\}$ if the sum-and-distance set $A^*B$ comprised of all the absolute values of the sums and distances $a_i\pm b_j$ contains either the consecutive odd integers $\{1,3,5,\ldots 4m^2-1\}$ or with the inclusion of the set elements themselves, the consecutive integers $\{1,2,3,\ldots,2m(m+1)\}$ (an inclusive sum-and-distance system). Sum-and-distance systems can be thought of as a discrete analogue of the union of a Minkowski sum system with a Minkowski difference system. We show that they occur naturally within a traditional reversible square matrix, where conjugation with a specific orthogonal symmetric involution, always reveals a sum-and-distance system within the block structure of the conjugated matrix. Moreover, we show that the block representation is an algebra isomorphism. Building upon results of Ollerenshaw, and Br\'ee, for a fixed dimension $n$, we establish a bijection between the set of sum-and-distance systems and the set of traditional principal reversible square matrices of size $n\times n$. Using the $j$th non-trivial divisor function $c_j (n)$, which counts the total number of proper ordered factorisations of the integer $n= p_1^\ldots p_t^$ into $j$ parts, we prove that the total number of $n+n$ principal reversible square matrices, and so sum-and-distance systems, $N_n$, is given by \[ N_n = \sum_^ \left( c_j(n)^2 +c_(n)c_j(n) \right)=\sum_^ c_j^(n) c_j^(n). \] \[=\sum_^ \left(\sum^j_(-1)^ \prod_^t \right ) \left ( \sum^j_(-1)^ \prod_^t \right), \] where $\Omega(n)=a_1 + a_2 + \ldots + a_t$ is the total number of prime factors (including repeats) of $n$. Further relations between the divisor functions and their Dirichlet series are deduced, as well as a construction algorithm for all sum-and-distance systems of either type. Superalgebra structures relating to the matrix symmetry properties are identified, including those for the reversible and most-perfect square matrices of those considered by Ollerenshaw and Br\'ee. For certain symmetry types, links between the block representation constructed from a sum-and-distance system, and quadratic forms are also established.
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Darpö, Erik. "Problems in the Classification Theory of Non-Associative Simple Algebras." Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-9536.
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In spite of its 150 years history, the problem of classifying all finite-dimensional division algebras over a field k is still unsolved whenever k is not algebraically closed. The present thesis concerns some different aspects of this problem, and the related problems of classifying all composition and absolute valued algebras. A tripartition of the class of all fields is given, based on the dimensions in which division algebras over a field exist. Moreover, all finite-dimensional flexible real division algebras are classified. This class includes in particular all finite-dimensional commutative real division algebras, of which two different classifications, along different lines, are presented. It is shown that every vector product algebra has dimension zero, one, three or seven, and that its isomorphism type is determined by its adherent quadratic form. This yields a new and elementary proof for the corresponding, classical result for unital composition algebras. A rotation in a Euclidean space is an orthogonal map that locally acts as a plane rotation with a fixed angle. All pairs of rotations in finite-dimensional Euclidean spaces are classified up to orthogonal similarity. A description of all composition algebras having an LR-bijective idempotent is given. On the basis of this description, all absolute valued algebras having a one-sided unity or a non-zero central idempotent are classified.
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Bongiovanni, Alex. "Problems with power-free numbers and Piatetski-Shapiro sequences." Kent State University / OhioLINK, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=kent1618331559201676.
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Torshage, Axel. "Non-selfadjoint operator functions." Doctoral thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-143085.
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Spectral properties of linear operators and operator functions can be used to analyze models in nature. When dispersion and damping are taken into account, the dependence of the spectral parameter is in general non-linear and the operators are not selfadjoint. In this thesis non-selfadjoint operator functions are studied and several methods for obtaining properties of unbounded non-selfadjoint operator functions are presented. Equivalence is used to characterize operator functions since two equivalent operators share many significant characteristics such as the spectrum and closeness. Methods of linearization and other types of equivalences are presented for a class of unbounded operator matrix functions. To study properties of the spectrum for non-selfadjoint operator functions, the numerical range is a powerful tool. The thesis introduces an optimal enclosure of the numerical range of a class of unbounded operator functions. The new enclosure can be computed explicitly, and it is investigated in detail. Many properties of the numerical range such as the number of components can be deduced from the enclosure. Furthermore, it is utilized to prove the existence of an infinite number of eigenvalues accumulating to specific points in the complex plane. Among the results are proofs of accumulation of eigenvalues to the singularities of a class of unbounded rational operator functions. The enclosure of the numerical range is also used to find optimal and computable estimates of the norm of resolvent and a corresponding enclosure of the ε-pseudospectrum.
Books on the topic "Mathematical division problems"
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Big truck and car word problems starring multiplication and division. Enslow Publishers, 2009.
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Running for class president: Represent and solve problems involving division. PowerKids Press, 2015.
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At the food fair: Represent and solve problems involving division. Rosen Classroom, 2015.
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Food from around the world: Represent and solve problems involving division. PowerKids Press, 2015.
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Super sums: Addition, subtraction, multiplication and division. Children's Press, 2018.
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I, Plotnikov Pavel, ed. Small divisor problem in the theory of three-dimensional water gravity waves. American Mathematical Society, 2009.
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50 Fillin Math Word Problems Multiplication Division. Scholastic Teaching Resources, 2009.
Find full textBook chapters on the topic "Mathematical division problems"
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Fukuda, Mari, Emmanuel Manalo, and Hiroaki Ayabe. "The Presence of Diagrams and Problems Requiring Diagram Construction: Comparing Mathematical Word Problems in Japanese and Canadian Textbooks." In Diagrammatic Representation and Inference. Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-86062-2_36.
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AbstractIt is generally considered beneficial for learners to construct and use appropriate diagrams when solving mathematical word problems. However, previous research has indicated that learners tend not to use diagrams spontaneously. In the present study, we analyzed textbooks in Japan and Canada, focusing on the possibility that such inadequacy in diagram use may be affected by the presence (or absence) of diagrams in textbooks, the kinds of diagrams that are included, and whether problems requiring the construction of diagrams are provided in those textbooks. One set each of Japanese and Canadian elementary school textbooks were analyzed, focusing on the chapters dealing with division. Results revealed that the Japanese textbooks contain worked examples and exercise problems accompanied by diagrams more than the Canadian textbooks. Furthermore, the Japanese textbooks often use line diagrams and tables that abstractly represent quantitative relationships and they include more problems that require students to use diagrams. However, to encourage students to use diagrams spontaneously, it may be necessary to include problems that scaffold the use of diagrams in a step-by-step manner in both the Canadian and the Japanese textbooks.
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Barbeau, E. J. "Evaluation, Division, and Expansion." In Problem Books in Mathematics. Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-4524-7_2.
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Peixoto, Jurema Lindote Botelho. "The Meaning of Division for Deaf Students in the Context of Problem-Solving Situations." In Inclusive Mathematics Education. Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-11518-0_18.
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"Solutions of the extension and division problems." In Translations of Mathematical Monographs. American Mathematical Society, 2002. http://dx.doi.org/10.1090/mmono/211/05.
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Fried, Katalin, and Judit Török. "Überbringen Störche Babys? Dinge, die wir Kindern nicht erzählen (können)." In Komplexer Mathematikunterricht. Die Ideen von Tamás Varga in aktueller Sicht. WTM-Verlag Münster, 2020. http://dx.doi.org/10.37626/ga9783959871648.0.12.
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One of the main goals of the Complex Mathematics Education Experiment set by Tamás Varga was the following: “That is, the knowledge we provide fits the closest developmental zone and developmental level of the children; and yet is mathematically correct and forward-thinking. We do not tell stork tales.” (Varga, 1974, p. 1984.) We give some examples from the topic of number theory, where we cannot avoid telling “stork tales”, no matter how hard we try. In section 1.3 we describe some of the sources of disturbance. In section 2 we deal with the conflict of the different interpretations of some concepts occurring in primary/secondary school and university education, such as: “divisibility”, “divisor”, “common divisor”, “greatest common divisor”, “division with remainder”, the perceived or real special properties of zero, and “prime number”. We believe that it is important to make prospective teachers aware what facts they hide and why when teaching. In section 3 we present problems that can be discussed with children of different ages and different abstraction levels. Classification: D70, E40, F60, U60. Keywords: misconceptions and student errors, concept formation, treatment of mathematical concepts and definitions in mathematics education, number theory, educational games.
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D'Agostino, Susan. "Reach for the stars, just like Katherine Johnson." In How to Free Your Inner Mathematician. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780198843597.003.0006.
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“Reach for the stars, just like Katherine Johnson” tells the story of the mathematics and mathematician behind NASA’s 1961 Apollo spacecraft flown by astronaut John Glenn. When Glen grew concerned that NASA had switched to an inanimate computer for checking computations regarding re-entry into the atmosphere, he insisted that human computer and mathematician Katherine Johnson check the numbers. Johnson needed to consider drag, aerodynamic lift, vacuum perigee altitude, the spacecraft’s center of gravity, and more to ensure a safe reentry corridor. The discussion is illustrated with numerous hand-drawn sketches. Katherine Johnson, whose biography is summarized, was an African American woman who worked in NASA’s Research Flight Division at the Langley Research Center in Hampton, Virginia. Mathematics students and enthusiasts are encouraged to reach for literal and metaphorical stars in mathematical and life pursuits. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.
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"The Additive Divisor Problem." In Translations of Mathematical Monographs. American Mathematical Society, 2005. http://dx.doi.org/10.1090/mmono/004/04.
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Dall’Aglio, Marco. "Optimization Problems in Fair Division Theory." In Handbook of Analytic Computational Methods in Applied Mathematics. Chapman and Hall/CRC, 2000. http://dx.doi.org/10.1201/9781420036053.ch20.
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Dall’Aglio, Marco. "Optimization Problems in Fair Division Theory." In Handbook of Analytic-Computational Methods in Applied Mathematics. Chapman and Hall/CRC, 2019. http://dx.doi.org/10.1201/9780429123610-20.
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Selikowitz, Mark. "Arithmetic." In Dyslexia and Other Learning Difficulties. Oxford University Press, 1993. http://dx.doi.org/10.1093/oso/9780192622990.003.0014.
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Specific difficulties have been described in a number of areas of mathematics, but difficulty in arithmetic has received the most attention. This may be because all children are required to do arithmetical calculations in the early years of school, but can choose alternative subjects later, and it probably also reflects the fact that arithmetical calculations play an important part in everyday life. Another reason may be that arithmetical difficulty following brain damage in adulthood (dyscalculia) is a well-recognized and well-studied entity. This chapter will focus on specific arithmetic difficulty in children, that is, unexplained, significant delay in arithmetic ability. Although specific arithmetic difficulty was once considered rare, there is now evidence that it is not as uncommon as was previously thought. The psychologist may obtain sufficient information about the child’s arithmetical ability from the Arithmetic section (sub-test) of the Wechsler Intelligence Scale for Children (WISC-IV). This is a commonly used intelligence test that can be used for children from 6 years to 16 years 11 months. This test does not require the child to write down the answers. The problems are timed and they relate to various arithmetical skills. Addition, subtraction, multiplication, and division can all be tested. Some problems also require memorized number facts and subtle operations, such as seeing relevant relationships at a glance. The emphasis of the test is not on mathematical knowledge as such, but on mental computations and concentration. The WISC-IV will also give the psychologist information about other abilities, which may shed light on the child’s difficulties. In the Digit Span sub-test, the child’s ability to remember numbers for a short period is tested. In the Comprehension sub-test, verbal reasoning is involved. If, for example, a child has high comprehension but low arithmetic scores, this may suggest that reasoning ability is adequate in social situations, but not in situations involving numbers. If the psychologist wants further information on arithmetic ability, there are a number of tests that specifically test mathematical skills and allow these to be compared with those of other children of the same age.
Conference papers on the topic "Mathematical division problems"
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Bakhtiyarov, S. I., G. M. Panakhov, E. M. Abbasov, A. N. Omrani, and A. S. Bakhtiyarov. "Polymer Adsorption Phenomena in Porous Media Filtration Problems." In ASME 2009 Fluids Engineering Division Summer Meeting. ASMEDC, 2009. http://dx.doi.org/10.1115/fedsm2009-78551.
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This paper represents the results of the mathematical modeling of the chemical flood process in oil reservoirs. The mathematical model of a problem resulted in solution of the differential equations in private derivatives with initial boundary conditions. In non-uniform net area the problem is replaced with a discrete problem by means of combination of obvious and implicit-difference schemes increasing accuracy order and is solved by double-sweep method. The effective decision algorithms of one-dimensional problem as adsorption and convectional diffusion equations are obtained.
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Mgebrishvili, Nikoloz, Guri Sharashenidze, Manana Moistsrapishvili, et al. "Mathematical Justification of the New Method of Determination of Wheel Pair’s and Rail’s Damage." In ASME 2009 Rail Transportation Division Fall Technical Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/rtdf2009-18007.
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Development of modern railways largely depends on the control of intactness of the wheel pairs and rails. It is impossible to ensure safety and increase the speed of railway traffic without providing such control. Detection of the wheel pairs’ and rails’ wear and damage is one of the key problems of railway science. Many scientific works are accomplished and lot of devices are elaborated in this field, but the mentioned problem still remains urgent. In order to increase the traffic safety, the group of authors has proposed a constructional scheme of the mobile device for detection of wheel pairs’ and rails’ wear and damage, which will have increased accuracy of measurement. Increasing of accuracy of measurement is carried out on the basis of development of mathematical model, which provides the realization of equal wear of wheel pairs’ as well as ones no equal wear. By installation of the proposed device on each wheel pair of each railcar of the rolling stock, the automatic control of wheel pairs’ and rails’ condition will be achieved. Namely: -Detection of the worn out wheel pair and determination of wear degree; -Detection of the damaged wheel pair; -Identification of the worn out or damaged wheel pair. Besides, on the basis of elaborated mathematical model: -Detection of the worn out rail; -Detection of the damaged rail; -Identification of location of the worn out or damaged rail. The obtained information will be constantly connected to the locomotive computer system. Therefore, for checking rolling stocks, there will no longer be need to build expensive stationary systems, to move trains great distances for their inspection and in the result, the time lost for stoppage will be saved. So, with the help of proposed mobile device, the traffic safety increases and at the same time expenses for detection of wheel pairs’ wear and damage decrease, which results in significant economic effect.
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Sharashenidze, Guri, Nikoloz Mgebrishvili, Tengiz Nadiradze, and Pavle Kurtanidze. "Improved System of a Braking Lever Transmission for Rail-Cars." In ASME 2008 Rail Transportation Division Fall Technical Conference. ASMEDC, 2008. http://dx.doi.org/10.1115/rtdf2008-74006.
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Operating lifetime of rail-cars greatly depends on fitness of braking systems and on effectiveness of their constructions. Various braking systems are being elaborated in this trend where lever transmissions are used. Designing of a braking lever transmission for rolling stocks is a very urgent problem. A transmission should be simple and should contain minimum number of levers and hinges. Increased reliability and operating lifetime of braking lever transmissions for rail-cars are achieved in the work. A new construction of rail-car braking lever transmission is elaborated where number of levers and braking joints are decreased significantly and it ensures increase of traffic safety and economic efficiency. A calculating dynamic model of a new construction is elaborated in the work, which obey the basic provisions of mathematical analysis. Dynamical loads in joints of the construction are calculated taking into account increased wear and stresses. Accomplished work provides with possibility for solution of urgent problems. Introduction of the mentioned construction will significantly increase reliability and operating indices of modern rail-cars.
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Schiffer, Wilfried, and Jean Jenzer. "3-D Shafting Calculations for Marine Installations: Static and Dynamic." In ASME 2003 Internal Combustion Engine Division Spring Technical Conference. ASMEDC, 2003. http://dx.doi.org/10.1115/ices2003-0590.
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Due to the particular geometry of the crankshaft of two-stroke diesel engines and the rotational degree of freedom torsional vibrations may be calculated with a 1-D mathematical model, but all other kinds of vibrations have to be calculated with a 3-D model. This concerns also static problems. In this paper the influence of coupled vibrations on ship vibration is explained and illustrated with an example for axial vibrations and an example for external vertical forces and moments. Another example for the use of the 3-D model is the calculation of crank deflection and jack load. It can be shown that such calculations are suitable for daily work.
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Kumar, V., C. Ramana, S. Afrin, J. Ortega, Neelam Agarwal, and Victor Udoewa. "Touchpad in Education: Dynamic Learning Framework Assessment and Content Development for the Undergraduate Fluid Mechanics." In ASME 2013 Fluids Engineering Division Summer Meeting. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/fedsm2013-16257.
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This paper presents a dynamic learning framework (DLF) based on dynamic course contents and assessment methods using latest web-based technologies with keeping in mind the recent advancement in touchpad computing devices (such as IPAD and Android based tablets). In the DLF framework, the effectiveness is assessed via evaluating the learning outcomes of increasing the learnability of high level concepts in the Bloom’s Taxonomy of cognitive learning. It proposes to address the challenges is creating a fluid mechanics module that incorporates all levels of the Bloom’s cognitive taxonomy. This is achieved via integration of mathematical, conceptual and visual contents. The lower level concepts (i.e., Remembering, Understanding, and Applying) are computerized and tested using Computer Adaptive Testing (CAT) algorithm. Our targeted audiences are from a predominantly Hispanic cultural setting and in undergraduate mechanical engineering courses. To capitalize on unique cultural setting and linguistic needs, the assessment is prepared in bi-lingual (Spanish and English) with localized problems. A pre-assessment of students’ learning styles was performed to assess their learning preference and the presentation was tuned to average audiences. It was observed that about 10% of the students used bi-lingual instructions in the exam which was conducted as an extra-credit option to paper based exam in order to assess the DLF framework. Students were also asked to contribute questions to generate a question database with localized problems.
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Stanko, Milan, Andrea Shmueli, Miguel Asuaje, et al. "CFD Simulation of the Submerged Cofferdams Effect on the Operation of the Future Tocoma Hydroelectric Power Plant." In ASME 2009 Fluids Engineering Division Summer Meeting. ASMEDC, 2009. http://dx.doi.org/10.1115/fedsm2009-78265.
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The Tocoma hydroelectric power plant, currently under construction, is located on the lower basin of the Caroni River in Bolivar State in Venezuela. This power plant will have 10 Kaplan turbines in its powerhouse that will generate approximately 2160 MW of hydroelectric power. During its construction, two cofferdams designated “A” and “B” will be built and afterwards will remain submerged. The main purpose of this experimental-numerical study is to analyze the possible future hydrodynamic effects of these structures on the operation of the Kaplan turbines. The presence of the submerged cofferdams could originate tridimensional hydrodynamic behaviors that could produce energy looses and operational and functional problems to the turbines. Two mathematical steady state single phase models using Computational Fluid Dynamics (CFD) Techniques and applying the commercial software ANSYS-CFX were developed. The first model represented the hydroelectric power plant reservoir that was quantitatively and qualitatively calibrated with a Froude Similarity 1:80 Scale Physical Model. Hydrodynamic flow patterns near to the intakes were found in the first model. Those patterns showed a non-uniform velocity profile in the unit’s intakes nearest to cofferdam “B”. The second mathematical model represented the study of the intake, the semi-spiral case and the Kaplan turbine. This model considers the non-uniform velocity profile that was found in the first model as an inlet boundary condition. Two methodologies were used to develop this model: one using two simulations with two overlapping physical domains, and the other one using the whole geometry. It was found that using overlapping domains in order to reduce the computational cost of the total simulation is a good way to obtain physical results with fair accuracy. The general results reported that the velocity profile at the intake of the powerhouse does not produce any stationary non uniform behavior on the velocity and pressure profiles in the unit compared to the uniform velocity profile case. This result could be an indicator that the non uniform condition at the intake of the Kaplan Turbines at Tocoma will not affect the normal operation conditions of the unit.
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Mgebrishvili, Nikoloz, Marina Tatanashvili, Tengiz Nadiradze, and Ketevan Kekelia. "Increase of Railway Transportation Safety by a New Method of Determination of Wheel Pair and Rail Wear and Damage." In ASME 2007 Rail Transportation Division Fall Technical Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/rtdf2007-46024.
Full textAbstract:
As a result of interaction between the wheel pairs and rails during rolling stock movement, lateral, longitudinal and torsional creep forces are created. Over the long term these forces cause wear and damage of the wheel pairs and rails. Accordingly railway safety decreases and the probability of derailment increases. Development of modern railways largely depends on controlling the proper functionality of wheel pairs and rails. It is impossible to ensure safety and increase the speed of railway traffic without providing such control. Detection of the wheel pair and rail wear damage is one of the key problems of railway science. Many scientific studies have been performed and a number of devices have been developed in this field, but the mentioned problem still remains urgent. In order to increase railway safety, the authors of this paper have proposed a conceptual design of a mobile device for detection of wheel pair and rail wear and damage. Such a device does not currently exist. By installation of the proposed device on each wheel pair of each railcar, the automatic control of wheel pair and rail condition will be achieved. Namely: • Detection of the worn out wheel pair and determination of the degree of wear; • Detection of the damaged wheel pair; • Identification of the worn out or damaged wheel pair. Also possible, on the basis of elaborated mathematical model: • Detection of the worn out rail; • Detection of the damaged rail; • Identification of location of the worn out or damaged rail. Information obtained from the device would be constantly connected to the locomotive computer system. Therefore, for checking rolling stock, there could no longer be a need to build expensive stationary systems or to move trains great distances for their inspection. As a result, the time lost due to equipment stoppage will be saved. Also, with the help of the proposed mobile device, railway safety could improve, while at the same time expenses for detection of wheel pair wear and damage could potentially decrease. This could result in significant economic savings.
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Kumar, V., A. Castellanos, J. Ortega, et al. "Dynamic Learning Framework: Adaptive Assessment Development for the Undergraduate Fluid Mechanics." In ASME 2014 4th Joint US-European Fluids Engineering Division Summer Meeting collocated with the ASME 2014 12th International Conference on Nanochannels, Microchannels, and Minichannels. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/fedsm2014-21718.
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This paper presents a dynamic learning framework (DLF) for engineering courses with rich mathematical and geometrical contents. The word “dynamic” implies that there are several moving components in the course contents and assessments. Moving contents are enabled by random-number generators to select text/paragraph from a database or chose a number between two ranges within engineering bounds. Dynamic contents are usually missing in traditional form of instructions such a fixed format book-type problem or static online material. The framework leverages on the computing resources from the recent advancement in touchpad computing devices (such as IPAD and Android based tablets) and web-based technologies (such as WebGL/SVG for virtual-reality and web-based graphics and PHP based server level programming language). All assessments are developed at four increasing levels of difficulty. The levels one through three are designed to assess the lower level learning skills as discussed in the “Bloom’s taxonomy of cognitive skills” whereas level four contents are designed to test the higher level skills. The level-one assessments are designed to be easiest and include guiding materials and solved examples. To lessen the impact of disinterests caused by mathematical abstractions, the assessment and content presentations are strengthened by integrating the mathematical concepts with visual engineering materials from real-world and local important applications. All problems designed to assess the lower level skills are computerized and tested using the Computer Adaptive Testing (CAT) algorithm which enabled the instructor to focus on the higher level skills and offer the course in partially flipped classroom setting.
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Coppo, Marco, Claudio Dongiovanni, and Claudio Negri. "Numerical Analysis and Experimental Investigation of a Common Rail Type Diesel Injector." In ASME 2002 Internal Combustion Engine Division Fall Technical Conference. ASMEDC, 2002. http://dx.doi.org/10.1115/icef2002-507.
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A production common rail type injector has been investigated via numerical simulation and experimentation. The functioning principle of the injector has been carefully analysed so as to obtain a mathematical model of the device. A zero-dimensional approach has been used for modelling the injector, thus considering the variables as function of time only. The analysis of the hydraulic part of the injector resulted in the definition of an equivalent hydraulic scheme, on which basis both the equations of continuity in chambers and flow through nozzles were written. The moving mechanical components of the injector, such as needle, pressure rod and control valve have been modelled using the mass-spring-damper scheme, thus obtaining the equation governing their motion. An electromagnetic model of the control valve solenoid has also been realized, in order to work out the attraction force on the anchor, generated by the electric current when flowing into its coil. The model obtained has been implemented using the Matlab® toolbox Simulink® and solved by means of the NDF (Numerical Differentiation Formulas) implicit scheme of the second order accuracy, suitable for problems with high level of stiffness. The experimental investigation on the common-rail injection system was performed on a test bench at some standard test conditions. Electric current flowing through the injector coil, oil pressure at the injector inlet, injection rate, needle lift and control valve lift were gauged and recorded during several injection phases. The mean reflux-flow rate and the mean quantity of fuel injected per stroke were also measured. Temperature and pressure of the feeding oil, as well as pressure in the rail were continuously controlled during the experimental test. The numerical and experimental results were compared. The model was then used to investigate the effect of control volume feeding and discharge holes and of their inlet fillet, as well as the effect of the control volume capacity, on the injector performance.
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Giacomelli, Enzo, Jun-Xia Shi, and Fabio Manfrone. "Considerations on Design, Operation and Performance of Hypercompressors." In ASME 2010 Pressure Vessels and Piping Division/K-PVP Conference. ASMEDC, 2010. http://dx.doi.org/10.1115/pvp2010-25040.
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The production of LDPE may require huge power, heavy-duty reciprocating compressors, provided with opposite cylinders and special frames to withstand the loads, deriving from the operating pressures. In many new projects larger capacities and higher performance requirements, renew the challenge of engineers to handle the mechanical, thermodynamic, chemical-physical, process and operational aspects. Safety, performance, operation and reliability are usual expectations needing a thorough evaluation of the service and the machine selection is based on positive results in similar applications. Valve, packing and cylinder performance is noticeably influenced by the design, operation and maintenance activities. The cylinders are compound pressure vessels, excluded by design codes, but their design and construction have to consider the extreme internal pressures and the nature of the process gas. The ability to withstand the high fatigue stresses and the need to avoid any leakage of gas around the compressor area implies solutions to minimize such occurrences including abnormal operations and emergencies. The design must foresee all possible failure modes of each component, to have safe and smooth operation. Innovative methods of simulation and modeling, like FEA, CFD, and others, are very important tools for the design of cylinder components. Valves are simulated by mathematical models optimizing performance and ensuring reliable operation, to reach a correct mechanical behavior with minimum energy consumption. The available technological improvements are taken as a base, resulting from the R&D of manufacturers and long experience of End Users. Also the pulsation and related vibration of the piping have to be investigated to keep the plant in operation without hazard. The operation is the stage where all the parameters have to be kept under control and incipient problems have to be identified to minimize shut down and arrange various maintenance works. Automation systems, together with new monitoring and diagnostic systems, allow very high safety levels, availability and optimized maintenance interventions.