Division Of Fractions Research Articles

초등학생들의 분수 나눗셈 문제해결 방법에 대한 예비교사들의 지식 분석

Because the role of the teacher is important for the education to actualize the goals of the curriculum, the interest about the teacher’s knowledges has been addressed as an important research topic. Among them, the pedagogical content knowledge is the knowledge that can emphasize the professionalism of the teacher. In this study, I analyzed the elementary pre-service teachers’ the problem solving methods that they imagined the methods that elementary school students can think about fraction division. Pre-service teachers who participated in this study were completed all of the mathematics education courses in the pre-service teachers’ education courses. The research was conducted using the four type-problems of fraction division. The results showed that elementary pre-service teachers responded in the order of equal sharing problem-measurement division-partitive division-context of determination of a unit rate problem. They presented significant responses not only with typical algorithms but also with pictures or expressions. On the basis of this research, we take an interest in the necessity of sharing and recognizing various methods of fraction division in pre-service education.

Investigating Using Lesson Study

Our middle school mathematics department used lesson study to investigate how to introduce fractions division to our sixth-grade students. We highlight our learnings during the Study and Plan phases, describe our observations during the lesson, and provide tips for educators interested in using lesson study to study their own content.

PARTITIVE FRACTION DIVISION: REVEALING AND PROMOTING PRIMARY STUDENTS’ UNDERSTANDING

Students show deficient understanding on fraction division and supporting that understanding remains a challenge for mathematics educators. This article aims to describe primary students’ understanding of partitive fraction division (PFD) and explore ways to support their understanding through the use of sequenced fractions and context-related graphical representations. In a design-research study, forty-four primary students were involved in three cycles of teaching experiments. Students’ works, transcript of recorded classroom discussion, and field notes were retrospectively analyzed to examine the hypothetical learning trajectories. There are three main findings drawn from the teaching experiments. Firstly, context of the tasks, the context-related graphical representations, and the sequence of fractions used do support students’ understanding of PFD. Secondly, the understanding of non-unit rate problems did not support the students’ understanding of unit rate problems. Lastly, the students were incapable of determining symbolic representations from unit rate problems and linking the problems to fraction division problems. The last two results imply to rethink unit rate as part of a partitive division with fractions. Drawing upon the findings, four alternative ways are offered to support students’ understanding of PFD, i.e., the lesson could be starting from partitive whole number division to develop the notion of fair-sharing, strengthening the concept of unit in fraction and partitioning, choosing specific contexts with more relation to the graphical representations, and sequencing the fractions used, from a simple to advanced form.

분수 나눗셈 지도 방법의 변천과정 분석

To analyze how fraction division is used in Korean mathematics textbooks, this study examined the various situations in which fraction division is applied and how the fraction division algorithm is developed. In the textbooks from the 1st curriculum to the 2015 revised curriculum, we analyzed how and in what situations fraction division is being developed, and present the results of the analysis and discussion points.

Design division mixed fractions materials using PMRI and lesson study

One of the problem in modern mathematical learning prosess is the mathematics concept presented by the teacher to students such as machines. So, the concepts are used by students as finished product without knowing the origin and how to reconstruct them. The effort that can be done by the teacher to overcome this problem by designing learning wherever possible starting with presenting realistic problems to drive and support students understanding abstract idea in mathematics. In this article we will discuss the design mixed fraction division based on PMRI, which is a type of design research. Design research has three cycles, namely preliminary design, classroom experiment, and retrospective analysis. This Design is aimed for supporting students understanding the concept division mixed fraction then use it into problem solving. In this article only the preliminary study phase will be discussed, which includes literature study activities, curriculum analysis, and the design of the initial prototype based on non-formal trials conducted at the SMP Srijaya Negara Palembang. An HLT (Hypotetical Learning Trajectory) has been produced regarding division of mixed fraction using image representation. This material not only designed using PMRI but also Lesson Study as learning system. So the design contain three activities, which are two activities about sharing task and one activity about jumping task.

BITERNARY OPPOSITION AS AN ELEMENT OF MULTIPOLAR SOCIO-CULTURAL SYSTEM DESCRIPTION LANGUAGE IN POSTSTRUCTURAL ANTHROPOLOGY

The target of research is the structuralist and poststructuralist paradigm of describing and explaining cultural constants through binary or ternary semiotic codes. The author’s hypothesis, based on anthropological theories of V. Turner and Cl. Levi-Strauss as well as those of contemporary South American anthropologists E. V. de Castro and E. Kohn, is that socio-cultural oppositions must be considered not as binary but as ternary ones. The main aim is to trace the influence of structuralism and poststructuralism on the construction of binary, ternary, tetrary schemes of coordination and subordination of elements of complex «neuro»-systems in Ethnology and Social Anthropology. The originality of the study lies, first, in clarification of the formal components of binarism and its derivatives (such as «monodualism», «monotrialism», and «bitrialism», the latter being introduced into scholarly circulation for the first time) as an approach that claims to be a universal explanation of the existence of cultural constants, and, secondly, in author’s suggestion to replace the discrete horizontal and vertical planar projections of the «primitive ontology», produced by binary and ternary oppositions, with the one integrating them into three, namely tridimensional, «bitriplet». Conclusion: in social sciences logical binarism of structuralism together with post-structuralism were replaced by tetrary, monodual, bitriple semiotic schemes with a large number of poles and fractional scale divisions of the states of analyzed systems that are able to capture and describe not only two-dimensional but three-dimensional statically-balanced and dynamically-non-balanced states of complex metaphysical, mytho-ritual, and social systems.

What the Volume of Matrices Says?

We know matrices and their transposes and we also know flip matrices. In my previous paper Matrices-One Review, I introduced transprocal matrix. Flip matrices are transpose of transprocal matrices. Now I would like to introduce water image of four matrices said above and properties of such matrices. Also we know, determinant of sum of matrices is not equal to sum of determinant of matrices. Why can’t we get equal value on addition side and additive side of determinant of matrix addition and subtraction? This question triggered me to find the reason. The basic algebra of mensuration gave ideas to retreat determinant of matrix addition and subtraction. I extent that ideas for matrices sum. Further, in 1812, French mathematician Jacques Philippe Marie Binet described how to multiply matrices. Matrices are defined on addition, subtraction and multiplication but not in division. By the inspiration of Binet, I would like to describe how to do divisions on matrices. This idea is derived from division of fractions. In division of fraction, reciprocal of divisor fraction multiplies with dividend fraction. I do the same in division on matrices with some modifications. By this way, we could find quotient matrix and remainder matrix which satisfy division algorithm. So we could say, determinant of division of dividend matrix and divisor matrix is equal to division of determinant of dividend matrix and determinant of divisor matrix.

Asked & Answered

The Asked & Answered department shares excerpts from discussion threads on the online MyNCTM community. In this issue, featured threads highlight responses to members' questions regarding 1st grade number sense, multiplication and division of fractions, issues of definition and precision related to circles, and the value of rationalizing denominators.

Математическое моделирование гидроочистки дизельного топлива

The principles of mathematical modeling of Hydrotreating diesel fuel in the representation of raw materials in the form of a set of narrow fractions in which the total content of various organosulphuric components is considered as a pseudocomponent are considered. As a result of the analysis of schemes of reactor blocks of Hydrotreating plants the most perspective two-reactor systems characterized by separate desulfurization of streams are revealed. It is shown that the preliminary fractionation of Hydrotreating raw materials into light and heavy fractions with the choice of the optimal boundary of fraction division can minimize the loading of the catalyst into the reactor unit. An algorithm for solving this problem, including experimental and computational fragments, is presented.

An Intermittent Frequency Synthesizer With Accurate Frequency Detection for Fast Duty-Cycled Receivers

An intermittent frequency synthesizer for fast duty-cycled receivers is presented in this paper. Different from state-of-art techniques which try to eliminate the initial phase error that degrades the intermittent frequency detection, a new frequency detector is proposed to maintain an accurate frequency detection regardless of the initial phase error. Moreover, an averaged fraction division scheme (AFDS) is integrated in the synthesizer to improve the frequency resolution. The frequency synthesizer is implemented and verified by using discrete devices and an FPGA. The measurement results show that the frequency detector provides a frequency detection accuracy of 0.88%, and the synthesizer achieves a frequency resolution of 0.625 MHz when controlled by a duty-cycled signal with a 6-μs period and a 2-μs time duration. The frequency synthesizer can perform an intermittent frequency hopping from 160MHz to 140MHz while requiring a period of 430 μs without disturbing the VCO's oscillation.

The Shmit Bequest

This article is part of a larger study that deals with Bolshevik finances during the years 1905–1914. Since Martov (Spasiteli ili uprazdniteli, 1911), most accounts of the finances of the Bolshevik fraction of the RSDRP have formed part of an anti-Leninist narrative in which the methods of the Bolsheviks have been likened to those of a Camorra. I focus instead on the technical (including legal) aspects of the origin, management and mismanagement of Bolshevik finances, and on the connection between funding, policy, and fractional divisions. In many accounts of the Shmit affair, it is claimed that Nikolai P. Shmit intended his bequest for the RSDRP as a whole, and that his sisters, Ekaterina and Elizaveta Pavlovna, were naive accomplices in the diversion of the funds towards the Bolsheviks. I argue, on the contrary, that in the absence of a written document, both sisters strove conscientiously to execute their brother’s will as they understood it, even when, on occasion, this meant defying the wishes of the Bolshevik beneficiaries. In this article, I attempt to provide a precise characterization of the “Shmit bequest”, and draw a distinction between the “bequest” and the entire estate of Nikolai Pavlovich (a distinction that the Bolsheviks were reluctant to acknowledge). Concentrating upon the realization of the portion of Ekaterina Pavlovna, I describe the priority claims that she made upon the bequest, and explain why recourse was ultimately made to an arbitration panel (treteiskii sud). I examine the extent to which the proposals of the arbitration panel were implemented, and conclude with an estimate of the proportion of the Shmit bequest that was eventually realized by the Bolsheviks. Realization by the Bolsheviks of the portion of Elizaveta Pavlovna, which extended over several years, requires separate attention.

A Fractional-$N$ PLL With Space–Time Averaging for Quantization Noise Reduction

This article presents a space–time averaging technique that can realize instantaneous fractional frequency division, and thus, can significantly reduce the quantization error in a fractional- $N$ phase-locked loop (PLL). Spatial averaging can be achieved by using an array of dividers running in parallel. Their different division ratios are generated by using a fractional $\Delta \Sigma $ modulator (DSM) and a dynamic element matching (DEM) block. To reduce the divider power, this article also proposes a way to achieve spatial averaging using only one divider and phase selection. A prototype 2.4-GHz fractional- $N$ PLL is implemented in a 40-nm CMOS process. Measurement results show that the proposed technique reduces the phase noise by 10 and 21 dB at the 1- and 10-MHz offset, respectively, leading to a reduction of the integrated rms jitter from 9.55 to 2.26 ps.

MULTIPLICATION AND DIVISION OF FRACTIONS: NUMERICAL COGNITION DEVELOPMENT AND ASSESSMENT PROCEDURES

Number and its basic operations can be conceptualized within a general system of relationships. Children need to build a number system within which they can add, subtract, multiply, and divide any rational number. Products and quotients can be defined in terms of general relational schemes. In this study, we examine whether elementary school children can construct a number system such that multiplication and division of fractions are based on the construction of general relational schemata. Student groups are not homogeneous and children progress at different rates. For reliable assessment, teachers need methods to examine developmental and individual differences in cognitive representations of mathematical concepts and operations. A logistic regression curve offers a visualization of the learning process as a function of the average grades. Item analysis of multiplication and division of fractions shows an improvement in the probability of a correct answer, especially for students with a higher average score.

Mixed methods study of middle school mathematics teachers’ content knowledge in usa and russia using sequential nested design

The sequential nested mixed methods study focused on comparative analysis of middle school mathematics teachers’ content knowledge in two countries. The study consisted of two stages: (1) quantitative study of teacher content knowledge; (2) qualitative study of teacher topic-specific content knowledge. The initial sample for the first stage included lower secondary mathematics teachers from the U.S. (grades 6–9, N=102) and Russia (grades 5–9, N=97). The Teacher Content Knowledge Survey (TCKS) was applied to assess teacher content knowledge based on the cognitive domains of Knowing, Applying, and Reasoning, as well as addressing the lower secondary mathematics topics of Number, Algebra, Geometry, Data and Chance. The second stage – an interpretive cross-case study – aimed at the examination of the U.S. and Russian teachers’ topic-specific knowledge on the division of fractions. For the second stage, N=16 teachers (8 – from the U.S., and 8 – from Russia) were selected for the study using non-probability purposive sampling technique based on teachers’ scores on the TCKS. Teachers were interviewed on the topic of fraction division using questions addressing their content and pedagogical content knowledge. The study revealed that there are explicit similarities and differences in teachers’ content knowledge as well as its cognitive types. The study results may inform the field on priorities placed on lower secondary mathematics teachers’ knowledge in the USA and Russia. It also suggests close comparison and learning about issues related to teacher knowledge in both countries with a potential focus on re-examining practices in teacher preparation and professional development.

A Fully Synthesizable Ultra-${N}$ Audio Frequency Multiplier for HDMI Applications

This brief develops a highly programmable and fully synthesizable large-N frequency multiplier. It is based on Time-Average-Frequency Direct Period Synthesis (TAF-DPS) and its purpose is to generate audio frequencies in HDMI (high definition multimedia interface) applications in which frequency multiplication ratio is extremely large. The proposed multiplier is a full digital feed-forward frequency control loop. The output frequency is produced by performing a fractional frequency division of the oscillator clock with TAF-DPS. The source of electrical oscillation in the TAF-DPS is a free-running inverter ring that consists of inverters. The novel structure is implemented in a TSMC 0.18 μm 1P6M CMOS process. The implementation is fully digital HDL coding → simulation → synthesis → place & route without any need for analog simulation. The hardware cost is equivalent to 3150 two-input NAND gates. The proposed structure can achieve fast frequency switching within two input reference clock cycles. The multiplication ratio is evaluated experimentally to be as high as 106 with an input signal of 50 Hz; this is the largest ratio and lowest input frequency to be reported to the best of our knowledge. Besides generating an audio clock for HDMI applications, this frequency multiplier is very suitable for frequency measurements and system-on-a-chip applications.

Demonstrating Conceptual Understanding of Fraction Arithmetic: An Analysis of Pre-Service Special and General Educators’ Visual Representations

To provide effective fractions instruction, teachers need deep understanding of fraction concepts (i.e., content knowledge) and skill in using visual representations of fractions to support student learning (i.e., pedagogical content knowledge); yet, research indicates that many pre- and in-service teachers lack knowledge in both areas. The present study examines the performance of 55 pre-service teachers (PSTs) on tasks assessing their procedural and conceptual knowledge of fractions, including a test of their ability to generate visual representations and story problems to model fraction multiplication and division. PSTs demonstrated high levels of procedural and conceptual knowledge of fraction addition and subtraction, moderate procedural knowledge for fraction multiplication and division, and relatively weak conceptual knowledge of fraction multiplication and division. Weak conceptual knowledge was related to inability to accurately model fraction multiplication and division using visual representations and story problems. Implications for research and practice are presented.

The identification of students’ didactic obstacles in learning fractions based on the form of the problems

This study aims at knowing whether there are didactic obstacles in learning fractions of students in Yogyakarta based on the form of the problems. Those problems include visual, narrative, and symbolic forms. Materials of fractions consist of the concepts of fractions, equivalent fractions, the order of fractions, decimals, percents, addition of fractions, subtraction of fractions, multiplication of fractions, and division of fractions. The method of this study is survey with expost facto approach. The populations are students of Junior High School grade VII in Yogyakarta. The samples are 151 students from four schools in Yogyakarta. The choice of sample is done by stratified proportional random sampling technique. The instrument is diagnostic test with 27 questions that divide into 9 visual forms, 9 narrative forms, and 9 symbolic forms. Based on the result, students have didactic obstacles in visual forms and narrative forms. While in symbolic forms, students did not have didactic obstacles.

Enhance elementary school students to learn mathematics: Case of the use of mother tongue-base language (MTBL) students

The study of using Mother Tongue-Based Language (MTBL) in remediation on Least Learned Competencies in Elementary Mathematics to two sections in Grade VI at Tambo Central School was carried out. Both groups were exposed to team-assisted individualized remediation. The data needed for this study were obtained through 30-item achievement test and seventeen (17) Learning Materials constructed by the researcher which were construct-validated and evaluated by mathematics experts. Identified competencies of the materials were based on the National Achievement Test result and in the K to 12 Curriculum on Rational Numbers. The findings of the study revealed that the use of MTBL in team-assisted individualized remediation evidently improved the performance of the respondents in the three competencies on fractions namely comprehension of fractions; multiplication of fractions and its applications; and division of fractions and its applications.

Fourth to Sixth Grade Teachers’ Invented Real World Problems and Pictorial Representations for Fraction Division

Teaching fraction operations for conceptual understanding is a challenging task. For the topic of fraction division especially, teachers need support because teachers find this difficult to teach and elementary and middle school students struggle to learn this concept. Well-structured professional development can assist teachers in exploring their own understanding as well as how to incorporate multiple representations in their instruction. This naturalistic inquiry study investigated fourth to sixth grade teachers’ invented real-world problems and pictorial representations for fraction division. The teachers completed a written assessment focused on conceptual understanding of fraction operations through multiple representations at the beginning and conclusion of a weeklong professional development experience. The questions on the assessment were taken or modified from prior research studies. The results in this paper focus specifically on a task in which the teachers had to develop a real world story problem for 2½ ÷ ¾ and solve the problem by drawing a pictorial representation. The teachers showed statistically significant improvement on developing appropriate real-world problems and using pictorial representations to solve their problems. Implications for teachers’ usage of multiple representations are discussed along with needed structures for professional development targeted on fractions.

Pre-Service Teachers Making Sense of Fraction Division with Remainders

This study reports an analysis of how pre-service teachers (n=34) made sense of fraction division with remainders using pictorial modeling strategies, and how small-group and whole-class discussion helped them develop conceptual understanding. One and a half class sessions were video recorded, and 12 interviews were conducted. Results indicate that pre-service teachers can develop a conceptual understanding of fraction division with remainders using modeling strategies, and their understanding emerges in three levels: a) level one: ignoring the remainder or labeling it incorrectly; b) level two: interpreting the remainder in the original unit but not relating it to the new unit; and c) level three: interpreting the remainder both in the original unit and the new unit flexibly.