Um estudo sobre a aprendizagem de números irracionais no ensino médio

The present study describes theoretical practical relationships between development and application of activities in Mathematics education. It s proposed a methodological approach to Mathematics in the first grade of Ensino Médio, supported by an experiment involving Irrational Numbers education by using constructive activities, applied obeying an educational sequence. Constructivism is used as an important theoretical reference in teaching learning process of Mathematics. The methodological intervention was done with two classes of students of the first grade of Ensino Médio, in two public schools, a state one and a federal one, located on the city of Natal, Rio Grande do Norte. The development, application and testing of the activities used on this experiment led us to think more profoundly about the value of constructivism ideas and understand that the use of activities that obey an educational sequence favors the learning. It s also discussed the research results, commented on a way to contribute to the advances of the proposal and it s more constant use. The participation and testing of the students were analyzed and judged using Skemp s Instrumental Understanding and Relational Understanding concepts. The results of the research were considered good, so we believe this methodological intervention can be used more frequently in the classes of Ensino Médio and also be applied to teachers in courses of initial education and continuous formation

De solutione problematum diophanteorum per números integros : o primeiro trabalho de Euler sobre equações diofantinas

The present dissertation analyses Leonhard Euler´s early mathematical work as Diophantine Equations, De solutione problematum diophanteorum per números íntegros (On the solution of Diophantine problems in integers). It was published in 1738, although it had been presented to the St Petersburg Academy of Science five years earlier. Euler solves the problem of making the general second degree expression a perfect square, i.e., he seeks the whole number solutions to the equation ax2+bx+c = y2. For this purpose, he shows how to generate new solutions from those already obtained. Accordingly, he makes a succession of substitutions equating terms and eliminating variables until the problem reduces to finding the solution of the Pell Equation. Euler erroneously assigns this type of equation to Pell. He also makes a number of restrictions to the equation ax2+bx+c = y and works on several subthemes, from incomplete equations to polygonal numbers

A eficácia na execução do planejamento estratégico no poder judiciário do Rio Grande do Norte

This paper presents an analysis of the current strategic plan of the Judiciary of Rio Grande do Norte emphasizing the evaluation of strategic indicators verifying the effectiveness in implementation, since the implementation of the Balanced Scorecard as a tool for performance evaluation of strategic management. The research presents the strategy map and the evaluation indices of strategic performance reporting on the effectiveness. After literature review and documentary, is making the measurement of indicators that are treated from the standpoint of an exploratory and descriptive in strategic planning used by the judiciary Potiguar. The problem was evaluated qualitatively and quantitatively using statistical techniques for data analysis comparing them between Judiciaries of Brazilian States. With respect to data collection was used performance indicators extracted from the data of Justice in Numbers provided by CNJ the period from 2004 to 2011, and the information sought in the Sector Strategic Planning TJ / RN. The main results of this study are as follows: Acquisition of insight into what level is the strategic planning of the judiciary of Rio Grande do Norte and the evolution of its performance indicators comparing them with the states of RS, CE, SE and the National Judiciary

Análise estatística das emissões de CO e HC produzidas por gases da exaustão veicular oriundos de gasolina, GNV e mistura álcool/gasolina

The vehicles are the main mobile sources of carbon monoxide (CO) and unburned hydrocarbons (HC) released into the atmosphere. In the last years the increment of the fleet of vehicles in the municipal district of Natal-RN it is contributing to the increase of the emissions of those pollutants. The study consisted of a statistical analysis of the emissions of CO and HC of a composed sample for 384 vehicles with mechanization Gasoline/CNG or Alcohol/Gasoline/CNG of the municipal district of Natal-RN. The tests were accomplished in vehicles submitted to Vehicular Safety's Inspection, in the facilities of INSPETRANS, Organism of Vehicular Inspection. An partial gases analyzer allowed to measure, for each vehicle, the levels of CO and HC in two conditions of rotation of the motor (900 and 2500 rpm). The statistical analysis accomplished through the STATISTICA software revealed a sensitive reduction in the efficiency of the converters catalytic after 6 years of use with emission average it is of 0,78% of CO and 156 (ppm) of HC, Which represents approximately 4 (four) times the amount of CO and the double of HC in comparison with the newest vehicles. The result of a Student s t-test, suggests strongly that the average of the emissions of HC (152 ppm), at 900 rpm, is 40% larger than at 2500 rpm, for the motor without load. This result reveals that the efficiency of the catalytic conversion is limited kinetically in low engine speeds. The Study also ends that when comparing the emissions of CO and HC considering the influence of the fuels, it was verified that although the emissions of CO starting from CNG are 62% smaller than arising from the gasoline, there are not significant differences among the emissions of HC originating from of CNG and of the gasoline. In synthesis, the results place the current criteria of vehicular inspection, for exhaust gases, in doubt, leading the creation of emission limits of pollutant more rigorous, because the efficiency of the converters catalytic is sensibly reduced starting from 6 years of use. It is also raised the possibility of modifications in the test conditions adopted by the current norms, specifically in the speed engine, have seen that in the condition without load the largest emission indexes were registered in slow march. That fact that allows to suggest the dismissal of the tests in high speed engine, reducing the time of inspection in half and generating economy of fuel

Estudo da reidratação do feijão verde (Vigna unguiculata L. Walp) desidratado por micro-ondas com e sem pré-tratamento osmótico

The green bean has organoleptic and nutritional characteristics that make it an important food source in tropical regions such as the Northeast of Brazil. It is a cheap source of protein and important for nutrition of rural population contributing significantly in subsistence farming of the families from Brazil s northeast. It is consumed in entire region and together with the dry meat and other products composes the menu of typical restaurants, being characterized as an important product for economy of Northeast. The green bean is consumed freshly harvested and has short cycle, being characterized as a very perishable food, which hampers your market. The drying method is an alternative to increase the lifetime and provide a reduction volume of this product making easier your transportation and storage. However is necessary to search ways of drying which keep the product quality not only from the nutritional standpoint but also organoleptic. Some characteristics may change with the drying process such as the coloring, the rehydration capacity and the grains cooking time. The decrease of drying time or of exposure of the grains to high temperature minimizes the effects related with the product quality loss. Among the techniques used to reduce the drying time and improve some characteristics of the product, stands out the osmotic dehydration, widely used in combined processes such as the pretreatment in drying food. Currently the use of the microwaves has been considered an alternative for drying food. The microwave energy generates heat inside of materials processed and the heating is practically instantaneous, resulting in shorter processing times and product quality higher to that obtained by conventional methods. Considering the importance of the green beans for the Northeast region, the wastefulness of production due to seasonality of the crop and your high perishability, the proposal of this thesis is the study of drying grain by microwaves with and without osmotic pretreatment, focusing on the search of conditions of processes which favor the rehydration of the product preserving your organoleptic characteristics. Based on the analysis of the results of osmotic dehydration and dielectric properties was defined the operating condition to be used in pretreatment of the green bean, with osmotic concentration in saline solution containing 12,5% of sodium chloride, at 40°C for 20 minutes. The drying of green bean by microwave was performed with and without osmotic pretreatment on the optimized condition. The osmotic predehydration favored the additional drying, reducing the process time. The rehydration of dehydrated green bean with and without osmotic pretreatment was accomplished in different temperature conditions and immersion time according to a factorial design 22, with 3 repetitions at the central point. According to results the better condition was obtained with the osmotically pretreated bean and rehydrated at a temperature of 60°C for 90 minutes. Sensory analysis was performed comparing the sample of the green bean in natura and rehydrated in optimized conditions, with and without osmotic pretreatment. All samples showed a good acceptance rate regarding the analyzed attributes (appearance, texture, color, odor and taste), with all values above 70%. Is possible conclude that the drying of green bean by microwave with osmotic pretreatment is feasible both in respect to technical aspects and rehydration rates and sensory quality of the product

Análise da salubridade do meio urbano com base na utilização de índices ambientais: aplicação na bacia de drenagem XII da cidade do Natal-RN

The growing accumulation of people in urban centers caused chronic problems of the cities to begin to take an increasingly unsustainable. Primarily related to lack of infrastructure coupled with sanitation and lack of investment in critical sectors such as health, education, housing and transportation, these problems start to deteriorate markedly the quality of life of city dwellers and put into test management policies of the spaces urbanized. To reverse this situation, shows is essential to the use of tools (highlighting this harvest rates and environmental indicators) that help in assessing the current conditions and may assist in predicting future scenarios. From the information listed above, now put the research seeks to present an index called ISBA Environmental (Sanitation Index) which looks at the four urban systems (water, sewer, solid waste and urban drainage) from the viewpoint of application in a geographical cutout specific - in this case the Drainage Basin XII, defined by the Plan of Urban Drainage Stormwater in the city of Natal, capital of Rio Grande do Norte. This index, together with analysis of other factors sought to trace the current conditions of the basin and thus, assist in proposing the best solutions. For the preparation of the index was applied a questionnaire with a sample of 384 (three hundred eighty-four) households that aimed to study two variables: access to services and satisfaction of the population in relation to these. The ISBA has shown that the system is the most deficient collection and disposal of effluents (ICE = 47.66%), followed by the drainage of rainwater (IDAP = 54.17%), water supply (AAI = 61, 36) and solid waste collection (IRS = 78.28). With the ISBA was possible to verify that the qualitative data shows whose subjectivity is evident (as is the case of user satisfaction) can be of great importance when an assessment, since we obtained the correlation coefficient between the variables "Access" and " Satisfaction "equal to 0.8234, showing a strong correlation between the existence / quality of service offered and the impressions of the population that receives them

Impacto do abastecimento irregular de água nos altos índices de dengue

Dengue fever is an infectious disease that causes thousands of deaths each year in Brazil and other tropical countries. This work demonstrates that the irregularity in the supply of water by public water supply systems is a major factor that contributes to the proliferation of breeding sites of the mosquito that transmits dengue, due to the impossibility of removing water storage tanks. 31 points in the water supply network in thirteen districts of Natal, Brazil, were monitored by the installation of pressure gauges type Datalogger. The data about pressure showed deficiency in water supplies in many neighborhoods, forcing residents to accumulate water in tanks at ground level. In addition, It was observed that in neighborhoods with regular water supply, Infestation Index per type of container (ITR) of type A2 (Deposit ground level) was 0.00% and where there were failures in the supply of water , the ITR was high (above 50%). We believe that policies to combat dengue in Brazil should be reassessed so that more resources can be directed to the improvement of water supply systems and supply companies should be blamed for the problem too

Números complexos: uma proposta de mudança metodológica para uma aprendizagem significativa no ensino médio

This work presents a proposal of a methodological change to the teaching and learning of the complex numbers in the Secondary education. It is based on the inquiries and difficulties of students detected in the classrooms about the teaching of complex numbers and a questioning of the context of the mathematics teaching - that is the reason of the inquiry of this dissertation. In the searching for an efficient learning and placing the work as a research, it is presented a historical reflection of the evolution of the concept of complex numbers pointing out their more relevant focuses, such as: symbolic, numeric, geometrical and algebraic ones. Then, it shows the description of the ways of the research based on the methodology of the didactic engineering. This one is developed from the utilization of its four stages, where in the preliminary analysis stage, two data surveys are presented: the first one is concerning with the way of presenting the contents of the complex numbers in math textbooks, and the second one is concerning to the interview carried out with High school teachers who work with complex numbers in the practice of their professions. At first, in the analysis stage, it is presented the prepared and organized material to be used in the following stage. In the experimentation one, it is presented the carrying out process that was made with the second year High school students in the Centro Federal de Educação tecnológica do Rio Grande do Norte CEFET-RN. At the end, it presents, in the subsequent and validation stages, the revelation of the obtained results from the observations made in classrooms in the carrying out of the didactic sequence, the students talking and the data collection

Uma análise dos questionamentos dos alunos nas aulas de números complexos

This work presents a contribution for the studies reffering to the use of the History of Mathematics focusing on the improvement of the Teaching and Learning Process. It considers that the History of Matematics, as a way of giving meaning to the discipline and improve the quality of the Teaching and Learning Process. This research focuses on the questions of the students, classified in three categories of whys: the chronological, the logical and the pedagogical ones. Therefore, it is investigated the teaching of the Complex Numbers, from the questions of the students of the Centro Federal de Educação Tecnológica do Rio Grande do Norte (Educational Institution of Professional and Technology Education from Rio Grande do Norte). The work has the following goals: To classify and to analyse the questions of the students about the Complex Numbers in the classes of second grade of the High School, and to collate with the pointed categories used by Jones; To disccus what are the possible guidings that teachers of Mathematics can give to these questions; To present the resources needed to give support to the teacher in all things involving the History of Mathematics. Finally, to present a bibliographic research, trying to reveal supporting material to the teacher, with contents that articulate the Teaching of Mathematics with the History of Mathematics. It was found that the questionings of the pupils reffers more to the pedagogical whys, and the didatic books little contemplate other aspects of the history and little say about the sprouting and the evolution of methods of calculations used by us as well

A difícil aceitação dos números negativos: um estudo da teoria dos números de Peter Barlow (1776-1862)

The present study seeks to present a historico-epistemological analysis of the development of the mathematical concept of negative number. In order to do so, we analyzed the different forms and conditions of the construction of mathematical knowledge in different mathematical communities and, thus, identified the characteristics in the establishment of this concept. By understanding the historically constructed barriers, especially, the ones having ontologicas significant, that made the concept of negative number incompatible with that of natural number, thereby hindering the development of the concept of negative, we were able to sketch the reasons for the rejection of negative numbers by the English author Peter Barlow (1776 -1862) in his An Elementary Investigation of the Theory of Numbers, published in 1811. We also show the continuity of his difficulties with the treatment of negative numbers in the middle of the nineteenth century

O uso pedagógico de uma seqüência didática para a aquisição de algumas idéias relacionadas ao conceito de números complexos

The aim of the present work is to contribute to the teaching-learning process in Mathematics through an alternative which tries to motivate the student so that he/she will learn the basic concepts of Complex Numbers and realize that they are not pointless. Therefore, this work s general objective is to construct a didactic sequence which contains structured activities that intends to build up, in each student s thought, the concept of Complex Numbers. The didactic sequence is initially based on a review of the main historical aspects which begot the construction of those numbers. Based on these aspects, and the theories of Richard Skemp, was elaborated a sequence of structured activities linked with Maths history, having the solution of quadratic equations as a main starting point. This should make learning more accessible, because this concept permeates the students previous work and, thus, they should be more familiar with it. The methodological intervention began with the application of that sequence of activities with grade students in public schools who did not yet know the concept of Complex Numbers. It was performed in three phases: a draft study, a draft study II and the final study. Each phase was applied in a different institution, where the classes were randomly divided into groups and each group would discuss and write down the concepts they had developed about Complex Numbers. We also use of another instrument of analysis which consisted of a recorded interview of a semi-structured type, trying to find out the ways the students thought in order to construct their own concepts, i.e. the solutions of the previous activity. Their ideas about Complex Numbers were categorized according to their similarities and then analyzed. The results of the analysis show that the concepts constructed by the students were pertinent and that they complemented each other this supports the conclusion that the use of structured activities is an efficient alternative for the teaching of mathematics

Euler e os números pentagonais

The present investigation includes a study of Leonhard Euler and the pentagonal numbers is his article Mirabilibus Proprietatibus Numerorum Pentagonalium - E524. After a brief review of the life and work of Euler, we analyze the mathematical concepts covered in that article as well as its historical context. For this purpose, we explain the concept of figurate numbers, showing its mode of generation, as well as its geometric and algebraic representations. Then, we present a brief history of the search for the Eulerian pentagonal number theorem, based on his correspondence on the subject with Daniel Bernoulli, Nikolaus Bernoulli, Christian Goldbach and Jean Le Rond d'Alembert. At first, Euler states the theorem, but admits that he doesn t know to prove it. Finally, in a letter to Goldbach in 1750, he presents a demonstration, which is published in E541, along with an alternative proof. The expansion of the concept of pentagonal number is then explained and justified by compare the geometric and algebraic representations of the new pentagonal numbers pentagonal numbers with those of traditional pentagonal numbers. Then we explain to the pentagonal number theorem, that is, the fact that the infinite product(1 x)(1 xx)(1 x3)(1 x4)(1 x5)(1 x6)(1 x7)... is equal to the infinite series 1 x1 x2+x5+x7 x12 x15+x22+x26 ..., where the exponents are given by the pentagonal numbers (expanded) and the sign is determined by whether as more or less as the exponent is pentagonal number (traditional or expanded). We also mention that Euler relates the pentagonal number theorem to other parts of mathematics, such as the concept of partitions, generating functions, the theory of infinite products and the sum of divisors. We end with an explanation of Euler s demonstration pentagonal number theorem

Os artigos de Euler sobre os números amigáveis

Among the many methodological resources that the mathematics teacher can use in the classroom, we can cite the History of Mathematics which has contributed to the development of activities that promotes students curiosity about mathematics and its history. In this regard, the present dissertation aims to translate and analyze, mathematically and historically, the three works of Euler about amicable numbers that were writed during the Eighteenth century with the same title: De numeris amicabilibus. These works, despite being written in 1747 when Euler lived in Berlin, were published in different times and places. The first, published in 1747 in Nova Acta Eruditorum and which received the number E100 in the Eneström index, summarizes the historical context of amicable numbers, mentions the formula 2nxy & 2nz used by his precursors and presents a table containing thirty pairs of amicable numbers. The second work, E152, was published in 1750 in Opuscula varii argument. It is the result of a comprehensive review of Euler s research on amicable numbers which resulted in a catalog containing 61 pairs, a quantity which had never been achieved by any mathematician before Euler. Finally, the third work, E798, which was published in 1849 at the Opera postuma, was probably the first among the three works, to be written by Euler

Diagnóstico ambiental das águas do estuário Jundiaí-Potengi pela determinação dos índices de qualidade da água e toxidez

This study is an environmental diagnosis of the Jundiaí-Potengi/RN estuarine system waters, using calculations of pollution indicator indices such as the Water Quality Index (WQI) and the Toxicity Index (TI). The samples were collected at twelve points on the estuary, at high and low tide, between August and November 2007, over four campaigns. The study area, located in a high impact region, has various activities on its banks such as: discharge of untreated or undertreated domestic and industrial sewage, shrimp farming, immunizer stabilization lakes, riverside communities, etc. All the parameters analyzed were compared to the limits of CONAMA Resolution No. 357 of 2005 for healthy and saline Class 1 waters. The results found prove the impact caused by various activities, mainly the parameters related to the presence of organic material, such as DQO, DBO, COT and thermotolerant colliforms. The IQA for most of the collection points was of medium quality. For the metals, although values above the Resolution limits were found, most of them were lower than the detection limits of ICP-OES used, indicating that they tend to be transported by the dynamic of the tides or rainfall and are deposited in bottom sediments, resulting in a TI of 1.0 in this water, when they are absent, which occurs in most cases, or 0.0, when heavy metals are found in these waters

Monitoramento das águas da bacia hidrográfica pirangi pela determinação dos índices de qualidade da água-iqa e índice de toxidez-it

The problems of water supply in Northeast Brazil are severe and require more focused studies. This work was intended to assess water quality in the watershed Pirangi, located in the Northeastern state of the newborn using the Water Quality Index, AQI associated with the Index of Toxicity-IT. The data presented in this study were collected in November 2008, June 2009 and March 2010 at eight sampling stations distributed throughout the basin. The study covered nine parameters, based on guidelines established by CETESB, and seven members of Metal Toxicity index-IT. These waters are framed in the classification between GOOD and BAD showing AQI 41.34 minimum and a maximum of 76.23. Virtually all seven metals analyzed were below the detection limits of ICP-OES giving IT a water equal to one when they are absent and 0 when there are levels of trace metals