Uma proposta alternativa para o ensino progressões relacionadas a funções

Generally, arithmetic and geometric progressions are taught separately from ane and exponential functions, only by the use of memorized formulas and without any concern of showing students how these contents are related. This paper aims at presenting a way of teaching such contents in an integrated way, starting with the definition of ane and exponential functions relating them to situations from the daily life of the students. Then, characteristics and graphics of those functions are presented and, subsequently, arithmetic and geometric progression are shown as a restriction of the ane and exponential functions. Thus, the study of the progressions is introduced based on the functions mentioned above using situations from students daily lives as examples

O estudo das funções quadráticas e sua relação com o cotidiano

Across the centuries, Mathematics - exact science as it is - has become a determining role in the life of man, which forms to use suprir needs of their daily lives. With this trajectory, is characterized the importance of science as an instrument of recovery not only conteudstica, but also a mathematician to know that leads the apprentice to be a dynamic process of learning ecient, able to find solutions to their real problems. However, it is necessary to understand that mathematical knowledge today requires a new view of those who deal directly with the teaching-learning process, as it is for them - Teachers of Mathematics - desmistificarem the version that mathematics, worked in the classroom, causes difficulties for the understanding of students. On this view, we tried to find this work a methodology that helps students better understand the Quadratic functions and its applications in daily life. Making use of knowledge Ethnomathematics, contextualizing the problems relating to the content and at the same time handling the software GeoGebra, aiming a better view of the behavior of graphs of functions cited

Sistema de numeração posicional e a lógica da divisão de inteiros

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

Probabilidade no ensino médio

This thesis aims to show teachers and students in teaching and learning in a study of Probability High School, a subject that sharpens the perception and understanding of the phenomea of the random nature that surrounds us. The same aims do with people who are involved in this process understand basic ideas of probability and, when necessary, apply them in the real world. We seek to draw a matched between intuition and rigor and hope therebyto contribute to the work of the teacher in the classroom and the learning process of students, consolidating, deepening and expaning what they have learned in previous contents

Função Afifim: Teoria e aplicações

In this work we present a proposal to contribute to the teaching and learning of affine function in the first year of high school having as prerequisite mathematical knowledge of basic education. The proposal focuses on some properties, special cases and applications of affine functions in order to show the importance of the demonstrations while awaken student interest by showing how this function is important to solve everyday problems

Um estudo dos sistemas de amortizações SAC e francês no ensino médio apoiado na construção de planilhas Eletrônicas

The aim of this work is to provide a text to support interested in the main systems of amortization of the current market: Constant Amortization System (SAC) and French System, also known as Table Price. We will use spreadsheets to facilitate calculations involving handling exponential and decimal. Based on [12], we show that the parcels of the SAC become smaller than the French system after a certain period. Further then that, we did a comparison to show that the total amount paid by SAC is less than the French System

Uma justificava da validade do teorema fundamental da álgebra para o ensino médio

Among several theorems which are taught in basic education some of them can be proved in the classroom and others do not, because the degree of difficulty of its formal proof. A classic example is the Fundamental Theorem of Algebra which is not proved, it is necessary higher-level knowledge in mathematics. In this paper, we justify the validity of this theorem intuitively using the software Geogebra. And, based on [2] we will present a clear formal proof of this theorem that is addressed to school teachers and undergraduate students in mathematics

Área e volume da esfera

In this dissertation, we present a study on the teaching of volume of the sphere and the area of spherical surface. On this topic, a quali-quantitative was taken survey with the objective of identifying how these topics are addressed. For this, we made 14 questions to 30 teachers of Natal and the results of this survey are presented and discussed. After that, we present alternative ways to derive the formulas of the volume of a sphere and the are of a spherical surface

A relação de Euler para poliedros

In this paper we analyze the Euler Relation generally using as a means to visualize the fundamental idea presented manipulation of concrete materials, so that there is greater ease of understanding of the content, expanding learning for secondary students and even fundamental. The study is an introduction to the topic and leads the reader to understand that the notorious Euler Relation if inadequately presented, is not sufficient to establish the existence of a polyhedron. For analyzing some examples, the text inserts the idea of doubt, showing cases where it is not fit enough numbers to validate the Euler Relation. The research also highlights a theorem certainly unfamiliar to many students and teachers to research the polyhedra, presenting some very simple inequalities relating the amounts of edges, vertices and faces of any convex polyhedron, which clearly specifies the conditions and sufficient necessary for us to see, without the need of viewing the existence of the solid screen. And so we can see various polyhedra and facilitate understanding of what we are exposed, we will use Geogebra, dynamic application that combines mathematical concepts of algebra and geometry and can be found through the link http://www.geogebra.org

Equações de segundo grau e mudança de variáveis

In this work are presented, as a review and in a historical context, the most used methods to solve quadratic equations. It is also shown the simplest type of change of variables, namely: x = Ay + B where A;B 2 R, and some changes of variables that were used to solve quadratic equations throughout history. Finally, a change of variable, which has been used by the author in the classroom as an alternative method, is presented and the result of this methodoly is illustrated by the responses of a test that was done by the students in classroom

As dunas eólicas de Natal/RN: datação e evolução

The main purpose of this study is to present complementary information on eolian systems that occur in the Center-South portion of the city of Natal, Rio Grande do Norte State, Brazil. The investigation focused on the identification of eolian units, absolute luminescence dating of eolian sediments, to geophysical analysis by Ground Penetrating Radar (GPR), as well as qualitative identification of the use and terrain occupation in the study area. A map was elaborated in order to separate eolian geological units, and describe their deposicional and erosional characteristics, by aerial photo-interpretation. The study lead to the identification of six main units: Unit I - Blowouts (BW), Unit II - Active dunes field (DA), Unit III - Fixed dunes field (DF), Unit IV - Parabolic dunes field (DP), Unit V - Red parabolic dunes field (DV) and Unit VI - Devastated dunes surface (SD). Were analised by Termoluminescense (TL) Method Eight samples from units I, III, IV and V. The largest number of samples with similar or close resulted yielded ages about 15,000 years. Two GPR profiles were obtained in areas of the Dunes Park, near the "Natal s Conventions Center". Results showed the migration a new dune generation over on older one, the geologic contact between dunes and the Formação Barreiras (FB), and the groundwater level inside the dune unit. The qualitative study on the use and terrain occupation of the soil in some dunes fields in the study area, it was possible identify some use and occupation, as the following ones: the construction of residential and commercial buildings, dunes fields cutted for construction to access road and garbage deposition

Estratigrafia de sequências do intervalo aptiano ao albino na Bacia do Araripe, NE do Brasil

This study has as a main objective to make a detailed stratigraphic analysis of the Aptian-Albian interval in the east part of Araripe Basin, NE of Brazil which correspond, litostratigraphically, to Rio Da Batateira, Crato, Ipubi and Romualdo formations. The stratigraphic analysis was based on three different stages, the 1D, 2D and 3D analysis; these ones were adapted to the sequence stratigraphy concepts in order to create a chronostratigraphic framework for the study area within the basin. The database used in the present study contains field and well information, wells that belong to Santana Project, carried out by the Ministério de Minas e Energia- DNPM- CPRM from 1977 to 1978. The analysis 1D, which was done separately for each well and outcrop allowed the recognition of 13 sedimentary facies, mainly divided based on predominant litologies and sedimentary structures. Such facies are lithologically represented by pebble, sandstones, claystones, margas and evaporates; these facies are associated in order to characterize different depositional systems, that integrate from the continental environment (fluvial system and lacustre), paralic system (delta system and lagunar) to the marine environment (shelfenvironment). The first one, the fluvial system was divided into two subtypes: meandering fluvial system, characterized by fill channel and floodplain deposits; the facies of this system are associated vertically according to the textural thinning upward cycles (dirting-up trend pattern in well logs). Lacustrine environment is mainly related with the lithotypes of the Crato Formation, it shows a good distribution within the basin, been composed by green claystone deposits and calcareous laminated. Deltaic System represented by prodelta and delta front deposits which coarsening upward tendency. Lagunar system is characterised by the presence of anhydrite and gypsum deposits besides the black claystone deposits with vegetal fragments which do not contain a fauna typically marine. The marine platform system is composed by successions of black and gray claystone with fossiliferous fauna of Dinoflagellates (Spiniferites Mantell, Subtilisphaera Jain e Subtilisphaera Millipied genre) typical of this kind of depositional system. The sedimentary facies described are vertically arranged in cycles with progradational patterns which form textural coersening upward cycles and retrogradational, represented by textural thinning dowward cycles. Based in these cycles, in their stack pattern and the vertical change between these patterns, the systems tracks and the depositional sequences were recognized. The Low System Track (LST) and High System Track (HST) are composed by cycles with progradational stack pattern, whereas the Trangessive System Track (TST) is composed by retrogradational stack pattern cycles. The 2D stratigraphic analysis was done through the carrying out of two stratigraphic sections. For the selection of the datum the deepest maximum flooding surface was chosen, inside the Sequence 1, the execution of these sections allowed to understand the behaviour of six depositional systems along the study area, which were interpreted as cycles of second order or supercycles (cycles between 3 and 10 Ma), according to the Vail, et al (1977) classification. The Sequence 1, the oldest of the six identified is composed by the low, transgressive and high systems tracks. The first two system tracks are formed exclusively by fluvial deposits of the Rio da Batateira Formation whereas the third one includes deltaic and lacustrine deposits of the Crato Formation. The sequences 2 and 3 are formed by the transgressive systems tracks (lake spreading phase) and the highstand system track (lake backward phase). The TST of these sequences are formed by lacustrine deposits whereas HST contains deltaic deposits, indicating high rates of sedimentary supply at the time of it s deposition. The sequence 4 is composed by LST, TST and HST, The TST4 shows a significant fall of the lake base level, this track was developed in conditions of low relation between the creation rate of space of accommodation and the sedimentary influx. The TST4 marks the third phase of expansion of the lacustrine system in the section after the basin´s rift, the lacustrine system established in the previous track starts a backward phase in conditions that the sedimentary supply rate exceeds the creation rate of space accommodation. The sequence 5 was developed in two different phases, the first one is related with the latest expansion stage of the lake, (TST5), the basal track of this sequence. In this phase the base level of the lake rose considerably. The second phase (related to the TST5) indicates the end of the lacustrine domain in the Araripe Basin and the change to lagunar system ant tidal flat, with great portions in the supratidal. These systems were formed by restricted lagoons, with shallow level of water and with intermittent connections with the sea. This, was the phase when the Araripe Basin recorded the most several arid conditions of the whole interval studied, Aptian Albian, conditions that allow the formation of evaporitic deposits. The sequence 6 began its deposition after a significant fall of the sea (LST6). The sequence 6 is without any doubtlessly, the sequence that has deposits that prove the effective entrance of the sea into the Araripe Basin. The TST6, end of this sequence, represents the moment which the sea reaches its maximum level during the Aptian Albian time. The stratigraphic analysis of the Aptian Albian interval made possible the understanding that the main control in the development of the depositional sequences recognized in the Araripe Basin were the variations of the local base level, which are controlled itself by the climate changes

Interpretação de dados de GPR com base na hierarquização de superfícies limitantes e na adaptação de critérios sismoestratigráficos

Due to its high resolution, Ground Penetrating Radar (GPR) has been used to image subsurface sedimentary deposits. Because GPR and Seismic methods share some principles of image construction, the classic seismostratigraphic interpretation method has been also applied as an attempt to interpret GPR data. Nonetheless some advances in few particular contexts, the adaptations from seismic to GPR of seismostratigraphic tools and concepts unsuitable because the meaning given to the termination criteria in seismic stratigraphy do not represent the adequate geologic record in the GPR scale. Essentially, the open question relies in proposing a interpretation method for GPR data which allow not only relating product and sedimentary process in the GPR scale but also identifying or proposing depositional environments and correlating these results with the well known Sequence Stratigraphy cornerstones. The goal of this dissertation is to propose an interpretation methodology of GPR data able to perform this task at least for siliciclastic deposits. In order to do so, the proposed GPR interpretation method is based both on seismostratigraphic concepts and on the bounding surface hierarchy tool from Miall (1988). As consequence of this joint use, the results of GPR interpretation can be associated to the sedimentary facies in a genetic context, so that it is possible to: (i) individualize radar facies and correlate them to the sedimentary facies by using depositional models; (ii) characterize a given depositional system, and (iii) determine its stratigraphic framework highligthing how it evolved through geologic time. To illustrate its use the proposed methodology was applied in a GPR data set from Galos area which is part of the Galinhos spit, located in Rio Grande do Norte state, Northeastern Brazil. This spit presents high lateral sedimentary facies variation, containing in its sedimentary record from 4th to 6th cicles caused by high frequency sea level oscillation. The interpretation process was done throughout the following phases: (i) identification of a vertical facies succession, (ii) characterization of radar facies and its associated sedimentary products, (iii) recognition of the associated sedimentary process in a genetic context, and finally (iv) proposal of an evolutionay model for the Galinhos spit. This model proposes that the Galinhos spit is a barrier island constituted, from base to top, of the following sedimentary facies: tidal channel facies, tidal flat facies, shore facies, and aeolic facies (dunes). The tidal channel facies, in the base, is constituted of lateral accretion bars and filling deposits of the channels. The base facies is laterally truncated by the tidal flat facies. In the foreshore zone, the tidal flat facies is covered by the shore facies which is the register of a sea transgression. Finally, on the top of the stratigraphic column, aeolic dunes are deposited due to areal exposition caused by a sea regression

Beachrocks do Rio Grande Do norte: correlação entre os depósitos costeiros e os de zona costa-afora com base na faciologia, petrografia e diagênese

Beachrocks são rochas sedimentares formadas pela cimentação de sedimentos praiais por carbonato de cálcio em especial, calcita e/ou aragonita em zona de estirâncio. A ocorrência dessas rochas é bastante comum em diversas partes do globo, sobretudo em regiões com latitudes inferiores a 40º. O Rio Grande do Norte possui grande quantidade de beachrocks, os quais afloram tanto em regiões costeiras quanto em zona costa-afora. Os depósitos de beachrocks de zona costeira do referido estado têm sido estudados por diversos autores, os quais abordaram os mais variados temas desde o início do século XX. Por outro lado, os depósitos de zona costa-afora apesar de terem sido estudados por poucos autores têm ganhado atenção apenas nos últimos anos. Porém, nenhum trabalho até o presente fez algum tipo de estudo comparativo de cunho geológico entre os corpos de beachrocks presentes em ambas as zonas: costeira e costa-afora. Sendo assim, a presente dissertação teve o intuito de correlacionar os corpos de beachrocks que afloram em zona costeira aos que estão atualmente dispostos em zona costa-afora, próximo a isóbata de 25 m, levando em consideração seus aspectos petrográficos, diagenéticos e sedimentológicos. Para isso, foram percorridos cerca de 260 km de litoral, correspondendo ao trecho entre os municípios de Extremoz e Tibau, em busca de afloramentos de beachrocks. Seções colunares foram confeccionadas e amostras coletadas em estações de amostragem representativas da zona costeira, ao passo que da zona costa-afora apenas seções delgadas foram analisadas. Trabalhos disponíveis na literatura sobre o tema e área em pauta também foram utilizados. A partir dos dados levantados, observou-se que os beachrocks são formados por diferentes camadas ao longo de um perfil vertical. Estas camadas são claramente identificadas em afloramento pela diferença existente na composição, textura e estruturas sedimentares peculiares a cada uma delas. Seções delgadas foram confeccionadas e analisadas a partir de amostras coletadas nas diferentes camadas de diversos afloramentos. Um afloramento foi escolhido como afloramento modelo sendo este o de São Bento do Norte por apresentar a maior espessura de rocha aflorante (1,9 m). Este tem sido muito bem estudado tanto no corrente trabalho quanto em trabalhos de outros autores. A este foram comparados todos os outros afloramentos analisados. A partir da análise micropetrográfica, foram identificadas 03 microfácies para os beachrocks do Rio Grande do Norte, sendo elas: Quartzarenítica (< 2,9% de bioclastos), Quartzarenítica Bioclástica (entre 3 e 9,9% de bioclastos) e Bio-quartzarenítica (> 10% de bioclastos). Associando essas microfácies às análises sedimentológicas realizadas foi possível propor que as microfácies Quartzarenítica e Bio-quartzarenítica foram depositadas em zona de estirâncio enquanto que a microfácies Quartzarenítica Bioclástica foi depositada em zona de face litorânea superior. A história diagenética dos beachrocks estudados é marcada por quatro principais processos: compactação mecânica, cimentação, dissolução e geração de porosidade secundária, e oxidação. Dentre esses, o processo de cimentação é o mais importante, sendo caracterizado por precipitação de cimento de calcita rica em Mg sob cinco morfologias, a saber: cutículas criptocristalinas, franjas prismáticas isópacas, calcita espática microcristalina, calcita espática equante e agregados pseudo-peloidais. Todas estas morfologias foram formadas durante o estágio de eodiagênese, nas zonas freática marinha ativa ou freática meteórica ativa, corroborando assim com a idéia de que beachrocks têm sua litificação completa a pequenas profundidades. Associando as análises microfaciológicas às diagenéticas foi possível sugerir que a sucessão vertical de camadas vista em alguns beachrocks costeiros representam registros de variações de mais alta frequência do nível do mar durante o Holoceno. A partir daí, baseando-se em informações obtidas através de curvas de variação do nível do mar relativo no Holoceno para o Rio Grande do Norte, disponíveis na literatura, e na correlação aqui realizada entre os beachrocks costeiro e aqueles de zona costa-afora, foi possível inferir que estes últimos representam uma antiga linha de costa formada a idades relativas superiores a 7.000 anos A.P

O esgotamento sanitário e o uso do território em Natal/RN (1969 - 2009)

Natal é uma cidade bastante antiga, data desde o ano de 1599. Durante seus três primeiros séculos de existência ela praticamente não cresceu, porém teve um significativo aumento populacional e crescimento urbano na década de 1940, a partir da Segunda Guerra Mundial. Nos últimos anos, Natal tem passado por um intenso processo de urbanização, que exigiu da cidade um acompanhamento na sua infraestrutura, principalmente por ser uma cidade turística; o que aumenta bastante a sua demanda. O esgotamento sanitário entra como um dos problemas que Natal tem que resolver. A cidade possui apenas cerca de 32% do seu território servido por um sistema de coleta de esgotos. O serviço de esgotamento sanitário é de extrema importância para a manutenção da saúde e da qualidade de vida da população, assim como os demais serviços que compõem o saneamento básico. Tal serviço tem sido um grande desafio para a população e para os administradores públicos. Este trabalho tem como principal objetivo analisar os impactos socioespaciais causados pelas diferentes formas de esgotamento sanitário existentes em Natal. Durante a elaboração deste trabalho, foram realizadas pesquisas bibliográfica e de campo através de estudos documentais, visitas aos órgãos públicos, entrevistas, aplicação de formulários junto à população e observação direta. O embasamento teórico parte dos estudos de Milton Santos sobre território usado, adotando também, ao longo do desenvolvimento do trabalho, as contribuições de outros autores. O território usado foi escolhido como categoria de análise deste trabalho, por entendermos que as questões do esgotamento sanitário passam pelo uso do território, e este se dá pela sociedade como um todo