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

Equações de segundo grau

Apresenta a revisão de tópicos de matemática elementar do ensino fundamental com visão do ensino superior. Na subunidade 5 são abordados os seguintes tópicos: resolução de Equações do Segundo Grau com exemplo de cálculo, estudo das Raízes da Equação de Segundo Grau, resolução de Equações Biquadradas com exemplo de cálculo e Equações Irracionais com exemplo de cálculo.

Dificuldades na resolução de equações de 2º grau dos alunos do 8º ano

Relatório da Prática de Ensino Supervisionada, Ciências da Educação (Mestrado em Ensino da Matemática), Universidade de Lisboa, Instituto de Educação, 2014

Utilizando processos geométricos da história da matemática para o ensino de equações do 2º grau

The present work had as principal objective to analyze the, 9th grade students understanding about the solutions of an equation of the 2° degree, using geometric processes of the History of the Mathematics. To do so, the research had as base the elaboration and application of a group of teaching activities, based on Jean Piaget's construtivism. The research consisted of a methodological intervention, that has as subjects the students of a group of 9th grade of the State School José Martins de Vasconcelos, located in the municipal district of Mossoró, Rio Grande do Norte. The intervention was divided in three stages: application of an initial evaluation; development of activities‟ module with emphasis in constructive teaching; and the application of the final evaluation. The data presented in the initial evaluation revealed a low level of the students' understanding with relationship to the calculation of areas of rectangles, resolution of equations of the 1st and 2nd degrees, and they were to subsidize the elaboration of the teaching module. The data collected in the initial evaluation were commented and presented under descriptive statistics form. The results of the final evaluation were analyzed under the qualitative point of view, based on Richard Skemp's theory on the understanding of mathematical concepts. The general results showed a qualitative increase with relationship to the students' understanding on the mathematical concepts approached in the intervention. Such results indicate that a methodology using the previous student‟s knowledge and the development of teaching activities, learning in the construtivist theory, make possible an understanding on the part of the students concerning the thematic proposal

Equações de 1.º grau : estratégias e erros na resolução e simplificação de equações de 1.º grau

Relatório da prática de ensino supervisionada, Mestrado em Ensino da Matemática, Universidade de Lisboa, 2011

Equações de primeiro grau

Trata-se da revisão de tópicos de matemática elementar do ensino fundamental com visão do ensino superior. Na subunidade 3 são abordados conceitos de cálculo algébrico, conjunto universo e conjunto solução de uma equação, equações do primeiro grau e inequações do primeiro grau com resolução de problemas. A subunidade 4 engloba a definição dos conceitos de monômios ou termos algébricos e polinômios e suas propriedades. Como complemento a teoria abordada apresenta exemplos de cálculo do mmc de polinômios e de equações fracionárias de primeiro grau com uma incógnita.

A legitimidade da jurisdição de primeiro grau na concretização dos direitos fundamentais: proposta de um novo modelo jurídico-administrativo para o segundo grau de jurisdição

The current study is about the legitimacy of lower court jurisdiction as a way of exercising basic legal rights, proposing, therefore, a new legal-administrative model for appellate court. In order to achieve that, a demonstration of the importance of basic legal rights in the Brazilian legal system and an open interpretation in light of the Constitution, as a way to affirm said rights, among which are accessibility to the justice system and proper legal protection, is required. As a result, the legitimacy to access the legal system resides in the Constitution, where the interpreter should seek its basic principles to achieve basic legal rights. It is observed that the lack of credibility regarding lower court decisions comes from the dogmatic view of truth born from power, and therefore, that the truth resides in decisions from appellate court and not from lower court judges. A lower court judge holds a privileged position in providing basic legal rights for citizens, considering his close contact to the parties, the facts, and the evidences brought forth. Class action suit is presented as an important instrument able to lead the lower court judge to provide basic legal rights. Small Claims Courts may be used as paradigm to the creation of Appellate State Courts formed by lower court judges, reserving to higher jurisdiction courts and Federal Circuit Courts, the decisions of original competency and the management and institutional representation of the judiciary system. Instilling an internal democratization of the judiciary is also required, which means the participation of lower court judges in electing their peers to chief positions in the court system, as well as establishing a limited mandate to higher court judges.

Função quadrática

In general, the study of quadratic functions is based on an excessive amount formulas, all content is approached without justification. Here is the quadratic function and its properties from problems involving quadratic equations and the technique of completing the square. Based on the definitions we will show that the graph of the quadratic function is the parabola and finished our studies finding that several properties of the function can be read from the simple observation of your chart. Thus, we built the whole matter justifying each step, abandoning the use of decorated formulas and valuing the reasoning

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

Técnicas de parametrização para o fluxo de carga continuado desenvolvidas a partir da análise das trajetórias de soluções do fluxo de carga

Pós-graduação em Engenharia Elétrica - FEIS

O tratamento ortodôntico em uma perspectiva a longo prazo: avaliação da estabilidade oclusal e do grau de satisfação dos pacientes

This study aimed to evaluate factors associated to orthodontic treatment stability and patient satisfaction in the long-term. A total of 209 patients (88 class I and 121 class II) treated with straight wire fixed appliance were selected at least 5 years post treatment. Six hundred twenty seven dental casts were examined with the PAR Index at pretreatment (T1), end of treatment (T2), and at long-term follow up (T3, mean 8.5 years post treatment). At T3, a Dental Impact on Daily Living questionnaire was used to assess patient satisfaction with the dentition in the long-term. Friedman test and multiple regression analysis were used to evaluate changes among the time points and factors associated with stability and patient satisfaction. Predictive factors used to exam the occlusion were: PAR Index at T1 and T2, age at T1, the amount of time without retainer, length of Hawley retainer wear, length of follow-up, sex, extraction and third molar status. To assess patient satisfaction were considered: changes produced by the orthodontic treatment (PAR T2-T1), post treatment stability (PAR T3), age at the start of treatment (T1), length of treatment (T2-T1), gender, and extraction. Orthodontic treatment produced a significant improvement of 94.2% in the PAR Index (T2-T1), but this change was not associated with the level of satisfaction when the patient was questioned at T3. No significant change was observed between T2 and T3. However, when the sample was divided according to the level of finalization (PAR T2), it was observed that well-finished patients experienced some deterioration (P<.001), whereas the less well-finished ones showed some improvement (P<.05). Even with the deterioration, the well-finished patients still had a better PAR Index at T3 compared to the less well-finished ones (PAR T2- T3). Regression analysis showed that PAR Index at T1 and T2, age at T1, and length of retainer wear had a slight association with occlusal stability (R2 = 0.27). Patient satisfaction was significantly associated only with PAR Index at T3 (r2=0.125, P<.0001). We can conclude that, even thought orthodontic treatment is quite stable, not so well-finished treatments tend to show some improvement and well-finished ones deteriorate some in the long-term. Despite of that, well-finished patients still have better occlusal characteristics. Patient satisfaction is not related to the result of orthodontic treatment; nevertheless, there is a slight association with dentition in the long-term

Agrupamento de países segundo indicadores de padrão de vida

Foram estudados 125 países avaliados por um conjunto de 26 indicadores básicos, de saúde, econômicos e educacionais, usando-se três métodos estatísticos multivariados: Análise de Agrupamento, Análise de Componentes Principais e Análise de Variância Multivariada. As variáveis mais discriminatórias foram a expectativa de vida, as taxas de mortalidade infantil e de menores de cinco anos, as taxas de natalidade e de fertilidade e a taxa de matrícula no segundo grau para o sexo feminino. Os países foram ordenados de acordo com um índice de padrão de vida e separados em cinco grupos.

GeoGebra e o método de Briot & Bouquet para a resolução gráfica de equações cúbicas

In the mid-nineteenth century, french mathematicians Briot and Bouquet have proposed an intriguing graphical method for solving cubic equations "depressed" - the third degree equations that do not have the quadratic term. The proposal is simple geometric construction, though based on an ingenious algebra. We propose here the verification and testing graphical method through an instructional sequence using the software GeoGebra also present the ingenious algebraic development that resulted in this graphic method for determination of real roots of a cubic equation of the type x³ + px + q = 0 where p and q are real numbers. The method states that these solutions are summarized in the abscissas of the points of intersection of the circumference containing the origin and the center C (-q/2, 1-p/2) with the parable y = x².