This work has the general objective of developing the Taylor series and Euler's equation to
obtain trigonometric transformations in comparison with the textbook of basic education and
application. The methodology is bibliographic in nature, based on authors (ÁVILA, 2008; B.
BOYER, 2012; GUIDORIZZI 2014; IEZZI, 1978, 2010 and 2016), STEWART, 2006) who
represent the basis for the development of formalisms with the use of derivatives included in
the Taylor series for expansion of transcendental functions (sine and cosine) in polynomial
expressions, resulting in Euler's equation. Other bibliographic references that support the
applications and history of trigonometry, such as (MORSCH, 2019; ANDRADE, 2019;
DIEDERICHSEN, 2023; MARQUES, 2023). These authors, among others inserted throughout
this work, consolidate the study for a better deepening of History, mathematical development
and applications. Regarding the results obtained, it appears that there are two ways of studying
trigonometry, the first through the Taylor and Euler series and the second through textbooks
that use the trigonometric cycle to obtain transformations and trigonometric tables. In this sense,
the work comes, as a result for teachers, to present a way of knowing the development of
trigonometry considering the derivatives, the Taylor series and the Euler equation, considering
that this formalism is not found in textbooks. The research is concluded considering that the
trigonometric cycle and Euler's equation present two different methodologies to develop
trigonometric transformations, the second of which is fully analytical, with the Taylor series
being crucial to truncate the functions and apply them in situations of everyday phenomena,
mainly with regard to trigonometric tables, using transcendental functions in the form of
polynomial functions

This research work addresses the new Poh-Shen Lo method to solve quadratic equations. The motivation was given by the restlessness arising from the classroom experiences, together with the debates with colleagues in the profession. The objective of this work is to draw a comparative study between the method of solving traditional quadratic equations and the method developed by the writer Poh-Shen Lo, in order to establish whether there are advantages in the didactic use of this second degree. To assist in this discussion, arguments from authors such as Boyer (2003), Fragoso (2000) and other scholars who outline opinions on the subject were sought, in addition to official documents such as the PCN (BRASIL, 1997; 1998). In the description of the bibliographic research, we tried to highlight the importance of history in mathematics classes, recalling some civilizations and mathematicians that contributed to the discovery of formulas and methods of practical solution. The content in question was developed using formulas to solve questions in algebra and we analyzed the time series of facts, identifying several men who are involved in the development of mathematics and to develop a practical way for them to have contributed to the development of such equations. As a methodology, in addition to the bibliographic research, this work was dedicated to the development of a case study in the classroom of the first year of high school in a school in Pará, so that lesson plans, practical mathematical tests and questionnaires were developed. with the objective of measuring not only the difference in the teaching effectiveness of the analyzed methods, but also the students' level of satisfaction in doing so. Finally, as a result, there was a considerable increase in the arithmetic mean of the class when taught using the Poh-Shen Lo method compared to the traditional one. There was also a considerable gain in students' self-esteem in relation to their mathematical knowledge and a greater generalized achievement.

Este trabalho tem como objetivo apresentar uma sequência didática sobre o estudo de Análise Combinatória, com o auxílio do jogo de xadrez como ferramenta educacional, visando melhorar o processo de ensino aprendizagem em matemática; a relevância deste trabalho está no fato deste conteúdo se fazer presente na grade curricular do Exame Nacional do Ensino Médio (ENEM), e os alunos apresentarem uma certa dificuldade em sua assimilação e até mesmo uma aversão a este conteúdo, por esta razão, aulas alternativas como a utilização de jogos se fazem necessárias, para tornar este momento mais atrativo e dinâmico melhorando a qualidade do ensino, e consequentemente obtendo uma resposta na aprendizagem. Além disso o conteúdo de Análise Combinatória tem a sua importância para a formação dos alunos, pois estuda vários formas e métodos que possibilitam a resolução de problemas referentes a contagem e auxilia nas tomadas de decisões. Para a realização deste trabalho foi utilizado a sequência didática de Dolz et al. (2004), e aplicada na turma da 2ª série do ensino médio da Escola de Educação Básica Agropalma, que está localizada na PA-150, km 74 próximo ao município de Tailândia, foi realizado duas avaliações durante este processo, a primeira ocorreu antes da aplicação dos módulos, com o objetivo de verificar o grau de entendimento da turma, em relação ao conteúdo de Análise Combinatória. Já a segunda avaliação ocorreu após o fechamento dos módulos com o intuito de verificar o desempenho dos alunos referente às duas avaliações, os dados obtidos foram coletados e registrados por meio de gráficos, também foi apresentado resultados de pesquisas em escolas que adotaram o jogo de xadrez em seu sistema pedagógico, mostrando que a utilização do jogo não somente traz benefícios para matemática, como também para as diversas áreas de conhecimento.

O presente trabalho tem como objetivo encorajar professores da Educação Básica a introduzir os conceitos básicos iniciais de Limites e Derivadas no terceiro ano do ensino médio, utilizando como ferramenta pedagógica o software Geogebra. Quer-se mostrar que é possível introduzir o estudo do cálculo na educação básica, e não apenas isso, que se pode fazer de uma maneira lúdica, mais ilustrativa e perceptiva, tornando assim esse estudo muito mais atraente para o aluno. Fazendo uso desse software, tem-se a intenção de tornar mais fácil e ágil não só a introdução dos conceitos, mas também a manipulação, interpretação e análise dos mesmos por parte do aluno. Tem-se a ideia de tornar o estudo do cálculo na educação básica um estudo com melhor visualidade e intuitivo do que formal, visando promover um primeiro contato do aluno com este conteúdo. Dessa forma, se acredita que ao se deparar com a disciplina de cálculo em algum curso de nível superior o aluno terá um melhor aproveitamento da disciplina.

The main objective of this work is to provide support material and also to deepen the discipline of Plane Geometry, more specifically in the study of problem solving involving areas of plane figures, and in this way contribute to Basic Education teachers and students who participate or intend to participate in Olympic competitions such as the Brazilian Mathematics Olympiad for Public and Private Schools – OBMEP, also serving as support material for students of the Professional Master's Degree in Mathematics – PROFMAT in the MA 13. This work will address unusual properties and theorems. in the curriculum of the subject of mathematics in Basic Education, more of great importance in the study of Plane Geometry and in the context of the current Olympics, such as the Theorems of Stewart, Ceva and Menelaus, which refer to remarkable segments in a triangle, and which are fundamental for the solution to many problems in this area.

This research work approaches the counting methods studied in high school. With an emphasis on bibliographical research in books and scientific articles, and based on the use of a activities sequence, it was possible to build a teaching proposal using problem solving that seeks to value logical thinking, the mounting of strategies, rather than by the use of standard examples solved mechanically. The motivation for proposing this approach arose from concerns derived from the teaching experience in basic education and from the knowledge acquired in the Professional Master's Degree in Mathematics in the National Network (PROFMAT), which has enabled a meaningful teaching vision from the dialogue with other works of textbooks that are used daily at schools. In this sense, we seek to investigate and understand the counting methods, starting from a proposal that directs the use of problems, stimulating the ingenuity and understanding of the situation described. The propose addressed favors the use of the Basic Principle of Counting (BPC), with which problems can be solved regardless of the type of grouping involved (arrangement, permutation and combination).Thus, four sequences of activities are presented, along with  methodological proposals for teaching and learning of counting methods. The construction occurred in line with the guidelines of the BNCC and with the contributions of analyzes of the works of Hazan, Dante and Morgado. The research was concluded taking into account that the approach of counting methods based on problem solving enables significant learning.

This work aims to verify whether textbooks satisfactorily address the Incognita in the knowledge and resolution of 1st Degree Equations. Thus, it seeks to analyze how textbooks approach the study of Equations of the 1st Degree, especially the Incognita, in the seventh year of Elementary School, according to the dimensions of Algebra presented by the PCN and the competencies and skills proposed by the BNCC for the that year. Thus, initially, a historical approach was made about Algebra and the emergence of Equations, based on the works of several authors. This work also reinforces the importance of using letters as Incognita for learning Equations, and highlights the student's need to understand the algebraic language as well as the meaning of Incognita for a better knowledge and resolution of 1st Degree Equations. For this purpose, textbooks from the seventh year of elementary school were analyzed in order to check whether they satisfactorily contributed to the algebraic development of the students.

This work addresses the study of the complex number and application in the area of geometry, with the aim of showing that such knowledge can be relevant in basic education. Historically, the development of complex numbers took about 300 years to formalize as a set of complex numbers where it was possible to obtain a solution for equations of the type and another with similar characteristics. The solutions of this type of equation were described by Euler as being of the type and , thus enabling the construction of the set of complex numbers, where $ i $ is called an imaginary unit of the number $ z $. However, this set came to be better understood when K. F. Gauss and Jean-Robert Argand independently discovered that Z could be represented geometrically. It is a fact that the number z is not a priority in basic education in Brazil, largely because of the new tendency of content flexibility that appears less and less in the entrance exams and is not required in the National High School Exam. However, it is important that the elementary school student knows the advantage of using the set to better understand non-real 2nd degree equation roots. The formalism of set z is treated in undergraduate courses with various applications in the areas of mathematics, physics, engineering and etc. Although not useful in basic education, it is necessary to approach the concept of the number z as an algebraic operator responsible for rotating a vector 90º. From this approach it was possible to apply some problems in the area of geometry, such as in the lunar movement (application of the 2nd Moivre formula), angle between lines, cube root of a non-real number, systems solutions to two variables involving complex numbers with emphasis on plane geometry. In this sense, there is a great advantage in considering the applicability of complex numbers in other areas of knowledge and due to this advantage, it could be part of the content at the level of basic education. Thus, taking into account the algebraic and geometric aspects in the solution of several problems in which it is necessary to know the set z, the teacher would undoubtedly have one more resource to address some contents that are intrinsic in basic education and that requires looking for methods that facilitate learning. It is concluded with the research that the set of complex numbers has the advantages of expanding the learning using the concepts of complex number as an algebraic operator and with the advantage in applications in geometry and other segments of scientific knowledge.

The objective of this work was “To propose, through Active Methodologies, the teaching of Constant Amortization Systems - (SAC) and French Amortization System - (SAF) using the application“ Capitalizacao ”, created on the block programming platform MIT App Inventor” aimed - there is a need for better financial education for the Brazilian population linked to a teaching strategy that involves new methodological trends. In this sense, it was guided in this dissertation to make a proposal involving hybrid teaching and meaningful learning using the technological tool, the “capitalization” application. Therefore, it was necessary to seek previous studies with methodological experiments using new trends for the teaching of financial mathematics, thus the reading of dissertations related to financial mathematics and the use of technological tools in the teaching methodology was carried out. It was observed in the literature abstracts present in this dissertation that the methodological proposals are similar to the objective chosen in this work. Still as information, research was done on financial education seeking to give meaning to the teaching of the formal content of financial mathematics which was presented and used for the elaboration and structuring of four activities for the teaching of financial mathematics, emphasizing the SAC and SAF. Thus, based on the information collected in the aforementioned research, a comparison was made with the proposal chosen in this dissertation, where it was found that the methodological proposals of the analyzed dissertations show similarities to the objective of this work. This fact provided an indication of the proposal's validity. carried out in this dissertation which must be applied later in the face-to-face.

Mathematics has always been present in people's daily lives, being, therefore, a fundamental instrument for understanding the world and the phenomena surrounding it, in addition to contributing to intellectual development and the formation of citizens conscious and capable of understand their reality. But in the contemporary world, a problem has been identified that involves learning mathematics by students. It is in this context that this work aims to present the applicability of Cardano's formula to high school students, through an algebraic method for resolutions of third-degree polynomial equations taking as a universe the Set of Complex Numbers. In the methodological scope, the work consisted of a qualitative research, whose analysis was based more on the interpretation of information and theoretical discussions than on statistical indicators. A bibliographic survey was also conducted through readings, which reflect on the theme in question. The study pointed to the fact that Cardano's method is an additional instrument in the resolutions of cubic equations, presenting a different strategy to solve certain problems involving cubic equations, which consequently, it can insert the student to study mathematics through the search for historical facts, also contributing to the formation of more qualified citizens and aware of their role in society.