This work initially presents a systematic literature review of theses and dissertations that deal with the use of graph theory in mathematics classes in schools. The research was carried out in the BDTD databases (Brazilian Digital Library of Theses and Dissertations) and in the PROFMAT collection (Professional Master's Program in Mathematics on a National Network), from 2013 to 2023, in Portuguese, seeking to understand how Graph theory has been used in schools and what are the purposes of using this theory. The selected studies show that there are countless applications and everyday problems that can be addressed in the classroom, relating mathematics to graphs, thus contributing to student learning. However, some gaps were found in these researched works, such as the absence of computational thinking in graph applications. Therefore, the present work proposes an integration between computational thinking and graph theory in the context of mathematics classes in basic education. By presenting fundamental definitions of these areas and suggesting specific activities, it is expected that the teacher can develop computational and logical skills in students, in addition to promoting their autonomy. The tools explored from graph theory include Dijkstra's algorithm, Eulerian cycle, directed graph and semi-Eulerian cycle. In relation to computational thinking, the set of skills from the Computational Thinking in K-12 Education Leadership toolkit was used. Finally, other suggestions for applications aimed at schools are given using new tools and variations of the proposed activities.

This work is based entirely on a scientific article with the same title, written by the authors Daniele Alves Souza, Valdinês Leite de Sousa Júnior and Erica Boizan Batista. In it, we present an approach to using geometric magic squares as an innovative pedagogical tool in mathematics education. The aim is to explore the characteristics of geometric magic squares of order 3, investigating their history, properties, and, most importantly, how these squares can be introduced into elementary classrooms to promote meaningful and engaging learning of mathematical concepts. By replacing traditional numbers with geometric shapes, geometric magic squares offer a unique opportunity for students to explore visual patterns and develop mathematical skills while having fun. Additionally, a pedagogical practice is proposed that includes the construction, analysis, and solving of problems related to geometric magic squares, stimulating critical thinking and teamwork. This interdisciplinary approach allows for the integration of concepts from geometry, algebra, and arithmetic, providing a holistic and enjoyable learning experience for students.

Assessment is of great importance for verifying the quality promoted in the teaching and learning process. Therefore, external evaluations aim to evaluate the pedagogical work of schools and generate public policies. As a result, schools feel the need to coordinate in order to prepare their students for these exams, the vast majority of which are Portuguese language and mathematics subjects. However, the form of preparation used can negatively affect the meaningful learning of knowledge objects present in the curriculum of these subjects. Therefore, this work aims to provide a theoretical overview based on other researchers on the topic and discuss the possible implications that external assessment policies can have on basic education and especially in the subject of mathematics. As a consequence, some notes will be made on the precautions that teachers and education managers should take when deciding to include preparation for these assessments in the classroom routine.

The research developed in this work sought to answer the following question: has the relationship between gender equity and the teaching of Mathematics in Basic Education been the focus of investigations in academia? To seek answers, a survey of the main research on the topic was carried out in order to obtain a general overview of what has been addressed. The objective was to establish an overview of the main national research published in dissertation or thesis format, between 2013 and 2023, related to the teaching of Mathematics and the issue of gender in Basic Education. For this, the Systematic Literature Review technique was used, a qualitative research of an exploratory nature, with searches on prestigious online platforms. It was found that the amount of research that met the defined criteria was quite small, highlighting the need to expand the discussion on this topic in academia. Furthermore, the selected research indicates that the use of practices to promote gender equity in Mathematics classes has a positive impact on students' self-esteem and empowerment. As a complement, a guide was created with tips and suggestions to help promote gender equality in Mathematics teaching, aimed at educators who wish to reduce gender disparity in their classes.

The present work, titled “Challenges and Counting Techniques in Mathematics Teacher Selection Exams for Federal Institutes,” explores the importance and usage of counting problems in public selection exams for teaching positions at Federal Institutes (IFs). The research investigates recent exams from these competitions, highlighting the frequency and types of counting problems, such as permutations and combinations, that are commonly addressed. The study also covers Pascal’s Triangle and Newton’s Binomial, fundamental tools in high school mathematics, analyzing their properties and applications. Additionally, it includes combinatorial and algebraic demonstrations of some properties of Pascal’s Triangle and presents practical applications of these properties in the classroom. Moreover, the work proposes solutions to counting problems that have been featured in exams for mathematics teaching positions at IFs. The dissertation is structured into five chapters, ranging from basic concepts of counting methods to more in-depth discussions on the use of these concepts in educational contexts. The ultimate goal is to provide useful study material for candidates preparing for these exams, emphasizing the relevance and complexity of counting problems in teacher selection. Thus, the work not only contributes to the technical knowledge of the field but also offers practical support for the development of teaching and learning strategies focused on combinatorics.

With the technological advancements of the past decades, society has undergone profound changes in various sectors, such as communication, leisure, health, work, among others. Education is not immune to this process.Therefore, it is crucial to adapt teaching methodologies to this new reality, accompanying the demands and possibilities provided by technology. In the context of mathematics, various applications are already used as didactic resources. However, it is observed there is little emphasis on the field of statistics. This work aimed to develop automated spreadsheets using the VBA programming language for allow dynamic and interactive exploration of key topics in Statistics covered in high school. The main achievements of this work are four spreadsheets, which cover subjects such as frequency tables, graphs, measures of central tendency and dispersion, entirely based on the Statistics curriculum of the National Common Core Standards and the Reference Matrix of the ENEM (Brazilian National High School Exam). The spreadsheets showcased their functionalities through practical examples of activities that can be conducted in classroom.

This work is based entirely on a scientific article with the same title, written by the authors Damião Jeferson de Araújo Souza, Valdinês Leite de Sousa Júnior and Erica Boizan Batista. In it we present an approach for teaching some properties of plane Euclidean geometry, using concepts and properties of analytic geometry. The authors adopt a method known as barycentric coordinates, introduced by the mathematician Möbius in 1827, which allows solving problems and demonstrating theorems of conventional geometry using area concepts, instead of similarity and congruence of triangles, which are commonly used in basic education in Brazil . Some classic theorems are presented, such as the Inner Bisector Theorem and Ceva’s Theorem, both proved using a purely analytical approach. In addition to exemplifying the use of the method, the work also presents solved problems to help students better understand the concepts presented.

Access to quality education is still a very distant reality for a large part of 

the people around the world. For this very reason, much can be done to 

ensure the objectification of this right. Open Education proposes to this, 

offering in addition to other instruments, the Open and Massive Online 

Courses – MOOC. The present work is the production of a course of this

 nature, which has as its theme the Introduction to the Study of the Parable 

through the Quadratic Function. The course offers slides, video-lessons 

and handouts that will provide the student to appropriate this content that 

is studied in Basic Education, is required in competitions, and is also a 

prerequisite for the disciplines of Calculus of Higher Education. It will be

available on the site https:/poca.ufscar.br/, and the goal is to give as many 

people as possible access and appropriation of this content. 

Mathematics plays a fundamental role in our daily lives, even though it often goes unnoticed. In recent years, communication through the internet, whether via cell phones, computers, or other devices, has become commonplace. The messages exchanged by users are sent encrypted, ensuring that the content cannot be read if someone tries to intercept the communication. This process is more discreet nowadays. However, this practice has existed for a long time, dating back to the era of kings and queens, many years before Christ. During that time, kings communicated with their distant allies through messages sent by messengers. Those who managed to keep their messages solely between the sender and the recipient, using secret writing, could keep their kingdoms intact even if their messages were intercepted. On the other hand, those who used weak secret writing allowed their enemies to obtain valuable information contained in the messages, which could lead to the ruin of their kingdoms. Mathematics played a crucial role in deciphering these encrypted messages in the past and has become essential in modern cryptography. This work addresses the history of cryptography, with various methods and the use of mathematics to decipher and encrypt messages. The end product is activities that can be used in the classroom to make the application of mathematics more relevant and engaging for students, demonstrating the importance of mathematics in ensuring secure communication.

In this study, we objectively discuss two Active Methodologies applied to probability

teaching, the Flipped Classroom and Game-Based Learning, motivating teachers to

apply them concomitantly with the Traditional Method in their classes. In addition,

we present three didactic sequences with the digital tools Mathematical Multimedia

Wordwall and Kahoot that, combined with the Active Methodologies mentioned

above, can serve as support to stimulate learning and combat student demotivation. 

In each sequence, we provide step-by-step instructions on how to access and/or

build each tool, facilitating the practice for the reader. At the end of this work, we

approach the importance of continuing teacher education and the relevance of this

TCC for the teaching community of the state network of Ceará through the Foco

na Aprendizagem initiative with an experience report.

The female presence in mathematics has historically been marginalized and underrepresented. In this context, this paper presents a historical review of how women have gained access to education, and more precisely to mathematics teaching, highlighting the participation of some mathematics in important scientific contributions. In addition, a survey is made of the main scientific awards in the field of mathematics and of the women who managed to win some of them. Also highlighted are some programs, projects and initiatives that promote participation and female recognition in the field of mathematics in Brazil. The research is based on quantitative data provided by the CNPq promotion agencies, Capes, INEP and the Department of Education of the State of Ceará through the BR Speech.Finally, a workshop on math women is presented within the scope of the UFCA extension project, Women Scientists in the Cariri (MC ²), at the José Bezerra Menezes High School in Juazeiro do Norte-Ce.

The present work aims to study and expand the properties of Archimedes’ Arbelos and the Pappus

Chain. An Arbelo is a structure composed of three semicircles that are pairwise tangent and have

the diameters on the same line. Such a structure was studied by the mathematician Archimedes,

who found several interesting geometric properties. Initially we will have an introduction

followed by a brief historical account of the life and work of Archimedes of Syracuse and Pappus

of Alexandria. In the preliminary concepts, we present configurations and results about circles

and ellipses that will be of extreme importance for the good understanding of this work. In

the section dedicated to the results, we present an extension of two properties of the arbelos to

the ellipses, provided that the ellipses involved have the same eccentricity. We prove Pappus’ 

Theorem using only plane geometry and analytic geometry, which differs substantially from

what is traditionally done: construction using the inversion technique, normally not seen in High

School. Finally, as an educational product, we present a didactic sequence devoted for teaching 

of concept of circles and ellipses, such activities can be applied to all levels of high school.

Astronomy is the natural science that studies celestial bodies and the phenomena that occur in
them. As in other sciences, for example in Mathematics, in Astronomy, a certain distancing from
the daily lives of students is common. The knowledge and practices in Astronomy originated
from the first civilizations, passing through several stages of evolution, with that there was an
approach to other areas, especially Mathematics. For the understanding of astronomical aspects
such as distances and relative sizes of stars, as well as the orbits of the planets, scholars of
antiquity already used mathematical results, especially of Geometry as the similarity of triangles
and ellipse properties. Given the relationship between the two sciences, we seek in this work
to highlight mathematical aspects that influenced the evolution of Astronomy. In this research,
we present as an educational product a didactic sequence based on Didactic Engineering, the
suggested activities can be applied in the classroom as a way to foster interdisciplinarity and
students’ curiosity both in Mathematics, as in Astronomy. Lastly, we hope that the work will
contribute positively to the teaching of Mathematics and Science, in order to make the lessons
more attractive and participatory for students.

The continuous advance of society imposes constant changes on all sectors, demanding at
all times that the forms of work accompany this progress. In the educational field, this is
even more evident, since contact with new generations of students happens practically
instantly. Given this fact, it is necessary to search for new teaching methodologies that fit
this new profile and support the good development of education, as well as technologies
that can help in the application of these methodologies. This work aims to evaluate the
scope of the Gamification methodology in mathematics teaching in Brazil and identify
the most commonly used technological tools for the application of this methodology.
The methodological approach used was the Systematic Literature Review, having as a
search interval the years from 2016 to 2021. The results obtained indicate that the use of
Gamification as a teaching strategy was quite satisfactory when it was desired to improve
student engagement in carrying out the proposed tasks. In addition, it was possible to
observe that currently there are few works in the area aimed at high school students, which
motivated the creation of a proposal for a gamified activity aimed at this audience.

This work aims to explore the domino game as a didactic resource for the learning of probability
and statistics in high school mathematics. For that, a bibliographic research was carried out in
academic and didactic works involving concepts of probability, statistics and the domino game.
The main contribution of this research is to provide a didactic product that helps high school
mathematics teachers to reflect and innovate their teaching practice in the classroom in teaching
the concepts of probability and statistics. The proposed didactic product is to explore the domino
game to help understand the mathematical issues proposed here through problem situations that
use domino pieces and analysis of domino games between two players. It is hoped that this work
promotes dialogue in the scientific and school community about the importance of rethinking
engaging, captivating teaching practices that demystify the look of mathematics as a discipline
that is difficult to understand and disconnects from other areas of knowledge and life.

The conditions inherent to the contemporary technological world have demanded from citizens
more and more knowledge and capacities for abstraction and analysis. All this raises the
importance of Mathematics Education in general and, more specifically, of Statistics Education.
The latter enables the individual to make better decisions in situations involving uncertainty.
Among the academic mechanisms adopted to encourage mathematics education, the Mathematics
Olympiads stand out. They originated in Hungary in 1894 and in Brazil in 1979. In particular,
the Brazilian Public School Mathematics Olympiad (OBMEP) was created in 2005 with the
aim of encouraging the study of mathematics and identifying talents in the area. It is a national
project whose target audience is basic education students, from the 6th year of elementary school
onwards. In its current format, OBMEP is aimed at Brazilian public and private schools. In order
to seek evidence of the importance attributed to Statistical Education in the OBMEP exams, this
work seeks to analyze issues related to Probability and Statistics. Although there are already
works with a similar theme, they do not bring the teaching of probability and statistics from the
perspective of the skills to be developed together with the students and that are included in the
National Common Curricular Base (BNCC) for the solution of these problems, as well as they
do not discuss a path solution, from definitions, axioms and theorems. The questions analyzed
were those of the 1st and 2nd stage tests from 2005 to 2019, establishing a parallel with the
basic education content necessary for the student to solve the issue at hand. The contributions
in relation to the existing literature are then: (i) the emphasis on the perspective brought by the
BNCC (with regard to skills related to Probability and Statistics), (ii) an analytical treatment
(from the frequency with which each one of them is required in the OBMEP) and (iii) systematic
(from the discussion of a generic way of developing solutions), allowing to glimpse points of
improvement. The methodology was based on bibliographic and documental research, of a
qualitative and quantitative nature, with a descriptive study.

Mathematics learning can be developed using several different methodologies. Playfulness
is an increasingly important concept in the student’s learning process, including
in mathematics. After observations in the classroom, it was noticeable that
one of the difficulties in learning mathematics is due to the non-absorption of basic
knowledge, such as basic arithmetic. One of the ways to develop the learning of
basic arithmetic is through repetition and this is often applied through the multiplication
table, but the traditional teaching methodology is not at all attractive to
students, especially in the digital world we live in, which may not be efficient for all
students. Thus, one way to approach teaching through repetition is to implement
playfulness in the process. The gamification strategy, contained in games in general,
is one of several forms of playfulness and is well received by most students. In this
work, a digital game capable of assisting in the process of learning basic arithmetic
was developed. The game was developed using the Unity 3D platform. This game
addresses the methodology of learning by repetition, to, in a playful way, establish
basic knowledge in the development of learning in mathematics. This work presents
the traditional teaching methodology and the practice of repetition and memorization
as part of the student’s teaching and learning process. In addition to showing
the concepts of gamification, what digital and educational games are, the importance
of arithmetic for learning mathematics and the use of multiplication tables as
an indispensable tool in the construction of knowledge. The characteristics of the
game will be presented, how the game was designed and which methodologies and
pedagogical techniques are being addressed in the use of the application. A strategy
for applying the application will also be suggested and, finally, an overview of the
information presented in the work and the expected results in the implementation
of the product created as the object of study presented in this work will be made, in
addition to future improvements and implementations noted in the course of building
the product. After development, the application was published on the "Play
Store" for the appreciation of all users of the Android operating system, where it
is expected that it will be well accepted, especially by the student community, and
can truly contribute to the construction of knowledge base, needed in mathematics.
This information may be observed in future research

In this work we present a contextualized approach to Combinatorial Analysis, with the

objective of explaining this matter in a practical and objective way, through the explora-
tion of problems. For that, we resort to the detailed resolution of problems extracted from

competitions, entrance exams and books, and through them, we cover the most important
topics of the combinatorics taught to us at school. The text is structured in three chapters.
In the rst one, we introduce the most basic topics, such as the Fundamental Principle
of Counting, Permutations, Arrangements, Combinations and others. In the second, we
deal with more technical topics, such as the Pigeonhole Principle, Newton's Binomial,
Pascal's Triangle, Principle of Inclusion and Exclusion and Other Forms of Counting.
In the third and nal chapter, we provide the reader a list, with answer key, containing
ninety problems extracted from competitions carried out by several exemination boards.
Throughout the rst two chapters, we have thoroughly resolved several questions extrac-
ted from competitions, entrance exams and books. As the reader evolves in the text, the
problems become more interesting and demand more and more of him/her; however, the
structure remains independent of each problem, exempting the reader from the need to
return to previous pages to review another problem that serves as a basis for what is
being analyzed. Everything to be useful as study material for students of competitions
and entrance exams or as a supporting text for teachers of the high school students use
in their classes. In full, the work leads to the conclusion that combinatorics learning is
only attainable through involvement and hard work, that is, putting into practice what
is being studied.

This paper is result of an extensive analysis, that took into account 100 of compe-
tition test for mathematics educator, distributed in 13 contest institution. Bearing

in mind that civil service exam is the means of access to the teaching career, the
objective of the qualification in contest incited the need for a material like this, that
would offer a mathematical support for candidates who want to apply for a teacher
vacancy. This dissertation makes a study about functions, covering the general

properties, affine function, quadratic function, modular function, exponential func-
tion, logarithmic function, trigonometric function and basic notions of differential

and integral calculations the maintaining the focus in the civil service exam. The
methodology used is based on theoretical aspects followed by the application of the

theory found in the selective process. For this second part, 147 contest questions an-
swered involving the topic of functions. Some information about the analysis of the

evidence was presented, such as data from the recurring questions, and discussion
about the difficulty level of some institution contest. Therefore, having done the
study of functions in the context of public tenders, this work is able to contribute
to the preparation of students.
Key words: functions, Civil service exam, application.

Sometimes it is difficult to illustrate the importance of mathematics to high school
students, especially when dealing with fundamental concepts, used only to estab-
lish a certain mathematical theory. The lack of clarity when trying to relate a
mathematical theory with its applications usualy discourages students, who end
up losing interest on relevant topics to our daily lives, such as mathematical logic.
This gets worse when the teacher decides to adopt traditional teaching approaches,
considered tedious by students overall. A much more promising strategy employs
computer softwares that allow to construct dynamic simulations of mathematical
problems in a very intuitive way. Still, this does not allow students to have concrete
experiences, so important during the whole teaching-learning process. Besides stim-
ulating students’ curiosity, engagement, interest in science, autonomy and the ability
to communicate and express their ideas, such concrete activities allow to establish
a direct link with the real world. In this context, it is proposed a didactic sequence
for teaching mathematical logic fundamentals through practices where students can
construct and debug their own digital logic circuits. The sequence is divided into
three modules, sorted in ascending order of difficulty. This work offers to the high
school teacher a complementary teaching proposal that promotes active teaching
and evoke student skills of observation, investigation, reflection e desision making,
in accordance with the basic principles of the Brazilian common core state stan-
dards. It emphasizes that the field of Mathematics behaves like a lively science that
contributes to solve real world problems and forms the foundation of humanity’s
most influential discoveries.

This work presents an approach on the Jacobsthal Sequence, its properties and applications. In this sense, solving a problem with essential issues, such as the Infinite Induction Principle, Sequences, Recurrences and Matrices. That done, we formally define a Jacobsthal Sequence to later show some types of generalizations of that sequence. Next, we present the Jacobsthal Matrix and show a specific generalization that allows us to relate to the Jacobsthal Numbers using a matrix that we denote by Fs. Finally, we present a
didactic sequence proposal based on the Zabala Sequence for the study of matrices using the Jacobsthal matrices.

The National High School Exam − ENEM − is currently the main means of access to
Higher Education in the country. Therefore, students in the final years of high school, or
when completing it, who intend to enter Higher Education, need to dedicate themselves to
preparing for the ENEM tests, as they have levels of knowledge considered low, according
to data from the ENEM itself at each year and other assessments adopted by the Govern-
ment. Given this and thinking about contributing to the studies of those who aspire to
a place in Higher Education, this work presents some basic knowledge of Flat Geometry.
First, an analysis of the ENEM Reference Matrix and the tests carried out, from 2009
to 2019, was carried out, where we sought to outline the profile of Geometry questions.
This analysis is presented in chapter 2 and shows that of the mathematical questions that
correspond to a quarter of the tests, 32 % of them are charged with geometric knowledge,
and that the part of Geometry, charged in the tests, is very comprehensive. Thus, trying
to establish a didactic sequence, we present some basic notions of Geometry, fundamental
concepts of angles and a study of the triangle. With this, we hope to provide the reader
with material that contemplates geometric contents addressed in a considerable part of
the ENEM exam questions and that serves as a basis for the study of subsequent Ge-
ometry subjects, also collected in these exams. The theoretical part of the contents is
presented in chapter 3, with definitions and important results followed by some examples
and we end the chapter with a section of questions, collected from ENEM.

This dissertation aims to describe the maximization of areas with minimization of perime-
ters in Euclidean geometry. We sought to answer five problems of area maximization and

perimeter minimization in triangles and convex polygons, reaching a proof of the isoperi-
metric inequality for polygons. The results contained herein were extracted from articles

[5] and [8]. Aiming to enrich such approaches and ensure a better understanding of what
is presented, we will take a brief approach to Euclidean geometry, seeking to visualize and
understand basic properties of triangles, polygons and circumferences as well as notions
of geometric demonstrations, using results of several authors of great reference in these
subjects . In the end, we worked out some classical problems of maxima and minima in
Euclidean geometry.

This paper presents an approach on the Padovan Sequence, its properties and applications. In this sense, the initial chapters deal primarily with the study of numerical sequences and calculation of recurrences, in a generalized way. Were highlighted some relevant sequences, in order to bring familiarity with the proposed theme, namely: Fibonacci, Pell, Lucas and Jacobsthal sequences. Then the work emphasizes the Sequence of Padovan, its convergence to the Plastic Number and its import architecture proposed by Hans van Der Laan. By m, a pedagogical proposal of application of the theme, using the technique of Didactic study of proportion in Elementary School. Keywords: Sequence. Sequence of Padovan. Following teaching. Proportions. Plastic Number.

Geometry is considered the science of spaces because through it we can visualize abstract concepts with the eyes of the mind. It is able to stimulate a large capacity for deduction and reasoning. In this work, we will focus on the study of tangent circumferences. For this, it is important to know how to find analytically a point of tangency using the equations of the same and how we can study the problem of the circumference tangency using the geometric construction, based in their centers and by means of a common line. In addition, through an analysis of textbooks adopted in basic education we can have a sense of how such works present and develop the study of circumference. In a comparative table, we will observe by means of criteria related to the study of the circumference, that Each book deals with the subject. Focusing on the geometric approach, we will review some fundamental concepts such as the construction of the perpendicular bisector of a segment, the bisector of an angle, a line perpendicular to another among other basic concepts. We will do a study on Apollonius, his history and his contributions to the study of geometry, and some solutions to his problem to find a circumference tangent to three objects, we can work several different ways to construct it. The development of this work will demonstrate that different approaches, analytical or geometric, can lead to the same result and that it is important for the student to have access to and master both for their greater achievement in school.

In this work we present a brief report of the main educational reforms from Brazil Colônia until the present day, relating the proposed changes with the historical con- juncture of the period. We present the modifications in the Teaching of Geometry, comparing the curricula proposed in the Francisco Campos and Gustavo Capanema Reform, through the modifications with the implementation of the National Curricular Parameters and analyzing the proposal of the National Curricular and New High School. Showing the impact of the reforms in the construction of the curriculum and in the teaching - learning methodology of Geometry through the exploration of the questions proposed in each period studied. Next, we present methodological suggestions, in agreement with the BNCC, for the Teaching of Geometry in High School as an aid to Dynamic Geometry (Poly Pro, GeoGebra, Cinderella) and Concrete Material (Geolig)

Seeking better methods for explaining or understanding important subjects developed in the classroom, this paper proposes to present and stimulate the use of simple, visual proofs or even formal demonstrations of results and mathematical formulas in the scope of elementary education and medium. Since understanding from visualization is more meaningful, and encourages students to seek and acquire knowledge, we present some visual evidence of some basic results, sometimes accompanied by algebraic arguments to explain the illustrations. Next, we highlight some geometric representations of numerical sums and we present some geometric demonstrations of irrationality. In order to enrich such approaches and to involve the reader in the search for other results, in each chapter we propose some exercises whose solution is similar to previous discussions. The results presented in this work were extracted or inspired mainly in [2] and [6].

The present work aims to present the use of concrete materials in mathematics teaching so that the educator and the student have it as a tool in the Teaching-Learning process making the teaching applicable and dynamic. This teaching tool meets the desire of learners to make classes more dynamic and with greater student participation. The use of concrete material develops the student's reasoning, interactivity with other students, stimulating mathematical logical thinking. The interconnection of the abstract to the concrete provides the student with a verification of the theory presented in class, often restricted only to the theoretical part. For the student the teaching brings a greater sense as it is applicable thus making a direct relation of the abstract to the concrete. It is believed that the use of these materials is a strategy for teaching mathematics, as well as facilitating teacher development and student learning, being able to develop significant knowledge in what is taught. The materials Concrete works serve as a mediator in order to facilitate the teacher / student / mathematical knowledge relationship. The use of this pedagogical practice facilitates the use of scientific knowledge applicable to the student's daily life, providing support for the teacher to come to reconcile theory and practice

The objective of this work is to perform a study on the conic sections (ellipse, parabola and hyperbola) from different perspectives, to construct with the software of dynamic geometry, Geogebra, Learning Objects to support these approaches and to model some applications resulting from the properties that each curve has. More precisely, the most relevant historical aspects are discussed, the description of each conic based on the focal definition, the geometric places from the General Second Degree Equation, the conic equations based on the definition of eccentricity and their parametrizations. It is also analyzed in different approaches the reflective properties of the conics involving tangent and normal lines, finishing with an overview about Geogebra and the description of the process of construction of each Learning Object.

In this work we present an introductory text of the non-Euclidean geometries focusing on the study of the differences with Euclidean geometry, approaching all the development of the same and its main exponents. Euclid of Alexandria is considered the father of geometry, thanks to his ability to write and expose geometric concepts, being the first to use rigor in mathematical demonstrations. We discuss the emergence of non-Euclidean geometries with the study of the fifth postulate of Euclid, the only one of the five that was unclear about its validity. Next, we show two educational software of dynamic geometry, Cinderella and Geogebra, used to facilitate the understanding about the models that we use in non-Euclidean geometry. Finally, we propose activities for teachers and students that will allow the appropriation of the concepts of these geometries.

This work is based on the idea of presenting the mathematical knowledge present in the
format, operation and installation of the parabolic antennas, whose main study is the
parabola, whose rotation, around a fixed line, generates the parabolic surface also known
as paraboloid. For this, we will start with a historical approach on the conic sections.
After this chapter, we will see some basics about Flat Geometry that will be very useful
in the following chapters. In sequence we will present the quadratic functions whose
graph is a parabola and the reflection property of the parabola which justifies the antenna
format and finalize the work presenting the importance of mathematics in the operation
and installation of the parabolic antennas.

As important as knowing the practical procedures is knowing the mathematics behind
them. In this work, we present the mathematics behind computed tomography, that is,
we study the mathematical fundamentals used in computed tomography. For this, we will
make the mathematical model of the computed tomography using matrices and linear systems.
We will see that these linear systems are equivalent to the image search generated
by the CT. To obtain such systems, mark the algorithms of Kaczmarz and Cimmino, we
will see that these algorithms approach a solution of these systems. It is worth pointing
out that this work has as its main objective the material for the teacher of the physical,
algarbrica and computational form.

This work is dedicated to the study of the Pythagorean Theorem. Pythagoras and
his Pythagorean School had great importance in the development of knowledge in several
areas, especially in Mathematics. Although not the rst to perceive the relationship
between the measurements on the sides of a right triangle, the Pythagoreans were the
rst to demonstrate it. We are presented with various demonstrations of this theorem,
as well as of its reciprocal and also of its generalization. We conclude this work with
the suggestion of some activities to be applied in the classroom.

We present two important properties of the Mercator projection: the connection with
the complex exponential function and the intimate relationship that can be established
between the projection and the tractrix curve. These properties allow us to present the
complex exponential function and the tractrix curve using the projection of Mercator as
a motivation, thus providing a way of presenting these abstract mathematical concepts
with the practical extreme appeal of cartography. This work was strongly inspired
by the article textit Some properties related to Mercator projection, by Wim Pijls,
published in volume 108 of The American Mathematical Monthly.

The main purpose of this work is to present modular arithmetic, a content commonly seen in higher education, and show through examples of applications that this topic could be taught in basic education. We will show a little of the history of modular arithmetic, the theory needed to understand the basic ideas of the subject, and see some basic textbooks that address divisibility criteria beyond MMC and MDC. The presence of such subjects in books of this level could be used as a motivation for the introduction of more general concepts of modular arithmetic. We will also show some problems of admission exams and olympics that can be solved using modular arithmetic. In addition, we have listed a number of applications of the theory developed in the work and the study of how to apply them to understand the creation of ISBN and CPF numbers and why the traditional `` ninth proof '' has fallen into disuse. Further examples of division of polynomials using modular arithmetic will be seen. In the end, we hope that this paper has shown some of the benefits that teaching modular arithmetic in basic education can bring to students.

A Matemática é uma ciência que vem sendo estudada pelo homem a milhares de anos
devido a sua enorme utilidade e as suas inúmeras áreas de aplicação. Pode-se encontrar
Matemática em praticamente tudo o que existe a nossa volta e sua abrangência é tão
grande que podemos vê-la em um simples relógio digital assim como em uma estação espacial.
Sendo a Matemática a base de toda a tecnologia, no mundo moderno ela tornou-se
ainda mais importante e, devido a isso, ela vem sendo desenvolvida dia após dia por estudiosos
em todos os cantos do mundo. Paradoxalmente a tudo isso, vemos hoje em dia, nas
escolas brasileiras, um grande desinteresse por parte dos alunos em estudar e aprender
esta ciência tão importante. Grande parte deles acredita que a Matemática básica, que
é estudada nos ensinos fundamental e médio, não possui utilidade ou aplicações no cotidiano.
Entretanto a Matemática básica é extremamente importante e seus tópicos mais
elementares possuem muitas aplicações interessantes no dia a dia. Para provar tal ponto,
este trabalho mostrará algumas aplicações interessantes de Matemática básica.

O Cálculo Diferencial e uma disciplina básica da Análise Matemática, a qual versa sobre Derivadas.
O conceito de Derivada esta associado às ideias de retas tangentes à uma curva e à taxas de variação.
Matemáticos como Issac Newton (1643-1727) e Pierre de Fermat (1601-1665) contribuíram para o desenvolvimento
do Cálculo, porém deve principalmente a Gottfried Wilhelm Leibniz (1646-1716) a relação
entre derivada e integral. O Cálculo Diferencial contribui com a construção de conceitos e resultados das
ciências em geral, em especial, a Engenharia; Medicina; Economia; Administração; Física; Química; Biologia
e etc. Desta forma, este trabalho apresentará resultados sobre limites e derivadas, entre eles um estudo
sobre máximos e mínimos de funções, de forma a construir uma base sólida de conhecimentos básicos, pois
os mesmos serão necessários na resolução das aplicações propostas no  capítulo 04. Tais aplicações
estão distribuídas em dois grupos, algumas ligadas a prática, outras um pouco mais teóricas, e tem como
propósito despertar o interesses de alunos, participantes de olimpíadas de matemática, pela disciplina de
Cálculo .

Um número real pode ser aproximado por um núumero racional, isso e comumente
feito através de uma representação decimal, ou seja, uma sequência de aproximações
por racionais cujos denominadores são potências de 10. Contudo, a representação
feita por frações contínuas produz aproximações muito melhores. Se por um lado, a
representação decimal e prática, por outro as frações contínuas não deixam de ser
simples e natural. Neste trabalho, abordamos esta útima forma de representação
dos números reais. Para isto, é introduzido o conceito de fraçõoes contínuas, com
destaque para as frações contínuas simples, resultados mais relevantes e a representa
ção dos núumeros racionais e dos núumeros irracionais por expansão de frações
contínuas. Como aplicação, apresentamos o modelo para construção de calendáario.
Vale ressaltar que frações contínuas e um tema de pesquisa com ampla aplicação em
Matemática Pura ou em Ciências Aplicadas, apesar de seu carater elementar.

Apresentar demonstrações em Matemática torna-se um desafio para o professor em sala
de aula, pois para que o aluno possa compreendê-las, é necessário atenção e maior compreens
ão da matemática em estudo. Saber o que deve ser provado e o que pode ser
utilizado na prova de um resultado é fundamental para se obter êxito. Tendo em vista
que os números naturais são essenciais em todos os ramos da Matemática e é estudado
em todos os níveis de ensino, neste trabalho, temos como propósito o estudo do conjunto
dos números naturais e o Princípio da Indução Matemática ou Método da Indução, uma
técnica de demonstração aplicada em situações que envolvem números naturais. Vale
ressaltar que nem toda demonstração que envolve números naturais podem ser provadas
com esta técnica e que outras formas de demonstração também foram utilizadas para dar
segmento ao trabalho, porém o foco foi o Método da Indução. Para o desenvolvimento do
trabalho zemos um estudo sobre as propriedades deste conjunto e destacamos resultados
que podem ser demonstrados com o Princípio da Indução, como o Teorema Fundamental
da Aritmética e algumas fórmulas clássicas e desigualdades apresentadas em disciplinas
do Ensino Básico que são úteis para a compreensão de conteúdos no Ensino Superior.
Finalizamos com algumas aplicações lúdicas clássicas, que fazem uso desta técnica, como
a Torre de Hanoi, a qual pode ser trabalhada em sala de aula como motivação para a
compreensão desta técnica de demonstração.

Estudar aritmética é fundamental para compreensão matemática. Este trabalho visa resgatar
o estudo de divisão de números inteiros positivos através da aplicação e análise da
prova dos “noves fora”. Este método utilizado pela escola antiga e que hoje ficou relegado
ao esquecimento traz ferramentas poderosas de divisibilidade, sitema de numeração e congruência.
Assim é abordado neste trabalho o sistema de numeração decimal e mudança de
base, algoritmo da divisão, critérios de divisibilidade e congruência. É mostrado o método
prático da prova dos noves para as operações de adição, subtração, divisão e multiplicação
bem como as demonstrações necessárias. Por fim, conclui-se que a regra não consiste
apenas de uma prática social e sim de um instrumento aritmético capaz de desenvolver
aprendizagem significativa em matemática.

Neste trabalho abordamos topicos de Geometria Euclidiana ministrados no Ensino Basico: Semelhanca de Triangulos e as relacoes metricas no triangulo retangulo. Para isto, inicialmente fizemos um breve historico da Geometria Euclidiana, em sequencia apresentamos alguns resultados sobre Congruencia e o Teorema de Tales. Tal resultado e de fundamental importancia para dar continuidade ao trabalho, pois a partir dele obtemos os criterios de semelhancas e consequentemente as relacoes metricas no triangulo retangulo. Finalizamos o trabalho com algumas aplicacoes e breves comentarios sobre textos da literatura matematica do Ensino Medio que abordam o assunto estudado.

O calendario como o conhecemos hoje(gregoriano) sofreu algumas alteracoes importantes, e muita resistencia de paises, reinos e ate mesmo de bispos de regioes mais afastadas, ate ser aceito como uno, reconhecendo os erros do tomado anteriormente (juliano). Ate a reforma promulgada pelo Papa Gregorio XIII, apenas seguia-se a regra de acrescentar um dia a fevereiro a cada 4 anos, e isso gerou uma defasagem, por exemplo, na data de inicio das estacoes, que nao mais coincidiam com solsticios e equinocios astronomicos, fazendo com que a pascoa fosse festejada fora do periodo determinada no Concilio de Niceia e conseguinte outras festividades moveis. Apos 1582, o proclamado Ano da Reforma, houve desavenca e desestruturacao da Igreja Catolica e tambem com nacoes do mundo. Tanto e que Grecia e Turquia, por exemplo, mantiveram-se com o antigo calendario ate a ultima decada de 20. A influencia da igreja garantiria a nivel mundial, a adesao ao novo calendario, e, alem disso, a propria evolucao da sociedade, cada vez interligada e interdependente, levaria a adocao do mesmo, cedo ou tarde. O calculo dos anos bissextos e bem claro e simples, ate na Bula Papal do proprio Gregorio, e assim como ele, a regra da determinacao do domingo da Pascoa sao facilmente encontrados, e este ultimo com o desenvolvimento da informatica, acelerou ainda mais tal computo. Este trabalho visa, sob o olhar da problematica das datas moveis e a historia do calendario gregoriano, apresentar a Formula de Gauss que determina a data da Pascoa de qualquer ano apos 1600, e "traduzi-la" uma linguagem de programacao C++. Para isso, contera os topicos necessarios da Aritmetica para a construcao do mesmo, em especial a aritmetica modular, alem do instrumento basico de programacao para a confeccao deste programa. Logo, um aluno do Ensino Medio com conhecimento basico ou em um curso tecnico de informatica, com leitura deste texto, tera os fundamentos para a implantacao da regra em outras linguagens(ate em planilhas eletronicas), baseando-se na matematica por tras deste processo.