https://repositorio.ufu.br/handle/123456789/33431 2026-08-23T09:16:08Z https://repositorio.ufu.br/handle/123456789/49625 Title: Fatoração de expressões polinomiais e decomposição em frações parciais Abstract: This dissertation investigates the factorization of polynomials and the decomposition into partial fractions within the context of Basic and Higher Education, articulating algebraic foundations with proposals aimed at teaching practice. The study begins with the extension of number systems, emphasizing the introduction of complex numbers as a natural extension of R, preserving its algebraic properties while enlarging the set in which solutions exist. It is shown that the solution of the quadratic equation ax^2+bx+c = 0, with real coefficients, maintains the same structure expressed by Bhaskara’s formula, the essential difference lying in the numerical set in which the discriminant is interpreted. In C, regardless of the sign of Δ = b^2 − 4ac, the square root of Δ always exists, ensuring the existence of solutions. It is further emphasized that algebraic properties valid in R, such as special products and completing the square, remain valid in C, helping to clarify frequent difficulties both in the factorization process and in solving polynomial equations. Classical methods are presented, including Euclidean Division for polynomials, the Briot-Ruffini algorithm, and the Rational Root Theorem, highlighting that the factorization of a polynomial depends on the coefficient ring. The case of x^2 + 1 in R[x] and C[x] is discussed, showing that in C[x] every polynomial can be factored into linear factors, according to the Fundamental Theorem of Algebra. This fact structurally simplifies the partial fraction decomposition of rational functions with real coefficients when analyzed in the complex setting. As an educational product, a Workbook of Commented Activities is presented, in which these structural observations are explored didactically through examples and proposed activities. The material is intended for high school students, early undergraduate students, and mathematics teachers, contributing to a more structured understanding of Algebra and its articulation with Calculus. 2026-03-26T00:00:00Z https://repositorio.ufu.br/handle/123456789/48980 Title: Estroboscopia digital: aplicações de interpolação e ajuste de curvas na educação básica Abstract: This dissertation develops and validates an original educational product: a video-based kinematic analysis web app for inquiry-based teaching of Physics and Mathematics in Basic Education. It addresses the disconnect between the abstraction of school content and the concrete phenomena experienced by students through an interface that trans- forms audiovisual records of real motions into quantitative data, tables, graphs, and mathematical models. The study brings together Mathematics Education, mathemati- cal modeling, metacognition, digital technologies, and Design-Based Research, starting from the hypothesis that interaction with the software promotes investigation, hypothe- sis validation, and an integrated reading of multiple representation registers. From a mathematical standpoint, the system is grounded in the NyquistShannon Sampling Theorem, the Mean Value Theorem, polynomial interpolation, cubic splines, and the SavitzkyGolay filter, which underpin temporal discretization, trajectory reconstruction, and data smoothing. The product was developed and refined in iterative cycles and compared with the Tracker software in projectile motion experiments with different frame rates. The results show that the digital environment provides accuracy compara- ble to well-established tools, enhancing pedagogical accessibility, the transparency of mathematical procedures, and its potential for use in schools. It is concluded that the web app is a relevant contribution to inquiry-based teaching by integrating concrete ex- perience, mathematical modeling, and technological mediation, aligning with students scientific and mathematical education. 2026-06-18T00:00:00Z https://repositorio.ufu.br/handle/123456789/48949 Title: Modelagem via fusion 360 e impressão 3D no ensino de geometria Abstract: Mathematical modeling seeks to bridge the gap between theoretical content and concrete objects. In this context, Autodesk Fusion 360 stands out as a powerful geometric modeling tool, enabling detailed study and visualization of solids from multiple perspectives. Coupled with 3D printing, this methodology transforms theoretical perception into manipulable physical objects. This paper presents instructions for 3D construction and modeling via Fusion 360 and the slicing of these solids, exploring geometric definitions and Boolean operations to analyze volumetric relationships—such as the volume of a sphere via the Anticlepsydra and the decomposition of a triangular prism into three pyramids of equivalent volume. In addition to algebraic demonstrations, the text encourages the use of digital tools for creating functional objects. The proposal was applied in a short course for high school students, integrating the teaching of 2D and 3D geometry into a real-world modeling and 3D printing process. The final stage of this work consisted of developing GeoEscola, a Python-based addin for Autodesk Fusion 360 designed to optimize geometric modeling in educational environments. This tool eliminates manual processes, such as 2D sketching and extrusions, replacing them with a direct input interface. By entering numerical variables—suchas height, length, radius, angles, and the number of edges—GeoEscola triggers algorithms that automatically generate pre-defined solids, including the anticlepsydra, pyramids, and cone shells, among others. 2026-03-20T00:00:00Z https://repositorio.ufu.br/handle/123456789/47307 Title: Cubo mágico e sala de aula: uma parceria de sucesso Abstract: The Rubik’s Cube is much more than a simple toy; it’s a three-dimensional puzzle with complex solutions. This work examines this three-dimensional game, linked to a didactic-pedagogical activity, highlighting positive aspects of student learning, along with the development of various skills addressed by the BNCC. In addition, a pedagogical workshop was carried out with 9th-grade students, in which the layer-by-layer method was taught as a solving strategy. The activity enabled practical experience and the enhancement of cognitive and socioemotional competencies, reinforcing the potential of the Rubik’s Cube as a pedagogical resource in Mathematics education. 2025-08-30T00:00:00Z