Discrete-Time Fractional Variational Problems

We introduce a discrete-time fractional calculus of variations on the time scale $h\mathbb{Z}$, $h > 0$. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler-Lagrange and Legendre type conditions are given. They show that solutions to the considered fractional problems become the classical discrete-time solutions when the fractional order of the discrete-derivatives are integer values, and that they converge to the fractional continuous-time solutions when $h$ tends to zero. Our Legendre type condition is useful to eliminate false candidates identified via the Euler-Lagrange fractional equation.Comment: Submitted 24/Nov/2009; Revised 16/Mar/2010; Accepted 3/May/2010; for publication in Signal Processing

Activities with desmos and geogebra for formative and automatic feedback

The study developed here was carried out within the scope of a National Meeting “19.º MatViseu” with secondary school mathematics teachers in a workshop entitled Desmos and GeoGebra: activities with automatic feedback and formative”. The objectives were to articulate the mathematics curriculum and essential learning program, and to explore activities that use some of the strengths of Desmos and GeoGebra to provide feedback to students and develop some activities with feedback appropriate to the level of education they teach. This National Meeting was organized with plenary sessions (invited speakers) and parallel sessions (workshops). In-service teacher training is an essential element to help them to adapt to different curricula and new methods and resources in their professional careers. This study aims to understand how the use of these two platforms can allow a different approach in their classes. The research methodology adopted consists of a case study, relating to a group of teachers who worked autonomous in this workshop. Essentially, a qualitative approach was adopted through the interpretation of data collected through observation, interaction in these platforms and a brief questionary. We can conclude that using the GeoGebra platform requires prior preparation, essentially with GeoGebraScript and with the Desmos platform it was easier to work with the Computation Layer language, although GeoGebraScript and Computation Layer language are new for most of them. Thus, it is recommended that in future training, the creation of strategies and dynamics of preparation for work in the GeoGebraScript and in the Desmos language, to carry out a complete work. However, it is important to keep in mind that the GeoGebra platform was known to most of the participants, but Desmos was not and that the sample is not significant to draw general conclusions.publishe

An experience with Desmos in the study of the quadratic function

In this paper we present a didactic experience in the subject of Mathematics carried out in a distance learning context, on the topic Quadratic Function, using the digital and free platform Desmos. The use of this tool was determinant for the teaching and learning of quadratic function since its teaching took place in distance education, due to the pandemic situation. In a pandemic context, the use of tools to gauge student learning was a necessity, but practices such as the one described in this paper should be incorporated into a normal classroom environment, promoting discovery through graphical and algebraic manipulation.publishe

Calculus of variations involving Caputo-Fabrizio fractional differentiation

This paper is devoted to study some variational problems with functionals containing the Caputo-Fabrizio fractional derivative, that is a fractional derivative with a non-singular kernel

An optimization method for the best fractional order to estimate real data analysis

In this paper we consider fractional differential equations, with dependence on a Caputo fractional derivative of real order. Using real experimental data of Blood Alcohol Level we obtain a system of fractional differential equations that model the problem. A numerical optimization approach based on least squares approximation is used to determine the order of the fractional operator that better describes real data as well as other related parameters. We prove that it describes better the dynamics than the classical one

Active learning during the COVID-19 Pandemic: a triple experiment

In recent times, the debate on active methodologies has been intensified with the emergence of strategies that can favor students' autonomy. The active, dynamic and constructive environment can positively influence the perception of teachers and students. One of the ways to achieve this is through the use of technologies that enhance learning. Due to the pandemic situation, the use of technologies was imperative and essential. Three higher education institutions were involved in an active learning project, involving diversified technologies and methodologies to enhance the learning of mathematics. In the University of Aveiro, the experiment involved all the 1st-year students enrolled in the course “Complementary Mathematics II” during the second semester of the academic year 2020-21. These are students of a master’s program for pre-service teachers, preparing to become primary school teachers and mathematics and science teachers, grades 5 and 6. At the Institute of Engineering in Coimbra, the students who attended this experiment were students of Mathematical Analysis I, from the undergraduate degree in Electrical Engineering. In the Polytechnic Institute of Viseu, the students involved in this experiment were students from the Preparation Course in Mathematics (which aims to prepare candidates over 23 who wish to apply to an undergraduate degree in Business). The methodologies used were applied during three distinct phases and according to the students’ profile, related to each course. The methodologies used were very diverse: team-based learning; flipped classroom; peer teaching; think-pair-share and exposition followed by exercise practice; problem-based learning and inquiry-based learning. The technologies were also very diversified: Desmos, Kahoot!, Quizizz, GeoGebra, Excel, OneNote, Google Docs, Zoom, Miro, Wolfram alpha, PowerPoint, calculators and videos. The use of a wide range of methodologies and technologies made it possible to teach during the lockdown imposed by the pandemic situation, and simultaneously motivated the students. In addition, they provided opportunities to adapt the learning process to the diverse knowledge and learning styles of students and increased their involvement.publishe