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To enable collaborative modeling activities online digital tools are essential. In this paper we present a holistic and adaptable concept for the development and implementation of modeling activities – which could especially be fruitful in times of homeschooling and distance learning. The concept is based on two digital tools: Jupyter Notebooks and a communication platform with video conferences.We carried out this concept in the context of two types of modeling activities: guided modeling days, where the students work on previously prepared and didactically developed digital learning material, and modeling weeks, in which the students work on open problems from research and industry very freely. In this paper the usage of Jupyter Notebook in modeling activities is presented and illustrated with the example of the optimization of a solar power plant. On top, we share our experiences in online modeling activities with high-school students in Germany.

In this paper we show how modeling helps us to motivate the Graph Theory's students. We use the well-known game of “rock, paper, scissors” and its generalizations to show the applicability of mathematical concepts studied.

This paper describes, with a case study, the teaching methodology in which a research perspective is introduced in a master program. Throughout the semester, students are asked to apply and analyze the results obtained with the implementation of different heuristic algorithms for the optimal design of a concrete structure. As a final report, students should implement a heuristic algorithm different to those implemented in class for the optimal design of the concrete structure. The results obtained should be presented in the format of a scientific paper. The case study presented in this paper consists on the optimization of the mix design of simply supported beams made of high-performance concrete (HPC). The optimization algorithm implemented is an hybrid strategy based on a variable neighborhood search and threshold acceptance algorithm. This paper begins by defining the optimization problem. It will then go on to present the results of applying the optimization algorithm. Finally, a parametric study is undertaken, analyzing beams with lengths varying from 10 to 20 m. The application of this teaching methodology has supposed an aditional challenge to postgraduate students, as they not only have to adquire the technical competences of the subject but they also have to apply them in a practical example and publish the results in the format of a scientific journal.

This paper belongs to the mathematical modeling used as a teaching tool field, essential in teaching linear algebra to future telecommunications engineers. We present several technical examples in which linear algebra plays an important role, and describes how these examples can be used to achieve a better understanding of algebraic concepts discussed. At the end of the paper, some of the results obtained in the experiment conducted in the ETSEEI La Salle (Ramon Llull University) are discussed, and some significant conclusions can be drawn regarding the influence of technical examples on the final performance of students.

In this paper we propose and evaluate different mathematical parameters extracted from the experimental semivariogram for land use/land cover classification using high-resolution images and cadastral mapping limits for the definition of the objects of analysis. First, we describe the process of calculating the semivariogram from the gray level values in an image object. In order to optimize the computation time we present two pixel selection techniques that preserve the original shape of the semivariogram. Several parameters are then extracted from the semivariogram. Finally, we use various statistical techniques to select the most discriminant parameters. Last section shows the results obtained using aerial digital images of an agricultural area on the Mediterranean coast of Spain. The study of the practical application presented in this paper facilitates the understanding of the relationship between the behaviour of the experimental semivariogram and the variability of the intensity values in a digital image. In order to follow the development of this work, the reader should know some basis of classification methods and digital image processing techniques.

The paper deals with biomedical modeling in general and in particular how modeling techniques can be used to improve the training of clinical practitioners in the use of new medical devices. The Bioheat Equation is given as an example to show how, even though mathematical models are based on complex equations, the terms of the equations represent phenomena which can be intuitively understood. The paper also describes examples in which mathematical modeling was used to study the performance of medical devices, especially those related with radiofrequency ablation.

This paper describes different possibilities to answer the question of automatically creating a choreography for musical fountains using mathematical modeling. The possibilities for an implementation range from a short project in the middle school, which has large parts in mathematics, to long-term projects, which deal with the physical background, an implementation with the help of the computer and in the highest expansion stage even a technical realization. In the article different variants are described, supplemented by experiences from practice and extension possibilities are pointed out. All in all, the readers should be enabled to design and practically implement their own project according to the individual local conditions such as time frame, learning level of the target group and possibilities for interdisciplinary cooperation. (This article is a translation and update of the German first edition Bracke, M. Capraro, P. (2021)).

Due to developments in the financial marketplace, digitalization of finance and easy access to financial products, financial literacy is becoming increasingly important. The paper at hand introduces parts of a learning environment for mathematics classes that focuses on mathematical modelling and financial literacy. After theoretical considerations, two worksheets are presented, which deal with influences of the interest rate on the repayment of loans. Based on the description of a test run, the usability of the worksheets in class is concluded.

Heterogeneity can be taken into account in the classroom by using self-differentiating tasks. From a mathematical-didactic point of view, modelling tasks are of particular interest in this regard because of this. The paper focuses on the heterogeneity dimension of mathematical achievement and modelling competence of 15-year-old students. First results of a case study will be described in how far the processes of a modelling task vary with regard to the aspect of achievement levels.

This paper is practice-oriented and reports on a mathematical modelling unit specifically developed for first-year engineering students in a South African context. The main idea with the unit was to foster students’ mathematical modelling competency development. This idea supports an essential goal of mathematics teaching, that is to enable students to solve real - world problems by means of mathematics. The unit consists of five lessons and several tasks, carefully planned to consider students’ mathematical pre-knowledge, the demands of the first-year mathematics (calculus) curriculum and the intended competency development. The unit was linked to the mathematical topic of functions and taught for different groups of students according to two different teaching designs, similar to the designs used in the German DISUM project. 144 first year engineering students from the extended curriculum programme of a large public university were divided in three groups and exposed to the unit. An empirical evaluation of the intervention (with a pre-post-test design) showed a significant competency growth for all groups, with substantial differences, dependent on the teaching design. Some strengths and shortcomings of the unit will be identified and implications for future practice will be discussed.

Mathematical modeling can be promoted in regular lessons as well as in project form. Both options have advantages and disadvantages and provide different challenges, but also opportunities for students and teachers. Since 2001, teachers in Hamburg have the opportunity to participate with their students in modeling projects organized by members of the University of Hamburg. In this paper, the perspective of teachers, some of whom have been participating in modeling activities for more than five years, on the opportunities, but also on the difficulties of participation are presented. In addition, the views of over 200 students regarding their participation in the modeling days are outlined.

Since 2000 the Zentrum f¨ur Mathematik in Bensheim (ZFM) and the Technical University Darmstadt (TUDa) have organized an annual autumn school for mathematically highly talented pupils in their last year of school. For one week, the participants work in groups on application problems, which first need to be transformed into mathematics and then be solved by mathematical analyzation and numerical approximation and implementation on a computer. Finally, the results are presented. The problems are very realistic. Only minor details may be skipped, to make the problem easier to handle and to focus on the main issue at hand, but only if they do not affect the basic result. For some of the problems there exist similar, but much simpler, counterparts that can be solved in a shorter time period even in a usual school setting. Solving those simple problems opens the outlook to the hard problems if they are mentioned before. This paper examines two applications that led to mathematical problems from analysis and geometry.

In Germany, there is quite a tradition of modeling projects, carried out by different universities all over Germany. Since 2001, different modeling projects has been conducted by the working group on Didactics of Mathematics of Department of Education of the University of Hamburg. In this paper, the current project, the Hamburg Modeling Days, is described in detail. One goal of this project is to enable students to gain experience in working on modeling problems and to acquire modeling competencies. Furthermore, prospective teachers are supported in their professionalization for teaching mathematical modeling, especially with respect to the supervision of students working on modeling problems. Finally, practicing teachers are in the implementation of modeling problems in mathematics classes.

In this work we present an application case of high school level probability concepts to solve an interesting real problem. In particular, we try to shed some light on uncertainties faced, at the beginning of the school year 2020-2021, by infant and primary schools and families with children in these educational stages in relation to possible temporary confinements of bubble groups that may occur as a result of positive COVID19 cases in them. These uncertainties lead centres and families to live in fear and anxiety the likelihood of suffering these confinements. In this paper, we present a simple probabilistic model using only high school level concepts and based on data about bubble group confinements during the first four weeks of the school year in nursery and primary schools in the Valencian Community (Spain), which allows us to make a prediction of the likelihood of centres and families to be confined for the remainder of the academic year. According to our model, the probability that centres will have to apply one or more bubble groups in the remainder of the course is quite high, while for families it is not too worrying, especially if they are not numerous. Also, we estimate the productivity impact related to working parents needs to look after their confined children which, as we show, is very low.

In this paper, we present a software tool entitled E-aplan Express (version 2018), with free access for educational and commercial use, to the modelling and resolution of aggregate production plans generating medium-long term production planning, based on a forecasted demand in that period. E-aplan tool models the aggregate production plan though a mixed integer linear programming model (MILP). The LP solver optimization engine generates the planning by adjusting all the optimization variables with the least possible error. Finally, it is presented an illustrative example that considers a collaborative aggregate production planning, in a two-echelon supply chain. Different scenarios are modelled in order to simultaneously consider the planning objectives of both enterprises of the network.

The new curricular transformation carried out in the electrical engineering career, responds to the requirements at international and national level for a graduate of this career. For that reason, the main question remains under which conception and with what procedures to perform the fundamental changes to develop a communicative, independent and creative educational teaching process. This paper takes as an example the lecture of Electrical Systems I, given in the first semester of the fourth year of the Electrical Engineering degree, at the Technological University of Havana, CUJAE. The objective of this work is, from the use of a model for showing the behavior of transmission lines, to characterize the teaching-learning process as one of the components of the educational process. The analysis is carried out from a communicative dimension, so that the didactic treatment propitiate in the practice, an interrelation between professors and students and between students. This allows the students to be involved in their own learning process. The present didactic proposal considers, in the general framework of active learning, the experimental activity in the classroom as a systematic and organized working tool, where it is necessary to articulate the methodological use of the laboratory-class to improve the teaching-learning process and the construction of knowledge.

The new curricular transformation carried out in the electrical engineering career, responds to the requirements at international and national level for a graduate of this career. For that reason, the main question remains under which conception and with what procedures to perform the fundamental changes to develop a communicative, independent and creative educational teaching process. This paper takes as an example the lecture of Electrical Systems I, given in the first semester of the fourth year of the Electrical Engineering degree, at the Technological University of Havana, CUJAE. The objective of this work is, from the use of a model for showing the behavior of transmission lines, to characterize the teaching-learning process as one of the components of the educational process. The analysis is carried out from a communicative dimension, so that the didactic treatment propitiate in the practice, an interrelation between professors and students and between students. This allows the students to be involved in their own learning process. The present didactic proposal considers, in the general framework of active learning, the experimental activity in the classroom as a systematic and organized working tool, where it is necessary to articulate the methodological use of the laboratory-class to improve the teaching-learning process and the construction of knowledge.

The aim of this paper is to show a methodology, based on the so-called Lognormal Model, to describe the dynamics of underlying assets by taking into account the uncertainty of nancial markets. In spite of its simple formulation, the Lognormal Model is a valuable tool from a formative standpoint becauseit provides an excellent basis to study more advanced models. The proposed approach has been put into practice by the authors in their teaching in both undergraduate and postgraduate studies. We present the model and its application to describe the dynamics of real data of an asset traded in the Spanish stock market index IBEX35.

The main aim of this paper is shown a mathematical model about the social networks that enhances the interest of the Mathematics' students that are in rst course. This model is based on Markov chains. Its approach and resolution permit to illustrate the applicability of the discrete equation systems and toemphasize the concepts studied about eigenvalue problems and diagonalization. Furthermore, all results are computed with the Mathematica software.

In this paper we show how to model and to analyse the solubility of some puzzles, brainteasers and other mathematical toys. We make use of basic group theory concepts and the computational algebraic system GAP (Groups, Algorithms and Programming). The main purpose is to motivate beginner students in this algebraic theory, and also to show the playful applicability to advanced students.

As the relationship between both students (teachers) and information technology evolves, new tools are required to improve learning (teaching) in social sciences. Economics, business and finance are mainly based on data and dealing with data requires specific skills and techniques such as computer programming in order to get full potential of most quantitative models. In this paper, we propose a coding oriented learning method based on Python Notebooks which is specifically designed for students of degrees in economics, business and finance. We follow a learning-by-doing strategy that encourages students to implement economic models as a suitable way to improve the understanding of fundamental concepts. As an illustrative example, we also describe a case study in which Python Notebooks are the key tool to teach cash management in a Master in Business Administration program. Since students of today are the decision-makers of tomorrow, a further advantage of the use of a programming language as a teaching tool is the possibility to connect theory to practice by enabling students to implement their own decision support tools.

This paper presents two experiments that take advantage of smartphones pressure sensor, using it to determine the value of the pressure within in equilibrium water in a test tube. In a first experiment, aimed at students of secondary and higher levels, the data collected with the pressure sensor of a smartphone at different depths allow a linear fit, thus checking the validity of Stevin's Law. In addition, based on this law and the parameters of the adjustment it has been possible to determine the density of the water. The density values obtained indirectly by this procedure differ between 2% and 4% of the result of dividing the mass and volume of a sample. In a second experiment, more focused on students of the first university courses of Physics subjects, the submerged smartphone oscillates suspended from a spring. The measurements of the pressure sensor allow, through Stevin's Law, to obtain the time series of the pressure in the corresponding damped oscillation, whose parameters are determined by a non-linear adjustment. The comparison of the free pulsation obtained from these values differs by 1.5% from that obtained from the direct measurement of the period outside the water. In both cases, the results obtained indicate the suitability of the current methodology in teaching contexts.

In this paper we present a comparative analysis of the resolution process of three modeling tasks performed by secondary education students (13-14 years), designed from three different points of view: The Modelling-eliciting Activities, the LEMA project, and the Realistic Mathematical Problems. The purpose of this analysis is to obtain a methodological characterization of them in order to provide to secondary education teachers a proper selection and sequencing of tasks for their implementation in the classroom.

In this paper, a comparison between the results obtained in laboratory experiments and those calculated by a numerical simulation of shallow water equations in an open channel is performed, considering the energy losses that occur as it passes through a local narrowing of the cross section. The mathematical model that simulates this physical phenomenon is governed by a partial differential equations system whose solution provides the water depth and the flow rate per unit of width, which is related to the velocity of the water. Such movement is controlled primarily by the force of gravity, being fundamental the relationship between it and the inertial forces. In the present study we have also taken into account energy losses caused by friction of the water with the contours and local losses caused by obstacles or changes in the width of the channel. A numerical scheme based on a high-order finite volume method has been used for obtaining the solutions of such model. Two type of laboratory tests have been simulated. The first type represents a slow transition regime, upstream and downstream of a narrowing in the channel. The second type represents a subcritical flow upstream, a narrowing that works as a control (regime change) and a downstream supercritical flow. Numerical-experimental comparison demonstrates the importance of adequately modeling of the different physical phenomena involved in the process, and the proper imposition of the boundary conditions of the problem.

The concept of random variable is a mathematical construct that presents some theoretical complexity. However, learning this concept can be facilitated if it is presented as the end of a sequential process of modeling of a real event. More specifically, to learn the concept of discrete random variable, the Monte Carlo simulation can provide an extremely useful tool because in the process of modeling / simulation one can approach the theoretical concept of random variable, while the random variable is observed \in action". This paper presents a Research and Study Course (RSC) based on series of activities related to random variables such as training and introduction of simulation elements, then the construction of the model is presented, which is the substantial part of the activity, generating a random variable and its probability function. Starting from a simple situation related to reproduction and survival of the litter of a rodent, with random components, step by step, the model that represents the real raised situation is built obtaining an \original" random variable. In the intermediate stages of the construction of the model have a fundamental role the uniform discrete and binomial distributions. The trajectory of these stages allows reinforcing the concept of random variable while exploring the possibilities offered by Monte Carlo methods to simulate real cases and the simplicity of implementing these methods by means of the Matlab© programming language.

The loads applied on a linear structural element generate internal forces in the cross sections which, in turn, result in stresses along the element. The nature, extent and shape of stress distributions are required parameters to compute the strength of structural elements or machinery components in order to its analysis or design. In this work, it is presented a virtual laboratory which allows to obtain different stress distributions in an isostatic beam, prismatic or cylindrical, subjected to axial forces, shear forces and bending moments. The virtual laboratory permits a great interactivity, allowing the simulation of various real situations in which the user can modify the magnitude and direction of acting loads, and also the boundary conditions of the beam. The ultimate goal of this paper is to present a tool aimed to support the learning and teaching of subjects related to Elasticy and Strength of Materials that are found in bachelor university degrees.

The aim of this paper is to present a proposal for teaching linear algebra based on heuristic of emergent models and mathematical modelling. This proposal begins with a problematic situation related to the creation and use of secure passwords, which leads first-year students of engineering toward the construction of the concepts of spanning set and span. The proposal is designed from the results of the two cycles of experimentation teaching, design-based research, which give evidence that allows students to progress from a situation in a real context to the concepts of linear algebra. This proposal, previously adapted, could have similar results when applied to another group of students.

In this work an experience using Virtual Learning Environment (VLE) in a secondary education school is described for third and fourth courses of secondary compulsory education, thus substituting standard books in class. The aim of this paper is showing the results of our study about the effects of the use of this new tool through the analysis of the survey filled in by the students. Several features of the sample of students are given to better understand the conclusions arising from the poll. Lastly, advantages and disadvantages arising from the use of VLE in secondary education classrooms are analyzed in light of the results of the survey. We can conclude that VLE are well seen by the students and can be used to improve the motivation of the students and their academic results.

Mechanical vibrations as subject can be found within many Engineering and Science Degrees. To achieve that the students understand the mathematics and its physical interpretation is the objective we should get as docents. In this paper we describe how to create a simple graphical model of a single degree of freedom vibrating system allowing us to visualize concepts like above concepts damping, resonance or forced vibrations. For that, we use the popular symbolic software Wolfram Mathematica with which, without an excessive programming complexity, we can obtain a very satisfactory visual model capable to move itself, controlled by parameters. In addition, the model incorporates the curve-response, something that links the mathematical results with reality

The aeroelastic modelling of aircraft structures is a fundamental area for the students of Aerospace Engineering Degree. This subject has a strongly multidisciplinary character and involves other several subjects like mechanics, vibrations, aerodynamics, structural analysis. Consequently, the students find stimulating the challenge of merging their knowledge at different areas. In this paper, a teaching experience on the solution of the aeroelastic problem of a 3D-wing through six different computer tasks is presented. The main objective is to attempt a relatively complex problem using a simple version of the Finite Element Method with only four degrees of freedom. The students begin creating the shape functions of the discrete model and finish solving the flutter instability problem.

Mathematical models are important tools for environmental management and risk assessment. Predictions about the toxicity of chemical mixtures must be enhanced due to the complexity of eects that can be caused to the living species. In this work, the environmental risk was accessed addressing the need to study the relationship between the organism and xenobiotics. Therefore, ve toxicological endpoints were applied through the WTox Model, and with this methodology we obtained the risk classication of potentially toxic substances. Acute and chronic toxicity, citotoxicity and genotoxicity were observed in the organisms Daphnia magna, Vibrio scheri and Oreochromis niloticus. A case study was conducted with solid wastes from textile, metal-mechanic and pulp and paper industries. The results have shown that several industrial wastes induced mortality, reproductive eects, micronucleus formation and increases in the rate of lipid peroxidation and DNA methylation of the organisms tested. These results, analyzed together through the WTox Model, allowed the classication of the environmental risk of industrial wastes. The evaluation showed that the toxicological environmental risk of the samples analyzed can be classied as signicant or critical.

In a social-environmental modeling course, students need to learn complementary skills that include the conceptualisation of a model, different modeling paradigms, computer pro- gramming, and the process of rigorously converting ideas and data into a computational program using a given toolkit. Such topics need to be taught in parallel in order to keep a heterogeneous audience motivated. Based on the experience with multidisciplinary au- diences, this paper describes a socio-environmental modeling course that explores three modeling paradigms: System dynamics, Cellular automata, and Agent-based modeling. We also present a small tutorial with some examples developed for the course.

In this paper we describe the process of designing and developing of interactive digital resources adapted to each student. It is shown the process which the authors have followed during the last years in order to generate virtual learning objects for the learning and teaching of Statistics. It is also shown the main features that the digital resources must fulfil and the different possibilities of use. Finally we describe the advantages of the designing of digital resources using author tools, and the integration of several tools as LaTeX, R, GeoGebra, etc. which let us obtain educational resources adaptable for its use in massive environments or big groups.

The Engineering Education requires that students acquire an appropriate knowledge on a mathematical computational language as well as on a numerical simulation procedure. The computational language of mathematics usually is taught in advanced courses, once that the curriculum mathematical education is mainly completed; in addition, the numerical simulation is usually located late or even in doctoral studies. In this paper, we propose that the Computational Fluid Dynamics (CFD) become to be a teaching-learning tool, instead of a strategic resource only. CFD can be regarded as a transversal skill i.e., as a useful educational tool for the Hydraulic Turbomachines learning, which achieves to overcome some epistemological obstacles of students. We develop a teaching-learning method in which the Tutor Facilitator plays an important role.

In this paper we present an outline of an ongoing research in which we intend to investigate on the teaching of the modelling process and families of functions in the context of a course in Economics and Management. Specifically, we introduce some of the theoretical foundations that support the research and we present an example of one of the activities that we are designing for teaching.

In this paper we present three examples that can be used to motivate and introduce classical notions of a course of Linear Algebra (and its connection with other subjects such as Probability and Computing) in an Engineering degree. More specifically in this article we shown how the use of Linear algebra allows us to study odds and winning strategies by means of three board games: one of Coins, the game called Snakes and Ladders© and finally the well-known Monopoly©.

In this paper, a mathematical model describing the dynamic behavior of a commercial piezoelectric ceramic, vibrating in thickness, is presented. This model solves, by finite differences, the piezoelectric equations which relate he mechanical and electrical phenomena produced in this kind of material. The behavior in time domain of some physical variables, as the electric field and particle velocity, is simulated too. In addition, we have implemented a GUI in Matlab that allows simple and affordable simulations for all users. The implemented interface provides two kind of study: Broadband and single frequency excitation. The output parameters, obtained after time domain simulation, are piezoelectric constants of the ceramic and the electric response in frequency domain, i.e. admittance and impedance curves. The results allow predicting vibration modes, electrical response and understanding the effect of piezoelectricity on the mechanical response.

A large number of fractals known, as Julia fractals and Mandelbrot, can be generated from an iterative method. In this paper we present a virtual laboratory developed as a Graphical User Interface (GUI) of Matlab that allows us to study and visualize in real time the relationship between Newton iterative methods of two variables and the generation of fractals. The main objective is to allow Technical School students in Numerical Computation subjects to acquire the skills to generate fractals and interpret their plots in terms of the convergence or divergence speed of the sequence of iterated.

The purpose of this paper is to develop and investigate a non-linear model for analysing the reaction of a self-detachable cabin monocable ropeway exposed to a sudden deceleration and wind forces. The First and Second Newton's Law and Differential Equations are the basic tools for building the model. Furthermore a few basic considerations have been made about the air “dragging and lifting" forces that induce oscillations and vibrations in mechanical systems alike. All the numerical data used for the simulation was taken from a ropeway in the skiing site of Ravascletto-Zoncolan in the North- East of Italy.

We present a classroom task that consists to model the path of the road between two paraboloid surfaces. The purpose of this paper is to teach the differential calculus using mathematical modeling as a teaching tool. The context of the engineering design of a highway serves as a vehicle for teaching and learning the mathematical contents necessary for driving the task.

Linear elasticity is a discipline that studies the elastic solid under external forces that generate a mechanical response in linear elastic regime. In particular, it is of special interest to study the stresses and strains at any point of the solid, which is known as the elastic problem. Moreover, the problem of plane stress state -that is, in two dimensions- it is very common in engineering. Prior to its study, it is necessary to first obtain the so-called stress state in the differential vicinity around a point of the elastic solid, then calculating the different values that can reach the tensions at that point in any direction. This paper shows a virtual laboratory based on the graphical method called Mohr's Circle -attributed to engineer Otto Christian Mohr (1835-1918)- which allows to represent the stress state at a point from a user-defined set of variables. The virtual laboratory MOHR, implemented with the MATLAB package through its graphical user interface (GUI), allows full interactivity and can obtain the values of the stresses in any direction. This application is an excellent tool to support the teaching and learning process of mechanics-related subjects in engineering studies.

This paper presents the generation of encryption keys using the local oscillating properties of the partial sums of Weierstrass-Mandelbrot fractal function. In this way, the security key can be replicated if the parameters used to obtain it are known. Therefore, these parameters can be sent instead of sending the key. This procedure reduces the amount of information to be sent and prevents possible interception of the key. Moreover, the key can not be affected by data loss or pollution. The effectiveness of the Weierstrass-Mandelbrot keys were demonstrated by computer simulation in a 4f optical encryption system and the double random phase encoding technique. These keys allow us to encrypt and to retrieve the information, which is the evidence of their effectiveness for data security. Furthermore, it is shown that if the encrypted data is obtained under an attack, it is not possible to access the information without the security key.

The growing importance of skills in our education system emphasizes the need not only of methodological changes in the teaching of mathematics in compulsory secondary education, but also to design new tools to develop the basic competences in general and mathematical competence in particular our students. The problems of modelling, in the context of realistic mathematics education, become an important tool for this purpose. In this paper we describe the design and implementation of a pilot activity modelling with students aged 15-16 as a first step in the introduction of this type tasks in the programming course.

Clothoids, also known as radius of curvature or Cornu spirals, take their name from the greek word Klothó, meaning, weaver. They are also widely used as transition curves in railroad or highway engineering for connecting and transiting the geometry between a tangent and a circular curve (to join straight sections with curve transitions or to connect two circular sections of different curvatures). This is their most relevant characteristic, since the radius of curvature diminishes in inverse proportion to the distance run over it, and that allows the driver to adapt slowly to the change in the trajectory. The new highways are designed including a series of clothoids with a large curvature which involves a greater visibility in the distance and easy adaptation to the track. In this paper we take into account the geometric property which defines clothoids and we obtain its equation from Fresnel integrals. Secondly, we represent the properties derived geometrically and obtain a numerical approximation of the clothoid. Finally, we use these approximations to present an example.

The capacity and computing power of programs like Matlab enables first year students of a college degree are able to model using the computer both topographic maps and roads on them, as acoustic maps. In other words, moving from paper to digital model. This article shows several collaborative works made over many years by Gandia Campus students of UPV. In the models developed is essential to use matrices and operations with them, and interpolating surfaces.

In this paper, we present a software tool entitled ADGIP (version 2012), with free access under the Creative Commons licence, for the modelling and resolution of decision trees for the multi-stage decision making and that contemplates the evaluation of the incorporation of additional information by using the Bayes theorem. Finally, it is presented an illustrative example of a basic tree decision and a method of easy replication of the tree parts for considering the sucesive stages in a serial way of the evaluation of the acquisition of additional information.

Mathematical modeling is a fast and low cost tool used as a first approach to solve certain problems in real contexts. In this paper we present an example of how a real clinical problem, in the context of cardiac surgery, can be simulated and responded to through mathematical models. Specifically, the resolution of these models, which are based on a coupled electric-thermal problem, allows us to assess in detail the temperature distribution and the geometry of the thermal lesion created in the ventricular wall after radiofrequency ablation in order to eliminate cardiac arrhythmias.

In this paper, we present an example of linear optimization in the context of degrees in Economics or Business Administration and Management. We show techniques that enable students to go deep and investigate in real problems that have been modelled using the Excel platform. The model shown here has been developed by a student and it consists in minimizing the absolute deviations over the average expected return of a portfolio of securities, using the solver tool that it is included in this software.

In the present paper we consider to determine the relative position between a point and a simple polygon in a plane. For this purpose we build a model of the polygon, which manipulation carries us, in a very natural way, to the solution of this problem and it allows us to arise a more ambitious problem based on the same model: the determination of the polygon area. This geometric example is useful to illustrate the creation process in Mathematics. It is especially interesting the way in which the formalization of a simple problem with a successful notation and a great dose of algebraic exploration results in a rediscovery of an interesting geometric result which is much more important than the original objective.

In this paper a didactic model for the control of the speed of the vehicles in a road is presented. Based on empirical assumptions, the model provides an answer to the problem of finding the best road from the point of view of the time that is needed for going from the top of a hill to a certain point in the valley under the assumption that certain restrictions for the speed are imposed. Thus, we construct a simple model for representing the time dependence of the traffic in a road as an exercise of mathematical modelling adequate for a basic level of a civil engineering career.