Proposing a Class with Mathematical Modeling. Biembengut and Hein Model (2007)

ORIGINAL ARTICLE

MALAQUIAS, Helbert Santana ^[1]

MALAQUIAS, Helbert Santana. Proposing a Class with Mathematical Modeling. Biembengut and Hein Model (2007). Revista Científica Multidisciplinar Núcleo do Conhecimento. Year 06, Ed. 03, Vol. 14, pp. 75-84. March 2021. ISSN: 2448-0959, Access link: https://www.nucleodoconhecimento.com.br/education/mathematical-modeling

ABSTRACT

This article proposes a class for students between 13 and 14 years of elementary school applying mathematical modeling and using the mathematical model of Biembengut and Hein (2007). This article aims to present the construction of a mathematical model, through the calculation of plans of two cell phone operators operating in the metropolitan region of Belo Horizonte, to identify which plan will be more into account for each student, according to their profile. This work can be developed with elementary school students and uses the mathematical model of Biembengut and Hein (2007), which proposes the stages of interaction (recognition and familiarization with the problem situation), matematization (formulation and resolution of the problem), creation of the mathematical model (interpretation of the model) and verification of its suitability (validation). It also serves as a proposal for a lesson plan for the elementary school mathematics teacher in the classes of functions and inecdation of the first degree. It is worth mentioning that this class is a proposal, and it can be used with other types of situations and other contents. The methodology adopted in this study was bibliographic review. With this research, in addition to discovering the cell plan with the best cost-benefit, according to the profile of each student, how many minutes each student uses monthly, one can develop the contents of functions and inecities of the 1st degree, contents of 8th and 9th grade of elementary school. Thus, it is observed that students will be more happy to study these contents, using Mathematical Modeling.

Key Words: Mathematical modeling, cell phone plans, elementary school.

1. INTRODUCTION

This article proposes a class for students between 13 and 14 years of elementary school applying mathematical modeling and using the mathematical model of Biembengut and Hein (2007). It is a class of functions and inecdation of the first degree, aiming to identify the cell phone plan with the best cost-benefit according to the profile of each student. From this perspective the understanding of mathematical modeling described by Granger (1969) apud Biembengut and Hein (2003), explains it as an art, when formulating, solving and elaborating expressions that are worth not only for a particular solution, but that also serve, later, as support for other applications and theories. In this article we will use the modeling steps presented by these authors in the proposition of a modeling on possibilities of cell phone plans available in the market today. Thus, in the proposition of models in the classroom, it is possible to understand that the use of models as a representation of an object or interpretation of a reality is used by man in the constitution of the knowledge of humanity from very remotely. Researchers, scholars and engineers use models to perform simulations, observations, and constructions. Models or templates for artwork are examples of this use. In the session on teaching-learning mathematics and the use of modeling, a brief discussion on teaching-learning mathematics and the use of mathematical modeling in school. It is noteworthy that there are several types of difficulties with regard to the teaching-learning of mathematics in elementary school, both on the part of students and by teachers, thus being able to describe as easily identifiable the students’ disinterest in the discipline; the difficulty in abandoning the paradigm of exercise; the existence of the strong influence of the traditional school (teacher who owns knowledge); the absence of teaching strategies and more dynamic and contextualized methodologies and others more. In the construction of the model in cell phone plans, the purpose is to research on Mathematical Modeling and propose a class using mathematical modeling techniques, elementary school content sums and information from cell phone companies that are part of the student’s social context.

Thus, this article is an invitation to observe a model that can facilitate the teaching learning of elementary school students and assist the teaching staff. We conclude with the identification of the benefits and adequacy of each plan to different user profiles that can be constructed and explored by students from the construction of mathematical function concepts.

2. DEVELOPMENT

3. THE PROPOSITION OF MODELS IN THE CLASSROOM

In conceptual terms the model can designate the representation of something, a pattern or ideal to be achieved, in a production can be a particular type within a series. For Granger (1969) apud Biembengut and Hein (2003) model is an image that is formed in the mind, at a time when the rational spirit seeks to understand and express intuitively a sensation seeking to relate it to something already known, making deductions. In this case the model does not always refer to a physical object, but it can represent a structure of “mathematical symbols and relationships that seeks to translate, in some way, a phenomenon in question or problem of real situation”.

From this perspective the understanding of mathematical modeling described by Granger (1969) apud Biembengut and Hein (2003), explains it as an art, when formulating, solving and elaborating expressions that are worth not only for a particular solution, but that also serve, later, as support for other applications and theories. In this article we will use the modeling steps presented by these authors in the proposition of a modeling on possibilities of cell phone plans available in the market today.

The proposal is to develop a model with elementary school students.Mathematical modeling as a teaching learning methodology is recognized by authors such as D’Ambrósio (2002), Bassanezi (2002) and others.

It follows a brief discussion on teaching-learning mathematics and the use of mathematical modeling in school. In the third part we present the stages of modeling according to Biembengut and Hein (2007) and the proposal of the model of cell phone plans.

4. TEACHING-LEARNING MATHEMATICS AND THE USE OF MODELING

There are, of course, innovative pedagogical work initiatives in mathematics such as those presented by Ole Skovsmose (2006); Ubiratan D’Ambrosio (2002); João Pedro da Ponte (2003) and others; with emphasis on the contextualization of mathematical knowledge, the exploration of mathematical situations, investigative practice and mathematical modeling.In this work proposal, we decided to research a fact of the student’s social context (cell plans) and unite with the school context, using a content of elementary school mathematics using the proposal of mathematical modeling.

D ́Ambrosio (2002) argues that the cycle of knowledge acquisition is triggered from the facts of reality; the construction of mathematical knowledge and can be more efficient if it emerges from phenomena that originate in reality. Mathematical Modeling allows establishing a relationship between the mathematics of school programs and the reality of the student.

Thus, we elaborate the construction of a model for students to know which cell plan is cheaper when acquiring. It is worth mentioning that this subject is of interest, not only of the students, but of all of society, because it brings economy and less annoyances.

Using mathematical modeling, we will propose a class with the construction of a model that approximates the student’s social context and the school context. The elaboration of the model can serve as an example of a lesson plan for the elementary school mathematics teacher, who can develop in other subjects and other contents.

Beatriz D’Ambrosio (2005) states that “Mathematical Modeling is characterized as a way to break the dichotomy between formal school mathematics and its usefulness in real life”. That is why it serves as an incentive for elementary school students and teachers. But for this, the researcher of Mathematical Modeling, according to his authors Biembengut (2000), needs to go to the field to recognize the problem situation, for familiarization with the theme to be modeled. Most of the time, try to understand facts, elaborate and assign meanings to the models, using mathematics for this, regardless of whether it is the teacher or if it is the students who choose the theme. Choose a theme of the student’s reality and apply it to the content studied, or, use the content to solve a frequent problem in our reality. Mathematical Modeling allows establishing a relationship between the mathematics of school programs and the reality of the student.

Thus, the interest of the student arouses, because it deals with matters of interest. According to Bassanezi (2002) modeling can be a way to arouse greater interest of the student in learning mathematics.

5. THE CONSTRUCTION OF THE MODEL IN CELL PHONE PLANS

According to D ́Ambrosio (2002) the cycle of knowledge acquisition is triggered from the facts of reality; the construction of mathematical knowledge can be more efficient if it emerges from phenomena that originate in reality. Mathematical Modeling allows establishing a relationship between the mathematics of school programs and the reality of the student, which is above all a perspective of mathematical matematization of reality and methodology for the pedagogical practice of the mathematics teacher, something to be explored that focuses on reality and mathematical knowledge.

In the course of this work, it is possible to elaborate a class with Mathematical Modeling, using data from the social context of students of cell phone plans, which is now something very common among them, and a content of Mathematics, where we will answer a problem: which plan will I pay cheaper at the end of the month? We will research, through this lesson plan, how Mathematical Modeling can help the mathematics teacher to work mathematics in a more contextualized way.

It aims to research relevant facts that can motivate us to work mathematical modeling, of course, we should not forget that according to Mathematical Modeling is one of the methods to learn and teach Mathematics in a contextualized way, an option.

Cell phone plans are common among students, due to their access to cell phones today. Because it is a subject of interest to all, it brings economy and is part of the social context of the student. Thus, the social context of the student and the school context can be joined. It also aims to identify the mathematical model that solves the problem and responds which cell plan is cheaper; propose a class with Mathematical Modeling that assists the elementary school mathematics teacher, using the content worked.

5.1 MODELING STEPS ACCORDING TO BIEMBENGUT AND HEIN (2007).

• In the first column are the steps performed in the Biembengut Hein model (2007), in mathematical modeling. Steps according to these authors that are steps to apply modeling.
• In the second column is the methodological choice of our study. This frame can be used in any situation we use Mathematical Modeling.

The following table presents the scheme proposed by Biembengut and Hein (2007, p. 15).

Table 1: Scheme proposed by Biembengut and Hein (2007)

Source: Stages of Modeling elaborated by Biembengut and Hein (2007, p.15).

5.2 PLANNING THE PROPOSED LESSON

The theme will be developed in two classes proposed for classrooms of 8th and 9th grade of elementary school, whose average age is between 13 and 16 years. Two 50-minute classes each will be used. We will divide 3 groups of 5 students each. In this lesson, we will use the modeling steps according to Biembengut and Hein (2007):

1. Interaction – recognition of the problem situation

1.1 The class proposal aims to identify the best cell phone plan for today’s elementary school adolescents.

1.2 Familiarization with the subject to be modeled; ask students to research two cell phone companies to find out the amounts of fees charged per minute and fixed fees. The companies Vivo and Oi celular were searched and the following rates were found:

Plan 1: Vivo; Fixed fee R $ 42,00 with the right to 50 minutes of connection, each minute of excess connection, will be charged a value of R $ 0.72 per minute of connection.

Plan 2: Oi; Fixed fee R $ 51,90 with the right to 60 minutes of connection, each minute of excess connection will be charged a value of R $ 0.69 per minute of connection.

2. Matematization

2.1 Problem formulation – Hypotheses.

2.2 Resolution of the problem at the end of the model. By evaluating rates and profiles, we build the following formula to identify the best plan.

Plan 1. Vivo: f(x)= 42+(x-50).0.72

Plan 2. Oi: f(x)= 51.90+(x-60).0.69

In the formula, we call the variable x minutes of binding.

The variable y or f(x), we call the total amount to be paid.

We identified a 1st degree function for each carrier, because the total amount to be paid is depending on the amount of minutes spent (x).

3. Mathematical model

the. Interpretation of the model. By comparing the two functions, such as:

1) When Vivo’s plan 1 will have the lowest cost-benefit.

Vivo                           Oi

42+(x-50).0.72 < 51,90+(x-60).0,69

42+0,72x-36 <    51,90+0,69x-41,40

0,72x-0,69x < 51,90+36-41,40-42

0,03x < 4,50

X < 150

2) When Oi’s plan 2 will have the lowest cost-benefit ratio.

Vivo                           Oi

51.90+(x-60).0.69 < 42+(x-50).0,72

51,90+0,69x-41,40 < 42+0,72x-36

0,69x-0,72x < 42+41,40-36-51,90

-0,03x < -4,50 . (-1)

X > 150

6. VERIFICATION OF ITS SUITABILITY – VALIDATION

The model above meets the objective of the class, answers which plan is most cost-effective according to each student’s profile. The students used function, function comparison, inecdation of the 1st grade, which should be concepts already studied in the previous grades.

7. FINAL CONSIDERATIONS

By the model above and aware that the operator with the lowest cost-benefit will be the best option and observing the profile of each student, we conclude that: if the student spends less than 150 minutes of connection per month, the plan 1 of the Operator Vivo will have a better cost-benefit. If the student spends more than 150 minutes of connection per month, Oi carrier plan 2 will have a better cost-benefit. If the student spends exactly 150 minutes of connection per month, any of the options will do. It is worth mentioning that this class is a proposal, and it can be used with other types of situations and contents. In addition to discovering the best cost-benefit plan, it was possible to develop the contents of functions, inefactors of the 1st grade, contents of 8th and 9th grade of elementary school, thus, it is observed that students will be more happy to study these contents, using mathematical modeling.

REFERENCES

BIEMBENGUT, M. S; et. al, Modelagem matemática no ensino – 4ª ed. – São Paulo: Contexto, 2005.

BIEMBENGUT, M. S; HEIN, N. Modelagem matemática no ensino. São Paulo: Contexto, 2007.

D`AMBROSIO, U. A matemática nas escolas. Educação Matemática em Revista, ano 9 no 11A, edição especial, abril de 2002.

CIDADE, C.; FIOREZE, L. A. Modelagem Matemática na Conta de Luz. 2008. Disponível em: http://arquivo.sbmac.org.br/eventos/cnmac/xxxi_cnmac/PDF/459.pdf

BASSANEZI, R.C. Ensino –aprendizagem com modelagem matemática: uma nova estratégia. São Paulo: Contexto, 2007.

SCANDIUZZI, Pedro Paulo. Água e Óleo: Modelagem e Etnomatemática? In: Bolema, Ano 15, nº 17, 2002, PP.52 a 58.

^[1] Postgraduate in Instrumentalization for the teaching of Mathematics; graduated in full degree in Mathematics; Bachelor of Theology. Professional and Self Coach.

Submitted: February, 2021.

Approved: March, 2021.