PARITARY PRIVATE HIGH SCHOOL
DECREE N.338 MITF005006
DECREE N.1139 MITNUQ500H
DECREE N.2684 MIPMRI500E
IT

MATHEMATICS - GENERAL OUTLINES AND SKILLS

MATHEMATICS - GENERAL OUTLINES AND SKILLS


At the end of the course of the humanities high school (economic-social option), the student will know the elementary concepts and methods of mathematics, both internal to the discipline itself considered and relevant to the description and prediction of phenomena, both in the classical sphere of the physical world and in the social and economic sphere. He or she will be able to place the various mathematical theories studied in the historical context within which they developed and understand their conceptual significance.
The student will have acquired a historical-critical view of the relationships between the main themes of mathematical thought and the philosophical, scientific and technological context. In particular, he or she will have acquired the sense and scope of the three main moments that characterize the formation of mathematical thought: mathematics in the Greek civilization, the infinitesimal mathematics that arose with the scientific revolution of the seventeenth century and led to the mathematization of the physical world, the turning point that took its cue from Enlightenment rationalism and led to the formation of modern mathematics and a new process of mathematization that invested new fields (technology, social sciences, economics, biological sciences) and changed the face of scientific knowledge.


Hence the groups of concepts and methods that will be the focus of the study:


1) the elements of Euclidean geometry of the plane and space within which the procedures characteristic of mathematical thinking (definitions, demonstrations, generalizations, axiomatizations) take shape;
2) the elements of algebraic calculus, the elements of Cartesian analytic geometry, a good knowledge of the elementary functions of analysis, the elementary notions of differential and integral calculus;
3) the basic mathematical tools for the study of physical phenomena, with special emphasis on vector calculus and differential equations, especially Newton's equation and its elementary applications;
4) elementary knowledge of some developments in modern mathematics, especially the elements of probability calculus and statistical analysis;
5) the concept of a mathematical model and a clear idea of the difference between the view of mathematization characteristic of classical physics (unambiguous correspondence between mathematics and nature) and that of modeling (possibility of representing the same class of phenomena by different approaches);
6) construction and analysis of simple mathematical models of classes of phenomena, including using computer tools for description and computation, with special emphasis on social-economic modeling;
7) a clear understanding of the characteristics of the axiomatic approach in its modern form and its specifics with respect to the axiomatic approach of classical Euclidean geometry;
8) an acquaintance with the principle of mathematical induction and the ability to know how to apply it, also having a clear idea of the philosophical significance of this principle ("invariance of the laws of thought"), its diversity with physical induction ("invariance of the laws of phenomena"), and how it constitutes an elementary example of the not strictly deductive character of mathematical reasoning.

his articulation of themes and approaches will form the basis for establishing conceptual and methodological connections and comparisons with other disciplines such as physics, natural and social sciences, philosophy and history.
At the end of the course, the student will have deepened the procedures characteristic of mathematical thinking (definitions, demonstrations, generalizations, formalizations), will know the basic methodologies for the construction of a mathematical model of a set of phenomena, and will be able to apply what has been learned to the solution of problems, also using computer tools of geometric representation and calculation. These operational skills will be particularly developed in the area of mathematical modeling of social and economic processes. The student will deepen the critical evaluation of the advantages, difficulties and limitations of the mathematical approach in such a highly complex field.
The computer tools available today provide suitable contexts for representing and manipulating mathematical objects. The teaching of mathematics offers numerous opportunities to become familiar with such tools and to understand their methodological value. The course, when appropriate, will encourage the use of these tools, also with a view to their use for data processing in other scientific disciplines. The use of computer tools is an important resource that will be introduced critically, without creating the illusion that it is an automatic means of problem solving and without compromising the necessary acquisition of mental calculation skills.
The wide range of content covered will require the teacher to be aware of the need for good use of available time. While maintaining the importance of the acquisition of techniques, dispersion in repetitive technicalities or sterile case studies that do not contribute significantly to the understanding of problems will be avoided. In-depth study of the technical aspects will never lose sight of the goal of in-depth understanding of the conceptual aspects of the discipline. The main indication is: a few fundamental concepts and methods, acquired in depth. 

SPECIFIC LEARNING OBJECTIVES

FIRST TWO YEARS

ARITHMETIC AND ALGEBRA

The first two years will be devoted to the transition from arithmetic to algebraic calculus. The student will develop skills in calculating (mentally, with pen and paper, using tools) with whole numbers, with rational numbers both in writing them as fractions and in decimal representation. In this context, the properties of operations will be studied. The study of the Euclidean algorithm for determining the MCD will provide a deeper understanding of the structure of whole numbers and an important example of an algorithmic procedure. The student will gain an intuitive knowledge of the real numbers, with special reference to their geometric representation on a straight line. The demonstration
of the irrationality of 2 and other numbers will be an important opportunity for in-depth
conceptual. The study of irrational numbers and the expressions in which they appear will provide a
significant example of the application of algebraic calculus and an opportunity to address the subject.
The student will acquire the ability to perform calculations with literal expressions both to represent a problem (by means of an equation, inequalities or systems) and solve it, and to demonstrate general results, particularly in arithmetic.

GEOMETRY 

The first two years will focus on knowledge of the fundamentals of Euclidean plane geometry. The importance and meaning of the concepts of postulate, axiom, definition, theorem, and proof will be clarified, with special emphasis on the fact that, beginning with the Elements of approximation. The acquisition of methods of calculating radicals will not be accompanied by excessive manipulative technicalities. The student will learn the basic elements of literal calculus, the properties of polynomials and the simplest operations between them. Euclid, they have permeated the development of Western mathematics. Consistent with the way it has presented itself historically, the Euclidean approach will not be reduced to a purely axiomatic formulation. Special attention will be devoted to the Pythagorean theorem so that both its geometrical aspects and its implications in number theory (introduction of irrational numbers) will be understood, insisting above all on its conceptual aspects. The student will acquire knowledge of the main geometric transformations (translations, rotations, symmetries, similarities with special emphasis on Thales' theorem) and will be able to recognize the main invariant properties. Elementary geometric constructions will be carried out using both traditional tools (particularly the ruler and compasses, emphasizing the historical significance of this methodology in Euclidean geometry) and geometry computer programs. The student will learn to make use of the Cartesian coordinate method, at first limited to the representation of points and lines in the plane and properties such as parallelism and perpendicularity. The intervention of algebra in the representation of geometric objects will not be divorced from a deepening of the conceptual and technical scope of this branch of mathematics.

Relations and functions Objective of study will be the language of sets and functions (domain, composition, inverse, etc.), also to construct simple representations of phenomena and as a first step to the introduction of the concept of mathematical model. In particular, the student will learn to describe a problem with an equation, inequality or system of equations or inequalities; to obtain information and derive solutions of a mathematical model of phenomena, including in contexts of operations research or decision theory. The student will study functions of the type f(x) = ax + b, f(x) = |x|, f(x) = a/x, f(x) = x2 both in strictly mathematical terms and as a function of describing and solving applied problems. He will know how to study the solutions of first-degree equations in one unknown, associated inequalities, and systems of linear equations in two unknowns, and will know the techniques necessary for their graphical and algebraic resolution. He/she will learn the elements of the theory of direct and inverse proportionality. The student will be able to move easily from one register of representation to another (numerical, graphical, functional), including using computer tools to represent data.

Data and Predictions The student will be able to represent and analyze in different ways (including using computer tools) a set of data, choosing the most suitable representations. He/she will be able to distinguish between qualitative, discrete quantitative and continuous quantitative characters, operate with frequency distributions and represent them. Definitions and properties of mean values and measures of variability will be studied, as well as the use of computational tools (calculator, spreadsheet) to analyze data collections and statistical series. The study will be done as much as possible in conjunction with other disciplines, including in areas within which data are collected directly by students. The student will learn the notion of probability, with examples drawn from classical contexts and with the introduction of notions of statistics. He/she will learn to draw simple inferences from statistical diagrams.The concept of a mathematical model will be explored rigorously, distinguishing its conceptual and methodical specificity from the classical physics approach.

Elements of computer science The student will become familiar with computer tools, for the primary purpose of representing and manipulating mathematical objects, and will study ways of representing elementary textual and multimedia data.
A fundamental theme of study will be the concept of algorithm and the development of strategies of algorithmic resolutions in the case of simple and easily modeled problems; and, also, the concept of computable function and computability and some simple related examples.

SECOND TWO YEARS

ARITHMETIC AND ALGEBRA

The student will learn to factor simple polynomials, will be able to perform simple cases of division with remainder between two polynomials, and will explore its analogy with division between integers. He/she will learn the elements of vector algebra (sum, multiplication by scalar and scalar product), and understand its fundamental role in physics. The study of the circumference and the circle, the number π, and contexts in which exponential growths appear with the number e, will enable a deeper understanding of the real numbers, with regard to the subject of transcendental numbers. Through an initial understanding of the problem of formalizing real numbers, the student will be introduced to the problem of mathematical infinity and its connections with philosophical thought. He or she will also acquire the first elements of approximate calculus, both from a theoretical point of view and through the use of computational tools.

GEOMETRY

Conic sections will be studied from both a synthetic and analytical geometric point of view. In addition, the student will deepen the understanding of the specificity of the two approaches (synthetic and analytic) to the study of geometry.
He/she will study the properties of the circumference and the circle and the problem of determining the area of the circle. He will learn the definitions and elementary properties and relations of circular functions, theorems that allow the resolution of triangles and their use in the context of other disciplines, particularly physics.
Will study some significant examples of geometric locus.
It will address the extension to space of some themes and techniques of plane geometry, also in order to develop geometric intuition. In particular, he/she will study the reciprocal positions of lines and planes in space, parallelism and perpendicularity.

Relations and functions The student will learn the study of quadratic functions; to solve second-degree equations and inequalities; and to represent and solve problems using second-degree equations.

He/she will study the elementary functions of analysis and their graphs, especially polynomial, rational, circular, exponential and logarithmic functions.
He/she will learn to construct simple models of exponential growth or decrease, as well as periodic trends, also in relation to the study of other disciplines; all this in both a discrete and continuous context. The acquisition of particular skill in solving equations and inequalities in which these functions appear will not be required; this skill will be limited to simple and significant cases. The student will also know how to use a logarithmic or semilogarithmic reference system.
The student will learn to analyze graphs of major functions, identify and analyze characteristics of functions, work with compound and inverse functions, and make qualitative reasoning about functions. He/she will learn the notion of average growth and the concept of rate of change of a process represented by a function.
Data and predictions The study of conditional and marginal double distributions, the concepts of standard deviation, dependence, correlation and regression, and sample; and also conditional and compound probability, Bayes' formula and its applications, and the basic elements of combinatorial calculus will be addressed.
In connection with the newly acquired knowledge, the concept of a mathematical model will be deepened.
The use of mathematics in social and economic disciplines According to a modeling approach will be deepened. An important theme in this high school will be the mathematical foundations of microeconomic theory, the fundamentals of utility theory, and the basic elements of the Keynesian macroeconomic model.

 

FIFTH YEAR

GEOMETRY

The student will learn the first elements of analytic geometry of space and the analytic representation of lines, planes and spheres. Relations and functions. The student will deepen the study of the fundamental functions of analysis including through examples from physics or other disciplines. He/she will acquire the concept of limits of a succession and a function and learn to calculate limits in simple cases. The student will acquire the main concepts of infinitesimal calculus-particularly continuity, derivability and integrability-also in relation to the problems in which they originated (instantaneous velocity in mechanics, tangent of a curve, calculation of areas and volumes). No special training in calculus techniques will be required, which will be limited to the ability to derive functions already studied, simple products, quotients and compositions of functions, rational functions, and the ability to integrate integer polynomial functions and other elementary functions, as well as to determine areas and volumes in simple cases. Above all, the main objective will be to understand the role of infinitesimal calculus as a fundamental conceptual tool in describing and modeling physical or other phenomena. In particular, the general idea of optimization and its applications in numerous areas, particularly in the economic and social fields, will be explored.

Data and predictions The student will learn the characteristics of some probability distributions (in particular, the binomial distribution and some examples of continuous distributions).In connection with the newly acquired knowledge, including in the area of the relations of mathematics with other disciplines, the student will have further deepened his understanding of the concept of mathematical models and developed the ability to construct and analyze examples of them. He will also have deepened his knowledge of the elementary foundations of microeconomic theory (marginal utility, general equilibrium and its mathematical formalization), macroeconomics and econometrics.

 

 


 


S. Freud Paritary Institute - Private School Milan - Paritary School: IT Technical Institute, Tourism Technical Institute, High School of Human Sciences and High School
Via Accademia, 26/29 Milano – Viale Fulvio Testi, 7 Milano – Tel. 02.29409829 Virtuale fax 02.73960148 – www.istitutofreud.it
Milan High School - Private IT School Milan
Milan Private Tourism School - Human Sciences High School, Social and Economic Address Milan
Liceo Scientifico Milano
Contact us for more information: [email protected]

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