OVERVIEW

The course has the aim to give basic mathematical tools ; it is developed in the first semester, with six hours of lessons each week. the students come from different types of schools and in various cases they show serious lacks as regards competences and abilities. For these reasons, the initial part of the course is devoted to recall, re–elaborate and deepen some important background concepts.

AIMS AND CONTENT

LEARNING OUTCOMES

Aim: to provide basic mathematical tools suitable to the construction of "models" for solving problems

AIMS AND LEARNING OUTCOMES

Aims

1) To give basic instruments directed to the understanding, the planning and the solution of problems, to the construction and recognition of models (especially graphics).

2) To lead towards a scientific approach to the learning.

3) To see as Mathematics gives "unifying" language and methodologies through models and structures.

Learning outcomes

At the end of the course one expects that the student reaches the ability:

1) to understand the text of an exercise/problem

2) to sketch out solutions of problems through suitable tools (choosen among the ones provided in the course)

3) to solve exercises/problems in a "non mechanical", reasoned and autonomous way

4) to express results and conclusions in a clear and precise way

5) to develop "scientific" way of thinking more than to acquire technical tools.

TEACHING METHODS

Usually, the classroom–taught lessons consist of theoretical part and exercises, that in some cases get the students involved in the subject in an operative way. On the contrary, because of the state of emergency due to COVID–19 and according to the rectoral indications for teaching "phase 3", in the school year 2020–21 the Mathematics course lessons (which are developed in the first semester) will be performed by remote mode, on the online platform TEAMS.

SYLLABUS/CONTENT

Introduction of the ideas of "structure" and "model", with a “short trip” in history. Sets and structures, in particular: natural, integers, rational and real numbers. Function of a real variable, with various examples of physical or chemical character. Graphs. Cartesian coordinate systems in the plane. Recognition of special "symmetries”. Overview and motivation on elementary functions: linear, quadratic, irrational, trigonometric, exponential and logarithmic, absolute value, presented as the suitable models to describe the behavior of some aspects of the nature. Functions obtained as sum, product, ratio or composition of elementary ones. Problems related to inequalities. Resolution of some types of inequalities, and their graphical interpretation. Concepts of "continuity" and "limit" .Derivatives: historical reasons, geometrical interpretation, derivatives of the above considered functions. Applications to the study of the graph of a function and to optimization problems.

Integration theory. Concept of "defined integral", connected with the problem of computing areas in the plain. Primitives, "fundamental theorem of the integral calculus." Main methods of integration. Examples of differential equations of the first order and related physical and chemical problems.

Fundamentals of probability theory. Sample space, probability of an event in the "discreet" case. Probability of exclusive, conditioned, independent events. "Binomial" probability. Breaf reference to the "continuous" case and Gaussian distributions.

RECOMMENDED READING/BIBLIOGRAPHY

Appunti di Matematica per CTF ( G.Tamone, "appunti interni" )

MATEMATICA DI BASE Esercizi svolti, testi d'esame,richiami di teoria, di A.M.Bigatti e G.Tamone, Editrice Esculapio (2016)

Further reference

Matematica di base , di A.M.Bigatti e L.Robbiano , Editrice Ambrosiana  ( 2014 )

TEACHERS AND EXAM BOARD

Exam Board

EMANUELA DE NEGRI (President)

GRAZIA TAMONE (President)

ANNA MARIA BIGATTI

FABIO DI BENEDETTO

Anna Maria MASSONE

LESSONS

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

At the present time, because of the state of emergency due to COVID, we cannot predict whether (or when) it will be posssible to return to "face–to–face" exams.                                                                                                                      – If it will be necessary to continue with the remote mode, the new exam modalities will be defined as soon as possible and communicated on Aulaweb.                                                                                                                         – If it will be possible to carry out "face–to–face" exams, we will return to the usual methods listed below under the title "FACE–TO–FACE EXAMS".                                                                                                                                              – Anyhow, from time to time before each exam session the modality willl be confirmed/updated well in advance.

FACE–TO–FACE EXAMS

The usual examination consists of a written exam, composed of three exercises, regarding the main parts of the syllabus, that must be solved with suitable explanations, within two hours. After the correction, the papers are analysed and discussed with the students.

If the written exam has obtained a mark ≥ 18:                                                                                    

•  an oral exam is mandatory only in the following cases:

(a) one of the exercises has reached less than half of the assigned score                                        

(b) the given explanations are completely or partially very lacking and generate doubts on the real understanding                                                                                                                    

•  no oral exam is required except cases (a), (b)                                                             

•  in any case, the student can ask also an oral exam.

If the written proof has obtained an unsatisfactory mark between 13 and 17,  there are two possibilities:           

• to apply for an oral exam ( usually consisting of  "exercises" and managed by two tenured teachers).                                                                                                                            

• to apply to one of the other scheduled exam sessions

If the written exam has obtained a mark < 13 : the student must apply to another session, but not earlier than 28 days.

During the written exam it is allowed to consult a sheet of paper of notes (size A4, two sides) and the use of a calculator: other types of notes, mobil phones, smartphones, personal computers, books are not allowed.

The sheet of the notes can be handwritten, typesetted, coloured, photocopied, according to the personals choice. It is suggested to prepare it personally.

For each session, the deadline for the registration is 5 days before the date of the exam, as one can deduce from the procedure of the online registration, which is mandatory . No exception will be allowed and the not registered students will not be able to apply to the exam.

Further, as above written, the students which have obtained a mark less than 13 in one session, are not allowed to apply to another session earlier than 28 days.

ASSESSMENT METHODS

Besides the scheduled examinations, during the year there are dispensed some practice exercises, for self–evaluation. The examination must check the students have learned the required expertise, and are able to make use of these comptences and to express them with correct terms, as explained in the following.

The written exam will judge the capability to solve exercises related to the main topics developed in the course, by giving suitable explanations on the procedure and clearly expressed conclusions. The oral exam (when needed or requested) will essentially deal with exercises, with particular reference to the possible lacks or difficulties put in evidence by the written exam; further the oral exam will judge the capability to understand and solve a problem with critical thinking , to use suitable tools and language.

Criteria of judgment:                                                                                                                                                                                              – ability to think critically and with coherence                                                                                                                                                       – speaking ability through a sitable "scientific language"                                                                                                                                     – time needed to give a satisfying answer                                                                                                                                                             – level of autonomy

Exam schedule