MATHEMATICS BACKGROUND

The course is mainly aimed at students enrolled with additional educational obligations (OFA). That is, students who have not reached the minimum threshold in the TOLC-I entrance test (https://www.cisiaonline.it/area-tematica-tolc-ingegneria/home-tolc-ingegneria/ ). However, the course is STRONGLY RECOMMENDED for all first year students to review the basic knowledge necessary for the subsequent Calculus course.

The aim of the course is to briefly review the mathematics knowledge useful for facing the teachings of the first year of the degree course and successfully pass the TOLC-I test (see the mathematics sections of the TOLC-I syllabus https: //www.cisiaonline.it/area-tematica-tolc-ingegneria/struttura-della-prova-e-syllabus/).

By the end of the Mathematics Background course, students will have a firm grasp of the concept of function and how it is applied in various areas of mathematics. They will be able to identify and define rational and irrational functions involving modulus and fractions, as well as logarithmic, exponential and trigonometric functions. They will have completed exercises to strengthen their understanding of calculational procedures and techniques in preparation for the first-year logic and mathematics exams. They will also have practised linking the properties of functions to their graphical representation.
Furthermore: a number of specific metacognitive activities are planned in order to strengthen students’ study method for mathematics and problem solving.

Knowledge
- Properties of numbers (natural, integer, rational, real) and of the main operations;
- Principles for solving polynomial, fractional, exponential, logarithmic, trigonometric, irrational equations and inequalities;
- Properties of elementary functions;

Skills:
- Simplification of expressions with powers, exponentials, logarithms, trigonometric functions;
- Solving polynomial, fractional, exponential, logarithmic, trigonometric, irrational equations and inequalities;
- Graph elementary functions;
- Interpret and solve problems of logic and verbal understanding

Judgment skills:
- Evaluate your own preparation through self-assessment tests.

Basic mathematics at lower secondary school level (e.g. see the mathematics sections in the Syllabus TOLC-I https://www.cisiaonline.it/area-tematica-tolc-ingegneria/struttura-della-prova-e-syllabus/ ).

NOTE that the duration of each "chapter" may change, according to the needs of the students attending the lectures.

1. Rational functions (five hours)
Knowledge
Definition of real function of real variable
Methods for solving polynomial, rational equations and inequalities, with and without absolute values
Ability to solve basic problems
Plot the graph of first-, second- and third-degree polynomial functions, with and without absolute values
Problems for developing mathematical skills
Study the domain and sign of rational functions with fractions and absolute values. Draw the areas of the plane in which their graph lies
Problems for developing personal, study and updating skills
Build and adapt formula sheets and summaries for learning and training purposes.

2. Irrational functions (five hours)
Knowledge
Schemes for solving irrational equations and inequalities
Definition of composite function
Ability to solve basic problems
Solve systems of inequalities and solve irrational inequalities
Study the domain and sign of irrational functions
Problems for developing mathematical skills
Study the domain and sign of fractional or irrational functions, with and without absolute values. Draw the areas of the plane where their graph lies
Problems for developing personal, study and updating skills
Build and adapt formula sheets and summaries for learning and training purposes.

3. Exponential functions (five hours)
Knowledge
Definition of the power of a number
Properties of powers of real numbers
Definition of exponential function and its invertibility
Ability to solve basic problems
Plot the graph of exponential functions
Apply definitions to solve equations and inequalities containing exponential functions
Problems for developing mathematical skills
Study the domain and sign of exponential functions and define the areas of the plane in which their graph lies
Draw the graph of composite functions related to exponential functions
Problems for developing personal, study and updating skills
Build and adapt formula sheets and summaries for learning and training purposes.

4. Logarithmic functions (five hours)
Knowledge
Definition of logarithm and logarithmic function
Properties of logarithms
Ability to solve basic problems
Plot the graph of logarithmic functions
Apply the definitions and properties of logarithms to solve logarithmic equations and inequalities
Problems for developing mathematical skills
Study the domain and sign of logarithmic functions and defining the areas of the plane in which their graph lies
Draw the graph of composite functions related to logarithmic functions
Problems to develop personal, study and updating skills
Build and adapt formula sheets and summaries for learning and training purposes.

5. Basic trigonometry (ten hours)
Knowledge
Measuring angles: degrees and radians.
The operational definition of sine, cosine and tangent of an angle in the first quadrant.
Extending trigonometric functions by symmetrys and periodicity
Ability for solving basic problems
Measure angles using degrees and radians
Plot the graph of trigonometric functions
Determe the period and amplitude of a trigonometric function
Solve trigonometric equations and work with trigonometric identities
Problems for developing mathematical skills
Draw graphs of composite functions related to basic trigonometric functions
Manipulate trigonometric formulas
Problems for developing personal, study and updating skills
Build and adapt formula sheets and summaries for learning and training purposes.

Material provided online through the Moodle platform

At the conclusion of the course and during each examination period, a final assessment is administered to fulfill the requirements of the 'OFA'.
The test consists of a 60-minute written exam containing six open-ended questions: one for each of sections 1–4, and two for section 6. Credit is awarded for achieving 70% of the total score.

written

The training requirement is fulfilled after passing the verification test (or not fulfilled, otherwise) .

The course is articulated into:
a) classroom lectures;
b) classroom exercises;
c) individual study.
Students are strongly encouraged to actively attend classes, by reading the teaching material before the class and by performing exercises similar to those taken into account in the classroom after the class.
Online lectures are available in moodle.

The course is coordinated with that of Mathematics in order to optimize the recovery of the basic mathematical skills.