B018753 - ELEMENTARY MATHEMATICS FROM THE HIGHER POINT OF VIEW

Italian

Geometric transformations of the Euclidean plane. Isometries: reflections, translations, rotations, glide reflections. Similarities and their properties. Homotheties. Fixed points. Circle inversions. Homographies of the Riemann sphere and their properties. Euclid’s fifth postulate. Construction of a model of hyperbolic geometry. Trigonometry in the hyperbolic plane. Elliptic geometry of the sphere. Spherical Trigonometry.
Analysis of italian mathematical school programs.

Maria Dedo’ – Trasformazioni geometriche –1999 – Decibel Zanichelli
H. Meschkowski - Noneuclidean Geometry - 1964 - Academic Press Inc

The course is aimed to provide knowledge and to learn about issues related to the historical and logical foundations of geometry, also in order to identify the fundamental principles of teaching and learning of geometry in secondary school. A further purpose is to use the acquired knowledge in order to solve theoretical problems, to prepare educational activities for secondary school and to develop critical competences by confirming the basic skills.
The course contributes to the development of a flexible scientific attitude, useful to critically analyze mathematical problems, and sets geometry and, more in general, mathematics in a wider cultural context. Particular emphasis is given to the development of interdisciplinary knowledge and skills in the school context.

The knowledge of elementary algebra and geometry acquired in high school and during the first and second year of the bachelor's degree in mathematics are sufficient.

Lectures: presentation of the theory described in the course program, with teacher-student direct interaction, to ensure a full understanding of the subject.

Periodically, exercises and theorical problems are posed in order to
• improve critical competences and communication skills of the students, by means of an appropriate use of mathematical language;
• get students used to analyze critically the steps of the proof, by identifying the minimal hypothesis.
During class, most interesting problems are solved.

CFU: 9
Total hours of the course: 225

Hours reserved to private study and other indivual formative activities: 153

Contact hours for: Lectures (hours): 72 (possibly including 2-4 laboratory hours to introduce educational softwares)

Frequency of lectures, practice and lab: not compulsory, but recommended.
Teaching Tools: http://donatopertici.wordpress.com/
Office hours: by appointment
Dipartimento di Matematica e Informatica "Ulisse Dini"
Viale Morgagni, 67/a
50134 FIRENZE
Tel: 055 4237125
Email: donato.pertici@unifi.it

The examination consists only of an oral test: it starts with some exercises in order to evaluate the skills and the acquired mathematical techniques.
Afterwards, a number of questions are posed, in order to evaluate the degree of understanding of the theory presented in the course. Special attention is paid to communication skills, critical thinking and appropriate and correct use of mathematical language.

Geometric transformations of the Euclidean plane. Isometries: reflections, translations, rotations, glide reflections. Structure theorem. Fagnano’s Problem. Classical triangle centers. Similarities and their properties. Homotheties. Fixed points. Circle inversions. Homographies of the Riemann sphere and their properties. Euclid’s fifth postulate. Construction of a model of hyperbolic geometry: Poincare’ half-plane (or Poincare’ hyperbolic disk). Intersecting, parallel and ultraparallel lines. Hyperbolic isometries. Trigonometry in the hyperbolic plane. Elliptic geometry of the sphere. Spherical trigonometry. The polar triangle. Analysis of italian mathematical school programs.