1. SylabUZ
2. Wydział Matematyki, Informatyki i Ekonometrii
3. semestr zimowy 2022/2023
4. Mathematics - pierwszego stopnia z tyt. licencjata
5. Elementary Geometry

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Elementary Geometry - opis przedmiotu

Cel przedmiotu

This course aims to introduce the main theorems and methods of elementary geometry, as well as the theoretical and practical training in solving advanced geometrical problems. Students should attain the skills in geometrical constructions, solving exercises, and demonstrating geometrical theorems using open-access computer programs. Besides that, they should train their future geometry teachers' roles and the skills of facultative tutoring. Finally, students should learn the didactic application of supporting computer programs.

Wymagania wstępne

Zapisz zmiany

It is required that the participants have knowledge and expertise in linear algebra, analytical geometry, and mathematical logic. Besides that, they should have basic computer skills.

Zakres tematyczny

Lecture:

1. Axiomatic methods in geometry; axiomatization of Euclidean geometry and the formulation of the parallelism concept (2h)
2. Isometric maps in the Euclidean plane: invariants, symmetries, symmetries' compositions, and classification of isometries (2h)
3. Groups corresponding to Euclidean space symmetries in 1, 2, and 3 dimensions (2h)
4. Similarity transformation and its property (2h)
5. Inversive geometry in a plane: circumference and line examples (2h)
6. Euclidean plane' affine transformations (2h)
7. The geometry of triangles: Menelaus's, Ceva's, and Steiner-Lehmus theorems (2h)
8. The geometry of quadrilaterals: incircle and circumcircle; Ptolemy's theorem, Brahmagupta-Fibonacci identity, and Euler's quadrilateral theorem (2h)
9. The power of a point; the intersecting secants theorem; the radical axis of two non-concentric circles; Euler's theorem in geometry (2h)
10. Vector calculus in classical geometry (2h)
11. Geometrical constructions: ancient problems and their solutions, modern approaches (2h)
12. Regular polygon constructions, golden ratio, Mohr-Mascheroni, and Poncelet-Steiner theorems (2h)
13. Convex polyhedra, Platonic solids, and Euler's polyhedron formula (2h)
14. Hyperbolic plane and its models (2h)
15. Spherical geometry (2h)

The exercises are focused on practicing the topics introduced during the lecture. Students train in the applications of the learned definitions and methods. They learn the problem-solving skills and techniques of formulating the claims needed for a mathematical teacher. Besides that, students are asked to solve lists of exercises, the solutions of which are discussed during the course. Since the first meeting, the necessity and practical skill of using open-source geometrical applications in teaching geometry are explained (for instance, WolframAlpha - www.wolframalpha.com and GeoGebra - www.geogebra.org).

Excercises:

1. geometric maps; maps' invaariants; isometries: their examples, properties, and related groups; homothety and similarity; axial affinity (4h)
2. geometry of triangles; a triangle and its properties; theorems concenring the sides and angles in triangles; circumcircle, incircle, and excircles; the nine-point circle and its properties; significant points regarding triangles; law of sines and law of cosines; Menelaus's theorem and Ceva's theorem (4h)
3. geometrical proofs in GeoGebra; experimental proofs (4h)
4. geometrical constructions; constructional problem formulation and its solution; ancinet constructional problems; geometrical constructions of polygons and the golden ratio; Mohr-Mascheroni and Poncelet-Steiner constructions (4h)
5. selected geometrical constructions in GeoGebra (4h)
6. polygons; experimental proofs for the polygons area fomulas
7. convex polyhedra; Euler's polyhedron formula; Platonic solids (2h)
8. elements of projective and spherical geometry - the fifth postulate problem; examples of non-euclidean geometries (2h)
9. test (2h)

Metody kształcenia

lecture: problem sessions, conversations, multimedia presentations
excercises: recitation sessions, teamwork, discussion

The lecture has a combined form. It consists of a multimedia presentation and a classical blackboard tutoring, where the instructor writes down the lecture's content. Auxillary materials supporting the course are given in advance.
The exercise section takes the form of problem-solving meetings with a supplementary use of tutoring computer programs and other geometry-related software. This section aims to help learning and understanding students' knowledge attained during the lecture. The teacher regularly provides additional explanations and moderates the discussion during the meetings. Learning results are verified in a test.

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

Passing the exercises section requires obtaining more than a threshold of points in a test.
The passing requirement for the lecture is a positive grade obtained in a written exam.
The final grade is the mean of the results from the exercises (50%) and the exam (50%)

 

Literatura podstawowa

• H. S. M. Coxeter, Introduction to Geometry, Wiley
• Berger M., Geometrie, Nathan, Paris

    

   

Literatura uzupełniająca

• I. Agricola, T. Friedrich, Elementary Geometry, American Mathematical Soc.
• D. Brannan, M. Esplen, J. Gray, Geometry, Cambridge University
• M. Berger, Geometrie, Nathan, Paris
• H. S. M. Coxeter, S. L. Greitzer, Geometry revisited, Toronto New York

Uwagi

Zmodyfikowane przez dr Ewa Sylwestrzak-Maślanka (ostatnia modyfikacja: 07-02-2024 14:42)