| What are completely integrable Hamilton systems | 1$-$14 |
Abstract
This paper is aimed to undergraduate students of mathematics and mechanics to draw their attention to a modern and exciting field of mathematics with applications to mechanics and astronomy. We cared to keep our exposition not to go beyond the supposed knowledge of these students.
Keywords: Hamilton system; completely integrable system.
| A problem from the PISA assessment relevant to calculus | 15$-$29 |
Abstract
In this paper, one of the multiple choice tasks from the PISA test is considered. This particular task deals with the relation between a water tank, in which water is poured at a constant rate, and the given graphs of functions showing how the height of water in that tank changes in time. For the purpose of better understanding the types of tasks that resemble this one, a few similar tasks have been considered. These tasks could be used on various levels of education. We also hope that our paper could exemplify the case of a research project to be assigned to the best students and that the paper can also be useful for all those who teach introductory calculus.
Keywords: PISA tests; the height of water surface in a tank.
| The maximum number of rectangular islands | 31$-$44 |
Abstract
In this paper we consider the combinatorial problem of rectangular islands by elementary means. The topic of islands and the methods for its investigation is suitable also for high school students, although some of the corresponding results are quite new. The arising questions need no advanced mathematical knowledge. Because most of the problems are of finitary type, experimental mathematics with computer support proves to be useful for the formulation of general conjectures related to the bounds of the number of islands in particular configurations.
Keywords: Rectangular island; information theory; lattice theory.
| Teaching envelopes in secondary school | 45$-$55 |
Abstract
The paper presents an educational module, in the spirit of inquiry based method, which is concerned with the case of some simple envelopes. The inclusion of dynamic geometry system applets in presenting the topic allows the shape of the envelope of one-parameter family of curves to be established by examining dynamic constructions. Starting with an example for the Euler formula, further motivation of introduction of envelopes belongs to the field of kinematics. Two main problems are under consideration: the misty sprinkler and the wet wheel. The introductory part of the kinematics problems does not go out of the common school practice.
Keywords: Dynamic geometry system; parabola; envelopes.