Techniques in solving mathematical problems
| Autor: | Posel, Anja |
|---|---|
| Přispěvatelé: | Arambašić, Ljiljana |
| Jazyk: | chorvatština |
| Rok vydání: | 2022 |
| Předmět: |
Cauchy-Schwarzova nejednakost
nejednakosti rješavanje matematičkih problema Chinese remainder theorem arithmetic heuristic teaching method kineski teorem o ostatcima PRIRODNE ZNANOSTI. Matematika aritmetika heuristička nastava solving mathematical problems inequalities George P'olya NATURAL SCIENCES. Mathematics Euclidean algorithm George P’olya Cauchy-Schwarz inequality Euklidov algoritam |
| Popis: | Ovaj diplomski rad sastavljen je od tri poglavlja s naslovima Rješavanje matematičkih problema, Nejednakosti i Aritmetika. Prvo poglavlje započinje opisom pojma matematički problem te nekoliko različitih klasifikacija matematičkog zadatka. Opisan je nastavni sustav problemska nastava te su prezentirane smjernice za izvođenje nastavnog sata ovog tipa. Ukratko je opisana i metoda rješavanja matematičkog problema koju je predstavio George P’olya u svom djelu “Kako ću riješiti matematički zadatak”. Na kraju poglavlja proučen je i nastavni sustav heuristička nastava te su nabrojane i opisane heuristike koje se koriste za rješavanje matematičkih problema. U drugom i trećem poglavlju rješavaju se matematički problemi iz područja nejednakosti i aritmetike. Prije rješavanja navedenih problema, potrebno je prisjetiti se nekoliko najvažnijih i najpoznatijih teorema iz ovog područja. Među njima su Cauchy-Schwarzova nejednakost te nejednakosti između aritmetičke, geometrijske, kvadratne i harmonijske sredine brojeva. Nadalje, nakon prisjećanja osnovnih pojmova vezanih za djeljivost brojeva, teorema o dijeljenju s ostatkom i Euklidovog algoritma, dokazan je osnovni teorem aritmetike te uveden pojam kongruencije i dokazan kineski teorem o ostatcima. Time su usvojene određene tehnike koje omogućuju da se zadaci koji su možda teško rješivi, riješe na znatno jednostavniji način. This master thesis is composed of three chapters with the titles Solving Mathematical Problems, Inequalities and Arithmetic. The first chapter begins with a description of the term mathematical problem and several different classifications of a mathematical task. The problem-based teaching method is described and the guidelines for performing this type of lessons are presented. Also, the method of solving a mathematical problem, presented by George P´olya in his book “How to solve it”, is briefly described. At the end of the chapter, the heuristic teaching method is explained, and the heuristics used to solve mathematical problems are listed and described. In the second and third chapters, mathematical problems in the field of inequality and arithmetic are solved. Before solving the mentioned problems, it is necessary to recall several of the most important and well-known theorems in these fields. Among them are the Cauchy-Schwarz inequality and inequalities between arithmetic, geometric, quadratic and harmonic means of numbers. Furthermore, after summarizing the basic concepts related to the divisibility of numbers, the divison theorem and Euclidean algorithm, the fundamental theorem of arithmetic was proved, the concept of congruences was introduced and the Chinese remainder theorem was proved. Thereby, certain techniques have been mastered, which enable mathematical tasks that may be difficult and demanding to solve, to be solved in a much simpler way. |
| Databáze: | OpenAIRE |
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