Steinerov problem

master's thesis

• Title (english): Steiner's problem
• Author: Katarina Novosel
• Mentor: Slaven Kožić (mentor)
• Committee member: Slaven Kožić (predsjednik povjerenstva)
• Committee member: Dijana Ilišević (član povjerenstva)
• Committee member: Zrinka Franušić (član povjerenstva)
• Committee member: Filip Najman (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2024, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Steiner je postavio pitanje: “Na koliko najviše dijelova n pravaca može podijeliti ravninu?”. Iako pravce možemo smjestiti u ravninu na razne načine, kako bismo dobili najveći broj dijelova ravnine potrebno je slijediti sljedeća pravila: svaka dva pravca sijeku se u točno jednoj točki i nikoja tri pravca ne sijeku se u zajedničkoj točki. U ovom radu proučavamo Steinerov problem i njegove moguće primjene u osnovnoškolskom i srednjoškolskom obrazovanju.U prvom poglavlju, slijedeći ranije spomenuta pravila te korištenjem raznih metoda, došli smo do rješenja Steinerovog problema. U drugom poglavlju prezentirali smo tri različita pristupa rješavanja Steinerovog problema. U sljedećem je poglavlju obrađena Eulerova formula s posebnim naglaskom na njezinu vezu sa Steinerovim problemom. Osim spomenutog problema, Steiner je uveo i problem određivanja maksikalnog broja dijelova ravnine koji dobijemo smještanjem kružnica u ravninu, kojeg smo proučili u poglavlju 4. Na kraju rada prikazani su razni primjeri kako Steinerov problem uvrstiti u osnovnoškolsko i srednješkolsko gradivo. Uz pomoć raznih aktivnosti i zadataka učenici se mogu upoznati s ovim problemom dijeljenja ravnine na maksimalan broj dijelova te ujedno uvježbati ili otkriti nastavne sadržaje koji su propisani kurikulumom.

Abstract (english)

Steiner posed the question: ”Into how many parts can n lines at most divide a plane?” Although lines can be arranged in a plane in various ways, to obtain the maximum number of plane sections, it is necessary to follow the following rules: every two lines intersect at exactly one point, and no three lines intersect at a common point. In this thesis, we explore Steiner’s problem and its potential applications in primary and secondary education. In the first chapter, by adhering to the previously mentioned rules and using various methods, we found a solution to Steiner’s problem. In the second chapter, we present three different approaches to solving Steiner’s problem. The following chapter covers Euler’s formula with a special emphasis on its connection to the Steiner problem. In addition to the mentioned problem, Steiner introduced the problem of determining the maximum number of plane sections obtained by placing circles in a plane, which we have studied in Chapter 4. Finally, the thesis provides various examples of how to incorporate Steiner’s problem into primary and secondary school curricula. With the help of various activities and tasks, students can become familiar with the problem of dividing a plane into a maximum number of sections and, at the same time, practice or discover the topics prescribed by the curriculum

• Keywords: osnovnoškolsko obrazovanje; srednjoškolsko obrazovanje; Eulerova formula
• Keywords (english): primary education; secondary education; Euler’s formula
• Language: croatian
• URN:NBN: urn:nbn:hr:217:329451
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar edukacije matematike (sveučilišni magistar edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-02-02 13:25:24