Mohr-Mascheronijeve konstrukcije i primjena u školskoj nastavi matematike

master's thesis

• Author: Lidija Cindrić
• Mentor: Željka Milin Šipuš (mentor)
• Committee member: Željka Milin Šipuš (predsjednik povjerenstva)
• Committee member: Rudi Mrazović (član povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Committee member: Slaven Kožić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-07-21, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu naglasak je na geometrijskim konstrukcijama izvodljivim samo šestarom. Prije iskaza i dokaza Mohr-Mascheronijevog teorema, koji kaže da je svaka geometrijska konstrukcija izvodljiva ravnalom i šestarom također izvodljiva i samo šestarom, navedeni su aksiomi planimetrije euklidske geometrije, a nakon toga i kratki povijesni razvoj geometrije. Jedan dokaz Mohr-Mascheronijevog teorema uključuje preslikavanje ravnine pomoću inverzije pa je u radu ukratko opisana teorija o inverziji. Osim pomoću inverzije, teorem smo dokazali i direktno. Postupak koji provodimo u direktnom dokazu je vrlo sličan postupku koji provodimo prilikom konstruiranja samo šestarom. Zbog toga je potrebno znati dokazati Mohr-Mascheronijev teorem i na ovaj način. U drugom dijelu diplomskog rada, na konkretnim smo primjerima mogli vidjeti kako provesti konstrukcije isključivo šestarom. Naveli smo i primjere koji se javljaju u školskoj matematici. Osim toga, predložili smo način na koji se ova tema može obraditi u školi, kao dio istraživačke nastave. Navedeno o istraživačkoj nastavi potkrijepili smo didaktičkom pozadinom.

Abstract (english)

Emphasis in this thesis is on geometric constructions carried out only by compass. Before a proof of Mohr-Mascheroni theorem, which says that every geometric construction carried out by compass and ruler can be done without ruler, we provided axioms of plane Euclidean geometry and stated short historical development of geometry. One proof of Mohr-Mascehoni theorem includes plane transformation called inversion, so in this thesis we shortly introduced specification of the theory of inversion. Except by inversion, we proved theorem directly as well. Process that is carried out in the direct proof is very similar to process that is used in construction only by compass. That is why it is necessary to know how to prove Mohr-Mascheroni theorem in this way. In the second part of this thesis, we saw concrete examples how to carried out constructions only by compass. There are also examples that we can find in school mathematics. Besides, we suggested method of implementation of this topic in school, for example, as a part of inquiry based mathematics teaching. Everything about this way of teaching, that is stated in this thesis, is confirmed by didactic background.

• Keywords: geometrijske konstrukcije izvodljive samo šestarom; Mohr-Mascheronijev teorem; nastava matematike; istraživačka nastava
• Keywords (english): geometric constructions carried out only by compass; Mohr-Mascheroni theorem; teaching mathematics; inquiry based mathematics teaching
• Language: croatian
• URN:NBN: urn:nbn:hr:217:631978
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2020-12-21 09:31:51