Matematično preiskovanje poliedrov v osnovni šoli

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Language:	Slovenian
Year of publishing:	2012
Source:	Ljubljana
Typology:	2.11 - Undergraduate Thesis
Organization:	UL PEF - Faculty of Education
Publisher:	[P. Svetlin]
UDC:	51:373.3(043.2)
COBISS:	9215049
Views:	948
Downloads:	152
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Secondary language:	English
Secondary title:	Mathematical investigation of polyhetra in elementary school
Secondary abstract:	In Slovenian elementary schools the curriculum addresses only the simplest polyhedra, such as prisms and pyramids. In my thesis I intended to show that also complex polyhedra can be the object of study in elementary schools. In the first, theoretical, part the definition and some properties of polyhedra are considered. Polyhedra are grouped for easier treatment. There are five Platonic solids, their surface consists of regular polygons. Archimedean solids are composed of different regular polygons, with the same number of polygons meeting in the same order in each vertex. Johnson solids, Kepler-Poinsot solids, antiprisms and pyramids are also described. Among the characteristics of polyhedra I highlighted the most interesting ones, especially the ones used in my empirical work. One of these properties is the configuration of the vertices of polyhedron, for it is possible to describe regular and semi-regular polyhedra with just a few numbers. For Platonic solids it is simple to measure the dihedral angle, i.e. the angle between adjacent polygons on its surface. Besides having tabulated these angles I calculated myself the dihedral angle for the dodecahedron. I also pointed out the Euler characteristic and I proved the Euler formula for convex polyhedra. Euler formula describes the relation between the number of vertices, edges, and faces of a polyhedron. I also considered the total angular deficit of a polyhedron, which is an analogy to the sum of external angles of a polygon. Angle deficit was calculated for several different polyhedra and it was proved that it has the same value for all convex polyhedra. As the introduction to the empirical part of my thesis, I described mathematical investigations, the role of teacher and pupil in investigations, and the organization of work in the classroom. In the empirical part of the thesis, three mathematical investigations for grade 8 of elementary school are presented. Since complex polyhedra are not a part of Slovenian elementary school curriculum, the investigations were carried out by a group of good achievers from an elementary school. My intention was to show that grade 8 students are able to work out investigations on polyhedra through their own reasoning processes, that the problem situations that I had chosen for investigations are suitable for elementary school students, and that mathematical investigation is an appropriate method for exploring complex polyhedra in elementary school. Every investigation was carefully planned. For every lesson I prepared the necessary material. During the investigation the conversation between students (that worked in pairs) was recorded and afterwards analysed, together with the students’ writing.
Secondary keywords:	mathematics;primary school;matematika;osnovna šola;
File type:	application/pdf
Type (COBISS):	Undergraduate thesis
Thesis comment:	Univ. Ljubljana, Pedagoška fak., Matematika in fizika
Pages:	91 str, XVII str. pril.
Type (ePrints):	thesis
Title (ePrints):	Mathematical investigation of polyhetra in elementary school
Keywords (ePrints):	polieder
Keywords (ePrints, secondary language):	polyhedron
Abstract (ePrints):	V osnovni šoli se po učnem načrtu obravnava le najbolj preproste poliedre, kot so prizme in piramide. V diplomski nalogi bom pokazala, da se lahko v osnovni šoli obravnava tudi kompleksnejše poliedre. V prvem, teoretičnem delu je zapisana definicija in nekaj lastnosti poliedrov. Poliedri so zaradi lažje obravnave razdeljeni v skupine. Platonskih teles je pet, njihovo površje sestavljajo skladni pravilni mnogokotniki. Arhimedska telesa so sestavljena iz različnih pravilnih mnogokotnikov, pri čemer se v vsakem oglišču stika enako število mnogokotnikov v enakem vrstnem redu. Opisana so tudi Johnsonova telesa, Kepler-Poinsotova telesa, prizme in antiprizme ter piramide. Med lastnostmi poliedra sem izpostavila najzanimivejše, predvsem tiste, ki sem jih uporabila v empiričnem delu. Ena izmed teh lastnosti poliedrov je konfiguracija oglišč, saj lahko z le nekaj številkami opišemo pravilne in delnopravilne poliedre. Pri platonskih telesih je preprosto izmeriti diedrski kot, to je kot med stičnima ploskvama. Ta kot je med vsemi stičnimi ploskvami enak. Med tabeliranimi diedrskimi koti sem za dodekaeder ta kot tudi sama izračunala. Izpostavila sem tudi Eulerjevo poliedrsko formulo in jo dokazala za konveksna telesa. Eulerjeva poliedrska formula opisuje odnos med številom oglišč, robov in ploskev poliedra. Tako kot meri vsota zunanjih kotov mnogokotnikov , imajo tudi poliedri »vsoto zunanjih kotov«, ki jih imenujemo skupni kotni primanjkljaj. Kotni primanjkljaj sem izračunala za nekaj različnih poliedrov in dokazala, da je enak za vse poliedre, za katere velja Eulerjeva formula. Kot uvod v empirični del diplomskega dela je opisano matematično preiskovanje, vloga učitelja in vloga učenca pri preiskovanju ter organizacija dela v razredu. V empiričnem delu so zapisana tri matematična preiskovanja za 8. razred devetletne osnovne šole. Ker poliedri niso del učnega načrta v osnovnih šolah, sem k sodelovanju povabila učno uspešnejše učence pri matematiki iz osnovne šole Livada in jim v izziv ponudila pripravljena preiskovanja iz področja poliedrov. Moj namen je bil pokazati, da so učenci zmožni priti do rešitev s pomočjo lastnih miselnih procesov, da so teme, ki sem jih izbrala za preiskovanje primerne za osnovnošolske učence, in da je preiskovanje primerna metoda za raziskovanje poliedrov v osnovni šoli. Vsaka ura preiskovanja je bila skrbno načrtovana, zato je za vsako uro zapisana zgradba učne ure, po izvedeni uri pa zapisan potek in analiza. Ob koncu pa je zapisana še celotna analiza in razmišljanje o eksperimentalnem delu.
Abstract (ePrints, secondary language):	In Slovenian elementary schools the curriculum addresses only the simplest polyhedra, such as prisms and pyramids. In my thesis I intended to show that also complex polyhedra can be the object of study in elementary schools. In the first, theoretical, part the definition and some properties of polyhedra are considered. Polyhedra are grouped for easier treatment. There are five Platonic solids, their surface consists of regular polygons. Archimedean solids are composed of different regular polygons, with the same number of polygons meeting in the same order in each vertex. Johnson solids, Kepler-Poinsot solids, antiprisms and pyramids are also described. Among the characteristics of polyhedra I highlighted the most interesting ones, especially the ones used in my empirical work. One of these properties is the configuration of the vertices of polyhedron, for it is possible to describe regular and semi-regular polyhedra with just a few numbers. For Platonic solids it is simple to measure the dihedral angle, i.e. the angle between adjacent polygons on its surface. Besides having tabulated these angles I calculated myself the dihedral angle for the dodecahedron. I also pointed out the Euler characteristic and I proved the Euler formula for convex polyhedra. Euler formula describes the relation between the number of vertices, edges, and faces of a polyhedron. I also considered the total angular deficit of a polyhedron, which is an analogy to the sum of external angles of a polygon. Angle deficit was calculated for several different polyhedra and it was proved that it has the same value for all convex polyhedra. As the introduction to the empirical part of my thesis, I described mathematical investigations, the role of teacher and pupil in investigations, and the organization of work in the classroom. In the empirical part of the thesis, three mathematical investigations for grade 8 of elementary school are presented. Since complex polyhedra are not a part of Slovenian elementary school curriculum, the investigations were carried out by a group of good achievers from an elementary school. My intention was to show that grade 8 students are able to work out investigations on polyhedra through their own reasoning processes, that the problem situations that I had chosen for investigations are suitable for elementary school students, and that mathematical investigation is an appropriate method for exploring complex polyhedra in elementary school. Every investigation was carefully planned. For every lesson I prepared the necessary material. During the investigation the conversation between students (that worked in pairs) was recorded and afterwards analysed, together with the students’ writing.
Keywords (ePrints, secondary language):	polyhedron
ID:	8310047