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[{"key": "dc.contributor.advisor", "value": "\u00c4kkinen, Tuomo", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Ristil\u00e4, Katariina", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2018-06-08T11:33:59Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2018-06-08T11:33:59Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2018", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/58471", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on tutustua yhteen ep\u00e4euklidiseen geometriaan, pallogeometriaan, ja verrata sit\u00e4 koulumatematiikasta jokaiselle tuttuun euklidiseen geometriaan. Yksinkertaisuudessaan pallogeometria on geometriaa pallon pinnalla. Tutkitaan, mit\u00e4 esimerkiksi suora ja kolmio tarkoittavat pallon pinnalla, ja mink\u00e4laisia ominaisuuksia niill\u00e4 on. Pallogeometriaan tutustuminen syvent\u00e4\u00e4 avaruuden ja aksioomaattisen j\u00e4rjestelm\u00e4n ymm\u00e4rt\u00e4mist\u00e4. Yhden aksiooman poisj\u00e4tt\u00e4minen m\u00e4\u00e4ritt\u00e4\u00e4kin useita erilaisia geometrian malleja. Pallogeometria on k\u00e4sitteen\u00e4 samaan aikaan hyvin et\u00e4inen ja hyvin arkip\u00e4iv\u00e4inen; harva tiet\u00e4\u00e4, mik\u00e4 on isoympyr\u00e4n m\u00e4\u00e4ritelm\u00e4, mutta jokainen tarvitsee joskus karttaa, joka on joko pallo tai sen approksimaatio euklidisessa tasossa. \nTutkielman alussa m\u00e4\u00e4ritell\u00e4\u00e4n yksikk\u00f6pallo, joka toimii koko tutkielman pohjana, ja kaikki tutkielmassa esitetyt asiat tehd\u00e4\u00e4n yksikk\u00f6pallolla. Sen j\u00e4lkeen m\u00e4\u00e4ritell\u00e4\u00e4n antipodi, isoympyr\u00e4, isometriat ja kolmiot. N\u00e4iden lis\u00e4ksi tarkastellaan tuttuja trigonometrisi\u00e4 funktioita ja muotoillaan sini- ja kosinilauseiden lis\u00e4ksi my\u00f6s Pythagoraan lause pallolle. Lopuksi viel\u00e4 tarkastellaan, voidaanko palloa approksimoida euklidiseen tasoon ja pohditaan, voivatko tasokartat olla tarkkoja. \nPalloon tutustumisen lis\u00e4ksi t\u00e4m\u00e4n tutkielman ohella on tehty pallogeometriaa havainnollistavia ty\u00f6kaluja GeoGebraan, joka on lukiolaisille arkip\u00e4iv\u00e4inen oppimisymp\u00e4rist\u00f6. N\u00e4m\u00e4 ty\u00f6kalut esitell\u00e4\u00e4n viimeisess\u00e4 luvussa. Nykyisten s\u00e4hk\u00f6isten ylioppilaskirjoitusten vuoksi lukion oppimateriaalit hy\u00f6dynt\u00e4v\u00e4t paljon s\u00e4hk\u00f6isi\u00e4 oppimisymp\u00e4rist\u00f6j\u00e4 ja ty\u00f6kaluja, kuten GeoGebraa. Ty\u00f6kalujen tekemisen lis\u00e4ksi tutkielmaan on haastateltu GeoGebraa oppimisv\u00e4lineen\u00e4 tutkinutta Jyv\u00e4skyl\u00e4n yliopiston lehtoria Markus H\u00e4hki\u00f6nieme\u00e4, joka kannustaa opettajia ja oppilaita rohkeasti tutustumaan GeoGebraan. Pallogeometria on monipuolinen ja erilainen n\u00e4kokulma geometriaan koulumaailmaan tuotavaksi. Se on oppilaille ja opiskelijoille uudenlainen esimerkki ymp\u00e4rist\u00f6st\u00e4, jossa kaikki ei menek\u00e4\u00e4n niin kuin on euklidisessa tasossa totuttu.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2018-06-08T11:33:59Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2018-06-08T11:33:59Z (GMT). No. of bitstreams: 0\n Previous issue date: 2018", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "57", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "pallogeometria", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "pallo", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "ep\u00e4euklidinen geometria", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Hilbertin aksioomaj\u00e4rjestelm\u00e4", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "paralleeliaksiooma", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "GeoGebra", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "GeoGebra-ty\u00f6kalut", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Seikkailu pallogeometriaan ja ty\u00f6kaluja GeoGebraan", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-201806083126", "language": "", "element": "identifier", 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