20.500.12556/RUL-86814 Teoretični in didaktični vidiki verižnih ulomkov Theoretical in didactical aspects of continued fractions Verižni ulomki so v matematiki znani predvsem zaradi omogočanja natančnejše predstavitve racionalnih in iracionalnih števil. Verižni ulomki racionalnih števil so končni, medtem ko je treba za predstavitev iracionalnih števil vpeljati neskončne verižne ulomke. Teoretični del magistrskega dela bo poleg predstavitve osnovnih pojmov verižnih ulomkov, konvergence in iracionalnosti verižnega ulomka zajemal tudi natančnejšo obravnavo števila e, ki se ga da enostavno razviti v neskončni verižni ulomek in tako pokazati njegovo iracionalnost. Iracionalna števila namreč lahko na enoličen način predstavimo z neskončnim verižnim ulomkom, delne vsote neskončnega verižnega ulomka pa predstavljajo odlične racionalne približke števila, ki jim pravimo tudi konvergenti. V nadaljevanju bomo obravnavali periodične verižne ulomke in pokazali, da je neskončen enostaven verižni ulomek kvadratična iracionala, če in samo če je periodičen. Zapisana teoretična vsebina bo, skupaj s predstavitvijo osnovnega algoritma računanja verižnih ulomkov, vpeljava v empirični del. Empirični del bo predstavljal projekt, v katerem bomo verižne ulomke predstavili kot obogatitveno vsebino pouka matematike v osnovnih in srednjih šolah. Pokazali bomo, kako z različnimi učnimi pristopi obravnavati verižne ulomke z nadarjenimi učenci in dijaki. Continued fractions in mathematics are mainly known due to the need for a more detailed presentation of rational and irrational numbers. Continued fractions of rational numbers are finite, while for presentation of irrational numbers, it is necessary to introduce infinite continued fractions. The theoretical part of the master's thesis will, in addition to the presentation of the basic concepts of continued fractions, convergence, and irrationality of a continued fraction, also include a more detailed discussion of the number e, which can be easily expanded into an infinite continued fraction in order to demonstrate its irrationality. We are able to represent irrational numbers in a unique way with an infinite continued fraction, with its finite approximations representing an excellent rational approximation of the number, also called convergents. Hereinafter, we will address periodic continued fractions, and show that an infinite simple continued fraction is a quadratic irrational, if and only if it is periodic. This theoretical content, together with the presentation of the basic algorithm of the computation of continued fractions, will be an introduction to the empirical part. The empirical part will represent the project by means of which we will present continued fractions as an enrichment of the content for teaching mathematics in elementary and secondary schools. We will show how to use different learning approaches in teaching continued fractions to gifted students. verižni ulomek continued fraction true false false Slovenski jezik Angleški jezik Magistrsko delo/naloga 2016-11-08 02:25:19 2017-09-05 13:45:41 2025-04-17 04:52:40 0000-00-00 00:00:00 2016 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 11288137 Magistrska_naloga_Katja_Skubic_26.9..pdf Magistrska_naloga_Katja_Skubic_26.9..pdf 1 4A37B60A55D9E4E4FA68DFD751F44568 98f7c0ea9a9cd74006bf3e2422bafeee2446226d1f285308d2cef46c4d52c489 41a58ad5-1aac-11f0-b232-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=205049 Pedagoška fakulteta 0 0 0