METODIČKE TRANSFORMACIJE SADRŽAJA U UNIVERZITETSKOJ NASTAVI MATEMATIKE
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1 ISSN (p) , ISSN (o) X ISTRAŽIVANJE MATEMATIČKOG OBRAZOVANJA Vol VIII (06), Broj 5, 8 Stručni rd METODIČKE TRANSFORMACIJE SADRŽAJA U UNIVERZITETSKOJ NASTAVI MATEMATIKE Miomir Anđić Fkultet z informcione tehnologije, Univerzitet Meditern Bulevr Josip Broz Tit bb, Stri erodrom, 8000 Podgoric, Crn Gor E-mil: ndjicm@t-comme Aleksndr M Anđić Studije sjber bezbjednosti, Univerzitet Donj Goric Donj Goric bb, 8000 Podgoric, Crn Gor E-mil: leksndr@ndjicme Sžetk: Neuspjesi student u mtemtici i nedovoljno znnje koje pokzuju nkon zvršenog školovnj dobrim su dijelom posljedice činjenice d se nstv većinom izvodi n nižem nivou, gdje se suviše insistir smo n usvjnju grdiv, zpostvlj se viši nivo nstve Rzlog zpostvljnj leži u činjenici d su z viši nivo nstve mtemtike potrebne zhtjevnije nstvne metode zsnovne n heurističkoj i problemskoj nstvi U univerzitetskoj nstvi se nedovoljno koristi svremen metodik nstve mtemtike, koj pruž rzne mogućnosti z rješvnje problem uvođenjem student u smostln i istrživčki rd, rzvijnje njihovih spososbnosti z rješvnje problem ko i rzvoj njihovog stvrlčkog mišljenj i sposobnosti Rd koji slijedi n prvi pogled izgled jednostvno, međutim, njegov cilj je d ukže kko se u univerzitetskoj nstvi mtemtike mogu vršiti metodičke trnsformcije konkretnih sdržj, npr nlitičke geometrije, geometrije i mtemtičke nlize i uspješno primjenjivti principi postupnosti i očiglednosti, misone opercije nlize i sinteze i nlogij ko oblik zključivnj Ključne riječi i frze: mimoilzne prve, hiperboloidn površ, rotcij kocke, Kvlijerijev princip, zpremin rotcionog tijel Abstrct: Filures of students in mthemtics nd lck of knowledge tht they express fter grdution re lrgely due to the fct tht teching is mostly performed t lower level, where the focus is only on the cquisition of content, nd the higher level of eduction is neglected The reson for neglecting lies in the fct tht demnding teching methods bsed on heuristic nd problem teching re necessry for higher level of mthemtics The university teching insufficiently uses contemporry methods of teching mthemtics which provide vriety of options for solving problems by introducing students to independent nd reserch work, developing their bilities to solve problems, nd developing their cretive thinking nd competences This pper my seem very simple t first glnce, but its im is to show how methodicl trnsformtion of specific content cn be mde in university teching of mthemtics, such s nlytic geometry nd geometry nd mthemticl nlysis, nd how the principles of grdulism nd obviousness, thoughtful opertions of nlysis nd synthesis, nd nlogy s form of resoning cn be successfully pplied Keywords nd phrses: skew lines, hyperboloid surfce, cube rottion, Cvlieri's principle, rottionbody volume
2 IMO, Vol VIII (06), Broj 5 Nije dovoljno smo znti; treb i primijeniti Nije dovoljno htjeti; treb i učiniti Johnn Wolfgng von Goethe N dvočsu vježbi iz Mtemtike n Fkultetu z informcione tehnologije Univerzitet Meditern u Podgorici, n krju semestr, d bih ngžovo kompletnu grupu student, predložio sm sljedeću igru s mtemtičkim socijcijm Igr počinje tko što pišem n tbli krtku rečenicu s nekom mtemtičkom sdržinom Igrči studenti treb d smisle sljedeću mtemtičku rečenicu, čij se sdržin ndovezuje n sdržinu prethodne rečenice Rečenicu njhitrijeg igrč npisćemo n tbli, ispod prve rečenice Poslije tog će studenti formulisti treću rečenicu, veznu z sdžinu prethodne rečenice, p četvrtu rečenicu s socijcijm s trećom rečenicom i tko dlje Cilj igre je d se n ovj nčin obuhvti što više grdiv Z početk odbro sm mtemtičku rečenicu: U prvouglom Dekrtovom koordintnom sistemu u prostoru dte su prve (,0,0), (0,, ) l A (,0, ), D (0,0, ), pri čemu je 0 l A C i Prve rečenice student, podstknute prethodnom izjvom, bile su: Vektor prve l je s (0, 0, 0) (,, ), njen jednčin x y 0 z 0 l : 0 0, tj x y z 0 x y z Vektor prve l je s (, 0, 0), njen jednčin l : 0 0 Treć rečenic je bil: Prve l i l su mimoilzne, jer je 0 0 AA s s ( ) Uslijedil je i četvrt rečenic: Njkrće rstojnje između mimoilznih prvih l i l iznosi d () Zist, AA ( s s) s s i j k 0 ( ) ( ) d 0 0
3 IMO, Vol VIII (06), Broj 5 Ztim je usijedil i pet rečenic: Ugo između prvih l i l je rccos To je ugo između njihovih vekor s i s i rčun se po formuli cos s s (,, ) (,0,0) s s ( ) ( ) 0 0, odkle je cos () Poslije tog jedn od student je uočio: Dte tčke A, C, A, D su tjemen kocke ABCOABC D O(0,0,0), B(,,0), C(0,,0), B (,, ) Ncrtli smo n tbli koordintni sistem u prostoru s tčkm A, B, C, O, A, B, C, D (slik ) Slik Stizle su i konsttcije: Rstojnje tčke D od dijgonle AC iznosi Zist, iz prvouglog trougl ADC je DM () AD D C AC D M P DM DM Rstojnje tčke A od dijgonle AC kocke je tkođe Rotcijom prve l oko mimoilzne prve l nstje površ hiperboloid
4 IMO, Vol VIII (06), Broj 5 Time se ukzl zgodn prilik d se studentim istkne potreb z dokzom nekih mtemtičkih istin Neko vrijeme smo ćutli i rzmišljli, ond su Filipu oči zblistle i uzviknuo je: J sm uvjeren d iskz vži, jer j to vidim! N moj zčuđen pogled Filip mi je pokzo ovj crtež (slik ) N slici su predstvljene dte mimoilzne prve l i l Filip je uočio njihovu zjedničku normlu PQ, pri čemu je već izrčunto d je PQ d Pri rotciji tčk prve l oko prve ose l kružnic njmnjeg poluprečnik dobij se pri rotciji tčke Q Filip je dlje n prvoj l uočio tčku R, rzličitu od tčke Q i normlu RS iz tčke R n prvu l Uočio je prvu koj sdrži tčku Q prlelnu prvoj l i prvu koj sdrži tčku S i prleln je prvoj PQ Oznčio je s T presječnu tčku tih prvih S y je oznčio rstojnje RS, s z rstojnje PS i RQT s Trougo RST je prvougli s ktetm ST d i RT z tg Slik Kko je već izrčuno d je cos tg cos, p je RT z Dlje je RS ST RT, tj y d ( z) y z d y z y z N ovj nčin je dobio jednčinu hiperbole, pri čijoj rotciji se dobij ist površ, ko pri rotciji prve l oko prve l - ose rotcije Zključio je d se rotcijom prve l oko mimoilzne prve l dobij površ koj se zove hiperboloid Kko su njegovi presjeci s rvnim prlelnim rvni Oxy kružnice, to je njegov jednčin x y z (4) Ovj Filipov rezultt je podstko nekoliko student n rzmišljnje kko d riješe problem koji je, prem izjvi, zokupljo njihovu pžnju već duži niz godin Riječ je o 4
5 IMO, Vol VIII (06), Broj 5 nlženju zpremine tijel nstlog rotcijom kocke ivice dužine oko svoje dijgonle Bez Filipovog rezultt određivnje zpremine tog tijel nije trivijln zdtk Mtild je rdosno uzviknul: J znm kko izgled to tijelo i prikzl g crtežom (slik ) Slik Čuje se nekoliko uzvik: Sd ovo tijelo treb smjestiti pogodno u prvougli koordintni sistem u prostoru i ond nije teško izrčunti njegovu zpreminu, koristeći integrlni rčun Mtild kže: Nek dijgonl kocke AC bude n z - osi, njeno središte u koordintnom početku, ko n slici 4 Slik 4 Slik 5 (Presjek tijel s rvni x 0 ) Ovdje je z os usmjeren duž ose rotcije (glvn dijgonl kocke) Tijelo se sstoji od gornjeg i donjeg konus i srednjeg dijel ogrničenog pomenutom hiperboloidnom površi 5
6 IMO, Vol VIII (06), Broj 5 y z Lko je sd nći koordinte tčk K i L hiperbole, ko i koordinte tčke M Tržene koordinte su K,0, L, i M 0, Zpremin konus je VK ( ML) MC, tj Zpremin tijel ogrničenog hiperboloidom je z VH y ( z) dz dz 0 0 V K (5) 9, odnosno V H 5 (6) 9 Ukupn zpremin tijel nstlog rotcijom kocke oko njene glvne dijgonle je 4 5 V V K VH, tj 9 9 V (7) Studentim sm postvio pitnje: D li se zpremin središnjg dijel ogrničenog hiperboloidom može izrčunti n elementrn nčin, bez primjene integrlnog rčun? Kko su studenti bezuspješno pokušvli d nđu postupk, uputio sm ih n Kvlijerijev princip i sliku Međutim, i pored dtih input nijesu se snšli, p sm im pomogo u određivnju zpremine Iskoristili smo sliku s drugčijom numercijom Pri rotciji prve l oko mimoilzne prve l (ose rotcije), ko što smo vidjeli, nstje hiperboloidn površ Nek je OA zjedničk norml mimoilznih prvih l i l i nek je OA d (slik 6) 6
7 IMO, Vol VIII (06), Broj 5 Slik 6 Nek su i dvije rvni normlne n osu rotcije l i nlze se s rznih strn tčke O n rstojnju, redom h i h Oznčimo s B i E tčke n prvoj l, tkve d je B ( O, B, E), pri čemu je OB x i OE h i s ugo između prvih l i l Uočimo prvu koj sdrži tčku A i prleln je prvoj l i prvu koj sdrži tčku B i prleln je prvoj OA Nek je presjek ove dvije prve tčk D Postvimo rvn, normlnu n l, n rstojnju x od tčke O Poluprečnik BC tog krug je d x tg, njegov površin inosi ( d x tg ) Zpremin V tijel ogrničenog hiperboloidnom površi i rvnim koje su u tčkm O i E normlne n osu rotcije l, prem Kvlijerijevom principu jednk je zbiru zpremin cilindr visine h i poluprečnik osnove d i konus visine h i poluprečnik osnove h tg To slijedi iz ( d x tg ) d x tg, 0 x h p je V d h ( h tg ) h (8) Slično se nlzi d je zpremin V tijel ogrničenog hiperboloidnom površi i rvnim koje su u tčkm O i G ( B ( G, O, E) ) normlne n osu rotcije l, jednk zbiru zpremine cilindr visine h i poluprečnik osnove d i konus visine h i poluprečnik osnove h tg, tj d iznosi V d h ( h tg ) h (9) Končno, zpremin tijel ogrničenog hiperboloidnom površi i rvnim i je VH V V d ( h h ) h tg h tg (0) 7
8 IMO, Vol VIII (06), Broj 5 S slike 5 zključujemo d je h h, tj d je hh, već smo izrčunli d je tg i d, što zmjenom u (0) dobijmo d je V H 5 9 Filipove i Mtildine slike su ns oduševile očiglednošću i motivisle d i n elementrn nčin nđemo trženu zpreminu U to se dvočs zvršio Imo sm utisk d smo od ove mtemtičke igre svi imli koristi, kko studenti, tko i j Ponovili smo niz detlj iz prethodnog grdiv, utvrdili smo veze između rznih oblsti Iznend smo se sukobili s nlženjem jednčine hiperboličke površi i n krju nšli smo tri izvnredno lijep dokz S obzirom n činjenicu d se rdi o studentim informcionih tehnologij, ovj dvočs ih je motiviso d o ovom problemu rzmišljju i u slobodno vrijeme, p su pretrživnjem internet sdržj došli do dekvtne simulcije rotcije kocke oko svoje dijgonle Pomenut nimcij se može pogledti n dresi: Litertur [] G Poly, Kko ću riješiti mtemtički zdtk (prevod USA), Školsk knjig, Zgreb, 966 [] Б С Каплан, Н М Рогановский, Н К Рузин, А А Столяр, Практикум по педагогике математики, Вишэйшая школа, Минск, 978 [] M Anđić, Mtemtik (teoreme, definicije, primjeri,zdci), Fkultet z informcione tehnologije, Univerzitet Meditern, Podgoric, 009 ISBN [4] M Anđić, Mtemtik (teoreme, definicije, primjeri,zdci), Fkultet z informcione tehnologije, Univerzitet Meditern, Podgoric, 009 ISBN [5] Simulcij rotcije kocke dostupno n: Primljeno u redkciju 0706 Dostupno online
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