1.º PORTUGAL - ITÁLIA EM GeCla
Transcrição
1.º PORTUGAL - ITÁLIA EM GeCla
ATRACTOR No âmbito de uma colaboração entre a Gazeta e o Atractor, este é um espaço da responsabilidade do Atractor, relacionado com conteúdos interactivos do seu site www.atractor.pt . Quaisquer reacções ou sugestões serão bem-vindas para [email protected] 1.º PORTUGAL - ITÁLIA EM GeCla Em 2009, o Atractor concluiu o DVD “Simetria – apresentação dinâmica”, GHVHQYROYLGR DR ORQJR GH FHUFD GH FLQFR DQRV RQGH SURFXURX WUDQVPLWLU FRPIRUWHDSRLRGHLPDJHQVHDQLPDo}HVHVSHFLDOPHQWHFULDGDVDVLGHLDVEDVH GRV PpWRGRV GHVHQYROYLGRV SRU : 7KXUVWRQ SDUD R HVWXGR GD VLPHWULD GH IULVRV H GH SDGU}HV GXSODPHQWH SHULyGLFRV QR SODQR 3RXFRWHPSRGHSRLVGHXLQtFLRDRGHVHQYROYLPHQWRGHRXWURPDWHULDOLQWHUDWLYRVREUHRPHVPRWHPDPDVGHVWDYH]GLULJLGRDXPS~EOLFRPDLVYDVWR FDGDSDGUmRRXIULVRHVWiDVVRFLDGRXPFDULPERVyGH- A pendente da simetria desse padrão ou friso, que permite FDULPEDURSODQRUHFULDQGRDVVLPWRGDVDVVLPHWULDVGRSDGUmRRXIULVRLQLFLDODSDUWLUGHXPPRWLYREDVHVHPVLPHWULD &RPHFHPRVSRUYHUDOJXQVH[HPSORVFRQVWLWXtGRVSHORVTXDWURWLSRVGHFDULPERVPRVWUDGRVQDÀJXUDXPFRQHURODQGR DSLQWDUXPDURViFHDXPFLOLQGURDFDULPEDUXPIULVRVyFRP WUDQVODo}HVXPDWLUDGH0RHELXVDSLQWDUXPIULVRFRPUHÁH[}HVGHVOL]DQWHVHXPWULkQJXORUHWkQJXORLVyVFHOHVFRPERUGRV HVSHOKDGRV D FDULPEDU XP SDGUmR FRP UHÁH[}HV QR SODQR SRGHPYHUVHPDLVLPDJHQVHP>@ 3RUTXHVmR~WHLVRVFDULPERV"eTXHSDUDFRQWDURVGLIHUHQWHVWLSRVGHVLPHWULDEDVWDFRQWDUWRGRVRV´ERQVµFDULPERV8PDH[SOLFDomRGHWDOKDGDGRTXHpXP´ERPµFDULPER Figura 1 H GH FRPR VH SRGHP HQFRQWUDU WRGRV HOHV p WDPEpP DSUH- XWLOL]DGDVFRPSURYHLWRSRUSURIHVVRUHVSDUDWUDQVPLWLULGHLDV VHQWDGD QR '9' LQWHUDWLYR QmR H[LJLQGR GRV VHXV XWLOL]D- matemáticas importantes a alunos mais interessados. GRUHV R FRQKHFLPHQWR WpFQLFR QHFHVViULR j OHLWXUD GRV WH[WRV Pouco tempo depois de iniciada a difusão do DVD, PDWHPiWLFRVRULJLQDLV(HPERUDR$WUDFWRUQmRWHQKDYLVDGR R $WUDFWRU FRPHoRX R GHVHQYROYLPHQWR GH RXWUR PDWHULDO com este DVD criar um instrumento didático para ser usado LQWHUDWLYR VREUH R PHVPR WHPD TXH XWLOL]D FRPSRQHQWHV UHJXODUPHQWHSRUDOXQRVDOJXPDVGDVVXDVVHFo}HVSRGHPVHU GR'9'PDVpJXLDGRSRUXPDÀORVRÀDFRPSOHWDPHQWHGLIH- ATRACTORr1.º Portugal - Itália em GeCla 03 UHQWHRSURJUDPDGHYHULDVHUDFHVVtYHOHDSHODWLYRDRVDOXQRV de uma larga faixa etária e procuraria ser útil no estudo e na FRPSUHHQVmRGDVTXHVW}HVGHVLPHWULDHQWUHWDQWRLQWURGX]LGDVQRVFXUUtFXORVHVFRODUHV (VVHQRYRSURJUDPD²R*H&OD>@²IRLDIHUUDPHQWDXVDGDQXPHYHQWRUHFHQWHTXHUHXQLXYLD,QWHUQHWDOJXQVDOXQRV de uma escola portuguesa com outros de uma escola italiaQD QXP WRUQHLR FRP FRQWRUQRV LQpGLWRV ( IRUDP R VXFHVVR GRHYHQWRDDGHVmRHRHQWXVLDVPRGRVDOXQRVSDUWLFLSDQWHV H D VDWLVIDomR GDV SURIHVVRUDV HQYROYLGDV TXH QRV OHYDUDP jHVFROKDGHVWDQRWtFLDFRPRWHPDGDFROXQDGR$WUDFWRUQR presente número da Gazeta, na esperança de que mais pro- Figura 2 IHVVRUHVYHQKDPDPDQLIHVWDULQWHUHVVHHPWRPDULQLFLDWLYDV análogas com os seus alunos. &RPHoDUHPRV SRU GHVFUHYHU UHVXPLGDPHQWH R *H&OD DQWHV GH GDUPRV PDLV DOJXQV GHWDOKHV VREUH R UHIHULGR WRUQHLR 2 *H&OD WHP XP gerador ÀJXUD TXH SHUPLWH FULDU XP GHVHQKR DEUHYLDWXUD GH IULVRSDGUmRURViFHD DSyV D HVFROKDGHXPPRWLYRVHPVLPHWULDHGHXPWLSRGHVLPHWULD Figura 3 LQGLFDGRSHORUHVSHWLYRFDULPER3RUH[HPSORSDUWLQGRGH LFRPRPRWLYRSDUDGHVHQKDUXPIULVRTXHVyDGPLWDWUDQVODo}HVEDVWDHpQHFHVViULRHVFROKHURFDULPERFLOtQGULFRXP FLOLQGURDRURODUGHVHQKDH[HPSODUHVGRPRWLYRREWLGRVSRU VXFHVVLYDVWUDQVODo}HVXQVGRVRXWURVÀJXUD2SURJUDPD SHUPLWH JXDUGDU R GHVHQKR REWLGR FRPR ÀFKHLUR QR IRUPDWRMSJHQHVVHÀFKHLURRSURJUDPDJUDYDWDPEpPVREIRUPD HQFULSWDGDLQIRUPDomRVREUHWRGDVDVVLPHWULDVGRGHVHQKR 'H PRGR VHPHOKDQWH SDUD GHVHQKDU XPD URViFHD RX XP SDGUmR EDVWDUi HVFROKHU HP YH] GR FLOLQGUR RXWUR FDULPER Figura 4 FRUUHVSRQGHQWH DR WLSR GH VLPHWULD GHVHMDGR 3RU H[HPSOR FRPXPWULkQJXORLVyVFHOHVFRPRVGRLVODGRVLJXDLVHVSHOKDGRV H ID]HQGR HVVHV ODGRV XP kQJXOR GH REWpPVH XPD URViFHDFRPVHLVUHÁH[}HVHVHLVURWDo}HVÀJXUDFRPXP WULkQJXORUHWkQJXORLVyVFHOHVDJRUDFRPRVWUrVERUGRVHVSHOKDGRV REWpPVH R SDGUmR UHSUHVHQWDGR QD TXDUWD LPDJHP GDÀJXUD$OpPGRJHUDGRUR*H&ODFRQWpPXPFODVVLÀFDGRU ÀJXUD D VXD IXQomR p D GH SHUPLWLU XPD SHVTXLVD VLVWHPiWLFDGHWRGDVDVVLPHWULDVGHXPGHVHQKRHFKHJDUDVVLP j GHWHUPLQDomR GR VHX WLSR GH VLPHWULD H SRUWDQWR GR FDULPER DVVRFLDGR 6H R ÀFKHLUR GR GHVHQKR TXH VH SUHWHQGH FODVVLÀFDU WLYHU VLGR SURGX]LGR SHOR JHUDGRU GR *H&OD 04 GAZETA DE MATEMÁTICA r171 Figura 5 R FODVVLÀFDGRU WDPEpP p FDSD] GH OHU D LQIRUPDomR HQFULS- 'HSRLVXVDQGRRPRWLYRJXDUGDGRSRGHJHUDUFRPRFDULP- WDGDTXHIRLJUDYDGDDRJHUDURGHVHQKR(OHÀFDDVVLPDFR- ERHQFRQWUDGRXPQRYRGHVHQKRTXHWHUiDVSHWRLJXDODRGD QKHFHUWRGDVDVVLPHWULDVGRGHVHQKRHRSURJUDPDSHUPLWH IRWRJUDÀD RULJLQDO PDV DJRUD R ÀFKHLUR FRQWHUi D LQIRUPDomR TXHHVVDLQIRUPDomRVHMD~WLODRXWLOL]DGRUVREXPDIRUPDTXH HQFULSWDGDVREUHDVVLPHWULDVH[LVWHQWHV)RLDVVLPREWLGDDLPD- HOHWHQKDSUHYLDPHQWHHVFROKLGRHPPRGRGHajuda máxima, JHPTXHHVWiDVHUPRVWUDGDHPIDVHGHFODVVLÀFDomRQDÀJXUD R SURJUDPD SRGH UHFXVDU LPHGLDWDPHQWH TXDOTXHU HVFROKD 8PDOXQRTXHYHQKDDXVDUHVWHQRYRÀFKHLURSRGHXWLOL]DU TXH FRQVLGHUH LQFRUUHWD QR GH ajuda mínima, pode deixar DDMXGDGRSURJUDPDYHUÀJXUDSDUDFODVVLÀFDURGHVHQKR DYDQoDUWRGDVDVHVFROKDVFRHUHQWHVGDQGRDSHQDVQRÀPXP LGrQWLFRDRGDIRWRJUDÀDHHQFRQWUDUXPPRWLYRYHULPDJHQV UHODWyULR GH HYHQWXDLV HUURV FRPHWLGRV (VWH DVSHWR p PXLWR GDÀJXUD1RSRUWDOGR$WUDFWRUHQFRQWUDVHXPDJUDQGHYD- ~WLOGLGDWLFDPHQWHpQDWXUDOHVFROKHUajuda máximaQRLQtFLR ULHGDGHGHGHVHQKRVWRGRVFRPLQIRUPDomRHQFULSWDGDVREUHDV da aprendizagem e, após alguma experiência, começar a re- VLPHWULDVH[LVWHQWHVDOJXQVGHOHV>@IRUDPREWLGRVDSDUWLUGH GX]LURJUDXGHDMXGD IRWRJUDÀDVSHORSURFHVVRGHVFULWR (RTXHDFRQWHFHVHTXLVHUPRVFODVVLÀFDUXPGHVHQKRREWL- $OpPGRJHUDGRUHGRFODVVLÀFDGRUFXMRVHVTXHPDVGHIXQ- GRSRUH[HPSORIRWRJUDIDQGRXPDSDUHGHGHD]XOHMRV"2EYLD- FLRQDPHQWR DFDEDP GH VHU EUHYHPHQWH GHVFULWRV R SURJUDPD PHQWHQHVWHFDVRRÀFKHLURQmRFRQWpPQHQKXPDLQIRUPDomR FRPSRUWD HQFULSWDGDVREUHDVVLPHWULDVHRFODVVLÀFDGRUQmRSRGHDMXGDU 8PDVHFomRTXHSHUPLWHDRXWLOL]DGRUWHVWDUSRUTXDOTXHU QDVHVFROKDVGRXWLOL]DGRUH[FHWRSDUDH[LJLUFRHUrQFLDQDLQIRU- RUGHPHVHPTXDOTXHUUHJLVWRYLVDQGRXPDFODVVLÀFDomR PDomRTXHOKHpWUDQVPLWLGD0DVVHRXWLOL]DGRUQmRPDUFDU GRGHVHQKRVHXPDGDGDLVRPHWULDpRXQmRXPDVLPH- QHQKXPDVLPHWULDHUUDGDPHQWHQHPHVTXHFHUQHQKXPDGHGX] WULDGRGHVHQKRYHUÀJXUDFRQVWLWXLXPDX[LOLDUGLGi- corretamente qual o tipo de simetria e o correspondente carim- WLFR~WLOVREUHWXGRQDIDVHLQLFLDOGHDSUHQGL]DJHPSRLV ER1XPDVLWXDomRFRPRHVWDXPSURIHVVRUTXHHVWHMDLQWHUHV- RDOXQRTXHVHFRQIURQWDFRPRSUREOHPDGHLGHQWLÀFDU VDGRHPSHUPLWLUTXHRVVHXVDOXQRVFODVVLÀTXHPFRPDMXGD DV LVRPHWULDV TXH FRQVHUYDP R GHVHQKR SRGH FRP HVWD GRSURJUDPDRGHVHQKRGDIRWRJUDÀDSRGHVHPSUHWRUQHDUD IHUUDPHQWDYHULÀFDUIDFLOPHQWHDYDOLGDGHGDVFRQMHWXUDV GLÀFXOGDGH EDVWDOKH FODVVLÀFDU FRUUHWDPHQWH R SDGUmR VHP TXHDVXDLQWXLomROKHVXJHUH 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Outra secção que permite a realização de uma competi- Figura 6 Figura 7 Figura 8 ATRACTORr1.º Portugal - Itália em GeCla 05 Figura 9 Figura 10 omR HQWUH GRLV MRJDGRUHV GXUDQWH D TXDO FDGD XP JHUD GRLVWLSRVQDIDVHVHJXLQWHFDGDMRJDGRUWHPSRLVDSHQDVGH XP FRQMXQWR GH GHVHQKRV FRP PRWLYRV j VXD HVFROKDH GHVFREULUTXDLVRVIULVRVJHUDGRVSHORDGYHUViULRTXH´DGPL- XWLOL]DQGRGLIHUHQWHVFDULPERVHGHSRLVWHQWDFODVVLÀFDU WHPXPDUHÁH[mRKRUL]RQWDOµRTXHpYLVXDOPHQWHPXLWRIiFLO RVGHVHQKRVJHUDGRVSHORVHXDGYHUViULRQRÀQDORSUR- YHUÀJXUD8PDYH]IDPLOLDUL]DGRFRPRIXQFLRQDPHQWR JUDPDSURGX]XPUHODWyULRTXHLQGLFDTXDLVDVFODVVLÀFD- GRSURJUDPDDHVWHQtYHOPDLVHOHPHQWDUpSRVVtYHORMRJDGRU o}HVFRUUHWDVHTXDLVDVLQFRUUHWDVDSRQWDQGRSDUDFDGD DYDQoDUSDUDQtYHLVVHJXLQWHVSRUH[HPSORFRPIULVRVDGPL- uma das incorretas, o primeiro erro cometido. O progra- WLQGRWDPEpPUHÁH[}HV´YHUWLFDLVµRXURWDo}HV PDSHUPLWHDHVFROKDGRQ~PHURGHGHVHQKRVDSURGX]LU 2*H&ODWHPSUHVHQWHPHQWHLQVWUXo}HVHPTXDWUROtQJXDV SRUFDGDMRJDGRUHGHXPDJDPDPXLWRYDULDGDGHQtYHLV LQJOrVLWDOLDQRHHVSDQKRODOpPGRSRUWXJXrVHR$WUDFWRU GH GLÀFXOGDGH 8PD YH] IHLWDV DV HVFROKDV R SURJUDPD WHPFRQKHFLPHQWRGHTXHIRLXWLOL]DGRFRPVXFHVVRHPDXODV JHUHDSURJUHVVmRGRMRJRDWpjVXDFRQFOXVmR8PDYHU- HP3RUWXJDOHHP,WiOLD(IRLSUHFLVDPHQWHGHXPDSURIHVVRUD VmRSUySULDSDUDRVPDLVMRYHQV²R*H&ODPLQLÀJXUD LWDOLDQD TXH VXUJLX HP DEULO GHVWH DQR D SULPHLUD SURSRVWD ²QHPVHTXHUSHUPLWHRSWDUSHORVQtYHLVPDLVDYDQoDGRV SDUDDRUJDQL]DomRGHXPDFRPSHWLomRFRPR*H&ODYLD,Q- DOpPGLVVRWHPXPDDSUHVHQWDomRHXPFRQMXQWRGHPR- WHUQHWDOJRTXHpSRVVtYHOQDV~OWLPDVYHUV}HVGRSURJUDPD WLYRVIHLWRVDSHQVDUHPIDL[DVHWiULDVPDLVEDL[DV e de uma forma por ele inteiramente controlada. O Atractor GLYXOJRXHVWDSURSRVWDHQWUHSURIHVVRUHVTXHWLQKDPSDUWLFL- 9HMDPRV XP H[HPSOR H[WUHPR DGDSWDGR D DOXQRV GR SDGRHPVHVV}HVGHWUDEDOKRVREUHR*H&ODRUJDQL]DGDVSHOR FLFORTXHQXQFDWLYHUDPFRQWDFWRQHPFRPRSURJUDPDQHP Atractor, alguns dos quais usam regularmente o programa FRPDVTXHVW}HVGHVLPHWULD&RPR*H&ODPLQLHHVFROKLGR FRPRVVHXVDOXQRV&KHJRXDVHUHQFDUDGDDSDUWLFLSDomRGH RSULPHLURQtYHOGHGLÀFXOGDGHQDFRPSHWLomRFDGDMRJDGRU XPDHVFRODFXMRVDOXQRVFRQKHFLDPR*H&ODPDVSUREOHPDV QDIDVHHPTXHWHPGHJHUDUXPGHVHQKRGLVS}HGHDSHQDV ORJtVWLFRV QD HVFROD OLJDGRV WDPEpP DR IDFWR GH Mi VH HVWDU GXDV RSo}HV RX XP IULVR Vy FRP WUDQVODo}HV ÀJXUD RX GHPDVLDGRSHUWRGRÀQDOGRDQROHWLYRQmRSHUPLWLUDPTXHVH XPIULVRTXHWHPDSHQDVXPDUHÁH[mR´KRUL]RQWDOµDOpPGDV FRQFUHWL]DVVHHVVDSDUWLFLSDomR2HYHQWRYLULDDVHUJDUDQWLGR WUDQVODo}HV&ULDGRVSHORVGRLVMRJDGRUHVYiULRVIULVRVGHVWHV quase in extremisSRUXPDSURIHVVRUDTXHHVFROKHXXPJUXSR 06 GAZETA DE MATEMÁTICA r171 Figura 11 WFigura 12 GHDOXQRVTXHQmRFRQKHFLDPR*H&ODPDVPDQLIHVWDUDPXP HQRUPH HQWXVLDVPR SHOD LQLFLDWLYD (P SRXFR PDLV GH XPD VHPDQDKRXYHTXHHQVLQDUDRVDOXQRVRPDQXVHDPHQWRGR SURJUDPDPDVDSHVDUGDVFRQGLo}HVHPTXHGHFRUUHXDSUHSDUDomRGRVSDUWLFLSDQWHVSRUWXJXHVHVDLQLFLDWLYDIRLXQDQLmemente considerada um enorme sucesso. As duas professoUDVTXHÀFDUDPPXLWRFRQWHQWHVFRPRHQWXVLDVPRFRPTXH RVUHVSHWLYRVDOXQRVHQFDUDUDPDH[SHULrQFLDHFRPDIRUPD FRPRHODGHFRUUHXSXEOLFDUmRQRXWURORFDOXPUHODWRGHWDOKDGR3DUDRVDOXQRVRDVSHWRLQWHUQDFLRQDOGRHYHQWRIRLEDVWDQWHYDORUL]DGRHSDUHFHTXHRchat que o programa permite IRLXVDGRLQWHQVDPHQWH1DÀJXUDHVWmRDOJXPDVGDVQXmerosas imagens geradas pelos alunos durante a competição. 'RSRQWRGHYLVWDGR$WUDFWRUQmRIRLGDGDrQIDVHDRDVSHWR´FRPSHWLWLYRµHRVFHUWLÀFDGRVSDVVDGRVYHUÀJXUD apenas atestam a participação dos alunos. 0DV QDWXUDOPHQWH QmR p VXUSUHVD TXH RV SDUWLFLSDQWHV Figura 13 WHQGDP D YDORUL]DU PDLV R DVSHWR GH FRPSHWLomR GR TXH RV organizadores… $V GXDV SURIHVVRUDV À]HUDP SUHYLDPHQWH DOJXQV WHVWHV SDUDHQVDLDUDVOLJDo}HVHR$WUDFWRUHVWHYHGLVSRQtYHOSDUDR REFERÊNCIAS DSRLRTXHOKHIRLVROLFLWDGRFRPRHVWDUiSDUDLQLFLDWLYDVDQi- >@http://www.atractor.pt/publicacoes/entrada2.htm ORJDVTXHYHQKDPDVHUWRPDGDVQRIXWXURHQYROYHQGRXPD >@ http://www.atractor.pt/mat/GeCla RXPDLVHVFRODVDQtYHOQDFLRQDORXLQWHUQDFLRQDO >@http://www.atractor.pt/mat/GeCla/im4.html ATRACTORr1.º Portugal - Itália em GeCla 07
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