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Editora UFPR ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR SHORXVRGDUpJXDGHFiOFXORSDUHFHEHPDIDVWDGRGRXVRGDVFDOFXODGRUDV PDVJXDUGDVHXLQWHUHVVHQDPHGLGDHPTXHFRORFDHPHYLGrQFLDXPDSUiWLFD GHFiOFXORTXHGHVHPSHQKRXXPSDSHOLPSRUWDQWHQDKLVWyULDGDPDWHPiWLFD 1RVVDSURSRVWDHVWiDSRLDGDQRXVRGHXPDSURSULHGDGHGRFRQMXQWRGRVQ~PHURVUHDLVPDLVHVSHFL¿FDPHQWHDGRLVRPRU¿VPRGRJUXSRDGLWLYRGRFRQMXQWR GRVQ~PHURVUHDLVVREUHRJUXSRPXOWLSOLFDWLYRGRFRQMXQWRGRVQ~PHURVUHDLV HVWULWDPHQWHSRVLWLYRV f IIR + o IIR +* u f a + b = f a u f b 3HUFHEHPRV TXH D PDLRULD GRV OLYURV GLGiWLFRV GD )UDQoD H GRV SDtVHV DIULFDQRVGHOtQJXDIUDQFHVDDSUHVHQWDVLWXDo}HVHPTXHVHSURFXUDPIXQo}HV TXHWrPDVHJXLQWHSURSULHGDGH f IIR +* oIIR tal que f xy y = f x + f y 2REMHWLYRGHQRVVDSURSRVWDSDUDLQWURGX]LURFRQFHLWRGHORJDULWPRp SUHYDOHFHUVH GD SURSULHGDGH ORJDUtWPLFD GH XPD IXQomR HVWULWDPHQWH FRQWtQXDQRFRQMXQWRGRVQ~PHURVUHDLVSRVLWLYRVHQmRQXORV³f transforma toda proporção geométrica em uma proporção aritmética´'RSRQWRGHYLVWDGD WUDQVSRVLomRGLGiWLFDKiXPDGLIHUHQoDQRWiYHOHQWUHDVDERUGDJHQVSURSRVWDV QRVOLYURVGLGiWLFRVEUDVLOHLURVDQDOLVDGRVHDVSURSRVWDVGDPDLRULDGRVOLYURV GLGiWLFRVIUDQFHVHVHGDÈIULFDGHOtQJXDIUDQFHVD&RPRGHVWDFDPRVRVOLYURV GHOtQJXDIUDQFHVD)UDQoDHÈIULFDGHOtQJXDIUDQFHVDGH¿QHPDIXQomRORJD1 ULWPRQHSHULDQRFRPRDIXQomR)SULPLWLYDGDIXQomRIGH¿QLGDSRU I [a I[ = [ [RXVHMD VHQGR ) VHQGR ) QRV OLYURV EUDVLOHLURV \ RORJDULWPRpGH¿QLGRFRPRQ~PHUR\WDOTXH log a x = y a = [ 3URSRPRVRXWUDPDQHLUDGHLQWURGX]LUHHVWXGDUDQRomRGHIXQomRORJDULWPRXWLOL]DQGRVHGDSURSULHGDGHDFLPDFLWDGDDSDUWLUGHXPDVLWXDomRTXH SHUPLWHHVWXGDUWRGDVDVSURSULHGDGHVGHVVDIXQomRUHVROYHUHTXDo}HVORJDUtWPLFDVHVERoDUDFXUYDHDUHWDWDQJHQWHHPXPSRQWRGDFXUYDGHVVDIXQomR $SRLDPRQRVQRWUDEDOKRSURSRVWRSHOR*URXSH0$7+,5(0GH3DULV9,, $GDSWDPRVDVLWXDomRSURSRVWDSRUHVWHJUXSRGHSHVTXLVD Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR 199 ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR Enunciado da situação-problema estudada2 $SUHVHQWDPRVGXDVGH¿QLo}HVLPSRUWDQWHVSDUDDUHVROXomRGDVLWXDomR 1. Quatro números reais positivos e não nulos a, b, c, d dizem-se em proporção geométrica se, e somente se ad = bc 2. Quatro números reais a, b, c, d são em proporção aritmética se, e somente se a + d = E + c . 4XHVWmR6HMDIXPDIXQomRHP50RVWUHTXHVHIWUDQVIRUPDWRGDSURSRUomRJHRPpWULFDHPXPDSURSRUomRDULWPpWLFDHQWmRIWUDQVIRUPDWRGDVHTXrQFLD JHRPpWULFDHPXPDVHTXrQFLDDULWPpWLFDSRGHVHYHUL¿FDUTXHVHXnpXPDVHTXrQFLD JHRPpWULFDGHUD]mRq z 0HQWmR1, q, un , un+1VmRHPSURSRUomRJHRPpWULFD $JRUDTXHUHPRVWUDoDUDFXUYDUHSUHVHQWDWLYDGHXPDIXQomRIFRQWtQXDH estritamente crescente em IR+* TXHYHUL¿FD DVHJXLQWHSURSULHGDGH³ORJDUtWPLFD´ f transforma toda proporção geométrica em uma proporção aritmética. Para ID]HUHVVDUHSUHVHQWDomRHVWXGDUHPRVDOJXPDVGDVFDUDFWHUtVWLFDVGHVWDIXQomR &RPR-2]DQDPHVFROKHPRVHVWXGDUDIXQomRIWDOTXHI HI $JRUDTXHWHPHVVDVLQIRUPDo}HVUHVSRQGDjVVHJXLQWHVTXHVW}HV 4XHVWmR Calcule : n f(100, f(1000) f(10.000) f(10 ) 4XHVWmR a) Mostre que f(15) = f(3) + f(5), HTXHSDUD[H\HOHPHQWRVGH5+* I[\ I[I\ E&DOFXOHSDUD[HOHPHQWRGH5+* RYDORUGH em IXQomRGHI[ F'HWHUPLQHSDUDDHEHOHPHQWRVGH5+* f ( ab ) HPIXQomRGH IDHIE 4XHVWmR 5HVROYDQR5+* DVHTXDo}HV DI[ ò EI[ FI[ 8PDVLWXDomRSUREOHPDpDHVFROKDGHTXHVW}HVDEHUWDVRXIHFKDGDVQXPDVLWXDomRPDLV RX PHQRV PDWHPDWL]DGD HQYROYHQGR XP FDPSR GH SUREOHPDV FRORFDGRV HP XP RX HP YiULRV FDPSRVGHFRQKHFLPHQWRVPDWHPiWLFRV$IXQomRSULQFLSDOGHXPDVLWXDomRSUREOHPDpDXWLOL]DomR LPSOtFLWDGHSRLVH[SOtFLWDGHQRYDVIHUUDPHQWDVPDWHPiWLFDVHPQRVVRFDVRDVSURSULHGDGHVGD IXQomRISRUPHLRGHTXHVW}HVTXHRDOXQRVHFRORFDQRPRPHQWRGHVXDSHVTXLVD 200 Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR 4XHVWmR&RPSOHWHDVHJXLQWHWDEHOD x 1/10 f(x) -1 1 -3/4 -1/2 -1/4 0 10 10 1/4 1/2 3/4 1 5/4 3/2 7/4 4XHVWmR(PXPVLVWHPDFDUWHVLDQRRUWRJRQDOGHXQLGDGHFPWUDFHD FXUYDUHSUHVHQWDWLYDGDIXQomRI 4XHVWmR &RQVLGHUHRVSRQWRV0iGDFXUYDGHRUGHQDGDV D 5HSUHVHQWHRVSRQWRV0i¶GHFRRUGHQDGDV \ M i - &XLGDGR $XQLGDGHpFP ( ) l E &RQVWUXDDVUHWDV M i M i ' l F ,GHQWL¿TXHDUHWD M i M i ' TXHDSUR[LPDPHOKRUDFXUYD4XDOpR FRH¿FLHQWHDQJXODUGHVVDUHWD"(QFRQWUDI[ 1. Análise matemática da situação-problema 'H¿QLPRVDVQRo}HVGHSURSRUomRJHRPpWULFDHGHSURSRUomRDULWPpWLFD (VWDVGXDVQRo}HVVmRIXQGDPHQWDLVQDUHVROXomRGDVLWXDomRDVVLPFRPRQD XWLOL]DomRGDSURSULHGDGHORJDUtWPLFDGDIXQomRI $ SULPHLUD TXHVWmR WHP SRU REMHWLYR GHPRQVWUDU TXH D IXQomR I WUDQVIRUPD WRGD SURJUHVVmR JHRPpWULFD HP XPD SURJUHVVmR DULWPpWLFD 3DUWLQGR GH XPD SURJUHVVmR JHRPpWULFD Xn GH UD]mR T WHPRV TXH T u n u n +1 HVWmR HP SURSRUomR JHRPpWULFD SRLV 1u n +1 = qu qu n /RJR p g 1 q un un +1 f 1 + f un +1 = f q + f un 6HMDt n = f u n e t n +1 = f u n +1 SRUWDQWR t n +1 = r + t n RQGH r = f q f 1 1DVHJXQGDTXHVWmRSHGHVHFDOFXODUDVLPDJHQVGHFHUWRVQ~PHURVSRU I OHYDQGR HP FRQVLGHUDomR DV VHJXLQWHV FRQGLo}HV f 1 = 1D ® ¯ f = 1 UHVROXomRGRVGLIHUHQWHVLWHQVGHVWDSUHFLVDPVHUXWLOL]DGDVSURSRUo}HVJHRPpWULFDVHDSURSULHGDGHDOJRUtWPLFDGDIXQomRI a)&DOFXOH f f f e f n p g 1 f = f f 1 + f = f = p g 1 f = f f 1 + f = 1 + = p g f = f f 1 + f = 1 + = 3RULQGXomRFRPSOHWDGHPRQVWUDVHTXH n IN IN f nn = n 3DUDQ I I Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR 201 ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR +1 6XS}HVHTXH n IN IN f nn = n HGHPRQVWUDVHTXH f nn+1 = n + 1 +1 p g 1 n n +1 f nn+1 = f f 1 + f nn = 1 + n 3RUWDQWRSDUDWRGR n IN nn = n SULQFtSLRGDLQGXomRFRPSOHWD IN f b) D Demonstrar que f = f + f p g f = f f 1 + f = f + f SRLV f 1 = * 'HPRQVWUDUTXHSDUDWRGRV[H\GHI IR + WHPVH f xy xy = f x + f y x, y,\ [\ xy f xy xy = f x f 1 + f y = f x + f y S g [ f xy xy = f x + f y 3URSULHGDGHIXQGDPHQWDOGHI E 'HGX]DSDUDWRGR[GH IR IR +* f 1 e f x HPIXQomRGH f x x 1 1 1 f 1 = f x = f x + f = o f = f x x x x 1 f x = f x x = f x + f x = f x f x = f x IR + f J &DOFXOHSDUDWRGRDHEGH IR * abab HPIXQomRGH f a e f b 1 1 1 f a + f b = f a + f b $WHUFHLUDTXHVWmRWHPSRUREMHWLYRHVERoDURJUi¿FRGDFXUYDGDIXQomRI f ab ab = f a b = f a + f b = x f x 202 1 -1 1 1 1 1 1 1 1 Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR &XUYDUHSUHVHQWDWLYDGHI ),*85$*5È),&2'$)81d2)('$65(7$60i0¶i D0¶10¶0¶0¶0¶ 0¶0¶0¶ EFI)LJXUD F$UHWDTXHPHOKRUDSUR[LPDDFXUYDGHIpDUHWDTXHSDVVDSHORVSRQWRV 6HXFRH¿FLHQWHDQJXODUpLJXDOD $QiOLVHGLGiWLFDGDVLWXDomRSUREOHPD $VLWXDomRSUREOHPDIRLHODERUDGDOHYDQGRHPFRQVLGHUDomRDVVHJXLQWHV FRQGLo}HV 2VDOXQRVFRPSUHHQGHPIDFLOPHQWHRVGDGRVHSRGHPHQJDMDUVHQD H[SORUDomRGHVVHVGDGRVFRPRVFRQKHFLPHQWRVGLVSRQtYHLV3RGHP FRQFHEHUFODUDPHQWHRTXHpXPDUHVSRVWDSRVVtYHOHSHUWLQHQWHj TXHVWmRFRORFDGD $VLWXDomRSUREOHPDHQYROYHXPFDPSRFRQFHLWXDORFDPSRFRQFHLWXDOGRORJDULWPRTXHGHVHMDPRVHIHWLYDPHQWHH[SORUDUHHPTXH Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR 203 ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR VHVLWXDPDVDSUHQGL]DJHQVYLVDGDV 2VFRQKHFLPHQWRVDQWLJRVGRVDOXQRVVmRLQVX¿FLHQWHVSDUDDUHVROXomRLPHGLDWDGRSUREOHPD 2VFRQKHFLPHQWRVREMHWRVGDDSUHQGL]DJHPIRUQHFHPDVIHUUDPHQWDVDGHTXDGDVSDUDREWHUDVROXomR $VLWXDomRSUREOHPDHQYROYHRVVHJXLQWHVTXDGURVDOJpEULFRJHRPpWULFRQXPpULFR 2IDWRGHVXSRUTXHDIXQomRIpGH¿QLGDHPIR IR +* UHPRQWDjKLVWyULDGRV ORJDULWPRV$IXQomRIpFRQWtQXDHHVWULWDPHQWHFUHVFHQWH(VWDSURSULHGDGH JDUDQWHDXQLFLGDGHGDVVROXo}HVGDVHTXDo}HVORJDUtWPLFDV6DEHVHTXHXPD IXQomRFRQWtQXDHFUHVFHQWHHPLQWHUYDORpELMHWRUD3RUWDQWRDHTXDomR IR e x IR f x = m m IR IR +* DGPLWHXPDVROXomR~QLFDRTXHSHUPLWHFRPSOHWDU DWDEHOD As igualdades f 1 = HI VmRGXDVFRQGLo}HVUHODFLRQDGDVFRP DSURSULHGDGHGDIXQomRHVWXGDGD2VQ~PHURVHVmRSDUWLFXODUPHQWHLPSRUWDQWHVQDHVFROKDGDVSURSRUo}HVJHRPpWULFDVFXMDVLPDJHQVSRUISHUPLWHP UHVROYHUSUDWLFDPHQWHWRGDVDVTXHVW}HVGDVLWXDomR 3DUDMXVWL¿FDUDHVFROKDGHXVDVHDHTXDomRGDWDQJHQWHjFXUYD GHIHPXPSRQWR[ RXVHMD\ I¶[[±[I[6HI[ ORJ[HQWmR DHTXDomRGDWDQJHQWH7jFXUYDGHIHPXP SRQWRGHDEVFLVVD[p7 PDV VH[ WHPVH SRUWDQWRy=ymo-0,43RQGH \ M = I[ 4XDLVFRQKHFLPHQWRVRVDOXQRVSRGHPPRELOL]DUQDUHVROXomRGDVLWXDomR" 'L]HPRVTXHXPDOXQRSRVVXLXPFRQKHFLPHQWRPDWHPiWLFRVHHOHIRUFDSD] GHXWLOL]iORGHIRUPDH[SOtFLWDQDUHVROXomRGHSUREOHPDV1DVLWXDomRSURSRVWDR DOXQRSUHFLVDPLQLPDPHQWHPRELOL]DURVVHJXLQWHVFRQKHFLPHQWRVDSURYDSRULQGXomR FRPSOHWDTXHSHUPLWHGHPRQVWUDUTXHf(10n)=nDSURSULHGDGHORJDUtWPLFDGDIXQomRI HDVFDUDFWHUL]Do}HVGDVSURJUHVV}HVJHRPpWULFDVHDULWPpWLFDVTXHSHUPLWHPGHVYHQGDU DVSLVWDVTXHOHYDULDPjVROXomRGDVLWXDomRSURSRVWDDVSURSULHGDGHVGDUDL]TXDGUDGD GHXPQ~PHURSRVLWLYRTXHVmRLPSUHVFLQGtYHLVDRFiOFXOR f x e 1RWDPRVDLPSRUWkQFLDHRSDSHOGDVPXGDQoDVGHTXDGURQRWUDWDPHQWRGDV TXHVW}HVGDVLWXDomR(VVDVPXGDQoDVSHUPLWHPPXGDUGHSRQWRGHYLVWDHWUDGX]LU DOJXPDVGDVTXHVW}HVGDVLWXDomRGRTXDGUR'28$'<DOJpEULFRDRTXDGURGD JHRPHWULDDQDOtWLFDFRPD¿QDOLGDGHHVSHFt¿FDGHPRELOL]DUDVIHUUDPHQWDVDGHTXDGDV 204 Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR jUHVROXomRGHVVDVTXHVW}HV$VLQWHUDo}HVHQWUHHVVHVTXDGURVFRQVWLWXHPXPSRGHURVR LQVWUXPHQWRSDUDDFRQVWUXomRSRUSDUWHGRDOXQRGRFRQKHFLPHQWRVDEHUHPMRJRQD VLWXDomRSUREOHPDDSDUWLUGHVHXVFRQKHFLPHQWRVGLVSRQtYHLV Uma breve análise da fase experimental $H[SHULrQFLDTXHUHODWDPRVIRLUHDOL]DGDHPXPDWXUPDGHVpULHGR(QVLQR 0pGLR&LrQFLDV([DWDVGH0DOL(VWDH[SHULrQFLDIRLUHDOL]DGDSRU6LDND.RQDWp 3DUWLFLSDUDPGDSHVTXLVDGDVLWXDomRSUREOHPDDOXQRVTXHDLQGDQmR HVWXGDUDPRFRQFHLWRGHORJDULWPR2VDOXQRVWUDEDOKDUDPHPJUXSRJUXSRVGH TXDWURDOXQRVHJUXSRVGHDOXQRVTXHGHQRPLQDPRVGH*1* GG** GDOXQRV *e GFRPSRVWRVSRUDOXQRV2WHPSRSUHYLVWRSDUDDUHVROXomR GDVLWXDomRIRLGHK 1DPDLRULDGRVJUXSRVKRXYHFRQIXVmRHQWUHDSURSRUomRDULWPpWLFDHDSURJUHVVmRDULWPpWLFDH[FHWRQR*TXHMXVWL¿FRXVXDUHVSRVWDGDVHJXLQWHIRUPDYQ IXQIT±I UHWIXn YnSRUWDQWRYQ YnU 3DUDRFiOFXORIVDEHQGRTXHI HI IRLFRQVWDWDGRTXHRV DOXQRVQmRVDELDPSRURQGHFRPHoDUDVROXomRGRSUREOHPD3UHFLVRXVHLQGDJiORV VREUHDSRVVLELOLGDGHGHHQFRQWUDUXPQ~PHUR[WDOTXHRVQ~PHURV[H HVWLYHVVHPHPSURSRUomRJHRPpWULFD(VWDSHUJXQWDSHUPLWLXDRVDOXQRVLQLFLDUD UHVROXomRGDTXHVWmR3DUDRFiOFXORGHInDPDLRULDGRVJUXSRVIH]XPDFRQMHFWXUDPDVVHPGHPRQVWUiOD3DUDLOXVWUDUHVWHIDWRpDSUHVHQWDGDDVHJXLU)LJXUD H)LJXUDDSURGXomRGRJUXSR* ),*85$5(62/8d2$35(6(17$'$3(/2*5832.21$7eS Educar em Revista, Curitiba, Brasil, n. 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Editora UFPR ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR ),*85$352'8d2'2*5832.21$7eS Conclusão )RLSURSRVWRGRSRQWRGHYLVWDGDWUDQVSRVLomRGLGiWLFDXPPpWRGRSDUD LQWURGX]LUHHVWXGDURORJDULWPRDSDUWLUGDVQRo}HVGHSURSRUomRDULWPpWLFDH JHRPpWULFDVHPH[SOLFLWDUDOHLGDIXQomR$OpPGLVVRFRQVHJXLXVHDQDOLVDU GHWHUPLQDGRVFRPSRUWDPHQWRVGRVHVWXGDQWHVEHPFRPRDVGL¿FXOGDGHVTXHHQFRQWUDUDPQDVHTXrQFLDGH.RQDWp(YLGHQWHPHQWHDVLWXDomRSUREOHPD H[DPLQDGD QmR SHUPLWH HVWXGDU WRGRV RV HOHPHQWRV GR FDPSR FRQFHLWXDO GR ORJDULWPR&RPRSRGHVHUSHUFHELGRHODIRFDOL]DRORJDULWPRGHEDVHGHFLPDO +iSRUWDQWRDQHFHVVLGDGHGHFRQVWUXomRGHRXWUDVVLWXDo}HVFXMRSURSyVLWR VHULDRGHHVWXGDUDIXQomRORJDULWPRGHEDVHGLIHUHQWHGH$OpPGLVVRKiD QHFHVVLGDGHGHHVWXGDUDUHODomRTXHH[LVWHHQWUHDIXQomRORJDULWPRHDIXQomR H[SRQHQFLDO Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR 209 ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR REFERÊNCIAS $/028/28' 6 Fundamentos da didática da matemática &XULWLED (GLWRUD GD 8)35 %52866($8 * )RQGHPHQWV HW PpWKRGHV GH OD GLGDFWLTXH GHV PDWKpPDWLTXHV Recherche en didactique des mathématiques*UHQREOHYQS &+(9$//$5'<-2+68$0$La transposition didactique*UHQREOH/D3HQVpH 6DXYDJHeGLWLRQV '28$'<5-HX[GHFDGUHVHWGLDOHFWLTXHRXWLOREMHWRecherche en didactique des mathématiques*UHQREOHYQS *5283( 0$7+ ,5(0 GH 3DULV 9,, 0DWKpPDWLTXHV DSSURFKH SDU GHV WH[WHV KLVWRULTXHVRepères-IREM3RQWj0RXVVRQ7RSLTXHV(GLWLRQVYS .21$7(6L’enseignement et l’apprentissage des logarithmiques dans nos classes de terminales0pPRLUHGH'($GH'LGDFWLTXHGHV0DWKpPDWLTXHV%DPDNR0DOL 8QLYHUVLWpGH%DPDNR 9(55(70Le temps des études3DULV+RQRUp&KDPSLRQ 210 Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR