II NGHIEN cuu & UNG DUNG ar SIT DUNG BIEU DIEN 601 TRONG DAY HOC KHAI NiEm HAm s o m TS Nguyin Danh Nam, m Thj Hi£m Tru&ng Dgi hgc Suphgm Dgi hgc Thdi Nguyin SUMMARY This paper presents the method of[.]
II NGHIEN cuu & UNG DUNG ar SIT DUNG BIEU DIEN TRONG DAY HOC KHAI NiEm HAm s o m TS Nguyin Danh Nam, m Thj Hi£m Tru&ng Dgi hgc Suphgm - Dgi hgc Thdi Nguyin SUMMARY This paper presents the method of using multiple representation in teaching the concept "fitnction" at the secondary schools The history of developing this concept showed the diffirent types of representations like geometric curves, algebraic expressions, tables of values, and graphs Therefore, using multiple representation would make a contribution to help students understanding the nature of the concept and apply various strategies in approaching a solution In particular, students know how to determine and choose the best method of representation in order to solve mathematical problems Keywords: Representation, Multiple Representation, Function Ngdy nh^n bdi: 1/8/2014; Ngdy duy^t d&ng: 20/8/2014 Khii nifm bilu dien bpi Bilu diln bOi li sy kit hpp cua nhilu d^ing bilu dien nhim md ti mOt hifn tupng, tinh hudng nio dd vi lim cho nd ed y nghTa Bilu diln bOi dupe chia thinh hai \o^\ ca bin: bilu diln bOi ngoii vi bilu diln bOi Biiu diin bgi ngoai md ti eie md hinh vit ly, dd thi, bing bilu (bing cic gii tri), ngdn ngft, ki hifu, bilu thurc dai sd (hoic phuang trinh) vi hinh inh (so dd, bilu dd, hinh vg) Bilu dien bOi giiip HS bilt lya chpn lo?ii bilu diln tdt nhit cho mOt khii nifm toin hpc vi ed thi mft sd HS ed thi nhfn thiy d^ng bilu diln nio dl hilu ban nhtog HS khic TCr dd, HS bilt dinh gii uu dilm, nhupc dilm ciia cic lo^i bilu dien khic nhau, phit triln ning Ifp luin vi tim tdi eie phuang phip giii toin khic Khii nifm him s6 chuong trinh mon Toin phi thong Him sd li mOt khii nifm quan trpng, gift vai trd trung tim ehuang trinh mdn Toin phd thdng Lich sir phit triln cua khii nifm him s6 cung cho thiy eie d?ng bilu diln khic cua khii nifm niy Nd dupe hinh thinh tir vin dl nghien eto vl sy phy thufc lln cua hai d?i lupng liy gii tri eie t|lp hpp hto h?n vi rdi r^ie, die bift li cic d^i lupng lifn quan din v§n tdc, thdi gian, quing dutog, gia tdc, lye, thdi ki diu, tinh phy thuOe gifta eie d^i lupng dupe md ti bing Idi thdng qua eie bilu diln hinh hpc mi khdng dl e?p din quan hf sd lupng Din thi ki VXIII, bilu dien tuang quan him sd dugc chuyin tft trye giie hinh hpc sang bilu thiirc giii tich ll • TAP CHf THlfTBJGlAO DMC-sd 109-9/2014 Tuy nhign, tu tutog ddng nhit him sd vdi mft bilu thiire giii tich sau dd da dupe thay bdi bilu diln bing bing sd, dd thi, cdng thiirc hoic tdng quit hon li bing mOt phuong tifn nio dd cho phdp xic dinh sy tuong tog gifta hai d^i lupng D^e bift ly thuylt tip hpp ddi, him sd nhu mOt quy tie tuong tog hay quan hf gifta cic phin tiJr cua hai t$p hpp thda mft sd dilu kifn nio dd Nhu vf,y, nghign eto lich sir phit triln cua khii nifm him sd giiip ta nim vtog ddng thdi cic die trung vi eie cich bilu diln ciia nd, cich chuyin ddi giOa cic cich bilu diln vi die bift li vife ip dyng nd vio vife giii quylt eie bii toin niy sinh thyc tiln euOe sdng Trong qui trinh nghifn eiiru, chtog tdi d$t ciu hdi phdng vin HS Idp 10 THPT: "Trinh biy nhtog hilu bilt ciia em vl khii nifm him sd?" MOt sd ciu tri Idi nh^n dupe li: "Him so ed d?ing y = ax + b, y = ax hoic y = ax^", "Him sd ed dd thi li m^t dutog thing ho^e mOt parabdn", "Him sd li quy tic vdi mii gii tri ciia x ed tuang tog mOt gii tri ciia y", "Him sd li y, biln sd li x v i dugc cho bdi edng thiirc y = f(x)", vi nhilu HS khdng md ti dupe khii nifm him so mi chi don thuin lift kf mOt s6 him sd cy thi dudi d^ng eie bilu thiire giai tich (him bfc nhit hoic him b^c hai) Hiu hit eie em it quan tam din cic d$e trung ciia khii nifm him sd nhu tuong tog, phy thuOe, biln thif n, miln xic dinh, miln gii tri Cic tinh chit ciia him sd ed thi dupe nhgn bift thdng qua cdng thiirc ham sd vi dd thi Dd thi li mft phuang tifn bilu diln thi hifn rd cic die trung ciia him sd Do v^y, HS ein bilt "dpe dd thi", ed nghTa li NGHIEN CUU & UNG DUNG dya vio nhtog dd tW da bilt hoac thdng qua cic phdp bign ddi tW dk hilu dupe cic tinh chit cua him sd That viy, khii nifm him sd ddng biln, nghich biln, him sd chin, Iiim sd le, him sd tuin hoin, khdng phii li nhtog khii nifm khd nhung nlu giio vign (GV) khdng giiip HS bilt cich phign dich vi chuyin ddi gifta ba d?ing giii tich, dai sd vi hinh hpc thi cic em sg gip nhilu khd khan vife vin dyng cic khii nifm niy (bing 1) Nhu v|y, sur dyng bilu diln bOi day hpc khii nifm him sd tao dilu kifn cho HS kiln tao tri thiire bing nhtog dutog khic theo ly thuylt da tri tuf Vi thi, qui trinh day bpe GV cin thilt kl cic tinh hudng, eie bii tip md t i ding difu dd thi him sd, chuyin ddi eie d?ing bilu diln giii tich, ^ i sd, hinh hpc nhim giiip HS bilt sur dyng bilu diln bOi qui trinh giii quylt vin dl Su dyng bilu diln bpi d^y hpc khii nifm him s6 Trong dgy hpc Toin, GV ein quan tim din eie dgng bilu diln thutog gip dudi ^ y : ki hifu, bilu thiire, ngdn ngft, hinh vg, so dd, db thi, bing bilu, md hinh thyc tl Nhu vgy, dk giiip HS hilu siu mOt khii nifm toin hpc, GV cin bdi dutog cho HS ning lye bilu diln, bilu dien bOi, phidn dich vi chuyin ddi linh boat giiia eie dang bilu diln khic ciia etog khii nifm toin hpc dd Tir dd, chtog tdi dl xuit quy trinh dgy hpc khii nifm him sd d trutog phd thdng gdm bude sau diy: - Bude 1: Siir dyng bilu diln don giiip HS tilp e$n khii nifm him sd Bude 2: Su dyng bilu diln bOi giup HS phit triln sy thdng hilu khii nifm him sd Bude 3: Tgo mdi lifn hf gifta cic dang bilu dien, tich hpp vi ehuyin ddi linh hogt gifta eie logi bilu dien giup HS etog ed, vgn dyng khii nifm him sd cic tinh hudng khic II Tmh hulng Su dyng bilu diln bOi dgy hpc khii nifm him sd bic nhit Trude hit, GV sir dyng bilu diln don (nhu bing gii tri hoac tuong tog) nhim giup HS hilu dupe quy tic cho tuong tog vdi mdi gii tq ciia x mOt vi chi mOt gii tri ciia y Sau dd, GV sir dyng bilu diln bOi (nhu tpa dO, tuong tog, bing, dd thi) dk minh boat mOt sd tinh chit cua him sd nhu: mdi quan hf gifta gii tri ddi sd vi him sd, dd thi vi bing gii tri ciia him sd, lim thi nio dk xic dinh dupe miln gii tri cua him sd tir bing gii fri, dd thi hoic phuang trinh bilu dien ciia him sd Cudi etog, GV thilt kl eie bii tip phign djch vi chuyin ddi giiia cic dgng bilu diln vdi nhu: Tim phuang trinh ciia iiim sd di cho six dyng dd thi cua nd hoac bing gii tri; xic dinh gii tri Ito nhit cua him sd bing phuang phip dd thi hoic bing gii tn; lift kg eie dgng bilu dien khic ciia mOt him sd; sijr dyng dd thi cua him sd di cho dk xic dinh bing gii tri him sd vi ngupe Igi; diln eie gii tri cdn thilu vio eie cOt tpa dO, tuong tog, bing gii tri, dh thi cua mOt him sd cy thi (bing 2) Nhu v|ly, hilu mOt khii nifm toin hpc ed nghTa li HS cd thi nhin khii nifm dd cic cich bilu diln khic vi ed thi phign dich tft cich bilu diln niy sang cich bilu diln khic Tinh huong Hiy lift kf vi phien djch cic dgng bilu diln tinh don difu cua him sd y=-jc' x^ - 2ac+2 trgn khoing (-1; 2) Phign dich li thugt ngft dtog dl md ti qui trinh suy nhtog y tutog ciia bilu diln bpi Nd dl egp din qui trinh nh|ln thiire lifn quan din vife chuyin tft dgng bilu dien niy sang dgng bilu dien khic Vi dy, tft dgng edng thftc, phuong trinh, bing gii tri sang dgng dd thi vi ngupe Igi Kit qui d bang dudi diy cho thiy eie dgng bilu diln khic thi hifn tinh don difu ciia him sd trdn v i y tutog thilt kl eie hogt dOng Bdng 1: Bdngphien dich cdc dgng biiu dien gidi tich dgi sd, hinh hgc Dang giii tfch Him s6 f(x) ding biln Him so f(x) nghich biln Dang dai so x,f(x,)f(x,) Him so f(x) li him s6 chin f(-x) = f(x) xtog Hims6f(x)lihims6lg f(-x)=-m Do thi him so nhin trye hoinh lim trye doi xtog Ding difu ciia thi him so lip Igi sau moi chu ki Him s6 f(x) li him si tuan 3T>0,f(x) = f(x + T) hoin Dang hinh hpc D I thi ciia him so di Ign Dd thi cua him sd di xudng Do thi him so nhin trye tung lim trye ddi TAP CHf THI^BIGlAO DMC-sd 109-9/2014 • 2} II NGHIEN CUU & UNG DUNG Bang 2: Biiu diin bdi cua ham sd bgc nhdty = 2x + Tpa dp Tuong tog (x;y) (-i;-i) (0;i) (i;3) (2; 5) chuyin ddi cic dgng bilu dien him sd nghich biln Thdng qua tinh hudng niy, GV rgn luyfn cho HS eie thao tie phign dich vi ehuyin ddi giOra cic dgng bilu diln khic thi hifn tinh chit don difu ciia him s6y = ^x' x" -2^+2 trgn khoing (-1; 2) nhu dgng ngdn ngft, dang ki hifu vi dgng dd thi giup HS cung cd vi vfn dyng khii nifm him sd nghich biln Hon niia, phign dich vi ehuyin ddi eie dgng bilu diln bing gii tri, phuong trinh, vi dd thi ciia him sd giiip HS hilu mdi lign hf gifta sd hpc, dgi sd vi hinh hpc Tft dd, giiip HS xiy dyng y tutog quan trpng ciia khii nifm him s6 li hf s6 gde (dgo him ciia him sd) vi giao dilm cua dd thi vdi trye hoinh (giii phuang trinh) Nhu v$y, phign dich eie dang bilu diln li kT ning co ban dl phit triln nang lye giii quylt vin dl vi tinh linh hogt phign dich cic dang bilu dien li nhin td quan trpng ciia ning lye tu Hilu dupe moi lign hf gifta hinh hpc vi dgi s6 giiip HS dl ding chuyin tft tilp c^n eie ddi tupng bing phuang phip hinh hpc sang phuang phip dgi sd vi ngupe Igi Tft dd giiip HS thiy dupe y nghTa ciia ttog dgng bilu diln vife dl xuit eie phuong phip giii cic dgng bii toin khic nhu phuang phip dd thi hoic phuang phip dgi s6 Tinh hulng Vdi gii tri nio cua m thi phuang trinh x* - 2x2 - - m = ed y^^ nghifm phin bift? Hiy giii bii toin trgn bing hai phuong phip khic Vdi bii toin niy HS ein cd kha ning khai thie bilu diln dgng ki hifu (bilu thiirc dgi sd) vi bilu diln dgng dd thi dl dl xuit hai phuang phip giai khic nhau, dd li phuang phip dgi sd vi phuong phip dd thi 24 • TAP CHf THI^BIGlAO DMC-sd - / Dithi Bing X y -1 -1 * Phuang phip dgi so: Dit t = x^ Dg phuong trinh x* - 2x2 - m = ed nghifm phan bift thi phuang trinh t^ - 2t - m = phii ed hai nghifm duong phin bift, dilu niy tuong duong vdi: 4+w>0 A >0 5>0 2>0 o - < w < P>0 -3-m>0 V§y vdi -4 < m < -3 thi phuang trinh da cho cd nghifm phin bift * Phuang phip dd thi: Sft dyng dgng bilu diln dd thi, bii toin da cho dupe ehuyin thinh bii toin "Vdi gii tri nio cua m thi dutog thing y = m cit dl thi him sd y = x" - 2x2 - tgi ^jgjj, pj^^^ jjj^t» If •^ a -a -1 "0 i 13 fn»-1 i > i » ' (0.-3) ^•^ -4 A » (.1 -A) / C • (1, -41 Tifp egn bii toin theo phuang phip dd thi, GV hutog din HS vg dutog thing y = m vi dd thi him sd y = X* - 2x2 - j ) ^ ^^J^ yj ^^f tuong ddi cua chiing, HS ed thi dl ding xic dinh dupe vdi dilu kifn -4 < m < -3 thi dutog thing y = m cit dd thi him sd y = x* - 2x2 - tgi ^jgm phan bift Nhu v§y, thft nhit, bilu diln bOi ddng vai trd NGHIEN CUU & UNG DUNG II Bdng 3: Biiu diin bgi thi hiin tinh dan diiu cua ham sd nghich biin Dang ngdn ngir D^ng ki hifu Dang dl thj A Vx,,X2e(-l;2) H ^ s l y = -x^—x2-2x +2 nghich biln tren khoing (-1; 2)., hoic D I thi him so "di xuong" khoing (-i;2) /(x)