JOURNAL OF SCIENCE OF HNUE
Educational Sci. 2011, \bl. 56, No. 4, pp. 3-12
REN LUYEN VA PHAT TRIEN NANG LlJC
KHAI THAC BAI TOAN CHO SINH VIEN SU PHAM TOAN
Bi!ii Duy Hitng
Tntdng Dat hoc Su' pham Ha Noi E-mail: biiidu>huii^tb^iyalioo.com.x'ii
Tom tat. \'ier ren luyou cho sinh vien sir pham toan giai mot each sang tao cac bai toan trong chuong trinh Toan d tnrdng trung hoc ph6 thong la mot trong cac bien phap gop phan nang cao ch4t luong dao tao cua trudng Dai hoc Su pham. Bai bao de xuat cac hudng khai thac bai tap toan trung hoc pho thong va each td chtre ren luyen phuong phap khai thac, dao sau cac bai toan cho sinh vien su pham toan tren cac gid hoc chuyen dl tu chon.
1. Md dau
Dg nang cao chat krong dao tao giao vien giang day Toan d cac trudng ph6 thong, ngoai viec trang bi cho sinh vien su pham toan cua cac trudng Dai hpc Su pham (DHSP) tri thiic vg phuong phap day hoc, cSn ren luyen va phat triin cho ho kha nang va phudng phap giai mot each sang tao cac bai tap thuoc chuong trmh mon toan trung hoc phQ thong (THPT). Qua khao sat thuc te, chung toi nhan thay rang nhigu sinh vien su pham toan chua van dung thanh thao cac tri thiic co ban trong chuong trmh toan THPT dl giai cac bai tap, con nhigu lung tiing khi gap phai cac bai toan kho trong sach giao khoa mon Toan, dac biet hku nhu chua biSt phuong phap khai thac, tim toi sang tao tii cac bai toan. Di khac phuc nhiing ton tai do chiing toi cho rang can boi dudng nang luc giai cac bai tap ^•a ren luyen phuong phap khai thac cac bai toan cho sinh vien su pham toan thong qua mOt he thdng cac bai tap toan THPT cimg vdi cac bien phap su pham phii hop.
Trong bai bao nay chiing toi de xuat cac hudng khai thac bai tap toan va each td chiic ren luyen va phat trien nang luc khai thac bai toan cho sinh vien su pham toan d trudng DHSP
2. Noi dung nghien cu'u
Tim higu viec day hoc Toan d trudng THPT cho thay mot thuc tg la viec day hoc mot bai toan thudng dudc kgt thuc khi hoc sinh da cd dudc mot Idi giai cho bai toan. Nhigu giao vien toan chita chu trong tdi viec khai thac dao sau cac bai toan,
Biii Duy Hung
nham phat trign nang luc (ri tuc noi chung va kha nang sang tao ciia hoc sinh ndi rieng. Chiing tdi nhan thay can lam cho sinh vien su pham toan, tnrdng DHSP nhan thay su can thiet va hieu qua ciia vice khai thac, dao sau bai toan trong day hoc;
trang bi cho ho phUdng phap khai thac, dao sau bai toan; ap dung cac bien phap can thici S ren luyen cho ho tliUc hanh thdng qua mot he thong chon loc cac bai toan thudc chUdng tiiiili mon toan THPT. Lam dUdc nhu vay se dat dUdc muc tieu kep. Thii nhat, thong qua viec giai va khai thac bai toan thi sinh vign se nam chac kien thiic mdn toan THPT, hifni sau sric nhiing noi dung khd trong sach giao khoa.
trudc nirit giang da\' tot, trong cac ddt tliUc tap sU pham d trudng phd thdng. Thii hai, giiip cho cac giao vien toan THPT tUdng lai biet each khai thac, dao sau mot bai toan. tir dd cd the hudng dan, to chiic hoc sinh thi/c hien cac cdng \'icc khai thac dd. gdp phan phat trien tU duy sang tao cho hoc siiih va nang cao hieu qua day hoc bo mdn Toan.
Viec khai thdc bai toan co the ditac thuc hien theo cdc hudng sau:
- Nhin nhan bai toan dudi nhieu gdc dp khac nhau dg tim ra nhieu Idi giai ciia bai toan, tii dd tim Idi giai hdp li nhat.
- Tign hanh cac hoat ddng dac biet hoa, tuong tu hda, khai quat hoa dg tim ra cac ket qua mdi, de xuat cac bai toan mdi.
- Tign hanh lat ngUdc van de, de xuat va nghign ciiu bai toan dao.
- Bien ddi bai toan va phat bilu chiing dudi cac hinh thiic khac nhau dl tao su linh hoat va mem deo ciia tu duy hpc sinh, gdp phan hinh thanh cho hp cac phim chat tri tuf.
Can phai chii y rang, cac ket qua nhan dUdc nhd suy doan bang tUdng tu.
bang khai quat hda hay lat ngupc van de chi la cac gia thuygt. cd thg diing, co thi sai, can dupe kiem nghiem, chiing minh hay bac bd. \'di mdi bai toan cd t h i khai thac dao sau theo nhiing hudng khac nhau dl thu dupe nhung ket qua mdi, ddc dao.
tii dd dg xuat dupe cac bai toan mdi.
Chiing tdi da tign hanh trang bi va ren luyen phudng phap khai thac, dao sau bai toan cho sinh vien su pham toan trgn cac gid hpc chuyen de tU chon theo quy trinh gdm cac budc nhu sau:
Budc 1. Trang bi tri thtic
Giang \ien trang bi cho sinh vien nhung tri thiic li luan cd ban nhat vg khai quat hda, dac biet hda va tUdng tu, nhiing dang suy doan thudng gap trong day hpc tpan a trudng THPT.
Budc 2. Nam viing phuang phdp
Gidi thieu chp sinh vign cac hudng khai thac, dap sau mdt bai toan. Td chiic cho sinh vign tap luyen hoat ddng khai thac, dao sau cac bai toan theo tiing hudng rigng re nhu tiep can bai toan theo nhigu hudng khac nhau, khai quat hda, dac biet hda, tUdng tu hda, lat ngupc van dg, chuyin ddi bai toan sang dang khac.
Budc 3. Thuc hanh theo nhdm
Ren luyen va phat triin nang luc khai thdc bdi toan cho sinh vien...
Giang vign lua chpn cac bai tpan tiem nang thudc mdt so dang diln hinh trong sach giao khoa mdn Toan THPT. Cac nhdm sinh vien tien hanh khai thac, dao sau cac bai toan dd. Hp thao luan, ban bac vdi nhau \T' each thiic tien hknh cdng viec, ve cac kgt qua dat dupe, cimg dg xuat \a giai qii>'e( cac bai toan mdi.
Budc 4- Tap luyen tdng h0p
Giang vien td chiic \a dieu hanh cho sinh vien cac nhdm bao cao cac ket qua khai thac, dao sau cac bai toan da dupe chu;in bi. Trong budc nay, viec khai thac dao sau bai toan dupe tien hanh mot each toan dien, he thdng, tuan tu theo cac hudng da neu d tren, giiip cho sinh \'ien hilu ky. khric sau va dAii thanh thao viec khai thac, dao sau bai toan, nang cao kha nang gi;ii va sang tao ve toan THPT.
Sau day chiing tdi trinh ba>- mdt so bai toan \'a cac ket (lua khai thac, dao sau tir cac bai toan dd.
.Bdi toan 1. Ch.vr.ng nji.vh. rdng trong hmh binh hanh ABCD cd ddng thiic:
AB^ + EC- + CD' + DA^ = AC' + BD'^ (2.1)
* Thvc nhdt. Cd the chiing minh ding thiic (2.1) theo cac hudng sau:
Sir dung cdng thiic trung tuyen.
I)' + c^ a' 1
ml = — hoac b'^ + c~ = -a' + 2ml vao AABC va ACD.
z ~± z
Sir dung cac bilu thiic vec to, dl bign ddi ve phai cua ding thiic (2.1) nhu sau:
AC + BD' = AC + BD'
= (AB + BCy + (BC - DC\
2
AB + BC] + BC -AB
Sii dung cdng thiic Cosine trong AABC va AABD.
Ap dung dinh li Pitago sau khi ha cac dudng vudng gdc AH va DK Ign dudng thing BC.
* Thii hai, khai thac bai toan bang dac biet hda: Chuyin tii hinh binh hanh ve cac hinh vudng hoac hinh thoi. Ket qua thu dupe thi hien trong bai toan sau.
Bdi toan 1.1. Chiing minh rang trong hinh thoi canh a, tdng cdc binh phuang hai dudng cheo bang Aa'
Day la mdt bai toan dg vdi hpc sinh ldp 10, cd t h i ap dung dinh li Pitagp dl giai bai toan ma khdng can tdi dinh li Cosine. Tuy nhien ciing ngn ngu ra dl sinh vign thay dupe mpi quan he ciia cac bai toan.
* Tha ba, khai thac bai toan bang each phat bilu va giai bai toan dao.
Bill Duy Hung
Bdi toan 1.2. Ch/fmg minh rdng neu tit. gtdc ABCD thod mdn dang thiic (2.1) thi ABCD Id hinh bhiJi hanh.
Vice giai true tiep bai toan 1.2 c|ua la khdng dg dang ngay ca vdi hpc sinh kha, gioi ciia ldp 10 THPT, chiing ta se trd lai bai toan nky d phin sau.
* Thit tu, khai thac bai toan bing khai quat hda, chuyf'ii tir hinh binh hanh sang tii giac bat ky. Khi nghien ciiu tii giac ABCD ta cd tlu'' ap dung cdng thiic tdng binh phUdng hai canh ciia AABC va AACD nhu da ap dung trong hudng thii nhat.
Gpi J\LN tUPng ling la trung dilm cac dudng cheo AC va BD ciia tii giac.
Khi dd ap dung cdng thiic dudng trung tuyen cho AABC va AACD cd:
AB' + BC' - -AC^ + 2BM' DA' + DC' = l^AC' + 2DM'' Cdng hai ding thiic dupc:
AB' + BC' + CD' + DA' = AC' + 2{BM' + DM') (2.2) TuPng tu, AMBD cd:
BM' + DM' = ^BD' + 2MN' (2.3) Thg (2.3) vao (2.2) dupe:
AB' + BC' + CD' + DA' = AC' + BD' + 4MX' (2.4) TCr ket qua nay chiing ta cd thi phat bilu bai toan cho tii giac nhu sau.
Bdi toan 1.3. Chiing minh rang trong tit gidc bdt ky ABCD vdi trung diem hai dudng cheo Id M vd N ludn cd ddng thitc:
AB' + BC' + CD' + DA' = AC' + BD' + 4MN'
Nhan thay ring khi dac biet hda tii giac ABCD bing each cho dilm M triing vdi dilm A'" thi tii giac ABCD trd thanh hinh binh hanh va he thiic (2.4) thanh he thiic (2.1). Tii dd nhan dupc kit qua: tii giac ABCD la hinh binh hanh khi va chi khi thpa man he thiic (2.1). Vay bai toan 1.2 dupc giai nhd van dung bai toan 1.3.
Tiep tuc xem xet ding thiic (2.4), nhan thiy dai luong AMN' > 0. Tii dd de xuat dupc bai tpan
Bdi todn 1.4- Chiing minh rang trong tii gidc bdt ky, tdng cdc binh phuang bdn canh khdng nhd han tdng cdc binh phuang hai dudng cheo cua nd.
Ren luyen vd phat trien nang luc khai thdc bdi todn cho sinh vien...
* Thii ndm, khai thac bing tUPng tu tir Hinh hoc phdng sang Hinh hoc khdng gian.
Trong Hinh hoc khdng gian, hinh hop thudng dupc coi la hinh tUdng tu vdi hinh binh hanh trong Hinh hoc phdng. Kha nhilu tinh chat cua hinh binh hanh nhd phep tUdng tu dupe chuyin thanh cac- tinh chat ciia hinh hop. Ngudi giao vign toan can nim dupc dieu nay, tir dd cd each hudng dan hpc sinh ldp 11 dUa vao ket qua da bilt ciia hinh binh hanh de giai bai toan trong hinh hop. Tinh chat dUdc de cap trong bai toan 1 dupc chuyen thanh bai toan sau (la\'. cd mat trong cac sach giao khoa Hinh hoc ldp 11 THPT
Bdi todn 1.5. Chdng minh rdng tdng b)nh ph.uang cdc ca/iih. riia mdt hinh hop bang tdng binh phuang bon dudng chro cua nd.
N'iec giai bai toan 1.5 giup hpc sinh ciing cd tri thiic ve hinh binh hanh va thay ctudc mdi lien he giua cac tri thiic trong Hinh hoc phdng va Hinh hoc khdng gian.
Ngoai ra, giang vien ciing gpi y dl nhiing sinh vien cd thi tiep tuc khai thac bai toan 1.3 theo cac hudng nhu xet tii dien ABCD, khai quat hda trung dilm M, N ciia cac doan AC BD thanh cac dilm chia cac doan dd theo ti so k nao dd.
Bdi todn 2. Cho hinh chdp diu S.ABC cd canh ddy bang a, canh ben bdng b. Tinh the tich khdi chdp S.ABC.
\'igc giai bai toan kha ddn gian vdi da so hpc sinh ldp 12, chi can tinh dien tich tam giac diu ABC canh a vh dudng cao SH nhd dinh li Pitago, kit qua thu duoc la:
V = — 0V352 - a'- (2.5) 1
12 ^ ^ Khi dac biet hda bai toan 2. chiing ta cho a — b thi hinh chdp deu trd thanh tii dien deu va chiing ta nhan dupc bai toan sau:
Bdi todn 2.1. Tinh the tich khdi tU dien diu cd cdc canh bdng a.
Kit qua nhan dupc thi tich khoi tii dien deu canh a la
V - ^ (2.6) Khai thac bai toan theo hudng thay hing bdi bien
Chiing ta cho mdt yeu to nao dd cua hinh chdp diu S.ABC thay ddi va giii nguygn yen to khac, khi dd thi tich V ciia khoi chdp se la mdt ham sd ciia mdt bien nao dd.
Trudc het chiing ta co dinh canh day hinh chdp bing a, cho canh bgn SA — x thay ddi. Khi dd t h i tich khdi chdp S.ABC tinh dupc la
V = 2 ^ ^ ^ f ^ = /(.) (2.7)
7
Bin Duy 11 Ung
Thi tich \ ciia khdi did]) la mdt. ham so t.lieo bien .r thudc khoang ( - ^ : +ooj.
Tinh dao ham ciia hc\m sd, nhan thiy /'(./) ludn dUdng, tii do suy ra ham /(./;) dong biln tifMi timg khoang xac dinh ciia nd. Din day cd le cluing ta chua thu dUdc ket qua thii vi nao.
Theo ludl hudng khac, t,a (o djnh canh ben SA = b va cho ('anh day ciia hinh chdp thay ddi, dat AB = .1 Theo cdii^^ thiic (2.5) In cd dUdc thd tich khdi chdp la:
.r'/Mi'
^2 ^=^'^''^ ^^'^^
Till tich khoi chdp bAiig gia tri ham sd f{:r) vdi bien r thudc khoang (0 : v ^ 6 ) . Tinh dao ham /'(,;) va lap bang biln thign ciia f{.r) nhan dUdc gia tri Idn nhat ciia the tich V la: max V --- gia tri Idn nhit dat (huu- khi .r = \/2b.
6
TCr ket ciua nhan dupc ta dl xuat bai toan ci.rc tri nhu sau.
Bdi todn 2.2. Cho hinh chdp diu S.ABC cd canh day AB = -/ canh Ijcn SA = b. Tinh thi tich V cua khoi chdp S.ABC theo .r vd h. Khi r thay ddi, hay tlrn gid tri Idn nhdt cua V
Mot hudng khai thac bai toan 2 la bien ddi nd thanh bai toan md nhu sau Bdi todn 2.3. Cho hinh chdp diu S.ABC cd canh ben S.-\ = b. cdc yeu to cdn lai thay doi. Tim GTLN cua thi tich V khoi chdp S.ABC
Bai toan tren cd tac dung tot trong viec ren luyen va phat trien tu duy hpc sinh THPT, dac biet la tu duy sang tao, bdi \'i thudng cd nhieu hudng tiep can de tim Idi giai bai toan. O day cd ba hudng ma giang vien can gdi y cho sinh vien thi.rc hien tim Idi giai ciia bai toan.
Hudng 1: Dat AB = .r va tiln hanh nhu giai bai toan 2.2.
Hudng 2: Dat SH = r
Tinh duoc: V — — x i h ' - ./"). TCr dd cd max \ = — khi .r =
4 ^ ^ 6 3 ^ Hudng 3: Gpi gdc giua canh ben 5"/! va mat day bing a. Khdng khd dl hpc sinh tinh dupc thi tich khoi chdp la:
\/3 6^ cos-n siiid \/3 6'^(l — sin" .r) sin.r
Sau dd dat t — sin a. nhd lap bang biln thign cua ham so y = t — t^ vdi t G (0; 1) nhan dupc kit qua nhu da cd d tren.
Ngoai ra chung ta cd t h i tiep can theo nhiing hudng khac niia, chang han dat AH = X, gpi gdc giiia canh bgn 5.4 va dudng cap SH bang /5, hpac gdc ASB = 2a.
Chuyen tCr bai toan tinh toan thanh bai toan chiing minh b i t ding thiic (BDT).
Ren luyen vd phat triin ndng luc khai thdc bdi todn cho sinh vien...
Ti-d lai bai toan 2, sau khi tinh dupe thi tich khoi chdp deu S.ABC theo cdng thiic (2.5) ta biln ddi bilu thiic \' nhu sau:
\ -~^s/{2a^{2^i^){:\l>^ a^) (2.10) Ap dung BDT giiia trung binh cdng va trung binh nluin cho ba so duong dupe:
'>\ -i
( 2 a - ) ( 2 r r ) ( 3 6 - - ( / - ) < f ~ j -^ (a'+ b')' (2.11) TCr (2.10) va (2.11) nhan dxMc: V < ~- /(P + h^f'^ Dfiu ding thiic xay ra khi 1 a = b. \'ay bai toan ban dau dupc cliu>ln thanh bai toan \e BDT hinh hoc sau day:
Bdi todn 2.4- Khdi chdp deu S..\BC cd canh. day AB — a, canh ben SA ^ b vd cd flic tich V Chiing minh: \ < -~ y/J/r + b'^)'-^
Din day giang \-ien khuyin khich sinh vien bing each biln ddi kheo leo va cung ap dung BDT dg tim them mdt so bai toan tUdng tu neu sau day:
Bdi todn 2.5. Khoi chdp deu S.ABC vdi canh ddy AB = a. canh ben SA = b, cd tilt tich \' Chdng minh: ]' < — \ / ( o " + 26-)'^
Bdi todn 2.6. Khoi chdp diu S.ABC vdi canh ddy AB = a, canh ben SA = b.
cothiUchV Ch^ng m,nh: V < ±,/i3i?TW
Xgoai cac hudng da xet d trgn, giang vign cdn gdi y di sinh vign thuc hien cac khai thac, dao sau tUdng tu cho hinh chdp tii giac diu, ngii giac deu va khai quat hda cho hinh chdp n-giac diu.
Bdi todn 3. Cho ba sd duang a, b, c tuy y. Chdng minh rdng:
ab be ca , , ^.
— + — + —> a+ b + c (2.12) c a b
Dl chiing minh (2.12) cd thi ti('>n hanh theo nhieu hudng khac nhau. Dudi day la mdt vai hudng phu hpp vdi cac hpc sinh ldp 10 THPT vdi trinh dp trung binh.
Theo hudng thii nhat, ta chi dung cac phep biln ddi dai so ddn gian, nhu quy dong miu sd vl trai, chuyin vl va thuc hien nhdm cac so hang phu hpp nhu sau:
(2.12) <^ a'b' + b'c^ + c'a' > abc{a + b + c)
^ 2a'b' + 2b'c' + 2c'a' - 2abc{a + b + c) > 0
^ {ab - be)' + {be - ca)' + {ca - abj' > 0 (2.13)
be
(1
ca
1 -• > 2r b -
en (lb - + - > 2(1
b r "
Biii Duy Hung
BDT (2.13) diing, vay BDT (2.12) dupc chiing minh.
Dau d i n g thiic xay ra khi \a chi khi: ab = be = en <^ n -= b =- c.
Theo hudng thii hai, ap dung BDT giiia trung binh cdng va t r u n g binh nhan ciia hai so dUcJiig cho tiing cap so iiang o vl trai ciia (2.12)
ab he _.^ (ib l)c -.+.-- >2\ = 2b.
<• (I V (' a
Cdng \e vdi \e ba BDT va nit. gpn ta dupe BDT (2.12).
, , , ab be en
Dau dang thiic \;i\- la khi va chi khi cd: — = — --_ — <=> a r^ 0 — c.
(• a 0
Theo hudng I hii ba, tir BDT quen thudc: .;'- + g' + ::' > rg + yz + ::.i\ vdi ./. V, c G M, nhd viec dat: — = :r~ — = y\ - - -= z' ta nhan duoc (2,12).
(• a ' b
Tiep theo cluing tdi chi ra cac hudng cd the khai thac. dao .sau (2.12), de xuat cac bai toan phii hpp \di chudng trinh mdn toan ldp 10 T H P T
- Trong bai toan 3 cho gia thiet a. b, c la cac .sd dudng va \eu cau chiing minh (2.12). Theo hudng lat ngupc van de, ta de nghi hpc sinh tim dieu kien c i n \ a dii ciia cac so n.b,c khac 0 d l (2.12) diing. D l tim can Ira Idi can trP lai xem xet cac Idi giai d trgn cua bai toan 3. Theo hudng thii nhat tim dupc d i l u kien tfch abc phai la so duong, theo hudng thii hai thi khdng t h i tim dupc dieu kien dd.
- Theo hudng dac biet hda bai toan 3 cd the bd sung dieu kien nao dd ciia ba so dUdng a, b, c. C h i n g han, hoac la cho c = 1, hoac la cho a + 6 4- c = 1, ciing co
ah be ca
the cho 1 h -;- = 3 ta nhan dUdc cac bai toan sau c a b
Bdi todn 3.1. Cho hai sd duang a. b. Chunq minh: ab -\ 1— > a + b + 1 a b
Bdi todn 3.2. Cho ba so duang n.b,c thod man a + b + c — 1. 7"///) gid tri nhd , . , ,„ ab be ca
nhat cua: 1 — 1 1
c (I b
Bdi todn 3.3. Ch.o cdc so duang a. b, c thoa man (fb' + b'c' + c'a' = 3abc. Tim gid tri Idn nhdt cua biiu thdc P — a + b + c
- Khi xem xet tim kilm cac BDT tUdng tu vdi (2.12) c i n chii y tdi moi lign he giiia so mii ciia cac thCra sd cd mat d tu thiic. d m i u thiic ca trong ve trai va ve phai ciia BDT nay.
Khdng khd d l cd t h i de xuat cac bai toan tUdng t u nhu sau:
Bdi todn 3.4- Cho cdc so duang a, b, c. Hay chiing minh BDT
a'b b'c c'a
- ^ + - y + - 7 ^ > a + 6 + c 2.14 c^ a^ b^
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Ren luyen vd phat triin ndng luc khai thdc bdi todn cho sinh vi&n....
Bdi todn 3.5. Cho cdc sd duang a,b,c Hd.y chiing minh BDT a'-^b b\' r\i
-T + ~-T + T r >(' + l> + <• (2.15)
f' cr' b^ ' Bdi todn 3.6. Cho cdc so duang a,b.c Chdng ininh.:
a% b'^r r\i
-,T ^ - - . +--;;3 > " + / H - r (2.16) Cd t h i chiing minh (2.14), (2,15), (2.16) b i n g each .sii dung BDT gifta trung
binh cdng va trung binh nhan cho cac so duong. C h i n g han chiing minh BDT (2.14) nhu sau:
Ap dung BDT giua trung binh cdng \a trung binh nhan cho 3 so dudng.
a'b b'c ^ ^, a'bl)'c „ a'b b'c
-V + ^-.i+c> 3 \ - - - ,y.c = lib =^ --- + - . , - > 3b - c c- a- V c- a- c- cr
„ , Ire ca ^ ^ ^ ca a'-b
Tudng t u co: - - r + TT" > 3c - n \-a —:- H > 3a - b a- b- b^ c~
Cdng ba B D T theo cac ve \'a riit gpn dupc (2.14).
Ta ciing neu ra dupc BDT tUPng tu sau da}- vdi cac so dudng a.b.c:
a'b b'c c'a , , , , . , ^ ^
— + — + -—>a' + b' + c' 2.17) c a b
Day la mot B D T cd hinh thiic kha dep, tuy nhign khi b i t ta}- vao chiing minh nd chung ta mdi nhan ra r i n g hoan toan khdng de dang. Cac pliUdng phap chiing minh thdng thudng dupc dem ra ap dung d i u khdng dat kit cjua. Khi dd tu nhign nay sinh mdt nghi vin: hay la BDT (2.17) la sai? Ta thii tim mdt bd ba so dUdng a, b, c ma (2.17) sai. Sau mdt hdi thii vdi cac bd ba so dudng khac nhau, chiing tdi tim t h i y vdi a = 1 . 6 = 2,c = 7 thi ve trai ciia (2.17) b i n g 52 H . cdn ve phai b i n g 54. \ ay (2.17) la sai vdi a = 1, 6 = 2. c = 7. Nhu vay (2.17) khdng la BDT dung vdi mpi a.b,c dUdng. Tu}- nhign chiing ta lai dat ra can hdi: \'di dieu kien nao cua cac so dUdng a, 6, c thi (2.17) la niOt BDT diing? DA}' la vin d l hay va khd, d l nghi cac sinh vign kha, gioi tiep tuc nghign ciru.
Bay gid ta chuyin sang tdng quat hda BDT (2.12) theo cac hudng sau: Thii n h i t , tang so biln, nghia la tang so cac sd dudiig cd mat trong (2.12) tCr ba so thanh n so, vdi n nguyen dudng Idn hdn 3.
Thii hai, tang so mii cua cac thCra so cd mat trong cac vl ciia (2.12) tCf 1, 2, 3 len n.
Cd t h i nhan dUdc mOt so BDT tdng quat cua (2.12), mdt trong sd dd la:
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Biii Duy Hu'ng
Bdi l(>d.n. 3.6. Ch.o cdc so duang aj),c vd n l.ii. so nguyen. duang tuy y. Chdng
mil inli:
a"b b"c r"a , , , , /.. 1o^
-I -^ > a I- /; -I- c (2.1«j r/' //
C.iang vim de nghi sinh vien chiing luinli cac BDT d tien va tiep tuc nghien eiiu giiii (luyet trpn veu nhrmg \aii de da dat ra ma \iee nghien ciiu cdn cdn dd dang.
3. Ket luan
Qua thirc te giau^; dav chuyen de Giai t(;aii T i l l ' I cho sinh vien chiing tdi nhan thay lAiig: \'iec ren lu}en vh phat trien nang lue khai tliac dao sau cac bai tocin T H P T cho sinh vien su pham toau. trudng DHSP la can thiet d l nang cao chat llrpn,^ dao tao giao vif-n loan. \'iec dd cd till tliUc hien duPc thdng qua mdt he thdng cac bai loan va cac bien i)hap sU pham phii hdp tren cac gid hoc chuyen M^ (U chon hoac cac budi sinh hoat ciia nhdm sinh vien veu thich giai toan THPT Qua dd gdp phan lam cho sinh vien nang cao kha nang giai toan. sang tao cac bai toan va them yen thich cdng vice giang da}' Toan sau na}- d tnrdng T H P T .
T A I L I E U T H A M K H A O
[1] Polia G. 2010. Sdng tao todn hoc. Nxb Giao due, Ha Xdi.
[2] Polia G, 2010. Todn hoc vd nhitng suy ludn cd ly. Xxb Giao due, Ha Ndi.
[3] Hoang Chiing, 1969. Ren luyen. khd ndng sdng tao todn hoc a trudng pliu thdng.
Xxb Giao due, Ha Xdi.
[4] Xguyln Canh Toan, 1992. Tap duat cho hoc sinh gidi toan lain quen ddn vdi nghien ciiu todn hoc. X^xb Giao due. Ha Xdi.
A B S T R A C T
Training and d e v e l o p i n g a b i l i t y of e x p l o i t i n g m a t h e m a t i c a l p r o b l e m s for p r o s p e c t i v e t e a c h e r s of M a t h e m a t i c s
Training pre-servicc- t(>achers of Mathematics for solving mathematical prob- lems creati\'ely at Upper Secondar}- Schools is a \va}' to develop the qualit}- of training in uni\'ersities of education. This article discu.sses orientations of exploiting Mathe- matics exer(is(\s at Upper Secon(lar\- Schools and solutions to developing abilities of soh-ing matluMiiatical i)roblems for prospective- teachers of Mathematics in optional lessons in the Facultv of MatluMiiatics in Hanoi Xational I'niversitv of Education.
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