SANG TAD Bill T O J l N i l I I I BAI TDAN BAN DAU VtBATOiNG IHOC NHAM BtNlUYlNTUBUYDPGlAP SANGTAD CHD HOC SINH TRDNG HOC PDOTHONG O ThS N G U Y I N S O N HA" R6n luy^n hi duy d ^ l§p vd sdng Igo cho[.]
Trang 1SANG TAD Bill TOJlNil III BAI TDAN BAN DAU VtBATOiNG IHOC NHAM BtNlUYlNTUBUYDPGlAP.SANGTAD CHD HOC SINH TRDNG HOC PDOTHONG
O ThS N G U Y I N S O N HA"
R6n luy^n hi duy d ^ l§p vd sdng Igo cho
hpc sinh (HS) Id m$t trong nhOng nhl^m vi,
quan trpng trong dgy hpc mdn Todn 6
trudng phd' thdng Phfin Idm nhOng bdl todn (BT|
6 pha llidng hl^n noy Id BT cd kS't lu^n rS rdng,
ygu cdu chvng minh m^t k^t qud ndo dd ho$c
tim m$t dd'l tupng c\f th^; tu nhihig BT dd, gido
vlSn (GV| cd thi thoy dd'l cdch phdt bllu d l duo
ro BT mdl B^t ddng ihljc (BDT) Id m^t dpng todn
tllm long nhllu khd ndng phdl trlln tu duy cho
ngudi hpc Bdi bdo trinh bdy m^t so' cdch sdng
tpo BT mdi thdng qua BT ban i6u v l BDT nhdm
r^n luv^n tu duy d$c l$p, sdng too cho HS d trung
hpc pnd' ttidng
1 GV gie(u di phdp todn cda BDT, yiu c&u
HS so sdnh gid trj cOa cdc bilu ihuc, sau dd
HS tv thay d6\ dllu ki^n cua BT d l Hm ra kit
qud mdt
BT han ddu: Cho A, 8 to cdc bliu ihuc chiia
b/6i ChOng minh BDT A > B
Trong BT ban ddu ndy, bllu thi>c cdn chung
minh do du<?c phdt bilu mdt cdch tucmg minh
GV glau di phep todn cua BDT d l HS phdl phdn
dodn, phdt hl|n kit qud cdn chiing minh GV
y6u cdu HS chO ddng thay dd'l dllu ki«n cOo BT
d l flm ro kit qud mdl Vi vdy, HS cd ihl phdt
bllu k^i BT nhu sou:
BTmdl: Cho A, B Id cdc bliu ihlic chiia blin
So sdnh gid Irl cua hal hliu ihuc A, B hliu ihay
doi dliu klSn cua blin ihl kii qud ihay dil nhu
ihi ndo?
Vidv h BTban diu: Chiing minh rdng:
x^ + 2x > 3x^ V X > 2
Brmi«.Chox>2
a} So sdnh gid tri cDo hoi bllu thiic
A=!i? + 2xvdB = 3>f
bl Nlu thay d^l dllu klSn dcfl vdi x thi kit
qud Ihay d l l nhu i h l ndo?
ehdnllch:A-B=.i^ + 2x-3x''x(i^-3x*2l
=.4x-l)lx-2l
aj Vi X> 2n6nxfx-;)(x-2)>0=>A-B>0=> A>B
bl X6t dSu Clio bllu thiic xlx-](x-2l:
X
X x-1
x - 2
-co 0 1 2 KO
- 0 + 1 +
0 + + +
- 1 - 0
>t(x-lXx-2) 1 - 0 + 0 - 0 +
To cd: v6l X e {0;1;2) thi A-B > 0 => A=B: v6l
i s ( ^ ; 0 ) u ( l ; 2 ) thi A-B < 0 => A<B; vdi
r e (0; 1) u (2; +«) thM-B > 0 => A>B Nhdn xSt: Trong qud Irlnh x6t dd'u cOo bllu
thiic A - B, HS cd thi tv dS xud't cdc dllu kiSn khdc doi X vdi vd duo ro kit qud mdi
VIdu 2: BTban dau: Chiing minh rfing: aix-1
> a+b, v6l^a,b > h
BTmdl: Cho cdc sd thi/c o, b> L
al So sdnh hal bleu thiic M = ab+ 1 vd
N = a+b _
bj Nlu thoy doi dieu kiSn dd'l vol cdc so' a, b,
kit qud thoy dot nhu ihl ndo?
Phan tich: ab+ha-b = (a-ljlb-lj
a) Nluo>/;b>? lhla-l>0,b-l>0=»la-lllb-J) >0=> ab+1 > a+b
b)Ta xet cdc trudng hpp sou: - Nlu mpt trong
hoi sd'o, t bdng 1 ihl (a-))(b-l) = 0 => ob+! =
a+b; - Nlu cd hal s l a, b nhd hon 1 thi: o- J < 0;
b-1 <0=>la-l)lb-ll>0=>ob+l >a+b;-N4u
hal s l a, b cd diing mot s l Idn hon 1 vd mdt so'
nhd hon 1 thi (a-Ijlb-1) <0=>ob+] < a+b ^
Nh^n xil: Trong qud trinh x6t dd'u cuo bllu
thiic A - fl, HS cd thi tv d l xud't cdc dllu kiSn khdc dd'l vol hot s l a, b vd duo ra kit qud mdl
VIdv 3: BTban dau: Cho neN', xeR, Hx) =
2x" + I, gtxl = >e + 2 Chiing minh rSng: f{xl >
glxl,'^x> 1
BTmdl:Cho neH',xsR
' TnWlfTHPTCIlliyil t)l tgc Wplm Hi N«l
Trang 2+ 1 v d g W x " + 2 k h l x > /
b; Nlu Ihay d l l dllu kl^n cHa BT d l l v6l x vd
n thi kit qud thay d l l nhu thi ndo?
Phdn Hch: a] Hxj • gW - x"-1 Vdi V x>) - > x»
>;->ffx;>gW;
W N l u x - I t h l x " }->xr-
l'0->Hx}-glxl'0=>Hxl-glxl;
' N l u X < /, x6t cdc trudng hpp sou cOa n:
Trudng h<^ /; n Id s l nguyin duong 16, x<J
=> x"</ =•> X"-; < 0 - > fix) • gix) <0-> fix) <
Trudng hgp 2: n Id s l nguyin duong chfln,
x" - 1 = 0 o X = ±1 To cd bdng xlt ddu:
Jt
x T - l
-00 - 1 1 -MO
+ 0 0 +
Bdng trln cho ihSfy: + Vdi x - - 1 ; n chgn:
Hx) - glxj - 0 - > Hxl - glxl; + Vdi x < • 1; n chSn:
fix) - gW > 0 => fixl > glxl; + Vdt - I < x < 0/
n chSn: fix) - gW < 0 - > llxl < gix)
Kil ludn: - Nlu n l l : + Hxl = gix) khi x - ];
+ Hx) > glxl khi X > I; + Hxl < gix) khi x < / ;
- N l u n c h a n : + / i ' x ; g M « i l i s ( - l ; l } , + / i ' x ; >
gW khi xe(-a);-l)u(l;-H»); + Hx) < gix) khi
xsl-l;l)
HS dd Sli di/ng tinh chSn l l cuo n, kit hpp vdi
cdc khodng cua blln d l so sdnh gid trj hal bllu
thiic VlSc xet cdc trudng hpp cy thi cua n vd x
khong cd trong yiu cdu cua bdi todn; tuy nhlln,
vol cdu hdl 'md', HS phdl tv Hm ra cdc dilu kl|n
lien quan cuo n vd x M$t khdc, HS cd thi khdm
phd ro BDT Hx) > gW khdng chi dung khi x > ;
md cdn diing khi X < - J trong trudng hpp n Id s l
chSn
2 Chuyin ti> chi/ng minh m|t BDT sang
chiing minh nhilu BDT cd i h l c6
BT ban dSu: Chiing minh BDT A 2 B
BTmdl: Cho cdc bllu thiic A,B,C,, = B +
mlA-B),C^ = B + n^A-S; Hay l0p td't cd cdc BDT cd thi
cd glua cdc bilu thiic A,fi,C,^C„ vd chiing minh
cdc BDT dd
rtion llch:A-C 'A-B- mlA-B) =
(hm)tA-B),B-C_ = -mlA-B), C^-C„ - lm-n)IA-B) Vi|c l^p
cdc BDT mdl dupe dva vdo kit qud xlt dd'u cOa
bllu thiic A-B Nhu v§y, lii BOT A>fi, GV cd i h l
y8u cou HS tim mil liln h$ glua cdc bllu thiic
V6I m3l s l thvc m khdc 0 tuong ling s3 cd m$t
thiic A, B; tuong ling v6l hoi s l thvc phdn bijt m,
n khdc 0, GV cd thi yiu cdu HS so sdnh glOa C,
vd C, Cdch phdt bllu cOa BT mdl glCip ngudi hpc rIn luyln tu duy d$c ldp, sdng tpo nhllu hon
so vdi v i ^ chiing minh BDT ban ddu
Vf dv 4: BTban ddu: Cho x,y eR, A - x'+y', B
• 2xy Chiing minh rdng: A 2 B
BT mdi: Cho x,y€R, A - x' + /, B - 2xy,
c ('*>•)' n^(£+Ml±(Zl2£2l
° 2 ' 16
Hdy ldp cdc BOT cd i h l cd giOa cdc bllu thiic
A, B, C, D vd chiing minh cdc BOT dd
Phdn Kch: A-a = x' y'-2xy.-ix-yT^m^AiS
A-IJ
! " ) • > '
B
c-0 16 -Ix-yfsO^CSD
d BT ban ddu, GV dd rIn luy?n cho HS kr
ndng v$n dvng kit qud (x-y)^ ^ d l chiing minh BDT Tuy nhlln, vdi BT mdl, GV dua HS vdo linh hulng tv kiln tpo ra cdc BDT; trong dd, HS phdl
van dvng mpt cdch nhudn nhuyin kit qud
Ix-yf £0 vdo chiing minh cdc BOT khdc
VI dll 5: BTban ddu: Cho X£(I;2I, A=if + 2,B- 3x Chiing minh rdng: A < B
Brmdl:Choxef'.a,•*-;<' +2, B 3 x , C
2 , 2i
— x ' + — X
5 5 Jx»- I2x+ (0, c =
Hay ldp tdt cd cdc BOT cd i h l cd giOo cdc
bllu thiic A, 6, C, D Chiing minh cdc BDT dd
Phdn Hch: A-B-x'*2-3i-U-nx-2)<l>^A<B
»2-l5i'-l2«10).-<(i'-3i»!l.-<(i-ll(x-!)>0!=.(>C
14 7,
A-C~
«-C=3x-lSr-l2i+I01=-5(i-l)li-2)>Oafl>C
«-0.).-l4>'*T"-T>-|"'-!"Ml'-l«'-21'»='<'' 5 5 5 5 5 5
21 27, Khi HS gldl BT ban ddu, GV cht kllm tra dupe
d HS kr ndng v§n dgng BDT (x- l)lx-2l < 0 (dllu
Trang 3mdl, G V cfuti HS vdo tinh hu^ng ti/ l0p ra vd
chiing minh c6c BDT mdl, trong dd, HS phol v^n
dyng mpt c6ch nhu6n nhu/en k^t qud (x- l)(x-2)
< 0 (vol dllu kiSn 1<x<2\
3 TCr BT ban d6u, HS c6 t h i duo ra nhllu BT
mdl vd "dn phdi dv dodn kit qud hodc mit phin
ndo dd cOa kit qud, h/ minh khdm phd kiin ^uc
hang khd ndng cd ihi, h/ khdm phd tdl mdc td'l
da h-ong nhffng hadnh cdnh cv thi" [ 1); nhOng BT
nhu v^y dux?c gpl Id BT md M^t s6' d^c dllm cOo
BT md d h-udng ph^ thdng: - BT tim tdl thi dliu
p/id/ Hm khdng dugc niu lin mdl ed^ h/dng minh,
ngudi hgc phdl tim td't cd cdc kil qud cd thi ed;
-Niu Id BT ehung minh, ngudi hgcf^di p/idn dodn,
phdt hlin cdc kit qud cin chdng minh vd ehung
minh dliu dd; - Ngudi hgc cd thi thay dil mdt
so dliu kl$n dedua ra kit qud mdl
V6I BT md, HS khdng b} phy thu$c vdo kiln
thijc dd cd md dutjrc tham gla vdo qud trinh
sdng tqo BT mdl, cdc em cd t h i h/ gldl quylt
van d l vd thu duge nhOng kit qud mdl h> dllu
t(nh mim d6o cOa tuduy sdng too, khd ndng dp dyng linh hopt kit qud dd cd vdo gldl quylt cdc
BT khdc nhau, d^c bi^t Id cdc em r^n luy^n dupe tinh nhudn nhuySn, mdt tinh chd't quan trpng cuo
tu duy sdng tpo Q
(l)G.Polya.S4ngt90to6nhpc.t9p3 NXB GWorffic, H.1976
Tdi li^u tham khito
I, Ton Than Xdy ditng H th6ng cdu hdi bdi ldp nhdm
bdi du&ng m^t s^yiu t^ciia tuduy sdiig igo cho hqc sinh khd vd gidi Todn dtradrng trung hgc cosd Vifi
Nam Lu9n fin Ph6 ti£n sT IChoa tiQc tarn li Vi$n ktioa
tiQCgiiiodvc, H 1995
SUMMARY
The article summarizes some characteristics of the open problems In schools and offers some creative measures to open thepfoblem of Inequality contents These measures will help teachers from the Inequality has to create problems on the ability to promote In-dependent and creative students
Thu€ trang sir dung ki thuaL
(Tiip thea trang 14)
De khdng djnh vol trd cua PPDH vd KTDH
dd'l vdi mdn G D H , chung tdl phdng vd'n mot sd
GV V kiln dupe cd V.T.LA neu ra Id «trong xd
hpl ngdy noy con dudng de cho ngudi hpc chilm
Ifnh tri thi/c nhanh nhd't Id con dudng h/ hpc
Md d l kich thich dupe hung thu nhdn thue cuo
cdc em khdng cd con dudng ndo khdc dd chinh
Id vi^c «n6n" vd «cdn phdl" su dyng cdc PPDH
vdKTDH"
3 Qua thi/c l l nghlSn ci>u, chung tdl nhdn
thdy, G V dd rd't no li/c duo PPDH vd KTDH hi^n
dpi vdo trong bdl gidng cOa minh Tuy nhlen, so'
lupng ndy khdng coo vd vl#c si> dyng ndy cdn
mang tinh chd't phong trdo chua dCing vdi quy
trinh cdch thue tlln hdnh thyc hi#n Mdt so G V
d w a thyc sy tdm huyet khi khi ISn Idp Cd nhllu
nguySn nhdn dnh hudng den vlSc G V v^n dyng
PPDH vd KTDH vdo mdn G D H , nguyin nhdn cd
v l phia chu quon Idn khdch quan: ndng li/c td
chuc: 35,7% G V chpn sdung", thdi gian: 71,4%
chon adung", SV chua thich ung vdi si; thay d^l
PP hpc tdp: 78,d% sd' G V chpn
ftp Chi filao due s6 2 9 5 (fci i - io/aoia>
D l ndng coo chd't lupng vd hl&u qud DH mdn GDH, GV cdn phdi vung v l chuySn mdn, ch^c v l PPDH vd KTDH vd thudng xuyin hpc hdl khdng ngOng cd t h i ddp ung dupe nhung yeu cdu, ddi hdi cua xd hpl •
( D T r i n Ngpe Lan - D$ng H6ng Hitfu "la thujt d^y
hpc - MOt c i p do ciia ptiuong phSp dgy hpc" 7<jp cM Gido difc, SO' 205 (Ki i, 1/2009)
Tk\ li$u ttiam khdo
1 NguySn Cudng Phinmg ti$n kl thu$t vd dA ditng d^y hpc (Churnig trinh gido dMc d^i hpc) BO
GD-DT, H I 995
2 Dp an Vi$t - Bi Tdi li^u tdp hudn dgy vd hpc tich
c(/c.H200I
3 NguySn K5' Phumig phdp gido di^c tfch cvc Id'y
ngudi hpc ldm trung tdm NXB Cidodifc, H 1995
4 Le Nguyen Long Thi> di tim nhOng phuwng phdp
d^y hpc hifu qud NXB Gi&od^c, H 2000
5 Trin H6ng Qudn "Phuemg phip d?y hpc tich cpc
mOt phuong phip vO cang qu^ biu" Tgp chi Nghiin
ciiugidodftc 12/1994
SUMMARY
Through the Invertlgotlon process the situattlon,
we realized that: teachers have access to the
mod-em methods and techniques of teaching but not high