MDI 80 BIEN PHAP BOl D U S N I ; Til DDY THDAN N G H I C H CHD HDC SINH '''' TRONB DAY HDC GlAl BAI TAP TDAN H T B D I I N G PHllTHiiNG ThS THAI THI H O N G LAM Trong day hpc toan, dd''''i vdi hpc sinh (HS)[.]
Trang 1MDI 80 BIEN PHAP BOl DUSNI; Til ĐY THDAN NGHICH CHD HDC SINH
ThS THAI THI H O N G L A M
Trong day hpc toan, đ'i vdi hpc sinh (HS), cdthe
coi hoatđng giai bai tap toan l i hinh thifc chu
yéu cua hoatđng toin hpẹ Cae bai toin (BT)
la mdf phuong tien rat cd hieu qua frong viec giiip HS
nam vifng tri thdc, phit trien tu duy, hinh thanh ki
nang, kixao, biet iJng dgng toan hpc vao thuc tlln Bai
viet de cap mdf sd bien phip boidudng tuduy thuin
nghjch cho HS frong day hpc giai bii tap toan dtrudng
phdthdng
1 Hudng dan HS thuc hien cac thao t i c t u
duy phan tich va tdng hpp nham thu nhan them
thdng tin, ho tro cho viec tim each giai BT; đng
thdi, HS biet khai thac md'i quan he giira phep
phan tich va tdng hpp khi tim each giai va trinh
bay Idi giai mpt BT
Phan tieh v i td'ng hgp la hai hoat đng tri tue trii
ngutyenhung lai l i hai mat cua mot qua trinh thdng
nhát Chung la hai boat đng tri tue co ban eua q u i
trinh tu duy, nhifng boat đng tri tue khie deu diin ra
tren nen tang cua p h i n tich va td'ng hgp De phat frien
tri tue eho HS trong day hpc toan, giao vien (G V) can
chii frpng van de ren luyen kha n i n g phan tieh va
td'ngh(;pcho HS
Cd the hiéu, trong hoat đng giai toin, thao tie tu
duy phin tich va td'ng hgp dugc the hien: - P h i n tieh
yeu tđa cho, yéu to can tim va can chdng minh ;
- Thue hien p h i n tieh va tdng hgp xen ke; - Tieh BT
da cho (thudng l i khd ban) thanh nhieu BT thanh
phan, BT d i e biet, don gian hon, cud'i cung td'ng hgp
lai déthu dupe ket quạ
Vidu 1: Cho tam giae ABC Gpi (0,), (0^), (O3),
(0 J la cic dudng trdn bang nhau, trong đ mdi dudng
trdn (0,), (Oj), (O3) tiep xue vdi hai canh eua tam giic
v i eung tiep xue vdi (0 J Chdng minh ring: cae diém
Ộtam dudng trdn npi tiep, t i m dudng trdn ngoai tiep
tam giae ABC thing h i n g {hinh 1)
Day lamdfBTkhakhdGia thief eiia BT chda nhieu
yéu td va HS rait khd djnh hudng de tim cac m i t xich
lien kef gjifa gia thief vdi ket luan GV ed the td chdc,
hudng d i n HS phan tich vatdng hgp degiai BT
54 Tap ehi Gido due so 325
Hinh 1
Phan tich 1
(phin tich gia thief):
Hudng 1: T u
gia thiefmdi dudng trdn (9,), (0,), (O3) tiep xue vdi hai canh cua tam giic va bing nhau fa suy ra dieu gỉ
Kef qua mong mud'n: 1) I = AÔ nBÔ r^CO, {V);2)OflJI/^B;Op^ilBC\OpjlAC{2')
Hudng2{\6r)g hgp thanh phan 1): Td(1') v i (2>),
ta thu dugc dieu gỉ
T u ( r ) , ( 2 - ) ^ ^ ^ ^ ( 3 ' )
Hudng 3: Td gia thief cac dudng trdn (0,), (0^),
(O3) eiing tiep xue vdi (OJ, ta suy ra dieu gỉ
T d 0 , 0 , = 0^0^= 030^=>0, la tam dudng trdn ngoai tiep A0,0203 (4')
Den day, neu đ'i tugng HS la khi gidi, GV cdthe tiep tge hudng din cie em khai thac ketqua thu dupe tdcac he thdc (3'), (4') va sudgng phep vj tutim I de giai BT Tuy nhien, hudng tiep can nay l i khd đ'i vdi
HS Lue nay, GV nen dinh hudng cho HS phan tieh kef luan eua BT
Phan tich2{phar\ tich kef luan): Déchdng minh
1,0,0^ thing hing, ta can chifng minh dieu gỉ Mgc dieh eua eau hdi niy la yeu eau HS xem xet kef luan, neu dugc mdt so cich thudng sd dgng de
chifng minh ba diém thing hang "Khi tap trung tu tudng vao muc dieh vi hudng tát ei nghi luc vao viec thuc hien muc dieh đ, ehung ta di nhin tháy cac phuong tien deditc^ketqui"{^; tr 121)
Hudng 1: ziOfi = 180°; Hudng 2.-1, 0 , 0 , la
anh cila 3 diem thing h i n g qua mdt phep bien hinh
(phep đng dang, hoic phep đi hinh); HudngS:^^^
* Khoa Toan -TnTOngDai hoc Vinh
(kil 1 /2014)
Trang 2tai phep v j t u V e d tam lampttrong 3 die'm 1,0,0^
v i 2 die'm cdn lai l i cap anh va tao anh tuong ifng;
Tong hqp 1: Vdi 0 l i tam dudng trdn ngoai tiep
AABC, fa can chifng minh ton tai V ,: 0 ^ 0 ,
Phan tich 3:1 on tai V,,,,: 0 - > 0 ' <=tdn tai V,,,.:
A-^0,
B->0,
C^0„
10, = klA, lO, = kIB, 10, = kIC< :cae
bd3 diem (A, 1,0,); (B, 1,0^); (C, 1,0,) thing hing,
<:=Tif(3')va(1') (td'ng
• u ' - , - /O, lO, lo,
c u n g t h u f u v a ^ = ^ = ^
hpp thinh phan 2')
Tif dd, GV hudng din HS trinh biy Idi giai BT tren
CO sdcac hoat dpng phin tich, td'ng hpp dtren
Khigiaimdt By, su phin tich can hudng vao mge
dieh tim ra eic mit xich logic nd'i giifa gia thie't vdi ke't
luan De'tap luyen cho HS kining khaifhie md'i quan
he giifa phep phan tich va td'ng hgp trong giai toan,
GV ed the thue hien mdt sdthao tic nhu: - Thief ke
cac BT m i ktii giai HS phai su dgng phep td'ng hpp;
•Cho HS giaicicBT (edcung yeu ciu neu cdthe) m i
neu sudgng phep tdng hgp eic em se gap khd khin
Td dd, ggi y eho HS su dgng phep phan tich; - Yeu
cau HS giai BT sudgng phd'i hgp giifa phep phan tich
va td'ng hgp (bien dd'i td hai phia)
2 Giup HS hieu dugc y nghia cua viec dat BT
trong moi lien he hai chieu vdi cac BT khac "gan"
vdi nd (BT to'ng quat, BT dac biet, BT tuang tu,
BTngu(7c, )nhlm khaithac,tim each gialBTva
tao BT mdi
_ De'tao eho HS thdi quen xem xet mdf BT trong
md'i lien he vdi BT khac, ching ban vdi BT ngupe, G V
can: - Khai thie triet de'eic BT eg the, cac dang toan
khigiaidua vio md'i lien he vdi BT ngugc;-Trude moi
BT tren, yeu cau HS giaj bing cich lien he vdi BT
ngupe; - Sau khi HS giai xong mdf BT cd khai thac
md'i lien he vdi BT ngugc, GV nen nhan manh hieu
qua cua BT ngugc dd'i vdi viee giai BT da eho
Cie boat dpng tren thue cha't la goi ddng coket
thuchoaX ddng Theo Nguyin Ba Kim: "Goiddng eo
keithuccung cd tie dung nang cao tinh tugiae trong
hoat ddng hqc tap nhu cic each goi ddng co khac
Mac diu nd khdng cd tic dung kich thieh ddi vdi nqi
dung da qua hoic hoat dong da thuc hien, nhung nd
gdp phan goi ddng co thuc day hoat ddng hqc tap ndi
chungvanhieukhivieegqiddngcoketthue dtrudng
help nay iai ia su chuan bi gai ddng comd dau cho
nhung trudng hop tuong tusau nay"{2; fr.141 -142)
Vidu2: Mpt tmdng tieu hpe cd 50 HS dat danh
(kil 1/2014)
hieu chiu ngoan Bae Hd, trong dd ed 4 cap anh em sinh ddi Can chpn mpt nhdm 3 HS frong sdso em nay di duDai hdi chiu ngoan Bic Ho sao cho trong nhdm khdng cd cap anh em sinh ddi nao Hdi cd bao nhieu each chpn?
A//jan xet: Giai true tiep BT nay HS phai xet ri't
nhieu trudng hpp,^viec phan chia cae trudng hop la khdng d l dang bdi ra't d l mie sai lam Cg the, HS phai xet eac trudng hpp sau: chpn 3 em frong sd42 HS; chpn 2 em trong 42 HS va 1 em frong 8 em cua
4 cap sinh ddi; chpn 1 em frong 42 HS va 2 em trong
8 em (luc nay phai chia eac trudng hgp nhd ifng vdi 2
em khdng d trong mdt cap); chpn 3 em trong 4 cap sinh ddi (phai ehia nhieu trudng hgp nhd) Chinh vi su phuc tap cua viec giai true tiep BT, budc HS phai thay dd'i eacli lam bing viec xet BT ngupc vdi yeu cau la:
Tinh sd each eu3 HS ma trong do cd mot cap anh em sinh ddiva them mqtHS nua Bing viee xem xet BT
ngupc, HS cd Idi giai BT nhu sau:
Ldi giii: Ta cd cj = 19600 each cd 3 H S tiiy y tu
50 HS
Cd 4 each chpn 1 cap anh em sinh ddi Vdi mdi each chpn 1 cap anh em sinh ddi cd 48 each chpn them mdf ngudi nifa (eho du 3 ngudi) Do dd, ed4.48
= 192 each cd 3 ngudi ma trong do cd diing mpt cap anh em sinh ddi Vi vay, sdcach chpn ra 3 HS ma frong nhdm khdng ed cap anh em sinh ddi nao la: 19600-192 =19408 (each)
3 Giup HS ludn y thuc duac viec khai thac md'i quan he giifa quy nap va suy diin khi tim each giai BT
Quy nap la loai suy luan di tif nhifng trudng hpp
rieng den cai ehung, tdnhifng trudng hpp eg the rut ra
kef luan td'ng quit Suy dien la suy luan di tdcai chung
den cai rieng, fi/quy luat phd'bien den trudng hpp cg the Do dd, kef luan thu dugc ludn diing Trong toan hpc, suy luan suy dien logic ddng vai trd chu ye'u Trong qua trinh tu duy, quy nap va suy dien cd md'i lien he mat thie't vdi nhau.Khdng the cd suy diin neu khdng ed quy nap, suy dien la di tdcai chung den cai rieng Oedi den cae luan de chung lim eo sd cho suy dien, con ngudi phai ehuyen td nhiing quan sit rieng le do tri giae cam nhin dugc sang viec khii quit cae quan sit rieng le ay, trong qua trinh nay thi suy luan quy nap giuvaifrdchu ye'u Trifhucthu dugc nhd suy luan quy nap thudng khdng day du, khdng hoan
chinh va ed tinh cha't du doin Nhd the, quy nap cd the duqc kiem nghiem bing suy diin Mac du quy nap va suy dien la hai phuong phap nhan thuc cd chieu hudng ddi lap nhau, nhung cd lien he huu co vdi nhau, iam tien decho nhau, cai nay ddi hoi cai kia va
Tap chi Gido due so 325 55
Trang 3bo sung cho eai kia Do dd, khdng nen tich rdi quy
nap vdi suy diin, eudng dieu phuong phap nay miha
thi'p phuong phip kia v i ngugc lai
^ l'/du3;Tim tip hgp cae tmng die'm M eua doan
thing AB di ddng sao cho hai dau cua doan thing tua
tren hai dudng thing a, b cheo nhau (hinh 2)
Thdng thudng, HSraf"e ngai" dd'i vdi dang toin f/m
quy tich, nhat l i cae BTtim quy tich trong khdng gian
Nhieu HS khidpe dang toin nay khdng djnh hinh dugc
cich lam, khdng biet huy ddng kien thdc nao da hpe de
giai.Od'ivdi nhifng BTquyfich,detim dupe hudng giai
quyet, HS can dudoin dugequyfich Mpt frong nhifng
each lam dedudoin quy tich ddlaxef mdtsd trudng
hpp die biet, mpt sdvj tri suy bien, nghia lasddgng quy
nap Vdi finh than dd, d BT niy, G V cd the hudng din
HS tim dudng ldi giai BT nhu sau:
- Hay xet mdt vi tri die biet cua doan thing AB Vdi
trudng hqp dd, hay tim diem M7
HS: Dd la doan vudng gdc chung IJ cua a v i b
Khi dd M = 0 (vdi 0 la trung die'm eua doan IJ)
- Khi A sIvaB di ddng tren b thi M didqng tren
dudngnao?
HS: Do finh chi't dudng trung binh eua tam giic,
suy ra M thupc dutjng thing b' song song vdi dudng
thing b vadi qua 0
- KhiB=JvaA di ddng tren a thi M di ddng tren
dudngnao?
^ HS: M thude dudng thing a'song song vdidudng
thing a vadi qua 0
- Td eac ket qui tren, hay du doan quy tich eac
diem liJIl
HS: Tap hpp cac die'm M l i mp (a) di qua 0 v i
song song vdi cic dudng thing a v i b
- Hay chung minh dudoin trenl
Dua vao kef qua du doin ve quy tich eua diem M
lamp (a) da girfyeho hudng chtJng minh, dd la chuyen
BT tim quy tieh eua die'm M trong_ khdng gian ve BT
tim quy tieh die'm M trong mat phing Tddd, ggi cho
HS sudgng phep chieu song song len mp(a) va de'
BT don gian hon_, fa chpn phuang chieu la IJ
4 Huong din HS thuc hien hoat dong dao
ngupc de tim each giai BT va sang tao BT
Viec thue hien hoat ddng dao ngugc ed the' giup
HS linti hoat trong suy nghi, trong each giai quyet van
de
l^/rfu4;Trong he toa dp Oxy eho diem M(- 3; 2)
Tim hai die'm A, B tren frge Ox sao cho AB = 5 va M A
•I-MB be nha't
Viec tim gia trj nhd nha't cua bieu thdc MA H- MB =
7(a + 3)'+ 4+^(6+3)'+ 4; trong dd: A{a;Qi), B{b;<y)
thda man |a-6| = 5 la khdng d l ding vdi HS Theo
56 Tap ehi Gido due so 325
thdi quen, HS hay su dgng khao sif ham sohoiceaebat ding thile quen thude de' giai
BT Tuy nhien, dd'i vdi BT niy,
su dgng cac cdng eg dd la khdng thuin Igi; HS can ed nhifng phit hien mau chdt de tim dugc BT d i biet nhim hd trg eho qui trinh giai BT ban dau GV ed the ggi y HS
bing cic cau hdi nhu: Chung ta da gap BTnio gin gidng vdi BT nay khdn^ Neu HS vin khdng phit hien dugcGV edthedln daXlhem: Hay nghiden BT quen thude ed eung ket luan vdi BT da cho?
Mge dieh m i GV hudng tdi la HS can phat hien
dugc BT ngugc: "Cho hai diem cddinh A, B (cd the cho bang tqa do) Tim diem M tren true Ox sao cho I^A + I^B be nhat"{a day da dao ngugc vai trd eua
cic ye'u to trong bai) Day la BT quen thudt: d i dupe day tudng minh frong chuang trinh (giai bing phep dd'i xifng frge frong trudng hgp A, B eung phia dd'i vdi frge Ox) G V tiep tge dua ra nhifng eau hdi djnh huting giup HS tim ra each bien ddi BT ban dau de'vin dung dugc BT ngugc
5 Tap cho HS phan tich dac diem cua dir kien, lam xuait hien cac lien tudng giira dCf kien dd vdi nhung ddi tugng quen thuoc, khai thac lien tudng nham thuc day viec dinh hudng, tim each giai quyet BT
Theo {3), dd'ituong trong hoatddng nhan thuc luc dau tdn tai doe lap vdi chu theHS, HS cin dung cae hoatddng tri tue, cac thao tie tuduy dua tren tri thdc, kinh nghiem died detham nhap vao ddi tuong nghien Ciiu thdng qua phan tich mdi quan he, lien he chda trong dd'ituong, detudd thain loi cho viec lien tudng, huy ddng kien thdc giiiS T
Vidu5:G&i phuong trinh (PT): sin' x -1- cos' x = 1
DasoHSdalim nhu sau:
PT o (sin x + cos j:)(sin ^ )c - sin AT cos )c -I- cos^ J:) = 1
Bing each thay bieu thdc sin^x+cos^x bdi 1
„ , , „ „ (sin.t + cosj:)=-l l'-\
s u y ra sinjcos;r = = - ^ , , = sinx + coSX,
ta bien dd'i PT da eho ve PT bic 3 an t ed the nha'm dugc nghiem
Neu GV ggi y HS thay sd 1 d ve phai bing sin';c+cos'x khi dd, PT da eho trd thinh: sin'x + cos'x = sin';c+cos'x- Luc nay, HS cd the
(kil-1 /2014)
Trang 4chuyen ve, dua ve PT tich hoac dung phuong phip
danh gii: s i n ' x < s i n - x , c o s ' x < c o s " j r = >
sin'x-Hcos'Ar<sin^x + cos-j: Tif đ, díu '=" xay
Tuđfaed:(''+ir + M+C- = 2/i 5/ic+2t
rakhivachikhi: sm r = sin X
= 2i;r
= - + 2t;r
2
Vdi phuong phap lien tudng, thay the sd 1 bdi
sin^x + cos^ 1- va dung phuang phip danh gii, HS
se khdng "choing nggp" khi G V yeu eau giai eac^PT
tuang tu khi bic ciia sinxva cosxcao hon; ching
han, PT sin'x +cos'x = l,mabing cae phuang phap
giai khie se rít khd khin
6 Ren luyen eho HS thdi quen ra soat lai, tu
danh gia qua trinh giai toan
Trong đi thudng, ngudi ta thudng khuyen "nghi
di roi phii nghi iaf thi dieu đ va viee ehung minh
"thuan, dio" trong toin hqc cd tic dung qua lai vdi
nhaụ HS kem thudng bd sdt phan dio, nen trong
nhieu BTve giii PT, cic em hay bo sdt nghiem hoic
dua nghiem ngoai iai vao; vdi cie B T quy tich thi khdng
xac dinh gidi han cua quy tich, BTve bien iuan thi bo
sdt nhieu tnrdng hqp khdng xet den (4, tr 124) Cd
the ndi, hoatđng giai toan laco hdi eho HS thehien
kien thue, kining eua ban than mdf each rd nhait Neu
HScdyfhucrisoiflai,tudanhgiaquatrinh giai toan
cua ban thin nghia l i c i e em da tu thíy dugc nhirng
cai dung, eii sai, cai dugc va chua dugc trong kien
thdc, kining eua minh
Wduff.-Trong khdng gian vdi hefpa đOxy, cho
m i f e a u ( 5 ) : ( x - l ) ' + ( : i ' - 2 ) ' + ( r - 3 ) ' = 9 v a d u d n g
thing A : i ^ = ^ = ^ Viet PT mat phing (P)
di qua M(4; 3; 4), song song vdi dudng thing A va
tiep xue vdi mat eau (S)
CdHSgiainhusau:Gpi^(o;A;c)(fl'+Á+ểiO)
la vecfophip tuyén cua (P) Khi đ, PT eua mat phing
(P) •.ăx-4)-^ b(y-3)->- c(i -4) = 0 Vi dudng thing
Asongsong vdi mat phing {P)=>n,,.u^ = 0, frong
đ Wi(-3; 2; 2)la vecfochi phuong eua dudng thing
ASuyra: -3a + 26+2c = O o o = —^—^
Gpi ^ 1 ; 2; 3), R=3 lan lugt la tim va ban kinh cila
mat cau (S) Mat phing (P) tiep xiie vdi mat cau (S)<=>
\-3a-b-±_^
''(''(''»='^«;^?:P^
-Vdic = 0=>b = a = 0(loai) Vdi e ?t 0 ta ed: 2b2 5bc + 2c^
« 2 -5-+2 = 0 o - = 2 hoac - = -
c c • c 1
- Vdi - = 2, chpn b = 2, c = 1, suy ra a = 2 PT mat phing (P):2x-i-2y + z - 1 8 = 0(*)
- Vdi - = - , ehon b = 1, c = 2, suy ra a = 2 Khi đ
c 2 • '
(P):2x+y + 2 z - 1 9 = 0 ( " ) Oen day, nhieu HS két luan cd 2 mat phing (P) can tim^cd PT (*) va (**) dtren Nhung thúe te, chi cd mat phing (P) vdi PT (**) thoa man BT, cdn vdi PT (*) khdng thoa man vi (P) chifa Ạ Ldi nay khdng phai la
do so suit dan thuan, bdi HS hau nhu khong cd thdi quen thd lai neu nhu chua dugc GV nhan manh Nguyen nhan sai lam d day cd the la: HS sd dgng luan cukhdng diing, da cho ring (/>)// A <=> n^.w^ = 0 (do khdng nam vifng kien thifc, khdng phan biet dugc dieu kien can va dieu kien du); hoac da trao luan de
ma hai luan de nay lai khdng tuang duong, đ la cac
em van suy ra dugc tif (/>)//A => n7-"I = 0 • nhung lai
diing két qua n^,.ii^ = o de két luan (P)//A
* * *
Tren day la mdt sdbien phap su pham G V edthé
su dgng de giiip HS trong viec tim tdi each giai BT va nang cao nang lye giai toan, đng thdi gdp phan hinh thanh va phat trien cho HS each suy nghi theo hai chieu ed xu hudng trai ngugc •
(kil 1/2014)
(1) G Polyạ Giai bai toan nhy the naỏ NXB Giao
diic.H 1997
(2) Nguyen Ba Kim (chii bien) Phmmg phap day hpc
mon Toan NXB Dai hoc supham, H 2004
(3) Dao Tam (chu bien) - Tran Trung Td chirc hoat
đng nhan thirc trong day hpc mon Toan or trinrng trung hpc phd thong NXB Dai hpc supham, H 2010
(4) Nguyen Canh Toan Phmmg phap luan duy vat
bien chirng vdi viec hpc, day, nghien cuu toan hpc,
tap 2 NXB Dai hpc qud'c giạ H 1997
SUMMARY
Fostering and developing thinking for students are important issues in teaching mathematics in schools This paper mentions some measures to fos-ter reverse thinking for students in teaching how to solve mathematical problems at high school
Tap chi Gido due so 325 57