Mpt vai kinh nghiem trong qua trinh day va ligc Toan

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S62-2011 Dmji^I^

u b

Mpt vai kinh nghiem

trong qua trinh day va ligc Toan

L

d gido vien khong gi vui bdng m'mh dgy md hoc sinh hiiu bdi vd biet van dung kien thuc hoc duoc.De cd duoc dieu do thi ngudi thdy ngodi vdn kiin thirc cdn phdi cd nghe thugt supham nhu gidng gidi de hieu, ldi cuon hoc sinh... Sau day tdi manh dgn trao doi mot vdi kinh nghiem.

I. PHAN TICH, TONG HOfP VA MO R Q N G 1. Mff rong va phat trien

VDI: Day v6 cong thuc tinh dien tich cac hinh Sau khi hoc sinh da hoc cong thuc tinh dien tich cua tam giac G'V dua ra mot so bai tap mang tinh ma rpng va phat trien sau:

Bai toan 1; Tinh dien tich hinh tam giac vuong co 2 canh goc vuong lan lugt la b, c.

Bai toan 2; Tinh dien tich cua hinh chiJ nhat co kich thuoc la a va b(hinh binh hanh, hinh thoi...).

Bai toan 3; Tinh dien tich cua hinh tron co ban kinh R.

Vcri bdi todn I: HS phai xac dinh chieu cao chinh la mot canh goc vuong tir do c6 cong thuc moi ve tinh dien tich cua tam giac vuong

V&i bdi todn 2: HS phai biet each chia hinh chu nhat thanh hai tam giac vuong da biet each tinh Tu do rut ra cong thuc tinh dien tich hinh chu' nhat

V&i bdi todn 3: Voi bai toan nay tuy c6 khac vi la hinh tron co hinh dang khac voi hinh tam giac nhung hoan toan co the tinh dugc niu chia hinh tron thanh nhieu tam giac cong OAB, OBC,

. . . . O M N (trong do AB, BC... MN la cac cung) va neu cac tam giac nay nho thi coi cung chinh la doan thang khi do dien tich ciia hinh tron bSng tong dien tich ciia cac tam giac

1

MAI VAN NAM TrUdng THCS Yen Nhan - Yen Mo - Ninh Binh

S = ~R.AB 2

-R.BC + ... + -R.MN 2 2

Nhu vdy v&i kien thirc ca bdn co the ma rong vd phdt trien sang cdc kien thitc m&i mot cdch tyc nhien de hieu de ndm hdt han vd viec ghi nh&

cUng nhe nhe han.

VD2: Khi day vh so do cac goc trong tam giac, GV CO thi dua ra cac bai tap phat trien va mo rong timg budc nhu sau:

Bai 1; Tinh tong so do cac goc ciia tir giac Bai 2; Tinh tong so do cac goc ciia ngu giac Bai 3: Tinh tong so do cac goc ciia luc giac ...

- De giai quyet bai toan 1 HS phai biet each chia tir giac thanh hai tam giac bang each ke them duong cheo.

- Voi bai toan 2 HS co the chia ngu giac thanh 3 tam giac hoac I tam giac va 1 tir giac.

- Voi bai toan 3 HS co the chia luc giac thanh 2 tir giac, 1 ngii giac va 1 tam giac,...nhu hinh ve duoi day;

< I

R{.\B + BC +... + MN) = - R.ITTR = TTR'

- Tir do hoc sinh co thi tim ra each tinh tong s6 goc ciia da giac n canh la: (n-2). 180

2. Bat chude sang tao

Toi r§t tam die voi cau noi ciia GS. Nguyen Canh Toan

DatjvaHpc

SO2-2011

GV khong chi la day hpe sinh bdt chude ma con phai day cac em hoc each sdng tgo.

VDI: Khi day GV co thi dua ra bai tap sau:

Tinh tong:

5' = 1.2-h2.3-f3.4-f-...-^99.100

Sau khi hoan thanh loi giai neu la HS thong minh se nghT d6n ngay tim cong thirc tong quat cho bai toan d6ng thoi bdt chuac Ai dua ra bai toan moi.

Tinh tong sau: , ,• - i

5, = 1.2.3-i-2.3.44-3.4.5-^...-f-«("+ !)(« +2) Cir nhu vay GV huong dan cac em tir do HS se co thoi quen doi voi cac bai toan khac

3. Phan tich, tong hop

Cai lan do nhilu cai be hap lai ma thanh, nhieu cai rieng hgp lai thanh cai chung, cai kho la truong hgp mo rong ciia cai don gian, Vi vay voi nhieu bai toan ta phai mo se hoac tong quat lai.

VDI: Co bao nhieu hinh vuong trong mgt ludi gom n X n 6 vuong.

De don gian ta cir xet cac ldi nho han xem sao.

- Xet ludi cap 1x1: C6 ngay 1 hinh vuong: 1=1^

- X6t ludi cap 2x2: Co 5 hinh vuong (4 hinh vuong capl va mgt hinh vuong cap 2): 5 = l^-i-2"

- Xet ludi cap 3x3: Cd 14 hinh vuong (9 hinh vuong cap 1, 4 hinh vuong cap 2 va 1 hinh vuong d p 3): 14 = P-^2^-f 3 l

Tir dd ta thay so hinh vuong se la:

V=l2+2--l-3^-i-...-^n^

VD2: Cd 5 cap vg chong gap nhau va bat tay nhau. Hoi cd bao nhieu cai bat tay biet rang cac ong chong khong ai bat tay vg minh va cac ba vg khong bat tay nhau;

- Ong chong thir 1 se cd 8 cai bat tay - Ong chong thir 2 se cd 7 cai bat tay - Ong chong thir 3 se cd 6 cai bat tay - Ong chong thir 4 se co 5 cai bat tay

- Ong chong thir 5 se eo 4 cai bat tay vdi 4 ba vg nhung khong co vg minh.

S6 cai bat tay la 8-1-7-1-6-^5-^4 = 30

Mot cdu hdi dgt ra Id neu cd n cap va chdng nhu tren thi cd bao nhieu cdi bat tay? Lap luan nhu tren cd cdng thirc sau:

Tdng quat neu cd n cap nhu tren thi s6 cai bit tay l a — ^ —

Nhu vgy viec xet cdc trudng hap nhd le ldm ta di hieu han de tit do xdy dung duac cdng thitc tdng qudt. Nguac lgi khi cd cdng thitc tdng qudt rdi ta

.:'.. .h'i ""' """h nhanh chdng.

VD3: Rut ggn bieu thirc sau;

a. A = 4^+lM + V4-2V3

45

b. B = V546 - 84V42 + V253 - 4V63 Vdi y a rat don gian. Nhung vdi y b thi hgc sinh gap khd trong viec dm ldi giai do sd ldn.

Ta cd the tim ldi giai tong quat cho bai toan.

De rut ggn bieu thirc A-yla± b4c ta tim each dl dua A ve dang

A = -^a + b^c = 4{4m±4nY = |VOT ± V«| (*) Van de dat ra la phai tim dugc m va n

Til' (*) khai trien va dong nhat he sd ta co:

m-i- n = a mn b^

4

Suy ra m, n la 2 nghiem ciia phuong trinh X'-aX + — = On

4

Ap dung vao bai toan ta riit ggn dugc

Tir nay viec rut ggn bieu thirc A - yja ± b-^jc chuyen thanh di giai PT (*).

II. MEG NHOf, MEG HOC 1. Cach nhd lau cac cong thirc

D6i vdi HS kha, gioi thi viec ghi nhd cac cdng thirc that don gian. Nhung ddi vdi HS trung binh, yiu thi viec ghi nhd cac cdng thirc la ca mgt van dl... Vi vay GV can huong dan cac em mgt so meo de ghi nhd.

VDI: Khi day vl ti sd lugng giac trong tam giac vudng vdi cac khai niem nhu: sin, cos, tg, cotgyk cgnh doi, canh huyen, cgnh ke.. HS thudng hay nhim lln. Khi dd GV cd the huong dan HS meo nhd nhu sau:

- Sin di hoc [Cd nghTa la di (d6i)tren hoc (huyen)]

- Cos khong hu [co nghTa la ke (khong) tren huyen (hu)]

- Tan doan kit [cos nghTa la doi (doan) tren ke (ket)]

- Cot ket doan.

VD2: Khi hgc vl phln lugng giac ldp 11 nao la cong thirc cdng, cong thirc tong thanh tich, tich thanh tong, nhan doi, nhan ba, goc doi, goc phu,g6c bii... tat ca chdng ca mat GV hudng dan HS cau ndi cira mieng(Cos doi, sin bii, phu cheo) cd nghTa la;

Ap

GV khdng chi la day hgc sinh bdt chuac ma con phai day cac em hgc each sdng tgo.

VDI: Khi day GV cd the dua ra bai tap sau:

Tinh tong:

5 = 1.2-h2.3-+-3.4-^...-H99.100

Sau khi hoan thanh ldi giai neu la HS thong minh se nghT den ngay tim cong thirc tdng quat cho bai toan dong thdi bdt chuac de dua ra bai toan mdi.

Tinh tong sau:

5, = 1.2.3 + 2.3 A -h 3.4.5 +... + n{n + \){n + 2) Cir nhu vay GV hudng dan cac em tir do HS se cd thoi quen ddi vdi cac bai toan khac

3. Phan tich, tong hop

Cai Idn do nhieu cai be hop lai ma thanh, nhieu cai rieng hgp lai thanh cai chung, cai khd la truong hgp md rong ciia cai don gian, Vi vay vdi nhieu bai toan ta phai mo se hoac tong quat lai.

VDT. Co bao nhieu hinh vuong trong mot ludi gom n X n 6 vuong.

De don gian ta cir xet cac ldi nho hon xem sao.

- Xet ludi cap 1x1; Cd ngay 1 hinh vuong: 1 = 1"

- Xet ludi cap 2x2; Cd 5 hinh vuong (4 hinh vuong capl va mgt hinh vudng cap 2): 5 = l"-i-2' - Xet ludi cap 3x3; Co 14 hinh vuong (9 hinh vuong cap 1, 4 hinh vudng cap 2 va 1 hinh vuong cip 3): 14= 1--I-2-4-3-.

Tir do ta thay sd hinh vuong se la; ' '' V=l--^2--^3--^...-^n-

VD2: Co 5 cap vg chong gap nhau va bat tay nhau. Hoi co bao nhieu cai bat tay biet rang cac ong chong khong ai bat ta>' vg minh va cac ba vg khong bat tay nhau;

- Ong chong thir 1 se co 8 cai bit tay - Ong chong thir 2 se cd 7 cai bit tay - Ong chong thir 3 se cd 6 cai bit tay - Ong chdng thir 4 se co 5 cai bit tay

- Ong chong thir 5 se co 4 cai bit tay vdi 4 ba vg nhung khong co vg minh.

So cai bat tay la 8-i-7-i-6-i-5-i-4 = 30

Mdi cdu hdi dgt ra Id neu cd n cap vg chdng nhu tren thi cd bao nhieu cdi bdt lay? Lap luan nhu tren c6 cong thirc sau;

Tong quat neu co n cap nhu tren thi s6 cai bit tay

2

Nliu vdy viec xet cdc truang hap nhd le ldm ta de hieu han de tic do xdy dimg duoc cdng lime Idng qudl. Ngtcac lgi khi cd cdng ihicc tdng qual rdi ta cd tra lai cdu hoi moi cdch nhanh chdng.

SO 2-2011 VD3: Rut ggn biiu thirc sau:

Dmj ^\vaHpe

a. A = 4^-ir24i +4'\-24l

b. B = V546 - 84V42 + ^253 - 4 7 6 3 Vdi y a rit don gian. Nhung vdi y b thi hgc sinh gap khd trong viec tim ldi giai do s6 ldn.

Ta CO the tim ldi giai tdng quat cho bai toan.

De riit gon bieu thirc A = 4a±b4c ta tim each de dua A ve dang - '.! .f r ^ P.l -•-''.[

A = ia±b4c = -ii^m+^n)- = | V ^ ± V ^ | (*) v i n dl dat ra la phai tim duge m va n

Tir (*) khai trien va ddng nhat he sd ta cd:

m + n = a b'c mn

Suy ra m, n la 2 nghiem cua phuong trinh X'~aX + — = 0{*) • •'• ' " '

4

Ap dung vao bai toan ta riit ggn dugc

A = yj{7^-6^ +^f(&J7^ ^iS-]

Tir nay viec riit gon bieu thirc A = ^a^b^c chuyen thanh di giai PT (*).

II. MEO NHO, MEO HQC ' ^ 1. Cach nhd lau cac cong thiirc ' Ddi vdi HS kha, gidi thi viec ghi nhd cac cdng thirc that don gian. Nhung ddi vdi HS trung binh, yeu thi viec ghi nhd cac cdng thu'c la ca mdt vin de... Vi vay GV can hudng dan cac em mot sd meo de ghi nhd. - • • ^' •

VDI: Khi day ve ti sd lugng giac trong tam giac vuong vdi cac khai niem nhu: sin, cos, tg, cotg va cgnh ddi, cgnh huyen, cgnh ki.. HS thudng hay nham lln. Khi do GV cd thi hudng din HS meo nhd nhu sau:

- Sin di hoc [Co nghTa la di (d6i)tren hoc (huyen)]

- Cos khdng hu [co nghTa la Ke (khdng) tren huyen (hu)]

- Tan doan kit [cos nghTa la ddi (doan) tren kl (kit)]

- Cot ket doan.

VD2: Khi hoc vl phln lugng giac Idp 11 nao la cong thirc cong, cong thirc tdng thanh tich, tich thanh tdng. nhan ddi, nhan ba, goc doi, gdc phu.goc bii... tit ca chong ca mat GV hudng dan HS cau ndi cira mieng(Cos doi, sin bii, phu cheo) cd nghTa la;

DatjvaHpc

' i ' « - r ^ SO2-2011

47

+ Cos{-a) = Cosa cdn lai diu am vdi cac ham sin, tan, cot

+ s i n ( ; T - a ) = s i n a cdn lai diu am vdi cac ham cos, tan, cot

TV

+ sin( or) = Cosa va ngugc lai.

VD3: Khi day vl phep cong trir cac sd nguyen khac diu HS ylu,tb thudng hay nham lan khi dd GV gin vao thuc tiln dl day: Tinh (-5)-i-10 =?(ng 5 cho 10 de tra cdn bao nhieu) khdng nhat thiet cir phai thuoc timg tir trong djnh nghTa.

Z.Tudng tugng

- Khi day ve cdng thirc lugng giac va cac PT luong giac co ban GV thudng dua ra cac cong thirc, Nhieu HS rat vat va co gang hgc thuoc long cac cong thiic nghiem ciia cac PT luong giac co' ban nhung riit cuoc van nham lan lung tung vi vay GV hudng dan cac em tudng tugng tren vong tron luong giac.

- Doi vdi hinh hoc hudng dan cac em tudng tugng trong khong gian de ve diing hinh ve (dudng net, xac djnh thiet dien...). ,,

3. Tu chirng minh cac cong thuc

- Khi hoc ve cdng thu'c nhan ba cac em thuong hay nham lan hay khi cac em chi nhd mang mang khong chinh xac thi cac em cd the kiem tra lai.

sin 3A: = sin(x-i- 2x) - sinx.eos2x-i-sin2x.eosx -= sinx(l - 2 s i n ^ x)-i- 2sinx(l - s i n ^ x)

= 3sinx-4sin"' x

- Khi khong nhd ham so lugng giac ciia gdc dac biet (30°, 45° ) c6 the tinh dugc dua vao dudng trdn va dinh li pytago

4. Phan loai

- Phan loai cac bai toan, cac dang toan giiip ta hieu kT va nhd lau hon, ghi nhd de dang co he thong (Vi nhu mot can phdng neu ta xep moi thir ngan nap theo loai thi de duoc nhieu va de lay khi can con neu vat ldn xon thi dung duoc it va khi can ta lai mat thdi gian de tim).

III. TINH NHAM, TINH NHANH 1. Nhan so 2 chir so voi s6 co hai chir s6 VDT. Thuc hien phep tinh 25.32

- D I thuc hien nhanh phep tinh ta tim each chuyin vl phep nhan vdi so cd mot chCr s6 hoac nlu khong thi nhan vdi sd co hai chCr so nhung tron chuc.

25.32 = 750+50 =800 ( 0 day da tach 32=30 + 2) cung CO the tach 25= 20-1-5 hoac bing 3 0 - 5

ie hon phep trir.

Nhu vay ta khong can phai dat nhap ciing chang can phai bam may cung cd the tinh nhanh ra ket qua.

Trong khi hgc sinh dang bam may hay dat nhap thi tdi da tinh ra ket qua rdi.

Hien nay hoc sinh rat ngai lam tinh luon dua y lai vao may tinh. Chi khi nao tinh toan vdi sd Ion, so le thi mdi nen diing may tinh, ddi vdi phep tinh don gian bam may mat thdi gian tham chi bam nham cho ket qua sai.

Co rat nhieu hgc sinh khi co may thi tinh diing, rdi may thi tinh sai ngay.

Khi cdng cac phan so cac em khong biet quy dong deu dimg may de bam ra ket qua roi ghi vao dan den sai sd.

2. Nhan vol so cd phan thap phan

VD2: Thuc hien phep tinh 34.2,6 Thuc hien nhu VDI.

- 34.2,6 = 34(2-^0,6)=2,6(30-i-4) = 34.26:10 cac em tu duy tren deu dua ve nhan, chia vdi so c6 mpt chu so hay sd trdn true:

l^Di: 42.1,5=?

- Ngoai each tu duy nhu tren, ta co the tu duy nhu sau 42.1,5 = 42.(H0,5) = 42-t- 42/2 = 63 tir day ta CO nhan xet sau:

Nhan xet 1: Mot sd nhdn vai 1,5 bdng sd do cong v&i nita sd do.

Chua hit ta lai cd 1,5 = 3/2 nen 42.1,5 = 42.3:2

=63 tir day ta lai cd nhan xet sau;

Nhan xet 2: Mgt sd nhdn v&i 1,5 bang lay sd do nhdn v&i ba sau do chia cho hai

Nhu vay la nhan vdi sd co hai chir so da chuyen ve a chir sd.

Nhu vay ta hoan toan co the nhan nhanh cac sd co 2 hoac 3... chu' so vdi khong chi 1,5 vdi 2,5 va vdi 3,5 ....ma khong can phai dat tinh.

Cin luu ly 0,5 = 1/2, 0,25=1/4, 0,125 = 1/8 Ren luyen each tinh nham la rat can thiet vi du khi ta di chg mua ban thudng phai tinh toan vdi cac con so le liic do khong phai liic nao cung cd may tinh ke ben hay la giay nhap ben minh vi vay ta phai biet each tinh sao cho nhanh.

3. Khi binh phuong cac chu- so

Cac ban hoc sinh Idp 9 khi giai phuong trinh bac 2 thudng phai tinh binh phuong ciia cac so.

VD4: Tinh 23" Neu khong diing may tinh ta van CO the nham nhanh dugc ra ket qua.

- 2 3 - = 23(20+3) = 529

- 23- = (20+3)- = 400 + 120 +9 =529 (hing ding thirc) - AB^ = A^2ABB^ (trong do AB la so co hai chir so) ta lam nhu sau:

+ binh phuong chir so B dugc ket qua dat d cuoi (3-=9) 9

48

^{so 2-2011}

^{Dmjv^J^}

+ hai lln tich 2.AB dugc ket qua dat d giira (2.2.3=12) vilt 2 nhd 1 29

+ binh phuong chti so A dugc ket qua dat d dau , nhung CO nhd (2^=4) nhd 1 la 5 529

Vay 23- = 529

Cac ban co thi tinh nhanh 57^ = 3249.Nhu vay sir dung cdng thirc tren ta cd the nham dugc binh phuang ciia cac so cd hai chir sd.

Dac biet vdi cac chCr so co hai chir so tan ciing la 5 va 1 CO the tinh nhanh nhu sau:

- A5Ml0A+5)^=100A^+100A+25=100A(A+l)+25 tir day ta nhan xet:

Binh phuang cua so tdn ciing Id 5 ta ldm nhu sau lay so ddng tru&c sd 5 nhdn v&i sd han nd 1 dan vi rdi them 25 vdo ket qud.

VD: 45^= 2025 (liy 4.5=20 vilt them 25

A P=( 10A+1 )-= 100A^20A+1 tir day ta cd nhan xet sau:

Binh phuang cua sd tdn cung Id 1 ta ldm nhu sau hdng dan vi dgt Id 1, tru&c hdng dan vj Id 2 ldn sd tru&c sd 1(2A) tru&c hdng chuc Id binh phuang cua so tru&c sd l(A^).

VD: 31^=961, 51 "=2601 (5+5 bing 10 vilt 0 nhdi, 5.5 nhd 1 la 26 kit qua la 2601).

A I ^ = J : A . A ) ( A + A ) 1 (CO nhd).

Ngoai ra con phai luyen tap rat nhieu tinh nhanh, viet nhanh de cd kT nang, kT sao nhieu khi tinh toan.ve hinh bang 6c . Cai gi ciing vay luyen tap nhieu thanh quen

IV. QUAN SAT VA SUY NGHI

Nhieu HS khi doc de xong la cam cui giai ngay.

Nhung neu ta bd vai giay ra de quan sat thi doi khi ta se tim dugc nhieu ldi giai mdi tham chi ldi giai con doc dao bat ngd day sang tao.

VDT. Cho a, b la cac so khong ani thoa man a + b =2.

Chimg minh rang:

a a' +b' >2 b. a^ +b' >2 Cdch 1:

Tir gia thiet c6 b = 2-a thay vao cd a' +{2-a)' >2

<=> (a - 1 ) - ( a ' - 2a + 7) > 0 (dung) Cdch 2:

Ta thay dau bang xay ra khi a = b = 1 ta co each giai sau;

a ' ' + 1 + 1 + 1 > 4a va ft"* + I + 1 + 1 > 4/> cong 2BDT tren ket hgp vdi gia thiet ta dugc dpcm.

2x = 2y-

4

-y --x'

X

1 y

(1) (2) Nlu ap dung each 2 vao y b bai toan nhanh va dom gian va nhanh hon so vdi each 1.

VD2: Giai he PT sau:

Day la bai giai HPT dii ximg loai nen khi dge de HS thudng chgn each giai la trir hai PT cho nhau.

Nhung nlu quan sat ta thiy x, y > 0 nen ap dung BDT tir (1) cd x>y, tir (2) cd y>x. Do do he CO nghiem x = y = 1.

VD3: Cho x, y la 2 sd thay ddi thoa man . , - I I V5 X -f ^y = 1. Chimg mmh rang: |x - 7| < . Cdch 1:

Dat ^ = x - > ' O x = >' + ^ thay vao gia thiit co iA + y)'+4y- =1 <^ 5y^ +2A.y +A^ -\ = 0 PT CO nghiem khi A^ - 5{A^ - 1 ) > 0 o ^^ < -

h a y | x - > ' | < (dpcm). ' I ' l - ' ' .

Cdch 2: N I U quan sat gia thiet ta nghT den BDT bunhiacopki tir do cd ldi giai sau:

^5

p^-y

1 . X - - 2 V

2

<J(l + i ) ( x ^ + 4 / ) = - ^

Cdch 3: Neu ngam kT mdt chut ta thay gia thiet mang dang dap ciia bieu thirc lugng giac viet lai gia thiet dudi dang x^ + i2yy = 1

dat X = s i n 0 , 2 7 = c o s a suy ra:

c o s a A = X- y = sma

<=> 2 s i n a - c o s a - 2A PT cd nghiem khi

2 - + ( - l ) ' >(2Ay

hay |x - ^1 < (dpcm).

c^A'<-