22 sa 11 2012 DjPfe j f ty Phat trien tif duy linh hoat cho hpc sinh tieu hpc thong qua mpt so bai toan giai blng phû ofng phap dung chuf thay so TRUCQXJYNH Trucmg Ti8u hoc s6 1 Quang Vinh Quang Di6n[.]
Trang 122 sa 11-2012 D j P f e j f t y
Phat trien tif duy linh hoat cho hpc sinh tieu hpc thong qua mpt so bai toan giai blng phu^ofng phap dung chuf thay so
TRUCQXJYNH
Trucmg Ti8u hoc s6 1 Quang Vinh - Quang Di6n - tinh Thira Thien Uui
MQt trong nhirng muc tieu quan trong ciia
chiroTig trinh toan a Tieu hoc la phat
trien kha nang tu duy cho hoc sinh Trong thirc te
giang day b6i duong cho hoc sinh, chung toi nhan
thiy rdng, cac bai toan giai bfing phuong phap chiJ
thay s6 co iru the dac biet trong viec phat trien cac
thao tac tir duy cua hoc sinh Day la mot trong
nhirng dang toan kho va co ngi dung phong phu vi
vay chting thirong xuat hien trong cac ki thi chpn
hgc sinh gioi hay thi tuyen vao cac truong chuyen
Dang toan nay thucmg sir dung vcri cac doi
tugng la hgc sinh gioi cac lop cuoi cap d Tieu hoc
Viec dung phirong phap dung chir thay so trong
viec giai cac bai toan a tieu hgc hien nay van con
nhieu y kien khac nhau Nhieu giao vien cho rang: six
dung phuong phap diing chif thay so la qua tai voi
hgc sinh vi thirc chSt cua phucmg phap nay la viec
giai cac phuong trinh se dirge hpc trong chucmg
trinh a cac lop tren Viec trang bj cho hpc sinh mgt
cong cu giai toan qua manh nhir vay se lam so ciing
kha nang tu duy cOa hgc sinh Ben canh do, mgt so
giao vien giai thich rang: viec van dung phuong
phap nay van phu hpp voi trinh dp nhan thirc ciia
hpc sinh Tieu hgc neu biet van dung linh hoat cac
kien thirc ma hgc sinh dupc hpc trong chuang trinh
Trong bai viet nay, chiing toi xin gioi thieu mgt
so bai toan phu hpp voi doi tugng hpc sinh gioi cac
lop cuoi cap bang each van dung linh hoat phirong
phap diing chu thay s6, De giai tot cac bai toan bang
phuong phap dimg chif thay s6, hpc sinh c§n nlm
vifng mpt so kien thirc sau:
2 Mot so bai toan ve so tu nhien va tong, hieu, tich,
thiroTigcac chir so
Bai 6: Tint so co hai chir so, biet rdng so do gap 8
Idn tong cdc chir so ciia no
Cdch 1: Gidi Goi so cdn lim Id ab (khac 0) Theo bai ra ta co:
^ = 8 X (a + b) (3)
a x l O + b = 8 x a + 8 x b (cau t&o s6, nhan mgt so voi mpt tong)
a x l O - a x 8 = 8 x b - b (ciing trii 8 x a va b)
a X (10 - 8) = b X (8 - 1) (nhan mot s6 vai mpt hi?u)
2 x a - 7 X b
* a X 2 dat gia tri Ion nh§t la 18 khi a = 9, do do: 7 x
b ciing d^t gia tri Ion nh5t la 18 Vay b < 3 Mat khac a x 2 la s6 chin do do 7 x b cung la so chan nen b la so chan Vay b = 2
Suyra: a = 1 4 : 2 - 7 Thir lai: 72 = 8 x (7 +2) (dung)
Vay so phai tim la 72
Cdch 2: SOr dung phuong phap diing so do doan thang
Quan sat (3) ta co sa do sau:
(a+b):
Ta CO A = ab - (a+b) la mpt s6 chia h6t cho 7 hay
A = a x 10 + b - a - b = 9 x a mpt so chia het cho 7;
ma 9 khong chia het cho 7 nen a chia het cho 7 Vai
a = 7 thi A - 7 x 9 - 63 Do do gia trj ctia mgt phan hay (a + b) la: 63: 7 = 9.Suy ra: b = 9 - 7 = 2 Ta dupc so 72
Thu lai: 72 = 8 x (7 +2) (dung)
V^y so can tim la 72
Bai 7: Tim mot so co hai chii' so, biet rang khi
chia so do cho tong cac chir so cua no thi duqtc
thuong la 5 va du* 12
Cdch I: Gidi
Trang 2DaiMHpc s6^i.i 23
ab =(a + b ) x 5 + 1 2
ab- = 5xa+5xh+ 12 (nhan mpt s6 vai mgt tong)
a x I0 + b = 5 x a + 5 x b + 12
5 x a + 5 x a + b = 5 x a + 4 x b + b + 1 2
5 x a = 4 x b + l 2 (cung trir cac so hang giong nhau)
5 x a = 4 x ( b + 3) (nhan mgt s6 voi mot t6ng) (4)
- Ta co: a x 5 chia h6t cho 5 va 4 nen a x 5 chia het
cho 20 Vay a X 5 = 20 ho^c a x 5 = 40 (vi a <10)
• a X 5 = 20 nen a = 20: 5 = 4
V|y b + 3 = 20: 4 = 5 hay b = 5 - 3 = 2
Tadupc s6: 42
Thir lai: 42: (4+2)= 7 (lo^i)
• a X 5 = 40 nen a = 40: 5 = 8 thi b + 3 = 40: 4 =10
hay b = 10 - 3 = 7 Ta dupc s6: 87
Thir lai: 87 = (8 + 7) x 5 +12 (dimg)
Vay s6 thoa mSn bai toan la 87
Cdch 2:
Sir dyng phircmg phdp chdn Iren, chdn du&i, kit hgp
vdi nhdn xel:s6 chia ldn hem sd dir vd dieu kien
a,b<10 de tim gid tri ldn nhdt nhd nhdt cita long (a
+b) Sau do la tien hdnh thd chgn
Gigi
ab = ( a + b ) x 5 + 1 2 Taco: a + b > 12
Matkhaca + b < l 8 ( v i 18 x 5+12 = 102 > a ^ )
V$y:12<a + b < l 8
Xet cac truang hgp sau:
• a + b = l 3 thi 1 3 x 5 + 12 = 77 ( M : vi 7 + 7= 14
khac 13)
• a + b =14 thi 14 X 5 + 12 = 82 (loai: vi 8 + 2 =10
khac 14)
• a + b = l 5 thi 1 5 x 5 + 12 = 87 (chpn:vi 8+7= 15
bang 15)
• a + b =16 thi 16 X 5 + 12 = 92 (loai: vi 9 + 2 = 1 1
khac 16)
• a + b=17thi 1 7 x 5 + 12 = 97 (loai: vi 9 + 7= 16
khac 17)
Vjiy so thoa man bai toan la 87
Cdch 3: Vgn dyng ddu hi^u chia hit cho 5
Phdn ddu nhir cdch 2
Xet biiu thirc (4) 5 x a = 4 x (b + 3)
Ta c6 ab chia cho 5 dupc thuang la (a+b) va dir
12 Do d6 ab t|n ciing bang 2 ho$c 7 hay b = 2
> b = 7 thi a = 40:5 = 8 Ta dugc so: 87 (chpn)
> b= 2 thi a - 20: 4 = 2 Ta dupc so: 42 (loai) Vay so thoa man bai toan la: 87
Cdch4: 'ab - ( a + b ) x 5 + I 2
Ta CO ab chia cho (a+b) duac thuang la 5 va du
12 Vay ah tan ciing bang 2 hoac 7 Mat khac: a +
b > 12 (vi so chia lon hon so du) Neu b = 2 thi a
> 10 (d6 a + b > 12) Vay b = 2(loai) Chpn b = 7 thi
al la so le ma 12 la so chan nen (a + 7) x 5 la s6 le;
do vay a phai la so chan va a > 4 (de a + 7 > 12) Xet cac truang hpp:
> a = 6 va b = 7 thi ta duoc so: 67 (loai vi: 67: 13
= 5(du2))
> a = 8 va b = 7 thi ta dupc so: 87 (chgn) Vay so can tim la: 87
Bdi 8: Tim sd cd hai chir so, biet rang so do gdp 21 lan hieu cdc cda so cua nd (chu so hdng dffn vi be hon chir so hdng chuc)
Cdch} Gidi
Goi so cdn lim Id ab (a khac 0, a > b), Theo
bai toan taco: ab = ( a - b ) x 2 1
Ta co: so chia khac 0, do vay a khac b M^t
k h a c : a - b < 5 ( v i 9 8 : 2 I < 5)
Xet cac trudng hgp sau:
> a - b = l thi: I x2l =21 (chpn)
> a - b = 2 thi: 2 x 2 1 = 42 (chgn)
> a - b = 3 thi: 3 x 2 1 = 63 (chgn)
> a - b = 4 thi: 4 x 2 1 = 84 (chpn)
Vay cac so thoa man de bai la: 21; 42; 63; 84
Cdch 2: Goi so can tlm la ab (a khac 0, a > b)
Theo bai toan ta co: ab = (a -b) x 21
a x l 0 + b = 2 l x a - 2 1 x b (nhan mpt so voi mpt hi?u)
21 x a = a x IO + b + 2] x b (tim so bj trir)
21 x a = a x 10 + 22 x b
11 x a + 1 0 x a = 1 0 x a + 2 2 x b (nhan mpt so vai mgt tong)
11 X a = 22 X b (ciing trir so giong nhau)
a = b X 2 (cimg chia cho 11) Taco bang sau:
b
a
ab
1
2
21
2
4
42
3
6
63
4
8
84
Trang 324,
Vay CO 4 s6 thoa man bai toan: 21; 42; 63 va 84
3 M^t so bai toan v^ tdng cua s^ tu- nhien va cac
chir so cua no
Bdi 9: Tim mgt so, biit rdng s6 do co hai chir so
va s6 do cong v&i tdng cdc chir si cda no thl dirffc
kit qud bdng 84
Cdch 1: Gidi
Gpi s6 cSn tim la ab (a khac 0, a,b <10) Theo bai
ra ta co: ab + a + b = 84
a x l 0 + b + a + b = 84 (c§u tao so)
11 x a + 2 x b = 84 (nhan mgt so vai mgt tong)
aa + 2 X b = 84 (cau tao s6) (5)
2 X b = 84 - aa (tim s6 hang chua biet)
Ta co: a < 8 va b X 2 la so chan nen a phai la so
chin Vay a = 2; 4; 6 Xet cac trirang hpp:
> a=2thi2xb = 84-22 = 62hayb=22:2=lI(loai)
> a = 4thi2xb = 84-44 = 40hayb=40:4-10(loai)
> a = 6 thi 2 x b = 8 4 - 6 6 = 18 hay b = 18:2 = 9
(chgn), Ta dupc so: 69
Thif lai: 84= 69 + 6 + 9 (diing)
Vay so c^n tim la: 69
Cdch 2:
Phan dau nhu each 1
Xet (5): aa + b x 2 - 8 4
Ta thay a < 8 va a chan (vi b x 2 va 84 la so chan)
Neu lay gia tri lon nhat ciia a + b = 9 + 8 = l 7 ( v i a
va b khac nhau) thi ab = 84 - 17 = 67
Do do: ab > 66 (vi 66 + 17 < 84) Vay 66 < ab <
84, ma a < 8 va a chan nen chpn a = 6 do do;
b = (84 - 66): 2 = 9 Ta dupc so: 69 (chpn) Vay so
can tim la 69
Bai 10: Tim so cd bon chir si, biit rdng tong cdc
chit so cda no thi bdng hi^u 1990 vd so phdi tim
Cdch I: Giai
Ggi so can tim la abed (a khac 0)
Theo b^i toAn ta co: 1900 - abed = a + b + c + d
hay abed + a + b+ c+ d = 1990 (tim so bj trir)
Ta c6: d + a + b + c + d < 9 X 5 = 45
V§y neu ph^p cpng a hang dan vj c6 nho sang
hing chgc nhieu nhkt \k 4 va do v§y sS khong c6
nhil sang hdng trSm hay:
ab = 1 9 T a c o : \9cd + 1 + 9 + c + d = 1990
hay 1910 + cd +c+ d = 1990 (c5u tao s6)
1910 + c x l O + c + d + d =1990 hay 1 9 1 0 + c c + 2 x d ^ 1990
ec + 2 X d ^ 80 Ta co: c < 8 va c la so chan
N6u ISy gia tri Ion nhat ciia d=9 thi cc = 80 - 1 8 = 62 Vay 5 < c < 8 do do c = 6
Voi c = 6 thi d = (80 - 66): 2= 7
Ta dirge so: 1967
Thir lai: 1967 +1+9+6+7=1990 (diing)
Vay so can tim la 1967
Cdch 2
Theo bai toan ta co: 1900 - abed = a + b + c + d
abed + a + b+ c+ d - 1990 (tim s6 bi trir)
Ta CO the viet:
Xet phep cgng ben, ta c6 ab = 19
Dodo: 1900+ C £ / + l + 9 + c + d = 1990 Lap luan tuang tu tim each 1 tim ra c, d
Qua mpt so vi du, ta nhan thay cac bai toan a dang
nay kha phong phu va da dang Moi bai toan, tiiy theo yeu cau, se co each nhieu giai phii hpp; ta kho long dua ra mgt cong thifc chung hay dang mau Cac ki§n thiirc dupc six dung a day hoan toan phii hpp vai trinh do hpc sinh tiSu hpc Tuy nhien de giai tot cac bai toan a dang nay doi hoi hpc sinh khong nhirng nam vifng kien thirc ma con doi hoi phai van dung chiing mpt each linh hoat tiiy theo yeu cau bai toan Thuc hanh giai nhieu bai toan dang nay se giiip hpc sinh phat trien kha nang l$p
lu^n, thao tac phan tich, linh hoat trong tu duy &h
lira chpn phucmg an hpp 11
Tai lieu tham khao:
1 Phucmg phdp dgy hoc Todn - Tap 2: Thuc hdnh
giai todn D6 Trung Hieu, Nguyen Hirng Quang,
Ki6u Dire Thanh NXBGD 2000
2 10 chuyen de bdi du&ng hpc sinh gidi Todn 4- 5
(t^p 1) Tran Dien Hien NXBGD 2008
3 123 bai todn sd va chir sd D6 Tmng Hi$u