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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[.]
22 sa 11-2012 D j P f e j f t y Phat trien tif 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 Quang Vinh - Quang Di6n - tinh Thira Thien Uui Qt nhirng muc tieu quan ciia chiroTig trinh toan a Tieu hoc la phat trien kha nang tu 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 viec phat trien cac thao tac tir cua hoc sinh Day la mot nhirng dang toan kho va co ngi dung phong phu vi vay chting thirong xuat hien cac ki thi chpn hgc sinh gioi hay thi tuyen vao cac truong chuyen Cdch 1: 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: Thu lai: 72 = x (7 +2) (dung) V^y so can tim la 72 Bai 7: Tim mot so co hai chii' so, biet rang chia so cho tong cac chir so cua no thi duqtc thuong la va du* 12 Cdch I: Gidi M Gidi Goi so cdn lim Id ab (khac 0) Theo bai ta co: ^ = X (a + b) (3) a x l O + b = x a + x b (cau t&o s6, nhan mgt so voi mpt tong) a x l O - a x = x b - b (ciing trii x a va b) a X (10 - 8) = b X (8 - 1) (nhan mot s6 vai mpt hi?u) x a - X b * a X dat gia tri Ion nh§t la 18 a = 9, do: x Dang toan thucmg sir dung vcri cac doi b ciing d^t gia tri Ion nh5t la 18 Vay b < tugng la hgc sinh gioi cac lop cuoi cap d Tieu hoc Mat khac a x la s6 chin do x b cung la so Viec dung phirong phap dung chir thay so chan nen b la so chan Vay b = viec giai cac bai toan a tieu hgc hien van Suyra: a = : - nhieu y kien khac Nhieu giao vien cho rang: six Thir lai: 72 = x (7 +2) (dung) dung phuong phap diing chif thay so la qua tai voi hgc sinh vi thirc chSt cua phucmg phap la viec Vay so phai tim la 72 giai cac phuong trinh se dirge hpc chucmg Cdch 2: SOr dung phuong phap diing so doan thang trinh a cac lop tren Viec trang bj cho hpc sinh mgt Quan sat (3) ta co sa sau: cong cu giai toan qua manh nhir vay se lam so ciing kha nang tu cOa hgc sinh Ben canh do, mgt so (a+b): giao vien giai thich rang: viec van dung phuong phap van phu hpp voi trinh dp nhan thirc ciia Ta CO A = ab - (a+b) la mpt s6 chia h6t cho hay hpc sinh Tieu hgc neu biet van dung linh hoat cac A = a x 10 + b - a - b = x a mpt so chia het cho 7; ma khong chia het cho nen a chia het cho Vai kien thirc ma hgc sinh dupc hpc chuang trinh Trong bai viet nay, chiing toi xin gioi thieu mgt a = thi A - x - 63 Do gia trj ctia mgt phan so bai toan phu hpp voi doi tugng hpc sinh gioi cac hay (a + b) la: 63: = 9.Suy ra: b = - = Ta lop cuoi cap bang each van dung linh hoat phirong dupc so 72 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 gap Idn tong cdc chir so ciia no Goi so cdn lim Id ab (khac 0) Theo bal ta co: DaiMHpc s6^i.i ab =(a + b ) x + ab- = 5xa+5xh+ 12 (nhan mpt s6 vai mgt tong) ax I0 + b = x a + x b + 12 x a + x a + b = x a + x b + b +12 x a = x b + l (cung trir cac so hang giong nhau) x a = x ( b + 3) (nhan mgt s6 voi mot t6ng) (4) - Ta co: a x chia h6t cho va nen a x chia het cho 20 Vay a X = 20 ho^c a x = 40 (vi a b = thi a = 40:5 = Ta dugc so: 87 (chpn) > b= thi a - 20: = Ta dupc so: 42 (loai) Vay so thoa man bai toan la: 87 Cdch4: 'ab - ( a + b ) x + I Ta CO ab chia cho (a+b) duac thuang la va du 12 Vay ah tan ciing bang hoac Mat khac: a + b > 12 (vi so chia lon hon so du) Neu b = thi a > 10 (d6 a + b > 12) Vay b = 2(loai) Chpn b = thi al la so le ma 12 la so chan nen (a + 7) x la s6 le; vay a phai la so chan va a > (de a + > 12) Xet cac truang hpp: > a = va b = thi ta duoc so: 67 (loai vi: 67: 13 = 5(du2)) • a X = 40 nen a = 40: = thi b + = 40: =10 > a = va b = thi ta dupc so: 87 (chgn) hay b = 10 - = Ta dupc s6: 87 Vay so can tim la: 87 Thir lai: 87 = (8 + 7) x +12 (dimg) Bdi 8: Tim sd cd hai chir so, biet rang so gdp 21 Vay s6 thoa mSn bai toan la 87 lan hieu cdc cda so cua nd (chu so hdng dffn vi be Cdch 2: hon chir so hdng chuc) Gidi Sir dyng phircmg phdp chdn Iren, chdn du&i, kit hgp Cdch} vdi nhdn xel:s6 chia ldn hem sd dir vd dieu kien Goi so cdn lim Id ab (a khac 0, a > b), Theo a,b 12 Xet cac trudng hgp sau: Matkhaca + b < l ( v i 18 x 5+12 = 102 > a ^ ) > a - b = l thi: I x2l =21 (chpn) > a - b = thi: 2x21 = 42 (chgn) V$y:12 a - b = thi: 3x21 = 63 (chgn) Xet cac truang hgp sau: • a + b = l thi x + 12 = 77 ( M : vi + 7= 14 > a - b = thi: x = 84 (chpn) khac 13) Vay cac so thoa man de bai la: 21; 42; 63; 84 • a + b =14 thi 14 X + 12 = 82 (loai: vi + =10 Cdch 2: Goi so can tlm la ab (a khac 0, a > b) khac 14) Theo bai toan ta co: ab = (a -b) x 21 • a + b = l thi x + 12 = 87 (chpn:vi 8+7= 15 a x l + b = l x a - x b (nhan mpt so voi mpt hi?u) bang 15) 21 x a = ax IO + b + 2] x b (tim so bj trir) • a + b =16 thi 16 X + 12 = 92 (loai: vi 9+2=11 21 x a = ax 10 + 22 x b khac 16) 11 x a + x a = x a + 2 x b (nhan mpt so • a + b=17thi x + 12 = 97 (loai: vi + 7= 16 vai mgt tong) khac 17) 11 X a = 22 X b (ciing trir so giong nhau) Vjiy so thoa man bai toan la 87 a = b X (cimg chia cho 11) Cdch 3: Vgn dyng ddu hi^u chia hit cho Taco bang sau: Phdn ddu nhir cdch b Xet biiu thirc (4) x a = x (b + 3) Ta c6 ab chia cho dupc thuang la (a+b) va dir 12 Do d6 ab t|n ciing bang ho$c hay b = ho$c b = Xet cac truimg hpp: a ab 21 42 63 84 Chpn Chpn Chpn Chpn s6u-20iz D^lfydHpC 24, Vay CO s6 thoa man bai toan: 21; 42; 63 va 84 M^t so bai toan v^ tdng cua s^ tu- nhien va cac chir so cua no Bdi 9: Tim mgt so, biit rdng s6 co hai chir so va s6 cong v&i tdng cdc chir si cda no thl dirffc kit qud bdng 84 Cdch 1: Gidi Gpi s6 cSn tim la ab (a khac 0, a,b a=2thi2xb = 84-22 = 62hayb=22:2=lI(loai) > a = 4thi2xb = 84-44 = 40hayb=40:4-10(loai) > a = thi x b = - 6 = 18 hay b = 18:2 = (chgn), Ta dupc so: 69 Thif lai: 84= 69 + + (diing) Vay so c^n tim la: 69 Cdch 2: Phan dau nhu each Theo bai toan ta co: 1900 - abed = a + b + c + d abed + a + b+ c+ d - 1990 (tim s6 bi trir) Ta CO the viet: Xet phep cgng ben, ta c6 ab = 19 Dodo: 1900+ C £ / + l + + c + d = 1990 Lap luan tuang tu tim each tim c, d Qua mpt so vi du, ta nhan thay cac bai toan a dang kha phong phu va da dang Moi bai toan, tiiy Xet (5): aa + b x - theo yeu cau, se co each nhieu giai phii hpp; ta kho Ta thay a < va a chan (vi b x va 84 la so chan) long dua mgt cong thifc chung hay dang mau Neu lay gia tri lon nhat ciia a + b = + = l ( v i a Cac ki§n thiirc dupc six dung a day hoan toan phii hpp vai trinh hpc sinh tiSu hpc Tuy nhien de va b khac nhau) thi ab = 84 - 17 = 67 giai tot cac bai toan a dang doi hoi hpc sinh Do do: ab > 66 (vi 66 + 17 < 84) Vay 66 < ab < khong nhirng nam vifng kien thirc ma doi hoi 84, ma a < va a chan nen chpn a = do; phai van dung chiing mpt each linh hoat tiiy theo b = (84 - 66): = Ta dupc so: 69 (chpn) Vay so yeu cau bai toan Thuc hanh giai nhieu bai toan can tim la 69 dang se giiip hpc sinh phat trien kha nang l$p Bai 10: Tim so cd bon chir si, biit rdng tong cdc lu^n, thao tac phan tich, linh hoat tu &h chit so cda no thi bdng hi^u 1990 vd so phdi tim lira chpn phucmg an hpp 11 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 < X = 45 V§y neu ph^p cpng a hang dan vj c6 nho sang hing chgc nhieu nhkt \k va v§y sS khong c6 nhil sang hdng trSm hay: Tai lieu tham khao: Phucmg phdp dgy hoc Todn - Tap 2: Thuc hdnh giai todn D6 Trung Hieu, Nguyen Hirng Quang, Ki6u Dire Thanh NXBGD 2000 10 chuyen de bdi du&ng hpc sinh gidi Todn 4- (t^p 1) Tran Dien Hien NXBGD 2008 123 bai todn sd va chir sd D6 Tmng Hi$u Nguyin Ang NXBGD.2010 ... toan la 87 a = b X (cimg chia cho 11) Cdch 3: Vgn dyng ddu hi^u chia hit cho Taco bang sau: Phdn ddu nhir cdch b Xet biiu thirc (4) x a = x (b + 3) Ta c6 ab chia cho dupc thuang la (a+b) va dir... each tim c, d Qua mpt so vi du, ta nhan thay cac bai toan a dang kha phong phu va da dang Moi bai toan, tiiy Xet (5): aa + b x - theo yeu cau, se co each nhieu giai phii hpp; ta kho Ta thay a < va... hang giong nhau) x a = x ( b + 3) (nhan mgt s6 voi mot t6ng) (4) - Ta co: a x chia h6t cho va nen a x chia het cho 20 Vay a X = 20 ho^c a x = 40 (vi a