BOI DirdNG Id II

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NGHIEN c a u & ONG DUNG

II

BOI DirdNG Id NANG THVTC HANH

XAY DUNG BAI TOAN M0I CHO SINH VIEN H i TIEU HOC

TS. Nguyin Manh Cliung - ThS. Ld Ngoc Hod TrUdng Dai hgc Hong Dtic

7 , trfldng pho thdng, day I I toan la day boat ddng V ^ toan hpc cho hpc sinh (HS) va hinh thflc chu yeu cua boat ddng toan hpe la boat ddng giai toan. Vi the cd the ndi rang td chflc tdt viec day giai bai tap toan (boat ddng giai toan) cd tac dung quyet dinh ddi vdi chat Iflpng day hpc (DH) toan va de td chflc tdt viec day giai bai tap toan, ngfldi giao vien (GV) can phai biet Ifla chpn he thdng bai tap thich hpp, cung nhfl can phai biet xay dflng cac bai toan mdi phfl hpp vdi chflc nang cua bai tap toan v^ muc dich cua bai hpc mdi cd the nang cao hieu qua DH toan d trfldng Tieu hpc (TH). De nang cao chat Iflong dao tao d cac trfldng sfl pham, viec bdi dfldng ki nang thflc hanh xay dflng bai toan mdi la mdt viec lam can thiet va can phai dat ra thfldng xuyen d moi co sd dao tao giao vien tieu hpc (GVTH).

Bai viet nay, gidi thieu mpt sd van de ve bdi dfldng ki nang thflc hanh xay dflng bai toan mdi cho SV nganh giao d u c T H .

1. Dinh hfldng xay dflng bai toan mdi d TH

Theo quan diem "DH trong boat ddng va bang hoat ddng" va dfla vao ndi dung DH, cac bai toan d TH can

dflpc xay dflng dfla tren cac dinh hfldng sau:

- Thdng qua DH giai bai tap toan, tang cfldng ren luyen cho HS nhflng boat ddng toan hpc phflc bpp. Dd la nhflng boat ddng nhfl chflng minh, tim tdi, dfl doan, giai bai toan cd ldi van (toan don, toan hpp)...

- Thdng qua DH giai bai tap toan, tang cfldng ren luyen eho HS nhflng boat ddng tri tue phd bien trong toan hpc.

Dd la nhflng boat ddng nhfl lat ngflpc van de, xet tinh giai dflpc (cd nghiem, nghiem duy nhat, nhieu nghiem), phan chia trfldng hpp, boat ddng tfl duy bam, md hinh hoa va the hien.

- Thdng qua DH giai bai tap toan, tang cfldng ren luyen cho HS nhflng boat dpng tri tue chung. Dd la cac boat ddng nhfl phan ticb, tdng bpp, so sanh, tflong tfl boa, trflu tflpng hoa, khai quat boa, dac biet boa, cu the boa...

- Thdng qua DH giai bai tap toan, tang cfldng ren luyen cho HS nhflng hoat ddng ngdn ngfl. Dd la nhflng boat ddng dflpc tien hanh khi HS dflpc yeu cau phat bieu de toan, menh de toan hpc, giai thich hoac bien ddi mdt menh de toan hpc, phat bieu bai toan...

2. Mot so ki thuat xay dflng bai toan mdi d TH

2.1. Phdn tich - tong hdp Nang Iflc phan tich - tdng hpp dflpc the hien trong nhflng tinh hudng sau:

a. Khi sfl dung cac phep chflng minh: suy xudi, suy ngflpc tien, suy ngflpc lui.Thdng thfldng, ngfldi ta dung phflong phap suy ngflpc lfli (ket hpp vdi suy ngflpc tien) de tim phflpng phap chflng minh va dung phflpng phap suy xudi de trinh bay phep chflng minh.

b. Khi phaii chia trfldng hpp.

c. Khi lat ngflpc van de.

d. Khi tim tdi va trinh bay ldi giai cfla bai tap toan (dung phan tich de tim tdi ldi giai cfla bai tap toan va dung tdng hpp de tinh bay ldi giai cua bai tap toan).

2.2. TUdng tii hod: Tflpng tfl hda dflpc xet trong cac tinb hudng sau:

Tflong tfl ve each giai cua bai toan; Tflpng tfl ve ndi dung cua bai toan, nhflng cd nhflng each giai khae nhau.

2.3. Khdi qudt hod: Khai quat boa dflpc xet trong cae tinh hudng: - Khai quat tfl mdt sd trfldng hpp; Khai quat tfl mdt trfldng hpp.

2.4. Dqc biet hod: Dac biet hoa dflpc xet trong cac tinh hudng sau:

TAP CHi THifTBI GIAO DMC-So 71,7/201! • I I

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II NGHIEN CCru &

O N G D U N G Dac biet hoa de dfla ra mpt tinb chat manh hOn; Dac biet hoa de dfl doan tim ra each giai quyet van de (each giai).

2.4. Cu the hoa: Cu the boa dflpc xet trong cac tinh hudng cu the gan gui vdi HS.

3. Mot sd phflOng phap xay dflng bai toan mdi d TH

3.1. Xdy dUng bdi todn mdi tUdng tu vdi bdi todn dd gidi

Vi du: Yeu eau SV thflc banh giai bai toan sau:

a. Giai bai toan sau: Cho tam giac ABC. M va N la hai diem tren canh BC va CA sao cho MB = 2 X MC; NC - 3 x NA. Hai doan thang AM va BN cat nhau d O; CO keo dai cat

PA AB d P. Tinh ti sd .

PB

b. Cd nhan xet gi ve tich:

MB NC PA ^ . . , tfl do neu eia MC NA PB

thuyet va chflng minh gia thuyet.

c. Dat ba de toan mdi tflOng tfl bai toan tren cho HS TH.

Ket qua la SV cd the giai bai toan tren nhfl sau:

PA 1

thang AM va BN cat nhau d O;

CO keo dai cat AB d P. Tinh ti . PA

a. Tinh dfloc PB b. Nhan xet:

MB NC PA

= 1 suy ra gia MC NA PB

thuyet.

c. Dat dfloc 3 de toan nhfl sau:

Bai toan 1: Cho tam giac ABC. M va N la hai diem tren canh BC va CA sao cho MB = 4 X MC; NC = 5 X NA. Hai doan

so . PB

Bai toan 2: Cho tam giac ABC. M va P la hai diem tren canh BC va AB sao cho MB = 4 X MC; PA = 2 X PB. Hai doan thing AM va CP cat nhau d O;

BO keo dai cat AC d N. Tinh ti so . .NC

NA

Bai toan 3: Cho tam giac ABC. M la trung diem cua BC va N la trung diem CA. Hai doan thing AM va BN cat nhau d O; CO keo dai cat AB d P.

Chflng minh ring: PA = PB.

3.2. Xdy diing bdi todn mdi ngUdc vdi bdi todn dd gidi

Vi du: Yeu cau SV thflc hanh giai bai toan sau:

a. Cho tam giac ABC, tren cac canh AB, AC va BC lan Iflpt lay cac diem E, D va M sao cho sao cho AB = 3 X AE; AC = 3 x AD; BM - MC. Chflng minh rang ba dfldng thang AM, BD, CE ddng qui (cat nhau tai mdt diem).

b. Hay phat bieu bai toan ngflpc (bai toan dao) cua bai toan tren va giai bai toan dd.

Ket qua la SV giai dflpc bai toan a va cd the dfla ra dflpc bai toan dao cua bai toan a nhfl sau:

Bai Toan: Cho tam giac ABC, gpi M la trung diem cua BC, O la trung diem cua AM.

BO keo dai cat AC tai D, CO keo dai cat AB tai E. Chflng minh rang: AB = 3 x AE; AC = 3 x A D .

3.3. Xdy dting bdi todn mdi bdng cdch ghep noi cdc bdi todn ddn, cdc bdi todn dim htnh lai vdi nhau

Vi du 1: Yeu cau SV thflc hanh giai bai toan sau:

Anh (chi) hay phat bieu hai bai toan, trong dd cd mpt bai toan ve tim hai sd khi biet tdng va hieu, mdt bai ve bai toan ti le thuan, sau dd hay ghep lai de dflpc mdt bai toan mdi.

Ket qua la SV cd the dfla ra dflpc 2 de toan nhfl sau:

Bai Toan 1: Tim hai sd khi biet tdng va hieu: "Ca hai bao gao dflng dflpc 96 kg. Bao ldn dflng nhieu bon bao nhd 10 kg.

Hdi mdi bao dflng dflpc bao nhieu kg gao?"

Bai toan 2: Bai toan ti le thuan:" Cd hai bao dflng gao.

Bao ldn dflng 53 kg, bao nho dflng 43 kg. Sd gao trong bao ldn ban dflpc 159000 ddng. Hdi sd gao trong bao nhd ban dflpc bao nhieu tien?"

Bai toan mdi: Ghep bai toan 1 va 2: "Ca hai bao gao dflng dflpc 96 kg gao. Bao ldn dflng nhieu hon bao nhd 10 kg. Sd gao trong bao ldn ban dflpc 159000 ddng. Hdi sd gao trong bao nho ban dflpc bao nhieu tien?"

Vi du 2:. Yeu cau SV thflc hanh giai bai toan sau:

Anh (chi) hay phat bieu hai bai toan, trong dd co mdt bai toan ve tim hai sd khi biet tong va ti sd, mdt bai ve tim hai so khi biet hieu va ti sd, sau dd hay ghep lai de dfloc mpt bai toan mdi.

Bai Toan 1: Tim hai sd khi biet ti sd va tdng: "Tudi bd gap 5 lan tudi con. Tdng sd tudi cua hai bd eon la 42. Tinh tudi bd va tudi con"

Bai toan 2: Tim hai sd khi biet hieu va ti sd :"Tudi bd gap 3 lan tudi con. Hieu sd gifla tudi

18 • TAP CHi THIFT BI GIAO DUC-So 71,

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N G H I E N

cCru & CTNG

D U N G

II

bd va con la 28. Tinh tudi bd va tudi con."

Bai toan mdi: Ghep bai toan 1 va 2: "Hien nay, tudi bd gap 5 lan tuoi con, tdng sd tudi cua bai cha con la 42. Hdi sau bao nhieu nam nfla thi tuoi bd gap 3 lan tudi con?

3.4. Xdy diing bdi todn mdi td mgt ddy tinh ggp

Cho day tinh sau: 28 - (28 - 24) X 5.

Anh (chi) hay dat 2 de toan, trong dd cd mdt bai ve giao due dan sd, mdt bai ve giao due mdi trfldng, cd ldi giai nhfl day tinh tren.

Bai toan 1: Khu phd A cd 28 gia dinh. Trong phong trao van ddng gia dinh khdng sinh con thfl ba, 24 gia dinh da thflc hien tdt, cac gia dinh khdng thflc hien tdt tinh nguyen gdp mdi gia dinh 5 trieu de xay dflng nha van hda. Theo tinh toan nha van hda can phai chi phi 28 trieu. Hdi khu phd A cdn phai lap ke hoach xin nha nfldc bao nhieu tien nfla?

Bai toan 2: Dpi thanh nien tinh nguyen phd B nhan trdng cay xanh tren dfldng pho. Dpi cd 28 ngfldi, theo tinh toan moi ngfldi trdng 1 cay la se du.

Vi cd ke hoach dot xuat, 24 ngfldi cua dpi da dflOc cfl di lam viec khac, sd ngfldi cdn lai mdi ngfldi da trdng 5 cay. Hdi cdn thieu bao nhieu cay chfla dflpc trdng?

3.5. Xdy dting bdi todn mdi bdngcdch khdiqudthod

Tinh tong:

1 1 1

+ -I- -I-

-I--

1x3 3x5 5x7 1 --I--

7 x 9 9x11

sau dd neu nhan xet khai quat rdi phat bieu bai toan tdng quat cua bai toan tren va giai bai toan dd.

Ket qua la sau khi tinh dflpc tdng tren, neu nhan xet khai quat, SV cd the dfla ra dflpe 2 de toan nhfl sau va giai dflpc cac bai toan nay:

Bai toan 1: Tinh tdng

c 1 1 1 1

1x3 3x5 5x7 7x9

--H...-h- I

9x11 '•• (2xrt-Hl)x(2xrt + 3) vdi n la sd tfl nhien.

Bai toan 2: Tinh tdng:

1 1

l x ( l + *) {\ + lc)x{\-^-k-\-k) 1

{\-i-k + k)x{\-\-k-\-k-\-k) 1

--h

-f... + -

( l + rtX^)x(l + /7X^-|-A:) vdi n la sd tfl nhien.

Ngoai cac phflOng phap neu tren, chflng ta cd the bdi dfldng cho SV ki nang thflc hanh xay dflng bai toaii mdi bang nhflng phflong phap khac nhfl:

- Xay dflng bai toan mdi dfla tren each giai bang day tinh cua 1 bai toan cu

- Tdm tat bai toan bang bang ke d, so dd, bieu do rdi dfla vao sd lieu cho trong bang, so dd, bieu dd dd de dat de toan mdi:

+ Dung so dd doan thang de tdm tat de toan rdi yeu cau phat bieu de toan tren sd lieu da cho tren so dd;

+ Dung bieu do Ven de tdm tat de toan rdi yeu cau phat bieu de toan tren sd lieu da cho bdi bieu dd Ven dd;

-I- Lap bang sd lieu de tdm tat de toan rdi yeu cau phat bieu de toan tren sd lieu da cho trong ban sd lieu do...

- Xay dflng bai toan mdi chfla nhflng ndi dung thflc te da dinh trfldc...

(Do khudn khd bai viet chflng tdi khdng the dfla ra cac vi du minh hpa cho cac phflpng phap nay).

De dat dflpc muc tieu giao due toan hpc cho HSTH thi xay dflng bai toan mdi la viec lam can thiet cfla ngfldi GVTH. Vi the cd the ndi rang td chflc tdt viec bdi dfldng ki nang thflc hanh xay dflng bai toan mdi cho SV nganh giao due TH ngay trong qua trinh dao tao la gdp phan nang cao chat Iflong dao tao d cac trfldng sfl pham theo dinh hfldng bdi dfldng ki nang nghe cho SV d mdi co sd dao tao GVTH.

Tai lieu tham khao

1. Dd Trung Hieu, Do Dinh Hoan, Vu Dflong Thuy, Vu Qudc Chung, PhUdng phdp DH mon Todn d TH, Nxb Dai hpc sfl pham, Ha Ndi, 2009.

2. Nguyen Thanh Hflng, PhUdng phdp DH mon Todn d TH, Nxb Giao due. Ha Ndi, 2009.

3. Nguyen Ba Kim (chu bien), Vu Dflong Thuy, PhUdng phdp DH mdn todn, Nxb Giap due, HaNdi, 1997.

4. Sach bdi dfldng Sinh vien va Giao vien ve ddi mdi Chflong trinh va SGK Toan TH.

TAP CHi THIFT BI GIAO DMC-So 71,7/2011 • 19