LUYEN TAP CHO HOG TRONG DAY HDC HINH« HOAT ODNG KET NOI TRI THUE iTRll i lNG TRUNG HOC PHOTHONG ThS PHAN THANH HAI* Abstract In this article, the author presents some points of view of connecting know[.]
Trang 1LUYEN TAP CHO HOG
TRONG DAY HDC HINH«
HOAT ODNG KET NOI TRI THUE iTRllilNG TRUNG HOC PHOTHONG
ThS P H A N THANH HAI*
Abstract: In this article, the author presents some points of view of connecting knowledge andactimes
of connecting knowledge The students must mobilize knowledge to solve problem, discover knowledge u\ learning geometry These acliuilies should be applied in teaching mathematics at high school Keywords: knowledge connection, geometry, students
Ngdy nhgn bdi: 22/03/2016; ngay sija chua- 25/03/2016; ngdy duy$t ddng: 25/03/2016
Phathien kien thire mdi la thudc ITnh vue tim tdi tri
tueduoc mptso nha khoa tipc quan tam nhu:
M.Cnjgliac;V.A.Cnjchetxki; Nguyen Ba Kim;
Dao Tam; Bui Van Nghj; Cao Thi Ha; Trong hoat
dpng tim toi kien ^ d c mdi hpc sinh (HS) gap nhung
ktid khan chu yeu sau day: - Khd khan trong viec phat
hien cac mau thuan npi tai khi gap cac tinh hudng cd
van dede tudd tim each giai quyet cac mau thuan
ni\'m kham pha tri thuc mdi; - Khd khan trong viec
huy dpng diing kien thue de lam sang to nhiem vy
nhan thirc, lam bpc 16 cac ddi tupng can kham pha;
- Kho khan giai dap eau hoi: Dua tren co sd nao de
huy dpng diing kien thdc da biet nh&m lam sang to
van de.lap luan cdcan eirdechiem ITnh trithircmdi?;
- Kho khan bien doi thdng tin trong cae tinh hudng m(^
dechuyenvedgngquen thupe, tudocothehuy ddng
cac ki^'n thuc da biet nham xdli thong tin can tim,phat
hien kien thdc mdi,
Cac hoat dpng khSc phye nhOnp kho khan neu
tren thupc pham tru hoat dpng ket ndi tri thUese duoc
lam sang to qua noidung baivietnay, nh§m giiJp HS
phat trien kien thuc mdi tron^ viec tim toi kien thdc toan
hpc eung nhutrong thue tien
1 Mpt so quan diem ve ket nd'i tri thuc
l.l.Tdgoc do triei hgc cho thay:y\ecketn6\]j'\
thup dacd vdi tii thuemdi thdng qua cae ptiuong thuc
chuyeu sau: -Tochuc eho HS khao sat eae trudng
hpp rieng, cac mdi lien he dac biet de tddd nhd hoat
dong khai quat hoa chiing ta cd nhung tn thuc mdi
tong quat hPn; - Phat hien mdi lien he nhan qua gida
tri thdc da CO cua HS vdi tri tilde mdi can tim de dinh
hudng each giaiquyetvan de Khidung tn/de mptvan
de can^giai quyet, chiing ta sudyng mdi hen he nhan
qua de hudng HS tim tdi cae tn thuc cpi nguon, iam
tien de cho viec chung minh, lam sang to van de can
giaiquyet,
481 Tap chi Gido dye so 381
1.2 TIJ goc do tam Tihgc lien tuang cho ^^^
CalmdidUde phathien thdng qua hoatdoig lientifSfig
va ehuyen hda cac lien tudng tu" ddi tupng nay sang ddi tuong khac Thieu kha nang lien tudng, HS se gap khd khan trong hoatdpng huy dpng kien thiJcde giai quyet van de, khd khan chuyen hoa cac van de khd sang nhdng van de quen tiiupe, kho khan quy la vequen
Nha su pham G Polya dac biet coitiipng hoatdgng lien tudng n h i m huy dpng tdi da cac kien thiJc da hpc lien quan den gia thietva ket luan eua bai toan, chgn Ipc cac nhdm tri thde can thiet eho viec thyc hien cic hoatdpng giai bai toan mdi Chii trpng lien tuong bai toan can giai vdi cac bai toan gde quen tiiupc maHS
da biet, tim each bien ddi bai toan ve bai toan goc, TiJ ddhuy dpng kien thdc nh^m ket ndi gia thiet vaketpi cua bai toan, nh&m giup HS d i d a n g huy dong kill thirc de giaiquyetvan dedat ra trong batoan.G,P^a cung coitrpng ket hpp gida dac biet hoa vakb&pl hda, chu trpng khuyen HS xem xet cae trucmg hdp rieng de tddd c d c p s d khai quathoa tdi bai toan tong quat han, tao eo hpi dedang ttiiet lap mdi Pen hegiOa kien thue can tim vdi cae kien thtJc da cd
M.Cnjgliac da nhan manh: Nhung tri thdc da finh hpi dupc lai tham gia vao qua trinh tu duy nhu lam^ yeu tdcua tu duy de tiep thu nhung tri thdc men khac [1;tr 64-65]
Nhu vay, tu duy di tCrhe thdng tri thiic da bietjJai cactrithucmdican tim.Noi each khac.tu duy dak^no
he thdng tri thdc da biet den cac tri thuc can biei D^ day ehiing taxetmptvaividy trong day hpc toan
2 Ket no! tri t h u c Tren eosdphantich cac quan diem tri^thoc,larn
li hpc ve hoat dpng ket ndi tri thdc, trong baivietnay
* Tnrong Trung hoc phd thong Tnrdng Chinh - Dah Nong
Trang 2moi can phat hienlrong qua trinh tim toi tri tue la viec
chgn Ipc CO tinh quy luat cac tri thuc da CO va to Chile
Chung voi tucach de dir doan cac van de, van dung
Chung de lap luan lam sang to nhiem vu nhan tliilc
Ihong qua cac tinh huong l<ham phajri tiiiic moi hay
van dung tri thiic toan hpc vao thuc tiln
Nhu vay, ket noi tri thiic doi hoi tim ra droc he
thong cao tri thiic da co lien he phu thupc lan nhau
dupc Ilia chon theo mpt trinh t u n h j m de'djnh hucmg,
dieu chinh qua trinh lap luan phat hien tri thirc mdi
Trong thuc tiin day hpc toan, viec xac djnh moilien he
gJSa tn thiic da co va tri thiic can tim I<h6ng phai lijc
nao cung dedang nhan thay ngay.nhieukhi^xuathien
cac ho ngan each, xuat hien cac mau t h u i n va cac
chucmg ngai doi hoi HS phai bien doi cac tinh huong
tri thiicm6ivedangdldang xac djnh cac moi lien he
tri thiic da co v6i tri thiic can tim
Chiing ta co the nhan thay dieu nay thong qua
viec phan tich vi du sau day;
V i d u 1 : Chiing minh rSng trong tam giac ABC
bat ki, ta luon CO X f + B 5 [ + C C ; = 6, Irong do AA,;
BB,; CC, lan lupt la cac ducmg trung tuyen xuat phat
tii cac dinh A B ^ C
Cach 1: Dang thiic vecto can chiing minh;
^4i + BB^+Cc: = 6 lien quan I6i ba dudng trung
tuyen va trpng tam G cua tam giac gpi cho HS lien
tudng tfllmenhdequen thupc; a 4 + G B + G C = 6 S i i
dung kien thiic
da biet ^
neufl = 0 thi f
-a = 0 Khi
do he thong tri *
thiic dupc ket
noi nhim giai
baitoancothe
mo ta Iheo so do sau; {hinh 1)
GB.Gr = 2G^
ll-io.BO.CGJ.i
GA*GB*CC^O AGi-BC*Ca^O
-:*G+|BC+|CG-O
A^»Bfl,*cc,^a
Cac/72.-ViecChung minh AA, +BB, +CC, = O gdi
cho HS lien tudng tdi AA^, BB^, CC, la^dp dai ba
canh cua mottam giac AA^I\/I, tddd co the dyng tam
gi^c cd dp dai ba canh la AA^, BB^, CC, nhu sau:
SjW song song va
bang BA, (hinh 2)
Khi do ta cd ti) giac e-4,Me, la hinh binh hanh
T u do, nhd tinh chattnjng dian cua doan tiling vatinh chatcua dudng trung binh suy ra eac tu'giac>l,S,WC,/1C,C/W la cac hinh binh hanh va tudo suy ra tam giac edba canh lan
luat b^ng AA,, BB^ CC,
Khidd ttieo quy t&c ba diem ta ed: ^^+¥B, + CC,
= AA,+Aji^ + MA^AA=Q
Nhu vay, viee ket ndi tri ttidc d day dupe diln ra thdng qua cac hoat dpng lien tudng tdi quy tac ba diem, ddi vdi phep toan cpng veetp, tinh chat hinh binh hanh, dudng tnjng binh trong tam giac Nhdcac tinh chatnayvatiidng qua hoat dpng dung hinh, suy
ra dupc fam giac AA,M co dp dai ba canh lan lupt
b&ng dp dai ba dudng trung tuyen, Cae tri thdc duoc ket ndi d tren bao gom tri thde gde: Tong ba vecto djnh tren ba canh cua mpt tam giactiieomptchieu quay nao do ludn b^ngvectp 0
3 Hoat dpng ket ndi tri t h u c Chung toi quan niem hoat dpng ket ndi tin tuc la nhOng hoat dpng cua HS tim kiem eac tri thuc cpi nguon, cac tri ttidc tnjng gian duac ket ndi thanh mpt
he thdng theo cac moi lien he nhan qua, lien he phy thupc nham lam sang td nhiem vu nhan thdc, de ehu thexam nhap vao ddi tupng, xam nhap vao van dede chiem Hnh tri thdc mditrong toan hpc cijng nhutrong thuc tien Tren ca sd phan tich cac quan diem triet hpc, tam llhpcve hoat dpng ket ndi tri thdcvakhaithae cau tnJc cua hoatdpng tim toi tritue, chung tdi dua ra mptsdhoatdpng ttianh phan eua hoatdpng ketnditri thdc trong ITnh vue tim tdi tri tue n h i m phat hien va chiem finh cac kien thuc mdisau day:
3.1 Hoatdgng xac dfnh cac mau thuin trong cac tinh huong tri tht/c mdi nham lam bgc 16 nhiem vu nhan thdc trong qua trinh chi the hoat dgng ttr duy, xam nh$p vao do) tugng nghien CIIU, xam nhap vao tinh huong mdi
Xuat phat td gdc dp tam li hoc nhan thdc; tam li hoc tri tue va td gdc dp tu duy bien chdng cijng nhu phuong phap luan nhan thuc toan hpc, cd the thay ring: hoat dpng nhan thdc ndi ehung, nhan thdc toan hpc noi rieng dupc b i t nguon tuviee phat hien eac mau thuin de tudd tao dpng lue cho hoatdpng giai
Tap chi Gido dye so 381 49
Trang 3hoc toan nay sinh cac nhiem vu nhan thdc, cac doi
tupng cua hoat dong, ddi tuong cua tu duy thuc day
hoatdpng Xac dinh mau thuan trong cac tinh hudng
tri tti ucmfflva viee giaiquyet cae mau thuan do ddi hoi
giaovien dmh hudng eho HS huy dongcactn thucva
kinh nghiem da biet, de tirdo tim each khic phyc eac
mau thuan, phathien cac tnfhUc mdi,
Mptsdmau thuin thudng bieu hien cua HS trong
day hpctoan:-l\/lauthuanhJnguyen nhan HS khong
chu trpng ve sucan ddi gida hai matcii phap va ngu
nghia cua cac ddi tupng quan he toan hoc; - Mau
thuin thong qua khao sat HS tuang tac vdi tinh
hudng tn thuc phuang phap mdi; - Mau thuan giua
tri thuc phuang phap da co eua HS khong tuong
thich vdi phuong phap van dung trong tinh hudng
dupc khaiquat,
K/dt/^.-Cac til thuc phuang phap xac dinh khoang
each giua hai dudng thang cheo nhau a, b duoc neu
qua hai quy trinh chu yeu sau:
Quy trinh /."Bao gom cac budc
- Xae dinh mat phang (P) ehua b va (P) // a;
• Tinh khoang each td mot diem M thupc a
den (P)
Quy trinh ^.'Gom cac bude
- Dyng mat phing (P) ehua a va (P) lib;
-Dung mat phang (O)chLfa dva{0)//a;
- Tinh khoang each giua hai mat phing "(P)
va(0)
Van dyng quy trinh tren de giai bai toan sau:
Cho hinh lap phuong ABCD.A 'B'C'D'canh bing
a Tinh khoang each giua hai dudng thing BD
waAS:
Theo quy trinh 1,
ehung ta cb the xac
dinh mat phing
(AB'D)chuaAB'va
(ABV) // BD Do
trong hinh lap
phuang dudng thing
CA'l{ABD)raiH ,,„,,
-nen khoang each
can tim bing dp dai doan 0J,OJ LAO\^ e A0\
trong do 0' la tam cua mat A'BC'D', khi dd
Q i = - c / / = i r r = : ^
501 Tap chi Gido due so 381
Cho hinh hop chu nhat ABCD.A'B'C'D'ziickYyi\ thudc AB = a; AD = 'b;AA'=c Tinh theo a;6;c
khoang each giua hai dudng thing SDva/lS'.' Mau thuin nay sinh khi HS tiep can vdibaitoan tdng quat nay la: HS khong the ap dgngtri/cliep quy trinh da xet bai toan tren, noi each khac, quy trinh tren khong tuong thich vdi bai toan md'i nay
la: CA 'khong phai la phuang vuong goc v6i (/IS'D),
tronjkhid6(/ie'D)//00 Cdthe khic phuc mau thuan nay nhd dinh hudng eho HS sudung gian tiep the tich cua hinh chop D'/4fie'c6 dien tich day
s = -BABB- = -ac khi do the tich hinh chop D'ABB'\aV = ^abc
6
Tu do khoang each can tim bing do dai di/arg
cao h vetddinh Sden mat phing (AB'D)va(i\er\k\\
tam giac y4S'£>'ia S, va khi dd ^ = y = ^ Bai loan
quy ve tinh dien tich tam giac AB'D' theo
a;b;e
3.2 Hoat dgng bien ddi thong tin vecac do! tugng, cac quy luat can kham pha theo ntiieu hudng khac nhau, nhim gitip chu the huy dong kien thuc theo nhieu each khac nhau
Bien doi thong tin toan hpe la hoatdpng cua chu the lam thay doi hinh thdc dien dat cua thong lin de
CO the huy dpng kien thuc da CO lam bpc !p cac npi dung, cac mdi quan he an chda trong cac thong tin
do de tiep nhan tn thue mdi mot each hieu qua [12;tr43-44]
Vi du 3: Cho ba diem A, B, C cd dinh tren ma! phang (a) va A la diem di dpng trong khong gian,
S z (a) GpiJ fa trung diem cua ;4Cva/lacacdiBm
th6amanhethu'c:3M^-2/B + 7c-oQ.XacW giao tuyen eua mat phang {S>1i)va mat phang (SSJ)
HS da CO vdn kien thuc ve khai niem giao tuyen
cua hai mat phing phan biet la dudng thang&
qua haidiem chung cua haimatphangddhoacia dudng thang di qua mgt diem ehung va co vecld chi phuang a
Haimatphing (S/^/)va(SSJ)phan bietvacoSIa diem chung, ta ean xac dinh diem ehung thu'liai Vol
bai toan nay, ndu giir nguyen giathidt{*) khongt^^
doigithiHSseratkhdkhantrongvleetimdiinchvng
Trang 4dpng bien ddi thdng tin, hudng dan HS tdng bude
bien ddi gia thiet {*) de lam boc Id cac ttiuoe tinh, quan
he an ehua ben trong ddi tupng nhusau:
(') '^2/A^AB+AB-~AC<o2JA=AB+CB
<f>2iA=-{BA+BC)= iBJ => 'AJ = BJ- Vay glao
tuywi can tim la
dudngthanga
di qua S va
song song vdi
BJ{hinh4)
Nhd hoat
ddng bien dot
thong tin giup
cho HS kha
nang chuyen
ddi giua hinh thue va npi dung cua ddi tupng de tim
thay mdi lien he giua tn thuc mdi vdi nhung tn thuc da
CO eua HS,tddd tim ra hudng giaiquyetvan de
3.3 Hoat dgng chuyen hda cac bai tap trong
sach giao khoa thanh cac bai toan md, tao cohgi
Cho HS kham pha kien thdc mdi tren cosd huy
dgng tdi da cac tri th dc da cd
Hoat dpng ehuyen bai toan trong sach giao khoa
Ihanh bai toan md nham tang eudng hoat dpng phat
hiwi tim toi kien ttiirc mdi cua HS, qua dd phat tnen
nang lyc tuduy sang tao,tao cahpi cho HS huy dpng
twda cae kien tfiLfc dadupc hpc.Baig thditich cue hda
hoatdmg cua HS, khPiday kha nang tylap, chudpng,
sang tao cua HS Nham nang eao nang tuc phat hien
vagiaquyetvan de, tac ddng den tam li,tinh cam, dem
lai niem say m e va hung thu hoe tap eho H S
Wdu^.'Tucong thiic iupng giac ca ban daduoc h x '
|cosj+sin-T]<V2 hay (cosjc+sin.v)' ta co the
giup HS the phat trien thanh eac bai toan bat dang
ttiue hinh hoc lien quan tditam giae ^
T d b a t d i n g thuc daisdlren, cdtheap dung trong
tam giac / I S C vdi cac goc la A, B, C:
Ta cd (sin.-i-t-cos /}• < 2 {*), mat khac
cos ^ = — ^ ' ~" va sin -J=—, ttiay vao {') ta
6\s<3c-2bc "^
{be Ihc )
Nhu vay, ta cd the phat trien thanh bai toan:
Chung minh ring trong tam giac ABC^& luon c6
\H^+c''a'+^S\<2^bc (1)
<:^A = 4 5 ^
Tiep tue bai loan tren, eo the cho HS tim ldi giai
bai toan: Cho tam giae ABC thoa man dieu kien
1 />• + c' - o- + 4S1= 2V2ftf hay tinh gde A eua tam giae
do
Ap dung nhuvay vcfi cac ketqua (sin B +cos S ) ' < 2
va (sinC + cosC)- < 2 ta lan luptthu duoe:
\c-+a b ^AS\<2^ca (2)
\a- + b'-c \-AS\<2j2ab (3)
Cdng theo ve cae bat ding thuc (1), (2)va(3)
ta suy ra dupe:
a-+/j-+ c'+12.9 < 2>^(a6 + k-+ Cfl) • Cdthephatbieubaitoan mdi: ChuTig minh ring trong fam g i a e / l f i C t a ludn c d ^ - + 6 - + c ' + I 2 S <
2^{ab + hc +
ca)-Tiep tuc ta eon thu dupc cac bai toan khac nda Nhdhoatdpngchuyen hda cac bai tap trong sach giao khoa thanh cac bai toan md, HS co ca hpi huy ddng eac kien thde nhudinh li Cdsin, edng thde tinh dien tich tam giac, da duoc hpc de kham pha eae kien ttiuc mdi
3.4 Hoat dgng huy dgng cac trithi^ coi nguon thong qua phat hien moilien he nhan qua hoac kha nang lien ttiong cac tri thdc Hen quan nham giijp HS quy la vequen khi giai quyet cac van de toan hgc
Thong qua hoat ddng lien tudng se giup HS goi nhd lai cac tn thuc cdi nguon, de td dd ehu the eo the dinh hucffig viec bien doi ddi tuong, tao thuan ipi cho
HS giai quyet van de mptcach dan gian hon
Viec xac dmh cac tri thuc cpi nguon de lam sang to quan he nhan qua giua tri thifc da biet va tri thuc can tim, Oac biet cac tn thuc cpi nguon dong vai trd dinh hudng, dieu chinh hoatdpng xam nhap vao ddi tupng, xam nhap vao cac van delam sang to cac ddi tuong toan hpc, eac quan he can kham pha.Bd la eosdde
HS hoatdpng chiem finh kien thue
l';'du5;Cho hinh iap phuang ABCD.A,B,C,D,
Gpi cac diem M.'W.'Plan lupt la cae trung diem cua
cac canh AD, BB^ C,D, Hay dung thiet dien eua hinh lap phuang tao bdi mat phang (MNP)
Khi giai bai toan nay HS gap khd khan la xae djnh giao cua mat phang (M/VP) vdi mpt mat cua hinh lap
Tap chi Gido due so 381 51
Trang 5cacmatconlai
Tri thuc cpi nguon da chuan bjcho HS bao gom:
- Quy trinh tim giao tuyen cua hai matphang.^
- Quy trinh tim giao diem cua mpt dudng thang
v6imptmatphlng
• Cach xac dinh matphang; Matphang diqua ba
diem; mat phlng xac dinh boi hai dudng thing cjt
nhau; matphing xac djnh boi hai dudng thang song
song; mat phang chua mpt dudng thang vampt diem
khong'thupc dudng thing dd TiJ do co the giiJp HS
huy dpng cac kien thi>c noi tren thdng qua he thong
cac cau hoi hoac dera cac yeu cau dpi vdi HS.cuJhe
nhusau;-Neu quy trinh dung giao cua dudng thang
MN vdi mat phing {A,B,C,D,)7; - Xic djnh mat
phang (a)chija MN, dola matphing chua haidudng
song song fiS, va M/V,, trong do M^ la trung diem
cua canh 4,0^;-Xac djnh giao tuyen cua matphing
(a)vdimatphangday(/1,B,C,D,),d6ladudngthang
S,M|; - Trongmat phang (a) xac djnh giao diem S
cua dudng thing M/Vva B,M,7
Tut do yeu cau HS «
xac djnh giap cua mat
phing (MNP) va mat
cua hinh lap phuong
Nhd cac kien thiJc da
bietHS tim dudc giao
la dpan PQJhinh 5)
Cac giao diS'm cdn lai
HS CO the ti^ xac djnh
nhd he thcng cac kien " ' " " *
th lie dabjetneu Iren vatim dupc thietdien la lucgiac
deu MTNQPR, trpng dp 0 ; R; T lan lupt IS trung
diem cua cac canh S, C,, D,D va XS
3.5,Khaosatcacm6hmhthticti§n nham khac
sail vaitrd ling dung tri thiic toan hgc
Vidu 6: {Tro chal vol nhiing dongxu) Hai ngudi
Choi mot tro chpi nhu sau; Lan lupt dat nhung dong
xulencaibanhinhchunhat,neuailanguc(c6thedat
dupc dcng xu cuoi cunglen ban thingudi dose thSng
cuqcchaigiasi/caibandunhovasoltrangdongm
dudeditkin len mai ban] Bai toan dat ra la tim quy
tac dat de ngudi choi thing cupc
^ Ta nhan thay, neu ngucC choi tmdc dat dong xu
dau lien vao chinh giua ban Sau do, dat thep quy tSc;
datdvjtridoixungvdidongxucuangudichoithilhai
vi)a datqua dong xu dohinh gifla ban, thi ngudi choi
52! Tap chi Giuo due so 381
cd tam doi xiing nen bat kimotvj tri nao trenmatbin thi ta luon tim duoc vi tri doi xiing vdi vj tri ay quafim cua mat ban Qua each phan tich nhutren tatiay Ira choi nay CO thechoi tren bat kimatgl, miln lamatiJo phaicdtamdo'ixdng
Trong pham vi bai viet nay, chiing toi trinh bay mpt SP quan niem veketnoi tri thiic va mpt sodang hpat dpng ket ne'i tri thiic trcng linh vuc tim tdi tri tue, qua dp luyen lap cho HS kha nang huy dSngliiin thiic n h i m djnh hudng, dieu chinh hpat dong giai quyet van de, phat hien tri thiJc mdi trcng day iiot hinh hpc Chung tpi che r i n g cac heat dpng nayco the van dung vao day hpc cac npi dung tpan Iigc khac dtrudng THPT P
Tai lieu tham khao
[1] M Alecxeep - V Onhisuc - M Cnjgliac (1976)
Phdt iri^n tu duy hoc sinh NXB Giao due
[2] Cruchelxki V A (1973) Tdm linUnglireToinhoc
cua hgc sinh NXB G i i o due,
[3] NguySn BS Kim (2006) Phuang phip dfy Im
mdn Todn NXB Dai hpe Sir pham
•[4] BOi v a n Nghi (2009) Vdn dung ii ludn va Ihfc
tiin vdo day hqc mdn Todn & trudng phSthdng NXB
Dai hpc Suphiim
[5] J, Piagel (\'i'>(i).Tdm lih</cvd gido due hoc.fm
Giao due
[6] G Polya (2010) Sdng tgo todn /ioc(Ngu6i d | *
Doan) NXB Giio due Vif t Nam
[7] Dao Tam (chii bien) - TrSn Trung (2010) TtcW
hog! ddng nhdn thuc trong dgy hgc mdn Todn ft trudng trung hoc phd thdng NXB D^i hpe Siiphgm
[8] Dao Tam (chu biCn) - U Hiin Duong (2008) Tlfp
cdn cdc phuong phdp dgy hgc khdng truyin 0ni [rong dgy hgc todn d trudng dgi hgc vd trudng pItS Ihdng NXB Dai hpe Sir pham
[9] Diiorsm {20fi) Phdt hiin vdsii dung hinh Inrni gian d/ lim rdi Id'i gidi bdi todn hinh hgc Tap chi
Toan hgc & Tudi Ire
[lOJ NguySn Canh T o i n (1997), Phuang phdp kfi
duy v(tf biin chung vdi vi^c dgy hgc nghiin ciiu lodn
ftpc (Up 1) NXB Dai hpc QuiSc gia Ha N6i
1111 Cao Th| Ha (200J) Dgy hgc mdt si chi ti Hi*
hgc khdng gian (Hinh hgc 11) theo quan Mm Ai/n
rao Luan an tiin sr Giao dye, Vifn Khoa hoc giao due
[12] L e T h i H\xm% ilOM) Bdi dudng cho hge M
trung hgc casd ndng luc biin ddi thdng Ufilodn hoc trong qud trinh dgy hgc mdn todn Lu^n dn lii'n si Gido due hgc, Trirpng Dai hoc Vinh