Mô hình hóa toán học trong bối cảnh học tập dựa trên giải quyết vấn đề

MO HINH HOA TOAN HOC
TRONG BOI CANH HOC TAP
Dl/A TREN GIAI QUYET VAN OE
Ld Hdng duang
Trudng THPT Xudn Giang, Sdc San, Hd Ndi
TOM TAT

P^'^dungmdUnhtodnZcrZ^d7d^'^,^'''l^"«^"g'^>-O'^" -'«« cin
Nghiin aiu di xuit quy trinh ShnhT^f^^
cua phuang ph^ pdng day cua minh
todn hgc vd thuc tiin
" """ '"" """'g 1"' "•'"* g'"' I'vil vin di
tiint21,T''''''''^'^''^'^--^'^'^'^>-''*'^.-Oidiqioi,vindi:7huc
Ngdy nhgn bdi: 8/9/2016; Ngdy duyil ddng bdi: 25/9/2016.

I. fiat vin dl

tiong su thay d l ^ T ^ d ^ f r
I ' . ' ^ - " " ^ "^^"S gia ttog Toto hp
ntog'tofa h ^ V e d n 7 S d i r u i " d b t " ' * " ' ^ " " ' ''" ^^"« ' ° ^ '
toto m ^ h md nhu: x i y ^ i l i ^ t f i l ^ i t h t h n
• " ™ / ' " ' ' " ^ " ^ ' " " h "nh
djnh vin dd, phdi hop s u d ^ g ^ o L i 14 mdt n i r ' K " * " ^ * " ? ' ^'^
nfag cua ngudi hpe, no ngiy cto„ aua„ i o „ K L ^ ' " ' " ^hit tiita k h i
tirit cin tttilt dd ttiinh c^ng C
df » L ^ ^ ^ " " i ' t " " " ^ " ^ «'^° "'^P '^
vto eto ttudng phd ttidnfed toh h u f r , r " " . ^ u ' ' '"*"• h"= ^^ ™ 8 dvng
tiong ttiue tiln cudc sdng
™« ' " h "^("^ <" hfa ehe nhitag khd khin

TAP CHfTAMLyHOC, Sfi 10(211), 10-2016

89


Hpi che cua day hpc ttuydn ttidng li hpat ddng hoc tap didn ra tiong
khong gian lop hoc vi giao vidnttiuftngttinhbiy mdt loat eto buftc x i yeu ciu
hpc smh pbil Iim theo vfti mdt hinh ttiuc tirang tu. Viec niy vd tinh Ito h^in
che tiem nang cua hpc suih ttong vide giii quylt vin dl. Giai quylt bii tofa
ttieo cach ttiuc dd vto cd hita qua nhat djnh ttong chuong timh hoc. Tuy nhidn,
no khong hieu qui ttong viec chuto bi cho hpe sinh nhihig kinh nghiem. kjl
nang giai quyet vto dd toin hpc cd lien quan din cac tinh hudng ttiuc tidn.
_ Mdt sd nhi nghidn ciiu chi ra ring, cich thilc giii quylt bii toin thuc
tien, gli thuc tidn trong khudn khd lop hpc thuftng Ii ap dat. Hinh ttitic niy la
khong phu hop dd ttien khai kien ttiuc loan hpc. Dilu dd din din su thilu hilu
biet kinh nghiem ttiuc tidn cua bpc sinh. Theo thfti gian, vidc hoc nhu v?y Iam
cho hpc sinh trd nen tbu dpng, hpc mpt each co hoc (hanh vi). Nhimg no Ivc
eai each mfti day huftng tfti giup cho hpc sinh ky nang suy nghi mdt ctoh sau
sac vi co kha ntog ip dung nhiing tu duy d trinh dd cao ttong mpt mdi ttuftng
Cfa giai quyet btog toin hpc (ttnh hudngttiuctiln).
, Mpt ttong nhung cich tidp cto la phuang phap day hoc dua vio giai
quyet van de. Nhiem vu cua nghien ctiu niy li giai thich md hinh hda toin hpc
vi giai quyet vfa de theo quy trinh thdng qua giii quylt vin dl. Ddng ttifti
nghien cuu viec thidt ke cto nhidm vu (bii gitog) trong md hinh hda loan hoc
Phuong phap nay cd Ipi ich li ren cho hpc sinh phat huy duac tu duy tpin hoc
cung nhu khi nfag toan hpc va thiu hilu dat vto dl (cd khi ntog dit cto hdi
va glii quyet vin dd).
'
Uu diem ndi bft cua day hpc dua vio giii quylt vin dl la nhin manh vai
tto cua nguai thidt ke bii gitog tiong vide din dit hpc sinh tilp thu kiln thuc

Kh, hpc s,nh qufa sit dupe md hinh hda toto hpc tuang ung thi hpe sinh sd cd
dupc rat nhieu k,nh nghiem ttong vide ip dung nhttng kiln thuc toin hpc vip
thuc tten kh, do, d,ta voi mpt van dd cu thd nio dd.
eita till" -'"'' "*'•"'*" *"' ™ "^^y h^^.-^Va vto giii quylt vin dl Ii nhiing
cich tidp Cta mm co mdi quan he mat thilt vfti nhau.
2. Md hinh hda toan hpe Irong hoat d^ng giii quylt van de
,h„. , " *r-"'^'' "^''•"h h™ 1^ i"?' '™g "101 sd ho,t dpng hpe tap dua trta
md^I- ^° uZ ""'". 'ã'" "^^ ô'*" hda mpt tinh hudng (hue tl phuctop,'tao ta
tnpt mo hmh, Iam viec toto hpc vfti nd vi giii ttlich kit qui tiong tinh hinh'tfiuc
td. Fox, Waners (2005) tai ra md hinh tofa hpc dupc su dgng dl giii ,h eh ttnh
huong thvc td hoto cie tinh hudng phi toto hpc trong cie dhih d!ng toin ho"
v, du, b,eu dd,^ bang bidu va cto phumig tiinh dupc su dpng dl md hinh vi
Ok K?'^L! I ' r ' l ^ "at """^ he phuc tap giOa cto hita nrpng khic nhau.
Ok K, Kang, J.bwa Noh (2012) dua ra khii niem md binh hda li mpt qui trinh
^

TAP CHf TAMLy HPC Stf 10(211), 10-2016


dongh^l^<en::s.'^i^t:^ir ' ' t r * ^'^ "'>•'
- ^ H ^ ^ m , ta vi Phuang phip v

^

^

t

^

" ^ ^ ^ ^ X

s^.iS^"l^tlSSs:f;s:ss,;t-'''h-'-'''Oc
ngufti day ei, ta ky ning dilft^f^ 1 ^ , " ^ " ' ' " * '"'°"S '« "h^- D" ^^
ttinhtir:diln d ^ t . t h ^ t i l t o L T ^ ' l J i t S h S S h a r ' ' ^ ^ * ^ ' '

Cic khS'^acthiu ::^'mni'',rr 'vtrt "ã" -" ^'"h xuit pbitô

dinh ting kiln ttiuc toto h ^ ^ a h o t s L h T t *"/• " " " " h - "^y W>ing
dong vi ttii nghiem thuc t? Vide p h / t t f a kS,*"tti*''d " ' " ^ ""^ " ' ' h'"h
h ^ dpng cua mpt ci nhto ddi 4 n ^ S ? h , v ,
-"'" " " " "h" '* " 9 '
Hpc smh phi, tiiln cac md hin^ bao Z t ^ T ^ ! T . T " "^""^ ' ^ "h^'
quan hd giOa eto ytirftng,ttiuchidn v n l r r ^ ttropg. cto kha, n,dni, cto tide,
dat cu ttti tfidng qua eithhTth^lnT^^-.^t'l"'"
h«* "^^ '° *« 0"Ve diln
cic biiu dd. pblilai^in," 1 * " S K a f ?
• ' M"*' **' ""'h- ^^
hpc sinh ttiam gia vto eto hoat dfln«Tai.*- ^ '°'^- ^°' "^"^ '*^':- "^i
tiong ra bta ng'ii Z^t.^Jr^l'l^^^^J:^!"''"^
" h - ".'en, bta
t ^ g ttidng qua ngdn ngii k'v hidu ddThrh-^ ^ ^ ^ ^ " " ^ ^ "'*' "-^h "
gSn day chi ra ring, hoc"'iS cd L °.^'' "1^° <'" ''°>- ^ ' « " "ghita euu
md hinh hda, dto b! t cMng cd„ c ^ S n P n ' ""l ''"* ""'"^ ' ' ' " '"'^" "«"
Chung lam rd hmi cic khi, ^idm
^ ^ " " ' " " " ^ " ^ ' " " •"» hinh giup

l™ cho vide gili q u K dl
t a ? ""*ãô"
^ ^^'-""-8

"^^
tt,uit.
ue tua nta
nen eeo' T> nnghta
va co sue" ^manh
ve m|t hpc

g^h^^i^tirm ::;;ii:s,ta iiht^?^'^p -^ «"h hudng nham

cto hpc sinh cin hop tic Ito vJeTJ* nh, f
l ' " " ^ ""'• ^ ^ <•' ^* "° yen
xem lai kit qui.
" * ' ™ "h^"''™g "hidm vu. phit ttita, kita, tia.


Giio vita dupc xem nhu la ngudi huftng dan vd mjt nhfa ttiuc. Giio vidn
cd ttic ttiam gia vao mdi tiirftng bpc. Didu niy gitip giio vien hilu dupc hpc
sinh can vi cd kha nfag lim dupc gi. Giio vidn cd ttid lim nhidm vu hd tip khi
can thidt vi mft rpng suy nghT cua hpe smh ttieo hudng siu ban, huftng tfti mpc
tieu bai giii. Do do, day hpc dua vio giii quylt vin dl giup hpc sinh cd kha
ning hr dinh huftng tiling viec hpc. Giio vien khdng cin phii cung cip y tirdng
cho hpc sinh vi en ttil ving mat ttong Iuc hpc sinh giii quylt nhidm vu. Day
hpc dua vio giii quylt vin dl la cich ttiuc cung cip mdi tiuftng hpc ttiuit,
tiong dd nang cao vai tid hpc sinh hr nhto thuc.
3. Quy trinh md hinh hda va thilt kl nhifm v(i
Md hinh li dai dien cua mpt he thdng hay mdt qui tiinh. Xay dung md
hlnh giup xdc djnh cac vto dl, td chuc nhttng suy nghl ciia ngufti hpc, hilu dii
lieu, giao tilp vi kiim tra su hilu bilt dd cung nhu dua ra du doin. Do dd, md
hlnh li mpt cdng cu huu ich cua tri tiid. Cheng (2000) ttinh biy vi du vl hinh
ttitic cic qua ttinh md hinb hda toto hpc va dupc sil dung trong Iftp hpc ft cic

tiirftng trung hpc d Singapore. Nhft dd, phit ttiin nhOng y tuftng vi kliai ni^m
toto hoc CO bfa, bd sung nhilu ky nang diy thach thiic vi ttiu vj mdi trong khi
ngufti hpc phat trien cic md hinb. Dilu niy ttiuftng bj bd qua trong day hpc
tpin hpe truyen thdng.
Van de the gioi th^^c

Triru tugng (1)

He thong loan hgc

Cic thi nghiem vi
dirdoSn(r)

Ket quitfiirctiln

Van dung tri thdc
Toin hgc (2)

Giai thich kit qui
Ket qua toin hgc

Mg rpng van dl
the gidi thuc (4)
Mdrpng van dl trong
cugc song
Saddl:

Quy trinh md Mnh hda todn hpc

Qua trinh xay dung mp hinh ttr ttil gifti ttnrc li nhidm vii ttid vj bfti qui

titah niy nhto giii quylt cac vin dl pbit sinh trong cudc sdng, ngufti xiy d(mg
92

TAP CHl TAMLy HPC Stf 10(211), 10-2016


md hinh ein huy ddng kiln ttnic bto tiiin, kinh nghidm ttiuc tl vi ^i ,h • ukm. Mdhmhttiuftng lita yiandin ti.nlJlu lin MaTtton; mdrch^.^ c h t "

Mo la quy trinh mS htnh hda di xudi tai Sir

ddi

toto h^\^r;ac"'q:y' Z V ^ t ^ ' ' ' ^ '"l " ' '^ "•'' ?"* ^
>^' " ' ' dvng ^ p h t o l o Z I ' ^ ™ ^ ^ ^ c ^ t ^ d T a c Z ^ t a ^ i " ' " ^ ' " * " ' ' '^^
lua tapn nhung dai dita tao eac'tSittog c l i t t S t
"""^ ' " * "'•"" ''

gita cie^Si-s:t
ttiuc cua cto ddi hrong Thtog qurv1tani;to'',T """ T " ' '^™« *« «™
qufa. Phto tita vi ttiuc hien cto hoat d S / d ^ " ' ' ^ ' " « '"*" hoe lita
kit luto; nlu vidc tfiuc hiln S e W d' * t?- "" ' ^ ' •""''"""' h^ -l^ ™ ' "
ddi cto lua chon cu7cto Winlu'cfti 2 1 " ^ ° " ? *.' ""*" ""'"h" ''" '° ™^
ta cd kl, luto toto hpe
" ^ ' ""'' '''"S cto md hinh. Do dd,

nta du dofa kit Into vl'ttilSh ttiuc d u t h
™ ' ' ''^ *™« ' " ^ hpc
thich hpp duoc tt,u'?hi^„ 1? v a ^ S ' S t ^ ' l V " " " ! '*J ' ' ' ã""ã " ^ ' ô " '
bfa diu

•'™ Sia, u„ch ket qui cua tofa hpc ve tinh hinh

titi J^l^^ttZ

^ttofh"''' ""^'^ '^' ^^ '' "h^ 8'* "h^^ ^^

.hue ,1 cung cip cicTtg™ ng e u T l t i l i : ^ T'' ' " " " " ^'"h' ^^ <"ã
dupettonglftphpccdlid^quan^tttoftX^irvrxihdi'^"'"^''''
Thiet ki nhif m vy
chuan^:i^l^S::'SSfvlS^tr'^'^'^^''^^H^vô
khang nang nl vd ,ri'nh biy,'grchrp cd t,!^ thti; ,^° T '^'' " ' ''* '"«''
Thep Jonathfa Borwein Kei h n „ r
J
^ ' h *""= (8P' m*). ">« la.
dpng md hinh h d X ' i ^ Sifeleh ' ' T^ "8hidp (2009), thdng qua hoat
giii quylt vin dl ttong t ^ u t t t S U t j ^ o ^ f ' ^l '-'" " ' "'"'• '°-'^ hpc dl
td, sti dpng cto phuang Phin bTlnd^S ^ong. Hpc s,nh cfa xu ly cto sd lita tt,vc
Cie cdng cp vi phu^g phin Toto h i I T ' ^ ? ' "h^"" "?^ 'hpn va ap dpng
tu ehinh cie tmh hurnl tittg^h^tig.""" ""^ * ^ " ''"^'' hii toto nfy stah
yiu ciu Chung phdi ddm bdo sdu nguyin tic:
- Ddm bSo tinh thi;c t i
TAP CHfTAMLyHOC, Sfi-10(211), 10-2016

93


- Dim bao vd xay dyng md hinh; cie tinh hudng dat ra cd cin phai kidm
tta, dieu chinh, mft rpng hay khdng?
- Tu dinh gii: Tinh hudng ditracd cin yta ciu hpc sinh tu danh gia hay khdng?
- L t o rd y tirftng: Tinh hudng dat ra cd cin ydu ciu hoc sinh Iim rd

nhung suy nghT, y nrdng cua minh hay khdng?
- Dan gifa: Md hinh cd don gito khdng, cd dd sii dung khdng?
- Tdng quit hda: Md hinh niy cd ip dung vio cac tinh huong khic nhau
hay khdng?
Vidu I (Bdi loan ve the tich): Ban cua em ndi ring: "Muc nuoc co lidn
he true tiep vol thd tich cua iiuoc rdt vao". Em hay giii thich rd tai sao khing
dinh eua bfa ehua du td,? Ndu cin, hay giii ,hich cu ,hl vl mdi quan hd eiOa
chieu cao vi ,hi,ich dl,huylt phuc ban.
Lien he ttiuc td, nhi truftng hita cd mdt bl chua nuftc da xay dtmg tu rit
lau nen khong ai ed dupe sd lita cua bd. Chi bilt ring, bd cd hinh tru dung vi
bin kinh diy cung nhu chilu cao cua bl cd ttld do dupc. Hay tinh thi tich cua
be,ttiftigian miy bom nuftc cd thi bam diy nuftc vao bl.
Phdn Itch
Truftc tien, gido vidn hudng hpc sinh kidm tra sau nguyen tic cho thilt
ke nhiem vp di dua ra:
- Thuc td: Vi du, yeu ciu hpc sinh tim ra chting cti dl thuyit phyc ci
Iftp; hpc sinh da quen thudc ft nhi va trii qua kinh nghidm.
- Xay dung md hinh: Hpc sinh phii tu dit ra ciu hdi cho ban thto Ii muc
nuoc vi the tich ed mdi lien be hay khdng? Ndu co thi mdi lien he dd cd gidng
nhau tiong ed ba truftng hop khdng.
-J u dinh gia: Hpc sinh cin phii dat mpt sd gii dinh, vi du, chiing phai
gli thiet mpc nuftc thay ddi nhu tbl nto nlu ctog mpt luang nuftc duoc rdt
vap ba binh?
- Lim rd ytirftng:Hpc sinh cp tbl trinh bay duoc su [bay ddi cda muc
nuoc duo, djng bing hoto biiu dd.
- Don giin: Hpc sinh cd tbl md ta duoc mdi quan hd giua muc nuftc vi
ttie tich ttong batiirftnghop.
- Tdng quit hda: Nlu ba truong hpp lim dupe thi vfti mdt hinh dane nio
do hpc stnh cung cd thd Iam hrang tu.

TAP CHf TAM Ly HPC Stf 10 (211), lo - 2016


De td chuc boat ddng md hinb hda nhiem vu d vi du niv oiin via™
huong dto hpe sinh thuc htenttieoquy ttinh dl xuit
tita.
^' ®
Thuc hien cdc budc md hlnh hda
( 0 Day li mpt bii toto mft, dilu kien bar din ^h,™ , - •

r^

.

cung ttog nhrniE chiln en lrhs„„,- „.
•
''^S ™ chteu cao
nhau vi dd S j e t cfa b i ^ ° 2 f r f
t ™ ^ " " ' ' "* '^ h^"* '^"^ hinh khto

nj.... .
ã ""=ã'^^ã''-'^''ôThongsododupc* = 2.5m * = 3Im
Cto h d n r s ' " " ' '':^-r'<^^^'<^-'> = « . " ' - ' - Hon ntta. hpc sinh tim hilu : i
s'^it 2 0 0 i ^ , 3 : V d ^ a T d i y t o i ^ - ^ ^ ^ h ^ ^ ^ ' - ''" '''^ • ''^^' ' ^ "
25mm. Hpc s.^h t t a h X „ t b ^ ' l i t a mc m m? t"* ""I;'"ãô ""' ^^ ^^
l^'^ằyi=2ô2.giftsdbldiybi^irrcdtdl'nIcZr'^'^ô^ta t m h ' t r f h e ' o t i c l d n l t u '^T''''

T

" " ^ ' ^ "h^ ^ ^ "h^^" hon taung

cong ti:^^':stiik^r tilit'^iiiS-^ s ;:f r r r ^ cau tinh ttie tich eua mdt vat thi r^„ „
'i.u- I
™"* han khi cd yeu
cdng ttiuc tinh. Hoe toi do fhd T j Z ^ T ^ ã""' ''^ "'"S "ô' h'?' ^d
mang bang ddng hay tinh tbl n>h K
^ ^ *"' ' " * *« "ch cua mdt pbo
stia vi c a i L b i S ,^i* kl h, d™ '"'n"'.' " " * ^'-"^ ^ ' "'^y ''"' •** >"
xiy ra, hpc sinh cin ^ I t f a R . "df "'^ J^
""'' ™" " ' 'hue tidn thuang
'"fa- Kb, do, quy ttmh md hinh dupe lap Iai. Vin dl
TAPCHfTAMLynp^^^^^^^^i^^TloT^


niy huong hoc sinh bilt each xto djnh ttil ti'ch mpt vat ttid bit ky ttidng qua ttil
Uch bi chiem chd.

n du 2 (Chieu cao cua cing dgng Parabol): Tai Truftng D ^ hpc Bich
Khoa Hi Ndi, co nhidu cong ra vio nhi tiuftng nhung c6 mdt cdng c6 htah djc
btpt dang parabol co bd lom quay xudng dufti. Em hiy tmh chilu cao cda c6ng
(khotog each tu diem cao nhat cua cong dta mat dit).
Phdn tich

Tuang t\r nhu vi dvi Uift nhit, giao vidn huftng hpc sinh tfti kiim tta 6
nguydn tic chpttiietkd nhi|m vp da dua ra.
Thtrc td: Hpc smh di gjp nhilu md hinh djng parabol nay ft cu0c sdng
nhu: htah inh diy hp cSt ngfag, vlt quJf djo cda phi lap khi vjn dpng vidn
phdng Iao, cong AcxoftUianh ph6 St. Louis (My),
^ y dting rad htah: C6ng djng parabol c6 thi xem Ii dd ttij ciia h t a so
b|c hai, ctaeu cao cua cdng hrang ftng vfti d0 cao tft dtah cia parabol xu6ng

dit. Hpc smh c6 tfie do djc vj tti 2 chto cia c6ng, xic dtah dupc mdt s6 vj tti
tidn cdng, I$p btog sd lidu, biiu do tinh toto thft nghidm.
^ Tti dtoh gii: Hpc sinh cin phii dit m^t s6 gii dinh, vi dp, vj tri chto
cong tfiay dot nhung vail gift htah djng parabol tfii dd efa cia e6ng ttiay dli
ttld nio, phvttiuOcvio yeu td nio?
L t o rd yttrftng:Hpc sinh cd ttil titah biy dupc sir ttiay ddi dd cao cia
cdng dufti dang bfag hpto biiu do.
Don gito: Hpc stah cd Uil chpn h? tpa d0.Oxy sao cho g6c tpa dd tring
VItilmpt chfa cdng.
= T v
6
Tong quit hda: Nlu hpc sinh hilu dugc moi lien h | gifl-a cic ylu ,6 lidn
(JUfa den dinh cia parabol ttii hpc sinh cfing cd tfil giii quydt nhidm vu vfti cic
hnh huongttronghr.

96

TAP CHlTAMLyHpCSd 10(211), 10-2016


Dd td ehue boat dpng md hinh hda nhidm vu ft vi du nay. giao vidn huftne
dan hpc smh ttiuc bitatfieoquy trinh dl xuit ttta.
Tltuc hiin cdc budc md hinh hda
Be giai quylt bii toto niy cin phii tim hilu dd xic dinh cic giattiidtctia
bi. toan cho cu Uid ban. Do vay, gito vita khuyin khita hpe smh Uito luto vd
nhmig so lieu cfa ttaet cfa ttiu Uiip dd dmi gifa boa bii toto. Nhto ttiiy hinh
ft. COttietim m dp cao cua c6ng. Hoc sinh giai ttiita ring, cto didm ttdn cdng
settioamfa mpt tinh chit eia parabol.
*
^ Hpe sinh bilt ring h t o sd bto hai cd djng y = ax'- +ix.^e, (a*0) Vay

muon do th, him sd bto hai niy tiung vfti hmh dtog cdng, tiuftc tita hoc sinh
can chpn hp ttue tpa dp vudng gdc. Dl Unh tofa ftuto tidn. chon g& o' t ^
nTit s ' d i t r , ' ™ * ' ^ ' ' ' ' f f l""™^ " ^ '"^ "° ^ phai xto linb d u r h
h L^ i i ^ '^?^' ' h ^ 8 hjn O(0;0), B(Xk;0), M(x„;y„). Hoe sinh'tiln
hfah do dje de tim sd lidu cto ttiidt.
xic dinf 11*^0'^^*^''^ ^^' " ^ "^•'^
^°'- ' * ° ^ 8 ctoh gifta hai chto vi
WtnteT-^^d•^'ZJ'T^:
'h?"S hfa: khotog cita hai tato cdng Ii
10m, tire xt - 10; diem M bat ky ttdn edng ta dp xa so vfti ttuc ttting dftn|tii
g d c 0 1 i x „ = l v i d i l m M e a t a m i t d i t y „ = 3.Khidd:^ = d^=+l£;t.Nhu
3

3'

viy, cdng parabol cao ;- = — » 8,33 m
« n h ™ ° '"^- i ''^'''™' ^^° ''*'""" ™"8 " h ^ ^ "Sy dvmg c6ng ttii clng
eao hon so va, t,nh toto mi hpe sinh dua ta.khotog 7cm. Dilu niy dtfpc ly g,i!
eto cong,
c L lhon
' " nfta
> " cong
^ " cdn
? 'phii
^ " *tinh
' ' "dp
' " diy
'^"^
"''"'
M "^eiu' «ttic

' * clng.
mating
cua
cua"' Idp
bd tdng
dinh . ^ " ' ^t' !;^L''"^"Shi«n cftu ttiuc nghidm tao ttiiy, hiu hit hoe sinh diu
^
p i mo hmh^hda cic vto dd tfiuc tita li bii totokhd. Theo Nguyta Dfah
toto en til h ' u"^ T If" ^•P ^° " ^ ^ * hudc diu tien, dd Ii flm Wlu bai
toto cutfienbieu hpe sinh nhat, xet rfag bii toto khdng di do lita dl giii, khdng
It^l til "",? T^l"
" ' ' '^ ^^ '"^^•- •• Chinh vi vay, nhilu hoc stah da gto
S x t diTd
K ' " ?' *''^*' "•' "'y " ^ hii toto. Thuc tiring, khi ho^
sinh xiy d ^ g dupc bai tototfncdtfidgiii duoc bii toto dd. Vi till hpc smh X
vm;t quattftngj, toto hpc hda vin dltttuctiln vi bilt giii ttiita k t q u T
4. Kit luin
.™t„ 1 ° " ""'l^ "'*.''"h hda hoto toto cd ttil duac vto dmig ttong dav hoc
toto a ttuong ttung hpc phd ttidng dua ttieo quy ttinh tiic gii df xuit'lS^'hiS
TAP CHf TAM Ly HOC, SS 10 (210,10-2016


oan hpc va da> hpe dua trta giai quylt vin d l da duac chung minh cd mdi
lien he bien cbung. Md hinh li mdt phin cua day hoe dua trdn'aiai quvIt vin
de ma toan hpc la thinh phan quan trpng cua bai toto md hinh hda xi'nd dpa
tren eac y luang toan hpc quan trpng. giup hpc sinh cd thd dao sau v.i phat tridn
su thong hieu toan hpc. Nhieu nghien cftu truftc day gifti thidu c i c md hinh
toan hpc cho thay chien lupc nhung chua quan tam thda dana ddn phuana phip
djy hpc dua tren giai quyet vto de. Nghien cuu nay. phin n i o minh chuifa cho
nhung Ipi ich cua vide djy hpc md hinh hda trong giii quvIt vin dd. aiup hoc

sinh phat tnen cic ky nang giai qu)ci vto dd. kv nana hpp tic. kha niing van
dung til thuc toin hpc. kinh nghidm tich Iilv bin than v i o giii quylt nhftng
nhiem vu thuc tidn.
T i i lieu tham k h i o
1. Farad Bahmaei. Mathematical modelling In primary .school, athantages and chtilknges
Journal ol Mathematical Modelling and Application, Vol. I. No. 9, 2011.
2. Lesh R. & Doen- ll.M. (Eds.). Beyond conslruclivism: A models & modelling perspective
on mathematics problem solving learning & teaching, Mahwah. NJ: Eribaum. 2003.
PubTsh*"'200™'''"' ^"^

" " ' ' " ' ^•'''"•'•'"™"'''*^'"'"'"""''"-. Speetmm Academic

Ltitoc?Eta::,t,'^:fK.ls:'^^^"""""""" •"'"'"""'''
ofR'omZtTllN:'*™"''"'"'"*''""™'"^^^

° = ' ' ° ' h r a International Journal

Ihong. Tap chi Khoa hpe. Dai hpc Qude gia Ha Npi, Tjp 31, Sd 3. 2015.

TAP CHfTAM Ly HPC S>-) 10(211), 10-2016