NGHIEN COrULlLUAN
Day hoc LiTdng giac
trong Chifdng trinh Toan phd thong hien hanh
Pham Minh Phildn9 TnJflng Tmng hgc pho thong Chuyen Oai hgc Si/ phgm - Tmflng D31 hpc Si/ pham Ha N61 136 ftjan ThLiy, CSu Gi3y, Hi Ngi, Vi^ Nam Email, thaygiaopfiuong@gmail cnm
TOM TAT: Bii viit lam rd tiin trinh day hgc Luang giac d cap Trung h<?c cdsdva Trung hgc phi thdng trong chUdng trinh mdn Toin hign hanh thdng qua vi$c phim tieh nhi/ng uu diim vi han chi cua bin giai dogn dgy hgc Lugng giic cua ChUdng trinh mdn Toan phi thdng hi$n nay. Trdn cd sd dd, da chl ra m^i si diem cin chu y trong day hgc ndi dung Lugng giac chUdng trinh mdn To6n mdi. Trudc hit ta thong nhit cung mdt quan diem xay di/ng him si lugng gi6c, xuydn suit td gia tri lUgng giac cua gdc din ham si lugng giic biin si thi/c;
Hai la. bo sung cac khai nidm gdc (cung) dii nhau. pdc (cung) bit nhau. gdc (cung) phu nhau, gdc (cung) hdn kem nhau n va tong, hiiu cua gdc (cung) lugng giac trudc khi xiy dgng cac cdng thUc biin dii; Ba li, tSng cddng c6ch tiip can trUc quan khi day hgc cic ndi dung lugng giac nhu: Tgp xie djnh ci)a cac ham y^taux va ham >'=cott, phuong trinh lugng giic cd bin: su«=a, COM=O, tanj«r=a, coix=a; Bin la, tSng cUdng gin kit cac ndi dung d$y hgc lUgng gi&c vdi nhdng vin di thUc Uen nhU: Do dgc, tinh to^n, cac chuyin ddng trong V^t If,... Cach tiep can nay se tang cddng higu qui va chat lugng dgy hgc ndi dung lugng giac cua chuong Irinh Toan phi thdng dip Ung ydu ciu doi mdi glAo dye ctia nude ta hien nay.
T D KHOA: Chifdng trinh mon Todn hl$n h^nh; chifdng Irinh mon Todn mdi; Lifglng gide.
•* Nhan bai 7/3/2019 -> Nhan kH quS phin bi^n vS chinh sila 10/4/2019 •> Ouy^ dSng 25/5/2019,
1. D^t van de
Chucmg trinh (CT) mon Toan trong CT giao dye pho thong (GDPT) moi (ban hanh ngay 26 thang 12 nam 2018) da xac djnh n^i dung mach ''Lupng gide" a cap Trung hoc CO scr (THCS) va cap Trung hpc pho thong (THPT) [1].
Dong thai, CT mon Toan trong CT GDPT mai ciing nhan m^inh quan diem "chii trgng ke thira va phat huy nhiing uu diem ciia CT mon Toan hien hanh, dong thai van dung co chon Ipc nhirng kinh nghiem tien tien ciia the gicri" [2] De thi^c hifn daac quan diem tren, truac het, chung ta can phan tich ro vi^c d?y hpc Lu(7ng giac trong CT mon Toan hi?n hanh. Bai \ lel nay lam ro tien trinh d^y hpc Lupng giac a cap THCS va THPT trong CT mon Toan hi^n hanh [3], tir do neu len mpt so diem can chii y trong day hpc noi dung Lugng giac trong CT mon Toan moi.
2. Npi dung nghien citu
2.1. Tien trinh day hpc Lupng giac trong chifdng trinh mon Toan hi^n hanh
Trong CT mon Toan pho thong hi?n hanh, Lupng giac dupe dgy hpe theo bon giai d o ^ sau.
- Giai do?n 1 • Ti so lupng giac cua goc nhpn (d\ra vao ti so giiJa dp dai cac canh trong tam giac vuong).
- Giai do^in 2: Gii trj lugng giac ciia goc tir 0" den 180°
(dgra vao niia duang tron don vj).
- Giai dogn 3: G6c lupng giac, gia tri lupng giac ciia goc lupng giae (dya vao duang iron lupng giac).
- Giai doan 4, Ham lupng giac hien so thyc, phuang trinh lupng giac
66 TAP CHI KHOA HOC GIAO DUC VlfT MAM
Wk mat tign trinh day hpc, caeh tiep c?n nhu CT mon Toan hien hanh la tuang doi hpp li, phii hpp vai nhan thirc cua HS, phii hpp vai lich su hinh thanh ciia Lupng gik, dl tiep nhan d6i vai hpc sinh (HS). Tuy nhifin, each tiepc^
6 mpt s6 giai doan con chua hpp li. Duai day, chung ta si phan tich ro hon ve timg giai dosin.
2.1.1, Giai doan 1
Lupng giac dupe dua vao tam giac vuong, gSn lien voi tam giac vuong la buoc ke thira ciia phan tam giic dong dang dupe trinh bay truoe do.Truac do, cuoi 16p 8, HS da biet rang, hai tam giae vuong dong dang neu chung co mpt goc nhpn bang nhau thi ti 1? giua cac canh bang nhaa Ngupc lai, neu hai tam giac vuong co li 1^ giiJa cac canh bang nhau thi dong d^ng va do do goc nhpn bSng nhau (xem Hinh 1).
Hinh I
Nhu vgy, ti 1^ giihi cac canh eiia mpt tam giac vuong ^
Knam Minn Kni/ong
pc v a o d p lan cua goc nhpn: N^u hai tam giac vuong C va A ' B ' C CO cac g o c ^ = ^ ' = 90", 5 = 5 ' = a thi 4B A'B' A£^A;C^ AC AT 3C B'C' BC B'C' AB~ A'B'
ix do, dan den cac khai nipm ti s6 lupng giac ciia goc
ranh dot anh ice Cinh ke
cpih dm' catiA taqwn canh huyen' canh ki lie ti so luong giac cua goc nhpn noi tren da giup tra lai I hoi ve bai toan "Giai t a m giac vuong": Neu mpt tam c vudng bigt 1 canh va 1 goc nhpn thi hoan toan xac dinh
the tinh dupe cac canh con lai).
]dch tiep can n h u tr6n la h o i n loan hpp li, phii hpp vai in thirc ciia H S , phii hpp vai nhiem vu cu t h i a giai doan I, dd l i " G i i i tam giac vuong". N g o i i ra, each tiSp can r tren eon phii hpp v a i ljch su hinh thanh L u a n g giac, m i ban dau L u p n g giac d u p e hinh thanh de phuc vu nhu I do dgc: D o chieu cao, d o khoang each...
'1 du, de do chigu cao ciia mpt c i i cay, ta t i l n hanh n h u h du6i day (xem H i n h 2):
Hinh 2
Ta diing 6 vj tri A tren mat dat c i c h goc cay mpt khoang Ing X (c6 the dii dung bong ciia cay tren mat dat, diem l i bong cua ngon cay). Sii dyng thiet bi d o goc, do goc [hieng 9. Khi do: h = x.tan9 .
Hoic de d o chieu cao ciia mpt ngon thap, ta tien hanh n h u nh du6i day (xem Hinh 3).
t
_, Angle of Elevation
Fig a Hinh 3
Ta co: h = BC + BD^ AB.tan 0 + BD.
2.1.2. Giai doan 2
L u p n g g i i c d u p e m a rpng tir ti so lupng giac cua goc nhpn sang gia tri l u a n g ciia c i c goc tir 0° den 180°. Wk mat t i l n trinh la hpp li, sau khi "giai tam giac v u o n g " thi nhu c i u t i t yeu la "giai tam giac t h u a n g " . Dieu do cung phii hpp v a i lich su hinh thanh ciia lupng giac, ngoai nhu c i u d o dac chi diing den tam giac vuong thi con c6 nhiing bai loan do dac thyc tien gan voi viec " g i i i tam g i i c thucmg".
Tuy nhien, C T hien hanh khong x u i t p h i t theo logic tren ma xuat phat tir nhu cau m a rpng k h i i niem fi so lupng giac de phuc vy cho viec day hpc tich vo h u o n g cua hai vecta. Caeh tiep can n h u the da tao ra sy khong t h i n g n h i t trong day hpc L u p n g giac. Trong C T hien hanh, de m a rong khai niem ti so lupng giac cua goc nhpn sang gia t n lupng cua cac g o c t u 0" den 180° sach g i i o khoa hien hanh su dyng nua d u o n g tron d a n vi: Vai m6i goc nhpn a se tuong ling voi mpt diem M tren n u a d u a n g tron d a n vi sao cho
xOM = a (xem H i n h 4).
C2r
sin a - > ' , „ , c o s a =x,^, tana=—^, cot a y^
Uu diem: Cach tiep can tren bao h i m k h i i niem ti so l u a n g g i i c cua goc nhpn da d u p e dinh nghTa truac do, n g o i i ra eon thuan loi cho viec xay dyng k h i i niem g i i tri lupng g i i c cua cung (goc) l u a n g giic sau nay.
Hgn che: Chuyen tiep dot ngpt tii dinh nghTa hinh hpc (ti so dp dai) sang djnh nghTa ham so (gia tri luong giic). Dieu d o gay kho khan cho HS khi tiep nhan kien thirc. Ve mat lich su, k h o n g dung vai ljch sir p h i t trien ciia L u a n g giac:
N h u cau m p rpng khai niem ti so lupng giie cua goc nhpn sang g i i t n lugng giac ciia goc tir 0" den 180" phuc vy cho viee giai tam giac t h u a n g . Ngoai ra, trong cich tiep can nay tinh chat hinh hpc v i tinh img dung yeu. Ngay trong sach giao khoa hien h i n h , viec tinh gia tri lupng giac eua mpt so goc dac biet lan hon 90" nhu: 120", 135°, 150° cung dupe d u a ve tinh gia trj lupng g i i e eiia cac goc bii voi no la 60°, 45°, 30°.
2.1.3. Giai doan 3
Lupng giac d u p e m o rpng tir g i i tri lupng giac ciia goc t u 0" d i n 180° sang gia tri lupng giae ciia goc (cung) lupng g i i c b i t ki thong qua d u a n g tron lupng g i i e .
Sd 17 thdng 5/2019 67
NGHIEN ciiru Lf LUAN
i/u diem.
Vi?c mo rpng khai ni?m goc hinh hpe sang goc lucmg giie la cin thilt, phii hpp vai lich sir hinh thanh ciia Luang giie, khi nhu ciu do dac, tinh loan trong Vat li, Thien van...
doi hoi phai mo rpng cac goc hmh hpc thinh eac goc luong giac. Viec xie dmh gia tri lucmg giie ciia goc lupng giac thong qua duang Iron lupng giac li hpp li.
Mpi vdi hgn che.
Hgn chi thu nhat, de ma rpng khai niem goe luong giac, sieh giao khoa iheo chuong trinh ehuan chpn giii phap xay dyng cung lupng giac, sau do dinh nghTa goc lupng giie qua cung lupng giic: "Tren ducmg tron djnh huong (chpn truac mpt chieu chuyen dpng gpi la chieu duang, chieu ngupc lai li chieu am) cho hai diem A va B. Mpt diem M di dpng tren duong iron luon theo mot chieu (am hoac duong) tir A den B tao nen cung lupmg giac co diem diu A, diem cuoi B".
"Tren duong tr6n djnh huong cho mpt cung lupng giae CD.
Mot diem M chuyen dpng tren duang tron tir C den D tao nen cung lupng giac CD noi tren. Khi do tia OM quay xung quanh goe O rtr vi tri OC den vi tri OD. Ta noi tia OM tao ra mpt goc luprng gide, co tia dau li OC, tia cuoi la OD".
De mo rpng khai niem gia tn lupng giac cua mpt goc tit 0° den 180° sang gii tri lupng giic ciia goc lupng giac, saeh giao khoa theo CT chuan chpn giii phap xay dung gia tri lugmg giac ciia cung lupng giie (xem Hinh 5).
Hinh 5
"Tren dudng tron lupng giac cho cimg ^Mco s6 do bang a. Tung dp y = OK ciia dilm M gpi la sin ciia a vi ki hieu la sina : sina = 0K .
Hoanh dp J: = OH cua diem M gpi li cosin ciia a va ki hi?u la cos a : cosa = OH .
Neu cosa ^0 thi ti so gpi la lang ciia a va ki hieu cosa
,. sina la tan a : tan a = .
Neu sin a * 0 thi ti s6 gpi li cotang eiia a vi ki sina
hi^u la : cot a = . cola sina
Cic gii trj sina, cosa. tana, cola gpi la cic gii tii lupng giic ciia cung a".
68 TAP CHIKHQA HOC GlAO DUC VIETNAM
Sau do, chi co mpt chii y ve ti so lugng giac eiia goc lupng giac: "Cic djnh nghia tren cung ip dyng cho cac g6c lugng giie". Truoc het, each tiep can nhu v§y ta khong dam bio tinh thong nhat, khong nhit quin: Mach trien khai phai tir ti so lupng giac ciia goc nhpn, qua gii tri lupng giac ciia goc tir 0° den 180° den gia tn lupng giic ciia goc lupng giic (khong phai den gia tn tugng giae cung lugng giae), Ngoai ra, each tiep can nhu vay li lan giO'a giii phap kT thugt voi myc tieu.
Cach tiep can ciia saeh giao khoa theo CT nang cao tranh dugc dilu nay. Sach giio khoa theo CT nang eao chpn giai phip dinh nghTa goc lupng giac, sau do dinh nghia cung lupng giic qua goe lupng giac:
"Cho hai tia Ou, Ov. Neu tia Om quay chi theo chilu duong (hay ehi theo ehieu am) xuat phit tij tia Ou den trimg vai tia Ov thi ta noi: Tia Om quet mpt goc lugng giac tia dau Ou, tia cuoi Ov".
"Ve duong tron tam O, bin kinh R. Gpi giao ciia cac tia Ou, Ov v6i duong tron li U, V va giao ciia tia Om voi duong tron la diem M. Khi tia Om quel nen goc lupng giac (Ou, Ov) thi diem M chay tren duang tron luon theo mpi chieu tir diem U den diem V. Ta noi diem M vach nSn mgt cung lupng giac miit dau (diem dau) U, miit cuoi (diem cuoi) V, tuong ling voi goe lupng giic (Ou, Ov)".
Hgn che thir hai. Van de ve mien xie djnh ciia tancr va cola .
Sieh giio khoa dua vao phan he qui sau dinh nghTa:
" tan a xie djnh voi mpi a^ — + k;i[ksZ)."
Sau do, ehiing minh nhan xet tren:
" tan a khong xie djnh khi cos a = 0 , tiic Ii diem cuoi W ciia cung,^Mtriing vai B hoac B', hay a = — + A;r {A € Z)"
Trong chiing minh tren, tir nhan xet diem cuoi M ciia cung AM triing vcri B ho$c B' nit ra a = — + *«• {A:eZ) la 2 •
qua duong dpt, thieu tinh trvc quan, gay kho khan cho ' HS. Can nhin xet diem cu6i 5(0; 1) tuong ung vdi cic cung ' lupng giac CO s6 do la
a=^ + k27r{keZ)
Diem cuoi S'{0;-l) ling voi cae cung lugng giic c6 s^
do la
3;r •k27nk€Z\ = 2 Ket hpp hai hp tren ta dugc hp a=~+k7i{k^Z).
Tuong ty vai dilu ki^n xie djnh cua c o t a . Thyc ti ^ hifc o nha tnrcmg pho thong cho thay, nhieu HS kbdng tfci hilu duge cie buoc I^p luan tren. d$c bi^ ti qui trinh kfe hgp hai hp n g h i ^ thinfa mgt hg nnSfi.
2 ^k27t{keZ) = ^ + {U + l):r{keZ)
Phgm Minh PhiJdng
Hgn chi ihir ba: Vl cie cong thiic lugng giic.
Sach giio khoa neu cong thiie cpng, ching hgn nhu:
cos(o-6) = cosa.cos6 + sina.sinA, cos(ti + b) = coso.cos/j - sin a.sinb
Tuy nhi§n, ea hai bp sach giao khoa diu khong dl cap toi hai niem tong va hieu cua hai cung (goc) lupng giie. Dilu 6 la khong hpp logic toin hgc vi gay kho khan cho HS ong tiep thu kien thue.
2.1.4. Giai doan 4
Lugng giae dugc nghien ciiu tren phuang dien him so.
^am so lugng giac bien so thuc dugc xiy dyng thong qua
;ia trj lugng giic cua goc lugng giic vdi s6 do radian bing
;ia trj cua so thuc. Cich tiep can nhu vay la tuong doi hpp i, phii hpp v6i nhan thiic cua HS.
Vi dy, sach giio khoa nang cao "Dai s6 va Giai tich 11"
Iinh nghia him 7 = sin i :
"Quy tac cho tuong img moi so thue x voi sin ciia goc ugng giac co so do radian bing x gpi li him so sin, ki hi?u a _>' = sinj;."
Ham s6 :»' = COSJ: dugc dinh nghTa tuong tu. Cac ham y = tanx vaham _>' = cotx dugc dinh nghTa qua ginx va
ma eosx?iO, tiic la
"Vdi m5i so thyc
:^ —+ A;T{A:eS),taxacdinhdupcs6thuc tanx = .
2 cosx Dat D, = R \ j —+ 4;r|A:eZ I. Quy tic dat tuong ling m6i
i r. ;.: i . Sin X . . , , , . X so xsDy vai so thuc tan x = dupe gpi la ham so
cosx lang, ki hi^u la y = tanx".
Ham so y = cot x dugc dinh nghTa tuong ty. Sau khi dinh ighTa him lugng giac, sach giao khoa hien hinh khong ihac tdi cac tinh chat, cong thiic bien doi lugng giac... cho :ic so thyc ma ngam hieu chiing dugc chuy6n tuong iing ir cic tinh chit, cong thirc bien doi lugng giac ciia cac gdc cung) lugng giac sang. Dieu do li thieu chit che.
Hep theo, sach giio khoa trinh bay eac phucmg trinh trgng giac cabin: smx = a, cosx = a, tanx = a, cotx = a 'a mOt s6 phuong trinh lupng giic thudng g|ip.
2.2. Mot SO nhan xet
Nhihig phin tich trgn day cho chiing ta thay dl day hpc liinh cong npi dung Lugng giac trong CT mon Toin mdi
thi ehiing ta can tdiac phuc nhiing diem han che trong day hpc Lupng giic d cap THCS va THPT trong CT mon Toan hi|n hinh. Cu the, chiing ta cin thuc hien cac viec sau:
1/ Thong nhat eiing mgt quan diem xiy dung ham so lugng giac, xuyen suot tir gii tri lugng giic cua goc den ham so lugng giac bien so thuc.
2/ Can bo sung cic khii niem gdc (cung) dii nhau, gdc (cung) bii nhau, goe (cung) phu nhau, goc (cung) hon kem nhau % va tdng, hieu ciia gdc (cung) lupng giac trudc khi xay dung cac cdng thiic bien doi.
3/ Tang cudng each tilp cin true quan khi day hpc cae npi dung lugng giac nhu: Tap xac dinh ciia cic ham y = tanx vi ham >' = cotx; phuong trinh lugng giac co ban: sin x = a, cos x = a, tan x - o, cot x = a .
4/ Tang cudng gan ket cac ngi dung day hpc lupng giac vdi nhiing vin dl thuc tiln nhu: Do dac, tinh loan, cac chuyin dpng trong Vat li,...
3. Ket ludn
Viec day hpc lugng giac d CT mon Toin phd thong hien nay theo tien trinh 4 giai doan. Caeh tiep can niy da bpe 16 nhirag yeu diem cin khac phuc. 0 giai dogn 2, vi?c chuyin tiep dot ngpt tir dinh nghTa hinh hoc (ti so dp dai) sang dinh nghTa ham so (gii tri lugng giac) gay kho khin cho HS khi tilp nhin kiln thiic, O giai doan 3, dl md rpng khai niem gdc lugng giie, sach giao khoa theo CT chuin chgn giii phip xay dung cung lugng giac, sau do dinh nghTa gdc lugng giic qua cung lupng giac. Cach tiep can nhu vay li khdng dim bio tinh thdng nhat, khong nhat quin, trong khi do sach giao khoa theo CT nang cao chpn giii phap djnh nghia gdc lupng giic, sau dd djnh nghTa cung lupng giic qua gdc lupng giic da trinh dupe dieu nay.Thuc tl day hpc d nha trudng pho thong cho thay, nhieu HS khdng the hieu dupe eac budc lap luan khi de cap den van de mien xie dinh cua tan a va cot a , dgc bi?t li qui trinh kit hgp hai hp nghiem thinh mpt hp mdi. Ve cic cdng thiic lupng giac, ci hai bg saeh giio khoa deu khdng de cap tdi khii ni^m tdng va hieu cua hai cung (gdc) lugng giie, dd Ii dieu khong hpp logic toin hpc vi giy khd khin cho HS trong tilp thu kien thiic. 0 giai dogn 4, dieu thieu ch^t che la d ch5 sau khi dinh nghTa ham lupng giie, sieh giao khoa hien hinh khong nhac tdi cic tinh chit, cdng thiie biin ddi lugng giac... cho cic sd thuc mi ngam hieu chiing duge ehuyen tuong ling tii cac tinh chat, cdng thirc bien ddi lugng giic ciia cic gdc (cung) luong giac sang. De khac phyc nhung thilu sdt neu tren, can thiet phii cd sy thay ddi each tiep c|n trong day hpe npi dung lugng giic trong CT Toan phd thong mdi.
a U^u thain khio
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Sd 17 thing S ^ t g 69
NGHIEN curu LfLUAN
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Ha Npi,
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ON TEACHING TRIGONOMETRY IN THE CURRENT MATHEMATICS CURRICULUM
Pham Minh Phuong High School tor Gifted students - Hanoi National University of Education 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam Email: [email protected]
ABSTRACT: The paper clarifies the process of teaching Trigonometry at secondary and high school levels in the current Maths curriculum by analyzing the advantages and limitations of four stages in teaching Trigonometry in the current Maths Program. On that basis. Some points to be noted in teaching Trigonometry in the new f\Aaths Program have been emphasized.
Firstly, agreeing on building trigonometric functions: from the trigonometric value ol the angle to the trigonometric function of the real variable; Secondly, adding the concept of opposite angles (arc), the supplementary angles (arcs), the complementary angles (ares), the reference angles (arcs), their sum and difference of trigonometrical angles (arcs) before built^ing transformation formulas; Thirdly, enhancing the visual approach when teaching trigonometry such as: the certain set of functions y-tanx and function y=coic; basic trigonometric equations: sint^a, coit=a, tanr-a cou=a; Foudhly, strongly integrating trigonometric teaching content with practical issues such as measurement, calculation, movement in Physics, ... This approach will enhance the efficiency and quality of teaching trigonometric content of the New hAath program to meet the education innovation requirements of out country.
KEYWORDS: Current Math Curriculum; new Math C u r r i c u l u m ; triflonometry strand al the Current Math Curriculum.
70 TAP CHI KHOA HOC GIAO DIIC VIETNAM
