Geometriya darslarini tashkil etishda matematik paket dasturlaridan foydalanish

12-SEMINAR MASHG’ULOT

GEOMETRIYA DARSLARINI TASHKIL ETISHDA MATEMATIK PAKET DASTURLARIDAN FOYDALANISH.

Reja:
1.Planametriya bo`limini o`rganishda matematik paket dasturlaridan foydalanish

2.Matematik paket tushunchasi.

Tayanch iboralar: matematik paket, axborot texnologiyalari, planimetriya, dastur, koordinata
Planametriyaning mantiqiy tuzulishi. Geometriya real hayotdagi predmetlarning miqdoriy ko`rsatgichlari va fazoviy shakllarini o`rganadigan fandir. Narsalarning boshqa xossalarini boshqa fanlar o`rganadi. Agar biror narsa o`rganilayotganda, uning fazoviy shakli va o`lchamlari hisobga olinsa, unda geometrik shakl deb ataluvchi abstrakt obyektga ega bo`lamiz.

Geometriya- yunoncha so`z bo`lib, “yer o`lchash” degan ma`noni bildiradi. Maktabda o`rganiladigan geometriya qadimgi yunon olimi Evklid nomi bilan Evklid geometriyasi deb ataladi. Geometriya ikki qismdan: planametriya va stereometriyadan iborat. Planametriya – tekislikdagi, stereometriya esa fazodagi geometrik shakllarning xossalarini o`rganadi.

Planametrik masalalar va ularning tasvirlari:

1. 
   Rasmdan foydalanib BOC uchburchak burchaklarini toping.
   

Yechim: Rasmdan ko`rinadiki, < AOC=48, AB diametr

2. 
   Uchburchakning uchlari radiuslari 6 sm, 7 sm va 8 sm bo`lgan va jufti-jufti bilan urinadigan uchta aylana markazlarida yotibdi. Bu uchburchakning perimetrini toping.
   

Yechim: Chizmadan ko`rinadiki, a=AB=13 sm, b= BC=15 sm va c=AC=14 sm P=a+b+c=42 sm

3.Teng shakllar deb birini ikkinchisiningustiga aynan ustma-ust tushadigan qilib qo`yish mumkin bo`lgan shakllarga aytiladi.

Rasmdagi shakllardan qaysilari ustma-ust tushadi?

Geometriyaning ba’zi masalalarini yechishda axborot texnologiyalarini qo‘llash. Ta`lim sohasi samaradorligini oshirish uchun bir qancha amaliy dasturlar qo‘llanilib kelinmoqda. Shunday amaliy dasturlardan biri kompyuter algebrasi Maple12 paketidir. Bu amaliy dastur asosida algebra va matematik analiz asoslari hamda geometriya masalalarini hisoblashlardan boshlab yechimini, tasvir holatlarini ham ko‘rish imkoniyati bor. Biz quyida kompyuter algebrasi Maple12 paketi orqali geometriya masalalaridan biri bo‘lgan tekislikda uchburchak masalasini yechishni taqdim etamiz.

Masalani yechishdagi har bir bajarilgan amalni Maple12 paketi orqali bajarilishiniko‘rsatib boramiz.

Tekislikda A(10;3), B(-4;7), C(4;8) nuqtali koordinatalar bilan berilgan,uchburchak.

Quyidagilarni toping.

1) ABC tomonlarining tenglamalarini;

2) AB va AC tomonlar orasidagi burchakni;

3) tekislikda ABC ni quring.

Yechish:

1) ABC tomonlari tenglamalarini berilgan ikki nuqtadan o‘tuvchi to‘g‘ri chiziq

tenglamasi: = formulasiga asosan topamiz.
AB: = , = , 2x+7y-41=0 , = ;
with (geometry) :

point (A,10,3), point (B,- 4, 7) :

:= line (AB, );:= AB

Equation (AB, ); 82-4x-14y=0

AC: = , bundan 5x + 6y – 68 = 0,= ;
> with (geometry):

> point (A,10,3), point (C,4,8):

> :=line (AC, ); :=AC

> Equation (AC, ); 68-5x-6y=0

BC: = , bundan x-8y+60=0, = ;
> with (geometry):

> point (B, - 4,7), point (C,4,8):

> :=line (BC, );

> Equation (BC, );

ABC tomonlarining uzunliklari:

> with (geometry) :

> triangle (ABC,:

> distance (A,B) : distance (A,C) : distance (B,C):

> sides (ABC);

2) AB va AC tomonlar orasidagi burchak

tg

formula asosan topiladi:= ; = ;

tg= = ,  =arctg ( ) , =arctg ( )4215`
> with (geometry) :

> _ EnvHorizontal Name :=’x’: _EnvHorizontal Name :=’y’:

> line ( AB, 82- 4*x-14*y=0) , line ( AC, 68-5*x-6*y=0):

> phi := FindAngle (AB, AC); arctg ( )

> phi := evalf (phi) ; 416418617;
3) Koordinatalari bilan berilgan uchburchakni quyidagicha quramiz.
> with ( geometry ):

> triangle (T, ):

> draw ( );

Yuqorida yoritilgandek talabalarning ishlagan masalalarini kompyuter algebrasi Maple12 paketi orqali tekshirish quyidagilarga imkon beradi:

- o‘zlashtirilayotgan materialni mustahkamlash masalalar yechish malaka va ko‘nikmalarini hosil qiladi;

- ob’ektlarni kuzatish, o‘qitish jarayoniga bo‘lgan qiziqishni va talabalar faolligini oshiradi;

- o‘z-o‘zini va o‘zaro tekshiruvni amalga oshirishda.

Misollar:

>with(plots):

spacecurve([cos(t),sin(t),t],t=0..4*Pi);

spacecurve({[sin(t),0,cos(t),t=0..2*Pi],[cos(t)+1,sin(t),0,numpoints=10]},

t=-Pi..Pi,axes=FRAME);

spacecurve({[t*sin(t),t,t*cos(t)],[4*cos(t),4*sin(t),0]},t=-Pi..2*Pi);

knot:= [ -10*cos(t) - 2*cos(5*t) + 15*sin(2*t),

-15*cos(2*t) + 10*sin(t) - 2*sin(5*t), 10*cos(3*t), t= 0..2*Pi]:

spacecurve(knot);

helix_points := [seq([10*cos(r/30),10*sin(r/30),r/3],r=0..240)]:

spacecurve(helix_points);

spacecurve({helix_points,knot});

Maple 9.5 da geometric figuralarning tasvirlarini hosil qilish mumkin, buning uchun foydalanuvchi quyida keltirilgan misoldagi komandalardan foydalanishiga to`g`ri keladi. >with(geometry):

triangle(T,[point(A2,0,0),point(A1,2,4),point(A3,7,0)]):

circumcircle(C,T,'centername'=OO):altitude(A2A22,A2,T,A22):

altitude(A3A33,A3,T,A33):

altitude(A1A11,A1,T,A11):orthocenter(H,T): centroid(G,T):median(A1M1,A1,T,M1):

median(A2M2,A2,T,M2):

median(A3M3,A3,T,M3):dsegment(dsg1,OO,H):

dsegment(dsg2,H,G):

dsegment(OM1,OO,M1): dsegment(OM2,OO,M2):

dsegment(OM3,OO,M3):

triangle(T1,[M1,M2,M3]):

AreCollinear(OO,H,G);testeq(distance(H,G) = 2*distance(G,OO));

draw([C(color='COLOR'(RGB,1.00000000,1.00000000,.8000000000),filled=true), T(color=blue),T1,A3M3,A2M2,A1M1,A2A22,A3A33,A1A11,

dsg1(style=LINE,color=green,thickness=3),dsg2(thickness=3,color=green),OM1, OM2,OM3],axes=NONE);


>with(geometry):Angle := table():

n := 8:

for i to n do Angle[i] := 2*Pi*i/n end do:

dsegment(dseg,point(A,0,0),point(B,4,0)): point(o,0,0):

circle(c,[o,1]):

homothety(c1,c,3/2,point(M,-1,0)):

homothety(c2,c,2,point(M,-1,0)):

homothety(c3,c,5/2,point(M,-1,0)):

translation(t,c,dseg):

translation(tt,c1,dseg):

translation(ttt,c2,dseg):

translation(tttt,c3,dseg):

for i from 1 to 8 do

rotation(t||i,t,Angle[i],counterclockwise,o);



Tavsiya qilingan adabiyotlar ro’yxati:

1. R.K Otajonov. “Geometrik yasash metodlari.” Toshkent o`qtuvchi – 1986 – yil

2.A.V. Pogorelov. “Geometriya.” 7 – 11 – sinflar uchun. Toshkent o`qituvchi 1991 – yil.

Nazorat savollari:

1. 
   Matematik paket deganda nimani tushunasiz?
   
2. 
   Geometriya darslarida qanday axborot texnologiyalaridan foydalaniladi?
   

3.Planimetriya darslarida qanday matematik paketlardan foydalaniladi?

4. Dastur deganda nimani tushunasiz?