Ensiklopedia_Matematika

8/8/2019 Ensiklopedia_Matematika

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MatemtikaHOMAN HO ORU (LAFATIK, LUHU, MAMA-FATIN BALU)

Forma bziku: Oru

Iha lafatik no buat seluk neeb halo ho homanhanesan nee, oru mak forma neeb bziku liu.Forma nee nia naran matemtika maklozangu. [LI: belah ketupat] Lozangu makparalelograma neeb espesil, ho ninin haatneeb hanesan ka igul.

Oru nia ngulu

Oru laran iha ngulu haat. ngulusira neeb se oin ba malu ninia valr

mak hanesan. ngulu rua neebkiik mak grau 60, no ngulu ruaneeb boot mak grau 120 . Nunee,forma hotu neeb halo ho oru uzangulu rua nee deit.

Tringulu Ekilteru: meiu oru

Dala ruma bainhira homan ho oru,bele ms hetan tringulu ekilteru

maib la barak. Tringulu ekilterumak tringulu neeb iha ninin toluhanesan, no ngulu tolu hanesan.Tringulu ekilteru mak meiu oru.

Tringulu Isoseles: meiu oru oin seluk

Iha lafatik balu nia laran, besik ninin, bele hetantringulu isoseles, neeb ms meiu oru. Trin-gulu isoseles mak tringulu neeb iha ninin rua

deit, no ngulu rua deit, neeb mak hanesan.

Forma rua seluk neeb bziku: fitun no heksagonIha lafatik laran, forma rua mak mosu beibeik, ihanaran fatin ida. Ida mak fitun, neeb halo ho oruneen. Ida seluk mak heksagon, neeb halo ho orutolu. Naran pontu ida mak fitun ka heksagon ida

nia klaran.. Bainhira foin haree, ema balu hareedeit fitun, ema seluk haree deit heksagon. Neenaran persepsaun.

Heksagon Fitun


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Forma sira selukBele ms hetan forma oi-oiniha lafatik laran. Buka tookhetan forma sira-nee noforma seluk tan.

Ro Mitsubishi

Matan Ai-Funan

Fitun rua Deltakiik

Delta boot Manu semu

Kutun Eskalera

Iha infinitu tan. Haree took!


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Bele halo oru husi kartuHalo oru husi karta hanesan tuirmai nee: Silu kartu ba sorin hanesan iha foto, no tesi.

Silu fita ida ba ida seluk. Hanesan nee bele halo ngulu 60 graus.Pontu iha klaran mak importante: baluk tolu tenke hamutuk.Labelehalo hanesan foto rua tuirmai nee.

Silu tan too rohan, atu hamosu tringulu ekilteru.

Loke fali. Buka oru rua, tesi tiha.


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Bainhira hetan oru ida kapas, bele uza hodi halo tan.

Tesi too oru sanulu hodi halo forma barak.

Koko atu halo forma neeb hetan iha lafatik.Ho oru kartu, bele koko halo forma neeb ita hetan iha lafatik. Purezemplu:

Ro Mitsubishi

Matan Manu semo

Heksagon Fitun


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Oru nia pozisaun ba malu

Iha lafatik laran, oru ruanunka mosu pontu sorin ha-mutuk ho pontu tutun. Maibho oru kartu bele tau pontusorin hamutuk ho pontu tutun.Nunee ho oru kartu, bele haloforma neeb la bele hetan ihalafatik. Purezemplu, foto ruanee:

Homan fita ida tan, kr seluk, atu bele hetan forma seluk tan

Lafatik nia laran iha oru deit.Maib ho fita seluk belehoman tan forma seluk.

Purezemplu:Forma sira-nee maski foun,ms iha ngulu oin rua deit:60 graus no 120 graus.

Forma nee balu mak narantrapeziu. Trapeziu mak forma honinin rua neeb paralelu no ninin ruala paralelu. Trapezio iha nee makhanesan oru no tringulu ekilteruhamutuk.

Oru nia diresaun mak tolu deitHo forma heksagon no Mitsubishi, bele haree diresaun tolu ba oruiha lafatik laran. Nee katak fita neeb halo lafatik ms dada ba dire-saun tolu. Bele pinta ho lapiz atu bele haree.


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rea bele sura ho unidade heksagon deitBele foti rgua no sura fatin kabeer nia rea uza frmula deit. Maib iha

buat ruma neeb homan, bele ms uza unidade bziku liu atu sura ninia rea.Iha buat neeb homan ho oru, purezemplu lafatik ka luhu, unidade bziku liumak oru. Depois iha forma rua tan neeb bziku: fitun (oru 6, pontu 6) no hek-sagon (oru 3).

Fitun mosu iha naran fatin, maib sempre iha heksagon iha fitun tolu nia la-ran. Nunee fitun labele taka rea hotu, ka, se taka hotu, ms taka malu. Hek-sagon ms mosu iha naran fatin, no mosu hamutuk deit la iha fatin entre hek-sagon rua neeb hamutuk. Ms, heksagon neeb hamutuk la taka malu.Nunee, ita bele uza heksagon hodi sura rea.

Tenke kuidadu: konta heksagon mak susar. Diak liu marka pontu klaran baheksagon ida-ida neeb atu konta. Depois bele sura rea. Iha foto ida-nee nialeten, rea mak heksagon neen.

Heksagon espesilHeksagon balu hamutuk bele halo heksagon boot, ho relasaun espe-sil. Husi foto nee bele haree heksagon espesil tolu: Ida kiik, oru tolu deit. Ida natoon, heksagon rua ba ninin ida, hamutuk heksagon 7. Ida boot, heksagon tolu ba ninin ida, hamutuk heksagon 19.Bele ms pinta ida boot liu, heksagon haat ba ninin ida, hamutuk 37.

Ba heksagon espesil sira-nee, bele halo tabela hanesan nee:

Bele haree no hetan konkluzaun hanesan nee: L liu sempre impar

L sempre aumenta rua ba rua A sempre impar Maski N sai boot ida ba ida, A sai boot lailaisA sempre sai nmeru prim. Prim katak nmeru neeb bele fahe ba nmeru 1 deit.

Bele ms buka frmula hanesan nee:L=N+(N-1) = 2N-1

Atu hetan frmula ba rea, bele konta heksagon deit. Purezemplu, ba heksagon ho N=3, hexagon kiik hamu-tuk, lia ba lia husi leten too kraik mak:A=3+4+5+4+3 ka

A=5 + 2(3+4), nunee

Heksagon iha ninin ida

N

Heksagon iha fatin luan liu

L

rea A

(heksagon hamutuk hira)1 1 1

2 3 7

3 5 19

4 7 37

5 9 61

)(21

+=

L

N

xLA


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Bainhira N=3, L=5

A=5+2(3+4) = 19

Se troka L ba N, (L= 2N-1), frmula haree hanesan nee:

Agora bele haree tanbasa L no A sempre impar: tanba L no A nia forma bziku mak2x 1 no 2x + 2y 1

no frmula rua nee nia resultadu sempre sai impar.

Lafatik tomak mak heksagon.Bele haree katak lafatik tomak hanesan heksagon ida. Ninia pontu (kantu) neennia ngulu iha grau 120.

Uza oru hodi sukat la-fatik nia ninin.Bele sura heksagon nia rea ho

rgua, uza sentmetru, maib msbele uza oru hanesan unidade.Iha lafatik iha foto nee bele kontaoru sia iha ninia ninin ida-idak.

Bele fahe lafatik ba oru tolu, tringulu

ekilteru neen.Hanesan heksagon hotu, bele fahe lafatik ba oru tolu. Oru ida belefahe ba tringulu ekilteru rua. Se liga tringulu ekilteru nia pontuida ba ninin kontrariu nia fatin klaran, lia nee naran ninia altura.Se pinta altura rua ka tolu, bele ms hetan tringulu nia pontu klaran.

Sura oru ida nia rea.Oru ida mak tringulu ekilteru rua hamutuk. Nunee, oru ida nia reahanesan ho rea tringulu ekilteru nian dala rua. Tringulu nia reaA mak ninia ninin naruk N vezes ninia aas (altura) a, fahe ba

rua:

)(2515

3

+= xA

)(254

3+= xA

)(212)1(2

+=

N

N

xNA

2

NaA =


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Sura heksagon nia rea tomak.Heksagon ida iha oru tolu iha ninia laran. Nunee, ninia rea

mak oru nia rea vezes 3:

Sura fitun hooru neen nia rea tomakFitun ida iha oru neen iha ninia laran. Nunee, ninia rea mak oru niarea vezes neen:

Bele ms haree katak ida-nee mak heksagon nia rea vezes rua.

Primeiru, ita buka a:

Tuir frmula Pitgoras nian, ninia aas mak:

22

2

2a

NN +

=

22

2

2 a

N

N=

222

4

4a

NN=

22

4

3a

N=

aN=43

Nunee, tringulu ekilteru nia rea mak:

Oru nia rea mak AT vezes 2:

2

NaAT =

4

32NAT =

4

32 2NAO =

2

32NAO =

2

332NAH =

33

2

36

2

2

NA

NA

F

F

=

=


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Heksagon no fitun nia distnsia ba malu.Haree pontu ihafitun nia klaran no buka pontu sira neeb besik liunia. Distnsia entre pontu rua nee mak rua tuirmai nee:*Iha diresaun tolu, lia metan iha foto ida-nee, distnsia mak N (orunia ninin ida).*Iha diresaun tolu seluk, lia mean iha foto ida-nee, distnsia makoru nia diagonl boot

Haree pontu iha heksagon nia klaran no buka pontu sira neeb besikliu nia. Distnsia entra pontu rua nee mak hanesan deit ho pontuiha fitun nia klaran:Nno

Buka fitun espesil, neeb la forma ho oru neen.Bele hetan fitun balu neeb iha oru hanesan tuirmai nee:7 (lafatik, mamafatin)5 (lafatik)3 (mamafatin)

PARA LAI!! KETA HAREE FOTO SIRA NEEB TUIR!! BUKA RASIK FITUN ESPESIL HANESANNEE, FOIN LEE TAN NO HAREE FOTO NEEB TUIR!!

32322NNaD ===

3N


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Bainhira fitun la halo ho oru neen, nia labele kabeer. Se iha menus,nia tenke hamriik. Se iha barak liu, nia tenke silu ba kotuk uitoan.Homan nain bele uza buatnee hodi halo forma barak.

Haree didiak fatin ho fi-tun neeb la forma ho oruneen. Baibain, heksagon ihafatin nee norml, la iha prob-lema. Fitun deit mak troka.

Maib heksagon ms bele troka no sai fali oin seluk. Haree took fotoida-nee: Heksagon sira iha ninia ninin laos heksagon ona maib orurua deit. Buat nee labele halo kabeer; nia tenke hamriik.

Halo fitun oioin ho orukartuBele aprende tan konaba buatnee se halimar ho oru kartu.Halo tiha oru kartu hitu noliga sira hamutuk ho fita kolahanesan foto ida-nee.Silu no loke atu halo fitunoioin, hanesan iha foto sira-nee. Bele tau iha homan fatinatu kompara sira-nia forma.


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Sura ngulu iha fitun klaranBele sura ngulu iha fitun nia pontu klaran nia leet. Oru ida-idak nia pontu tutun iha 60 graus. Nunee, fitunnia ngulu total mak sai hanesan nee:Oru 6: 6x60=360Oru 7: 7x60=420Oru 5: 5x60=300Oru 4: 4x60=240Oru 3: 3x60=180

Oru 2: 2x60=120 (maib ida-nee kabeer deit.)

Ida-nee konaba jeometria nao-Euklidian, katak jeometria neeb lamosu iha fatin kabeer. Purezemplu, jeometria neeb mosu iha rai-klaran nia superfisia mak nao-Euklidian. Se halo tringulu boothanesan iha foto ida-nee, ninia pontu nia ngulu mak 90 hotu.Nunee, hamutuk iha ninia laran iha 270. Tringulu baibain (iha fatinkabeer) nia ngulu iha 180 deit.

Ezemplu ida tan mak bola tebe. Ninia superfisia iha heksagon no pen-tagon (ho ninin lima). Nee kapas ba bola, maib labele halo buatkabeer ho forma rua hanesan nee. Koko took! Hanesan iha lafatiklaran, atu halo buat ruma kabeer, uza heksagon deit; labele hatamapentagon.

Pentagon no tringulu hamu-tuk ms bele halo bola kabuar,hanesan bola iha foto rua nee.


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HOMAN RAGARaga hanesan foto iha sorin nee halo ho ai-kesak neeb falun ho fitaakadiru-tahan. Rezultadu mak kuadradu barak. Kuak nia forma notamau ms hanesan deit kuadradu sira seluk. Kuadradu sira hotuliga ba malu ho sira-nia ninin deit, maib haree didiak interesanteliu.

Tringulu haat ms ihaBainhira haree raga, baibain ema haree kuadradu barak, maib belems hetan tringulu haat. Buka took.

Kuadradu kiik haat, neeb liga ba sorin-sorin,forma kuadradu boot idaBele haree, iha naran fatin ida, kuadradu kiik haat hamutuk halo kuad-radu boot ida.

Pinta iha biti ka surat-tahan milmetruIha biti laran, bele haree kuadradu deit. Bele pinta atu hetan padraunbziku husi raga. Uza surat tahan milmetru ms bele.

Buka kuadradu boot neeb

bziku liuForma hanesan raga nee naran padrauntanba nia repete. Padraun nia baze simples liumak ida neeb repete, no bele uza nia atuhalo forma kompletu. Iha raga, padraun bazemak 5x5 kuadradu. Ho parte ida-nee, belehalo hotu. Ho parte kiik liufali kuadradu 5x5, labele.

Tesi kartaun atu halo padraun raga nianBele tesi kartaun ida atu ninia naruk hanesan ninia luan no hetan kuad-

radu. Bele halo boot ka kiik.Halo barak no tau iha mezahodi halo padraun raga. Beletau kuadradu balu deit,

ka halo kuadradu ida neebbziku liu, hanesan foto ihasorin karuk nee, 5 x 5.


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Bele aumenta lia, tau barak.

Haree hakleuk hodi haree hanesan raga.

Laran: fita ida tuir diagonl bootIha raga nia laran, fita sira ba diagonl, husi sikun ida too sikun kon-trariu.

Okos: haree lia ida neeb tuir ninin no liga ba kuak nia pontu (1)Bele halo lia ida neeb tuir deit

raga nia ninin, no liga ba kuak ida-idak nia sikun. Lia nee mak krazl iha foto iha sorin karuk nee, no 1iha dezeu sorin loos iha nee.

Okos: haree lia ida tuirdiagonl no liga ba kuaknia pontu (2)

Iha raga nia okos, fita halai diagonl, hanesan lia mean iha foto leten nee, no lia 2 iha dezeu leten nee.Maib nia tuir kuadradu deit no la ba husi sikun too sikun kontrariu. Ninia ngulu ho lia 1 bele hetan hotrigonometria; lia rua nee halo tringulu, sorin ida mak kuadradu ida, sorin seluk mak kuadradu rua.Nunee, tan = 2/1, =63.4 Tanba tringulu sempre iha 180, =26.6

Okos: lia ida neeb tuir loos diagonl (3)Se halo lia tuir diagonl sikun too sikun, hanesan lia lotuk metan ihafoto leten ka lia 3 iha dezeu leten nee, nia forma tringulu ida seluk ho

lia fita nian (1): sorin ida mak kuadradu ida, sorin se-luk mak kuadradu tolu. Nunee, tan = 1/3, =18.4.

Bele haree katak ngulu sira + = 45. Bele hareengulu iha dezeu ida-nee ms 45.
http://videos%20compressed2/kuadru1.jpghttp://videos%20compressed2/kuadru2crop1.JPG


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MAMAFATIN NIA DIDIN:FATR MULTIPLIKASAUNMama fatin nia didin iha dezeu neeb ema homan. Atu halo dezeuida-nee, presiza hatene matemtika. Naran dezeu ida neeb repeteita temi padraun ka sekuensia, husi lia-portuges. Foto iha sorinnee, dezeu nia aas mak fita 13, luan mak iha fita 24. Dezeu ida-nee repete dala lima iha mamafatin nia didin. Nee katak fita neeb

dada husi leten ba kraik iha mamafatinnia didin hamutuk fita 120.

Maib se Ita konta fita iha mamafatin nia kidun, Ita sei hetan fita 60 deit. Seharee didiak, bele hetan katak fita ida-idak iha mamafatin nia kidun fahe ba ruabainhira nia silu sae ba didin.

Nunee, se ema hakarak halo padraun kasekuensia neeb repete iha mamafatin

nee nia didin, nia tenke buka fatr 120nian. Entaun, posibilidade sira mak hane-san iha tabela iha sorin nee:

Padraun iha mamafatin iha fotoliman karuk nee okupa fita

ualu, no mosu dala 15.

Padraun iha mamafatin iha foto liman karuk nee la hanesan, maibms okupa fita ualu, no mosu dala 15.

Mamafatin iha foto liman karuk nee iha padraun neeb okupa fita12, no mosu dala 10.

Fatr 120

Padraun okupa

fita hira

Padraun mosu

dala hira

2 60

4 30

5 24

6 20

8 15

10 12

12 10

15 8

20 6

24 5

30 4

60 2

120 1


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Koko took:*Buka mamafatin oioin no haree diak. Konta fita iha sorin no kidun. Konta padraun ida uza fita hira foinrepete.

*Bele halo padraun ka sekuensia ho koto. Foti koto no halo padraun hanesan ezemplu tuirmai nee:>Mean-Mutin,Mean-Mutin,Mean-Mutin (Padraun okupa koto rua,repete dala tolu.)

>Mean-Mutin-Mutin, Mean-Mutin-Mutin, Mean-Mutin-Mutin(Padraun okupa koto tolu, repete dala tolu.)

>Mean-Mutin-Malahuk-Mutin-Mean, Mean-Mutin-Malahuk-Mutin-Mean, Mean-Mutin-Malahuk-Mutin-Mean(Padraun okupa koto lima, repete dala tolu.)

Bele koko forma padraun husi buat seluk tan, purezemplu fatuk, ai-tahan, ai-funan, no seluseluk tan.

*Nmeru 24 ms iha fatr barak: 2, 3, 4, 6, 8, no 12. Koko took halo padraun ka sekuensia neeb repete falidala hira iha 24 nia laran.Purezemplu, padraun tuirmai nee okupa iha fatin haat (XYZX) no repete dala neen:X Y Z X X Y Z X X Y Z X X Y Z X X Y Z X X Y Z X

1 2 3 4 5 6

Padraun tuimai nee okupa fatin neen (XYZZYX) repete dala haat:X Y Z Z Y X X Y Z Z Y X X Y Z Z Y X X Y Z Z Y X

1 2 3 4

Koko took halo padraun ho fatr 2 no 12, no 3 no 8.

*Dala ruma mamafatin laos kabuar, hanesan foto iha kraik nee. Ninia sorin haat iha fita 14, tutun rua iha fita6. Nunee: (14x4)+(6x2)=68. Ninia fita ms fahe ba rua bainhira sae ba leten, entaun iha ninia didin iha fita136. Padraun uza fita ualu, no repete dala 17: 17x8=136


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KATUPAKatupa iha oin barak. Bele analiza sira nia forma no karakteristiku atukomprende liu tan.

Bele konta pontusikun [LI:titik sudut, LP: vertce]: fatin nebe meikoitoan

Bele konta oin [LI:bidang]: fatin kabeer iha pontu tolu ka liutannia klaran Bele konta ninin [LI:sisi, LP: aresta]: lia entre pontu rua Bele mos konta kuadradu ida-idak iha katupa nia oin hotu.Uza lapiz hodi marka ba katupa atu bele konta fasil oitoan.

Ami nia grupu iha UNTL analiza tiha katupa oin 5, iha foto iha sorin, no halo lista hanesan nee:

Oktahedron Baibain nian Bero

Alu nian Nafalun

? katak ami la konkorda malu tanba susar atu sura loos.

Ami nia grupu iha UNTL hetan buat barak nebe interesante hanesan tuir-mai nee:

Katupa Alu deit mak iha oin impar, ninin impar. Katupa seluk hotu

iha oin no ninin par. Kuak fatin atu hatama foos mak sempre susar atu analiza. Belehatama fali fita nia rohan, hanesan atu homan fali ho nia, atu bele kom-prende kuak. Haree foto iha sorin nee.

Sura Oktahedron Baibain nian Bero (se hanehan,sai kabeer deit)

Alu nian Nafalun

Pontu sikun 6 8 4 6 7

Oin 8 8 2 5 ?

Ninin 12 14 4 9 14?

Kuadradu 32 28 ? ? ?


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Susar atu konta oin no ninin ba katupa Nafalun. Dala ruma oin ladun kabeer, no dala ruma ninin ladunmoos. Katupa nia pontu sikun bele oin tolu:

kona kuadradu haat (kabeer), kuadradu tolu (kroat oitoan), ka kuadradu rua (kroat liu).

Bele hatudu konseitu ida nee ho kartu baluk kuadradu liga hamutuk ho fita kola:

kuadradu haat (kabeer), kuadradu tolu (kroat oitoan), no kuadradu rua (kroat liu).

Bele mos halo ida ho kuadradu lima, maiba ida nee baibain mosu deit iha homan oin seluk, hanesan foto ihakraik nee:

Katupa nia forma balu mak hanesan deit manu nia knuk. Ema uza fita kiik balu deit hodi halo katupa no fita bootbarak ba manu nia knuk.

Katupa Oktahedron mak forma basiku iha geometria laran. Bele halo ho pirmede rua, ho base kuadradu. Niaiha simetria ba leten-kraik; foti nia fila an, ninia forma hanesan deit. Halo komparasaun entre katupa Baibain nian no katupa Oktahedron: Oktahedron nia oin hanesandeit, pontu sikun no ninin aumenta rua, maibe kuadradu menus haat.


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KOKO TOOK:*Estudante hotu bele lori katupa nebe sira nia familia hatene halo, no halo fali aktividade hanesan ida ihaleten: konta pontu, oin, ninin, no area superfisie. Halo komparasun no analiza.

*Iha katupa oin barak tan. Katupa laos buat natural, maibe ema mak deskobre oinsa halo katupa ida-idak niaforma. Koko took aprende halo ida foun ka inventa ida tan!


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FORMA DEKORASAUNAmi-nia grupu iha UNTL hetan matemtika barak iha forma de-korasaun. Ami hili ona balu hodi analiza.

Poliedru PlatnikuPoliedru mak forma ho dimensaun tolu neeb iha ninin neeb loosno oin neeb kabeer. Poliedru iha foto lima iha sorin nee mak espe-sil ho naran Poliedru Platniku. Poliedru lima nee nia ninin narukhotu hanesan, no oin hotu nia forma ms hanesan. Tanba ema hetanforma sira-nee desde tempu uluk (tinan 2000 resin liuba) iha Grsia,sira-nia naran husi dalen Grega.

Ami-nia grupu matemtika iha UNTL hetan Poliedru Platniku toluiha forma dekorasaun. Ami konta ida-idak nia oin, pontu sikun, noninin. Ami ms buka sira-nia rea superfisie, liuhusi maneira rua:

#1 buka rea ba oin ida-idak depois soma hamutuk, ka#2 konta deit kuadradu hotu iha ninia superfisie.Kubu mak fasil atu konta, maib Tetrahedron no Oktahedron mak

susar uitoan. Atu hetan oin ida nia rea, haree tringulu sira neebami halo iha surat-tahan milmetru nian, hanesan diagrama sira ihasorin. Tringulu hotu nia reamak

maib dala ruma susar atuhetan ninia aas eh altura a.Atu sura, ami uza unidadeN neeb fita ida nia luan.rea sempre ho unidade

kuadradu, nunee: N2.Bele ms halo poliedru sira-nee ho ai-kesak sate no fehuk roupa

baluk kiikoan.

Tetrahedron

*Oin 4, Pontu sikun 4, Ninin 6
http://videos%20compressed2/MVI_4532.wmvhttp://videos%20compressed2/Gabrielson%20Doc10001.jpghttp://videos%20compressed2/Gabrielson%20Doc20001.jpg


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*rea superfisie ho oin:Tetrahedron nia oin haat mak oin ida deit, ho rea neeb hanesandiagrama ida-nee, A=7.5N2

Nunee, ninia rea superfisie mak 4A = 30N2. Konta atu hatene loos.

Kubu (Heksahedron)Forma ida-nee ami la hetan loos, maib ami hetan forma dekorasaunbalu neeb hanesan liman fuan naruk liu fali kubu, no ms ami hetanida diak nuudar mamafatin kuadradu nia ain. Haree foto iha sorinnee.

*Oin 6, Pontu sikun 8, Ninin 12*rea superfisie ho oin:Kubu mak simples liu. Ita bele haree ka-tak ida neeb kiik liu ninia aas makkuadradu 1, luan mak kuadradu 1, narukmak kuadradu 1. Bele ms halo ida ho

aas, luan no naruk kuadradu 2, ka kuadradu 3, ka konforme. (Tetrahedron no Oktahedron ms bele halo boot

ka kiik liu, maib susar atu haree tanba sira-nia estrutura komplikadu. )Kubu mamafatin nia ain iha aas, luan no naruk kuadradu 3. Ninia oin ida nia rea mak 3Nx3N=9N2. Se

karik nia hamriik ketak husi mamafatin, ninia oin mak neen, nunee, ninia rea superfisie sura hanesan nee:6x9N2 = 54N2.
http://videos%20compressed2/MVI_4526.wmvhttp://videos%20compressed2/Gabrielson%20Doc60001.jpg


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Oktahedron(Maski oktahedron nee laos regular, katak nia ninin sira-nia naruklaos ezatamente hanesan, no ninia oin ualu laos ezatamente hanesan,ita sei bele f naran oktahedron ba nia.)

*Oin 8, pontu sikun 6, ninin 12*rea superfisie ho oin: Oktahedron niaoin mak oin rua, ida luan, ida kloot,hanesan foto kraik nee:

Oin Luan (#1) Oin Kloot (#2)

Oin rua nee nia rea hanesan

hatudu iha diagrama rua ihakraik nee.rea Oin Luan (#1) mak4.5N2, no rea Oin Kloot(#2) mak 3.5N2.

(Oin Kloot nia rea susar atu hetan tanba ninia ngulu hotu ladn klaru, no ninia aas (altura) ms susar atu

hetan. Tenke uza trigonometria.)

Oktahedron iha Oin Luan (#1) haat no ms Oin Kloot (#2)haat. Nunee,rea superfisie AS = 4(Oin Luan) + 4(Oin Kloot)AS = 4(4.5) + 4(3.5)AS =18 + 14AS =32 Konta atu hatene loos.
http://videos%20compressed2/Gabrielson%20Doc30001.jpghttp://videos%20compressed2/Gabrielson%20Doc20001.jpg


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Iha Poliedru Platniku rua tan neeb ami la hetan iha formas dekorasaun:Dodekahedron Icosahedron

Oin 12 Oin 20Se hatene homan diak, bele koko took haloforma rua nee.

Bele ms halo poliedru oi-oin, neeb laos Platniku, ho ai-kesak no fehuk roupa, no ms ho homan.Purezemplu, katupa iha foto sorin karuk hanesan ai-kesak poliedru iha foto sorin loos.

ManuHaree manu nia okos. Fatinnee kuadradu, maib fita badiagonl. Bele sura ninia reauza unidade kuadradu. Belems uza maneira nee ho biti,kohe ka okan hodi sura reaboot liu. Bele ms uza surat-tahan kuadradu milimtriku.
http://videos%20compressed2/MVI_4523.wmv


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Hakerek lia diagonl ba kuadradu. Haree foto iha sorin nee, nobuka kuadradu sira-nia rea liuhusi frmula no ms liuhusi kontakuadradu, hanesan iha tabela tuirmai nee:

Bele haree katak maneira rua sei hetan resposta neeb hanesan.

Brinkus, tringulu bokarIda-nee mak homan neeb kiik liu no bziku liu. Beleanaliza hanesan tuirmai nee:*Oin: Konta brinkus nia oin depende ba oins ema haree nia: Se hanehan husi diresaun tolu bele hetan oin tolu. Se hanehan ba pontu rua neeb la meik bele hetan oin

neen maib oin sira-nee mak kiik dala rua fali kuadradu

ida.Nunee, ninia rea superfisie mak kuadradu tolu.

*Pontu: Nia iha pontu lima hanesan tuirmai nee: Tolu neeb meik: fita rua deit hasoru malu Rua neeb ladn meik: fita tolu hasoru malu

Ramus oin idaHoman fita rua ba malu atu hetan ramus hanesan foto iha sorin nee. Bainhirahusik nia tara-an, kuadradu ida-idak neeb mosu hakleuk uitoan. Se marka

sorin ida, bele haree katak ramus ida-nee silu dala 10 ou 11 foin hakleuk too360 graus.

Diagonl hira iha

sorin ida

a

Frmula:

kuadradu

Konta kuadradu:

A=kuadradu + tringulu/2

(tringulu nia rea mak kuadradu nian)

4 A=2x16= 32 kuadradu *(3x4)+(4x3)=24 kuadradu

*(4x4) = 16 tringulu = 8 kuadradu

A=32 kuadradu

3 A=2x9=18 kuadradu *(2x3)+(3x2)=12 kuadradu


A=18 kuadradu

2 (manu nia kidun

iha foto)

A=2x4=8 kuadradu *(1x2)+(2x1)=4 kuadradu


A=8 kuadradu

1 (lia kuadradu

iha manu nia

kidun iha foto)

A=2x1=2 kuadradu *(0x1)+(1x0)=0 kuadradu


A=2 kuadradu

Frmula geral: A=2a2

kuadradu *2((a-1)a)+(a(a-1))=2(a2-a) = 2a

2-2a kuadradu

*(4xa) = 4a tringulu = 2a kuadradu

Hamutuk:

A=(2a2- 2a)+2a = 2a

2kuadradu


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HOMAN BITI, KOHE, NO OKAN

Kuadradu deitHoman hanesan nee uza kuadradu deit. Se halo metin, fita nia naruk tenkehanesan ninia luan iha kuadradu ida nia laran. Estudante sira bele uza kuadradunee hanesan surat-tahan milimtriku atu pinta buat ruma ho kuadradu deit. Ida-nee halimar diak matemtika nian. Ida-nee ms ezemplu ida diak ba tela

[LI: layar] vdeo, hanesan televisaun ka komputadr nian. Tela vdeo ms haloho kuadradu kiikoan, barak tebes, neeb naran piksel. Imajen hotu neebmosu iha tela laran mak halo ho piksel deit, balu lakan, balu nakukun.

Tringulu iha ninin, sikunBele ms hetan tringulu iha ninin, no tringulu rua hamutuk iha si-kun.

MultiplikasaunIha biti nia laran, simples atu esplika multiplikasaun no rea.Purezemplu, foto sorin nee hatudu rektngulu neeb luan 2, naruk 4.Ninia rea mak 8.

Foto sorin nee nia luan 3, naruk 5, rea 15.


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Potnsia 2Se rektngulu nia luan hanesan ho ninia naruk ida-nee naran kuadradu, no beleuza potnsia 2 nian. Iha foto nee, kuadradu kr mean nia rea = 12 = 1 aumenta kuadradu kr kinur sira no bele hetan nia rea = 22 = 4 aumenta kuadradu kr azl lalehan sira no bele hetan nia rea = 32 = 9 aumenta kuadradu kr azl tasi sira no bele hetan nia rea = 42 = 16 no, aumenta kuadradu kr matak sira no bele hetan nia rea = 52 = 25

Lia naturlBele pinta lia tuir fita iha biti laran. Rezultadu mak lia rua neeb ngulu bamalu mak 90. Bele ms pinta lia rua tan diagonl tuir fita nia sikun.


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MATEMTIKA AI-MANULIIN

Jogu naran Ai-manuliin uza matemtika barak. Ida neeb moos liu mak grupuida-idak tenke sura ninia pontu. Baibain ema la hakerek pontu neeb manan,maib sura iha kakutak deit. Labarik ho edade kiik aprende sura husi labarikho idade boot. Iha maneira barak atu hetan pontu, no dala barak la simples idahodi aumenta pontu sira-nee.KOKO TOOK:

*Sura pontu sira-nee, la uza surat-tahan:4+3+5+2+10+1+20+3+4+5+100+8 =2+2+4+15+3+1+3+9+20+3+5+20+12=

Karik iha maneira barak hodi joga Manuliin, tanba ema iha fatin-fatin joga lahanesan. Maib iha maneira pelu menus tolu atu simu pontu bainhira sukit kabaku ai-oan no sukat distnsia entre ai-oan nia monu fatin too iha fatuk rua karai-kuak.1. Uza ai-inan (neeb naruk) hodi sukat. (Bainhira baku direitamente husi fatuk ka rai-kuak.)2. Uza ai-oan (neeb badak) hodi sukat. (Bainhira baku ai-oan neeb soe tiha.)3. Uza ai-inan hodi sukat, depois vezes 100. (Bainhira baku ba leten dala rua depois baku ba oin.)Maneira #1 no #2 mak ezemplu ba unidade espesil. Baibain ita uza unidade hanesan metru ka sentmetru kakilometru hodi sukat distnsia, maib iha jogu nee, ema uza unidade ai-inan ka ai-oan. Bainhira sukatho unidade boot, nmeru rezultadu sai kiik. Nee duni, sukat ho ai-inan atu hetan nmeru kiik. Bainhirasukat ho unidade kiik, nmeru rezultadu sai boot. Nee duni, sukat ho ai-oan atu hetan nmeru boot. Baibainai-inan nia naruk mak boot liu ai-oan dala rua. Nunee, ai-oan nia nmeru rezultadu sei sai boot dala rua faliai-inan nian.

KOKO TOOK:*Foti ai rua, ida naruk no ida badak. Ida naruk ninia naruk ezatamente dala rua ida badak. Sukat buat ruma,hanesan kuartu nia luan, premeiru ho ida naruk, depois ho ida badak. Se halo didiak, nmeru rezultadu ba ida

badak sei sai boot liu dala rua nmeru rezultadu ida naruk nian.Entaun, rezultadu husi #2 sei sai boot liu dala rua rezultadu husi #1. Maib rezultadu husi #3 sei sai boot dala100 rezultadu husi #1. Nunee, se ema bele uza maneira baku #3 hodi baku ai besikdeit, rezultadu sei belesai boot liu. Purezemplu, se ai monu iha distnsia hanesan ai-inan nia naruk ida deit, rezultadu ho maneira #3sei sai hanesan deit ho ai neeb baku distnsia ai-inan naruk 100 ho maneiru #1, ka distnsia ai-oan naruk 50ho maneiru #2.KOKO TOOK:*Iha 100 sentmetru iha metru ida nia laran. Nunee, metru ida nia naruk mak sentmetru nia naruk dala 100.Foti metru fita no sukat meza ida nia naruk ho metru (rezultadu karik purezemplu 2.4metru). Depois sukat hosentmetru (rezultadu iha ezemplu ida-nee mak 240 sentmetru). Rezultadu ho sentmetru mak boot dala 100

fali rezultadu ho metru. Nunee, sukat ho sentmetru hanesan uza maneira #3, sukat ho metru mak hanesanuza maneira #1. (Maneira #2 mak hanesan sukat ho meiu metru, ka 50 sentmetru.)

Iha ms maneira tolu atu hetan pontu bainhira simu ai-oan neeb ema baku semo.1. Simu ho liman rua 10 pontu2. Simu ho liman ida 20 pontu3. Simu depois ema baku dala tolu 100 pontu.

KOKO TOOK:*Ema tenke simu ho maneira #1 dala hira atu hetan pontu hanesan deit ho pontu neeb hetan bainhira simudala ida ho maneira #2?

*Ema tenke simu ho maneira #2 dala hira atu hetan pontu hanesan deit ho pontu neeb hetan bainhira simudala ida ho maneira #3?*Ema tenke simu ho maneira #1 dala hira atu hetan pontu hanesan deit ho pontu neeb hetan bainhira simudala ida ho maneira #3?


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JOGU KARTAUN Bobak (tahan tolu)

Iha jogu naran tahan tolu, ema simu kartaun tolu no halo komparasaunho ema seluk atu haree se mak manan. Diak liu mak hetan 2,2,2.

Bele hakerek probabilidade [LI: kemungkinan] nuudar frasaun neebnmeru iha leten mak posibilidade neeb ita hakarak, no nmeru ihakraik mak posibilidade hotu.

Atu sura probabilidade hetan 2,2,2 tenke sura probabilidade ba hetan 2 ba kartaun primeiru (3/3), vezesprobabilidade hetan 2 ba kartaun segundu (2/2), vezes probibilidade hetan 2 ba kartaun terseiru (1/1). Atuhalo ezemplu simples, hanoin katak ema ida deit mak joga, no iha kartaun tolu deit: 2,2,2

Iha ezemplu nee, ema sempre hili 2,2,2, nunee probabilidade mak 100%.Se karik iha kartaun haat 2,2,2,3 no ita hili kartaun tolu, probabilidade atu hetan 2,2,2 sei sai hanesan tuir-mai nee:

Se karik iha kartaun sanulu 2,2,2,2,A,A,A,A,K,K no hili tolu, probabilidade atu hetan 2,2,2 sei sai hanesantuirmai nee:

Se karik iha kartaun 52 hamutuk, ho 2" haat hanesan deit realidade, no hili nafatin tolu, entaun probabili-dade atu hetan 2,2,2 sei sai hanesan tuirmai nee:

Ida-nee ema hotu hatene: baibain ema la foti 2,2,2 buat nee susar liu atu mosu. Se Ita haree buat neemosu, karik ema joga foer.

Bainhira ema nain rua ka liutan mak joga, rezultadu sei sai hanesan deit, tanba frmula ida-nee ba ema idaatu simu kartaun 2,2,2. Ema seluk nia kartaun la halo influnsia ba Ita nian. Ms, ema seluk nia probabili-dade atu simu 2,2,2 mak hanesan deit Ita nian.

Koko took:*Kalkula probabilidade atu simu kartaun 2,2,2 se uza kartaun balu deit: husi A too 8; entaun kartaun 32deit.


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*Nee duni, 2,2,2, ladn mosu, maib valr selukmosu beibeik. Purezemplu, ita bele haree maneirahotu hodi hetan valr sia:1. A,A,72. A,2,63. A,3,54. A,4,45. 2,2,5

6. 2,3,47. A,8,(10, J, Q, ka K)8. 2,7, (10, J, Q, ka K)9. 3, 6, (10, J, Q, ka K)10. 4,5, (10, J, Q, ka K)11. 9, (10, J, Q, ka K), (10, J, Q, ka K)12. 2,9,813. 3,7,914. 3,8,815. 4,6,916. 4,7,8

17. 5,9,518. 5,8,619. 5,7,720. 6,6,721. 9,9,A(3,3,3 la konta sia maib valr aas liu.)Hamutuk iha maneira oin 21 atu hetan sia.

Atu hetan valr ualu:1. A,A,62. A,2,53. A,3,44. 2,2,45. 2,3,36. A,7,(10, J, Q, ka K)7. 2,6,(10, J, Q, ka K)

8. 3,5,(10, J, Q, ka K)9. 4,4,(10, J, Q, ka K)10. 8,(10, J, Q, ka K), (10, J, Q, ka K)11. 9,8,A12. 9,7,213. 9,6,314. 9,5,415. 8,8,216. 8,7,317. 8,6,418. 8,5,5

19. 7,7,420. 7,6,521. 9,9,(10, J, Q, ka K)(6,6,6 la konta ualu maib valr aas liu)Hamutuk iha maneira oin 21 atu hetan ualu.

Atu hetan hitu:1. A,A,52. A,2,43. A,3,34. 2,3,25. A,6,(10, J, Q, ka K)6. 2,5,(10, J, Q, ka K)7. 3,4,(10, J, Q, ka K)8. 7,(10, J, Q, ka K),(10, J, Q, ka K)9. 9,A,710. 9,2,611. 9,3,512. 9,4,413. 8,A,8

14. 8,2,715. 8,3,616. 8,4,517. 7,3,718. 7,4,619. 7,5,520. 6,5,621. 9,8,(10, J, Q, ka K)(9,9,9 la konta hitu maib valr aas liu)Hamutuk iha maneira oin 21 atu hetan hitu.

Atu hetan neen:1. A,A,42. A,2,33. A,5,(10, J, Q, ka K)4. 2,4,(10, J, Q, ka K)5. 3,3,(10, J, Q, ka K)6. 6,(10, J, Q, ka K),(10, J, Q, ka K)7. 9,A,68. 9,2,59. 9,3,410. 8,A,711. 8,2,612. 8,3,513. 8,4,4

14. 7,2,715. 7,3,616. 7,4,517. 6,4,618. 6,5,519. 8,8,(10, J, Q, ka K)20. 7,9,(10, J, Q, ka K)21. 9,9,8(2,2,2 la konta neen maib valr aas liu)Hamutuk iha maneira oin 21 atu hetan neen.


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Maneira atu hetan zero ms barak.:1. A,A, 82. A, 2, 73. A, 3, 64. A, 4, 55. 2, 2, 66. 2, 3, 57. 2, 4, 4

8. 3, 3, 49. A, 9, (10, J, Q, ka K)10. 2, 8, (10, J, Q, ka K)11. 3, 7, (10, J, Q, ka K)12. 4, 6, (10, J, Q, ka K)13. 5, 5, (10, J, Q, ka K)14. 2, 9, 915. 3, 9, 816. 4, 9, 717. 4, 8, 818. 5, 9, 6

19. 5, 8, 720. 6, 8, 621. 6, 7, 7((10, J, Q, ka K) dala tolu la konta 0 maib valr

aas liu.)Hamutuk iha maneira oin 21 atu hetan 0.

Nunee, iha posibilidade no probabilidade hanesan atuhetan valr ida-idak. Ida-nee hanesan deit se foti kar-taun ida arbiru: Probabilidade atu besik hanesan atusimu naran kartu ida.(DIAK LIU, LABELE HATUDU TABELA SIRA-NEE HO INFORMASAUN HOTU BA ESTU-DANTE SIRA, MAIB HARUKA SIRA BUKA RA-SIK.)

Koko took:*Buka maneira hira atu hetan 5,4,3,2, no 1.

*Bainhira joga Bobak haree took valr ida neeb makmosu beibeik


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Konke (tahan hitu)Konseitu probabilidade [LI: kemungkinan] ms tama beibeik iha joga konke.Tenke hanoin beibeik probabilidade ba kartaun ruma neeb atu mosu, no bain-hira soe kartaun, sempre soe ida neeb ho probabilidade kiik atu hetan niniamaluk (par.)

Purezemplu, se Ita iha kartaun hanesan tuirmai nee: 3, 4, 8, 8, 2, 2,K, no foin dada K, agora presiza soe fali ida. Kartaun hotu iha posibilidade,maib diak liu, soe ida neeb mak probabilidade kiik liu. Atu hetan kartauntahan tolu neeb bele hatun iha meza, iha possibilidade oin haat:#1: 4, 5 presiza deit 3, ka 6#2: 8, 8 presiza deit 8, ka 8#3: 2, 2 presiza deit 2, ka 2#4: K, K, presiza deit K, ka KNunee, probabilidade atu hetan kartaun tolu neeb bele hatun mak hanesandeit ba posibilidade ida-idak. Ema bele hili naran ida deit atu soe. Ho lian matemtika bele hakerek prob-abilidade ba posibilidade ida-idak #1 - #4 hanesan tuirmai nee:

Ida-nee mak rezultadu se karik ita foin hahu halimar, nee duni, Ita hatene kartaun ualu, maib seluk Ita sei-dauk hatene. Ema seluk ms kaer kartaun balu, maib tanba Ita la hatene sira-nian, labele konta.

Maib se karik ita joga ona tempu uitoan, no kartaun balu mak soe tiha ona iha meza, probabilidade mstroka tiha. Purezemplu, se karik kartaun ualu soe tiha ona iha meza, hotu neeb ita la bele uza, tenke hasaisira husi frasaun nia kraik, hanesan nee:

Nunee, diak uitoan. Maib probabilidade sei nafatin hanesan deit ba posibilidade hotu. Entaun, bele soenaran ida deit.

Maib se karik ema seluk soe tiha ida neeb diak, no ita la bele foti, ida-nee bele hamenus probabili-

dade. Purezemplu, se karik entre kartaun ualu neeb soe tiha ona iha meza inklui K, maib Ita labele foti,agora probabilidade atu hetan K tolu (posibilidade #4) tun:

Nunee, lalika hein tan ba K. Probabilidade atu hetan sira seluk mak diak liu. Nunee, bele soe K no K.Frmula sira-nee mak probabilidade atu hetan kartaun tolu neeb bele hatun ba posibilidade #1 - #4. Atu

hatene probabilidade atu hetan naran ida neeb bele hatun, tenke soma probabilidade haat nee hamutuk. Baezemplu primeiru, frmula sei sai hanesan tuirmai nee:

Nunee, probabilidade diak atu hetan buat ruma neeb bele hatun.Se karik ita kaer kartaun hanesan nee: 8, 8, 4, 5, A, 3, K, probabilidade ladn diak. Iha posi-

bilidade tolu deit:#1: 8, 8 presiza 8 ka 8#2: 4, 5 presiza 3 ka 6#3: A, 3 presiza 2Hamutuk, se Ita foin hahu joga, no ema seidauk soe kartaun iha meza, probabilidade katak possibilidade idahusi sira iha leten bele sai mak hanesan tuirmai nee:

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