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Hei Whakarāpopoto
Ko te kaupapa nei, he tirotiro i ngā tauira o roto i te tapatoru a Pascal, me tētahi rautaki hei tātai i te tapeke o ētahi tau piri tata.
Achievement Objectives
8. Ka whakaatu pānga rārangi: mā te whārite whai taurangi (ka whakaoti hoki i te whārite); mā te ture; mā te kauwhata.
AO elaboration and other teaching resources
Te Hononga ki te Marautanga
Taumata 4
Te Tau me te Taurangi (Te Tauira me te Pānga)
Whāinga Paetae 8:
Ka whakaatu pānga rārangi:
Ngā Whāinga Ako
Kia mōhio te ākonga ki te:
  • kimi mai i ngā tauira i tētahi raupapatanga tau
  • whakamahi tauira ki te whakaroa i tētahi raupapatanga tau
  • whakamārama i ngā tauira i tētahi raupapatanga tau
  • kimi mai i te huarahi māmā hei tātai i te tapeke o ētahi tau piri tata
  • whakamārama i te huarahi māmā hei tātai i te tapeke o ētahi tau piri tata
Ngā Rauemi

Whārangi Tārua 1

Whārangi Tārua 2

Whārangi Mahi 1

He Rārangi Kupu
hauroki diagonal
pānga taurangi algebraeic relationship
pūrua the power of two
tau piri tata consecutive numbers
taupū exponent, power
taurea multiple
tūtohi table
Hei Raupapa I Ngā Mahi Ako
  1. Ko te mahi tuatahi, he hanga, he tūhura i te tapatoru a Pascal.

 

Hei Mahi mā te Pouako He Tauira Kōrero mā te Pouako
Hoatu te Whārangi Tārua 1 (PDF, 133KB) ki ngā ākonga. Aratakina rātou ki te kimi i ngā tau o te tapatoru a Pascal. Titiro ki te whakaahua i ā koutou pepa. Ko tā tātau mahi, he hanga i tētahi tauira tau.
E hia ngā ara mai i te tīmatanga kia tae atu ki te porowhita ‘A’? Kotahi anake.
Nō reira, me tuhi atu te 1 ki taua porowhita.
E hia ngā ara mai i te tīmatanga ki te porowhita ‘E’? Kotahi anake.
Nō reira me tuhi atu te 1 ki taua porowhita.
E hia ngā ara rerekē mai i te tīmatanga kia tae atu ki te porowhita ‘I’. E rua. Ko tētahi huarahi mā te porowhita ‘A’, ko tētahi mā te porowhita ‘E’.
Nō reira, me tuhi atu te 2 ki te porowhita ‘I’
Kia pērā tonu te mahi kia tuhia he tau ki ngā porowhita katoa. Tuhia ngā tau ki te tapatoru porowhita i raro o te wharangi. Kua oti kē ētahi o ngā tau te tuhi atu. Titiro ki te tapatoru porowhita tuarua i ā koutou pepa. Kua tuhia te 1 ki te porowhita tīmatanga. Kua tuhia hoki he tau ki ētahi atu o ngā porowhita. Māu e tuhi te tau e tika ana ki ngā porowhita e wātea tonu ana.
Ko tāu mahi, he āta kimi i te maha o ngā ara rerekē mai i te porowhita tīmatanga ki ērā atu o ngā porowhita katoa.

 

Ka mahi takirua ngā ākonga ki te kimi mai i ngā tauira e kitea mai ana i te tapatoru porowhita nei. He nui ngā tauira e kitea mai ana i ngā tau i te tapatoru porowhita kua oti nei i a koutou.
Anei tētahi. Pānuihia mai ngā tau o te rārangi tuarima, mai i te taha mauī. 1, 4, 6, 4, 1.
Pānuihia mai i te taha matau. 1, 4, 6, 4, 1
He ōrite ngā tau, ahakoa ka pānuihia mai i te taha mauī, mai i te taha matau rānei.
Koirā tētahi o ngā tauira. Mā kōrua ko tō hoa e kimi ētahi atu tauira, kātahi ka whakawhitiwhiti kōrero tātou.
  1. I konei ka tūhura tonu i ngā tauira tau o te tapatoru a Pascal.

 

Hei Mahi mā te Pouako He Tauira Kōrero mā te Pouako
Hoatu te Whārangi Tārua 2 (PDF, 63KB), ka tono ai i ngā ākonga ki te whakamahi i ngā tauira ki te whakaoti i ngā rārangi katoa.
Whakawhitiwhiti kōrero ki ngā takirua, ki te akomanga katoa mō ngā tauira tau. Tonoa rātou ki te whakamārama i ngā tauira i kitea, i whakamahia.		 Titiro ki tēnei tapatoru porowhita. Tekau ngā rārangi porowhita. Ka mahi takirua anō kōrua ko tō hoa ki te tuhi i ngā tau e tika ana ki ia porowhita.
Āta tirohia, āta whakamahia ngā tauira kua kitea e kōrua ko to hoa, kua whakawhitiwhiti kōrero tātou.

diagram.

Anei ētahi o ngā tauira hei whakawhiti kōrero me ngā ākonga.

  1. He ‘1’ kei ngā rārangi hauroki e rua ki ia taha o te tapatoru.
  2. He tatau ā-tahi ngā rārangi hauroki tuarua kei ia taha o te tapatoru.

diagram.

  1. E pēnei ana te raupapa o ngā tau kei ngā rārangi hauroki tuatoru:
    1 +2 3 +3 6 + 4 10 + 5 15 + 6 21 + 7 28 + 8 36 ….
  1. E pēnei ana ngā taurea o te 3:

    diagram.

    He takitoru te puta mai o ēnei taurea o te 3. Arā:

    Ki te tāpirihia ngā tau e rua i te rārangi runga, ka rite ki te tau i te rārangi raro.
     

  2. E pēnei ana ngā taurea o te 5:

     

    He takitoru anō te puta mai o ēnei taurea o te 5. Arā:

    Ki te tāpirihia ngā tau e rua i te rārangi runga, ka rite ki te tau i te rārangi raro.

Tonoa nga ākonga ki te whakamahi tonu i ngā tauira ki te tuhi i ētahi rārangi e rua anō ki te tapatoru porowhita.

diagram.

  1. Ka haere tonu te mahi tūhura i ngā tauira o te tapatoru a Pascal.

 

Hei Mahi mā te Pouako He Tauira Kōrero mā te Pouako
Tuhia ngā rārangi e 6 o te tapatoru a Pascal ki te papa tuhituhi, ka tīmata ai ki te tāpiri i ia rārangi:
 Anei ngā rārangi e 6 o te tapatoru a Pascal. Kua tuhia te tapeke o ia rārangi ki te taha.
He aha te tapeke o te rārangi tuawhā? Ko te 8 (1 + 3 + 3 + 1 = 8).
He aha te tapeke o te rārangi tuarima? Ko te 16 (1 + 4 + 6 + 4 + 1 = 16).
Tuhia he tūtohi hei whakaatu i te rārangi me te tapeke o taua rārangi. Hei tauira:
te rārangi 1 2 3 4 5 6
te tapeke 1 2 4 8 16 32
Whakawhitiwhiti kōrero mō te tauira e kitea mai ana i te tūtohi.		 Tā tātou mahi, he whakaatu i ēnei tapeke ki tētahi tūtohi. Kia rua ngā rārangi o te tūtohi. Whakaaturia te rārangi o te tapatoru, arā, rārangi 1, 2, 3, 4, haere ake ki te 6.
Whakaaturia hoki te tapeke o ngā tau o taua rārangi.
Āta tirohia te raupapa mai o ngā tapeke. Arā, 1, 2, 4, 8, 16 …
He aha te tauira? Rearuatia te tau i mua. Rearuatia te 1, ka 2. Rearuatia te 2, ka 4. Rearuatia te 4, ka 8. Ka pēnā tonu te haere o te raupapa.
Tonoa ngā ākonga ki te whiriwhiri i te tapeke o ngā rārangi 7 ki te 10. Ka tuhia ēnei tapeke ki te tūtohi. Whiriwhiria te tapeke o te rārangi tuawhitu. Ko te 64.
He aha te mahi i puta ai te 64? Ka whai tonu i te tauira. Arā, ka rearuatia te tau o mua. Rearuatia te 32, ka 64.
Tuhia te tapeke o ngā rārangi 7 ki te 10 ki te tūtohi, kātahi ka hoki anō ki te tapatoru ki te tāpiri i ngā tau kia kitea ai mēnā e tika ana, kāore rānei.
rārangi 7 8 9 10
tapeke 64 128 256 512
Whakawhitiwhiti kōrero mō te kimi i te te tapeke o te rārangi nama 100.
Whakamāramatia te pānga o te nama o te rārangi me te tapeke o taua rārangi.		 Me pēhea te kimi i te tapeke o te rārangi nama kotahi rau? He mahi nui ki te whakaroa ake i te tūtohi mai i te rārangi nama 10 ki te rārangi 100.
Āe, he mahi nui, nō reira me kimi tētahi atu huarahi. Ko te mahi nui, he kimi i te pānga o te nama o te rārangi me te tapeke o taua rārangi.
Anei te tīmatanga o te pānga.
rārangi tapeke Whakamārama
1 1 20 = 1
2 2 21 = 2
3 4 4 = 2 x 2 = 22
4 8 8 = 2 x 2 x 2 = 23
5 16 16 = 2 x 2 x 2 x 2 = 24
6 32 32 = 2 x 2 x 2 x 2 x 2 = 25
He aha te tauira o te pānga e kitea mai ana? Ko te nama o te rārangi i mua, koirā te taupū o te rua hei whiriwhiri i te tapeke. Mō te rārangi tuawhitu, ko te taupū 6 o te 2 te tapeke. Mō te rārangi tuawaru, ko te taupū 7 o te 2 te tapeke.
Nō reira, me pēhea te tātai i te tapeke o te rārangi 100? Ko te taupū 99 o te 2. Arā, 299. Whakamahia te tātaitai. He tau tino nui rawa atu!
  1. Ko te mahi i konei, he tirotiro i tētahi atu pānga taurangi. Mā te whakamahi i tēnei pānga e tātaihia ai te tapeke o ētahi tau piri tata.

 

Hei Mahi mā te Pouako He Tauira Kōrero mā te Pouako
Whakaaturia ētahi tau piri tata e 3 ki tētahi tūtohi, me te tapeke ki te taha.
Whakawhitiwhiti kōrero mō te tātai i te tapeke o ia huinga tau.		 Titiro ki te tūtohi nei. E 3 ngā tau piri tata e whakaaturia ana ki ia rārangi. Ko te tapeke o aua tau kei te taha matau.
Ngā Tau Piri Tata Te Tapeke
1, 2, 3 6
3, 4, 5 12
9, 10, 11 30
Me mahi takirua koutou. Tuhia ētahi atu huinga o ētahi tau piri tata e toru. Tuhia te tapeke o aua tau ki te taha matau.
Āta kimihia tētahi huarahi māmā hei tātai i te tapeke o ia huinga tau. Mēnā ka whakareatia te tau o waenganui ki te 3, ka hua mai ko te tapeke o aua tau.
Whakamātauria te huarahi māmā ki te whakaoti i te tūtohi nei:
Ngā Tau Piri Tata Te Tapeke
8, 9, 10  
19, 20, 21  
24, 25, 26  
29, 30, 31  
99, 100, 101  
999, 1000, 1001  
 
  1. Hei Whakawhānui:
  • Tuhia he whārite hei tātai i te tapeke o ngā huinga tau o te Whārangi Mahi 1.(PDF, 81KB)  Hei tauira: t = 3w hei tātai i te tapeke o ētahi tau piri tata e 3 (w = tau waenganui, t = tapeke)
  • Tirohia ētahi atu pānga taurangi, ka kimi ai i te ture e hono ana i ētahi huinga tau e rua. Hei tauira:
    1 2 3 4 5 6
    1 4 6 8 10 12

    ture: whakareatia ki te 2

    1 2 3 4 5 6
    1 5 7 9 11 13

    ture: whakareatia ki te 2 ka tāpiri ai i te 1

    1 2 3 4 5 6
    4 5 6 7 8 9

    ture: tāpirihia te 3

    1 2 3 4 5 6
    1 4 9 16 25 36

    ture: pūruatia

    1 2 3 4 5 6
    1 5 10 17 26 37

    ture: pūruatia ka tāpiri ai i te 1

  • Rangahaua ngā kōrero mō te tangata nei a Pascal.
Attachments
HePangaTaurangiWM1.pdf81.43 KB
HePangaTaurangiWT1.pdf132.98 KB
HePangaTaurangiWT2.pdf63.48 KB
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Taumata 4