Hangaia Ki Te Māti

Hei Whakarāpopoto

Ko te kaupapa nei, he whakamahi māti hei hanga i ētahi tauira, he whakaroa i ngā tauira, he tirotiro hoki i ngā pānga kei roto i ngā tauira māti.

Achievement Objectives
6. Ka whakamahi rautaki tāpiripiri ki te whiriwhiri i tētahi tau o roto i tētahi tauira tau.
7. Ka whakamahi rautaki āhuahanga ki te whiriwhiri i tētahi āhua o roto i tētahi tauira āhua.
8. Ka whakamahi rautaki taurangi māmā ki te whiriwhiri i tētahi huānga o roto i tētahi tauira raupapa.
Te Hononga ki te Marautanga
Taumata 3
 Te Tau me te Taurangi (Te Tauira me te Pānga)
Whāinga Paetae 6:
Ka whakamahi rautaki tāpiripiri ki te whiriwhiri i tētahi tau o roto i tētahi tauira tau.
 
Whāinga Paetae 7:
Ka whakamahi rautaki āhuahanga ki te whiriwhiri i tētahi āhua o roto i tētahi tauira āhua.
 
Whāinga Paetae 8:
Ka whakamahi rautaki taurangi māmā ki te whiriwhiri i tētahi huānga o roto i tētahi tauira raupapa.
Ngā Whāinga Ako
Kia mōhio te ākonga ki te:
  • tautohu tauira āhuahanga
  • whakaroa i tētahi tauira āhuahanga
  • kimi mai i te ture hei whiriwhiri i te maha o ngā māti kei roto i tētahi wāhanga o te tauira māti
  • whiriwhiri i te wāhanga o te tauira e hāngai ana ki tētahi rahinga māti

whakamārama i te tauira me tōna ture

Ngā Rauemi
he māti
He Rārangi Kupu
taurangi algebra, variable
tauira mokowā spatial pattern
pānga relationship
Hei Raupapa I Ngā Mahi Ako
  1. Ko te mahi tuatahi, he hanga i tētahi tauira tapatoru ki te māti, he tirotiro i te pānga o te tapatoru ki te maha o ngā māti.

Te mahi a te pouako He tauira kōrero mā te pouako
Whakamahia he māti hei hanga i te tauira tapatoru nei. Anei tētahi tauira tapatoru. He tauira raupapa. Kua tapaina ia wāhanga o te raupapa ki ngā ingoa nei ‘Tapatoru-1’ ‘Tapatoru-2’ …
diagram.
Tonoa ngā ākonga ki te hanga i te wāhanga o te tauira e whai muri ana. Whakawhitiwhiti kōrero mō te maha o ngā māti hei hanga i ia wāhanga o te tauira raupapa. Hangaia te tauira tapatoru-4.
Hangaia te tauira tapatoru-5.
Tuhia ngā tauira nei ki ā koutou pukapuka. Tuhia te maha o ngā māti hei hanga i ia wāhanga o te tauira ki raro i ngā pikitia.
diagram.
Tukuna ngā ākonga ki te mahi takirua ki te kimi i te maha o ngā māti hei hanga i te wāhanga o te tauira ‘tapatoru-20’. He aha tētahi rautaki hei kimi mai i te maha o ngā māti hei hanga i te tapatoru-20?
He mahi nui te hanga i ngā tauira ki te māti, he tuhi rānei ki te pukapuka, nō reira, me āta kimi tētahi huarahi poto.
Me mahi takirua kōrua ko tētahi hoa ki te kimi mai i tētahi huarahi poto.
Tonoa nga akonga ki te whakamārama i ā ratou rautaki, ka whakawhitiwhiti kōrero ai.
Anei tētahi o ngā rautaki:
He māmā te kite atu i te tauira me te whiriwhiri i te maha o ngā māti mēnā ka pēnei te whakarite:
pikitia wāhanga o te
tauira
maha o ngā
māti
diagram. tapatoru-1 1 + (2 x 1)
diagram. tapatoru-2 1 + (2 x 2)
diagram. tapatoru-3 1 + (2 x 3)
  tapatoru-4 1 + (2 x 4)
Mō te tapatoru-20, ka whakareatia te 2 ki te 20, kātahi ka tāpiri i te 1. Arā, ko te 41.
Anei anō tētahi o ngā rautaki: He māmā anō hoki te kite atu i te tauira me te whiriwhiri i te maha o ngā māti mēnā ka pēnei te whakarite:
pikitia wāhanga o
te tauira
maha o ngā māti
diagram. tapatoru-1 3
diagram. tapatoru-2 3 + (2 x 1)
diagram. tapatoru-3 3 + (2 x 2)
  tapatoru-4 3 + (2 x 3)
Mō te tapatoru-20, ka whakareatia te 2 ki te 19, kātahi ka tāpiri i te 3. Arā, ko te 41.
Hoatu ētahi pātai pēnei i ēnei. Whakamahia te rautaki tuatahi ki te whiriwhiri i ēnei pātai:
E hia ngā māti hei hanga i te tapatoru-21
E hia ngā māti hei hanga i te tapatoru-10
E hia ngā māti hei hanga i te tapatoru-50
E hia ngā māti hei hanga i te tapatoru-100 …
Whakamahia te rautaki tuarua ki te whiriwhiri i ēnei pātai:
E hia ngā māti hei hanga i te tapatoru-22
E hia ngā māti hei hanga i te tapatoru-15
E hia ngā māti hei hanga i te tapatoru-30
E hia ngā māti hei hanga i te tapatoru-1000 …
Tonoa ngā ākonga ki te tuhi i tētahi tūtohi hei whakaatu i te tapatoru me te maha o ngā māti. Anei te tūtohi hei whakaoti mā koutou:
tapatoru 1 2 5 10 25 67 83 146 328
māti                  
Whakamōhiotia atu te maha o ngā māti, ka tono ai i ngā ākonga ki te whiriwhiri i te maha o ngā tapatoru ka taea te hanga.

39 ngā māti a Hinewai. E hia ngā tapatoru ka taea e ia te hanga?
He mahi nui ki te hanga i ngā tapatoru ki te whiriwhiri i tēnei rapanga. Āta kimihia e kōrua ko tō hoa tētahi huarahi poto/māmā.
He Tauira Whakautu: Mēnā ka hanga i te tapatoru tuatahi, e toru ēnā māti, 36 kei te toe mai:
diagram. E rua atu anō ngā māti hei hanga i ia tapatoru:


.....diagram.

Nō reira 18 atu anō ngā tapatoru ka taea te hanga ki ēnei māti 36. 19 katoa ngā tapatoru.

Hoatu ētahi anō rapanga pēnei hei whakaoti mā ngā ākonga. 55 ngā māti a Hinewai. E hia ngā tapatoru ka taea e ia te hanga?
243 ngā māti a Hinewai. E hia ngā tapatoru ka taea e ia te hanga?
73 ngā māti a Hinewai. E hia ngā tapatoru ka taea e ia te hanga?
405 ngā māti a Hinewai. E hia ngā tapatoru ka taea e ia te hanga?
  1. Ko te mahi tuarua, he hanga i tētahi atu tauira ki te māti, he tirotiro i te pānga o te tauira ki te maha o ngā māti.

Te mahi a te pouako He tauira kōrero mā te pouako
Whakaaturia tētahi tauira māti ki ngā ākonga, ka tono ai i a rātou ki te āta whakaaro i tētahi huarahi māmā ki te tatau i te maha o ngā māti.

Titiro ki tēnei tauira kua hangaia ki te māti. Me mahi takirua kōrua ko tētahi hoa, ka āta whakaaro i tētahi rautaki hei tatau i te maha o ngā māti. Kaua e tatau takitahi i ia māti, engari ka āta kimi i tētahi huarahi poto, māmā hei tatau i te maha o ngā māti.
diagram.  

He tauira whakautu

  1. Tatauria te maha o ngā tapawhā rite. E 4 ngā māti hei hanga i te tapawhā rite tuatahi, e 3 ngā māti mō ia tapawhā rite i muri mai. E 2 ngā māti hei tāpiri atu mā ia tapatoru.

    diagram.
  2. E 3 ngā māti hei hanga i te tapatoru tuatahi. Tatauria te maha o nga tapawhā rite. E 3 ngā māti mā ia tapawhā rite. E 2 atu anō mā te tapatoru whakamutunga.

    diagram.
Tonoa ngā ākonga ki te whakamahi i ā rātou rautaki ki te whakaoti i te tūtohi.

Anei anō ētahi tauira:
diagram.  

Whakamahia tētahi o ngā rautaki tatau ki te whakaoti i te tūtohi nei:
 

te maha o ngā tapawhā rite 7 5 3 2 1 4
te maha o ngā māti            

Ko te mahi mō te roanga atu o te wiki, he tirotiro i ngā pānga o ētahi atu tauira māti. Ka whāia te raupapa mai o ngā mahi i runga nei. Koia nei ētahi (me tono hoki mā ngā ākonga e hanga ētahi tauira māti):

He tauira tapawhā rite:

diagram. diagram. diagram.

Ngā ture:

  • Whakareatia te nama o te tapawhā rite ki te 3, ka tapiri ai i te 1. Ka hua mai ko te maha o ngā māti

tapawhā rite 1 2 3 5 10 15 100
ture 1 + (3 x 1) 1 + (3 x 2) 1 + (3 x 3) 1 + (3 x 5) 1 + (3 x 10) 1 + (3 x 15) 1 + (3 x 100)
māti 4 7 10 16 31 46 301
  • Tangohia te 1 i te nama o te tapawhā rite, ka whakarea ai ki te 3. Kātahi ka tāpiri i te 4, ka hua mai ko te maha o ngā māti.

tapawhā rite 1 2 3 5 10 15 100
ture 4 + (3 x 0) 4 + (3 x 1) 4 + (3 x 2) 4 + (3 x 4) 4 + (3 x 9) 4 + (3 x 14) 4 + (3 x 99)
māti 4 7 10 16 31 46 301

He tauira whare:

diagram. diagram. diagram.

Ngā ture:

  • Whakareatia te nama o te whare ki te 5, ka tapiri ai i te 1. Ka hua mai ko te maha o ngā māti

tapawhā rite 1 2 3 5 10 15 100
ture 1 + (5 x 1) 1 + (5 x 2) 1 + (5 x 3) 1 + (5 x 5) 1 + (5 x 10) 1 + (5 x 15) 1 + (5 x 100)
Māti 6 11 16 26 51 76 501
  • Tangohia te 1 i te nama o te whare, ka whakarea ai ki te 5. Kātahi ka tāpiri i te 6, ka hua mai ko te maha o ngā māti.

tapawhā rite 1 2 3 5 10 15 100
ture 6 + (5 x 0) 6 + (6 x 1) 6 + (5 x 2) 6 + (5 x 4) 6 + (5 x 9) 6 + (5 x 14) 6 + (5 x 99)
māti 6 11 16 26 51 76 501

He āhua uaua ake tēnei:

He tauira ara whakapiki

diagram. diagram. diagram. diagram.

Ngā ture:

  • Whiriwhiria te maha o ngā tapawhā i te tuatahi: Whakareatia te nama o te ara whakapiki ki te 2, ka tango ai i te 1.
  • Kātahi ka whiriwhiri i te maha o ngā māti: Whakareatia te maha o ngā tapawhā rite ki te 3, ka tāpiri ai i te 1.

ara whakapiki 1 2 3 4 10 15 100
tapawhā rite 1 3 5 7 19 29 199
ture (mō ngā tapawhā rite) (2 x 1) – 1 (2 x 2) – 1 (2 x 3) – 1 (2 x 4) – 1 (2 x 10) – 1 (2 x 15) – 1 (2 x 100) – 1
māti 4 10 16 22 58 88 598
ture (mō ngā māti) 1 + (3 x 1) 1 + (3 x 3) 1 + (3 x 5) 1 + (3 x 7) 1 + (3 x 19) 1 + (3 x 29) 1 + (3 x 199)
  • Whiriwhiria te maha o ngā tapawhā i te tuatahi: Whakareatia te nama o te ara whakapiki ki te 2, ka tango ai i te 1.
  • Kātahi ka whiriwhiri i te maha o ngā māti: Whakareatia te maha o ngā tapawhā rite ki te 3, ka tāpiri ai i te 1.

ara whakapiki 1 2 3 4 10 15 100
tapawhā rite 1 3 5 7 19 29 199
ture (mō ngā tapawhā rite) (2 x 1) – 1 (2 x 2) – 1 (2 x 3) – 1 (2 x 4) – 1 (2 x 10) – 1 (2 x 15) – 1 (2 x 100) – 1
māti 4 10 16 22 58 88 598
ture (mō ngā māti) 4 + (3 x 0) 4 + (3 x 2) 4 + (3 x 4) 4 + (3 x 6) 4 + (3 x 18) 4 + (3 x 28) 4 + (3 x 198)

Hei Whakawhānui:

Tuhia he whārite hei whakaata i te ture o te pānga. Hei tauira:

5w + 1 = m (w = te maha o ngā whare, m = te maha o ngā māti)
Whakareatia te nama o te whare ki te 5, ka tapiri ai i te 1. Ka hua mai ko te maha o ngā māti.


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