He Pānga Riterite

Hei Whakarāpopoto
Ko tā te kōwae ako nei he tūhura i ngā rapanga, pānga riterite, ngā rautaki hei whakaoti i ēnei momo rapanga, me te reo e hāngai ana.
Te Hononga ki te Marautanga
Taumata 4
Te Tau me te Taurangi (Te Mātauranga Tau me ngā Rautaki Tau)
Whāinga Paetae 5:
Ka kōwhiri, ka whakamārama i te rautaki e tino whaihua ana hei whakaoti rapanga e whai wāhi mai ana te tauoti, te hautau, te tau ā-ira, te ōrau, te ōwehenga, te tau tōpū me te taupū māmā:
  • rautaki tatau;
  • rautaki tāpiripiri;
  • rautaki whakarea;
  • rautaki pānga riterite māmā.
 
Whāinga Paetae 6:
Ka mārama ki te whaitakenga o te otinga o tētahi rapanga.
Ngā Whāinga Ako

Kia mōhio te ākonga:

  • ki te whakatairite ōwehenga
  • ki te whakaoti rapanga e whai wāhi mai ana te ōwehenga
  • ki ngā rautaki whakarea hei whakaoti rapanga pānga riterite
Ngā Rauemi
Whārangi Tārua 1

Whārangi Tārua 2

Whārangi Tārua 3

He Rārangi Kupu

kāneihana
carnation
ōwehenga
ratio
parani
daisy
rapanga pānga riterite
proportional problems
rautaki whaihua
useful strategy
variable
tuaina
string
table (of data)
Hei Raupapa I Ngā Mahi Ako
  1. Ko te mahi tuatahi, he tirotiro i te rerekē o te whakaaro tāpiripiri me te whakaaro pānga riterite (whakarea).
    Ngā tohutohu He Tauira Kōrero Mā Te Pouako
    Tuhia te rapanga nei ki te papa tuhituhi, ka pānui tahi ai me ngā ākonga.

    E rua ngā putiputi e tipu ana, he kāneihana tētahi, he parani tētahi. Ka inea e Haki te teitei o ēnei putiputi. E 40 cm te teitei o te kāneihana, e 20 cm te teitei o te parani. I te paunga o te rua wiki, ka inea anō ngā putiputi nei. E 60cm te teitei o te kāneihana, e 40 cm te parani. Ko tēhea te putiputi he rahi rawa te tipu i roto i te rua wiki nei?

    Whakawhitiwhiti kōrero mō te whakaatu i ngā kōrero matua i tēnei rapanga ki tētahi tūtohi.
    He aha ngā kōrero matua i roto i te rapanga nei? Ko ngā inenga a Haki i te teitei o ngā putiputi.

    Me pēhea te whakaatu i ngā inenga nei? Whakaaturia ki te tūtohi.

    Tuhia te momo putiputi ki runga o te tūtohi. Whakaaturia ngā inenga ki te taha mauī.

      kāneihana parani
    te inenga tuatahi 40 cm 20 cm
    te inenga tuarua 60 cm 40 cm
    Tonoa ngā ākonga ki te whakaatu i te tipu o ngā putiputi ki tētahi kauwhata pou. Me mahi takirua koutou ki te whakaatu i te teitei o ngā putiputi nei ki tētahi kauwhata pou.

    Ko tēhea o ēnei kauwhata te mea pai hei tirotiro i te rahi o te tipu i roto i te rua wiki?

    graph.

    graph.

    E pai ana te kauwhata tuarua nā te mea e noho kotahi ana ngā pou mō te inenga tuatahi me te inenga tuarua mō ia putiputi. He māmā ake te kite i te rahi o te tipu o tēnā me tēnā o ngā putiputi.
    Whakawhitiwhiti kōrero mō te pātai ko tēhea te putiputi he rahi rawa te tipu. Me hoki tātou ki te rapanga. Ko tēhea te puptiputi he rahi rawa te tipu?

    He tauira whakautu:
    1. He ōrite te rahi o te tipu o ia putiputi, arā, e 20 cm te tipu o te kāneihana, e 20 cm hoki te tipu o te parani.
      graph.

       
    2. He rahi ake te tipu o te parani. E 2 whakareatanga te tipu o te parani. Kotahi me te haurua whakareatanga te tipu o te kāneihana.
      diagram.
    Whakaaturia te whakautu tuatahi hei tāpiritanga.

    Whakaaturia te whakautu tuarua hei whakareatanga.
    Kei te pai ngā whakautu e rua. Ko tētahi e whakaatu ana i te rahi o te tipu, koia anake. Kāore i arohia te teitei o te putiputi i te tīmatanga. Arā, he tāpiripiri te whakaaro:

    kāneihana 40 cm + 20 cm = 60 cm
    parani 20 cm + 20 cm = 40 cm


    Ko te whakautu tuarua i āta arohia te teitei o te putiputi i te tīmatanga. Arā he whakaaro whakarea:

    kāneihana 40 cm x 1 1/2 = 60 cm
    parani 20 cm x 2 = 40 cm
    Kei Whārangi Tārua 1 (PDF, 72KB) ētahi atu rapanga pēnei hei tūhura mā ngā ākonga. He mea nui te:
    • tuhi pikitia, tuhi tūtohi, tuhi kauwhata rānei hei whakaatu i ngā kōrero matua o roto i te rapanga
    • whakawhitiwhiti kōrero mō te rautaki i whāia hei whakaoti i te rapanga. Arā, mēnā he whakaaro tāpiripiri, he whakaaro whakarea rānei, ā, ko tēhea te mea e tika ana, e hāngai ana rānei.


     
  2. Ko te mahi tuarua, he tirotiro i te pānga o te whānui me te roa o ētahi tapawhā hāngai.
    Ngā tohutohu He Tauira Kōrero Mā Te Pouako
    Tuhia te rapanga nei ki te papa tuhituhi, ka pānui tahi ai me ngā ākonga.

    Ko tēhea o ēnei tapawhā hāngai e pātata nei te āhua ki te tapawhā rite:
    • 1 cm mā te 4 cm
    • 2 cm mā te 5 cm


    Whakawhitiwhiti kōrero mō te whakautu.
    Me mahi takirua ki te āta whakaaro i tēnei rapanga. He aha tētahi rautaki matua? Ko te tuhi pikitia.

    diagram.

    diagram.

    He pātai ārahi: Mō te tapawhā hāngai tuatahi, e hia te rahinga ake o te roa, tērā i te whānui? E 3 cm te rahinga ake (1 + 3 = 4).

    Mō te tapawhā hāngai tuarua, e hia te rahinga ake o te roa, tērā i te whānui? E 3 cm te rahinga ake (2 + 3 = 5).

    Nō reira kei te tika, kāore i te tika rānei te whakautu he ōrite te āhua o ngā tapawhā hāngai nei, kāore tētahi e pātata atu i tētahi ki te āhua o te tapawhā rite? Kāore i te tika. Titiro ki te āhua o ngā tapawhā hāngai. He whāiti te āhua o te mea tuatahi, he whānui ake te āhua o te mea tuarua. Ko te mea tuarua e pātata atu ana ki te āhua o te tapawhā rite.

    Mō te tapawhā hāngai tuatahi, e hia te whakareatanga ake o te roa, tērā i te whānui? Whakareatia te whānui ki te 4, ka rite ki te roa (1 x 4 = 4)

    Mō te tapawhā hāngai tuarua, e hia te whakareatanga ake o te roa, tērā i te whānui? Whakareatia te whānui ki te 2 1/2 , ka rite ki te roa (2 x 2 1/2 = 5)

    Titiro ki te tūtohi nei:
      whānui roa te whakareatanga
    tapawhā hāngai 1 1 4 x 4
    tapawhā hāngai 2 2 5 x 2 1/2
    tapawhā rite 2 2 x 1


    He aha te tauira e kitea mai ana i te tūtohi?
    He rahi ake te tau whakarea mō te tapawhā hāngai whāiti (tapawhā hāngai 1).
    Kāore i te pērā rawa te rahi o te tau whakarea mō te tapawhā hāngai āhua whānui nei (tapawhā hāngai 2)
    Ko te 1 te tau whakarea mō tētahi tapawhā rite.
    Hoatu ētahi atu rapanga pēnei. Mēnā e mārama ana te ākonga ki te pānga riterite o te whānui me te roa o tētahi tapawhā hāngai, kāore he take o te tuhi pikitia. Anei anō ētahi tapawhā hāngai takirua. Ko tēhea o ia takirua te mea e pātata atu ana ki te āhua o te tapawhā rite?

      whānui roa
    tapawhā hāngai 1 3 cm 9 cm
    tapawhā hāngai 2 2 cm 6 cm


      whānui roa
    tapawhā hāngai 1 5 cm 10 cm
    tapawhā hāngai 2 10 cm 18 cm


      whānui roa
    tapawhā hāngai 1 4 cm 15 cm
    tapawhā hāngai 2 2 cm 8 cm


      whānui roa
    tapawhā hāngai 1 6 cm 15 cm
    tapawhā hāngai 2 3 cm 7 1/2 cm


     
  3. Ka tūhuratia te whakatairite i ētahi ōwehenga e rua.
    Ngā tohutohu He Tauira Kōrero Mā Te Pouako
    Tuhia te rapanga nei ki te papa tuhituhi, ka whakawhitiwhiti kōrero ai mō te rautaki tino whaihua hei whakaoti.
    E $9 te utu mō ngā pukapuka e 3. E hia ngā pukapuka ka taea te hoko mō te $45.
    Anei anō tētahi rapanga. E pai ana te whakaatu i ngā kōrero ki tētahi tūtohi:

    te utu te maha o ngā pukapuka
    $9 3
    $45 ??


    He aha ētahi rautaki hei tātai i te maha o ngā pukapuka e taea te hoko mō te $45?

    He tauira whakautu:
    1. Tātaihia te utu mō te pukapuka kotahi ($9 ÷ 3 = $3). E $3 te utu mō ia pukapuka. E hia ngā $3 kei roto i te $45 (3 x ? = 45)? Whakareatia te $3 ki te 15, ka eke ki te $45. Nō reira 15 ngā pukapuka ka taea te hoko mō te $45.
    2. Wehea te utu ki te 3, hei tātai i te maha o ngā pukapuka. (9 ÷ 3 = 3, nō reira 45 ÷ 3 = 15). 15 ngā pukapuka ka taea te hoko mō te $45.
    3. I whakareatia te utu tuatahi ki te 5 ($9 x 5 = $45), nō reira whakareatia te maha o ngā pukapuka ki te 5 (3 x 5 = 15). 15 ngā pukapuka ka taea te hoko mō te $45.


    Ko tēhea te rautaki tino māmā? He āhua māmā ngā rautaki katoa.
    I tēnei o ngā rapanga kāore i te pērā rawa te māmā o ngā rautaki katoa. Kotahi anake pea te rautaki e tino whaihua ana. Ko tā ngā ākonga, he āta whiriwhiri i te rautaki e māmā ake ana.

    E $2.40 te utu mō ngā pene e 3. E hia te utu mō ngā pene 10?
    Me mahi takirua kōrua ko tō hoa ki te whiriwhiri i te rautaki e tino whaihua ana hei whakaoti i te rapanga, ka whakamārama mai ai ki te akomanga.

    Tuatahi, ka whakaaturia ngā kōrero o te rapanga ki tētahi tūtohi:

    te utu te maha o ngā pene
    $2.40 3
    $?? 10


    He māmā te kitea atu mēnā ka whakareatia te maha o ngā pene ki te 80 hēneti, tērā ka rite ki te utu (3 x 80c = $2.40). Nō reira mēnā 10 ngā pene e hokona ana, ka whakareatia te 80 hēneti ki te 10 hei tātai i te utu (80c x 10 = $8.00).
    I tēnei o ngā rapanga kāore i te pērā rawa te māmā o ngā rautaki katoa. Kotahi anake pea te rautaki e tino whaihua ana. Ko tā ngā ākonga, he āta whiriwhiri i te rautaki e māmā ake ana.

    $15 Aotearoa hei hoko i te $13.50 Ahitereiria.
    E hia tāra Aotearoa hei hoko i te $54 Ahitereiria?
    Me mahi takirua kōrua ko tō hoa ki te whiriwhiri i te rautaki e tino whaihua ana hei whakaoti i te rapanga, ka whakamārama mai ai ki te akomanga.

    Tuatahi, ka whakaaturia ngā kōrero o te rapanga ki tētahi tūtohi:

    $ Aotearoa $ Ahitereiria
    $15.00 $13.50
    $?? $54.00


    He āhua māmā te kite atu e 4 te whakareatanga ake o ngā tāra Ahitereiria (13.50 x 2 = $26.00, $26.00 x 2 = $54.00). Nō reira me whakarea anō hoki ngā tāra Aotearoa ki te 4 ($15.00 x 4 = $60.00). E $60 Aotearoa hei hoki i te $54 Ahitereiria.
    Hoatu te Whārangi Tārua 2 (PDF, 67KB)hei whakaoti mā ngā ākonga. He mea nui kia āta tonoa rātou ki te whakaatu i ngā kōrero matua o roto i te rapanga ki tētahi tūtohi, kauwhata, pikitia rānei, me te tono hoki kia āta whakamāramahia te rautaki i whāia hei whakaoti.


     
  4. Ka haere tonu te whakatairite i ētahi ōwehenga e rua.
    Ngā tohutohu He Tauira Kōrero Mā Te Pouako
    Tapahia tētahi tuaina mā ia takirua kia ōrite te roa. Tonoa ngā takirua ki te āta whakaaro i tētahi waeine hei ine i te roa o te tuaina (ki te takiwā o te 1 mita pea te roa o te tuaina).

    Whakawhitiwhiti kōrero mō te pānga o ngā waeine i whakamahia e ia takirua.
    He mahi takirua tēnei. Anei he tuaina mā ia takirua. He ōrite te roa o te tuaina a ia rōpū.

    Whakaarohia tētahi waeine hei ine i te roa o te tuaina. Ākene pea he poro rākau Cuisenaire, he pene rākau, te tapa o tētahi pukapuka rānei hei waeine.

    Whakamahia tā kōrua waeine, ka ine ai i te roa o te tuaina.

    E Mere he aha te waeine i whakamahia e kōrua? Ko te pene kahurangi nei.

    E hia pene te roa o te tuaina? Tekau mā rima (he tauira whakautu).

    Rōpata, he aha te waeine i whakamahia e kōrua? Ko te rākau Cuisenaire pango nei.

    Titiro ki ngā waeine e rua nei, te pene i whakamahia e Mere, me te rākau pango i whakamahia e Rōpata. Pēhea nei te rahi ake o tētahi i tētahi? Kei te takiwā o te haurua o te pene te rahi o te rākau pango.

    Ki ō koutou whakaaro, e hia rākau pango te roa o te tuaina? Tuhia ki te tūtohi nei:

    te waeine pen. block.
    te roa o te tuaina 15 ???
    Kei te takiwā o te 30 ngā rākau pango te roa o te tuaina, nā te mea e rua ngā rākau pango ka rite ki te roa o te pene. Nō reira me whakarea te 15 ki te 2 hei tātai i te roa o te tuaina.

    Rōpata, e tika ana tērā? E pātata ana ki te 30 ngā rākau pango tā kōrua inenga i te tuaina?
    Kia pērā tonu te whakatairite i ngā waeine me ngā inenga mō te roa o te tuaina.


     
  5. Mō te roanga atu o te wiki ka tūhura ngā ākonga i ngā rapanga kei Whārangi Tārua 3 (PDF, 168KB). He mea nui kia āta tonoa rātou ki te whakaatu i ngā kōrero matua o roto i te rapanga ki tētahi tūtohi, kauwhata, pikitia rānei, me te tono hoki kia āta whakamāramahia te rautaki i whāia hei whakaoti.

Hei Whakawhānui.

Tūhuratia te whakamahinga o te taurangi hei whakaoti rapanga pānga riterite. Hei tauira:

te roa o ngā tapa o tētahi tapawhā rite 1 cm 2 cm 3.5 cm 6.25 cm 10 cm h cm
te paenga o te tapawhā rite 4 cm 8 cm 14 cm 25 cm 40 cm 4h cm

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