Wehenga Hautau

Hei Whakarāpopoto

Ko tā te kōwae ako nei he tūhura i te wehenga o tētahi tau mā tētahi hautau. Ka tuhia he pikitia hei whakatauira i ngā wehenga, ā, ka tūhono hoki ki te ture mō te whakawehe hautau, arā: a/h ÷ e/k = a/h x k/e

Te Hononga ki te Marautanga

Taumata 4
Te Tau me te Taurangi 
Whāinga Paetae 5:

Ngā Whāinga Ako

Kia mōhio te ākonga:

  • ki te ture mō te whakawehe hautau
  • ki te whakamārama, ki te whakatauira i te wehenga o tētahi tau mā tētahi hautau
He Rārangi Kupu
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pakitau number story
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tau hanumi mixed number
tauoti whole number
tau taupoki reciprocal
tūtohi table
whakaputa derive, produce
whārite taurangi algebraic equation
Hei Raupapa I Ngā Mahi Ako
  1. Ko te mahi tuatahi, he tirotiro i te wehenga o te kotahi mā tētahi hau-tahi, pērā i te 1 ÷ ¼. Ka tirohia hoki te whakareatanga e whai pānga ana (¼ x 4 = 1)
  2. Ngā tohutohu He Tauira Kōrero Mā Te Pouako
    Tuhia te rapanga nei ki te papa tuhituhi, ka pānui tahi ai me ngā ākonga.

    Kotahi te poro tiakareti a Mareta. E hiahia ana ia kia wehea tana poro tiakareti, kia haurua te rahi o ngā wehenga. E hia ngā wehenga ka hua mai?

    Aratakina ngā ākonga ki te whiriwhiri i te otinga.

    He aha te whārite e hāngai ana ki te rapanga nei? E wehea ana te kotahi mā te haurua, arā, 1 ÷ ½ = □

    He tāhei tēnei hei tohu i te poro tiakareti:



    He tāhei tēnei hei tohu i te haurua poro tiakareti:



    Mēnā ka whakatakotoria ngā haurua ki raro i te kotahi, he mārama te kitea, ina wehea te kotahi mā te haurua, ko te rua te otinga.



    Nō reira, he aha te otinga o te whārite? 1 ÷ ½ = 2

    He aha te whakareatanga e hāngai ana? E rua ngā haurua kei roto i te kotahi, arā, ½ x 2 = 1

    Tukuna ngā ākonga ki te tūhura i ngā rapanga nei. Tonoa rātou ki te:
    • tuhi i te whārite whakawehe e hāngai ana;
    • tuhi i tētahi pikitia hei whakaatu i te rapanga;
    • whakaoti i te wehenga;
    • tuhi i te whakareatanga e hāngai ana.
    1. Kotahi rita (1l) te rōrahi o te wai ārani, ka wehea ki ētahi ipu ¼ rita te kītanga. E hia ngā ipu ka taea te whakakī?
    2. Kotahi mita te roa o tētahi aho. Ka tapahia kia hautekau mita te roa o ngā tapahanga. E hia ngā tapahanga ka hua mai?
    3. Kotahi te parehe (pīta), ka tapahia kia hauwaru te rahi o ngā wehenga. E hia ngā wehenga parehe ka hua mai?
    4. Kotahi te poro tiakareti, ka tapahia kia haurima te rahi o ngā wehenga. E hia ngā wehenga ka hua mai?
  3. Ko te mahi tuarua, he tirotiro i te wehenga o te tauoti mā tētahi hau-tahi, pērā i te 5 ÷ ¼. Ka tirohia hoki te whakareatanga e whai pānga ana (¼ x 20 = 5)
  4. Ngā tohutohu He Tauira Kōrero Mā Te Pouako
    Tuhia te rapanga nei ki te papa tuhituhi, ka pānui tahi ai me ngā ākonga.

    E 2 mita te roa o te poro rākau a Hinewai. E hiahia ana ia kia tapahia tana poro rākau, kia hautoru mita te roa o ngā tapahanga. E hia ngā wehenga ka hua mai?

    Aratakina ngā ākonga ki te whiriwhiri i te otinga.

    He aha te whārite e hāngai ana ki te rapanga nei? E wehea ana te rua mā te hautoru, arā, 2 ÷ ⅓ = □

    He tāhei tēnei hei tohu i te poro rākau:

    Mēnā ka tapahia te poro rākau kia hautoru mita te roa o ngā wehenga, e 6 katoa ngā wehenga ka hua mai:

    Nō reira, he aha te otinga o te whārite? 2 ÷ ⅓ = 6

    He aha te whakareatanga e hāngai ana? E ono ngā hautoru kei roto i te rua, arā, ⅓ x 6 = 2

    Tukuna ngā ākonga ki te tūhura i ngā rapanga nei. Tonoa rātou ki te:
    • tuhi i te whārite whakawehe e hāngai ana;
    • tuhi i tētahi pikitia hei whakaatu i te rapanga;
    • whakaoti i te wehenga;
    • tuhi i te whakareatanga e hāngai ana.
    1. E 3 rita (3l) te rōrahi o te wai ārani, ka wehea ki ētahi ipu ¼ rita te kītanga. E hia ngā ipu ka taea te whakakī?
    2. E 50 mita te roa o tētahi aho. Ka tapahia kia haurua mita te roa o ngā tapahanga. E hia ngā tapahanga ka hua mai?
    3. E 5 ngā parehe (pīta), ka tapahia kia hauwaru te rahi o ngā wehenga. E hia ngā wehenga parehe ka hua mai?
    4. 320 mita te roa o tētahi aho. Ka tapahia kia hautekau mita te roa o ngā tapahanga. E hia ngā tapahanga ka hua mai?
  5. I konei, ka tūhura i te wehenga o te tauoti mā tētahi hautau, ina he tauoti te otinga, pērā i te 3 ÷ ¾ = 4. Ka tirohia hoki te whakareatanga e whai pānga ana (¾ x 4 = 3)
  6. Ngā tohutohu He Tauira Kōrero Mā Te Pouako
    Tuhia te rapanga nei ki te papa tuhituhi, ka pānui tahi ai me ngā ākonga.

    E 3 ngā kapu maramara tiakareti a Hone hei tunu keke māna. E ¾ kapu hei tunu i te keke kotahi. E hia ngā keke ka taea e ia te tunu?

    Aratakina ngā ākonga ki te whiriwhiri i te otinga.

    He aha te whārite e hāngai ana ki te rapanga nei? E wehea ana te toru mā te toru hauwhā, arā, 3 ÷ ¾ = □

    E toru ngā kapu maramara tiakareti e whakaaturia ana i tēnei pikitia. Kua wāwāhia ki ngā wehenga ¾ kapu te rōrahi. E 4 katoa ngā ¾ kei roto i te 3:

    Nō reira, he aha te otinga o te whārite? 3 ÷ ¾ = 4

    He aha te whakareatanga e hāngai ana? E whā ngā toru hauwhā kei roto i te toru, arā, ¾ x 4 = 3

    Tukuna ngā ākonga ki te tūhura i ngā rapanga nei. Tonoa rātou ki te:
    • tuhi i te whārite whakawehe e hāngai ana;
    • tuhi i tētahi pikitia hei whakaatu i te rapanga;
    • whakaoti i te wehenga;
    • tuhi i te whakareatanga e hāngai ana.
    1.  E 6 rita (6l) te rōrahi o te wai ārani, ka wehea ki ētahi ipu 3/5 rita te kītanga. E hia ngā ipu ka taea te whakakī?
    2. E 72 mita te roa o tētahi aho. Ka tapahia kia toru hauwaru mita (3/8 m) te roa o ngā tapahanga. E hia ngā tapahanga ka hua mai?
    3. 12 ngā parehe (pīta), ka tapahia kia 2/3 te rahi o ngā wehenga. E hia ngā wehenga parehe ka hua mai?
    4. 100 mita te roa o tētahi aho. Ka tapahia kia 2/3 mita te roa o ia tapahanga. E hia ngā tapahanga ka hua mai?
  7. I konei, ka tūhura i te wehenga o te tauoti mā tētahi hautau, ina he tau hanumi te otinga, pērā i te 2 ÷ ¾ = 2 ⅔. Ka tirohia hoki te whakareatanga e whai pānga ana (¾ x 2 ⅔ = 2)
  8. Ngā tohutohu He Tauira Kōrero Mā Te Pouako
    Tuhia te rapanga nei ki te papa tuhituhi, ka pānui tahi ai me ngā ākonga.

    E 2 ngā poro tiakareti a Hone hei wehewehe māna kia ¾ te rahi o ia wehenga. E hia ngā wehenga tiakareti ka hua mai?

    Aratakina ngā ākonga ki te whiriwhiri i te otinga.

    He aha te whārite e hāngai ana ki te rapanga nei? E wehea ana te rua mā te toru hauwhā, arā, 2 ÷ ¾ = □

    E rua ngā poro tiakareti e whakaaturia ana i tēnei pikitia:

    Kua wāwāhia ki ngā wehenga ¾ te rahi. E 2 ⅔ ngā ¾ kei roto i te 2:

    Nō reira, he aha te otinga o te whārite? 2 ÷ ¾ = 2 ⅔

    He aha te whakareatanga e hāngai ana? E rua me te rua hautoru ngā toru hauwhā kei roto i te rua, arā, ¾ x 2 ⅔ = 2

    Tukuna ngā ākonga ki te tūhura i ngā rapanga nei. Tonoa rātou ki te:
    • tuhi i te whārite whakawehe e hāngai ana;
    • tuhi i tētahi pikitia hei whakaatu i te rapanga;
    • whakaoti i te wehenga;
    • tuhi i te whakareatanga e hāngai ana.
    1. E 3 rita (3l) te rōrahi o te wai ārani, ka wehea ki ētahi ipu 2/5 rita te kītanga. E hia ngā ipu ka taea te whakakī?
    2. E 4 ngā kapu maramara tiakareti a Hone hei tunu keke māna. E ¾ kapu hei tunu i te keke kotahi. E hia ngā keke ka taea e ia te tunu?
    3. 10 ngā parehe (pīta), ka tapahia kia 3/8 te rahi o ngā wehenga. E hia ngā wehenga parehe ka hua mai?
    4. E 5 mita te roa o tētahi aho. Ka tapahia kia ⅔ mita te roa o ia tapahanga. E hia ngā tapahanga ka hua mai?
  9. I konei, ka tūhura i te wehenga o te tau hanumi mā tētahi hautau, ina he tau hanumi te otinga, pērā i te 2 ½ ÷ ¾ = 3 ⅓ . Ka tirohia hoki te whakareatanga e whai pānga ana (¾ x 3 ⅓ = 2 ½)
  10. 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 me te haurua rita (2 ½ l) te rōrahi o te wai ārani, ka wehea ki ētahi ipu ¾ rita te kītanga. E hia ngā ipu ka taea te whakakī?

    Aratakina ngā ākonga ki te whiriwhiri i te otinga.

    He aha te whārite e hāngai ana ki te rapanga nei? E wehea ana te rua me te haurua mā te toru hauwhā, arā, 2 ½ ÷ ¾ = □

    He tāhei tēnei e tohu ana i te 2 ½ rita wai ārani:

    Kua wehea te 2 ½ ki ētahi ipu e toru hauwhā rita (¾) te rōrahi. E 3 ⅓ ngā ipu ka taea te whakakī:

    Nō reira, he aha te otinga o te whārite? 2 ½ ÷ ¾ = 3

    He aha te whakareatanga e hāngai ana? E toru me te kotahi hautoru ngā toru hauwhā kei roto i te rua me te haurua, arā, ¾ x 3 ⅓ = 2 ½

    Tukuna ngā ākonga ki te tūhura i ngā rapanga nei. Tonoa rātou ki te:
    • tuhi i te whārite whakawehe e hāngai ana;
    • tuhi i tētahi pikitia hei whakaatu i te rapanga;
    • whakaoti i te wehenga;
    • tuhi i te whakareatanga e hāngai ana.
    1. E 3 ½ rita (3 ½l) te rōrahi o te wai ārani, ka wehea ki ētahi ipu 2/5 rita te kītanga. E hia ngā ipu ka taea te whakakī?
    2. E 4 ¾ kg mīti a Hone hei whāngai i tana kurī. E ⅔ kg mīti hei kai mā tana kurī i ia rā. E hia ngā rā ka taea e ia tana kurī te whāngai, kia pau rā anō te mīti?
    3. 2 ½ ngā parehe (pīta), ka tapahia kia 3/8 te rahi o ia wehenga. E hia ngā wehenga parehe ka hua mai?
    4. E 5 ⅔ mita te roa o tētahi aho. Ka tapahia kia ¾ mita te roa o ia tapahanga. E hia ngā tapahanga ka hua mai?
  11. Ināianei, ka hoki ki te tirotiro i ngā wehenga hautau katoa kua oti i mua ake nei, me te tūhura anō i te ture mō te whakawehe i tētahi tau mā tētahi hautau.
  12. Ngā tohutohu He Tauira Kōrero Mā Te Pouako
    E hoki ki te tirotiro i ētahi o ngā wehenga kua oti i tēnei kōwae ako, me te whakaatu anō i te hatepe hei whakawehe i tētahi tau mā tētahi hautau. Koia nei tētahi o ngā wehenga i oti i a tātou, me te pikitia e hāngai ana:
    2 ÷ ⅓ = 6

      

    Ka taea ngā tauoti o te whārite te huri hei hautau:

    te whārite te huri i ngā tau katoa o te whārite hei hautau
     2 ÷ ⅓ = 6  2/1 ÷ 1/3 = 6/1

    Ināianei, ka hurihia te wehenga hei whakareatanga, ā, ka huri taupoki i te hautau tuarua o te whārite. Ka puta tētahi tauira, arā, ka tika tonu te whakareatanga:

    te whārite te huri i ngā tau katoa o te whārite hei hautau te huri i te ÷ hei x me te huri taupoki i te hautau tuarua
     2 ÷ ⅓ = 6  2/1 ÷ 1/3 = 6/1  2/1 x 3/1 = 6/1

    E kitea ana te tika o te whakareatanga.

    Kia pērā anō te huri i ētahi atu o ngā wehenga kua oti i tēnei kōwae ako. Kua oti ētahi o ngā whārite i raro nei hei tauira:

    te whārite te huri i ngā tau katoa o te whārite hei hautau

    te huri i te ÷ hei x me te huri taupokii te hautau tuarua

     3 ÷ 1/4 = 12  3/1 ÷ 1/4 = 12/1  3/1 x 4/1 = 12/1
    50 ÷ 1/2 = 100 50/1 ÷ 1/2 = 100/1  50/1 x 2/1 = 100/1
    5 ÷ 1/8 = 40  5/1 ÷ 1/8 = 40/1  
    320 ÷ 1/10 = 3200  320/1 ÷ 1/10 = 3200/1  
    6 ÷ 3/5 = 10    
    72 ÷ 3/8 = 792    
    12 ÷ 2/3 = 18    
    3 ÷ 2/5 = 7 1/2    
    4 ÷ 3/4 = 5 1/3    
     4 3/4 ÷ 2/3 = 9 1/2  2/3 ÷ 3/4 = 3/4  
     2 1/2 ÷ 3/8 = □    
     5 2/3 ÷ 3/4 = □    
    Āta tirohia te tauira e puta ake ana i ngā whārite, ka whakawhitiwhiti kōrero mō te ture hei whakaoti wehenga. I te pou tuatoru o te tūtohi, i hurihia ngā wehenga hei whakareatanga, ā, i huri taupoki anō hoki te hautau tuarua o te whārite. He tika katoa ngā whakareatanga i te pou tuatoru nei? Āe, kei te tika katoa.

    Nō reira, kua kitea he huarahi hei whakaoti wehenga. Arā, hurihia te wehenga hei whakareatanga, me te huri taupoki anō i te hautau tuarua o te whārite. Hei tauira:

    5 ÷ 1/2 = □ 5/1 x 2/1= 10/1 = 10
    3 ÷ 4/5 = □ 3/1 x 5/4 = 15/4 = 3 3/4
    Hoatu ētahi atu rapanga wehe hei whakaoti mā ngā ākonga. Tonoa hoki rātou ki te tuhi pakitau e hāngai ana ki te whārite. Hei tauira:
    6 ÷ 1/3 = □ 15 ÷ 1/4= □ 20 ÷ 2/3 = □
    2 2/3 ÷ 1/4 = □ 1 3/4 ÷ 2/5 = □ 2/3 ÷ 3/4 = □
  13. Ko te mahi whakamutunga, he whakaputa i te ture mō te whakawehe hautau.
  14. Ngā tohutohu He Tauira Kōrero Mā Te Pouako
    Āta whakamārama i te whakaputanga o te ture mō te whakawehe hautau.

    Me āta whakamārama ia hīkoi:
    Hei tauira atu anō o te huri wehenga hei whakareatanga:
    8 ÷ 2 = □, nō reira, 2 x □ = 8

    Mēnā ka ōrite te rāwekeweke i ia taha o te tohu ‘=’, ka tika tonu te whārite. Hei tauira:
    4 x 2 = 8 → 4 x 2 x 3 = 8 x 3
    4 x 2 = 8 → 4 x 2 + 3 = 8 + 3

    Anei tētahi wehenga hautau, hei tirotiro mā tātou:

    3/4 ÷ 5/6 = □

    Ka rāwekeweke haere tātou i te whārite nei, kia kitea he aha e tika ai te ture nei mō te whakawehe hautau:
    ‘hurihia te wehenga hei whakareatanga, me te huri taupoki anō i te hautau tuarua’.

    Tuatahi, ka hurihia te wehenga hei whakareatanga:

    5/6 x □ = 3/4

    Tuarua, ka whakareatia ia taha o te whārite ki te 6/5. Ko te 6/5 te tau taupoki o te 5/6:

    6/5 x 5/6 x □ = 3/4 x 6/5

    Ko te take i whakareatia te 5/6 ki te 6/5, kia huri i te tau whakarea i te pouaka ki te tahi. Arā:

    30/30 x □ = 3/4 x 6/5

    He ōrite te 30/30 ki te 1, nō reira:

    □ = 3/4 x 6/5

    Kia tīkina atu te whārite wehe i te tīmatanga:

    3/÷ 5/6 = □

    Nō reira:

    3/÷ 5/6 = 3/4 x 6/5

    Tukuna ngā ākonga ki te whai i tēnei rāwekeweketanga mō ētahi atu whārite wehenga hautau.

    Whakawhitiwhiti kōrero mō te tika o te ture, ahakoa he aha te whārite wehenga hautau.

    Kia pērā anō tō rāwekeweke i ngā whārite wehenga nei:

    2/3 ÷ 1/2 = □

    5/3 ÷ 3/4 = □

    Ahakoa te whārite, e kite ana tātou i te tika o te ture mō te whakawehe hautau, arā:
    ‘Hurihia te wehenga hei whakareatanga, me te huri taupoki anō i te hautau tuarua o te whārite’.

Hei Whakawhānui:
Tirohia te rāwekeweketanga o te wehenga hautau taurangi nei:
a/h ÷ e/k
hei whakaputa i te ture hei whārite taurangi:
a/h ÷ e/k = a/h x e/k


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