# Wehenga Hautau

The Ministry is migrating nzmaths content to Tāhurangi.
Relevant and up-to-date teaching resources are being moved to Tāhūrangi (tahurangi.education.govt.nz).
When all identified resources have been successfully moved, this website will close. We expect this to be in June 2024.
e-ako maths, e-ako Pāngarau, and e-ako PLD 360 will continue to be available.

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

Achievement Objectives
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 pān
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
 hātepe algorithm hau-tahi unit fraction pakitau number story pou column rāwekeweke manipulate rōrahi volume tāhei bar, strip 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. 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? Kotahi mita te roa o tētahi aho. Ka tapahia kia hautekau mita te roa o ngā tapahanga. E hia ngā tapahanga ka hua mai? Kotahi te parehe (pīta), ka tapahia kia hauwaru te rahi o ngā wehenga. E hia ngā wehenga parehe ka hua mai? 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. 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? 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? E 5 ngā parehe (pīta), ka tapahia kia hauwaru te rahi o ngā wehenga. E hia ngā wehenga parehe ka hua mai? 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.  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? 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? 12 ngā parehe (pīta), ka tapahia kia 2/3 te rahi o ngā wehenga. E hia ngā wehenga parehe ka hua mai? 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. 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? 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? 10 ngā parehe (pīta), ka tapahia kia 3/8 te rahi o ngā wehenga. E hia ngā wehenga parehe ka hua mai? 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 ⅓ 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. 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? 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? 2 ½ ngā parehe (pīta), ka tapahia kia 3/8 te rahi o ia wehenga. E hia ngā wehenga parehe ka hua mai? 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: 3/4 ÷ 5/6 = □ Nō reira: 3/4 ÷ 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