He Ara Tere hei Whakatau i ngā Raru Raupapa
He momo pātai noa ngā pātai raupapatanga (raupapa tau) i roto i ngā whakamātautau whakauru kura, ngā whakamātautau whiriwhiri mahi, tae atu ki ngā whakamātautau pūkenga taketake. He māmā noa te whakaaro: ka tonoa koe kia kimi i tētahi tauira i roto i tētahi raupapa tau, kātahi ka whakatau i te tau e whai ake nei (te tau ngaro rānei). Ahakoa he ahua uaua, ka taea te whakaoti tere i ngā raupapatanga mēnā he "mahere" mārama āu o ngā hikoinga. Ka matapakihia e tēnei tuhinga me pēhea te whakaoti tere i ngā rapanga raupapatanga mā te whakamahi i ngā rautaki mahi, ngā momo tauira tino noa, me ngā tohutohu hei karo i ngā tauira teka.
1. Kia mārama ki te kaupapa o te rapanga raupapa
Ko te tikanga, ka whakamātautauhia tō pūkenga ki te mōhio ki ngā whanaungatanga i waenga i ngā tau mā ngā pātai raupapa. Ka uru atu pea ēnei whanaungatanga ki te tāpiri, te tango, te whakarea, te wehewehe, te whakakotahitanga o ngā mahi, tae atu ki ngā tauira, ngā rōpū rānei e hangai ana ki te tūranga.
Ko te tikanga kei roto i te puka ngā pātai:
– Whakatauhia te tau e whai ake nei: 2, 4, 8, 16, …
– Tāutuhia ngā tau e ngaro ana: 3, 6, __, 24, 48
– Kōwhiria te whakautu tino tika mai i ngā kōwhiringa e wātea ana.
Ko te kī tere ko: kaua e whakaaro noa. Whakamahia ngā mahi pūnaha e whai ake nei.
2. Ngā mahi tere (tikanga whānui) hei whakaoti i te raupapa
Koinei te raupapa o ngā huarahi whakaaro whai hua e whakamahia ana i roto i ngā whakamātautau:
1) Tirohia te rerekētanga (rerekētanga) i waenga i ngā tau
Tātaihia ngā rerekētanga e whai ake nei: a2–a1, a3–a2, me ērā atu. He maha ngā raupapa māmā e whakamahi ana i ngā rerekētanga pumau (ngā raupapatanga tātai) ngā rerekētanga rānei e hanga ana i tētahi tauira.
2) Tirohia te ōwehenga (wehenga)
Ki te kore e mārama ngā rerekētanga, ngana ki te wehewehe: a2/a1, a3/a2, me ētahi atu. Mā tēnei ka āwhina ki te kimi i ngā raupapatanga ā-ira, i ngā tauira whakarea rānei.
3) Tirohia te tauira whakauru (e rua ngā taahiraa/whakawhitiwhiti)
He maha ngā raupapa e whakamahi ana i tētahi tauira tauutuutu: tāpiri, whakarea, tāpiri, whakarea. E rua rānei ngā raupapa e tauutuutu ana (he tauira kotahi ngā tau tauhou, he tauira anō ngā tau ōrite).
4) Tirohia te tauira taumata tuarua (rerekētanga rerekētanga)
Mena ka rerekē tonu te rerekētanga, tatauhia te rerekētanga o ngā rerekētanga. He maha ngā wā ka puta tēnei i roto i ngā raupapa tapawhā, i ngā tauira piki haere rānei i ia taahiraa.
5) Tirohia ngā tauira motuhake: tapawhā, pūtoru, matua, Fibonacci, whārite
He maha ngā whakamātautau e whakaatu ana i ngā tauira "tohu". Mena he āhua mōhio ngā tau (1, 4, 9, 16…) he tapawhā pea.
6) Tirohia ngā tauira matihiko
I ētahi wā, kāore te tauira i roto i te uara, engari i roto i te mati: te tapeke o ngā mati, te whakahurihanga, te tāpiri o ngā mati, te tauwehenga rānei o te 9/11.
Mā tēnei raupapa, kāore koe e moumou taima. Me whakaharatau noa koe kia aunoa rā anō.
3. Ngā tauira raupapa e tino kitea ana, me te huarahi tere ki te mahi i tēnei
A. Raupapa pāngarau (rerekētanga pumau)
Tauira: 5, 9, 13, 17, …
Te rerekētanga: +4, +4, +4.
Nā, ko te tau e whai ake nei: 21.
He mahi tere: kia kitea rā anō te rerekētanga kotahi e rua ngā wā karapīpiti, me haere tonu taua tauira.
B. Raupapa āhuahanga (ōwehenga pumau)
Tauira: 3, 6, 12, 24, …
Ko te ōwehenga: ×2, ×2, ×2.
Panuku: 48.
Tohutohu tere: ki te piki haere, ki te heke rānei ngā tau, whakapaehia te whakarea/wehenga.
C. Raupapa rerekētanga kua whakarōpūtia (rerekētanga piki haere auau)
Tauira: 2, 5, 9, 14, 20, …
Rerekētanga: +3, +4, +5, +6.
Te rerekētanga e whai ake nei +7 → 27.
Tohutohu tere: ki te "piki ake te rerekētanga takitahi", he tohu tēnā o tētahi tauira taumata-papa.
D. Raupapa tapawhā me te raupapa pūtoru
– Ngā tapawhā: 1, 4, 9, 16, 25, … (n²)
– Pūtoru: 1, 8, 27, 64, … (n³)
He mahi tere: maumahara ki ngā tapawhā 1–15 me ngā poraka 1–10. Mā tēnei ka tere ake te tautuhi i ngā mea katoa mā roto i ngā hēkona.
E. Raupapa Fibonacci me ōna momo rerekētanga
Fibonacci Matarohia: 1, 1, 2, 3, 5, 8, 13, …
Ia tau = te tapeke o ngā tau e rua o mua.
Ngā rerekētanga e puta pinepine ana:
2, 3, 5, 8, 13, 21, … (tīmata mai i te 2 me te 3).
He mahi tere: tirohia mēnā ko a(n) = a(n-1) + a(n-2). Mēnā ka uru ki roto i ngā taahiraa 2–3, koinā te whakautu i te nuinga o te wā.
F. Raupapa tau matua
2, 3, 5, 7, 11, 13, 17, …
He mahi tere: maumaharahia ngā tau matua tae atu ki te 50. He maha ngā raruraru ka mutu i taua awhe.
G. Raupapa Whakatauranga
1, 2, 6, 24, 120, … (1!, 2!, 3!, 4!, 5!)
He mahi tere: tirohia mēnā ka taurite ngā whakareatanga: ×2, ×3, ×4, ×5.
H. Tauira tauwhitiwhiti (ngā mahi e rua)
Tauira: 2, 6, 7, 21, 22, 66, …
Tauira: ×3, +1, ×3, +1, ×3 …
Panuku: 67.
He tinihanga tere: ki te kore he tauira kotahi, ngana ki te wehewehe i ngā mahi "whakarea me te tāpiri".
I. E rua ngā raupapa taupatupatu (he rerekē ngā tauira o ngā tau tauhou me ngā tau taurite)
Tauira: 1, 4, 2, 8, 3, 12, 4, …
Ngā tau tūnga tauhou: 1, 2, 3, 4 (piki +1)
Tau tūranga ōrite: 4, 8, 12 (piki +4)
Panuku (tūnga ōrite): 16.
He mahi tere: wehea kia rua ngā rarangi: (1, 3, 5,…) me (2, 4, 6,…).
J. Ngā tauira matihiko
Tauira: 10, 11, 13, 17, 25, …
I ētahi wā ko te tauira te tapeke o ngā mati, ko te "tau e whai ake nei = tau o mua + tapeke o ōna mati".
Hei tauira, ko te ture: a(n+1) = a(n) + (te tapeke o ngā mati o a(n)).
He tinihanga tere: ki te mea he "āhua rerekē" ngā tau, ā, kāore e uru ki te rerekētanga/ōwehenga, tirohia ngā mati.
4. Ngā tikanga hei karo i ngā mahanga tauira rūpahu
I roto i ngā whakamātautau, he maha ngā wā ka whakaratohia e ngā kaiwhakamātautau ngā raupapatanga e "āhua" ana he rite ki ngā tauira maha. Me pēhea te karo i tēnei:
1) Whakamātauria te tauira i runga i te iti rawa kia 3 ngā whakawhiti
Kaua e makona ki te kotahi noa te taurite. Kia tino taurite ngā takirua tau maha.
2) Kōwhiria te tauira māmā rawa, ā, tino ōrite hoki.
Mena e rua ngā mea ka taea, ko te tauira e whiriwhiria ana ko te tauira kāore i te tino uaua, ā, e pā ana ki ngā tau katoa.
3) Kia aro ki ngā kōwhiringa whakautu
I ngā pātai whiriwhiri maha, ka taea e koe te whakamātautau i tētahi o ngā tauira, kātahi ka kite mēnā ka puta te hua i roto i ngā kōwhiringa. Ki te kore, he hē te tauira.
5. Ngā tauira whakaharatau poto (ara tere)
1) 7, 14, 28, 56, …
Tirohia te ōwehenga: ×2 kātahi ka → 112 e whai ake nei.
2) 3, 8, 15, 24, 35, …
Rerekētanga: +5, +7, +9, +11 (ake 2) → panuku +13 = 48.
3) 1, 3, 6, 10, 15, …
Rerekētanga: +2, +3, +4, +5 → muri +6 = 21 (he raupapa tapatoru tēnei).
6. Ngā tohutohu mō te ako tere ake
– Hangaia he rārangi o ngā tauira me maumahara: tapawhā, pūtoru, tuatahi, whārite, Fibonacci.
– Whakaharatauhia kia 10–20 ngā pātai i te rā: he pai ake te mahi auau i te maha engari he onge.
– Whakamahia he wati mutu: kia 20–40 hēkona te roa mō ia pātai mō te taumata taketake.
– Tuhia ngā tauira e pohehe pinepine ana koe: ka waiho hei "pūtea hapa" ka tere ake te whakapai ake.
Te Katinga
Ehara i te mea ko te pūkenga te take o te whakaoti rapanga raupapatanga, engari ko te rautaki me te tikanga. Tīmata ki ngā mahi tino noa (te rerekētanga me te ōwehenga), neke atu ki ngā tauira tauutuutu, ngā tauira honohono rānei, kātahi ka tirohia ngā tauira motuhake pēnei i ngā tapawhā, ngā tau matua, ngā tau Fibonacci, ngā tau whakarea rānei. Mā te nui o tō whakaharatau, ka nui ake tō taunga ki te kite i ngā tauira i roto i ētahi hēkona noa iho. Mēnā he rite tonu tō mahi, ka māmā ake, ka tere ake hoki te whakaoti i ngā rapanga raupapatanga i te tuatahi he pōhēhē.
Ki te hiahia koe, tukuna mai kia 5-10 ngā tauira o ngā rapanga raupapa kei a koe, ā, māku koe e āwhina ki te matapaki takitahi i aua rapanga mā te whakamahi i te tikanga tere rawa atu.
