Te Poutama Tau: He Whakaaturanga mā te Kaiako Te Whakaaro Whakarearea 1

Te Poutama Tau:He Whakaaturanga mā te Kaiako

Te Whakaaro Whakarearea 1

Ngā whāinga mō tēnei akoranga:

• Kia mōhio ki ngā āhuatanga tātai o te whakarea me te wehe.

Calculationproperties

• Kia mārama ki te whakamahinga o ēnei āhuatanga tātai i roto i ngā rautaki matua mō te whakarea me te wehe.

• Kia mōhio ki ngā ngā kupu matua mō te whakarea me te wehe, me te whakamahinga o aua kupu i roto I ngā rerenga kōrero pāngarau.

Ngā kupu mō te whakarea me te wehe:

Ko te whakarea me te whakarau ngā kupu e rua e whakamahia ana mō te ‘multiply’.

Nā te nui o te whakamahinga o te ‘rau’ hei ingoa tau (100, 200 ...), kua riro ko te whakarea’ te kupu e tino whakamahia ana mō tēnei paheko tau (kia kore ai e pōhēhē te ākonga ko tēhea o ngā ‘rau’ e kōrerohia ana).

Numberoperation

Ko te wehe te kupu e tino whakamahia ana mō te ‘division’.

Ngā kupu mō te whakarea me te wehe:

Ehara i te mea he kupu hou ēnei ki tō tātou reo. Kei roto ēnei kupu i te papakupu o Wiremu, he mea whakamahi hoki i roto i ngā tuhituhinga tawhito a ngā mātua tīpuna.

Tā Wiremu: rea – multiply, numerous wehe – detach, divide

He aha ngā kupu e whakamahia ana i tōu kura?

Ko te mea nui pea kia ōrite te kupu a tēnā kaiako a tēnā kaiako i roto i te kura kotahi.

Ngā kupu mō te whakarea me te wehe:

Ko tā te whakarea, he tātai i te maha o ngā mea katoa kei roto i ētahi rōpū (huinga). Kia ōrite te maha o ngā mea kei ia rōpū.Hei tauira ...

E toru ngā rōpū (huinga).

E rua ngā mea kei roto i ia rōpū.

Ngā kupu mō te whakarea me te wehe:

Multipicand

He tauwehe te rua me te toru o te ono.He aha ngā tauwehe katoa o te 12?

Multiplier

2 x 3 = 6tau e whakareatia ana

tau whakarea

otinga

Ko tētahi hei whakaatu i te maha o ngā mea kei roto i ia rōpū (huinga). Ka kīia tēnei ko te ‘tau e whakareatia ana’Ko tētahi hei whakaatu i te maha o ngā rōpū (huinga). Ka kīia ko te tau whakarea tēnei.

FactorE rua ngā tauwehe o tētahi whakareatanga.

Dividend

Divisor

18 ÷ 6 = 3tau e wehea ana

tau whakawehe

otinga

Mō te wehe:

Ngā kupu mō te whakarea me te wehe:

Te whakahua me te whakaahua i te whakareatanga:

E rua ngā whakahuatanga matua mō te whakarea:

Expressing,representing

1.“Whakareatia te rua ki te toru, ka ono.”(“E rua, whakareatia ki te toru, ka ono”)

Me pēhea te whakaahua i tēnei whakareatanga?

2 x 3 = 6

Ko te 2 te rōpū e whakareatia ana. Kia 3 ngā rōpū o te 2.

2.“E rua ngā rōpū (huinga) o te toru, ka ono.”(“E rua ngā toru, ka ono”)

Me pēhea te whakaahua i tēnei whakareatanga?

2 x 3 = 6

Ko te 3 te rōpū e whakareatia ana. Kia 2 ngā rōpū o te 3.

Te whakahua me te whakaahua i te whakareatanga:

Expressing,representing

Kua riro ko tēnei hei tikanga matua mō te whakarea i roto i te reo Māori:

“Whakareatia te rua ki te toru, ka ono.”

2 x 3 = 6

Te whakahua me te whakaahua i te whakareatanga:

Expressing,representing

The usual convention (in English) is that 4 x 8 refers to four sets of eight, not eight sets of four. There is absolutely no reason to be rigid about this convention. The important thing is that students can tell you what each factor in their equation represents. (Van de Walle, 2007. p154)

“E rua ngā rōpū (huinga) o te toru, ka ono.”(“E rua ngā toru, ka ono”)

2 x 3 = 6

Engari, kia mōhio hoki te ākonga, kei te tika hoki tēnei:

Te whakahua me te whakaahua i te whakareatanga:

Expressing,representing

Te whakahua i te rerenga whakareatanga:

Whakawhitiwhiti kōrero mō te tika, te hapa, te mārama rānei o ēnei rerenga kōrero.

He aha ngā rerenga kōrero mō te whakarea e rangona ana, e whakamahia ana i roto i tōu kura?

Expressing

E whai ake nei ētahi o ngā whakahuatanga rerenga kōrero mō te whakarea e rangona ana i roto i ō tātou kura.

Te whakahua i te rerenga whakareatanga:

5 x 3 = 15

Expressing

Rima whakarea toru rite tekau mā rima.

Whakarea te rima me te toru, ka tekau mā rima.

Whakareatia te rima mā te toru, ka tekau mā rima.

Rima toru ka tekau mā rima.

E rima ngā toru ka tekau mā rima.

Whakareatia te rima ki te toru, ka eke ki te tekau mā rima.

Whakareatia te rima ki te toru ka rite ki te tekau mā rima.

E rima ngā rōpū toru ka tekau mā rima.

E rima ngā huinga o te toru, ka tekau mā rima.

Te Whakaaro Tāpiripiri me te Whakaaro Whakarearea:

Ka pau i te whānau Horomona te $96, hei hoko hāngi. E $8 te utu mō ia tākaikai hāngi.

E hia ngā tākaikai hāngi i hokona e rātou?

Additive thinking,Multiplicative

thinking

Whakaarohia te rapanga nei:

Anei ngā rautaki a ētahi ākonga tokorua:

Ka tangotango haere au i te $8 i te $96.

E hia ngā tangohanga o te $8 kia pau katoa

te $96?

Manahi

Te Whakaaro Tāpiripiri me te Whakaaro Whakarearea:

Additive thinking,Multiplicative

thinking

96 - 8 = 8888 - 8 = 8080 - 8 = 72

Te Whakaaro Tāpiripiri me te Whakaaro Whakarearea:

Additive thinking,Multiplicative

thinking

Whakawhitiwhiti kōrero mō ngā rautaki e rua.

Ko wai te mea e whakaaro tāpiripiri ana? Ko wai te mea e whakaaro whakarearea ana? Ko tēhea te rautaki e tino whaihua ana mō tēnei rapanga? He aha ai?

8 x 10 = 8080 + 16 = 96

nō reira…

Tekau ngā $8, ka $80. $16 atu anō kia eke ki te $96.

Nō reira ...

Awhina

Ngā Āhuatanga Tātai mō te Whakarea:

E toru ngā āhuatanga tātai matua mō te whakarea:

āhuatanga tātai kōaro āhuatanga tātai tohatoha āhuatanga tātai herekore

Calculation properties

Commutative property

Distributive property

Associative property

Multiplicative strategies

He mea whakamahi ēnei āhuatanga tātai i roto i ngā rautaki whakarea.

Mēnā he mārama te ākonga ki ēnei āhuatanga tātai mō te whakarea, he māmā anō tana tūhura i ngā rautaki hei whakaoti whakareatanga.

Ngā Āhuatanga Tātai mō te Whakarea:

Calculation properties

Kāore he take kia mōhio te ākonga ki ngā kupu nei (āhuatanga tātai kōaro, āhuatanga tātai tohatoha, āhuatanga tātai herekore), engari ...

Kia mōhio ia ngā tikanga o ēnei āhuatanga tātai.

Kia āta tūhura tātou i ēnei āhuatanga tātai ...

Te Āhuatanga Tātai Kōaro o te Whakareatanga:

Kāore he take o te raupapa mai o ngā tauwehe o tētahi whakareatanga ki te otinga o taua whakareatanga.

Commutativeproperty

Factor

Whakamārama atu ki tō hoa he aha e ōrite ai te otinga o te 4 x 5 me te 5 x 4.

Te Āhuatanga Tātai Kōaro o te Whakareatanga:

He ōrite te otinga o te 4 x 5 me te 5 x 4.

Commutativeproperty

Representation

Anei te kapa kotahi o te 4

Anei ngā kapa e 5 o te 4. Hei whakaahua tēnei i te 4 x 5.

5

4

Te Āhuatanga Tātai Kōaro o te Whakareatanga:

Commutativeproperty

Ināianei e 4 ngā kapa o te 5. Hei whakaahua tēnei i te 5 x 4.He ōrite te otinga o te 4 x 5 me te 5 x 4.

Hurihia te mahere tukutuku, he ōrite tonu te maha o ngā pūkeko, kāore i tāpirihia tētahi i tangohia tētahi rānei.

4

5

Te Āhuatanga Tātai Kōaro o te Whakareatanga:He ōrite te otinga o te 4 x 5 me te 5 x 4. Hei whakaahua anō:

Commutativeproperty

Representation

E 5 ngā rourou. E 4 ngā āporo kei ia rourou. Hei whakaahua tēnei i te 4 x 5.

Tohaina ngā āporo o tētahi o ngā rourou ki ērā atu o ngā rourou.

Ināianei, e 4 ngā rourou, e 5 ngā āporo kei ia rourou.Hei whakaahua tēnei i te 5 x 4.

Te Āhuatanga Tātai Kōaro o te Whakareatanga:

Commutativeproperty

Kāore i tāpirihia tētahi āporo, kāore i tangohia tētahi rānei. Nō reira he ōrite te otinga o te 4 x 5 me te 5 x 4.

Te Āhuatanga Tātai Kōaro o te Whakareatanga:

Commutativeproperty

. . .

Mēnā e mārama ana te ākonga ki te āhuatanga tātai kōaro o te whakarea, kāore he raruraru ki a ia te whakaoti i ngā whakareatanga pēnei i ēnei ...

3 x 100

Te Āhuatanga Tātai Kōaro o te Whakareatanga:

Commutativeproperty

Ka huri kōarohia te whakareatanga:100 x 3

He māmā ake te whakaoti i te 100 x 3, tērā i te whakaoti i te 3 x 100.

Reverse order

Easy number

Compensation

Hei tauira anō:3 x 99 = 99 x 3 = (huri kōaro)(100 x 3) = 300 (tau māmā)300 – 3 = 297 (tikanga paremata)

Hei tauira anō:5 x 398 = 398 x 5 = (huri kōaro)

400 x 5 = 2,000 (tau māmā)

2,000 – 10 = 1990 (tikanga paremata)

Reverse order

Easy number

Compensation

Te Āhuatanga Tātai Kōaro o te Whakareatanga:

Commutativeproperty

Te Āhuatanga Tātai Kōaro o te Whakareatanga:

Commutativeproperty

Whakaarohia tēnei rapanga:

E 78 katoa ngā ākonga o te kura i mau mai i te $4 hei utu i te pahi kawe i a rātou ki te konohete kapa haka.

Whakaarohia ngā rautaki a ngā ākonga tokorua nei. Ko tēhea e mārama ana ki te āhuatanga tātai kōaro o te whakarea?

. . .

He māmā ake te tātai i te $78 x 4

Whakareatia te $4 ki te 78 ...

Manahi Awhina

Te Āhuatanga Tātai Kōaro o te Whakareatanga:

Commutativeproperty

Te Āhuatanga Tātai Kōaro: Commutativeproperty

He aha te otinga o te 4 ÷ 2?

He aha te otinga o te 2 ÷ 4?

E whai ana te wehe i te āhuatanga kōaro, kāore rānei?

Tuhia he pikitia, ka whakamāramatia atu ki tō hoa.

Te Āhuatanga Tātai Tohatoha o te Whakarea:

Distributiveproperty

E whakareatia ana te $5 ki te 36. Arā te whārite $5 x 36 =

Ka huri kōarohia: 36 x 5 =

Whakaarohia tēnei rapanga:

E $5 te utu mō te pukapuka kotahi. E 36 ngā pukapuka i hokona e Whaea Mihi.E hia katoa te utu?

Equation

Whakaarohia tēnei rautaki hei whakaoti i tēnei whakareatanga:

Tuhia tēnei rautaki hei whārite.

36 x 5 = (30 x 5) + (6 x 5) = 150 + 30 = 180

36

5 30 x 5 = 150 6 x 5 = 30

Te Āhuatanga Tātai Tohatoha o te Whakarea:

Distributiveproperty

Te Āhuatanga Tātai Tohatoha o te Whakarea:

Distributiveproperty

I whakamahia te āhuatanga tātai tohatoha o te whakarea i roto i tēnei rautaki.

I wāwāhia te 36 ki ētahi wāhanga māmā (te 30 me te 6).

5

30 6

Te Āhuatanga Tātai Tohatoha o te Whakarea:

Distributiveproperty

DistributedKātahi ka tohaina te whakareatanga ki te 5 (x 5) ki ngā wāhanga e rua, arā, te 30 me te 6.

5

30 6

30 x 5 = 150 6 x 5 = 30

Te Āhuatanga Tātai Tohatoha o te Whakarea:

Distributiveproperty

Factor

Anei tētahi anō tauira o te āhuatanga tātai tohatoha o te whakarea:

8 x 7 = 8 x 4 + 8 x 3

Ko tēhea o ngā tauwehe i wāwāhia?

Te Āhuatanga Tātai Tohatoha o te Whakarea:

Distributiveproperty

Hei tauira anō:

Ko tēhea o ngā tauwehe i wāwāhia i konei?Tuhia te whārite e hāngai ana. Equation

Te Āhuatanga Tātai Herekore o te Whakarea:

Associativeproperty

Kia hoki tātou ki te rapanga nei:

E $5 te utu mō te pukapuka kotahi. E 36 ngā pukapuka i hokona e Whaea Mihi. E hia katoa te utu?

E whakareatia ana te $5 ki te 36. Arā te whārite $5 x 36 =

Ka huri kōarohia: 36 x 5 =

Doublingand halving

36 x 5 = 18 x 10 (te haurua me te rearua) = 180

Whakaarohia tēnei rautaki hei whakaoti i tēnei whakareatanga:

Te Āhuatanga Tātai Herekore o te Whakarea:

Associativeproperty

36

5

18

10

Te Āhuatanga Tātai Herekore o te Whakarea:

Associativeproperty

Partition

Factor

He mea whakamahi te āhuatanga tātai herekore o te whakarea i roto i tēnei rautaki. Tirohia, whakaarohia ...

36 x 5 =

= (18 x 2) x 5 (i wāwāhia te 36 kia rua ngā 18)

= 18 x (2 x 5) kāore he take o te raupapa mai o ngā tauwehe – (he ‘herekore’ te tātai)

= 18 x 10= 180

Te Āhuatanga Tātai Herekore o te Whakarea:

Associativeproperty

Cube

He tauira anō tēnei o te āhuatanga tātai herekore o te whakarea.

Whakaarohia tēnei rapanga:

E hia ngā mataono rite paku hei hanga i tēnei āhua:

Te Āhuatanga Tātai Herekore o te Whakarea:

Associativeproperty

Layer

E toru ngā raupapatanga o te whakarea hei whakaoti i tēnei rapanga:

1. (3 x 5) x 4 E 4 ngā paparanga o te 3 x 5.

Te Āhuatanga Tātai Herekore o te Whakarea:

Associativeproperty

2. (4 x 3) x 5 E 5 ngā paparanga o te 4 x 3.

3. (4 x 5) x 3 E 3 ngā paparanga o te 4 x 5.

Te Tikanga Paheko Kōaro: Inverseoperation

Ko te wehe te kōaro o te whakarea.Hei tauira ...

2 x 5 = 10

Huri kōarotia:10 ÷ 5 = 2

Ko tētahi atu āhuatanga matua o te whakarea me te wehe, ko te tikanga paheko kōaro.

Tuhia ētahi atu tauira o te tikanga paheko kōaro o te whakarea me te wehe. Whakamāramatia atu ki tō hoa.

Ko te whakarea te kōaro o te wehe.Hei tauira ...

8 ÷ 4 = 2

Huri kōarotia:2 x 4 = 8

Te Tikanga Paheko Kōaro: Inverseoperation

Te tikanga paheko kōaro o te whakarea me te wehe:

Inverseoperation

He tino rautaki te ‘huri kōaro’ hei whakaoti whakareatanga, hei whakaoti wehenga rānei.

Whakaarohia tēnei rapanga:

E 70 ngā āporo i wehea ki ētahi pēke 14, kia ōrite te maha o ngā āporo ki ia pēke. E hia ngā āporo ki tēnā, ki tēnā o ngā pēke?

Wehea te 70 ki te 14.70 ÷ 14 = ???Aue, kei hea taku tātaitai?

Manahi

Te tikanga paheko kōaro o te whakarea me te wehe:

Inverseoperation

Wehea te 70 ki te 14.70 ÷ 14 =

Ka hurihia kōarohia hei whakareatanga

14 x = 70Whakareatia te 14 ki te aha,

ka 70?Whakareatia te 14 ki te 10,

ka 140, nō reira…

Awhina

E 280 ngā pou a tētahi kaimahi pāmu. E 8 ngā pou hei hanga i te iari kotahi. E hia ngā iari ka taea e ia te hanga?

Yard

Whakaarohia tēnei rapanga me te rautaki a tēnei ākonga:

Multiplicative properties

Whakaarohia tēnei rapanga me te rautaki a tēnei ākonga:

He aha ngā āhuatanga tātai o te whakarea e whakamahia ana e tēnei ākonga?

Wehea te 280 ki te 8.280 ÷ 8 =

He rite tēnā ki te whakareatanga 8 x = 2808 x 30 = 240

E 40 atu anō kia eke ki te 2808 x 5 = 40

Hui katoa, e 35 ngā 8 kia eke ki te 280.

Whakaarohia tēnei rapanga me te rautaki a tēnei ākonga:

E 8 ngā kapa whutupōro kei roto i te whakataetae. E 22 ngā kaitākaro o ia kapa. E hia katoa ngā kaitākaro i tēnei whakataetae?

Multiplicative properties

22 x 8 = 22 x 2 = 44 44 x 2 = 88 88 x 2 = 176

He aha ngā āhuatanga tātai o te whakarea e whakamahia ana e tēnei ākonga?

Hei whakakapi ...

Kia matatau ia ki te whakamahi i ēnei āhuatanga tātai o te whakarea i roto i ā rātou rautaki whakaoti rapanga.

Commutative properties

Distributive properties

Associative properties

Kāore he take kia mōhio te ākonga ki ngā kupu nei:• āhuatanga tātai kōaro• āhuatanga tātai tohatoha• āhuatanga tātai herekore

Engari ...