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

2. Ngā whāinga mō tēnei akoranga:
• 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.
• Kia mōhio ki ngā āhuatanga tātai o te whakarea me te wehe.
Calculation properties
Nga whainga mo tenei akoranga
• Kia mārama ki te whakamahinga o ēnei āhuatanga tātai i roto i ngā rautaki matua mō te whakarea me te wehe.

3. 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, ), 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).
Number operation
Ko te wehe te kupu e tino whakamahia ana mō te ‘division’.

4. 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.

5. 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.

6. Ngā kupu mō te whakarea me te wehe:
E rua ngā tauwehe o tētahi whakareatanga.
Factor
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.
tau e whakareatia ana
2 x 3 = 6
otinga
Multipicand
tau whakarea
Multiplier
He tauwehe te rua me te toru o te ono.
He aha ngā tauwehe katoa o te 12?

7. Ngā kupu mō te whakarea me te wehe:
Mō te wehe:
tau e wehea ana
18 ÷ 6 = 3
otinga
Dividend
tau whakawehe
Divisor

8. Te whakahua me te whakaahua i te whakareatanga:
Expressing,
representing
E rua ngā whakahuatanga matua mō te whakarea: 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
Animation required.
For first diagram, show one group of 2, then 2, then 3 groups of 2. As this is happening the number line jumps are happening.
For second diagram show one group of 3 then 2 and as this happens the jumps on the number line appears
Ko te 2 te rōpū e whakareatia ana. Kia 3 ngā rōpū o te 2.

9. Te whakahua me te whakaahua i te whakareatanga:
Expressing,
representing 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.

10. 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, p154)
Kua riro ko tēnei hei tikanga matua mō te whakarea i roto i te reo Māori:
“Whakareatia te rua ki te toru, ka ono.”
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, p154)
2 x 3 = 6

11. Te whakahua me te whakaahua i te whakareatanga:
Expressing,
representing
Engari, kia mōhio hoki te ākonga, kei te tika hoki tēnei:
“E rua ngā rōpū (huinga) o te toru, ka ono.”
(“E rua ngā toru, ka ono”)
2 x 3 = 6

12. Te whakahua i te rerenga whakareatanga:
Expressing
E whai ake nei ētahi o ngā whakahuatanga rerenga kōrero mō te whakarea e rangona ana i roto i ō tātou kura.
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?

13. Te whakahua i te rerenga whakareatanga:
Expressing
5 x 3 = 15
 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.

14. Te Whakaaro Tāpiripiri me te Whakaaro Whakarearea:
Additive thinking,
Multiplicative thinking
Whakaarohia te rapanga nei:
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?

15. Te Whakaaro Tāpiripiri me te Whakaaro Whakarearea:
Additive thinking,
Multiplicative thinking
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?
= 88
= 80
= 72
Manahi

16. Tekau ngā $8, ka $80. $16 atu anō kia eke ki te $96. Nō reira ...
Te Whakaaro Tāpiripiri me te Whakaaro Whakarearea:
Additive thinking,
Multiplicative thinking
Tekau ngā $8, ka $80. $16 atu anō kia eke ki te $96. Nō reira ...
8 x 10 = 80
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?
= 96
nō reira…
Awhina

17. Ngā Āhuatanga Tātai mō te Whakarea:
Calculation properties
E toru ngā āhuatanga tātai matua mō te whakarea:
 āhuatanga tātai kōaro
āhuatanga tātai tohatoha
āhuatanga tātai herekore
Commutative property
Distributive property
Associative property
He mea whakamahi ēnei āhuatanga tātai i roto i ngā rautaki whakarea.
Multiplicative strategies
Mēnā he mārama te ākonga ki ēnei āhuatanga tātai mō te whakarea, he māmā anō tana tūhura i ngā rautaki hei whakaoti whakareatanga.

18. 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 ...

19. Te Āhuatanga Tātai Kōaro o te Whakareatanga:
Commutative property
Kāore he take o te raupapa mai o ngā tauwehe o tētahi whakareatanga ki te otinga o taua whakareatanga.
Factor
Whakamārama atu ki tō hoa he aha e ōrite ai te otinga o te 4 x 5 me te 5 x 4.

20. Te Āhuatanga Tātai Kōaro o te Whakareatanga:
Commutative property
He ōrite te otinga o te 4 x 5 me te 5 x 4.
Anei te kapa kotahi o te 4
Anei ngā kapa e 5 o te 4. Hei whakaahua tēnei i te 4 x 5.
Representation 4 5

21. Te Āhuatanga Tātai Kōaro o te Whakareatanga:
Commutative property
Hurihia te mahere tukutuku, he ōrite tonu te maha o ngā pūkeko, kāore i tāpirihia tētahi i tangohia tētahi rānei. 5 4
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.

22. Te Āhuatanga Tātai Kōaro o te Whakareatanga:
Commutative property
He ōrite te otinga o te 4 x 5 me te 5 x 4. Hei whakaahua anō:
Representation
E 5 ngā rourou. E 4 ngā āporo kei ia rourou. Hei whakaahua tēnei i te 4 x 5.
Animation required.
After the line ‘Tohaina ngā āporo ...’ one apple from the basket on the right flys off to each of the other baskets to make 5 apples in those 4 baskets and the fifth basket is empty and greyed out (but still visible in the background)
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.

23. Te Āhuatanga Tātai Kōaro o te Whakareatanga:
Commutative property
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.
Animation required.
After the line ‘Tohaina ngā āporo ...’ one apple from the basket on the right flys off to each of the other baskets to make 5 apples in those 4 baskets and the fifth basket is empty and greyed out (but still visible in the background)

24. Te Āhuatanga Tātai Kōaro o te Whakareatanga:
Commutative property
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
. . .

25. Te Āhuatanga Tātai Kōaro o te Whakareatanga:
Commutative property
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.

26. Te Āhuatanga Tātai Kōaro o te Whakareatanga:
Commutative property
Hei tauira anō:
3 x 99 = 
99 x 3 =  (huri kōaro)
Reverse order
(100 x 3) = 300 (tau māmā)
Easy number
300 – 3 = 297 (tikanga paremata)
Compensation
Hei tauira anō:
5 x 398 = 
398 x 5 =  (huri kōaro)
Reverse order
400 x 5 = 2,000 (tau māmā)
Easy number
2,000 – 10 = 1990 (tikanga paremata)
Compensation

27. Te Āhuatanga Tātai Kōaro o te Whakareatanga:
Commutative property
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.

28. He māmā ake te tātai i te $78 x 4
Te Āhuatanga Tātai Kōaro o te Whakareatanga:
Commutative property
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
. . .
Manahi
Awhina

29. Te Āhuatanga Tātai Kōaro:
Commutative property
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.

30. Te Āhuatanga Tātai Tohatoha o te Whakarea:
Distributive property
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?
Animation required
The 36 coloumns of 5 pūkeko partitions into 30 columns of 5 and 6 columns of 5
30 x 5 = 150 appears on that group of pūkeko and 6 x 5 = 30 appears on that group of pūkeko
E whakareatia ana te $5 ki te 36. Arā te whārite $5 x 36 = 
Ka huri kōarohia: 36 x 5 = 

31. Te Āhuatanga Tātai Tohatoha o te Whakarea:
Distributive property
Whakaarohia tēnei rautaki hei whakaoti i tēnei whakareatanga: 36 5
30 x 5 = 150
6 x 5 = 30
Tuhia tēnei rautaki hei whārite.
Equation
36 x 5 = (30 x 5) + (6 x 5) =
= 180

32. Te Āhuatanga Tātai Tohatoha o te Whakarea:
Distributive property
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). 30 6 5
Animation required
Same animation as previous slide

33. Te Āhuatanga Tātai Tohatoha o te Whakarea:
Distributive property
Kātahi ka tohaina te whakareatanga ki te 5 (x 5) ki ngā wāhanga e rua, arā, te 30 me te 6.
Distributed 30 6 5
30 x 5 = 150
6 x 5 = 30
Animation required
Same animation as previous slide

34. Te Āhuatanga Tātai Tohatoha o te Whakarea:
Distributive property
Anei tētahi anō tauira o te āhuatanga tātai tohatoha o te whakarea:
8 x 7 = 8 x x 3
Animation Required
For first diagram. There are 7 groups of 8 pūkeko. These partition in to 4 groups of 8 and 3 groups of 8
For the second diagram. There are 7 groups of 8 pūkeko. These partition off into 7 groups of 3 and 7 groups of 5
Ko tēhea o ngā tauwehe i wāwāhia?
Factor

35. Te Āhuatanga Tātai Tohatoha o te Whakarea:
Distributive property
Hei tauira anō:
Ko tēhea o ngā tauwehe i wāwāhia i konei?
Tuhia te whārite e hāngai ana.
Animation Required
For first diagram. There are 7 groups of 8 pūkeko. These partition in to 4 groups of 8 and 3 groups of 8
For the second diagram. There are 7 groups of 8 pūkeko. These partition off into 7 groups of 3 and 7 groups of 5
Equation

36. Te Āhuatanga Tātai Herekore o te Whakarea:
Associative property
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 = 
Animation required
Starts off with 36 columns of 5 pūkeko. Then 18 columns of 5 from the right hand side seperate off and move to sit underneath the 18 columns of 5 on the left to make a grou pof 18 columns of 10. The numbers appear at the top and left of the columns to show how many.
Ka huri kōarohia: 36 x 5 = 

37. Te Āhuatanga Tātai Herekore o te Whakarea:
Associative property
Whakaarohia tēnei rautaki hei whakaoti i tēnei whakareatanga: 18 36 5 10
36 x 5 = 18 x 10 (te haurua me te rearua)
= 180
Doubling and halving

38. Te Āhuatanga Tātai Herekore o te Whakarea:
Associative property
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)
Partition
= 18 x (2 x 5) kāore he take o te raupapa mai o ngā tauwehe – (he ‘herekore’ te tātai)
Factor
= 18 x 10
= 180

39. Te Āhuatanga Tātai Herekore o te Whakarea:
Associative property
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:
Cube

40. Te Āhuatanga Tātai Herekore o te Whakarea:
Associative property
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.
Layer

41. Te Āhuatanga Tātai Herekore o te Whakarea:
Associative property
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.

42. Te Tikanga Paheko Kōaro:
Inverse operation
Ko tētahi atu āhuatanga matua o te whakarea me te wehe, ko te tikanga paheko kōaro.
Ko te wehe te kōaro o te whakarea.
Hei tauira ...
2 x 5 = 10
Animation required
First diagram: Starts off with 5 groups of 2. Each group flys off one at a time to make one large group of 10
Second diagram: Starts off with one group of 10. Each group of 2 fly out of the set to make 5 groups of 2.
Third diagram: Starts off with one group of 8. Each group of 2 fly out of the set to make 4 groups of 2.
Fourth diagram: Starts off with 4 groups of 2. Each group flys off one at a time to make one large group of 8
Huri kōarotia:
10 ÷ 5 = 2

43. Te Tikanga Paheko Kōaro:
Inverse operation
Ko te whakarea te kōaro o te wehe. Hei tauira ...
8 ÷ 4 = 2
Huri kōarotia:
2 x 4 = 8
Tuhia ētahi atu tauira o te tikanga paheko kōaro o te whakarea me te wehe.
Whakamāramatia atu ki tō hoa.

44. Aue, kei hea taku tātaitai?
Te tikanga paheko kōaro o te whakarea me te wehe:
Inverse operation
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

45. Te tikanga paheko kōaro o te whakarea me te wehe:
Inverse operation
Wehea te 70 ki te 14.
70 ÷ 14 = 
Ka hurihia kōarohia hei whakareatanga
14 x  = 70
Whakareatia te 14 ki te aha, ka 70?
Whakareatia te 14 ki te 10, ka 140, nō reira…
Awhina

46. Whakaarohia tēnei rapanga me te rautaki a tēnei ākonga:
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

47. Whakaarohia tēnei rapanga me te rautaki a tēnei ākonga:
Wehea te 280 ki te 8.
280 ÷ 8 = 
He rite tēnā ki te whakareatanga 8 x = 280
8 x 30 = 240
E 40 atu anō kia eke ki te 280
8 x 5 = 40
Hui katoa, e 35 ngā 8 kia eke ki te 280.
He aha ngā āhuatanga tātai o te whakarea e whakamahia ana e tēnei ākonga?
Multiplicative properties

48. 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?
22 x 8 =  x 2 = 44
44 x 2 = x 2 = 176
He aha ngā āhuatanga tātai o te whakarea e whakamahia ana e tēnei ākonga?
Multiplicative properties

49. Hei whakakapi ... Kāore he take kia mōhio te ākonga ki ngā kupu nei:
• āhuatanga tātai kōaro
• āhuatanga tātai tohatoha
• āhuatanga tātai herekore
Commutative properties
Distributive properties
Associative properties
Engari ...
Kia matatau ia ki te whakamahi i ēnei āhuatanga tātai o te whakarea i roto i ā rātou rautaki whakaoti rapanga.