Lero, tidzakutsogolerani kuti muphunzire Masamu a 4th grade ndiye werengerani. Konzani masewero olimbitsa thupi ndi masewera olimbitsa thupi kuti muphatikize chidziwitso.
1. Malangizo a masewera a masamu a sitandade 4 kuti muwerenge ndikuwerengera
1.1. Chitsanzo 1: Khazikitsani kuwerengera ndi kuchulukitsa.
Kuchulukitsa motsatana kuchokera kumanja kupita kumanzere tili ndi:
- 2 kuchulukitsa 4 ndi 8, lembani 8
2 kuchulukitsa 2 ndi 4, lembani 4
2 kuchulukitsa 1 ndi 2, lembani 2
- 4 kuchulukitsa 4 ndi 16, lembani 6 (pansi pa 4) kumbukirani 1
2 kuchulukitsa 4 ndi 8 onjezani 1 ofanana ndi 9, lembani 9
4 kuchulukitsa 1 ndi 4, lembani 4
- 1 kuchulukitsa 4 ndi 4, lembani 4 (pansi pa 9)
1 kuchulukitsa 2 ndi 2, lembani 2
1 kuchulukitsa 1 ndi 1, lembani 1
6 kuphatikiza 4 ndi 10, lembani 0 ndikukumbukira 1
2 kuphatikiza 9 ndi 11, 11 kuphatikiza 4 ndi 15, onjezani 1 kukhala 16, lembani 6 kumbukirani 1
4 kuphatikiza 2 ndi 6, onjezani 1 kukhala 7 kulemba 7
Pansi 1
Kotero 124 x 142 = 17608
M’chiwerengero pamwambapa:
- 248 imatchedwa eigenproduct yoyamba
- 496 imatchedwa eigenproduct yachiwiri. Eigenproduct yachiwiri imalembedwa ndime imodzi kubwerera kumanzere kwa eigenproduct yoyamba. Chifukwa izi ndi 496 makumi
- 124 imatchedwa eigenproduct yachitatu. Eigenproduct yachitatu imalembedwa ndime imodzi kubwerera kumanzere kwa eigenproduct ya 2. Chifukwa ichi ndi 124 mazana.
1.2. Chitsanzo 2: Khazikitsani mawerengedwe kenako werengerani ndi magawidwe.
Gawani mwadongosolo kuchokera kumanzere kupita kumanja. Masitepe motsatana: kugawa – kuchulukitsa – kuchotsa. Tili ndi :
- 144 yogawidwa ndi 17 ikufanana ndi 8, lembani 8
8 kuchulukitsa 17 ndi 136, 144 kuchotsa 136 ndi 8
- Pansi 5 ipeza 85, 85 yogawidwa ndi 17 ikufanana ndi 5
5 kuchulukitsa 17 ndi 85, 85 kuchotsa 85 ndi 0
Chifukwa chake 1445 : 17 = 85 ndi yogawa
1.3. Chitsanzo 3: Khazikitsani mawerengedwe ndikuwerengera ndi kuwonjezera.
Lamulo: Kuti tiwonjezere manambala awiri achilengedwe, titha kuchita izi:
– Lembani liwu limodzi pansi pa linzake kuti manambala a mzere womwewo afole pamzere womwewo.
– Onjezani manambala pamzere uliwonse kuchokera kumanja kupita kumanzere, mwachitsanzo, kuyambira magawo ogwira ntchito mpaka makumi, mazana, masauzande, …
Kuwerengera molingana ndi lamulo lomwe tili nalo:
1.4. Chitsanzo 4: Khazikitsani mawerengedwe kenako werengerani podula.
Lamulo: Kuti muchotse manambala awiri achilengedwe, titha kuchita izi:
– Lembani liwu limodzi pansi pa linzake kuti manambala a mzere womwewo afole pamzere womwewo.
– Chotsani manambala pamzere uliwonse kuchokera kumanja kupita kumanzere, mwachitsanzo, kuchokera kumagulu ogwira ntchito mpaka makumi, mazana, masauzande, …
2. Zochita zolimbitsa thupi
2.1. Masewera olimbitsa thupi
Phunziro 1: Khazikitsani kuwerengera kenako kuwerengera ntchito yochulukitsa.
a) 253×172
b) 146 x 160
c) 46×14
d) 1837 x 725
g) 147848 x 3
Phunziro 2: Khazikitsani mawerengedwe ndikuwerengera magawano
a) 125462: 9
b) 50562: 6
c) 2475: 36
d) 37125: 99
e) 4375: 175
g) 73645 : 416
h) 8000: 160
Phunziro 3: Khazikitsani mawerengedwe ndikuwerengera kuchotsa
- Khazikitsani mawerengedwe kenako werengerani ndikuchotsa musakumbukire .
a) 82959-10547
b) 564383 – 460532
c) 27458-6324
d) 7578-534
- Khazikitsani mawerengedwe ndikuwerengera ndikuchotsa ndi kukumbukira .
g) 567283 – 468496
h) 36270 – 13758
m) 64763 – 5697
n) 9370-999
Phunziro 4: Khazikitsani mawerengedwe ndi kuwerengera kuwonjezera
- Khazikitsani kuwerengera ndikuwerengera ndikuwonjezera osakumbukira
a) 3682 + 5217
b) 41280 + 37619
c) 28475 + 1524
d) 184759 + 413210
- Khazikitsani kuwerengera ndikuwerengera ndi kukumbukira kukumbukira.
g) 3421 + 2847
h) 17492 + 2649
m) 683992 + 28490
n) 93756 + 758
2.2. Yankho
Phunziro 1:
Kuwerengera motsatana kuchokera kumanja kupita kumanzere tili ndi:
- 2 kuchulukitsa 3 ndi 6, lembani 6
2 kuchulukitsa 5 ndi 10, lembani 0 ndikukumbukira 1
2 kuchulukitsa 2 ndi 4 onjezani 1 ofanana ndi 5, lembani 5
- 7 kuchulukitsa 3 ndi 21, lembani 1 (pansi pa ziro) kumbukirani 2
7 kuchulukitsa 5 ndi 35 onjezani 2 ofanana ndi 37, lembani 7 kumbukirani 3
7 kuchulukitsa 2 ndi 14 onjezani 3 kukhala 17, lembani 17
- 1 nthawi 3, lembani 3
1 kuchulukitsa 5 ndi 5, lembani 5
1 kuchulukitsa 2 ndi 2, lembani 2
0 kuphatikiza 1 zikufanana ndi 1, lembani 1
5 kuphatikiza 7 ndi 12, kuphatikiza 3 ndi 15, lembani 5 kumbukirani 1
7 kuphatikiza 5 ndi 12 onjezani 1 ofanana ndi 13, lembani 3 kumbukirani 1
1 kuphatikiza 2 ndi 3 kuphatikiza 1 ndi 4, lembani 4
Kotero 253 x 172 = 43516
- 0 kuchulukitsa 146 ndi 0
- 6 kuchulukitsa 6 ndi 36, lembani 6 kumbukirani 3
6 kuchulukitsa 4 ndi 24 onjezani 3 ofanana ndi 27, lembani 7 kumbukirani 2
6 kuchulukitsa 1 ndi 6 onjezani 2 kukhala 8, lembani 8
- 1 kuchulukitsa 146 ndi 146
- Pansi 6
7 kuphatikiza 6 ndi 13, lembani 3 kumbukirani 1
8 kuphatikiza 4 ndi 12 onjezani 1 ofanana ndi 13, lembani 3 kumbukirani 1
Otsika 1 ndi 1 akufanana ndi 2
Kotero 146 x 160 = 23360
- 4 kuchulukitsa 6 ndi 24, lembani 4 kumbukirani 2
4 kuchulukitsa 4 ndi 16 onjezani 2 kukhala 18, lembani 18
- 1 kuchulukitsa 46 ndi 46
- Pansi 4
8 kuphatikiza 6 ndi 14, lembani 4 kumbukirani 1
1 kuphatikiza 4 ndi 5 kumbukirani 1 ndi 6, lembani 6
Choncho 46 x 14 = 644
- 5 kuchulukitsa 35 ndi 5, lembani 5 kumbukirani 3
5 kuchulukitsa 3 kukhala 15 onjezani 3 kukhala 18, lembani 8 kumbukirani 1
5 kuchulukitsa 8 ndi 40 onjezani 1 kukhala 41, lembani 1 kukumbukira 4
5 kuchulukitsa 1 ndi 5 onjezani 4 kukhala 9, lembani 9
- 2 kuchulukitsa 7 ndi 14, lembani 4 (pansi pa 8) kumbukirani 1
2 kuchulukitsa 3 ndi 6 onjezani 1 kukhala 7, lembani 7
2 kuchulukitsa 8 ndi 16, lembani 6 kumbukirani 1
2 kuchulukitsa 1 ndi 2 onjezani 1 ofanana ndi 3, lembani 3
- 7 kuchulukitsa 7 ndi 49, lembani 9 (pansi pa 4) kumbukirani 4
7 kuchulukitsa 3 ndi 21 onjezani 4 ofanana ndi 25, lembani 5 kumbukirani 2
7 kuchulukitsa 8 ndi 56, lembani 6 kumbukirani 5
7 kuchulukitsa 1 ndi 7 onjezani 5 kukhala 12, lembani 12
- Onjezani 9185 + 3674 + 12859 = 1331825
Choncho: 1837 x 825 = 1331825
- 3 kuchulukitsa 8 ndi 24, lembani 4 kumbukirani 2
- 3 kuchulukitsa 4 ndi 12 onjezani 2 kukhala 14, lembani 4 kumbukirani 1
- 3 kuchulukitsa 8 ndi 24 onjezani 1 kukhala 25, lembani 5 kumbukirani 2
- 3 kuchulukitsa 7 ndi 21 onjezani 2 kukhala 23, lembani 3 kumbukirani 2
- 3 kuchulukitsa 4 ndi 12 onjezani 2 kukhala 14, lembani 4 kumbukirani 1
- 3 kuchulukitsa 1 kukhala 3 onjezani 1 kukhala 4, lembani 4
Chifukwa chake 147848 x 3 = 443544
Phunziro 2:
Tikuchita division tili ndi:
- 12 yogawidwa ndi 9 ikufanana ndi 1, lembani 1
1 kuchulukitsa 9 ndi 9, 12 kuchotsa 9 ndi 3
- Pansi 5 pezani 35 kugawa 9 ofanana 3, lembani 3
3 kuchulukitsa 9 ndi 27, 35 kuchotsa 27 ndi 8
- Pansi 4 ndi 84 ogawidwa ndi 9 ofanana ndi 9, lembani 9
9 kuchulukitsa 9 ndi 81, 84 kuchotsa 81 ndi 3
- Pansi 6 pezani 36 kugawa 9 ofanana 4, lembani 4
4 kuchulukitsa 9 ndi 36, 36 kuchotsa 36 ndi 0
- Pansi 2, 2 sigawidwa ndi 9, lembani 0 yotsala 2
Kotero 125462 : 9 = 13940 (zotsalira 2)
- 50 yogawidwa ndi 6 ikufanana ndi 8, lembani 8
8 kuchulukitsa 6 ndi 48, 50 kuchotsa 48 ndi 2
- Ochepera 5 apeza 25 kugawidwa ndi 6 kukhala 4, lembani 4
4 kuchulukitsa 6 ndi 24, 25 kuchotsa 24 ndi 1
- Pansi 6 pezani 16 kugawidwa ndi 6 kukhala 2, lembani 2
2 kuchulukitsa 6 ndi 12, 16 kuchotsa 12 ndi 4
- Pansi 2 pezani 42 kugawidwa ndi 6 ofanana ndi 7, lembani 7
7 kuchulukitsa 6 ndi 42, 42 kuchotsa 42 ndi 0
Ndiye 50562 : 6 = 8427
- 247 ogawidwa ndi 36 ofanana ndi 6, lembani 6
6 kuchulukitsa 36 ndi 216, 247 kuchotsa 216 ndi 31
- Pansi 5 pezani 315 kugawa 36 ofanana 8, lembani 8
8 kuchulukitsa 36 ndi 288, 315 kuchotsa 288 ndi 27
Kotero 2475 : 36 = 68 otsala 27
- 371 yogawidwa ndi 99 ikufanana ndi 3, lembani 3
3 kuchulukitsa 99 ndi 297, 371 kuchotsa 297 ndi 74
- Pansi 2 pezani 742 kugawa 99 ofanana 7, lembani 7
7 kuchulukitsa 99 ndi 693, 742 kuchotsa 693 ndi 49
- Pansi 5 pezani 495 kugawa 99 kukhala 5, lembani 5
5 kuchulukitsa 99 ndi 495, 495 kuchotsa 495 ndi 0
Ndiye 37125 : 99 = 375
- 437 yogawidwa ndi 175 ikufanana ndi 2, lembani 2
2 kuchulukitsa 175 ndi 350, 437 kuchotsa 350 ndi 87
- Pansi 5 pezani 875 kugawidwa ndi 175 kukhala 5, lembani 5
5 kuchulukitsa 175 ndi 875, 875 kuchotsa 875 ndi 0
Ndiye 4375 : 175 = 25
- 736 yogawidwa ndi 416 ndi 1, lembani 1
1 kuchulukitsa 416 ndi 416, 736 kuchotsa 416 ndi 320
- Pansi 4 pezani 3204 kugawa 416 ofanana 7, lembani 7
7 kuchulukitsa 416 ndi 2912, 3204 kuchotsa 2912 ndi 292
- Pansi 5 ndi 2925 yogawidwa ndi 416 ikufanana ndi 7, lembani 7
7 kuchulukitsa 416 ndi 2912, 2925 kuchotsa 2912 ndi 13
Kotero 73645 : 416 = 177 otsala 13
- 800 yogawidwa ndi 160 ikufanana ndi 5, lembani 5
5 kuchulukitsa 160 ndi 800, 800 kuchotsera 800 kufanana ndi 0
- 0 yogawidwa ndi 160 ikufanana ndi 0, lembani 0
Ndiye 8000 : 160 = 50
Phunziro 3:
Kuwerengera motsatana kuchokera kumanja kupita kumanzere tili ndi:
- 9 kuchotsa 7 ndi 2, lembani 2
- 5 kuchotsa 4 ikufanana ndi 1, lembani 1
- 9 kuchotsa 5 ndi 4, lembani 4
- 2 kuchotsa 0 ndi 2, lembani 2
- 8 kuchotsa 1 ndi 7, lembani 7
Chifukwa chake 82959 – 10547 = 72412
- 3 kuchotsa 2 ndi 1, lembani 1
- 8 kuchotsa 3 ndi 5, lembani 5
- 3 kuchotsa 0 ndi 3, lembani 3
- 4 kuchotsa 3 ikufanana ndi 1, lembani 1
- 6 kuchotsa 6 ndi 0, lembani 0
- 5 kuchotsa 4 ikufanana ndi 1, lembani 1
Chifukwa chake 564383 – 463032 = 101351
- 8 kuchotsa 4 ndi 4, lembani 4
- 5 kuchotsa 2 ndi 3, lembani 3
- 4 kuchotsa 3 ikufanana ndi 1, lembani 1
- 7 kuchotsa 6 ndi 1, lembani 1
- pansi 2
Chifukwa chake 27458 – 6324 = 21134
- 8 kuchotsa 4 ndi 4, lembani 4
- 7 kuchotsa 3 ndi 4, lembani 4
- 5 kuchotsera 5 kufanana ndi 0, lembani 0
- pansi 7
Chifukwa chake 7578 – 534 = 7044
- 13 kuchotsa 6 ndi 7, lembani 7 kumbukirani 1
- 18 kuchotsera 9 kufanana ndi 9, 9 kuchotsera 1 kufanana ndi 8, lembani 8 kumbukirani 1
- 12 kuchotsa 4 ikufanana ndi 8, kuchotsa 1 ikufanana ndi 7, lembani 7 kumbukirani 1
- 17 kuchotsera 8 akufanana ndi 9, kuchotsera 1 akufanana ndi 8, lembani 8 kumbukirani 1
- 16 kuchotsera 6 kufanana ndi 10, kuchotsera 1 kufanana ndi 9, lembani 9 kumbukirani 1
- 5 kuchotsera 4 ikufanana ndi 1, 1 kuchotsera 1 ikufanana ndi 0
Kotero 567283 – 468496 = 98787
- 10 kuchotsa 8 ndi 2, lembani 2 kumbukirani 1
- 7 kuchotsa 5 ikufanana ndi 2, 32 kuchotsa 1 ikufanana ndi 1, lembani 1
- 12 kuchotsa 7 ndi 5, lembani 5 kumbukirani 1
- 6 kuchotsa 3 ndi 3, 3 kuchotsa 1 ndi 2, lembani 2
- 3 kuchotsa 1 ndi 2, lembani 2
Kotero 36270 – 13758 = 22512
- 13 kuchotsa 7 ndi 6, lembani 6 kumbukirani 1
- 16 kuchotsera 9 ndi 7, 7 kuchotsera 1 kufanana 6, lembani 6 kumbukirani 1
- 7 kuchotsera 6 ndi 1, 1 kuchotsa 1 ndi 0, lembani 0
- 14 kuchotsa 5 ndi 9, lembani 9 kumbukirani 1
- 6 kuchotsa 0 ndi 6, 6 kuchotsa 1 ndi 5, lembani 5
Chifukwa chake 64763 – 5697 = 59066
- 10 kuchotsera 9 ndi 1, lembani 1 kukumbukira 1
- 17 kuchotsera 9 ndi 8, 8 kuchotsera 1 kufanana ndi 7, lembani 7 kumbukirani 1
- 13 kuchotsera 9 ndi 4, 4 kuchotsera 1 kufanana 3, lembani 3 kumbukirani 1
- 9 kuchotsa 0 ndi 9, 9 kuchotsa 1 ndi 8, lembani 8
Kotero 9370 – 999 = 8371
Phunziro 4:
Kupanga kuwonjezera kuchokera kumanja kupita kumanzere tili ndi:
- 2 kuphatikiza 7 ndi 9, lembani 9
- 8 kuphatikiza 1 ndi 9, lembani 9
- 6 kuphatikiza 2 ndi 8, lembani 8
- 3 kuphatikiza 5 ndi 8, lembani 8
Chifukwa chake 3682 + 5217 = 8899
- 0 kuphatikiza 9 ndi 9, lembani 9
- 8 kuphatikiza 1 ndi 9, lembani 9
- 2 kuphatikiza 6 ndi 8, lembani 8
- 1 kuphatikiza 7 ndi 8, lembani 8
- 4 kuphatikiza 3 ndi 7, lembani 7
Chifukwa chake 41280 + 37619 = 78899
- 5 kuphatikiza 4 ndi 9, lembani 9
- 7 kuphatikiza 2 ndi 9, lembani 9
- 4 kuphatikiza 5 ndi 9, lembani 9
- 8 kuphatikiza 1 ndi 9, lembani 9
- pansi 2 pansi
Chifukwa chake 28475 + 1524 = 29999
- 9 kuphatikiza 0 ndi 9 kulemba 9
- 5 kuphatikiza 1 ndi 6 kulemba 6
- 7 kuphatikiza 2 ndi 9, lembani 9
- 4 kuphatikiza 3 ndi 7, lembani 7
- 8 kuphatikiza 1 ndi 9, lembani 9
- 1 kuphatikiza 4 ndi 5, lembani 5
Kotero 184759 + 413210 = 597969
- 2 kuphatikiza 9 ndi 11, lembani 1 kukumbukira 1
- 9 kuphatikiza 4 ndi 13 onjezani 1 ofanana ndi 14, lembani 4 kumbukirani 1
- 4 kuphatikiza 6 ndi 10 kuphatikiza 1 ndi 11, lembani 1 kukumbukira 1
- 7 kuphatikiza 2 ndi 9 kuphatikiza 1 ndi 10, lembani 0 kumbukirani 1
- pansi 1 kuphatikiza 1 zikufanana ndi 2, lembani 2
Chifukwa chake 17492 + 2649 = 20141
- 2 kuphatikiza 0 ndi 2, lembani 2
- 9 kuphatikiza 9 ndi 18, lembani 8 kumbukirani 1
- 9 kuphatikiza 4 ndi 13 onjezani 1 ofanana ndi 14, lembani 4 kumbukirani 1
- 3 kuphatikiza 8 ndi 11 onjezani 1 ofanana ndi 12, lembani 2 kumbukirani 1
- 8 kuphatikiza 2 ndi 10 kuphatikiza 1 ndi 11, lembani 1 kukumbukira 1
- Pansi 6 ndi 1 ikufanana ndi 7, lembani 7
Kotero 683992 + 28490 = 712482
- 6 kuphatikiza 8 ndi 14, lembani 4 kumbukirani 1
- 5 kuphatikiza 5 ndi 10 kuphatikiza 1 ndi 11, lembani 1 kukumbukira 1
- 7 kuphatikiza 7 ndi 14 onjezani 1 ofanana ndi 15, lembani 5 kumbukirani 1
- Pansi 93 ndi 1 ikufanana ndi 94, lembani 94
Kotero 93756 + 758 = 94514
3. Zochita zoyeserera: Khazikitsani kuwerengera kenako werengerani mafunso asanu a mayeso a semesita yoyamba ya masamu a sitandade 4.
3.1. Masewera olimbitsa thupi
Mutu 1: Khazikitsani mawerengedwe ndikuwerengera
a) 1998: 14
b) 235 x 19
c) 104562 + 572820
d) 864937 – 364024
Mutu 2: Khazikitsani mawerengedwe ndikuwerengera
a) 365852 + 25893
b) 57395 – 49375
c) 308×563
d) 7564: 72
Mutu 3: Khazikitsani mawerengedwe ndikuwerengera
a) 75995 + 50248
b) 437520 – 58038
c) 576949 x 4
d) 9603: 7
Mutu 4: Khazikitsani mawerengedwe kenako werengerani
a) 9172: 653
pa 56×92
c) 7539 + 8290
d) 8493-7493
Mutu 5: Khazikitsani mawerengedwe ndikuwerengera
a) 36075: 925
b) 28403 x 5
c) 57760 + 30149
d) 674029 – 521007
3.2. Yankhani
Mutu 1:
a) 142 (zotsalira 10)
b) 4465
c) 677382
d) 503913
Mutu 2:
ndi) 391745
b) 8020
c) 173404
d) 105 (otsala 4)
Mutu 3:
ndi) 126243
b) 379482
c) 2307796
d) 1371 (otsala 6)
Mutu 4:
a) 14 (otsala 30)
b) 5152
c) 15829
d) 1000
Mutu 5:
a) 39
b) 142015
c) 87909
d) 153022
Masamu a 4 giredi ndikuwerengera ndi njira yowerengera ya 4 kuchulukitsa, kugawa, kuwonjezera, ndi kuchotsa. Kuti muphunzire bwino masamu awa, muyenera kuchita khama komanso kutitsata pafupipafupi kuti muwonjezere chidziwitso chanu.
