Ukuba sele uqalile ukufumana imithambo evela kwi Ipropathi yokutshintshiselana, kwaye awuyazi ukuba yintoni kunye nendlela esetyenziswa ngayo. Emva koko ufike kwinqaku elichanekileyo, kuba apha siza kuchaza yonke into ofuna ukuyifumana ukuze ufunde ngokulula ukwenza le misebenzi yemathematika. Kuya kuba mnandi!

## Ipropathi yokutshintsha

Ipropathi yokutshintsha yinto yemathematika apho sizama ukufumana ukulingana kwisiphumo ngokudibanisa izinto zaso, sibonisa ukuba ngaphandle kokulandelelana kwazo, imveliso yokugqibela iyakufana.

## Ipropathi yokuguqula ngokudibanisa

Kule mi sebenzi eyongeziweyo, nangona iyahluka ngokulandelelana kwayo, iya kusinika iziphumo ezifanayo. Masenze umthambo njengomzekelo:

a) 6 + 5 = 5 + 6

11 = 11.

Siyabona ukuba xa sitshintsha i-odolo yezinto, sifumene iziphumo ezifanayo. Emva koko oku kulandelayo kuyasekwa:

A + B = C kunye no-B + A = C.

#### Ipropathi yokutshintsha: imizekelo

Masisebenzise eminye imizekelo ukuze siyiqonde ngcono:

b) UMaria unebhokisi enemibala eli-12 kwaye wadibana noJuana oneminye imibala eli-10. UJuana ubeke imibala yakhe kwibhokisi kaMaria. Mingaphi imibala xa iyonke ebhokisini?

Siqhubeka noku-odola idatha yengxaki, ngolu hlobo lulandelayo:

Sibeka kuqala imibala eli-12 kaMaria kunye nemibala eli-10 kaJuana.

b) 12 + 10 =?

Ukuba sidibanisa amanani omabini, isiphumo siya kuba: 22.

Kwaye kwenzeka ntoni xa sitshintsha iodolo abanayo kwaye siyibeka ngolu hlobo:

b) 10 + 12 =?

Ukongeza amanani omabini, isiphumo siya kuba ngama-22.

Kulula kakhulu, akunjalo?

Singaqhubeka nokubeka eminye imizekelo:

c) UPedro uneebhaluni ezingama-50 ezinemibala kwaye uLuis unama-30. Zingaphi iibhaluni uPedro noLuis abanazo zizonke?

Sibeka iibhaluni ezingama-50 zikaPedro kuqala kunye neebhaluni ezingama-30 zikaLuis, ngolu hlobo lulandelayo:

50 + 30 = 80

Ukuba sidibanisa omabini la manani sifumana: 80 ngenxa yoko.

Kwaye ukuba sitshintsha iodolo:

30 + 50 = 80.

Sinemveliso yokugqibela efanayo nomsebenzi wokuqala.

Emva koko sinokukufumanisa oko, ukuba sitshintsha iodolo yezongezo sifika kwisiphumo esinye. Jonga yonke le mithambo:

d) 35 + 45 = 45 + 35 e) 25 + 17 = 17 + 25 f) 22 + 51 = 51 + 22

80 = 80. 42 = 42 73 = 73

Ilula kakhulu kwaye iyonwabisa.

## Ipropathi yokuguqula ukuphindaphinda

Ngendlela efanayo, sinokwenza imisebenzi nge ukuguqulwa kwepropathi yokuphindaphindwa Izinto. Masenze lo mzekelo ulandelayo:

a) 5 x 3 = 3 x 5

15 = 15.

Siyabona ukuba xa sitshintsha imeko yezinto eziphindaphindayo, sifumana iziphumo ezifanayo kumacala omabini. Ke, sinokuseta oku kulandelayo:

A x B = C kunye B x A = C.

#### Ipropathi yokutshintsha: imizekelo

Masibeke omnye umzekelo ukuze siyiqonde ngcono le nkqubo:

b) UMonica unevenkile ethengisa ii-ayisi khrim kwaye bacele i-ice cream ze-2 zeetafile ze-4 zeetafile ezi-XNUMX. Zingaphi ii-ayisikhrim ekufuneka zenziwe nguMonica?

Siqhubeka noku-odola idatha yengxaki, ngolu hlobo lulandelayo:

Kuqala ii-ice cream ezi-2 kunye neetafile ezi-4, ezinje:

b) 2 x 4 =?

Ukuba siphinda-phinda isi-2 ngesi-4, sifumana eli nani: 8.

2 x 4 = 8.

Kwenzeka ntoni ukuba sitshintsha iodolo ngoku, kwaye sibhale:

Iitheyibhile ezi-4 kunye ne-ice cream ezingama-2.

4 x 2 =?

Xa siphinda-phinda omabini amanani, iziphumo esizifumanayo ziya kufana: 8.

4 x 2 = 8.

Singatsho ke ukuba uMonica kufuneka alungiselele ii-ayisikhrim ezisi-8 zizonke ukusasaza kwiitafile ezi-4.

Umsebenzi omhle!

Masisebenzise eminye imizekelo:

c) Abantu aba-6 bahlala endlwini kaJosé kwaye kufuneka alungiselele isidlo sangokuhlwa, kufuneka enze izinja ezi-2 ezishushu kwilungu ngalinye losapho. Zingaphi izinja ezishushu ekufuneka ezenzile uJosé?

Masiqale ngoku-odola izinto zengxaki ngolu hlobo lulandelayo:

Sibhala kuqala abantu aba-6 kunye nezinja ezi-2 ezishushu, ngolu hlobo:

c) 6 x 2 =?

Kufuneka siphindaphinde kwaye iziphumo ziya kuba: 12.

Kwaye ukuba siyazitshintsha izinto, siza kubhala ngolu hlobo:

Izinja ezi-2 ezishushu kunye nabantu aba-6.

2 x 6 =?

Siphinda-phinda ngoku kwaye sifumana: 12.

2 x 6 = 12.

Sigqiba ngokuthi uJosé makalungiselele izinja ezishushu ezili-12 ekhaya zelungu ngalinye losapho. Kakuhle kakhulu!.

Makhe sijonge kule mithambo ingezantsi:

d) 8 x 6 = 6 x 8 e) 12 x 5 = 5 x 12 f) 15 x 2 = 2 x 15

48 = 48. 60 = 60. 30 = 30.

Kulula kakhulu ukwenza umthambo wepropathi eguqukayo, kufuneka uzilolonge ngokwaneleyo ukuze uzenze njengomdlalo.

### Yintoni na?

La commutative kunye nepropathi yokudibanisa ziyafana, kodwa zahlukile kwinani lamanani owasebenzisayo ukwenza imisebenzi yakho.

Ukusukela, ukuguquguquka kusetyenziswa amanani amabini kunye nokutshintshiselana ngesikhundla sabo, ngelixa indibaniselwano isebenzisa amanani amathathu nangaphezulu kwaye itshintsha iodolo. Kuzo zombini iipropathi injongo kukufumana ukulingana okuqulethwe kumanani, ukufumana inani elifanayo nemveliso yokugqibela, ngaphandle kokulandelelana kwezinto.

Kuqhelekile ukubuza ukuba zeziphi ezi ndawo? Ewe, ngaphandle kokuba yimisebenzi eyonwabisayo yokwenza, inokusetyenziswa kubomi bemihla ngemihla, ukubonelela ngeziphumo kwimizuzwana embalwa, ngaphandle kwesidingo sokubhenela kwimithambo ebanzi enokuhlala ixesha elide.

Ukuba ufuna ukufunda ukongeza ngaphezulu kwamanani amabini kwaye, ufuna ukwenza ulwazi lwakho malunga nepropathi yokuzibandakanya kunye nokudibanisa, kwaye ndiyakumema ukuba ufunde eli nqaku linomdla: Funda ukongeza amanani angaphezu kwamabini.

### Ukuthabatha kunye nokwahlulahlula

Kubalulekile ukuba uqaphele ukuba kwipropathi yotshintsho kuhlala kumiselwe ukuba "ukulandelwa kwezinto akuyitshintshi imveliso." Ngumgaqo-siseko ekufuneka usoloko uthathelwa ingqalelo, kuba ukuba sizama ukusebenzisa le propati kwimisebenzi enjengokuthabatha kunye nokwahlulahlula, asiyi kufumana ukulingana.

Oko kukuthi, ukuba sitshintsha iodolo yezinto esizenzayo (ukuthabatha okanye ukwahlula), isiphumo asizukufana, ke ngoko, ipropathi ayinakusetyenziswa kuba akusayi kubakho kulingana kwimveliso yokugqibela.

Ukukhupha: A - B = C ≠ B - A = D.

Icandelo: A ÷ B = C ≠ B ÷ A = D.

### Umlando omncinci

Ingcali yezibalo yamaGrike kunye nejometri uEuclides; Uthethe ngomsebenzi wakhe "Elements" malunga nokutshintsha kwepropathi. Sithetha ngenkulungwane ezi-3 ngaphambi kukaKrestu.

Kodwa, kwakudala kwe-Egypt kwakusekho ulwazi malunga nepropathi eguqukayo kwaye yayisetyenziselwa ukufumana iziphumo ezikhawulezayo nezisebenzayo ukubala iimveliso ezazithengiswa ngelo xesha. Kuyamangalisa!

Nangona yayisetyenziswa ukususela kumaxesha amandulo, kufuneka sikhankanye ukuba yayingasetyenziswa ngokusesikweni de kufike umfundisi waseFrance kunye nengcali yezibalo egama linguFrancois Servois; Owabhala ngo-1814 kwi-calculus eyahlukileyo esebenzisa le migaqo okokuqala. Inomdla!.