Enkalala z'emiramwa gy'Ekibalangulo

Gakuweebwa Charles Muwanga !! 

Emiramwa gy’Ekibalangulo emigandawaze

1.	Ekibalangulo(Mathematics)

Lino “sessomo” eriwagala obwongo nga lisengeka ebirowoozo  nga lyogeza   namba. Essomo lino n’olwekyo liyamba okukulaakulanya  “ EkkyO”(kisomwa  “Ekkyu”)  y’omuntu.  Ekibalangulo mu bumpi kiwandiikibwa “ekibalo”(maths). Gino gye gimu ku mibalanguzo( operational concepts) egyetaagisa mu sessomo lino:

a)	Ekibalangulo		(mathematics)

b)	Ekibalo 		(maths)

c)	Namba=erinnya   	(number)

d)	Nambiso	=kabonero	(numeral)

e)	Ekikunizo		(mathematical problem(question)

f)	Omweyoreko		mathematical expression

g)	Okusonjola  		(to simplify)

h)	 Okubalangula		(to calculate)

i)	okubalanguza		(to calculate)

j)	Okubaza 		(to solve)

k)	Ekibazo			(solution to mathematical problem)

l)	Ekibalanguzo		(formula)

m)	Omubalanguzo/akabalanguzo		(mathematical operation)

n)Sessomo (Academic discipline)

o) Essomo(Topic, subsdiscipline, subfield of study)

p) Essomeso  (Lesson)

q)Okuteekateeka essomeso /Enteekateeka y'essomeso (Lesson planning)

r)Essomesso etteeketeeke (Lesson plan)

2. Amasomo agali mu  Sessomo ly’Ekibalangulo

Sessomo ly’ekibalangulo lirimu amasomo gano wansi:

i)	Ekibalirizo	(arithmetic ,calculations using    numbers only )

ii)	Aligebbula	       (algebra,calculations using both numbers and letters)

iii)	Omugereeso gwa namba			(number theory)

iv)	Emisingi gya namba(Emisimba)		(number bases)

v)	Omupimo oba essomampimo			(geometry)

vi)	Emikutule				(fractions)

vii)	Emitonnyeze				(decimals)

viii)	Okukunuukiriza n’okuzingako namba	(estimation and rounding off numbers)

ix)	Nambuluzo n’Okulambulula namba	(Factors and factoring numbers)

x)	Namba ennambulukufu			(composite numbers)

xi)	Namba ezitali nambulukufu		(prime numbers)

xii)	Ennyingo ne Namayingo		(Terms and Polynomials)

xiii)	 Emigereko 				(Sets)

xiv)	Emikwanaganyo n’emikwataganyo	(Relations and functions)

xv)	    Ebikwataganyo				(coordinates)

xvi)	Emigerageranyo			(Ratios)

xvii)	Ebigerageranyo			(Rates)

xviii)	Emigendaganyo			(proportions)

xix)	Kalonda 					(Data)

xx)	  Ekisengekakalonda / Essomakalondomusengeke			(statistics)

xxi)	Ekibalo ky’Obusuubuzi		(business math)

3.	Omugereeso gwa Namba
         (Number theory)

Mu mugereeso gwa namba mulimu ebika bya namba eby’enjawulo  omuntu  anabeera omubalanguzi bye yetaaga okunnyonnyoka . Bye bino:

a)	Namba eza kibazo			(cardinal numbers)

b)	Namba ezibala				(counting numbers)

c)	Namba eza ndagakifo 			(ordinal numbers)
	
d)	Namba eza ndagalinnya			(nominal numbers )

e)	Namba enzijuvu				(Whole numbers)

f)	Kibalirampuyibbiri /Yintegya:				(integers)

g)	Namba ez’ensibo				( natural numbers)

h)	Namba za kyegabanya			(Even numbers )

i)	Namba eza kigaanira			(odd numbers)

j)	Emikutule /Emwenkutule 			(Fractions)

k)	Namba ez’omugerageranyo 	 	(Rational Numbers)

l)	Namba  ezitali za mugerageranyo	 	(Irrational Numbers)

m)	 Namba Zennyini				( Real numbers)

n)	Namba Ez’omuteeberezo			(Imaginary Numbers)

o)	Namba za kyebiriga			(square numbers )

p)	Namba za kyesatuza			(cube numbers)

3. Emisingi gya namba oba emisimba
         (Number bases)

Omusingi gwa namba(omusimba) eba “nsengekera ya namba”(number system) esimbira mu digito ez’enjawulo.

•	Emisingi gya namba			(number bases)

•	Omusingi gwa namba  ogwa bbiri		(number base two)

•	Omusingi gwa namba ogwa Satu		(base three)

•	Omusingi gwa namba ogwa nnya		(base four)

•	Omusingi gwa namba ogwa ttaano		(base five)

•	Omusingi gwa ogwa mukaaga		(base six)

•	Omusingi gwa namba ogwa musanvu		(base seven)

•	Omusingi gwa namba ogwa munaana		(base eight)

•	Omusingi gwa namba ogwa mwenda		(base nine)

•	Omusingi gwa namba ogwa kkumi		(base ten)

N’okweyongerayo. Omusingi gwa namba ogwa bbiri gusimbira  mu digito bbiri (0 ne 1) , omusingi gwa namba ogwa satu  gusimbra mu digito satu (0,1, ne 2), omusingi gwa namba ogwa nnya gusimbira mu digito nnya(0,1,2 ne 3) ate omusingi gwa namba ogwa kumi gusimbira mu digito kumi(o.1.2.3.4.5.6.7.8. ne 9),

Mui bufunze “omusungi gwa namba”  ka nguyite  “omusiimba”,Buli musinga gwa namba kw’egyo gyendaze waggulu era guyinza okuwandiikibwa mu bufunze nga wano wansi:

Emisingi gya Namba(Number Systems or number bases)	Mu bufunze	English short form

2	Omusungi gwa namba ogwa bbiri	Omunabbiri	(binary)

3	Omusingi gwa namba ogwa satu	Omunassatu	(ternary)

4	Omusingi gwa namba ogwa nnya	Omunannya	(quaternary)

5	Omusingi gwa namba igwa ttaano	Omunattaano	(Quinary) 

6	Omusingi gwa namba ogwa mukaaga	Omunekaaga	 (Senary)

7	Omusungi gwa naba ogwa musanvu	Omusenvu	( Septenary)

8	Omusingi gwa namba ogwa munaana	Omunenna	(octal)

9	Omusingi gwa namba  ogwa mwendas	Omunenda	(nonary)

10	Omusingi gwa namba ogwa kkumi	Omunekkumu	decimal)

Emisingi gya namba okuva ku  gw’ekkumi n’ogumu okweyongera waggulu oyinza okugisoma nga si mifunze bwoti:

Omusingi gwa namba ogwa kumi n’emu	(undenary)

Omusingi gwa namba ogwa kumi na bbiri	(duodecimal)

Omusingi gwa namba igwa kumi na satu	(hexadecimal)

Omusingi gwa namba ogwa kumi na nnya 	(vigesimal)

Omusingi gwa namba ogwa kumi na ttaano	(sexagesimal)

Omusingi gwa namba ogwa kumi na mukaaga	

Nga omubalanguzi atandika ,  manya bino:

•	Mu nsengekera y’omuwendo gw’ekifo, buli musimbalaala (column) gubaamu digito ezikiikirira obufunza bwa namba ey’ekikolo obw’omudding’anwa. 

•	Mu nsengekera y’omunekkuni, ekikolo(ensingi) kiba 10. 

•	Namba meka ezikozesebwa mu Nsengekera ya Namba? Ensengekera ya namba ey’omunekkumi (omusingi gwa namba ogwa kkumi) gwe tukozesa buli lunaku guzimbirwa “obutaffaali”  buno kkumi  {0,1,2,3,4,5,6,7,8,9} ekintu ekiguyisa omusingi ogwa -10. 

•	Digito ez’omunebbiri zisobola kuba 0 oba 1, ekiziyisa omunebbiri (omusingi ogwa-2).
Yadde nga tusobola okuba n’emisingi gya namba (emisiimba) egy’enjawulo mingi , wano essira tugenda kulissa ku misiimba  esatu egisinga okukozesebwa mu bulamu ne mu sayansi. Emisingi gya nmab gino esatu gye :

(i)	Omunakkumi(=Omusingi gwa namba ogwa kkumi)

(ii)	Omunabbiri (Omusingi gwa namba ogwa bbiri)

(iii)	Omusimba ogwa kumi na mukaaga.

Mu lufuutifuuti , omunnakkumi gwe guyitibwa “base 10” ate omunabbiri ne guba   “binary base”. Omusingi gwa namba ogwa 16 gwe guyitibwa  “hexadecimal”

4. Okulambulula Namba 
    (Factoring numbers)

Okusobola okunnyonnyoka essomo ly’ekibalangulo eky’okulambulula namba obulungi,  wetaagisa okumanya egimu ku miramwa egikozesebwa naddala  emiramwa nga :

•	Enkubisa			(coefficients)

•	Enkubise			 (multiples)

•	Entakyuka			(constants),  

•	entakoma 			(infinite)

•	enkyuso				(variable), 

•	engabiso			(divisor)

•	omugabo				(dividend)

•	Ennyingo			(terms), 

•	Mufunza				(exponentiation)

•	Kyekubisa			(logarithms).

•	Namba ennambulukufu		(composite numbers)

•	Namba ezitali nambulukufu	(prime numbers)

•	Enfikkizo			(remainder)

•	Nambuluzo 			(factors)

•	Nambuluzo Eyawamu 		(Common factors)

•	Enkubise Eyawamu   		(Common Multiples)

•	Enkubise Eyamu Esembayo 	(Lowest Common Multiple)

•	Nambuluzo  Eyawamu Esingayo (Greatest Common )

•	Okulambuluza namba ezitali nambulukufu (prime factorization)

5. Emikutule 
     (Fractions)

Emiramwa egyetaagisa okusobola okubaza oba okusonjola emikutule gye gino: 

(i)Ebitundu by’Omukutule(Parts of a Fraction)

 (a) Kinnawansi ne  Kinnawaggulu. Mu mukutule guno  namba eri waggulu eyitibwa kinnawaggulu ate eri wansi kinnawansi:  11	
                       13	 

 Kinnawaggulu      (Numerator)

 Kinnawansi	  (Denominator)

 (b) Kinnawansi Eyawamu(Common Denominator)

 (ii) Nambuluzo. Muno mulimu :

(a)	Nambuluzo ey’awamu(Common Factor)

(b)	Nambuluzo Eyawamu Esingayo (Greatest Common Factor)

(c)	Enkubise Eyawamu	(Common Multiple)

(d)	Enkubise Eyawamu Esembayo(Lowest Common Mutlple)

Okugatta emikutule egirina kinnawansi y’emu kuba kulya mungu buteesokoola naye bwe  giba ne kinnawansi ez’enjawulo oba olina okusooka okuzuula “kinnawansi ey’awamu”(common denominator). Okuzuula kinnawansi ey’awamu  weyambisa enkubise ey’awamu era eno gy’osooka okunonnya.

6. Emigereko 
    (sets)

•	Omugereko				 (set) 

•	Omuteeko gy'emigereko   		                 (Union of sets)

•	Veniggulaamu   			(Venn Diagram)

•	Entabiro oba Amasang’azzira  g’emigereko		(Intersection of sets)

•	Enjawuzo y’Emigereko ebiri		(The difference of two Sets)
	
•	Ekitundu ky’Omugereko 			(Sub set)

•	Endaga y'emigereko			(Set Notation)

•	Embaranguza z’Emigereko 		(Set Operations)

•	Omugattiko				( union) 

•	Amasang’azzira				( intersection) 

Omugereko guyinza okubaamu erementi (bamemba) ez’entakoma (infinite) nga Omugereko gwa namba ez’ensibo (namba z’obutonde) oba ne gubaamu bamemba ab’ekkomo (finite ) nga Omugereko gwa waliifu z’oluganda. 

Omugereko guyinza obutabaamu bamemba, ntegeeza obutabaamu kintu kyonna era Omugereko nga guno guyitibwa Omugereko omwangaala (empty set). Kino kiragibwa n’akabonero    , ekisomwa (fi). Ebintu oba bamemba ababeera mu mugereko A balagibwa nga n (A), ekitegeeza nti n ( ) = 0, ekitegeeza nti Omugereko A teguliimu erementi yonna. Omugereko gulimu emiramwa gino :

7. Emikwanaganyo n’Emikwataganyo
          (Relations and Functions)

•	Omukwanaganyo			(relation)

•	ebifulumyo 			(outputs)

•	ebiyingizo 			(inputs)

•	emigago emisengeke 			(ordered pairs) 

•	Omusooka 			(domain)

•	Omufulumyo			(range)

•	Omukwataganyo 			(function)

•	Kifuulannenge w’omukwataganyo” 	(the inverse of a function)

 		
Singa omukwataganyo “m” gukwataganya “k” ku “g”, kino kitegeeza nti kifuulannenge w’omukwatagenyo, ekiwandiiki bwa nga M-1, akwataganya k ku g.

8. Ennyingo n'Ekibalo ky'ennyingo(Ekiyingo)
            (Terms and Polynomials)

Emiramwa egyetaagisa:

(i)	Ennyingo 				(term (in algebra)

(ii)	Ennyingo ezifaanagana   	(Like Terms)

(iii)	 Nnyingo ezikwatagana		(Like Terms)

(iv)	Ennyingo ezitafaanagana 	(Unlike Terms)

(v)	Nnyingo ezitakwatagana	(Unlike Terms)

(vi)	Namayingo oba Ekiyingo oba Ekibalo ky'Ennyongo				(Polynomials)

(vii)	Nnyingemu			(Monomial)

(viii)	Nnyingobbiri			(Binomial)

(ix)	Nnyingosatu			(Trinomial)

(x)	Enjawuzo eya kyebiriga ebbiri (The difference of two squares)

9. Ekibazamukisa=Ekibalangulo ky’Omukisa=EKM(Probability , Mathematics of Chance)

Mu kibalo  ky’Omukisa(Mathematics of chance, prpbability), emiramwa egyetaagisa gye gino:

•	Ekibalo ky’Omukisa(EKM)	         ( Mathematics of chance(probability)

•	Ekituukiriro oba ekituuko	 (event)

•	Ebisoboko			  (possibles,outcomes)

•	Omugereko gw’ebisoboko	         ( probability space)

•	Ebituuko eby’ejjuuliriza        ( complementary events)

•	Ebituuko ebyetengerevu	        (  independent events)

•	Ebituukiriro ebitali byetengerevu  (dependent events)
	
•	Ensatuza			(dice)

•	Ekinusu				(coin)

•	Omutwe(M) n’ekikira(K)				(head and tail)

•	Ebituukiriro ebitasobola kubaawo mu kiseera kye kimu	( mutually exclusive events)

•	Ebituukiririo ebisobola okubaawo mu kiseeta kye kimu		(mutually inclusive events)

10. Emigerageranyo ,Ebigerageranyo, n’emigendaganyo
                       (Ratio, rate, and Proportions)

•	Omugerageranyo			(ratio)

•	Emigerageranyo			(ratios)

•	Ekigerageranyo			(rate)

•	Ebigerageranyo			(rates)

•	Omugendaganyo			(proportion)

•	Emigendaganyo			(proportions)

11. Ebikyusaganyo
        (Transformations)

•	Ekifaananyi			(Picture)

•	Ekifaananyo			(Image)

•	Ekifaanaganyo			(reflection)

•	Ekiseetuko			(displacement)

•	Ekyetoloozo			(rotation)

•	Eppeto				(angle)

•	Eppetero			(bearing)

•	Akaserengeto			(slope)

•	Ekiserengeto			(gradient)

•	Obuserengefu			(steepness)

12.Kalonda ne Ekisengekakalonda / Essomakalondomusengeke 
                  (Data and Statistics)

•	Kalonda 			(data)

•	Okusengeka kalonda		(to arrange data)

•	Kalonda omusengeke		(statistics)

•	Emisengeko gya kalonda	      ( statistical patterns)

•	Okukung’anya kalonda		(data collection)

•	Okutaputa kalonda		(to interpret data)

•	Kalonda ow’endagabungi	      (  quantitative data)

•	Kalonda ow’endagamutindo	(qualitative data)

       Omusengesi wa kolonda  (Statician)   

12. Ekibalangulo eky’obusuubuzi 
       (Business mathematics)

•	Ekibalo ky'obusuubuzi                 (Business maths)
        
         Ekyewolo				(Credit)

•	Ebbanjo					(loan)

•	Omuganyulwo				(interest)

•	Omuganyulo				(interest)

•	Omukendeezo				(discount)

•	Commission				(bwasiisi)

•	Omugeranyakikumi				(percentage)

•	Ekigambululo kya Bbanka  		( bank statement)

•	Omujjuulirizo				(subsidy)

•	Enzijuulirirzo				(subsidy)

13.Nakyenkanyampuyi ne Nakyekubira
       ( Equations and Inequalities)

•	Nakyenkanyampuyi		(equation)

•	Nakyekubira			(Inequality)

•	Musikiza/Ekisikiza			(substitution)

•	Okusikiza namba			(to substitute a variable with a number)

•	Akenkanyanjuyi			(the equals symbol)

14. Emifunza ne Kyekubisa
       (Exponents and Logarithms)

•	Ekifunza=Ekibalo ky'emirundi emifunze		(exponentiation)

•	Kyekubisa		(Logarithms)

•	Emufunzo=Emirundi emifunze		(exponents)

•	Akafunza		(the power of a number)

•	Ekikolo 		(the base or the number at the base of an exponent)