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!!PRE%ALGEBRA!!
! Afrikan!mathema-cs!! Prac-cal!mathema-cs!!for!real3world!applica-ons!! Common!sense,!everyday!mathema-cs!! Ancient!Afrikan!problem3solving!techniques!! Instruc-on!in!english!and!Akan/Twi!! Educa-on!for!independence!
!!WORKBOOK!AND!PROBLEM%SOLVING!GUIDE!
1ST!EDITION!
WORKBOOK&AND&PROBLEM.SOLVING&GUIDE&
PRE.ALGEBRA&&
1ST&EDITION&&
Yaw&Agbede&&&&
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Copyright*©*NKALA*Environmental*|*M:*919:628:4208*|*E:*education@nkalaenvironmental.com*|*W:*www.nkalaeducation.com*
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&This& publication& is& designed& to& provide& accurate& and& authoritative& information& in& regard& to& the& subject&matter& covered.& It& is& sold& with& the& understanding& that& the& publisher& is& not& engaged& in& rendering& legal,&accounting,&or&other&professional&service.&If&legal&advice&or&other&expert&assistance&is&required,&the&services&of&a&competent&professional&should&be&sought.&&&&&&&&&&&&&&Copyright&©&2014&NKALA&Environmental&&All&Rights&Reserved.&&No& part& of& this& book& may& be& reproduced& or& used& in& any& form& without& written& permission& from& the&publisher.&The&original&purchaser&is&hereby&granted&permission&to&print&or©&the&pages&for&use&with&the&original&purchaser’s&children/students.&&&&&&&&&&&&&&&&Published&by&NKALA&Educational&Services&&P.O.&Box&1781&Durham,&North&Carolina&27702&+1.919.628.4208&www.nkalaeducation.com&
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&Copyright*©*NKALA*Environmental*|*M:*919:628:4208*|*E:*education@nkalaenvironmental.com*|*W:*www.nkalaeducation.com*
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&&&&&EMU$NSƐM$ $ iii*
NNIANIMU$ $ v*
TI$1* Akan/Twi$Numbers$1.1*BASIC*AKAN*PHRASES**1.2*COUNTING*IN*AKAN**&
1&1*7$
TI$2$ Number$Systems$2.1*ORIGINS*OF*NUMBER*SYSTEMS*2.2*BINARY,*DECIMAL,*AND*HEXADECIMAL*NUMBERS*2.3*DECIMAL*PLACES*AND*ROUNDING*2.4*ADDING*AND*SUBTRACTING**2.5*MULTIPLYING*2.6*DIVIDING*2.7*FACTORING*2.8*GREATEST*COMMON*FACTOR*2.9*LEAST*COMMON*MULTIPLE*2.10*SIMPLIFYING*FRACTIONS*2.11*CONVERTING*BETWEEN*FRACTIONS*AND*DECIMALS**
1&11*21*27*31*41*47*51*55*59*63*69**
TI$3$ Percent&3.1*CONVERTING*BETWEEN*FRACTIONS,*DECIMALS,*AND*PERCENTS*3.2*FINDING*PERCENT*3.3*PERCENTS*OF*CHANGE*3.4*TAX,*TIPS,*DISCOUNTS,*AND*MARKUPS*3.5*SIMPLE*AND*COMPOUND*INTEREST*&
1&74*80*85*89*95*
TI$4$ Algebra$Basics&4.1*VERBAL*EXPRESSIONS*4.2*ORDER*OF*OPERATIONS*4.3*DISTRIBUTIVE*PROPERTY*&
1&101*107*113*
$$ $ 1&
EMU * N S Ɛ M *
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*Copyright*©*NKALA*Environmental*|*M:*919:628:4208*|*E:*education@nkalaenvironmental.com*|*W:*www.nkalaeducation.com*
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TI$5$ Algebraic&Equations$&$Inequalities&5.1*SOLVING*ALGEBRAIC*EQUATIONS*5.2*GRAPHING*SINGLE:VARIABLE*INEQUALITIES*5.3*SOLVING*ALGEBRAIC*INEQUALITIES*&
1&119*131*137**
TI$6$ Exponents$and$Roots&6.1*DIVISIBILITY*RULES*6.2*EXPONENTS* *6.3*SCIENTIFIC*AND*EXPONENTIAL*NOTATION*6.4*ROOTS*&
1&145*151*157*163*
TI$7$ Proportions$and$Similarity$7.1*PROPORTIONS*7.2*SOLVING*PROPORTIONS*7.3*SIMILAR*FIGURES*&
1&171*177*185*
$TI$8$ Linear$Equations$and$Inequalities$
8.1*COORDINATE*GRAPHING*8.2*MIDPOINT*FORMULA*8.3*SLOPE*8.4*GRAPHING*LINEAR*EQUATIONS*8.5*WRITING*LINEAR*EQUATIONS*8.6*GRAPHING*LINEAR*INEQUALITIES*8.7*GRAPHING*SYSTEMS*OF*EQUATIONS*8.8*SOLVING*SYSTEMS*OF*EQUATIONS*BY*SUBSTITUTION*8.9*SYSTEMS*OF*EQUATIONS*WORD*PROBLEMS*&
1&193*203*209*217*233*243*257*263*267*
$TI$9$ Polynomials$
9.1*IDENTIFYING*LIKE*TERMS*9.2*ADDING*AND*SUBTRACTING*POLYNOMIALS*9.3*MULTIPLYING*POLYNOMIALS*&
1&273*275*279$
APPENDIX$A$ Iahmesu$Mathematical$Papyrus$Problems$ &APPENDIX$B$ Suggested$Readings$ &&
WORKBOOK&AND&PROBLEM.SOLVING&GUIDE&&
Copyright*©*NKALA*Environmental*|*M:*919:628:4208*|*E:*education@nkalaenvironmental.com*|*W:*www.nkalaeducation.com*
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NNIANIMU&&
Ti&9&&
Adesua&1&&
This&workbook&is&written&for&Afrikan&people&−&homeschool&students,&parents,&adult&learners,&public/private&school&students,&unschoolers,&and&most&importantly&those&seeking&Afrikan¢ered&resources.&This&book&is&best&used&as&a&supplement&to&an&Afrikan¢ered&mathematics&course&(or&tutorial)&or&as&a&refresher&guide&for&those&already&familiar&with&the&topics&inside.&Each§ion&contains&a&brief&explanation&of&how&to&solve&the&problems&within,&but&does¬&go&into&detail.&Success&in&each§ion&is&dependent&on&the&student's&ability&to&look&at&the&example(s),&find&the&patterns&and&apply&them&to&similar&problems&within&a&different&context.&&&Although&this&book&does&include&much&information&about&the&correct&context&of&Afrikans&in&the&development&of&mathematics&I&want&to&emphasize&that&this&is¬&a&history&book.&The&focus&is¬&on&"what&we&did"&but&on&"how&we&did&it."&For&that&reason&I&have¬&gone&into&great&detail&on&the&history.&For&more&information&on&the&historical&accomplishments&of&Afrikans&in&the&development&of&mathematics,&see&the&reading&list&in&Appendix&B.&&&Subsequent&editions&will&be&made&available&entirely&in&the&Akan/Twi&language.&The&author&strongly&recommends&that&students&learn&to&speak&about&and&solve&math/engineering&problems&in&various&Afrikan&languages.&The&development&of&Afrikan&languages&to&be&used&for&science/engineering&problem&solving&applications&is&very&important&work&to&be&continued&by&future&generations&of&pan.Afrikan&nationalists.&&Solutions&to&all&of&the&problems&are&available&for&download&at&www.nkalaeducation.com.&&−Yaw$Àgbedẹ""(223"years"after"the"start"of"the"Ayitian"Revolution)"&
NUMBER'SYSTEMS'
ORIGINS'OF'NUMBER'SYSTEMS''
Ti'2''
Adesua'1''''
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2.1$Number$Systems!"!Origins!of!Number!Systems!
'A"first"group"of"eight"signs,"called"the"secret"count,"reproduces"the"first"seven"numbers,"which,"according" to" a" [Bamana]"myth,"were" engraved" in" space"when," through" [the" supreme" being's]"word," the"creator" formed"the"universe." It" is"said"that"they"contain"the"whole"universe," for"they"are"an"arithmetical"summary"of"the"creator"and"[the"creator’s]"work."The"number"one"represents"the" primordial" thought" that" formed" the"world;" two" symbolizes" the" splitting" in" two" of" the" first"principle."The"number"three"corresponds"to"the"element"of"fire"and"the"masculine"principle;"it"is"the"origin"of"life,"movement,"and"time."The"number"four"symbolizes"the"female"principle,"born"of"..." nature" and" the" element" of" water." Number" five" is" the" synthesis" of" the" creator" and" [the"creator’s]"work;"six"represents"male"as"well"as"female"twinship;"seven,"which"is"the"sum"of"three"and"four,"represents"the"couple,"the"person"(both"male"and"female),"intelligence,"fecundity,"and"the"earth.1"
!
!!
In!the!same!way!that!the!spiritual!observations!of!Afrikan!people!were!used!to!create!the!foundations!of!astrology,!astronomy,!agriculture,!geometry,!geodesy,!geography,!and!cartography!these!same!observations!were!used!in!the!formation!of!the!branch!of!mathematics!called!arithmetic.!!!Arithmetic!is!the!branch!of!mathematics!that!deals!with!quantity,!numbers,!and!the!combining!of!numbers!through!the!operations!of!addition,!subtraction,!multiplication,!and!division.!When!Afrikan!women!discovered!that!their!bodies!operated!on!a!monthly!cycle!that!was!in!sync!with!the!moon!they!developed!the!first!number!system!as!a!way!to!measure!and!chart!that!cycle!as!well!as!to!measure!and!chart!the!different!stages!of!their!pregnancy.!!!
The!oldest!piece!of!physical!evidence!that!we!have!of!this!calendar/chart/arithmetic!connection!is!the!Lebombo!bone.!This!37,000!year!old!bone,!the!fibula!of!a!baboon,!was!found!in!the!mountains!between!Azania!and!Ngwane2.!This!bone!had!29!distinct!notches!cut!into!it,!which!were!used!to!chart!the!29!days!of!the!lunar!month.!!
!
''''''''''''''''''''''''''''''''''''''''''''''''''''''''1 Diop, CA. Civilization or Barbarism. Brooklyn: Lawrence Hill Books, 1991, pg. 317 2 Two countries in southern Afrika. Also known as “South Afrika” and “Swaziland” respectively
''
2.1$Number$Systems!"!Origins!of!Number!Systems!
The!most!well!known!mathematical!artifact!is!the!Ishango"bone.!This!was!also!the!fibula!of!a!baboon3!that!was!found!near!Lake!Rutanzige!in!the!fishing!district!of!Ishango!near!the!border!of!the!Kongo!and!Uganda!at!the!source!of!the!river!Nile.!This!25,000"year"old!artifact!has!several!groups!of!markings!(both!front!and!back!views!shown!below)!that!indicate!knowledge!of!a!decimal!(base!10)!number!system,!prime!numbers,!and!a!method!of!multiplication!by!doubling.!The!Ishango!bone!was!also!used!as!a!lunar!calendar!and!it!is!believed!to!have!been!used!as!a!writing/engraving!tool!as!there!is!a!piece!of!quartz!stone!embedded!into!one!end!of!the!bone.4!Can!you!find!some!patterns!in!the!groupings!on!the!Ishango!bone?!!!!!
!!!!!!!! !
''''''''''''''''''''''''''''''''''''''''''''''''''''''''3 It is interesting to note that both the Lebombo and the Ishango bones were fibulas belonging to baboons. It is probably no coincidence then that in the Nile Valley the NTR Djehuity – who is associated with the moon and with baboons (baboons are intelligent, nocturnal animals who “sing” to the moon at night) and is credited with the creation of mathematics and time amongst other things. Djehuity is often depicted as having the head of an ibis, which is a bird has a beak that resembles the crescent moon. 4 In a neighboring district other artifacts were found such as harpoon heads, which have been dated back 70,000 years ago. Interestingly enough, these same kinds of artifacts, as well as the same method of multiplication by doubling is seen further north along the Nile river amongst the Nile valley civilizations. This is yet another piece of evidence which speaks to the central Afrikan origins (as opposed to european or non-Afrikan origins) of ancient Kemetic civilization.
''
2.1$Number$Systems!"!Origins!of!Number!Systems!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!
!As!the!development!of!mathematics!continued!these!ideas!and!techniques!were!compiled!into!mathematical!textbooks.!There!are!several!mathematical!textbooks!that!we!know!of!which!demonstrate!mathematical!knowledge!ranging!from!tables!of!Kemetic!unit!fractions!to!the!foundations!of!integral!calculus.!!!One!textbook!includes!84!problems!involving!arithmetic,!algebra,!geometry,!trigonometry,!and!more.!This!book!was!a!copy!of!an!earlier!textbook.!The!author!of!the!book!is!a!scribe!who!was!called!Iahmesu!(commonly!referred!to!as!Ahmes),!and!he!wrote!this!textbook!around!1834!bce!during!the!middle!kingdom!of!KMT.!!!
!!!!!
Baboon,'Djehuity’s'sacred'animal'
Djehuity,,lunar'NTR,of'mathematics,'time,'medicine,'writing,'kwk'
''
2.1$Number$Systems!"!Origins!of!Number!Systems!
The!title!page!of!the!Iahmesu!mathematical!papyrus!is!a!very!interesting!one:!
!Rules" for" conducting" research" into" nature," for" the" purpose" of" knowing" all" that" exists," each"mystery"and"each" secret." For" that"purpose," this"papyrus" roll"was" copied" in" the" year"33," fourth"month"of"Akhet,"under"the"Majesty"of"the"King"of"Upper"and"Lower"[KMT],"AaRWserRRa,"may"he"live!" And" this"was" accurately" copied" from"an" ancient" text" produced" in" the" time" of" the" King" of"Upper"and"Lower"[KMT],"NeRMaatRRa,"by"the"scribe"Iahmesu"(Ahmes),"who"inscribed"this"copy.5"
!This!tells!us!that!ancient!Afrikan!people!envisioned!mathematics!as!a!method!that!would!allow!us!to!better!understand!the!nature"around!us.!The!Akan!adinkra!symbol!hwεmudua!can!offer!us!some!insight!as!to!the!nature!of!the!Afrikan!concept!of!“research.”!The!proverb!associated!with!hwεmudua!is!as!follows:!Sε!wohwehwε!mu!a,!na!wohu!mu;!sε!wopusu!mu!a,!na!wote!ne!pampan.6!This!symbol!teaches!us!of!the!importance!of!critical!examination!in!all!that!we!do.!Indeed,!this!was!the!same!type!of!critical!examination,!or!nhwehwεmu!that!early!Afrikans!engaged!in.!$ $
''''''''''''''''''''''''''''''''''''''''''''''''''''''''5 Ahmes, Ahmes Mathematical Papyrus 1316 AFK (1834 BCE) 6'Akan'(Twi)'“When'you'examine'it,'its'contents'will'be'known;'and'when'you'shake'it,'you'will'know'its'smell.”'
''
2.1$Number$Systems!"!Origins!of!Number!Systems!
Kemetic$Numerals$!The!letters!and!numbers!that!we!are!familiar!with!were!not!created!by!Afrikans.!However,!the!images!below!are!our!symbols!!!
!! !
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2.1$Number$Systems!"!Origins!of!Number!Systems!
!
!!Kemetic!Numerals!!1!!!!10!!!!100!!!!1000!!!!10,000!!!!100,000!!!$$
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2.1$Number$Systems!"!Origins!of!Number!Systems!
Kemetic$Fractions!!!!2/3!!!!1/2!!!!1/3!!!!1/7!!! !
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2.1$Number$Systems!"!Origins!of!Number!Systems!
!
LINEAR'EQUATIONS'AND'INEQUALITIES'
SLOPE''
Ti'8''
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8.3$Linear$Equations$and$Inequalities!"!Slope!
'Rules&for&finding&the&slope.&&The'slope'of'a'linear'function'is'a'measure'of'the'line's'steepness.'A'positive'slope'indicates'"climbing'uphill"'from'left'to'right,'and'a'negative'slope'indicates'"climbing'downhill"'from'left'to'right.'The'closer'the'value'of'slope'is'to'zero,'the'more'horizontal'the'line'is.'Any'line'with'a'slope'='0,'is'a'horizontal'line.'The'farther'the'value'of'the'slope'is'away'from'zero'the'more'vertical'the'line'is.''
''
'''
Slope'(usually'called'"m")'is'the'rise'over'the'run'of'the'line.'That'is,'the'change'in'the'YPvalues'(the'rise)'divided'by'the'change'in'the'XPvalues'(the'run)'between'any'two'points'on'the'line.'The'slope'is'the'same'throughout'the'entire'line.''
! = ∆!∆! =
!! − !!!! − !!
= !! − !!!! − !!
'
Graph'with'a'slope'near'zero.' Graph'with'a'large'positive'slope.'
Graph'with'a'slope'equal'to'zero.' Graph'with'a'negative'slope.'
''
8.3$Linear$Equations$and$Inequalities!"!Slope!
Example:''We'can'find'the'slope'between'any'two'points'as'follows.'(11,21)'and'(P14,P24):''
! = ∆!∆! =
21 − (−24)11 − (−14) =
4525 =
95 = 1.8'
'The'slope'of'the'line'that'passes'through'the'points'(11,21)'and'(P14,'P24)'is'm=!!.'This'means'from'any'point'on'this'line'if'you'rise'9'units'in'the'YPdirection'and'then'run'5'units'in'the'XPdirection'you'will'still'be'on'the'line.'
'
'
'!'9'
5"'
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8.3$Linear$Equations$and$Inequalities!"!Slope!
'Slope'is'only'one'part'of'a'linear'equation.'The'other'part'is'called'the'YPintercept.'This'is'the'point'where'the'graph'of'the'line'crosses'(or'intercepts)'the'YPaxis.'If'you'know'the'slope'and'the'YPintercept'of'a'line'you'can'write'an'equation'for'that'line'as'follows:''
! = !" + !''This'is'known'as'the'"slopePintercept'equation"'for'a'line'where'm'is'the'slope,'b'is'the'YPintercept,'x'is'any'xPcoordinate,'and'y'is'the'yPcoordinate'that'corresponds'to'that'xPcoordinate.'Another'form'is'the'standard'form'for'the'equation'of'a'line:''
!" + !" = !''Finally,'another'useful'form'is'the'"pointPslope'form"'of'a'linear'equation.'You'can'use'this'formula'to'determine'the'equation'of'a'line'based'on'only'having'the'slope'and'one'point'(x1,y1)'on'the'line:''
! − !! = !(! − !!)''Example:&&Find'the'slope'of'the'line:'2x+4y=12''Convert'the'equation'in'standard'form'to'slopePintercept'form'like'this:''
2! + 4! = 12'2! + 4! − 2! = 12 − 2!'
4! = 12 − 2!'4!4 = 12 − 2!
4 '
! = 124 − 2!4 = 3 − 12 !'
! = ! − !! !''Based'on'this'slopePintercept'form'of'the'original'equation'we'can'see'that'the'slope'of'this'line'is'! = − !
!.'''''
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NKALA!Educa:onal!Services!|!PO!Box!1781,!Durham,!NC!27702!usa!|!+13919362834208!
TOPICS!! Number!Systems!! Algebraic!Equa-ons!!! Algebraic!Inequali-es!! Exponents!! Percent!
! Linear!Equa-ons!! Linear!Inequali-es!! Factoring!! Propor-ons!and!Similarity!! Polynomials!
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