DOCUMENT RESUME 0779-115 ,s,10_16_, 447; TITLE Mathematics 74£3_:FiaridbOtik, "1973Reprint. INSTITUTION New`York- StateEdUcatiOn Albany.__Bureau of SeCOridaryCurriculumDevelopment. _ -DATE_- 69 EDRS PRICE 'MF--$6. 65 HC9._81 *CurricUlUm; Curriculum Guides; C.ade7; Grade8; *Instruction;MatheMaticalEnrichment; Mathematids Education; *SecondarySchool Mathematics;_*Teaching Gliidet AbstAliat= Thit handbookp'i.Ovidet: suggestions fcirteaching tdPidtin -seventh and eighth, grade.Mathematict =and,-wat intended. to be -used.as a supplement- toithe =seventh- andi eighth; -grade syllabus(see- SE 116 446) . -Optimalmaterials,_ and.-approaches- are -suggested- 'for following_ topiCt;sets;' _numerationsystems; natural nUMberS, Vhole numbers, rational numbers, integers, and:real nuMbers; pkoporticin,_percent,,and'Variation; ,geometry; statistics;_ and probability.- ADT)- U S. DEPARTMENT OF HEALTH. ;EDUCATION & WELFARE- NATIONAL INSTITUTE OF EDUCATION THIS OOCUMENT HAS SEEN-REPRO OUCEO,E XACTLY-AS - RECEIVED FROM -THE PERSON OR ORGANIZATION ORIGIN- AriNG IT, POINTS OF VIEW OR OPINIONS STA TEC'00 NOT,NECESSARILV-REPRE 'SENT OFFICIAL NATIONAL INSTITUTE OF- .EOUCATION POSITION OR POLICY The Unhiersity:ofithe-State Of,iO4v801A- TfieSTATEIEDUCATION DEPARTMENT SU foal,- Of -Sileo0464;_-_Curriiuliiini6evjloprnelit- Albbny,-NoWI'O*122214_/4949_, THEMAT 7 - 8 1973Reprint The-1,niVersityi-of the ttate=-Of:NeW. York THE:STATE- EDHCATION7,pEPARTMENT of -Secondary_ :Orel-oil Um Development t Nevi Yptk-12224/1965- THE_DNIVERITY_OE THE St/kit:DE NEW _YORK Regents Of_ the UM iie-r5i ttiye#s-- when- terins expire), New Ctiaikel lOr. _ . -., _ _ D.C.S--- -1985 iElierett,J._ __ 'jenny .B.C'S Vice_Chancello.i..'-_ _ _ _.' ,Witite,Oliins 1978_Aleiandet,J..TAllani Jr., LL:D., 'Litt.D. -: -Tioy -1973-Ckailei-.1(._-Millifd; Jr., 4.3,-,_,LL:b.,_-[:11: D, Buffalo. _1987--01--uri-.--P-fcifiliOitief, Ji.,-- -4B.,14:13.A.-, .'D- .; -,11.11:1). Pufeliaie 1975;:Edviaid!,11. /1._'=itaibilig-,_,B- ,=.1.-..D. D. _'-._ ,-tieii-;'York 1977.-J-e-Oh'_1%._-King,LL.B,_ -.Que-eri_ 1s74,JOSeptt,-_Cs._ In-delitato,:-.11:D._ Orooklyd :1976 Mi..._iHelek-Z.B., Poiier-,- -A:B.,_ Liti.D.,..111..11:D., LL.D.. --_------Rochestei 1979_ FranCis='W._McGinfey-i_B.S., Jip,,__LL.D '------_-- Gleni:Falls =1980 Max .-.J. RubiniLL.B.i=1:11.1); i -. ,' -1,-: --_- Nei.: york- 1986-_Keinieth-B.,Clatki-_-A:B.,_41.S. f Ph.b.-LL:p:, Hastings on --:Ittidscili -1982 Ste-06kt% :Bailey,.A.13., -B.A., Ph.D.,, LLD._ , Syracuse 4983 -lia-rold,EL.NeilCOMbi B.A. 1981: ThendafeAt4.-,_Bliek,.,A4-., -Litt:D:, Saild.s Point_ Pfeil dent _okE_theitni y_and ok EduCat ion Ewald_B. Nyquzst ExeentiVe Deputrtnimnissiorier,of.,EdUcat Gordon Deiiuty:ComMissiOner for_atiententaryi-Sec-ondaiy,...-and-_tontinuing Education- Thomas-IL- Sheldon; A.4§OCiateCominiiolieflof :William Bitner 411 Assitaiit.iCornmissioner_for_iithindtiokal_-_Services -,(deneral :Education) Bernard,-F:'-itaake- - ---- bireCtoi---,_-Divi-Sfon_ of ,SChool,_ SupervisiOn Gordon=E-.-,:Nan.ttooft _Iitireatirok_SeCandaijr_--durii-cnium,iDeVelopnient =bireetoi,- of =denefal,EdtiCatiOn, $- -Ted T. Grenda-. L -Chiefs:Bdreaii_-Of_idattfeliatias:EitiCatink Frank -S. Hawthorne r W1 CONTENTS ;Page Fir'ewdrd:' Lyn -Unit 1. -SETS f *r09q:--,,__. t ldentitiOtion,df" Sets The4mpik--Sef Finite-,diut:Infiilite, getS....:... : .... -. ..-..:::....._: ....... EA Ual--SetS----and:Equiiialent-TSeiS ,. I: -- , _ Sul4etS - = : -Union -ancrIliterseet ion Of Set'S: 10: lieliii -OragraMs -_ 12 . Unit 41, -§Y§ITM-of'-N1.10RAtfON- 47 T;Illying-- -- .._ _ ,, _ _ . V Egyptian?Numeral s .-.. -: .... .--. ; ... .... :-.. : .....: . : .. ._. _,_ __ _- _ _ _ . __ __ ..,_:--..-...-, __ _ 18.* Roman -Ntullerals _ . _ , 18 ThelHilidur=Afabid, of' Dee rear =SYS tiO 19 Place Vaige-iu -the ,lileCiiial .y-_§:fai, 2o , Writingiing,- -a= a- gumerar,iS a :15elyholliiaL_aild -in Positional: Notation : 21- -Wilting-, -Reading, -Aild4e 11 ing_ NiMiera 1-s in Dec* i -NOtatidn_-- - 22 'Factors-; Exponents _23 'Digits in Various_-BaSeS 26- F lade -Ata lbe- Tables-- :- 2G Trans lati!ig- Frei, Aor_ BaSe To-Any= Other- WaSe- -_ 30 :Addition. in---yariOUS RaseS, - '31 -Sdh t ratt i olii n- V arioUs- Rasesl: -- 32 Multiplication :in itatiOugases 33 Additional '_ Enriehinelit Activities-ties- 34: Unit _III . THE :SET. OF -NA,TORAL ;NUMBERS- = 34- Propgrti OS =of thye _Natdral _NOrthers 39 nit_ -IV. TOE- SET . OF :WHOLEJNUMOERS 44- theNdlither,Line - e--- 48 IlathefliatiCal' Sentences -- 503 :Divisibpity: '52: Primes , -54' -0iael= -df ,Operat ions , -SS Magic Squares T SG FigOrate -NUilberS- :58- Mi-Siing Digits 0,- AlphalietiCs- -60_ 111: 1 P.* -,Unit. V._ THE 'SET OF POSITIVE-RAT-1ONAIS i 63 Subsets of the ,64 Fractions 65 Increasing -the. Terms -Of- a +Fra ctio n: 67 Operatien-§ -With5FraCtiOn§ Properties of Pogiiiire-,,kationdi§: - 76 -Unit VI,TH-5ET'OF INTEGERS.- 1'34: -Addition 85 Subtraction 86 Multiplication 88- Division $ 1 Al?§Oitite yaluo 92 , - -uplt -VII:_ THE:damftEtE_Et:Of kATIoNAt§,,,,;.. ,i .., ..- . -...f 93 Addition _okliatioiiii§ , _ _ _ -. 94- 'Snlitfietleil of ,Ra-tioliais r -,0,-, 14tiltiplient-ion_: -95 DiVigTOnnfilatienal.............:.. , .. ,-.. -.,...-..: -........ 96, Negati_ye,:EZproniknts: , - 97: _SeientifiC.:,Notatioli- 98 :Measurement,: 98 Significant Digits. , 101 -Stpare -koot - '102 - Unit'vIII -..THE- °SET`OF.REAL =NUMBERS .: 105r linit_ ix:, RATIO; -PROPORtI011,_ :OER:CENT, VARIATION-- 108 -. , . Ratio- _ .- -108 3rOlortiOn ..-..-:-.--i: ..:.. .. ..,,-..,._199 -per :Cent 110- .Variation 117 Unit-X.. GEOMETRY 120.- `Undefined`Undefined'' Terms . '120' 13-oint- 121 Line- ., 121- -_ Plane _andAines in- a'Plae 123_ Afigio --- 124: Measurement--of- Angles ' :126 Angle§%a§ -Api5iied'-to -140§- and_ Planeg '126 'oljigoli 128 Triangle -_- 129 Quadrilaterals-. 131 Cirale ,. _ "42, Measurementand-Ldon"gttiidtiag 134 iteasureineni-Ok-TbistanCes- 140. -Meakirement. Of- Afea--.--,- 141. '!13.folinitions as `Aiiplied'tb±Similnr Triangle"§::- 145' -OOlyhedrcins- _ _ , , _ _ _ __ _ - 149 P4ge PfismS, 152 15-y'fami (1§ - 154, Coile§ 156, 6'siiii49f- 8156_ Spheres 15a SurfaceAtea§ of '5o1i0- 159 -Volume§ of 5oit& 160; Space. Visualization 161-= -CoOkii hate -GeOniceiy-igifig:StIie Ntiinbefc-.I.ine- 161 '46-okaina-te-46-mcfrk,-Tbsi-ng --Vie, ,Isitiiiiboi ,Pralie-- .. drafilnit:o090--$6-iitCiles 166 -Graphing- = ri,:ciiialiti-e-§: 167 Graphing- .,-InectUalitie§--- y 168 Glii-s§tirk of -Terms fat. deoitletik,- -1.71 'Unit XI: --51N11611,CS-- 10: labtliatioh of-Data- 183. Itopieson-tat-iori of Data ---- .- -: 1 187 Interpretation -__of. ---Ciii-tili§::::-.,..:. :,..,..-...- .. -: . ; . ,,.._...,.._.. ;.-i- -190; Measures of Ce-iiti-al l'en-deficy 191 Data and -Data-,Collection 193 ;Unit XI I-: PIt011AilILITY- .195 t. 7 FOREWORD *NS handbook- -,haS -been _prepared in- orderteprOvide teachers ofseventh year and -eighth year thatheifia-tic,with suggeStiens for teaching Various aspects-of -the-course.It is 'intended' that it _be used with and supplement _the pepartinent Ittathematiee gourses for the Seventh Tear and- Eighth Year: A committee of teachers andiDepartment-persOnnel.met in- May 1966 to- doelop an- -out -line 'for seventh and eighth, grade mathematics: 'this- commit =tee designed a--recoMMended :Nndamental course,- -the scope and ..ofgilliization, of WhiCh reflected several igniding principles: (I)the fraMework must be consistent and---reaSobah lei- -(2)'the- content should= be ba§cd- on the =1k- and= 8",i( -material's as :modified',-=by -evalbations and-xecomniendations resulting - -from their use,: and: (3) the course. =mnst:proVide good _articulation and _cOntilutity_ with -both Mathematics K - 6,_ A Propoged Program and IlinthiYear Nathema= tics, -Couree F = Algebra,, as- reViSe(lin 1965.: The ebiline,,- Alath enzatics- courseci for_ the Seventh Year and Eighth Year, which ,this-- comMittoe constructed,contains-no-optional For_ -suggestions in thiS area reference was' -made in the Syllabus to 7X and-8X -materials-, -sone-ef-whichAre-.now: -61it-of print!. In order -to-_=proVide'teciehers .iof seventh and-eighth -grade MathethaticS with a_ bactg-roubd -Of Optibnal _material's ,and, a 1Se,to-ficSh Out_he "chat " of the otitiine With, _Sonicsuggestions_concerning the 'how," .1 this handbook the suggestions should -prove particularly -helpful- to beginning teachers_ of mathematics.Such _teachers ,shouldrealiie, however, , that the-Methods, find :illUstraii-onS- used ,afe--net prescribed. proce= duress -teachers will-want-- tO-eXperiMent =with alternate apprOache'S, as dies's' come -to -their attent-ion-. Nor Should:=teacherS- feel that everything in thiS -hail-606k, Must be "covered " in-, every class.A number of areas, which-,are lesignated' as optional_ have-been -included in deyelopMent., "ironger pupils - may be led: to ,explare some of theSe -optional topics, -but'- t- is probable -that.net all of these 'topics can be touched-upon by- -any -one .group.It is-the-intent of thiS -handboOk to make_ Such explorations, profitable--wheb, opp_ortunitioS =for -thiS -adtivity' arise:The ,optiOnal topics open windows upon fascinating viStas , 'but this additional 'Material should not be introdbced where-this iS ,done- at the expense of adequate treatment -,of the reqUired-topics . -Experienced, teaCherS, 'too;, will welcome' new suggestions cifi -presenting-, old= and new -topics.SuCh -bnekgrannd-supplementation - will- available to thein wider- variety of approacheS-, and ideliniqUeS Ito-vary their- presentations,:Some of the _suggestions will _provide the teaChor procedures Which_ willenablegreater:OptiaSiSto-