A DOCOBEIT RESUME ED 1-76 955 SE 027 924 TITLE Expermental Units fer Grades Seven and Eight. INSTITUTION", Stanford Univ., -Calif. Schodl Mathematics Study Group.' SPQNS AGENCY National Science Foundation, Washington,E.C. PUB pATE 59 NOTE 20p.; Contains occasional 3i.ght ancrbroken type / .Ep-Rs.PRICE MF01/PC12 Plu,s.Postagee ,DESCRIPTORS' CurriCdlut; Decimal Fractions; Fracticns;*Geometry; Grade 7; CradC 8; *Instrgction; *MathemAcal' Applicat s; MatAmatics Education; Maasureient; *Number C cepts; Probability; Secondary EduCation; -*Secondary School Mathematics; Sta'tistics;' *Textbooks IDENTIFIERS *School Mathematics Study Group - ABSTRACT- This is an experimental SMSG mathematics,,ext for junir h4h sch6ol students. Key ideaS emphasizedare st cture of ar etic from an Algebraic viewpoint, the real number stés,as a pro ssing development, and metl'ic and won-metric mela cns in geotetry. Chapter topics include why study 110hematics eciwal and nom-decrital nuserationt.the natural numberS ard zero, 4ctoring and pripes, divisibility, unsigned rationals, nort-metric'gOmetry, .measurement, info mal geometry, approximation.; the le ro.statistics, chance, and tinit mathePatical systems. (0) , k a 1 *4********************************************************************' Reproductions supp49 by EDRS are the best that caD, be made *; from the original document. U S DEPAHTMENT'OV HL AL H EDUCATIONS. WkLF- AWE- NATIONAL iNSTITUTE OF LDUCATIPN OtP, ,).4^1 NI fitI N F x.' tA',i4t '.k At F,()N G T PC/A", i,t VS10}4 OPNE(P;eN N(.)TNFL k Y F k Pitt Ak AL N.Ailt)NAL Nt !.)V .AN P(v,,,oryA. FA AY 'PER ISSION TO REPRODUCE THIS 3 MATE IAL HAS BEEN GRANTED BY . r 1\1750. TO TkiE EDUCATIONAL RESOUIRCES INFORMATION CENTER (ERIC) EXPERIMENTAL U ITS. :FOR GRApES SENT ANb:EIGHT. N EXPaIMENTALUNITS . FOlf, GRADES SEVkiN AND EIGIIT CI. Prepare by the SCHOOL MATHEMATICS STUDY GROUP Under a grant front the NAT I ON A L SCIENCE FOUNDATION 9 , CapyrAght 1959 by tale Unlversi 4 L 4 ° vb. a. Printd In U.S.A. 4 TItRODUCTfisiON ,As a part of the activity of ths writinggroup of the_ School MatheMatics Study Group, yOrkingat Yale University in the summer of 1958, the Committeeon Grades 7 and 8 prepared g number of units to be tfiledoUt by a:large number of teachers in these grades duringthe-academic year 1958-59. A number of the units areon contenk which has rarely if ever beeu treated in grades 7 and 8; othersare basedon the more traditionalontent. but developed froma new poinof, view ahd with newHemphasis. A Teacheyls.Guide,' prepared for each'unit, icludeSa statement of the purpose 'of the. indication of preyious pupil mathematicalexperiences which are assumed, teaching auggestlons and,backgrondinformaliion for the teacher. T .. The tdpics of these units alie: f I 1, An IptroduCtory Unit on,'WhYStudy Mathematics?" 2. Decimal And Non-Dozimal Numeration. 3. The Natural NumbePs and Zero 4.. Factoring and Primes f a. Divisibility (A Supplementar Unft) 5. Unsigned Rationals 6. Non-Metric Geometry 7. Measurement 8. InfoVmal Geometry I. pIngle Relationships) 9. .Informal Geometry II. Congruent Triangles, POrpendicular.Bisqctors, - Parallelograms, Theoremof Pythagoras) a ro. Approximation .70 11. Mathematics in Science-- The Lever (A Supplementary Unit) 12. Statisticd- A Unit on Mathetatics in the social StrieS . 13. Chance t . 4 14. Finite Mathematical Systems (A:SupplementaryUnit) , The ComOtittee makes noreco endations about' which of the two grades 7 and 8, woula bemore suitable for teaching these units. Also, the order of teathing theunits is not of greae importance, but thereare, some e?5ceptions. For eXample, it would be preferablehatUnits 3, 41and 5 be :in sequence,as numbered. Also, itt would be preferablethat the four units on geometry (6,7, 8-, and 9) Ie taught inteat order. Units 4a, 11, and 14 are labelled a's,supplementary'. The COmmittee does not prbpose tnatthese units be usea wina' all.students in a class.) ° , The Committee has made free use of materiala prepared at the University of Maryland Mathematics Project (Junior High School). In some cases (Units 3,_4, and 14) the Maryland units haVe been used with only slight modification. It must be understood that all 61 the imits are ex- ploi-atbry and that, ;the Committee for Qrades 7 and 8 has made no attempt to.otitline a complete mathematics sequence fOr these grades nor ta identify any particular topics that must be included in the mipthematics of these gtades: Decisi- ons like these will be based on teaching experienCe with the new materials during the academiC year 1958-59. NOTE: Unit 7.1s not indluded intlhis 'Volume,because defays in editing and printing. AIR Ao. 4 OS UNIT I WHAT IS MATHEMATICS kND WHY-YOU NEED TO KNOW IT . "Once, on a train, I feLkinto conversationwith the man next to me. He asked me what kind of t work,I do. I'told him I was a:mathematician. -He , exclaimed, 'you are!, Don't yoU get tired of adding columns of figures all day long?' e' I had to explain . J. to,him that adding numberscan be-done by a machine. , My ,job is mainly logical reasoning." .( What is this mathematics that'so many peopleare talking about,these days? Is it'counting and computing? Is itimeasuring and drawing? Is it.a language that uses symbols, like)a code? No, mathematic& is much more than'this. Ilathematics.isa way . of thinking, 4 w6Y of re'asoning., 'It isa science of.reasoning 6 called.deductive T.easoninlic. ,1 Letus take'an. example of reasoning. Suppose there are ;thirty pupils in your claii-sroom. How can you prove that there -7'Ete at least two pup'ils who have birthdaysdUring the same mohth? Toprovethis,s#4.4tee4you don't needto know the birth- days of ihe pupils inyour groom. Instead, you reasbn like this. 4 )Imagine 12 boxes, one for each month of Ntheyear. Imaginegialso that your eeapher writes each pupil'sname41a slip of .paper, then puts each slip intd theproper box aócording to the pupil's birthday.'' If nO\poxhas more than onename, then there -could no'tbe mor'e than//l2 names. Since there are-30 nal;t s then gat ,least one box must contmin more thanone name. This is what a mathematician does. He provei,statements. 'Ninglogical reasoning. NOtice exactly what you proved. You proiredi that if there are 10,pupils inyour room then there are at lea'st two pupils wild werek b9rn theisameOmonth. It is not the Mrathematician's jOb tofind-out kih'ther there are 30 pupils in ,your room. This can be found out by observation; the teacher can count the pupils. The mathematic,ian's business is4the --then" statement. By reasoning he tries to-prove that if something is true, then something else must 15etrue. In arithmetic you have learned how to:prove simple state- ments.about numbers. You can tell without seeing the bags of peanuta.and without counting, that if you have 7 tAgs with 5 peanuts in each bag, then you have 35 peanUts altogether. You cansolv,more difficult problems reasoning instead of . calculation.. Exerclse: 1. What is'the least n'upber of students that could be enrolled in a school,so that one, could be sure,there are' at least two students with the aame birthdate? What is the largest number of students that could be enrolled in ql-schoolso that one could be sure that no two students have the same birthdate? 364? .365? Less than 364? If there are 12-movie houses'in a certain town,, how many people have to go to the show before you can be sure,that there will be at least.twó people in one. show?- 4 G. " 4. If there are five movfe houses in a *own what is the smallest number of people that would have to go,to the movies before you cag be sure that at least two,people"will see the same show? 5. What'is the le'n.st numbeiC,af peciple that could go to the movfe hoUses SO that tou cduld beilere thatno tWo people see the same show? If 8-candy,bars are to be divided among 5 boys, how many boys aan receive three candy bars if each boy is to receive at least 'one bar? A 200 pound man'and his two sons each -Neighing 100 pounds want-to cross.a river. IZ they have only one boat which can safely carryon1y400 pounds, howc thfy cross-the .river? 0 8 A farmer w4nts to"ake a goose, a fox, (and a bag of corn 'across. a ri2er4...,1,Vleft aloge the fox will eat thegoose imvor the goose will eat the' -corn. If he has only one boat l'arge enough to carry him and one of the others, how does he cross the rivbr? IPAINBUSTER 9. Three cannibals and three miszionaries want to cross a .river./ They must share a boat which is large enough to 8 carry only two people'. At no't e may the cannibals out- : number the missionaries, but the missipnarie'smay outnUMber' the cannibals. 1,If they work tOgether,how can they all crbsa the river? , BRAINBUSTER 10. Eight marbles all'have the sameisize,color, and shape. Only one marble differs in welghts Using a:balance scale, can you find the heavy m.arble if STou make only twoweighings? , rs. One way to solve problems is to try-allpossible ways to f see if any ofthATworks. This method,,is called the'experimental( S. method. Wheny6uuse this gethod you experiment likea ScientS'Ist in a laboratory. In many problems Suchan experimental method work's. Canyou think of some problems that have been solvedin . this way? A good'example is the story ofhowEdison invented ,the eledtric lamp.