tt, OCUB,B)IT.RBSUBE- v.19 'tst tt- I ,AnOison, Rrep.; And'Others ,,.tathemati:cstalii"Junior,High School., Commentar* for `,Tvaohers. VoImme II (Pgrt.3). Preliminary E ition . StatifordiAniv:.;4afiff. School Mathematics S udy Gmoup. iohal Science ndat Wadhington D.C. 60 ea _ . ED 130, 877; Cont,ains anal light type EDESPRICr ..MF01/PC04.Pius PoMtaqm.._ _ DrkIPTORS7 .-/ ICUri,iculumiTurrj,Oulum-puldoeoWri ,,In4ructiOnunioxiigh SCriOolsi,,Mathemttics ___, --EdUtatipriMeasqement; sOOontlary Educat.ionl. v$econdarrSO4opl:Mathematiez;-*Splidnqmetry_ IDENTIFIE'R SdhOol ''MatheMatiOS Study GroOp ABSTRACT This is -.part three of a three -part manual for teachers_ using _MgG junior high sChbol text .mate ials. Each chapter .contains.an,introduction,and a'collection of 'sample test questions. .Each section contains a disdussion xelat4d to the topic,at.handand, anSwers-to all the-extrtiSeS. Chapter:topics include:(1) ncinmettic ----ge-osetT14-()--Ar IMmes-and-surface-areas;L(I)--the:--sphere; and_ (4) relative error_ -: ((MP) e roductions supplied by EDES ire the best. that Can be made' fpqm the originalocuen ***************** ******** *** xttt 1.1-11 1 - ' 't";- ' 4t;"V 3 I r 5 ';',';'`-* ','it t1' ,V0,, 1 4- 4 r , g, A i M 1'Ti', 1 1 ,r. r I ''fi", ' "1.+,i 1, I I, ,- 5., . .,, 1 W414 ., --1:7,--1,1,19,I ,, 11.%,,e 4UU 71- ''0,4 ttt \ .V .,7,11,..---1.7','-.7411 ii ' l'i , \' ,' 1.7 I 5 , % ,1,,Aag, - 1., -,5 1 I. I . t . t, IL a , U. r'rt'r:=- 1 ttl ii 1,A ,r.'1 11- t 1,1, , 1 , ' 1. 7 ' , . ___. i ,' 1 ,..t2' ,,:i . 7`.1!, i - ,i _ . - - ,-..41.,- ,,,, r , . _ I V4,.1 L''' 1 ,'''--1 '' ,L',,,U-.1,; ;.., ';',: 1. '; '-' 9---10,(;)14. -.:1`',,,,,.,:'' ,%,.,,,',::,:.; r1C, 1.'1,, =. TA EM A4ICS FOR- ji....JN1(;)R ITIG.14. p-,, . ,, 1 1..-,I -,..'. , . , UUU , ,..''' ',I .,,, _..., , rFt:el I,S. t VD.' t 1,1 -1 tl--, 11 it' , .p, )-' 1:7' ----,-,'-,..--74 -,,, '' Itg I'. 'II Nmentary or,Teaehers',T :. :,,..,,, ..., U i't . ,. , . U , ', ' " ,- .', f '' :k,,r,,,t , I ..., L l' 1+ I 1, VQIUMe ti (Part 3) ! , .,,,, , ., \ j' '1' -(preliminaly edition) 4%)1 . 1-4 ; :F"r4iArcid under the supervision of the lanel on 7th- and 8th Grades of theSchool- .Mathernatics Studyrou ftr,D. Anderson, Louisiana State -.. A.7 Brown, University of Delaware 'Lenore ,.University of Chicago-1 .-g. W. JO s,University of C orado I_ :I- p ,.. .. ..-1 ,P. & _Jones, Uthve,rsity of Michigan v J. W. Mayor, AmericanAssociation for the Advancemtht of Science. :C.--RecieniSieforn, -University of Minnesota eryi Schult, Supervisor of Mathematics, Washington, D.C. I V d-R4-Wrtaee_z '_Fan = support ;fled by the- , National Science Fort on.- Cpprigkt 196qyYak rove ysity, chRINTED IN 'ME Models o use 0.9 Polyhedrons. One-Dimensional. -Polyhedrons Twok2Dimensional. Polyhedrons Three .,DimensionaliPolyhedrons ,COunting Veptices,' Edges, andFace The,t.11iers Formula . ,.. VOLUMES An SURFACE AREAS 5 . :. ,315 ,` `,' generalRemiasics, .317 11-1. Areas of Plane Figures Pl/anesandLines.... 819 11 -. 2. : 320 11- a.- .Right pria'ms.. 6 11:- Obkique Prisms 21 11- Pyramids. 322 : 1 _ Volumes of .Pyram ds' .,..,. 323 11- 7.Cones . , Sample 'QUestionS -,_ Ch THE SPWERE . 12- 1.Introduction ,-, . -12- 2,Great andSMal.:1Circles 12- -3.Properties krGreet. C4rc1es ..- . - 12- -4..Lodating- ointsonthe Surface of 7 the E a r /. - .. .. .. 3374 Volume and Area of 04, SphericalSolid 342 12- 5. 345 ' 12- 6.Finding Lengths of SmallCircles'? Sample Questions -- Chapter12 . 347 RELATIVEERROR; 355 General Re 355, 13-- 1.Greatest Pissibl e Error 0 362 132.Precision and Significant Digits 363' 13- 3.Relativerror, Ace and, Percen ofError. 464 Adding and -SubtractingMea.sUred Multiplying and Dividing. Measures . 7, -Sample stions for :Chapter _13 NcIE ,TO pEAcHERS experience of over100,junior high- 1.;.& all parts of the,country'and the=estimates _hpol,-teaclh_et - high school teaoher au e visions (including , junior Chapter ApproxiMate -number or days Total 43. _ which Thriva iut the robleMs, 'topic's andsetion t. are:indicated by an . re' Aesignpd_fOr-the_ett r students shOuld be used or asterisk (*) . Items starred in this manner scheciu* omitted' ats a means ofadjustinrthe approximate time Chapter. 10 NON-METRIC GEOMETRY.: s chapter will=be new to almost allpupils.. opt of it has been( ed in a number of .sighn, gra rather surprising success.There are a number of, reaso luding It in theighth grade curriculum.Among these are It. helps .de elopspatial 'Intuition and understanding. It er a hasizes inanother context the-.rC4e of thematic nreducing. things to theirsimPlest elements. It affords .other ways'off .looking :at ``oh Teets- inthe world about us. and. fundamental_ questions- kbouttheml- ofmalthematical (gebmetricreasoni- It illustrates types N i inand approaches t` It gives aninteresting in Sight intohe meaning of dtmension. The general 'purposes Chapter areim l r . to tIlase''for Chapter 4 of. Volume I.It suggested thal t teddherP-read 'the ..beginning of the Commentary Tpachers of that chapter In this -Chapter. thereis ipre emphasion, the use ofmodels understanying spatial geometry.There iscprresponciinkly,,mss than emphasis onisetin_3 aa such diapter is more geomebr sets.- Hot;iever. theteRminologyof.setS, intersections, andunions d..thrplIghoUt thip 'chapter,The'students shOuld ;alreadyhave s _ , these Ideas.-A qUick.run-7over of theideas some familiartywitki -o- .volume vul-fie.. asbackground for most ... soMe- ge6- students,.However, if sets,...intersections, unions, and', Metrical 'term's', can be:explained as ypu go along,this 'Tnaterial- -should-16e teachable- kwithoUt'muCh. s'peditic-- background-on the' part. of students At*, , -'use of terms "simplex" 'and the OiSta.hction8 be ween'.0 0.r 1-37.-diteftaibnal :polyhedrons- should ct5ritvibute preettison of hou -lan 0. Reading. e4t is It ten will not .follow:asilY. PulAls ,6hOuld- ,be encouraged to --beygndltheae pas 'sages when they occur, and thes go .back and, study Teachers will Sinit profitable to Bead some tpematerial With the puPils. - v -Materials: *-.in g subject at4er like this o junior. high.)school._ rgt 614,4-are of, cdhaiderable use. and fability with, models' should increase"-teaching of web Well as itapove basic understandings: The students should be en- GOraged to use cardboard or oak- tag for. -,their models of teiflahe- drops 'and cUbes. They will 120 asked, to draw on the surfaces of these later. It is suggested ttlateachatudent c6ntribute one model of di'egular -6etrahedroh and* one. ub lf67Tidtedi7-diabs-- use.- They will be needed later in thejstudy of Polyhedrons. student shouiii; Xeep a model of c1.4.10e:a44. a .model of a tetrahedron at _ome for use in homework; In this Commentary for Section 40-4 and: lb-7 there are several sliggestions for making.-modele from blobkik0i4 wood, The boys in' the cldisShould: be -encouraged- make these. Time'. The material of this Unit recommended for -study- for 0:period of about two weeks. I cla nterest is' high, some extra. - t 0 could Profitably be used. 4)F A.Word of Warning. Inc prepari_ themselves on thismateria to should not exact to ;taster it at one reading. T mi- nofogyand-points -of - view can be asgimilated.gradually- -not- take .the attitude' of- exploring the mpterial together wih the students? 9 . ProbablY -desirable'or. the: teacher ;t6 folloy the ns-in thd text to makein:cladseither '_a model-of tetrahedron or a model of a , hon-reg -iartetrahedrono -o- _ 'ddent-leave -one- edro4 to. -,uSer at home yethem keep-one modal of a tetr models oftetr4hedroAsby ' Some bdyg might like to, m and -blocks :edgers of (PQRS )" are The faces Of (P S) ' are Answers tc Exercises 10-1 1. (Construction 2. (Construction The measure of the newangle SACnus nbt be morethan the sum- of the meares of dhgle andBAD nor les (This than their -dif-f_e _noes. Otherwise, itwo fit . could be tried in class- . SIMR1exeS in'Chapter 4 weregarded points, lines, andplanes as..our of:lines and "building-blocks" : Here (with_some.prior notion and Jp-lanes) we will use points,segments, triangular ,regions -dialid_tetrakiedrons. ad our"buildingiblocks".__ The disdussion about"takingjpoints-between" and- dimension' Aftdr one %probably will needt(7)-be read mor6.-than once by pupils. A class gets the general. idea, thematerial is quite readable. discUdliorp-of twIda4-15i-tbrtb-thi readineof-the-text-by the puPilsr-tay-be-a-good_idea.. The discussion of "betweenneSs"illustratesan important fac about reading ,some Mathematics. On'e.,Sometime hasa-easier tune µ-:. reading ,for'serieral .ideas and the rereading detail' The better s might be encouraged to draw figures for eat of\these problems as: The vertices- of a pleas are blue,. Theedges- willbered. The interior 'will No. Just two. 4rerticesof tie are blue. Theedgeswill'be- red. The faces- will be green. The .solid tet-ra4e ron. Note: When we say "color points" what we mean, of course, "color the- representationscifthe points on the paper or ers o ,Exercises Constructions There_. are a_numbr of - ,POsSAJ4e_ answers PABD)arid(P130). betwo of -the if one face is labeled asonthe right. 24 =snip1exe. 24 Yes,youcan make the comparison. 'The- top vertices of the six pyramids would correspond. _ to_the_ six poirkts in the middle ofthefaces of the cube. 19-4 Po3_ In class it would be a good ideato hold or fastenmodels of tetrahedrcins together to indicate various3-dimensional pol,yhedrorie; Also, blocks of wood With somecorners and edges sawed off maybe .with Wedge-shaped piece removed) aregoOdmodels- o 3-dimensional polyhedrons.' Models of tetrahedrons should beheldtogether tlo indicate howthe intersection of twocould' be exactly an edge of one but t an edgeof the other. (They would not intersect"nicely."' Other ex_amp-es can also beShown here. swexi e ce- 0j-1- o, one bossible ,answer 305 pages 405,406-409.