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Lat diagrammea tuivantt llluttrent la mothode. 1 2 3 1 2 3 4 5 6 MIOOCOfY RESOIUTION ItSI CHART (ANSI ond ISO TEST CHART No. 2| 1.0 [tK i^ II 1^ m A APPLIED IIVVIGE ^^ 1653 East Moln Streel g'-S Rochester, Ne« York 14609 US, ■^— (716) 482 - OJOO - Phone ^= (716) 28B - 5989 - Fa« TEACHERS' JHAWJ/tL TO ACCOMI*ANV ARITHMETIC FOR THE tSRABES CANADIAN EDITION TEAOllKRt^' MANUAL tcill JIM IIKKM 1>!N1I ARITIIMKTIC F )l THE (JKADKS CANA1>IAX KDITION TORONTO THE C'OPP, CLAIIK COMF'ANY, LIMITED 1902 Kiitered ac-corclint; to Ait of tlit> I*.irlianieiit of Ciuindii, in the .V4'ar oiif tlionsiuxl iiiiic huiiHret) aiiy Tub Coi'i', Clark C'umhasv, Limitku, Toronto, in the Ortlce of the Minister ol Ayricuituru. PREFACE. CovsiPERivo the amount of time given to the subject of Arith- metic in our schools, it must be ailmitted that tlie results are lamentably poor. It is fair to presume that the faulty cliaraeter of the text-books in general use is accountable in part for these results. With few exceptions American text-books in Avithmetio seem to leave the teacher out of the account. Definitions, rules, exphmations, illustrative problems — in short, everytliing which should be tauglit by tlie teacher is placed before tlie pupil to be learned. It is true that the best teachers generally ignore tliese false aids ami ask tlie pupils not to refer to tliem. Yet, because tliey are constantly before tlie pupils, the forliiddcn aids are often- times unwisely used, and prolilei.is are i)erformeil slyly or tliought- lessly according to the rule or model solution. While these pra.^tices are carried on to some extent in good scliools, desiiite tlie care of leachers, tliey are universally pursued in scliocds whose teacdiers are guided by the text-book. Anotlier fault of text-bo„ks in com- mon use is the insufficiency of problems for practice, and teachers are compelled to make up the deficiency by placing,' upon tlie black- board original prolilems or problems taken from Arithmetics other than tlie" regular text-book. The objections to tliis practice are : (1) The danger for want of time of giving to the pupils unsuitable or poorly-arranged problems ; (2) the liarni done to pupils^ eyes by close and long-continued looking, frecpiently before a lighted window; (3) a loss of tlie teacher's time in copying. The series of books of which this Manual is a part is designed to meet the above objections. It will be readily seen that the pupils' books are not merely books of problems intended to supple- ment the use of an arithmetic already in the hands of pupils, but ' ' PREFACE. Mo™ 't1 '" '" T' '""'"^""'''''y - *'- -ly t-t-book. needed. Moreover tl,e very large number and variety of nroblen.s warr-mt the as.ert,on that tl,e book, eontain all theVobL. that™ needed lor drih work. The books for j,„i,il« are eigbt in number, .arranged somewhat on he Imes of elassilieation in City yrad.-d schools. The first tWj or I,ree books ar,. inten.led for use i„ ]>rimarv sehools, tlie last book ,n advaneed (Jramniar sehools or High sehools. The subjects are divided as follows : ^ Booh I. : Numbers from 1 tn 20. Book II. : Xunibers from 1 to 100, 7« ///..- Integers to K.OOOOO. Fnu-tional P.arts of Numbers U. t>. Money, Common Weights and .Measures. JiofJV- Whole Numbers unlimited, Common Fractions to Iwelfths, Decimal Fractions to Thousandths, Jleasuremeuts, Busi- ness rransaetions, Denominate Numbers. Book v.: Common and Decimal Fractions, Jlensnration, Denomi- iiate Numbers, business Transactions. linok VL : .Mensuration, Denominate Numl,ers. Jfetric .Svstem lereentage and .Simple Apjilications, Business Transactions and Accounts. Book VIL: I'roiit and Loss, Commission, Insurance Taxes Duties, Interest, Hanking, Stocks and Bonds, Exchange, Business Accounts, Geometrical Exercises and Measurements, Katio and i roportion. Book nil.; Miscellaneous questions involving the making of dehnit.ons, rules and formubas ; Algebraic Exercises, Involution and Evolution, Exercis, 3 in Geometry and Mensuration, Book- keeping. It has been the aim of the author to include in the books all subjects that are likely to be needed in any school, rather than limit them to the possible needs of a certain class of schools. A selection of sul)jects, tlierefore, as well as a selection of |)rol)lenis under a given s\ibje(tt may be made to suit existing conditions. The following advantages are claimed for the use of tlie Manual and text-books : 1. The separation of teachers' and pupils' books, wliereby pujiils may be tanglit i)roperly and may not be given too great assistance. Suggestions as to methods of teaching and drilling, as well as the illustrative processes, exi)lanations, rules, and definitions which belong to the teacher to develop analytically are put into the Teacliers' Manual, wliile in the ]>upils' books are jiresented only such exercises as are needed for i)ra(!tiee. 2. The careful gradation of problems, by which puiiils acquire inductively a knowledge of a ithnictical ridations and principles, and skill in arithmetical processes. This is in recognition of the well-known pedagogical principles of proceeding from tlie known to the unknown, and from the simpli^ to the coiiii)lex. It is advised that tills plan be kept constantly in mind by the teacher, ami tliat whenever a process is not understood or is not readily jjerformed, the pupils be taken back to proi^esses which are well known and whi(^h can be performed readily, and then be le"""« ^I'l'lc'.neutary drill 6. I'ractiealriess of -n-iirk ;„ '-;s; -d the so.uti:":;.rrt;:?"r'-°^''-i-^ l-Me„..,,Meiw.e,„o,stlikcl,t^,c,; ^ """Z'"^"' '° ^ive g've the,„ i,. a ,,,,,ti,.,,, ,,,■>- ^^"^"^ ^--T-Jay life, and to ..roblems serenade bv .nechanic Lk ""■^"•"'^"^<'"'' review 7 "' '--'"'"^ oo^iitions^s ;;i:s^ t~r'"' "^•' ^^^"' ^ much useful iuformatio;,. '^ ''"'"'" '° S^in incidentally JJ.;eu. of drill table, and otW device., to save tl.eti.e Of reparation of pupils- ,,e,,i'l?°"''^ 7 ™""-''>«'> there i. the -'■a' ou the lines of ,.ra, a o^ n,'^ ", """ ""^» ''"""^^ «on.e. -.gement there are', tr-'.tt'' '"'"'''' '"''"""^' '^^ «- «■- ec'ono™, of .ear than' n the ^ ^o ^ri:';'™:" "^ '""""'"^ ™'> ine author is aware tint- n, i'"l'ii- Pauyin, ,„a„ual in t": t^l, ^ ;";;'•- ^^^ -ith aceon. Essentially the same plauharbeenT," ""' ''" -^l>«"'"ent, years, and it is confiLntly believed ";" '" '^'''™-'y f- "'any readily recognise its n.erit, "* '^""'"'■''" '^achers will J~«U' J=';:r:;:r r ; ■-- of e^rclses has business men, and mechados I-l "'""'';' '"'""■'•^ <"' ^^'achers, author's re,,uest for assistance ' e" fT'T"'' "^"""'""^ *« «,e service should be made to Masses Sirn'r''"''"'"'^"'^ ^"^ ^"'''' Practice Departmen. onnected liH. f. ^ ' ''"' ""'' ^^•■"•'^"' «f "'« School ; Miss A. Koof and Jr B w" ^;^'"""^'"'"' 'f'^- ^^'»™al •^^- ^'■''^«' °f VValtham, Mass.; rBEFACE. Vll and Mr. L. F. Warren, of West Newton, Ifass. Obligatiims for valuable assistance in conecti i „' [iroof slieets aie also due U> Jliss K. A. Conistock, lit' the l!ri(lj,'cwatcr .Normal Seliool, anil .Miss .\nielia Davis, of the Framingham Normal Sehool. Suggestions for improving the woik and oorrections of errors, either in the Manual or i'upils' Books, will be gratefully received by the author. CONTEXTS. rtErTTOK I. Gt^neral Sugspstions PAOES Knds — ('our*' cif Slmly — Tlio Hccitutiiin — Mclluicls cif Drill _ Ui'Vii'Ws — 'riL-iks ami Di'sk-Wurk— lllustnitimia — Reatliiij: I'rolilcms — Analysts ainl Kxphinations — I'l-nh lenis anil ImUi'ati'il ojn'ratiniis — Fnvccastiiii; Answcis- Lanjiua^^'. II. First Stpps in Nunilu'i Groupili).' ■ if I'lipils — f '.mntiTS— Mi'thinla — Fai-ls of Num- bers to 'IVn — I'ii'liirin^' of I'nibk'Uis— I'so of Fisuri'S — Number of Stories. III. Notes for lioolc NuuLlier Oia^ IV. Notes for rnKik Nunilier Two V. Notes for rioiil< Xuinlier 'i' ree VI. Notes for lioolc Nuiiilier 1 \\- VII. Notes for liool; Nunilier Five VIII. Notes for I!oi>lc Nuiulier Six IX. Notes for Rooli Nunilier Seven X. Notes for Buulv Nuuibor Eiglit »-l:l l: -21 til -M 3.- -."iti ,-.( -HI 8L'- 111 111- 13!) l:i9- -178 170- -225 SECTION T. GENERAL SU00KSTION8. Endi. Arithmetic like most subjects of instruction may be said to have two values — its practical value as an aid in carrying on the ordinary affairs of life, and the value it has as a means of mental discipline. The ends of teaehini,' Arithmetic therefore are knowledge and imwer. It is well fti tlie teacher to know at the outset the relative iuiportanci of tliesi: ends for the purpose of making a wise selection of subjects and of knowing how they should be taught. If in our teaching of Arithmetic we keep only in mind the direct use that it has as a preparation for business, we shall find that a small fraction of t-he time usually given to this branch of instruction is sufficient. If, on the other hand, we con- sider the possibilities of a proper presentation of the i.rious topics of Arithmetic in t\w cultivation of the jmwers of observation, imagination, retloction, and reasoning, and note the rare oppor- tunity which the recitation in this branch has for teaching accuracy and clearness of expression, we shall be inclined to place the dis- cipliniivy side of the study far above that of the direct assistance it has in earning a living or in performing the common duties of life, and give to Aritlimetic a jirominent place in the Course. Another end of arithmetical instruction which perhaps is in- volvtd in the ends already named, is skill or facility in the manipulation of numbers. The practical as well as the disciplinary value of the study is greatly enhanced by such facility as will e.i- able the pupils to perform accurately and quickly the computations involved in the problems. It is the purpose, therefore, of good teachers of Arithmetic to lead the pnjiils to become accur.ate and quick computers, and to secure this end there is needed much practice in the application of priuciples already known. The URADEU AKITHMETIC. pupils can adwh'd(,'e by tlii.ir own efforts. Little shouhl be told the pupils in tliis priH-ess. The teacher's i)art is Hrst to lead the pujnls to recall ideas already in the mind which have some relation to the new principle or prooess to be tauftht. Then foil nvs the presenta- tion of the object of thouj;ht in such a w ^ as to awaken new ideas in the learner's mind, or to readjust and reinforce old ones. The object of thou.,'ht may be a phv V ' ,d)ject, as a meter-.stiek or a bond, or what represents an et, as a picture or dia-ram. Sometimes, as in teaching a ru, or dcHniti.m, the te.-ieher has only to help the pui)ils to recall . ul rearrange their ideas in an iirilerly way, and to make a general statement from them. Some processes also may be taught by leading the pupils to recall what they have previciusly learned, and make the needful applicaticm. Such purposes nmy be aeeompli,-,hed frequently by skilful ques- tioning. When the matter in hand is taught, ami ideas concerning it are clear in the pupils' ndnds, they need to be fixed by mueh re|,eti- tion. This is done in the recitation by drill and in the practice which the pupils -et in study. Little time is needed for teaching compared t' the time that is needed for jiractice. A jioriion of nearly every recitation may well be given to drill, h.ith upon what has been tan -ht recently and iipon wh.at has been taught days or weeks before. Thus it is that three mean.; of conducting the recitation in Arithmetic should be used by the teacher: first, examination to I TKAC11KU8 MAMUAU t«8t the pupils' knowledge of the to]ilc or toi>ie8 ; secondly, teaeli- in^' to awaken new idi'us in tlic impils' luiiids; and tliirdly, driliiiiK to tix and Htrciii^'tlicn tin' |iM|>ilH' kimwli'dge uS wliat lias bern tanj^ht. These priHM'ssrs ;ii'«' in)t idwiiys Heparatcd in tlie rteilation. Sonu'tinics in tlic ti'acliiii},' exen'isti there is niiire or less of exaniinatidn and drill, and sdnietinies in the examiiiation there is neci'ssarily miii'h useful drill. Hut they all have tlieir place ill the reeitatiun, and nu one of them shimld lie nejjlected. Kethodi of Srill, — The etteetiveness of tlie drill exercise will dcpenil lar','cly upon the methods employed. Tlie prime olijcct of tlie drill i-., as lias heeii said, to lix in the mind what has U'cn tauijht. Til f,'aiu this ohject, there shonld 1k' a repetition of the mental act which was occasioned when the principle or jiroeess was taught. In the drill exercise, the teacher should have in mind ex.u'tly V hat acts of the mind are to lie repeated, and he slionld also aim to have every pupil employed durin(,' the exercise. Drill which has not a delinitc ohject, or which consists eneils ihnvn."' The teacher then gives the answer, and asks all pupils who have tliat answer to hold up the .slate or ]iaper so that the teacher can see the answer. It will be necessary for the ligures of the answer to be written large 6 OBADED AKITHMETIC. and in a given place, that tliey may be seen readily. Properly managed, tliis method of drill will afford opportunity for giving niueli praetiee in a short perioil of time. This method will serve for examination as well as for dietation. Written drill work may be given from die .tion or from tlie booi<. If from tlie liook, a nnndier of exercises may be given out for pupils to perform as many as they can witliin a certain time, or tlie exercises may be given singly to members of tlie class, eacli ])U|iil ])crforming sucli jiroblems as are most needful or most nseful for him to perform. Seviews, — For the sake of making pupils familiar with certain imjiortant ])rinciples or processes, and also for the sake of connect- ing togi'tlier kindred subjects, fre(iuent reviews should be made. In giving sucli reviews, let tlie ti'aclier have in mind a well-defined purpose, and not force pupils to work simply for tlie sake of work- ing. Xo exercise should be given tliat is not needed, eitlier for fixing in the mind wliat has been taught, or for testing the pupils' knowledge of a given subject, or for seiairing skill and facility in the use of numbers. Tasks and Desk-work. — The lessons set for practice or study should, as nearh' as possililc, servo the needs of every pupil in the class. So far as the character of tlie work is concerned, those sul)- jects only shonhl be given for tasks which are really needed to be revieweil or which will serve as a test of the jjupils' knowledge. The given lesson also should be of sufficient length to employ nil the iiupils of the class a reasonable annuiiit of time. To accom- modate the work to the different degrees of ability and ijuickness in members of the class, it may be well to give a lesson which all can do, and in addition, extra work for those who are able to do more. Illnstrations. — I'roblems involving measurements, and other problems whose conditions are not quite clear, should be illustrated by means of plans or diagrams. Care should be taken, however, that the attention of pu]iils be not diverted from the real purpose of the drawings by their uiccty or elaborateness, and that illustra- '— iSWfe^l TEACHEKS' MANUAL. 7 tions be not roquirod wlion the problems are fullj understood, or wlien tiie ].rol,l,.ins ciiu be jierforined without tliem. Reading Problems — Let the pupils form the habit from the outset of r,.adi.,g over thoughtfully every problem before they begin the solution. The points to bo noted in the readiii- are • What IS re,p,ire,I ? Wliat steps are to be taken ? About how lame will the answer be ? Analyses and Explanations. - Oral analyses or explanations in the lower p-ades should be very simple. In these grades short and direet st,at..iueuts as to what lias l)een or is to be don,, in the solution of a given problem may be made. The statements should as nearly as possible rej.reseiit exactly the pupil's thought and should consist therefore of words largely of his own choi... and arrangement. In the mid.Ue and uj.per grades more and more minute analyses m,ay be insiste.l upon, the pupils being re.juired as tlieir imn.ls develop to give reasons for the stej.s taken. But in no part of the eourse should pui.ils be required to give formal and re,i,ly.made explan.ations whieh do not represent their own thinking .Some statements whhdi explain to the teacher may actually bewihler the pupils and deaden their thinking powers. Problems and Indicated Operations. - From a very early period in the course in Arithmetic the pupils should be led to see and to indicate operations that are involve.l in problems. In the first year pupils are encouraged to make story problems from equations and soon after to make the equations from the story pilblems. Thi work .should go on nil through the course until in the later grades much of the work in Arithmetic ni.ay well consist of written oror.al statements of processes that are involved in given problems. Forecasting Answers. -Every teacher knows what surprising and unreasonable answers to problems are given by pupils, being m .some cases many thousand times too large or small. To prevent such niistakes and to encourage thoughtful attention to the condi- tions of problems, it is well by means of questions first to give an api.roxim,ate estimate as to what the answer will be. .Suppose for example the problem is : "How many bushels of potatoe will 8 GRADED ARITHMETIC. be gathered from a lot of land 100 feet square if it yields at the rate of 200 busliels to the acre '! '' The teacher may ask : " How many scj. ft. in the lot ? What part of an acre in tlie lot '! If the lot is a little less tlian one quarter of an acre, less tlian how many bushels will the lot yieWl"' Or if the problem is : "How many bushels of wheat can be jiut into a bin 6 ft. 4 in. long, 4 ft. 2 in. wide and 3 ft. di^ep," tlio teacher may ask : " About how many cu. ft. does tlie bin contain ? How do a bnshtl and a cub! ''mt compare ? Will the bin contain a few more or less tlii . 72 bushels then ? " Gradually the teacher niMy ,'ivc less anu less assistance in this direction until tli(^ jjnpils will be able to forecast the answers independently and form the habit of so doing. Language. — The good teacher is quick to see the useful and constant opportunities which Arithmetic aifords for teaching lan- guage. From the very first year when eliildren are encouraged to tell litthi number stories to the highest grade of the Grammar School when explanations, rules and definitions are required to be made by the pupils themselves, is there constant occasion for the clear and accurate ex])ression of thcmght. The very accuracy of the thought required in Arithmetical ])rocesses and definitions makes accuracy of expression necessary both in the choice and in the arrangement of wurdii. SECTION II. FIRST STEPS IS NUMBER. Before the book is put into the hands of children, it will be well to teach the numbers to ten by the aid of objects without the use of figures. Tliis may occupy a greater or less time according to the ability and previous training of the children. Most children who enter school at five years of age know very little of numl)er, while otheiti who enter at a later period, or who have had the TEACHERS MANUAL. advantage of a good inlicritanre or hpljifiil homp and kindorgartpn training, can recognize numbers and make all combinations and separation of numbers to four or six. For tlie saki^ of a jirojier adaptation of tlie work to tlie capacity of the cliildren, the separa- tion of tlio children into groups will be found advisable. The division made need not be lixed. Then! is as much ditl'erence among children in the growth of their intelligence as tliere is in the intelligence itself. Accordingly pujiils should be transferred from one group to another as soon as they are discovered to be ahead of or behind their fellows. Counters. — On some accounts the variety of objects used for teaching numbers in the first lessons should not be great, being limited to the splints or wooden blocks. If blocks are used, they might be either of three sizes, viz.: culjcs (1X1X1 in.), half-cubes (1 X 1 X .} in.), quarter-cidies (1 X i X -J in.). Tlie whole cubes are preferred by many teachers. Splints have the advantage of being easily handled. Hethods. — For teaching number to young children, there should be a table about which tlie children can stand in full sight of the counters in f"ont of the teacher, who sits at the head of the table. (-)r, if there is no table, the children may sit in their seats and the teacher use an inclined table in sight of the children. If the number three and their combinations are known by the children, ask them tc jiluee three blocks tcjgether. .\fter this is done by all, the teacher and children jiut one block with the three blocks, and the new number is named by the teaclier. Ti:iirher: Three blocks and one block are liow many blocks? C/iUdren (answering singly): Three blocks and one block are four blocks. T. (taking away (me block): Four blocks less one block are how many blocks? C/t. (taking away one block as the teacher did): Four blocks less one block are three blocks. 1\ (moving tlie blocks): Two blocks and two blocks are how many blocks? Four blocks less two blocks ? One block and three blocks ? Four blocks less three blocks ? Two twos ot blocks ? One four of blocks ? One-half of four blocks ? One-fourth of four blocks ? 10 GKADED AltlTHMETIC. How many twos of blocks in four blocks? How many ones ot blocks n, four blocks ? Answers to tlicse (jucstions are to be given m onler, tbe objects bein- uscl in the manner indicate,! Tlie work witli objects sliould be contiuucil until the cliihlren have i clear and deHnite idea of the combinations. These ex,.rciscs may be followed by a comparison of four blocks with three blocks, and two blocks and one block, the children being led with tbe gr„„|,s before them to answer the following questions : Four blocks are how many more than three blocks ? Tliree blocks are how many ess than four blocks ? Four blocks are how many more than two b ocks ? Two blocks are how many less than four blocks ■/ Four blocks are how many more than one block ? One block is how many less than four blocks ? In the same way numbers to ten may be taught, no abstract memory work being required or expected during these lirst lessons. Ihe following facts of numbers to ten should be tau-ht • 6: f4+l; 5-1; ,3 + 2; 5-2; 2 + .3 ; 5-3; 1 + \5 — 4; Ifive; 5ones;5-i-l; JnfS. i; rr, + l; G-1; 4 + 2; fi-2; 3 + 3; r,-S; 2 + 4; 6 : j (! - 4 ; 1 + 5 ; 6 - 5 ; 1 six ; 6 ones ; 0^1; ,3 twos , L 6 -f- :. ; 2 threes ; H~3; J of ; J of ; J of C. r; 10-.-,; :; + 7; 10-7; L' -|- 8 ; 10-8; !+',»; 10 — 0; Iti'ij; 10 ones; 10 4-1; .".twos; 10 -^L'; 2 lives; 10 -^ ."i ; J of 10 ; J of 10 ; ,'„ of 10. 10: Pictaring^ of Problems. — Tlie ]>icturiug of problpms m:iy some- tiiiifs bo given foi- lmsy-\v(jik in the lower grades, one olijeet being to provide a pleasiint indiieenjent lor the ehildren to review wliiit h:is been tanglit. Tliis snggests the iidvis;ibility of teachers' .show- ing jiupils hov.- the drawings nniy be niiule> and allowing them a certain degree of freedom for tlie e.Yereise of their inventiv(! powers, lint eare should be taken not to carry the pietnring too far. It slionld be remembered tli;it sn(di work is a means and not an end. Properly used, the representiitions will serve to m:ike eleiir tlie rehitions whieli are not fully grasi)ed :ind to fix in the mind com- binations and processes wliieli are not (jnite ideiu' ; Imt carried to the extent of diverting tlie attention from the nniiu pur[iose of the exercise, they are to be deprecated. Use of Figures. — Early in the year figures to rc^present numbers should be used, first in having the children read wluit is written liy tlie teacher, and afterwards in writing the figures. Such work iiuiy proceed in the following order : (1) The objects counted and placed in a group by e:ich child. (2) The name of tlie nunilier given by the teacher and repeated by the children. (;?) The figure written on the board and read by the children. (4) The figure copied or written from dictation on shite or jiajier by each child. The teacher should be careful to present only good models before the children for imitation. The writing of equations to represent combinations of objects 12 ORADED ARITHMETIC. aud thn intor,.rfitati.,n „f repics,,,!,.,! .■oml,in!di„i,s niiistitutf a useful t,.atiiiv cf iminlK.r w„ik. I'l.r ,.xain,,l,., thr .-hil,!,.,.,, aiv tol.l to jmt ;i stK.ks witli ;i stu'ks, u.ul alter tcllinf. lunv many in all the teacher wnt.« on the board ;i + ;i- (i, which the ehil.lren read The same is done after all operations witli sticks in addition, sub- traction, niultiidication and division. At another time the teacher writes on the board an expression, as 4 + 1', and :.sks the children to do with the sticks what that says. This exercise will 1.™! to a good kind of busy work in number. Number Storie..-For purpos,.s of lan-uaKe as well as for the purpose of atfordin}; a ],leasa.it means of drill in number, pupils should be led to tell stories involving' the combinations which have been taught. The general plan of snch work mav be .as follows • lirst h't tlie teacher make a story, each pupil using the sti.'ks as called lor. Then each puj.il may tell a storv and a.s h,- talks, he with the other members of the class, uses the objects for illustra- tion. After this has been done with the ais supposed that pupils need only the :::;;s ~;r'~*' '" ••"■"- -■' -^'■' -— » The use of lines and tlio i-ou"h dnivin.r. c !■ ^ • ing problen,s is shown on p ^ ^^ ^^^ J™'^ 'Y'-f '■■- 1^ used constant,- in the 1^.:. ^1^^: .S':^^ ^^^ ~:: :;::;;:;:;"'™''^- -^ ^"^ -.'—»'"- - i- thei^ ext.:; means of reviewing what they have had d o |:r S^ :^t;i!:r!;r^^r "Tr^^"'" 7^^ r " "^ '^''^^--^ = II quite trcU^ Ihe sign of multiplication as used in 1.12] Tr.A(;iiKi:s mani'au 16 thPSfi books is supposed to be read " iimltipIiiMl by " ratht'r tliim "tiiiu's" for two ri'asoiis : 1st, for tlit' sakr of uiiiforiuity. Tim signs H ami -v- all iuilicatc tliat tliii iiuinlH'r pruir by the jiupils. If desired the expression 4X1' may be reail U times 4. The pages opposite the illustrative W(Jrk are designed for busy- work and for rei'itation. In recitation the children slionld lie led to read the sentences sui)plying the ellipses as they read. I'i^tii These exercises are to be used lor busy-work and also for recitation. If necessary the children may be permitted to use the splints in finding the answers, but in the recitation they should be expected to give the answers jiromjitly after the statement, thus " 4 and 3 are 0," " :i and ."> are 8." 1(1 A good exercise is to jiut jiairs of numbers on the boanl as here indicated and have the children give the sum instantly. This will assist in subseipu'ut work in adding columns. 17 The children blumld be eneiuraged to make illustrations of processes as here indicated. 18-19 The children wiHi a little assistance can lie led (o make origiiud stories like the foUowing : '• Jly fatlic" had S ducks and he sold 4 of them. He had then 4 ducks left." 30 Good illustrative work is here shown wliicb tlie children can occasionally do by themstdves for busy-work. 31— tj5 A variety is here given for drill both in !iu.S3--work and reeitatiim. 26-28 With little effort on the part of the teacher, the children can be led to illustrate problems for themselves. But in 16 OKADKD AltlTHMETIC. [t. 2» '■I'iMr... who an. sit.i„./,t tl„.i ' '" " '■"'"'""■ ''''"' '>l'-k.s there ,uv .,„, ,„,„., ..h f, ' " ». '"'^^ ' ' "'"">• ;';..''';s-.n.,,.,c,..,,.M;:Mr'rt;: '^ ''I'-ks a,.,I o.,.. l,Wk, .M;.n;- Sus,,. '""' '"■»■ '"^"'y ''■tve you ■/ Kiev,.,, 1,1, IfdW l„aiiy iiuiiyljI,„.ksi,iotpii '■ti-- 'nik(^ a«ay ,„„. I,],,,.^ ■'<■■* ''■■'*■'* ""'■ lil'«'k are I,,,-, "'-»>■ Llo-ks r' otf;,.r faets i,, w'T ,' '"T '''""'' ^'"' '"•■^• 5 bloeks are « l.loek - ^.^ ? "I' "^ ""•>■ -^ - ^ I'l-.'k^ less ...any left over r'°„,, ... ,,;i„"' "'j .'"T '-">• «-- -"' Low l».w many l,.ft over •^■' / if,.,. *, '" " ''"'^ "'"".^' *'""■■>* «■"! urs, threes, a,.,l twos, a Lu ■!:',,, on f'''-f n '" '"""''" """ sl'OuUl be „.a,le; thus in •„ s -er t . " '" '"""'""■^ ^'"^"^'■ ..-thanoi,,,..,,,,,. ii.;:;::.;:::';;^r;j''''"''k--. with a rul,ber hand an^ ^^^ ^:;:^'T": ^I' 'i'' ""■'■■ *'>«''"- tens of sti,.ks iu eleven amllmv '"'''••^- '' ^^ '"^"'y be le,l to bin,, the iek T' """'■ ^ '^'^ -^''i'''- ^'-uM -ay thaMu 11 tL:rt'r:;:;^irrt;n"--^ I. a?] TKACHEItS MANUAL 17 ill liguri'S tliis inmilii'r tlii' ti'ii ImumIIi' may Im; pliipod on the lodge of tlir UiickliMaiil ami tin' <'l]iiilrrii Ijc li'il tii wiiti' 1 iilmvc it lor 1 tell. Tlii^ 1 sti.'k may In' plafi'il lii'siilc tin' tiMi and aiiotliiT li^'iiri' 1 1h' plai-rd aliovf tlir 1 stii'k. making 1 ten and l.orU. Tliis I'xcirisf slionld hv worki'd ont viMy slowly and carididly. ami if nri'fsmiry it slionlil Ix' rc|pi'at,i'il scviTal times, the imrixwi' l.cing to Ifad the ('Idldrcn to liavc a clc ■: idiM of tin' di'i'imal systrni at tin' (put.srt. Altrr this work with olijcrts has lirrn (hmi', filiation stati'nn'nts may \i<' nivrn liy thi' impils in answer to i|iii'stionM ^ivcn liy the tfaidii-r ':;■ nad Inini tin' hook. Thns all oinM-ations that havi- hct'ii IHTfoiancd with the ulijccts aii' now ^ivi'n orally withont ohjccts. Then fcillows tht! writing,' of tlic' cipiations liy tin' childri'ii. Tliis will constitute a jiart of the linsy-work of the childri'ii, who copy from the book tlic (lucstions and coinph'tc tlic ('c|nations. The story iiroldi'iiis which naturallv lollow may lie tidil and solved in the recit,ition and also at the seat. The order of procedure in all this work sliouhl be noticed — (1) operathras with objects, (J) yivin;,' of oral eipiations, (3) },'"''ng of written ('([nations, (4) (,'ivin},' of story jii'obleins. Sometimes drawing's may be made in illustration of operations directly after the objects have been used, and sometimes the drawinj,'s may be used instead of the objects, l-'or review, tln^ onh'r above yiveii may be cliaiifjcd or inverted. Tlic oral or writtiMi eiiuations may bo given by the eliildven, wlio will illustrate or demonstrate the equa- tions by the use of objects or drawings ; or the eiiildrcn may give the story problem, afterwards solve it by objects or drawings and finally make tin; oral and written eciuatioiis. The order above given with such variation as circumstances will determine, sliould lie foUowed in treating each new number. HT—'.Ht Xearly all the (pie.stions on tlicsf pages should be S(dved by means of simple drawings as shown on page .'17. Some care will have to be taken to show the children just wliat is expected of them. Slow progress in picturing |iroblcms at this stage must be e.\pected. From what the teacher does upon the blackboard in 18 "■'■W.KI. AUITIIMCTIC. [I. 40 rr;:::;::;;7r;L7-'f-:.":rsr -«- =::::;l:i;;f ':r-=^™rvj.S "J;r"- ■■^"" ''=.-;■« ;;-:,r£ '" .'/• I must ,,ut t,.., witl, ti / "' •"'" " '«-'.-.. , *.„, irom thesH cons?" ..(.| i- , •'^ ^' " 'lo w 1 Hn.l >,; -«:'^r;f::;?i:r:;;:,;::.;-.; -.. given w.,:a.;:,:c^e'r:fi;;::::. "'"— -'-la ^e 1. ni] TKACIIERS MANUAL. 19 nt^ii'i Thn form nf jiroMi'ms .itnl answiTs in Kxorciscs 12 and 13 is a iiiihIi'I I'lir oriKiiial |>i'c>lili'iiis wiiicli hIhmiIiI In' );ivi'ii l>y tlii' rliiUlivii ori'asiiiiially. Tlii'.si' may In' nivcti with ami witliiuit c ijc"ts. !H Tlic pKililcms iiivdlviiiK days in tin' svcfk, wi-cks in the niniith, ami units in tin; dozi'ii, slnmld lie cxtcndrd liy the ti'aclii'r ami l>y tlif I'liildrcn witli a» (jrcat varii'ty as possililc. rt.T Intiiuluci' till' mca.snri's lii'i'c, usin^ watiT i>i' dry saml. l.i't the cliildicn measure tin' water iir sand until t)ie facts lie<'{inm fully understoinl. The measures nIiouM also he '.seil in teaeliin^ proU lems similar to those in the lower jiart of this pane. This is followed hy picturiuj; the prolilems as shown in the euc. The ehildren should nuike the drawin^;s with as little assistaiux' from the teacher as possihle. Such work will he louuil very useful ami interesting' to t\w chihlren. The illustrations should always lie made whenever tlic children find ditlieulty in makiii),' a mental imaRe of the conditions, and in performinf,' the prohlem. Three or f(mr days may hi' protitahly spent upon proldi'ins similar to those given mi this paj;e anil the pa^e followiu);. 57 The measurinf; strip called for should he cut from stiff paper; jiastehoard or cardhoard wcjuld he lictter. If the children cannot cut and mark tliis strip accurately, the teacher or older children should do it. .Several exercises with this measure and a yard-sti<'k, similar to tho.se on the lower part of the ]>af,'e, should be given. 5S A teaching exercise shoidd precede drill upon this page. Teacli ri'ctangle and square, hy showing that rectangles have sijuare corners and that squares are rectangles whose sides are equal. l'a|)er cutting and fidiling should accompany the soluticui of e, (I, e,f, and jr. Other similar prohlems nmy he given. an If the children carnot ]ierf.'rm these prohlems readily, objects anil drawings si. i d he used. Let tlie answers be in entire sentences, thus : '■ Tn two quarts there are four pints," or '•Tliere are four pints in two quarts.'' Do not insist upon one form of answer to these jiroblems. If the statement of the mm no OliADKl, AllVniilEcLV. [I. oa r;i"—-*- .*«,. „,,,„„,„:,,, ''^ H.nsly, numi,:,,. i„,,i„„.,, '''";• -^-'■",v pairs a. well cokmn ,aa,v 1.. a.Mnl, (i, ,s, , ,, J.',, ,!, ,. ';;'";","^- '^'''"« ti'e iir.st i-iS*'" J!';;;: •:x:f :;,:;,/;;:;;; "V''''^"-^^ «'-'" ..ot ,. ""' c'"a.-. A good ft,„u ,a V : 'V'" IT'""''' '' ""T -e hfteen cents, an,I I «,„„ „„.^, '"-■'■"' 1"«'« I sl.all have -•■"ts"; aa,l in A: " Two ,.,v , 4 ."\ """■'' '" ''•■'-« seventeen ;-'-!««. T.„ ,„,,, a„„ t , ;^i:;';^-"'" ''- -lo,. have . ''-'■ J^'HT legs a„,l t„-elve le... a,e Xe T ."' '"«■' ■'""' '"'^l'- -'" -rite witl.oat a e„„^, it ,^ . t"''" ^'^ ""^ "InWren '-'f'-,^e for tl.en, to ,vri e ti, i ,s .f" T""'' "' ^"'"'•■' ""■J™";;;«-eitethe,nintl,ecL;r "' '''"'' '"'''' ''^"' »- t;;>" '""^I-'-i^^I'I;,::?;^";; ":;;; "'™'-'.P^'^- I-tie„Iar atte„. 1« of the .Manual. '""' ""' """» "^ -sl.ovvn on page thJ^iM;::;:;:':::;:;'^ tz t;:-.,nj't7"'^'""" ''" -«' p--- repeating a part of the .p.e.tion Tl I ""''' ""•" '''"«"•«'■ ''V -t.r« .staten,ents, .ith n'o p e bef re"*;""" ''"""" '^'' '""-" '" tl- children n,i,,,t gj,,, th a „-er " m" """"'■■ ''"'^'•'-onallv '^";k1« H-or,l. '"""" 'l''"^kl,v to eaeh question l,y a '3 Sight exereise.s for drill T .. ., P~.n,,tly, thus: 7, JO, 8 ., etc '■''"'''"" ^''^-'^ ^'-S'-lt-s --. 's::,;:;;:::":.'-;;- ^o^- he given m entire «.. before they are understood by the ellLLr" '"" '" ^"^ "''^'"-'^ 1. 78] TEACHEKS MANUAL. 21 78 Show by bundles of sticks the expression of two tens and no units. 8 1 A iiortion of tliis pnge only ]nay be giwn at a time. Observe the order of reiiiesontation. Keview frequently wl>at has been taught. 84-8(J The problems of these pages will suggest a kind of work that nuiy be done iu connection witli nature lessons and language. Lead the children to give the answers iu entire sen- tences. Thus, in 26, page 84, the answers m.ay be : («) 3 pansies liave ti times ij petals. 3 times "> petals are 1". jiotals. (i) I pay () times 2 cents for two-cent postage stamps, (i times 2 cents are 12 cents. I pay n cents for the paper. I ]Kiy 12 cent.-- and r, cents for the stamps and jiaper. 12 cents and 5 cents are 17 cents. Let tlie statements he made by tlie children in re]dy to questions. Gradually they will he able to give the answers without assistance. in proper form SECTION IV. NOTES KdH HOOK NLMBER TWO. For a brief statement of the purposes of this book and for some general directions see the Xote to Teachers given in Hook No. 2. It may be said further, that objects should be used in teaching every new fact or process. The illustrative work here given is intended to supplement and not to rake tlie place of such teaching. The objects needed will be pasteboard squares and sticks for counters, common weights and measures, and toy money. The kind of illustrative blackboard work which mav be done by teachers will ajipear in the ncjtes. After the various ojierations have been taught, mucli drill in- volving a rejietition of tlie mental act in learning them will be found necessary. There is given, accordingly, in this book a great variety and number of diUl problems. For variety of class-work J UADKIl AKITHMETIC. [ir. 1 it may be w,.ll t,, ,ivo .xordsos fn.m tho blackboard from ti.no lo ^0.1. It tl»,c.b,l,l,<.n cannot perfonn tbcm witl. sonu- dc™>e of P om,,tnc.s,s, a >s advis,.,, that needed ..oH, ,ns of tl,e jn-evi bo L , W " ,"''; ■, ""''''^' ""■" "•'" "^ '-'->-! °"ly ™'- oral or 1 Tliese figures represent all pos- silile pairs of units that can be made. There is no combination m simjile addition tliat may not be referred to them. It is for this reason tliat a thorough knowledge of tliese combinations should be had. t2] TKACIIKltS MAMAI,. 23 13-13 If ni'cpssiiry, Iwron' giviiif,' these lessons from the Ijook. tfui'li tlie writiiiL; of iiuml>ers to L'O with tlie iiiil of stieks as sni,'ge.steil on (layelTor tlie Aluiuial. Further drill similar to tliat ^'iveii on jiage 1- iiiiiy he },'iv('n. 14-10 The teai'liing of iiaiiihers to 1(10 hy means of stieks sliouhl jireeeile anil areoiiniany tliese exercises. Tlii^ iinrjiose of sueli teaeliin;; is to ^'ive the eliililren a good foundation knowledge of numl)ers and their expression. As before, every ten of stieka slionld 1m^ lionnd into a bundle, and tlie numher of sueh bundles in a nuiiiber should lie ealled .so many tens, wliile the single stieks should lie e;illed ones or units. For cxainiile, if twenty-six is Jie number to be taught, let the teaeher eount out ten and jilaec a ruljber band about it, proeeeding as follows: "How many stieks liave r here ■' We will eiill it a wliat ? Ves, a ffii. Let us count out ten more. How many tens liavo I now in my hand ? How many niinv stieks liave J '.' J low many stieks in all? Two tens of stieks and six stieks are liow many stieks '.' Xow let us write this nuinber on tlie board ((daeing tlie two bundles together and six stiol.., togetlicr on the leilge). Wlio will plaee above the tens bundles tlie figure wliieh tells how nniiiy tens ? Who will place above the single stieks tlio figure whieli tells the number of ones or units •." Who will read tlie whole number ? '' The same method should be pursued witli otlier numbers in tlie twenties and tliirties, and then tlie ehildren shoulil lie given sti.'ks and bands to eount out and exjiress given numoers as far as forty. The childK n are now ready to do the work called for on page 14. The same course sliould lie followed in teaching the numbers to 100. Several days may lie profitably spent upon this work, tlie book being used for review in the recitation and for busy-work. Lead tlie children to draw the squares very carefully witli a ruler or other straight edge. 20 A review which sliould be performed without objects. 31-20 Observe carefully the order of these problems, taking up new work only when tlic old is well understood. Lead the children to perform the work of addition and subtraction first with 24 GRADED ARITHMETIC. [II. 27 o.k becaus,. the ..luldix-n can jju-e answers to the ,,nostions. One >' "■■•'■'"' l-n-o ,„ u.i,„ oi.jeets at this stage is' to give a good louiKhition lor suhscciuciit work. Ti„. prohh.n,., on i„ge 2;i are 'int..n,lea for sight-work, Imt it n,ny ■ ...II to h,-ne the ehihhvn repoat the prohlen. first hetWe gi ^ , i ftj-s>.x less twenty are tl.irty-si.x, ete. If the ohihl e„ he itate .n g,v,„g answers, go l,aek one step ; for e.xan.ple, if tl,ev en nol Kive an answer at ouee to the .jnestion i. +. 40 gh-e probleZ , tlie a^UUfon of tens alone. .■!0 + 2„, 40 + ;„), oO-f 40 ete The same course shonM be pu.sued in snhtraetion ' The qnestions on page 1-4 shouhl also be repeated thns : Five and wo are seven, fifteen an,I two are seventeen, ete. Ad,ling so a m.ke even tens, an.l subtracting so as to bre.ak np even en sho,dd he fnlly sl,own by objects and drawings. I'oJsiblv in,; work w.ll have to be given by drill fron, the board. ^7-2H The children sl,onld read these problems silently and repeat a porfon of thenr in the answer; thns in 4, page ir' le answer nnght be: Mf I sleep ten hours of the day I shali e awake twenty-four hours less ten hours. Twenty-four hon^fle n hours are fourteen hours." Of course other for,.s of answers should be permitted. >iis«ers .-iTZ^l- '^!"''' f"-'""''''^ ""» '« porfornu-d by the children wthout objects, and they will not be found very difficult ■ but™ wdl be far better to use the sticks for a day or'two up .'.s h examples both for the expression of the nun.bers and ft e proeess „ adding or subtracting. This will be especially nee , sary ,n .subtraction. Thus to take 00 fron> 84. the chihbVn take 8 tens o sticks and 4 sticks, and express that nund.er in w it , as 84. Ihey are then asked to take (J tens of .sticks or «0 sti,.ks from the n,,,.„ .,.,.. They do this and lind .4 st.ks ren.ai i ' Thu process they represent by figures, writing one number u der I I I II. 32] TEAOIIElts' MANUAL. 25 tho other, drawing a lino umlor tlic lower number, r •' writing; tlie reniailiiler below. All this is a good prejianition nut only lor llie problems of these tlire(^ jiages lint also I'or the jirolilems wliieii follow. 32 — 3j> Here is taken a new ami important stejt in a(i(lilinn and snhtraetion — the adding of units whose sum is more thiiu ten, and the subtracting of units iVom a ten and units, the units of the whole being less than tli(^ units of the p.art. The children Iiave learned to jierform these operations to twenty, and U'lW they are to apply the knowledge tluis gained to larger numbers. Sticks shimld bo used as before, first by tho addition of L' to 0, making 1 ten and 1 unit, and afterwards by the achlition of the sanu' nundier to 111. -1), .'10, etc. T,et the children stop in oa4 will be good busy-work for the ehihlrcii. Tlie answers may first be given by repeating tlie numbers to be added or subtracted, and afterwards at sight by giving the result alone ; tl)US, in 3. liage .'{.'! : Eleven less two are nine, twenty-one less two are nineteen, etc. Xine, nineteen, twenty-nine, etc. The eliihliTU will be alile by repeated rlrill to add and subtract by twos and threes very rapidly. 30 — 4(> There is no now princi]de involved in the problems of these pages, and the same course should be fcdlowecl in teaching and drilling which was pursued in the iirevious fcair pages. ])o not neglect to use the objects and drawings whenever a new number is to be added or subtracted. A repetition of tho mental act whi(di aceomp.anies the work with objects is as neeess.iry as the rciietition of the mental act in recalling former impressions. Very much drill is needed, both with and without objects, to li.x the combina- tions in the minds of the children, so that they will be given with perfect accuracy and lu-omi.tness. Occasional drill-wcuk on the board may be found useful. For busy-work, lead the children to 26 GRADED ARITHMETIC. [a. 47 ^n-ittfM ,v„rk. ' ^ "'" '"" ■"'' tl"'i" iii future 47 Asim],I,.ext,.nsi„n of tli,.,.ro(.,....,l,. i , cut o„ this page let tl,e ol.i,,,,;, ,,,, ^^ ' " ^'J";'"- I" t],e »U"li a way as to .sl.ow •' tl„-ees •; tl i ' "'" "l"""''^ "' T<.Muu,tip.iea,,.,ou,oH:::;: »;,:;:;:;;''■' '■'"'■ "'---e, ^•.il.'n.n, and ,s,,o„M le ,1^'^ •■"'''"■-'"' -'■'< *-' tl.e .sl.oukn,ei„„„e,iatelvfo,ow' ' ' !'"■ ""^ "''J""'"'^' «'"-k l-ol.l-nwitIu,uttlK/oW™^,^^^^^ i-ne„ that - of 1. „i ij 4 iz:;:::^'^;:'": r^ ""'""■"■' """- S'luaivs, they should be aske.l rl„. '^ " "I"'"''''* "■■'• '^ twelve? two-U.inls of tweh' , ' """'"""' " "■''^" ^ --tlm-.l of The division of muidjers In- nuinhei-s i. ti tion, and n,ay be ta„,ht dim ' * ' T""^' "'' ■'"■"il''-''- -unting out 24 stiek^, h " „ , n"""""" "'"' '^- ■^"- groups of 3. Then ask • ifou "'" '"l"'™'" *'"■"' "'to ". t«-e„ty-fV,ur stiek;;- "' o/^r^^'V"'^ '" ^"''^^ "- ""■- I'ow manv tin.es-' The Un,e I '" '"•""'■^-'■""■- »ti,-ks, ^.-..es, lettin, the .^nr: ^ ^ : "l';; ^^^''' "■'■ "-- "-so. and the nneove.-ed squares the kXd '"''''"'" ""' upo,: :1s i:::;::::;,;";:!;';;:': •'" ' '"- "■ '"■ '-■«■" -^ """•■" solution of appiie,, p Ob „ ,r"r,r'''"''^f ^ ''^' •■■ I'^'f-'- «- -ith a consi,iible de^ I^:^ ^ ^^i;:' «''"'"" '"' "''!« to give on these pages. l'>on,,,tne.ss answers to all examples to do anything for the el d.t vh , u' '' "'"" ,^« --'"' -t ^or example, in 8, page 5., ;Z::'n^, '!Z:Zu :T'''T represent eggs, and you may count .'.ut one do n H '"^^^ W yon, WUUe. How many have y^nZy'^lrVZ II. 55] TEAOHEKS' MANIA!,. 27 hons lay four egfjs pvpiy day. liow many ilays will it tako tliciii tci lay the ilozoti." Dii imt distiivli tlic rliililii'ii in tlicir tliinkin^'. until you see that the jjroccss is net I'li.jr. II it is not clrar say ; " If tlipy lay four CfJKS in a day cdtint out li5(> Not nuieh time is needed for the multiidieation and division by iive, and yet it is not well to neglect any jiart of the objective work. The addition and subtraction work indicated on page m may be extended if it is found that the children are losing their hold of such work. 57 It nill probably not be necessary to teach any of these problems. If not, give them to the children us they an; to work out with and without drawings. flS-SO In teaching and drilling, dwell particularly ujion those examples which are most ditliodt to remember, as for example, 6 X 7, X U, 42 -^ G, 54 -^ 0. For review in addition and subtraction 28 fiKADKI) AUITFIVKTrc. [ir. oo a.s .'iS, wl,i,.h the l,iM ;,,, '-^ ""?' "■ ''""'■ ''.^- ^-i"*.-' -.-ni,,.,, an««e.. ....„, ,„, ,.,,,.; -l"""'"' t', ,livi,„. 1,, r, ,„. ,. ■,,„■ -;;;;• y.:.,ow;jv:;:j:,:;:: :!"'•-•' -•■•'- - -una\:;:;-;;^;:~;i;;:;r''''^'''-' - '-■- tlie ino,I,.I slu.wn i„ 1. ' ^""""""^ H'-'s.' ].n,l,l,.,„s a,r„nliug to ...0^ "'nil-le <^r;:,:.^i""'„;,;:^ ;; ;;^; f"h- ^ ..v,. tl,o ..,uI,l..on lore. slun,I,l be ,„ost persist, h'""' '■";"'''"='"""'^' "'™- ana (i l,ave 1„.,.„ ]„a,„e,l , 'h , ' "'""il'>i™ti(,ns by 4 for oval drill. ' ''■^' ''- ""'''" '«' '■^t''ml-l .un^iderabiy 63 A little assistance mav l,„,.„ t„ i ''rforetl,eyea,M™,.kouttl,e xe ,"""" *" "'« '-''il'I'''" t>.-.u with this „.o,.k. a,:; Se "':;::,::;" ■'■^'"■^^ ^"^ ■"" "^'^^™ to dra«-, let him us,. ,„a,-ks ,.,. ,." ^ '" f' "" "''J^"^' '"" 'M-Ht probably be too ,liffie„lt for el, Id,.. , '1",' ''"""\""'' ^''i'''-' "ill G 8ee suggestions for oa-,. GJ -S o^te^^^iM^:;: :.:i"T r:f""^- -' "-"■ - -^- whieh are most diffieult ,, llrn 'l !'" '"""'"''■' "'' *''« table the others. ''■"'" ^" ^"-'tt*'" "' '"'avier lines than fcf ■"" ''-'-' '•^'" --^'- ^"- '-y-w.,r. and t.r reeita- II. 71] TEACHERS' MANt'AL. 29 71-73 AVoifjlits ami mp;isiircs new to tlio I'hililron sliould lie taught with tlie objects. Tlii! .simpli'i- ijnilili'ijis also slic.iilil lie fiist tau[,'lit with oliji'cts. AltiTwanls, pii-tiu icprrsi'iitin,^' th« weights and measui'es may liu used. Siiuaies or oliloiiy.s ol' aji- jiroxiimitely correct rchitive size would siLitiee I'or this iiMrpose. For examjile, iu 11, a sciuare whose edge; is 1 in. coidd re|ircscnt the bushel, marked off in four eiiual jiarts to represent jiecks. Opliosite the spaces the prices couhl be placed, and from them the re(piired answers given. I'roldems involving the use of jiounds and dozens also can be illustrated in the same way. The children if ])ermitted will give some very interesting grajihic sidutions of these problems. 'When the children have a (dear idea of the jiro- cesses by the aid of objects and drawings, they should be expected to solve the problems ]irom]itly without aid of any kind. 74 When the cliildren know thoroughly and can tell at sight all ciunbinations to 1(H», using any number for the adding, sub- tracting, nmltiplyingi or diviiling number to lit, they are ready for the work of this section, which includes the addition and subtrac- tion of any nundier to KM), and the nndtiplication anil division of numbers to KM) by nuiidiers to 20. To accom)ilish this it will be nei'cssary to follow slowly the order indicated, and to give frecpu'ut reviews. 75 — 83 If the work is found too difficult at any jioint. observe the preceding ste]is. For example, if the children cannot perform readily the work on page 7(!, let tlieni first add by tens and units separately ; thus, iu 10 : 30 and 40 are 70, and it are 70 ; oO and 30 are 80, and 2 are 82, etc. After a rapid review of this kind they will be ready to analyze and add mentally. .Subtraction will be found nu)re ditticult than addition, and more tinu' shoidd be spent upon it. Tlie sejiaration of the number to be subtracted into tens and units may be freiiuently necessary ; for cxamjilc. in 2, page 81 : "it less 30 are 45 less 9 are 3(>. etc A com])lete mastery of the subtraction of any number from 100 is especially desirable, since such subtraction will be found very convenient iu making change. There is no more reason for adding coins piece by piece 30 GRADED AlilTHMETIC. [II. aa -..:.;...L,,,,:r:',;;;;!;::-:;:;;r;:r:;:;;-;:::";,;;;™ 85-80 An almost wnliinit,.,! ,u,ml.,.r of examplo, mav 1». given in connoetion with these t.,l,l„. T^ 7^°'P"^'' """y be the n„n,l,ers are arran." J fi en ., 'V '"' "^""""^ "'^' A,J,i & 4. 1 . ti»»^'^*i HI i.ihie A, linos s, 7). and » Ad 8 to nnn,be,.s ,n lino j> as far as A A,hl 18 as f. ™ ' AajS as far as / Do the san.e to nnmbers in ,,. Subt ae ?' ai::^::r^.t'--^-:,;---.the.^^^ columns , ., . etc. Snbt.et 14-^:;::^ i^ I^T r^"' ^.muing with . m Table C. subtraet 14 froj .^m^ *!-. 11. H7] T1:A(I1KI!S' MANIAL. 81 columns r, 0 Ability tn nudti|il,v numbers liv any nnndier to 1(1. and to divide by numb.-rs to 2(1, woul.l seem tc be id' snttieient eonv.'n- ieiuv in everyday life to Kivcsnme attention to these processes in the lower Rrades. The use of the drill tables on the two preeedint,' pa^'es nmy help to supplement these pa:,'es in KiviiiK the needed drill. I'laee >ii>i)n the boaril other drill taldes. eon.sistint,' of the multiples to 100 of all nundn'rs as far as L'0.-.. 01 Some attention to the form of answ.Ts to ].roblenis should be (,'iv''n. with tlie understamlintr always that a I'orreet form in itself is not an end. but only a nmans of showing that tlie )iroees3 of Miiuking is eorreet. Kirst be sure that the eliild understands a process, and then lead liini to express the steps ..f tlie process in correct languas,'e. This page oi problems, and others winch b>llow, are supposed to furnish nnidels of simide explanations. To lix some of these forms in the mind, it nUKbt lie well to },'ive other numbi'rs than those stated in a problem. l-"iu- exampb'. in 9, after the problem is sidved as (,'iven, the teacher mis,'ht say, "What will four books cost?" or. " Supjiose six books cost eiL;liteen dollars, what woidd one book cost ? two books ? " etc. If the thoui,'ht is not clearly I'xpressed, do not i-orrect the form of answers until it is clear that the process is understood. Objects or drawinsis should freiiuently be used, both in testing the child's knowledge of a process, and in teaidiing him the process. 82 OKADKI) AKITII.MKTIC. [11. »!l 08 Somn of thos.. ,,rnl,I,.,„s may liavn t., 1«. il|„str„t,.,I l„.r„r,. t .e stat,.n....,t, ..f st..,,, a.,,1 uns«-,.,» ■,„■ .-nvn, l„,t la tl,. .hiluivn llustmt,. tlu.m if tl„.y ,„„ ,vitlM,ut ussista,,..... Wul, tl„- ill,.sl>.a. t.ouH l„..,„... ,1,., ,.,,il,l,,„, M„. t 1,.,,. „,,y ,,k. as i,. 5, ••..,„. anpio lli.'.. th,. ,,r.,l,l,.„, ..,.„ 1,.. snlv..,! i„ tl». f,„„, ^Mv,.,,. t„ l„. f„ll„«v,l fy tlH. us., of „tl„.r nu>ul...,s. Tl„. .stat,.,„,.„t nf ^t,.,,s iu s|„„.t sont.,,,.™, as in 7 aa.l 8. will 1. f,„„„l „,..,,., „,„, „,it,,,,,„ ,,^ busy-wnrk. It ,„ ,u,.v ras.. tl,« n„n,l„.r.s soria t,.o la,.;,.. „h,. s,„.,11,.,. on.s ; as i„ 3. tl,.. t l,..r ,„„, ,„,, .. ;vi,at will it ..t at that'n.,.. l"f'^ '"■" ''"'^ "■••'«'»"1 ■' f"»r e„IIa,s'.' ..i^ht ..nlla.sy f.,,,,- collars aii,l two cuffs •' •' ,.t,.. vis^ll'l f*"" '"■'■"'"'"''^•>- 'inosti„„i„« ,„. t,.a,.l,i„„ ,„av l,o a.l- v,sabe .,..f,,„, so,,,., of tl„.s,. ,„.ol,l s a,-.. kIv,.,, , f,„. Vxa,,,,.!,., p..hta,l.. ri,« ,„. SS..S i„v„lv,.,l i,. 8 1 9 will I,.. r™,lilv umlemoo.1 ,t s,„,ilar prol.lems witl, s.aall,.,- „,„„1„„, an, li.-.'t 94 T„ l..ar,, o„or fo,. all tl,o „„,„l,er of .lays in oaH, ,ao„th is qmte ,™porta„t, an,l ne..,l not 1„. so ,liffi,.„lt as n.any s„|,pos.. The vorse .-riurty .lays l.ath S..pt,.„,l„.,.," ,.o„l,, ,„ l„a„,e.l, ,t it woul. , ,„n.. to ,..a..„ tl,„ fa,.ts tl.at fo„r .,f th.- n.onths hav.. 30 rest ,a^e ,il. lays. H.,.,i„.,„t appli,.ati.,n.s o ,;,.-,, , ;, i. in little Sr^n" n " *'""' '" ""■ '"""' ■'"• "'^" '■"•>■ -" ■"" '- "- «,e ch,l.h-..„ are .l.rected to n,ak.., will 1,.. a nev..,-e,>.li„. source of pleasure ami jjrotit. 95 This work shouW be continued until tl„. chil.h-en have a tolerably aecu^xte i.lea of short distances 96 Reauetion, first witli the ai.l of a n.casure, an.l afterwanls as far as possible w,th no aid, will be found a,> ..xclh.nt prepara- tion for subselO-l Must of till' fiirtu cnntiiincd in tliPSP prnMi'ins slinuM liavv \hvn tauKlil pri'vioiwly. II ill iiiiv ]ii()lili'm tliiTi' is a fact nr piiici'ss i(iiiti' new to the clillclri'li. it slioiiM !«■ tauj;lit cilijrctivi-l> , till' nilr liiMiik' "IwiTVi'il iliat iiDthiliK' slu>iilii lii' told tin' child whii'h hf can find mit for liiiusclf. Thi' iUu>liati<)ii of iirclilciuH hy ilic children i.s advisi'd whciinvcr the processes si'cni tmi dillicnlt, hut the same or siinihir pnihlciiis should afterwards be peil'urmed without the aid ul ol.ji'cts or dr.wiuys. Blackboard Drill. — Vor the sake of variety, it may he well sometimes to (,'ive a drill iii addiiif;, sahtraetiii},'. multiplyiii),', and dividing from the hlacklioard. If it is found, for esamph', that the jiupiU are shiw in adding and suhtr,actiii(,', put ou the hoard eolumiis hesjinniii;: with (Uily ones and twos, and iuereasinn; 1H'W numhiu's to he added lu- subtracted (gradually. I'ractiee sho\dd continue until as j;reat lacility is had in adding,' and suhtractiuj,' T, 8, and '.t. as in addii 'K and suhtr iclini; 1, o , ml ;!. Th( V iluiiins will MM^ ar as ollows : 1 1 .3 '» .■) ;j 4 .'i 4 r» .■f 8 (t 8 i) 1 '> 1 1 o ^> ."> l> :i (> H ;i 8 U fi 1 i; •■ 1 I o - o 1 4 ■ 1 s '.) *> I 4 1 ."i ft ',( (> .*{ ,, 1^ ., ^( ( .> ;i o t\ s - •\ J 4 1 1 1 L* ;! 4 ." ^y i " 7 4 :\ t 1 o ;i i 1 .") li (I ;! 4 s ■'i s 8 o ;j 4 ;; \i 4 L' .•i .") ,", i; 8 1 11 1 L' - 1 .") ;! ;[ 4 4 r, ;; 7 8 »> 1 ;i 4 1 ') "i 11 1) t ,s r> o !) O ;i 1 ."> o ~f (; o •"• Ci i) it " _ Entirely new combinations may he made by suhstitutiiif; a num. ber in place of the first number id' the column to be added, or by placing a new number below or above the column, and beginning with that number to add. I 84 '■'•AI.KI. AUIT„.MKT„ i':;:i '■-'"'"'-S.:!'™;;;;,-:-" ■■■■■" Tl.o sauu. ,„. • '"""•'•' I'l'iwrt above I III. 1] TEACUJiUS MANUAIi. 8£ SECTION V. I NOTES KOU JiO(JK .NLMKEK TIIIiEE. By far the greater ]iart of tlie work laiil out in this book is with numbers under 1000, and if a year is to be given to tlie Imok, at h^ast seven niontlis may well be sjjent uiion this j.art of it. Tlioroughness in the use of small numljers means a saving of time and increased power to apijly what is learned in the higher grades. The apiiaratus needed for teaehing and illustrating these exer- cises is a large number of short wcjoden sticks witli rubber bands, and all the eommou weiglits and measures, sueh as the foot rule, yard stick, rod line, gill, cjuart, gallon, peck, bushel, and balance for weighing. Eead carefully the Note to Teachers on pages iii and iv, particu- larly whiit is said in tln' jiaragraph marked 2. 1-11 These exer('ises are a partial review of vhat has been given in ]5ook II. Pupils should be able to i)erform them with facility before Section II. is Iiegun. If for any reason there is difficulty in performing or in iin"."•- on t„o ,„.,.„ o -siJ ;'.,'J:;'; . ^"'V- wnto tl.t fon,. ; n,nety..seven. .Sl„„v n.e 1 ten n ! ^ '"S'">-«-^« ! -xty- ■^; one h„n.h-,.a written? Why^^h"''"'^'"'' ""- "'••"■™; ^^•« -ill p.,t the 10 ten ' V" '""^ "'"'-^ -"1 tens "'any l,„n.h.ej» i„ this I,„n,„„ ,'"''* *'"■■• "' » ''"■"He. Ir„„. "K"t - a„. ,.ane„ two h,.,; ■ ft r;,;?; - """'^'■'■"- -^Vse «-r.te fan,. ,,u,.j..„, „„ ^,_^, ,„f '':""J<"' ' ^ I'un.Ired. A\-ho will -' tens i., eall,.,l wl.at •.' ;vint he, I "'" ' '"""'^^'' -"' 2 tens "o- win ,„„ „,,it, 1 ,,„ , ;^ ' ", - the whole nn„.ber ealle.l ? ■'^ '"- .- 2 hunj..e.l.s an,I C. t ns \v, ;r;, """' ''"""-1 *"■<'"*>•? How shall ,ve write it :' " I„ !,„•, . ''" ''"■ '"" "''» ■»""1-.- '.' ;■•« the hun,Ire.is, then the hun ."fi l""'" ^"''^ ™' *-•''■ '"'■"'■•'■d^ tens, an,l units, until tieth .'"' ""'' '^"""^ "'« oo.nes the .IriU be^nn on pa!e V' '.-"'■""' " ^^'"•'""^- '^'-- l'a.'es indieate the order wh oh m.; be f T'"'''?'',' ' *''''''^" "" "'"^e be sufficient in nnn.ber for son,e ,Z,-,! """'''"' •^'" "'^^ "'"y "ot lSo"Th:r"""""^'-'^"'^""i'"— it. an-vers eannot be readily gi'" ".f''.^ ""' am" ease the 5;™.. I'-or example, in U,^:"i,/f,"; "'tern.ediate steps be «.i^'ti:i:,:n;r:;r wijr r:'r' - ^^^^ - «-' i I III. 17] teachers' manual. 87 numbers on paper or slate. They then put together the 3, 4, and 4 sticks. 10 of these they slmuhl put into a ten bundle and say, "3 units and 4 units and 4 units are 11 units, eipial to 1 ten and 1 unit. I write tlie 1 unit in tlie place of units, and add the 1 ten to the tens." Then |mtting the bundles of tens together they say, '•1 ten and * tens and 2 tens and 3 tens are 10 tens, e([ual to 1 hundred and tens. I write the 1 hundred and tens in their places, and have for an answer 101." When a sufficient amount of drill with the sticks has been given, the pupils sliould place in colnnins the numbers beginning with 3, page 17, and proceed in order until all exercises on these pages are performed. It is i)ermitted sometimes for coTivenience in " proviu" " answer- to write the " carrying " figure above the column in whi(di the addition is made ; but it is not well for pupils, either in addition or in nudtiplication, to depend upon writing down the carrying figure ; thus, 37, page 17 ; 11_ 314 1.11 2()8 733 A good way of preserving the carrying figure for proof, is to write the result of each addition, thus : This is found especially useful in adding long columns. 314 ir)i 268 13 12 6 733 Do not insist upon formal and elaborate "explanations." It is enough for the i)upils to express what they recall of the process of unitnig t units and add the 1 ten with the tens. 1 and 4 and 1 ami 8 tens are 14 tens, e(pial to 1 hundred and 4 tens. I write the 4 tens and add the 1 hundred with the Imndreds. 1 ami I ;iiid 2 and 3 linndr..ds are 7 hundreds, which I write! Answer: 745." Uf course this form or any set form should not be 88 GUAI)KI> AUITH.MKTIC. [III. lO put before them as a model. Wliat is ili^sircd of tliem is simply to recall impressions which were made in the olijecti\e work, and to express what they think. 19 Anxirers: 1 707. 2 SGI. 3 71S. 4 S41. 5 690. 6 781. 7 705. 8 807. 9 7<.»r>. 10 !l<.)8. U 807. 12 83.",. 13 'J71. 14 840. 15 81o. 16 Mi. A large number of drill exami..es may be made by asking the pupils to jdaee given numbers above or below in the eohimns, or the teacher could dictate to the pupils the numbers here given and one other numlier. The answers could lie found by adding the extra nunilxT to the answers above given. In adding aloud, permit only results to be given, and lead the children to i:.i.l by pairs. If difficulty is found in this, put pairs of numbers on the board for them to add (piickly at sight as f sticks, say, ''O units from 10 units are 4 units. I write 4 in the units iilaee. I took 1 ten from the 7 tens, and there are left (! tens. J cannot take 8 tens from tens, so I take one of the hundreds.'' This he does, and after resolving it into tens and placing them witli the 6 tens, says, " 1 hundred equals 10 tens. 10 tens and 6 tens equal 10 tens." Hi^ then goes on with the subtraction, writing after each result is obtained, and telling what is done. As in addition, tljc statements should be a simple and natural expression of the pupiKs thought. At this point teach the ti'rms iiiiininiil, siihtnihi-nil, and remaini/e.r, and lead the pupils to use them in reading the exercises and answers. III. SZ] TEACHEUS MANUAL. 39 23 Answers: 1 nOl. 2 861. 3 961. 4 748. 5 966. 6 900. 7 '17. 8 8'.);i. 9 90;i. 10 752. 11 957. 12 979. 13 7.-)9. 14 101'7. 15 866. 16 857. 17 988. 18 778. 19 714 495 60.">. 20 3l>7 350 159. 21 L'70. Tlii.s ilrill tablo may be usi'd in sujijilying a large number of examples in adilitinn ami .subtractinn. For example, the pupils may be tolil to iiiUl hi from A ir K or from 1> to 1', or the samif work may be dictated to the pii]iils. (.)tlier eolunins as eb, dc, ote., could be given in tlie same way. I'raetice also in adding by lines could be given from tlie table. Otlier columns than those given for subtraction could be given as dch, ale, etc. 23 swers: 1 509. 2 L'Ki 491.' 307 l.'?9 219. 3 176 317 252 517 1S7 1S6. 4 292 704 237 179 173 124. 5 61fi 3,32 174 203 288 .381. 6 206 337 186 291 144 69. 7 278 184 289 249 336 186. 8 303 163 226 340 203 197. 9 667 58;i 83 387 626 287. 10 963 cd. 11 217 sheep. 12 625 hours. 13 71 highest grades 176 lowest gr,ades. 24 Ansirers : 1 226 194 292 393 98. 2 643 188 419 361 206. 3 60 83 294 147 139. " oo OgO 361 335 228. 5 348 266 719 304 422. t 370 206 606 445 91. 7 220. 8 268. 9 508. .-w 319. 11 342. 12 287. 13 31)6. 14 302. 15 305. 16 276. 17 321. 18 244. 19 29(1. 20 ;J72. 21 612. 22 426. 23 104 72. 24 124 120. 25 164 82. From tliis table a large amount of drill-work may be given, the pupils taking different combinations. 25 Answers: 1 873 bu. corn 102G bu. wheat. 2 388 pear trees. 3 742 mi. 4 27. 5 986 bu. 6 1 14 bu. 7 008 yds. 8 400 yds. 9 432 yds. 10 H».'i ft. 11 402 years. Some of these prol)lertis suggest a kind of wtf.k that mav be given from statistics found in the latest almunuc. 40 GRADED ARITHMETIC. ''• 2 $ai)6. 3 «!6S. fill. 28 4 837 lb. 86 Answers: X 36" 5 1091. 6 3.10 mi. abandon ti.e objects because ,,u , e^. ^d «''"'""' ''""' "" ""' without tliem. ' ' "" *""' "'« CK'-i'ect answers 30 Austt'f'i'i 6 798. 7 «7' 13 80.5. 14 i,(),T 1 90. 8 ."iOO. 2 8S4. 9 (>78. 20 771'. 26 9L'8. 32 988. 38 880. 44 903. 15 9:.'8. 21 07(1. 27 38;';. 33 588. 39 872. 45 942. 3 004. 10 320. 16 8U 22 819. 28 978. 34 774. 40 948. 46 828. 4 424. 11 780. 17 748. 23 77(). 29 621. 35 980. 41 957. 47 828. 5 024. 12 84(». 18 984. 24 891. 30 820. 36 995. 42 804. 48 567 19 900. 25 936. 31 915. 37 861. 43 649 49 910. have four grouj.s of stieks to ut \ ' '" ' ""•' '""'"^ ^''™'<1 hun,lred.b„„,ne: 8 ten-bundk.^ TuX ^'7' T '""'''' *" ''-" ^ the single sticks or units tl,e pu'piu 1, "'',', ' ^" '"'""'S together are 28 units''; an.I then in ,.utti ,71)! 7 T'- ^ *'""■'' ' ""'*^ tens shonhl a.hl : -'s unit fl o V"'""*-^' '"'" t-° '""Klles of the 8 units in the plao „ ".f ,*""f , '''r' « '""'- I write r''"J.-t." Then puttin/to et r th ; ''" ' "" ' '™''' *" "'" "-' « -.s are 32 tens „h,s 2 te^;:: ^^^r.^ttr"'" '''■"' '^^^^ Q1 . '"'"^ '""' "'plaining problems. 2 " 400; 4 072; „ 525 ',. 574 ' / ™« ^ "^ ^^^^ ' « 936. «920. 5«610; i52S. .836 ,U'^'' " ^''' '^635; 534 623 712 801 890 AS';.:' " '"'• ^ « 44.5 C265 318 371 424 4;7 sfo , ^ V r' ''' '^■' ' •^" ^ *'' JJO, «!400 552 644 736 828 III. 32] teachers' manual. 920. 7 a 1,30 infi 1,S2 I'O.S 712 801 890 ; ,■ .■«.-, 4{iL' fl.'ii) 651 744 «;i7 9;i0. 8 » 410 h 465 558 051 744 8.37 !).'{(); 670; rf 425 510 595 (!.S() IK 016 60.S 770; A 405 .558 651 602 688 774 800; ,1 1,35 10 ffl 380 450 5.32 008 084 4.32 480; c 400 .552 044 7,30 828 9"0 532 008 684 700. 11 „ ;!85 4<;2 h 340 408 470 544 612 080 ; 570; r, ■ em; /289. 40 546 735 833. 41 594 3<5 3 552. 10 864. 16 836. 22 621. 28 840. IS 21 27 98(1. 966. 986. [III. 3fl b 592; c 832; '; 323 J r/ 408 ; 513 910. ^4"«s7' ^'^.f-'"' '•"" '""■ 2«-'3 21 254 343. U 918. 12 903. 13 870. 14 83" 17 9.i6. 18 848. 19 806. 20 870 23 638. 24 700. 25 744. 26 910 ' 29 882. 30 864. These should he perfonned without ohjects, the pujuls .iviu- i„ clear form statements of process. ^ The ohjeetive work in division of lar^e >u ■„h,.rs hore hec-un shonld «ouia be . " i of 8 tens of sticks is l„nv many V J of 4 sticks is how many ^ of 84 sticks is how many , •' 'L, *,^J, ' J , tnnes called ",.,..,>/„„" t„ distinguish it fron, •■ J/,-/.,/., •' or t I 36 Lead the pupils to use the ohjeets and to make statenn-nts as they d.vide. The form of questions used for c.ereiscs on age 3o may be changed to asking how n,any tin.es one nun.ber will t contained ,n .another numb,.r. For exan.ple, in 1.- •■ How ™,v tunes are 2 sticks contained in 200 sticks ? 1I„„- ,„,„, ^ , ' ?e 2 shcks contained in 20 sticks'/ Il„,v many times are 2 si," contained in 220 .sticks ■.'" -"'-sikks If multiplication has been taught by objects thoroughly, it will «reat exTer^It '">""" "'"'' "'^ "'^"''^ "' ^^^^" '" ' great extent. It may be assumed, for example in 6, that the pupils know that 2 is contained in 16 tens 8 tens times d I similar examples may be so explained without the aid of object n 8. .a new step is taken, and should be taught witli objects thus '■ '2 sticks are contained in 13 tens of sticKs how manv ens tin s' and how many tens remainder ? 2 sticks are eontaincd in 1 te " ; 10 how many times? What is the answer ? " This princi 1^ is Ik III. 37] TEACIIKRS MANl-AI.. 48 further applied in 11, ami a similar iMuirso of ti'adiint,' slioiild Im taken. • After tlio examples of tliis pa),'e have Imm'ii performed in tlie way indieated, the same examples may he perfornn-d as if stieks and not times were (sdh-d for. Tliis is a more simple proee.s to teach. For examph'i in 1, tlie teaeher should lead the pupils to get J of 2 hundreds, then J of 2 tens, tlms liiidint; 1 of 22(1 to he 1 hundreil anil 1 ten or 110. A rapid oral review without ol)jeets should hu given before the next page is taken. 37 This page of exereises may lie taken, without olijeets, orally. In sueh exereises as 23 n, tie' [iiipils may be led to say first: "4 in 12(1, 30 times. 4 in l;i2. .i.'i times.'' Afterwards they may perform tlie exereises witliout analysis. 38 The same course with objects is to be ]inrsued here as was advised for tlic^ exereises on page ;}(!, with the added feature of expressing with figures the results as tliey are found. Teach the terms dicideml, i/irixor, and ijiint'ii'iit, and lead the pupils to use them. 39 Long division, if thought best, may be begun with small numbers, and nniy be performed without objects. A gooil form of long division is to place the divisor at the left of the dividend, and the quotient above the dividend, as shown lielow. In teaching long division, the jiupils should be led to see that there are no new processes to Icaru. but tliat it is the same as short division, with the products written out instead of carried in the mind. The following order of teaching is suggested, it being understood that as the pupils answer the (piestions the numbers are written. 2 is contained in 4 tens bow many tens times '.' 17 is contained in 83 tens how many tens times '.' How shall we find tlie remainder that is not divided by 17 '.' ilulti- plying 17 by 4 tens and subtracting, what have we '! This numlier of tens and 3 units is what number? 17 is contained in li">;i how many times ? 17 multiplied by 9 is what? What remainder is there ? What is the answer 'I 49 S33 08 1.-.3 l.W 44 OnADED AUITHMKTIC. fUI. 40 The teacliing of partition in wlii,l, ti,,. t, ,■ , number i. fo,„„l is 1„ to »„. ta « t „ l,","" '"'' "' " th. Han.,.. T,,., ,,t,t,.n„.„t of s m V k ' , ,"'"' •"' "'"'' '^ be at first very ,in,„l.. foil,,. , "*-' ''''•"""" "'""'I'' 40 Ailsuvn ; 1 •'•'\* 9 1114 I ooi,. _ ■ '1"' " *■ .i^- 3-1;',;:. 4 -'I'r/. 5 I'lVj. LM.i. 24 l(!,v 30 2oijj. A 42,», ; rf 21|J. 36 no,!", 40 62]. 46 49|. 6 2"1V. 7 2(.;t;|. 8 20v;. 9 o,„. ,. „, „ „ _. .. -'i,v «x -.11 ,?. 22 L'l'a- 2^ ic-.i 25 2qj 26r,,, 27 2^:, 28 J.ir SS " -Ji't! '^ 4, ). 33,, ,y , . , , I. ' 47 30,. 48 -TJ 49 L'.^.v 50*7 " Lead the pupils to .livi.le in .some cases hv -l iy\■,^ r ■ Slating of the first fi.mrp nf n .■ •* '' '''^''sor. «'"- times the divisor eon l,e.i:t;'r'"'/''r''' '"""" '"'«' '-y helpful when larger rCa:!' uti:"^""'""- "''" ^"' ''^ "-« 6 W5. 7 11.1^ 8 . .' ^9 n,;^- , *,.:''^- 5 204. 12 112. i3i4 1-.;-- ?s ■. ; "'•'"'■ "*•■'*• 18 13.1^ '"• " '"^*- 16 442. 17 5,,^,^. it is well to hav t, rtSi Z^^ "'■""'"■ '" """"i'"™«™. ^ ^y,-.nuitip,yi„g the lotL '^ z^XLr:;:i ",r "^r' remainder to the amount. "'''''"S ""^ «= "It wm ta.e . man, Oa.s to bur^ y^o'lt iir'a™ III. 43] lEACHEIls' MANlAr,. 45 spvpiis in 3"> " ; or, ' It will tiiki' us niunv duys as 7 rnnls l« ■on- tiiineil in .'t'l ciirdH '; or, "To linrl liow many clays, 1 iliviili o/i cnrils liy ~ ccircls." 43 Li'ad till' pupils into tlii' lialiit of writing; out tin- stfps of \m .lili' ill jjooil form, ami also of maikiiij,' tlir ilcuomination of each numlicr. Two forms may 1m> miuirccl. - form in wlilrh tlit^ sti'ps ar« only iniliratnl, ami anotliiT form in wliiih all tliu steps aru worked out ; thus, iu 1 : 1^ of 3i)6 uii. = avi'rago nunilwr of mi. in 1 da. IH^^'.Ml mi. in IX da. » nii. in 1 da. 44 Kxtcnd the work in makiii},' and snh iii;.,' orijjinal ]ircilili'ms. It will 1)1' fcnind good -vork fur rcvii'W. 45 Some ti'iudii uf the in Ci lonpy should precede these exercises. Show liow tw mces, eij,'lit pounds, six dollars, may he expressed by tiu'ures and words, and also by titjnres and ablireviatinns. (iive the si},'n lor dollars, and show its u,se by examples. I.eail the iiu|jils to .see that a nuuilier of cents less than one Imndred is written at tin' rij^lit of the point, the numlier of dimes expressed by the iirst liynre at the ris,'ht of the point, and the number of cents less th.m ten expressed by the second figure. After a little of sueli work, the pupils ought to be ready for the exercises on this page. 46 If the pupils have a thorough knowledge of writing numbers in Can. money they shouhl have little ditliculty in adding, sub- tracting, multiplying, and dividing snch niunbers. ]!efore the exercises of this page are given it would be well to have .some simjile exercises in ]iointing off, such as the fidlowing: "In 2(10 cents how many dollars ? Where shall I place the point to show the number of aollars ? I'oint off the dollai'S in .'iOfI cents ; 2i'0 cents ; 640 cents ; 104 cents. How should these numbers be written for addition and subtraction? " In multiplii'ation, lead the pupils first to multiply dollars, then cents, and finally dollars and cents, such as tlie following : $84 X 4 ; $(il X 8 ; §7u X « ; $0M X 2 ; 46 OKAUKI) AIMTII.MKTIC. [III. 47 $0.7riX4; fl.fiOxfi; f :.'.24 X .1 ; .fl.nSx4; .'B1.;18 X (1. In tliia wui'k, insist iijiiin the ixiiiit Uint; niadc in nil cases, and Icaii the children to hiw thiit, it' the ninlti|ili('and is I'l'iit.s the |irudu('t will bu cents, and that it must Im^ jniinteil iitT iiccorilingly. It is well in the written statement iif the solntiim of ])rul)li'ms to write what is given and what is re(|uired to lie t'ounil ; thus, in 1: Given the c t of 1 cow and I horse. To find tlie ojst of (i cows and ;i horses. $04 cost of 1 cow, $140 cost of 1 horse. 6 S 9'SSi " " 6 cows. $41'<) " " ;! horses. 420 " " 3 horses. $804 " " 6 cows and 3 horses. 47 Hefore giviuf? 5, 6, 7, and 8 to the pupils, lead them to see that in all such problems the dividend and divisor must he of the same demimination. This may be done by showing that 'J cents is contained in G dimes or (I dollars more than 3 tinu's. Oive a few exercises like the following ; '• 3 cents in 3 dimes how many times? C cents in 3 dimes? 2 dimes in 1 dollar? 4 dimes in 2 dollars ? 5 cents in 2 dollars ? 4 cents in 1 dollar ? 4 cents in 8 dollars ? " 9 and 10 should be performed and exphained by partitioii, in which the answer is the same denomination as the sum divided; thus; i of $-1 = $•>■, J of $8.25 = . § !.(>.-). In these and similar exercises pay particular attention to pointing off. In exercises like the last of 10 it ni.'iy lie well to reduce the sum divided to cents before dividing ; thus : " ,'j of 408 cents is 34 cents." 48 Let the pupils pcM'form as many of these orally as they can, 1, 2, and 4 to be done by partition, and 3 by division. A simple statement of process should be made to represent the difference between these two oper.atiims ; thus, in Irt: "1 qt. of kerosene will cost i as nmch as 4 qt. i oi 3C>^ = 'il^." And 3: -Since 1 book can be bought for 2!>iliar t,. tl.o pupils, Kiv- s,.m,. si...,,!.' work witl. l.ln.'ks, as i of IG bl.Kks, i I.t 1.1 l.l.Hks, , „t I'O .,l,„.ks. l.,s,.s o>- .lraw,n,» ,„rtofn„,nLstl,at.^ Um.v,.„1v f,mn,l ; tl.,s : A .1 . ..r.-Vs. 40-51 X..n..w,.nnripl,Msinv„lv..,lintl,..s....x,..Tis..s. Mat.- mentB like tl.o f..ll..wintj may !..■ pbu-.l s..n...ti,..,.s l„.f.,.v ,,i-..l.l.n.s. so as to make tlu-ir c.m.liti..ns more d.'ar t.. tl... pupils, tl.us ; 30 onii.m's cost ""-Y 1 oraiit;.' costs '.' It is well occasionally t., ask sn.'h cpi.-stions as: -8 11.. ^v ill est how many times as n.ncU as 1 11..V l,..w n.any ti.u,.s as much as 2 lb.r how many tim.. as n,n,.h as 4 11. . V AN .at ,nvrt as n. as le 11. •'" l'upil» s''"»l'' '"' •'''''■ '" '''''"K'uze fnmi the hist tU.it afmctionalpart of the given nnn.b.T is taken when the .■e.imre, answer is of the san,.' .lenomination as the nnml«T ,l.vu e.l, ami that division ..f one nnmln-r by another is performed when the number of parts is re.piired. sa Great care shonhl be tak.^n in making th.'se bills, dir..cti..ns being given as to r.ding, date, rc.ipt, et... The pnpils shonhl learn to make their own rulings. 33 Toy monev may be tirst used in perforn.ing these exer,.>ses, if necessary. I'npUs should he able to t.dl inslauUy tl... .blh.-.-n..- between one dollar .and any sum of n.otiey b,.low that an.ount, thus avoiding the necessity of adding in making .d.ange. 54 As many of the problems in this s..cti.>n as possibh- sh,.uhl be performed orally. Some of then, may be writt..n ..ut lor the sake of practice in written analysis. The n.aking .d ong.i.al prob lems may be considerably cxteniled. 55-64 Weights and measures shnidd be used m performing the problems of this section when nee.led. It will U' necessary to use them eonsid..rably if they have not been us.-d be ore ; and if they have been used before, the pupils should be led to v;.i. 48 i GRADED AltlTHMETIO. [III. 58 w h the measures or drawings of the measures whenever the eon- litions of a problem are not ..learly understoo,]. Xearly all of hese problems should be performed without K,„res; but 1 ,r p J. fee .„ stafn« the stepo of the solution, sont of 'them „ a be vr,tteu out ,„ full. Jt is advisable for the teachers to spend L' n e >„ show,ng the pupils agood forn. of .olutiou. The following written solutions are suggested : ^ Hxcri'ise 10, jmij,- SS. Given the numbe. of bushels of grain on hand. Xo hnd the number of bushels more to hll an order. 1000 bu. of grain ordered. ^'^'^ " " " on hand. 265 " « « required to fill the order. Exenise 6, pngi: G2. Given the an.ount of paper a bookseller had and sold. lo hud tlie amount he had left. 10 reams 12 quires = 212 quires, an>t. of paper on hand. 8 " i " =104 .< « . „ ,„ij_ *** " " " " remaining. fol" X™::':"''' ^"" "' ''- ^"""'"" "->' "^ ™'"-^ o« - the Jij-erclse 13, pay,; OS. HEgeiHKi). SOLITION. Cost of Sugar. 30 lb. in 1 tub. 28 " "1 " 68 " " 2 tubs. $0.1,'{ cost of 1 lb. X 58 104 65 $7.54 cost of 58 lb. Answkh. $7.54 III. 65] TEACHERS MANUAL. 49 65-07 The pupils hnvn liad some practice in measuring, but it should not be presumed tliut tliey do not need more practice of the same kind. In addition to tlie writing out of stejis in the solution of problems, tlie ])U|)ils should be led sometimes to draw a diagram representin;; tlie distances given and reijuired. Tliis should be done in sueli problems as 4, 6, 8, 10, 11, 12, and 16, page 67. 68—70 Do not permit pupils at this stage to find tlie scpiare contents of a reetangli^ by nmltiijlying the length by the width. Lead them always to think of the number of units in a row to be multiplied by the number of rows. In the exercises on jiage 69 the ri'nuiri'd measurements should be made, but tlie fraction of a;i inch or foot may be omitted. Wlierever it can be done, a plan of the surface should be drawn to scale. This should be done in nearly all of tlie iirobleuis on page 70. Oral and written state- ments of processes should be required. Forexam|ile, in 7, page 70, the oral statement niiglit be: "In the passage-way there are 10 rows of blocks and 12 blocks in a row. .Since in 1 row there are 12 blocks, in 10 rows there are 10 times 12 blocks, or 120 blocks." 71—73 If numbers to 1000, including the four fundamental rules, have been thoroughly taught by objects, the knowledge thus gained will serve as a foundation for the teaching of numbers in the higher orders. At tiiis jioint the pupils are supjiosed to know that ten of any order make one of the next higher order, and that tlie value of the number exjiressed is determined by the position that tlie figure has with reference to the decimal point. Proceeding upon this basis, the pujiils count the thousands jireci.sely as the units were counted, until a thousand thousand is reached, when the number is called one million. The period of thousands, both in reading and writing, should be treated exactly as the jieriod of units was treated, ten of each order making one of the next higher. Extend tlie work here given if necessary. 74 .liisifei-s: 9 7703. 10 15044. 11 17788. 12 11125. 13 l.'!.S88. 14 180."i4. 15 2(t08'J. 16 20058. 17 16987. 18 13587. 19 25204. 20 11806. 50 GUADEU AKITHJIETIC. Liii. 7r. The combinations in tliousands will be found easy, if tlie corrp- sponding conibinatiims in the period of units are well understood. The lirst eight t'xeieises can doubtless be jjerfornied mentally bv the pupils, but for practice in expression the numbers should be written in full. For convenience the fifth and sixth orders should be named ten-thousand and hundred-thousand. The reason for carrying one for every ten of a lower order shoidd be given in the same way as was tauglit for numbers below 1000. For example, in 15, the pupil is led to say : "Sand 3 and 9 hundreds is L'O hun- dreds, equal to 2 thousands, to be added with the thousands. 2, 8, 12, 20 thousands, expressed as 2 ten-thousand, and thousand," etc. 75 Ansiirrs: 1 10421. 2 loZltt. 3 19378. 4 14574. 5 8003. 6 l.">7r)0. 7 2332C9. 8 93533. 9 54877. 10 110981, 11 51720. 12 674073. 13 733329. 14 884770. 15 7(>5277. 16 G7688. 17 08185. 18 155813. 19 15403. 20 1G042. 21 9558. 22 10G88. Let the statement of steps in adding be continued until the pupils have a clear idea of the jjrocess. Give fre(pient exercises in dictating numliers for aihlition. This is best for seat-work, so as to be sure that all work independently. 70 Answers: 1 218495. 2 11084. 3 n 12733; h 16404; c 25809 ; / 20045 ; g 22148 : h 25484 ; I 18801 ; m 22302. 4 96513. 8 1047. 9 7742. 10 6942. 12 3250 2173 3760 .3236. 9099 3215. 14 2468 3239 8176 575 256 4165 3046 5995 8387 704 c 21729 ; i 35362 ; 5 549. 11 2290 13 2417 8189 7058 312. 1045 1653. d 15131 ; j 14794; k 17256; 6 ]:!78. 7 2.'?21. 3244 1480 19.30. 751 8290 86 6048 5686 70(»5 9012 1021. 15 267 820 4();!9 4423. 16 8392 6401 8355 672 17 3685 517 254 833 1158 48. 18 8401 6355 715. 19 4915 3289 786 20 1007 5127 7564 69 859 6444. Lead the pupils to give a statement of steps in the process of subtraction until it is clearly understood. If the process is not understood, use objects in the subtraction of numbers to 1000, L. III. 77] TKACHEIts' MANUAL. 61 T7 Amwers : 1 \r,i^2. 2 rMX 3 652. 4 642. 5 604. 6 3264. 7 5,177. 8 7(i77. 9 6.i'.l8. 10 2.!.-.. IX 32;!7. 12 2353. 13 2318. 14 21(11. 15 5(i65. 16 5(i5i). 17 294. 18 465!). 19 i)6!J. 20 84411. 21 11456. 22 12122. 23 268(18. 24 28539. 25 23!);i9. 26 6588. 27 23175. 28 317!). 29 187(144. 30 27;!7()6. 31 !)3341. 32 588880. 33 17562!). 34 302357. 35 ;{01245. 36 176586. Let tliR pupils i)rove tlii'ir auswcrs. Show, liy "se of siiuiU numbers, why the, sum of the subtrahend and remainder ought to be the same as the minui'ud. 78 .l».s"vT.s-.- 1 " .333; ^<213; r 25 ; ,/ 1225 ; .192; / 873 ; ff 656(! ; h 92 ; i 5275 ; J 2569 ; />■ 666. 2 « 138 ; b 3002 ; '• 25 ; ./ 244 ; e 733 ; / 2587 ; y 1316 ; /. 91 ; ; 029 ; j 609 ; k 177. 3 'i 2445 ; I, 252 ; '■ 6352 ; ,1 .307 ; .86; /579; 'J U ; A .5.52 ; / 2239 ; ./ 1326 ; /. 212. 4 81208. 5 251.5(). 6 7!)92. 7 8925. 8 77611. 9 92316. 10 11102. 11 696(!. 12 17212. 13 10628. The proof by sulitra(^tion is only given for praetiee in subtrac- tion. Tlie best proof in addition is made by adding eolumns down if they have been added up. 79 Answer...- 1 " 2.5483; h 7253; e 20319; ress the work of some of them in figures. Sueli exer- cise will prepare the pupils fur bubsequeut work. 52 GRADED ARITHMETIC. [III. 81 81 Answrrs: \ ,'M74. 5 810858. 10 lT>,',:',rA. 15 4l.'.S4(l. 20 44.-)l'(). 25 1444L'. 30 L'.-.,S:'(;. 35 ri.->47(!. 40 G(;(;!Mi. 6 l'8. 33 .UlMl. 38 4(i;!ir.. 43 7ooor>. 4 102848. 9 112208. 14 r>7;{i2. 19 31)702. 24 17088. 29 1102.'5. 34 .'i3.-)79. 39 .'i8i)HJ. 44 08;)!)2. 2 12474 7 81402. 12 82708. 17 27.S7(). 22 181)75. 27 20S28. 32 28()0(i. 37 0072(1. 42 2;{;!74. I{cf„iv multiplying I,y tons .-uul units as givon in tl,e exercises beg„,n,nK wit), 12, let tl.e ,m,,ils l,ave .s.nne review- i„ nniltiplvinK smaller uun^bers «itli objects. t„ show that the right-haml fcun. i„ multiplyn.^ by tens is in tlie tens' coliinni. « **,o.„: '"■"''""■■ ^ ^''"'- 2 214... 3 .'iOOOO. 4 80850. 5 lO.iOO. 6 20040. 7 -.1880. 8 80480. g .t-KiO 10 S!10(.S.2,5. 11 ,<),S4.ill. 20 .?.-.()0. 21 »<)"U *!4'm''''«" ^ *"^'-"" 24 .' 64880.40 $7225.35 $.1575.40 *8..,1,(L $,„.).{.,„ ,S!00S7,84 !«5.;88..».i )|!;0285.75 $0.)02..il ; h .'?2i;i5..;.) $2481.44 .27.;.48 $(i9;!0 $.;722.88 ; $42(15.7.) $;i881.25 $3014.40 (' #.'i877.80 $57;i.).45 $5215.02 $4511.22 $4985.25 $4832.07' {,!:!'^?" J?' *"''l *^^''-''- *"'*"'-*^^ «fi'i«^-^o «^"43.(i8 $48,„S.12 ,^,.)4.,..,(, .S51,S2.22; y $,X!l.-,.80 ,$4890.95 .!;44.59 "2 $....14.14 $.-.O.S.S.75 ,«;47;;8.,SS $;i,S57.42 $4262.75 $413177 25 $228. 26 $10,5);o. 27. *1. 3080. _ 83 .Some of these exercises should be performed bv the aid of figures, nn.l each step explained, as for ex.ample, 14, in which the pupils may be le,l to say : '-10 units in 48 units, 3 times ; 10 units in 48 tens, 3 tens or 30 times ; JO units iu 48 hundreds, 3 huudreda III. 84] teachers' MANITAL. 58 or 300 times ; 1(! units i„ 48 tl,o„san,ls, .1 thousands or 3000 times ' h«,n,. „t the exercises sl„mld be periormed by partition as in 1 i of 40, ^ (,f 400, ete, 84 Anxiiviv : 1 871 6 70(i. 12 07011. 17 12562. 22 4!).".3. 28 172. 34 40}}. 23 r)Soo. 2 1414. 3 l;-)9.'!. 4 2347. 5 1982 7 121.0. 8327I;. 9 14.-.21. 10 17400. 11 4r,81 13 2*.01»>. 14 ,!4U(i. 15 1220. 16 13801. 18 ..lOj. 19 8201. 20 9102. 21 12439 24 809S. 25 27(!. 26 .568. 27 34" 29 112. 30 2,-.8. 31703. 32 54. 33 6,3' « .-, ^® "'^"**'*- ^ ^"-""-i"- 37 1"27J§. 38 692A 44 S^^^S r? 'T/?.-- "''^'■'■"- "fo.V.. 43 i *4 OB. 45 01. 46 2.5. 47 "OP" 50 3917Ji. 51 I54O5 55 093,-. 56 10.3. S7 104,V 53 ll,,8 , 60 -17,»,. 61 1264j,j. 62 10.". 65 48. 66 89,3,",. 67 715|J. 8.'> Answers.- 1 10070. 2 6410. 3 168 5 3280. 6 r.768. 7 148. 8 10 1030. 11 2004. 12 470. 13 3005 16 26. 17 ,368. 18 275. 19 26 22 ,38,S8(!. 23 2,S(i,S4. 24 l(!435(i. 27 9008. 28 6912. 29 1719 32 1019.5. 33 11070. 34 57631. 48 11.5J0 ■^5,r 52 12;i4j,^ 53 493JJ. 57 104,\. ■•'A- 68 627 63 27. 49 84J;-;. 54 799Jg. 59 923§«. 64 2,306. 4 3027. 9 19140. 15 1020. 21 13030. 26 14692. 31 3280. 1227. 14 1008. 20 1090,15. 25 21 8.34 . 30 2440. 35 17.526. If the Kn,li„K of fnu-tio„al parN of numbers is not understood teach the jiroeess ivitli objects, u^ - small numbers. 8<» Aiiswri-s: 1 958. 2 14N9M. 5 32713. 6 ,3472. 7 10200. 8 'o.S 11 Sil3.50. 12 $49^2,. 13 ,?125.37. 16 105 y,ls. 17' l;i7(; acre 19 897 quires. 9 125i-'i' ■. 14 .S45.75. 18 8.'i7 pairs, 4 1093^»j. 10 •'S0.7.S. 15 .S.3.5.27. ifiO.70 left. _ Call attention to the fact that to have an entire nuud.er of parts in the quotu'nt, the dividend and divis denomination j and that i •isor must l)e of the same ill getting the fractional part of a number 64 GUAIJEI) AUITHMKTIC. [III. 87 the answer is of the same di'iiomination as tlm number wrouji'it upon. I'roblcms like 10 to 15 niiiy be performed by takinj; the fractional part, tlius : ,'iii74.1() = cost of!)", bu. „V, of $74.10= " " 1 bu. = ?^Ji^ = .?;n.r8. 7 L'O ko,<,'s. 8 70 T. 12 l.i T. 14 11). 10 oz. 87 Aiisirri-s.- 6 $;H.L'r> $4n.(>L>;-;. 9 CiOOO pk.t;. 10 $l.'L'l.L'->. 11 IjtO.T." 13 17 T. '.) II). U oz. Eucoura!;e oriijiiialitv in the solution of problems like 6. Some pupils may timl tlip eost of 1 ewt., and tlien of o ewt. Others may re.i,'ard tlie 5 ewt. as J of a ton and nuiltijily by (1^. Lead them to take the shorter way wlienever it i.s clearly underKto.in. 4 $r)0.5.;s5. 2.S<.)4 ft. 9 U yr. 3 84 casks. 4 8 48 tubs. 357 »0 Aiixirers .■ 1 .fio8().L'.''>8. 2 .SI 750. bars. 5 245 yds. 6 2297. 7 »3841.25. Some of the most difficult of tlie problems on the last pages of this section may need to be talked about in the recitation before they are given as a lesson tor the pupils to learn. Such questions as the following for 2, page 90, m.iy be lielpfnl in inducing the pupils to think, and in leading them to give a good written analysis: " What is given in this problem '' What is requireil ? Do you know the whole number of through passengers ? How can you HI. 01] teachers' mantal. 65 How ran jdu find tlin wliole amount paid by find till! number \ tlit'sc passenffi'i's '.' !)t-H)l Wl,..n,.ver the .•„nditi„ns „f a problem are not dearly -."'.•.stuod, l..ad the pupils to ...-asp ti.em by the use of questions ami dlustrafons, 1 not by any set form of reasoning. Let the questions be suel, as t,. make the pupils think. In sueh prob- lems as 7, pa^e !)1, .s.une i,reliniinary questions like the followiui; may be lu-lpful : '• 10 pounds cost what ,,art as much as 20 pounds '^ 4 pounds est wh.at j.art as nuieh as 20 poun.ls? Do you see now any way of getting the cost of 8 pounds'"' If the pu,,il is still uncertain, ask bim wh.at 4 poun.ls cost, and then what 8 pounds cost. In fin.ling the eo.t of 25 pounds, the pupils may be led to hnd the cost of 1 poun,! tirst, and then of 2,". pounds. There is some advantage in having pupils form a habit of working through be unit m Hudiug the ...st of a given nmuber. But in sueh prol> oms as 6. page 1.1, it is better to work by multiples. To perform tins pr,,blem the pupils should be led to see that 12 peaches will cost () times as much as 2 peaches. In some problems it may be well, if the pupils find difficnltv to lead up to the required result by caretuUv-..:rad<.d steps • for cxaiuple, in 3, ,,age -.2, to ask how many egg^ uould pay fo'r 40 cents worth of butter, HO cents' worth, etc. In 1, page 95 the questions might be: "How many can I make in 1 h.mr? in 3 hmirs •.' ' And in 5 : •• How many times can the measure be filled from a quart can? from a gallon can •.' from a tw.vgallon can'" In such problems as 4. page 98, and 2. page 99. it will be found nselul to give the same conditions with small numbers. Form.al oral -exphiuations'' of problems should n^ ' be required at this time Statements of ,,rocesses, however, i y be made, but care shoul.1 be taken that the words exactly represent the thought of the speaker with little reference to the form of lan-uage Continued attention should be paid to the written analysis in the solution of problems. The following; analvses of the last three problems on page 101 may suggest good forms for the pupils- 56 GUADKn AltlTHMKTlO. [III. lOl (livrn tlip iiiinilxT (if l)u. ill 4 Itins. Tu liuil the iiuiiibtT ut' lb. ill 4 liiiis 7"> 1)11. (iO llj. in 1 Im. 4.S " IMS 90 " ■ 480 jn '• • 240 248 •' ill 4 bins. 1 20 14880 lb. ill 248 bu. Given the licifjlit of an icpbern iibove water. To find tlie whole hei^lit of the iceber);. 012 ill. = height of iceberg above water. 8 48t)() ill. = •• " " under " CI 2 in. .').")08 in. = entire height of iceberg. Ciiven the cost of 1 ton of coal. To find the cost of 87 tons. $7.2"> cost of 1 T. 87 5075 5800 $0.'«>.75 cost of 87 T. SECTION VI. NOTES KOK HOOK MMnKH FOT'R. A mastery of the subject. esented in Hook No. 4 will give nearly all the practical kiiowi' Ige of Arithmetic needed for tl.i^ ordinary affairs of life, besides fnrnisliiug a gnoil foundation for sub.sequeiit work. It may not be necessary for )iu|iils to perform all the examples and jiroblems here given ; but before any consider- able uuiuber of them are omitted, the teacher should be sure that rv. 1] teachers' MANITAL. 57 they are not needpd, pittipr for tljo jmriKisc of fixinR tlif piiticiiili's and jirocesses wliicli liiive I..tii tiiuijlit, cr for tlir |nir|,ost. of iiiciit;il disoijiliiif. To more ('li'arly iiiulcrstimd tlic ri^'lit us,, of the 1 k. tciu'ljcMs are advised to read tlie Note to 'n'acliers, in wliieli are given its distinctive features and some liints as to possilili' dangers. ApplianOM. — Some of tlie ajiidianees neeiled for teaidiing the various subjects may he supplied by tlie teaidier and jiupils as they are needed, but it would be well to have at the ontsi't all the common weights and measures, and plenty of eardboard or ohl pasteboard boxes from which discs anil scinares may be luiuh'. Analyiii of Problemi, — Xo formal explanations or riMsons should be insisted ujmn, but the method of solution should be freiiueiitly called for, both of oral ,and of written prcjlilems. The aim should be first, to have the jmpils think as they solve tlie problem, and secondly, to have them express their thimghts in their own words. Good written forms of analysis should be reipiircd. SeTelopment Work. — It has been the aim to give many simijle exercises leading up to dittieult processes or jirinciples. If any problem is found too difficult for the pupils to perform, instead of attempting to "explain" the problem in words, teaidi the part not understood by the use of illustrations, or h'ad up to it by simple (pu'stions involving small numbers. 1-7 A few of the.se problems m.ay bo founil too difficult to be performed orally, but let the pupils try them in that way before the pencil is taken. Xot much development work imght to he needed for these review problems. Possibly in such iiroblcms as 7, page ,5, some questions like the following may be asked : ■■ [f there were .3 rows of trees and 4 trees in a row, and the trees were placed 20 ft. apart, how long and wide would the lot be in which the trees are planted? How many feet of fencing would be needed for such a lot ? Sujipose there were S rows and (i trees in a row, how long and wide would the lot be ? If ow many feet of fencing would be needed?" Such ipiestioning may not be needed, especially if the pupils are required to draw an illustrative 68 GRADKD ARITHMKTIC. [IV. 8 diaRTam. 5, page 6, is of a difforc'iit kind, and may need sudi questions an: -How many niili's in 1 lioui'.' in ;i liouis / in H hours? How many liours would it takr liini to walk ;) miles? 6 miles? 12 miles?" Simple, natural statements of proeesses sliould be expected, but see that tliey are Mot too wor.S(l. 9 HL'14. ■Sir.ii.j.iL'. .fL'd.dd.'iS 8 !Si.-,.l,s. 2 ISO ft. 3 1460!} S(i. yd. 7 .*!;i;i.(iO Kain. 8 .liHolil.'!)!. iiirr.u'K. 8 AiiKiivm .- 1 .«i;iip;{,;!l 5 «U!)2.34. 6 !S(i;i(i;i.(;H. 7 .«i;iL'!i;i.,s;!. 8 .'Si.-.(i«i.i;i. 10 i74.-)!)(i. u !iiL'.-);«.s;i. 12 ...10. 9 Si9.r,2. 10 .«!(l.24. 11 ,1i;2. 12 .«;(;(l.7.-. 13 .'Sr.4.4(t 14 9.46 bbl. 15 $lir..;i!l, 16 292(( 11, ] T. 920 lli. 10 Ansuvm; 1 2.J110s(i, ft. 4 «I697. S $").-«->. 6 .'S0. 3 10."i20. 4 42900. 5 44125. 6 oloO. 7 ;i(l!)00. 8 28.-.04. 9 .MS.'i^. 10 .'i'lU. 11 12()r). 12 ;i()8. 13 (ivi, 10 mo. 14 24:i. 15 .ii!0.;{.-.. 16 96 bu. 17 $64..W. 18 .-.70 lb. 19 ,1i(!(;.(;4. 20 *7;i90.5(). 21 $4.44. 18 Ansu-ers: 1 $12.'{.70. 2 17600 yd. 3 .$54.40. 4 1.50 times 960 times. 5 5(t00 min. 6 55j da. In all problems on tlie last five i)a{,'es of this ueetion. require the pupils to write out statements of jiroeesses, to draw 'iagrams when needed, and to carefully label eaeb result. 13 A brief exercise from the board to teach orders and periods may be given before the jnipils answer the (juestions given on this page. The exercises on this jiage will suggest the kind of work to be given. 14 Much drill similar to this should be given from the board and by dictation. Observe the omission of the word "and" ia IV. IB] TEACHERS' MANUAL. 60 readinR integers. Const.int care in thi.s roR.ird will prevent con- fusion in the reading of mixed lirciinals. 3 44S()l(i:;i. 6 od.'.KiKi, VMVMl. 15 Aiimivrs : 1 44.'i7!)20. 4 llH««y(!;i, 5 7i'iii.54;i. 8 9!l(IU!M)'.»".tl. If tlie puiiils have been thonrnghly drilleil in writinR nunil>era, and if they know well the onler.s and peiidda, there will he little difficulty in performinB these exercises. 10 Anmrera: 1 7!»C15204. 2 L'L'!»!)711. 4 710r)3!«)(!. 5 l.-,(M)(i.).-,. 6 174(i71('.l!)4. 8 448;ii«).'tl. 12 11448(1. 16 5()0.1(>i'5. 20 8.'>94nG. 24 (>747()(). 9 3r>7(i(Ki,S7!». 13 I8;i7;i(i. 17 I'.'i8j-;i84. 21 ,'il 10(100. 25 218772. 10 r>(i4!iii(;(;i. 14 4.'i( 1(1(18. 18 i'r)"i8l(J. 22 ir>48;». 26 3.->0.-.20. 3 r);i!»L'«4'j. 7 17;i!)I047. 11 27i;iC.8. 15 l'(i(MHlS8. 19 4l.'il()(i. 23 4(>.'>88o. If pu])ils thoroughly understand and can exjdain the four funda- mental proee.s.ses to millions, no explanation of tho.se jirocesses in the higher orders need bo recpiiied. 17 Answers: 1 'J74(;y,S8 181744L'4 2 3 4 5 6 7 8 9 184. 14 1287Jg 293!)00(;4O 9940860 217914192 230799840 5740388(i4 58.'54C2987 l.'>r>(>70762 10 127,>j. 15 8r.4;.\. 408789072 ll;f.-)7820 27441 0404 11414722.'!,') 558977328 r),W881990 112)o,S854 11 174^. 16 .S79S^ 21 08:i5|j{ 210.-|,SO4O. 008285204. 22109520. 05O5 112176. 341044.">OO0. 10201078(18. 797022405. 351392574. 12 l:io|5. 13 O.'ii'j. 17 64,")i. 18 802,^,,. 22 4290|{. 23 90:i: 19 9445»§. 20 63,34J3. 24 20000. 25 2O0O. 26 200. 27 .'ioo. 28 300. 29 8ol 30 KM.';. 31 1907]jS. 32 nolOj;^. 33 N.,!^^- 34 414,s,v„. 35 2115,%. 36 1922/,"„. 37 8.-.2;|,1S- 38 8r7.-VV 39 6392JJJ. 40 18673ej. 41 15014i!]3. 42 10290.^,. 60 43 l'."42jiJJ. 51 l.s;ii!i;!. 55 !«M2;f,ij. UlIADKIl AltlTH.MKTIC. 44 TVJllii 48 4;!l,',v„. 52 ''^•'iiij ;)■;. 56 -an^tiil 45 ■'t-"><>la,",''. 49 llii.y"-,. 53 sni ;;;;;'', ' 57 -ii;;!^;;. [IV. 18 46 .WtiiT,',*. 50 l.J-i 58 Ttil'J'r,'. IN A'i„rm: 1 II.nM.rxjOs.]. nil. 2 »3n,fl2S,C35 ; 833,9(iii 295 • «l.48L',5;-i4; «:i(i,9 15,405 ; r,-<30,«.l; ?I,8,'J|L'; S5C3,;!(;i. 3 .*'l04 - 705,394; iB«4,779,-,59 ; ?.i,799,54- ; «99,09I,t(55; S59,700' Hi-'-UTV $1,827,810. 4 ri6,509,4i'9. 19 A>,m.- 1 ,?50,ni:,a95. 2 «82,91.l,(l38. 3 S78 OfiO 6(l-> ■ «109,556,573 ; «53,8«:.M43, 4 #30,810,191. B .W:i,9S9',:)L'3' 6 «9,173,032. 7 «l,7e9,20S; «n2ll,339 ; .«7 48r) 1 7 ■ 8-'29 149 ■ •247,002; $140,273; «1759,.^58. 8 S9,777,L>.i7. %0 A,mo>rH: I 199,892. 2 07.582,021. 3 289,210 men: 280,398 n.-c. omcers nn,l p,.iv„te.s. 4 4,'i50 quiiiUT hours. 8 1210.93 l.u.sl'icl.s. 6 31 ten-cent piccp.s. 7 $:!,204. 8 43 niilcs. 91 Jn.,«v.„; 1 ,*600. 2 3 aces. 3 .'.3 pupils; 159 pupils 4*37.90. 6 8120.00 Kain. 6 (») 3M04,!>75 p.,p„kti,m ; 12(1,137 area ; (A) Knglaii.l has piipulaU.iii ^r|.o,,ter than \Vali..s l,y 25,9(U,455 ■ than Srctland by 2,3,457,843 ; th.i.i hvlaml l.y 22,778,7^0 ; (.•) Kng- land has greater area than Wales l,y 43,522 ; than Smlland Uy 21.000; than Ireland l.y 19,241; ( s- .. th.) disks upon it. There ini^'ht he n.a(h. grooves up.,n the surface to pernut the ilisks to remain in place. In t^'ach- iny, sometimes th« teaclnu- pi, observe, and sometimes the "up;l direction of the teacher. 'I ;.• ^:,., pupils in i)re(iar:,ti(;n y ■ ' -.rs also can he used liy tho '■■: I, ' exercises on this jiage ■■ • ' dch are a.skeil hy the • li . :•. this circle into two t called (holdinj,' nji one of the parts) ? this ],art (hohlin.i; up the otla'r part) ■.' How manv halves in one? i and i ..pials what? i taken out of one e,|uals what? If 1 take the half two times, what is the result ? How mai.v times is the half <'outained in the whole one? Now. to rovhnv i + J? 1 h'ss i? 'J times i? 1 divided l.y iV The impils ought to be ready now for the exercises on this pa-e. If they find difficulty with the problems in imdies ami half inches, similar ..xer. cises with measures cut from cardboard or pajier mi-ht be i-ivcu 62 GRADED AlilTHMETIO. [IV. 24 hi 24-37 These for rapid oral work. Do not leave tlieiii until the iMipils are al)le to perform tliem rapiillv. ISelore 9, iia;,'e L'l, is attemjited, a .sh.prt teaeliiuf; exereise slioulil lie f,'iven. It may be necessary to perform some of tlie exercises at lirst with tlie aid of measures. Other exercises, lilie 9, jiage 'J'l. may liave to he illustrated hy sticks or marks ; liut wlieii tliis is done they should be reviewed wHhout aids "f any kind. 28-30 Tlie teachinj; exereise liere may lie as follows: "I will cut tliis circle, as you see, in -1 eipial jiarts. One (d' these parts is called one fourth. Wliat is tliis part called (luildiii),' up one of tlie jiarts)? What is this jiart called (holding up another part/.' How many fourtlis in the whole ciivle V How many fourths here (poiutiiij,' to two jparts arranged in the form id' a hall'-cireh')? How many fourths here (pointing to tliree ]iarts)'.' j plus \ plus i liUis i are what (putting the ]«irts t..gctlier in the form of a circle)'.' 1 less ;J- is what (taking away one oi the parts from, the circle)? i less j';' 1 less J? How many limes have I t.iken ;}■ (putting two of the parts together)? L' tinu'S i is what ? How many times have I taken i ■ ■ (putting three of the fourtlis togc'ther/.' What is three tiiiu's J .' 1 times I ? How many times must I take J to make J ? to make '}? to make a whole one? \ is contained in } how iiiaiiy times? in J how many times? in a wliide one how many times? What else may we call this part of the circle (put- ting two fourths together)? How luiiuy fourths is ^ ? J le.ss ^ is wliat ? 'J times ^ is what ? How many times is | contained in i ? i of I is what? " (lo on in tliis way comparing i with J and J j, also ^ with 1 1 and U. In teaching the expression of the fraction, lead the iiuiuls to see that thi' denominator expresses the number of jiarls into which the unit is divided, and the numerator the number of parts taken, (iive the (lupils much practice in this. Finally, they sliouhl be able to answi'r the questions : What does the denominator exprt■^^ ? What does tie' numerator express ? They should also be able to illustrate liy objei^ts or marks their answers, 30 I'V",- or many of these exercises should be perfonucd by the IV. 31] TKACHKlis" MANTAL. 68 aid of disks accordinK to tlio pupils' iindorstandinR of tliem. 11, 12, and 13 can best 1k> pcrfonn,.,! l>y the aid „f stirks, marks, or dots. Kliow tlio pnpils liow this may lie doni', Afti-r tlivse liave been performod ohjoctively the jiupiis nii-ht he led to ^-ive little explanations of the s(dution thus : •• J „f ,S is 1'; J „f ,s is .'! times 2, or (; ; C is J of 2 times (1 ; :> tinn'S (! is VJ. l'| is i „{ l times •'+• 4 times 2 is 8, and 4 times -j is 1 ; 4 times 2} is !»." Explain 'by examples what is meant Ijy •• rediieing to lowest terms " as re,|uired in 7. 31-3a These exereises oUf,dit to be jierformed readily without the aid of objeets or preliminary .luestioiiiii- unless in S(mie of the applied problems the measures are not familiar. Th.' sti^iis of the solution .sliould b,. j^iven. Thus, in 9. pa-e .■|1 : -i |r„i,i -I I eaniiot take; so I take J from 1|. leaviiiK :,'. I look f from «. leavinjr 7; 1 from 7 leavi's 6. J,w«vr. OJ." ,\,„1 i„ g, j.a-e 3'J : "The horse eats ;i times ,} peek or 11 ],e,-ks in 1 day. It wiU take as many days for him to eat 4.} peeks as U is e,uitaine,l times in 4J, 1^ is contained in 4^. ,'{ times. .l«.s-»-,r, :>, days. Since 4.J bushels is 4 times as niucli as 4^ jiecks. it will take'liim 4 times ;i days, or VJ days." Probably some teaidiing will be necessary for tills ]u'oblem. Imt the jiupils sjuuild be permitted to try to solye the problem b.'fore any help is f;iyen. If they solye it by reducing the pecks to iruarts. aei-ept the solution. 3.*J-34 For teachInK ei,!,'htlis and their ivlations to fourtli.s and hah-es teachers are referred to the note in wliieji a method for teaehinf,' f(mrtlis was shown (pa-e IVJ of Manual). Kssentially the same ]dan sluuild be imrsued here, .\fter the teacher has tauKht by objects all the important fa.'ts, short dictation exercises miKlit be tfiyeu for the pupils to sohe with disks at their seats Such exercLsesas the following will .snj;;;,.st the kind of work which ■nay be giyen : i + i + l ,,,|„„i „.|,at " i + i ■■ .Always expert tile an.swer to be -iven in the sim].le.st form, and if Ihev aiv i„,t so giyeu. ask ipiestions till the dcsircil answer is obtain, •,! ) i -f- i + i + i? i + r- i+i + y.' i + i:' i + i-i^r: i + ,i + i'' i + r.- i + r: i + y.- j + ^v vi + ]v ]» + - j',,„;i 64 GKAUKI) AUITHMKTIC. [IV. 35 t-ikojj 1-J? ^_jj j_.j., ^_^5. ^_j,, ^_^, 4-t'^' U-iV ]i-i'--' IJ-S'-' .Multiplv 4 hv -, ; fx-'" ixi';' fxr.' jx.r.- jx4-.' j xi".' ' i x" .•!■.- |x4- ixo? ixc'.' ioti'.' i„u-v j.,r -;;■.' i„uj? i„fir.' 1 [-30 Host of those exercises should lie iierform.'d without objects, and the steps of the soluti(m shoidd be reciiiired, as in 12, page 3.-,: -1' times H = H1; Jof ,S = 3; I(i + ;) = 10. S mul- tiplied by 2f = l9." And iu 15 on the same page: --'i^-.- i';fj- = l; i"f |=t: Jof| = Vage .'17: (liven the i To find the (I ''11: ist mC f bu. a|iples. 'ost (d 1' liu. = eoHt of 3 bu. = •• " i bu. .f!l.(iO = e(]st of 1 bu. ^ -' bu. ;vji> = 9. page ;;)*. (id,! yd. = entire jiiece. ICi yd. X ■■! = \K} yd. = yards cut off. any yd, - t.S^ vl. = 1 1 I yd. remaining. Aiuiweiv .■ 1 VJSi (12§ 42} .-.Si f,!(J. 39 6 4800 13.J3#. lO.lf IGl^. 3 10 179f 2L'9| 11, Sf. 534f 1G.'!J !H5 8 CO J r,2i I'.-.lf 7-'0 33. 10 t.->L'i 107f Iti 7875 1714^. 12 80^ 14 $303 J. 15 98 lots. IHdJ 1 TfiJ 8f.,S 5 131 1 .-,.v 712^, 7 U-^8 322 4(i|. 9 2 'j.-.f 27(;(ij 2:y2 27 i 380 09,^ 7:;«. OOfiJ. 4 .■0 2.-)9| 36::* ,-,0f 43J 220i 49f. 97.5g- 11 .-{OS (iOi 13 4.-.tii lb. IV. 40] TKACHKUS' MANUAL. 66 40 Answers: 1 82 books. 2 VJ-IO^ bu. 3 JB0.7fi. 4 HO ini 5 Oi 1.11. 6 TJi mi. Kl-fil mi. 7 .f^O?.;,!. 8 4 times ■ •^/. 9 Hors.., $(Ur, Ciin-m-e, JjiC'l.l. 10 .Slolc'f 11 .fl"«j' 12 «i:..-,(ii. 13 .SL'7.."S. 14 .*i^O4.0(ii, 15 9100.78+' 16 .f...i:; :si. 17 .Iji.'iiii. 187.014. 19 «!i.-,,u 20 .'-iK^u 21 .«i;!4.41'.V. t *u ii'-i. Good lorn.s „f written u.mly.si.. .ire suKKested l,y tl,e following: 3 .S0.I.-4 sidlin,!,' iirii.e of 1 -id. 4J :m) 00 $0.r)I selling i)rire of 42 g.al. ■^.7o cost " '• k. «i0.70 giiin. 9 .?8(i0 = post of lioi-so mid oarriag,. = 4 times eost of earrliiije. Cost of fiirriage = | of .«i800 = .S21 ;•>. 41-43 One or more teacldng e.xereises .slionld ),rpcet.s and reasons. For e.xamide, the pupils .should be led t,) show that ,™ can be nuMtiplied bv '• by mereasing tl,o number of parts while the size of the part.s remains the same or by increasing the si/e of the parts while the number of parts remains the same. Tl„.y may afterwards make th,' state- ment that a fr,aetion may be multiplied by a number bv multiplvin.^ the numerator or diviiling the denominator by that number. 43 Kxereises for pr.actice, which should be jierf.irmed .at si-ht If the impils cannot perform them readilv at first, let them work for a while with the disks. Thi.s will be goo,i desk-work 66 GRADED AUITIIMETIC. [IV. 44 44 Care shoiilil l)o taken to m.-ikd tlie work in divisinii very simiili! at this Mngv iif tlie imjiils' progress, Wlien the divisor is n, fraction it sli(]iihl be eontiiineil in tile ilividend an even niiniher of times. Lead the iiupils to rediiee tlie ilivisor and dividend to th(^ same ileiuiniinatioii liel'ore dividing, l-^ti, '.—li, ete., in 4 shonld lie solved by getting ,'.. 4f, etc., of the given numliers. When twelfths are tanght, let the ]mpiU have mueli praetieo in finding the relation of twelfths to si.\th.s, thirds, halves, and fourths. 45 Show the pu)]ils how this diagram eaii bo nsi'd in the s(du- tioii of the prolilenis, anil give similar ones. \\'hi>n they are nnder- stood h't them he performed without aids of anv kiml. 4«> The work in division here may In- too diftieult without some preliminary teaehing. llesides the ilUistntion given on page t,"> for teaehing division iiv a fraetion. diskt or sipiai-s may be used in whieh the divisor and ilivhlend are made to !»■ of the same, denomination. Thus. ' :- i, ,,r .H-; ^ ,;■, may U- represented olv jeetively as tli(^ division ,,f si.\ths b- si.xtlis. I'litil the pupils are entirely familiar with the jirocss of division and of tinding the fi-iuttional parts of iiumbiis, le.ul them to illustrate with objects or drawings the operations called for. 47 Lead the ]iiipils to state ivasons in their own language first. Afterwards they ran be h'd to iiiipruve their statements by simie questions or suggestions from the teadiin- ; or different pupils may be .-ailed upon t» p» a.UUMl to IL'S is l;i(ij.-- L,,,,. , i,,|,„,,.,l ,.x|,la,mti() A tahle tli:it can l,r uspil for ilr-ill in rpvitnv at any time. Other exorcises than those indicateil cim !»■ siven, :is A'Xii, ('XVJ I~ i, etc. In ,l,.n..niinate nnmhcrs the talih' can 1«! nsed 'in many ways an.l to :in nlnn.st iinlimitcl ,.xtent. ]int in nsin;,' drill tables care shonld he t:,ken not to prai'th-e with them ,oo much On account of the ease with which drill ea., he .tjiven from th. m, the tem|,t:,t,on ,s to use them f,,r a lnn,i;.'r lime th:in is reallv needed lor lacility in the use of i.uuiiiers. 5 !-.->•> Xe:uly all of thr.se prohlems ou-ht to he performed orally; hut lor the puipnse of l,,u-iiiTi,- ;, .,.„„d furm of written analysis the pupils should l,e :,.sked to write out in full the steps tluit are taken ,n the soluti.ui of the most ditti.Mdt lu-ohlems 53 Aiis 12 2i»l 17 :.'.si,-„. 22 ;i7i'i. "■'■'■■■'■■ 1 111. 2 -s.'.y,. 3 m. 4 2Vl 5 ^^ll. 7 .•i.H- 8:11;. 9.ti'|i. 10 .-,(;^. 11 io;ii. 13 :msi 14 rm?,. 15 gkh,;;. le 2t>^ 18 .'ire J. 19 ,Ti,sJ. 20 41 fi. 21 4704 23 .il'L'l. 24 :M2I. 25 41-4]. It IS not nec..ss:,ry in all cases to write out in full the solution of these prohlems, Imt only such parts of them as cannot he per- formed witliout the aid of fij;ures. Thus, in U the pui.il c.nild ""y ■'■•;, -hi V. and ,4, = j^. Thinls, lialvcs. and sixths can he reduced to twelftlis. ,-'-'-/= I " -4- 4 - 14 j_ „ _..„ p . ... , , 1 - I -• \'i' ' •>■! I ii T 1 a — { J, -)- ,"^ = I ,, — -i. I write the if and add the two, etc." 54 .f«.s»w 7 .Si. 8 .-.i 14 (VI 20 !IJ. 26 XVi 1 CJ- 9 15 i;i<. 21 i;i,-v 27 '.f',." 32 50,-;. 33 04 ji-. 2 5J. 3 1.1. 'J- 10 (Vi. 11 16 73. 17 10,:^. 22 i. 23 .S', 28 OL'i. 29 J-,t, 34 I'll. 35 44- 4 4i. 5 il ■'i- 12 7J. 18 l!»f. 24 4ov;. 30 -..-.;:. 36 512. 6 4,.V 13 ^. 19 27 1. I,. 25 1,V 31 «,!„. 37 I'S-i, 68 GRADED ARITHMKTIC. \ [IV. 66 38 66?. 39 ir.ilj. 40 8r.f. 41 141,ij. 42 223J. 43 r,33,3„. 44 ->07t. 45 l-'t'Of. 46 IHSJ. 47 l'K6.j. 48 S-T^'j. 49 31'j;. SO y.S!t,>,. 51 1(i;ii!. 52 .•i.ViJ. 53 C5i;,->o. Tlie same may Iji' said dI tlie solution of tliesc problems as was said of tlie solution of tlie iin)l)lenis on the last page. Kcquire the jiupils to write out only .sucli parts of the solution as cannot lie performed oriiUy; thus, in 35 the pupil nuiy .say, as he [lerform.s thi^ prolileni t "J from ^ I cannot take; » from \i or V = ii ; I write the ,■; ; 1,S from (!!.', 14. .t„.i,rn; U;,." And in 41 : ••'jfr..mj. I can reduce them to twelfths. ,»., from ,"., = jij, etc." 55 Keasons for stcjis taken should lie -.'iven ; thus, in 2: '• There are I fourtlis in I ; there are as many ones in l!)(l fourths aoi four fonrtlis are contained times in I'.M) fourths or 47A." Tlie •»rritten form of solution at Hrst may be : ^ *"'"'t''«ll!»"' '"«>•"'« 47A This a K""ii time to look upon tlie fraction as only aiMjther form of division, the numerator lieiuj; the dividend and the denominator lieiuf,' the divisor. With this view the only exidanation neeiled is that one numlier is divich'd liy another nmnher. The reduction of mixed liiuuliers to fractional numlie- may be explained thus : '•There arc > S in 1: in ,S there an' .-< limes ', S = VS, + ,■■;, = »,",,i." If written, the lorni may be siiiiph .s,', = imj. -)- .^, = l..l, In the solution of tlie excrci^.'s in multiplication on this page write out only such parts as cannot lie carried easily in the mind ; thus, in 33: '■■'! times 1'^ = .Si, i of 2» = ] ii^ ; ,s^ + 1 i-'j = 9J^. The written form which apiiears on slate or paper is ■ft" 5<; .i,.v«-,,-, i1o,sij, 2 l'.V>ti3. 3 14.-).-i^. 4 ii;i.-.-, 5 i,-,s:, 6 i;f'.».S{i. 7 Si-jin. 8 l'.)li;{ 9 1'l'--4-;. 10 l',S77,^. 11 L';i,Sl^'. ^Ijl ..;,«,/■< 12 ''.I'-C! 21 iri. 22 i.-s;,. 23 iii>;. 24 11'. 25 is. 26 L'li. 27 ll,\. 28 .S(i,:.,. 29 ]9L',»,. M i-r**. 31 MiH. 32 :~.\>:!,. 33 1771. 34 I'lllJ. 35 14.-ij. K -*^S. 37 7H,v IV. 57] TEACHEKS' MANl'AL. 68 Pursue the same course witi, tl.ese exercise, as was suRgested for previous exerc.es. Ul.en necessary tl.e divisor and dTwdenJ s ou^, e red..ed to t,,e san.e denon,inati,,„. Instead of ^S m full these -lenonunatu.ns, the ,,„pils n.ay, after the first few exercises, wnte only the numerator ; thi.s iu 24 : 2 7 ^ or Hi 1- Aiisi/ 4 -I- 7 = Y = 12 ,/„.,„.„. 57 The answers to some of the exercises contain fractions Sin:: ;:;"r"r"7 '-■«-■ '"- ^^^ -t it is understood cz le ; ■> ;' "'"■" '■'"'"'"■'' '" ""■ ^""- '>""-"nation n,av be treated as wh.dc nunihers in the ,livisi„n. 13 . ( ,„.„,,i., 14 .J,.- ,j^^ ,jj.. ^^^^^^__^^ ^1^^ ^^^.^ _ ^,?. ' 'U'":, "^ -M l-y earns 40C ;« i„v, »,>■ 4th boy I-'IUMV the pupils t„ ,„:,ke drawi,u.-s „f the solution of proln en. ,J.ne..r,t,s possible to do so. .na to wnte out n.fuUU steps of each Illustrated s,dut,„„. Th- principle involvcl in 16 s,,p,.scdto ...e heen ,au.l,t. but ir n.ay r^uire son.e leadi^ to «e the pup.ls t„ dustrate ,he probleu, withdo.sas ,t should be At h,st use snnple deuondnations, as halves or thirds, .-raduanv •-l.n.np to the work re.p.ired. The finished dra«„; s , ■p.-csent the n„nd„.r ., een.s that each boy h.as earned, 17 , an M>er or.ned ,0 t.„ w.ys t„ iirst lind the value of th,. wh.de lot . .' be i „ ,t : „,. ,„ ,„„,,,,,,, ^,.,. ^.^,,,_,. ^,,, ^^^ hrst lendn.;; the pupils to s,.,- the nitio of 4 t„ ,, ^ ' n 1 J* ~*H .lfis,rr,-s . 1 17I7.nb. $i^'.< + * '\l- 5 SI.;;?.!. 6 l:!4 mi. 1^ h. bSj mi. 9 I'ju 1,11, ^.jj i,^ 3 l'411jmi. Xi-r,} mi. 7 10 mi. Ijj h. 8 :i h. 70 OUADKD AUITHMKTIO. [IV. no Such proliloins as 9 can Iw illiistiiitcil l)y Hni's on the board drawn to seaU', I incli to a inilf, iilacinn tho distance aliove the line, and time Ixdow. 5J) Ansu-eiv; 2 -C) li. lo rain. 3 12 in. entire 2J in. 1 in. 6 $5.1(>S. 6O-01 Stiff pajier or cardboard should be used for tliese exer- cises. Some jH'elimiuarv exercises may liave to be given to lead the iiupils IJ see clearly the reliitiv(^ size of the larjje square, the! strips, and suiall siiuares. and to be able tu exi)reBS the size in decimals as required. Such exercises as tlie following may be useful: "How many strips in the lar^e scpiare i" Wliat part of the large sejuare is eacli strip '.' 2 strips are what )iart of the large square? ;i strips are what jiart of the large square? Hold up 4 tenths of the large square. Hold up 7 tentlis of tlu! large square. Hold up !» tenths of the large square. Kach strip is how divided? Eiich small sipiare is what part of a strip? How many of the small S(puires are there in the large square? What ]iart of the large square is each small s<|nare ? Cut otf 1 hundredth of the large square. 1 more huns? 6 .strips? 1 small S(piare? .'i small sipiarcs ? 7 small squares? Hold up 4 ti'Uths and C liundredths. How many hun- dredtlis in 1 tentli ? ill .'i tentlis ? in <> tentlis ? How many liundredths in 2 tciitlis and .'i hundredths '.' in 4 tenths and 7 liundredtlis? Write 1 tciitli in the common form. Wliat is tlie denominator ? Another way of writing 1 tenth is to write the niiiiicrator 1, and placf. a dot caUed a decimal point before it, thus. .1. A fraction wliose denominator is ten. oi some power of ten, is called a decimal fraction. In this decimal fractiim which we have writti-n what only is expresseil? How may we know that the figure one stands for 1 tenth ? Read these decimals : IV. oaj teachehh' man cm.. 71 .7; .8, 3, How shouM «•« express 10 tenths 7 Holdup the l«irt of 1 ,vl„eh these lruetio„s expres.s : .^ ; .7 ; .6. AVhe.i we wish to express h„„,l,v.ltl,s we write the numerator in the second p aee to the n«l,t of the .leeinul point, thus, .04. This expresses wha iruefonv „,« expresses how n,.ny tenths and how , nan l".n.lre.lths? Kea.l the fraetion as hnn.lredths - ^ Alter sneh an e.xerei.so the pupils should he ready to take up the wo,.k pven on these pa«es. The last five exercises of pa,e . »h..u d he exten.led until the reading and writing of d,.ei „als to hundre,lths an. thorouj-ldy understood. Oa Th,M,s..„dths sho,dd he tau.^-ht in a way .sin.ilar to that sug- gested ahove lor teaehing lu.ndredths. (in.at ..are .should he taken mn,arkM,gande„ft>ng the squares and strips, each expression of figures representing what the pupils actually see to he the fr^ie tuinal part of the given unit. un!!!;st^::ri;."^"'" "'"'' "•■'■ '"■'■•■ '''"' -'^ *"" ^-^'^^ ^-^-^-'y nr i"*,- !',! * '"'!'' *'" ''"''"' *" "■" *''"* * "■• * -'f 1 i^ t''" «an,e as | «■• i "M ,,:: ">• \ Zi- i .d- 100 hundredths is To hundredths • how express..d ;' ete. The redueth.n ask.'d for in 5 should he perforn.ed as s^ply as possiide; f.„. exan,ple: "2..0 thousandths is how .any hundred hs •.' -J, In,„,lredths is the same as what fraction ?" he pupds he.tate. .sk what .-,0 hunlredtl.s is e,ual to, and th.'n J. hun.lrnl.hs. in 6. 7. an.l 8 lead the „„,,i,, to ge the ^u,re,l fraetn,md part of 100 hundredths . . :0M, .housondths as betor, („ve other exer,.,ses sinular ... XO, th.l tl,.. principle in- volved may ho fully ^inderstood. ' i'i '. .0.(147. (W -/h.-/w,s-.. 10 r.lO,S7.(;4.", 11 •7,',. 13 lL';f.7L',S. 14 407..-,();i. The li, t four exen-ises are im,,ortant, an.i nn.y he extended I-h ...,ly. Addition of d,.ein,als ought not to he .of^.„. , ,T^u who have he,.,, carefully traine.l previ„„sly, l.,.t U,e expiau on l^ as snnph g.ven as was advised for addition of whole .o'll'T The su,u .'t thousandths to he t,eated as hundredths, and Ih-^ 72 (iKAOEO ARITHMETIC. [IV. oo I fit Hnnilths mijfht to lie as well iimliTstouil iis so many tens, ti> Iw treated as liumlrcils anil tens. aU Ansitrrs: 12 :..«.".«. 13 .06.1. 14 ir.Sai. 15 27.2f>l 74.(160. 16 (>4(l.'.l.;«)8. 19 2!KI.;il7 L'C.O!)!!. If necessary e.\ten(l the olijective work in subtraction ui deei- mals. Sinijile explanations of ste|is in written work should bu made l)y tlie pupils; tlius. in 15: "0 tlioiisauiltli.s from 1 thou- sandth is 1 thousaTullli ; I is eipial to 1(1 tenths, anil 1 tenth is equal to !(• linndredtlis ; ") hundredths from 10 hundredths is fi hundredths; 7 tenths from U tenths is 'J tenths; from 1*7 is 1'7. Ansitrr, i;7.-fl1." 3 .405 T. 4 371.25 gr. «7 Jiisim-s .- 1 10.27(i T. 2 .51 5 27.4 Kal. 6 70.255 A. If necessary, let the objects l>e used in nuiltiplying. 08 .Im«v,-.v.- 1 ..■!2. 2 .0.'iL'. 3 .21(i. 4 1.1«4. 5 2.422. 6 1.0. 7 (1.75. 8.552. 9 1.21S. 10 .'i.084. 11 ,->.715. 12 5,201. 13 5.272. 14 (i.dS. 15 5.022 16 0.048. 17 i;i.;i44. 18 lo.SO. 19 18.2.52. 20 20.025. 21 25,0.5. 22 ;«».08H. 23 7;!.104. 24 102.042. 25 151.648. 26 255,!M,s. 27 268.584. 28 405.405. 29 691.01. 30 5918.4. 311168.50. 32 2577.748. 33 46840.8. 34 2l;i8.752. 35 292;i8.4S. 36 1762..502. 40 3()45.;{2. 37 4;i50.675. 38 1865.528. 39 5650.316. 09 Aiisiirrs: 1 5040.156. 2 4750.395. 3 2200.44. 4 679().098. 5 7315.9;!4. Aniiwers to nearly all the remaining problems sliould be written without any figures of solution. In all cases where the nuiltiplier is 50 or 25, first multiply by 100 by removing the decimal point two places to the right, and divide by 2 or 4. 70— 71 Dividing any number by an integer is simply getting a fractional part of the number, the product being of the same deuomiuatiuu as the luultiplicaud. The so-called process of IV. 7a] TEAOIIEHh' MANt aL. 78 not <.l..arb. .......Tst 1 I,,, t,„. ,„„„„ ,.,„ ,,,j....,, ,, i,„,,.,J.„ 7j. in ,.v,Ty „|,n„t.„n will, tl.r ohjorts l,.t tl„. ,„■,„•,.,, .„„1 -«w.tl.o.,j..,.t.t..atwl„.„ tl,.. .livis,,.. is a f,-.,,i„„ ■,.„„, ,i,,,:, ."..1 d,v.,„r an- to .« .,f t,,.. «.,„.. ,I..„.,n„„ati„n ..f, ,,, ,/' 20, 22, ..-0,1 not b,. .,„lv. bj..,.tiv,.lv. If tl„. ,l,vis„r i« .„ ,„,.,-, w, I be su«i,.,e,.t to g., U„. fra,.,i , ,„„, .„• ,„., „„„„„:,; o te, tbs, ,„„,lr...lths, .,r tl.„„.s,„„Ul.H, a,„l if t ivis„r i., a ra '""• '" "'•■"'■ '"'"' ''i^--'"'-"! ••""• 'liviso,. to t an, ,1,.,, „ i" -..•tbus,,„ 10,b,. „„„i,, „„^. , H,ousa„,,tb. : tenth. Ih,.se solutions would U- writt.-u as follows : .oo4 )(in.()on .1 1 til 3010 .SOlO ifi..') 4n,'t.,'i, '> ft 2 2.1(5 7S0 -'.5. 4 .04 I'.-,.;-; • I''- 6 i.'ni.r, lOKi HT.r, 10.25 9 L';;.oii,s 1 :.'(i,;i.v. '>i-'7j :yj5 11)^.114. 74 Ansireiv •■!4<> 200 .00. 3 7.-,5 fiOO.l lol.r, f.O - ■»'"» ■'■>■ 5 .'t.^ .18 ICO. 17.« I-'.).4« .4;t8 •_M(!«.4 0550. 7 UTG.. --.7.'iS. 8 Ul'.dtS •j;UM 'JJ ;(., ;;„„., !':".» 2800 ;io.(i(i(;. 10 -44.004 175" H I3.(!.i5 702 24'J.«(; 24 lOOO. The ,.„|,ils so. in „u,lti,,lyi„K that tho n,ulti,,li..ati, f tonth, ," / , '"'■"'"•'"'•■^ ""■ tl.">.san,ltl,s; an,l fn,,,. thos,. faots ..- able to njako an.l follow a siu.pl,. rulo f,,,. pointing; off. |,. h^ 1^.0 way „.U.s :aado for ,.oi„ti,„ off i„ division u.; bo f„,U,wod, 'mt „ all oa.sos the pujuls shouhl be ,.„„dv to ,,ivo ro .sons for i:^^; r'''V^ "'''■'•""'"••''' ''^ ■"■■■^—^^^^^ MiaOCOFY lESOlUTION TEST CHAIIT (ANSI ood ISO TEST CHART No. 21 ^ -APPLIED IIVHGE ^^L '653 Eost Moin Street ^^^ Rochester. New Vork 14609 iit;' F:S (^'6) 482 - UJOO - Phone !^^ (716) 288- 5989 -Fax 74 GKADED AUITHMKTIO. 75 Thpsfi exercises should he pei-foriiicMl urallv. To 000 fi4,S0O 13 4.08. 14 792 CO. 15 702 it. 107.2.-, It. ;io;j 00 ft' 16 4 rd. 86 rd. These problems should be perforuied witliout niucli figure work, unless the steps of the solution are rerjuired. 77 After distances are known by nu^asurement, they should bi> placed upon the board and kept tliere I'or .■ouveuicne.. of com- parison. Other known distanc's, sucli as the distance from one town or city to anotlier, might also be posted for reference. „ Ifr /'""■''"''■ 7 2 mi. 1 mi. 2 mi. 4 mi. 3 mi. 40 r.l 8 275 ft. 17160 ft. ;U680 ft. 74.25 ft. 9 83.. it 204 75 ft 9240 ft. 235jft. 10 4135 yd. 6600 yd. 20 yd. 11 l-^» mi.' Ij^jini. 17Jmi. 14|gmi. 12 61 1 ft. 201- yd. $2.56." Most of these probh.ms eau l,e i,,.rfor,uc,l ,„allv bv tlie pupils Steps and reasons for stejis shonhl be .-alled f,u- oeeasiouallv After a problem is performed, an,l th.. correct answer given questions might be asked ; as f„v example in 12: '• What do voii find first ? How do you find it '.' 1 low iv.lmv to vards " Wliv ■■ How do y,m find the cost V Why ',' " Oenerallv, however, a f'ul'l statement of the process should be given bv tlic pupil without interruption. 79 In finding the scpiare contents of a surface, lead tlu" ].u],ils to think first of the number of units in a row, and to multij.Iy this number by the number of rows, as was shown in ]!ook III 80 If tlie pujiils have not had im-vious practice in drawing to scale in conneetion with other studies, s«me time n.av wciri,,. spent upon it here. These exercises will suggest other work of the same kind whicli may be given. An explanation of 10 mi.d.t be somewhat as follows: "There are 12 ft. in 144 in. and 8 ft"in IV. 81] teachers' MAXITAI,. 75 9fi in. In a s„rfa,.« 12 ft. ]„„g ,,„( i a. ,vi,le tl,ere aro ]" .n ft 81 In such problems us 1 5 anil K l..-„i n ■, ,. r-'ikr. H.n ,.„„ V ' in.iny ini nils .-is can, .1: 't;r;r::r:,,;::""' ««- -«- 83 yljMicen,.. 1 320 step.s. 2 (!!' trees ■? •^iv— «37^s..y^ 4.io.s,,a sl^w^a. 'eKo!: ■ - _tt. 7 23«;)4| .sq. ft, C2,5t ft. 8 ,^:!;!.;i f,; s- A 9o,r.,. 10 27H.ni. 86. n,i. ^4* mi. \l 2"!;;^ '^■;: Besides drawing a plan for the solution of each of these problems' 1 imp.lsshonhl be expected to write out a brief analys™ "he following solution of the last part of 8 ne,v . , ^"'- ' ''" of what may be done . ^ ^ " "^ "' '■'" ''^"'"P'" 1"8J ft., length of lot. 2_ 3r>7 ft., length of two sides. J^^ ft., widtli of lot. S)mi ft., length of two si.les and end. i(j^■^ya., length of fence. 83^,«„...^113is„.ft 2GJs,.ft, 12^s,.ft. 8 in,, in. =" t-lj It. 10 li) ft. 11 90 ft 19 1'imply ask the question at first, and give directions for findiuL- the answer only as they are found to be necessary. The statement IV. 86] TEACHEUS' MANUAL. 77 HV, -U.,rrrs.- 4 I'ClMd s,|. It. 5 ,s,^ ,. ,,^ .>c sq. ft. 6 :> it. 7 l.so s,j. ft. i.„ „,. „|. ■ <; tt, 8 I'll : 4 ft. 1- .V.I. 9 80 yd. N,..sHista„oo sl..,„l,l be given tl... ,,„,,ils 1,.,- , „,„t ;, „;,„„ m 1. t w„„M 1.,. wWl to giv,. s,.v,.,,,l j„.„hl..,„s in ,i,„lin. t area of fuu.gles. It will he pn.tit.il,!,. an 1 a..-,....,Kl • u to giv,. then. ,„.obI,.,us sin.il^ t„ 5. (Jiv,' a ,,,'"'" ' %-s 1,.., station ;«,.t,.t tin. Hgn,.s:;;:;:t:r^ scale, an,l ask for the area. The fo'lou-i,,.- ,li,.t.,ti " serve as a nn,„el . .>I,raw a line ., / [j ""i ".'"'Tr" '"^'^ ^, per,.en,l,eular to M^, .Iraw a line ;, I , ^^ ' '; , ';'; '"'■"! onhisHnen .Toiny.a Kn„ U. A: luJ^^l^J^l from ../, draw a line paraU,.] to J/l. M,,,< the e,,,! r' r , j-n^^,,,.,„,,n.eiine.t..,.eal:',n:a :,:,';:,:: to an inch, find the area of eieh H.r,,,.,. i .. . . a™.n on the board, i.e,,r:;:it;;i:i. r''':;;'T interesting work to impils. - Mh.. n.s, ,t, ., wiH g,v,. * C sq. ft. 5 108 sq. ft, 102 sq. ft. A teaebing exereise may be neeessarv before the ,.,„.n . .■ , the areas of „, /,, and /,. It niav be sutf ei to !l ' ' " \ liKl.t dotte,l Iine.s .so as to mak leh ,1, ■ ""''" a reelangle and a right trian, w • , "f'T ' '""' "' both as drill and teJi exereis:i- ,h ;;;;:'■ I! V'^'^^ '- '"'■'' possible to the pnpi.s in finding these e ' 2:;::Tl "' thMiuinls shonld write oat in fnn the soUiUr;;rr^^^^^^ 8 22500 sq. ft. * '• ^ ® " "''■ "■ ' -'i «'!• ft. The pupils should draw a jdan illustratin.r ti, , .1 tiou of the lot upon the street,'in the slit^' ''''"' ""' '""" '"I'isq. yd. et till pupils 78 GUADKI) AIIITHMKTIO. [IV. 8,> be free tn d.noso tl,,. .,„„litin„s of v.qmro.l originnl ,,rol,l,.ms Entourage tl...„, to giv. as .littk-ult ,,i.obU.,„s as th<.y an. ulJo to soivi'. 8!) Show to tlie pajiils as many of these coins as can bo ob- tam,.,l. The g,,!,! ,.,„■„» „.,,| „„, the United States >in. SlO ■--". .■..•ul^ the Knglish sovereign. Pupils shouhl also li„„w the Unite.l .States silver an.l nickel coins. The number of cents in imuths, eighths, fifths, thirds, ami sixths of a .loUar, shouhl be remembere,!, an. problems involving these sums should be „er- lornietl orally. »0 This account should l,e eojued, and each entrv e.xolained before the account of the following week is written. "]!aUnee on liand m 2. i>l.r,r>; in 3, •§]] l.S!). 91 Re.iuire the pupi's to make the ruling for a bill ■ also to copy and finish the bill given on this page. K.xplain the forn. of dating and receij.ting bills. The amounts of bills are as follows ■ fi IwoV-r ^ *'"''•'•'• 3*^-3.o0. 4 12<)4.o0. 5 m4«, 92 J,is,rers: 1 $141-J.S(I. 2 $4(110.02. 3 $.W,o 00 4.S4(i4.r5 $SU2Ml 5S1.'1.87. 7 25,,. 72,, 8 4 '' '^' «3.G4o ,$,.29 $l,.4!m $42.r.2.-, ,<;r,9.77S $h,M^r,. As many of the English coins as ,.an be obtained should be ...■ought into the class, an,l their value given. Show the greater con- venience of our decimal .system both in writing ami in reckoning •M Most of these problems can be ,,erformed orally bv the pupils. Reasons for multiplying or dividing in reduction may or may not be given. ■' 94 J„s,,rr. : 7 10 lb. 5 oz. 8 lo lb. fi oz. 9 20 „, ,i „,. l2« -r,-, "^ «"■'-'" "' 12 11cwt. 13 4cwt.20 1b. iB^T.:.^;- ,15^ ''••«"• 'I'- 16 10T. 17 9 T. 100 lb. 18 1:r.J00n,. 19 2 lb. 10 oz. 20 lib. 12 oz 21 1 lb. 9 oz. 2 4 lb. 6 oz. 23 1700 lb. 12 cwt. GOO lb. 24 14 owt IV. 05] TK.1CHKRS' JIANL-AL. ""'Oil'. 25 1 T. Ifi nvt. 28 lib. 29 III,. 8oz. 32 1 T. 4 owt. 26 1' T. I I cHt. 30 1 CHt. 79 27 I T. 11 ,.«l 31 1' cwt. 4(» 11,. i.;.4r';::,:;;::'.;:™i;:-r"":'- ■ sul)tnicti„i, of siinpl,. i,wiiil„.,'.s. as ill ii(liliti(,ii aii,l J.> ./««»•,«.• 5 'jr, 11,. 8 oz. 6 14 8 11 11,. 4 (,z. 9 (i(i 11,, ]^, „^ jQ ^.j 1,^ 12 14L' lb. 8 oz. 13 l;i t. 14 .-i;; i 16 3;{ T. 4 (nvt. 17 37 T. lo ,nvt. (Jd II, 19 05 T. 11 c»t. L'O lb. 22 4 11, S , 24 2 lb. lOJ oz. U <,z. ,•! 11, 1.. „, ■ . ., 2 T. 1,-!^ CHt. 1 T. 5 (■« t. h) T. 1 11'- 7 l.'i lb. II oz. '< <'z. 11 1],S lb. -J oz. 4c«t. 15 1.-5 T. l,S(»(»lb. '• 18 .'irT. ]l',.,vt. (14 11, 23 4 lb. 8 oz. cvt. 3 cwt. 60 lb. ' I'wt. oO lb. 25 1' T. U cwt. 30 $120. 31 e(i 11,. 12 oz." An,l iu 24 : •• i „f -111, - ■' H T- ""■"■' 96 ri,,.se are oral ex,,.r,.i.s,.s, «-l,i,.l, e.n .Iso 1„. yiv..,, for ,lesk 97 J«,wm-.- 1 obu. 3i,k. 2 6bu0r)k Tui i , 8 3 gal. 3 qt. 1 pt. 1 gl 9 11 1 1 \, ,/, ,f' "• ;^*/,l"- 1 I'k. qt. 11 , bu. 2 i,k. r. ,,t. 12 1 1„ 13 1 pk. 2 pk. 4 qt. 3 pk. (i ,|t. 14 " b, 3 Pk. 16 2 bu. 2 pk. 17 3 pk. (5 qt. IHI. 1 Jik. 3 qt. U jit. - I'k- 15 2 bu. 18 4 jral. 3 qt. 80 OBADED ABITHMKTIC. 7 qt. 28 7 pk. 4 ([t. 29 [IV. 98 6 bu. 2 qt. 26 8 gal. 27 30 25 gal. 2 qt. ou2T r"^'r f ■"" ^"""'"'■" J-k-u„rk, altl.ouKh the pupils ouglit to l)e .ible to perforin tlipin orally. 98 A„.,n^:l l.'J bu. 2 34 gal. L- qt. 3 80 bu. 1 pk. * -1 fc'.il. 5 1..3 gal. ;! qt, 6 'Jll bu. 7 , i.ould not be insisted upon .®i*,^;7"-- 1 •»• 2 $15.50. 3 «8. 4*1.02. 9 1(,0se.. 405sec. 10 40 nnn. 45 n.in. 210 n.in. 11 18 h. 9 h. 117 h. 14 3 h. 30 niin. lOO-lOl Tbese problems, which may be recited orally in recitation, might be given for desk-work. In finding the .Uffer- ence of time in years, niontlis, and days, let the time be given first m entii-e years or months, for example, in 1, page 101 . ..From April 22, 1.564 to April 22, IGlfi, it is ..2 years, and f, , .A,,,.;. 103 Answers : 1 7J r. 20 r. 2(iS 500 q. 120 sheets. 308(iJ slieets. S 480 q. 6 24 r. 7 .$17.50. 4J doz. 10 44 doz. 105 doz. 13 $75.60. \ q. 375 q. 10 r. 3 08 r. 15 q. 8 $1705.32*. 11 30/. 12 2 48 r. 4 500 r. 9 15 doz. 24 boxes. 103 Answers: 10 $;i.'505.76. 11 $10374.67. 12 $2283.85. _ If answers to exercises from 1 to 9 cannot be quite readily given, drill the pupils upon the exercises on page 1. rv. 104J TEACHERS' MANUAL. 104 JiiKHrra.- 1 SL'. 2 T." 3 S'^ 13 72. 14 .„. 15 -„, ig 20 4-... 214.... 22 ,W. 23 .^:;. 24r,7 33 408. 34 ,-.;(., 35 4(14. 36r,",-i 39^*^1.. 40;:s8. 414./;. 42 4o;i: 81 4 77. 5 78, 11 ■■'l- 12 47. 18 4!l. 19 4,-;. 25 (i!l7. 26 -.97. 31 r.4<». 32 (i2(). 37 43S. 38 ,31)2. ,, ., , — 43 r>o-,. ■lui'il.s .shciulil lii> riMiiiiivd ti. ..,1,1 1 i- — "'.I-'. b .>.J_'.G1 over .::!',' ■'''■"•' ® *^'-"-^ «-*-»«»■ 10 .¥.V'4 «1,r 11 .MO.,!.'ii ,^121.,;,;.. 12 .?1(;.80 21 .1. 13 m 8(.(, il„z. 9 24/. 10,92240. "-'"k- 7 I'rulay. 105) Aiisicrri: 3 .'>i2!)!).2j"i. mi. lai'Kcr. 6 37,-,3 ft. I.ij^hVr. 10 7.2 mi. 8^ mill. 110 Aiisin'rx : 1 .«I04 Ml o in picket.. 9 LIS. .s,,. ^.r ^ -**" ^'"''ks. 7 3.576 »,,. ft. 8 1192 5 4-;(ift B •ii—iT-i, \^V 2*8,-0. 3 ,S3312. 4 200y,] 9 3902 poles. 10 lOOOU C0.00M 340-J.., mi '*-^"i* ^• 4 «!n.70. 5 0. Sen 42900 .sr, 7 «l.^-iO. 8 70,,",. 9 jg;j(i„. 82 OUAl).:iJ AlilTIJJIETlC. [V.X SHCTIOX VII. MPl'HS l-DIt lioilK MMIiKli FIVE. Hcfiim t'lkiiif; u]! till' wuik (.niljr;ic..,l in tliis liook, the pupils aiv siip|«,.s,.,l to l.uv,. ;i tlioruiiKl. kni.wlclK,. „f (•(.ii>i„„„ Imctious to twflltlis iinil (,f ilfciiiKil Inicticms U> tliiiusiuidtlis. They iini also »U]ip(,s(.a to liavu Ijail iM.nsiilc.'alilo iJi-acticv in HikHmk th(. airad of pal•all(■lo^'|■alns and triangles, and in piTfomiinw proM.Mns in- volving' the coninion widftlits and mcasniTs. TcacdnTs arc advised to look ov.T liook l\-. to s,... if so p.,rt.s oftl.at book • -ay not be iTvii-wcd before *akinK up the work of tliis 1 'ok. Teacber.s are also referred to the Note to Tea(diers of Hook V. for general sugt,'estions. 1-a There shonlil be practice iiium these exercises until answers are Kivin with a go )d degree (,f promptness. ,\t tirst it may be necessary to solve soi.n. ,,f them by steps ; for example, in 3, page 'J, the pn]iils may le hd to s,ay. in multiplying 47 by « : ";iL'(), r.ti' ;!7r,." Or in 14, page I : ...1(1(1, ,S.-,, .t.S.-, ; '.'KIO, ];i.-., 4;i.-,." «„', by degrees answers should be given at sight, or the step, shcmhl be so ipiickly taken as to m;ike the exercises practically sight exercises. Previous jirac ti<.e shoidd have made the jiupils familiar with tli i:; n'cognized as the j.roduct of VJ and S. The reverse' operations in division sliould be eiinally familiar. Some analysis may also be neci'ssary in the solution of the exer- ci.ses in fractions, but such analysis should be as limited as pos- sible; for examph', in 19, page 2. the ]iupil may think and s.ay " «, *• h 'i'i-" After a while ii less number of steps than are here given will be needed. 3-« Short and natiirally-expresscd exjilanations of these exer- cises should be made by the pupils. The following will serve as V.7] TEiiCllElis' MANl-AI,. 88 widow-. 18 $:.'i'i,s:^. ^' *-"•'•'*• '*""! •^I'^^O'i, :^-'-' ^-'--. .".."»■ t,,,. L:.u; r.r. ::; r:;,."'"' ''■'• half, an extra c-nt is a'"P". J(II.;i ; Asia, o?.:- bo A o.... ; Polar regions, .OC. 5 $11.25 «15 $10 35 6 ^i' - *lu..50 ^,;i,, 7«.7 moo ^7. 8 S^ O^^' f* -I pencil 7(10 '^'l.S40 $-s:m. 12 §54 .sc Is '.H»0 jjeneils oi>.-,() jieneiLs 'i>:u-e,:i: 1 $4(I0() ,„.„(ij sujfgests any. pupils So. 5 8U,V bbl 3 IGOO iges. 6 «93. 7 i.45^>.i. 8 4 127J 4435.33. 84 9 1.3000.22. 13 liiiiK).:!:.'.'!. 16 llidll I'll! ;iu.(».s>s. OltADKU \ltITII.METlC. 10 111. fin:. 11 fi(io4.!)n.-,. 14 i'S(;ii,s l;.M.-,sr...-,(! 2m,s(;,-,i,7:'. •"-• 17 171(1 IT.SH .|',I.">1). [V. 10 19 .".Kir.:; j.s.inii .•iu.-.i'.Hdi. 20 : 10 Aiimi-iTi : 1 M.7\i 3 .'f!».t;7 ;i'.i(i7 ;;'.((;.7. 4 I lilil. .*i!).7.".. 7 ,1. .. 12 ;!7.S() «-liit,.s Jlltc,.l„r.'(l ll'li C.mp. 13 174 .si,....,.. 14 4aMl„^. 15 .*:'!.:' U. 16 04;MI.„. 1-Jilii. 18 .Ti,4.(;(iit. 19 .*iO.;-;o. 20 i!i7;!,s. 21 ,s..-iL'(i T. 11 Example of tho t.Tn.s ,„« „„„,/,,,, ,,,,,„ „„„,,,n; f„ctor ,,r,m. f„.t.„; nn,l,;,,k; .,,.1 l„,.t ,■„,„,„„„ „n,lt!,,l,; will have t„ be giv..., W.l.,r,. tl,,. ,..x..,...i,s,.s inv..lvi„f;tlms,. t.T.us .are kIv,.,,. A L-.od m- l>o.l IS t.. wnt.. ui„„, tl... l..,:.,..l ox:u„,,le.s of wlu.t is .losiml to b.. taM,^'l,t, ,,n.l th..„ t.. ask th.. pupils t,. giv.. oth.T ...x,t,„pl..s. After t be ,,„p,ls get a .■l,.ar i,l..a of tb., t..,.„,s tb.-y ,„ay be aske.l to .lefino tl„.m iM tb..n- .,»•„ w.,r.Is. Tbe following .letinitions will suggest to ..aebers tbe kiu.l ..f illustn>ti.„,s an.l -luestious tbat may !« ..s,.,l. An ey,.„ nu.ul.er is a number tb.-it i.s exaetlv .livisible by 2 An O.I.1 nu,ul.,.r is a nunib..r tbat is not exaetlv .livisibl.. by " .V ia..tor of a uun.b.T is its .livisor. A prin.e fa.ior is a .lin^.r tb.at IS a i,r,ni,. n,Mnb,.r. A niultipb. of a nunil,..r is anv number wbieb It v,-,ll ..xaetly .IivnI... A eommon multiple of two or more numbera IS any number wbi.-b ea..]. of tbem will exaetlv .livide The least '•';^"".<>n nuiUiple of two or m.u-o numb..rs « tbe least number wliicb eaeb ..f tbi^m will exaetly .livi.l... li Tbe exercises of tbis ,,ag.. sl,.,ul.l be i.r.aetieed upon until the pupils can g,ve tho r,..,uir,.,l ansvvers readily. The con.posite taetors may be i;iv..n tii'st if n.^.-.'ssarv. 13 I., t..aebing b.,w t., find the prime fa.'t-.rs of large numbers use .small numl...rs first. Two ways of fac^toriug may 1... us...l, J hrst to get tbe composite factors, uud then the prime factors of V. 14] TKACdKiis' MANtTAL. tl'"""; tip.., tho prim,, ,,,.(,„, , ti.;;,-mi>.. »........,,,, ,^,,„,,- :,:;:„ J,r^--'>- to „ivi„., -Ill-Ulrl /<."i>nn„.,.,.t..,::;'\,l;; :;'r:'r!:-r'"': ""• pupils l,v ,.x;„„,,,„, ,„ ^ -•-;-' - "■! ... I.,.,„l t''"^ in '. Is. T),.. S ',:;:;:;:''''■'■'. ^•'*^i-i--t,,,. «uff.',;st t„ t,.,u.|„.,. ,,,,„ ,,„„,'',' '-'■■■'■'-« l-.x.Kiv,.u „.i„ revi..,v fm,,„,,th, ■'' ''""■"*■'' -^'""b- ^>t tl.is point, an.l '•' ''"' tli(ir(iii"li lf„„„.i 1 ■""^■>' I'r.U'ti.v, will ,„,,, t,,„ , '""' ''? "''.l"'t'™ t.aH,in,. a,„l f various .l,.no,„i„ati„„.. t ;; ""■■■'■'""™ si., of fractions f<"-m any .^ivn part of „„. „,;' , j " '" "■vpress i„ fractional ti.. n,.nl„.r of snW, „„it,. J ni " ' ' ''■''"■"""■■'' ""'' •■""l ■•■'■'' ■•'■■" 1...IV sl.own, l..a,l tl„.n '""■'''•'"""■^ ''■"1 ""mparisons .sncl, ''^l>n..ssion, tln..siz,. of tl,.. JL *''"* '" ■•""■'■ •' ''l.ang., of "'■ P-'ts l,as .i<.cn.as..,, l,r : ' : ""■'■'T"' '''"''' "^ ""'"b..r """ "I"- «"'ns-s .similar to 17 , ' ' ^^ '"" '■' ""' "''■■'"^'-l. «'« r..,.il» -an t,.ll at si„.t , "['Z '"■ '^ '""'' '"■ S"-, nntil ---..i.ms,.i.o..,,.nomil;:r;;~^^ -';'! !;;7:r3.*;::7;:'::;r--'-- - --'---^^^^^ ''ver, the denominator, are krge let T '"',""'' '" ''-'"• ^'■' '"'»'- J-^ge, let their least common multiple 86 GRADED ARITHMETIC. [V. 19 w be found by finding all the differpiit jiriiiio factors, as shown in the following' ilhi.strativo .solutions : What is thi' least I'oiunion ninltiplc of 81t, 72, and GO? 80 = 1.' X L' X 2 X L' X r, '. 2 = J> X ^ X ^ X .! X a (iO = » X ^ X J X 5 ?ox;i ;,;! = = 720 =:L. CM. By tliLs method tlic L. C. If. is the largest nnndier ninltiplicd by tlie factors of the otljer nundjers not found in the largest number. 80 72 CO 40 .■id ;io 20 1.S Ij-i 10 'J 15 10 .". ") y 1 ' X 2 X ;i X .") X 1 Hy tliis method the numbers arc divided by any prime number that will divide two of tliem witliout a remaindi'r. The divisors, renuiin- in,":; quotients, and undivided numbers are the factors of tlie least common multiple. X .-i = 720. Constant drill upon the oral exercises given on these three pages should be given. A good method of drill is to let the pupils rejieat the steps orally; thus, in 24, page 10: "These fractions can be reduced to twelitlis. -,»j and ^'.^ are jj, and \rq« 19 3K',^l 24 r,s ,' »■■ 29 15 21 ;j 20 ;ioo^: 25 m.-i. 26 V.>\fi . 4 2i. I :. 11 1, 16 7;),',j 21 2.T4.'5 U 30 '' 6 2^' (0- " -T21). 12 O 5 1 17 8213,1 . 22 193,^5;!. 27 lOj^j. A- 31 ," ,V„ .;,U- 32 13y 243" 39)5,V 33 198ii'j lOSjg 23'5. 34 1115 14/, 7i»- 35 .32{? 164jU 215|. 36 24- ISji; V. 20j TEACHEIt.s' MANCAL. tl.n solution of pZhUn. ''^ '""' «"'"■'- ■'« I"'««iUc. in 10,'. = 1 ,-, 30 Jiisin'n-.- 1 ;jni.i7 9V!--., aM „, „, '" ^* a !■ 26 2(;,'„L. 27 2i> 1 , 1 1-'-. 1 V. -'" ^'"■'- ZO «J l! 3.1 1 4j '-" '-•li ^iilV 29 4». tR-t fi.. o -* S1-' iS l''r 211;*^ 32 (;!»!;^ fill.. .,s"4 cot' "^ ^''5ii: 'i^j^ ^Ji^'i 80? « 2'>,W 11^ 1^ IS.:. 4-■ 39 sss' 40 28 1 ^l J, 35 8 = ^^ 36 l'8i'.7 37 liO,'!.'. 38 4.^'.;,^ •'S? lojn. 41 C'.H SO! .11 48,1!'' 00,4, 42 8.;^i .,4, *3 12U1 30/^ ^4^ s ;i(i,7, ;!(iii ;iOvW- (i(iA niisju 041? Si'iV^- 44 32,"^ *«]!;? •''45-3J seigj,! 45 7:i,ti'. 1<^'A'« ^*l!li '^>i?;l, 46 ;«i>«5 ;5i<;i!; 47i^i! nuiij 47 77,»J', llllJSl WiS;;^ ST'-i) II l-OJli!" l"-Us ^"sS" •'•'••ill 1<>-1."m',I- 49 ll-ird'i l-'-lJ'ri', llliViT l''«>35;- l'>!»il K'-'J liri;;,'! 12(»,V„',i. " 50 i.-..\rv<, l^^S^VJff i;iOtSJ§ 14()S'5 121, 'V llNv'nA, 14^11il!ii 171).;V\A- 51 A li no ans. J- ,'\i\, ^J ,% ^V^- 52 v',', „«, iSi «". i .UJ .A vVVs"*- 53 ■?• ,', >5 «;; ^y, pj. ti 5;?S- 54 ;i«-? iTui, fiSS 25j^ 7|>, ;!ji;!j r,v> 7i}!i. 55 3iy 17i!J 5'85 2,V7'r '5 -/A ^J ■Ji- 56 TSJi} 204. li'iWr IIU ISiVff 20,V;(V 1!>^5U. 57 7^ 18,VA If J 17^ 2535 ISr^;.. 58 i/,?^ l"iSy 3,^-M, U,1j 20,l. 59 27iU ami i^i'4 31K3 30,',^. 60 27,Vi -'9iV« 1«bVo 2S?|S 48iVVs 65 77H3 66 mm 31 Aiisirers: 1 2r)r) 4 $.75. 5 |41(;i)J>. 9 4335 T. 10 $4^. 14 ,K. -^ 1 fid 11?^ 45J'; ;!!.» 29jji 20n;;jj. 61 aSaVSVi 28H 24J la,",-';,. 62 2()/,-'^ :mnU -^Hl i*,%% 4,V, 2i«Jii.. 63 •124.5, 64.^241 50,V 44JS}S .12,»„'i,. 64 29U§ 48ie?Vi! 46}y 30g«J 32J',V. ''Sii'SS «3352a 59,VA "Hi '-liVo 4fng 34fg3,\ lOl-gg 5^j 21/,7„ 20 1,2 00411 7 ~ " 7 iMI (U33. i A. 2 3154|?- hit. 3 583/:, .\. 6 *(>2(ij;;. 7 *14;!. 8 .'520,i,|;. 11 21/,, It. 12 557 J} ft. 13 .*,«?„. Let the pupils iierfonii orally such problems as tlioy eaii poi'l'ori in that way. 33 Answers: 1 jjj. 2 194 J J, lb. 3 6 9 h. 15 niin. 7 .321^3°. 8 ,30i yd. 11 618ii mi. 12 f 16|. 13 677H- . 4 .'J9;? bu. 5 8,,',, mi. 9 2,>, yil. 10 S8,'„. 14 698tJ,'^j gr. V. 23] teachers' .manual. 89 Analyse, both oral and written sho.u.l be require.!, the ste,„ to be elearly .nd.eated, and what eaeh result .stands lor. T e u I <.f .solufon ., son,e cases .night be indi,.ate,l i„ .ne olaee ■ d 1 work in another, as, for example, 4 : 100 lni.-(16» bu.+l'5J bu. + 41)|i, bu.-127 i,k.) = anit. of corn to till the bin. 10 J ■■!" 41) j J ii »» +(!i2=iiii;)=9iii io()-co.=;ii,. i,u, ;;„:-^^,^ 23 L<.t the pupils praetice upon this illustrative work until .o.y ean clearly see and state the two ways of n.ultiplving a tion by a whole number. ' • *' "" th J*1 ^T 1- ""■ "'r"^ '""'" '""'"^ ^"^ '' '■"■'•'"•' ^"t it is advised th.it .he objeehve work bo continued until the pupils have a clear Afttr tl e hues have been used in the solution of problen.s the .same solutions should be reviewe,! without olijeets n s.k s'f ments .as the following (14): " J of ;• = . or ' ■ ^ J\l ' 1 1 — r, " Ti,,v 111^,, - ii?"' ii> II "I '.: = o times J-„. Plus eonld be followed by rapid silent .solutions the answers only being given. '""oiib, ti.e 2-. IMvidmg by a whole number is only another form of ex- press.ou for getting tiie fractional part of a nu,nl,er. -J o /or J -^- is obtained ,n two way.s, as sliown. U-t the pupiK „raAice ^th^with illustrations until they can tell rea.ii,; J,,!;::;;: nf'^r'r^^ P'""',''"" ''^'•' f™'t'"" >''^'y also be taught by the aid o disks or by the drawing of lines or squares. For sil sthe . lustra ,ve teaching see page OG of the Manual. Th me ktd ot,llustr.at,onsm,aybe u.sed for division in other fraeti T,n bers to show, «rst, that the number divided and tlj iW , a ^ hrst subdivided into parts of the same size, or reduced to t e ■ me denomination For a time this method of division n.av b u lo -T Afterwards the pupils, knowing that the quotient depend upoi 00 GRADED AlilTIIlIETIC. [V. 28 the sizp of Hip divisur. may anal,vz(^ jirdlili'ins as fnllows (19, page 1'7) : •• g -I- 1 = ;; j ■; -|- J of 1 = r, times J or -'/' i V "^ S = i of ^»' "I- 5i' = li.'4-" •'^"'''' analysis slioulil not he fjivfU to tlie pupils, lint made by tliem in answer to (piestions from the teacher. After a time the i)U)iils will see for theiiiseives that the qmjtient ohtained by ilividinfj by any fraction is the same as the product obtained by multiplying by the same fraction inverteil. There san be no harm in such a method so long as the pupils understand the process. In a linal review of these pages let the pupils say, in such exercises as 12, ]iago t'7: -^ of 41=^ of .Y' = |,"' etc. And in such exercises as 15, pagi^ L'7: -"-I- .^' = 7 X 3 = -.' =4)." The processes of the solution may Ix^ made silently and only tlie answers given, thus : •■ 5 -^- ^ = ]."« ; ^ -^- ^ = ;/." Si8— 30 Let the analysis of thesa jiroblems be simjily expressed, and, as nearly as clearness will permit, in the pupils' own words. The analysis of some of the problems may have to ba i)receded by questions; thus, in 2, page 29: '• Oy/c third of a yard will cost what part as much as ta-,) thirds? If you know tlie price of one thir' of a yard, how can you fiiul the price of a yard?" And in 15, page L'O: "5 lb. will cost how many times as much as 1"J lb.? IJlb. will cost what part as niucli as I'l lb.?'' 31 271'i 5 'J 7 A -wem: 1 8« 80^ 184.-. 7r>;!i!. 3 11,', KM)-' L'di 45 5 A'o 67,-,,', i- 7 2i)?? 4101 J IL'.'i-;. 10 l;-i4,i„ l'.542» 19452 IL'Of ;j;^i;-i. 12 84' 13 187-1 7333 888» 1851 8.->i. 15 152 241 9 J 17 1009i«5 195 40A' ' 391| (>5(iS. 2 SIJ 335? 7H 100 125^,,. 4 7| S ,V 7 74 1.1 117 R u e 1 1S(I0 •>>!,'! .Ill 500' " TTIlT 31 S 179J 102i,;-|;! 104*3. 8 232JI 2G1J 9 28;,; 4.55:5 194:!^ 7().-)f 42()|. 2004J ;i539,v. 11 lli'i ;i08i: 41()f J 42031 7359 = ;! 2(l94i?5 773,V 242;f 901. 14 281,-',, 27a.ij 2350 2535 1515 4091;] 4581,!. 16 128i 5VV 1354;^ 199.3^. 17 l:iO),» 1304..', 444}f 18 11.', 18,' 20 19 l.iCjJ 444f 246ii 523iV I'm mU -It'.J 103181 86948 89^1 ijj 10/j 13§J 40/,, ."Jjij. 21 90 105^5 315 J 5 687) V. 32] teachers' manual. 2S3n^«, (!;!,'/, 91 Sll L',S71 8.'0.''.. 24 ,;!,., iH),«^ 1'.-,. |,-,7 «803o^j. hospital. 7 «1S.1',S|. 8.. sjjj 02 » Hi!! i%'«-i8^i" 20 9A T? •03.?n !47^,-! 9n^, rn. 2 82J fljjj •*»4:! ':,;?., ioi(y. 4 2:)}. 5 sV ? -'A- 6 141^ Oj 92 GRADED AKITHMF.TIO. 75^j 53 1,V(,. 10 i1,'„ !),«j, 23 12 i; [V. 34 9 59-)/, I,-'., 1,>., ins 3» 70' •"■-<:! i,y;,- 15 2JJ ,„ ,_„ ■ ■ ■■■» - !!• 17 fiL'.'iJ- •l.')7,"i- 18 4..0. 19 'J7i. 20 14;ii. 21 K/iO.v 22 4!v^ i^ 34: Ansicers ; 1 5 $.7Gf. 6 27. 9 i^iV '.lu. apples «125. 2 .«i'8. 3 $2M). 4 22A bu 7 $1.43 loj 11,. 8 20 bu. $12ijt. 00 bu. 10 222,5 yd. 10 suits 4l'v,l UIOIJ^T. ^«.J^ X2...02i .00f:;L. 13 « b 'i^ 8 breadtlis. 14 7 bu. 8 bu. r>i bu. 15 ,f 2 Rain 16 « " $112J. 17 283Jgal. $6G.04i. 18 $("^51 ^ ^'^^ The usual forms of oral and written a„aly,sis of tliese concrete problems may be made, and afterwards tboy may be written out on a hne. As the pupil proceeds witli the oral analysis, he writes the number above or below the lint ; thus, in 10 the pu„il may be taught to say: "My answer is to be in yards, so 1 wrL 100 ya. as the number to be wrought upon. It will take V. as many yards to make 1 suit as it will to n.ake IS suits. I write 18 below the hue as divisor. It takes so many yards to n>ake 1 suit. To make 40 suits ,t will take 40 times as many yar,ls as it takes for 1 ...it. I place 40 above the line." The prollem when nnished appears thus : 20 100 yd. X 40 2000 T$ = -TT y''- = 222? yd. 3.-, Some work with objects may help the pupils to learn this new principle of finding the whole when a part is given Tiie Illustrative work given will show what should be done with the objects. Give the pupils a certain number of counters, as 6, and tell them to show you i of the counters; ^ of them; .| of them Then give to each pupil 4 counters, and say that they have f as many V. 36] teachers' MANUAt. 08 or .narks will J l 1'" ^'^.""'l '""»'™t--" work witl. .lot» -.M,a::.^:::;—:t;^;s-;:------^^^ eo.mi„. va^ eonsiaeraM, and ni^itarJLu! L^;"" "'" '"" PuS lyZ:ZZ:J'tT^ % "™"'"- '^ «'-" - '" 9- '<'t the before caHi,.; th ^^ . t T ''"'"""''"^' '" ''^'''^'' "^ -•"" as they .,,n,e li^.^ I 'f^;";;; ""I'^f ''""■ " ""T say, part wa ief ^,",0 , IT' """'•' «""™^ "^"•'-- '^" '-""l -1-t 8 -'J yd. 9*17 in «•..„■ «.^^ '*''''• 7 30 peaches. 2f4. ~- ''*^""- ^«*^"-- "^S in.licated a,?/ e eh result .' 'T^ ''"''''''" "■'"' '-" ^'^'P «^-nJ„_the JwinSLt?™^ '^'"^^ '^^ "-"""■^ - cost of 2J lb. " " 41b. 192^ 24 20 cost of 1 lb. 18^ lb. tl X ' = «4.S2. 94 OBADED ARITHMETrO. [V. 3« 8 iii48000 4i3(iOOO, « idow. 12 G weeks. 13 15717 T, 6 *440(H) .$1;j7-,ii. 9 $--Ml 10 *.(;!),.■>,»,. 39 Jiixinm: 1 7,', Ji. 2 lOlJ \ 5 $l(W,Vii J!. ««»l'(ij- 6 154,vui. 8 3 da. 9 2 da. 10 .5 da. 7} J da. 13 $14(lJ. 14 7(i si,.„.,.s j.j4_ 40 3 $.80. 11 iS7n(i. 7 40 pupils. 11 «!S.4(). 4 .'S(i()480. 7 11. '{41 mi. 12 $2iW. Aiis„-er.i: 1 $753.81^. 2 .?().! T2« 3 «17/, 8 $8«.{;L'i. 4 $79J. 9 House, 5 $157<.J. 6 $L'17(;,.,. 7 $L'5(,v S!r05Sj Land, «4235l. 10 6S T n tr /i , , , 16*40.26. 17 27.,^ jars. 18 ? " * " *^ "'■ 41 .4««„-s™.- 1 nj. 2 $38.76+. 3 249-^ A 5 $l;i.;2j. 6 6G;|« yd. 7 C.4.4l|. s ;!7j- 1,; 10 'Si- ll $42 ,V. 1,, ,- „ ; ..^ • 12 lOObu. 13 .'ti4(J8j. II.U0.+ 2yr.4m„.+ 15 41j... le $Hi. 42 A„s,re,:: 1 ,|;J0(;. 1. 3 .!!1127' loss. J 4 $4S!j. 9 $6.50. 14 3yr. 82.50+. 4 Pc, 500 mi.; Kan., 9r..7mi. ; Yel., 1000 mi. ; Red ]"00 mi • 5 1,011). 8|j lb. 6 20yr. 7 100ft. 8 S"' ral cm. 9 To Sa,.., 180 mi. ; to K., 360 mi. ; to K, 48ol- 7to A 7C S;i'r""" '"'-'''''-'-^ toaL.c.,i8oomi.t'toa^.: 43 A review of ti.e objective work in decimals to thousandths for the work here R.ven. If the pupils clearly understand the expess,o„„ud use of decin.als to thousandths, there need be no 6 n 7 r; hI" "'"""^ ;f " '"^"'^ ^-o-inations. Dwell upon 6 and 7 untU thev are well un.lerstood. If 9 is found too difficult It may be omitte, for the present. u 100 aimcult V. 44] TKACHEHS' MAN LAI,. 96 *J.;^::!::;:':s:::;^;-™:f"-..p ix -»ii. »_',!. 22 'J!(i(i. 19 .si7(;.i07i'. 23 !»L'47.8;-.4. 27 (i.os.sjs. 31 .O0,-J2(i7 16 .iM'.)-)H'j:i. 20 (;7.i'(;s. 24 1(;4.4(;7. 28 2470.71; 17 .24(iO!>102. 21 22,2ar,(;2. 25 2;i70.1()4, 29 4.8. •06« oi.-n- 44 Joo.; a?, "f:'*^""^"- 42 10000.84 ;* ^"'-^ ^l-t- 45 1312.fi47fi 48 6S4,39;i78. 49 (ji,.,^;i., 52 .2(..!77a 53 .58.'il;iSoO. ' 47 .1 II sice IV. ■ 1 88 7S7 4 878.17221. 5 2UC.478. 46 44S.';o.:)i02. 50 KiOKJO. 54 r,2().i4. 6 525.5491. 39 20.50.;;<)3,r,. 43 .087(175. 47 61(i,0472. 51 .87575G7. 3 1424.08476. 7 7852.547. M ORAUEU AUITHMETIC. [V. 47 8 214.9854. 9.7871.0072. 10 r.214.T821. 11 2r)r.l.fi27S. 12 104()(l(i.r)212. 13 C3;t.87H7. 14 l(»r.2.(>2201. 15 .lll«ir>lL'. 16 3iMo8:'i7'J. 17 '.»«l. 18 70(11 ;.;j. 19 loo.ic. 20 ■>l8.i,ir.7. 21 48.S.0H(),!:,'. 22 ;!70:i.'J4(IH. 23 L'.V.i;!. U. 24 «(I011'.(1. 25 17.!8.8 v.l. 26 •'!i81.'i4.G7ri. 27 ll");t.70."nni. 28 iSlli;.447.'.. 29 «tO;!8.7l4. 30 «i;i7.047(i. 31 «i34.8;!2. 32 »2044.so. 33 !iii.i,-.48.lL'. 34 .l!i:ir..-,74..-,!(4,iri. 35 .2 .01: .004 .4 ;i.4 .00004 3..');!4 .0(MI,!4 .000(l(»04. 36 20 L'OO L'OddO .L' .02 300000 .003 aoOOOOO 3000. 37 .000 .0 10 .01 .OCl .0001 1 .1 ».(!. licfDVO the exercises in divisidii of decimals are begun, there shouhl be nmeh jinietice ujiou such work as tlie IdllcwiiiK : •• Tentlia of Inindreiltlis t;ivu\vliaf/ hundredths of tentli,< . liundredtli ■ of huuilredtlis? teutlis of tliimsandths ? thousandtlis of lumdredtlis'.' tliousandths of tliousandtlis ? tentlis of tens? hundredths of tens? tenths of hundreds? liundredths of liundreils ? etc. Tenths divided by tenths give what? liundredths by hundredths? units by tenths ? tenths by hundredths ? hundredths by teiith.s ? thousandths by hundredths ? thousandths .ly tenths ? '' etc. In division of decinuils there are three jiossible eases, viz. : (1) Division in which the divisor and dividend are of the same denomination ; e.g. .6 4- ..3. (2) Division by a decimal in which the divisor is .lt : ,:; ::;;""jr" 7 "7"-" ''^ t, .y '"•'ll>.tl.e,,ui,>lssl,„„l,ll,„,,,,„,.i„„(|,. fmmliar ,vitl, .imltiplirati,,,, i„ h„. v.nn,,, ,1 , ■ .• 48 ,J„,sv,-,r,,, 1 „ .SOUL'; /, .'il'dOd. , 2„ .m>o.:i7r.., ,, io.!(u;.i.i+ ; ..„:;,;+. 4 -JCmO ; ,. ,„js,| . ' l> 4-.94G; ,. 2.(i.s' A lOOOd; " ' ' /' 900; .(l,S; 3 « 10(l(!.(i(i(i + 4 " 1 l(!.r.l'(i;i ; 5 a ].;!;);j,'{+ ; 6 n !)()ijO; 7 " .S ; /, S;i4.4(«» ; ,/ ;i . ,. ,y ,->() yd. 7 12 lit. 1<»J .[t. .'iL'iit. 2(10(1 .|t, 8 7(1.1. '.t2(la. SOo'ihi! 9 »4.;iO. 10 .■S4.20. 11 8.2+ h. 12 4 l.(i Mii. 13 «.-.() A. 14 00pr. soo |,r. IF 2t0 cents. 16 2;i .lu. G0.-...-+ ,Iu. 17 2^)0(1. 18*1.50 "i0.7">. 50 AnsKvrx: 1 120 casks. 2 fiO'.MIO yd. 3 fi.H4. 4 r.(ll.(i Luxes. 5 ■l2.,-> A. 6 .7.!! A. 7 ."> .7.-> .12.-. X,'1T, .870 ..•17r>. 8 .o.l .07". .08 .0025 .07.-. .l.V 9 .1,-. .10 ..•180!),-,+ .187,-; .0.;7.-. .4218+. 10 1.4 l.'c:-; 1.8 2.02.-, 1.5025. 11 1.75 2.875 14.7(;i!)+ 21.!).)75 40 015 12 1.-,.3125 20.4 l.-,..-;!125 27.187.-;. 13 J V,', ^l, .,.3, ^, ,iiT li'J- 14 11,;!,, ,;;,^„ T^,. 18 ,„^„, ;i:i;i ,;[,;' ^:;;;;""ii !,',„. 19 2.;i5. 20 7.'.)l,s. 21 10..-.78,-,. 22 .§1.728. 23 iS.Si .«!110.,s;!+. 24 ll,", yd 111 eliaiigiii- ('oiniiiiiii '■i-actinns tc dfciuials lead l\w imjiils to think iif the rnieti..!, .-is aiK.ther lo.m of express! m lor divisu.u- thiis, -J = ;) -^4 ci- ^ „r ;i, ' 51 J/,.«'v,-.< .• 1 <).(;875 yd. 2.Snsii. rd. 40 scj. rd. (JO sip rd. 100 sip rd. 3 4 A. 120 s,p rd. 12 s.p yd. ,«„ sip ft. 7. sip yd. (i sq. ft. 100.2 sij. ill. ;!8 A. '.14 sip 1 i. 18 sq. yd. 1 sip ft. .",(',4 sq. in. II .\. 12.S .sip rd. !.-,(• A. 112 sip rd. 4 2.0225. 5 .■i2.5t5 mi. 6 .s.-,(i5.,S8 .Sl2f.O. 7 .'i*2(;;!.4;375. 8 (17 Ih. 1 («. 12 pwt. 12 tjr. 9 .-.017.05 11,. 10 1500. 11 2..S(;:;+ (,/. 12 ;, % ,i', ,<, ,i'Jl^. 13 . ': .142857 .1(J .(»;} .201(1. 14 1,', ,'„ , ,^i,„ !■, 1?' In teaidiini,' ifpetiMids let the pnj.ils set' liy examples that the lijtiires ,d a repetend express the inimerator (if a eommim fmi'tion liaviriji as many nines in the deniiminator as there are fignres in the repetend ; thus, .10 = .1§, and .iu8= \l%. ^fT V. 82] TKACHKIW' MANUAL. 99 53 Atlfirrrii : 2 1 ■lun.os » MI.VAf;-, s 7II.-4.IIIIIHMI 4 .(HIST.'l s wi.0(«i()7() ti .IHMIN I .IKHMPTlW It .IKIIIIidI !) .l!lll 10 •l.'ilP.dlKIOM- II .lllll.-,() U 7il.(i7i) 1.1 ..■il5 n Him.(i7 IS lINIl.OOO 16 ;ioOI)(HI. I 17 .»0i)7il 18 M.imimii) I'J '>I- 10:!.4-.>0 34ni):iO.:i05 .oniin 11.00001 47.:i49t 600.402 13 450, 71 528, (J. 70, 80. :i 700, 14, 440, 'i GO. 700, 07;t, :!40n00, in! 40, 590, 0045 (WOOli .02725 ,010:170 ,o:i:i2 .0008202 ,700:iO(i 228 010;!27 .0205 .008 .5:i75 .80:) .010 .:ii)2 ,2402 ,07001 ,(i:J09 ,7 14 .:i5 .110 .:i5 8.25 202.00 811.4 .78 .105 480.0 80.70 .008 7.05 8OO0. 70(1.08 9005.8 700.05 10.000 80.40 7.200 10.08 16 :t5. 7:!. It 0. l((,s.05t :!5. .■i210. S25. 108.018 20200. 1.4520 8040. 1 140.774 78. 54.01 (i2 10.5 12011.4 48000. 250.52 1 8070. 1.4.5:iOS .8 51.:i78 705. lS.o:i(i 000(10. .1:15 70008. :i.720 00580. 541.02 70005. 72.141 10(IO.(i 4.:i:!(i2 8040. .l:i2 720.0 141.5118 1008. 1814.40 s 6 7 S .9 10 11 1^ 13 H la 10 17 IS 10 go 17 4.0008 .800075 7.08(i:i()06 .0000875 .80000070 .000008 .000000708 .00000004 .0010(1 4..)0(100087 .O0100.)0 .70070 .00545 8.0007 10.040(19 3500.004 .000007 .20((0(I010 .48070 7.003 18 50.14872 8.7208175 77.175270.54 .00095:i75 8.720008284 .OO10(i82 .0(1(10077172 .000(1058:10 .02071 49.05000948:) .0100545 7.0:)70:) .050405 87.2O70:) Ifl0.4:)(i981 ;)81,5(l.o4:30 .0000070:) 2.18000109 5.2:ioe3 70.354 19 .1428 .030018 6.3 4.0797 1.0:iO0242 043.54572 .2340702 7.3584 092.02908 .05221776 .0024108 .70(iU 0.0045 14.401056 27098.4.522 280.58004 .21104454 .10290 5.78049000 101.6064 tMi'"^ fiaj TEAOHElis' JIA.VrAi,. s 4 s (1 7 8 n 10 11 1-1 13 U IS in 17 18 m 20 16.] 0280 Ammn 24.781001)1 •»""ai87S lei«.481.5:j-,(i-,ii .07Sr!)2 ■OOOOO.-;,-,').) ( 3''''!(.i>oo702(il'i •"OOOSOl " 4».:i!)!)3-, 4;i0.;ti7 5«011.:!00,-,8 n'»41!)().4«-,-)'> -'4301778.002 .00070012 1«0.8000801 700.101 101 23 4.0008 ■80007o 7."80aoofi •0000875 .80000070 .000008 .000000708 .00000001 .001.1 4.r,ooooo87 .001005 .7007 .00515 fi.0007 10.04009 •■i.J00.004 .000007 .2000001 .4807 7.005 5 G 7 8 10 11 Ig IS U 15 16 17 IS 19 SO 24 460080. S0O07.5 "080:io.ort 8.75 80000.O70 0.8 .0708 .601 100. 4"i0000.087 100.5 70070. •545. 800070. looioo.o. 350000400. .7 20000.01 48070. 700500. 25 ]:i6. 200.1 (I'OOO. 201.2 2.00 2008.1 100.o:i 2a300. 480,0 2.0!I2 100.7 ••i:i.l .25 O.f) ]oo;i. l:;:i.O K.o:) .8 207.07 JiiOO. 26 70. 12. 70. 1050. 40112. 10080. 150. 21. .06120. 16I,-,2. 1.0 I4I0. 1001200. MoOUi. lS01I(;o. Moolo. 2001.2 16080. 3441 " 2010. 27 116.5714+ 1000.5 5142.8571 + 7.;jl0:j + .0(W0 + 0.0555 + 38.47:i0+ 66742.857+ ..i .01 3776.25 1.4212 + .000000 + .0020 + .0.{.il + .0572+ .2407+ .0020+ n.ii.io+ luy. ^. _.... 102 GRADED ARITHMETIC. rv. 53 3 238.065 A. 6 17J 1(».S, 53 Atisieers: 1 1449.0129. 2 .'5.040150. 4 9.85. 5 21 .i yd. 9i yd. SI,',, yd. 2(1 yd $7.;{G8. 7 28.7 T. 8 $84.00+ $414.98+ $16988.81. 9 16.5008;i24. 10 $ll.S5.3;i75. 11 33.4 da. 12 $3048."..(i() $17419.20. 13 $4().426 $1,946. 14 65.78+ da. 15 $5404.20. 16 26666.06+ sq. ft. 120000 sq. it. 54 .Ijwows: 1 $4.11375. 2 48. 3 $14,748+. 4 U bbl. 5 $7.3125. 6 $.0002 2,50000. 7 146(>.542 bu. 8 .$21.82.5. 9 $42.59025. 10 $5,405+. 11 $1.53. 12 357.88830. 13 $27,133+. 14 697. 15 1366.326 bbl. 16 9| mi. 55 Ansu-ers: 4 10659 ft. 5 102 ft. 1224 iu. 9 15 yd. 2 in. 10 1 mi. 4720 ft. 11 2 mi. 269 vd. i yd. 12 ,;„ V.i- 19 3732 in. 20 2654 ft. 21 10989 ft. 22 1 mi. 280 rd. IJ mi. Tlie pui>il.s are supposed to have liad some practice in reduction before beginning tliis section. (See Section VI., Book IV.) Jlany of these problems should be jierformed orally, but the sohiti(Uis may be written out in full for the sake of learning a good form of written work. 5G -l«.w('y.«; 1 .">|mi. 2 264 paces 2112 paces. 3 30.6 mi. 4 1485 paces. 5 198,'^ rd. 6 1 mi. 208 rd. 2 yd. 2 ft. 7 2O8/5 times. 8 15 rd. 3 yd. 1 ft. $5.85+ 9 $1,947. 10 $12. 11 stions or by comparison with a correct form. The following definitions may aid teachers in drawing from tha })upils accurate statements ; A rhondius is an obli(pie-ang'ed parallelogram whose sides are equah A rhomboid is an oblique-angicd parallelogram whose ojtposite sides only are equal. A trapezoid is a quadrilateral which has only two sides parallel. A trapezium is a quadrilateral which has no two sides jiarallel. fil Answers. ■ 1 320 sq. ft. 90 sq. ft. 101^ sq. ft. 3 yd. 4 in •tL'JJ sq. yd. 8;ii Sep ft. 123^ S(i.' rd. (! A. 40 scp rd. 2 A. 158 sq. rd. 13G.5 sq. ft. 40 A. 2 A. 130 sq. rd. 3JJ rd. 21/^ rd. V. 63] TEACHERS' MANUAL. 105 way lie iiorforinwl 1,„ fi. 7- „ ""•'•'■'•'gam. "" a l,„e, tl,e solution woul,I l,e : '" ^ '^- I'^rfonu,.,! $240 X (- 3 30 rd. 4 10 ft. " t£ v,t ^ « ■ - ^ "^ ^1- y^ »i «a. yJ j;~ir:i;:-;!;r:;n,f-jr™rrr'^'''~- **-"-** HI- X 4 m, X 2 in, * •^-'«0.2.. 5 «.lr.80 ' 8 "170^ ? ^^- "■ -«^" ^H- ". taught previou.sIy, In,t t,,e ^ o ^ Il^'Tr" "' "''"''" ''"■^ ''-» «'-" I'ofore the pupil.s can fi , the . ? "' "'^'^ '''»" '" I'" «>"'ihu. to thi.s n,a; he „s M fo n 7" "''• ""'"' — '«'« -vor ,vhe„ theycLparealt r* H% J'" '"'"''-^ "-= "'- ■■"' »•''» as the size, ( eter.ni,. T , 1 ' "'''■''"■ "^ *'"' '"t- '-1'" solve 6, the giv'e.. o ^m I 'r"^'"' °*' «'« '-"'ia,,- lin.. ™cta„gles, ancUlie nee. 1,1 , Til 1 ''"■"'"'' "'*" '"•^"^■"■s "r not elo... than the si.;eent,;:f';rtj^ ""'™-'"'"^ "*' "-' ««'-« f-.r2.iro;:;:;i:;;:;;--''.^ee^ The ^'^'^'I'd.) It as bounded by six faces. The 106 GRADED ARITHMETIC. faces arp equal to eaeli other. Eaeli fare is in tlie form of a s(iuari'." All tliesi' fai'ts may lie gatluTcd into one .stati'ment I'ormin},' a (U'li- iiition, tliu.s : A ciilie i.s a soliil Iponiiilcil liy si.x cipial .siiuari^ faces. The liloeks useil in 2 sliiuihl lie 1-inch cnlies. Tlie work indieateil in 3 slamkl lie iierlornn'il objectively with tlie inch eulies. The rows, hiyer.s, and nunilier of layers sliouhl he oh.served by the liupils, and the nnmber of enbie inches fonnd by mnlti[ilyinf,' tlie number of cubic inches in a row by the nninber of rows, and this product by the nnmber of layers. Tliis should be repeated until it is well understood. 65 Ansirers: 1 CO cu. in. 2 1728 on. in. 3 4147L' en. in. 4 20736 cu. in. 5 i'''AC> cu. in. 2.(! m. ft. 6 lODli en. in. 7 72 en. ft. 8 109 en. ft. 9 27 en. ft. 10 8 en. yd. 20 cu. yd. S.-'il cu. yd. 11 li . u. yd. 22 cu. ft. 2;! cu. yd. 1!) cu. ft. 12 2 cu. ft. o44 cu. in. (i cu. ft. 1712 m. in. 13 1152 cu. in. 12!)(; cu. in. 2;«2^ cu. in. 14 ^, f. 15 7 in. by 5 in. by 3 in. l(t,"> cu. in. 16 208 sij. in. 142 sq. in. There are two ways of expressing the operation of changing a number from one denomination to another. Thus, in 10, the pupil may be led to say: " Tliere is ,j', as nitiny cu. yd. as there are cu. ft."; or, "There are as many cu. yd. in 210 cu. ft. as tliere are 27's in 210 " ; or, •' as many cu. yd. as 27 cu. ft. i= contained times in 216 cu. ft." But in all such ex]ilanations care should be taken that the jiupil sees the reason for the step taken. Sonu'- times it is well in such exercises to ask it the answer is to be less or more than the given number, and then to ask what part as many- or how many tin;"s as many. 66 Ansii-ei:i : 1 128 cu. ft. = 1 cd. 8 cd. ft. = 1 cd. 10 cu. ft. = 1 cd. ft. 3 04 cu. ft. 4S cu. ft. 90 cu. ft. 4 SO cu. ft- 56 cu. ft. 88 cu. ft. 5 4 (xl. ft. 0} cd. ft. 24 cd. ft. 6 i tV i- 1 li cd. \{\, ."J. 8 S;!.12.1. 9 ;!i»(i cu. ft. 24^ cd. ft. 3/,, cd. 10 $40i. 11 S!130.67. 12 1248 packages. 13 2027.5 papers. V. 67] TE.VCHERS' JIAJfUAL. 107 To give a ploarpr imnrr"■ It. 7 32,V tons fi!>-70 For information as tn TT « •■'»<' "otc thereon i„ t.,e Jlal ,1 Jw'"; 7 ''""' '' '' I"''«^' «»' pages sliouUl be solved orally '""'''""'* "" ""■'^^ 71 ^«wf™.. 1 $j, 2 SI 48 4- o -^' 8 »-'.4« Fuller, $8.32 " o„ ."'.s'l i; ^■T'^ ^'-'^' *«-«^'^ •511.42 Hall, $11.70 Wood «li .-; ' *^*'-'-^ ^^' «""«'. #■8.49. "^°'"'^°' *"3'S iiellei, *11.81 Heed, 108 flRADEI) AUrTHMETIO. [V. 72 Le.ld tlie pupils to employ sliortpr processes when tliey ,ire eiisily lUKlei-stooi! ; tlms, in 1, liy ri-ilneing ITi lli. iind iVIj 11). U> thirds the pujiil cim he h'd to si'u thnt ■<," will cost * us nineh as V- J of a dollar = 44} c. The pupils should lie told that gener- ally the Iraetion of a eent is dropped if it is less than half, but that ill this and all similar cases the extra cent would proliahly he charged. Fractions should he usM whenever the process is shortem'd thereby; thus, in 3 and 5: (>\ M. anil '.'.S^ M. should be used instead of GoOO and L'8L'")(). 73 Answers: 1 $8.00 $4.25 $1.70 $fi.;i8 $l.->.8(!. 2 $«L'.nO $(i2o $31L'.r>() $ir,(!.2r,. 3 O?*^. 4 48^ $;i8.4() .1i!28.8(). 5 $1.50 $4.fi() $5.(12.1. 6 $175 $;!5 $140 $l;!1.25. 7 $;i.CO $;S(iO $180 $270. 8 $7.08 gain. 9 $4.20 gain. 10 $3f. 11 $19i. 12 $20.;U. 13 . '4 lb. 088 gal. 14 10.40+11.. 15 \it. 16 223.25 dollars. 17 5 lb. SJ oz. Most of these exercises can be performed orally. Lead the pupils to work through 100 or 10 when more convenient, as in lto7. 73 Answers: 120. 2 1 lb. 4^ oz. 3 144 2970 6090. 4 31 J spoons. 5 57J cu. in. eil.OSijt. 7 2150.4 cu. in. 537.6 cu. in. 14515.2 cu. in. 0182.4 cu. in. 8 17.14 (jt. 9 348.348 bu. 10 598.4 gal. 3111.9 gal. 72.3 gal. 11 18977.14 gal. 12 1386 cu. in. larger. 13 37.23 (it. 14 1.6 qt. 74 Ansivers : 5 72 sq. rd. 12.8 sq. rd. 5.6 sq. rd. 261.33 sq. ft. 310 sq. rd. 11 4 rd. 5 yd. 4 rd. 3 rd. 1 ft. 6 in. 51232 rd. 12 sq. ft. 23 sq. yd. 8 sq. rd. 18 .25 rd. .4,ij rd. 19 .0025 mi. .027 + mi. 20 .1 A. .0037 + A. 29 .459 T. 13600 lb. Have as many of these performed orally as possible. A few whose answers are not given may have to be performed by the aid of figures. For the sake of practice in written analysis some of the easier solutions may be written out in full, as, for example : V. 7rt] •*4 n- yd. 2.41' sq. yd. TEACHERS' MANPAL. 4 9 .sq. ft ;i.7« ' sq. yd. ,•{ .sq. ft, 112.3L' S(|. ft. sij. m. '44 sq. in. _.7S 1 WJ JOHN llL'..'iL' sq. in. 100 Ann. 60 sec.)(|200 sec. Kia (I'O .sec). 1 !'• 4.'5 n,in. 20 sec. fiOmin.)]o;i niin. 1 (-l.'i uiin.). «6 ^«,,„.,,.,, 2 }ialanco,«!;il44 3 7M ^ ana whiC, to tl,e edit sid'e l^t tZl^'''"' *" ""^ "••^'' -'« f""n.s ,.re easily obtained Z^^ i ' ;" '"^ "'™"'' '''"■ ^"^"k -' ^ a.„e at any ti,„e to;rt^.:l':^:t,;: ^^"" '^^ ""^ ""'"'^ ■'• * JiiUrtnee ilue June 1, $•>•] 04 ^ ^ ^l- all that he does or J' ou' gT'"'' ""'.""'* '" '^ "-"'"■' f- this point. ^ ^ *• ^"' ''^^«^al «^"«ises illustrating 110 (iliADKt) AllITHMF.TIO. [V. «0 80 AnnifrrM : 1 liiilaiwi' (liics |!44.L'7. 2 Halnncf lUic $l.'!".;Ci. 3 S. L. CliiMs Dwes Asii Ilottliind, isillP'.l.li;. 4 Sl">.7r., ciif lum: *1S'.»(). 5 *7.«;i:j. Ill the lust niiswor tii 4 tlio assiiiiiiitinii is thai IL'O iiii'ii lire cmployt 1 tlie cntiiT tiiiif yivi'ii, witli im lust tiiii<'. 81 AHsin;:i: 1 «ll."i4()ri.(ir). 2 *1'(IL'0S.14. 3 $14:!T!).'.M1. 4 .«iiL';i(K).oi;. s lHio" L'''i'0(i (iiMMXKi ;(7oi»oo. 6 03000. 5 to 10 should 1«! inn-foriiidl orally, ami as rajiiiUy us imssilile. It luav lie well to Iiavi' tlio inijiils go over tlicso ext'ri'isfs two or tliri'i' times for tlii' sake of aciiuiriiij; facility in iiiultiplyiiig or dividiiij,' liy the fractional jarts of 10 and 100 Sa Allsirrrs.- 20 * l.'iO.lT,. 21 Sllo.dO. 22 $r.r.!).L'4. 23 .1iil".l7.ir). 24*lf;o.«4. 25 .'Sl'4'J.84. 26 .*i'.)..">r>. 27 JSloH.M. The first 1!) cxeroisBs aro for iiniek oral prai'ticp. In niultiplyiiij,' by a nnmlier two or three niiits h'ss than 100 or 1000, lead the piijiils to make two inultijilieations and siihtraet ; thus, in 4 : \)'S X 100 = iKiOO; !>;'. X L'= 180; il.'iOO — 18C. = '.»114. After a little .raetice such solutions may be made silently, the answer only bein;; given. In 9, first multiiily by 10, ."iO, and GO, and add to the products the multiiilieaiids. liCt tlie jmiiils perform orally as many of the exercises from 20 to 27 as they can. 83 ,f«.s»r™; 1 $1.02. 2 .IJiL'l.lHr,. 3 .fillS.fiL'S. 4 $42.5)8. 5 .fiSl.l.-). 6.50.7.-1. 7 .•S4..V.). 8 .liill.Olif Total. .«!2,S!».;!.-)li'J. 9 91). 10 l.Hri:!.!,|S. J 7 iS'.IOOCiO.OOf. 18 .filOl-.'), val. of estate $o('i25, wife $112."), son. 84 Ansim-s: 3 .'"Jl A. l.'W.JJJ sq. rd. 4 (>,■', lb. 5 2.V s.]. rd. 1 .V. 70 sq. rd. 6.1)131.08. 8 2075. 9 287.V;. 10 .$6.40. 12 r,r,ff. $11.00. $r 4 8r> A iisir. IS : 1 20 ] loles. 2 ■ *0 iiosts. 3 99 posts. 4 1782 jiiekets. 5 2070 stones 6 3'JG,", sheets. 7 $0.00. 8 $7. -..-■+. 9 145 ti ees. 10 3vy , s.]. r, . 11 140 "So sif rd 12 Hi- 13.31;\5. 14 00H. 15 0|^ 16 8^/. 17 800 doz. 18 $1 5.00. 19 131.84 pts. • V. 80] TEACUEHS' MANUAL. Ill *iil«r"'''"\;',^,-7V3...;.,.. ,,, ls->-w I . *6 ^ili' planks. 17 "I -.MI 14 .<;.!l>4,S7,S,S!Mi(. '■2U,.iy,l 13 -l.-,n(H. „n,Mg,..s. RECTIOX Mir. NOTES F,m lio.,K Xr.MHF.n SIX ""'";:: "rr':;:::r;;;!:;::;,, :;-■■-■"- ■■ '■'■'■ ■» « ..-. :;i:::i "":";;,;;";;';;"■ " "■'■ "■»«■•■ ■.nio.i V. I,. 5i„,, „, , ,,"'''"' " "I"" •■Ji,,.i„i ...":«7S:;:;:;;:--tr"™-i..-. it will not L .eU C;S iuS " "'"' "" "" '""'""^^- "•'""" pup>U t, overcome. la the apiJieution, 112 (iltADKD AltlTH.MKTIC. [VI. 1 iif poropntaKO, also, wliicli iinliiilr all kiiiilN of proMcnis that an' liki'ly to ipii'ur ill till' cvrn-dav lili^ nl' a iinclianic m larnicr, tlicic will 1k' Nome (litliralti.'s liolli in tlic uiiilfrstandiii^' nf |jiiiici],li'.s anil ill tin: |iiiJcM'ssi's inviilvi-il iiiilrsH tin' luanilatiiiii is rari'liillv laid. Till! di'vclii|p|iii'iit I'Xc'i-i'isi's |iifiTdin,i,' tin' |irartii-al iinililfnis sliimld 111' carc'fiilly and s,vstcmaticall\ loUowrd if Kmid results am to lio I'XiM'rti'd. As till' Work luofjrrssi'a Ivss attrntiou ni'i'd liv '/wvu to the iiu'ohaiiii'al oi»'i'ations with iininlii'rs if tlir iirrvloiis work has licrii t!ioroiij;hly iloiii', while' a i staiitly iiicrrasini; I'mjihasis should be |Mit ii|ioii till' soliilion of |iiolili'iiis wliirh ri'iinim i-loso and careful thiiikiii),', Iiirri'asi'd atti'iitiou also should liii (jivi'ii to tin' explana- tion of processus and tin' fonniilation of rides a- d delinitions. With this chanKi'd work of the jiiipils eonies a corresiiondiii',' change in the eharaeter of tlie teaehini,' and ilrillin-. 'Whilu the use of olijeets and drawing's in niakiiii,' elear a new pri ss or prin- ciple ia always desirable, there is not soeoin|ilete dependence n| them as the iiu]iils come to have a fuller power id' yeneralization and reasoning. Other hints of a Kcneral nature will be found in the Note to Teachers, which precedes the first section of tlie book. 1 .tiisiiri:i; 1 1 lilO. 6 Ills. 7 I.J.U. 12 TG'J. 13 (107. 18 uiW. 19 471. 24 611. 25 5.i,-;. 30 673. 31 700. Encourage the ]iupils to add by jiairs, tlius (li: 11, !'(!, 47, etc. As a convenience it may be well to have the sum of each . ,,-, column jilaccd below the line. For example, the jiartial and -..,- total sums ill 1 would appear as follows : Tiiiii 3 Jnsii-ei:^; 1 .•S4'.l.!(7. 2 .IJiod.r.C. 3 .501.9.3. 4 .'5i.->r).44. 5 $'>r>.S'. 6 )jiu6.58. 7 .'8i54.(;!). 8 .'8;40.L'0. 9 Sol.. '14. 10 !5l4'J.50. 11 $40.79. 12 $35.01. 13 $41.15. 14 $47.63. 2 i;:'. ■J. 3 l.-.l's 4 11(11'. 5 l.W.s. 3 1.171. 9 l.V.7. 10 l:i(ll'. 11 0.!;!. 14 on I. 15 OL'4. 16 Oil'. 17 490. 20 oil. 21 oos. 22 711. 23 o.-i,-.. 26 471. 27 .-.91. 28 071'. 29 004. 32 471. 33 7.". 34 .IIS. _.- M VI. ;,j 19 *i'.-,im;.,s:'. 23 «l!i(i.-,.(io. 27 $l'l(ii.N.!. 31 .«il(ifi(ii.i:' «.'i.Sl».Sl ■■s.'.ri.d.-,. TKACHEUS' MANUAL. 16 K.'M. 20 .*i'i.ss.,s.';, 24 *Il'l(i.((i. 28 *llr7..V,. 32 ■Slociii.ij 34 17 .'«ias;-.n4. 21 •*ii'.'!.-,i;„-is. 25 .•5ti';i!).-.;:-. 29 .«i I, •!?(;. II ■*■ 39 .M.,a'..-,( $i.,„o.,„ 40 .«i.l!IO.,S,; .4117,;,,.,., 41 $'M'JS.()-j 42 .SJllLM!) •*i»!(il.ri *(iv'>lt. (.■! $iim.r,t *!-'!» 1. 7;; •■Si'iri'.'ji- 8HilMi.(i;i $u;n.:i $I.'1.0( 113 18 jS.'iS.I.OI. 22 •'Jinn.i:. 2ii .*(j.-.,s. (1 30 .Si'i;),s.ii. *i.s:io.7r. Sl-'!L'l..S(i. sii;".i;.,s(i, *'|m;i'..-i. .'S:.'ll(;.i7 *l."ioi.(;(i *:'lo|.,si) •SlU(i.l,s *i.sr,.i7 43 .Sl'l.-i.-,,;)o 44 .'«ii'(to,s.(i7 ........ *Sa "-"'■- *--' •" $l'7..s() ,'o.iir,. $U(u;M:i .^.S77;i.) SI".,... «1.SC.(« ,S]|i-'i *'■•'•*•■'•'• 54 55 $l(l„,,.;..s $144,1.77 »i;ii'!>.:.'7 .SI 1 11,1 1. _-, ^,,«--., ., •Siiii.i.:."!) ^■■^I'.n.'.H) •*llo;!,N..; •S.l.".e...|;) •S-'-'iH.:(.-, •?1!I7;!,(;l' .S-.'iO."i,.-.l $'M).r,(\ 49 ..:w 56 .jillS7..-;i -1 .«ii!);i.:!i .■Si 7: Mi I •SllJ.S..-,!' •'SdLV. 17 §-1141.111 .*!r4.!((!. •SI7(i7,l'(i .Siitci;.!!') )S:.'o,r..();i. ■'ii7!P,S.7ii 3 •-'"■■'"■'■'•■«.■ 1 .si'i»,-,i)4(i,-i()(;.(),s laOU1.0 44ai«40U u,450U0 l>IISllH',s.S. 2 .«(i'<».jn(;+. TIS'J'JOOO 9 7i.-;i)i'(i 1'1477(;U(I0. 114 GUA1)EI> AltlTEIMKTK;. [VI. 4 10 i:.'.">:'2t() KiCii'siiro, 4771 KWiCiO. ,S7.S(llli:!L'0. ;!,s'.ii4;!L'.-.. 71 l.'U'.ll'iMi. 4(;!»7j,t;!. 1.")71)(l-l!,, M,;i|-. 4'J7ii./„Y; L'dOL'Cil ,' ir.d'.M'.M 24(>44.S(I() L'(il8r>l()0 l.'(>;i(!'.t<.»."(; 11 loi'L'.vw r.(i(;.-i!t24J ri8or)i744 r)U),s(»M4ii(i 12 ;!7.wl.".IO(» 7L'471'L'2(I 5!)4tll,S(;(l (IIIO.IIS.-.IMMI 13 i;:;.'!."L'tt(i ,'!,s1'.)(il'."i(i 4J(),SL'i;ri() .•!44S7'.).s.") 14 ."iCiiir.'iiiioo r);!iiL'ii7.sr) 4ii(;()'.iL'7sr. ,"iS,s2l'i.'(ii<50S5;; 16 ;i()(i()2.s! iii7.".s(;:;; (;2.-)()r)i 7«ir.,'i7?,'^ .■il7.s2 .;,::;■! 17 l2r)(MH) ,s(i(ii)() ir),s7;!()i;; 22727,-, 42(il(i,'vi, 18 ."i72.")72; 1;i;!(10(),! S.-ladOi ;il«(l!l.-)J 71m;,S2,'„'., 19 lcS'.l477'J,'., !M7oS',»j!; 7»!lS1,Sf;'; r,;jr.4>,", . 47;i2;;i7i5 rroliliniis siiiiiliu- to tlic fnlldwiiif,' iiiiKlit be given froiu-tlie table: Wljut was tlie (lilfei-enee in amuuiit of imports in ISDl ami 1!S',I2 from Knj^laml '.' from Franee ':' from (ieianany '.' f I'om lirazil '.' from ilexiciiV from Culja'.' Wliat was the diffei'enee in exports in IS'.tl and 1S'J2 to England ■.' to France ? ete. liy how nuieh did the exports to England exceed the imports from tliat conutry in IS'.ll ? in ]S'.»2'' AVliat can you say of the difference lietween the exports to iM-ain'c and the impcn-ts from tliat country in ISlll V in ]S!I2? eti'. I'iud the total imports in 1891 from England, France, (terniany, lirazil, Jlexico, and Cuba; etc. 4 .l».»«v«; 1 25s J. 2 l;)5JJ. 3 12. 4 114;J3;!JO§^ 7 §S83. Let the jiupils pra<'tiee upon exercises in cancellation until they can recognize readily the common factors !)■,'. 67 i;ov'„. 68 2!l.i:. In 1 to 20 Icail the pupils first to find by inspection tlic least common dciiouiinator, and then to add as tlie reduction is made. VI. OJ TEAOHiSKS' MANUAL. Thus, in 10, tlie pupils would sav: "T'..' v^ + .^ = l5; 1^ + ,^ = 115 li';r,t rr.rmion (loiuinii- iiator is lli = n = -!i-' lVssil,ly 21 t„ 31 ,nav I.avo to b. , • ■ . . , . „ ., hgures, but ti.e pupils sl.ould bo ' ' '"''"" •' ""' ""' "*' solution onillv. 6 Jiisirerx : 1 m i .i fe'iv<'ii an 0]ip(,itiinit,v to try their 21 l;i,!(;i,-,:.' 22 lil'7.;il".M' -'i "*. 16 (;4 23» 4,\ 4(;.j 23 L'7.()7(i;!. 24 lsi'r.,si'|!i. Bl.oul,ll,ep,.,.fo,.„,..do..:f" """"'""" ^••'^'■"■'•^- - "- I-«'^ .- 1^ rd;r^:;:;:;.'t:;;:"'? :---••--• -• given to the pupils' explanations " " ' '""""' ^'"'""'"" "^« ol/^"'-'^ all ot these revie. probleu. s 1„ be perto™..! lO sl,i.i„vi-s .■ 2 ](K) rcl. .'U-" ,-,l ■? ■ >. - 10,5(iin. 5 .•Siri'ri+ ic -;.-, ^ i"'"'- 4 .. r,I. .■! ft. 18.iorOs,pin. ■l9,i.v. ^T :7'\o 'Jf"^^'l-ff- r , •?-t-l-(l..M) .l-'l;jy(l. 4 $:mi> $-J.UH) s-.f In plaees where this method of measurement is , , exerci.ses similar to 3 in,! 4 1 i > ,"'''""^"""' '" '-'Miunon, niauv of land found . eds 11^ ' '"-' ^""'"- J''-"l'ti""-s of lots the class and fully :x;d:t:: '""""" '''■' ^"""''' '^'^ "^^^^''t """ 9 10 s,,. ft.* "i ft ^ "lo 'o ft « •^*'>^ »'i- "• l'>iS eouri.s. in:i=i:a:;::'^f5ri::iS'':;ir-.T?- --- 116 GRADED ARITHMETIC. [VI. 13 13 Aiiswem: 7 261;! sq. ft. 8 l.'3V'„V„ s')- ri are curvilinear figures. In teaching the definitions of the various figures, first present the figure and call attention by questions to its essential characteristics. Kor exanqile, after pre- senting a triangle, the teacher says: "What kind of a figure is this? ]{y how many sides is it bounded ? Define triangle." The ])upil is led to say: " A triangle is a phine rectiliue:ir figure bounded 1)3' three sides." If his wording of tlic definition is not quite as good as it should be, lead him l)y (piestioning to make tlie desired correction. The following definitions may be developed in the Game way: A i)olygon is a plane rectilinear figure having three or more sides. A cpiadrilateral is a poh'gon of four sides. A pentagon is a polygon of five sides. Ktc. A S(puare is a riglit-angled parallelogram ' having its sides e(pial. A rectangle is a right-angled parallelogram having only its op- positi^ sides e-angled parallelogram having its sides ecpial. ' This term is supposed to have beer taught previously. Sec Manual, p. 70. VI. 14] TKAcHlou.s- MA.NTAL. 117 117 A rJinmboil] U -in ni.i; i'P'«it,. ,M,. „:„.;;;; "•"■'i--.lo.l ,«u.an,>,o„.a,„ haw,,, on,, it, 111" walk i„..„tioi,o,i i„ 11 ; ■-^■"1 then t.M„„lti,,ly by fl„. „■';;, "■''- '™fe'"' "f "» the boards, '■^- 2 Mo. Id. 3 ^,;f^ ,, -^'t'T tl... „U|,ils l,.,v,. , ."•4 4.,|.s,i. It. 3. >.;^..t,.,,,,,L'.;:,::::;:-,^ -;>'.-,.«,,, t,,o. Ob "":'-^tl. -fa ..,k o. t,.,.„ ,. '; i'^-^ "':;' ''3-asbo..tp,.oe,.,, :f^r"v ^-^'"-' '"t, .„., also s :,";;'" ;"^'"" •-'<- of 'f t "^ i"t- To ti,„i tb. ]„„,,„ of , ;: ■ '"'" °" "'« ""*«"'« -"'•' f-". tl.o ,„.,.,,„,.,., „, ,';';; f""' fm;. the ,vi,ltb of t|,e 7'^ >f ma,!,, on the o„tsi,!.. of ,,,,.; 'f "' f '' "'" ''"".^t], of a "f tbo walk to tl,e ,,..,.in„,,.,. ofTb. b ' ' ^"'" ''""■'^ "'" -i'itb J" tb(, .solution of 4. ),..„, ti,„ , •; ''"«..! linos, an.l to ,i,„l • ."'"'■^ *" ""'"""•' ti.e .ornors by ^^ ;' K00.1 ,„..a„s of ,„„;.s,„.,„„.:,, ■ , :" "'"r "'"" "^ '' ™"! ^^-iii '"■"""- "-■ -'-b,. points to 1,0 a , , , ""'■"•"'";'>' "'"l<"-tan,l tb. ;'- ^'« follow.s: 1. Ti,o ,„„.b.s . t t , ' '" "•■"■'' "^ ""-■ ^■''"'•^i-- '■>■ "— «.,.i„b snbtena U.:.:^ 3' '^ ,:'^ ;' '' -•'■'-■ a- meas„„.„ -f v.-a.io„s ,„a..,it,„b.s. 3 To' nnk '""'''■' "^ "'"""ff ■'">Sles ""'^'™- '-'t Kn AlUTHMKTIC. [VI. 16 three angles of a triansle is oq.u.l t,> tw„ right angles. Tl,e s,„u ,.f the angles o£ a i^nlvgcu is equal t,. two right angles taken as many times as the figure \iis sides less twii. It" is net desirable at this time for the pupils t,. spend murh time in formulating the facts. The in.portaut point is lor then, to diseover the faets by observation and eonstruel.ion. K; .!„..« .,.■ 6 L'.-8 scp ft. 7 14;i.S0+ s,i. rd. 8 14(>.2«.).' S(i. rd. 9 Kxai-t area = -'7Sl.lo.S4 scp ft. Proceed verv slowlv with the first four ex.nvises, an.l review fre.iuently so that the nu^aning of the terms may be fixed lu the mind. . , The pidygon »/«■,/-■ is a regular ],olygon, because it has e rolls. (Chimney not reckoned.) Let the pupils carefidly measure the lines whose distances are not indicated. Aec.rding to the measurement given of the hall, one breadth of the carpet will just fill the narrow hall-way. lour VI. 18] TKACIlrCIIS- MASVAL. 119 ;"-"•'■'■« "ithcit .i,.,i„i„. i„.t,,„.,i, " ■""'"?"'-^' " ■ ''"■ ""-'"H U-,,Iu,.,.,l „i,h r,.|,.,v..,.,: ,"""■"'"'"■"•""• -ainsn-t will ' ^'>-- '^ '"■i-n-. ::;::;;;/'■■ ;''''^'T'-' -">-i" «":H..t;::■;,::;:;tl^:;-'::,,7^''^•-.•.■.......is,...^ lu , ' '•"''' ''""111. etc. -;:,'"■ "'-^ '■"■ ';-~rr"„siL^ "^™r':.r'i;:;^n;",;r;;,:";;;;:'r-"r" ""-■■ "■""■™' '"■» a.™,,,r,M,: ;:; ;;::;,:;,,,,;';'-' j-'—h 5 $ J «(>i-^;/;;is .;;':'"'■'"■ ^rijjs,,^. 4 ir;;,is „,.;,, '•'^-''i"''t.,:;;;l;t:;::;:::;;:r--^^™M,.,^ ''rf «-ifl' at first, and wl.on ph'... ■' „ '""i"" ""'^' ''"'"''' ''" l-Hit. ot resemblance in all prisons 120 OnADF.l) ARITHMETIC. [VI. 20 are : (1) Th" two bnscs :iro lariillrl (!') Tlin liitcriil si.li-s are IiiivallcloKiaiiis. Tlii' actiiiitiou, then'loiv, cil' a inisia whu-h may he uiailt! liy tlic pupils is: '-A Holicl wlios.^ basrs aiv i.aiallfl ami wliost! lati'i-al Kiilt'S an' parallclii,i,'rains." Only the apiiroxiinatc I'nnteiits can lie givi'ii in aiisw.T to tlir quostinns ask.'d in 8 and 9, uwin.: to tin' fact tliat the exact iliincii- siuiis of tlic prisms caiiiiut lie rcmml Inmi tlic cuts. In cxplainin;,' the process of tinilin',' tlie contents of prisms. Icail the pnpils at tii-st to use the nietlioil yiven on ;iai,'e 1011 of the .Manual. '>0 Jiisirri:-< : 1 lit- iMi. in. .'ilU en. in. 2-lr, en. ft. 3 7.") sq. ft. LV.dC, en. ft. 4LMlls.|. ft. :'M) cu. ft. 5 ."Hi s.]. ft. .-'.SS.Sgal. 8 ■■'■sii.lt. L'.-|.7S+ M|. in. (;1K.7:'+ eii. in. 9 .31i) + en. in. ."i.io.-.l si[. in. The wofk ealleil ior in 1 shonld be done veiy carefully. Tf the models are uukIo of oanlhoanl. let the puiiils tirst cut the outline as given, and then cut half thvough th.e eardlioaril where the edges are to be. The figures when folded can be held in jdaee by means of mucilage or pastil Jlockding may be eontinu:Ml witli profit to the pupils, besides furnishing models for future nieasurene iits. Tlie following facts concerning the cylinder sht)id I be taught objectivelv and detinitions developed : A cylinder (right cylinder) is a .s.ilid luunidrd by a curved sur- face ami by two eipial parallel eirchs. The curved >urlace is .■alled the convex surface, and the other two sides an' called tlu> Iimms, To timl the convex surface of a cylinder, nudtiply the circ\ini- fereuce of the base by the height. To find the vcdume of a cylinder, nuiltiply tin' surface of one of the bases by the height. ai A„«,re,'s: 1 .94(1.71'+. 2 !i?1i:i.lO-. 3 7(m.8C, cu. in. 4 1800 cu. in, -. in. 5 8 in. 6 10 ft. 7 -' ft. 8 4J. ft. 9 Kift. Sin. 10 ;!011..'«-1. in. 11 «;; sip yd. 12 .'tll7.7<.)-. 13 Si cu. ft. 14 30:i0 cu. ft. 15 U40 sq. ft. $9.60 $5.1-'. 16 ITH yd. VI. 2aj TEACflKlis- MANUAL. "3 AllKWrr.i ■ 1 M; „ , *"r i,u.tl,o,l of (l„,li„,, tl,„ a,„„.nvl . S be perfo,-,„,.,l bv that niotl,,,.! V "'' '"'"" ^^''- ^-t ess in nn 1 ^^ '" m.iicate tl,,. pr„c. »-:«=e:;r,^i;r" "-«"««. Ill o '■ 2 40 ft. 5 S mils. 10 IS y,|. Ill ii:8. ior examjile, iilxi£^~^- ^'- ""'"'*■•• y =sn,«int(-.l. 32 JO or, 104 sq. ft. 21(1 101 1-M S'l. yrl ;a(i. 4)] = s(j. ft. in,jo,,,,,,^^ 14 216 ^ 112 n\^ — ~~ .E'tW Of thes,. solutions o,. an, „tS'^''"- ^^'^'^ '^^^ -^ ■'• ^ aceepte, ana .a, ,e folWea I ^'^an^Snlt!"" "'""'^ 9 -^l-yti-" floor unpainted. \-2-2 OKADi'.i) ai;itiimi;ti('. [VI. aa If tlie iiuiiils MP iKit consiilcriMl n'x\y lor supli work as tlmt iintcl above, they should U- led to write out tlie Miluthm. so as t„ iudieate that thry uu,lei>taua even ^lep. The l.dhwiu},' is a sample ot whiit sljcuhl 1 xiieeteil: 1,S IS li. - 1 II. = 1 I It., leu^'th of unpaiuted part. 12 \J It.- I rt.= .S It., wiilth •■ IJld si|. It. in eutne lloor. 1 I 111' unpaiiited part. _ S 104 <• " ■' border. H- su- 't- i" uui.aiiited part. !) .scj. It.l lll.' sip It. ll';! = uo. sip yd. in unjiaiiiteil part. '>:t Jnx.n;:-:: 12(14,', loads. 8 .S;!"!;..'!-'- 9 -•'••+ pailluls. 10 17L';i 173.r>7+ bu. lit b'u. .-. -r. >S ewt. 11 4700.ir, gal. 12 34.-'7+ rd. 'j:).i>j+ sip rd. a4 ,ii,>:inrs: 1 L'.-Ji ill. 2 ". It. (LSI in. 3 .",(! sq. ft. 4 70-J- sq. ft. 5 i!ir,(i.i() .i?.'!:.'.';;.;!;'.. Dictation e.xereises similar to 6 Avill bo found interesting .and profitable to the pupils. Original problems of the same kind should be made and lirought into the elass. •45-27 if the measures have been proiierly used in previous grades, tiiev need not be used in the solution of these proldems, most of wh'ieh should be performed orally. Let the pupils perforin the proldeins in the shortest anil most direet way. Correet .and lead by questioning rather than by giving outright the .sborter method. For examjile, if the pujiils attempt to perform 24, page L'!i, in a long or roundabout way, ask sueli questions os : ■•What short way of getting the eost of L'.". bu. at 7,-|,«.' a bushel? (Probably two ways will be given : one, i of I'lO times 7.")/, and the other, 2u timJs J of a dollar.) How inueli is it? ,S ]ik. 6 qt. is how many quarts less than a bushel ? AVhat part of a bushel is it? 3 pk. 6 qt., then, will cost liow mueli less than To/?" After answering these questions the pupils can easily perform the VI. 28] TKACIIKIIS- MANTAL. 123 -• ^■-■-t.?r^■■o.. :;;::;: 'J:--:'"^!':^''-^^"- ->■ 1....S tl,u„ 7.-,.- , !:;-:;'"""■'• ^""' "'" tiM.n.lo,.eLt ,.. !.f Inun 7.-./ ,.,j,„i, ;, ,,"• ,,,, ' ■ ^^ ' '■"'"■^ ^'"'1 a ln„.ti,m, siil.tra.-t,.,! -«' "f -^ '".. o ,,k. ,; ,,t ;; ^;' !':•/■ ,^l^-- is SI!....,;, th.. ■"-t "f th,.,,. ,.„„ ,„ ,„„,,,,„„„, , ;• ' ;^" -''. -f %...., i„,t '■"ins not c.mtain,.,l l,..,v will 1,„ r ' V • '"^""■"""' •" "Wud to Vnr.; also on „a,e ,4 Ik , ' ' I" *'"■ •^'■'"■■"'- "'■ ^'ook n"t.sthe,.c.on in il,.. Manual "'"' '"'^"' ''''' ^'""k V., an.l 31 .4«.s„w.«.- 10 ll'll,, 4,„ „ ,„ ^ -t.«oii.. 13 ..T..,..,j,„'j,v ;;:"'-'• ""''•"' 'i <«. 1' ,„rt. 16 25. 17 ., ,.. "., ''• ■' '"■ ' l'"t. 15 4 lb. 20 4 T. 18 ,.„.t. 21 V't , , " ^^ -^ -■' ".'-5 l'i'1. 23.313-)Kk, 24-,i',:;;':'^^'"- 22='-4,,u.t. 25 1 i. 1 .nvt. ,;(. 11, ,.; .,. J. ,.,,.^ ^,,^ j';'- ' "■• ' "X. .i ,1,, 1 „.. •lo'i,: "'4 ;■;;■■ ^^.^ '!'■ ;j- /;:/^,t/3- 3 27 t. ^s .„.t. 1-' ewt. 80 lb. 6 3 lb " 5 q in ' '"''■ " "'• '** "^- '!-' T. jj;:-t.'.n„..^o. Jo..4tti';''^-:: "^^^'''^. 13 ,>oz. l.-i j,„.t. 12 ,r, 14,,-,. 12 •'•-...) llw. .i^s.dC. 17 $20o. 18 I ,^-, ,.f ^* '■ . 15 l;;j lb. ig ^,4,, 22 2,S, and 40 ,,ivt. left. ^^ """ I""''- •'•iO j,ill.s. ''q. ft. A- ,oi,,r„ .,, ft ;•'•;;';, '.' f ^0 .-'l. 3 ./ 871.0 124 OKAUEU AUITinrKTlO. K l;t20 yil. ]' U'M\ yd. M ()72 cil. 12 ./ IMiO )i O .->79L' Ki. /. L'04(>0 ft. [VI. an T. OCi'it. 6./ Ills. <) L'.."i,-. /' 7..-|.«. ^,* iL'.llw. },'[•. .1/ .'llfCI ;;r. .V 117(111 j;,.. !.■. 8 ./ .',', A. K i\:j.,'\. " ^V^v^. A. /■ j;VV„ A. y. -,:., T. .1/ J,v^ T. ;! , f, ■; „ T. /■ 1 ,■'„-„'„ T. (,i .'! i'',', T. 10 •/ .s«(» \i 1 . XlKMIyil. .)A 7;i:iiV'l. .V KILT,?, y,l. O '.i;!SJ y,l. yllll^yil. -■• ,/ L'.'iL'A.-d. /\VM\\r,l y 5(;:'.| cil. o 7.'i7i cd. /' ch;:; cii. . A' Ull-'d f,'i. /, 1.'.SSI) -i. .1/ IKkS -i. /' ."lOSS 1,'i. V -lol'd },'i. 13 .A 1 l."')L'0 It. J/ .'J.V.Illl It. .V l.".)7l"> It. '* Sircti it. O 33 qt. /• 44 '[t. Q nr, .[t. 5 -/^ ,S7(i i'(> It. 14 -/ i.'.->.(i Hf- '»' !'■'•- 'if .V 1.60 (it. I (it. /" IJ ([t. V 111. I.itl ... 15 ./ .'iL'O A. A' 480 .V. I. 400 .\. .1/ I'CO .\. lOC, s,i. 1,1. LSI.), s.]. ft. .V .■!7:i A. ■>;{ 3(1. rd. i'7i;i si{. ft. f> .'lu .\. ->;> s(|. ni. i;7:'i s(i. n. /• r>4i A. y 40") A. .">;i S(i, V(l. l.'71.'i s(i. ft. 16 ./ 1000 ll,. A' 1. -,0(1 II,. /. lL'r>(i lb. ,1/ .s;i;i ll,. -> oz. .V IICO lb. lo^; m. O KlCO lb. 10| oz. 1' 1700 lb. V 1-'''' 11'- l"i} <«• 17 •■' **" !''• A' (!(l lb. L 25 lb. J/ 75 lb. X 5 lb. O ll.'..') 11). /' a7.5 lb. y (KI.S lb. 18 ./ -T)!! ril. 7\"lllL'i(l. /. .SO id. .)/l.'40ril. A' IC, iil. ^>40i(l. PlL'Onl. y 104 r,l. ft. l'.8S ill. 19 ,/ 1 1 (it. A'7.it. Al.-)(it. Ipt. M l.'7(it. 1;,; gi. .V L'LNit. 1 pt. <) lli(it. -Si gills. /' .'id ,^t. 2|gi. y 14(it. li'j gi. 20-/.-i4(,/. K \: put. M 7.'i oz. 10 pwt. y 101 oz. L' pwt. 17.L'8 gr. (I 7(5 oz. ll.' pwt. 15.;i0 gr. I' 121 oz. 1 ],\vt. 14.4 gi-. V 210 «/.. 1 pwt. 4.S gr. 21 ./ 13 eu. ft. 8()4 (.'u. in. K 20 cu. ft. 4;!2 on. in. /. Ki cu. It. 1512 cu. in. M 11 on. ft. 4;',2 ou. in. .V 15 on. ft. 121ICi on. in. O 14 on. ft. (101! oil. in. /' 22 on. ft. 1041;^ ou. in. (,> 17 on. ft. 172| cu. in. 22 ./ 52 rd. K 75 rd. /. 42 rd. M 01 id. X 52 rd. GO rd. P 84 rd. Q 82 id. 23 -A l."!;? T. A' 258 l". 10 owt. L 2?S T. .1/ ;;4G T. K; cwt. '.V 271 'l'. owt. ,",-,4 T. 1800 cwt. P 271 T. 1400 cwt. Q 418 T. 800 cwt. 24 J 102 rd. 4 yd. 4J ■•>. ^'•'^^ TKA.IIIOI.S- MANC.v,.. ,.,. '•-^''■.l.:^vW.7,. ;,!';:':,;;,■•■:'• -'^-■■'■:^v,l.. „.;;„.. •'•">"• /Mni. / i.r,,,, :,;,,"■'"• ''"■-■■-1... .::!>. •i M. o' 1' ,,t. 1.- „i ,., ., , , '■"■ ;- -'-".^'.ii. /•■i',,t. 30 •' -'" ^.. nl. A-,;o , •■"■ ;, ' •■ «^'1>. > p.. 0.x ,i. »-«i.k. /,.■..,. i, .,>"-;!■ ■„''';';< -'■'■■^ 31,/.,,,,, 36 .Asi7,ss-. v^xTkikI /;:;-■?: ^' *'--'■•■'• -v$;^ Hid. 10 (Ih yr. 2 i,„). 2!> (la nio. i; ,1a 7 4(10 ,1a. 47;; ,|, ly, July, 4 .S.-i4(i 6 T yr. I. 29,S3 da. llit! (lUADKi) auitmmktk;. [VI. m 8 $;U'.'.08. 10 .■*! vr. 7 hid. li'J dii. 11 April l."i. IHIl.".. 12 (i(M>tlir : H'J \r. I< 111". -•'! ilii. I.i'iiKlVlli'w, 7.'. yi'. inn. I'.'i ila. 1-iitliiT, CiL' >r. ;! inii. H (la. Kruiikliii. SI \r :'. iimi. (I ilii. Sliakr- spfaii', ."il! JI-. Napcili'iin I., ."il yr. H iii". -*' ila. ISuitis, ."7 yi'. fl mo. -li ilM, .Miltiin. (i."p yr. In la^. .'in da. Til.' uumlH'i- (if (lays in cacli iiKintli of tlic year sIkmiM lie rc- jicatcd nntil it can lie tiild as siidu us tlic iiaiiui iif tlic iiiiinlli is given. A gddd nictlidd (if tiudiiin tl inntlis and days I'ldin one date t(i andtlicr is illnstratcd liy the lollmvinK' s(dnti(iii cf 5: Knan All;,'. !■" til l>fc. l;! is I mil.; I'nnii Dec l;i tn l>rr. L'd is 7 da. ,I/M»vc.- 4 mo. 7 da. Kinm •lime .'In t" An.u'. .'Hi is L' mo.; I'n.m Ailt;. -'ill to Scjit. r. = I day in .Viigiisl + r, days in Sc|il. =11 da. .(»«»■,■/•.■ L' mo. (i da. And in 12: Imohi 17l'.l to Is.-.l is .S'.'yr.; from An;,'. I-'S to Kcli. L'S is 11 mo.; from l''i'li. L'S to .M:ii-. I'L' is L'!i da. .Iiixinr: Slj \ i. (1 mo. L'L' da. A (■(jmnioii custom in li;iiiks is to ic','aid .'id days for every montli in estimating tiiiu'. Km- e\;iiiiple. in tindin;,' the time lietween .Tune .'id and Sept. ."> tlie process would lie: l-'nmi .lune .'iO to Sept. .'iO = ;! mo. — -Tl da. (the time from Sejil. ."i to Se|it. ;j(l) = 'J mo. "i ilii. In performiiit,' the latter p;irt of 4, let the jiupils tind tlie iimouiit saved if the year liegiiis on a week-day, and. ai,'aiii. if the year begins oil Sunday; ;il.so if the year is ii leap year. To tind the exact number of days from one date to another (7): Kroni Sejit. 1(1, IWH. to Sejit. 1(1. IH'.IL', is ;!(1(; da,; It more d:iys ill Sept. 4- -It in (let. = .'ll d;i., wliiidi, added to ;i(i(l, eipuils 4(1(1 da. 3.~» J,isin;v: 4 ."7.S' 7.".4'. 5 l.SL'll" L'.-.,-iS". 6 1(I,'<4.S" 144:.'4", 7 r>° 17° 7J'. 8 "> of a degree .(1 of a degree .L'7r. (if a degree ^f. 9 L'L'" 11". H 1C° :!.T Jid"; 8f,l.0C miles. 13 21 GOO geographical miles; 24840 stiUule miles. :,?jp»? hi''- w VI. :mii TKA(Fll:i!.s' MAM Al,. '■'■.mI the iiniiils tii use tl,,. „l i, i "-^ ^- '^'•'-i.n,:ij,t XI ;;';::''''''';■''■: '""^ '■' ii. l.^nwll, „r t 1 ,„, , " "■""'" l'"tl.,.,l,|),.,.. «-■'• -.ni. ,,.,.„,.. J': ,r;' ;" ^^ ^ ""''- •111 .\il.iii;i-x ■ 1 |il'> ,,,: 1.1 - I 10 .i.l.L',s+ ,,t n ■ It ;•'• , 8 ''■''^■-+ 1'"- 9 IT.:! + i,„. '•"• 'f- Tl„. .„.„.,.,. :,l,„v,. „,;• '-'"■■- '■"i'<-.H,„ U io^i.u, n,,;;::;;;;/.,/.;!';"- 8 *<--.-... "s'.i.. 13 71. 1 ,, .. -.!'"' -■"" l""-l T. 14-10 lb. 5 lUiOO lli. Ti'J^''^ bbl. 6 .'!!;!(;..S0. 7 •'S-'.7;!,V. 8 •■SSLM' mi. 9 '.I A. M7 sq. rd. 24 sij. jd. i'>i sq. ft. 10 1(10 rd. 6 li. 4;!^ lain. 11 .'ill mi. L'2(!| mill. 14Jmiii. 12 9."i|<-. 13 rM times. 12 L'l-VSO'". 13 $5.74. 14 70001 jioles. Similar exercises may be given for class drill if needed. Show two ways of reading metric numbers ; tliiis, O'J.8;!'" may be read sixty-nine and eighty-three hundredths meters, or, sixty-nine meters and eiglity-thrce eentinieters. Let the pupils practice in reading numbers in both ways. 41 Jiiaicn-x : 2 lOOOOi™. 3 lOOOOOO""'. 5 .OOOl"''""' .000001 1"^'" .001 'i"^'"'. 6 10001™ 1001" l()Oi>". 7 !).00.".'.)7(ri'*"'. 8 8.097.144 1'". 11 0849002.11'". 12 f)00o000.8i">. 13 201.1251'". 14 2000001"'. 15 12500 bricks. 16 5" .5* 500 « .05". 17 200" 4'' .08'' 6000 ». The sign for square is sometir.ies wi'ittcn sq. instead of q. Call attention to the fact that, as 100 units of one denomination make a unit of a denomination next larger, two places of iigures are allowed for each denomination. TKACHEtIs' MANUAL. VI. 421 129 42 Aiisirerx : 1 .S.K!7.50. 2 S.'iOd •? anon • ^ „^ " ■*■•--'+ • 13 1:^0.. 14 .s.. ' is ^.,,s^ "'^• 43 /«.sv,v...... 3 ,S(KM (i<,. o.,s> 4' !.oo'. 4«K. o-s^-. ^IKKMI". 6 0.01>1. 7 f,(MI(|l GO"l I'ie itor ,„ea»,u.o is „„e .lecimot,.,- long, wi.lo, an.l ,1..,.,, Tf hJ ^c h„o ,s „„t supplie,. wit,, a lit,., ,„.'eau b.' ,aall J , r.i::r£';;;::;:;;;::;;l™;;;:: ;:;;;-»""..- volume, capacity, a„,l w.i.ht l,av. witli „„,. an„tl„.,- a llr t.>e aistane. oa a ,a..i.ia„ ,i:,,. uJ,,.;:t::;\:r:;;r''' '"'" '" 44 ./«.«/■,«.- 1 400 B ,soooB 3 CO;»..S01«. 5 .s(;..)7.. Si' 9 12000 ■>■». 10 24000 ->. The last four oxordscs on th. pag,. are important onlv as thev »erve to give the pupils an i,l,.a of th,- value o the ,., , ^ .neasures in measures familiar to h, i, t • , "" """■" for all these e.ereises to be ,ive,i t , un, i , "''"'"'" ,-.'>' 11 u. .nioinpii^ii tlj(> ]iurpose. 4:ii Aiixin-i.s : 2 2% lir T 'So/io CM., - ... 'i -J "I- J ij-'oo .s21..,o. 4 1;i41?'"' 8 2;;2:;.2'". 9 7211001. lo 2.-.''.s« S!;iliO]l).20. 13 74,;,;j.,n, 5, ,j,; !»00(1^- .S0« COB. 2 S;!I,1I7|': 6 SOI.'. looo>;. 3 j^Ka^ 5 l.'i.SS'^--. 7 ;!(i7,-,;!(;oo *■ 11 ^V". 12 .0 1,1021 ■"<"' 14 31)<)400'i'". 15 (jiion,. 4« .1,1.. 4 ,';i.072™ J2.!)r(;74». 8 I 12 7112+ liair-,lol •Mif-''-'''- 2-'"""' 3 •-■'■'■ 3000. •■"•■'-was. 5l0hr. 6 IK 7 .I2:.7.074'n" 9 J..S'". 10 ;j T ooo K.-. u ]2o'''s 13 4.-.20.S.,S«. 14 .s;7i,s,s. 15 :^l.;,:^ .r ':;::;:: ;:;:r",;;,r:;l';;i;;t?;r"--- 130 GUAUED AUITHMETrO. rVI. 47 the groat saving of tinip wliich its ailoption would occasion. It it is desired that pupils shall be acmiaintcd with tiie deiioiiiiuations of the system well enimgli to perform prolileiiis at any time, fre- quent reviews will be necessary. 47 — 4:S) Kolhiw carefully the oriler of exercises liere given, and let tlie pupils |U'actice upon tluMU until tliey can i)erform tlicm readily. Their inulcrstaudiug of subsecpieut work will depend largely ui)on tlie thorongliiiess with which they take this prelim- inary work. 50 A„M-r,:t: 1 O.I)'.)(! L'.71:!(i:^ Tr>.li->2. 2 I'OO.l 44r.-i;;,i O.O;5;!02(;. 3 Sd'.i.CS 4o.0,S 44..")4^. 4 i3;;!."i.8:.':i4 17.4147^ 2.jo(l.(iL'|. 5 .'iii44..SS'.)i U;!4.;!;>!»2 88(>.(«;{ mi. 6 L'DOd.S.Sli .Sir.4.22$. 20 $04.8284 l.;j()()8§ 0.()7.i;. 21 2;S.40.'> (l.;!!t(12425 (l.(l(>,\.. 22 ().lli;;!(;ir) o.(Mj2.\i .■!.->.(i;;2r). 23 22();«.Gr) iuso.CkS 5."..s72;{J. 24 •Sll.l722<> .S U>.2o."i2r) (I.0tl4;. 25 (>.(lO:!,i, 217.87.") n.OO;!;54;:25. 26 21(l2.118 1ir. 0.012./, l.-).772:i28. 27 4a()720f 40r)i';!).44:f 0.247ii. 28 O.0."i,L (l.dOdO;;:; liHWiO. 5 I Xoarly all of tliese pr()l)lenis sjiould be perfornu'd orally, but for tlie sake of learning a gnod I'orm of written analysis, the pupils might write o\it the solution of scuue of them. Tlie following form of solution is suggesteil : $;!o,5d.00 cost of farm. ^H $118y.;!;>^ gain. .'Jo.lO.Od $47-)3..'».'iT\- selling price. 2 $.0807. .'!!.l.'i21. 3 ®14r)('). 6 07! liu. 7 Sfi.lO. 8 SSl.Sl. 11 «21i loss. 12 .S3u9.13. 14 (iiven : Cost of f;irni. Ee(iuired : The selling price. 52 .liixir,;:i : 1 .«r);!7r). 4 SoOihS, 5 ;!8:!r)20 111. '.Illni, 9 ;!d.'11.2 lilil. 10 .'i82oO. 13 $49.93. VI. S3] teachers' manual. 131 as m„c I, ,t.s posM ,1,.. Ju tlus i,robI..m they would say: " There are ^^.' gu Ions ,„ all. If 8% of tl... ,.,„h.s.ses was lost ami 4.4 If I "?i..;t «Io:^;'r' I'™^ "' '^ «""""' ■■"""j'Hed by the nu.nh.. JSuVl^r-V ^■'"- 7.?0600000. 8 787828.29 tons. 1^ ifH.jI.HO. 13 pounds. 14 Sl.i.m. When these prohlen.s have been solved an,l solutions in good fonu .•r.tton out, the pupils n,i,,l,t praetiee iu writin,- the solution on a line, usin,- hundredths iov per r.ut, as in 12, as follows : !).3 2 p X rHd X .(It , Si X i.sco X Gil ; + - .- r— = * 1450.80. 4 /(>() xw 54-.,i, A new principle is involved in the.se exercises. In prevu.us exereises the base an,l rate were given to find the per- centage In tl,ese whi,.h follow, the base ar.d percentage are given to hnd the rale. These tern.s need not be given to the pupils at present. 'I hey should be led to reeogni^e the eondition.s after sutheu.„t praeti,.e. ^N'henever the rate per cent that one number .s of another ,s ealh.! f,n-, let the pupils first get the conunon frac- tional part and then rclui^e the fraction to huudredth.s, or per cent Great care should l,e taken to leail the pnpils to recogui.e tlic hase, or the nundier of which .another number is a ..art The form .,f question should be variously given, so that the pupils will not work mechanically. Sonu'times the base is the first number (riv*.i» 111 +1..1 1.: ii-t . IS 4 'i and sometimes ! pen^-ntage is the first numlier giv tl Sufficient jiraeth tl given, iis, 4 is what part of 8 ? •e in both forms should be given, so as to enable it is placed iu the !io pupils to rucoguizo the base wherever questiou. 132 GRADEn ARITHMETIC. [VI. r>«i 50 Ansiiri-s: 1 ^2.rt'j'„ 2 67.5% 1.;% 40% 800% ]:!i% 15^5% 71Ji% 113% l^iVii'i,'^ 6SJj:i%. 11 50% 35% i;s% «04ij% 0;^% 100% ;««% 101;]% 5% 7!%. 17 30i%. 13 Canada, 5§gi;jU%i l^-S., Gy53ii;;si%. S? ^Hy»v/-,<; 1 (•.•24-2.S5 "9.14; ; 2.').1"V. 2 Moli., 5.8 + ;/ P.riili.. 20iii;/ Hiul. 40 . 3 'Tio . 4 Cnl., 1:1+ Jiitl., ^i + / Whites. si;.I+. 5 87.:i+ . 6 1.4 + /. 7 5/'. 8 ^iVj/. 9 14;V. 10 15';. 11 10.1 Xotico the wording of 3 both kinds of cxeroisi's. Tlie term base niiglit be introduced lici-e .is lieing the amount on whieh the rate is estimated. Some introduetiou will h.ive to be given to show what the base is in .sueli exeri'ises as 7, 8, 9, and 11. 58 On tliis ]jage and the two following ]iages are simple appli- cations of tlie two principles of percentage already taught. The pupils should perform the inoblems by analysis, recognizing in the varying conditions wliat princi]ile is to be applied. Of the two methods of finding tlie interest of a given sum for a period greater or less than a year, it is better on some accounts to find the interest first for one year, and tlicn for the given time, as indicated in 8. Let the an;ilysis be simple and direct, no set form being required. 5J* By "rate of interest," referred to in 1. the jmiiils should know that the rate jii-r i/nir is meant. The pujiils should be told that the per cent of gain in' loss in business transactions is always estimated im the cost nnles.s otherwise specified, Some talk about VI. (M»J teachers" mantal. 133 insurance, commission, savings banks etc will !>„!,,,,• neotion with ,.ro,„c„.s ..nn.i!;, i„ tl.o t;.. ' p^ '£1 '" ""' ing these subjects sec Jiook VI r nn i -^C" *acts concern. Manual " '""' "'•'•""Panying notes in the -V be iritu-n „:;x:: iris!' ^^' """'-" *'^^' "'^ ''-^^- i''o"(, of the number = ;« ■f *" " " " = I'.i of 32 f" " " '• = Vo" of 32 = 80 32 is 40% of 80. 134 (iUADKD AUITHMBTIC, [VI. 03 percpiitago, let them snbstitut- small iiumbi-rs in place of luvge ones, aud cuninic.n fractions in j)lacc of vcrccntagcs. The substi- tution of aliiinot iiarts should be made loi' percentages whenever simplicity is };aineil. but for useful pra.'tice some of the percentages on this page should be used ; thus, in 4, the analysis may be : "J!!'^ = the rcpiired number; 4o = V'.Mi ^ ^}u, = . ">■ 3-^'; l;!!! =1<'" '""«'* 3-^' or *;52." 63 .(««»■.«.• 1 :!08i 2ir„-,5 iS(i4i)3 f.-iSOSOO $12000.25. 2*42500 n;«li',t..ns 2401»5:it.ms ;!(i5 da. 1!)405;]. 3 $11552.;»4 «81834 $1402s'.5.-.-t- $H728.>S8-l-. 4 8,s;. 5 214^ 6 420. 7 8'>2'-' 8 405J. 9 345. 10 IDS. 11 108(tO. 12 120. 13 111 J. 14 2741 J. 15 28000. 16 252. 17 U- 18 100. 19 03 98-1- 20 $2517.482-1-. 21 $2500. 22 1225 bn. $2858.;;;!^. 23 $113.35. 24 $205.07. 25 $155928.88-1-. 26 531i. 27 $4.04, cost per bbl. Tlie f(dlowing form of written analysis for these exercises is suggested : Cost of house = ^38 of itself. Selling price or $400 = US »* cost. $400 ' lie 21 liiT "f '^"*' = I"" of cost = , ... $400 X 100 = $344.83. 64 The miscellaneous exercises on this and the following pages of the section involve all the principles of percentage winch have been taught. Do not give the pupils any direct assistauc(> m then- solution ; nor should any assistance be given until the pupils have made a strone effort to do the work by themselves. If the condi- tions of a problem are not understood, make them clear by judicious questioning. VI. en] TKACIIKU.s' MA.NIJAL. Cii AiLvivrs: 1 ,'ir)2.S,';A 9 15 7317,:.,. 13,) 3 30fi ,, . • - 8 K^ini, 11 VMM. 12 1V,, 13 ,«,.,. ,,f 16 3..ll} .7HUIM1 .>y3o3i 10 >;ii,s.sn *'-0(»o, Hostoii .{si'iod:*. x.v «!> .l«.s-„-o-.s-.- 1 Sum ],„i,l, ,S7.->()|) sreater. 4 .'ji3r>(;.L',-;, |.,„t of iiLsiiiMiicc. 8 !.>/„ -■'•"45? 11 SU'37..T(I, *a3;{3i, H„„. 3 ^llo, |)r(i. 6 $37. coin. 8 .$400. oost. cost. $1'!)J. , $1000 8 l'(»(to 11 «'Hl'/;. 12 $;->(!, $1443. rctui-n..(l 9 $1'0. CO],,. 10 $l!)(il».8(), cost oin. 12 o.;i(;f^. 1 (Kic 7 $4(1 collected 11 2 $]281'8.57. 5 $8.1(1, com. 13000, 71 F, 'A- 2 1(18 11,. (i44 ji, ^.00/«ou$,. 4 $.40. 5 4.,(Hin, '■ «"<'nui^i, ^iccoiints, d blanks 186 GKADKl) AUITHMK, [VI. 72 may hp provirled for tlip jMipils to till out. The pupils, howevor, should not (lf])Oii(l upon tlii'se Icu'Uis. hut h'liru to wvitf them with- out aids of any kind. It will 1k> noticed that two forms of bills arc given, either form heiiiK .ilhnvahle, excei.t for hills for services, which should be in the form given ou page 7L', except that the word "For" instead -f -To" may be used if desired. 73 .t,i.iin;v: 2 Amount due, .Sl.Sll.40. 3 I'.alance dms !i?:.M!8. 73 A„«;t.7n. 3 $<11.S0. 4 Kal.inee. .1ii!).S8. 75 AiiHirnv : 2 Halau.c due from U., .Sl.in. 3 Balance due from Holmes, .S0.20i. 70 .U,«,r,rs .■ 1 .Vmount due Jan. L'4, 1894, li>\".(>P.. 4 Amount of bill, $34(!.5.-> Cash payment, .«il(i'.i.,Sl. 5 Halanoe due I'arker, $5.16 Balance due Dexter, $2.32 lialanee due the merchant, f2..">9. Explain, in 4 the custom of giving credit, that is, of giving a cer- tain time in which payment may be made. Also explain the custom of making a discount when money is paid before it is .50. 3 34iSdoz. 4 138Uyd. 5 3 da. 6 Hi da. 7 $2.70. 8 1079.!)+ pesos. 9 $009. 10 Mobile, 92304000 cu. in. ;i99.-.,S4-t- gal.; St. I'aul, 40032000 cu. in. 173299- gal.; Boston, (iOSlOOOO cu. in. 2,S9247- gal.; Chicago, 403G8000 cu. in.; 200727-^ gal. 11 27* 11). Several of these problems should be performed orally. 6 will doubtless cause some difficulty. Ask such questions as the follow- ing if necessarv: ^-At the end of three days how many days would it take 9 men to tinish it ? Suppose all had left but one man, how many days would it have taken to finish the work ? If it takes one man so long, how long would it take 4 men?" Give similar exercises varying the conditions. VI. 78] teachers' mandal. 137 4 «,{i4.(,.{. 5 f UO.oa 6 $27.;{4 7,«i4'{"0 a i t ^mmr,. 10 ..;,, ,.a,s. u J«*t r^ «*"''"""* 17 8 ft. wide. The 6 mo. in 13 is to be regarded as ISO days. 17 i< t.oal problem ,n finding tbe greatest common divisor. 12 $;:4,'i(i. 16 Hit J 3 ft. IS a prac- 79 .4n.wm.' 1 f, A, 74 rd. 9,S.5 sn. ft 2 Cl •» «i ro i78.40 " 13 2mo. 2112->lll.0KX 14 $41.60 ZU2 ' ""\^''""""« Sundays, the answers would be .i4.iJ9840 copies, and .SOSOrtOCSO. 8«-. 9 1.8,.,,,,,. 10 840 1b. 11 $320 $9120. 12 403 ..f \ ^"nU^ "'•''''+ ''"• 2 50.892+ bu. 3 88 furrows ;"t.„;-^ *^^ '*'''• 5 «-'398.,50. 6 $4000. 7 27,«V The answers to 8 are found on the supposition tliat 1 ru ft of wa er . ,, , ,,00 ... T.t the pupils perform the problen.; ree ! ;:— Lirr^"— '■'^^""'•^"- -r"'^^-"- 82 Answers. ■ 1 $19220. 2 $28.9.5. 3 $ 1" „er ..•,] 4 $15.39. 5 10000. 6 3200 1b. 7 5;«3+ ,b 3 ..-'".^ u'tJe'M "/'to''- ^0 l-'^'H panes 132% stt 11 20106.24 sq. rd. 8293.824 ft. 12 $63. 13 $1080. IM (iitAiiKn AiiiTn>rF,Ttr. 138 83 Aiisirn-s: 1 -r,c. 2 Iimk;, SI], ft. 6 SUKHMI. 7 .'11111.1'+ ivv. boxes. 10 «(• mils IL'-'l mil crate. 13 ifil'-KMIO. 14 .*l.-.:!.(iH. [VI. 8!» 3 .'«1, '!.'!', 2ri». 5 .'11"' 8 '.1,1, "111,-:,",. 9 tl.'.■-':''+ -■'' ■'•" i-i'riis. riiysiciuns' presnviptions are friMpiciitly made in tlic maimer indicated iu 11. The Kiveii (|nantity slmuM lie read H drams and 2 sernples. 84 Jiisin-rs : 1 .'t!10440.,SO. 2 ".MlllJ liu. 3.'?I.S(>. 4 -Mini- day Wednesday. 5 S\.''r, naiii. 6 .«i.'!.;i.s-;{. 7 .SKIL'O. 8 lOHlKI sq. ft. 121 ft. 9 . pills. 15 22.7il bid. 16 8(>.7S+ bu. 80. .•)h.v«vt.s-,- 1 §G72. 2 8 ISO. 3 12S2Jeu. ft. 4 243^ yd. 5 77igA. 6 18 rd. 7 .•SlC.;'..7 1 + . 8 1." = ^ ''d. 9 27b. 201). 10 12''da. 4 da. 11 l;)() men. 12 1.^ bu. 13 i. 14 iS30000 ,'™; 1 l<.>a2brh-l;s 11(101 + . 2 31(1* rev. .•Ui'j-nii. 3 3!);■;%. 7 Chicaj^'o ahead 211,7;. 8 .S2.SI2..-.0. 9 Hi,';,'/. 10 101,1yd. 11 Sll.Ol '. 12" .$(18. l(t. 13 1 ;,'„ da. 14 $2'J(;0?, U '237:1J- K. 15 ;i%. 10 lfi.22%. 11 28.16 gal. 12 420''X 13 20178885 bbl. 14 18!»1770S3i bu. 7 Abimt ,', 8 4.;i8'i 9.0.1 VI. 80] teachehm' MANITAI,. liiU ".l sk.n ,.l.,s., a,.eun.t,. ,v„rk .sl,ouI,l n„t l„. insistcl „„„„ itt tlM> im,.,ls „„.a«,rc l.y tl.c ai.l of tl,. seal,, of ,„iks o 'l f -,Ms ...aw„, ana also 1^ t,. aia of panm,.,s a;:;; ;::;;,::;;? "'^ J»0 J»™vvw.- 1 .■!i.(!:>,s.l,s:>4 11, l;ilil.)4r,"ll, a .,..o„o 10 .01.4+ ..d. u%'j', «•"•'•''"■'■■ 9 1-^'-'.8+rd. SECTIOX IX. NOTE.S FOR It()„K MMHKit SEVEN. For general suggestions as to the kind of work peculiar to advanced grades .n Aritl.netie, teaehers are referred to'the Xo e lea hers on pages iv and v, partienlarly to ,vhat is said of n les, dehn.fons, and explanations of problems. In these 1, of expression speeial attention should be g.ven to JLl"lZ s atenn.nt, and by aeeuraey is meant not me.tly a e osf d nee to any one parfeular forn,, but an exaet expression of a lo "a proeess. Loose staf^uents whieh earlier in the eourse were alio", ed A rule in Arithmetic is a general statement of the method of 140 GRADED ARITHMETIC. [Vll. A definition is a gciipral statomont nf the ]>pciiliar and oharap- teristic prnpcrtii'S of tlif huIijitI (Idiiird. An analysis of tlir subject to be d('fiiit'd sliinild !)(' made, and frcun tlir analysis tlir di-linition should 1k' di'duri'd, if possibli', liy tlio pupils, with such con('<'tion as thry may bo abh' to niakc. Kor cxiinipli', if it is di'sircd to teach the detinition of addition, the tcaclipr should lead the pupils to perform or to reeall the pmeess of addition, and in that proeess to sei that they have found a nnndier which contains is tuany units as all the given nunilwrs taken together. Aftc:' ' ■ uik told that this process is addition, the impils will say : '■ Aildition is the process of finding a nundxT whiidi contains as many units as all the given numbers taken together." On being told that the nundier found is called the sum, tliey will ecu ret tlie detinition given, and say: "Addition is the proeess of iinding the sum of two or more numbers." If this detiniHi'; is still thought to be im uniplete, attention may lie called by c.\ inples to the fact that oidy numbers of the same kind can be added, as, 8 pounds and pounds. 8 pounds and 6 ounces might be put together to express a denominate number, as 3 lb. G oz., but such jiutting together is not addition. The pupils will therefore further correct their definition, and say: "Addition is the process of iinding the sum of two or more numbers of the same kind." An explanation of a proldem should be made in such a way as to be easily understood, and be so comprehensive as to include a statement of the jirocess of solving tlie problem and reasons for each step of the process. It is not advised th.at :ill or many of the problems performed be explained, but as many of each class of proWeniB sliould be explained as to give assurance that they are thorong'ly understood. In this and the following book it is recommended that the processes only of many of the problems he indicated, either orally or in writing, it being assumed that the mere computations may be accurately performed. In the work of previous books the attention of teachers lias been called repeatedly to the importance of illustrations by plans or diagrams. The illustrations should be continued in connection Vll. 1] TEACHEUs' MANUAL. 141 «.tl. all .l.ffi,.,>lt prol,l..,„s, l,„t for tho s„l,„i„„ ,„ ,,,,.1,1,.,,. „„t V.T .nm,.u It tl„. ,,u,,|,s „,, ,„„ „,.„,„, ,J,, ,' - .shm.I.l b. ,-,.a,lv t,> ,„ak. tl„. „e,.,l,.,l illuMr.,ti.„,.s ^ ,.„,"''''," ""''"'"" "'■ """«^ '•"^•»'"' l"-"l'l'-'"s will K'lv.. tl„. „„„ils l.ttle ,lm„.,.lty ,f tho „r..vio,..s «„,.k l,u.s 1„...„ „.,.ll ,,,„„.. \ ^ -■-.•. l.tt.H,lty ,. fo,„.,l, l,.a,l tl,.. ,„,,,il.s l,y .,„,.sti.„,,s ,„ .s .;.,,. «l.at .. r..,,„„.o,l, a„.l wl,ut »t,.,.s are „..,.,.ssarv i„ tl„. s„l„,i„„' id .■xa.n,,l,. ,1 tl,. ,.„,,iU li,.,l .lim,..>lty i„ |,..,.l„,„,i„,, 7, ,„■,... ;! .sk tlH-qu,.st>o„s: ..B,..„re yn„ ,,,„ «„,, the w,„.,l, „r tl„. n.n.aia.l,'., what m„st ym, Hn,r.' JI„„. ,„„ tl,:,t 1„. ,1..„.,," If tl„. ,, K choose t„l,„.l i„ a ,„,.„,lal,o„t way tl„. „„„,1„„. „f ,,„,„„ J,„ ■ ;. n«>t the,,, so ,i,,a it; l,,.t after it is .,,,,,,, ask the,,, if the IS not a shorter way. So.ae bright ,„„,il will s,.,. that 1'.; .jt. = CI Bal and that th.s a,hl,.,l to ];;| ,..1. is m ,al., wl,i,.l, .s„ tra,.te. Irum JHi «al. f,Mves SJ ,..1. 1„ .,,.,1, ,.,„,.„is..s as 8, l-a^e ;! h.t the pumls .aake a .liagra,,, l.efor,. the .soluti,,., is „,a,l! f ; " f A f • . ^ *•"■'"• * - '■'•• « >•''• 5 7.i v.l. ,S i„. 2^4'^"^^ r''''""- ■'''''^-' "«i'rx Ui^ .7 $400.74. 12 11'()eu,,s. 13 "'l' b„. ;! ,,k. 14,,, 1.. ,\ MVo 3Si'. 15 $.-.0 .«ilJl.'.-,() .*;..,.i-, ^isr,.-M '" In the solntion of U, lead the ,„,,,il.s t"o H,'„l the nu„,ber of enbie inehes or leet tl,at a eental or l,„ndre,lwei:,ht oee„,,ies. ( )„„ husheJ or 60 11,., oe,.ui„es :'1.^<..4 e„. i,,.; loo n, ,v„„ld o.-enpv .■■" ot .'I0O.4 or 3.-„S4 11,. The solution of the iirst part of the ox';; Kise would be ludieateil thu. : »l.lf>X a X 10 X (!_X 1; X li X 3584 ?) 8 A,i,,rers.- 1 L'OIA. L'O s.j. rd. 2 11 A. 55 sq. rd. iH^^q-ft. 3 5 A. 17«L'7.2+ sq. ft. 4 837/, ft. 5 50.20+ s.i ft. 6 l''.7+ i 7 fl3«0. 8 y56.a5' 9 |il5200. 10 («) $524.38; (I,) S;2855.25i 142 GRADED ARITHMKTI(\ [VII. o 13 $40uOO. 3 m'/r- 7 $41,S $.14'.)+. (f) ir>9 posts; (r/) $r,r.04; (<■) .$.">1.07; !8!L'4. 5 «;i Si;?.oc $2.70 «2.4->. 6 i3oj loaves 4r>.'J+yyd. $320.11. 12 $1743.0'J. la Aitsircrs . 1 ly; da. 2 $4870. 3 ."♦ mo. 4 $18,375. 5 2.75. 6 $14,375. 7 25.43+ sec. 8 .33j;i T. $07.80. 9 507J cii.yd. 10 2255'/c. 11 180 mi. 12 .")2'J.21b. 13 $04 $14.80. 14 $1,125 $3.00 $2.30. 15 $44.84. in Answers: 6 2m^. 7 itjf . 8 $1."'.04 $21.16. 9 $6'J.12 $4.53.12. Before giving these exercises in Profit and Loss, it might be well to tell the pupils something of the common i)ractice of mer- chants in receiving a profit for the trouble of buying and selling goods, and to give a few examples. 14 Answers: 5 3'Ji% 33^%. 6 11),%. 7 11,' % on sugar 165 7; on kerosene. 8 77?,%. 10 22ii''/o. 11 14ia.%,. Lead the pupils to see that the per cent of gain or loss depends not only upon the amount ipf gain or loss, but also upon the cost. This is illustrated in 5. In 9, the supposition is tliat the part unsold is worth 4 of what was paid for the farm. lu this case the VII. 15] TEACHEUS' JIANt'AI,. 143 ans„-or wnukl bo 25^„. A.k tho ^nostion : •• What wa. H.o por ,v„t to use „,n„ „...s as «L'..(K), io. th. cost .,f tl„. fan,,. The LJt of h. part sohl wouhl the, ho Sl^OO ; an.l if it woro .sold for Sl'dOO the i,a„Mvoul,l 1,0 3;!i^,. Aj;ai,. ask: •■ What wo„hl ho tho p,'.. ..o„t of Kai„ ,t I shouhl soli th,. othor fo„rth at tho sa.n,. rato"" Altor tho i„,pils aro fa„,iliar ^vith tho pn.ooss, tho oo„„„o„ f,,,o. t.onal part of tho oust may l,o oiaitto.l, tho por .vnt of ..st hoi,,,, foiiiid (liroctly. '' iii J,ix,rrr,: m\if ;!^^. ^ ^_^f^ 6 «!' '.(l 7 S"! -1 12 SUiiKif.;!/!. 13 $u>Ti«. ^ ■■'■ In oxplaiaiug theso probloms, tho pupils should 1,0^1,, with tho statomont that the cost is ..aHod for, a.,d that ,.,p,als ■ "" of itsolf The rest of the oxplanatio,, may l,o givo,, as i„,lioatod i„ th.^ wr,tto„ analyses the reason for eaoh stop being ad.lod ; th„s, in l''t^" !"•"■'"= i" »f cost. ^l„ of eost = ,l„ of ?::;.;"'*■'• *'' = tJ. "f"-t. l:;Sofeost = 100times^i;;or After the pupils aro fa„,iliar with the solution of this kind of lu-ob ems, the use of hundredths as give,, i„ tI,o analysis „,av be om.tted, and s„upler fractious may be substituted; tl,us, in 11 tiie solution miglit be : ; > »* *-•"' = l---";i "!■ i of cost of 1st horse. i of $m) = $lm=z cost of 1st l,O,'S0. $■200 = T.-,oi ,„. I „f post of I'd horse. * of Si20(, -^ 8J()(1J = cost of 2d l,o,-so. %(.(if , loss on 2,1 hors,. - .erce7itnge is the number found by taking a certain per cent of the base. The percentage in I'rofit and Loss is the amount of gain or loss. The amount is the number which includes the base and percent- age, and is the selling price in Profit and Loss when the percentage is the amount of gain. The remainder is the difference between the base and percentage, and is the selling jjrice in Profit and Loss when the percentage is the amount of loss. The rules called for in 3 to 5 should be deduced by the jmpils from the corresponding processes, and may be as follows : To find the gain or loss, multiply the cost by the rate per cent. To find the rate per cent, divide the gain or loss by the cost, carrying the division to hundredths. To find the cost, divide the selling price by 100, increased by the rate per cent of gain, or decreased by the rate per cent of loss, and multiply by 100. 1 7 Answers ; 3 4/3% loss. 1 $11093.20 154i'o%. 4 $1820 30%. 40i% 5^\%. VU. 18] TEACHERS' MANUAL. 145 Some of the exercises from 5 to 20 will have two sets of answers aceordmsas the percentage is gain or loss. Let the ,,„i,iirKke ev/*, '^'r t';'''""'"*'"" "'"'"''"'^ '" 1 "'•''>■ ''■-'ve to be enlarge,! or ^^.ll ass St the ,,u,„ls to understan.l more Inllv the l,usi„..ss „ commuss.on. Besides giving the names of the 'consignor ^,1 con -gnee, the pnpils should he led to answer in genenU la .to , '?"•"■ :\' - 1^--^-' «•>'" -nds the goods t,. another,- and ..tie to,m,jne. ,s the person to whom the goods are sent.- P.I!!.yr"" ""' """" ' '■"^""- "'■ "'"'-^^ "f S-'l'^ -'ts is the caiSd r:::;;'"' ' ^"•"'^'^'"' '- '^-^ - -" --"^ ^- -other is ti j;::!z- r ""'"' *° ^" ^^""' '- "^^'^^ -" -'"■■« «-« ^^ Attention should l,e called to the fact that tl,e base in ,.on, X2l.?;-....5(? 9*^. 3 •«i4(t. 4 Sl'U.i;}. 7 .liS.'iOOOO. 8 1' + '/,. 9 .SIOOO. 12 *ir.7r.. 13 *") less. -.• 1 19- 6 SlisiL'. 11 .1il23y.So. ai Anxii; 5 $2(i(;(;|. 10 .^ITjOO. aa Aii.-:iivi-s: 3 li% .Hi.OlLT). 4 .l!i8S..S(l. 5 *1L' «.(>11' S4(».cS0. 6 ii^i'ir/'/o ■1*74.57. 8 •fflO.lIlL' .1ii71.S.7t;. In the jireliminary teaching lesson, use familiar examples, both in showing the iiurjiose of taxation and the meaning of tlie various terms used. I'roiii recent reports of assessor", give actual facts of the vahintion ;iiid rate of taxation of the city or town ill wliich the pupils live. Real f.tfiite is fixed ]iroperty, such as lands and houses. Pei-mntil jiniperti/ is projierty that is movable, such as money, stocks, cattle, furniture, etc. A j)i)!l fu.r is a tax upon the person of a citizen. Ansi'iisiirs are oHicers whose business it is to estimate the value of pro])erty, and to apportion among individuals tlie sum to be raised. Tn finding tlie method of assessing a tax, get from the local assessors the inuthoJ tliul is actually employed by them. In some places the assessors are paid a regular salary, and in assessing the VH. 23] TKACIIKIIS' JIANIAL. 147 "r-i"' ;::;;"■ " " ;■■" ■ «- -I-* »?', li> !<»(•. Jrom this s„,„ suhtrnct tl,,- „„11 tax i,,,! ,livi,l„ .1 , v than .,%, tl,,. hrst ,l,v.snr must 1, ,rr,.sj,o„,lingly Lss." 23 Aii«irny ; 2 .S;11.'.-,(P. 3 Sr.;! 4(1 emr... 7 *2l:;7.;;!.7:/. a mH.: In 7, .leduct tlu. „„colle,.til,k. tax befor.. ti„,,i,„ 4 ■*'l.'i4.1o. 5 .«!l,s.7;';. ■ tlie cost of 6 .SGr)(;.(;.i. 7«L1r""o'cf *"'--'^- **^''-i"- 5 $100. 7 »180.48. 8 .$llo. 9 ,$4:... In teaching this n.bjcct show the reasons which Kovon.nu-nts ha^c for tax,ng nnporto.l goo.ls. Tl... following faefs n i 'l t h, presented in one form or an<,tl,er ■ ° '"' taJeS'!:: ''"'■?"■", ""''^ '•"""•""' ^' ti.e custom-house,. --o^i.r'^;iu.^:-gi:-;tnr-^^^ J. g ™ : i """ " T"""""^^ "'■'""^'■''' «'^^''»« *'-' ". '",''■''•,■'"',;"""■"■■ "■"""'" *" "-'""^ the allowance for pedfi :;""' "" *'"' ■'T"^"'-'^ °f '-"'- ^'"l- otherwise «4.866. The .- long ton - (2240 Ih.) is used at tlu. Custon, House. 25 Aiisiivi-H. 5 |i;i.59. 7 1 ••5370. ■il.2(i. 9 2 .?10.04 3 .«!42.44. 750 lbs. 10 4i28.72. 4 ■*.197r,.25. 148 OKADED ARITHMETIC. [VII. 30 26 yl».wf™; 1 $40.-0 !i;2C7!».30. 2 432,". bu. 3 «4.-.1.50. 4 ai40.".L"J2.50 .•!i;!7131«.12i. 5 $60. 6 $r.00 1oh.s. 7*2209.80. 8 $1.00.s. 9 1%. 10 $12406.02. 11 $61.>..90. In 8, if 2240 lb. are taken as a ton the answer is W^. 27 Amirers .■ 1 $326.40 loss. 2 $3,323. 3 $.003473 $40.65. 4 $20.02. ' 5 $2SS0.15. 6 $8;iS.50. 7 $059.50. 8 1% 3% «% 9 $1000.59. 10 18 machines. 38 Answers: 1 li%. 2 .'$10,S0.22. 3 Silk («) $.'...'-.7 Kibbon (a) $.205 Uwe @ .Ifl.ie. 4 $5860J. 5 $044.;!;!. 6 $18025. 7 U%. 8 $8.-.;!.i;!. 9 A, .$20.53 H, $23.08 C, $42.63 D, $12.03. 10 $095.70. 11 $5580.08. 39 These problems should be ])erformed orally and by the shortest way. Where the way is not indicated in the (juestion, let the pupils choose the method of solntion. 30 Aimtrers : 4 .142. .(»91. 9 .097. 10 15 .(t37it. 20 .127J. 25 $1H.07. 30 $23.75. 38 $411.25. .041 J. 11 .OO;!. 12 6 .1815. 7 .o;i{; .o;ioj. 13 .i;!5j. 16 .211;?. 21 .2285. 26 $4;i.05. 31 $21.08. 17 .246J. 22 .302'. 27 $,33.37. 32 $109.12. 18 .093 J. 23 .178,l'. 28 $.38.15. 33 $37.02. k- 8 .0715. 14 .002^. 19 .279. 24 .109|. 29 $180.49. 34 $20.05. 39 $278.85. While occasionally the method of casting interest is determined by )ieculiar (conditions, it is generally best to adopt and follow one method. A metliod very generally pursued by business men is wh.it is called the 00-day method. By this method the interest is asi'crtained first tor 00 days by getting .^J^ of the principal, and for other times from this sum. For example, if it is reciuircd, as in 25, to tind the interest of $200 for 1 yr. mo. 20 d,a. ,at 6%, first divide by 100 to find the interest for S mo., and multiply that sum by 9 to find the interest for 18 mo. 20 da. is i- of 60 da., and therefore divide the interest for 2 mo. by 3. Add results lor the Jinal result. The solution will appear as follows : VII. 30J TKACIIKKS- MANLAL. U'J $2n0.nn IVincipal. _2M} Int. lor 2 mo. IS-""* " " IS .no. (2 mo. X 0) -—% " " -'" <1^MA '.f «. ,1a.) fractional parts. T ,.t , 30 «"",": "'"' "' "'"''"« ""^ "-i-1 the inten..st for - ,„o l,v 5 to «, , h * '"^- ^" =^3. mnlti|,ly of the infrost fo 1 ,: to '1 U "'!'"' ''"' ' '""• ^^'^ A solution will appear as folL;. " ""''""' ^'^^ "^ '^^ ™^ Interest of $632.2!) for 1 1 mo. 27 ,1a. ? __6.322 Int. for 2 nio. .316 *''l''' ^"'■'■"■•l"">"(2mo.X5). 9 :r'^ " " ^ '""• oo- fro,,, J),,,. 2, '90 to May 27, '02. (n ',%. 17 mo. 25 da. (n, 6% on $l>907. 1« mo. 232.56 160 GRADED AUITHMETIC. [VII. 31 llf > lit I The above mothod is a sli^lit m..aifi<'ation of what is cal od the six per cent method, liy this n.etliod the interest of *1 or the giv..,. time at .-.7; is nmltiplied l,v the nuud.er ., do hus lu the priueiial, and if the rate is .,tl,er than .;%. the ^"^^^^fj^'^"^ is increase.1 or din.inished as re,,uired. The .nterest o *1 <,. 1 vr. at (1% is #.0C.; for U mo., *...! ; for 1 nu.. *..)0o ; tor (.da «i.001 : and for 1 da., J of a mill. In general, the interest oi *1 is found hy takiui; six times as many cents as there are years, one- half as many cents as there are m.,nths, an.l one-s,xth as many mills as there are days. By a little practice one is able to give tin- interest of «1 at (>% for any ti«>e very quickly, and it the f./„ method is pursued it will be good practice for pupils to do this. 31 .l„.„,vT.; 1 »-'13.;i3. 2 .Slll.8«. 3 .K-l.-'.-.. 4 $20.35. sSm 6«(i0.7.. 7«4«.30. 8«H.l.. 9mo. 10 »138.03. 11 *H.o.J. 12 «:'5.S(i8.4'J. 15 Apr. 10 and Oct. 10 of each year to Oct 10, 18.J8,eacl> payment being $43.75. The last payment .*1.. 3., o, including principal and six months' interest, was made Ort. 0, 18-J3. 16 *L',W.3(). 17 «04 ».lTr.+ »12.80 «2.,.9... 18 $8.17. 19 *20.',)0. 20 $35.55. 21 $34.14. In finding the time between two dates for business purposes, there is a difterenee of practice am.mg business men. A co.nmon practice in estimating time for casting interest, is to regard each month as consisting of 30 days; thus, in finding the fame from Sept 20, 1801. to Aug. 1, 1802, it is 11 months to Aug. 20, a.iu to An-. 1, it is 19 days less than 11 moiitlis, or 10 mo. 11 da. M the time of purchase (Ex. 15), Mr. Brown reeeu-ed from Mr. Smith a deed of the property, and at the same time gave to Mr. Smith two papers signed by him, one u note promising to pay Mr Smith $1750 with interest at 5% payable semi-annually, and the other a «9 73 108 .06 0.48 _58.'i2__ 73)590.10(8,160+ 584 121 _73 486 438 480 8.17 Ans. To find the exact number of days between July 1 and Oct. 1, lead the pupils to say : «3o more days in July, ;ji ;„ Augu.st, 30 in September, and 1 1" October. Adding, we I'ave 92 .lays." The explana- tion of the problem is simply • "Multiply «540 by .06 to get the interest fori year. Divide this product by 365 to find the interest for 1 day, and multi- ply l«y 92 to find the interest for 92 days." 5>'- leiyr.lmo lol .,\ "3-*^ + %- 15*210. IS r. 7 mo Ida an r ''V' ^ '""• " » '"o- 22 da. 22 - yr. 5 „,o da ^"^ T' " '" ^1 1 yr. 6 mo. 21 da. 25 2 mo. 17 da. 26 4 Jr 8 .. ,"** ' ^^^ ^ ■""• ^^ '^-• 28 3 yr. 11 „.o. 9 ,,, ^^ ^ J r. 8 mo. 20 da. 27 4 mo. 4 da. Draw reasons as well as results from the n„„ii, ,,.,,. „ say in explanation of 1 • .. Since nt ' ' '"» "'™ *" tlie given time is SI to • , ''" <"'"' "'« ""«''<'st for time IS «1 to yield an interest of $20, it will take as 162 OKADED AKITHMBTIC. [VII. 38 many per cent as 91 is contained times in .f20, wl.ioh is 20. Aimwei; 20%." The form of explaiiatioii may vary soiimwliat ; for example, tin- reason for dividint; may W expressed : " To yield »2(» interest the rate nnist ] vr "i vr q i 3 March 20, 1893. 4" 10* y'r '^ vr U. '""« -'^'•''- ^ ''"^''■ S«?-! 12fc'300. 13 $14400 $2190(..- 14 .,t XO -r.il i) loss. ■*•* * /&• In_15 by "average expen.so " is ,„..ant avora^o yarlv ,..,„.„,<. oJ» Answers; 3 ,«;i()0 4. en .11 ,. 1 . j, 7 Si;i4r,;{..4 a «i--T;., t " ^ •*-'"*■ ^ .■i!»4.(.,s,s+. ^ '■ *• «» *1m.6.3!». 9 .'?ilr>7,S <)47 given .sun, in the gav,. time and rl T "* *" '^ from the pnpils may b.. ,3): "T^ am l.t ^f' "r'"'" ''■•""■" and time is .fl (),-, It will ,■ ' ""' *-'"■''" ™''' *10o a.s $1.0., i„ contamcMl m «!1(.5, which is 100. ..,„,„.„, . ^„ „ ." 36 A, swers : 1 .fi2'J4.12 latter offer, #41.14 better « To ..1 •■■♦(■"'>. 4 I en.. «,. ' * ^* uttter. 5 Less tlian t™.. v..il„.. c e-. S-02 $808 9 $588. 10 $22 7 $(io(i.88. ,30. 8 $1188. an true valuf. 6 .«l79(i 11 $450. 12 $15 Less $1199.88. ORADKIl AUITHMCTIC. [VII. 37 164 hot the pupiU see, in 8uph prc-bl-ms as 3, that th. answor givon i, the ,>r..HH>t gain, the prrsont worth of the imte g.v..n \-e.ng reBanU'il us the lutiial cost in cash. In finding the tme vah.e of the note (6) Maroh 1, the prineipal i, fonn,l, whieh, put at interest, ^vonl.l an.ount to *8lay 1 (4 mo.). Amount, Payment M:iy 1, New principal. Interest to Aug. 1 (.3 mo.), Amount, Payment Aug. 1, New principal. Interest to .Tan. 1, '94 (5 mo.) Amount due, .lam 1, '94, $100.00 2.00 102.00 40.00 62.00 .93 G2.y3 20M ~"42.y3 1.07 38 Answers.- 1 $254.44. 2 $112.81. 5 $132.20. 6 $258.43. 7 $164.36. $44.00 3 $255.30. 4 $88.24. VII. .-Mi] Til.' metli,,(I TEACIIKHh" MAXtAI,. ].W by lmM,„..s.s l„,u;,.s uT ''""'■ «'•""■•" I l.v l"ll"».,! i^n... s.t;:t;.i '::;^;:;tt;:;;:'''"' ;;''■;•-'' ,^^''"' states rul... ()„c. (Wf,,,^ „,■ •l"'iW"r.. ,„11,.,1 ,|,.. fnit,..! be taken to iiui'iiient tl„. . ,.i ■ i , iiite.rst must ,i„t toward .hseharK'inK tl.e ,,ri,„.i,,.l." T,.ll tl , , ?^'^ show ho,v it w,,uM,,e„p,,,,,,A,. 7 i/ :,;:,; ;-,;i>-^ and January payments l.ad ]«.en #1(. an,, i i „ 1 , ' "':"' same with other exereises. •'»"«"'■.», tiv,n. ,,„,,„. When i,artial jmyments are ,na,h. on a,..,„i„t. ..„ . runnu.g a short j,en.,a of time the i„t f ■ "" "'"" by the followin^rule : Fi,^ H:^ '! ;^,:";"':'"-';'"'>''"'" ti.ne it l.«i„s to draw interest to ti, .^ s e i:'"'. ' 'h''" amount subtract the sum of tl„. , .,,-, «' "»'"t. iTom H,is the tin.e of payment to t e < "' """™'^ '"'"" cHants-... The ..,h..i„;;:i::;:: 1^ -;:--;-;;;=-' "- Principal, on interest from April 1, '7t, 1.05; a yr. (o) 3%, 1.092727; 3"yr. (ii: r.%, 1.10702.-) ; 4 yr. («< 4%, 1.1098r.9 ; o yr. (a 3%, 1.159274; 5 yr. (^i, 5',;, 1.2702S2 ; (i yr. (<(, ::%, 1.194052; (i yr. @ 4%, 1.2().->319; 7 yr. (,j, Hrf^. 1.229874; 7 yr. (n, 5%. 1.4071; 8 yr. (nt 4%, l.;!()8.">09 ; 8 yr. (a^ .".%, 1.477455; 9 yr. (a; 3%, 1.304773 ; 10 yr. (n) 5%, 1.628895. 40 Jn.iin;:i.- 2 $14.->8. 3 $9(;0.4r>. 4 80.'i(> $C.74.1(1 .SIO! $007.48 $850.48. 5 $178('..20 $1200.45. 6 $2044.73. In the instance given in 1, tlie account stands as follows, if no interest is jiaid until maturity of note : Principal, .?100.00 Total simple interest (3 yr. (« 0%), 18.00 .Interest on 1st annual interest (2 yr.), $0.72 "2d " " (lyr.), 0.30 _^08 Amount due at the end of 3d year. .1^119.08 The amount liy compound iiitep'st for tlie same time and rate is $119.10, oidy 2 cents nuire tlian the amount by annual interest. In 4, by tlie process indicated, $074.10 would bi^ due Sept. 1, 1886. If the note were ]iaid at this time, $101 should be deducted from $674.16. The simiilest solution of this problem is to find the amourt due as if no payment had been made, and then subtract $100 pins the interesi of $100 fnmi .Tuly 1, 1886, to time of settle- ment. 5 s'lould be performed in the same way. Let the pupils 'mm\t^kMSi, VII. 41] TEACHBKS' MANUAL. 157 £■:: J tn:3T;^ ''- ;r"' ^^"*- -"""'- -■• "-■ method " «'''•" '''"'™ '» "l"''i"-' l-y tin. l.tter 3 $1(11) ■■.i\,-r. 41 ^«.™. ... X $9«oO. 2 1 n.o. 1 da. VJ v. « ,„o. 6 $.4o.8o. 7 3 y,.. 5 ,„o. 22 ,1a. 8 C,..sh o({,v -s'j-, . - , 13 $111.^.073 $1104.,o .$l"Jc ^■'-»'^'- "*-' 43 \ational Hanks, before i«.sui„,- any bills of tl.eir o^-n ...e wr-:n;=i" ::;;:;: t';i;;;:s'r;rr;: ^ :7f"^-::r:,;;::::r;::;:-;;-i vlu.bearhsu...kl,ol,le,.own.s. The books .n,l papers of ^^.t , ' Ua..ks arepenodieallyins, te.I I,- b„„k ex h.e s ,,, o^i J SL!;::r";:r- ■'■": ^"'■■'' " "- '-- •-'"- '>!-"!. pre.,,, , nt, .,,sl„er. and. ,t need-d, otl„.r o(ti,.iuls, are s« sed to atteml to the general ,n.„a.^e,ne„t of the bank. -.™,,Hn.n..tes,.n:;i;:,:;;':;:;,i:::,:;:;;:;:::-^^^^ n.ent bond» and other seenrUie.s, .,„, i, ,„„,„„ „, ,„.,;:- .elbn, property. Th. .xpea^e. are the tuxes „ i.as ,„ ;:;iZ 158 GRAllIl) AKITHMKTIO. [VII. 44 general government and to the state and loeal governni'-nts, and the running expenses, such as rents, sahiries, eti'. Tlie above farts, and otliers which may lie gained from bank authorities, shf the note forth.' time before maturity. The/r(ce of tte note is the sum of money written in the body of the note. VII. 4BJ teachers' MANI'AL. 159 Themafunty of the note is tl.e tin.e at wl.ioh the note is ,l„e Hank l>..count is the allowance „,a.le t„ a bunk lor the pavn.e ,' ot a note before it becomes due. ■ The ,«„X.,.of the note is the person who si^ns his name to the note. The,.,,,e. ,s the person to whon, payn.ent of ,„onev ,s pronuse.l the note ' " ' """"' "'''" """■' ''''^ '"""^' "" ''"« '"''k "f due. If the last day of grace falls on Snndav or on a legal ' .oli- Jay the note nn>st be paid on the dav previon; I.et pupilt i, days ot grace „, States where no grace is allowed bv law I'ract.ce varies in ..stnnating th.. time of maturiu- and period of discount o notes. Some banks in estin.ating the t,n,e of maturitv of a note dne ,n DO days regard tl,.. tin.e as tlnve calendar -no.d ' but nuire commonly such time is taken in exa.^t .lavs In , ,sconnt' ...K notes son,e banks always reckon the time before maturity in' exact number of days, others in months and davs. and still ot , ' .n e..aet nun.ber of days if the un..xpired tin.e ,s (i(» days or less and .n months and days if for more than 00 davs. Tl„. problen.s of tlMS sectnn. are reckoned on the basis of n.onths if the tin.e ffu-en >n the note is n.„nths, an,l of exact nund.er of ,lavs if the nnc g,ven .s .n days. For exan.ple. the n.atnrity of a note ,Iat d duly 15, due ,n 3 n.onths, is Oct. U + , days of grace, or . let. IS d". <>c . .i + ,3 d.ays, or Get. l(i. If the first of these notes is lisconn ed at date of note or any time snl,se,p,..ntlv. the tin,e of d.s..ount ,« reckoned as months and days. The di'seonnt of the second note is reckoned in exa.'t nnmb,.r of days 45 .■/«.,«-,. 1 ,f 4!i;i.,S7. 2$-m.7(>. 3 .S-.IM 7.-, 4.«7.><7,;o *(..'(i.-U .».,li'..,;i. 13 .S4,-.i;.KS ,$47..-, 17 - t?.:- or™ "■^*""' *'-""■ 2 *"•'"■ 3«1"""- 4 $.110 4.! 5 .'51,o.,.n9. 6 «.-.•!]..-.(. ,f;j«.l(i. 7 .*,i„,v I,, »691.S0. 9 #G07.S5 «7. 85. 10 $JOOU .*il'j7'J. 11 8 SS.L'o 160 GRADED ARITHMETIC. [VII. 47 .'S1253.05. 12 «4S2 $.'544.«2. 13 SiioO SaitT.r.'J.V 14 »r..r.(Ml $420.79. 15 $ir).!)7. The pxiilaiiiitum of such imilih^ms :is 1 may h-' : -If a note lr, as ,«i.il.S'.».-, is ,.ontaine,l •■imes in SILS.',)."), which is 1(10. Aiismr: .S1'"l." 47 The deposit slip, with the bank hook, is passed to the teller, who writes in the bank book the amoant of lU'posit. Tlie elieck by which .'S.SO is drawn out is signed by tlie ix'rson who wants the n'loney, and is made jiayable to himself or t.. his ord.T. The check given to James Smith may pass through varicms hands until finally it comes to the Kmporia Hank. It would be well for tlie teacher at this point to explain the bank check and to show its uselulness to busim^ss men and others. To do this it will be n 'Ssary to show tli.^ necessity of lirst making a deposit oi money at the bank, and to lell exactly how it may be used in paying' persons wlio live at a distance as well as persons near the bank upon which the clieck is drawn. .Vn example may be given of James Itobinson living in Albany, and wishing to pay a <'reditor in Omaha. If he has a deposit in a bank in Albany, he has only to till out a blank similar to die blank check represented on this jiage and .send it to the .■re.litor in Omalia, who carries it to a bank there for payment. If he i- known, he will receive the face value of the check ; if he is m.t known, he must be "Kb'ntiiicd" by some one iieicuc he I'an get tlie check "cashed." The banks all have d.^alings with one another, iiid therefore tlie check is taken by the Dmalia bai.k and retunid to Albany. It llr. Robinson has no de|».Mt in a bank, lie may make a special deposit of a sum e.pial in amount to tb- lace value of the check which he wishes t.i scud t<. Omaha, nr he may for a small amount buy a "cashier's check" for the desired ;...i.,unt, ])avable to the the former. Mr. Kobiusou writes. "Pay to the order of .Inhn Hmwn," and signs his name. Find out from bankers the exa.-l working of a '■Cnearing House." and explain the ]iroerss to the pupib 4S Ansivcrs; 1 i>ti(i2.4T. 2 $sOJ.17. VII. 4»-no] TKACllKIiS' MAMM,. Kil ConporativP l«nks l.nvo I,,.,.,, vctv ..xtonsivolv .-.staLlislu.,!, an,l are likoly tn 1„. ,„„,,. ,.„„„iion i„ lutiir,.. K,,,- tliis n-a-s,,,,, an.l b.-cau.s.. It will l„. „s,.tul r,„. |„.„,,!,. t„ know th.M,- |„u-|K,s,. ai„l i,».tho,ls „f <>|"Mati„n. tl.t.y shouM 1„. ,|„. s„l,j,...t „f stmly in s,.l,„„l TLdr nature an,l .„,.tl„Ml« vary s„ ,„,„■!, ,„ ,|iir,.,.,,,t ,...,ts „f tl„. oountrv *'"'' '♦■ ^'■■"' ' " tli"u-l.t n„t l»..st t.. ■ntn«lur,. i,, ,!„. t,.xt-l,o„k iix.re tl,a„ a statement ol tl„.ir t;,.,,,.,,,! |.,,,,„,,., ,j ^,,,„,,, |,,. ^^,^.„ to Kot from tho offi,...,.s „f tl„. n.an.st ,.„„|„.n,tiv,. l,a„k it. m,.tl,o,l "t ni„M-atmM ami jji™ tn tl,,- |,„|,ils ],rol,l,.,„« «-l,i,.l, are likHv tn ™nio tu patrons „t tl„. l,;,nl<, l,„tl, as l.Mnl..r ami as borrower IVs,.,-.!,,. (ally its a,hantaf„.s, :„,.! nniko the ,,rol,lems as praetieal as [lossible. 4!»-.V* In na.nin,- fl.e .iiffeivnt kinds of e,n-porations, only tl..,s,. th.r ,,n. best known need be .iven, sueli as, banks, railroa.ls and lai-Ke niannlae„.ri,.s. It w.,„ld be well wliile discnissi,,.. these snbjeets t,. sh„wlhe jmpils the various ,loeume„ts that are „sed s.ieh as, eharter. bon.l with eonpons. and boml eertilieate ; also "■rtain blanks whieh banks and other luisini.ss houses have to till "..t Lead tin. pnpils to see .dearly th,. .lilferenee between bon.ls and sl„,d<.. b„n,lh,dders and stoekholders, interest and dividemi Answers to the most ditHeult qnestions given on tln-se pa.-es. and some others that ou,d,t to l,e ans«ered.,„e e„ntained in the fol- Iowiiil; brief statement : A ,•„,/„„.„//„„ i, .,„ ,,,gani,„H.,„ ,,|,i,|, |,:,s a rharter to do liusmess under th,. laws ol' ,he state. Its ,.„,„V„/ is divnled into "''.'■"■'■^ "" ' ''^- ""■ ■-''"•/■■/'"/'/,.,■„.. eaeh or whom holds a -o.^»„/,. ■.J ^''■^'^^^■"-'^ .-). lo' ■Si.'oio:!.7.-.. 11 .si()oo:i,i;!. 12 .S:ioc.ii..')(i $l.-.:!l7.ri(i. 13 7',; bonds 14 ISoncls abnuti'/f bitter. 18 $1920. ,>;{ KNpliiHi til tlie pujiils the great conveni. ncc -f the use ol draits or checks, and lioH drafts are exchanged in different jiarts of tbr cBttt!' , Sliuw th:.l liusiness tiri»r> and private imuviduais in on-- fart ■■( tl mv; -we iiartiei. in aiiiitl«rr ]iart. and tiat by i system i.f exclidiige tlie^.^ il«-bt* i&iv I"', paiil without s.-n,lmg the Monev Illustrate tlii« l-v ithe -«jniple indicatisi n, ts«i> l«ige. M. « !..-„ _•__ u=_ ... i-sa=^=an owes Jislu! SiaitU ol KyatJJtt >.2S9;. 6 4 811. 7 8. 8 Stock J ;;% better. 9 (iOO sliaros. H'/'c -iV/a ■"'i'/o- 12 Stock 5% better VII. a4] teachers' manual. 1«3 »4S0.90. He Roes to Mr. Uph.a.n, a Lankor in Chicago, and l,„v, of h.m an or.ler, eallea a draft, on Tl.os. S. Appleton & Co., banli.Ts of Host.m, who have an account with Mr. Unl.ani Mr Ma,.un.l,..r writes on the back of the .haft, •■ I'ay to the or.ler of J..hn Sni.th, • an.l sisj„s hi., nan.e. This he sends to Mr. Smith who .fraws the n, y f.nni .\,,ph.t..n & Co. .at the given time. If he dra t ha.l r,.ad t.-n days affr sight, it sh.udd first he present.-.! .. Applet,,,. .S. C„. l„r a....cpta...e, which is done by their writing the word •■a,-..,.pte.i 'an.l signing the firn.'s name aeros.s the face "i the note, IVn .lays. + ;; ,l^ys „f g,,,,,, ^ft„ acceptance the .-mcy ,s paya de. Th,s ,s ..alle.l a ..time draft," to .listinguish It Ironi a •. sight ilraft." whh-h is p:,val,le '• at sight " If in two .iilferent parts .,f th,. ,„,„.try, as, for example, Hoston an,l ,San l-ranos,-.,, th.. a,u.,unt .,f ,lr..lts .Irawn in ea.^h citv on h.. other .s nearly ..q„al. th,. huyr .,f a .Iraft in ..itl,er ..itv'wiil have to pay l.ttl,. or n„thi,„ l„,v 1 the face vah.,.. liut if the am.,unt of drafts in K,sr„n „„ .San Fra«..isco is far in ex.-ess .,f the amount drawn in .Sum Fraiu-is..., on H„„ton, then the buver of a .Iratt ,n Boston will have to pay a premium t., pay f„r th,. risk and exp,.n.s,. .,f semling ,„o,„.y to Sau Fram-isco, an,l the buver of a dr., It in San Kran..i.s..., may be abl,. to g.-t it at a .ILscount. " r.4 .l«.w,.,v..s-.- 1 .«:i„i.-, s;i,„, ,,-,,). 2 $U(u,.r^>. 3 ,«;,joo 4 .«.s:u.r..). 5 $.-;ii..i2. 4 i'.'iliZ 14.«. f)!/. Show the jiupils that essentially the same princijile is involved in foreign excliange that has already been explained in domestic exchange. The intrinsic value of the standard coins of two countries determines tlie par of exchange between those countries. Exchange is at a premium or discount when the market value is more or less than the par value. Three bills of the " same tenor and date " are sent by different mails, so that if one is lost the other may be jiresentcd. In tile bill of excliange given, Mr. Mason is the biit/er or remiftrr who iiKhirsps the bill |iayable to tlie order of Mr. ISrown. As in domestic excliange, foreign bills or drafts are made jiajable botli on time and at sight. Time ilrafts are usually ijuotcd at a less jiriee tlian sight drafts. 50 Aiisim's; 1 t'.->87 ll.v. 4+1207..j5. 8 «>72.i.S4. 9 *ti79.72. 10 40G.30 $402.21. i-iy«>K '•'K'il' VM. nn] TEACIIEK8' MANUAI,. 58 105 Conn. It.ver, »1..7.r(i;{. (2fi) "OMp. ,*■'*"' ''-^- ^-■») ^ ,lj!^f—l"^"-e March 31. W9..7. 2 l..,.,.. Xtl.ron.n-e,l, tlu.t,vosi,l,.sof tl,o..asl. ■iccoimt may ho ,,I,„.,.,I on Ti,„ f •••—1..)-. 3 Halancf, «<)l>4 j.-j I- 1'. WALKER. til. ;(lni\f. soiial : 7'tt Uttlfniff^ 1 no 17 ' fIJ 1 /: 00 *:j ^.WMBPft:^^mW '^ . r^ ■'^ ^ r M- 166 QBAJMCl' A '"■''"*"'■"'"• [VII. ea :.r,(;im... 3 '■';'"";■:'.''''• o T,,;, r.-oo m... H ...0. 6 ;< ...... .'i .la. 7 •>.; . 1 • " ; ■ .,,. ,,av...-nt» i» tl.o ti....> KxpUi.. to the l,..i..l.s 1 , *' ;\ '^^^.......t ti,.„, ,„ay be i.ui.l .,t wUicl. s.>ve.al it.Mi.s of 'l.'l't <"«• •'' without Kui.. or h,s« to ..ith...- l-avty. A tV.r... of writt.... aolutio.i 1...- 6 .s as foUovvs . 2iuo. X 4i;i>= S4()mo. 3..10. X 31.-.= OIJ.""- 4..,o.Xj^=21!£j""- 12G0 ):iH8r. (:i n.o- 105 «o,hv. xin-._.lino,i^=:2ida. ^7i»M.er.' 3 mo. 3 da. ,,H„au.«th..;..;;t..t.......he„.H.^^^^^^^^^^ a ,lay is o"- ''^'H' ">■ ■"'"'■• '"'■'"'" ' ' regar.1 it. ^^^^.,„^ ^,.„, m at the ti....- In 9, as.sm..i..;,' that th t«.. >"" ,, ^ t,„, „«« of the first su„. was d..o, tl.. --"If ;^ ^ th is ..,,.ivale..t to th.. hiss of the ..se ol .. i „,,^„„ta,.t,..er ....t t.. lose any- $:.,.00 9 .lays. '■-•-" 7;;';,^\,"'C ,, aavs aft... .U.. 20, or thill", th.. .*iL!000 sho.ihl 1).' 1^'M't ".^ until .I.....' •-'!'• s..pt. LT,. 3 K<.h. 11. Wh,.„ ea..h ..f tw., 1-;^;-^.; j" ;\, ,,y <,iputi,.g the i„. t,.„.st. l.i 4. .t .,..;,. t 1 .. . ^ ,,,;,. .i-l,e exi.hiiiat..... may ,,.,,„„„t ..f .■a.'h i.ui.il Nv.th . . h. ^ """""/^ ^ .^,,, „i,..„ in the „., a,-awn f,-o... th,. ,,u,.ils ''^ -■'' '-^t H, i:,;!;.,.. pn.bh.m exev..ise. Aft....war.ls the pupils i..av be h .1 to n us foU.iws : „,.,,, I i„u. 1 and $50, „ ,„ this a..ou,it Spauhb,.B -^ ■"^*''" ; ''\V"'l ,,,,. to find due July 2V. I owe Spauldmg *100, .u.,: July U. VII. «4] TKACHEKS' MANirAL. 167 the tunn when he may pay tl„. Lalatwo («r.(), duo me without Rair, or l.«s to either of „,. We will assume July -1 us the time of payment of all the sums. If 1„. pays „„. the «10() ,lue July I o„ the I'lstnf July, I shall los.. the use of ^l.H, from Julv f to rZ u\'" ■■"■ ''"t", '"""' '' "' *""• '■'"■ -■" ''"y ••" '*■'% '" •i-i*^. It h. pays me July L'l .ji.-.o, ^-liieh is due July I'l I slnll neither ,a,u nor lose. If I pay l,i„. j„,y ., ^i„K whi.-h i.s .I'ue «1 H) or 1.. days ,,, l.i^. If payment of the halanee due i.s made .I..1V 21. my net los.s is KIJo. That I may not gain or lo.se, he must pay me the «o(. as many days hefore July I'l as it takes $50 to KaiM l.,i<, or -() day.s. I'O days before July -1 is J.K 1 when the liahmce shouhl be ]iaid." SpanldiuK o^yes me : I owe Spauldin« : Assumed time of «10() due July 1. 9Um due July II. p,v„„.nt, ..uly ''1 ull '• July 21. ■ • " Loss, int. of $100-0 da. =..•(.■!*, ,ai„, int. of $U,i, lOda ^ UM Net loss, .!(!!). ■ "' Interest of.«l.lo J da. - .oos^. ' .„;^ , ,„,^^ _ ^O. July I — :.'0 il.i. =July I. Ky introduoinK the l.st question a n.-v problem is made. Assume that the last lull of goods. lik« the oth..rs. i.s due i„ 1 m„. U-t Aug. 1 be the a.ssunuMl tin,,- of payment. The written solution by a good method may b(. as follows : 1>||«' .Xiiniiiiit ] Tfiiu- ,/li/i/ 1 ' ^floo ' .It .1,,. JiiliJ Sil , 51) 11 ,l„, A III/. 1 100 ; $lou\ lllt.Tt'Ht lost .oui I .1107 il ..IS I .\iiioinit Time Iiitcnut I unliMtl /"/'/ 11 , a cr ».• 10_ - ^^ ^^ ^.^^ ^^ ^ ,„ia Au,. 1, 1 l-« tbe ."tevest ot »' 'J ",, ^,; interest of $100 ;,,,, July ^1. 5^^';,r ;:v :; .t ^e'ie Au. 1, n.y .eUoss is 21 .la., or *.A.. K all I">"' ^ ,,, ,Uoul,l [.ay the bal- $.257. That 1 may noitl.er ^.un > or lo ^ ^^^^_ __^__^^^^^_^. ^^^ ;„ee (§inO) due me as n.auy -lays^l- fou. W ^^^^ .^^^^^^^^ aavs it .-ill take for $100 to ga," *.-•>. • /^^^.^^.J.,,, a„,. l is ot"$15l)forl. lay the quotient IS 10. July lil, t'-,7"'^*";V"":;;..,„in. accounts may be solved best by Many problems m ''-^^S'-V, „„,,,,. .,f days or mouths, solved as follows : SpauhUng owes me : i owe Spanlding : $100 due July 1. $100 due July 11. 50 " July 21. If settlement is made July -l, „ I gain the use o£ I lose the use of ^^ ^ j,^^ „,« „{ $1 $100 for 20 da. = the use of $1 *100 ^^^^ ^^^ 2000 da. IJetlossofuseof $1 for 1000 da. „ u u " " $60 " 20 da. July 21 — 20 da. = July 1. 9 Tipc 3 3 Balance, $260, aue March 31. 4 J-. ., , 80 ^^^^^^^ /'-^ •;?Srdr:::S"latest may be taken as the — -'■"-^:r':fr"r:alancedueJune22,lS01. IVilauce due, $11.74 ^,^^^^^.^^ ^^ ^ ,.,,^ *''' ^ )'"'TS 1 ':lt:: i ^eSllue U «„« tU^ Uas the in the surface of stiU water. VII. 08] teachers' manual. 169 direction of a plumb line. These lines need bo drawn only approxi- mately in the li-ht diiTcticni. Taiallel lines may be drawn by the aid of a square or ruler, care being taken to draw them so tliat they shall h: tlie same diLitanee apart tlirougliout their lengtli Other ways of drawing i.arallel lines will be shown later. A line may be .livided int,> 2 equal parts by means of a measure. Aiiotlier w.ay is to divide it with the aid of compasses, as follows : 'With ^ as a '•'. ' , •'■' centre, draw the arc ^^), and with II as a /j\ centre, dra«- tlie arc EF, cutting the arc ,/ j \ CD. TJnitu the points of intersection of / j \ the two arcs, and the dividing line divi.les "^ 1 — fM the line AB at G into 2 equal parts. ICach \ j / of the two parts can be divided in the same \ j .■■'' way into 2 equal parts, thus dividing 'j/ the original line into 4 equal parts. ■" ''^ 68 Jlost of these exercises are a review of previous work 69 The conclusion in 1, after several similar experiments, is, that the sum of two sides of a triangle is greater than the third side. The two ways of finding by exjieriment the .sum of tlie angles of a triangle are: (1) Cutting off the corners of a paper triangle and measuring, and (2) Measuring by means of a protractor. Several conclusions may be made from the measurements made m 17; such as : The two diagonals of a parallelogram divide each otlier into two equal parts. In a square the diagonals are equal. In a square the diagonals are perpendicular to each other. In a rliombus tlie diagonals are iierpendieular to each other. 70 Aimers: 1 $80. 2 129600 sq. ft. 3 J 25 A. sq. rd. l'J6G lb, 6 -3IJ A. 9 2244 sq. ft. 10 21f,(»o's'.i. ft. 5 ^Ul^ 11 1 T. 12 176.50+ ft. 13 85J rd. 14 19 planks $G.O,S. In 9, t!,e walk is supposed to be on the border of the garden. 71 Anstverg.- 1 2;{i yd. 2 $3.30. 3 20^ sq. ft. 4 869 .^ sq. ft. S -Hr,r>.3+ sq. ft. $116.24. 6 «46.78 «3.92 (full allow- ance for openings) $24.75 460^=,, sq. ft. BllSjJsq.rd. 9140ft 170 GRADED ARITHMETIC. [Ml. 72 For a good practical method of finding the number of ro.ls of paper required for a room, «ee page 119 of the Manual. ,a Answer.: 1 20000 sq. ft. $.'U43.53. 6 2 A. 4 sq. rd. 7 1 A. 126.1+ sq. rd. 11 43.5.0 ft. The land taken by the railroad company in 1 is a parallelogram whoL i::: i7J whose amtude is equal to the d.stance U For solution of 2, see Book IV., page 8o The dotted line called for in 8 is perpendicular to the base since the llttuTe is the perpendicular distance from the vertex to the base Let the statements be as concise as possible. sq.ft. 7 900 sq.ft. 8 920sq.rd. The line required to be drawn 60 ft. long iu 7 is supposed to be horizontal. , _ o, t^ •* 92400 sn. ft. 46200 sq. ft. sq.ft. 7 1450 sq.ft. . 19 Bnnk VI will suggest a solution tor 1. rryrSS— u^^- aftei^ansformmg a^t.p.oid into an equivalent rectangle ; also from the formula ^ — y- X A. - ,. tt o 19600 sr ft 11 36.34 so. ft. .;^.^^^i^^?r^t.^^s:-.- 17 9..S.A. "s !give helps to the solution of these exercises will be found on pages 14-18 of Book VI. ^^^.^^^ Use protractor or compasses in marKing on off equal arcs of the desired length. VII. 70] TEACHEIis' MANUAL. 171 U-t tl„. i,u,,ils ,livi,l.. i«,,,,.r ,.ir,.l,.s i.ito triMngl.s l„.foro ,l,awi„.. It thoy liiivr nut ulrcaily ,I„m. so (s,.,. jiaj,-,. l,s, Ji„„k VJ.). ft. '.»4.'.4,S .s-i. ft. 4 .?.5(M»;{. 5 4!.!' A. M.-.J+ .s.j. „1. In 1 it i. .suppos,.,! tl,at the row is t,.tl„.,-e,l in su.^h a way as to en.abl,. Lor to g.aze at a .listanoe ..xactly forty f.vt fvon> tl„. j,ost. t-omewliat sunilar work witli ,,risn,s is .alU.,! for on u-.L 19 Book ^ I For suggestions iu teaHnng tl.ese forn.s, .soe Mruu.al,' page 1-0. Ihe following stat,.nient.s n,ay be drawn from the pupils : The lateral faces of all right pricnis are in (],e form of reetandes Ihe l,a.ses of a triangular jirisui are triangles. Tin. bases of ,nn,l ■■angular prisms are cpunlrilaterals j ete, X j..„,,„,„„„i .„.;,,„ is a pnsm having pentagons for its bases. A ./,//,/ jln.,,, is a prism whose latc.ral e,lges are peri,..n,licular to the bases. An n.sm. IS a prism whoso lateral edges are inclined to the ba^es The -Irawing on page 64, Book V., will suggest the kind of work called for m 8. 77 Anxw,,:.- 1 245 eu. ft. 2 .-J en. in. 3 57?" bbl 8 2().5«32 gal. 9 8 ft. .i.l241- in. ... ft. 0.43, in. 10 l!,'- 5o+ . Sep ft. "' ' The altitude of a x'lirmnid is tlie perpendicular distance from the vertex to the base. The .hn,t M.jU „f „ ,-1;T- 9 lH--' sq. ft. Paper will have to be pasted over the joined edf,'es of these m>,lels, that th.'y may hold the sand. Let the ]H.pil measnve or „-ei"h the sand as aceurately as possible. The ndes ^vhn■ll they slKHild discover are as follows : Tlie volnnie of any pyranud is equal to one third ot the volume of a prism having the same base and altitude. The volume of a cone is ecpial to one tliird the volume of a cylinder having the same base and altitude. Iron, these rules may be derived the following: The volume of a pyra- mid or of a cone is e.iual to i of the area ot the base X the altitude. The convex surface of a pyramid is the sum of the lateral ^aces The convex surface of a con- is its curved surface. Lead the pupils to give the rule for finding the area of the convex surface of pyramids and cones. Tlie lule is: Multiply the per- iraeter of the base of the i.yramid or cone by one halt tlie slant height. 79 Ans>re,-,: i a 414.6912 sq. ft.; h 1910.0928 sq. ft.; c 13,Sl.;i2+ sq. ft. 5 a 175.92 cu. in.; /- 2,SG5.Ki cu. ft. 8 l-''-^-<^^J'l; "'; 2827.44 sq. in. 5 sq. ft. 13..S+ sq. in. 530.93+ sq. in. 9 1.«.08 )9 cu. in. 4.1888 cu. ft. 904.7808 cu. in. 10 $1884.90. 11 oU bullets 110592 bullets. The frustum of a pi, ram id is the portion of a pyramid included between the base and a plane made by cutting tlie pyramid tlirougli the lateral faces, parallel to the base. The slant height of the frustum of a pyramid or of a cone is the perpendicular distance between the sides of the upper and lower bases. The total surface of a frustum is found by adding the surface of the bases to the lateral, surface. The lateral surface is found by multiplying half the sum of the perimeters of the bases by the slant height. VU. 70] TEACHEKS' MANUAL. 173 Tl,c volnmn of tl,o frustun, of a ,,yr,„„i,l or rou. is o,,ual to tl,o .lifferen.v l,et»-,.,.n tl.e original pyran.i.l or ,•„„,. an,l M,.. |,vr„„i,i or cono cut olf If tl.e lu.igl.t of tl„. original , ,s n,!t\.iv..n, tl.e volume ot tl... fru.stnni is fonn.l l,y tl,.. tollou in,- nil. ■ i, tl.J sum of tlu. s.juan.s of tl.e .liana.t,.rs of tl,e t«-o k.s.s ud,l tl,,. ,.,-o,l. net of the t»-o .liann.t,.rs. MultiiJy this snn. hy the ,,ro.ln!.t of i of the heiglit anil ."Sol. With an orange or woo.len hall t,.aoh t'.,. following iletinitions ■ A sy,/„..„ ,s a solid boun.le.l by a .-urn.a sarfaee/evrv ,,oint of «-h.oh 13 equally .listant from a point within .-all,.,! tho c'ntre Iho ,l.n„.f.-r of „ ,jj,,,, i, ,, ,„,,iy|,t ]i„, ,„,,i„,, t,„.„ ,,, „„. eentre anil having its extremities in the .suHace. A grn,t .-inle of a sphere is u section ma,Ie l.y entting tl.e spher,. through the centre. ' A goo,l method of finding the surface of a spher,. is co compare It Avith the surfa,-e of a right cylinder whose height and diameter of base are exactly equ.al to t'le dianieter of sphere. ]!y wra„„in.. the sphere and cylinder with narrow waxcl tape, and after u,," wrapping them, comparing the amounts of tape, it will be observed tha the areas of the surface of the sphere and the lateral surface of the cylinder are alike. Since tlie lateral .surlaee of a cylinder is found by multiplying the circumference of the base by the altitude 1 will be seen that the surface of a sphere is found by multiplying the circumference by the diameter. To find the volume of a splicre, cmparc the contents of the sphere with the cmtents of the envch.ping cylinder. This ma, be done by making a ball which will exactly lit a cvlinder. Fiil the cylinder with water and immerse the ball. Comp^r,. the depth of water before the ball was immersed with the dcj.tl, of water after the ball is taken out, and it will be found that two thirds of the water has been displaced. Therefore, the volume of a sphere IS equal to two thirds of the volume of a cylimhT whose .li.meter of base and height are equal to the diameter of sphere. From this feet may be derived the rule : Multiply th,. cube of the diameter by .o.36i or, by conceiving a series of pyramids formed from the 174 OltAI>EI) AltlTll.MKTIC. [VII. 80 spliiTP, having tlioir vpiiox at the cciitiT, it will hi" soon that thii viiluiiio ot tho pvraniiils iipcsinf,' the sphor.' must ociual the siirlaco of their l)ases "nmltiiilied by ^ of their radius ; henee the rule ■. To tinil the volume of a sphci-o, multiply the surfaee hy i of tlie radius. HO Ans,rn:i : 3 'JO' (J0° .W. 5 00,1 mi. 6 ll-'lMd.S mi. 7 C'J8.1 mi. L';i01'.4 mi. 107(;.->.8 mi. L'l.Wl.C) mi. 8 About (lUJ mi. 9 47° SL'odJ mi. 10 4.'i° 1".»74J mi. If necessary, review previous work in uireles and degrees, pages 14, l.">, Book VI. 81 Aiifirei's: 2 240' 14400". 3 040'. 4 l.-.OOO". 5 40.V 40.V 10' Oiid'. 6 4'.'00" ].-)(),S0" SSS'Jn" «J0O" 10". 7 10' 12" 2" 7' 30". 8 4' fV 40" 3' 12" 31)' 21" 9 jVo" ^o" I'jo"- 10 .10.^° .oiii '. 11 .7.-.,'', .oir. 12 10° -.0' 4c". 13 .'Hi'- 31' .30". 14 9° 34' 44". 19 23° 10'. 20 12° 20'. 21 Sy ,-).-.' 22". 82 .-1h.w«-.s; 1 C°. 2 113 da. 3 735;< h. 4 2° ,31'. 5 7' .Wji;?,". 9 300° 15° l.">'. 10.3° 4r)°. 13 4(»min. 26 rain. 1 h. 41 min. 20 sec. 14 Slower 4 iiiin. 15 ll.O.'i 1"..m. l.r)7 i-.m. 16 132° 10' E. 83 Jimrers: 2 N.Y., 7 h. 3 min. .58 sec. a.m. Chi., C h. 9 min. 28+ sec. A.M. N.O., Jl h. 59 min. 40+ see. a.m. London, 11 li. 59 min. 37 sec. a.m. Pafis, 12 h. 9 min. 20+ see. r.M. Boston, 7 h. 15 min. 40 see. a.m. Wash., C h. 51 min. 57 sec. A..M. Eoine, 12 h. 49 min. 48+ tec. e..v:. Berlin, 12 h. .53 min. 35+ sec. p.m. San F., 3 h. .50 min. 20+ sec. a.m. Cal., 5 h. 53 min. 20 sec. p.m. St. L., 5 h. 58J min. a.m. 3 Rome, 9 h. 57 min. 50+ sec. p.m. Paris, 9 h. 17 min. 22+ sec. p.m. Cal., 3 h. 1 rain. 22 sec. a.m. 5 15°. 6 36° 15'. 7 5° East. 8 9 h. 3 min. 8^\ sec. 9 5 h. .50 min. 85 sec. 10 20°. 11 93° 20' West. 12 1.3° 22' 6". 84 Aiisim-s: 2 Ind., 12 o'clock. Pitts., 1 o'clock p.m. Denver, 11 o'clock A.M. San. P., 10 o'clock a.m. X. 0., 12 o'clock, liosto 1 o'clock P.M. Umahu, 12 o'clock. 3 Sau Jose, 1 o'clock p.j VII. 8S] TEACHKIIS' MANUAL. 175 M..I0 ■„„....>,. Sa,..,7 1,. l-i„nn...M. 5 Ci,,., - ,„i„. Ill .,v. \. . 11 m,„ 'lO s,.,.. St. |-„ !•' „un. l'(. .s,.,.. 7 1 m,.,. l.i srr. past 4 :.'8 nun. I'O sec. past I. 85 Tl,„ ....lati,,,, „,■ rati,, „f C, l,l„,ks tu 2 l.lo.-ks is f,mn,l l,v 6 l,l,„.k., . > l,lo,.k,s. Tl,.. a„t,.,.,.,l,.„t of this ratio is <; 1, ks, a,„l the eonsciiufiit is :.' hlorks. l..a.lthe ,,u,,ils to make tho foUowi,,,, statements from exa.„|,Ies caiiecl tor in 11 : ' If 1.0th terms of a ratio are multiplie.l In tl,e same numher. tl,e terms are lai-fror, Imt the ratio irmains unehan If both terms of a ratio are aivi,le,l 1,v the same numher, the term.s are smaller, but the ratio remains un,.han.'e,| 86 A proportion is the e,,ualitv of ratios. The first an,l fourth terms of a proportion are eall,.,l the e.vtremes. The seeoml an,l ^h ird erms are ealle,l the means. Krom ob.servation the p„pi,s vull diseover and .,tate the eoiielusion that the prodnet of the fourth term by the ,„,ml,er of units in tlie first term equals tie product of the third term by the number of units in tli seeom, term; -r more briefly, the prodnet of the e.xtremes emials the prod^t of the means From this faet it will appear t,.at 'the pro! uc o the extremes divided by a mean e,p,als ,he other mVai, , and that the product of the means divided by an e.xtivn.e emials the other extreme. ' 87 J«s,re,:«: i „ $8 ; b $90 ; $.2:, ■ d $ r/n- ,. 10 „, . Vf^'' "f ^' ''*^'"'' ««•!•''; /«-'0<»; ySlb.; Aoex.; ;«,,;: J$M). 6 11.'/. 7.W.8O. 8. §.80. gydoz. 10 .§..^1. The last five problems on this page ami all the problems on the five follo«-ing pages are intended to be performed bv analvsis and by proportion. The solution by analysis should be either oral or on a line. The solution by proportion may be ,, follow, ,'6 liie ratio of 2 apples to C apples uuist be the 'same as the cos of 170 ORAOED AKITHMETIC. [VII. H» 2 apples to tlip rost of 6 iippl<'s. It is ii.(|iiiroi! tn lii.d tlic cist of (1 appli'S 1 thoiTloro we m;ikc 4 (••■iits. tlir cost of L' iqipl's, tlif ;{,l tiTiu. Siucf tlif KiciitiT tlif (piiuitit thr Kivat.T til.; cost, w.,' make (i tlic M term ami 1' tl.f 1st Iciiii. Tlic pL-opoitiou is 2 : ri = .|^ . r. Miiltiplyiiin tlic t Uy (1 and divi.liii;,' tlic proiluct by 5, wc liave for the fourtli term U'. Aiisin;; VJf. 88 .Im-wc™.- 1 20 apples. 2 *«'•. 3 2." Im. 4 $14.2.-,. 5 cm mi 6 i!!22.40. 7 Sirjr.. 8 Mr,,^ .\. 9 IISS Im. 10 4oOnl. 1124(H.T. 12 .Viiwk. 13 41, da. 14 *l;52:i(.. IS 2(1 ft. 16 1113- 17 480 men. 18 2'.l,i| mi. 19 10;i,'f ft. Ti.u solution on a line may be as foUo'.vs (_13;: » da. X 12 HO 5 ^ 4J da. If 12 men mow the meadow in 8 day^. it will take 1 man 12 times 1-: many davs to mow it as it takes 12 men. and 20 men will mow it in one twentieth as many days as it takes : man. Mnltiplying and dividiug, we have an answer of '.} days. 89 Jnxwerx: 1 4(il,'., mi. 2 $2;i4(;i. 3 1.'!; yd. 4 12j^. 5 «;llor,.!l2. 6 tmooo times. 7 277 da. 18 li. 4't mm. 8 2..8000 men 9 101? rolls. 10 IH:} lb. 11 11"'-' mi- K'4 l'- 12 $(177.00+. 13 9.S20O0. 14 444^ lb. 15 1 lb. 11+ u/. «)0 AiiKinTs: 1 4< and 8/. 2 ."O vr. and 10 yr. 3 ,^„. 4 A .«;8; ]!. $10. 5 A,.'i?12; P., $10 : C, $8. 6 11/, 22^, M-t^. 7 ,i!iis(i| «280, $.-!7:ii. 8 A. $100S ; 1!, $-.04 ; C. $108. 9 12, 8 4 10 ."15 and .-.2.-.. 11 18 rd. and 24 vd. 12 Each danghter, $8000 ; each son, $0000. 13 If. rolls. 14 oMSOl,;', U'- 15 Og A. 16 43r,2 lb. 17 123 lb. In such problems as 6, the whole number will be the sum of the parts 1, 2, .S, and may be re,>rcsented by fi. 'I'lie proportions will be, 6 : 1 = (50 : 11 ; : 2 = f.G ; 22 ;^G ; ;! = 0(5 : 33. By analysis the solution is, 06 = I ; 5 = 11; 1 = 22; §—33. VII. 1»1] TKAI.HKUS .MAMAI,. 177 n..::nC:::;:v """"■' ■■" — --'■'^ In 7 lull, ,„;,k„ „„, , |i,j„„^ ,_,^.,,^^.^^_ 1^ , ,,,,,„„,„,„ ].'- ■"- ' 1 'l.n. .M.a lis si,;,,., was ; „„,, , ,,,,. ^^ ^,, " ""'••"■^ ■■MM„.y sl,„„l,l 1,0, tl„.r,.ron., !:.;,; „r .^^js,,,,. ^l_;^u 5.,..,,, 6$-JU>. 7 1IMn. a/sni-nl. gsi^'u!; <./ and 1 „f s/ ;i of (if ,„„1 r, of s/. Tl...se ,„.oi,l..,„s sl:ouM 1,.. j„.,.forn,o,l l,v a„alvsis on a lino, .at ;,.,■ tl.an l,y c.„n,|,o„n,l propoition, Tl„. solution of 8 .nav l>e as follows : ;o 5 1*0 ni. X ^0 X no ■ ~rrx"lJ~=''""J?"'- !) 3 My answer is to I,o in rods, therefor.- SO rods is the nun.her to '1»"1 "K l.y 18; 40 n,..n w.ll hnild 40 tin.,.s as n.any rods as ] ...an fo„n,l by n,ulti,,lyin,, l,y 'O. If they ean hnihl so n.anv rods n l,n, ,y,4.and ,n oO day. they will '.niM 50 times as n.any as they hn.h ,„ , ,,ay, lo„„,, ,,, „„,Ui,,,yi„, ,,3. „, Si.„,,IU l.Mii^' hy iM- rllation we have .•!7ov., ,.,] ' 03 X,.a , all of these probkius should he ..e,fo,-n,ed orally »4 A„.„..-rs. 7 ll)f^S,-,|a..Vii„„.s. 8 lJ74S,'n.l„'. !.,« 10 \ "0- I, 30/^ J, 20% K, U K, 47.3/ '0/^. U, 40/. J I, -M-j.r/ 178 UllADEl) AUrrllMKTll'. [VII. on 05 A„.,n-,-s; i^:>^:; It. 2 ,'./;■ 3'M \'.h -M ^r,>;r/„ .jihk;^ 5 r. ;i -'oiki .(mcj. 6 .os ;! .k l'Kio. 7 -0(ll„.lts%Hl(ll,olts -IIIH. holts .«i;l «!.-!.l(i .*..-.S. 8 •«l,.Mt s.-ii.i',) si'.M-.'.i. 9 SUI.7.-.. 10 SatM;;!. 11 sir.i;i..-.i. 12 711'' 13 SHH't"!- 14 lir. sliari's (.'ijlL'M.r.d Icl't). 15.'il.'l+%- 16 r.l.'slMn's(.*'s() ami $1111.01.4. 11 .$10008. 12 .«ilOO'.l'.'.7.">. j>7 . 1 /,.«"•,■,•.«.■ 1 .1?1."oo. 2 JiicifiLMio. 10 .11. P.M. 5 .«i;ioi..-.o. 6 .'iiiii,s.-..H.i. 3 .STlilMKMMI. 4 4 ll. 7 .■;,i'/. 8 .iiiU7;i.7'.l :'l d:i. 11 Silt«7.(i5. SIllL' $VA:JI. 9 .'Sill.'-':!. 10 1 vr. "»8"!l'».s»,.™: 1 .«.00.;iO. 2 .$707.4;i. 3 ,$4000.00. 4 ,«i4,soo 5 wr/,.. 6 •$o'.'..';o. 7 .ooo,ss. 8 $i;ii.'.-' $1100 9U", $i.m 9 .$o74i..'!(;. 10 .*ii-'..-ii+ .$7.1 .^..•!+%. JM) An.n-r.: 1 .*S.l.'l.'. 2 S'-V,. U. 3 SIKU". 4 5 5 $490.13. 6 .IKoo. 7 »1(h;..'-..-.,v 8 .*:!8..., 1 j. 9 900 ston..s. lOL',-, .la. 110.H.V.1. Sl-'L'.nO 12 ••«0 ... „>.. 13 4'U-'>» s.'c A littli- nioiv tliaii i nt a swoiul. 14 9>^i.— l,V, il,> 20ili <•". y.l.i ('■) $107.1.'4+; 01) $'M.U; tA%-,ii-09140,S(»00O000O ,nilos. 5 10 lb. 6 20 lb. 7 .'UOO lb. 7000 lb. 8 2iaudl. 9 5ilb. 10 8;^ lb. 11 If ft. from the end. VIU. 1] TliACUKHS MANUAL. 179 SW'TIOV X. NOTKS rmi iKHiK MMllKII KlflirT. t'nlcss tl.o priiinpl,., an.! process™ of ,„. various snl.j.,.t can be expr,..s.sed by a single figure ; thus, .fi, .4. .;{, .2 ; and that "i and J can be expressed by two figures ; thus. AH and .10. The origin of the words expressing numbers h, twelve may be found in an nuabridg,.! dietionary. Other names are derivatives, and tlieir origin will lie ]dainly seen. The definitions e.aJled for on tliis page are as follows: A unit is anything which is cousidered as one. 180 GUADED AKITHMETIC. [VIII. 1 i ! An intMiml unit is one or a collection of ones regarded as an undivided whole. A fmctiiniil unit is a part of one regarded as an undivided whole. A niimbrr is a miit or a collection of units. An iiittfjml number is an integral unit or a collection of integral units. \ fmctionul iium/>,'i- is a fractional unit or a collection of fractional units. Number is a quality of objects which answers the question, "how many," and arises from distinguishing one from more than one. Anthmetk is that knowledge which has for its object the ex- pression, the operations, and tlie rel.ctions of numbers. The sum of two or more numbers is their united value. Addttim is the process of finding the sum of two or more numbers. Subtmi-tion is the process of taking away a part of a number to find how many units are left. The tnlnuend is the number separated. The subtrahend is the number taken away. The remtiinder is the part left. Multijtlautlon is the jirocess of finding the united value of two or more eipial numbers. Tlie multlpUcund is the number multi- plied. The multiplier is the number which shows how many times the multiplicand is taken. The product is the result obtained by multiplication. The factors of a number are the numbers which, multiplied together, produce that number. Diriiioii is tlie process of finding one of the equal parts of a inuiiber, or of finding how many times one number is contained in another. Tlie diridcud is the number divided. The dirisor is the number by which we divide. The quotient is the result obt-ined by division. ' An mid number is a number wliicli cannot be separated into two eipial integral parts. An ecen number is a number which can be separated into two eipial integral parts. A prime nundier is a number whose only integral factors are itself or one. A eom/iosite number is a number which is composed of other integral factors besides itself or one. A prime f'letor is a factor which is a prime number. VIII. 2] TEACIIKHS' .MA.NTAL. 1«1 ■ --Hi,., of J,, ,,,,,;:::-■ '2 -.r a „,.,,.,,„ .w,i,,. . two .„• ,„,„■,. n»,nl».,-s is tl,e Ipn^f , '■"""""" """'">"'« of a/actor wl.i,.,, ,„.,o,„. ,, each of th., '-,?""■""""""" '^ '/"•'»„. or fart,,,- „f two ,„■ ,„o,... ; . ^'''■'"'•■" '■"""""" "■'"'■'' l-'o'^^.^ to „a,.,; If ;;„,';;""• ""'""'"-^ - «- S-ato.st factor ■ .irts tak,.„. ■■ '" ""'"«'•"""• .» tl.e number of ,.,,ual To Chans,, iractions to eonivalcnt f,-, f "omnun, .Icnonunator . Divi,., „,", '•"'^"""' ''^""S the least Jenonunators l,y tl,e ,lonon,i„, , of TT'" '"""''■'« "*' "'« botlUcnnsof the fractio:h:r;:ft ™:f '•"""" '"" -"''"'^ -10 add fractions: Clianw tl,.. f. ^ 'savins a connnon denominator ad U '" "'"'"'■''""' ^■'"^"'"■■^ s.™ over the connnon dencnnin.'.tor "'"■""'"• ''"'• »"'« "- ^-'-S;::i !:::;:;:;,:;;':•-;''« ^-^ion. to e.,uiva.ent of the nun^erat^rs ov "c n "'i '"'' "'" """'^ ""^ "i'^— To nuiltinlv f !""""'""" ''•-■""iiiMnator. '". ..4:^:.;r ,::.:;:;:,;;;•''"*-«»=.. I. ■""' *■■* " i" .■ .:..■> J ^IZnZT"" •"""'"■'■ ™ m miittipliration '^'' "" ''i^'w iiiW prori.,vl ~ :.:::;■;■,*;;: ;r™:ii°r"r """'»'-•■« 182 GKAUEn AlUTHMETIC. [VIII. a W 4 .t tUe foot point. .itU „tU. .livision. at tent,, ana hun..eaths '^::;:::,.,.„,ts.nU.afo.n.yU.Vo-n,Unti,,^TaU^^^ end ot Hoou V... <. '" ^'^ ':^^'::;;::::': u^ ',;;.,..,. .1.^ eon. i^lli^ n..n. . tUe ^,;on -^^^ el,aavon. soii,..tinu-s used m >"^^^"""' ,^^,,ij a stone, Barrels, l,oKsheads, tieves, and VU-es -" " -^^ > >^ J .on>eti,nes used n. nK^snrin, J^^ -^^i.^^Ju ,„etvic n.asures The meter is the «tand.u-d urn' " ^ ^^.,„„i„„tu part o£ the "•■''"'^"\nr^rtluJ:,^itoti.epole. Lead t.,e distanee on a men.lian no i ,„etrie system - ,,,,ilsto see the eo.mee Jon a 1 ^pa^ ^ ^^^,,„_ ^„, , a euhic eentimeter ot «ater (at .>.> ; euhie deeimeter of .-ater '"-'^•'""^ \ '*;'^^, ^i,, earth to nralce a Asnhr r-a'- '^^ «»•■ ^'"-■'^'' *""" '' , t iU-. da 5 h. 48 min. complete revolution around ti.e ^l^^^^^':,, ti.retore 40 see. Tlds, U wil be -;>• ^ ,J; !„ f'" ..^ ',,; this reekoning "- r™ w^: rs :^^ " - ^-^^^^ ^^- -^^ ''--";i we sliould gain neauj . „„„„f,.,i m ordinary years. It the eentennial years are conunon J s ,,^^^^ ^^^^^^^_ ^^^^'^^^^-'T'T^^Z^^X^^ centennial years a day in four hundred yea s lie e j ^ ^^_^.^ .^ ,,,,U,,e "y.^"<;;- ;— 1;L X^:i .1- this new style of tliis connection to tell tue pupii -rt:;%:::irt;:f7ri.et.een t.o ne. moons, and is '^:d^:T^ a nuud.er which is a certain numher of hundredths of anotlier number. VIH. 5] TEACHEIiS' MANUAL. 183 ^^J J-'^« is the „u,„ber of i,„n<,rodths which the pero..„tugo is of other iioiiits referred h, „„ fi,; IX. of the .Manual. ' ''"''" "« '^"P'""'-' "' «--tiou « 111 2, the formiilii is y,' = 7>-u/,' t„ , ,. Wliile it is not advisable srener.llv f /< =,/ -^ (1 + v.._). i" P.'rcenta,e by fo, n L r ' '"! '™'"'-'' '" '""■-°"" l-^W'-ns forn. then, I tl >t w^t^, ;. j^/::' '™'^'";r '-'---"y t" l-r- tlie impils. ^' "'""•'"> ^^ '»■" tlie 'ormuks are made by 'i;... the jn.,...,:s iiii^Jn'l:^ . ' ;>:;, t'";V ""r '" '"' ■' If there are n.any nnn.bers to be .11' IC ' t" ''*"""'"' •'•^''^■ sections and the se, , '^' ""^''" '^" '""''«• "'to mimber between 91 and i)!., mnlti.dv n- on ' ^, ' , ^^ thi^r;;;;''^ "-*"-' --^i-'-^t-i"t"oo, and — .n, , ,, ,.,, ,,,,j,^ ^,,; wie':;:i:;^s::nd'r;;:: 48 37 2-, 04 4.5 A- 184 GRADED AlUTHMETIC. [VIII. product add i of the ..m ^ *\ :':;^ a T T^ ^J = ^5 Multiply the whole nuu.bers, an.l to the Fo,lu t add that^ ja t:S then. exp.essed hy the su,„ of the f----;;;^^ ^^ polihh-tnr ; the shortest method. Ut the ,upds p.act.ee upon them until facility is acquiwl. „ , ... lovr ".no. 12da. 2*62.50. 3 f ',)0 more. 8 Answers. 1 ->.■ - m ..^, . _.^_^ ^ ,,^„ ,^,,^,^^ 4 6487.1.")+ bu. 5 ii'iVo- " •"'" or about G,\%- 8 '^t'"'"' ^'^ ^'""' ''^' ^*^''^" 75 1b. 80 1b. 4ttlb. lO Ansirers: 1 "^ ^PunTiitaae Pcroentage of the ;Nu,„lH.r„f Per.o.,. spoken by. PBri;-"»S>', W hole. LanguitiifS. English 201,-2O.1flO 314o0000 303-.>000O 15070000 28100000 7480000 30770000 French German Italian Spanish Portuguese Russian 1 Total . 2 $2750. T36rTj^ii^^21^5r23min.30scc.p.M. 6 About 12 acres. 1890. n now mo uliOOOUO 75200000 33400000 42800000 I 13000000 ! 75000000 hwrwiso ill' EiKlity-oiiH' Ytars. 1801. ISIK). 441.4 ' 12.7 i 27.7 62.7 19.4 12.7 115.03 18.7 18.7 121.63 9.3 8.3 03.42 10.2 10.7 73.70 4.7 3.2 143.74 19.0 18.7 148.26 100.0 100.0 5 .$24.53. vin. 11] teachers' manpal. 185 11 Atiawem; 10%. 2 00%. 3 $25500. 4 $7521; 5(1 5 $25000. eSff. 7 $12000. 8 $12315 $;J5(;S5. 9$''4,-!"0 10 $9500.62i $6179.74. 11 C. & S. M., 2S%. 12 $1<.IU00. 12 .dnsu-Prn: 1 $2900.87. 2 SI50G.54. 3 2(5(i(!f lb s„f;uv cane ; 4000 11). beet root; 5000 lb. wheaten flour. 5 100' C ISfC. 26fC. 08° F. 6 6.1. 2 A. $100 li, $.. C f 150. 3 Kol...rt, 40 WilU^un, L'O Tl..mm», IL'l). 4 J 7 S lb. 8 4 11,. M.K.ha 12 lb. Java. 9 A, .*(,.")0 1!, $MiW C, .$1000. 10 A lias «i()()0. The following form of analysis is sn-gested for this class of problems : 1 Let J' = cost of cow. 4a- = " " horse. 5^_- ii 11 " aiul cow. 5x = f;i60. X = 172, cost of cow. ix — .$'-'88, cost of horse. The pupils should learn to select the number that can be most conven ently represented by .'. For exan.ple, in 2, the numlK^r ot dollars that B has, and in 3, the nnn.ber of marbles that ^^ .llnun has, is the most convenient unit of representation. In simplifying 11 to 15, lead the pupils *>; -'^ ''7 «- l^':; theses may b.^ remc.ved in such examples as : 1- + (4 + -); < t" ■ ; + (f. + 2); (8-0) + (8 -4); 10 - (8 - 0) ; l"-*' + -2, , Ly he shown that the 10 - (8 - .;) is cjual to 10-8 + by first subtractin., 8 from 10 as indicated, and calhng at ention to the fact that we have subtracted a r..mber too large by (., and therefore we must ad,l to the differen,... In the expression 10 - (0 + 2) bv subtracting instead of + 2, we have a subtra- hend too small by 2, and therefore we must subtract 2 from the difference. 16 The four axioms involved in 5 to 8 are as follows : If the same quantity or equal quantities be adtied to equal quantities the sums will be equal. If the same quantity or equal quantities be subtracted from equal quantities the remainders will be equal. If equal quantities be divided by the same quantity or equal quantity the quotients will be equal. VIII. 17] TKACllElts" MANTAI.. 1.S7 Two ,,,ui„titios ™d, equal t,. a tl.inl .[.mntity ar.. en,,,,! to oa.^l, otlicr. ' The kiii.l of illnsirativ,: i„-ol,l,-,„,s ,.m11..,1 f„r is .sl,o,v„ i„ the lollowing : .' 'HIPS lias 18 iTnts, whmh is ,T c.nts moir tluii ,I„hn |,as ](„„• many cents lias Jolm •/ Let ./• = tlic number of eents that Jcjhu has.- »■ + •'' = 1'**. the uumliei' of eents that James has. Subtractinsj 3 from these equal quantities and tlieiv is -iveu : j+.-i — ;i=l,S_;i. ..■ = 1.V L^ad the pupils to give and t,. .solve similar |,rol,I,.u,s illustratiuL- the tour iirinciples. 17 Jnsim-s; 16 8 G Uk 17 VJ 10 4 18 P' 'i o 19 12 18 30. 20 8 'Jl ir,. «>->■> .«. In removing the parenthese.s (11 to 15), I.^ad the imi.ils to see the reason for changing the sign by asking the following questions • I' In 11, what IS to be subti-aeted frou.l'O'.' If Calou,. is subtraete,! IS the result larger or smaller than the requiri'd answer •' Wh,t more must be subtraetcl ■/ How nuiy both pr. sses be indieated - " In the same way proceed with other exereis,.s until tin. j.upils can see why parentlieses may be removed by changing the signs of all e.fcept the first (juantity. 18 Answers: 1 Corn, ,$1.42 ; wheat, .SI. of. 2 James "(i vr • sister, 16 yr. 3 Robert, IL'/; James, L'4/. 4 .'U ",S -d ' s'ls' 18, 24. 6 8 lb. (»: 7r, VJ lb. (,1: ->/. 7 7 in. H in. etc. 8 l(i an,i 24. 9 24. 10 48. 11 40. 12 l{„bert. 0(1/; IJalol, :■',■ 13 180 A. 14 18 yr. 15 10,8. i • - ■ 19 Answers: 1 Father, 4.'; vr. $2500. 3 ./, .«40()0 ; n, «!( 6 800. 7 43,',. 8 640. 9 40. IQ & yr.; son. ISyr. 2 $1200. .§1800. MlOO ; V. .■SOOOO. 4 .52.70. 5 1.!;^ lb.' 11 »f'. 12 .S800. 13 $4800. 14 4^%. 15 .f!)00. 16 «70. 17 §4000 188 GHAI)KI> AltlTHMKTlC. [VIII. 20 [ 1 SO The theory of nogativc iiuantitios inny lie shown liy tlic device iiiilicati'il iit the liciid of tlic iiii^'c. In tin' rcliitive srrii's indicated, all tlie (luaiititics may lie said to iniTrase liy 1, or by a i'luni left to right, and to decrease liy 1, or liy a i'mm right to left, the positive qnantities being at tlie right of zero, ami the negative quantities being at the left of zero. The device may be extended in illustrating 1, the jinpils being asked to begin at the zero jmint and ]iass the jiencil to the right 4 spaces. This would be indicated by + 4(( ; then 'A mo-o s|iaces. Tlie distance from zero would now be indicated by + In : and by moving the pencil over 2 more spaces the distance from zero woidd be indicated by +!)('. 2 may be illustrated in the same way, the pencil in each case jiassing in the opposite or negative directioh. The spaces jiassed over in all would be indicated by— G". In 3, the pencil starts from zero, as before; passes to the right, .as indicated, 4 spaces ; then to the left 3 spaces. The pencil now is 1 sjiace to the right of zero, and the distance would be indicated by + 1 n, or + "• Thi! pencil moves to the right - spaces, and tin! distance would be indicated by +3'(, which is tbc required answer. Cont.;aering siditraction as the jirocess of finding the difference between two quantities, the minuend and subtrahend may b(! indi- cated in the relative series at the right or left of zero, and the difference by the distance between them. Thus, in i, the distance between + id and + •'« '» + "• 1" 5, the question is, what quan- tity added to —'Z/r '^■''~-^\ + 10 ir2 + 4.i-// + 4// !)j" — .■iOj7/ + ;'.fi/. 15 1 +■"'•'■ + •"•'■' + •'' 8 J-' + 12 xV + v. .nf + / 27 J'' - 54 .,•'' + •">4."/, + 18/A 20-7..' + l.-./Hy.'' + 2 + 12^'-/ - ;«) ..•,/. .V=U.r-l'J. 6 x = '^^Lzll/ u= 4.,— 20 4„-_10„ ,'/ = .'/ = .'/ = L'.s -:.'.,. ■'• = ^^ _40 — ;;,,■ ,_-" + ■■!.'/ y = 20-l'^. 8 .'=-4// 7 r = ifi.,-„ _8,-,;,, + ^^ „=.""-^+l+4 • = "-•♦// y = — 6 ..^"/Z ■'' (7 11.,. = ,^. i3,_s -••_. 10.- = .. , = .,. ^=5. 16 x=,', .= 1. 17 !i;" ■"=;■• ^='=^ y=10. ■ A^J-'J y = L'. 18 .,=;{ i ,'/ = l<». 4.=a „ = 2 5,-,. ■' - 3' = ' y »• 14 .1-8 y = s. 15 .,. = 12 y = 4 192 OHADRt) ARITHMKTIC. [VIII. 30 if m 16 a-— 18 ^ = 8. 17 j' = L'0 // = l.'l. 18 r = !',•, ,/ = -2. 19 J- = 40 ^ = 1.'4. 20 .'■ = .Ml //=I8. 21 . = lli| // = 4S Jt-aOj. 22 J- = 11' !/ = 'J i--=l(). 23 j='.» -/ = :'(( '.-. = (;. !fO y/ "««•«/■.<.• 1 j=l.'. 2 .'/ = 44. 3 // = •!. 4 // = !'. 6 j' = H , = 0, 6 J- = 'J //=iii. ?.'■ = .") // = «. 8 .'■ = H j^ = 3. a ■' = — •■'it .'/=4:i(. 10 .'■ = '.» // = ;!. 11 j=« 1/ = 1'J. 12 J=7 (/ = !'. 13j=1L' //=lti. 14.. =1.". !/=in. 15 J=;t4f .'/=-'!'./,. I6j-=t|; //=-i,',. 17 j- = r) //=l(i. 18 J- = (IT. 5 i/ = AC. 2 x = r. // = !(>. 3 j=ll' y=---l'(). 4 J- = 9 .v = 8. 5 r=ll>. // = !'. 6 .'=4 ,'/=lCi. 7 a' = 20 ,y=ir>. 8 J- = 1S // = 7. 9. .■ = 7,-,, //=14,''',. 10 x = 28j ,/ = _]4i(. 11 .,= :i;; // = :;(p. 12 .'=11! // = 10. 13 .. =1.'(1 //=!S. 14j = 104 ,/ = — i;tj5. 15 j-=(l ,'/= 1.'). 16 .'■ = -'» .'/=li>. 17 .<■ = -'! I } //= 1. •!,',. 18 .'■ = »> .'/ = -'4. 19 ..• = .s //=1L'. 20 .'■ = 1-' «/ = '.'. 21.'='.) // = 4. 22 .-■ = i;i.',i, .'/=Ifi,*, 23 a- = L'l}j i/ = 20J|. 24 .' = 11' // = 1'0. 25 .'=1; 2^ = 8 .- = 10. 26 J=5 y = ;i() .•. = ](). 88 Answers; 1 ^r-j-A — 1' 11 — h — r n — h — e a — h -\- r „+h — ,-—,l + r. 2 n'-\-'2nl, + l? .,■' — /,'. 3 (!"» — 5r<4-|-2".' -(W,^-hl';W'r — l'0 ,•■-'. i or'~r,r+-,<;/+\n.). 9^S^^^ J£^. 10 (« + /')■•' "H ''" + -"'' ("-''/ "' — /''' (" + ^') X (" — ?') ('i+hy. 11-4 -6 «/> — c c — '/ a a — b 12— H ti,+ii «+r 120 ^±^ I 13 9 12 2 nb 14 4 + ac VIII. 2111 TKAdlKlls' .MANI'Ar,. ifta „v, — ,„■ _ -■ ;, .Ml. 14 ().>() iijiiili.s. 30 ./,M,,-..... 1 ,^,80. 2.SniO. 3.V,no „„,,, 4.-«i,Ia. » -.4 h. 6 40 prisons. 7 A, .¥7 li, .<(-.. 8 I'lLVon -Mi ... ..„ .,0/. 9 1(. shoop, ,0 .„v,..,. 10 .,40,, n :,,.; S-'-.0 Saval,, .fl(«) J.;il..„, .«;ir,o .M„v, ,*.,„, « ].T *. 9 is in,l,.tornnnat... (Itli-.r answers ar., .T slu.,.p an.l 17 c-.lvea ■ 15 sl.m.p an,l l;i . alv... ; I'O .sl„...p a„,l (i ..alvrs. 3 1.HM,, 4<,lli. 5 10.-,, 7S5. 6«iinoor,Ml./. 7 8 ii.oo-,':^';';/'''''^"^'- •'''^^'•'"- io*.4i;4- •1Mj)00 m i>s. 11 ,i-.. 12 4s 8 is in.letrnninato, any „.„■ of forty answers hdnf; eonei.t. 3a /™«.,.,,,. 1 \il„, .'jooo „„l,.s I);,nul„.. IfiOO mi'.Ps .\,„a. -n aiOO n,....s. 2 H.-a„,lon, 106 feet Ott.w.. .,, f" Londoii, .'i.'lo feet 1 ■{ I, 1|. » ,. . *ir.oo. 5 1 ill,, s.s.ia. 7(ii]i. 8^'-^:^:±'. 9_iiL "-^- " :7,T-. ,i~, ;;^ 11 „(iii;^|: 20. 15 V: li. yi; ''f — ad 13 i5/« + (, 14 IJ 194 GKAPED ARITHMETIC. [VIII. 33 33 Aiisirrrx: 12 102. 13 lOK. 14 0. 17 1J. 18 .7-".). 19 .OlMKiOT.Sll'r). 20 J. 23 lli,. 24 .O-T.. 25 .244,,",. 26 15 32 21 KW. fl 1 (i rt B ;13fl00. 29 (!<.)«<)00 30 SI TOG. 31 34 ;il()40n2,"> 38 32708 16 2J. 22 2K;t. 27 5^- 100270 20.1)704. 35 1.10271001. 39 88.121125. 28 1000 70."iO 0801 10000 474721 008(101 KMIOOOO. 32 .O0O0.-)(;2.-|. 33 27.270901. 36 4i;i71l."8.;-|010. 37 049'J;1. 40 ">801-;i ;.;;!. 41 ;{8;!;i2Si. 34 The for ula lor tlic ^xtriiction of the square root of a number can be i. .md by tindint; the square of the root: t+i(. I'upils who have taken the al{,'ebraic exercises will have no difti- culty in this. Others will have to be tauyht. If the symbols representing any root be found too difficult to work with, let figures representin;,' the root of a given power be used. e.>j., the square of 2". = 20= + 2 X (20 X r>) 4- ',■'. 35 7 40. 14 04. 24 r..(i. 31 7.0. 37 280, 43 500 49 •■!.-> I 54 »i.:;i 59 4.0 64 2.1 69 5.2 8 30. 15 80. . 25 4.7 1 32. 2 43. 3 51. 4 02. 5 73 9 57. 10 7(1. 11 07. 12 74. 16 07. 20 4.2. 21 0.3. 22 5.4. 26 5.8. 27 0.0. 28 8.8. 29 0.0. 32 7.0. 33 2.n. 34 342. 35 254. 38 523. 39 475. 40 270. 41 034 44 078. 45 0(>0. 46 804. 47 700. ,0. .' + . .'4. 2 + . 50 405.8. 55 15.55+. 60 2.r)0+. 65 30.01104 70 .04 + . 51 50.70. 56 37.14 + . 61 2.84 + . 66 2.2;:+. 71 .87+. 52 60.10. 57 40.02+. 62 12.0(;+. 67 3.01 + . 72 73 6 35. 13 83. 23 3.5. 30 0.9. 36 304. 42 488. 48 2,32.1. 53 1.41 + . 58 1.S7+. 63 5.72+. 68 10.83+. .52+. 30 .lH«'vr,,; 1 74 r,l. 2 ISO ft. 274 ft. 511.23+ ft. 3 12.040+ rd. 0..">24+ rd. 4 0.8+ rd. on two sides. 5 254.13+. 6 884.84 ft. 7 148 blocks 08J ft. square. 8 05087J paving stones. 9 1 1.14+ in. 10 24 It. by 10 ft. 11 12.17+ ft. on longer side 11.01 + It. on shorter side. 12 154.0+ rd. 13 10 and 15. 14 Ji^ 14 1.40. 15 5.25+ ft. vui. :{7 1 TEACHERS JIASITAL. lOf) 37 ./Hs«r™.- 7 80. 8 32. 9 41. 10 4,5. 1168. 12 CJ. 13 9!). 14 97. 15 4.7. 16 .'i.'i. 17 12.7. 18 .89. 19 ■JC/J. 20 41.-;. 21 472. 22 90.!. 38 J .»v™.- 1 1.58+. 2 ;!.91+. 3 r,M+. 4 ll.(J9+. 5 l.ii,s+. 6 2.4;:+. 7 1.89+. 8 4..sl'+. 9 i.<)i+. 10 l'.;;,s+. 11 .79+. 12 .42+. 13 .7(i+. 14 .4.'!+. 15 .;i9+. 16 .74 + . 17 13 in. Sain. 18 87 in. 72 in. 19 12.9+ in. 47.55+ in. 20 73 in. 28.4+ in. 21 1.41 + It. 22 lO.OC. + ril. by 15.09+ id. 23 12.14+ ft. 24 ('> ft. louij. 3 ft. wid,.. 1| ft. deep. 25 8.07+ ft. 26 IGO rd. 27 88.4 + in. 132.G + in. 170.8 + in. 28 l".0.26 + rd. 3S) The fiiUdwinf; fiicts should li,. cinuvn fmni tlin iiui)ils by teacliini;. us before shown : Mutter is anything we ■{ct a knowledge of tliiimgh the .sen.ses. A lini/i/ is a limited portion of matter. Sj)(ire is the room a body oeeupies, and th<^ room that is around a body. A riihiiiir is a limited portion of spaee. A volume may bo repre- sented by a m}l!,l, ivhi(di has tlirei; dimensions, length, breadth, and thiekness. A KKr/iiir is tlie limit of a vobime, and has only two dimensions, leiigtli and bri-adth. .V Hilt' is tlie linut of a snrfaee, and has only one dimension, length. A /mhif i.s tlie limit of a line. It has position only. It can be represented In' a ilot. A Ktriitiiht Hue is a line which has the same direction throughout its entire length. A nirrnl line, is a line that ecmstantly changes its direction. (For definitions of linri-ontul line and rertieid line, see JIanual, page 1(18.) Lines are punillel when tliey have the same direction. However far pridonged, tliey can never meet. When one line meets another lino so as to make the adjacent angles equal, the lines are said to be j>erjK'iii/i'ii/i(i' to each other. li-- 196 GRADKI) AIIITHMKTU;. [VIII. 40 •-.w ^•-,. 9 A line m;iy be dnuvn jiarallol to ancitlu'r line as prpviously shown, or h\ tlic following waj- : To draw through the jioint B a line parallel to the given line 01'. From the point C, with a radius equal to CR, draw the scmi-pircuinference JliJi. From It as a eenter, with a radius ecjual to jn. cut the oivcumferenee at .S'. Join JiS, and we have a line parall(d to OJ'. (Let tiie jinpils sliow why t!.- lines are i)arallel.) 11-12 For a method of dividing a line into 2, 4, or 8 ecpial parts, see Maiuial, page 109. The following method of dividing a line into any number ol -,_ ' parts may be taught : /"~~--,, To divide a line JB into 7 — _ equal parts. Draw tlie line AO of any ^^ length, and lay off on that " " ■*' line parts of any conven- ient equal lengtli, .Toin BC, and tlirough the points of division on AO draw lines parallel to BC. These lines divide AB into equal parts. The standard unit of length in this country is tlie yard, the same as tlie imperial yard of Great Britain. Its length is 3§^g§S of the length of a pendulum ivliich vibrates seconds in a vacuum at tlie level of the sea at 02° Fahrenheit in tlie latitude of London. Other interesting fai'ts concerning how and where the standard yard is kept can be gatliered from a cyclopedia and given to the pupils. 40 Teach, as before, tlie following definitions : An aii!/!i- is the difference of direction of two lines in the same plane. The point wliere tlie two lines meet is the irrfe.!' of tlie angle, A rlii/it iiiii/!i- is the difference of direction half as great as oppositeness ; or. it is an angle formed by two lines extending periiendicularly from each other. An obtuse angle ia au angle VIII. 41] TEACHEHS' MANUAL. 197 All wutii ,in,jlo is .an .angle less than greater tluan .a right angle 13 To dra". an angle eq ,, to a given angle. The pujiils will .see that the proof of the e,,ualitv of H,.,=„ , rests upon the faet that eanal'.res suhteii,! ^^t^l . ' ThTsl! ^ .1 what they have le.arned ahont tl meLl; Jnt '^ ^f an"e. ''° '"" "' "''"" '""""' """ ""■"« *'"'- "'« -« of a given Im te.icliiiig the ahove prohlems, .-is well as all tle.t f„li .v. 41 J.,y„v,/„„y,.,,i,„ „„g,,3 that have a common vertex and tha have sides extending in opposite direetions ' vent™! '::f ^^ ""' ' '" ""; '^""> ^^-^ -»^'- "-' '-- the veicex and one side eommon, and wliose other «ides .,>.,. „ P--ts of the same straight line, rit wiU t .^ , ^ thaf " element of adjaeent angles is left out in 2.) ft'- 108 OUADKl) AKITHMKTIC. [VIII. 4a 4 T.. nrovo that tlie siur. of tw., udjacent angle, is rqual to two right angles. Draw DO pprpcndicnlar to AB. j0C+JU)C=.i(>l>+H0D. A 01) + II0I> = - right angles. . JOC+ HOC =2 right angles. 1,. 7, lead the pupils to see and to say that the angles « and /, (in ihe figure, 10) ithe angles I. an,l ,-. Tak.ng away the eon.- nion angle l, the angle a = the angle -•. l" 10. the four angles ,■, <^ «, and . are called ,«..•«„/ .^^ be anse thev lie between the parallel lines;, and the four angle ! r and ; are called e.f.n,.,l ,^''rl<.>- angles The angles . S : and the angles ,/ and . are ,-.. .-U..r.. angk-s. Lea. the pupils to discover ^vh.at angles are e,p,al, and wh t ey are ir The lines AB a-.d CD are dra. . parallel to ..ael> ot^.er Tphu,e .surtWe is such a surface that, if any two p.m.ts n >t heto;— d'hy a straight line, that Hue will lie wholly .n the "tHhe definitu.ns called for in 12 b.. eoncise.>nd co.nprehensive. rFor method of teaching, sec Manual, pages 104 a-.d 1 1(..) ^l.> Let the solution of the theore.ns a,.d pro.,le...s be n.ade by „.,;:.,.en.e,.t a..d eo,.struetion, and also by de„.on.^ra ,o..s so l^,r " the pupils ca.> be led to make then or to understa..d th, ... The follow i,.g fignv..s a,.d hints n.ay suggest demonstrat.ons of the more difficult p''opositions : 3 Draw CE II to AB. » Prolo.ig A C to /'. ' ^ ECF + HCII + A CB = 2 right angles. A = liCi''- B = ECB. VIII. 43] TEACHER.S' JIANUAL. 199 Let tlie order of demonstrating 5, 6, and 7 l,e reversed. 6 Bisect angle C. AC=CB. CD eonimon, (• = d. .-. ,1 = 4. From the fact proved in 6, it can be si.own that the angles of .in equilateral triangle are equal. ^ AC= CB. AI> = Dli. Join AB. CAB = ABC. BAI> = AIW. CAD = CUD. anlTuir""" '"" " '" " "" '" '* P"'°™^'^ ^•■"^ P™'-"'- 43 Tlie pupils may be able to solve 3 by demonstration. that tlie angle at tlie vertex is bisected by the hue drawn from the middle point of base to vertex, and that the base is bisected. This figure will suggest a method of bisecting the angle A. Lead tlie pupils to prove that the tri.angle ./J/0 = the ,gle .LVZ), and that tlicrefdiv the angle A .,ect' 1. For a metho.1 of hisectin- ..ne, sue Manual, page 109. i-; 200 OBADEI) AUITHMETIC. [Vlll. 44 The followiiiK figures contained in 7 : TO will suggest ways of solving the problems n .-■' ~ll It would be well to apply the principles n.volved in the la t .X CIS s of the par. by actually n>easuring distances m a held. TWs ly be done with line, and stakes or with stakes alone. 44 The definitions called for on this page are supposed to have ^ir;:?r-ssi^. -^-r--^::: n C preceding propositions. The fol- lowing hints might be given if necessary : What angles do you know to be equal? What lines? Compare the size of the two triangles. If . tn ,.rolon.' one of the sides, what two angles are equal r^^Hght .nS^ ^Vhat other two angles must be equal to 2 ri^'ht angles ? „,_ . 7\< What angles can you prove to be equal ? What lines? What triangles? \\hat other lines? VIII, 45] teachers' manual. 201 Ihc pupils may bo toM tli;it an is,.scolo.s trapezoid is a trapozoi.l whose sides between tlio two parallel lines are ..qual. Ti.e propo- sitions eontaiiied in 13 to 16 can be easily proved. 45 A polygon witb two reentrant angles : Exercises may be given to show how the area of such a polygon may be found. Let the jiupils discover two ways, and wliat dimensions must lie known. The simplest way to eonstruct a reguhir poly- gon IS from the circle. Sliow to the jmpils that bv means of rom- ])asses the eircuniferenee of a rircle can be divided into any number of equal parts, and that the straight lines connecting the points of division are the sides of a regular iKdvgon. The formulas called for in IS and 16 are, .S' = «/,; „ = ,y-^-; /.-^^^;„ = V^ ' "^' 46 The triangles ADH; and JBC may be called similar, because they are of the same sliape. Later (page 50) the jiupils will learn moie dehnitidy what similar polygons are. Triangl.-s ar ^,i^■a- lent If they have the same size, and eciual if they have tlie same shape and size. The following formulas should be made by the pupils from their knowledge of finding areas ; Tna„r,re, S = "-^; trai,e,ohl,'^xh; rnjuhn- p.,hj.j,m, S = '^f Tiie transformation of polygons into e-iuivalent i.olvgons of any required shape can be readily made if tlie method of fiu.liug the areas is thoroughly understood. 47 Auswe,-,: 1 {gsp, .V. 2 1728 .sq.ft. 4290 so. ft. 2.5'^:,S4 sq.ft. 3 272ift. 4, SO sq.ft. 201(;s,i.ft. 5 1ol2s„it fil640 sq. ft. 6 217t ft. 7 147.5+ ft. 8 1.13 sq. ft 9 "i ft' 8.3 -t- in. 10 344.6-1- ft. 11 141.7+ ft. 12 4380 so. ft. 13 .' 14 17+ bundles. "' 48 Answers : 5 5 in. 13 in. 6 8 in. 13.41 + ft. 7 25.08 + yd. 202 GRADED AHITHMETIC. [Mil. 40 The pupils should be encourased to give other proofsjlm. those here given. Tlie f ormulas to be given are: A = V6' + /' i h = \fli'-i;'; p = >|l|'-li'■ ^9 Aimvers: 1 100 ft. 2 -'.T ft. 5 62.4+ ft. 6 l-'O rd. 7 1!(.:.9+ yd. sii. ft. 12 L'u.4.-.+ ft. 13 7'.>.T1+ ft. 15 4r>.Gi)+ ft. 16 30.98+ ft. 19 25.45+ rd. 20 2G.07+ ft. 3 43.8+ rd. 4 CO'". 11 34.C()+ ft. 61(3.2 + 14 24.08+ ft. 32.8+ ft. 17 36.76+ ft. 18 65.28+ rd. 21 722.9+ ft. 50 The term rorreyiond mg may be used instead of homologous, if preferred. The proof called for in 6 may be experimental rather than demonstrative. It follows what is supposed to be done in previous exercises. The proof called for in 7 may be made from measurement and comparison of the sides of similar polygons. 51 Ansivers: 1 9 times as large. 25 : 1. 5 468f sq. ft. 6 94.8+ ft 8y the methcid sliowu in 5, page ',2. A.,- is the same fra<'tioniil |part of AX that J// is of -11'. The two triangh'S .(.ry and (.VTare sijuilar, and A.r : A V = «/ : XY. The following definitions may lie taught as previously sliowu : A rin-h: is a plane figure hounded by a eurved line, all |ioints of which are ecjually distant fnjni a jioint within, called the center. The ein-uiiifei-eiKv of a circles is the line which hounds it. The itmiM't,;- is a straight line passing through the center and teruunat- ing at the circumference. The rialhix is a straight line connecting the center with any point in the circumference. .\n ,„■«■ is any por- tion of a circumferem-e. A rh.ml is a straight Hue connecting the extremities of an arc. A s^^jmnit is a portion of a circde Ixnnided by a chord and its arc. A sertnr is a i)c-tion of a circle bounded by two radii and the included arc. A sniii-rirric is a portion of a circle bounded by a diameter and half tlie circumference. Draw lines as indii ited in 10, and show that every point of the lierpendicnlar is ecpiidistant from the ends of the chord. 11 and 12 can be readily shown from 10. r>4: In 1, join the points by lines, wliicli mtiy be regarded as chords of the reciuired circle, and erect ptu-pendiculars froui the middle points. In drawing an inscribed angle (4), there are three possible con- ditions. The center may be in one of the sides of the angle, or, between the sides of the angle, or, withiiut the sides of the'angle. The first case should be proved first, the proof resting upon the 204 OKADED AltlTHMETIC. [Viii. n4 facts, (1) that the a,.«le nOC is oau.l tn tl.o s.„„ of tUo .mglcs ar(. ciual. From tl.is piuof thr otl.or cases may be easily provwl. A ,,'r„nt is a straife'lit line .■utting tl.o cir- cumference of a circle in t\v.> points. A i„n;/ent is a Inie which t.mclics a circum- ference in one point without cutting it. The proof of 6 rests upon the facts, (1) that n H„. from the l.oint of contact to tlie center is t\ie shortest .lis- : : tZ^^ tangent and t.,e center , ami ,0 t.,at the siK,H.st „e between a point and a straiglit line is the p,.rpen,l uMil.n luu. I„^ CO nee the point with tlie center of the circle ami upoa th^l^^:" a .liametlr. describe a circle wl.hdi wiU cut t^ie c™ ference of the first circle at two points. Join these " I" ^ with the Riven point outside of the circumterence, and the l.iu Stdrawnare the tangents required, '.nis can be easily proved from what has been proved before. Tefle giving 9 to 12, show that a circle is inscribed in a joly- «on when its ctrcuniferenee touches every si.le of the polygon, ad ?,:at a lie is circumscribed about a l-ol^l-'-.u - en its circum- ference passes through all the corners of the poly gon *"l!!to,draw such perpendiculars to radii as will intersect and foi™ a tnangle. ^^ ^^^^^ ^^^^^^^ .^ .^ ^^^^^^.^^^^^^^ wStors of the andes of a regular polygon meet m a point equally distant f'-o- all the sides and all the comers. This can be proved from the equality of the triangles. VIII. saj TEACHEHS' MANUAt. 205 In 16, tho pupils can readily find tho approximate ratio li^- measur.-ni.Mit. They can also estimate by oalciili.ti.,11 the piTinietir of a regular polygon inscr.ued in a circle haviiii; a given la.lius, and can see that the greater the number of sides of the inseribe.l' polygon, the nearer the perimeter apjiroachea tlie hiigtli of tli.' eiienmference. an From what has licen done previously (s.e Hook VI., page 1,S) tlie foriiiiilas in 1 to 5 eau \>i\ illustrated. 6 Tlie seetor COD may he diviili'.I into any number of trian^'les whi.si' altitu.Ie is A'Caiid the sum of whosi- buses is CD. 7 The area of the sc<,'im'iit CJ-.'I) is e.puil to the si'ctor COD —tho triangle COD, i.e. CA'DX^?- CD X To find tlio area of a ciienlar zone or eir- cular ring, subtract f:om thi' aiva of the circle tlie part not included in tlie zme or ring. L,-t the pupils determine what dimensicms must b.' given. Let the iiroofs called for in 10 to 13 lie made from construction and comparison of measurements. Pupils may be led to use what they already know of similar polygons. To draw an ellipse, stick two pins or tacks into tlie surface upon which the ellipse is to be drawn, and ti,' to tli.'iii a string longer than the distance between tliem. Describe witli a pi'iieil tlie eurv.., keeping the string constantly stretelie.l to its full lengtli. Tin' points A and D in tlie figure are the foci, CD the majo" „.ns, JJl tlie minm- axis. The c,','entr!vil ,j is the ratio that tli.'" minor a.xis bears to the major axis. An ellipse is a ]dane tigure l,„iimh>d bv such a curv9d line that if from any point in it straight lim-s be drawn to two puints within, called the foci, tlieir su.n will 1„. a constant quantity. 66 Answers: 9 175.14+ sq. in. 124.o6+ eu. in. 007.62+ sq. ft. 541+ cu. ft 10 600 sq. in., 1000 en. in. 'i:,:\\ s,j. it., '>',-{% cu. ft. 80J sq. ft., 49ift. cu. ft. 11 \\T, in. 12 4.4+ ft. 183S4 811. in. 19 23.34+ cu. ft. 20 493i cu. in. 206 (iltADKI) AltlTIIMKTIC. [VIII. m li 1 This iind th.> fi.UowiiiK paRo of cxcrcisrs sIumiM )«• t!V\i>{lit laiRcly r,„i„ the l.lorks. Till) fc.Uowini,' notes apply t.. soin.. of tlie iiMiru (litHi'\ilt points : A ,/;Af./i.// ./«'//'• is tliH openinK iM'tw.^.n two int.TSfctMiK planes. \ ^•;/-.v/,•.,/ ,,»;//« is th« opening of thieo planes whieh ni.'et at a eoninion ]ioint. \ ^■^•-,/„■,/,■„« is a solid lioun.ie,! by four l>lanes. A m„il">' pnl.j- l„;h:m U a polyluMlron having ecpial an.l reKular faees and equal polvliedral angles. A /-/ -" is a polyliclron l.oun,le,l l.y parall.'lo- Rfams au,l two e.p.al an.l paiall,'! polvfjons, .-alle.l liases. A n.jht ,,rhM is a prism whose lat.Tal e,li,'es are periieuilieular to the liases. A ,„inilM,.j,;i.r,l is a Iirism all of whose laces are parallehii,'ranis. \,H,mn,hl is a polvheilrou lioUBc.e.l liy triauKles that hav.' a eoni- mon vertex, and by a poly-on, ealled the base. A ,:;p,h,t i,;i'-«m„l is a pyramid whose has., is a regular poly-on, and whos.i vertex is directly above the center of the base. A ,i,„„ln„,d is a pyrami.l whose base is a ,piadrih,teral. Kor other su-nestions relating' to pyramids, etc., see Manual, page 171, and for rules :ulh i for, see Appendix, page ll.">. r,7 Am,nr»: 11 207.;U-.r. sq. ft. 22r,.1952 cu. ft. 12O0.,3 + so in 4021.248 en. in. 12 1IM72 sq. ft. ;n.7(i.-iO + sq. it. 13 ;i ,-,;i4:! eu ft. 41.451 + cu. ft. 14 2(in4.r,02 eu. iu. 15 S04.24i)G sq in 2144.tif..-.r. cu. in. 22(;.9S()(i scp ft., .•)21.r,,>,H.-. cu. ft. 2.^,8.:? so ft 12:i(l8.S04+eu.it. 16 2.-,.4 + in. 17 C sq. tt. 2..(.+ U. 8.G70 + ft. 18 10.4 + tt. 19 2.01 + in. 20 1797.8 + gal. A v^hn-e is a volume that may be generated by the rotation of a semi.cir.de upon the diameter as an axis. The dU,».U;- of a suhere is a straight line passing through the center an.l haying its extremities in the surfac... A .j,-ent rurle of a sphe.^ :s a s,.ctiou made bv cutting the sphere through the center. A smal eMe oi a sphere is a section made by cutting th.. sphere outside the center. A ,,/,/»-■/.•„/ ..,„« is the surfa..e of a spher.. luclu. ed between the circumfer..nces of tw.. parallel cut1..s, .ir between the circumference of a circle and a parallel plane tangent to the sphere. VIII. nH] TKACIIEliS' MANTAL. 207 k »i,lur!,„l „.,jni,nt is ii port i. ill of .1 s|ili('re inclu.li.l lictwci'ii two |.ar;ill,a oiiclfs, iir lii'twi'i'ii u circl.' iiiid 11 ]pi,iall.'l pi;,!!.' taiiK..|it to thu spliiTC. X Hphniiil sirhir \* tlui | .,iti„ii iif a .spliiTc wliirli may be KriMTatcd hy tlii' Mtatioii of a .rtipf ,,f n .iivl.. uiiim -a dianii'tiT of tlif Hplien^ as an axis. Notes a|)|iU-ilin to other cxerei.ses on this pa},'e will lie found on panes 17-' anil 17;i of the Manual. 58 Annirvm: 5 I'.t'.I.J lb. 1(1S4.8 lb. 1.1C,+ |t, L'. .->:>+ ft 6 8:1. 7. I ft. BL'ft. 9 2renl)..s. 10 ll'.(,+ tt. u"l l.-Jl bn' 10 ft. 12 l.i.;i+ ft. 13 i.;i+ ft. X (i.i+ ft. Al» A,mn;v.- llO.lHn+ft, l.l.O.'J + ft 'I •'! + ft 7 V + ft a (") 80 ft.; ,/,) 14;il' + eu. ft.; (,•) N.;t.J + .,1s. 3' Jol'dr,'' (00 sq. mi. l'.-.(!7,!(i.->78o(l()() sq. mi. 4 U time.s. L'7 ti s 5.\|„„i,= .01'0o708L'.l of earth Karth = „,}„, of the son. 6 0.1+ in. 7 Surtaee, 432 si), in. Volume, Gl(l..-.(i+ en. i,,. 3 ■.\Mr>+ in. 2.+ in.,2.5+iii.,10.4+in. 9 Diameter, 1 1.0+ in. Deptli ."id.'l+in 10 2481 balls. 11 ;«)40+ cu. in. 12 0.075 T. 13 .-,,!.i)+ .......' ; ('/) II,',; ft. 4 l.siii.,-, it. of op,.||in;rs^. {'■) 11)08 ft. ; 00 Alisiivn. 1 („; 12 ft.; (/,) 12 ft.; ,,-,. .'.;, |t 2 («) l.'S^^lt.; (I,) 12j3; (,■) 20 > 5 ft. 3 *41.s'-,. 5 $27.18. 6 Kli bundles .«i2.8;i.^ (•allowin- \ 7 ".8 Imndle.s. 8 (") 24.->13 ft.; (/,) 221 !,• ft. ■ ('/) «;74.87; (e) $l!tr,.8-, «1 .lH.sHvra.- 1 4.-)l».S(i. ft. 2700 sq. ft. 2 2i,^-'''C 3 •> ,, 7 C. 14,V„ C. 4 083 J. 5*42. 6 1i;icd. i512.'!'o(,' 7 no'j bricks ,«236 bricks. 9 30.35+ gal. ii'i Amwers: 1. («) 123040 brieks ; (/,, .Sl2lC,.41»- (,|lli-i + Imndle.s; (,/) .|S00 s,,. ft. 2 (.<) S:8.17 ; (/,) .SO. l.S ; (,.) i<\:, 44 ■ (rf)18i'olls; (,.) ;!9,i, yd. 3 igl8.,-,0. 4 31 ■ vd •"1.t,,i' 5 15 rolls .§23.94 .'ii;5.;iO. " ' ' " " fi3 Ansiivrs: 1 15 T. 2 2 T. 3 51 T 5 9i?ft. 6 422od. 1200 bn. I2'' eu vd •WOO gal. 8 90 bbl. 9 155 lb. 10 42 C. ■«i4(i, 3555J sq. yd. 12 $92.09. 4 111.9+ ft. 7 750(1 iiid. 11 09(; '1'. 208 OIIADED AUITHMKTIC. [vm. c»4 M.t,„ l«l.ft. 4*li).(i(». 5:i:^J b,l. it. «}-, ./«.s-»rr.v.-l;-ir,Hs,i.tt. «4..->() .l,S^jis,i.lt. ;!(.72 iiarkagCB. 2 848 boxcri. 3 41.") S(i. ft. $11.14. em Jn.irers : 2 t;:i+ it. lOJ. 8 4a(.+ sq. ft. 8/. 07 J«s(m'.s-.- 1 14.18+ .SCI. ft. 00.1.!+ sq. ft. 2 5U.l.'(i+ »!. in. 2(>1.(m;+ sq. rd. 3 14.i;8+ rd. -T-.TSW- rd. 4 I'.Mi.'io sq. It 10 ;.(.+ sq.ft. 4.;«i+ sq.ft. 17.45+ sq.ft. 5 80rd. 6 flOO.O.. 7 5,>,Vjft. 8 60y.l. ,-,S.7 + yd. 9 Kacli sidc,.sr,.;{+ft. 10 L.^ngtli of perpendicidar lino 72 ft. A.va, IKlt sq, ft. and 'MMi s,]. ft. 11 1413.72 Ml. ft. 12 S114(l.(t2. 13 31.8 lb. 14 51.!) ft. 68 A,mrer,: 1 (1(1 ft. 2 2400 sq. yd. 3 l(i07+ sq. ft. 4 Eacb side 50 vd. Altitude, 4S vd. 5 Sui-fa<^es, 1 to 4 ; voluni.'S, 1 tos' 6 1" 7-'i"- 8 H times as much. 9 «5.S8. 10 2.004+ in ' 11 58.8+ ft. 12 157.08 sq.ft. 13 17.7+ ft. 14 42.42;!+ ft 15 Ti'iau-le, 007.0+ sq. ft.; siiuaro, 120C. sq. ft.; circle, 1G50.1+ sq. ft. 16 «00.80. 17 Volumeof tnistum. ;!J of com-. 69 Ansu'e,.: 1 502C,,.-,0 sq. ft. 11300.7(i .sq. ft. -!S:i.a-.+ sq. ,„ 2 7.1+ ft. 3 2004.1+ sq. ft. 4 10.8+ ft. 5 4.0+ tons. 6 12.4+ ft. X 38.2+ in. X 22.0+ in. 7 93S251+ sq. ft. 34(;4343+ eu yds 8 12 sldngles 8 sbingles (14+ bundles. 9 8,48o+ rd. 10 1022,0+'". 11 C4 rd. 0+ ft. 12 10 mi. 3 hr. 32+ mm. 70 Ansurrs: 1 G.1,45 eu. in. 2 20.(1+. 3 ^-^^J^ 4 10(1 7 ft. 5 170.3 ft, 6 $44,44. 7 420,(1+ ft. 2((.l, 1 .,8+ ft 8 3^' 'ft. 9 inches. 10 400.1 ft. 4 A. 13,.,>„V «'l- ■''i- 11 22G1.95+ sq. ft. 1131.98+ sq. ft. 71-73 Nearly all these exercises are a review of business exercises given in Hooks VI. and VII. head tl.e impils to giv-e concrete examples before definitions. In some cases, as in descnb- ingthe various features of a promissory note, it would be well to have tlie examples written out in Ml. In some Slates no days of grace are allowed, and lu some t,tates days of grace are allowed only under certain circumstances, ihe VIII. 74:-7r.J TKACHEUS' MANUAL. M9 liiH- and iiraoticc in this regard, and I'kg, lionds, oi- dtlicr projiert^- teacher sliould ascertain tli' exphiin to tlie jmpils. A (■(illntKi-iil note is oni gi\ i n v. i as security, enipoweriuc thr [.a y, ,,. t. sell the same if the' n.rte is not paid ^vlien it heeonip;. mie. Xv. a,;;,i,nm,t„t!nn n„tc is one fcn- which t\»- maker receives no consideration. It is ^Mven for the purpose .,f Icdin,;,' credit to the payee. Forn.s of tliese notes can oe obtained at a hank. Tlie endorser of a note makes himself resiK.nsihle for its pay- ment. unless lie writes the words "without recour.se" before his name. Ill answer to 7, page 7.'!, there may be given examples of general Iiartuersluj,, ,n which the partners hav,. the sam,^ or ditfereiit capital, and examples of limited partnership, in which the n'spoii- sibility of one (n- more of the partners is limited to the amount invested. The general eu.stom of nu'rehants as well as htws of the State regulating partnerships shouhV U; ascertained and explained to the jpiipils. 74 -75 The rulings of the cash account should be made .-is indicated. After the form given on page To is carefully looked over by the pupils, they may be as;ced to write out the account in lull. 'I he balance Sept. L'l is .S24().(;i. 7«S-78 Tliese items should be used in writing cash and iicr- soiial aeeonnts, as previously shown. Lead the pupils bv abundant e.xamidcs to h.arn the use of the terms deliit and credit, debtor and creditor. A j.erson who receiv,.s may be called a debtor, an.l one who gives, a creditor. It may be helpful tor pupils in determining which side of the ivasli a.Tount certain transactions shall be placcl to aj.ply the .sam,! distincticm of receiving and giving to the money- drawer or cash-box. What is in the box in opening the account and what is put into it are to l,e debited to the account ; what is taken oat of the box aiul what remains in balancing the account are to \m: credited. _ The items given on page 70 should be written out by the pupils in the ioUuwinj; form .■ •< i »- 210 GUADED ARITHJIETIC. [VIII. 70 ^ '^ ■S }3 '^ "-» 'N C|5 *^ -s a *1 - - :; 'i ^ \ \ ' - i^ i.^ •^ *■> ^ ^ 'K 3) ci :o "^ "--^ "! 1=^ '■1 '^l Cl '^ ^ ii l"^ 5 rt 5 •^ -^. '••^ -i -o 5 =1 j ■> v-i > ?. 1 ji ^ '--'i g "^ 5 ?:^ ^ "^ 0^ -^ =<; ^ --"h >, .-i *, -^ -^ ", i4 ' ' ' ' S S*" Cj Oi'l » ■1 '§ fc •? fc 1 fe •^ .s 5 Ji 1 ^ ■1 J I: ^ C 1 ^ -c e 1 in ■1 &~, ^ ^ -4 '^^ .^ 3 3 s ■^ g i "S i ^ ^ li i 8 0^ ^( 00 eS : : - ' y - - ^ ' :: e:^^ - - :: - ^ ^ ,^ . 'T "^ c - ^ i '^ :: 1 '- '-1 -^ •- ' "S. - ^ - . - 2 3 J . - - ^^ :: :: ; :: - - «; i i VIII. 77] TKACHKIts' .MANUAL. 211 |5 <-■>•. -^ , ^. ~, ^ ,, „ ,, ^ , J 212 OUADKl) AIUTHMETIC. [VIII. 77 '■; '■- a cv '■: o y ■? "-. I t^ s-i "-. --1 '^1 '-^ '1 "1 I- li b4 ^ >" s *» ^ ^ SI S| rl "^ to "^ ■c = ■§ II I - - c H -S i ! 1 I !■ 5, ^ C :| S ^ ,, „ ^ ^ o, ., -, < ^ . = : . . — a, : : ; i i - i «3 ■ to ^7 "-O *- "^ i« m C^ "^ >."« »1 *, G( IN to "^j to_ 1«H ""H ©J *i 5^ si 1 II ^ C-; 0, ©^ lO ei •^t •-■ §. .. = : ; .• Ct to :; s?i J S - r = = = ■ = - -- r. < \~ -"4 * . VIJI. 78] 78 Tlie items i TKACHKUs' MAXUAL. 213 ■Sl'10,12, .„.,ki„g a total looting ,f S:.r3 58 Tl V "•""."' :s,j;;L:te;r;s::, "-^^^ '- ^-'--^ -- -- -- Tlie follon-iiig are tlie missiii;; in the order ouiitted : items of pages 82 aiul S3, TomdsP soon 1 V ^>-"'"» -'-a^'iK'e ])r., Jan. 12, j-u inube., ^^.'JU. Amos Laurence Ti- Ti., i - ij i i?!'*) 00 • Jan ■>', V.r r. ^ ■ V • ' '' ^^^ ''■'"*'' "" account, v-"uu, Jan. J,) Ijy cash m ful .*>(! 0(! T?,.n(-!, j- . rence's account, M>. '"'= °^ '""°^ ^^'^• Charles Smith Dr., Jan. 1.1, To mdse., .«1 ."lO • T,„ 1st. , U.r,n. Charles Smith Cr Jan 17 ,,:,'" ''!■'": 1'^' To rndse, w..od, ,13..0 ; Jan. «;, B;;'h;i:ni'j2;; So"'"' *•"•■ ' '^"'- ''' ^' Balance Sheet Dr., Jan. 31, To casl. on hand, ,$,-1.02. .Tan ^ JlSOO r ' t"''"^-, ^■^'■'- S'-t Cr., Jan.3i,r,y A.C; ' »1S.00; By net ra].ital, .?14.-fill • -urown, Kct capital Jan. 31, §14W.ll. ' Net capital Jan. 1, $1319.71. S4 On the following four pag«s are the Day Book and Tb,1~ accounts f ron, these items, ^^"8*^ 214 GKAUUll AlllTIIMlVnC. Day Book. [Mil. H4 Went Svii-tw, M"!l 1- •'*^'''- A. A. Etuis, T, I.' ijiil. X. <)■ iiwl'ims, " 4 lb. F '".f"^' 1 " I lv>x sardiiux, '' J lb. .Stni/rnajins, " lu lb. IS. II. smjar. Ci: ■i TSij ..' dair' lalmr, ! " ;.' loudH tjravel. MO i O'l J..' "" 5 U '■"-' ;..' 110 Dr. Dr. .vs 2. fir .i:;i 00 ,10 i 1 VIII. 84] TEACH KI!S' .MA.NLAL. Day Book. jl/'cv* \i;ftoil, Muy hj, JSHO, • 1. A. Kfiiti.t, To ..'I Ih. M,„l,ii rnfee, •*>' III, Liitt'iimn rirc, -1 lb. Miilitiiu rui.iiiifi — : 17 — iririim Viirtft; Til .1 III. I'lirrmts. I j "I m. fit. l,„:i.ijl,ii,r, i'"- 4 j Bij caahim "/c I " .'I "'. rheatiuil wivil. nr. .IIJ nr. Its ^4 A, A. KvitHH^ C.B. ■ [ Jill otM nil n/^ 4 I IHriiiii Curler, By .' iliiijH- liiliur, Cr. Cr. J.:M , S David Grant, To ,.' rfoj. Imnmvis, " S gal, P, H. moliaaes, Cr, I 5 j 2;j, j.i; g„i_ rineijar, ~'S C.B. Uimm Carter, j 4 To mah on "/^ nr. ■■'" 1 1 (III Dr. LJ .-,) 2-10 (iUAUKI) AKITHMKTIC. fVIII. H4 ;-. •M ? i •«> ~^l ~ Cl -1 '5';^ = *??S^^§•Sv!gSK; ft ''T\ ai a? ti » 4j ^ .||<3^Ci^||:|=-=||?*. ~2: H~i"^^-^;"3|ci^^5, B= ^^r S'' VIIl. 84] TEACirKKs' MANCAL. 217 ^ ii_ss l?i|3 sss;-: \-^\ ■■ '' .^z i fi — ."^ ■= ^ I?; § w '.^ •n ;, -, .; « s ;i s .? r^ Is I ■S- t (5- -^ £^ i : C) !■:: 218 OUADED ARITHMETIC. Lviii. Ha a tf ■- a 'i < .} 8 1 •= 5 5^5 ■S Ji .t -^ c tc 3 t: ■S - E = 2 — = _ = " n B •r o X ,, S 5 £ ' ■s i -^ I 7 '7- ^ « 5 3 C' 4, . - - r. '^ :5 ■" 'V ^^ 'a^ -— C -^ O O ^ i- - -S a- " = S « , s c i j! " S: 2- ^ C "^ o s ^i ^ ^ « .A - . C 3* S 2 t O >i i- J= ^ 0) .2 H 5 r- 5: - ^ = H . '^ ^ ^ S ' — *■£. o •" .3 5 ■s -_. ~- «j o c H IS X C -3 o : 5 g g 8 g *^ - » -s' s "^ s e C- — w^ -? -^^ ?: 2 VIII. 8<»] TEACllK[!.s' MAXrAU 219 ,_ 80 Tills iiccmint is siip|H.scil to ,>\tr\u\ fi i May 1.' tn May I'D. Till' last items of the tirst j.agu of tlir Day lidck slimild follows : apiii'ai' as !— I) George n. Gales, ^ lif/ ■! f'Jiiriijieiin Iftrrfi, 4 " ..' Wm-on!*'ni wiUnw Cr. J.IJ Hi: C.B. 4 To C'lah on "/o .1 i:i) On the second page ot the Day Bock tlierc will be tin' accmints of (Jatfs, Slin.'s; Hart, L' linos ; Ciatrs, 4 lines ; Katon, L' lines; Ho.kI, 4 lines ; and Gates, 4 lines. Tin; eash account on the tldnl page of the sheet will oceui.y 1':.' lines, and give room on the pa-e for the Ledger accounts of Jsaac Hart and A. JI. Katoii. Cash is credited with the folhjwing amounts: $4..1(), SVKU! SO S" lO «!3.04, S10.48, ,«i2..-,2, ,«il(l, $4, $I1.C8, SI..",-, *;!(), k^.V,, h's'.'M), i!ji4.74, $1.70, Sl.lM, §.58, and lialauee on hand Jfay 20, .«i21i8.4;i. On the fonrtli page of the slieet will apiiear tla. Led,!,'er acc.amts of Horace Hood, (! lines, and G. G. Gates, lines, and the lialance Sheet. The debit side of the liahmee Sheet will l.o : JMse. on hand, $1000 ; Cash on hand, $21)8.4;! ; Horaces Hcjod, .«I'.I8.,"0. The credit side will be: A. M. Eaton, $50.25; G. G. Gates, $27.71; Net capital May 20, $1300.82, making a total footing of S1;J!J0.81. The net gain ($13y.57) slamld bu noted as before. 220 a; r^ V .t tt. ■*-» *■ 'T z 5 -a ••P =, -^ 7Z rt .^ ^ , ki 'r tft 5 :=: ►J £ ^ 4i -* r - ■s Si- - "5 "= 2 g & = c " = " ^ « I -^' 7 2 o i 1 1 ? > := rt - i * i; ^ — ~ ■" i i i « c •= ? i ^- a* 2 r tfl O; 3: '3:: ' i I OHAnKl) AltlTIIMKTIC. [VIII. M7 ' i- *s - r •- ■> '"■: i i 1 - IS ^ - ■ ;>l! ~ -7. 't. "■ ~ . "■ li — ~ - Jl ■ • "// ■^ '^ i 3 i 7 - "^ •- -= "r — - X — ♦- ~ - s ^ 1 -r -7 ^ '''- ^' jt " c £ '£ i " 5 ''• "^ t~ S -J" ■" ;^ r " if £ c V- t ■- _^ •" ^ -- ;x X *j r .^ *" — ■ 2 *" it — tf. — » -?3^^^ > ;_ ^ It- , ^ ' ^ it:--. 5 - -2 ^ ■ 1 ■^ Z 5 '" .- in - ^ II C "Z; v: tX S- ^ .1- ji* ^^ "T c .— 1 '- C" ^ '- 1 ^T 5 ^ |^> S. -s ■/ I = •• ! < 't^ .^ — i; » " .. 'J 1 . • s: [ ^ >-- 1 - 1 ^ * t; i 1 C 1 1 ^ ■ --' -^ W^^i? •< • 'i2 i -z :> c - r : a * fe CS 1. C a: * '^ ■i Js C ^ 3 ■^ ' - ^ ' " »> i ac i 1 - _; i ^" 1^ '' *?■=:— = * ■^ .5 £ j: i; tlq !^ VIII. «»] TEACIIEHS' MANfAL. 2-21 Kl> Att.T till' (iist four items uiuI.t thf first diitv, Apr. 1, tin- I'l.llowiiis vutnrs .If the Dixy |!„„k should he na.h' ii. onh^r ■ Kirst patfe; C,l' items; A, a items; 11, a it,. ins ; A, 1 item; (.', 1 it,.,,,. Second pane ; II, ii items ; (J, 3 items ; A, L' items ; (;, ;j it, i„s ; \, 1 iteui ; A, .'i iti'iiLS. In the Cash aeeouiit, there will h.- 1) ,.|itri. ^ on the ih.jiit sidi! and 13 t.ntries on the i.re,lit shl,-. ( |,,.,|,.,.r ai'i.oitnt will !ilso h,. on the thinl |i,iKe of th,. si t. Itahin,.,. Sheet Dr.: Mils,., on liaii.l, .*i;i(«l(l.(P(l ; C^isl, ,,ii Im,,,!, $l'17.'i.;W; Amos I.a\v|.,.|ii.e (A), .SL'Sd.lU); liiUs i;e..,.iv,,Mi.. .S:.'.",(l. Or.; Charles Smith (Ci. . (-1 3 (I) (■■"') __(«! 1 1 'J 11 4 8 12 10 20 21 30 L'7 •>\ '>\ 1,S 1.-. (') 1.1 I.-! 2.S .')2 12 !) Loail the jaipils to diseoviT the fa,.t that in eai'h .,f the :il,nve series of nnml),.rs there is a constant diflVreni.e hetw,.,.n thi' ,.,.n- .spcutive terms, and tc make a ilelinition like the folhnvin-: Arilh- nn.ti,.al [u-o-ression is a seri..s of nnnil„.rs which im-ri'iise or decre;ise hy a constant dilTereiii.e. Any tcih, ,if an aiilhmein'al series ..my he fouml hy nmlti|,lving the common difference by the number of terms which iireccded it. 222 GRADED AUITHMETIC [VIII. 03 The other rules may be readily fouml from tlie given examples. To teaeh the rule lor tiiulius; the sum of an arithmetical series, place ujjon the board any series, as : 3 « 9 12 15 IS and, directly below it, the same series reversed. The two series will appear as follows : 3 G 9 12 15 18 21 IS 15 12 9_ 6 24 + 24 + 24 4- -'■! + -'+ + -1 = twice the sum of the series. Therefore the rule : The sum of an arithmetical series is e(iual to the product of one half the sum of the first and last terms nndtiplied by the number of terms. 93 Ansurrs: 6 2430. 7 00}. 8 5. 10 9S;i04 131070. 11 |25.()0 §51.15. 12 . 959049 .'i;S8573. 13 )i!ilT9.08. 14 9yr. 2 da. 15 $107;i7418.23. First ])lace upon the board two or more series of numbers in geometrical progression, as follows : 2 C 18 54 40 20 10 5 The pupils will see that each of the above series of numbers increases or diminishes by a constant ratio, (ieometrical progres- sion, therefore, is a series of numbers wliich increase or diminish by a constant ratio. To show liow any term or ratio of a geometrical serii'S may be found, write the factors of each term of the series upon the board ; thus : 2 6 18 54 2 2X3 2X3X3 2X3X3X3 By cjuestioning tiie pupils upon the above numbers, the following rules may be dwveloped : The last term of a geometrical series is equal to the first term multiplied by the ratio raised to a power whose degree is one less than the number of terms. The first term is equal to the last VIII. 04] TEACHEUS' MANUAL. •223 term diviJed by the ratio raised to a power whose degree is one less than tlie number of terms. The ratio is ecjual to tlie root wlioso index is one less tlum t)u. number of terms, of the ijuotient of the last term dividi-d by the lirst term. The sum of tlie series may be found as iudicated in 9. Otlier formulas may be expressed as follows : 94 AHsim-s: 1 Oii/. 2 41,;,/. 3 2 lb. (S !»/, 1 lb. @ 6/ 4 lb. @ (!/, 4 11,. (r,, <)/. 4 1 11,. (,,, r,()^, 1 lb. at (1.V, I'i II.. (ni $1 5 15 lb. 6 20 lb., 20 lb., no lb. 7 12J ^,1. 8 1 lb. 2,? oz. 9 lib. lloz. 11 ,,wt. r>.2.Sfrr. 10 ;?.9,S«. 11 (i oz. 12 3 oz Spwt. 13 .S2.4n. 14 2t%. 15 2,3..-.2 oz. 4!) lb. Many of the cpiestions and answers given on tlie remaining pages were given by business men, mecbanies, or specialists. ar* Jimrei-s: 1 !S;?0110. 2 JiiiOfiO los.s, 4 ^.^Ol 07 5 S.-.4.-5.4;!. 6 S21«.()l. 7 8-, lb. 8!) lb. 8 .Ian. 15. 9 About H%. 10 5% stoek i% better. 96 Ansiims: 1 .$110.08. 2 .'S172.72. 3 $392 20 4 .iSlin $7r, 5.;i47%. 5 §8.22. 6 .«i2.i.44. 7 27083 b"ri,.ks. 8 5281.44144 ft. 5276.97984 ft. 9 $21.50. 10 .S.12S +. 97 Ansii-ers: 1 ,155.472. 2 Kivets, gib.; burrs. J lb. 3 O.W yd. 4 f 172.40. 5 $,S50. 6 7:^'% gain .•(..^■,% loss. 7 $11,19 23 rolls 44 yd. 98 Ansim-s: 1 („) 8200 ft.; (i) 7520 ft.; (<■) 4700 ft.; (,/) 151 bundles; (,■) 120 bundles ; (/) 600ft,; (-,) .$,591..S9. 2 $879.30 51 3 .V, $9000 H. $101.-5 C, $7875. 4 $811.69. 5 125 bu. 140 bu. 6 (") 374 yd.; (4) lA.96sq.rd.; (<•) 80 sq, rd, ; (,/) 108 sq. rd. ; (e) 68 sq. rd. 9ft Answers : 1 2^g I'd. §5 cd. 2 105° 15'. 11 840 ims.s- books. 12 Jler., 5j times as lart u'ge as tli£ 1633^ sq. iu. 15 8.944+ rd. 8 rd. 15.576- moon. x« liiuu sq. ir r tx 224 OUADED vlUTHMKTIC. [VIII. 100 i 100 Amwm: 1 66 ril. 2 ft. 3 A. n sq, rd. 12 sq. yd. 7 sq. ft. "•2 sq. in. 2 8527.21. 3 443«3 perches. 4 KIDI bricks. 5 81 ;; 23/ 1.8,^ 3.7% 37% 7.3% 12« %. 6 H .strips 13 rolls $2.44 37 yd. 7 646.5 + l)bl. 101 An.^irers: 1 .SIOISO. 2 253 S(i. ft. 18 sq. in. 3 $120. 4 4096 bullets. 5 A, .$512.82 I!, 8687.18. 6 -'I) yd. 7 July 16. 8 }>3 pi M/y iM ara. 9 11 i% gain 11,',,; los.s 8% loss 15.1% gain 42?% gain. 10 41.4 + bl.l. 11 56.57 + gal. 108 Answers: 1 Eggs, .49, .49, .37, .13, .066, .05.5, .05C,, .16, .017, .118, .3.5, .06, .23, .189, .13, .0011, .21, .28, .09: Butter, .16, .17, .418, .0.54, .11, .41, .17, .18, ..58, .024, .77, .75, .22, .158, .:i6, .5, .98, .023, .8; Cheese, .415, .0017, .17, .12, .14, .18, .05, .25, .0013, .05, .014, .22, ,15, .15, .015, .02, .051, .2, .045. 2 1881. 3 1897. 4 1881. 5 1881. 6 22651554 2.3.596124. 103 Answers: 1 18 h. 37 min. 10 sec. 2 5:i;isS A. 3 illi Eng. mi. 4 4071 mi. 5 Wheit, SSOS bu. ; apples, 442 bu. ; beets, 442 bu,; carrots, 442 bu.; potatoes. 412 bu. ; corn. 221 bu.; salt, 230? bu. 6 80 perches. 7 567,;, 6171 T. 8 Wheat, 33J bu.; rye, 35? bu.; oats, 571 bu. ; barley, 41:-; bu. ; .salt, 40 bu.; potatoes, 331 bu. ; coal, 25 bu. '9 107! bu. 10 611b, 11 826.67, 12 810 gain, 13 600 lb. 14 .346.25. 15 H. 120. in. 16 D 5 ft. In 5, the approximate answers are given, the imshel being reckoned as containing IJ en, ft.; heaped Imsliel, U cu. ft., and 2 bu. ears as equal in bulk to 1 bu, shelled corn, 104 Amii-ers: 1 Pop, Sweden, 4784350; pop, r,S„ 02784000 ; children, Belgium, 829850; rate p.c, Bavaria, 21.2 ; rate p.c, Neth., 14.4. 2 £3906 5s. 3 8467.50. 4 About 13 m. 5 22 + %. 6 219%. 7 .82800. 8 4 h. 6.54 + min. P.M. 9 5(1% 5 . iQ:i Answers: 1 $504.07. 2 2d is greater discount l)y 2.85%; $22.80 .saved, 3 8'J,; 5 g.ain, 4S436-A 8163,',, 5 5',i,4S + gal, 6 16.56 mi, 7 10sh, 8 2096,11 81855,59 $2511,22, 9 831ft. 10 150% premium. \m Ans,ners: 1 (1) 12.08+ min. 21.6+ min. 31.6+ min. (2) 104,9+ min. 166.2+ min. (3) :i',>3.3+ rain. 2 78.5+ bu. 133.5 + bu 307.9+ bu. 3 1302.3+. bu. 4Su0.1+gah 5 1692.05+ gab 35 7+ in. 6 91.19+bbl. 9 .019 + -"". 10 1,655012""' 1.04+%. VIII. 107] TKACHKHS MANUAL. 107 An.um-s: 1 .40 X'.W/^.. 2 304.-> meters. 3 f>9.7!) times. 4 47..'U in. <).77r. in. 5 O.-'iKi sei'. 6 144.72 ft. 303.52 ft. 4(;(;.;f2 ft. 7 u(;i8 ft. 8 l.S8.'.t4 ft. ijcr sec 108 An.iirei:i: 1 4 ft. 2 Hi in. from end where ihe Kilo- gram weight is. 3 Of in. from miilille on siunc siile as 2 and :i. 4 129.]«)«. 5 « lb. on balance near tlie weight, ano 4 lb. on th.' other. 6 11 lb. on b.alance near the 12 lb. woijjht ; 10 lb. on other balance. Many solntions are i)o.ssil)lc for 7; <:;/. (!0 lb. in middle, and 40 lb. 2J- It. from nniu ; or, 40 lb. in middle, and 00 lb. .'ij ft. from man. 1 Jiixiivrs: 1 40+ lb., /.<■. anytliing in c. e.ss of 40 lb. 2 Anything in excess of J of jicrson's H-cij;lit. 3 2.T+ lb. 4 2(1 + lb. 5 73.5+ lb. 6 18849000 lb. 22.S25(;,S.75 lb. 7 .,+5 /■,- ^. 110 An.iimv; 1 (iO lb. 2 5.S.02 lb. 3 .84 in. 4 12 boys 12 boys. 5 Bo.xwood, .."i;!7+ ; majile. .10;)+. 6 7.20. 7 1.125. 8 2.5G 2.08""". 9 1.1:!+. 10 .OOOO"'". 11 20.;i2 ft. 111 Aii.iirei:s.- 2 .."i()5+. 3 ..'!70. 4 .,'i07. 5 204 207 ;i;i(l 352 .'iOO 440 405 528. 6 .Vbcmt 43'. 7 Abont 20 millions of millions of miles. 8 12. 7 + . TliC coniimted res\dt (jf 1 is 18.00+'"'. The difference ni.ay be disregarded, as it is not greater than jiroliable error in reading .scale used. In 8, sonml is reckoned as traveling 1120 ft. per second. 113 Ai/.sircr.s: 1 A little more than 11G°. 2 20.78 in. 3 3.144 ohms. 4 100.01 ohms 40.24 ohms. 5 4- its weight at surface i its weight at surface! 0. 6 2S00 nii. from centre. 7 25 lb. nj lb. 4 1b. 11,;, lb. Oilb. 8 24000 mi. 113 AiiKirrfK.- 1 20.8+ .sec. 2 1 JJ times as loud. 3 4.