? POLIO: = 2 ; 4 % %, % . BA ‘ my ae ? my % ada ‘ ; % THE UNIVERSITY OF ILLINOIS LIBRARY Sy Be : s 3 oi | MATHEMATICS LIBRAR} ee es . J. ) Practical Writing Portfollo, onetstiae: of the eae! § ap- - proved form of Writing-Book, combined with Penholder and Pen, Pen- eiper, Blotter, and Ruled Slope Lines, all in one book. The exclusive manufacture of this popular wr iting book has been transferred to me. . Five numbers, each......... ote ee” eee ON Cec edb eves dni we eerererse 25 « 4 Extra paper, per set of 12sheets. see eeet%es Sse erercvescos ee. eeeseeseeeersese 04 Capies, per set of 24 for each number... 2.0.0.0... 0. .cc cece cccccccvesscuces 05 | Buckham ( feary 8.) Handbooks for Y oung Teachers. Burst Steps. “Gloth, ENE N S RANE aa iotae gna e ao 18Iea Hig sis kine a Wa oe wie R 5 ee Ree wes 5 lst Bes a —~ The "Song. udget. A collection of Songs beh Music for schools and . 50m cational! gatherings. . Paper, sinall 4t0, pp. 72. 0... 2.00 cee cece eee ee 15 he School-Room Chorus, _A collection Of 200 cone suitable for Public and EPR Le SONOS, DOATKS, Sia alO, PP ld baseaic vo bos onde ace wee capac M35) Dickinson (J. W.) Limits of Oral Teaching. Paper, 8vo, pp. Ag She ape ORAS 15 Diplomas, printed to order from any design furnished, Specimens sent. (a) Bond paper, 14x17, for Bis see ae ae y aheiola Me din's ann wes ya Oh wR SE bie ath 5 00 50 “ewnaee Pewee eee score Cee eererere cee sees soni eee 6 50. (b) sf Ad 16x20, ty Dkccers wun ve ae cua 6 26 aU elednek Ree caaiey eeeeos 5 50 x ; Ae “ eee : ay BU, ccc atee wis adie coe oa ate A Sinan eeeee eerenees 8 50 “(c) Parchment, 5 | POR Se bicidas com Rage pe tbe AES © dvaants . 6 00 PLAGI MOG ITLOD GL (COPY: cd's scan nie sant dele bg dm tps lorem eetnss os Mire ie eae Db “Emerson (H. P.) Latin in High Schools. Paper, Syor DR US iti ooh a dela 25 Farnham (Geo. L.) The Sentence. Method of teaching Reading, Writing, Z and Spelling. A Manual for Teachers. Cloth, 12mo; pe. DON Sete ae enee 50 Fitch (Joshiua G.) The Art of Questioning. 3d Edition. laper, 12mo, pp. 86. _15 — The Art of Securing Attention. Paper, 16mo, pp. 48. Second edition. 15 Giffin (Wm. M.) How Not to Teach; or, 100 Things the Teacher should Rien PaO SAOTRON TID. 81 sp 5 acs 5 ose 'po sce bese vg e's vice t ben's PEP RCP TC oad Hailmann (W.N.) Kinder parten Manual. Pr tard Helps... . 95 — The New Education. A summary of Kindergarten Principles “and Methods. 8vo, pp. 146. Two series.. Each..... be acre ae evar ania Aly 2 00 Hendrick (Mary F.) A series of Questions in English and American Litera- ees te pal ee a for Class Drill and Private Study. Third Edition Revised. 12mo, pp. 100, interleaved. ............... BSL ES wore) aa. Goes ~ Hough. (F. B.) The Thousand Islands of the St. Lawrence. Cloth, 16mo, o pp. SPSCSHSHHS HLH HOS HOOD HOOSOHOHSOOD OOHS OOOH HOT HOD HOSHOHES AF EDMOEES SE OBE OEE SS [se eT TE may 5 BOOKS FOR EVERY TEACHER, + _» This series of Books for Teachers began with the issue in 1875 of ~~. - Common School Law for Common School Teachers. Within nine yearsmore — | - than one hundred books were issued, with an ageregate saleexceeding _ five hundred thousand copies. That no teacher’s library is fairly com- plete without at least several of these books is commonly admitted, and the titles of some of the more important are hereto appended... Besides his own publications, the undersigned deals largely in all Teacher’s and School Supplies of every kind. He also makes a speciality _ of works on Pedagogy; in other words, of works intended for the use of — Teachers, as distinguished from Educational Text-Books, His cata-. logue of over 8500 such works will be sent for two three-centstamps; and he will endeavor to fill promptly and cheaply ordersfor any Americanor English publications of this character. It is his intention to keep con- 4 stantly in stock every reputable pedagogical book now published; and he also keeps close watch of auction sales, both in this country and ~~ abroad, in order to secure such works as are now ‘‘out of print,” but which have present or historical value, Correspondence is solicited, and will receive attention. C. W. BARDEEN, Syracusn, N.Y. . . Agalite Slating for Blaci-boards. From an entirely new recipe and manu- factured exclusively for C. W. Bardeen, It is nota liquid buta powder, and is sent in tin boxes POST-PAID without extra charge. To Gover 50 TE6L Olle COAL. S. Bebe svi s ee anes poke tse Slate ares ee Pe at ss 00 Aids to School Discipline, containing 80 certificates, 120 Checks, 200 Cards, ~ 100 Single and Half Merits, Per box.;...............%5 a gh Sed chou, whe Supplied separately; per 10", Merits, 15 cts; Half Merits, 15 cts; Cards, 15 cts; Checks, 40 ets; Certificates, 50 cts.» AP: THe Ban ta abet Alden (Joseph) First Principles of Political Economy. Cloth, 16mo, pp.1538. 95 Bardeen (C. W.) Common School Law. A digest of. statue and common — law as to the relation of the Teacher to the Pupil, the Parent, andthe” ~— District. With 400 references to lezal decisions in 21 different states. To which are added the 1400 questions given at the first seven New York Examinations for State Certificates. 7th thousand. Cloth, 12m0, pp. 188 hoe ane Appendix. «ci. ee as "laiibaf ati « alverolory vet eiciwarst Fee pes ies cote ei poi Seite oc ale Hume, The story of a New York Teacher. Cloth, 16mo, pp. cS ih SET ee UN pine e My oe SA ei Ue “eer esee eeee eoaeihe Same ep res eee eee eereee « Seas Gal VERBAL PITFALLS. A manual of 1500 misused words, compiled from . _ ~ Jeading authorities. Cloth, 16mo., pp. 223... vie — Some Facts about our Public Schools. An argument for the Township — Ose es eee sess ntosn esos oF reese eens _ System. AO, DDe B2e . oacceoan raion vim & Raton as oss Genie eats ee, pee beet thers — Tancational ournalism. 8V0, PP. 80... 22.004. MMC Ae eat Bryce tee — The School Bulletin Year Book: Educational directory of the State of New York for 1879. . 8vo, pp. 40. with map. ....... goes Bee ar wade ‘Bassett (J. A.) LATITUDE, LONGITUDE AND TIME. Embracing a com- rehensive discussion, with over 100 illustrative questions and examples. ANN. AGMO, DPe 4 Qe a9 as ee a ea a oe a geawitel ths OO Beebe (Levi N.) First Steps among Figures. A Drill Book in the Funda- .— mental Rules of Arithmetic, based upon the Grube Method. Teachers’ — Edition: Cloth; 16mo, ppysen. 9. Pe ek Dae Lae Race 04003 vias etereacene : Pupils’ Edition. Cloth, 16m0., pp. 148. 5...5... . 6 fees oe eet AP a Reha Fe '. Beesau (Amable) The Spiril of Education. 16m0,, pp. 325...,.....+. ivecmeete a Bennett (Prof. C. W.) National Hducation. Paper, 8vo., pp. 28.......... 15 + Bible Lhe, in the Public Schools... Vols. 1, I... .... oa) oss Sepden & deel Oe 5 2 id ’ Bradford (W. H.) The Thiity Possible Problems of Percentage. Flexible Cloth, 16mo, pp. 34 Crew oesr ete weer aeese eeeseeore ee ; eenvee e. ye ~~ v4 be | he , ~~ * r & 4 1 si e es “ = FIRST STEPS AMONG. _ Bréures. A ae ag 7 Fundamental Rules of Arithmetic. TEACHERS ION. We / - sts B BE UPN . BEEBE, | He WV, / CANANDAIGUA, N. Y.. SEVENTH EDITION, ENLARGED AND CAREFULLY REVISED, SYRACUSE, N. Y.: C. W. BARDEEN, PUBLISHER. 1884. Sas , ny Lee ey oo " ost : ee Libran ee PREFACE TO TEACHERDS’? EDITION. In putting this work before the-public the author disclaims any ambitious schemes or “reat expectations,” but he wishes to have the book for the use of his assistant teachers both as to methods and examples. The author has used some parts of it for many years and feels confident that excellent results may be obtained by using it. The aim of the book is to give so much practice as to fix each method in the pupil’s inind, rather than to deal with the philosophy of each operation. leaving any teacher who believes that no step should be taken unless the pupil understands the reasoning process by which that step may be reached, to give it in his own way. It is possible that a few who see this book may have found that 7 times 8 are 56 by actual addition, yet those who have never added it may know the fact just as well for all practical purposes. If no one were to eat until he understood how food nourishes the system there would be a deal of hunger in the world. 4 PREFACE TO TEACHERS’ EDITION. This book deals only with the fundamental rules of arithmetic. ‘The intention is that they shall be so thoroughly mastered that much less time will be required for the remainder of the subject of arithmetic than would otherwise be needed. The teacher is to use the Teachers’ Edition © for one to two years before the pupil has advanced enough to use the Pupils’ Edition or in fact any book on arithmetic. It is recom- mended that teachers begin to teach numbers as given in the first part of this book after pupils who have the alphabet and words to learn have been in school four to six months. In each new operation the examples are very easy ; as more problems are given they gradually increase in difficulty. By teaching the four operations of addition, subtraction, multiplication, and division, from the first, the examples are of such a kind as to compel some thoughtfulness on the part of the pupil. Much pains has been taken to make exam- ples of a sort to interest the youngest pupils. Those teachers who wish to. teach only addition and subtraction at first can designate those examples involving multiplication or division by some mark, and omitting them ~ may return to them afterwards and so secure the variety of examples so essential to a pupil’s real progress. It has been found, however, by actual trial. that pupils may learn the four operations from the first without serious difficulty. PREFACE TO TEACHERS’ EDITION. é — The first pages are devoted to what is known as the “ Grube Method.” If the teacher pre- fers it, the schedules may be omitted, and, in passing through the first time, the multiplica- tion and division also, as before stated. The author hopes that teachers into whose hands this work may come will give it a thorough examination. Special attention is called to the treatment of numeration and notation. The examples are not all given in one place, to be forgotten, but are so placed as to review the subject often. Attention is called to the examples for rapid solving and the illustration of the easy exam- ples given under each rule. Also to the method of teaching long division and to the definition of addition. . The method of teaching the addition, sub- traction, multiplication and division tables is believed to be entirely new, so far as being published is concerned. The author dis- covered and used the method about ten years ago, and in his school has found it invaluable. To hear a recitation of a large class in tables and make the questions to each pupil promiscuous, and yet fuil enough to satisfy the teacher that the pupil has a thorough knowl- edge of the tables gone over, is not only very wearying to the teacher but is exceedingly dificult also. By the old method a pupil frequently acquires the habit of saying the 6 PREFACE TO TEACHERS’ EDITION. table from the beginning to find the result of any combination, as 7 times 6. To enable any one to make new series like those here given, I insert the method. The following is for g’s and review. In the given lines of ‘figures there is one more figure in the upper line than in the lower one: 3.94 56 ORS 4 5 On aRO mG If the upper line be written several times and the lower line in the same way as follows: 3.4 5.6.7 ° 8 9 3345 Oyo 4 5 6-7-8 9. 4°95 07. 3 ee 3.4 °5.6 9°83 9 - 3°45) 0 ee 6° 7 8 9 4 5 6 .Y 626 4 oe 3.4 5 6 7.8 9 3 4.95 G6 eRe 8.9 4.5 6 7 8) 9 4° 5y 0 ye The first 9 in the lower line comes one place before the 9 in -the: upper lines sia. second g in the lower line comes two places before the second g in the upper line, and so on until it has been under every figure i in the upper line. If. written farther, the series will be repeated as shown above, where 4 and 3 occur again at the end as they did at the beginning. The upper line of figures must be written one less number of times than there are figures In it. This series may be used for addition or multiplication, thus: 4 and 3 are 7, 5 and 4 are 9, &c., or 4 times 3 are 12, 5 times 4 are 20, &c, PREFACE TO TEACHERS’ EDITION. . 7 This arrangement is objectionable for most of the results in addition vary only by 2 or 4 and are not as promiscuous as they should be. By disarranging the upper line of figures we have 6, 9, 5, 8, 4, 7, 3. Re-writing this for the upper line and writing the lower line as before we have momo. 47° 3° 6 9-5 8 4:7 3 peer 7) oO 4. 5 6 78 Oo 4 5 nse) 42707360 9° 5 8 47 3 Meee 4 5-6-7 8. O04 5 67 Ppeseeo 4.07.3) 6955.8 4.7 3 Bepcmeaees O72 8 9 4°75 6.7 8-9 which, like the other series, contains every combination between 4 and 3 and g and 9g inclusive, and none repeated except by inver- sion as 4+5 and 5+4, but unlike that series it is entirely promiscuous. To make a series for subtraction, write the series as above, and write the sums of the numbers above, thus: Pominertets P2207 11.15 12 16 13° &c. Sogo 401 °3° 6 9 5.8 4°73 &e. ieperc 7) 5 9° 4-5 6 7 8 9. &e. Then copy for the minuends the upper num- bers, and for the subtrahends the lower ones and the series becomes Soi aei i i5-t2 16° 7 11 15-12 16.13°° &c. mesmo eS. 94.5560 °7 & 9 &c. In this book those for subtraction have been still further disarranged so that the results 8 | PREFACE TO TEACHERS’ EDITION. willnot be 695 84 7 3 and soon lest the pupils notice it and recite that instead of subtracting. For division find the products for the upper line instead of the sums. For division with remainders, which is an excellent preparation for short division, after having written the products above as before, add to each one of them a number less than the lower number in that column and write for the upper line these sums and for the lower line the lowest line of figures. In the first series of division with remain- ders, the remainders are very small, that it may be as easy as possible, In the first series the combinations do not go as far as g— that is 2 and g, 9 times 2, &c., but only to combinations of 2 3 4 5 and 6 with 2 3 4 5 and 6. The examples which follow immediately after the learning of any table involve only what is contained in the table. It is believed there is a very large amount of work for practice, both in the Pupils’ Edition and in the Teachers’ Edition, more than twice as much as in other works of the kind. The greatest care has been taken that they may proceed from the easiest to those involving every difficulty which pupils should meet at the age for which this book is designed. As anything is learned it is immediately put into use. ; LEVI N. BEEBE Canandaigua, N. Y., July, 1877. _ FIRST STEPS AMONG FIGURES. 9 Aur oa. gd Bee ie eet Ls 1 ONE, (See Appendix, page 185.) 1. Be sure the pupil has the idea of ove thing in distinction from ¢wo or more things. Illustrate by objects as much as possible, using small sticks, or square blocks % inch square and ¢ inch thick of different colors, or bright cents. Only ten of each are needed and if the teacher has a// these he can add interest to the exercises. An abacus, or numeral frame, is almost indis- pensable as a further help. Show the pupil that taking one article (as a bean, a cent or a block) one time, or putting it into a box or upon a book or table makes one article there, which is the interpretation of “once one is one.” —Let the fupz/ place the article, and thus impress his mind more thor- oughly with the idea once 1 is 1, written-1 xX 1 =Tty *In this book the multiplier is uniformly placed on the right of the sign XX: thus 2 times one are 2 will be written [oR es ie See Appendix, pp. 185-192, id FIRST STEPS AMONG FIGURES. 2. The idea of division may be taught in the following way: The teacher may place a pile of 2 blocks on a table or book and ask the pupil “ How many times one block have I in this pile?” Pupil: ‘“‘Two times.” Teacher: ‘One block in two blocks how many times?” Pupil: “Two times.” Teacher: ‘One in two how many times?” Pupil: “Two times.” (The pupil may use in each answer the word “twice” instead of the words “two times.”) Teach that this is written 2+-1=2 and should be read by the youngest pupils, 1 in 2 twice. * Schedule : o | tX1=1. (Read once one is one.) | I-+1=1. (Read one in one, once.) What can you find once in the school-room, in your pocket, on your face, at home? &c. What is there that moves on one wheel? TWO. 3. Teach in counting that the second of two things is not of itself tzevo, but one. In teaching number and in operations on numbers use objects for some time —at least *The schedules being written on the blackboard, the pupils are to be taught to read them, and eventually to make them themselves, FIRST’STEPS AMONG FIGURES. II three months to six months, until the pupil is thoroughly familiar with the composition of numbers. 4. Teach pupils to count to 11 and continue to teach counting daily until the pupil can count 100. Schedule : o | 1+1=2. (Read one and one are two.) 0 1x2=2. (Read twice one are two.) | 2—1=1. (Read one from two leaves one.) | 2-1=2. (Read one in two twice.) 2 is one more than what number? I is one less than what number? 2 is the double of what number? 2 is twice what number? 1 is one-half of what number ? 1 and 1 are*? 1 from 2 leaves? (1 from 2 leaves 1, because 1 and 1 are 2.) Mary has 2 sticks of candy ; she gives away 2 sticks ; how many sticks has she left ? 2 from 2 leaves? Henry had 2 marbles ; he gives none away: how many has he? Nothing from 2 leaves ? or ‘‘how much” in such examples, according to the sense. Show the pupils that 1 block placed on the table, and then another, make 2 blocks there, hence 2 is 2 times I. 12 FIRST STEPS AMONG FIGURES. What is there that moves on two wheels? Hold up two fingers. What have you on your head of which there are 2 and only 2? In the school room? At home? &c. What animals walk on 2 legs? 5. What is 2 of an apple? (Let the pupil take an apple and cut it into halves and ask him what one piece is called. Show him that if he takes one-half of the apple there is left as much as he takes.) What is 3 of 2 apples? Placing 2 apples on the table, let one of the pupils take half of them by leaving as much as he takes. (Vary the exercise by taking 2 of a stick of candy, 3 of 2 sticks, &c ) 6. 2 is the double of what number? Of what number is 1 one-half? What number must I double to get 2? I know a number that is 1 more than 1 ; what number is it? 7. What number must be added to 1 to get 2? Fred had 2 dimes and bought peaches with 1 dime. How many dimes had he left? (No analysis of these examples is expected ; simply a prompt answer.) FIRST STEPS AMONG FIGURES. 1 A slate pencil costs t cent, how much will 2 slate pencils cost? Charles* had a marble, and his sister had twice as many. How many did she have? How many slate pencils can you buy for 2 cents? How many 2-cent stamps can you buy for 2 cents? How many ti-cent stamps? (Both these and the following examples should be gone over many times, taking them in a different order each time and often giving them promiscuously. ) 8. Teach that there are 2 pints in a quart by pouring a pint cup full of water twice into a quart cup. ! What cost a quart of milk at 1 cent a pint? THREE. Schedule : 9g. Measuring by 1. | Iet, ft 93) 71 1+14+1=3. Pro xe = 3! ite 71 3—1—1=1, for 3—1=2 and 2—1 ie. *In such examples it will interest the class to use their names instead of those given, 14 FIRST STEPS AMONG FIGURES, Measuring by 2. Tei p22 (2213: II 2X1+1=3. (To be réeadpencesg III 3 and 1 are 3, or once 2 plus I are 3.) 3—2=1, 3—1=2. (To bereada2 from 3 leaves 1 and 1 from 3 leaves 2.) 3--2=1 (and 1 rem.) (To be read 2 in 3 once and 1 re- mainder. ) The pupils should read these schedules many times each, until they are familiar with the lan- guage. 1o. Illustrate by a pile of 3 blocks. How many times have I 2 blocks in the pile? Once. Take them away once then. How many are left, or how many remain? One. 2 blocks in 3 blocks how many times? Once and 1 re- mainder. 2 in 3 how many times? Once and I remainder. 11. To be written on the blackboard for pupils to bring written with the answers to reci- tation. 1 Loe 3—1+1=? I+1=? I+2=? 24+2=? FIRST STEPS AMONG FIGURES. 18 g—2=—° 3-1 2—2=? 2+1=? atit+i=? «m1X2+1=? 2--2=? 3+2=? 373=? 12. 3 is rt more than? 1 is t less than? 3 is 2 more than? 2 is 1 less than? 2 is t more than? 1 is 2 less than? . 3 is 3 times? 13. To illustrate tell a pupil to put one block or one cent on the desk and then another. Show the pupils that a block has been put upon the desk twice and that there are two blocks there; hence 2 times 1 block are 2 blocks ; also 2 times 1 orange are 2 oranges, and 2 times 1 pencil are 2 pencils, &c. 2 times any one thing are two of those things. 2 times I are 2. Show the pupils that 1 block taken 3 times or placed on a table 3 times makes 3 blocks there, hence 3 is 3 times 1. 14. This form of illustration may be used fot any multiplication. How many pints in a quart ? Teach pupils to write numbers as high as 20, 16 FIRST STEPS AMONG FIGURES. ~ It may be well to teach the writing of 12 before 10 or 11. Show the pupil by the abacus or otherwise 12 objects and show him that they are 1 ten and 2 ones. Show him that we can- not write 12 by any ome of our figures ; then teach him about ten’s place and one’s place. Do not use the word units for several weeks yet. 2 15. Teach pupils to count by 2’s from 2 to 6 and back to 2,-thus: 2, 4, 6: ‘6; 4ie2. Explain that 1 and 1 are equal numbers, that is equal to each other ; 1 and 2 as well as 2 and 3 are unequal numbers. 16. Give the pupils much practice in exam- ples like the following: 3—1—1+12% Tobe read, how many are 3 less 1 less 1 plus 1? or 3, subtract 1, subtract 1, add 1; or 3 minus 1 minus 1 plus 1.* *These are to be read by the teacher, thus: 3, add 2, subtract 1, divide by 2, multiply by 3. The examples may be read through and those who can answer raise the hand; the teacher call upon one most — unlikely to be correct for the answer ; if incorrect call upon — another until the correct answer be given, It may be best at first and perhaps often to have the result of the first step given by one pupil, the next step by the next, &c EE. g. teacher, 3,add 2, 1st pupil says ‘‘5,” Teacher, FIRST STEPS AMONG FIGURES. 17 2—1+2—1x1=? I+2—1—1x2+1=? 2+1—2xX3—1+1=? 14+1X1—1+2-1=13 Read with as much rapidity as the class can follow silently and give the answer at the end, ‘the rapidity being increased as the pupils have more practice. : 17, From what number can you take one and have one left ? ; Count by 2’s from 2 to Io. What number is twice 1 ? } | 18. I write a number once, and again, to get 23 what number did I write twice ? How many cents must you have to buy a 3-cent stamp ? _ Mary had to get a pound of tea. for $1 ; her mother gave her $3 ; how much money ought she to bring back? Henry learned 1 line in his primer, and his ? “subtract 1,”” 2d pupil says **4,”” Teacher, ‘divide by 2.” 3d pupil ‘¢ 2,” Teacher, ‘‘ multiply by 3.” 4th pupil ‘* 6,” Call on an inattentive pupil at any step in these examples for the answer. Usually, the teacher reads the whole example and the pupils give only the final answer, The foregoing examples are not written so as to be cor- rect for solving from the written or printed form for in that case 3-2 x 2 would mean, take 2 x 2 from 3, but it is to be read: 3 subtract 2, multiply by 2. 2 18 FIRST STEPS AMONG FIGURES. sister learned 1 line more than he did; how many did she learn? If 1 slate pencil cost 1 cent what will 3 slate pencils cost ? Anna found 3 roses in the garden; how can she divide them between her father and mother ? Can she give them an equal number ? How many roses must she have had in order to give her father 1, and her mother 1 also ? Count by 2’s from from 2 to 12. The counting may be sometimes in concert, oftener rst pupil say 2; 2d pupil, 4; 3d, 6 and so on, and perhaps oftenest one pupil give the whole series. Teach pupils to count rst, 2d, 3d, &c. FOUR. Schedule : 20. Measuring by 1. ES) as OM 71 1+1+1+1=4. (Because 1+1=2, 2+1=3, 3+1=4.) 7s 1X 4=>—4. 7 it 4—-1—1—1=1,0r 4—-I—I—1I—1=0. 7 i 471=4. 44 FIRST STEPS AMONG FIGURES. 19 Measuring by 2. fetee2. §2-+2—. 2X 2=4. Ii 2. 4—-2=—2, or 4—2—2=0. Itt 40 4--2=2. Measuring by 3. III 3. 3+1=4,1+3=4. 3Z34+1+1=5. ietie 4-3 —1,'4—I—. ———_——— IIIr 4. 4+3=1 (and 1 remainder.) : a1 3—2=? 2xX1+2=?2 4+3=? ai’? 3+1=? 3--2=? 3—1I=? 2ti=? 3+1=? 2+2=? Sem eet 4—3=? 3-2=7? 2xX1+1=? 4-I=? 4—2>=? 2x11? 4—4=? ae mad 4—2—2=? 42>? 4+4=? 22. Name animals with 4 legs ; with 2 legs. Name wagons and vehicles with 1 wheel ; 2 wheels ; 3 wheels; 4 wheels. Compare them. (For instance a wagon with 4 wheels has how many more wheels than one with 2 wheels? &c.) 23. 41S I more than ? t is 1 less than ? 40) FIRST STEPS AMONG FIGURES, I more than ? 1 less than ? 1 more than ? 2 less than ? r less than? 2 more than ? 2 less than? 2 more than? 1 is 1 more than? (Nothing.) 4 is 4 times? 24. Solve rapidly the following :. 2X2—3+2xX1+1—2xX2=? 4—I—1+1+1—3=how many less than 4? 3—2+3—1—1xX2—1=how many times 1? I+2—1-+2+2—1=how many more than 2? Teach to count by 2’s from 2 to 20 and back to 2. 25. I1X2—I X 3—2—2= how many less than 3? 3—2+1X2—1—2+1=? 4—2—1X3—1IX2—-1=? 26. What number must I double to get 4? Of what number is 4 the double ? Of what number is 2 one-half? What number can be taken twice from 4? What number is 2 more than 1? What number must I add to 2 to get 4? KR b® NH HNO S&S FW DN w FIRST STEPS AMONG FIGURES. 21 What number is 1-2 of four? How many less than 3 is the half of 4? 27. Minnie had 4 pinks which she neglected sadly ; one day 1 of them withered, the second day another, and the following day 1 more. How many fresh ones had she then ? How many $’s are $2+ $2? How many apples are 3 apples and 2 apples ? 28. Teach that there are 4 quarts in 1 gal- lon, taking a gallon measure and filling it by pouring a quart measure full of water into it 4 times. Nellie bought a gallon of milk; how many quarts did she buy P She paid x dime for each quart; how many dimes did she pay for the gallon? If 2 qts. of milk cost 2 di, can you get a gal. for 3 di.? How much can you get for the 3 di.? If I drink a quart of milk in 2 days, what part of a qt. do I drink in 1 da.?— 29. William having 4 apples, ate half of them and one more, how many had he left ? What number is 1 more than half of 4 ? Ann had 3 apples ; she gave an equal number to her mother, father and brother; how many did she give each ? 22 FIRST STEPS AMONG FIGURES, Sarah cut 1 apple into 2 equal pieces; what would you call one of the pieces ? Teach to count by 2’s from 2 to 40 and back to 2. By marking off paper or pasteboard, or bet- ter a thin board, and cutting, according to the following directions, an excellent aid in teach- ing notation and numeration may be obtained. By ruling both ways, mark off into squares 10 squares in a row and 21 rows, as shown below. Cut off one row or strip of 10 squares and then cut up the strip into single squares. After- wards cut off 10 strips of 10 squares, which will leave a large square containing 10 rows of small squares with 1o oe in each row. Babs | | Eek BEM eee BRE: (aaa | de ae Baba, Beha: Beeke The ro small squares cut up may be used to illustrate ones or units, and the strips, tens, FIRST STEPS AMONG FIGURES. 28 while the large square will represent hundreds. With them it will be easy to show that 10 ones equal a ten, and ten tens one hundred, and the teacher will show where the ones, tens, and hundreds are written in numbers. FIVE. Schedule ; 30. Measuring by r. ls 92S Be 71 1+1+1+1+1=5. Aamee Tax 5S, 71 5—1—I-1—-1=1, 71 5+1=5. 71 Measuring by 2. Pele 2s 62 -b 2-95. II 2. 2x2+1=5. (See note p. 23.) I 2. 5—2—2=1. IIIT 5. 5-~-2=2 (1 remainder.) By 3 IrIr 3. 3+2=5,2+3=5. It 2.- 3X1+2—5. - Prune 5— 42. 5+3=1 (2.rem.) 24 FIRST STEPS AMONG FIGURES. By 4. IIIs 4. 4+1=5, 1+4=56 I Ir 4X1I+I1=5. IIIII = 5—4=1. 5+4=1 (1 rem.) eT 3—1=7 IX3+1=? 5—2—2=% 5-2=? 2x2+1=7 5 ibaa 4—2=12 5—2=% 5+5=—? Sy Gt gt+i1+i1=? ~ 5-4! A+3=? 4—2-—2=—1 3x1+2=3 5—3=? 3+3=7? 4+1=? AA wood 5+4=7 2+2+1=? 5+3=? 2+2=? 32. 5 is I more than? 2 is 1 less than? 3 1s 2 less than? 4 is 2 more than? What number added to 2 will make 5? 5 is 4 more than ? 3 is 1 more than? 5 is 2 more than? 2 is 2 less than? 3 is 3 more than? 1 is 2 less than ? 5 is how many times 1? FIRST STEPS AMONG FIGURES. 25 (Since 2 is added twice and 1 is added to the result to get 5, 2 times 2+1=5 and since 2 may be subtracted from 5 twice and 1 will remain, 2 is contained in 5 twice and 1 remainder, or 5-2=2 [1 rem. ].) 33. Teach pupils to write and read Roman notation to V. Teach pupils to count by 2’s . from 2 to 50,and back to 2, and from 1 to 11 and back. 34. For rapid solving. 5—2—3+2X2—-3+2=? Ans. 3. 2X2+1—3X1X2—3+3=—?. Ans. 4. 4—1+2—3X2—1—2X3—1=? Ans. 2. B-be--1—2X2—3--1=—!, Ans. o. 5-3 +2—-3%3—1+2=?. Ans. 4, 2+2—3xX2+3—2+1—2=? Ans. 2. 2+1—2xX3—3+2X2 is how much more than 1 ? Ans. 3 more. 3+2—1—2+1—2x3—1=? Ans.2 Review these frequently. 35. Review counting by 2’s. Teach to count by 2’s, commencing with 1, to 21. 36. How many must I add to 3 to get 5? How many must be taken from 5 to get 3? Why? (Because z added to 3 makes 5.) 26 FIRST STEPS AMONG FIGURES. 37. How many times 2 must I add to 1 to get 5? I have taken away twice 1 from a certain number and 2 remains. What number was it? _ I have taken 2 from a certain number and 1 remains. What number was it? I have added 2 to a certain number and have 3. What number was it? How many gallons are 2 quarts. Ans. None. John had 5 dimes ; he bought 2 copy books, each of which cost 2 dimes. How many dimes did he keep? (Illustrate, using dimes.) George read a lesson once. Helen read it as many times as he did and two times more. How many times did she read it? A father had 5 peaches and gave them to his 3 children ; he gave the oldest 1 peach, and gave to each of the others an equal number ; how many did each of the younger children receive? A boy has 2 cents, he finds 2 cents; how many will he have to earn to have 5 cents? 1 is 2 of what number? James has 5 marbles, he loses 2 ; how many more than 2 has he left ? 2 boys are passing my house, and each boy FIRST STEPS AMONG FIGURES. 24 is driving 2 goats ; how many goats are pass- ing my house? David rode 1 horse from the pasture to the barn and at the same time led 2 others ; how many horses did he bring to the barn? Jane had 5 chickens. A rat ate 1 of them, and then a cat ate half of what were left and I more ; how many lived? A boy, having 4 pockets, has 2 apples in 1 of them ; one pocket is empty, and he has r apple in each of the other pockets : how many apples has he? 38. Teach counting by 2’s from 1 to 21 and back. Review former counting. (Do not teach all of this before giving other exercises, but require some of this kind of exercise daily until as much as is denoted above has been ac- complished. These directions apply to future countings. ) SIX, Schedule s 39. Measuring by 1. (Teach pupils to make these schedules.) 1 2 OE OS 23° FIRST STEPS AMONG FIGURES. I «1+1+1+1+1+1=6. Tee tae O? I 6—1I—-1—iI—I—i=1, I 6+1=6. I I sn Box Uitnn, mg ee oy » SD Oo 2 2+2+2=6. 29 (2 iteew. 2 6—2—2=2, 6—2—2—-2=0 6 O24, Sra 2eb eee Ge UT 3 eee 0 eee 6-0-3 350 3a Oz Gas Toad scl et As eet Greets Ii) 2.” xX +26, ZII3Iii 6. 6—4=2, 6—2==4: 6+4=1 (2 rem.) By 5. Il: ir 5. S+1=6,1-+52e6- Tors 25 Xa reo. 6. 6—5=1, 6—1=5, 6— 5—I=o. 6--5=1 (1 rem.) =a Se Carat Pes ke FIRST STEPS AMONG FIGURES. 29 4O. B—2>?r 2X2+1=? 4-—2=>? ae eet kA fr e+t? axe 6 6-2? 6—2—2—2=? I+4+1? 3+2=? A-1—?. eee — es SIS oe xXe4+2=? aoe §.6-— 257 Roa t 6—2—2=? 1xX5=—? 4—1—1—1==? 41. 6 is 3 more than? What is = of 6? 6 is 4+how many? 6 is 2 times what number ? How many times can you take 4 from 6? 6 is 3 times what number? 4 less than 6 is? 6isz+? What number is 3 less than 6? What number is half of 4? 6isi1+? 42. For rapid solving. It+2+2=? 3+2—1+2=—? Iti+i+2=—? 2+2+2=? 3+2—-1+2x3=? §—24+3—2—3+1xX2+1—3+1? Ans. 3. eee tot Sl — 2X 2— 2 - aol Ans. 2. 3 f, 36 FIRST STEPS AMONG FIGURES. 2+3—2xX2—5xX3+2—3xX1+2? Ans. 4. 4—3+2-—-1X2+24+2+2+4+2+2+2? Ans. 16. 5t+2+24+ 24+2424+24+2+2+2+42+42? Ans. 27. 43. Count by 3’s from 3 to 12. Review counting. Count by 3’s from 3 to 18. Review the counting already taught. Teach Roman nota- tion to X. 44. For rapid solving. 6—2—2+3—2x2+2+2 7%? Ans. lo. 2X3—4+3—1+2+2+2+2+2+2+2! Ans. 18. §—3t+1+3—2424+3+2+27242F2 +2+2? Ans. 19. +3 —24+3+173+2+2+2+2+2=?% ANSo 12, 5—2—2X2—14+4—3X34+3+34+2+2 +2=% Ans. 18. 2+3—24+37242 $2724 2m ¢. Anse i197. 4—3+2+2—3x2+2+2+/? Ans. 8. 6+3+3—2 X 24343 +3 eee +22 Ans. 23. 45. Examples in addition may be written on FIRST STEPS AMONG FIGURES. 31 the board, a few for the pupils to solve each day on their slates and bring to class; and ad- ditional examples may be read to the class and solved on the slates or on the board.* 46. 3 cts. is 2 cts. more than John has; how many has he? 3 cts. is 1-2 of what money Jane has; how much has she P 2 cts. is 3 cts. less than an orange cost; how many cts. did it cost? 1 is how many less than 6? 6 cts. will buy how many 3 cent. stamps? 2 ct.stamps? 2ct. stamps? 5 ct. stamps? George had 1-2 of 6 cents; how many had he? Clara had 6 flowers; she gave them to her father and mother. If she gave each of them the same number, how many did her father get? Her mother? Charles had 6 cts.; he lost 2 of them; how many had he left P Carrie had 6 peaches: she gave her father 2 of them and her mother 2; how many did she keep ? 3 cts. is 3 cts. more than Byron’s money; how much money has he? " See T. Ed., p. 84, and P. Ed., p. 33. 32 FIRST STEPS AMONG FIGURES, 47. 3+2—1+2—5+3—2? Ans. 4. 1-2 of 6+2—3 is what part of 4? Ans. 1-2. 1+2+2—3—1+3—2=% Ans. 2. 3+2—1—34+1X34343+3+2+2=? Ans. 19. 6—4+2 —1X2-— 24243737 o eee 2? Ans, 18. I put down a number once and again and again to get 6. What is the number ? 48. Count by 3’s from 3 to 24. Review the counting. From what number can you take 2X2 and keep 1? What number must I double to get 4? What number is one less than 5? What is 1-2 of 4? What is 14 of the number 1 less than 5? What number is one less than 4 of 4? Augusta had 5 cents. She lost 1 of them, and spent % of what she had left, and then found 3 cents. How many had she then? (Solve one step at a time.) 49. For the method of making series like the following, see preface. For method of use see — pages 42 and 43. For addition.* * Teach each series thoroughly before taking the next. FIRST STEPS AMONG FIGURES. » P38 a b Cc d e dag tt ce tates aL se Nh 4 1 2 For subtraction. a b Ps d e f 2 Bee > 3 54/3 1163 4 Pees | 2 22 {2 1 [3.4 1 For multiplication. a b Cc d e f g Riinleze Oblate (iy 2. Ort 9 4 TeeetOn te u200'| 12. }0 <1] 2.0 For division. a ae Cc d aes 6 2 ear OG: Tra eo Lee: : PSE ee For addition. (Re-arranged.) a b £ d e x 8 h 4 3 2113.0 Z 2 ve eeereisert | 3. 2) | 1; 3 2 : 2 For subtraction. Bese ed a b Cc d e f g | h 5 6 hy SE ee Baths 2 2 : 2 Ze sel 50. Teach Roman notation to XVIII. Review counting by 2’s commencing with 2, and also with 1, to 60 and 61. | Teach counting by 3’s commencing with 3 to 30. 3 34 FIRST STEPS AMONG FIGURES, The examples given under previous numbers should be frequently reviewed, so that the pupil may become quite familiar with the formation and use of numbers. SEVEN. 51. At this stage pupils should make the schedule from their memory on the plan of those already given. Schedule : Measuring by tr. pie Ges ME? eee Fah a Gp 1 hes Rens wit cet! ce 9 Weg as Bis NY domme Bap Gk 7—I—1—I—1—1—1=1, or 7—1—1—1 I—I—I—I=0, 7=1=7 By 2. II. 2. -2+2--2+ 107, L-E 2.) 2x34 1==7; II 2 7—2—2—2==1. 1. 7-9.. 7--2=3 (1 rem.) Teach pupils to write Arabic to 199. * FIRST STEPS AMONG FIGURES. Pera) Peetet 3. 3X 2+ Tae, I I Oe ea Gate B By 5. Pepreeie TE 6 5 2=7, It 2 5XI+2==7. Pelee tet tf 7.) >7—coez, 7-5==I (2 rem.) ITri¢i31: 6. 6+1=%. It 6X1+1==9. Seer let Io 727 7 —6==1. 7--6=1 (1 rem.) 52. 5ti=? 7—5=? 6+-4=—=? 6—2=? IX5+I=? 47—3=—? 35 4 36 FIRST STEPS AMONG FIGURES. SI=? 3X2+I—? 2X24+2—? 2X2—? 6—2—2=? 7—2—2—2em 3t+2=? 2X2? IX5+2=—? 4—2=? 7+4=? 5—I=? 6+1=? 3X2=—? 73 2xX2+1=?2xX2+3=? 53- 7 1S 3 more than? What is 4 of the number 1 less than 7? 7 is 1 more than twice what number? What number is 3 less than 7? What number must be added to 3 to get 7? How many times can you subtract 2 from 7? 7is 4+? 3 is 4 of? I added 3 to a certain number and got 5; what number was it? I pear is what part of 7 pears? 2 pears are how many times 1 pear? How much must be taken from 7 to leave 3? What number must be added to z to get 7? How many times 3 must I add to 1 to get 7? Count by 3’s from 3 to 42. Review the counting. How many times can you subtract 5 from 7 ? 54. 61s double what number ? 3 is 2 less than? What number is one less than 7 ? What is % of the number 1 less than 7 ? FIRST STEPS AMONG FIGURES. 55. For addition. a b Cc d e f 314 | 203 | 120 | 312 | 0131 | 420 123 | 412 | 312 | 342 | 3441 | 234 For subtraction. a b Cc d e f 456 | 277 | 615 | 454 | 353 | 2643 432) |-143 | 412 | 312-| 341] 2312 For multiplication. Bin O81 | 102 | 50 ee ia 2 132 | Te For division. a hae Perc d e 26 | o1 40 | 36 023 Ree feret |) 21 |_ 32° 322 Same re-arranged. * For addition. a bofiec d |e f 241 | 302 | 413 | 021 | 302 | 4130 123 | 412 | 341 | 231 | 234| 1234 | For subtraction. aaab- Db Cc d @ f 435 | 256 | 534 | 634 | 741 | 6375 AZ2 |-143'| 412 | 312 | 341 | 2312 For multiplication. geebelec t djive f 20 | 12 | 03] 12] 031 12 | 12| 31] 23] 123 * For review. 1 ochayag Waleed Bikace lis tad as 2 37 nr 38 FIRST STEPS AMONG FIGURES, For division. a 1 bc | didseseme 32 | 60; 40 | 63 | 201 (123) 223i 240aes 56. After earning 3 cts., Fanny had § cts. ; how many had she before ? How many must you add to 3 to make 7? Nellie has 2 pencils, and Sarah has 1 more than Nellie ; how many have Jo¢/ of them ? Marcus has 4 marbles and Arthur has 2 less ; how many have both ? Mary has 6 pins and Stella has 2 ; how many more has Mary than Stella? 57. When the pupils become listless or rest- less, or a minute or two of spare time is at com- mand, the following examples and like exam- ples given through the book will be found both useful and interesting. Use them often. 34+242—-34+242—-1X2+ 37 2h ee Ans. 4. §—3 +1 X2+3+2—-34-4—2 3 2 +1? Ans, 7. 4+ 3+-2—-3+1%2—3++2-—8ix oe —2? Ans. 5. 7—2—-344—243—5 +1 Ke 3oaeee —2? Ans. °. FIRST STEPS AMONG FIGURES. 39 2+5—3--2—2+3X2—2+2x3+2-11 ams. 2, ee ey 5 A 2-3 1X3 — 4+ 38+ 4—47% Ans. 3. 6— 44-3 +2—1-3 X2+3-—44+24+24+3+ eet! Ars 216. 59. Teach the pupils to read numbers to root. Show them that the figure in the third place represents hundreds (or is named hundreds ) Read the following numbers : toe 2 2. 395. awe Uy AS i 5: 341. 6.1704: y Pew AY Ds OLA, Ga 7oue 10. Sgt, Lkee Oo, 12, 260. Someta 14.) 750, 15. 981. LOs8 700; Teach the pupils that the fourth place rep- resents thousands (or is named thousands.) Teach the pupils to write a comma between hundreds and thousands before reading a num- ber. Read: BeOS eld. 374%) LO75410) 420. 7308. BeeOtoOen 227.7075. 2le Al18.. :.245:.9400. 2eam7504.. 20. 1000, 27. 7050. -28. 8004. 60. Teach pupils to write numbers to 1,000 in Arabic, and in Roman to XX, ~ 37RTB 40 FIRST STEPS AMONG FIGURES. 1. Write two hundred forty-five (in Arabic.) 2 “* five hundred sixty-one. 3 ‘¢ one hundred thirty four. 4 ‘¢ seven hundred twenty-one. 5 ‘¢ three hundred eighty-six. 6 ‘“¢ four hundred sixteen. 7 ‘“¢ nine hundred twenty-one. 8 “¢ six hundred thirty-two. 9. ‘¢ eight hundred seventy-nine. 61. Give more examples like the above until the pupils write them readily. Then give the fol- lowing : 10. Write three hundred nine. * Pt: “eight hundred forty. Vip “‘ four hundred fifteen. r3: ‘* six hundred thirty-seven. 14. “ two hundred ninety. 1a “sixty-four. 16. “« seven hundred two. ay Fe ‘“* five hundred six. 18. “* three hundred thirty. IQ. ** four hundred one. Review these often. 62. g+24+343 4+ 24+2434+3+3 245 ee Ans. 31. FIRST STEPS AMONG FIGURES. 4i Mae Stet oto tots F242 242 pare acns, 27. mg poe t2t+3+3+3+3 F242? Ans. 28, oP HG oe i BPS eae ps Be Sd i SB Sogrias i PANS. 13. 6+2—1X3—44+1X2+3+34+3+2+2 +3437: Ans. 25. Vee aeie te 1 ot St 3+ 24 2 + 2-23 3+2? Ans. 29. Sage 3t3t2t2t+2+2+3+3+3 + 2? Ans. 33. fee 3 2 tes +2 te 3 45 St 3 +2+2% Ans. 25. Some boys are sliding down hill. There are 3 sleds and two boys on each sled, how many boys are there ? James had 7 apples; he ate one and gave his sister half of the rest. How many did he give his sister ? John had two apples ; he cut each of them in halves. How many halves had he? How many horses in 2 two-horse teams ? A stingy boy had 5 sticks of candy; he would neither eat any nor give any away. How many did he keep? 42 FIRST STEPS AMONG FIGURES. A generous boy had 3 sticks of candy; he gave his sister 2 sticks and he ate half a stick. How many had he left? Ralph had 5 peaches ; he gave 2 of them to his little sister, I to his father, I to his mother, and ate I himself. How many had he left ? 64. Count by 3’s from 3 to 5o. Count by 2’s from 2 to 60. Count by 2’s from I to 61. =. Teach Roman notation to XXVIII. Write in letters.19, 13. 21, 14, 11, $,)8O,; 2m, 12,:20,:177,01025, 27smeos 66. Count by 3’s from I to 22. Review counting. EIGHT, 67. Pupils may make the schedule like pre- vious ones. 68. Give only a small part of these series each day and give with it slate examples from pp. 93 to 94, and oral exercises in the exam- ples following the series. By this variety much mo.e work may be accomplished without the weariness resulting from too much sameness. The series should on each succeeding day be FIRST STEPS AMONG FIGURES. 43 ey reviewed.* For a review after completion take the re-arranged series. | For addition. a b c d e f g Bryne 135.2 4 3214135455 a AsY Meee a 120) 2-4 1 | 9273 T)}-2 2.434 1 234 For subtraction. a b C d e f- | 2 O5310350/8521/8571476)47.5|4 feeeeteeteseii2 2.1 j.4 3-213 41) 2 3-4 1:3 For multiplication. a b ¢ d e 2 GARE OP TaD Ore Reg ee ier | 02" 2" 14? 3 For division. a b fe d Meet tao 251-0. 3021.6 28 Gemeente a Ios 122-2 354 For addition. Re-arranged. a b é d Creole b g WA S144 2531042 153%) 425): 3 Peres | AS 19.471) 2.3 Ail 3T. 236) 4 Fcr subtraction. a b C fa gd fates 2 345|627|665/484/875]567) 3 213)213/412)342/3411432 1 * Leave on the board the previous day’s lesson in series and add to it as much more ofthe series as can be mastered with the review. 44 FIRST STEPS AMONG FIGURES, 69. Teach Roman notation to XXX, What 2 equal numbers make 8 ? What is half of 8? What number is one less than half of 8? What number can you double and get 8 ? From what number can you take 2x3 and have I left? Count by 3’s from 1 to 4o. I write a number four times and add._ I get 8, what is the number ? Henry had half of 8 cents. How many had he? 8 cents will buy how many I-cent stamps ? 4-cent stamps (I. Revenue) ? 2-cent stamps? 5:cent stamps? 3-cent stamps? 3 lemons is 3? of how many lemons ? Lewis brought 6 eggs from the barn ; he broke half of them. How many whole ones were left 2 George has 3 cents, he finds 2 cents ; how many must he earn to have 8 cents ? William had 2 sticks of candy, he ate half of a stick, and his sister half a stick ; how much candy had he left? What number is I less than half of 6 ? Jane’s bird hatched 3 young birds and there FIRST STI PS AMONG FIGURES. - 45 were 2 eggs which did not hatch; how many eggs in the nest at first ? What is half of the number I less than 7 ? William bought 3 marbles at 2 cents each ; how much should he pay for them? How many tops at 3 cents apiece can Ed- ward buy for 8 cents? For 7 cents ? 6 cents is 4 cents more than Robert’s money, how much money has he ? 3X2—4+3—2X1—1X3+1=? Ans. 7. 7—3+2+2—1%X4—3 is how many less than 8? Ans. 3. 2+3—I+2—1X6+2+5—2=3 is + of what number? Ans. 4. 6+2—5+1+2+1x2. is 1 less. than? TIS; 7 5+2—-153X4724+34+3+3+3+3+2+ Pia AUS, 23. | 3723 3-1X34+3+3+3+3+3+2+2+ ef ANS. 25, j—a—teavqtatitgigt2tatet . at a2 ANS. 3%. §+3> 4x3¢+3t3+3+3t2+2t+3+1 +3+2% Ans, 31, Seton L 2 2st s+3+2+2+ 3+ gt 34243? Ans. 29. 46 - FIRST STEPS AMONG THIGUEES. Seer Wer a + 3 aa 3a Ans. 27. 44+3+24+24+34+34+3+24+24+34+3+2+ S72 Fv ANS, 24. 643 +2 ieea-eess eae pate sac ANSie3 3; 44+2434+34+3+2+2434+3+2+2? Ans. 29. 7+34+-34¢34+3+24243 73550 Ans. 30: 442434343 4.224223 ee 3-133? SAnseetos 6+3+2+2—3 X% 24343735 34 5eeee 2t2+2? Ans. 27. Oral exercise. Place the figures of any of these examples which involve only addition in a column on the board, let one pupil add them upward. then another downward, then another add them upward, but commence with I and so get a result I greater than before, add down- ward in the same way commencing with one ; then use 2 instead of I and also 3 instead of 2. This makes 8 different examples instead of one and gives excellent practice. ; How many feet have 2 pigs ? 8 is 2 more than what number ? FIRST STEPS AMONG FIGURES. 47 8 is 2 more than twice what number ? How many legs have 3 hens ? What number is 5 less than 8? How many times can you substract 3 from 8? How many legs have a rabbit and a bird to- gether P 8 peaches are how many times 2 peaches P 1 pencil is what part of 8 pencils? How many wheels have 4 sulkies ? What number must I add to 5 to get 8? 4 is half of what number ? From what number can you take 2 times 2 and have 3 left ? 2 is how many less than 6? Count by 2’s from 1 to 62. 6é 6“ 3's iT4 3 to 60. Emeeass, ‘4 2 t0.60; ame o3'sy oS P ta Gt, Teach Roman notation to XXXIX. _ Write in letters 26, 18, 34, 9, 16, 22, 37, 19, 35) 21, 36, 39, 17. 7 is how many more than 5? John had 7 sticks of candy; he ate 3 and gave away % of the rest. How many did he give away? How many did he keep? Count by 3’s from 2 to 62. | 48 FIRST STEPS AMONG FIGURES. Teach Roman notation to L. Write in letters 34, 16, 25, 10,540, 20m 17, 26, 33, 46, 37, 22, 50, 44, 27, 33, 49. Write in figures XVIII, XXXIV, XV, XL, XXVI, XII, XXXVI, XIV, XVI NINE. Pupils make a schedule. For addition. aoe. ab Cc ps 24 yet ah | g 41513241153 at 136/241 |362 12314121345 31412|345|234 h i j 451/532 511] 235 For subtraction. a b c d e Bad 736 | 498 | 756 | 948 | 585 | 384 | 757 412 | 345 | 125 | 312 | 341 | 232 | 345 viene ha Kk cas 672 | 69 327 tei For multiplication. a={ bel vce edie esl ee ee h i 24. | 101] 21)30)21 | 30 |s2a ae {25/231 (521401 ht eel sarees 234 FIRST STEPS AMONG FIGURES. 49 For division. wears: |.d- psecpf fogs | oh i 30 | 44 | 30 | 61 | 84 | 94 | 42 | 062 Papieea tie.) 31) 22, | g2 |.¥2 |321 Re-arranged and including o. For addition. a b c d e f g h 526 | 301 | 452 | 831 | 740 | 523 | 214 | 365 123 | 435 | 123 | 051 | 226 | 341 | 723 | 614 I | 2201014 529 | 34 For subtraction. a b Cc d e f go. fy h 268 | 643 | 485 | 756 | 796 | 467 | 967 | 759 123 | 412 | 453 | 215 | 430 | 213 | 465 | 123 i k l 835 | 6839 404 | 3215 For multiplication. | Serepeec. |) die |e fojig. | h pk Aaeta20 (si. | 20 (31 | 24 | 14-| 20 Popieaz. | 23.| 12°] 34 |-22°) 11} 14 For division. epimDatecet d)|.e.|)f Jogo) hf 32 | 09 | 40 | 61 | 86] 04 | 82 | 343 Geer su 4F 2 ties | ar | 42°} 125 4 5° FIRST STEPS AMONG FIGURES. How many 1’s make 9? From what number can I take 9g 1’s and have nothing left ? How many 3 cent stamps can you buy for 9 cents? 2 cent stamps? 1 cent? 5 cent? 4 cent? How many oranges at 9g cts. each can you buy for g cts. 2 How many times can I take 2 from g and have 1 left? George had 9g peaches ; he ate one and gave you half of what were left, how many did he give you 2 What number taken 3 times will make 9 ? Henry gave each of his 3 playmates 3 plums. How many did he give away? What is half of the number 1 less than 7 ? William has 9 wheels. He has wheels for how many three-wheeled velocipedes ? 5 and how many make 9g ? What number taken twice and 3 added makes 9? What is half the number 1 less than g ? Theodore had g marbles, he lost 4 of them, how many had he left ? 44-35 f=¢. FIRST STEPS AMONG FIGURES. §t 8 is 3 and how many? 1-3 of 6 and how many make 6? Count by 4’s from 4 to 20. Review. What number is 1 less than 4% of 8? g=6+? 5 from g leaves? What cost 4 lemons at 2 cts. each ? What cost 2 marbles at 3 cts. each ? 5+3—4—24+3+4-—6=? Ans. 3. 34+4—2+4—34+3+2+1=? Ans. 4. 64+3—5+4-—1X2+7+2=? Ans. 9. oo = Sree eg r4 e455 St Ans. 8. | Meera Sy 3K Sti 4 Pf Ant 3. Oe Ramer SP ws wn ld we SL 20. st5r4tas et2t24343t2te+ 34+3+2° Ans. 25. ee eo ee ee + 3+ 33 +3+2+2—-4? Ans. 21. eee eto eo oot +2+3° Ans. 41. Beato see 332+ 3a 2 +2+3? Ans. 39. eee o 7 2 e+ 24343434243 es 2p Ans. 42. §2 FIRST STEPS AMONG FIGURES. 3+242+-34- 2 £3 434S ree 2. Maan toy St2t+5+3t2424:3 13 sage ees +2% Ans. 38. 8+3+2+24+-34+2+3 + 32 eeeaee 3-2 De ANS. 4e78 a 44-:34243.53 5 34n2 aetna $244 2-2! (Ans. i324; Review these often. Count by 2’s, commencing with 2 and with 1, to 60 and 61. : Count by 3’s, commencing with 3, 2 and 1, to 51, 53 and 52. Count by 4's, commencing with 4, to 32 and back. James having 9 apples ate 1, and gave the rest to his sisters, giving them 2 each; how many sisters had he ? George had 9 oranges; he ate one of them ; if he were to give you half of what were left, how many. would you get ? Charles having 9g pears sold 3, and gave his sister half of what he had left ; how many did he give his sister? Teach Roman notation to LXX. Write in letters 67, 44, 36, 59, 62, 46, 28, 16, FIRST STEPS AMONG FIGURES. 53 Write in figures XVII, LXV, LXVII, LXIV, LXIX, XXIV, XIX. Pupils read the following, which should be copied on the black board, and the pupil who reads any number to point off the periods him- self before reading. TeG205- 2.7020. 3.8005. 4.9400. 5.1641. 6.6780. 7.5416. 8.8605. 9.5400. 10.7508. 11. 4870. 12.5718. 13.5851. 14. 6504. 15.5790. 19. 1432. 17.9007. 18. 5000. 19. 7400. 20. 8040. 25.4037, 22. 5919. 23. 7990. ‘24. 7803. 25.7001. Teach Arabic notation to 10,000. Teach the pupils carefully, as being of the utmost importance, that they should place a comma after the number expressing thousands and before they write the units period. Teach them that units period takes three places, and show them that when the number does not fill the three places, the places on the //¢ must be filled with ciphers. The following are to be read by the teacher and the pupil is to write them in Arabic: Re- view these often. 1.4,573 2+ 3,240 3.5,296 4.7,315 5. 2,324 6.7,560 7.1,427 3. 3,670 9. 7,305 10. 5,741 If, 2,816 12. 51980 Iz $407 1 4-4,300 54 FIRST STEPS AMONG FIGURES. 15. 9,700 16. 4.315 17.8,590 18. 1,731 19. 7,800 20. 5.004 21. 3 060 22. 5,104 28. 6,003 24. 8,600 25. 9,419 26. 8,040 27.5,900 28. 4,307 29. 5.009 30.9,016 31. 9,300 32. 9,070 33- 7,049 34- 3,900 35- 7,008 36. 7,080 37. 5,700 38. 7.000 39. 55075 40. 3,716 41. 9,060 32. 5,800 43. 6,904 44. 6,008 45. 7,600 46. 8,009 Cut an apple into 3 equal pieces and teach the pupils that we call one piece oné third. Break a stick of candy into 3 equal pieces and so illustrate the same thing. In the same way teach one-fourth by 4 divi- sions’ instead of 3; then one-fifth, one-sixth, &c.’ When the pupils are familiar with this show them that of any 3 equal things, one of those things is one-third; of 4 equal things, one of them is one-fourth. Do not leave this subject until the pupils are very ready with their answers to the following questions : An apple is cut in 5 pieces; what do we call 1 piece’? | An apple is cut in 3 pieces ; what do we call Te pieces An apple is cut in 6 pieces ; what do we call I piece? FIRST STEPS AMONG FIGURES. sl 5 An apple is cut in 10 pieces ; what do we call 1 piece ? An apple is cut in 4 pieces ; what do we call I piece? One apple is what part of 7 anes One apple is what part of 9 apples ? One apple is what part of 6 apples? One orange is what part of 4 oranges? | One pencil is what part of 8 pencils? A boy having 5 apples gave away one of them ; what part of his apples did he give away? A little girl had 6 peaches ; she gave one- third of them to her brother. How many did she give him ? (Teach the pupils that they get one-third of a number by dividing by 3; 4 of a number by dividing by 4, &c.) A boy gave away one-fourth of his 4 mar- bles ; how many did he give away ? How many did he keep ? If I divide 6 apples equally among 3 boys, what part of them do [ give each boy? How many do I give each boy ? | TEN. Pupils make schedule. 56 FIRST STEPS AMONG FIGURES. For addition. a1 bo jecr id 536|247|503 247 234|5021364 623 For subtraction. ate Da Cyd 1098/758|676/6107|1074'389 1085 623|406|223/6 45] 532/054] 323 For multiplication. aot Desc | d |e) f tg 3521313|240/132 421/521] 20 123/012 415|231|203|124| 53 For division. b|c 842/920,616 412 313/213 Re-arranged. For addition. ac DwECtiad 635/264,635|207 23415021345 1652 For subtraction. a |b co [id sce tie heat i 8410 839|796|869'5710/897/75917 105 86510 52 31406|543|223| e!| f-|:o 20S taaee 535/245 623,043/6234 14501235|236|742|3405 e re go. | hoieas 983/5969 643|/2345 a ete 1038 035 212|231 f 44 | 24 | e 463 345 Fo seath 257/430|524 120314641235 i 3524 2346 54 § 623 235/0 42.345 6 For multiplication. a | b-}-c°| d)|- ete t.|2o7| Sie 42] 14:|-24 | 3£ |-2§ 4 02 | 31 |) oneness 12 | 12] 30 | 12:|\.4r-|)367)0 237 eee FIRST STEPS AMONG FIGURES. 57 For division. Meeiecc|.dj.¢| fg | h |i 18 | 60 | 32 | 85 | 44 | 02 | 106 |} 310 | 94 Searle.) At | 12 |31 | 22 | 3 5 1-34 What is 3 of 10 buttons? Lucy had g pins; she lost 4 of them and then found 1 ; how many had she then? 4 and how many make Io? 10 boys were out in a sail boat; 1 more than half of them were drowned ; how many were drowned ? Minnie had 9 cents ; she spent 5 of them and lost 4 ; how many had she left ? 10 cents will buy how many 3 cent stamps? How many 2 cent sticks of candy can you buy for to cents ? How many cents will 4 two-cent marbles cost ? Lucy had 5 cents and her mother gave her 4 cents, how many cents had she then? Show the pupils that we call 2 and 2 equal numbers, that they are equal to each other, also 1 and 1 are equal numbers, 3 and 3, &c. ; 2 and 1, or 2 and 3, or 1 and 3 are unequal numbers. What 2 equal numbers make 4? What 2 unequal numbers make 4? 58 FIRST STEPS AMONG FIGURES, What equal numbers make 3? What unequal numbers make 3? What 2 equal and 1 unequal numbers make -§? (Two answers.) How many wheels have 5 sulkeys? How many wheels have 3 three-wheeled velocipedes ? Mary was bringing in 10 eggs in her apron, she broke 2 less than half of them; how many did she break ? How many were unbroken? John had a string ro yards long and William had one three yards long; how much longer was John’s string than William’s. 6 and how many make 10? What number taken from 10 leaves ae What cost 5 two-cent stamps ? Henry had 10 miles to walk, he has walked 4 of them ; how much farther has he to walk? 1o beans are how many times 2 beans? 8 boys were playing “snap the whip,” 6 of them kept hold of hands; how many were there that did not let go? ; What number added to 3 will make 10? . Count by 4’s, commencing with 4, to 60, FIRST STEPS AMONG FIGURES. 0) Review counting by 2’s and 3’s. For rapid solving. AF3Z—-UF2X3+ISHSX4+4+ 243 + 3 te Gee ee eae ANS. 24. ee Ae 2X2 +3434 3.4342 +24+34+3+4+4+4=? Ans. 4o. Bees ho aga ot 2s It 4 a 4 4 +4+4=? Ans. 44. Sees 4444-47 3434-3 +2 +2 332-3 ?. Ans. 48. mnene ho 2 2X 8+ et At Aa 4 +3 pe 3 3. oe 2 Ans.36. fees O41 4 AT 43 2b 2 +3 ee Gt Bae th asa Te Seas a 4 23 t3t34r2e +2 + 37+3 +3+3+2+2+3+3=—7?. Ans. 38. 40-3 Sao TSt3 t 3+ 2h 4447444 ee h2b2+3+3—) Ans: 55. Poa at aes 2-2 2 X38 oo 3 2 2+ 34+1+4+4+4=7? Ans. 29. Beeson 4 A AAT AAT 4 tr 3 +3 +3+2+2+2+2=—7 Ans. 57. iweeres wot 2 Ad 4 4.4.3 4-3: 3412 +2+3+4+4+3=2? Ans. 55. §+34+44+4+4 + 44+3+2+3+ 44848 +4+8+2+8=—7 Ans. 74. bo FIRST STEPS AMONG FIGURES. 4t+3t2ti1+4t+4+4+4+3+2434+3 +2+34+3+3+3=? Ans. 51. For more practice see pp. 51 and 45. Count by 4’s from 2 to 34. (T9 (75 4’s 46 4 75 60. cc éc 3's 6c 3 ce 60. 66 ce 3's cc I 73 6r. 6c (79 3S 6¢ 2 (73 62. T3 (79 2’s cc 2 ce 60. ce Ce 2’s 6s I éc 60. Write in letters 36, 41, 16, 64, 56, 47, 69. Write in figures XLV, XV, LXIV, LXXXV, XXVII. What equal numbers will make 6? (3 ans.) What unequal numbers will make 6? (Several answers.) What 2 equal. numbers and 1 unequal number will make 6? The following numbers are to be copied on the blackboard and the pupils are to be re- quired to point them off in periods and read them. I. gI0or7. 2. 86700, . 3: 90007. ay faogae 5. 70000. 6. 50010. 7. 38419. 8. 74058. 9. 60800. 10. 16040. II. 3000. [2. 7014. 13. 10061. 14.-- 3020. 15. : 7003: *165 6600: r7, 17500. 18. 3540. 19: 67374)020moggem FIRST STEPS AMONG FIGURES. bi What 2 equal numbers and 1 unequal num- ber will make 7? (3 answers.) What 3 equal numbers and 1 unequal num- ber will make 7? (2 answers.) What is 1-3 of 9? I is what part of 7? Teach Arabic notation to 99,000. Show the pupils that they should place a comma after the figures that express thousands before writing the unit period, and a period at the end of the number. The following numbers are to be read by the teacher and written upon the blackboard or slates by the pupils. beeeye300., 2) «53000, 3.°.2,050. 4. 10,091, Remos0102" 6; 4,000. 7. 12,090. 3. 50,700. OmGs,070. 010) 45,0132 11. $0.000. 12. 80.010. 13. 15,061. 14. 40,002. 15. 79.500. 16. 81,018. 17. 30,600. 18. 60.060. 19. 90,004. 20. 8,050. G7 5.060, 22, 74.695. 23.-31,280. 24. 13.300. 25. 14 O41. 26. 10,010. Teach Roman notation to C. Write in Roman 1. 64. 2. 49. 3.97. 4. 76. Write in Arabic 5. XIX. 6. LXXXIV. 7. Doese VIL, 9. XVIII. 10, XX1X. Write in Roman 11. 17. 12. 56. 13. 83. 14. 49. 62 FIRST STEPS AMONG FIGURES. Write in Arabic, 15. LX VID Gee ee If the teacher prefers it, the pupils can buy their books (the Pupils’ Edition) at this stage, and do more slate work than the teacher could have time to dictate to them, or copy upon the board for the pupils to copy and solve. If the pupils do not have their books the teacher will assign daily lessons from page 1, 2, &c., of the Pupils’ Edition, doing it in connection with this work and thus carrying on that work together with the following work. The teacher should now begin to give the par- allel work of the Pupils’ Edition in connection with that of the Teachers Edition. The parallel pages are denoted by the numbers at the bottom of the pages in each. FIRST STEPS AMONG FIGURES. 63 se aa ae Ne Counting two or more numbers into one number is called Adding, or Addition. The number obtained by counting two or more numbers into one number is called the sum .of those numbers. For addition and multiplication. (5 and rev.) a b cf speadap hse f g Be to 4 | P35. 4401 35.2.) 4 1 ee oe | 452-1345 | 234 1.523 £4 5 For subtraction. a b Cc d e ii g 7410 | 746 | 758 | 693 | 696 | 857 | 8 5 Paes fo | to | 345 1432 [5 “4 For division. a b Cc d e 41s10} 6165 | 20 g2 | 20815 | 610 8 Seem imontont 43201) 154.3 ||) 255? f g Breen 4 t2 2) AeA Bea) See Pupils’ Edition, p. 5. Og FIRST STEPS AMONG FIGURES. It is well to require pupils to bring a written analysis of an example to recitation and to give the solutions of other examples orally in class in the same form, but there should be a large number of examples given, of which ond the answer is to be given and that as soon as possible after the reading. 1. Susan had 4 cents and her mother gave her 3 more ; how many had she then? Solution. She had the sum of 4 cents and 3 cents, or 7 cents. 2. John has 5 marbles and James has 4 mar- bles ; how many have both? 3. Lulu has 3 eggs in one hand and 2 in the other ; how many has she in both? 4. Walter bought some candy for 4 cents, and some raisins for 5 cents; how many cents did he spend? | 5. Martha read 4 pages in the forenoon and 2 in the afternoon ; how many did she read that day? 6. A boy had 3 pencils in one pocket and 5 in another ; how many had he in both? 7. If a top cost 4 cents and a marble cost 2 cents, how many cents must a boy have to buy a top and a marble ? FIRST STEPS AMONG FIGURES. 65 8. Jane bought 2 books; she had 3 before. How many has she now? g. Henry walked 4 miles before dinner, and 4 after dinner ; how far did he walk that day? to. There are 3 barrels of apples under one tree, and two under another ; how many under both? Ask the pupils to bring examples of their own to recitation different from those given them. The teacher also will make additional examples, using pleasant facts about the school room, or the pupils, or their homes, something they have seen. Count by 4’s from 2 to 62. Taking one number from another number is called subtracting,,or Subtraction. The number obtained by taking one number from another number is called the Remainder or Difference. 1. Joseph had 8 cents; he spent 5 cents for an orange. How many cents had he left ? Solution: He had left the difference be- tween 8 cents and 5 cents, or 3 cents. 2. Mary had a cake which she cut into 10 pieces; 7 were eaten. How many were left? 3. My knife has 6 blades; 2 of them are open. How many are closed? See P, Ed., p. 9. b6 FIRST STEPS AMONG FIGURES. 4. James bought a paper for 5 cents ; he gave the newsboy 1o cents. How much change should James receive ? 5. Samuel put 9 peaches on the table and his sister took 5 of them ; how many were left ? 6. A man owed $7 ; he paid $3 ; how many $’s did he then owe? 7. Willis took 9 cents to buy candy with ; he lost 4 cents. How many had he to buy candy with? . 8. Henry bought a pencil for 4 cents and sold it for 7 cents; how many cents did he gain? 8. Matthew bought one pencil for 4 cents and another for 5 cents; what did both cost him ? | to. Susan bought 3 spools of white thread and 6 spools of blue thread ; how many spools did she buy? 11. Jane tried to solve 6 examples ; she had 4 of them correct. How many were wrong? 12. Walter had 8 pencils ; he broke 3 of them. How many whole ones had he? 13. Fanny had 6 needles ; she found 4 more. How many had she then? 14. May is 7 years old and her brother FIRST STEPS AMONG FiGURES. 67 Frank is 4 years old ; how much older is May than Frank? 15. George had a stick g inches long ; he cut off 3 inches of it. How long was the stick then? 16. A farmer having 8 turkeys, sold 4 of them ; how many had he left? 17. John paid 3 cents for candy and 5 cents for marbles ; how many cents did he spend? 18. A little boy had 3 fingers cut off ina machine ; how many had he left ? 1g. Silas had 3 marbles in one pocket and § in the other ; how many had he in both? 20. How many wheels have a sulky and a wagon together ? 21. There are 3 girls on the front seat of a carriage, and 5 girls on the back seat ; how many girls in the carriage ? 22. Jesse had 8 sticks of wood to bring in ; he has brought in 3 ; how many more has he to bring in? 23. An orange cost 6 cents, and a peach cost 3 cents; how much more did the orange cost than the peach ? 24. If a pear cost 4 cents, and a lemon cost 5 cents, what will a pear and a lemon cost? 25. There were 6 eggs in a nest and 4 of See P, Ed., p. II. 68 FIRST STEPS AMONG FIGURES. them were broken ; how many whole ones were there ? ; 26. Ellen’s father gave her g cents ; she bought a doll with 5 cents. How many cents had she left ? 27. There are 4 boys riding in a sleigh and 2 riding behind on the runners ; how many boys with the sleigh ? 28. Ella has 5 roses on her bush, and 5 in her hand ; how many has she? 29. There were g chickeas in a coop and a rat ate 3 of them; how many were left ? 30. There are in the class 4 girls and 3 boys ; how many pupils in the class? 31. A little boy bought 10 sticks of candy ; he ate 3 of them and gave away the rest. How many did he give away ? These 31 examples should be reviewed and others given, until the pupils know at once in such simple problems whether they are to find the sum or the difference. Review the series also. Count by 4’s from 1 to 17. Copy the following examples one at a time on the blackboard ; require a pupil to point one off into periods and read it. Erase it, then FIRST STEPS AMONG FIGURES. 69 write another and require another pupil to point it off and read and so with the others: I. goo16. 2. 45378. 3 461340. 4. 908714. 5. 876341. 6. 608790. 7. 379000. 8. 75608. Q. 40713. 10. 100740. It. 98716. I2. 900000. 13. 800601. 14. 200003. 15. 761300. 16. 500000. 17. 700300. 18. 60050. 19. 700060. 20. 600000. 21. 200361. 22. 500700. 23. 40010. 24 900007. Teach Arabic notation to 999,000. To be read by the teacher for pupil to write upon slates or blackboard. Write in Arabic the following : I. 1,040. 2. 3,506. 3. 10,016, 4. 8,400. 5- 9,350. 6. 7,518. aa Lye 8. 10,010. 9. 40,070. ore7 5-501." 11. 36,000. 12. 90,090. 13. 100,100. I4. 702,940. 15. 900,070, 16. 816,902. 17. 49,049. 18. 860,705. 5 407,017" 20. 791,486.21. 21. 10,020. Write these, or similar numbers on the board and require the pupils to read them. Write in Roman the following: ta 0: 2. 96. 3. TIO. 4. 47. Sori: 6. 134. See P. Ed., p. 26, 56 | FIRST STEPS AMONG FIGURES. _ Write in Arabic: Walia a. oe be g. CXC. 10° CXLIV. 11. :\CLXXXV Ista sais Write in Roman: 13.0192... ¥4. 136. 915..1000 Beaten A short method of adding equal numbers is called Multiplication 5 or, Taking a number a certain number of times is called Multiplication. The number obtained by multiplication is called the Product. 1. John bought 5 pencils at 4 cents each; what did they cost? Solution: They cost 5 times 4 cents, or 20 cents ; or if one pencil cost 4 cents, ve pencils will cost 5 times 4 cents, or 20 cents. 2. If 1 orange cost 5 cents, what will 4 oranges cost ? 3. What cost 4 marbles at 3 cents each ? 4. How many quarts in 2 gallons? Solution: In one gallon there are 4 quarts, in 2 gallons there are 2 times 4 quarts, or 8 quarts. 5. How many pints in 3 quarts ? 6. How many quarts in 5 gallons? 7. Ilow many wheels have 3 wagons? FIRST STEPS AMONG FIGURES. 71 8. A lady gave 3 little girls 5 bunches of grapes each ; how many bunches did she give them all ? 9. How many feet have 4 hens. to. How many feet have 3 cows? 11. What cost 5 books at 4 shillings each ? ~12. What cost 3 lead pencils at 5 cents each ” 13. If a lead pencil cost 6 cents and a mar- ble cost 3 cents, what will both cost ? 14. What cost a doll worth 4 cents, and a spool of thread worth 6 cents ? 15. What cost 3 pencils at 4 cents each ? 16. The boys are riding down hill on sleds ; > there are 4 sleds and 2 boys on each sled ; how many are riding down hill ? 17. ‘A boy bought a sled for 8 shillings and sold it for 5 shillings ; how many shillings did he lose ? 18. A boy had 8 cents in his pocket, but he lost 4 of them through a hole in his pocket ; how many had he left ? 19. A boy paid 4 cents for candy and 5 cents for nuts; how much money did he spend? Review these examples carefully. Count by 4’s from 1 to 29. For rapid solving. See P. Ed., p. 19. 72 FIRST STEPS AMONG FIGURES, 12=- 4434313444444 4 ees +2+2=? Arnis. 41. 34+44+413+313 +3 +24 25) coe +2+3+4=? Ans. 44 3+3+2+2+2+3 +3+3 +293 tee +3+3+3=? Ans. 39 - 34+44+41+4+3 +3 | 2 +27 +2+2=? Ans. 41. 24343734225 2er ies tees +3+4=? Ans.-37- 3+2+2124+343+ 24+ 2 +1 3a +3+-7— 2 Ans. 22 4+3+4+4— 3+4—-3—-2 + 444-342 +3 ARS de 24+-34744444—-342-— 3 — 260.00 “312 SANSA TFs 2+4+3+4+3 —4—-3 —4+3+31+414 +3— ? Ans, 22, 5+3—-2—-44+3+ 3 +24443144+3—-2 —3=? Ans. 19. 314724 45-24-2— fe teehee | +4=? Ans. 13. s g+2+4-3-2+ 3.434343 fae —4+3+2=2 Ans. 16. 4+3+24+3+3+4+ 3—4-4+1—-34+4 —3—4—4=? Ans. 5. FIRST STEPS AMONG FIGURES. 73 20. What cost 5 marbles at 4 cents each ? 21. What cost a marble at 3 cents and a pencil at 4 cents ? 22. James bought 3 books at 4 shillings each ; what did they cost him? 23. David had 8 apples when he started for school, but he ate 3 on the way; how many had he when he got to school ? 24. Sarah ate 3 crackers at breakfast and 5 at dinner; how many did she eat at both meals? 25. How many horses in 3 four-horse teams? Finding how many times one number is contained in another is called Division. The number which shows how many times it is contained is called a Quotient. 1. How many pears at 2 cents each can be bought for 8 cents? Solution: If 1 pear cost 2 cents, for 8 cents you can buy as many pears as 2 is con- tained times in 8, or 4; or, aS many as there are 2’s in 8, or 4. 2. John has 15 cents ; how many marbles at 3 cents each can he buy? 3. Willis spent 20 cents for oranges at 5 cents each; how many oranges did he buy ? see P. Ed., p..25- 74 FIRST STEPS AMONG FIGURE: 4. If one pig cost $4, how many pigs may be bought for $12? 5: How many lead pencils at 4 cents each can be bought for 20 cents? 6. If 1 doll cost 3 shillings, how many such dolls can be bought for r2 shillings ? 7. to shillings will buy how many knives at 2 shillings each ? 8. How many balls at 3 shillings each may be bought for 6 shillings ? g. When pears are 2 cents each, how many can you buy for 8 cents? Count by 4’s from 1 to 61. 6“ 6“ 4’S 66 2 & 62. 6é 66 4’s 66 4 “ 60, 3 6c 4’s 6 3 6“ 63. ro. What cost 4 pineapples at 2 shillings each ? 11. How many pencils at 4 cents each can be bought for 16 cents? 12. A boy walked 5 miles 1 day, and 3 miles the next day; how far did he walk in the 2 days? 13. How many pairs of mittens at 3 shillings a pair can you buy for g shillings ? 14. 20 shillings will buy how many purses at 5 shillings each? FIRST STEPS AMONG FIGURES. 75 15. Henry earned g cents on Monday and 5 cents on Tuesday ; how many cents more did he earn on Monday than on Tuesday ? 16. Louisa had 5 cents and she found 3 cents more ; how many cents had she then? 17. If one sied cost 5 shillings how many sleds can you buy for 20 shillings? ‘8. What cost 4 vests at $8 each? © ig. How many neckties at 3 shilings each can you buy for 12 shillings ? 20. 5 little boys each have a pair of copper- toed boots ; how many boots have they ? Review these examples. For addition and multiplication. (6 and rev.) a b c d e Poneeeael 1 362 (415-4 [0241 eee 2345163451623415623 | 2456 f g h Betetyy) 362.143 1'5 2345|623/456 For subtraction. a b | Cc | d fegoaro| 7 11 -7-8.| 3210 8 6.\-4 7.9.8 SiO re4 | 5.15 4.3 1-2: 6-61) |-3 12.4.5 e f g h Peereyets 080.5 6 | 9 10°5-| 5.9 6 O45 432).6 541234 ce Ls, pies, 76 For division. FIRST STEPS AMONG FIGURES. a 80 127584 52 6n253 d AVi2gVaees 2.3.4.5 (G)- (2) 32 33 13 21 3 2 21 30 13 13 33 32 LIS 131 (7) (8) Oo 23 23 33 33 of 12 22 31 31 23 33 33 22 185 194 b c 10-20 15 36 | a 2erceeeee 2:4 § 6 | *4aieeeG e f 18 12°6 16 175 @osgeTe 623 415 62 3 g h 8 20 6/9 24 I0 4, 5.9:1.30 eis (3) (4) (5) a6) 2 8 33 Jt oe 33 3! 23 23 3t 23 33 33 I2 13 23 21 22 22 32 23 31 32 13 134 152 154 155 (9) (10) (xx) (x2) fed 3 3t ee a 23 33 32 23 ot 23 +S 3 23 39 33 Ba 33 22 21 23 of St 32 33 oY 33 23 75 166 203 177 See P. Ed., p. 16. od FIRST STEPS AMONG FIGURES. iy 13. Add 32, 21, 3, 33, 23, 31, 23, 33- 14. Add 33, 23, 30, 21, 2, 33, 23, 22. Meme ln, 34922, 31, 23..32, 12;-23. Merrodest, 20, 22, 33°21) 32, 22, 32. 17. Add 23, 13, 32, 23, 3, 13, 32, 33- Count by 5’s from 5 to 60. For rapid solving. 374744343 42144+3+3 +2+4+4 +3? Ans. 42. Pepe 4a oT oT OE L834 4+ 4+ Bees 20) ANS..ST. statatststatatstststat2t 4+4+4¢ Ans. 57. eee 3 toe eg + 4+2+443 +-2+3+5?. Ans. 46, Rages oot 43 3 t 2 FA 3 +4 +3+3+4? Ans. 50. 4+3+4+3—2—-2+5—3+2+4? Ans, 18. 2+453X5—4X% 4-3-3434 6+4X6 +4+3—2? Ans. 23. 3+64+44+44+4+34+4—-6—4—2—-5+3 +4-—4? Ans. 14. 34+5>2X5—3+4—-2—47+5 x6+4+3 so53+31 Ans. ft. 78 FIRST S11 PS AMONG FIGURES. BR cote ea X54 ee iatreg +5+2+4+37 Ansiaeee 3+6+4—1 — 3x6—3—4-3 +4=6x5 4 Pre 4g ei Seecns §5+6—3444+3413 +3 +4731+474-2 —3+4-P3'-4 5 9 Ase e Do not use all of these at once, but use them occasionally and in connection with a lesson in examples of another kind, or to wake up the whole school» sometimes when they are listless. 1. What cost a pencil at 6 cents and a mar- ble at 4 cents ? 2. Mary had 5 peaches and her brother gave her 6 more; how many had she then ? : What Egat 5 books at 6 shillings each ? 4. 3 boats are on the lake; each has a pair of oars, how many oars have the 3 boats? 5. There are 9 bovs in a class, and 6 of them recite well ; how many do not recite well? 6. How many baskets, at 4 cents each, can be bought for 24 cents? 7. Charles spent 18 cents for candy at three cents an ounce ; how many ounces did he buy ? 8. Jane had 10 needles, she lost 7 of them ; how many had she then? FIRST STEPS AMONG FIGURES. 79 g. A lazy boy brought his mother 3 sticks of wood at one time and 4 at another ; how many sticks did he bring her ? Io. Six boys can sit on this seat, how many boys can sit on four such seats? ir. In a school room there are 6 keys hang- ing on a nail, 2 keys for each door ; how many doors are there? 12. There are 18 words in the sp:lling les- son, 3 words for each pupil ; how many pupils in the class? 13 There are 5 piles of books and 6 books in each pile ; how many books in the 5 piles? 14. One stormy day George cleared the path of snow 4 times in the forenoon and 5 times in the afternoon ; how many times did he clear the path? '15. Amelia had 11 cents and spent 5 of them; how many had she left ? 16. Arthur had 7 buttons on his jacket; how many had he after losing 2 of them ? 17. How many quarts in 6 gallons? 18. Mr. Smith has a quart of maple syrup; how many times can he fill a pint cup with it? 19. How many skates at 6 shillings each can you buy for 24 shillings? 80 FIRST STEPS AMONG FIGURES. 20. What cost a knife at 5 shillings and a saw at 6 shillings ? 21. If 2 oranges cost 12 cents, what will 1 orange cost ? Solution: If 2 oranges cost 12 cents, I orange will cost + of 12 cents, or 6 cents; or 1 orange will cost + of 12 cents, or 6 cents. Before giving examples like the above teach the pupils carefully that if 2 things of equal value cost a certain sum, 1 of them will cost 4 of that sum ; if 3 cost that sum, 1 of them will cost 1-3 of it; if 5 cost any sum, 1 of them will cost 1-5 of it; if 9 of them cost any sum, 1 of them will cost 1-9 of it, &c. Question on this subject until it is thorough- ly mastered. Show the pupils that to get 4 of 12 apples (or marbles or pencils) they may be placed in 2 equal piles, and they will find that 3 of 12 apples is.6 apples. Show them that to get.4 of 12 pencils, they may be placed in 3 piles, and that 4 of 12 is 4. Show in the same way that + of 12 is 3. When this is thoroughly understood, show them that 3 of 12 may be obtained by dividing 12 by 2— the result in each case being 6; show that 4 FIRST STEPS AMONG FIGURES. 81 of 12 may be obtained by dividing 12 by 3; teach 4 of 12 in like manner. Illustrate also by + of 6 and 3 of 6 and ¢ of 8. Then teach in general terms that 4 of any number may be obtained by dividing the num- ber by 2; +, by dividing by 3; 7, by 7, &c. 22. What cost 1 pear if 4 pears cost 8 cents? 23. If 3 knives cost 15 shillings what will 1 knife cost ? 24. If 2 pencils cost 16 cents, what will 1 pencil cost ? 25. If 4 stools Fie 12 legs, how many legs will 1 stool have? 26. If 6 boys earn 18 cents, how many does 1 boy earn? 27. If 5 cords of wood cost $25, what will 1 cord cost? 28. How many pounds in 1 box of honey if . 4 boxes contain 24 pounds? 29. At 5 cents each, how many oranges can be bought for 30 cents? | 30. If 3 lemons cost 18 cents, what costs :1 lemon ? 31. How many pounds of butter will last a family 1 week if they use 12 pounds in 4 weeks? 32. How many days will 18 apples last a boy who eats 3 apples each day ? 6 82 FIRST STEPS AMONG F GURES. 33. A blacksmith shod 5 horses each day ; how many did he shoe in 6 days ? 34. 6 boys are skating on the ice, and 4 boys are sliding on the ice without skates ; how many boys on the ice ? 35. Nine boys were riding down hill on sleds ; 3 of them went home. How many continued to ride down hill? 36. If 15 yards of cloth will make 5 pairs of trowsers, how many yards will it take to make 1 pair of trowsers ? 37. George has 4 books, and Mary has 5 books ; how many have both ? 38. 12 cents will buy how many marbles at 3 cents each ? 39. If 5 marbles cost 10 cents, when will 1 marble cost ? 40. What cost 1 apple if 5 apples cost 10 cents? . How many dolls at 4 shillings can you Sg ae zo shillings ? Review the last 21 examples until the pupils solve them readily and can distinguish when they divide and when get one-half or one-third, &c. Count by 5’s from 1 to 61. FIRST STEPS AMONG FIGURES. 83 Count by 4’s from 1 to 61. Count by 4’s from 2 to 62. Count by 3’s from 3 to 60. Count by 3’s from 1 to 61. Count by 3’s from 2 to 62. To be read by the teacher. Write in Arabic: Peey7050.. 2,010,002. 92. 40,200. 425,209. 5s (10,010. ~ 6.7 300,040. 7- 9,610, 684-3 1.6; g. 215,000. 10. 80,090. II. 600,000. 12. 809,740. 13. 100,010.14. 916,008. 15. 835,941. £02970,000." 14. 90,005. / 18. 5,016. 19. 213,033. 20. 30,000. Write these or similar numbers on the black- loard and require pupils to read them. For rapid solving. 64+4+3+2+4+4—-3=+4x6—3—3+4 +4? Ans, 32. Beet tes 1 3 1423 > 5 XA 4 1+ 313 4+ 3t5 +5? Ans. 3r. Coe te dt 4s +34 tat 25 +2 +3+3+4? Ans. 53. Ae es 8a ec 23-34-4342 +3+3+22% Ans. 44. eg io 21 44 4 + 3-2 +334 +4 feeat satis 4c eh ANS. 52. 84 FIRST STEPS AMONG FIGURES. 34+24+44+34+54+3+2+4+4+1+244 +14+3+1+2? Ans. 44. 44+-343744263 +263 +3 tegen ee toi 2 Domai 34+24+41-4434+3+2+24 35 +3+2+1? Ans. qt. 1794+4+3761615—3--3 460 3-5%X3! Ans. 12, 9+44+3+2+3 X5—4—3—2+4—1+6 X3+3+2 2. Ans, 17: 16+4+31+3+3+2—1+5+3+3-3% 5—-3+4—17%4? Ans. 5. In giving the following examples as well as those “ for rapid solving” the teacher should be very careful that pupils do not acquire a pernicious habit of counting instead of adding at sight or as soon as heard. Slate examples. (t.) (2) (3) (4) (5) (6) (7) (8) (9) ml wwe Heenan I ta | Yel ay Cy eerep ~~ N m= WS NO = BO = = NY = “SAWN edcyeps SUC ee TO ee eee ert PG Tea eS Wooo iN ieee Ca artnet i TN Gy eae — BS, | == N NY = NN NN A a pa i an Le | oul 85 (10) (11) (12) (13) (14) (15) (16) (17) (18)(19) FIRST STEPS AMONG FIGURES. (24) = (25) (23) ) (22 (21) (20) 20 27 ? *9 29 #3 86 FIRST STEPS AMONG FIGURES. (26): (27) (28): ; (29) (30) (gee Z I 2 I 2 2 2 I 2 I 2 I I ce | 2 I 2 I 2 2 2 2 1 2 2 | 2 2 I 2 2 I I 2 2 2 2 I 2 2 I I 2 t I I I 2 2 2 2 I 2 I I I 2 I 2 I I Z I - 2 2 2 2 2 2 2 I 2 I 2 I I 2 2 2 I 2 2 2 I I 2 2 I I 2 v2 2 I 2 2 2 2 2 2 —_— I I I a 2 2 a4 my = 7 24 ee. its 21 24 21 25 26 FIRST STEPS AMONG FIGURES. 87 (33) (34) (35) (36) (37) (38) (39) (49) . I 2 I 2 I 2 I 2 2 I 2 I 2 I 2 I 2 2 I 2 I 2 I I 2 I I I 2 2 I I I I 2 I 2 2 I 2 2 I I 2 2 I 2 2 I 2 2 2 I 2 Zee tt I 2 2 I 2 2 Ins) 2 2 2 2 2 2 2 2 2 2 2 I 2 2 2 2 2 2 I I I I Det I 2 I I I 2 2 I 2 I 2 2 I 2 2 te | 2 I I 2 I I 2 2 I 2 2 2 2 2 2 I 2 I I — 2 7 I — marae. Se 23 a. Tae Bie tik Oo ZO TA «23 27.5227. 25 If the teacher is careful that the pupils do not keep any of the solutions of the foregoing examples, they may be given 2 or 3 times over, first solving them all, then solving them aJl again and so on 8% FIRST STEPS AMONG FIGURES. (41) (42) (43) (44) (45) (46) (47) (48) (49) 2 23 #39 413 632.) 643) Sao 3 32 23 .2T <13 033° $20 1 632 «6 3E 33" 327. 30" 833 Q° 2k 12. 10. “21% 12.0023 1.39 23 32 6°33 «4.33 SS Ea 3.12. 32) 13° $20, 225) 3 eae % 63% -33. “225 32 431 oc ee e 2°23 2b 31). 23 5513S cee 3. 33 29 23° 33 3305933) eee 20 217 225 198 239 230 233 2205205 (so) (5t) (52) (53) (54) (55) (56) (57) 32.21 31. 23 © 30° 23 agmmeeme Phe Kee oe 31.. 23° 3) 2i0eeee 33. 20 32 33 «610 | -23)segaieee 220 ts eso 30. 22-530 Rai gues 12332) 13 22 31, 220). 20a 3h. 33. 31 - 93. 33> SG 33 a1 235. 312 23 (jase same 23 13 32 23 I2 21 vik 23 30°. 32, 33. 305133.) (33ceuntG mune 23 31 23 23 22 23 32 33 13°33: 33. (33. 320. 320) Samu 274 282 274°° 202 “271 (283°. 260qmeun See P. Ed., p. 33. a eee -_—— ees —_—— eres FIRST STEPS AMONG FIGURES, 89 (58) (59) (60) (61) (62) )63) (64) (65) (66) 21 33 20 13 2 23 32 21 3 32 23 31 12 = 306 33 23 31 20 32 13 oo FANE 32 12 33 21 23 —— 327 23 31 13 32 20 33 21 13 32 Ze 30 13 32 315 30 23 33 20 13 32 23 I2 ZZ Ae) 30 21 33 326 33 21 13 30 23 32 II 33 23 32 13 22 31 33 21 I3 ar 33 20 13 oy 23 31 32 13 31 23 33 ZI 13 To 32 31 23 32 31 21 33 32 23 30 22 33 12 32 23 31 13 3 33 22 31 23 31 12 23 32 31 23 r3 33 ee, 317 325 328 339 Teach pupils to prove every example in ad- dition by adding both upward and downward, and in this way they will get more practice— just what is needed. _ If the pupils have their books—P. Ed. —the following examples are intended to be given at recitation for immediate solution, while those in the P. Ed. may be solved by the pupils at their seats and brought to recitation, go FIRST STEPS AMONG FIGURES. ee If the pupils have not got their books these examples may be written on the board or read to pupils to solve at their seats, a few daily: (TSE) (3) (4) (5) (6) 231 321 333 331 132 133 322 233 223 223 223 332 122 213 Bie 113 330 i. sign 333 332 233 322 233 322 212 322 213 232 213 331 223 ae. e198 223 131 332 213 232 322 132 303 213 329 221 222 a2 233 233 a 2,096 2,109 2,091 1,888 1,910 1,808 (7). (8) ©) (9) (10) 2911 |e 22 eye 213 123 2a5 22 203 re a oo. 37s 332 ZOE aat 222 232 124 P2SCr Bae 233 233 L2n 232 333 333 SOF eo 20a 212 232 23 312 323 233 See cea 234 233 ee aay Toke! 250 ee oe DES tse 223 130 2d5 4e5 Las es: 331 412 321 312 ai 212 2,445 ~2,663) 2,022 2)543°> 2552 Sue Review these if need be; in any case be sure the pupils can add such examples as the above readily and accurately. See P, Ed., p. 36, FIRST STEPS AMONG FIGURES. Examples in subtraction. 13. 69758 6035 63,723 17. 75860 2330 733530 14. 97856 74239 23,626 18. 79685 4242 75443 Multiplication. 21. 32032 2 64,064 25. 23103 2 46,206 22.123103 3 69,309 26. 14023 2 28,046 15. 69587 20152 495435 19. 58796 25062 331734 23. 24130 2 48,260 27. 32023 3 96,069 29. 40312 2 80,624 30. 31203 3 93,609 ob 16. 68059 6040 62,019 26. 96807 4200 92,607 24. 31402 2 62,804 28. 20312 3 60,936 92 Division. 31. 2)48206 24,103 34. 2)28460 14,230 37. 2)84602 42,301 35. 38. FIRST STEPS AMONG FIGURES. - 3)90396 mn 30,tt2 3)39069 13,023 3)30960 10,320 (1) 324 431 243 432 444 342 g32 2,448 (2) 434 431 444 213 332 441 424 25759 (3) 23 431 344 244 432 421 444 25339 See P. Ed., p. 38. (4) 43 324 442 243 134 424 233 1,843 ee 33- 3)69306 23,102 36. 2)60482 30,241 39. 3)93600 31,200 (5) (6) 342 5-0 4at 434 343 4457) 242 321. 434 443.,...gnhd 22 Ael 314550 Gee 1,900 2,429 FIRST STEPS AMONG FIGURES. (3) 31 42 44 4 31 43 (9) 422 344 (10) (1x) ‘(12) ich hie Shoe Y 323 44 44 See el See oe See cae 2,573 2,953 232 2,834 94. FIRST STEPS AMONG FIGURES. (28)~(29) (30) (31) (32) iggy 243. «21 «44. 423° 343 gee 444. 432 21 341 . 444 5025 eeeae 231. 6 344). 34,444 fea 4 44 422.211. 43, 332 4432 (eeaqeeeete 444 321 AZ 2 t feel 22 22 233 432.31 441 343 eee 423. 343° 24: 3824. 444 S039 344. 444 32 432 (434° 9 140 gee 212 323° 43 344 | 232 5a geen 440. 342 44.9 443 9 347 4 eee 3,436 3,213 359 3,737 3,166 291 347 (35) (36) (37) (38) (39) (40) 343. 423° «14 «314° 232, Br ge 444 442 3 432 443 “429443 321 341 40 341 444 43 344 432° 234 34.¢ 134 $9324 eee 344 434 . 42° 244 | 437 ages 344 6340 > 447-440 3435 4 234.423 +. 34, 342. 234 gee 421. 412 . 31 444 444 Sea 432 344 > 23. 233 . 323 42a 444° 431. 42) 132450242 eae 3,759 3,825 308 3,249 3,460 317 2,911 See P. Ed., p. 42. FIRST STEPS AMONG FIGURES. 95 (42) (43) (44) (45) (46) (47) (48) Beemesa ear 24... 234 Pe EY, Perm aes 4s dar yy, dg 231 BimAd ae gage? Gy 422 eee 4 a4 324. 24) < 334 344 1:42 123 Heett > BLA 244 Seen 6.442 )) 33-6 442.6 44 6413 eran ad 3440 24 7134 2 43-423 Brioemmsz2 301° 394 3:853 ~ 373 3,952 b a Cc d e 4735|2473|/2647/3526/4735 345617456134561734510734 f g h 2647)352/665 gH 7314551737 For subtraction. a b c d 119812]105914|10697]|1211 107 O74 51 43° 7} 6345| 5 7°34 e f g 10711 8]129138]9810611 13 Reagereo G7 OA Si3a7 4 5 6 96 FIRST STEPS AMONG FIGURES. For division. a b Cc I2 25 12 21 | 28 36 49 24 | 9 20 10 42 65 4.31 7 © 7.6) 354esee d = f 8.35 15:8 | 30 21 42 6 | 2420 16 35 3.734155 7. 63:54 s5Reae oa h 16 14 12 | 28 15 30 4.07 34 4 5e5an8 (1). (2) 343) Se eee 342 24. 2432 404 23 421 32 434 34 34 34 44 on 2a s ead Be 44 43° 3 343 342 3 143 42 Se ae 43 432 33-2234 aaa 4 341 4. 444 342 44 ee 13 321 244 32 4,604 452 4,296 4,774 476 See P. Ed., P- 40, FIRST STEPS AMONG FIGURES. 97 6. Add 342, 234, 344, 421, 342, 24, 431, 231, 4, 423, 344, 434, 324. 7- Add 24, 43, 24, 32, 44, 34, 42,4, 34, 21, 43, 14, 34, 23, 43. 8. Add 43, 233, 424, 341, 3, 434, 4, 342, 434, 342, 243, 414. g. Add 2, 34, 44, 41, 32, 43, 34, 3) 44, 34, 23, 42, 34, 41. How many quarts are there in three gallons? Solution: In ove gallon there are 4 quarts, in three gallons there are three times 4 quarts or 12 quarts. The following examples may be given during recitation : 1. How many feet have 5 horses ? 2. Arthur was paid 5 cents for doing an errand and his sister gave him q cents ; how many had he then? 3. If a carpenter can drive 3 nails in a minute, how many minutes will it take him to drive 18 nails ? 4. Charles had 10 snow balls in a pile ; he threw 4 of them at his playmates. How many remained in the pile? 7 98 FIRST STEPS AMONG FIGURES. 5. Mr. Smith paid 4 dollars for the cloth for a pair of pants and 2 dollars for making them ; what did the pants cost him? 6. How many vests at $4 each can be bought for $24? 7. At 4 shillings a pair, what cost 5 pairs of scissors ? 8. If a hat cost $3 and a pair of boots $10, how much more do the boots cost than the hat ? g. Andrew has a pair of ponies, how many feet have they ? 1o. How many more feet than eyes has a four-horse team ? tr. How many less heads than feet hasa _ three-horse team ? 12. A boy spent 21 cents for marbles at 3 cents each; how many marbles did he get? 13. Barton has 17 cents ; how many pencils at 2 cents each can he buy and keep 3 cents? gt5+44+5+5+3t4t5+3+4+2+3 +2+4—% “Ans. 52. 5St5 734324141543 4244 ee +4+5+3+4=? Ans. 60. Examples like the above, having only addi tion, may be given both forward and back- ward, thus they will make 4 examples instead See P. Ed., p. 45. HI lo FIRST STEPS AMONG FIGURES. 99 of 2. Still more may be made by commenc- ing with another number, as 12, and adding 12 to the answer. Thus: 12+ (5+5+313+4444+54+34+2+4+ I+ 5+4+54+3+4)=60+12=72. 8+4+5+4>3X5+4+445—-3—4—5 —A—5—3-—32? Ans. 2r. I5+44+4—-5>345+5+4+4+7+3+4 r5+5¢ Ans. 29. Seems 4 ter t4atsot3ters 5 +4+4+3? Ans. 58. See StS tits t4teretsgto+4 c+ 4+4ts? “Ans. 62. $4+51+44+512+4+4%X3—5+4X8—4 —5—5—4? Ans. 14. Re O47 4-77 X65 3-5 — 4+2+5=6+5? Ans. 9. I8+5+4+4+5—-3—-4—-5+4+5+5+ eee ADS, 27. gee 5 344 42 5 + 34 +5+3+3+2+4? Ans. 61. Seo 4 3-4 X75 —2—-3—4—-5 3 —5—4X3+3+6? Ans. 4. 4424+5434345+444+5+5+3+2 eae tet 4-1-5 3 © Ans, 07. Bt4+44+5t+24+543434+4+5 +444 fe Sates tose sa ApS. 68, 100 FIRST STEPS AMONG FIGURES. 34415 437475751474 4555 +5+3+5+44+5? Ans. 69. 5445444541176 +375 Ans. 12. 744457514474 13--5 34S eee 54 Reon ae —4x3? Ans. 21. §+34+5+47473 13757533 eee —5—5-4—4—-3? Ans. 6. (x) (2)... (3) (4): © {SC 35 5 04 4 59 5a eee 54. 43.-°35 235 (S45 © 53aeae 5.54. 43 4: \45> . 24°50NSs 43°. 935 5.53. .335. 45a 24... 42. 52 45> 4 Slee 555i 4g” 24 244 Gee 34° 35 9525 55 > 32)” 54a § 22> 64 G5 Soe 5855 45 45 21) 44.340 eee 347 364 332 341 385 344 361 Examples in subtraction: 1. 67,548 2. 69,584 3. 75,897 4. 97,867 43:235 345331 545353 52,343 24,313 34,253 21,544 45,524 5. 64,786 6. 79,684 2,432 72953 62,354 72,631 See P. Ed., p. 47. FIRST STEPS AMONG FIGURES. rot The teacher may use his own judgmeni as to teaching subtraction when some figures uf the subtrahend are greater than the corresponding figures of the minuend. — A purely mechanical method is here given with the idea that the method of doing many things may properly precede the veason for the method. If the following method be used, after one or two years or in a larger book the reason of the method should be fully explained to the pupil, and he should then be required to give the reasoning himself. 32,413 — 5,667. Solve by separating the figures of the minuend, as in the line below, and then when any figure of the subtrahend is smaller that the figure above it, write 1 before era T4153 1.3 it thus : 5 6 6 ¥ and then subtract, 2 Se ly pea oa ae being careful when 1 is prefixed to the upper figure to add 1 to the next left hand figure of the subtrahend. The following examples are so arranged that no figure of the subtrahend is greater than 7, the tables having been learned only so far. After solving the above example the pupil should say 5,667 from 32,413 leaves 26,746. 102 FIRST STEPS AMONG FIGURES. ee 7. Qto201 8. 423423 g. 621423 56454 45466 53667 853747 377957 567756 As soon as the pupil understands the mechan- ical work, he should not be allowed to write the 1’s in the minuend, but imagine them to be there. 10, 831,242 “II. 513,423) 2912. 430pees 63,567 46,647 262,637 767,675 466,776 167,576 13. 731,420 14. 342,031 15. 532,514 54,654 25,266 65,251 676,766 _ 316,765 467,263 16. 731,420 17. 624,091 18. 831,042 61,265 62,035 63,415 670,155 562,056 767,627 See P. Ed., p. 48. FIRST STEPS AMONG FIGURES. 103 (2) (2) are t) ce es) (9) eyes 32. 523 249 823555544 Boe 455. 454 Sad 3251 S52 pom 44. 342-44.) 534.) 435 ee 503-535 Bonaeere 55 2534 meee 434. 453 Ripe ots. 6253 Ree ss5i 544 fs - 45. 424) 6 545 43 Been 325 5 4ue ak hang 324 See 545. 454 Bh ESA Ge: poess4) 543 Agee 444e vys43 335 3,205 4:173 = 384 3,876 3,962 More examples may be made from these by reading them from the center each way,thus giving new combinations, or by giving two ad- ditional numbers, one above the upper number and one below the lower one; in this way the combinations will be different whether the pupil add upward or downward. Of course the teacher must add the sum of these two numbers to the answer in the book to get the answers of the new example. Exam- ples in Pupils’ Edition may be treated in the same way. 7. 30,142 8. 23,103 9. 42,301 10. 24,130 2 3 2 2 — oy —= —> — ——_ —_— 60,284 69,309 84,602 48,260 104 FIRST STEPS AMONG FIGURES. Show the pupils that in multiplying, as in adding, if any result is greater than 9, the left hand figure is added to the next result, which is of the same kind. II. 63,524 12. 36,546 13. 36,426 14. 63,524 3 2 3 4 190,572 73,092 109,278 254,096 15. 26,463 16. 64,524 17. 53,625 4 4 5 105,852 258,096 268,125 18. 26,463 19. 46,035 20. 26,304 5 4 6 132,315 184,140 157,924 21. 25,036 22. 50,264 4 6 100,144 301,584 Caution: Do not allow the pupil to write any- where what he is to add to the next product. See P, Ed., p. 52, FIRST STEPS AMONG FIGURES. 105 For addition and multiplication. (8 and rev.) d f g 84736 36258|47362 67834 45678|34567 i 362 678 Cc 73625 78345 h 5847 8345 (e 25847 56783 a 4 58473/62584 34567 \83456 For subtraction. a b | c Berspicowt4 5 Sera eee 3) 4503 0 454) 3° 8 7 6 6 d e f rs Is PaeOnTO | Feb 1e.4T2 ieee "3 (ems 415.6 7 8 16 54. 3 g h i o-oo Beas. 0 719.3145) 67 8 For division. a b é 24 49 24 18 8|40 20 6 48 21/42 20 32 15 56 Seems 4|-5 + 43°98. 71.65 4 3 8 Tees e f H24,10 30 56/32 28:12 64 35/1230 12 21 14 geese 7} >. 43°98 7[' 6 5 4°37 g h 1 eGmmeeteet0124\40-15 42/15 16-48 28 Pepe A 3/96 7),5 8 6 7 106 (24) 345 234 532 424 352 443 345 532 314 31521 (30) 545 434 55 lee: 224 532 454 534 325 43 3,199 FIRST STEPS AMONG FIGURES. (25) (26) 235 25 543. 54 455 43 345 55 A32n42 Soqueed 345 54 553 65 345 54 3,807 383 (31) (32) 435 45 DA. 345 23 534 54 242 2045 555 33 3430 654 543 45 254 54 442 33 4,247 390 See P. Ed., p. 59. (27) 435 543 a4 455 543 235 352 544 535 3,876 (33) 543 355 ot 542 345 554 435 543 343 a0 4,449 (28) 345 523 454 325 543 454 535 234 555 3,968 (34) 231 545 354 434 543 355 332 435 543 345 4,117 FIRST STEPS AMONG FIGURES. 107 The following tables are an excellent prepa- ration for short division. Before the pupils solve the examples on page 120, give them a review of this page. Division with remainders. *o5’s (and review. ) a b c d e SGntO si1\22 9 .6..13/5 14.4, 7118 13. Tz 3.42 31424 3l2. 43214 3 2 6’s (and review.) a b Cc d fagioos 20)11 26.11 Oj21°10 19 7\33 149 5 Weel 5 4° 3 O15 4° 3 21-5 6 2 3 e f g h i Pimento 71% 5 35 27/20 23 Siig 7 32 pemeee set 3 2) 9.67,95| 6) 23) 4°56 1. What cost 8 dozen buttons at 7 cents a dozen ? 2. What cost a pair of boots at $7 anda hat at $5 ? 3. How much more does a reader cost at 6 shillings than a speller at 2 shilliags ? *These are to be recited as follows: 3 in 16, 5 times and F remainder, &, 108 FIRST STEPS AMONG FIGURES. 4. There were 9g birds in a flock, and a hunter killed all but 4; how many did he kill? 5. How many knives at 7 shillings each may be bought for 35 shillings ? 6. A boy spent 21 cents for marbles at 3 cents each; how many marbles did he buy? 7. There are 8 pigs in one pen and 5 in another ; how many in both pens? 8. A boy earned 8 cents on Monday, 7 cents on Tuesday, and 6 cents on Wednesday ; how many cents did he earn in the 3 days? 9. Henry bought 8 marbles at 4 cents each ; what did they cost him? Io. 13 boys were skating ona pond ; during the afternoon 8 of them fell upon the ice. How many of them did not fall? 11. Fred had 15 cents, he spent 5 cents for oranges and 1 cent for candy; how many cents had he left ? 12. George had a bank into which he put 7 cents, his father 8, and his sister 4; how many cents had he in his bank ? 13. Charles has 6 cents and his sister has 2 cents more than he; how many cents have both ? 14. Henry had $8 for Christmas and his sister half as many ; how many had both ? pee P. Ed., p. 56. ' FIRST STEPS AMONG FIGURES. 109 15. William bought an orange for 4 cents, a fig for 1 cent and some candy for 2 cents. He sold them all for 12 cents. How much did he gain? 16. A boy received 6 cents a bushel for picking hops ; he earned in this way 48 cents in one day. How many bushels did he pick? 17. Carrie whispered 3 times in 1 day, for each time she whispered she had to remain after school 5 minutes ; how long did she have to remain ? 18. 20 cents are to be divided equally among 5 boys; how many cents should each boy receive ? 19. If 6 pieces of tape cost 24 cents, how much did one piece cost ? 20. If a boy earned 28 shillings in-7 days, how much did he earn in 1 day? 21. Samuel walked 28 miles in 4 days; at that rate how far would he walk in 1 day? 22. How many quarts of milk at 6 cents a quart can be bought for 36 cents ? 23. If 4 gallons of molasses cost 28 shil- lings, what cost 1 gallon ? 24. 56 cents will hire how many boys for an hour, if each boy is to have 7 cents for an hour’s work ? 110 (1) 52 45 3! 53 44 25 53 44 a5 34 55 od 471 4,640 4,481 (11) 544 354 435 543 431 303 454 FIRST STEPS AMONG FIGURES. (2) 242 525 343 451 535 434 453 345 523 2 5+ 535 (3) 543 244 554 343 234 455 542 435 ore 344 535 (4) (5) 42 54 35 43 51 45 34 34 25 45 she 34 (6) (7) 535 344 454 553 342 435 535 234 453 544 341 345 355 453 434 423 544 345 355 534 433 453 (15) (16) 345 432 53 544 434 434 545 542 354 344 534 453 435, G42 543 Sao See P. Ed., p. 59. 5,212 5,200 5,086 4,729 5,300 FIRST STEPS AMONG FIGURES. Il!t Pupils read : I. 3040321 2. 30245000 3. 463317030 4. 500030261 5. 2000032 6. 15000000 7. 320030000 8. 674346537 9. 42000 10. 400320219 II. 3605000 I2. 50000018 13. 463308260 14. 75000341 15. IOIOOIO Read the following: » 67345768 2. 476347854 3. 74000037 4. 735400005 5. 900007000 6. 316000140 7. 80370000 8. 700000004 9g. 7020500 10. 86000045 11. 800006000 12. 90000007 13. 8060700 14. 90430000 15. 735468371 Teach pupils to write Arabic to billions, that is including 999,999,999. Teach the pupils to numerate by periods to the right as well as to the left. Thus: units, thousands, millions; millions, thousands, units, until they are per- fectly familiar with it. Method: Suppose the number ten million ninety thousand three is to be written. Instruct the pupil to write the number of millions first with a comma after it, and that the first period at the left does not need to be filled to three places by prefixing ciphers. For the above t12 FIRST STEPS AMONG FIGURES. number the pupil will write ro, at first. Teach- er ask 10 what? Pupil, 10 million. Teacher: What period is next to right of millions? Pu- pil: thousands. ‘Teacher: How many thou- sands are there (in this number)? Pupil: ninety. Teacher: Write it after the comma, and, as it fills but two places, place a cipher at the left of the 90 and a comma after the go. The number will now be 10,090. - Teacher : You have now millions and thousands; what period is next? Pupil: units. Teacher: How many units are there in this number? Pupil: three. Teacher: Place the 3 to the right and prefix two ciphers to it to fill the three places of the periods. Place a period at the right because it is the end of the number, and you have 10, 090, 003. Teach the pupil when writing numbers at the blackboard to turn directly away from it as soon as units and the period are written, for he should be sure that the number is correct with- out numerating to the left. Teach the pupils of course when there are no thousands, to write three ciphers and treat unit’s period in the same way. When teaching to write billions, trillions, etc., follow the same method. FIRST STEPS AMONG FIGURES. 113 Teach the pupils that for the word hundred, you will write on the board hun. ; for thousand, th. ; for million, mil. ; and when you get so far, for billion, bil. ; for trillion, tr. ; for quad- rillion, quad., etc. This method will save the teacher much labor and much space on the blackboard. Thus the teacher may write upon the board: ** Write in Arabic five mil. forty th. six.” To be written on the blackboard for pupils to bring to recitation written in Arabic: 1. Write in Arabic four th. fifteen. fs) a pe ** twenty mil. three hun. ae f) nine milforty th. Caution: Teach pupils to put a comma only after each period, except the last. ‘Thus in Nore.—The following diagram may assist pupils in writing numbers, but after being used a few weeks the pupils should write numbers without using it: | Millions. Thousands, Units. 35 | 058 The teacher may draw a diagram like the above and allow 003 the pupils to write numbers in it, as the number 35,058 003 is placed there. Teach the pupils that the third period represents millione, and that each period is read as if it stood alone, only that its name is given, 8 1i4 FIRST STEPS AMONG FIGURES. three hundred seventy-five million, four hun- dred eight thousand seven hundred forty, the pupil may have an idea that he should put a comma for hundreds, whereas the above num- ber should be written 374,408,740. When no © name is given to a number it is supposed to be units; ¢. g., two thousand three hundred eight. Eight here means eight units, and three hun- dred eight, (which has no name given to it) is 308 in units period. The following examples may be written upon the board and the pupils required to bring the answers to class: 1. Write in Arabic, thirty million, eight thousand, three hundred fifty-one. 2. Write in Arabic, two hundred fifty thou- sand. 3. Write in Arabic, one hundred sixteen million two hundred twenty. 4. Write in Arabic, three hundred million, sixty thousand, five hundred, seven. 5. Write in Arabic, five hundred thousand. 6. Write in Arabic, one million, one thou- sand, one. 7. Write in Arabic, seventy million, six hundred thousand, eighty. See P. Ed., p. 61, FIRST STEPS AMONG FIGURES. 4.5 8. Write in Arabic, one hundred fifty-four million, two hundred sixty-one thousand, five hundred forty-eight. g. Write in Arabic, eight hundred million. to. Write in Arabic, three million, three. 11. Write in Arabic, ten thousand, ten. 12. Write in words, 809271300. 13. Write in Arabic, five hundred, four million, forty. 14. Write in Arabic, ten million, ten thou- sand. 15. Write in Arabic, one hundred one million, one hundred one. The Roman notation uses the following Pe eieeeee) ——1,.4V 52 X— 102 Ls, C=— 100, D=500, M=t1o00. To read a number expressed in the Roman notation : *Rule: Add the values of the letters, observ- ing that when a letter is followed by one of greater value than itself, the difference between the two is to be taken in making up the sum. 16. Write in Roman, three hundred forty- five. 17. Write in Roman, one hundred seventy- four. * From Olney’s Elements of Arithmetic, 116 18 19 FIRST STEPS AMONG FIGURES, . Write in Roman, four hundred sixty-two. . Write in Roman, six hundred ninety-six. nine, 21 22 23 20. Write in Roman, eight hundred ninety- . Write in words, 245306341. . Write in words, 32743642. . Write in Roman, three hundred eighty- seven. For rapid solving. I. 44+34+54+2+64+44+54+64+34+5+4 +3+6+6+5=? Ans. 67. » 34+54+446434+6454+24+64+5+4+5 +4+6+6-2-b4— i Ansa ~5+64+64+44+34544+3+6+246 +6-+5 +64+-2--4= fA sae © 4434675 +24+-64-64-'5 + Anes ae +5+4+6+6+5=? Ans. 76. -54+64+344+64+5+64+64+5+4+3 +6+6+6+5+4=? Ans. 80, © 16 +657 X54+35-3XS8 35a 7 oe +5+3—5—6=?. Ans. 29. - 19+4+5+4x6—6—5—6+5 X8—4 +6+5—6x7—4=17 Ans. 31. . 7X6+5+6—4—6—6—5—SxO75 +6—3—6—5+6=? Ans. 27. . 6x8+ 6—2—5—6—5—4+4X7—5 43 OU aaa, Ans, 8, 10. II. BY 13. er 4s re 16. 17. 18. 19. 20. FIRST STEPS AMONG FIGURES. EY7 84+77+6+5+5 +475 x6—5—6—6 —6—3+8+5=? Ans. 7. e505 — 6-4-5 +045 +4+6X5 —6—5—6+5=? Ans. 23. 47+5—6—4—5—2+5 x 8—5—6—6 +3+6+8+6=? Ans. 21. 2a2-5 0-4. * 0+5--0-5=-7.%5—6 —6—-6x8—6=7 Ans. 18. 16-4 x6+5—4—6—5+6+51+6+4 7X S—4—A—5 = ¢ AnSy 27. Gee 0 — 4-3-3 040-4 5-6-5 —6— 2-7-5 =? 2Ans. 3: Boe O 4-0 Be 4 a-O 5 0-4-4 X'S —=6—5-+-38x7= 7? Ans, 21. 61—6—6—5—5—4—-6—3-—6+5 x8 —5—6+7+6=? Ans. 9g. Bate O55 tt 3 6 54 Ot 37717 +5+5+6+6=? Ans. 33. 16+5+4+3+5—6—3+8x5—6+4 Han 5-6 —4— +) yAnS. 13. 27+5+4~6x8—6—4—5—3—6—5 —4+3xX6—4—4—6=? Ans.16. The teacher is advised to give a few exam- ples in subtraction each day, and with them a few in multiplication and perhaps in division also, 118 I. FIRST STEPS AMONG FIGURES. Subtraction. 3,423,056 654,368 2,768,688 - 4,320,032 543,054 3776,978 - 7,360,042 2,500,075 4-859,967 10. 8,340,050 762,034 _7;578,016 13. 6,343,520—656245= ? 2 i fe 834,657 6,517,386 . 63,140,052 6,572,036 56,568,016 . 61,420,035 549,257 60,879,778 64,230,051 510,765 63,719,286 » 75352,043 3. 9,635,024 eae 548 8 8,888,476 - 8,400,314 50,248 8,350,066 - 5,304,036 202,356 5,101,680 12. 75,310,040 230,076 75,979,964 Ans. 5,687,275. * 14. 94530 062—8,240,277=? Ans. 86,289,785. See P. Ed., p. 64. FIRST STEPS AMONG FIGURES. 119 * Multiplication. 1. 648,057 2. 746,805 3. 470,685 © 8 7 6 5,184,456 eee 2,824,110 4. 8,640.753 5. 680,574 6. 358,407 _ 7 8 6 60,485,271 5.444.592 2,150,442 7m055,740 X4— 2 Ans. 2,742,960. 8. 7,406,853 x7=7 Ans. 51,847971. The teacher should solve an example in which there are two figures in the multiplier. G.0)364.%23= 2 Ans,.146,142. Boast 402 % 32> ¢- Ansi-3,7.10;754. ot. 20 A25°% 34= ? Ans. 1,238,450. Berets 2.025 x 43=— 4 ¢ Ans. 15,205,875. Peese.524 x O5—' -Ans..36.629,060. 14. 350,264xX36=7% Ans. 12,609.504. Beer so. 204 x 34 —- Ans, 17,004,336. 16. 640,536x64=2 Ans 40.994,304. 17. 4675 X45=—? Ans. 210,375. *The numbers used in these examples are pointed off in periods for convenience in copying to blackboard or slate, 126 FIRST STEPS AMONG FIGURES. Before taking up examples in short division review division with remainders, p. 107. In teaching pupils short division when there are remainders during the operation, write the figures of the dividend well apart ; thus, in the : ste 43.7 19 35 33 11 example 379531~4, write wera 6S write the remainder before the next figure as in the example given. After solving 2 or 3 exam- ples in this way, write the figures closely, in the usual form on the blackboard, and let a pupil divide orally, the teacher using the crayon, one pupil telling how many times it is contained and what remainder, the next stating what the next partial dividend is and how many times the divisor is contained and what remainder, etc. Let the pupils first solve the examples with- out remainders, given in Pupils’ Edition, and the following 4 examples : : I. 24129318—-3=? Ans. 8,043,106. 2. 281,683,220—4=? Ans. 70,420,805. 3. 12,246,921 ~-3=? Ans. 4,082,307. 4. 322,412,836+4=?P Ans. 80,603,209. With remainders. 5. 83,923-3=? Ans. 27,9741. 6. 182,539+4=? Ans. 45,634.34. oo) fg) Wear aie and FIRST STEPS AMONG FIGURES. I2I 7- 1,373:224>3=% Ans. 457,7418. 8. 91,354.328+4= ? Ans. 22,838,582. Oma73.701-5—!f, Ans. 54,756". 10. 1,035,879+6=? Ans. 172,646°. T1. 2,683,507+4=? Ans. 670,876%, 12. 3,921,278--6—7? Ans. 653,546%. Method of teaching pupils to add numbers like 46 and 7. In adding 46 and 7, ask the pupil what he should add first, and either get from him or show him that 6 and 7 are to be added first; that it makes 13, of which the right hand figure is 3, which will be the right hand figure of the sum of 46 and 7, and that 46 and 7 are 53. Persevere in this plan upon the following numbers or until the pupil in adding such num- bers as 37 and 8, will say at once “the right hand (or least) figure will be 5 ; 37 and 8 are 45+6? 67+5? 58+7? 65+8? 86+5? 58+6%2 34+7? 57+52 26+8? O57 & a7 +82 78+5 2 86+7? S74 58+8 ? 63+7? 47+7? 28+4? 76+5? 53+8? 7 t:0;? 764672 67+3? 384+7? apron 3515 20 47-462 28+3? ¥22 FIRST STEPS AMONG FIGURES. 43 +7! 64+4° 45+8? 26+5? 4720 38+6! 533s 7447 % $5ck 42 he yOnrRoy 37+5?2 58+2? 5376? 24+3? 65 -+7< 86+ 4? 27+8? 68+5?2 83+2? 54+6? 65+3? 36-472 sy 4a 68+8? 43 +5? 54+2 2 35+6; 16+3? Ag 4 38+4? 53+8? 24+5% 45+2? 86+6? ~47+3? 58+7? 4944200 3443209 075-5 eae Hy Bw 48+3? The exercise above should be most used un- til the pupils are perfectly familiar with the combinations given, which embrace all between 8 and 3 inclusive and some are given twice. It may first be given to the pupils in the order above, then commence in the middle of the ex- ‘ercise and go each way. It should not be written upon the blackboard but recited orally from the reading of the examples to the pupils. 1. A boy bought a top for 18 cents and sold it so as to gain 7 cents; what did he sell it for ? ‘ 2. James’s mother gave him 30 cents with which to buy oranges. At 6 cents each how many could he buy? : FIRST STEPS AMONG FIGURES. 123 3. Willie said he had 4 cents ; John said he had 4 times as many; how many had John ? 4. George had 8 sticks of candy and his sis- ter had 7 ; how many did both have? 5. 18 ripe peaches were on a tree and a bad boy stole 7 of them ; how many were left? 6. A flock of 18 birds lit upon the ground ; a hunter shot rz of them; how many were left? Ans. 11. 7. How many fingers have 4 boys? 8. Charlie’s mother gave him 7 cents, and his sister gave him enough to make 13 cents ; how many did his sister give him? g. John had 10 cents, one of his sisters had 7 cents and the other had 6; how many cents did both the sisters have ? 10. Mary. bought 2 yards of calico for a doll’s dress ; she gave 8 cents a yard; how much did the dress cost? 11. A tired school teacher struck a naughty boy five times upon each hand ; how many times did she strike him? For addition and multiplication. g’s (and review.) b 74859 94567 d e f | g 6374 | 85963 |74859|62748/59637 8945 | 67894)56789|49678|94567 See P. Ed., p. 74. a 85963 45678 Cc 124 FIRST STEPS AMONG FIGURES. h i j 4859 6374 8945 6789 For subtraction. a b c Il, IO631 13 13 15 ;104t2 It 16 14 12 8 99 5 7. © 5.4: 500 yee d e f Llei eagle Octhr7zie IO “LONtiees 4.138.750 7 5aeO ates 6 8 6 g h 1 8 15 13 18 TO clades © 9 13 15 IO 5 Ps] Bee 4-94" s 69 8 7 j k I2 14 14 12 17°9 3557 8 94 For division. ea b Cc 42 54 25 32 24 63 16 40 30 63 48 ThE ord. S09 5 7 d e f 27420472 025 48 20 54 12 35 24 64 9 4 7 659 4 57 0ra8 g | h i 28 42 18 24 81 40 56 45 36 30 7 9 4 9 B27 9 4 5 FIRST STEPS AMONG FIGURES. 7 (1) 36 j 18 49 32 72 6 8 k 18 36 56 Se Sa palpi oe(3)s2 "05 (4) 654 245 45 343 634 56 565 453 43 654 546 64 345 563 36 4,056 4,797 421 See P. Ed., p. 77. 426 FIRST STEPS AMONG FIGURES. (12) (13) (14) (15) 45 546 54 354 66 66s 36 - 236 45 565 56 653 (16) 544 654 | 565 666 325 456 302 2,019 2.210 (3) (4) 24 15 45 344 54a se 53 544 44 355 35189243 33 5aei25 42 544 Chae at 33. 554 2 Leaeaas BA eetz3 35 22345 507 4,800 5,880 mee 1.) id... p. i 70s 5.245 5,048 (17) (18) 45 53 6 64 55 55 63 46 34 = 333 65 56 268 307 (6) (7) 36 26 56° 353 63 66 25 32 66 3865 54 66 46 26 35 63 63 66 56 2755 6-43 63 66 45 35 614 662 FIRST STEPS AMONG FIGURES. 127 The answers to the examples are at the end of the book. They are placed there so that if the teacher wishes any pupil of the class to copy the examples on the blackboard for him, it may be done without the pupil’s knowing what the answer is. If the teacher prefers to have the answers with the examples he can copy them from the end of the book. In adding long columns of figures it is well to write the sum of each column 56 separately, as follows, so that in 34 adding each way for proof the 120 sum of each column may beseen = 57 at a glance. 69396 Show the pupils that when g is added to any number the unit figure of this sum will be one less than the unit figure of the number, thus: 9 +37 is 46, 74+9 is 83, &c. 34+6? 75+9? 26+5? 4748? 18+4? 47+9%? 24+3%? 85+6? 36+9? 67+5? 48+8? 294+4? 64+72 45+3? 26+6? 87+9? 68+5? 59+8? 74+4? 8517? 564+3% 47+6? 78+9?% 69453 128 FIRST STEPS AMONG FIGURES. 74482. 35442 56471 G7aeoe 78+62 5o+9? 75+4? 45+8? 26+4% 574+7% 78+3?% 59+6? 84+92 65+5?% 4648? 8744? 68+7? 494+3? Give much practice on the above exercise ; it will be of great use to the pupil in all additions. 8. 463,075 x 465 =? 9. 640,753 X 3061 10. 560,423 X 204= ? 11. 67,052 X224e= 12. 574,003 X27=8 13. 867,534 X 56=? 14. 680,574 X 78=? 15. 475,306 X 406=? Before taking the next examples give the pupils a thorough drill in division with re- mainder. Division series with remainder. (6 and rey.) a b c 23 10 3125-5 2Zslled 19 27 2207an7 4 32 66 5 4 23 45 tae d e f 21 817 20 532i 7a | 6 13 15 26 65 4 3 2-6 5.2 |g g 1549 39 12 432 6 5 See P. Ed., p. 60. FIRST STEPS AMONG FIGURES, 129 7’s (and review) with remainder. a b | Cc 22 19 42 34 69 19 14 39 28 62 17 39 Bae 5. 6 Hee SM i ye A or | e | f 32 23 53 13 34 29 57 47 | If 31 21 52 es | 45 67 4 5 P| ages: | 193 51 28 27 17 47 32 26 21 49 39 25 (ly py SU) URS ie ae ad On, a 8’s (and review) with remainder. a | b c 55 204524 | 3416 31 53 29 43 21 29 Bee | 404° 3 8) 7 65 4 3 d e | f 41 69 13 31 20 39 28 53 | 76 47 22 38 Leow se + Sy ie Siete & O85 g | h | i 26 28 63 32 49 33 61 39 | 58 31 14 23 aeoeey | 25 6-7 8 6 4 4 3 j k 19 39 69 | 44 2617 On) 8053154 I. 1,396,897 -3=? 3- 2,144,698 4=? 5. 22,539,018 +6= ? ? 2. 1,621,287--5=f 4. 39,164,794+6= 6. 27,622,523 +6= ? ? 130 FIRST STEPS AMONG FIGURES. 7. 10,763,483+-7=% 8. 45,171,547>7=? 9. 31,752,200+7=? I0. 508,019,899--8= ? 11. 34,856,549+-8=? ‘12. 61,476,243—-8= ? ’ Unless the pupils solve the foregoing examples readily, they should review them at once. LONG DIVISION. The teacher may say to the pupils that when the divisor is a large number the method of short division is too difficult, illustrating by an example. Teach the pupils that the first step in solv- ing an example in which the divisor is greater than 12, is to place a comma after the first figure in the divisor as in the example, 5,02)73245. As in short division we cannot divide the whole of a large dividend at once, so we cannot in long division. The next step is to find how much of the dividend we will divide at first. See if the first figure of the divisor is less than the first figure of dividend or whether it is greater. In the example given it is less, (5 being less than 7). Teach the pupils that when it is less they are to count as many figures in the left of the diyi- FIRST STEPS AMONG FIGURES. —13t dend as there are figures in the divisor and place a comma after the last figure. In the given example there are three figures in the divisor, so count 3 figures in the left of the dividend and writing a comma there, the example becomes 5,02)782,45. Require the pupils to take these two steps, (and no more) with the following examples, first placing them in form for dividing: 76345 +4321. 5738+ 49. 9458875. 387598858841. 748567+2145. 57881 ~468. 76854+68. 38768456+25864. Teach the pupils that if the first figure of the divisor is greater than the first figure of the dividend we count one more figure in the divi- dend than there are figures in the divisor, in order that the part we take may be large enough to contain the divisor. * In the example 687531 —7342 the first figure of the divisor, 7, being greater than the first figure of the dividend, 6, we count one more figure in the dividend than the 4 figures there are in the divisor and the example with these two steps taken becomes 7.342)68753,1. Require the pupils to take these 2 steps with the following examples and pevee re, 10., D.. Oo. 132 FIRST STEPS AMONG FIGURES. ee ee ee ee more if they are needed to make the entire class familiar with these steps: 57281456 — 71235. 4232145347. 34216052543. 684771. 5875643 +643. 68475032 +934674. 47325684+ 3145. The next step is to count the number of figures at the right of the comma in the divisor and count the same number of figures at the ieft of the comma in the dividend, and place a comma before the one counted which is farthest to the left: thus, in the example 34276404, the first step is 4,03)34276; the second, 4,03)3427,6 ; the third, 4,03)34,27,6. Require the pupils to take these steps (and no more) with the following examples : 67453275342. 75694857 +845 321. 546327 +643. 345367471 +75382. 47346 = 23. 43264726351. FIRST STEPS AMONG FIGURES. 133 8345376-+702. 56341 68, 763542 + 8547. 57345 =, 493: In the example 63166 3-+201, which after the three steps is 2,01)6,31,663, the next step is, see how many times the number at the left of the comma in the divisor is contained in the number at the left of the first comma in the dividend. 2 is contained in 6 three times. The example becomes 2,01)6.31,663(3. Next multiply. the divisor by this quotient’ figure, placing the first figure of the product under the figure before the last comma, thus: 2,01)6,31,663(3 603 Next step see if you can subtract. (Teach the pupils to look at the left hand of.the num- bers to see whether they can subtract. If the pupils ask what is to be done when you cannot subtract, tell them you will show them in the first case in which they cannot subtract, which will not occur in the examples given for some time.) Next subtract. (Show the pupils that the remainder should be less than the divisor.) 134 _—«¥FIRST STEPS AMONG FIGURES. Next see that the remainder is smaller than the divisor. (Do not show the pupils what to do when the remainder is larger than the divisor until a case . occurs in their work.) Write the next figure of the dividend at the nght of the remainder. 2,01)6,31,663(3 603 286 Next step count as many figures from the right of the partial dividend as there are at the right of the comma in the divisor and the ex- ample becomes 2,01)6,31,663(3 603 eee 2,86 Divide as at first and so continue the opera- tion. The steps are: rst. Write the divisor and dividend in the proper form. 2d. Point off in the divisor. 3d. Place the right hand comma in the divi- dend. FIRST STEPS AMONG FIGURES. 135 4th. Place the left hand comma in the div- idend. 5th. Divide. 6th. Multiply. 7th. See if you can subtract. 8th. Subtract. gth. See that the remainder is less than the divisor. toth. Write the next figure of the dividend. t1th. Point off. Repeat steps 5, 6, 7, 8, 9, 10 and 11 until the example is solved. The teacher should solve the following three examples zz*h the pupils, before any are given them to solvealone. The teacher taking the crayon, the pupils will tell what is to be done, one pupil describing the first step, another the second and so on, or better yet, one of them take one of the steps then another pupil take another and so on. First solve twice the example ANG in the foregoing illustration. * For method: * The pupils should erase the commas which divide the number into periods before pointing off, that there be no confusion. The commas for the operation -of di- viding may be placed above the number instead of beneath it, if preferred, 136 FIRST STEPS AMONG FIGURES. I. 5,450,204 403 = 2. 3: 162,479,845 + 3042= 3 20,913.844>604=? Do not give the pupils more than one or two examples each day until you are sure they understand the method. 4. . 16,312,418--5,013—=* — Ans. 3,254 - 273,785,577 + 60,345 = % . 26,308,025 4,002-— . 189,771,597 +50,364= 4 . 524,601,734+6,047= 2? . 391,838,602 + 80,675 = . 619.307,367 + 80,597 = % . .278,696,736+6.075 = # . 45+3603+456+542 +0 300 aa on nui 4,920,352-+2,023=? Ans. 2,432"° 116 454+ 32 +46 +553 +636145= - 465+564+324+646+4+ 553 bc ,. 632 +665'+356+43 +655 +6— F . 426 +563 +365 +6344+545 +643 +356 | 266 +633+56+445+54+63=? 1. A fox caught 5 geese which were 1-3 of the farmer’s flock; how many geese in the flock ? 2. A hen had 15 chickens ; a cat caught 4 of them and a hawk 3. How many were left ? See P. Ed., p. 82, FIRST STEPS AMONG FIGURES. 1373 3. Arthur was. paid 14 cents for doing an errand ; he lost 5 cents and his sister gave him 7. How many cents had he then? 4. A squirrel carried 4 nuts home one day 5 5 the next day and on the third enough to make his number 16. How many did he carry home the third day ? 5. James has 18 apples to divide equally among 3 boys; how many shall he give to each ? 6. How many yards of tape at 2 cents a yard can I buy for 15 cents and have 1 cent left ? 7. Fred gave 5 cents for an orange, 18 cents for figs and 4 cents for a lead pencil. How many cents did he spend? 8. In an orchard the trees.were set 16 in a row ;-7 in each row died. How many living trees in each row? g. A boy sold a pair of doves for 25 cents and bought as many marbles at 3 cents apiece as he could for the money. How many marbles did he get and how many cents left? 10. A boy earned g cents one day and 12 cents the next. How much more did he earn the second day than the first ? c 138 FIRST STEPS AMONG FIGURES. 11. Frank had 16 rabbits ; he sold 3 to one boy and 4 to another. How many did he keep ? 12. How long will it take a miller to grind 42 bushels of grain if he grinds 6 bushels in an hour ? 13. Willie bought 2 pass-books at 5 cents each ; a lead pencil for 6 cents, and 3 oranges at 4 cents a piece. What did he pay for all? 14. Willie keeps rabbits to sell; he has 20 and has 4 little houses for them. How many does he keep in each house? 15. If 4 bags contain 8 bushels of grain, how many bushels will 9 bags hold? 16. If 5 cords of wood cost $30, what will 3 cords cost? 17. If it cost 15 cents to ride 5 miles on the cars, how much will it cost to ride 7 miles ? 18. How many bushels of oats will 3 horses eat in a week, if 6 horses eat 42 bushels ina week ? | 1g. If 6 brooms cost 18 shillings, what will 5 brooms cost? 20. If a barrel of flour will last 2 men 6 months, how long will it last 1 man? 21. If 2 men consume 6 barrels of flour ina See P, Ed., p. 86. FIRST STEPS AMONG FIGURES. 139 certain time, how much will 1 man consume in the same time ? 22. If 4 horses eat 12 bushels of oats in 3 days, how many bushels will 1 horse eat in the same time? 23. If 3 teams will plow a certain field in 6 days, in how many days will 1 team plow it? 24. If 4 men can dig a certain ditch in 8 days, how long will it take 2 men to dig it? 25. If 3 men cut 6 cords of wood in a day, how much will 9 men cut in a day ?, 26. If it take 3 men 6 days to cut a pile of weod, how long will it take 9 men? 27. A man lost $6 by selling a cow for $37 ; what did the cow cost him? 28. A boy sold 4 pencils at 2 cents each, and 3 marbles at 3 cents each ; how much money should he receive ? 29. What is the wheat in 7 bags worth at $2 a bushel, if there are 2 bushels ineach bag? Read the following : I. 70000580030. 2. 68000¢50000. 3. 680507415371. 4. 756847597547. 5. 76500000068. 6. 67459800009, 140 FIRST STEPS AMONG FIGURES. Teach pupils to write billion. 1. Write in Arabic, seventy million three thousand forty. 2. Write in Arabic, five billion four mil- lion nineteen. 3. Write in Arabic, ten billion three hun- dred million fifty thousand. 4. Write in Arabic, fifteen ‘billion nine thousand. 5. Write in Arabic, two hundred billion ten million twenty. 6. Write in Arabic, forty million twenty thousand. 7. Write in Arabic, nine hundred forty bil- lion one hundred six thousand five hundred. 8. Write in Arabic, sixteen billion sixteen. g. Write in Arabic, five billion forty mil- lion. to. Write in Arabic, nine hundred billion nine. 11. Write in Arabic, eight billion ninety thousand four. Teach Roman to 1880. 12. Write in Roman, one thousand three hundred forty-one. 13. Write in Roman, nine hundred seventy- SIX. See P, Ed., p. 88. FIRST STEPS AMONG FIGURES. I4!I 14. Write in Arabic and in words MDCCL- XXV. 15. Write in Arabic, ten billion. 16. Write in Roman, 1876. 17. Write in Roman, nine hundred forty-nine. 18. Write in words 761308260017. 20. Write 6 units of the 8th order, 8 of the 5th, 3 of the 3d and 5 of the rst. 21. Express the following number by naming the units and their order, beginning at the left : 70900048010. For practice in subtraction, to be given to the pupils oradly and recited as the series have been. CN ie) +e (To be read, 45 from 49 ?) as Ne ae Moeeo e520 35. 62)..35> 66 19 51. 22 37.96 45 34.55 29. 65 14 43 19 45 82 56 32 59 36 51 85 31 56 AomOOmeAO. 26 ~53 ° 29°40. G6. 25°. 54 Been to) 35°<72' 4647 , 63.30. 57 Beeb yre 10552564 43° 38 59 23 54 Ages 2) 32-04 -46 61 67 48.54 39 Beee4oez 7 507° 45 54.63 °39- 48 37 142 FIRST STEPS AMONG FIGURES. 3L-.70 34 50 24° 51-67) 6h eaeneD 26 64 25 46 17 48 59 63 79 48 39-905 27 86 For rapid solving. (To be read.) I. 3+5+9+7+8+4—6xX9+94+5+7+ 6—3+7x9—8-8—=3 37 ee X8-+9—4—-5-—-7= ee 2. 657-5 5-3 19%) —-0 7 X9+ 9+-8+7%9+8-6 27 ee —5—5~4x8=? Ans. 64. 3..61— §-9—5—7+941 > Ocean +6—7—8+3 x 8—g—7—9— 5=+7 %4+7 £4—6= 2 2 Ans, 20 4. 9x8—6—5—7=90X4-8 X67 Ge 7+3+9xX6—274xX6—7+7x8+2 +6X4+4+8=? Ans. 4. 5. 4X34+8+4x7—-348X5+476X7+ 8-9 X6+6+5x3+2+4+4+2+648 +5+o=? Ans. 17. 6. 5X7+7+6X4+7+7X6—4—5 +3 X 4+4+8x51+6+6+4x6—3=+5 x3 +8+5+9=? Ans. 16. 9. 5X7—-3+8 X64+6+5 x8—3+9x6— 6+-4xX7—6—8+7X5+6—1+543 x7—2+6—=? “Ans. 9, FIRST STEPS AMONG FIGURES. 143 8. 4X6+8+8 x9—4+8 x 9—9+34+7= eras? ANS, 4. 9. 6X34+74+44+7+9X44+5+3 x 6—6+ 6x8+6+9 x 7—8—6+4 x 3+3+8 mee SO 7 7 ANS: 16, 10. 44+7+6+94+8—7+34+8+48+5 x6— 6—3=-4.%8~-7--6—7=-3.xX9+9+5 Xg9—7—8=? Ans. 66. Problems for the slate involving Addition, Subtraction and Multiplication : 1. There are 320 rods in a mile ; how many rods in 79 miles? 2. * William has 75 cents and Charl:s has 68 cents more than William ; how many cents have both boys ? , 3. David had 123 cents ; he spent 47 cents for a ball and 39 cents for marbles. How riany cents had he left ? 4. The larger of two numbers is 916 and their difference is 43 ; what is the less number ? 5. If the drive wheel of a locomotive turn around 352 times in going 1 mile, how many times will it turn around in going from Canan- daigua to Rochester, the distance being 29 miles ? * Tn the different steps of such an example it is im- portant as a help to mark each result. wee P..Ed,, p. 90, 144 FIRST STEPS AMONG FIGURES. 6. A farmer having 239 sheep, sold 99 of them and then bought 113 ; how many had he then ? 7. There are 5280 feet in a mile ; how many feet in 357 miles? - 3744-645 +57+ 767+4364+543 +675 + 7+454+765+577+456—=? » 454+7764+567+457+7344+475 +674 7544+5+6764+547+375=? - 64347+75+657+746 +773+464+457 +347+675+766+577=? - 5764745 +457+6744+77+556+473 + 765+7+657+564+705+76=? » 5464+375+657+744 765 +257+6+75 +743+6564+ 7771+467+762=? » 647+7564+475 +367+6364+753+77+ 345 +676+45+576+767 + 654+365=¢ . 6.314,532—521,987=? - 463,524—39,043= ? - 653,425 —64,287=? - 475,067 —36,543= 3 - 688,045 —95,387=? - 4,760,352 —376,534= ? . 4,630,024—921,045=? . 34,000,435 — 2,700,518? See P. Ed., p. 92. ror nun = Oo ON AM HRW NN SS VN i fe) FIRST STEPS AMONG FIGURES. . 4,500,375 —760,187= 2 ‘ 354,000,253 —272,102,437= 4% - 530,024—543,052= 3 . 89,756 xX 96= ? - 364,758 x 356=? . 638,497 X68=? - 498,675 x97=? maD.o47 < 70a £ . 796,805 X 705=? . 807,009 x 608=? m0,0590 % 740—> ¢ . 94,786 X 7,.968= ? © 603,405 —612,134=? CT, 6,53 7,065 aes 743,987 =1 Division series with remainder, g’s and review. a | b 32 15 68 39 61 33 49 26 5 4,9 8. 7 6:54 Cc d 19 37 21 46 78 | 52 53 27 Mesure 031/77. 8 9 Ox 5° | e Brat 28 54 29 34 61 | 35 58 41 PGT 288s bet 4G. 7 145 146 + Ha ODO ON AM RW ND * = N FIRST STEPS AMONG FIGURES, h g 29 18 40 68 43 21 88 54 25 : AG 6 BE 49 8 7 i 47 23 34 6.5 4 « 33,256,023 +7? . 39,044,761+6=? - 34,352,839+5= 3 . 27,832,074-6=? - 445,941,095+7= 1 » 27,516,279+4=? . 449,181,483+6=? - 487,959,992+7=? . 28,556,208 -6=? . 15,228,723+4,058=? . 251,776,292+70,486= ? . 386,124.633+60,578=? 133 14. 41,847,116+9,048—? 520,613,471 70,697= ? Teach the pupils that when any partial div- idend (after writing the next figure of the div- idend at the right of the remainder) is less than the divisor, they must write a cipher in the quo- tient, just as they do in short division. Next See P. Ed., p. 96. FIRST STEPS AMONG FIGURES. 147 erase the comma made in pointing off the par- tial dividend, write the next figure of the divi- dend at the right of the partial dividend, point off and proceed as before. 15. 276,265,200+4,036= ? 16. 42,865,597-+6,075 = ? 17- 397,706,673 +6,053=? 18. 44,153,419,619+70,386= ? Ig. 15,086,456--4,023=? 20. 4,437,512,234+60,289= ? In the preceding examples the second figure from the left of the divisor has in each case been a cipher, and the examples have been so constructed that the divisor is_ con- tained as many times in each partial dividend as it appears to be. A new difficulty will arise in the following examples since the second figure of the divisor is a significant figure. Show the pupils that when the second figure has value, the divisor is often not contained in the partial dividend as many times as it ap- pears to be, since in multiplying the divisor by the quotient figure there will usually be some- thing to add to the product of multiplying the first figure, coming from the product of multi plying the second. ‘Teach the pupils to observe 148 FIRST STEPS AMONG FIGURES. how much there will be to add to this first product and to allow for it. | Teach them also that when any product is greater than the partial dividend, it shows that the quotient figure which gave that product is too large, and that the partial product and that quotient figure must be erased and a smaller quotient figure used. Use the following examples in illustration to the class, or at least as many examples as will make the matter clear to the class: 267,142+-352=? 2,659,478 — 461—? Use too large a quotient figure in some in- stances so as to show the pupils what to do when they use one that is too large. Teach them to use much care in finding the quotient - figure and so save themselves much work. Show that the divisor never should be con- tained in the partial dividend ro times. - 4,367,695 +673=? » 220,396--254= ? - 307,627+354=? . 4,925.151-+-694= ? . 6,310,318-+781= ? . 3,280,381 + 482= ? See P. Ed., p. 98. Nm Rh & NN 7. 8. | FIRST STEPS AMONG FIGURES. 149 239,294,268+3,642=2? | 334,990,037 +574= ? For rapid solving. (To be read and to be answered without use of slate.) I. To. e 6x6—8+4xX6+8—2+6x4+8+5 x9 —9+9X6+8+9=2? Ans. 59. 9xX6—5+7 X6—6+4X7+5+4-9+9 +64+8—4+3=2? Ans. 9. mai O—O— 7-3 X 5+ 7 +6+5--9 X60 +3 [5xX3+9—7—! Ans. 29. » 8X7—9—5+6X 3—-3+3X74+3+5 x 6 —6—6+6x4=? Ans. 28. Ox O—6—7-=-5 X$8444-357X% 449745 +8+7+5+9=? Ans. 27. .4X9+84+4+6 XxX 4—5+3 X6—8 —4+6 x5+8+4=? Ans. 47. 7X O—5—9+4x 8—8—6—2'—8 X54+9 +7—8=? Ans. 33: mOx 44-8—-8X%9016—7X% 841-7 xX9—4 —3-—-7+9——? -Ans. 17. - 9X8—6—9—1-7X54+57+9x6+84+4 pO eOrl yO XA O40. i 5 OX 6 +7+2+9x8—8—5+9=? Ans. 3. 7*x*6—6+9+5 x 6—8'-9+5+7 x 8—6 oo Geeta ok Rg ear Pa Tie mtorr 150 FIRST STEPS AMONG FIGURES. X 4+94+8+9xX7+84+5+8x7=7 Ans. 42. Ir. 9X6—8—7+6+9 X6—6+-3X7+38—7 +6+7 X4+9+8—7—4+6X8—7 —6—7+44+74+8+ 8 +4 X74+8=? Ans. 64. 12, 8X84+6+3—7—8—4+6 xX 7—8—6+7 x 3—8=+6xX090+3—8—45-0 +o oe 7+6+9—7+5+8—7=2? Ans. 45. 13. 6xX8+3— 7 —8+9X7+94+8275 x64 2+7X9—7—9 —8+8 xX 7—8—7+3 +7+8+7=? Ans. 31. 14. 7X9—8— 7 +8X9+8—5—6—4—5 = 6x8+94+97+90+9+ 90 +8 tee —7+8X5+8+9+8=? ‘Ans. 60. 1. How far will a boy walk in 7 days, walking g miles each day? z. How far will a boy walk in 2 days, walking g miles the first day and 7 milés the ‘second day? 3. In how many days will a man earn 48 shillings, at 8 shillings a day? 4. A boy earned 45 cents Monday, and 53 cents Tuesday ; how much more did he earn Tuesday than Monday ? See P, Ed., p. 99. FIRST STEPS AMONG FIGURES. 15 5. If 4 peaches cost 8 cents, what cost 9 peaches ? | 6. If a merchant sells 9 spools of thread in 3 hours, at that rate how many would he sell in 1 hour? 7. If 3 girls can make 6 aprons in a day, now many can one girl make in a day? 8. If 2 girls can do a piece of work in 8 days, in how many days can 1 girl do it? g. A boy bought 9 marbles and lost all of - them but 3 ; how many did he lose? 10. A farmer bought a pig for $6 and sold it for $9 ; how much did he gain ? 11. There were 8 cows in a field and 6 more were put in; how many were in the field then ? 12. There are 17 girls in a class and g boys ; how many pupils in the class ? 13. There are 16 caps in the entry and 7 bonnets ; how many more caps than bonnets in the entry? | 14. If John is well how many days should he come to school in 4 weeks ? 15. A man may rightly work how many days in 3 weeks ? 16. James has 5 cents and his sister has 2 cents less than twice as many; how many have both ? ‘ 152 FIRST STEPS AMONG FIGURES. Examples for slate in Addition, Subtraction, sda eats and Division: . A hasg fields, containing in all 197 acres ; B Ny 13 fields, containing 239 acres ; C has 17 fields, containing 298 acres ; D has 6 fields, containing 85 acres; how many fields and how many acres have all ? 2. How many horses at $185 each can be bought for 25 cows at $37 each ? 3. If 69 acres of land cost $6,486 what will 207 acres cost P 4. I borrowed of Mr. Rawson at one time $697, at another $1,748, and at another $456; I paid him $975; how much do IJ still owe him P 5. What is the sum of eighteen thousand three, nine million twenty thousand, eight hun dred six, seven thousand sixty, 95 thousand seven hundred, twenty-one million five hundred seventy-six, and ten million ten ? 6. Henry’s kite was up in the air 375 feet, it then fell 98 feet and then rose 268 feet; how high was it then? . 7. Three men bought a hotel for $25, Boon the first paid $6,790, the second twice as much, See P. Ed., p. 102. FIRST STEPS AMONG FIGURES. 153 and the third the remainder ; how much did the third pay ? 8. The earnings of a father and his 3 sons for a year amount to $2175 ; their expenses are $957; if the balance is divided equally, how much will each have ? g. If four dresses of 15 yards each are cut from 78 yards of calico, how many yards will be left ? 1. 7+ 46 +744354+47+73+644+57+75 +64+77+451+63+57+464+75= 2. 3764+ 455 +757+463+375+747 +654 +374+576+ 346+775+7+464+ 647 +356+565=? 3. 475+647+756+765 +437 +674+575 +756 +647+ 567+456+743 +357 +556+463+756=? 4. 764+354+8+47+ 85 Geg750587-%75 +84+ 08=? §- 58+ 765 —485+ 678 +537+6+753+ 488 + 846+ 537 +755= ? 6. 678+845 + 784 + 326+487+856+678 +588 + 865+478+756=? 7. 74+58+84+56+78+644-87+6+58+ 75+86+68+75:=? 154 Io. II. I2. Nn WwW NO FIRST STEPS AMONG FIGURES. . 648+ 785+ 874 +688 +5764+845 +688 +786 +75 +847+687+58+6=? - 7584875 +684+ 768 +475 +886 + 744 +358 +652+ 887+ 5464785 +648 eee 8+57+68+84+75+7+58+76 +88 + 47+63+78+86+55+67+88=? 368+ 475 +638+857 +583 +646+878 57 + 645+ 768+582 +7+6764+3848 5387+ 766=°? 67 +788 +856 +475 +687 + 878 + 564 + 87 +656+478 + 880+ 567 +375 +688 856+785=? - 7,963,034—546,573=? . 758,600.341 — 79,423,275 = 4 . 8,460,075 —987,286= ? . 658,000,468 — 35,030,273 = ? » 43,750,078 —44,345,621=? Impossible 6,475,000,374—293,030,596= . 97,806 X 59= ? - 97,865 x 896=? . 96,897 X 6,978=? WS 4exX 7, SOO . 746,800 X 9,000= ? . 470,900 X 70,580= ? See P. Ed., p. (10, ___ _FIRST STEPS AMONG FIGURES. 155 7. 869,070 X 670,900= ? 8. 790,600 x 806,700=? 9. 62,802,889+9=? 10. 71,262,955—-8=? II. 538,908,792+7=? 12. 376,571,086+-7=? 13- 339,253,657+9=? 14. 324,763,528+7=? 15. 446,217,169+-9=? 16. 54,284,406+8,047= ? 17. 25,534,849 +7,197=? 18. 45,126,612+914=? 19. 4,368,565 9,168 ? 20. 63,008,141-+-5,274=? 21. 47,230,943+-79= 1 22. 9,290,055,741 + 4,869= ? 23. 33,279,851 — 48,600 ? 24. 68,643,216-+87,000 ? 25. 76,845,678 100= ? 26. 3,921,534,261-+-486,000= ? 27. 60,064,175 —8,000= ? 28. 90,700 X 50,700=? 29. Subtract 3 billion 6 thousand 750 from 4s billion 1 million seven hundred sixty-three thousand 4 hundred. 30. 284,553,437+3,790=? | See P. Ed., p. 1163 16 FIRST STEPS AMONG FIGURES. | 31. 4,167,300,326-+4,790,000= 2 32. 5,074,000 X 68,070= ? 33: 3:424,330,021-+ 497,000= ? 34. 2,468,576,216+10,000—=? 12’s (and review.) For addition and multiplication. a b Cc 9 12 8 Io 7 II ‘9 T2 8 10-7 11 SiO ulOmEr 1252 O"GgTO rt :12°8. 9 d e i Ee Lote be 7 TAv Ow ie 8 10-9 ai T2aaL ibs 2 ae O.1OgL tae 8-9 MOwia g h 1 6 12° 5 12]. 9 12 8 10-7911 |> Ogtgeeee Ee Ae 12 6 eae 12-6650 210 Lia2 For subtraction. = 0° a1 18 20 17320716 11,(10 O° sos teeme d 20 24.16 19 17 22 Il 12.6 (Qptowas a 22 19 22 18 Ig 19 [2 TIZIO/ Oy oes Cc 16 °21023710, 2015 9-10-11 tO0 Oso e 21° 20/17 20° 23 1874 12°76 0 $102 2071 See P. Ed,, p. 140. ~_—— FIRST STEPS AMONG FIGURES. 154 For division. © aoe 3 Gay b 132 77 80 110 84 48 21 10 2411 £2: i592 a 72 121 108 96 72 I00 feed t2.°S- 9 10 Cc 120 108 72 132 90 99 Peer. o Tr 10 «=O d 56 88 120 81 96 80 Seid 2t0). 9:12.°8 f 63 110 99 144 64 90 Ce LO rity ie 725.41 9.7 e 60 36 eat 2 . 799,896 xX 12=? - 6,347,435 X12=? - 7,968,473 X11=? 73,540,247 X 122 . 989,769 xX 12=? - 799,958 xX 12>? . 8,989,978 X 12=? NI Am BW ND 12’s (and review.) Division with remainders. b II5 71 96 137 106 85 10°-9 Toft ye tar 8 d 116 102 78 105 86 139 | 124 88 94 95 117 89 pam Oe LO, Tl) 12°} 10): 9, 8 12°+1I 10 seein sy Da IL 7. a 127 7694 70 150 108 migecesgr Oo: 412° 11 Cc S 115 78 96 130 63 107 g Sir 12 8 §g For more practice. For division. a 86 152 40 68 47 ifs 12a Oley ao C 1627S whl AOI eek 12S... meaa0 ¢ sOmmeey c 69 54 102 89 45 | 10 2S rye g i” O27 OmnsOnG! | 8 Goel Tanti2 8 09.587,888,171 ~i2==? gy. 95.62:.647+12= Yo. 813,764.564-+-11= ? fi. 83,645,840°>13—— q 2 see P, Ed; 158 FIRST STEPS AMONG FIGURES. b 78 130 94 62 34 QT 2125 eee d 52 55 88 64 107 S : Gari ame f 142 86 52 78 48 fz tt Gloag, h 57 41 70 107 ME a «5 p- 118. FIRST STEPS AMONG FIGURES. 159 For rapid solving. (To be read and to be answered without the use of slate.) I. 17+4+3X91+8+7+6+12x9—8—6 yay 6 O+9O+6+9 KI +04847 179 4 104-34-90+0—24722—1 Ans. 18. 2. 144--2+-2--9X8X 2—147X 1243+ 6-6 X12X2+3+2xX3+9+9-3+ 2+6+7X4+10X8+4x3+38=? Ans. 9. 3. 9X 12-2+3X2—47+2X3+124+7xX 12-3 X2>4+1272X31+6+ 3X2 +775 X8+3X2+3X47+8+9+8 +9X3+6+9=? Ans, 12. Pee 2 + hee] X 12-2 X 3 12 X 1143 X4+12X8+8—24+373x2+8+6 BAGS esa a mh 2 T2- 3X 5 4X 10-42 xX4—4+12=? Ans, 8. Re Oe 7 O XS PK 2—- 3X 247 +7+6 + 2—10X 3—8+5+7xX4xX 2-415 XQX2X3—9+3X4+11 x38+3=—2 Ans. 32. 6. 14X2-4x8+2xX3+7X8+3+2%x4q4 +82 xX 3—-2+-3X2X2—12xX9+8 +9—6+10+3~+2+9+3x2+4=? Ans. 8. ae es 160 FIRST STEPS AMONG FIGURES. 7. I2X11+-3+8—+—2+4+8—2 X 3—3+2+4 X9+3+3X4+24+8+2x343 +12 X9+-3—274X9+ 3X2>-3X278 Xa 2s=7) Ansir6, 8. 19K 24+4+3X 2+4X 124+8+2+245 2X 3+-2X4~712xK8+2x3+4+8 X9+134X3+5X2+8 2 eee =P Ans 00: 9. 9X 3—5 +2X343 4 27 90KN6 oe +5~+-2xX3+9+8X3+ 2X3+2X3 ~-9X4+2xX 5+2+5xX6+3=? Ans. 18. 10. 16X4—-2—2X3+12+4X3-5X4+2 X4-2—6>2X5+25x124+8-+ 12 25 4XI0OS 227 X15 — 50 3—25-5=? Ans. 5. 1. If 4 lemons cost 7 cents, what cost 20 lemons ? | Explanation: Teach the pupils that if a cer- tain quantity of anything cost a certain amount, 3 times that quantity will cost 3 times as much; 4 times that quantity will cost 4 times as much, etc. Solution: If 4 lemons cost 7 cents, 20 lemons, which are 5 times 4 lemons, will cost 5 times 7 cents, or 35 cents. See P. Ed., p. 119. FIRST STEPS AMONG FIGURES. 161 2. If 3 oranges cost 10 cents, how many oranges may be bought for 30 cents ? Explanation: Teach the pupils that if a certain sum of money will buy a certain quan- tity, 3 times that sum will buy 3 times that quantity, etc. Solution : If ro cents will buy 3 oranges, for 30 cents, which are 3 times 10 cents, you can buy 3 times 3 oranges, or 9 oranges. 3. What cost 18 spools of thread at the rate of 2 spools for g cents ? . 4. If 2 knives may be bought for 5 shil- lings, what will 20 knives cost ? 5. If 2 men cut 5 cords of wood in a day, how many cords will ro men cut ina day? 6. If 3 bushels of wheat cost $6, what will 8 bushels cost? 7. If 2 bushels of wheat cost $3, how many bushels may be bought for $18 ? 8. 36 cents will buy how many marbles at 3 for 4 cents ? g. If 3 boys can do a certain work in 6 days, how many days will it take 1 boy to do the same work? 1o. If 2 men can hoe a field of corn in 4 days, how many days will it take 1 man to do it? pte v1 162 FIRST STEPS AMONG FIGURES, 11. If 3:men can cradle 6 acres of grain ir a day, how many acres can 1 man cradle in a day? 12. If 2 men can build a wall in 6 days, how many men can build it in 1 day? 13. If 4 men can dig a ditch in 12 days. how many men can dig it in 1 day? 14. If 2 men can dig 8 rods of ditch in 1 day, how many rods can 1 man dig in a day? 15. If 3 men can dig a ditch in 12 days, how many days will it take 4 men? Call attention of pupils to the difference between the 15th example and the 16th, and teach them to find about 1 of the kind the question asks about. For instance the 15th asks about 4 men, hence find out how many days it will take 1 man. In the 16th it asks about 6 days, hence find about 1 day? Solution of 15th: If 3 men can dig it in 12 days it will take t man 3 times r2 days, or 36 days; and four men can dig it in ¢ of 12 days, or 3 days. 16. If 3 men can dig a ditch in 12 days, how many men can dig it in 6 days? Solution: If 3 men can dig it in 12 days, to dig it in 1 day, it will take r2 times 3 men or See P. Ed., p. 120, .< : ‘ 5 i ] FIRST STEPS AMONG FIGURES. 163 36 men ; and to dig it in 6 days it will take 1-6 of 36 men, or 6 men? 17. A boy lost 4 marbles, then bought 6, and losing 10 he has 35 ; how many had he at first ? 18. What number multiplied by 3 will give race 19. What number subtracted from 7 will leave 4? 20. How many days will it take 8 men to do a work that requires 6 men 12 days? 21. How many men will do a work in 25 days that takes 5 men Io days? 22. What cost 9 suits of clothes at $14 for each coat, $2 for each vest, and $4 for each pair of pants? 7 23. How many oranges at 6 cents each can be bought for 4 cents and 5 lemons at 4 cents each ? 24. A boy gave 10 marbles worth 7 cents for 3 figs worth 2 cents each ; how much did he lose? , 25. What cost 60 eggs at 12 cents a dozen? 26. A boy has 33 cents, how many marbles at 3 cents each can he buy and keep 6 cents ? 27. A boy has 39 cents, how many must he earn that he may buy a dozen oranges at 4 cents each? 164 FIRST STEPS AMONG FIGURES. 28. In Mary’s garden are 8 roses, twice as many pinks and a dozen daisies; how many flowers in her garden? 29. If 3 pounds of sugar cost 24 cents, what will half a pound cost? 30. Mary has 8 cents, her sister has 6 cents, and their brother has half as much as both of them ; how many have the three children? 31. If I buy 60 chickens at the rate of 5 for $2, and sell them at the rate of 12 for $5, how much will I gain? 32. How far apart will 2 men be in 7 hours, if they start from the same place, and travel in opposite directions, one 6 miles an hour and the other 4 miles an hour? How far if they travel in the same direction ? 33. A man who drives 9 miles an hour is trying to overtake a man who is 24 miles ahead of him and who goes 6 miles an hour; in how many hours will he overtake him ? © 34. How many ducks at the rate of 7 for $6 can I buy for $29 and have $5 left ? The pupils should mark each answer, and also its denomination. They should be required to mark not only the denxomination of each re- sult in the process of solving problems, but FIRST STEPS AMONG FIGURES. 166 ——— what it represents, that is whether it is cost, selling price, gain or loss, A’s number, B’s number, &c. In this way they will succeed with many problems on which they would otherwise fail. EXAMPLES FOR THE SLATE. 1. A had $8,948 to which he added $2,284, and then he lost $1,632 when he used all he had in buying 38 village lots; how much did each lot cost ? 2. B bought 265 acres (of land). for $22,- 790 ; sold 169 acres of it at $97 an acre and the rest at cost. Whole gain? 3. A horse and 16 oxen are worth $1439 and the horse is worth $175; what are the oxen worth. What is each oxen worth ? 4. Paid 36 barrels of flour for 60 yards of cloth at $6 a yard; how much was the flour a barrel ? 5. If the front and rear walls of a house each contain 37,390 bricks, and the other two walls each 49,758; how many bricks in the four walls ? 6. If 15 boys walk goo miles in 60 days, how far will they walk in 2 days? See P. Ed., p. 125. 166 FIRST STEPS AMONG FIGURES. 7. Add forty-five million nine thousand ten, fifty thousand eight hundred, nine million nine hundred thousand seven hundred nine, ninety million ninety thousand seven, and six hundred seventy-eight. 8. A sold one horse for $185, another for $165, and another for $187; what was the average price of a horse? g. Divide the product of 6580 and 7900 by their sum. 10. A bought 300 acres of western land for 1,200 ; B bought 275 acres for $175 less, and C 125 acres at $4 an acre ; how many acres did the 3 men buy? How much did they pay ? 11. A grocer bought 279 pounds of butter at 27 cents a pound and 98 pounds at 26 cents a pound ; he sold the whole at 32 cents a pound ; how much did he gain? : 12. The weight of a number of hogs was as follows: 250 pounds, 245 pounds, 260 pounds, 257 pounds, 273 pounds and 293 pounds ; what © was their average weight? 13. A man wishes to buy a piano for $375 ; he lays up $5 a week for a year, or 52 weeks; how much more must he save to get the piano ? 14. Two men start from the same place at See P. Ed., p. 128, FIRST STEPS AMONG FIGURES. 167 the same time and travel in the same direction, one at the rate of 35 miles a day and the other 44 miles a day ; how far apart are they at the end of 4 days? How far apart if they had traveled in opposite directions ? 15. Divide the product of the sum and differ- ence of 364 and 93 by the difference between their sum and difference. 16. A farmer bought one cow for $34, another for $43 and another for $61 ; what was the average price of the cows? 17. The product of two numbers is 1,017,702 and one of them is 2,758; what is the other ? 18. What is the sum of seventy thousand nine, nineteen thousand six hundred forty-nine, nine million seven hundred thousand, six hundred thousand nine hundred eight, fifty million sixty, and three hundred seventy-nine thousand eight hundred ninety-eight ? 1g. The remainder is 713, the quotient 579, the divisor 2758 ; what is the dividend ? 20. A man bought 5 horses at $165 each and 6 more for $902 ; what was the average price paid? 21. A woman left her four children $15,000 ; the eldest received one-half of it, and the re- 168 FIRST STEPS AMONG FIGURES. mainder was divided equally among the other children ; what was the share of each ? 22. A, Band C sold 20 village lots for $14,- 600; A received twice as much as B, and B $200 more than C, whose share was $3,500 ; what did A receive? B receive? 23. A company of 14 miners sell a mine in 1,245 shares at $210 per share; what does each receive ? I. 49 + 7,068 +9,847 +958 +37 + 489 + 8,- 956 +9843 = ¢ 2. 946+378+ 795 +849 + 696 + 784 +359 “+ 4304-775 7.000. 3- 547+397+ 484+ 758 +969 +847 +958 + 497+3844+947+358+596=? 4. 567+ 498 + 948 + 397 + 846 +372 +458 +796 + 389+ 486+ 958 +347+598 =i, 5. 123+456+789+987 + 456+ 321+743 + 398+ 476 +395 +948+767 + 496 324-1 6. 578+ 397 +956 + 789 + 437 +4964 875 +749+ 658 +976+345+876+ 901 =? 7- 947 +643 +358+ 895 +769 +5764 348 +954+ 847 +659 +438 +987 + 648 320 FIRST STEPS AMONG FIGURES. 169 8. 756+395 + 408 + 347+579+943 +658 +547+ 892 +675 + 487+949 +673 +246+987=? 9. 391+849+327 +496 + 327 +843 +659 +742+869+ 324 +496+932+783 +468 +-579+453=? 10. 938+ 493 +745 +679 +548 +987 + 765 +899+624+ 345 +879+354+497 +384+947+ 486+ 849+435=? PAGE. NO. 126- 1. 5,048. 507. - 4,800. 5,245, 5,880 614, 662. . 215,829,875. . 196,070,418. . 114,326,292. . 15,690, 168. . 21,269,931. . 48,581,904. . 53,084,772. . 198,177,236. . 465,632 13, . 364,257 25, . 536.174 24, . 6,527,465 +8, . 3,756.486 26, 5. 4,603,753 58, . 1,537,640 37, . 6,453,078 17, . 4,536,028 47, . 63,502,487 28, . 4,357,068 *8, . 7,684,530 #8, | 13,524 92, 53,412 541, . 34,625 344, 2,432 118, 3,254 116, » 4,037 312, ANSWER. PAGE, NO. . 6,475 100, . 3,768 45, . 86,754 296, . 4,857 127, . 7,684 19, 12. . 4,197. . 5,648. . 5,049. 143- 1. . 218 cts. — he OOM ee Or He Oo $00 22D OUR 99 DOH OIE OT G9 D9 Poi eRe EDUPION. ANSWER. 45,876 36, 25,280 rds. " 10,208 times, 144- 6. . 1,884,960 ft. 253 sheep. 5,756. 5,478, 5,472. 6,332. 6,160. 7.189 . 5,792,545. . 423,881. . 589,138. . 438,524. . 592,658... . 4,383,818. . 8,708, 979. . 31,299,917. . 3,740, 188. . 81,897,816. . Impossible, _ DO WIA MP WWE SwoeAH orp gow 146- KEY TO BEEBES 8,616,576. 129,853,848. 43,417, 796. 48,371,475. 43,079,676. 561,747,525, 490,661,472. 5,180,582. 755, 204, 848, Impossible, 5,793,078. 4,750,860 3-7. 6,507,460 1°, 6,870,567 +5, 4,638,679. 63,705,870 57, 6,879,069 3-4, 74,863,580 36, 69,708,570 27, 4,'759, 368. . 8,752 3107, . 8,572 300, . 6,374 461, . 4,625 116, . 7,364 783, 15. 68,450 1,000, . "2,056 397, 17. 65,704 361, . 627,304 275, 8,750 208, 20. 73,604 678, . 6,489 598, . 867 178, 869 1. 7,096 527, 8,079 619, 6,805 371, 65,704 300, 583,607 219, 45 fields; 819 A, 5 horses. $19,458. $1,926 . 40,142, 155. 545 wf 153- 8. $5,430. $304 24, 9. 18 yds. 847 7,600. 9, 630. 7,563. . 9,066, . 1,005, . 9,881, . 9,687, . 7,416,461. . 679,177,066, . 1,472,789. . 622,970,195, . Impossible. . 6,181,969, 778, . 5,770,554, . 87,687,040. - 676,147,266, . 43,050,000. . 6,721,200,000. . 83,236, 122,000. . 583,059,063, 000. 8.:637, 777,020,000, . 6,978,098 7. . 8,907,869 *8, . 76,986,970 27. . 58.795, 869 27," . 87,694,850 7°, . 46,394,789 ©7, . 49,579,685 +, . 6,745 7801, . 8,547 7,090, . 49,372 604, . 476 4597, . 11,946 4,937, . 597,860 3, . 1,908,000 3742, i 684 37,451, . 789 26, = (Si) Ay — fpane beennerr es 5. FIRST STEPS AMONG FIGURES. 173 . 768,456 78, _ 8,069 261, 7,508 275, 4,598, 490,000. - 42,001,756, 650. 0. 75,080 237, 870 326 32. 345,387,180,000. _ 6,890 21, | 246,857 6216, . 9,598,752. 76,169,220. . 87,653,208. . 42.554.964. 11,877,228, 9,599,496. ‘ 107, 879, 736. 798, 990, "680 11712, 7,968,470 7-12, 73.978,596 11, 6,970,486 &12, $252 24-38, $1,859. : $1, 264; $79. $10 174, 296 bricks. 30 mi. 166- 7. 167-15. 145,051, 204. $179, if 3,589 13,280-14,480 . 700 A.; $2,725, » 1,983. cts. . 263 tbs. $115. . 86 mi; 316 mi. 665 157-186, ‘ S157 [ $2, 500 each. 21. Eldest $7,500; oth’s 169- 10. $0 90's > OUP CO BO r= ¢ . A, $7,400; B, $3,700, $18,675. eet er = ENpD OF KEY TO TEACHERS’ EDITION, PUPILS’ EDITTORs PAGE. NO. ANSWER. PAGE, NO. ANSWER. 33-1. 251. 34. 317. 2, 272. 37-35. 21.752. 8. Q71. 36. 24,853. 4, 239, | 37. 63,545, 5. 234, 38. 22.544, 6. 239. 39. 93,733. 7, 262. 40. 62,592. 8. 324. 41. 66,651. 9, 252, 42. 77,444, 10. 260. 43. 28.345. 34-11. 296. 44, 72,353. 12, 295. 45, 62,156. 13. 308. 46. 63,234. 14, 335. 1. 46,206. 15. 314. 2. 90,693. 16. 325. 3. 39,606, 17. 313. 4, 42,064. 18. 327. 38-5. 93,069. 19, 324. 6. 24,640. 35-20. 316. 7. 60,369. 91. 317. 8. 69,396. 22. 346. 9. 62,840. 23, 319. 10. 96,908. 24. 302, 11. 60,846. 95, 845. 12. 69,369. 26. 307. 13. 32,341. 97, 207. 14. 31,208. 36-28. 325. 15. 10,243. 29. 328. 16. 20,413. 30. 308. 17. 18,028. 31, 293. 18. 41,302. 32. 323. 19. 14,032. 33. 303. 20. 80,218. FIRST STEPS AMONG FIGURES. . 13,024. . 40,132. . 20,312. . 2,429, . 2,463. » 2,544. . 2,420. . 2,349. . 2.587, 2,917. * 21996. 2. 2,364. _ 2,967. . 3,178 . 3,184. . 34,892. . 36,775. . 85,561. . 35,673. . 85,680. . 4,655. 4.514, 4.615. . 4,300. A522. . 4,837. . 49,044. . 48,573. . 48.874. . 49.083. . 51.204. . 4.258. 4.011. . 4,741, 4,743. . 4,893. 3 4 5 6. 48-7. 8 9 10 . 563,135. . 741,027: . 411,303. . 623,024. . 71,634. . 44,252. . 443,043. 2. 476,446. . 556,467, . 375,667, . 77,567. . 766,475. . 876,737. . 875,565. . 666,567, TUT 657. . 267,146, . 365,662, S80T TT: . 176,515. . 262,607, . 421,256. . 277,465, . 219,173. . 373,676. . 378,074. | 257,156, . 471.465, BLTTTS, . 76,516, . 311,856. . 677,172, - 376,046. . 562,567, . 40,895, . 42,073, - 41,217, 44,286, 42.308, 44,509, 41,808, . 46,765, - 46,558, 48,802. - 4,628, 175 176 KEY TO BEEBE’S 12. 8,462 2. 73,276, 956. 13. 4,028 3. 79,097,578. 14. 60,248 4, 71,178,967. 15. 63,906 5. 87,819,971. 16. 68,402 6. 3,765,687. 17. 46,082 7. 68,797,784. 18. 70,492 8. 5,748,957. 19. 127,068 9. 73,287,865. 20. 92,704 10. 86,995,768. 21. 106,926 11, 80,773,478, 22. 109,356 12. 26,579.678. 23. 79,359 65-13. 7,170,658. 24. 92,704 14, 94,880,045. 25. 193,578 15. 860,697,867. 26. 214,496 16. 47,159,685. 27, 198,572 17. 32,660,068. 53-28. 213,704 18. 6,697,868. 29, 258,144 19. 77,995,682. 30. 71,284 20. 90,859,567. 31. 139,056 21. 77,706,580. 32. 145,704 22. 93°730,145. 33. 105,738 23. 35,764,946. 34. 218,712 24 70,299,847. 35. 219,276 25. 77,660,078. 36. 387,156 67- 1. 246,722. 37. 231,760 2. 296,492 38. 254,096 3. 213,852 39. 140,168 4, 492,241 40. 202,680 5. 249,494 41. 182,772 6. 281,968 42. 230,17 7. 459,704 43. 157,824 8. 211.476. 44. 150,216 9. 368,464. 45. 312,384 68-10. 5,480,296. 59- 1. 52,279 11. 3,395,469. 2. 45,897 12. 5 142,276. 3. 57,437 13, 2,548,282, 4. 50,384 14. 2,548,546. 5. 51,459 15. 29,126,064. 60- 6. 47,931 16. 53,085,480. 7. 54,123 17. 1,483,264. 8. 54,794 18. 1,295,866. 9. 57,547 19. 84,320. 10. 56,480. 20. 15,674,618. 64- 1. 7,867,676. 21. 1,586,880. aoslet adalat atts FIRST STEPS AMONG FIGURES. . 2.298.402. . 24,593,992. . 41,303,168. . 4,828,052. . 313,225. ”, 4,252,262. . 22,517,838. . 3,186,721. . 3,430,458. . 11,993,364. 2. 27,078,591. . 47,923, 624. . 2,011,205. . 16,799,022. . 53,208,409. . 2457 ¥8, . 3,543 14, 5 ATB 23, . 274,658 3, . B45, 734 84, . 265.435 *5, . 238,636 24. . 1,887,677 ©. . 6,543,346 *, 54,764 35, . 154,264 46, 64,357 14. 623,542 26, 75,246 *5, 543,452 47. 974,658. 642,455 &7, 646,819 2. 786.538 "4, . 24,635 37. . 3,456,827 *¢, . 63,287. . 61,205. , 63,924. . 66,698. 5. 65,307. . 55,344. . 52,771. . 57,783. . 61,074. . 60,493. . 69,274. . 69,883. . 68,111. . 71,276. . 67,740. mags 438, 656. naa: 971, 752. ; 244,806,912. . 382,698,325. . 23,698,382. 26,155,500. . 2,595,136. . 4,188,375, . 5,965,938. 3,235,848. . 55,044,555, - 17,508,384. . 26,435 *4, . 635,241 25, . 854,624 14, . 8,746,254 34, . 657,342 1-4, : 4,635,246 +, . 7,562,435 24, 20. 437,526 46, rw) oO . 4,537,264 37, 2. 5,743,263 **, . 5,762,474 &6, . 2,605 4, 6,370 524 #7, ~ 423,177. . 6,258,085. . 3,604,756 *7. 177 KEY TO BEEBE’S - 4,653,702 &8, - 7,086,534 26, . 5,768,430 47. - 6,573,048 38, . 2,540,867 &8, . 4,352 143, . 3,624 152, 4,235 214, - 4,236 1,022, 5,342 647, 24,353 1408, G45 1125, . 3,624 231, . 3,425 240, aul Oa . 3,426 2,120, . 3,564 3182, . 6,457 341, . 54,673 1,342, 2) 4. TG: 9, 0005 . 57,643 216, . 3,745 198, . 4,576. . 74,656 991, . 46,576 397, . 6 534 279, - 76,487 1329, . 7,458 8. 8. 6: 47, "664 2.388, . 174 marbles. . 632 bu. . 189. © @ oS ! SD WED OUP G9 2 [@r) . $12,250. . 127 marbles. . 105 cts. . 1,781 steps. $85 . 7,682 cts. . $362. . 92 da. . 425 . $668. . 91 marbles, . $1,920. . 25,092 cts. . 59 bu. . 15 ets. 2. 874 bu. 3. 152 bu. . 1,784,910. 64,482, 94-11.3,608, 157. 12. 658,280. . 616,178. : 266.078. . 6,050 993. . 391.648. . 407,708. . 278,079. . 277,164. . 391,667. . 622,678. 2. 578,238. . 39, 213, 566. . 670, Tal. . 3,764,877. . 79,088. . 24,147, FIRST STEPS AMONG FIGURES, 179 28. 42,869,909. H 12 5,746,387 +7, 29. 69,059,978. 13. 7,468,576 7°. 30. 37,523 976. 14. 67,580,760 ©. 31. 3,189,825. 15. 4,637,586 7-8. 2. 36,970.172. 16. 65.748,760 37. 1. 70 919.205. 17... 796,859,809 *¢, 2. 47,009,130. 18. 697,879,680 3-7. 95- 3. 5,449 025. 19. 64,859 760. 4. 302,421,261. 20. 748,695 47. 5. 31,269.744 21. 479,568 !- 7, G. 392. 262,525. 97-22. 840,967 *°8. 7. 3,756,608. 23. 639,408 7-8, 8. 726.525. 24, 4,253 526, 9. 187,044. 25. Omit this example. 10. 7,221,816. 26. 35,246 108, 11. 5,498,265. 27. 5,264 2243, 12. 5,448, 946. 28. 6,423 1-243, 13. 4,231,810. 29. $63 12,217, 14. 35,677,072. 30. 4,536 1036, 15. 64,367,975. 31. 64,352 341, 16. 49,849,081. 32. 48,372 345, 17. 610,079,611. 33. 57,362 3.782, 18. 308,940,676. 34. 41,572 395, 19. 28,681,464. 35. 3, 754. 1,000, 20. 379,544,763. 36. 4,827 314, 21. 35 695,072. 37. 36,472 247, 22. 68,097,323. 38. 64 727 4.6%, 23. 58,390,182. 39. 468,752 4,306, 24. 379,239,951. 40. 74,635 325, 25. 6,257,507,544, 41. 4 605 325, 26. 6,328,767. 42, 70,534 3-041, 96-27. 2,816,929. 43. 4,653 1124, 28. 708,709. 44. 64,075 38, 29. Impossible. 45. 736,502 128, 463,057 37, 98-46. 475,630 2345, 634,750 *8. 47. 4,607 3,762, 475.307 3°, 48. 60,835 31, 5,463,060 +9, 49. 72,506 119, 564,037 29, 50. 43,072 3.060, 374,675 &9, 51. 4,073 19-301, eres 3-8, oy 47,260 479, . 6.5 rye-e . 58,240 749, 6,870,657 &7, 54. 47,050 78, 4, 857,680 69, 55. 70,648 2876, . 6,578, "648 28, 56. 4,756 127, “EH SOM MMO mR ONe © fmt pe 180 KEY TO BEEBE’S « 697 247, 10S 457, - 579 433, 196 527, Reed Oued “1 108 143, « 687 541, » 80,743; . 8,607 801, . 6,908.74. » 8.097, 47°, . 7,906 473, . 47,650 230, - 76,305 2°, . 67,057 9.246, 2. 57,642 787, if 592. 109 sheep. . Lost $900. . 1,796,256 apples. $4,543. . 59,888,884. . 136 tons, 8. 337,022. $19. 20. 191,284 yd. ; Lost $50. . 14 horses an 3 | 421 A.; $37,309. fs $6 54-135, 6. 105-27. 28. 29. rem. $50 $4,456. 3,743,520 ft. 33 mi, Nothing. 30. dl. 106- 1. 86 yrs. 840,413,978. ; 389, 385, 828, “i bs 165, 668. 29.730,808.. . 7,080,807. . 291,864,906. . 77,927,921. . 730,868,068. . 20,679,828. . 7,779,807... . Impossible. . 89,299,876. - 4214769,927. . 670,796,919. . 6,479,979, 692. . 4,585,944. ; 772, 926, 042. ‘379,239,951. | 550,842 699. 6,953, 140,080. 113- FIRST STEPS AMONG FIGURES, . 490,619,928. 5. 4,311,281,464. 78,650,000. ’ . 6,800. . 6,320,000. . 87,500,000. . 172,500,000. . 16,205,000. . 444,500,000. . 18,700,000,000. . 4,336,400,000. . 17,983.00. . 51,300,000. . 6,887,200. . 59,073,000. * . 40,843,000. . 8,407,454,000. . 452,276,640, 000. . 3,589,638,200,000. . 4,827,581,000. . 6,409,800,000. . 4,798,697 5-9, . 87,896,790 7%, . 769,589,079 *8, 9.687,890 2-7, - 769 809,780 &9, . 6,589,679 5-8, - 978,697,089 6-9, . 76,898,697 4-6. . 7,049,680 5-7, . 4,906,704 7-9, . 79,684,796 4-7. - 958,007,980 8-8. . 893,798,400 7-9. . 65,870,486 4-7. . 870,956 5-6, - 75,680,390 57, 903,780 5-8, - 9,586,090 7-2. . 89,607,980 3-8, . 5,869.759 +7, » 4,375 6001, 5,736 621, 865 216, 116- OUP 6929 . 147,283 7.069, . 6,485 70°. . 46,372 528, . 14,735 499, . 17,399 321, . 68,591 7457, - 147,964 948, . 9,608 4579. 2. 13,568,274, . 49,807 291, . 138,769 3,700, 79 3,468, 376 8°, 23 47,600, 762,196 48, OL 2,938,641, 486 3211, 6,857 12,131, 796 75.300, 96 21 352, 873 4758, 98 30 000, 468 2,300, 2,005,600,000. 7,958 123, 68,050 759, . 7,960 2°. _ 5,920,005,998. . 48,690 20,172, - 79,080 790, 1,785 76,495, 6 60,970 28. . 9,780 58, Impossible. i 740 157,000, _ 8,069 37,281, . 6,095 2°, . 405,025,600. . 78,096 231, 796,000. . 947,604. 767,679. . 9,058,032. . 10,149,216. 749,042,756. 181 KEY TO BEEBE’S . 7,043,688. 78,956,244. 833,855,616. 104,154,369. . 5,755,896. 1,171,075, 764. 2. 83.710.764. . 837,443,940. . 1,136. 156,268. . 11,758,752. . 7,543,756. . 105,587 . 598,772,364. . 1,175.855,832. . 95 879,820. . 98, . 49,688,132 92, - 68,374,989 #12, 748, 790,948 1-12, 7,076, 859 7-12, "786,547,997 922, . 869,897,046 311, . 37,998,060 1012, . 79,684,968 92, . 74,869,740 512, 30. 675,846,090 211 125-1. : . $34,995. - 490,000. 8,552 $760. . $14,352. [& $27 left. 1742797A.; or 174 A. . 614 mi. : 22,999, 800,925. 2,250. $2,010 17 yrs. 236 A.; $74 136.236, 15, 148. : 3.066. ius . 22,032 solid ft. . $4,340. » $8,380. 129-29. 0. 69 horses, an . 42 times, 133 21. 22. 23. 4 24, 25. 26. au. 28. $45 118 79-158, rem. $53 . 398 lbs. 130-38. . 49 animals. . 6 yrs. . 1,440 Ibs. 2. 3,536. . $772. ae .6 horses anc . 1,274 bags. 131-46. 15 sheep. $40 32 teachers & $175 remaining, . 14 cows. . 949 cts. . Gained 80 cts. . 70 half dimes. . 87 bags ; 2. 1382 birds. 132-53. 54, 55. 4 518 cts. 1,162,568 880-1,090 sec, $4,730. 8. Omit ** more.” 56. . $86. . 6,784. . 76 Co’s and 15 men . af yds. . 1 215rdez 9,051 ra . 12 yrs. . 158, Lost $685. [left. FIRST STEPS AMONG FIGURES. 64. 8,589,783. [ 902. 10. 74,617. 65. Quo. 105; rem. 3,- | 13'7-11. 90,190. 66. 2,392. 12. 95,621. 135-1. 70, 169. 13. 93,522. 2. 67,005. 14. 89,651. 3. 58,369. 15. 93,822. 4, 74,595. 138-16. 104,643. 5. 74,473. 17. 168,075. 136- 6. 76,967. 18. 98,259. 7. 81,025. 139-19. 111,157. 8. 73,892. 20. 111,236. 9. 72,772. APPENDIX. DETAILED METHODS IN ARITH- METIC. FROM THE COURSE OF STUDY PREPARED FOR THE PUB- LIC SCHOOLS OF SAN FRANCISCO. I. Lessons FOR BEGINNERS. Grube’s Method in Number. THE following are in substance some of the most important principles given by Grube for his method in teaching beginners to compre- hend numbers and their relations. Principles. ““r. Each lesson in Arithmetic must also be a lesson in language. The teacher must insist on readiness and correctness of expres- sion. As long as the language for the number is imperfect, the idea of the number will be defective. “2. The teacher must require the scholar to speak as much as possible. ‘3. Answers should be given occasionally 185 186 APPENDIX. by the class in concert, but usually by the scholar, individually. “4. Every process must be illustrated by means of objects. ‘““s. Measure each new number with the preceding ones. “6. Teachers must insist on neatness in making figures.” ORDER OF STEPS. first Step. Ulustrate the required combi- nations by means of counters, such as blocks, splints, or shells, in the hands of the children themselves, and by other objects in the hands of the teacher. 3 Second Step. Express the. same combina-’ tions on the blackboard or on slates with marks, Third Step. Take the same combinations mentally with abstract numbers. Fourth Step. Practical problems in applied numbers. HOW TO BEGIN. [=s~ The time required for the work will depend upon the age of the children, as also APPENDIX. 184 somewhat upon their natural ability. Some children may require a year to complete the work which others may master in a term. RIae NOMBER ONE. 1. Hold up one counter, one hand, one finger, one slate, etc. On your slate make a straight mark, one dot, one cross, etc. On the blackboards make one mark, one dot, one cross, etc. 2. Place one counter in the middle of the desk; take it away; how many have you left ? Make one mark on your slate ; rub it out ; how many marks are left ? 3. Send the class to the blackboards and let them make the mark for one thus, | ; and also the figure thus, 1. 4. Proceed very slowly. Much time should be given to those who do not learn easily. Il. THE NUMBER TWO. 1. Each of you take one counter and place it by itself on your desk ; now take another, 188 APPENDIX. = wy and place close to it ; how many counters have you? (Require the answer in a full sen- tence.) Make one straight: mark on your slate; make another close to it; how many have you now? Go to the blackboard : make one mark ; another, close to it ; how many now ? Clap your hands once; again; how many claps ? Rap on your desk once ; again; how many raps? 2. Counting —Place one counter on your desk,* ; a little way off from the first one, place two counters close together, thus* *. Count, one two ; two, one. On your slates make marks thus, | ||, and count forwards and backwards. 3. Addition.—I. Place one counter on the desk ; place another counter close to it; how many have you now? 8 54 Se 3: © Gree J: IO’ 200827 5 Ae 2. See h 25. T2ueeas 5 Omaae 5) aes See Teachers’ Edition, p. 95. FIRST STEPS AMONG FIGURES. 33 To the teacher. Read carefully the pre- faces to both the Pupils’ and the Teach- | ers’ Editions, and also the Special Notice which precedes the latter. Solve the following examples in ad- dition :-— Meee 03 314 31) 5, 32. 6-21 31 23 33 13 13 13 55 31 22 12 e3 32 20 20 30 33 21 33 13 13 23 30 12 20 32 23 Ia 22 32 32 23 32 32 31 23 3 31 23 31 13 13 31 33 31 23 23 32 31 12 33 33 31 23 23 7iss2 8. 33 O37) 3107 32 23 31 12 23 3! 23 33 31 33 33 22 23 21 12 31 13 13 23 23 31 22 ral 12 20 33 32 od 33 31 23 32 23 23 33 23 31 ——— — et — Some of these examples are to be given daily. See Teachers” Edition, p- 88. 34 FIRST STEPS AMONG FIGURES. Solve the following examples in ad- dition :-— 11.232 21 13 Bo 30 2 23 31 23 ee S33 23 16. PL RFS 21 32 35 13 32 3 12 30 21 33 32 Suet: 23 23 30 23 31 22 32 21 32 33 Ighize By: 31 22 33 21 13 2I Be Be 13 33 2 eGert oO: 22 30 21 13 Los 33 22 31 23 22 31 14.63 39000 fens 22 23 31 30 23 33 20 22 32 31 31 ra 23 33 33 20: 31 3 23 31 oa 33: 217510. 33 23 32 31 23 23° 33 30 ze 33 aa 21 23 13 32 32 13 32 33 23 31 32 Some of these examples are to be given: daily. See: Teachers’ Edition, p. 89. FIRST STEPS AMONG FIGURES, 35 Solve the following examples in :id- dition :— 20232 13 23 31 33 23 31 21 12 32 33 32 24. 33 21 30 13 23 3I 30 23 31 LZ 23 32 21. 25. 33 21 33 32 22 23 ei 32 13 at 23 35 23 BI 22 30 3 32 52 23 33 31 32 33 22. 206. 31 23 a3 31 22 a2 23 21 32 33 32 aye al 23 32 21 13 33 Se 20 13 32 33 23 ope 31 23 oe 30 23 12 33 23 32 om 23 27.513 53 21 a2 33 23 33 20 12 33 32 12 Some of these examples are to be given daily. 36 FIRST STEPS AMONG FIGURES, Solve the following examples in ad- dition :— 28. 32 20.) 32 30. 21 213% 20 2% 32 23 33 32 23 30 23 33 33 22 I2 21 Pes | i Ye 33 13 220 33 31 33 33 21 Pes 20 13 21 33 22 23 13 32 31 33 33 21 33 21 30 33 23 32 22 22.230 3321 34s 23 33 32 12 12 13 31 23 23 33 32 32 22 31 33 32 32 23 23 20 32 31 II 30 33 32 31 22 23 23 31 33 22 Some of these examples are to be given daily. FIRST STEPS AMONG FIGURES. OL After these examples have all been solved once, unless the pupils are very ready in adding, let them commence at the first example and solve them all again. It is of the utmost importance that the pupil should learn to add quickly and correctly, for in practical life he will use addition a dozen times where he will use fractions once. *Examples in Subtraction. 35. 75986 36. 96897 37. 68795 54234 72044 5250 38. 69587 = 39. 95786 ~—s go. 68795 47043 2053 6203 41. 86975 42. 79586 43. 68579 20324 2102 40234 a 44. 76859 45.97586 46. 68579 4506 35430 5345 *Examples in Multiplication. ie 92 130,23 F 3.°13,202 4. 21,032 2 3 3 2 See Teachers’ Edition, p. 9I. 38 FIRST STEPS AMONG FIGURES. So 31,023 6.0n tence 20,123 3. 2RPS2 3 2 6 3: 9 31,420 10. 32,301 If. 30,423 12. 23,123. 2 3 2 3. *Examples in Division. 13. 2)64.682 14. 3)93,609 I5. 2)20,486- 16 2)40,826 17. 3)39,069 18. 2)82,604 19. 2)28,064 20. 3)90,639 21. 2)26,048 22. 2\80,264 23. 3)60,936 See Teachers’ Edition, p. 92. FIRST STEPS AMONG FIGURES. 3 = *Examples in Addition. 24. 342 25. 234 . 260. 434 231 423 244 423 344 342 443 244 444 334 432 322 234 343 434 422 443 324 275,213 28. 444 29. 431 432 231 242 444 443 344 344 232 423 231 424 344 414 344 332 342 231 Agios os ——— —— Some 31. 2439 320-4 442 424 342 321 331 234 434 212 23 444 443 341 343 344 4.2 212 424 444 243 341 341 444 234 213 * The teacher is to show the pupils how to solve such exarr ples. 4o G7 246 FIRST STEPS AMONG FIGURES. 33. 36. 134 423 344 103 422 344 441 343 413 2342 4324 3432 2343 3423 2343 3434 4232 1443 3234 4342 See Teachers’ dition, p. 93. 34. 37- 324 431 344 222 431 444 324 25 444 4323 2433 4344 4224 3433 1234 4322 2443 3244 4432 2343 35. 38. 432 341 2235 212 434 244 443 332 423 3 eee eee At FIRST STEPS AMONG FIGURES. 39. 3424 40 4234 4132 3442 3434 2323 1234 4342 4343 1234 2432 3421 1243 4343 3324 2422 4432 3234 3243 4343 4432 2342 AzyBAUE 43.431 234 343 443 234 444 321 321 443 213 444 442 324 344 242 431 441 342 304 214 323 404 444 321 4B 42 FIRST STEFS AMONG FIGURES. A421 5. 4s act 40. 423 441 232 244 334 444 231 234 323 444 443 213 324 342 441 431 134 344 344 423 422 444 444 343 321 324 243 423 443 412 344 22 333 442 203 431 322 47. 3424 48. 4234 49. 4324 4323 3342 3432 32.43 4433 4243 4434 1234 2324 2332 3243 4433 3443 4324 3442 4324 2442 2334 2423 3334 4343 4244 4234 2434 3423 2442 2343 3342 3223 4123 4324 4324 3434 3423 3443 4442 2342 4321 3223 See Teachers’ Edition, p. 94. FIRST STEPS AMONG FIGURES. 43 50. 3423 51. 3443 3244 4234 4334 2343 2433 4324 4242 4432 3423 2444 4344 4321 1234 3443 4422 4242 4342 4433 3423 3444 4444 2344 3444 4323 2331 3434 I. Richard has 7 cents and Oliver has 6 cents, how many have both ? 2 Cora bought 5 sticks of candy and Hattie 4, how many did both buy ? 3. Herman had 11 cents, but he lost 3 of them, how many had he then? 4. Herbert had 7 marbles, he found 5 more, how many had he then? 5. Ella had 8 new needles, she broke 5 of them, how many whole ones had she then ? 6. Clara solved 12 examples while Kitty solved 7, how many more did Clara solve than Kitty? See Teachers’ Edition, p. 97. 44 FIRST STEPS AMONG FIGURES. 7. Anna had 10 oranges, she gave away 4 of them, how many had she left? 8. Frank had 4 pencils, he bought 3 more ; how many had he then? 9. What cost 7 lemons at 4 cents each ? 10. If an orange cost 5 cents, what will 3 oranges cost ? 11. If an orange cost 6 cents, and a lemon cost 5 cents, what will both cost ? 12. How many quarts in 3 gallons ? 13, Henry had 13 apples, he gave his brother 7 of them ; how many had hethen ? 14. At 5 shillings a bushel, what will 6 bushels of apples cost ? 15. What cost 7 pencils at 3 cents each? 16. A boy walked 4 miles in the morn- ing and 3 miles in the afternoon. How far did he walk ? 17. Samuel has 7 cents in one pocket and 5 cents in another, how many in both ? 18. Carlos had 9 cents in his pocket, he lost 4 of them through a hole in his pocket, how many had he then? 19 Mary spelled 8 words correctly and Emma spelled 6 words correctly, how many more did Mary spell correctly than Emma? 20. At 3 cents each, how many marbles. can you buy for 15 cents? 21. 12 boys were sliding on the ice, 7 of them fell, how many remained standing? See Teachers’ Edition, p. 98. FIRST STEPS AMONG FIGURES. 45 22. If a boy earn 7 cents a day, how many days will it take him to earn 42 cents ? 23. If one lead-pencil cost 6 cents, how many can be bought for 30 cents? 24. James had 12 cents, he spent 5 of them and lost 3 more, how many had he left ? Solution: He disposed of the sum of 5 cents and 3 cents or 8 cents. He had left the difference between 8 cents and 12 cents or 4 cents; or, ifhe spent 5 cents and lost 3, he disposed of the sum of 5 cents and 3 cents or 8 cents. If he had 12 cents and disposed of 8 cents he had left the difference between 12 cents and 8 cents or 4 cents. 25. Sarah had 10 needles; she gave 3 of them to Nellie and 4 of them to Martha. How many had Sarah left ? 26 Herbert had 6 cents, he earned 7 cents and spent 5 cents; how many had he then? 27. Joel had 7 cents, he earned 5 cents and found 4 cents ; how many had he then ? 28. How many days will it take Walter to walk 28 miles, if he walks 4 mileseeach day ? 29. An orange cost 6 cents and a cocoa- nut 3 times asmany. How much did the cocoanut cost? 2* 46 f1RST STEPS AMONG FIGURES. 30. Mary had 11 cents, she lost 5, and earned enough to make her number 9. How many did she earn ? 31. There are 4 columns in John’s spell- ing lesson and 5 words in each column. How many words in his lesson ? 32. Some boys are out flying kites, the wind blows down two kites and 7 less 3 re- main. How many at first in the air? 33. A boy hoed corn for 4 cents a row and earned 24 cents. How many rows did he hoe? 34. A mancan walk amile in lominutes, he starts from his home and walks to town in 5 minutes. How far from town does he live ? 35. [had 1ocents. I bought 2 two-cent stamps, and gave 4 cents to a poor little boy for bread ; how many left for candy ? 36. There are g ten o'clock scholars this morning and one more than one third of them left their books athome. How many of them brought their books ? 37. Willie had 13 cents and he spent 6 of them for candy ; how many had he left? 38. How many sponges at 6 cents each may be bought for 36 cents? - 39. What cost a pencil at 7 cents and a pint of peanuts at 5 cents? FIRST STEPS AMONG FIGURES. 47 40. What cost 7 pencils at 5 cents each? Count by 5’s from 2 to 62. Review counting by 5’s from I and 5 to 61 and 60. 1. 345 452 553 Review counting by 4’s. Count by 5’s from 3 to 63. Count by 5’s from 4 to 64. 4 553 a5 454 542 135 523 454 542 5 352 534 — 435 _—— - 505 453 344 Ei 413 aia 435 214 543 345 3. 405 352 544 255 533 441 355 523 344 555 434 6. 305 | 543 444 354 . 532 445 354 543 455 544 353 ——_—— 48 FIRST STEPS AMONG FIGURES. Examples in Subtraction. ifs II. 13. 15. 764,358 201, 79,687 64,252 - 79,689 43,204 795,869 43,543 96,758 54.428 223 18. 20. 2222 O12 46, 647,094 24,070 74,087 30,435 566 16. 764,037 23,010 613,021 56,554 23. 8. 96,487 31,343 10. 687,985 34,532 12. 47,685 5,232 14. 5,879 2,343 17. 423,703 12,40C 19. 74,087 3,053 21. 695,047 252,004 24. 431,024 55,357 -_-——- ——_—_ See Teachers’ Edition, p. 101. rae Ay 28 34. 37: 40. B43. 46. FIRST STEPS AMONG FIGURES. 26. 634,210 56,643 942,031 66,406 . 312,403 45,257 213,042 36,527 324,102 46,637 430,221 52,147 534,210 16,436 703,524 20,352 29. ao 35. 38. AT! 44. 47. 820,132 53,657 731,203 64,636 420,314 54,652 314,253 51,646 231,430 12,257 320,413 63,257 130,241 53,725 423,102 47,056 27 30. 36. 50: 42. 45. 48. 49 431,201 54,4604 841,310 63,653 : 3415532 34,354 453,621 32,365 425,301 51,625 534,102 62,637 342,013 30,157 624,130 OTs503e> 50 FIRST STEPS AMONG FIGURES. If more practice is desired at this stage the foregoing examples may be reviewed. Count by 6’s from 6 to 60. I. 4532 2. 3541 3. 5423 3215 4325 4352 5453 5432 3544 3344 3254 5435 2535 5545 4334 4253 3251 2453 5432 4314 3545 2354 2443 4334 4235 5425 5442 5542 4543 2355 4. 3524 5. 4325 4352 3552 3445 5445 5334 3453 4523 4524 5452 5335 3445 3452 4334 5344 5523 2325 4354 4553 See Teachers’ Edition, p. 103. FIRST STEPS AMONG FIGURES. Ti 6. 3542 4354 2435 5043 3530 354 ‘4245 5432 3554 4322 2453 5245 245 4532 2454 5321 3543 4254 5435 2353 540 3254 4342 5535 Ome 5 442 3534 4253 5425 343 4534 2345 5234 4553 3425 5342 2435 ET 10. 3452 5544 2°45 4532 5453 1324 5045 3453 51 52 FIRST STEPS AMONG FIGURES. Multiply : TI2(2, 314 eee and 13.12,014 2 2 2 14. 30,124 * 15. 21,302: 2 3 16. 34,201 17. 23,041 2 2 The teacher should show the pupils how to solve the following examples: 18. 35,246 19. 42,356 2 3 20. 46,352 21. 35,042 2 3 22. 36,452 23. 26,453 24. 46,352 3 3 2 25. 64,526 26. 53,624 27. 64,524 3 4 3 See Teachers’ Edition, p. 104. FIRST STEPS AMONG FIGURES. 53 28. 53,426 29. 64,536 30. 35,642 4 4 2 31. 46,352 32. 36,426 33. 35,246 3 4 3 34. 36,452 35. 36,546 36. 64,526 6 6 6 37. 46,352 38. 63,524 39. 35,042 5 4 4 40. 40,536 41. 30,462 42. 46,035 5 6 5 43. 26,304 44. 25,030 45. 52,004 6 6 6 Count by 6’s from I to 61. Count by 6’s from 2 to 62. ‘Count by 6’s from 3 to 63. ‘Count by 6’s from 4 to 64. Count by 6’s from 5 to 65. 54 FIRST STEPS AMONG FIGURES. ADDITION TABLE. 3’s and review. OlwNnN co; MINA g Olu og —_ mq | CO} Ww WwW} ON OV} CO ~ CO}W Un 5 penne 780 7 ae 1341 1015 ago wir}; COON NY | O07 WW |B Cols —" O};}uUIm OC} ON N ioe) Oo’ aH —_ SUBTRACTION TABLE. S’s and review. a LOM? Jal Omens 55 43 OLO ae, 6 O21126,8.:10 Stas Onto es Oe Sa Go Bees Ss 96.12.13 8. I0 10°97 . 12 sigameeie §3°:7. 54.3. 83 4. 35 Oee 43 5 84° °7 263 9 Fea See Teachers’ Edition, p. 105. FIRST STEPS AMONG FIGURES. 55. - MULTIPLICATION TABLE. S’s and review a b C Wee 6307 4 8 5 2.6 2 Smee 38 45 (8 7 8 3 PAO eee O 14 A509 28 20 48. 35 16 18:12" a e ye fe eS. 2) 0°30 7.2 4, 8 5 2 Bee 3A 6 6 7. 8 3 ANS ieee tee O4 158 30 18 40. 32 24 20 10 g : h z eee 5 2-6 3 A 8 5 2 6 pees 4 5. 027.8 (02057. 83.4 SOrz 502 32 25 12 4224.24 5640624 DIVISION TABLE. S's and review. a b C 20°09 45 14:30 4016 21.20 48.35 160 18 12 ee Oe 5 4 38 5 O87 83 Bom e ee, 8 4) 7 48 5 2 0. 3 ad 6 2 é 35 24564215 24640 Aes TO 5 2 4 6 Sumo neo 7 8 5 5 8 5 56 FIRST STEPS AMONG FIGURES. 1. George paid 15 cents for a knife, and after breaking it sold it for 8 cents. How many cents did he lose? 2. Lewis bought a reader for 6 shillings and an arithmetic for 4 shillings ; how much money did he spend for both ? 3. Anna is 13 years old and Mary is 5 years old. How many years older is Anna than Mary? 4. John had 8 cents; he lost § of them, how many had he left ? 5. What cost 4 books at 8 shillings each ? 6. How many oranges at 5 cents each can you buy for 40 cents ? 7, Walter spent 16 cents for pears at 2 cents each ; how many pears did he get? 8. Jane had 15 needles; she lost 3 of them and broke 5 ; how many had she left ? 9g. Byron rode down hill 4 times one afternoon and his father twice as many times. How many times did Byron’s father ride down hill? 10 William had 26 cents ; he lost two of them and spent the rest for marbles at 3 cents cach. How -many marbles did he buy ? 11. Charles was sent to the store to buy 6 spools of thread at 8 cents a spool. He took 50 cents with him, how much change should he take home ? See Teachers’ Edition, p. 108, FIRST STEPS AMONG FIGURES. 57 12. A boy having 8 cents earned 5 cents and his sister gave him 2 cents, how many had he then ? 13. A boy set 2 traps. in the woods; 2 rabbits: went into one trap and twice as many went-into another. How many went into both? 14 A gun carriage has four wheels ;how many wheels have 7 gun carriages ? 15. If it takes 6 horses to draw one can- non, how many horses will draw 8 cannons ? 16 There are a sergeant and 6 privates at one picket post, and a corporal and 4 privates at another. How many soldiers at both? 17. If 3 oranges cost 12 cents what will one orange cost? Solution : If ¢ivee oranges cost 12 cents, one orange will cost one-third of 12 cents or 4 cents. We get 4 of a number by dividing it by 3, 4 of a number by dividing it by 4, etc. What is one-fourth of 24? What is one-third of 21? What is one sixth of 42? One-fifth of 30? One-fourth of 32? One-seventh of 56?. One-half of 14? One-sixth of 18? 18. If 8 horses eat 24 bushels of oats in a week, how many bushels will one horse eat? 58 FIRST STEPS AMONG FIGURES. 19. If 42 cents is the price of 7 marbles, ‘what is the price of one marble? 20. 4 boys have 28 cents, and each have an equal number. How many cents has each boy? 21. A farmer has 18 pigs in 3 pens and the same number ineach pen. How many pigs in one of the pens? : 22. If 8 pears cost 16 cents, what cost I pear ? 23. At 4 cents each, how many lemons can be bought for 28 cents? 24. If three’ tops cost 15 cents, what cost I top? 25, WV Dateecost eat pieeel of oats, if 7 bushels cost 35 shillings ? 26. 12 dollars will buy how many birds at 2 dollars each ? 27. If 5 hens cost 20 shillings what will 1 hen cost? 28. How many peaches at 2 cents each can you buy for 16 cents ? FIRST STEPS AMONG FIGURES. f. 5304 4243 3524 5452 SBE 4343 3425 5254 - 2542 4335 3454 5543 4325 4 See Teachers’ Edition p. 110. 4305 3450 5234 2543 5455 3213 4543 2355 5432 4343 2524 3455 3532 2. 453 304 5035 4540 3235 2424 5353 4545 3424 4353 an32 4245 5454 3. 5243 4524 3452 5335 3543 4254 5435 3542 5354 4435 3423 5354 3543 —e 5. 3544 5434 4355 3243 5424 4552 2345 5433 2253 4534 3445 2343 4554 59 60 FIRST STEPS AMONG FIGURES, 6. 4305 7 2345 8.5435 3453 5334 554 435 4243 3443, 5044 3545 5545 3453 5434 4353. 4545 3552 5534 5432 3345. 3444 3354 5453 5554 — 3545 4245 2345 34 3524 4235 - 5432 4305 5453: 4355 5353 3544 4544 3445 5355. 9. 3452 10. 3452 2345 5435 5443 4544 4534 3353 4455 5445 5243 3234 3524 5544 4355 4353 5533 2455 4432 4434 5454 5554 3345 5432 5432 3245 ——— FIRST STEPS AMONG FIGURES, 61 Count by 6’s from 2, 3, 4 and 5, to 62, 63, 64 and 65. Read the following numbers: I. 750000748 2. QOO000047 3. 680000740 4. 700746000 i 1 2007000 6. 750908716 7. 801000071 8. 679374819 9. 715016390 10. QOOO60000 II. 7000000 12. 800000005 Addition : Finding how many units there are in two or more numbers and expressing them in one number is called addition. The number found by addition is called the sum or amount... Proof. Add the columns both upward and downward, and if the results agree they are probably correct. 11. Write in Arabic fifteen mil ten th. ninety. 12. Write in Arabic three hun fitty mil. nine hun th. 13. Write in Arabic eight mil. five hun. forty-three th. seven. 14. Writein Arabic onehun mil six hun. thirty-two. 15. Write in Arabic seventy-fye mil. three hun th. 62 FIRST STEPS AMONG FIGURES. 16. Write in Arabic fifty mil fifty th. fifty. 17. Write in Arabic two hun. mil. sixty th. four. 18. Write in Arabic one hun. five mil. one hun. five. 19. Write in Arabic one hun. nineteen mil. forty th. 20. Write in Arabic three hun. eight mil. thirteen th. two hun. eighty-one. — 21. Write in Arabic twenty th. 22. Write in Arabic fifty mil. fifty. 23. Write in Arabic three th. two hun. forty-five. 24. Writein Arabic nineteen mil.five hun, th. 3 25. Write in Arabic eight mil. three hun. . 26. Write in Arabic four hun. mil. 27. Write in Arabic seven hun. sixty-one mil. five hun. sixteen th. twenty. 28. Write in Arabic six hun. th. 29. Write in Arabic fifty th. forty. 30. Write in Arabic three hun. eight. 31. Write in Arabic nine th. 32. Write in Arabic six hun. mil. five. hun. 33. Write in Arabic sixty mile, four hun. th. three hun. 34. Write in Arabic three mil. fifteen th. thirty. See Teachers’ Edition, p. 114. FIRST STEPS AMONG FIGURES. 63 a *35. Write in words 13,004,020. 36. Write in words 300,216,000. 37. Write in words 232,341,519. 38. Write in Roman fourhun. sixty-three. 39. Writein Roman eight hun. forty-four. 40. Write in Arabic DCLXXVII. | 41. Writein Romantwo hun. eighty-nine. 42. Write in Arabic XCVIIL. 43. Write in Arabic CDXI. 44. Write in Roman seven hun. nineteen. Subtraction: — Taking one number from another num- ‘ber is called subtraction. Remainder or difference. The number found by taking one num- ber from another is called the difference or remainder. In subtraction the number to be sub- tracted is called the subtrahend, and the number it is subtracted from is called the minuend. ‘The result in subtraction is called the difference or remainder. ProoF. Add the remainder and the sub- trahend, and if the result equals the min- uend the work is probably correct. The pupils should often be required to * Notice that such compound words as seventy-five, forty-one, sixty-nine, etc., require a hyphen. 64 FIRST STEPS AMONG FIGURES. write the name of each number in examples in subtraction, as follows: Subtraction. (1) 8,342,053 474:377 —— (4) 71,420,035 241,068 (7.) 74,200, 35 2 5,402,568 (10.) 94,002,531 7,006,763 321 Minuend. 45 Subtrahend. 276 Remainder or dif. (2.) 73,520,031 243,075 ——, (5.) 93,520,014 5,700,043 (8.) 6,314,025 565,068 (AT) 81,350,024 576,546 (3) 81,400,253 2,302,675, (6.) 4,531,024. 765,337 (9.) 73,500,241 212,376: ——s (12.) 34,200,156 7,620,478 See Teachers’ Edition, p-. 118. FIRST STEPS AMONG FIGURES. 65 (13 ) 71:352,034 181,376 —=—— (16.) 93,510,042 46,350,357 (19.) $3,001,425 5,005,743 (14.) 95,300,421 420,376 (17.) 35,200,416 2,540,348 (20 ) 95,320,041 4,460,474 (22.) (15) 863,005,241 2,307,374 (18.) 71300,425 602,557 (21.) 85,241,300 71534,720 94,300,52 I—570,376=? (23.) 42,5 30,014—6,765,068=? (24 ) 71,300,524— 1,000,677=? (25.) 82,400,153—4,740,075=? 66 FIRST STEPS AMONG FIGURES. The teacher will choose between the two following sets of definitions: Multiplication. A short method of adding equal numbers. | is called multiplication. Multiplicand. | One of the equal numbers is called the multiplicanda. Multiplier. The number which shows how many of the equal numbers are used in adding is. called the multiplier. | In the example 542+542+542, 542 is the multiplicand and 3 is the multiplier, and we solve the example as follows: 542 Multiplicand, 3 Multiplier, 1,626 Product. Or the following definitions may be used: Multiplication. Taking a number a certain number of times is called mzultzplication. Multiplicand. The number taken, (or multiplied,) is called the mzultiplicand. FIRST STEPS AMONG FIGURES. 67 Multiplier. The number by which we multiply, (or which shows how many times the multipli- cand is taken,) is called the szzltzplier. Product. The result of the multiplication is called the product. Proof. I. Multiply the multiplier by the multi- plicand, and if the product equals the pro- duct first obtained, the work is probably cor- rect. II. Or divide the product by the multi- plicand, and if the work is correct the re- sult will be the multiplier, or divide by the multiplier and get the multiplicand, ~ Multiplication. Mueeseeg 2. 42,356)" 3. 35,642 7 7, 6 4. 57,463 5.- 35,042 6. 35,246 7 7 8 7. 57,463 8. 35,246 9 46,058 8 6 8 See Teachers’ Edition, p. 119. 68 FIRST STEPS AMONG FIGURES. ‘ 10. 685,037. II. 485,067 8 13. 12. 857,046 6 637,058 14. 364,078 4 7 15. 3,640,758 x 8=? 16. -7,593,04G, gaan When there is more than one figure in the number by which we multiply, the right hand figure of the result is placed under the figure we multiply by. 17. 46,352 18. 56,342 nd 20. 21. 22. 23; 24. ons 26. 2: 32 23 364,526 43=? A2: 502 xiha —- 534,652 x 40=? 645,362 x 64=? 63,527 x 76=? 4675 xO7=7 57,463 x 74=? 19. 2,635 3 FIRST STEPS AMONG FIGURES. 6g 28. 357,426x63=? 29. 47,563 X67=? 30. 63,527 x 54=? 31. 352,746 x 34=? 32. 475,063 x 57=? 33. 630,574 x 76=? 34. 57,463 x 35=? 35. 357,420 47=? Division. Finding how many times one number is ‘contained in another is called adzvision. Dividend. The number which contains the other is called the azvidend. (If preferred, the number divided is called the adzvzdend.) Divisor. The number which is contained in the other is called the dzvzsor. (If preferred, the number by which we divide is called the divisor.) . OQuotzent. The number found by dividing is called the quotient. 7°? FIRST STEPS AMONG FIGURES. Proof. Multiply the quotient by the divisor, and add the remainder if there be one; if the work is correct the result should equab the dividend. Divisor, 7\4321 Dividend. Quotient, 617—2 Remainder. Short Division. (1.) (2.) (3.) 2)6,846 3)612,921 4)3,281,224 — oe (4.) (5.) - (6.) 3)9,182,715 4)2,082,836 5)5,304,535- (7.) (8.) (9) 3)24,152,721 2)128,166,414 4 836,284,820 (10.) ae (12.) 3)21,186,912 412,032,828 2)106,416,81& See Teachers’ Edition, p. 120. FIRST STEPS AMONG FIGURES. 7® a With remainders. * 13. 7,372+3=? 14. 14,173+4=? 15. 16,427---3=? 16. 823,975+3=? Meeate2,020>4—? 18. 1,327,178+5=? 19. 954,540+4=? 20 5,663,032+3=? 21.-20,173,385-4=? 22. 273,823+5=? 23. 925,588+6=? 24. 257,429+-4=? Bea 741,254--6—? 26 376,233+5=? 27. 3,804,168+7=? 28. 4,873,290+5=? 29. 4,497,190+7=? 30. 3,880,916+6=? 31. 3,146,153+4=? 32. 172,448+7=? 33. 20,740,905 +6=? Count by 7’s from 7 to 42. * Teach the division series with remainders in Teach-- ers’ Edition, p. 107, before solving these examples. 72 FIRST STEPS AMONG FIGURES. ADDITION TABLE. 9’s and review. C ies 518 4g 9 4 7h One 107 LL 15 12;1G0 separ Saat Pe 15412 6 9 4 5 10141 b 6 9 3220 eee 5 6 rE 3 5. St 2°75 ome 6 9 44967 89 21I9 13 17 14 12/1113 1014 18 ie h So 37> 4iyen sae Ae 35> OF Saas OI 3uIO 44s bE 5 6 I a e 8 eG 9 me 9 Ae, oe 8 16 I Sit 7 “SUBTRACTION TABLE. Q’s and review. I I 8. cere C7’ ®% O He 17, 13:16 526 9: 87 8 5 9 63 1418 9 12 14)17, 812.9013 89.5 8400918 AS ies 6 94 8 5194 73 6 h 14 1 5 9 See Teachers’ Edition, p. 123. FIRST STEPS AMONG FIGURES. 73: MULTIPLICATION TABLE. 9’s avd review. C Ow WEI STOVE eo RA O)\ ses cn DIVISION TABLE. | G's and review. a 30 45 24 56 OF) 12.3 79549 4 8 3 3 2 8 4 Oo | OF vn Aiwa Oo & 42 18 35 16 72/40 63 36 72 45 |3 ps A 9) 8 7 48 81 49 24 40 18 5/ Behe 54 (2-.S Oey Ar8 9 6 See Teachers’ Edition, p. 124. 8 6 Ta. FIRST STEPS AMONG FIGURES. 1. In a school there are 8 recitation ‘rooms, each room has 7 benches; how many benches in these rooms ? 2. There were 60 freight cars on a road .and 6 of them were destroyed in a collision ; _ how many remained? 3. If a man can draw 7 loads of sand in 1 day, how many days will it take him te draw 42 loads? 4. Ina class-room there are 6 seats, and -each seat will hold 8 pupils; how many pupils can be seated in the room? 5. Six sheep were put into a flock con- taining 87; how many then in the flock ? 6. Pineapples are 6 cents each; how many can be bought for 48 cents? Count by 7’s from 7 to 70. 7. If Nellie pays 5 cents for candy and has 8 cents left, how many cents had she -at first? 8. How many legs have 7 cats? 9. How many quarts in 8 gallons? 10. How many quarts in 12 pints? 11. If John buys some candy for 18 “cents, some peanuts for 7 cents and an orange for 6 cents, how many cents does -he spend? See Teachers’ Edition, p. 123. FIRST STEPS AMONG FIGURES. 75 12. * If 8 sheep cost $32, what will one sheep cost? 13. How many fingers and thumbs have © boys? 14. Willie bought 2 pounds of crackers at 8 cents a pound, and half a pound of cheese at 14 cents a pound; how much money did he spend ?. 15. Harry had a ten-cent piece, a five- cent piece and two three-cent pieces ; how much money had he? 16. Annie had 60 cents; she spent 30 cents for a doll, and received 10 cents and 2 more from her mother for doing an errand ; how many had she then? 17. Thomas had 13 chickens and II little turkeys; the cat caught 5 of the ‘chickens and the rats caught 4 of the little turkeys ; how many of both were left ? Count by 7’s from I and 2 to 71 and72. 18. William had 7 cents and John had twice as many; how many had both the boys? 19. I have 18 pupils in a spelling class; 4 of them misspell some of their words ; how many recite perfectly ? *The character $ means dollars and $32 is read thirty- two dollars, 76 FIRST STEPS AMONG FIGURES. 20. George had 7 glass agates, 8 china. and 6 common marbles ; how many marbles. did he have? 21. If 24 apples be equally divided among 8 boys, how many will each have? 22. Walter took 56 cents to the store to: buy sugar at 8 cents a pound; how many pounds could he get? 23. Katie’s mother gave her g cents, her father 7 cents, and her aunt 4 cents; she bought 3 oranges at 4 cents each and spent. the rest of the money for candy at 2 cents. a stick; how many sticks of candy did. she get? I, 5435 2. 3454 3. 5342 4354 5323 3455 3343 2345 5534 5425 4534 2345 2342 5453 4453 3454 3225 5225 ao a0 4544 3542 4344 2345 4354 5432° 5432 5335 2345 4543 4543. 4554 1234 3254 3435 5432 5432 5342 3555 1234. 4554 4343 4555 3343 5443 5321 See Teachers’ Edition, p. 126. FIRST STEPS AMONG FIGURES. 4. 3452 5324 4543 5435 3354 5445, 4554 3425 5432 2345 4553 5434 4423 3545 5434 6. 4356 3645 5234 6563 3656 4345 5456 6563 3625 5356 6545 7. 3564 6453 2566 4035 5 364 6532 3056 4304 5635 6556 3440 5. 4532 3254 4325 5543 3454 3434 2345 5432 4545 3454 5235 4523 5445 4354 5432 cea eee 8. 6345 2436 6563 3656 5365 6534 4655 5362 6656 3545 6666 es a, 78 II. 12. 13. 14. I5. 16. 17. 18. FIRST STEPS AMONG FIGURES. 9. 3456 10. 6354 6563 5665 5635 6536 6366 3456 3556 6365 6645 6653 4536 5546 5663 3665 6636 6236 6355 6463 5663 3554 684,632—47,757=? Ans. 636,875. 43,120—2,765—? Ans. 40207 935,043 —77,387=? 754,231—20,154=? 836,425 —68,268=? 364,135 —71,543=? 463,845 —98,536=? 763,843 —46,452=? Ans. 858,256. Ans/725,0774 Ans. 768,157. Ans. 292,592. Ans. 365,309. Ans. 717,391. 19. 79 ——— FIRST STEPS AMONG FIGURES. 6453 20. 3456 = 21. 3645 3566 6566 6556 4635 5635 2663 6326 4364 6350 5663 5662 5505 4566 6556 4636 3656 6636 6353 6305 3665 5064 5636 5326 6656 6656 6553 3533 5625 3466 6665 5564 6530 3456 4563 5402 6363 22. 5663 23. 6543 6656 2345 3546 6234 6635 5623 5366 4562 4653 3456 6636 6566 3565 5335 6366 46606 5436 5656 6563 6535 3666 6666 6525 3553 Sce Teachers’ Edition, p. 126. 80 FIRST STEPS AMONG FIGURES. When there is a cipher or ciphers in the multiplier between significant* figures, do not use it in multiplying, since nothing times any number is nothing ; but be care- ful to write the first figure of each product under the figure you multiply by. . 57,364 x 304=? - 47,563 x 504=? 246,75 20700—1 » O20,A7 5800 7—1 50,740 x10 7a. . 46,375 x 564==? P2315 PO3 7320 =e . 6,375 X657=? NOB 74205 A 7AT,580 x O8—? 119647,5 530-05" 12. 364.758 x 48=? 13. 105,743 +4=? 14, 3,176,207 = 5—=¢ 15. 1,418,407 -4—? 16. 22,477,527-+-6=? 17, 2,620,3690+4=? 18. 23,176,234+5=? to OO ON AN BW DN * The significant figures are 1, 2, 3, 4, 5, 6, 7, 8 and g, See Teachers’ Edition, p. 128. FIRST STEPS AMONG FIGURES. 81 19. 30,249,742+4=? 20 2,625,160+6=? Glee 00,850)-7 =? 22) 934,459,582 -6 =? When the divisor is not contained in the partial dividend, write a cipher in the quo- tient and the partial dividend will be the remainder to be prefixed to the next figure of the dividend. The teacher will illustrate by the following examples: | BAe O,4 215 A? 25. 44,593,072+7=% 20, e2, 115.085 =5=? 27. 37.548,510+6:=? 28. 25,233,2905+7=? 29. 37,220,621+8=? 30. 42,519,206+6=? 31. 40,379,014+7=2 22, 52,594;387+8=? 33. 20,326,941+8=? I. 874,895 + 20I—? 250728,550-201=? . 12,756,024+ 3,012? 8,604,338+2,031=? *L Ww CONT OVM FIRST STEPS AMONG FIGURES. . 16,154,855 +3,024=? . 40,486,701 +2,032=? . 2,595,900+4,023=? . 1,460,703+403=? . 10,357,440+ 3,024=? . 14,244,539+3,014-=? . 13,822,604+-4,034=? . 25,079,486 — 7,036=? . 38,968,336+6,035=? . 278,881,300 —5,040—2 . 334,654,184 ~ 70,364=? . 349,143,867 +6,057=? . 226,396,593 ~60,453=? . 18,651,776 —4,076=? . 526,026,567+7,046=? , 236,326,931 — 5,074=—2 » 39,361,095 +6,024=? . 540,611,445 ~7,068=? . 450,299,132+60,378=? . 327;040,029 ~ 5,048=? - 413,535,490+ 70,485 =? . 241,993,12 + 5,086=? . 546,695,551+80,574=? . 465,406,942 + 7,068=? . 289,561,188 +6,075 =? See Teachers’ Edition, p. 136. 30. FIRST STEPS AMONG FIGURES. 4563 3656 6366 5665 4656 6546 6665 5656 3456 6543. 5655 6366 5621 33. 31. 5465 32. 4536 6536 5665 6653 4354 3566 6635 6632 5666 5 366 6563 6556 6356 3465 3065 6634 5636 5663 6565 2556 3663 6665 6356 5326 6565 5634 34. 3456 6565 6536 3456 5665 6563 3566 4050 6343 5665 4656 6363 3565 5636 6666 3565 5350 6366 4663 5636 6536 6653 5465 3546 6323 83 84 FIRST STEPS AMONG FIGURES. 1. A boy had 15 marbles and lost all but six of them; how many did he lose? 2. Mary comes to school 5 days ina week; how many days does she come in 8 weeks ? 3. There are 55 sticks of candy ina jar; if 8 little girls each buy a stick, how many sticks will be left in the jar? 4. How many marbles can a boy buy for 27 cents at three cents apiece ? 5. Fanny had 8 cents and Julia had 9g cents, how many did both girls have ? When an example involves several opera- tions the pupil should give but one at a time. 6. How much more will 6 oranges cost at 4 cents each, than 7 peaches at 2 cents each ? Solution : If one orange cost 4 cents, 6 oranges will cost 6 times 4 cents or 24 cents. If one peach cost 2 cents, 7 peaches will cost 7 times 2 cents or 14 cents. If the oranges cost 24 cents and the peaches I4 cents, the oranges cost as much more than the peaches as the difference between 24 cents and 14 cents or I0 cents. 7. Henry had 25 cents;he gave 3 cents each to his brother and sister, spent 5 cents for an orange and 2 for candy; how many cents had he left ? See Teachers’ Edition, p. 137. FIRST STEPS AMONG FIGURES. 85 8. Harvey had a twenty-fivecent piece, a ten-cent piece, a five-cent piece and a three-cent piece ; how muchmoneyhad he? g. If I had 4 apples and found as many amore, and ate two of them, what part of a dozen had I then? What are they worth at 12 cents a dozen? 10. ] have a clock that strikes every quarter hour ; how many times will it strike in g hours? 11 William spent 12 cents, James spent one third as many and three cents more; show many did James spend ? 12. There are 2 little dogs passing ; how many eyes, ears and feet have they ? 13. Three men each take three bags of wheat to mill, and each bag contained 2 bushels; how many bushels did the men ‘take to the mill? 14. When milk is 6 cents a quart, how Many quarts can you get for 42 cents? i5. When milk is 4 cents a quart, how Many pints can you get for 20 cents? 16. In a school-room there are 7 rows of ‘seats, and 6 seats in each row; how many ‘seats are there in the room ? 17. A lady made 7 squares of patch- work, and her little girl sewed so many that “one-half of what both sewed was 10; how amany did the little girl sew? 86 FIRST STEPS AMONG FIGURES. 18. How many boxes of wafers at 6 cenits: a box may be bought for 9 sheets of paper at 2 cents a sheet? 19. How many barrels of apples at $3 a barrel can be given for 6 yards of flannel at $2 a yard? 20. How many four-horse teams can be: mers from 20 horses? Three fields have each 3 trees, Werte: see tree are 3 cows; how many cows in the three fields ? 22. A man bought a duck at 9 cents a- pound and paid 54 cents for it; how much did the duck weigh? 23. If 6 oranges cost 24 cents, what cost. 8 oranges? Solution: If 6 oranges cost 24 cents, ome orange will cost one-sixth of 24 cents, or 4 cents, and 8 oranges will cost 8 times 4. cents, or 32/ cents. 24. If a boy walks 15 miles in 3 days, at the same rate, how far will he walk in 4. days? 25. If it takes 16 yards of cloth for 2 suits of clothes, how many yards will it take for 6 suits ? 26. If a boy goes 8 feet in stepping 4. times, how far will he go in stepping 7 times ? See Teachers’ Edition, p. 138. FIRST STEPS AMONG FIGURES. 87 27. If 3 men can cut Q acres of grain in: one day, how many acres can 6 men cut in a day? Poet it takes 12° buttons for 3 vests, how many buttons will it take for 8 vests? 29. How many yards of cloth at $3 a yard can be bought for 4 barrels of flour at £€6 a barrel ? 30. If 3 men can build a wall in 6 days, how long will it take one man? 31. If 5 men can mow a field of grass in 10 days, how long will it take one man ? 32. If 4 men cut 8 cords of wood ina day, how many cords will 1 man cut ina day ? 33. If 3 mencut a pile of wood in g days, how long will it take one man? 34 If 3 mowing machines will cut 27 acres of grass in one day, how many acres. will 7 mowing machines cut in one day? 35. If a boy earn 63 cents in 7 days, how much will he earn in 6 days? 36. How many books at 4 shillings each can you buy for 8 dozen eggs at 2 shillings. a dozen? 37. A teamster drew 8 loads of stone each day for 7 days; how many loads did he draw ? 38. A boy gained 7 cents by selling a. knife for 42 cents ; what did it cost him ? 58 FIRST STEPS AMONG FIGURES. 39. William worked 8 hours at 2 shillings an hour, and Henry worked 3 hours at 3 shillings an hour ; how much did both earn ? 40. Four girls have each 2 hens, and each hen has 6 chickens; how many chickens have the four girls? | Read the following numbers: . 750406300. . 45706000043 . 860000307. . 15000045001. - 7845078437. 37147415006. 47583000000. 370015 300. . 40000036700. 7 LOOOIOO00O. OO ON ANALWN Write in Arabic the following numbers : 1,, Fifteen mil tenth three 2. Two hun. eight bil. one hun. th. 3. Three bil. twenty mil. six. 4. Thirteen bil. nine th. seven hun. ‘forty-five. 5. Ninety-one bil. one mil. one th. one. 6. Four bil. seven hun. fifteen. 7. Two hun. sixty mil. 8. One bil. three hun. sixty mil. two ‘hun, th. See Teachers’ Edition, p. 140. FIRST STEPS AMONG FIGURES. 8q 9. Five mil. ninety. 10. Write in Roman nine hun. thirty- 11. Write in Roman seven hun. forty-six. hematite in. Arabic DCCCXCVII. 13. Write in words 709460371000. 14. Write in Arabic ten th. thirty. Teach the pupils that the figure at the right expresses units of the first order, the next figure to the left, units of the second order, the next figure, units of the third. order, and so on. 15. Write 7 units of the 5th order, 4 of the 3d and 1 of the Ist (in one number.) 16. Write 3 units of the 8th order, 5 of the 7th, 9 of the 3d and 4 of the 2d. It may aid the pupil in solving the follow- ing examples, to put small numbers in the place of the ones given, and see how it would be solved without the slate, then solve in the same manner. Simple problems for the slate, involving. Addition, Subtraction and Multiplication : 1. Mr. Rogers had 746 bushels of wheat and sold 197 bushels of it; how much had‘ he left ? 2. George had 295 cents and his father gave him 75 more ; how many had he then ? “gO FIRST STEPS AMONG FIGURES. 3. Mr. Smith had 96 bushels of oats -and Mr. Jones had 9 times as much; how many bushels had Mr. Jones? 4. Lewis has 87 marbles and John has just as many; how many have both boys? 5. Mr. Howard drew 8 loads of oats to market, and there were 79 bushels in each load ; how many bushels did he draw to market ? 6. From the sum of 79 and 268, take 158. 7. How much will a teacher’s salary amount to in 14 years, at $875 a year? 8. James has 47 marbles less than John, and John has 174; how many has James? g. John lost 15 cents by selling his knife for 90 cents ; what did it cost? 10. Miles took 2341 steps in going to school. and Marcus took 560; how many more steps did Miles take than Marcus ? 11. Mr. Decker borrowed $150 and paid $65 of it; how much does he still owe ? 12. What will 46 bushels of barley cost at 167 cents a bushel ? 13. A clerk received a salary last year of $1000. He spent $260 for board and $378 for clothing and other expenses ; how much money did he save? 14. There are 30 days in June, and 31 each in July and August; how many days 4n these three summer months ? See Teachers’ Edition, p. 143. FIRST STEPS AMONG FIGURES. gI 15. There are 168 acres in Mr. Fox’s farm, and Mr. Norton's farm contains 8g acres more than Mr. Fox’s; how many acres in both farms ? 16. A man put $950 in the bank; he drew out $78 at one time, $45 at another, and $159 at another; how much had he left in the bank ? 17. John had 39 marbles and Ezra had 13 more than twice as many; how many had Ezra? 18. Mr. Brown bought a farm for $8460 and sold it for $10380; how much did he gain? 19. What cost 369 bushels of oats at 68 cents a bushel? 20. I have 216 bushels of potatoes in 3 bins ; there are 59 bushels in one bin anc 98 bushels in another; how many bushels in the third bin? 21. A boy having 85 cents, bought a top for 18 cents, aball for 25 cents,andsome oranges for 27 cents; how many cents had he left? 22. If there are 76 bushels of corn ina bin that will hold 950 bushels, how many more bushels of corn may be put into it? 23. A farmer filled at one time 29 bags with oats,and at another 47 bags. If he put two bushels in each bag, how many bushels of oats were put in all the bags ? 92 FIRST STEPS AMONG FIGURES. 24. From one million eight hundred thousand take fifteen thousand ninety. 1. 4507 2. 7045 3. 4756 7654 5774 3567 6347 6547 6635 7576 5732 7456 4757 7665 6573 5674 4576 5746 7505 5766 7455 4757 7537 6567 5664 6452 7730 7575 3776 6564. 2346 6565 6475. 4. 7546 5. 7654 4757 6537 5675 5465 6757 77506: 7577 4675. 4664 3456 7357 6747 5675 7504. 4567 6475. 7654 7746 4735 4653, See Teachers’ Edition, p. 144. FIRST STEPS AMONG FIGURES. Co 6. 3456 7653 6575 5747 7356 5676 4747 5635 4567 7476 5745 6574 7. 6754 5673 7566 4375 7047 5774 6757 7577 4652 5767 7760 5473 8. 5746 7457 6575 7067 5734 4757 7676 4567 6735 5657 4576 7465 93 FIRST STEPS AMONG FIGURES. II. 4,362,516—754,359=? 129.730.0522 7S .07 28 13. 642,534—26,356= ? 14. 352,432—86,354=? 15. 6,425,314—374,321=? 16. 463,524—71,876= ?. 17. 425,362—17,654= ? 18. 364,253—86,174=? 19. 4603,521—186,357? 20.. 483,654—91,087=? 21.:635,245—12,567? 22. 837,524—259,280= ? | 23. 43,452,431 —4,238,865 =? 24. 756,324—85,543=? 25. 4,738,536—973,659= ? 26. 86,357—7,269= ? 27. 34,023—0,876=? 28. 45,300,435 —2,430,520= ? 29. 74,200,032—5,140,054= ? 30, 43,250,001 —5,726,025=? 31. 3,400,564—210,739= ? 32. 45,700,325 —8,730,153=? I. 796,845 x 89=? 22 cA Toma Sete FIRST STEPS AMONG FIGURES. . 68,975 x 79=? . 647,583 x 467=? . 68,574 456=? bASO,075 X S07? 59,007. x O64—? BOO 7 07.5 = 2 - 4,790 x 39=? . 85,974x 84= 2 070,035 x Go= ? . 68,974 x 79= ? AO, / 00% OF fo . 59,068 x 604= ? 170,008:x 907=? ee7O; 200 x 7OO='¢ . 860,479 x 709= ? . 759,068 x 407? . 47,096 x 609= ? . 748,609 x 507= ? , 58,709 x 608= ? . 00,047.x 7090= ? PO7,500:< 507— ? - 79,689 x 4,759= ? . 896,748 x 6,978=? 7,500,005 — 1,25.1,:2908= ? See Teachers’ Edition, p. 145. 95 to \O — = me Ne OW ON AM AW WN QO = = = = = = 00 ON AwM DB W NS tt FIRST STEPS AMONG FIGURES. - 3,740,683 —923,754= 725 O20 520 ee : 7,430,246 —7,503,472= ? . 3,241,402+7=? . 5,079,005 = 62 3,802,457 O=-0 . 49,167,544+9=? . 5,076,335+9=f » 3,372,081 +9=? . 3:725,801+8—2 . 52,301,166+8=? . 48,0904,605+-7=? . 43,719,125+9=? . 52,620,186+8=? (40,224,713 eet 507,21 7710 Le sat . 608,226,845 -9=? . 37,100,695 +8=? . 460,241,323+7=? . 4,781,158,859+6=? . 4,885,157,701+7=P . 389,15 8,560+6=? . 5,240,869 -7=? » 3,356,977+7=? See Teachers’ Edition, p. 146. FIRST STEPS AMONG FIGURES, 97 2G, 727,741 Oo? Mpst 5.271 Ss! . 21,307,398+5,024=? . 3,722,901+607=? . 14,204,241 ~403=? . 158,032,783 = 30,135? . 38,693,395 —6,024=? . 38,464,305 +50,396=? . 31,970,764~7,048=? . 3,891,687,541+60,475= ? . 2,925,490,533+60,479= ? . 462,857,740+8,069= ? . 2,934,401,497+70,586= ? . 226,663,766+60,379= ? - 194,513,933+40,297—? $2184,110,003=-5,048= ? 700. 701s5d0+6,037= 7 . 32,688,027,778+70,486= ? . 674,.476,820+9,037=? . 27,971,095 +6,074= ? . 355,212,265+5,030=? - 23,377,790+5,024= ? 452,113,508 +7,056= ? . 299,019,935+406= ? 98 FIRST STEPS AMONG FIGURES. . 2,408,592,665 +5,064=? 47. 36,979,544 8,026= ? 48, 426,696,721 —7,01d=" 49. 437,791,338+6,038= ? 50, 2106,526,004-+-5,027=? 51. 2,464,085,695 + 604,978 ? 52. 238,049,090+ 5,037= ? 53. 4,110,929,380+70,586= ? 54. 283,899,778+6,034= ? 55. 4,980,403,784+ 70,496= ? 56. 52;202)15 Sea 57. 251,864+361=? 58. 434,801+573=? 59. 268,500+463=? 60. 463,799+582—=! 61.* 356, 116— 365=? 62. 357,243+465=? 632: AO3j12 oe Sober C4. 414,017+473=? 65. 4,174,696+485 =? * When the left hand figure of the divisor is equal to- the left hand figure of the dividend, if the next figure of the divisor be greater than the next figure of the divi- dend, point off as if the left hand figure of the divisor were greater. The divisor (in one step of the opera- tion) never is contained more than nine times. See Teachers’ Edition, p. 148. FIRST STEPS AMONG FIGURES. 99 66. 5,499,513+796=? 67. 5,538,824+684=? 68. 7,084,249+896=? 69. 30,734,480+645=? 70. 43,722,966 573=? 71. 270,578,240 4,035 = Bae 2005430, 50 > 4,657== 1. William paid 54 cents for 6 doves; what did each dove cost? } 2. In an orchard there are 6 rows of trees and 7 trees in each row; how many trees in the orchard ? é 3. Henry has 8 cents in one pocket and 9 Ronis in the other ; how many cents has he? 4. James has 8 apples and his sister has 6; how many more has James than his sis- ter? 5. What cost 9 knives at 7 shillings each ? 6. A boy paid 25 cents for a ball and sold it for 18 cents ; how many cents did he lose? 7, If 1 pencil cost 4 cents, what will 8 pencils cost ? 8. George bought a knife for 8 shillings, a ball for 5 shillings, and a bat for 2 shil- lings ; what did he pay for all ? 9g. Marcus spent 8 cents for lemons at 4 cents each; how many lemons did he buy? See Teachers’ Edition, p. 150. e [00 FIRST STEPS AMONG FIGURES. 10. If 5 men cut ten cords of wood ina day, how many cords will 7 men cut? 11. If 2 men can dig a certain ditch in4 days, how long will it take one man to dig it ? 12. If 4 men can cradle 12 acres of grain in one day, how many acres will one man cradle in a day? 13. How many rods of wall will one man build in a day, if 3 men build*g rods in one day? 14. If 3 boys can pick the stones from a meadow in 9 days, how many days will it take one boy to pick them ? 15. How many weeks in 35 days? 16. If 4 pounds of sugar cost 36 cents, what cost 8 pounds ? 17. When a pineapple costs 18 cents and an orange costs 6 cents, how much more does the pineapple cost than the orange ? 18. If a boy can walk 12 miles in 4 hours, how far can he walk in 5 hours? 19. If 4 men can do a piece of work in 8 days, how long will it take one man? 20. Jane bought 5 figs for 3 cents each, anda yard of cloth for g cents; how much did she pay for all? See Teachers’ Edition, p. 151 FIRST STEPS AMONG FIGURES. IOl 21. Mary sews 4 hours each day, how ~-many hours does she sew in a week ? 22. How much will a man’s board for a aveek cost at 4 shillings a day ? 23. How much will a man earn in a week, if he gets 9 shillings for a day’s work ? 24. If 15 cats are on a wall and every third cat jumps off, how many are left? 25. There are 8 quarts in a peck, how many pecks in 32 quarts? 26. How many quarts in 3 pecks? 27. A boy picked 16 quarts of beans and ‘sold them at 25 cents a peck; how much money should he receive ? 28. Charles has 7 cents and his brother 3 more than twice as many ; how many have both? 29 On Monday morning Mary had 20 ‘sticks of candy; she ate 2 each day, how ‘many had she left the next Monday night ? 30. Arthur had 11 peaches, he ate 3 and gave his sister half of the rest; how anany did he keep? 31. How many marbles, 2 for 4 cents, «an you get for 18 cents? 102 ' FIRST STEPS AMONG FIGURES. EXAMPLES FOR THE SLATE. If the pupil will use small numbers in- stead of the large ones in the following examples, and think carefully how he would solve them if they were not for the slate, and then do the same with the numbers. given, using the slate as a help, he will be Bede ascisted: A man had $3,210, he spent $978 for ee and $749 for corn ; how much money: had he left ? 2. What is the product of 9,687 and 75 ? | 3° From the sum of 3796 and 4279, take their difference? 4. If a farmer have 256 gallons of cider, how many barrels holding 36 gallons can he fill ? 5. Froma cistern holding 743 gallons, 98° gallons were pumped out and afterwards 39. gallons poured in; how many gallons were then in the cistern ? 6. What cost 37 carriages at $185 each ? 7. If a ship sail 7289 miles in 37 days,. how many miles does she sail per day ? 8. A miller paid $169 for 78 bushels of wheat, $97 for oats and $395 for corn 3. ° what did he pay for all of the grain? g From the sum of 397 and 6708, take 69. % See Teachers’ Editon, p. 152. FIRST STEPS AMONG FIGURES. 103 10. The difference between two numbers: is 347, and the less number is 79, what is. the greater number ? 11. A man died leaving $5600, of which he gave his wife $2,800, his son $900, one daughter $850 and the rest to another daughter ; how much did the second daugh-. ter receive? 12. A man bought 75 sheep at one time, and 169 at another ; he sold 86 of them to one man and 49 to another; how many had he left ? 13. Mr. Wilson bought one house for $4150, and afterward another for $3750 ; he sold both of them for $7000; did he gair or lose, and how much? 14. There is an orchard consisting of 24 rows of trees, and 36 trees in each row; how many apples in the orchard, allowing an average of 2079 on a tree? 15. A man owing $7165, gives in pay- ment 39 cows valued at $48 each and $750 in money ; how much does he still -*¢? 16. Add 16 thousand 20, fifty millio. *t, thousand nine, 79 thousand 847, and 9 m - lion 79 thousand 8." 17. How many tons of hay at $18 a ton must be given for 16 horses at $153 each? 18. 6394+91,758+9,347 481,731 $9,342 + 35,4464 8,237+12,849+87,677=? 104 FIRST STEPS AMONG FIGURES. 19. A grocer spent $881 for molasses and sugar; he paid $368 of the money for molasses, and the rest for 27 barrels of sugar; how much did the sugar cost a barrel ? 20. How mary yards of cloth in 68 bales, each bale having 97 pieces, and each piece containing 29 yards? ; 21 Paid $6 each for 75 sheep, and sold the flcck for $400; did I gain or lose,and how much? 22. How many horses at $165 each can he bought for $2360? 23. How much is gained by buying 48 cows at $37 each, and selling them at $45 each? 24. Mr. Dixon has 225 acres of land worth $97 an acre, and Mr. Taft has 196 acres worth $79 an acre ; how many acres have the two together, and what is the value of the whole? 25. A man sold a farm of 96 acres at $9 ar e-:<, and with the money received for it » 2nt a farm of 135 acres; what did he yay an acre for the latter farm? 26. A teacher had his life insured for $2500. At the time of his death he owned a house and lot worth $1850 and furniture worth $475. He owed debts to the amount FIRST STEPS AMONG FIGURES. 105. of $369; how much did he leave his. family ? 27. There are 5280 feet in a mile; how many feet in 709 miles? 28. A man starts from New York on Tuesday morning and travels at the rate of 57 miles a day; another starts from the same place Wednesday morning and follows on at the rate of 69 miles a day; how far apart are they Thursday night? 29. James sold a grocer 96 eggs at 15 cents a dozen, and received 120 cents ; how much does the grocer still owe him ? 30. If there were 365 days in each year, how many years would there be in 31390 days? 31. Add seventy million nine hundred thousand, two hundred six thousand eight sixty thousand sixty, seven thousand nine hundred, ten million ten thousand ten, and. seven hundred fifty-nine million two hun dred thirty thousand. 106 FIRST STEPS AMONG FIGURES. - 7057 4775 7777 3456 6735 5677 7750 3457 6574 5767 4575. ae 5676 2345 4. 4756 7577 6645 7734 5675 6756 7497 4575 5047 7777 6452 3567 6776 5643 2, 2456 5707 7475 4567 7756 5647 6775 7564 4677 3450 5797 7534 6467 3. 7567 4756 3675 6777 7546 5735 6767 7476 3456 3456 7763 4567 2475 5. 7654 4775 54607 7556 6775 3457 5747 7674 6577 7754 TFT, 7777 4564 5675 See Teachers’ Edition, p. 153. 6. 4508 3785 8678 5786 8678 8888 8888 5678 8765 4876 5487 FIRST STEPS AMONG FIGURES, 107 7, 6758 8. 8765 8547 4658 7808 7777 8778 3456 8888 6778 8888 7507 3576 8878 6758 7797 8087 8585 5678 UY Ufos 6786 3456 10. 3748 8675 7887 4567 8888 8888 4507 PLAT ZELE 4505 8486 108 FIRST STEPS AMONG FIGURES. II. 6758 7584 4676 8778 3456 8888 88838 6547 3754 6678 7563 5835 14. 4786 8657 6578 7865 4576 8888 3456 ewer 8765 3578 8657 4768 12:, 7658) 9 1 Stegone 4576 4578 6785 8057 5467 6785 8878 8888 4057 8888 8765 3450 6578 7777 7857 5678 5686 8753 8578 4576 6785 8687 15. 7684 4578 3456 8888 8888 7654 3567 WK: 5648: 7385 4637 8386: See Teachers’ Edition, p. 154. 16. FIRST STEPS AMONG FIGURES. Book | 1A 4037, 18> 4587 8546 7584 8635 7685 8758 6754 4868 5875 8888 5784 8888 8888 FATE 8888 6754 6548 6753 7578 3675 4584 4785 8888 it | 1 ge Vf 8888 5678 5768 5678 8765 8654 8563 3857 4585 5785 6586 7848 8678 8465 5686 19. 8476 20. 7586 5768 4767 4567 8658 TEAM 4875 sede Fe 8888 8654 8888 3568 7654 8888 3867 8888 TIT 4567 4685 8765 8568 4478 ERG 7586 4825 8357 8674 £10 FIRST STEPS AMONG FIGURES. . 4,570,365 — 323,456=2 - 9,374,056—636,587=! 685,700,305 — 296,314,537 =? . 76,400,235 — 3,234,567=? . 38,500,684—8,769,876=? . 7,460,683 —370,876=2 . 375,600,735 —83,735,829=? . 83,640,574—5,71 2,053 - 794,600,435 —63.732,367=? ~ 74,300,375 —5 3,620,547 = ». 8,750,043 —070,230=2 : 6,7 3,021 O07 3.21 eat 1°48; 300.5 03 0, COO 7 ==5 . 487,500,564—65,730,637=? . 756,000,375 —85,203,456=2 . 6,847,0C0,346— 367,020,654=? 79,068 x §58=? 80,479 x 74=? A NSS S808 ae 968.570 «FOS —=c - 79.689 x 4,759=? - 79,807 *6,507—? VOSGosocel Oe 49,678 x 9,876=? . 497.896 x 8.659=? See Teachers’ Edition, p. 154: eit ON AM PW HN & at NO = Oe ee ee | 00 ON Aw L&W by NY NN OY Oa) aS oy er FIRST STEPS AMONG FIGURES. Ill Since moving a figure one place to the left increases its value ten fold, and moving it two places, ten times ten fold or one hundred fold,—to multiply any number by 10, 100, 1000, &c., annex as many ciphers to the multiplicand as there are in the multiplier. 379 X 100=37,900 Ans, B02 7 805 %.10,000=7 Cas 7210! pee 65 < 100? 201 0;320 x 1000=? 30 875 < 100,000=? When there are ciphers at the right of either the multiplier or multiplicand, or of both, place the multiplier under the miulti- plicand so that the significant figures farthest to the right shall come under each other. After multiplying by the significant figures and adding, write as many ciphers at the right of the product as there are at the right of the multiplier and multiplicand together. (These directions are given very minutely but are not to be committed to memory.) II2 FIRST STEPS AMONG FIGURES. 34200 34000 1368 1026 I, 162,800,000 . $337 50 440 000-5 46,300 X25 O==¢ . 635,000 X 700=? . 27,500 X 680,000=? . 586,000 x 7,400=? 400% BOy700—t . 6,840 x 7,500? 8,609 x 800=? . 67,900 x 870=? . 8.690 x 4,700=? . 480,600 X 7,090=? . 70,580 x 6,408,000=? . 706,900 X 5,078,000=? . 68,090 x 70,900=? . 640,980 x 10,000=? I. 43,188,278+9=? 7OT,O71, 017-0 ak FIRST STEPS AMONG FIGURES. 0, 150)712,035--3=? . 67,815,232+7=? GO 20;260,026—.9 8 295 2:717;437-=- 8 =? BEOtO08,27 25507 = O—=? fei, 202;100= O—7 - 49,347,765+7=? - 44,160,343+9=? + 5575793,570+7=! . 7,664,063,843+8=? . 8,044,185,607+9=? . 491,093,400+7=? 5522557410? . 520,762,735+7=? 97230,;2453=? S027 4;817'- O—? 710,863,843+8=? BeriOps,.17~ 7! . 30,884,751 +7,058=? . 34,600,073 +6,032=? w1,890,186-5,178=? . 1,203,161,896+8,169=? 5,279,490 814=? MeO 20.220 725-7 See Teachers’ Edition, p. 155. 113 Ris FIRST STEPS AMONG FIGURES. 27. 92,197,3604+6,257—0 28... 7:038, 402-3430 se 29 511,764,908— 7,461=? 30. 946,526,656 -6,397=? 31. 65,790,555 : 6,847=? 32°C, 303,000 =7400=ar 33. 18,827,247-+378=? 34. 526,493,286 3,794=? Since moving a figure one place to the right diminishes its value ten fold, and two places, ten times ten fold or one hundred fold,—to divide any number by 10, 100, 1000, &c., cut off by a vertical line as many figures on the right of the dividend as there are ciphers at the right of the divisor. The number at the left of the vertical line will be the quotient, and the number at the right of it the remainder. Illustration : 78634+100=? Solution: 786 | 34 the quotient is 786 and 34 is the remainder. I. 793,468 10,000=? . 37,680 = 100=? . 2,347,600+ 100,000=? 76,210,048+ 100 =? . 372,938,641 + 10,000,000=? uy B WwW WN FIRST STEPS AMONG FIGURES. Il5 To divide by any number with ciphers at the right. 78,673 700=? Divide both dividend and divisor by 100, and cutting off the 2 figures at the right, and the example becomes— 7|00)786|73 I12—273 rem., or 273 Il 2—— 700 Divide the number at the left of the vertical line in the dividend, by the number at the left of the vertical line in the divisor, and to the remainder annex the figures of the dividend cut off. 45|000)612|370(13 quo, 45 162 135 ———e 2IRTOTeM: £16 FIRST STEPS AMONG FIGURES. Hence the rule: To divide by any num- ber with ciphers at the right, cut off the ciphers at the right of the divisor by a vertical line, and also as many figures at the right of the dividend. Divide the remain- ing number in the dividend by the remain- ing number in the divisor, and to the remainder annex the figures cut off from the rightof the dividend for the true remainder. . 18,228,211+37,500=? . 5,142,762,131 - 750,000=? . 546,927,300+687,000=? 976,514,152 >-706,500-—¢, . 8,734,758+ 10,000==? 350,870,000+ 3,580,000=? » 3,278,300 7,000—¢e . 87200 X22 000—2 . 7,162,323 ~QoO=? . 394,690,750~ 5,800=? . 20,85 0,010=> 3;750—2 =~ OO OCNANAWDN = — 12. From six billion six thousand six, take eighty million eight. 13. 27,752,320,172~5 70,000= P I4. 2,910,144,700+ 36,800= P See Teachers’ Edition, p. 155. 15. 16 Li 18. FIRST STEPS AMONG FIGURES. I17 178,576,495 + 100,000=? 365,820,038 — 6,000=? 475,308,056 — 48,600=? Subtract forty-five million, one thousand ten, from forty-two million seven hundred thousand. 19. 20. oe oan: 23 24 441,937,000+597,000=? 30,500,857,231-+3,780,000=? B05 7 200 000-1 47,096 x 8,600=? 70,286,631+900=? 7900 x 10O=? When the multiplier is less than 13 the pupil should be taught and vreguzred to multiply but once through, multiplying by II or 12 as he has already been taught to multiply by 4, 5 or 6. An Bw nN pvo;007% 12—? 69,789 x I1=? 545030 x 12=2 845.768 x 12=? . 68,094,796 x II=? m5oO,074 x. 12—? See Teachers’ Edition, p. 157. 118 FIRST STEPS AMONG FIGURES. . O579,087-K1 22 60/487,005 412-28 10;408;5 70 01 face . 479,058 x 12=? ; 07,500 04 7aale ae 1-0,075.007eele ae 2-69;786:005 XA2—6 . 94,079,689 x 12=? . 979,806 x 12=? 2-085. 700 4 tee , 8:798,070 x22=? 49,897,697 X 12=? . 97,987,986 x I2=? | 7. O8Q.O85 20Igat . 1,185,401,377-—- 12s (540,500.40 Loa iler . 820,499,872+12=? . 1,044,922,315+12=? . 9,438,575,909 + 12=? . 9.568,867,509+11=? . 455,976,730+12=? . 956,219,025 12=f . 898,436,885 + 12? . 7,434,306,992 ~ 11=? FIRST STEPS AMONG FIGURES. II9g mr SSSSSSEESeeSeSSesesSsS FOR ORAL RECITATION, I. A boy having 25 cents, bought mar- bles at 4 cents each, keeping 5 cents of the money ; how many marbles did he buy ? 2. Jane lost 10 cents on her way to the post-office, and spent the rest of her money for 10 3-cent stamps; how much money had she when she started ? 3. * Mark earned 8 cents, lost 5 cents, and then found 10 cents, when he had 25 cents; how much money had he at first ? 4. What cost 12 pounds of sugar if 7 pounds cost 63 cents? 5. A man spent $5, then earned $7, and after giving away $6 found he had $15 ; how many dollars had he at first ? 6. If 6 apples cost 2 cents, what cost 18 apples ? Solution: If 6 apples cost 2 cents, 18 apples, which are 3 times 6 apples, will cost 3 times 2 cents or 6 cents. 7. If 4 marbles cost 3 cents, what cost 24 marbles? * If he had 25 cents after finding 10 cents, before he found it he had the difference between 25 cents and Io cents, or 15 cents. If he had 15 cents afer losing 5 cents, defore he lost it he had the sum of 15 cents and 5 cents or 20 cents. If he had 20 cents after earning § cents, before he earned it he had the difference between 20 cents and 8 cents or I2 cents. See Teachers’ Edition, p. 161. 120 FIRST STEPS AMONG FIGURES. ~ 8. What cost 20 figs, if 5 figs cost 2 cents? g. If 3 oranges cost 12 cents, what cost 7 oranges ? 10. What cost 12 lemons, if 5 cost 25 cents ? 11. How many apples can be bought for 15 cents, at the rate of 5 for 3 cents? 12. If 3 oranges cost 10 cents, how many may be bought for 40 cents? 13. 30 cents will buy how many apples, at 9 for 6 cents? 14. How many figs may be bought for 24 cents, at the rate of 3 figs for 2 cents? 15. If 4 marbles cost 5 cents, what cost 20 marbles? 16. If 5 lemons cost 20 cents, what cost 9 lemons? ' 17. What cost 30 pears if 3 pears cost 5 cents? 18. If 4 peaches cost 3 cents, what will 24 peaches cost ? 19. At the rate of 2 oranges for 9 cents, how many may be bought for 18 cents? 20. If 3 men cut 6 cords of wood in a day, how many cords will 7 men cut ina day? 21. If 3 men dig a ditch in 12 days, how long will it take one man? See Teachers’ Edition, p. 162. * FIRST STEPS AMONG FIGURES. 1 | 22. If 4 men harvest a field of wheat in 8 days, how many days will it take 1 man to harvest it ? 23. If 12 men can dig a field of potatoes in 13 days, how many men will do it in I day? 24. How many men can load a car in I . hour, if 2 men can load it in 4 hours ? 25. If 4 men can doa piece of work in 12 days, how long will it take 3 men to do it? 26. How many days will it take 6 men to earn $32, if it takes 4 men 6 days to earn it? | 27. If 4 men can doa piece of work in 9 days, how many men can do it in 6 days ? 28. If 6 men can doa piece of work in 4 days, how many men will it take to do the work in 3 days? 29. If 6 men can doa piece of work in 12 days, how long willit take 4 men todo it ? . 30. If 4 men can build 12 rods of wall in a day, how many rods can 6 men build in a day ? 31. How many men will build a wall in 12 days, if 6 men build it in 8 days ? 32. If 3 men cut 7 cords of wood in a day, how many cords will 12 men cut ina day? 122 FIRST STEPS AMONG FIGURES. °33. A gitl took 7 pins from a paper and then put on 9; her brother afterwards took off 6, and left in it 24; how many on the paper at first? 34 How many men will do a work in 6 days that 9 men do in 4 days? 35. If 8 men do a work in 6 days, how many men will do it in 12 days? 36. If 4 men do a work in 12 days, how long willit take 6 men? 37. If 3 pencils are worth 11 cents, how many pencils can be bought for 33 cents? 38. A girl having a paper of candy, ate 7 pieces ; then her brother gave her § pieces, atter which she gave her mother g pieces. She had left 27 pieces ; how many pieces had she et first ? 39 What number divided by 2 will give 6? 40 If 4 cords of wood cost $20, how many cords can be bought for $35 ? 41. If 6 vests are worth $24, what are Q vests worth? 42. If 5 cords of wood cost $24, what will 15 cords cost ? 43. What number divided by 3 will get Loe 44. At 10 cents a pint, what willa gallon of molasses cost? 45. How many bushels of potatoes at 4 FIRST STEPS AMONG FIGURES. 123 shillings a bushel may be bought for 3 ‘bushels of wheat at 12 shillings a bushel ? 40. A boy gave 8 marbles worth 2 cents apiece, for 7 pencils worth 3 cents each; how much did he gain? 47. How many eight-gallon cans will be required to hold 56 gallons of milk? 48. If 15 bushels of wheat will make 3 barrels of flour, how many bushels will make 8 barrels? 49. How many yards of cloth at $6 a yard will pay for 9 tons of coal at $8aton? 50. When flour is $7 a barrel, how many barrels can be bought for $8, and 9 bushels of wheat at $3 a bushel? 51. 96 eggs are how many dozen? 52. If 8 horses eat 48 bushels of oats in 2 weeks, how many bushels will 5 horses eat in the same time? 53. Our school has a recess in the fore- noon and also in the afternoon. If there are one hour of school before each recess and two hours after each recess, how many hours of school in a week? 54. If two apples cost one-half of 10a cents, how many can be bought for 15 cents? 55. How many three-cent stamps can be ‘ought for 27 cents? 56. A boy caught some fishes ; he threw (24 FIRST STEPS AMONG FIGURES. away 7, then caught 3 and bought 2, when he had 14; how many did he catch at first ? 57. Kate lives two miles from school, and does not go home at noon; how far must she walk in a week if she loses no time at school ? 58. Frank has 6 five-cent pieces, 4 three- cent pieces and five two-cent pieces; how many cents has he ? 59. John has 7 cents, his brother 8, and .their sister has 4 more than both of them ; how many have they all ? 60. Which costs the more, 3 lemons at 4 cents each or 6 pears at 2 cents each? 61. A boy went to the grocery with 25 cents, and bought 2 pounds of sugar at 9 cents a pound ; how much change should he bring back if he has 2 cents for doing the errand ? 62. If 2 barrels of flour will last 3 men 6 months, how long will it last 9 men ? 63. Bought some peaches for 24 cents, at the rate of 5 for 2 cents, and divided them equally among 6 boys; how many did each boy receive? 64. If 7 bushels of clover seed are worth $42, how many bushels of wheat at $2 a bushel will 3 bushels of clover seed buy ? FIRST STEPS AMONG FIGURES. 125 65. If 2 men start from the same place and travel in the same direction, one 6 miles an hour, and the other 3 miles an hour, how far apart will they be in 9 hours ? 66. In how many hours will a man who drives 8 miles an hour overtake a footman who is 60 miles ahead, and walks at the rate of 3 miles an hour? 67. A man bought a span of horses for g100, paid $60 for their keeping, and sold them for $200; what did he gain on each horse ? 68. How many turkeys can I buy for $43, at the rate of 3 for $5, and have $8 left ? EXAMPLES FOR THE SLATE, 1. In a certain church 28 pews rent at $35 each, 19 at $25 each and 37 at $15 each ; for how much do they all rent? 2. A railroad 18 miles long cost $452, 682 for labor, and $177,228 for other ex- penses; what was the cost per mile? 3. One half of the inhabitants of Con- stantinople are Turks, 150,000 Greeks, 30,000 Armenians, and 65,000 Jews; how many in all? See Teachers’ Fdition, p. 165. 126 FIRST STEPS AMONG FIGURES. 4. If a man earns $960 a year, and spends yearly $688, in how many years will he lay up $4,624. 5. A man bought a farm for $17,600 ; he sold half of it for $9,322, at the rate of $79 an acre; how many acres did he buy ? How much did he pay an acre? 6. From the sum of 7574 and 10746, take their difference. 7, A lady having $125, paid $37 fora set of furs, and $2 a yard for 23 yards of silk ; how much money had she left ? 8. 133416 emigrants arrived in New York in 1867, which was 9,731 more than arrived in 1866 ; how many arrived in 1866? Q James and George started together, and traveled in the same direction. James walked 2 miles an hour and George 4 miles an hour ; how far apart were they at the end of 19 hours? 10. In six boxes of crayons there are 864 pieces ; if 864 pieces cost 360 cents, what will one box cost ? 11. There are two numbers, the greater of which is 37x96, and their difference is 18 x 27; what are the numbers? 12. Aearns $45 a month, and B earns 13 times as much lacking $490 ; how much does B earn in 8 months? FIRST STEPS AMONG FIGURES. 127 13. If a house is worth $1,800 and the farm on which it stands five times as much lacking $36, and the stock one-third as much as the house and farm, what is the value of the whole? 14. A man sold his farm of 245 acres at $69 an acre and bought some land at $97 an acre; how many acres could he buy ? 15. Mr. Smith was 968 miles from home; he traveled toward home 137 miles one day; 119 the next day, and 98 the third day; how far was he from home then ? 16. From thirty billion ten thousand, take seven billion two hundred nine thou- sand seventy-five. 17. A man bought 325 bushels of barley for $500; 450 bushels of oats for $250; 625 bushels of corn for $150 more than he paid for the oats ; 300 bushels of beans at $2 a bushel, and some wheat for $100 more than he paid for the corn ; how much did he pay for all? 18. How many solid feet of earth can be removed in 36 days by two carts each carrying 9 loads a day, and 34 solid feet at a load? 19. A man having $0,840, gave each of his two sons $2,750 and the remainder to his daughter; how much did he give his daughter ? 128 FIRST STEPS AMONG FIGURES. 20. $26,250 is 3 times what A gave for his farm, and he gave $370 more for it than it was worth; how much was the farm worth ? | 21. I sold a horse for $375 which cost me $295 ; how much did I gain? 22. I sold a cow for $65 and by so doing lost $15 ; what did she cost? 23. A man began business with $3,850, and in 7.years he was worth $10,465 ; how much did he make each year ? 24. How many days would 36 horses live on an amount of food that would keep 24 horses 288 days? 25. A merchant received $248 on Mon- day and $396 on Tuesday; what was the average receipts per day? 26. Two men start from the same place and travel tn opposite directions, one at the rate of 54 miles a day, and the other at the | rate of 45 miles a day ; how far apart will they be at the end of 6 days? How far apart if they travel in the same direction? 27. A man bought 478 bushels of corn ; all but 136 bushels were sunk in a boat; how much was saved ? 28. A merchant bought 46 yards of cloth for $93, and sold it at $3 a yard; how much did he gain? 3 See Teachers’ Edition, p. 166. FIRST STEPS AMONG FIGURES. 129 29. Divide the product of 79 and 237 by their difference. 30. At $135 each, how many horses can be bought for $9,368 ? 31. How many times can 317 be sub tracted from 13,314? 32. There are 3 bins containing 856 bushels of wheat ; I contains 376 bushels, another contains 297 bushels; how many in the third bin? 33. A farmer sold 13 tons of hay at $16a ton, and 24 cords of wood at $5 a cord; he divided the money received among four credi- tors; how much money did each receive ? 34. A has 18 barrels of flour of 196 pounds each ; if a family of 9 persons use 49 pounds of flour a week, how long will the flour last them? 35. If Mr. Long’s sheep were put into 6 fields, 96 in a field, there would be 5 sheep remaining ; how many sheep has he? 36. A grocer bought 2 cheeses, one weighing 68 pounds and the other 75 pounds, at 14 cents a pound; how many cents would he gain by selling them at 17 cents a pound? 37. A man killed four hogs, one weigh- ing 368 pounds, one 412, one 379 and one 433; what was their average weight ? od 130 FIRST STEPS AMONG FIGURES. 38. There were 84 sheep in four pastures; there were 30 in the first and 24 in the second; if there were an equal number in each of the others, how many in each ? 39. If a man paid $500 for four horses, $200 for 5 cows and $175 for 40 sheep, how many animals did he buy ? 4o. If a man earns $685 a year, and spends $496 a year, in how many years will he save $1,134? 41. How many pounds of coffee at 27 cents a pound will pay for three hogsheads of sugar, each containing 1080 pounds, at I2 cents a pound? 42. What is the sum of the difference and sum of 1768 and 987 ? 43 Aman deposited in bank at different times $397, $459 and $568 ; he drew out at one time $275 and at another $368 ; how much remained in the bank ? 44. A man sold 26 cows at $35 each; how many horses at $145 each can he buy with the money received ? 45. A dealer shipped 500 bushels of beans in 250 bags, 600 bushels of wheat in 280 bags; he used 136 less bags in which to ship 300 bushels of corn than he did for the wheat; he put 1200 bushels of oats in bags holding 2 bushels each; how many bags did he use for all the grain ? FIRST STEPS AMONG FIGURES. 13) 46. The income of a man who “struck oil” is $75 a day; how many teachers would this employ at $850 a year ? 47. A farmer having $1397, bought 9 tons of hay at $16 a ton, a horse for $185, 155 sheep at $4 each, and spent the rest of his money for cows at $32 each ; how many cows did he buy? |. 48. A fisherman caught 2 dozen fishes ; he sold one-half of them at 25 cents each ; the other half for 26 cents each, except one which, weighing 33 pounds, he retailed at 11 cents per pound; how much did he get for his fishes ? 49. A boy paid 100 cents for 5 quires of paper (24 sheets each) and sold it at the rate of 2 sheets for 3 cents ; did he gain or lose, and how much ? 50. How many half dimes in 350 cents? 51. A miller ground 34 bushels of wheat, 18 of corn, and 22 of oats; how many bags holding 2 bushels each, held the grain ? What did the grinding cost at 7 cents a bushel ? 52. 24 boys attended the same school, but in three different rooms ; 5 were in one room, and 8 in another, and if the number of boys in the third room be multiplied by 12, the product will equal the number of blackbirds they saw on their way to school ; how many did they see? 132 FIRST STEPS AMONG FIGURES. 53. Sound travels at the rate of 1090 feet in a second; at this rate how long would it take the report of a cannon to reach the moon, which is 240,000 miles away (1 mile is 5280 feet) ? 54. An estate of $14350 was divided between a widow and two children; the widow’s share was $5450, the son’s $1280 less than the widow’s, and the daughter had the rest; how much did the daughter have? 55. The product of two numbers is 36288, and one of them is 756; what is the other ? 56. A man bought 145 acres of land for $9,850, and 95 more acres at $45 an acre; he sold the whole at $56 an acre; did he gain or ’ lose, and: how much ? 57. A farmer bought 47 acres of land for $4,416, and 34 acres at $75 an acre; what was the average price per acre? 58. The sum of two numbers is 7568, and one of them is 784; what is the other? 59. How many military companies of 98 men each, can be formed from 7,463 men? 60. How many yards of cloth at 24 cents a yard, will pay for 26 dozen eggs at 14 cents a dozen, and a jar of butter worth 284 cents? FIRST STEPS AMONG FIGURES. 133 61. George and Lewis start from the same place at the same time, and travel in the same direction, George at the rate of 714 rods an hour, and Lewis at the rate of 579 rods an hour; how far apart are they at the end of g hours? How far apart in 7 hours, if they had traveled in opposite directions ? 62. If I receive $40 a month and spend $32 a*month, in how many years will I save $1,152? 63. Subtract the difference between 79 and 2300 from their sum. 64. What is the sum of ten thousand ninety, seven thousand nine hundred, eight million nine hundred eighteen, five hun- dred thousand, seventy thousand seventy- five, and eight hundred. 65. The dividend is 7365092, the divisor is 6978 ; what is the quotient and remainder ? 66. The remainder is 658 and the sub- trahend 1734; what is the minuend ? Pupils make and solve the following ex- amples: 67. Given a multiplicand of 4 figures, a multiplier of 3 figures, required the product ? 68. Given the minuend and remainder, find the subtrahend. 69. Given the subtrahend and the re- mainder, find the minuend. 134 FIRST STEPS AMONG FIGURES. 70. Given the sum of three numbers and two of them, find the third. 71. Given the difference between two numbers and the less number, find the greater. 72. Given the divisor, quotient and re- mainder, find the dividend. 73. Given the product of two numbers. and one of them, find the other. 74. Given the difference between two numbers and the greater number, find the less number. 75. Given whole price, number of arti- cles, find the price of a different number of articles. 76. Given the cost and selling price, find the gain. 77. Given the selling price and the loss, find the cost 78. Given the cost and the gain, find selling price. 79. Given the selling price and gain, fine cost. So Given the cost and the loss, find the selling price. FIRST STEPS AMONG FIGURES. 135 » 6745 2. 5847 3. 9337 5678 9576 4598 9867 4684 3765 6543 5967 9458 7698 8439 7695 4759 4785 4739 5978 9478 5345 9047 4567 4869 8458 8975 3765 4790 | 4687 4798 4. 8451 5. 5747 6759 9835 7846 4696 9567 8739 6976 5684 7569 7938 8427 6456 9508 8845 3753 7587 136 FIRST STEPS AMONG FIGURES. 7. 4768 8. 5869 9535 9768 9849 4352 7697 4675 4857 9538 9564 4678 5739 5765 5647 4976 9788 9738 5347 4657 8234 9876 10. 7465 es 25 4795 8649 3578 9756 6549 8732 5685 7839 FIRST STEPS AMONG FIGURES. £37 meaese . 12.99038 |) 13.9768 9764 5796 7596 3797 4579 4879 8436 8947 9087 9768 + 6896 4739 4593 . 9758 8645 8945 4563 3787 7037 9755 9568 4895 6879 4837 7537 5768 8795 4885 8597 4568 5938 4856 9756 9457 9589 | 6897 14. 6795 15. 5896 4569 9748 8769 7635 9245 5864 8876 9787 6789 4538 3954 7087 8450 4859 3789 5321 8375 4978 9999 9654 3478 7987 6457 9868 ——— ee 138 16. FIRST STEPS AMONG FIGURES. 17. 9768 4593 7846 5937 + 9876 2345 3527 8498 6549 7856 7387 9765 3849 9674 3758 6847 eee 18. 4879. 9763 4598 8756 4975 68 14 7589 4837 5692 7859 4537 6895 4576 3844 7989 4596 FIRST STEPS AMONG FIGURES. 139 19. 4532 20. 9476 7856 3869 4978 5458 9456 7507 3379" 4835 4856 8769 7995 7654 8447 8579 9568 6432 4789 4976 6435 9845 7896 5637 4967 7948 8538 8654 7689 6739 9876 4798 140 FIRST STEPS AMONG FIGURES. MULTIPLICATION TABLE. 72’sand review. a b € 8. 910 I1 12. 8 OF IO igo 9 12.8 10:7 11. 9 125350 eee 72 108 80 110 84 88 81 120 88 120 56 99 d vp é IO IL 12° 8 9Q- FO.IT 4220 One 9 12 810.7, Il Oe 12aGstO eee 9O 132 96 80 63 I10 99 144 64 64 70 121 £ h 3.5212. 8 -O ole a1 Sige 12. 12).-.0.12 “8.310% 7 ie 36 00.108 96 72 100 77 1320,R ae DIVISION TABLE. a b c 70 I2I 108 96 72 100 132 77 80 110 84 88 LO. -LIlfpsl anes 20 pete 12 II 10 1 hy or RIL: 9123-10. 1427 See ad e 120 108 72 132 90 99 56 88 120 81 96 80 I2 ‘9. 8 1110 9 8 Ii 103g 10.12 9 12: QL! 7 6) ee Z 72 48 63 I10 99 144 64 90 36 60 12 12.9. 10 11512) 65)0nig aim 64°77 "11 “9% 42-8) toa See Teachers’ Edition, p. 156. «loose (James H.) Studiesin Articulation: a study and drill-book in ; _ betic elements of the English language. 4th thvuxand. Cloth sa On the Province of. Methads of Teaching. A professional study. © ~ es ot ioe i=) oy = ie) er =) ~ Co oI 5 o 3 a - oS eF © oh & ) & ag bag -) 7 C. > et =d 5 0) — Pestalozzian Series of Arithmetics. First Year Arithmetic, Teacher's of public schools. Based upon Pestalozzi’$ method of teaching eleinen-— tary number. Boards,.16mo, i i a ee ee ee 16mo, pp. YS Depa ‘ Jackson (E. P.) Class-Record Cards. 9\'white and 10 colored cards. Johnsen’s Chart of Astronomy. On blue enamelled cloth; 40x46 inehes..... 3 Keller (C.) Me Report Cards. 2%x4 inches. Per hundred. ......... ‘The Philosophy of School Discipline. Paper, 16mo, pp. 23 Lawrence (E. (.) Recreations in Ancient Fields. Cloth, 12mo, pp. 200.. Kennedy (John.) sete nee seseeee Locke (John.) Some Thoughts concerning Education. Paper, 2imo. pp. 192. _ Maps for the Wall. New York State, revised to the present time, colored by Bilas and towns. 2'4x3 feet, mounted on cloth tse e ee see fee ee Hemispheres....¢.... The We orld, Mercator’s Projection......... Europe 3. ee AID? OT GO DD pat South AMMEVICO OCS a sive van dan Secon cee et Foes coneee eeceoeosecs . POND aslh ae dct we HEN e Ewes bay ae: FEPCH Cos eeE EHH OGe FEC HERE DAES EH EES vs LEIOR yah, o uhton he Ulan . SCCOCSHO SHH SEE OHEH SORES H EEE EH EE ES ee eoen sevens or ee a COSC OCCHHLS SOE SOHO CEHEHE HEHE ESCH ESEHH+ COREE Cr eee ne Pe seccrese Beeesaetessseieuecs eeeeccecosen —— DISECTED MAPS. United States sawn into States.... Mersveceae as an optional in the place of Geometry. Flexible Cloth, 16mo0 Cece een ese ree eee eeees SOCCER EEO EHH OH ESES OHHH ES CO8O8 ehevreceesronevresneceeseseeseve eee eeeeeeSeezreseseeecaetessoces THE SAME, 68x74 inches, cloth, the finest State map ever issued........10 United States, 8%4x5 feet, with map of the world on back. paper...... Johnston’s National Maps, 42x50 inches, cloth, colored on copper, eee Cee eepeces eetocese 5 5 5 eeeoeas 5 5 eee cess ’ CRCDS B55 cael bec cu deiboge-s oc» 0 dcuine RIETIE EEE bats aaewe. ayy en sd . pp. 150. . Northam (Henry C.) Civil Government for Common Schools. Cldth, 16mo. ; p . 185 a ee ee ee ee a ar ents ewe weer ee cece ececcecses — Pesing the Facts of American History. Cloth, Mes pp, 800..2.,. eoseoece ' eaeeceees . Pp S00 6010) 9:50 SOL e OS FOS Pee SS ee eee eee Payne (Joseph.) Lectures onthe Art of Education, with other lectures and ‘Payne (W. H.) A Short History of Education. Being a reprint of the PP e eH esate s reese eeseereaseeeerense Pete w tn eear ere esrensews sense secre eene pp. 105 Bret Canes _ Periodicals. The School Bulletin and New York State Educational Journal. —— Bound Vols. I-IX. Cloth, 200 pp., eath..r...-.2..-.c suse Bee tye —— The School Room. A Monthly Journal of Practical Help to- eoes cere eeee see ~*~ Young on eee. 2nS8ss & Sas 1 1 2 2 “5 5 5 * RURRETTSESSSSSsss ss 8 VN Sharsed Bound Volumes. -T, O, eloth, pp. 222. “Each beweesseon Ns pate be @osreeeere 1 30 Phillips (Phillip). Song Life. Illustrated. Boards, pp.176:......-ecese0+. 50 | . Pooler (Chas. 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The Regents’ Questions from the first examina- tion in 1866, to June 1882, when publication ceased. Being the Questions for the preliininary examinations for admission to the University of the State of New York, prepared by the Regents of the University, and . ~ participated in simultaneously by more than 250 academies, forming a , mar, Key to Grammar, and Spelling... Each............2+++. name eet nes - Thousand Regents’ Questions in Arithmetic, printed on 500 slips of card- © WAU Wiel DOOR 5.5. os. bs Weiss wale 8 CON Wp ek Aun ye hyd ee ee 1 00 : Grammar and Key. The 2,975 questions in Grammar, with complete _ .» Key, and references. Cloth, 16mo, pp. 198............ RS RES ES ir cede chael aoe Richardson (B. pe Learning and, Health. Paper 16m0, pp.89.....e000006 15 — )} A Workin Number. Cloth, 12mo, pp. 160........ ee cas mom Roget (P. M.) -Thesaurus of English Words. Cloth, 12mo, pp. 800...... 2 00 ‘Russell (Hattie Sanford.) Half a Hundred Songs. Boards, 12mo, pp. 103. 85 Ryan (G.W.) School Record, (18x23),112 blanks. Each set of Seven..... a) ‘Sanford (H. R.) The Word Method in Number. 5 i