^^m^. £2 m MEMOmAM H« J •Boone. I)e (Levi N.) First Steps among Figures. A Drill Book in the 'undamenlal Rules of Arithmetic,based upon the Grube Meth- common law as to the relations of the Teacher to the Pupil, the Parent, and the District. With 400 references to legal (li( i^ions in 21 dilTerent states. To which are adfU'd the 1 10(1 (jucstions iriven at the tirst seven New York Examina- tions for State Ccrtiticutes. 7th thousand. Cloth, 12mo., pp. lyS and Appendix Roderick Hume. The story of a New York Teacher. Cloth, 16mo.. pp. 295 Bulletin Blank Speller. Designed by Principal H. B. Buckham. Buffalo Normal School. Boards.SJxTi, round corners, pp. 40 Book-Keejnng lilanks. Day- Book, Journal, Ledger. Cash Book. Sales Book. In sets or singly. Press board, 7x8^, pp. 28 Compmtion Book. Designed by Principal H. B. Buckham, Buffalo Normal School. Manilla, 7x9, pp. 34 ClasH Register. Desiraed by Edward Smith, Superintendent of Schools, Syracuse, N. Y. Press board covers, T\co Sizct. {a) 0x7, for terms of twenty weeks, (b) 5x7, for terms of fourteen weeks. W/ien not othencise specijied the smaller size is always Rent. Pp.48 School Ruler, marked on one side by inches and metres, and containing on the back an immense amount of condensed sta- tistical information. Tico Styles, (a) Manilla, 12 inch, (6) Card- board, 6 inch. Each 3 cts. Per hundred Colored Crayon, for Blackboard, per box of one dozen, nine different colors— Red, orange, yellow, green, lake, brown, light brown, blue, lilac Common School Thermometer, in box, post-paid Cooke (Sidney G.) Politics and Schools. Paper, 8vo., pp. 23 Craig (Asa H.) The Question Book, A general review of Com- mon School Studies, to be used in schools in connection with text-books. Invaluable to teachers as a means of giving a Normal Training. 42d Thousand, Cloth, 12mo., pp. 340.^.. De Graft' (E. Y.) PRACTICAL PHONICS. A comprehensive study of Pronunciation, forming a complete guide to the study of the elementary sounds of the English •Language, and con- taining 3000 words of difticult pronunciation, with diacritical marks according to Webster's Dictionary. Cloth, 12rao., l>p. 108..... .-. POCKET PRONUNCIATION BOOK, 'containing the 3000 words of dillicult pronunciation, with diacritical marksaccord- ing to Webster's Dictionary. Manilla, Itimo., pp. 47 7 he School Rt>oin Gvide, embodying the instruction given by tlie author at Teachers' Institutes in New York and other States, and especially intended to assist Public School Teach- ers in the practical work of the school room, lenth Edition, with many additions and corrections. Clotii. 12mo., pp. 449.. The Stuff Budfjet. A collection of Songs and !Music for schools and educational gatherings. Paper, small 4lo., pp. 72 -. Digitized by tlie Internet Archive in 2008 witli funding from IVIicrosoft Corporation littp://www.archive.org/details/firststepsamongfOObeebricli FIRST STEPS Among Figures. A Drill Book in the Fundamental Rules of Arithmetic. TEACHERS' Edition. BY LEVI N. BEEBE, CANANDAIGUA, N. Y. Sixth BsmoN, SKLAResD and CABBrt7Li.T Rxyissd. SYRACUSE, N. Y. : C. W. Bardeen, Publisher. 1881. B4- Ck)pyright, 1877, by L»ti N. Binn. EDUCATION DEFT- Oca PREFACE TO TEACHERS* EDITION. In putting this work before the public the author disclaims any ambitious schemes or *• great 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 mind, rather than to deal with the philosophy of each operation, leaving any teacher who believes that no step should be taken unless the pupil untlcrstands 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. f,4!.<45 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. 5 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. 'i'he 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 full 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- difficult also. By the old method a pupil frequently acquires the habit of saying the 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 9's and review. In the given lines of figures there is one more figure in the upper line than in the lower one: 3456789 456789 If the upper line be written several times and the lower line in the same way as follows : 34567893456789 4567 8 9 4 5 6 7 8 9 4 5 3456 6789 7 4 8 5 9 6 3 4 5 789 6 4 7 8 9 5 6 7 3456 8945 7 6 8 7 9 8 3 4 5 9 4 5 6 6 7 8 9 7 8 9 The first 9 in the lower line : comes one place before the 9 in the upp€ IT line ; the second 9 in the lower line comes two places before the second 9 in the upper line, and so on until it has been under every figure in the upper line. If written farther, the series wil) be repeated as shown above, where 4 and 5 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, &€., or 4 times 3 are 12, 5 times 4 are 20, &c. 6 9 5 8 4 7 3 6 9 5 8 4 7 4 5 6 7 8 9 4 5 6 7 8 9 4 6 9 5 8 4 7 3 6 9 5 8 4 7 6 7 8 9 4 5 6 7 8 9 4 5 6 6 9 5 8 4 7 3 6 9 5 8 4 7 8 9 4 5 6 7 8 9 4 5 6 7 8 PREFACE TO TEACHERS EDITION. ^ 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 3 5 3 7 3 9 which, like the other series, contains every combination between 4 and 3 and 9 and 9 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 ot the numbers above, thus : 10 14 II 15 12 16 7 II 15 12 16 13. &c. 69584736958473 &c. 456789456789 &c. Then copy for the minuends the upper num- bers, and for the subtrahends the lower ones and the series becomes 10 14 II 15 12 16 7 II 15 12 16 13 &c. 456789456789 &c. In this book those for subtraction have been still further disarranged so that the results 8 PREFACE TO TEACHERS' EDITION. will not be 6958473 and so on 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 10 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 ot 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 9— that is 2 and 9, 9 times 2, &c., but only to combinations of 2 3 4 5 and 6 with 2345 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 thje 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. BEEB& Canandaigua, N. Y., July, 1877. FIRST STEPS AMONG FIGURES. PART I. ONE. (See Appendix, page 185.) I. Be sure the pupil has the idea of one thing in distinction from two 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 ^// 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 pupil place the article, and thus impress his mind more thor- oughly with the idea once i is i, written i x i • *In this book the multiplier is uniformly placed on the right of the sign X : thus z times one are 2 will be written I X 2 = 1. See Appendix, pp. 185-192. lO 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 limes 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, i in 2 twice. * Schedule: T X 1 = 1. (Read once one is one.) 1-^-1 = 1. (Read one in one, once.) What can you find once in the school room, in your pocket, on your face, at home t &c. What is there that moves on one wheel ? TWO. 3. Teach in counting that the second of two things is not of itself Hvo, hxiXmie. In teaching number and in operations on numbers use objects for some time — at least *Thc schedules being written on the blackboiard, the pupils are to be taught to read them, and eventually to make them diemselves. FIRST STEPS AMONG FIGURES. 1 1 three months to six months, until the pupil is thoroughly familiar with the composition of numbers. 4. Teach pupils to count to ii and continue to teach counting daily until the pupil can count 100. Schedule : 1 + 1 = 2. (Read one and one are two.) 1 X 2 = 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 nrjore than what number } 1 is one less than what number .? 2 is the double of what number? 2 is twice what number? I is one-half of what number? I and I are* ? i from 2 leaves ? (i from 2 leaves i, because i and i 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 ? •The teacher will supply the words " what Qumber '* or " how much " in such examples, according to the sense. Show the pupils that i block placed on the table, and then another, make 2 blocks there, hence 2 is 2 times I. 12 FIRST STEPS AMONG FFGURES. 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 i 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 i 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 J of a stick of candy, * of 2 sticks, &c ) 6. 2 is the double of what number? Of what number is i one-half? What number must I double to get 2 ? I know a number that is i more than i ; what number is it ? 7. What number must be added to i to get 2 ? Fred had 2 dimes and bought peaches with I dime. How many dimes had he left ? (No analysis of these examples is expected; simply a prompt answer.) FIRST STEPS AMONG FIGURES. A slate pencil costs i 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 i-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 i cent a pint? THREE. Schedule : 9. Measuring by i. I I I 3. / I i + i -1-1=3. ^i iX3=3- [=1. /j_ 3 — 1-1 = 1, for 3 — 1 = 2 and 2 — I s 3-^1=3. ♦In such examples it will interest the class to use tkti* names instead of those given. 14 FIRST STEPS AMONG FIGURES. Measuring by 2. I I 2 2 + 1=3. II 2 X I + I =3. (To be read once 2 III 3 and 1 are 3, or once 2 plus i are 3.) 3—2 = 1,3 — 1 = 2, (To be read 2 from 3 leaves i and i from 3 leaves 2.) 3-1-2 = 1 (and I rem.) (To be read 2 in 3 once and i re- mainder.) The pupils should read these schedules many times each, until they are familiar with the lan- guage. 10. 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 i 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. 3-i-i=? iX2=? 3 — 1 + 1=? I-fl=? 1+2= ? 2|2=? FIRST STEPS AMONG FIGURES. 15 3 — 2=? 3-1=? 2 — 2=? 2+I=? 2+1+1=? IX2+I=? 2-J-2=? 3^2=? 3-^3=? 12. 3 is I more than? 1 is I less than ? 3 is 2 more than ? 2 is I less than ? 2 is I more than ? I 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 i block are 2 blocks ; also 2 times i orange are 2 oranges, and 2 times i pencil are 2 pencils, &c. 2 times any one thing are two of those things. 2 times I are 2. Show the pupils that i block taken 3 times or placed on a table 3 times makes 3 blocks there, hence 3 is 3 times i. 14. This form of illustration may be used foi any multiplication. How many pints in a quart? Teach pupils to write numbers as high as 20. l6 FIRST STEPS AMONG FIGURES. It may be well to teach the writing of 12 before 10 or II. Show the pupil by the abacus or otherwise 12 objects and show him that they are i ten and 2 ones. Show him that we can- not write 12 by any one of our figures ; then teach him about ten's place and one's place. Do not use the word units for several weeks yet. 15. Teach pupils to count by 2's from 2 to 6 and back to 2, thus : 2, 4, 6. 6, 4, 2. Explain that i and i are equal numbers, that is equal to each other ; i and 2 as well as 2 and 3 are unequal numbers. 16. Give the pupils much practice in exam- ples like the following: 3 — i — i + i? To be read, how many are 3 less i less i plus i ? or 3, subtract i, subtract i, add i ; or 3 minus i minus i plus i.* *Thesc arc to be read by the teacher, thus: 3, add a» subtract 1, divide by 1, 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 j 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 E. g. teacher, 3, add 2. 1st pupil says •' 5." Teacher, FIRST STEPS AMONG FIGURES. 17 2—1+2 — 1X1=? 1-1-2 — 1 — 1X2 + 1=1 2+1—2X3—1+1=? 1+1X1—1+2—1=1 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 10. What number is twice i 1 18. I write a number once, and again, to get 2 ; what number did I write twice ? How many cents must you have to buy a 3cent 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 i line in his primer, and his "subtract 1." ad pupil says "4." Teacher, "divide by 2." 3d pupil " a." 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 * would mean, take 2 x * from 3, but it is to be read : 3 subtract 2, multiply by 2. 2 1 8 FIRST STEPS AMONG FIGURES. sister learned i line more than he did ; how many did she learn? If I slate pencil cost i cent what will 3 slate pencils cost ? Anna found 3 roses in the garden; how can she divide them between her father and mother 1 Can she give them an equal number ? How maqy roses must she have had in order to give her father i, and her mother i also ? Count by 2's from from 2 to 12. The counting may be sometimes in concert, oftener ist pupil say 2 ; 2d pupil, 4 ; 3d, 6 and so on, and perhaps oftenest one pupil give the whole series. Teach pupils to count ist, 2d, 3d, &c. FOUR. 20. Measuring by i. I I I I 4. /i 14-1 + 1 + 1=4. (Because i + i=a, 2 + i=3» 3 + 1=4-) / I 1 X4=4. /i 4 — I — I — 1 = 1, or 4— I — I — I — 1=0. / i_ 4-M=4. S 4 FIRST STEPS AMONG FIGURES. 19 Measuring by 2. 112. 2 + 2=4. 2 X 2=4. 112. 4 — 2 = 2, or 4 — 2— 2=a I III 4. 4-^2 = 2. Measuring by 3. Ill 3. 3 + 1=4, 1+3=4- 3 + 1 + 1=5- !_ i: 4-3 = 1^4-1=3- nil 4. 4-+3 = i (and i remainder.) 21. 2 X I + 2= ? 3 — 2=? 4x1 = 3-1 = 2 + 2 = 3-^2 = 4-2 = 1x3 = 4-7-2 = 3-i=? 2 + I=? 2X2=? 4-^3 = 3-^2 = 3 + 1 = 4-3 = 4-1 = 4-4=? 2 X 1 + 1= ? 3x1+1=? 4 — 2 — 2= ? 4-f-4=? 22. Name animals with 4 legs ; with 2 legs. Name wagons and vehicles with i 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. 4 is I more than ? I is I less than ? 20 FIRST STEPS AMONG FIGURES. 2 is I more than ? 3 is I less than ? I is I more than ? 1 is 2 less than? 2 is I less than ? 2 is 2 more than ? 2 is 2 less than? 4 is 2 more than ? I is I more than ? (Nothing.) 4 is 4 times ? 24. Solve rapidly the following: 2x2—3+2x1+1—2x2=? 4— I — I + I + I— 3 = how many less than 4? 3"-2 + 3— I — I X 2— i=how many times i ? 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. 1x2 — I X 3 — 2 — 2 = how many less Ihan 3 ? 3—2+1 X2— 1—2+1= ? 4—2—1 X3— 1X2— I=? 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? W^hat number is 2 more than i ? What number must I add to 2 to get 4? 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 i of ihem withered, the second 3 213 41 2 342 341 ' 432 1 * Leave on the board the previous day's lesson in seriea and add to it as much morf ofthc 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 tiumber 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 1 Count by 3's from i to 40. I write a number four limes JUid add. I get 8, what is the number ? Henry had half of 8 cents. How manv had be? 8 cents will buy how many l-cent stamps ? 4-cent stamps (I. Revenue)? 2-cenl starapsf 5cent stamps ? 3 cent stamps ? 3 lemons is * of how many lemons ? Lewis brought 6 eggs from the barn ; he broke half of them. How many whole ones were left? 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 S>T. PS AMONG FIGURES. 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 bou> 60. " *• 2'S " I " 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 i 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. 91017. 2. 86700. 3- 90007. 4. 14071. 5- 70000. 6. 50010. 7- 38419- 8. 74058. 9. 60800. 10. 16040. II. 3000. 12. 7014. 13- 10061. 14. 3020. 15. 7003. 16. 8500. 17- 17500. 18. 3540. 19- 67374. 20. 86000. FIRST STEPS AMONG FIGURES. 6f What 2 equal numbers and i unequal num- ber will make 7 ? (3 answers.) What 3 equal numbers and i 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. I. 7.300. 2. 5,006. 3. 2,050. 4. 10,091. 5 8,016. 6. 4,000. 7. 12,090. 8. 50,700. 9.65,078. 10.45,913. 11.80000. 12.80010. 13. 15,061. 14.40.002. 15. 79,500. 16.81,018. 17. 30,600. 18. 60060. 19. 90,004. 20. 8,050. 21. 75,000. 22. 74.695. 23.31,280. 24. 13.300. 25. 14 041. 26. 10,010. Teach Roman notation to C. Write in Roman i. 64. 2. 49. 3. 97. 4. 76. Write in Arabic 5. XIX. 6. LXXXIV. 7. XLI. 8. XXVII. 9. XVIII. 10. XXIX. Write in Roman 11. 17. 12. 56. 13. 83. 14. 49. 62 FIRST STEPS AMONG FIGURES. Write in Arabic, 15. LXVIII. 16. XCII. 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 i, 2, &c., of the Pupils' Edition, doing it in connection with this work and thus canying on that work together with the following work. TAf teacher should how 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 9/ the pages in each. FIRST STEPS AMONG FIGURES. 63 PART II, 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 c d e f g 352 413 524 135 241 352 4 I 234 523 452 345 234 523 4 5 For subtraction. a b c d e f g 7410 746 758 693 696 857 85 32 5 432 523 452 345 432 5 4 For division. a b c d e 41510 6165 20 92 20815 610 8 4 5 2 3 45 4 32 54 3 2 5 « f g 31225 4 12 3 4 5 2 3 See Pupils' Edition, p. 5. 64 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 only 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 t 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 ? 9. Henry walked 4 miles before dinner, and 4 after dinner ; how far did he walk that day ? 10. 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 iheir 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 niany 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? Sec P. i:«l., p. 9. 06 FIRST STEPS AMONG FIGURES. 4. James bought a paper for 5 cents ; he gave the newsboy 10 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 wiih ? 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 } 10. 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. Waller 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 9 inches long ; he cut off 3 inches of it. How long was the stick then ? 16. A farmer having 8 turkeys, sold 4 o^ 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 in' a machine ; how many had he left ? 19. Silas had 3 marbles in one pocket and 5 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 V. Ld., p. 11. 68 FIRST STEPS AMONG FIGURES. them were broken ; how many whole ones were there ? 26. Ellen's father gave her 9 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 9 chickens 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 slicks of candy , he ate 3 of them and gave away the rest. How many did he give away 1 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 i to 17. Copy the following examples one at a time on the blackboard ; require a pupil to point one off into oeriods 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. 90016. 2- 45378. 3 461340. 4. 908714. 5. 876341. 6. 608790. 7. 379000. 8. 75608. 9. 40713 10. 100740. II. 98716. 12. 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. 7. 3,761. 8. 10,010. 9. 40.070. 10. 73,801. II. 36,000. 12. 90,090. 13. 100,100. 14. 702,940. 15. 900,070. 16. 816,902. 17. 49»049- 18. 860,705. 19. 461,017. 20. 791,486.21. 21. 10020. Write these, or similar numbers on the board and require the pupils to read them. Write in Roman the following : 1. 79. 2. 96. 3. no. 4. 47. 5- '9- 6. 134. See P. Ed., p. 26. yo FIRST STEPS AMONG FIGURES. Write in Arabic : 7. CLx. 8. xcii. 9. cxa lo. CXLIV. II. CLXXXVI. 12. CIX. Write in Roman : 13. 192. 14. 136. 15. 168. 16. 154. A short method of adding equal numbers is called Multiplication; 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,_;fz^r pencils will cost 5 times 4 cents, or 20 cents. 2. If I 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. How many wheels have 3 wagons \ FIRST STEPS AMONG FIGURES. Jl 8. A lady gave 3 little girls 5 bunches of grapes each ; how many bunches did she give them ail ? 9. How many feet have 4 hens. 10. 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 i to 29. For rapid solving. See P. Ed., p. 19. 72 FIRST STEPS AMONG FFGURES. i2-r-4 + 3+3+3-f-4 + 4-l-4-+-4 + 3-+"3 + 3 + 2 4-2= ? Ans. 41. 3-1-4 + 4 + 3+3 + 3 +3 -'-2 + 2+2+3+3 + 2 +3+4= ? Ans. 44. 3 + 3 + 2 + 2 + 2 + 3+3+3 +2+3 + 2 + 2 + 3+3 + 3=? Ans. 39 3+4 + 4 + 4 + 31-3 - 2 +2+4 + ^ + 3 + 2 f 2 r 2— ? Ans. 41. 3+3+3+3 + 2+2 + 2 + 2 +3+3 + 2+3 + 3+4=? Ans. 37. 3 + 2 + 2 + 2+3+3+ 2+ 2 +1+3 -f- 2 + 3 + 3 + 1=? Ans. 32. 4+3 + 4+4— 3+4-3-2 + 4 + 4-3+2 + 3= ? Ans. 21. a + 3 + 4 + 4 + 4-3 + 2 -3 - 2-3 + 4 + 2 + 3= ? Ans. 17. a+4 + 3+4 + 3 -4-3 -4 + 3+3+4 + 4 + 3= ? Ans. 22. 5 + 3-2-4 + 3+3 + 2+4 + 3+4 + 3-2 —3= ? Ans. 19. 3+4 + 2+4 + 2 + 2—4—4+3 — 2—4 + 3 + 4=? Ans. 13. 3 + 2+4-3-2+3 +3+3+3 + 4-3-2 —4 + 3 + 2=1 Ans. 16. 4+3-«-2+3 + 3+4+3 -4-4 + 1-3 + 4 —3-4-4=? Ans. 5. FIRST STEPS AMONG FIGURES. 73 20. What cost 5 marbles at 4 cents each 1 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? B'inding how many limes one number is contained in another is called Divis'on. 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 i 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 FIGURES. 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 I doll cost 3 shillings, how many such dolls can be bought for 12 shillings ? 7. 10 shillings will buy how many knives at 2 shillings each ? 8. How many balls at 3 shillings each may be bought for 6 shillings ? 9. When pears are 2 cents each, how many can you buy for 8 cents ? Count by 4's from i to 61. « u ^^'s «t 2 " 62. " " 4's " 4 " 60. « « 4's " 3 " 63. 10. \Vhat 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 i 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 9 shillings ? 14. 20 shillings will buy how many purses at 5 shillings each ? FIRST STEPS AMONG FIGURES. 75 15. Henry earned 9 cents on Monday and ^ cents on Tuesday ; how many cents more did he earn on Monday than on Tuesday ? 16. Louisa had 5 cents and she found 5 cents more ; how many cents had she then ? 17. If one sled cost 5 shillings how many sleds can you buy for 20 shillings? 1 8. What cost 4 vests at $8 each ? 19. How many neckties at 3 shillings 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 1 ( : c I e 3624 2345 1362 6345 4154 6234 6241 5623 5536 2456 f 2415 2345 g 362 623 h 3 1 5 45 6 For subtraction. a b c d 9 7 6 10 5634 7 II 7 8 3 10 8 6 5 543 2 661 4798 3245 e 12 4 : 6 22 1 1 ; 6 68 5 4 f 58 3 2 BCP. 9 6 Ed 1 I >] 05 54 3. 28, 5 2 h 9 3 6 4 76 FIRST STEPS AMONG FIGURES. For division. a b c 30 12 8 3 10 20 1 s 36 12 15 2 24 5 6 2 3 2 4 5 6 4 3 2 6 d e f 4 12 4 25 18 12 6 16 5 30 6 18 2 3 4 5 6 2 3 4 5 6 2 3 g h 8 20 6 9 24 10 456 3 4 S <0 (2) (3) (4) (5) (6) 32 33 2 23 33 31 13 21 32 33 31 23 3 2 23 31 23 33 21 30 33 12 13 23 13 13 21 22 22 32 33 32 23 31 32 »3. "S 131 134 152 154 155 {7) (8) (9) (10) (II) (12) 30 23 32 3 31 23 23 33 30 23 33 32 33 30 23 31 23 13 12 22 3 23 30 33 31 31 31 33 22 21 23 33 23 32 31 32 33 22 33 21 33 23 '85 194 17s 166 203 177 See P. Ed ., p. 16. FIRST STEPS AMONG FIGURES. 77 13. Add 32, 21, 3, 33, 23, 31, 23, 33. 14. Add S3^ 23, 30, 21, 2, S2, 23, 22. 15. Add 2, 33, 22, 31, 23, 32, 12, 23. 16. Add 31, 30, 23, s:^, 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. 3-f4-f4 + 3 + 3 + 2+4 + 3+3 + 2+4+4 + 3 ? Ans. 42. 2+3+4+3+3+5+5+3+3+4+4+4+ 3 + 3 + 2 ? Ans. 51. 3+4+4+3+3+4+4+5+5+5+3+2+ 4 + 4 + 4? Ans. 57. 3 + 4 + 3+3+3 + 2+2 + 3 +4 + 2 + 4 + 3 + 2 + 3 + 5? Ans. 46. 5+*3 + 4 + 2 + 3+4 + 3 + 3+ 2 +4+ 3 +4 + 3 + 3+4? Ans. 50. 4 + 3 + 4 + 3-2-2 + 5-3+ 2 +4? Ans. 18. 2 +4-+3X 5-4x4 -3-3-^3 +6-^4x6 + 4 + 3 — 2? Ans. 23. 3 + 6 + 4 + 4 + 4 + 3+4-6-4-2-5 + 3 + 4—4.^ Ans. 14. 3 + 5 + 2X5-3 + 4-2-4H-5 X6 + 4 + 3 ■+5 + 3 + 3^ Ans. 11. 78 FIRST STEPS AMONG FIGURES. 2 + 5 + 4+1-^4+1 X5-4-I- 3 4 4+1-^4 + 5 + 2+4 + 3? Ans. 20. 3 + 6 + 4-1 -3x6-3-4-3 +4^6x5 + 4 + 3+4 + 3? Ans. 29. 5 + 6—3+4 + 3+3 -^3 + 4 + 3+4 + 4-2 — 3+4 + 3 + 4? Ans. 27. 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 ? 3. What cost 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 boys 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 9. A lazy boy brought his mother 3 sticks of wood at one time and 4 at another ; how many sticks did he bring her ? 10. Six boys can sit on this seat, how many boys can sit on four such seats ? 1 1. 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 i8 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? 8o 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 i orange cost ? Solution : If 2 oranges cost 12 cents, i orange will cost i of 12 cents, or 6 cents; or I orange will cost i 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, i of them will cost J of that .sum ; if 3 cost that sum, i of them will cost 1-3 of it; if 5 cost any sum, i 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 J of 12 apples (or marbles or pencils) they may be placed in 2 equal piles, and they will find that ^ of 12 apples is 6 apples. Show them that to get i of 12 pencils, they may be placed in 3 piles, and that i of 12 is 4. Show in the same way that i of 12 is 3. When this is thoroughly understood, show them that i of 12 may be obtained by dividing 12 by 2 — the result in each case being 6 ; show that i FIRST STEPS AMONG FIGURES. 8 1 of 12 may be obtained by dividing 12 by 3 ; teach i of 12 in like manner. Illustrate also by J of 6 and i of 6 and i of 8. Then teach in general terms that i of any number may be obtained by dividing the num- ber by 2 ; i, by dividing by 3 ; }, by 7, &c. 22. What cost I pear if 4 pears cost 8 cents? 23. If 3 knives cost 15 shillings what will i knife cost ? 24. If 2 pencils cost 16 cents, what will i pencil cost ? 25. If 4 stools have 12 legs, how many legs will I stool have ? 26. If 6 boys earn 18 cents, how many does I boy earn ? 27. If 5 cords of wood cost $25, what will i cord cost ? 28. How many pounds in i 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 i lemon ? 31. How many pounds of butter will last a family i week if they use 12 pounds in 4 weeks 1 32. How many days will 18 apples last a boy who eats 3 apples each day ? 6 82 FIRST STEPS AMONG FIGURES. 33. A blacksmith shod 5 horses each day ; how many did he shoe in 6 days 1 34. 6 boys are skating on the ice, and 4 lx>ys 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 i 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, what will i marble cost ? 40. What cost I apple if 5 apples cost 10 cents ? 41. How many dolls at 4 shillings can you buy for 20 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 i to 61. FIRST STEPS AMONG FIGURES. 83 Count by 4's from i to 6i. Count by 4's from 2 to 62. Count by 3's from 3 to 60. Count by 3's from i to 61. Count by 3's from 2 to 62. To be read by the teacher. Write in Arabic : I. 7,050. 2. 10,003. 3- 40,300. 4. 5.209. 5- 10,010. 6. 300,040. 7- 9,610. 8. 4,316. 9- 215,000. 10. 80,090. II. 600,000. 12. 809,740. 13- 100,010. 14. 916,008. 15. 835.941. 16. 70,000. 17. 90,005. 18. 5,016. 19. 213,033. 20. 30,000. Write these or similar numbers on the black- board and require pupils to read them. For rapid solving. 6 + 4 + 3 + 24-4 + 4-3-^4x6—3-3 + 4 + 4? Ans. 32. 3 + 5 + 4 + 3 + 3+4 + 3+-5X4-4 + 3+3 -4 + 3 + 5+5? Ans. 31. 4+3+4+4+4+2+3+4+4+2+5+2 + 3 + 3+4-^ Ans. 53. 3 + 2+4 + 3+44-3 + 2+3+3+4 + 3 + 2 + 3+3 + 2 1 Ans. 44. 2+3+3+2+4+4+3+2+3+3+4+4 + 2 + 3 + 3+4 + 3^ Ans. 52. 84 FIRST STEPS AMONG FIGURES. 3+2+4+3+5+3+2+4+4+1+2+4 + I+3 + I + 2? Ans. 44. 4+3+3+4+2+3+2+3+3+3+2+4 + I + 2 ? Ans. 39. 3+2+4+4+3+3+2+2+3+4+2+3 + 3 + 2 + 1? Ans. 41. 17 + 4 + 3-^6 + 6 + 5-3 + 3x6 + 3-4 — 3 + 5x3* Ans. 12. 9 + 4 + 3 + 2^3><5-4— 3-2 + 4— 1-^6 X3+3 + 2 ? Ans. 17. 16 + 4 + 3 + 3+3 + 2 — 1-^5 +3+3^3 X 5-3+4-1-^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. (,.) (2) (3) (4) (5) (6) (7) (8) (9) 2I22I2I2I 1 2 I22I2 12 2 2 II222 22 2 I 22122 22 1 2 12222 22 2 2 22I2I 21 I 2 22222 22 I 12222 12 II I I II 21 13 12 16 14 16 15 16 15 FIRST STEPS AMONG FIGURES. 8j (io)(.. )(") (I3)(«4)(i5)(i6)(.7)(i8)(i9) 2 I 2 2 2 2 I I 2 I I 2 I I 2 I 2 2 2 2 2 2 2 2 2 I 2 2 I 2 2 2 2 2 I 2 I 2 2 I 2 2 I I I 2 2 I 2 I 2 2 2 2 I I 2 2 2 I 2 2 I T I I 2 2 2 I I 2 2 I 2 I I 1 2 2 2 2 2 2 I 2 2 2 2 2 I I 2 2 3 2 2 2 2 I I I I 2 a 2 2 I 2 2 I 2 I I I I I I 2 2 I I i8 i8 21 22 22 21 24 20 19 19 (20) [21) (22 ) (23) (24) (25) 2 I I 2 2 I 2 2 I I 3 2 2 2 2 2 2 I 2 2 2 I 2 2 I I I I I I I 2 2 I 2 2 I 2 I 2 2 I 2 2 2 2 2 2 2 I a 2 I I I 2 I 2 I I I 2 I I 20 22 20 19 19 22 86 FIRST STEPS AMONG FIGURES. (26) (27) (28) (29) (30) (31) (32) 2 2 I 3 3 I I 3 I 2 3 X 3 3 9 I I I 3 3 3 3' I I 3 3 I 2 I I 3 3 9 9 X 3 I 2 3 3 I 2 I I 2 3 3 3 a 3 2 2 ». I I I — 3 2 94 21 24 91 25 26 FIRST STEPS AMONG FIGURES. 87 (33) (34) (35) (36) (37) (38) (39) (40) I 2 I 3 I 3 I 3 3 I 2 I 2 I 3 I 3 2 I 3 I 3 I I 2 I I I 2 3 I I I I 2 I 3 2 I 3 3 I I 3 3 I 3 3 I 2 2 2 I 3 3 I I 2 3 I 3 2 I 3 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 3 2 I I I I I 2 I 2 I I I 3 3 I 2 I 2 2 I 2 3 2 I 2 I I 3 I I 2 3 I 2 2 3 3 2 2 I 2 I I 3 I I — — — 23 — _ _ 23 26 24 23 27 27 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 all again, and so on. 88 FIRST STEPS AMONG FIGURES. 41) (42) (43) (44) (45) (46) (47) (48) (49) « 23 30 13 32 23 32 33 13 3 32 23 21 13 33 20 21 21 1 12 31 33 32 30 33 13 33 2 21 12 10 21 12 23 23 23 J 30 23 32 33 33 31 32 32 3 12 32 13 20 22 23 33 U 3 3» 33 22 32 31 32 22 22 2 23 21 31 23 13 II 12 31 3 33 20 23 33 s^ 3^ 31 20 20 217 225 198 239 230 238 220 208 (so) ($■) (5^) (S3) (S4) (SS) (S6) (S7) 32 21 31 23 30 23 33 23 21 33 23 31 23 31 21 3f 33 20 32 33 10 23 32 23 23 13 20 30 22 30 13 32 12 32 13 22 31 22 30 33 31 33 31 13 33 13 ^^ 12 33 21 23 31 23 32 33 31 23 13 32 23 12 21 21 23 30 32 13 30 33 33 13 32 23 31 23 23 22 23 32 33 13 33 33 33 32 32 33 23 274 282 274 292 271 283 283 296 See P. Ed., p. 33. FIRST STEPS AMONG FIGURES. 89 (S8) (59) (60) (6.) (6--) ;63) (64) (65) (66) 21 33 23 3' 33 »3 32 21 33 33 23 31 23 21 22 13 33 22 20 31 13 33 13 31 31 32 31 '3 20 32 20 30 33 23 23 23 32 32 .20 13 23 21 33 30 31 23 13 33 32 32 13 21 22 12 32 33 2 1 23 II 32 13 33 23 21 21 13 12 33 33 10 12 32 13 32 32 22 23 20 32 32 3» 32 12 22 33 32 13 31 23 23 23 33 30 30 33 32 23 31 13 31 21 U 21 21 23 32 13 33 12 23 32 33 33 31 31 23 32 306 327 315 326 338 317 325 328 339 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. 90 FIRST STEPS AMONG FIGURES. If the pupils have not got their books these examples may be written on the board or reacJ to pupils to solve at their seats, a few daily : (>) (2) (3) (4) (5) (6) 231 321 333 331 ^32 135 322 233 223 223 223 332 122 213 3'2 "3 331 125 333 332 233 322 233 322 212 322 213 232 213 331 323 123 322 13' 332 215 332 332 132 303 213 321 221 233 323 233 233 33 2,096 2,109 2,091 1,888 1,910 i,8c8 (7) ^ (8) (9) (10) (") (12) 33 313 213 123 221 ^31 203 232 2 331 323 333 332 231 321 322 332 123 123 323 233 233 121 232 333 333 332 123 203 313 212 232 23 312 323 233 332 321 231 233 232 321 323 233 133 331 ^23 335 223 133 213 223 333 231 331 312 321 312 312 312 2,445 2,663 2,022 2,543 2,523 2,663 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. 9 1 Examples in subtraction. 13- 69758 M- 97856 15- 69587 16. 68059 6035 74230 20152 604a 63.723 23,626 49»435 62,019 [7. 75860 18. 79685 19. 58796 2C. 96807 2330 4242 25062 4200 73»530 75^443 33,734 92,607 Multiplication. 21. 32032 22. 23103 23. 24130 24. 31402 2322 64,064 69,309 48,260 62,804 25. 23103 26. 14023 27. 32023 28. 20312 2 2 3 3 46,206 28,046 96,069 60,936 29. 40312 30. 31203 2 3 80,624 93.609 ^2 FIRST STEPS AMONG FIGURES. Division. 31. 2)48206 32. 3)90396 33. 3)69306 24.103 30,132 23»I02 34. 2)28460 35- 3)39069 36. '. 2)60482 I4»: 230 13,023 30.241 37. 2)84^ )02 38. 3)30960 39- : 5)93600 42,301 10,320 31,200 (0 (2) (3) (4) (5) (6) 324 434 23 43 342 431 431 431 431 324 434 343 243 444 344 442 44 242 432 213 244 243 321 434 444 332 432 134 443 244 342 441 421 424 2 42 1 232 424 444 ^33 3H 314 2^48 2,719 2,339 I '^43 i»90o 2,429 See P. Ed., p. 38. FIRST STEPS AMONG FIGURES. 93 (7) (8) (9) (10) 00 (12) (i3> 424 31 422 431 21 34 414 434 42 344 323 44 44 445 342 44 142 444 14 21 302 241 4 433 444 23 30 244 424 31 424 123 42 44 431 332 43 344 334 44 23 444 2,197 ^95 2,109 2,099 188 196 2,27* (14) (15) (16) 07) (18) (19) (20) 214 44 44 342 244 21 23 424 213 23 444 434 34 41 444 321 32 32 423 43 34 123 444 41 421 342 21 44 434 3 44 344 231 44 3' 341 432 31 432 444 32 23 344 424 22 444 234 43 42 2,324 1 ,88 1 237 2,459 2,352 238 238 (21) (22) (23) (24) (25) (26) (27) 444 21 423 432 34 442 24 344 43 444 444 42 434 43 213 24 144 324 33 324 21 422 44 231 131 23 413 4 343 32 432 423 44 32 32 214 24 344 442 3 424 20 444 33 432 343 22 321 44 321 44 123 414 31 444 23 2,745 265 2,573 2,953 232 2,834 2 94 FIRST STEPS AMONG FIGURES. (28) (29) (30) (31) (32) {33) (34) 243 21 44 423 341 44 34 444 432 21 341 444 21 42 231 344 34 444 34 4 44 422 211 43 332 432 44 13 444 321 43 213 121 32 22 233 432 31 441 343 12 41 423 343 24 324 444 33 44 344 444 32 432 434 14 34 212 323 43 344 232 43 31 440 342 44 443 341 44 42 3,436 3,213 359 3,737 3,166 291 347 (35) (36) (37) (38) (39) (40) (41) 343 423 14 314 232 21 32 444 442 3 432 443 42 443 321 341 41 341 444 43 344 432 234 34 134 324 14 214 344 434 42 244 431 44 423 344 341 44 441 343 21 4 234 423 34 342 234 13 234 421 412 31 444 444 44 342 432 344 23 233 323 42 444 444 431 42 324 242 33 431 3»759 3^825 308 3,249 3,460 317 2,9 See P. Ed., p. 42 II oiiii^'iE^.. FIRST STEPS AMONG FIGURES. 95 (42) (43) (44) (45) (46) 432 343 231 424 344 243 412 424 243 432 234 34 41 4 23 42 13 24 44 33 21 43 341 423 444 34 123 442 344 431 233 342 444 24 43 32 44 4 33 24 41 43 34 42 234 421 444 324 413 442 134 224 442 431 344 (47) (48) 32 231 44 443 43 422 24 31 44 43 24 31 44 13 334 244 413 423 341 444 413 244 3,762 322 3,601 364 3,853 373 3,952 For addition and multiplication. (7 and rev.) e 7 3 5 47 3 5 345 6 b 47 3 456 f 2647 5673 For subtraction, b c 264 34 5 II 9 8 12 6745 10 5 9 436 d 3 5 2 7 3 4 h 665 7 3 7 7 3 10 7 S3 II 8 45 12 9 13 8 7376 c [069 7 6345 g 9 8 10 53 7 12 6 107 34 13 6 96 FIRST STEPS AMONG FIGURES. For division. a 12 2,S 12 21 6 5 d 4 3 i8 35 158 3 7 34 28 36 49 24 7676 30 21 42 6 5 7' 63 9 20 10 42 3 4 5 7 f 24 20 18 35 •4565 g 16 14 12 4 7 3 h 28 15 30 456 (I) (2) (3) (4) (5) 342 24 243 404 23 421 32 434 34 34 34 44 241 423 44 z 13 343 342 3 40 • 234 424 43 423 34 413 334 21 342 21 324 41 34 244 44 34 423 44 321 32 443 342 32 143 42 3 333 43 432 33 234 423 4 341 4 444 342 44 132 13 321 244 32 324 44 243 344 43 431 32 342 321 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, 2i,43» 14, 34, 23,43. 8. Add 43, 233, 424, 341, 3, 434, 4, 342, 434, 342, 243, 414. 9. 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 one 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 4 cents ; how many had he then ? 3. If a carpenter can drive 3 nails in a minute, how many minutes will it tiake 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 ? 9. Andrew has a pair of ponies, how many feet have they ? 10. How many more feet than eyes has a four-horse team ? 11. How many less heads than feet has a three-horse team ? 1 2. A boy spent 2 1 cents for marbles at 3 cents each ; how many marbles did he get ? 13. Barton has 17 cents ; how many pencil^ at 2 cents each can he buy and keep 3 cents? 3+5+4+5+5+3+4+5+3+4+2+3 , -+-2+4= ? Ans. 52. 5+5+3+3+4+4+5+3+2+4+1+5 + 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. 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+3+3+4+4+5 + 3 + 2-1-4+ i + 5+4+5+3+4)=6o+ 12=72. 8 + 4 + 5+4^3x5+4 + 4+5-3-4-5 —4—5—3—3^ Ans. 21. 15 + 4 + 4-5-^3 + 5 + 5+4^4 + 7 + 3+4 + 5+5? Ans. 29. 4+5+3+4+2+5+4+5+3+2+5+5 + 4 + 4 + 3? Ans. 58. 3+4+5+3+1+5+4+5+2+3+5+4 + 5+4 + 4 + 5? Ans. 62. 8 + 5+4 + 5 + 2 + 4+4x3—5-^4x8—4 -5-5-4? Ans. 14. 16 + 4 + 5 + 3 + 4-4-7x6 + 5-3-5- 4 + 2 + 5+6 + 5? Ans. 9. 18 + 5+4 + 4 + 5-3-4-5^4 + 5+5+- 4x5+4 + 4+4? Ans. 7. 3 + 2+4 + 5 + 5-^3+4 + 4 + 2+5+3 + 4 + 5+3 + 3 + 2+4? Ans. 61. 4+5+4+3^4x7+5-2-3-4-5-3 —5-4x3 + 3+6? Ans. 4. 4+2 + 5 + 3+3 + 5+4 + 4 + 5+5+3-1-2 + 4+1+5+4 I 5 + 3? Ans. 67. 3+4+4+5+2+5+3+3+4+5+4+4 + 5 ^5+5+3 + 4? Ans. 68. lOO FIRST STEPS AMONG FIGURES. 3+4+5+3+4+5+5+4+4+3+5+2 + 5 + 3 + 5+4 + 5? Ans. 69. 5-r4 + 5+4 + 5 + i-6 ^-3 + 5 +4 + 5-3 —5 — 1-1-4x7—5—4? Ans. 12. 7+4+5+5+4+4+3—5-4-5-3-4 —4 X3? Ans. 21. 5 + 3 + 5 + 4 + 4 + 3 + 3 + 5 + 5-3-37-4 -5-5-4-4-3 • Ans. 6. (I) (2) (3) (4) (5) (6) (7) 35 5 54 50 34 45 32 54 43 35 35 54 53 45 5 54 43 4 45 24 53 43 35 5 53 33 45 34 24 42 52 45 54 51 15 55 5^ 44 24 44 35 52 34 35 25 55 32 54 41 52 54 53 31 55 5 35 45 45 21 44 34 32 54 347 364 332 341 385 344 361 Examples in subtraction : I. 67,548 2. 69.584 3. 75,897 4. 97^867 43^235 34.331 54.353 52,343 24,313 34,253 21,544 45*524 5. 64,786 6. 79,684 2,432 7,053 62,354 72,631 See P. Ed., p. 47. FIRST STEPS AMONG FIGURES. lOI The teacher may use his own judgmeiii. as to teaching subtraction when some figures of 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 reason 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 i before 3 I 2 I 4 I I I 3 it thus : 5667 and then subtract, 26746 being careful when i is prefixed to the upper figure to add i 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. I02 FIRST STEPS AMONG FIGURES. 7. 910201 8. 423423 9- 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 I's in the minuend, but imagine them to be there. 10. 831,242 II. 513-423 12. 430^213 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 63.415 670,155 562,056 767,621 See P. Ed., p. 48. FIRST STEPS AMONG FIGURES. 1 05 (0 532/ (3) (4) (5) (6) 24 523 24 355 544 55 455 454 55 432 352- 32 44 342 44 534 435 44 503 535 32 255 534 53 434 453 53 543 253 25 355 544 45 424 545. 43 3 325 54 354 324 5 545 454 34 535 432 54 334 543 43 444 543 335 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 2322 60,284 69,309 84,602 48,260 I04 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 IS- 26,463 4 16. 64,524 4 17. 53»625 5 105,852 258,096 268,125 18. 26,463 5 19- 46,035 4 20. 26,304 6 132,315 184,140 157.824 21. 25.036 4 22. 50,264 6 100,144 301.584 Caution : Do not allow the pupil to write any^ wher€ 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.) alb c I d I e I f I g 58473 62584 73625 !84736!2s847j36258|47362 3456718345678345167834156783145678134567 h i I 5847 362 8345 678I For subtraction. a b 1 c 9 13 10 »4 5 56783 9 13 8 12 6 9 II 14 10 453443876 14 6 d e • f II 15 10 6 10 78534 7 II i; 12 139 12 8 II 5 6 7 8 7I6 5 4 3 13 8 g 10 7 II 8 I 3456 h i i 2 16 7 II 8112 9 10 783 4 5! 6 7 8 For division. a 24 49 24 18 8 67834 b 1 c 40 20 6 48 21I42 20 32 15 5 438 7i 6 5 4 3 1 d 1 e 9 24 10 30 56132 28 12 6435 3456 7I 84387 f 12 30 12 21 6543 14 7 36 15 35 16 24 65543 h 40 18 42 15 8675 i [6 48 28 867 Io6 FIRST STEPS AMONG FIGURES. (24) (25) (26) (27) (28) (29) 345 235 25 435 345 5+ 234 543 54 543 523 4S 532 455 43 234 454 25 424 345 55 455 325 54 352 432 42 543 543 24 443 554 21 235 454 55 345 345 54 352 535 32 532 553 35 544 234 54 314 345 54 535 555 43 3,521 3,807 383 3,876 3,968 38+ (30) (30 (32) (33) (34) (35> 545 435 45 543 231 4S 434 554 4 355 545 53- 55 345 23 234 354 4+ 53 534 54 542 434 53 224 242 45 345 543 2S- 532 555 33 554 355 44 454 343 54 435 332 2S 534 543 45 543 435 5+ 325 254 54 343 543 45 43 442 33 555 345 55 3,199 4,247 390 4,449 4,117 439 See P. Ed., p. 59. FIRST STEPS AMONG FIGURES. I0> Tlie following tables are an excellent prepa- jation for short division. Before the pupils solve the examples on page 120, give them a review of this page. Division with remainders. *5's (and review.) a I b c I d I e i6 lo 3 ii\22 9 6 13 5 14 4 718 13 II 3 42 3, 424 32 432I4 3 2! 6's (and review.) a 1 b j eld 14 16 3 26 II 26 II 921 10 19 733 14 9 5 4326:54361543215623 II 22 18 7 2454 3 5 38 27 g I h 20 13 8 19 7 32 326562 3! 456 1. What cost 8 dozen buttons at 7 cents a dozen ? 2. What cost a pair of boots at $7 and a hat at $5 ? 3. How much more does a reader cost at & shillings than a speller at 2 shilli.igs? ♦These are to be recited as follows : 3 in 1 6, 5 times and I rapiainder, &c. I08 FIRST STEPS AMONG FIGURES. 4. There were 9 birds in a flock, and a hunter killed all but 4 ; how many did he kill ? 5. How many knives at 7 shilling's 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 it 4 cents «ach ; what did they cost him ? 10. 13 boys were skating on a 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 i cent for candy ; how many cents had he left ? 12. Georije 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 1 See P. Ed., p. 56. FIRST STEPS AMONG FIGURES. IO9 15. William bought an orange for 4 cents, a fig for I 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 i 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 eacb 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 i day ? 21. Samuel walked 28 miles in 4 days ; at that rate how far would he walk in i 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 I 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 FIRST STEPS AMONG FIGURES. (0 (2) (3) (4^1 (5) (6) (7) 52 242 543 1 2 34 535 344 45 525 244 54 53 454 553 31 343 554 35 4 342 435 53 451 343 43 25 535 234 44 535 234 51 40 453 544 25 434 455 45 32 341 345 53 453 542 34 44 355 453 44 345 435 34 53 434 423 35 523 252 23 45 544 345 34 254 344 45 44 355 534 55 535 535 53 35 433 453 471 4,640 4,481 459 409 4,781 4,663 S6 (11; (12) (13) (14) (15) (16) 544 543 443 532 345 432 354 441 355 344 53 544 435 355 534 434 434 434 543 434 433 453 545 542 431 542 341 345 354 344 303 344 432 524 534 453 454 354 544 343 435 341 435 425 342 234 543 535 544 543 455 555 345 344 432 254 544 443 254 455 435 434 345 344 342 532 353 543 432 535 545 344 5,263 5^12 5,200 s,o86 4,729 5,300 See P. Ed., p. 59. FIRST STEPS AMONG FIGURES. Ill Pupils read : i. 3040321 2. 30245000 3- 463317030 4. 500030261 5. 2000032 6 15000000 7. 320030000 8. 674346537 9- 42000 10. 400320219 II. 3605000 12. 50000018 13. 463308260 14. 75000341 ^5- lOIOOIO Read the following : .. 67345768 2. 476347854 3. 74000037 4- 735400005 5. 900007000 6. 3 16000 1 40 7. 80370000 8. 700000004 9. 7020500 lo, 86000045 II. 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 4)laces by prefixing ciphers. For the above lit FIRST STEPS AMONG FIGURES. number the pupil will write lo, at first. Teach- er ask lo what? Pupil, lo 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 90. 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. I 13 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. 2. " " twenty mil. three hun. 3. " " nine mil. forty th. Caution : Teach pupils to put a comma only after each period^ except the last. Thus in Note. — 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. 35 Thousands. Units. 003 058 The teacher may draw a diagram like the above and allow the pupils to write numbers in it, as the number 35,058 003 is placed there. Teach the pupils that the third period represents millions, and that each period is r»ad as if it stood alone, only that its name is given. 8 114 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 ; e. ^., two thousand three hundred eight. Eight here means eight units, and three hun- dred eight, (which has n6 name given to it) is 308 in units period. The following examples may be written upon the board and the pupils required to bring ^he answers to class : 1. Write in Arabic, thirt}' million, eight thousand, thre? 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. ^^'rite in Arabic, seventy million, six hundred thousand, eighty. See P. Ed., p. 61. FIRST STEPS AMONG FIGURfeS. i»5 8. Write in Arabic, one hundred fifty-four million, two hundred sixty-one thousand, five hundred forty-eight. 9. Write in Arabic, eight hundred million. 10. Write in Arabic, three million, three. 11. write in Arabic, ten thousand, tep. 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 letters: I = i, V = 5. X=io, L=:5o. C = 100, D = 5oo, M = iooo. 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. Il6 FIRST STEPS AMONG FIGURES. 18. Write in Roman, four hundred sixty-two. 19. Write in Roman, six hundred ninety-six. 20. Write in Roman, eight hundred ninety- nine. 21. Write in words, 245306341. 22.. Write in words, 32743642. 23. Write in Roman, three hundred eighty- seven. For rapid solving. 1. 4-H3 + 5 + 2 + 6 + 4 + 5+6 + 3 + 5+4 + 3^-6 + 6 + 5=? Ans. 67. 2. 3 + 5-f44-64-3 + 6 + 5 + 2 + 6 + 5-F5 + 4 + 6 + 64-2 + 4= ? Ans. 72. 3. 5 + 6 + 6 + 4 + 3 + 5 + 4 + 3 + 6 + 2+6 + 6 + 5 + 6 + 2+4=1 Ans. 73. 4. 4 + 3 + 6 + 5 + 2 + 6 + 6 + 5+4 + 3 + 6 + 5 + 4 + 6 + 6 + 5= ? Ans. 76. 5. 5 + 6 + 3 + 4 + 6 + 5 + 6 + 6 + 5+4 + 3 + 6 + 6 + 6 + 5+4=? Ans. 80. 6. 15 + 6-^7x5 + 3-^3x8 + 3+5-^7x4 + 5+3-5-6=? Ans. 29. 7. I9 + 4+5-^4x6-6-5-6-^5x8-4 -4-6 + 5 — 6 X 7—4= ? Ans. 31. 8. 7x6 + 5 + 6 — 4—6 — 6 — 5-8x6 + 5 + 6—3—6—5+6= ? Ans. 27. 9. 6x8 + 6— 2-5-6-5-4-J-4X7-5 —4—3 — 6 — 6+4=? Ans. 8. FIRST STEPS AMONG FIGURES. II7 10. 8 + 7+6 + 5 + 5 +-4-^-5x6-5-6-6 -6-3-^8 + 5=? Ans. 7. 11. 53-6-5-6-4-5+6 + 5+4-^6x5 —6 — 5 — 6 + 5=? Ans. 23. 12. 47+5—6—4—5-2-4-5x8 — 5-6—6 + 3-7-6+8 + 6= ? Ans. 21. 13- 23-5-6-4x6 + 5-6-5-=-7X5-6 —6-6x8 — 6=1 Ans. i8. 14. 16-7-4x6 + 5—4 — 6 — 5 + 6 + 5+6+4 -^7X 8— 4-4-5=.? Ans. 27. 15. 6x7-6 — 5-4— 3-T-3 + 5+6 + 6 + 4 + 5-6 — "5- 6-2-^5= ? Ans. 3. i6. 28 + 6 + 6 + 5 + 4— 6—5— 6— 4-H4X5 —6—5-7-8x7=? Ans. 21. 17. 61—6—6 — 5—5—4-6—3-6^5x8 — 5 — 6-5-7 + 6=? Ans. 9. 18. 34-6-5-2^3x5-4-6 + 3^7 + 7 + 5+5 + 6 + 6= ? Ans. 3^- 19. I6 + 5+4 + 3 + 5-6-3-^8x5-6 + 4 + 5+5-6-4=^ Ans. 13. 30. 27+5 + 4+-6X8-6-4-5-3-6-5 —4-7-3x6-4—4-6=? Ans. id The teacher is advised to give a few exam- ples in subtraction each day, and with them a few in multipHcation and perhaps in division also. Il8 FIRST STEPS AMONG FIGURES. Subtraction. I. 3»423»056 2. 7.352»043 3- 9»635»024 654.368 834,657 746,548 2,768,688 6,517,386 8888,476 4, 4,320,032 5. 63,140,052 6. 8,400,314 543.054 6,572,036 50^248 3.776,978 56,568,016 8,350,066 7. 7,360,042 8. 61,420,035 9. 5,304,036 2,500,075 540,257 202,356 4.859,967 60,879,778 5,101,680 10. 8.340050 II. 64,230,051 13. 75.310,040 762,034 5^0'765 230,076 7,578,016 63,719,286 75,079,964 13- 6,343.520-656245- ? Ans. 5.687,275- M- 94,530,062-8,240,277 = ? Ans. 86,289,785. See P. Ed., p. 64. FIRST STEPS AMONG FIGURES. 1 19 •Multiplication. I. 648,057 8 2. 746,805 7 5^227,^35 3. 470685 6 5.I84-456 2,824,110 4 8,640753 7 5- 680,574 8 6. 358,407 6 60,485,271 5 444,592 2,150,442 7. 685,740x4= ? Ans. 2,742,960. 8.7,406,8^3x7=? Ans. 51.847971. The teacher should solve an example in which there are two figures in the multiplier. 6,354^23=? Ans. 146,142. 53,462x32=.? Ans. 1,710,784. 36,425 X 34= ? Ans. 1,238.450. 353.625 X 43= ? Ans. 15.205,875. 563.524 X 65 = ? Ans. 36,629,060. 350,264 X 36= ? Ans. 12,609.504. 526,304x34=.^ Ans. 17.894,336. 640,536x64=? Ans 40.994,304. 4675^45= - Ans. 210,375. •The numbers used in these examples are puinted off !n periods for convenience in copying to blackboard or slate. I20 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 example 379531 -^4, ^fite ♦>3 7 19 35 33 " ^^^j 94003^ 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 : 1. 24129318-^3= ? Ans. 8,043,106. 2. 281,683.220-7-4=? Ans. 70,420,805. 3.12,246,9214-3=? Ans. 4,082,307. 4. 322,412,836-^-4=? Ans. 80,603,209, With remainders. 5. 83,923—3= ? Ans. 27,974>i. 6. 182,539^4= ? Ans. 45'634^. See P. Ed., p. 71. FIRST STEPS AMONG FIGURES. 121 7.1,373,224-3=? Ans. 457.74ri S- 9^354 3^8-^-4= ? Ans. 22,838,582. 9. 273.78i-=-5=? Ans. 54,756'. 10. 1,035,879-5-6= ? Ans. 172.646'. 11. 2,683,507-4-4=? Ans. 670,876'. 12. 3,921,278^6= ? Ans. 653,546V 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-" 45 +6. > 67 + 5? 58 + 7-'' 65+8? 86 + 5? 58 + 6? 34 + 7? 57 + 5? 26 + 8? 65 + 7? 37+8? 78 + 5? 86 + 7? 37+4? 584-8.^ 63 + 7? 47 + 7? 28-H4-^ 76 + 5? 53+8? 37 + 6? 76 + 6.? 67 + a? 38 + 7? 64 + 8.^ 35 + 5? 47 + 6? 28 + 3.? 122 FIRST STEPS AMONG FIGURES. 43 + 7 64 + 4? 45 + 8? 26 + 5? 47 + 2: 38 + 6I 53 + 3? 74 + 7? 45 + 4^ 76 + 8? 37 + 5? 58 + 2? 53 + 6: ' 24 + 3? 65 + 7? 86 + 4? 27 + 8: 68 + 5? 83 + 2? 54 + 6? 65 + 3: ' 36 + 7? 57 + 4? 68 + 8? 43 + 5-' 54 + 2? 35 + 6; 16 + 3? 47 + 7.' 38 + 4? 53 + 8? 24 + 5^ 45 + 2 86 + 6? 47 + 3? 58 + 7? ' 43 + 4 ? 34 + 8? 75 + 5? 56 + 2? 27 + 6? 48-^3? The exercise above should be rriost 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. 1 23 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 11 of them; how many were left? Ans. II. 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 ? 9. 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. 9's (and review.) a I b 85963 748';9 45678 1 94567 c I d I e I f I gr 6374 j 85963 I74859I62748 59637 8945 67894,56789,49678194567 Sec p. Ed. , p. 74. 124 FIRST STEPS AMONG FIGURES. h 4859 8945 6374 6789 For subtraction. II 16 ii 13 8995 d II 17 12 14 4876 8 15 13 18 5789 3 15 10 12 7654 9 15 7 '2 5 9 4 5 h 10 14 13 II 4 9 4 5 II 16 14 12 6789 f 10 16 II 13 6876 9 13 15 10 6987 .1 12 14 14 6 5 7 k 12 17 9 894 For division. g 28 42 18 24 7694 a b c 42 54 25 32 7654 24 63 16 40 8945 30 63 48 6 7 8 d e f 27 28 72 35 9487 48 20 54 12 6594 35 24 64 5 6 8 h 81 40 56 9 8 7 45 3^ 30 9 4 5 FIRST STEPS AMONG FIGURES. 25 j k 1 18 49 32 72 18 36 56 2 I 36 45 6 7 8 9 298 7 6 5 (I) (2) (3) (4) (5) 36 654 245 +5 456 53 343 634 56 546 46 565 453 +3 323 52 654 546 H 654 64 432 635 55 65 45 543 364 6 556 34 56 655 +6 343 65 435 536 55 656 395 3,682 4,068 370 3o99 (6) (7) (8) (9) (10) 65 356 654 +5 256 6 645 565 56 643 35 546 634 +3 345 64 356 362 56 656 23 465 556 i 55 562 46 321 665 26 234 52 456 342 53 54S 46 566 456 ^5 634 35 345 563 , 36 563 372 4,056 4.797 421 4,438 See P. Ed., p. 77. 1 26 FIRST STEPS AMOMC nGURE& («») (■3) (•4) (•S) (16) (■7) (18) % 65^ ♦S 546 544 45 S3 566 54 354 654 6 64 626 345 66 66s 565 55 55 356 636 36 ^36 666 63 46 465 563 45 565 3^5 34 3i SS6 454 56 653 456 65 56 3,022 3,216 302 3.019 3,210 268 307 (I) (») (3) (4) (5) {^) (7) 354 453 24 «5 546 36 26 45 343 45 344 365 56 S3 533 545 34 23^ 653 63 66 542 434 53 544 546 -s 32 324 455 44 355 53 66 65 453 44 35 243 466 54 66 545 532 33 425 232 46 26 343 341 42 544 665 35 63 45^ 553 54 43' 536 63 66 344 434 33 554 462 56 55 533 345 21 345 355 6 43 234 25 54 423 646 63 66 543 544 35 345 355 45 35 5,245 5,048 507 4,800 s,88o 614 662 Sec P. Eee P. Ed., p. 80. FIRST STEPS AMONG FIGURES. 129 7's (and review) with remainder. a 22 19 42 34 3456 b 69 19 14 39 7 3 4 5 c 28 62 17 39 6734 d e f 32 23 53 13 5673 34 29 57 47 4567 II 31 21 52 3456 g 51 2827 17 7 3 4 '5 h 47 32 26 21 6734 i 49 39 25 5 6 7 8's (and review) with remainder, a I b 55 26 45 24 I 34 16 31 53 8765 I 4387 4' 69 13 31 7834 g 26 28 63 32 4387 20 39 28 53 3 4 5^ h 49 33 61 39 5678 29 43 2 1 29 6543 f 76 47 22 38 8765 i 5831 14 23 6443 19 39 69 5 6 7 k 44 26 17 8 3 4 I. 1,396,897-3=? 2. 1,821,287-7-5=? 3. 2,144,698-4=? 4. 39,164,794^6=? 5. 22,538,618-7-6=? 6. 27,6^2,523^6=? 9 130 FIRST STEPS AMONG FIGURES. 7. 10,763,483-^-7=? 8. 45.171 547-7=? 9- 3i,752,20o-i-7= ? 10. 508,019,899-^-8=? II. 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 divi- FIRST STEPS AMONG FIGURES. I3I 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 : 763+5-^4321- 5738-^49- 9458-^875. 875988^58841. 748567^-2145. 57881-468. 76854-68. 8768456 -i- 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 • See P. Ed., p. 98. 132 FIRST STEPS AMONG FIGURES. more if they are needed to make the entire class familiar with these steps ; 57281456-71235. 423214-^5347. 342165-^2543. 6847-71. 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 left of the comma in the dividend, and place a comma before the one counted which is farthest to the left: thus, in the example 34276^404, 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 : 674532-^5342. 75694857-^845321. 546327-^643. 345367471-^75382. 47346-T-23. 4326472-^6351. FIRST STEPS AMONG FIGURES. I33 8345376-H702. 56341-5-68. 763542^8547. 57345^493- In the example 631663-7-201, 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 Tiumber at the left of the first comma in the "dividend. 2 is contained in 6 three times. The example becomes 2,01)631,663(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- ibers 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 right 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 2,86 Divide as at first and so continue the opera- tion. The steps are : I St. 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. 9th. See that the remainder is less than the divisor. 10th. Write the next figure of the dividend. nth. 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 7///M ' the pupils, before any are given them to solve alone. 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 0/ them take one of the steps then another pupil take another and so on. First solve twice the example given 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. 1. 5,450,204-7-403=? 2. 162,479,845-3042=1 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- 4.920,352-7-2,023=? Ans. 2,432*'^ 5. 16,312,418^5,013=? Ans. 3,254"* 6. 273,785,577-^60,345=? 7. 26,308,025-7-4,063=? 8. 189,771,597-^-50,364=? 9. 524,601,734-^6,047= ? 10. 39 1,838,602 -f-80,675 = ^ 11. 619.307,367^80,597=? 12. 278,696,736^6.075=? 13- 45 + 363 +456 + 542+6 -f- 356 -f- 663 + 454 + 32+46 + 553 + 636+45= ? 14- 465+564 + 32+646 + 553+465+566-f- 632-^665+356-^43+655+6= ? 15. 4264-563 + 365+634 + 5454-643+356 266 + 633 + 56 + 445+54 + 63= 't I.. 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. 137 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 J 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 i 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 ? 9. 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 9 cents one day and 12 cents the next. How much more did he earn the second day than the tirst .' 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 oransjes 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 130, 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 in a week ? 19. 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 i man ? 21. If 2 men consume 6 barrels of flour in a See P. Ed., p. 86. m FIRST STEPS AMONG FIGURES. 1 39 certain time, how much will i man consume in the same time ? 22. If 4 horses eat 12 bushels of oats in 5 days, how many bushels will i horse eat in the same time ? 23. If 3 teams will plow a certain field in 6 days, in how many days will i team plow it ? 24. If 4 men can dig a certain ditch in & 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 wood, 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 al 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 in each bag ? Read the following : 1. 70000580030. 2. 68000C50000. 3. 680507415371. 4. 756847597547. 5. 76500000068. 6. 67459800000. 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. 9. Write in Arabic, five billion forty mil- lion. 10. 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. I4I 14. Write in Arabic and in words MDCCL- XXV. 15. Write in Arabic, ten billion. 16. Write in Roman, 1876. 1 7. Write in Roman, nine hundred forty-nine* 18. Write in words 761308260017. 20. Write 6 units of the 8ih order, 8 of the- 5lh, 3 of the 3d and 5 of the ist. 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 orally and recited as the series have been. 49 (To be read, 45 from 49?) \\ \\ 5| 45 69 52 38 62 35 66 19 51 22 37 66 45 34 55 29 65 14 43 19 45 82 56 32 59 36 51 85 31 56^ 38 66 46 28 53 29 46 66 25 54 38 73 18 30 72 46 47 63 30 57 33 67 16 25 64 43 38 59 23 54 43 50 32 64 48 61 67 48 54 39 25 48 27 56 45 54 63 39 48 37 142 FIRST STEPS AMONG FIGURES. 31 70 34 50 24 51 67 65 84 56 26 64 25 46 17 48 59 63 79 48 30 93 27 86 For rapid solving. (To be read.) I. 3 + 5 + 9 + 7+8-4-4-6x9 + 9 + 54-7-1- 6-3-+7X 9-8-8-3-7 +8 + 5-i-3 X8 + 9 — 4— 5 — 7=? Ans. 41. J. 65-7-5-5-3^9x7-9-7-4 + 5 X9+9— 8 + 7x9+ 8—6 — 2-6x7 —5-5+4x3=? Ans. 64. 3. 6I-5-9-5-7 + 9 + I+-9 + 7+-3X9 + 6— 7-8+3X 8-9-7-9-5-7 X4 + 7+4-'6=? Ans. 29. 4. 9 x8— 6 — 5 — 7-^9 X4 — 8 X6 + 6+-4X 7 + 3+9x6 — 2-^4x6 — 7-4-7x8 + 2 -+6x4 + 4-8=.' Ans. 4. 5. 4x3 + 8-5-4x7-3^8x5 + 4-6x7 + 8-7-9x6 + 6 + 5x3 + 2+4 + 2+6 + 8 -+5 +9= ? Ans. 17. ^. 5X7 + 7-^6x4 + 7^7x6-4-5+3X 4 + 4-^8x5+6 + 6-^4x6—3-^5x3 + 8-4-5 + 9= ? Ans. i6. 7- 5x7-3^8x6 + 6+5x8—3-9x6 — 6-4-4x7—6-8^7x5+6-1-4-5+3 X7 — 2H-6=.' Ans. 9. FIRST STEPS AMONG FIGURES. I43 8. 4x6 + 8-^8x9 — 4^-8x9—94-3-1-7-5- 4= ? Ans. 4. 9- 6X3 + 74-4 + 7-5-9X4 + 5-^3><6-6-7- 6 X 8 + 6-T-9 X 7— 8— 6-r-4 X 3 -t-3-i-8 X9--2-^5 x6 — 7 — 7= ? Ans. 16. 10. 4-f-7-}-6 + 9 + 8-7-f-3-f-8-h8-r-5x6- 6-8-T-4X8— 7-6 — 7^3 X9 + 9-J-5 X9 — 7— 8= ? Ans. 66. Problems for the slite 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 Charbs 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 many 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 i mile, how many times will it turn around in going from Canan- daigua to Rochester, the distance being 29 smiles? • In the different steps of such an example it is im- .portant as a help to mark each result. See 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 j how many feet in 357 miles? 1. 374 + 645+57 + 767+436 + 543+675 + 7 + 454 + 765 + 577 + 456=? 2. 45 + 776 + 567 + 457 + 734 + 475 + 67 + 754 + 5 + 676 + 547 + 375= ? 3- 6 + 347 + 75 + 657 + 746+773 + 46 + 457 + 347 + 675 + 766 + 577=? 4- 576+745 + 457 + 674 + 77 + 556 + 473 + 765 + 7 + 657 + 564+705 + 76=1 5- 546 + 375 + 657 + 74 + 765 + 257 + 6 + 75 + 743 + 656 + 777 + 467 + 762=? 6. 647 + 756 + 475+367 + 636 + 753 + 77 + 345 + 676 + 45 + 576 + 767 + 654 + 365=^ 7- 6,314,532-521,987=? 8. 463,524-39-643= ? 9. 653,425-64,287=? 10. 475*067-36,543=? 11. 688,045-95,387=? 12. 4,760,352-376,534=? 13. 4,630,024—921,045=? 14. 34,000,435 — 2,700,518? See P. Ed., p. 92. FIRST StEPS AMONG FIGURES. MS 15- 4,500^375-760,187= ? 16. 354,000,253 — 272,102,437=1 17. 530,024-543,052=? 1. 89,756x96=? 2. 364.758x356=? 3. 638,497 x68=? 4. 498.675x97=? 5. 60,847 X 708= f 6. 796,805x705=? 7. 807,009 x6o8= ? 8. 6,859x748= ? 9. 94,786 X 7,968= ? 10. 603,405 — 612,134=? II- 6,537,065-743,987=? Division series with remainder. 9*s and review. a b 32 15 68 39 61 54987 33 49 26 46 77 65489 c d 19 37 21 46 78 45678 52 53 27 39 61 96549 e f 28 54 29-34 61 87678 35 58 41 70 43 96789 146 FIRST STEPS AMONG FIGURES. g 29 18 40 68 43 45675 21 88 54 25 4987 47 23 34 6 5 4 1. 33.256,023-7-7=? 2. 39,044,761-^6=1 3- 34,352.839-^5='^ 4. 27,832,074-6=? 5. 445.941,095-^7=1 6. 27,516,279-5-4= ? 7. 449,181,483-^6=? «. 487>959'992-^7=? 9. 28,556,208-6=? 10. 15,228,723^4,058=? 11. 251,776,292-^70,486=? 12. 386,124 633-^6o,578=? 13. 41,847,116^9,048=? 14. 52o,6I3,47I-^ 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. I47 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,i53,4i9,6i9-=-7o,386=f 19. 15,086,456-7-4,023= ? 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 pupilsto 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 10 times. 1. 4.367,695^673=? 2. 220,3964-254= ? 3. 307,627^354=? 4. 4,925.151-^-694=? 5. 6,310.318-^781=? 6. 3,280,381-^482=? See P. Ed., p. 98. FIRST STEPS AMONG FIGURES. 149 7. 239,294,268-^3,642=? 8. 334/;90,637 4-574= ? For rapid solving. (To be read and to be answered without use of slate.) 1. 6x6 — 8-1-4x6 + 8 — 2-7-6x4+8-7-5x9 —9+9x6 + 8 + 9=? Ans. 59. 2. 9x6 — 5-+7 x6 — 6-r-4X 7 + 5 + 4-9+9 + 6 + 8-4-7-3=? Ans. 9. 3. 5X8-6-7+-3X5 + 7+6 + 5+9X6+3 ^5x3 + 9 — 7= ? Ans. 29. 4. 8x7-9-5-+6x3-3 + 3X7 + 3 + 5x6 — 6 — 6-j-6x4=? Ans. 28. 5. 6x8-6-7-^-5x8 + 4 + 3 + 7x4 + 7+5 -8 + 7 + 5+9= ? Ans. 27. 6. 4x9 + 8 + 4^-6x4—5+3x6—8-4+6 x5 + 8 + 4=.? Ans. 47. 7. 7 x6 — 5— 9-+4X8— 8— 6 — 2— 8 X5 +^ + 7-8=? Ans. 33. 8. 6x4 + 8-^8x9 + 6 + 7x8 + 1-7-7x9-4 -3-+7 + 9=^ Ans. 17. 9. 9x8 — 6—9 — 1-:-7 x5 + 5-^9 X 6 + 8+4 -^6x8 + 7 -^9x4 + 6 + 9 + 5+8 X 6 + 7 + 2-7-9x8-8 — 5+9=? Ans. 3. 10. 7 x6— 6 + 9-+5 X 6— 8— 9+5-7-7 x8— 6 -5 -5 +-8x9 + 9-7 + 6+9+3+8 150 FIRST STEPS AMONG FIGURES. X 4 + 9 +8-^9x7 + 8 + 5-^8x7=? Ans. 42. II. 9x6 — 8 — 7 + 6+9x6 — 6+3x7 + 8 — 7 + 6^7 X 4 + 9 + 8 — 7— 4+-6x8 — 7 -6-7 + 4 + 7+8+ 8 +4 x7+8=? Ans. 64. la. 8x8-+ 6 + 3 — 7 — 8— 4+-6X7-8— 6-1-7 X8-8+6X9 + 3— 8— 4+-9+8 + 9 + 7 + 6 + 9 — 7 + 5 + 8 — 7=1 Ans. 45. 13. 6x8 + 3-7-8 + 9x7 + 9 + 8 + 5x6 + 2 + 7x9 — 7—9 —8-8x7— 8 — 7-+3 + 7 + 8 + 7=? Ans. 31. 14. 7x9-8-7-4-8x9+8-5-6-4—5 + 6X8 + 9 + 7+-9 + 9+ 9 +8 + 2-^4x7 — 7-^8x5+8 + 9 + 8=? Ans. 60. 1. How far will a boy walk in 7 days, walking 9 miles each day ? 2. How far will a boy walk in 2 days^ walking 9 miles the first day and 7 miles 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 1 See P. Ed., p. 99. FIRST STEPS AMONG FIGURES. 151 5. If 4 peaches cost 8 cents, what cost 9 peaches ? 6. If a merchant sells 9 spools of thread in 5 hours, at that rate how many would he sell in 1 hour ? 7. If 3 girls can make 6 aprons in a day, how 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 i girl do it? 9. 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? 1 2. There are 1 7 girls in a class and 9 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, Multiplication and Division : 1. A has 9 fields, containing in all 197 acres j B has 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 ? 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 I still owe him? 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. Henr)''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 $25,800; 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 $2175 ; their expenses are ^957 > 'f ^^^ balance is divided equally, how much will each have ? 9. If four dresses of 15 yards each are cut from 78 yards of calico, how many yards will be left ? 1. 7+46+74 + 35+47 + 73+644-574-75 + 6 + 77 + 45+63 + 57 + 46 + 75=? a. 376+455 +757+463+375 + 747+654 + 37 + 576 + 346 + 775 + 7-^464 + 647 + 356 + 565=? 3. 475 + 647 + 756 + 765+437 ^674 + 575 + 756 +647+ 567+456 + 743 +357 + 556 + 463 + 756= ? 4. 76 + 35 + 8 + 47+ 85 +37 + 56 + 87 + 75 + 84 + 68= ? S- 58+765 -485+ 678 +537+6 + 753 + 488 + 846 + 587 + 758=? 6. 678 + 845 + 784 + 326 + 487+856 + 678 + 588 + 865+478 + 756=1 7. 7 + 58 + 84 + 56 + 78 + 64 + 87+6 + 58 + 75+86 + 68 + 75=? 154 FIRST STEPS AMONG FIGURES. 8. 648 -I- 785 4- 874 +688 + 576 + 845+68$ + 786 + 75 + 847+687+58 + 6=? 9- 758 + 875 + 684 + 768 + 475 + 886 + 744 + 358 +652+ 887+ 546 + 785+64$ 10. 8 + 57 + 68 + 84 + 75 + 7 + 58 + 76 + 88 + 47 + 63 + 78 + 86 + 55+67+88= ? 11. 368+475 + 638 + 857 + 583+646 + 87$ 57 + 645 -^ 768 + 582 + 7 + 676 + 848 587 + 766=? 12. 67 + 788 + 856 + 475 + 687+878 + 564 + 87 + 656 + 478 + 880 + 567 + 375 + 688 856 + 785=? 13- 7,963^034- 546,573=? 14. 758,600341-79,423,275=1 15. 8,460,075—987,286= ? 16. 658,000,468-35,030,273=? 17- 43»750'078— 44.345,621= ? Impossible 18. 6,475,000,374—293,030,596=1 1. 97,806x59= ? 2. 97,865x896=? 3. 96,897x6,978=? 4- 5*740x7,500=? 5. 746,800 X 9,000= ? 6. 470,900 X 70,580= ? See P. Ed., p. iia FIRST STEPS AMONG FIGURES. 1 55 7. 869,070 X 670,900= ? 8. 790,600x806,700=? 9. 62,8o2,889-=-9=? 10. 71,262,955-8=? 11. 538,908,792-4-7=? 12. 376.57i,o86-j-7=1 13- 339,253.657-^9=? 14. 324,763,528^7=? 15. 446,2i7,i69H-9= ? 16. 54,284,406-7-8,047= 1 17- 25,534,849-^7,I97=? 18. 45,126,612-4-914=? 19. 4,368,565-4-9,168 ? 20.63,008,141-7-5,274=? 21. 471230,943-^79=? 22. 9,290,055.741-^4.869=1 23- 331279,851-48,600? 24. 68,643,216^87,000? 25. 76,845,678-7-100= ? 26. 3.921,534,261-7-486,000=1 27. 60,064,175-^8,000=? 28. 90,700 X 50,700= ? 29. Subtract 3 billion 6 thousand 750 from 45 billion I million seven hundred sixty-three thousand 4 hundred. 30. 284,553,437^3,790=? See P. Ed., p. 116. '56 FIRST STEPS AMONG FIGURES. 31. 4,167,300,326-5-4,790,000=? 32. 5,074,000X68,070=? 33- 3-424 330.02 » -^ 497iOoo= ? 34. 2,468,576,216-^10,000=? 12's (and review.) For addition and multiplication, b 7 9 12 8 10 8 9 10 II d 3 12 8 12 12 II 12 4 g 12 5 12 3/26 12 7 9 II 9 8 9 e II 9 12 10 II 12 II 9 12 8 10 7 12 8 9 10 II 12 8 10 7 II 1 1 12 8 9 I' 8 10 7 II 8 9 10 II i 6 12 4 12 5 12 For subtraction. a 22 19 22 18 19 19 12 II 10 9 8 12 c 16 21 23 19 20 15 9 10 II 10 9 8 21 18 2t 17 20 18 II 10 9 d 8 12 8 20 24 16 19 17 22 II 12 8 9 10 II 21 20 17 20 23 18 12 8 9 10 12 II See P. Ed., p. 140. FIRST STEPS AMONG FIGURES. ^57 For division. a 72 121 108 96 72 100 12 u 12 8 9 10 c 120 108 72 132 90 99 32 77 80 110 84 4g 12 1 1 10 1 1 12 12 d 56 88 120 81 96 80 9 8 1 1 10 9 811 10 9 12 S e 60 36 12 12 f 63 no 99 144 64 90 9 10 II 12 8 9 I. 799,896 X 12= ? 2- 6,347,435 X 12= ? 3. 7,968,473 X 11= ? 4. 3,546,247 X I2=? 5. 989,769 X 12=/ 6. 799.958 X 12= ? 7. 8,989,978 X 12= ? 12's (and review.) Division with remainders. 127 76 94 70 150 108 11 10 9 8 12 II c 116 102 78 105 86 139 12 8 9 10 II 12 115 71 96 137 106 85 10 9 10 II 12 8 d 124 88 94 95 117 89 10 9 8 12 II 10 See P. Ed., p. 117. «58 FIRST STEPS AMONG FIGURES. IIS 78 96 130 63 107 9 8 II 12 8 9 For more practice. For division. ^6 152 40 68 4- II 12 6 7 8 c 118 75 114 61 S^ 12 II 987 69 54 102 89 45 67876 g 76 93 70 139 81 689 II 12 78 130 94 62 34 9 1 1 12 7 6 52 55 88 64 107 6 8 9 II 12 f 142 86 52 78 48 12 II 9 8 7 h 57 41 70 107 6789 8. 9,587,888,171 -fia=»t 9. 95,621,647^12='? 10. 813,764.564-4-11=? 11. 83,645,840-12=1 See p. Ed., p. 118. FIRST STEPS AMONG FIGURES. I59 For rapid solving. (To be read and to be answered without the use of slate.) I. 17+4^3x9 + 8 + 7-1-6-^12x9—8—6 + 7-^7x6 + 9 + 6-^-9x11+9+8 + 7 + 7-7-9 X 10-^-3 + 9+94-2 + 7-7-2 = ? Ans. 18. a. 144-^2-7-2-^-9 x8x 2 — i-i-7 X i2-i-3 + 6-^6 X 12 X2+-3+-2 X3+9 + 9 + 3-r- 2 + 6 + 7 X 4-7- 10 x8 -5-4x3-7-8= ? Ans. 9. 3. 9Xi2-2-^3X 2—44-2 X3 4- 12 + 7 X 12-7-3 X 2-^4+ 124-2x3 + 6-^3x2 + 74-5x8-^3x24-3x44-8 + 9+8 + 9x3 + 64-9=? Ans. 12. 4. 19x2 + 4 + 7x12-7-2x34-12x114-3 X44-12 X 8 + 84-2 +34-3 X2+8-T-6 XI2-^3X2 — 2 4-3X2 + I2 4-3X54- 4X 104-4+2 X4— 4+12= ? Ans.8. S- 13x3+8 + 7-^6x84-3x2-^-3x2 + 7 + 7 + 64-2 — 10 X3—84-5 + 7X4X 24-44-15x9x2x3-94-3x44- II x84-3=? Ans. 32. 6, 14X 2+4x84-2 X34-7 x8-^3 4- 2 X4 + 84-2x3-24-3x2x2-1-12x9+8 + 9—6 + 10 + 3 + 2-1-94-3x2+4= ? Ans. 8. l6o FIRST STEPS AMONG FIGURES. 7. i2Xii-^3-f8-j-24-8-r-2X3~3-i-2-^4 X9-f3-T-3X 4-^2 + 8-^2X3+3 4- 12 X9-r3 — 2 + 4X9+3X2 + 3X2 + 8 X4X2=? Ans. 16. 8. i9X2 + 4-r3X2+4Xi2 + 8+2-i-2 + 5 -4-2X3+2x4+12x8+2X3+4+8 X 9+ 1 +4X3 -4- 5X2 + 8 -2-4X4 = ? Ans. 60. 9- 9X3-5^2x3+3^24-9X18+9+8 + 5-^2X3+9 + 8X3-i- 2x3-2x3 -7-9X4-^2x5 + 2+5x6 + 3=? Ans. 18. 10. 16x4+2+2x3+12-^4X3-^5X4+2 X4-r-2 — 6+2X5-r-25X 12 + 8+ 12 + 2-7-4X10+2+7x15-50x3X2 -J-3-25+5=? Ans. 5. I. 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. l6l 2. If 3 oranges cost lo centg, 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 10 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 rale of 2 spools for 9 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 10 men cut in a day ? 6. If 3 bushels of wheat cost $6, what will 8 bushels cost ? 7. If 2 bOshels 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 } 9. If 3 boys can do a certain work in 6 days, how many days will it take i boy to do the same work ? 10. If 2 men can hoe a field of corn in 4 days, how many days will it take i man to do it ? IX 1 62 FIRST STEPS AMONG FIGURES. 11. If 3 n^^n can cradle 6 acres of grain in a day, how many acres can i man cradle in a day ? 12. If 2 men can build a wall in 6 days, how many men can build it in i day ? 13. If 4 men can dig a ditch in 12 days, how many men can dig it in i day ? 14. If 2 men can dig 8 rods of ditch in r day, how many rods can i 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 i6th. and teach them to find about i of the kind the question asks about. For instance the 15th asks about 4 men, hence find out how many days it will take i man. In the i6th it asks about 6 day.s, hence find about i-day? Solution of 15th : If 3 men can dig it in 12 days it will take i man 3 times 12 days, or 36 days ; and four men can dig it in i 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 i day, it will take 12 times 3 men or See P. Ed., p. 120. 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 Josing 10 he has 35 ; how many had he at first ? 18. What number multiplied by 3 will give 12? 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 10 days ? 22. What cost 9 suits of clothes at $14 for each coat, $2 for each vest, and $4 for each pair of pants ? 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 1 27. A boy has 39 cents, how many must he earn that he may buy a dozen oranges at 4 cents each ? 1 64 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 cent.% 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 1 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 denomination of each re- sult in the process of solving problems, but FIRST STEPS AMONG FIGURES. 165 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. I. 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 1 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 1 6. If 15 boys walk 900 miles in 60 days, how far will they walk in 2 days ? Sec P. Ed., p. 125. 1 66 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 1 1 65, and another for $187 ; what was the average price of a horse ? 9. 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 ^2 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? 19. 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- t 168 FIRST STEPS AMONG FIGURES. mainder was divided equally among the othei children ; what was the share of each ? 22. A, B and 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 1 1. 49 + 7»o68 + 9,847+958 4-37-f-489+8,- 956+9843=? 2. 946 + 378 + 795+8494-696 + 784 + 359 + 436 + 775+898=? 3- 547 + 397+484 + 758 + 969+847+958 + 497+384 + 947+358 + 596=? 4. 567 + 498 + 948 + 397+846+372+458 + 796 + 389 + 486+ 958 +347 + 598 5. 123 + 456 + 789 + 987+456 + 321+743 + 398^+ 476 +395+948 + 767 + 496 324=? 6. 578 + 397+956 + 789 + 437+496 + 875 + 749+ 658 +976 + 345+876 + 901 7. 947 + 643 + 358 + 895 + 769 + 576 + 34-8 + 954+ 847 +659 + 438+987+648 + 326=? FIRST STEPS AMONG FIGURES. 169 8- 756 + 395+468 + 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=? TEACHERS' EDITION, fXOM. NO. ANS-WKR. PAOB. Na AKsmn. 126- 1. 5,245. 7. 6,475 »«•. 2. ,\048. 8. 3.768 ". 3.507. 9. 86.754 «••. 4. 4,800. 10. 4.a57 »«. 5.5.880 11. 7.684 >». 6. 614. 12. 45.876 »•. 7. fi62. 13.4,197. 128- 8. 215.829,875. 14. .5.W.S. 9. 196,070.418. 1.5. 5.049. 10. 114.826,392. 143- 1. 25.280 rds. 11. 15.600,168. 2. 218 cts. 12. 21.260.981. 3. 37 cts. 13. 48,.'i81.904. 4.873. 14. .53.084.772. 5. 10.208 times. 15. 193.177.236. 144- 6. 253 sheep. 7. 1.884.960 ft. US- 1. 465.632 '•». 2. 3ft4.257 »». 1. 5.7.56 3. 536.174 >^. 2. .5.478. 4. 6.527.465 «"•. 3. .5.472. 5. 3,756.486 >«. 4. Q.3»Z. 6. 4,603,758 •-•. 5.6.160. 130- 7. 1,537.640 >->. 6.7.139. 8. 6.4,53.078 '•\ 7. 5.792.545. ^. 4..536.028 *^». 8. 423,881. 10. 63.502,487 «. — 11.4.a57,068K 9. 589,188. 10. 438,524. 12. 7.684.530 ". 11. 592.658. 136- 1. 13,.534 ».' 12. 4.383.818. 2. 58.412 M'. 13. 3.708.979. 3. 34.«V.'r, :««. 14. 31.299.917. 4. 2.4:i2 ■■*. 145-15.3,740.188. 5. 3.2W "•. 16. 81.897,816. 6. 4.587 »'». 17. Impossible. 172 KEY TO BEEBE S 1. 8,616. 5T»). 7. 2. 129,85:i.i>48. 153- 8. 8. 48,417.?J<>. 9. 4.48.371.475. • 1. 5. 43.079.076. 2. 6.561.747.525. 3. 7 490.661.4?2. 4. 8. 5. 130. .532. 5. ». 755.a54.848. 6. 10. Impossible. 7. 11. 5.7«J.a78. 154- 8. 146- 1. 4,750.880 »». 9. 8. 6,507,400 >^ 10. 8. 6,870,567 **. 11. 4. 4,688.679. 12. 5. 68.705,870 •■'. 13. 6. 6,879,069 *■*. 14. 7. 74,868,580 »^. 15 8. 69,708.570 *^. 16. 9. 4.759,368. 17. 10. 3.75a >••«■, 18. 11. 3.5?a »<«. 1. 12. 6,374 «««. 2. 18. 4.625 "«. 8. 14. 7,364 T«». 4. 15. 68,450 >.«»•. 5. 147-16. 7,056 »»\ 6. 17. 65.704 »««. 155- 7. 18. 627,304 «^». 8. 19. 3,750 «»•. 9. 20. 73.604 ««. 10. 148-1. 6,489 •w. 11. 2. 867 "•. 12. 8.869'. 13. 4. 7,096 •«. 14. 5. 8,079 •»». 15. 6. 6.805 «T». 16. 149- 7. 65.704 »<». 17. 8. 583,607 '»». 18. 152-1. 45 fields; 819 A. 19. 2. 5 horses. 20. 8. $19,458. 21. 4. $1,936. 22. 5. 40,142, 155. 23. 6. 545 ft. 24. $5,430. $304 "^ 18 yds. 847. 7.600. 9,680. 658. 5.961. 7,841. 803. 7,568. 9,066. 1,005. 9,881. 9.687. 7,416,461. 679,177,066. 7.4?2,789. 622,970.195. Impossible. 6,181.969,778. 5,770,5&L 87,687,040. 676,147,266. 43,050,000. 6, r21. 200,000. 33.236,122.000. 583,059,063,000. 6:37,777,020,000. 6,978,098 '-•. 8,907,869 ". 76,986,970 *^, 53.795,869 «^ 37,694,850 »-•. 46,894,789 ", 49,579,685 *"•. 6,745 '.»»i. 3,547 T.OM, 49,372 •«. 476 «.'". 11,946 «.»w, 597,860 «. 1,908,000 «.'*>• 684 «.«»i, 789 «•. FIRST STEPS AMONG FIGURES. 73 25. 768.4S6 '% 166- 7. 36. 8,060 »••. 8. 27. 7,50S "». 9. 28. 4.598.490,000. 10. 29. 42, 00 1.756, two. 11. 80. 75,080 "-. 12. 156-81. 870 ««. 13. 82. :^5,387, 180,000. 14. 83. 6.890 ". 167-15. *4. 346.857 • a»«. 16. 157- 1. 9,598.752. 17. 2.76.160.220. 18. S. 87,653,203. 19. 4. 42.5.54.964. 20. 5. 11,877,228. 21. 6. 0,59'.),496. 168-22. 7. 107,8?J,736. 23. 158- 8. 71>8,91>0.680 »>». 1. 9. 7,968,470 • '2. 2. 10. 73.978.596 o-^K 3. 11. 6,970,48») '^'a. 4. 165- 1. #2,52 "3-. 5. 2. $1,850. 6. 8. «l,26l; $79. 7. 4. $10. 169-8 5. 174,296 bricks. 9. 6. 30 mi. 10. 14.5,051,204. $179. 8..589 IS4M, 700 A.: $2,725! 1.988 cts. , 263 lbs. «U5. 86 mi.; 316 ml. 665 >"•«»«. «4*i. 360. 60,770.524. 1,597,595. $157 f$2,.500each. Eldest $7,500 ;oth'i A, $7,400; B, $3,700. $18,675. 37,247. 6,916. 7.742. 7.660. 7,079. 9,033. 9.395. 10,102. 10. 11,854. End of Key to Tsachers' Edition. PUPILS' EDITION. rAOB. NO. ANSWER. PAQB. NO. ANSWCm. 88- 1. 251. 34. 317. 2.272. 8T-35. 21.752. 8.271. 36. 24,853. 4.239. 37. 63,545. 6. 284. 38. 22,544. 6.289. 39. 98,788. 7.262. 40. 62,592. 8.324. 41. 66,651. 9.2152. 42 77,484. 10.260. 43. 28,345. 34-11. 296. 44. 72.353. 12. 295. 45. 62,156. 18.308. 46. 63,234. 14.335. 1. 46,206. 15. 314. 2. 90,698. 16.325. 3. 39,606. 17. 318. 4. 42,064. 18.327. 38- 5. 93.069. . 19. 324. 6. 24,640. 85>20. 316. . ri, 60,360. 21. 317. 8.69,896. 22.346. 9. 62,840. 23. 319. 10. 96,908. 24.302. 11. 60,846. 25. 345. 12. 69.369. 26.307. 13. 32,341. 27.297. 14. 31,203. 36-28. 325. 15. 10,243. 29.328. 16. 20,418. 30.308. 17. 13,028. 31.293. 18. 41,302. 33.323. 19. 14,032. 33.308. 20. 80,218. FIRST STEPS AMONG FIGURES. 75 21. 13,024. 15. .563,135. 22. 40,132. 16. 741,027. 23. 20,312. 17. 411,303. 3»-*24. 2.429. 18. 623,024. 25. 2,463. 19. 71,634. 26. 2,544. 20. 44,262. 27. 2,420. 21. 443,043. 28. 2,340. 22. 476,446. 29. 2,637. 23. 5.%,467. 30. 2,917. 24. 375,r)67. 31. 2.996. 49-25. 577, .567. 32. 2,3W. 26. 760,475. 40-33. 2,967. 27. 376,737. 34. 3,178 28. 875..%5. 36. 3,184. 29. 6<56,.567. 86. 34,892. 30. 777,6.57. 37. 36,775. 31.267,146. 38. 35.561. 32. 365,662. 41-39. 35,673. 33. 307,177. 40. 35,680. 34. 176,51.5. 41. 4,655. 35. 3<32,6()7. 42. 4.5J4. 36. 421,2.56. 43. 4,615. 37. 277,465. 43^4. 4,300. 38. 219,173. 45. 4,522. 39. 373,676. 46. 4.8.37. 40. 37»,074. 47. 49.044. 41. 2.57,1.56. 48. 48.573. 42. 471,46.5. 49. 48.874. 43. 517.774. 43-50. 49,083. 44. 76,516. 61. 51.204. 45. 311,8.56. 4T- 1. 4.258. 46. 677,172. 2. 4,011. 47. 376,046. 8. 4,741, 48. .W2,567. 4. 4,743. 50- 1. 40,895. 6. 4,883. 2. 42,073. 6. 4.872. 3. 41,217. 48- 7. l.'3.4.%. 4. 44,286. 8. 6r).144. 6. 42,:i08. 9.86,425. 51- 6. 44,.')09. 10. 663,453. 7. 41,808. 11. 762.326. 8. 46,765. 12. 42.4.'>3. 9- 46,.558. 13. 42.3*t. 10 4.8,803. 14. 8,636. 52-11. 4.628. 176 KEY TO BEEBE'S 12. 8,462. 2. 73,276,956. 33. 4.(»28. 3. 79,097,578. M. 60,248. 4. 71.178.967. 15. 6;i906. 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. 73.287,866. 20. 92,704. 10. 86,995.768. 21. 106.926. 11.80,773,478. 22. 109.a'i6. 12. 2G.579.078. 23. 79..T59. «6-13. 7,170,658. 24. 92.704. 14. 94,8iS0,045. 26. 19:^.578. 15. 860.097,867. •26. 214.496. 16. 47,159,685. 27. 193,572. 17. 32,660,068. 53-28. 213.704. 18. 6,697,868. 29.258,144. 19. 77.995,682. 80. 71,284 •20. 90,a59,567. 31. 139.066. 21. 77,706,580. 82 145,704. 22. 93,730,146. 33. 105,738. 23. a'>,764.946. 84. 218,712. 24 70,299,847. 86, 219.276. 25. 77,660,078. 36. 387,156. 67- 1. 246,7*2*2. 37. 231,760. 2. 290,49*2. 38. 254,096. 3. 213,a=i2. 39. 140.168. 4. 402,241. 40. 202.680. 5. '249.494. 41. 182.772. 6. 281,968. 42. -230.175. 7. 459,704. 43. 157,824. 8. •211.476. 44. 150,216. 9. 368,464. 45. 312,384. 68-10. 5,480.296. 60- 1. 52. ^^79. 11. 3,595,469. 2. 45.897. 12. 5 142.276. 3. 57.437. 13. 2,548,23*2. 4. 50,384. 14. 2,548,546. 5. 51,459. 15. 29.126,064. «0- 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. FIRST STEPS AMONG FIGURES. 177 22. 2.i98.402. 32. 24.ft35 ^^ 23. :.'4„V.)8,9i)2. m. 3.456.827 »^. 24. 41.;KW3,168. 76-1. 63,237. 25. 4,828,(152. 2. 61.205. 28. 313,225. 3. 63.924. 27. 4,252.262. 77- 4. 66,698. 69-28. L'2,517.838. 5. 65,307. 29. 3,186,721. 6. 55,344. 30. 3,430,458. 7. 52,771. 31. 11.993,:3ftl. 8. 57,783. 82. 27.078..591. 78-9. 61,074. 33. 47.923.624. 10. 60,493. 34. 2,011,205. 79-19. 69,274. 35. 16.799,022. 20. 69.883. 70- 1. 3.4;,'8. 21. 68.111. 2. 2(»4.:^i7. 22. 71.276. 3. 820 300. 23. 67,740. 4. .^,060.905. 80- 1. 17,438,656. 6. 520,709. 2. 23.971,7.52. 6. 1.0G«».5H)7. ?,. •",' -"'-"i:.'. 7. 8,O.'>0,907. 4. :>. 8. 64,08;i,207. •'». 9. -209.071,205. 0. ;,'i;.irj.:).5W. 10. 7.062..3t)4. :. 2,59,5,186. 11. 508,207. 8. 4.188,375, 12. 53,208,409. 9. 5,965.938. 71-13. 2,4.')7»». 10 3.235.848. 14. 3,543 ' *. 11. 5.5.044,.5.'>5. 16. 6.475 »». 12. 17.508.384. 16. 274,658 >». ^3. 26,4.35 ^■*. 17. 545,734 ". 14. i].%:2n a-». 18. 265,435 »». 15. ;{54.(;24 > «. 19. 238,636 ". 16. 3.74G.254 »^. 20. 1,887,677 ''. 17. (157.;U2 >*. 21. 6,543.^46 >"•. 18. 4.(kJ.-).246 «■». 22. 64,764 »^». 81-19. 7.502.4.35**. 23. 154,264 ♦^. 'Al 437.520 *•'. 24. (M.357 >•*. .21. 4..5.37,264»-». 26. 623.542 «. 22. 5 743J.'63 «-«. 26. 75.246 ". 2;j. 5,762,474 ". 27. 543.452 *\ 24. 2,6t»5 »•*. as. 974,058. ^. 6..370 524 «-». 29. 642.455 ". 26. 423.177. 80. 046,819 »•. 27. 6,258.086. 81. 786.538 'V 28. 3,004.756 ^. 178 KEY TO BEEBE'S 29. 4.a>3.:02 ^. 7. $12,250. "^30. 7 0S6.534 «. 8. 127 marbles. 31. - '- * •'♦\ 9. 105 ct8. /3L' «. 10. 1.781 stept. , 3;i. -*. 11. $85. 1. laiii '^ 12. 7.682 ct8. 2. 3,624 >«. lo. ^m-Jt 3. 4.235 «»*. 14. 92 da. 4. 4.236 «.•«. •1-15. 425 A. S2- 6. 5..J42 •*». 16. $068 6. 24.353 »•«••. 17. 91 marbles. 7. 645 ' «". IX. * 1.920. 8. 3.624 «>. 19. 2,5.092 cU. 9. 3.425 «««. 20. 59 bu. 10. 4.r26 "». 21. 15 cts. 11. 3,428 « »»•. 22. 874 bu. 12. 3,564 «•'". 23. 152 bu. 13. 6,457 «>. 92-24. 1,784,910. 14. 54.673 '.»". 1. 64.482. 15. 4 756 «.«>«. 2. 68,035. 16. 57.643 ««. 3. 69,530. 17. 3,745 »«. 4. 66,904. 18. 4.576. ' 5. 6><.728. 19. 74,656 »". 93- 6. 7L-207. 20. 46,576 »". 7. 75,781. 21. 6 534 ^: 8. 74.612. 22. 76,487 » »». 9. 72,156. 23. 7.458 «. 10. 69.201. 24. 64.786 >•'. 94-11.3.608.157. 25 5.867 >. 12. a58,280. 26. 4,758 "*. 13. 016.178. 27. 6,785 ••». 14. 266,078. 28. 65,847 ***. 15. 6,050 993. 29. 47.664 >•»«•. 16. 391 648. 83-30. 71,414. 17. 407,708. 31. 71.083. 18. 278.079. 32. 72,225. 19. 277,164. 33. 70.304. 20. 391.667. 34. 68.796. 21. 622.678. 89- 1. 549 bu. 22. 578.238. 2. 370 ct8. 23. 39.213,566b »0- 3. 864 bu. 24. 670.781. 4. 174 marbles. 25. 3,764,877. 6.632 bu. 26. 79.088. 6. 189. 27. 24,147. f IRST STEPS AMONG FIGURES. 179 28. 42,869.909. -12 5,746,387 ♦-^ 29. 09.059.978. 13. 7.468,576 '■•. 30: S7J)Zi 970. 14. 67.580.760 •-». 81. 3.189.82.5. 15. 4,0157,586 '-». 32. 36.970.172. 16. 65 748.760 »-'. 1. 70 919.205. 17. 796.859,809 »-•. 2. 47.009.1:50. 18. 097.879,680 »-^ 9B- 3. 5.449 025. 19. 04,859 760. 4. 302.421.261. 20. 748:695 ♦-^ 6. 31.269.744 21. 47i^.508 '-■>. 6. 392 202.525. 97-22. 840.-967 »-«. 7. 3.756,008. 2:5. 639.-408 T-«. 8. 726.525. 24. 4.253 "«. 9. 187.044. 2.5. Omit this example. 10. 7.-221.816. 26. 35,246 »<". 11. 5.498.2r>5. 27. 5,264 3'". 12. 5.448,946. 28. 6,423 »•"». 13. 4,231.810. 29. 763 '»"^ 14. 35,677.072. 30. 4.536 «.o". 15. 04,:W7.97S. 31.64,:552"'. 16. 49.849,081. 32. 48,372 '". 17. 610,079,011. 33. 57,362 s"'. 18. 308.940.670. 34. 41,572 so*. 19. 2H.().K1,4(>4. 35. 3,754 i-<««'. 20. 379.514.76.3. 36. 4.827 "«. 21. 35 69.5,072. 37. 36,472 "^ 22. 68.097.323. 38. 64 727 *••«. 23. 68,390,182. 39. 463.7.52 ♦.»«•. 24. 379,239,951. 40. 74,635 "». 26. 6.257.507,544. 41. 4 605 "». 26. 6.328,767. 42. 70,5:54 » •«». 96-27. 2.816.929. 43. 4,6.53 '.'". 28. 708.709. 44. 64,075 "•. 29. Impossible. 45. 736,502 »". 1. 463.067 »^ 2. 684,756 ". 98-46. 47.5,630 '•»♦•. 47. 4,607 »-^«'. 8. 475.307 ' «. 48. 60.835 »«. 4. 6.463,060 « ». 6. 6Wite7 ' •. 49. 72,506 "». 60. 43.072 •<»•. 6. 374,675 • •. 51. 4,073 »o»<». ^7. 466,738 »■•. 62. 47,280 «". - 8. 6,537,645 •«. 63, 58.240 »««. 9. 6,870,657 • ^ 64. 47,060 ". 10. 4,867,680 » ». 56. 70,848 «."•. 11. 6,678 648 »•. 66. 4,766 »". i8o KEY TO BEEBE'S 67. 697 »«. 30. 86 vrs. 68. 758 **\ 31. 840.413,978. 59. 579 ««. 106-1. 79,984. 60 796»«. 2. 76.908. 61. 975 »«. 3. 72.1116. 62. 768 ««. 4. 87,047. 63. 687 »*'. 5. 89,229 m. 876 »••. lot- 6. 74.077. 65. 8.607 >«. 7. 81,212. 91M6. 6.908 '«*. 8. 76 464. e7. 8.007 «'•. 9. 81.195. 68. 7.906 «». 10. 75,826. 68. 47.660 »•. 108-11. 79,456. 70. 76,306 >•«. 12 82,270. 71. 67,067 ^»^. 13. 84,588. ri. 67,642 '•». 14. 78,351. 102- 1. ♦1,483. 15. 78,548. 2. 728.525. 100-16. 95,050. 3. 7,502. 17. 97,295. 4. 7 bbl.; 4 gal. left. 18. 97,187. 5. 684 jral. 6. t6,8i5. 10. 98.616. 20. 97,489. 7. 197 mi. 110-1. 4,246,909. 8. $661. 2. 8,737,469. 9. 7.126. 3. 389,385 828. 103-10. 426. 4. 73,165,608. 11. $1,060. 5. 29,730,808. 12. 109 eheep. 6. 7.080,807. 18. Lost $900. 7. 291,864.906. 14. 1.796,266 apples. 8. 77.927,921. 15. $4,543. 9, 730,868.068. 16. 59.888.884. 10. 20,679,828. 17. 136 tons. 11. 7.779,807. 18. 337,022. 12. Impossible. 104-19. $19. 13. 39,299.876. 20. 191,284 yd. 14. 421.769,927. 21. Lost $50. rrcm. 15. 670,796.919. 22. 14, horses and $50 16. 6.479.979,692» 23. $384. 17. 4,585,944. 24. 421 A.; $37,309. 18. 5,955,446. 25. $6 »« '». 19. 1,.540,080. 26. $4,456. 20. 772.926.042. 105-27. 3.743,520 ft. ^ 21. 379,239,951. 28. as mi. 22. 550,842 699. 29. NotbiuK. 23. 6,953,1 40.080i FIRST STEPS AMONG FIGURES. i8i \U. 490,019,928. 24. 147,283 '■'^. 25. 4.311.281,464. 25. 6,485 "0 1 1 1-2U. 78,(>.-iO,000. 26. 46.372 •". 27. 5.730. 114-27. 14,735 ♦••. 28. 0.800. 28. 17.399 "'. 29. 0,320. OtK). 29. 68.591 ''*". .JO. 87..-)O0.O00. 30. 147.904 M«. 112-31. 172..')OO.0O0. 31. t».008 *■"*. 32. 10.205.000. 32. 13,508,"*. 33. 444..500.000. ;«. 49,807 »<». M. 18.700,000,000. 34. 1.38,709 »•»••. 35. 4,XW,4OO.0O0. 1. 79 »•«". 30. 17.983 000. 2. are »«. 37. 51.300,000. 3. 23 "••00. 38. 6.887.200. 4. 702.196 ♦«. <». 59,073,000. : 5. 37 a»38««. 40. 40.^43,000. 116-1 486 '.»•». 41. 3.407.4;%4.000. 2. 6.a57 "1". 42. 452.27(5,040 000. 3. 790 ".»<><>. 43. 3,5^9.038,200,000. 4. 90 «.»", 44. 4.827.581,000. 6. 873 «•"•'. 45. 6,409.800.000. 6. 98 so.ooo. ^ 1. 4.798.687 *-». 7. 468 '.'oo. 2. 87.806,790 '-*. 8. 2 005,600,000. 113-3. 709.589,079 »-8. 9. 7,968 »«». 4. 9.687.890 a-^ 10. 68,050 ^M. 5. 769 809,780 ••. 11. 7,960 ». 6. 6,589.679 »^. 12. 5,920.005.998. 7. 978,697.089 •». 13. 48.690 «>■"». 8. 76.808,697 *-«. 14. 79.080 TOO. 9. 7.049,680 » •^ llT-15. 1.785 TO"'. 10. 4.906,704 T-». 16 60.970". 11. 79.684,796 «-^ 17. 9.780 M. 12. 958.007,980 *-». 18 Impossible. 13. 893,798,400 »». 19. 740 1*7000. 14. 65.870,486 «-'. 20. 8.1K59 "«». 15. 870.956 •-«. 21. 0,095 «oo. 16. 75.680,390 •-». 22. 40;>.026,600. 17. 903.780 *^. Si. 78,096 «»>. 18. 9.586,090 »-•. 24. 790.000. 19. 89.607.980 •-•. 1. 947,004. 20. 5.809.759 *-\ 2. 707.079. 21. 4,375 « <«'. 3. 9,058,032. 22. 6.736 »»'. 4. 10,149.216. 23. 365 «'•. 6. 749.042,756. l82 KEY TO BEEBE'S a 7,043,688. 118-7. 78.JJ56.244. 8. SSIJ.S.'iS.eie. 9. 104.154,369. 10. 5,7.'i5.896. 11. 1 171 (>::, 764. IJ ;. IS. ;o u .08. If. IG, 17 ,^. 18 L 19. i,it.t. 1^*^,632. 20. 95 879.820. 21. 98.790.948 »-». 22. 49.688.182 •-». 23. 68.874,989 «->«. 24. 87,076.850 '•". 26. 786.647.997 ►>«. 28. 869.897.046 >^*. 27. 87,998,060 »«^««. 28. 79,684,968 ••". 29. 74.869.740 "«. 30. 675,846.090 ^»» 12&- 1. $2,010. 2. $34,995. 8. 490,000. 126-4. 17vr8. 6. 236 A.; $74 '»•.««•. a 15,148. 7. $42. 8. 123,685. 9. 38 mi. 10. 60 cU. 11. 3,553; 3,066. 12. $760. 1 27-13. $14,353. [A $27 left 14. 174 "-"A.; or 174 A. 15. 614 mi. 16. 22,999,800,925. 17. $2,250. 18. 22,032 solid ft. 19. $4,340. 128-20. $8,380. 21. $80. 22. $80. 23. $945. 24. 192 da. 25. $322. 26. .584 mi.; 64 mi. 27. 136 bu. 28. $45. 129-29. 118 »•■»»•. [rem. 30. 69 horses, and $58 31. 42 times. 32. 183 bu. ai. $82. 34. ?2 weeks. 35. 581 sheep. 36. 429 cts. 37. 388 lbs. 13- 1 ' > v T \ ij LE. j's and review. " UJt repeated, long remembered.** To be recited •• 5 in 25 five times," "2 in 6 three times." etc. a b c 25 6 10 16 3 10 9 8 12 5 2 3 5 2 4 3 2 3 4 3 5 4 I 5 3 2 4 d e 2 15 20 6 8 4 2 5 4 3 2 2 4 4 I I 3 5 / ir 20 4 IS ] [2 5 5 2 3 4 5 425 31 "5 in 25 five times" may be written 25-^-5 = 5, and is read "25 divided by 5 equals 5." I. At 3 cents each how many lemons can you buy for 1 2 cents ? See Teachers Edition, pp. 63, 73- FIRST STEPS AMONG FIGURES. 25 • Solution : If one lemon costs 3 cents, for 12 cents you can buy as many lemons as 3 is contained times in 12, or 4. If preferred, use the following: As many as there are 3's in 12, or 4. 2. 10 cents will buy how many marbles at 2 cents each ? 3. Joseph spent 12 cents for oranges, paying 4 cents for each orange ; how many did he buy ? 4. A boy sold a pair of rabbits for 25 cents ; how many oranges at 5 cents each can he buy with the money ? 5. Mr. Brown paid a boy 12 shillings for work, at the rate of 2 shillings a day ; how many days had the boy worked ? 6. Lottie spent 16 cents for candy ; she gave 4 cents an ounce ; how many ounces did she buy? 7. At 3 cents each how many marbles can Edward buy for 1 5 cents ? 8. Mary's brothers gave her 16 cents, each giving her 4 cents ; how many broth- ers had she ? 9. Harry has 8 dollars in the bank ; his father has put 2 dollars there for him each birthday ; how many birthdays has he Been ? 10. If one pineapple costs 2 shillings^ how many can you buy for 10 shillings ? 26 ' FIRSI STEPS AMONG FIGURES. 1 1. How many ink wells at 3 cents each can you buy for 9 cents ? 12. If the fare on the cars is 4 cents to a certain village, how much is the fare for both ways? 13. Eight little girls were in the woods looking for violets ; only 3 girls found any ; how many found none ? 14. How many quarts in 3 gallons? 15. How many pailfuls of beans will it take to fill an eight-quart basket, if each pail holds 2 quarts? 16. Fred has 5 apples, John has one, and Harry has 3 ; how many have all ? Read the following numbers: (17.) 321468; (18.) 108320; (19.) 716381; {20.) 400750; (21.) 604025 ; (22.) 700006 ; (23.) 800000; (24.) 70016; (,25.) 215000; {26.; 380500; (27.) 50000. It is an excellent exercise for the class to write the foregoing numbers in words. Write in Arabic : 28. Forty-nine thousand seven hundred sixty. 29. Ten thousand ninety. 30. Three hundred seventeen thousand nine hundred thirty-one. 31. Nine hundred thousand one hun- dred one. 32. Four hundred thousand forty. FIRST STEPS AMONG FIGURES. 27 33. Six liundred thousand. 34. Twt) hundred ninety-one thousand rfive. 35 Thirty thousand ten. 36. Add 32. 21. 13, 3, 32, 22, 12, 33, 32, 13, 23, and 2. 37. Add 3, 31. 23, 3, 33. 12, 2. 33, 21, 32, 22, 33, and 23. 38. Add 33. 21, 23, 31, 12, 3. 23, 13, 32, 20, 32. and 12. 39. Add 21, 33, 3, 12, 30, 23. 33. ^^ J3, 33, 21, and 33. 40. Add 32, 3, 23, 33, 13, 21. 23, 33, 30, 20, 12, 2, and 21. 41. Add 23, 33, 2, 30, 33, 13, II. 32, ^3, 33, 21. and 32. 42. Add 43, 21, 4, 14, 32, 23, 42, 34, 24. 42, 33. and 4. 43. Add 4, 23, 42, 21, 3, 44, 43. 23, 31, 14, 23. and 31. 1. What cost 3 books at 4 shiUings •each '" 2. Mary rode in the swing five times, and Jane 4 times ; how many times did they both ride ? 3. How many marbles at two cents each can you buy for 6 cents ? 4. Jane saw five doves on the ground ; three of them flew away ; how many re- mained on the ground? See Teachers' Edition, p. 69. 28 FIRST STEPS AMONG FIGURES. 5. Bessie went to school 5 days, and Mary went 3 ; how many more days did Bessie go than Mary ? 6. How many times can I take 2 mar- bles from a pile of 8 marbles? 7. A little girl had 20 cents ; how many four-ceni lead pencils can she buy with her money ? 8. A boy walked 2 miles each day for 4 days ; how far did he walk ? 9. A boy having three five-cent pieces lost two of them ; how many cents had he left? 10. A little girl ate 3 buckwheat cakes for breakfast, and her brother ate 4 ; how many did both eat ? 1 1 . Lewis gave 4 boys 4 marbles apiece ; how many did he give them all ? 12. Jennie spent 5 cents for raisins, 3 cents for candy, and 2 cents for a stick of gum ; how much did she spend ? 13. On the east side of a house there are 5 windows ; 3 of them are open ; how many are closed ? 14. A good boy carried 8 pails of water for his mother on Monday, and 5 on Tues- day ; how many more did he carry on Monday than on Tuesday? * 15. How many quarts in 4 gallons? 16. Among how many children can I FIRST STEPS AMONG FIGURES. 29 divide 15 plums that each may receive 3 plums? 17. There are 16 towns in Ontario County ; if you learn the names of 4 of them each day, in how many days will you learn the names of all of them ? 1 8. Charles had 4 marbles and his broth- er gave him 5 ; how many had he then ? 19. Henry had to stay after school 5 minutes for whispering, and 2 minutes to solve an example ; how long did he have to stay ? 20. If John gets 5 cents for husking one bushel of corn, how many bushels must he husk to earn 20 cents ? 21. If a boy traded a knife worth 10 cents for a top worth 5 cents, how much did he lose ? 22. How much will a boy earn in 4 days at 3 shillings a day ? 1)D iTioN Table. fs and review. 6 3 a 3 7 5 4 5 6 b 2463 7 3 4 5 9 7 12 II 9 7 10 & See Teachers' Edition, p. 74. 30 FIRST STEPS AMONG FIGURES. C d 7.524 7637 6734 3567 13 12 58 10 II 9 14 ^ / 5246 3524 3456 7456 8 6 9 12 10 9 7 10 g h 637 524 7 3 4 567 13 6 II 10 8 II Subtraction Table. fs and review, a b 13 12 5 8 10 II 9 7 6734 3465 524 7732 c d 14 9 7 12 II 9 7 10 7 3 4 5 ^734 7637 5246 FIRST STEPS AMONG FIGURES. 3 1 e / 8 9 10 13 6 10 6 10 5 4 7 7 3 6 4 5 3 5 3 6 3 4 2 5 1? 9 8 II 8 II 6 5 3 5 6 7 645 624 Multiplication Table, fs and review, a b ^375 2463 3456 7345 18 12 35 c 30 14 12 24 d 15 7 6 5 2 7 3 4 4 7 3 21 6 3 5 6 30 18 7 7 42 35 6 16 49 5 3 e 2 4 4 5 6 6 3 7 / 5 2 4 5 4 6 15 8 20 36 21 20 10 24 32 FIRST STEPS AMONG FIGURES. 6 3 7 3 7 4 5 5 h 2 4 6 7 42 9 28 25 [2 28 Division Table. 7'^ r tf «^ review. 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, if he 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 miles each day } 29. An orange cost 6 cents and a cocoa- nut 3 times as many. How much did the cocoanut cost? 46 URST 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 man can walk a mile in 10 minutes, he starts from his home and walks to town in 5 minutes. How far from town does he live > 35. I had 10 cents. I bought 2 two-cent stamps, and gave 4 cents to a poor little boy for bread ; how many left for candy ? 36 There arc 9 ten o'clock scholars this morning and one more than one third of them left their books at home. How many of them brought their books ^ 37. Willie had 13 cents and he spent 6 of them tor 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 $ to 61 and 60. Review counting by 4*3. Count by 5's from 3 to 63. Count by 5!s from 4 to 64. 345 2. 35 3. 405 452 454 352 553 542 544 245 135 255 523 523 533 345 454 441 521 542 355 453 5 523 45 352 344 35^ 534 555 424 435 434 345 5- 505 6. 305 532 453 543 454 344 444 325 535 354 543 413 532 355 553 445 324 435 354 513 214 543 454 543 455 545 345 544 353 553 353 48 FIRST STEPS AMONG FIGURES. Examples in Subtraction. 7. 7<^,6^7 8. 96.487 64.252 31.343 9. 79.689 10. 687,985 43.264 34,532 II. 795.869 12. 47.685 43.543 5.232 13. 96.758 14. 5.879 • 54.428 2,343 15. 764.358 16. 764.037 17. 423.703 201,223 23,010 12,40c 18. 647,094 19. 7A>^^7 24.070 3,053 20. 74,687 21. 695,047 30.435 252,004 22. 52^,012 23. 613.021 24 431,024 46,566 56.554 55.357 See Teachers' Edition, p. lOI. FIRST STEPS AMONG FIGURES. 49 25. 634,210 26. 820,132 27. 431,201 56,643 53/657 54464 28 942,031 29. 731,203 30. 841,310 66,466 64,636 63,653 31. 312,403 32. 420,314 33- 341.531 45.257 54.652 34.354 34. 213,042 35. 314.253 36. 453.621 36,527 51.646 32,365 37. 324,102 38. 231,430 39. 425.301 46,637 12,257 51,625 40. 430,221 41. 320,413 42. 534.102 52,147 63,257 62,637 43. 534,210 44- 130,241 45- 342,013 16,436 53.725 30,157 46. 703.524 47- 423.102 48. 624,130 26,352 47.056 61,563 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 M 3524 5. 4325 4352 3552 3445 5445 5334 3453 4523 4524 5452 5335 3445 3452 4334 5344 5523 2325 4354 4555 See Teachers' Edition, p. 103. FIRST STEPS AMONG FIGURES. 5r 6. 3542 7- 245 8. 5342 4354 4532 3534 2435 2454 4253 5043 5321 5425 3530 3^43 343 354 4254 4534 4245 5435 2345 5432 2353 5234 3554 540 4553 4322 3254 3425 2453 4342 5342 5245 5535 2435 9- 3045 10. 3452 4532 5544 5450 2^45 3345 4532 2534 5453 350 13H 4003 5045 5435 3453 3542 4234 4354 4553 5425 5345 4543 3522 52 FIRST STEPS AMONG FIGURES. Multiply: II. 2,314 12. 4,231 13. 2,014 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 31. 35.642 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 38. 53,426 29. 64,536 30. 35>642 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. 25,036 45. 52,064 6 6 6 Count by 6's from I to 6 1. 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. S's and review. a b C 74852 34 5 ^7 63 74 8 5 263 7 8345678345 10 8 13 II 9 14 6 II 9 14 12 10 9 7 12 d 37485 56783 / 2637485263 4567834567 8 13 II 16 8 6 II 9 14 12 II 9 7 12 10 g 74 8 5 83 4 5 // i 2 6*348 526 6 7867 834 15 7 12 10 8 13 II 10 15 13 5 10 Subtraction Table. 8's and revienr. a b c 10 12 7 15 8 5 43 86 13 II 9 7 12 10 II 12 14 7 845 67387 58472 63526 3847 d 9 II 6 8 10 ] 65434 / 14 5 13 8 13 II 9 14 II 16 63856 77848 36256 82537 42678 g 96 12 13 8 5 3 7 5 4 h i 10 10 9 7 12 10 15 II 3834 5676 43 5 84 7263 7485 See Teachers' Edition, p. 105. FIRST STEPS AMONG FIGURES. SS Multiplication Table. (5"j and ' review. /? ^ c 7 4 8 5 2 63 7 4 8 5 2 6 3 3 4 i6, 5 6 4030 _7 14 83 4 5 6 7 35 8 3 16 18 4 21 48 9 28 20 48 12 ./ e / 7 3 7 4 8 52 6 3 7 4 8 5 2 5 5 6 1542; 7 8 2864 34 5 6 15 830 18 7 49 8 3 4 5 35 12 24 20 10 ^ h i 6 3 7 4 8 5 2 6 3 4 . 8 52 6 _2_ 8 3 _4 5 6 7 8 6 : _z 83 _4 3621 56 12 32 2512422424 5640624 28948 1430 43 8 7 6 b C 40 16 21 2048 35 16 18 12 54356 7834 73625 84748 5263 35245642 15 56765 / 24640 28 64 15 8 21 36 43 8 7 8 34 7 <^ 74873 62548 5236 g 1020243249 54387 h i 18303225 12 422456 12 65456 7883 25847 36852 6374 See Teachers' Edition, p. 105. 56 FIRST STFPS 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 Maiy } 4. John had 8 cents ; he lost 5 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 ? 9. 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 each. How many marbles did he buy ^ 1 1 . 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. io8. 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 ///r^^ oranges cost I2 cents^ 18. If 8 horses eat 24 bushels of oats in a week, how many bushels will one horse eat? 58 FIRST STEPS AMONG FIGURES. 10. 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 in each 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. What cost I bushel of oats, if 7 bushels cost 35 shillings ? 26. 12 dollars will buy how many birds at 2 dollars each ? 2y. If 5 hens cost 20 shillings what will I hen cost ? 28. How many peaches at 2 cents each can you buy for 16 cents ? FIRST STEPS AMONG FIGURES. 59 1. 5304 2. 453 3- 5243 4243 304 4524 3524 5035 3452 5452 4540 5335 535 3235 3543 4343 2424 4254 3425 5353 5435 5254 4545 3542 2542 3424 5354 4335 4353 4435 3454 2532 3423 5543 4245 5354 4325 5454 3543 4 4305 5- 3544 3450 5434 5234 4355 2543 3243 5455 5424 3213 4552 4543 2345 2355 5433 5432 2253 4343 4534 2524 3445 3455 2343 3532 4554 See Teachers* Edition p. no 6o 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. 6t Count by 6*s from 2, 3, 4 and 5, to 62, 63, 64 and 65. Read the following numbers : I. 750000748 2. 90000047 3. 680000740 4. 700746000 5. 12003000 6. 750908716 7. 801000071 8. 679374819 9. 7 1 50 1 6390 10. 900060000 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 sii7n or amoimt. 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. fifty mil. nine hun. th. 13. Write in Arabic eight mil. five hun. forty-three th. seven. 14. Write in Arabic one hun. mil. six hun. thirty-two. 15. Write in Arabic seventy-five mil. three hun th. 62 FIRST STEPS AMONG FIGURES. i6. 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. Write in Arabic nineteen mil. five hun. th. 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. 6^ •35. Write in words I3,(X)4,020. 36. Write in words 300,216,000. 37. Write in words 232,341,519. 38. Write in Roman fourhun. sixty-three. 39. Write in Roman eight hun. forty-four. 40. Write in Arabic DCLXXVII. 4 1 . Write in Roman two hun. eighty-nine. 42. Write in Arabic XCVIII. 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 subtra/iend, 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 in subtraction, of each number as follows : in examples 321 Minuend. 45 Subtrahend. 276 Remainder or di£ Subtraction. (O (2.) (3) 8»342.053 474.377 73.520,031 243.075 81,400,253 2,302,675 (4) (5.) (6.) 71,420,035 241,068 93,520,014 5,700,043 4.531.024 765.337 (?•) . (8.) (9.) 74,200,352 5,402,568 6,314.025 565,068 73.500.241 212,376 (10.) (II.) (12.) 94,002,531 7,006,763 81,350,024 576,546 34,200,156 7,620,478 See Teachers' Edition, p. u8. FIRST STEPS AMONG FIGURES. 65 (13) (14.) (15) 7»352,o34 95,300,421 863,005,241 181,376 420,376 2,307,374 (16.) (I7-) (18.) 93,510,042 35,200,416 7.300,425 46,350,357 2,540,348 602,557 (19.) (20.) (21.) ^3,001,425 95.320,041 85,241,300 5,005,743 4460,474 7.534.720 (22.) 94,300,521—570,376=? (23.) 42,530.014—6.765,068=? (24) 7i»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 muliiplication. Multiplicand. One of the equal numbers is called the viultiplicand. 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 isihe 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 multiplication. Multiplicand. The number taken, (or multiplied,) is called the multiplicand. FIRST STEPS AMONG FIGURES. 6/ Multiplier. The number by which we multiply, (or which shows how many times the multipli- cand is taken,) is called the multiplier. Product. The result of the multiplication is called \^^ 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. I. 35,246 2. 42.356 7 7 3 35.642 6 4- 57.463 7 5. 35.642 7 6. 35.246 8 7. 57.463 8 8. 35.246 6 9. 46,058 8 See Teachers' Edition, p. 119. 68 FIRST STEPS AMONG FIGURES. 10. 685,037 II. 485,067 12. 857,046 % 7 6 13. 637,058 14. 364.078 4 7 15. 3.640,758^8=.? 16. 7.583.640x7=? 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 19- 2.635 32 23 32 20. 364,526x43 = ? 21. 35.264x45=? 22. 4^563>^54=? 23. 534,652x46=? 24. 645.362x64=? 25. 63.527x76=? 26. 4675x67=? 27. 57.463x74=? FIRST STEPS AMONG FIGURES. 69 28. 357»426x63=? 29- 47063x67=? 30. 63,527x54=? 31. 352,746x34=? 32. 475.063x57=? 33. 630,574x76=? 34. 57.463x35=? 35. 357426x47=? Division. Finding how many times one number is contained in another is called division. Dividend. The number which contains the other is called the dividend. (If preferred, the number divided is called the dividend.) Divisor. The number which is contained in the other is called the divisor. (If preferred, the number by which we divide is called the divisor?) Quotient. The number found by dividing is called the quotient. 70 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 equal the dividend. Divisor, 7)4321 Dividend. Quotient, 617 — 2 Remainder. Short Division. (I.) (2.) (3.) 2)6,846 3)612,921 4)3,281,224 (4.) (5.) (6.) 3)9,182,715 4^2.082,836 5)5.304.535 (7.) (8.) (9) 3)24,152,721 2)128,166414 4)836,284,820 (10.) (II.) (12.) 3)21,186,912 4)2,032,828 2)106,416,818 See Teachers' Edition, p. 120. FIRST STEPS AMONG FIGURES. J I With remainders. ♦ 13- 7.372-3 = ? 14. 14.173^4=? 15. 16,427-^3=? 16. 823,975-3 = ? 17. 2,182,939-^4=? 18. 1,327.178-^-5=? 19. 954.546^4=? 20. 5•663,032-^3=? 21. 26,173,385-^4=? 22. 273.823-7-5=? 23. 925,588-7-6=? 24. 257.4294-4=? 25. 3,741,254^6=? 26 3/6.233--5=? 27. 3.804.168-7-7=? 28. 4.873.2904-5=? 29- 4.497.190^7=? 30. 3.880.9164-6=? 31. 3,146.1534-4=? 32. 172.4484-7=? 33. 20.740,9654-6=? Count by 7*s from 7 to 42. • Teach the division series with remainders in Teaciv era' Edition, p. 107, before solving these examples. 7^ Hksl slLI'b AMONG FIGURES. Addition Table. g's and review. a b c 695847369584736 4 5 6 7 8 . 94 5 6 7 | 8 9 45 6 101411 15 121167 II 13 121613 11812 d e f 9584 7|3 695 8| 27369 7 8 9 4 5 i 6 7 3 9 4 1 96789 i6 13 17 8 1 2!9 13 17 14 1 2! 1 1 13 10 14 18 5 8 4 4 5 6 7 3 7 8 9 13 10 14 II 15 9 5 8 4 7 3 4 5 6 7 8 9 13 10 14 II 15 12 Subtraction Table. p'.f and review, a be 13913 10 12 15 II 14 II 98768 3 7 4 8 3 d 14 18 9 12 14 89549, 15 13 10 14 9 4 5 7 10 6 r 6 9 5 7 4 5467 85 7 3 / 7 8 12 9 1317 13 16 12 8 8456 7! 98765 6 94 8 5 94 73 II 13 16 12 15 II 498765 748596 6' 8 5 963 h 7 16 10 14 II 15 12 4945678 3769584 See Teachers' Edition, p. 123. FIRST STEPS AMONG FIGURES. 75 Multiplication Table, gs and review. 69584 45678 73695. 94567 c 8473 8945 63 12305435 !64 36 28 15 e f 8275 4967 2445305632 d 69584! 73695 6789 4 I 56789 36 61 40 72 1635 18 42 72 45 32 18 42 21 695847316958473 89456789456789 48 81 20 40 24 49 24 154 36 25 48 28 56 27 (/s and review. Division Table. a b c 32 563045 2463 12 305435 1643628 18 87654 48596 94567 73695 5 d e 42 1835 1672I4063 367245 76549 ; 87689 6 3 7 4 81 5 9 6 9 5 8949 8 4 7 / 2 32 2042 21 4 5 6 7 8 4 7 3 // 48 81 49 2440 18 5/ 89765 2 9 69748 96 362448 28 5627 25 4867895 9384735 Sec Teachers' Edition, p. 124. 74 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 u mained ? 3. If a man can draw 7 loads of sand in I day, how many days will it take him to -draw 42 loads ? 4 In a class-room there are 6 seats, and €ach seat will hold 8 pupils ; how many pupils can be seated in the room ? 5. Six sheep were put into a flock con- taining Sy ; 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 11 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 y's from i and 2 to 71 and 72. 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 $31 U read thirty- two dollars. y6 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 i Tiother gave her 9 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 5535 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 5. 4532 5324 3254 4543 4325 5435 5543 3354 3454 5445 3434 4554 2345 3425 5432 5432 4545 2345 3454 4553 5235 5434 4523 4423 5445 3545 4354 5434 5432 6. 4356 7 . 3564 8. 6345 3645 6453 2436 5234 2566 6563 6563 4635 3656 3656 5364 5365 4345 6532 6534 5456 3656 4655 6563 4364 5362 3625 5635 6656 5356 6556 3545 6545 3446 6666 78 FIRST STEPS AMONG FIGURES. 9- 3456 JO. 6354 . 6563 566s 5635 6536 6366 3456 3556 636s 6645 6653 4536 5546 5663 3665 6636 6236 6355 6463 5663 3554 11. 684,632-47,757=? Ans. 636,875. 12. 43,126-2,765 = .^ Ans. 40,361. 13- 935.643-77.387=? Ans. 858,256. 14. 754,231-26,154=? Ans. 728,077. 15. 836,425-68,268 = ? Ans. 768,157. 16. 364.135-71.543 = ? Ans. 292,592. 17. 463.845-98,536=? Ans. 365.309. 18. 763,843-46,452 = ? Ans. 717,391- FIRST STEPS AMONG FIGURES. 79 19. 6453 20, • 3456 21. 3645 3566 6566 6556 4635 5635 2663 6326 4364 6356 5663 5662 5565 4566 6556 4636 3656 6636 6353 6365 3665 5664 5636 5326 6656 6656 6553 3533 5625 3466 6665 5564 6536 3456 4563 5462 6363 22. 5663 23 6543 6656 2345 3546 6234 6635 5623 5366 4562 4653 3456 6636 6566 3565 5335 6366 4666 5436 5656 6563 6535 3666 6666 6525 3553 See Teachers' Edition, p. 126. 8o 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. 1. 57.364x304=? 2. 47*563 X 504=? 3. 346,752x706=? 4. 630,475 X 607=? 5. 50.746x467=? 6. 46.375x564-"=? 7. 3.526x736=? 8. 6.375x657=? 9. 68,574x87=? 10. 47,586x68=? 11. 647,583x85=? 12. 364.758x48=? ^ 13 105,743-4=? 14. 3,176.207-5=? 15. 1,418,497-4=? 16. 22,477,527-^6=? 17. 2,629,369^4=? 18. 23,176,234-^5=? ♦The significant figures are i, 2, 3} 4* 5, 6, 7, 8 and 9. See Teachers' Edition, p. 128. FIRST STEPS AMONG FIGURES. 8l 19 30,249,7424-4 = ? 20 2,^25,160-^6 = ? 21. 31.760,850 7-? 22. 34,459.582 6 =? 23. 34,574,849 63? 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 : 24. 10,421^4 = ? 25. 44,593.672^7=t 26. 2,115,885-^5=? 27- 37.548,510-^-6=? 28. 25,233,295-^7=? 29. 37,229,621^8=? ^30. 42,519,206-^-6=? 31. 40,379,014-7=? 32. 52,584.387^8=? 33. 20,326,941-^8=? 1. 874,895-^201=? 2. 728,556^201=? 3. 12.756.034-^-3,012=-? 4. 8,604,338-7-2,031=? 4* 82 FIRST STEPS AMONG FIGURES. 5. 16.154,855-- 3,024 = ? 6. 49,486,701-^2,032=? 7. 2,595,960^4,023 = ? 8. 1,460,703-^403 = ? 9. 10.357,440-3,024=? 10. 14,244,539-=- 3,014--=? 11. 13,822,604^4,034-? 12. 25,079,486-7,036=? 13. 38.968.336-^6.035=? 14. 275.881,300-5,046=? 15- 334.654.184-70,364=? 16. 349,143,867-^6,057=? 17. 226,396,593-60,453=? 18. 18,651,776-4,076=? 19. 526,026,567^7,046=? 20. 236,326,931-5.074=? 21. 39.361.095^6,024=? 22. 540.611,445-7,068=? 23. 450,299. 1 32--6o,378=? 24. 3 2 7.040,029 - 5 ,048 = ? 25. 413.535496-70.485 = ? 26. 241,993,12-5,086=? 27. 546,695,551-^80,574=? 28. 465,406.942^7,068=? 29. 289,561,188^6,075 = ? See Teachers' Edition, p. 136. FIRST STEPS AMONG FIGURES. 8^ 30. 4563 31 . 5465 32. 4536 3656 6536 5665 6366 6653 4354 5665 3566 6635 4656 6632 5666 6546 5366 6563 6665 6556 6356 5656 3465 3665 3456 6634 5636 6543 5663 6565 5655 2556 3663 6366 6665 6356 5621 5,326 6565 33. 5634 34 • 3456 6565 6536 3456 5665 6563 3566 4656 6343 5665 4656 6363 3565 5636 6666 3565 5356 6366 4663 5636 6536 6653 5465 3546 6323 FIRST STEPS AMONG FIGtJRES. 1 . A boy had 1 5 marbles and lost all but six of them ; how many did he lose ? 2. Mary comes to school 5 days in a week ; how many days does she come in 8 weeks ? 3. There are 55 sticks of candy in a 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 9 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 14 cents, the oranges cost as much more than the peaches as the difference between 24 cents and 14 cents or 10 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-five cent piece, a ten-cent piece, a five-cent piece and a three-cent piece ; how much money had he? 9. If I had 4 apples and found as many more, and ate two of them, what part of a dozen had I then ? What are they worth at 12 cents a dozen ? 10 I have a clock that strikes every quarter hour ; how many times will it strike in 9 hours ? 11 William spent 12 cents, James spent one third as many and three cents more; how 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 ? 15. 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 many did the little girl sew ? 86 FIRST STEPS AMONG FIGURES. 1 8. How many boxes of wafers at 6 cents 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 32 a yard ? 20. How many four-horse teams can be arranged from 20 horses ? 21. Three fields have each 3 trees, under each 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, ond 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 9 acres of grain ii> one day, how many acres ca 1 6 men cut in a (lay ? 28 If 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 lor 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 in a day, how many cords will i man cut in a day ? 33. If 3 men cut a pile of wood in 9 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 ? 88 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: 1. 750406300. 2. 4576000043 3. 860000307. 4. 1 5000045001. 5. 7845678437. 6. 37 1 474 1 5006. 7. 47583000000. 8. 370015300 9. 40C00036700. 10. 71000100000. Write in Arabic the following numbers : 1. Fifteen n^l. ten th 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. 89 9. Five mil. ninety. 10. Write in Roman nine hun. thirty- four. 1 1. Write in Roman seven hun. forty-six. 12. W^rite 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 ne.xt 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 i of the ist (in one number.) 16. Write 3 units of the 8th order, 5 of the 7rh, 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 } ^O 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 Sy 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? 9 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 -$6$ 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 ^ICXX). 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 in these three summer months ? See Teachers' Edition, p. 143. FIRST STEPS AMONG FIGURES. 9 1 15. There are i68 acres in Mr Fox's farm, and j\Ir. Norton's farm contains 89 acres more than Mr. Fox's ; how many acres in both larms ? 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 ant 98 bushels in another; how many bushels in the third bin } 2 1. A boy having 85 cents, bought a top for 1 8 cents, a ball for 2 5 cents, and some oranges for 27 cents; how many cents had he left? 22. If there are y6 bushels of corn in a 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 mUlion eight hundred thousand take fifteen thousand ninety. I. 4567 2 . 7645 3. 4756 7654 5774 3567 6347 6547 6635 7576 5732 7456 4757 7665 6573 5674 4576 5746 7565 5766 7455 4757 7537 6567 5664 6452 7736 7575 3776 6564 2346 6565 6475 4. 7546 5 . 7654 4757 6537 5675 5465 6757 7756 7577 4675 4664 3456 7357 6747 5675 7564 4567 ► 6475 7654 7746 4735 4653 See Teachers' Edition, p. 144. FIRST STEPS AMONG FIGURES. 93 6. 3456 7- 6754 8. 5746 7653 5673 7457 6575 7566 6575 5747 4375 7^7 7356 7647 5734 5676 5774 4757 4747 6757 7676 5635 7577 AS^7 4567 4652 ^7}>S 7476 5767 S^SJ 5745 77^6 AS7^ 6574 5473 7A^S 9 4576 10. 6457 7654 7564 5773 5735 7566 4657 6757 7576 5674 4657 7365 3456 4537 7573 3456 5746 6765 7757 7S7^ 4567 4457 3456 94 FIRST STEPS AMONG FIGURES. 11. 4,362,516-754,359=? 12. 736,952 — 78,672? 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. 463,521-186,357? 20. 483,654-91,987=? 21. 635,245-12,567.? 22. 837,524-259,286=? 23. 43,452,431-4.238,865=? 24. 756,324-85.543-=? 25. 4.738,536-973.659=? 26. 86,357-7,269=.? 27. 34.023-9,876=.? 28. 45.300,435-2.430,526=? 29. 74, 200,03 2 — 5,1 40,05 4 = ? 30. 43,250,001-5,726,025=? 31. 3,400,564-210,739=? 32. 45.700,325-8.730,153=? 1. 796,845 <89=? 2. 479,685 x98=? FIRST STEPS AMONG FIGURES. 95 3. 68,975x79=? 4. 647.583x467=? 5. 68.574x456=? 6. 486,075 X 807= ? 7. 58,697x64=? 8. 9.687x75= ? 9- 4.796 X 39= ? 10. 85,974x84=? 1 1. 79,685 X 69= ? 12. 68.974x79=? 13. 49,786x85=? 14. 59,068 X 604= ? 15. 70,968x907= ? 16. 70,309 X 709= ? 17. 860,479x709=? 18. 759.068 X407 ? 19. 47,096x609= ? 20. 748,609 X 507= ? 21. 58,709x608= ? 22. 96,047 X 709= ? 23. 97,806x597=? 24. 79.689x4,759=? 25. 896,748x6,978= ? 26. 7,580,065-1,251,298=? See Teachers' Edition, p. 145. 96 FIRST STEPS AMONG FIGURES. 2T. 3,740,683-923.754=? 28. 735»035 -26,326= ? 29. 7430.246- 7,503,472=? I. 3,241,402-^7=? 2. 5,678,005-7-8 = ? 3. 3,802,457^8=? J^ 49,167,544^9=? 5. 5.076.335-9=? 6. 3»372,o8i-^9=? 7- 3.725.891-8=? 8. 52,301,166-^8=? 9. 48,094,605^7=? 10 43.719.125-9=? 11. 52,629,186^8=? 12. 40,224,713-7=? 15. 67,217,191-9=? 14. 608.226,845-9=? 15. 37.100.695-^8 = ? 16. 460.241.323^7=? 17. 4,781,158,859-^6=? 18. 4.885,157.761-7=? 19- 389,158,560-^6=? 20. 5,240.869-7=? 21. 3.356.977-7=? See Teachers' Edition, p. 14& FIRST STEPS AMONG FIGURES. 97 22. 6.727741-8=:? 23. 5,115,271-8 = ? 24. 21,367,398-^5,024=? 25. 3,722,901^-607 = ? 26. 14,204,241-403 = ? 27. 158,632,783-30,135=? 28. 38.693,395-6,024 = ? 29. 38,464,365^50.396 = ? 30. 31.970,764-^7.048 = ? 31. 3,891,687.541^60,475=? 32. 2,925.490,533^60,479=? 33. 462,857,740^8.069= ? 34. 2,934.401,497-^70,586=? 35. 226,663,766^60.379= ? 36. 194,513.933^40.297=? 37. 1 84, 1 1 0,903 -^ 5,048= ? 38. 390.761,546-6,037= ? 39. 32.688,027,778-5-70,486=? 40. 674,476,820-^9.037= ? 41. 27,971,095-6.074=? 42. 355,2I2.265-^5,036=? 43- 23.377.796-5.024= ? 44 452,113,508-^7,056=? 45. 299.019,935-^406=? 98 FIRST STEPS AMONG FIGURES. 46. 2,408,592,665 ^ 5,064= ? 47. 36,979.544-^8,026=? 48. 426,696,721-7,014=? 49- 437.791.333-6.038=? 50. 216,526,004-^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. 2,202,155-^463=? 57. 251,864^361=? 58. 434,801-573 = ? 59. 268,500^463=? 60. 463,7994-582=? 61.* 356,116-365=? 62. 357,243^465=? 63. 403,123^586=? 64. 4i4,6i7--473 = ? 65. 4,174,696^485=? • When the left hand figure of the divisor is equal to the left liand figure of the dividend, if the next figure of the divisor be greater than the next figure of the di\a- 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^ ^' 5499.513^796 = ? 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 = ? 72. 268,439,581^4,657-? 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 ? 3 Henry has 8 cents in one pocket and 9 cents 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 I pencil cost 4 cents, what will 8 pencils cost ? 8. George bought a knife for 8 shillings, aball for 5 shillings, and a bat for 2 shil- h'ngs ; what did he pay for all ? 9. Marcus spent 8 cents for lemons at 4 cents each ; how many lemons did he buy ? See Teachers' Edition, p. 150. lOO FIRST STEPS AMONG FIGURES. lo If 5 men cut ten cords of wood in a day, how many cords will 7 men cut ? 11. If 2 men can dig a certain ditch in 4 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 ? 1.3. How many rods of wall will one man build in a day, if 3 men build 9 rods in one day ' 14. li 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 ? 1 7. 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 ? 30. Jane bought 5 figs for 3 cents each, and a yard of cloth for 9 cents ; how much did -she pay for all? See Teachers' Edition, p. 151 FIRST STEPS AMONG FIGURES. lOI 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 week cost at 4 shiUings 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 1 1 peaches, he ate 3 and gave his sister half of the rest ; how many did he keep ^ 31. How many marbles, 2 for 4 cents, can you get for 18 cents? I02 FIRST STEPS AMONG FIGURES. EXAMPLES FOR THE SLATE. If the pupil will use small numbers in* stead of the large ones in tlie 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 greatly assisted. 1. A man had $3,210, he spent $978 for wheat and $74^ 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. From a 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 ; what did he pay for all of the grain } 9 From the sum of 397 and 6798, take 69. See Teachers' Editon, p. 152. FIRST STEPS AMONG FIGURES. IO5 10. The difference between two numbers is 347. and the less number is 79, what is the greater number } 1 1. A man died leaving ^5600, of which he gave his wife $2,8CK), 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 gain 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* '»«^ thousand nine, 79 thousand 847, and 9 n*. lion 79 thousand 8. 17. How many tons of hay at $18 a ton must be given for 16 horses at $153 each? 18. 639+91. 758+9.347 + 81.731 -i-9.34^ + 35,446+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 ot sugar; how much did the sugar cost a barrel ? 20. How mai/y yards of cloth in 6S bales, •each bale having 97 pieces, and each piece containing 29 yards ? 2 1 Paid $6 each for 75 sheep, and sold the flock for $400 ; did I gain or lose,and how much ? 22. How many horses at $165 each can *>e 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 $g 3r if - :, and with the money received for it ^'" ^nt a farm of 135 acres ; what did he ^4y an acre tor 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 ft^orth $475. He owed debts to the amount FIRST STEPS AMONG FIGURES. I05 of ^^369 ; how much did he leave his family ? 27. There are 5280 feet in a mile ; how many feet in 709 miles ? 28. A man starts from New York on Tuesday morning and travels at the rate of 57 miles a day ; another starts from the same place Wednesday morning and follows on at the rate of 69 miles a day ; how far apart are they Thursday night .'' 29. James sold a grocer 96 eggs at 1$ 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. I06 FIRST STEPS AMONG FIGURES. I. 7657 2 . 3456 3 7567 4775 5767 4756 7777 7475 3675 3456 4567 6777 6735 7756 7SA^ 5677 5647 5735 7756 6775 6767 3457 7564 7476 6574 4677 3456 5767 3456 3456 4575 5767 7763 7717 7534 4567 5676 n't aT. 6467 2475 2345 5 4- 4756 • 7654 7577 4775 6645 5467 7734 7556 5675 6775 6756 3457 7467 5747 4575 7674 5647 6577 7777 7754 6452 7777 3567 7777 6776 4564 5643 5675 See Teachers' Edition, p. 153. FIRST STEPS AMONG FIGURES. I07 6. 456S 7. 6758 8. 8765 3785 8547 4658 8678 7868 7777 5786 8778 3456 8678 8888 6778 8888 8888 7l<^7 8888 3576 8878 5678 6758 77^7 8765 8687 8585 4876 5678 7777 5487 6786 3456 8888 10. 3748 8888 8675 7654 7887 4765 4567 7777 8888 7777 8888 5678 4567 8765 7777 4567 7777 8778 4565 7658 8486 Io8 FIRST STEPS AMONG FIGURES. II. 6758 12. 7658 13. 7865 7584 4576 4578 4676 6785 8657 ^77^ 5467 678s 3456 8878 8888 8888 4657 8888 8888 8765 3456 6547 6578 7777 3754 7857 5678 6678 5686 • 8753 7563 8578 4576 5885 67ZS 8687 4. 4786 15. 7684 8657 4578 6578 3456 7865 S^Z^ 4576 ^^^^ 8888 7654 3456 3567 7777 7777 8765 5648 3578 7385 8657 4637 4768 8386 See Teachers' Edition, p. 154. FIRST STEPS AMONG FIGURES, IO9 16. 3687 17 . 4837 18. 4587 8546 7584 8635 7685 8758 6754 4868 5875 8888 5784 8888 8888 7777 8888 6754 6548 6753 7578 3675 4584 4785 8888 7777 7777 8888 5678 5768 5678 8765 8654 8563 3857 4585 5785 6586 7848 8678 8465 5686 19. 8476 20, . 7586 5768 4767 4567 8658 7777 4875 7777 8888 8654 8888 • 3568 7654 8888 3867 8888 7777 4567 4685 8765 8568 4478 7777 7586 * 4825 88^7 8674 no FIRST STEPS AMONG FIGURES. 1. 4,570.365-323»456=? 2. 9»374.056-636,587=? 3. 685,700,365-296.314,537=? 4. 76.400,235-3,234.567=? 5. 38,500.684-8,769,876=? 6. 7.460,683-379,876=? 7. 375»6oo,735-83.735,829=? 8. 83,640,574-5,712,653-=? 9. 794.600.435-63.732,367=? 10. 74.300.375-53.620.547=? 11. 8.750.043-970,236=? 12. 6,713,021-6,873.213=? 13. 48,300.563-9,000,687=? 14. 487.500,564-65.730,637=? 15. 756,000,375-85.203.456=? 16. 6,847,000,346-367,020,654=? 17. 79,068x58 = ? 18. 80,479x74=? 19. 4,185x368=? 20. 968,579x798=? 21. 79.689x4,759=? 22. 79,867x6.897=? 23. 85,765 X 8 1.072=? 24. 49.678x9,876=? 25. 497^896x8,659=? See Teachers' Edifion. p. 154- FIRST STEPS AMONG FIGURES. HI 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, lOO, looo, &c., annex as many ciphers to the multiplicand as there are in the ihultiplier. 379 X icx)=37,900 Ana. 26. 7,865 X 10,000=? 27. 573 X io=? 28. 68 X ioo=? 29. 6,320 X 1000=? 30. 875 X 100,000=? When there are ciphers at the right of either the multiplier or multiplicand, or of both, place the multiplier under the multi- 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.) 112 FIRST STEPS AMONG FIGURES. 34200 34000 1368 1026 1,162,800,000 31. 3,750x46,000=? 32. 46,300x350=? 33- 635,000x700-? 34. 27,500x680,000=? 35. 586,000x7,400=? 36. 490x36,700=? 37. 6,840x7.500=? 38 8,609x800=? 39. 67,900x870=? 40. 8.690x4.700=? 41. 480.600x7,090=? 42. 70,580x6,408,000=? 43. 706,900x5,078,000=? 44. 68,090 X 70,900=? 45.640,980x10,000=? I. 43,188,278-^9=? 2. 791.071,117^9=? FIRST STEPS AMONG FIGURES. II3 3. 6,156,712,635-4-8 = ? 4. 67,815,232^7 = ? 5. 6,928,288,028-9 = ? 6. 52.717.437^8=? 7. 8,808,273,807^9=? 8. 461,392,186-4-6=? 9. 49.347765-7=? 10. 44,160,343^-9=? Ji- 557793.576^7=? 12. 7,664,063,843^8=? 13. 8,044,i85,6o7-=-9=? 14. 461,093,406^7=? 15. 5,225,741^6=? 16. 529^762,735-7=? 17. 7,230,245^8-? 18. 86,274,817-^9=? 19. 716.863,843-4-8=? 20. 41,088.317^7=? 21. 30,884,751-^-7.058=? 22. 34,600.073^-6,032 = ? 23. 1,890,186-^5,178 = ? 24. 1,203,161,896^8,169=? 25. 5.279.490-814 = ? 26. 33,620,328-4-725=? Sec Teachers' Edition, p. 155. 114 FIRST STEPS AMONG FIGURES. 27. 92.197.364^6,257 = ? 28. 7,638,482^439 = ? 29 511,764.908-7,461=? 30. 946,526,656-6,397 = ? 31. 65.790.555-^6,847 = ? 32. 0,363,666^469 = ? 33. 18,827.247-4-378 = ? 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 nun^ber 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. 1. 793,468 10,000 = ? 2. 37,680^100 = .'* 3. 2,347,600^100,000=? 4 76,219,648-5-100=.^ 5. 372,938,641-^10,000,000=? FIRST STEPS AMONG FIGURES. 11$; To divide by any number with ciphers at the right. 78»673^700=? Divide both dividend and divisor by lOO, and cutting off the 2 figures at the right, and the example becomes — 7|oo)786i73 112—273 rem., or 112 — 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|ooo)6i2|37o(i3 quo. 45 162 135 27370 rem. i;i6 FIRST STEPS AMONG FIGURES. Hence the rule : To divide by any num- ber with ciphers at the right, cut off the ciphers ^t 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 cutoff from the rightof the dividend for the true remainder. 1. 18,228,211 -5-37,500=.!* 2. 5,142,762,131-750,000=? 3. 546,927,300-^687,000=? 4. 70,514,152^796,800=? 5. 8,734.758^10,000=? 6. 350.870,000^3,580,000=? 7. 3,278,300-^7,000=? S. 87. joo X 23,000=? 9. 7,162,323-900 = ? 10. 394.690.750-5,800=? 11. 29,850,010^3.750=? 12. From six billion six thousand six, take eighty million eight. 13. 27,752,320,172-^570,000=? 14. 2,910,144,700-7-36,800=? See Teachers' Edition, p. 155. FIRST STEPS AMONG FIGURES. I I 7 15. 178,576,495-- 100,000 = ? 16. 365,820,038-6,000 = ? 17. 475,308,056-48,600 = ? 18. Subtract forty-five million, one thousand ten, from forty-two million seven hundred thousand. 19. 441.937.000-^597.000=- 20. 30,500,857,231^3,780,000=? 21. 3,657,200^600=? 22. 47,096 X 8,600=? 23. 70,286,631^900=? 24 7960 X 100= ? When the multiplier is less than 13 the pupil should be taught and required to multiply but once through, multiplying by II or 12 as he has already been taught to multiply by 4, 5 or 6. 1. 78,967 X i2=? 2. 69,789/ II =? 3. 754.836 X I2 = ? 4. 845,768 X I2 = ? 5. 68,094,796 X 1 1 =? 6. 586,974 X I2 = ? See Teachers' Edition, p. 157. 1 1 8 FIRST STEPS AMONG FIGURES. 7. 6,579.687 X I2 = ? 8. 69,487,968 X I2 = ? 9. 9,468.579 X II=? 10. 479,658 X I2 = ? 11. 97.5^9.647^ I2 = ? 12. 6.975,897 X 12 = ? 13. 69.786995 X I2 = ? 14. 94,679.689x12=? IS- 979.896 X I2 = ? 16. 685.796 X II=? 17. 8.798,979 X I2 = ? 18. 49,897,697 X I2 = ? 19. 97,987,986 X I2 = ? 20. 7,989,985 X 12 = ? 21. 1,185,491,377-7-12 = ? '22. 546,569.461-5-11=? 23. 820,499,872-^-12 = ? 24. 1,044,922,315-^12 = ? 25. 9.438,575.969-12 = ? 26. 9,568.867,509-^11=? 27. 455.976,730^12 = ? 28. 956,219,625-7-12 = ? 29. 898,436,885-^12=? 30. 7.434.306,992-^11=? FIRST STEPS AMONG FIGURES. II9 FOR ORAL RECITATION. 1. A bov 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 y pounds cost 63 cents ? 5. A man spent $s^ ^^^en 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 i& apples ? Solution: If 6 apples cost 2 cents, iS 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 a/t^r hnding 10 cents, before he found it he had the difference between 25 cents and 10 cents, or 15 cents. If he had 15 cents a//^r losing 5 cents, ^/art he lost it he had the sum of 15 cents and 5 cents or 20 cents. If he had 20 cents a/ttr earning » cents, before he earned it he had the difference betweea ao cents .ind 8 cents or 12 cents. Sec Teachers' Edition, p. 161. 120 FIRST STEPS AMONG FIGURES. 8. What cost 20 figs, if 5 figs cost 2 cents ? 9 If 3 oranges cost 12 cents, what cost 7 oranges ? 10. What cost 12 lemons, if 5 cost 25 cents ? 1 1 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 ? 1 7. 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 in a 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. 121 22. If 4 men harvest a field of wheat in 8 days, how many days will it take i 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 do a 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 do a piece of work in 9 days, how many men can do it in 6 days ? 28. If 6 men can do a piece of work in 4 days, how many men will it take to do the work in 3 days ? 29. If 6 men can do a piece of work in 12 days, how long will it take 4 men to do 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 in a day? 122 FIRST STKPS AMONG FIGURES. 33. A girl 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, bow long will it take 6 men ? 37. If 3 pencils are worth 1 1 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 5 pieces, after which she gave her mother 9 pieces. She had left 27 pieces ; how many pieces had she at 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 9 vests worth ? 42. If 5 cords of wood cost ^24, what will 1 5 cords cost ? 43. What number divided by 3 will get 12? 44. At 10 cents a pint, what will a gallon of molasses cost ? 45. How many bushels of potatoes at 4 FIRST STEPS AMONG FIGURES. 1 23 shillings a bushel may be bought for 3 bushels of wheat at 12 shillings a bushel? 46 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 $S a ton ? 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 10 cents, how many can be bought for 15 cents ? 55. How many three-cent stamps can be bought for 27 cents ? 56. A boy caught some fishes ; he threw 124 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 I42, how many bushels of wheat at $2 a bushel will 3 bushels of clover seed buy ? FIRST STEPS AMONG FIGURES. 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 lootman who is 60 miles ahead, and walks at the rate of 3 miles an hour ? 67. *A man bought a span of horses for $100, paid S6o for their keeping, ajid 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 $S left ? EXAMPLES FOR THE SLATE. 1. In a certain church 28 pews rent at $35 each, 19 at $23 each and 37 at $1$ 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 for a set of furs, and $2 a yard for 23 yards of silk ; how much money had she left ? 8 13341*6 emigrants arrived in New York in 1867, which was 9.731 more than arrived in 1866 ; how many arrived in 1866? 9 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 37 x 96, and their difference is 18 X 27 ; what are the numbers ? 1 2. A earns $45 a month, and B earns 13 times as much lacking $490 ; how much does B earn in 8 months ? FIRST STEPS AMONG FIGURES. 1 27 13. If a house is worth $i,8oo and the farm on which it stands five times as much lackinj^ $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 ;^ioo 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 $9,840, 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 $29$ ; 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 in 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 wilJ 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 1 36 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 ? See Teachers' Edition, p. 166. FIRST STEP.^ ...»i«7.>^. FIGURE^S. I 29 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 1 3 tons of hay at $ 16 a ton, and 24 cords of wood at $$ 2l 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 ; it 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 ? 130 FIRST STEPS AMONG Hv.uKhS. 38. There were 84 sheep in fourptisturesp . there were 30 in the first and 24 in the second; if there were an equal number irk 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 ? 40. 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 2J cents a pound will pay for three hogsheads- of sugar, each containing 1080 pounds, at 12 cents a pound ? 42. What is the sum of the difference and sum of 1768 and 987 ? 43 A man deposited in bank at different times $397, $450 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 ir> 280 bags ; he used 136 less bags in which to ship 3(X> 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 1 1 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 ; bow 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 (i mile is 5280 feet) ? 54. An estate of $14350 was divided between a widow and two children ; the widow'?; share was 55450, 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 ^,850, and 95 more acres at $45 an acre ; he sold the whole at $$6 an acre ; did he gain or lose, and how much ? 57. A farmer bought 47 acres of land for 54.416, 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 ajar of butter worth 2^4 cents ? FIRST STEPS AMONG FIGURES. I33 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 9 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 736592, 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 difierence 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. TJ. Given the selling price and the loss, find the cost 7S. Given the cost and the gain, find selling price. 79. Given the selling price and gain, find cost. 80 Given the cost and the loss, find the selling price. FIRST STEPS AMONG FIGURES. I 35 I. 6745 2. 5847 3- 9337 5678 9576 4598 9867 4684 3765 6543 5967 9458 7698 8439 7^95 4759 4785 4739 5978 9478 5345 9647 4567 4869 8458 8975 3765 4796 4687 479« 4. 8451 5. 5747 6759 9835 7846 4696 9567 8739 6976 5684 7569 7938 8427 6456 9568 8845 3753 7587 5679 8946 36 FIRST STEPS AMONG FIGURES. d 5768 7. 4768 8. 5869 9576 9535 9768 4869 9849 4352 8753 7697 4675 4968 4857 9538 7495 9564 4678 ^478 5739 5765 4956 5647 4976 9875 9788 9738 5637 5347 4657 8592 8234 9876 ^ 7648 10. 7465 3752 5876 9746 4795 3859 8649 8432 3578 9594 9756 3745 6549 9458 8732 4637 5685 8358 7839 3543 5693 FIRST STEPS AMONG FIGURES. 137 II. 4538 i: 2. 9638 13. 9768 9764 5796 759^ 3797 4579 4879 8436 8947 9687, 9768 6896 4739 4593 9758 8645 8945 4563 3787 7657 9755 9568 4895 6879 4837 7537 5768 8795 4885 8597 4568 5938 4856 975<^ 9457 9589 6897 14. 6795 15. 5896 4569 9748 8769 7635 9345 5864 8876 9787 6789 4538 3954 7687 8456 4859 3789 • 5321 8375 4978 9999 9654 3478 7987 6457 9868 138 FIRST STEPS AMONG FIGURES. L 6587 17. 9768 18. 4879 4759 4593 9763 5896 7846 4598 9537 5937 8756 4678 9876 4975 4957 2345 6874 8436 3527 7589 6758 8498 4837 4957 6549 5692 8796 7856 7859 4587 7387 4537 7948 9765 6895 6595 3849 4576 4769 9674 3844 9487 3758 7989 5896 6847 4596 FIRST STEPS AMONG FIGURES. 139 19- 4532 20. 9476 7856 3869 4978 5458 9456 7567 3279 4835 4856 8769 7995 7654 8447 8579 9568 6432 4789 4976 6435 9845 7896 5637 4967 7948 8538 8654 7689 6739 9876 479« Dnvls (W. W.) FRACTIONAL APPARATUS, consisting of eight wooden balls, three inches in diameter, one whole, and the others divided respectively into halves, thirds, quarters, sixths, eiirhths. tenths and twelfths. In wooden box, 4 00 A MANUAL 6P^ SUGGESTIONS FOR TEACHING • • FRACTIONS, specially designed to accompany a Fractional Apparatus for developing the idea of Fractious. Paper, 12nif PP 43 Farnliain (Geo. L.) THE SENTENCE METHOD of teaching R(':u I in ir. Writinsr. and Spelling. A Manual for Teachers. 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