• • •• . .:.♦:••• 
 
 TEACnEllS' MANUAL"-^" • 
 
 FOR TEACHERS USING 
 
 ARITHMETIC BY GRADES 
 
 BY 
 
 JOHN T. PRINCE, Ph.D. 
 
 AUTHOR OF "COURSES OF STUDIES AND METHODS OF TEACHING," "METHODS OF 
 INSTRUCTION AND ORGANIZATION OF THE SCHOOLS OF GERMANY," ETC. 
 
 BOSTON, U.S.A. 
 PUBLISHED I;Y GINN & C().MPANY 
 
 1895 
 
Copyright, 1894 
 By JOKN T. PKINCE 
 
 ALL RIGHTS RESERVED 
 
 
PREFACE. 
 
 Considering the amount of time given to the subject of Arith- 
 metic in our schools, it must be admitted that the results are 
 lamentably poor. It is fair to presume that the faulty character 
 of the text-books in general use is accountable in part for these 
 results. With few exceptions American text-books in Arithmetic 
 seem to leave the teacher out of the account. Definitions, rules, 
 explanations, illustrative problems — in short, everything which 
 should be taught by the teacher is placed before the pupil to be 
 learned. It is true that the best teachers generally ignore these 
 false aids and ask the pupils not to refer to them. Yet, because 
 they are constantly before the pupils, the forbidden aids are often- 
 times u.nwisely used, and problems are performed slyly or thought- 
 lessly according to the rule or model solution. While these practices 
 are carried on to some extent in good schools, despite the care of 
 teachers, they are universally pursued in schools whose teachers 
 are guided by the text-book. Another fault of- text-books in com- 
 mon use is the insufficiency of problems for practice, and teachers 
 are compelled to make up the deficiency by placing upon the black- 
 board original problems or problems taken from Arithmetics other 
 than the regular text-book. The objections to this practice are : 
 (1) The danger for want of time of giving to the pupils unsuital)le 
 or poorly-arranged problems ; (2) the harm done to pupils' eyes 
 by close and long-continued looking, frequently before a lighted 
 Aviiidow; (J)) a- loss of tlie teacher's time in copying. 
 
 The series of books of which this Manuvd is a juirt is designed 
 to meet the above objections. It will be readily seen tluit the 
 pupils' books are not merely books of problems intended to supple- 
 ment the use of an arithmetic already in the hands of pupils, but 
 
 54 I /36 
 
IV PEEFACE. 
 
 are intended to be used independently as the only text-books needed. 
 Moreover, the very large number and variety of problems Avarrant 
 the assertion that the books contain all the problems that will be 
 needed for drill work. 
 
 The books for pupils are eight in number, arranged somewhat 
 on the lines of classification in City graded schools. The first two 
 or three books are intended for use in Primary schools, the last 
 book in advanced Grammar schools or High schools. The subjects 
 are divided as follows : 
 
 Book I. : Numbers from 1 to 20. 
 
 Book II. : Numbers from 1 to 100. 
 
 Booh III : Integers to 1000000, Fractional Parts of Numbers, 
 U. S. Money, Common Weights and Measures. 
 
 Book IV. : Whole Numbers unlimited, Common Fractions to 
 Twelfths, Decimal Fractions to Thousandths, Measurements, Busi- 
 ness Transactions, Denominate Numbers. 
 
 Book V. : Common and Decimal Fractions, Mensuration, Denomi- 
 nate Numbers, Business Transactions. 
 
 Book VI. : Mensuration, Denominate Numbers, Metric System, 
 Percentage and Simple Applications, Business Transactions and 
 Accounts. 
 
 Book VII. : Profit and Loss, Commission, Insurance, Taxes, 
 Duties, Interest, Banking, Stocks and Bonds, Exchange, Business 
 Accounts, Geometrical Exercises and Measurements, Eatio and 
 Proportion. 
 
 Book VIII. : Miscellaneous questions involving the making of 
 definitions, rules and formulas ; Algebraic Exercises, Involution 
 iiiul Iv.(iluiii)ii, Exercises in Geometry and Mensuration, Book- 
 keeping. 
 
 It has been tlie aim of the author to include in the books all 
 subjects that are likely to be needed in any school, rather than 
 
PREFACE. V 
 
 limit them to the possible needs of a certain class of schools. A 
 selection of subjects, therefore, as well as a selection of problems 
 under a given subject may be made to suit existing conditions. 
 
 The following advantages are claimed for the use of the Manual 
 and text-books : 
 
 1. The separation of teachers' and pupils' books, whereby pupils 
 may be taught properly and may not be given too great assistance. 
 Suggestions as to methods of teaching and drilling, as well as the 
 illustrative processes, explanations, rules, and definitions which 
 belong to the teacher to develop analytically are put into the 
 Teachers' Manual, while in the pupils' books are presented only 
 such exercises as are needed for practice. 
 
 2. The careful gradation of problems, by which pupils acquire 
 inductively a knowledge of arithmetical relations and principles, 
 and skill in arithmetical processes. This is in recognition of the 
 well-known pedagogical principles of proceeding from the known 
 to the unknown, and from the simple to the complex. It is advised 
 that this plan be kept constantly in mind by the teacher, and that 
 whenever a process is not understood or is not readily performed, 
 the pupils be taken back to processes which are well known and 
 which can be performed readily, and then be led forward by easy 
 steps until the desired end is reached. 
 
 3. Frequent reviews, and such an arrangement of exercises as 
 will enable pupils to have needed practice in the applications of 
 each principle, first by itself, and afterwards in connection with 
 other principles which have been learned. 
 
 4. The large amount of oral work, or work which may be done 
 without tlie aid of figures. Three objects of Mental Arithmetic 
 are sought in these exercises : (a) Illustration of principles and a 
 preparation for written work, (b) Development of the logical 
 powers, (c) Cultivation of ability to work with large numbers by 
 short processes. 
 
 5. The great number and variety of problems. The aim has 
 been to give the largest number of problems that will be needed 
 
VI PREFACE. 
 
 for teaching and for drilling in all grades. For this reason, and 
 because the forms of expression are varied, being taken from many 
 sources, there will be no necessity of giving supplementary drill 
 lessons on the blackboard. 
 
 6. Practicalness of work in respect to the character of the jjrob- 
 lems, and the solution of them. Care has been taken to give 
 problems which are most likely to be met in every-day life, and to 
 give tliem in a practical form. Many of the miscellaneous review 
 problems were made by mechanics, clerks, accountants, etc., with a 
 view of presenting conditions most likely to occur. 
 
 7. The introduction of statistics and facts of physics, astronomy, 
 history, geography, etc., thus enabling pupils to gain incidentally 
 much useful information. 
 
 8. The use of drill tables and other devices to save the time of 
 teachers. 
 
 In addition to the above features, some of which are distinctively 
 new so far as American text-books are concerned, there is the 
 separation of pupils' exercises for practice into small books some- 
 what on the lines of gradation in City graded schools. By this ar- 
 rangement there are gained greater convenience of handling and 
 economy of wear than in the use of a large book which is intended 
 to be used for several years by the same pupil. 
 
 The author is aware that the use of tliese books with accom- 
 panying manual in the manner proposed is not an experiment. 
 Essentially the same plan has been pursued in Germany for many 
 years, and it is confidently believed that American teachers will 
 readily recognize its merits. 
 
 The work of bringing together so large a number of exercises has 
 been materially lessened by the valuable contributions of tea'chers, 
 business men, and mechanics, who generously responded to the 
 author's request for assistance. Especial acknowledgments for such 
 service should be made to Misses Smith, Dale, and Barber of the 
 Practice Department connected with the Framingham, Mass., Normal 
 School ; Miss A. Roof and Mr. B. W. Drake, of Waltham, Mass.; 
 
PREFACE. VU 
 
 and Mr. L. F. Warren, of West Newton, Mass. Obligations for 
 valuable assistance in correcting proof sheets are also due to Miss 
 F. A. Comstock, of the Bridgewater Normal School, and Miss Amelia 
 Davis, of the Framingham Normal School. 
 
 Suggestions for improving the work and corrections of errors, 
 either in the INIanual or Pupils' Books, will be gratefully received 
 by the author. 
 
CONTENTS. 
 
 SECTIOX 
 
 I. General Suggestions 
 
 Ends — Course of Study — The Recitation — Methods of 
 Drill — Reviews — Tasks and Desk- Work — Illustrations — 
 Reading Problems — Analyses and Explanations — Trob- 
 lems and Indicated Operations — Eorecasting Answers — 
 Language. 
 
 II. First Steps in Number 
 
 Grouping of Pupils — Counters — Methods — Pacts of Num- 
 bers to Ten — Picturing of Problems — Use of Figures — 
 Number of Stories. 
 
 III. Notes for Book Number One 
 
 IV. Notes for Book Number Two 
 V. Notes for Book Nnmber Three 
 
 VI. Notes for Book Number Four 
 
 VII. Notes for Book Numlier Five 
 
 VIII. Notes for Book Number Six 
 
 IX. Notes for Book Number Seven 
 
 X. Notes for Book Number Eight 
 
 PAGES 
 
 1-8 
 
 8-13 
 
 13-21 
 
 21-34 
 
 35-5G 
 
 56-81 
 
 82-1 1 1 
 
 111-139 
 
 139-178 
 
 179-225 
 
,' > » ' ; 
 
 SECTION I. 
 
 GENERAL SUGGESTIONS. 
 
 Ends.-— Arithmetic like most subjects of instruction may be 
 said to have two values — its practical value as an aid in carrying 
 on the ordinary affairs of life, and the value it has as a means of 
 mental discipline. The ends of teaching Arithmetic therefore are 
 knowledge and power. It is well for the teacher to know at the 
 outset the relative importance of these ends for the purpose of 
 making a wise selection of subjects and of knowing how they 
 should be taught. If in our teaching of Arithmetic we keep only 
 in mind the direct use that it has as a preparation for business, we 
 sliall find that a small fraction of the time usually given to this 
 branch of instruction is sufficient. If, on the other hand, we con- 
 sider the possibilities of a proper presentation of the various topics 
 of Arithmetic in the cultivation of the powers of observation, 
 imagination, reflection, and reasoning, and note the rare oppor- 
 tunity which the recitation in this branch has for teaching accuracy 
 and clearness of expression, we shall be inclined to place the dis- 
 ciplinary side of the study far above that of the direct assistance it 
 has in earning a living or in performing the common duties of life, 
 and give to Arithmetic a prominent place in the Course. 
 
 Another end of arithmetical instruction which perhaps is in- 
 volved in the ends already named, is skill or facility in the 
 manipulation of numbers. The practical as well as the disciplinary 
 value of the study is greatly enhanced by such facility as will en- 
 able the pupils to perform accurately and quickly the computations 
 involved in the problems. It is the purpose, therefore, of good 
 teachers of Arithmetic to lead the pupils to become accurate and 
 quick computers, and to secure this end there is needed much 
 practice in the application of principles already known. The 
 
•?f'2 : : GRADED ARITKMETIC. 
 
 puipjls ciife add columns, ascertain measurements, and cast interest 
 accurately and quickly only by the repeated performance of the 
 same mental act, and for all such work abundant practice is 
 necessary. 
 
 Course of Study. — Although the comparative unimportance of 
 the practical side of Arithmetic may be recognized, the mistake 
 should not be made of spending much of the time given to the 
 study upon unpractical subjects. The constant aim of the teacher 
 should be to emphasize essentials, and by essentials is meant those 
 subjects and processes which are in some degree a preparation for 
 life or for further study. There is abundant opportunity for useful 
 disciplinary practice in some of the common and simpler subjects, 
 such as denominate numbers, fractions, and the applications of per- 
 centage that are made in business. 
 
 To rightly judge what the essentials of Arithmetic are, a con- 
 sideration of interests other than those of direct utility is necessary. 
 The age, capacity, and, so far as circumstances permit, the probable 
 future occupation of the pupils should be considered. Tavo kinds 
 of practice with numbers are required of pupils from the beginning 
 of the course : one kind which has to do with the mechanical side 
 of problems — the computations with what are sometimes called 
 abstract numbers ; and another kind which has to do with the 
 applications and use of principles, and which involve some logical 
 processes. The amount of the latter work is relatively small in 
 the lower grades, and gradually increases in amount and complexity 
 as tlie pupil grows in maturity, while the amount of simple com- 
 putations for exactness in the four fundamental rules gradually 
 decreases until, in the latter part of the course, nothing but con- 
 crete or applied work is done. 
 
 So far as this series of books is concerned, it is not siipposed 
 that all schools will follow the exact order prescribed, or tliat the 
 books will exactly correspond with grades of pupils in all places. 
 In the amount of work to be given also there will be a great 
 variety of practice depending upon the age of admission of pupils, 
 the requirements in otlier departments of study, and the intentions 
 
teachers' manual. 
 
 of pupils as to the probable extent of their attendance at school. 
 While in no one school or system of schools will it be needful to 
 take up every subject presented, or to give to pupils all the exercises 
 of certain subjects, it is hoped that both subjects and exercises are 
 sufficient in number and variety to meet the needs of all, and that 
 from them a wise selection may be made. 
 
 The Recitation. — One prime purpose of the recitation is to 
 develop the power of self-direction in the pupils, — to lead them 
 to depend upon themselves not only in accuracy of computations 
 but also in all reasoning processes. For the purpose of encouraging 
 pupils to proper effort in study, and of seeing that certain prin- 
 ciples or processes are understood, the teacher finds it necessary to 
 examine their completed tasks, or to test them upon what they 
 have done. But to accomplish the objects named, it is not neces- 
 sary or well to spend the entire recitation time in examination, A 
 glance at the work performed, and a comparison of the pupils' 
 answers with the correct answers, are generally sufficient to in- 
 sure faithful effort in study. To give the lessons learned such 
 prominence as to make them alone determine the pupils' promotion 
 or standing in the class, is to put temptation in their way to copy 
 solutions or answers, or to seek assistance that will not be helpful. 
 Moreover, the detailed explanations and solutions of problems 
 already performed, frequently mean a tedious recitation and much 
 needless waiting on the part of some members of the class. If it 
 is desired to ascertain whether the pupils understand thoroughly a 
 principle or process which has been taught, a few problems which 
 involve the principle or process may be given the class as a test. 
 These problems should have small numbers, so that the difficulties 
 of computation Avill not distract attention, and that the pupils' 
 understanding of the question may appear in the answers they give. 
 For example, if the subject of Profit and Loss has been taught, and 
 the pupils have broTight to the recitation the problems which they 
 have solved in two of the cases of that subject, the teacher, after 
 glancing at the work and leading them to correct errors in their 
 answers, might give as a test the following problems : 
 
4 GEADED ARITHMETIC. 
 
 A horse whicli cost $250 is sold for what to gain 20 <^ ? 
 
 I buy a house for $1000 and sell it immediately for $750. What 
 was the gain or loss per cent? Suppose I had sold it for $1200. 
 What would have been the gain or loss per cent ? 
 
 From answers to these and similar questions it will be known 
 whether tlie subject is understood, and time will be left for other 
 purposes of the recitation. 
 
 When any subject or process is not understood, it should be 
 taught, and by teaching is meant, leading the pupils to acquire 
 knowledge by their own efforts. Little should be told the pupils 
 in this process. The teacher's part is first to lead the pupils to 
 recall ideas already in the mind which have some relation to the 
 new principle or process to be taught. Then follows the presenta- 
 tion of the object of thought in such a way as to awaken new ideas 
 in the learner's mind, or to readjust and reinforce old ones. The 
 object of thought may be a physical object, as a meter-stick or a 
 bond, or what represents an object, as a picture or diagram. 
 Sometimes, as in teaching a rule or definition, the teacher has 
 only to help the pupils to recall and rearrange their ideas in an 
 orderly way, and to make a general statement from them. Some 
 processes also may be taught by leading the pupils to recall what 
 they have previously learned, and make the needful application. 
 Such purposes may be accomplished frequently by skilful ques- 
 tioning. 
 
 When the matter in hand is taught, and ideas concerning it are 
 clear in the pupils' minds, they need to be fixed by much repeti- 
 tion. This is done in the recitation by drill and in the practice 
 which the pupils get iu study. Little time is needed for teaching 
 compared to the time that is needed for practice. A portion of 
 nearly every recitation may well be given to drill, both upon what 
 has been taught recently and ujjon what has been taught days or 
 weeks before. 
 
 Thus it is that three means of conducting the recitation in 
 Arithmetic should bo iised by the teacher : first, examination to 
 
teachers' manual. 5 
 
 test the pupils' knowledge of the topic or topics ; secondly, teach- 
 ing to awaken new ideas in the pupils' minds ; and thirdly, 
 drilling to fix and strengthen the pupils' knowledge of what has 
 been taught. These processes are not always separated in the 
 recitation. Sometimes in the teaching exercise there is more or 
 less of examination and drill, and sometimes in the examination 
 there is necessarily much useful drill. But they all have their 
 j)la(H' in the recitation, and no one of them should be neglected. 
 
 Methods of Drill. — The effectiveness of the drill exercise will 
 depend largely upon the methods employed. The prime object of 
 the drill is, as has been said, to fix in the mind what has been 
 taught. To gain this object, there should be a repetition of the 
 mental act which was occasioned when the principle or process 
 was taught. In the drill exercise, the teacher should have in mind 
 exactly what acts of the mind are to be repeated, and he should 
 also aim to have every pupil employed during the exercise. Drill 
 which has not a definite object, or which consists of a rejietition of 
 words only, or which permits inactivity on the part of some 
 members of the class, cannot accomplish all the desired ends. 
 
 The drill may be botli oral and written. If oral, the given 
 problem may be read by one pupil, all the rest of the class reading 
 it silently at the same time ; or it may be dictated by the teacher. 
 In either case all should solve the problem together, either follow- 
 ing the oral solution of a l)U})il, or else performing it silently and 
 indicating the answer on slate or paper. By the latter plan it 
 would be well to have all tlie pupils work and write very promptly 
 exactly at the same time. For example, the pupils have before 
 them paper or slate and pencil. The teacher says, "pencil in 
 hand," and dictates (not from a book) the problem. After giving 
 sufficient time for all to solve it, the teacher says, "Avrite the 
 answer," and then after just enough time for the pupils to write 
 the figures of the answer, says, " pencils down." The teacher then 
 gives the answer, and asks all pujjils wlio have that answer to hold 
 up the slate or paper so that tlie teacher can see the answer. It 
 will be necessary for the figures of the answer to be written large 
 
6 GRADED ARITHMETIC. 
 
 and in a given place, that they may be seen readily. Properly 
 managed, this method of drill Avill afford opportunity for giving 
 much practice in a short period of time. This method will serve 
 for examination as well as for dictation. 
 
 Written drill work may be given from dictation or from the 
 book. If from the book, a number of exercises may be given 
 out for pupils to perform as many as they can within a certain 
 time, or the exercises may be given singly to members of the class, 
 each pupil performing such problems as are most needful or most 
 useful for him to perform. 
 
 Reviews. — For the sake of making pupils familiar with certain 
 important principles or processes, and also for the sake of connect- 
 ing together kindred subjects, fre(j[uent reviews should be made. 
 In giving such reviews, let the teacher have in mind a well-defined 
 purpose, and not force pupils to work simply for the sake of work- 
 ing. No exercise should be given that is not needed, either for 
 fixing in the mind what has been taught, or for testing the pupils' 
 knowledge of a given subject, or for securing skill and facility in 
 the use of numbers. 
 
 Tasks and Desk-work. — The lessons set for practice or study 
 should, as nearly as possible, serve the needs of every pupil in the 
 class. So far as the character of the work is concerned, those sub- 
 jects only should be given for tasks which are really needed to be 
 reviewed or which will serve as a test of the pupils' knowledge. 
 The given lesson also should be of sufficient length to employ all 
 the pupils of the class a reasonable amount of time. To accom- 
 modate the work to the different degrees of ability and quickness 
 in members of the class, it may be well to give a lesson which all 
 can do, and in addition, extra Avork for those Avho are able to do 
 more. 
 
 Illustrations. — Problems involving measurements, and other 
 problems whose conditions are not quite cleav. should l)r illustrated 
 by means of ])lans or diagrams. Care should be taken, however, 
 that the attention of pupils !)(> not diverted fi'om the real purpose 
 of the drawings by their nicety or elaborateness, and that illustra- 
 
TEACHEKS MANUAL. 7 
 
 tions be not required when the problems are fully understood, or 
 when the problems can be performed without them. 
 
 Reading Problems. — Let the pupils form the habit from the 
 outset of reading over thoughtfully every problem before they 
 begin the solution. The points to be noted in the reading are : 
 What is required ? What steps are to be taken ? About how large 
 will the answer ho ? 
 
 Analyses and Explanations. — Oral analyses or explanations in 
 the lower grades should be very simple. In these grades short 
 and direct statements as to what has been or is to be done in the 
 solution of a given problem may be made. The statements should 
 as nearly as possible represent exactly the pupil's thought and 
 should consist therefore of words largely of liis own choice and 
 arrangement. In the middle and upper grades more and more 
 minute analyses may be insisted upon, the pupils being required as 
 their minds develop to give reasons for the steps taken. But in 
 no part of the course should pupils be required to give formal 
 and ready-made explanations which do not represent their own 
 thinking. Some statements which explain to the teacher may 
 actually bewilder the pupils and deaden their thinking powers. 
 
 Problems and Indicated Operations. — From a very early period 
 in the course in Arithmetic the pupils should be led to see and to 
 indicate operations that are involved in problems. In the first 
 year pupils are encouraged to make story problems from equations 
 and soon after to make the equations from the story problems. 
 This work should go on all through the course until in the later 
 grades much of the work in Arithmetic may well consist of written 
 or oral statements of processes that are involved in given problems. 
 
 Forecasting Answers. — ^ Every teacher knows what surprising 
 and unreasonable ansAvers to problems are given by pupils, being 
 in some cases many thousand times too large or small. To prevent 
 such mistakes and to encourage thoughtful attention to the condi- 
 tions of problems, it is well by means of questions first to give an 
 approximate estimate as to what the answer will be. Suppose 
 for example the problem is : " How many bushels of potatoes will 
 
8 GEADED ARITHMETIC. 
 
 be gathered from a lot of land 100 feet square if it yields at the 
 rate of 200 bushels to the acre ? " The teacher may ask : " How 
 many sq. ft. in the lot ? What part of an acre in the lot ? If the 
 lot is a little less than one quarter of an acre, less than how many 
 bushels will the lot yield?" Or if the problem is : "How many 
 bushels of wheat can be put into a bin 6 ft. 4 in. long, 4 ft. 2 in. 
 wide and 3 ft. deep," the teacher may ask : *' About how many 
 cu. ft. does the bin contain ? How do a bushel and a cubic foot 
 compare ? Will the bin contain a few more or less than 72 
 busliels then ? " Gradually the teacher may give less and less 
 assistance in this direction until the pupils will be able to forecast 
 the answers independently and form the habit of so doing. 
 
 Languag^e. — The good teacher is quick to see the useful and 
 constant opportunities which Arithmetic affords for teaching lan- 
 guage. From the very first year when children are encouraged to 
 tell little number stories to the highest grade of the Grammar 
 School when explanations, rules and definitions are required to be 
 made by the pupils themselves, is there constant occasion for the 
 clear and accurate expression of thought. The very accuracy of 
 the thought required in Arithmetical processes and definitions 
 makes accuracy of expression necessary both in the choice and in 
 the arrangement of words. 
 
 SECTION II. 
 
 FIRST STEPS IN NUMBER. 
 
 Before the book is put into the hands of children, it will be well 
 to teach the numbers to ten by the aid of objects without the use 
 of figures. This may occupy a greater or less time according to 
 the ability and previous training of the children. Most children 
 who enter school at five years of age know very little of number, 
 while others who enter at .a later period, or who have had the 
 
TEACHEES' MANUAL. 9 
 
 advantage of a good inheritance or helpful home and kindergarten 
 training, can recognize numbers and make all combinations and 
 separation of numbers to four or six. For the sake of a proper 
 adaptation of the work to the capacity of the children, the separa- 
 tion of the children into groups will be found advisable. The 
 division made need not be fixed. There is as much difference 
 among children in the growth of their intelligence as there is in 
 the intelligence itself. Accordingly pupils should be transferred 
 from one group to another as soon as they are discovered to be 
 ahead of or behind their fellows. 
 
 Counters. — • On some accounts the variety of objects used for 
 teaching numbers in the first lessons should not be great, being 
 limited to the splints or wooden blocks. If blocks are used, they 
 might be either of three sizes, viz.: cubes (1X1X1 in.), half-cubes 
 (1 X 1 X |- in.), quarter-cubes (1 X ^ X -g- in.). The whole cubes 
 are preferred by many teachers. Splints have the advantage of 
 being easily handled. 
 
 Methods. — For teaching number to young children, there should 
 be a table about which the children can stand in full sight of the 
 counters in front of the teacher, who sits at the head of the table. 
 Or, if there is no table, the children may sit in their seats and the 
 teacher use an inclined table in sight of the children. 
 
 If the number three and their combinations are known by the 
 children, ask them to place three blocks together. After this is 
 done by all, the teacher and children put one block with the three 
 blocks, and the new number is named by the teacher. Teacher: 
 Three blocks and one block are how many blocks ? Children 
 (answering singly): Three blocks and one block are four blocks. 
 T. (taking away one block): Four blocks less one block are how 
 many blocks ? Ch. (taking away one block as the teacher did) : 
 Four blocks less one block are three blocks. T. (moving the 
 blocks) : Two blocks and two blocks are how many blocks ? 
 Four blocks less two blocks ? One block and three blocks ? 
 Four blocks less three blocks ? Two twos of blocks ? One four 
 of blocks ? One-half of four blocks ? One-fourtli of four blocks ? 
 
10 
 
 GRADED ARITHJMETIC. 
 
 How many twos of blocks in four blocks ? How many ones of 
 blocks in four blocks ? Answers to these questions are to be given 
 in order, the objects being used in the manner indicated. The 
 work with objects should be continued until the children have a 
 clear and definite idea of the combinations. These exercises may 
 be followed by a comparison of four blocks with three blocks, and 
 two blocks and one block, the children being led with the groups 
 before them to answer the following questions : Four blocks are 
 how many more than three blocks ? Three blocks are how many 
 less than four blocks ? Four blocks are how many more than two 
 blocks ? Two blocks are how many less than four blocks ? Four 
 blocks are how many more than one block ? One block is how 
 many less than four blocks ? 
 
 In the same way numbers to ten may be taught, no abstract 
 memory work being required or expected during these first lessons. 
 The following facts of numbers to ten should be taught : 
 
 ^ r4 + l; 5-1; 3 + 2; 5-2; 2 + 3; 5' 
 \ 5 — 1 ; 1 five ; 5 ones ; 5 -i- 1 ; ^ of 5. 
 
 3; 1 + 4; 
 
 r^ + i 
 
 6 — 4 
 6 — 2 
 
 6 — 1; 4 + 2; 6 — 2; 3 + 3; 6 — 3; 2 + 4; 
 
 1 + 5 ; 6 — 5 ; 1 six ; 6 ones ; 6 -i- 1 ; 3 twos ; 
 
 2 threes ; 6^3; i of 6 ; ^ of 6 ; ^ of 6. 
 
 : -< 
 
 6 + 1; 7-1; 5 + 2; 7-2; 4 + 3; 7-3; 3 + 4; 
 
 7 — 4; 2 + 5; 7 — 5; 1 + 6; 7 — 6; 1 seven ; 7 ones ; 
 7^1; } of 7. 
 
 8:^ 
 
 7 + 1; 8-1; 6 + 2; 8-2; 5 + 3; 8-3; 4 + 4; 
 8-4; 3+5; 8-5; 2 + 6; 8-6; 1 + 7; 8-7; 
 1 eight ; 8 ones ; 8 -i- 1 ; 4 twos ; 8 -j- 2 ; 2 fours ; 
 L8-^4; ^of 8; 1 of 8; ^ of 8, 
 
TEACHERS MANUAL. 
 
 11 
 
 10 
 
 f8 + 1; 9-1; 7 + 2; 9-L>; C. + 3 ; 9-3; 5 + 4; 
 
 9-4; 4 + 5; 9-5; 11 + G ; 9-G; 2 + 7; 9-7; 
 
 1 + 8 ; 9 — 8 ; 1 nine ; 9 ones ; 9^1; 3 threes ; 9 -i- 3 ; 
 . 1- of 9 ; ^ of 9. 
 
 9 + 1 ; 10 - 1 ; 8 + 2 ; 10 - 2 ; 7 + 3 ; 10 - 3 ; T) + 4 ; 
 
 10 — 4 ; 5 + 5 ; 10 — 5 ; 3 + 7 ; 10 — 7 ; 2 + 8 ; 10 — 8 ; 
 1 + 9 ; 10 — 9 ; 1 ten ; 10 ones ; 10 -4- 1 ; 5 twos ; 
 10^2; 2 fives ; 10 -^ 5 ; ^ of 10 ; l of 10 ; -jL of ] 0. 
 
 Picturing of Problems. — The picturing of problems may some- 
 times be given for busy-work in the lower grades, one object being 
 to provide a pleasant inducement for the children to review what 
 has been taught. This suggests the advisability of teachers' show- 
 ing pupils hov/ the drawings may be made and allowing them a 
 certain degree of freedom for the exercise of their inventive powers. 
 But care should be taken not to carry the picturing too far. It 
 should be remembered that such work is a means and not an end. 
 Properly used, the representations will serve to make clear the 
 relations which are not fully grasped and to fix in the mind com- 
 binations and processes which are not quite clear ; but carried to 
 the extent of diverting the attention from the main purpose of the 
 exercise, they are to be deprecated. 
 
 TJse of Figures. — Early in the year figures to represent numbers 
 should be used, first in having the children read Avhat is written 
 by the teacher, and afterwards in writing the figures. Such work 
 may proceed in the following order : (1) The objects counted and 
 placed in a group by each child. (2) The name of the number 
 given by the teacher and repeated by the children. (3) The figure 
 written on the board and read by the children. (4) The figure 
 copied or written from dictation on slate or paper by each child. 
 The teacher should be careful to present only good models before 
 the children for imitatioij. 
 
 The writing of equations to represent combinations of objects 
 
12 GRADED ARITHMETIC. 
 
 and the interpretation of represented combinations constitute a 
 useful feature of number work. For example, the children are told 
 to put 3 sticks with 3 sticks, and after telling how many in all, the 
 teacher writes on the board 3 -[- 3 = 6, which the children read. 
 The same is done after all operations with sticks in addition, sub- 
 traction, multiplication and division. At another time the teacher 
 writes on the board an expression, as 4 + 2, and asks the children 
 to do with the sticks what that says. This exercise will lead to 
 a good kind of busy work in number. 
 
 Number Stories. — For purposes of language as well as for the 
 purpose of affording a pleasant means of drill in number, pupils 
 should be led to tell stories involving the combinations which have 
 been taught. Tlie general plan of such work may be as follows : 
 First let the teacher make a story, each pupil using the sticks as 
 called for. Then each pupil may tell a story and as he talks, he, 
 with the other members of the class, uses the objects for illustra- 
 tion. After this has been done with the aid of sticks, stories may 
 be told from spoken or written examples or equations, and finally 
 they may be told with no assistance or suggestion from any one. 
 In all this work the aim should be to make it so interesting that 
 all members of the class will pay close attention. The following 
 lesson will illustrate the plan above given : 
 
 Teacher. Show me six sticks. Sliow me four of them. How 
 many sticks are four sticks and two sticks ? Call the sticks books. 
 ChildreM. Four books and two books are six books. T. I will tell a 
 story and as I tell the story you may work with the sticks. I have 
 four apples and my brother has two apples. How many apples have 
 we both together ? C. Both have four apples and two apples. Four 
 apples and two apples are six apples. T. Who can make a story 
 about five and one ? Mary. M. I had five cents and my mother 
 gave me one cent. How many cents have I now ? T. Charlie may 
 tell. C. You have five cents and one cent. Five cents and one 
 cent are six cents. 
 
 (The " story " may be given in the form of a question, as above, 
 or the children may be led to give the solution in a statement, 
 
teachers' manual. 13 
 
 thus : If I have 5 cents and my mother gives me 1 cent more, I 
 have 5 cents and 1 cent. 5 cents and 1 cent are 6 cents.) 
 
 T. Who Avill tell a story about 3 twos ? Jamie. J. If I buy 3 
 pencils and pay 2 cents apiece for them I pay 3 times 2 cents. 
 3 times 2 cents are 6 cents. T. Willie, another. W. 3 rabbits 
 have 3 times 2 eyes. 3 times 2 eyes are 6 eyes. 
 
 In all this work it is supjjosed that every member of the class is 
 following with or without the objects the stories as they are told. 
 More elaborate stories may be told and more extended combinations 
 may be made if the attention can be held. Such questions as the 
 following may not be too difficult provided the objects are used 
 and progress is made by easy steps. 
 
 Charlie was given 10 cents to buy something for his mother. 
 He bought 2 oranges at 3 cents apiece and a yeast-cake for 2 cents. 
 How many cents did he bring back ? 
 
 I had 6 cents and my brother gave me 2 more. With the money 
 I had then I bought apples at two cents apiece. How many 
 apples did I buy ? 
 
 From given stories children may be led to perform the required 
 operations with objects and to make the proper equations. The 
 finding and writing of equations from problems constitute an im- 
 portant part of luimber work later in the course and may well be 
 begun in the lowest grade of schools. 
 
 SECTION III. 
 
 NOTES FOR BOOK NUMBER ONE. 
 
 (The numbers at the head of the page indicate tlie number and page of the 
 Pupils' Book for which notes are made : thus [II. 18] means Book No. 2, 
 18th page. Figures in heavy type on a page of the Manual denote the page or 
 pages of the book for which notes are made.) 
 
 The exercises of Section I. are intended as a review of what has 
 been taught with objects. It may be necessary still for objects to 
 be used occasionally, both by pupils at their seats and by the 
 
14 G:fcADEt) ARITHMETIC. [1. 2 
 
 teacher and pupils in recitation, but the aim should be to lead 
 pupils to perform promptly the exercises of this section without 
 objects before the second section is begun. If an example or 
 problem is not readily performed, the want of promptness is due 
 to one of two causes : either the pupils do not understand the 
 operation involved, or else, understanding it, they have not facility 
 in performing it. If an operation' is not understood, it should be 
 taught, and the best method of teaching ideas of numbers and their 
 operations is by the use of objects. Facility is gained through 
 continued practice in repeating the operation in which facility is 
 needed. This practice may be gained in the recitation and in the 
 busy work of the pupils at their seats. If the previous lessons 
 have been well taught, it is siipposed that pupils need only the 
 drill of busy work and recitation practice in exercises of Section I. 
 to prepare them for the work of Section II. 
 
 For counters use the blocks or wooden sticks which have been 
 used previously (see Manual, page 9), or small pasteboard squares 
 which can be made with little difficulty. If the recitation table 
 could have upon it lines indicating squares of the size of the 
 counters, some advantages for placing and comparing the counters 
 would be gained. 
 
 The use of lines and the rough drawings of objects in perform- 
 ing problems is shown on pages 20 and 26. Such work should 
 be used constantly in the lower grades, care being taken that in 
 representing numbers and operations by drawing the attention of 
 children be not diverted from the real purpose of the exercise, 
 either in the complexity of the representations or in their extreme 
 accuracy or nicety. 
 
 Other suggestions as to the scope and use of the book will be 
 found in the "Note to Teachers," page 1. 
 
 2—11 The design of these exercises is to present graphically 
 before the children the various combinations and to give a pleasant 
 means of reviewing what they have had with objects. Let the 
 children practice upon the illustrated pages until the combinations 
 can be given quite freely. The sign of multiplication as used in 
 
I. 12] teachers' manual. 15 
 
 these books is supposed to be read "multi})lied by " rather than 
 " times " for two reasons : 1st, for the sake of uniformity. The 
 
 signs -| and -7- all indicate that the number preceding the sign 
 
 is to be operated upon, 2 + 3 means that 3 is to be added to 2 and 
 that the expression should be read 2 and 3 or 2 plus 3 ; 6 — 4 
 means that 4 is to be taken out of 6 and it should be read G less 4 ; 
 8-r-4 means that 8 is to be divided by 4 and should be so read. 
 In the same way 4X2 means that 4 is to be multiplied by 2 
 and that it be read 4 multiplied by 2. Another and stronger reason 
 for placing the multiplier after the sign is convenience in the 
 analysis of problems. This will appear later when the written 
 analyses of problems are made by the pupils. If desired the 
 expression 4X2 may be read 2 times 4. 
 
 The pages opposite the illustrative work are designed for busy- 
 work and for recitation. In recitation the children should be led 
 to read the sentences supplying the ellipses as they read. 
 
 13-15 These exercises are to be used for busy-work and also 
 for recitation. If necessary the children may be permitted to use 
 the splints in finding the answers, but in the recitation they should 
 be expected to give the answers promptly after the statement, thus 
 "4t and 5 are 9," "3 and 5 are 8." 
 
 16 A good exercise is to put pairs of numbers on the board as 
 here indicated and have the children give the sum instantly. This 
 will assist in subsequent work in adding columns. 
 
 1 7 The children should be encouraged to make illustrations of 
 processes as here indicated. 
 
 18—19 The children with a little assistance can be led to make 
 original stories like the following : " My father had 8 ducks and 
 he sold 4 of them. He had then 4 ducks left." 
 
 20 Good illustrative work is here shown which the children 
 can occasionally do bv themselves for busy-Avork. 
 
 31—35 A variety is here given for drill both in busy-work and 
 recitation. 
 
 36—38 With little effort on the part of the teacher, the 
 children can be led to illustrate problems for themselves. But in 
 
16 GRADED ARITHMETIC. [I. 29 
 
 all this work observe the caution given on pages 11 and 14 of the 
 Manual. 
 
 39—36 Let the teaching of each number above 10 precede the 
 recitation from tlie book. Thus in teaching 11, place upon the 
 table in sight of all the pupils 10 square blocks in a column. The 
 children who are sitting at their seats or standing around the table 
 do the same with their squares. The teacher asks how many 
 blocks there are and proceeds as follows : "We will place one 
 more block with the ten. Now we have eleven blocks. How many 
 blocks have you, Charley ? Eddie ? etc. How many blocks are ten 
 blocks and one block, Mary ? Susie ? etc. Take away one block 
 and how many have you ? Eleven blocks less one block are how 
 many blocks ? " Other facts in 11 developed in the same way are 
 shown on page 29. Where there are two subtractions, as in 
 11 — 5 — 5, lead the children first to combine 5 -\- o -\-l, leaving 
 the blocks as indicated, and then to remove a little way from the 
 collection 5 blocks, covering them up as they say : " 11 blocks less 
 5 blocks are 6 blocks " ; repeating the process they say : " 6 blocks 
 less 5 blocks are 1 block, 11 blocks less 5 blocks less 5 blocks are 
 1 block." The groups of blocks are now in position to divide by 5, 
 the question being : " In 11 there are how many fives and how 
 many left over ?" or, '-5 blocks in 11 blocks how many times and 
 how many left over ? " After the separation of the number into 
 fours, threes, and twos, a comparison of 11 with all numbers below 
 should be made ; thus, in answer to questions, the children are led 
 to say: ''11 blocks are 1 more than 10 blocks ; 11 blocks are 2 
 more than 9 blocks, etc. 11 is 1 more than 10," etc. 
 
 The writing and reading of numbers should be carried on in 
 connection with objects. Sticks may be the most convenient 
 objects for this purpose. The teacher takes 11 sticks and after 
 asking the children how many there are, binds 10 of them together 
 with a rubber band and says : This is 1 ten of sticks. How many 
 tens of sticks in eleven and how many more ? The cliildren should 
 be led to bind the 10 sticks into a ten as the teacher has done and 
 to say that in 11 there is 1 ten of sticks and 1 more. To express 
 
1. 37] teachers' manual. 17 
 
 in figures this number the ten bundle may be placed on the ledge 
 of the blackboard and the children be led to write 1 above it for 
 1 ten. The 1 stick may be placed beside the ten and another 
 figure 1 be placed above the 1 stick, making 1 ten and 1, or 11. 
 This exercise should be worked out very slowly and carefully, and 
 if necessary it should be repeated several times, the purpose being 
 to lead the children to liave a clear idea of the decimal system at 
 the outset. 
 
 After this work with objects has been done, equation statements 
 may be given by the pupils in answer to questions given by the 
 teacher or read from the book. Thus all operations that have been 
 performed with the objects are uoav given orally without objects. 
 Then follows the writing of the equations by the children. This 
 will constitute a part of the busy-work of the children, who copy 
 from the book the questions and complete the equations. The 
 story problems which naturally follow may be told and solved in 
 the recitation and also at the seat. 
 
 The order of procedure in all tins work should be noticed — 
 (1) operations with objects, (2) giving of oral equations, (3) giving 
 of written equations, (4) giving of story problems. Sometimes 
 drawings may be made in illustration of operations directly after 
 the objects have been used, and sometimes the drawings may be 
 used instead of the objects. For review, the order above given may 
 be changed or inverted. The oral or written equations may be 
 given by the children, who will illustrate or demonstrate the equa- 
 tions by the use of objects or drawings ; or the children may give 
 the story problem, afterwards solve it by objects or drawings and 
 finally make the oral and written equations. 
 
 The order above given with such variation as circumstances will 
 determine, should be followed in treating eacli new number. 
 
 37-39 Nearly all the questions on these pages should be solved 
 by means of simple drawings as shown on page 87. Some care 
 will have to be taken to show the children just what is expected of 
 them. Slow progress in picturing problems at this stage must be 
 expected. From what the teacher does upon the blackboard in 
 
18 GRADED ARITHMETIC. [I. 40 
 
 illustrating a process, and from little corrections that are made, 
 the children will gradually get the idea of representing the condi- 
 tions and the solution of problems by means of drawings. 
 
 40 In teaching 13, iirst use the squares and blocks in the order 
 indicated, somewhat as is given on pages 29 and 33. With some 
 suggestions and directions, the children can be led to draw the 
 squares and to indicate what process is shown. Dwell upon this 
 page until a tolerable degree of familiarity with the combinations 
 is had. 
 
 4:1—43 In reciting these exercises the children may read only 
 the answer thus : in a, page 42, " Twelve boys and one boy are 
 thirteen boys," etc.; in e, "In twelve there are 6 twos," etc.; and 
 in g, " I must put ten with three to make thirteen." In the story 
 problems let the children first read the problem and then give the 
 answer in a good statement, tlius : in /, page 43, " The little boy's 
 sister found eight eggs and three eggs, or eleven eggs." 
 
 44 A teaching exercise with the coins should precede the 
 recitation of these problems. 10 one-cent pieces, 5 two-cent pieces, 
 2 five-cent pieces, and a dime will be needed to show all the facts 
 given. The children already have some knowledge of the value 
 of these coins, and the exercise with them may be made very 
 interesting. Thus, the teacher may say, " Who will find six cents 
 from these coins ? " " Charlie has a dime. Who will take from 
 the two-cent pieces enough to have as much money as he has ? " 
 The children will also enjoy buying, selling, and making change 
 with the coins. 
 
 45 The combinations indicated on this page should be taught 
 first with objects, and then shown by squares as given on pages 29 
 and 33. The children should also be expected to make the draw- 
 ings with as little assistance as possible. 
 
 46—47 In the column addition, lead the children to add in 
 pairs, thus: in Ex. 4 : 6, 8 ; 7, 11, etc.; and in Ex. 5: 6, 13 ; 
 8, 12, etc. 
 
 49 First teach with objects as before. The answers should be 
 given with and without the aid of squares. 
 
1. 51] teachers' manual. 19 
 
 51-53 The form of problems and answers in Exercises 12 and 
 13 is a model for original problems which should be given by the 
 cliildren occasionally. These may be given with and without 
 objects. 
 
 54 The problems involving days in the week, weeks in the 
 month, and units in the dozen, should be extended by the teacher 
 and by the children with as great variety as possible. 
 
 iiiy Introduce the measures here, using water or dry sand. Let 
 the children measure the water or sand until the facts become fully 
 understood. The measures should also be used in teaching prob- 
 lems similar to those in the lower part of this page. This is 
 followed by picturing the problems as shown in the cut. The 
 children should make the drawings Avitli as little assistance from 
 the teacher as possible. Such work will be found very useful and 
 interesting to the children. The illustrations should always be 
 made whenever the children find difficulty in making a mental 
 image of the conditions, and in performing the problem. Three or 
 four days may be profitably spent upon problems similar to those 
 given on this page and the page following. 
 
 57 The measuring strip called for should be cut from stiff 
 paper ; pasteboard or cardboard would be better. If the children 
 cannot cut and mark this strip accurately, the teacher or older 
 children should do it. Several exercises with this measure and 
 a yard-stick, similar to those on the lower part of the page, should 
 be given. 
 
 58 A teaching exercise should precede drill upon this page. 
 Teach rectangle and square, by showing that rectangles have 
 square corners and that squares are rectangles whose sides are 
 equal. Paper cutting and folding should accompany the solution 
 of c, d, e, f, and g. Other similar problems may be given. 
 
 59 If the children cannot perform these problems readily, 
 objects and drawings should be used. Let the answers be in 
 entire sentences, thus : " In two quarts there are four pints," or 
 "There are four pints in two quarts." Do not insist upon one 
 form of answer to these problems. If the statement of the 
 
20 GRADED ARITHMETIC. [I. 62 
 
 pupil expresses a clear thought of the number combinations let 
 it pass. 
 
 62 The children should be encouraged to illustrate these prob- 
 lems in as many ways as possible. After the processes have been 
 shown by drawings, the problems should be read and full state- 
 ments of answers should be given. Here also one particular form 
 of statement should not be insisted upon. 
 
 64 Some of these problems, such as d, f, g, and li, should be 
 taught before they are given to the children. Add by pairs as well 
 as singly, numbers indicated in the columns. Thus the first 
 column may be added, 6, 8, 11, 13, or 6, 11, 13. 
 
 68-69 While a particular form of answer should not be 
 insisted upon, tlie answers given should be corrected if they are 
 not clear. A good form may be given by the teacher, as in a, 
 page 68 : " If I have a dime and a five-cent piece I shall have 
 fifteen cents, and I shall need two cents more to have seventeen 
 cents " ; and in h : " Two boys have four legs and three dogs have 
 twelve legs. Two boys and three dogs have four legs and twelve 
 legs. Four legs and twelve legs are sixteen legs." If the children 
 can write without a copy, it may be good exercise in written 
 language for them to write the answers as they have been ac- 
 customed to recite them in the class. 
 
 70 In teaching to write each new number, pay particular atten- 
 tion to the expression of the ten and the units as shown on page 
 16 of the Manual. 
 
 7 3 These should be given at sight. In recitation do not permit 
 the children to read the question first and then give the answer by 
 repeating a part of the question. The answers should be given in 
 entire statements, Avith no pause before the answer. Occasionally 
 the children might give the answer quickly to each question by a 
 single word. 
 
 73 Sight exercises for drill. Let the children give results 
 promptly, thus : 7, 10, 8, 5, etc. 
 
 74 Let the answers to these problems be given in entire sen- 
 tences. The solution of some of them may have to be pictured 
 before they are understood by the children. 
 
I. 78] teachers' manual. 21 
 
 78 Show by bundles of sticks the expression of two tens and 
 no units. 
 
 81 A portion of this page only may be given at a time. Observe 
 the order of representation. Keview frequently what has been 
 taught. 
 
 8-4-80 The problems of these pages will suggest a kind of 
 work that may be done in connection with nature lessons and 
 language. Lead the children to give tlie ansAvers in entire sen- 
 tences. Thus, in 26, page 84, the answers may be : (o) 3 
 pansies have 3 times 5 petals. 3 times 5 petals are 15 petals. 
 (J>) 1 pay G times 2 cents for 6 two-cent postage stamps. G times 
 2 cents are 12 cents. I pay 5 cents for the paper. I pay 12 cents 
 and 5 cents for the stamps and paper. 12 cents and 5 cents are 
 17 cents. 
 
 Let the statements be made by the children in reply to questions. 
 Gradually they will be able to give the answers in proper form 
 without assistance. 
 
 SECTION IV. 
 
 NOTES FOR BOOK NUMBER TWO. 
 
 For a brief statement of the purposes of this book and for some 
 general directions see the Note to Teachers given in Book No. 2. 
 It may be said further, that objects should be used in teaching 
 every new fact or process. The illustrative work here given is 
 intended to supplement and not to take the place of such teaching. 
 The objects needed will be pasteboard squares and sticks for 
 counters, common weights and measures, and toy money. The 
 kind of illustrative blackboard work which may be done by 
 teachers will appear in the notes. 
 
 After the various operations have been taught, much drill in- 
 volving a repetition of the mental act in learning them will be 
 found necessary. There is given, accordingly, in this book a great 
 variety and number of drill problems. For variety of class-work 
 
22 GEADED ARITHMETIC. [II. 1 
 
 it may be well to give exercises from the blackboard from time to 
 time, varying with the needs and progress of the children, but for 
 seat-work it is believed that the given exercises will furnish 
 abundant material. 
 
 1—11 These problems are a review of work given in Book 
 No. I. If the children cannot perform them with some degree of 
 promptness, it is advised that needed portions of the previous book 
 be taken up. Possibly there will be needed only some oral or 
 blackboard drill. It may be well to place upon the board the 
 following for drill in addition : 
 
 111111111 
 123456789 
 
 22222222 
 23456789 
 
 3 3 3 3 3 3 3 
 
 3 4 5 6 7 8 9 
 
 4 4 4 4 4 4 
 
 4 5 6 7 8 9 
 
 5 5 5 5 5 
 
 5 6 7 8 9 
 
 6 6 6 6 
 
 6 7 8 9 
 
 7 7 7 These figures represent all pos- 
 
 7 3 9 sible pairs of units that can be 
 ~ ~ ~ made. There is no combination 
 
 8 8 in simple addition that may not 
 8 9 be referred to them. It is for 
 
 this reason that a thorough 
 knowledge of these combinations 
 should be had. 
 
II. 12] teachers' manual. 23 
 
 13—13 If necessary, before giving these lessons from the 
 book, teach the writing of numbers to 20 with the aid of sticks 
 as suggested on page 17 of the ManuaL Further drill similar to 
 that given on page 12 may be given. 
 
 14-19 The teaching of numbers to 100 by means of sticks 
 should precede and accompany these exercises. The purpose of 
 such teaching is to give the children a good foundation knowledge 
 of numbers and their expression. As before, every ten of sticks 
 should be bound into a bundle, and the number of such bundles in 
 a number should be called so many tens, while the single sticks 
 should be called ones or units. For example, if twenty-six is the 
 number to be taught, let the teacher count out ten and place a 
 rubber band about it, proceeding as follows : " How many sticks 
 have I here ? We will call it a what ? Yes, a ten. Let us count 
 out ten more. How many tens have I now in my hand ? How 
 many more sticks have I ? How many sticks in all ? Two tens 
 of sticks and six sticks are how many sticks ? Now let us write 
 this number on the board (placing the two bundles together and 
 six sticks together on the ledge). Who will place above the tens 
 bundles the figure which tells how many tens ? Who will place 
 above the single sticks the figure which tells the number of ones 
 or units ? Who will read the whole number ? " The same method 
 should be pursued with other numbers in the twenties and tliirties, 
 and then the children should be given sticks and bands to count 
 out and express given numbers as far as forty. Tlie children are 
 now ready to do the work called for on page 14. The same course 
 should be followed in teaching the numbers to 100. Several days 
 may be profitably spent upon this work, the book beiug used for 
 review in the recitation and for busy-work. Lead the children to 
 draw the squares very carefully with a ruler or other straight 
 edge. 
 
 20 A review which should be performed without objects. 
 
 31—26 Observe carefully the order of these problems, taking 
 u]) new Avork only when the old is well understood. Lead the 
 children to perform the work of addition and subtraction first with 
 
24 GRADED ARITHMETIC. [II. 27 
 
 sticks, afterwards by means of drawn squares, and finally without 
 the aid of objects. Do not proceed too quickly to the abstract 
 work because the children can give answers to the questions. One 
 important purpose of using objects at this stage is to give a good 
 foundation for subsequent work. 
 
 The problems on page 23 are intended for sight-work, but it may 
 be well to have the children repeat the problem first before giving 
 the answer ; thus : Twenty and ten are thirty, and four are thirty- 
 four ; thirty and twenty are fifty, and three are fifty -three, etc. 
 Fifty-six less twenty are thirty-six, etc. If tlie children hesitate 
 in giving answers, go back one step ; for example, if they cannot 
 give an answer at once to the question 53 -j- 40, give jDroblems in 
 the addition of tens alone, 30 + 20, 40 + 30, 50 + 40, etc. The 
 same course should be pursued in subtraction. 
 
 The questions on page 24 should also be repeated thus : Five and 
 two are seven, fifteen and tAvo are seventeen, etc. Adding so as to 
 make even tens, and subtracting so as to break up even tens, 
 should be fully shown by objects and drawings. Possibly similar 
 work will have to be given by drill from the board. 
 
 27 — 28 The children should read these problems silently, and 
 repeat a portion of them in the answer ; thus in 4, page 27, the 
 answer might be : " If I sleep ten hours of the day I shall be 
 awake twenty-four hours less ten hours. Twenty-four hours less 
 ten hours are fourteen hours." Of course other forms of answers 
 should be permitted. 
 
 29 — 31 These examples can be performed by the children 
 without objects, and they will not be found very difficult ; but it 
 will be far better to use the sticks for a day or two u})on similar 
 examples, both for the expression of the numbers and for the 
 process of adding or sul)trafting. This will be especially neces- 
 sary in subtraction. Thus to take GO from 84, the children take 
 8 tens of sticks and 4 sticks, and express that number in writing 
 .as 84. They are then asked to take 6 tens of sticks or ()0 sticks 
 from the iiuiiil)('r. They do tliis and find 24 sticks remaining. 
 This process they represent by figures, writing one number under 
 
II. 32] teachers' manual. 25 
 
 the other, drawing a line under the lower number, and writing the 
 remainder below. All this is a good preparation not only for the 
 problems of these three pages but also for the problems which 
 follow. 
 
 33 — 35 Here is taken a new and important step in addition 
 and subtraction — the adding of units whose sum is more than ten, 
 and the su.btracting of units from a ten and units, the iinits of tlie 
 whole being less than the units of the part. The children have 
 learned to perform these operations to twenty, and now they are to 
 apply the knowledge thus gained to larger numbers. Sticks should 
 be used as before, first by the addition of 2 to 9, making 1 ten and 
 1 unit, and afterwards by the addition of the same number to 19, 
 29, 39, etc. Let the children stop in each case to put the band 
 about the new bundle of ten, and express in figures what they have 
 done. The same course should be taken in subtraction. Tlie book 
 may now be taken for the solution of problems by the aid of the 
 cuts. The drawing of the squares in the manner indicated on 
 pages 32 and 34 will be good busy-work for the children. The 
 answers may first be given by repeating the numbers to be added 
 or subtracted, and afterAvards at sight by giving the result alone ; 
 thus, in 3, page 33 : Eleven less two are nine, twenty-one less two 
 are nineteen, etc. Nine, nineteen, twenty-nine, etc. The children 
 will be able by repeated drill to add and subtract by twos and 
 threes very rapidly. 
 
 3(> — 40 There is no new principle involved in the problems of 
 these pages, and the same course should be followed in teaching and 
 drilling which was pursued in the previous four pages. Do not 
 neglect to use the ol)jects and drawings Avhenever a new nuMiber 
 is to be added or subtracted. A repetition of the mental act which 
 accompanies the work with objects is as necessary as the repetition 
 of the mental act in recalling former impressions. Very much 
 diill is needed, both with and Avithout objects, to iix the c(iHd)i!i;;- 
 tions in tlie minds of the children, so that they Avill be given with 
 perfect accuracy and promptness. Occasional drill-work on tlie 
 board may be found useful. Fur busy-work, lead the chiUlren to 
 
26 GRADED ARITHMETIC. [II. 47 
 
 write out problems in the addition and subtraction of 5, 6, etc., in 
 the manner indicated on page 37. Tliis will aid them in future 
 written work. 
 
 47 A simple extension of the process already learned is required 
 here. Use objects and drawings of lines, dots, or squares. In the 
 cut on this page let the children place a card over the squares in 
 such a way as to show 2 threes, 3 threes, 4 threes, etc. Read in 6, 
 Two multiplied by four, or four times two. 
 
 48 The getting of fractional parts of numbers by the aid of 
 objects and drawings is very interesting and useful work for the 
 children, and should be frequently done. This objective work 
 should be immediately followed by a repetition of the solution of the 
 problem without the objects. For example, after the children have 
 learned that i of 12 dots is 4 dots, and that § of 12 squares are 8 
 squares, they should be asked the questions, '' What is one-third of 
 twelve ? two-thirds of twelve ? " 
 
 The division of numbers by numbers is the reverse of multiplica- 
 tion, and may be taught directly in conpection with it. After 
 counting out 24 sticks, have the children separate them into 
 groups of 3. Then ask, '■'■ How many threes of sticks are there 
 in twenty-four sticks ? " or, " Three sticks in twenty-four sticks, 
 how many times ? " The same may be done with the drawn 
 squares, letting the column or line of squares represent the 
 divisor, and the uncovered squares the dividend. 
 
 49 — 51 Partition and division by 4 are to be taught and drilled 
 upon as was indicated for the same processes by 3. Before the 
 solution of applied problems, the children should be able to give 
 with a considerable degree of promptness answers to all examples 
 on these pages. 
 
 53 — 54 The most difficult of these problems should be taught 
 with objects in the recitation, but in teaching them be careful not 
 to (h) anything for tlie children which tliey can do themselves. 
 For examph;, in 8, l)age 52, you say first, "We will let the sticks 
 represent eggs, and you may count out one dozen. How many eggs 
 have you, "NN'illie V How many have you, Mary ? Now, if my 
 
II. 55] teachers' manual. 27 
 
 hens lay four eggs every diiy, liuw many days will it take them to 
 lay the dozen." Do not disturb the children in their thinking, 
 until you see tliat the process is not clear. If it is not clear say : 
 "If they la}' four eggs iu a, day covnit out how many they would 
 lay in tw^o days. oSTow see if you can find how many days it will 
 take the hens to lay a dozen." Probably uo further help will be 
 needed. Tlien proceed with the larger number recpiired, varying 
 the number, and calling upon children singly. After the most 
 difficult of the problems are taught, let the children " picture " 
 them, and others which have not been taught. Thus, in the above 
 problem, the children could be led easily to illustrate it as follows : 
 
 1 (lav. 1 clay. 1 day. 
 
 and to write below the pictured solution : " It will take 3 days for 
 the hens to lay a dozen eggs if they lay 4 eggs a day." If it is 
 thought best to have the representation of objects more realistic, 
 ovals instead of lines could be drawn. 
 
 After the problems have been taught and pictured, the children 
 should be ready to solve them without aids, and give the answers 
 in entire sentences, as in 5, page 53 : " From the fourth to the 
 twenty-second of June there are eighteen days. In eighteen days 
 there are two weeks and four days." 
 
 55 — 5G Xot much time is needed for the multiplication and 
 division by five, and yet it is not well to neglect any part of the 
 objective work. The addition and subtraction work indicated on 
 page 55 may be extended if it is found that the children are losing 
 their hold of such work. 
 
 57 It will probably not be necessary to teach any of these 
 problems. If not, give them to the children as they are, to work 
 out with and without drawings. 
 
 58 — 59 In teaching and drilling, dwell particularly upon those 
 examples which are most difficult to remember, as for example, 
 6 X 7, 6 X 9, 42 -^ 6, 54 -i- 6. For review in addition and subtraction 
 
28 GRADED ARITHMETIC. [II. 60 
 
 as well as in division, practice considerably upon such examples as 
 10 to 13 and 16, page 59. This may be done by giving a number, 
 as 38, which the cliildren are exj^ected to divide by 6 or 5. The 
 answer would be, " There are 6 sixes and 2 in 38," or, " 6 is con- 
 tained in 38, 6 times and 2 remainder." 
 
 CO If the previous work lias been thoroughly done, the children 
 will find little difficulty in performing these problems according to 
 the model shown in 1. 
 
 61-63 7 X 6, 7 X 8, and 7X9 are likely to give the children 
 most trouble to remember. Drill upon these combinations, there- 
 fore, should be most persistent. The other multiplications by 4 
 and 6 have been learned previously, Avith the factors inverted. 
 The last four numbers of page 62 might be extended considerably 
 for oral drill. 
 
 63 A little assistance may have to be given to the children 
 before they can work out the exercises as indicated. Do not hasten 
 them with this work, and when any child finds an object too difficult 
 to draw, let him use marks or dots. Chairs, horses, and children will 
 probably be too difficult for children to draw, and any attempt to 
 do it might divert their attention from the numerical operations. 
 
 64 Dwell with special emphasis upon the most difficult com- 
 binations in both multiplication and division. Frequent additions 
 and subtractions by eights, beginning with 8 and 9G, will lielp to 
 fix the tables in the minds of the children. 
 
 65 These problems should be performed first by picturing as 
 called for, and afterwards without aids of any kind. 
 
 66 See suggestions for page 64. 
 
 67 This is a review of what may profitably be done in various 
 stages of the cliildren's progress. Let the numbers of the table 
 which are most difficult to learn be written in heavier lines than 
 the others. 
 
 68 — (>9 Review drill exercises for busy-work and for recita- 
 tion. 
 
 70 To be performed without aids if possible. It may be neces- 
 sary to give some assistance in 15. 
 
II. 71] teachers' manual. 29 
 
 71—73 Weights and measures new to tlie cliildren sliould be 
 taught with tlie objects. The simpler problems also should be 
 first taught with objects. Afterwards, pictures representing the 
 weights and measures may be used. Squares or oblongs of ap- 
 proximately correct relative size would suffice for this purpose. 
 For example, in 11, a square whose edge is 1 in. could represent 
 the bushel, marked off in four equal parts to represent pecks. 
 Opposite the spaces the prices could be placed, and from them the 
 required answers given. Problems involving the use of pounds 
 and dozens also can be illustrated in the same way. The children 
 if permitted will give some very interesting graphic solutions of 
 these problems. When the children have a clear idea of the pro- 
 cesses by the aid of objects and drawings, they should be expected 
 to solve the problems promptly without aid of any kind. 
 
 74 When the children know thoroughly and can tell at sight 
 all combinations to 100, using any number for the adding, sub- 
 tracting, multiplying, or dividing number to 10, they are ready for 
 the work of this section, which includes the addition and subtrac- 
 tion of any number to 100, and the multiplication and division of 
 numbers to 100 by numbers to 20. To accomplish this it will be 
 necessary to follow slowly the order indicated, and to give frequent 
 reviews. 
 
 75 — 82 If the work is found too difficult at any point, observe 
 the preceding steps. For example, if the children cannot perform 
 readily the work on page 76, let them first add by tens and units 
 separately ; thus, in 10 : 30 and 40 are 70, and 9 are 79 ; 50 and 
 30 are 80, and 2 are 82, etc. After a rapid review of this kind 
 they will be ready to analyze and add mentally. Subtraction will 
 be found more difficult than addition, and more time sliould be 
 spent upon it. The separation of the number to be subtracted into 
 tens and units may be frequently necessary ; for example, in 2, 
 page 81 : 75 less 30 are 45 less 9 are 36, etc. A complete rnastery 
 of the subtraction of any number from 100 is especially desirable, 
 since such subtraction will be found very convenient in making 
 change. There is no more reason for adding coins piece by piece 
 
30 GRADED ARITHMETIC. [II. 83 
 
 in making change for a dollar than in making change for a dime. 
 One good method of drill in subtraction is to place a number upon 
 the blackboard, and give orally a number with the expectation that 
 the children will name anotlier number, which added to the given 
 number will make the number on the board. For example, 
 the teacher writes 89 on the board, and says : Forty-six. Pujnls. 
 Forty-three. T. Twenty-four. P. Sixty-five, etc. Rapid individ- 
 ual work of this kind will be found useful. 
 
 83 — 84: Tlie previous work of this section is expected to be 
 performed witliout the aid of figures. For variety, and for the 
 purpose of keeping fresh in the children's minds the decimal 
 system of notation, these written exercises are given. If the 
 processes required are not fully understood, let two or three of the 
 examples be performed with sticks, as was previously shown. Then 
 let the children add, explaining the process of "carrying" as they 
 add. The explanation may be very simple ; thus, in 36 : 9 units 
 and 4 units are 13 units, equal to 1 ten and 3 units. Write 3 units 
 in the place of units. 1 ten and 5 tens and 3 tens are 9 tens. 
 Write 9 tens in the place of tens. Answer: 9 tens and 3 units, 
 or 93. 
 
 85 — 86 An almost unlimited number of examples may be 
 given in connection with these tables. It will be observed that 
 the numbers are arranged differently in the three tables, and that 
 they can therefore be used at any time for a particular purpose. 
 For example, if it is found that the combination 8 + -"^ ii^ connec- 
 tion with large numbers is especially hard for the children to 
 remember, drill may first be given in Table A, lines s, j^, and n. 
 Add 8 to numbers in line p as far as i. Add 18 as far as g. 
 Add 38 as far as /. Do the same to numbers in .s. Subtract 5, 
 15, etc., from line n. The same may next be done with Table B, 
 lines 0, p, and n. 
 
 If it is desired to drill with 14 as a subtrahend, the order of 
 drill-work would be : In Table B, subtract 14 from numbers in 
 columns c, d, e, etc. Subtract 14 from numbers in lines I, m, etc., 
 beginning with c. In Table C, subtract 14 from numbers in 
 
II. 87] TEACHERS* MANUAL. 31 
 
 columns c, d, e, etc. Subtract 14 from numbers in line I beginning 
 with e, from numbers in ///, n, etc. 
 
 If it is ilesired to add or subtract a number from numbers whose 
 tens are the same and whose units are unlike, the columns in tables 
 B and C, could be used ; and if it is desired to add or subtract a 
 number from numbers whose tens are unlike and whose units are 
 the same, the lines of A and B could be used. Other uses of the 
 tables will appear for multi})li('ation and division. 
 
 For methods of blackboard drill, see Manual, pages 33 and 34. 
 
 87-l)t) Ability to multiply numbers by any number to 10, and 
 to divide by numbers to 20, would seem to be of sufficient conven- 
 ience in everyday life to give some attention to these processes in 
 the lower grades. Tlie nse of the drill tables on the two preceding 
 pages may help to sup|)lement these pages in giving the needed 
 drill. Place ujion the board other drill tables, consisting of the 
 multiples to 100 of all numbers as far as 20, ■ — one with the 
 multiples set in order, and another with the multiples not in any 
 regular order. Drill u})un these until the multiples are readily 
 recognized. There is no reason why 72 cannot be as quickly 
 recognized to be a multiple of 18 as of 9. 
 
 91 Some attention to the form of answers to problems should 
 be given, with the understanding always that a correct form in 
 itself is not a,n end, but only a means of showing that the process 
 of thinking is correct. First be sure that the child understands a 
 process, and then lead him to express the steps of the process in 
 correct language. This page of problems, and others which follow, 
 are supposed to furnish models of simple explanations. To fix 
 some of these forms in the mind, it might be well to give other 
 numbers than those stated in a problem. For example, in 9, after 
 the problem is solved as given, the teacher might say, " What will 
 four books cost ? " or, " Suppose six books cost eighteen dollars, 
 what would one book cost ? two books ? " etc. If the thought is 
 not clearly expressed, do not correct the form of answers until 
 it is clear that the process is understood. Objects or drawings 
 should frequently be used, both in testing the child's knowledge of 
 a process, and in teaching him the process. 
 
32 GRADED ARITHMETIC. [II. 92 
 
 92 Some of these problems may have to be ilhistrated before 
 the statements of steps and answers are given, but let the children 
 illustrate them if they can without assistance. With the illustra- 
 tions before the children, the teacher may ask, as in 5, " One apple 
 costs what?" " Eight apples will cost how many times two cents ?" 
 Then the problem can be solved in the form given, to be followed 
 by the use of other numbers. The statement of steps in short 
 sentences, as in 7 and 8, will be found useful and i)rofitable as 
 busy-work. If in any case the numbers seem too large, use smaller 
 ones ; as in 3, the teacher maj say, " What will it cost at that rate 
 to get two collars washed ? four collars ? eight collars ? four 
 collars and two cuffs ? " etc. 
 
 93 A little preliminary questioning or teaching may be ad- 
 visable before some of these problems are given ; for example, 
 3 and 6. Problems similar to 4 will be found interesting and 
 profitable. The processes involved in 8 and 9 will be readily 
 understood if similar problems with smaller numbers are first 
 given. 
 
 94 To learn once for all the number of days in each month is 
 quite important, and need not be so difficult as many suppose. 
 The verse, " Thirty days hath September," could be learned, but it 
 would be better to learn the facts that four of the months have 30 
 days, that one has 28 in all years except leap years, and that all the 
 rest have 31 days. Erequent applications of these facts in little 
 problems will fix them in the mind so that they will not be for- 
 gotten. Children never tire of measuring, an-d the rod line which 
 the children are directed to make, will be a never-ending source of 
 pleasure and profit. 
 
 95 This work should be continued until the children have a 
 tolerably accurate idea of short distances. 
 
 90 lleduction, first with the aid of a measure, and afterwards 
 as far as possible with no aid, will be found an excellent prepara- 
 tion for subsequent work. Paper of different sizes should be 
 counted out in sheets and quires, and some simple problems given 
 with the paper in sight of the children. 
 
11. J>7] teachers' imanual. • 33 
 
 97-104: Most of the facts contained in these problems should 
 have been taught previously. If in any problem there is a fact or 
 process quite new to the children, it shouhl be taught objectively, 
 the rule being observed that nothing sliould be told the child which 
 he can find out for himself. The illustration of problems by the 
 children is advised whenever the processes seem too difficult, but 
 the same or similar problems should afterwards be performed 
 without the aid of objects or drawings. 
 
 Blackboard Drill. — For the sake of variety, it may be well 
 sometimes to give a drill in adding, subtracting, multiplying, and 
 dividing from the blackboard. If it is found, for example, that 
 the pupils are slow in adding and subtracting, put on tlie board 
 columns beginning with only ones and twos, and increasing ; new 
 numbers to be added or subtracted gradually. Practice should 
 continue until as great facility is had in adding and subtracting 7, 
 8, and 9, as in adding and subtracting 1, 2, and 3. The columns 
 will appear as follows : 
 
 1 
 
 3 
 
 o 
 
 3 
 
 3 
 
 4 
 
 3 
 
 4 
 
 5 
 
 3 
 
 8 
 
 G 
 
 8 
 
 9 
 
 2 
 
 1 
 
 1 
 
 o 
 
 2 
 
 3 
 
 6 
 
 3 
 
 G 
 
 8 
 
 3 
 
 8 
 
 9 
 
 6 
 
 1 
 
 2 
 
 3 
 
 1 
 
 4 
 
 5 
 
 9 
 
 9 
 
 r^ 
 
 i 
 
 4 
 
 r* 
 
 1 
 
 7 
 
 8 
 
 9 
 
 2 
 
 2 
 
 4 
 
 3 
 
 5 
 
 3 
 
 4 
 
 
 4 
 
 5 
 
 G 
 
 9 
 
 G 
 
 3 
 
 2 
 
 3 
 
 2 
 
 4 
 
 1 
 
 9 
 
 3 
 
 9 
 
 Ad 
 
 G 
 
 8 
 
 o 
 
 5 
 
 
 4 
 
 1 
 
 1 
 
 1 
 
 9 
 
 3 
 
 4 
 
 5 
 
 5 
 
 
 r- 
 ( 
 
 1 
 
 4 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 4 
 
 5 
 
 G 
 
 G 
 
 • > 
 
 4 
 
 8 
 
 3 
 
 8 
 
 8 
 
 2 
 
 3 
 
 4 
 
 3 
 
 2 
 
 4 
 
 2 
 
 3 
 
 5 
 
 5 
 
 G 
 
 8 
 
 7 
 
 9 
 
 1 
 
 2 
 
 2 
 
 1 
 
 5 
 
 3 
 
 3 
 
 4 
 
 4 
 
 G 
 
 3 
 
 7 
 
 9 
 
 8 
 
 2 
 
 1 
 
 3 
 
 4 
 
 1 
 
 9 
 
 
 6 
 
 G 
 
 7 
 
 8 
 
 G 
 
 5 
 
 9 
 
 2 
 
 3 
 
 1 
 
 3 
 
 3 
 
 r> 
 
 G 
 
 5 
 
 3 
 
 G 
 
 5 
 
 5 
 
 
 
 Entirely new combinations may be made b}- substituting a num- 
 ber in place of the first number of the column to be added, or by 
 placing a new number below or above the column, and beginning 
 with that number to add. 
 
34 
 
 GRADED ARITHMETIC. 
 
 The same columns may be used for subtraction by placing any 
 number above the column, from which successive subtractions are 
 made. Thus, if 50 were placed above the first column, the children 
 would be led to say, 49, 47, 46, 44, etc. 
 
 By the same method, correctness of work in addition may be 
 proved. For example, if tlie pupils in adding the first column 
 have an answer of 17, they can begin with that number and sub- 
 tract successively the numbers of the column. If in subtracting 
 the last number the remainder is nothing, the answer is presumed 
 to be correct. 
 
 For drill in multiplication and division the same columns may 
 be used, the number to be multiplied or divided being placed above 
 the column. 
 
 The same progressive drill-work may be given by placing the 
 numbers in a circle, as follows : 
 
 The pupils could add in either direction to any given number, 
 or from any number placed in the centre, subtractions to zero 
 could be made. Numbers could also be placed in the centre for 
 multiplication and division. 
 
III. 1] teachers' manual. 36 
 
 SECTION V. 
 
 NOTES FOR BOOK NUMBER THREE. 
 
 By far the greater part of the work laid out in this book is with 
 numbers under 1000, and if a year is to be given to tlie book, at 
 least seven months may Avell be spent upon this part of it. 
 Thoroughness in the use of small numbers means a saving of time 
 and increased power to apply what is learned in the higlier grades. 
 
 The apparatus needed for teaching and illustrating these exer- 
 cises is a large number of short Avooden sticks with rubber bands, 
 and all the common weights and measures, such as the foot rule, 
 yard stick, rod line, gill, quart, gallon, peck, bushel, and balance 
 for weighing. 
 
 Eead carefully the Note to Teachers on pages iii and iv, particu- 
 larly what is said in the paragraph marked 2. 
 
 1—11 These exercises are a partial review of what has been 
 given in Book II. Pupils shoidd be able to perform them with 
 facility before Section II. is begun. If for any reason there is 
 difhcidty in performing or in understanding any of the exercises, 
 they should be tauglit and illustrated as previously recommended. 
 If the difficulty lies in the computations involved in the four 
 fundamental rules, drill is needed until the difficulty is removed. 
 For a good method of drill upon simple computations to 100 see 
 Book II., pages 85 and 86 ; also Manual, pages 33 and 34. 
 
 If there is difficulty in making the applications, the use of small 
 numbers and a graphic representation of the processes by drawings 
 are advised. 
 
 12—14 The new grouping of hundreds to make the thousand 
 should be taught by objects as before, and if the objects have 
 previously been sticks in bundles of ten. the same olijfcts should 
 now be used. First review what has been taught, — 10 units or 
 ones of sticks to make a ten, and 10 tens of sticks to make a 
 
36 GRADED ARITHMETIC. [III. 15 
 
 liundred. Let the pupils work witli these tens, and units until 
 their use is quite familiar. Lead the }mpils to put into bundles 
 and represent 2, 3, 4, etc., hundreds, varying the order of number- 
 ing and of representing. For example, the teacher puts before the 
 pupils piles of single sticks and bundles of tens, saying, ''Show 
 me 6 tens. • How many units in 6 tens ? Who can write that 
 number on the board ? ShoAv me seventy-six ; eighty-three ; sixty- 
 four ; ninety-seven. ShoAv me 10 tens. What is that called ? How 
 is one hundred written ? Why ciphers in the units and tens 
 places ? We Avill put tlie 10 tens together in a bundle. How 
 many hundreds in this bundle ? Show me 2 hundreds. These 
 together are called two hundred. How shall Ave write the number 
 two hundred ? Show me 3 hundred ; 4 hundred. AVho will 
 write four hundred on the board ? ShoAv me 1 hundred and 2 tens. 
 2 tens is called what ? What then is the whole number called ? 
 How will you write 1 hundred and 2 tens, or one hundred twenty? 
 Show me 2 hundreds and 3 tens. What shall we call this number ? 
 How shall we write it ? " In this way the teacher goes on teach- 
 ing the hundreds, then the hundreds and tens, and finally the 
 hundreds, tens, and units, until the thousand is reached. ISTow 
 comes the drill begun on page 12. The exercises given on these 
 pages indicate the order which may be followed, but they may not 
 be sufficient in number for some pupils. 
 
 Teach the term .min, and lead the puj)ils to use it. 
 
 15—16 These exercises are supposed to be performed with and 
 without the aitl of objects and without figures. If in any case the 
 answers cannot be readily given, let the intermediate ste})s be 
 given. For example, in 11, page 15, let the pupils say : 660, 760, 
 790 ; 380, 580, 630 ; and in 7, page 10 : 850, 650, 41)0 ; 720, 420, 
 380. 
 
 1 7—1 S These exercises should be performed on ])a])er or slate, 
 first with sticks and afterwards without sticks. Lead the jiupils 
 to pvit together the sticks in each case' before the residt is written ; 
 thus, in 19. page 17, the ]mpils should first ])nt in columns the 
 tens and units of sticks to be added, and write the corresponding 
 
III. 17] teachers' IklANUAL. 37 
 
 numbers on paper or slate. They then put together the 3, 4, and 
 4 sticks. 10 of these they should put into a ten bundle and say, 
 " 3 units and 4 units and 4 units are 11 units, equal to 1 ten and 
 1 unit. I write the 1 unit in the place of units, and add the 1 ten 
 to the tens." Then putting the bundles of tens together they say, 
 '' 1 ten and 4 tens and 2 tens and 3 tens are 10 tens, equal to 
 1 hundred and tens. I write the 1 hundred and tens in their 
 places, and have for an answer 101." 
 
 Wlien a sufficient amount of drill with the sticks has been given, 
 the pupils should place in columns the numbers beginning with 3, 
 page 17, and proceed in order until all exercises on these pages are 
 performed. 
 
 It is permitted sometimes for convenience in " proving " 11 
 answers to Avrite the " carrying " figure above the column 314 
 in which the addition is made ; but it is not well for pu})ils, 151 
 either in addition or in multiplication, to depend upon 208 
 writing down the carrying figure j thus, 37, page 17 : 733 
 
 314 
 151 
 
 A good way of preserving the carrying figure for proof, 268 
 is to write the result of each addition, thus : 13 
 
 12 
 This is found especially useful in adding long columns. 
 
 733 
 
 Do not insist upon formal and elaborate " explanations." It is 
 enough for the pupils to express what they recall of the process of 
 uniting together ten of one denomination to make one of the next 
 higher. For example, in 1, page 18, the pupils might be led to 
 say : "2 units and 7 units and 6 units are 15 units, equal to 1 ten 
 and 5 units. I write the 5 units and add the 1 ten with the tens. 
 1 and 4 and 1 and 8 tens are 14 tens, e(|ual to 1 hundred and 4 
 tens. I write the 4 tens and add the 1 hundred with the hundreds. 
 1 and 1 nnd 2 and 3 hiindreds are 7 .hundreds, which I write. 
 Answer: 745."' Oi course this form or any set form should not be 
 
38 GRADED ARITHMETIC. [III. 19 
 
 put before them as a model. What is desired of them is simply to 
 recall impressions which were made in the objective work, and to 
 express what they think. 
 
 19 Answers: 1 707. 2 861. 3 718. 4 841. 5 690. 
 
 6 781. 7 765. 8 807. 9 795. 10 998. 11 867. 12 835. 
 13 971. 14 840. 15 815. 16 854. 
 
 A large number of drill examples may be made by asking the 
 pupils to place given numbers above or below in the columns, or 
 the teacher could dictate to the pupils the numbers here given and 
 one other number. The answers could be found by adding the extra 
 number to the answers above given. In adding aloud, permit only 
 results to be given, and lead the children to add by pairs. If 
 difficulty is found in this, put pairs of numbers on tlie board for 
 
 them to add quickly at sight as follows : ^ 1 Q i J ®^^- ^^ 
 
 will not be difficult after some practice to add in the manner 
 desired ; thus, in 1, the answers should be 15, 26, 39, 46, etc. 
 
 30-Jil These exercises should be performed first with objects, 
 the pupils being led to use the sticks as indicated before the results 
 are written ; thus, in 12, page 20, tlie pupils place before tliem 6 
 bundles of hundreds and 7 bundles of tens, and after writing on 
 paper or slate 670 above 286, say, " I wish to take 286 from 670." 
 Having no units they take a ten-bundle, and after seimrating into 
 units and taking from them 6 sticks, say, " 6 units from 10 units 
 are 4 units. I write 4 in the units place. I took 1 ten from the 
 
 7 tens, and there are left 6 tens. I cannot take 8 tens from 6 tens, 
 so I take one of the hundreds." This he does, and after resolving 
 it into tens and placing them with the 6 tens, says, " 1 hundred 
 equals 10 tens. 10 tens and 6 tens equal 16 tens." He then goes 
 on with the subtraction, writing after each result is obtained, and 
 telling what is done. 
 
 As in addition, the statements should be a simj^le and natural 
 expression of the pui)il's thought. 
 
 At this point teacli the terms minuend, suhtrdJiend, and remainder, 
 and lead the pupils to use them in reading tlie exercises and answers. 
 
III. 22] teachers' manual. 39 
 
 22 A}mvers: 1 591. 2 861. 3 961. 4 748. 5 976. 
 6 900. 7 717. 8 803. 9 OO;?. 10 752. 11 957. 12 979. 
 13 759. 14 1027. 15 866. 16 857. 17 9S8. 18 778. 
 19 71-1 495 603. 20 307 350 159. 21 270. 
 
 This drill table may he used in supplying a large number of 
 examples in addition and subtraction. For example, the pupils 
 may be told to add ba from A to K or from I) to P, or the Sailie 
 work may be dictated to the pupils. Other columns as cb, dc, etc^, 
 could be given in the same' way. Practice also in adding by lines 
 could be given from the table. Other columns than those given 
 for subtraction could be given as deb, edc, etc. 
 
 23 A7iswei-s: 1 509. 2 216 492 307 139 219. 3 176 
 317 252 517 187 186. 4 292 704 237 179 178 124. 
 5 616 332 174 203 288 381. 6 206 337 186 291 144 
 69. 7 278 184 289 249 336 186. 8 303 163 226 
 340 203 197. 9 607 583 83 387 626 287. 10 963 cd. 
 11 217 sheep. 12 625 hours. 13 71 highest grades 176 lowest 
 "Trades. 
 
 & 
 
 2-4 Answers: 1 226 194 292 393 398. 2 643 188 419 
 
 361 206. 3 60 83 294 147 139. 4 39 280 361 335 
 
 228. 5 348 266 719 304 422. 6 593 670 206 606 
 
 445 101. 7 157. 8 268. 9 508. 10 319. 11 342. 
 
 12 287. 13 306. 14 302. 15 305. 16 276. 17 321. 
 
 18 244. 19 290. 20 372. 21 612. 22 426. 23 104 
 72. 24 124 120. 25 164 82. 
 
 From this table a large amount of drill-work may be given, the 
 pu})ils taking different combinations. 
 
 25 Answers: 1 873 bu. corn 1026 Im. wheat. 2 388 pear trees. 
 3 742 mi. 4 195 ft. 5 930 papers. 6 674 papers. 8 284 yrs. 
 1492, Dis. 1620, Tilgrims 1776, Dec. of Independence. 
 
 Some of these problems suggest a kind of work that may be 
 given from statistics found in the latest almanac. 
 
40 GRADED ARITHMETIC. [III. 26 
 
 36 Answers: 1 365 d. 2 $396. 3 $65. 4 837 lb. 
 
 5 1091. 6 350 mi. 
 
 The making of original problems is good practice in language. 
 Useful local statistics may be used profitably for this purpose. 
 
 37—29 Observe the order of work here required, and do not 
 abandon the objects because pupils can find the correct answers 
 without them. 
 
 30 Answers: 1 99. 2 884. 3 604. 4 424. 5 624. 
 
 6 798. 7 872. 8 590. 9 678. 10 720. 11 780. 12 840. 
 13 805. 14 905. 15 928. 16 815. 17 748. 18 984. 
 19 960. 20 772. 21 976. 22 819. 23 776. 24 897. 
 25 936. 26 1088. 27 385. 28 978. 29 021. 30 820. 
 31 915. 32 988. 33 588. 34 774. 35 1050. 36 995. 
 37 861. 38 880. 39 872. 40 948. 41 957. 42 804. 
 43 649. 44 963. 45 1062. 46 1188. 47 1228. 48 567. 
 49 910. 
 
 Let work with objects accompany the description of the process 
 and precede the writing of results ; thus, in 17, the pupils should 
 have four groups of sticks to put together, each group to have 1 
 hundred-bundle, 8 ten-bundles, and 7 units. In putting together 
 the single sticks or units the pupils should say : "4 times 7 units 
 are 28 units "; and then in putting the twenty into two bundles of 
 tens should add : " 28 units equal 2 tens and 8 units. I write 
 the 8 units in the place of units, and add the 2 tens to tlie next 
 product." Then putting together the tens the pupils say : " 4 times 
 8 tens are 32 tens plus 2 tens are 34 tens," etc. 
 
 Teach the terms multiplicand, multijdier, and product, and lead 
 the pupils to use the terms in reading and explaining problems. 
 
 31 Answers: 1 a 1044 ; Z. 1216 ; c 768 ; fZ 984 ; c 936. 
 2 a 406 ; 6 672 ; c 525 ; d 574 ; e 679. 3 a 783 ; h oi^5 ; 
 c 828 ; d 342 ; e 657. 4 a 1296 ; h 1496 ; c 864 ; d 1016 ; 
 e 1472. 5 a 616 ; b 528 ; c 836 ; c^ 352 ; e 737. 6 a 445 
 534 623 712 801 890 ; h 380 456 532 608 684 760 ; 
 c 265 318 371 424 477 530 ; d 460 552 644 736 828 
 
III. 32] teachers' ]MANUAL. 41 
 
 920. 7 a 130 15G 182 208 234 2G0 ; b 445 534 623 
 712 801 890 ; c 385 462 539 616 693 770 ; d 465 558 
 657 744 837 930. 8 a 410 492 574 656 738 820; 
 h 465 558 657 744 837 930 ; c 335 402 469 536 603 
 670 ; d 425 510 595 680 765 850. 9 a 385 462 539 
 6J 6 693 770 ; h 465 558 657 744 837 930 ; c 430 516 
 602 688 774 860; d 135 162 189 216 243 270. 
 10 a 380 456 532 608 684 760; h 240 288 336 384 
 432 480; c 460 552 644 736 828 920; d 380 456 
 532 608 684 760. 11 a 385 462 539 616 693 770; 
 b 340 408 476 544 612 680 ; c 285 342 399 456 513 
 570 ; d 210 252 294 336 378 420. 12 a 445 534 623 
 712 801 890; b 380 456 532 608 684 760; c 465 
 558 657 744 837 930; d 420 504 588 672 756 840. 
 13 620 570 654 566 876 732 1026. 14 a 636; b 768 
 c 990 ; d 1092 ; e 1134 ; / 1182. 15 a 609 ; b 672 ; c 518 
 d 266 ; e 574 ; / 679. 16 a 448 ; i 384 ; c 576 ; d 744 
 e 520 ; / 768. 17 a 513 ; ^^ 783 ; c 324 ; d 738 ; e 882 
 /774. 18 a 539; ^. 726 ; c 913; (Z 836 ; e 638 ; / 737. 
 19 70 41 51. 
 
 33 Ansxvers: 1 a 787 ; ^^ 808 ; c 1064. 2 a 964; b 1078; 
 c 1023. 3 a 992 ; b 1067 ; c 988. 4 « 288 ; b 207 ; c 463. 
 
 5 a 150 ; ^» 501 ; c 688. 6 a 118 ; Z; 146 ; c 574. 
 
 33 The exercises of this page should be performed with and 
 without the use of objects. The statement of steps and process 
 given should be the same in one case as in the other. 
 
 34 Ajiswers: 1 736. 2 676. 3 594. 4 852. 5 812. 
 
 6 645. 7 525. 8 946. 9 720. 10 756. 11 546. 12 931. 
 13 782. 14 702. 15 528. 16 432. 17 1157. 18 1358. 
 19 646. 20 1288. 21 1548. 22 931. 23 885. 24 741. 
 25 1472. 26 1144. 27 425. 28 1148. 29 836. 30 1170. 
 31 1066. 32 688. 33 476. 34 1022. 35 840. 36 a 1064; 
 6 700 ; c 1022 ; c^ 728 ; e 1204 ; / 1008. 37 a 795 ; b 915 } 
 
42 GRADED ARITHMETIC. [III. 35 
 
 c 1125 ; d 1035 ; e 855 ; / 570. 38 a 464 ; b 592 ; c 832 ; 
 0^ 624 ; e 720 ; f 288. 39 a 663 ; Z^ 765 ; c 323 ; (/ 408 ; 
 e 527 ; / 289. 40 1246 1035 1343. 41 1314 1273 910. 
 
 35 Answers: 1 1134 300 235 1186. 2 623 21 254 
 
 343. 3 552. 4 884. 5 837. 6 782. 7 770. 8 768. 
 
 9 902. 10 864. 11 918. 12 903. 13 870. 14 832. 
 15 980. 16 836. 17 936. 18 848. 19 806. 20 870. 
 21 966. 22 621. 23 638. 24 700. 25 1054. 26 970. 
 27 1073. 28 840. 29 882. 30 864. 
 
 These should be performed without objects, the pupils giving in 
 clear form statements of process. 
 
 The objective work in division of large numbers here begun should 
 be very careful and methodical. In 33 another form of statement 
 would be : " \ of 8 tens of sticks is how many ? i of 4 sticks is 
 how many ? i of 84 sticks is how many ? " This process is some- 
 times called " partitio7i " to distinguish it from " division/' or the 
 process of finding how many times one number is contained in 
 another numl)er. 
 
 3G Lead the pupils to use tlie objects and to make statements 
 as they divide. The form of questions used for exercises on page 
 35 may be changed to asking how many times one number will be 
 contained in another number. For example, in 1 : " How many 
 times are 2 sticks contained in 200 sticks ? How many times are 
 2 sticks contained in 20 sticks ? How many times are 2 sticks 
 contained in 220 sticks ? " 
 
 If multiplication has been taught by objects thoroughly, it will 
 not be necessary to continue with the objects in division to a 
 great extent. It may be assumed, for example in 6, that the 
 pupils know that 2 is contained in 16 tens 8 tens times, and all 
 similar examples may be so explained without tlie aid of objects. 
 In 8'' a new step is taken, and sliould be taught with objects tlius : 
 "2 sticks are contained in 13 tens of sticks how many tens times, 
 and how many tens remainder ? 2 sticks are contained in 1 ten or 
 
 10 how many times ? What is the answer ? " This principle is 
 
III. 37] TEACHEES MANUAt. 43 
 
 further applied in ll, and a similar course of teaching' should be 
 taken. 
 
 After the examples of this page liave been performed in the way 
 indicated, the same examples may be performed as if sticks and 
 not times were called for. This is a more simple process to teach. 
 For example, in 1, the teacher should lead the pupils to get ^- of 
 2 hundreds, then J of 2 tens, thus finding ^ of 220 to be 1 hundred 
 and 1 ten or 110. A rapid oral review without objects should be 
 given before the next page is taken. 
 
 37 This page of exercises may be taken, without objects, 
 orally. In such exercises as 23 a, the pupils may be led to say 
 first: "4 in 120, .30 times. 4 in 132, 33 times." Afterwards 
 they may perform the exercises without analysis. 
 
 38 The same course with objects is to be pursued here as was 
 advised for the exercises on page 36, with the added feature of 
 expressing with figures the results as they are found. Teach the 
 terms dividend, divisor, and (piotlcnt, and lead the pupils to use 
 them. 
 
 39 Long division, if thought best, may be begun with small 
 numbers, and may be performed without objects. A good form of 
 long division is to place the divisor at the left of the dividend, and 
 the quotient above the dividend, as shown below. 
 
 In teaching long division, the i)upils should be led to see that 
 there are no new processes to learn, but that it is the same as 
 short division, with the products written out instead of 
 carried in the mind. The following order of teaching is ^^ 
 
 suggested, it being understood that as the pu])ils answer 17)833 
 the questions the numbers are written. 68 
 
 2 is contained in 4 tens how many tens times ? IT is 153 
 
 contained in 83 tens how many tens times ? How shall 153 
 
 we find the remainder that is not divided by 17 ? Multi- 
 plying 17 by 4 tens and subtracting, what have we ? This number 
 of tens and 3 units is Avhat number ? 17 is contained in 153 how 
 many times ? 17 midtiplied by 9 is Avhat ? What remainder is 
 there ? What is the answer ? 
 
44 GRADED ARITH^NIETIC. [ill. 40 
 
 The teaching of partition in "wliich the fractional part of a 
 number is found is also to be taught, although the form of Avork is 
 the same. The statement of steps taken in long division should 
 be at first very simple, following somewhat closely the order of 
 teaching as given above. 
 
 40 A7isu-ers: 1 22i|. 2 14 
 
 e- 3 2H 
 
 3- 4 20H- 
 
 5 21i|. 
 
 6 20i-o. 
 
 7 20§f. 8 20§|. 
 
 9 20§f. 
 
 10 21^^,. 
 
 11 21H. 
 
 12 20§|. 
 
 13 21§f 14 22if. 
 
 15 21^V- 
 
 16 22||. 
 
 17 20i|. 
 
 18 21|J. 
 
 19 19|-|. 20 21if. 
 
 21 23||. 
 
 22 22||. 
 
 23 16fi. 
 
 24 16^^,. 
 
 25 26J1- 26 14||. 
 
 27 20/^. 
 
 28 16i|. 
 
 29 19|§. 
 
 30 25|f. 
 
 31 a 16H ; i^ 25U ; 
 
 c 43/^ ; 
 
 d 50/5. 32 a 231 1 ; 
 
 b 42/, ; 
 
 c 29/, ; cl 47if 
 
 33 a 62/,: 
 
 ; ^ 49/^ ; 
 
 c 50/5 ; 
 
 d 2111. 
 
 34 a ll^V ; ^' lOtI ; 
 
 c 8||; cZ 
 
 lOH- 35 
 
 76 24/,. 
 
 36 50/^ 
 
 52/,. 37 56}<i 31 
 
 /,. 38 41/, 463A,. 
 
 39 126. 
 
 40 621. 
 
 41 4(»|. 42 25|. 
 
 43 18. 
 
 44 15. 
 
 45 73. 
 
 46 49f. 
 
 47 36i. 48 27|. 
 
 49 25/0. 
 
 50 27. 
 
 
 Lead the pupils to divide in some cases by a trial divisor, con- 
 sisting of the first figure of the divisor, to see about how many 
 times the divisor is contained in the dividend. This will be more 
 helpful when larger numbers are used. 
 
 41 A?iswers: 1 28|. 2 51i. 3 78|. 4 73*. 5 294. 
 
 6 105. 7 115|. 8 123|. 9 llof 10 63if 11 68. 
 
 12 112. 13 113f. 14 67/^. 15 134|. 16 442. 17 50/^. 
 18 1351. 
 
 Exercises 19 to 53 are for drill, either in recitation or in study. 
 For the sake of accuracy and of securing practice in multiplication, 
 it is well to have their Avork in these exercises proved in the visual 
 way, — multiplying the quotient by the divisor and adding the 
 remainder to the amount. 
 
 43 Some review Avork Avhich the pupils slioiild be able to do 
 without lielp of any kind. Let the statement of steps and answers 
 be a simple and natural expression of the pupils' thought ; thus, in 
 JL2 : " It Avill take as many days to burn 35 cords as there are 
 
III. 43] teachers' IMANUAL. 45 
 
 sevens in 35 " ; or, '^ It -svill take as many days as 7 cords is con- 
 tained in 35 cords " ; or, '' To find how many days, I divide 35 
 cords by 7 cords." 
 
 43 Lead the pupils into the habit of writing out the steps of a 
 problem in good form, and also of marking the denomination of 
 each number. Two forms may be required, • — one form in Avhich 
 the steps are only indicated, and another form in which all the 
 steps are worked out ; thus, in 1 : 
 
 ^^ of 396 mi. = average number of mi. in 1 da. 
 
 18 ) 396 mi. in 18 da. 
 22 mi. in 1 da. 
 
 44 Extend the work in making and solving original problems. 
 It will be found good work for review. 
 
 45 Some teaching of tlie writing of numbers in U. S. money 
 should precede these exercises. Show how two ounces, eight 
 pounds, six dollars, may be expressed by figures and words, and 
 also by figures and abbreviations. Give the sign for dollars, and 
 show its use by examples. Lead the pupils to see that a number 
 of cents less than one hundred is written at the right of the point, 
 the number of dimes expressed by the first figure at the right 
 of the point, and the number of cents less than ten expressed by 
 the second figure. After a little of such work, the pupils ought 
 to be ready for the exercises on this page. 
 
 46 If the pupils have a thorough knowledge of writing numbers 
 in U. S. money they should have little difficulty in adding, sub- 
 tracting, multiplying, and dividing such numbers. Before the 
 exercises of this page are given it would be well to have some 
 simple exercises in pointing off, such as the following : " In 200 
 cents how many dollars ? Where sliall I place the point to show 
 the number of dollars ? Point off the dollars in 300 cents ; 220 
 cents ; 640 cents ; 104 cents. How should these numbers be written 
 for addition and subtraction ? " In multiplication, lead the pupils 
 first to multiply dollars, then cents, and finally dollars and cents, 
 such as the following : $84 X 4 ; |61 X 8 ; $75 X 6 ; $0.80 X 2 ; 
 
46 GRADED ARITHMETIC. [III. 47 
 
 $0.75 X 4 ; $1.50 X 6 ; $2.24 X 3 ; $1.08 X 4 ; $1.38 X 6. In 
 this work, insist upon the point being made in all cases, and. lead 
 the children to see that, if the multiplicand is cents the product 
 will be cents, and that it must be pointed off accordingly. 
 
 It is well in the written statement of the solution of problems to 
 write what is given and what is required to be found ; thus, in 1 : 
 
 Given the cost of 1 cow and 1 horse. 
 To find the cost of 6 cows and 3 horses. 
 
 $64 -cost of 1 cow. $140 cost of 1 horse. 
 
 6 3 
 
 $384 '' " 6 cows. $420 " " 3 horses. 
 
 420 " " 3 horses. 
 $804 " '' 6 cows and 3 horses. 
 
 47 Before giving 5, 6, 7, and 8 to the pupils, lead them to see 
 that in all such problems the dividend and divisor must be of the 
 same denomination. This may be done by showing that 2 cents 
 is contained in 6 dimes or 6 dollars more than 3 times. Give a 
 few exercises like the following : " 3 cents in 3 dimes how many 
 times ? 6 cents in 3 dimes ? 2 dimes in 1 dollar ? 4 dimes in 
 2 dollars ? 5 cents in 2 dollars ? 4 cents in 1 dollar ? 4 cents 
 in 8 dollars ? " 9 and 10 should be performed and explained by 
 partitioTi, in which the answer is the same denomination as the 
 sum divided; thus: i of $4 = $2; I of $8.25 = $1.65. In these 
 and similar exercises pay particular attention to pointing off. In 
 exercises like the last of 10 it may be Avell to reduce the sum 
 divided to cents before dividing ; thus: " jL ^^ '^^^ cents is 34 cents." 
 
 48 Let the pupils perform as many of these orally as they can, 
 1, 2, and 4 to be done by partition, and 3 by division. A simple 
 statement of process should be made to represent the difference 
 between these two operations ; thus, in "L <( : "1 qt. of kerosene 
 will cost i as much as 4 qt. ^ of 36/ = 9/." And 3: "Since 1 
 "book can be bought for 25/, as many books can be bought for 
 $4.25 as thei'e are twenty-fives in 425, or 17 books"; or, '<as many 
 books as 25/ is contained times in 425/." 
 
III. 49] I'EACHERS' MANUAL. 47 
 
 If the use of fractions, such as are given in 6 and 7, is not 
 familiar to the pupils, give some simple work with blocks, as ^ 
 of 16 blocks, f of IG blocks, |- of 20 blocks. Discs or drawings 
 of circles, squares, or lines may be used in finding the fractional 
 part of numbers that cannot be evenly found ; thus : ^ of 4 circles, 
 
 1 of 4 circles, ^ of 6 circles, f of 6 circles, etc., may \w taught. 
 49-51 No new principle is involved in these exercises. State- 
 ments like the following may be placed sometimes before problems, 
 so as to make their conditions more clear to the pupils, thus : 
 
 . 36 oranges cost 72/ 1 yd. costs 14/ 
 
 1 orange costs ? ? " cost 224/ 
 
 It is well occasionally to ask such questions as: "8 lb. will cost 
 how many times as much as 1 lb. ? how many times as much as 
 
 2 lb. ? how many times as much as 4 lb. ? WJiat part as much 
 as 16 lb.":'"' Pupils should be able to recognize from the first that 
 a fractional part of the given number is taken when the required 
 answer is of the same denomination as tlie number divided, and 
 that division of one number by another is performed when the 
 number of parts is required. 
 
 53 Great care should be taken in making these bills, directions 
 being given as to ruling, date, receipt, etc. The pupils should 
 learn to make their own rulings. 
 
 53 Toy money may be first iised in performing these exercises, 
 if necessary. Pupils should be able to tell instantly the difference 
 between one dollar and any sum of money below that amount, thus 
 avoiding the necessity of adding in making change. 
 
 54 As many of the problems in this section as possible should 
 be performed orally. Some of them may be written out for the 
 sake of practice in written analysis. The making of original j'rob- 
 lems may be considerably extended. 
 
 55-64 Weights and measures should be used in performing 
 the problems of this section when needed. It Avill be necessary 
 to use them considerably if they have not been used before ; and 
 if they have been used before, the pupils should be led to Avork 
 
48 
 
 GRADED AEITHMETIC. 
 
 [III. 58 
 
 ■with the measures or drawings of the measures whenever the con- 
 ditions of a problem are not clearly understood. Nearly all of 
 these problems should be performed without figures ; but for prac- 
 tice in stating the steps of the solution, some of them may be 
 written out in full. It is advisable for the teachers to spend some 
 time in showing the pupils a good form of solution. The following 
 written solutions are suggested : 
 
 Exercise 10, page 58. 
 
 Given the number of bushels of grain on hand. 
 
 To find the number of bushels more to fill an order. 
 
 1000 bu. of grain ordered. 
 735 " " '' on hand. 
 2r7 '^ " '' required to fill the order. 
 
 Exercise 6, page 62. 
 
 Given the amount of paper a bookseller had and sold. 
 To find the amount he had left. 
 
 10 reams 12 quires = 212 quires, amt. of paper on hand. 
 8 " 4 " =164 " '' '' " sold. 
 
 48 
 
 remaining. 
 
 If thought best, the parts of the solution may be ruled off in the 
 
 Exercise 13, page 63. 
 
 following manner 
 
 Required. 
 
 Solution. 
 
 Answer. 
 
 
 30 lb. in 1 tub. 
 
 
 
 28 '' «' 1 '' 
 
 
 
 58 '' '' 2 tubs. 
 
 
 Cost of 
 
 $0.13 cost of 1 lb. 
 
 $7.54 
 
 Sugar. 
 
 X58 
 104 
 65 
 
 
 
 $7.54 cost of 58 lb. 
 
 
IIT. 65] teachers' manual. 49 
 
 G5-07 The pupils have had some practice in measuring, but 
 it should not be presumed that they do not need more practice of 
 the same kind. In addition to tlie writing out of steps in the 
 solution of problems, the pupils should be led sometimes to draw 
 a diagram representing the distances given and required. This 
 should be done in such problems as 4, 6, 8, 10, 11, 12, and 16, 
 page 67. 
 
 6^^—70 Do not permit pupils at this stage to find the square 
 contents of a rectangle by multii)lying the length by the width. 
 Lead them always to think of the number of units in a row to 
 be multiplied by the number of rows. In the exercises on page 69 
 the required measurements should be made, but tlie fraction of 
 an inch or foot may be omitted. Wherever it can be done, a plan 
 of the surface should be drawn to scale. This should be done in 
 nearly all of the problems on page 70. Oral and written state- 
 ments of processes should be required. For example, in 7, page 70, 
 the oral statement might be : " In the passage-way there are 10 
 rows of blocks and 12 blocks in a row. Since in 1 row there are 
 
 12 blocks, in 10 rows there are 10 times 12 blocks, or 120 blocks." 
 71-73 If numbers to lOOO, including the four fundamental 
 
 rules, have been thoroughly taught by objects, the knowledge thus 
 gained will serve as a foundation for the teaching of numbers in 
 the higher orders. At this point the pupils are supposed to know 
 that ten of any order make one of the next higher order, and that 
 the value of the number expressed is determined by the position 
 that the figure has with reference to the decimal point. Proceeding 
 upon this basis, the pupils count the thousands precisely as the 
 units were counted, until a thousand thousand is reached, when 
 the number is called one million. The period of thousands, both 
 in reading and writing, should be treated exactly as the period of 
 units was treated, ten of each order making one of the next higher. 
 Extend the work here given if necessary. 
 
 74 Answers: 9 7703. 10 15044. 11 17788. 12 11125. 
 
 13 13888. 14 18054. 15 20089. 16 20958. 17 16987. 
 18 13587. 19 25294. 20 11896. 
 
75 Ansiv 
 
 ers : 
 
 1 104 
 
 5 
 
 8003. 
 
 6 1 
 
 5750. 
 
 10 
 
 110981. 
 
 11 
 
 51720. 
 
 15 
 
 765277. 
 
 16 
 
 07688 
 
 20 
 
 10042. 
 
 21 
 
 9558. 
 
 50 GRADED ARITHMETIC. [III. 75 
 
 The combinations in thousands will be found easy, if the corre- 
 sponding combinations in the period of units are well understood. 
 The first eight exercises can doubtless be performed mentally by 
 the pupils, but for practice in expression the numbers sliould be 
 written in full. For convenience the fifth and sixth orders should 
 be named ten-thousand and hundred-thoasand. Tlie reason for 
 carrying one for every ten of a loAver order should be given in the 
 same way as was taught for numbers below 1000. For example, in 
 15, the pupil is led to say: "8 and 3 and 9 hundreds is 20 hun- 
 dreds, equal to 2 thousands, to be added with the thousands. 2, 8, 
 12, 20 thousands, expressed as 2 ten-thousand, and thousand," etc. 
 
 I. 2 15749. 3 19378. 4 14574. 
 
 7 233269. 8 93533. 9 54877. 
 
 12 674673. 13 73,3299. 14 884770. 
 
 17 68185. 18 155813. 19 15403. 
 
 22 10688. * 
 
 Let tlie statement of steps in adding be continued until the 
 pupils have a clear idea of the process. Give frequent exercises 
 in dictating numbers for addition. This is best for seat-work, so 
 as to be sure that all work independently. 
 
 76 Ansivers: 1 218495. 2 11084. 3 a 12734; h 16404 
 c 21729 ; d 15131 ; e 25809 ; / 20045 ; y 22148 : h 25484 
 i 35362 ; J 14794 ; k 17256 ; I 18801 ; m, 22300. 4 96513 
 5 549. 6 1378. 7 2321. 8 1047. 9 7742. 10 6942 
 11 2290 3144 1480 1930. 12 3250 2173 3760 3236 
 13 2417 751 8690 86 6048 9099 3215. 14 2468 3239 
 8189 5686 7005 9012 1021. 15 267 820 8176 575 
 7058 4639 4423. 16 8392 6401 8355 672 25(5 4165 
 312. 17 3685 517 254 833 1158 48. 18 3046 5995 
 1045 8401 6355 715. 19 4915 3289 786 8387 704 
 1653. 20 1007 5127 7564 69 859 6444. 
 
 Lead the pupils to give a statement of steps in the process of 
 subtraction until it is ck^arly understood. If the process is not 
 understood, use objects in the subtraction of numbers to 1000. 
 
III. 77] teachers' manual. 51 
 
 77 Ans7vers: 1 1G82. 2 5343. 3 652. 4 642. 5 604. 
 
 6 3264. 7 5377. 8 7077. 9 6398. 10 235. 11 3237. 
 12 2353. 13 2318. 14 2161. 15 omn. 16 rAuA). 17 294. 
 18 5659. 19 969. 20 8449. 21 11456. 22 12122. 
 23 26808. 24 28539. 25 23999. 26 6588. 27 23175. 
 28 3179. 29 181044. 30 273706. 31 93341. 32 588680. 
 33 175629. 34 302357. 35 301245. 36 176586. 
 
 Let the pupils prove their answers. Show, by use of small 
 numbers, why the sum of the subtrahend and remainder ought to 
 be the same as the minuend. 
 
 78 Ansivers: 1 a 633; h 203; c 25 ; d 1225; e 192; 
 / 873 ; (/ 6566 ; 7i 92 ; i 5275 ; j 2569 ; k 666. 2 a 138 ; 
 b 3002 ; c 25; (I 244 ; e 733 ; / 2587 ; f/ 1316 ; h 91 ; 
 i 929; ./ 609; A- 177. 3 a 2445; ^- 252 ; c 6342; d 307; 
 e 86 ; / 579 ; rj U ; h 552 ; i 2239 ; j 1326 ; /.: 212. 
 4 81208. 5 25156. 6 7992. 7 8925. 8 77611. 9 92316. 
 10 11102. 11 6966. 12 17212. 13 10628. 
 
 The proof by subtraction is only given for practice in subtrac- 
 tion. The best proof in addition is made by adding columns down 
 if they have been added up. 
 
 79 Ansicers: 1 a 25483; h 7233; c 20319; d 22857 
 e 7866; / 38134; r/ 21684 ; A 14280 ; i 8098 ; y 1582 
 2 / 4074 ; g 7123 ; h 5003 ; i 2110 ; _/ 20. 3 /" 2229 
 y 5175; h 3922; i 1618; ,/ 122. 4 / 1774; r/ 435 
 h 1046 ; I 9!)4 ; J 381. 5 / 5185 ; y 5269 ; h 2287 
 i 2249 ; ,/ 1. 6 (' 2066 ; b 5115 ; c 2169 ; d 3608 ; e 3592 
 
 7 " 2509 ; h 289 ; c 1542 ; d 3023 ; e 41. 8 « 2912 
 I> 119; c 2423 ; d 383 ; e 345. 9 " 2282 ; ^. 192 ; r 1688 
 (Z2301; e53. 10 2918 mi. 1152768. 12 30961. 13 164932. 
 
 80 After these exercises have been performed orally, it may 
 be well to express the work of some of them in figures. Such exer' 
 cise will prepare the pupils for subsequent work, 
 
52 
 
 GRADED ARITHMETIC. 
 
 [III. 81 
 
 81 Answers: 1 3474. 2 12274. 3 20032. 4 102848. 
 
 5 816458. 6 28452. 7 81492. 8 94542. 9 112208. 
 
 10 753354. 11 37536. 12 82768. 13 102816. 14 57312. 
 
 15 42840. 16 45474. 17 27870. 18 127224. 19 39702. 
 
 20 44520. 21 11572. 22 18972. 23 17955. 24 17088. 
 
 25 14382. 26 24783. 27 26828. 28 36868. 29 11025. 
 
 30 25826. 31 48802. 32 28006. 33 34941. 34 33579. 
 
 35 55476. 36 21924. 37 66720. 38 46315. 39 38016. 
 
 40 66690. 41 49068. 42 23374. 43 76665. 44 68992. 
 
 Before multiplying by tens and units as given in the exercises 
 beginning Avitli 12, let the pupils have some review in multiplying 
 smaller numbers with objects, to show that the right-hand figure in 
 multiplying by tens is in the tens' column. 
 
 83 Answers 
 5 19300. ( 
 
 10 $1008.25. 
 14 $2.88. 
 18 $4148.16 
 22 $137.19. 
 $8131.02 $7503.75 
 h $2135.60 $3155.90 
 $2484.44 $2745.50 
 
 I 4760. 2 2140. 3 30000. 4 86850. 
 5 29940. 7 5160. 8 80480. 9 32100. 
 
 II $4191.50. 12 $85.68. 13 $1402.56. 
 15 $9011.25. 16 $1083.06. 17 $4567.50. 
 
 19 $5984.94. 20 $506. 21 $92.64. 
 
 23 $48.90. 24 a $4889.40 $7225.35 $6575.46 
 
 $6987.84 $5688.06 
 
 $3551.48 
 
 c $5295.20 
 
 $2872.04 
 $2661.14; 
 $7710.72 
 
 $3401.10 
 $3151.35 ; 
 
 $5542.08 
 
 $6276.48 
 
 >972.16 $8280 
 
 d $2529 $3737.25 
 
 $2942.10 $3251.25 
 
 $6448.74 $5951.25 
 / $4158.80 $6145.70 $5592.92 
 
 $4838.12 $5346.50 $5182.22; 
 
 $5514.14 $5088.75 $4738.88 
 
 25 $228. 26 $10560. 27 $231904. 
 
 ;6285.75 $6092.61 ; 
 
 ^3277.50 $3052.16 
 
 $7972.80 $7255.68 
 
 $6836 $6722.88 ; 
 
 $4205.70 $3881.25 $3614.40 
 
 e $3867.80 $5730.45 $5215.02 
 
 $4511.22 $4985.25 $4832.07; 
 
 $6916.04 $6382.50 $5943.68 
 
 g $3305.80 $4899.95 $4459.22 
 
 $3857.42 $4252.75 $4131.77. 
 
 83 Some of these exercises should be performed by the aid of 
 figures, and each step explained, as for example, 14, in which the 
 pupihs maybe led to say : "16 nnits in 48 units, 3 times ; 16 units 
 in 48 tenSj 3 tens or 30 times ; 16 units in 48 hundreds, 3 hundreds 
 
III. 84] teachers' manual. 53 
 
 or 300 times ; 16 units in 48 thousands, 3 thousands or 3000 times." 
 Some of the exercises should be performed by partition as in 1 : 
 ^ of 40, ^ of 400, etc. 
 
 84: . A?i.mwrs : 1 S71. 2 1414. 3 1593. 4 2347. 5 1982. 
 6 70(). 7 1299. 8 3276. 9 14521. 10 176(50. 11 4581. 
 12 <'.7()9. 13 2509 J. 14 3466. 15 1226. 16 13801. 
 17 12562. 18 316f 19 8201. 20 9102. 21 1243. 
 22 12439. 23 5S00. 24 .S098. 25 276. 26 568. 27 342. 
 28 172. 29 112. 30 258. 31 763. 32 54. 33 63. 
 34 40||. 35 514-^:j. 36 1058^9^.. 37 112||. 38 Si'l/^-. 
 39 474/g. 40 4652j\. 41 15742^. 42 79O5!'.. 43 5J. 
 44 88. 45 61. 46 25. 47 201f. 48 1151. 49 84§-3. 
 50 39] 711. 51 1540i§. 52 1234ff 53 493?i. 54 799a^. 
 55 693fi. 56 163. 57 104^^. 58 1138JL. 59 923§|. 
 60 71735^. 61 1264if. 62 1055/g. 63 27. 64 2306. 
 65 48. 66 S%\%. 67 715|f. 68 627J5. 
 
 85 Answers: 1 10070. 2 6410ffi. 3 168|i. 4 3027^- 
 5 3280||. 6 5768i§. 7 148|^. 8 1227||. 9 19140;fV 
 
 10 1030. 11 2004. 12 470. 13 3005. 14 1608. 15 1020. 
 16 26. 17 368. 18 275. 19 26. 20 lOOOjf. 2I 13030. 
 22 38886. 23 28684. 24 164356; 25 21834. 26 14692. 
 27 9068. 28 6912. 29 1719. 30 2440. 31 3280. 
 32 10495. 33 11076. 34 57631. 35 17526. 
 
 If the finding of fractional parts of numbers is not understood, 
 teach the process Avith objects, using small numbers. 
 
 80 Answers: 1 958. 2 1480. 3 121522^^. 4 1093^V 
 5 32713. 6 3472. 7 10200. 8 68. 9 125. 10 fO.78. 
 
 11 1^13.50. 12 $493^. 13 $125.37. 14 $45.75. 15 $35.27. 
 16 105 yds. 17 1376 acres. 18 837 pairs, $0.70 left. 
 19 897 quires. 
 
 Call attention to the fact that to have an entire number of parts 
 in the quotient, the dividend and divisor must be of the same 
 denomination ; and that in getting the fractional part of a number 
 
54 GRADED ARITHMETIC. [III. 87 
 
 the answer is of the same denomination as the number wrought 
 upon. Problems like 10 to 15 may be performed by taking the 
 fractional part, thus : 
 
 $74.10 = cost of 95 bu. 
 
 1 
 
 ^5 
 
 of $74.10= " '' 1 hn. = ^'l =$0.78. 
 
 9o 
 
 87 Answers: 6 $3.25 $40,625. 7 20 kegs. 8 70 T. 
 
 9 6000 pkg. 10 $221.25. 11 $6.75. 12 13 T. 14 lb. 10 oz. 
 13 17 T. 9 lb. 11 oz. 
 
 Encourage originality in the solution of problems like 6. Some 
 pupils may find the cost of 1 cwt., and then of 5 cwt. Others may 
 regard the 5 cwt. as ^ of a ton and multiply by 6^. Lead them 
 to take the shorter way whenever it is clearly understood. 
 
 Lead the pupils to see that the same principle is involved in the 
 addition of compound numbers as in the addition of simple num- 
 bers, the only difference being in the system of notation. 
 
 88 Ansivers : 1 10 lb. 2 7 lb. 1 oz. 3 10 lb. 3 oz. 4 8 T. 
 5 10 T. 100 lb. 6 15 T. 7 1058v§. 8 996 mi. 9 293662. 
 
 10 $4166.66|. 11 357866. 12 2286 ft. 634 ft. 365 ft. 
 11383 ft. combined height. 13 3280 ft. 14 177 ft. less. 
 
 89 Ansivers: 1 $2724. 2 $3364.28. 3 $95.19. 4 $595.35. 
 5 $46494. 6 $11319. 7 5401 gal. 8 2894 ft. 9 6 yr. 
 10 152 mi. 11 $1552.96. 12 $234. 
 
 90 Answers: 1 $586.2.58. 2 $1750. 3 84 casks. 4 357/^ 
 :bars. 5 245 yds. 6 2297. 7 $3841.25. 8 48 tubs. 
 
 Some of the most difficult of the problems on the last pages of 
 this section may need to be talked about in tlie recitation before 
 they are given as a lesson for the pu])ils to learn. Such questions 
 as the following for 2, page 90, may be helpful in inducing the 
 pupils to think, and in leading them to give a good written analysis: 
 "What is given in this problem? What is required? Do you 
 know the whole number of through passengers ? How can you 
 
III. 91] teachers' manual. 56 
 
 find the number ? How can you find the whole amount paid by 
 these passengers ? " 
 
 91—101 Whenever the conditions of a problem are not clearly 
 understood, lead the pupils to grasp them by the use of questions 
 and illustrations, and not by any set form of reasoning. Let 
 the questions be such as to make the pupils think. In such prob- 
 lems as 7, page 91, some preliminary questions like the following 
 may be helpful : " 10 pounds cost what part as much as 20 pounds ? 
 4 pounds cost what part as much as 20 pounds ? Do you see now 
 any way of getting the cost of 8 pounds '/ " If the pupil is still 
 uncertain, ask him what 4 pounds cost, and then what 8 pounds 
 cost. In finding the cost of 25 pounds, the pupils may be led to 
 find the cost of 1 pound first, and then of 25 pounds. There is 
 some advantage in having pu})ils form a habit of working through 
 the unit in finding the cost of a given number. But in such prob- 
 lems as 6, page 91, it is better to work by multiples. To perform 
 this problem the pupils should be led to see that 12 peaches will 
 cost 6 times as much as 2 peaches. 
 
 In some problems it may be well, if the pupils find difficulty, 
 to lead up to the required result by carefully-graded steps ; for 
 example, in 3, page 92, to ask how many eggs Avould pay for 40 
 cents' worth of butter, 80 cents' worth, etc. In 1, page 95, the 
 questions might be : " How many can I make in 1 hour ? in 3 
 hours ? " And in 5 : " How many times can the measure be filled 
 from a quart can ? from a gallon can ? from a two-gallon can ? " 
 In such problems as 4, page 98, and 2, page 99, it will be found 
 useful to give the same conditions with small numbers. Formal 
 oral '< explanations " of problems should not be required at this 
 time. Statements of processes, however, may be made, but care 
 should be taken that the words exactly represent the thought of 
 the speaker with little reference to the form of language. 
 
 Continued attention should be paid to the written analysis in 
 the solution of problems. The following analyses of the last three 
 problems on page 101 may suggest good forms for the pupils : 
 
56 GRADED ARITHMETIC. [III. 101 
 
 Given the number of bu. in 4 bins. 
 To find the number of lb. in 4 bins. 
 
 "5 bu. 60 lb. in 1 bu. 
 
 48 " _248 
 
 90 '' 480 
 
 35 '■' 240 
 
 248 '^ in 4 bins. 120 
 
 14880 lb. in 248 bu. 
 
 Given the height of an iceberg above water. 
 To find the Avhole height of the iceberg. 
 
 612 in. = height of iceberg above water. 
 8 
 
 4896 in. = " •< " under " 
 
 612 in. 
 5508 in. = entire height of iceberg. 
 
 Given the cost of 1 ton of coal. 
 To find the cost of 87 tons. 
 
 $7.25 cost of 1 T. 
 
 87 
 5075 
 5800 
 
 $630.75 cost of 87 T. 
 
 SECTION VI. 
 
 NOTES FOR BOOK NUMBER FOUR. 
 
 A mastery of the subjects presented in Book No. 4 will give 
 nearly all the practical knowledge of Arithmetic needed for the 
 ordinary affairs of life, besides furnishing a good foundation for 
 subsequent work. It may not be necessary for pupils to perform 
 all the examples and problems here given ; but before any consider- 
 i;bh r'lir.ber of them are omitted, the teacher should be sure that 
 
IV. 1] teachers' manual. 57 
 
 they are not needed, either for the purpose of fixing the principles 
 and processes which have been taught, or for the purpose of mental 
 discipline. 
 
 To more clearly understand the right use of the book, teachers 
 are advised to read the Note to Teachers, in which are given its 
 distinctive features and some hints as to possible dangers. 
 
 Appliances. — Some of tlie appliances needed for teaching the 
 various subjects may be supplied by the teacher and pupils as 
 they are needed, but it would be well to have at the outset all 
 the common weights and measures, and plenty of cardboard or 
 old pasteboard boxes from which discs and squares may be made. 
 
 Analysis of Problems. — Ko formal explanations or reasons should 
 be insisted upon, but tlie method of solution should be frequently 
 called for, both of oral and of written problems. The aim should 
 be first, to have the pupils think as they solve the problem, and 
 secondly, to have them express their thoughts in their own words. 
 Good written forms of analysis should be required. 
 
 Development "Work. — It has been the aim to give many simple 
 exercises leading up to difficult processes or jDrinciples. If any 
 problem is found too difiicult for the pupils to perform, instead 
 of attempting to "explain" the problem in words, teach the part 
 not understood by the use of illustrations, or lead up to it by simple 
 questions involving small numbers. 
 
 1—7 A few of these problems may be found too difficult to be 
 performed orally, but let the pupils try them in that way before 
 the pencil is taken. Not much development work ought to be 
 needed for these review problems. Possibly in such problems as 
 7, page 5, some questions like the following may be asked ; " If 
 there were 3 rows of trees and 4 trees in a row, and the trees 
 were placed 20 ft. apart, how long and wide would the lot be in 
 which the trees are planted ? How many feet of fencing would 
 be needed for such a lot ? Suppose there were 8 rows and 6 trees 
 in a row, how long and wide would the lot be ? How many feet 
 of fencing would be needed?" Such questioning may not be 
 needed, especially if the pupils are required to draw an illustrative 
 
58 GRADED ARITHMETIC. [IV. 8 
 
 diagram. 5, page 6, is of a different kind, and may need such 
 questions as : " Hoav many miles in 1 hour ? in 3 hours ? in 8 
 hours ? How many hours would it take him to walk 3 miles ? 
 6 miles ? 12 miles ? " Simple, natural statements of processes 
 should be expected, but see that they are not too wordy and 
 labored. 
 
 8 A7iswers: 1 1303.31. 2 $643.38. 3 $899.20. 4 $1235.80. 
 5 $1192.34. 6 $6303.68. 7 $3293.83. 8 $5681.13. 9 44214. 
 
 10 174596. 11 $2533.83. 12 $2266.11. 13 $1565.12. 
 14 $1892.34. 
 
 9 Ansu-ers : 1 $59.40. 2 $317. 3 $535.70 a mo. $20.6038 
 a day. 4 $657. 5 $336. 6 $57.80. 7 $195.50. 8 $5.18. 
 9 $9.52. 10 $0.24. 11 $2. 12 $60.75. 13 $54.40. 
 14 9.46 bbl. 15 $115.39. 16 2920 lb. Ifa T. 
 
 10 Aiiswers: 1 25110 sq. ft. 2 180 ft. 3 1422f sq. yd. 
 
 4 $697. 5 $5.55. 6 $66.24. 7 $33.60 gain. 8 $1561.91. 
 9 $981.34. 10 $1080.11. 11 $1977.29. 
 
 11 Answers: 1 8580. 2 2630. 3 10520. 4 42900. 
 
 5 44125. 6 5150. 7 30900. 8 28504. 9 3483^. 10 3914. 
 
 11 1265. 12 368. 13 6 yr. 10 mo. 14 243. 15 $0.35. 
 16 96 bu. 17 $64.50. 18 576 lb. 19 $66.64. 20 $7390.50. 
 21 $4.44. 
 
 13 Answers: 1 $123.70. 2 17600yd. 3 $54.40. 4 150 
 times 960 times. 5 5000 min. 6 55| yd. 
 
 In all problems on the last five pages of this section, require 
 the pupils to write out statements of processes, to draw diagrams 
 when needed, and to carefully label each result. 
 
 13 A brief exercise from the board to teach orders and periods 
 may be given before the pupils answer the questions given on this 
 page. The exercises on this page will suggest the kind of work 
 to be given. 
 
 14 Much drill similar to this should be given from the board 
 and by dictation. Observe the omission of the word ''and" in 
 
IV. 15] 
 
 teachers' manual. 
 
 59 
 
 reading integers. Constant care in this regard will prevent con- 
 fusion in the reading of mixed decimals. 
 
 1 4437920. 
 5 7261543. 
 
 2 44201673. 
 6 5051010. 
 
 3 
 7 
 
 3348952. 
 1201991. 
 
 15 Ansivers 
 4 11888983. 
 8 990990991. 
 
 If the pupils have been thoroughly drilled in writing numbers, 
 and if they know well the orders and periods, there will be little 
 difficulty in performing these exercises. 
 
 16 Ansivers: 1 79615294. 
 
 4 71053996. 
 8 44839931. 
 12 114480. 
 16 5005625. 
 20 359406. 
 24 674700. 
 
 5 1500635. 
 9 357000879. 
 
 13 183736. 
 17 2385184. 
 
 21 306000. 
 25 218772. 
 
 2 2299711. 
 6 174679994. 
 10 56490661. 
 14 430008. 
 18 255816. 
 22 154830. 
 26 350520. 
 
 3 5392649. 
 
 7 17391047. 
 
 11 271368. 
 
 15 2609088. 
 
 19 413100. 
 
 23 465885. 
 
 If pupils thoroughly understand and can explain the four funda- 
 mental processes to millions, no explanation of those processes in 
 the higher orders need be required. 
 
 17 Answers: 1 9746388 18174424 
 
 9 184. 
 
 14 
 
 19 
 
 2 293900640 
 
 3 9940860 
 
 4 217914192 
 
 5 230709840 
 
 6 574038864 
 
 7 585462987 
 
 8 155670762 
 10 127^V 
 
 1287i| 
 
 24 
 30 
 35 
 39 
 
 15 854|i 
 20 
 200000. 25 
 1045. 31 
 
 9445f| 
 
 6334|3. 
 2000. 
 
 -|QA714 
 
 408789072 
 11357820 
 274410464 
 1141472235 
 558977328 
 553881996 
 112108854 
 
 11 174-1*. 
 16 879ig. 
 
 21 6835f§. 
 
 21658640. 
 608285264. 
 22169520. 
 6505142176. 
 3416445000. 
 1620167808. 
 797622465. 
 351392574. 
 12 130§f. 13 93af 
 17 645^. 18 862/^ 
 22 4296i6j- 
 
 23 96332f. 
 
 26 200 
 32 
 
 3010fg 
 
 6392UI. 
 
 36 
 
 1922 9 
 
 40 
 
 1867311^. 
 
 27 3000 
 33 
 37 
 
 41 15014-11 f 
 
 851/A 
 8o2§tf 
 
 28 300. 
 34 
 
 29 801. 
 
 4148^-^0- 
 38 877^3^^. 
 42 10290^^^. 
 
60 GRADED ARITKMETIC. [IV. 18 
 
 43 2o742§f 0. 44 7121i|j. 45 SSeifff. 46 333f§^f. 
 
 47 442H§4. 48 431tVtV- 49 410^^2^- 50 92f|3. 
 
 51 483i§|. 52 830f34o. 53 SJO^U- 54 UOS^b^. 
 
 55 901fej. 56 401§ff|. 57 2423iif|. 58 762f ^sa. 
 
 18 Ansivers: 1 11514500 sq. mi. 2 1940162 3571392 
 2994002. 3 774 mi. 1278 mi. 525 mi. 4 35.2 mi. Omaha, 
 12 h. 26 m. P.M. Tuesday ; Cheyenne, 3 h. 38 m. a.m. Wednesday ; 
 Ogden, 8 h. 38 m. p.m. Wednesday; Palisade, 5 h. 17 m. a.m. Thurs- 
 day; Reno, 1 h. 16 m. p.m. Thursday; Sacramento, 5 h. 38 m. p.m. 
 Thursday; San Francisco, 8 h. 11 m. p.m. Thursday. 
 
 Other problems similar to these and to the problems on the 
 following pages should be given, the object being to combine the 
 getting of useful information with work in Arithmetic. The in- 
 formation may be gathered from the regular text-books, almanacs, 
 newspapers, and railroad guides. 
 
 19 Answers: 1 $162065. 2 $121481. 4 $1545777816. 
 5 $2708393886 $567934781 $84886022 $236244642 $53642805 
 $129756975 $145141038 $54989193 $1162616070. 
 
 For areas referred to in 3, see tables given on pages 20 and 21. 
 30 
 
 .• 1 Area, 162038 
 
 Population, 
 
 17401545. 
 
 2 268620 
 
 
 
 8857921. 
 
 3 610215 
 
 
 
 " 11150675. 
 
 4 753540 
 
 
 
 22362279. 
 
 5 1601103 
 
 
 
 3059408. 
 
 6 3395516 
 
 
 
 62831828. 
 
 7 107.3 33.3 
 
 18.2 
 
 29.6 
 
 1.9. 
 
 Other problems may be given from this table, particularly those 
 relating to the area and population of the pupils' State in compari- 
 son with those of other States. 
 
 21 Ansivers: 2 Portugal, 138.3 England, 540.9 France, 
 187.8 Germany, 236.7 China, 289.3 Italy, 261.8 Japan, 271.3 
 Spain, 88.7. 4 $137238663 4919764. 5 951 h. 43/^ d. 
 
IV. 22] teachers' manual. 61 
 
 23 Answers: 170200000. 2 ITO/a^g. 3 6122423 15251345 
 115050 11597412 12G7G044. 4 G39148|a. 5 $3174 gain. 
 6 51003321. 
 
 23 Before each set of examples and problems dealing with a 
 new fraction, one or more short teaching exercises should be given 
 in Avhich the fraction should be taught both by itself and in rela- 
 tion to fractions already known. One of the best means for teach- 
 ing fractions is the disks made of wood, cardboard, or pasteboard. 
 Those for the teacher might be six or eight inches in diameter, 
 and those for pupils two or three inches in diameter. If they 
 are not provided by the school authorities, the pupils, doubtless, 
 would be able to assist the teacher in cutting them from old boxes. 
 Means for marking and cutting should also be provided. If the 
 pupils to be taught are few in number, a table could be used by 
 the teacher for placing the disks in teaching the various operations ; 
 in case a large number have to be taught at one time, the teacher 
 could use a flat surface, inclined in such a way as to permit all 
 the pupils to see the disks upon it. There might be made grooves 
 upon the surface to permit the disks to remain in place. In teach- 
 ing, sometimes the teacher places the disks for the pupils to 
 observe, and sometimes the pupils place the disks they have by 
 direction of the teacher. The small disks also can be used by the 
 pupils in preparation of a lesson. 
 
 The preliminary teaching lesson for the exercises on this page 
 is indicated by the following questions, which are asked by the 
 teacher as he places the disks : " I will cut this circle into two 
 equal parts, as you see. What is this part called (holding up one 
 of the parts) ? this part (holding up the other part) ? How many 
 halves in one ? ^ and ^ equals what ? ^ taken out of one equals 
 what ? If I take the half two times, what is the result ? How many 
 times is the half contained in the whole one ? Now, to review, 
 i + i? 1 less i? 2 times i? 1 divided by i?" The pupils 
 ought to be ready now for the exercises on this page. If they find 
 difhcvdty with the problems in inches and half inches, similar exer- 
 cises with measures cut from cardboard or paper might be given. 
 
62 GRADED ARITHMETIC. [IV. 24 
 
 34—27 These for rapid oral work. Do not leave them until 
 the pupils are able to perform them rapidly. Before 9, page 24, 
 is attempted, a short teaching exercise should be given. It may 
 be necessary to perform some of the exercises at first with the aid 
 of measures. Other exercises, like 9, page 25, may have to be 
 illustrated by sticks or marks ; but when this is done they should 
 be reviewed without aids of any kind. 
 
 28—39 The teaching exercise here may be as follows : "I 
 will cut this circle, as you see, in 4 equal parts. One of these 
 parts is called one fourth. What is this part called (holding up 
 one of the parts)? What is this part called (holding up another 
 part) ? How many fourths in the whole circle ? How many fourths 
 here (pointing to two parts arranged in the form of a half -circle)? 
 How many fourths here (pointing to three parts)? ^ plus ^ plus ^ 
 plus ^ are what (putting the parts together in the form of a circle) ? 
 1 less ^ is what (taking away one of the parts from the circle)? 
 1 less I ? 1 less f ? Hoav many times have I taken ^ (putting 
 two of the parts together)? 2 times ^ is what? How many times 
 have I taken ^ here (putting three of the fourths together)? AVhat 
 is three times ^ ? 4 times ^ ? How many times must I take ^ to 
 make | ? to make f ? to make a whole one ? ^ is contained in | 
 how many times ? in f how many times ? in a whole one how 
 many times ? What else may we call this part of the circle (put- 
 ting two fourths together)? How many fourths is ^ ? ^ less ^ is 
 what ? 2 times |- is what ? How many times is ^ contained in -^ ? 
 ^ of |- is wliat ? " Go on in this way comparing ^ with f and 1^, 
 also f with 1^ and 1^. 
 
 In teaching the expression of the fraction, lead the pupils to 
 see that the denominator expresses tlie number of parts into wliich 
 the unit is divided, and the numerator the number of parts taken. 
 Give the pupils much practice in this. Finally, they should be 
 able to answer the questions : What does tlie denominator express ? 
 What does the numerator express ? They sliould also be able to 
 illustrate by objects or marks their answers. 
 
 30 Tew or many of these exercises should be performed by the 
 
lY. 31] TEACHEKS' MANUAL. 68 
 
 aid of disks according to the pupils' understanding of them. 11, 
 12, and 13 can best be performed by the aid of sticks, marks, or 
 dots. Show the pupils how this may be done. After these have 
 been performed objectively the pupils might be led to give little 
 explanations of the solution thus : ":^ of 8 is 2; f of 8 is 3 times 
 2, or 6 ; 6 is ^ of 2 times 6 ; 2 times G is 12. 2^ is ^ of 4 times 2^; 
 4 times 2 is 8, and 4 times i is 1 ; 4 times 2^ is 9." Explain by 
 examples what is meant by '' reducing to lowest terms " as required 
 in 7. 
 
 31 — 3*3 These exercises ought to be performed readily without 
 the aid of objects or preliminary questioning, unless in some of 
 the applied problems the measures are not familiar. The stej^s 
 of the solution should be given. Thus, in 9, page 31 : "^ from ^ 
 I cannot take ; so I take ^ from 1^, leaving f. I took 1 from 8, 
 leaving 7; 1 from 7 leaves 6. Answer. 6f." And in 6, page 32: 
 " The horse eats 3 times ^ peck or 1|- pecks in 1 day. It will take 
 as many days for him to eat 4^ pecks as 1|- is contained times 
 in 4-^, 1-^ is contained in 4-^, 3 times. Answer, 3 days. Since 4|- 
 bushels is 4 times as much as 4|- pecks, it will take him 4 times 
 3 days, or 12 days." Prol)ably some teaching will be necessary 
 for this problem, but the pupils should be permitted to try to solve 
 the problem before any help is given. If they solve it by reducing 
 the pecks to quarts, accept the solution. 
 
 33 — 34: For teaching eighths and their relations to fourths 
 and halves teachers are referred to the note in which a method 
 for teaching fourths was shown (page G2 of Manual). Essentially 
 the same plan should be pursued here. After the teacher has 
 taught by objects all the important facts, short dictation exercises 
 might be given for the pupils to solve with disks at their seats. 
 Such exercises as the following will suggest the kind of work which 
 may be given : -g^ + -g- + ^ equal what ? -g + ^ ? (Always expect 
 the answer to be given in the simplest form, and if they are not 
 so given, ask questions till the desired answer is obtained.) -g- + ^ 
 + i + i? i + i? i + i + i? i + i? i + i+i? i + i + i? 
 i + f? f + i? t + i? f+i? If + i? If + f? From 1 
 
64 GKADED ARITHMETIC. [IV. 35 
 
 takef; 1 — f? i-^? f-i? i— i? i-f? | — i? 
 f-f? li-i? li-f? li-f? Multiply i by 5 ; fX2? 
 fX2? fX4? fX3? fX4? |X2? fX3? fX4? 
 fX6? |X6? iofi? iofi? ioff? ioili? iofli? 
 How many times is ^ contained, inf? f-T--|-? ^~r- ^? i~i~8'^ 
 
 a_L.l? 1i_L.X? 5.-1.3? 
 4'8* -^¥-8- 4-F- 
 
 The first 18 exercises on page 34 ought to be performed first 
 with the aid and afterwards without the aid of the cut at the top 
 of the page. 
 
 35 — 36 Most of these exercises should be performed without 
 objects, and the steps of the solution should be required, as in 
 12, page 35 : "2 times 8 = 16; f of 8 = 3 ; 16 + 3 =19. 8 mul- 
 tiplied by 2|- = 19." And in 15 on the same page: "2^ = ^; 
 iofi = i; iof f = f; f off = V-orli" 
 
 37—38 These exercises are supposed to be performed orally; 
 but it may be well for the sake of clearness and the acquirement 
 of a good form of analysis for future use to have the pupils write 
 out the separate steps of some of the most difl&cult, as, for example, 
 in 14, page 37: 
 
 Given the cost of f bu. apples. 
 To find the cost of 2 bu. 
 
 $0.60 = cost of % bu. $1.60 = cost of 1 bu. 
 
 0.20 = " '' ^ bu. 3.20 = " "2 bu. 
 
 9, page 38. 60^ yd. = entire piece. 
 
 16i yd. X 3 = 48f yd. = yards cut off. 
 60^ yd. — 48f yd. = llf yd. remaining. 
 
 39 Anstvers: 1 128^ 65f 42f 58} 69|. 2 8i 43^ 
 519f 105f 161^. 3 15^ 189| 176^ 868 996f 4 226f 
 62f 179f 229i 118^. 5 131f 55^ 50 259f 49f. 
 6 4800 534f 163f 1050 712^. 7 16f 96f 58f 353f 
 1353f. 8 60| 52i 251f 322 46f 9 2362^ 49^ 975f 
 720 33. 10 152^ 107f 95f 2766| 232. 11 30f 66} 
 16 7875 1714^. 12 80^ 27^ 380 99^ 72^. 13 456^ lb. 
 14 $303f . 15 98 lots. 
 
IV. 40] teachers' manual. 65 
 
 40 Answers : 1 82 books. 2 635^ bu. 3 $0.76. 4 30 mi. 
 5 6^ mi. 6 22^ mi. 1676^ mi. 7 $267.75. 8 4 times ; 
 2/. 9 Horse, $645 Carriage, $215. 10 $1592^. 11 $128f. 
 12 $7.56^. 13 $27.57f. 14 $254.00i. 15 $100.78+. 
 16 $562.18f. 17 $311|. 18 7.60^. 19 $1.56^. 20 $1172^. 
 21 $34.42f 
 
 Good forms of written analysis are suggested by the following : 
 
 3 $0.15|- selling price of 1 gal. 
 42 
 
 30 
 60 
 
 21^ 
 
 $6.51 
 
 5.75 
 
 $0.76 
 
 selling 
 
 price 
 
 of 42 
 
 gal 
 
 cost 
 
 a 
 
 a 
 
 a 
 
 gain. 
 
 
 
 
 9 $860 = cost of horse and carriage = 4 times cost of carriage. 
 Cost of carriage = ^ of $860 = $215. 
 
 41—42 One or more teaching exercises should precede this 
 work. For suggestions as to how they may be conducted, see 
 directions for teaching fourths and eighths. As soon as the pupils 
 can discover for themselves that there are two ways of multiplying 
 a fraction by a whole number and also two ways of dividing by 
 a whole number, let them show objectively how this may be done, 
 and afterwards state the facts and reasons. For example, the 
 pupils should be led to show that | can be multiplied by 2 by 
 increasing the number of parts while the size of the parts remains 
 the same or by increasing the size of the parts while the number 
 of parts remains the same. They may afterwards make the state- 
 ment that a fraction may be multiplied by a number by multiplying 
 the numerator or dividing the denominator by that number. 
 
 43 Exercises for practice, which should be performed at sight. 
 If the pupils cannot perform them readily at first, let them work 
 for a while with the disks. This will be good desk-work. 
 
66 GHADED ARITHMETIC. [IV. 44 
 
 44 Care should be taken to make the work in division very- 
 simple at this stage of the pupils' progress. When the divisor is 
 a fraction it should be contained in the dividend an even number 
 of times. Lead the pupils to reduce the divisor and dividend to 
 the same denomination before dividing. 1 -i- 6, f -l- 3, etc., in 4, 
 should be solved by getting ^, ^, etc., of the given numbers. 
 
 When twelfths are taught, let the pupils have much practice 
 in finding the relation of twelfths to sixths, thirds, halves, and 
 fourths, 
 
 45 Show the pupils how this diagram can be used in the solu- 
 tion of the problems, and give similar ones. When they are under- 
 stood let them be performed without aids of any kind. 
 
 46 The work in division here may be too difficult without some 
 preliminary teaching. Besides the illustration given on page 45 
 for teaching division by a fraction, disks or squares may be used 
 in which the divisor and dividend are made to be of the same 
 denomination. Thus, ^-i- ^, or 8§ -I- 1, may be represented ob- 
 jectively as the division of sixths by sixths. Until the pupils are 
 entirely familiar with the process of division and of finding the 
 fractional parts of numbers, lead them to illustrate with objects or 
 drawings the ojDcrations called for. 
 
 47 Lead the pupils to state reasons in their own language first. 
 Afterwards they can be led to improve their statements by some 
 questions or suggestions from the teacher ; or diiferent pupils may 
 be called upon t© perform the same problem to see who will give 
 the simplest and clearest explanation. By substituting whole 
 numbers for fractions the conditions of a problem may sometimes 
 be better understood ; thus, in 3, if the pupil hesitates or does not 
 know whether to multiply or divide, say : "At $2 a yard how many 
 yards of cloth can I buy for $8 ? " 
 
 49 In all cases where the problem can be solved at sight, do 
 not permit the steps to be given. Answers to such exercises as, 
 ^-i-^, 4 -I- -^j and -^^ X 2, should be given at sight. In giving the 
 steps of a solution some of the most obvious ones should be omitted, 
 as, for example, in adding 8^ and 4-j7^, it is enough for the pupil 
 
IV. 22] teachers' manual. 67 
 
 to say: "y^^ cand -/^ is ll, added to 12 is 13^." And in multi- 
 plying IGy'V by 8 to say: "S times 16 is 128, 8 times ^''(^ is 2f, 
 added to 128 is 130f ." Long and labored explanations should be 
 avoided. 
 
 50 A table that can be used for drill in review at any time. 
 Other exercises than those indicated can be given, as -E" X 6, C X 12, 
 /-^-■^, etc. In denominate numbers the table can be used in many 
 ways and to an almost unlimited extent. But in using drill tables 
 care should be taken not to practice with them too much. On 
 account of the ease with which drill can be given from them, the 
 temptation is to use them for a longer time than is really needed 
 for facility in the use of numbers. 
 
 51—52 Kearly all of these problems ought to be performed 
 orally; but for the purpose of learning a good form of written 
 analysis the pupils should be asked to write out in full the steps 
 that are taken in the solution of the most difficult problems. 
 
 53 Ajistvers: 1 ll^. 2 23^- 3 
 
 ISf 4 2 
 
 1| 
 
 5 21§ 
 
 6 34ii. 7 35§. 8 32. 9 42H- 
 
 10 56*. 
 
 
 11 103J^ 
 
 12 24"8*. 13 388|f. 14 309f. 
 
 15 610f|. 
 
 
 16 272f 
 
 17 281^7^. 18 376f. 19 348f. 
 
 20 41l/. 
 
 
 21 470f 
 
 22 372f. 23 322i. 24 372|. 25 
 
 424i. 
 
 
 
 It is not necessary in all cases to write out in full the solution 
 of these problems, but only such parts of them as cannot be per- 
 formed v/ithout the aid of figures. Thus, in 11 the pupil could 
 say : " f = f , | = ^, and ^^ = ^. Thirds, halves, and sixths can be 
 reduced to twelfths. j\ + j% == }|, + j% = }|, + j% = f §, + ^% = 
 ft "^ -i- -^ write the ^ and add the two, etc." 
 
 54 A7iswers: 1 6^. 2 5^. 3 2f. 4 4^. 5 4^. 6 4^-^. 
 7 8^. 8 r^i. 9 5^. 10 6f. 11 5f. 12 7^. 13 8f. 
 14 6f. 15 13f. 16 7f. 17 10/^. 18 19|. 19 27}^. 
 20 9|. 21 13/2. 22 I 23 lOi. 24 45f. 25 l^^. 
 26 33f. 27 Of 28 62i. 29 12^^. 30 55^. 31 Sj\-. 
 32 50 1 . 33 24f. 34 21^. 35 44^. 36 51f. 37 28|. 
 
68 GRADED ARITIOIETIC. [IV. 55 
 
 38 66f. 39 159f. 40 86f. 41 141^^. 42 223^. 43 533^3^. 
 44 607|. 45 220f. 46 188|. 47 2863. 48 377^72. 
 49 319|. 50 389tV 51 163f. 52 353|. 53 652^3^. 
 
 The same may be said of the solution of these problems as was 
 said of the solution of the problems on the last page. Require 
 the pupils to write out only such parts of the solution as cannot 
 be performed orally; thus, in 35 the pupil may say, as he performs 
 the problem : <' | from ^ I cannot take ; | from 1-^ or y- = f 5 I 
 write the |; 18 from 62, 44. Answer, 44|." And in 41 : "| from f. 
 I can reduce them to twelfths. y\ from ^^ = -jL, etc." 
 
 55 Reasons for steps taken should be given ; thus, in 2: ''There 
 
 are 4 fourths in 1 ; there are as many ones in 190 fourths as four 
 
 fourths are contained times in 190 fourths or 47-^." The written 
 
 « ri.- J. n J. 1-4 fourths) 190 fourths. ^, . . 
 
 term of solution at nrst may be : ^ This is 
 
 47^ 
 
 a good time to look upon the fraction as only another form of 
 
 division, the numerator being the dividend and the denominator 
 
 being the divisor. With this view the only explanation needed is 
 
 that one number is divided by another number. The reduction 
 
 of mixed numbers to fractional numbers may be explained thus : 
 
 ''There are || in 1; in 8 there are 8 times }f = f|, + -jS^ = -yji.." 
 
 If written, the form may be simply 8^^^ = f f , + y5_ = j^o^l 
 
 In the solution of the exercises in multiplication on this page 
 
 write out only such parts as cannot be carried easily in the mind ; 
 
 thus, in 33: "3 times 2| = 8i, ^ of 2| = 1/^; 8^ + 1^^ = 91^. 
 
 The written form which appears on slate or 
 
 paper is 2| 
 
 X3i 
 
 56 Answers: llOSli. 2 1956f. 3 1455^. 8^ 
 4 1615f. 5 I5873J5. 6 1398f. 7 2525f. 1^% 
 
 8 1916i. 9 ISlOf. 10 2877/^. 11 2389f. 9}^ A7mver. 
 
 12 6197H- 21 11^. 22 13^. 23 10§. 
 24 12. 25 18. 26 21f. 27 41/^. 28 80^V 29 192^^ 
 30 228. 31 346^. 32 339J. 33 177|. 34 219f. 35 145f. 
 36 248^4. 37 79^^. 
 
IV. 57] teachers' MAiJUAL. 69 
 
 Pursue the same course with these exercises as was suggested 
 for previous exercises. When necessary the divisor and dividend 
 shouhl be reduced to the same denomination. Instead of writing 
 in full these denominations, the pupils may, after the first few 
 exercises, write only the numerator ; thus in 24 : 
 
 3^42 
 
 2 2 or -8/ -f- 1 = V = 12 Answer. 
 
 7~[84 
 
 12 Answer. 
 
 57 The answers to some of the exercises contain fractions 
 having denominators larger than 12 ; but it is understood that the 
 divisor and dividend when reduced to the same denomination may 
 be treated as whole numbers in the division. 
 
 Answers: 1 -6-j<y>- ^2° \¥- ¥/• 2 ^ 2^\ 3^%\. 3 13^ 
 3/t- Iff 4 lOfa 4^V4 ^2%%- 5 921f. 6 lUh 7 146. 
 8 405. 9 880f 10 12 steps. 11 240 da. 12 39f da. 
 13 500 people. 14 365 da. 313 working da. $61^\. 
 
 15 6m "^i"- 16 i 2d boy earns 40/ 3d boy, 30/ 4th boy, 
 30/. 17 $1440 11080 $1620. 
 
 Eequire the pupils to make drawings of the solution of prob- 
 lems whenever it is possible to do so, and to Avrite out in full the 
 steps of each illustrated sohition. The principle involved in 16 
 is supposed to have been taught, but it may require some leading 
 to get the pupils to illustrate the problem with dots as it should be. 
 At first use simple denominations, as halves or thirds, gradually 
 leading up to the work required. The finished drawing should 
 represent the number of cents that each boy has earned. 17 can 
 be performed in two ways : to first find the value of the whole lot 
 and then ^ of it ; or, to multiply the value of j\ of the lot by 4, 
 first leading the pupils to see the ratio of -^ to J^. 
 
 58 Answers: 1 1717^ lb. $4.29+. 
 
 3 2411^ mi. 3375f mi. 
 
 4 7|.\. 5 $4,371. 6 13^ h. 14- h. 
 
 7 10 mi. 1^ h. 8 3 h. 
 
 18f mi. 9 195 mi. 2| h. 
 
 
70 GEADED ARITHMETIC. [IV. 59 
 
 Such problems as 9 can be illustrated by lines on the board 
 drawn to scale, 1 inch to a mile, placing the distance above the 
 line, and time below. 
 
 59 Ansivers : 2 26 h. 15 min. 3 12 in. entire 2f in. 1 in. 
 6 $5.16|. 
 
 60-61 Stiff paper or cardboard should be used for these exer- 
 cises. Some preliminary exercises may have to be given to lead 
 the pupils to see clearly the relative size of the large square, the 
 strips, and small squares, and to be able to express the size in 
 decimals as required. Such exercises as the following may be 
 useful : " How many strips in the large square ? What part of 
 the large square is each strip ? 2 strips are what part of the large 
 square? 3 strips are what part of the large square? Hold up 
 
 4 tenths of the large square. Hold up 7 tenths of the large square. 
 Hold up 9 tenths of the large square. Each strip is how divided ? 
 Each small square is Avhat part of a strip ? How many of the 
 small sqiiares are there in the large square ? What part of the 
 large square is each small square ? Cut off 1 hundredth of the large 
 square. 1 more hundredth. How many hundredths have you cut 
 off ? Cut off as many small squares as will make 7 hundredths 
 of the large square. Hold up 1 tenth and 2 hundredths of the 
 large square. Hold up 2 tenths and 4 hundredths of the large 
 square. Calling the large square one, Avhat may we call 1 strip ? 
 
 5 strips ? 6 strips ? 1 small square ? 3 small squares ? 7 small 
 squares ? Hold up 4 tenths and 6 hundredths. How many hun- 
 dredths in 1 tenth ? in 3 tenths ? in tenths ? How many 
 hundredths in 2 tenths and 3 hundredths ? in 4 tenths and 7 
 hundredths ? Write 1 tenth in the common form. What is the 
 denominator ? Another way of writing 1 tenth is to write the 
 numerator 1, and place a dot called a decimal point before it, 
 thus, .1. A fraction whose denominator is ten, or some power of 
 ten, is called a decimal fraction. In this decimal fraction which 
 we have written what only is expressed? How may we know 
 that the figure one stands for 1 tenth ? Kead these decimals : 
 
rv. 62] teachers' isianual. 71 
 
 .7 ; .8 ; .3. How should we express 10 tenths ? Hold up the 
 part of 1 which these fractions express : .4 ; .7 ; .6. When we 
 wish to express hundredths we write the numerator in the second 
 place to the right of the decimal point, thus, .04, This expresses 
 what fraction ? .38 expresses how many tenths and how many 
 hundredths ? Kead the fraction as hundredths." 
 
 After such an exercise the pupils should be ready to take up 
 the work given on these pages. The last five exercises of page 61 
 should be extended until the reading and Avriting of decimals to 
 hundredths are thoroughly understood. 
 
 62 Thousandths should be taught in a way similar to that sug- 
 gested above for teaching hundredths. Great care should be taken 
 in marking and cutting the squares and strips, each expression of 
 figures representing what the pupils actually see to be the frac- 
 tional part of the given unit. 
 
 63 Dwell upon each step here given until the pupils thoroughly 
 understand it. 
 
 64 In 4 lead the pupils to see that f or f of 1 is the same as | 
 or I of jg§ or |g§§. f of 100 hundredths is 75 hundredths ; how 
 expressed ? etc. The reduction asked for in 5 should be performed 
 as simply as possible ; for example : " 250 thousandths is how 
 many hundredths ? 25 hundredths is the same as what fraction ? " 
 If the pupils hesitate, ask what 50 hundredths is equal to, and 
 then 25 hundredths. In 6, 7, and 8 lead the pupils to get the 
 required fractional part of 100 hundredths or 1000 thousandths as 
 before. Give other exercises similar to 10, that the principle in- 
 volved may be fully understood. 
 
 65 Answers: 10 54087.645. 11 376.848. 12 656.647. 
 13 123.728. 14 407.503. 
 
 The first four exercises are important, and may be extended 
 profitably. Addition of decimals ought not to be difficult to pupils 
 who have been carefully trained previously. Let the explanations 
 be as simply given as was advised for addition of whole numbers. 
 The sum 24 thousandths to be treated as hundredths, and thou- 
 
72 GRADED ARITHMETIC. [IV. 66 
 
 sandths ought to be as well understood as so many tens, to be 
 treated as hundreds and tens. 
 
 66 Answers: 12 5.858. 13 .063. 14 15.731. 15 27.251 
 74.066. 16 640.991 76.324. 17 65.147 78.591. 18 85.201 
 86.308. 19 290.317 26.699. 
 
 If necessary extend the objective work in subtraction of deci- 
 mals. Simple explanations of steps in written work should be 
 made by the pupils ; thus, in 15 : "0 thousandths from 1 thou- 
 sandth is 1 thousandth ; 1 is equal to 10 tenths, and 1 tenth is 
 equal to 10 hundredths ; 5 hundredths from 10 hundredths is 5 
 hundredths ; 7 tenths from 9 tenths is 2 tenths ; from 27 is 27. 
 Answer, 27.251." 
 
 67 Answers: 1 10.276 T. 2.525. 3 .495 T. 4 371.25 gr. 
 
 5 27.4 gal. 6 79.255 A. 
 
 If necessary, let the objects be used in multiplying. 
 
 68 Answers: 1 .32. 2 .032. 3 .216. 4 1.184. 5 2.422. 
 
 6 1.9. 7 6.75. 8 552. 9 1.218. 10 3.084. 11 5.715. 
 12 5.201. 13 5.272. 14 6.68. 15 5.022. 16 9.048. 
 17 13.344. 18 10.89. 19 18.252. 20 20.925. 21 25.05. 
 22 39.088. 23 73.104. 24 102.942. 25 151.552. 26 255.948. 
 27 268.584. 28 405.405. 29 691.01. 30 5918.4. 31 1168.50. 
 32 2577.748. 33 46840.8. 34 2138.752. 35 29238.48. 
 36 1762.592. 37 4350.675. 38 1865.528. 39 5659.316. 
 40 3645.32. 
 
 69 Answers: 1 5049.156. 2 4750.395. 3 2200.44. 
 4 6796.998. 5 7315.934. 
 
 Answers to nearly all the remaining problems should be written 
 without any figures of solution. In all cases where the multiplier 
 is 50 or 25, first multiply by 100 by removing the decimal point 
 two places to the right, and divide by 2 or 4. 
 
 70-71 Dividing any number by an integer is simply getting 
 a fractional part of the number, the product being of the same 
 denomination as the multiplicand. The so-called process of 
 
IV. 72] TEACHEKS' MANUAL. 73 
 
 " niultipl^'ing by a fraction " is also getting the fractional part of a 
 number. Thus, 8.4 X .9 may be read 9 tenths of 8.4, If this is 
 not clearly understood let the pupils use objects as indicated. 
 
 73 In every operation with the objects let the process and 
 result be represented by figures. Pupils will readily see in work- 
 ing with objects that when the divisor is a fraction both dividend 
 and divisor are to be of the same denomination before dividing. 
 
 73 Problems in which large numbers are used, as in 10, 19, 
 20, 22, need not be solved objectively. If the divisor is an integer 
 it will be sufficient to get the fractional part of the number reduced 
 to tenths, hundredths, or thousandths, and if the divisor is a frac- 
 tion, to reduce both dividend and divisor to the same denomina- 
 tion ; thus, in 10 the pupils may say 4 thousandths in 60000 
 thousandths 15000 times. And in 22 : ^l^ of 120 tenths is 1 
 tenth. These solutions would be written as follows : 
 
 .004 )60.000 ' 120 )12.0 
 
 15000. 
 
 74 Answers: 1 6.1 3010 3010 16.5 
 340 200 .09. 3 755 600.1 101.5 50 
 2 400 .9. 5 3.2 .18 160. 17.8. 
 120.48 .438 2166.4 9556. 7 11760 
 27.738. 8 142.048 2360 27 3.2 3002. 
 .979 2890 39.606. 10 244.664 175 
 11 13.635 792 249.66 9.1224 1000. 
 
 The pupils see in multiplying that the multiplication of tenths 
 by units gives tenths, that tenths of tenths are hundredths, and 
 that tenths of hundredths are thousandths ; and from these facts 
 are able to make and follow a simple rule for pointing oft". In the 
 same way rules made for pointing off in division may be followed, 
 but in all cases the pupils should be ready to give reasons for 
 pointing off as they do. 12 and 19 should be performed by re- 
 moving the decimal point to the left. Answers only of such 
 problems need be written. 
 
 
 1 
 
 
 403.5. 
 
 2 2. 
 
 46 780 
 
 2.5. 
 
 4 .04 25.5 
 
 .15. 
 
 6 
 
 1541.5 
 
 1010 
 
 87.5 
 
 10.25 
 
 9 23.008 
 
 120.355 
 
 3.772 
 
 325 
 
 193.04. 
 
74 GKADED AEITHMETIC. [IV. 75 
 
 75 These exercises should be performed orally. To divide by 
 50 or 25 divide by 100 and multiply the quotient by 2 or 4. 
 
 76 Anstvers: 1 110 yd. 1.1 yd. 2 20 paces. 3 $3 15/. 
 4 20/ $1.28. 5 20. 6 5. 7 72. 8 $15. 9 $1300. 
 10 $55.04. 11 2000 pencils. 12 $36000 $44000 $4800. 
 13 4.08. 14 792 &Q,. 15 792 ft. 107.25 ft. 303.60 ft. 
 16 4 rd. 86 rd. 
 
 These problems should be performed without much figure work, 
 unless the steps of the solution are required. 
 
 77 After distances are known by measurement, they should be 
 placed upon the board and kept there for convenience of com- 
 parison. Other known distances, such as the distance from one 
 town or city to another, might also be posted for reference. 
 
 78 Answers: 7 2 mi. 1 mi. 2 mi. 4 mi. 3 mi. 40 rd. 
 8 275 ft. 17160 ft. 31680 ft. 74.25 ft. 9 83^ ft. 204.75 ft. 
 9240 ft. 235^ ft. 10 413| yd. 6600 yd. 20 yd. 11 1|| mi. 
 l^V^ mi. 17^ mi. 14f | mi. 12 61-^ ft. 20^ yd. $2.56. 
 
 Most of these problems can be performed orally by the pupils. 
 Steps and reasons for steps should be called for occasionally. 
 After a problem is jjerformed, and the correct answer given, 
 questions might be asked ; as for example in 12 : "■ What do you 
 find first ? How do you find it ? How reduce to yards ? Why ? 
 How do you find the cost ? Why ? " Generally, however, a full 
 statement of the process should be given by the pupil without 
 interru2:)tion. 
 
 79 In finding the square contents of a surface, lead the pupils 
 to think first of the number of units in a row, and to multiply this 
 number by the number of rows, as was shown in Book III. 
 
 80 If the pupils have not had previous practice in drawing to 
 scale in connection with other studies, some time may well be 
 spent upon it here. These exercises will suggest other work of 
 the same kind which may be given. An explanation of 10 might 
 be somewhat as follows : " There are 12 ft. in 144 in. and 8 ft. in 
 
rv. 81] teachers' manual. 75 
 
 96 in. In a surface 12 ft. long and 1 ft. wide there are 12 sq. ft. 
 In a surface 12 ft. long and 8 ft. wide there are 8 times 12 sq. ft., 
 or 96 sq. ft." 
 
 81 In such problems as 1, 5, and 6, lead the pupils first to 
 reduce the dimensions to like terms ; thus : " 20 yd. = 60 ft. 
 60 sq. ft. X 20 = 1200 sq. ft." This problem would be explained 
 as previously shoAvn. In 7, let the pupils find first the number of 
 square yards that will exactly cover the seat, and afterwards the 
 amount which would be required if in covering there were a waste 
 of 6 in. in the length and width. In 9, let as many pupils as can, 
 make the proper allowance for corners in finding the area of the 
 walk. Kequire plan to be drawn, 
 
 83 Answers: 1 320 steps. 2 62 trees. 3 30375 sq. ft. 
 3375 sq. yd. 4 5103 sq. ft. 5 351 sq. yd. 6 170 sq. ft. 
 57 ft. 7 238941 sq. ft. 625| ft. 8 $3361.16 $32.77. 
 9 5| rd. 10 27^^ mi. 86^ mi. 64f mi. 11 25 sq. yd. 25 yd. 
 
 Besides drawing a plan for the solution of each of these problems, 
 the pupils should be expected to write out a brief analysis. The 
 following solution of the last part of 8 may serve as an example 
 of what may be done : 
 
 178i ft., length of lot. $0.20 cost of 1 yd. 
 
 2_ 163| 
 
 357 ft., length of two sides. $3260 
 
 134^ ft., width of lot. 16f 
 
 3) 491^ ft., length of two sides and end. $32.76f cost of fence.- 
 163| yd., length of fence. 
 
 83 Answers: 1 13^ sq.ft. 2 6i sq. ft. 12^ sq.ft. 8 lO,?^ in.. 
 9 41f ft. 10 15 ft. 11 90 ft. 12 128.125 sq. ft. 432.25 
 sq. ft. 14.2361 sq. yd. 
 
 84 A teaching exercise upon angles and right angles should 
 precede these exercises. If the teacher's definition of angle is the 
 difference of direction of two lines, he places upon the board pairs 
 of lines extending in different directions, and shows that the angles 
 
76 GRADED AEITHIMETIC. [IV. 85 
 
 are of different sizes. In all these examples the lines forming the 
 angles should be represented as meeting, and the point where the 
 lines of the angle meet is called the vertex. Proceeding, the teacher 
 draws two lines crossing each other, thus, +, and says, when two 
 lines cross each other so as to make four equal angles, the angles 
 are called right angles. The pupils are led to describe an angle as 
 the difference of direction of two lines, and a right angle as a square 
 corner. Angles similar to those in 1 are drawn, and the pupils 
 are led by questioning to say that an obtuse angle is an angle 
 greater than a right angle, and that an acute angle is an angle less 
 than a right angle. 
 
 Before parallelogram is taught, parallel lines are drawn, and 
 described as lines having the same direction, or, as lines which 
 are as far apart in one place as in another, or, as being the same 
 distance apart. Then a figure is drawn whose opposite sides are 
 parallel, and the figure is described as it was drawn, the descrip- 
 tion being drawn from the pupils by questioning. The terms base 
 and altitude are also taught, and the pupils are led to point out 
 the base and altitude of several parallelograms. Parallelograms 
 with right angles and those that are not so formed should be 
 drawn, and the pupils should be told that parallelograms with 
 right angles or square corners are rectangles. They should be 
 led to see that all figures whose square contents they have thus 
 far found are rectangles. They are now ready for the work desig- 
 nated in 2 and 3, which should be repeated until they see that 
 the area of any parallelogram can be found by niultij)lying the 
 length by the width, or the base by the altitude. 
 
 85 Applications of the principle developed in the last lesson 
 should be made \intil the pupils can readily find the area of any 
 parallelogram. The "diamond" used in playing baseball is a square 
 whose edge is 90 ft. 
 
 As the pupils proceed to find the relative size of a parallelogram 
 and triangle of the same base and altitude, do not assist them. 
 Simply ask the question at first, and give directions for finding 
 the answer only as they are found to be necessary. The statement 
 
rv. 86] teachers' ]\IANUAL. 77 
 
 in answer to the question contained in 5 should be made by the 
 pupils themselves. 
 
 86 Answers : 4 26400 sq. ft. 5 8 ft. 6 ft. 6 ft. 4 ft. 
 36 sq. ft. 6 9 ft. 7 180 sq. ft. 20 sq. yd. 8 20 sq. yd. 
 9 80 yd. 
 
 No assistance should be given the pupils beyond what is given 
 in 1. It would be well to give several problems in finding the 
 area of triangles. It will be profitable and agreeable to the pupils 
 to give them problems similar to 5. Give measurements of such 
 figures by dictation ; say that the figures are drawn to a given 
 scale, and ask for the area. The following dictation exercise may 
 serve as a model : " Draw a line AB 1 inch long. From the point 
 A, perpendicular to AB, draw a line 3 inches long. Mark the end 
 of this line C. Join BC. From a point D in the line AC, 2 inches 
 from A, draw a line parallel to AB. Mark the end of line last 
 drawn JS. Supposing these lines to be drawn to a scale of 12 rods 
 to an inch, find the area of each figure drawn." Similar figures 
 drawn on the board, representing fields, gardens, etc., will give 
 interesting work to pupils. 
 
 87 ^/m^ers.- 2 $16.80. 3 224 ft. 1536 sq. ft. 107| sq. yd. 
 4 6 sq. ft. 5. 108 sq. ft. 162 sq. ft. 
 
 A teaching exercise may be necessary before the pupils can find 
 the areas of a, b, and h. It may be sufficient to ask them to make 
 light dotted lines so as to make each of the figures to consist of 
 a rectangle and a right triangle. Work similar to 1 may be given 
 both as drill and test exercises. Give as little direct assistance as 
 possible to the pupils in finding these areas. In addition to plans 
 the pupils should write out in full the solution of the last four 
 exercises. 
 
 88 Ansivers: 1 15 ft. 2 3600 sq. ft. 3 27^ sq. yd. 
 4 60 sq. yd. $14.40. 5 128^ sq. ft. 6 22 sq. ft. 7 9i sq. ft. 
 8 22500 sq. ft. 
 
 The pupils should draw a plan illustrating the shape and loca- 
 tion of the lot upon the street, in the solution of 8. Let the pupils 
 
78 GRADED ARITHMETIC. [TV. 89 
 
 be free to choose the conditions of required original problems. 
 Encourage them to give as difficult problems as they are able to 
 solve. 
 
 89 Show to the pupils as many of these coins as can be ob- 
 tained. Of the coins mentioned the gold three-dollar piece, the 
 gold dollar, and the silver half-dime are not now coined. The 
 silver three-cent piece also is not coined. The number of cents in 
 fourths, eighths, fifths, thirds, and sixths of a dollar, should be 
 remembered, and problems involving these sums should be per- 
 formed orally. 
 
 90 Tliis account should be copied, and each entry explained 
 before the account of the following week is written. Balance on 
 hand in 2, $2.55; in 3, $111.89. 
 
 91 Require the pupils to make the ruling for a bill ; also to 
 copy and finish the bill given on this page. Explain tlie form of 
 dating and receipting bills. The amounts of bills are as follows : 
 
 1 .$17.71. 2 $1338.70. 3 $413.50. 4 1294.50. 5 $69.46. 
 6 $1801.75. 
 
 92 A?isi(fers: 1 $1444.80. 2 $4015.92. 3 $3465.60. 
 4 $464.75 $302.40. 5 $21.87. 7 25s. 72s. 8 4 32 25. 
 
 9 £3 £4^. 10 138s. 175s. 11 5 sov. 12 $.243. 13 $1,458 
 
 $3,645 $7.29 $17,496 $42,525 $59,778 $10.8135. 
 
 As many of the English coins as can be obtained should be 
 brought into the class, and their value given. Show the greater con- 
 venience of our decimal system both in writing and in reckoning, 
 
 93 Most of these problems can be performed orally by the 
 pupils. Reasons for multiplying or dividing in reduction may or 
 may not be given. 
 
 94 Answers: 7 10 lb. 5 oz. 8 15 lb. 6 oz. 9 26 lb. 11 oz. 
 
 10 18 lb. 7 oz. 11 7 cwt. 50 lb. 12 11 cwt. 13 4 cwt. 20 lb. 
 14 8 cwt. 70 lb. 15 4 T. 800 lb. 16 10 T. 17 9 T. 100 lb. 
 18 17 T. 300 lb. 19 2 lb. 10 oz. 20 1 lb. 12 oz. 21 1 lb. 9 oz. 
 22 4 lb. 6 oz. 23 1700 lb. 12 cwt. 600 lb. 24 14 cwt. 
 
IV. 95] teachers' manual. 79 
 
 1600 lb. 25 1 T. 16 cwt. 26 2 T. 14 cwt. 27 4 T. 11 cwt. 
 28 1 lb. 29 1 lb. 8 oz. 30 1 cwt. 31 2 cwt. 40 lb. 
 
 32 1 T. 4 cwt. 
 
 As many as possible of these problems should be performed 
 orally, but for the sake of exercise in analysis the solution may 
 be written. Explanations should be the same as in addition and 
 subtraction of simple numbers. 
 
 95 A7isivers : 5 25 lb. 8 oz. 6 14 lb. 7 13 lb. 11 oz. 
 
 8 11 lb. 4 oz. 9 66 lb. 12 oz. 10 43 lb. 5 oz. 11 118 lb. 2 oz. 
 
 12 142 lb. 8 oz. 13 13 T. 14 33 T. 4 cwt. 15 15 T. 1800 lb. 
 16 33 T. 4 cwt. 17 37 T. 10 cwt. 60 lb. 18 37 T. 12 cwt. 64 lb. 
 19 95 T. 11 cwt. 20 lb. 22 4 lb. 8 oz. 23 4 lb. 8 oz. 
 
 24 2 lb. lOf oz. li oz. 3 lb. 12 oz. 2 T. 5 cwt. 3 cwt. 60 lb. 
 
 25 2 T. 6 cwt. 2 T. 13^ cwt. 1 T. 5 cwt. 10 T. 17 cwt. 50 lb. 
 30 $120. 31 $69. 
 
 Explanations may be as in simple numbers ; thus, in 9 : "12 times 
 
 9 oz. = 108 oz. = 6 lb. 12 oz. I write the 12 oz. and add the 6 lb. 
 to the next product. 12 times 5 lb. = 60 lb. + 6 lb. = 66 lb. Answer, 
 66 lb. 12 oz." And in 24 : " ^ of 24 lb. = 2 lb., and 6 lb. remainder 
 = 96 oz. I of 96 oz. = lOf or lOf oz. Anstver, 2 lb. lOf oz." 
 
 96 These are oral exercises, which can also be given for desk- 
 work, to be brought into the class. Explanations of processes 
 might be given in the recitation. 
 
 97 Anstvers: 1 5 bu. 3 pk. 2 6 bu. pk. 3 11 bu. 1 pk. 
 4 8 bu. 2 pk. 1 qt. 5 12 gal. 6 13 gal. 3 qt. 7 11 gal. 2 qt. 1 pt. 
 8 3 gal. 3 qt. 1 pt. 1 gi. 9 11 gal. 1 qt. pt. 3 gi. 10 6 bu. 
 
 2 pk. 6 qt. 11 7 bu. 2 pk. 5 qt. 12 1 bu. 1 pk. 3 qt. 1|- pt. 
 
 13 1 pk. 2 pk. 4 qt. 3 pk. 6 qt. 14 2 bu. 2 pk. 15 2 bu. 
 
 3 pk. 16 2 bu. 2 pk. 17 3 pk. 6 qt. 18 4 gal. 3 qt. 
 19 3 gal. 3 qt. 20 5 gal. 1 qt. 21 4 gal. 2 qt. 1 pt. 
 22 1 gal. 1 qt. 3 gal. 1 qt. 23 3 pk. 4 qt. 3 pk. 5 qt. 
 24 1 pk. 1 pt. 25 3 gal. 1 pt. 1^ gi. 3 gal. 1 qt. i pt. 
 
80 ' GRADED ARITHMETIC. [TV. 98 
 
 26 8 gal. 27 7 qt. 28 7 pk. 4 qt. 29 5 bu. 2 qt. 
 
 30 25 gal. 2 qt. 
 
 These may be given for written desk-work, although the pupils 
 ought to be able to perform them orally. 
 
 98 Answers : 1 13 bu. 2 34 gal. 2 qt. 3 80 bu. 1 pk. 
 
 4 21 gal. 5 153 gal. 3 qt. 6 99 bu. 7 61 bu. 8 152 gal. 
 2 qt. 9 1 pk. 2 qt. 1^ pt. 10 2 qt. 1 pt. 1^ gi. 
 11 5 bu. 3 pk. 12 7 gal. | qt. 13 8 gal. 1 qt. 14 2 bu. 1 pk. 
 15 7 pk. 4f qt. 16 2 bu. 25 1 qt. 17 2 gal. 1^ qt. 18 9 qt. 
 1 pt. 2i gi. 
 
 For variety of form the pupils might be led to say in such prob- 
 lems as 21 : " If I give to one person 3 pecks, I can give 3 bushels 
 or 12 pecks to as many persons as there are 3 pecks in 12 pecks. 
 There are four 3 pecks in 12 pecks. So I can give 12 pecks to 4 
 persons." But this or any one form should not be insisted upon. 
 
 99 Answers: 1 $36. 2 $15.50. 3 $8. 4 $1.92. 
 
 5 $11.71|. 6 $3.53^. 7 5 bu. 3 pk. 6H qt. 8 90/. 
 9 160 sec. 405 sec. 10 40 min. 45 min. 210 min. 11 18 h. 
 9 h. 117 h. 14 3 h. 30 min. 
 
 100—101 These problems, which may be recited orally in 
 recitation, might be given for desk-work. In finding the differ- 
 ence of time in years, months, and days, let the time be given first 
 in entire years or months ; for example, in 1, page 101 : " Erom 
 Feb. 22, 1732, to Feb. 22, 1799, it is 67 years. From Feb. 22 to 
 Nov. 22 it is 9 months. From Nov. 22 to Dec. 14 it is 8 + 14 days 
 or 22 days. Ansiver, 67 years, 9 months, 22 days." 
 
 103 Answers: 1 'i v. 20 r. 26| q. 375 q. 10 r. '2 48 r. 
 500 q. 120 sheets. 3086f sheets. 3 98 r. 15 q. 4 500 r. 
 5 480 q. 6 24 r. 7 $17.50. 8 $1795.32-*. 9 15 doz. 
 4f doz. 10 44 doz. 105 doz. 11 36/. 12 24 boxes. 
 13 $90. 
 
 103 Answers: 10 $3305.76. 11 $10374.67. 12 $2283.85. 
 If answers to exercises from 1 to 9 cannot be quite readily 
 given, drill the pupils upon the exercises on page 1. 
 
IV. 104] teachers' manual. 81 
 
 104 Answers: 1 82. 2 75. 3 82. 4 77. 5 78. 
 6 75. 7 73. 8 78. 9 70. 10 82. 11 51. 12 47. 
 13 72. 14 57. 15 50. 16 G3. 17 62. 18 49. 19 45. 
 20 49. 21 49. 22 52. 23 55. 24 67. 25 697. 26 597. 
 27 669. 28 657. 29 639. 30 543. 31 539. 32 620. 
 33 408. 34 560. 35 464. 36 523. 37 438. 38 392. 
 39 389. 40 388. 41 455. 42 463. 43 505. 
 
 Pupils should be required to add by lines as they are, and not 
 write the numbers in columns before adding. 
 
 105 A7istvers: 1 1800 bbl. 2 $4412657.81+. 3 $1272.90. 
 4 $112. 5 $346.18. 6 $222.61. 
 
 106 Atistvers: 7 $1.38. 8 150 overcoats. 9 73f bu. 
 
 10 $214.08. 11 74 yd. $68.45. 
 
 107 Answers: 9 $14,024- $4488. 10 $3.24 $135. 
 
 11 $10.33i $121.66|. 12 $16.80 21 da. 13 330 paces. 
 
 After working through the unit, as the pupils should be expected 
 to at first, they might afterwards be led to get the desired result 
 directly, as, for example, in 3 : "It will take 60 men ^ as long to 
 do the work as it takes 20 men. -^ of 180 days is 60 days." 
 
 108 Answers: 1 3 yr. 4 mo. 5 yr. 6 mo. 2 2112 yr. 3 8^ 
 S?>i 330. 4 7|f oz. 5 $19.98. 6 25 wk. 7 Friday. 
 
 8 66 doz. 9 24/. 10 $2240. 
 
 109 Amwers: 3 $299.25. 4 $11.70. 5 C. Sea 42900 sq. 
 mi. larger. 6 3753 ft. higher. 7 $1.50. 8 76^%. 9 $360. 
 10 7.2 mi. 8^ min. 
 
 110 Ansivers: 1 $94.50. 2 40 mi. 5 h. 45 min. 3 $2.40. 
 
 4 $84560. 7 3576 sq. ft. 8 1192 pickets. 9 133^ sq. yd. 
 
 111 Answers: 1 134^ sq. ft. 2 $850. 3 $3312. 4 200 yd. 
 
 5 456 ft. 6 $573.60. 7 18 min. 121^ min. 8 1904213|| T. 
 
 9 6748 poles. 10 10000 cocoons 3409jV mi. 
 
82 GRADED ARITHMETIC. [V. 1 
 
 SECTION VII. 
 
 NOTES FOR BOOK NUMBER FIVE. 
 
 Before taking up the work embraced in this book, the pupils 
 are supposed to have a thorough knowledge of common fractions 
 to twelfths and of decimal fractions to thousandths. They are 
 also supposed to have had considerable practice in finding the areas 
 of parallelograms and triangles, and in performing problems in- 
 volving the common weights and measures. Teachers are advised 
 to look over Book IV. to see if some parts of that book may not 
 be reviewed before taking up the work of this book. Teachers 
 are also referred to the Note to Teachers of Book V. for general 
 suggestions. 
 
 1— 2 There should be practice upon these exercises until answers 
 are given with a good degree of promptness. At first it may be 
 necessary to solve some of them by steps ; for example, in 3, page 2, 
 the pupils may be led to say, in multiplying 47 by 8 : " 320, 56, 
 376." Or in 14, page 1 : "400, 85, 485 ; 300, 135, 435." But by 
 degrees answers should be given at sight, or the steps should be so 
 quickly taken as to make the exercises practically sight exercises. 
 Previous practice should have made the pupils familiar with the 
 products of all numbers to 20 by all numbers to 10, i.e., 136 should 
 be recognized as the product of 17 and 8 as readily as 96 is 
 recognized as the product of 12 and 8. The reverse operations 
 in division should be equally familiar. 
 
 Some analysis may also be necessary in the solution of the exer- 
 cises in fractions, but such analysis should be as limited as pos- 
 sible ; for example, in 19, page 2, the pupil may think and say '■' |, 
 h f ' "^f-" After a while a less number of steps than are here given 
 will be needed. 
 
 3i-G Short and naturally-expressed explanations of these exer- 
 cises should be made by the pupils. The following will serve as 
 
V. 7] TEACHEES' MANUAL. 83 
 
 examples of what might be expected of them : 26, page 3 : " 62^/ = 
 f of a dollar. 10 bushels will cost 10 times f of a dollar or $ \°-. 
 As many dozen eggs will be worth this as $f is contained times in 
 $Y-, OJ" l<5f dozen." 18, page 4: ''3 rods = 49^ ft. A surface 
 49^ ft. long and 1 ft. wide will contain 49^ sq. ft. A surface of 
 the same length 6 feet wide will contain 6 times 49^ sq. ft. = 
 240 + 54 + 3 = 297 sq. ft." 16, page 5 : "30 apples will cost 10 
 times as much as 3 apples. 10 times 2 cents = 20 cents, cost. 
 The boy wovdd get 15 times as much for 30 apples as he gets 
 for 2 apples. 15X3 cents = 45 cents. He gained 45 cents less 
 20 cents, or 25 cents." 
 
 7 Answers: 1 $20415.16. 2 $30446.15. 3 $23953.38. 
 4 $6807.89. 5 $2685.25. 6 $543.75. 7 $278.64. 8 $476.28. 
 9 $980. 10 $14.72. 11 $216. 12 $94.50. 13 $22.68. 
 14 $31.65^. 15 $118.71. 16 $294. 17 $2033^, son; $18300, 
 widow. 18 $2218|. 
 
 Let the pupils understand that in all final results of cost or 
 selling price, when the fraction of a cent is less than half, the 
 fraction is dropped, and when the fraction is one half or over one 
 half, an extra cent is added. 
 
 8 Answers: 1 Total area, 51250800 sq. mi.; Total population, 
 1467920000. Population per sq. mi.: Europe, 101.3; Asia, 57.7; 
 Africa, 11.03 ; Australasia and Pacific, 1.4 ; N. A. and W. I., 13.8 ; 
 So. A., 5.3 ; Polar regions, .06. 5 $11.25 $15 $10.35. 6 $117 
 $175.50 $130. 7 $57 $49.59 $42.75 8 $50.62^ $70.87^ 
 $54. 9 $224i $243f $260. 10 $4009.50 $3766.50 $3098.25. 
 11 $14700 $21840 $28560. 12 $54 $66 $32. 13 $180 
 pencils 900 pencils 2250 pencils. 
 
 Several problems may be made of 2, 3, and 4. Let the pupils 
 make as many as they can before the teacher suggests any. 
 
 9 A7iswers : 1 $4000 profit. 3 1600 oranges. 4 127f bu. 
 $76.65. 5 89^^ bbl. 6 $93. 7 14452.1. 8 4435.33. 
 
84 GRADED AEITmHETIC. [V. lO 
 
 9 13960.22. 10 111.607. 11 5604.935. 12 479.917. 
 
 13 6066.925. 14 28608 123586.56 288654.72. 15 100 10 1. 
 16 1600 240 32. 17 1710 4788 4959. 18 6.88 54544 
 35.088. 19 5407.2 48.064 3652.864. 20 7.68 3.84 .768. 
 
 10 Anstvers: 1 36.72 3.672 4.08. 2 2.38 357 23.8. 
 3 39.67 3967 396.7. 4 9.6 ft. 5 10 panes. 6 $3.00, cost of 
 
 I bbl. $9.75. 7 $.20 $2.95. 8 $1522^. 9 5 da. 10 20 bbl. 
 
 II 2000 pencils 166| doz. 12 3780 whites 294 colored 126 
 Chinese. 13 174 sheep. 14 495 doz. 15 $21.21f 16 643ibu. 
 21| loads $129.95^ profit. 17 2 yr. 2 mo. 7 da. 1 yr. 9 mo. 
 15 da. 18 $74.66|. 19 $.95. 20 $738. 21 8.526 T. 
 
 11 Examples of the terms odd number, even mimber, factor, 
 prime factor, tmdtiple, and least common multiple, will have to be 
 given before the exercises involving those terms are given. A good 
 method is to write upon the board examples of what is desired to 
 be taught, and then to ask the pupils to give other examples. After 
 the pupils get a clear idea of the terms they may be asked to define 
 them in their own words. The following definitions will suggest 
 to teachers the kind of illustrations and questions that may be 
 used. An even number is a number that is exactly divisible by 2. 
 An odd number is a number that is not exactly divisible by 2. A 
 factor of a number is its divisor. A prime factor is a divisor that 
 is a prime number. A multiple of a number is any number which 
 it will exactly divide. A common multiple of two or more numbers 
 is any number which each of them will exactly divide. The least 
 common multiple of two or more numbers is the least number 
 which each of them will exactly divide. 
 
 12 The exercises of this page should be practiced upon until the 
 pupils can give the required answers readily. The composite 
 factors may be given first if necessary. 
 
 13 In teaching how to find the prime factors of large numbers, 
 use small immbers first. Two ways of factoring may be used, — 
 first to get the composite factors, and then the prime factors of 
 
2 
 2 
 2 
 2 
 3 
 
 240 
 
 120 
 
 60 
 
 30 
 
 15 
 
 V. 14] teachers' manual. 85 
 
 these ; thus the prime factors of 240 may be found by finding the 
 prime factors of 24 and 10, or 12 and 20. Another way is to divide 
 the feumber successively by prime numbers thus : 
 
 The prime factors of 240 are 2, 2, 2, 2, 3, and 5. Lead 
 the pupils by examples to see that the common divisor of 
 two or more numbers exactly divides each of the numbers ; 
 thus, in finding the greatest common divisor of 8 and 12 
 the question should be asked : " What number will exactly 
 divide 8 and 12 ? Is there any other common divisor ? 
 Which is the greatest common divisor ? " Other examples should 
 be given of the same kind, the numbers increasing in size as the 
 lesson proceeds. The kind and order of exercises here given will 
 suggest to teachers what should be done before each new set of 
 exercises is given. Let the work proceed slowly at this point, and 
 review frequently. 
 
 14—15 The thorough knowledge of common fractions to 
 twelfths, which pupils have acquired by objective teaching and 
 much practice, will help them to use, intelligently, fractions of 
 other denominations. By means of the cut at the head of the 
 page, pupils may learn something of the relative size of fractions 
 of various denominations. Lead them to express in fractional 
 form any given part of the unit, indicating the fractional unit and 
 the number of such units. By illustrations and comparisons such 
 as are here shown, lead them to express a fraction by smaller 
 terms. Lead them to see and to say that in such a change of 
 expression, the size of the parts has increased while the number 
 of parts has decreased. In the same way teach the expression of 
 a fraction by larger terms, and why the value is not changed. 
 Drill upon exercises similar to 17, page 15, should be given until 
 the pupils can tell at sight to what common denominator any two 
 or more fractions, whose denominators are less than 20, may be 
 reduced. 
 
 16—18 Frequently the common denominator of such fractions 
 as are given in 1 and 2, page 16, can be named at sight. If, how- 
 ever, the denominators are large, let their least common multiple 
 
86 GRADED ARITHMETIC. [V. 19 
 
 be found by finding all the different prime factors, as shown in the 
 following illustrative solutions : 
 
 What is the least common multiple of 80, 72, and 60 ? 
 80 = 2X2X2X2X5 By this method the L. C. M. is the 
 
 72 = ^X^X^X3X3 largest number multiplied by the 
 
 60 = ^X^X$X^ factors of the other numbers not 
 
 80 X 3 X 3 = 720 = L. C. M. found in the largest number. 
 
 80 
 
 72 
 
 60 
 
 40 
 
 36 
 
 30 
 
 20 
 
 18 
 
 15 
 
 10 
 
 9 
 
 15 
 
 10 
 
 3 
 
 5 
 
 By this method the numbers are divided by 
 any prime number that will divide two of them 
 
 Q Tn 7\ — Tk without a remainder. The divisors, remain- 
 
 ^ -T7. 7, ^ mg quotients, and undivided numbers are the 
 
 — — factors of the least common multiple. 
 
 2X2X2X3X5X2X3 = 720. 
 
 Constant drill upon the oral exercises given on these three pages 
 should be given. A good method of drill is to let the pupils repeat 
 the steps orally; thus, in 24, page 16: "These fractions can be 
 reduced to twelfths. -^^ and -j^ are \^, and i| are f ^, and ^% are 
 f|- = 2y\ or 2^." By degrees the pupils may be led to state only 
 results; thus in 47, page 16 : ''§§, ^1, ||, -^^Y = 2§§ or 2|"; or 
 even shorter than this : " 30, 38, 68, -^^%P- = 2|." Eeduction to sim- 
 pler forms and to mixed numbers may sometimes be made in the 
 midst of the work ; thus, in 7, page 18 : "f + ^- less | = f." ^'-2 5 
 + f^^lg-fj less ^§ = 1^"^ or Ij." Short exercises of this sort in 
 which every pupil is actively thinking should be given frequently. 
 
 19 A7isivers : 1 2^. 2 4f. 3 2i^. 4 2^. 5 5§g. 6 2^^^. 
 7 Hf 8 Iflf. 9 U- 10 1^\. 11 lAV 12 2/^V 
 13 Ifa- 14 55|fl3. 15 2135. 16 737V 17 82||^. 
 
 18 51f a. 19 383tW 20 300^^. 21 254^1^ 22 193§|f. 
 23 176f-f|. 24 55|f 25 484f 26 19}|f. 27 10^|. 
 28 ^^^ If U- 29 T%\ ^%\ 13. 30 ^V m m- 31 t\^ 
 ^V^ Ul- 32 13|| 243f 39|-^. 33 198|§ 109|f 23^. 
 34 ll^f 14/t- 7^. 35 32^ 164^^ 2l3|. 36 24^ 45^ 
 
V. 20] teachers' manual. 87 
 
 8^t. 37 50,«T 99|| 193|. 38 21,^ G^Uh ^hh 39 18^ 
 ll§i 10. 40 14^f IT-Ht 1^ 2r>f|. 41 85tYo 6^ 11^^ 
 
 41 5 4^ .S.S7 9 10043 0131 43 43 1113 139 4f)3 1 
 
 10'i 4 1 
 
 Some of the above answers can be obtained without the aid of 
 figures. Encourage the pupils to use as few figures as possible in 
 the solution of problems. 
 
 The following forms of written solutions are suggested : 
 
 25 27 
 
 1fi2,-\ 29 _1_110_L1244_12 3 3 86 998 949 
 
 J-"3 f T4¥ > Tl4 T^ T44 I T44 T44 ^T44 J'2 
 
 Sjf l + 7 + 2fa=:10f|. 
 
 20 Ansu-ers: 1 30}f|. 2 SStV^. 3 8/^. 4 5|f 5 lOSif. 
 
 6 1 ■'5 7 17 a 2,^5 Q 5919 10 .Sli 11 15 643 
 
 12 14^4/^. 13 If'o- 14 31//^. 15 241. ig 71^3. 17 351. 
 18 85i-i||. 19 28tV 20 57Ji- 21 39ff. 22 224i|i. 
 
 23 1 27 04 18 59 01^ 107 PR 9« 11 07 2 9 1 5 41 
 
 254 67 83 3 55 969 OQ 61 11 2 3 147 29 ll 
 
 273^ T5^ T4(y TT7? T573- '^O 7^ ^^ 24 -^?T? 40 •'^5 5 
 
 1_2 5 1 5 9_ OQ 4.9 IS 2 3 fil9 .Q1370 861 4 3J1_ fil3 
 
 8 7 2 30 883 9129 183 4 10811 1147 18 29 9fi725 
 
 ^T5T- •'^ "2T(r ^-'^Sl -'-''35 •'■'^1547 J--'^T20 -'-''12^ -'"TOOH 
 
 102 3 59 31 '>S 6 7 304 7 987 9 ^9 5 5 4r)16 9 39 9 Ql 5_ 
 
 9118 29 32 fiOlS 802 5 482 4 A8 7 86 58 9 48709 58101 
 
 80 2 3 5 33 15 13 317 147 II 171 137 14 5 3 
 
 34 415 18487 A31 4205 811 462 fi43 8 72 31^ 8247 
 
 91151 18 563 9076 11 5 1 8 1 1 •'> 2789 104321 3fi 2847 
 
 30102 1 9813 1 n3 2 5 4f; 9 39 2 5 31 5 9 99 54 5 37 (50 5 1 
 
 802 6 3 48 7 1 fiO 3 1 582. 40 17 501 •'5 4 81 53 7 3« 4205 
 
 189 ^61 4751 87 5 61 71 04 .3Q 822 2125 lOiii 
 
 20 5 2 12 10 29 97 G 7 20 IT 7 40 984 3 3010 903 3 5318 7 
 
 4fil7 33 7_ 312 9 99 9 1 41 104 4 SOl^l 483 7 9 fiO 277 
 
 585 40123 5 50 5 9 81 123 1 40 8fi3 1 04262 8.3103 70142 
 
 00^ *<^54B¥ '^"^102^ "^-"-^TT^^?- *^ '^"45 '^^■M5 "'^7"50 ' ^3TS' 
 
 fi,3i 1 70i£o 873335 00 4 43 19215 30 73 24271 2.320 
 
88 GRADED ARITHMETIC. [V. 21 
 
 191 22Hf 32t«A 27§H- ** 32//^ 48i§| 34«H 56^11^ 
 
 54^ 36-rV 36H SO^W- 45 73/JV It^^uVu 54i|^ 72^11^ 
 
 66f 52fHi 64H 89^3^V 46 36if| Sliff 47i|f Tli^* 
 
 57^ 50^ 57/5^:j 41HM- 47 77//^ llOifi 66|02 87if§^ 
 
 69i 66|f J 85^5^? lOOHfi- 48 97^3^7^ 119HI 76|f|i 
 
 1201195 10413 S0209 SQ31 1021877 4Q 111221 194.323 
 
 111 3 1 1.S03 8 7 lOQll 10*^1 117455 190287 t^f) 15523 
 
 1,S3_9-3J7 1Q049 19 14A6 19 1 91 5 US 429 1 15 8 9 _ 170205 
 
 -'-'-''^504(y •^'^'-'T¥20 -"-^"sOS -^"^l^ -'-^"1120 "'^^'^^228 -'-*^2024- 
 
 51 _3 11 rin flTT? 188 53 31 92 527 I^O 3 5 8 
 
 **■*■ 16" 45^ •'^^ '*'^''- 2 73 T2(T 7 TST T5 7 3- *''^ T44 'g'S 
 
 181 8 1 131 7_ 5_5__9 9_ Ik'i 31 i 14 4GX6 J7J!_ 19.1 
 
 B^B^O B^5 B" 308 2^7 T25?4* *'*^ 72 8 40 5733 120 220 
 
 141 _8_305 54 ,SG5 17297 529 93A1 711 ,^235 5lO 7140 
 
 T^tf 10B45* ^'^ '-'95 -^'^30 "^30 -^357 *T^ "^364 ^2T *l43- 
 
 55 3125 1731 51G7 9 887 74 Q63 5l 7 5fi 7209 904 1 
 
 183j6_ 1Q_30_59 1111 IS 11 9(^269_ 101271 57 729 904 
 
 -^"105 -'■•^1082^ -'--'-80 -'-"120 -^"1008 •'-^3S72- ^' ' 5i ''^^ 
 
 18-a3_ 1« 500 10^ 17 59 95 67 5 iqi29 5Q 1 8 9_ Q 7 
 
 12^12. 151292 Q2754 14-7 90 1 50 273 1 90 6 7 
 
 9Q3 8 3 51137 4.59 81102 80 13 fiH 97 8 9 90 6 1 9Q 7 
 
 ^'^7 2 ^-"^S-^? ^^T4 ^-"-TT^ '-*^T0 8- "^ -*T44 ^'^T^H -^24 
 
 512093 1510 8119 2931 20123 9 fil 24318 1153 99407 
 
 •^-•-6174 *'^2T '^-'-44 ^^^V '^'^T^ge- ^•*- ^^T603 •'■-'-55 —'720 
 
 18 8 53 8SII34 2Sll 243 18 9 5 fiO 20 3 5 816 7 1 
 
 ^^1^^^ '->°3577T ^"^5^ ^^^4 ^'-'BBS- "^ ^'-'432 "T782 
 
 10-9 8*^1017 1 8417 14-8 5 449 2184 2 5 63 fi42 2 1 
 
 ■^^5^^ '^-'TT6B2 "-"^^T -'^^4TJ5 ^T44 ^e5'g2i- "'* "^2 24 
 
 717^^ 42^15 64iafi 50^^ 4411^ 52//^. 64 60}|| 
 
 fi742 5 2917 9 18163 10 Ifi 1 8 5 80263 8*^3 17 (>119 503 
 
 "'452 '^•^3 85 *"T'9T59 ^^222 ^'^6 7 2 ^*'244K "•'-8905e- 
 
 fi5 77118 792 56 8 f;83 7 63 50 46 2 5111 51 7 9 fiO 5 2 3 3 
 
 66 IfiiOl 4.1549 842 23 1 10209 5_1_ 21 -2_9_9_ 28^* 2 1_7_ 
 
 00 J^"l05 *TB?0 ^^7 92^ -'-'^SOS ^12 '^-'-1120 -^"^546 84 
 
 1 716 8 1 9 
 -^'2 2 2^4" 
 
 21 Answers: 1 260/5 ^- 2 31o4ia bu. 3 583^5^ A. 
 
 4 $.75. 5 $416911. 6 $6262|. 7 ^143. 8 f26if. 
 
 9 43|| T. 10 $439. 11 21^9^ ft. 12 557if ft. 13 $j%%. 
 14 ^\. 
 
 Let the pupils perform orally such problems as they can perform 
 in that way. 
 
 23 Ansivers: 1 |^. 2 194iJ lb. 3 if. 4 39? bu. 5 8.,Vmi. 
 6 5 h. 15 min. 7 32f a°. 8 30^ yd. 9 2J^ yd- 10 f 8,^- 
 11 6I815 mi. 12 $15f. 13 o77|i. 14 69863^5 gr. 
 
V. 23] teachers' manual. 89 
 
 Analyses both oral and written should be required, the steps to 
 be clearly indicated, and what each result stands for. The method 
 of solution in some cases might be indicated in one place and the 
 work in another, as, for example, 4 : 
 
 16| 2 
 
 251- 2 7 
 
 100 bu.-(16| bu. + 25f bu. ^^^^ ^^ 
 
 + 4011 l^u.-127 pk.)==amt. of ^^, = 1^=91^ 
 corn to fill the bin. ^ ^^^ '^^ ^^J^ 
 
 100 — 60^ = 39| bu. 60^ bu. 
 
 3,3 Let the pupils practice upon this illustrative work until 
 they can clearly see and state the two ways of multiplying a frac- 
 tion by a whole number. 
 
 24 Some of the work here called for is review, but it is advised 
 that the objective work be continued until the pupils have a clear 
 idea of the method of finding the fractional part of any number. 
 After the lines have been used in the solution of problems, the 
 same solutions should be reviewed without objects in such state- 
 ments as the following (14) : " ^ of f = ^3^ or ^ ; s of | = 5 times 
 ^ = fl." This could be followed by rapid silent solutions, the 
 answers only being given. 
 
 35 Dividing by a whole number is only another form of ex- 
 pression for getting the fractional part of a number. |- of f or 
 J -^ 2 is obtained in two ways, as shown. Let the pupils practice 
 upon this with illustrations until they can tell readily which pro- 
 cess to use. 
 
 36 — 37 Division by a fraction may also be taught by the aid 
 of disks or by the drawing of lines or squares. For suggestive 
 illustrative teaching see page 66 of the Manual. The same kind 
 of illustrations may be used for division in other fractional num- 
 bers to show, first, that the number divided and the divisor are 
 first subdivided into parts of the same size, or reduced to the same 
 denomination. For a time this method of division may be employed. 
 Afterwards the pupils, knowing that the quotient depends upon 
 
90 GRADED ARITHMETIC. [V. 28 
 
 the size of the divisor, may analyze problems as follows (19, 
 page 27): ''|^l=f; f-^^ of 1 = 5 times f or ^^^ ; ^-^1 = 
 ■J of -2/ or II = 1^^4." Such analysis should not be given to the 
 pupils, but made by them in answer to questions from the teacher. 
 After a time the pupils will see for themselves that the quotient 
 obtained by dividing by any fraction is the same as the product 
 obtained by multiplying by the same fraction inverted. There 
 can be no harm in such a method so long as the pupils understand 
 the process. In a final review of these pages let the pupils say, 
 in such exercises as 12, page 27: "-^ of 4f =-J of -3_2 = i^" etc. 
 And in such exercises as 15, page 27: "7 -^ § = 7 X ^ = %^ = 4i." 
 The processes of the solution may be made silently and only the 
 answers given, thus : "5-i--^ = 15; i^i=: 2." 
 
 38 — 30 Let the analysis of these problems be simply expressed, 
 and, as nearly as clearness will permit, in the pupils' own words. 
 The analysis of some of the problems may have to be preceded by 
 questions; thus, in 2, page 29: "One third of a yard will cost 
 what part as much as two thirds ? If you know the price of one 
 third of a yard, how can you find the price of a yard ? " And 
 in 15, page 29: "5 lb. will cost how many times as much as 2^ lb.? 
 l;^lb. will cost what part as much as 2^ lb.?" 
 
 31 Answers: 1 8f 80^ 1062 391f 656f. 2 81f 335a 
 272| 1845 753f. 3 ll^V 20^ 7U 100 12/^. 4 7i 8 
 
 23i 45 .5-7- 5 _2JL 217 74 13 117 fil7 6 1 
 
 ^Og to UjQ. i' 100 T8 2 85 3^0 5^00- " TSIJ ^K ^ 
 
 iU f 7 24|| 67/3V 1791 162J|^ 164||. 8 232^ 261| 
 292f 4101 1 123457^. 9 28 j^ 455f 1943^ 705f 426|. 
 10 134iV 2542| 1945f 2604^ 3539i72- H 1-19^ 398^ 410f 
 120f 33|g. 12 847|f 420§5 735023 2094/^ 773^1^. 
 13 187i 733| 888 « 242f 96^. 14 281^^^ 273^ 2356 
 IS^f 85^. 15 152 253| 151f 409j3 458^^. 16 128^ 
 24193- 1775/7 13545 1093^. 17 ISOif 1364^^ 444|2 
 1009,V 1055^. 18 inn IS^^'V 1311 iejH 41'^ 108\U 
 40,Vo- 19 13611 444f 246?^ 523/^^5 869|a 89j|§. 
 20 2«3 10^^, 1323 40/^ 2931^. 21 90 1055»g 3151^ 687f 
 
V. 32] teachers' manual. 91 
 
 82^5^. 22 88^^^- 188,5f} 13f 14511^ 23 05,? 814 2871 
 8202«. 24 73^^ OO^^V 25^ I573V 25 35 ^/v' 63 J^ 93e 
 
 11717 1 
 
 33 Answers : 1 $44G8f . 2 345 qt. 3 2602| ft. 8G7f yd. 
 
 4 $17.12f|. 5 $508800. 6 |G4280|, wife $24105^, son 
 
 $8035t-V, hospital. 7 $18.28^. 8 )ii8.202V 9 fl057i. 
 
 10 $1197^^. 11 -IFGOll. 12 |3.06i. 13 f313±|. 
 
 14 $14.87|. 15 |29.56i. 16 $57.49i\. 17 $83.43. 
 
 18 $5488GiV 19 ^'^Ul 20 $446i. 21 $8G7. 
 
 22 $75.G3iJ. 23 $78.72a. 24 $15341f. 25 ^^^^ ^V 
 
 Qi ,S5 1 ^_1_ 2fi 2^5 4 1_ .3 7 4 J. 07 14.1 G 3 
 
 12 7 2_2J)^ 5 _ 1 OQ _8 3 11055 L 13 21 
 
 ■55^5" T080 15^!»4"4 T2?* "*' T50000 -^ T 5 U ff TiJ2(J 7400 
 
 2§a||. 29 280 408 8f 47f ^V^o- 30 .1G|- 20 IGei 
 40tV 873§. 31 40f 58f 158if" 10|f 24i-f. 32 2|| 
 
 1145 1 5Z JL5_ 143 
 
 -^24 7 4 7 5 12 2 -^8f " 
 
 Let the jmpils be accustomed to multiply by a multiplier placed 
 in any position ; thus, in 1, the solution is written : 
 
 G|- Cost of 1 ton. 
 650 $G|, cost of 1 ton, X G50 = $3900 
 
 3900 or, + $5G8f = $44G8f, cost of G50 
 
 5G8f tons. 
 
 $44G8f Cost of 650 tons. 
 
 In these solutions the logical multiplier is 650, but the number 
 actually used is 6^. The multiplication should be made with the 
 multiplier in either position. 
 
 Cancellation of common factors in both dividend and divisor 
 may be made when convenient ; thus, in 25 : 
 
 ^0 
 
 6f ^ 400 : 
 
 3 X0i^ ^"' 
 20 
 
 
 33 Ans7vers: 1 8| 155a 
 
 5U| %% 7||. 
 
 2 8|| em 
 
 Hu 6-tf un- 3 tV? 
 
 ll"i§ 4fi| 7^1^ 
 
 16ff. 4 23^ 
 
 $.16^^^^ l.lSfH •03§§| $.30|£^. 5 ^\ f 2/^ 
 
 . 6 U\t 6f 
 
92 GRADED ARITHMETIC. [V. 34 
 
 S1015 7J3 71 115 ft1l3 129 69 0_ Q 5Q5_5 1 0.S 
 
 #iUy^. / ^'oTT 'T? -"-TF- " ^T4 -'■^B^ TOOT- ^ •-'^'^^T -"-^O 
 
 <t)yY t^*-* J^^TO- •*■" 300 ^T^5 '^T32 -^^147 -^^T04* 
 
 11 QII6 I.SS22 .Si 8 5 Qfil Q117 12 18 1-3 11 59 
 
 1Q1191 49 2 47 14 588 212 QQi 1 46 T^ 9 1 1 
 
 565^ l;54ff. 16 lofii lejif 2686f. 17 G2.S|- 457Ty5 
 2||f. 18 450. 19 97^. 20 143^. 21 1756t;1i. 22 4ifia 
 23 275. 
 
 34 ^nsu-ers ; 1 125. 2 H- 3 •'i'2.40. 4 22^ bu. 
 5 $.76f . 6 27. 7 $1.43 15f lb. 8 20 bu. $12,3^. 
 9 25 /j bu. apples 60 bu. 10 222f 10 suits 4i| yd. 
 11 10^^ T. $30f. 12 $5.62| 200 francs. 13 6 breadths. 
 8 breadths. 14 7 bu. 8 bu. 5f bu. 15 $2 gain. 16 1/^ 
 $112^. 17 283^ gab 166.04^. 18 $68f| 22| da. 
 
 The usual forms of oral and written analysis of these concrete 
 problems may be made, and afterwards they may be written out 
 "on a line." As the pupil proceeds with the oral analysis, he 
 writes the number above or below the line ; thus, in 10 the pupil 
 may be taught to say : " My answer is to be in yards, so I write 
 100 yd. as the number to be wrought upon. It will take J^ as 
 many yards to make 1 suit as it will to make 18 suits. I write 
 18 below the line as divisor. It takes so many yards to make 1 
 suit. To make 40 suits it will take 40 times as many yards as it 
 takes for 1 suit. I place 40 above the line." The problem when 
 finished appears thus : 
 
 20 
 100 yd. X 40 ^ 2000 
 
 X$ 9 J • -^ -9- 
 
 9 
 
 35 Some work with objects may help the pupils to learn this 
 new principle of finding the whole when a part is given. The 
 illustrative work given will show wliat should be done with the 
 objects. Give the pupils a certain number of counters, as 6, and 
 tell them to show you -J of the counters ; f of them ; § of them. 
 Then give to each pupil 4 counters, and say that they have f as many 
 
v. 36] teachers' manual. 93 
 
 as you have in your hand. Ask them to point out -^ as many as 
 you have, and then ask them how many more counters they must 
 take to have as many as you. Much illustrative work with dots 
 or marks will help to fix the principle. To be sure that they 
 understand it, give them work of both kinds — a part given, to find 
 the whole, and the whole given, to find a part. 
 
 Let the analysis, even of those problems which the pupils can 
 perform orally, be written out in full. It will be seen that the 
 conditions vary considerably and need careful thinking. 
 
 3G When any new feature or condition is given, as in 9, let the 
 pupils try the problem first before attempting to assist, or even 
 before calling their attention to the new condition. If they say, 
 as they will be likely to at first, that 20 gal. is f of the whole 
 number, simply ask them how many gallons were left and what 
 part was left ; and so lead them to think what part of the whole 
 20 gal. is equal to. 
 
 37 A7iswers: 1 $.51. 2 20| wk. 3 A, $45 B, $15. 
 
 4 A, $20 B, $120. 5 .4 T. .096 T. 6 32 bbl. 7 30 peaches. 
 8 24 yd. 9 $17. 10 $21i. 11 $.06 $4.52. 12 $.52|. 
 13 $.23^. 14 $8000. 15 $200. 16 $1000. 17 $333^. 
 18 $47i|. 
 
 The pupils should be asked to make written solutions on a line 
 with oral analysis, or to make written analyses with each step 
 indicated and each result designated. These two methods are 
 shown in the following solutions of 11 : 
 
 8) $0.64 cost of 2| lb. 
 
 .08 
 
 i( 
 
 " ilb. 
 
 3 
 
 
 
 $0.24 
 
 u 
 
 " 1 lb. 
 
 18| 
 
 
 
 192 
 
 
 
 24 
 
 
 
 20 
 
 
 
 4 
 
 $ 
 0^/ X $ X 113 
 
 t 
 
 = $4.52. 
 
 ^4.52 cost of 185 lb. 
 
94 GRADED ARITHMETIC. [V. 38 
 
 Encourage the pupils to perform orally as many of tliese and the 
 following problems as they can. 
 
 38 Ansivers: 1 | cd. Ij da. 2 2| da. 2 da. If da. 3 $.30 
 $.36. 4 $96. 5 $4.20. 6 $44000 $13750. 7 40 pupils. 
 8 $48000 $36000, widow. 9 $.34f. 10 $.69tV5. 11 $8.40. 
 
 12 6 weeks. 13 157U T. 
 
 39 Answers: 1 7^^ h. 2 191^- A. 3 $.80. 4 $60480. 
 5 $16853^ B, $8426f. 6 154/^ lb. $19.29. 7 113f mi. 
 
 8 3 da. 9 2 da. 10 \% da. 12|f da. 11 $750. 12 $4800. 
 
 13 $140^. 14 76 shares $14. 
 
 40 Ansivers: 1 $753.81^. 2 $63x1^. 3 $17t'VV- * ^^H- 
 5 $1579f. 6 $2176585. 7 $2508. 8 $86.62^. 9 House, 
 $7058-^- Land, $4235i. 10 6§ T. 11 16 s. (| yd. left). 
 12 $25116|. 13 $15,073+. 14 | lb. 15 .07 lb. 
 16 $45.26. 17 27i| jars. 18 f j%. 
 
 41 Answers: 1 f|. 2 $38.76+. 3 249| A. 4 $48f. 
 5 $13621. 6 66|| yd. 7 $4.41|. 8 37f|f h. 9 $6.50. 
 
 10 $5f |. 11 42if I h. 12 162^ bu. 13 $468f. 14 3 yr. 
 
 11 mo.+ 2 yr. 4 mo.+ 15 41if^. 16 $8^. 
 
 43 Answers: 1 $306^1. 2 $11271- gain. 3 $82.50+. 
 
 4 Po., 500 mi. ; Kan., 900 mi. ; Yel., 1000 mi. ; Ked., 1200 mi. ; 
 Ark., 2000 mi. ; St. L., 2200 mi. ; Nile, 3300 mi. ; Am., 3600 mi. 
 
 5 1761b. 8f lb. 6 20 yr. 7 100 ft. 8 32 j/:^ gal. 65^^ lb. 
 
 9 To Sar., 180 mi. ; to R., 360 mi. ; to B., 480 mi. ; to A., 720 mi. ; 
 to L., 900 mi. ; to J., 1500 mi. ; to S. L. C, 1800 mi. ; to S. F., 
 2250 mi. 
 
 43 A review of the objective work in decimals to thousandths, 
 given in Book IV., will be found useful in giving a good foundation 
 for the work here given. If the pupils clearly understand the 
 expression and use of decimals to thousandths, there need be no 
 use of objects in dealing with higher denominations. Dwell upon 
 
 6 and 7 until they are well understood. If 9 is found too difficult 
 it may be omitted for the present. 
 
V. 44] 
 
 I'EACHERS MANUAL. 
 
 95 
 
 14 19.9991. 
 
 18 111.8909. 
 
 .03 .64 1.33 
 
 44 Lead tlie pupils to see why the annexing of a cipher at the 
 right of decimals does not change their value. 
 
 45 Answers: 1 7576.13558G57. 2 !i?99.'.)99 $99.9995. 
 3 210.962803. 4 3.37079 in. 5 60.8533^ rd. 6 59.0135 A. 
 7 390.95347. 8 652.6939. 9 837.363. 10 1.0425 .1953. 
 11 280.4274. 12 9.9191. 13 1005.991201. 
 15 1.94. 16 114.194. 17 94.61359. 
 19 72.7699. 20 4.2 .42 .042 7.2 1.62. 21 
 .021 6.4. 22 .0101 .1836 .624 .576 .02108. 
 
 Some preliminary exercises may be given to show that hundredths 
 of hundredths give ten thousandths, and that thousandths of hun- 
 dredths give hundred thousandths, etc. This may be shown by 
 common fractions. 
 
 It is not necessary after the first few exercises to explain each 
 step in addition and subtraction. Work of this kind in one part of 
 the decimal system ought to be no more difficult than similar work 
 in any other part. 
 
 46 Ansivers: 1 35.77. 2 .3431. 3 3.9786. 4 12.353536. 
 5 14.404968. 6 2432.5. 7 84.78384. 8 26.3088. 9 8298.63. 
 
 10 
 
 74.2118. 
 
 11 8.64838. 
 
 12 49.0245. 13 81.2448. 
 
 14 
 
 793.074. 
 
 15 3.10156. 
 
 16 .0095823. 17 .24609102. 
 
 18 
 
 .976050. 
 
 19 8476.4072. 
 
 20 67.268. 21 22.25562. 
 
 22 
 
 9.322987. 
 
 23 9247.854. 
 
 24 164.467. 25 2370.104. 
 
 26 
 
 181.80208. 
 
 27 6.0888. 
 
 28 2470.713. 29 4.8. 
 
 30 
 
 1564.9236. 
 
 31 .003267 
 
 .93654 .0772101 2.228094. 
 
 32 6.1509 .061509 6.09609. 33 .040068 .0941598 2.1005649 
 .060102. 34 .0144684. 35 369.84414285. 36 .108 1.08 
 .01188 .0108108 10.8. 37 100040. 100040.10004. 38 .6501755 
 
 .0650390 .0078520 .262626 
 40 803.8024. 41 3.5000056. 
 44 10924.914. 45 1312.6476. 
 48 684.39378. 49 61.2612. 
 52 .203775. 53 .58313850. 
 
 47 Ansivers: 1 88.787. 
 4 878.17221. 5 296.478, 
 
 .0135395. 
 42 10000.84. 
 46 44850.3102. 
 50 160160. 
 54 520.96. 
 2 94.261544. 
 6 525.5491. 
 
 39 2050.3935. 
 
 43 .087675. 
 
 47 616.9472. 
 
 51 .8757567. 
 
 3 1424.08476. 
 7 7852.547. 
 
96 GEADED ARITHMETIC. [V. 47 
 
 8 214.9854. 9 7871.0072. 10 52413.821. 11 2551.6278. 
 12 104066.5212. 13 633.8787. 14 1052.62201. 15 .1984512. 
 16 32.1082179. 17 981. 18 7006.3. 19 109.46. 20 518.1357. 
 21 488.08032. 22 3793.9408. 23 2593.44. 24 80012.6. 
 25 1738.8 yd. 26 $8154.675. 27 1153.705 mi. 28 $112.4475. 
 29 $1038.744. 30 $37.0476. 31 $34,832. 32 $2044.80. 
 33 $3548.12. 34 $35574.59435. 35 .2 .02 .004 .4 3.4 
 .00004 3.334 .00034 .0000004. 36 20 200 20000 .2 .02 
 300000 .003 3000000 3000. 37 .006 .6 10 .01 .001 
 .0001 1 .1 9.6. 
 
 Before the exercises in division of decimals are begun, there 
 should be much practice upon such work as the following : " Tenths 
 of hundredths give what ? hundredths of tenths ? hundredths of 
 hundredths ? tenths of thousandths ? thousandths of hundredths ? 
 thousandths of thousandths ? tenths of tens ? hundredths of tens ? 
 tenths of hundreds ? hundredths of hundreds ? etc. Tenths 
 divided by tenths give what ? hundredths by hundredths ? units 
 by tenths ? tenths by hundredths ? hundredths by tentlis ? 
 thousandths by hundredths ? thousandths by tenths ? " etc. 
 
 In division of decimals there are three possible cases, viz. : 
 (1) Division in which the divisor and dividend are of the same 
 denomination ; e.g. .6 -r- .3. (2) Division by a decimal in which 
 the divisor is of a lower denomination than the dividend ; e.g. 
 .8 -f- .2 ; .8 -f- .04. (3) Division by a decimal in which the divisor 
 is of a higher denomination than the dividend ; e.g. .04 -^- .2 ; 
 .008 -i- .04, The first of these cases ought to give no trouble to 
 the pupils. It is readily seen that 4 tenths will be found as many 
 times in 8 tenths as 4 quarts in 8 quarts, etc. That is, they see 
 that when the divisor and dividend are of the same denomination 
 the quotient is a whole number (of times). Neither will there be 
 difficulty in dividing by a whole number. The pupils see at once 
 that .08 -^ 4 is :J- of 8 hundredtlis = 2 hundredths, which is the 
 same denomination as the dividend. All other classes of problems 
 in division may be easily resolved into one or the other of the 
 classes already understood. For example, if it is required to find 
 
V. 48] teachers' islanual. 97 
 
 how many times 4 lumdredtlis is contained in 2 tenths, the operation 
 would be expressed, .2 -~ .04, or .04) .2. The pupils know that a 
 cipher or ciphers placed at the right of decimals do not change 
 
 their value, and therefore express the operation thus : " ■^ . 
 
 5 
 
 All such problems may be performed in a similar way. Again, if 
 
 it is required to find what part of .2 is .04, the operation would be 
 
 expressed, .04 -i- .2, or .2) .04. The pupils should know, that when 
 
 the divisor and dividend are multiplied or divided by the same 
 
 number, the quotient is the same. Therefore, by multiplying the 
 
 divisor and dividend in this problem by ten, the operation may be 
 
 x2.) X 0.4 
 expressed thus : ''^ '— • Really, the pupils should be sufficiently 
 
 familiar with multiplication in the various denominations to know 
 that, when the dividend is hundredths and the divisor is tenths the 
 quotient must be tenths, or when the dividend is thousandths and 
 the divisor is tenths the quotient must be hundredths, etc. 
 
 After several problems have been done in this way the pupils 
 will see that they can point off as many decimal places in the 
 quotient as the number of decimal places in the dividend exceeds 
 the number of decimal places in the divisor. 
 
 48 Answers : 1 a .8002 ; I 32000 ; c 834400 ; rf 3 ; e .9. 
 
 2 a 3000.375 ; h 1066.666+ ; c .0754+ ; d .00301 ; e .00008. 
 
 3 a 1066.666+ ; h 26900 ; c 6680 ; d 255 ; e .0433+. 
 
 4 a 116.5263 ; h 47.946 ; c 2.08 ; d .2726+ ; e .084. 
 
 5 a 1.3333+ ; i 16000 ; c 1600 ; cZ 82000 ; c 146.3902 + . 
 
 6 a 9050; Z. 960 ; c .0463 ; fZ 9.26 ; e 1027.7279+. 
 
 7 a .8 ; Z> .08 ; c 80 ; rZ 800 ; e 8000. 8 a 1402.5 ; 
 h 140.25 ; c 140.25 ; d 7.0125 ; e 140.25. 9 a 2090 ; Z. .02 ; 
 c 2000 ; d 4.06 ; e .5236. 10 402.3739. 11 200170. 
 12 48.63794+. 13 .0092. 14 2980. 15 a 1.4634+; 
 h .0047+; c 499.5004+ ; d .7976; e 14.1843; / 50000 ; 
 ^133.3333 + . 16 a 54.225+; i 13.8910+; c 548.5301 ; 
 (Z 196176.470+ ; e 496761.904+ ; / 30.2641 ; ^333.3333+. 
 17 a 4073; h 27900+ ; c 17.2558+; d 90.065; e 9008; 
 / 344.5783. g .1568+. 
 
98 GKADED ARITHMETIC. [V. 49 
 
 49 Answers: 1 a .4673; h 2.0925; c .00007+ 
 d 502588.94+ ; e 8.014. 2 a .0000686+ ; b 59.0277 + 
 c 49.975+ ; d 1795.537+ ; e 8612.8345 + . 3 a 2.2168 + 
 b 63.8297+ ; c .5602+ ; d .00541+ ; e 4664.067+. 4 100 yd 
 1000 yd. 110 yd. 5 12000 apples. 6 20 yd. 200 yd. 250 yd 
 7 12 qt. 19f qt. 32 qt. 2000 qt. 8 7 da. 92 da. 800 da 
 
 9 $4.30. 10 $4.20. 11 8.2+ h. 12 44.6 mi. 13 650 A 
 
 14 90 pr. 800 pr. 15 240 coats. 16 23 da. 605.7+ da, 
 17 25 cd. 18 $1.50 $150.75. 
 
 50 Ariswers: 1 120 casks. 2 50900 yd. 3 5.84. 
 4 504.6 boxes. 5 42.5 A. 6 87.9 A. 7 .5 .75 .125 .625 
 .875 .375. 8 .05 .075 .08 .0625 .075 .15. 9 .15 .16 
 .38095+ .1875 .9375 .4218 + . 10 1.4 1.16| 1.8 2.025 
 1.5625. 11 1.75 2.875 14.7619+ 21.9375 40.015. 
 12 15.3125 20.4 15.53125 27.1875. 13 i f^ ^k tI? 
 
 7 1 1Q1 14.143 _547_ 4_2_43_ _2_5_5J[_ 15 304-2-^-3- 
 
 ¥0 ?TJO -'^•^^- -^^ ^^T~S^ 50000 ^4 000 2500 0* ■^*' ""^5 
 
 ^44 57 11 1 100 1 16 _!_ _1_ 1_3„ ^9_ J _401_ 
 
 "^500^ -'-^400 20000- •*■" 80 5O0 "^4 ^00 2000 50000" 
 
 17 11 12 1 1_ 18 T 323 ^3_ _^3Ji. 11. 
 
 ■*•' ¥ ■J^ T5 ^ ¥00 TSO- ■*■" 8000 ^30 ?00 400 "^4 
 
 T-1^. 19 2.35. 20 7.918. 21 10.5785. 22 $1,728. 
 
 23 $8f $110.83+. 24 14.6 yd. 
 
 In clianging common fractions to decimals lead the pupils to 
 think of the fraction as another form of expression for division ; 
 thus, f = 3 -^ 4 or i of 3. 
 
 51 Ansivers: 19.6875 yd. 2 80 sq. rd. 40 sq. rd. 60 sq. rd. 
 100 sq. rd. 3 38 A. 94 sq. rd. 18 sq. yd. 1 sq. ft. 50.4 sq. in. 9 A. 
 128 sq. rd. 150 A. 112 sq. rd. 4 A. 126 sq. rd. 12 sq. yd. -^% sq. ft. 
 7 sq. yd. 6 sq. ft. 100.2 sq. in. 4 2.0225. 5 52.545 mi. 6 $565.88 
 $126. 7 $263.4375. 8 67 lb. 1 oz. 12 pwt. 12 gr. 9 5017.95 lb. 
 
 10 15. 11 2.803+ oz. 12 i I ^*, -^S ^\%. 13 .3 
 .142857 .16 .09 .2916. 14 l^V tjV tWit itV l^'^- 
 
 15 yV :jV 555 7j- 
 
 In teaching repetends let the pupils see by examples that the 
 figures of a repetend express the numerator of a common fraction 
 having as many nines in the denominator as there are figures in 
 the repetend; thus, .16 = .If, and .108 = Jgf. 
 
V. 52] 
 
 TEACHERS MANUAL. 
 
 99 
 
 53 Answers 
 
 1 
 
 2 
 3 
 
 5 
 
 6 
 
 7 
 
 S 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 U 
 15 
 16 
 17 
 18 
 19 
 20 
 
 460 
 80 
 
 708 
 
 80 
 
 450 
 
 70 
 
 800 
 
 1004 
 
 350000 
 
 20 
 
 48 
 
 700 
 
 2 
 
 3 
 
 .08 
 
 .75 
 
 .0075 
 
 .60()6 
 
 .03006 
 
 .375 
 
 .00875 
 
 .3125 
 
 .000076 
 
 .5555 
 
 .0098 
 
 .5833 
 
 .0000708 
 
 .7333 
 
 .000604 
 
 .3895 
 
 .190 
 
 .7083 
 
 .000087 
 
 .6333 
 
 .10050 
 
 .425 
 
 .070 
 
 .18 
 
 .545 
 
 .3166 
 
 .07 
 
 .2 
 
 .009 
 
 .0583 
 
 .4 
 
 .0622 
 
 .00070 
 
 .0045 
 
 .000010 
 
 .127 
 
 .070 
 
 .059 
 
 .50 
 
 .001 
 
 ^TTTJ 
 
 3 
 
 ZTiV 
 
 7 
 
 ^TJTT 
 
 33 
 
 ¥(T 
 
 mu 
 
 85^5 
 
 39 
 
 ■5T)(T 
 
 2 1 
 
 ■J0I)TJ 
 
 48/^ 
 
 8 1 » 
 
 1 
 
 T25Ty 
 
 141 
 
 2(J^ 
 
 800f 
 
 70Tk 
 
 900211 
 
 so 
 
 '^^hho 
 
 1 3 
 
 -'sooo 
 
 ^i. 
 
 3 fi 3 
 
 ST)Ty(T 
 
 1 1 
 
 4.115 
 6.009 
 180.035 
 6.861 
 20.2867 
 88.083 
 3.0789 
 700.8105 
 62.478 
 8.15676 
 3.0218 
 1.707 
 800.6075 
 
 70.215 
 930.67 
 74.013 
 1.2415 
 8.064 
 8.7507 
 101.800 
 
 .785 
 .6726 
 .41 
 1.1375 
 20.7615 
 8.6233 
 .8113 
 .4 
 48.7683 
 8.7093 
 .4258 
 .885 
 800.9106 
 
 70.208 
 900.6383 
 70.0672 
 1.0051 
 8.167 
 
 .7802 
 1.009 
 
 4.83 
 6.6096 
 180.375 
 0.3485 
 .6362 
 80.6263 
 3.7342 
 701.1895 
 15.1263 
 .71409 
 3.446 
 1.182 
 .3241 
 .407 
 30.1483 
 4.0702 
 .2454 
 .151 
 8.0897 
 100.801 
 
 1 
 
 2 
 3 
 
 4 
 
 5 
 6 
 / 
 
 8 
 9 
 10 
 11 
 12 
 13 
 
 U 
 15 
 16 
 17 
 18 
 19 
 20 
 
 460. 
 80, 
 
 708. 
 
 100, 
 
 8, 
 
 48, 
 458, 
 
 70, 
 
 801, 
 
 870, 
 
 1904, 
 
 350070, 
 
 1, 
 
 28, 
 
 48, 
 
 701, 
 
 8 
 
 115 
 
 0135 
 
 06506 
 
 83375 
 
 200076 
 
 0498 
 
 0780708 
 
 011104 
 
 25 
 
 ,076087 
 
 1013 
 
 775 
 
 145 
 
 078 
 
 .589 
 
 405 
 
 0013 
 
 04001 
 
 7906 
 
 508 
 
 404 
 
 80 
 
 888 
 
 6 
 
 80 
 
 80 
 
 3 
 
 700 
 
 14 
 
 450 
 
 3 
 
 71 
 
 800 
 
 1034 
 
 350004 
 
 20 
 
 56 
 
 801 
 
 .16 
 .0105 
 .03006 
 .04475 
 080776 
 .0528 
 .0009708 
 800604 
 608 
 
 080847 
 1215 
 .072 
 .5525 
 277 
 099 
 .408 
 .2416 
 .02401 
 .1001 
 .30 
 
 10 
 
 460.83 
 
 80.6741 
 708.40506 
 .32125 
 80.55576 
 .5931 
 .7333708 
 .390104 
 .8983 
 450.633387 
 .5255 
 70.25 
 .8616 
 800.27 
 1004.0073 
 350000.4622 
 .0052 
 20.12701 
 48.1290 
 700.501 
 
 11 
 
 4.045 
 
 5.997 
 179.905 
 .5.211 
 20.12.53 
 72.003 
 2.9229 
 700.7895 
 33.642 
 7.99524 
 3.0202 
 .297 
 800.5925 
 
 69.801 
 870.49 
 65.997 
 .7597 
 8.016 
 7.3095 
 99.792 
 
100 
 
 GRADED ARITHMETIC. 
 
 [V. 62 
 
 12 
 
 1 460.045 
 
 2 80.0015 
 
 3 707.99.506 
 
 4 .81625 
 
 5 59.794076 
 
 6 8.0302 
 
 7 .0779292 
 
 8 .009896 
 
 9 47.87 
 
 iO 441.924087 
 
 11 .0997 
 
 i^ 69.365 
 
 13 800.055 
 
 i.^ 730.062 
 
 i5 103.429 
 
 16 349930.395 
 
 17 .9999 
 
 18 11.96001 
 
 19 47.3494 
 ^0 699.492 
 
 13 
 
 456 
 74 
 
 528 
 
 6 
 
 79 
 
 80 
 
 3 
 
 700 
 
 14 
 
 449 
 
 2 
 
 69 
 
 799 
 
 973 
 
 349996 
 
 19 
 
 40 
 
 599 
 
 0045 
 .03006 
 02725 
 919376 
 .0332 
 0008292 
 .799396 
 .228 
 919327 
 .9205 
 068 
 5375 
 863 
 919 
 392 
 2402 
 97601 
 .0399 
 .7 
 
 14 
 
 .35 
 
 .06 
 .35 
 
 8.25 
 202.06 
 80.4 
 .78 
 .105 
 480.6 
 80.76 
 .008 
 7.05 
 8006. 
 
 700.08 
 9005.8 
 700.05 
 10.006 
 80.40 
 
 7.206 
 10.08 
 
 15 
 
 35. 
 
 6. 
 
 35. 
 
 825. 
 
 20206. 
 
 8040. 
 
 78. 
 
 10.5 
 
 48060. 
 
 8076. 
 
 .8 
 
 705. 
 
 800600. 
 
 70008. 
 
 900580. 
 
 70005. 
 
 1000.6 
 
 8040. 
 
 720.6 
 1008. 
 
 16 
 
 73. 
 
 108. 
 
 3240. 
 
 108. 
 
 1. 
 
 1440. 
 
 54. 
 
 12614. 
 
 259. 
 
 1. 
 
 54, 
 
 18. 
 
 3 
 
 541, 
 
 72, 
 
 4, 
 
 144. 
 
 1814, 
 
 44 
 054 
 
 648 
 
 4.526 
 
 774 
 
 0162 
 
 4 
 
 524 
 
 45368 
 
 378 
 
 036 
 
 135 
 
 726 
 
 62 
 
 144 
 
 3362 
 
 432 
 
 5418 
 
 40 
 
 1 
 
 2 
 2 
 
 4 
 5 
 6 
 
 7 
 3 
 9 
 10 
 11 
 12 
 13 
 
 U 
 15 
 16 
 17 
 18 
 19 
 20 
 
 17 
 
 4.6008 
 .800075 
 
 7.0803006 
 .0000875 
 .80000076 
 .000098 
 .000000708 
 .00000604 
 .00190 
 
 4.50000087 
 .0010050 
 .70070 
 .00545 
 
 8.0007 
 10.04009 
 3500.004 
 
 .0000007 
 
 .20000010 
 
 .48070 
 
 7.005 
 
 18 
 
 50.14872 
 
 8.7208175 
 77.17527654 
 .00095375 
 8.720008284 
 .0010682 
 .0000077172 
 .000065836 
 20.71 
 
 49.050009483 
 .0109545 
 7.63763 
 .059405 
 87.20763 
 109.436981 
 38150.0436 
 
 .00000763 
 2.18000109 
 5.23963 
 76.354 
 
 19 
 
 .1428 
 .036018 
 6.3 
 
 4.9797 
 1.6306242 
 643.54572 
 .2340702 
 7.3584 
 692.92908 
 .65221776 
 .0024168 
 .70641 
 6.0045 
 14.491656 
 27098.4.522 
 280.58004 
 .24104454 
 .19296 
 5.78649006 
 101.6064 
 
V. 52] 
 
 TEACHERS MANUAL. 
 
 101 
 
 3 
 
 k 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 U 
 15 
 16 
 17 
 IS 
 19 
 20 
 
 16, 
 
 24! 
 
 1616, 
 
 9, 
 3634 
 
 49! 
 
 436, 
 
 5601 L 
 
 904190, 
 
 24501778, 
 
 160, 
 34 
 
 706, 
 
 20 
 
 10280 
 
 480045 
 
 7810521 
 
 00721875 
 
 049535656 
 
 078792 
 
 0000055224 
 
 000006342 
 
 1314 
 
 200702612 
 
 0000804 
 
 39935 
 
 327 
 
 30056 
 
 46522 
 
 002 
 
 00070042 
 
 8000804 
 
 639242 
 
 104 
 
 21 
 
 .0035 
 .0006 
 .0035 
 .0825 
 
 2.0206 
 .804 
 .0078 
 .00105 
 
 4.806 
 .8076 
 .00008 
 .0705 
 80.06 
 
 7.0008 
 90.058 
 
 7.0005 
 .10006 
 .8040 
 .07206 
 .1008 
 
 22 
 
 23 
 
 40.8 
 
 4.6008 
 
 60.03 
 
 .800075 
 
 1800. 
 
 7.0803006 
 
 60.36 
 
 .0000875 
 
 .807 
 
 .80000076 
 
 800.43 
 
 .000098 
 
 30.009 
 
 .000000708 
 
 7008. 
 
 .00000604 
 
 144.18 
 
 .0019 
 
 .8076 
 
 4.50000087 
 
 30.21 
 
 .001005 
 
 10.02 
 
 .7007 
 
 .075 
 
 .00545 
 
 2.07 
 
 8.0007 
 
 300.9 
 
 10.04009 
 
 40.08 
 
 3500.004 
 
 2.409 
 
 .000007 
 
 .24 
 
 .2000001 
 
 80.301 
 
 .4807 
 
 1008. 
 
 7.005 
 
 24 
 
 1 460080. 
 
 2 80007.5 
 
 3 708030.06 
 
 4 8.75 
 
 5 80000.076 
 
 6 9.8 
 
 7 .0708 
 
 8 .604 
 
 9 190. 
 
 10 450000.087 
 
 11 100.5 
 
 12 70070. 
 
 13 545. 
 U 800070. 
 
 15 1004009. 
 
 16 350000400. 
 
 17 .7 
 
 18 20000.01 
 
 19 48070. 
 
 20 700500. 
 
 25 
 
 136. 
 
 200.1 
 6000. 
 201.2 
 2.69 
 2668.1 
 100.03 
 23360. 
 480.6 
 
 2.692 
 100.7 
 33.4 
 .25 
 6.9 
 1003. 
 133.6 
 8.03 
 .8 
 267.67 
 3360. 
 
 26 
 
 70. 
 
 12. 
 
 70. 
 
 1650. 
 
 40412. 
 
 16080. 
 
 1.56. 
 
 21. 
 
 96120. 
 
 16152. 
 
 1.6 
 
 1410. 
 
 1601200. 
 
 140016. 
 
 1801160. 
 
 140010. 
 
 2001.2 
 
 16080. 
 
 1441.2 
 
 2016. 
 
 27 
 
 116.5714+ 
 1000.5 
 5142 
 
 7 
 
 9 
 
 38, 
 
 66742 
 
 3776 
 1 
 
 11 
 
 100 
 
 8514 + 
 3163 + 
 0039 + 
 95.55 + 
 4730 + 
 857+ 
 3 
 
 01 
 25 
 
 4212 + 
 000009 + 
 0029 + 
 ,0334 + 
 0572 + 
 2407+ 
 0029 + 
 1436+ 
 
102 GRADED ARITHMETIC. [V. 53 
 
 53^w5fc'e?.s; 1 1449.0129. 2 5.049150. 3 143.285 A. 
 
 4 9.85. 5 21i yd. 91 yd. 84/^ yd. 20 yd. 6 17^ 108-rV 
 $7,368. 7 28.7 T. 8 $84.69+ $415,181 $16988.81. 
 9 16.4008324. 10 $1135.3375. 11 33.4 da. 12 $30483.60 
 $17419.20. 13 $46,426 $1,946. 14 65.78+ da. 15 $5404.20. 
 16 26666.66+ sq. ft. 120000 sq. ft. 
 
 54 A7istvers: 1 4.11375. 2 48. 3 $14,748+. 4 Is^^^ bbl. 
 
 5 $7.3125. 6 $.0002 250000. 7 1466.542 bu. 8 2'l.825. 
 
 9 $42.59025. 10 $5,405+. 11 $1.53. 12 357.88839. 
 13 $27,133 + . 14 697. 15 1366.326 bbl. 16 9| mi. 3000 mi. 
 
 55 Anstvers : ^ l^bil\ it. 5 102 ft. 1224 in. 9 15 yd. 2 in. 
 
 10 1 mi. 4720 ft. 11 2 mi. 269 rd. 5 yd. 12 ^V t'o- 
 19 3732 in. 20 2654 ft. 21 10989 ft. 22 1 mi." 280 rd. 
 1|- mi. 
 
 The pupils are supposed to have had some practice in reduction 
 before beginning this section. (See Section VI., Book IV.) Many 
 of these problems should be performed orally, but the solutions 
 may be written out in full for the sake of learning a good form of 
 written work. 
 
 56 Answers: 1 5f mi. 2 264 paces 2112 paces. 3 30.6 mi. 
 4 1485 paces. 5 198t\ rd. 6 1 mi. 208 rd. 2 yd. 2 ft. 
 
 7 208t»9 times. 
 
 8 15 rd. 2 
 
 yd. 4 ft 
 
 . $5. 
 
 87^. 
 
 9 $1,947. 
 
 10 $12. 11 
 
 66 ft. 792 in. 
 
 12 7. 
 
 92 in. 
 
 
 
 57 Answers 
 
 : .• 1 396 in. 
 
 617.76 
 
 in. 
 
 2 5 
 
 ch. 80 ch. 
 
 3 3659.04 in. 
 
 304.92 ft. 4 
 
 498 ch. 
 
 32868 ft. 
 
 5 83952 in. 
 
 6 70604.88 in. 
 
 7 2 li. 4.16 in. 6 li. 
 
 2.48 in, 
 
 8 
 
 17 li. 5.36 in. 
 
 75 li. 6 in. 
 
 9 2 ch. 40 li. 
 
 6 ch. 
 
 80 li. 
 
 10 
 
 16 ch. 84 li. 
 
 11 1 ch. 7 li. 
 
 2.56 in. 1 ch. 
 
 26 li. .( 
 
 }8 in. 
 
 12 
 
 16 ch. 50 li. 
 
 13 23 li. 7.84 
 
 in. 2 ch. 27 li. 
 
 2.16 in. 
 
 14 
 
 1 ch. 
 
 15 12 ch. 
 
 16 541.2 ft. 
 
 17 1056 ft. 
 
 18 2811.6 ft. 
 
 53 mi. 3jJ'j min. 
 
 19 2f min. .35+ mi. 
 
 
 
 
 
 Measurements by surveyor's measure are not generally made 
 now, and the problems here given may be omitted if thought best. 
 
V. 58] teachers' manual. 103 
 
 Exercises with the surveyor's measuring tape may be substituted. 
 On this tape are given the feet, and tenths and hundredths of feet. 
 , Pupils are supposed to have been taught angle, triangle, and 
 rectangle, and to have had some practice in finding areas. (See 
 pages 84-86, Book IV.) Some of the descriptions here called for, 
 however, are new, and the figures may have to be taught again. 
 Let the descriptions be made by the pupils, and let them be made 
 from the objects or illustrations. For example, the rectangle may 
 be shown, and attention be called to the number of sides of the 
 figure and to the kind and number of angles. From these facts 
 the pupils are led to say that " a rectangle is a four-sided figure 
 having four right angles." 
 
 5S In explaining the process of finding the area of a rectangle 
 the pupils should be led to multiply the number of square units in 
 a row by the number of rows, as previously shown. (See Manual, 
 pp. 49, 75.) 
 
 59 A7iswers: 1 10 A. 2 1 A. 140 sq. rd. 3 3 A. 93^ sq. rd. 
 4 106f sq. rd. 32261 sq. yd. 5 104544 sq. ft. 6 585446.4 sq. ft. 
 13.44 A. 7 1424025.1872 sq. ft. 32.69 A. 8 4 yd. ft. 
 9 45.375 ft. 10 21a yd. 11 7^} in. 12 f 762.30. 13 ^64. 
 14 6422 sq. yd. 112.L yd. 241 sq. ft. 15 21a yd. 2§ rolls. 
 16 ;$38.25+. 17 11 yd. 18 I846.28+. 
 
 If the pupils find difficulty with exercises in which the area and 
 one dimension are given to find the other, lead them to see the 
 process from simple illustrations. (See page 83, Book IV.) The 
 drawing of plans in illustration of problems should be required 
 when needed. 
 
 60 Ansivers : 8 432 sq. ft. 9 5400 sq. ft. 10 6 rd. 
 11 10 ft. 12 8712 ft. 13 4356 sq. ft. 14 10. 15 4- i ^i^. 
 16 ^% 25 i.o|. 17 .25 .01875 .058^. 18 .0075+ 
 .1628+ .339+. 19 i ^V U- 20 .00625 .875. 21 1.8. 
 
 The terms horizontal and vertical as applied to lines should be 
 described as having a certain direction. Comparisons with the 
 
104 GRADED ARITHMETIC. [V. 61 
 
 surface of still water and with a plumb line will lead to a good 
 description of horizontal and vertical lines. The figures given 
 should be recognized as quadrilaterals, and the reason why they 
 are quadrilaterals should be given. Parallel lines and parallelo- 
 grams, which have been described in connection with exercises in 
 Book IV., should be described again. 
 
 The descriptions called for in 4 may be given, if thought advis- 
 able, in the form of a definition. In teaching a definition the object 
 or process to be defined should be brought into the presence of 
 the pupils, and by questions they should be led to observe those 
 features of the object or process which must be named in the 
 definition. For example, in teaching the definition of a square, a 
 square surface, preferably of a cube, is presented. The pupils are 
 led to see that it is a i^lane figure (that having been previously 
 taught), that it is hounded hy four straight lines, that the sides are 
 equal, and that it has four right angles. From these facts the 
 following statement may be drawn from the pupils : <' A square 
 is a plane figure which is bounded by four equal straight lines, and 
 which has four right angles." If a rectangle has been already 
 defined, the pupils might say that the square is a rectangle whose 
 sides are equal. Whenever the language of a pupil's definition is 
 faulty, lead him to see the faulty or bungling construction of the 
 definition by questions or by comparison with a correct form. 
 
 The following definitions may aid teachers in drawing from the 
 pupils accurate statements : 
 
 A rhombus is an oblique-angled parallelogram whose sides are 
 equal. 
 
 A rhomboid is an oblique-angled parallelogram whose opposite 
 sides only are equal. 
 
 A trapezoid is a quadrilateral which has only two sides parallel. 
 
 A trapezium is a quadrilateral which has no two sides parallel. 
 
 61 Answers : 1 320 sq. ft. 90 sq. ft. 101^ sq. ft. 3 yd. 4 in. 
 42f I sq. yd. 83^ sq. ft. 123^ sq. rd. 6 A. 40 sq. rd. 2 A. 
 137 sti. rd. 189f sq. ft. 40 A. 2 A. 130 sq. rd. 3^^ rd. 24^^ rd. 
 
V. 62] teachers' manual. 105 
 
 19| rd. 5542922.88 sq. ft. 2 $25 |60 $156.25. 3 $378. 
 4 $1158.75. 5 $7224.35. 6 $.005. 7 $1155.75 gain. 
 
 5 may be performed by finding first the square chains, and then 
 dividing by 10, the niimber of square chains in 1 A. Performed 
 on a line, the solution would be : 
 
 $240 X («M8^iil«) 
 
 • G2 A7isu'ers : 1 25 sq. yd. 27 yd. 2 lOf sq. yd. 9^ sq. yd. 
 
 3 30 rd. 4 10 ft. 5 29^ yd. 6 7iJ sq. ft. 7 36f sq. ft. 
 8 933^. $11.66f. 9 30. 1451-. 10 $180 cheaper by sq. ft. 
 11 $5.31. ^ 12 3 10 6 20. 13 20 sq. yd. 14 39.98+ A. 
 15 $35.50. 
 
 Nearly all of these problems should be illustrated by plans. In 
 1, assume that the width of the carpet is 1 yd. or f yd. In 2, 
 let the pupils find for themselves how to make allowance for 
 corners. A properly-drawn plan will give all needed assistance. 
 The bricks mentioned in 12 are of common size. 
 
 63 Answers : 1 180 sq. ft, 2 360 sq. ft. 2640 sq. ft. 
 
 4 $2450.25. 5 $217.80. 8 2176 sq. ft. 9 4416 sq. ft. 
 
 Finding the area of the figure in 1 is a review of what has been 
 taught previously, but the hint of a dotted line may have to be 
 given before the pupils can find the correct area. Other exercises 
 similar to this may be useful for practice. The pupils will dis- 
 cover, when they compare answers to 4, that the shape of the lot, 
 as well as the size, determines the length of the boi;ndary line. 
 To solve 6, the given polygon must be divided into triangles or 
 rectangles, and the needed dimensions be determined by measure- 
 ment. In 7, let the pupils estimate the dimensions of the figures 
 not closer than the sixteenth of an inch. 
 
 64 A teaching lesson from the cube should precede 1. The 
 following facts should be observed and stated by the pupils : " The 
 cube is a solid. (They are told that anything which has length, 
 breadth, and thickness is a solid.) It is bounded by six faces. The 
 
106 GRADED ARITH]SIETIC. [V. 65 
 
 faces are equal to each other. Each face is in the form of a square." 
 All these facts may be gathered into one statement forming a defi- 
 nition, thus : A cube is a solid bounded by six equal square faces. 
 
 The blocks used in 2 should be 1-inch cubes. The work indicated 
 in 3 should be performed objectively with the inch cubes. The 
 rows, layers, and number of layers should be observed by the 
 pupils, and the number of cubic inches found by multiplying the 
 number of cubic inches in a row by the number of rows, and this 
 product by the number of layers. This should be repeated until 
 it is well understood. 
 
 65 A7iswers : 1 60 cu. in. 2 1728 cu. in. 3 41472 cu. in. 
 4 20736 cu. in. 5 4536 cu. in. 2.6 cu. ft. 6 4096 cu. in. 
 7 72 cu. ft. 8 169 cu. ft. 9 27 cu. ft. 10 8 cu. yd. 
 20 cu. yd. 3.51 cu. yd. 11 14 cu. yd. 22 cu. ft. 23 cu. yd. 
 19 cu. ft. 12 2 cu. ft. 544 cu. in. 6 cu. ft. 1712 cu. in. 
 13 1152 cu. in. 1296 cu. in. 23328 cu. in. 14 ^V f- 
 15 7 in. by 5 in. by 3 in. 105 cu. in. 16 208 sq. in. 142 sq. in. 
 
 There are two ways of expressing the operation of changing a 
 number from one denomination to another. Thus, in 10, the pupil 
 may be led to say : '' There is | as many cu. yd. as there are cu. 
 ft."; or, « There are as many cu. yd. in 216 cu. ft. as there are 
 nines in 216"; or, "sls many cu. yd. as 9 cu. ft. is contained 
 times in 216 cu. ft." But in all such explanations care should be 
 taken that the pupil sees the reason for the step taken. Some- 
 times it is well in such exercises to ask if the answer is to be less 
 or more than the given number, and then to ask what part as many 
 or how many times as many. 
 
 66 A7iswers: 1 128 cu. f t. = 1 cd. 8 cd. ft. = 1 cd. 16 cu. 
 ft. = 1 cd. ft. 3 64 cu. ft. 48 cu. ft. 96 cu. ft. 4 80 cu. fd. 
 56 cu. ft. 88 cu. ft. 5 4 cd. ft. 6i cd. ft. 24 cd. ft. 6 i 
 /^ f. 7 li cd. lt\ 3f. 8 $3.12i. 9 396 cu. ft. 24f 
 cd. ft. 3^\ cd. 10 $40|. 11 $136.64. 12 1248 packages. 
 13 2027.5 papers, 
 
V. 67] teachers' JVIANtJAL. 107 
 
 To give a clearer impression of a cord of wood, splints 4 in. long 
 might be piled between stakes. The pile should be 8 in. long and 
 4 in. high. In the same way 1 cord foot could be shown. Piles 
 of other dimensions could be made for measurement. The pupils 
 will find this interesting as well as profitable work. 
 
 67 Ansivers: 1 227 p. 9051 p. 617 p. 2 30 sq. rds. 3 6 
 strips. 4 $76.56. 5 1023^ ft. 544^ ft. 6 150 ft. 22315-^ 
 •sq. rds. 7 $24.80. 8 60 posts 1530 pickets. 9 $13.22f 
 529f sq.ft. $1.68. 10 198^bu. 11 1770^ sq. ft. 12 $6.52^. 
 
 In the first part of 1, if the pickets are flush with the ends of 
 the fence, one space will be only -^ in. wide. Let the pupils see 
 this by dividing the space in inches between one end of the fence 
 and the inside edge of the picket on the other end by 3|-, the 
 number of inches that one picket and one space occupy. Make 
 no account of corners in the last part of 1. 
 
 68 Answers : 1 252^238^ cu. ft. 2 |3f a^ or $3,878 +. 3 5§f cd. 
 
 4 30f §-!§§. 5 4^B/^; 16|ff 6 2H cu. ft. 7 32 J^ tons 
 $176.34f. 
 
 The making and solving of original problems in which the pupils' 
 own measurements are given will be found especially valuable. 
 Lead the pupils in such work to apply all the principles that they 
 have learned. 
 
 69—70 For information as to U.S. coins see Book IV., page 89, 
 and note thereon in the Manual. Most of the problems on these 
 pages should be solved orally. 
 
 71 A7iswers: 1 $|. 2 $1.48+. 3 $117. 4 |78.62f 
 
 5 $233.0625. 6 $123.12 gain. 7 9f bu. 50 days. 8 $2.46. 
 9 $51.84. 10 Brown, $10.50 Eaton, $8.79 Black, $8,925 
 Fuller, $8.32 O'Connell, $10.00 T. Smith, $10.74 A. Smith, 
 $11.42 Hall, $11.70 Woods, $11,375 Kelley, $11.81 Reed, 
 $8.49. 
 
108 GRADED ARITH^SIETIC. [V. 72 
 
 Lead the pupils to employ shorter processes -when they are 
 easily understood ; thus, in 1, by reducing 15 lb. and 6f lb. to 
 thirds the pupil can be led to see that -y>- will cost | as much as 
 4^5. I of a dollar = 44f /. The pupils should be told that gener- 
 ally the fraction of a cent is dropped if it is less than half, but 
 that in this and all similar cases the extra cent would probably 
 be charged. Fractions should be used whenever the process is 
 shortened thereby; thus, in 3 and 5: 6|- M. and 28;^ M. should be 
 used instead of 6500 and 28250. 
 
 72 A7iswers: 1 $8.50 $4.25 $1.70 $6.38 $15.86. 2 $62.50 
 $625 $312.50 $156.25. 3 67^/. 4 48/ $38.40 $28.80. 
 5 $1.50 $4.50 $5,625. 6 $175 $35 $140 $131.25. 7 $3.60 
 $360 $180 $270. 8 $7.68 gain. 9 $4.20 gain. 10 $3f. 
 11 $19^. 12 $20.34. 13 3i lb. 274^ qt. 14 10.46 lb. 
 15 Iff 16 223.25 dollars. 17 5 lb. 8| oz. 
 
 Most of these exercises can be performed orally. Lead the 
 pupils to work through 100 or 10 when more convenient, as in 
 Ito 7. 
 
 73 Answers: 1 20. 2 1 lb. 42 oz. 3 144 2976 6096. 
 4 31^ spoons. 5 57f cu. in. 6 11.63 qt. 7 2150.4 cu. in. 537.6 
 cu. in. 14515.2 cu. in. 6182.4 cu. in. 8 17.14 qt. 9 348.348 bu. 
 10 598.4 gal. 3111.9 gal. 72.3 gal. 11 18977.14 gal. 12 1386 
 cu. in. larger. 13 37.23 qt. 14 1.6 qt. 
 
 74 A?iswers: 5 72 sq. rd. 12.8 sq. rd. 5.6 sq. rd. 261.33 
 sq. ft. 310 sq. rd. 11 4 rd. 5 yd. 3 rd. 1 ft. 6 in. 17 yd. 
 512^2 rd. 12 6 sq. ft. 23 sq. yd. 8 sq. rd. 18 .25 rd. .4 rd. 
 19 .0625 mi. .027 + mi. 20 .1 A. .0037 + A. 29 .459 T. 
 13600 lb. 
 
 Have as many of these performed orally as possible. A few 
 whose answers are not given may have to be performed by the aid 
 of figures. For the sake of practice in Avritten analysis some of 
 the easier solutions may be written out in full, as, for example : 
 
V. 75] teachers' manual. 109 
 
 4 
 
 
 .8 sq. rd. 
 
 
 30i 
 
 
 2.42 sq. yd. 
 
 9 
 
 9 
 
 60)6200 see. 
 
 3.78 s(i. ft. 
 
 60)103 min. 20 sec. 
 
 144 
 
 1 h. 43 min. 
 
 1152 
 
 Ans.: 1 h. 43 min. 20 sec 
 
 1008 
 
 
 112.32 sq. in. 
 Ans.: 2 sq. yd. 3 sq. ft. 112.32 sq. in. 
 
 75 Most of these exercises can be performed orally, advantage 
 being taken of convenient fractions whenever possible. 
 
 Frequent practice should be had in writing receipts and bills. 
 
 76 Answers: 2 Balance, f 31.43. 3 Balance, $89.25. 
 
 Ask the pupils to explain each item of these accounts, and lead 
 them to distinguish readily which items belong to the debit side 
 and which to the credit side of the account. 
 
 Encourage them to keep a cash account of their own in the form 
 given, and to balance the account at the end of every week. 
 
 77 A?mvers: 1 Balance, $1063.45. 2 Balance, $91.36. 
 3 Balance, $11.40. 
 
 78 Point out the difference of form in these two bills of sale, 
 and show the use of the term debtor (Dr.) in the second bill. Blank 
 forms are easily obtained, and yet it would be well for the pupils 
 to be able at any time to rule their own forms. 
 
 79 ^wmws.- 1 Balance, $166.07. 2 Balance, $4.87^. 3 Bal- 
 ance, $28,705. 4 Balance due June 1, $23.24. 
 
 Explain fully the use of the term creditor (Cr.). Show that a 
 man is debtor for all that he receives, and that he is creditor for 
 all that he does or pays out. Give several exercises illustrating 
 this point. 
 
110 GRADED ARITHMETIC. [V. 80 
 
 80 Answers : 1 Balance due, $44.27. 2 Balance due, $137.35. 
 
 3 S. L. Childs owes Asa Howland, $109.12, 4 $15.75, one man; 
 $1890. 5 $7.90. 
 
 In the last answer to 4 the assumption is that 120 men are 
 employed the entire time given, with no lost time. 
 
 81 A)isu-ers: 1 $15465.05. 2 $20208.14. 3 $14379.90. 
 
 4 $12300.02. 5 11600 25200 696000 375000. 6 63000. 
 
 5 to 10 should be performed orally, and as rapidly as possible. 
 It may be well to have the pupils go over these exercises two or 
 three times for the sake of acquiring facility in multiplying or 
 dividing by the fractional parts of 10 and 100 
 
 82 Ansivers: 20 $130.26. 21 $115.60. 22 $559.24. 
 23 $297.15. 24 $165.84. 25 $249.84. 26 $9.55. 27 $108.54. 
 
 The iirst 19 exercises are for quick oral practice. In multiplying 
 by a number two or three units less than 100 or 1000, lead the puj^ils 
 to make two multiplications and subtract ; thus, in 4 : 93 X 100 = 
 9300 ; 93 X 2 = 186 ; 9300 — 186 = 9114. After a little practice 
 such sokitions may be made silently, the answer only being given. 
 In 9, first multiply by 10, 30, and 60, and add to the products the 
 multiplicands. Let the pupils perform orally as many of the 
 exercises from 20 to 27 as they can. 
 
 83 A?iswers: 1 $1.92. 2 $21,675. 3 $118,625. 4 $42.98. 
 
 5 $81.15. 6 $6.75. 7 $4.59. 8 $11.66f Total, $289.35||f 
 9 9.5. 10 1.853i|a. 17 $90666.66|. 18 $10125, val. of 
 estate $5625, wife $1125, son. 
 
 84 Answers: 3 31 A. 139i|i sq. rd. 4 Gj\ lb. 5 2^ 
 sq. rd. 1 A. 70 sq. rd. 6 $31.68. 8 207J. 9 287' f 10 $6.53^. 
 
 12 55/. $11.00. $54 
 
 85 Answers: 1 26 poles. 2 70 posts. 3 99 posts. 4 1782 
 pickets. 5 2970 stones. 6 396^9^ sheets. 7 $6.00. 8 $7.55+. 
 9 145 trees. 10 3-^j s(i. rd. 11 146fgor; sq. rd. 12 8|f. 
 
 13 31-^7. 14 69^}. 15 6|§. 16 8^/. 17 800 doz. 18 $25.00. 
 19 131.84 pts. 
 
V. 86] teachers' manual. Ill 
 
 The pupils may be able to perform some of these problems- 
 orally. Let plans be drawn for all problems in mensuration. 
 When the steps of a process are not clearly seen, make the reason- 
 ing clear by questioning, and use small numbers. 
 
 86 Answers: 1 24.99 sq. rd. 2 .025 A. 3 $16.02. 4 28001b. 
 $13104. 5 $240.07i. 16 844if planks. 17 2150.4 cu. in. 
 18278.4 cu. in. 
 
 87 Answers: 1 4.821 bu. 2 336.294 bu. 20177.64 1b. 3^ 
 400 ft. 37i ft. 10 $9,074- $16,335 $12.10. 11 46| cu. yd. 
 12 463 lb. 5i oz. 10502f lb. 13 f . 
 
 88 Aiisivers: 6^ U 1300 1b. 1700 1b. $10. 7 $857} 
 $1714f $3428i. 8 5184 lb. 9 128 flagstones. 10 1536 
 bottles. 11 29.403 lots. 12 166§ yd. 13 45000 oranges. 
 
 14 $62487889.60. 
 
 89 A?iswers: 1 1 mi. 109 rd. 6^ ft. 2 li|| sec. 3 231 rd. 
 3i ft. 4 59 lb. 6 oz. 5 750 lb. 6 18600 ft. 3|| mi. 
 7 3 mi. 220 rd. 5/^ ft. 8 1200 times. 9 ij week. 8^\^ hr. 
 10 Mf- 11 '^2.50 3tV yd. 12 llllfl cu. ft. 14 $9.53. 
 
 15 $1.50 $3.75 $2 $2.30. 
 
 SECTION VIII. 
 
 NOTES FOR BOOK NUMBER SIX. 
 
 Before taking up the exercises included in this book, the pupils 
 are supposed to have a tliorough knowledge of common and deci- 
 mal fractions and of some of the simpler and more important 
 processes in mensuration and denominate numbers. If these sub- 
 jects are not well understood, it is advised that a more extended 
 review of Book V. be given than is found in Section I. 
 
 It is not unlikely that some parts of mensuration may be thought 
 too difficult ; but if the previous work has been well done and each 
 new principle is taught objectively, there will be no difficulty which 
 it will not be well for the pupils to overcome. In the applications 
 
112 GRADED ARITHMETIC. [VI. 1 
 
 of percentage, also, which include all kinds of problems that are 
 likely to occur in the every-day life of a mechanic or farmer, there 
 will be some difficulties both in the understanding of principles 
 and in the processes involved unless the foundation is carefully 
 laid. The development exercises preceding the practical problems 
 should be carefully and systematically followed if good results are 
 to be expected. 
 
 As the work progresses less attention need be given to the 
 mechanical operations with numbers if the previous work has been 
 thoroughly done, while a constantly increasing emphasis should be 
 put upon the solution of problems which require close and careful 
 thinking. Increased attention also should be given to the explana- 
 tion of processes and the formulation of rules and definitions. 
 With this changed work of the pupils comes a corresponding 
 change in the character of the teaching and drilling. While the 
 use of objects and drawings in making clear a new process or prin- 
 ciple is always desirable, there is not so complete dependence upon 
 them as the pupils come to have a fuller power of generalization 
 and reasoning. 
 
 Other hints of a general nature will be found in the Note to 
 Teachers, which precedes the first section of the book. 
 
 1 Answers: 1 1490. 2 1392. 3 1528. 4 1402. 5 1588. 
 6 1448. 7 1534. 8 1371. 9 1537. 10 1362. 11 633. 
 12 769. 13 607. 14 664. 15 624. 16 612. 17 492. 
 18 599. 19 474. 20 644. 21 608. 22 714. 23 532. 
 24 611. 25 535. 26 471. 27 591. 28 672. 29 624. 
 30 673. 31 700. 32 471. 33 757. 34 548. 
 
 Encourage the pupils to add by pairs, thus (1) : 14, 26, 47, etc. 
 As a convenience it may be well to have the sum of each ^.^ 
 column placed below the line. For example, the partial and -i ok 
 total sums in 1 would appear as follows : 1490 
 
 2 A7iswers: 1 $49.97. 2 $50.56. 3 $61.93. 4 $55.44. 
 5 $55.87. 6 $56.48. 7 $54.69. 8 $46.20. 9 $51.34. 
 10 $49.50. 11 $46.79. 12 $35.01. 13 $41.15. 14 $47.63. 
 
VI. 3] teachers' IVIANUAL. 113 
 
 15 $34.44. 16 32.98. 17 $384.94. 18 $385.04. 
 
 19 $2506.82. 20 $2188.85. 21 $2352.38. 22 $1710.47. 
 23 $1905.60. 24 $1216.46. 25 $2395.32. 26 $258.41. 
 27 $2101.83. 28 $1177.55. 29 $1376.14. 30 $2138.41. 
 31 $10864.12. 32 $10664.12. 33 $848.32 $340.39 $788.07 
 $389.81 $508.91. 34 $62.84 $355.95 $888.56 $381.19 
 $574.05. 35 $315.83 $611.81 $0.44 $676.29 $339.05. 
 36 $323.71 $600.23 $318.33 $49.61 $269.06. 37 $427.83 
 $664.71 $331.89 $525.47 $36.59. 38 $84.73 $665.29 
 $650.43 $30.34 $722.74. 39 $1642.54 $2100.91 $2191.90 
 $1691.54 $1830.77. 40 $2490.86 $1760.52 $1403.83 
 
 $1291.73 $1321.86. 41 $2428.02 $2116.47 $2292.39 
 
 $1672.92 $1895.91. 42 $2112.19 $1504.66 $2291.95 
 
 $1096.63 $1656.86. 43 $2435.90 $2104.89 $1973.62 
 
 $1146.24 $1825.92. 44 $2008.07 $1440.18 $2305.51 
 
 $1671.71 $1862.51. 45 $776.34 $412.37 $6.42 $289.56 
 $13.04 $249.92 $500.10. 46 $72.54 $520.22 $12.39 
 $623.76 $294.80 $701.80 $613.92. 47 $131.55 $529.81 
 $22.44 $553.41 $185.47 $8.11 $688.88. 48 $45.90 $165 
 $27.86 $265.10 $484.55 $4.33 $757.41. 49 $703.80 
 $932.59 $18.81 $334.20 $307.84 $451.88 $113.82. 
 
 50 $59.01 $9.59 $34.83 $70.35 $480.27 $709.91 $73.96. 
 
 51 $85.65 $364.81 $50.30 $288.31 $299.08 $3.78 $68.ry:^. 
 
 52 $835.35 $402.78 $41.25 $219.21 $493.31 $459.99 
 $575.06. 53 $352.18 $2379.36 $179.61 $1707.20 
 $1106.63 $877.39 $1554.93. 54 $1128.52 $1966.99 
 $186.03 $1417.64 $1093.59 $627.47 $2055.03. 
 55 $1055.98 $1446.77 $198.42 $2041.40 $798.79 
 $1329.27 $1441.11. 56 $1187.53 $1976.58 $220.86 
 $1487.99 $613.32 $1337.38 $2129.99. 
 
 To add in lines as well as in columns is good practice, since 
 additions have to be so made sometimes in business. 
 
 3 Ansu-ers: 1 $295946306.08. 2 $20,456+. 9 715920 
 1909120 14318400 9545600 71592000 214776000. 
 
114 
 
 GRADED ARITHMETIC. 
 
 [VI. 4 
 
 10 1232240 
 
 10628070. 
 
 477146360. 
 
 878640320. 
 
 38944325. 
 
 714319200. 
 
 4697111. 
 
 1578fi. 
 
 14611-ff 
 
 4970 4 9 ■'> 
 
 '±Vt V^Q Jg^. 
 
 200261t\V 
 
 1509494 24644800 26185100 
 
 11 1622538 50659242 58651744 
 
 12 37552900 72472220 59401860 
 
 13 2335200 38190250 42082250 
 
 26369936 
 516808400 
 690583200 
 
 34487985 
 5882220190 
 
 9698|§ 
 31782^1 
 
 6070751811 
 
 14 569754600 539211785 496692785 
 15 1127461 72157|i 9019/^0 474143 
 
 16 3000281 197386§| 62505^- 7653' 
 
 17 125000 80000 158730i§ 22727 iSj- 42016//^ 
 18 572572f 1336001 83500^ 318095f 79682^^% 
 
 19 1894779tV 947389i§ 761818|f 535481/^ 
 151117t*vV- 20 103621421^1 8232936f§ 
 
 47323171 1} 13976S3|§ 146729^^^. 
 
 Problems similar to the following might be given from the 
 table : What was the difference in amount of imports in 1891 and 
 1892 from England ? from France ? from Germany ? from Brazil ? 
 from Mexico ? from Cuba ? What was the difference in exports 
 in 1891 and 1892 to England ? to France ? etc. By how much 
 did the exports to England exceed the imports from that country 
 in 1891 ? in 1892 ? What can you say of the difference between 
 the exports to France and the imports from that country in 1891 ? 
 in 1892? etc. Find the total imports in 1891 from England, 
 France, Germany, Brazil, Mexico, and Cuba ; etc. 
 
 4 A7isivers: 1 25ff. 
 
 195|f 
 
 13- 
 
 1143331^2^4 
 
 Let the pupils practice upon exercises in cancellation until they 
 can recognize readily tlie common factors of dividend and divisor. 
 Answers to 5 and 6 should be given as rapidly as possible without 
 
 the aid of figures. 
 
 
 5 Ans^vers: 52 603^^,. 53 681*3. 59 317. 
 
 60 49tV 
 
 61 543 1. 62 107§§. 63 106i|. 64 m. 65 46||. 
 
 66 693 
 
 67 60^3_. 68 29i. 
 
 In 1 to 20 lead the pupils first to find l)y inspection the least 
 common denominator, and then to add as the reduction is made. 
 
6 Ansivers : 
 
 ■ 1 lll^VV 2 13 
 
 03- 
 
 3U 4 39f. 
 
 12 800 9 7 
 
 1( 
 
 2i li. 16 
 
 G4 6}^ 4^ 46f. 
 
 
 23 27.0763. 
 
 24 1827.8249. 
 
 
 Vi. 6] teachers' manual. 115 
 
 Thus, in 10, the pupils would say: "The least common denomi- 
 nator is 12 ; ^9^ + /j=14; 11 + ^2^ = ^1; H + T\ = tl; tl + A 
 
 = fl = 2i" 
 
 Possibly 21 to 31 may have to be performed by the aid of 
 figures, but the pupils should be given an opportunity to try their 
 solution orally. 
 
 3 3p. 11 &% 21 
 
 3 5. 13 ii,v iM n 
 
 21 13.9352. 22 627.3294. 
 
 7 For the sake of practice in pointing off, the solution of 1 to 7 
 may be written out. All the remaining exercises on this page 
 should be performed orally. 
 
 8 All these problems, with the possible exception of 1, 2, 11, 
 and 13, should be performed orally. Let careful attention be 
 given to the pupils' explanations. 
 
 9 Nearly all of these review problems should be performed 
 orally. 
 
 10 Ansivers: 1 100 rd. 34f rd. 3 fgoo. 4 5 rd. 3 ft. 
 10.56 in. 5 $13.93+. 16 32670 sq. ft. 17 5080| sq. ft. 
 18 6076 sq. in. 19 1^ A. 6^ A. ^ A. 20 fW A. ^%%% A, 
 21 If f f f 22 $298.07. 23 6352^ sq. ft. $1030.75. 
 
 11 Ansivers: 1 $4420.80 318 yd. 4 $3840 $2400 $512. 
 
 In places where this method of measurement is common, many 
 exercises similar to 3 and 4 should be given. Descriptions of lots 
 of land found in deeds, advertisements, etc., should be brought into 
 the class and fully exj^lained. 
 
 13 Ansivers: 5 32 yd. breadthwise. 6 29 yd. 6 in. 27 yd. 
 
 8 in. 7 f ft. i ft. 1|- in. 8 60^^^ sq. ft. IG^f courses. 
 
 9 10 sq. ft. 20 sq. ft. 10 32 sq. ft. 
 
 If necessary use strips of paper to illustrate 5 and 6. Several 
 illustrations similar to that given in 12 will assist the pupils to 
 discover how the average width of a board may be found, and in 
 general how the area of any trapezoid may be found. 
 
116 GEADED ARITHMETIC. [VI. 13 
 
 13 Answers: 7 26}| sq. ft. 8 23yVg\ sq. rd. 9 160 sq. ft. 
 10 11^ sq. yd. 11 142f sq. yd. 12 384 sq. ft. 13 576 sq. ft. 
 113^ sq. yd. 
 
 The description of plane figures here called for should be more 
 carefully made than that made in answer to the same question 
 in Book V. (page 57). Tlie description should be as nearly as 
 possible in the form of an accurate definition. The pupils should 
 be led first to see that a plane surface is a surface in which a line 
 connecting any two points will lie wholly in the surface, and that 
 a plane figure is a portion of a plane surface bounded by one 
 or more lines. These lines may be straight or curved. Figures 
 bounded by straight lines are rectilinear figures, and figures bounded 
 by curved lines are curvilinear figures. In teaching the definitions 
 of the various figures, first present the figure and call attention by 
 questions to its essential characteristics. For example, after pre- 
 senting a triangle, the teacher says: "What kind of a figure is 
 this ? By how many sides is it bounded ? Define triangle." The 
 pupil is led to say: "A triangle is a plane rectilinear figure bounded 
 by three sides." If his wording of the definition is not quite as 
 good as it should be, lead him by questioning to make the desired 
 correction. The following definitions may be developed in the 
 same way: 
 
 A polygon is a plane rectilinear figure having three or more 
 sides. 
 
 A quadrilateral is a polygon of four sides. 
 
 A pentagon is a polygon of five sides. Etc. 
 
 A square is a right-angled parallelogram^ having its sides 
 equal. 
 
 A rectangle is a right-angled parallelogram having only its op- 
 posite sides equal. 
 
 A rhombus is an oblique-angled parallelogram having its sides 
 equal. 
 
 1 This term is supposed to have been taught previously. See Manual, p. 76. 
 
VI. 14] teachers' manual. 117 
 
 A rhomboid is an oblique-angled parallelogram having only its 
 opposite sides equal. 
 
 The walk mentioned in 13 is supposed to be included in the 
 garden. If tlie blackboards in the room be of the same width, 
 lead the pupils to find first the entire length of all the boards, 
 and then to multiply by the width. 
 
 14 Answers: 2 $10.40. 3 $60 $44. 4 43^ sq. ft. 
 
 After the pupils have drawn plans illustrating the solution of 
 3, lead them to show how they can always find by a short process 
 the length of a walk or trench if made on the inside border of 
 a given rectangular lot, and also its length if made on the outside 
 of the lot. To find the length of a walk if made on the inside 
 border of a rectangular lot, subtract four times the width of tlie 
 walk from the perimeter of the lot ; and to find the length of a 
 walk if made on the outside of such a lot, add four times the width 
 of the walk to the perimeter of the lot. 
 
 In the solution of 4, lead the pupils to connect the corners by 
 dotted lines, and to find by measurement the altitude. That the 
 altitude may be as exact as possible, see that the line measured is 
 exactly perpendicular to the base. The square edge of a card will 
 be a good means of measurement. Let the figures made for 5 be 
 exchanged among members of the class and answers be compared. 
 Let the radius, angles, etc., be designated by letters ; thus, ao, aob, 
 etc. 
 
 15 Slow and careful measurements are needed here. Do not 
 take up a new exercise until the pupils thoroughly understand the 
 jDrevious one. The points to be developed in each of the exercises 
 are as follows : 1. The angles at the centre of a circle are measured 
 by the arcs which subtend them. 2. A simple way of finding angles 
 of various magnitudes. 3. To make a protractor for measiiring 
 angles. Let this be most carefully made. 4. Practice in esti- 
 mating magnitude of angles. 5. The sum of all the angles made 
 about a given point is equal to 360°. 6. The opposite angles made 
 \)j two lines crossing each other are equal. 7. The sum of the 
 
118 GRADED AEITHJMETIC. [VI. 16 
 
 three angles of a triangle is equal to two right angles. The sum 
 of the angles of a polygon is equal to two right angles taken as 
 many times as the figure has sides less two. 
 
 It is not desirable at this time for the pupils to spend much 
 time in formulating the facts. The important point is for them 
 to discover the facts by observation and construction. 
 
 16 Answers: 6 258 sq. ft. 7 143.86+ sq. rd. 8 46.7694 
 sq. rd. 9 Exact area = 2781.1584 sq. ft. 
 
 Proceed very slowly with the first four exercises, and review 
 frequently so that the meaning of the terms may be fixed in the 
 mind. 
 
 The polygon abode is a regular polygon, because it has equal 
 sides and equal angles. The j^upils are supposed to have com- 
 passes in constructing the figures mentioned in 3. These two 
 ways of constructing regular polygons should be repeated until 
 they are well understood. 
 
 From the questions and illustrations of 4 the pupils can easily 
 discover the process of finding the area of any regular polygon. 
 The pupils should be told that the line fo is the apothem or less 
 radius. The rule for finding the area of a regular polygon should 
 be made by the pupils, and may be as follows : Multiply the pe- 
 rimeter by half the apothem, or multiply half the perimeter by the 
 apothem. The exact length of the apothem of the regular pen- 
 tagon mentioned in 5 is 8.2584 ft. (.6882 X 12). The measured 
 distance will be, of course, only approximately correct. The ratio 
 of the apothem to the side of a regular polygon is as follows : 
 triangle, .2887; pentagon, .6882; hexagon, .866; octagon, 1,2071; 
 decagon, 1.5388 ; dodecagon, 1.866. 
 
 17 Answers: 1 $5628. 2 Neither 71^ yd. 3 12f yd. 
 49.6 yd. 28| yd. 4 53^ sq. yd. 37f sq. yd. 21^ sq. yd. 21^ 
 sq. yd. 5 611 sq. ft. 25 rolls. 
 
 Let the pupils carefully measure the lines whose distances are 
 not indicated. According to the measurement given of the hall, 
 one breadth of the carpet will just fill the narrow hall-way. Four 
 
VI. 18] teachers' manual. 119 
 
 strips, therefore, will be needed, one strip 20 ft. long and 3 strips 
 each 6 ft. long. 
 
 In estimating the amount of paper that ■will be needed for the 
 walls, only an approximate estimate can be made, as more or less 
 may be needed for matching, etc. A good method is, first to find 
 the number of strips required, then to find how many strips can be 
 cut from a roll. The whole number of strips required divided by 
 the number that can be cut from a roll will give the number of 
 rolls required. Let the pupils perform 6 by this method. The 
 pupils' answers to 6 can be but ajiproximately near the correct 
 answers without definite instructions as to how the wainscot will 
 be placed with reference to chimney, and whether the window in 
 the bay extends below the wainscot. The widest part of the 
 recess of the bay window is supposed to be 10 ft. wide, and the 
 narrowest part 4 ft. wide. Depth of bay, 4 ft. 
 
 The given floor plan may be xised for other exercises, such as 
 finding the number of sq. ft. of flooring in each room, the cost of 
 painting a border for the floor of each room, etc. 
 
 18 ^nsM-ers; 3 314 sq.ft. 4 40.13 sq. rd. .2508+ A. 5 1925 
 sq. ft. 8 314.16 sq. ft. 1256.64 sq. ft. 9 3.183 ft. 15.9154 ft. 
 2.758 yd. 10 1.76+ sq. ft. 9.621+ sq. ft. 11 420.168+ 
 times. 
 
 Let the pupils try both ways of finding the area of a circle. 
 The rule for finding the area which the pupils should be led to 
 give, from the questions in 1, is : " Multiply the circumference by 
 half the radius." And it is by this rule that the pupils should at 
 first be required to find the area of a circle. 3.1416 may be used 
 as the ratio of the circumference to the diameter. 
 
 19 Answers: 2 384 sq. in. 3 7^^| sq. ft. 4 17568 sq. in. 
 5 $180 12926?f gals. 
 
 The cube and other right prisms should be presented to the 
 class in teaching these exercises. Right prisms only should be 
 dealt with at first, and when prism is referred to in this book, a 
 right prism is meant. The points of resemblance in all prisms 
 
120 GRADED ARITHMETIC. [VI. 20 
 
 are : (1) The two bases are parallel. (2) The lateral sides are 
 parallelograms. The definition, therefore, of a prism which may 
 be made by the pupils is : "A solid whose bases are parallel and 
 whose lateral sides are parallelograms." 
 
 Only the approximate contents can be given in answer to the 
 questions asked in 8 and 9, owing to the fact that the exact dimen- 
 sions of the prisms cannot be found from the cuts. In explaining 
 the process of finding the contents of prisms, lead the pupils at 
 first to use the method given on page 106 of the Manual. 
 
 20 Answers : 1 192 cu. in. 364 cu. in. 2 24| cu. ft. 3 75 
 sq. ft. 27.06 cu. ft. 4 24 sq. ft. 330 cu. ft. 5 311f sq. ft. 
 3888 gals. 8 3 sq. ft. 25.78+ sq. in. 618.72+ cu. in. 9 .319+ 
 cu. in. 5.1051 sq. in. 
 
 The work called for in 1 should be done very carefully. If the 
 models are made of cardboard, let the pupils first cut the outline 
 as given, and then cut half through the cardboard where the edges 
 are to be. The figures when folded can be held in place by means 
 of mucilage or paste. Modeling may be continued with profit to 
 the pupils, besides furnishing models for future measurements. 
 
 The following facts concerning the cylinder should be taught 
 objectively and definitions developed : 
 
 A cylinder (right cylinder) is a solid bounded by a curved sur- 
 face and by two equal parallel circles. The curved surface is called 
 the convex surface, and the other two sides are called the bases. 
 
 To find the convex surface of a cylinder, multiply the circum- 
 ference of the base by the height. 
 
 To find the volume of a cylinder, multiply the surface of one 
 of the bases by the height. 
 
 31 Answers: 1 $40.72+. 2 $113.10-. 3 706.86 cu. in. 
 4 1800 cu. in. 5 in. 5 8 in. 6 10 ft. 7 2 ft. 8 U ft. 
 9 ir, ft. 8 in. 10 3911.38+ in. 11 8f sq. yd. 12 $17.79-. 
 
 13 ^ cu. ft. 14 3630 cu. ft. 15 640 sq. ft. $9.60 $5.12. 
 16 171^ yd. 
 
VI. 22] teachers' manual. 121 
 
 After solving several simple problems similar to 5, let the rule 
 be given for finding one dimension of a solid when the cubic con- 
 tents and two dimensions are given. Whenever any point is not 
 clear, use models and small numbers. Let the pupils illustrate 
 all processes by drawings when it can be done. 
 
 33 Answers: 1 56 sq. yd. 62* sq. yd. 24 yd. 2 40 ft. 
 320 sq. ft. 3 486 sq. ft. 54 sq. yd. 4 35^ sq. yd. 5 8 rolls. 
 8 104 sq. ft. 12f sq. yds. 9 272 sq. ft. $13.60. 10 18 yd. 
 11 26f yd. widtlnvise of the room. 
 
 For method of finding the approximate number of rolls of paper 
 required to paper the walls of a room, see Manual, page 119. Let 
 5 be performed by that method. Great care should be taken in 
 the solution of all these problems to desig- 
 
 18 
 
 nate what each concrete number stands for. 
 Another good way is to indicate the proc- 
 ess in one place and perform the indicated 
 operations in another place. For example, 
 in 8. 
 
 [(16 X 2) + (10 X 2)] X 2 = sq. ft. in border. 
 
 q' 
 
 (18 X 12) - 104 
 
 •^^ ~ = sq. yd. m fioor unpainted. 
 
 32 
 20 
 
 52 
 lOl sq. ft. ^^^ ''I- y^- 
 
 216 
 104 
 9)112 
 
 or, 
 
 (18 X 12) — [(18 - 4) X (12 - 4)] = sq. ft. in border. 
 
 14 
 
 14 X 8 , . ^ . , ^ 216 8 
 
 sq. yd. m floor unpainted 
 
 9 ^ ' ' ^ 112 9 )112 
 
 104 sq. ft. 12f sq. yd. 
 
 Either of these solutions or any other correct solution should 
 be accepted, and may be followed by an oral explanation. 
 
122 GRADED ARITHMETIC. [VI. 28 
 
 If the pupils are not considered ready for such work as that 
 noted above, they should be led to write out the solution, so as 
 to indicate that they understand every step. The following is a 
 sample of what should be expected: 
 
 18 18 ft. — 4 ft. = 14 ft., length of unpainted part. 
 
 12 12 ft. — 4 ft. = 8 ft., width " " " 
 
 216 sq. ft. in entire floor. 14 
 
 112 '' " " unpainted part. __8 
 
 104 " " " border. 112 sq. ft. in unpainted part. 
 
 9 sq. ft. )112 sq. ft. 
 
 12| = no. sq. yd. in unpainted part. 
 
 23 Ans7rers : 1 20^ bd. ft. 2 19 iV bd. ft. 3 96 ft. 4 2652 ft. 
 $42.43. 5 1421^ bd. ft. $21.32. 6 12/^ loads. 7 1564^ 
 loads. 8 $376.32. 9 2.9+ pailfuls. 10 172j 173.57+ bu. 
 144 bu. 5 T. 8 cwt. 11 4700.16 gal. 12 34.27+ rd. 93.46+ 
 sq. rd. 
 
 34 Answers : 1 23^ in. 2 5 ft. 6.84 in. 3 36 sq. ft. 4 70^ 
 sq. ft. 5 $56.10 $3234.33. 
 
 Dictation exercises similar to 6 will be found interesting and 
 profitable to the pupils. Original problems of the same kind 
 should be made and brought into the class. 
 
 35-37 If the measures have been properly used in previous 
 grades, they need not be used in the solution of these problems, 
 most of which should be performed orally. Let the pupils perform 
 the problems in the shortest and most direct way. Correct and 
 lead by questioning rather than by giving outright the shorter 
 method. For example, if the pupils attempt to perform 24, 
 page 26, in a long or roundabout way, ask such questions as : 
 "What short way of getting the cost of 25 bu. at 75/ a bushel? 
 (Probably two ways will be given : one, \ of 100 times 75/, and 
 the other, 25 times f of a dollar.) How much is it ? 3 pk. 6 qt. 
 is how many quarts less than a bushel ? What part of a bushel 
 is it ? 3 pk. 6 qt., then, will cost how much less than 75/?" 
 After answering these questions the pupils can easily perform the 
 
VI. 28] teachers' manual. 123 
 
 problem, and will say: "1 bu. costs f of a dollar ; 25 bu. will cost 
 25 times f of a dollar or ^^, equal to $18.75. 3 pk. 6 qt, is 2 qt. 
 or ^^ of a bushel less than a bushel, and will therefore cost ^^ of 
 75/ less than 75/. ^^ of 75/ is 4 cents and a fraction, subtracted 
 from 75/ equals 71/. This sum added to $18.75 is $19.46, the 
 cost of 25 bu. 3 pk. 6 qt. of wheat at 75/ a bushel." 
 
 28 — 30 If possible, bring to the class the various coins referred 
 to in these exercises, and talk about their absolute and relative 
 values. Some of the solutions may need the aid of figures, but 
 most of them can be performed orally. Information in regard to 
 coins not contained here will be found in the Appendix of Book 
 VIII. ; also on page 89, Book IV., and page 69, Book V., and 
 notes thereon in the Manual. 
 
 31 Answers: 10 12 1b. 4 oz. 11 12 cwt. 20 1b. 12 6 T. 
 
 10 cwt. 30 lb. 13 16 T. 5 cwt. 50 lb. 14 1 lb. 7 pwt. 15 4 lb. 
 2 oz. 2 pwt. 16 2i. 17 9 /§ 4/3 lOm. 18 8 cwt. 91 lb. 
 13 oz. 9 T. 18 cwt. 8 lb. 12 oz. 19 101 2 3 29 6/§ 16iTl. 
 20 4 T. 18 cwt. 21 3 T. 11 cwt. 82 lbs. 22 9 oz. 4 pwt. 
 23 2S 13 23 17 gr. 24 43 29 5 gr. 7 lb. 7 oz. 3 dr. 1 sc. 
 25 1 T. 1 cwt. 60 lb. 3 T. 12 cwt. 20 lb. 
 
 33 Answers: 1 4 lb. 51. 2 9/§ 4/3. 3 27 T. 3 cwt. 
 60 lb. 4 8 lb. 1 oz. 15 pwt. 5 9 T. 17 cwt. 2 lb. 8 oz. 112 T. 
 
 12 cwt. 80 lb. 6 38 lb. 19 10 gr. 148 cong. 4 0. 7 19 cwt. 
 551b. 8| oz. 8 3 oz. 10 pwt. 3^gr. 9 3 3 29 5 gr. 10 2 3 16ni. 
 
 11 9 cwt. 89 lb. 5i oz. 5 oz. 2 pwt. 18 gr. 12 537.5 lbs. $8.06. 
 
 13 3 oz. 15 pwt. 12 gr. 14 l\ T. 15 |gj lb. 16 $240. 
 17 $2.05. 18 1/5 2/3. 19 7 lbs. 1 oz. 16 pwt. 16 gr. 
 71 lb. 6 oz. 6 pwt. 16 gr. 20 $89.08. 21 200 pills. 360 pills. 
 22 28, and 37 pwt. left. 
 
 33 Answers : 2 J 640 rd. K 1920 rd. L 960 rd. M 1280 rd. 
 N 320 rd. 2880 rd. P 2560 rd. Q 2240 rd. 3 J 87120 
 sq. ft. K 261360 sq. ft. L 130680 sq. ft. M 174240 sq. ft. 
 N 43560 sq. ft. 392040 sq. ft. P 348480 sq. ft. Q 304920 
 sq. ft. 4 </ 48 qt. K 60 qt. L 40 qt. M 52 qt. N 24 qt. 
 
124 GRADED ABITHMETIC. [VI. 33 
 
 32 qt. P 44 qt. Q 50, qt. S J U qt. K 192 qt. L 96 qt. 
 M 128 qt. N 32 qt. , 288 qt. P 256 qt. Q 224 qt. 6 J^ 16s. 
 K 12s. L 5s. M 15s. N Is. 2.5s. P 7.5s. Q 12.16s. 
 7 J 1320 gv. K 840 gr. L 1860 gr. M 3264 gr. i\r 2700 gr. 
 2320 gr. P 3640 gr. Q 1696 gr. 8 J j%\ A. /f ^3^9/^ A. 
 L ^\S A. M ^^^ A. AT ^V'.i^ A. if^S% A. P ^WsV A. 
 ^ t¥A a. Q J ^%% T. a: IV^x T. L 2i^ T. Jf 2j-«_6_ t. 
 AT 2/A«^ T. 3^ ef T. P 1 J'^s.T^^ T. Q 3^% t. 10 J 880 yd. 
 JT 1320 yd. L 1100 yd. TIf 733^ yd. N 10261 yd. 938| yd. 
 P 1496 yd. Q 1114f yd. 11 J 232^ cd. A^ 4961 cd. L 5010 cd. 
 Jf 672 cd. A^ 563f cd. 151\ cd. P 496f cd. Q 876 cd. 
 12 J 960 gi. a: 1920 gi. A 2880 gi. Ji" 1168 gi. A^ 3344 gi. 
 
 5792 gi. P 5088 gi. Q 4320 gi. 13 J 14520 ft. AT 9240 ft. 
 L 20460 ft. Jf 35904 ft. A^ 29700 ft. 25520 ft. P 40040 ft 
 Q 18656 ft. 14 ./ 25.6 qt. K 19.2 qt. L 8 qt. J/ 24 qt. 
 X 1.60 qt. 4 qt. P 12 qt. Q 19.456 qt. 15 J 320 A. 
 A 480 A. L 400 A. J/ 266 A. 106 sq. rd. 1811 sq. ft. A^ 373 A. 
 53 sq. rd. 2721 sq. ft. 341 A. 63 sq. rd. 272^ sq. ft. P 544 A. 
 Q 405 A. 53 sq. rd. 272^ sq. ft." 16 J 1000 lb. K 1500 lb. 
 L 1250 lb. 3f 833 lb. 5^ oz. X 1166 lb. lOf oz. 1066 lb. 
 10| oz. P 1700 lb. Q 1266 lb. lOf oz. 17 J^ 80 lb. X 60 lb. 
 i 25 lb. M 75 lb. A^ 5 lb. 12.5 lb. P 37.5 lb. Q 60.8 lb. 
 18 J 256 rd. a: 192 rd. L 80 rd. Jf 240 rd. A^ 16 rd. 40 rd. 
 P 120 rd. Q 194 rd. 9 ft. 2.88 in. 19 ^ 11 qt. K 7 qt. L 15 qt. 
 
 1 pt. M 27 qt. If gi. A^ 22 qt. 1 pt. 19 qt. 2f gills. P 30 qt. 
 2| gi. () 14 qt. 1 tV gi- 20 ^ 54 oz. K 75 oz. A 52 oz. 16 pwt. 
 3f 73 oz. 10 pwt. A" 101 oz. 2 pwt. 17.28 gr. 76 oz. 12 pwt. 
 15.36 gr. P 121 oz. 1 pwt. 14.4 gr. Q 216 oz. 1 pwt. 4.8 gr. 
 21 J 13 cu. ft. 864 cu. in. K 20 cu. ft. 432 cu. in. A 16 cu. ft. 
 1512 cu. in. 31 11 cu. ft. 432 cu. in. A"^ 15 cu. ft. 1296 cu. in. 
 14 cu. ft. 6911 cu. in. P 22 cu. ft. 1641| cu. in. Q 17 cu. ft. 
 172|cu. in. 22 c/52rd. A' 75 rd. A 42 rd. 31 91 vd. A^52rd. 
 60 rd. P 84 rd. Q 82 rd. 23 J 133 T. A' 258 T. 10 cwt. 
 L 236 T. 31 346 T. 16 cwt. A^ 271 T. 9 cwt. 354 T. 1800 cwt. 
 P 271 T. 1400 cwt. Q 418 T. 800 cwt. 24 J 162 rd. 4 yd. 4.^ in. 
 
VI. 34] ' teachers' manual. 125 
 
 K 241 rd. 4 yd. 4^ in. L 203 rd. 4 yd. 2 ft. 5^ in. M 140 rd. 
 
 2 ft. 2f in. N 192 rd. 1 yd. 1 ft. 9f in. 175 rd. 2 yd. 2 ft. 3 in. 
 P 279 rd. 3 yd. 7| in. () 206 rd. 2 yd. 2^ in. 25 .7" 13 A. 100 
 sq. rd. if 30 A. 130 sq. rd. L 25 A. 80 sq. rd. M 37 A. 96 sq. rd. 
 N 29 A. 29 sq. rd. C) 46 A. 138 sq. rd. P 32 A. 134 sq. rd. Q 50 A. 
 124 sq. rd. 26 J 1 bn. 3 pk. 5 qt. 1.2 pt. K 2 bn. 5 qt. 0.4 pt. 
 L 1 bu. 1 pk. ;; (^t. 0.4 pt. M 2 bu. 1 pk. 1 qt. N 2 bn. 5 qt. 
 0.048 pt. O 1 bu. 2 pk. 7 qt. 0.176 pt. P 2 bu. 3 pk. 4 qt. 1.44 pt. 
 Q 5 bu. ;5 (jt. 0.992 pt. 27 ./ 16 en. ft. 432 cu. in. K 22 cu. ft. 
 L 20 cu. ft. 1296 cu. in. M 18 cu. ft. 86.4 cu. in. N 21 cu. ft. 
 648 cu. in. 19 cu. ft. 403.2 cu. in. P 1 cu. yd. 3 cu. ft. 3240 
 cu. in. Q 20cu. ft. 1094.4 cu. in. 28 A 4 rd. B 3rd. C 20 rd. 
 D 120 rd. E 2110 rd. F \ rd., or 4 ft. U in. ^ 0.2 rd., or 3 ft. 
 3.6 in. H 1 rd. / 1.75 rd., or 1 rd. 4 yd. 4^ in. 29 A 5 gal. 1 qt. 
 B 12 gal. 2 qt. C 52 gal. i) 150 gal. E 2950 gal. P 2 qt. 
 
 3 gi. 6^ 2 qt. 1.2 gi. ZT 3 qt. 1 gi. 7 5 gals. 1 pt. 0.8 gi. 
 30 J 40 S(i. rd. a: 60 sq. rd. L 68 sq. rd. M 11^ sq. rd. 
 N 47ird. 33^ sq. rd. P 63 sq. rd. Q 33^ S(i. rd. 31 ./ 2 bu. 
 a: pk. L 2 bu. M 6 bu. 2 pk. N 5 bu. 2 pk. 5 bu. P 7 bu. 
 1 pk. Q 3 bu. 32 ./ 123.5 sq. rd. K 89.75 sq. rd. L 35.6 
 sq. rd. M 113.875 sq. rd. N 428 sq. rd. 13.614 sq. rd. P 49.91 
 sq. rd. Q 79.275 sq. rd. 33 ./ 7^^. K IS^s. L ^s. M l^%s. 
 N Q^^^s. 5,f ,s-. P 9tV. Q 9^2_5. 34 j l^?^ qt. K 11 -^\ qt. 
 L 11/t- qt. J/ 15/^ qt. N 6^4 qt. 9iJ qt. P 12f qt. 
 Q 16if qt. 35 ./ f 180. K $375. Z $140.80. M $477.75. 
 N $387.69-. $332.07+. P $736.57. Q $1224.34. 
 36 J $17.88-. A' $11.38-. L $25.19-. M $44.20. 
 AT $36.56+. $31.42-. P $49.29+. <) $22.97-. 37-7$30.61+. 
 JT $45.92-. X $38.27-. Jf $25.51-. A^ $35.72-. $32.65+. 
 P $52.04+. Q $38.78-. 
 
 34 Answers: 1 Apr., June, Sept., Nov. ; Jan., Mar., May, July, 
 Aug., Oct., Dec; Feb. 2 365 52 wk. 1 da. 366 da. 4 -$546 
 or $547.75. 5 4 mo. 7 da. 6 mo. 10 da. 2 mo. 6 da. 6 T yr. 
 1 mo. 29 da. 22 yr. 2 mo. 29 da. 7 400 da. 473 da. 298? ia. 
 
126 GRADED ARITHMETIC. [VI. 35 
 
 8 $342.08. 10 67 yr. 9 mo. 22 da. 11 April 15, 1865. 
 12 Goethe : 82 yr. 6 mo, 23 da. Longfellow, 75 yr. mo. 25 da. 
 Luther, 62 yr. 3 mo. 8 da. Franklin, 84 yr. 3 mo. da. Shake- 
 speare, 52 yr. Napoleon I., 51 yr. 8 mo. 20 da. Burns, 37 yr. 
 5 mo. 26 da. Milton, 65 yr. 10 mo. 30 da. 
 
 The number of days in each month of the year should be re- 
 peated until it can be told as soon as the name of the month is 
 given. 
 
 A good method of finding the months and days from one date 
 to another is illustrated by the following solution of 5 : From 
 Aug. 13 to Dec. 13 is 4 mo.; from Dec. 13 to Dec. 20 is 7 da. 
 Answer: 4 mo. 7 da. From June 30 to Aug. 30 is 2 mo.; from 
 Aug. 30 to Sept. 5 = 1 day in August + 5 days in Sept. = 6 da. 
 Ansiver: 2 mo. 6 da. And in 12 : From 1749 to 1831 is 82 yr.; 
 from Aug. 28 to Feb. 28 is 6 mo.; from Feb. 28 to Mar. 22 is 
 22 da. Ansu-er : 82 yr. 6 mo. 22 da. 
 
 A common custom in banks is to regard 30 days for every month 
 in estimating time. For example, in finding the time between 
 June 30 and Sept. 5 the process would be : From June 30 to 
 Sept. 30 = 3 mo. — 25 da. (the time from Sept. 5 to Sept. 30) = 
 2 mo. 5 da. 
 
 In performing the latter part of 4, let the pupils find the amount 
 saved if the year begins on a week-day, and, again, if the year 
 begins on Sunday; also if the year is a leap year. 
 
 To find the exact number of days from one date to another (7) : 
 From Sept. 16, 1891, to Sept. 16, 1892, is 366 da.; 14 more days 
 in Sept. + 20 in Oct. = 34 da., which, added to 366, equals 
 400 da. 
 
 35 Answers: 4 378' 754'. 5 1820" 2558". 6 10848" 
 14424". 7 5° 17° 7^°. 8 .3 of a degree .6 of a degree 
 .275 of a degree §|°. 9 22° 12'. 10 New York, 74° W. 
 
 Baltimore, 76° 35' W. Chicago, 87° 35' W. San Francisco, 
 
 122° 25' W. 11 16° 33' 30". 86I3IJ5+ "^i- 13 21600 geog. mi. 
 24840 statute mi. 
 
VI. 36] teachers' manual. 127 
 
 Lead the pupils to use the globe and outline map in estimating 
 distances. Show by means of the globe the reason for the differ- 
 ence in length of the degree of longitude in various latitudes. 
 Some of the distances are given in the Appendix of Book VIII. 
 
 36 Answers: 1 192 mi. 12.5 hr. 5 da. 5 hr. 2 21 da. If hr. 
 3 312 lb. 8 oz. 4 3 T. 750 lb. 5 537.605 cu. in. G7.201 cu. in. 
 6 19.286 bu. 7 247.76 bu. 8 198.2 bu. 9 97.71+ bu. 
 10 64.28+ qt. 11 (a) 1.75 M. (b) 3.50 M. (c) 1.29/ (d) 1.29/ 
 (e) 25.8/ (/) 14.902. 
 
 For all practical purposes, in estimating the capacity of bins, 
 the bushel, stricken measure, may be regarded as containing 1:|- 
 cu. ft. The answers above given are found by exact measure. 
 Show that a bin or box will hold f as many bushels by heaped 
 measure as it will by stricken measure, on the supposition that 
 the heaped measure will hold ^ more than the stricken measure. 
 
 37 Answers: 13^^/. 2 $1104. 8 $65.75 gain. 9 $12. 
 10 $120. 11 7560 bricks 648 bricks less. 12 158 rd. 6.6 ft. 
 13 7H loads. 14 733^ loads. 
 
 The solution of problems on a line sometimes shortens the pro- 
 cess, as shown in the following solution of 1 : 
 
 .60 
 $.00 
 
 $6.00 ,, ,, „,,, $0.00X^X3 _^^,^ 
 
 49 
 
 The tons mentioned in 8 are short tons. Performed on a line, 
 the solution is : -j^qq 
 
 $5.25 X^g^- ($5.20X128.3). 
 
 112 
 Bricks vary greatly in size, the dimensions here used for 11 
 being 8^" X 4" X 2^". Let the pupils see how the bricks must 
 be placed in both instances to make a pile of the given dimensions. 
 
128 GRADED ARITHJVrETIC. [VI. 38 
 
 Tlie common bricks are placed on edge, and the Milwaukee bricks 
 are laid flat. 
 
 38 A7isicers : 1 4 T. 1800 lb. 2 70|§ bbl. 183f | bbl. 
 
 89if bbl. 3 17.5 T. 4 25 T. 1440 lb. 5 11600 lb. 59^% bbl. 
 6 $36.80. 7 $2.73^5. 8 3824i mi. 9 9 A. 87 sq. rd. 24 
 sq. yd. 6i sq. ft. 10 160 rd. 6 mi. 230 rd. 11 36 mi. 226f mi. 
 141 mi. 12 95f/. 13 556 bu. $444.80 $300.24. 14 114f lb. 
 2 bu. 1J| qt. 15 34 cu. ft. 1483xV cu. in. 
 
 It is assumed in 7 that the flour bought is whole Avheat flour. 
 
 31) In addition to the helps here given, all the common meas- 
 ures of the metric system should be used for the exercises of this 
 section. Much time should be given to actual measurements before 
 the exercises are taken. 
 
 If a meter-stick is not provided, each pupil can easily make one 
 by cutting a stick exactly ten times as long as the decimeter shown 
 in the cut. The stick can be divided by cross-lines into decimeters 
 and centimeters. 
 
 40 Answers: 7 8.64^™ 86.4 1^™ 864^^"'. 9 8570.704°^. 
 10 9030.043™. 11 312.5 times. 12 2280'". 13 $5.74. 
 14 70601 poles. 
 
 Similar exercises may be given for class drill if needed. Show 
 two ways of reading metric numbers ; thus, 69.83™ may be read 
 sixty-nine and eighty-three hundredths meters, or, sixty-nine meters 
 and eighty-three centimeters. Let the pupils practice in reading 
 numbers in both ways. 
 
 41 Answers: 2 10000 ^«™. 3 1000000 1™. 5 .0001 1 ^''" 
 .000001 qJ^'" .001 "Km_ e 10001™ lOO^™ lOO^™. 7 9.603<)76i'^"\ 
 8 8.697544 <i™. 11 6849602.1 1™. 12 5005000.8 1™. 
 13 251.125 <i™. 14 2000001™. 15 12500 bricks. 16 5* 
 .5^^ 500^ .05 ^ 17 200* 4=^ .08 '^ 6000 ^ 
 
 The sign for square is sometimes written sq. instead of q. Call 
 attention to the fact that, as 100 units of one denomination make 
 a unit of a denomination next larger, two places of figures are 
 allowed for each denomination. 
 
VI. 42] teachers' manual. 129 
 
 43 Answers: 1 $407.50. 2 $500. 3 $320 gain. 4 20001™ 
 20 ^ 6 0.001 <■»'« 0.000001*^"'" 0.000000001'""'. 7 Oue 
 
 thousand times as large. 8 One million times as large. 
 
 9 8048000™'™. 100.0080'""'. 11 357*=""> 142.8 loads. 
 
 12 1.4476 + ''""\ 13 120 «'. 14 58.5 «'. 15 $294.84. 
 
 43 Aiiswers: 3 800 1 GO^ 0.8 ^ 4^ 900 1. 4 S'^'' 0.78 '^'^ 
 4'^''. 5 450 '»! 0.8 '^1 50-11 400OO'". 6 0.02 1. 7 0000 ^ GO"'. 
 
 The liter measure is one decimeter long, wide, and deep. If the 
 school is not supplied with a liter, one can be made of wood or 
 tin. This measure, filled with water at a temperature a little 
 above freezing, weighs a thousand grams, or one kilogram. These 
 facts should be given to show the relation that the measures of 
 volume, capacity, and weight have with one another, all depending 
 upon the meter, which is supposed to be the ten-millionth part of 
 the distance on a meridian from the equator to the pole. 
 
 44 Avswers: 1 400^' 8000 s 9000 « 80 « G()s. 2 834.971 «. 
 
 3 G09.80lg. 5 8G.97S 84 s. 6 80^. 1000^. 8 l^^". 
 9 12000 Ks. 10 24000 i^X 
 
 The last four exercises on the page are important only as they 
 serve to give tlie pupils an idea of the value of the common metric 
 measures in measures familiar to them. It may not l)e necessary 
 for all these exercises to be given to accomplish the purpose. 
 
 45 Anstvers: 2 2| hr. 3 $200 $21.50. 4 1341f"'. 
 5 13.88'^'"'. 7 36753G0K". g 2323.2^1. 9 720001. ^.0 25.8 «^ 
 
 11 5"'. 12 .04502411^'" $36019.20. 13 71GGf'"" $48. 
 14 399700'!'". 15 goO"'. 
 
 46 Aiisivers: 1 $20.34. 2 270001. 3 S.G'^'*™ 3G00i. 
 
 4 51.072'^"'" 31.92 loads. 5 10 hr. 6 l^^X 7 4297.674 'i"! 
 42.97674 ^ 8 9^*. 9 4.8"\ 10 3 1' 447 1^»^. 11 12.5i^X 
 
 12 7112+ half-dollars. 13 45208.8 s. 14 $7188. 15 30.35 ^ 
 One important end should be gained in work upon the metric 
 
 system, and that is, to give pupils a clear idea of the simplicity 
 of the system in comparison with our system in common use, and 
 
130 GRADED ARITHMETIC. [VI. 47 
 
 the great saving of time which its adoption would occasion. If it 
 is desired that pupils shall be acquainted with the denominations 
 of the system Avell enough to perform problems at any time, fre- 
 quent reviews will be necessary. 
 
 47—49 Follow carefully the order of exercises here given, and 
 let the pupils practice upon them until they can perform them 
 readily. Their understanding of subsequent work Avill depend 
 largely upon the thoroughness with which they take this prelim- 
 inary work. 
 
 50 Answers: 1 0.696 2.7130f 75.1152. 2 296.1 447.43^ 
 0.033026. 3 923.68 45.08 44.54^. 4 $35.8234 17.4147^ 
 2550.62^. 5 $44.889i 1134.3392 886.033 mi. 6 2000.882 
 $154.26^ 142.47^. 7 0.0039501 0.21375 47.66625 A. 
 
 8 9.945 0.5899 29.41. 9 0.01^^ ^.437 0.684988. 10 2.80^ 
 0.0027. 0.0001369|. 19 1.852 0.03538 0.002f. 20 $64.8284 
 1.3008i 0.07^f 21 23.465 0.3962425 0.06/^. 22 0.063615 
 0.002^^ 35.6325. 23 22633.65 1980.68 55.8723f. 
 24 $0.17226 $16.25525 0.004f. 25 0.003^ 217.875 
 0.0033432^. 26 2102.118 hr. 0.012^6^ 15.772328. 27 43.6720f 
 40559.44f. 0.247f 28 0.05^ 0.000032 19630. 
 
 51 Nearly all of these problems should be performed orally, 
 but for the sake of learning a good form of written analysis, the 
 pupils might write out the solution of some of them. The following 
 form of solution is suggested : 
 
 $3550.00 cost of farm. 
 
 .334- 
 14 Given : Cost of farm. $1183 33t^ 
 
 Required: The selling price. 'irrr\ac\ 
 
 $4733.33i selling price. 
 
 53 Ansivers: 1 $6375. 2 $.0807. $.1521. 3 $1456. 
 4 $5098. 5 3835201b. 940 bu. 6 97f bu. 7 $630. 8 $84.81. 
 
 9 3031.2 bbl. 10 38250. 11 $21^ loss. 12 $359.13. 
 13 $49.93. 
 
Yl. 63] tejlchers' manual. 131 
 
 Lead the pupils in such problems as 13 to shorten the process 
 as much as possible. In this problem they Avould say: "There are 
 332 gallons in all. If 8*}^ of the molasses was lost and 45"^ of it 
 was sold, there would be left 47% of it. 47% of 332 is 156.04 
 gallons. 32 cents, the price of a gallon, multiplied by the number 
 of gallons is $49.93." 
 
 53 Answers: 1 7 cwt. 9.5 lb. 2 )$2523.17. 3 1.64| yd. 
 4 $112.50. 5 $1827.50. 6 $117. 7 $2100000. 8 787828.29f 
 tons. 9 640485.449 tons. 10 1099962. 11 1515111.54. 
 12 $1450.80. 13 6 lb. 14 $43.80. 
 
 When these problems have been solved and solutions in good 
 form written out, the pupils might practice in writing the solution 
 on a line, using hundredths for per cent, as in 12, as follows : 
 
 93 2 
 
 $3 X XH(t> X M , $4 X I860 X 6 «.-,,. ^^^ 
 
 li — ^< — m + ~5 — X — m ^ ^^^^^•^^• 
 
 54—55 A new principle is involved in these exercises. In 
 previous exercises the base and rate were given to find the per- 
 centage. In these which follow, the base and percentage are given 
 to find the rate. These terms need not be given to the pupils at 
 present. They should be led to recognize the conditions after 
 sufficient practice. Whenever the rate per cent that one number 
 is of another is called for, let the pupils first get the common frac- 
 tional part and then reduce the fraction to liundredths, or per 
 cent. Great care should be taken to lead the pupils to recognize 
 the base, or the number of which another number is a part. The 
 form of question should be variously given, so that the pupils will 
 not work mechanically. Sometimes the base is the first number 
 given in the question, as, What part of 8 is 4 ? and sometimes 
 the percentage is the first number given, as, 4 is what part of 8 ? 
 Sufficient practice in both forms should be given, so as to enable 
 the pupils to recognize the base ifhererer it is placed in the 
 question. 
 
132 GRADED ARITHMETIC. [\1. 56 
 
 • 
 
 56 Answers: 1 12.5% S.GO^i-r^ 197i|§% 16&Ui% 25/^%. 
 
 2 67.5% |f% 46t\% 11% 34713%. 3 lJ^J^j% f|% 
 14^% Ml% 114111%. 4 40/^% 2^y,% 266^1^% 29|gf% 
 4§ix%. 5 24ff% 131% 18i% 12^,% 39-}!. 6 l§if% 
 90x|«% 30if% 5^U% 33^h- 7 30|e% 389^i,% 74if^,% 
 554i|M% 562ifff%. 8 « 12^%; ^-111%; c 33%; .7 12|%; 
 e7i%;/88%; yl2i%; A 20%; i i%; J U^. 9 a 6%; h 30%; 
 c7i%; fZ5%; e50%;/16%; ^74%; A 15//g % ; fl5%;il4f%. 
 10 6|% 26|% 200% 800% 13^% 15i;f% 7-}3.% n^cf^ 
 15t\V^% Hm%- 11 50% 35% 28% 80411% 93^% 
 100% 38|% 10i|% 5% 71%. 12 36i%. 13 Canada, 
 5MMU%; U.S., 69H^§H% 14|%. 
 
 57 Ansivers: 1 Mass., .22+% KY., 1.36% Cal., 4.45+% 
 Texas, 7.49+%. 2 Moh., 5.^,+ % Brah., 20-if% Bud., 40%. 
 
 3 650%. 4 Col., 13+% Ind.,f+% Whites, 86.1+%. 5 87.3+%. 
 6 1.4+%. 7 5%. Q^^%S%. 9 14.2%. 10 15%. 1119.2%. 
 12 li%. 
 
 Notice the wording of 3. Ask the pupils to give examples of 
 both kinds of exercises. 
 
 The term base might be introduced here as being the amount on 
 ■which the rate is estimated. Some introduction will have to be 
 given to show what the base is in such exercises as 7, 8, 9, and 11. 
 
 5S On this page and the two following pages are simple appli- 
 cations of the two principles of percentage already taught. The 
 pupils should perform the problems by analysis, recognizing in the 
 varying conditions what principle is to be applied. Of the two 
 methods of finding the interest of a given sum for a period greater 
 or less than a year, it is better on some accounts to find the interest 
 first for one year, and then for the given time, as indicated in 8. 
 Let the analysis be simple and direct, no set form being required. 
 
 50 By ''rate of interest," referred to in 1, the pupils should 
 know that the rate per year is meant. The pupils should be told 
 that tlie per cent of gain or loss in business transactions is always 
 estimated on the cost unless otherwise specified. Some talk about 
 
A"I. 60] teachers' manual. 133 
 
 insurance, commission, savings banks, etc., will be helpfiil in con- 
 nection with, problems bringing in those terms. For facts concern- 
 ing these subjects see Book VII., and accompanying notes in the 
 Manual. 
 
 60 Some of these problems will need to be solved by the aid 
 of figures. The analysis of the most difficult ones may be written 
 out if thought best. 
 
 61 In these exercises the base is asked for. Analysis in the 
 use of common fractions should first be made, such as : '' Four is 
 one half of some number ; two halves of the number is two times 
 four, or eight"; and again: "Twenty-four is two thirds of some 
 number ; one third of the number is one half of tAventy-four, or 
 twelve. If one third of the number is twelve, three thirds of the 
 number is three times twelve, or thirty-six." 
 
 From such reasoning the step to similar work in percentage will 
 be easy. The pupils will readily see that ^^J'^, or 50%, may be 
 treated exactly as ^ is treated, and that 12 is 50"^ of 2 times 12. 
 When the required number cannot be obtained directly by multi- 
 plying or by using an equivalent common fraction for the' given 
 per cent, 1% of the nunVber is first found, and then 100*^; thus, 
 in 10, the pupil will say: "Six dollars is six per cent of some 
 number ; one per cent is one sixth of six dollars, or one dollar. 
 If one per cent of the number is one dollar, one hundred per cent 
 of the number is one hundred times one dollar, or one hundred 
 dollars." 
 
 A similar analysis should be made in solving all the problems 
 on this page. To assist in making the solution clear, the analysis 
 may be written out ; thus, in 13 : 
 
 j\<\y of the number = 32 
 
 100 4 Ui o^ 
 
 |o a u u = j_o_o of 32 = 80 
 
 32 is 4:0% of 80. 
 
 62 Use the common fractional form in the solution of 1; and 
 generally, when the pupils find difficulty in solving a problem in 
 
134 GRADED ARITHMETIC. [VI. 63 
 
 percentage, let them substitute small numbers in place of large 
 ones, and common fractions in place of percentages. The substi- 
 tution of aliquot parts should be made for percentages whenever 
 simplicity is gained, but for useful practice some of the percentages 
 on this page should be used ; thus, in 4, the analysis may be : 
 "l§o=the required number; AS = ^%% ; t^tt = ^n of ^5, or ^; 
 igo = 100 times -J, or 50." And in 8 : " \^^ — the required num- 
 ber. $40 = i§5; ^^^ = ^^^ of $40, or 32/; |n = 100 times 32/, 
 or $32." 
 
 63 A7isivers: 1 308^ 2155| $6493 $380800 $12006.25. 
 2 £42500 1739^^^ tons 2409||tons 365 da. 19405|. 3 $11552.94 
 $8183^ $14028.55+ $8728.88+. 4 88|. 5 214|. 6 420. . 
 7 822|. 8 405f 9 345. 10 105. 11 10800. 12 120. 
 13 111^. 14 2741^ 15 28000. 16 252. 17 H 18 100. 
 19 93.98+. 20 $2517.482+. 21 $2500. 22 1225 bii. 
 $2858.33^. 23 $113.35. 24 $265.07. 25 $155928.88 + . 
 26 531i. 27 $4.04, cost per bbl. 
 
 The following form of written analysis for these exercises is 
 suggested : 
 
 21 10 = cost of house. 
 
 1^6 :=: selling price of house = $400. 
 
 , . , $400 
 T^o of cost = ^^. 
 
 ,nn f ^ $400 X 100 ^...Q„ 
 
 |o of cost = — j-^ = $o44.83. 
 
 64 The miscellaneous exercises on this and the following pages 
 of the section involve all the principles of percentage which have 
 been taught. Do not give the pupils any direct assistance in their 
 solution ; nor should any assistance be given until the pupils have 
 made a strong effort to do the work by themselves. If the condi- 
 tions of a problem are not understood, make them clear by judicious 
 questioning. 
 
VI. 65] teachers' manual. 135 
 
 65 Ajiswers: 1 352.83^. 2 39 /j. 3 306. 4 1.41. 
 5 42 lb. 2f oz. 6 1%. 7 0.525. 8 100%. 9 180%. 
 
 10 0.01G65. 11 4900. 12 25. 13 |2.247. 14 4if%. 
 15 TSlTj^ men. 16 3.G9G. 17 774|f. 18 55^^\%. 
 
 19 87^%. 20 3140f. 21 5375. 22 14904. 23 1100. 
 24 83(;Ubii. 25 tIst- 26 74/3^^%. 31 ^236^ $1111^. 
 32 $8250 $18250. 33 1^/ per lb. $22.50 entire gain 
 
 20 lb. for $1. 
 
 66 Ansirers: 1 2634.016. 2 $5.40. 7 $3 40%. 8 15% 
 gain 15% loss. 9 6.62+%. 13 65ia-f/. 15 $300. 
 
 67 Ansicet-s: 1 $8888| $13333^. 2 361-6/. 3 5023.14|f. 
 
 4 Lose $50. 5 $653.33^. 6 $129.16f. 
 
 68 Ansivers: 1 30%. 2 $4. 3 $13403/^. 4 $6.11^. 
 
 5 $970.31. 6 $160. 7 $10. 8 2^%. 9 $1400 $7000 
 $3500 $14000 $9333^ 10 $118.81^. 11 $1237.50. 
 12 $6000. 13 $2115 pre. $2500, Lon. $3333^, Hon. 
 $2000, Boston $2166|, KY. 
 
 69 A?isiaers: 1 Snm paid, $7500 greater. 2 $12828.58. 
 3 $115, pre. 4 $356.25, cost of insurance. 5 $8.10, com. 
 
 6 $37, com. $1443, returned. 7 $40 collected 25%. 
 8 $400, cost. 9 $20, com. 10 $1969.80, cost. 11 $3900, 
 cost. $29^, com. 12 9.36%. 
 
 70 Ansivers: 1 60%. 2 108 lb. 64± lb. 3 $5.80 on 
 $1000 $.0058 on $1. 4 $240. 5 4000 1b. 7 82f%. 
 8 2000 lb. wood 37| lb. ashes. 9 4400.97+ lb. 10 83^%. 
 
 11 65f\%. 12 $56. 13 6%. 14 $672 left $11.76, int. 
 
 71 Four of the most important kinds of business papers are 
 presented in this section, and the teacher should in as practical 
 and attractive way as possible lead the pupils to be familiar with 
 their form and to be able to write them without assistance. Trans- 
 actions which are most likely to occur, and which involve the 
 making of bills, promissory notes, receipts, and personal accounts, 
 should be given in addition to those here given. Printed blanks 
 
136 GRADED ARITHMETIC. [VI. 72 
 
 may be provided for the pupils to fill out. The pupils, however, 
 should not depend upon these forms, but learn to write them with- 
 out aids of any kind. It will be noticed that two forms of bills 
 are given, either form being allowable, except for bills for services, 
 which should be in the form given on page 72, except that the 
 word ''For" instead of ''To" may be used if desired. 
 
 73 Answers: 2 Balance due, $186.40. 3 Balance due, $2.68. 
 73 Answers: 1 Amount of receipt, $153.75. 4 Balance, $9.89. 
 
 75 Answers: 2 Balance due from K., $1.91. 3 Balance due 
 to H., $0.26. 
 
 76 Answers: 1 Amount due Jan. 24, 1894, $177.6.3. 4 Amount 
 of bill, $340.55 Cash payment, $169.81. 5 Balance due Parker, 
 $5.02 Balance due Dexter, $1.39 Balance due the merchant, 
 $2.59. 
 
 Explain in 4 the custom of giving credit, that is, of giving a cer- 
 tain time in which payment may be made. Also explain the custom 
 of making a discount Avhen money is paid before it is due. 
 
 It would be interesting for members of the class to open an 
 imaginary account with each other, using slips of paper for money, 
 and making out the necessary papers that pass between them. 
 
 77 Answers: 1 $7.00. 2 $76.50. 3 34f doz. 4 138iiyd. 
 5 3 da. 6 Hi da. 7 $2.70. 8 1079 pesos. 9 $669. 10 Mobile, 
 92304000 cu. in. 399584 -F gal. ; St. Paul, 40032000 cu. in. 173299- 
 gal.; Boston, 66816000 cu. in. 289247- gal.; Chicago, 46368000 
 cu. in.; 200727+ gal. 11 27^ lb. 
 
 Several of these problems should be performed orally. 6 will 
 doubtless cause some difficulty. Ask such questions as the follow- 
 ing if necessary: "At the end of three days how many days would 
 it take 9 men to finish it? Suppose all had left but one man, 
 how many days would it have taken to finish the work? If it 
 takes one man so long, how long would it take 4 men ? " Give 
 similar exercises varying the conditions. 
 
VI. 78] teachers' manual. 137 
 
 78 Answers: 1 19+ da. $19.96. 2 43 bbl. 166^^ lb. 3 $153. 
 4 $324.63. 5 S140.63. 6 $27.34. 7 $43.20. 8 i of a pint. 
 9 $54,085. 10 26y2^bags. 11 £16 8s (9^0- 12 $3430. 
 13 119 T. 1531 lb. 14 $21.78. 15 $141.86. 16 194^| ft. 
 17 8 ft. wide. 
 
 The 6 mo. in 13 is to be regarded as 182^ days. 17 is a prac- 
 tical problem in finding the greatest common divisor. 
 
 79 A7isu-e)-s: 1 6 A. 74 rd. 93.5 sq. ft. 2 |i. 3 $1.52 
 $18.18f. 4f|^cd. $5,188 + . 5 67|fgal. 562^1b. 6 S18.487f 
 8 50.2656 sq. rd. .31416 A. 9 12/^ cu. ft. 10 250 lb. 11 1333^ 
 lb. 12 1130.976 cu. ft. 13 21120 ^ 211.2" 21120^X 14 $41.60 
 $78.40. 
 
 In 7, counting Sundays, the answers would be : 40033200 copies, 
 and $800064 ; not counting Sundays, the answers would be : 
 34329840 copies, and $686596.80. 
 
 80 Ansicers: 1 $1402.50. 2 $37.50 per mo. 3 $571.10. 
 4 $1.74. 5 63 bu. 6 $198.45. 7 1§| mi., or 1 mi. 223 rd. |-ft. 
 8 $17. 9 138JV bu. 10 840 lb. 11 $320 $9120. 12 405 
 steps. 13 ll^fif f- Ha. 381,Wu- 1* $8.802|. 15 $1152. 
 
 81 A7isu-ers: 1 63.1504+ bu. 2 51.130+ bu. 3 88 furrows 
 511 A. 4 $12 less. 5 $2398.50. 6 84000. 7 27yV^. 
 8 133.67+ oz. 33.41+ oz. 16.7+ oz. 9 $10. 10 8800. 11 $.02 
 80%. 12 7i m. past 2 p.m. 13 41 yd. 15 in. 14 2.581 m. 
 15 $36666|. 
 
 The answers to 8 are found on the supposition that 1 cu. ft. of 
 water weighs 1000 oz. Let the pupils perform the problem, reck- 
 oning 1 lb. of water as measuring 27.7 cu. in. This is a more 
 accurate measurement. 
 
 82 Ansu-ers: 1 $21600. 2 $28.95. 3 $.12 per gal. 
 4 $15.39. 5 10000. 6 3200 lb. 7 5333^ lb. 8 36if cu. ft. 
 0^^^ T. 9 124.2 T. 10 131|f panes 132 posts. 
 11 20106.24 sq. rd. 8293.824 ft. 12 863. 13 $1080. 
 
138 GKADED ARITHMETIC. [VI. 83 
 
 83 Ansivers: 1 75/. 2 100%. 3 $133^ 26f. 5 300 
 sq. ft. 6 $16000. 7 560.2+ rev. 8 9^\ 6O/5V 9 118 qt. 
 boxes. 10 80 rails 4224 rails. 11 173 pills. 12 $1 per 
 crate. 13 $24000. 14 $33.68. 15 $358.06+ 6tVW4V5*V% 
 $53369557.71. 
 
 Physicians' prescriptions are frequently made in the manner 
 indicated in 11. The given quantity should be read 8 drams and 
 
 2 scruples. 
 
 84 A7isivers: 1 $10440.80. 2 969i bu. 3 $1.80. 4 Mon- 
 day Wednesday. 5 $1.15 gain. 6 $3.38§f. 7 $1620. 
 8 10890 sq. ft. 121 ft. 9 $8,121. 10 800 da. 333^ da. 
 
 11 Cheaper by the dozen $1 cheaper. 12 13fi|J A. 
 13 $4142 gain. 
 
 85 Answers : 1 864 books. 2 6 widths 30 yd. 7 widths 
 37^ yd. 3 Apr. 10, 1876. 6 15 lb. 1 gal. 1 gi. 7 3 oz. 
 
 6 oz. 8 493i ft. 8/7 mi. 9 8 lots. 10 80 doses. 11 88 
 bottles. 12 1411 yd. 13 $2.19. 14 205 pills. 15 22.79 bbl. 
 16 86.78+ bu. 
 
 86 Ansivers : 1 $672. 2 $480. 3 1215 cu. ft. 4 243^ yd. 
 5 77^1 A. 6 48 rd. 7 $163.74+. 8 13|| cd. 9 27 b. 20 b. 
 
 10 12 da. Ida. 11 136 men. 12 l^\)u. 13^. 14 $30000 
 $64000. 15 $3.46. 16 $1309^^- 
 
 87 Ansivers: 1 19712 bricks 14601+. 2 316| rev. 34^^™!- 
 
 3 396 tiles. 4 i% of 5000. 6 5^^%. 7 Chicago ahead 
 211%. 8 $2812.50. 9 IIUI/- 10 ^O^a yd. 11 $14.91|. 
 
 12 $68.40. 13 1/0 tla. 14 $30000 D $2373^ E. 15 960 
 steps. 16 $4118.40. 
 
 88 Answers: 2 i^^ (whole, 3184). 3 lyW^ ft. '^i^S ^^^ 
 
 4 50166869 62633336 42.09%. 6 About ^\ About 2\ times. 
 
 7 389+ times. 8 4.38% 9.05%. 9 11.65%. 10 16.22%. 
 
 11 28.16 gal 12 420^5. 13 2017888§ bbl. 14 189177083^ 
 bu. 
 
VI. 89] teachers' manual. 139 
 
 89 The answers to these exercises will be approximate only, 
 and will therefore vary greatly. As the purpose of the work is 
 more to get the pupils to measure on the map and to learn valuable 
 information in geography than to acquire arithmetical knowledge 
 and skill, close, accurate work should not be insisted upon. Let 
 the pupils measure by the aid of the scale of miles to which the 
 map is drawn, and also by the aid of parallels and meridians. 
 
 90 A7iswers: 1 32G2S.1824 lb. 13111.9452 1b. 2.23998. 
 3 294148.008 lb. 374520 lb. 4 588060 lb. 5 30000 lb. 
 6 3 oz. 17 pwt. 12 gr. 7 34 ft. 8 339.87"'. 9 1699.35'" or 
 337.89257 rd. 10 1 mi. 237.52+ rd. 11 37.88 sec. 
 12 202.73 rds. 
 
 SECTION IX. 
 
 NOTES FOR BOOK NUMBER SEVEN. 
 
 "For general suggestions as to the kind of work peculiar to 
 advanced grades in Arithmetic, teachers are referred to the " Note 
 to Teachers" on pages iv and v, particularly to what is said of 
 rules, definitions, and explanations of problems. In these forms 
 of expression special attention should be given to accuracy of 
 statement, and by accuracy is meant not merely a close adherence 
 to any one particular form, but an exact expression of a logical 
 process. Loose statements which earlier in the course were allowed 
 to pass should now be challenged and corrected. The pupils should 
 be led by judicious questioning to detect inaccuracies of their own 
 as well as of others' statements. 
 
 A rule in Arithmetic is a general statement of the method of 
 solving a given class of problems. A rule should therefore be 
 derived from the process of solution, and, as far as possible, be 
 made by the pupils themselves. The rule thus made by the pupils 
 may be amended by means of criticism and comparison with a 
 correct form. 
 
140 GRADED ARITHMETIC. [Vll. 
 
 A definition is a general statement of the peculiar and charac- 
 teristic properties of the subject defined. An analysis of the subject 
 to be defined should be made, and from the analysis the definition 
 should be deduced, if possible, by the pupils, with such correction 
 as they may be able to make. For example, if it is desired to 
 teach the definition of addition, the teacher should lead the pupils 
 to perform or to recall the process of addition, and in that process 
 to see that they have found a number which contains as many 
 units as all the given numbers taken together. After being told 
 that this process is addition, the pupils will say : " Addition is the 
 process of finding a number which contains as many units as all 
 the given numbers taken together." On being told that the number 
 found is called the sum, they will correct the definition given, and 
 say : " Addition is the process of finding the sum of two or more 
 numbers," If this definition is still thought to be incomplete, 
 attention may be called by examples to the fact tliat only numbers 
 of the same kind can be added, as, 8 pounds and 6 pounds. 8 
 pounds and 6 ounces might be put together to express a denominate 
 number, as 8 lb. 6 oz., but such putting together is not addition. 
 The pupils will therefore further correct their definition, and say: 
 "Addition is the process of finding the sum of two or more numbers 
 of the same kind." 
 
 An explanation of a problem should be made in such a way as 
 to be easily understood, and be so comprehensive as to include a 
 statement of the process of solving the problem and reasons for 
 each step of the process. It is not advised that all or many of 
 the problems performed be explained, but as many of each class 
 of problems sliould be explained as to give assurance that they 
 are thoroughly understood. In this and the following book it is 
 recommended that the processes only of many of the problems be 
 indicated, either orally or in writing, it being assumed that the 
 mere computations may be accurately ])erformed. 
 
 In the work of previous books the attention of teachers has been 
 called repeatedly to the importance of illustrations by plans or 
 diagrams. The illustrations should be continued in connection 
 
VII. 1] teachers' manual. 141 
 
 with all difficult problems ; but for the solution of problems not 
 very difficult the pupils should not depend upon such helps, but 
 imagine and state the conditions as nearly as possible. If, how- 
 ever, their statements are questioned or are not understood, they 
 should be ready to make the needed illustrations. 
 
 1-6 The solution of these review problems will give the pupils 
 little difficulty, if the previous work has been well done. When- 
 ever difficulty is found, lead the pupils by questions to see exactly 
 what is required, and what steps are necessary in the solution. For 
 example, if the pupils find difficulty in performing 7, page 3, ask 
 the questions: ''Before you can find the worth of the remainder, 
 what must you find ? How can that be done ? " If the pupils 
 choose to find in a roundabout way the number of gallons remain- 
 ing, let them so find it ; but after it is found ask them if there 
 is not a shorter way. Some bright pupil will see that 26 qt. = 6-^ 
 gal., and that this added to 13^ gal. is 19f gal., which subtracted 
 from 28^ gal. gives 8f gal. In such exercises as 8, page 3, let the 
 pupils make a diagram before the solution is made. 
 
 7 A7iswers: 1 /^ ^V% ?\ tVVtt Ml ^h- 2 $32 $184.73 
 $5.94 $140.04. 3 $52.50. 4 2 rd. 3 yd. 5 76 yd. 8 in. 
 6 155rd. l^ft. $927.61 $1902.27. 7 10 yd. 8 80 rd. 120 rd. 
 200 rd. 288 rd. 9. 11875 mi. .16098+ mi. 10 $1760. 11 $389.78 
 $406.74. 12 120 cups. 13 22 bu. 3 pk. 14 tIt tVt ^i 
 ^3^2^7^ ||5. 15 $50 $112.50 $9.15 $185.38. 
 
 In the solution of 11, lead the pupils to find the number of 
 cubic inches or feet that a cental or hundredweight occupies. One 
 bushel, or 60 lb., occupies 2150.4 cu. in.; 100 lb. would occupy Yts' 
 of 2150.4, or 3584 lb. The solution of the first part of the exer- 
 cise would be indicated thus : 
 
 <37 X 19 X 6 X 1728\ 
 
 $1.15 X (' 
 
 2 X 3 X 3584 
 
 8 Answers : 1 201 A. 20 sq. rd. 2 11 A. 55 sq. rd. 181^ sq. ft. 
 3 5 A. 17642.'^+ sq.ft. 4 837^^:^ ft. 5 50.26+ sq. ft. 6 12.7 + A. 
 7 $1360. 8 926.25 ^ 9 $15200. 10 (a) $524.38; (b) $2855.25; 
 
142 GRADED ARITHMETIC. [VII. 9 
 
 (c) 159 posts; ((I) $57.04; (e) $31.07; (/) 1027^^ eu. yd. 11 36 
 ft. 12 5624.64 oz. 13 5j%% cu. ft. 1296 cu. ft. 14 llJ cu. 
 yd. l^^VA cu. yd. 15 3^%% cords. 16 41 ft. 2*f in. 
 
 9 Answers: 1 (a) 519 sq. ft.; (b) 177i sq.ft.; (c) 19f i sq. yd. 
 (<;) $16.54; (e) 38-1 sq. ft. ; (/) across the room, widtliwise. 2 99-i-J- 
 cu. ft. 4 6 ft. 5 6J| cords. 6 1615f f gal. 180 sq. ft. 
 7 $560 $120. 8 18150 cu. ft. $2637.42. 9 $264. 
 
 10 Answers: 1 4800. 2 35.2+ qt. 3 115f^|cu. ft. 4 $2.69 
 $1.35 $.68 $7.35 $24. 5 $3 $3.06 $2.76 $2.45. 6 130f 
 loaves 45.9+%. 7 166f % $.093|. 8 $14.24. 9 828.8 T. 
 417.2 T. 10 $5180. 11 405 bu. 12 3if i. 13 $40500. 
 14 T%% Ui $44.60. 
 
 11 Answers: 1 Amt. of bill, $383.20. 2 $6120. 3 12^^;^. 
 
 4 46|% lost. 5 12038- rd. 6 1 h. 6 min. 9 sec. 7 $418 
 $33,589+. 8 4|%. 9 73^ planks. 10 $19,125 $.149+. 
 11 220/^ yd. $326.11. 12 $1743.09. 
 
 12 Ansivers: 1 If ff da. 2 $4876. 3 5* mo. 4 $18,375. 
 
 5 2.75. 6 $14375. 7 25.43+ sec. 8 33J§ T." $67.86. 9 567| 
 cu.yd. 10 22ff%. 11 180 mi. 12 529.21b. 13 $64 $14.80. 
 14 $1,125 $3.60 $2.30. 15 $44.84. 
 
 13 Answers: 6 28|/. 7 9|/. 8 $15.64 $21.16. 9 $69.12 
 $453.12. 
 
 Before giving these exercises in Profit and Loss, it might be 
 well to tell the pupils something of the common practice of mer- 
 chants in receiving a profit for the trouble of buying and selling 
 goods, and to give a few examples. 
 
 14 Answers: 5 33^% 33^%. 6 11^%. 7 llJ% on sugar 
 16f% on kerosene. 8 77|%. 10 22|%. 11 U^\%. 
 
 Lead the pupils to see that the per cent of gain or loss depends 
 not only upon the amount of gain or loss, but also upon the cost. 
 This is illustrated in 5. In 9, the supposition is that the part 
 unsold is worth ^ of what was paid for the farm. In this case the 
 
vn. 15] teachers' manual. 143 
 
 answer would be 25%. Ask the question : " What was the per cent 
 of gain on the part sold (33^%)?" If necessary, lead the pupils 
 to use numbers as $2000, for the cost of the farm. The cost of 
 the part sold would then be f 1500 ; and if it were sold for $2000, 
 the gain Avould be 33^%. Again ask: "What would be the per cent 
 of gain if I should sell the other fourth at the same rate ? " 
 
 After the pupils are familiar with the process, the common frac- 
 tional part of the cost may be omitted, the per cent of cost being 
 found directly. 
 
 15 A7iswe)-s: 4 ICf/ 3^/. 5 |40. 6 $2.50. 7 $3.75.. 
 8 $90. 9 $5 $5.75. 10 $4f. 11 Lost $26f., 
 
 12 $16304.35. 13 $1.074§. 
 
 In explaining these problems, the pupils should begin with the^ 
 statement that the cost is called for, and that it equals i§^ of itself.. 
 The rest of the explanation may be given as indicated in the 
 written analyses, the reason for each step being added ; thus, in 
 3 : '( $360 = selling price = |-§§ of cost, ^-i^ of cost = ^i^ of 
 $360 or $3. $3 = t1o of cost, laa of cost = 100 times $3 or 
 $300." 
 
 After the pupils are familiar with the solution of this kind of 
 problems, the use of hundredths as given in the analysis may be 
 omitted, and simpler fractions may be substituted ; thus, in 11 
 the solution might be : 
 
 $200 = 125% or -| of cost of 1st horse, 
 f of $200 = $160 = cost of 1st horse. 
 
 $200 = 75% or f of cost of 2d horse. 
 
 of $200 = $266f = cost of 2d horse. 
 
 $66f , loss on 2d horse — $40 gain on 1st horse = $26f = net loss. 
 
 Problems may also be performed by a short process on a line ; 
 thus, in 12 : 
 
 $10000 XlOO X3 , n -u . 
 — — ^^ = cost of ship. 
 
 Pupils should always be able to explain problems performed in 
 such a way. The explanation of the above solution would be, " If 
 
144 GRADED ARITIOIETIC. [VII. 16 
 
 the loss was 8%, the selling price would be 92% of the cost. 
 $10000, the selling price = 92% of the cost of f of the vessel. 
 1%=^!^ of $10000, found by dividing by 92, and 100% = 100 
 times this quotient. $10000 -^ 92 X 100 = cost of f of the vessel. 
 ■^ of the vessel cost ^ of this sum, found by dividing by 2, and f , 
 or the cost of the vessel, is 3 times this quotient." 
 
 16 Answers: 6 25% gain. 7 86^/. 8 30/. 9 72 baskets. 
 10 $3 $2.72t\. 11 48§|% gain. 
 
 Lead the pupils to define these terms in the manner indicated 
 on page 140 of the Manual. The following definitions may assist 
 teachers in securing concise and comprehensive statements : 
 
 The base in Profit and Loss is the number of which the per cent 
 of gain or loss is to be found. It is usually the cost. 
 
 The rate per cent is the number of hundredths that is taken of 
 the base. 
 
 The percentage is the number found by taking a certain per cent 
 of the base. The percentage in Profit and Loss is the amount of 
 gain or loss. 
 
 The amount is the number which includes the base and percent- 
 age, and is the selling price in Profit and Loss when the percentage 
 is the amount of gain. 
 
 Tlie remainder is the difference between the base and percentage, 
 and is the selling price in Profit and Loss when the percentage is 
 the amount of loss. 
 
 The rules called for in 3 to 5 should be deduced by the pupils 
 from the corresponding processes, and may be as follows : 
 
 To find the gain or loss, multiply the cost by the rate per cent. 
 
 To find the rate per cent, divide the gain or loss by the cost, 
 carrjnng the division to hundredths. 
 
 To find the cost, divide the selling price by 100, increased by 
 the rate per cent of gain, or decreased by the rate per cent of loss, 
 and multiply by 100. 
 
 17 Ans^vers: 1 $11095.20 154^\,%. 2 G8^^% 40f% 5^^%. 
 3 45«3% loss. 4 $1820 30%. 
 
VII. 18] TEACHERS* MANUAL. 145 
 
 Some of the exercises from 5 to 20 will have two sets of answers 
 according as the percentage is gain or loss. Let the pupils make 
 these problems, having the percentage either profit or loss as they 
 choose. 
 
 18 The explanation included in 1 may have to he enlarged or 
 extended by the teacher. Possibly other examples than these given 
 will assist the pupils to understand more fully the business of 
 commission. Besides giving the names of the consignor and con- 
 signee, the pupils should be led to answer in general that <'the 
 Consignor is the person who sends the goods to another," and " the 
 Consignee is the person to whom the goods are sent." 
 
 The person for whom a buyer or seller of goods acts is the 
 PrincijKil. 
 
 The person who is authorized to buy or sell goods for another is 
 called an agent. 
 
 The allowance made to an agent for buying and selling goods is 
 the Commission. 
 
 Attention should be called to the fact that the base in com- 
 mission is the cost or selling price, according as the agent is a 
 buyer or seller. 
 
 19 Amwers: 3 1840. 4 $2500.00. 5 13947.33. 6 $49.07. 
 7 21%. 8 $2125. 9 $5625.20. 10 $27.60. 11 $880. 
 12 10279.50. 
 
 The pupils should be able to tell after reading over a problem 
 whether the base is given or not. Custom varies somewhat as to 
 Avhat the base should be for estimating the commission in case of 
 a selling agent who pays expenses of cartage, storage, etc. When 
 not otherwise specified the selling price is regarded as the base. 
 
 ^30 Answers: 4 $45. 5 $75 $2075 (if premium is counted). 
 6 $1500. 9 $79.50. 10 $2637.50 more. 11 $47.93. 
 
 Bring to the class insurance policies of various kinds. Blanks 
 of policies may be procured from insurance agents, or policies 
 owned by the parents of pupils may be borrowed. Explain all 
 the terms that are used in the policies, and the purpose and plan 
 
146 GRADED ARITHMETIC. [VII. 21 
 
 of carrying on the business of insurance. The methods of organ- 
 izing a joint stock company and some features of such a company 
 are shown on pages 49 and 50, Book VII. Teach, as previously 
 shown, the following definitions : 
 
 A iwlky is a written contract, insuring security against loss be- 
 tween two parties. 
 
 The 'premium is a sum paid for insurance. 
 
 The undenvriter is the insurer. 
 
 Refer to the fact that mutual companies usually charge a higher 
 rate of insurance than stock companies, but that the excess is off- 
 set by dividends declared. Give problems involving insurance in 
 both kinds of companies to ascertain which would involve the least 
 cost, taking into account the interest of the excess paid to a mutual 
 company. 
 
 21 Answers: 1 f%. 2 $1936. 3 $40. 4 $241.13. 
 
 5 $2666|. 6 $2842. 7 $30000. 8 2%. 9 $1000. 
 
 10 $2500. 11 $23985. 12 $1675. 13 $5 less. 
 
 23 Answers: 3 1^% $.0125. 4 $88.80. 5 $12 $.012 
 $40.80. 6 1tW5% $74.56. 8 $16,912 $718.76. 
 
 In the preliminary teaching lesson, use familiar examples, both 
 in showing the purpose of taxation and the meaning of the various 
 terms used. From recent reports of assessors, give actual facts of 
 the valuation and rate of taxation of the city or town in which 
 the pupils live. 
 
 Real estate is fixed property, such as lands and houses. 
 
 Personal properttj is property that is movable, such as money, 
 stocks, cattle, furniture, etc. 
 
 A poll tax is a tax upon the person of a citizen. 
 
 Assessors are officers whose business it is to estimate the value 
 of property, and to apportion among individuals the sum to be 
 raised. 
 
 In finding the method of assessing a tax, get from the local 
 assessors the method tliat is actually employed by them. In some 
 places the assessors are paid a regular salary, and in assessing the 
 
VII. 23] teachers' manual. 147 
 
 tax their salaries are not taken into account. The amount of tax 
 uncollectible cannot be known before the tax is collected, but 
 frequently allowance is made, based upon the experience of past 
 years. The following is a practical rule for finding the rate of 
 taxation where the overlay is not more than 5*^ of the sum to be 
 raised on the property: 
 
 Divide the net sum to be raised by 96, and multiply the quotient 
 by 100. From this sum subtract the poll tax, and divide the differ- 
 ence by the amount of taxable property. If the overlay is greater 
 than 5^, the first divisor must be correspondingly less. 
 
 23 Answers: 2 $52.50. 3 $53.40. 4 $134.10. 5 $18.75. 
 
 6 $98.70. 7 $212763.72. 8 $4500. 
 
 In 7, deduct the uncollectible tax before finding the cost of 
 collecting or net proceeds. 
 
 34 Ansivers: 3 $188.31. 4 $46.19. 5 $100. 6 $656.64. 
 
 7 $338.62. 8 $145. 9 $42. 
 
 In teaching this subject show the reasons which governments 
 have for taxing imported goods. The following facts might be 
 presented in one form or another : 
 
 In addition to the usual duties collected at the custom-house, a 
 tax called tonnage is levied upon vessels according to the number 
 of tons they can carry. In some places duties are paid upon 
 articles of export. An invoice of goods, called also a manifest, 
 sometimes gives a wrong valuation. To guard against this fraud, 
 a certificate from the consul is sometimes required, stating that the 
 prices given in the invoice are the prevailing market prices. The 
 terms leakage and breakage are used to denote the allowance for 
 waste of liquids and for the breaking of bottles. Unless otherwise 
 specified, the value of a pound sterling in U. S. money is reckoned 
 at $4,866. 
 
 35 Ansivers: 1 $888. 2 $17.53. 3 $18.19. 4 $2881.25. 
 5 $97.18. 6 $1687.50. 7 $2.13. 8 $252.76. 9 300 lb, 
 10 $45.68. 
 
148 GRADED ARITHMETIC. [VII. 26 
 
 36 Anstvers: 1 $40.70 |2679.30. 2 4325 bu. 3^451.50. 
 4 $405292.50 $371518.12|-. 5 $60. 6 $500 loss. 7 $2209.80. 
 8 $1,008. 9 1%. 10 $12466.02. 11 $6155.85. 
 
 In 8, if 2240 lb. are taken as a ton the answer is 90/. 
 
 21 Answers: 1 $326.40 loss. 2 $3,323. 3 $.003473 
 
 $46.65. 4 $29.62. 5 $2866.20. 6 $838.50. 7 $659.50. 
 
 8 1% 3% 1% 9 $1900.59. 10 18 machines. 
 
 28 Answers: 1 U%. 2 $1680.22. 3 Silk @ $5.57 
 
 Ribbon @ $.295 Lace @ $1.03. 4 $5866f. 5 $644.32. 
 
 6 $18625. 7 li%. 8 $853.13. 9 A, $20.51 B, $23.67 
 C, $42.60 D, $12.62. 10 $695.76. 11 $5585.08. 
 
 39 These problems should be performed orally and by the 
 shortest way. Where the way is not indicated in the question, let 
 the pupils choose the method of solution. 
 
 30 Answers: 4 .142. 5 .091. 6 .1815. 7 .036^. 8 .0715. 
 
 9 .097. 10 .041|. 11 .063. 12 .036f. 13 .135^. 14 .062^. 
 15 .037^. 16 .211|. 17 .2461. 18 .093^. 19 .279. 
 20 .127f. 21 .2285. 22 .362|. 23 .178,1. 24 .109|. 
 25 $18.67. 26 $43.05. 27 $33.37. 28 $38.15. 29 $180.49. 
 30 $23.75. 31 $24.08. 32 $109.12. 33 $37.62. 34 $20.05. 
 38 $411.25. 39 $278.85. 
 
 While occasionally the method of casting interest is determined 
 by peculiar conditions, it is generally best to adopt and follow one 
 method. A method very generally pursued by business men is 
 what is called the 60-day method. By this method the interest is 
 ascertained first for 60 days by getting -j^i^ of the principal, and 
 for other times from this sum. For example, if it is required, as 
 in 25, to find the interest of $200 for 1 yr. 6 mo. 20 da. at 6%, 
 first divide by 100 to find the interest for 2 mo., and multiply that 
 sum by 9 to find the interest for 18 mo. 20 da. is ^ of 60 da., and 
 therefore divide the interest for 2 mo. by 3. Add results for the 
 final result. The solution will appear as follows : 
 
VII. 30] teachers' ISIANUAL. 149 
 
 $200.00 Principal. 
 
 2.00 Int. for 2 mo. 
 18.00 " " 18 mo. (2 mo. X 9) 
 .67 " " 20 da. (^ of 60 da.) 
 
 $18.67 " " 18 mo. 20 da. 
 
 Not all problems can be so easily performed by this method, but 
 a little practice will give the pupils skill in finding the desired 
 fractional parts. Thus, in 30, find the interest for 2 mo., then for 
 15 da. by dividing the interest for 2 mo. by 4, then for 2 da. by 
 dividing the interest for 00 da. by 30 (^ of ^i^). In 33, multiply 
 tlie interest for 2 mo. by 2 to find the interest for 10 mo. Divide 
 the interest for 2 mo. by 2 to find the interest for 1 mo. Get ^^^ 
 of the interest for 1 mo. to find the interest for 27 da. This 
 solution will appear as follows : 
 
 Interest of $632.29 for 11 mo. 27 da.? 
 6.322 Int. for 2 mo. 
 $31.61 Int. for 10 mo. (2 mo. X 5). 
 .316 3.16 " " 1 mo. (^ of 2 mo.). 
 9 2.84 " " 27 da. (j% of 30 da.). 
 
 2.844 $37.61 " " 11 mo. 27 da. 
 
 When this method is familiar to the pupils, only the times need 
 be written after each partial interest. The following solutions, 
 furnished by business men, illustrate this method of casting interest 
 actually used in business : 
 
 Int. of $2907 from Dec. 2, '90, 
 
 Int. of $4120 from Aug. 26, to May 27, '92, @ 5%. 
 '90, to May 27, '92, @ 5%. 17 mo. 25 da. @6% on $2907. 
 
 21 mo. 1 da. @ 6% on $4120. 
 ^ per mo. 
 
 20 mo. 
 
 412. 
 
 1 mo. 
 
 20.60 
 
 1 da. 
 
 .68 
 
 
 433.28 
 
 >ss ^ 
 
 72.21 
 
 $361.07 
 
 16 mo. 
 
 232.56 
 
 1 mo. 
 
 14.54 
 
 20 da. 
 
 9.69 
 
 3 da. 
 
 1.45 
 
 2 da. 
 
 .97 
 
 
 259.21 
 
 less ^ 
 
 43.20 
 
 $216.01 
 
150 GKADED AEITEBIETIC. [^'"H* 31 
 
 The above method is a slight modification of what is called the 
 .six per cent method. By this method the interest of $1 for the 
 •given time at 6^^ is multiplied by the number of dollars in 
 .the principal, and if the rate is other than 6%, the interest found 
 is increased or diminished as required. The interest of $1 for 
 1 yr. at 6% is $.06 ; for 2 mo., $.01 ; for 1 mo., f.OOo ; for 6 da., 
 $.001 ; and for 1 da., i of a mill. In general, the interest of $1 
 is found by taking six times as many cents as there are years, one- 
 half as many cents as there are months, and one-sixth as many 
 mills as there are days. By a little practice one is able to give the 
 interest of $1 at 6% for any time very quickly, and if the 6% 
 .method is pursued it will be good practice for pupils to do this. 
 
 31 Answers: 1 $213.33. 2 $111.88. 3 $61.25. 4 $20.35. 
 5 $29.89. 6 $66.72. 7 $46.39. 8 $8.15. 9 $19.70. 
 10 $138.03. 11 $8.59. 12 $25.95. 13 $792.25. 
 
 14 $868.49. 15 Apr. 10 and Oct. 10 of each year to Oct. 10, 
 1893, each payment being $43.75. The last payment, $1793.75, 
 including principal and six months' interest, was made Oct. 10, 
 1893. 16 $250.30. 17 $64 $.175+ $12.80 $25.95. 
 18 $8.17. 19 $20.96. 20 $35.55. 21 $34.14. 
 
 In finding the time between two dates for business purposes, 
 there is a difference of practice among business men. A common 
 practice in estimating time for casting interest, is to regard each 
 month as consisting of 30 days ; thus, in finding the time from 
 Sept. 20, 1891, to Aug. 1, 1892, it is 11 months to Aug. 20, and to 
 Aug. 1, it is 19 days less than 11 months, or 10 mo. 11 da. 
 
 At the time of purchase (Ex. 15), Mr. Brown received from Mr. 
 Smith a deed of the property, and at the same time gave to 
 Mr. Smith two papers signed by him, one a note promising to pay 
 Mr. Smith $1750 with interest at 5^ payable semi-annually, and 
 the other a deed, called a mortgage deed, providing that the 
 property shall come into full possession of Mr. Smith in case the 
 terms of the note are not fulfilled. This deed must be acknowl- 
 edged before a Justice of the Peace or Notary Public. 
 
VII. 32] 
 
 TEACHERS MANUAL. 
 
 151 
 
 To find the accurate interest, find the interest for the exact 
 number of days, regarding 365 days as the year. The following 
 solution of 18 will illustrate the method : 
 
 30 
 31 
 30 
 _1 
 92 
 
 rind the accurate interest of $540 for 92 da. at 0%. 
 108 
 ^$M X .06 X 02 
 
 ■m 
 
 73 
 
 108 
 
 .06 
 
 6.48 
 
 92 
 
 12 96 
 
 . 583 2 
 
 73)596.16(8.166+ 
 584_ 
 121 
 73 
 486 
 438 
 
 8.17 A 
 
 us. 
 
 480 
 
 = .1f;8.166+ 
 
 To find the exact number of 
 days between July 1 and Oct. 
 1, lead the pupils to say : " 30 
 more days in July, 31 in 
 August, 30 in September, and 
 1 in October. Adding, we 
 have 92 days." The explana- 
 tion of the problem is simply: 
 "Multiply $540 by .06 to get 
 the interest for 1 year. Divide 
 this product by 365 to find the 
 interest for 1 day, and multi- 
 ply by 92 to find the interest 
 for 92 days." 
 
 32 Answers: 3 $15.30 4% 6.84+%. 4 6% 5% 3|-+%. 
 5 4^% 5.611%. 6 4%. 7 13i+%. 8 4i%. 9 6.53+%. 
 10 6%. 11 4^f%. 12 3.45+%. 13 3.41+%. 15 $210. 
 5 yr. 16 1 yr. 2 mo. 12 da. 17 1 yr. 3 mo. 18 9 mo. 22 da. 
 19 1 yr. 7 mo. 1 da. 20 5 mo. 9 da. 21 1 yr. 6 mo. 21 da. 
 
 22 2 yr. 5 mo. 8 da. 23 4 mo. 24 3 yr. 3 mo. 17 da. 
 
 25 2 mo. 17 da. 26 4 yr. 8 mo. 20 da. 27 4 mo. 4 da. 
 
 28 3 yr. 11 mo. 9 da. 
 
 DraAV reasons as well as results from the pupils, leading them to 
 say in explanation of 1 : " Since at ojic per cent the interest for 
 the given time is $1 to yield an interest of $20, it will take as 
 
152 GRADED ARITHMETIC. [VII. 33 
 
 many per cent as $1 is contained times in $20, whicli is 20. 
 Answer, 20^." The form of explanation may vary somewhat ; for 
 example, the reason for dividing may be expressed : " To yield $20 
 interest the rate must be as many times one per cent as there are 
 times $1 in $20, which is 20." 
 
 The form of written solution of this class of problems may be as 
 follows : 
 
 6 Find the rate per cent. 
 
 $480 X .01 X 3 = $2.40 int. at 1%. 
 $9.60 -^ $2.40 = 4. 
 Answer, Aofo. 
 
 As in finding the rate per cent the interest at one per cent is 
 made the divisor, so in finding the time the interest for one year is 
 used in a similar way. The explanation of this class of problems 
 may be as follows : " Since in one year $400 Avill gain at the given 
 rate $20, it will take as many years for it to gain $60 as $20 is 
 contained times in $60, which is 3. Answer, 3 years." The 
 written solution of 17 would appear as follows : 
 
 $800 X .05 = $40 int. for 1 yr. 
 $50 -^ $40 = li. 
 
 Aiiswer, 1 yr. 3 mo. 
 
 The common custom is to drop the fraction of a day if it is less 
 than ^, and to add a day if it is ^ or more than ^. 
 
 33 Answers: 3 $266.67. 4 $150. 5 $810. 6 $1425. 
 7 $2306.49. 8 $1370.30. 9 $3378.64. 11 $4. 12 $16. 
 13 $620. 14 $1021.504+. 15 Int., $75.95 ; Amt., $915.95. 
 16 Time, 6 mo. 12 da.; Amt., $412.80. 17 Kate, 4|% ; Amt., 
 $350.60. 18 Prin., $713,958 + . 19 Time, 7 mo. 27 da.; 
 Amt., $8700. 20 Rate, 4.39+ % ; Amt, $1700. 21 Prin., 
 
 $360.54+ ; Amt., $387.04 + . 22 Int., $13,848 ; Amt., $294,248+. 
 23 Time, 9 yr. 2 mo. 10 da.; Amt., $1480.40. 
 
 Essentially the same principle is involved in the solution of 
 these problems as in the solution of problems in which the rate or 
 
VII. 34] teachers' :manual. 158 
 
 the time is sought. A principal of one dollar is taken when it is 
 desired to find the principal. The following written solutions of 
 6 and 12 will illustrate a good method : 
 
 Find the principal. 
 
 II X .01 -^ 2 = $.005 int. of $1 for 1 mo. 
 $.005 X 9 = $.045 '' " '' '- 9 mo. 
 $.005 -^ 5 X 3 =: $.003 " " '' " 18 da. 
 $68.40 4- $.048 = $1425. 
 
 Answer, $1425, principal. 
 
 Find the principal. 
 
 $1 X .04^ X 2|- = $.ll^ int. of $1 for 2 yr. 6 mo. at 
 
 $1.1125) $17.8000 (16 
 11125 
 
 
 
 
 
 6 6750 
 
 
 
 
 
 
 
 
 6 6750 
 
 
 
 
 
 
 
 Answer, . 
 
 ^16, principal. 
 
 
 
 
 
 34 Answers : 1 
 
 5 yr. 2i 
 
 yr. 3 yr. 1 mo. 15 
 
 da. 
 
 2 6% 
 
 3 
 
 1 yr. 8 mo. 4 
 
 16| yr. 25 
 
 yr. 14f yr. 
 
 5 59^ 
 
 1- 
 
 6 $36000 
 
 7 
 
 H%- 
 
 8 Jan. 
 
 1, 1897. 
 
 9 $652.63. 
 
 10 
 
 May 20, 1894 
 
 11 July 1, 
 
 1891. 
 
 12 $2500. 
 
 13 $14400 
 
 $21600. 
 
 14 4% 
 
 15 $375 loss. 
 
 In 15, by " average expense " is meant average yearly expense, 
 
 35 Ansxvers: 3 $100. 4 $100 cash. 5 $200. 6 394.088+. 
 7 $345,394. 8 $177,639. 9 $1578.947. 
 
 No new principle is involved in these problems. The present 
 worth is the principal, which, put at interest, will amount to a 
 given sum in the given time and rate. The explanation draAvn 
 from the pupils may be (3): "The amount of $1 for the given rate 
 and time is $1.05. It will require as many dollars to amount to 
 $105 as $1.05 is contained in $105, which is 100. Answer: $100." 
 
 36 Answers: 1 $294.12. 2 $21.06. 3 $123.53. 4 The 
 latter offer, $41.14 better. 5 Less than true value. G '3796 
 $.02 $808. 7 $657.64. 8 $1188.12 Less $1211.88+. 
 9 $588. 10 $22.50. 11 $450. 12 $745.29. 
 
154 GRADED AEITHMETIC. [VII. 36 
 
 Let the pupils see, in such problems as 3, that the answer given 
 is the present gain, the present worth of the note given being 
 regarded as the actual cost in cash. 
 
 In finding the true value of the note (6) March 1, the principal 
 is found, which, put at interest, would amount to $800 in 1 mo. 
 at 6^. Sometimes more than one discount is made from the list 
 price. Show this to the pupils by giving an instance, as, for 
 exainple : The list price of tacks is $16 per cwt., the cash price 
 being at a discount of 25% and 10% of the list price, or, as the 
 seller might say, 25 and 10 off. To find the cash price, first deduct 
 25% from the list price, and then 10% of this result. Thus, 25% 
 offl6 = $4; 116 — 14 = $12; 10% of $12 = $1.20; $12 — $1.20 = 
 $10.80, cash price. It might be well to give several problems of 
 this kind to the pupils. 
 
 37 Anstvers: 1 $53.04, due Sept. 1. 2 $44, due Jan. 1, 1894. 
 3 $218.85. 
 
 Pupils should be led to write out fully and clearly an analysis 
 of each problem, and to give an explanation of each item. The 
 following written solution of 2 may suggest a good form for 
 teachers to present as the given questions are answered by the 
 pupils : 
 
 Principal, Jan. 1, 1893, $100.00 
 
 Interest to IVIay 1 (4 mo.), 2.00 
 
 Amount, 102.00 
 
 Payment May 1, 40.00 
 
 New i)rincipal, 62.00 
 
 Interest to Aug. 1 (3 mo.), .93 
 
 Amount, 62.93 
 
 Payment Aug. 1, 20.00 
 
 New principal, 42.93 
 
 Interest to Jan. 1, '94 (5 mo.), 1.07 
 
 Amount due, Jan. 1, '94, $44.00 
 
 38 Answers: 1 $254.44. 2 $112.81. 3 $255.30. 4 $88.24+. 
 5 $132.20. 6 $258.42+. 7 $164.29+. 
 
Til. 38] teachers' ISIANtTAL. 155 
 
 The method above shown of finding the balance due on a note 
 "when partial payments have been made, is quite generally followed 
 by business houses. It is in the main tlie method adopted by the 
 United States Supreme Court, and is therefore called the United 
 States rule. One feature of this rule provides that, "if any pa}^- 
 ment be less than the interest, the surplus of interest must not 
 be taken to augment the principal, but interest continues on the 
 former principal until the period when the payments, taken to- 
 gether, exceed the interest due, and then the surplus is to be applied 
 toward discharging the principal." Tell this to the jKipils, and 
 show how it would be applied in 7 if the payment made Sept. 25 
 had been $10. Show how it would be applied if the September 
 and January payments had been $10 and $6 respectively. Do the 
 same with other exercises. 
 
 Wlien partial payments are made on accounts and on notes 
 running a short period of time, the interest is sometimes computed 
 by the following ru.le : Find the amount of the principal from the 
 time it begins to draw interest to time of settlement. From this 
 amount subtract the sum of the payments and their interests from 
 the time of payment to the time of settlement, and -the difference 
 will be the balance due. This rule is sometimes called the mer- 
 chants' rule. The following solution of 5 is by this rule : 
 Principal, on interest from April 1, '92, $300.00 
 Interest to March 1, '93 (11 mo.), 16.50 
 
 Amount, 
 Payment June 1, '92, 
 
 Interest to March 1, '93 (9 mo.). 
 Payment Oct. 1, 92, 
 
 Interest to March 1, '93 (5 mo.). 
 Payment Dec. 1, '92, 
 
 Interest to March 1, '93 (3 mo.). 
 Payment Jan. 1, '93, 
 
 Interest to March 1, '93 (2 mo.). 
 Sum of payments and interests. 
 
 Balance due March 1, '93, $131.90 
 
 $316.50 
 
 $50.00 
 
 
 2.25 
 
 
 60.00 
 
 
 1.50 
 
 
 30.00 
 
 
 .45 
 
 
 40.00 
 
 
 .40 
 
 
 
 184.60 
 
156 GRADED ARITHMETIC. [VII. 39 
 
 Ask the pupils to perform 6 and 7 by the above method, and 
 compare answers with the answers obtained by the United States 
 rule. 
 
 39 Answers: 2 $195.06+ $197.49 + . 3 $1427.33. 
 4 $747.34+. 5 $248.94 + . 6 Simple int. $30.14 greater. 
 
 In 1, draw from the pupils the fact that compound interest is 
 interest on interest. Savings banks generally allow compound 
 interest on deposits. In 3, interest is calculated for five separate 
 times, the last being 8 mo. 
 
 The answers to question calling for blanks to be supplied are as 
 follows: 1 yr. @ 4%, 1.04; 1 yr. @ 5%, 1.05; 3 yr. @ 3%, 
 1.092727; 3 yr. @ 5%, 1.157625 ; 4 yr. @ 4%, 1.169859 ; 5 yr. @ 
 3%, 1.159274; 5 yr. @ 5%, 1.276282; 6 yr. @ 3%, 1.194052; 
 6 yr. @ 4%, 1.265319; 7 yr. @ 3%, 1.229874; 7 yr. @ 5%, 
 1.4071; 8 yr. @ 4%, 1.368569; 8 yr. @ 5%, 1.477455; 9 yr. @ 
 3%, 1.304773 ; 10 yr. @ 5%, 1.628895. 
 
 40 Answers: 2 $1458. 3 $166.46. 4 $636 $710.16 $101 
 $730.20. 5 $1786.20 $1257.64. 6 $2668.73. 
 
 In the instance given in 1, the account stands as follows, if no 
 interest is paid until maturity of note : 
 
 Principal, $100.00 
 
 Total simple interest (3 yr. @ 6%), 18.00 
 
 Interest on 1st annual interest (2 yr.), $0.72 
 
 " 2d " " (1 yr.), 0.36 1.08 
 
 Amount due at the end of 3d year, $119.08 
 
 The amount by compound interest for the same time and rate is 
 $119.10, only 2 cents more than the amount by annual interest. 
 In 4, by the process indicated, $609.16 would be due Sept. 1, 1886. 
 If the note were paid at this time, $101 should be deducted from 
 $609.16. The simplest solution of this problem is to find the 
 amount due as if no payment had been made, and then subtract 
 $100 plus the interest of $100 from July 1, 1886, to time of settle- 
 m«nt. 5 should be performed in the same way. Let the pupils 
 
VII. 41] TEACHEKS' MANUAL. 167 
 
 perform 6 in two ways, one by the United States method, and the 
 other as if the interest at 6%, payable annually, was unpaid until 
 maturity of note. The answer given above is obtained by the latter 
 method. 
 
 41 Answers: 1 $9000. 2 1 mo. 1 da. 12 yr. 6 mo. 3 $100 
 gain. 4 3 yr. 11 mo. 1 da. 2 yr. 11 mo. 8 da. 5 80%. 
 
 6 $545.85. 7 3 yr. 5 mo. 22 da. 8 Cash offer, $45.45 cheaper. 
 9 $64800. 10 $1243.92. 11 First by 2/^%. 12 $2790.62. 
 
 13 $1115.075 $1104.50 $1115.36. 
 
 42 Answers: 1 Tlie second. 2 $13333^ $35555f 
 3 $1949.85. 4 5i%. 5 $15238^^- 6 $444.05. 
 
 7 $5214.285. 8 $13.50 9 16/ 18/. 10 22-| yr. 
 lllfyr. 11 $1459.44. 12 $480000. 13 $887.97. 
 
 14 $307.80. 
 
 43 National Banks, before issuing any bills of their own, are 
 obliged to })ut a certain portion of their capital into government 
 bonds, which are held in trust for the banks by the Treasury 
 Department at Washington. The bills " payable to bearer " " on 
 demand " are printed at Washington, and cannot exceed in amount 
 90% of the face value of the bank's deposit in the Treasury 
 Department. The stockholders of a bank are its owners, and may 
 share all profits and losses as partners in proportion to the number 
 of shares they hold. If a bank lessens its capital, or is unable 
 from any cause to pay its debts, the stockholders are obliged to 
 make up the deficiency not exceeding the par value of the shares 
 which each stockholder owns. The books and papers of National 
 banks are periodically inspected by bank examiners appointed by 
 the government. The board of directors, besides electing the 
 president, cashier, and, if needed, other officials, are supposed to 
 attend to the general management of the bank. 
 
 The income of a bank is derived from the amount received in 
 discounting notes, in collection of notes, in the interest on govern- 
 ment bonds and other securities, and in profits of buying and 
 selling property. The expenses are the taxes it has to pay to the 
 
158 GRADED ARITHMETIC. [VII. 44 
 
 general government and to the state and local governments, and 
 the running expenses, such as rents, salaries, etc. 
 
 The above facts, and others which may be gained from bank 
 authorities, should be given to the pupils. 
 
 44: The method of borrowing money at a bank should be shown 
 in a familiar and practical way, the example here given being 
 taken as a basis. By questioning and telling, the following facts 
 should be brought out : It is supposed that the payee goes to the 
 bank to-day to get the note discounted. He has first to indorse 
 the note, that is, to write his name on the back of it. By this 
 indorsement he guarantees the payment of the note. If the officials 
 of the bank require greater security, another person must indorse 
 the note. The payee receives from the bank in cash $6.15 less 
 than $300. This is called the avails, and is really less than the 
 true value of the note. If, instead of presenting the note to-day, 
 the payee should present it one month later, the bank would have 
 to wait but 2 months before receiving $300. It therefore discounts 
 the interest of $300 for 2 months and 3 days. At the end of 3 
 months, or within 3 days after maturity of the note, Mr. Douglas 
 is supposed to pay $300 to the bank. If lie does not do this after 
 due notice, a notice called a " protest " is sent by a notary public 
 to the payee, and, if he does not pay, tlien to the other indorser. 
 The indorser must pay the amount at once, so that the bank 
 may not lose, but he can liold the maker responsible for payment 
 afterwards. 
 
 Blanks illustrating the business of a bank can be easily obtained 
 and shown to the pupils. The officials will be willing to give all 
 needed information. 
 
 The following definitions may be evolved from actual examples 
 given : 
 
 The avaih or j^^oceeds of a note is the sum that the bank pays 
 for it after deducting the interest on the face of the note for the 
 time before maturity. 
 
 The face of the note is the sum of money written in the body of 
 the note. 
 
Vll. 45] teachers' ]MANITAL. 159 
 
 The maturity of the note is the time at which the note is due. 
 
 Bank Discount is the allowance made to a 1)ank for tlie payment 
 of a note before it becomes due. 
 
 The niakevoi the note is the person who signs his name to the note. 
 
 The 2Hii/ee is the person to whom payment of money is promised. 
 
 The indorse)' is the person who writes his name on the back of 
 the note. 
 
 Days of grace are days after a note is payable before it is legally 
 due. If the last day of grace falls on Sunday or on a legal holi- 
 day, the note must be paid on the day previous. In the state of 
 New York no grace on notes, etc., is allowed by law. 
 
 Practice varies in estimating the time of maturity and period of 
 discount of notes. Some banks in estimating the time of maturity 
 of a note due in 90 days regard the time as three calendar months, 
 but more commonly such time is taken in exact days. In discount- 
 ing notes some banks always reckon the time before maturity in 
 exact number of days, others in months and days, and still others 
 in exact number of days if the luiexpired time is 60 days or less, 
 and in months and days if for more than 60 daj's. The problems 
 of this section are reckoned on the basis of months if the time 
 given in the note is months, and of exact number of days if the 
 time given is in days. For example, the maturity of a note dated 
 July 15, due in 3 months, is Oct. 15 -|- 3 days of grace, or Oct. 18. 
 If the note should read "90 days after date," the note would fall 
 due Oct. l."> + 3 days, or Oct. 16. If the first of these notes is 
 discounted at date of note or any time subsequently, the time of 
 discount is reckoned as months and days. The discount of the 
 second note is reckoned in exact number of days. 
 
 45 .l».s-?m-.s-; 1 $493.87. 2^590.70. 3 $594.75. 4 $787.60. 
 5 $444.75. 6 $296.85. 7 $670.65. 8 $1267.63. 9 $182.48. 
 10 $343.67 $346.53. 11 $674.05. 12 Nov. 15, 1893 $628.01 
 $620.44 $612.53. 13 $452.92 $455.21. 
 
 4.Q Answers : 1 ^100 $1200. 2 $750. 3 $1000. 4 $510.46. 
 5 $1755.91. 6 $561.59 $28.57. 7 $1012.42. 8 $8.20 
 $591.80. 9 $607.85 $7.85. 10 $2000 $1979. 11 $6.35 
 
160 GRADED ARITHMETIC. [VII. 47 
 
 $1253.65. 12 «5482 $344.82. 13 $600 $597,525. 14 $5.51 
 $420.79. 15 $15.97. 
 
 The explanation of such problems as 1 may be : " If a note for 
 one dollar will give proceeds of $.9895, it will take a note for as 
 many dollars to give proceeds of $98.95 as $.9895 is contained 
 times in $98.95, which is 100. A7isiver: $100." 
 
 47 The deposit slip, with the bank book, is passed to the teller, 
 who writes in the bank book the amount of deposit. The check 
 by which $80 is drawn out is signed by the person who wants the 
 money, and is made payable to himself or to his order. The check 
 given to James Smith may pass through various hands until finally 
 it comes to the Emporia Bank. It would be well for the teacher 
 at this point to explain the bank check and to show its usefulness 
 to business men and others. To do this it will be necessary to 
 show the necessity of first making a deposit of money at the bank, 
 and to tell exactly hoAv it may be used in paying persons who live 
 at a distance as well as persons near the bank upon which the check 
 is drawn. An example may be given of James Robinson living in 
 Albany, and wishing to pay a creditor in Omaha. If he has a 
 deposit in a bank in Albany, he has only to fill out a blank similar 
 to the blank check represented on this page and send it to the 
 creditor in Omaha, who carries it to a bank there for payment. If 
 he is known, he will receive the face value of the check ; if he is 
 not known, he must be "identified" by some one before he can get 
 the check "cashed." The banks all have dealings with one another, 
 and therefore tlie clieck is taken by the Omaha bank and returned 
 to Albany. If j\Ir. Robinson has no deposit in a bank, he may 
 make a special deposit of a sum equal in amount to the face value 
 of the check which he wishes to send to Omaha, or he may for 
 a small amount buy a "cashier's check" for the desired amount, 
 payable to the order of Mr. Robinson or to his creditor. If to 
 the former, Mr. Robinson writes, "Pay to the order of John Brown," 
 and signs his name. Find out from bankers the exact working of 
 a "Clearing House," and explain the process to the pupils. 
 
 48 Answers: 1 $683.41. 2 $803.08. 
 
VII. 49-50] teachers' manual. 161 
 
 Cooperative banks have been very extensively established, and are 
 likely to be more common in future. For this reason, and because 
 it will be useful for people to know their purpose and methods of 
 operation, they should be the subject of study in school. Their 
 nature and methods vary so much in different parts of the country 
 that it has not been thought best to introduce in the text-book 
 more than a statement of their general features. It would be well 
 to get from the officers of the nearest cooperative bank its method 
 of operation and give to the pupils problems whicli are likely to 
 come to patrons of tlie bank, both as lender and as borrower. 
 Describe fully its advantages, and make the problems as practical 
 as possible. 
 
 49-50 In naming the different kinds of corporations, only 
 those that are best known need be given, such as, banks, railroads, 
 and large manufactories. It would be well while discussing these 
 subjects to show the pupils the various documents that are used, 
 such as, chartei-, bond with coupons, and bond certificate ; also 
 certain blanks which banks and other business houses have to fill 
 out. Lead the pupils to see clearly the difference between bonds 
 and stocks, bondholders and stockholders, interest and dividend. 
 Answers to the most difficult questions given on these pages, and 
 some others that ought to be answered, are contained in the fol- 
 lowing brief statement : 
 
 A corporation is an organization which has a charter to do 
 business under the laws of the state. Its capital is divided into 
 shares owned by the stockholders, each of wliom holds a certificate 
 of stock stating the number of shares owned by that person. The 
 stockholders elect a board of directors, who attend to the general 
 management of the corporation. The par value of tlie shares is 
 the original or face value. If for any reason the shares are sold 
 for less than their par value, they are said to sell below par, or 
 at a discount ; if for more than their par value, they are said to 
 sell above par, or at a premium. The profits of the business are 
 divided among the shareholders in proportion to the number of 
 shares that each one holds. These profits are called a dividend. 
 
162 GRADED ARITHMETIC. [Vll. 51 
 
 If for any reason a corporation desires to increase its capital, it 
 can make assessments on the stockholders, that is, oblige them 
 to pay a certain amount for each share owned, or it can borrow 
 money and issue promissory notes called bonds, which are given 
 to persons as security for the money they lend. They are made 
 payable at a certain time or at regular times, and bear a specified 
 rate of interest. Attached to each bond are printed slips stating 
 that so much interest will be paid the bearer at a certain time. 
 These slips are called coujwns, and can be cut off as they are 
 needed. If the business of the corporation is good, and the in- 
 terest promised is large, the bonds are likely to sell above par, or 
 above their face value. If the interest promised is very low, or 
 if there is danger of not pa^dng the interest, the bonds are likely 
 to sell below par, or at a discount. 
 
 A stock-broker is a person who buys and sells stocks and bonds. 
 If this is done for other people, he charges a commission called 
 brokerage. The brokerage is generally estimated upon the par 
 value of the stocks or bonds. 
 
 51 Ansicers: 1 $1080 $60. 2 $2210 $100. 3 25 shares 
 $4762.50 4/^\%. 4 $4400 $300 (S^j%. 5 $60 on a thousand 
 5.28%. 6 4 80. 7 8. 8 Stock *§% better. 9 600 shares. 
 10 118375. 117^% 6i% 4|% 5f%. 12 Stock 1% better 
 Stocks lif§% better. 13 $18987.50. 
 
 52 Ansivers: 1 80 120 133^ 75 70. 2 4%. 3 6%. 
 4 25%. 6 $1672.50. 7 $5400. 8 $8010. 9 $5835. 
 10 $26193.75. 11 $10003.13. 12 $3069.50 $15347.50. 
 13 7% bonds. 14 Bonds about 1% better. 15 $2(100. 
 
 53 Explain to the pupils the great convenience of the use of 
 drafts or checks, and how drafts are exchanged in different parts 
 of the country. Show that business firms and private individuals 
 in one part of the country owe parties in another part, and that by 
 a system of exchange these debts may be paid without sending the 
 money. Illustrate this by the example indicated on this page. 
 Mr. Macomber, who lives in Chicago, owes John Smith of Boston, 
 
VII. 54] teachers' manual. 163 
 
 $480.90. He goes to Mr. Upliam, a banker in Chicago, and buys 
 of him an order, called a draft, on Thos. S. Appleton & Co., 
 bankers of Boston, who have an account with Mr. U})ham. Mr. 
 j\[aconiber writes on the back of the draft, " Pay to the order of 
 John Smith," and signs his name. This he sends to Mr. Smith, 
 who draws the money from Appleton & Co. at the given time. If 
 the draft had read ten days after sight, it should first be presented 
 to Appleton & Co. for acceptance, which is done by their writing 
 the word '* accepted" and signing the firm's name across the face 
 of the note. Ten days, + 3 days of grace, after acceptance the 
 money is payable. This is called a "time draft," to distinguish 
 it from a " sight draft," which is payable " at sight." 
 
 If in two different parts of the country, as, for example, Boston 
 and San Francisco, the amount of drafts drawn in each city on 
 the other is nearly equal, the buyer of a draft in either city will 
 have to pay little or nothing beyond the face value. But if the 
 amount of drafts in Boston on San Francisco is far in excess of 
 the amount drawn in San Francisco on Boston, then the buyer of a 
 draft in Boston will have to pay a premium to pay for the risk 
 and expense of sending money to San Francisco, and the buyer of 
 a draft in San Francisco may be able to get it at a discount. 
 
 54 A7iswers: 1 $1015 fl004..50. 2 f 1465.50. 3 $609. 
 4 $831.60. 5 $531.06. 6 $732.17. 7 $879.30. 8 $762.02. 
 9 $900.10. 10 $804.42. 11 $609.14. 12 $5940.27. 
 
 13 $8074.83. 14 $1962.17. 15 5iio%. 
 
 In 1, the time draft costs less than the sight draft, because the 
 banker has the use of the money for 63 days, including 3 days of 
 grace. 
 
 $1015 — $10.50, the interest of $1000 for 63 days = $1004.50. 
 
 Another method of solution is to find the cost of $1 of exchange 
 and multiply by $1000, thus : 
 
 $1 + $.015 = $1 .015 $1,015 - $.0105 = $1.0045 
 
 $1.0045 X 1000 = $1004.50. 
 
164 GRADED ARITHMETIC. [VII. 55 
 
 Another solution of 9 may be as follo^\'s : $1 of exchange can be 
 bought for $1.01. As many dollars of exchange can be bought for 
 $1000 as $1.01 is contained times in $1000. $1000 ^$1.01 
 = 990.10. Ansiver, $990.10. 
 
 The time draft mentioned in 9 would cost .9945 of the face. 
 $1000 -^ .9945 = face. 
 
 The solution of 10 is as follows : The cost of $1 of exchange is 
 found by subtracting $.0155, the interest on $1 for 93 days, from 
 1000 = $9845. To this add $.01, the premium = $.9945. Since 
 one dollar of exchange can be bought for $.9945, as many dollars 
 of exchange can be bought for $800 as there are times $.9945 in 
 $800, etc. 
 
 55 Answers : 2 $488.25. 3 $334.62. 4 £367 145. 5d. 
 
 Show the pupils that essentially the same principle is involved 
 in foreign exchange that has already been explained in domestic 
 exchange. The intrinsic value of the standard coins of two 
 countries determines the par of exchange between those countries. 
 
 Exchange is at a premium or discount when the market value is 
 more or less than the par value. Three bills of the " same tenor 
 and date " are sent by different mails, so that if one is lost the 
 other may be presented. 
 
 In the bill of exchange given, Mr. Mason is the buyer or remitter 
 who indorses the bill payable to the order of Mr. Brown. As in 
 domestic exchange, foreign bills or drafts are made payable both 
 on time and at siglit. Time drafts are usually quoted at a less 
 price than sight drafts. 
 
 56 Answers: 1 £587 15s. 2d. 2 $1679.69. 3 $1104. 
 4 14392 francs. 5 $2367.47. 6 35 bonds, $95.85 surplus. 
 7 $24.50. 8 7%. 9 $11520. 10 $454.78. 11 $325.03. 
 12 $84.50. 13 $2 more for buying land. 14 $162.49. 
 
 The note refcri-cd to in 11 is supposed to bo discounted at a bank. 
 
 51 Answers : 1 $2VJ.rA) 8%. 2 U\i%. 3 $787.60 6.09+%. 
 4 7.15+% 0.09+% 9.21+% 4..55+%. 5 18.4+%. 6 $304.50. 
 7 $1267.55. 8 $723.84. 9 $679.61. 10 $406.20 $402.21. 
 
VII. 58] 
 
 TEACHERS MANUAL. 
 
 165 
 
 58 It will not be necessary for the pnpils to perform many of 
 the problems included in 1. A few answers are given below : 
 
 (1) 1500 Atch., f!1038.75. (3) 2000 C. B. & Q., $2250. (8) 2000 
 Or. Sh. Line 6's, 111811.33. (16) 42 B. & M., f 5717.25. (24) 5 
 Conn. Eiver, )i;i075.63. (26) 20 Mex. Cen., $126.25. (37) 235 
 Bos. & Mont., $4141.88. (46) 96 Pullman, $13240. 
 
 59 Answers: 1 Balance March 31, $58.27. 2 Balance Jan. 31, 
 $182.21. 
 
 If preferred, the two sides of tlie cash account may be placed on 
 the opposite pages of a blank book. 
 
 60 A7isivers : 1 Balance, $221.52. 3 Balance, $1049.43. 
 
 The form suggested in 4 is given below. It has the advantage 
 over other forms in the ease with which the balance is found, and 
 also in the greater convenience for making out monthly bills. 
 
 L. P. WALKER. 
 
 189Jf. 
 
 
 
 
 
 Jan. 1 
 
 To Balance, 
 
 
 $18 
 
 60 
 
 u a 
 
 By Cash, 
 To Balance, 
 
 
 10 
 
 00 
 
 
 8 
 
 60 
 
 « s 
 
 " 10 gal. K. Oil, 
 
 11 
 
 1 
 
 10 
 
 « 5 
 
 " 2 Ih. Coffee, 
 
 32 
 
 
 H 
 
 te i( 
 
 " 3^ lb. Cheese, 
 
 12 
 
 
 J^ 
 
 " 9 
 
 " 3 gal. Molasses, 
 
 4^ . 
 
 1 
 
 26 
 
 (( « 
 
 '' 1 hhl. Flour, 
 
 
 5 
 
 60 
 
 
 17 
 
 62 
 
 (( (( 
 
 Btj Cash, 
 To Balance, 
 
 
 15 
 
 00 
 
 
 2 
 
 62 
 
 (jl— 0'^ In uuiking the cash account, use the form already 
 given. The other accounts may be given in the simple form given 
 above. The poultry account should be made precisely as the per- 
 sonal accounts are made, debiting all items of expense incurred 
 
166 GEADED ARITHMETIC. [YII. 63 
 
 on account of poultry, and crediting all items of income from the 
 poultry. 
 
 63 A7iswers : 1 8 mo. 18 mo. 200 mo. 2 18 mo. 60 mo. 
 800 mo. 3 In 2 mo. 11 da. 4 2 mo. from date. 5 82^ da. 
 6 3 mo. 3 da. 7 Oct. 14. 8 2 mo. 1200 mo. li mo. 
 
 Explain to the pupils that the average of payments is the time 
 at which several items of debt due at different times may be paid 
 without gain or loss to either party. 
 
 A form of written solution for 6 is as follows : 
 2 mo. X 420 = 840 mo. 
 
 7Ut da. 
 
 3 mo. X 315 -- 
 
 4 mo. X 525 : 
 
 = 945 mo. 
 = 2100 mo. 
 
 365 X 30 
 1260 
 
 10950 , 
 = 1260 ^"- 
 
 1260 
 
 ) 2885 (2 mo. 
 2520 
 
 
 
 365 Jnsiver : 2 mo. 8 da. 
 
 In finding the equated time, when in the result the fraction of 
 a day is one half or more, reckon it one day; if less than ^, dis- 
 regard it. 
 
 In 9, assuming that the two sums owed were paid at the time 
 the first sum was due, the manufacturer would lose the use of 
 $1200 from June 20 to July 5, or 15 days, which is equivalent to 
 the loss of the use of $1 18000 days, or tlie loss of the use of 
 $2000 9 days. Therefore, for the manufacturer not to lose any- 
 thing, the $2000 should be kept by him 9 days after June 20, or 
 until June 29. 
 
 04 Answers : 1 June 16. 2 Sept. 26. 3 Feb. 11. 
 
 When each of two parties is a debtor, one way of finding the 
 equated time of paying the balance due is by computing the in- 
 terest. In 4, it might be assumed that the account given is the 
 account of each pupil with J. li. 8i)aulding. The explanation may 
 be drawn from the puj^ils by such questions as are given in the 
 exercise. Afterwards the pupils may be led to explain the problem 
 as follows : 
 
 '■'■Xn this account Spaulding owes me $100, due July 1, and .*!50, 
 due July 21. I owe Spauldiug $100^ due July 11. I wish to find 
 
VII. 64:] 
 
 TEACHERS MANUAL. 
 
 167 
 
 the time when he may pay the balance ($50) due me without gain 
 or loss to either of us. Wi will assume July 21 as the time of 
 payment of all the sums. If he pays me the $100 due July 1 
 on the 21st of July, I shall lose the use of $100 from July 1 to 
 July 21, or 20 days. The interest of $100 for 20 days at 6% is 
 33-J/. If he pays me July 21 $50, which is due July 21, I shall 
 neither gain nor lose. If I pay him July 21 $100, which is due 
 July 11, I shall gain the use of $100 10 days. Tlie interest of 
 $100 for 10 days is lOf /. If payment of the balance due is made 
 July 21, my net loss is 16f /. That I may not gain or lose, he 
 must pay me the $50 as many days before July 21 as it takes $50 
 to gain lOf/, or 20 days. 20 days before July 21 is July 1, when 
 the balance should be paid." 
 
 Spaulding owes me : I owe Spaulding : Assumed time of 
 
 $100 due July 1. $100 due July 11. payment, July 21. 
 
 50 " Jidy 21. 
 Loss, int. of $100 20 da. =^-- .33^. Gain, int. of $100 10 da. = .16f . 
 
 Net loss, .16|. 
 Interest of $50 1 da. = .008^. .16| -^ .008^ = 20. 
 
 July 1 — 20 da. = July 1. 
 
 By introducing the last question a new problem is made. Assume 
 that the last bill of goods, like the others, is due in 1 mo. Let 
 Aug. 1 be the assumed time of payment. The written solution by 
 a good method may be as follows : 
 
 Due 
 
 Amount 
 
 Time 
 
 Interest 
 lost 
 
 Due 
 
 Amount 
 
 Time 
 
 Interest 
 gained 
 
 July 1 
 July 21 
 Aug. 1 
 
 $100 
 
 50 
 
 100 
 
 31 da. 
 
 11 da. 
 
 
 
 $.516 
 .091 
 
 July 11 
 
 $100 
 
 21 da. 
 
 $.35 
 
 
 250 
 100 
 
 .607 
 .35 
 
 
 
 $150 
 
 .257- 
 
 ■^ .0025 = 1 
 
 
 
 
 Aug. 1 — 10 daijs = Jul^ 2U 
 
168 GRADED ARITHMETIC. [VII. 65 
 
 Explanation : If $100 due me July 1 is paid Aug. 1, I lose the 
 interest of $100 31 da., or $.516. If $50 due me July 21 is 
 paid Aug. 1, I lose the interest of $50 11 da., or $.091. If I 
 pay, July 11, $100 that is due Aug. 1, I gain the interest of $100 
 21 da., or $.35. If all payments are made Aug. 1, my net loss is 
 $.257. That I may neither gain nor lose, he should pay the bal- 
 ance ($150) due me as many days before Aug. 1 as the number of 
 days it will take for $100 to gain $.257. Dividing by the interest 
 of $150 for 1 day the quotient is 10. 10 days before Aug. 1 is 
 July 21, the equated time. 
 
 Many problems in averaging accounts may be solved best by 
 reducing the gain or loss of $1 for a number of days or months, 
 and then dividing by the balance due. For example, 4 could be 
 solved as follows : 
 
 Spaulding owes me : _ r, i t 
 
 *100 due July 1. iT,'''T '"f,^ 
 
 50 " July 21. «100 due July U. 
 
 If settlement is made July 21, 
 
 I lose the use of I gain the use of 
 
 $100 for 20 da. = the use of $1 $100 for 10 da. = the use of $1 
 
 2000 da. 1000 da. 
 
 Net loss of use of $1 for 1000 da. 
 
 << " " " " $50 '' 20 da. 
 
 July 21 — 20 da. = July 1. 
 
 65 Armvers: 1 June 15. 2 Nov. 18. 3 Balance, $260, 
 due April 1. 4 Oct. 27. 
 
 These problems are essentially the same as the problem solved 
 above. The earliest date instead of the latest may be taken as 
 the assumed time of payment if thought best. 
 
 06 Answers : 1 June 4, 1893. 2 Balance due July 16, 1894. 
 3 June 13, 1894. 4 March 2, 1895 $891.09. 5 May 19, 1879 
 Balance due, $44.74. 
 
 67 A horizontal line is a line having the direction of a line 
 in the surface of still water. A vertical line is one that has the 
 
VII. 08] teachers' manual. 169 
 
 direction of a phimb line. These lines need be drawn only approxi- 
 mately in the right direction. Parallel lines may be drawn by the 
 aid of a square or ruler, care being taken to draw them so that 
 they shall be the same distance apart throughout their length. 
 Other ways of drawing parallel lines will be shown later. 
 
 A line may be divided into 2 equal parts by means of a measure. 
 Another way is to divide it with the aid 
 of compasses, as follows : With A as a '■■•l--''^ 
 
 centre, draAV the arc CD, and with Jj as a /l\ 
 
 centre, draAV the arc EF, cutting the arc / ; \ 
 
 CD. Unite the points of intersection of . /I • 
 
 ^ ... . . . d ; ' G : B 
 
 the two arcs, and the dividing line divides 1 T i " 
 
 the line AB at G into 2 equal parts. Each \ : / 
 
 of the two parts can be divided in the same \ ! / 
 
 way into 2 equal parts, thus dividing X 
 
 the original line into 4 equal parts. 
 
 68 Most of these exercises are a review of previous work. 
 
 09 The conclusion in 1, after several similar experiments, is, 
 that the sum of two sides of a triangle is greater than the third 
 side. The two ways of iinding by experiment the sum of the angles 
 of a triangle are : (1) Cutting off the corners of a paper triangle 
 and measuring, and (2) Measuring by means of a protractor. 
 
 Several conclusions may be made from the measurements made 
 in 17; such as : The two diagonals of a i:»arallelogram divide each 
 other into two equal parts. In a square the diagonals are equal. 
 In a square the diagonals are perpendicular to each other. In a 
 rhombus the diagonals are })erpendicular to each other. 
 
 70 Answers: 1 $80. 2 129600 sq. ft. 3 HI ^- 5 11^^^ 
 sq. rd. 6 T^V ^^- 9 2244 sq. ft. 10 21G00 sq. ft. 11 1 T. 
 1966 lb. 12 176.59+ ft. 13 85* rd. 14 19 planks $6.08. 
 
 In 9, the Avalk is siipposed to be on the border of the garden. 
 
 71 Answers: 1 23^ yd. 2 $3.30. 3 20^1^ sq. ft. 4 869.5 
 sq. ft. 5 7855.8 sq. ft. $116.26. 6 $46.78 $3.92 (full allow- 
 ance for openings) $24.75 460^^^ sq. ft. 8 118-^^^ sq. rd. 9 140 ft. 
 
170 GRADED ARITHMETIC. [VII. 72 
 
 For a good practical method of finding the number of rolls of 
 paper required for a room, see page 119 of the Manual. 
 
 73 Answers: 1 20000 sq. ft. $3443.53. 6 2 A. 4 sq. rd. 
 7 1 A. 126.2+ sq. rd. 11 435.6 ft. 
 
 The land taken by the railroad company in 1 is a parallelogram 
 whose base is ef and whose altitude is equal to the distance bd. 
 
 For solution of 2, see Book IV., page 85. 
 
 The dotted line called for in 8 is perpendicular to the base, since 
 the altitude is the perpendicular distance from the vertex to the 
 base. Let the statements be as concise as possible. 
 
 '73 Answers: 1 5e\^s(i. ixl. 1936 yd. 4.54.^^^^ £c. 4826| sq. 
 ft. 16 A. 3^ sq. ch. 45 A. 8J| sq. ch. 15f rd. 2 1200 sq. ft. 
 3 1212^ sq. ft. 4 1732^ sq. ft. 5 6383^ sq. ft. 6 1758f 
 sq. ft. 7 900 sq. ft. 8 920 sq. rd. 
 
 The line required to be drawn 60 ft. long in 7 is supposed to be 
 horizontal. 
 
 74 Answers: 2 8^ sq. ft. 3 22400 sq. ft. 46200 sq. ft. 
 5 2 inches. 6 Area of triangle, 2400 sq. ft.; of trapezoid, 2100 
 sq. ft. 7 1450 sq. ft. 
 
 The cut on page 12, Book VI., will suggest a solution for 1. 
 
 4 may be solved by construction, after transforming a trapezoid 
 
 7? -4- /) 
 into an equivalent rectangle ; also from the formula A = — - — X h. 
 
 At 
 
 75 Ansimrs: 1 6 sq. ft. 2 12600 sq. ft. 11 36.34 sq. ft. 
 14 25.1328 ft. 15 4.77+ ft. 16 314.16 sq. ft. 17 9.58+ A. 
 18 $4.36. 
 
 Suggestive helps to the solution of these exercises will be found 
 on pages 14-18 of Book VI. 
 
 Use protractor or compasses in marking off arcs of various 
 degrees. 
 
 A regular polygon is a polygon having equal sides and equal 
 angles. In drawing regular polygons, draw circles first and mark 
 off equal arcs of the desired length. 
 
VII. 76] teachers' manual. 171 
 
 Let the pupils divide paper circles into triangles before drawing 
 if they have not already done so (see page 18, Book VI.). 
 
 76 AnsH-ers: 1 502.656 sq. ft. 3 22 ft. 138.2304 ft. 175.9296 
 ft. 932.48 sq. ft. 4 $41.30. 5 492 A. 145.4+ s(i. rd. 
 
 In 1, it is supposed that the cow is tethered in such a way as to 
 enable her to graze at a distance exactly forty feet from the post. 
 ' Somewhat similar work with prisms is called for on page 19, 
 Book VI. For suggestions in teaching these forms, see Manual, 
 page 120. The following statements may be drawn from the 
 pupils : 
 
 The lateral faces of all right prisms are in the form of rectangles. 
 The bases of a triangular prism are triangles. The bases of quad- 
 rangular prisms are quadrilaterals ; etc. A jJcntaffonal j-jm-m is a 
 prism having pentagons for its bases. A rirjht prism is a prism 
 whose lateral edges are perpendicular to the bases. An oblique 
 2))'is>ii is a prism whose lateral edges are inclined to the bases. 
 
 The drawing on page 64, Book V., will suggest the kind of 
 work called for in 8. 
 
 77 Answers: 1 245 cu. ft. 2 1152 cu. in. 3 57f|| bbl. 
 8 20.5632 gal. 9 8 ft. 6.1241- in. 21 ft. 9.4379 in. 10 79.5872 
 
 sq. ft. 
 
 The altitude of a jjp'amid is the perpendicular distance from the 
 vertex to the base. The slant height of a reyular pyramid is the 
 perpendicular distance from the vertex to any side of the base. 
 (First show that a regular pyramid is a pyramid which has a regu- 
 lar figure for the base, and the vertex directly above the center 
 of the base. All pyramids referred to in this book are regular 
 pyramids.) Teach and ask the pupils to describe triangular, quad- 
 rangular, pentagonal, hexagonal, and octagonal pyramids. 
 
 For suggestions in answering 5 and 6, see page 20, Book VI. 
 
 The altitude of a cone is tlie perpendicular distance from the 
 vertex to the base. The slant height of a cone is the distance from 
 the vertex to any point in the circumference of the base. (All 
 cones referred to in this book are right circular cones. A right 
 
172 GRADED AKITHMETIC. [VII. 78 
 
 circular cone is a volume that may be generated by tbe revolution 
 of a right-angled triangle about one of its legs as an axis.) 
 
 78 Ansum-s: 3 128 cu. ft. 4 93537284 cu. ft. 5 386.34 
 sq. in. 6 Of in. 7 15.919 cu. ft. 8 1.005' J T. 9 141.372 
 sq. ft. 
 
 Paper will liave to be pasted over the joined edges of these 
 models, that tliey may liold tlie sand. Let the pupil measure or 
 weigh the sand as accurately as possible. The rules which they 
 should discover are as follows : The volume of any pyramid is 
 equal to one third of the volume of a prism having the same base 
 and altitude. The volume of a cone is equal to one third the 
 volume of a cylinder having the same base and altitude. From 
 these rules may be derived the following : The volume of a pyra- 
 mid or of a cone is equal to ^ of the area of the base X the 
 altitude. Tlie convex surface of a pyramid is the sum of the 
 lateral faces. The convex surface of a cone is its curved surface. 
 Lead the pupils to give the rule for finding tlie area of the convex 
 surface of pyramids and cones. The rule is : Multiply the per- 
 imeter of the base of the pyramid or cone by one half the slant 
 height. 
 
 79 Answers: 4 « 414.6912 sq. ft.; Z* 779.1168 sq. ft.; c 1381.32225 
 sq. ft. 5 a 175.92 cu. in.; h 2865.13 cu. ft. 8 1256.64 sq. in. 
 2827.44 sq. in. 5 sq. ft. 13.3+ sq. in. 534.83+ sq. in. 9 135.0899 
 cu. in. 4.1888 cu. ft. 904.7808 cu. in. 10 $1884.96. 11 512 
 bullets 110609 bullets. 
 
 The frustum of a ^^i/ra^nid is the portion of a pyramid included 
 between the base and a plane made by cutting tlie pyramid through 
 the lateral faces, parallel to the base. 
 
 The slant heiglit of the frustum of a pyramid or of a cone is 
 the perpendicular distance between the sides of the upper and 
 lower bases. The total surface of a frustum is found by adding 
 the surface of the bases to the lateral surface. The lateral surface 
 is found by multiplying half the sum of the perimeters of the bases 
 by the slant height. 
 
VII. 79] teachers' ilANUAL. 173 
 
 The volume of the frustum of a pyramid or cone is equal to the 
 difference between the original pyramid or cone and the pyramid 
 or cone cut off. If the height of the original cone is not given, 
 the volume of the frustum is found by the following rule : To the 
 sum of the squares of the diameters of the two bases add the prod- 
 uct of the two diameters. Multiply this sum by the product of 
 i of the height and .7854. 
 
 With an orange or wooden ball teach the following definitions : 
 
 A sphere is a solid bounded by a curved surface, every point oi' 
 which is equally distant from a point within called the centre. 
 
 The diameter of a sphere is a straight line passing through the 
 centre and having its extremities in the surface. 
 
 A great circle of a sphere is a section made by cutting the sphere 
 through the centre. 
 
 A good method of finding the surface of a sphere is to compare 
 it with the surface of a right cylinder whose heiglit and diameter 
 of base are exactly equal to the diameter of sphere. By wrapping 
 the sphere and cylinder with narrow waxed tape, and, after un- 
 wrapping them, comparing the amounts of ta})e, it will be observed 
 that the areas of the surface of the sphere and the lateral surface 
 of the cylinder are alike. Since the lateral surface of a cylinder is 
 found by multiplying the circumference of the base by the altitude, 
 it will be seen that the surface of a sphere is found by multiplying 
 the circumference by the diameter. 
 
 To find the volume of a sphere, compare the contents of the 
 sphere with the contents of the enveloping cylinder. This may 
 be done by making a ball which will exactly fit a cylinder. Fill 
 the cylinder with water and immerse the ball. Compare the depth 
 of water before the ball was immersed with the depth of water 
 after the ball is taken out, and it will be found that tAvo thirds of 
 the water has been displaced. Therefore, the volume of a sphere 
 is equal to two thirds of the volume of a cylinder whose diameter 
 of base and height are equal to the diameter of sphere. From 
 this fact may be derived the rule : Multiply the cube of the diameter 
 by .5236 j or, by conceiving a series of pyramids formed from the 
 
174 GRADED ARITHMETIC. [VII. 80 
 
 sphere, having their vertex at the center, it will be seen that the 
 volume of the pyramids composing the sphere must equal the surface 
 of their bases multiplied by -^ of their radius ; hence the rule : 
 To find the volume of a sphere, multiply the surface by ^ of the 
 radius. 
 
 80 Answers: 3 90° 60° 30°. 5 69j mi. 6 12430.8 mi. 
 7 598.1 mi. 2392.4 mi. 10765.8 mi. 21531.6 mi. 8 About 
 69^ mi. 9 47° 3250| mi. 10 43° 2974i mi. 
 
 If necessary, review previous work in circles and degrees, pages 
 14, 15, Book VI. 
 
 81 Ansjvers: 2 240' 14400". 3 640'. 4 15600". 5 495' 
 405' 10' 950'. 6 4200" 15630" 22320" 900" 10". 7 600" 
 
 12" 7650 
 
 '. 8 
 
 14800" 
 
 192" 2340' 
 
 " 21" 
 
 9 J3 7 ° 
 ^360 
 
 1 
 
 80 
 
 180 
 
 10 lOf 
 
 .061°. 
 
 11 .75/g° .01i°. 
 
 12 
 
 10° 50' 40". 
 
 
 13 36° 
 
 31' 30". 
 
 14 9° 
 
 34' 44 
 
 19 23° 
 
 10'. 
 
 20 12° 20'. 
 
 
 21 89° 
 
 55' 22". 
 
 
 
 
 
 
 
 
 S2 Ansivers : 
 
 1 6°. 
 
 2 llf da. 
 
 3 73|a h. 4 2° 
 
 31' 
 
 '. 5 7' 
 
 59f||". 
 
 9 360^ 
 
 ' 15° 
 
 15'. 10 3° 
 
 45°. 
 
 13 40 min. 
 
 26 min. 
 
 I h. 41 min. 20 sec. 14 Slower 4 min. 15 11.03 p.m. 1.57 p.m. 
 16 132° 10' E. 
 
 83 Ansivers: 2 N.Y., 7 h. 3 min. 44 sec. a.m. Chi., 6 h. 9 min. 
 28+ sec. A.M. KO., 5 h. 57 min. 23+ sec. a.m. London, 11 h. 
 59 min. 37 sec. a.m. Paris, 12 h. 9 min. 33+ sec. p.m. Boston, 
 7 h. 15 min. 46 sec. a.m. Wash., 6 h. 51 min. 57 sec. a.m. Eome, 
 12 h. 49 min. 4+ sec. p.m. Berlin, 12 h. 53 min. 34+ sec. p.m. 
 San F., 3 h. 50 min. 26+ sec. a.m. Cal., 5 h. 53 min. 20 sec. p.m. 
 St. L., 5 h. 59 min. a.m. 3 Rome, 9 h. 57 min. 50+ sec. p.m. 
 Paris, 9 h. 17 min. 22+ sec. p.m. Cal., 3 h. 1 min. 22 sec. a.m. 
 5 15°. 6 21° 15'. 7 5° East. 8 9 h. 3 min. 8/^ sec. 9 5 h. 
 50 min. 8| sec. 10 20°. 11 93° 20' West. 12 13° 22' 6". 
 
 84 Ansivers: 2 Ind., 12 o'clock. Pitts., 1 o'clock p.m. Denver, 
 
 II o'clock A.M. San. E., 10 o'clock a.m. N. 0., 12 o'clock. Boston, 
 1 o'clock p.m. Omaha, 12 o'clock. 3 San Jose, 1 o'clock p.m. 
 
VII. 85] TEACaERS' MANUAL. 175 
 
 Portland, 4 o'clock p.m. Springfield, 3 o'clock P.M. 4 Charleston, 
 10 h. 15 min. a.m. Sac, 7 h. 15 min. a.m. 5 Cin., 22 min. 19 sec. 
 Wil., 11 min. 30 sec. St. P., 12 min. 20 sec. 7 1 min. 16 sec. 
 past 4 28 min. 20 sec. past 1. 
 
 85 The relation or ratio of 6 blocks to 2 blocks is found by- 
 dividing 6 blocks by 2 blocks = 3. The ratio is expressed thus: 
 6 blocks : 2 blocks. The antecedent of this ratio is 6 blocks, and 
 the consequent is 2 blocks. 
 
 Lead the pupils to make the following statements from examples 
 called for in 11 : 
 
 If both terms of a ratio are multiplied by the same number, the 
 terms are larger, but the ratio remains unchanged. 
 
 If both terms of a ratio are divided by the same number, the 
 terms are smaller, but the ratio remains unchanged. 
 
 86 A proportion is the equality of ratios. The first and fourth 
 terms of a proportion are called the extremes. The second and 
 third terms are called the means. From observation the pupils 
 will discover and state the conclusion that the product of the 
 fourth term by the number of units in the first term equals the 
 product of the third term by the number of units in the second 
 term ; or, more briefly, the product of the extremes equals the 
 product of the means. Prom tins fact it will appear that the prod- 
 uct of the extremes divided by a mean equals the other mean ; 
 and that the product of the means divided by an extreme equals 
 the other extreme. 
 
 87 Answers: 4 « $8 ; h $90; c f .25 ; d $.52|; e 12 lb.; 
 / 16| lb.; rj $625; h $2.13^; i $.47^; ./ $18.75. 5 a $.10-}^; 
 b $.13^; c $li; d $1.50; e $.15; /$200; r/ 31b.; h 6 ex.; i $.07|; 
 j $.60. 6 12/. 7 $4.80. 8 $.80. 9 3 doz. 10 $.50. 
 
 The last five problems on this page and all the problems on the 
 five following pages are intended to be performed by analysis and 
 by proportion. The solution by analysis should be either oral 
 or on a line. The solution by proportion may be as follows (6) : 
 The ratio of 2 apples to 6 apples must be the same as the cost of 
 
176 GRADED ARITHMETIC. [VII. 88 
 
 2 apples to the cost of 6 apples. It is required to find the cost 
 of 6 apples ; therefore we make 4 cents, the cost of 2 apples, the 
 3d term. Since the greater the quantity the greater the cost, we 
 make 6 the 2d term and 2 the 1st term. The proportion is 
 2:6 = 4/ : x. Multiplying the 4 by 6 and dividing the product 
 by 5, we have for the fourth term 12. Answer^ 12/. 
 
 88 Answers: 1 20 apples. 2 $80. 3 25 bu. 4 $14.25. 
 5 66| mi. 6 $22.40. 7 $1120. 8 145^ A. 9 1488 bu. 
 10 450 rd. 11 2400 T. 12 5.']^ da. 13 4i da. 14 $13230. 
 
 15 20 ft. 16 lllj. 17 480 men. 18 29iV mi- 19 l^-^A ^^■ 
 
 The solution on a line may be as follows (13) : 
 2 
 
 % da. X 12 _ ^^ , 
 
 ^TT = V =" H da. 
 
 5 
 
 If 12 men mow the meadow in 8 days, it will take 1 man 12 times 
 as many days to mow it as it takes 12 men, and 20 men will mow 
 it in one twentieth as many days as it takes 1 man. Multiplying 
 and dividing, we have an answer of 4| days. 
 
 89 Answers : 1 461-/3 mi. 2 $2346|. 3 13| yd. 4 12|/. 
 5 $1105.92. 6 GGOOO times. 7 277 da. 18 h. 40 min. 8 258000 
 men. 9 lOia rolls. 10 18f Ih. 11 1152 mi. 16f h. 
 12 $677.00+. 13 $32000. 14 444^ lb. 15 1 lb. 14+ oz. 
 
 90 Answers: 1 4/ and 8/. 2 30 yr. and 10 yr. 3 ftr- 
 4 A, $8; B, $16. 5 A, $12 ; B, $10 ; C, $8. 6 11/, 22/, 33/. 
 7 $186|, $280, $373^. 8 A, $1008 ; B, $504 ; C, $168. 9 12, 
 8, 4. 10 315 and 525. 11 18 rd. and 24 rd. 12 Each daughter, 
 18000 ; each son, $6000. 13 18 rolls. 14 3389|a lb. 15 6| A. 
 
 16 4352 lb. 17 125 lb. 
 
 In sucli problems as 6, the whole number will be the sum of 
 the parts 1, 2, 3, and may be represented by 6. The proportions 
 will be, 6 : 1 = 66 : 11 ; 6 : 2 = 66 : 22 ; 6 : 3 = 66 : 33. By 
 analysis the solution is, 66 = f ; | = 11 ; | = 22 ; | = 33. 
 
VII. 91] teachers' manual. 177 
 
 In 9, reduce the fractions to a common denominator and nse the 
 nnmeratorSj as in 6. 
 
 91 Ansivers : 1 A, ^^00 B, .flSOO. 2 A, 14800 P>,- f 2880 
 C, $4320. 3 A, $G35yV T., $204jf. 4 A, $1800 B, $2700. 
 5 Hall, $888 Eeed, $5i)2. 6 A, $3777?, B, $5(3G()| G, $7555^ • 
 7 A, $G26,2^ B, $1173M. 8 A, $()4,^, B, $43J.j C, $32^3. 
 9 A, $49t^V B, $30|!i. 10 A, $15600 B, $20400 C, $24000. 
 11 A, $3840 B, $4800 C, 3300. 
 
 In 7 to 11, make the conditions uniform before the proportion 
 is made. A's share of the work done was equal to the service of 
 192 men 1 day, and B's share was 360 men 1 day. A's share of 
 the contract money should be, therefore, i^f of $1800. 
 
 92 Answers: 1 5^ da. 2 2 gal. 3 9^ weeks. 4 320 mi. 
 21? da. 5 5 mo. 6 $240. 7 Hi da. 8 370i? rd. 9 272^12 
 I765L. 10 T)! rd. 11 8/ 2/ 6 of 40,^ and 4 of 50/. 12 3 of 
 6/ and 1 of 8/ 3 of 6/ and 5 of 8/. 
 
 These problems should be performed by analysis on a line, 
 
 rather than by compound proportion. The solution of 8 may be 
 
 as follows : 
 
 40 5 
 
 $(l) rd. X M X 50 -„ , , 
 n :r: — = 3 i ^K rd. 
 
 X$ X M 
 
 9 3 
 
 My answer is to be in rods, therefore 80 rods is the number to 
 work upon. 1 ]nan will build ^l as many rods as 18 men, found by 
 dividing by 18 ; 40 men will build 40 times as many rods as 1 
 man, found by multiplying by 40. If they can build so many rods 
 in 24 days, in 1 day they will build ^\ as many rods, found by 
 dividing by 24, and in 50 days they will build 50 times as many 
 rods as they build in 1 day, found by multiplying by 50. Simpli- 
 fying by cancellation wo have 372V ^'^■ 
 
 93 Nearly all of these problems should be performed orally. 
 
 94 A7isive)'s: 7 1988515.54 T. 8 311694000001b. 15584700 T. 
 9 $31684702.80. 10 W., 46.66% CI., 56.89% N.Y., 44.06% 
 
178 GKADED ARITHISIETIC. [VII. 95 
 
 Bos., 72.13% L., 39.34^0 Ph., 62.71% Bal., 25% S.L., 41.37% 
 Br., 62.71% Cin., 55.17% Pitts., 46.77% Cli., 47.36%. 
 
 95 . A7iswers : 1 257^7 ft. 2 ^hVo- 3 ^6 IQi 30. 4 66f% 
 33^% 400%. 5 5 3 2000 .002. 6 .08 .0003 .00001 
 2400. 7 200 bolts 4000 bolts 21100 bolts $3 $3.46 $.58. 
 
 8 $24.50 $58.19 $19.69. 9 $24.75. 10 $50.63. 11 $1513.51. 
 12 70%. 13 $10200. 14 95 shares ($235 left). 15 34.6+%. 
 16 52 shares ($80 left). 17 $88500. 18 $300. 19 8f %. 
 
 96 Answers: 1 116 bales. 2 $3037.50. 3 $7507.58. 
 4 $3461538.46. 5 $25.35. 6 $69.23. 7 42 shares. 8 28+%. 
 
 9 $7087.50. 10 $10675 $97.50 and $154.32. 11 $10008. 
 12 $40099.75. 
 
 97 Ansivers: 1 $1500. 2 $362.09. 3 $7200000. 4 3 h. 
 44 m. P.M. 5 $301.50. 6 $807.50. 7 3^%. 8 $4473.79 
 $192 $134.21. 9 $111.23. 10 1 yr. 21 da. 11 $987.65. 
 
 12 $100. 
 
 98 Ansivers: 1 $393.89. 2 $685.94. 3 $4125.72+. 
 
 4 $4800. 5 80%. 6 $925. 7 .00948. 8 $1550 $1475 
 $1400 $1325. 9 $6744.36. 10 $112.31+ $75 5.3+%. 
 
 99 Answers: 1 $8.22. 2 $26.44. 3 $110.10. 4 5. 
 
 5 $496.13. 6 $60. 7 $106.55/j. 8 $385.71f 9 960 
 stones. 10 2-j\ da. 11 6oi yd. $122.50. 12 380 sq. mi. 
 
 13 4945| sec. A little more than ^ of a second. 14 $87.22. 
 
 100 A7iswers: 1 $432.56. 2 5616 tiles 2990 tiles. 3 $3.62^ 
 4 $1441.19. 5 $1438.25. 6 76/. 7 54^6_ ^^k. 8 $18.13. 
 9 (a) $2450.25; (b) 20f|f cu. yd.; (c) $167.24+; (d) $36.16; 
 (e) $58.58 ; (/) 2.019+ bu. 10 2538.0864 gal. $45.24. 
 
 101 Ansivers: 1 6f| cu. yd. 2 20 yd. 3 72000 cu. ft. 
 120 ft. 4 25 42.24 '^"\ 5 2.891 + '" .18™'". 6 1.8° P. 
 |°C. 5fC. 25° C. 68° P. 95° P. 7 37^° C. 8 20° C. 9 38400 
 .cu. ft. 512000 cu. ft. . 10 .655+ gal. 11 8.6+ qt. 
 
 103 Answers: 1 548 ft. 2 12.04+ sec. 3 1.27+ sec. 
 4 23209140800000000 miles. 5 10 lb. 6 20 lb. 7 3400 lb. 
 7000 lb. 8 2| and 1. 9 5^ lb. 10 83^ lb. 11 If ft. from 
 the end. 
 
VIII. 1] teachers' manual. 179 
 
 SECTION X. 
 
 NOTES FOR BOOK NUMBER EIGHT. 
 
 Unless the principles and processes of the various subjects of 
 Arithmetic are familial*, it will be necessary to give some prelim- 
 inary reviews before the pupils are able to give the definitions and 
 rules called for in this book. The exercises themselves suggest 
 some desirable kinds of review work, as well as some forms of 
 definitions which may be made. For suggestive hints as to methods 
 of teaching definitions and rules see Manual, pages 139, 140. 
 
 Some portions of the section on Geometry may seem too difficult 
 for pupils of Grammar grades, but if the geometrical exercises 
 of previous books have been fairly well performed, pupils of the 
 highest grades should be able to take all the inventional and con- 
 structive work, much of which is a review, and the easier portions 
 of the demonstrative work. The most difficult exercises might 
 be given to such pupils only as show a special aptitude for the 
 study of geometry. In this, as in other subjects, the aim should 
 be to give work that shall be sufficient, both in amount and kind, 
 to fully tax the powers of every pupil. 
 
 1 In 4, lead the pupils to discover the fact that the decimal 
 system of numbering depends upon the principle that ten units of 
 one order equal one unit of the next higher order. Other systems 
 should be illustrated by examples. In the duodecimal system, for 
 example, the pupils should be led to see that ^, -J, ^, and I, can 
 be expressed by a single figure ; thus, .6, .4, .3, .2 ; and that ^ and 
 I can be expressed by two figures ; thus, .18 and .16. 
 
 The origin of the words expressing numbers to twelve may be 
 found in an unabridged dictionary. Other names are derivatives, 
 and their origin will be plainly seen. 
 
 The definitions called for on this page are as follows : 
 
 A unit is anything which is considered as one. 
 
180 GRADED ARITHMETIC. [VIII. 1 
 
 An integral unit is one or a collection of ones regarded as an 
 undivided whole. 
 
 A fractional unit is a part of one regarded as an undivided whole. 
 
 A number is a unit or a collection of units. An integral numher 
 is an integral unit or a collection of integral units. K fractional 
 numher is a fractional unit or a collection of fractional units. 
 Number is a quality of objects which answers the question, ''how 
 many," and arises from distinguishing one from more than one. 
 
 Arithmetic is that knowledge which has for its object the ex- 
 pression, the operations, and the relations of numbers. 
 
 The sum of two or more numbers is their united value. 
 
 Addition is the process of finding the sum of two or more 
 numbers. 
 
 Subtraction is the process of taking away a part of a number to 
 find how many units are left. The minuend is the number 
 separated. The subtrahend is the number taken away. The 
 remainder is the part left. 
 
 Multij)lication is the process of finding the united value of two 
 or more equal numbers. The multi^jlicand is the number multi- 
 plied. The mtdtiplier is the number which shows how inany times 
 tlie multiplicand is taken. The pi'oduct is the result obtained by 
 multiplication. The factors of a number are the numbers which, 
 multiplied together, produce that number. 
 
 Division is the process of finding one of the equal parts of a 
 number, or of finding how many times one number is contained in 
 another. The dividend is the number divided. The dinisor is the 
 number by which we divide. The quotient is the result obtained 
 by division. 
 
 An odd number is a number which cannot be separated into two 
 equal integral parts. An even numher is a number which can be 
 separated into two equal integral parts. 
 
 A prime mimher is a number whose only integral factors are 
 itself or one. A composite number is a number which is composed 
 of other integral factors besides itself or one. 
 
 K prime factor is a factor which is a prime number. 
 
VIII. 2] teachers' manual. 181 
 
 A multiple of a number is any whole number of times a number. 
 A common multiple of two or more numbers is a number wliich is 
 a multiple of each of the numbers. Tlie least common multiple of 
 two or more numbers is the least number wliieh is a multiple of 
 the numbers. 
 
 A common divisor or common factor of tAVO or more numbers is 
 a factor which belongs to each of them. The greatest common 
 divisor or factor of two or more numbers is the greatest factor 
 which belongs to each of them. 
 
 The denoyninator of a fraction is the number of equal parts into 
 which the unit is divided. The numerator is the number of equal 
 parts taken. 
 
 To change fractions to equivalent fractions having the least 
 common denominator : Divide the least common multiple of the 
 denominators by the denominator of each fraction and multiply 
 both terms of the fraction by the quotient. 
 
 To add fractions : Change the fractions to equivalent fractions 
 having a common denominator ; add the numerators, and write the 
 sum over the common denominator. 
 
 3 To subtract fractions : Change the fractions to equivalent 
 fractions having a common denominator, and write the difference 
 of the numerators over the common denominator. 
 
 To multiply a fraction by an integer : Multiply the numerator 
 of the fraction by the integer and place the product over the 
 denominator ; or, divide the denominator of the fraction by the 
 integer and place the quotient under the numerator. 
 
 To find the fractional part of a fraction : Write the product of 
 the numerators over the product of the denominators. 
 
 To divide an integer or fraction by a fraction : Change the 
 numbers to equivalent fractions having a common denominator, 
 and divide as in Avhole numbers ; or, invert the divisor and proceed 
 as in multiplication. 
 
 The surveyor's chain, often called Gunter's chain, has largely 
 gone out of use, having been replaced by the steel ribbon, com- 
 monly 100 feet long, with the principal divisions and markings 
 
182 GRADED ARITHMETIC. [VIII. 3 
 
 at the foot points, with other divisions at tenths and hundredths 
 of feet. 
 
 Other points called for may be found in the Tables given at the 
 end of Book VII. or in the Appendix of Book VIII. 
 
 3 The standard unit of dry measure is the bushel, which con- 
 tains 2150.42 cu. in. This bushel is sometimes called the Win- 
 chester bushel, which was the standard bushel of England before 
 the imperial bushel was adopted. The imperial bushel contains 
 2218.192 cu. in., or 80 lb. of distilled water. The standard unit 
 of liquid measure is the gallon, which contains 231 cu. in. A 
 chaldron, sometimes used in measuring charcoal, contains 36 bu. 
 Barrels, hogsheads, tierces, and pipes vary in capacity. A stone, 
 sometimes used in measuring iron and lead, is 14 lb. 
 
 The meter is the standard unit from which all metric measures 
 are derived. It is intended to be the ten-millionth part of the 
 distance on a meridian from the equator to the pole. Lead the 
 pupils to see the connection of all parts of the metric system — 
 a cubic centimeter of water (at 39° F.) weighing 1 gram, and a 
 cubic decimeter of water measuring 1 liter. 
 
 A solar year is the average time it takes the earth to make a 
 complete revolution around the sun, or about 365 da. 5 h. 48 luin. 
 49 sec. This, it will be seen, is about 365:^ days, and therefore 
 one extra day is counted every fourth year. But by this reckoning 
 we should gain nearly a day in one hundred years, and therefore 
 the centennial years are commonly counted as ordinary years. If 
 all the centennial years were thus counted, we should lose nearly 
 a day in four hundred years ; therefore, only the centennial years 
 divisible by 400 are counted as leap years. It would be well in 
 this connection to tell the pupils how and when this new style of 
 reckoning the years was adopted. 
 
 The lunar month, is the time between two new moons, and is 
 about 29 da. 12 h. 44 min. long. 
 
 All other measures and values referred to on this page may be 
 found in the Appendix. 
 
 4 The percentage is a number which is a certain number of 
 hundredths of another number. 
 
VIII. 5] teachers' manual. 183 
 
 The hoie is the number of whicli a number of hundredths is 
 taken tu hnd the percentage. 
 
 The rate is the number of liuudredths which the percentage is of 
 the base. 
 
 The amount is the sum of the base and percentage. 
 
 Tlie remainder is the difference between the base and percentage. 
 
 Lead the pupils to find these terms in Profit and Loss, Insur- 
 ance, Duties, Interest, etc., and to give examples as required in 
 2 to 4. 
 
 Other points referred to on this page are explained in Section 
 
 IX. of the Manual. 
 
 5 In 2, the formula is B^P-^B. In 3, B = A ^ (1 + ^). 
 "While it is not advisable generally for pupils to perform problems 
 in percentage by formulas, it is good practice occasionally to per- 
 form them in that way, especially when the formulas are made by 
 the pupils. 
 
 6 Numbers may be added in pairs instead of one at a time ; 
 thus, in adding 28, 36, 43, 84, 68, 74, either in a column or in a 
 line, the process might be 12, 19, 33 ; 10, 24, 31, 33. Ansiver, 333. 
 If there are many numbers to be added they might be divided into 
 sections and the separate sums added. Two columns may be 
 added at once by first adding the column of the higher 
 denomination ; for example, in adding the numbers of this 48 
 column the addition might be made as follows : 45, 105, 37 
 109, 129, 134, 164, 171, 211, 219. 25 
 
 In 6, let the pupils see that multiplication by the aliquot 64 
 parts of 10, 100, and 1000 may be made by annexing one, 45 
 two, or three ciphers and dividing. To multiply by any 
 number between 91 and 99, multiply by 100 and subtract. The 
 same process may be followed in multiplying by 99, 999, etc. 
 
 In dividing by the aliquot parts of 100, first divide by 100, and 
 then multiply. 
 
 To add fractions having 1 fur the numerator, place the sum of 
 the denominators over their })roduct. To multiply any number 
 containing ^, by itself. Multiply the Avhole numbers, and to the 
 
184 
 
 GRADED ARITHIVIETIC. 
 
 [VIII. 7 
 
 product add ^ of the sum of the whole number and ^ ; for 
 example : 16^ X 16^ = 16 X 16 = 256 + (^ of 32) + ^ = 272^. 
 From this process lead the pupils to make a general rule for the 
 multiplication of mixed numbers in which the fractions are alike. 
 To multiply mixed numbers when the whole numbers are alike: 
 Multiply the whole numbers, and to the product add that part 
 of one of them expressed by the sum of the fractions, and to this 
 sum add the product of the fractions ; thus : 16f X 16:^^ 16 X 16 
 = 256 ; f of 16 = 10 ; f X i = ^\. 256 + 10 + ^^^ = 266 j\, A?is. 
 
 7 These exercises are intended to be performed with the fewest 
 possible figures by the shortest method. Let the pupils practice 
 upon them until facility is acquired. 
 
 8 Atisivers : 1 2 yr. 2 mo. 12 da. 2 $62.50. 3 $90 more. 
 4 6487.15+ bu. 5 7^^^%. 6 $300. 7 $4565.34 e^W/^W, 
 or about 6^%. 8 Atch. the better by ^ff %. 
 
 9 Answers: 1 $8666f. 2 30%. 3 200% 20 men 7 men. 
 4 9.45 cu. in. 5 59f| qt. 5la qt. 6 538f f gal. 
 8 2m ft. 9 8i ft. 10 820i T. 11 4712^2_9_ p^t. 
 75 lb. 80 lb. 
 
 10 Answers 
 
 7 lOf f in. 
 12 66|lb. 
 
 44| lb. 
 
 Languages. 
 
 Number of Persons Spoken by. 
 
 Percentage 
 
 of 
 
 Increase in 
 
 Eighty-nine 
 
 Years. 
 
 Percentage of the 
 Whole. 
 
 1801. 
 
 1890. 
 
 1801. 
 
 1890. 
 
 English 
 
 20520000 
 31450000 
 30320000 
 15070000 
 261(»0()00 
 7480000 
 30770000 
 161800000 
 
 111100000 
 51200000 
 75200000 
 33400000 
 42800000 
 13000000 
 75000000 
 
 441.4 
 
 62.7 
 115.03 
 180.91 
 157.0 
 171.7 
 
 71.8 
 
 12.7 
 19.4 
 18.7 
 
 9.3 
 16.2 
 
 4.7 
 
 in.o 
 
 27.7 
 12.7 
 18.7 
 
 8.3 
 10.7 
 
 3.2 
 18.7 
 
 French 
 
 German 
 
 Italian 
 
 Spanish 
 
 Portuguese! 
 
 Russian 
 
 Total 
 
 401700000 
 
 248.26 
 
 100.0 
 
 100.0 
 
 
 2 $2750. 3 36. 4 June 21, 5 h. 23 min. 30 sec p.m. 5 $24.53. 
 6 About 12 acres. 
 
VIII. 11] TEACHEKS' MANUAL. 185 
 
 11 Ans7vers: 1 0%. 2 60%. 3 ^25500. 4 $7522.50. 
 5 «i25000. 6 3/. 7 $12000. 8 if! 12315 i8;35685. 9 $243.20. 
 10 $9500.624- $5179.74. 11 C. & S. M., 2f %. 12 $19600. 
 
 12 Ansivers: 1 $2900.87. 2 $506.54. 3 2666| lb. sugar 
 cane ; 4000 lb. beet root ; 5000 lb. wheaten flour. 5 100° C. 
 33^° C. 44f C. 68° F. 6 61.952 gr. 7 732. 8 Uf C. 
 9 Germany, 27.6+% England, 12.3+ % Ireland, 12.5+% Nor- 
 way and Sweden, 10'.6+%. 10 2.2+% 42.2+% 17.8+%. 
 
 13 The pupils should be led to see by exercises similar to those 
 given on this page that symbols denoting quantity may be expressed 
 by figures and by letters, and that operations may be denoted by 
 signs. The knowledge that pupils possess of the expression of 
 numbers by figures and their operations by signs should be used in 
 leading them to acquire a knowledge of the expression of algebraic 
 quantities and operations. Such knowledge will help them to make 
 generalizations in number and to solve problems that cannot be 
 solved easily by the aid of figures. 
 
 14 Pupils will probably be able to perform the first ten exer- 
 cises with little difficulty. If any difficulty is found, let figures 
 be substituted for letters in representing number, and let the num- 
 ber of exercises be increased. 
 
 In clearing equations of fractions, lead the pupils to see the 
 principle involved by using figures instead of letters. Tlie fact 
 that multiplying both sides of the equation by the same number 
 does not affect the equality may be shown by multiplying equal 
 numbers by the same number and letting the pui)ils see that the 
 products are equal. 
 
 In 9, the axiom that equal quantities divided by the same or 
 
 equal quantities give equal quotients may also be shown by the 
 
 use of figures ; thus : 
 
 20 + 10 30 
 20 + 10 ==30 — -^ — = ^ 
 
 Other axioms may be shown in a similar way as they are needed. 
 
186 GRADED AKITHMETIC. [VIII. 15 
 
 15 Answers: 1 Horse, $288 Cow, $72. 2 A, $100 B, $50 
 C, $150. 3 Eobert, 40 William, 20 Thomas, 120. 4 9 
 oranges and 9 bananas. 5 12. 6 Father, 48 yr. Son, 24 yr. 
 7 8 lb. 8 4 lb. Mocha 12 lb. Java. 9 A, $6000 B, $3000 
 C, $1000. 10 A has $G00. 
 
 The following form of analysis is suggested for this class of 
 problems : 
 
 1 Let X = cost of cow. • 
 
 4 03= " '' horse. 
 5 .X = " " <' and cow. 
 5 a; = $360. 
 
 X = $72, cost of cow. 
 4.r = $288, cost of horse. 
 
 The pupils should learn to select the number that can be most 
 conveniently represented by x. For example, in 2, the number of 
 dollars that B has, and in 3, the number of marbles that William 
 has, is the most convenient unit of representation. 
 
 In simplifying 11 to 15, lead the pupils to see how the paren- 
 theses may be removed in such examples as : 12 + (4 + 2); (8 + 5) 
 + (6 + 2); (8 -6) + (8 -4); 16 -(8 -6); 10 - (6 + 2). It 
 may be shown that the 10 — (8 — 6) is equal to 16 — 8 + 6 by 
 first subtracting 8 from 16 as indicated, and calling attention to 
 the fact tliat we have subtracted a number too large by 6, and 
 therefore we must add 6 to the difference. In the expression 
 10 — (6 + 2) by subtracting 6 instead of 6 + 2, we have a subtra- 
 hend too small by 2, and therefore we must subtract 2 from the 
 difference. 
 
 16 The four axioms involved in 5 to 8 are as follows : 
 
 If the same quantity or equal quantities be added to equal 
 quantities the sums will be equal 
 
 If the same quantity or equal quantities be subtracted from equal 
 quantities the remainders will be equal. 
 
 If equal quantities be divided by the same quantity or equal 
 quantity the (piotients will be equal. 
 
A^lli. 17] teachers' manual. 187 
 
 Two quantities each ('(^iial to a tliii-d (piniitity are equal to each 
 other. 
 
 The kind of illustrative problems called for is shown in the 
 following : 
 
 James has 18 cents, which is 3 cents more than John has. How 
 many cents has John ? 
 
 Let X = the number of cents that John has. 
 
 X -{-S = 18, tlie number of cents that James has. 
 
 Subtracting 3 from these equal quantities and there is given : 
 
 a; + 3 — 3 = 18 — 3. 
 .r ^15. 
 
 Lead the pupils to give and to solve similar problems illustrating 
 the four principles. 
 
 17 A7iswers: 16 8 6 15. 17 12 10 4. 18 12 8 9. 
 19 12 18 30. 20 8 24 15. 
 
 In removing the parentheses (11 to 15), lead the pupils to see 
 the reason for changing the sign by asking the following questions : 
 " In 11, what is to be subtracted from 20 ? If 6 alone is subtracted, 
 is the result larger or smaller than the required ansAver ? What 
 more must be subtracted ? How may both processes be indicated ? " 
 In the same way proceed with other exercises until the pupils can 
 see why parentheses may be removed by changing the signs of all 
 except the first quantity. 
 
 18 Ansn-ers: 1 Corn, $1.42 ; wheat, f 1.54. 2 James, 26 yr.; 
 sister, 16 yr. 3 Robert, 12/; James, 24/. 4 34,28,20. 5 18, 
 18, 24. 6 8 lb. @ 7/, 12 lb. @ 5/. 7 7, 8, 9, 10. 8 16 and 
 24. 9 24. 10 48. 11 40. 12 Robert, 96/; Ralph, 72/. 
 13 180 A. 14 18 yr. 15 10, 8. 
 
 19 Answers: 1 Father, 45 yr.; son, ]8yr. 2 $1200, f!1800, 
 $2500. 3 .4, $4000; 5, $6000 ; C, $!)0()0. 4 $.250. 5 13^ lb. 
 6 800. 7 43t\. 8 640. 9 40. 10 $5. 11 8/. 12 $800. 
 13 $4800. 14 $8000. 15 $900. 16 $70. 17 $4000. 
 
188 GRADED AHITHMETIC. [VlII. 20 
 
 20 The theory of negative quantities may be shown by the 
 device indicated at the head of the page. In the relative series 
 indicated, all the quantities may be said to increase by 1, or by a 
 from left to right, and to decrease by 1, or by a from right to left, 
 the positive quantities being at the right of zero, and the negative 
 quantities being at the left of zero. The device may be extended 
 in illustrating 1, the pupils being asked to begin at the zero point 
 and pass the pencil to the right 4 spaces. This would be indicated 
 by -j" 4a ; then 3 more spaces. The distance from zero Avould now 
 be indicated by -|- 7a; and by moving the j^encil over 2 more spaces 
 the distance from zero would be indicated by -\-9a. 2 may be 
 illustrated in the same way, tlie pencil in each case passing in 
 the opposite or negative direction. The spaces passed over in all 
 would be indicated by — 6 a. In 3, the pencil starts from zero, 
 as before ; passes to the right, as indicated, 4 spaces ; then to the 
 left 3 spaces. The pencil now is 1 space to the right of zero, and 
 the distance would be indicated by +1«, or + "• The pencil 
 moves to the right 2 spaces, and the distance would be indicated 
 by -\-3a, Avhich is the required answer. 
 
 Considering subtraction as the process of finding the difference 
 between two quantities, the minuend and subtrahend may be indi- 
 cated in the relative series at the right or left of zero, and the 
 difference by the distance between them. Thus, in 4, the distance 
 between + 4a and + 3a is + a. In 5, the question is, what quan- 
 tity added to — 2 a will equal + 4 a. Beginning at — 2 a, the pencil 
 must move to the right, or in a positive direction, G spaces before 
 it reaches +4 a. The difference is, then, indicated by+Ga. In 
 6, the pencil must move 7 spaces to the left from + 3 a to reach 
 the point — 4 a. The difference is therefore indicated by — 7 a. 
 
 After several similar illustrations are given the pupils should 
 be ready to apply the principle expressed in 7. In Algebra, the 
 multiplier indicates the number of repetitions either of addition 
 or subtraction that is made of the multiplicand. For example, 
 (-}- 4 a) X (-|- 2) means that -}- ia is to be added 2 times, and 
 (-}- 4 a) X ( — 2) means' that + 4 a is to be subtracted 2 times. The 
 
VIII. 21] teachers' manual. 189 
 
 answer in the first case is -\- 8 a, and in the second case — 8 a. 
 Again, adding — 4 a 2 times, we have for a result — Sa, and sub- 
 tracting — 4« 2 times, Ave have + 8 '-f^j '•t'- (~ ■!'') X (+ 2) = — Sa, 
 and (— 4 a) X (— 2) = + 8 a. 
 
 After this explanation is fully understood l)y the pupils, and 
 can be given by them in multiplying other quantities, they may 
 make use of the following rule : 
 
 The sign of the product of two numbers is plus if the signs of 
 the numbers are alike, and minus if the signs are unlike. 
 
 Since Division is the reverse of Multiplication, and the quotient 
 is that number which, multiplied by the divisor, will give the divi- 
 dend, the way of finding the sign of the quotient will readily 
 appear. The rule for both Multiplication and Division in brief 
 will be : 
 
 Like signs give plus ; unlike signs give minus. 
 
 31 Some teaching may be necessary for a few of these exer- 
 cises, especially those which involve the midtiplication and division 
 of powers of tlie same quantity. No attempt should be made to 
 simplify results by factoring at this time, but later, if it is thought 
 desirable, the results may be reduced to their simplest forms. In 
 some cases of expressed division, as in 14, (6ab) -^ (ftc), the common 
 factor a may be removed, as in similar cases of Arithmetic. 
 
 23 Answers: 1 ((x-\-kr f'!/-}-^!/ — <f-T — % — ('.'/ — b>/- 
 2 «.i' -|- ki' -\- ((I/ -\- hij (tx -\- hi/ — It (/ — /yy. 3 ".«' — hx -\- a ij — hij 
 
 ax — /m — <i tj -\rliij. 4 cr -f" 2 r? -|- 1 "' — 1 1 — ""• 5 6 ax 
 
 -\-A,h.i — S^'cc — Q>axi/ — Altx;/ -\-?>cxy. 6 4<f^j'-f- Ga"'' — 8a 
 
 4 a\r + (; (i\v — 8 ax 4 a''x -f G a^ — 8 a — 2 ax — 3 fl^ -f 4. 7 4.a^b 
 
 + 2 a%'' + a/r 4 a'% + 2 aV- -f ah^ — 4 a% — 2 ah' — //"' — 4 a'-h 
 
 — 2 «2^2 _ ,,/,! _^ 4 ,,2/, _^ 2 ah' -f h^ 8 (r + 2 ah + h' + ar + he 
 a' + ac — h' — hr a- -}- 2ah + h' + 2 ac -\- r' -\-2hc. 9 2x^ — 43;^ 
 
 +x/ 2.rV-4xy + uy-f2xV-4V + / 2x'>/-4:xy + x2/ 
 
 — 2x'i/-\-Axi/' — y^. 10 x'-\-2xi/-\-y' x' — if x^-\-x^y-\-x'f-\-'i^ 
 x^-\-xSi — xxf — if, 11 x^y-\-x'ij-\-x^f-\-xxf x^f-\-x'if-\-x^y 
 
 — xyx'f-\-x'f xhf-^x'f. 12 a;" — 2a;^ — 13x'' + 14ic + 24. 
 
190 GEADED ARITHMETIC. [A'lII. 23 
 
 13 x-^2xi/-\-if x- — 2xy-\-ir .r^ + 2,>' + l. 14 ,r--2,r + l 
 
 4.r- + S.ry-f'±.'/ 4.r2 — 8.ry + 4/. 15 a" + ?> a'- -\- 5 air ^ b" 
 
 a'-t^a-b — 'iab' — lf' rr^ + 3«- + 3« + l. 16 a-b'- + 2 abed + vM- 
 
 a%'' + 2ab + l ./■^ + 2.r-y+//. 17 2a- + 2a. 18 2x''+2>f. 
 
 1:Q 2crb + ^aJr + 2lf. 20 b-\-r + rl ^^±j±^ ^+^^ 
 
 ^ + ''-^'K 21 6rt + 12/> + lS (/ + 2/; + 3 fr + 2r^5 + 3«. 
 
 « 
 
 22 1 ^^ + ^' «- + 2rY/> + /r. 23 a — b a- — 2al> + lr. 24 r^ + ^' 
 
 1. 25 « — /> ^^ + A. 26 r/2 + 2.//> + /r r/ + /;. 27 2r + l 
 
 2x — 1. 28 1 — 2.r 2.r — 1. 29 /> — r — /> + r' a. 
 
 30 r + (/. 31 2 + 3j' — r>.r-. 32 a- — lr a- + b\ 33 4a; 
 
 + 3// + 4. 34 3.r — 2y+l. 35 3.r + 2^. 
 
 23 Ajxsu-ers: 1 2(5 ./ + A + r 6«. 2 25 « + /v~c 8 a. 
 3—10. 4 rt — 4Z> + c+r/. 5 .r — 3.r-. 6 7a:- + 4a; + ^-. 
 
 7 8.r + 5ia;l 8 5.r- + .r + 2y — 15. 9 .t" + 3/ — 2a' + 2. 
 
 •t r\ ^ A i\ 90 40 900 Ofio oro 12 2 O (O 
 
 10 x' X* y^ x-y^ x*y x-ifrJ x-y'z- x- -\- lx]i-\- y x- — 2xij-\-y 
 a;2 + 2 a; + 1 x* -\- 2 xhf- + y' 9 a- + 12 ah + 4 //I n Sum of their 
 squares + twice their product Sum of tlieir squares — twice their 
 product. 12 x^ x^if x^yh^ x'^y^z^ x^-\-ox-y-\-Sxy~-\- y^ 
 
 x' - 3 xV + 3 xy'- — ir 8 x^ + 24 a;2 + 24 x + 8 27 j^ - 2 7 ./- + 9 x - 1 
 64a^ + 96a2^<-f 48«i'2 + 8/>l 13 q^^^ ^f ^he 1st + 3 times the 
 square of the 1st into the 2d + 3 times the 1st into tlie square of 
 the 2d + the square of the 2d. Cube of the 1st — 3 times tlae 
 square of the 1st into the 2d + 3 times the 1st into the square of 
 the 2d — the cube of the 2d. 14 4 ,/■- + 12 xy + 9 //- 9 x- — 2\x 
 + 16 a-- + 4.r// + 4y 9«2 — 3G.7.7/ + 36/. 15 1 + 3./' + 3,r^'+.r3 
 
 8 a;'' + 12 xhj + 6 xf + //'^ 27 x' — 54 x~ + 3(3-8 x"" — 9 x''y + 27 .r// 
 
 — 27/. 16 16x^-\-8ax + a' 8a- — 36a-2 + 54.r - 27 IG^r 
 + 2Aab-]-9b' 8«^— G0«2 + l50rt — 125. 17 5Qxy. 18 30 ^r 
 
 — Aab — 15 «-. 19 54 a'^b + 18 ^z". 20-7 .r" + 15 x'- + 9 ,'• + 2 
 -{-12x-y — 30xy'+7y'. 21 2 X 2 X a X a 2X2X2Xa XaXa 
 2x2X2x2XaXaXb 2 X 2 X o X a X a X a X b X b X c 
 ((( + 1; (a + 1) X /// . 22 a (x + y) a (ax — y) x X x X x {a- + a 
 + l;(a-l) aXa{b-x). 23 (a + &)(«- 1) {a-\-b){a + b) 
 
VIII. 24:] teachers' INIANITAL. 191 
 
 (x + 1) (x + 1) (a- + 1 ) (x - 1). 24 (<' + ^0 (a + h) (a + /^ 
 
 (.^— /;)(r^2_|.,,^,^^^2^ Q,_^l,s^(^^,2_^^l^_^f'2y 25 (2 ^/. + 3 />) (2 rt 
 
 + 8 b) (•• ! .r + D) (.'5 ,r + 9) (3 ,r + '.)) (;^> .r - <J) . 262X2 
 
 X ab (2 ^r' — <>/> + ;; rr') 2 X 2 X 2 X a X(iXaXbXb(^2 — a}r ^1n). 
 21 2(a- + yH2.r + l) 4(.r + 2^)(2 + y). 
 
 10 — « 
 34 Alls ire rs : 1 .t^8 — // x = — x = 12 + 2y 
 
 x = 2(i/ + zy 2 .T = 4//-2 ^.^ 48-o// ^^3^5 _2^. 
 
 o , , , 3./ + 2Z. A5 — 2a ^ ^. ^, 
 
 3 a' = 8 + /; + a ir = y. x = — — 4 .r = SA — a — 2b 
 
 b lo 
 
 a" = ^ • x=^4a — ,>ao. 5 .r = lb — // y = lh — aj 
 
 (5 
 
 = 2+y y/ = ;r-2 x = ^^^ ^ = 28-2a; a'-^^ + ^ 
 
 a; 
 
 
 32 — 2v/ 32-3x 20 + 3// 
 
 v/ = 2.r-12. 6 .'■ = ^ y = 2 ^= 7 
 
 4;r — 20 40 — lOy 40 — 3. r 20 — y 
 
 2/=^^ ^=~^r^ y=~-r~ ""^-^ir 
 
 o r> ''' + 1-'/ 16.X' — (/. . 
 
 ,; = 20-2r. 7 :^-= j^ y= ^^ a. = a-4y 
 
 r? — .r 8y — 6rt + i — 1 Ga — ^ + l+4a; 
 
 •'^ - ^^ -^^^ = "^ 4 ^ ^ 8 
 
 6 6y Ix Zay — 16 a 
 8. . = -4^ v/ = -~ .■ = - ,/ = - :« = j^^— 
 
 y = 12.r + 16r^ 9 ./■ = 4 r).*' = 20. 10 .'=5 y/ = 4. 
 
 11 a^ = 5. 13 ./■ = 8 // = 2. 14 .r = 5 // = (;. 15 .'■ = 7 
 
 ?/ = 5. 16 x = o y = ^. 17 a; = 9 y = 2. 18 .c = 3 
 i/ = 10. 
 
 25 Anawers: 1 ?/ = 6 .r = 8. 2 ./■ = 6 y = 2. 3 a- = 4 
 2/ = 10. 4 .r = 3 y/ = 2. 5 .*■ = (; y = \. 6 a- = 5 ?/ = r). 
 7 a; = 8 y = 3. 8 ./-^lO y = %. 9 .r = 9 ^ = 7 
 
 10 a; = 12 .y = 4. 11 x = 9 ^=12. 12 .r = 10 y = Vl. 
 13 a; = 15 y = 9. 14 a; = 8 ^ = 8. 15 x = Vl y = 4. 
 
192 GRADED ARITHMETIC. [VIII. 26 
 
 16 ic = 18 y = 8. 17 ic = 20 y = 24. 18 ^- = 2^8^ y = — 2. 
 
 19 a^ = 40 ?/ = 24. 20 .'k = 30 v/ = 48. 21 ic = 14f 3/ = 48 
 s = 26f. 22 .c = 12 y = 2 « = 10. 23 x- = 9 y = 20 2; = 6. 
 
 26 Answers: 1 x==2. 2 y = A^. 3 y = 6. 4 ?/ = 2. 
 
 5 x = 8 y = 6. 6 x = 9 y=10. 7 :t' = o ?/ = 6. 8 .f = 8 
 y = 3. 9 .« = — 3a // = 4f. 10 x = 9 y = S. 11 a' = 8 
 2/ = 12. 12 a=7 ^ = 2. 13 .r = 12 ?/ = 10. 14 .): = 15 
 2/ = 18. 15 a. = 34f ^ = 22^. 16 x = l{f l/ = -^\. 
 
 17 x = 6 ?/ = 16. 18 ir = 67i y = 4(3^. 19 rc = 42 y = 3. 
 
 20 x = 6 i/ = 25. 21 « = 6 .r = 8 y = 4 2; = 9. 22 « = 18 
 a; = 8 v/ = 12 « = 6. 
 
 27 Answers: 1 ,t = 8 ?/ = 5. 2 a; = 6 ?/ = 10. 3 x = 12 
 2/ = 20. 4 x = 9 7/ = 8. 5 .r = 12. y = 2. 6 x = 4 ^ = 16. 
 7 a; = 20 ij = 15. 8 .r = 18 ?/ = 7. 9 x = 7j'-g ^ = 14^2-. 
 10 ii: = 28f y/ = — 14f. 11 .r = 33 y = 30. 12 :i' = 12 
 3/ = 10. 13 a: = 20 y = 18. 14 a- = 10| y = — 13if 
 15 x = () v/ = lo. 16 a' = 20 y = 10. 17 .r = 2l'if 
 y/ = 13_j7_. 18 if = 30 y = 24. 19 x = 8 y = 12. 20 * = 12 
 ?/ = 2. 21 a' = 9 y = 4. 22 a- = 22^1- ^/^I^tV 
 23 a; = 21|f v/ = 20|f. 24 .r = 12 2/ ==20. 25 .r = G 
 y = 8 ,^ = 16. 26 .r = 5 y = 30 « = 10. 
 
 28 Ansivers : 1 a-\-b — c a — h — c a — b — c a — b-\-c 
 a-{'b — e — d + e. 2 a'' + 2ab + b^ d- — V\ 3 ^a^ — ^ab-\-2iW 
 — 6/r + 23Z'c — 20('l 4 56"2 — r)c + 5crf + 13r/. 5 c + rf a — b. 
 
 6 a^^y a^ — irh^a%''—ah''-^b\ 7 « + i — e + ^/ — 3^/ + A. 
 
 2 //72 I 7 2\ 4 7 
 
 8K«-1) ^(-^-1) «(^-2y + 3..). 9 \.Zy.^ -^^~^,' 
 10 (« + &)' «2+^r + 2«Z- («-^^)' «' — ^' (« + ^>) X 0? — />) 
 
 {a-\-hf. 11—4 TTT j— — — :• 12 6 f 
 
 ^ ^ a 12 — n m-\-n a-\-\ a 
 
 _6 ±Z1 iizi. 13 9 12 2 ^. 14 7^ - 
 
 a a — b c 0-f-c ac 
 
VIII. 29] TEACHEKS' MANUAL. 193 
 
 '^'^-"^ ;^^3^, ^^^'Uh 15 22i 5'- ^ ^ 12.247+ 
 
 39 Jwsu-ers.- 16 8. 2 18 8. 3 20 8. 4 John, 12/ 
 James, 18/. 5 Apple, f / orange, 5|/. 6 6f , 2|. 7 15 25. 
 8 Man, 50 wife, 40 daughter, 10. 9 20 yr. 10 James, 18 
 John, 12. 11 Coffee, 24-i<V/ tea, 65/^^/. 12 lilOO. 13 A, 
 $29 B, $11. 14 656 apples. 
 
 30 A7isirers : 1 $480. 2 $540. 3 A, 50 B, 30. 4 36 da. 
 5 2.4 h. 6 40 persons. 7 A, $7 B, $5. 8 Pigeon, 40/ 
 chicken, 50/. 9 10 sheep, 20 calves. 10 $400. 11 Jane, 
 $250 Sarah, $400 Ellen, $150 Mary, $200. 12 15 8. 
 
 9 is indeterminate. Other answers are, 5 sheep and 27 calves ; 
 15 sheep and 13 calves ; 20 sheep and 6 calves. 
 
 31 Ansivers: 1 Apple, 2/ pear, 1/ orange, 4/. 2 $36000. 
 3 13^ da. 4 6i li. 5 105} 78f. 6 $1500 @ 6%. 7 8, 
 12. 9 A, 9|1 da. B, 15|i da. C, 8|a da. 10 $3500 in 4's 
 $1500 in 5's. 11 ^'^. 12 48. 
 
 8 is indeterminate, any one of forty answers being correct. 
 
 32 Ansivers : 1 Nile, 3000 miles Danube, 1600 miles Ama- 
 zon, 3600 miles. 2 St. Peter's, 442 ft. Bunker Hill, 215 ft. 
 Washington, 549 ft. 3 3 h. 16/j- min. 6 h. 32^^ min. 4 $1400 
 
 $1600. 5 14| h. 6 8 da. 7 6i li- 8 ^"^^~ ■^ - 9 -^, 
 
 (till al am amo / lOO + » \'" 
 
 ^;r+i' j»+i VI + 1 w(m+o' "v 100 / ■ 
 
 ant (d — c) hm (d — c) _ _ ah -, a -to 
 
 12 x = —r^ j^ y = --r^ f-' 13 J— r — mo. 14 12 
 
 he — ad he — ad bm -\- an 
 
 20. 15 12 h. 32^8^ min. 
 
194 BEADED ARITHMETIC. [A^III. 33 
 
 33 Ansu-er.^: 12 192. 13 IDS. 14 729. 15 S192. 16 2^. 
 17 11 18 .729. 19 .0000078125. 20 f. 21 108. 22 283. 
 
 23 Iji^. 24 .0225. 25 .244/^. 26 ^t^Vittt- 27 ^1^. 
 28 1600 7056 9801 10000 313600. 29 698900 106276 
 474721 998001 1000000. 30 81796. 31 20.9764. 
 32 .00005625. 33 27.210701. 34 31640625. 35 1.10271001. 
 
 36 41371138.5616. 37 94a||. 38 32768. 39 88.121125. 
 40 5861fg|. 41 383328f 
 
 34 The formula for the extraction of the square root of a 
 number can be found by finding the square of the root : t -\- u. 
 Pupils who have taken the algebraic exercises will have no diffi- 
 culty in this. Others will have to be taught. If the symbols 
 representing any root be found too difficult to work with, let figures 
 representing the root of a given power be used, e.g., the square of 
 25 = 20- + 2 X (20 X 5) + 5^. 
 
 35 A7mcers: 1 32. 2 43. 3 51. 4 62. 5 73. 6 35. 
 7 46. 8 36. 9 57. 10 76. 11 67. 12 74. 13 83. 
 14 94. 15 89. 16 07. 20 4.2. 21 6.3. 22 5.4. 23 3.5. 
 
 24 5.6. 25 4.7. 26 5.,S. 27 6.9. 28 S.S. 29 0.6. 30 9.9. 
 31 7.9. 32 7.9. 33 231. 34 342. 35 254. 36 364. 
 
 37 286. 38 523. 39 475. 40 279. 41 634. 42 488. 
 43 596. 44 678. 45 969. 46 804. 47 709. 48 232.1. 
 49 354.6. 50 465.8. 51 50.76. 52 60.19. 53 1.41 + . 
 54 6.32+. 55 15.55+. 56 37.14+. 57 40.92+. 58 1.87 + . 
 59 4.02+. 60 2.69+. 61 2.84+. 62 12.66 + . 63 5.72+. 
 64 2.10+. 65 30.009+. 66 2.23+. 67 3.61 + . 68 10.83+. 
 69 5.22+. 70 .64 + . 71 .87 + . 72 .73+. 73 .52 + . 
 
 36 Answers: 1 14 rd. 2 186 ft. 274 ft. 511.23+ ft. 
 3 12.649+ rd. 6.324+ rd. 4 9.8+ rd. on two sides. 5 254.13+. 
 6 884.84 ft. 7 148 blocks 98| ft. square. 8 056871 paving 
 stones. 9 14.14+ in. 10 24 ft. by 16 ft. 11 12.29 ft. on 
 longer side 12.11 ft. on shorter side. 12 154.5 rd. 13 19 
 and 15. 14 1141.40. 15 $141.40. 
 
Till. :J7] TEACHEKS' ISIANUAL. 19o 
 
 37 Ansirers: 7 80. 8 o2. 9 41. 10 lo. 11 68. 12 63. 
 13 99. 14 97. 15 4.7. 16 33. 17 12.7. 18 .89. 19 362. 
 20 41o. 21 472. 22 903. 
 
 :^H Jiisn-ers: 1 1.58+. 2 3.91+. 3 5.64+. 4 11.69 + . 
 5 1.6.S + . 6 2.43+. 7 1.8!) + . 8 4.82+. 9 1.91+. 10 2.38+. 
 11 .79+. 12 .42+. 13 .76 + . 14 .43 + . 15 .39+. 16 .74+. 
 17 13 in. 35 in. 18 87 in. 72 in. 19 12.9+ in. 47.55+ in. 
 20 73 in. 28.4+ in. 21 1.41+ ft. 22 10.06 + rd. by 15.09+ rd. 
 23 12.14+ ft. 24 6 ft. long, 3 ft. wide, 1^ ft. deep. 25 8.07+ ft. 
 26 160 rd. 27 88.4 + in. 132.6 + in. 176.8 + in. 28 150.26 + rd. 
 
 39 The following facts should be drawn from the pupils by 
 teaching, as Jaefore shown : 
 
 flatter is anything we get a knowledge of through the senses. 
 
 A Ikh/i/ is a limited portion of matter. 
 
 Space is the room a l)ody occupies, and the room that is around 
 a body. 
 
 A volume is a limited portion of space. A volume may be repre- 
 sented by a so/id, wliieh has three dimensions, length, breadth, and 
 thickness. 
 
 A surface is the limit of a volume, and has only two dimensions, 
 length and breadth, 
 
 A line is the limit of a surface, and has only one dimension, 
 length. 
 
 K. iwlnt is the limit of a line. It has position only. It can be 
 represented by a dot. 
 
 A straight line is a line which has the same direction throughout 
 its entire length. 
 
 A curved line is a line that constantly changes its direction. 
 
 (For definitions of horizontal line and vertical line, see Manual, 
 page 168.) 
 
 Lines are parallel when they have the same direction. However 
 far prolonged, tliey can never meet. "When one line meets another 
 line so as to make the adjacent angles equal, the lines are said 
 to be perp>endicidar to each other. 
 
196 GRADED aeith:metic. [VIII. 40 
 
 9 A line may be drawn parallel to another line as previously 
 shown, or by the following way : 
 
 To draw through the point It a line 
 parallel to the given line OF. 
 
 From the point C, with a radius equal 
 
 to CB, draw the semi-circumference ARB. 
 
 From 5 as a center, with a radius equal 
 
 to AB, cut the circumference at S. Join BS, and Ave have a line 
 
 parallel to OP. (Let the pupils show why the lines are parallel.) 
 
 11-12 For a method of dividing a line into 2, 4, or 8 equal 
 
 parts, see Manual, page 169. The following method of dividing 
 
 Q a line into any number of 
 
 "'~~--.,^^ f> parts may be ttiught : 
 
 / ~-~-~-, ^ To divide a line AB into 
 
 / ?'~~*--.^ equal parts. 
 
 / """7-^. Draw the line AO ot any 
 
 ^/ ^ / '''---..^ ^ length, and lay off on that 
 
 ■B ^ ^ line parts of any conven- 
 
 ient equal length. Join BC, and through the points of division 
 on ^0 draw lines parallel to BC. These lines divide AB into 
 equal parts. 
 
 The standard unit of length in this country is the yard, the 
 same as the imperial yard of Great Britain. Its length is §f y§§§ 
 of the length of a pendulum which vibrates seconds in a vacuum 
 at the level of the sea at 62° Fahrenheit in the latitude of London. 
 Other interesting facts concerning how and where the standard 
 yard is kept can be gathered from a cyclopedia and given to the 
 pupils. 
 
 40 Teach, as before, the following definitions : 
 
 An anf/le is the difference of direction of two lines in the same 
 jdane. The i)oint where the two lines meet is the vertex of the 
 angle. A 7'if/ht (nu/le is the difference of direction half as great 
 as o])]iositeness ; or, it is an angle formed by two lines extending 
 perpendicularly from each other. An obtuse angle is an angle 
 
VIII. 41] 
 
 TEACHERS MANUAL. 
 
 197 
 
 greater than a right angle. An acute angle is an angle less than 
 a right angle. 
 
 An angle is measured on the circumference of the circle whose 
 center is at the vertex of the angle. The unit of measui:e is a 
 degree, which is ^^^ of the circumference of a circle. The follow- 
 ing figures will suggest a method of making an angle equal to a 
 given angle, and also for making angles equal to twice or three 
 times the size of a given angle : 
 
 13 To draAV an angle equal to a given angle. 
 
 The pupils wall see that the proof of the equality of these angles 
 rests upon the fact that equal arcs subtend equal angles. This is 
 implied in Avhat they have learned about the measurement of 
 angles. 
 
 14 To draw an angle twice and three times the size of a given 
 angle. 
 
 In teaching the above problems, as well as all that follow, the 
 teacher should give as little direct assistance as possible, leading 
 them on slowly by suggestions and questions. 
 
 41 Vertical angles are angles that have a common vertex, and 
 that have sides extending in opposite directions. 
 
 Adjacent angles (a and h in the figiire) are angles that have the 
 vertex and one side common, and Avhose other sides are opposite 
 parts of the same straight line. (Tt will be observed that one 
 element of adjacent angles is left out in 2.) 
 
198 
 
 GKADED ARITHMETIC, 
 
 [A III. 42 
 
 4 To prove that the sum of two adjacent angles is equal to two 
 right angles. 
 
 Draw DO perpendicular to AB. 
 AOC + BOC = ADD + BOD. 
 A0D+B0D = 2 right angles. 
 -r .•.A0C'i-B0C=2 right angles. 
 
 B 
 
 
 
 In 7, lead the pupils to see and to say tliat the angles a and b 
 (in the figure, lOj = the angles b and c. Taking away the com- 
 mon angle b, the angle a = the angle c. 
 
 In 10, the four angles c, d, n, and m are called internal angles, 
 because they lie between the parallel lines ; and the four angles 
 a, b, 0, and j) are called external angles. The angles a and m and 
 the angles b and 7i are exterior-interior angles. The angles c 
 and m and the angles d and n are alter^iate-interior angles. Lead 
 the pupils to discover what angles are equal, and why they are 
 equal. The lines AB and CD are drawn parallel to each other. 
 
 A. plane surface is such a surface that, if any two points in it 
 be connected by a straiglit line, tliat line will lie Avholly in the 
 surface. 
 
 Let the definitions called for in 12 be concise and comprehensive. 
 (For method of teaching, see Manual, pages 104 and 116.) 
 
 43 Let the solution of the theorems and problems be made by 
 measurement and construction, and also by demonstrations so far 
 as the pupils can be led to make them or to understand them. 
 The following figures and liints may suggest demonstrations of the 
 more difficult propositions : 
 
 3 Draw CE II to AB. 
 Prolong J C to F. 
 EOF + ECB + ACB = 2 
 angles. 
 
 A = E( 'F. 
 B = EC 'J. 
 
 right 
 
VIII. 48] 
 
 TEACHERS ]\LA.NUAL. 
 
 199 
 
 Let tlie order of demonstrating 5, 6, and 7 be reversed. 
 
 6 Bisect angle C. 
 AC=CB. 
 CD common. 
 
 .'. a = b. 
 
 From the fact proved in 6, it can be shown that the angles of 
 an equilateral triangle are equal. 
 
 AC= CB. 
 
 AD = DB. 
 
 Join AB. 
 CAB = ABC. 
 BAD = ABD. 
 CAD = CBD. 
 
 The exercises from 11 to 15 are to be performed with protractor 
 and ruler. 
 
 43 The pupils may be able to solve 3 by demonstration. 
 
 In 4, lead the pupils to prove that the two triangles are equal, 
 that the angle at the vertex is bisected by the 
 line drawn from the middle point of base to 
 vertex, and that the base is bisected. 
 
 This figure Avill suggest a method of bisecting 
 the angle A. Lead the pupils to prove that 
 the triangle Jlfi) = the triangle AND, and 
 that therefore the angle A is bisected. 
 
 For a method of bisecting a line, see Manual, 
 page 169, 
 
200 
 
 GRADED ARITHMETIC. 
 
 [VIII. 44 
 
 The following figures will suggest ways of solving the problems 
 contained in 7 : 
 
 Ey, 
 
 c 
 
 ■'R 
 
 \ 
 
 F 
 
 
 H 
 
 .B 
 
 B 
 
 It would be well to apply the principles involved in the last 
 exercises of the page by actually measuring distances in a field. 
 This may be done with lines and stakes or with stakes alone. 
 
 4:4 The definitions called for on this page are supposed to have 
 been taught previously (see Manual, page 104). 
 
 The simplest solution of 3 may be made by dividing a quadri- 
 lateral into 2 triangles. 
 
 The propositions on this page can all be performed by measure- 
 ment or construction. As many of them should be demonstrated 
 by the pupils as possible. They will not be found difficult for 
 
 those who have demonstrated the 
 preceding propositions. The fol- 
 lowing hints might be given if 
 necessary : 
 
 What angles do you know to 
 be equal ? AVhat lines ? Compare 
 the size of the two triangles. If 
 we were to prolong one of the sides, what two angles are equal 
 to 2 right angles ? What other two angles must be equal to 2 
 right angles ? 
 
 li :r- 
 
 What angles can you prove to be equal ? 
 Wliat lines? What triangles? What 
 other lines ? 
 
 ''E 
 
 B 
 
VIII. 45] teachers' manual. 201 
 
 The pupils may be told that an isosceles trapezoid is a trapezoid 
 whose sides between the two parallel lines are equal. TJie propo- 
 sitions contained in 13 to 16 can be easily proved. 
 
 45 A polygon with two reentrant angles : 
 Exercises may be given to show how the area 
 
 of such a polygon may be found. Let the pupils 
 discover two ways, and what dimensions must 
 be known. 
 
 The simplest way to construct a regular poly- 
 gon is from the circle. Show to the pupils that by means of com- 
 passes the circumference of a circle can be divided into any number 
 of equal parts, and that the straight lines connecting the points of 
 division are the sides of a regular polygon. 
 
 The formulas called for in 15 and 16 are, S=ah; a = S'T--x', 
 
 46 The triangles ADE and ABC may be called similar, because 
 they are of the same shape. Later (page 50) the pupils will learn 
 more definitely what similar polygons are. Triangles are equiva- 
 lent if they have the same size, and equal if they have the same 
 shape and size. 
 
 The following formulas should be made by the pupils from their 
 knowledge of finding areas : 
 
 Triangle, S = ~] traipezoid, — ^r-^XA; regular polygon, 8^=—' 
 
 The transformation of polygons into equivalent polygons of any 
 required shape can be readily made if the method of finding the 
 areas is thoroughly understood. 
 
 47 Answers: 1 i ooo a. 2 1728 sq. ft. 4290 sq. ft. 252T8f 
 sq. ft. 3 272i ft. 4 80 sq. ft. 2016 sq. ft. 5 1512 sq. ft. 
 61640 sq. ft. 6 217| ft. 7 147.5+ ft. 8 153 sq. ft. 9 3 ft. 
 8.3+ in. 10 344.6+ ft. 11 141.7+ ft. 12 4380 sq.ft. 13 -}. 
 14 17+ bundles. 
 
 48 A7iswers: 5 5 in. 15 in. 6 8 in. 13.41+ ft. 7 25.98+ yd. 
 
202 GRADED ARITHMETIC. [VIII. 49 
 
 The pupils should be encouraged to give other proofs than 
 those here given. The formulas to be given are : h = V*^^ + j^"^ ; 
 
 in 
 
 49 Ansii-ers: 1 100 ft. 2 25 ft. 3 43.8+ rd. 4 60 
 5 62.4+ ft. 6 120 rd. 7 19.59+ yd. 11 34.66+ ft. 693.2+ 
 sq. ft. 12 25.45+ ft. 13 70.71+ ft. 14 24.08+ ft. 32.8+ ft. 
 15 45.69+ ft. 16 30.98+ ft. 17 36.76+ ft. 18 65.28+ rd. 
 19 25.45+ rd. 20 26.07+ ft. 21 722.9+ ft. 
 
 50 The term covresponding may be used instead of homologous, 
 if preferred. The proof called for in 6 may be experimental rather 
 than demonstrative. It follows what is supposed to be done in 
 previous exercises. 
 
 The proof called for in 7 may be made from measurement and 
 comparison of the sides of similar polygons. 
 
 51 Answers: 1 9 times as large. 24:1. 32:1. 45:1 
 25 : 1. 5 438f sq. ft. 6 94.8+ ft. 7 5250 sq. rd. 8 222-| 
 sq. ft. 9 242.4+ ft. 10 2.4+ ft. . 11 35f ft. 12 36 ft. 
 
 These exercises should be performed by proportion. It would 
 
 be well for pupils to write the proportions and statements in full ; 
 
 thus, in 5 : 
 
 (40 ft.)2 : (50 ft.)2 = 300 sq. ft. : number of sq. ft. in larger triangle. 
 
 300 X 2500 . „ „ . , ^ . , 
 
 ■ ■=^ number of sq. ft. m larger triangle. 
 
 and in 6 : 
 
 4000 sq. ft. : 10000 sq. ft. = 60^ : the square of the side of larger 
 
 hexagon. 
 
 10000 X 3600 . ,1 . . n 1 . 
 
 - = the square of the side of larger hexagon. 
 
 4000 
 
 4 
 
 10000X3600 .-, ^, 
 
 - = side of larc:er hexagon. 
 
 4000 
 
 LC,V.X iiV^^lVj^V. 
 
 52 Answers : 1 33 it. 2 93^ ft. 3 27f. 4 29T\ft. 6 86 § ft, 
 7 43^ ft. 
 
VIII. r>3] teachers' manual. 203 
 
 In 8, a line may be tlraAvn downward from 7> parallel to AX, 
 and one from A parallel to BX, so as to meet the first line at K 
 Thus Avould be found two equal triangles, two angles and included 
 side of one triangle being equal to two angles and included side of 
 the other. 
 
 In 9, extend the line XI to a point F. Draw FG perpendicular 
 to FX. Join GX. From the point A draw a line .47/" parallel to 
 GX. Thus are formed two similar right-angled triangles, and 
 FIT : FG = FA : FX. 
 
 53 In 1, the lines ^LY and ^-iFare found by the method shown 
 in 5, page 52. Ax is the same fractional part of AX that Ai/ is 
 of A Y. The two triangles Axy and AXY are similar, and Ax : AX 
 = xy : XY. 
 
 The following definitions may be taught as previously shown : 
 A circle is a plane figure bounded by a curved line, all points of 
 which are equally distant from a point within, called the center. 
 The circuiiifeyence of a circle is the line which bounds it. The 
 diameter is a straight line passing through the center and terminat- 
 ing at the circumference. The radius is a straight line connecting 
 the center Avith any point in the circumference. An arc is any por- 
 tion of a circumference. A chord is a straight line connecting the 
 extremities of an arc. A segment is a portion of a circle bounded 
 by a chord and its arc. A sector is a portion of a circle bounded 
 by two radii and the included arc. A semi-circle is a portion of a 
 circle bounded by a diameter and half the circumference. 
 
 Draw lines as indicated in 10, and show that every point of the 
 perpendicular is equidistant from the ends of the chord. 11 and 
 12 can be readily shown from 10. 
 
 54 In 1, join the points by lines, which may be regarded as 
 chords of the required circle, and erect perpendiculars from the 
 middle points. 
 
 In drawing an inscribed angle (4), there are three possible con- 
 ditions. The center may be in one of the sides of the angle, or, 
 between the sides of the angle, or, without the sides of the angle. 
 The first case should be proved first, the proof resting upon the 
 
204 GEADED ARITHMETIC. [VIII. 54 
 
 facts, (1) that the angle i?OC is equal to the sum of the angles 
 
 A and C ; and (2) that the angles A and C 
 are equal. From this proof the other cases 
 may be easily proved. 
 
 A secant is a straight line cutting the cir- 
 cumference of a circle in two points. 
 
 A tangent is a line which touches a circum- 
 ference in one point without cutting it. 
 
 The proof of 6 rests upon the facts, (1) that 
 a line from the point of contact to the center is the shortest dis- 
 tance between the tangent and the center ; and (2) that the shortest 
 line between a point and a straight line is the perpendicular line. 
 
 In 8, connect the point with the center of the circle, and, upon 
 this line as a diameter, describe a circle which will cut the circum- 
 ference of the first circle at two points. Join these two points 
 with the given point outside of the circumference, and the lines 
 thus drawn are the tangents required. This can be easily proved 
 from what has been proved before. 
 
 Before giving 9 to 12, show that a circle is inscribed in a poly- 
 gon when its circumference touches every side of the polygon, and 
 that a circle is circumscribed about a polygon when its circum- 
 ference passes through all the corners of the polygon. 
 
 In 9, the point of intersection of the lines bisecting the angles 
 is the center of the inscribed circle. To prove that the sides of 
 the triangle are tangents to the circle, draw lines from the center 
 perpendicular to the sides of the triangle, and show from the 
 equivalence of the triangles that the perpendicular lines are equal, 
 and are therefore the radii of tlie circle touching all sides of the 
 triangle. 
 
 In 10, draw such perpendiculars to radii as will intersect and 
 form a triangle. 
 
 11 and 12 can be readily done when it is understood that the 
 bisectors of tlie angles of a regular polygon meet in a point equally 
 distant from all the sides and all the corners. This can be proved 
 from the equality of the triangles. 
 
VIII. 56] teachers' manual. 205 
 
 In 15, the pupils can readily find the approximate ratio by 
 measurement. They can also estimate by calculation the perimeter 
 of a regular polygon inscribed in a circle liaving a given radius, 
 and can see that the greater the number of sides of the inscribed 
 polygon, the nearer the perimeter approaches the length of the 
 circumference. 
 
 55 From what has been done previously (see Book VI., page 18) 
 the formulas in 1 to 5 can be illustrated. 
 
 6 The sector COD may be divided into 
 any number of triangles whose altitude is 
 EG and the sum of whose bases is CD. 
 
 7 The area of the segment CED is equal 
 
 to the sector COD — the triangle COD, i.e. 
 
 ^T-,T. EO ^_, FO 
 CED X -^ CD X — • 
 
 To find the area of a circular zone or cir- 
 E cular ring, subtract from the area of the 
 
 circle the part not included in the zone or ring. Let the pupils 
 determine Avhat dimensions must be given. 
 
 Let the proofs called for in 10 to 13 be made from construction 
 and comparison of measurements. Pupils may be led to use what 
 they already know of similar polygons. 
 
 To draw an ellipse, stick two pins or tacks into the surface upon 
 which the ellipse is to be drawn, and tie to them a string longer 
 than the distance between them. Describe with a pencil the curve, 
 keeping the string constantly stretched to its full length. Th(^ 
 points A and D in the figure are the foci, CD the major axis, A]> 
 the 7ni7ioi' axis. The eccentricity is the ratio that the minor axis 
 bears to the major axis. An ellipse is a plane figure bounded by 
 such a curved line that if from any point in it straight lines be 
 drawn to two points within, called the foci, their sum will be a 
 constant quantity. 
 
 56 Ansivers : 9 175 sq. in. 124 cu. in. 10 600 sq. in., 1000 
 cu. in. 253| sq. ft., 274f cu. ft. 80f sq. ft, 49^^ cu. ft. 114ifin. 
 12 4.4+ ft. 18 384 sq. in. 19 23.34+ cu. ft. 20 493^ cu. in. 
 
206 GEADED ARITHMETIC. [A'III. 57 
 
 This and the following page of exercises should be taught largely 
 from the blocks. The following notes apply to some of the more 
 difficult points : 
 
 A dihedral angle is the opening between two intersecting planes. 
 A trihedral angle is the opening of three planes which meet at 
 a common point. 
 
 A tetrahedron is a solid bounded by four planes. A regular pohj- 
 hedron is a polyhedron having equal and regular faces and equal 
 polyhedral angles. A prism is a polyhedron bounded by parallelo- 
 grams and two equal and parallel polygons, called bases. A right 
 jorism is a prism whose lateral edges are perpendicular to the bases. 
 A parallelopiped is a prism all of whose faces are parallelograms. 
 A pyramid is a polyhedron bounded by triangles that have a com- 
 mon vertex, and by a polygon, called the base. A regular pTjramid 
 is a pyramid whose base is a regular polygon, and whose vertex is 
 directly above the center of the base. A quadrangular pyramid 
 is a pyramid whose base is a quadrilateral. For other suggestions 
 relating to pyramids, etc., see Manual, page 171, and for rules called 
 for, see Appendix, page 115. 
 
 57 Answers: 11 207.3456 sq. ft. 226.1952 cu. ft. 502.656 
 sq. in. 4021.248 cu. in. 12 10.472 sq. ft. 28.4925+ sq. ft. 
 13 3.5342 cu. ft. 41.4517 cu. ft. 14 2869.0858 cu. in. 15 804.2496 
 sq. in. 2144.6656 cu. in. 226.9818 sq. ft., 321.5575 cu. ft. 2576.896 
 sq. ft., 10761.79392 cu. ft. 16 25.4+ in. 17 6 sq. ft. 2.76+ ft. 
 8.676+ ft. 18 10.4+ ft. 19 2.01+ in. 20 32.127+ gab 
 
 A sphere is a volume that may be generated by the rotation of 
 a semi-circle upon the diameter as an axis. The diameter of a 
 sphere is a straight line passing through the center and having 
 its extremities in the surface. A great circle of a sphere is a 
 section made by cutting the sphere through the center. A small 
 circle of a sphere is a section made by cutting the sphere outside 
 the center. A spherical zone is the surface of a sphere included 
 between the circumferences of two parallel circles, or between the 
 circumference of a circle and a parallel plane tangent to the sphere. 
 
VIIT. 58] teachers' manual. ^0? 
 
 A spherical segment is a portion of a sphere included between two 
 parallel circles, or between a circle and a parallel plane tangent 
 to the sphere. A spherical sector is the portion of a sphere which 
 may be generated by the rotation of a sector of a circle upon a 
 diameter of the sphere as an axis. 
 
 Notes applying to other exercises on this page will be found on 
 pages 172 and 173 of the Manual. 
 
 58 Anstvers: 5 499.2 lb. 1684.8 lb. 1.11+ ft. 2.52+ ft. 
 
 6 8:1. 7 5 ft. 8 2 ft. 9 27 cubes. 10 12.7 ft. 11 118|ibu. 
 G ft. 3.4+ in. 12 13.3+ ft. 13 4.3+ ft. X 6.4+ ft. 
 
 59 Ansivers: 1 10.122+ ft. 15.69+ ft. 12.4+ ft. 10.5+ ft. 
 2 («) 80 ft.; {h) 1432+eu. ft.; {<■) 14.9+ cords. 3 2010624000000 
 sq. mi. 25673657856000000 sq. mi. 4 9 times 27 times. 5 Moon 
 = .001801824 of earth Earth = i^^\^^^ of the sun. 6 9.4+ in. 
 
 7 Surface, 432 sq. in. Volume, 610.56+ cu. in. 8 3.04+ in. 
 2.+ in., 2.5+ in., 10.39+ in. 9 Diameter, 11.6+ in. Depth, 5.6+in. 
 
 10 624 balls. 11 3040+ cu. in. 12 6.075 T. 13 17.7 + ™™ 
 
 GO Ansicers: 1 (r/) 12 ft.; (Ji) 12 ft.; (c) 8* ft.; ((/) llyV ft- 
 
 2 {a) 13ift.; {b) 12i|; {<■) 29^1 ft. 3 )i!41.S5. 4 4819.5 ft. 
 5 $27.18. 6 10^ bundles $2.83^ (allowing h of openings). 
 7 58 bundles. 8 (<i) 2401f ft.; {h) 2202i ft.; (^•) 1908 ft.; 
 {(1) $74.78; (e) $193.36. 
 
 61 Ansivers: 1 450 sq. ft. 2700 sq. ft. 2 2i|§a C. 
 
 3 14-/U)j C. 4 58fg. 5 $42. 6 113 cd. 7 7104 bricks 
 $99.46. 8 195840 bricks 172836 bricks. 9 36.35 gal. 
 
 63 Ansicers: 1 {a) 109494 bricks; {h) $1067.57; (c) 130 
 
 bundles; {<!) 8816 sq. ft. 2 {a) $8.17; {b) $6.48; {(■) $15.44; 
 
 {d) 18 rolls; (e) 39 J^ yd. 3 $18.50. 4 311 yd. 29|| yd. 
 5 15 rolls $23.94 $5.30. 
 
 63 Answers: 1 About 15 T. 2 2.15+T. 3 5.4+T. 4 10.9+ 
 
 ft. 5 9iaft. 6 422.4 cd. 1 267.2 bu. 422.4 cu. yd. 7 7480 
 
 gal. 8 97.9+ bl)l. 9 154.9+ lb. 10 About 42 C. $297.60. 
 
 11 697.5 T. 3555| sq. yd. 12 $24.92. 
 
208 GEADED ARITHMETIC. [VIII. 64 
 
 64 Answers: 2 38| bcl. ft. 3 82^ ft. 4 935ft. 5 37|ift. 
 
 65 A7isivers: 1 57 sq. ft. $4.50 47| sq. ft. 2938 packages. 
 2 848 boxes. 3 41.5 sq. ft. $11.14. 
 
 66 Ansivers: 2 63+ ft. 10^ 8 49.6+ sq. ft. 8/. 
 
 67 Ansivers : 1 14.18+ sq. ft. 60.13+ sq. ft. 2 50.26+ sq. in. 
 201.06+ sq. rd. 3 14.28+ rd. 25.72+ rd. 4 19.635 sq. ft. 
 10.90+ sq. ft. 4.36+ sq. ft. 17.45+ sq. ft. 5 80 rd. 6 $70.20. 
 
 7 5392 ft. 8 60 yd. 58.7+yd. 9 Each side, 35.3+ft. 10 Length 
 of parts, 96 ft. and 54 ft. Area, 1944 sq. ft. and 3456 sq. ft. 
 11 1413.72 sq. ft. 12 $1140.02. 13 31.8 lb. 14 51.9 lb. 
 
 68 Answers : 1 60 ft. 2 2400 sq. yd. 3 1692 sq. ft. 4 Each 
 side, 50 yd. Altitude, 48 yd. 5 Surfaces, 1 to 4 ; volumes, 1 
 to 8. 6 if . 7 2 in. 8 8 times as much. 9 $5.18. 10 25.1328 
 in. 11 58.8+ ft. 12 157.08 sq. ft. 13 17.7+ ft. 14 42.423+ 
 ft. 15 Triangle, 996 sq. ft. ; square, 1296 sq. ft. ; circle, 1574.48 
 sq. ft. 16 $90.80. 17 Volume of frustum, fj of cone. 
 
 69 Ansu-ers: 1 5026.56 sq. ft. 11309.76 sq. ft. 485.43+ sq. 
 in. 67.5+ sq. ft. 2 7.0+ ft. 3 2901.6+ sq. ft. 4 1890.7+ ft. 
 5 4.9+ tons. 6 12.4+ ft. X 38.3+ in. X 22.1+ in. 7 794596 
 sq. ft. 8 12 shingles 8 shingles 64+ bundles. 9 8.485 rd. 
 
 10 1022.9+'". 11 64 rd. 6 ft. 12 10 mi. 3 hr. 32 min. 
 
 70 A7iswers: 1 65.45 cu. in. 2 20.6+ . 3 1.49+ sq. in. 
 4 196.7 ft. 5 170.3 ft. 6 $44.44. 7 51.6 ft. 24510 ft. 
 
 8 3§f ft. 9 15.5+ in. 10 460.1 ft. 4 A. 135jy^- sq. rd. 
 
 11 2261.95+ sq. ft. 537.21+ sq. ft. 
 
 71-73 Nearly all these exercises are a review of business 
 exercises given in Books VI. and VII. Lead the pupils to give 
 concrete examples before definitions. In some cases, as in describ- 
 ing the various features of a promissory note, it Avould be well to 
 have the examples written out in full. 
 
 In some States no days of grace are allowed, and in some States 
 days of grace are allowed only under certain circumstances. The 
 
VITT. 74:-75] TEACHERS* MANtJAL. 209 
 
 teacher should ascertain the hiw and practice in this regard, and 
 explain to the pupils. 
 
 A collateral note is one given with stocks, bonds, or other property 
 as security, empowering the payee to sell the same if the note is 
 not paid when it becomes due. An accommodation note is one for 
 which tlie maker receives no consideration. It is given for the 
 purpose of lending credit to the payee. Forms of these notes can 
 be obtained at a bank. 
 
 The endorser of a note makes himself responsible for its pay- 
 ment, unless he writes the words " without recourse " before his 
 name. 
 
 In answer to 7, page 73, there may be given examples of general 
 partnership, in which the partners have the same or different 
 capital, and examples of limited partnership, in which the respon- 
 sibility of one or more of the partners is limited to the amount 
 invested. The general custom of merchants as well as laws of the 
 State regulating partnerships should be ascertained and explained 
 to the pupils. 
 
 74—75 The rulings of the cash account should be made as 
 indicated. After the form given on page 75 is carefully looked 
 over by the pupils, they may be asked to write out the account in 
 full. The balance Sept. 21 is $246.61. 
 
 76—78 These items should be used in writing cash and per- 
 sonal accounts, as previously shown. Lead the pupils by abundant 
 examples to learn the use of the terms debit and credit, debtor and 
 creditor. A person who receives may be called a debtor, and one 
 wlio gives, a creditor. It may be helpful for pupils in determining 
 which side of the cash account certain transactions shall be placed 
 to apply the same distinction of receiving and giving to the money- 
 drawer or cash-box. What is in the box in opening the account 
 and what is put into it are to be debited to the account ; what is 
 taken out of the box and what remains in balancing the account 
 are to be credited. 
 
 The items given on page 76 should be written out by the pupils 
 in the following form : 
 
210 
 
 GtiADEt) Arithmetic. 
 
 [VIII. 70 
 
 ^ 
 
 d <S Its 1^ "^ ^ 
 
 Ci OS ^ Qi to Ci 
 
 ho 
 
 
 1^ 
 
 
 d , Co 00 <3-J <i! 00 
 to ^ '-( ^ SO 
 
 i^ 
 
 to to 
 
 '--t to 
 
 to ^ 
 
 to "sj 
 
 so 
 
 
 '^ so 
 00 -to 
 
 "^ 
 
 
 to 
 to 
 
 to 
 
 
 o 
 
 
 to 
 
 
 a 
 
 
 ©•J 
 
 -^o 
 
 •'!>^ 
 
 O 
 
 
 ~> 
 
 -o 
 
 o 
 
 o 
 
 s. 
 
 , 
 
 S 
 
 
 
 -« 
 
 w 
 
 so 
 
 e 
 
 o 
 
 l-< 
 
 •cO 
 
 s 
 
 '■-^i 
 
 fO 
 
 >J 
 
 O 
 
 O 
 
 •■^ *^ ^ 
 
 ~ o S 
 
 »0 so S 
 
 » 5i -2 
 
 K) - 
 
 CO 
 
 C} - "^ e^ <?5 ^> 
 
 1-^ »-H 
 
 & 
 ^ 
 
 Co to '^ 
 *-H <5^ ^ 
 
 
 to "^ ^ to to 
 to J^ Co to to 
 
 
 to 
 to 
 
 to 
 
 U5 to 
 
 ^^ to 
 
 to to to 
 i^ to Co 
 
 
 so to to to 
 to Co lo So 
 
 to v^ lo "^ 
 to li^l ^ to 
 
 to 
 
 to 
 
 s<( >~i 00 to lo 
 to "o '-s 
 
 ■^ 
 
 to 
 
 CO 
 
 to 
 
 '-H t^ 
 
 So *~< '3'} 
 
 
 Co So to 
 So S>J >-i 
 
 '-H 
 
 1^ 
 
 to 
 
 to 
 
 ■5* 
 
 Cc '^5 ^ to 
 iti So Q-i so 
 
 (5i 
 
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VIII. 77] 
 
 TEACHERS MANUAL. 
 
 211 
 
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212 
 
 GEADED ARITHMETIC. 
 
 [Yin. 77 
 
 
 
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VIII. 78] teachers' manual. 213 
 
 78 The items on the debit side of the cash account are $137.50, 
 $2.40, $3.70, $2, $12.75, $r)2.50, $1.96, $3.60, $1.26, $27.90, 
 $3.72, $6.60, $.54, $1.15, and $3. The items on the credit side 
 are $13.50, $3.36, $6.60, $7.38, $5, $27.62, and a balance of 
 $210.12, making a total footing of $273.58. The balance in A's 
 account is $62.28 (debit side), and in B's account, $13.50 (credit 
 side). The last questions may be postponed until after other 
 accounts have been written. 
 
 79-" 83 In this exercise, and in each of the seven following 
 exercises, it will be found convenient to have the pupils use a 
 sheet of letter size paper having twenty-nine lines, the first and 
 second pages to be used as Day Book, and the third and fourth 
 pages as Cash Book and Ledger. It may be well to go over 
 the items of this exercise and explain each one not fully under- 
 stood. Wiien this is done, let the pupils rule their papers and 
 make out the account in full before comparing it with the account 
 given. 
 
 The following are the missing items of pages 82 and 83, given 
 in the order omitted : 
 
 Cash Dr., Jan. 15, To A. Lawrence on V^j $20.00 ; Jan. 25, To 
 A. Lawrence, $6.96. Casli Cr., Jan. 31, By balance, $51.02. 
 Footing of Cash account, $109.77. Amos Lawrence Di"., Jan. 12, 
 To mdse., $2.90. Amos Lawrence Cr., Jan. 15, By cash on account, 
 $20.00 ; Jan. 25, By cash in full, $6.96. Footing of Amos Law- 
 rence's account, $46.96. 
 
 Charles Smith Dr., Jan. 13, To mdse., $1.50 ; Jan. 18, To mdse., 
 $4.55. Charles Smith Cr., Jan. 17, By labor, $5.25 ; Jan. 18, By 
 wood, $13.50 ; Jan. 31, By balance, $23.09. 
 
 Balance Sheet Dr., Jan. 31, To cash on hand, $51.02 ; Jan. 31, 
 To C. Smith, $23.09. Balance Sheet Cr., Jan. 31, By H. Brown, 
 $18.00 ; By net capital, $1456.11. 
 
 Net capital Jan. 31, $1456.11. Net capital Jan. 1, $1319.71. 
 
 84 On the following four pages are the Day Book and Ledger 
 accounts from these items. 
 
214 
 
 GRADED ARITHMETIC. 
 Day Book. 
 
 [VIII. 84 
 
 JVest Newton, May 1, 1889. 
 
 C.B. 
 
 Mdse. on hand. 
 Cash " 
 
 A. A. Evans, 
 
 To 15 gal. N. 0. 7nolasses, 
 " 4 i^- Formosa tea, 
 
 Hiram Carter, 
 
 To 2 Ih. Yt. butter, 
 " 4 lb. Persian dates, 
 
 Cr. - 
 
 C.B. 
 
 C.B. 
 
 By cash on "/c 
 
 A. A. Evans, 
 By cash un "/c 
 
 8 
 
 Dr. 
 
 To 2 bu. potatoes. 
 
 11 
 
 Hiram Carter, 
 
 To 3 lb. Rio coffee, 
 " 1 box sardines, 
 " 3 lb. Smyrna figs, 
 " 10 lb. B. H. sugar, 
 
 Cr. - 
 
 By 2 days'' labor, 
 " 2 loads gravel, 
 
 Dr. 
 
 .60 
 
 .75 
 
 Dr. 
 
 .25 
 .15 
 
 Cr. 
 
 .80 
 
 Dr. 
 
 .25 
 
 .20 
 .08 
 
 1.87 
 .63 
 
 500 
 
 42 
 
 00 
 00 
 
 9 
 
 3 
 
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 50 
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 75 
 50 
 60 
 SO 
 
 74 
 26 
 
 542 
 
 00 
 
 12 
 
 00 
 
 10 
 
 00 
 
 00 
 
 60 
 
 2 
 
 65 
 
 00 
 
VIII. 84] 
 
 TEACHERS MANUAL. 
 Day Book. 
 
 215 
 
 West Newton, May 15, 1889. 
 
 A. A. Evans, 
 
 To 2\ lb. Mocha coffee, 
 " 8 lb. Rangoon rice, 
 " 4 lb. Malaga raisins, 
 
 — 17 — 
 
 Iliram Carter, 
 To 3 lb. currants, 
 " 1 bbl. St. Louis flour, 
 
 . Cr. 
 
 C.B. 
 
 By cash on Vc 
 " 1^ cd. chestnut wood, 
 
 18 - 
 
 C.B. 
 
 A. A. Evans, 
 By cash on Ve 
 
 20 
 
 Iliram Carter, 
 By 2 days'" labor. 
 
 22 
 
 David Grant, 
 
 To 2 doz. bananas, 
 " 3 gal. P. R. molasses, 
 
 Cr. ■ 
 
 By 25 gal. vinegar. 
 
 28 
 
 C.B. 
 
 Iliram Carter, 
 To cash on Vc 
 
 Dr. 
 
 .32 
 .10 
 .18 
 
 Dr. 
 
 .15 
 
 5.00 
 
 Cr. 
 
 Cr. 
 
 1.38 
 
 Dr. 
 
 .50 
 .65 
 
 .19 
 
 Dr. 
 
 
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 7 
 
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 4 
 
 
 
 12 
 
 32 
 
 20 
 
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 76 
 
 95 
 
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216 
 
 GKADED ARITHMETIC. 
 
 [VIII. 84 
 
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VIII. 84] 
 
 teachers' IVIANUAL. 
 
 217 
 
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218 
 
 GRADED ARITHMETIC. 
 
 [VIII. 85 
 
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VIII. 86] 
 
 TEACHERS MANUAL. 
 
 219 
 
 86 This account is supposed to extend from INIay 2 to May 20. 
 The last items of the tirst page of the Day Book should appear as 
 follows : 
 
 C.B. 
 
 
 q 
 
 
 4 
 4 
 
 George G. Gates, 
 By 3 European larch, 
 " 2 Wisconsin willow. 
 
 Cr. 
 
 1.12 
 .62 
 
 4 
 
 To cash on "/c 
 
 
 
 
 3 
 1 
 
 36 
 24 
 
 
 
 4 
 
 10 
 
 GO 
 
 00 
 
 On the second page of the Day Book there will be the accounts of 
 Gates, 5 lines ; Hart, 2 lines ; Gates, 4 lines ; Eaton, 2 lines ; Hood, 
 4 linos ; and Gates, 4 lines. The cash account on the third page 
 of the sheet will occupy 22 lines, and give room on the yjage for 
 the Ledger accounts of Isaac Hart and A. M. Eaton. Cash is 
 credited with the following amounts : $4.50, $9.66, f 3, |2.50, 
 $3.04, $10.48, $2.52, $10, $4, $1.68, $1.52, $30, $2.16, $3.90, 
 $4.74, $1.76, $1.24, $.58, and balance on hand May 20, $298.43. 
 On the fourth page of the sheet will appear the Ledger accounts of 
 Horace Hood, 6 lines, and G. G. Gates, 9 lines, and the Balance 
 Sheet. The debit side of the Balance Sheet will be : Mdse. on hand, 
 $1000 ; Cash on hand, $298.43 ; Horace Hood, $98.30. The credit 
 side will be: A. M. Eaton, $59.25; G. G. Gates, $27.74; Net 
 capital May 20, $1309.82, making a total footing of $1396.81. 
 The net gain ($139.57) should be noted as before. 
 
220 
 
 GRADED ARITHMETIC. 
 
 [VIII. 87 
 
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VIII. 89] teachers' ]vianual. 221 
 
 89 After the first four items uiuler tlu; first date, Apr. 1, the 
 following entries of the J)ay Book should be made in order: First 
 page : C, 2 items ; A, 3 items ; B, 3 items ; A, 1 item ; C, 1 item. 
 Second page : B, 3 items ; C, 3 items ; A, 2 items ; C, 3 items ; A, 
 
 I item ; A, 3 items. In the Cash account, there will be 9 entries 
 on the debit side and 13 entries on the credit side. C's Ledger 
 account will also be on the third page of the sheet. 
 
 Balance Sheet Dr.: Mdse. on hand, $3000.00; Cash on hand, 
 $2473.38 ; Amos Lawrence (A), |280.()0 ; Bills Receivable, $250. 
 Cr.: Charles Smith (C), 1761.25; Net capital, $5242.73. 
 
 90 This account is supposed to extend from March 1 to March 19. 
 The cash balance March 9 is $113.04, and May 19, $199.00. The 
 Balance Sheet is as follows: Dr.: Cash on hand, $199.60; Mdse. 
 on hand, $400 ; Byron Waters, $40.00. Cr.: Dennis Rand, $4.78 ; 
 Net capital, $634.82. 
 
 92 Answers: 3 26 56. 4 /± [r/ X (« - 1)]. 5 50 39 10. 
 
 6 '7 = -^— 4,or,^^^; w = ^^, or,-^-^- 7 5. 8 6. 10 108. 
 n. — 1 71 — 1 a a 
 
 II s='^^Xn. 12 420. 13 $1.45 $19.50. 
 
 It would be well to place upon the board several series of num- 
 bers like the following : 
 
 (1) (2) (3) (4) (5) (6) (7) (8) 
 
 13 5 7 9 11 13 15 
 
 4 8 12 16 20 24 28 32 
 
 30 27 24 21 18 15 12 9 
 
 Lead the pupils to discover the fact that in each of the above 
 series of numbers there is a constant difference between the con- 
 secutive terms, and to make a definition like the following : Arith- 
 metical progression is a series of numbers which increase o.- 
 decrease by a constant difference. 
 
 ..Any term of an arithmetical series may be found by multiplying" 
 the common difference by the number of terms which preceded it. 
 
222 GRADED ARITHMETIC. [VIII. 93 
 
 The other rules may be readily found from the given examples. 
 To teach the rule for finding the sum of an arithmetical series, 
 place upon the board any series, as : 
 
 3 6 9 12 15 18 
 and, directly below it, the same series reversed. The two series 
 will appear as follows : 
 
 3 6 9 12 15 18 
 21 18 15 12 9 6 
 24 + 24 + 24 + 24 + 24 + 24 
 
 = twice the sum of the series. Therefore the rule : The sum of 
 an arithmetical series is equal to the product of one half the sum 
 of the first and last terms multiplied by the number of terms. 
 
 93 Ansivers: 6 2430. 7 60f. 8 5. 10 98304 131070. 
 11 $25.60 $51.15. 12 $59049 $88573. 13 $179.08. 14 9yr. 
 2 da. 15 $10737418.21. 
 
 First place upon the board two or more series of numbers in 
 geometrical progression, as follows : 
 
 2 6 18 54 
 
 40 20 10 5 
 
 The pupils will see that each of the above series of numbers 
 increases or diminishes by a constant ratio. Geometrical progres- 
 sion, therefore, is a series of numbers which increase or diminish 
 by a constant ratio. 
 
 To show how any term or ratio of a geometrical series may be 
 found, write the factors of each term of the series upon the board ; 
 thus : 
 
 2 6 18 54 
 
 2 2X3 2X3X3 2x3X3X3 
 
 By questioning tlie pupils upon the above numbers, tlie following 
 rules may be develo])ed : 
 
 The last term of a geometrical series is equal to the first term 
 multiplied by the ratio raised to a power whose degree is one less 
 than the number of terms. The first term is equal to the last 
 
VIII. 94] teachers' manual. 223 
 
 term divided by the ratio raised to a power whose degree is one 
 less than the number of terms. 
 
 The ratio is equal to the root whose index is one less than the 
 number of terms, of the quotient of the last term divided by the first 
 term. 
 
 The sum of the series may be found as indicated in 9. Other 
 formulas may be expressed as follows : 
 
 94 Jnsivers: 1 6f/. 2 41 /^Z. 3 2 lb. @ 9/, 1 lb. @ 6/ 
 
 4 ]b. @ 6/, 4 lb. @ 9/. 4 1 lb. @ 50/, 1 lb. at 65^, 2^ lb. @ $1. 
 
 5 15 lb. 6 20 lb., 20 lb., GO lb. 7 12^ gal. 8 1 lb. 2| oz. 
 9 1 lb. 11 oz. 11 pwt. 5.28 gr. 10 3.98 s. 11 G oz. 12 3 oz. 
 5 pwt. 13 $2.45. 14 31%. 15 23.52 oz. 49 lb. 
 
 Many of the questions and answers given on the remaining pages 
 were given by business men, mechanics, or specialists. 
 
 95 Answers: 1 $39110. 2 $3600 loss. 4 I3G1.07. 
 5 $545.43. 6 $216.01. 7 85 lb. 89 lb. 8 Jan. 15. 9 About 
 4^%. 10 5% stock 1% better. 
 
 96 Answers: 1 $116.68. 2 $172.72. 3 $392.20. 4 $112^ 
 $75 5.347%. 5 $8.22. 6 $26.44. 7 29531 bricks. 
 8 5276.97984 ft. 9 $21.56. 10 $.128 + . 
 
 97 Aiiswers: 1 $5,472. 2 Rivets, f lb; burrs, ^ lb. 3 650.4 
 yd. 4 $172.40. 5 $850. 6 5hi% gain 3^^% loss. 7 $11.12 
 18 rolls 14 yd. 
 
 98 Answers: 1 (a) 8200 ft.; (h) 7520 ft.; (c) 4700 ft.; (d) 151 
 bundles; (e) 108 bundles ; (/) 660 ft.; (y) $591.39. 2 $87936.51. 
 3 A, $9000 B, $10125 C, $7875. 4 $811.69. 5 125 bu. 
 140 bu. 6 (a) 374 yd.; (b) 1 A. 144 rd.; (c) 80 sq. rd.; (d) 108 
 sq. rd. ; (e) 116 sq. rd. 
 
 99 Answers: 1 2^9^ cd. If cd. 2 105° 15'. 11 720 pass- 
 books. 12 Mercury, 5f as large as the moon. 14 1350 sq. in. 
 1633^ sq. in. 15 8.944+ rd. 8 rd. 15.576+ ft. 
 
224 GRADED ARITHMETIC. [VIII. lOO 
 
 100 Answers : 1 66 rd. 5 ft. 3 A. 5 sq. rd. 12 sq. yd. 7 sq. ft. 
 72 sq. in. 2 $408.41. 3 $44.12. 4 1051 bricks. 5 75.2 
 65.2 32.2 31.2 27.2 16.22 15.01 11.94. 6 41 strips 13 
 rolls $2.44 37 yd. 7 646.5+ bbl. 
 
 101 Answers: 1 $10150. 2 253 sq. ft. 18 sq. in. 3 $120. 
 
 4 4096 bullets. 5 A, $512.82 B, $687.18. 6 20 yd. 7 July 16. 
 
 8 2^s pi' pgy ptt ara. 9 11^-% gain 11^% loss 8% loss 
 15.1% gain 42f% gain. 10 41.4+ bbl. 11 56.57+ gal. 
 
 103 Answers: 3 57+% 28+% 68+%. 4 KY., 1123385932. 
 
 5 S. Dakota, 1048+%. 6 1880: 16983034673; 1890: 25265581794. 
 
 103 Ansivers: 1 IS h. 37 niin. 10 sec. 2 5§||aA. 3 4|fi 
 Eng. mi. 4 4071 mi. 5 Wheat, 530| bu.; apples, 442 bu.; 
 beets, 442 bu. ; carrots, 442 bu. ; potatoes, 442 bu. ; corn, 221 bu. ; 
 salt, 2304 bu. 6 80 perches. 7 567 J^ 6171 T. . 8 Wheat, 
 33^ bu. ; rye, 35| bu. ; oats, 57^ bu. ; barley, 41| bu. ; salt, 40 bu. ; 
 potatoes, 33-J bu. ; coal, 25 bu. 9 107Lbu. 10 6^ lb. 11 $26.67. 
 12 $10 gain. 13 600 lb. 14 $13.14. 15 Radius, 120 in. 
 
 In 5, the approximate answers are given, the bushel being reck- 
 oned as containing 1^ cu. ft.; heaped bushel, 1^ cu. ft., and 2 bu. 
 ears as equal in bulk to 1 bu. shelled corn. 
 
 104 Ansivers: 1 Pop. Sweden, 4784675 ; pop. U.S., 62622250 ; 
 children, Belgium, 827928 ; rate p.c, Bavaria, 21.2 ; rate p.c. Neth., 
 14.4. 2 £3906 5.S. 3 $500.89. 4 About 13 m. 5 18+%. 
 
 6 44+%. 7 $1600. 8 4 h. 6.54+ min. P.M. 9 50% 5%. 
 
 105 Answers: 1 $504.07. 2 2d is greater discount by 2.85%; 
 $22.80 saved. 3 $9.75 gain. 4 $436^=*^ $163i7t-- 5 59.48+ 
 gaL 6 1650 mi. 7 10,;; h. 8 2096.11 $1855.59 $2511.22. 
 
 9 83^ ft. 10 15(;l% premium. 
 
 100 Ansivers: 1 33 min. 27.5+ sec. 15 min. 40.1+ sec. 33 
 min. 16.8+ sec. 2 78.5+ bu. 133.5+ bu. 307.9+ bu. 3 1302.3+ 
 bu. 4 890.1+ gal. 5 1692.02 gal. 35.7+ in. 6 91.19+ bbl. 
 9 .019+'"'". 10 1.655012™'="' 1.04+%. 
 
VIII. 107] teachers' manual. 225 
 
 107 Ajiswers; 1 .49 3.94%. 2 3045 meters. 3 59.79 
 times. 4 47.31 in. 9.775 in. 5 ().316 sec. 6 144.72 ft. 
 305.52 ft. 466.32 ft. 7 3618 ft. 8 188.94 ft. per sec. 
 
 108 Ansa-ers : 1 4 ft. 2 11^ in. from end wliere the Kilo- 
 gram weight is. 3 6|- in. from middle on same side as 2 and 3. 
 4 129.16^. 5 8 lb. on balance near the weight, and 4 lb. on the 
 other. 6 11 lb. on balance near the 12 lb. weight ; 10 lb. on other 
 balance. 
 
 Many solutions are possible for 7; e.g. 60 lb. in middle, and 
 40 lb. 2^ ft. from man ; or, 40 lb. in middle, and 60 lb. 3-^ ft. from 
 man. 
 
 109 Answers: 1 49+ lb., i.e. anything in excess of 49 lb. 
 2 Anything in excess of 1 of person's weight. 3 25+ lb. 4 20 + 
 lb. 5 73.5+ lb. 6 18849600 lb. 2282568.75 lb. 7 -^\ ^^ i. 
 
 110 Answers: 1 601b. 2 58.92 1b. 3 .84 1b. 4 12 boys 
 12 boys. 5 BoxAvood, .537+; maple, .463+. 6 7.29. 7 1.125. 
 8 2.56 2.08""". 9 1.13+ . 10 .0000™^. 11 26.32 ft. 
 
 111 Answers: 2 .565+. 3 .376. 4 .307. 5 264 297 
 330 352 396 440 495 528. 6 About 43°. 7 About 20 
 millions of millions of miles. 8 12.7+ . 
 
 The computed result of 1 is 18.99+*^™. The difference may be 
 disregarded, as it is not greater than probable error in reading 
 scale used. 
 
 In 8, sound is reckoned as traveling 1120 ft. per second. 
 
 113 Answers: 1 A little more than 116°. 2 20.78 in. 3 3.144 
 ohms. 4 100.61 ohms 40.24 ohms. 5 -J- its weight at surface 
 ^ its weight at surface 0. 6 2800 mi. from centre. 7 25 lb. 
 11^ lb. 4 lb. llj lb. 6i lb. 8 24000 mi. 
 
 113 Answers: 1 20.8+ sec. 2 Ifi times as loud. 3 4.99+ 
 qt. 4 12.58+ pk. 5 34.37+ ft. 6 2500 lb. 1.55+ ft. 7 1 ft. 
 1.1+ in. 8 About 117 bbl. 8 2160° F. 10 2880° F. 11787.92 
 ft. 12 3.61+ sec. 7.82+ sec. 13 9.775 in. 156.4 in. 14 2.9+ 
 times a second. 
 
ADVERTISEMENTS 
 
16 
 
 HILL'S LESSONS IN GEOMETRY. 
 
 •' 
 
 HILL'S LESSONS IN GEOMETRY. For the Use of Beginners. By G. A. Hill, 
 A.M..,T^\\i\\oxoii\\eGeo7nc•t^yforIyc'ffil!l!t:rs. Illustrated. i2mo. Cloth. 190 pages. 
 
 For introduction, 70 cents. Aiiswcrs, in pamphlet form, can be had by teachers. 
 
 In France and Germany, says the President of Harvard Univer- 
 sity, they begin geometry at ten years of age, and in its right place, 
 namely, in connection with drawing. Geometry, pursued in the 
 proper way, ought to be introduced in grammar grades, and within 
 a few years it doubtless will be. The right method is known, and 
 is embodied, it is believed, in this book. The subject is presented 
 objectively and practically, instead of abstractly and theoretically. 
 The central purpose is intellectual training, — teaching by practice 
 how to think correctly and continuously, — but at the same time the 
 main facts and principles of geometry are taught, and also a great 
 deal of the most useful knowledge. The exercises, involving the use 
 of simple instruments and drawing to scale, are of great interest and 
 educational value. 
 
 ^EOMETRY taught in this way is interesting, disciplinary 
 and practically valuable in a very high degree. 
 
 Gives real and living knowledge. 
 
 For giving to students who have never 
 studied geometry a real and living knowl- 
 edge of the subject and a command of its 
 more important applications, I know of no 
 book equal to this. — C. C. Rounds, /"/-/«. 
 State A'orDial School, Plymouth, N.H. 
 
 Full of inspiration. 
 
 It is a delightful little work, and full of 
 inspiration. A teacher who gets the spirit 
 of that book into him cannot fail to teach 
 well. In the hands of the pupil I know of 
 nothing that approaches it. — Corwin F. 
 Palmer, Si<pt. of Schools, Dresden, Ohio. 
 
 HILL'S DRAWING CASE. Prepared expressly to accompany Hill's lessons in Geome- 
 try, s.x\d containing, in a neat wooden box, a seven-inch rule with a scale of millimeters; 
 pencil compasses, \\\\.h. pencil and rubber; ^triangle; zxid & protractor. Retail price, 
 40 cents; for introduction, 30 cents. 
 
 A specimen copy of the Lessons in Geometry, with the Drawing Case, will be sent, post- 
 paid, to any teacher on receipt of ^i.oo. 
 
 FRACTIONS. A Teachers' Manual of Objective and Oral Work. By Helkn F. Page, 
 State Normal-Training School, Willimantic, Conn. 8vo. Boards, iv + 47 pages. 
 Introduction price, 30 cents. 
 
 Pupils' Edition. Containing over three hundred e.\amples, illustrated with color- 
 diagrams. Svo. Boards. 52 pages. Introduction price, 30 cents. 
 
 PRIMARY NUMBER CARDS. Prepared by Miss Isabel Shove, of the George 
 Putnam School, Boston. Printed on cardboard, and boxed in sets of 60. Price, 
 25 cents. 
 
BOOKS FULL OF LIFE AND THOUGHT 
 
 WEIiTWORTH'5 ARITHMETICS 
 
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 MONTGOMERY'S 4MERIC4N HISTORY 
 
 A panoye.ms. of tl^e kc^clii??' facb Fl^eir caiU5cs 
 s^nd Vf7eir resalts 
 
 STICKNEY'S REAPERS 
 
 Best ip ide^ and plap.best ip n^atler AT)d n?akc 
 
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 firrr; biodip^, locy prices 
 
 STICKHEY'5 WORD BY WORD 
 
 Af) improved spelling coarse in tuuo numbers, 
 conservative and original 
 
 BLAiSDELLS PHYSIOLOGIES 
 
 Endorjed by the pKysicievn^, tKe ;ciei\h7ic rqen. 
 tl:]e rT)orali5l-5, t\\c kacher^ 2vT]d ti\e W.C.TU. 
 
 T4RBELLS LESSOR'S in L4n(3y>ACJE 
 
 Expressi'09 \\roUi^h ajriWer] forrrjs n7s.de cxs 
 r)2>,^aYa\ as \]rioa<^h\: ^.rjd speeciy 
 
 THE MEW mTIOMAL MU5IC COURSE 
 Studied by n^orc pupils \\)^y) a.11 other 
 re^alsvr c caries to^etF^er 
 
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