FsliMi^^ y ri V | l „l ' II I I I E II== 1 cKool Problems A JoumaJ for the PracticjJ Use of Teachers CONTENTS THE TEACHING OF READING Margaret Orvis McCi,osk;ey i General Supervisor of Soliools, Newark, N. J. Plan Phonic Lessons, First Month 5 Plan Lang:uage Lessons, First Semester 9 8th Year. THE TEACHING OF ENGI.ISH COMPOSITION JNO. J. BuRKE 12 P. 8. 16, Man. Term Plans, 8B, English Composition 15 8A, English Grammar 16 7tli Year. HANDLING OF TECHNICAL GRAMMAR B. M. F. Caui,FIELD 18 P. 8. 42, Bronx Parsing for Grades 7th and 8th Year 20 Term Plans. 7B, Outline Grammar 21 7A, Outline Grammar 25 6th Year. APPRECIATIVEJREADING Henry Davidhofp, P. S. 18, Man. 28 Term Plan. 6th Year, English Composition 32 5tii Year. RULES FOR SPELLING, PREFIXES AND SUFFIXES M. E. Bonn 34 P. 8. IS, Man. Term Plans. 5B, Prefixes and Suffixes 37 5 A, English Composition 38 4th Year B. THE TEACHING OF PUNCTUATION. ...May F. Ci,ark, P. S. 38, Bronx 39 Term Plans in the Teaching of Punctuation 40 4th Year A. ORAL ENGLISH Anna K. PERGOUE, P. S. 40, Bronx 41 Term Plan. English Composition 42 3rd Year. TEACHING THE PARAGRAPH Anna G. BuTler, P. S. 5, Queens 43 Suggestions for Plan in Article 2nd Year. DRAMATIZATION CATHERINE Maddock, P. S. 6, Queens 46 Suggestive Stories. Term Plan ist Year A. THE TEACHING OF THE SENTENCE f Anna CooTE \ 49 ist Year B. ORAL REPRODUCTION 1 Jamaica Training School / 50 Plans in Article for Teachers KINDERGARTEN ENGLISH Catherine Maddock, P. S. 6, Queens 54 Term Plan in Article BOOK REVIEWS $1.00 per year'in advance BI-MONTHLY, EXCEPT JULYi 25 cents the copy Foreign and Canadian.ftperAjrear in advance, $1.25. 30c. the Copy 3O Volume I SEPTEMBER-OCTOBER, 1909 Number 1 p PUBLISHED BY THE EDUCATIONAL PRESS, 123 East 23d Street, NEW YORK D : n riffht 1909 Application filed for entry as >econd-clast matter at N. Y. Pott Office vi^ % STANDARD TEXT-BOOKS LAWLER'S PRIMARY HISTORY OF THE UNITED STATES LAWLER'S ESSENTIALS OF AMERICAN HISTORY Thorouglily modern, progressive, and practical treatises on American history. FRYE'S FIRST STEPS IN GEOGRAPHY FRYE'S GRAMMAR SCHOOL GEOGRAPHY These are the latest revised editions of fthis epoch-making series. AITON'S DESCRIPTIVE SPELLER SPAULDING AND MILLER'S GRADED SPELLERS," BOOKS I -VII An unrivalled series of spellers for ^the eight yearj|"{requirements. THE MCCLOSKEY PRIMER The new method of reading that has met with such wonderful success in the class-room. A manual for teachers gives detailed suggestions concern- ing the best method of using the book. GINN & COMPANY, EDUCATIONAL PUBLISHERS NEW YORK BOSTON CHICAGO PHILADELPHIA 70 FIFTH AVENUE 29 BEACON STREET 2301 PRAIRIE AVENUE 16th and CHESTNUT STREETS The Right Books for Schools STEPS IN ENGLISH Book One — 40 Cents Book Two — 60 Cents Teach the child how to express his thoughts in written and spoken language rather than furnish an undue amount of grammar and rules. Mark out clearly the dally work for the teacher and relieve her of much that is wearing and unnecessary. Picture study, study of literary selections and letter writing are important features. HUNT'S PROGRESSIVE COURSE IN SPELLING Complete, 20 Cents. Parts One and Two, each 15 Cents. Thlsbooknotonly teaches the child to spell, but develops a wide knowledge of the commoner English words. In all there are some 9,000 different word forms in exercises grouped according to the phonetic, topical, grammatical, antithetic, synthetic and sentential methods, and furnishing work in word building, word analysis, etc. Syllabication, accent and diacritical markings and spec- ial exercises in pronunciation receive attention. HICKS'S CHAMPION SPELLING BOOK Complete, 25 Cents. Parts One and Two, each 18 Cents. This book embodies the method of teaching spelling which, after two years' use, enabled the pupils of the Cleveland Schools to win the victory in the National Education Association Spelling Contest of 1908. The lesson for each day consists of ten words from the spoken vocabulary of the pupU. Two of these words are selected for Intensive study, and given at the beginning of the lesson in heavy type. The remaining eight words follow in smaller type. This constant hammering on a few words brings results and in the entire book some 6,000 words are mastered. AMERICAN BOOKCOMPANY NEW YORK CINCINNATI CHICAGO irW.'^. 1926 40 SEP 1 3 ^^909 Vol.. I. robl^ms ^ Journal for the Practical Use of Teachers SEPTEMBER-OCTOBER, 1909. No. 1. THE TEACHING OF READING. Model Lesson. By Margaret Orvis McCloskey. THE following lesson is based upon the conception that all studies having as a common end knowledge of the mother-tongue, are members of one organism and that the preservation of this or- ganism in teaching is the first essential to success. Accordingly, in any system system literature, reading, oral and written composition, writing, spell- ing, and phonics are closely related and co-ordinated. Each one is suppor- ted and strengthened by the other. The major element, from first to last, is literature through which is naturally developed an interest in reading and a foundation for that study. The best loved nursery rhymes — rhymes of the cumulative type — supply the spirit that giveth life and kindle a powerful motive for learning to read. At the same time they present, in their pecul- iar structure, the formal problem re- duced to its lowest terms. Some idea of the plan in detail may be gained from the brief outline of lesson on the Hebrew version of the House That Jack Built. The Kid. A kid, a kid, my father bought For two pieces of money: A kid, a kid. Then came the cat, and ate the kid. That my father bought For two pieces of money : A kid, a kid. Then came the dog, and bit the cat, That ate the kid, That my father bought For two pieces of money: A kid, a kid. Then came the stick, and beat the dog, That bit the cat. That ate the kid. That my father bought For two pieces of money: A kid, a kid. Then came the fire, and burned the stick. That beat the dog, That bit the cat, - That ate the kid, That my father bought For two pieces of money : A kid, a kid. Then came the water and quenched the fire, That burned the stick, That beat the dog, That bit the cat, That ate the kid. That my father bought ■ For two pieces of money: A kid, a kid. Then came the ox, and drank the water. That quenched the fire, THE TEACHING OF READING. That burned the stick, That beat the dog, That bit the cat. That ate the kid. That my father bought For two pieces of money: A kid, a kid. Then came the butcher, and killed the ox, That drank the water, That quenched the fire. That burned the stick. That beat the dog, That bit the cat. That ate the kid. That m)^ father bought For two pieces of money : A kid, a kid. Preparatory Work. Nature Study. — The kid. The ox. See live objects, if possible, and, if not, use pictures. Oral Treatment of Story. 1. Tell story through in conversa- tional prose, to give the idea of the whole. 2. Tell story more slowly with at- tention to parts, until pupils know the incidents in their right order. 3. At last, tell story rapidly to bring parts together again. Telling the story must not be made a school exercise over which the teacher and the pupils struggle, but if keen interest is aroused and the teacher proceeds indirectly, it is not too much to expect that each pupil will be able to give his version of the story be- fore written symbols are presented. The text is not committed as a prepa- ration for reading, but after the story has been read, it should be remembered exactly and repeated, always with fresh interest, until it has become for each pupil a permanent acquisition. For suggestions on oral work with stories, see : DeGarmo's Essentials of Method, pages 94-107; McMurry's Method of the Recitation, pages 26-29 ; McMurry's Special Method in Primary Reading, pages 1-46; Bryant's How to Tell Stories to Children ; Corson's Aims of Literary Study. One week may be devoted to this preparatory work, and, if the pupils have not had kindergarten work, the teacher may extend this time, filling in the session with well chosen hand work, music, drawing, games, etc. Introdttction of Script Forms. . First day, A. M. — Suggest the delight of being able to read stories and the ease with which reading may be learned. By questions, secure the thought and form of the following and write upon the board in large hand with generous spacing, the words as far apart as they can be without appearing unrelated: A kid, a kid, my father bought For two pieces of money. Secure smooth, confident reading of the above by several pupils. Because they know what the teacher has written, the pupils will repeat this smoothly without knowing the individual words. Through questions, secure a partial analysis of the sentence. What is the story about? A kid. The teacher shows the word kid by underscoring or encircling it in the sentence. Then other words are suggested and distinguished as follows: Who bought the kid? Father. What did he give for it? Money. How many pieces ? Two. As the teacher is doing this, she will stop to have the two, then the three, and finally the four words pointed to, in the sentence, and named. At the end the words will be marked as follows : A kid, a kid, my father bought For tivo pieces of money. MODEL LESSON. AftOT several pupils have named the four words in the sentence, write them, one at a time, in a column and have pupils name them by comparison with the words in the sentence. Then re- move dependence upon position by cautiously erasing the sentence and having the words named independently. Naming the words independently siiould be done rapidly by each pupil as he volunteers. Each child who names them all once through without a mis- take returns to his seat to look at a book or to do hand work. His work on this exercise is done. Any pupil who says at last that he does not know all the words is asked to point to the ones he does know, and the teacher makes a written note of the name and words known. The large number of words given and the individual tests are for the purpose of securing a basis for grouping. If all pupils cannot be tested in one period, the test may be completed at any con- venient time during the day. At the end of this work, temporary groups are made as follows : 1st group. All who learn four words in one period. 2nd group. All who learn three words in one period. 3rd group. All who learn two words in one period. From now on, suggested lessons are for the first group. The second and third groups proceed as rapidly as they are able. It must not be supposed that class grouping is an essential feature of this system. This plan can be used, as well as any other, without, grouping, but grouping is earnestly advised, because in a well-grouped class taught by any method, the results are incomparable to those obtained by the same teacher and method in a similar class taught en masse. The first grouping is, of course, ten- tative. Frequently pupils work in a short time from the third group to the first. Pupils of the higher groups who are long absent or who develop any un- expected weakness, should be placed, at least temporarily, in the lower groups, because nothing can atone for the lack of a solid foundation. In large schools having many be- ginning classes, it is possible and de- sirable to so assign the pupils that each class can be divided into two even groups. At this point word cards are made. On drawing paper or thin cardboard (about nine inches by five inches) the words are written, one on each card, for rapid review. As new words are taught, they are placed on cards and ad- ded to this collection, 1st day. P. M. Review and complete the arrange- ment of groups. 2nd day. A. M. Reading. — Review sentence. Add last line, A kid, a kid. Pupils should recognize and read this unaided. Words : Distinguish as before the word bought. Add this word to those known and reviewed. 2nd day. P. M. Reading. — Review Avith questions. Words. — Pieces. Review. If the teacher prefers, she may teach all of the words in this sentence before going on. But the words repeat so fre- quently that it will be quite safe and probably more interesting to go on and gradually add the small words tem- porarily passed. All new words are secured hj questions on the story dur- ing the reading lesson. No words are taught before the reading as a pre- THE TEACHING OF READING. paratory excerise. The words are fixed by a little drill following, not too closely, the reading lesson. Words. — Rat. Review. 3rrf day. A. M. Reading. — Review. Words. — Cat. Review. Zrd day. P. M. Reading. — Then came the cat, and ate the kid. ^th day. A. M. Reading. — Finish second stanza: Then came the cat, and ate the kid. That my father bought For two pieces of money. A kid, a kid. Words, came. Review. All repeated phrases should be recognized and read without assistance. ^th day. P. M. Reading. — Review from beginning. Words, then. Review. ^th day. A. M. Reading. — Then came the dog, and bit the cat. Words, dog. Review. hth day. P. M. Reading. — Finish 3rd stanza. Words, bit. Review. ^th day. A. M. Reading. — Then came the stick, and beat the dog. Words, stick. Review. %th day. P. M. Reading. — Finish fourth stanza. Words, beat. Review. ''ith day. A. M. Reading. — Review from beginning. Words, my. Review. 1th day. P. M. Reading. — Individual tests should be given with new arrangement as. The cat came and ate the kid. The stick beat the dog, and the dog bit the cat. Sentences which take the mind away from the story are not given. The teacher does not write, for example. The dog ate the kid, because the pupils must not be confused on the events of this narrative. As the early work progresses this ex- ercise should be given just frequently enough to prevent rote work and to de- velop the power to grasp thought from written symbols. Later, the new stories will constitute sufficient and better tests. From the first, pupils should be trained to see words in phrases. Examples. — My father bought a kid. A cat ate the kid that my father bought. If the teacher or the pupil points to the sentence while reading, the pointer must glide smoothly under a phrase. As a rule, each sentence should be read silently by every pupil in the group before it is read orally. This exercise is a better form of word drill than the use of word cards. ^th day. A. M. Reading. — Then came the fire, and burned the stick. Words : fire, burned. Wi day. P. M. Reading. — Finish fifth stanza. Words, that. Review. Three words are suggested from now on because the third word has been re- peated so frequently, but the teacher will not give three, if undue eflFort is re- quired to gain them. 'dth day. A. M. Reading. — Then came the water, and quenched the fire. Words : water, quenched. mi day. P. M. Reading. — Finish sixth stanza. Words : a. Review. Teach pupils to say "a" and "the" when the words stand alone and to make both vowels obscure, a kid, the cat (not u) when part of a phrase. MODEL LESSON. The correct rendering of "a" and "the," in phrases, is a matter of accent rather than of vowel sound. In each word the vowel has the sound of a in an unaccented syllable. Examples : apart, away, again. At the outset no elaborate effort is required to train the pupils to slight the article in reading ex- actly as they do in speaking, but if they once form the habit of making the article as prominent as the word it modifies, much determined vigilance is required to substitute the correct habit. 10th day. A. M. Reading. — Then came the ox, and drank the water. Words : ox, drank. 10th day. P. M. Reading. — Finish seventh stanza. Words : the. Review. 11th day. A .M. Reading. — Then came the butcher, and killed the ox. Words : butcher, killed. 11th day. P. M. Reading. — Finish eighth stanza. Words : and. Review. 11th day. A. M. Reading. — Review from beginning. Words : of. Review and individual tests. Total. — 29 words. Time: About two and one-half weeks. This work may be done in less time than allowed, but there is no cause for discouragement if more time is re- quired. The order of word teaching and the number of words at a lesson are only suggestive. All teachers are expected to make any changes which, in their judgment, seem wise. This outline supposes two fifteen minute periods daily for reading and one or two five minute periods for word drill. No written language, spelling, pen- manship, or phonics are done until a later stage in this work. Emphasis is placed on nature, handwork, stories, reading, music, and games. There should be constant attention to the de- velopment of spontaneity, freedom, and skill in oral language. Outline of Phonic Lessons. 1^^ day. When fifty or sixty words are thoroughly known at sight, there is probably established in the minds of the pupils a tendency to see words first as wholes and then it is safe to begin phonic analysis. A strong motive can be created by calling the attention of the pupils to the fact that they must now depend up- on someone to tell them all new words and suggesting that they learn how to make out new words. The teacher writes the word man on the blackboard and has the pupils name it. She asks the class to listen sharply while she says the word slowly. The teacher pronounces the word slowly and more slowly until she has separated the word into m — an. The pupils are then asked if they can say it as the teacher did. Volunteers are allowed to ivy and when several have succeeded, the teacher asks, "Who can give the first part, m, alone? The second part, an ?'• The next step is to associate the sounds they have heard and made with the letters that represent these sounds. The teacher takes a piece of stiff paper or cardboard and covers an, telling the pupils that she is showing the part that tells them to make the first sound, m. Then she covers the letter m and the children make the second sound, an. Beginning 6 THE TEACHING OF READING. with the best in the group each child then sounds m — an and pronounces the word. If correctly done on first trial the pupil returns to his seat. If a mis- take is made, the pupil waits to try again after others have succeeded. It is important that each pupil should sound the word correctly. No concert work should be allowed. The words are not divided on the blackboard. All temporary divisions are made by covering part of the word to concentrate attention upon the other part. The first lessons in phonics usually re- quire more time than should be allowed later. After a few lessons have been given, the work outlined for each day can be done in five minutes. This exercise may be combined with the drill on sight words or given at a sepa- rate period, as the teacher prefers. '2nd day, Srd day. Phonics. — Rapid review, make writ- ten signs. The teacher writes the word man slowly in large hand, after which she asks if any pupil can write the word after her word is erased. If no pupil volunteers, the teacher erases the word and writes again and again, if neces- sary, until some pupil says that he can write. The teacher then erases the word and the pupil takes his position as far from the board as he can stand and yet reach the board easily. He holds loosely a piece of long crayon with the large end running under all his fingers and not between the thumb and fore- finger as a pencil is held. Pupils are not allowed to reach above or below the level of the shoulder until they are quite skillful in making written forms. The teacher is careful not to concen- trate the attention of the child upon his position. She secures the right po- sition as unobtrusively as possible while the pupil is intent upon what he is to write. If he has a clear image of the word as a whole and the process of making, the pupil will write confidently, rapidly and correctly. The teacher watches carefully, and if the child be- gins to go wrong, she stops him gently, but instantly, and writes again for him. If the pupil makes all the elements in their right order, he is discreetly praised and excused without regard, at this point, to the penmanship. From the first, however, the word must be writ- ten through. The crayon must not be lifted until the word is finished. Dur- ing this exercise no child writes the word twice except for some definite correction. If at the end of the class the teacher finds any children who are nervous or greatly lacking in muscular control, they should be allowed to wait and take this work later with the second or third group. After the word as a whole has been written, the teacher asks who can make m alone and then an alone. During the first writing the teacher pronounces the word man as a whole. In asking for the written signs of the parts she sounds the parts de- sired. If necessary, she rewrites the word, covers all but the required part, and then erases or covers the entire word before the pupils begin to write. Several periods may be required to enable each pupil to write the word man under the direct supervision of the teacher, but this should be finished be- fore any other phonic exercise is attempted. If more than one period is required, the teacher calls at each succeeding period only those pupils who have not written the word. 4:th day. Phonics. — Rat. Hear and make sounds r — at. MODEL LESSON. Associate sounds with written signs, that is, distinguish the letter r as stand- ing for the first sound and the group at as standing for the second. Close by having each child sound r — at, rat. 5th day. Qth day. Phonics. — Write the word rat in accordance with suggestions given for the word man. Write r when that sound is made and at when the soimd of the group is given. 7th day. Phonics. — Mat. The new phonic exercise requires the recognition of known elements in a new relation. Write the word man and have pupils re-sound the m. Write rat and review at. Then write mat and ask the class to find in this the letters whose sounds they know. They will readily find and sound m — at, but they may have difficulty in blending the sounds to make the word. In that case the teacher will help them to make the sounds more rapidly as they listen care- fully. If they still fail to grasp the word, the teacher may blend this word for them. Some pupils are able to blend the first v/ord for themselves, but others require two or three blended for them before they grasp the idea. 8th day. 9th day. Phonics. — Write the word mat. After the word has been sounded and pronounced, ask if the pupils can write m.at. If they think they cannot, remind them that they can write man and rat and that there is nothing new in the word mat. If they still hesitate, the teacher may say, "You can write m and at (sounding those letters) and, of course, you can put them together to make mat." Some child will write now, if the class is made of normal pupils, and if not, the teacher may write the word and erase as in the first instance. If the pupil regards each word as a new task to be mastered, the work is very difficult and its chief value is lost. From the beginning the pupil must rea- lize that effort to remember sounds and written symbols is required only for the first zvord in each column. This effort, too, is reduced to a minimum by deriving all elements from well-known sight-words. Very gradually the pupil learns to sound and write eleven words requiring eighteen letters, and by sim- ple re-arrangement, he pronounces and writes one hundred and forty-one other words, making a total of one hundred and fifty-two words. During this time the pupil should be conscious of the ad- vantage and the pleasure derived from receiving interest on an investment of intellectual effort. IQth day. Phonics. — Ran. Distinguish and make sounds. Associate sounds and written representation. 11th day. Phonics. — Ran. Write word as a whole and phonic elements r — an, as before. In giving new words to be written, the teacher sounds them exactly as the pupil does in his phonic analysis, e. g., r — an, ran. The group sounds an, ail, ire, etc., are not separated. As soon as the pupils indicate the power to see and hear the phonic elements of a word without separation, the words should be pronounced as wholes. 12^/z day. Phonics. — Tail. Hear and make sounds, t — ail. Associate sounds with written signs. 13^/^ day. Phonics. — Tail. Write the word as man and rat were written. If the pupil forgets to dot the i, the teacher does not tell him what is wrong 8 THE TEACHING OF READING. or allow him to add the dot as an after- thought. Instead, she erases the word and writes again, if necessary, with a little emphasis on the dot, after which the pupil writes again with the dot as a part of his image of the word. 14:th day. Phonics. — Mail, rail. Sound and pronounce. Hear and write. By this time all new words should be analyzed and pronounced unaided, and they should be written with compara- tive ease. Pupils must not be permitted to guess. . If any pupil forgets the sound of a letter or the written symbol of a sound, he should return to the sight- word containing the needed element and analyze again. If the pupils indicate the slightest confusion by substituting the sound of one letter for another, or by sounding the letters correctly and then saying the wrong word, the teacher is urged to proceed more slowly and carefully. Nothing can atone for the lack of a solid foundation. 15th day. Phonics. — Tan. Sound and pro- nounce. Hear and write. IQth day. Phonics. — Fire. Have the pupils say this word slowly and see if they cannot distinguish unaided the sounds, f — ire. By this time alert pupils are able to analyze any simple phonetic word known as a whole. 17th day. Phonics. — Mire, tire. Sound and pronounce. Hear and write. Difficult words should be omitted with foreign pupils and with English pupils having limited oral vocabularies. Objects, actions, or questions should bring to consciousness ideas which will make the words more than a mere col- lection of sounds. 18th day. Phonics. — Fail, fat, fan. Sound and pronounce. Hear and write. Idth day. Phonics. — Will. Proceed as with the word fire. 20th day. Phonics.— Mill, rill, till, fill. Sound, pronounce, and write. 9,lsf day. Phonics. — Win, wail, Sound, pro- nounce, and write. '22nd day. Phonics. — Horn. Sound and write. 23rc? day. Phonics. — Complete the list with orn and h. Since all known elements, both vocal and written, are continually repeated, no reviews, as such, have been assigned. 24:th day. Phonics. — Bit. Sound and write. 25th day. Phonics. — Complete the list with it and b. 2^4 h day. Phonics. — Lay. Sound and write. Complete the list with ay. The word play is the first instance of blending two initial consonants. These sounds must not be separated. The pi is said with one effort, and pr, tr, br, fl, etc., are given in the same way. The pupils should be able to combine two familiar consonants without assistance. 21th day. Phonics. — Complete the list with 1 and the accompanying words having two initial consonants. 28th day. Phonics. — Soon. Complete list with oon and ,y. MODEL LESSON. 9 FrCfTn this date the amount given in one day will depend upon the time available for phonics rather than upon the power of the class. A bright class will now be able to grasp easily an en- tire group of words. Slow pupils should be allowed the time required to do the work thoroughly. It is no longer necessary for each pupil to write every word, providing the teacher knows that every word can be written. An even group may now write in con- cert, but no oral work is done in con- cert. ^9th day. Phonics. — Complete list in ing and k. The word string introduces the blending of three initial consonants. The str should not be separated. dOth day. Phonics. — Complete list with ing as a suffix. Outline of Langfuage Lessons, Scope of Work for First Semester. Narration based upon literature, in- formal nature study, and the social ex- perience of the pupil. Time of Beginning. When the third story has been read, the highest group may begin written lan- guage. The second and third groups should wait until they have accom- plished the same amount of work in reading. The first group usually be- gins about the middle of the second month, but this work may be postponed, if the teacher preferes. Oral Treatment of Story. 1. One of several stories read is told through by the pupils. 2. Certain parts are re-told for im- provement in thought and form. 3. The story is told through with attention to the suggestions given. Without a struggle for different ways of expressing the same thought, each pupil's telling should be his own, unless the pupils are frankley committing something which is worth memorizing. Introduction of Written Work. Pupil's purposes. — Suggest the de- light of sharing this story. Pupils tell the story at home. How share with persons at a distance? Perhaps books cannot be sent, but pupils can write the story and send it to a relative, a friend, or a child in a hospital. First sentence. — By a question the teacher secures thought and oral ex- pression of introductory sentence. Example. This is the house that Jack built. Fixing images of written symbols and process of making. Standing with her right side to the board, so that every movement may be seen, the teacher slowly writes the sentence in a large hand. She asks if any pupil can write the same after the teacher's work has been erased. If no one volunteers, she repeats the writing until some one is ready to try, Reading the story has done much to fix the required images, and the written phonic work has developed ability to reproduce written symbols. Pupil's "writing. — From the volun- teers select the most hopeful. One suc- cess arouses ambition and gives confi- dence. If image is clear in pupil's mind, writing will be confident, easy, and accurate. Mistakes. — No mistake is passed. If possible, anticipate and prevent. If pupil beging to go wrong, he should stop immediately and watch the teacher write again. In the event of many mis- takes, try other pupils and give the one 10 THE TEACHING OF READING. who failed greater opportunity to ob- serve. Specific mistakes and their treatment. 1. Failure to carry thought shown by omission of words or wrong order of words. The pupil should stop and repeat orally the sentence he is trying to write. 2 Failure to secure complete image of symbols shown by omission of let- ters or their elements; substitution of one letter or element for another; or omission of punctuation marks. In outline for writing phonic words, see suggestions for specific mistakes illustrated by treatment of failure to dot an i. 3. Failure to secure correct image of process shown by beginning at the wrong points to make a letter pro- ceeding in the wrong direction dur- ing the making of a letter, stopping in the middle of a word to dot an i or cross a t; separating or patching the letters of any word as a result of lifting the crayon before the word is finished. It is most important to train pupils to write words and sentences through without erasures and without hesita- tion. During the writing of a word there should be no lifting of the crayon followed by an attempt to join the sepa- rated letters, no retracing part of a word, and no erasing, except that done by the teacher. To avoid these errors the pupils must have clear, accurate images, and they must concentrate their attention. The teacher may be en- couraged to vigilance by the fact that children love exactness. Time required for writing. The pupils write one at a time in the order of their supposed ability. Each child who has finished the sentence re- turns to his seat to do hand work or something approved by the teacher. During the first fifteen or twenty minute period devoted to written language, it may happen that only three pupils successfully finish the sentence. At the next language period the teacher calls only those who have not yet writ- ten the sentence. Readiness in writing usually increases with the number of times the pupil has seen others write the exercise, and many teachers who devote one period to the first three pupils, complete the work with a group of twenty in three periods. Grouping. As a result of the exercises in writ- ing phonic words the pupils are probably grouped satisfactorily, but if any pupil shows unexpected weakness in the sentence writing, he should be transferred to the second group in language. Second sentence. The thought and the oral statement are secured in the same way as the first sentence, but the teacher does not write this sentence. Instead, she asks if the pupils can write it, and, if not, what part they cannot write. At this point the pupils must distinguish what they know from what they do not know. The unknown words are then written by the teacher. Usually it is necessary to show how the first word is written by way of securing the capital letter. The plan for writing the second sen- tence is continued thereafter until the story is finished. New forms of spell- ing capitalization, and punctuation are shown at the time they are needed and erased before the pupils attempt to write. All repeated forms should be written without assistance. MODEL LESSON. 11 Writing on Paper. Time of hegining. — When each pupil in the group can write, accurately and easily, a short paragraph on the board, writting on paper may be introduced. Material. — Unruled, uncalendered paper, about twelve inches by nine in- ches is best. Pencils should be large and soft. Position. — Teachers are urged to ex- ercise great care to secure reasonably correct writing positions at the desks. If necessary to the formation of right habits, this work should, at first, be done individually. Arrangement on Paper. — By showing papers on which work is properly placed and giving careful attention to this feature, the teacher can train pupils from the beginning to leave correct margins, to indent paragraphs, and to place effectively material which is not to fill the sheet. On filled sheets of the paper recommended, the top, right, and left margins should be about one and one-fourth inches and the lower margin about one and one-half inches. In all cases the lower margin should be slightly greater than the top. Names of pupils should be placed upon the hack of the sheet. Teachers are urged to prevent the concentration of the pupils' attention upon formal details. No mention of margin or paragraph is at first made to the pupils. Gradually the teacher in- troduces these terms when speaking of the work, just as she uses the names of the letters of the alphabet without re- quiring the pupils to name them. In a short time the pupils begin to use these terms quite as easily and intelli- gently as they use automobile, elevator, etc. Development of the story. — The new sentence is sometimes added to the paper on which the pupil wrote before, and sometimes the story is written from the beginning with the new sentence added. When writing from the beginning, the pupil must not try to remember mechanically the order of the sentences. The sentence sequence must be de- termined always by the knowledge of the events in the story. Examination of Papers. — Every paper should be carefully examined during the period in which it is written, and no careless or inaccurate work should be accepted. If a pupil is aware of a mistake, he takes his paper to the teacher. If the paper can be corrected without serious injury, the teacher erases the wTong form ; if not, the pupil re-writes the exercise. Writing without supervision. — Pupils who have acquired a reliable degree of power may, if desired, write while the teacher is giving lessons to others. In such cases the exercise should be written only once except for some definite correction. Owing to the time required for in- dividual work, about one month is allowed for the completion of the first written story. During this period other stories are read, told, and discussed by the pupils. Many interesting personal experieneces suggested by the literature and the nature work are related by the pupils. Every lesson should improve the oral language power. Lost time, if any, should be deducted from that de- voted to the written work rather than the oral. Some teachers desire and secure in- dividual construction in the first re- production, but the degree of difficulty is increased by the fact that the pupik have less opportunity to obserre ^e writinsf of others. 12 THE TEACHING OF COMPOSITION. The Teaching of English Composition. :eiGHTH YBAR. By John J. Bukke. OTHING but perfunctory work in the teaching of English com- position can result from incor- rect standards, mistaken aims and faulty methods, for the aim and the method necessarily follow the stand- ard. Although the study of English is a science and writing is an art, the higher and broader aim of all compo- sition work should not be to teach the science or to cultivate the art, but rather "to train the mind to the acquisition and expression of ideals."* The teaching of the facts and principles of language should be subordinated to this broad aim, for to exaggerate the critical faculties is to inhibit the activity, but to emphasize the thought side of the work is to increase the pupil's ability in the gathering and ordering of material and to make for ease in orderly writing. Method. In the eighth year the logical faculties of the pupil are developed enough to enable him to understand and appreciate the synthetic method of pre- secitation. He is familiar with the sen- tences as a unit of thought, and as the paragraph is also a unit of thought, that is, a group of logically connected sentences, the five laws of the para- graph should now be reviewed, allowing one lesson a week for each law. Thus (1) First law : Indention. (2) Second " : One topic to a paragraph. *Teacibing of Bnglish, Carpenter. (3) Third law (4) Fourth " (5) Fifth " Unity in the paragraph. Plan in the paragraph. The topic sen- tence. Following the synthetic method the pupil is now able to grasp the idea that a composition is a number of logically related paragraphs, that narrate, de- scribe or explain. For the material of all composition work naturally falls un- der three heads, which are here arranged in the order of their im- portance : (1) Narration. — Covering the world of story (both history and fiction.) (2) Description. — Covering the world of external facts. (3) Exposition. — Covering all ex- planations and interpretations. Letter writing (social and business) and current events may fall under any one of the three above-named heads. The pupil should now be taught that "every true composition must con- tain at least three main parts :" * (1) Introduction. (2) Body. (3) Result, conclusion or inference. The pupil is now ready to take up the composition work of the grade in a broad, comprehensive way. Model Lesson in Narration* To preserve the balance between analysis and synthesis is not easy, but a good working rule is to see that the * Graded Composition Work, McKeon, p. 96. THE TEACHING OF ENGLISH COMPOSITION. 13 oral work supplements the written work, for in the eighth year there should be more written work in com- position than in any preceding year and the tendency is to slight the oral work. Begin with the simplest and easiest form, Narration, and select a subject from the grade work in History. It need not be a concrete subject, as in the lower grades, but it should be both adaptable and interesting and well within the intellectual grasp and judg- ment of the pupils. History and cur- rent events are fine fields for the select- ing of topics in Narration. "The Youth of Washington," "Lincoln's Boyhood," "The Peninsular Campaign," or "The Secession of South Carolina" would be suitable to begin with. To illustrate, let the subject be "The Peninsular Campaign." The pvipils delve into their histories for the facts and take brief notes of the time, con- ditions, etc. The teacher then calls for working plans, the best of which are written on the blackboard. One pupil might suggest: (1) Purpose of campaign. (2) The campaign and its failure. (3) The far-reaching effects of the failure. Another pupil, with more inventive- ness might suggest: (1) Brief statement about the cam- paign and its failure. (2) Details of campaign. (3) How the campaign might have succeeded. After five or six good plans have been placed on the blackboard (the number of plans will vary with the nature and comprehensiveness of the subject), the pupils individually select the plan that most appeals to them and the class is set to work. In this way genius and invention are given free range. The above is an outline of a method that has been found to be most effec- tive in teaching composition in the eighth year. It is a method that does not cramp the individualit}'- of the pupil, but rather serves to encourage deep thoughts and judgment. Correctinf^ Compositions. The personal correcting of the in- dividual work of the pupils is a veri- table bugbear to the teacher, but a method by which the pupils correct their own work individually is not diffi- cult to find. And as quality not quan- tity is the criterion in all composition work, the fact that all the compositions have not been corrected during any one month, need not cause anxiety. On the day following the writing of a compo- sition, a portion of the time assigned to English should be given to correcting it. Let the compositions be distributed to their respective owners and hang upon the blackboard the following chart : Code of Sig:ns. ===== and used too often. «—«»—" Misspelled word. Wrong word. Omission. Error in use of capital. " " punctuation. ( ) " " grammar. [ ] Poor construction. The teacher then allows the pupils a few minutes to correct their respective compositions, as to ( 1 ) Paragraphing. (2) Mechanics and spelling (here the chart is called into use). The pupil corrects the paragraphing and indicates (in the one-inch margin on the left) the errors in composing. 7 14 THE TEACHING OF COMPOSITION. After a little practice, the compositions of an entire class can be corrected in four or five minutes. The teacher then selects promis- cuously from the roll-book the names of eight or ten pupils to be heard and marked for that day. As the names are not selected alphabetically any pupil may be expected to be called up and marked. The selected pupils read their several compositions aloud and the cor- rections are made orally by the pupils under the direction and supervision of the teacher. When a selected pupil has made his corrections orally, he takes his seat and makes the corrections in writing. And after the oral correct- ing is finished, those who were not called on to recite, are allowed a few minutes to make the indicated cor- rections in writing. This entire lesson should not require more than fifteen minutes, and if ten pupils are heard in a week, a class of forty can be heard and marked in a month. The teacher should keep a list of the most common errors made by the class as a whole, and give special attention to these errors in all recitation and written work. Choice of Subject. In order to eliminate vague and crude expression, the subject selected should be both attractive and adaptable. Narrative composition work correlates with the history of the grade. Descrip- tive composition work with the grade work in geography, science and picture study, and Expository composition is of use in interpreting the arithmetic, history and gymnastic games. As the term advances, the selected subjects should be such as to call forth the reason, judgment and inventiveness of the pupils, otherwise the interest will flag. By the end of the year, the pupil should be able to delve into any subject, select an appropriate name for it, formulate a plan to cover it and write out a fairly good composition about it. Original CompositiorLS* The purpose of original compo- sition work is to get the pupil to ex- press his own thought and his own opinion about the topic assigned. To do this the pupil must be familiar with his subject, and the more familiar he is, the better will be his work in orig- inal composition. The selection of topics for original compositions is not an easy task, so a list of subjects suit- able to the seventh and eighth years is here appended: My Dog. My Motto. My Ambition. My Uncle John. What I did during Vacation. Why I wish to be a Merchant. Description of a Picture. Music at Home. One of my Friends. My I^ibrary. My Seat-Mate. My Grandfather. Some of my Troubles. The Baby at Home. Our Circle. My Club. We Boys (or Girls.) My Friends. Skating. My Favorite Study. My Photograph Album. My Study. One of my Friends. My Visit to the Park. Cheerfulness. Taking Exercise. My Den. My Favorite Game. A Walk in the Woods. My Vacation Plans. TERM PLAN IN ENGLISH COMPOSITION. 15 Transcription, This helpful form of composition work is very often neglected in the eighth year. "To write out prose para- graphs and poems," says Laurie, "with due attention to legible writing and punctuation, gives linguistic material. It helps the pupil who is deficient on the formal side." The copying of some of the speeches of Portia, Shylock, Caesar, etc., is very helpful and corre- lates with the Reading and Penmanship. Reproduction. This means the reproducing (in the pupil's own words) of poetical and prose selections. It serves to acquaint the pupil with tlie best literature and helps to form his literary style. A use- ful form of it is the summarizing (by the pupil) of the contents of a dictated paragraph. In this, brevity and accuracy are the ends to be aimed at. Letters. The writing of letters (social and business) is an important part of the composition work in the eighth year. In social letters pad-size paper (9 in. X 6 in.) should be used and the correct way of folding the paper (in three folds) should be taught as well as the addressing of the envelope. In business letters the regular letter-size paper (10 in. x 8 in.) should be used and the correct way of folding it, etc., should be shown. Letters of applica- tion come under the head of business letters. The plans, methods and suggestions given above have additional value in- asmuch as they have been tried out in the class room and excellent work has resulted from their adoption and use. Term Plan in English Composition. GRADB 8B, John J. Bukke. First Month. First Week. Paragraph work. Lesson I. Inden- tion and unity of topic. Pupils to write a paragraph to illustrate both laws. Second Week. Paragraph work. Lesson II. The Unity of the paragraph. Third Week. Paragraph work. Lesson III. The Plan in a paragraph (not every para- graph admits of a plan.) Fourth Week. Paragraph work. Lesson IV. The Topic Sentence (not every paragraph needs a topic sentence.) Second Month. First Week. Naration. — Begin with a simple narrative composition, a transcrip- tion (of poetry into prose) or a repro- duction. Second Week. Description. — "Civil Governmettt in the Thirteen Colonies." Thiee Week. Exposition. — "My Favorite Game," "How to Build and Manage a Float," * or other suitable topic. See Graded Work in Composition. and Johnston. Maxwell 16 TERM PLAN IN ENGLISH GRAMMAR Third Month. First Week. Narration. — "Legend of William Tell," or narrative current event. Second Week. Description. — "The U. S. Senate," or "The House of Representatives." Third Week. Exposition. — "How Congress makes laws." Fourth Week. Business letter. — How to fold the paper, etc. Fourth Month. First Week. Narration. — "Origin of our two Great Political Parties." Second Week. Description. — "Lincoln's Boyhood," or picture study. Third Week. Exposition. — "How to build a camp- fire." Fourth Week. A book review, a letter or an oral re- port. Fifth Month. First Week. Narration. — Original work. Select a subject from list (see art. on eighth year composition), or tell a story, ask for plans and tell the pupils to work from their own individual plans, after they have been approved by the teacher. Second Week. Description — Original work, "My Seat Mate," "My Study," etc., or pic- ture study, as The Angelus, etc. Third Week. Exposition. — Original work. "How I study," "Why I BeHeve in Work," etc. Fourth Week. Business letter. — How to fold the paper. Term Plan in English Grammar. GRADE 8A. Arranged in Weeks. John J. Burke. First Month. Assign Lessons by Topics. First Week. Define sentence, clause and phrase. Classify as. declarative, interrogative, imperative and exclamatory-declarative, exclamatory-interrogative, exclamatory- imperative. Pupil should be able to classify any sentence according to the above scheme. Drill any standard Grammar. Second Week. Teach the simple sentence. Define and classify nouns (two classes, com- mon and proper.) Singular and plural numbers. Regular and irregular plurals. Three persons. Three cases. Analysis. Easy, simple sentences. Third Week. The verb. Define and classify. No- tional and auxiliary; transitive and in- ^transitive; regular and irregular. The TERM PLAN IN ENGLISH GRAMMAR. ir pupil should be able to classify any verb on sight and state in his otvn words the reason for so classifying it. Drill any standard Grammar. Analysis. Simple sentences involv- ing verbs of the various classes. Fourth Week. Define 'and classify pronouns. Five classes. The pupil should be able to classify any pronoun on sight. Analysis. Simple sentences involv- ing the above. Second Month. First Week. Teach the complex sentence. Define it. Emphasize the grammatical de- pendence of the dependent clause on some word or words of the leading clause, also the possibility of the de- pendent clause being used as the sub- ject, object or attribute of the leading clause. Define dependent clause. Give sentences showing the various type- forms of the complex sentence. Call for complex sentences of the common type- forms. Drill thoroug.hly. Analysis. Typical complex sentences. Second Week. Define declension. Declension of nouns. Teach rule for apostrophe in possessive plural. Declinsion of pro- nouns. Decline the five personals, the four relatives, the three interrogatives, the two demonstratives, also the five compound personals and the three com- pound relatives and interrogatives. Analysis. Simple and complex sen- tences. Parsing. Nouns, pronouns and verbs. Third Week. Teach the compound sentence. De- fine it. Define co-ordinate and subordi- nate clauses. Point out the distmction between a dependent clause and a sub- ordinate clause: i. e., a logical depend- ence is involved in the latter and a grammatical dependence in the former. Analysis. Compound sentences. Parsing. See second week above. Fourth Week. Define antecedent. Concord of pro- noun and antecedent. Apposition. Explain it. Drill thoroughly. Define the three genders. The three ways of indicating the feminine gender, viz., boy, girl ; host, hostess ; he-goat, she- goat. Give a few examples under each. Analysis. Sentences, all kinds. Parsing. Nouns, pronouns and verbs. Third Month. First Week. Teach the four moods. Use syn- thetic method, that is, ask the pupils to give short sentences involving the different moods. In teaching the in- finitive mode, show that a true infini- tive mood must always have an objec- tive case for a subject, c. g., I told him to go. Analysis. Sentences, all kinds, but more involved. Parsing. Verbs. Second Week. Teach the six tenses. Show the difi'erence between the three simple tense and three perfect tenses. De- fine a perfect tense. The pupil should be able to classify the tense of any verb on sight. Analysis and parsing. See first week. Tpiird Week. Teach the conjugation of a regular verb (from copied or printed forms in the hands of each pupil.) Teach also the two emphatic forms and the pro- 18 TECHNICAL GRAMMAR IN THE ELEMENTARY SCHOOLS gressive forms. Drill thoroughly, using the synthetic method. Analysis. See first week. Parsing. Verbs, their mood, tense and agreement. Fourth Week. Teach comparison of adjectives and adverbs. The three degrees and the three kinds of comparison. Give ex- amples to show the three kinds. Analysis. See first week. Parsing. Complete syntax of nouns, pronouns, verbs, adverbs anfl adjectives. Fourth Month. First Week. Teach the participle. Three kinds. Call for the participles of regular and irregular verbs. Analysis. Involved sentences taken from the literature of the grade. Parsing. Nouns, pronouns, verbs, adverbs, adjectives and participles. Second Week. Synthetical study of the phrase. Three kinds: Noun phrase, adjective phrase and adverbial phrase. Ask the pupils to construct sentences exempli- fying them. Analysis and parsing. See first week above. Synthetical study of the clause. Three kinds: Noun clause, adjective clause and adverbial clause. Ask the pupils to construct sentences exempli- fying them. Analysis and parsing, as above. Fourth Week. Teach the interjection. Also drill the parts of technical grammar in which the class, as a whole, are deficient. Analysis, involved sentences from the literature of the grade. Parsing. The syntax, etc., of any word that is called for. Fifth Month. First Week. Review the work of the first month, synthetically. Second Week. Review the work of the second month, synthetically. Third Week. Review the work of the third month, synthetically. Fourth Week. Review the work of the fourth months, synthetically. Technical Grammar in the Elementary Schools. SEVENTH YEAR. By Bridget M. F. Cauleield. BEFORE one can determine the best method of teaching gram- mar, a sense of the nature of the subject is necessary, an appreciation of its value, absolute and relative, and a clear conception of the aim. Grammar, as it can be presented to children of the elementary school age is best de- fined by Carpenter, — "a systematic de- scription of the essential principles of TECHNICAL GRAMMAR IN THE ELEMENTARY SCHOOLS 19 a language." The word "description" may be misleading, in that it appears to suggest a deductive method, whereas, as the following plans will show, the method should be eclectic. It is truly a description, however, evolved from the old Greek glossaries through the Latin, and entering English curriculum centuries after our language had been developed in its present form. No one should attempt to instruct in grammar without a full realization of its value as a study in the total of the subjects of instruction, as an im- portant sub-division of language work, and in its function in the development of power and the enriching in content of the mental equipment of children. I can not do better than refer the reader to the paragraph on grammar in the Report of the Committee of Fifteen which has laid the foundation of our Course of Study, and to the Introduc- tion to Maxwell's "Advanced Lessons in English Grammar." The principle of self -activity, I think, will support my view that each teacher will do her most effective work, if, upon her conception of the nature and value of her subject, she formulate her own particular aim, in accordance with the principles of correlation in its sub- divisions, co-ordination, articulation and concentration, for whch we are so greatly indebted to the Committee of Fifteen. In the 7A grade the method should be mainly inductive, the teacher supply- ing many examples for comparison, contrast and abstraction, and by ques- tion and discussion the pupils should be led to formulate generalizations in rules, definitions and classifications. The Herbartian steps preparation, pre- sentation, formulation and application should appear in every lesson. The student of grammar is usually interested in the subject, so the teacher need not go far afield for material for the prepara- tory step. It will be found best, as a rule to use the former lesson, or a re- lated rule or classification for this first step. However, other illustrations may sometimes be introduced, as the spirits in the "Christmas Carol" to develop the idea of tense. By basing new lessons upon old and upon the work of earlier grades, gram- mar will be made a united whole, arti- culated grade with grade, and within each grade. Thus the noun, and the noun phrase prepare for the noun clause, the comparison of adjectives for the com- parison of adverbs. Next to the teacher of composition, the teacher in grammar should be blessed with the "many-sided interest," for all subjects should contribute to the supply of material for analysis and synthesis, that the pupil may not see in grammar a study of dry bones. "The Wrights are experimenting upon flying- machines," is as good an illustration of the progressive form of the verb as is "John is reading a story." In teach- ing voice, verbs which will cause the pupil to image should be used, so that the definitions of active voice and pas- sive voice may be life-like and meaning- ful. It will be noted that the work of 7B repeats largely the work of the previous grade, but it need not be a repetition for the pupils. Rather should there be a progression leading to clearer con- ception of relations, organization and unification. The maximum of analysis in 7A should be replaced by a gradual increase of synthetical work. The text- book, used mainly for application and review in 7A, should contribute to the presentation in 7B. Correlation with '0 TECHNICAL GRAMMAR IN THE ELEMENTARY SCHOOLS other subjects of the curriciULim, as with literature, composition, English history should enrich the grammar lesson. There may be greater variet}/ in the teacher's methods of presenta- tion, a deductive lesson often afifording clearness and economy. It would be very unjust to lay hard and fast rules for method for our New^ York schools, because the language sense of the pupils varies greatly in different localities, and even w^ith differ- ent classes of the same locality. The teacher, therefore, must build to the best advantage upon the pupils' native ability and previous training. Grammar must not be subordinated to other divisions of English, but its relation to composition, literature and v,'ord-s'(.vidy should be shown. Sen- tences should increase gradual!}^ in the number and kind of difficulties pre- sented, both for analysis and synthesis. In the use of text-books, the Max- well grammars, graduated through the grades, are clear and logical, and save the teacher in the assignments for the application of rules, and for the reviews of lessons. For diagramming sentences (a process which may be defended by the multiple-sense theory) an excellent method is found in Reed and Kellogg's "Lessons in English." Many grammars should be used by the teacher for examples of kinds of phrases, clauses and sentences. Parsing for Grades of Seventh and Eighth Years. (For the teacher: — To parse a word is to give its classification, its properties and its syntax.) Noun. — 1. Class. 2. Properties — Person. Number. Gender. Case. 3. Syntax (use or relation.) (N. B. — In English there is really no "gender," though the distinction is made in the best text-books.) Pronoun. - -1. Q Class. Antecedent Person. ^ Number. Gender. 8. Case. J 4. Syntax. ['Properties. form. Principal parts. 2. Classification as to Adjectives. Adverb. 1. Class. 2. Degree of Comparison. 3. Syntax. Verb. 1. Classification as to | Regular. 5. I Irregular. Transitive. . . . . , Intransitive. use, voice if transitive, j Copulative. 3. Mode, tense. 4. Agrees with subject — in — person, and — number. Preposition. Shows the relation between its ob- ject — , and the word — which the phrase modifies. Conjunction. 1. Class and sub-class. 2. Syntax. TECHNICAL GRAxMMAR IN THE ELEMENTARY SCHOOLS 21 Punctuation. For punctuation of simple, complex and compound sentences, see Max- well's "Advanced Lessons in English Grammar," pages 271 to 277 and corre- late with grammar and composition. Definite assignments should be made for development and drill in grades 7A then 8B, with systematic review. New Pai'sing in 7B Infinitive. — 1. From the verb — x — . 2. It is used as a — noun. — adjective. — adverb. vei •^% 3. Syntax, i. e., use in sentence. Participle. — 1. Verb from which de- rived, giving principal parts. 2. Transitive or intransi- tive voice, tense. 3. Syntax (modifies noun — X — .) Gerund. — 1 and 2 as for participle. 3. Syntax, — case and reason. ine in Grammar for 7B Fjest Week. Analysis and Synthesis. — General re- view. Syntax. — 1. Each, every, no, etc. (See 7A.) 2. The pronoun agrees with its antecedent in person and num- ber (In 7B a greater variety of antecedents should be used, and pronouns whose antecedents are indefinite may be given, as "(He) who steals my purse, steals trash ;" "Heaven helps those who help themselves ;" "Who can number the sands of the sea?" x^lso show, by easy examples, that phrases and clau- ses may be represented by pro- nouns, as "To extend British dominion was the army's pur- pose, but it cost many a British life," etc., etc. Second Week. Analysis. — Complex sentences, con- taining adjective clauses. (In- clude adjective clauses which re- late to pronouns.) Use of "whose" as relative. Parts of Speech. — Noun, plurals in .y and es. Irregular plurals. Familiar foreign. Syntax. — 1. Position of adjective clauses. 2. Use of li'ho, which, that, as de- termined by gender of ante- cedent ; use of "whose" and "of ivhich," as in "Those books, whose titles are black- lettered, must be read during the term." "These drawings, the measurements of which are slightly inaccurate, may mislead you in the shop." Analysis and Synthesis. — Adjective clauses introduced by conjunctive ad- verbs. Parts of Speech. — Conjunctive ad- verbs. Noun, gender. Three ways of distinguishing gender. Call pupils' attention to such Christian names as John, Johanna, (Teuton Johan, John) Joseph, Josephm^, 22 OUTLINE IN GRAMMAR FOR 7B. (Josef) William, Wilhelmina (Wil- helm) etc., etc. (Correlate with migrations to Bri- tain.) Syntax. — Avoid changing gender of the pronoun when referring to same antecedent. ( Correlate. — Autobiographical com- positions.) When the subject is a rela- tive pronoun, the verb should agree in person and number with the antecedent of the pronoun. Fourth Week. Analysis and Synthesis. — Complex sentences as before. Parts of speech. — Noun, person, case. Case. — Subjunctive or nominative. Objective or accusative. Explanatory nouns. Attributive nouns (as "It is he.") Syntax. — Case of subject and of at- tribute, especially in use of pronouns. Case of explanatory nouns. Transitive verbs and prepositions govern the objective case, as "Write often to your mother and me" etc., etc. Fifth Week. Analysis and Synthesis. — Complex sentences; adjective clauses containing compound relatives (or noun clauses introduced by compound relatives.) Parts of Speech. — Relative pronouns, simple, compound. Personal pronouns, simple, com- pound, reflexive, emphatic. The infinitive. Syntax. — The subject of an infinitive is in the objective case. The verb to he takes the same case after it as before it. Sixth Week. Analysis and Synthesis. — Compound sentences. Parts of Speech. — Noun, possessive case. 1. Of singular nouns. 2. Of plurals ending in "s," Of plurals not ending in "s.' 3. Of compound nouns, as man- of-war, son-in-law, etc., etc. 4. Of nouns connected; joint possessive, several possessive. 5. Of explanatory nouns. Syntax. — Possessive case and equiva- lent expressions. Seventh Week. Analysis and Synthesis. — Compound sentences. Parts of Speech. — Conjugation of ■'he." Conjugation of see, go, etc. Co-ordinate conjunctions and sub- classes (See Maxwell's "Advanced Lessons in English Grammar.") Parse conjunctions. Syntax. — Use of pronoun them. " " adjective those. " "■ " these " " each other. Each other is applied to two things ; when more are referred to, one, the other, one another should be used. Use of hetween. " " among. Eighth Week. Analysis and Synthesis. — Complex sentences containing adverbial clauses of time, place, manner. Parts of Speech. — Conjunction ad- verbs. Subordinate conjunctions giv- ing the idea of time {while, until, etc.) Parse both. Syntax. — Review previous rules. Agreement of verb with its subject. (Examples should give variety of kinds of subjects.) OUTLINE IN GRAMMAR FOR 7B. 23 Ninth Week. Analysis and Synthesis. — Adverbial clauses of degree. Summarize kinds of adverbial clau- ses, after Maxwell's "Advanced Les- sons in English Grammar," p. 236 and following. Parts of speech. — Adjectives of quality, adjectives of quantity, demon- strative. Parse adjectives. Syntax. — An adjective agrees in number with the noun it modifies. (Avoid these kind for this kind, those sort for that sort, etc.) Tenth Week. Analysis and Synthesis. — Adverbial clauses of reason or cause; end or pur- pose. Parts of Speech. — Conjunctions which express reason, cause, end, pur- pose. Syntax. — The tense of the verb in one clause must not conflict with the tense of a verb in another clause. (Apply to succession of sentences in composition.) Eleventh Week. Analysis and synthesis. — Complex sentences containing adverbial clauses of supposition, condition, doubt. Parts of Speech. — Subordinate con- junctions expressing supposition, con- dition, doubt. Syntax. — Use of the subjunctive mode. Twelfth Week. Analysis and Synthesis. — Simple sentences containing infinitives, as 1. Adjective. 2. Adverb } ^f^"^"^^ ^ ""J^' . ( adjectives. 3. Noun < Subject. Object. Attribute. Explanatory. Apposition. Parts of Speech — { Verbals. Parts of Speech. — Verbals. 1. Infinitive. 2. Participles. 3. Gerunds. Special attention to the infinitive. Review conjugation, adding the in- finitive. Syntax. — Two uses of the present tense are (1) to express present facts. (2) to express unchangeable truths. Thirteenth Week. Analysis and Synthesis. — Simple sen- tences to study participial, gerundial and infinitive constructions. {^ 1 . Participles I 2. Gerunds (noun participle or verbal noun.) Infinitives, all uses. Syntax. — A participle or a participial phrase is usually placed immediately after the noun whose meaning it modi- fies. When there is no doubt as to the noun it modifies, it may be placed at the beginning of the sentence. Fourteenth Week. Analysis and Synthesis. — Noun phra- ses as subject, object, attribute and ex- planatory. Noun clauses as attribute and as ob- ject. Parts of Speech. — Adverb, of time. " place " degree. " manner. Interrogative. Conjunctive. 24 OUTLINE IN GRAMMAR FOR TB. Comparison of adverbs. Parse adverbs. Syntax. — An adverb should be used to express the manner of the action ; an adjective, to express the quality or state of the subject. "The heavy night hung dark The hills and waters o'er." "The sun shone bright on the aut- umn fields." Fifteenth Week. Analysis and Synthesis. — Noun clause as subject, and in apposition. Note kind of words to which a clause may be in apposition.) Parts of Speech. — Verb, conjugation in the passive voice. Add participles, infinitives and gerunds. Auxiliary — definition dififerentiate have as auxiliary and as a verb ; also do, will, can. Analyze verb-phrase, as "may have been made" in simple sentence. Syntax. — Use of shall and will. Sixteenth Week. Analysis and Synthesis. — Sentences containing independent constructions, as (1) nominative by direct address; (2) nominative absolute; (3) inde- pendent phrase. Parts of Speech — Verb, conjugation, active and passive. Syntax. — Review of rules previously given. Seventeenth Week. Analysis and Synthesis. — Simple, complex and compound sentences. Parts of Speech. — Adjectives com- pared. 1. Adjectives of one syllable. 2. Adjectives of more than two syllables. 3. Irregular comparisons. 4. Show that some adjectives cannot be compared, and give idea of reason. Examples. — Perfect, superior, in- ferior, absolute, supreme, ex- treme, full, complete, sufficient. (No 4 may be ommited.) Syntax. Use comparative degree when comparing two objects. 2. Use of superlative degree. 3. Avoid double comparatives. 4. Avoid double superlatives. (Avoid comparison of adjectives whose meaning will not admit of different degrees.) Eighteenth Week. Analysis and Synthesis. — Sentences in systematic review ; classification tabu- lated; relations, comparisons, contrasts; definitions. r Sentence <( /■ Simple. Form J Complex. (. Compound. Use r Interrogative. } Declarative. (. Imperative. 1. Word. Sentence units ^ 2. Phrase. 3. Clause. Parts Subject Word. Phrase. Clause. «*^< Predicate 5"^^^^ Sen-I (Conn tence r Word. (Complement i Phrase. C Clause. Connecting words. Independent elements. GRAMMAR OUTLINE BY WEEKS.— 7A. L^a 1 Nineteenth Week. ''verb Adverb Nonn Adjective J Prr Twentieth Week. } Notional. ir'rononn ] Word elements. . School Problems. BOOK REVIEWS. 59 Book Reviews. By Jno. J. BuKKE. U'lde Awake Primer. By Clara Murray. Boston : Little, Brown & Co. 30 cents net. The Wide Awake First Reader is intended for pupils' use during the first year. In typography and make-up il is similar to the primer, but has a larger number of illustrations. Wide Aivake First A'eader. By Clara Murray. Boston : Little, Brown & Co. 30 cents net. THE past half-century has shown a wonderful development in the art of text-book making. And if we compare the old Sanford reader with the artistic, multi-colored reader of to-day, the contrast is so marked that we wonder why the children of this generation do not appreciate the great advantages they have over their forefathers. The Wide Awake Series is a marvel of the printer's art. The type is large and clear and the many colored illustrations aid greatly in illus- trating the subject-matter. The Wide Awake Primer is in- tended for the earliest years of school- life. It is copiously illustrated in color and has a novelty in the line of vocabu- laries, namely, a list of words used in the primer, arranged numerically according to the pages on which they appear. This enables the teacher to as- sign the words for spelling, before the daily reading-lesson is taken up. Wide Awake Seco7td Reader, By Clara Murray. Boston : Little, Brown & Co. 35 cents net. The Wide Awake Second Reader con- tains a half -hundred short reading- lessons beautifully illustrated by Heyer. Selections and adaptations from Susan Coolidge, Laura Richards, Mary Wilkens et al. are interspersed with anonymous tales, which are really as interesting as the adaptations, all stories being extremely interesting. The work outlined is intended to dove- tail with the first year's work, and in- stead of a vocabulary, there is a word list and key to pronunciation. In the latter the diacritical nomenclature of Webster is used. Wide Awake Third Reader. By Clara Murray, Boston : Little, Brown & Co. 40 cents net. In the Wide Awake Third Reader the mechanics of reading are subordi- nated to learning facts by reading, thus preparing the pupil for the task of studying. The same plan of selection and adaptation is followed and the illustrations are numerous and ex- cellent . It has a word list and a pro- nouncing key as in book two. In this book the same harmony of method and thought as is evident in the previous volumes, charms both teacher and child. Taken all in all this series of read- ing is a very excellent one and deserves a wide circulation in the schools. A First Book in American History. By Ger- trude Southworth. New York : D. Appleton & Co. List price, 60 cents. A biographical history of the United States from Columbus to Dewey and Edison complete in one volume seems an impossible task, yet such is the work accomplished by Gertrude Southworth in "A First Book in American History." Instead of giving the outline of the events in each epoch, as do the majority of our school-book historians, the writer gives a chapter each to Colum- bus, Cabot, John Smith, Miles Stand- ish, Hudson, Penn et al. and around 60 BOOK REVIEWS. the life-history of these men as pivots, is woven the story of the growth and development of America. Thus the bare facts of history are made subordi- nate to the characters that were instru- mental in furthering or bringing about the events. From a pedagogical stand- point, this is excellent. There is no better way to teach history. The ethical as well as the purely historical matter is more easily retained by the pupils. The book is written in conversational form and has a goodly proportions of half-tones. It makes a good text-book and an excellent supplementary reader. Progressive Course in Spelling. By J. M. Hunt, American Book Co., N. Y. I,ist price, 20 cents. Although the strictly pedagogical way to teach spelling is to select the words from the reading material of the grade, there is something to be said for the teaching of spelling from a graded text-book. For there is a large number of words in common use that do not occur in the grade readers. To get good results in spelling, both methods should be used. The Progressive Course in Spelling is in two parts ; part one covers the first three years and part two, the next three years of the elementary school. As to grouping, the words are arranged (a) phonetically and (b) topically. In ad- dition there is a well-selected list of pre- fixes, suffixes, synonyms and homo- phones. Accents and syllables are indi- cated and for pronunciation, the Web- ster system of diacritical marks are used. Steps in English, Book One. By Morrow, McLean & Blaisdell. American Book Co., New York. Price 40 cents. A good text-book for the use of pupils is not always a practical book for teachers' use. But (Steps in English, American Book Co.) is an excellent book for both teacher and pupil. The composition work is so dovetailed into the grammar work that it makes an ex- cellent teachers' manual as well as a pupils' text-book. Steps in English, Book One, covers the third, fourth and fifth years of the elementary course; the topic for the third year being the mechanics of zvrit- ing, for the fourth year the sentence and for the fifth year the parts of speech. For those who like a book that combines the study of grammar with that of composition, it is an ex- cellent text-book and manual. It has a good index and the composition work in picture study is particularly good. Steps in English, Book Two. By Morrow, McLean & Blaisdell. American Book Co., New York. Price 60 cents. Steps in English, Book Two, covers the sixth, seventh and eighth years of the elementary school, the last three years preparatory to the high-school. It is divided into two parts : part one being devoted to grammar and part two to composition. It is intended that the number of minutes given to English should be divided equally between part one and part two, thus covering the two subjects, grammar and composition. As in book one, the twenty-eight rules outlining the principles of composition are given. In the composition work the importance of grammar is emphasized. It is not only a good text-book but also an excellent reference book. Hicks's Champion Spelling Book. By Warren E. Hicks. American Book Co,, New York. Price 25 cents. "The boys that you send us can't spell" is the cry of business men and the investirations of various educa- BOOK REVIEWS. (51 tional commissions re-enforce the cry, Hicks's Champion Spelling Book covers six school years, from third to the eighth inclusive, and contains about 6,000 words in all. Of these, 1,800 are selected for intensive study, two being made prominent in each lesson. The pronunciation, syllabication, derivation, phonetic properties, oral and written spelling, and meaning of these are all to be made clear to the pupils, who are to use the words in intelligent sentences made by themselves. The subordinate words are arranged in helpful group- ings. Systematic reviews, and frequent oral and written spelling contests, are provided for throughout. Supplemen- tary lessons teach such helpful subjects as abbreviations, prefixes, suffixes, and word building. The work is laid out in such detail that no teacher will have the least difficulty in securing satisfac- tory results from the use of this thoroughly practical book. This book was prepared in direct re- sponse to the cry of business men in general. It embodies the method of teaching spelling which after two years' use enabled the pupils of the Cleveland schools to win victory in the National Education Association spelling contest of 1908. The Summers Readers. Primer, 114 pages. Price 30 cents. First Reader, 160 pages. Price 36 cents. Second Reader, 186 pages. Price 42 cents. Manual, 50 cents. Frank D. Beattys & Co., Publishers, 225 Fifth Avenue, New York City. One of the first things that attracts you as you pick up the Summers Readers is the mechanical excellence of the books. Tastefully bound, well illustrated and printed, with due regard to all the demands of the psychologists with regard to type and illustration. Miss Summers utilizes in this valuable contribution to the "Readers" of to-day in the building of her series the child's love for action and play. Every lesson is built around the activi- ties so dear to the heart of the normal child. The books are three, the Primer, First Reader and Second Reader, which are accompanied by the Manual, a most necessary book for any teacher hand- ling this series. In the Manual the aims, methods and principles of teach- ing elementary reading are outlined. Miss Summers follows faithfully the outlines here mentioned and makes the lessons themselves teem with life. In the Manual Miss Summers out- lines board work for the first eight weeks, beginning with action words like run, hop, jump, sing, the general method being to let the child look at the word and then perform the action. Later on the connective is taken and the words combined, as Rover can run and jump. Gradually new words are brought in by appealing to the child's own store of knowledge, as, for in- stance, I have a baby brother. He has brown eyes. He can laugh. He can walk. His name is Robert. This is written on the blackboard ; the children observe and read to themselves, then with backs to blackboard, each in turn repeats the story in his own words. After this work then com.es the Primer. Initial consonants are first taught, then final consonants, then medial, and finally blended consonants, through it all the training of the ear receives strict attention through the- medium of the words in the Primer. In the First Reader the phonic lessons are centered upon the vowel sounds, the- 62 BOOK REVIEWS. diphthong and equivalent vowel sounds, while the Second Reader extends to initial and final syllables as phono- grams. All through the First Reader we have the child's play instinct appealed to, e. g., "Making Maple Sugar," "The Engine," "The Miner," "Reap the Corn," etc. In the Second Reader the lessons are grouped as to months, March for in- instance being The Four Winds. The Bag of Winds. The Feast of the Dolls. The Feast of Flags. Franklin's Kite. Boats Sail on the Rver. The Story of a Water-Drop. The words of Longfellow, Emerson, Field, Hawthorne, Stevenson and many others are called upon to supply these little stories. Particular attention is called to the illustrations in these books by Lucy Fitch Perkins, they are very fine and aptly illustrate each lesson. The author has built up a splendid system, which will find its place among the leading systems of reading of the day. Cooper's I/ast of the Mohicans, adapted for school reading by Margaret N. Haight, cloth 1 2mo., 142 pages. Illustrated. Price 35 cents. American Book Company. Adventures of the Pathfinder, same author, 144 pages. Illustrated. Cloth, i2mo. Price 35 cents. These are additions to the Eclectic Readings, and like their predecessors are well adapted for the schools as supplementary readers. «» % » | «» J «» | «« ^ «» ^ «» | «» J «» ^ «« ^ «» | «> ^ «» ^ «« | «« ^ «» | «» ^ «» J «» | *» I Modem English Course t I m TWO BOOKS 4i By Henry P. Emerson, Superintendent of Education, Buffalo, 4* N. Y., and Ida C. Bender, Supervisor of Primary Grades, Buffalo, N. Y. 4 Book One. i2mo. Cloth. ix+ 238 pages. 35 cents ?z'^vv'.v'-v ^^ School Problems Will furnish teachers, free of charge, any information they may want with regard to any Text Book or Professional Book published. Where an immediate personal reply is desired, stamp should be enclosed. Write as often as you like. We are at your service. THE EDUCATIONAL PRESS 123 East 23rd Street, - - - . New York M1 .ss^>>^ss^=??=f;f:=^ ?=^'=^ ^vv^^^ I When vn-iiiixg to advertisers please mention School Problems. 64 SCHOOL PROBI.EMS. Five Himdred Adoptions in Southworth's Builders of Our Country An American History on the Biographical Plan. Used in many Catholic Schools. Book II, Sixth Graoie CORRMSPONDENCM INVITED Co. 35 W. 32nd St., New York City SCHOOL PROBLEM! is Your Journal THE NEW IDEA IN AS EMBODIED IN THE SUMMERS READERS By MAUD SUMMERS The Literature of CtiildHood present- ed in the Lang\iage of Ohildhood Pi'ofuaely and beautifully illustrated from original drawings by LUCY FITCH PERKINS These Readers, lirst of all, establish a knowledge of words that relate to the life and action of the child. The beginner really lives and acts through his newly acquired vocabulary. Thus he learns to read as he grows in general activity— a natural,, all-round development. Very soon the memory and imagination are called into play in a most powerful and direct way. The plan of the series as a whole ie to utilize the child's most vital ex- periences in his aoquisitiou of a vocabulary, con- tinually relating his development and growth in language to the things that interest and attract him most. The Manual provides daily lessons worked out in detail for the guidance of the teacher. Reading lessons and phonic lessons are given in orderly sequence. PBIUEB - 114 papes Price 30c. By mail S6e FIBST READKB 160 naireK PrJce S6o. By mail 42c SBCO.XI) BEADI,R 186 pagee Price 42c. By mail fiOc MAKUAIi F0BTEACi«K.B8 Price 60c. By mail 66c FRANK D. BEATTYS & CO., Publishers 225 Fifth Ave,, New York City CORRESPONDENCE INVITED A. C. McClurg & Co., Chicago, Western Depository We would like to hear from subscribers for auy preferences they may have with regard to arti(;les, methods of treatment, etc. We will always be pleased to consider articles suitable for publication in this Magazine. plI^MMIMlJMMlllliMiilMlMMlMlI^MlMMii When writing to advertisers please mention School ProNe7ns. The American Normal Readers, By May Louise Harvey Illustrated in color and in black and white. First Book, 144 pp. Second Book, 168 pp. Third Book, 224 pp. Books Four and Five in preparation AN IDEAL SERIES OF READERS Special Features of Excellence: They exemplify the best pedagogical principles. Their simple Anglo-Saxon style renders them exceptionally helpful in language work. The material is delightful, vivid, stimulating. "I consider 'The American Normal Readers' far superior to any book of their kind in the market to-day. They are the perfect work of an experienced teacher."— /Sr. If. Fidelis,Prin. Gomaga School, Washington,D.O. Silver, Burdett & Company New York Boston Chicago THE WIDE AWAKE READERS ^ well'graded primary series THE WIDE AWAKE PRIMER (all pictures in color) - 30 cents THE WIDE AWAKE FIRST READER (all pictures in color) 30 cents THE WIDE AWAKE SECOND READER (with colored plates) 35 cents THE WIDE AWAKE THIRD READER (with colored plates) 40 cents Careful grading, unflagging child interest, abundant reading matter, and artistic excellence characterize this new series throughout. The Second and Third Readers are rich in copyrighted— and therefore unhackneyed— material. The books are already in use in all parts of the country, including the cities of Washington, Boston, Chicago and New York. The series, in part or as a whole, has recently been adopted for the schools of Oklahoma, Virginia, New Mexico and Arizona. ^^.^J^J^J^J^,^ LITTLE, BROWN & CO. 34 BEACON ST., BOSTON 378 WABASH AVE., CHICAGO j?^M^i^%Ni%(»Ma / ||W M ^»^^^ ^ * AI| | W wn> m < ^ |iW % ^»^^< »^ >'V U «»* " > 'W^ »^^*lA»W| SCHOOL PROBLEMS ^ Journal for the Practical Use of Teachers $1.00 per Year in Advance Bi-Monttaiy, except July 25 Cents the Copy Efficiency in Class Instruction can only result when the grade work is carefully planned in detail. TMs is our Creed, and if you are a Routinist, your work will lack vigor if not Efficiency, But if you are a Progressive Teacher, you will need a copy of wWcli presents practical plans and details of grade work, series of model lessons, and articles based upon sound pedagogical principles. The aim will be to make every article so positively practical that it will show not only what ought to be done but how to do it. All the articles are working plans of actual teaching practice — plans that have been proven successful. THE EDUCATIONAL PRESS 123 £ast 23rcl Street New YorK City WE PAY FOR MATERIAL N.B. Stamps skould be enclosed if it is desired that manu- script be returned. •«*VI^'*'''*'^vif^^''^^'s#^>^^>4 5>4 45^ divided by (Read, 5/2.) (6) 2T456.001>4. \* J 800 (8) .900 The Forty-five Combinations. — The first necessity of the primary teacher in the matter of addition is to drill the children to recognize at sight the sum of each of the digits combined with each of the others. There are forty-five pos- sible combinations ; and the mastery of these is as much a necessity for future success in addition, as the learning of the multiplication table is to skill in multiplying. Many devices for teaching addition could be given, but the writer intends to give these in another article devoted entirely to methods. Easy and Difficult Combinations. — There are really twenty-nine easy com- binations and sixteen difficult ones. THE FORTY-FIVE COMBINATIONS. Teachers should not spend too much time on the easy combinations, for pu- pils rarely fail on them. It is in the sixteen difficult combinations that pu- pils generally fail in column addition, and there should be drill on these until they can be recognized at sight. The 29 Easy Combinations. ( a ) Those which add 1 : 234 5 6789 1 1 (b) The "Doubles": 123456789 123456789 (c) Those whose sum is 10: 6 7 8 9 4 3 2 1 (d) Those in which the sum is less than 10: 2222 2 3334 34567456 5 The 16 Difficult Combinations. (a) 7 4 7 5 7 6 6 5 (b) 8 3 8 4 8 5 8 6 8 7 (c) 9 9 9 9 9 9 9 2 3 4 5 6 7 8 Drills in rapid addition should be re- quired in every grade, and interest will be stimulated if one problem in column addition is given in written weekly or monthly tests. Subtraction and addi- tion are inverse operations and should be taught together ; for it is much easier for children when they are mastering the addition tables to learn the sub- traction tables. If taught as shown here the pupil will learn the siun and differ- ence of each pair of figures and thus save time and energy : 3 4 5 6 7 8 9 3 3 3 3 3 3 3 The Multiplication Table. — This table is one of the most useful tools of education and is used constantly through life. Its use should be made automatic as soon as possible. Drill until children : (1) Can write the tables. (2) Can give product when teacher names any two factors. (3) Can give both factors when teacher names product. (4) Can give missing factor when teacher' gives product and one factor. (5) Can apply tables to easy prob- lems, both oral and written. "OtzX" and '* Mental*' Arithmetic. Many teachers make a distinction be- tween oral arithmetic and mental arith- metic. By oral work is meant the analysis of the reasoning" process in words; by mental arithmetic is meant the silent method of reasoning, which requires the answers to be written by the pupil. In all oral work the formula method should be avoided, for it destroys orig- inality ; and the pupil should be obliged to invent his own explanations. For the purpose of drill in oral composition the explanation of processes by the pu- pil is excellent; but correct and com- plete sentences should be required. The first lesson in every class at the beginning of a new term should be on the new work of the grade. The pupil 8 ORAL AND MENTAL ARITHMETIC. who has just been promoted is all in expectation. He has a right to suppose that in a new grade he will get new work; and how deep must be his dis- appointment and disgust, when he finds that his promotion is but a name and a delusion. What is the use of promotion if the work is the same old grind with which he is familiar? The good teacher plunges right into the new work, and then works back to the known just far enough to connect the unknown. In this way the atmos- phere of the new grade is kept fresh and new from the first day of the term ; interest in the subject matter is not for- feited ; the labor of the teacher is dimin- ished, and the effectiveness increased. While something new is to be pre- sented every day, there should also be a daily review of some portion of the work in the lower grades. Pupils never become proficient in arithmetic unless there is constant review. The vast field of fractions, of decimals, of weights and measures, and of the three cases of percentage must be reviewed again and again, with ever-varying exercises while the pupil is engaged in working applied percentage and other topics. These reviews should be thought drills and not rote-drills. The best re- sults are accomplished by the use of mental problems, involving small num- bers, but applying principles in every conceivable way. The problems should be carefully constructed by the teacher out of school hours; for no one is able to invent intelligent problems in suffic- ient quantity and variety in the presence of the class. A Correct Method of Presentation. — A good method requires each pupil to do the work, even if only one or two are called on to give the result. Let the pupils be supplied with pencil and paper. Then let the class solve the problem mentally. At a given signal each pupil writes his answer, and lays down his pencil. Thus, all are alert, all are obliged to do the work, and the suc- cess of the teaching is easily ascer- tained. After the answers have been written, one or more pupils may be re- quired to state in his own way how he obtained his result. This he may do by means of a few figures on the black- board. For example, if the problem is to find J^% of 37 apples, the pupil might write the following: 1% of 37 apples=.37 apples. >4% of 37 " =.185 " Methods to Avoid. — (a) One poor method in mental arithmetic consists in giving a problem to one pupil in the class and requiring him to solve and ex- plain it according to some set formula given by the teacher, while the other pupils of the class get into disorder. In a class of thirty-five pupils, these methods train one child in arithmetic and thirty-four in the habit of inatten- tion. (b) Another poor method consists in giving a problem to the entire class, who solve it in silence. At length, one pupil is called upon to give his result. If this is correct, the teacher says: "How many have the same ?" Up goes the hand of nearly every child in the class. It is certain that such a method offers a strong temptation to deceive, particularly if the teacher is a little easy. The teacher who uses this method is tempting her pupils to tell lies. Such a teacher preaches in vain the duty of honesty. Actions speak louder than words. An immoral method of teach- ing does more harm than the teacher MEASUREMENTS. 9 using it can ever undo by formal les- sons in morality. In presenting new work, it is better in all cases to begin the topic in arith- metic by making the problems so sim- ple that the operations may be per- formed mentally. In this way the new principle is firmly impressed on the mind. All inexperienced teachers and many other teachers are apt to neglect and under-value mental arithmetic. To guard against this error, Principals should ascertain the results of number teaching by giving oral tests as well as written ones. About one-third of the arithmetic period each day should be devoted to oral and mental arithmetic work. Measurements. — The introduction of human affairs into the science of num- ber is important ; for arithmetic has little value, particularly to small chil- dren, except as it is practically applied. When we introduce weights and meas- ures, buying and selling, percentage and interest into arithmetic, we are simply translating abstract number relations into concrete forms that are related to the child's experience. The fundamental thing is to induce judgments of relative magnitude. It is the relation of things that makes them what they are. "A living apprehension that mathematics deals with definite re- lations of magnitude does away with artificial distinctions between a fraction and an integer, by presenting each as a relation. Thus, three is the relation of a unit to another one-third as large ; and one-third is the relation of a unit to another three times as large." Measuring and constructive work should play an important part through- out the work of the grades : first by ac- tual measurement and then abstractly as an application of geometry. The following outline of work will be found helpful: 3A Grade. (1) The connection of square and linear units in measuring areas of ob- longs. (2) How to find areas of oblongs having integral dimensions. (3) The names and ideas of the more common standards for measuring distances, areas, boxes, liquids, bulks, weights, time, and value. 3B Grade. (1) How to find the areas of rect- angles having integral dimensions. (2) How to find the areas of forms that can be expressed in rectangular parts. 5A Grade. (1) The connection between the areas of a triangle and of a rectangle having the same base and altitude. (2) How to find the area of simpler forms of triangles from their bases and altitudes. (3) How to solve problems in which the area and one dimension of a rect- angle are given to find the unknown dimension. (4) The terms and forms of the square, the rectangle, and the triangle. (5) The new ideas and words: for time, — names of months and seasons; for distance, — the rod, and the mile; for areas, — the square rod and square mile; for liquids, — the barrel; the standard abbreviations of all units used in both Parts I and II, and the use of p. M. and A. M. for after noon and before noon. 10 MEASUREMENTS. 5B Grade. (1) How to make and to read scale drawings. Areas of rectangles, parallelograms, and triangles in sq. ft., sq. yd., sq. rd., and sq. mi. (2) How to apply simple scales to drawings to ascertain actual dimen- sions, areas, etc., of the objects repre- sented. (3) How to find unknown parts of areas, objects, etc., from data given in denominate units. (4) How to use the square and rect- angle to picture operations with frac- tional numbers. 6A Grade. (1) Making drawings to scales of moderate difficulty. (2) Reading and interpreting scale drawings. (3) Scale drawings, and graphs of thermometer readings, and of statistical data. (4) Drawing developments of regu- lar rooms. (5) The laws and terminology used in the mensuration of rectangles, par- allelograms, triangles. (6) Use of -{- and — as signs of quality in thermometer readings. (7) The necessary language and laws of the rectangle, the parallelo- gram, the triangle, the circle, and the square box (or prism.) (8) How to calculate and to express quantities of lumber in board feet. (9) How to calculate cost of fencing fields and lots. (10) How to analyze irregular fig- ures into triangular parts. (11) How to graph thermometer readings and statistical data to show the laws of change embodied in such data. (12) How to make and to interpret scale-drawings of some complexity. (13) How to represent the funda- mental operations with lines and areas, and to show operations with fractional number with divided rectangles. (14) How to represent and to deal with such signed numbers as occur with thermometer readings, or opposing forces. 8A Grade. (1) Areas and perimeters of com- mon forms. (2) Construction of developments and models of solids, and nature of surfaces of solids. (3) Mensuration of the trapezoid. (4) Mensuration of cube, right and oblique prism, the pyramids, the cylin- der, the cone, and the sphere. (5) Fundamental concepts of per- pendicularity, parallelism, bisection of angles, learned through paper-folding. (6) Classifications of quadrilaterals. (7) Angles learned through the turning of clock-hands. (7a) Enough land measure to enable pupils to read intelligently the descrip- tions of land in deeds, to locate the land and to calculate the areas, when tract is regular, from the description,. (8) Measurement of angles with the protractor. (9) Angle-sums of triangles, paral- lelograms, etc. (10) Land measure, townships, sec- tions, fractions of sections, lots, merid- ians, base lines, ranges, and how to interpret legal descriptions of locations of farms. (11) That the square on the hypo- tlienuse of a right triangle equals the sum of the squares on the other two sides. (12) Practical uses of similar tri- ansfles. METFIODS FOR REVIEW IN 8B. II Methods for Review in 8B. John D'Arcy McGee, M.A. THE course of study proposes in the second part of the eighth year a review of the entire course in arithmetic. The principal advantage of this re- view seems to be to fit the graduate of the elementary school for efficiency in the office or the shop. To review thoroughly in five months all the subjects presented in seven and a half years must seem to the conscien- tious teacher an appalling task. If, however, he bear in mind the chief ob- ject of the review, the feat becomes less difficult. Many of the subjects in arithmetic proposed in grades lower than 8B have little, if any practical value, and seem to have been retained in the curriculum in deference to tradition or from the desire to impart a certain amount of culture. The teacher in touch with the world outside the class-room may readily select those topics of the mathematical course of study which bring the child into closer contact with life, and upon these spend his energy, lightly touching upon the less vital subjects. If, in review, he find any of the topics well understood, it will be the part of wisdom and in the interests of economy of time to pass on to other topics. Methods. Oral work will be found of even greater value than written in fixing the principles of the work. If the oral prob- lems are so constructed as to let the mind of the pupil dwell rather upon the principle, than upon the mechanics of the problem, a much stronger and last- ing impression is produced upon the child's mind. Of course, no earnest teacher proposes oral problems which he has not solved beforehand. Much blackboard work is important. Fortunate is the teacher of arithmetic who has a large area of blackboard at his disposal — or rather at the disposal of his class. Here can be detected errors that may easily escape the keen- est if committed on the individual paper. If carefully supervised, such solutions often lay bare misconceptions of principles and ignorance of proper methods hitherto unexpected. Drills in approximate answers prevent pupils from arriving at absurd conclusions and enable them to gain perspective. This of course presupposes that the conditions of the problems are not in themselves absurd or contrary to the conditions of life; e. g., a problem whose solution gives 40 cents as the price of a horse, or $12 as the cost of a pound of butter. Approximations should be a means, not an end. Instances are cited in which an over-conscientious teacher has rejected exact answers to problems, be- cause they were not approximate!. The review should begin with nota- tion and numeration of numbers, but the use of extraordinary and impracti- cal numbers is not useful even as a drill. An explanation of the principles of Roman notation, and practice in writ- ing a few simple numbers will be found sufficient. The four fundamental operations 12 METHODS FOR REVIEW IN 8B. should be a matter of daily drill. They require no marked intellectual ability. Here "practice makes perfect." Simple methods of proving the results in these operations should be used, but much valuable time may be lost in insisting upon elaborate proofs. Definitions should not be committed to memory until the operations have been understood. Then the definitions should be formulated, as far as possi- ble, in the language of the child's own experience. In the solution of problems a careful distinction should be made of (1) the thing given; (2) the thing required; and (3) the relation between 1 and 2. Here again there is danger of over-em- phasizing the formal part of the prob- lem. It will be sufficient if the child recognize the three factors in the prob- lem without specifically pointing them out in recitation or written work. In teaching of fractions the finding of the G, C. D, and L. C. M. can be re- viewed without dwelling upon the mathematical explanation of these pro- cesses. The meaning of the terms Proper and Improper Fractions, Mixed Numbers, Numerator and Denominator should be recalled to those, and they are many, to whom these words are sounds of forgotten signification. In fraction problems the numbers should be such as are met with in life. Ifi for example, has significance to few children and doubtless to not many more teachers. If the child can add, subtract, multi- ply and divide with ease, using only the so-called "business fractions" he may be said to have attained skill in fractions. Complex and compound fractions may be taught incidentally as a drill upon fractional operations. though their use in practical problems is extremely rare. Decimals, when un- derstood as a form of fractions, de- mand only a short time for review, in- cluding the changing of fractions to decimals, and vice versa. Percentage. — The recognition on the part of the pupil that percentage con- sists merely of operations in decimals in which the denominator is invariably 100 simplifies review in this topic. While per cent, may etymologically sig- nify "by the hundred," as some text- books explain it, the child gains a clear- er understanding of the relation of this term to decimals if the teacher insists upon translating it "hundredths." A thorough drill upon the "business fractions," translating them into deci- mals and percentage and inversely, will prepare the pupil for rapid and accur- ate work in the "Cases of Percentage." When the pupil has thoroughly grasped the "Cases," he should be led to see, step by step that the "Business Applications" (except interest and its applications) are the same operations under different names. Whether or not the teacher use the terms Base, Rate, Percentage, he can readily bring out the fact that these factors are em- ployed in problems of the "applica- tions." In trade or commercial discount a slight variation is found in the fact that sometimes more than one rate is used. In commission a problem such as "sent my agent $1,260, which includes his commission at 5%, to buy sugar. How much sugar did he buy?" is op- posed to common business practice, and useful only, if then, as a drill. The distinction between the basis of computation in fire insurance and other insurance upon property and that METHODS FOR REVIEW IN 8B. IH in life and accident insurance should be noted. The former only employ the principles of percentage. Taxes are generally based, not upon the real value of property, but upon as- sessed value. The rate is generally ex- pressed decimally. Trade Discount. — In practice each separate discount is subtracted from the preceding net cost. Thus : "On a bill of $1,000 are discounts of 5, 10, and 20%." 1000 less 5% 20 980 less 10% 98 882 less 20% 17.64 For the sake of economy or to test the above method the problem may be stated thus : 95 90 80 1000 X — X — X — = 100 100 100 Interest. — Any good method of com- puting interest may be used, but most pupils seem to prefer the cancellation method if they have been well trained in its use. The oblique cases of interest are most easily solved by the equation method. Exact interest can be employed only when the time is expressed wholly or in part as the fraction of a year. If the time is expressed in years only, ex- act interest coincides with common in- terest. Find Time Between Dates. — In prac- tice time between dates is calculated by the use of tables. Many advertising calendars give not only the date of the month, but the date of the year. By these the table-method of finding the difference between dates can be ex- plained. In compound interest care should be taken to use numbers easily handled. The use of tables for both common and compound interest should be explained. In bank discount the learner's chief difficulty appears to be to find the term of discount (the time for which the note is discounted.) Bank discount is merely interest demanded of the seller of a note for the use of the buyer's money from the date of sale (discount) until the date when the buyer has to collect the note. If this be thoroughly understood the oft-dreaded problems of this subject become as simple as any other in interest. Proportion should be preceded by a careful study of ratio. Problems in proportion may be easily solved by the equation method. Foreign money may be introduced in problems in duties. Approximately equivalents for the monetary units of the leading European countries should be memorized. Longitude and time is best taught in connection with geography. The use of a globe is practically indispensable. The problems should be easy. The metric system is advantageously reviewed a little at a time during the term. In the applications of measurements, as carpeting, papering, plastering, etc., the text books, as well as the workers in these trades differ to such an extent that any suggestion on these topics is practically useless. 14 TYPICAL PROBLEMS.— 8B. Typical Problems.— 8B. John D'Akcy McGee, M.A. I. What number, diminished by 253i^ leaves 84M ? 2„ Divide ^ of f by f of \^. 3. Reduce .00375 to a common frac- tion. 4. Change fu to a decimal. 2i of 6i 6. Add 342/5 thousandths, 219>^ hundredths, 743V3 tenths, and Vs of a ten-thousandth. 7„ How many square feet in a piece of ground 43.24 ft. long and 18.8 ft. wide? 8. Read LXXIX, MDXC, XLIV. 9. Express in Roman notation, 66, 58, 101, 1909. 10. How much is 224.64 plus 75% of itself ? II. A ball team has played 28 games and lost 6. What is its percentage of games won? 12. Bought $334.80 worth of shoes and sold them at an advance of 30^. Find gain. 13. A class of 36 boys increased its register 33^%. It then decreased 25%. What was its register then? 14. A tailor charged $12.25 for 3 yards of cloth which had cost him $3.50 a yard. What was his per cent, profit ? 15. A merchant lost $495 in business in 1908, which was 15% of his capital at the beginning of the year. What was his capital at the beginning of the year? 16. An agent bought 500 bbls. of ap- ples at $2.75 a bbl. and received a com- mission of dy^%. What was his com- mission ? 17. An affent who charged a commis- sion of S%, received $324.60 for sell- ing goods. How much did he sell? 18. What rate was charged by a real estate agent who received $18 for col- lecting $400 of rents ? 19. On a bill of goods amounting to $190 are discounts of 33^ and 10%. What is the total discount? 20. To what single discount are 25 and 10% equal? 21. In a town whose assessed valua- tion is $4,200,000, the expenses of the coming year are estimated at $84,000. What is the tax rate ? How many mills on the dollar? 22. A man who lives in a town where the tax-rate is 5^%, pays taxes on $27,500 worth of property. How much does he pay yearly? 23. What is the assessed valuation of a town in which a tax of 7^ mills on the dollar yields a tax of $22,500 ? 24. Find the interest of $470 from June 12, 1900, to Dec. 21, 1903, at 5%. 25. I loaned a man $3,000 for which he paid me $3,050 at 5%' interest. How long did he keep the money? 26. A loans B a certain sum of money for 2 years, 6 months at 4%. The interest is $500. How much is loaned ? 27. If I receive $548 for loaning a man $4,000 for 2 years, 3 months, 12 days, what rate have I charged? 28. What is the exact interest of $500 for 230 days at 5%? 29. Jan. 2, 1908, I deposited $1,000 in a savings bank paying 4% annually and compounding interest every six months. How much am I entitled to draw Tan. 2. 1909? BANKING. 15 30. A note dated March 5, 1909, to run for 4 months, was discounted May 28. Find the term of discount. 31. Find the discount on a note for $500 dated May 3, 1909 ; time 60 days, discounted June 20, at 6%. 32. What are the net proceeds of a note for $2,000 dated Sept. 15, 1909, for 2 months, bearing interest at 6%, and discounted Nov. 2, at 6% ? 33. Solve by proportion : — If it cost $1.98 to paint a floor containing 24^ sq. ft., how much will it cost to paint a floor of 46>4 sq. ft. ? 34. If Philadelphia be taken as 75° W., and Berlin as 15° E., when it is 2 P. M, in Philadelphia, what time is it in Berlin? 35. A wireless message sent from Philadelphia at 2:30 P. M. is received on a steamer at 3 P. M. In what longi- tude is the steamship? 36. What is the cost of covering with asphalt one-half mile of a 60-foot wide street, at $2.30 a square yard? 37. If the base of a triangle is 24 ft. and its area 144 sq. ft., what is its alti- tude? 38. If a bicycle wheel is 22 in. in di- ameter, how many feet will it travel in making 20 revolutions? 39. If water from a hose can be thrown 42 ft., how many square feet of lawn can be sprinkled from any one position ? 40. A cylindrical tank 10 feet in diameter and 15 feet high holds how many cubic feet of water? How many gallons ? 41. Two boys start from the same point. One walks directly north 600 feet, the other directly west 800 feet. How many feet are they apart? 42. What is the cost of carpeting a room 15 ft. by 21 ft. with carpet a yard wide at $1.50 a yard? 43. Express 302 m. as feet and inches. 44. A tank is 3 m. long, 2.5 m. wide and 90 cm. deep. How many liters will it hold? How many kilos? 45. Solve by using x: Two boys to- gether have $3.20. If one has three times as many as the other, how much has each? 46. Find the value of x and y: 5 ^+2 3;=22 5 x-{- y=^l By Charles H. Davis, B.S. Importance of a Bank Account. A BANK is an establishment for the safe-keeping, lending and ex- change of money. It is a safe place in which to keep one's treasure, "for thieves do not break through nor steal." It is important because it tends to habits of prudence and economy and induces people to save in time of plenty to provide against a time of need. Savings Banks* A savings bank is authorized by law to receive sums of money on deposit and to pay on the same compound in- terest at stated intervals. The intervals between the dates at which interest is paid is called the interest term. Interest is computed only upon the sum of money that has been on deposit during the entire interest term. The 16 BANKING. rate of interest depends upon circum- stances and varies in different banks and from time to time in the same bank. It is usually about 3^% or 4% per annum. The interest term in many banks is three months. In such cases the terms begin Jan. 1, April 1, July 1, and Oct. 1. In calculating the interest the cents are ignored in the principal. Compound Interest. If the interest is not paid when due it may be added to the principal. In or- dinary business transactions this is not allowed by law; but in the case of sav- ings banks it is an essential feature. The way in which savings banks in- terest is calculated may be shown by an Illustrative example: What will be the balance due on the following account on Jan. 1, 1903, in- terest being allowed quarterly at 4% ? Deposited Jan. 1, 1902, $100; Feb. 4, $75; June 29, $150; Sept. 15, $100; drew out Apr. 10. $65; June 5, $50; Dec. 15, $40. STATEMENT. Dates Deposited Drew Oat Interest Balances 1902 Jan. 1 100 100 Feb. 4 75 175 April 1 1 00 176 April 10 65 111 June 5 50 61 June 29 150 211 Julyl 61 211 61 Sept. 16 100 311 61 Oct. 1 2 11 313 73 Dec. 15 40 373 73 1903 Jan. 1 3 73 276 45 Banks of Deposit. Savings banks are useful chiefly as a convenient means for keeping one's money where it will be safe and where it will draw interest at the same time. But a bank is also serviceable as a medi- um of exchange. In the modern busi- ness nearly all payments are made by a system of checks on banks. By this means the handling of large sums of money is avoided. Banks engaged in this service are called banks of deposit. They do not usually pay interest on deposits, as their chief function is to supply a convenient method of ex- change. Use of a Bank of Deposit. On opening an account in a bank the depositor receives a check book which contains checks and stubs like the fol- lowing : BANKING. 17 w V - ^ « i OJ o c ^ >- ^ ^ > UJ ^, =^ z z § < O to o o V- c > THE C TWKNTI to S owe Side. L 6re«.^73ase x«lt.tUc(e PAR ALI^ELOGRAM . RHOMBUS. SQUARE. TRAPEZIUM. TRAPEZOID. 20 PICTORIAL ILLUSTRATIONS OF FORMULAS. TRIANGLE. EOUILATERAI, TRIANGI.E. CIRCLE. CIRCLE. Mental Problems. — SA. 1. Write in Roman notation the num- ber of hectometers in 25 kilometers. 2. What per cent, of a quadrant is a semi-circle ? 3. What will 2,750 shingles cost at $3 a hundred? 4. What is the net cost of a piano listed at $400, with discounts of 20% and 10% off for cash? 5. Divide $18 in parts in the ratio of 2 : 3 : 4. 6. What is the interest on $24 for 50 days at 6% ? 7. In what time will $200 gain $45 interest at 5% ? 8. What is the cost of a sight draft for $1,000 bought at J^% discount? 9. Paid $60 for sheep at the rate of v$10 for 3 sheep; how many did I buy? 10. What is 150% of $2? 11. What per cent, of 5^ is f? 12. A man sold f of an article for the cost of the whole ; what was his per cent, of gain? 13. If two places are 8 hours apart, what is their difference in longitude? 14. What is the interest on one mil- lion dollars for 1 day at 6%? 15. If 2.8 tons of coal cost $9,60, what will .7 tons cost? 16. If $300 gains $24 interest in 2 years, what is the rate? 17. 20 men can do a piece of work in 8 days ; how many men can do it in 5 days? PROBLEMS.— GRADE 8A. 21 18. What will it cost to fence a lot 40 ft. by 90 ft. at 10c. a foot? 19. Write the present year in Roman notation. 20. Find the value of x in 6:8=24:.r 21. Which is the better and how much, 7% stock at 110 or 5% at 75? 22. What per cent, of 1 cu. yd. are 9 cu. ft. ? 23. The cost of an article is $40, loss 5%; what is the selling price? 24. If the selling price of an article is $60, and the gain 20%, what is the cost? Algfebraic Problems. TWO UNKNOWN QUANTITIES. Find the value of each of the un- known quantities in the following: ■ i^ — y 2 \^ ' \ 12x — 41/ = 8 ^. S Gx + Uy = 20t '■'"1 to — 9j/ = 9 = 9 . =1 2x — '7y = 9 Sx-\-5y = 60 {•7x-\-'7z = 105 1 12a; + 2s = 68 16x — 4y = 12 9x — 8j/ = 22 S 5x + 2\y = 41 \ ISoj — 192/ = =^3 X 4- 17t/ = 10 8x — 7?/ = %% 5x — 7t/ = 4 7a; + 5i/ = 50 < 3x + lly = 49 ) 8x + 72/ = 86 5 92^ — 35 = 27 ^ 82/ + 60 = 76 Algebraic Problems. ONE UNKNOWN QUANTITY. 1. Divide 90 into two parts, so that one shall be twice as large as the other. 2. The sum of two numbers is 144, and the greater is 5 times the smaller ; what are the numbers? 3. The sum of $120 is to be divided among three children. The second is to receive two times as much as the first, and the third as much as the other two. Find the share of each. 4. A boy paid $1.00 for a speller and an arithmetic, the cost of the speller being one-fourth the cost of the arith- metic. Find the cost of each. 5. A man divides $360 between his two children, to the older he gives twice as much as to the younger. What is the share of each? 6. Divide $1.60 between two boys so that one shall have two-thirds as much IS the other. 7. A horse, wagon and harness cost together $390. If the horse is worth 3 times as much as the wagon and the wagon 3 times as much as the harness, what is the cost of each? 8. Divide 17 into two such parts that one part divided by the other part gives a quotient of 1 and a remainder of one. 9. Divide 53 into two parts so that the greater divided by the less equals 2 and a remainder of two. 10. Divide 150 into two such parts that one part shall be 2 less than 7 times the other part. 11. What number is that which is as much greater than 4 as its half is less than 5? 12. A man bought 60 head of cattle for $2,800 ; but 10 of them died. If he sells the remainder at the cost price, what does he receive for each ? 13. If a man gets 128 lbs. of sugar in exchange for 40 lbs. of cheese at 16c. a lb., what is the sugar worth per pound ? 14. If a water pipe discharges 48 bbls. of water in 2 hours and 28 min- utes, in what time will it discharge 108 bbls. ? 15. Find two numbers whose sum is 20 and whose difference is 8. 16. The rent of a house this year is $1,890,' which is 8% greater than it was last year : what rent was received last year? 22 PROBLEMS.— GRADE 8A. 17. There are three sisters whose ages together amount to 24 years, and their birthdays are two years apart. What is the age of each ? 18. A house and lot cost $8,500, and five times the value of the house was equal to 12 times the value of the lot; find the price of each. 19. A boy is ^ as old as his father and three years younger than his sister ; the sum of the ages of the three is 57 years. Find the age of the father. 20. Three boys together sell 270 papers. The second sells 10 more than the first and the third sells three times as many as the first and second to- gether. How many papers did each boy sell? Miscellaneous Problems. 1. Multiply nine and thirty-six ten- thousandths by one and seventeen hun- dredths, to the product add nineteen and eight-tenths and divide the sum by three hundred ninety-six millions. 2. f is what fractional part of -^ ? 6 11 is what per cent, of 4. 5yi% is equivalent to what com- mon fraction? 5. An agent bought hay for $8.50 per ton and received a commission of 5%. How much per ton did the hay cost his principal? 6. A merchant fails with liabilities amounting to $15,375.20 and assets in- ventorying $9,760.80 net. How much will a creditor receive on a claim of $1,040? 7. A man rowed down a river at the rate of a mile in 12 minutes, and re- turned at the rate of a mile in 20 min- utes, the trip occupying 2 hours and 8 minutes. How far down did he go? 8. Find the amount of $137.50 at simple interest for 1 year, 20 months and 13 days at 5% per annum. 9. Reduce the compound couplet !3:7 ) 22:5 > to a simple couplet in its low- 7i:33 S est terms. 10. 3 pecks, 3 quarts is what per cent, of 5 bushels, 2 quarts? 11. If 4 first-class workmen can paint a certain church in 18 days, and it takes 3 second-class workmen 32 days to do the same, how many days will it take the 7 men working together to paint the church? 12. Which is the greater ratio and how much, 8 : 9 or 7 : 8? 13. B and C start from two towns one hundred miles apart and travel toward each other. B traveling at the rate of seven miles in two hours and C at the rate of ten miles in three liours until they meet. What distance will each have travelled at the time of meet- ing? 14. Find the smallest number that will exactly contain 15, 18, 21, 24 and 30. 15. If 2 men dig 10>4 rods of ditch in 3^4 days, how many rods will S men dig in 4 days? 16. Find the sum of LXVI, CCIV, MDXIX, XVIII. Express your an- swer in the Roman notation. 17. If 3 men can do a piece of work in 4f days, how long will it take 5 men to do the same work? 18. An agent whose commission is 2% remits to his employer $2,808.19; what amount did he collect? 19. Express in Roman notation 84796. 20. Two-thirds of 12 is 12^ times i^, of what number? 21. Find the least common multiple of 153, 204 and 510. 22. Multiply 65.15 by 3.14159 and divide the result by 57.296, find the re- sult correct to three decimal places. PROBLEMS.— GRADE 8A. 23 23. Find the cost of 96 pieces of timber, each 3 in. by 4 in. by 12 ft., at $18 a thousand feet, board measure. 24. Find the proceeds of a note for .'I^ISO due :^Jarch 25, 1907, and dis- counted at a bank Jan. 31, 1907, at 6%. 25. Multiply 8 bushels, 2 pecks, 3 quarts by 74. 26. Define (a) improper fraction; (b) common multiple; (c) denominate number; (d) discount. 27. What fractional part of 5 rods is 3 yards, 2 feet, eight inches? 28. From 1 substract -|+-J+ 7 , di- vide the remainder by 5^ — 4^ and find the difiference between 1 and the quo- tient obtained. 29. A square field contains 40 acres, what will it cost to fence it at $ .60 a rod? 30. Divide $4.26 into parts that are to each other as f , |- and •^. 31. Successive trade discounts of 25% and 16f% are equal to what single discount? 32. The time from April 1, 1907, to June 17, 1907, was what fractional part of that year? 33. The owner of a corner lot 50 ft. by 150 ft. is taxed $1.65 a square yard for a walk 6 ft. wide extending around one end and one side of the lot. Find the amount of the tax. 34. Write (a) seven-eighths per cent; (b) four hundred-thousandths; (c) six hundred-millionths ; (d) a couplet expressing the ratio of two rods to eight feet. 35. A merchant bought 17.8 dekalit- ers of wine at 17.5 francs a dekaliter. He sold 95 liters for 2.15 francs a liter and the remainder for 2.5 francs a liter. How many francs did he gain ? 36. What will it cost to carpet in the most economical way a room 10' x 11', with carpet f of a yard wide, at SI. 25 a yard? 37. A watch was sold for $228 at a loss of 5% ; how much would have been gained b}^ selling it at a gain of 5% ? 38. A tank is 5.4 decimeters wide, 2.5 decimeters deep and 1.75 meters long; find the weight in kilograms, of the water it can hold. 39. At $22 per m. find the cost of 20 joists, each 4" x 6" and 16' long. 40. A box of 150 oranges is bought for $1.40; the oranges are sold at 20 cents per dozen ; find the gain per cent. 41. A cistern 2.5 m. by 3.6 m. con- tains 14 kiloliters of water ; how deep is the water? 42. A can do a piece of work in 3^ days, B in 3 days and C in 2f days; find how many days it will take them to do it if they all work together. 43. Simplify 17 ^ (f-l)(-i+U) 44. A boy bought oranges at 18c. a dozen and sold them at 5c. each ; find his gain per cent. 45. A grocer lost 20% by selling 120 gallons of oil for $1.62 less than cost; find the cost a gallon of the oil. 46. An agent sold 1,600 bushels of grain at 90c. a bushel and sent his em- ployer $1,418.40 as the proceeds of the sale; find the rate of the agent's com- mission. 47. Find the number of acres in a circular park whose diameter is 280 rods. 48. The net amount of a bill of goods after discounts of 15% and 20% is $516.80 ; find the list price. 49. A train runs at the rate of 5 miles in 6^ minutes; find how long it will take the train to run 42 miles. 50. Reduce ff|f to its lowest terms. 24 THE TEACHING OF THE METRIC SYSTEM. Nuts to Oack Dtjfingf the Study Fcnod, 1. A bin 3 meters, 5 decimeters long, 1 meter, 2 decimeters wide and 1 meter, 4 decimeters deep is full of corn which is sold at 9 francs a hectoliter; find in francs the selling price of the corn. 2. The net amomit of a bill of goods after discounts of 16% and 20% is $516.80. Find the list price. 3. Find the area of the largest circle that can be cut from a square contain- ing 272.25 square inches. 4. Find the proceeds of a note for $250, without interest, discounted at a bank for 3 months at 6%. 5. Find the number of acres in a cir- cular park whose diameter is 280 rods. 6. How high is a flagstaff whose shadow is 81 feet long when a flagstaff 30 feet high casts a shadow 36 feet long? Write the proportion. The Teaching of the Metric System. SEVENTH YEAR. By Jacob Theobald, Jk. IN preparing the following article on the teaching of the metric system, it has not been my purpose to divide into so many parts, the subject matter as presented in the syllabus, and then to distribute these parts over a like number of days or lessons. A mere skeleton outline of such a nature would be of practical value to but one person, and that is the one who originally thought it out, who had a particular object in view, and had particular con- ditions in mind. I refrain, then, from presenting a daily outline plan that would necessarily be a misfit elsewhere, and will suggest instead a method of procedure in the treatment of the metric system. In doing so I shall try not to be unnecessarily theoretical, and shall endeavor to descend to the realms of the practical as soon as possible. Elements to be Considered in the Treatment of a New Topic. In approaching any new topic, be it in Arithmetic, Grammar, Composition, or what not, we naturally ask ourselves just what we are trying to do. We find, as a rule, that there are three im- portant elements that enter into the question : First, there is in the course of study a certain more or less definite something which is to be taught; sec- ond, there is a boy who is to swallow, to digest, or to grow up into this ma- terial ; third, there is a teacher who is to bring boy and subject matter to- gether. Behind all three are the princi- pal and the superintendent, interpret- ing the syllabus, thereby making it more definite, and showing the teacher better ways of bringing the subject matter into the life of the child. Let us consider these elements in their application to the teaching of the metric system. The course of study lies open before me, and here at the bottom of the page are five innocent lines calling for: — "Table of linear measure, of volume, of weight; inter- relations emphasized. 1 cu. dm,=l liter; one 1. of water at its greatest THE TEACHING OF THE METRIC SYSTEM. 25 density weighs 1 kg. ; a five-cent piece is 2 mm. in thickness, 2 cm. in diameter, and 5 gr. in weight. Approximate equivalents: 1 m.=:39.37 in. (1 yd.); 1 L=l qt; 1 kg.=2i lb.; 1 km.=t mile." With all its inherent advantages this mass of matter is foreign to teacher and pupil alike, at least at the present time. It won't do to have the boy com- mit these facts to memory as one might so many Chinese symbols. There is to be a real understanding of the metric system. The ''emphasis on interrela- tions" is to be interpreted literally, not liberally. As interpreted not so many years ago, in the course of a uniform Graduating Class Arithmetic examina- tion conducted throughout this city, the work as outlined in the syllabus is to include such problems as.— volume of water, 3.5 m. by 42 dm. by 2.3 m. ; how many kilograms? Eas}^ enough, but certainly requiring a thorough under- standing of the system. As for the boy, the subject matter appears strange and unworldly to him; and much as we may talk about the ad- vantages of the metric system, he secretly wishes that we would leave him alone to cope with the disadvan- tages of his own. There is, at present, m the life of our boys little or noth- ing that is touched by the metric system. And now I turn around and look at myself. Yes, it is true ! I had for- gotten that the kilogram weighs 2-| lb., that the five-cent piece is 2 cm. in di- ameter, but with the aid of the syllabus T recall these figures now. To-morrow, how the boys will wonder at my erudi- tion! Shall I teach what is called for in the syllabus, and just how deeply shall I go into the subject? There can be but one answer. Whatever my own per- sonal opinion may be, it is my plain duty to teach, to the best of my ability and knowledge, all that is presented in the syllabus, plus what has been read into it as the result of official inter- pretations. In my teaching, however, I shall seize upon anything in the na- ture of the child that will aid me in bringing about a happy relation between him and the subject matter. Method of Attack, LINEAR MEASURE. My starting point will be the boy. Is there anything about the metric system he already knows? Almost nothing; but there is Tom, whose father is em- ployed in an importing house, he has heard his father talk about it ; William has seen "^ L" on something that I Avon't mention; John has just read about a ten kilometer flight by an avia- tor; and Henry tells us he has heard of a kilometer race. A kilometer! What does the strange word mean? The boys wonder and I appear to won- der with them. Now it so happened that, at the be- ginning of the term, I had "accidentally on purpose" placed upon the black- board a meter stick which the boys use now and then as a pointer and, perhaps, as something else when I'm not around. One day a month or so ago, while a number of the boys were at the black- board supposedly busy solving an in- terest problem, little James, who never will do as he is told, was calmly turn- ing this same stick over and over, and deliberately wasting his time counting the strange spaces on the other side. And now little James comes to the rescue. "Teacher," — they persist in calling me that despite my protesta- tions — "that ruler on the board is a THE TEACHING OF THE METRIC SYSTEM. meter." As a reward James goes to the board, draws a line one meter long, and writes "1 meter" below it. ''How many meters long do you think the board is? Write your guess on paper; and you, George, measure the distance for us." "Five meters and a little over," comes the answer. "How many of you guessed five? Now look at the meter stick, George, and tell us how many large spaces are marked off on it." "Ten," pipes little James, before George is half way through counting, "Mark off one of the spaces on the line that James drew on the board. What part of a meter is this distance?" "One-tenth," comes the chorus. "Write one-tenth of a meter in the space just marked off." Charles writes " ^o meter." I now say to the boys, "The people who use the metric system do not say one-tenth of a meter (I write italicized words on the board) but say one deci meter instead." As I pronounce the pre- fix deci, I draw a line through tenth of a and write above it in colored chalk, deci. "What does the prefix deci mean?" "A decimeter is what part of a meter?" "How many decimeters make one meter?" As the boys give me the an- swers, I write in three columns upon a side blackboard : — deci means f^o or .1 1 decimeter =.1 meter. 10 decimeters =1 meter. "Now (boys)," take out your arith- metics. Who can find the page on which there is a picture of a decimeter? James has it first, he may tell us the number of the page. "Take your papers and mark off one decimeter. Let's measure our desks; how many decimeters wide? Write your answers." Various distances are measured in this way merely to get all pupils to take an active part in the work, and to make them familiar with the term decimeter. Answers will be of the form, "5 deci- meters and a little over." "Look at 3^our arithmetics again, and let us see into how many large spaces the decimeter is divided. "Into how many, Fred?" "Ten." "Mark distance on the board. What part of a meter is it?" "One-hundredth of a meter." I tell them that a short way of say- ing this is one centi meter. One of the boys writes in the three columns al- ready on the board : — centi, means j\-^ or .01 ; 1 centimeter =.01 meter ; 10 centimeters = 1 decimeter. "Boys, I remember that centi means one-hundredth because one cent is one hundredth of a dollar." The drill in measuring outlined above, in regard to the decimeter, may be repeated here, including centimeters in the measurements. "I want to cover our bulletin board with denim. Will you tell us, Harry, just how much I will need?" Harry measures and Paul writes on the board as the length is called off : — 5 meters, 3 decimeters, 2 centimeters, and "What do you call them (he means those, but we let it go at that for the present) very small spaces that are on here? There are three of them." "Millimeters," I tell him. And some bright boy will usually be able to tell you that milli means .001 ; 1 milHmeter =: .001 meter ; 10 millimeters = 1 centimeter. We add THE TEACHING (>F THE METRIC SYSTEM. 27 this information to that ah-eady on the board. ■'Let us take our decimeter rulers and mark off the centimeters and milli- meters as is done in our arithmetics." With these improvised rulers the class measures various lengths, — the page in the history, the cover of the arithmetic, etc. We compare answers, and in disputed cases measure a second time. The answers assume the form 1 decimeter, 8 centimeters, 2 milli- meters. I have found this a convenient stop- ping point. Thus far the boy has be- come familiar with the language of the metric system, has had a little practice in measurement, and has probably no- ticed the decimal division. Drill and study are now necessary to fasten what he has learned. "For to-morrow's homework make a meter ruler and measure the length of ten objects found at home. Enter into your notebook the names of the objects and their lengths. Memorize: — (1) deci means ,1 centi means .01 milli means .001 (2) millimeter means foVo of a meter centimeter means joo of a meter decimeter means ^o of a meter (3) 10 millimeters =: 1 centimeter 10 centimeters = 1 decimeter 10 decimeters = 1 meter "If you notice any newspaper or magazine articles that refer to the metric system, bring them to school and I shall give you extra credit." The next day's work begins with rapid oral drill based on the tables memorized : — "Meaning of deci, milli, etc.? "What is a centimeter, millimeter, etc.? "How many centimeters in 1 deci- meter, etc. ?" We read a number of the answers to the objects measured, find quite a variety, and have a good laugh when one of our unconscious humorists informs us that his "Corn Flake" box is 2 meters, 2 decimeters deep. Several boys write their answers on the board and I find them to be of this type : — 3 meters, 5 decimeters. 4 centi- meters, 3 millimeters. "Now I'll show you a quicker way of writing answers of this kind. John, will you measure the length of the side closet? I'll take down the measure- ments." John calls out 2 meters, 5 decimeters, 7 centimeters, 4 millimeters. As he does so, I write 2.574 meters. Who can read this answer? Read as meters, decimeters, etc. ; also as decimal part of a meter. To make the pupils familiar with the decimal notation, I send ten or a dozen boys to the board and dictate rapidly, 5 m., 3 dm., etc., and pupils write as decimals of a meter. The common errors stand out so much more promin- ently at the board than they do on paper, that I can readily see them and apply a remedy. W^e then read the answers as decimal parts of a meter, and as meters and lower denominations. "But we do not yet know what a kilometer race is. In measuring dis- tances longer than those we have meas- ured thus far, we use larger units than the meter. How many meters shall we take to make the next larger unit?" The answer "ten" will probably be forthcoming; if not, let the class repeat the table as learned, and ask them to continue beyond meters. I tell them that Ave call ten meters a dekameter, and we add to our three columns on the blackboard : — '28 THE TEACHING OF THE METRIC SYSTEM. deka means 10 1 dekameter = 10 meters 10 meters = 1 dekameter "Five dekameters means five times ten meters, and may be written 50 meters. The five is in the ten's place." Boys are sent to the board and asked to virrite as decimals of a meter, quan- tities such as 7 dekameters, 3 meters, etc. The answers are then read as dekameters and lower denominations. The hektometer and kilometer are treated in the same way as the deka- meter. Drill is necessary to fasten the meaning of the prefixes hekto and kilo. We complete our table and the boys copy it for home study. I have found it well to proceed slow- ly with the table of linear measure, fix- ing the meaning of the Latin and the Greek prefixes firmly in the minds of the pupils. There are always some boys who will tell you that they have noticed that the Latin prefixes end in "i," and that the Greek prefixes have a "k" in them. If the boy finds such devices useful, why not let him use them ? As for abbreviations, it is a dull boy indeed who cannot, after a minute's study of the printed page, tell you that the abbreviations for units larger than the meter begin with capitals. "For to-morrow find out from your text-book how the meter was obtained, and to what it is equivalent in our own system. Construct a decimeter cube. Construct a centimeter cube." On the next day I send as many boys as possible to the boards, and we write the prefixes and meanings, and the table of linear measure. Abbreviations are permitted, and a time limit is set. Most of the boys will be able to tell you that the meter was obtained by taking f o (.qo ooo 1^^^^ °^ *^^ distance from the equator to the pole. Let us see how many inches this is, approx- imately. Calling a great circle 25,000 mi., we find ^, change to feet, to inches, divide by 10,000,000, and cancel. 25,000 X 5,280X12 qq n • 4X10,000,000 — ^^-^ ^"• If your boys have studied the home- work assigned, any number of them will be ready to tell you that a meter is 39.3T inches in length, and not 39.6 inches. "Of course our answer was only ap- proximate." Upon a corner of the blackboard we write with colored chalk 39.3/ inches, and we leave the information there for several days. "What measure have we that is about the length of a meter?" You will have no difficulty in getting the correct an- swer, one yard. "If we were using the metric system in this country, what would we buy by the meter? "How many feet are there in a kilo- meter ? "What part of a mile is a kilometer?" The problem is worked out by the class and the answers vary from .60 to .62. We decide to call a kilometer f of a mile. We clear up the mystery of the avia- tor's flight, and work numerous other problems involving the relations, — 1 meter = 39.37 inches, 1 kilometer =: f mile. The table for linear measure is now completed, and the equivalents called for are memorized. It remains for us to make a closer study of the decimeter and centimeter, as these will be needed in the tables of volume and of weight. I tell the boys that my middle finger is about one decimeter long. This state- ment invariably sends them to their metric ruler, and leads to two or three THE TEACHING OF THE METRIC SYSTEM. 29 minutes of animated discussion about the results in their own particular cases. Harry finds his middle finger just about a decimeter; William finds his index finger more nearly correct, etc. Then I tell them, "My smallest finger-nail is about a centimeter wide." This again leads to a test measure- ment upon the part of the class. "Let us now look at the cubes you constructed at home." The boys show the results of their labors. William didn't make any ; James had his crushed on the way to school, but after a little encourage- ment exhibits the remains ; John's cubic decimeter just about hangs together; Alexander has two fine models that we will keep upon the closet for future use. As the boys hold up their models a smile spreads through the room ; the "great big" cube and the "wee tiny" cube do look so "funny" when the}^ are side by side. There is nothing like the actual mak- ing and handling of the metric units, if you wish to make a lasting impres- sion. No amount of mechanical drill will be a substitute for this constructive work. I recall the story of a class in which a number of the pupils con- structed a cubic meter, and upon com- pleting it found it too large to get through the door. The hearty laugh that greeted them as they vainly en- deavored to bring in their prize must have helped to leave a vivid picture of the size of a cubic meter. For homework we take some simple problems from our arithmetic, prob- lems involving the use of equivalents learned thus far; we find the cubic con- tents of boxes, dimensions given in deci- meters, and contents in cubic deci- meters. The Liter. TABLE or VOLUME. We come now to the table of volume and the unit used, the liter. We re- turn to the cubic decimeter that we con- structed two days ago, and I tell the boys that in the metric system we call the contents a liter. "The liter is di- vided in the same manner as the meter. Let me see how many boys can build their own table?" If they can't do it, start them with, — 10 deciliters := 1 liter. It will be a dull class indeed in which some boy will not be able to give the table complete. L^pon our model cubic decimeter we write "/ liter." A number of the answers to the prob- lems on cubic contents of boxes are now written upon the board : — 60 cu. dm., 75 cu. dm., etc. "How many liters will each of these boxes hold?" I draw a line through cu. dm. and write liter in its place. "How can I find out how many liters an aquarium 3 dm. by 5 dm. by 4 dm. will hold?" Find the number of cubic decimeters and call your answer liters. A half dozen problems and most pupils will have caught the idea. But how about problems in which the dimensions are given in meters, deci- meters, centimeters, etc. ? In order to solve these it will be necessary to change the measurements to decimeters. The pupils must know that 3.3 m. = 33 dm. ; that 265 cm. — 26.5 dm., etc. The pupils have had drill in changing various denominations to meters and decimals of a meter. Thus they are already able to write 435 cm., 736 mm., etc., as 4.35 m., .736 m. We write a number of these upon the board and change to meters. ^'How many tenths 30 THE TEACHING OF THE METRIC SYSTEM. of a meter are there in 4.35 m. ?" The average boy will tell you ^-tenths. I tell him, "No, you forgot the 4 meters, there are 43 tenths of a meter." You will find it most economical to tell the boy to draw a vertical line after the tenths place and to call his answer decimeters : — 4.3 1 5 m. = 43.5 dm., 7.6|4 m. = 76.4 dm., etc. A little extra drill will be needed here to secure facility in changing the de- nomination from meters to decimeters. We are now ready to cope with the more difficult type of problem : — "Rect- angular tank, 5.75 m, by 7.35 m., by 175 cm. deep. How many liters will it hold?" The pupil changes the cm. to m., rewrites the dimensions as dm., finds the volume, and calls his answer liters. This part of the work is usually most difficult to the pupils and requires a large amount of practice. Permitting the pupils to make up their own prob- lems from actual measurements will add to the interest in the work. To fix the equivalent of the liter it is well to fill a cubic decimeter with sand and to compare with the dry and the liquid quarts. It is sufficient if they know that they are about the same in capacity. A discussion, in which the boys are the main contributors, as to what they would buy by the liter is in order. The Gram. TABLE OF WEIGHT. We proceed here as we did with the liter. Tell the boys that the weight of a cubic centimeter of water (at its greatest density) is called a gram. There will be little trouble with the con- struction of the table, and still less with committing it to memory. The pupils have constructed a cubic centimeter, and this may be used in de- termining the number of cubic centi- meters in a liter. If both models are at hand they will tell you at once that 1000 cu. cm. are equivalent to 1 liter. The equivalent of the kilogram, 2^/5 lb., may be written upon another side of the liter cube. To find the weight of a quantity of water (at its greatest density) it will be necessary to find the contents in cubic centimeters and to call the answer grams. After the work with the liter it will be comparatively easy to change various denominations to centimeters. Simple problems under this head will complete the requirements of the syl- labus. Perhaps at this point and not before it is well to show them the advantages of the metric system; the pupil cannot see its advantages until he has a basis for comparison, that is, has an actual knowledge of the metric system. Take two parallel problems: (a) Tank 3' 4" by 5' 6" by 2' 3" deep ; find contents in gallons, and weight. (&) Tank 3.4 m. by 56 dm. by .23 m. ; find contents in liters, and weight in grams. Let one-half of the class work (a) and the other half (&). See who will finish first. Or take the simpler problems of ad- ding various linear denominations in each of the systems, and apply the time test. Are we going to teach the are, the stere, and the quintal? Certainly not. Let the pupil use his text-book as a book of reference, if he is given a prob- lem involving these. Beware of enter- ing into the metric system too deeply ; if you do, your pupils will be hopelessly confused, and disgusted with it. By comparing the metric tables with PERCENTAGE.— 6B. 31 our own, all boys will see the compara- tive simplicity of the former. They are ready to listen to you when you tell them that you believe in the system, and they will be just as ready to ask you why it is not generally used in this country. You will have to explain the resistance to change on the part of the individual and the business implications involved in a change of standards. As for results. Is it not true that your boys forget most of this subject matter even before they leave school? Of course they will forget much of it. If you wish them to remember longer, review the metric system in connection with the mensuration and geometry in the seventh and eighth years. But after all, do they not really remember the es- sentials? — that there is a metric sys- tem; that it is constructed on the basis of ten; that the chief units are the meter, liter, and gram, and that the ap- proximate values of the meter and liter are the yard and the quart. Percentage. — 6B. John D'Aecy McGbe, M.A. IN this grade the child learns to make use of the knowledge of percentage which he has acquired in 5B and 6A. The pupil may be readily brought to see that the apparently "new" problems of this grade are in reality the old ones stated in new terms. A useful exercise for the pupil is to re-state orally a busi- ness problem as a problem in pure per- centage ; e. g., 'T bought a horse for ?200 and sold it at a gain of $24. What was the gain per cent.? Re-stated. What per cent, of 200 is 24?" Profit and loss presents little diffi- culty to the normal child as it deals with the world around him. The most common fault is assuming some other quantity than the cost as the basis of computation. An axiom of this grade, oft repeated, should be "Gain or Loss is based on Cost." Commercial or trade discount differs from the ordinarj-- cases of percentages in that sometimes more than one rate is given. In problems of this kind the base is constantly shifting, the net cost ob- tained from one discount becoming the base for the next discount. Commission may be explained by comparison with salary, wages on the one hand and with pront on the other. Show that unlike the first two, it is not based upon the kind or amount of work done, but like the third upon the amount of money involved. Such problems as, "Sent an agent $1,020 to be invested in cotton, after deducting his commission at 2%. Find amount of cotton bought," are of doubt- ful value, even as a drill. They are to- tally opposed to business practice. Taxes. The child who is a member of an or- ganization which subjects its members to payment of dues, can readily grasp the signification of the word taxes. He must be shown, however, that taxes, as an application of percentage, is based upo" value. While the early 32 TYPICAL PROBLEMS.— 6B. problems in taxes should involve opera- tions in formal percentage, it should be explained that the tax rate is usually based upon $1 or $100 and expressed as mills or decimally, often to five or six places. Duties are another form of tax, which in the United States can be levied only by Congress. One class of duties, the specific, since it is based upon quantity, has nothing to do with percentage. The other kind, ad valorem, is based upon the value of goods, and is ex- pressed in terms of percentage. On some classes of goods both kinds of duties are levied. The amount of taxes to be raised by duties has been a source of differ- ence between the two great political parties since the foundation of our government. The insurance contemplated in the course of study is that assumed upon property, fire, marine, etc. Life in- surance, based upon age, health, etc., is not computed by percentage. It should, however, be explained. Blank policies of both kinds of insurance may be easily obtained and should be shown to the class. Typical Problems. — 6B. John D'Arcy 1. A house bought for $3,200 was sold at a loss of 12>4%. What was the selling price? 2. What per cent, is gained by selling tea at 55 cents a pound, if it cost 40 cents a pound? 3. If I sell goods at a profit of $30 and gain 15%, how much did I pay for them? 4. A dealer sold a motorcycle for $144, and gained 20%. What was the cost? 5. A real estate agent ' collected $400 rent and charged 8% commission. How much did the owner receive? 6. How much worth of goods must an agent sell to earn $120, if his com- mission is 4% ? 7. If I buy a book marked $1.10 and receive a discount of 10%, how much should I pay for it? 8. An automobile is advertised at $4,800. Discounts of 25 and 20% are allowed. What does it cost the buyer? McGeb, M.A. 9. What rate of commission do I pay if I give an agent $20 for selling goods for $400? 10. If I insure a house for $5,000, at ^%, how much should I pay a year? 11. A yearly premium of $12.50 is paid for insuring property for $2,500. What is the insurance rate? 12. The owner of a house paid $4,80 yearly for insurance to a company which fixed its rate at ^%. For how much was the property insured? 13. A man's property is assessed at $10,000. The tax rate is 7^%. How much tax does he pay? 14. If the assessed valuation of a city is $50,000,000, and its expenses are estimated at $150,000, what is the tax-rate ? 15. A man pays $600 tax in a town where the rate is 3 mills on the dollar. For how much is his property assessed ? 16. The tariff on clocks is 40%. If 3'ou import $8,250 worth of these MENTAL PROBLEMS.— 6A. 38 articles, how much duty will you be expected to pay? 17. How many dozen of eggs can you import for a duty of $150 if the rate on eggs is 5 cents a dozen? 18. A merchant imported from Eng- land 450 pounds blankets weighing 1,350 pounds. The tariff is 30% ad valorem and 22 cents a pound. What is the total duty? Mental Problems. — 6A. By Charles H. Davis, B.S Tabic Drill. How many : 1. Ozs. make 1 lb. (Avoirdupois)? 2. Ozs. make 1 lb. (Troy)? 3. Lbs. make 1 T. ? 4. Lbs. make 1 long T. ? 5. Ins. make 1 yd.? 6. Ft. make 1 rd.? 7. Ft. make 1 mi.? 8. Units make 1 score? 9. Sheets make 1 quire? 10. Sheets make 1 ream? 11. Dimes make 1 dollar? 12. Cu. ft. make 1 cord? 13. Cu. ft. make 1 cu. yd.? 14. Qts. make 1 pk.? 15. Lbs. make 1 bbl. flour? 16. Cu. in, make 1 cu. ft? Cu. in. make 1 gal.? Sq. in. make 1 sq. ft.? Cu. in. make 1 bu. ? Units make 1 gross? 17, 18 19 20, Measurements. TABLE OF EQUIVALENTS 1 bu. =21 50.4 cu. in 1 gal. =231 cu. in f Approximately 1 bu.^^/^ cu. ft. 1 4 bu.=5 cu. ft. ( Approximately 1 cu. ft.=*/5 bu. I iy2 gals.=l bu. 7,000 grains =1 lb. Avoirdupois weight. 5,760 grains=l lb. Troy weight. Mental Problems. Reduce : 1. 20 cu. ft. to bu. 2. 2 cu. ft. to gal. 3. 60 gal. to cu. ft. 4. 10 gal. to cu. in. 5. 10 cu. ft. to gal. 1. Write 49 in the Roman notation. 2. If 1,000 boys each receive $0.25, how many dollars are distributed? 3. A boy bought 4 oranges and paid 6. 40 bu. to cu. ft. 7. 462 cu. in. to gal. 8. 10,752 cu. in. to bu. 9. 1 qt. to cu. in. 10. 1 pk. to cu. in. 4c. each for two, and 5c. each for two ; how much did he pay for all ? 4. Add ^/s of a score and J^ of a gross. 34 MENTAL PROBLEMS.— 6A. 5. 72 square inches is what part of a square foot? 6. How much larger is ^/g than J4 of a yard? 7. What is the cost of 7>^ yards of ribbon at 6c. a yard? 8. What is the difference between I6V3 and IIV9? 9. John is 20 years old. This is Vg of Henry's age; how old is Henry? 10. Bought 48^ gal. of molasses at 50c. per gallon; what was the whole cost? 11. Multiply 401.1 by 10 and divide the result by 1,000. 12. Draw a line 3^ inches long; erase l}i inches ; how long is the piece that is left? 13. If 12 bu. of apples are worth $9, how much are 8 bu. worth? 14. How many feet in 148 yards? 15. What will be the cost of >^ of a sq. yd. of cloth at $1.00 a sq. ft. ? 16. From 2>4 take ^. 17. If berries are 5c. a qt. what part of a peck can be bought for 25c. ? 18. If % of a bbl. of apples are divided among 6 persons, what part of a bbl. will each receive? 19. What is the cost of 2,500 feet of lumber at $4 per 100 feet ? 20. Write three fractions, each equal to J4. 21. A man owned Vt of a- farm and sold j4 of his share; what part of the farm had he left? 22. If 4 qts. of beans cost p/^, what will 7 qts. cost? 23. A man had $2^ and spent ^/g of it. How much did he spend? 24. How many weeks in a century? 25. Write one gross in Roman no- tation. 26. What will be the cost of a ton of hay at 2c. a pound? Written Problems. 1. (a) Find the G. C. D. of 235 and 685. (b) Find the L. C. M. of 3, 25, 60, 100. 2. Four cheeses weighing respec- tively, 321^ lbs., 41V6 lbs., 37^ lbs., and 51 lbs. were sold at $0.22 a pound. How much was received for them? 3. Vt of >^ a number is 75. What is the number? 4. Three persons, A., B., and C, bought a city lot for $10,400, of which sum A. paid $3,200, B., $2,400, and C, $4,800. What part of the lot belongs to each? 5. (a) From thirty-two and five- hundredths subtract nine and four- thousandths. (&) Divide seven hundred fourteen- thousandths by one and seven-tenths. 6. What is the value of a triangular plot of ground having a base of 128 rods and an altitude of 90 rods, at $30.75 per acre? 7. (a) How many acres in a field 242 yards long and 121 yards wide? (b) How many rods of fence are needed to enclose the field? 8. Find the circumference of a circle whose diameter is 22^ feet. 9. A farmer raised 125 bushels of potatoes. He kept 80 bushels, 3 pecks for his own use, and sold the remainder at 18 cents a peck. How much did he receive ? 10. A milkman sells 48 gallons, 3 quarts of milk, daily, at 8 cents a quart ; how much does he receive for milk in 15 days? 11. A farmer exchanged 12.35 cords of wood, worth $4.75 a cord, for cloth costing $2.47 a yard. How many yards did he get? 12. Suppose your teacher bought, to- day, from John Wanamaker, New WRITTEN PROBLEMS.— 6A. 35 York, the following goods : 12 yds. of doth at $1.37; 6 handkerchiefs at 25 cents; 1 pr. of gloves at $1.45; 1^ yds. of velvet at $2.40, and 1 pr. of shoes at $3.80, for which she pays cash. Make out a bill and receipt it as clerk for John Wanamaker. 13. Write a proper receipt to be given John Armstrong, who paid Frank Rogers $25, for October house rent, on October 2. 14. A tank is 4 ft. by 4>^ ft. by 5^ ft. deep. How many gallons of water will it hold? (231 cu. in.=l gal.) 15. When grain is worth $1^4 per bushel, how many bushels can be bought for $25 ? 16. Mr. Jones earns $26.25 a week and his son earns ^/^ as much. How much do both earn in 10 weeks? 17. What are the prime factors of 72? 18. (a) How many working days in October, 1909? (b) How much can a carpenter earn at $3,75 a day during that month? 19. If ^/j of Va of a piece of land cost $560, what is the value of the whole ? 20. A flour-dealer bought 125 bar- rels of flour at $6^4. He sold 97 bar- rels at $7^, but the remainder being damaged, brought only $5*/5 per barrel. What was his gain? 21. Change to equivalent decimals, and add ^, ^V^o, '/,,. 22. Find the cost of 75,500 bricks at $9.75 per M. 23. Divide .75 of 17^ by V^ of .035. 24. Multiply fourteen and twenty- three thousandths by five and seven- teen-hundredths. 25. A farmer having a flock of 1,800 sheep lost 37% of them. How many sheep did he lose? 26. What must be paid for 31 cwt., 45 lbs. of sugar at 5^ cents a pound? 27. From a cask containing 40 gals., 1 qt. of molasses there were drawn at one time, 12 gals., 3 qts., and at another, 5 gals., 2 qts. How much remained? 28. From one hundred and seventy- five and sixty-three hundredths, sub- tract ninety-four and two hundred fourteen thousandths. 29. Which is the greater and how much, 3/4 of 824 or .85 of 640? 30. Give an example of each of the following: (a) decimal fraction; (b) improper fraction; (c) mixed number. 31. Write in order of their value, "/g, 'A, n, Va, Vs. 32. ^ of a bin holds 150 bu. of grain. How many bu. does the bin hold? 33. A rectangle is IIV7 inches long and its width is 3^^ inches less. Find its area in square inches. 34. Using I6V3 as a divisor and y% as a dividend, find the quotient. 35. Add 34%, 168^7^2, 144V6, 36. A stone walk in front of a school-house is 4 rds. long and 3 ft. wide. What did the stone cost at 25 cents a sq. ft.? 37. (a) 8.001X100=? {b) 8.001-^100=? 38. A man who owned a farm of 68 a., 30 rds., sold 25 a., 100 rds.; how many building lots 40 ft. by 80 ft. can be made out of the remainder? 39. A man bought 3,644 sheep for $4,555. He afterwards sold 6% of them at a gain of 10%. How much money did he make on the sale? 40. I bought some goods for $1,625, and wish to gain 15%. For what must I sell them? 41. If a man uses 2 qt., 1 pt. of milk 36 MENTAL PROBLEMS.— 5B. a day, how many gallons will he use during the month of November? 42. Write four million three thou- sand two and six thousand three hun- dred two millionths. 43. (a) Write in words, 3002.10062. (b) From one take five millionths. 44. What will it cost to cement a cellar bottom, 24 ft. by 27 ft., at 96 cents a square yard? 45. John can do a piece of work in 8 days and Henry can do the same work in 6 days. In how many days can both do it working together? 46. Simplify 8>^-|-2X7— 32-:-4. 47. At $6.75 per C, what will 1,250 pineapples cost? 48. If Sj4 cords of wood cost $11.37>^, what will 12>^ cords cost at the same rate? 49. What is the altitude of a triangle whose area is 629 sq. in. and whose base is 37 in.? 50. If a man's pulse beat 300 times in 4 min., how many times will it beat in 8 hours? 51. Benjamin Franklin was bom Jan. 17, 1706, and died April 17, 1790. How old was he? 52. The War of 1812 began June 12, 1812, and ended by a Treaty at Ghent, Dec. 24, 1814. How long did the war last? 53. The Battle of Lexington was fought April 19, 1775, and Comwallis surrendered at Yorktown, Oct. 19, 1781. How much time elapsed? 54. How many days from April 25, 1908, to May 25, 1909? Nuts to Crack Dur mf the Study Period. 1. 14 bu., 3 pk., 6 qt. is what part of 239 bu.? 2. A grocer bought a cask of vinegar containing 52 gal., 2 qt., at 18 cents a gal. On transferring the vinegar to another vessel he found that 13 gal., 1 pt. had leaked away. How much must he charge a quart to cover the cost ? 3. How many cords of wood in a pile 24 ft., 10 in. long, 3 ft., 6 in. wide and 8 ft. high? 4. How many sacks will hold 35 bu., 2 pk., 7 qt. of grain, if one sack holds 1 bu., 3 pk., 7 qt., 1 pt. ? 5. Find the date midway between Independence Day and Columbus Day? Mental Problems. — 5B. By Charles H. Davis, B.S. 1. At $®/7 a box, how much must be paid for 35 boxes of candy? 2. At $% a yard, how many yards of silk will $70 buy? 3. A man spent $24, which was ^/j of his money. How much money had he? 4. Bought turkeys at $1J^ each. How many did he get for $36 ? 5. Product 27 ; multiplier ^ ; what is the multiplicand? 6. Cost of 550 pineapples at $2 per hundred ? 7. Write j4 cent in decimal form. 8. At $0.83^ a yard, how much will 8 yards of silk cost? 9. At $0.50 a yard, how many yards of silk can be bought for $42.50 ? MENTAL PROBLEMS.— 5B. 37 10. How many yards of binding will be required for a rug 16 feiet long and 14 feet wide? 11. ll+120-^2=? 12. If handkerchiefs are 12^ cents each, how many can be bought for $1 ? 13. What is the difference between 9 feet square and 9 square feet? 14. How many quart boxes can be filled from 3 pecks, 3 quarts of straw- berries ? 15. li silk is 83^ cents per yard, how many yards can you buy for $5.00? 16. A man sold ^/g of his farm and had 20 acres left. How many acres had he at first? 17. At $0.24 per doz. how many doz. eggs can be bought for $30.72? 18. How many pints of paint will it take to paint a floor 27 ft. square, if it takes I pint to cover 1 sq. yd. ? 19. At $57.60 per acre what is the cost of a field containing 14.6 acres? 20. From 3,968.3 take .928. 21. Reduce to decimals and add 03Vs 26V. 4476 T9V3 86V8 22. What will 7>^ yards of cloth cost at 12 cents per yard? Account Current. New York, May 1, 1906. HEZEKIAH HOPKINS, In Acct. with POTTER & PUTNAM, Publishers, 74 Fifth Avenue. AprU May June 7 16 30 22 34 1 10 26 Dr. To Mdse. per bill rendered. << (< << f< ti << << << It << << <( « << (< (< Cr. By cash " draft on G. M. Vail " cash Balance 46 103 125 69 238 12 58 00 76 60 572 535 95 May June 160 276 100 00 00 00 47 96 Copy and verify the above account. Personal Account. ARTHUR W. POTTER. 1909 Dr. Cr. March April May 2 26 7 1 To 42 yds. muslin, @ 6^c. 18 yds. Lyons silk, @ $2.26 By cash on acct. By 540 feet pine lumber, @ 8>ic. Balance (red ink) 1 2 40 31 50 15 18 8 90 91 43 81 42 81 38 WRITTEN PROBLEMS.— 5B. Open personal account with R. L. Putnam & Co., and include the follow- ing items and balance at the close: Jan. 7, 1906, received from them 150 bu. oats at 33 cents, and 40 bu. wheat at 80 cents; Jan. 27, received from them 400 bu. corn at 60 cents; Feb. 3, gave them a check for $150. Feb. 19, received from them 45 bbls. flour at $6.25; Feb. 27, received from them 10 bales hay at $4.25 ; Mar. 2, gave them cash $200; Mar. 17, gave them cash $150. Mar. 17, received from them 45 bu. barley at 50 cents, 30 bu. wheat bran at 10 cents, and 42 bu. white beans at $1.20; Mar. 30, gave them a note at 60 days for $200. Written Problems. 1. Write the decimal equivalents of: («) y& (b) 54 (c) Va (d) V, (e) ^. 2. 15.815 + 17.21+ 406.091 + .16 + 61 + 408.5= ? 3. 902 — 9.02=? 4. 9.3 X .0047 = ? 5. A man has a salary of $1,200, and saves 18% of it. How much does he save? 6. If 20% of the people in an elec- tion district were voters and there were 236 voters, what was the population? 7. V^xOf Vi5 0f 3/8=? 8. Reduce % in. to the decimal part of a foot. 9. If the average length of your steps is 2 ft. 2 in. and you take 425 steps. How far do you walk? 10. Find the area of the base of a box 20 in. long, 10 in. wide. 11. If there are 85 girls in a school and they are V9 oi the total number of pupils, how many pupils are there? 12. At 12>^ cents a yd., what will 8% yds. of ribbon cost? 13. Date, fill, foot and receipt the following bill; inserting the name and address of some purchaser and of some dealer whom you know: 6 chairs at $3.00 ; 2 arm chairs at $6.25 ; 1 rocking- chair at $5.50; 1 table at $27.75. 14. Dictate 75.001 976.478 39.008 1200.001 76.0432 7.G 9543.06 15. Change to decimals and add $4575 64V, 96V3 16. At $18.40 each, how many chairs can be bought for $496.80? 17. If a woman can make 5^ yds. of lace in one day, how many days will it take her to make 18^/3 yds.? 18. Cost of covering a floor 16 ft. 1. and 12 ft. w. with oilcloth at $1.50 per sq. yd. 19. If one man can build a wall in 24 days and another can build it in 18 days, how long will it take them work- ing together? 20. Find the cost of carpeting a room 9 ft. wide, 14 ft. long, with carpet 27 in. wide at $0.95 a yd. 21. Find the cost of sodding a plot of ground 27 ft. wide and 160 ft. long at $0.05^ a sq. yd. 22. If a farmer uses 15 bu., 3 pks., 1 qt. of oats, how many pts. does he use? 23. A merchant paid $10.60 per doz. for 5 doz. gloves. He sold them at $0.69 per pair, how much did he gain? WRITTEN PROBLEMS.— 5B. 39 24. If a grocer buys flour at $4.25 a bbl. and sells it at $5.15 a bbl, how many bbls. must he sell to gain $54? 25. Change ^^^/ujia to its lowest terms. Change "^/is to a mixed num- ber, and subtract the mixed nurnber from 100. 26. 2 is what part of 7 ? 3 pints is what part of 3 quarts? 4^ hours is what part of one day? V3 is what part of 2 ? 27. Change V^i to 99ths. Solve Va-^As- 28. A blackboard is 8 yards long and 3^ feet wide. How many square feet in the surface of the board? 29. Mr. Abbott paid V3 of this weeks' wages for groceries, V^ for meat, and ^/g for rent. He had $6.57 left. How much does Mr. Abbott re- ceive each week? 30. Find the cost of 3 pecks of strawberries at the rate of 2 quarts for 25 cents ; and 20 rods, 4 yards of wire at 5 cents a yard. 31. The difference between two numbers is 2,476, and the smaller num- ber is 4,389. What is the larger number ? 32. You went to the stors of Smith & Jones at your home yesterday and bought 4J^ yards of silk at $2.38 a yard; 8 yards of braid at 9 cents a yard and 5 spools of thread at 6 cents each. Make out a bill for the above purchase. 33. If one sack holds 3 pecks, 7 quarts of potatoes, how many bushels, pecks and quarts will 9 such sacks hold? 34. Which is the greater — the prod- uct ^/g multiplied by ^/^, or the quotient of Vg divided by ^/^ ? How much greater? 35. If 644 yards of silk cost $1.12 per yard, how much will ^4 ^s many yards cost? 36. It took James 10 hours, 49 min- utes to pick 8 barrels of apples. How long did it take him to pick one barrel of apples? 37. Write the table of square meas- ure. 38. At $1.15 each, how many geogra- phies can be bought for $48.30? 39. From a bin containing 596^ bu. of wheat they were sold at different times 196^ bu., 88^^ bu., 214^ bu. How many bushels remained? 40. What will it cost to fence a square field, 24 ft. on each side, at 83-J cents per foot ? 41. If ^ of a yd. of cloth costs $0.84 Avhat will 3V7 yds. cost? 42. From 8 subtract 4.006 and mul- tiply the difference by 5. 43. A man's salary is $1,320 and he saves 26% of it. How much does he save? 44. Change .0625 to a common frac- tion in its lowest terms. 45. Find the cost of SS^/g yds. of carpet at $1.17 per yd. 46. Add .0009, .0074, .078, .4, .0316, .9081, .0405. 47. How much will 12 men earn in 83^ days if each earns $4.50 per day? Nots to Crack During; the Study Period. 1. If the price of cabbage is $5 per hundred, what will fifteen heads cost? 2. If lumber is $18 per M., what will be the cost of inch lumber to build a tight board fence, 5 feet high, around a school-yard 400 feet square ? 3. Find the cost of carpeting a room 12 ft. by 16 ft. with carpet ^ of a yd. wide, at 95 cents per yard. 4. Express decimally 75%, 8>^%, 132%, 287^%, %%. 5. A strip of moulding is placed around a room 24 ft. by 16 ft. Find the cost of the moulding at 12j^ cents a ft. 40 MENTAL PROBLEMS.— 5A. Mental Problems. — 5A. By Charles H. Davis, B.S. 1. A boy had 30 cents, and found 10 times as much. How much money had he then? 2. What part of 2 yards is 1 foot ? 3. A grocer paid $^/3 for a bushel of nuts; how much did he pay fOr 6 bushels ? 4. Add ^ and ^■i. 5. Write in Roman notation, 88. 6. How many days in ^/g of a leap year? 7. A boy had $^, and then earned $^. How much did he then have? 8. How many quart boxes will be required to hold 2 bushels of berries ? 9. What is the sum of Vs o^ ^^ ^"d Vs of 27? 10. If 4 pens can be bought for 1 cent, how many can be bought for 20 cents ? 11. Bought lyz dozen lemons for 54 cents. What did 1 lemon cost? 12. Dictate for addition : 75.004 421.7506 9.7086 100.001 2.700 8.0009 900.6 13. Find the cost of 43 books at 25 cents each. 14. j4 oi the pupils in a class num- ber 36 ; how many pupils in the class ? 15. At $1.25 each what will be the cost of 1 dozen American histories? 16. A newsboy had 72 papers and sold ^/g of them ; how many did he return ? 17. In a certain class containing 48 pupils, ^/e of them skipped a class and ^/lo of the remainder were promoted; how many were left back? 18. What is the L. C. M. of 2, 4, and 5? 19. If % of a number is 49, what is the number? 20. What is the cost of 4 cans of corn at 12j^ cents a can? 21. What will be the cost of ^ a dozen handkerchiefs at 33^/3 cents each? 22. What is the difference between >4 and 7,2? 23. What is .I6V3 of 60? 24. Divide 9.65 by 100. 25. A man received $28 for his week's work and spent ^/^ of it; how much had he left? 26. 8 is what part of 12? 27. ^/s of a farm is 42 acres ; how many acres in the farm? 28. A boy had $*/5 and then earned 75 cents ; how much money had he ? "Written Problems. 1. Express in figures five million thirty-five thousand eight. Express in words 57,063. 2. Define (a) common fraction; (b) mixed number. (c) Give an ex- ample of each. 3. (a) Reduce 69^/9 to an improper fraction. (b) Reduce ^^^Vgg to a mixed number. 4. (a) Subtract $37.87 from $150. (b) Divide $556.11 by 111. 5. Change 9'^/^ to fifths; change }i to twelfths. 6. Find the sum of 13^, V3, 7, 18^. WRITTEN I'ROBLEMS.— 5A. 41 7. Find the difference between 2817, and eSV^. 8. A man had property worth $8,675. He willed Vo oi it to his wife and the rest to his son. How much money did he will to his wife and how much to his son? 9. Reduce 366 days to weeks, and 300 hours to minutes. 10. When rye is worth $1^ per bushel, how many bushels can be bought for v$45? 11. Find the total cost of dVo lbs. of tea at S0.95 per lb., 7 lbs. of sugar at $0.05^ per lb., and 15^4 lbs. of raisins at $0.17 per lb. 11a. Find the cost of 24 in. of silk at $1 per yd. 12. How many square yards are there in the surface of a blackboard 36 ft. long and 4 ft. high ? 13. Make out the following bill and receipt it : Mrs. George Grant bought of Henry Crawford, 11 yards of cloth at $0.75 ; >1 yard of velvet at $0.50 ; 15 yards of lawn at $0.13^. 14. Express (a) in figures, four mil- lion forty thousand and four; (b) in words, 28,082. 15. A farmer set out 6,480 tomato plants, and only ^^/^o of them bore tomatoes. How many plants bore no tomatoes ? 16. Solve (a) 2^+76. (b) I-V/4. (c) 3 is what part of 7? (d) )^ is what part of ^ ? 17. Change -/g to twentieths. Change '-/g to ninths. 38. Solve (a) j4XS. (b) 8-7e. id) 15X2^. 19. A butcher had a ham which weighed 14 lbs. He sold 3 J/2 lbs. to one customer, 4:^ lbs. to another, and 2^ lbs. to another. How many lbs. of the ham were left? 20. Find the whole cost of 4 lbs. butter at 23>^ cents a lb.; 2 lbs. of coffee at 32^ cents a lb., and 1^ lbs. tea at 48 cents a lb. 21. How much will 20 inches of rib- bon cost at 72 cents a yard ? 22. An oblong assembly-room is 72 ft. long and 46 ft. wide. How many yds. around the assembly-room? 23. Mr. Cobb earns $26.25 a week and his son earns -/, as much. How much do both earn in 10 weeks? 24. A merchant sold goods for $793, and thereby gained $89.65. How much would he have gained if he had sold the goods for $850? 25. Write in order of value, the least first, >4, Vs and j4- 26. If your answer paper is 12 in- ches long and 7^^ inches wide, how many square inches on one of its pages ? 27. Divide 340,061 by 231. 28. Name all the composite numbers between 10 and 40. 29. What are the prime factors of 60? 30. How many tons of coal at $6 a ton will pay for twelve cords of wood at $4.50 a cord? 31. A farmer bought two bushels of clover seed, and sold 75 of a bushel to one neighbor and % of a bushel to another. What part of a bushel had he left? 32. Two men bought a newspaper for $3,800, one paying f of the amount and the other -J-. How much did each pay? 33. How many vests, each containing ^ of a yard, can be made out of 24 yards of cloth? 42 WRITTEN PROBLEMS.— 5A. 34. Arrange according to value, be- ginning with the lowest, the fractions H-} Vl25 /l5' 35. A boy bought 5 pecks of cherries at 60 cents a peck and sold them at $0.10 a quart. How much did he gain? 36. At $12 a ton what is the cost of a stack of hay containing 9,000 pounds ? 36a. J. S. Gushing bought of H. A. Armstrong : 6 lbs. sugar at 7 cents ; 7 lbs. tea at 50 cents ; 11 lbs. coffee at 34 cents; 17 lbs. starch at 14 cents. Find the amount of the bill. 37. If 50 barrels of flour cost $260.40, what will 600 barrels cost ? 38. If the minuend is 70,000 and the remainder 9,899, what is the subtra- hend ? 39. How many yards of fence will enclose a rectangular garden 50 yards long and 33 yards wide? 40. How many square yards in a grass plot 45 feet long and 28 feet wide? 41. Make and receipt a bill of the following goods, sold by yourself to Joseph Cook: May 2nd, 15 pounds of sugar at 5^ cents; 9 pounds of coffee at 35 cents ; May 9, 7 pounds of cheese at 18 cents ; May 15, 3 gallons of syrup at 48 cents. 42. (o) Write in figures, these two numbers, twenty thousand nine and thirty thousand seven. (b) Find their difference. 43. I pay $30.50 for butter at 16f cents a pound, how many pounds did I buy? 44. How many bushels of potatoes at 75 cents a bushel will pay for 59 yards of cloth at $3 per yard? 44a. IOV9 times 3 plus 7^/3 minus 9^ equals? 45. If 1 acre of land costs $96, what will 2% acres cost? 46. If a cup holds ^ of a quart, how many cupfuls are there in 12^ quarts? 47. A man had 800 acres of land. He sold 49Y^ acres. How many acres had he left? Nats to Crack Daring the Study Period. 1. Change "/g and ^^/gg to decimals and divide the greater by the less. 2. A gentleman had 1,200 books in his library and gave away ^/g of them. He lost Ye of the remainder. How many books did he still have? 3. The floor of a room is 18 ft. long and 15 ft. wide. What will it cost to floor it at 62^ cents per sq. yd.? 4. Draw a square ^ of a foot on each side. 5. Find the value of 25>4 +573X3 —18^. DEVICES FOR RAPID DRILL. 43 Devices for Rapid Drill Work in Arithmetic. THIRD AND FOURTH GRADES. FIRST SECTION. By Adele Manning. to be the HAVE examples written on slips of oak tag, each slip bearing a number; as, 1, 2, 3, etc., accord- ing to the number of slips made. The teacher holds the corresponding num- bers in her hand. Pass rapidly down the aisle, handing out as many slips as the board space will allow. At a signal, the pupils pass to the board and work the examples. Any mistake being corrected by the pupils who remain in the seats. Sometimes, to enliven the interest, pupils who find the mistake are allowed to take the place and finish the example. At the end of a given length of time pupils stop work, face, and answers are called for by numbers. II. As many pupils are sent to the board as space will allow. The others are busily working at their desks. An ex- ample is given to those at the board to be worked in a given length of time. Each example worked correctly is represented by the figure 1 placed after the pupil's name. The child obtaining the greatest number of I's wins the game and receives some little reward. III. Contest in speed and accuracy. Send two pupils to the board to work the same example. The one who works the example cor- rectly in the shortest length of time wins. The interest of the other pupils is held by the fact that they are likely next two whose speed is tested. IV. Write an example on the board to be worked by pupils at the desks, having children stand when finished. Answers are read and the first one to obtain correct result, places the example on the board. V. In drilling for tables, have pupils line up on both sides of the room as for a spelling-match. The parts of the table are then given from side to side, the pupils failing to answer taking their seats. VI. Have parts of tables written on small cards, one by two inches, each card bearing a number. The teacher holds the corresponding numbers of the cards given out. Cards are mixed and then passed rapidly around the room. Card 1 is then called for, and the child hold- ing that card stands, reads, and gives the answer to the example; for in- stance, 8X9=72. Card 2 is then called for, and this is continued until all have been given. If the child hesitates to give the answer, the class is then called upon to give it. The above is given as a five or ten minute drill period. VII. Device for helping backward pupils. Divide the class into groups. Assign work to the brighter or up-to-grade pupils to be done at the seats. 44 DEVICES FOR RAPID DRILL. Take the backward pupils aside and work with them at the blackboard, al- lowing them to join Group I as soon as possible. It may not be possible to do this but once a week, but even that will prove of much benefit to the backward child. VII. Device for utilising small blackboard space. The following device has been suc- cessfully used with Fourth Grade pupils. It may, however, be used with any grade of pupils. The class, consisting of about forty pupils, is divided into two groups. One group has work assigned them to be done at the seats ; the other group, about twenty pupils, are sent to the black- board for rapid work in multiplication or division, as the case may be. The blackboard space may not allow more than eight pupils to work at the same time. Examples are placed on the board and eight of the twenty pupils are selected to work them while the remaining twelve stand immediately behind and observe the work, oflfering criticisms whenever necessary. Sometimes eight of the slower pupils are called upon to work at the board while the others stand back, ready to lend assistance when it is required. Children like to work in this way, for it gives them a feeling of independence and companionship. Even the dull child is roused to put forth an effort, from the fact that he is allotved to do and that his school-mates are observing his work and stand ready to lend a helping hand when necessary. When the class becomes familiar with this way of working, it is possible for the teacher to go to the front of the room after having started the class in the work, and call upon two or three of the pupils who stand just behind the workers, to look after the work. The class must be under good con- trol before attempting the above scheme, and the little people must be instilled with the feeling of helping one another, otherwise they are apt to be- come too critical. SECOND SECTION. General Considerations in Fourth Grade Numbers. The principal consideration in arith- metic is the ability to work accurately, with reasonable rapidity, and with in- terest, and to know how to apply num- bers to the ordinary affairs of life. To give all attention to acctiracy alone, to give all attention to speed alone, to have arithmetic without accuracy or speed, are things that we must avoid. An important point, then, is to com- bine accuracy and speed in the funda- mental operations and to get answers from a business man's standpoint. Great emphasis should be placed on the concrete work in which the teacher finds such an excellent opportunity for correlation, and for assisting the pupil in cultivating an interest in the uses of arithmetic. The text-book may be de- pended upon to furnish much of the material for abstract work. So often, in this grade of work, the question of analysis arises. Just how much should we require of the child? Just enough to prove that the child has a clear understanding of the problem or operation. To require a particular analysis and to insist upon a set form DEVICES FOR RAPID DRILL. 45 for the analysis may be the means of checking the originaHty of the child. We must keep in mind the common- sense, business reasoning that the child must use when he goes out into the world. To require a pupil to solve every ap- plied problem, in steps, is to encourage him to waste time; the pupils should be called upon to work rapidly, give the correct answer quickly as a business man would get it. A reasonable amount of written analysis should be required, but not limited in any notional way that would destroy the originality or make the solution longer than necessary. Devote much time to the funda- mentals in the Fourth Grade work. Time should be given each day to the working of an example in addition and one in long division, or to the working of an example in multiplication and one in subtraction, for it is principally through habit that a child acquires efficiency. Test results always and check long examples in addition in the following manner : Testing. 4283 (17) 867 (21) 48 (12) 18 18 10 4 5198 Add these numbers 18 18 10 4 50 Add horizontally: 17 21 12 50 The sums agree, therefore 5198 is correct. This should not be done with all ad- dition problems, but it excites interest and is a sure method of checking. Aim to train the child to think and work rapidly by obliging him to work an example in a given length of time. Ten minutes every day should be de- voted to mental arithmetic. Mental arithmetic should cover the work al- ready accomplished and precede new work being taken up. Small numbers should be used whose relationships are easily seen. No explanation of mental arithmetic should ever be required. The answer is sufficient, for the child cannot carry the process by which he got this answer in his head. For such explanations as are re- quired, written work is sufficient. For mental work it is best to have the prob- lems written on the board, covered till used, or use mimeographed copies, one copy in the hands of each pupil, or have old arithmetics cut up and pasted on cards. The blackboard, however, is the best because the work can be adapted by teacher to the child. It is well to have the answers written b)^ the chil- dren. The time limit should always be con- sidered. The work carried on as fol- lows : (1) Have problem read, either by the children or by the teacher, silently or aloud (thus giving an opportunity for variety). (2) Think. (3) Pencils. (4) Answers. Give several problems 46 DEVICES FOR RAPID DRILL. in this way and then read the answers, or ask children to read them and check them as read. The failures tell the teacher which kind of problems to repeat in following written lessons, where explanations are made. The above work is exceedingly valu- able for quick and accurate thinking, for number relationships, and for help- ing children to learn and use ingenious short cuts. Much attention should be given to writing dollars and cents in the Fourth Grade, for many of our pupils leave school at the end of this year's work. It is essential that they have some knowledge of business arithmetic to en- able them to write from dictation col- umns of figures both rapidly and accurately. The following device has proved successful : Turn the paper on the desk so that the lines run lengthwise, or from front to back, and have the pupils place the figures of the addends on the blue lines instead of in the spaces. The blue line acts as a guide to the child and he is very careful to keep his columns of ngures straight. Example : ::£5 O 00 O O to O p^ ,tf- "•<» GO bi 00 -1 O ;t o o o i;o The dififerent ways of putting a ques- tion are of great value in training the reasoning powers of a child. Examples : (1) From 8,247 take 6,795. (2) Take 6,795 from 8,247. (3) What is the difference between 8,247 and 6,795? (4) What is the difference between 6,795 and 8,247? (5) Find the difference between 8,247 and 6,795. (6) What must be added to 6,795 to make 8,247? (7) What must be taken from 8,247 to leave 6,795? (8) By how much is 8,247 greater than 6,795? (9) When the minuend is 8,247 and the subtrahend 6,795, what is the re- mainder ? The use of an authentic text-book, as a reader, has proved very profitable in connection with problem work. Be sure that the child understands what is required and not make the mis- take of thinking he doesn't understand processes. I have had pupils read a problem without having the least idea as to what is required, and so be unable to solve it. When I read the same problem the pupils readily understood and told the process involved without further question. Having discovered the difficulty, I experimented by cor- relating arithmetic and reading, as follows : During the regular reading period, the arithmetics were taken out and individuals were called upon to read a problem as they would a sen- tence from the reader. The pupils then discussed each problem as to what was required and the process involved. The problems may then be placed on the blackboard, or the solving of them may be deferred until another day. A good way to test pupils in problem work is to have a list of the problems studied during a certain length of time written on the board. Call upon a pupil to read and give the solution of the problem. Another pupil is called upon to work it on the blackboard. In this way the pupils' ability to read a problem is tested, as well as the knowl- edge concerning the process involved. The attention of the others is held, for it is not known who will be called upon DEVICES FOR RAPID DRILL. 47 to prove the solution given by working the problem. We speak of imagination being used in connection with geography, language and other subjects studied in our schools, but do we ever think of the wide field for imagination that we have in the subject of arithmetic? Consider, for instance, the imagination that the making of bills calls forth, the finding of the area of a grass plot that is to be mowed, the finding of the cost of two- thirds of a yard of ribbon when the cost per yard is known. Get down to the every-day life of the child. Encourage him to make out bills, examples, and problems of his own. Make out a bill of the groceries that he was sent to get this morning. Have them find the cost of hair ribbons, knowing the price of ribbon per yard, and that each braid requires two-thirds of a yard. Find the area of the school- room floor by actual measurements. In other words, lead the child to think and io do for himself. Fractions, Induce the habit of imagining equal parts of things wherever there is any reference to a fraction, by having pupils ase paper, paper- folding, apples, circles, blocks, and clay modeling. Use dia- grams of any kind that will bring out clearly to the child's mind the relation of one part to another and to the whole. Fractions cannot be seen from symbols. Three points in thinking of fractions should be clearly understood by the pupils in this year : 1. The relation of one part to the whole. 2. The relation of one part to another part, or one whole to another. 3. The relation that one magnitude bears to another consisting of other wholes. Put great stress upon the concrete work in the Fourth Year Deal with business fractions, those used in busi- ness arithmetic, and that are within the ready scope of the child. Very little abstract work should be done here. Practically all of the work of the 5A Grade can be well done here, but the problems are simpler and should always be solved by using the concrete illus- tration so that the child has a basis for the abstract work of the following term. He learns to see these difficult relationships only after a great amount of work with concrete material. The symbolic work that comes later means little if this foundation is not thor- oughly laid The Fourth Year is the place for much of this preparation. In touching on reduction of fractions let it be based as much as possible on what has gone before. The pupils are already familiar with terms used in liquid measure and know the relation that one quart has to a gallon, etc. I would, therefore, introduce the subject in the following manner: How many quarts in a gallon? One quart is what part of a gallon? Two quarts are how many fourths of a gallon? How many quarts in one-half of a gallon ? Is there any difference between one- half gallon and two-fourths gallon? I I I I I I I I I Each square in the above diagram is what part of the whole oblong? Two squares are how many eighths of the oblong? How many squares in one-fourth of the oblong? 48 DEVICES FOR RAPID DRILL. Compare one-fourth of the oblong with two eighths of it. Compare four-eighths of the oblong with two- fourths of it. With one-half of it. Vs=-V4 = >^. If both terms of the fraction are multiplied by two, what is the effect on the value of the fraction? The child sees that it has no effect on the value of the fraction and is ready to formulate the general rule: Both terms of a fraction may be multiplied by any number without changing the value of the fraction. By similar questioning he is led to see that dividing both terms of the fraction by the same number does not alter the value of the fraction, and is ready to formulate the rule: Both terms of a fraction may be di- vided by any number without changing the value of the fraction. Then may follow examples in chang- ing to higher and lower terms; as, ^f^^ to thirds ; ^Vao to fifths ; ^Va^ to sixths, etc. Va to twelfths ; % to eighths ; Vs to twentieths. Change to lowest terms: ^"As, ^Vs*' 21/ '^3/ gj-p /35) /80> *^'-'— Draw a line a foot long. Divide it into halves. How many inches in ^ foot ? Divide the line into fourths. How many inches in )4 foot? In ^ foot? Divide into thirds, and question in a similar manner. Divide into sixths. Draw a line 10 inches long and divide into fifths. Draw oblongs 3 inches long and 2 inches wide and divide them to shoM'^ that y2=-V^\ 73= Vo, etc. Draw oblongs to show that ^^Y^; ^=«A, etc. A. I I I B. I I I I D. I E. I Refer to the blocks drawn. Show Vg of A; Vg of A; Vs of A. Point to 34 of B ; % of B, etc. Show that >^+>4=M; Vg+Vg^L Show that l^Vs+Va ; Vs-Vs^V*. If we call A 1, what shall we call E, C, D, B? If we call B 1, what shall we call E, C, D? Show Vs of the above oblong; ^/^ of it. Vg is what part of Vg? Y,=: how many sixths? i/^= how many sixths? y2-\-^ / ^-=1 how many sixths? Y3-)-i/g=: how many sixths? Etc. Draw an oblong 6 inches long and 1 inch wide. Draw others Yg, Yz, Ys' -/g, Yc as long. Call the largest oblong 1 and name the others. In similar man- ner work with apples, circles, clay mod- els, etc., until pupils are so familiar with the fractional parts that they recognize at once the relation that one bears to another. Look at the blocks drawn. If we call D equal to 1, what shall we call E? DEVICES FOR RAPID DRILL. 49 C? B? A? If we call B equal to 1, what shall we call D? E? C? A? How many thirds are ^/s-hVaH" Va? Va^l ^^^ how many thirds? How many thirds are ^/g — -/g? Add: 3^2^,41. 81—5^=? A. B. In oblong A, point to Ve ; Ve ; V«- In oblong B, point to ^/g ; Vs; Vs. By using other diagrams, develop the terms numerator and denominator. Pupils are able to give at sight the sums of %-\-y2, ^/gH-/^, etc., and they know that before fractions can be added easily, they must be changed to similar fractions. By giving the example /4-\-^/s-\-}i, lead pupils by questioning, and by re- ferring to multiples previously learned, to give the smallest number that will contain each of 2, 3, 4 without a re- mainder. Give them L. C. D. Much concrete work should precede the ad- dition and subtraction of more difficult examples. From previous work the child gets the idea that the numerator of a fraction tells the number of parts taken, and that the denominator names the parts. Refer to oblongs A and B. Find ^/g of A. 2XV6=? 3XV6==?5XV«=? In like manner use other diagrams and the child will see for himself that to multiply a fraction by a whole number we multiply the numerator of the fraction. Multiplication with mixed numbers is based partly on the product of two numbers which the child has already had in the multiplication work of the preceding year, and partly on the point in multiplication of fractions just com- pleted. With few illustrations the child is led to see that the multiplication of a whole number and a fraction depends upon finding the product of a whole number and a fraction. v.. Count the number of parts into which the diagram is divided. Each division is what part of the whole figure? Into how many parts do the heavy lines divide the figure? Each of these is what part of the whole figure? VaOf y4=? VgOf 3A=? 3 Since Vs of H= , what is -/, 3X4 of }i? Find by diagram '/. of Vg, ^^ of >^ etc. Find without diagram V5 of -/^, ^ of -/g, etc., and lead pupils to discover that to find the product of two frac- tions, use the product of their numer- ators as the numerator of the result. and the product of the denominators as the denominator of the result. Measurements* In taking up the subject of areas, results should be obtained by actual measurements. Have pupils take a piece of paper an inch wide and fold over at one end a square inch for a measuring unit. Measure a pad-back and find out how many squares there are in the length of it. In the width. Then, how many squares can be cut from the pad-back. If a pupil does not see, let him find out by actually placing the squares and marking around them. Take larger measures, three inches by one inch, and three or four inches by two or three inches, and find out 50 PROBLEMS.— FOURTH GRADE. how many smaller areas can be made from the larger one. In this way, gradually, lead up to finding the areas with larger dimensions. Use variety whenever possible. Measure a book cover, the top of the desk, a door panel, the blackboard, the school-room floor, etc. A. B. Suppose 1 inch on this line to repre- sent a mile. A man goes from A to C, and back to B. How many miles does he travel? From B he goes to A and back to C. How many miles has he traveled ? A G F B C D B X Map of country road, starting from street AX, drawn to a scale of 1 inch to 3 miles. How many miles is it from the street to B? Plan of a garden, drawn to scale of of 1 inch to 12 feet. How many square feet in area B? How many square yards? Find the area of D in square feet. In square yards. Find the area of C. Find the area of A in square feet. Find the area of the whole garden in square yards. If 1 inch represents 3 yards, what is the area of each section in square yards? How many miles from the street to G ? How far is the road GF from the road DE? Etc. THIRD SECTION. Examples and Problems. Fottfth Grade. Give pupils about four examples like the following: $124.46, $74.24, $70.06, $145.23, $48.20, and require the correct results in three minutes. Have similar work in subtraction, multiplication, division. Add: $5.43, $6.48, $2.19. Add: $2.03, $9.86, $10.81. Add: $9.98, $22.14, $1.06. Subtract $7.45 and $5.03. Subtract $24.00 and $36.55. Subtract $7.11 and $6.45. Multiply $8.46 by 5. Multiply $7.08 by 9. Multiply $45.09 by 8. Divide $12.24 by 6. Divide $56.56 by 8. Divide $63.72 by 9. 1. Harry and his father went to a restaurant, and ordered as follows: 2 orders chicken soup at 15c. each; 2 orders blue-fish at 20c. each ; 2 orders small steak at 55c. each ; 2 orders sweet potatoes at 20c. each; 1 order mixed salad at 25c. ; 1 order vanilla ice-cream PROBLEMS.— FOURTH GRADE. 51 at 15c. ; 2 orders coffee at 10c. each. What was their bill? 2. Have pupils make out similar bills, using the one given as a model. 3. From a plot of ground 54 ft. long and 36 ft. wide, the sod is being taken. Each piece of sod measuring 12 in. long and 12 in wide. How many pieces of sod will be taken from the plot? 4. Price of furniture: Bedstead $24.00 Dresser 18.00 Common rocker T.OO Bookcase 16.50 Morris chair 12.50 Davenport 65.00 Dining-room chair 3.85 Kitchen table 4.75 Small desk 13.40 Kitchen chair 2.50 Refrigerator 10.95 Arm-chair 6.50 Sideboard 45.00 Hall chair 3.80 Large desk 48.00 Small range 25.00 Wicker rocker 8.75 Dining-table 24.00 5. At the prices given in the list, what is the cost of furnishing a kitchen with a small range, a kitchen table, a refrig- erator, and a kitchen chair? 5a. Find the cost of a dining-room table, a sideboard and 6 dining-room chairs. 6. Find the cost of furnishing a room with a bedstead, a dresser and two common rockers. 7. Find the cost of furnishing a room as you wish. 8. Mary has read 144 pages, which is five-eighths of the entire book. How many pages in her book? 9. The president of a bank receives $50,000 a year as salary. If he spends $85 each day, how much will he save in 4 years ? 10. What will be the cost of plaster- ing the ceiling of a room 24 ft. square, at 26c. a sq. yd. ? 11. The floor of a room is 324 sq. ft. One side of it is 4 yds. long. How many yards long is the other side? 12. Find the area of a tomato-bed that is 24 ft. long and 18 ft. wide. Draw a diagram, allowing i^ of an inch for a foot. 13. How many more sq. ft. in a plot of ground 32 ft. long and 28 ft. wide, that is devoted to the raising of corn? 14. Kate and Mabel went to cooking- school. They learned to make the fol- lowing breakfast for 6 persons and to find its cost. Muskmelons Post Toasties with Cream Broiled White Fish Fried Potatoes Cream Toast Graham Biscuit Coffee They used 3 melons at 8c. apiece; 4 pounds of fish at 10c. a pound, J^ package of Post Toasties at 12c. a package; 1 quart of potatoes at 18c. a peck ; half a loaf of bread at 5c. a loaf ; 2 pints of cream at 19y^c. a pint; ^ of a pound of coffee at 40c. a pound; 1 dozen graham biscuits at 12c. a dozen ; and 1^4 pounds of butter at 36c. a pound. What was the cost of the whole breakfast ? Cost for each person '^ 15. How many square yards of car- pet will be needed to cover a floor 27 feet long and 24 feet wide? 16. How many feet of fence will be needed to fence a garden 24 yards long and 18 yards wide? 17. A lot is 195 feet long and 53 feet wide. How many square feet does it contain ? 18. A garden is 46 feet long and 22 52 PROBLEMS.— FOURTH GRADE. feet wide. How many square feet does it contain? 19. A man sold 284 baskets of grapes, which was ^/g of his crop. How many- baskets ful did he raise? 20. There are 9 cars an hour on a street car line, and each carries 34 per- sons. How many persons ride on the road from 6 o'clock in the morning until 6 in the evening? 21. A conductor collected 29 five-cent fares, and 5 three-cent fares a trip, for 9 trips. How much did he collect? 22. The push-cart man pays 5 cents for 2 boxes of cracker jack, and sells it at 5 cents a box. How much does he make on 120 boxes of crackerjack that he sold in one day? 23. When 8 tons of coal cost $72.80, what will 18 tons cost? 24. H 7 yards of cloth cost $0.84, what will 24 yards cost? 25. What will 36 yards of cloth cost, when 8 yards cost $24.40? 26. If 3 tons of coal cost $36, what will 6^4 tons cost? 27. When 8 yards of cloth cost $3.60, how much will 9 yards cost? 28. Mr. Brown bought 24 dozen hammers at 66f cents each, and sold them at 70 cents each. How much did he gain? 29. Find the cost of 14 pounds of sugar at Qj4 cents a pound; 20 dozen eggs at $0.34^ a dozen; 2 bushels of potatoes at 80 cents a bushel. 30. At a fair held in town, 480 thirty-five cent tickets and 374 twenty- five cent tickets were sold. How much was received for admission? 31. Make a diagram of a blackboard, allowing 1 inch for a foot. The board being 15 feet long and 4 feet wide, how many square feet in it? 32. How many square feet in a floor 24 feet long and 18 feet wide? Make a diagram of it, allowing ^ inch for a foot. 33. The grass plot in front of your house is 24 feet long and 18 feet wide, how many square feet in it? Make a diagram allowing ^ of an inch for a foot. 34. John, Fred, and Jim sold 450 papers, John sold ^/^ of them; Fred sold ^/s of them. How many did Jim sell? 35. What will 474 acres of land cost at $64 an acre? 36. A boy went to a baseball game. He paid two nickels carfare, one quarter and a dime to get in, and fifty cents for a seat. How much did he pay to see the game? 37. What does the moulding of a room 22 feet square cost at $0.05 a foot? 38. In a barrel are 35 gallons of oil. After drawing out 18 gallons, how much is the remainder worth at $0.06 a quart? 39. If a bootblack earns $15 a month, how long will it take him to earn $645 ? 40. I bought a hat for $4.25, a pair of shoes for $3.75, and a coat for $12. I gave the man $25. What change should I receive? 41. 3 lbs. of tea cost $1.50. Find the cost of 108 lbs. of tea. 42. What is the value of 374 quarts of chestnuts at 12 cents a quart? 42a. A boy sold chestnuts at 8 cents a quart, receiving $18 for them. How many quarts did he sell? 43. A lady paid $18 for a cloak, $8.75 for a hat, and had $9.50 left. How much money had she at first? 44. Each of seven boys have one hundred forty-eight marbles. How many have all? 45. At five cents a quart, how much will eighty-six gallons of milk cost? MENTAL PROBLEMS.— 3B. 53 46. How many squares 1 inch on each side can be cut from a piece of paper 3 feet long and 8 inches wide? 47. At 2 cents a square foot, how much will 45 boards cost, each 15 feet long and 1 foot wide? 48. Find the difference between two surfaces, one 36 feet square and the other containing 1,034 square feet. 49. How many tiles 1 foot square are required to pave a hall 24 feet long and 8 feet wide? 50. How much sugar, at six cents a pound, will pay for fourteen dozen eggs at thirty-six cents a dozen? Mental Problems. — 3B. Charles H. Davis. 12 cents, what He bought 8 How much V5 1. If 3 pencils cost will 9 cost ? 2. James had $1.00. pears at 6 cents each money had he left? 3. At 25 cents a dozen what will of a dozen bananas cost? 4. A grocer gives T eggs for a quarter, how many eggs can be bought for SI. 00? 5. What will 12 bushels of potatoes cost at 80 cents a bushel ? 6. What are the factors of 49? 7. If 2 handkerchiefs can be bought for 25 cents, how many can be bought for $1.50? 8. A grocer had a case of eggs, which contained 360 eggs. How many dozen in the case? 9. What number multiplied by 9 makes 63? 10. What is the cost of a pound of tea if ^2 pound costs 28 cents? 11. Change 84 days to weeks. 12. A carpenter earns $3 per day; how much will he earn in two weeks ? 13. Bought 5 doz. oranges and ^/g of them were spoiled ; how many were good? 14. A boy had 54 cents and spent ^/^ of it ; how much did he spend ? 15. If your school-room is 8 yards and 2 feet long; how many feet long is it? 16. Helen paid 20 cents for candy; what part of a dollar did she spend? IT. Va+Vo=? 18. 7+7-=-7X2+l=? 19. Write 3 fractions that are of the same value as y^. 20. A piece of cloth measures 840 inches ; how many feet does it measure ? 21. If 3 pencils cost 7 cents, what will be the cost of 1 dozen? 22. At 60 cents a foot, what will it cost to build 30 feet of fence? 23. Drawing paper is 9 inches by 7 inches, how many square inches does it contain? 24. If Mary buys 5 gum-drops for a cent, how much will 30 cost? 25. How many thirds in 9V3? 26. Eggs are 2^ cents each; what will half-a-dozen cost? Written Problems. 1. Add: 4,726, 1,491, 3,208 and 6,407. 2. Subtract $1,280.50 from $5,000. 3. A piece of cloth measures 840 in- ches. How many feet does it measure? 4. Mr. Williams paid two bills of 54 WRITTEN PROBLEMS.— 3B. $276 and $498. How much change should he receive from a $1,000 bill? 5. A jumping rope is 27 feet long; what is Vs of ^ts length? 6. Find the total weight of 7 bales of hops, weighing respectively, 186 lbs., 205 lbs., 176 lbs., 198 lbs., 183 lbs., 199 lbs. and 217 lbs. 7. How long will it take a boy to earn $135, if his wages are $9 per week? 8. (a) How many thirds in five? (b) How many fourths? 9. Sold to one man 12^^ bushels of apples; to another 3^ bushels. How many bushels were sold in all? 10. Butter is worth 35 cents a pound, how many pounds can be bought for $5.95? 11. A boy earned 60 cents a day; he lost 15 cents and spent ^/g of the re- mainder. How much did he spend? 12. Divide 15,680 by 32. 13. At 48 cents a pound, what will be the cost of 4^^ lbs. of coffee? 14. Multiply 964 by 39. 15. What is Ve of $726? 16. When 2 apples cost 5 cents, what will 50 cost? 17. How many square inches in the cover of your book, if it is 8 inches long and 6 inches wide? 18. Two books cost 19 cents; what will be the cost of 30 books ? 19. Divide 4,968 by 72. 20. (a) Find % of 640 square in- ches, (b) Make a drawing to explain your answer. 21. Find the cost of 2 bushels of potatoes at 5 cents a quart. 22. Bought a house and lot for $9,750, and sold it so as to gain $2,275 ; how much did I receive for it? 23. Add: 7543 867 9089 86 438 7509 4327 24. (a) Draw a rectangle 4 inches long and 2 inches wide, (b) Divide it into thirds. 25. $36.89X78=? 26. $47.03— $29.67= ? 27. A lady paid $18.50 for dress goods, $6.94 for linings, $0.24 for thread, $14.68 for trimmings, and paid her dressmaker $16.50. Find the cost of the dress. 28. How many $20 bills does it take to make $960? 29. (a) 7,909—7,099=? (&) Prove your answer. 30. (a) Draw a rectangle 12 ft. by 6 ft., using a scale of 1 in. to 2 ft. (b) Divide it into fourths. 31. A lot is 100 feet square, find the cost of fencing it at 60 cents a foot. 32. Divide 67,845 by 31. 33. Multiply 8,764 by 329. 34. (a) Write 6,060, 6,600, 6,006, 666, 6,606. (b) Find their sum. (c) Prove your work. 35. Add 4,726, 1,491, 3,208, 6,407 and write your answer in words. 36. (a) Subtract $1,280.50 from $5,000. (b) Write your answer in words. 37. A farmer pays $1,500 for 20 acres of land; how much is that per acre? 38. 60>4+50^=? 39. 4834+24>^=? 40. A man earns $24 per week. If MENTAL PROBLEMS.— 3A. 55 he spends $2.50 per day, how much does he save during the week? 41. At 38 cents a pound, what will 56 pounds of butter cost? 42. Find the cost of 24 yards of cloth at $1.25 per yard. 43. Bought a house for $5,575 and sold it for $6,250. Did I gain or lose, and how much? 44. Ve of 7,218=? 45. How many square feet in a field 240 feet long and 84 feet wide? 46. How many feet of wire would it take to go around this field? Mental Problems. — 3A. Charles H. Davis. 1. If one orange costs 6 cents, what will 9 oranges cost? 2. 85 cents + 32 cents = ? 3. Yi of 48 books — ? 4. Multiply 125 by 3. 5. In a room are 42 seats ; 6 are empty. How many seats are filled? 6. Bought 4 quarts of milk; how many pints did I get? 7. A boy having 48 cents spent ^ of his money; how many cents had he left? 8. In a class library are 106 books. During one week 32 were loaned. How many were left? 9. A girl having a quarter, spent a dime, a nickel and 2 cents ; how much money had she left? 10. How many quarts in 9 gallons? 11. Bought 6 yards of ribbon for 42 cents. What was the cost of 1 yard ? 12. What is the area of a cardboard 6 inches by 8 inches ? 13. Which is the greater, ^/g or ^/g of an apple? 14. How many half dollars in $2.50? 15. If 2 oranges cost 6 cents; what must be paid for 10 oranges? 16. Change $20 to cents. 17. In one apple how many fourths? (o) Eighths? 18. How many thirds in 5? 19. How many inches in 6 feet? 20. Paid 8 cents for milk, 5 cents for bread and 15 cents for butter; how much did I spend? 21. A pupil learned 30 new words in 5 days; how many did she learn each day? 22. Had 1 yard of ribbon and cut off 10 inches; how much remained? 23. At $18 a half-dozen, what will 1 chair cost? 24. How many inches in 9 feet? 25. 2 oranges cost 6 cents ; what will half-a-dozen cost? Written Problems. 1. Write the following numbers: 6,547, 3,609, 8,796, 7,000, 2,704. Find their sum. 2. Add: $36.92 67.37 4.48 19.09 46.84 $461.35 —209.27 56 WRITTEN BROBLEMS.— 3A. 4. 586 X4 5. How many quarts in 144 gallons ;? 6. If one cow cost $48, what will 5 COWJ 5 cost? 7. What will one horse cost, if 6 horses cost $636? 8. From $10 take $0.10. 9. Add: $19.67 96.35 43.09 .68 29.16 16.29 .83 10. If 5 wagons cost $200, what will 6 wagons cost? 11. 7)4977 12. 6547 X8 13. Write in Roman notation, 69. 14. What is y^ of $450? 15. Add: 9648 3297 6003 4787 6194 3869 16. What will 8 yards of ribbon cost at $1.25 a yard? 17. 8)2728 18. A man had $400. He bought 4 cows at $29.50 each ; how much money had he left? 19. A boy bought a push-mobile for $7.50 and sold it for $10.25. How much did he gain? 20. How many gallons in 64 quarts? 21. A man bought a horse for $472 and sold it for $396. Find the man's loss. 22. Eight chairs cost $112 ; what was the cost of each? 23. Write 96 in Roman notation. 24. Bought a roll-top desk for $28.75 and sold it for $5.50 more than it cost ; what did I sell it for? 25. What is 34 of 40? 26. 9) $46.53 27. 6997 X9 28. In a class library are 6 shelves, each containing 57 books ; how many books in the librarj'^? 29. $93.24 —48.19 30. (a) 7)6314 (&) Prove your answer. 31. (a) 8329 X8 (&) Prove your answer. 32. Find the cost of 394 lbs. of flour at 4 cents per pound. 33. Write in Roman numbers : (a) 50; (&) 88. 34. Write in figures fifty-four dollars seven cents. 35. A druggist buys a drug at $1.00 per pound and sells it at 10 cents an ounce ; how much does he gain on each pound ? 36. Mary spent one dime, two nick- els and four pennies ; how much money did she spend? 37. Bought 3 chairs for $18 ; at that rate what will a dozen cost? 38. A man owed $320 and paid >4 of it; how much did he pay? 39. At $975 a lot, what will 8 lots cost? THE TEACHING OF NUMBER.— 2nd YEAR. 07 The Teaching of Number in the Second Year. Josephine MANY of the unsatisfactory re- sults in number in the primary grades may be traced to the following causes : L The nature of the processes in- volved is not understood by the pupil. He works with symbols with no idea of the things and relations that are symbolized. 2. The work is over-developed by the teacher and the children allowed to depend too long on the objective work. 3, Drill on the fundamental opera- tions is sufficient. 4.. A proper balance between ab- stract and concrete work is not pre- served. 5. Oral and written work is not carefully balanced and correlated. 6. Too few devices and schemes for drilling and fixing the combinations are prepared and used by teachers. 7. The problems used are not suffi- ciently interesting or varied. 8. The special work of the grade is not given enough time. Too much time is spent in reviewing the work of the preceding grade or in teaching work in the syllabus that is meant to be sub- ordinated to this special work. 9. There is a lack of knowledge of the course of study of the grades im- mediately below and above. Either poor correlation or none at all results. 10. Much of the time allotted to arithmetic is not used for arithmetic, but is lost in the following ways : (a) Too much time is used in pass- ing material, writing headings, rulings, etc. M. Lawlor. (b) Children are not taught to work on a time limit. (c) Every child in the arithmetic group is not working during the entire time. 11. The imagination is not trained or developed. The child works with sym- bols with no idea of the objects or relations symbolized. 12. The maximum capacity of the class is not tested often enough. The special difficulties of the entire class and of individual pupils do not receive enough attention and drill. It is important that the child should comprehend, as far as his development and mentality will allow it, the nature of the processes used and the proper number relationships involved. If this is not understood there will be great difficulty in the application of these operations both in the problems of the grade and in the problems of the follow- ingr errades. There is no educational value to this work if the child blindly follows rules or merely memorizes and repeats what he has heard. It is imperative to use objective work in introducing all four operations. The work should proceed from in- formal to formal work. This concrete work is necessary for developing the number relations. New ideas are learned only by meeting them in simple, fa- miliar forms. The child must do for himself in this work. He must answer all questions by reference to the objects used, and not imitate the teacher or his neighbor. This work must be con- tinued until relations can be imaged 58 THE TEACHING OF NUMBER.— 2nd YEAR. without the objects. Then the objec- tive work must be dropped, and not referred to again except in rare cases. The child must know that two or more numbers may be combined and a single new number result; that a single number may be divided and two or more numbers result; that unequal numbers are combined by the process of addition, and equal numbers by the process of multiplication. In the same way subtraction should be related to addition, and division to multiplication. The objects and quantities involved in all these processes must be handled by the children themselves. The objects used should be uniform, or nearly so, in shape, size and kind, and not too attractive. The following material is simple and easily obtained: Splints, sticks, buttons, paper for cut- ting and folding, beans, peas, corn, pegs, spools, counters, horse-chestnuts, acorns, toy money, real measures, as yard, foot, inch, peck, quart, pint and gallon measures, weights, cubes, etc. In teaching these operations the method must be suited to the various children, for in no other subject in the elementary school is there such a di- versity in ability and taste found. The best methods for teaching these processes are found in the New York City Course of Study, and are quoted below. Mathematics. INTRODUCTORY NOTE. Requirements. Both the course of study and the syllabus provide for min- imum requirements. Pupils capable of more rapid advancement should not be confined to the limits set in the syllabus for the grade. Methods. No attempt is made to dictate methods. The following sug- gestions are offered : Much oral drill and blackboard work should be given. Answers should be tested approxi- mately to ascertain whether they are probably correct. The results of addition, subtraction, multiplication, and division should be proved before they are declared. Excessive repetitions of forms of analysis and elaborate written explana- tions should not be required. Definitions should not be required until the meanings of the terms to be defined are fully understood. The Combinations. Special impor- tance is attached to the thorough mastery of the combinations in addition, subtraction, multiplication, and division. The following are the steps which should be followed in learning the com- binations of each table: Addition and Multiplication. 1. Care should be taken that the pupils appre- hend the nature of the required opera- tions. 2. The combinations that have been taught in the preceding grade should be reviewed frequently. 3. The results of the new combina- tions should be determined in addition by counting objects, and in multiplica- tion by adding the multiplicand as many times as there are units in the multiplier. 4. The entire table should be re- peated with the objects in view in ad- dition, and with the addends in view in multiplication. 5. The entire table should be re- peated without the objects or addends in view. 6. The results of combinations, mis- cellaneously presented, should be given without the aid of any form of mne- THE TEACHING OF NUMBER.— 2nd YEAR. 59 monies or external devices. If a pupil misses a combination there are two methods of correction: in addition he may be required to deduce the result multiplication to add numbers; or he may be required to deduce the result of the combination in question from the nearest combination whose result he knows. (Thus he may ascertain that 6X7=42 from 5X^=35, since 6 sevens are one seven more than 5 sevens.) The former method is ob- jectionable for two reasons: the per- formance of an isolated example will be of little value to aid association the next time the combination occurs; and the pupil is in danger of forming the habit of using his fingers as counters. The value of the reference to the nearest known combination consists in the association of the combination in its proper relations with the other terms of a series. 7. Exercises in finding the parts which constitute a number either as addends or as factors should follow the drills on tables. 8. The combinations should be ap- plied in the solution of simple problems. Subtraction and Division. 1. Each combination in subtraction and in di- vision should be related to its corre- sponding combination in addition or in multiplication. In subtraction the minuend is the sum and the subtrahend is one of the terms of a combination in addition ; in division, the dividend is the product and the divisor is one of the terms of a combination in multiplica- tion. 2. The results should be stated with- out the relation, in order that they may be given instantly. An error should be corrected by reference to the primary combination. The rest of the work is a matter of drill and device which tests the inge- nuity and skill of the teacher to the utmost. Care should be exercised in teaching the signs of the process. -|- and — present no difiiculties. X and -^ are more troublesome. Too much stress should not be laid on either, because they are not important in the business world. Teach with 15-f-3, 3)15; and 3 with 5X3, 5. The word times is likely to be confusing. It is better to think 5 multiplied by 3, or 5 3's. The latter is preferable. 2a. As a rule, primary teachers de- velop numbers well up to a certain point, but they carry the development to an extreme and leave the work with- out spending enough time in fixing and testing the work thus taught. The con- crete objects are left before the chil- dren too long and they become too de- pendent on such aids. Objective work is secondary to the process involved. It is merely a means to an end and is a hindrance to teaching wherever a mental picture has been formed by the child and the idea has been abstracted. It should then be put aside, for it has no further office to perform. 3. Too much emphasis cannot be placed on the necessity for drill in these grades. There must be repetition again and again until the results of all the combinations in all four operations can be given instantly without reference to the processes by which these results were obtained. A ready knowledge of these combinations is absolutely neces- sary for the work of succeeding grades, and can be taught only by repetition in interesting variation. To this end, drill charts, printed on oak-tag, on heavy THE TEACHING OF NUMBER.— 2nd YEAR. paper, and on the blackboard, as well as many devices and games should be used. 4. Since the aim in arithmetic is to train the child to be skilful in perform- ing the mechanical processes, and to give him the ability through developing the imagination and reasoning powers, to apply these processes in solving prob- lems, it is necessary to preserve a proper balance between the concrete and the abstract work. The abstract work is the important work in business. In business prob- lems, the relations are simple. Children tire quickly of abstract work unless properly handled. If it is not varied, it becomes very tiresome. This makes interesting devices necessary, for the abstract work must be done. Training which comes from constant practice is the only means of knowing that the child is able to apply his ab- stract work. One should always sup- plement the other. In the lower pri- mary grades the greater part of the time should be devoted to abstract work. 5. Both oral and written work should be used in every lesson. A great amount of oral work is justified. The effort made gives the child confidence, the solution involves small numbers and gives the child the sense of power which comes with accomplishment. Small numbers should be substituted and the work done orally whenever larger numbers confuse. The child per- ceives how the result is obtained and is thus able to solve problems containing more difficult numbers. Oral work takes less time. There is also an advan- tage in counting in rhythmic sequence, and in reciting the tables in proper se- quence aloud. Even concert work helps in this drill. In oral work there is also a chance for physical expression. Games can be used to great advantage. Written work should be used to teach the children forms of placing their work on paper, to give another method of fixing combinations in mem- ory, and whenever large numbers are used so that a pencil is needed as an aid to the head. An important use of written arith- metic is that in which the children are taught to take numbers from dictation and arrange them in columns for ad- dition. This is so necessary in business that children should be taught this as low as the Second Grade. Sketch and illustrate problems often. 6. Devices for fixing and drill work are absolutely necessary in the primary grades. Below are given a number that are commonly found in arithmetics or used by successful teachers. Each ingenious teacher will combine these and use them in various ways. I. Use. — This device can be used for all of the four fundamental operations, the number in the centre being variable. This central number can be used as an addend, multiplier, subtrahend or divi- sor. This device is equally serviceable for decade drills; c. g., have the num- bers on the circumference 11, 21, 31, THE TEACHING OF NUMBER.— 2nd YEAR. 61 41, 51, etc., and add the numbers in the centre to each of these. 1 II f '^^N -^-<\ \ 7 \^ Xd, w II. Use. — This is a variation of Device I. HI. Use. — This is similar to Device I, except that each number on the circum- ference is added to or multiplied by the four numbers on the four-leaved clover. IV. Use. — Each petal of the daisy can be erased as its number is added cor- rectly to the number in the centre or multiplied by it. The numbers can be varied for subtraction and division. •1 H H H H H H H V. Use. — This device may be called walking the railroad ties. Each pupil may see how quickly he can cover the distance from a to & without stumbling, by multiplying the numbers on the ties by 7, etc. /A VI. Use. — This device is called climbing the ladder. Each pupil may try to climb to the top of the ladder without missing a rung. If he misses a rung, his initials are put on it and he must climb down and learn the one he missed. Then he can trv as^ain. VII. Use. — This picket fence may be used for the bird to hop from one picket to another until he has gone the length of the fence. Each pupil may be a 62 THE TEACHING OF NUMBER.— 2nd YEAR. ilJLJiJiJ.A_i-ii_l^_iL .Lll U u u u rr [\ u Li \iu bird with a different number. When he misses he falls from the picket and flies away to learn the one he missed. yS /a vf /^ V? // '^ -3 /^ H- -t If- ^ J c /.I vin. Use. — These two running-tracks, one triangular and one oval, are used to run around without stumbling. If he misses he leaves the race and enters again when he is sure he will not stumble. As these devices are used for rapid drill only, the pupils give the answers without repeating the combination, as the teacher or a pupil points. Where there is floor space, these de- vices may he drawn with chalk upon the floor, and the walking, running, etc., may be actually done. These activities form a pleasing variation and therefore add much interest to the work. i- 1, 3 "^ 7 sS yo f /2- k If -f ^ /i L 5 V /o /•z. z r ? IX. Use. — Multiply each figure in the upper row by each figure in turn in the lower row Reverse the process. 3 T\ /i/^lf /» ^/;»v| J},^i3i(' y s /i /i Xo >y ^$ ?». ^^0 4f ^i ^ C> 7 f la // /a. X. Divide the first row by 3 and write the quotients. Divide the second row by 4. Count by 5's and write the re- sults in the third row. This scheme can be used in a variety of ways. The multiplication and ad- dition combinations can be put on a similar chart for drill work. XL Put proper signs with the following: 3 8 11 2 4 6 5 3 15 16 8 8 16 4 12 5 7 35 SS 5 7 Etc. THE TEACHING OF NUMBER.— 2nd YEAR. 63 XII. The teacher may give several sums, one at a time, and call on the children to give as many combinations as pos- sible that make up that sum. Example : 9— 8+1 1+8 2+7 3+6 4+5 5+4 Etc. xni. The same scheme can be used in multiplication and division to test fac- tors and knowledge of multiples. Example (1) : 24 12X2 2X12 4X6 6X4 3X8 8X3 2X2X4 2X2X2X3 Etc. Example (2). Factors of: 3 9 12 15 18 21 24 Etc. XIV. 5. Use. — For multipli- cation or addition, using 5 as multiplier or addend. XV. Games are useful. Choose sides in any drill work and use the tables, as spelling words are used in spelling matches. This can be done in three ways : 1. The one missing passes to his seat and the last one standing wins. 2. The class may stand in single file and the one failing passes to the foot. 3. The side losing on any combina- tion loses one member to the other side, which has the right to choose the quick- est pupil on the opponents' side. This appeals to the fighting instinct and is a strong incentive. XVI. Two rows can be pitted against each other in playing "three deep." A card containing a combination is laid on each pupil's desk. Each one in turn gives his combination. When the one in the rear seat has given his, each one moves into the seat ahead, the one in the front seat running to the rear seat. The results of the combinations on the desks are given as before and the changes made. One failing gets out of the line. The row finishing first with the loss of fewest boys, wins. XVII. Games for keeping score are popular. Several waste-paper baskets may be utilized as pockets. These are given values and are placed at diflferent dis- tances from the players. The object is to throw a bean-bag into the baskets counting most. Each child keeps his own score and learns to add in order that he may play. The same idea may be carried out with chalk-boxes and a rubber-ball. 1. Problems give the child a wider 64 THE TEACHING OF NUMBER.— 2nd YEAR. perception of the number space and number relations. They should be varied and interesting, largely local in color, and touching on many activities interesting to children. They are most interesting and valuable when corre- lated with work in and out of school. 2. Each grade has special work to do. In the 2 A it is the 45 combina- tions, in the 2B, multiplication tables through 5. From one-half to three- fourths of the time allotted should be spent in mastering this work. The re- maining time may be given to the rest of the work outlined in the course of study for the grade. Review work should be constant, but only in con- nection with the grade work, or when it is needed before a new step can be taken. No teacher should use her time to review the work of the preceding grade and allow her work to suffer. Such reviewing should be done where and when it is needed. 3. The arrangement of the course of study should be known so that each teacher may have a broader view of the work and be able to build on the work that has been learned, and prepare for the work of succeeding grades. If this is properly done the child loses little in time or effort. 4. The time given to arithmetic should be used for that subject and nothing else. All material should be ready before school. Little time, if any, should be spent in headings and rulings during this period. The routine work of distribution should be as me- chanical as is possible. A time limit should be insisted upon, so that the pupils may work with reasonable rapidity. Early training often fosters lazy habits of mind. Where habits of speed and accuracy are necessary, there should be the time limits set for the class, for individual pupils, for groups, a certain number of examples or prob- lems should be worked in 5 minutes, etc. No dawdling should be tolerated. In rapid work there is greater concen- tration, and more accurate work results. The child must make an effort. He likes to be made to measure up, so an added interest is gained. Rarely should a pupil be allowed to sit idle during this period. He will listen to the teacher for a short time and then his mind wanders. Enough work should be given to keep every child busy all of the time. No class should be kept waiting for a few slow pupils. No attempt should be made to keep the class together in each lesson. The brightest pupils should be given the greatest amount of work that they can reasonably and safely accomplish. 5. In the teaching of number it is absolutely necessary that the child's im- agination be developed and trained. It is a great error to underestimate its importance. In order that the child may be able to compare, perceive rela- tions, and make number judgments, he must be able to image the facts repre- sented by the symbols he is using, other- wise, he has no idea in his mind to use as a basis for comparison. The powers of attention, observation and vizualiza- tion to fix the number facts of the four fundamental operations, must be devel- oped through training the senses sys- tematically. In the first and second years this should proceed along the lines of eye, ear and touch training. Much drill should be given to the meaning and use of terms of comparison, such as higher, lower, wider, narrower, and to terms of position, as right, left, middle, east, west, to help develop the powers of observation, imagination and judgment. Since these powers PROBLEMS.— 2B. 65 grow by exercise, opportunity should be provided by : (1) Using quantities perceived by the senses. (2) Training the child in imaging these when they are not present to the senses. 12. Problems too difficult for the av- erage pupil of the class should be occa- sionally given to the bright child to test his capacity and give him an oppor- tunity to try his strength. The result is of value to both teacher and pupil. All combinations that present diffi- culties to the class as a whole should be carefully selected, presented in a variety of interesting ways, and drilled on. Equal time need not be given to all combinations. It is useless to drill too long on what the class knows. Certain combinations are more difficult than others for the individual child. The teacher should find out without delay what these are and take measures to teach and fix them before new combina- tions are taught, otherwise this weak link in the series will cause much trouble in all succeeding work. Problems. — zB. Josephine M. Lawlor. 1. James spent 5 weeks of his vaca- tion in the country. How many days was that? How many months? 2. I buy pears at the rate of 24 cents a dozen and sell them at the rate of 36 cents a dozen. What is the gain on 4 dozen ? 3. A newsboy bought 20 papers at 2 cents each and sold them at 3 cents each. What did he gain? 4. Journals are 5 for 3 cents, and are sold for 1 cent each. How much will Jack make in selling 15 papers? 5. Between January 1st, and June 1st there were 125 sunny days. How many cloudy days were there? 6. An excursion train made four stops. At the first one 90 passengers got off, at the third 150 got off; the train was emptied at the last station. If there were 400 people aboard when the train started, how many got off at the last station? 7. Henry had a dollar bill. He paid for three 25-cent tickets for a ball game. His change was — ? 8. Find the number of quart boxes needed for three pecks, two quarts of strawberries ? 9. Our school-room is 35 feet by 25 feet. How many feet of moulding will be needed for it ? What will the mould- ing cost at 5 cents a foot? 10. A man who was born in 1775 died in 1840 at the age of — ? 11. A man born in 1802 died when he was 87 years old. What year did he die in? 12. A man died in 1900 at the age of 87. When was he bom? 13. Mother is reading a book con- taining 406 pages. She has read 309 pages. How many more has she to read? 14. There are 150 pages in our arith- metic. How many pages in 3 of them? 66 PROBLEMS.— 2B. 15. The cook baked 3 tarts each for each card, how many buttons will be 9 little boys. How many tarts were needed for 5 cards? baked? 27. The basket-ball team plays 3 16. 2 pounds of coffee cost me 50 games a week. How long will it take cents. I received — as change from a them to play 15 games? How many (^Qlja,r. games do they play in a month? 17. Price list: 28. In basket-ball a goal counts 2 Eags 36 cents per dozen points, a foul 1 point. No. 17 shot 5 Rolls . 10 cents per dozen fouls and 4 goals. What is the score ? Oranges 5 cents apiece 29^ ^he other school made 5 goals Breakfast food .10 cents a box ^^^ ^ ^^^j ^^^^^ ^^^^^^ ^^^^ 3 Milk 10 cents a quart , , ^ Cream 12 cents a bottle l^^w much ? Find the cost of a family breakfast ^0. They played 15 minutes, rested if they ate the following: 6 eggs, 6 20 minutes and then finished the game rolls, 3 oranges, /a package of break- ^^ 15 minutes. How many minutes did fast food, 1 pint of milk, /^ bottle of the game last? How many minutes less than an hour was it? ""Ts" What will 4 such breakfasts ^l- A glazier has 48 panes of glass. ■^ If a window holds 4 panes, how many """"lO How many days in 5 weeks? windows can he fill with this glass? How many more days in 5 weeks than 32. Charles had 16 quarts of chest- in 3 weeks? What is the short way to ^uts. He sold 1 peck and gave 1 quart find out? (2XT days.) ^o each of 5 boys. He had — quarts 20. George walks at the rate of 4 ^^"• miles an hour, and walks for 3 hours, ^3. Harry had a dollar-and-a-half. then he rides for 3 hours at the rate of He spent a quarter and a dime. He 10 miles an hour. How far had he had — cents left. traveled when he came to his journey's 34. How many 4-ounce bags can be J ^ filled from 84 ounces of gum-drops ? 21. A farmer had 5 rows of apple ^5. Cake costs a baker 25 cents a trees, with 9 trees in a row. How many Po^nd. He sells it at 30 cents. What apple trees has he? Draw a picture does he gain on a dozen pounds? showing this, using dots for trees. 36. Find the cost of laying 9 rods 22. He has 4 rows of peach trees, 12 of gas-pipe at $5 a rod. in a row. How many peach and apple 37. A horse travels a mile in 10 trees has he on the farm? minutes. How far will he go in half 23. Mattie buys 5 yards of calico at an hour? In an hour? 6 cents a yard. How much change 38. A rug is 6 feet long and 3 feet should she bring back if she started wide. How many feet of braid will out with a half-dollar and a nickel? be needed to bind it? Show this by a 24. If oranges are 4 for 9 cents, how picture. How many yards is this equal many can you buy for 18 cents? For to? 36 cents? 27 cents? 39. A man earns 40 cents an hour. 25. How many marbles must be put He works 5 hours a day and earns — ? with 19 to make 34? 40. How many pint packages can be •26. If a dozen buttons are put on put up from 2 pecks, 2 quarts of peas? PROBLEMS.— 2B. 67 41. A farmer pays 60 cents a cord for sawing wood. A boy can saw 2 cords a day. How much can be earned in a day? 42. A basket of corn weighs 56 pounds. How much do 33 bushels weigh ? 43. 3 dozen pencils were bought at 35 cents a dozen. They were sold at 4 cents apiece. What was the selling price? What was the gain? 44. Change a five-dollar bill, using dollars, half-dollars and quarters. 45. A man counted his money and found that he had a five-dollar bill, 3 silver dollars, 1 half-dollar, 4 quarters, 2 dimes, 6 nickels and 4 pennies. How much money had he? 46. How many boxes are needed for caramels, if 4 oz. are put in each box, and there are 88 oz. to be put up ? 47. A quart of olive oil is worth 85 cents. What is a gallon worth? We use a gallon in 6 months. What is our oil bill for a year? 48. The grocer charges 10 cents for 3 eggs. What would 6 eggs cost at the same rate? 12 eggs? 15 eggs? 49. Draw a base-ball diamond. The distance from base to base is 90 feet. How far does a boy go who makes a home run? If he is caught at third base, how far has he gone ? At second ? H he circles the bases twice during the game, how far has he run? 50. There are 9 men in the team. If the average is 2 hits apiece, how many hits were made? 51. I purchased 2 bottles of ink for 5 cents, one eraser for 6 cents, a bottle of red ink for 8 cents and a pad for 3 cents. I have a quarter. Will it be enough to pay my bill? 52. Jerry bought 5 lemons for 3 cents each and sells them for 5 cents each. What is his gain? Use the shortest method (5X2 cents). 53. How many days in 2 weeks? How many weeks in 8 months? How many months in 5 years? 54. How old are you ? Find the year of your birth by subtracting your age from 1909. 55. If there are 8 caramels in a layer and 5 layers in a box, find the number of caramels in the box. 56. Divide 11 pounds of tea into quarter-pound packages. 57. It is 33 miles from Boston to Haverhill, If a horse can travel 11 miles an hour, how long will it take for the trip? 58. Bridge trip-tickets are 2 for 5 cents. Astoria ferry-tickets are 3 cents apiece. How much will be saved by 10 bridge tickets instead of 10 ferry- tickets ? 59. In group A there are 3 rows of pupils, 8 in a row. There are 5 ab- sentees. How many sheets of paper will be needed for the group? 60. At the rate of 5 yards of silk for $10, what will 25 yards cost? Use a short method (5X$10). 61. A ship travels 215 miles a day for 4 days. How far was that? How many more miles must it sail to finish 1,000 miles? 62. At $2.50 a rod find the cost of putting new fence on the end of a lot 4 rods wide. 63. At the rate of 3 oranges for a dime, find the cost of a dozen oranges. (4X10 cents.) 64. The class worked on an average of 10 examples a day for Monday, Tuesday and Wednesday. In order to finish 60 by the end of the week, how many must be worked on Thursday and on Friday? 65. 13 eggs are generally used for a 68 PROBLEMS.— 2B. setting. On an average of 9 chickens to a setting, how many will 5 hens hatch? 66. A man earns $4 a day for 4 days, and $5 a day for the other two. How much has he at the end of the week? 67. His son earns ^ as much. How much does he earn? How much do both together earn? 68. A ship makes a five-day trip. On the first day it covers 240 miles ; on the second, 300 miles ; on the third, 289 miles ; on the fourth, 263 miles ; on the fifth, 127 miles. How long was the trip? 69. On board the ship there were 390 women, 285 men, 105 children, and a crew of 120. How many people did it carry? 70. A 2-gallon jar is half full.' How many times must a quart measure be emptied to fill it? A pint measure? 71. The boys who went nutting gathered 2^ pecks of nuts. How many quarts was that? 72. One-half of a 2-gallon can of kerosene oil was lost by leakage. How many pint measures could be filled from what was left? 73. 7 Xmas-tree candles cost 35 cents. How much will be paid for 21 ? Use short method (3X35 cents). 74. 6 oranges cost 18 cents. Find the cost of 2 dozen. Short method (4X18 cents). 75. Two trains leave Syracuse at 10 A. M. One travels east at the rate of 40 miles an hour, the other west at the rate of 50 miles an hour. How far has each train traveled at the end of 3 hours? Show by a drawing how far each train is from the other at the end of this time. 76. We use 4 lbs. of sugar in 4 days. How many ounces is this a day? 76a. Mary went on an errand at 9 o'clock. She returned at 20 minutes past 10. How long was she gone? 77. A 15-lb. turkey lasted a family 3 days. On an average, how many pounds did they eat a day? 78. At 35 cents a pound what did their meat cost them for each of these days? 79. A chicken weighed ^ as much as the turkey. What was its weight? What was it worth at 25 cents a pound ? 80. John needed 85 cents, but could earn only 63 cents. How much more did he need? 81. A grocer sells 3J^ pounds of sugar for 20 cents. How much change should he return from half-a-dollar? 82. A man sold all but 13 lbs. of coffee from a 90 lb. sack. How much did he sell? 83. Jane has in her bank 3 dimes, 4 nickels and 4 cents. Carrie has twice as much. How much has Carrie ? How much have both together? 84. A milkman sells 5 gallons of milk at 5 cents a pint How much money should he receive for it? How many quarts does he sell? If he gains 2 cents on a quart, how much money does he clear? 85. Pencils can be bought at the rate of 3 dozen for a dollar. At this rate, what will 9 dozen pencils cost? Use short method (3X$1.00). 86. Oranges are 3 for a dime or — for 20 cents. Use short method (2X3). 87. A boy changed a ten-dollar bill at the bank. Give two or three answers telling how the bill was changed. 88. A boy steps 2 feet at a stride. How many steps will he make across your room? From the front to the back of the room? 89. The windows are placed 2 feet above the floor and are 7 feet high. PROBLEMS.— 2A. 69 How far above the floor is the top of the window? How many feet of muslin will be needed to make curtains for 1 window, allowing 2 pieces for a curtain? How many yards? If there are 4 windows, how many yards will be needed for all of them ? 90. A bicyclist travels at the rate of 20 miles an hour. How far will he travel in 3 hours? In 30 minutes? In a quarter of an hour? 91. In a street car there are 50 people sitting and 30 standing. How many fares should be rung up? If 20 of these fares are transfers, how many nickels should the conductor collect? How much money is this? 92. How many more than 5 is 25? How many times 5 is 25 ? 93. Compare 30 with 5 in two ways. 94. If a team of horses carries 5 loads of stone per day, how many will it carry in a week? 95. What will it cost per day to run a boiler which burns 2 tons of coke worth $2.50 a ton? 96. The postage on sealed letters costs 2 cents for an ounce or less than an ounce. What stamp shall I put on a letter weighing ^ ounce? What will it cost to send one weighing 3 ounces? 97. It cost me 8 cents to send a letter. How much did it weigh? 98. A boat sailed 20 miles from port and then turned back It sailed 3 miles on the return trip when it met with an accident. How far had it sailed ? How far was it from port? 99. I have to go 45 miles. My horse carries me at the rate of 5 miles an hour. How long will it take to make the trip? If I leave at 9 o'clock in the morning, when shall I reach my journey's end? 100. How many shoes are needed for 12 horses? Problems. — aA. Josephine M. Lawlor. 1. A boy riding his bicycle spent an hour in riding to his uncle's house, vis- ited with him for half an hour, then rode home in an hour. How long was he away from home on his visit? 2. If he left home at 1 o'clock in the afternoon, what time was it when he returned ? 3. In a class of 49 pupils, 7 missed words in spelling. How many were perfect ? 4. Frank solved 5 problems before dinner and 4 after dinner. How many did he solve? 5. If he solved the same number every school day in the week, how many did he solve by Friday? 6. If he had 37 perfect, how many did he fail on? 7. A coal train consists of four cars. The first carries 950 tons, the second 885 tons, the third 887 tons, the fourth 924 tons. What was the entire weight of coal on the train? 8. Find out the number of days in July, August, September and October. 9. If 95 of these days are pleasant, how many are unpleasant? 10. John has 159 lines to copy. If he has copied 138, how many more must he copy? 11. Annie had $1.50 in her purse when she went to the store. She bought 2 lbs. of sugar for 11 cents, an ounce 70 PROBLEMS.— 2A. of cloves for 10 cents, 3 cans of corn for a quarter, and 4 cents worth of candy. What was her bill? How much money should she bring home? 12. Jack lives 12 blocks east of the school and Henry 15 blocks west. How far apart are they? Show by a draw- ing. 13. Nellie lives 13 blocks east of the school and Hattie 9 blocks east. How far apart are theyf 14. A farmer raises 250 bushels of potatoes. He keeps 25 bushels for his own use and sells the balance. How many bushels does he sell? 15. There are 15 failures in spelling out of a class of 60. How many are perfect? 16. In a room seating 50 there are 34 pupils. How many pupils must be transferred from another room to fill all the seats? 17. A business man lives }^ hour's ride from his office. When will he reach home if he leaves his office at 5:30? 18. A boy bought three 20-cent tick- ets for a ball game. How many dimes can he get as change? 19. Edgar has a dime and Melie has a nickel and 3 cents. How much have they together? How much more has one than the other? 20. A boy received — change from a quarter, after paying for his lunch con- sisting of a sandwich at 5 cents, a glass of milk at 3 cents, and a piece of pie at 5 cents. 21. A step is 1 foot wide and 5 inches high. How high are you when you have climbed 8 steps? Show this by a picture, 22. A lot is 22 rods wide and 33 rods long. How many rods of fence would be necessary to enclose it? Use a picture. 23. George leaves his home at 8:30 A. M, to go to school. He reaches school at a quarter to nine. How long does it take him? 24. A 30-cent book was bought with half-a-dollar. How many dimes may be used in making the change? 25. Seeds are worth 10 cents a pack- age. How many quarters will pay for 5 packages? 26. Minnie purchased a book for 25 cents, paper for 15 cents, 2 pencils for 10 cents. She paid — for all. What 2 pieces of money would pay for them? What other money might she use? 27. A ship left New York on Mon- day at noon and reached London Satur- day noon. How many days did it take for the trip? 28. A train leaves one station at 2 :15 and reaches the next station J^ hour later. When is that? 29. Willie eats breakfast at 7 o'clock and has lunch at 12:30. How many hours are there between his breakfast and lunch? 30. A kite was flying 300 feet above the ground. The string broke, leaving 75 feet in the boy's hands. How many feet went with the kite? 31. A boy has 3 pieces of string of the following lengths : 35 feet, 81 feet, 46 feet. About how long a kite string can he make from this? 32. Laundry List: Collars 2c, Embroidered collars 2j4c, Cuffs 6c. Handkerchiefs 2c. Towels Ic, Shirts 12c, Make up your own laundry list and find out how much your bill will amount to. 33. A boy plays from 7 o'clock until 9 o'clock in the morning, and from 3 FIRST YEAR NUMBER. n o'clock until 4 o'clock in the afternoon. How many hours does he play in a day? 34. An Angora kitten was bought for $6 and sold for }^ more than it cost. How many more dollars was that? How much was it sold for? 35. A pistol costs 35 cents, caps must 5 cents. How much do they both cost? Find by counting how many boxes of caps could be bought for the price of the pistol. 36. Find by counting the cost of half-a-dozen flags at 5 cents each. 37. Find the score in a base-ball game if the following runs were made : First inning, 1 run; fourth inning, 2 runs ; fifth inning, 1 run ; sixth inning, 4 runs; ninth inning, 5 runs. 38. Find the cost of 7 two-cent pen- cils, a ten-cent engine, a three-cent marble. How much change would be received from 3 dimes? 39. 13 boys and 11 girls are playing Prisoner's Base. If the sides are equal, how many are on a side? 40. A car will seat 60 people. If 71 people are aboard, how many are stand- ing? 41. If 11 of these people give trans- fers, how many cash fares should be collected ? 42. If there are 4 cakes of paint in a box, how many boxes can be filled from 28 cakes ? Show this by counting or by a drawing. 43. 3 yards of ribbon are cut in 12- inch pieces. Show by a drawing how many pieces are cut off. 44. How much money will pay for 5 five-cent carfares and 5 three-cent ferry fares? 45. A milkman delivers 45 quarts of milk on Monday; 52 quarts of milk on Tuesday ; — quarts of milk on Wednesday; 39 quarts of milk on Thursday; 42 quarts of milk on Fri- day. Fill in the blank for Wednesday and find how many quarts are delivered in the five days. 46. In a nutting party the boys found 32 quarts and the girls 19 quarts. How many quarts? 47. Find the distance around a tri- angle whose sides each measure 5 rods. Show by a drawing. 48. There are 28 telephone poles on one side of the street and 23 on the other. How many poles on the street? 49. A fireman climbed 28 feet to a third-story window. He went 12 feet higher to the fourth story and still 12 feet higher to the fifth story. How high was he then above the ground? 50. The boys ran a race to a fence 215 yards away and back again to the starting-point. How far did each one run? First Year Number. Alice J. Christie. THE time assigned for number in this grade is necessarily shorter than that given to the much more important subject — reading; yet the few facts to be acquired must be taught, and so thoroughly that they become a part of the little pupil's being, just as much so as his circulation. Experienced teachers of the grade cannot agree with those educators who 72 FIRST YEAR NUMBER. claim that children learn the number facts by themselves. It is true that they delight in finding out new facts while using or playing with objects, and this is good busy-work, but unless it is well directed the only facts that the children will volunteer after such an exercise are that 5 and 5 equal 10, 100 and 100 equal 200, etc. They rarely tell you that 3 and 2 equal five, or other impor- tant facts of the grade work. Part of the work in this first year is to have the children become very familiar with the comparative values of 1 cent, nickel and dime; so to start them with a general idea of number, it is well to introduce their little experi- ences of life and have them buy 1-cent lollipops, 2-cent apples, 5-cent sodas, etc. (with real money if possible, or the substitute paper money) of a little storekeeper in the class. To start with, this gives life and a real interest in number. Children delight in drawing easy, familiar objects; as daisies, famiHar flowers, etc. They get a general idea of number by drawing the above in groups. They learn number more readily in this way than if the objects are placed horizon- tally : •• ••^ •• • Use the blackboard as much as pos- sible Find an easy, quiet way of hav- ing children run to the blackboard, two rows at a time. It rests them, develops the larger muscles and holds interest. Right here, it is essential to empha- size the importance of visualisation and not copying. The latter exercise is in- jurious to the eyes, and so many mis- takes are made; but if the teacher, while makins: her exercise on the board tells her little pupils it is going to "run away," they will be twice as alert, and more anxious to make it just like the teacher did, and therefore give better attention. By playing with money, making pic- tures, arranging sticks, the number whole is fastened up to 6. Next comes the ordeal of placing the number beside the picture. The figures 2 and 3 are very hard to make. It is not wise to waste much time on this part of the work, for the curious marks the little people make develop into real respectable-looking numbers before very long. There are many devices for busy- work from now on, and a hektograph is invaluable in this grade. Many little one-inch cards accompanying numbers I 4 and m interest and help fix these associations of number and that for which it stands. A box full of numbers is distributed among the pupils ; ask them to find numbers that look alike, 2 I 2 I 2 I 2 |5|5|5 etc. Large observation cards with spots on one side, and corresponding num- bers on the other are very useful for rapid drill around the class. Everything about number in the First Grade should be done rapidly, as this subject is also used for discipline in making the pupils alert. With very little effort the language FIRST YEAR NUMBER. 73 work can be helped at this point by insisting that the child shall speak in sentences. Using the above-mentioned cards, the pupil can form the habit of saying, "I see five spots," or, "That number is five." This is a good habit to fix while the work is going slowly, and will help when later on there is less time for correlating language with the number work. Little motion songs, gestures, etc., help to make the work very attractive. "Five Little Chicadees," though old, is always sung with pleasure. Children enjoy selecting four little birds, who may fly around the room twice. Having aroused a good, wholesome interest in number as a whole, the com- binations naturally follow. The chil- dren know that 1 and 1 equals 2, so no time need be wasted on that ; then three objects are given to them. Some one runs to the blackboard, makes the num- ber for which the objects stand, and if correct the children clap as many times as the number calls for. They next separate the group of objects, holding some in one hand, some in the other, and tell what they have. They like to name them: two boys were playing, then an- other boy came, so three boys were playing. If the lesson is well con- ducted, the children and not the teacher will give the following : 12 1 2 11 1 3 3 3 Place on the blackboard for the rest of the day, in sight of all: * 1 * 1 * * 2 In this same manner, teach the fol- lowing combinations : 2 3 1 4 13 2 4 4 4 5 5 5 5 Blackboard work being so helpful, it is wiser not to use pencil and paper until after the facts have been taught through 5. After many associations with the objects, it is very important to fix the combinations like machinery, so arranging them as to be most helpful. Write them upon the board, erase, and have children write what they saw. The following becomes most familiar: 121 312 4132 112 132 1423 2 3 3 4 4 4 5 5 5 5 After much practice, pupils should be sent to the board to write all the numbers that equal 4, 5, etc., without any help. That is the only way to find if they have acquired the facts taught. Counting must not be neglected in the meantime, and the numbers should be presented in groups through 10. o o o o o o 6 o o o o o o o ® o o o o o o o o o o 10 The little, alert minds will tell you that two 3's equal 6 ; that two 3's and 1 74 FIRST YEAR NUMBER. equal 7 ; that two 4's equal 8 ; that three 3's equal 9 ; and that two 5's equal 10 — if the spots are arranged as above. After many associations, it is time to arrange the numbers through 10 verti- cally and horizontally on the board for counting. After 10, it is hard to get 11, 12, 13 and 15; so those numbers should be taken up separately, pronounced slowly, e — leven, tw — elve (try to sound the two in it). It is a help to tell the pupil that every time he says teen there is a 1 on the left hand side of the number. This avoids confusion when teaching the numbers ending in ty. The horizontal arrange- ment of numbers is now most helpful : 133456789 10 11 12 13 14 15 16 17 18 19 20 It helps for later work when the pupil must count by 10 — not only 10, 20, 30, etc., but 1, 11, 21, 31 ; 5, 15, 25, 35, etc. Many teachers do not realize that the pupil must take each number through 10 and count by lO's with it. As we build or add a row each week, we fin- ally have: have all been presented carefully, will appear in the following form : 1 2 3 4 5 6 7 '8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Then the children should be sent to the board, each given a different num- ber, and they should be able to complete the row of numbers up to 100. They should arrange these numbers not only horizontally but vertically, for this is the way they will arrange the numbers later on for addition. Finally, the combinations, if they 5 1 1 5 4 2 2 4 6 1 1 6 5 2 2 5 6 6 6 6 7 7 7 7 7 1 1 7 6 2 2 6 8 1 1 8 7 2 2 7 8 8 8 8 9 9 9 9 9 1 1 9 8 2 2 8 9 2 2 9 10 10 10 10 11 11 If the class has been well drilled, they will be able to tell or write what makes or equals any number from 2 to 11. Besides cent, nickel, dime, the chil- dren must be taught, during the first five months, pints and quarts. As the peanut comes in the course in Nature Study in this grade, it is interesting ta correlate the two subjects. Of course, to buy them is also very interesting. It is a good investment to buy a few quarts and after playing with them (measuring and buying), have a little party where they are eaten. This makes a happy ending to a very interesting lesson. The children will tell you how these pint and quart measures are used, and the brighter ones will enjoy asking little problems about them. If a pint of milk costs 4 cents, what will 1 quart cost? Simple problems given for paper work may begin after the facts have been taught through 5. Small addition examples may be introduced at this time also: FIRST YEAR NUMBER. 75 problems on one side, and five written, on the other, leaving one space for dic- tation of numbers, another for a longer addition example and the rest for simple facts, in the oral, the answer alone is given; but in the abstract work, the whole combination is placed on the paper for practice. At the end of the first five months, the children may arrange their facts in tables : l-fl= 2 1+2= 3 2+1= 3 2+2= 4 3+1^ 4 3+2= 5 4+1= 5 4+2= 6 5+1= 6 5+2= 7 6+1= 7 6+2= 8 7+1= 8 7+2= 9 8+1= 9 8+2=10 9+1=10 9+2=11 Four sets of cards for rapid drill are helpful, interesting, and very much en- enjoyed by the pupils. They are made on oak-tag, very distinctly, so that every child in the room can see them. In the first set, we have pictures on one side and corresponding numbers on ^„A .-^ ioc<-. cVsnrt pr1<1\tion columns. The children become very expert p p p 5 p p 1 p the other. Some of the more immature pupils never get beyond that set. In the next set, we have cards without the pictures. Next comes the card with the second part of the combination omitted, with these cards, especially if there are enough of them made for each child to have one. Leaving out the little immature group and working with the more alert, the following game is a good exercise and good fun, too: Each child is given a card, which al- ways remains on the same desk. After each pupil has recited his combination, the first one in each row runs to the back of it, the other children taking a seat in front of their present one. In this way, each child has a new combina- tion. This changing of seats is contin- ued until the first children get back to their own seats. By that time each child has visited and recited at every desk in his row, giving at least seven combina- tions. If skilfully managed, the game will not last longer than ten minutes. I have seen it done in seven minutes in a class room of fifty pupils, forty-two of whom played the game, each pupil reciting seven times. If the facts taught have been firmly fixed for the little pupils, there is little time needed for the first subtraction work. It seems easier to start with ten. Using ten objects, take away nine, how many left? Take away one, how many left? Take away two, take away eight. In this way the following facts can be readily learned : 10 10 10 10 _1 _9 —2 —8 9 8 76 FIRST YEAR NUMBER. The cllildreii ^vill suijii Icain to Ubc tiic minus combinations of all the facts previously taught through association. They can readily tell what makes or equals the numbers up to eleven and will soon learn the separations. With little effort, the following can be taught : 11 11 —2 —9 10 10 10 10 —1 —9 —2 —8 etc. Going back to the number six, there is another combination to make, 3 3 6 We now have o 3 8 4 t 7 1 11 11 11 9 3 8 4 3 9 4 8 .2 12 12 12 9 4 4 9 13 13 With plenty of visualizing and drill, the above facts must be acquired. If these facts are thoroughly learned, the separations naturally follow, and it is a good plan to have them use a number, as 6, giving the combinations that make it, then the separations : 5 14 2 3 6 1 4 2 3 6 6 6 6 6 1 5 2 4 3 — — — — — 6 6 6 6 6 6 6 6 6 6 —1 —5 —2 —4 3 In this same way we learn that 3 4 4 3 7 7 must be added to our combinations that equal seven. 5 3 4 3 5 4 8 8 8 must be added to the eight combina- tions. 6 3 5 4 3 6 4 5 Associations help children with these abstractions more than anything else. Arranging the combinations in tables can readily follow the above work, but they should follow and not precede it. Having learned to take any number less than ten, and to write by tens to one hundred, makes another step in the work easy. A child may be given the number 4 and told to write by tens to 100. He will write it thus : 4 _ 14 _ 24 — 34 — 44 54 — 64 — 74 — 84 — 94 Then, if told to add five to each num- ber, the task is easy and pleasant : 4 14 24 34 44 4_5 _|_5 -[_5 4_5 -^5 10 10 10 10 9 19 29 39 49 NUMBER IN THE KINDERGARTEN. 11 64 64 +5 +5 74 +5 84 +5 94 +5 59 69 79 89 99 Counting by two's is very easy by starting with 1, 2, 3, 4, 5, 6, etc., and making every alternate number a differ- ent color from the next one to it. 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 may be written in yellow; 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 may be written in white or green. A device for counting by three's is also helpful: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 The children will soon learn the third row, and as readily the other two. The same device may also be used in count- ing by four's : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 There are many interesting ways of presenting inch and foot Colored inch sticks, and those one foot long may be used. Learn the names of each. Meas- ure things in the room. Use ruler and draw lines on board, etc. Dozen may be taught in an interesting way also. The keynote to successful work in the first year is drill and rapidity. The children must be wide awake in order to get good results. The number lesson following physical exercises or recess will be more of a success than one following reading, or a period that has been more of a tax on the little pupils. Number in the Kindergarten. Catherine Maddock. MOST young children on entering school for the first time are in possession of a large English vocabulary. This vocabulary, consist- ing of practical words used in every day life, is gained largely by the child in his desire to satisfy his wants and needs, and also to meet his social in- stincts. Among the words of his vocabulary we find the names of figures. Such names as one, two, three are very fam- iliar to him, because in satisfying his wants, he is not often content with one of anything, but requires more. So he is introduced into the world of number in a perfectly natural way, the concrete. If the child is fortunate enough to have older brothers and sis- ters who attend school and who play school at home, then through that play he becomes acquainted with the names of figures up to about ten. In fact, he may be able to count in consecutive order up to that number. Again, if before entering school, the child is taught the common nursery rhymes, such as, "One, two, three, The Bumble-bee, The Rooster crows, And away she goes." 78 NUMBER IN THE KINDERGARTEN. or "This little pig went to market, This little pig stayed at home, This little pig had bread and butter This little pig had none, This little pig cried, wee, wee, all the way home." ■or "One, two, buckle my shoe, Three, four, shut the door, Five, six, pick up sticks. Seven, eight, lay them straight. Nine, ten, a good fat hen. Eleven, twelve, who will delve, Thirteen, fourteen, maids a-courting. Fifteen, sixteen, maids in the kitchen, Seventeen, eighteen, maids awaiting, Nineteen, twenty, my stomach is empty, Please, mamma, give me dinner," he will have a fairly good speaking ac- quaintance with numbers. Now with this idea in mind, namely, that the child is not entirely ignorant of numbers, we must arrange our les- sons so as to give him an opportunity to apply his knowledge of number. That is, we must supply the material for him to count. To an experienced teacher this is a very easy matter. She knows that the young child is self -centered and that to him some part of his own little body is of more interest, and receives more concentrated attention than any outside material that she could furnish. So the natural thing is to begin with the fingers and get acquainted with them by point- ing to and naming them After he knows the names of his fingers, then he can name and count them in this way: Holding up the left hand, he touches the thumb and each finger in turn, with the fingers of his right hand. He re- peats this rhyme: "The thumb is one. The pointer two, The middle finger three, Ring finger four. Little finger five, And that is all you see." Then let the children dramatize this finger play. Ask them how many chil- dren are needed to represent the fingers. Have one child pick out five children and count them as she selects them. With this play associate the nursery rhyme, "This little pig went to market." After the rhyme has been played on the fingers, ask the children, "How many little pigs there are in that family." Or before you begin the rhyme, ask them how many little pigs we shall have and where we can find them. The children will immediately hold up their hands and tell you that we are to have five pigs. After this, allow some child to dramatize it. Ask how many children she will need to represent the little pigs, and tell her to select them from the ring. As soon as they are selected, ask her to count them, making sure that she points to each child as she counts. Then let them play the game. On the return of the little pigs, have them counted by another child, so as to be sure that all the little pigs have returned home. Five seems an easy number for the child to begin with because it can be applied to so many finger plays. These plays are very attractive to children and can be made more delightful and interesting by the children personally representing them in dramatization. Some of these plays are, "This is the Loving Mother," "This is Little Tom- my Thumb," "Mrs. Pussy, sleek and fat," etc. Here is a number game to be played in the circle. Take five red balls, or NUMBER IN THE KINDERGARTEN. 19 any color that you happen to be teach- ing, and place them in a straight line on the floor. Opposite this line of balls and some distance from them, make a chalk mark (X) on the floor. The child who is to roll, is to kneel on this chalk mark and roll a rubber ball straight across to the line of red balls and see how many he can knock out. Before he begins ask him, "How many red balls are there in that line?" Now roll. Then ask, "How many red balls have you knocked out?" "How many are left standing on the line?" "Now put the balls you have knocked out back on the line, count them again, and tell us how many there are." This game is also a good one for clinching color. If you insist on the child looking at the line of red balls and taking a good aim before he rolls the rubber ball, he will get a good picture of the line of red balls. When you are ask- ing questions about the number of balls, if you persist in naming the color of the balls in every question, you will then give him the auditory image of the balls. In this informal way he will get his visual image reinforced by his auditory image. This game may be played with any number of balls, allowing the child to give the combination by telling the num- ber of balls he knocked out, the number remaining and then the whole number. If spheres, cubes or cylinders are used instead of worsted balls, the children enjoy rolling a large croquet ball against them. The child's grasp of the ball is stronger and the aim straighter and more definite. But the pleasure lies in rolling a big ball like the ones the big boys and girls roll. Children naturally like to count, so allow them plenty of opportunities to indulge this liking. Let a boy count all the girls in the room, touching light- ly each girl on the shoulder as he counts. Then let a girl count the boys, touching each boy on the shoulder as she counts. Now select one child to count all the children in the room. This last should be a co-operative exercise, that is, all the children count in unison as the child selected touches the chil- dren. Let them count the chairs, the tables, and the material as it is passed out to them. Give the children plenty of problems to solve. Place six colored balls on the table, marching one by one. Have them counted. Send several children to the board to make the picture of these balls with colored chalk. Count them. Next have a child come and show another way that these balls can march. They are now marching one by one. Could they march as we do when we go to recess? He places them two by two. Ask him to count the balls. How many balls in each row? How many rows? How many balls altogether? Can you count them a short way? Then show him how to count by two's. Place six children in three rows of two each and have them counted by two's. Let the children place chairs in the same way and count them. I find it is better to use material that is near at hand, and that is familiar to the children, then they can concentrate on the number les- son more easily. Let the children using colored chalk draw on the board the picture of the balls, marching two by two. Count by two's. Rings, sticks and tablets arranged in border designs give excellent opportunity for counting by two's. The third and fourth gifts with their eight small divisions offer many prob- lems in addition, subtraction, multiplica- tion, division and fractions. Count the 80 NUMBER IN THE KINDERGARTEN. small cubes in the third gift as it stands as a whole. Now divide the cube in two parts. How many small cubes in each part? Make a whole cube again; how many small cubes in the whole cube? Now let us again divide it into halves. Can you tell the name of each part? How many cubes in each half? Make the whole cube. How many small cubes in the whole? Divide the cube in two parts, and separate each half into two parts. Now how many parts have we? Tell them that each part has a name and is called one- fourth. Then make the whole cube and count the number of small cubes in the whole. Next divide the cube into eight parts, calling each part an eighth. Make a whole cube and count. Of course, these lessons are all given through some playful exercise. Problems of this kind may be given. Build a wall two cubes high with half your cube. Build a wall four cubes high, using all your blocks. Count the number of small cubes. Count the short way by two's. Take two cubes off; how high is the wall now? How many cubes are in the wall? Put the two cubes on again. Now how high is it, and how many cubes have you used in all? Take four cubes off; how high is it now? How many cubes have you used in the wall? How many did you take away? Put the four cubes back ; how many cubes have you in all ? How many stories high is the wall? How many cubes in each story? How many cubes in the whole wall? There are also many problems in measurement that can be given. For instance, make a sidewalk one cube wide and eight cubes long. Make a path two cubes wide and four cubes long. Make a sidewalk four cubes wide and four cubes long. Now let us make a little garden and put a fence around it one cube high, making it square by placing two cubes at the back, two cubes at the front and two cubes at each side. This fence should be built on the squares on the table. Then they can make a fence three cubes back, three cubes front and one cube at each side. With the fourth gift consisting of the oblong shaped blocks, the children can work out more difficult problems of fence building. After the inch has been developed with the brick, the children can build fences with the dimensions given in inches. For instance, build a fence around a garden 4 inches long and 4 inches wide. Build another fence 6 inches long and 2 inches wide. They can also build houses, the dimensions being given in inches. With the fifth gift we get the com- binations of thirds and ninths. The Kindergartner must be careful and not make the number work abstract. It must all be given in concrete terms and in an informal playful way. Draw from the children, suggesting only what will give them an impetus to think fur- ther. Occasionally, after the lesson is over, give them some little problems, bearing on the lesson, to work out alone. Tell them it is something hard, such as the big brother has to do upstairs in his classroom and they will set to work with great glee and force. Children like to have something to think about and ponder over and often accomplish wonderful results. They like to be treated like reasonable beings and do not need to have things made too easy for them. A little mental effort will help the development of the brain cells and will get children into the habit of thinking and reasoning for them- selves. Do not be too sparing in praise of work well done, in fact find in every piece of work something to praise. Make the child feel that his work is growing with him; tell him what a big boy he has grown to be and how much better he works now than formerly. P,l@M^MlMM@M^^MIMlllMMiM@MliMMM^lI SCHOOL PROBLEMS is Your Journal We would like to hear from subscribers for any preferences tbey may have with regard to articles, methods of treatment, etc. We will always be pleased to consider articles suitable for publication in this Magazine. (^miiM^iii^piM^MMMPMnriii m] ^vvvvvv.^ SSc:^c^=?? ^vvvvvvv^^= ^^ F=^^^ E School Problems Will furnish teachers, free of charge, any information they may want with regard to any Text Book or Professional Book published. Where an immediate personal reply is desired, stamp should be enclosed. Write as often as you like. We are at your service. THE EDUCATIONAL PRESS 123 East 23rd Street, - - - - New York W^^VVVVVVX ^S ^ p3=?:? ^VVVVVVV ^ ^^VVVVVV SS^ s i SCHOOL PROBLEMS ^ Journal for the Practical Use of Teachers $1.00 per Year in Advance Bi^Monthly, except July 25 Cents the Copy. Efficiency in Class Instruction can only result -when the grade work is carefully planned in detail. This is our Creed, and if you are a RouTiNiST, your work will lack vigor if not Efficiency. But if you are a Progressive Teacher, you will need a copy of which presents practical plans and details of grade work, series of model lessons, and articles based upon sound pedagogical principles. The aim will be to make every article so positively practical that it will show not only what ought to be done but how to do it. All the articles are working plans of actual teaching practice— plans that have been proven successful. THE EDUCATIONAL PRESS 123 East 23rd Street New YorK City aMMi/|^W •mm^I^W* k WE PAY FOR MATERIAL N.B. Stamps sliould be enclosed if it is desired tbat manu- script be returned. ^ m mt i \l^iA V *J^ff*m A Journal for the Practical Use of Teachers Efficiency in Class Instruction is our Creed To this end, we want PRACTICAL LESSONS by PRACTICAL TEACHERS ^^e -will pay for suck material as we can use, at current rates. Address all Communications to The Educational Press,^SL'^vfri14* ««M«^|/w**Vlr**^' THE EDUCATIONAL PRESS I vV ^