5 ^^
 

 
 PRINCIPLES and METHODS 
 
 i uujjwu i miw * imm * J'.*ja* i! 
 
 or 
 
 Arithmetic Teaching 
 
 By WILLIAM B. CHRISWELL, Ph. B. 
 
 Published and Copyrighted by the Author 
 
 Potsdam, N. Y 
 
 1914 
 
 3 V , 1
 
 COPYRIGHT, 1914 
 By William B. Chriswell 
 
 Potsdam, N. Y. 
 Elliot Fay & Sons 
 
 1914
 
 A 
 \ :- 
 
 C 
 
 PREFACE 
 
 The following pages have been prepared for the 
 use of students in the Potsdam State Normal School. 
 
 Thanks are due to Professor L. D. Taggart, 
 Superintendent of the Training School, for his kind- 
 ness in reviewing the manuscript and offering valuable 
 suggestions. 
 
 VV. B. C.
 
 PART ONE 
 
 GENERAL PRINCIPLES. 
 
 It has been well said that we should be loyal to principles but 
 the slave to no man's devices. Only a few of the general principles 
 especially applicable to Arithmetic teaching can be treated in this 
 course, and these but briefly. For fuller treatment the student is 
 referred to Younr?, The Teaching of Mathematics; Smith, The Teach- 
 ing of Elementary Mathematics; Suzzallo, The Teaching of Primary 
 Arithmetic; McMurry, Special Method in Arithmetic; McLellan and 
 Dewey, The Psychology of Number; Brooks, Philosophy of Arithme- 
 tic; De Garmo, Interest and Education; and general works on Psychol- 
 ogy and Education. 
 
 I. REASONS FOR THE STUDY OF METHOD. 
 
 It is not so much whether a teacher is a good teacher as whether 
 she is a progressive teacher. The good teacher may deteriorate, the 
 progressive teacher never. Proper method is as essential to the 
 teacher as to the singer or the painter. Without some knowledge of 
 the technique of the profession, little progress can be made. 
 
 Teachers may be classified as follows: 
 
 I. UNINSPIRED. 
 
 1. The unskilled are those who have neither natural ability for 
 teaching nor knowledge of technique. The sooner such are eliminated 
 form the profession the better. 
 
 2. The skilled are those who have little natural talent for teach- 
 ing, but who have through the study of method mastered the tech- 
 nique of the profession. These are good teachers but lacking in 
 inspiration. 
 
 II. INSPIRED. 
 
 1. The born teacher is full of enthusiam and inspiration, but 
 lacking in knowledge of technique. Her methods are often faulty, 
 but this is more than counterbalanced by the inspiration imparted to 
 others. 
 
 2. The artists are the born teachers who have become masters 
 of technique. They are the Michael Angelos, the Rembrandts, the 
 Millets of the teaching profession. 
 
 4
 
 The purpose of the study of method is to transform into skilled 
 teachers those who would otherwise be unskilled, and into artists 
 those who are born teachers. To quote from Froebel: 
 
 "Step by step lift bad to good; 
 Without halting, without rest, 
 Lifting better up to best." 
 
 See Young, Chap. I, VIII, and pp. 254 and 256; Page, Theory and 
 Practice of Teaching, pp. 19-29; Suzzallo, Chap. Ill; Young, Teaching 
 of Mathematics in Prussia, Chap. IV. 
 
 II. AIMS OF ARITHMETIC TEACHING. 
 
 Various aims of arithmetic teaching have been given, the chief 
 being intellectual training, and utilitarian purposes. The reasoning 
 connected with the solution of problems fulfills the intellectual aim. 
 For practical use the essentials are knowledge of number facts; fund- 
 amental operations in integers, fractions and decimals; practical 
 problems including percentage; and absolute accuracy and a resason- 
 able degree of rapidity. Accuracy is the sine qua non. . "Careless 
 facility", says Professor Frank Hall, "is not merely useless, it is posi- 
 tively harmful." See Smith, Chap. II; Young, Chap. II; and pp. 203 
 to 209, 214 to 216; Suzzallo, Chap II; Home, Philosophy of Education, 
 pp. 115, 116; Spencer, Education, pp. 29-31; Brown and Coffman, How 
 to teach Arithmetic, Chap. III. 
 
 III. FUNCTION AND EXTENT OF OBJECTIVE TEACHING. 
 
 Clear perception is essential as a basis for future knowledge. At 
 first new ideas must be presented directly to the senses. Later these 
 ideas may be used as a foundation for an "apperceptive system." To 
 produce clear knowledge in any science in its initial stages requires 
 objective presentation. The extremes of no objective illustration and 
 ?ndless illustration should be avoided. A safe guide is the old rule to 
 proceed from the known to the related unknown. When there is no 
 known, objective illustration is essential. If there is clear knowledge 
 from which to proceed, it is a waste of time and a loss of power not 
 to use such knowledge. See Young, pp. 107, 209 to 212; Smith, pp. 
 101 to 100; McLellan and Dewey, p. 283; Bagley, Educative Process, 
 PP. 254, 255; Suzzallo, pp. 42-59. 
 
 IV. ESSENTIALS FOR EFFICIENT RECALL. 
 
 To commit to memory and to be able to recall our knowledge 
 when needed, certain mental laws must be observed. Among the 
 essentials are strong impression, association, repetition, and frequent 
 recall. 
 
 Few facts in arithmetic stand out so prominently in and of them- 
 selves as to make a strong and lasting impression at the first present- 
 
 5
 
 ation. One fact is as important as another. Hence little use can be 
 made of strong impressions. However, as the more concentrated the 
 attention, the stronger the impression, the teacher must require the 
 closest possible attention from every child. As the time of commit- 
 ting number facts is ordinarily during the age of passive or involun- 
 tary attention, and little reliance can be placed on forced attention 
 the teacher must appeal to the child's instincts. She must appeal to 
 his curiosity, satisfy his desire for new knowledge, for variety, for 
 activity, for rhythm, for success. By means of games she may associ- 
 ate the work with child life. >/ 
 
 Drill, repetition, and frequent recall will be found necessary. All 
 facts must be fully learned even at the cost of much drudgery. "It is 
 safe to say that the point will never be reached where pain and drudg- 
 ery can be entirely eliminated from the educative process", says Bag- 
 ley. "If the pupil does not sometimes find his school work disagree- 
 able, then something is radically wrong either with the pupil or with 
 the school or with both." See Young, pp. 91 to 96. 135, 136, 251; Mun- 
 scerberg, Psychology and the Teacher, Chap. XVI, and p. 122; De Gar- 
 mo, Essentials of Method, p. 60; James, Psychology, Chap. XVIII 
 Radestock, Habit, pp. 32, 74; White, School Management, pp. 160-166 
 Bagley, op. cit., Chap. XI; Smith, Systematic Methodology, pp. 37, 38 
 Chamberlain, The Child, pp. 341, 342. 
 
 V. TEACHING A NEW NUMBER FACT— DRILLS. 
 
 In teaching a new number fact, as many different cortical areas 
 as possible should be approached simultaneously or in quick succes- 
 sion, in order to make the proper nerve fiber connections in the brain 
 as a basis for future association of ideas and quick recall. Approach 
 visual, auditory, vocal muscular, and graphic muscular (strain) areas. 
 
 1. Teacher repeat. 
 
 2. Children repeat. 
 
 3. Teacher write on board. 
 
 4. Children read orally. 
 
 5. Teacher erase and children visualize. 
 
 6. Children repeat imaged fact. 
 
 7. Children copy written fact. 
 
 8. Children write imaged fact. 
 
 Repeat these processes to "form the habit" as it is commonly 
 expressed, but speaking with physical accuracy, to form the proper 
 nerve fiber connections. Have class repetition interspersed with in- 
 dividual repetition. Drill individuals found slow, then the class 
 again. Give very slow pupils outside drill. No other work of any 
 kind should be taken up after presenting new facts till these have 
 been thorougly drilled upon. As already said all facts must be fully
 
 learned even at the cost of much drudgery. Give a minute's absolute 
 rest, then drill again. 
 
 The teacher need not be surprised if the last facts presented are 
 forgotten before the next recitation. She must repeat the process 
 and drill again. As often as it is found that some fact has been for- 
 gotten, drill must again be given on this particular fact. Before pre- 
 senting new facts, be sure that the class know thoroughly all previ- 
 ously presented facts. Find what facts a child does not know and 
 drill him accordingly. Responses should be practically instantaneous. 
 "A child must not be allowed to forget old facts," says Munsterberg, 
 "in the effort to acquire new facts." See Thompson, Brain and Per- 
 sonality; Munsterberg,op. cit., pp. 117, 118, 140-147; Suzzallo, Chap. 
 VIII; Bailey, Teaching Arithmetic, p. 47; Bagley, pp. 122, 123, 328- 
 331; McMurry, pp. 46, 56; James, Psychology, pp. 134-150, 298; 
 Myers, Experimental Psychology, pp. 72-90, 112-116; Brown and 
 Coffman, op. cit., Chap. VIII. 
 
 VI. REVIEWS. 
 
 We have noted that one of the essentials to memory is frequent 
 recall. This is one, but only one, of the aims of reviews. There are 
 six purposes of reviews. 
 
 1. That the teacher may learn the child's needs. 
 
 2. To show the child his needs. 
 
 3. To fix more firmly in mind (by frequent recall). 
 
 4. To organize materials — the general review. 
 
 5. As a preparation for new related knowledge — the first step 
 
 of the Inductive Development Lesson. 
 
 6. To make knowledge usable. This, says Gordy, is the great 
 
 function of reviews. (Lessons in Psychology.) 
 
 According to Gordy there are three stages of knowing: 
 
 1. Implicit knowledge — The child knows but cannot express 
 
 what he knows. 
 
 2. Explicit knowledge — The child can tell what he knows, but 
 
 he cannot use his knowledge. 
 
 3. Usable knowledge. — The child can use his knowledge as well 
 
 as tell what he knows. 
 
 To show the importance of reviews, we may make use of the 
 illustration of the snow storm. 
 
 On a winter morning we look out upon the streets to see that a 
 great depth of snow has fallen. We watch the first team plod labo- 
 riously through to open a way. Another team follows, also with 
 great difficulty, but each succeeding team finds the task easier, till at 
 length the sleighs glide past with the greatest ease. But another 
 storm fills up the broken way. The process of breaking out must be 
 
 7
 
 repeated, but the difficulty is not so great as it was the first time and 
 soon the sleighing is better than before. So storm after storm re- 
 quires constant travel to keep the way open. If after the way has 
 been once broken out, several storms occur without the street's being 
 used in the meantime, the path may become utterly obliterated and 
 forgotten. 
 
 So it is with the brain. Children, like adults, will forget when 
 facts are allowed to remain unused for some time. The way must be 
 broken out and rebroken; or to speak literally, the nerve fiber connec- 
 tions must be strengthened by repeated use. 
 
 Drills, therefore, should be given not only in connection with 
 new work, but also in reviews. Drills should be quick and snappy. 
 In review drills when the child hesitates, call on the class. Work 
 rapidly while you work and waste no time. Ten minutes of rapid fire 
 recitation is worth more than a half hour of dawdling. 
 
 See Munsterberg, op. cit., p. 122 and Chap. XVI; Smith, pp. 143, 
 144; Young, pp. 129, 142, 225, 226; McMurry, pp. 56, 80, 81, 98, 127, 
 136, 137, 140; McMurry, Method of the Recitation, pp. 114, 115; 
 Radestock, Habit, p. 21; Bagley, op. cit. pp. 328-333. 
 
 VII. CONCERT RECITATIONS. 
 
 Beware of concert recitations. A skillful teacher may at times 
 use concert recitations successfully. 
 
 1. In drill work. Here what is desired is repetition with atten- 
 tion, hence time is saved by having several repeat simultaneously. 
 
 2. When the answer is short and one child fails, the rest of the 
 class may have the answer at their tongue's end. 
 
 3. When the answer is short and a pause is made after the 
 question to allow all to grasp the meaning and form the correct an- 
 swer. In recitation on number facts, for example, the teacher may 
 point to successive facts on the board, pause briefly, say "class" or 
 tap on the board with the pointer, and the class as a result of the 
 pause is ready to answer simultaneously. If, however, no time is 
 given for thought, one bright child will answer and the rest then re- 
 peat like parrots, some not even looking at the board to see what the 
 fact is. 
 
 Where there is a possibility of different wordings to the answer, 
 concert recitation is out of the question. See Page, Theory and 
 Practice of Teaching, pp. 151, 152. 
 
 VIII. ATTENTION. 
 
 Keep every child attentive as long as he is supposed to be atten- 
 tive. If impossible to hold the attention of the more advanced pupils 
 while drilling backward pupils, set them at other work. Do not try 
 to keep young children at high tension too long at a time. Allow 
 them to rest, then begin again. 
 
 8
 
 TO HOLD ATTENTION. 
 
 1. Secure interest if possible. In some way connect the work 
 with child life, (lames are often a great aid. But be sure to hold 
 attention, interest or no interest. One of the great aims of education 
 is the development of the power of voluntary, or active, attention. 
 "May it not be", asks Munsterberg, "that the most important aim of 
 education is just the power of overcoming the temptations of mere 
 personal interest?" "An education"', he says, "which simply follows 
 the likings and interests, leaves the adolescent personality in a flabby 
 and ineffective state." "There is no doubt that through the tendency 
 of our times to yield to this demand for interesting instruction, we 
 already feel the dangerous results of the crippling of the voluntary 
 attention." "Public life has to suffer for it." "But the great word 
 which is to control it is not pleasure but duty." Op. cit., pp. 16, 18, 
 190, 265, 269. 
 
 On the other hand, George Eliot says: For getting a fine 
 
 flourishing growth of stupidity there is nothing like pouring out on a 
 mind a good amount of subjects in which it feels no interest. Mill on 
 the Floss, p. 327. 
 
 2. Watch the class, not the pupil reciting. 
 
 3. Call on inattentive pupil when pupil reciting hesitates. 
 
 4. Call on the whole class when the pupil reciting hesitates, 
 provided the answer is brief. If but few respond, it may be a sign 
 of inattention. See De Garmo, Chap. XV; Munsterberg, op. cit., Chap. 
 XVIII; James, Psychology, Chap. XIII; Radestock, Habit, pp. 70, 71: 
 Puffer, Psychology of Beauty, pp. 160-168; Bagley, Classroom Man- 
 agement, pp. 137-187. 
 
 IX. THE ART OF QUESTIONING. 
 
 No teacher of Arithmetic should fail to make a thorough study 
 of the art of questioning. The mere instructor has little need of 
 questioning, but the true teacher must know how to question and 
 how to question well. "To question well is to teach well," says De- 
 Garmo. Those desiring to make themselves more proficient in this 
 art are referred to the chapter on "The Art of Questioning" in De- 
 Garmo's "Interest and Education". Also see Young, pp. 55, 67. 
 
 X. CLASS V. S. INDIVIDUAL RECITATION. 
 
 The teacher of power, like the conductor of the trained choir, 
 carries along his whole class. Here may be a solo and there a duet 
 followed by the ensemble, all being parts of one great production. 
 So the teacher may call on this individual or on that to recite for the 
 class and as a constituent part of the class, but the whole class is 
 carried along as if it were a single individual. Every one is attentive 
 and ready to respond if called upon. The individual is by no means 
 neglected. The teacher watches every move, every expression even.
 
 The slightest sign of incomprehension in a child's eye calls forth an 
 enlightening illustration or a thoughtprovoking question. 
 
 However this complete class unity is found neither in the first 
 recitation nor in the tenth. More or less individual drill must pre- 
 cede and accompany the choir practice. The quick ear of the conduc- 
 tor catches every weak point, and if he cannot strengthen it in gen- 
 eral practice, he does it in private. So the successful class room 
 teacher must supplement his class room work with individual instruc- 
 tion. This, in fact, is the strength of the Batavia System, in which 
 one teacher does the class instruction and another the individual 
 instruction. 
 
 But one might visit two adjoining rooms and hear the same 
 question put and practically the same answer given by individual 
 pupils; yet in one room the recitation might be class recitation and in 
 the other individual recitation. That is, in the one room the recita- 
 tion might be conducted in the manner described, while in the other 
 only the person reciting may be paying attention. The class as a 
 whole is receiving no benefit and the recitation consists of a series of 
 individual recitations, logically connected in the teacher's mind per 
 haps, but disconnected and unrelated in the pupils' minds. See 
 Young, 56, 57, 81 to 86, 138, 139; Bagley, Classroom Management 
 Chap. XIV; Young, The Teaching of Mathematics in Prussia, pp. 56, 
 57. 
 
 XI. THE INDUCTIVE DEVELOPMENT LESSON. 
 
 Inductive reasoning consists in passing from particular truths 
 to general truths. It is known as generalization. Induction is often 
 the method used in presenting new matter to children. The child, in 
 fact, begins his acquisition of knowledge by means of induction. 
 
 The inductive development lesson is generally considered as 
 having five steps: 
 
 1. Preparation. 
 
 Substep — Statement of the aim. 
 
 2. Presentation. 
 
 3. Comparison and Abstraction. 
 
 4. Generalization. 
 
 5. Application. 
 
 The preparation consists of a brief review of all matter necessary 
 as a basis for the new work to be presented, in order that the child 
 may have this knowledge fresh in his mind and thus readily take the 
 step from the known to the related unknown. Not the whole subject 
 should be reviewed, but only that necessary for the foundation. 
 Generally a few questions will bring out all that is required. See 
 McMurry, pp. 76, 79, 80; DeGarmo, Essentials of Method, pp. 46-51. 
 
 As to the statement of the aim, De Garmo says: "It would be 
 unpedagogical not to have the pupil understand from the beginning 
 what the aim of the lesson is." "A skillful dramatist never fully 
 
 10 
 
 1
 
 reveals his plot ahead of its unfolding, nor does he, on the other hand 
 ever allow any great but entirely unexpected culmination to occur." 
 Essentials of Method, pp. 48, 50. "It should state as clearly as pos- 
 sible the point that the lesson is intended to make," says Bagley. "It 
 should seize upon some need and show it may be satisfied." "The 
 aim really forms the connecting link between the old and the new." 
 Op. cit., pp. 291, 292. See McMurry 80, 81. However, many teachers 
 of successful experience deliberately omit this sub-step. 
 
 The presentation consists in placing before the class several 
 illustrations of the new matter to be taught, reaching it naturally 
 from the matter reviewed in the preparation. The new matter may 
 be brought out by questions, but occasionally it may be found best to 
 show the connection directly without questions. See De Garmo, op. 
 cit., pp. 51, 52. 
 
 The comparison and abstraction consists in noting what has been 
 done in all examples. 
 
 The generalization consists in stating in the form of a general 
 rule or formula how all similar examples should be worked. See 
 McMurry, p. 69. 
 
 The application consists in working examples by the rule that 
 has been generalized. This step as will be seen is deductive, though 
 a part of the inductive development lesson. See Smith, pp. Ill, 112; 
 Bagley, op. cit., Chap. XIX. For lesson plans, see Stamper, A text 
 book on the teaching of Arithmetic, Chap. VIII. 
 
 XII. TEACHING A NEW OPERATION. 
 
 In teaching a new operation, when possible, use the inductive 
 process as above. But aside from this, when the class is large, there 
 is a general method of procedure that will often be found advanta- 
 geous. 
 
 1. In developing the process the teacher should work several 
 easy examples on the board. 
 
 2. Send individuals to the board, the class observing and mak- 
 ing suggestions under the direction of the teacher. 
 
 3. Send several pupils or the whole class to the board. 
 
 4. If several have trouble, send all to their seats and attack the 
 difficulty again. 
 
 5. Send the class to the board again to finish the example. 
 
 Or in place of 4 and 5, as for example in teaching long division, 
 the teacher may ask the class to work as she directs. Suppose each 
 child has on the board ready for work the example 384 to be divided 
 by 12. Teacher — How many times is 12 contained in 38? Where 
 shall we place the 3? What shall we do with the 3 now? Where 
 shall we place the 36? Etc. Several examples may be worked in this 
 way until the class know all the steps. 
 
 6. Teacher give individual help to the few who still have trouble, 
 
 11
 
 7. Slow pupils should be dealt with out of class to prevent the 
 monopolizing of time that should be spent in class instruction. 
 
 It is important that but one difficulty, that is one new point, 
 should be introduced at a time. "The importance of cutting work up 
 into simple steps and taking them one at a time", says Young, "can- 
 not be overestimated." (p. 128.) See Young, pp. 134, 135. 
 
 XIII. THE DEDUCTIVE DEVELOPMENT LESSON. 
 
 Deductive reasoning consists in passing from a general truth to 
 a particular truth. We begin with rules and proceed to individual 
 cases. Our first duty is to see that the major premise is true. In 
 arithmetic we must apply the proper rule. 
 
 In Logic we have the Major Premise, the Minor Premise, and 
 the Conclusion. 
 
 Major Premise: All A is B. Minor Premise: C is A. Conclu- 
 sion: C is B. 
 
 The Deductive Development Lesson is considered as made up of 
 four steps: 
 
 1. Statement of data. 2. Statement of governing principles or 
 general rule. 3. Statement of the conclusion. 4. Verification. 
 
 It will be noted that the order of the logical sylogism and of the 
 deductive development lesson as sometimes given is here reversed, 
 the particular statement being given before the general. In promis- 
 cuous problem solving in arithmetic this inverted order is necessary, 
 as the data is first stated and the pupil must determine what general 
 rule applies to the particular case. He then draws his conclusion, 
 states the problem accordingly, and solves. The Verification may b-" 
 some check or reverse process, but it frequently consists in looking at 
 the answer or in receiving the instructor's approval. 
 
 Example: A merchant invests $160 in flour and sells it at a 
 profit of $40. Find the Rate of gain. 
 
 1. Statement of data: $160 is Cost or B. $40 is gain or P. 
 Wanted — the Rate of gain or R. 
 
 2. Statement of general rule: P divided by B = R. 
 
 3. Statement of conclusion: $40 h- $160 = 25 'v . Rate of gain. 
 Answer. % 
 
 4. Verification: 25 # of $160 = $40. 
 
 See Bagley, Educative Process, Chap. XX. 
 
 XIV. PROBLEM SOLVING. 
 
 There are several difficulties that the child will have to meet and 
 remove before he can solve problems readily. 
 
 1. He must be able to identify the data and what is wanted. 
 
 2. He must determine what rule or formula to use.
 
 3. He must determine what intermediate steps, if any, must 1"' 
 taken. Under this head he must determine: 
 
 (1) Is there a missing fact and what is it? 
 
 (2) Can the missing fact be found directly from the given 
 data? 
 
 (3) If the fact cannot be thus found, what can be found 
 from the given data, from which new data the missing 
 fact may be found? This is an appeal to the puzzle 
 instinct. 
 
 Much practice in identification of data in terms of Percentage. 
 Mensuration, etc., should be given, independent of solution. At first 
 it will be necessary to lead to identification by questioning. Then 
 the child should be led to ask himself the questions. For a time he 
 should be required to write on paper or on the board the data both in 
 terms of the problem and in terms of Percentage, if that is the topic 
 under study. 
 
 Even greater than identification of data may be the difficulty of 
 determining and finding the missing fact. 
 
 Problem: If 4 hats cost $12, what will 7 hats cost? Here the 
 missing fact is the cost of one hat. Question as to what must first 
 be known before we can find the cost of 7 hats. Can we find that 
 missing fact? How' 
 
 The following problem of exactly the same nature as the above 
 sometimes puzzles eighth grade pupils: If % of a bushel of wheat 
 cost *'\ of a dollar, what will -' :! of a bushel cost? Using the former 
 problem as a preparation determine the rule: divide the cost by the 
 number to find the cost of one, and multiply this by the number of 
 which the cost is desired. In both problems solve by cancellation. 
 
 Problem: What will 17 gallons of milk cost at 6 cents per quart? 
 Here the essential missing fact may be given as either the price per 
 gallon or the number of quarts. For further illustration, see Chapter 
 on Percentage. 
 
 XV. TEACHING HOW TO STUDY. 
 
 One of the greatest, if not the greatest end of intellectual edu- 
 cation, is to teach the child how to study independently of a teacher. 
 Sometimes it seems that altogether too much teaching is being done 
 and the pupil is becoming entirely dependent upon, instead of inde- 
 pendent of, his teacher. 
 
 "Dependence is not the preparation for independence - ', says 
 'lurry. "Indeed, preat skill on the part of a teacher in these 
 respects almost precludes such skill on the part of pupils. If allow- 
 ed prominence year after year, it so undermines self-reliance that 
 one's helplessness when alone is greatly increased." "By overlook- 
 ing the difference between studying with a leader and alone, there- 
 fore, the teacher overlooks initiative, and in consequence, she not 
 
 13
 
 only fails to develop that power, but she may easily undermine it by 
 accustoming her pupils to dependence upon her." (How to study and 
 teaching how to study, p. 290.) 
 
 Too much has the appeal been made to the ear. The child can 
 make little or nothing from his text book without the teacher's first 
 clearing up all difficulties for him in the "assignment". This is a 
 fine thing for the dull pupil, but the bright child is weakened. Many 
 of us were taught by the other extreme. The text book was placed 
 in our hands and we were compelled to dig out the assigned lesson as 
 best we could. This was fine for the strong, but woe be unto the 
 weak. If he could not work an example and was compelled to call on 
 his teacher, the latter would merely work out the example on a slate 
 and return it without a word of explanation. A medial course is the 
 best to follow. At first give sufficient assistance by means of sug- 
 gestion and illustration, gradually decreasing the amount till the 
 child is largely thrown upon his own responsibility. 
 
 A place where the child should early gain independence is in the 
 solution of problems. This should first be taught by means of the 
 Study Recitation, given for a time once a week perhaps. On other 
 days the difficulties may be cleared up in the assignment by means 
 of questions and illustrations as already indicated. But the child 
 should be taught as soon as possible to ask himself the necessary 
 questions without the aid of the teacher. "Power of initiative is the 
 key to proper study", says McMurry. Op. cit., p. 288. See Page, op. 
 cit., p. 45. 
 
 XVI. THE STUDY RECITATION. 
 
 The study recitation is at first given after the method of the 
 deductive development lesson. The problems should for a time be 
 placed on the board, but at length the regular text book lesson should 
 be taken. 
 
 1. The children read the problem silently. 
 
 2. Children write on paper a statement of data and what is to 
 be found. If in Percentage, identify in terms of Percentage. In the 
 early lessons the teacher will find it necessary to question as to what 
 each item in the problem is, for a time even having the answer given 
 orally. Then the questions will be merely asked by the teacher and 
 answered mentally unless some pupil indicates that he cannot identi- 
 fy a particular item. The teacher may ask how many can identify 
 this item or that, or who cannot identify it; or she may merely ask 
 whether there is anyone who cannot identify all the items. The 
 teacher's questioning should become less and less, the children asking 
 themselves the questions and gradually becoming more and more self 
 reliant. 
 
 The teacher finally does nothing but pass among the pupils, 
 glancing at their work and putting a question here and there as she 
 finds the necessity. In time the pupils will take this step mentally 
 
 14 
 
 ^/
 
 and put nothing on their papers, the great danger being that they 
 will do so before they are strong enough. But this must be the ulti- 
 mate aim. 
 
 3. Children determine how to find the result, whether it can be 
 found directly or whether some intermediate step is required. This 
 is the step in which the reasoning must take place. As in the former 
 step the teacher will have to direct thought by means of questions. 
 How do you find the result? Can you find it directly? What is the 
 essential missing fact? Can you find that? What must you find 
 first? Etc., etc. As before, the child should gradually learn to do his 
 own questioning, the teacher's suggestions becoming rare. 
 
 4. The children state the problem. 
 
 5. Solve and place result in the statement. 
 
 6. Verify by some check if possible. 
 
 The Study Recitation should be given during tfte regular recita- 
 tion period. For a time twice or at least once a week should be given 
 to such work. Though it may seem impossible to take so much time 
 from the ordinary recitation, yet in the end it will be found that much 
 time has been saved. 
 
 When the child is able undirected to read and solve miscella- 
 neous problems in Arithmetic, he has made a long step in the direc- 
 tion of one of the chief aims of school education-, ability to educate 
 himself from books without the actual bodily presence of a teacher. 
 He is becoming a student. "When the principles can be explicitly 
 stated and intelligently applied, the essential aim of arithmetic has 
 been reached." "The intellectual treasures of the past lie locked up 
 in books.. Proper school training unlocks this storehouse by accus- 
 toming one to their intelligent use." (McMurry, The Method of the 
 Recitation, pp. 6, 142, 143.) 
 
 . See Bagley, op. cit., pp. 316-322; Bagley, Classroom Manage- 
 ment, pp. 206-210. 
 
 XVII. THE ASSIGNMENT. 
 
 By the Assignment as used by writers on pedagogy is meant not 
 merely the lesson assigned for home study but the method of assign- 
 ing the lesson. Bagley says: "This is a preliminary clearing of the 
 road before the seat work begins." "The acme of a skillful assign- 
 ment is reached when the teacher reveals just enough of what is con- 
 tained in the lesson to stimulate in the pupils the desire to ascertain 
 the rest for themselves." "In general, the assignment will be much 
 more explicit and detailed in the intermediate grades, where the pupil 
 is just learning to use text books, than in the upper grades and the 
 high school, where some familiarity with the text book method may 
 be assumed. But in all cases the assignment, whether it be brief or 
 full, is an important step which should never be omitted." (Op. cit., 
 pp. 317, 318.) The purpose of the assignment, therefore, is to clear 
 up all insurmountable difficulties. 
 
 15
 
 For example, suppose the lesson to be assigned consists of prob- 
 lems and that the class has had little work of the kind. The assign- 
 ment should consist of such questions as are used in the Study Reci- 
 tation. The study recitation prepares for this. Only a sufficient 
 number of questions should be asked to enable the pupils to solve the 
 problems without aid during the study period. The ability of the 
 pupils to solve the problems will show the teacher how successful was 
 her assignment. Some pupils will understand, or think they under- 
 stand, under the questioning of the teacher; but when they come to 
 take up their lesson alone, they find they are unable to solve the 
 problem. Such pupils will need to be requestioned. As in the study 
 lesson, the number of questions will decrease as the pupils gain in 
 strength. This work should go hand in hand with the work of the 
 study recitation, and in fact it is an essential feature of the method 
 of teaching children how to study. 
 
 One danger that the inexperienced teacher is apt to fall into is 
 that in the questioning in the study recitation and in the assignment, 
 she will call on the brighter pupils, those who have little need of such 
 help. When the questions are answered orally, those called upon 
 should be the pupils meeting with difficulty. This is very different 
 from the method followed in the regular recitation lesson when the 
 aim is to learn how well the lesson has been prepared. In such reci- 
 tations, good, bad, and indifferent must be called upon indiscriminate- 
 ly. See Young, pp. 132, 133, 147; De Garmo, Essentials of Method, 
 p. 47; Bagley, op. cit., pp. 317- 319; Bagley, Classroom Management, 
 pp 192-206. 
 
 XVIII. DICTATION. 
 
 Have all members of the class take down dictation promptly, / 
 accurately, and neatly. Dictate slowly enough for all to keep to- 
 gether, constantly increasing the speed till the whole class can take 
 dictation rapidly. 
 
 XIX. ORAL AND SILENT MENTAL ARITHMETIC. 
 
 A large part of the time given to arithmetic should be devoted / 
 to oral work and to silent mental work in which only the answer is * 
 written. New work should generally be presented orally and the 
 first written problems in each new topic should be as simple as the 
 oral problems. In written work as few figures as possible should be 
 used, short processes and silent mental computation supplementing 
 the written. As a child works at the board, have him "chalk and 
 talk"; that is, state orally each step as he writes it. / 
 
 See Smith, p. 118; Young, pp. 134, 135, 230; Suzzallo, pp. 75-78; 
 McMurry, pp. 57, 59, 80, 82, 98, 107-109, 123, 124, 127, 128, 133; 
 Bailey, A Handy Book on Teaching Arithmetic, p. 50* Walsh, Meth- 
 ods in Arithmetic, pp. 25, 27; Young, Teaching of Mathematics in 
 Prussia, p. 57-64; Munsterberg, op. cit., 283-285; Smith, The teaching 
 of Arithmetic, Chap. VII. See Stamper, op. cit., pp. 228-233.
 
 PART TWO 
 
 Primary Arithmetic 
 
 I. FIRST LESSONS IN NUMBER. 
 
 The first lessons in number should consist in finding the child's 
 knowledge of indefinite relative magnitude and exact number, and his 
 ability to count. He should be drilled in the use and meaning of such 
 relative terms as short, shorter, shortest; long, tall; large, few, etc. 
 
 To find the exact knowledge of number, objects may be held up 
 and the child called upon to tell how many there are, or he may be 
 asked to bring a certain number of objects. He does not know the 
 number six till he can both select six objects from a larger number 
 of objects and also state correctly how many there are when six 
 objects are shown to him. The knowledge that the individual child 
 may have upon entering the first grade will depend upon his age, his 
 home surroundings and training, and his kindergarten training. First 
 grade teachers report varying degrees of knowledge from three to 
 more than ten. la order to have a starting point, the common knowl- 
 edge of the class must be learned, then the teaching may begin at this 
 point. 
 
 There is some difference of opinion as to whether we should 
 teach the numbers as far as ten, before teaching the figures repre- 
 senting them; or whether we should teach both number and figure at 
 the same time. The problem is largely determined for us, however, 
 with respect to the first numbers at least, by the child's learning the 
 numbers without knowing the corresponding figures. The more 
 common practice with regard to the remaining numbers is to follow 
 the principle laid down by Quintilian regarding the teaching of the 
 Roman alphabet, that is, to teach both name and figure at the same 
 time. 
 
 The question as to the desirability of abstract counting is also 
 largely predetermined. In school practice some of our best authori- 
 ties hold that counting, to be of value, should be concrete and should 
 follow the study of definite number so that when a child can count to 
 twenty-five, he also knows twenty-five objects. It does not seem to 
 be a matter of great moment; and whichever practice a teacher may 
 follow, she has good authority to support her. 
 
 Taking for granted that we are to follow the practice of teach- 
 ing all new numbers and the figures representing them at the same 
 time, we shall find that our first lesson in exact number must be 
 teaching the figures representing the numbers already known. This 
 may be done by adapting the word method of teaching reading. It 
 will doubtless be a saving of time to co-ordinate by teaching the writ- 
 
 17
 
 ten word and the figure at the same time. In teaching the figure 3, 
 supposing the class to be fully acquainted with the number three, 
 there might be some question as to the necessity of presenting three 
 objects. The question is whether the word "three" brings up a suffi- 
 ciently vivid concept of the number three. The same question is pre- 
 sented in the teaching of the word "cat". Is it essential to present a 
 picture of a cat, or does the word bring up a sufficiently vivid mental 
 picture to do away with the need of the physical picture? Opinions 
 of course will differ, but the question is worth considering. The gen- 
 eral practice of those who believe in objective teaching would indicate 
 a belief in the necessity of the physical picture. 
 
 In teaching a figure for a known number we may hold up the 
 number of objects, elicit the name (or ask the children to bring three 
 objects), write the figure on the board, have children name the figure 
 and write it and bring the number of objects indicated. Drill. 
 
 In teaching a new number, as eight, supposing that some mem- 
 bers of the class do not know the numbers beyond seven, teacher hold 
 up eight objects and ask how many there are? Bring out from those 
 who do not know the name that there is one more than seven. Havn 
 some child who knows tell how many this is. Write the figure 8 or 
 the board and drill as before, giving special attention to those wht 
 did not know the number, and having them bring 8 objects, etc. 
 
 In teaching zero, teacher hold up her empty hand and ask how 
 many objects she has in it. The class respond, "not any". Teacher 
 write "0" on the board and tell class that this is how we express "not 
 any" and that we call it naught or zero. Drill. Warning: Do not 
 say nor allow children to say "ought". 
 
 See McLellan and Dewey, pp. 144-195; Brown and Coffman, 131- 
 150; Smith, pp. 112-117; Walsh, pp. 33-45. 
 
 II. NUMBERS ABOVE NINE. 
 
 In teaching "ten" use the fingers on the two hands. Also use 
 splints, tying ten in a bundle. Have several children stand and c-.e 
 child tell how many "ten fingers" there are. Teacher write 10 on the 
 board and have children write it. 
 
 Teacher holding up one bundle of ten and one single splint, How 
 many splints have I? As in eleven we have one ten and one single 
 one, we write it thus, the left hand "one" tells us we have one "ten", 
 and the right hand "one" tells us we have a single "one", or eleven in 
 all. Similarly teach 12 and 13. If one ten and one is written thus, 
 one ten and two thus, and one ten and three thus; who can tell how 
 we shall write one ten and four? One ten and five? Etc. 
 
 Tell class that -teen means ten. If four and ten is called four- 
 teen; six and ten, sixteen; seven and ten, seventeen; what shall we 
 call eight and ten? Nine and ten? 
 
 Teacher holding up two bundles of ten, how many tens have I? 
 
 18
 
 Obtain or give name. Etc. Similarly with twenty-one, twenty-two, 
 and twenty-three. If twenty and one is called twenty-one; twenty 
 and two, twenty-two; twenty and three, twenty-three; what shall we 
 call twenty and four? Twenty and five? Etc. If twenty-one is writ- 
 ten thus; twenty-two, thus; and twenty-three, thus; how shall we 
 write twenty-four? Twenty-five? Etc. If one ten is written thus-, 
 two tens, thus; and three tens, thus; how shall we write four tens? 
 Etc. If four tens is called forty; five tens, fifty; and six tens, sixty; 
 what shall we call seven tens? Etc. 
 
 If the method of induction thus briefly sketched is used, the class 
 should soon learn to name and write new numbers without the teach- 
 er's aid except by means of questioning. As the numbers are given 
 by the child, the teacher should build up the following table. By 
 having the known numbers thus systematically before the child, he 
 will more readily determine how to write the new numbers. 
 
 1 
 
 11 
 
 21 
 
 31 
 
 41 
 
 51 
 
 61 
 
 71 
 
 81 
 
 91 
 
 2 
 
 12 
 
 22 
 
 32 
 
 42 
 
 52 
 
 62 
 
 72 
 
 82 
 
 92 
 
 3 
 
 13 
 
 23 
 
 33 
 
 43 
 
 53 
 
 63 
 
 73 
 
 83 
 
 93 
 
 4 
 
 14 
 
 24 
 
 34 
 
 44 
 
 54 
 
 64 
 
 74 
 
 84 
 
 94 
 
 5 
 
 15 
 
 25 
 
 35 
 
 45 
 
 55 
 
 65 
 
 75 
 
 85 
 
 95 
 
 6 
 
 16 
 
 26 
 
 36 
 
 46 
 
 56 
 
 66 
 
 76 
 
 86 
 
 96 
 
 7 
 
 17 
 
 27 
 
 37 
 
 47 
 
 57 
 
 67 
 
 77 
 
 87 
 
 97 
 
 8 
 
 18 
 
 28 
 
 38 
 
 48 
 
 58 
 
 68 
 
 78 
 
 88 
 
 98 
 
 9 
 
 19 
 
 29 
 
 39 
 
 49 
 
 59 
 
 69 
 
 79 
 
 89 
 
 99 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 60 
 
 70 
 
 80 
 
 90 
 
 100 
 
 It is im 
 
 iportant that only 
 
 such numbers as the cl 
 
 ass know be 
 
 placed before them in 
 
 the above table. 
 
 Similarly teach 
 
 the numbers 
 
 to 1000.' See Speer's 
 
 Elementary 
 
 Arithmetic; 
 
 p. 125 
 
 
 
 III. READING AND WRITING NUMBERS ABOVE 1000. 
 
 1. Instead of building up the numbers in regular order above 
 1000, begin by teaching the reading of such numbers as 371,371 and 
 follow this by the writing of such numbers. 
 
 2. Next teach reading followed by writing such numbers as 
 235,476; 46,840; 2,896. Emphasize that the three right hand digits 
 represent units and the next three digits to the left represent thou- 
 sands. Place the word thousand above the second period and units 
 above the first period and have the number read for a time as follows: 
 235 thousand, 476 units, soon dropping the word 'units". 
 
 3. Teach such numbers as 234,054; 5,023; 87,004; at first reading,' 
 and dictating as follows: 5 thousand, no hundred 23; 87 thousand, 
 no hundred no tens 4; thus emphasizing tbe ciphers and the fact that 
 there must be three places to the right of each comma. Frequent 
 drill in wi*iting and reading such numbers will be necessary for sev 
 eral terms after the work is first taught. 
 
 4. Teach millions as above. 
 
 19
 
 After a little practice of this sort, dictate in the usual way, at 
 first questioning as to what should be placed in hundreds place, etc. 
 
 IV. ADDITION. 
 
 In arriving at and learning the addition facts the child should 
 under no circumstances be allowed to count, therefore do not use 
 single objects in teaching these facts. Some teachers will cry out 
 against this as a great heresy in that it violates the principle of ob 
 jective teaching. This, however, is an error, but even though it were 
 true, the evils resulting from the objective method in this particular 
 instance far outweigh its benefits if there be any benefits. The child 
 ivho learns his facts by counting, generally never learns his facts, 
 paradoxical as that may sound. When he has a column of figures to 
 add, he continues to count through life. This is a most pernicious 
 habit and results in slow and inaccurate computation. 
 
 However, there is a means of discovering a fact without develop- 
 ing the pernicious counting habit. It is an adaptation of the method 
 used by Montessori. Have wooden rods of 1 inch, 2 inches, etc., up to 
 18 inches in length with the length written upon them in figures. 
 The rods should not be marked off in inch lengths as this would, as in 
 the case of single objects, lead to counting. To teach the sum of two 
 numbers, as 2 plus 3, have each child find a two inch and a three inch 
 rod and place them end to end and then find another rod equal in 
 length to the sum of the two rods. 
 
 2. Having thus discovered that 2 plus 3 equals 5, approach all 
 the cortical areas of th« brain as indicated in the discussion regarding 
 the teaching of new number facts. Teacher repeat, children repeat, 
 teacher write on the board, children visualize, children copy. Many 
 children have to repeat the facts before they can give the results. 
 They should be drilled to give answers at sight or sound as well as 
 by repetition. 
 
 3. Teach both 2 + 3 and 3 + 2 at the same time, as this is but ono 
 fact. 
 
 4. Teach both horizontally and in columns. 
 
 5. Do not try to teach any new facts till all the old facts hav« 
 been thoroughly learned. 
 
 6. The order of teaching facts should be determined by sums 
 rather than by tables. Teach two plus two, then all the facts whose 
 sum is five, then those whose sum is six, etc. 
 
 7. Preparation for subtraction should go hand in hand wit*' the 
 learning of the addition facts. See discussion of subtraction. 
 
 8. After a few facts have been learned, column addition with 
 sum not greater than nine should be begun. Also simple oral exam- 
 ples such as 2 plus 2 plus 2. During the second half year column 
 addition should be continued to sum 18, but in no case should the 
 column exceed the number facts already learned. For example in the 
 
 20
 
 column 3-6-6, if the class know the facts only so far as sum 15, they 
 could add downward but not upward; as in one case the greatest fact 
 is 9 plus 6 which they have learned, while in the other case the fact 
 is 12 plus S which is not even one of the 49 elementary number facts, 
 which include only the facts from 1 plus 1 to 9 plus 9. In column 
 addition do not point or allow children to point. This is important. 
 
 9. Carrying may alse be taught during the second term, the New 
 York State Syllabus advising no explanation. Always add the carry 
 immediately. Do not write «t at the top of the next column, but add 
 it to the top digit at once if adding down, and to the bottom digit if 
 adding up. 
 
 10. Each day the previously learned facts should be reviewed by 
 various means: by placing the fac f s on the board, the child to fill in 
 the results orally or on the board, or by copying on paper; by placing 
 a number in a hollow square or within a circle of numbers and adding 
 it in turn to each of the outer numbers without any pointing; by using 
 number cards; by means of ladders, steps, guessing games, and 
 other devices that may occur to the teacher. 
 
 As many of the class as possible should try to reach the top of 
 ladder or steps without missing a step. The number cards should 
 have on one side the reverse of the combination found on the other 
 side; that is, if 4 over 8 is on one side, 8 over 4 should be on the 
 other side. Thus the teacher can see the fact on the back of the card 
 as she brings it from the back of the pack to the front. . This is im- 
 portant. She should stand where all the class can see the cards with- 
 out effort and should manipulate them rapidly without fumbling. She 
 should vary the method of recitation; sometimes calling on the chil- 
 dren by name for successive facts, rarely calling on them in order, 
 generally having one child rise and recite several facts before calling 
 on another pupil. When one child misses, say "class"' or "tell', and 
 the child missing should pass to the board and write the fact in the 
 four ways for writing each new fact. Children should recite these 
 facts rapidly. DO NOT GIVE THEM TIME TO OBTAIN THE 
 RESULT BY COUNTING. Another device is to place the missed 
 fact back in the pack where it will occur again soon and thus give the 
 child another chance to recognize the fact. Another device is to hand 
 the card to the child to study. Very little time, perhaps from three 
 to five minutes a day, should be spent in this drill. 
 
 In the guessing game the teacher or one of the pupils says: "I 
 am thinking of two numbers whose sum is 14." The class then 
 guess as called upon by the teacher, "Is it 7 and 7 are 14?" "Is it 8 
 and 6 are 14?" etc., till the correct combination is guessed. It is 
 generally well for the child to tell the teacher what fact he has 
 chosen. As games take time, they should be used with discretion. 
 
 After carrying has been taught, see that all examples given be- 
 fore beginning series illustrate both carrying and not carrying, that 
 
 21
 
 the children may constantly be compelled to judge whether they must 
 
 carry or not. 
 
 25224 74476 34332 63648 34716 
 
 64454 21320 24023 34231 53243 
 
 23234 43553 85579 71587 98929 
 
 Have class rise, add down, and be seated as fast as the result of 
 a column is found. Calling on some child, often the one seated last, 
 ask the result. The child may answer in the first example, second 
 column, "ten". As if the teacher considered this result correct, she 
 should ask how many have ten. Why not, Charles? Because there 
 i.s one to carry. Add it again orally, Frank. See that the carry is 
 made at once whether beginning with the bottom or top figure. This 
 cannot be over emphasized. In the fourth column, Frank may this 
 time give the result as thirteen. Question as before, bringing out 
 that there is no carry this time. This method will keep the children 
 constantly on the alert to see whether there is a carry. After a few 
 minutes of this practice, send the class to the board to work examples 
 already on the board. As fast as two children have their results 
 correct, have them erase results and exchange examples, thus keep- 
 ing all busy. 
 
 One point of importance that all teachers should observe is to 
 have chalk and erasers ready at the board before class that no time 
 need be wasted in class. 
 
 See Brown and Coffman, op. cit., pp. 150-159; Bailey, op. cit., 
 Lesson 16; Walsh, Methods in Arithmetic, Chap. II; McLellan and 
 Dewey, pp. 195-200; Smith, pp. 114-117; McMurry, pp. 30-53. 
 
 V. SERIES AND COLUMN ADDITION. 
 
 There are three classes of series: 
 
 1. Sum of right hand digits less than ten. Illustration: sum of 
 right hand digits 7. 
 
 24 44 84 64 73 33 53 94 14 
 333344433 etc. 
 
 2. Sum of right hand digits ten. 
 28 plus 2, 38 plus 2, 42 plus 8, etc. 
 
 3. Sum of right hand digits more than ten. Illustration, sum 
 of right hand figures 12: 
 
 27 plus 5, 77 plus 5, 85 plus 7, etc. 
 
 The teacher may give the series orally, the children giving the 
 results; the series may be placed vertically on the board as above; or 
 the single digit number may be placed in the center of a series of two 
 digit numbers, all having the same right hand digit, the child giving 
 results in regular order from right to left or left to right, no pointing. 
 In the oral work the teacher should say 27-5. If the child hesitates, 
 the teacher should say 7-5, and the child should say 32. If he says 12, 
 
 22
 
 the teacher should repeat 27-5. If the child still hesitates, the teach- 
 er should return to 7-5, and on receiving the reply 12 again, the 
 teacher should say, what is the right hand figure? Then what is the 
 right hand figure in the sum of 27 and 5? 27-5? If the child still 
 cannot give the result place the numbers vertically on the board and 
 add as in column addition. 
 
 One purpose of series drill is preparation for column addition. 
 The two should therefore go hand in hand. Children who may know 
 readily that 7 plus 5 is 12, may not know without counting that 27 
 plus 5 is 32. In the column example 6-8-3 the child may start cor- 
 rectly by saying 14 (he should never say 6 plus 8 are 14, or even 6-14, 
 but only 14) and then, though he may know that 4 and 3 are 7, he 
 may not know that 14 and 3 are 17. The teacher should say "4-3". 
 If the child says "17", do not compel him to say "7", as 17 was the 
 answer wanted. If he says "7", you may be compelled to say, "Then 
 how much is 14 and 3"? If the series 4-3 is on the board at the time, 
 the teacher may merely refer to this. 
 
 During the third half year, series, sum less than 10 should begin. 
 Such work should at first be oral, followed with blackboard work sim- 
 ilar to the illustrations given above, and accompanied by appropriate 
 column addition. The work should continue as necessary through the 
 following grades. 
 
 The series should be built up something as follows: plus 1 to 
 plus 9; 1 plus 1; 1 plus 2; etc. to 1 plus 9; miscellaneous practice. 
 The child's readiness in the miscellaneous practice and column addi- 
 tion will show how successful the series has been. Follow this with 
 2 plus 2; miscellaneous practice applying all series hitherto drilled 
 on; 2 plus 3; miscellaneous; 3 plus 3; miscellaneous, etc., up to 4 plus 
 5. 
 
 As soon as scries 2 plus 2 has been learned, such examples as 
 
 the following should be given. The children should add from the top 
 
 down as follows: 12-14-15; 5-7-12-14; 9-12-13-14. 
 
 8 4 6 The addition should be oral that the teacher my know 
 
 3 2 6 how the pupil is arriving at the result. No other num- 
 
 15 2 bers or words than those used above should be used by 
 
 12 1 the children in adding. If the examples are added from 
 
 the bottom up, they will not be in accord with the series. 
 
 FIRST SERIES— SUM OF RIGHT HAND DIGITS LESS 
 
 THAN TEN. 
 
 The following examples are constructed progressively according 
 to the series: 
 
 347 245 274 789 456 264 639 3 57 
 653 755 726 211 544 736 361 643 
 876 318 539 423 543 589 986 518 
 
 23
 
 243 
 
 148 
 
 431 
 
 142 
 
 246 
 
 348 
 
 768 
 
 648 
 
 867 
 
 962 
 
 679 
 
 899 
 
 854 
 
 853 
 
 343 
 
 462 
 
 649 
 
 871 
 
 572 
 
 5X 6 
 
 765 
 
 175 
 
 678 
 
 631 
 
 110 
 
 16 
 
 1 10 
 
 111 
 
 111 
 
 1 10 
 
 110 
 
 118 
 
 579 594 908 870 
 
 643 426 294 948 
 
 222 222 272 124 
 
 103 212 111 12 
 
 876 
 
 543 
 
 375 
 
 294 
 
 206 
 
 458 
 
 902 
 
 980 
 
 456 
 
 777 
 
 777 
 
 938 
 
 885 
 
 884 
 
 399 
 
 339 
 
 223 
 
 233 
 
 213 
 
 322 
 
 141 
 
 213 
 
 231 
 
 222 
 
 10 1 
 
 112 
 
 211 
 
 110 
 
 323 
 
 111 
 
 123 
 
 117 
 
 987 
 
 876 
 
 567 
 
 452 
 
 378 
 
 597 
 
 397 
 
 958 
 
 3 5 7 
 
 555 
 
 777 
 
 777 
 
 666 
 
 645 
 
 824 
 
 463 
 
 122 
 
 211 
 
 122 
 
 413 
 
 222 
 
 424 
 
 44 1 
 
 24 1 
 
 21 1 
 
 124 
 
 211 
 
 124 
 
 211 
 
 100 
 
 1 14 
 
 1 14 
 
 767 898 858 375 
 
 574 225 172 848 
 
 322 243 333 413 
 
 113 311 313 131 
 
 397 
 
 894 
 
 836 
 
 975 
 
 789 
 
 597 
 
 34 5 
 
 738 
 
 823 
 
 629 
 
 696 
 
 54 7 
 
 743 
 
 655 
 
 667 
 
 589 
 
 532 
 
 232 
 
 225 
 
 235 
 
 235 
 
 423 
 
 245 
 
 331 
 
 125 
 
 122 
 
 121 
 
 121 
 
 111 
 
 1 12 
 
 521 
 
 121 
 
 489 
 
 638 
 
 543 
 
 178 
 
 837 
 
 9 " 7 
 
 839 
 
 847 
 
 954 
 
 696 
 
 779 
 
 977 
 
 586 
 
 585 
 
 694 
 
 598 
 
 334 
 
 323 
 
 225 
 
 322 
 
 224 
 
 235 
 
 214 
 
 322 
 
 1 10 
 
 101 
 
 231 
 
 301 
 
 121 
 
 101 
 
 13 1 
 
 1 10 
 
 876 593 956 147 
 
 567 749 568 996 
 
 434 242 224 341 
 
 110 214 141 414 
 
 849 
 
 849 
 
 748 
 
 847 
 
 958 
 
 678 
 
 567 
 
 596 
 
 773 
 
 273 
 
 498 
 
 389 
 
 264 
 
 655 
 
 6 ' 5 
 
 466 
 
 264 
 
 146 
 
 422 
 
 442 
 
 656 
 
 323 
 
 604 
 
 224 
 
 112 
 
 621 
 
 221 
 
 1 10 
 
 10 1 
 
 222 
 
 162 
 
 6 12 
 
 24
 
 847 
 
 876 
 
 638 
 
 938 
 
 538 
 
 678 
 
 587 
 
 596 
 
 683 
 
 349 
 
 677 
 
 294 
 
 746 
 
 746 
 
 765 
 
 847 
 
 353 
 
 313 
 
 223 
 
 153 
 
 57 1 
 
 44 1 
 
 53 1 
 
 245 
 
 1 15 
 
 35 1 
 
 351 
 
 5 13 
 
 133 
 
 123 
 
 1 1 3 
 
 2 1 1 
 
 847 
 
 948 
 
 657 
 
 847 
 
 673 
 
 537 
 
 586 
 
 5 09 
 
 465 
 
 484 
 
 746 
 
 674 
 
 668 
 
 777 
 
 636 
 
 894 
 
 447 
 
 237 
 
 324 
 
 301 
 
 531 
 
 513 
 
 473 
 
 454 
 
 230 
 
 220 
 
 272 
 
 177 
 
 127 
 
 172 
 
 203 
 
 132 
 
 976 
 
 849 
 
 1 73 
 
 438 
 
 838 
 
 489 
 
 473 
 
 821 
 
 678 
 
 573 
 
 987 
 
 974 
 
 786 
 
 856 
 
 792 
 
 E 99 
 
 312 
 
 216 
 
 433 
 
 225 
 
 214 
 
 342 
 
 428 
 
 243 
 
 33 
 
 361 
 
 306 
 
 362 
 
 161 
 
 312 
 
 306 
 
 326 
 
 678 
 
 678 
 
 483 
 
 459 
 
 543 
 
 678 
 
 594 
 
 73 1 
 
 954 
 
 542 
 
 534 
 
 497 
 
 537 
 
 853 
 
 4 " 
 
 947 
 
 121 
 
 434 
 
 438 
 
 432 
 
 464 
 
 434 
 
 429 
 
 286 
 
 245 
 
 245 
 
 543 
 
 501 
 
 454 
 
 24 
 
 523 
 
 34 
 
 SECOND SERIES— SUM OF RIGHT HAND DIGITS TEN. 
 
 At first the class should find any two successive digits whose sum 
 is ten and the teacher or child should connect them with a curve. 
 Then for a time the tens should be found before adding, but the 
 rurves should be dropped. As soon as possible the child should ac- 
 quire the ability of finding the combinations of ten as he adds. Note 
 that in these examples the passage from the teens to the twenties 
 and from the twenties to the thirties, etc., is made by a combination 
 of 10. Before beginning the column addition, give series drill as be- 
 fore on the combination to be introduced in the example. 
 
 The child should add rapidly as follows (see first two examples) : 
 15-20-23-33-37; 9-15-20-26-36-38; 12-22-25-30-38; 11-15-20-24-34-39; 
 11-21-25-30-38. 13-15-20-29; 12-15-20-27-37; 11-21-25-30-38; 9-15- 
 20-24-34-37; 11-15-20-28. 
 
 88968 
 
 46853 
 
 58947 
 
 78948 
 
 29498 
 
 54567 
 
 46555 
 
 85389 
 
 65698 
 
 14969 
 
 55555 
 
 4 "535 
 
 65464 
 
 65445 
 
 76743 
 
 44363 
 
 54252 
 
 42444 
 
 45345 
 
 35373 
 
 55555 
 
 25275 
 
 44444 
 
 44454 
 
 37373 
 
 75455 
 
 25554 
 
 46646 
 
 32454 
 
 33577 
 
 15434. 
 
 43855 
 
 35666 
 
 57365 
 
 43437 
 
 25
 
 49935 
 
 94886 
 
 95498 
 
 98657 
 
 85899 
 
 96778 
 
 67386 
 
 78888 
 
 69894 
 
 89279 
 
 34773 
 
 63788 
 
 24282 
 
 15219 
 
 29612 
 
 37577 
 
 42878 
 
 28272 
 
 86119 
 
 91981 
 
 56567 
 
 24836 
 
 88738 
 
 84778 
 
 72727 
 
 57344 
 
 54724 
 
 82822 
 
 24612 
 
 95393 
 
 73934 
 
 36372 
 
 52297 
 
 21491 
 
 13179 
 
 THIRD SERIES— SUM OF RIGHT HAND DIGITS 
 GREATER THAN TEN. 
 
 First give series and examples in which nine is added to various 
 numbers, calling attention to the fact that the right hand digit of the 
 result is one less than, and the left hand digit one greater than the 
 corresponding digits of the number added to nine. Then give the 
 series in the order illustrated in the following examples: 
 
 98788 
 
 48398 
 
 91398 
 
 99799 
 
 24898 
 
 79998 
 
 84638 
 
 28312 
 
 99789 
 
 68688 
 
 35989 
 
 56435 
 
 42777 
 
 12893 
 
 46656 
 
 99875 
 
 44574 
 
 67464 
 
 82286 
 
 73466 
 
 67998 
 
 47757 
 
 75669 
 
 78336 
 
 33347 
 
 93949 
 
 63763 
 
 96447 
 
 78844 
 
 64563 
 
 29489 
 
 66326 
 
 94744 
 
 37044 
 
 76663 
 
 42348 
 
 89299 
 
 78989 
 
 39769 
 
 99248 
 
 72689 
 
 75859 
 
 93948 
 
 88349 
 
 88278 
 
 78465 
 
 59874 
 
 76846 
 
 28885 
 
 47832 
 
 57558 
 
 71635 
 
 64765 
 
 55898 
 
 86866 
 
 79385 
 
 34646 
 
 87677 
 
 18684 
 
 83467 
 
 34627 
 
 89494 
 
 65183 
 
 88688 
 
 26545 
 
 57656 
 
 64475 
 
 64874 
 
 87858 
 
 88888 
 
 43438 
 
 99438 
 
 
 
 
 48768 
 
 45766 
 
 
 
 
 62342 
 
 88344 
 
 
 
 
 68687 
 
 98888 
 
 
 
 
 85455 
 
 86454 
 
 
 
 
 68787 
 
 58888 
 
 
 
 
 77878 
 
 88788 
 
 
 
 
 VI. SUBTRACTION. 
 
 Teach subtraction by the Austrian or addition process. Tne 
 following preparation should be given while teaching the addition 
 facts: 6-f-?=9; 3+?= 9. Arrange on board or cards both horizon- 
 tally and vertically. 
 
 Bring out the subtraction idea concretely. If you had 4 apples 
 and gave me 2, how many would you have left? How many Ho you 
 
 26
 
 have to add to 2 to have 4? Then 4 — 2=? If you have 5 marbles 
 and lose 2, how many do you have left? How many do you add to 2 
 to have 5? Then 5—2=? 
 
 Give several examples like the above that have come within the 
 actual experience of the child and bring out that we may find the re- 
 sult of subtraction by adding to the subtrahend a number that will 
 give the minuend. Make constant use of the terms subtraction, minu- 
 end, subtrahend, and difference, that the children may become thor- 
 oughly familiar with their meaning. It is much better to be able to 
 use the terms accurately without hesitation than to learn the defini- 
 tion without making use of the term itself in the work. Thus, in fact, 
 the child has learned the content of most of the words he uses. 
 Comparatively few words have been actually defined for him. 
 
 For practice give exercises as follows: 
 
 1. Each digit in the subtrahend equal to, or less than, the cor- 
 responding digit in the minuend. 
 
 47948635867208 
 — 1523061 2342104 
 
 Emphasize that we add down in subtraction; that is, we add the 
 lower number, or the subtrahend, to the number that we place below, 
 or the difference, in order to give the minuend, or the upper number. 
 At first the teacher will have to question the child as to how much 
 must be added to 4 to give 8, etc. Then the child should ask himself 
 these questions, but as soon as possible he should add as follows with- 
 out any extra words: 4 plus 4 is 8; plus is 0; 1 plus 1 is 2; 2 plus 
 5 is 7; etc.; putting down result as he repeats it. Even this should be 
 abandoned and the result seen at a glance and put down without a 
 word, Not much time should be spent on this step. 
 
 2. Digits in the minuend less than the corresponding digits in 
 the subtrahend. If the results of the first exercises are written be- 
 fore the subtraction begins, the method will be seen more clearly. 
 
 734695037826 
 — 17892 3 613917 
 
 55 5 771423909 
 
 As in the case of smaller numbers, we add the subtrahend and 
 difference to obtain the minuend. Thus: 7 plus 9 is 16. In addition 
 we should put down the 6 and carry the 1, but the 6 is already written 
 in the minuend, so all we have to do after setting down the 9 is to 
 carry the 1 exactly as we do in addition. Adding the 1 that we carry 
 to the 1 in the subtrahend gives us 2 and we say, 2 plus is 2. Just 
 as in addition we add the number carried to the top addend without 
 any extra words, merely giving the result of such addition, so here 
 we merely add the amount to be carried to the subtrahend, which in 
 this case is equivalent to the top addend. We then add this result to 
 
 27
 
 the difference, which is the equivalent to the next addend. Thus, 9 
 plus 9 is 18; 4 plus 3 is 7; 1 plus 2 is 3; 6 plus 4 is 10; 4 plus 1 is 5, 2 
 plus 7 is 9; 9 plus 7 is 16; 9 plus 5 is 14; 8 plus 5 is 13; 2 plus 5 is 7. 
 
 Call attention to the fact that when the digit in the subtrahend 
 is greater than that in the minuend, we add enough to give a number 
 whose right hand digit is the digit in the minuend. That is, we can- 
 not add to 7 to give 6 therefore we add enough to give 16. When we 
 do this, we must carry the 1. After going over several examples that 
 the teacher has solved on the board, place on the board exercises to 
 be solved. 
 
 54892070318 
 — 17925375236 
 
 The first 2 will be given without difficulty. Teacher question as 
 follows: Can we add to 3 to give 1? We must add enough to give 
 how much? Ans. 11. How much must we add to 3 to give 11? 
 Teacher or child sets down 8. As the figure in the minuend is only 1 
 instead of 11, what shall we do? Ans. Carry 1. To what shall we 
 add it? Then 3 and how many make 3? How many to carry? Why 
 not? What shall we say next? Ans. 5 plus 9 is 14. Is there any- 
 thing to carry? Etc. After working several thus with the class, 
 send individuals to the board to work under the direction of the teach- 
 er with suggestions by the rest of the class; then send the whole class 
 to the board, using the "chalk and talk" method. Be sure that the 
 voice does not indicate the answer to the question whether there is 
 anything to carry. 
 
 From now on every example should illustrate both carrying and 
 not carrying in irregular order, that the child may be compelled con- 
 stantly to determine whether there is a carry or not. 
 
 As soon as possible the class should add rapidly as follows: 
 G plus 2 is 8; 3 plus 8 is 11; 3 plus is 3; 5 plus 9 is 14; 8 plus 9 is 17; 
 4 plus 6 is 10; etc. At length the result should be seen at a glance. 
 
 The advantage of the addition method is that only one set of 
 facts has to be learned instead of two. If the teacher is not watchful, 
 however, one drawback will be found: in the solution of problems 
 requiring subtraction, the subtraction idea will be lost sight of. The 
 teacher must emphasize the fact that subtraction means take away 
 and that we merely find how much to take away by means of the 
 addition facts. Simple problems in subtraction should be given from 
 the first to obviate this difficulty. 
 
 See Brown and Coffman, pp. 160-163; Walsh, Chap. Ill; Bailey, 
 Lesson 17; Smith, pp. 121-122; McLellan and Dewey, pp. 200-206. 
 
 VII. MULTIPLICATION. 
 
 No attempt to teach multiplication should be made until all the 
 addition facts have been thoroughly learned. Teaching the four fun- 
 
 28
 
 damental operations simultaneously is the great weakness as well as 
 the chief feature of the Grube method. 
 
 McMurry (p. 54) suggests the following order of teaching the 
 multiplication tables: "10s, 2s, 5s, 4s, 8s, 3s, 6s, 9s, 7s." This is an 
 improvement over the old order. 
 
 In an excellent little monograph on teaching "The multiplication 
 tables", Gildemeister uses the following order: 2s, Is, 10s, lis, 9s, 
 6s, 0s, 3s, 4s, 6s, 7s, 8s, 12s. A study of this pamphlet will repay any 
 teacher of multiplication, though it proposes a too early introduction 
 of division. 
 
 At first the facts should be found inductively. To teach 2x3 = 6, 
 give 3 plus 3 is 6, that is, two 3s are 6, or 2 times 3 is 6. 2 plus 2 
 plus 2 is 6; that is, three 2s are 6, or 3 times 2 is 6. Teach 2x3 and 
 3x2 as one fact, writing both horizontally and vertically. See p. 6 
 for method of teaching a new number fact. 
 
 Concentrate upon one fact till that is known. Sooner or later 
 the child must learn each fact as an individual fact, or he will not 
 'earn it at all. As in addition, drill by means of oral questions, cards, 
 fi;cts on board or paper, circle, etc. Bring together such facts as 
 3x4 and 2x6. Give table of squares as 2x2, 3x3, etc. Prepara- 
 tion for division, as 3x?=6, should go hand in hand with multipli- 
 cation. 
 
 When assigned facts have been written out by the pupil, a chart 
 of the multiplication tables may be placed before him that he may 
 compare his results with it to see whether his are correct. He may 
 thus make his own corrections and study upon those missed, or skip- 
 ped because he was in doubt about them. The experience of many 
 teachers has been that when the only means taken to teach the table? 
 is; by hanging the chart before the class while examples are to be 
 worked, children generally depend entirely upon the chart and do not 
 commit the facts. As indicated above, the facts should first be writ- 
 ten and the chart merely used for correction. 
 
 As soon as a few facts have been learned, they should be used in 
 examples. The teacher should at first place the examples on the 
 board and write the results as the children give them. As soon as 
 2 X 3 is reached, the following examp'e may be used: 2x213. Send 
 class to board. Time may be saved by multiplying the result of one 
 multiplication by another multiplier. 
 
 As carrying has already been taught in addition, the idea ought 
 to present little difficulty here. As an aid to carrying, daily drill 
 should be given with such exercises as 4x7 + 5; 7x8 + 6. This drill 
 should continue till the most difficult operations in multiplication can 
 be performed with facility. 
 
 See that the children use readily the terms multiplication, multi- 
 plier, multiplicand, and product. From the beginning give simple 
 problems applying the principle of multiplication. At first oral prob- 
 
 29
 
 lems should be given. When written problems of any type are intro- 
 duced, the numbers used should be as small as those used in the oral 
 problems. The child can thus give his entire attention to the new 
 difficulty of expressing in writing what he has heretofore expressed 
 orally, without being distracted by the size of the numbers which may 
 be gradually increased after he has become familiar with the method. 
 That the first written solution of a problem and the use of large num- 
 bers in solutions are both serious difficulties will be the testimony of 
 any teacher of experience. Adhere rigidly to the pedagogical princi- 
 ple to introduce but one difficulty at a time. The violation of this rule 
 inevitably leads to confusion. In the working of problems emphasize 
 that the product is always the same in kind as the multiplicand. If 
 we multiply apples, the result is apples; if we multiply marbles, the 
 result is marbles; if we multiply an abstract number, the result is an 
 abstract number. 2 x $3 should be read 2 times $3. $3x2 should be 
 read $3 multiplied by 2. 
 
 Teach early multiplication by 10 by merely annexing a cipher to 
 the multiplicand. Give much practice in applying this short process 
 and allow no other. Later give similar practice in multiplying by 
 powers of 10. Never allow a child to multiply through by zero and 
 place a row of zeroes as a partial product. 
 
 Teach multiplication by the factors of a number in varied order. 
 30 X $25 = 2 x 3 X 5 X $25 = 3 X 5 X 2 X $25 = 2 X ~ X 3 X $25. 
 
 See Brown and Coffman, pp. 160-167; Bailey, Lesson 18; Walsh, 
 Chap. IV; McLellan and Dewey, pp. 207-220. 
 
 VIII. DIVISION. 
 
 Division and Partition. Whether or not the terms division and 
 partition are both employed, the distinction in ideas implied should be 
 made. That is, in multiplication the product is the result of multi- 
 plying together two factors, one of which may be concrete an 1 at 
 least one abstract. When the process is reversed in division the pro- 
 duct becomes the dividend, one of the factors the divisor, and the 
 other factor the quotient. If $15 is the product of 3 and of $5, then 
 $15 may be divided by $5 or by 3. $15 divided by $5 equals 3, ard 
 $15 divided by 3 equals $5. Much practice should be given in deter- 
 mining whether the quotient is abstract or concrete. 
 
 After division is well in hand, it is well to check examples by 
 multiplying the quotient by the divisor. Multiplication examples also 
 should be checked by dividing the product by the multiplier, not by 
 the multiplicand as the child may then merely copy the mistakes he 
 may have made in his multiplication. 
 
 For checks of various kinds, see Brown and Coffman, Chap. IV. 
 
 Simple division problems should be used from the first. Also 
 use and have children use terms division, dividend, divisor, quotient, 
 remainder, and trial divisor. 
 
 30
 
 Teach division as the reverse of multiplication. As a prepara- 
 tion while teaching the multiplication tables have the child for ex- 
 ample tell by what we must multiply 7 to give 28. 7x ? =28. 4 X ? 
 — 28. Then when we come to division, we refer to this and bring out 
 that 28 divided by 4 gives 7. As in subtraction, so here we must 
 guard against the danger of not grasping the division idea. To ob- 
 viate this difficulty, employ many simple problems within the experi- 
 ence and grasp of the child. 
 
 In division as in addition it is of the greatest importance to take 
 one step at a time.. The following series of progressive steps has 
 worked itself out as the result of class room observation. 
 
 IX. SHORT DIVISION. 
 
 In both short and long division, place the quotient above the div- 
 idend. The first quotient figure should be directly above the right 
 hand figure of the first partial dividend. Each succeeding figure in 
 the dividend should have a corresponding figure above it in the quo- 
 tient. Insist on exact placing of quotient figures. 
 
 First teach division within the multiplication tables, (a) Divi- 
 sor exactly contained in the dividend; as, 7)14, 7)21, etc. (b) Divi- 
 sion with a remainder; as, 7)16, 7)69. etc. Before giving more diffi- 
 cult examples, give thorough drill in dividing any number below 20 
 by 2, any number below 30 by 3, etc., up to 89 divided by 9. Write 
 the examples on board or give them orally, having child give quotient 
 and remainder orally, or send class to the board to work several ex- 
 amples as rapidly as possible. The teacher may place several exam- 
 ples at each child's place before class, or she may place them on some 
 conspicuous board or chart, numbering them in regular order. Num- 
 ber the children in irregular order to prevent copying; as, 1, 4, 7, 10, 
 2, 5, 8, 11, 3, 6, 9, 12. Have each child begin with the example cor- 
 responding with his number, and as soon as he works one example, 
 begin the next one without waiting. Those beginning with the higher 
 numbers should turn back to example 1 as soon as they finish the last 
 one given. The teacher should pass around giving a "C" for each 
 correct, and a "0" for each incorrect result. An honor roll may be 
 kept of those working a certain number without error. Another de- 
 vice is to place the divisor within a circle of dividends. This device 
 should be continued for rapid drill even after formal work in short 
 division begins. 
 
 After division within the tables is well in hand, give thorough 
 drill with examples of progressive difficulty as follows: 4)4881. 
 3)30639, 7)14707, 6)121824, 8)168056, 4)487, 6)105157,4)52804, 
 3)741. Give easy divisors at first, gradually increasing their difficulty. 
 For an abundance of examples illustrating most of these type exam- 
 ples, see Woodficld, A manual on the teaching of division. 
 
 As a preparation for long division, teach short division by 11 
 
 31
 
 and 12. Teach division by 10 by merely striking off the right hand 
 figure, the left hand figures being the quotient and the right hand 
 figure the remainder. 
 
 X. LONG DIVISION. 
 
 Never allow long division with divisors less than 11. Place the 
 result above the dividend. 
 
 1. Division by 11 and 12. As a preparation use short division 
 with divisors 11 and 12. Have a child tell how he worked a certain 
 example and the teacher put down all the work that the child has 
 done mentally, calling attention to the successive steps. After work- 
 ing several examples thus, send a child to the board to go through 
 the written steps under the guidance of the teacher. Finally send 
 the whole class to the board and use the "chalk and talk" method. 
 Suppose the example to be 12) 168. How many times is 12 contained 
 in 16, Mary? Once. Where do we write the 1? Above the 6. Write 
 it, class. 1 times 12, James? Where shall we write the 12? Write 
 it, class. What shall we do next, Anna? Bring down the 8. Class 
 brings down the 8. 12 is in 48, Harold? Etc. Continue this process 
 till the class can take the steps unaided. Use examples that will give 
 but two places in the result. Continue with divisors 11 and 12 till the 
 class is familiar with the order of the steps: 1) divide, 2.) multiply. 
 3.) subtract, 4.) bring down, repeat. 
 
 2. Each quotient figure found by dividing the first digit or first 
 two digits of the partial dividend by the first digit of the divisor, 
 failed the trial divisor. (See Shutts, Handbook of Arithmetic, p. 27). 
 The following examples illustrate this step: Divisor 13, numbers 
 from 148 to 149, 156 to 159, 260-269, 273-9, 286-9; divisor 14, 140-9, 
 154-9, 168-9, 294-9; divisor 21, 231-9, 441-9, 651-9, 861-9, 252-9, 462-9, 
 672-9, 882-9, 273-9, 483-9, 504-9, 714-9, 903-9, 924-9; 22), 242-9, 
 264-9, 286-9, 462-9, 484-9, 682-9; 23), 253-9, 276-9, 483-9, 506-9, 
 713-9, 943-9, 966-9; 24), 264-9, 288-9; 31), 341-9, 651-9, 961-9, 
 372-9, 682-9, 992-9, 401-9, 713-9, 961-9, 992-9, 32), 352-9, 672-9, 
 384-9, 416-9, 704-9, 736-9; 33), 363-9, 396-9, 693-9, 726-9; 34). 
 374-9, 680-9, 714-9; 35), 420-9, 735-9, 770-9; 36), 432-9, 756-9. 
 792-9: 37), 404-9, 444-9, 777-9, 814-9; 41), 451-9, 861-9, 492-9, 
 820-9: 42), 462-9, 882-9; 43), 473-9; 44), 484-9; 21), 1071-9, 
 1260-9, 1281-9, 1470-9, 1491-9 1113-9, 1134-9, 1302-9, 1323-9, 1554-9, 
 1743-9, 1932-9, 1974-9; 31), 1271-9, 1550-9, 1581-9, 1333-9, 1643-9, 
 :922-9, 2232-9, 2573-9, 2883-9; 41), 1271-9, 1681-9, 1722-9, 2132-9, 
 2296-9, 2337-9; 51), 1071-9, 1581-9; 52), 1092-9; 61), 1281-9, 
 1891-9; 71), 1491-9. For further examples and fuller development 
 of this topic, see Woodfield, op. cit., pp. 12-22. 
 
 3. Quotient figure one less than the number of times the trial 
 divisor is contained in the trial dividend. Here emphasize the two 
 principles: (a) The subtrahend should not be greater than the par- 
 tial dividend. If it is, it is a sign that the quotient figure should be 
 
 32
 
 smaller, (b) The remainder should not be greater than, or equal to, 
 the divisor. If it is, it is a sign that the quotient figure should be 
 larger. Examples: divisor 22), 400-417, 600-615, 800-813; 23), 
 400-413, 600-619, 800-804, 810-827; 24), 400-419, 600-619, 800-815; 
 25), 400-409, 600-619, 810-824; 26), 416-419, 600-619, 820-831; 32), 
 1230-1247, 1504-1535, 1810-1823; 31), 1510-8, 1820-8; 33), 1200-1220, 
 1230-1253, 1500-1517; 43), 1610-1633, 1650-1676; 44), 2120-2155, 
 2920-2947; 46), 3960-3999; 47), 2980-3007; 54), 3000-3023. 
 
 4. Quotient figure more than one less than the number of time? 
 the trial divisor is contained in the trial dividend. Here the divisors 
 from 14 to 19 should be used. As 19 is nearer 20 in value than it is 
 to 10, take 2 instead of 1 as the trial divisor. With divisors 14 to 18, 
 take both 1 and 2 as trial divisors and try as quotient figure some 
 digit between the two results of trial. Here it must be noted that in 
 dividing such numbers as 1288 by 14, 1 is not the trial dividend as 12 
 is less than 14. In such cases, as 9 is the largest digit, we try 9 or 
 some smaller number as the trial quotient. 
 
 5. More than two figures in the quotient. Promiscuous exam- 
 ples with two figures in the divisor may now be taken up. These may 
 be found in any text book or taken from such cards as Maxson's self- 
 keyed number cards. 
 
 6. More than two figures in the divisor. In dividing by such 
 numbers as 20, 300, 4000, etc., do not allow long division. Strike off 
 as a remainder as many places in the dividend as there are ciphers at 
 the right of the significant figure of the divisor, and divide by short 
 division. If there is an extra remainder after dividing by the digit, 
 place this before the remainder struck off. 
 
 124 178 8 
 
 6/) 744/0 4/00)712/45 8/000)65/721—1721 
 
 See Brown and Coffman, pp. 167-170; Bailey. Lessons 19, 20; 
 
 McLellan and Dewey, pp. 119-143, 220-240; McMurray, pp. 60-79; 
 
 Walsh, pp. 90-105. 
 
 X. ROMAN NOTATION. 
 
 Systematize the method of teaching the Roman notation. First 
 teach the following: 
 
 1=1, 10=X, 100=C, 1000 = M. 
 
 After this is thoroughly learned, give the following: 
 2=11. 20 = XX, 200=CC, 2000 = M\I. 
 
 3=111, 30=XXX, 300=CCC, 3000 = MMM. 
 
 Next give: 
 
 5 = V, 50 = L, 500=D, 5000 = V 
 
 Follow with: 
 
 6=VI, 60 = LX, 600 = DC. 6000=V1 
 
 7=VII, 70=LX\. 700 = DCC, 7000=VII 
 
 8=VIII. 80=LXXX. 800 = DCCC". 8000 = VI II 
 
 33
 
 This completes the method by addition. Now give the method 
 by subtraction: 
 
 4 = IV, 40 = XL, 400 = CD, 4000 =Tv 
 
 9=IX, 90=XC, 900=CM, 9000=IX 
 
 The above thirty-six numbers include every possible combination 
 below 10,000. When the child can write these numbers readily, he 
 should have no difficulty with combinations of two or more of them. 
 
 Example. Write 3469 in Roman notation. 3469 = 3000 + 400 + 60 
 + 9/ .3469 = MMMCDLXIX. 1912 = MCMXII. 
 
 Ones and fives have characters of their own. Twos, threes, 
 sixes, sevens, and eights are written by addition. Fours and nines 
 are written by subtraction. 
 
 Rule: Beginning at the left, write in Roman characters the num- 
 ber represented by each digit in turn. That is, while writing the 
 equivalent to one digit, do not pay any attention to the other digits. 
 
 XI. CANCELLATION. 
 
 Teach cancellation progressively as follows. See illustrations 
 below. 1. The same numbers in dividend and divisor. 2. All num- 
 bers except 1 cancelled. 3. Numbers in dividend divisible by numbers 
 in divisor. First strike out common numbers as in first step. 4. 
 Numbers resulting from cancellation cancelled. The order here (see 
 below) is 6 and 3 divided by 3, 8 and 2 divided by 2, 12 and 4 divided 
 by 4. 5. Numbers in dividend and divisor divisible by the same 
 
 number. The order is 6 and 3 divided by 3, 8 and 2 divided by 2, 12 
 and 4 divided by 4, 15 and 10 divided by 5, 14 and 2 divided by 2. 
 6. Result in fractional form. 7. Numerator of result 1. 
 
 /• z z - j* 
 
 1 I ivlvl 0X&X0 , ^A^ . ^ 
 
 5 6 1 
 
 2 1 
 
 ? 3 3 7 J Z Z 111 
 
 ^//-< .-XJ&XH L 65 $X0XW _ 4 JXW - ± 
 
 3A8X& ~ 3'XitXt£ J 0XfXJ& 4 
 
 # J2T f 5 2. Z 
 
 When possible, in the solution of problems time should be saved 
 by cancellation. State all successive multiplications and divisions 
 without actually performing them, then solve by cancellation. As in 
 other work, this should be built up by easy steps. 
 
 34
 
 1. If 4 hats cost $12, what will 7 hats cost? At first have prob- 
 lems analyzed. If 4 hats cost $12, one hat will cost one-fourth of $12 
 or $2. Have this placed on the board as below, emphasizing that the 
 number above the line is the dividend and the number below, the di- 
 visor. Continue analysis. If 1 hat cost $3, 7 hats will cost 7 times 
 $3 or $21. This will appear on the board as below before cancellation 
 takes place. 
 
 2. If 8 caps cost $4, what will 12 caps cost? Analize and cancel 
 as before. 
 
 3. If 12 bushels of apples cost $20, what will 9 bushels cost? 
 
 4. What will 7 turkeys cost if I pay $12 for 8 turkeys? 
 
 In later grades much more difficult problems should be solved by 
 cancellation. In each new subject admitting the use of cancellation, 
 practice should be given in the statement of problems. The difficulty 
 will be found in the making of the statement, not in the cancellation. 
 
 5. How many acres in a field 80 rods by 70 rods? How shall we 
 find the area of the field? 70 times 80 sq. rd. Indicate the operation 
 If there are 70 times 80 square rods in the field, how shall we find the 
 number of acres? Divide by 160 square rods. Indicate the division. 
 Now cancel. 
 
 6. What will 914 pounds of wheat cost at 90 cents per bushel? 
 What is the missing fact? The number of bushels. How shall we 
 find it? Indicate this. How shall we find the cost of so many bush- 
 els? Indicate this. Cancel. 
 
 $12 7x$12 12 X $4 9x$20 7x$12 
 
 4 4 8 12 
 
 70X80 sq. rd. 914x90 cents. 
 
 = 35.\35A. 
 
 160 sq. rd. 60 
 
 35
 
 PART THREE 
 
 Denominate Numbers and Practical 
 Measurements 
 
 I. DENOMINATE NUMBERS. 
 
 Denominate numbers should be illustrated by exact measure- 
 ments. The child should have in his hands the foot rule, the yard 
 stick, the pint, quart, and peck measures, the pint, quart, and gallon 
 measures, and the weights to be used with the balance scales. He 
 should discover by experiment the relation between the different 
 measures and build up his own tables as far as practicable. He 
 should also discover the difference between the dry and the liquid 
 quart. He should measure and weigh various objects. 
 
 As children often have trouble in determining whether to multi- 
 ply or divide in reduction of denominate numbers, the following 
 scheme may be employed for a time. 
 
 Example: Reduce 288 inches to feet. 
 
 Write the equivalent, then write the number of inches to be re- 
 duced directly below the 12 in. As the number of feet, 1, is less than 
 the number of inches, 12, the number of feet wanted must be less than 
 the number of inches, 288, therefore divide. 
 
 12 in. = l ft. 
 288 in. = ? ft. 
 12 in. = l ft. 
 ? in. = 288 ft. 
 
 Example: Reduce 288 ft. to inches. 
 
 Here evidently the number of inches must be greater than the 
 number of feet, therefore multiply. 
 
 In this work it is important that the child be not allowed to in- 
 dicate that 288 ft. multiplied by 12 gives 3456 in., nor that 288 ft. 
 multiplied by 12 in., nor 12 in. multiplied by 288 ft. gives 3456 in. He 
 must indicate that ] 2 in. multiplied by the abstract number 288 gives 
 the desired result. 288x12 in. = 3456 in. Ans. The first example 
 should be expressed as follows: 288 in.-:- 12 in. = 24. ".24 ft. Ans. 
 
 Example: Reduce 2 pk., 5 qt., 1 pt. to pints. 
 
 If this example is worked vertically, do not allow the work to 
 appear as if 2 pecks multiplied by 8 equals 16 quarts, nor that 2 pints 
 times 21 quarts equals 42 pints. Cancel the word "qt." after the 21, 
 thus abstracting the number and consider this as the multiplier and 
 the 2 pints as the multiplicand. If the example is solved horizontally, 
 do not allow the following false statement: 2x8 qt. = 16 qt. + 5 qt. 
 
 36
 
 = 21 qt.; for 2 times 8 qt. equals 16 qt. only, not 16 qt. plus 5 qt. as 
 the statement indicates. Either make several distinct statements; as, 
 2x8 qt. = 16 qt. and 16 qt. + 5 qt. = 21 qt.; or better state as below: 
 
 8qt. 2x8qt. + 5qt. = 21qt. 
 
 2 21 X 2 pt. + 1 pt. = 43 pt. Ans. 
 
 16 qt. 
 5qt. 
 
 21 (qt.) 
 2pt. 
 
 Draw a line through this "qt. 
 
 42 pt. 
 1 pt. 
 
 43 pt. Ans. 
 
 In connection with the above horizontal statements, if it has not 
 already been explained, it is well to instruct the class that it is a gen- 
 erally accepted mode first to perform all indicated multiplications and 
 divisions in the order in which they occur, then to perform the indi- 
 cated additions and subtractions in order. If parentheses are used, 
 operations indicated within should first be performed. 
 
 See Walsh, pp. 129-140; Bailey, Lesson 23; Brown and Coffman, 
 Chap. XII. 
 
 II. SQUARE MEASURE. 
 
 Example: Find the area of a surface 3 in. by 5 in. 
 
 3X5 sq. in. = 15 sq. in. Ans. 
 
 As there are 5 square inches in the upper row and there are 3 
 rows, there are 8 times 5 sq. in. in the whole surface. 
 
 37
 
 In solving such problems, have the class make the illustrative 
 diagrams until they thoroughly understand the process. Under no 
 circumstances allow them to say 3 in. times 5 in. 
 
 Example: If a rectangle 5 inches long contains 15 square inches, 
 how wide is the rectangle? 
 
 15 sq. inn- 5 sq. in. = 3. '.3 in. Ans. 
 
 If there are 5 sq. in. in one row there will be as many rows as 5 
 sq. in. is contained times in 15 sq. in. 
 
 In developing the number of square inches in a square foot, use 
 the above method and have each child make a diagram of a square 
 foot or 144 square inches on paper. Thus the square yard also should 
 be developed and placed on the board. 
 
 III. CUBIC MEASURE. 
 
 Have several inch cubes, several boxes with the inside dimen- 
 sions in exact inches if possible, and one box exactly one cubic foot 
 inside dimensions. 
 
 Take a box, say 3 in. by 4 in. by 5 in. Place n inch cubes in the 
 box in a row along one edge. Bring out that there would be 4 rows 
 of 5 cubic inches in one layer of blocks, or 4 times 5 cu. in. or 20 cu 
 in.; that in three such layers there would be 3 times 20 cu. in. or 60 
 cu. in. 
 
 3x4x5 cu. in. =60 cu. in. Ans. 
 
 Similarly develop the number of cubic inches in one cubic foot. 
 
 IV. PRACTICAL MEASUREMENTS. 
 
 In practical measurements the operations should when possible, 
 be reduced to a series of steps to be taken in regular order. These 
 steps in turn may often be reduced to a set form. 
 
 V. PERIMETER-FENCES. 
 
 Bring out that Perimeter means the measure around. To find 
 the perimeter of a rectangular figure 8 in. by 12 in., add the length of 
 two adjacent sides: 12" + 8" and multiply by 2. 2(12" + 8") =40". 
 the Perimeter. 
 
 Teach at once that a number before or after a parenthesis indi- 
 cates that the result of the operations within the parenthesis should 
 be multiplied by the number or numbers without. 
 
 In dictating examples in fencing, etc., have the class write the 
 following blank on the board to be ready to take down the numbers 
 as dictated: 2( -J-). As the example is dictated, the numbers are 
 placed in the blank spaces. This blank will be found useful in de- 
 termining the amount of fencing required to fence in a lot. 
 
 VI. AREA-ACRES. 
 
 The method of computing areas has already been given in con- 
 nection with square measure. 
 
 38
 
 How many acres in a farm 80 rods by 140 rods? 
 X 80 X 140 
 
 = .'. A. Ans. = 70. '.70 acres. Ans. 
 
 160 160 
 
 Making use of the above blank, the teacher should dictate many 
 examples for rapid computation. Insist on cencellation where 
 possible. 
 
 VII. PLASTERING. 
 
 Take a chalk box about 8 inches by 4 inches by 4 inches. Wha* 
 is the perimeter of this box? Open the box out so as to make the 
 four sides one plane surface. What is the length of the four sides? 
 The same as the perimeter, 2(8" + 4") =24". What is the area of 
 the four sides? 2(8 + 4)4 square inches. Dictate many examples for 
 practice in finding the areas of four sides of rooms. As the area of 
 these four walls has been found in square feet, how shall we find the 
 number of square yards of plastering? 
 
 Bring out the fact that the openings for doors and windows do 
 not have to be plastered, but state that on account of the difficulty of 
 plastering around openings plasterers generally deduct only half of 
 the openings. Only such deductions should be made as are directed 
 in each example. Develop the following blanks and have the class 
 place them on paper or on the board before any computations are be- 
 gun. Insist upon the placing of one equality sign directly under 
 another. 
 
 2( + ) — = .'. sq. yd. 
 
 9 
 
 X = .-. 
 
 X $ = $ Ans. 
 
 As the teacher dictates or as the child reads, he places the figures 
 in the appropriate blanks. The computations aside from mental pro- 
 cesses and cancellations should be performed below the line and the 
 results extended properly. 
 
 Example: At 30 cents per square yard what will it cost to 
 plaster a room 18' x 22' X 10', allowing one-half of 100 sq. ft. for the 
 openings? 
 
 2(18 + 22)10—50 
 
 =83 13 number of sq. yds. 
 
 =44 number of sq. yds. 
 
 9 
 18X22 
 
 9 
 Total 127 1 3 sq. yds. 
 
 127 1 3X$.30 =$38.20. Ans. 
 
 39
 
 In the second statement above, cancellation should be used when 
 possible. Give abundant practice. 
 
 VIII. CEMENT WALKS, PAVING, ETC. 
 
 Use the second statement in form for plastering. When, how- 
 ever, a walk is placed on two or more sides of a lot, an allowance has 
 to be made for one or more corners; one if on two sides, two if on 
 three sides, and four if on four sides. If the walk is on the lot, the 
 corners are deducted; if around the outside of the lot, the corners are 
 added. Illustrate by diagram. 
 
 Example: A. Find the number of square yards in a 4 foot ce- 
 ment walk on two sides of a lot 80 feet by 40 feet. B. Around two 
 sides. 
 
 A. 4(80 + 40 — 4)^-9 = 51 5/9 number of sq. yd. Ans. 
 
 B. 4(80 + 40 + 4) -^9 = 55 1/9 number of sq. yd. Ans. 
 
 IX. PAINTING. 
 
 For rectangular rooms use the same form as in plastering. For 
 irregular buildings find the total area in square feet before dividing 
 by 9. 
 
 X. PAPERING. 
 
 The length of a roll of wall paper is 24 feet. Of a double roll, 
 48 feet. The width is generally 18 inches. What is the area of a 
 single foil? Of a double roll? If we have the area of the surface to 
 be covered, and know the area of a roll of paper, how shall we find 
 the number of rolls? Is it possible to use all the paper of a roll? 
 Some paper hangers, allowing for waste, figure upon three single 
 rolls covering 100 square feet of surface. This is allowing about 33 
 square feet to the roll. Therefore divide the area to be covered by 33 
 square feet. For double rolls divide by 65 or 66. Salesmen of expe- 
 rience assert that when this method of computation is used, seldom 
 if ever is paper returned or more called for. The strip method of 
 computation is cumbersome and antiquated. 
 
 Example: At 75 cents per double roll what will it cost to paper 
 a room 18' X 22' X 10', allowing 100 square feet for openings? 
 • » 
 2(18 + 22)— 100 
 
 =10+. Ml rolls. 
 
 65 
 
 18X22 
 
 = 6+.'. 7 " 
 
 65 
 
 Total 18 rolls. 
 
 18X$.75 =$13.50. Ans. 
 
 40
 
 If an unusual width of paper is used, a different divisor must be 
 used. 
 
 Before beginning an example in papering, the following blank 
 should be placed on the board or paper: 
 
 2( + ) — 
 
 = .'. rolls. 
 
 33 
 
 X 
 
 rolls. 
 X $ = $ Ans. 
 
 For double rolls, substitute 65 or 66 for 33. Divisors 66 2 A and 
 33^5 would be evon better. To divide by 66^, divide by 2 /$ and poinr. 
 off two places. To divide by 33^, divide by 1/3 and point off two 
 places. 
 
 XI. CARPETING. 
 
 Though in many homes, for sanitary as well as for artisti'; 
 reasons, rugs are taking the place of carpets, capets are still gener- 
 ally used. 
 
 PREPARATION. 
 If this room is 18 feet wide, how many yards wide is it? How 
 many yard sticks placed end to end would cross the room? How 
 many measures two feet long? 6 feet long? How do you find the 
 number of measures each time? By dividing the width of the room 
 by the length of the measure. 
 
 STATEMENT OF AIM. 
 Let us see how to find the number of strips of carpet required to 
 cover a floor. 
 
 PRESENTATION. 
 Place a strip of figured wall paper on the floor. If this paper 
 were 2 feet wide, how many strips shou'd I lay side by side to cover 
 the floor? If it were 3 feet wide? 6 feet wide? 
 
 COMPARISON AND ABSTRACTION. 
 
 In each case how did you find the number of strips necessary? 
 By dividing tae width of the room by the width of the strip. 
 
 GENERALIZATION. 
 
 Then if you wished to carpet any room, how could you find the 
 number of strips required? To find the number of strips, divide the 
 width of the room by the width of a strip. Have children commit. 
 
 41
 
 APPLICATION. 
 
 If the room is 6 yards wide and the strips M yard wide, how shall 
 we find the number of strips? Divide 6 yards by }i yard. 
 6-- 34 =6 X 4/3 = 8. \8strips. 
 
 If a room is 17/3 yards wide and the carpet % of a yard wide, 
 find the number of strips. 
 
 Would a clerk cut a strip lengthwise for you? Then how many 
 strips would you be required to buy? Give several similar examples. 
 
 Note. As carpet is purchased by the yard, it is necessary to re- 
 duce the length of the room to yards, therefore for the sake of uni- 
 formity, reduce all dimensions to yards. As the width of the room 
 is generally reducted to an improper fraction before division by ?4, 
 leave this dimension in the form of an improper fraction of a yard. 
 
 If a room is 19 feet wide, what must we do with the 19 feet be- 
 fore we divide by Ya yard? Ans. Reduce 19 feet to yards. 
 
 19/3 yds. ^?4 yds. = 19/3x4/3 = 76/9 = 8 + .-.9 strips. Give 
 similar examples. 
 
 TO FIND THE NUMBER OF YARDS. 
 
 If there are 9 strips each 8 yards long, how many yards of carpet 
 will be required to cover the floor? Etc. 
 
 If the room is 24 feet long, what do you do to the 24 feet before 
 multiplying? Ans. Change it to yards. Give examples. 
 
 WASTE. 
 
 To illustrate that there is generally waste on all but the first 
 strip, use the figured wall paper. 
 
 If there is a waste of 1 foot on a strip, how many yards waste? 
 Ans. 1/3 yard. 8" waste? 8/36 yd. = 2/9 yd. Etc. 
 
 If there are & strips and a waste of 10 inches on a strip, how 
 much waste in all? 7x5/18 yds. = 35/18 yds. = l 17/18 yd. Etc. 
 
 If it takes 56 yards of carpet to cover the floor and there is a 
 waste of 2 2 /i yds. in matching, how much carpet must be purchased? 
 Etc. 
 
 ORDER OF STEPS. 
 
 I. Place on diagram all dimensions in yards. 
 
 II. Find the number of strips by dividing the width of the room 
 by the width of one strip. 
 
 III. Find the number of yards required to cover the floor by mul- 
 tiplying the length of a strip by the number of strips. 
 
 IV. Find the number of yards waste by multiplying the waste 
 on a strip by the number of strips less one. 
 
 V. Find the total number of yards that must be purchased by 
 adding the last two items. 
 
 VI. Find the cost. 
 
 Where there is no waste, omit steps IV and V. 
 
 42
 
 Before dictation or work on paper begins, the following blank 
 should be written: 
 I. 
 
 yd. 
 
 yd, 
 
 yd. 
 
 waste 
 
 yd. 
 
 II. 
 
 
 
 = .'. strips. 
 
 III. 
 
 X 
 
 yd. 
 
 yd. 
 
 IV. 
 
 X 
 
 yd. 
 
 yd. 
 
 V. 
 
 Total 
 
 
 yd. 
 
 VI. 
 
 X$ 
 
 
 = $ Ans 
 
 Dictation should proceed slowly enough to allow the change of 
 all dimensions to yards as the numbers are given. 
 
 Example: At $1.25 per yard what will be the cost of a 27 in. 
 carpet for a room 17 ft. by 23 ft., allowing 8 in. waste for matching 
 a strip? 
 
 OPERATION. 
 
 I. 
 
 17 3 yd. 
 
 23 3 yd. 
 
 3-1 yd. 
 
 waste 8 36 yd. =2 9 yd. 
 
 II. 
 
 17 4 68 
 
 
 
 3 3 9 
 
 = 7 + . ' .8 strip 
 
 III. 
 
 23 
 8X— yd. 
 
 3 
 
 = 61 1 3 yd. 
 
 IV. 
 
 2 
 
 7X— yd. 
 
 9 
 
 5 
 = 1- yd. 
 
 V. 
 
 Total 
 
 62 8 9 yd. 
 
 VI. 
 
 8 
 62- X $1.25 
 9 
 
 =$78.61. Ans 
 43
 
 XII. BOARD MEASURE. 
 
 To illustrate one board foot present two boards each 6" X 12" X 1. 
 Placing these side by side we have a board l'Xl'X 1", or 1 board 
 foot. Placing them end to end we have a board 2'x6"Xl" still 1 
 board foot. Placing one flat upon the other we have a board 1' X6" X 
 2" again 1 board foot. Thus develop rule: to find the number of 
 board feet multiply the number of feet long by the number of feet 
 wide by the number of inches thick, or multiply the length in feet by 
 the width in inches by the thickness in inches and divide by 12 to 
 reduce the width to feet. To find the number of thousand, point off 
 three places. It is generally best not to cancel the thousand. 
 
 XIII. TO FIND THE NUMBER OF BUSHELS IN A BIN. 
 
 Approximately 1 bushel equals 5/4 cubic feet, or 1 cubic foot 
 equals 4/5 of a bushel; hence to find the number of bushels in a bin, 
 multiply 4/5 of a bushel by the number of cubic feet in the bin. Ap- 
 proximately how many bushels in a bin 10 ft. by 4 ft. by 6 ft. 
 10x4x6x4/5 bu. = 192 bu. Ans. 
 
 Accurately 2150.42 cu. in. = l bu. ; hence to find the number of 
 bushels in a bin, reduce the number of cubic feet to cubic inches and 
 divide by 2150.42 cu. in. 
 
 (10x4x6x1728 cu. in.) -s- 2150.42 cu. in. = 192.85/. 192.85 bu. 
 
 XIV. TO FIND THE NUMBER OF GALLONS IN A TANK 
 
 OR CISTERN. 
 
 Accurately 231 cu. in. = l gallon. To aid in cancellation, have 
 children commit the factors of 231, i. e. 3, 7, and 11. In all of these 
 examples, the numbers in the curves should be placed above the line 
 and the divisor below the line and cancellation used where possible. 
 How many gallons in a tank 22 in. by 15 in. by 14 in.? 
 
 (22 x 15X14) -231 = 20/. 20 gal. Ans. 
 
 Approximately 1 cu. ft. = 7^ gallons. How many gallons of 
 water in a cistern 8 feet in diameter if the water is 7 feet deep? 
 
 22/7 X 4 x 4 x 7 X 15/2 gal. = 2640 gal. Ans. 
 
 44
 
 PART FOUR 
 A. Common Fractions 
 
 For suggestions regarding teaching fractions see Brown and 
 Ooffman, How to teach Arithmetic, Chap. XIII, Bailey, op. cit. Lesson 
 21, Walsh, op. cit. Chap. II, McLellan and Dewey, op. cit. Chap. XIII. 
 
 I. PROPER FRACTION TAUGHT OBJECTIVELY. 
 
 A. ]/>, l A, 14, etc. B. 2 /i, }i, etc. See especially Brown and 
 Coffman, pp. 183-4. The chief thing to be taught in class is the writ- 
 ing of the fraction and what is meant by the numerator and the 
 denominator, the denominator indicating into how many parts a 
 number has been divided, the numerator indicating how many of 
 these parts have been taken. Paper cutting will be found as good a 
 method of illustration as any. The terms numerator and denominator 
 should at once be defined and used. Emphasize that the denominator 
 is the name of the fraction, that is, the name of the parts, and tells 
 the size of the parts or into how many parts the number has been 
 divided. Show from the first that the larger the denominator the 
 smaller the parts: y 2 is larger than Yi, which, in turn, is larger than 
 Va, etc. 
 
 II. FINDING A FRACTIONAL PART OF A NUMBER. 
 
 Y> of $12, etc. 
 
 This step is given in connection with division and should dissi- 
 pate the notion that a fraction merely indicates one or more parts of 
 a single thing. Thus J4 of a number may indicate $6 as well as Yi 
 of a dollar. 
 
 HI. THE IMPROPER FRACTION. 
 
 Illustrate by piling books improperly, large upon small books. 
 
 IV. A FRACTION AN INDICATED DIVISION. 
 
 Emphasis should be laid upon the fact that a fraction is an indi- 
 cated division. That the dividend is always above the line and the 
 divisor below. Have children state which term of a given fraction is 
 the dividend and which is the divisor. Have them write in fractional 
 form 12 divided by three, etc. Give thorough drill till firmly fixed in 
 mind. Here also the principle that the larper the denominator the 
 smaller the fraction should again be emphasized and illustrated by 
 such fractions as 12/3 = 4, 12/4 = 3, etc. 
 
 45
 
 V. TO EXPRESS AN INTEGER AS A FRACTION. 
 
 Call on children to tell how many halves in 1, in 2, etc., using 
 ohjective illustration if necessary. 
 
 1 = 2/2 = 3/3 = 4/4, etc. 2 = 4/2 = 6/3 = 8/4, etc. 
 
 VI. REDUCTION OF A FRACTION TO AN EQUIVALENT 
 FRACTION WITH A LARGER DENOMINATOR. 
 
 Fold two equal strips of paper, one into halves, the other into 
 fourths. Call attention to the fact that the one half is equal to two 
 fourths and write on the board. 1/2 = 2/4. Similarly show that 
 1/2 = 3/6, 1/3 = 2/6, 2/5 = 4/10. 
 
 By comparison bring out that in each case both numerator and 
 denominator may be multiplied by the same number without chang- 
 ing the value of the fraction. Then generalize that multiplying both 
 numerator and denominator of a fraction by the same number does 
 not change the value of the fraction. Give thorough drill such as: 
 Multiply both terms of 2/3 by 4. What fraction results? What can 
 you say of the relative value of 2/3 and 8/12? Why? Etc. 
 
 SUB LESSON— TO REDUCE A FRACTION TO AN EQUIVALENT 
 FRACTION WITH A GIVEN DENOMINATOR. 
 Reduce 3/5 to an equivalent fraction with the denominator 20. 
 3/5=?/20. By what do you multiply 5 to obtain 20? Then if you 
 multiply the denominator by 4, what must you do to the numerator 
 in order not to change the value of the fraction. Give a large amount 
 of drill in such reduction. 
 
 VII. REDUCTION TO LOWEST TERMS. 
 
 Principle to be developed: Dividing both numerator and denom- 
 inator by the same number does not change the value of the fraction. 
 Either present objectively as in the last case, or reverse the former 
 process. 
 
 S/4 = 6/8/.6/8 = 3/4; 2/3 = 6/9.\6/9 = 2/3; 1/5 = 2/10.'. 2/10 = 1/5. 
 What do we do to 3/4 to obtain 6/8? Then what might we do 
 to the 6/8 in order to reduce back to 3/4? Do we change the value 
 of the fraction? Similarly for 6/9 and 2/10. Compare, generalize, 
 and drill. Also give drill in reduction to equivalent fraction with 
 given smaller denominator; as, reduce 8/12 to sixths. Also give drill 
 in reduction requiring first reduction to lower then to higher terms; 
 as, reduce 8/12 to ninths. 8/12 = 2/3 = 6/9. First reduce to lowest 
 terms; reduce result to required fraction. 
 
 VIII. ADDITION AND SUBTRACTION OF SIMPLE FRACTIONS 
 WITH A COMMON DENOMINATOR. 
 
 Two apples plus three apples = how many apples? Etc. Two 
 sixths + three sixths = how many sixths? 2/6 + 3/6 = 5/6, etc. 
 
 46
 
 IX. REDUCTION OF A MIXED NUMBER TO AN IMPROPER 
 
 FRACTION. 
 
 Preparation — Reduction of a whole number to a fraction, and 
 addition of fractions. Let us see how to reduce 3 2/5 to fifths. How- 
 many fifths in 3? How many fifths in 15/5 and 2/5? Then to how 
 many fifths is 3 2/5 equal? 17/5. Similarly for 2 1/3 and 5 1/2. 
 Compare, generalize, drill. 
 
 X. REDUCTION OF AN IMPROPER FRACTION TO A WHOLE 
 
 OR TO A MIXED NUMBER. 
 
 Preparation — A fraction is an indicated division, and division 
 with a remainder. 
 
 Presentation — Reduce 8/2 to a whole number, etc. Reduce 9/2 
 to a mixed number. 9/2 = 4 with a remainder of 1. What were the 
 nine? Ans. Halves. Then if one of the nine halves remains, what 
 is it? Ans. 1/2. Then 9/2 = what? Ans. 4 1/2. 
 
 XI. REDUCTION TO A COMMON DENOMINATOR WHEN ONE 
 DENOMINATOR IS A MULTIPLE OF THE OTHER AND 
 
 ADDITION AND SUBTRACTION OF SUCH FRACTIONS. 
 
 How do we add three qts. and one pt. ? Change 1/2 to fourths. 
 Let us see whether we can add 3/4 and 1/2. Why not? How did we 
 add 3 qts. and 1 pt. ? To what did you say 1 2 is equal? Can you 
 add it to 3/4 now? Similarly add 2/3 and 1/6, 1/5 and 3-10, etc. 
 Compare, generalize, and drill. Have examples of this type added 
 mentally as soon as possible. 
 
 XII. REDUCTION TO LEAST COMMON DENOMINATOR WHEN 
 TWO OR MORE DENOMINATORS HAVE A COMMON FACTOR 
 AND ADDITION AND SUBTRACTION OF SUCH FRACTIONS. 
 
 TWO FRACTIONS. 
 
 5/12 + 7/8. See illustration below. Divide 12 and 8 by 4, th? 
 largest number that will divide both, giving 3 and 2. Multiply the 
 numerator and denominator of the first fraction by the 2. giving thr 
 equivalent fraction 10/24; the numerator and denominator of thr 
 second fraction by 3, giving 21/24. In this and in previous cases 
 emphasize the idea of equivalent fractions. For a time it is well to 
 have children draw the connecting lines as guides in multiplication. 
 
 THREE OR MORE FRACTIONS. 
 5/12 + 3/16 + 7/18. Teach this in the old way. See illustration 
 below. Note that the divisor, in finding the least common denomina- 
 tor, should be a prime number unless it will divide all thr denomina- 
 tors. For example, if 4 had been used in the above example instead 
 of 2 and 2, the factor 2 would not have been eliminated from 18 and 
 
 47
 
 the common denominator 288 would have been found instead of the 
 least common denominator 144. Multiply 2x2x3x4x3 = 144, the 
 new denominator. Divide 144 by 12 and multiply 5 by the result, etc. 
 
 XIII. REDUCTION TO A COMMON DENOMINATOR WHEN ALL 
 
 DENOMINATORS ARE PRIME TO ONE ANOTHER AND 
 
 ADDITION AND SUBTRACTION OF SUCH 
 
 FRACTIONS. 
 
 A. TWO FRACTIONS. 
 2/3-j-i/5=? See illustration below. The ordinary method of 
 teaching this step is to multiply the 3 by 5, then to divide the 15 by 3 
 to find 5 which is already before us; and to divide 15 by 5 to obtain 3. 
 This is waste of time. Simply multiply 2 and 3 by 5, and 1 and 5 by 
 3. At first connect the numbers to be multiplied by lines and have 
 the children do the same, but this should soon be dropped. 
 
 M-'r^rs ukkkm+m+m 
 
 3^T 
 
 B. THREE OR MORE FRACTIONS. 
 5/7 -|- 4/9 _|- 2/5. Here instead of dividing the common denomi- 
 nator, 315, by 7, 9, and 5 in turn to get 45, 35, and 63, obtain these 
 numbers by multiplying together the appropriate denominators, 9 and 
 5, 7 and 5, 7 and 9. The numerators of the equivalent fractions may 
 be found thus: 5(9x5) =225, 4 (7x5) =140, and 2(7x9) =126. 
 
 XIV. ADDITION OF MIXED NUMBERS. 
 
 A. Sum of two fractions— 1. 17^ + 1934+12^. 
 
 Add mentally 2/3 + 1/3 = 1, 1 plus 3/4 = 1 3/4. Write -4 and 
 carry 1, etc. Drill should be given so that children will seek the 
 shortest method. 
 
 B. Simple fractions that can be added mentally. 5^+7 + 8^4, 
 
 etc. 
 
 C. More difficult fractions. 
 
 See illustrations below. Before beginning work, write the word 
 "Ans." to the right of the place where the result will finally appear. 
 To the right of this draw a vertical line to keep the numerators of 
 the equivalent fractions in a column. In the last example note that 
 the denominator 6 is contained in the denominator 12, therefore find 
 the least common denominator of 12 and 9. Divide either one of the 
 two by 3, the largest number that will divide both, and multiply the 
 other by the result, 9x(l2-^3)=36, or 12x(9-^3)=36. As soon 
 
 48
 
 as the least common denominator is found, place it below the line and 
 to the right of the word "Ans.", leaving room above it for the sum of 
 the numerators. Do not allow the pupils to place the mixed num- 
 ber 11 5/9 as equal to 20/36. They are not equal and it is a waste of 
 time to write the 36 more than once before placing it in the answer. 
 So, too, the numerators are more easily added if the denominators do 
 not intervene. Reduce 5/9 to 36ths and write the numerator 20 to 
 the right of the vertical line and opposite the numerator 5, and have 
 the fraction read as 20/36. Similarly write the numerators 21 and 6, 
 then adding the three numerators place the result above the common 
 denominator 36. The children may then write the equivalent of 
 47/36 in its simplest form. Place the fraction 11/36 in the space re- 
 served for the answer directly below the original fractions and carry 
 the 1 and add the integers. As much of the work as possible should 
 be performed mentally. If impossible to find the denominator men- 
 tally, place the denominators at one side and find the least common 
 multiple; however, there is little value in working with difficult de- 
 nominators. 
 
 19 6 7 18 11 5 9 20 
 
 12 2 3 14 16 7 12 21 
 
 17 5 7 15 9 1/6 6 
 
 5£ 5/21 Ans. 47/21=2 5 21. 37 1 1 36 Ans. 47 36 = 1 11 36. 
 
 XV. SUBTRACTION OF MIXED NUMBERS. 
 
 The following seven type examples are arranged progressively. 
 When possible have them solved mentally: 1. Nothing subtracted 
 from a fraction. 2. One denominator divisible by the other. 3. Two 
 denominators having a common factor. Divide 12 by the greatest 
 common divisor 4, and multiply both 16 and 7 by the resulting 3. 
 3x16 = 48 (least common denominator), and 3x7 = 21. The other 
 numerator may be similarly found: 5x (16-5-4) =20. 4. No common 
 factor. 5, 6, and 7 require "borrowing"; or add 1 to both minuend 
 and subtrahend. In 7, add 1 or 35/35 to 15/35 giving 50/35, from 
 which we subtract 28/35. Also add 1 to 401 making it 402 and then 
 subtract. 
 
 1. 2. 3. 4. 
 
 24 3/4 17 2 3 8 7/16 21 48 2 3 
 
 7' 5 5/9 5 5 12 20 17 12 
 
 17 3/41 Ans. 1I[12 1 9lAns. 3 1 48 Ans. 1 48. 31 1 6 Ans. 
 
 5. 6. 7. 
 
 35 
 
 44 44 411 3 7 15 
 
 50 
 3 5 17 3 5 4014 5 28 
 
 43 2,5 Ans. 26 2 5 Ans. 9 22 35 Ans. 22 35. 
 
 49
 
 XVI. MULTIPLICATION OF A FRACTION BY AN INTEGER. 
 
 A. Multiplication of the numerator. 
 
 2x3 apples = 6 apples, etc. 2x3 sevenths = 6 sevenths, etc. 
 2x3/7 = 6/7, etc. Principle generalized: Multiplying the numera- 
 tor multiplies the fraction. 
 
 B. Division of the denominator. 
 
 2x3/8 = 6/8 = 3/4, etc. Teacher erase the 6/8. What can you 
 say of the numerator of the original fraction and the numerator of 
 the resulting fraction? How can you obtain the 4 from the 8 and 2 1 
 Similar questioning for other fractions. Principle induced: Dividing 
 the denominator of a fraction multiplies the fraction. Induce and 
 have children commit thoroughly to memory: To multiply a fraction 
 divide the denominator or multiply the numerator. Always divide if 
 possible to save the time of reduction to lowest terms. Give much 
 drill in multiplication of simple fractions to develop quickness in de- 
 cision as to whether to divide or to multiply. This will later be a 
 great saving of time. 
 
 Sub step. . Multiplication of a fraction by an integer that is iden- 
 tical with the denominator of the fraction. 4x1/4 = 1, 4x3/4 = 3, 
 15x11/15 = 11, etc. Have the answer written at once without writ- 
 ing first 3/1 and 11/1. Give much drill till the children can give re- 
 sults without the slightest hesitation. 
 
 C. Cancellation. 
 
 Base on earlier work in cancellation of whole numbers. 12x5/16, 
 etc. Where the denominator is exactly divisible by the integer or 
 the integer is exactly divisible by the denominator, have the work 
 performed mentally without cancellation. 
 
 XVII. MULTIPLICATION OF AN INTEGER BY A FRACTION. 
 
 Teach by analysis. Find 2/3 of $6. Solve orally at first. 1/3 
 of $6 is $2; 2/3 of $6 is 2 times $2, (1/3 of $6), or $4. Give much 
 oral practice, then solve on the board, 2/3 X $6 = $4, accompanying 
 the solution with oral analysis. Rule developed: To multiply by a 
 fraction divide by the denominator and multiply by the numerator. 
 When the denominator is not exactly divisible, cancel if possible. The 
 order of these operations should be determined by the relation of the 
 integer to the terms of the fraction. In 3/4 of 8, divide first; in 3/5 
 of 8, multiply first. Emphasize that when 2/3 of 6 is found, 6 is 
 multiplied by 2/3. This understanding is essential to future progress. 
 
 XVIII. MULEIPLICATION OF A FRACTION BY A FRACTION. 
 
 Use previous cancellation as preparation. 
 
 XIX. MULTIPLICATION OF A MIXED NUMBER BY A 
 
 MIXED NUMBER. 
 
 This is the only case of multiplication where reduction to im- 
 proper fractions is preferable. But see Bailey, op. cit., p. 101. 
 
 50
 
 XX. DIVISION OF A FRACTION BY AN INTEGER. 
 
 Present as in multiplication of a fraction by an integer. 15 
 apples-^3 = 5 apples. 15/16-r-3 = 5/16. A. Principle: To divide tho 
 numerator divides the fraction. B. Principle: To multiply the de- 
 nominator divides the fraction. Refer to the principle: The larger 
 the denominator the smaller the fraction. If we make the denomi- 
 nator 4 times as large, how is the fraction affected? How many 
 times smaller? Then multiplying the denominator does what to the 
 fraction? 5/6 -f- 4 = 5/24. Develop and have children commit the 
 rule: To divide a fraction by an integer divide the numerator or 
 multiply the denominator. 
 
 Also develop the principles: A change in the numerator pro- 
 duces a like change in the value of the fraction, a change in the de- 
 nominator produces an opposite change in the value of the fraction. 
 Cive much practice in working examples requiring judgment as to 
 whether it is bettei to divide or to multiply numerator or denomina- 
 tor or whether to cancel. 
 
 XXI. DIVISION OF AN INTEGER OR OF A FRACTION 
 
 BY A FRACTION. 
 
 Here it is probably best, though not essential, to teach the in- 
 version of the divisor. With young children the dictum of the 
 teacher will doubtless be more effective than a philosophical expla- 
 nation. 
 
 XXII. DIVISION OF A MIXED NUMBER BY AN INTEGER. 
 
 A. When the mixed number is small, reduce to an improper 
 fraction and divide. 
 
 B. When the mixed number is large, divide directly and reduce 
 the remainder to an improper fraction and complete the division. 
 
 2)1729^=861 2)1729^=864^. The remainder 1 
 
 should mentally br reduced to 5/3 and this mentally divided by 2, 
 which gives 5/6. 4/3 divided by 2 should at once give 2/3, not 4/6. 
 In such examples never allow reduction to an improper fraction. 
 
 XXIII. DIVISION OF A WHOLE NUMBER BY A MIXED 
 
 NUMBER. 
 
 When the divisor is not an aliquot part, multiply divisor and 
 dividend by the denominator of the fraction and divide the resulting 
 numbers. Basic principle: To multiply or divide both dividend and 
 divisor by the same number does not change the quotient. This prin 
 ciple should be developed from the principle that multiplying both 
 numerator and denominator of a fraction by the same number does 
 not change the value of the fraction; and from the fact that a frac- 
 tion is an indicated division. 
 
 51
 
 In the example (1431-=- 175^), what can we do to both dividend 
 and divisor without changing the quotient? Then multiply both by 3 
 and divide results. 
 
 At first it may be well to place in fractional form till the relation 
 is clearly seen. Then place as below. At first allow the children to 
 put down the multiplier, but as soon as possible have them remember 
 that the multiplier is the same as the denominator of the fraction. 
 
 17^)1431 Multiply both by 3, placing result two lines below. 
 
 81 Ans. 
 53 )4293 
 424 
 53 
 53 
 
 XXIV. DIVISION OF A MIXED NUMBER BY A MIXED NUMBER 
 
 Change to improper fractions and divide. 
 
 XXV. MULTIPLICATION OF A MIXED NUMBER BY A 
 
 FRACTION. 
 
 Multiply by the numerator and divide by the denominator. 
 4782 1/7 x 2/3. See below. 
 
 XXVI. MULTIPLICATION OF AN INTEGER BY A MIXED 
 
 NUMBER. 
 
 Do not allow reduction to an improper fraction. 
 
 9564 2/7 434 279 
 
 4782 1/7 
 2/3 
 
 217 
 17 2/3 
 
 4783 3/7 
 93 
 
 3188 2/21 Ans. 
 
 144 2 3 
 1519 
 217 
 
 39 6/7 
 14349 
 43047 
 
 
 3833 2/3 Ans. 
 
 444858 6 7 Ans. 
 
 XXVII. DIVISION OF A MIXED NUMBER BY A FRACTION. 
 
 Time will generally be saved by inverting the divisor and multi- 
 plying as in multiplying a mixed number by a fraction. 28 3/7-*- 2/3 
 = ? Multiply by 3/2 as above. 
 
 At some period the following comparisons should be made: To 
 multiply a fraction by an integer, multiply the numerator OR divide 
 the denominator. To multiply by a fraction multiply by the numera- 
 tor AND divide by the denominator. 
 
 52
 
 B. Decimal Fractions 
 
 I. MEANING. 
 
 1/10 = . 1, 2/10 = . 2, 9/10 = .9, etc. 
 45/100 = .45, 1/100 = . 01, 7/100 = .07, etc. 
 215/1000 = . 215, 9/1000 = .009, 17/1000 = .017, etc. 
 There are as many places to the right of the decimal point as 
 there are ciphers in the denominator of the common fraction. 
 
 II. READING AND WRITING. 
 
 At first as the teacher dictates, the class should write on the 
 board in the common fraction form and then in the decimal form. 
 The ciphers will guide them in placing the decimal point. Give much 
 practice in reading and writing to thousandths. Beyond thousandths 
 "numeration" may be necessary. 0.001,786 — tenths, hun- 
 dredths, thousandths, ten-thousandths, hundred-thousandths, million- 
 ths — 1,786 millionths. See that the -ths is annexed in all cases. 
 Insist on correct wording when reading beyond thousandths. Do not 
 allow the use of the word "of" in reading .4,612. Read exactly as 
 the common fraction 4,612/10000 is read. Emphasize the use of the 
 word "and" to indicate location of the decimal point, and the use of 
 the decimal point to locate units. It is well to teach both of the fol- 
 lowing forms: 0.721 and .721, as both are sooner or later met with. 
 First teach reading, then writing of many such numbers as the fol- 
 lowing: 1,000,000.000,001; 78,000,004.06,078; 317,009,000.7,048; 
 2,002,002.2,000,002. After multiplication and division of decimals by 
 moving the decimal point has been taught, dictate numbers to be 
 written, have the class divide or multiply by 10; 100; or 1,000, and 
 write the result in words. See that such words as twenty-five and all 
 the parts of the denominator are separated by a hyphen; as, seventy- 
 eight million, four, and six thousand, seventy-eight hundred- 
 thousandths. 
 
 III. PLACE VALUE. 
 
 Analyze numbers as follows. 2987.456 = 2000 + 900 + 80 
 
 7 + .4 + .5 + .006. Show that in 33.33 each 3 is ten times as great in 
 value as the 3 to its right, and one-tenth as great in value as the 3 to 
 its left. 3 tens or 30 = 10x3. 3 = 1/10 of 30 or of 3 tens 
 
 3-v- 10 = 3/10 or .3. .3 or 3/10 = 10 X 3/100 (or .03). .03 = 1/10 of 
 .3. Etc. 
 
 IV. CIPHER AT THE RIGHT OF A DECIMAL. 
 
 Show that .20 may be read 20 hundredths and that it is equal to 
 .2. 20/100 = 2/10. '..20 = .2. Similarly .2000 = .200 = .20 = .2, etc. 
 
 53
 
 Generalization: Annexing ciphers to the right of a decimal does not 
 change the value of the decimal. Similarly show that 1.2 may be 
 read one and two tenths or 12 tenths. 1.2=1 2/10 = 12/10. Similarly 
 12.5 = 12 5/10 = 125/10 = 125 tenths. 4.73 = 4 73/100 = 473/100, etc. 
 
 V. REDUCTION OF A FRACTION TO A DECIMAL. 
 
 Preparation. A. A fraction is an indicated division. B. Division 
 of United States money. C. 3/4 = 75/100 = .75. Presentation. Re- 
 duce 3/4 to a decimal. As 3/4 means to divide 3 by 4, let us perform 
 the indicated division as we divide in United States money. 
 
 $.75 .75 
 
 4)$3.00 4)3.00 
 
 Therefore 34 =.75, the same result we found in c. Take similar 
 examples and draw the Generalization: To reduce a fraction to a 
 decimal, divide the numerator by the denominator and point off as in 
 United States money. 
 
 VI. REDUCTION OF A DECIMAL TO A FRACTION. 
 
 Write the decimal as a common fraction and reduce to lowest 
 terms. In the second example, multiply the numerator and denomi- 
 nator of the complex fraction by 3 and reduce the result to lowest 
 terms. 
 
 .75 = 75/100 = 3/4. .13*3 = 13^/100 = 40/300 = 2/15. 
 
 VII. ADDITION AND SUBTRACTION OF DECIMALS. 
 
 Principle: Only like numbers can be added, units only can be 
 added to units, tenths only to tenths, etc., therefore: RULE: Place 
 one decimal point under another, keeping units under units, tenths 
 under tenths, etc. Example, from 17 subtract .007. Emphasize that 
 a point is understood to the right of units. 
 
 VIII. MULTIPLICATION OF DECIMALS. 
 
 Multiplication of a decimal by an integer is easily developed by 
 referring to multiplication of United States money. Develop multi- 
 plication of a decimal by a decimal as follows: .5 X .7 = 5/10 X 7/10 = 
 35/100 = . 35. . 3 X. 08 = 3/10x8/100 = 24/1000 = .024. .45X.79 
 
 = 45/100 = 79/100 = 3555/10000 = .3555. Etc. Compare, abstract, 
 and generalize. 
 
 IX. DIVISION OF DECIMALS. 
 
 A. Decimal by an integer. 
 
 Base on division of United States money. Place the decimal 
 point in the result directly above the point in the dividend. 
 
 54
 
 2.39 .12 1.2 .012 
 
 2)4.78 144)17.28 144)172.8 144)1.728 
 
 It is well to place the point in the result before the division be 
 gins. The first significant figure in the result should be directly 
 above the right hand figure in the first subtrahend. All the figures 
 should be in strictly vertical columns. The teacher should insist upon 
 exactness in this respect, as it is essential to accuracy. 
 B. Division by a decimal. 
 
 Use the Austrian method. Principle: Division of both dividend 
 and divisor by the same number does not change the quotient. See 
 lesson on fractions, Section XXIII, for the development of this prin- 
 ciple. Method: Move the decimal point in the divisor far enough to 
 the right to make the divisor a whole number, move the point in the 
 dividend the same number of places to the right, perform the division 
 with the resulting numbers. 1.728 -=-1.44 = 172.8-=- 144. 172.8-=-. MI 
 = ? First move point in the divisor three places to the right, then 
 move the point in the dividend three places to the right annexing 
 sufficient ciphers, place the point in the result directly above the new 
 point in the dividend, divide placing the first figure 1 in the result 
 directly above the figure 2 in the dividend as the right hand 4 of the 
 first subtrahend 144 is also placed below the 2. In dividing by .40 
 move the point only one place to the right and divide by short division. 
 Divide $120 by $.40. In dividing by such numbers as 20, 300, 4,000. 
 etc., move the decimal point to the left of the ciphers in the divisor, 
 the same number of places to the left in the dividend, and divide by 
 short division. 476-:- 400. 
 
 1.2 1 200. 30 0. 1.19 
 
 1.44)1.728 .144)172.800- .4'0)120.0' 4*00)476 
 
 X. FRACTION AT THE END OF A DECIMAL. 
 
 .12K> =.125; A%=Aiy 4 =A12V 2 =.4125; 
 ^=.3^=.33H=.333^, etc. 
 ^=.6^=.6G^, etc. .16*$ =.166-.?, etc. 
 
 Give sufficient practice to familiarize pupils with these equiva- 
 lents, that they may be able to read and write them accurately. 
 
 XL APPLICATION OF DECIMALS IN DIVISION BY A 
 
 MIXED NUMBER. 
 
 Divide 7828 by 13 y 3 . Multiply both numerator and denominator 
 by 3, move the decimal points in the results one place to the left, and 
 divide by short division. 
 
 4.0)2348.4 
 
 587.1 Ans. 
 55
 
 XII. ALIQUOT PARTS. 
 
 Pupils should become so familiar with the following equivalents 
 that one immediately suggests the other. They should be encouraged 
 to use the equivalent most economical. 
 
 16ths 12ths llths lOths 9ths 8ths 7ths 6ths 5ths 4ths 3ds half 
 1/16 .06 1/4 
 
 
 1/12 
 
 
 
 
 
 
 
 .08 1/3 
 
 
 1 11 
 
 
 
 
 
 
 
 .09 1/11 
 
 
 
 1/10 
 
 
 
 
 
 
 .10 
 
 
 
 
 1/9 
 
 
 
 
 
 .11 1/9 
 
 2/16 
 
 
 
 1/8 
 
 
 
 
 
 .12 1/2 
 
 
 
 
 ] 
 
 ;7 
 
 
 
 
 .14 2/7 
 
 
 212 
 
 
 
 1/6 
 
 
 
 
 .16 2/3 
 
 
 
 2 10 
 
 
 
 1/5 
 
 
 
 .20 
 
 4 16 
 
 3/12 
 
 
 2/8 
 
 
 
 1/4 
 
 
 .25 
 
 
 
 3/10 
 
 
 
 
 
 
 .30 
 
 
 4/12 
 
 
 39 
 
 26 
 
 
 
 1/3 
 
 .33 1/3 
 
 6/16 
 
 
 
 3/8 
 
 
 
 
 
 .37 1/2 
 
 
 
 4/10 
 
 
 
 2/5 
 
 
 
 .40 
 
 8/16 
 
 6/12 
 
 5/10 
 
 4/8 
 
 3/6 
 
 
 2,4 
 
 1/2 
 
 .50 
 
 
 
 6/10 
 
 
 
 3/5 
 
 
 
 .60 
 
 10/16 
 
 
 
 5/8 
 
 
 
 
 
 .62 1/2 
 
 
 8/12 
 
 
 6/9 
 
 4/6 
 
 
 
 2/3 
 
 .66 2/3 
 
 
 
 7/10 
 
 
 
 
 
 
 .70 
 
 12/16 
 
 9/12 
 
 
 6/8 
 
 
 
 3/4 
 
 
 .75 
 
 
 
 8/10 
 
 
 
 45 
 
 
 
 .80 
 
 
 1012 
 
 
 
 5/6 
 
 
 
 
 .83 1/3 
 
 14/16 
 
 
 
 7/8 
 
 
 
 
 
 .87 1/2 
 
 
 
 9/10 
 
 
 
 
 
 
 .90 
 
 XIII. MULTIPLICATION BY ALIQUOT PARTS. 
 
 Considerable emphasis should be laid on this process: 
 
 A. By aliquot parts of 1. Multiply by the equivalent fraction. 
 .33^X465=(^X465); .14 2/7 X 212= (1/7 X 212) ; .66^ 
 
 X 516=^x516; .12^ X144=(^ X144). 
 
 56
 
 B. By aliquot parts of 10, 100, or 1000. Multiply by the appro- 
 priate fraction and also by 10, 100, or 1000 as the case may be; that 
 is, multiply by the appropriate fraction and annex one, two, or three 
 ciphers. 12^X144=^x100x144 = 1800. 66^x516=^X5160 
 = 34400. 1.25X 144= (% X 10x144) =180. 6.6^X516=^X5160 
 = 3440. 125 X 144= (% X 1000x144) =1800. 666^x516 = 
 
 5<3 x 516,000 = 344,000. In performing the operations, do not write 
 down the numbers within the curves above, but compute mentally. 
 Give much practice. 
 
 XIV. DIVISION BY ALIQUOT PARTS. 
 
 A. By aliquot parts of 1. Divide by the equivalent fraction. 
 252-.33^ = (252-J4)=756. 144-12J^ = (144- % ) =1152. 516 
 -^.66^=516x3/2 = 774. 
 
 B. By aliquot parts of 10, 100, or 1000. Divide by the appropri- 
 ate fraction and by 10, 100, or 1000; that is, divide by the appropriate 
 fraction and point off one, two, or three places. 
 
 1779-1634 = (1779-^6 ^100) =106.74. 
 
 279+- 333^ = (279- V 3 -1000) =.837. 
 
 7421 — 3^ =2226.3 144 — 12^4 = (144- y s -100) = 11.52. 
 
 516-66^ =5.16x3/2 = 7.74. 144-1.25 = 11.52. 
 
 516-6.66^=51.6x3/2 = 77.4. 144-125 = 1.152. 
 
 516-666^ = .516x3/2 = . 774. 
 
 C. Fractional Relations 
 
 I. TO FIND A FRACTIONAL PART OF A NUMBER. 
 
 Find Vi of $18. Find .7 of $250. Find .17 of $500. 
 Fraction x whole = part. 
 Factor X factor = product. 
 Y 3 X% 18 = $12. 
 .7X $250 = $175. 
 .17 X $500 = $85.00. 
 
 II. TO FIND WHAT PART ONE NUMBER IS OF ANOTHER. 
 
 Preparation: 
 
 Factor Factor Product 
 2x3 = 6 
 Given the product 6 and one factor 3, how do we find the other 
 factor? In general, how do we find the other factor when one factor 
 and the produce are given? 
 
 57
 
 Ans. Divide the product by the given factor to find the other 
 factor. 
 
 Factor Factor Product 
 
 y 3 X $18 = $12. 
 .7 X $250 = $175. 
 .17X$500 = $85.00. 
 Statement of aim: Let us see how to find what part one number 
 is of another. $12 is what part of $18? $175 is what part of $250? 
 $85 is what part of $500? 
 Presentation: 
 
 Statement of relation: 
 
 Factor Factor Product 
 ? x $18 = $12. 
 ? X $250 = $175. 
 ? X $500 = $85. 
 Given the product $12 and the factor $18, how shall we find the 
 other factor? The product $175 and the factor $250? Etc. 
 
 Ans. Divide the product $12 by the factor $18. Divide the pro- 
 duct $175 by the factor $250. Etc. 
 
 Statement for solution: 12/18 = 2/3 Ans. $175-^$250 = .7 
 Ans. $85 -h $500 = .17 Ans. 
 
 In finding what part one number is of another it is some times 
 preferable to place the division in fractional form and reduce to low- 
 est terms as above. 
 
 The two satements, the statement of relation and the statement 
 for solution may be used whenever these examples are to be worked; 
 or the teacher may develop the rule to divide the part by the whole 
 to find the fraction. 
 
 III. TO FIND THE WHOLE WHEN A PART AND ITS FRAC- 
 TIONAL RELATION TO THE WHOLE ARE GIVEN. 
 
 Preparation: The product divided by the factor gives the other 
 factor. 
 
 Factor Factor Product 
 y 3 x $18 = $12. 
 .17 X $500 = $85. 
 .7 X $250 = $175. 
 Statement of aim: Let us see how to find the whole when the 
 part and its relation to the whole are given. $12 is 2 /z of how many 
 dollars? 
 
 Presentation. 
 
 Statement of relation: 
 
 Factor Factor Product 
 Vi X $ ? = $12. 
 .7 X $? = $175. 
 .17 x $? = $85. 
 
 58
 
 Statement for solution: 
 
 $12 -5- Vi = % Ans. 
 
 $175 ■+■ .7 = $ Ans. 
 
 $85 ■+• .17 = $ Ans. 
 
 As in the preceeding case, the two statements may be used or 
 the teacher may develop the rule to divide the part by the fraction to 
 find the whole. Call attention to the fact that when "of" is used to 
 indicate multiplication, the whole always follows the word "of". 
 Give an abundance of drill on these three cases of fractional relations 
 as a preparation for Percentage. Have the children thoroughly un- 
 derstand that when the part is wanted they always multiply, hence 
 the part is the product. When the part is given it is always divided 
 by whichever factor is given to find the other factor. At first the 
 statements of relation and solution should both be written before the 
 example is worked, a blank space being left for the insertion of the 
 answer. 
 
 .7 X$? =$17.50. ?X$18 = $12. 
 
 $175--. 7 = $ Ans. $12/$18 = 2/3. Ans. 
 
 D. Oral Analysis 
 
 The old method of oral analysis should be used in the solution 
 of simple problems, though it should not be allowed to crowd out the 
 work in fractional relations which is a preparation for Percentage. 
 
 Example: Find Y, of $1200. Solution: \\ of $1200 is $300, 
 and Yi of $1200 is 3 times $300 or $900. 
 
 Example: $900 is Ya of how many dollars? Solution: % of 
 the money (which is l A of Y of the money), is l A of $900 or $300, 
 and 4/4 of the money is 4 times $300 or $1200. 
 
 As oral work should be given daily, a class should soon become 
 proficient in analysis. The statement within the curves is explana- 
 tory and should not be required in the solution, though the children 
 should be able to give it if requested. 
 
 59
 
 PART FIVE 
 
 Percentage 
 
 I. PREPARATION FOR PERCENTAGE. 
 
 As a preparation for Percentage, aliquot parts and fractional 
 relations should be thoroughly mastered. Much use should be made 
 of fractions and decimals with denominators of 100 and of the equiv- 
 alent fraction in its lowest terms. While decimals are a special case 
 in fractions with denominators of ten or powers of ten, Percentage ^ 
 is a special case in decimals with denominator 100. This should be 
 made clear to the class. The following questions will illustrate the 
 method of procedure: 
 
 12 is what part of 16? How many hundredths? write the frac- 
 tion as a decimal. What part of 15 is 10? How many hundredths? 
 What part of 7 is 6? Change to the decimal form expressing it as 
 hundredths. Etc. 
 
 Find .75 of $84. Find y 4 of $84. Compare the results. Find 
 .25 of $738. Find % of $738. Compare results. Find .66% of $18. 
 To what fraction is .66% equal? Find % of $18. Compare results. 
 Which is the shorter way? Find .16% of $72. By what simple frac- 
 tion may we multiply and obtain the same result? Find .07 of $17. 
 Is there any simple fractional equivalent of .07? Then what shall we 
 multiply by in this case? Etc. 
 
 $15 is .05 of how many dollars? $18 is M 2 A of how many dol- 
 lars? What simple fraction may we substitute for the decimal? Do 
 it and solve. Etc. 
 
 Drill until the children can readily solve simple problems in 
 decimals and fractions with denominators of 100, using the decimal 
 when there is no simple fractional equivalent, and exchanging frac- 
 tions and decimals without hesitation when it is advantageous to do 
 so. When the class understand the precess, tell them that they have 
 been working examples in "Percentage", a name applied to examples 
 in fractions and decimals in which the denominator is 100. 
 
 II. INTRODUCTION OF THE TERMS PER CENT, RATE PER 
 
 CENT, AND RATE. 
 
 As soon as the children are familiar with the method of working 
 examples in hundredths, the terms per cent, rate per cent, and rate 
 should be introduced. The class should become familiar with these 
 terms before the introduction of the term percentage that the two 
 terms may not be confused as sometimes happens when they are in- 
 troduced together. Tell the children that as the term per cent is used 
 
 60
 
 it means exactly the same as the word hundredths. Also teach the 
 symbol for per cent and have the class write the various equivah 
 
 75 percent = 7595 = s =. 75=75/100= H. 
 
 62^ percent = 62Y '/ = .62^ =.625= $|. 
 
 7 per cent =7'; =.07 = 7/100. Etc 
 
 Emphasize that as per cent means hundredths, when a per cent 
 is written as a decimal, two decimal places are necessary, (live much 
 practice in writing the per cent as a decimal of two places, then have 
 the fractional part of the per cent also written as a decimal. Read 
 Vi '.', , ' j ' < , etc., as l /i of one per cent, ' \ of one per cent, etc. 
 
 62^ </< =.6254 =.625. 7' , '; --.07' ', =.0725. 
 
 y 2 % = .oo y 2 = .005. y y, = .00 % = .0025. 
 
 Vi <A = .00 Vi = .0033 1 / 3 = .003333 Ys , etc. 
 
 Children should also acquire facility in changing such decimals 
 to the per cent form. To demonstrate to the class the decimal equiv- 
 alent of the fractional part of a per cent, use the following subtrac- 
 tion: 
 
 (a) 11 y, '■ =.ny 2 =.115. 
 
 (b) 179? =.17=.17. 
 
 (c) \i <:, =.00^ =.005. Subtracting b from a. 
 
 III. INTRODUCTION OF THE TERMS PERCENTAGE AND BASE 
 
 Review the words whole and part as used in fractional relations 
 and tell the class that in the study of Percentage new terms are used 
 for the words part and whole. For the whole we use the term Base, 
 and for the part, the term Percentage. Write on the board: 
 
 $8 is 25 % of $32. $8 is % of $32. $8 is 25^ of what quantity? 
 Find 25 </< of $32. $8 is what per cent of $32? 
 
 Ask such questions as: What is the rate? What is the whole? 
 Then what shall we call this in percentage? What is the part? 
 What is the new word that we have learned for part? Or in other 
 examples: Can you tell which is the percentage? If not, ask your- 
 self which is the part? Can you tell now? Which is the base? Or: 
 What do we call the 25'; ? What is the $32? Is it the part or the 
 whole? Then what shall we call it? What is called for? Etc. Con- 
 tinue such questioning till the children can tell at once which term to 
 apply to each number without first asking themselves which is the 
 '■art, etc. It is not necessary to have these examples worked as the 
 drill is in the application of terms. It is important that the class un- 
 derstand the terms and can apply them before they try to so 
 problems. 
 
 IV. GIVEN THE BASE AND RATE TO FIND THE PERCENT \(! 
 
 Preparation: 
 Find .17 of $59. Find .73 of $714. Find .08 of $216. What new 
 
 61
 
 way have we learned to express .17? Ans. 17 ',', Similar questions 
 for .73 and .08. What do you call the 17%, 73%, and 8%? The 
 Rate per cent. What do we call the $59, $714, and $216? The Base. 
 
 Sub step — Statement of aim. 
 
 Let us see now how to find the Percentage when the Rate and 
 Base are given. 
 
 Presentation: 
 Let us find 17% of $59. What did you say the $59 is? The 
 17% ? What did we find was the equivalent of 179c? And how did 
 you find .17 of $59? Then how shall we find 17% of $59? Ans. Mul- 
 tiply .17 times $59. Do it. What do we call the result? Ans. The 
 Percentage. How did we get it? Ans. We multiplied .17 times $59. 
 But what is the 17% that you multiplied by? The $59? Then how 
 did you get the Percentage? Ans. We multiplied the Rate times the 
 Base. Similarly work and question regarding two other examples. 
 
 Comparison. 
 How did you find the Percentage in each of the three examples 
 you worked? We multiplied the Rate times the Base. 
 Generalization: 
 Who will give me a rule for working all examples when the Rate 
 and Base are given and we wish to find the Percentage? Rule: To 
 find the Percentage when the Rate and Base are given, multiply th? 
 Rate times the Base. If we let R stand for Rate, B for Base, and 1 
 for Percentage, who will write this rule as a formula. 
 RxB = P. 
 
 This we shall call the Formula for Percentage and shall read it: 
 The Rate times the Base equals the Percentage. 
 
 Application: 
 Send class to board to write formula and interpret it. Each 
 member of the class should leave the formula written at the top of 
 the board and as the teacher dictates examples to be worked, the 
 numbers should be written below the appropriate letter in comp'ete 
 statement form. Example: The Base is $218, the Rate 15%, find 
 the Percentage. Each number is properly placed as dictation pro- 
 ceeds and the following should appear on the board before work 
 begins: 
 
 R X B = P. 
 .15 X $218 = $ Ans. 
 
 Insist on complete statement before work begins. The example 
 should then be worked below and the result placed in the blank left 
 in the statement. 
 
 Assignment: 
 Problems should be read from the text book, the numbers iden- 
 tified by the class in terms of percentage, and the method of solution 
 stated. The probliems should then be worked during the study period 
 
 62
 
 V. GIVEN THE BASE AND THE PERCENTAGE TO FIND 
 
 THE RATE. 
 
 Review fractional relations to find what part one number is of 
 another. Place formula on the board and below its appropriate letter 
 each number of the example. $12 is what per cent of $36? The class 
 should write the Formula, RxB = P, and below it the Statement of 
 relation, ? x$36 = $12. As the Percentage is the result of multiply- 
 ing the Rate and the Base, what is the Percentage of the Rate and 
 Base? Ans. The product. Then given the product $12 and the factor 
 $36, how shall we find the other factor, the Rate? State it. The ex- 
 ample should now appear on the board as follows: 
 
 (Formula) R X B = P 
 
 (Statement of relation) ? X $36 = $12 
 (Statement for solution) $12 $36 =1 '3=33 1 3>' Ans. 
 
 Example: $28.63 is what per cent of $409? 
 
 R X B = P 
 
 R X $409 = $28.63 
 $28.63-*- $409 = % Ans. 
 
 Solve and place the result in the blank. For a time give simple 
 examples and require the use of the formula. Then give problems 
 requiring identification of data. In time the class should make the 
 statement for solution as soon as the data has been identified. 
 
 VI. GIVEN THE PERCENTAGE AND THE RATE 
 TO FIND THE BASE. 
 
 Example: The Percentage is $35, the Rate 5%, find the Base. 
 Formula R X B = P 
 
 Statement of relation .05 x $B = $35 
 Statement for solution $35 -5- .05 = $ Ans. 
 After giving the three cases in Percentage, emphasize the fact 
 that when the Percentage is wanted, we multiply; when the Per- 
 centage is given, it is divided by the given factor. Dictate many ex- 
 amples merely to be stated on the board without being worked. Dic- 
 tation should be rapid, but not so rapid but that all members of the 
 class can write the first number before another number is given. 
 Unless it is the teacher's fault, do not repeat a number. See that 
 every member of the class writes every number as it leaves the 
 teacher's lips. If a child will not keep up, send him to his seat. He 
 will soon learn not to lag behind. It is an inexcusable waste of time 
 on the part of the teacher to repeat for an inattentive pupil. Such 
 repetition begets inattention. 
 
 Do not allow the pupils to hold the erasers in hand and erase 
 when they please. Leave all the work on the board until the spaces 
 are well filled, then say "erase" and have all erase at the same time, 
 thus saving valuable time. 
 
 63
 
 Many examples may be given to be stated without solving. The 
 formula and statement of relation should be written and the state- 
 ment for solution when this differs from the statement of relation. 
 The blank should always be left for the answer. Many simple oral 
 problems should be given to the class at their seats, also many diffi- 
 cult problems requiring identification of data should be read from 
 the text book, the numbers identified in terms of percentage, and the 
 method of solution indicated orally. 
 
 A point that should be brought out at this time is that as one 
 whole may be divided into several parts, so one Base may have sev- 
 eral Percentages and for each Percentage there is a rate. 
 
 Example: Mr. Knox had $190 and spent $40 for a suit, $35 for 
 an overcoat, $5 for a hat, and $7 for a pair of shoes. How much did 
 he have left? What per cent had he left? What per cent did the suit 
 cost? The overcoat? Etc. What per cent did he spend altogether? 
 Prove this result by adding the per cents spent for the several items. 
 Also add the per cents spent to the per cent left, the sum being 100%. 
 
 Example: Mr. Patterson had a salary of $150 per month. He 
 spent 20 % for rent, 8 l A % for insurance, 26 2 / 3 % for household ex- 
 penses, 33 Yi % for sundries, and saved the rest. What per cent did 
 he save? What was his rent? Etc. Check: the sum of the several 
 Percentages should equal the Base. 
 
 VII. AMOUNT. 
 
 James paid 40 cents for some papers and sold them at a gain of 
 'J.0 cents. What did he sell them for? This we call the Amount. 
 What is the 40 cents in terms of percentage? The 20 cents? And 
 what is the 60 cents called? Then how did you find the Amount? 
 Who will define Amount? The Base plus the Percentage equals the 
 Amount. We will write this as follows: B + P = A, and call it the 
 definition of Amount. Have children commit to memory. Give many 
 examples to find the Amount when the Base and Percentage are given. 
 Especially emphasize that the selling price when there is a gain is 
 the Amount. 
 
 When we have given the sum of two numbers and one of the 
 numbers, how do we find the other number? If 12 is the sum of 8 and 
 some other number, what is the other number? Mr. Ackerson paid 
 $150 for a horse and sold it for $180, what did he gain? What do we 
 call the cost? The selling price when there is a gain? The gain? 
 (The teacher should emphasize that the gain is always the Percent- 
 age.) Then how did we find the Percentage when the Base and the 
 Amount were given? Give several examples. 
 
 Mr. Hughes sold a watch for $30, thereby gaining $10, What 
 did the watch cost? What was the $30? The $10? The 20? Then 
 how did you find the Base when the Amount and Percentage were 
 
 64
 
 given? Give many examples illustrating the three kinds of examples 
 till the class can work them without hesitation. 
 
 VIII. GIVEN THE AMOUNT AM) THE BASE TO FIND 
 
 THE RATE. 
 
 Emphasize that the Rate is always found by dividing the Per- 
 centage by the Base, therefore, to find the Rate when the Base and 
 Amount are given, find the missing fact, the Percentage, and use the 
 formula for Percentage. This should be developed by means of 
 questions. 
 
 IX. GIVEN THE AMOUNT AND THE PERCENTAGE 
 TO FIND THE RATE. 
 
 By questioning, develop: To find the Rate when the Amount 
 and Percentage are given, find the missing fact, the Base, and use the 
 formula for Percentage. Give many examples illustrating the last 
 two cases. 
 
 X. GIVEN THE PATE AND THE BASE TO FIND THE AMOUNT. 
 
 To find the Amount when the Rate and the Base are given, at 
 first have the class solve by finding the missing fact, the Percentage, 
 then add this to the Base. But it is an advantage to be able to solve 
 by multiplying by 1 plus the Rate, also it is essential to an under- 
 standing of the next case. Before teaching the following lesson, the 
 fact should be emphasized that any number is 100 r '< of itself. $20 
 is 1009^ of $20. 100'; of $75 is $75. The purpose of the following 
 lesson is to teach the formula: (1.00 + R) xB = A. 
 Preparation: 
 
 RxB = P. The Base equals 100 of itself. Base plus Percentage 
 eouals the Amount. 
 
 RXB = P. 
 
 .05 x $60 = $3. .06 X $50 = $3. .07 X $80 = $5.60. 
 
 $60 + $3 = $63. A. $50 + $3 = $53 A. $80 + $5.60 = $85.60 A. 
 
 Statement of aim: We are now going to learn a better way of 
 finding the Amount when the Rate and the Base are given. 
 Presentation: 
 
 $60 is what per cent of $60? Ans. 100 'J of $60. Teacher writr 
 onboard: 100 ';', of $60 = $60. $3 is what per cent of $60? Teacher 
 write on board below last statement: 5'; of $60 = $3. How many 
 per cent of $60 is the sum of 100'; of $60 and 59i of $60? Ans. 
 105% of $60. Teacher write on board below the other statements. 
 How many dollars is the sum of 100'; of $60 and 5'; of $60? Ans. 
 $63. Teacher write on board. We now have the following: 
 100'; of $60 = $60 
 of $60 = $ 3 
 
 Adding 105' ; of $60 = $63 
 
 G5
 
 Send child to board to verify this statement by multiplying $60 
 by 1.05. What is the 5%? Ans. The Rate. What is the 105%. 
 Ans. 100% plus 5% or 100% plus the Rate. What is the $60? The 
 $63? Then how did we find the Amount? or What did we multiply 
 the Base by to find the Amount? Ans. 100% plus the Rate. Similar- 
 ly treat the other two examples used in the preparation. 
 
 Comparison: 
 
 In all these examples how did we find the Amount? Ans. We 
 multiplied 100% plus the Rate times the Base. 
 
 Generalization: 
 
 Give me a rule for finding the Amount when the Rate and 
 the Base are given. Rule: To find the Amount when the Rate and 
 the Base are given, multiply the Base by 100% plus the Rate. We 
 may write this as a formula thus: (1.00 + R) XB = A. 
 
 This we shall call the formula for Amount. It may be read 
 either 100 % plus the Rate times the Base equals the Amount, or I 
 plus the Rate times the Base equals the Amount. The use of the 
 curves in writing the formula is necessary. See that the class dif- 
 ferentiate between the formula and the definition. 
 
 Application: 
 
 Send class to board to write and interpret the formula. Then 
 dictate examples to be written below the formula in statements. At 
 first have examples stated only, then have some of them worked. 
 Example: The Base is $246, the Rate is 8%, state for Amount. 
 
 (1.00 + R) XB = A. 
 
 1.08 X $246 = $ Ans. 
 
 The addition of 1.00 and the Rate should always be made men- 
 tally and the example stated as above at once as the teacher dictates. 
 Assignment: 
 Problems from the text book should be read and the numbers 
 identified in terms of percentage and the method of solution indicat- 
 ed. They should then be worked during the study period. 
 
 XI. GIVEN THE AMOUNT AND THE RATE TO FIND THE BASE 
 
 Example: The Amount is $428, the Rate 7%, state for finding 
 the Base. 
 
 Formula (1.00 + R)XB=A 
 
 Statement of relation 1.07X$B = $428 
 
 Statement for solution $42S-h1.07=$ Ans. 
 
 Upon reaching this point the majority of a class will work this 
 example without help from the teacher except the suggestion that the 
 formula for Amount be used and the numbers placed under the ap- 
 propriate letters. 
 
 66
 
 XII. GIVEN THE AMOUNT AND THE RATE TO FIND 
 
 THE PERCENTAGE. 
 
 Develop: Find the essential missing fact, the Base, and subtract. 
 
 XIII. GIVEN THE PERCENTAGE AND RATE TO FIND 
 
 THE AMOUNT. 
 
 Develop: Find the essential missing fact, the Base, and add. 
 
 Many miscellaneous problems applying all the cases in percent- 
 age thus far covered should now be given for identification of data, 
 for statement, or for solution. Also give frequent special drills in 
 identification of data as follows: Send class to the board to write 
 the appropriate letter as the teacher dictates; the whole, the part, 
 the cost, the gain, the selling price when there is a gain, what you 
 have to start with, etc. Teacher should dictate rapidly and then have 
 all errors checked. The order of dictation should vary. 
 
 XIV. DIFFERENCE. 
 
 A dealer paid $12 for some goods and then on account of their 
 being damaged sold them at a loss of $3, what was the selling price? 
 What is the $12? The $3? Then how did you find the $9? Ans. We 
 subtracted the Percentage from the Base. We call this $9 the Dif- 
 ference because it is obtained by subtraction. Define Difference. 
 The Base minus the Percentage equals the Difference. We write this 
 thus: B — P = D, and call it the definition of Difference. If the Base 
 is $74 and the Percentage $14, what is the Difference? Etc. Empha- 
 sis should be laid on the fact that the selling price when there is a 
 loss is the Difference. 
 
 Mr. Jerome paid $35 for vegetables and sold them for $30, how 
 much did he lose? What is the $35? The $30? The $5? Then how 
 did you find the Percentage when the Base and the Difference were 
 given? Etc. 
 
 Frank sold his sister a tablet for $.10 thereby losing $.05, what 
 did the tablet cost him? What is the $.10? The $.05? The $.15? 
 Then how did you find the Base when the Difference and Percentage 
 were given? Etc. The class should become proficient in the three 
 kinds of examples given above. 
 
 XV. GIVEN THE DIFFERENCE AND THE BASE TO FIND 
 
 THE RATE. 
 
 Develop: Find the essential missing fact, the Percentage, then 
 find the Rate in the usual way. 
 
 XVI. GIVEN THE DIFFERENCE AND THE PERCENTAGE 
 
 TO FIND THE RATE. 
 
 Develop: Find the essential missing fact, the Base, again em- 
 
 67
 
 phasizing that the Percentage is always divided by the Base to find 
 the Rate. 
 
 XVII. GIVEN THE RATE AND THE BASE TO FIND 
 THE DIFFERENCE. 
 
 To find the Difference when the Rate and the Base are given, at 
 first have the class solve by finding the missing fact, the Percentage, 
 then subtract this from the Base. Later teach the following formula: 
 (1.00 — R)xB = D. See that the class differentiate between the 
 formula and the definition. 
 
 This lesson should be presented as the corresponding lesson in 
 Amount was presented. The examples in the preparation would be 
 like the following: 
 
 RX B= P 
 .05x$60 = $ 3 
 $60—$ 3 = $57 D 
 The questions in the presentation should lead to the following: 
 100% of $60 = $60 
 5% of $60 = $ 3 
 
 Subtracting, 95% of $60 = $57. 
 Verify and bring out that to find the Difference we multiplied 
 the Base by 100 '/r minus the Rate. In the application have class 
 write formula and statement, subtracting the Rate from 1.00 men- 
 tally. Example: The Base is $380, the Rate 12%, fiind the Differ- 
 ence. 
 
 (1.00— R) XB = D 
 
 .88 X $380 = $ Ans. 
 
 XVIII. GIVEN THE DIFFERENCE AND THE RATE TO FIND 
 
 THE BASE. 
 
 Example: The Difference is $399.50, the Rate 15%, state for 
 finding the Base. 
 
 Formula (1.00— R)X B = D 
 
 Statement of relation .85X$B=$399.50 
 
 Statement of solution $399.50-=-. 83=$ Ans. 
 
 XIX. GIVEN THE DIFFERENCE AND THE RATE TO FIND 
 
 THE PERCENTAGE. 
 
 Develop: Find the essential missing fact, the Base and subtract. 
 
 XX. GIVEN THE PERCENTAGE AND THE RATE TO FIND 
 
 THE DIFFERENCE. 
 
 Find the essential missing fact, the Base, and subtract. 
 
 68
 
 Examples in Profit and Loss, Commission, and Cash Discount 
 should be given to apply all the cases in percentage. Continue the 
 drill in identification by writing the appropriate letter on the board 
 as the teacher dictates. Now the terms dictated may include: what 
 you start with, the whole, the cost, the list or marked price, the sell- 
 ing price in commission, the principal, all of which are the Base; the 
 part, the gain, the loss, the commission, the discount, all of which are 
 the Percentage; the selling price when there is a gain, the sum sent 
 to an agent to pay for goods and commission, any sum after a gain. 
 Amount; the part left, the selling price when there is a loss, the 
 selling price when there is a discount, the net proceeds of a sale on 
 commission when there are no extra expenses, Difference. When the 
 Rate is called for, have class state what Rate; as, Rate of commis- 
 sion, Rate of gain, etc. Emphasize that the word after "of follow- 
 ing "Rate" indicates the Percentage. When the Rate is the Rate of 
 commission, the Percentage is the commission; when the Percentage 
 is the gain, the Rate is the Rate of gain; etc. Emphasize that the 
 selling price may be either Base, Amount, or Difference, bringing out 
 that it must be known whether goods are sold at a loss or a gain, on 
 commission, or at a discount, before the selling price can be identified. 
 Also after calling attention to the fact that in profit and loss the cost 
 is always the Base, bring out that in discount the Difference, which 
 is the selling price to the man selling the goods, is the cost to the 
 man buying. 
 
 In connection with the work in percentage, the teacher should 
 review the section on "Problem Solving"; and when the class are 
 writing down the identification of data either on paper or the board, 
 they should always indicate what Rate is given or wanted; as. the 
 "Rate of loss" or the "Rate of Comm.", not merely the "Rate". This 
 often is a material aid in identifying the Percentage itself. 
 
 XXI. SEVERAL SUCCESSIVE DISCOUNTS. 
 
 Example: The list price of a piano sold by Potter and Putnam 
 is $600 with 10%, 8#, and 5$ off. What is the selling price? 
 
 The example may be worked in either of two ways. Ordinarily 
 the quickest way is to use the following formula: 
 
 (1.00— Ri ) (1.00— R 2 > (1.00— R 3 ) etc.)XB=S.P. 
 
 The child should subtract the Rates mentally, stating the exam- 
 ple at once: 
 
 .90 X .92 X .95 X $600 = $471.96. Ans. 
 
 This method certainly has the advantage of compactness of 
 statement, while in the number of figures required in the operations 
 there is also a saving. The child working at the board to the teach- 
 er's dictation can state and work the example by this method in half 
 the time required by the method commonly in use. As the teacher 
 dictates, the child makes the mental subtractions and has the example 
 
 69
 
 stated as soon as dictation ceases. All that remains to be done is to 
 make the abstract computations and place the answer in the state- 
 ment. The constant oscillation between statement and work is 
 avoided. Give much practice in merely stating such problems with- 
 out solving them. 
 
 XXII. TO FIND A SINGLE RATE OF DISCOUNT EQUIVALENT 
 TO SEVERAL SUCCESSIVE RATES. 
 
 Use the following formula: 
 
 1.00— (1.00 -Ri ) (1.00— Ra ) <etc.) = R 
 Example: What single Rate of discount is equivalent to 8% 
 
 and 10% off? 
 
 1.00— .92 X. 90 = 11%%. Ans. 
 
 XXIII. MARKING GOODS. 
 
 Merchants sometimes wish to mark goods for a sale so that they 
 may be able to make a profit on the cost of the goods but at the same 
 time be able to offer a discount on the marked price. For example, a 
 dealer wishes to sell goods that cost $240 so that he will make a profit 
 of 25% on this cost, but still be able to discount his marked price 
 20%. To make 25%, he must sell for $300. He marks his goods 
 $375 and sells at a discount of 20% on this marked price, thus re- 
 ceiving the $300 he wishes. This involves a change of Base, the Base 
 being the cost in profit and loss, and the marked price in discount. 
 The selling price is the Amount in profit, and the Difference in dis- 
 count, therefore the $300 is the Amount in one case and the Differ- 
 ence in the other. We multiply the Base $240 by 1.25 to obtain the 
 selling price, then divide this selling price as the Difference in dis- 
 count by 1.00— R, or .80, giving us the other Base, the marked price. 
 That is, (1.00 + R, ) X B is the selling price, R t being the Rate of 
 gain and B the cost. This selling price as the Difference divided by 
 (1.00 — R 2 ) gives us the other Base, the marked price, R 2 being the 
 Rate of discount. Expressing this as a formula, we have: 
 
 (1.00 + Ri )XB 
 = Marked Price. 
 
 1.00— R 2 
 As the teacher dictates the problem, the class should add and 
 subtract the Rates mentally and state the problem promptly as fol- 
 lows: 
 
 1.25X$300 
 
 = $ Ans. 
 
 .80 
 
 As usual the teacher should give problems to be stated only, 
 sometimes giving the Rate of discount before the Rate of gain. 
 
 70
 
 If the cost is not stated, and the question is how must goods be 
 marked to gain 25 ',', and still allow a discout of 20 r J ? assume $1.00 
 as the cost. 
 
 1 ^5X$1 
 
 =$1.56 1/4. -.561 1 above coat 
 
 .80 
 
 But as 1 as a factor does not change the result, the $1 may be 
 omitted and the example stated as follows: 
 
 1.25 
 =1.56 1 4. .56 1 4V above cost. 
 
 .80 
 Put as a formula this would be either: 
 
 1.00 + Ri 
 
 = M. P. in per cent. 
 
 1.00— R 2 
 
 Or if the formula is to express the Rate above cost: 
 
 1.00-t-Rr 
 
 — 1 = R above cost. 
 
 1.00— R a 
 
 Example: How must pianos costing $850, $500, and $300 be 
 marked so that the dealer may give a discount of 20'; and still gain 
 25%? 
 
 1.25/.80 = 1.56J4. 1.56J4 X$500 = $ Ans. 1.56 '^ x $850 = 
 
 $ Ans. 1.56 ?4 X$300 = $ Ans. It will be seen that the 
 
 advantage of this method is that the dealer gets his Rate once for all 
 articles, and then mu'tiplies each cost by this Rate, instead of multi- 
 plying the cost of each article by 1.25 and dividing each result by .80. 
 
 XXIV. SIMPLE INTEREST. 
 
 As a preparation for Simple Interest, teach reduction of months 
 and days to years using the following progressive order: months to 
 years; years and months to years; days to years; years and days to 
 years; months to days; months and days to days and then to years; 
 years, months and days to years, first reducing the months and days to 
 days. Four months equals 4/12 of a year equals 1/3 of a year; 3 
 years, 4 months equals 10/3 years; 20 days equals 20/360 of a year 
 equals 1/18 of a year; 4 years, 18 days equals 4 1/20 years equals 
 81/20 years; 3 months, 10 days equals 100 days equals 5/18 of a 
 year; 2 years, 3 months, 10 days equals 2 5/18 years equals 41 L8 
 years. 
 
 Interest, which is money paid for the use of mony or capital, is 
 paid annually, hence we have a new element, Time. What will be the 
 Interest for one year on $450 at 5'/, ? 
 
 RX B = P 
 
 . 05 X $450 =$22. 50 Interest. 
 
 71
 
 What will be the Interest for 2 years? 3 years? Vz year? 
 
 2 X. 05 X $450. 00 = $45. 00 Interest for 2 years. 
 
 3X.05X$450.00=$67.50 Interest for 3 years. 
 
 1 2X.05X$450.00=$11.25 Interest for 12 year. 
 
 Letting I stand for Interest and T for Time in years, we have in 
 general: TxRxB = I. What will be the Interest on $300 for 3 yrs. 
 4 mos. at 6 r A ? (Use cancellation.) 
 
 T X R X B =1 
 10/3X.06X$300=$60. Ans. 
 
 What will be the Interest on $600 for 2 years, 12 days at 5%? 
 
 In computing Interest, one year is now generally treated as 360 
 days, and one month as 30 days; therefore 12 days equals 12/360 of 
 a year, or 1/30 yr. 
 
 61/30 X .05 X $600 = $61. Ans. 
 
 Find the Interest on $200 for 9 mo. 10 da. at 3%. 
 
 280/360 = 7/9. 7/9X .03 X $200 = $4.66%.\$4.67. Ans. 
 
 Find the Interest on $115 for 2 yrs., 1 mo., 10 da., at 5%. 
 
 19/9X.05X$H5 = $12.14. Ans. 
 
 XXV. GIVEN THE INTEREST AND TWO ELEMENTS TO FIND 
 
 THE THIRD ELEMENT. 
 
 Preparation: 2x3x5 = 30. 30- (2x5) =3; 30-5- (2x3) =5; 
 30-M3x5)=2. TxRxB = I. 
 
 Example: In how long a time will $470 placed at 4% Interest 
 produce $65.80 Interest? 
 
 T X. 04 X $470 = $65.80. 
 
 $65.80-^-(.04x$470)=3 I />-'-3 yr., 6 mo. Ans. 
 If the result were 2 7/9 years, 7/9 of 360 days is 280 days which 
 equals 9 months 10 days, giving 2 years, 9 months, 10 days. 
 
 Example: At what rate will $240 produce $32.40 interest in 2 
 years, 3 months? 
 
 $32.40-5- (2 % X $240) =.06. .-.6% Ans. 
 Example: What principle placed at 5% interest for 4 years will 
 produce $28.80 interest? 
 
 $28.80- (4 X. 05) =$144. Ans. 
 
 XXVI. SIX PER CENT METHOD. 
 
 The teacher should develop the method of subtracting dates and 
 of finding the Rate at 6% for the given time. For one month the 
 Rate would be one twelfth of .06 or .005; for one day one thirtieth 
 of .005 or .000^ or .001/6. 
 
 I. Find the Time by subtracting dates if necessary. 
 
 72
 
 II. Find the Total Rate for the given time: 
 Number of years X 06 
 Number of months X .005 = 
 Number of days X .000 ' /, = 
 
 Total Rate 
 
 III. Find the Interest: 
 
 r/6x Total RxB = I. 
 
 IV. Find the Amount: 
 
 B + I = A. 
 Multiplying the principal by the Total Rate for the given time 
 would give the Interest at §'/< . Dividing this by 6 gives the Interest 
 at 1%; and multiplying the Interest at 1% by r (not by R), gi 
 the Interest at the given Rate. Here r equals the number of per 
 cent. For example: if the Rate is 5',', , R = .05 and r=5. In practice 
 we do not actually find the Interest at 6 'A and then divide by 6 and 
 multiply by r. We at once cancel the 6 when possible and then multi- 
 ply in any order deemed shortest. Have the class place blank on the 
 board before beginning dictation. Insist on the vertical arrangement 
 and correct relative position of all work by all members of class. In 
 the end this is a time saving device. 
 
 SIX PER CENT BLANK. 
 
 X 
 
 .06 
 
 = 
 
 . 
 
 
 
 
 X 
 
 .005 
 
 = 
 
 . 
 
 
 
 
 X 
 
 .000% 
 
 = 
 
 • 
 
 
 
 
 $ 
 
 X $ 
 
 
 = 
 
 $ 
 
 $ 
 
 Int. 
 Amt. 
 
 Ans. 
 Ans 
 
 r/6x 
 
 $ + 
 
 Example: Find the Interest and Amount at 7 r f for a note for 
 $360, dated August 27, 1905 and paid December 8. 1911. 
 1911— 12--8 6X.06 =.36. 
 1905—8-27 3X.005 =.015 
 
 11 X. 000%=. 001 % 
 
 6 — 3-11 
 
 .376-% 
 7/6 X. 376 •% X $360 = $158.27 Int. Ans. 
 $158.27 + $360 = $518.27 Amt. Ans. 
 
 Give much rapid oral work in the solution of problems in Interest. 
 As the Rate at 6% for 60 days is .01, to find the Interest for 60 days, 
 move the decimal point two places to the left; to find the interest for 
 30 days, move the point two places to the left and divide by 2; to 
 find the Interest for 120 days, move the point and multiply by 2. For 
 other terms, various combinations of the above method may be used. 
 
 73
 
 The Interest on $240 for 60 days at 6% is $2.40; for 30 days, $1.20; 
 for 90 days, $3.60; for 120 days, $4.80. To find the Interest at 5%, 
 find H of the Interest at 6%. 
 
 XXVII. COMPOUND INTEREST. 
 
 Compound Interest, though it is illegal for general use, is still 
 employed by saving banks, which generally compute Interest semi- 
 annually and add this to the principal for a new principal upon which 
 the Interest for the next term is computed. Letting R equal the 
 Rate for the term of Interest, and t the number of terms, develop the 
 formula below: 
 
 Example: Find the Amount of $300 compounded annually for 
 3 years at 4%. 
 
 Example: Find the Interest on $400 compounded semi-annually 
 for 3 years at 4%. 
 
 (1.00 + RH XB= A 
 1.04 3x$300=$347.46 
 1.026 X$400=$450.46 
 $450.46— $400=$50.46 Compound Int. Ans. 
 
 Several simple examples should be worked by actually multiply- 
 ing them out, as in the first example above multiplying 1.04 by itself 
 three times and $300 by the result. Then more difficult example' 
 like the second should be worked by use of the following table. State 
 the example exactly as above, then in the table in the column for 2% 
 and opposite the year 6 will be found the Amount of $1.00 at 2% for 
 6 years, which is the same as the Amount of $1.00 for 3 years at 4% 
 compounded semi-annually. Multiplying this $1.12616 by 400 gives 
 the desired result, $450.46. 
 
 ; showing the amount of $1 at compound 
 
 interest for 
 
 Year 
 
 2 Per Cent 
 
 2% Per Cent 
 
 3 Per Cent &£ Per Cent 
 
 4 Per Cent 
 
 1 
 
 1.02000 
 
 1.02500 
 
 1.03000 
 
 1.03500 
 
 1.04000 
 
 2 
 
 1.04040 
 
 1.05063 
 
 1.06090 
 
 1.07123 
 
 1.08160 
 
 3 
 
 1.06121 
 
 1.07689 
 
 1.09273 
 
 1.10872 
 
 1.12486 
 
 4 
 
 1.08243 
 
 1.10381 
 
 1.12551 
 
 1.14752 
 
 1.16986 
 
 5 
 
 1.10408 
 
 1.13141 
 
 1.15927 
 
 1.18769 
 
 1.21665 
 
 6 
 
 1.12616 
 
 1.15969 
 
 1.19405 
 
 1.22926 
 
 1.26532 
 
 7 
 
 1.14869 
 
 1.18869 
 
 1.22987 
 
 1.27228 
 
 1.31593 
 
 8 
 
 1.17166 
 
 1.21840 
 
 1.26677 
 
 1.31681 
 
 1.36857 
 
 9 
 
 1.19509 
 
 1.24886 
 
 1.30477 
 
 1.36290 
 
 1.42331 
 
 10 
 
 1.21899 
 
 1.28009 
 
 1.34392 
 
 1.41060 
 
 1.48024 
 
 XXVIII. BANK DISCOUNT. 
 
 As an introduction to the subject of Bank Discount, the teacher 
 
 74
 
 should explain to the class the two chief functions of banking: deposit 
 and loan or discount. She should provide herself through some local 
 hank with the various blank forms used in banking: the pass book, 
 the deposit slip, the check book including checks and stubs, the bank 
 note, and the draft. 
 
 All banks receive money on deposit for safe keeping, sometimes 
 allowing Interest, sometimes not. Saving banks, which do not accept 
 deposits for checking purposes, allow Interest on all sums left on de- 
 posit a stated length of time. Commercial banks allow Interest on 
 sums left on deposit without being checked upon. When money [a 
 deposited merely for checking purposes, in some cases no Interest is 
 allowed; in others, Interest is allowed on unchecked balances. 
 
 The teacher should further explain the method and purposes of 
 checking; how personal accounts between individuals or firms are 
 balanced by the use of checks instead of money. It is well to explain 
 how such checks often pass from hand to hand, especially between 
 banks, in the payment of debts or the settlement of accounts, before 
 they are finally presented for payment at the bank upon which they 
 are drawn. In this connection the function of the New York Clearing 
 House should be explained. 
 
 Review Simple Interest and show that Bank Discount is nothing 
 more nor less than Simple Interest paid in advance by the borrower 
 to the bank for the use of the bank's money. Discounting notes is the 
 bank's method of loaning. When a person of good financial standing. 
 or one who can give good security for the repayment of the money, 
 wishes to borrow, say $400, from the bank; he draws up a promisory 
 note payable to himself or another at the bank in 30, 60, 90, or 120 
 days. The bank generally charges for this loan the legal rate of in- 
 terest for the given time and collects this charge in advance. Sup- 
 pose the time to be 60 days, the Interest charge, or the Bank Discount 
 will be $4. Thus the bank loans the borrower $400 and receives in 
 return the promisory note and $4 for the use of the loan. Or what is 
 the same thing, the bank deducts the $4 from the $400 and pays out 
 the balance, $396, called the Proceeds. Such examples differ from 
 Simple Interest only in that the Interest is subtracted to find the 
 Proceeds instead of being added to find the Amount due when the 
 note is payable. If the borrower in the above case had wanted the 
 use of the money for more than 120 days, instead of making out the 
 note for a longer time, he would have it renewed when it became due 
 and would again discount the $400 for the additional time. 
 
 Frequently business men receive from their customers in pay- 
 ment of accounts notes payable when the maker's crops will mature. 
 If such notes are long time interest bearing notes, they are often 
 deposited at a bank for collection, in which case no account is made 
 of the notes on the books of the bank, except to give the payee credit 
 when the notes are paid. When, however, these notes are short time 
 
 75
 
 non-interest bearing notes, the payee may in case of need of ready 
 money have them discounted at the bank at any time before they are 
 due. In such case, the discount is computed for the number of days 
 the note has yet to run and upon the amount that will be due when 
 the note matures. In the solution of such problems, the use of the 
 following blank will be found advantageous: 
 
 Date 
 Discounted 
 
 BANK DISCOUNT BLANK. 
 
 Due 
 
 t = 
 
 Due 
 
 ,'6X( X.001/6)X$ =$ B Dis. Ans. t 
 
 $ _$ =$ Pro. Ans. 
 
 When the term is 30, 60, 90, or 120 days, the appropriate rate 
 should be placed above the parenthesis at once without multiplication. 
 
 Example: Find the Bank Discount and the Proceeds of a $120 
 
 non interest bearing note dated June 3 and due in 90 days, if dis- 
 counted at the bank July 12 at 5%. 
 
 90 June 27 
 
 Date— June 3~ I t _ M ! Due— Sept. 1 July 31—19 
 
 J ; ; Aug. 31— 31 
 
 Discounted— July 12 Due Sept. 1— 1 
 
 90 51 t 
 
 5/6X(5lX.001/6)X$I20=$.85. B. Dis. Ans. 
 $120— $.85 =$119.15. Pro. Ans. 
 
 Before giving problems to find the Bank Discount, it would be 
 well to give drill in finding the date due and the term of discount. 
 First make a quick estimate as to date due. In the above example 
 the date due must be about Sept. 3. If in a 90 day note the date due 
 as finally found differs more than two days from the estimated date, 
 some error must have been made. To find the exact date, place the 
 number of days the note is to run, 90, below the column; after "June" 
 the number of days after June 3; after "July" and "Aug.", the num- 
 ber of days in those months respectively, and after "Sept." the num- 
 ber necessary to make the total 90. Do not add and then subtract 
 from 90, but adding say: 9 and 1 are 10, placing the 1 in the blank 
 opposite "Sept." Continuing say: 3 — 9. Thus getting the date 
 Sept. 1. 
 
 To find the term of discount, in the second column after "July" 
 place 19, the number of days after July 12, the date of discount. 
 After the remaining months, July and Aug., place the same numbers 
 as in the first column. Add and the result is the term of discount. 
 
 76
 
 Before even this work can be done accurately, each child must. 
 have at tongue's end the number of days in each month. 
 
 As the Rate of Discount at 6 f A for one day is .001/6, to find the 
 Discount for a given number of days, move the point three places to 
 the left, divide by 6, and multiply by the given number of days. Find 
 the discount on $1200 at G<'< for 17 days. Mentally moving the point 
 three places to the left and dividing by 6 gives $.20, the Discount for 
 one day, and multiplying this by 17 gives $3.40, the Discount for 17 
 days. 
 
 XXIX. DISCOUNTING INTEREST BEARING NOTES. 
 
 As Discount is computed upon the sum due at maturity, the 
 Amount of an interest bearing note instead of the face would be the 
 Base in Bank Discount; therefore find the Amount by the 6' ', method 
 and use this Amount as the Base in the Bank Discount blank. 
 
 Example: Find the Proceeds of a 90 day note for $400 bearing 
 Interest at 5%, dated Jan. 10, 1912, and discounted Mar. 5 at 6'. . 
 
 90 Jan. 21 
 
 Date— Jan. 10 1912 ~ 
 
 _ ls; | Due— Apr. 9. Feby. 29 
 
 * - 5 * i Mar. 31—26 
 
 Discounted— Mar. 5 Due Apr. 9 — 9 
 
 5'6X.005X$400 =$5 I. 
 
 $5 +$400 =$405 A. 
 
 6 6X(35X.001 6)X$405 =$ 2.36 B. Dis. 
 
 $402.64 Pro, Ans. 
 
 77
 
 ADDENDA. 
 
 PREVENTION OF COPYING. 
 
 1. Remove the possibility of copying by seating the class in a 
 manner to prevent it, and by giving different examples to neighbor- 
 ing children at the board. On tests with all teachers and at all times 
 with inexperienced teachers and generally in the lower grades, this 
 method is advisable, but it does not get at the root of the evil. The 
 tendency to copy remains and passive attention is relied upon, where- 
 as the power of active attention with effort should be developed. To 
 accomplish this end, the teacher must be constantly on the alert and 
 she must employ various means to prevent copying when the possi- 
 bility and temptation are always present. 
 
 2. Inflict some penalty, (a) Turn the erring child around with 
 his back to the board till the rest of the class finish the example, 
 (b) Mark him zero for the example, (c) Send him to his seat. But 
 all these appeal to a low motive and the child will still copy when 
 sure of avoiding the penalty. However he is given some discipline in 
 that he inhibits a desire to look at his neighbor's work and thus 
 strengthens his active attention. 
 
 3. Appeal to his sense of shame or pride or better his self res- 
 pect by making him see how other people regard such actions. Here 
 again he will continue to copy when he thinks he is sure of avoiding 
 detection. 
 
 4. Show the child the practical results of copying, that it pre- 
 vents his understanding the subject and keeps the teacher from 
 knowing when to give him proper help and hence he will probably 
 fail and remain in the grade another term, or that possibly he will be 
 demoted to a lower grade. This appeal too will be of only temporary 
 value, as when he comes to take a test, he knows that the practical 
 result is the other way around. 
 
 5. Make the child see that copying prevents mental development, 
 that instead of gaining mental power he is weakening his ability to 
 think in the future. This will appeal to the ambitious child but the 
 child who thinks only of the immediate present will receive litt'e 
 stimulus. 
 
 All of these fail in the end unless reinforced by some motive that 
 applies with equal force to all possible cases. Some of these means 
 will be found effective under some circumstances and ineffective un- 
 der others. By these the child who cannot be reached by a higher 
 appeal must be reached. 
 
 6. Appeal to the moral sense. It is not necessary to paint copy- 
 ing in black letters and brand it as cheating and stealing. Many 
 children do not have sufficient moral s^nse to realize that it is wrong, 
 
 78
 
 
 and the use of strong language is apt to do more harm than good. 
 If possible, however, the child should be made to see the evil, that it 
 is really wrong, that it is unfair to the rest of the class. How to ac- 
 complish this will depend upon the child and upon circumstances. 
 Sometimes the appeal should be to the whole class and at times to the 
 child privately. Many children if convinced of the wrong will cease 
 the evil practice, while of course others will deliberately continue to 
 copy knowing it to be wrong. With the latter some of the other 
 means discussed will be essential. 
 
 ONLY ONE DIFFICULTY AT A TIME. 
 
 Under the discussion regarding teaching a new operation, em- 
 phasis was laid upon introducing but one new difficulty at a time. A 
 few specific cases will here be given. 
 
 In column addition, which should be begun as soon as the child 
 knows the facts sum seven, the new difficulty is remembering the sum 
 of the first two digits and adding this to the third digit, with but one 
 of the two numbers in sight. As a preparation for this, it might be 
 well to place some number, as 2, on the board and then have the class 
 add to this any number from 1 to 5 dictated by the teacher. Suppose 
 the column to be 2-2-3, ask the child the sum of the first two digits, 
 then ask him the sum of this number and the last digit. Focus the 
 child's mind upon the thought that he is to add the sum of the first 
 two digits to the third, thus making the new idea as vivid as possible, 
 the strong first impression. At first the work will have to be oral, 
 the teacher questioning and the child answering, then the child doing 
 all the oral work alone, and finally adding mentally and giving the 
 final answer only. 
 
 The next difficulty in column addition is carrying and here again 
 the child's mind must be focussed upon whether there is a carry and 
 that he must add it as soon as he determines that there is. Of course 
 if any of the addition facts used are not thoroughly known, the new 
 difficulty will not be the only one that the child must contend with, 
 the attention will be distributed, and hence the first impression will 
 not be as vivid as it would be if the attention could be focussed upon 
 the one new idea. This emphasizes the importance of reducing the 
 number facts to habit. 
 
 The third difficulty is adding a column containing facts not in- 
 cluded among the forty-five elementary number facts. This has been 
 fully discussed under series and column addition, but under each new 
 class of series the teacher must emphasize and hold the child's atten- 
 tion upon the new idea till it is fully grasped and can be applied with- 
 out hesitation. As soon as this point is reached and no sooner, the 
 child is ready for another difficulty. It is the piling up of difficulty 
 after difficulty without mastering each in turn that creates havoc in 
 arithmetic teaching. 
 
 79
 
 In subtraction the first difficulty is the idea of subtraction itself. 
 As already explained, this idea should be brought out by use of simple 
 problems within the child's own experience and comprehension. Do 
 not introduce the Austrian method till sure that the subtraction idea 
 is fully grasped. So too the preparation for subtraction must be fully 
 in hand before formal subtraction begins. Then these two ideas must 
 be combined; the child finds the difference by applying the prepara- 
 tion. If very simple examples are first used, he will see the connec- 
 tion between the two ideas. Next is the carrying, now used in place 
 of the old borrowing. The method of focussing upon this new diffi- 
 culty has been fully discussed. 
 
 As in addition and subtraction, so in multiplication and division, 
 focus the child's attention closely upon each new idea in turn. The 
 steps have already been indicated, in some cases merely by progres- 
 sive type examples. These type examples should be closely studied 
 by the teacher to determine what is the essential new difficulty intro- 
 duced in each. Particularly note the general class of long division 
 examples in which the quotient figure cannot be directly found by 
 dividing the first digit or the first two digits of the dividend by the 
 first digit of the divisor. The new difficulty upon which to focus here 
 is that the subtrahend cannot be larger than the minuend. To be 
 sure, this difficulty may have come up incidentally before, due to a 
 child's carelessness, but then the teacher should merely have called 
 attention to the particular error, and the special difficulty should have 
 been left to a later time, as then some other idea was under focus. 
 Now the teacher places on the board several examples introducing 
 the new difficulty and questions the class regarding each step, and 
 then when the difficulty is reached in the example, she calls attention 
 to it and asks the class to watch hereafter for this trouble. After 
 the first example, the class should see the error as they are looking for 
 just this. After several examples have been worked at the board by 
 one child under the direction of the class and the supervision of the 
 teacher, the whole class should be sent to the board and an example 
 dictated with special instructions to watch for the new difficulty. 
 Until the class can work such examples without effort, the teacher 
 will find it necessary to remind the different individuals that they are 
 watching for something. 
 
 Similarly handle the principle that the remainder must be small- 
 er than the divisor. 
 
 In teaching reading and writing numbers above 1000, focus on 
 the fact that the digits between two commas or to the left of a comma 
 should be read as if they stood alone and then the name given as 
 thousand or million as the case may be. Then as a preparation for 
 the next step, focus on the fact that there must be three figures be- 
 tween any two commas or to the right of a comma. Emphasize this, 
 calling attention to it repeatedly. 
 
 80
 
 Then in reading numbers in which there are no hundreds or no 
 hundreds and tens, focus on the fact that the digits or digit are placed 
 to the right of the period and that the vacant places are filled in with 
 ciphers in order to make three places in the period. 
 
 To teach writing such numbers, the teacher should write on the 
 board 25,067 and have the class at the board copy this, then have 
 them write dictated numbers directly below it in the proper place. 
 In dictating 27,048, the teacher should say "27 thousand" and pause 
 long enough for the class to write 27 and place the comma after it. 
 Always hesitate thus and insist on the placing of the comma at once. 
 As soon as the class has written the comma, continue the dictation — 
 "48", letting the voice fall for the first time to indicate that the dic- 
 tation is complete, and question as to where the digits must be placed. 
 Ans. To the right of the period. Under what figures? Ans. Under 
 67. How many figures must there be in a period? Ans. Three. 
 Then what must you do to make three places? Ans. Write in a 
 naught. 
 
 Similarly teach numbers in which two naughts must be inserted. 
 After a time drop the questions but caution the class to be on the 
 watch for the new point just learned. If some member of the class 
 still has trouble and fails to insert the naught, question him as be- 
 fore. 
 
 In all work, fractions, denominate numbers, decimals, percent- 
 age, proportion, and square root, follow the same general plan of 
 focussing closely for some time upon the specific difficulty introduced 
 in each type example. In the end time will thus be saved. 
 
 In Proportion the order of steps would be first a preparation 
 showing that if the product of two factors equals the product of two 
 other factors, and one of the factors is missing, it may be found by 
 dividing the product of one pair of factors by the factor in the pair 
 from which the one is missing. Place the two factors to be multiplied 
 above the line and the single factor below the line and cancel. 
 
 Introduce proportion by means of diagrams illustrating the fact 
 that at a given time objects of different heights cast shadows of differ- 
 ent lengths; the higher the object, the longer the shadow cast. Also 
 use any other illustrations that will make the subject clear. 
 
 Next teach the statement of the proportion, emphasizing that 
 like numbers should be grouped in one ratio. The heights of the ob- 
 jects should be placed in one ratio and the lengths of the shadows in 
 the other. Give much practice in stating problems before attempting 
 solution. 
 
 Follow this with the principle that the product of the means 
 equals the product of the extremes, again giving practice in deter- 
 mining what terms should be multiplied together. Then basing upon 
 the preparation mentioned above, teach the method of finding the 
 required term. 
 
 81
 
 The next difficulty will be inverse proportion. Use a simple ex- 
 ample as: If one boy piles a certain amount of wood in 3 hours, how 
 long will it take two boys working at the same rate to pile the same 
 amount? Have the like numbers stated in one ratio, then place the 
 odd term in either the mean or the extreme in order that the result 
 may be larger or smaller than the given odd term as desired. That 
 is, if the result should be larger than the odd term, place the odd 
 term so that it will multiply the larger of the other two terms; if the 
 desired result should be smaller than the odd term, place the odd term 
 to multiply the smaller of the two other terms. Hereafter the first 
 thing to be considered in a problem is, whether in the new condition 
 the result should be greater or smaller than in the first condition 
 stated, then the example should be so stated that the result will be 
 greater or smaller as desired. 
 
 If compound proportion is to be taught, first put down the ratio 
 containing the unknown as a mean; then state each condition so that 
 the unknown will be increased or decreased as the condition requires. 
 Note that in determining how to state each ratio, its effect upon the 
 unknown alone is considered and all other conditions are ignored for 
 the time being. If the unknown will be increased by a given condi- 
 tion, place the larger term in the extreme and the smaller in the 
 mean; if the unknown will be decreased by the condition, place the 
 smaller term in the extreme. 
 
 Example: If it takes 50 men 12 days of 10 hours each to dig a 
 ditch 80 rods long, 6 feet wide, and 4 feet deep; how many men work- 
 ing 15 days of 8 hours each will be required to dig a similar ditch 96 
 rods long, 5 feet wide, and 6 feet deep? 
 
 Will it take more or less men working 15 days than it takes 
 when they work 12 days? Ans. Less. Then write the smaller nub- 
 ber, 12, in th extreme to multiply the 50, and the 15 in the mean to 
 divide the 12 x 50. 
 
 Will it take more or less men if they work 8 hours a day instead 
 of 10 hours? Ans. More. Then state so that the result will be more; 
 that is, multiply the extreme, 50, by the larger number 10, by placing 
 it in the other extreme. Where shall we place the 8? 
 
 Will it take more or less men to dig a ditch 96 rods long than to 
 dig one 80 rods long? Then place to obtain more. 
 
 Will a 5 foot ditch require more or less men than a 6 foot ditch? 
 Etc. 
 
 50 : X 
 
 Place all the extremes above the line and all the means below 
 and cancel. 
 
 15 
 
 12 
 
 8 
 
 : 10 
 
 80 : 
 
 96 
 
 6 
 
 : 5 
 
 4 
 
 : 6 
 
 82
 
 The following progressive steps in square root should be empha- 
 sized in turn: 
 
 1. Mentally extract the square root of such numbers as 16; 49, 
 81; 100; 144; 400; etc. 
 
 2. Mentally extract the square root of the largest square con- 
 tained in any number of one or two digits. The square root of the 
 largest square in 85 is 9; that is the largest square in 85 is 81, whose 
 root is 9. The square root of the largest square in 80 is 8. Etc. 
 
 3. Find the square root by factoring. The square root of 36 
 equals the square root of 4x9 = 2x3 = 6. The square root of 225 = 
 the square root of 9x25 = 3x5 = 15. Or the square root of 225 = 
 the square root of 3x3x5x5 = 3x5 = 15. Therefore to find the 
 square root of a perfect square, find its prime factors and take one 
 factor from each pair of identical factors as the factors of the root. 
 Find the square root of 3969. 3969= (7 X 7) X (3 X 3) X (3x3), 
 therefore the square root of 3969 is 7 X 3 X 3, or 63. 
 
 4. Teach pointing off into periods of two figures each both ways 
 from the decimal point. 567.5780; 5 '83 82. 60 00 00. 
 
 5. Develop the regular method of extracting the square root of 
 integers of three or four digits. Teach by means of diagram or rule. 
 See Stamper, op. cit., pp. 116-118. After developing method of solu- 
 tion, use "chalk and talk" method with class at board. 
 
 6. Square root of numbers of three periods. 
 
 7. Exact root of integer and decimal of two and four decimal 
 places. 
 
 8. Appropriate root found by annexing ciphers to the decimal. 
 
 MONTESSORI RODS. 
 
 Experiments in the first grade along the line suggested in 
 Myers, Experimental Psychology, pp. 72-90, indicate that a class will 
 learn a set of facts logically grouped together as soon as, if not 
 sooner than, it wili learn a single fact by itself. For example, to 
 teach the facts whose sum is six, have each child take a six inch rod 
 and find all the pairs of rods that placed together will be equal in 
 length to the six inch rod. Have them write these facts in the four 
 ways and give various drills. The next day have them write or tell 
 all the facts in the "six family" that they can remember. Then have 
 them find any forgotten facts by means of the rods. Before taking 
 up the "seven family", see that they thoroughly know all previous 
 "families". Children enjoy this work as it appeals to the duzxU 
 instinct.
 
 ERRATA. 
 
 Page 33. 124 8 
 
 6/0)744/0 8 000)65/721—1721 
 
 Page 49, above Section XV, inverted figure 7 (£) should be a cipher. 
 
 Page 41, fourth line from bottom, the. 
 
 Page 42, line 11. reduced. 
 
 Page 50, XVIII. MULTIPLICATION. 
 
 Page 57, line 4. 5160 should be 51600. 
 
 XIV. line 10. 11.52 should be 115.2. 
 
 Page 61, See that each .17 in (b) is directly below the corres- 
 ponding .17 in (a). 
 
 Page 71, line 5 from bottom, money. 
 
 Page 77, line 4 from bottom, .005 should be .015. Also 90 should 
 be below left hand column of dates and 35 below 
 right hand column. 
 
 34
 
 BIBLIOGRAPHY. 
 
 Bailey. A handy book on teaching arithmetic. Bailey, Yonkers, N. 
 
 Y. 1913. 
 Brooks. The Philosophy of arithmetic. Normal Publishing Co. 
 
 Philadelphia. 1876. 
 Brown and Coffman. How to teach arithmetic. Row Peterson and 
 
 Co. 1914. 
 Gildemeister. The multiplication tables. A Flanagan Co. Chicago. 
 
 1905. 
 McLellan and Dewey. Psychology of number. Appleton. 1895. 
 McMurry. Special method in arithmetic. Macmillan. 1905. 
 Maxson's self-keyed number cards. J. L. Hammett Co., New York. 
 Shutts. Handbook of Arithmetic. Ginn. 
 Smith. The teaching of elementary mathematics. Macmillan. 1900. 
 
 The teaching of arithmetic. Teachers College. 1909. 
 Stamper. A text book on the teaching of arithmetic. American 
 
 Book Co. 1913. 
 Stone. Arithmetical abilities and some factors determining them. 
 
 Teachers College. 1908. 
 Suzzalo. The teaching of primary arithmetic. Houghton, Mifflin Co. 
 
 1911. 
 Walsh. Methods in arithmetic. Heath. 1911. 
 Woodfield. A manual on the teaching of division. Flanagan. 
 Young. The teaching of mathematics. Longmans. 1906. 
 
 The teaching of mathematics in Prussia. Longmans. 1900. 
 Though many other works have been mentioned in the text, only 
 those dealing specially with the teaching of arithmetic have been in- 
 cluded in the above list. 
 
 85
 
 CONTENTS. 
 
 PART I. 
 
 General Principles. 
 
 I. Reasons for the study of method 4 
 
 II. Aims of Arithmetic teaching 5 
 
 III. Function and extent of objective teaching 5 
 
 IV. Essentials for efficient recall 5 
 
 V. Teaching a new number fact — Drills 6 
 
 VI. Reviews 7 
 
 VII. Concert recitations 8 
 
 VIII. Attention 8 
 
 IX. The art of questioning 9 
 
 X. Class vs. individual recitation 9 
 
 XI. The inductive development lesson 10 
 
 XII. Teaching a new operation 11 
 
 XIII. The deductive development lesson 12 
 
 XIV. Problem solving 12 
 
 XV. Teaching how to study 13 
 
 XVI. The study recitation 14 
 
 XVII. The assignment 15 
 
 XVIII. Dictation 1G 
 
 XIX. Oral and silent mental arithmetic 16 
 
 PART II. 
 
 Primary Arithmetic. 
 
 1. First lessons in number 17 
 
 II. Numbers above nine IS 
 
 III. Reading and writing numbers above 1000 19 
 
 IV. Addition 20 
 
 V. Series and column addition , 22 
 
 VI. Subtraction 26 
 
 VII. Multiplication 28 
 
 VIII. Division , 30 
 
 IX. Short division 31 
 
 X. Long division 32 
 
 XI. Roman notation 33 
 
 XII. Cancellation 34 
 
 86
 
 PART III. 
 
 Denominate Numbers and Practical Measurements. 
 
 I. Denominate numbers 36 
 
 II. Square measure 37 
 
 III. Cubic measure 
 
 IV. Practical measurements 38 
 
 V. Perimeter-fences 38 
 
 VI. Area — acres : ; >K 
 
 VII. Plastering 39 
 
 VIII. Cement walks, paving, etc 40 
 
 IX. Painting 40 
 
 X. Papering 40 
 
 XI. Carpeting 41 
 
 XII. Board measure 44 
 
 XIII. To find the number of bushels in a bin 44 
 
 XIV. To find the number of gallons in a tank or cistern. . . 41 
 
 PART IV. 
 
 A. Common Fractions. 
 
 I. Proper fraction taught objectively 45 
 
 II. Finding a fractional part of a number 45 
 
 III. The improper fraction 45 
 
 IV. A fraction an indicated division 45 
 
 V. To express an integer as a fraction 46 
 
 VI. Reduction of a fraction to an equivalent fraction 
 
 with a larger denominator 46 
 
 VII. Reduction to lowest terms 46 
 
 VIII. Addition and Subtraction of simple fractions with a 
 
 common denominator 46 
 
 IX. Reduction of a mixed number to an improper fraction 47 
 
 X. Reduction of an improper fraction to a whole or to 
 
 a mixed number 47 
 
 XI. Reduction to a common denominator when one 
 denominator is a multiple of the other and 
 
 addition and subtraction of such fractions 47 
 
 XII. Reduction to least common denominator when two 
 or more denominators have a common factor 
 and addition and subtraction of such fractions. . . 47 
 
 XIII. Reduction to a common denominator when all 
 
 denominators are prime to one another and 
 addition and subtraction of such fractions 48 
 
 XIV. Addition of mixed numbers 48 
 
 A. Sum of two fractions equal to 1. 
 
 B. Simple fractions that may be added mentally. 
 
 C. More difficult fractions. 
 
 87
 
 XV. 
 
 XVI. 
 
 XVII. 
 
 XVIII. 
 
 XIX. 
 
 XX. 
 
 XXI. 
 
 XXII. 
 
 XXIII. 
 
 XXIV. 
 
 XXV. 
 
 XXVI. 
 
 XXVII. 
 
 I. 
 II. 
 III. 
 
 IV. 
 
 V. 
 
 VI. 
 
 VII. 
 
 VIII. 
 
 IX. 
 
 X. 
 
 XI. 
 
 XII. 
 
 XIII. 
 
 XIV. 
 
 Subtraction of mixed numbers 49 
 
 Multiplication of a fraction by an integer 50 
 
 A. Multiplication of numerator. 
 
 B. Division of denominator. 
 
 C. Cancellation. 
 
 Multiplication of an integer by a fraction 50 
 
 Multiplication of a fraction by a fraction 50 
 
 Multiplication of a mixed number by a mixed number 50 
 
 Division of a fraction by an integer 51 
 
 Division of an integer or of a fraction by a fraction 51 
 
 Division of a mixed number by an integer 51 
 
 Division of an integer by a mixed number 51 
 
 Division of a mixed number by a mixed number. ... 52 
 
 Multiplication of a mixed number by a fraction .... 52 
 
 Multiplication of an integer by a mixed number. ... 52 
 
 Division of a mixed number by a fraction 52 
 
 B. Decimal Fractions. 
 
 Meaning 53 
 
 Reading and writing 53 
 
 Place value 53 
 
 Cipher at the right of a decimal 53 
 
 Reduction of a fraction to a decimal 54 
 
 Reduction of a decimal to a fraction 54 
 
 Addition and subtraction of decimals 54 
 
 Multiplication of decimals 54 
 
 Division of decimals 54 
 
 A. Decimal by an integer. 
 
 B. Decimal by a decimal 55 
 
 Fraction at the end of a decimal 55 
 
 Application of decimals in division by a mixed number 55 
 
 Aliquot parts — Table 56 
 
 Multiplication by aliquot parts 56 
 
 A. By aliquot parts of 1. 
 
 B. By aliquot parts of 10, 100, or 1000 57 
 
 Division by aliquot parts 57 
 
 A. By aliquot parts of 1. 
 
 B. By aliquot parts of 10, 100, or 1000. 
 
 C. Fractional Relations. 
 
 I. To find a fractional part of a number 57 
 
 II. To find what part one number is of another 57 
 
 III. To find the whole when a part and its fractional 
 
 relation the whole are given 58 
 
 D. Oral Analysis, 59. 
 88
 
 PART V. 
 Percentage. 
 
 I. Preparation for percentage 60 
 
 II. Introduction of terms per cent, Rate per cent, and Rate 
 
 III. Introduction of terms Percentage and Base 01 
 
 IV. Given the Base and the Rate to find the Percentage. . 61 
 V. Given the Base and the Percentage to find the Rate. 63 
 
 VI. Given the Percentage and the Rate to find the Base 6 
 
 VII. Amount 64 
 
 VIII. Given the Amount and the Base to find the Rate. . . 
 
 IX. Given the Amount and the Percentage to find the Rate 65 
 
 X. Given the Rate and the Base to find the Amount. ... 65 
 
 XI. Given the Amount and the Rate to find the Base .... 6(5 
 
 XII. Given the Amount and the Rate to find the Percentage <;7 
 
 XIII. Given the Percentage and the Rate to find the Amount 67 
 
 XIV. Difference 67 
 
 XV. Given the Difference and the Base to find the Rate. . . 67 
 
 XVI. Given the Difference and the Percentage to find the Rate 67 
 
 XVII. Given the Rate and the Base to find the Difference . . 68 
 
 XVIII. Given the Difference and the Rate to find the Base ... 68 
 XIX. Given the Difference and the Rate to find the 
 
 Percentage 68 
 
 XX. Given the Percentage and the Rate to find the 
 
 Difference 68 
 
 XXI. Several successive discounts 69 
 
 XXII. To find a single Rate of discount equivalent to 
 
 several successive Rates 70 
 
 XXIII. Marking goods 70 
 
 XXIV. Simple interest 71 
 
 XXV. Given the Interest and two elements to find the 
 
 third element 72 
 
 XXVI. Six per cent method 72 
 
 Six per cent blank 73 
 
 XXVII. Compound interest 71 
 
 XXVIII. Bank Discount 74 
 
 Bank discount blank 76 
 
 XXIX. Discounting interest bearing notes 77 
 
 ADDENDA. 
 
 Prevention of copying 78 
 
 Only one difficulty at a time 79 
 
 Montessori rods 83 
 
 ERRATA, 84. 
 
 BIBLIOGRAPHY, 85. 
 
 TABLE OF CONTENTS, 86. 
 
 INDEX, 90. 
 
 89
 
 INDEX 
 
 Accuracy, importance of 5 
 
 Acres, computing 38 
 
 Addition: 
 
 evils of counting in 20 
 
 use of Montessori rods. . . .20, 83 
 
 review devices 21 
 
 series and column addition 
 
 22-26, 79 
 
 of fractions 46-48 
 
 of mixed numbers 48 
 
 of decimals 54 
 
 Aims of arithmetic teaching. . . 5 
 
 Aliquot parts 56-57 
 
 Amount 64-67 
 
 Analysis 59 
 
 Area 37-38 
 
 Art of questioning 9 
 
 Assignment 15 
 
 Attention 6, 8, 9, 78 
 
 Austrian method: 
 
 of subtraction 26 
 
 of division 31, 54, 84 
 
 Bagley, quoted: 
 
 on drudgery 6 
 
 on statement of aim 11 
 
 on the assignment 15 
 
 Bank discount 74-77 
 
 blank 76 
 
 Base 61 
 
 Batavia system 10 
 
 Bins, finding contents of 44 
 
 Blanks: 
 
 perimeter 38 
 
 acres 39 
 
 plastering 39 
 
 papering 41 
 
 carpeting 43 
 
 6 % method 73 
 
 bank discount 76 
 
 Board measure 44 
 
 Bushels in a bin 44 
 
 Cancellation , 34 
 
 Cards for teaching facts 21 
 
 self -keyed number 84 
 
 Carpeting 41-43 
 
 Carrying 21, 22, 79 
 
 Cement walks 40 
 
 "Chalk and talk" method, 
 
 11, 16, 28, 32, 83 
 
 Chalk ready at board 22 
 
 Cipher at right of decimal 53 
 
 in writing numbers 81 
 
 Cisterns, finding number of 
 
 gallons in 44 
 
 Class vs. individual recitation. . . 9 
 
 Column addition 20, 22-26, 79 
 
 Commission 69 
 
 Concert recitations 8 
 
 Copying, prevention of 78 
 
 Counting 17 
 
 in addition 20-21 
 
 Cubic measure 38 
 
 Dates, subtracting 73 
 
 Decimal fractions 53-57 
 
 Deductive development lesson. .12 
 
 Definition vs. use of terms 27 
 
 DeGarmo, quoted: 
 
 on art of questioning 9 
 
 on statement of aim 10 
 
 Denominate numbers 36-38 
 
 Development lessons: 
 
 inductive 10, 41, 65 
 
 deductive 12 
 
 Dictation 16, 80 
 
 Difference 67 
 
 Difficulties, one at a time 
 
 12, 30, 79-83 
 
 Discount: 
 
 cash 69 
 
 bank 74-77 
 
 Division 30-33 
 
 of fraction & mixed numbers 51-52 
 
 of decimals 54 
 
 by a mixed number 55 
 
 by aliquot parts 57 
 
 Drills 6, 8 
 
 Eliot, George, quoted on interest 9 
 
 Erasing work discouraged 63 
 
 Errors corrected by child 29 
 
 90
 
 First lessons in number 17 
 
 Formula: 
 
 for percentage 62 
 
 for amount 65 
 
 for difference 67 
 
 for marked price 70 
 
 for successive discounts. .. .69-70 
 
 for interest 72 
 
 for compound interest 74 
 
 Fractional relations 57-59 
 
 Fractions 45-52, 55 
 
 Froebel quoted 5 
 
 Gallons in a tank 44 
 
 Generalization 11, 41, 66 
 
 General principles 4-16 
 
 Gildemeister quoted on the order 
 
 of multiplication tables.. ..29 
 
 Gordy quoted on reviews 7 
 
 Grube method, weakness of . . . .29 
 
 Hall, Frank, quoted on careless 
 
 facility 5 
 
 How to study, teaching 12-15 
 
 Indefinite relative magnitude.. .17 
 Individual vs. class recitation. .. 9 
 Inductive development lesson 
 
 10, 41, 65-66 
 
 Interest 6, 9 
 
 Interest, simple 71 
 
 6'r method 72 
 
 compound 74 
 
 Knowledge, kinds of 7 
 
 Logic 12 
 
 Marking goods 70 
 
 Maxson's self-keyed number cards 84 
 
 McMurry, Charles, quoted: 
 
 on aim of arithmetic 16 
 
 on the use of books 15 
 
 on order of tables 29 
 
 McMurry, F. M., quoted: 
 
 on dependence 13 
 
 on power of initiative 14 
 
 Montessori rods 20, 83 
 
 Multiplication 28-30 
 
 of fractions & mixed numbers50-52 
 
 of decimals 54 
 
 by aliquot parts 56 
 
 Munsterberg, quoted: 
 
 on old and new facts 7 
 
 on soft pedagogy 9 
 
 Number fact, teaching new . . . . 6 
 Numbers 17-19 
 
 Objective teaching 6 
 
 Operations, teaching new 11 
 
 Oral analysis 59 
 
 Oral arithmetic 16 
 
 Painting 40 
 
 Papering 40 
 
 Paving 40 
 
 Partition 30 
 
 Per cent 60 
 
 Percentage 60-67 
 
 terms learned before problems 61 
 statements of relation and 
 
 solution 58, 59, 63 
 
 Percentage : 
 
 the term 61 
 
 several for one base 64 
 
 a rate for each 64 
 
 Perception, clear, essential 5 
 
 Perimeter-fences 38 
 
 Place value 53 
 
 Plastering 39 
 
 Pointing 22 
 
 Practical measurements 38-44 
 
 Primary arithmetic 17 
 
 Problems: 
 
 size of numbers in first 
 
 written 16, 30 
 
 solving 12 
 
 cancellation in 34, 35 
 
 Proceeds, net 69 
 
 of a note 76 
 
 Proportion 81 
 
 Questioning, art of 9 
 
 Quintilian, quoted 17 
 
 Rate 60 
 
 one for each percentage 61 
 
 Reading numbers 17-19, 80 
 
 decimals 53 
 
 Reasons for study of method... 4 
 
 91
 
 Recall, essentials for efficient. . . 5 
 Reduction: 
 
 of denominate numbers. . . .36-37 
 
 of fractions 46-48, 54 
 
 of decimals 54-55 
 
 Reviews ... .- 7 
 
 Rods used in teaching addition 20, 83 
 Roman notation 33 
 
 Selling price: ^ 
 
 when there is a gain 64 
 
 when there is a loss 67 
 
 in commission and discount. .69 
 
 Series drill in addition 22-26 
 
 Silent mental arithmetic 16 
 
 Snow storm illustration 7 
 
 Square measure 37 
 
 Square root 83 
 
 Study: 
 
 teaching how to 13 
 
 recitation 14 
 
 Subtraction 20, 26-28, 80 
 
 of fractions 46-48 
 
 of mixed numbers 49 
 
 of decimals 54 
 
 Successive discounts 68-70 
 
 Syllogism 12 
 
 Teachers, kinds of 4 
 
 Terms, use of vs. definition . . . . 27 
 
 Walks, cement, on two or more 
 
 sides of lot 40 
 
 Waste in carpeting 42-43 
 
 Writing: 
 
 numbers 17-20, 80 
 
 decimals 53 
 
 Written problems, size of numbers 
 in first 16 
 
 Young, quoted on one step at a 
 
 time 12 
 
 M
 
 
 
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