SB 17 Outlines For Methods In Arithmetic MANCHESTER OUTLINES FOR METHODS IN ARITHMETIC RAYMOND E. MANCHESTER Head Dept. Mathematics, Kent State Normal School AUTHOR The Teaching of Mathematics Brief Course in Algebra John Citizen and His School Teaching Outlines, etc. DERRICK PUBLISHING COMPANY, PRINTERS Oil City, Pennsylvania \ COPYRIGHT 1922 R. E. MANCHESTER CONTENTS . PART I. General Outlines. Introduction. Course of Study. Lesson Types. Devices, Games, Drill Charts, Tests, Marking and Diagnosis Problem Analysis. PART II. Subject Matter Outlines. Fundamental Number Ideas. Number Appreciation (Sense Training). Language (Number System, etc.) The Fact Groups. Addition (Whole Numbers). Subtraction (Whole Numbers). Multiplication (Whole Numbers). Division (Whole Numbers). Introduction to Fractions. Equivalent Forms (Fractions). Addition (Fractions). Subtraction (Fractions). Multiplication (Fractions). Division (Fractions). The Three Problems in Fractions. Decimal Fractions. Percentage. Special Topics. 491083 INTRODUCTION As an introduction to the course in methods students are asked to consider the historical background of arithmetic, the values of the sub- ject and some of the modern aims and purposes. The following out- lines are arranged merely to serve as outlines for study. iiiimiiiiiiiiiiiiiiiiimiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiim 1 HISTORICAL : BACKGROUND Although^lUiS/tiot:neegsary; tor teachers of arithmetic to know the complete" history of the subject it is desirable that they know that arithmetic is one of the subjects of the common people and to know enough of its beginnings and development to appreciate its importance as a subject for study. In a course of this kind it is desirable to pre- sent the outstanding points connected with the rise of arithmetic and the needs all people of all times have had for it. It may inspire some teacher to know that she has something to give that has the justifica- tion of an historical background. OUTLINE FOR STUDY 1. Arithmetic developed among trading nations. 2. It has flourished when nations have been active. 3. It is a subject that has been kept alive through the interest of the common people. 4. It has been changed to meet the needs of the common people, 5. It has always been a subject for study. (Called one of the com- mon branches.) 6. It is now changing to meet new demands of society. 7. Teachers need to appreciate the necessity of meeting new needs. iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiim VALUES OK ^ Most of us accept arithmetic; j w]thou't' <}uestio^ as a- necessary sub- ject in the curriculum. This is bt^ca ; u^e j hcarly'^J^ti^tal experiences are connected in part with ideas of quantity. Arithmetic provides a way to express thoughts about quantity and its measurement. It pro- vides other things as well, such as training in useful habits in thinking, useful habits in oral and written expression, a group of facts useful in daily contact with common business practice, a part of oral and written language needed in ordinary conversation, an introduction to advanced study in other lines of mathematics and sciences, a group of facts need- ed in industry, etc. It is not only unnecessary but quite impossible to list all the values of the subject, but it is possible and also desirable to classify the values by groups and to discuss the relative values of the groups. Those who enjoy the study of mathematics and those engaged in vocations and professions, in the practice of which considerable knowl- edge rf mathematics is needed, would rate the values of arithmetic much higher than would those who derive less pleasure from and have less need for the subject. It is then reasonable to suppose that differ- ent groups of people would stress these values differently. It is good educational theory to accept this fact and arrange courses of study ac- cordingly. OUTLINE FOR STUDY 1. Provides an addition to our language that is necessary to ex- press fundamental ideas of quantity, form and position. 2. Provides training needed for appreciation of number relations. 3. Provides a group of usable facts about number relations and measurements. 4. Provides a group of processes through the use of which new facts may be discovered. 5. Provides training in methods of thought. 6. Provides special discussion of the common applications of facts and processes to daily affairs of life. 7. Provides mental pleasure. 8. Has traditional cultural values. 9. Provides a valuable group of mental and physical habits. iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiimin i} MODERN- PURPOSES The preser/t nees, of .soeiety for any subject are far different from the needs of e:vt$t a^w^a'rk'ajgq/ Every teacher knows that the per- fection of modern business methods and the invention of machinery have made unnecessary a complete knowledge of many of the processes and special topical discussions and have created needs for new and dif- ferent things. The teacher of arithmetic must be prepared to adapt herself to changing conditions and to select and stress those parts of the subject most useful to those who pass out from her instruction. OUTLINE FOR STUDY 1. A selection of facts on a basis of utility values 2. A discussion of the processes with stress on social needs for them. 3. A selection of applications based on the organization of modern business. 4. A discussion of the logical order of topics with reference to con- tinued study of mathematics. 5. A discussion of mental development values. 6. A discussion of the study of arithmetic from the point of view of the application. THE COURSE OF STUDY When considering a course of study the student should have in mind the educational theory upon which the course is based, the general plan followed in making the course, the plan of organization and ar- rangement, the purpose of the course and the selection, classification and distribution by grades of the subject matter. The following out- lines are arranged to guide the student in this discussion. iiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiimimiiiiiiimiimiiiiiiiiiiiiiiiiiiiiiin EDUCATIONAL THEORY 1. Education defined as a process of acquiring knowledge. Under such a theory the stress is upon the acquiring of facts and skills. It is a definition dominating many research schools as well as many of the technical and vocational schools. The effect of such a theory on a course of study in arithmetic is to make prominent the memorizing of facts and the development of speed and accuracy in the processes. Less attention is given to a study of the reasons underlying the steps in the process and more attention is given the learning of the steps themselves. 2. Education defined as a development of the mind. This definition of education stresses the development of the mental processes and makes the power to think the goal. Such a definition affects a course of study by placing stress on the thought processes in- volved in a study of number relation. Less importance is given to memorizing of facts and to speed and accuracy in operation and more importance is given to analysis, synthesis, and association of ideas. The general effect upon the course of study is to delay the opening up of topics, to give more time to their development and to keep a logical development of thought intact. 3. Education defined as adaptabilty. Many feel that the end of education is to provide one with enough general information to enable him to adapt himself to any situation. Exact and definite instruction is not given so much prominence as the classification of knowledge in usable shape. In arithmetic the effect is to stress the acquiring of general principles and the ability to find and use needed facts and process when needed. Less importance is given the knowing of facts and speed and accuracy in the operations and more importance is given the use of reference books and tables. Analy- sis is stressed since it serves adaptability. 4. Education defined as power to serve society. This definition is based on the consideration of an individual as a unit in the social group. It suggests the perfect individual as one who can serve the group best. A course of study based upon this definition stresses the development of initiative, inventiveness, and leadership. The effect on the course in arithmetic is to place less attention on the facts and processes for their own sake and more upon the use of them in the general social affairs of life. 5. Education defined as a preparation for immediate needs. Such a definition emphasizes the needs of the present rather than of the future. In place of a storing-up of knowledge or of power for future unknown situations the definition would suggest a solution of each problem as it arises. The effect on arithmetic is to eliminate greatly the time given the subject in the grades and to cut down greatly the subject matter of facts and processes. All special discussions would be left until such time as they may enter the life of the individual. IIIIIIIIIIIIIIIIIUIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIHIIIIIIIIIIIIIIIIIIIH GENERAL PLAN 1. A Topical Development. The older courses were arranged to develop each topic rather com- pletely before a new one was opened. This plan is successful only when all pupils complete the course and is not followed to any great extent. 2. A Spiral Development. This plan calls for the continuous development of many topics in each grade. Many topics are carried forward simultaneously. This plan is not used in pure form to any great extent. The spiral idea is used, however, in most courses. 3. A Combination Plan. Most courses are built on a merged plan of topical and spiral devel- opment. 4. Development on Basis of Pupils' Needs. This plan is one of the recently suggested ones and has not been sufficiently tried to prove its worth. iiiiiiiiimiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiifiiimiiiiiiiiiiiw ORGANIZATION I By Large Units. 1. Primary grades 1, 2, 3. Intermediate grades 4, 5, 6. Junior High School 7, 8, 9. Senior High School 10, 11, 12. 2. Primary grades 1, 2, 3, 4. Grammar grades 5, 6, 7, 8. High School 9, 10, 11, 12. 3. Primary grades 1. 2, 3. Intermediate grades 4, 5, 6. Grammar grades 7, 8. High School 9, 10, 11, 12. 4. Elementary grades 1, 2, 3, 4, 5, 6. Junior High School 7, 8, 9. Senior High School 10, 11, 12. II By Small Units. 1. By years. 2. By half years. 3. By months. 4. By weeks and days. iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiini mi minimi mmiimmiimmiimi mm iiimiimi miimmmmmiii miiimiimmmmm PURPOSE OF COURSE I. A course to be followed in detail by the teacher. 1. A manual of practice. 2. Definite subject matter outlined. 3. Definite method suggested. 4. Definite tests and examinations provided for. 5. Initiative on part of teacher limited. 6. Fixed possibility for success of teacher's work. This plan is a valuable one for inexperienced teachers and for use in a closely organized system of schools. The work of teachers and pupils is standardized. II. As a suggestive outline. 1. Subject matter suggested but not fixed. 2. Method suggested but not fixed. 3. Tests and examinations suggested. 4. Possibilities not limited but directed. 5. Initiative suggested. This plan is a valuable one for experienced teachers who have power for initiative and a desire to do excellent teaching. It is not usable in a system of schools so organized that standardization is neces- sary. III. To establish limits for work by grades. 1. Subject matter fixed in large units. 2. Selection within units in hands of teacher. 3. Method not fixed. 4. Tests and examinations fixed with reference in definite time requirements but not as to methods. 5. Possibilities fixed for large units but not within units. 6. Initiative partially limited. This third plan is useful for teachers who need only a general guide and who have power to organize the work within limits. A good plan for use in the smaller systems when experienced teachers are in charge. iiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiimiiiiiiiimiimiiiiiimiiiimiiiiiiiiiiiiiiiimiiiiiiiim SUBJECT MATTER I. Selection. (1) For later use. (2) For organization and development of the subject. (3) To serve immediate needs of the pupil. (4) For mental exercise. (5) For mental recreation. (6) To prepare for other subjects. II Classification. 1. Fundamental number ideas. 2. Number appreciation (sense training). 3. Development of language (number system). 4. Fact groups. A. Addition. B. Subtraction. C. Multiplication. D. Division. E. Measurement. 5. Processes. A. Whole numbers, a Addition, b Subtraction, c Multiplication, d Division, e Measurement. B Fractions. a. Changing to equivalent forms b. Addition. c. Subtraction. d. Multiplication. e. Division. f. Simplification. c. Solution of equations d. Substitution. f. Use of the formula. 6. Applications. a. Measurement. b. Buying and selling. c. Construction. d. Production. e. Communication. f. Transportation. g. Money and credit. h. Analysis and solution of problems. III. Distribution. 1. By departments. 2. By grades. 3. By topics. LESSON TYPES Although this topic is discussed fully in the classes in pedagogy it is desirable for those preparing to teach arithmetic to consider lesson types with reference to this particular subject. imiiiiiiiiiiiiiiiiimiiiiiiiiiiiimiiiiiiiii iiiiiiiiiiiiniii iniiiiiiiiiiiiiiiiiiiii i mini minim iiiiiiiiimiiiiinn minim Development Lesson (Inductive) Purpose: To aid the pupil discover facts, principles, etc., through his own efforts. The procedure is to develop the general truth from particular experiences. Development Lesson (Deductive) Purpose: To aid the pupil discover facts, prniciples, etc., through his own efforts. The procedure is to develop the particular truth from the general truth. Exposition Lesson Purpose: To present facts, principles, etc., to the pupils. Study Lesson Purpose: To help the pupil study and acquire facts, principles, etc., through independent effort. Drill Lesson Purpose: To fix facts in mind and establish habits of operation and of thought. Appreciation Lesson Purpose: To stimulate pupils to effort by inspiring them with ex- ample, by arousing interest through observation of the work of others, or by appealing to the aesthetic emotions. Recitation Lesson Purpose: To follow up assigned work of any kind. Review Lesson Purpose: To organize the material presented and to recall facts, principles, etc., to mind. Test Lesson Purpose: To test pupils for ability, performance and development. iiiiiiiiiiiiiiiiiiiiiiiiniiiiiiiiiniiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiuiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiin OUTLINES Inductive Development Lesson I. Purpose. 1. To help the student learn facts, principles, etc., by helping him organize judgments. 2. To teach the process of inductive reasoning. II. Steps. 1. Preparation. Statement of the problem. 2. Development of all particular facts, etc. Discussions of these facts. Development of the general truth. III. Application of the general truth to problem solution. All work cannot be developed in this way since the aim of some lessons is not to discover new truths. It is also true that all pupils have not had sufficient experience to make this type of lesson useful. IV. Discussion of the method used. iiiiiiiiiiiiiiimiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiimiiimiiiimmiiiiiiiim Deductive Development Lesson I. Purpose. 1. To help the pupil learn facts, principles, etc., by having him or- ganize his own judgments. 2. To teach the pupil the method of deductive reasoning. II. Steps. 1. Statement of the general truth. 2. Statement of particular truths. 3. Verification of particular truths through experiment. III. Application of the particular facts in the solution of problems. IV. Discussion of the method used. IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIM Exposition Lesson I. Purpose. 1. To present facts, principles, etc., to pupils. II. Steps. 1. Preparation. a. General The pupils must be mentally ready to attend to the presentation. The teacher hiust develop a desire to learn. b. Particular This preparation includes all review of needed material and all instruction leading up to the lesson itself. New words must be defined, class-room material distrib- uted, etc. 2. Procedure. a. The idea should be developed through a series of logical steps. b. Each step should be carefully taught before the next is at- tempted. c. Summary The points of the lesson should be discussed and organized. III. Drill. After the lesson has been taught and the results organized there should be a short drill to fix the lesson in the minds of the pupils. IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIINIIIIIIIIIIINIH The Study Lesson I. Purpose. a. To teach the pupils to work independently. b. Help the pupil to realize the importance of learning to study. II. Pupils should 1. Have a definite statement of the problem to be solved. 2. Have all data needed. 3. Establish a method of procedure. 4. Record results accurately. 5. Develop power of analysis, concentration, etc. III. The problem of securing independent work on the part of the pupil is two-fold. 1. Teaching the pupils to appreciate the factors involved in study 2. Developing interest and attention. IV. It is necessary to teach the pupil how to study and to furnish mo- tive for study. V. Factors that encourage study. 1. The teacher. a. Should state the problem clearly. b. Should aid the pupil with all suggestions necessary. c. Should furnish an incentive for studying the lesson. 2. The recitation. a. Should be a test of the work of the study period. b. Should stimulate the pupil to a desire to recite. c. Should recognize individual differences in ability. 3. The suggestion of individual interests and life purposes. HiiiiiimiiiimnmiiiiiiiiiiiiiiiiiiiiiHiiiiiiiiiiiiiiiiiiiiiim Drill Lesson I. Purpose. 1. To fix facts of the steps in a process in mind by repetition and establish habits of operation and of thought. II. Steps. 1. Giving a motive for repetition. 2. Repetition with attention on the particular fact. III. Means cf holding attention during drill. 1. Devices, games, contests, etc. 2. Placing a time limit and recording performance. 3. Appealing to emulation. IV. Development of accuracy in practice. V. Drill should be regular and organized. iiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimmiiiiiiiM The Appreciation Lesson I. Purpose: To please, to inspire and to stimulate interest II. Steps. 1. A careful preparation. a. General statements to develop interest and get pupils men- tally ready to appreciate the lesson. b. Particular statements to recall facts and principles needed to appreciate the lesson. 2. Presentation of the lesson. a. By lecture (teacher). b. By lecture (outside person). c. By trips to places of interest. d. Through class discussion and conversation. e. Through dramatization of application of processes. f. By use of pictures, charts and other material. g. By use of games. h. By reports given by students, i. By development of a project. 3. Discussion of lesson. III. Discussion of method. minimi n nun i ninii niinii iiuiiiii i iiiiiiiiiiiiiiiiiiiiiiiiini iiiiiiiiiiiiiiiiiiiiiitiinii inn in 1111 The Recitation Lesson I. Purpose. 1. To test the work done in the study period. 2. To test the ability of the pupil. 3. To test for facts. 4. To test the ability of the pupil to present knowledge. 5. To test for analysis and organization. II. Forms of the recitation. 1. Question and answer Oral and Written. 2. Discussion open to pupils. 3. Presentation by pupils. III. Questions and Answers. 1. Good questions. a. Concise and definite. They should arouse thought on the part of the pupil. b. Should follow in logical order and should suggest an answer in child's own words. 2. Questions should be given to all members of the class, before naming the pupil to answer it. 3. No set order in questioning pupils should be followed. IV. Discussion. 1. Should give pupil an opportunity to stand on his feet and speak. 2. Suggest the organization of subject matter. iiiiiiimiiiiimiiiimiiiiiiiiiiimiiiiiiimiiiiiiiiiiiiiiiiiim The Review Lesson I. Purpose. 1. To give an opportunity for organization of larger units of the subject than the daily recitations afford. 2. To test the mastery of the subject. 3. To give a broader perspective of the work. II. Kinds. 1. Short range review, as one topic. 2. Long range review, as the work of a term or year. III. Forms. 1. Organizing the facts learned around some central topic. 2. Giving the pupils a topic to discuss. 3. The application of the facts learned to some new central topic or situation. 4. The review should reveal to pupil and teacher. a. Growth of the formation of habits. b. Things that need be clearly understood. 5. Time should be taken after each review to note carefully the situation as outlined. iiiiiiiimimiiimiiiiiiiiiiiiiiiiiimimmiiiiiiiiimiimiiiiiiiiiiiimm Test Lesson I. Purpose: To test pupils for performance, ability and development. II. Plan. 1. Cover the ground. 2. Provide sufficient time. 3. State the aims clearly. 4. Provide definite plan for marking 5. Establish fairness. Tests should be arranged in such a way that pupils feel that an opportunity is offered for expression. DEVICES, GAMES, ETC. The following outlines are suggested for use in discussion of de- vices, games, drill charts, tests, marking systems and diagnosis of fail- ing students. IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIH DEVICES I. Needs for 1. An aid in presentation of new work. 2. An aid in drill. 3. Graphic presentation. 4. Saves teacher's time. II. Classification. 1. Blackboard drawings. 2. Charts to hang on wall. 3. Cards to use as perception cards. 4. Number arrangements on adjustable frame. 5. Graded tests, and exercises. 6. Blocks, cardboard figures, etc. 7. Measuring units. 8. Pictures. 9. Pegs, splints, balls, seat work cards, etc. 10. Sand box. 11. Level. 12. Store. 13. Bank windows, etc. 14. Business papers. III. Selection. 1. Provision for age of pupils. 2. Provision for rural or city schools, 3. Provision for community life. 4. Provision for possibilities of school room. 5. Provision for mental strength of pupils. IV. Purpose and Values. 1. To supplement presentation of new work. 2. To supplement drill. 3. To provide for variety of appeal. 4. To provide for motivation. 5. To provide for relaxation. 6. To provide for connection with life interests. 7. To provide for group activity. 8. To save teacher's time. 9. To stimulate initiative. 111,11,11 Ill, Illllllllllllllllllllllllllllllllllllll Illlllllllllllllllllllll Illllllllllllllllllllllllll I Illlllllllllllllllllllllllllllllllllllltlllllt GAMES I. Needs for. 1. To offer variety of appeal. 2. To bring arithmetic into play. 3. To provide for relaxation. II. Classification. 1. Counting games. 2. Games to fix number facts. 3. Games to stimulate sense training. 4. Games to develop processes. 5. Games to suggest applications. III. Selection. 1. Provision for age of pupils. 2. Provision for rural or city school. 3. Provision for community interests. 4. Provision for possibilities of school room. 5. Provision for materials available. IV. Purpose. 1. To supplement presentation of new work. 2. To supplement drill. 3. To develop appreciation for number ideas. 4. For individual students. 5. For group work. V. Values. 1. Variety of appeal. 2. Relaxation. 3. Connection with life interests of pupils. 4. Stimulates initiative and leadership. NOTE See list of games following this outline. uimiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiM ARITHMETIC GAMES 1. COUNTING. 1. There are many games used in the very early stages of count- ing, but the most commonly used one is, finger play. For example, a few of these finger plays are: A. Here is the bee hive. Where are the Bees? Hidden away, Where nobody sees. Soon they come creeping Out of the hive, One, two, three, four, five. In this game the closed hands is the bee-hive. B. Five little children sliding on the floor One tumbled down and then there were four. Four little children laughing with glee One tumbled down and then there were three. Three little children sliding toward you One tumbled down and then there were two. Two little children sliding for fun One tumbled down and then there was one. One little child sliding all alone He tumbled down and then there were none. Children should use their fingers to represent the number of chil- dren. C. One, two, three, four, five, I caught a hare alive. Six, seven, eight, nine, ten, I let him go again. 2. Another commonly used game in the early stages of counting is the following.: The children are asked to skip a certain number of times, and to count each skip. The same method may be used with hopping, jumping, clapping hands, or tapping on desks. 3. A very simple, but beneficial game for the children in their counting is as follows: The teacher taps oh board or table or floor with pointer. The children listen to the taps and count them. 4. Number touch is a very important factor in the early training of the child. The teacher may ask a child to close his eyes, then touch his hand a certain number of times, and have him state the number. Probably, to make this more interesting, the children could try this with one an- other. 5. Another very beneficial game, used to improve the child's count- ing is to play the blackboard is the sky. The children draw stars on it, playing it is just twilight, and the stars are just beginning to show. The children may count silently, and then in unison, telling how many there are. iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiimiiiiiiiiiiiiiiiiiiiim 6. The teacher may have envelopes containing various pieces of colored paper or the like. She writes on the outside (in words) the number of pieces they contain. It is impossible for a first grader to read this, but the teacher herself knows how many pieces are in the en- velope. She passes the envelopes and the children count and tell her the number. They may exchange envelopes, so giving each child a chance to improve his counting. 7. A game to be used in first grade is one where the children string articles such as corn, wooden beads, etc., and use them for counting. 8. Have two or more children extend their arms at the same time raising one or more fingers. Have all the children guess the number of fingers raised and see whose guess is the nearest correct. The teacher keeps score. 9. A certain number as "5" is chosen for "buzz." Have the chil- dren count either 1, 2, 3, 4, 5, 6, etc., or by 2's, 3's, etc., but each number containing 5 or its multiple is omitted and the word "buzz" used in its place. This is good training for multiplication drills. II. READING AND WRITING 1. Place several numbers in different places on board, point to them, and have children read them. Individual work is of more value. 2. Write names "four," "seven," etc., on the board. Have children copy names and write figures corresponding to the names. 3. Perception Cards Use number group cards. Hold one card at a time in front of the children just for an instant. The one saying the number first gets the card. When all cards have been used announce as "winner" the one holding the greatest number of cards. 4. Place a box upon a table and have one of the pupils count the objects in it writing their number on the board. The number of objects should be changed as each pupil counts. 5. Write the names of several children on the board and let the children count the letters. This can be done in groups until the names of all the pupils have been used. 6. Jack Homer Pie. Write numbers on quite a number of small cards and drop them all in something used to represent a "pie." (A box or basket will do.) Then choose a pupil to recite, "Little Jack Homer." When he comes to the line "He put in his thumb," let the pupil put his hand in the "pie" and draw out a number. When he says, "And pulled out a ," say the number drawn. This can be used in groups as well. After the children have become accustomed to the game, you can leave out the poem and only repeat the line, "He put in his thumb and pulled out a (number). 7. Draw several concentric circles on blackboard. Divide all cir- cles into equal parts by diameters. Number each space. While one child (who is blindfolded) is letting the pointer wander over the circles, the other children are saying in concert "Tic tac toe, round I go; hit or miss, I'll take this." As children say "I'll take this," the child is un- blinded and reads the number to which he is pointing. If his pointer is on a line he must read the number on both sides of the line. 8. Card Game. Cards for the "Jack Horner Game" are used. These cards are placed in a basket. One child must stand by the basket and pick out the numbers. The other children go to the board. A card is drawn by the child at the basket. As he reads it, those at the board write the number. The instructor corrects numbers at the board. imiimiiiiiiiiiimiiiimiiiiiiiiiiimimiiiiiiiiiiiiiiiiiiiiiimiiiim III. Many of the following drills are usable for addition drill, but they may be used in subtraction, multiplication and division, too. 1. When the addition facts are first being worked out, games may be useful. Cards made to represent dominoes are useful. Each of the children is furnished with a card. They come to the front in turn and show card long enough for other children to see it. The child quickly puts the card behind him and asks "How many?" 2. Instructor places cards in chalk tray in regular order. A card, having a number on it, as 6, is held up by the teacher. Child takes two cards from tray whose sum is 6. This must be continued until all cards of the sum 6 are found. Another number is then used and the game continues. 3. Combinations without answers are written on the board. Cards are placed in ledge, which contain single numbers. Teacher points to a combination of numbers and children go to cards and pick out answer on the cards. Child may point to the combination on the board after the teacher has pointed to a card. 4. Distribute cards on which are different numbers to the children. Pick out a number such as 5, which several children have on their cards. Call for this number and the children who have the 5 must place it on the board. 5. Single number perception cards are used. One number is placed on the board. A number placed on the board by the teacher is added to the number which the child has on his card. The number on the board is changed. If child cannot add this the class is called on to re- spond. 6. Write a combination on the board, let class see it and erase. Scores for correctness may be kept by rows. 7. When a list of combinations has been written on board pupils in turn tell the answers, then one child alone gives all the answers. 8. Have a list of combinations on board. Teacher names an an- swer and asks a pupil to point to the possible combinations that make this answer. 9. A picture of a wagon filled with hay or cakes of ice. Each piece has a written number combination on it. If one child can name all the results of combinations he may be horse to draw the wagon. 10. Draw a picture of creek with stones in it. Each stone has a number combination written on. it. If a child fails to give correct re- sults he has fallen in; that is, he has been unable to cross without get- ting the correct results. 11. Draw a picture of railroad track. Have stations all along the way indicated by number combinations. If a child can give correct re- sults of combinations he may be conductor all through the trip. 12. Make a list of different combinations as: 87264 43664 etc. Then let each child see if he can make a list after having observed what was written. 13. Write a variety of division, multiplication, subtraction and addition combinations on the board, each in a different color. Then give the children each a different color of chalk and let each one see how many combinations he can write. The one that gets the highest number correct is the winner. iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiN 14. Let one child stand before the class and have him think of a certain number. Then have the children guess by asking him ques- tions. Someone may say 4 plus 4. If this answer is not correct they guess until someone has guessed the correct answer, then the winner is the next one to stand before the class. 15. The teacher should place a number of combinations, plus, minus, times and divide, on the board. She tells one child to close his eyes, then she sends another child to the board and has him write a combination on the blackboard. He writes 20. The first child is told to open his eyes and guess whether 20 means 10 plus 10, or 10 times 2. If he does not guess the answer correctly the first time he does not get a very high score, but if he guesses 10 plus 10, and that is what the boy was thinking about when he wrote it on the board, his score is 10. 16. Place a row of combinations upon the board 2 3 7 8 8 5 C 2632424 etc. Two children are sent to the blackboard, one of the children starting with the first problem, and the other starting with the last and work- ing toward each other. When the teacher says "start" each child en- deavors to get more of the answers in a certain length of time than the other one. When the teacher says stop, she counts and sees which one has the highest number correct. This teaches the children to work rapidly. Some days one could have the boys' row compete with the girls' row, one row with another, or one division with another. 17. Place the children in a circle and give each one a certain num- ber. The teacher stands outside of the circle and says "4 plus 4" and as she says this number she throws the ball into the circle. The child that is the number catches the ball and throws the ball to the teacher. The children who do not catch the ball as their numbers are called, stand in the circle until they do catch the ball. 18. Bird Catcher Children may be arranged in a circle and a number card given to each one. One child stands in the center and gives problems, the answers of which are within the numbers assigned. He may say, "How many are 5 roses and 3 roses?" Then the person having the number 8 holds it up. Thus he has "caught the bird." 19. Number Tug Pupils are arranged as for a tug of war, each side having a captain. Then the captain of one side gives number com- binations to the other group, such as, "3 times 3" or "3 times 4." Each pupil takes his seat if he fails to give correct answer. 20. Circle Addition A circle of this sort may be drawn with in- terior and exterior numbers. The class may be divided into two groups with a captain for each side. Each captain may then call on those of the opposite side to give the sum of the interior and exterior numbers to which he points. Pupils are to sit down when they fail to answer quickly. The game may be used for multiplication, division, or subtrac- tion by changing the numbers to suit the process. 21. When the above game is used for the multiplication process the pupil may be asked to add a given number to the product, or to sub- tract a given number from the product. 22. A column of figures are placed on the board. The pupils may add each number to the number at the side. This game may be varied by having the child begin at the top and add down or begin at the bot- tom and add up. This game may also be used for subtraction by plac- ing a larger number~opposite the column; for example, "15." Then it may be asked, "This number and how many are 15?" iiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiimiimiiiiiiiiiiiiiiiiiiiiimim 23. Hide and Seek The teacher places statements on the board having one number hidden; such as, 3 plus ? equals 7; 5 plus ? equals 9; ? plus 4 equals 8. Pupils obtain good training in repetition of number forms through this game. 24. Make three arches of paper. Fasten the largest arch to the floor with thumb tacks. Place the two smaller arches on either side of the large arch and fasten with thumb tacks. Children stand at cer- tain distances from the arches and take turns trying to roll a small rub- ber ball under one of them. Place the score under the arch. After there have been two turns around add the scores. 25. Have an equal number of children seated in each row. Give the child in the front seat of each row a bean bag or an eraser. The children hold up their hands ready to receive the object when the sig- nal is given. The children pass the objects back over their heads. When the last child in the row gets the bag he runs quickly to the chalk tray and lays the bag in it. The child who gets there first wins the game for his row. The score is kept for each row. 26. Hoop Game Have a hanging hoop in which there is a bell. The children may throw bean bags through this hoop. If the bag hits the bell the throw counts only 2. Every ball that goes through the hoop without ringing the bell counts 10. In ten throws each, two pupils may have the following score: 2 10 10 10 2 10 2 Find the score of each and tell who wins the game. 27. Odd or Even This is a very old game. Two persons play this game. Each takes 10 peas or marbles. One of the children places his hands behind his back and arranges the objects to suit himself. He then stretches out his closed hand and says, "Odd or Even." If the other child guesses correctly he receives a maible and if incorrectly he pays one. The other child says in the latter case: "Give me one to make it odd or even (as the case may be)." 28. The teacher writes problems upon the blackboard in rapid suc- cessibn. One child is asked to stand and give results as rapidly as pos- sible. The answers are not written. As soon as one child makes a mis- take another child takes his place. 29. Thumbs Up One pupil acts as a leader in this game. Each player has been given a number. He sits with one thumb up. The lead- er says, "Simon says 15" at which the thumbs of 3 and 5 (factors of 15) must be turned down. If Simon says 12, then the thumbs 2, 3, 4 and 6 must be turned down. 30. A leader is chosen to give each child a number. After this is done, the leader names any number below any prescribed limit, such as 25. All children whose given number is a factor of the number named by leader must then change their seats. In case there is but one dis- tinct factor for the number named, as in the case of 25, the pupil whose number is that one single factor rises and bows. If a child fails to rise he is tagged. 31. This game may be used as a drill for either multiplication, division, or fractions. Draw a 2-foot square, which is then to be divided into 144 smaller squares. The products of the tables are then written promiscuously in these columns, and the teacher points to two num- bers, asking the pupils to divide, multiply, or get the fractional parts. 32: This is merely a simple drill, as: 4 times 4, minus 2, plus 4, times 4, divided by 3, minus 6, divided by 3, times 7, plus 7, divided by 7 equals ? ,,ll,ll,ll,,i,,,l,,l,llllllllllllll|||||||||||||||||||||l||||||||IU DRILL CHARTS I. Purpose. 1. To arrange drill on definite and systematic basis. 2. To provide teacher with a labor saving device. 3. To distribute time according to difficulty of work. II. Plan. 1. To provide for a number of days (perhaps 30). 2. To select subject matter. To arrange subject matter in order of difficulty. To make up chart so that all work is provided for. To provide for motivation of work. III. Classification. 1. Reading numbers. 2. Counting. 3. Facts. a. Addition b. Subtraction c. Multiplication d. Division e. Measurement 4 Processes. a. Addition b. Subtraction ,. Multiplication d. Division e. Measurement 5. Fractional equivalents. 6. Fractional operations. 7. Decimal equivalents. 8. Aliquot parts. IV. Value. 1. To pupil. 2. To class. 3. To teacher. ADDITION DRILL CHART 20 Days 1 2 11 12 13 14 11 12 13 14 5 7 8 9 6 5 6 5 15 16 3 1. 15 17 5 11 5 7 2 8 9 9 7 8 12 6 7 7 9 10 rr 1 3 5 3 2 2 4 6 7 o 8 8 6 4 10 5 10 3 6 4 3 1 6 2 5 3 13 14 15 16 4 14 15 12 11 5 6 9 > 8 7 7 5 17 13 6 8 14 13 1 9 10 8 7 3 5 8 0- 3 3 11 9 11 6 12 11 12 14 8 4 7 4 9 4 6 J_ 8 9 10 4 16 5 4 5 2 7 8 2 1 8 3 3 __: 5 11 12 13 14 6 13 14 15 17 6 5 5 9 6 6 9 8 15 16 4 11 16 13 2 12 7 7 7 9 7 8 9 12 13 18 13 11 8 10 5 4 4 9 9 2 5 2 9 10 7 4 6 6 7 4 8 5 2 4 1 2 3 2 2 3 4 2 7 17 11 11 12 8 14 15 15 ir 9 5 3 5 8 5 8 8 12 13 7 11 14 14 8 9 7 5 9 5 9 3 7 7 8 6 9 10 10 8 3 4 3 4 6 3 7 - 6 7 8 4 10 3 9 9 4 8 5 5 2 3 5 1 2 7 3 4 ADDITION DRILL CHART (Continued) 9 10 13 13 14 14 15 15 16 16 6 7 6 8 8 8 7 9 15 15 9 11 17 17 1 9 5 9 3 8 9 4 11 12 12 10 9 11 11 10 8 9 2 5 4 7 4 10 10 5 5 6 12 14 4 6 6 8 6 3 2 6 7 2 3 5 11 11 11 12 12 12 14 14 13 13 5 6 5 7 5 9 6 7 18 13 2 12 14 14 3 11 5 8 4 6 8 2 12 13 13 16 11 7 7 6 8 4 9 8 9 ? 4 4 IS 6 4 8 7 7 7 10 5 8 1) 2 3 4 6 2 5 5 3 1 tJ 15 15 15 15 14 17 17 11 11 5 9 7 8 8 9 5 6 1G 16 4 8 12 12 5 10 7 9 3 5 7 : ' 8 9 9 8 10 11 11 9 5 3 6 2 7 3 s 7 8 9 5 4 8 10 6 4 9 9 6 2 2 2 7 2 3 3 1 2 15 13 13 14 14 16 14 14 15 15 5 8 5 9 6 8 5 9 13 13 6 12 15 15 7 11 6 7 3 7 8 4 12 9 9 10 11 12 12 12 9 4 5 8 7 4 8 6 10 10 8 10 9 14 16 5 5 10 6 4 4 5 8 7 8 3 g e ADDITION DRILL CHART (Continued) 17 18 16 16 17 17 12 12 13 13 7 9 8 9 5 7 5 8 11 11 8 13 14 14 9 7 5 6 4 5 9 3 13 11 11 18 7 8 8 7 9 2 9 9 4 3 5 2 6 6 4 6 10 7 8 4 8 o 2 4 2 3 9 5 2 3 4 1 19 13 13 14 14 20 15 15 16 16 6 7 6 8 7 8 7 9 15 15 1 9 17 17 2 11 5 9 3 8 9 3 9 10 10 8 11 12 12 10 6 3 1 6 8 3 9 2' 9 9 10 5 3 10 10 5 4 3 2 7 5 3 1 8 6 2 2 2 iiitiiiiiiiiiimiiiiiiiiiiinii iiiiiiiiiiiiiiiiini iiiiim iiiiiiiiiitiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiniiMi tiiiiiiiiiii in TESTS I. . Needs for 1. To furnish information. 2. To provide incentive. 3. To provide basis for classification of students. 4. A means of associating ideas. II. Purposes. 1. Indirect purpose (General Information). 2. Direct purpose (To test for particular knowledge). III. Classification. 1. General information. 2. Performance, Accuracy, Speed. 3. Special information. 4. To test for development. IV. Plan in giving a test. 1. Cover the ground. 2. Provide sufficient time. 3. Clear aim. 4. Provide definite plan for marking. 5. Establish fairness. V. Values. 1. To the pupil. 2. To the class (as a basis for marking). :\. An aid to organization of work. IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIJIIIIIIM MARKING I. Needs for 1. To provide a record of work completed. 2. To stimulate pupils to effort. 3. To provide a basis for classification of pupils. II. Purposes 1. To record performance. 2. To record ability. 3. To record habits of study. 4. To record habits of accuracy. 5. To record habits of neatness. 6. To record attitude toward teacher, work and class. III. Classification 1. Daily class recitation. 2. Knowledge of topics. 3. Final knowledge of subject. 4. Attitude and attempt. IV. Plans 1. Five point scale. 2. One hundred point scale. 3. Combination scales. 4. Two point scale (all or nothing). V. Values 1. Temporary record. 2. Permanent record. 3. Motivation. 4. Basis for classification of pupils. iiiiiiiniiniiiiiiiiiiiiMiiiMiiiiiiiiiiiiiiiiniiiiniiiinHiiniiiiiiiiiiniiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiniiniiiiiiiiiiiiiiMiMniiiin iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiinii DIAGNOSIS I. Needs for 1. To provide basis for individual instruction. 2. To properly grade pupils. 0. To determine possibilities of pupils. II. Purposes t 1. To keep pupils advancing as rapidly as possible. 2. To protect individuals from class averages. 3. To aid teacher in classification of pupils. III. Plan 1. To arrange subject matter in order of difficulty. 2. To provide charts for recording the results of testing. 3. To provide careful tests. 4. To record results. IV. Values 1. To pupil. 2. To class. 3. To teacher. 111 iiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiim PROBLEM ANALYSIS First Read the problem. Second State what is to be found. Third State what is given. Fourth State the general method of Solution. Fifth State the steps in the solution. Sixth State the results of these operations. Seventh State the answer. 1. A man sells a farm for $4,300. The commission of the salesman is 5%. The deed costs $2.00, the abstract $7.00, and the incidental ex- pense amounts to $27.78. Find the net amount the man receives. 2. I am to find the net amount due the man who sells the farm. 3. I know the selling price is $4,300. I know the salesman is to re- ceive a commission of 5%. I also know that the other expenses are $2.00, $7.00 and $27.78. 4. The general method is to subtract the sum of the commission and the expenses from the selling price. 5. The first step is to find 5% of $4,300. The next step is to add the amounts, $2.00, $7.00 and $27.78. The next step is to subtract the sum of expense from $4,300. 6. 5% of $4,300 is $215. The sum of $215, $2.00, $7.00 and $27.78 is $251.78; $251.78 from $4,300 equals $4,048.22. 7. The answer is $4,048.22. PART II. Subject Matter Outlines iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiimiiiiiiiiiiiiiiiin FUNDAMENTAL NUMBER IDEAS All number processes are based upon a few very siniple prin- ciples. It is necessary that these be brought to the attention of pupils very early. This is done indirectly These principles are built upon some truths connected with quantity, form and position. Although these number ideas are so very simple that we are always inclined to disregard them all the processes are based on them. For example one of the simple ideas about quantity is that a quantity may be increased and upon this idea the processes of addition and multiplication are based. As suggested above the ideas we are concerned with when we build up the science of arithmetic are those connected with quantity, form, and position. Of these three groups of ideas we are especially concerned with the group relating to quantity. This explains why arithmetic is sometimes spoken of as a subject which provides an answer to the questions How much? or How many? OUTLINE FOR STUDY I. Quantity 1. A quantity may be a. Increased (by addition) (by multiplication) b. Decreased (by subtraction) c. Divided (by division) 2. Two or more like quantities may be a. Combined (by addition) b. Compared (by finding the difference subtraction) (by measurement division) II. Form (only standard forms are considered). 1. recognition 2. oral names 3 written names 4. classification 5. measurement III. Position (very little attention given to this topic). IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIINIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIH^ NUMBER APPRECIATION Although no definite attempt is made to give general sense train- ing it is desirable to give drills and presentations necessary to enable pupils to appreciate the number aspects of those things they see, hear and touch. This is not at all difficult and takes up very little time. The use of cards, pictures and other concrete material Is suggested. OUTLINE FOR STUDY I. Vision 1. Number groups 2. Recognition of measuring units 3 Estimation of value (length, weight, etc.) 4. Recognition of standard forms II. Hearing 1. Continuity of sound 2. Intensity of sound 3. Groups of sounds III. Touch 1. Number groups 2. Recognition of standard forms 3. Estimating weights LANGUAGE A part of the work in teaching arithmetic is devoted to the neces- sary addition to our language to make possible the expression of num- ber ideas. Not only must pupils learn to appreciate number truths but they must learn to express such ideas through speech and in writ- ing. Again they must learn to understand the expression of others. The most important part of this work is that connected with what we refer to as the Arabic number system. The Arabic number system is a part of our language. When teaching pupils the number system all principles of language teaching must be used. Another part of the language instruction in arithmetic is that given over to proper sentence constructions when number thoughts are to be expressed. Questions connected with a choice between two and one are three or two and one is three are questions of language. In the processes there are both oral and written forms to consider. Throughout all work in arithmetic children should be taught to make correct and complete statements. The use of language, both oral and written, is to give us an instrument to express our thoughts one to another. Much of the difficulty in arithmetic may be traced to care- less statements. Exact statements should be made not only in the solution of a problem but also in the reading of a problem. Often children do not understand what is asked for when a problem is given for solution. An answer in a problem solution must of necessity be correct or incorrect. To arrive at a correect solution, one must read the problem correctly and then speak or write correctly. IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIH THE LANGUAGE DEVELOPMENT One phase of the study of arithmetic is that connected with the development of the necessary language to express number thoughts. The mastery of the Arabic symbol system constitutes the main part of this addition to the general fund of language. OUTLINE FOR STUDY A. The number system a. Base symbols (0123456789) (1) Recognition (2) Oral names (3) Written forms (4) Order (5) Relative values b. Combination symbols (10-11-etc.) 1. Place value 2 Decimal (ten) law in classification (a) 2 digit numbers. (b) 3 digit numbers. (c) 4 digit numbers, etc. 3. Recognition 4. Oral names 5.. Written forms 6. Order 7. Relative Values (Note) The above outline is to be followed in a general way when studying Roman numbers. B Sentence arrangement (a) Choice of words (b) Structure of sentences C. Written forms for the processes. (a) Whole numbers. (b) Fractions. D. Oral forms for the processes. (a) Whole numbers (b) Fractions. E. Special language forms. This includes those language forms that do not classify in the above outlines. THE FACT GROUPS There are certain groups of facts the teacher must present. It is convenient for the teacher to have these groups arranged in the order of difficulty and in such form as to be accessible when needed. The common fact groups are those of addition, substraction, multipli- cation, division and measurement. In the following classification of the fact groups the arrangement suggested is based on the results of systematic study of work done in primary grades. The outlines are meant to be suggestive and they may be changed to meet the needs of special classes. The addition and subtraction groups are arranged in groups on a basis of difficulty and are to be taught together. The multiplication facts and division facts are arranged with ref- erence to products. They are also grouped according to difficulty. In the outline covering the measurement facts the suggestion is that the work should progress by grades with regard for the pupils needs and experience. iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiitiiiinii 111 iitiiiiiiiiiiiiiiiimiiiiiiMiiiiiiiiiiiiiiiiiiminiiiiiiiiiiiiimiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiiiiinii iiinn ADDITION FACTS (1) 111111111 123456789 (2) 2 2 3 *4 5 32345 (3) 222226678 456784678 (4) 23333334444 94567895789 (5) 000000000 123456789 (6) 5555666778 6789789899 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIN SUBTRACTION FACTS (1) 233445566778899 10 10 112131415161718 1 9 (2) 5 5 4 6 8 10 32234 5 (3) 6 6 7 7 8 8 9 9 10 10 10 10 12 14 16 18 24252627 2 86 4 6 7 8 9 (4) 11 11 7 7 8 8 9 9 10 10 11 11 12 12 9 9 2 9343536 3 7 3 8 3 945 11 11 12 12 13 13 474849 (5) 1234567 0000000 (6) 11 11 12 12 13 13 14 14 13 13 14 14 15 15 15 15 5657585967686978 16 16 17 17 7989 MULTIPLICATION AND DIVISION FACTS CHART I Group 1 4 6 8 9 10 15 2 2s 2 3s 2 4s 3 3s 2 5s 3 5s 3 2s 4 2s 5 2s 5 3s Group 2 12 14 16 18 2 6s 2 7s 4 4s 2 9s 6 2s 7 2s 2 8s 9 2s 3 4s 8 2s 3 6s 4 3s 6 3s Group 3 20 25 30 35 40 45 2 10s 5 5s 3 10s 7 5s 4 10s 9 5s 10 2s 10 3s 5 7s 10 4s 5 l.i 4 5s 5 6s 8 5s 6 4s 6 5s 5 8s 50 60 70 80 90 100 5 10s 6 10s 7 10s 8 10s 9 10s 10 10s 10 5s 10 6s 10 7s. 10 8s 10 9s Group 4 21 24 27 28 32 36 3 7s 3 8s 9 3s 4 7s 4 1 is 4 9s 7 3s 8 3s 3 9s 7 4s 8 4s 9 4s 4 6s 6 6s 6 4s Group 5 42 48 49 54 56 63 64 72 81 6 7s 6 8s 7 7s 6 9s 7 8s 7 9s 8 8s 9 8s 9 9s 7 6s 8 6s 9 6s 8 7s 9 7s 8 9s CHART II Group 1 22 24 33 36 44 48 55 60 2 11s 2 12s 3 11s 3 12s 4 11s 4 12s 5 lls 5 12s 11 2s 12 2s 11 3s 12 3s 11 4s 12 4s 11 5s 12 5s Group 2 66 72 77 84 88 96 99 6 11s 6 12s 7 11s 7 12s 8 lls 8 12s 9 lls 11 6s 12 6s 11 7s 12 7s 11 8s 12 8s 11 9s Group 3 108 110 120 121 132 144 9 12s 10 11s 12 10s 11 11s 11 12s 12 12s 12 9s 11 10s 10 12s 12 11s iHHumuiHHirHiiiiiiiiiimiiiiiiiiiiNimiiiiiiiimiiiimiiiiiiiiiiiiiiim MEASUREMENT UNITS There is a distinction made between a knowledge of the measuring units with their relationships and the operation of measuring. The following list shows a suggestive grouping of the measuring units with their relationships by grades. The work in measuring should follow the presentation of units and facts very closely. GRADE I Cent 5 cents 1 nickel dime 10 cents 1 dime 5 cent piece two nickels 1 dime pint 2 pints 1 quart quart inch foot 12 inches 1 foot GRADE II Quarter 25 cents equals 1 quarter Half Dollar 5 nickels equals 1 quarter Dollar Minute as used in conversation Hour Day Week GRADE III ounce 16 ounces equals 1 Ib. pound dozen 12 equals 1 doz. degrees of temperature To read thermometer yard 3 ft. equals 1 yd. 36 in. equals 1 yd. month 30 days or 31 days equals 1 month GRADE IV gallon 4 qts. equals 1 gal. peck 8 qts. equals 1 peck, bushel 4 pecks equals 1 bu. second 60 sec. equals 1 min. minute 60 min. equals 1 hour hour 24 hours equals 1 day day 7 days equals 1 week week 52 weeks equals 1 yr. month 12 months equals 1 yr. year 365 days equals 1 yr. rod 16% ft. equals 1 rod. mile ton 2000 Ibs. equals 1 ton GRADE V acre section Work out relationships board foot through problem work sq. in. sq. ft. sq. yd. sq. (100 sq. ft.) iiiiiiiiiiiiiiiiiiiiin iiimmmiiimiimmiimmHiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiin GRADE VI cu. in. Work out relationships cu. ft. through problem work cu. yd. Cord gill 4 gills equals 1 pt. mill 10 mills equals 1 cent barrel GRADE VII Degree of angle degree of arc minutes 60 sec. equals 1 min. seconds 60 min. equals 1 degree Use of tables IIIIIIHIIIIIIIIIIIIMIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIH WHOLE NUMBER OPERATIONS ADDITION In presenting addition one of several plans may be selected for arranging the steps but in any plan provision must be made to teach the oral and written forms, the carry, column addition and special points of difficulty such as adding when O is involved, adding money values, and adding named objects. In all work in addition the applica- tion to real things should constantly be made. In this connection the basic law of addition that only like things may be added should be stressed The only steps offering difficulty are those involving the carry and column addition. The carry idea can be made clear to pupils if they understand place values, and are able to make ah analysis of a number. ^ STEPS IN ADDITION Written Form Oral Form Step I. The addition facts Type Form Say or think 4 4 and 7 are 11 + 7 Suggestions for Rationalization Use objects 11 EXERCISES 5 + 8 4 + 3 6 + 8 4 + 9 5 + 2 7 + 3 1 + 4 5 + 6 IIIIIIIIIHIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIU Step II. To add a two-digit number and a one-digit number (with- out carry). Type Form 13 + 4 Say or Think. First Form 13 and 4 are 17 Second Form 3 and 4 are 7 Bring down the 1 Use objects. First Form is to be used in most drill work. The Second Form is to be used to show method followed in more com- plex exercises. 12 + 4 16 Extensions. First Type Form 22 + 4 26 32 + 4 36 42 4 etc. 46 Say or Think. There is no need for ob- 12 and 4 are 16 jective rationalization. 22 and 4 are 26 The work is largely 32 and 4 are 36 oral and the drill is car- 42 and 4 are 46 ried on in much the etc. same way the drill in Step 1 was carried out. Second Type Form 214 + 342 556 Say or Think 2 and 4 are 6 4 and 1 are 5 3 and 2 are 5 Many exercises of this type may be given since nothing new is involved except the new written form. This exercise is merely a group of three addition facts. EXERCISES. 18 + 1 2J 16 + 3 12 + 5 L6 2 17 + 2 13 4 14 + 3 13 + 5 23 + 5 33 + 5 etc. 1 1 3 24 + 3 34 + 3 etc. 241 435 268 411 357 132 342 546 567 312 Step III. Column addition sums less than 10. Type Form 3 Say or Think 2 1, 3, 5, 8 Use objects if necessary + !_ 8 EXERCISES. 5 4 2 2 4 6 2 3 1 2 2 2 1 4 2 1 1 + 1 + 1 + 1 1 +- H + 1 it iiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiitiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii itiiiiiiiiiiu Step IV. The carry. Type Form First 18 + 5 Second 18 + 22 40 Third 17 + 26 43 Think 5 and 8 are 13 Write 3, carry 1 1 and 1 are 2 Think 2 and 8 are 10 write 0, carry 1 3 and 1 are 4 Say or Think 6 and 7 are 13 write 3, carry 1 3 and 1 are 4 There is some difference of opinion concerning the best type form to use in opening the work on th: carry It is thought to be a good plan to use the first type form for many oral exercises of the form 18 28 38 48 etc. 5555 23 33 43 53 and follow this work with written exercises of the sec- ond type form. In this form the sum of the units column is 10. After these two forms have been used the third form may be used. Extensions 143 + 8 151 263 + 14 277 435 + 89 524 647 + 156 803 24 + 16 40 Say or Think 11, 5, 1 Say or Think 7, 7, 2 etc. etc. etc. It is desirable to eliminate all unnecessary language forms. As soon as possible., pupils should use the forms suggest- ed for these exercises. 14 8 16 + 9 EXERCISES. 37 26 1 4 + 23 + 28 13 + 47 29 442 7 361 16 297 47 625 365 4*7 + 603 276 -f 434 iiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiimiiiiiiiN Step V. Column Addition Type Form 6 3 4 r 7 8 33 Say or think 15, 20, 27, 33 Pupils should add continuously up or down. To allow pupils to look for easy groups develops a bad habit and encourages pupils to slight the more dif- ficult combinations. Extensions 14 63 + 14 91 Say or think 7, 11 Write 1 carry 1 2, 8, 9 Write 9 The teacher is to use her judgment as to the complexity of the exer- cises selected. 126 14 + 208 348 Say or think 12, 18 Write 8 carry 1 1, 2, 4 Write 4 3 Write 3 EXERCISES 62 14 23 41 25 62 73 241 62 16 146 25 216 143 641 18 16 217 iiiiiiHiiiiimiiiiiiiiiiiu iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiniiii iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiniiiiiiiii i mi iiiHiiiimiii Step VI. To add Money Values. Type Form 1st. $14 + $24 $38 Say or think Either 38, 38 dollars or 38 dollars The new point is the use of the sign $ 2nd 14c 8c 22c Say or think either 22, 22 cents or 22 cents The new point is the use of the sign c. Teacher to use judg- ment 3rd $1.14 2.61 .08 $3.83 Say or think same as above $21 $62 $25 $16 EXERCISES $172 $25 $14 17c 14c 25c 62c 62c 14c 8c $1.84 $ .25 $4.92 $1.73 $ .24 $4.16 $6.24 $ .04 $4.08 ADDITION DRILL CHART (20 days) 6 G 6 789 7 7 8 9 8 9 5 5 6 7 5 5 8 9 6 6 7 8 556 899 7 7 8 9 8 9 5 5 6 7 r 6 7 8 670 986 044 998 4 9 7 9 8 7 8 7 3 8 5 3 3 6 7 316 886 033 794 2 4 9 5 2 5 4 4 9 8 3 2 8 9 2 2 4 6 541 549 5 3 6 8 4 2 3 2 3 2 222 673 2 1 2 7 2 3 7 3 4 1 4 6 876 999 6 5 7 9 5 8 5 5 5 6 7 6 6 8 9 770 894 556 897 6 7 9 8 7 9 3 8 6 9 8 6 5 7 9 550 762 443 759 3 2 6 5 t\ 8 3 3 4 5 3 3 7 8 267 467 233 969 4 7 7 7 8 8 3 3 4 5 A 4 8 9 2 2 4 5 652 4 5 Z 1 5 2 3 2 3 1 4 945 945 1 3 2 2 6 3 1 2 2 556 787 677 989 2 1 9 8 6 9 8 5 5 9 8 550 768 776 989 6 5 7 8 5 7 8 7 9 8 6 6 8 7 5 5 9 8 344 757 2 2 7 8 6 3 4 3 3 3 4 5 3 3 8 9 6 7 6 7 033 667 4 4 8 9 2 8 2 4 9 3 3 5 8 3 2 9 4 4 1 4 1 8 5 8 5 2 1 C 9 692 492 3 1 3 8 2 2 6 7 4 5 4 5 1 7 766 997 5 5 9 8 5 7 8 7 9 8 6 5 8 8 5 5 7 6 766 997 5 5 9 8 5 7 1 8 7 9 8 6 5 8 9 550 863 033 945 3 4 7 5 2 6 5 6 4 7 9 4 4 8 7 3 2 8 4 233 945 3 2 6 8 6 7 4 7 3 3 7 8 3 4 9 5 6 8 6 8 922 932 1 6 2 2 6 7 3 4 3 4 1 5 521 534 9 2 9 2 3 1 3 3 776 989 6 5 7 9 5 7 8 6 9 8 G 5 7 8 550 762 766 898 6 5 7 7 5 6 8 7 9 9 6 6 9 7 550 989 044 557 4 4 8 9 4 4 8 7 3 3 9 7 2 2 4 8 023 794 3 3 5 6 2 6 3 3 7 8 3 4 9 5 2 2 6 7 222 567 4 5 4 5 6 2 4 3 2 1 o -j 2* 7 3 773 4 1 4 9 8 5 8 5 2 1 3 8 1 2 iiimiiiim iiiiiiiiiiiimiim mi i iiiiiiiiiiiiiiiiiiiniiiiiiiiii mini i i i iiiiiiiiiiiiiiiiiiiiiiiimntiiiiiiiiiiiiiiiiiii SUBTRACTION OPERATION All number processes are based upon a few simple principles. The following outline shows some of the simple principles upon which the four operations in whole numbers are based: 1. Quantity 1. A quantity may be A. Increased (by addition) (by multiplication) B. Decreased (by subtraction) C. Divided (by division) 2. Two or more like quantities may be A. Combined (by addition) B. Compared (by finding the difference subtraction) (by measurement division) It is evident that the process of subtraction is based upon the idea of decreasing a quantity and upon the comparison of two quantities by finding the difference. We express these ideas as The take-away idea (decreasing a quantity) The difference idea (comparing two quantities) We also have The adding-to idea (as used in making change) This last idea is not based on a fundamental number principle as outlined above. It is an outgrowth of the connection between the addi- tion and subtraction facts. Three Illustrative Examples Elsie has 5 apples and loses 2 apples. How many has she left? (The thought is 2 from 5.) Elsie has 5 apples and John has 2 apples. Who has more apples? (The thought is comparison by difference.) Elsie has 2 apples. How many more must she obtain to have 5? (The thought is what must be added to 2 to make 5.) Since any one of these three thoughts may come to the pupil it is not desirable that the teacher should use one only and submerge the other two. To further illustrate: 5 Pupil may say or think 2 2 from 5=3 (thinking take away) 5 Pupil may say or think 2 The difference between 5 and 2=3 (thinking difference) 5 Pupil may say or think 2 2 and 3 are 5 (thinking adding-to) iiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiimiiiimiiiiiiiiiiiiiiimiiimmiiiiM The statement of the problem always suggests the thought form and in turn the thought form always suggests the language form. Certain complications always arise. For example, a pupil may think "What is the difference between 5 and 2?" and use the adding-to method to discover the difference. Or a pupil may think "2 from 5 are how many?" and use the adding-to method to discover the answer. Or still again the pupil may think, "2 and how many are 5?" and use the take- away method to discover the result. Since the learning of the addition facts automatically gives the pupils the subtraction facts, it is desirable that pupils use this informa- tion. It is desirable to make every possible use of the fact knowledge acquired through a study of the addition facts. It is not desirable, however, to submerge entirely the take-away thought and the difference, thought simply because the adding-to method provides a possible way to discover a result. In all problem work the conclusion should always be stated in terms of the question that is asked. As the difficultly of examples and problems increases it becomes necessary to develop a mechanical process to aid the mind. Two me- chanical processes are in common use. The following examples illus- trate them: 81 9 and 2 are 11 81 9 and 2 are 11 39 4 and 4 are 8 39 3 and 4 are 7 42 42 In the first example 3 in the subtrahend is increased by one. In the second example 8 in the minued is decreased by one. Both processes may be explained if the teacher feels that it is necessary to make explanation. 81 = 8 tens 1 unit = 8 tens 11 units 39 = 3 tens 9 units = 4 tens 9 units 81 39 8 tens 1 unit 3 tens 9 units 4 tens 2 units 7 tens 11 units 3 tens 9 units 10 units have been add- ed to the minuend and 1 ten has been added to the subtrahend. The minuend has been written in an equivalent form. Results from careful testing fail to show that one mechanical pro- cess produces better results than the other. It seems to be the general feeling among those who have given the question attention that either mechanical process may be adopted. It is the feeling of teachers, however, that only one mechanical pro- cess should be taught. This does not mean that only one thought and language form must be used for all problems. The following outline shows some possible language forms with each mechanical process. 81 39 42 81 39 42 81 39 42 Say or think 9 and 2 are 11 4 and 4 are 8 9 from 11 are 2 4 from 8 are 4 The difference between 11 and 9 is 2 The difference between 8 and 4 is 4 81 39 42 81 39 42 81 39 42 Say or think 9 and 2 are 11 3 and 4 are 7 9 from 11 are 2 3 from 7 are 4 The difference 11 and 9 is 2 7 and 3 is 4 between miimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiin^ The general feeling is that 1. Teacher should select one mechanical process. 2. Teacher should use but one language form during that first stage of instruction when children are developing a skill in the per- forming of the mechanical process. 3. Teachers should use all three language forms after the skill is acquired 4. Teachers should always allow pupils to express the conclusion in a problem in terms of the question. It would seem that teachers and superintendents should take a reasonable attitude toward the question of teaching subtraction. Too much time is wasted in argument that has no base except the per- sonal prejudice of this or that person. iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiH The following outline of steps shows a possible choice of mechani- cal process and language form: Step. I Oral Form Suggestions for Bxplan- Written Form ation. 65 48 27 Build on the addition -3 -5 -4 and . subtraction facts. The purpose in this drill 62 43 23 is to develop a written form and the use of the subtraction facts in con- nection with the larger numbers. 73 2 EXERCISES 95 i 38 7 97 5 39 - 2 54 54 4 47 1 69 _ O Illllllllllllllllllllllllllllllll! llllllllllllllllllllllMlllllllllllllllllllllllllllllllimilllllHIIIHtllllllllllllllllllllllllllllllllllllllllllllllN 2 To take a one digit number from a two digit number: Written Form. Say or Think. Suggestions. 23 42 74 20 Same as above. Build _5 _9 -4 on addition and subtrac- tion facts and upon the 16 65 is series drill in addition. Perfect the written forms. Much time should be given to drills of this type. EXERCISES 83 42 55 22 35 46 _ 5 _3 _6 4 -6 -7 21 34 43 61 72 6 5 2 -4 iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiMiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiin 3 To take a two digit number from a two digit number: Written Form Say or Think. Suggestions. 34 9 and 5 are 14, Allow pupils to per- 19 write 5, carry 1. feet the mechanical op- 2 and 1 are 3, eration. Any explana- 15 write 1. tion that is attempted should be given with the thought of simply making the operation a reasonable one. EXERCISES. 88 75 93 86 27 19 16 14 27 16 55 84 72 52 49 29 18 68 19 26 iiiiiiiiiiiiiiini iiiiiiiiiiiiimiimiiii iHiiiiiiiiiiiiiiiiiiiiiiiiiiiiHiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii imiiiiiimiimiiiiimHiiiiiiiiiiiiiHiMiiiiiiiini 4 An extension of Step 3. Written Form. Say or Think 416 9 and 7 are 16, write 7, carry 1. 8 and 3 are 11, 237 write 3, carry 1, 2 and 2 are 4, write 2. 325 -186 EXERCISES. 387 198 Suggestions. Perfect the mechani- cal operation. Make the operation as- reasonable as possible without at- tempting rigid proofs. Build on the addition and subtraction facts. 783 595 476 -297 323 -198 456 -179 603 -494 805 167 234 156 iiiiiiiiimiiiHinuiiiiiiiiiiiHiiifiiiiHiiiiiiimimiiiiiiiimiiiim 5 To take any number from any other number: Written Form. Say or Think. Suggestions. 4123 1576 Same as above. Same as above. EXERCISES. 3781 6328 5342 1889 4789 3457 2776 7543 3586 -1889 2654 1798 5786 4080 7006 3877 2399 3687 Illllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllim 6 MONEY VALUES 14c $32 $2.15 7c $16 $1.67 7c $16 $ .48 EXERCISES 13c 27c 86c 64c 8c 13c 32c 19c $42 $19 $90 $88 $18 $ 8 $ 6 $32 $1.76 $9.49 $10.92 $ 38 $3.46 $ 6.48 iiiiniiMmiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiim MULTIPLICATION OPERATION. 1. Definition of multiplication as an addition process 2. Definition of X sign (to be read times). 3. Definition of terms Multiplicand, Multiplier and Product. 4. Fundamental law that the multiplier must be abstract and that the product is of the same kind as the multiplicand. STEPS. 1. To multiply a two-digit number by a one-digit number (without a carry). 23 Say or think: 2 3's are 6, write 6 x2 2 2's are 4, write 4 46 This step may be extended to the multiplication of a number of 3, 4 or more digits by a one-digit number (without a carry) . 223 14243 x 3 x 2 669 28486 Argument Base the operation on the multiplication facts, The purpose of this step is largely that of perfecting the written form. EXERCISES. 2 x 24 33 443 x3 x2 3 x 31 21 321 4 x 12 x3 x2 2 x 43 44 2134 x2 x2 3 x 23 32 1243 x3 x2 4 x 21 iiiiiiMiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimmiiiiiiiiimiimiiiiiiiiiiiiiiiiiiiiiiiN -. To multiply a two-digit number by a one-digit number (with a carry). 43 Say or think x 4 4 3's are 12, Write 2. carry 1 4 4's are 16, 16 plus 1 are 17, write 17 172 To multiply a three-digit number by a one-digit number (with a carry . 456 Say or think x f> 5 6's are 30, Write 0, carry 3 5 5's are 25, 25 plus 3 are 28, write 8, carry 2 2280 5 4's are 20, 20 plus 2 are 22, write 22 This step may be extended to the multiplication of a 4, 5. or more digit number by a one-digit number (with a carry). 6434 56243 x 7 x 6 45038 337458 Argument The base for the explanation is number analysis and knowledge of place value. Objects may be used if necessary. 4:: 4 tens 3 units 4 = 4 16 " 12 =17 tens 2 units = 172 EXERCISES 55 x2 343 x4 456 x3 3251 x4 16024 x4 67 \:: 6203 x5 41292 x7 37 x4 648 x2 6213 x5 23125 x8 iiiiiiiiiiiiiiiiiiiiiiiiiiimiHiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiwiiiimiiiiiiiiiim 3. To multiply a two-digit number by a two-digit number. 24 Say or think: 6 4's are 24, write 4, carry 2 x!6 6 2's are 12, plus 2 are 14, write 14 144 1 4 is 4, write 4 under tens 24 12 is 2, write 2, add 384 This step may be extended to the multiplication of 3, 4, or more digit numbers by a two-digit number. As in addition and subtraction all work should be connected with real problems as soon as the mechanical skill is acquired. Argument The new point in this step is the set-over. The teacher O A may show that .,* means 6 x 24 and 10x24 lo 6 x 24 = 144 10 x 24 = 240 Sum of products 384 This step may be extended as follows: 462 8434 x 14 x 49 1848 75906 462 33736 6468 413266 EXERCISES. 24 345 3845 x!7 x!7 x 36 37 426 6271 x!2 x!3 x 45 63 509 3678 x!5 x23 x79 inn iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiniiiiiiiii iiiiiiiiiiiiiiiiMiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiimiiiiiiiiiiiiiiiiiiiiiiiiiii 4. To multiply a number by a three digit number: 432 Say or think: x 178 8 2's are 16, write 6, carry 1 8 3's are 24, plus 1 are 25, write 5 carry 2 :',456 8 4's are 32, plus 2 are 34, write 34, etc. 3024 432 76896 This step may be extended to multipliers of 4, 5 or more digits as semis advisable. EXERCISES 365 x!27 3264 x!568 35792 X14653 28765 X10531 39607 X52613 432 x!29 6538 x!296 467 x235 3729 x!076 178 x!26 4291 x2305 429654 X132578 237 x964 3529 x!264 7960374 X1207123 IIIIMimilllllllllllllllllllllllllllllllllllllllllllllllllllllMIIIIIIIIIIIIIII Illlllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllliii 5 To multiply money values: $46.34 25 23170 9268 $1158.50 The language form would be the same as in previous steps. The point in the product should be relatively in the same position as it is in the multiplicand. $25.13 x!3 $82.96 x27 EXERCISES $365.27 x25 $387.91 x79 $4216.17 x36 $3781.27 x25 $35.68 x35 $525.75 x67 $4291.21 x206 $38.92 x!25 $57.68 x236 $468.18 x213 $721.46 xl-75 $7691.25 x357 $8219.14 x 1210 MULTIPLICATION DRILL CHART (20 days) 81 32 16 54 24 12 72 28 16 49 24 12 72 28 15 49 21 9 64 27 6 48 21 15 64 100 10 48 70 8 63 45 4 42 30 10 63 90 42 60 56 40 81 25 56 80 81 50 54 35 72 20 36 18 27 14 36 18 32 14 64 24 16 48 28 16 23 24 16 42 28 12 63 21 9 42 27 6 56 21 4 81 27 6 ( 56 45 8 81 50 4 54 60 8 12 40 9 54 35 72 70 49 80 64 30 49 25 64 90 48 100 63 20 32 18 36 18 32 18 32 14 56 28 16 81 27 12 54 28 16 72 30 12 54 36 10 72 32 15 49 27 10 63 64 6 49 100 15 64 70 9 48 45 8 64 56 4 48 90 63 60 42 40 24 25 42 80 56 50 81 35 36 20 21 18 24 14 32 18 21 14 49 32 6 64 27 12 48 27 16 63 28 12 48 50 10 63 28 4 42 21 8 56 24 4 42 70 36 56 60 9 81 45 15 54 30 16 81 90 54 80 72 40 49 25 n is 49 100 64 35 48 20 21 16 24 14 32 18 36 14 42 32 30 56 36 45 81 27 90 54 36 80 81 27 20 54 28 50 72 24 100 49 32 100 72 21 18 49 24 16 64 21 14 48 28 12 64 50 8 48 25 10 63 70 4 42 45 8 63 40 9 42 35 15 56 80 9 81 50 15 iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiin DIVISION OPERATION 1. Definition of division as partition and also as measurement. 2 apples 2 | 4 apples 2 Partition. Measurement. 5c | lOc In partition the thought is to divide a quantity into parts. In measurement the thought is to measure a quantity by another quantity of the same kind. 2. Definition of the -j- sign (to be read divided by). 3. Definition of the terms dividend, divisor, quotient. Partition. 2 oranges 3 | 6 oranges 5 yards 2 10 yards 4 trees 3 fT2 trees 2 marbles 4 | 8 marbles 2 dollars 2 fTdollars" 2 miles Measurement. 5 2 yds. | 10 yards _2 3 books | 6 books 2 2 chairs | 4 chairs _2 2c " 4c 4 2 pencils 8 pencils 5 | 10 miles Illllllllllllllllllllllllllllllllilllllllllllllllllllll lllllllllllllllllllllllllllltlltlllMIMIIIIIIMIMIIIIIIIIIIIIIIIIllllllinillllllMIIMMIIIMIIIIMIIIIIIillMMIIIIIIIIIIIIIIIII STEP I. 1. To divide a two-digit number by a one-digit number (without a carry and without a remainder). 23 Say or think: Ihere are 2 2's in 4, write 2 2 |~~46 There are 3 2's in 6, write 3 This step may be extended to cover the division of 3, 4 and more digit numbers by a one-digit number (without a carry and without a remainder). 123 2134 3 | 369 2 | 4268 The short division work is to be extended as follows: 2 62 72 2 ] 124 2 | 144 3. Say or think 37 Ihere are 3 2's in 7 and 1 over 2 j 74 There are 7 2's in 14 4. Say or think 32 There are 3 2's in 6 2 | 65 Rem. 1 There are 2 2's in 5 and 1 over Note: Later the form 32M> may be taught. 2 |~65~~ 5. Say or think 37 There are 3 2's in 7 and 1 over 2 | 75 Rem. 1 There are 7 2's in 15 and 1 over There aro 4 4'e in 17, tin equal parts (use objects). 2. To define a unit. (Use objects) 3. To define a fraction as one or more of the equal parts of a unit }- means 4 of the 5 equal parts of one thing (% of a pie.) 4. To show the distinction between a unit and a number. 5. To define a fraction as one of the equal parts of a number $.-, means 1 of the 5 equal parts of 4 things. (% of 4 pies). 6. To show that % of 1=4.S of 4 (Use drawings or objects.) 7. To show that the left over part when a measurement is made is the fractional part of the unit. 8. To show that the fraction sometimes is used to represent a division. % means 4-^5. 9. To show that this last meaning is connected with the defini- tion suggested in No. 5. 10. To define words numerator and denominator and to perfect all written forms. 11. To show that under the definition given in No. 3 the denom- inator shows the parts into which the unit is divided and the numer- ator shows the number of parts used. nMHiiimmiiiiiiiiiiiiiiiiiiHiiiiiiiiimiiiiiimiiiimiiiiiiiimiiiimiiiim EQUIVALENT FORMS. 1. To show that a whole number may be expressed as a fraction. (a) 1=%=% etc. Use objects or drawings. (b) 3=4=1% etc. 5^i(j=i^ etc. Use objects or drawings. EXERCISES 4, 8, 6, 9, 12, 7, 11, 10, 5, 14. 16, 18, wiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiim^ 2. To show that a fraction may be changed to an equivalent form by multiplying both numerator and denominator by the same number. (a) }=-%--...% etc. Use objects or drawings. Argument If the unit is divided into twice as many parts, each part is half as large. Therefore twice as many parts are used. (b) %=d%=%2, etc. Use objects or drawings. Argument same as above. (c) 5 /s= 1 %= 1 %, etc. Same as above. (d) Establish rule. (c) Drill. EXERCISES. % IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIM 3. To show that a fraction may be changed to an equivalent form by dividing both numerator and denominator by the same number. (a) % } Use drawings or objects. (b) %=% Use drawings or objects. (c) %=% Use drawings or objects. Argument If the unit is divided into half as many parts each part is twice as large. Therefore only half as many parts are used. C. Establish rule. D. Drill. Note: This is the base for cancellation. Since cancellation is used later, the teacher should present this process very carefully. EXERCISES (a) %=% (b) %=% (c) iiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiN 4 To change several fractions to equivalent forms having like denominators. (3) l._, 1;; 1, tO <',, %> %0 I'se rules established above. Verify if necessary with objects. Note: This step requires a discussion of the meaning of a multi- ple and practice in the selection of the common denominator. This is the time to develop the idea of the lowest common multiple. Only enough time should be devoted to the topic to make possible the above operation. (b) % % % tO l%o 15^ Sfa Use rules established above. Verify if necessary with objects. (C) % % % tO 1<> : , () l%o 7 %( , same as above Note Since the need for this process is in the preparation of frac- tions for addition and subtraction it is desirable to present this work when the pupils feel a need for it. (d) Develop procedure. EXERCISES (a) %, %, %, ',- IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIH 5. To change an improper fraction to a mixed number. (a) -%=--lV4 Use objects or drawings. >v\ Argument There are % in 1. Therefore there is one unit in % with 14 over. (b) %=2% Use objects or drawings. Argument In 1 there are %. Therefore there are 2 units in % with % over. (c) Develop rule Divide numerator by denominator. EXERCISES (a) 3/2=ii/ 2 ( b) %=2i/ 4 %--? %=? !%='? %=? %=? %=? 9fr=? %=? %-? %=? miiiiiiiJiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiN 6. To change a mixed number to an improper fraction. <3"SL;-(iL)^Al%('% and })=%. Use objects or drawings. Argument. In cne there are %. ?/> and \(> are %. (b) 3%= (% and %)=H$. Argument In 1 there are %. In 3 there are 3 X %=% (c) Develop rule Multiply the whole number by the denominator. Add the numerator to this product. Write the sum over the denom- inator. EXERCISES. I'-, Us 31 , 1% : 5%= 6%= 99.';= Illlllllllllllllllllllllllllllllllllllll Illllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll 7. To change several mixed numbers and fractions to equivalent fraction;having like denominators. y 2 , %, 3% Use objects or drawings if i i?y l2> 4. iiiiiiiiiimiiiiiiimiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiimim 2. To add fractions having like denominators. (a) % + 1/0 = % Use drawings or objects. Argument Same as above. 2 fifths 1 fifth 3 fifths (b) % + % = 94 T 2 Argument Same as above. Note It is desirable to use the following written form as well as the forms shown above. = 2 EXERCISES. 3 /T + % = + Vl + % niimiiiim iiiiiiiiiiiiiiiiiiiiiiiiniiiiniiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiimiimiiiinin iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiini minimi 3. To add mixed numbers having fractions with like denominators (no carry). 2 1,4 + 3J4 Remarks This may be made clear by arranging the parts as fol- lows: 2 and 14 3 and 14 5 and % 5% = & EXERCISES. 3% 1^ 4^ 7% 7V* miiiiiiimmiiiiiiiiiiiiimmiiiiiiiiiimiiiiiiiiimiimiiimiimmniiiiimiim 4. To add mixed numbers having fractions with like denominators (with carry). 2% + 4ft % - 7% Remarks } 5 = Ifo. Add 6 and 1. EXERCISES. 3% 5 2 /r> 6% 4% -4% 77/ 8 91/2 2i/ 4 3% 53/ 4 5% 4% 6% 2% Illllllllllimilllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllim 5. To add unit fractions having unlike denominators. & = .% Argument Only like things may be added. Change fractions to equivalent forms having like denominators, then add. EXERCISES. V* Vi % + % + K + % % % % + % + % + % iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiim 6. To add fractions having unlike denominators. 17 /12 = Argument Same as above. EXERCISES. % % V* % + % ' + % + immniiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiHiiiiiiiiiiiimiiiimniiH 7. To add mixed numbers having fractions with unlike denomina-. tors (no carry). 3% = EXERCISES. 4% 41/6 2y 4 + 31/6 +2% + 1% 3^ 31/g + 5% + 2% + 7% + 3y 4 + 3% iiimiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiimiHiiiiMim 8. To add mixed numbers having fractions with unlike denomina- tors. 4% 49^ + 2% = 21945 EXERCISES. 2% 5% 47/6 + 8% +- 78^ + 9% 7% 35^ + % + 4% -f llllltllillllllllllllllllllllllllllllllllllllllllllllllllllllllllllHIIIIIIIMIIIIIIIIIIlllllllllllllHIIIIIIIIHIIII IIIIHIIIIIIIHIHIIIIIIIIIIIIIIIIIinillll IIIIIIMIIIIIMIMIIM SUBTRACTION. 1. To subtract fractions having like denominators. Use objects or drawings. Language Forms. % 1. % and % are % ^ 2. y, from % leaves % 3. The Difference between % and % ls% Argument Similar to that used in addition. EXERCISES. % = % Mo HB - 7 /i2 = Mo 1 H 1 - Ve iimimiHiHiimiiiiiiwiiiwiiiiiiiitHitiimiiimiiwiMiiiiiiwiiimiiimiiiiiHim 2. To subtract mixed numbers having fractions with like denomi- nators. Language Forms. 4% 1. 2-> and % are $.v 2 and 2 are 4 - 2% 2. $-, from % leave $' 2 from 4 leaves 2. 3. The difference between % and % is %. The 2% difference between 4 and 2 is 2. Argument Similar to that used in addition. Use objects and drawings. EXERCISES. 2^ 7% 8% 3% 5.}:, 1 - 4% ft 6% 4% 5% 4-% 37^0 iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiniiiiiiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiniiiihiiii 3. To subtract fractions having unilke denominators. Argument Similar to that used in addition EXERCISES. %g/ ___ gy /9 70 5 /7 Hiiumiiiiiiiiiiiimiiiiiiiiiimiiiiimiiiiiii iiiiiiiiimiiiiiiiiiiiuiiiiiiiHiiiiiiiiiiiiiiMiMiHiiiiiiiiiiiiiiiiiiiiiiiiiiitiiiiiiiiiiiiiiiiiiiiiiiiiui.iiiHiiuiiiiiiiiiii 4. To subtract mixed numbers having fractions with unlike de- nominators (no borrow). 4% = 4i%o Argument Similar to that used in addition. 4% 6% 7% -2% -2% - 5% 3% 47/ 8 4% 1% 2% - 3% 4% 5% 5% MiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiM 5. To subtract mixed numbers having unlike denominators (with borrow). 5% = 5i% = 2% = 2i% Argument Since i% cannot be taken from i% one of the 5 units is changed to 20ths and added to i% . This leaves 4 units. Note This last step is difficult for children to grasp. The work should be developed very carefully. The above example may be presented as follows if the teacher feels it necessary to follow the general mechanical process used in subtrac- tion of whole numbers. * 2% = Argument Since i% cannot be taken from i% it is necessary to add 2%) to i% making 3% . Having added one unit ( 2 % ) to the minu- end we must add one unit to the subtrahend. Therefore we make the 2 a 3. = 3i% 2% 4^ 3% EXERCISES. 5% 3% 143/ 4 57/ 8 25% iiimmiiiiiiiimiiiimmmiiiiimmiiiiiniiimiiiiiiiiiiiiiiiimiiin iiiiniiiii MULTIPLICATION. 1. When X sign is read (times) the left hand number is the mul- tiplier. 2 X 1/6 is read 2 times }. 2. Fractional multiplicand: whole number multiplier. Sign read, "times." (a) 4 X ^ = % = 2. Use objects or drawings. Argument 4 times 1 apple = 4 apples J^ 4 time 1 pencil = 4 pencils 4 times 1 half = 4 halves Multiply numerator because it expresses the number of parts used. (b) 4 X % = % = 2% Argument Same as above. (c) 4 X % = 2% = 10 Argument Same as above. (d) 3 X 2% = 6% Argument 2 and i/ 5 3 6 and % Multiply the whole number and fraction separately. (e) 2 X 4% = 8% = 9% = 9i Argument Same as above. 94 = 1^8 and 1 are 9 (e) Develop short cancellation method % x % = 2. EXERCISES. (a) (b) (c) (d) 2 X % = 2 X % = 3 X % = 2 X 3% 3 X % = g vx 1 / 5 X % = 3 X 4i/ 4 (e) 2 X 31^ = 4 X 5% = 5X2% = 3X2% = 3X7% = 4X2^ = iiiiiiiiiiiiniiiiiiitiiiiiiiiiiiiiiiiiiiiiHiiiiiiiiiiiiiiiiiiiiiiiiiiiiitiiiiiiiiiiiiiiiii tiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiitiiiiiiiiiiiiiiiiiiiiiiiiiiuHiiiiiiiimiiiii 3. Whole number multiplicand; fractional multiplier. Recall the fact that the product is the same if factors are inter- changed. 2X3 = 3X2 Except in very simple cases interchange the multiplicand and mul- tiplier and proceed as in step 1. In such cases as % X 4 the plan should be to show that ^ X 4 = % of 4. After this work has been developed carefully use the short cancel- lation method, ^ J| 4 = % = 2. Argument If the teacher desires to use a special plan for this case her argument would be (a) % X 12 = % of 12 14 of 12 = 3 % of 12 = 3X3 = 9 (b) % X 8 = % of 8 1st plan J/5 of 8 = 1% % of 8 = 4 X 1% = 41% = 6% 2d plan % X 8 = % of *% % Of 4 9 i =: % % of 4 9 6 = 4 X % = 3% = 6% EXERCISES. (a) % X 6 = MX 8 = % X 15 = % X 30 = (b) % X 4 = % X 12 = ' % X 9 = % X 18 = % X 7 ft X 5 %i X 9 H X 8 % X 13 % X 3 % X 5 % X 10 iMiiiiiiwHwiimmiiiiiiiiiiiiiiiiiiiiiiiiHiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiw 4. Fractional multiplicand; fractional multiplier. (a) % X % = % of % = % Argument This is based on other steps, ^ of 4 fifths = 2 fifths. (b) % X % == % of % = % Argument % of % = % % of % = 2 X % = % (c) After this step has been covered carefully"the short method should be developed. Although a complete explanation is not desirable it is possible to so arrange the examples that the process seems a rea- sonable one. i x % = 2 /t may be worked out by the methods ^ X ty = ii4 = % suggested and then worked by rule and the answers compared. The pupil will observe that multiplying the numerators together and multiplying the denominators together produces the correct an- swer. When using the short method cancellation should be used. If the examples used in the first two steps are carefully selected the pupils will accept the short method for such an example as* % X % EXERCISES, (a) (b) (c) short method % x % % x % 94 x- % x % % x % % x % x % % x % % x miiiiiimiuiiiHiiimmiiiiiiiiHiiii n niiiiiiimiimn IIIIIIIIIHIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII iiiiiiiiiiniiiii iiiiiiiiiiiiiiiiiiiiiiimiiiiiiiin Should the teacher desire to avoid the short method she may change the multiplicand to an equivalent fraction having a numerator divisable by the denominator of the multiplier. % X fy = % of % Since % = % Then % of i%! may be written ft of i%! % Of 1%! % x # = ^x%= ;? ? = % X % = % X % = ^1 X % = ^ 4 x % = % X % = %o X % - 5 When the short method is used examples involving mixed num- bers may be arranged by changing the mixed numbers to improper frac- tions. % X 2% X % = % X 1% X % = % EXERCISES. HX3^X%= V 4 X2^X% = 1% X ?| X % = % X 3% X % = 41 X 2 X % = * X 1% X % - % X 3% X % = 4V 2 X 2 X % = iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiim DIVISION. 1. Define sign -r- to read, "divided by." 2. Divisor a whole number; Dividend a fraction. (a) % -r- 2 = ^5 Use drawings or objects. Argument 4 fifths ~ 2 = 2 fifths. The lead may be 4 apples -4-2 = 2 apples 4 pencils -=-2 = 2 pencils, etc. Divide numerator because it shows parts used. (b) 9.3 -T- 3 = 3,3 Use drawings or objects. Argument Same as above. (c) 4% -*- 2 = 21,4 Argument Divide whole number and fraction separately. 2 | 4 and % 2 and i /3 = 2i/ 3 If desired the mixed number may be changed to an improper frac- tion. yfo/ . O tA/ . O T/ Ol / 4% -.- 2 = 1% -f- 2 == % = 2% This plan is best when the example is like 5% -:- 2. 5% -*- 2 = 2^ _*. 2 = 1% = 2ft Note In some developments such examples as % -r- 3 are not dis- cussed until later. The development may show that 5/1 -5- 3 = i%! -5- 3 = %! if desired. EXERCISES. (a) (b) (c) % -*- 3 = % -*- 3 == 3% -s- 2 = 7 /8 -*- 2 = % -H 4 = 5% -4- 3 == ^--3= %-3= 6%-J-2 = 4 -s- 3 = iiiiiiifiHHiiuiiiiiiiiiiiiiiiiiiii ...... iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiinimi) ..... iiiiiiiiiiiiiiiiiiiiiiiiiniiii ....... iiiiniiiii ..... iiiiiiiiiiiimiiiinii 3. Show that 6 -r- 2 = ^ of 6. By induction show that dividing by a number is equivalent to multi- plying by its reciprocal. This may be done by using drawings and ob- jects. (a) 6 -*- % = 6 X % = 12 8 -4- % = 8 X % = 24 4 -*- % = 4 X % = 1% = 6 (b) % - % = % X % = % % -5- 2 = $ 3 X H = %o = % Mixed numbers are changed to improper fractions, and 1 is placed under each whole number. Use cancellation when possible. (c) % -4- 4 = % -s- ft EXERCISES. (a) (b) 8 - H = ? fi -H 2 = 12 - y 4 = ? % -5- 5 = 4 -*- % = ? 6 - = ? iiiiniiiiiiiiimmiiiiiiimiiiiiiimwwHHiiiiiiimiiiimiiiiiiiu 4. If the teacher does not use the short method she may change fractions to equivalent fractions having like denominators. (a) % -*- % = 2 Argument 4 apples -f- 2 apples = 2 4 cents -f- 2 cents = 2 4 fifths -T- 2 fifths = 2 Use the idea of division by measurement. (b) 4 -s- fc % -*- % = 8 Argument Same as above. (c) % *- % % * 10 /l5 = % = l%o = 1% Argument Same as above. (d) % H- 5 % * 2 % = %o EXERCISES. (a) (b) (c) (d) - 2 iiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiwim FRACTIONAL PROBLEMS. 1. To find a fractional part of a number. (a) % of 36 y 4 of 36 = 9 % of 36 = 3 X 9 = 27 (b) % of 14 % of 14 = 4% % of 14 = 2 X 4% = 8% = 9i (c) % of 3fc % of % = % of 3% % of 3 %o = %o % Of 3 %0 = 2 X 7^ Q = 1^ 1^ Q = 1% By the short method. (a) % of 36 = % X 3%'= 27 (b) % of 14 = % X i% = 2% = 9% (c) % of 3% = % X 7^ = i^ EXERCISES. (a) (b) ( C ) % of 24 = % of 7 = fc of 2% % of 30 = % of 13 = % of 3}| ^ of 28 - ^ of 25 = -% of 4$ % of 20 = % of 19 = ^ of 6% IIIIMIIIIIIllllllHIIIMtllllllltlllllllllllllllllllllllllllllllllllltllimillllllllllllllllllllllllllllUIIIIIIIIIIIIIIIIIIU 2. To find a number when a fractional part of it is known. (a) % of a number = 8 ^ of a number =M % of a number = 3 X 4 = 12 (b) % of a number == 8 y 4 of a number == 2% % of a number = 4 x 2% = 8% = 10% (c) % of a number = 3} $5 of a number = 19^ }$ of a number = % % of a number = *% = 8^ By the short method, sign X is read "multiplied by." (a) 8 -f- % = 8 X % = 2% = 12 (b) g -*- % = 8 X % = % = 10% (c) 3^ -^ % = 1% X % = so/ 6 = 8 % - 8^ EXERCISES. (a) Find number if (b) Find number if (c) Find number if % of a number = 9 % of a number = 7 % of a number = 2^ % of a number = 8 % of a number = 5 % of a number = 5^ % of a number = 6 % of a number = 6 ^ of a number = 4% To find the fractional relationship between two numbers. (a) 4 is what part of 20? Since there are 5 4's in 20,4 is \'- y of 20. This is expressed by the fraction % = };-, (b) By analogy. 5 is what part of 7? 5 is of 7 EXERCISES. (a) 3 is what part of 18 4 is what part of 24 7 is what part of 28 8 is what part of 64 6 is what part of 36 (b) 3 is what part of 5 2 is what part of 7 4 is what part of 21 6 is what part of 11 5 is what part of 12 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIMIIIIIIIIIIIIIIIIIIIIIIIIIMIIIIM DECIMAL FORMS 1. Show that a period may be placed at the end of a number. 264. 2. Develop the idea that fractions having denominators of 10, 100, 1000, etc., may be expressed in the decimal form. %o = -3 4 /io = -4 7 /ioo = -07 28/ 100 = .28 etc. 3. Show that some fractions may be changed to equivalent forms having 10, 100, 1000, etc., for denominators. % = 5 /io %o = 15 /ioo % = tfo etc. 4. Express the aliquot parts of 100. etc. EXERCISES. 392 25 1064 tio = ? 5 /io = ? 5 /ioo = ? 32 /ioo = ? % IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIH 5. Show that fractions may be expressed as equivalent fractions having 10, 100, 1000, etc., as denominators. %! Of 1^91oO == 27%! == '27%! IW Argument i/ii of 100 = 91/n %! of 100 = 273/!! Note Short method developed later. EXERCISES. tt % % % % % % iimiiiiiiiimiiiiiiiiimimiiiiiiiiMiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiim 6. Develop the method of multiplying and dividing decimal frac- tions by 10, 100, etc., by moving the point. .3 -5- 10 = .03 .3 x 10 = 3. 21.62 -=- 10 = 2.162 21.62 X 100 = 2162. etc. EXERCISES. .5 -f- 10 = ? A -5- 10 = ? .5 X' 10 ? .4 X 10 = ? 42.65 -r- 100 = ? 32.54 -5- 10 == ? 6.345 X 1000 = ? 32.54 X 100 = ? 34.6 -=- 100 = ? Illllllllllltllllillllllllllllllll lllllinilllllllllllllllllllllllHIUIIIIIIIIIIIIIIHIlllHNIIIIIIMIIIIIIIIIIIIIIIIinilliinililllUii.ilillllllillllllllllllHIIIIIIIUIIIIIIIIhllll 7. Show that to add and subtract decimal expressions the points must be placed one under another. .12 21.6 .03 1.08 .15 22.68 EXERCISES. .25 .32 27.8 63.4 2.54 04 .01 2.09 2.5 6.34 Illllllllllllllll ..... tlllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllltll ...... Illllllllllltllllllllllllltlllllllllllllllllllllllllllllllllllll 8. To multiply decimal expressions. (a) 21.6 3 64.8 Argument The carry may be shown by writing the example. 63is/ 10 = 64% = 64.8 (b) 251 .23 753 502 57.73 Argument .23 X 251 fc= -% )0 X 251 V 100 of 251 = 2.51 2% 00 of 251 = 23 X 2.51 = 57.73 (c) 103-32 Argument 16.4 = 6 X 16.4 and V 10 X 16.4 6.3 16.4 6 3/ 10 X 16.4 = 1.64 .y 10 X 16.4 = 3 X 1.64 = 4.92 98.4 98.4 4.92 103.32 EXERCISES. 26.4 8 264 .3 2.56 4 634 .25 .36 4 84.6 2 423 3.6 265 1.26 6.8 1.6 25.6 .14 6.23 .03 .634 2.5 iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiin linn mi i iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiini 9. To divide decimal expressions. (a) 2.31 2 | 4.62 (b) .03 | 369 Multiply both dividend and divisor by 100 12300 3 | 36900 Argument- Show that dividend and divisor may be multiplied by same number without changing their relationship. (c) 2.34 ] 64.6 Multiply both dividend and divisor by 100 27.606 + 234 I 6460.000 468 1780 1638 1420 1404 1600 1404 Argument Same as above. Note It is very important to perfect the mechanical process. 1. Multiply both dividend and divisor by 10, 100, etc., so that the divisor is made a whole number. 2. Place a point in the quotient directly over the new position of the point in the dividend. 3. Place each digit of the quotient directly over the last digit used in the dividend. EXERCISES. 3 | 64.3 .6 | 256 2.5 | 63.4 5 | .634 1.6 1 463 .43 | 624. 2 [ 4.36 .04 | 634 .24 | 1.634 II IIIIIIIINIIIIIIIIIIMIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIN 10. Develop the short method for changing a fraction to a decimal form. .2787 + 3/ n 11 | 3.000 22 SO 77 30 22 80 77 3 EXERCISES. % I Illllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIMIIIIIIIIIIIIIIIIMIIIHIIIIIIMIIIIII 11. The problems of decimal fractions: (a) To find .08 of 64 i lnu of 64 = .64 S IIM) of 64 = 8 X .64 = 5.12 (b) To find the number if .08 of it is 64 s | (l() of number = 64 ' ]uo of number = 8 io<) 100 of number = 100 X 8 = 800 (c) To show relationship between .08 and 64 .08 fractional relationship 64 .00125 .08 = = 64 fTOSOOiT 64 64 160 128 320 320 Argument A fraction represents a division. Short methods: (a) .08 of 64 = 64 .08 5.12 800 (b) 64 ~- .08 = .08 | 64 = 8 | 6400 .00125 (c) .08 = 64 I .08000 64 64 160 128 320 320 EXERCISES. Find Find Number If Find Relationship Between .06 of 48 .03 of it = 18 .03 and 1.4 .12 of 72 .45 of it = 90 25 and 1.06 .05 of 27 .15 of it = 75 1.6 and 24.6 iiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiimiiiiimiiiiiiiiiNH PERCENTAGE. 1. To define % sign. Word percent. 2% = %oo = -02 2. To show that fractions having denominators of 100 may be ex- pressed by using % sign. (a) Moo =3% 4y 100 = 41% (b) .04 = 4% .16 = 16% 3. Drill on aliquot parts of 100 4. Express fractions in percents (a) %! = %! of ioo/ 100 = 27% n hundred ths = 27/ n % (b) Short Method .27 + 3/ n = 11 I 3.00 22 80 77 3 Argument A fraction represents a division. (c) %! of 100% Mi of 100% = 9Mi% 3/ n of 100% = 3 X 9i/ n % = EXERCISES. iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiimiiiiiiiiiiimiiiiiiM 5. To find a percent of a number. (a) 5% of 86 = % o of 86 y 100 of 86 = .86 % 00 of 86 = 5 X -86 = 4.30 (b) Short method 86 .05 4.30 EXERCISES. 3% of 15 6% of 19 3% of 142 1% of 28 4% of 35 6% of 263 8% of 72 15% of 96 15% of 128 12% of 48 26% of 172 49% of 642 Illlllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll Illlllllllllll Illllllllllllllllllllllllllllllllllllllllllllllllllllllll 6. To find a number when a percent is known. (a) 86 = 5% of what number? 5% = 86 1% = 17.2 100% = 1720 (b) Short Method 1720. .05 j 86 = 5 | 8600. EXERCISES. Find Number If 8% of it = 16 5% of if = 14 10% of it = 50 8% of it = 27 5% of it = 35 14% of it = 46 6% of it = 66 70 % of it = 172 iiiiiiiiiiiiiiiminiHHiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiM 7. To find what percent one number is of another (a) 8 is what % of 86 8, 6 =^= fractional relationship % 6 of 100% ]e of 100% = 1.1627% % Q of 100% = 8 X 1.162 = 9.3016% (b) Short Method .093 + 86 ! 8.000 774 260 258 2 093 = 9.3% EXERCISES. Find What % 5 is of 25 14 is of 28 10 is of 60 8 is of 13 8 is of 48 7 is of 43 6 is of 90 56 is of 362 illlllllllllllllllllllllllllim IIIIIIIIIIHIIIHIIIIIIIIIIIIHIIIIIIHIIIIIHIIIIIIIIHIIIIIIIIIIIIIItllllMllinillllllllllllllllllllllltlllHIIIIIIIIIIIHIIIIIIIIIIIIIIIIHIIIIIIIIfl SPECIAL TOPICS Under the plan of accepted courses of study the pupils are to con* plete the work devoted to the common operations in whole numbers, common fractions, decimal fractions, and percentage in the first six years. During this time they are to study the common measuring units, their relationships, and perform all the ordinary measurements. The next logical course is one given over to the common appli- cations of facts and processes, and the ordinary measurements. The dominant feature of the work should be the composite problems in- volving the application of many topics. An example of such a prob- lem is the building of a house. Pupils study the smaller problems involved, the business problems connected with the buying of the lot, the measurements involved in the cellar excavation, the building of the wall, the labor costs, plastering, papering, painting, roofing, furnish- ing, and finally the business problems connected with selling and the housewives' problems connected with living- in the house. These composite problems should take the forms of: Buying and selling problems Construction problems Communication problems Transportation problems Money and credit problems Production problems Etc. In all such problems the center of interest should be on the ap- plication rather than on topics. The problems should draw from many topics. For example a problem in buying and selling must of neces- sity touch upon communication, transportation and money and credit. A construction problem may deal also with buying and selling, com- munication, transportation, money and credit and possibly production. A knowledge of general business procedure is needed by every- one regardless of special vocation or profession. The problems con- nected with banking are those which we are interested in; how to open an account, how to write a check, how to open a savings bank account, etc. The problems connected with taxes, insurance, bonds (such as liberty bonds) corporations, investments, etc., are interesting to more and more people. The war brought about much general edu- cation along these lines. A course of this sort is needed and it must be entirely different from the old topical courses offered in the eighth grade work. It must be given in connection with actual business pro- cedure. Modern business is so interlocking no person can carry on the ordinary activities of social life without knowledge of common busi- ness practice. It is not necessary or desirable to present topics as completely as has often been done in the past. The special technical instruction in any branch of business can better be given to those par- ticular students who enter it as a life occupation by those in charge of the special branch. For example a study of all the highly technical problems of in- surance is of little interest to society at large but all people should know the values of insurance, the kinds of insurance, how to select a policy, how to take out a policy, how to pay premiums and how to collect insurance. Or again in taxation the problems should cover the needs for taxes, the meaning of a tax rate, the kinds of taxes, how the tax rate is established, how to pay taxes, the penalties for not paying taxes, the value of a tax receipt, tax sales, how to make out an in- come tax form, etc. Illllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll Ill I Illllllllllllllllllllllllllllllllllllllllllllllllllfllfllllllllllllllllllllllllllllllllll After a course in which pupils have selected and applied the arithmetical knowledge they have, the classification of arithmetical knowledge in terms of local needs would seem advisable. For ex- ample there is a certain group of arithmetical facts and applications of interest especially to farmers and another group of facts and ap- plications of especial interest to those who live in cities. Pupils should study the arithmetical needs of society in terms of the needs of the larger groups. In this work attention should be given to the significance of so- cial problems as well as to individual ones. These problems differ from those of the individual in that they are connected with social welfare rather than individual welfare. For example when studying the relation of mathematics to rural life the student is interested not only in the problems of the farm but those of rural community life. A like plan should be followed when studying the relation of mathe- matics to urban life. There are problems of the home and problems of the community at large. In the construction work the students cover the definition, classi- fication, and mensuration of the common geometrical figures. This subject matter in connection with applied problems helps the students to acquire a fund of useful knowledge relating to the mathematics of form. After the general business applications are discussed the special mathematical applications needed in vocational lines should be taken up. Mathematics of wood-working, of farming, of shop practice, of automobile work, of cement work, of clerking, of dress-making, etc., should be taken up with the thought that boys and girls may get a spark of inspiration somewhere along the line. In some of the voca- tions they will find an interest. This course may seem to duplicate some of the subject matter but the stress is placed on the vocation. The thought is that stud ents should receive special preparation for entering some particular occupation. It is to be hoped that individual students will become especially interested in some particular phase of this presentation. In case they do this the teacher should encourage them to study appropriate problems. A well arranged reference library is very val- uable for use in this grade. There should be books in such a library on the mathematics of farming, retail trade, commerce, industry, clerk- ing, trades, salesmanship, construction, etc. Class work should be sup- plemented by talks by persons in the community engaged in various occupations and by visits to business houses. If possible, arrangements should be made to allow students to engage in the actual work they feel interested in. Note: The following outlines are arranged to suggest the import- ant points to develop under various topical headings. No attempt has been made to arrange complete topical outlines or to present a substi- tute for a text book. In upper grade work a text book together with reference books should always be used. The teacher should select and stress the sub- ject matter that is most closely connected with life problems and should present this subject matter in such a way as to appeal to the interests of students. The following two outlines on communication and transportation suggest the general plan to be followed when taking up life activities. iiimimiimniiiimiiiiiiiimiiiiiimiiimiiiiiiiimimiiimiiiiiH TRANSPORTATION 1. Railroad Freight Classification Rates Freight bills How to send How to receive Express Classification Rates Receipts How to send How to receive Parcel Post Classification Rates Limits of weight Insurance How to send How to receive 2. Shipping by boat Classification Rates Bills of shipment How to send How to receive 3. Shipping by motor truck Rates How to send How to receive 4. Air transportation Values Discuss development 5. Discuss all above tonics with regard for values and probable development. Work out problems dealing with each point. IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIMIIIIIIIIMIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIHIIIIIIIIIIIIIIII Illlllltlll Illlllllllllllllllllllllllllllllllllllllll COMMUNICATION Postal service Classification 1st, 2nd, 3rd class, etc. Limit of weight Regulations for sealing, etc. Rates on different classes. Exceptions, U. S. bulletin, etc. Foreign postage Registered mail. Special delivery. Discuss all above points and work out problems dealing with each. Telegraph and cable Classification day and night messages. day and night letters. Rates Cable rates Advantages of telegraph How to send message How to receive message Wireless Uses Discussion covering its development. Discussion covering equipment needed. How to send and receive messages. Air Postal service Uses Discussion covering its development Rates How to send and receive messages. Telephone Local and long distance. Uses, rates, etc. How to send and receive messages. iiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimiiiiiiiMN COMMON BUSINESS PAPERS 1. Discussion covering. Receipt Order. Statement Receipted bills Account Ledger Day book Contract Freight bill Etc. 2. Have all such papers for inspection and have pupils fill out blank forms. 3. Discuss common words and their abbreviations 4. Study the form of keeping an account and have pupils keep an account of simple transactions. Illlllllllllllllllllll Illlllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllltllllllllllllllllllllllllllllllllllllllllllllllllllll Illllllllllllll Illllllllllll COMMISSION BROKERAGE 1. Definition. 2. Uses. 3. Definition of Terms. The Commission. Rate. Net Proceeds. Principal. Agent. Broker. Etc. 4. Problems in Commission. A. To find a commission. To find 5% commission on sales amounting to $4,684.62: First Method 100% = $4684.62 1% = 46.8462 5% = $234.231 Second Method $4684.62 .05 $234.2310 B. To find principal when the rate and commission are known : 6% commission on sales yields $64.72. Find the amount of sales. 6% = $64.72 1% = $10.78% $1078.66% 100% = $1078.66% .06 | 064.7200 C. To find rate when. the principal and the commission are known : Principal = $643.70 Commission = $172.63 $64; .268 + 172.63 The relationship = 172.63 643.70 X 100% $17263.000 128740 438900 386220 = 26.81 + % 526800 514960 iiiiiiiiiiiiini iiiiimiiiiiiiimiiiiiiiiiiiiiiiiiiimiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiH COMMERCIAL DISCOUNT 1 Definition. ) 2. Uses. 3. Definition of Terms. Marked Price. Net Price. Rate of Discount. Successive Discounts. Trade Discount. Time Discount. Cash Discount. Etc. 4. Problems in Discount A. To find a discount when marked pric'e and rate are given: Marked price = $762.14 Rate = 6% First Method Second Method 100% = $762.14 $762.14 1%'= $7.6214 .06 6% = $45.7284 $45.7284 Note Net price = 94% = $716.4116 B. To find marked price when the rate and the discount are given : Discount = $17.50 15% = discount Second Method $116.66 .15 | $17.50 15 First Method - 15% = $17.50 25 1% = $1.1666% 15 100% = $116.66% 100 90 10 iiiiiiiiiinii iiiiiuii iiiiiiiiiiiiiiiiMiiiini!iiiiiiiiiiiuiiiiniiiiuiiiiiiiiiiiiuiiiiiiiiiiiininiiiiiiiiiiiMiMiiiiiiii!iiiiiiiiMiiiiiiiiiiiiiiiniiiiiiiiininiMiitiiin C. To find the rate when the marked price and discount are given: Marked price = $28.76 Discount = 4.36 .151 + 28.76 | $4.36000 2876 The relationship = - 436.% 4.36 - x 100% = 28.76 28.76 28.76 = 15.12% D. To find successive discounts: Marked price = $462.14 Successive discounts 14840 14380 4600 2826 1724 20%, 10% First Method 100% = $462.14 1% = 4.6214 20% = 92.428 80% = $369.712 100% = $369.712 1% = 3.69712 10% = 36.9712 90% = $332.7408 Second Method $462.14 .80 $369.7120 .90 $332.740800 iiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiijiiiiiiiiiiiiiiiiiiiimu iiiniii INTEREST 1. Definition of Interest. 2. Definition of Terms. Principal. Rate of Interest. Time. The Interest. Amount. 3. The procedure in finding interest. First Plan Find interest on $150 for 2 years 6 months at 6%. $ .06 interest on $1 for 1 year. 2*/ X $ .06 = $ .15 interest on $1 for 2y 2 yrs. 150 X $ .15 = $22.50 interest on $150 for 2% yrs. Second Plan $ .06 interest on $1 for 1 year. 150 X $ .06 = $9.00 interest on $150 for 1 year. 2% X $9.00 = $22.50 interest on $150 for 2% years. SPECIAL METHODS A $ . 06 interest on $1 for 1 year $ . 005 interest on $1 for 1 month $ . 000% interest on $1 for 1 day 2 X $ .06 = $ .12 interest on $1 for 2 years 6 X $ .005 = $ .03 interest on $1 for 6 months $ .15 interest on $1 for 2 years 6 months 150 X $ .15 = 22.50 interest on $150 for 2 years 6 months Note If the rate is 5% take % o*f this answer If the rate is 4% take % of this answer B 60 days = % of year At 1% the interest on $150 == $1.50 for 1 year Since 60 days = % of year and 6% =6 X 1% The interest on $150 for 60 days at 6% = $1.50 There are 15 60-day periods in 2i/> years 15 X $1.50 = $22.50 Note Same as above. Find interest on $56 for 80 days at 4% $ .56 interest on $56 for 60 days at 6% $ .18% (% of $ .56) interest on $56 for 20 days $ . 74% interest on $56 for 80 days % of $ .74% = $ .497/6 interest on $56 for 80 days at 4% 4. Compound Interest. A. Definition. B. Method of working out. C. Formula for compound interest. A = PCM Iff-) P (1 + RT)n A = amount. P = principal. R = rate. T = time in years, between the compoundings. N number of compoundings. iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiinii imiiiiiiiiiiiimi in IIIIIIIIIIMIIIIIIII iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiinii D. Problems. To find compound interest on $497.68 for 2 years at 6%, interest compounded semi-annually : 6% for 1 year = 3% for 6 months. $497 .1 $14.9304 interest for 6 mo. $497.68 $512.6104 amount after 6 mo. .03 $15.378312 interest 2d 6 mo. By Formula 1 + RT = 1.03 (1 X RT)4 == 1.125 + 1.125 X $497.68 = $559.88 + Note This may be carried out to greater accuracy by extending 1.125 + $15.38 512.61 $527.99 amount after 1 year .03 $15.8397 interest third 6 mo. 527.99 $543.8297 amount after 18 mo. .03 $16.314891 interest after 2 years $543.8297 amount after 2 years To find compound interest on $9040 for 4 years at 4%, interest com- ix unified annually: $9040 .04 $361.60 Int. for 1 yr. $9040. $9401.60 Amt. after 1 yr. .04 $376.0640 Int. after 2 yrs. $9401.60 $9777.6640 Amt. after 2 yrs. .04 $391.106560 Int. after 3 yrs. $9777.66 $10168.76656 Amt. after 3 yrs. .04 $406 . 7506624 Int. after 4 yrs, $10168.766 $10575.516 Amt. after 4 yrs. By Formula A == P (1 + RT)4 1 + RT = 1.04 (1 + RT)4 = 1.1698 4- 1.1698 4- X $9040 = $10574.99 Note This may be carried out to greater accuracy by extending 1.1698 + IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIM BANKING 1. A bank Provides A place to deposit money for safe-keeping A place to loan money not needed. (Saving Dept.) A place to borrow money. A checking privilege A way to send money (bank draft) A clearing house for checks on many banks, etc. 2. Classification National Banks State Banks. Private Banks. Postal Savings Banks National Reserve Banks Trust Companies Savings Banks 3. Definition of terms. Checking account. Savings account. Deposit slip. Check and check book, Draft. Note. Discount on note Security. Safety deposit box. Indorse Overdraft. Etc. 4. Problems. To open a checking account* To open a saving account To write a check. To draw out money from a savings account. To buy a bank draft. To borrow money. To sell a note. To rent a safety deposit box. Note: All papers should be made out by pupils. IIIJIIIIIIIIIIIIIIIIIIIII Illlllllllllllllllllllllllllll II I Illllllllllllllllllllllllllllllllllllllllllllllllllllllllll Illlllllllllllllllllllllllllllllllllllllllllll PROMISSORY NOTES 1. Definition. 2. Essential points. (a) A note should state the time and place when written. (b) A note should state the time and place when due. (c) A note should state the name of the person to whom payment is to be made. (d) A note should state the name of the person making the promise to pay. (e) A note should usually state that value has been re- ceived. (f) A note should state the rate of interest. 3. Definition of terms. Maker. Payee. Indorse i In blank) (In full) (In limited form). Negotiable. Maturity. 4. Values. Provides a record. May be transferred (sold to a bank for example). May be collected through the court. 5. Partial payment of promissory notes. The United States Rule. Kind the amount of the principal when the first payment is made. From this amount subtract the first payment. The remainder is the new principal. Continue until final payment is made. It' a payment is less than the interest, the payment is not sub- tracted. It is carried to the next payment. This procedure is follow- ed until a total payment is equal to or greater than the interest. ILLUSTRATIVE EXAMPLE . Note for $2500. 6% interest. Date July 15, 1919. August 15, 1919, $100 is paid. January 1. 1920, $10 is paid. June 15, 1920, $200 is paid. Find amount at this last date. Interest from July 15, 1919 to August 15, 1919=$12.50 Amount of note August 15, 1919=$2512.50. Amount of note less $100 $2412.50. Interest on $2412.50 from August 15, '19 to January 1, '20=$54.28 Amount of note January 1, '20=$2466.78. Payment not subtracted. Interest on $2466.78 from January 1, '20 to June 15, '20=467.83. Amount of note June 15, '20=$2534.61. Amount of note less $210=$2324.61. IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII1H STOCKS. 1. Definitions of corporations and capital stock. 2. Needs for corporations. A. Enables an organization to transact business as an in- dividual. B. Enables individuals to invest money without taking an active part in the activities of the company. C. Makes larger combinations of capital possible, which in turn makes greater projects possible. D. Offers the person having small capital an opportunity to join in the promoting of a large venture. 3. Definition of terms: Share of stock. Par value. Market value. Discount. Premium. Stock holder. Assessment. Dividend. Stock Broker. Brokerage. 4. Ways to buy stocks: A. Direct buying. B. On margins. C. Partial payment plan. 5. Examples: A. Find the cost of 40 shares of stock in a stove factory bought at 91 with brokerage at y 8 %. 40 X $91 = $3640 cost of stock at 91 40 X $ .125 = 5 cost of brokerage at y s % $3645 total cost of stock B. I must raise $950. How many shares of stock at must be sold to meet this amount with brokerage at }&%? $88.50 price of one share .125 brokerage $88.375 net price per share 10.749 + No. shares of stock. 88.375 |~950000T~ (H shares since one share cannot 88375 ' be divided.) 662500 618625 438750 353500 852500 Illllllllllllllllllllllllllllllllllll Illlllllllllllllllllllllllllllllllllllllllllllll I Illllllllllllllllllllllllllllllllllll Illlllllllllllllllil Illllllllllllllllll C. Find the cost of 450 shares of stock (par value 100) at 3% premium, brokerage %%. $100. cost of each share at par 3 . premium .125 brokerage on each share $103.125 total cost of each share 450 X $103.125 = $46406.25. Cost of 450 shares D. I buy 60 shares of stock (100 par value) for 88 If the stock pays 6% on par value what is the rate at 88? What is the total interest? 60 X $6 = $360. Total interest. $88! = cost of each share 6. == interest on each share .0681 rate at 88 88 | 6.00 5 28 720 704 160 E. A share of stock yields 15% when purchased at 70 (par value 100). What rate is this equivalent to on par value? 15% at 70 yields $10.50. .105 rate on par value 100 | 1~050 100 500 500 iiiiiiiiiiiiiiiiiiiiiiiiiimiiriiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiini iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiinii iiiiiiiiniiiiiiiii BONDS. 1. Definition. 2. Relation of a bond to a promissory note. 3. Relation of bonds to shares of stock. 4. Classification. U. S. Government. State. City. County. Industrial. Special. School. Bridge. Library. Paving. Etc. 5. Definition of terms. Mortgage. Premium. Discount. Coupon bond. Registered. Quotation. Bond holder. Commission. Brokerage. Par value. Market value. 6. How to buy bonds. 7. Examples: A. Find cost of 25 Liberty Bonds 4}4% at 92, brokerage 1 s- $ .125 brokerage on one bond. 25 X $ .125 = $3.125 brokerage on 25 bonds. 25 X $92 = $2300 cost of 25 bonds at 92. $2300 + $3.125 = $2303.125 total cost of 25 bonds. B. Find par value of 4% city bonds yielding $1200. $30000. par value of bonds. 04. 7Tl200007~ C. How much must be invested in bonds at 102 to yield $600 per year at 4i/,%, brokerage 1 8 . At 4}% it would take $13333.33 -f to yield $600. Or 134 $100 bonds. $13400 cost of bonds at par. 268 $2 premium on each. 16.75 brokerage. $13684.75 total cost of investment iiiiiiiiiiiiiiimiiiiiiiiiiiiiniiiiii iiMiiiiiiiiiiiuniiiiiiiiiiiiiiiiiniiiiiiiiiHiiniiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiniin D. If bonds yield 4y 2 % (100 par value) what percent will they yield if they can be bought for 90? $90 investment. $4.50 interest on each bond. .05 rate of interest. 90 I 4.50 4 50 E. A bond is at 32% premium, what rate does it pay if it bears 4U% (par 100) ? $132 cost of bond. $4.50 interest on bond. .0340 + rate of interest. 132 |~4750~ 396 540 528 120 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIM TAXES DUTIES INTERNAL REVENUES. 1. Definition. 2. Needs and uses of taxes, duties and revenues. To pay expenses of government. To pay for improvement of property. To buy property. To pay for protection, police, fire, accident, etc. To pay for education of children. To pay for care of insane, disabled and poor. To pay for care of criminals. To pay for care of roads and new roads. To pay for enforcement o.f special laws, prohibition, etc. To pay for harbors, dredging, canals, etc. To pay for conservation of natural resources, forestry, etc. To pay interest on old debts, war, etc. To pay army, navy, etc. '. Note This list may be extended by students. 3. Classification. Real estate. Personal property. Poll. School. Manufactured articles. Imported articles. Special taxes .Automobiles. Hunting and fishing. Income. Inheritance, etc. War times, etc. 4. Definition of terms. Assessor. Collector. Tariff. Ports of entry. Free list. Ad valorem duty. Invoice. Specific duty. Tare. Assessed valuation. Rate. Tax. 5. Problems in taxes. A. To find the tax when the assessed valuation and the rate are given. $4,500,000 Assess-ed valuation of city property. 13 mills on the dollar rate. $4,500,000 .013 13500000 4500000 $58,500.000 tax iiiiiiijiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiu B. To find the rate when the assessed valuation and the tax are given. $8000 = assessed valuation. $72 = tax. .009 rate 8000 | 72.000 72 000 C. To find the assessed valuation when the rate and the tax are given. $156 tax. . 014 rate. $11142.857 + assessed valuation. Vpj.4.1 156000 14 16 14 20 14 60 56 40 28 120 112 80 70 100 98 2 6. Method of assessing property. 7. Method of establishing rate. $456,000 assessed valuation. $26000 tax needed. .057 + rate 456,000 | 26000.000 2280000 3200000 3192000 8000 Method of paying taxes (value of tax receipt, etc ) IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIM INSURANCE. 1. Definition. 2. Values. \ A. Protection. B. Savings. "> C. Investment. 3. Classification. A. Property insurance. Fire. Tornado. Plate glass. Flood. Marine. Etc. B. Personal insurance. straight life policy. Life limited payment policy. endowment policy. Accident. Sickness. C. Other types. Insurance against rain. Insurance against frost. Etc. 4. Definition of terms. Policy. Premium. Rate. Paid up insurance. Endowment. Dividend. Loan on policy. Etc. 5. Taking out insurance. 6. Payment of premiums. 7. Method for collecting insurance. 8. Method for borrowing money on policy. IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIHIN LONGITUDE AND TIME. 1. Definitions. Meridfen. Equator. Prime Meridran. Longitude. 2. Relation between 360 of longitude and 24 hour day. 15 longitude corresponds to 1 hour of time. 15' longitude corresponds to 1 minute of time. 15" longitude corresponds to 1 second of time. 4 minutes of time corresponds to 1 longitude. 4 seconds of time corresponds to 1' longitude. Note Work out relationships. 3. Why do we estimate time from Greenwich? 4. Define International Date line. 5. Define Standard Time and explain time belts. 6. Examples. A. What difference in time exists between London and New York if longitude of London is 5' 48" w. and that of New York is 74 0' 3" w.? 74 0' 3" Longitude of New York 5' 48" Longitude of London 73 54' 15" Difference in longitude 73 longitude corresponds to 4 hrs. 52 min. time 54' longitude corresponds to 3 min. 36 sec. time 15" longitude corresponds to 1 sec. time 4 hrs. 55 min. 37 sec. time B. If Boston is in longitude 71 3' 50" w. and San Francisco is longitude 122 25' 42" w. what is the difference in time? 122 25' 42" Longitude of San Francisco 71 3' 50" Longitude of Boston 51 21' 52" Difference in longitude 51 longitude corresponds to 3 hrs. 24 min. time 21' longitude corresponds to 1 min. 24 sec. time 52" longitude corresponds to 3.46% sec. time 3 hrs. 25 min. 27.46% sec. time C. If two places h^ve 2y 2 hours difference in time what is the difference in longitude? 2i/ X 15 = 371/6 longitude. PRACTICAL MEASUREMENTS LENGTH 1. Common units. Inch. Foot. Yard. Rod. Mile. 2. Discuss measuring devices. Foot rule. Carpenters rule. Yard stick. 50 Foot tape. 100 Foot tape. 3. Work out relationships through actual measurements. iMiiiiiiiiiiiimiimimmimmmimiiiiiimiimiiiimmmmiiiimiiimiiiiiiiiiiim SURFACE 1. Common units. Square inch. Square foot. Square yard. Square rod. Square mile. Acre. 2. Study these units and \vork out table of relationships as far as possible. Use table for reference when needed. 3. Develop rule for finding area. 4. Special applications of square measure. Surface area in inches, feet, yards, rods, etc. Areas of floors. Areas of walls. Areas of walks. Areas of roofs. Etc. IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII1IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIH VOLUME 1. Common units. Cubic inch. Cubic foot. Cubic yard. Etc. 2. Study these units by making models, drawing pictures, etc. Work out relationships as far as possible. 3. Develop rule for finding volume of rectangular solids. 4. Special applications of cubic measure. Volumes^ rooms. *-.4 Volume^of bins Volumebf boxes. Etc. iiiiiiiiiiiiiiiiiiiiiiiiiwiiiiiiiiiiiiiiiiiiimiiiiiiiiiiiiiiiiiiiiiiiiiiiiiim SPECIAL CASES 1. Triangle Area= 1 /^ base X Alt. Circle Circumference=2f 3 . 1416l^rea%=3 . 1416R Trapezoid Area=i, (sum of bases) X Alt. Prism Lateral area Perimeter of base X Alt. Volume=Base X Alt. Cylinder Lateral area=Perimeter of Volume==Base X Alt. Pyramid Lateral area=Perimeter of base X slant height. Volume-=i & Base X Alt. Cone Lateral area=Perimeter of base X slant height. Volume=44 Base X Alt. Sphere Area>#3 . 1416jf*Vomme=% 3 . 1416* * 2. Work out areas and volumes using above formulae. 3. Special units. Board foot. Cord. Square (100 sq. feet). Etc. 4. All work in mensuration should be presented through the use of objects, drawings, tables for reference and actual measurement. A text book should be used as a guide. iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiin iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiitiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimmitiii THE EQUATION 1. Definition. 2. The four laws. (a) When equals are multiplied by equals the results are equal. (b) When equals are divided by equals the results are equal. (c) When equals are added to equals the results are equal. (d) When equals are taken from equals the results are equal. 3. The solution of equations when one unknown part is to be found. 4. The short method of solution by transposition. 5. Examples. (a) ix=3 x=6 Multiplying by 2. (b) 4x=12 x=3 Dividing by 4. (c) x 2=5 x=7 Adding 2. (d) x+3=7 x=4 Subtracting 3. (e; 2x 3=x+5 2x x= x=8 491083 UNIVERSITY OF CALIFORNIA LIBRARY