Y., under the act of August 24, 1912. Acceptance for mailing at special rate of postage provided for in section 1103, act of yee a ( cman’ 3, A917; authorized July 19, 19138 ~ Published Fortnightly ntered as bine chai, ieee Atigaok 2, 1913, ‘at the Post Office. at Albany, i) "ALBANY, Nai THE UNIVERSITY OF THE STATE OF NEW YORK Regents of the University With years when terms expire 1934 Cuxster S. Lorp M.A., LL.D., Chancellor - - Brooklyn , 1936 ApELBERT Moot LL.D., Vice Chancellor - - ~- Buffalo 1927 ALBERT VANDER VEER M.D., M.A., Ph.D., LL.D. Albany 1937 Cuartes B. ALEXANDER M.A., ey LUD. Eth, ei eee ee - - - - - Tuxedo 1928 Water Guest Kettoce B. LL. D. - - - + Ogdensburg 1932 James Byrne B.A., LL.B., LL.D. - - - - - New York 1931 Tuomas J. Mancan M.A. - - - - - - - Binghamton 1933 Witttam J. Wattin M.A. - - - - - - Yonkers 1935 Witt1am Bonpy M.A., LL.B., Ph.D., D. C. L. - New York 1930 Wititam P, BAKER BL. ne D. - - - - - Syracuse 1929 Ropert W. Hicpie M.A. - - - - - - - - Jamaica 1926 Rotanp B. Woopwarp B.A.- - - - - = = Rochester sd President of the University and Commissioner of Education FRANK P. Graves Ph.D., Litt. D., L.H.D., LL.D. Deputy Commissioner and Counsel FRANK B, Gitpert B.A., LL.D. Assistant Commissioner and Director of Professional Education Avueustus S. Downtne M.A., Pd.D., L.H.D., LL.D. Assistant Commissioner for Secondary Education James Suttivan M.A., Ph.D. Assistant Commissioner for Elementary Education Greorce M. Witey M.A., Pd.D., LL.D. Director of State Library James I. Wver M.L.S., Pd.D. Director of Science and State Museum Joun M. Crarxe Ph.D., D.Sce., LL.D. Directors of Divisions Administration, Ltoyp L. CHEenry B.A. Archives and History, ALEXANDER C, LICE M. AY Late Ds PRD: Attendance, JAMES D. SULLIVAN Examinations and Inspections, Avery W. SKINNER B.A., Pd.D. Finance, CLARK W. HALLIDAY Law, Inw1In Esmonp Ph.B., LL.B. Library Extension, Witt1am R. Watson B.S. School Buildings and Grounds, FRanK H. Woop M.A. Visual Instruction, ALFRED W. ABRAMS Ph.B. Vocational and Extension Education, Lewis A. WiILson FOREWORD This outline for the study of arithmetic in the elementary grades ei ; i bbs been prepared under the direction of the Commissioner of Edu- _ cation by the following committee: Dr H. DeW. DeGroat, principal, - State Normal School, Cortland; chairman; Frances Killen, super- visor of Elementary grades, Dunkirk; and Ina D. Porter, classroom ia Be Schenectady. This committee has been greatly assisted by the suggestions of successful teachers and supervisors throughout this and other states. At the request of many teachers some method work has been included. The suggestions for the work covering : _ the seventh and eighth years recognize that the point of view neces- sary for successful work in these higher grades is different from ae very helpful to he teachers of arithmetic throughout the State. GrorceE M. WILEY Assistant Commissioner for Elementary Education ‘al that of the lower grades. It is the hope that this outline will prove — TABLE OFF CONTENTS PAGE General introduction 233.505) in. Pelee oe bee oe ecaeel: ee 5 First grade: 30. 34 22205 sia ia teenie « a aol © aoe oe aie a aa ee 13 Second grade ss lic) 56 eis odes ates ore sln sO eaed ata ele Tin cee Och 24 Third grade oi. sess Fs fu sth ae ete See ee 42 Fourth: grade so. 3 5 os sic ee eine sce eels a Sas ciate ole lel eta ee ee cae 56 Putth ¢ Prades yes singh lek oleate cn Seo bse in veretor aie aheeta ie cakes Pat a tate en 72 Sixth gradep nis vases cern €0cs aie eee toe Mn BA clea © ne es eee ee ee 82 Introduction ‘to arithmetic in upper grades. 22. %o..s sie sca eee eee 91 Seventh: terddes.t. oyind Arai ee Sa eee Whe le hee le eel oe Ld 93 Eighth -grad@ cif sitio 2 < eiatae siete sce ates salen any te 104 University of the State of New York Bulletin Entered as second-class matter August 2, 1913, at the Post Office at Albany, N. Y., under the act of August 24, 1912. Acceptance for mailing at special rate of postage provided for in section 1103, act of October 3, 1917, authorized July 19, 1918 Published Fortnightly No. 815 ALBANY, N.. ¥: November 1, 1924 Syllabus for Elementary Schools ARITHMETIC GENERAL INTRODUCTION In the opening chapter to Thorndike’s The Psychology of Arith- metic! is the following statement regarding the functions of the elementary grades in the teaching of arithmetic: “What are the functions that the elementary school tries to improve in its teaching of arithmetic?” Other matters might well be considered in this connection, but the main outline of the work of the elementary school is now fairly clear. The arithmetical func- tions or abilities which it seeks to improve are, we may say: (1) Working knowledge of the meanings of numbers as names for certain sized collections, for certain relative magnitudes, the magnitude of unity being known, and for certain centers of nuclei of relations to other numbers. (2) Working knowledge of the system of decimal notation. (3) Working knowledge of the meanings of addition, subtraction, multiplication, and division. (4) Working knowledge of the nature and relations of certain common measures. (5) Working ability to add, subtract, multiply, and divide with integers, common and decimal fractions, and denominate numbers, all being real positive numbers. (6) Working knowledge of words, symbols, diagrams, and the like as required by life’s simpler arithmetical demands or by econom- ical preparation therefor. (7) The ability to apply all the above as required by life’s simpler arithmetical demands or by economical preparation therefor, includ- ing (7a) certain specific abilities to solve problems concerning areas 1 Thorndike. The Psychology of Arithmetic. Macmillan. p. 23. 6 THE UNIVERSITY OF THE STATE OF NEW YORK of rectangles, volumes of rectangular solids, per cents, interest, and certain other common occurrences in household, factory, and busi- ness life. It may be reasonable to assume that teaching of arithmetic in the elementary grades has at times failed of its purpose; first, because of the well-meaning but unfortunate attempt on the part of school authorities to include topics far beyond “ life’s simpler arithmetical demands,” second, through the failure on the part of the school program to give the necessary “attention to perfecting the more elementary abilities.” In the organization of the work presented herewith as an outline for the elementary grades the committee has been guided in large part by these principles as determining the general aims to be attained. Arithmetic that is taught purely as a mechanical procedure unrelated to social needs fails to function. This has been clearly demonstrated in much of our mathematics teaching. Arithmetic must be taught, not as an abstract subject independent of other interests, but as immediately and vitally related to all com- mon activities and interests of everyday life. This difference in emphasis between the work in the first 6 years of the elementary course and the work of the last 2 years is nowhere expressed better than by Klapper in his The Teaching of Arithmetic.* In the discussion of the aims of arithmetic for these two levels of the school course are found the following statements: Aim of the first six years: To develop skill in the fundamental operations as applied to whole numbers, common and decimal frac- tions, and to teach the necessary number facts. Aim of last two years: To develop power to understand arith- metical solutions and manipulate quantities in problems varying from the type. Not only are the aims of the work in these two periods clearly expressed but there is also much to be gained from the emphasis that is given to the importance of recognition of definite aims and objectives for the course. It is stated further in this connection regarding courses of study that “ There is little indication that those charged with the making of the course first formulate the aims of the course in arithmetic as an intelligent basis for selection of material.” The importance of such a step in the organization of any course of study can not be overemphasized. There has been no attempt to introduce the elements of algebra or geometry as such into the course in the higher grades. The algebraic elements that appear and the simple problems of mensura- 1 Klapper. The Teaching of Arithmetic. Appleton. p. 24. ELEMENTARY ARITHMETIC SYLLABUS fi tion involving geometric forms have been introduced as integral parts of a well-balanced course in elementary mathematics. The use of literal elements and the interpretation of simple formulas are used in the syllabus outline as‘a definite part of the suggested course of study in arithmetic. If there is any thought on the part of local school authorities in any community that they wish to introduce elementary algebra at the beginning of the second half of the eighth year, the work in arithmetic should be brought to a close at that point and the new subject of algebra introduced independently. At the very beginning of the course the teacher should make use of the experience of the child gained from his environment previous to his entrance to school. The school life itself and the pupil’s new experiences in his rapidly widening horizon will provide much interesting material for this purpose. With some modifications this ‘principle may well determine the procedure throughout the course in arithmetic. The teacher can and should so modify or add to the pupil’s experiences in number relationships and in the practical appli- cations of arithmetic to life as to secure a constant reaction of environment on the entire subject of arithmetic—the one thing in teaching tending most to vitalize the subject taught. It is not the thought that this syllabus should be used as an inflexible guide. It has been prepared rather as a suggestion in the hope that it will prove helpful and inspiring to teachers and will not be interpreted as a mere outline of topics. In some instances methods of presentation have been suggested. A variety of material has been presented that may be used or modified as may be determined by local school authorities. While some topics have been made entirely optional, others have been strongly recommended. There is therefore every opportunity for the use of the syllabus in the communities throughout the State with varying emphasis on different portions of the work. The committee has assumed that the primary purpose of arithmetic to be realized at the end of the sixth school year is to furnish the pupil with the tools with which to work, that is, a mastery of funda- mental processes in integers, fractions and decimals. In contrast with the work for the first 6 years, the emphasis in the plan for the seventh and eighth grades is on the application of these fundamental processes to the problems of home, business and community life. In the allotment of time to be given to the subject of arithmetic in each of the 8 years of the elementary course the practice will vary in different school systems. The committee has hesitated to make 8 THE UNIVERSITY OF THE STATE OF NEW YORK any recommendation on this point because of varying conditions in different localities. In the highly organized school system ample allotments can be given for this subject as well as for others through- out the grade. On the other hand in the small rural school where one teacher carries the entire responsibility for the elementary school program, grades must be handled in groups, and a reasonable time allotment is therefore much more difficult to meet. The following average number of minutes each week given to the subject of arithmetic in the elementary grades is taken from recently submitted programs in the cities of the State: Average number Average number of minutes of minutes each given weekly week from Doctor Grade toarithmetic! Holmes’ study? Le aaa etek Re TT SRO er ey eee 94 93 A PET On Ee ed PE MEY Pree LS /; 149 1 vad. Sopher Sah Pee Ss eee ee ee eee 195 203 ZS Te ORO DOE. od APN, TMNT A ATE ZS PS | CB Ss PAP eA etd Baile ine Pa es 088 216 Papa OU oeleane seek es aerae warte' st een ere 218 226 LSE ONES lee ae Cn ee 220 ZA Sia tSasitaial gin Meese erst ken aaterte Rae «lett es 220 Much of the work below the fourth grade will undoubtedly be oral. It is suggested that beginning with the fourth grade from 20 per cent to 25 per cent of the usual recitation period should be devoted to oral practice which might well include drill both on abstract work and on problem solution. The Solution of Problems In the lower grades all problems should deal with things with which the pupil is entirely familiar. Problems relating to things which the child possesses or which he may see every day should therefore be selected. A problem dealing with something familiar and near at hand is far easier than the same problem dealing with something that is strange or far away. The pupil should always understand clearly the language of the problem before he attempts to solve it. He should so thoroughly understand the situation that he can set it forth clearly in his own language. This he should frequently be required to do before attempting a solution. In the upper grades the data of the problem should be set forth by the pupil, and he should also be required to state what he is to 1 Twenty-three New York cities. ; 3 The Fourteenth Yearbook, National Society for the Study of Education, p. 21-27. ELEMENTARY ARITHMETIC SYLLABUS 9 find. He should also name the unit in which his answer is to be given. If the pupil can set forth the conditions of the problem up to this point he will be well prepared to undertake the solution. Frequently the teacher should require the pupil to state clearly in advance, step by step, how he proposes to solve the problem. It is also good practice for pupils to state how they would solve certain problems without having numerical data given. For instance, what would be the cost of a certain number of oranges at a certain price per dozen? The pupil would be expected to give the two processes involved in the solution. Pupils should be given much practice in making original problems both orally and in writing. The written problems will offer an excellent exercise in English. In order to make problems, pupils should have an intimate acquaintance with the vocabulary involved. This means not simply a knowledge of definitions, but facility to use terms with a full understanding of them. After an answer has been obtained, the pupil should frequently be required to determine for himself whether the answer is reason- able, first by checking up with his own knowledge of units of cost, or distance, or weights etc., second, by approximating the answer. The process of approximation is one in which the pupil should have much drill, and the checking of answers is a subject to which the teacher should give much attention. The problem should not be left until the pupil has made some effort by approximation, by proof, by a different method of solution, or by a repetition of the solution to satisfy himself that his answer is correct. The teacher should inspire his pupils with a zeal for correct answers, which will not leave them satisfied with anything less than correct and accurate work. This zeal may be fostered by giving deserved praise for originality shown in thinking and in working problems. The greater the variety in the solutions by members of the class, the greater satisfaction should the teacher feel and express. Too often both teachers and pupils approach a series of miscel- laneous problems with dread. Problems should be and can be made a pleasure if the teacher in the assignment, by a little wise explana- tion, anticipates the greatest difficulties, requires some systematic reasoning process, praises effort and encourages originality. Above all, teachers should remember that there can be no success in the teaching of problems, unless pupils are required to make conscientious preparation. 10 THE UNIVERSITY OF THE STATE OF NEW YORK The teacher should always feel free to omit from a list of prob- lems contained in a textbook any that are too difficult and any that may be poorly suited to the immediate purposes. Pupils often flounder in two-step problems in fractions simply because they are plunged into them with very little practice in the solution of two-step problems in integers. If a teacher finds that a class is weak in reasoning, he should make a close study to determine the reason. It may be because of a lack of practice in the solution of type problems. It may be because the pupil is so uncertain in his mechanical processes that he has little available mental power left to cope with the real difficulties of the situation. Often the explanation is the pupil’s inability to read. Whatever the cause of weakness in problem work, it is the teacher’s first responsibility to find it out. If the cause is lack of practice or mechanical inaccuracy, much easy practice will tend to correct the situation. Sometimes a set form of analysis may help. If the cause is the pupil’s inability to read, let the class in reading now and then use a textbook in arithmetic. Some skilful instruction in “silent reading ” ought to help the situation materially. Above all, the teacher should find poor work in the solution of problems a challenge to his best endeavor. A teacher enthusiastic about. his work in arithmetic is sure to find his enthusiasm met by both a like enthusiasm and a steady improvement on the part of his pupils. Tests It is recommended that to check up progress made and to deter- mine just what and how much instruction is needed in individual cases, standard tests be given from time to time. Printed instruc- tions for scoring and interpreting results should be ordered with the tests. Buckingham Scale for Problems in Arithmetic, Public School Publishing Co., Bloomington, II. The Courtis Research Tests, Series B (for tests covering the fundamental operations), Detroit, Mich., or similar tests Stone’s Standardized Reasoning Test in Arithmetic, Teachers College, Columbia University, New York City Woody-McCall’s Mixed Fundamentals, Teachers College, Colum- bia University, New York City Supplementary Material Flash cards or perception cards from any dealer in school supplies. More elaborate ones can be made. ELEMENTARY ARITHMETIC SYLLABUS 11 Number boxes Play money Kindergarten sticks Peg boards Weights and measures, weighing scales (from local dealers) Fassett Number Cards, Milton Bradley Studebaker Economy Practice Exercises, Scott Foresman Maxson Number Cards, J. L. Hammett & Co. Courtis Standard Practice Tests, World Book Company or any other standard tests The Rapid Calculator, Jones Compact Efficiency Drills, Iroquois Publishing Co. Sample packages of grocery store supplies Printed business forms Bibliography Brown and Coffman. How to Teach Arithmetic. Row Peterson Klapper. The Teaching of Arithmetic. Appleton Lennes. The Teaching of Arithmetic. Macmillan Lindquist. Modern Arithmetic Methods. Scott Foresman McLaughlin and Troxell. Number Projects for Beginners. Lippincott McNair. Methods of Teaching Modern Day Arithmetic. Badger Overman. Principles and Methods of Teaching Arithmetic. Lyons & Carnahan Smith, D. E. The Teaching of Arithmetic. Ginn Stone. How to Teach Primary Arithmetic. Sanborn The Teaching of Arithmetic. Sanborn Suzzallo. The Teaching of Primary Arithmetic. Houghton- Mifflin Thorndike. The New Methods in Arithmetic. Rand McNally Psychology of Arithmetic. Macmillan Wilson and Hoke. Howto Measure. Macmillan ' * . P .%) “4 ef vt. are Te Ode ane : ‘igs i. ve sors sat A ores F ? tea Leu Ye i i Wr, at bar WT oy, ; y ¢ F é ‘ Ps . a # ue ’ ¢ ‘ Fa Sa if r by Lis Th. vei J son ee Ry ef ry bs rl ‘ : ; At a * . shat ty it via ja pov a ie ’ Coney , a i 3 ' i } t) ahitt ywité ef Gite) - Vs qi reais : ] = : ‘ i A ret. 3, L oi | a4 a i” ine bits, on mie i, ; “ ee a? w t oeata ae ey a oe ELEMENTARY ARITHMETIC SYLLABUS 13 by LLABUSMINGARITEMETIC FIRST GRADE The teacher should read carefully the general introduction to the syllabus and should be familiar with the contents of the syllabus for the second year. Many children enter the first grade with some ability to count and make figures. There is need for some number work at this time in finding pages in the reader, in saving pennies, in playing games. It seems advisable, therefore, to do enough number work in the first grade at least to start the child correctly in counting, reading and writing numbers within a small compass, thus enabling him to use numbers with his games and directing his attention to value for his pennies. Wrong habits in counting and in making numbers are easily fixed at this time unless they are checked by teaching. After the first 10 weeks a definite time may be set aside for teaching number. This should mean serious work for the teacher but play and games for the child. Let the work grow out of the natural activities of the school room. Condensed Outline A Incidental number I Counting children, pages, papers etc. II Sense training II1Il Comparisons IV Concept training B Formal number I Counting — number scale 1 Counting 1 to 10 2 Recognizing and reading numbers 1 to 10 3 Writing numbers 1 to 10 4 Illustrating numbers 5 Counting to 100 6 Reading to 100 7 Writing to 100 II Combinations. (For combinations and suggestions as_ to teaching read carefully the expanded outline) III Measures 1 Cent, nickel, dime 2 Pint, quart 14 THE UNIVERSITY OF THE STATE OF NEW YORK Expanded Outline A Incidental number Incidental number should precede the formal work in number and should always be carried along with such work. I Use all number situations such as: 1 Counting children present and absent Numbering cloakroom hooks Keeping calendar record Counting out supplies used in class Having children number for playing games Reading pages of book or counting pages Paying for school lunches Applying number to handwork Counting numbers on clock 10 Keeping score in games LOM COS NISC eC tek CDS Il Sense training Use games for training the senses of hearing, sight and touch. IIf Comparisons — Steps preparatory to teaching number 1 Approximate comparisons Several lessons, the principal aim of which is to lead children to discern relations between quantities and to express them by indefinite terms of comparison, may pre- cede the counting lesson with profit. The following terms are suggested for comparison: larger — smaller longer — shorter — taller thicker — thinner higher — lower faster — slower wider — narrower farther-— nearer more — less — fewer This approximate comparison may be developed through the type lesson or by a game. The following game teaches more and less . From a pile of pegs the child who is “It” selects any number of pegs and the class guesses the number. If a child guesses 7 for instance, “It” answers: “I have ELEMENTARY ARITHMETIC SYLLABUS 15 more than 7,” or “less than 7,” as the case may be. As each guess is made the child responds with “more” or “less.” The one guessing correctly becomes “ It.” 2 Exact comparison For this step, lessons should be given to show that one number is more than another namely, 7 is 1 more than 6. For this the following game is good Bird Catcher Select two children as bird catchers, name each row after a bird. Mark off one section of the room for a cage, another for a bird’s nest. Two bird catchers stand in the front of the room. The teacher calls for one row of birds to fly to the nest. Bird catchers try to catch birds before they enter the nest. The birds who are caught must go in the cage. Teacher: “ Which has more birds, the cage or bird’s nest? Let us count them and see.” Bird catchers form a gate and as birds walk through, the children in seats count them. Teacher: “ There are 10 birds in the cage and 11 birds in the nest. How many more birds in the nest than in PMescage res Children: “ One more bird.” Teacher: “Eleven birds are how many more than 10 birds?” Children: “ Eleven birds are 1 more than 10 birds.” Application Teacher: “Eleven cents are how many more than 10 Gents tieuctG. IV Concept training The full meaning of number can not be acquired at this time. The following, from Doctor Thorndike, ex- presses the situation at this time: A number has not one meaning but several. Thus 8 means a certain point or place in the number series 1, 2, 3, 4,5, 6, 7, 8, 9, etc., which is 1 beyond 7 and 1 before 9. This we call the series meaning. Eight also means the number of single size meaning. Eight also means 8 times a certain unit, say a pint, whether isolated as 8 separate pints or combined together in a gallon. This we may call 16 THE UNIVERSITY OF THE STATE OF NEW YORK the quantity-size or ratio meaning. This ratio pce should not be neglected. It is not necessary or desirable to teach the full and exact meaning of a number all at once, for pupils learn more and more about the meaning of numbers by using them. The good teacher will make sure that, at any stage, the pupil knows the meaning well enough to use the numbers intelligently in those ways which are necessary but will be cautious about teaching any more elaborate meaning than that. In making a number objective, we should consider not only formal systematic presentation but such informal and incidental connections as can be made with objects and acts of daily life. The latter are more interesting and more surely understood. Nore. Training in number concept should precede formal num- ber and should be carried along for a time with formal number. Plenty of material should be available — kindergarten sticks, pegs, circles, toothpicks. 1 Count objects from 1 to 10 or higher 2 Play and illustrate number by use of pictures and objects 3 Drill on the numbers as follows: Example: To familiarize pupils with number “2” Place 2 blue pegs, 2 red pegs, 2 white pegs, 2 green pegs, on desk. Drill for other numbers in the same way. They may be arranged in geometrical forms as squares, triangles etc. or may be arranged to represent houses, barns etc. The following constructions have been used by the Rochester schools for objective representation of number by means of tooth- picks, kindergarten sticks of similar materials. Jax aX One stick boy, His tent, His flower bed, ye ti a ie hes His back yard, | His dog house, His dog, ELEMENTARY ARITHMETIC SYLLABUS 17 Be atc fe ae A, es coer ha His father’s barn, His kite, His back fence, His house. A good summary at the close of a concept lesson is to have the class give all the constructions that they can recall for a certain number, the teacher drawing them on the board as each pupil con- tributes. Later the figures should be substituted for the construc- tions and the work made abstract. In this way a rich content may be given to number symbols; eight should be more than seven and one; and the child may thus be introduced through his own activity to all the fundamental opera- tions. It is hoped that the germ thus transplanted may be nourished from time to time as occasion makes possible for further objective work and rationalization, in order that the child may be gradually led to an understanding of all the fundamental concepts and relations of number, and become a more independent worker in the upper grades. B Formal number outlined I Counting from 1 to 10 1 Counting objects 2 Counting without objects 3 Recognizing figures to 10 4 Making figures from 1 to 10 II Counting to 100 1 Counting objects 2 Counting from chart for recognition of symbols 3 Counting by 10’s using sticks tied in bundles of 10 4 Counting by 10’s from the chart 5 Counting from memory 18 THE UNIVERSITY OF THE STATE OF NEW YORK III Written number in connection with the counting C Formal number developed I To assist the counting have the following chart displayed in the room and a small one in the hands of the pupils 10. 20> 30: 405950" 60% 7/0 Seger 00 ay REE PASTRIES seek Rolie copy 22h 12°22" 32° Ae ees02" fe mee ee oe Lo 2333 AB ore OS moon 14 24 34 44 54 64 74 84 94 15. 25 435.45 oe 00 3 / oe aeons 16926 (36:46. 560.06 70° #365896 17 ©2737 4 RE OLAS, Sar IS 2838 °4 Skee OSt 7S ose es 19°) 29) 39.049 259. 0087S Soleo LG) OOS NIC ON Oia Co) Dee aS eo: Ga eo Gr _—" Value of chart a Count from chart for recognition of number symbols b Count from chart for organization of number in the serial relationship so child will see 5 after 4 and before 6 c Chart work makes recall easier 2 Work to be done from chart and from objects Note. It is suggested that kindergarten sticks be used singly and in bundles of ten. a Count by 1’s to 100 b Count by 10’s to 100 (row A) c Begin at 1 and count by 10’s to 91 (row B) d In the same way count each column by 10’s e Count by 2’s, emphasizing every second number, as one, two, three, four etc. f Count backwards g Seat work done with chart | (1) Use small cards from number boxes (2) Have children build the chart with these number cards and do various exercises from the chart (3) Copying numbers is good seat work at this time but should be limited in the amount required and should be supervised ELEMENTARY ARITHMETIC SYLLABUS 19 D Formal teaching of combinations I ‘Table of combinations 1 Be ee os aR 7 ed Lee eee wie esis le al wl 2 Violeta thaw 4 im Ldiogl CaS Pet a Oe fom OY (reversed) 5) al thd ni a kl pe 2 rel ANS Wl Se GSE bes Wile (mixed: Abe 3h ald das Sa oe ee. (oul eS } Optional 5 De Ame MG S/s5 co en oon 2. AOA e in CoMmbinattol) Fer MNS. EB MSitiae. Ssccieds 4 5 6 /7 (3 in combination) 4s 4 4 5 6 (4 in combination) Rows 5, 6 and 7 may be deferred until the second gerade at the option of the teacher. But if the child has use for these number facts, it is recommended that he be given the opportunity to acquire them. II Teaching 1 Develop combinations through pictures, objects, domino cards etc. 2 Give number symbols 3 Make the combination a matter of memory. Transfer combination to flash cards and drill. Do not allow child to keep resorting to objects until he depends upon them for counting. See that he depends upon his memory. 4 Problems for application as: Mary had 5 pennies and her father gave her 3 more, how many pennies had she? 5 Standards of speed and accuracy: Ten known combina- tions in about 15 seconds given orally from charts, lists on board etc. is a good standard. Optional 1 The more difficult combinations as given in the table. 20 THE UNIVERSITY OF THE STATE OF NEW YORK 2 The concept of subtraction 8 less 4, or of multiplication 2 times 4, or of division 2 fours in 8, or of % of 8 is 4 may be developed in the constructive work and expressed orally. It is recommended, however, that symbols expressing such ideas be deferred until second and third grades. First Grade Equipment Kindergarten pegs and sticks Rulers with inch divisions Yard stick Addition cards with combinations and answers on the back Domino cards with combination on one side Toy money Abacus Ringtoss Ring a peg Ten pins Bean bag For illustrating the number stories the domino cards are good. Have the domino illustration on one side, the number symbols on the other. been 0 | @) 5 O ) O O 3 O These can be used for giving the number idea, the symbol and games. For drill use the figures only. Devices and Games for Drills 1 Place flash cards along blackboard. Children run, take card and give combination. When all cards are chosen, children hold up cards at seats. One child takes place in front of room and says: “Bring me 2 and 3 are 5.” Then child hands the card to the leader saying: “I bring you 2 and 3 are 5.” Continue until all cards are gone. ELEMENTARY ARITHMETIC SYLLABUS Za) 2 Divide class into boys and girls. Leader calls for 2+ 3 are 5. A boy and a girl run up and hunt for 2-+3=5. First one may have the card. Keep score for boys and girls. If each score gives 2, this will give a good application for counting by 2’s, or counting by 10’s or 5’s may be used in same way. 3 Teacher: “I am thinking of a combination on the board” (or cards). Children: “ Are you thinking of 3 and 2 are 5?” Teacher: “ No, I am not thinking of 3 and 2 are 5.” Continue until a child gives the right combination. Teacher: “ Yes, I am thinking of 3 and 2 are 5.” 4 Draw lines on board to represent stalls. In each stall place a figure as 8, 5, 6, 4 etc. | Teacher passes out cards. Children run up and place cards on the chalk tray below the stall whose number equals sum of com- bination on card. 5 Ladder drill. Combinations written on a ladder, children climb ladder by giving combinations. In a similar way use fish in a pond and allow children to go fishing. Use fruit trees and allow child to pick fruit. Use brick wall with combinations on bricks. Children tear down wall. 6 Scoring games: (a) bean bag, (b) ringtoss, (c) nine pins. 7 Racing game. Place ten or more combinations low down on different parts of the blackboard. Let a boy and a girl be chosen to write the answers, each beginning at the word “go” to find which can write the answers the more quickly. If there is sufficient blackboard more than two may race. The winner chooses some other pupil to race against the loser unless the teacher desires to substitute another child. Store Projects 1 Material: Kindergarten table for counter, objects to sell made by children, money 1, 5, 10-cent pieces made by children, change drawer. The articles may include apples, pears, dolls, furniture, plasticene dishes, etc. 2 Grading difficulties as follows: a Counting to 10 cents, 1-5—10 (1) Purchasing single article, exact amount of money, but less than 10 pennies (2) Purchasing two articles using count — 10 pennies (3) Purchasing one article with change, using 5-cent piece 22 THE UNIVERSITY OF THE STATE OF NEW YORK (4) Purchasing one article with change, using 10-cent piece (5) Purchasing two articles with change, using 10-cent piece 3 Procedure The above steps are taken up in the following manner. This may be carried on any time in the year adapting the difficulties to the number knowledge which the children have before progressing to the next. a Arrange the articles for sale on a kindergarten table, each bearing the cost mark from 1 to 10 cents. Each child is given 10 pennies with the privilege of buying one article. Teacher selects a child to go to the store. Customer: “ Good morning?” Storekeeper: ‘Good morning.” Customer: “I will take a pear, 2 cents.” The customer counts out 2 pennies, hands them to the store- keeper who deposits them in the penny compartment of the money drawer. Continue as above. b For preparation, review combinations. First using the different serles as: D7 PG2T P2EIWS Su 0 lieei2 (d3° well eee SS rete When children grasp this step, skip around using combination in different series. Use the same arrangement of the store as given under a. Each child is given 10 pennies with the privilege of buying two articles. Cost of articles 1 to 9 cents. Teacher selects child to go to the store. Customer: “ Good morning.” Storekeeper: “Good morning.” Customer: “Iwill take a peat,@2 .cents, andsa doll Socents, 2 andsnpares/. 5 Customer gives the storekeeper 7 pennies. Storekeeper: “ The pear: costs 2 cents and the doll 5 cents, 2 andsocarer7. No muauces: c Arrange store as suggested under a. Each child is given a 5-cent piece with the privilege of buying one article. Articles are marked 1 to 4 cents. ELEMENTARY ARITHMETIC SYLLABUS we) Preliminary step: Review combinations at board as: (1) beeZ 4 osy3 es (2) ee Children run up and fill in the ee missing number. Saison (3) Lying fers aes 9 Customer: “ Good morning. Storekeeper: “ Good morning.” Customer: “I will take this chair, 3 cents.” storekeeper :. > Lhe chair costs:3 cents,..3.and.2.are.5..+2 tents change.” d Proceed with this step as under c. A greater range of com- binations will come in. e Arrange store as suggested under a, each article marked from 1 to 9 cents. Each child is given a 10-cent piece with the privilege of buying two articles with change. Customer: “Good morning.” Storekeeper: “ Good morning.” Customer: “I will take this fish, 3 cents, and this cup, 6 cents. 3and6are9. Qand1lare 10. 1 cent change.” 4 Other applications 7 a Suggestions: Christmas store, Christmas tree, party, picnic food, children’s lunches when served in school b Type problems of maximum degree of difficulty (1) Robert had 10 cents and bought at a toy store a top for 3 cents and a boat for 2 cents. How much change did he receive? (2) Mary’s mother gave her 5 cents. She bought some candy for 3 cents. How many pennies did she have left? (3) Arthur spent 5 cents for a kite and 2 cents for a ball. He gave the storekeeper 10 cents. How much change did he receive? | (4) Mary picked 4 apples and Frank picked 3 apples. How many more apples did Mary pick than Frank? Encourage the children to suggest original problems as far as possible. 24 THE UNIVERSITY OF THE STATE OF NEW YORK SECOND GRADE The teacher should read carefully the general introduction to the syllabus and should be familiar with the contents of We syllabus for the first and third years. Condensed Outline Note. The following outline and treatment are not to be interpreted by the teacher as a requirement. She should feel free to use other arrangement or treatment of topics if she believes it advisable. This plan of work has proved successful. 1 Reteach first grade number work a Counting, reading, writing numbers to 100 Use number chart to establish a good understanding of the number scale. Watch for the children who have immature number ideas. There is often such a group in the second grade. Use objects, pictures, domino cards for these children to illus- trate number but drop the objective work as soon as possible. b First twenty-five combinations taken up in first grade Here again use object illustrations, until the number idea is clear and then omit such work. See first grade outline for development lesson on 8. c Money taught in first grade, 1¢, 5¢, 10¢ d Problems by teacher on the above Original problems by children 2 Addition without carrying For teachers’ reference the combinations are classified in the following groups: a Combinations whose sum is 10 (group A, see page 27) _ 3 Addition carrying 4 Subtraction Use the ten combinations for drill on the addition form of subtraction thus 10, saying 5 and 5 are 10 (see page 30) —5 5 Addition continued Take up all the doubles (group B) one set at a time and give all drills as outlined (see page 32) 6 Multiplication j Table of 5’s (see page 28). Use in examples and applied problems 7 Addition (group C) using 9 endings with series and column drills (see page 34) ELEMENTARY ARITHMETIC SYLLABUS 25 8 Subtraction using 9 series with examples for continued drill and addition span (see page 36) 9 Multiplication Table of 2’s with examples and problems. Use same method as in table of 5’s It is suggested that the 2B work may be ended at this point. 10 Reading and writing to 1000 For groups D to K see pages 34-35 | 11 Addition (group D) using 8 endings with series drills, column addition and problems 12 Subtraction with changing minuend and subtrahend 13 Table of 10’s 14 Addition (group £) using 7 endings, with series, column addi- tion and problems This completes the groups of combinations used in series and column work. The remaining combinations are learned but not used in column addition until the 3d grade. 15 Subtraction completed — all groups 16 Addition (group F) using 6 endings without series drill or column addition Group G 5 endings, group H 4 endings, group L 3 endings, group J 2 endings, group K 1 ending, thus completing all com- binations 17 Subtraction continued involving all combinations 18 Multiplication tables of 2’s, 3’s, 4’s, 10’s 19 Fractions %, 1%, % of products learned in multiplication tables are optional Hints to the Teacher on Addition Addition can and should be acquired early. See that the child _ does not go through school poorly prepared, counting on his fingers, a slow process. Help him to be quick and accurate. He will then have a foundation for later work. Essentials Have a well-graded system of presenting the difficulties in addi- tion, and present a few at a time. Difficulties in Addition 1 Gaining a knowledge of the combinations 26 THE UNIVERSITY OF THE STATE OF NEW YORK bo SOR COO’ wm NO Applying the combinations to the higher decades or the series WOTk,.aS 2/1 4/ wos, Os Le BO Note. For the most gifted child the knowledge of 8+7 <= 15 prob- ably never insures the application to 38 + 7 = 45. For the less gifted child gaining facility in the decades involves much time and labor. : Learning to keep in mind the unseen result of each addition as involved, namely, column addition Adding this unseen result to the seen number Acquiring power to hold attention when any considerable number of digits in a column is involved Learning to skip 0’s Learning to skip empty spaces Keeping one’s place in the column Carrying Methods of Meeting the Difficulties Have system well graded as to difficulties Master a few combinations at a time a See that children early understand the combinations by use of objects b See that the use of objects is later discontinued and that children memorize the combinations c Look out for counting. Do not allow a child to stop for counting. If he hesitates, he does not know his combination result. Better give him the answer and drill upon it. As soon as a few combinations are known, introduce the work on the decades a After the first twenty-five combinations (see Ist grade) are known the work on the decades should begin b Make each series a matter of memory As soon as the combinations and the series are well started give plenty of practice on column addition. Column should con- tain only a small group of the combinations. a Have plenty of practice on the column. Let the child add aloud some column over and over if need be until he knows what is expected of him in smoothness and speed. b Addition is memory work. By the time he has memorized many addition lessons on columns he will have covered many dif- ferent applications of the same combinations. ELEMENTARY ARITHMETIC SYLLABUS 27 5 Advance pupils into new work very slowly so that each step may be done thoroughly 6 Teachers should keep a record book of all the combinations taught and the problems and examples containing these combinations. This will mean a saving of time as plenty of material will be available for drill. Courtis says: “ By practice on one example until it is learned by heart convince the child that it is quicker and easier to remember than to count.” 7 Children may know combinations and series and still fail on column addition. The difficulty is in the child’s power to hold the unseen number in mind and unite it with the next figure in the column. . Courtis says: “ Most children add steadily for 6 or 8 addi- tions. Some have an attention span of only 3 or 4 additions.” The remedy is practice. Practice only will train the attention and make column addition possible. 8 Carrying Teach the child to carry the figure immediately and not hold it in suspension in the mind until the end of the column. Detailed Procedure for the Second Year For each fundamental operation have some plan well-graded as to difficulties. Several plans have recently been made available in various courses of study. The following plan arranged after the plan in the city of Toronto Exercise Book is given here because it has proven very successful. The advantages are: 1 More easily memorized combinations are mastered first 2 A power to hold number in mind is given through column addition while the child is still working with the easy combinations 3 It grades the difficulties for subtraction Group A Deer Oe poe) hie Pmt ao a eo. eA 5 Slog he aie ae Rees 288 Eu Se el Area ot Ces Step 1 Review the above group. Arrange class into groups for drill. For those pupils that need objects resort may be made to some such work as is given for the first grade on teaching number concept. Follow with plenty of abstract drill. 28 THE UNIVERSITY OF THE STATE OF NEW YORK Step 2 Series work with each pair of combinations taking only one set at a time thus: a 55925 BS 5.4)d0: Saya 10 20 30 40 etc. (with answers) b 54 fly doo Dubbed {yi eed uO C cw ore etc. (without answers) 10 20 30 40 etc. (without second addend) Rapid drill on the series may be carried to 100. Recite as a table. This is also good drill. Teach at this time also the tens series with any digit thus: Des tele OeiiaO itt commie TO (20%, 30 LOL Ome ZO sera If a teacher desires, this lesson may be given with bundles of 10 sticks and single sticks thus: Take a 10 bundle and 4 pegs. Child says 10+ 4= 14. Take a 10 bundle and 6 pegs. Child says 10 + 6= 16. Take two 10 bundles and 7 pegs. Child says 20 + 7 = 27. Step 3 Column work Column of figures should be planned so that each column contains only known combinations and known series. Many columns should be given so that each step is completely covered by column work. Do much oral work. Require smoothness and accuracy in recitation. If a child has difficulty, he may be encouraged to practise on his column. Add columns from bottom to top. Do not allow grouping at this time. Purpose here is to give drill on these separate com- binations. owmunuiNy Y SEs) ee a | cnt U1 U1 U1 OD | orn U1 1 U1 Ut | WUT ON 1 Written work may be given. ELEMENTARY ARITHMETIC SYLLABUS 29 Step 4 Problems apply to above as counting nickels. Original prob- lems by children. Examples of original problems given by children. If I had 10 cents and spent 5 cents for a bag of peanuts and 2 cents for an eraser, how much change do I get back? If I buy a 5-cent tablet and three sticks of gum at a penny a piece and a 2-cent sucker, how much do I spend? Series Drills Take up each combination in group A and work through all the above steps. The following is work for 1+ 9 of Group A: Step 1 1+ 9,9-+-1; call attention to 0 ending. Step 2 Series l 9 11 2 20 30 50 40 (with answers) 11 0 | Hos (without answers ) Deb ah ae ee 10 20 50 30 (without second addend) Step 3 Columns involving 9+ 1,1+9,5-+5 mV moO em OS = Ore OO un mine Ore OO UwM\o mmr OWN wm Ute OO OV Ore uu OO W mmnNorreuut Make up many columns similar to above. Step 4 Problems, games for scoring, races. 2+ 8 Series Step 1 2+ 8,8+ 2; call attention to 0 ending. 30 THE UNIVERSITY OF THE STATE OF NEW YORK L222 SAP ase Re] S| ie Ye seas shales hal ate: (pide Sein Neh dem 3): abaiel sie he 2 8 10 20 30 50 40 (with answers) 8 2 (without answers) Bre aes 10 20 30 40 (without second addend) Subtraction 10 10 10 10 10 Se ee ee eS Step 3 Column work for 8+ 2,2+8,9+1,1+9,5-+5 6 7 Deel Z 5 A Sin eos) SHOE, DIE 25 Dit Duar eae ROBE eres Si dep ream nek eee ke Be we) Pee eee. ae Dee oo le Poems Beye SEL EES 82 oe De Wah Di Rie ie a et oes Step 4 Problems, games etc. 3+7 Series Step 1 7+ 3, 3-+7; teach O ending. Steps Fav 2 / TU os: ON 3 oa ees 10 30 20 40 (with answers) Stepao 3 gcd ees {32/6 33/4 (without answers) Tee /olaed, 10 20 30 (without second addend) Subtraction 10 abel Oe LO amen —7 —8 —9 —1 —3 —2 ae ELEMENTARY ARITHMETIC SYLLABUS 31 Column work for 7+ 3,3+7,8+2,9+1,5-+5 7 Gm 638 406 LP io raat iin $3 habit obec cy Wig Ad LE ZEN GN 2 Joe Soke oe Pk SAT 7ey ae oe Gee s5 Vee pas) \ td's of Bas) to kelp t Va, 3. fee 0 ot /e ee Le atc! Step 4 Application 4+ 6 Series Step 1 6+ 4, 4-6; teach 0 ending. GO lhe rea0 4 4 4 4 10 20 30 40 (with answers) Step 2 Ge lUSs 207/30 4 4 4 4 (without answers ) Oe LOmeZO 46 10 20 30 50 (without second addend) Column work for 6+ 4,4+-6,7+3,8+2,9+1,5-+5 Se Sea 3 PAG 5 Se. eed re Gee Cs forest Ae Oe ree, 80) AS ee tS ie Peete Gy ean ee Ge 4 on) Ce ere tL 47, (GMO bre 4D 2" 9 Ger Age. 9 eee: OR2 ESS HP Cy Loon) ae Fake fete Hints 1 Give column cf 4, 6, 8, 10 addends. 2 Call lower figure 24 or 34 or 14 to throw the drill into the upper decades. Step 4 Problems etc. This completes work on group A. Take up each group in the same way. a4 THE UNIVERSITY OF THE STATE OF NEW YORK Carrying At this point begin double columns and teach carrying. After carrying is established, written work may be given. DO Joe 7 PiOAn 7 Shh RDN ESS JO Le OomeLo MLS AD i Ome Be tO eee In making up double or triple columns take care to carry a figure to the next column which makes some known combination. These examples have been arranged for addition to begin at the bottom. Zh eB /Spig OS hie 37: A/a 136 43 53 64 Group B Take up each group in the following way. See Toronto Exercise Book. Ui a Ean) Ud Sra et il ane ONO 2/2 0 AG ee ea Step 1 9+9 Step. 2 Series iy 7 aes Olga sD DG BAO TESG 19 2 oa OM eA mee i 2 Of aD Oe ewe tO 2 tale oe eee Pal Cote eG) SPREE oh ERD OE ec at DELS OR BOO fas aia PAE AU GS Tid Comme nS 88 Dad sa CO 9 es eran Oe Os ek eee ae ELEMENTARY ARITHMETIC SYLLABUS Children will feel this new difficulty for a short time. 53 If they have much trouble on the 10’s, it will help very much to write the sums out at the side and allow a little time to child for study, thus: 9 2 40 ac eso Go 9 29 29 459 2 92 929 9 GOF US2 9 79 778 8 + 8 Series Step 1 8+8 mBtepe2 ~eries 8 18 28 48 38 cy Roy c kee eases Step 3 Columns | 5 Oe Pat cee hg Ae eases anne? ed’ DRAG OG 94s). 88 1O0 peo. 9 BO Rae As eer aR Oa Se Seo? LOmAIG ONT teRED A So Bie) Cua. aa 0 tk Oo Step 4 Problems 7+7 Series Step 1 7+7 Step 2 Ti OF PSS fey DAV ERP TA EENY, Step 3 Columns nO), oe 0 Deere AZ6 OT IN ae eee 9G Sees ee eg moro, Oh” Ofte tare ee et O0G/ Gye Ge MA GPAs 77/6 Atel See pee if 7 (hal Aatoe ESt Df, 546 . 34 THE UNIVERSITY OF THE STATE OF NEW YORK Step 4 Problems. Complete 6 + 6 (1) Through the series (2) Through single columns (3) Through double and triple columns Complete 4 + 4 (1) Through the series (2) Through single columns (3) Through double and triple columns Complete 3 -+ 3 (1) Through the series (2) Through single columns (3) Through double and triple columns Complete 2 + 2 (1) Through the series (2) Through single, double and triple columns Complete 1 + 1 (1) Through the series (2) Through single, double and triple columns Review of above 8 (0) 20.) 07 la See eendinewisee Review all other endings. Take up group C with each of the steps covered in Group A. Group D SHG Hil, Bom BE Aya Te Tate May isictabhateg shore’ Cee ELEMENTARY ARITHMETIC SYLLABUS 35 Group E Gimbare (hilt OadViapsine 2 hn le Boils Ze DSi ltOan Al vidn aOvendingist/ It is recommended that the second grade does not carry the series and column addition beyond this group. The following combinations should be taught thoroughly but it is recommended that they be not used in series and column addition: Group F Peek? & ©. ORR SO ee So 7 lS 2. Sgennineiss Group G Ie San Zvi 4 AS nye OD Ape cee So hOD le Zemie -Ovendifietis © Uy ae 04 ah aus 4 apa Pt A ibe Group H |RSS eet ss kee ee = Seno nah eos O< 41/4’ == 3/4" 3)>o) fourth 3 fourths. F Give many similar problems under each group. (Optional in fourth grade) Division, using only like fractions a How many times can you find D on B? How many times can you find D on A? How many times can you find C on A? b How many times can you find 1/4 in 2/4? How many times can you find 1/4 in 3/4? How many times can you find 1/4 in 4/4? c Repeat group 0 writing the expression. How many times can you find 1/4 in 2/4? 2/4—1/4=2 How many times is 1/4 contained in 3/4? 3/4~+1/4=3 How many times is 2/4 contained in 4/4? Write expression 4/4—-2/4=2 Fractional equivalents ; a Take up the 1/2 series with illustrations. Take up the 1/3 series with illustrations. ELEMENTARY ARITHMETIC SYLLABUS 67 Take up the 1/5 series with illustrations. Take up the 1/6 series with illustrations. Take up the 1/7 series with illustrations. Carry this as far as necessary to make clear the underlying principle. b Establish by illustration the series of names for 1/2 Dee? [4 =a /S = Of 1616432. c Establish the same way the series Pa 2/03 3/9 ele 1S. d Take the fraction and multiply both numerator and denom- inator by 2, by 3, by 4. What is the answer in each case? How do these answers compare in value? What is the effect on the value of the fraction of multiplying both terms by the same number? Establish that multiplying both terms of the fraction by the same number does not change the value of the fraction. e In the same manner, taking the fractions expressed in higher terms show that dividing both terms by the same number does not change the value of the fraction. f After much drill from illustrations and without illustrations in deduction of fractions, teach 9 Addition of fractions having unlike denominators 10 Cancellation with application of the principle that dividing both terms by same factor does not change value of fraction. 11 Fraction parts a Part taking 1/4 of 2/4 of etc. (1) Part taking This is application of table already taught. Let A=8 then find 1/4 of A, 2/4 of A, 3/4 of A is 6. b (Optional in fourth grade) Relation of part to part; 2/4 is what part of 3/4? Relation of part to whole; 2 is what part of 4? (1) Relation of part to part (a) Write letter names, point out D is what part of C? Answer: D is 1/2 of C. D is what part of BP? Answer: D is 1/3 of B. C is what part of B? D is what part of A? C is what part of A? 68 THE UNIVERSITY OF THE STATE OF NEW YORK B is what part of A? See page 64. (b) Find and point out (1) 1/4 of 2/4 one fourth equals what part of two fourths? (2) 1/4 of 3/4 one fourth equals what part of three fourths? (3) 2/4 of 4/4 two fourths equals what part of four fourths? (4) 2/4 of 3/4 two fourths equals what part of three fourths? (2) Relation of part to the whole using integers, as, 3 is what part of 4° etc. 12 Problems a Use social and economic environment of child for problems. Use problems relating to meat market, grocery store, depart- ment store, making Christmas gifts, scores etc. b Original problems by class c ‘Textbook problems d Projects worked out by class e Athletic records 13 Time allotment 1/3 oral work (drill work) including fundamentals 1/3 discussion of problems 1/3 written Teach children to check up their work for errors, by proofs; by looking over work the second time; by testing approxi- mate answers to see if results are reasonable. 14 Tests a To standardize the class work Courtis’ fraction tests may be used, in case no standards are obtainable. b In all drill in the various processes set up standards of speed and accuracy. | c Children should know what these standards are and teacher should endeavor to arouse enthusiasm of children to attain or surpass the standards. A Grocery Store Project (Arranged by Miss E. Scheyer, Dunkirk, N. Y.) Work up enthusiasm over a grocery store. Make children realize that it is necessary for someone to go to the store daily. Talk about ELEMENTARY ARITHMETIC SYLLABUS 69 what they give in exchange for groceries. 1 Get class to suggest project. 2 Motive in general: To teach children the industrial life by which they are surrounded and of which they are an active part. 3 Teacher’s aims a To help the children find vital problems b To teach them an appreciation of industry c To develop social instinct d To develop leadership e To develop cooperation f To give information g Yo develop thought and reasoning h ‘To increase the power of concentration 4 Children’s aims a To learn to make purchases b To learn United States money c To add, subtract, multiply and divide quickly and accurately d To learn fractional parts e To learn measures 5 Subjects a Language — oral and written b Arithmetic c Geography d Spelling e Penmanship f Drawing g Why can grocers accept a smaller margin of profit on a pound of sugar than on a pound of tea? Elementary meaning of overhead costs. 6 Outline by subjects a Language — oral (1) Class discussion of plans for store (2) Experiences in buying | (3) Value in paying cash as opposed to charging (4) Value in paying cash and carrying (5) Discussion of personal visit to store (6) Increase vocabulary b Language — written (1) Written record of purchases 70 THE UNIVERSITY OF THE STATE OF NEW YORK (2) Written advertisements for daily paper and windows. Secure names of groceries nationally and locally known. (3) Employment of boy in grocery — hour’s pay, what to know. (4) Food rationing and substitutes c Arithmetic (1) Study and use of United States money (2) Addition, subtraction, multiplication, division, and fractions (3) Sales slips, bills (4) Problems (5) Measures (6) Losses by poor accounts (7) Losses by perishable products Type examples 1 Ruth paid 58 cents for butter and 50 cents for tea. She gave the clerk $1.25. How much change did she receive ? 2 Mary has 25 cents. Eggs are 32 cents. How much more money does Mary need? 3 There were 100 bananas in a bunch. After 48 of them had been sold, how many bananas were left? 4 Find the cost of 2 dozen eggs at 32 cents a dozen. 3 Find the cost of 2%4 dozen eggs at 32 cents a dozen. 6 John has 40 cents. How many 5-cent measures of chestnuts can he buy? How many &-cent measures? 7 There are 4 quarts in a gallon. How many gallons in 80 quarts? 8 Mrs Smith bought 144 clothes pins, how many dozens did she buy? 9 If a gallon of maple sirup costs $1.12, how much should a quart cost? 10 A grocery sold 25 barrels of flour for $200. Find the price per barrel. 11 At 32 cents a dozen, what does the grocer receive for 3 eggs, 9 eggs? 12 When you buy 3 lemons for 10 cents what is the cost for one? 13 At 45 cents a pound what will 9 ounces of cheese cost? d Geography (1) Talk about oranges, spices, cocoa, cocoanut, and other things sold. (2) Tell where they come from. ELEMENTARY ARITHMETIC SYLLABUS 71 (3) Tell how they are obtained, bringing in a study of transpor- tation. (4) Tell what we give in return bringing in a study of exports, imports. (5) Talk about countries from which coffee, tea, spices, cocoa come. Study people in these countries and how they differ from our own people. e Spelling Difficult words used in subjects such as transportation, coffee, cocoa, oranges, export, import. f In connection with language Writing of sales slips g Health and hygiene (1) Clean grocery store (2) Supervision by health authorities (3) Supervision of weights and measures h Drawing (1) Cut pictures of groceries from magazines and mount on oak tag. (2) Make toy money from oak tag. (3) Make designs for advertisements. 1 Table of measurement (1) United States money (2) Liquid measure (3) Dry measure (4) Avoirdupois weight — ounce, pound Milk Situation Milk served in the school and sold to the children at 3 cents a half pint was made an interesting source of school work for a fourth grade by Miss B. Thompson, Dunkirk. A few of the sugges- tions are given here. 1 Geography a Source of milk, the dairy, local farms with cows, buildings, feed, shelter. New York State regions noted for dairying. United States regions noted for dairying. 2 Arithmetic — for discussion only a Some idea of capital invested, value of farm, stock, buildings, and implements 72 THE UNIVERSITY OF THE STATE OF NEW YORK b Overhead expense Farm help, bottles, delivery auto, etc. 3 For study a Pints, quarts, gallons b Fractions 1/2, 1/4, 1/8, 1/16 c United States money d Days, months, weeks e Addition, subtraction, multiplication, division of (1) United States money (2) Fractions f Keeping simple accounts Examples of Problems Involved 1 At 3 cents a half pint, find cost of 1 pint of milk, of 1 quart. 2 Find cost of milk for one school week for one child, for entire grade. 3 Find cost of milk for 1 month of 20 days. 4 At 3 cents a half pint, how many half pints can be bought for 15 cents, 30 cents? 5 A milkman pays 32 cents a gallon for milk and sells it at 12 cents a quart, how much does he make on one gallon? FIFTH GRADE The teacher should read carefully the general introduction to the syllabus and should be familiar with the contents of the sya bus for the fourth and sixth years. Condensed Outline First Half A General review of integers B Common fractions I Reduction II Addition III Subtraction TV Multiplication V Cancellation Second Half A Common fractions continued I Division of fractions II Fractional relationships III Decimals ELEMENTARY ARITHMETIC SYLLABUS 73 B Decimals I Numeration and notation II Reduction III Operations IV Aliquot parts C Denominate numbers (review of tables learned) D Business forms E Problems Expanded Outline Fifth Grade (First Half) A Integers A marked increase in speed and accuracy should charac- terize the fifth grade work in fundamentals. Use some standard tests and compare class with these standards. Follow up with much drill on whatever is needed. Some systematic drill such as is outlined in the lower grades is much better than scattered drill but such systems of drill as the Courtis Practice Tests published by the World Book Company; the Studebaker Drills published by Scott Fores- man and Company; Compact Efficiency Drills by Iroquois Publishing Company are time savers, and should be used if available. | Notation and numeration II Addition III Subtraction IV Multiplication V Short division VI Long division with 2 and 3 digits in divisor VII Review multiplication tables, factors, and multiples VIII Review division tables with remainders IX Review part taking tables X Problems. Two-step problems applying to the above (see Problem studies, pages 8-10) B Common fractions See fraction work of fourth grade. It is suggested that much objective work be given followed by abstract drill using small denominators. It is also suggested that some class activity or situation involving fractions be introduced 74 om II III iY V V — VII THE UNIVERSITY OF THE STATE OF NEW YORK at this time. As each difficulty is met and solved give much drill wherever needed. See activities at end of outline for fifth grade. Set standards of accuracy and speed for each process in fractions. Teach fraction idea in contrast to unit. Have both the unit and fraction before the child and draw comparison. Reduction 1 Intevers*to’ fractions 741 == 220242 i ci 2 Mixed numbers to fractions: 11/2 — 3/2, 21/2 = 5/2, CUC. 3 Fractions to whole or mixed numbers: 4/2 = ? 6/2=? FILE ete 4 Fractions to higher or lower terms: 1/2 =2/4= 4/8 = 8/16 etc. 5 Principles involved a Multiplying the numerator by an integer multiplies the fraction b Multiplying the denominator by an integer divides the fraction c Dividing the numerator by an integer divides the fraction d Dividing the denominator by an integer multiplies the fraction e Multiplying or dividing both numerator and denom- inator by the same number does not change the value of the fraction f Only like fractions can be added or subtracted Addition Subtraction Cancellation Divisibility by 2, 5, 10; factors and multiples as needed for the denominators in use Multiplication 1 Fraction by an integer 2 Fraction by an integer using cancellation 3 Mixed number by an integer 4 Fraction by a fraction 5 Fraction by a fraction with cancellation 6 Mixed number by a mixed number ELEMENTARY ARITHMETIC SYLLABUS 75 Sixth Grade (Second Half) Common Fractions Continued VIII Division It is suggested that no explanation of the process of division of fractions be attempted. Simply show that the result is obtained by inverting the divisor and multiplying. Teach as a mechanical process. IX Relation of numbers. Use small numbers in the two fol- lowing easy types: 1 One number is how many times another ? Problem application: if 3 lemons cost 10 cents what will a dozen cost? 2 One number is what part of another? Problem application: 2 examples missed out of 5 means 2/5 of the lesson missed. X It is recommended that in all work with fractions only the common business fractions be used. eel Se Waly ed Ae gues) ead heey ee) Onno Oi. OO On OFA) Orel PLL ems Ue eh ee C Decimal fractions I Review of United States money II Numeration and notation of decimals 1 Limit of 3 decimal places a Practice in reading decimals b Practice in writing decimals (1) Begin by writing as fractions 3/10 etc. (2) Write as equivalent decimals 3/10 = .3 etc. (3) Learn 1 place means tenths 2 places means hundredths 3 places means thousandths (4) Learn converse tenths means 1 place hundredths means 2 places etc. III Addition of decimals IV Subtraction of decimals V Reduction of decimals to common fractions. Write as decimals and rewrite as in form of regular fractions and reduce to simplest form. 1 Pure decimal as .25 2 Mixed decimal as 4.5 76 THE UNIVERSITY OF THE STATE OF NEW YORK 3 Decimals ending in fractions .02%. This form should be introduced just before taking up aliquot parts. VI Multiplication 1 In multiplying decimals develop the rule for pointing off by writing the decimals as common fractions and multiplying as such. Observe the denominators of the product. Finally teach that there will be as many places in the product as the sum of the places in the multiplicand and the multiplier. 2 Show that multiplying by 10 moves the decimal point one place to the right; 100 two places etc. Give plenty of drill on this. VII Division of decimals in 3 steps It is suggested that the pointing off in division of decimals be taught mechanically and the explanation as here developed be left until the sixth grade. Step 1. Decimal by an integer Teach pupils to place decimal point in quotient directly above the decimal point in the dividend before beginning to divide, then divide as in whole numbers. 2) 05 sD fe): Step 2. Integer by a decimal Rewrite the decimal so the divisor is a whole number. Then place decimal point in quotient as in step 1. For example: 27a rewritten 02.) Ase Step 3. Decimal by a decimal 1 Rewrite the decimals a Making the divisor a whole number b Moving the decimal point in the dividend to the right the same number of places as was contained in the divisor c Placing the decimal point in the quotient over the decimal point in the dividend and dividing as in whole numbers. Insist on placing the decimal point before dividing. It is suggested that instead of checking the decimal point or crossing out the ELEMENTARY ARITHMETIC SYLLABUS 77 decimal point to facilitate division of decimals, the example be rewritten when making the divisor a whole number. .04).156 rewritten 4)15.6 VIII Problems involving fractions and decimals IX Short processes in division Purchasing by 100 or by 1000 X Aliquot parts of $1 with Cecimal equivalents 1 The pupil should know how to find the aliquot part. 2 Learn the following and their equivalents: 1/2 LS rl fe oat / 8; SLO Re 58574505 /8;"5/8; TT AWA Other aliquot parts may be used but it is advised that only those of real value in class work be taught. 3 Application to problems a Sales when marked 1/3 off etc. b At 12 1/2 cents each, how many articles can be bought for a dollar? D Denominate numbers Collect denominate numbers previously learned into tables thus: De mea ELC: Lot Zac emis Sgt ec 2000 Ibs. = 1 ton (new) Depts aol ct. 100 lbs. = 1 cwt. (new) SSCS eee Pal, S qtsa=- 1 pk. Anokse:—al (bu, 5¢ = 1 nickel Ja ae LAK 10¢=. Ie dime 2A hr — lisday 50¢ = 1 half dollar 60 oming=* hr, 25¢ = 1 quarter dollar OD secanasnlermin, 100¢ = 1 dollar DAPI teen Yt: 10 dimes = 1 dollar 12s ee OLOSS Optional 1 1 doz. Surface measure E Business forms I Sales slips II Bills, integers or part-taking fractional used III Simple account keeping showing debit and credit side F Arithmetical information pertinent to fifth grade | 78 THE UNIVERSITY OF THE STATE OF NEW YORK I Knowledge of denominate numbers II Knowledge of various household costs 1 Common commodity values 2 Average cost of food, clothing, shelter for a child, for a week, a month or a year 3 Cost of school supplies for a child, for each grade, etc. III Budget idea, personal accounts, bills IV Knowledge in a simple way of what systematic saving amounts to 1 Through keeping a simple bank account 2 Through keeping a school bank account 3 Figuring interest so many cents on the dollar V Knowledge of post office rates, freight rates, etc. G Problems from text applying above. For suggestions as to. prob- lems see syllabus pages 8-10. I Review one-step problems to keep up recognition of process needed II Two-step problems III Teach children to think problems through in the steps of the solution, thus: 1 Given 2 To find 3 Solution; process A rigid insistence upon writing out every problem in the given steps is unnecessary. The easy problems will take care of themselves. For the difficult or more involved problems, the above steps will give the child a method of thinking the problem through. IV Problems from school activities V “The problems of out-of-school life of the pupil should receive primary consideration a part of the time.” 1 Activities that will give rise to such problems as a School or home gardening involving (1) Keeping accounts of costs and profits (2) Earning money by labor (3) Scale drawing and planning (4) Bills b Store sales (1) Remnant sales ELEMENTARY ARITHMETIC SYLLABUS 79 (2) Fractional discount sales (1/3 or 1/2 off) Have class bring in advertisements. | (3) Class sales of articles donated marked 1/2 or 1/3 off c Household accounts (1) Keeping account of groceries and meat bought for a week (2) Cost of food, clothing, shelter, fuel etc. for each child. Find high, low and average cost (actual or approximate prices may be used.) (3) Which costs more, to clothe a boy or a girl? (4) School cost accounts Approximate cost a month for supplies for each individual for grade and possibly for school. Seek cooperation in endeavor to reduce cost of supplies for one month by economy in use of supplies for 1 month. Compute saving for 1 month for grade, for entire school. d Budget idea (1) Plan a personal budget for a child e Charts for records Health, weight, height involving averages f Thrift accounts (1) School bank accounts (2) City bank accounts g Keeping a record of speedometer on bicycle or auto- mobile so that child might know how many miles have been traveled from one point to another and also have an idea of mile and part of mile h Installing a radio set t Automobile trip account A Type Fifth Grade Project The following garden project was worked out by Miss M. West at Dunkirk, N. Y. It developed from a child’s actual school garden account. Aim: To teach how to keep an expense account Frank was in the fifth grade and could at last begin to make plans for a garden under the direction of a supervisor. SO THE UNIVERSITY OF THE STATE OF NEW YORK 1 His father gave him a piece of land in a sunny place for his garden. Frank measured it and found it was 30 feet long and 15 feet wide. How many square feet of surface did he have to dig? What is another name for surface? 2 After he had spaded it, March 30th, the little chickens began scratching in it. He went to the hardware store to get some wire to go around it. How much did he buy? What did it cost at 31/2 cents a foot? 3 His father gave him fertilizer valued at $1. Frank wrote to his congressman at Washington for seeds, and received free on April 15th, flower seeds which were worth 75 cents. On May 30th he bought a dozen cabbage plants for 18 cents. On June 22d the supervisor advised him to get 1/2 pound of arsenate of lead for insects. It was 40 cents a pound. What were his total expenses? 4 He began to keep an account with his garden. He did it as follows: Garden Account Debit Credit Cost of garden Rec’d from garden Mar. 30! To chicken wire....!$3.15 |;|May 15] 5 bunches radishes May 20] To tomato plants....| .25 ||May 30 at Scents see ee $0.25 12 bunches radishes atiZ for i cents. <- .30 5 Finish the debit side of his account. 6 The last work was getting his vegetables and flowers ready for the school exhibit on September 19th. How many days did he have his garden? What do you think he could have planted as oo or earlier than March 30th, in your locality? 7 In March he worked 2%4 hours; in April 10 hours and 10 minutes ; in May 934 hours; in June 8 hours and 20 minutes; in July 5 hours; in August 2 hours and 15 minutes; in September 1 hour. What was his labor worth at 10 cents an hour? He said he would not charge this to his account because he thought he would get paid by developing his head, heart, hand and health, the motto of the junior project workers. 10 a 12 13 14 15 ELEMENTARY ARITHMETIC SYLLABUS 81 On May 15th he sold the grocer 25 cents worth of radishes. He put 6 in a bunch and received 5 cents a bunch for them. On May 30th he put the same number in a bunch but got only 5 cents for two bunches. The grocer paid him 30 cents for this lot. } How many radishes did he sell? How much did he receive for them? Write up his account for May. Notice which side of the account is larger. What does this mean? On which side did you put the money he received ? In June he sold a neighbor 15 heads of lettuce for the table at 5 cents a head, and 12 heads at 3 for 10 cents. What did he receive for his lettuce crop? With what do you think he replaced his lettuce and radish rows? Write up the account for June making up your own dates. Where is the balance now? His first tomato ripened on July 19th. He sold it for 10 cents because it was so early. On July 23d he sold 3 for 5 cents apiece; on July 24th, he sold 2 for 5 cents. On July 28th he sold % peck for 40 cents. On September 18th he sold YZ bushel of green tomatoes for 20 cents. Write up his account for his tomatoes. Has the balance changed sides yet? His flowers were his chief delight. He sold 3 bunches of sweet peas at 25 cents a bunch to a woman who would buy flowers every Saturday he had any to spare. He sold her 3 dozen Comet Asters at 25 cents a dozen. He gathered his everlasting flowers when they were still buds and hung them up to dry. In September he sold 200 at 10 cents a dozen. What was his flower crop worth? Make up the dates and enter his flower crop in his garden account. Has the balance changed sides yet? What does this condition mean? At the school exhibit in September, he received a prize of 50 cents for the best department garden record, a 50-cent prize for the best flower display, and 50 cents for the most perfect cabbage. He could not sell his cabbages because they were so plentiful, so he gave them to the chickens. 82 THE UNIVERSITY OF THE STATE OF NEW YORK 16 What did he gain on his garden project? Do you think his motto was a good one? For other activities see 20th Year Book of the National Education Association. A few are listed here. : Page 110. A ribbon sale for drill on fractions. Page 107. The play store or bank. Page 139. Learning thrift by keeping accounts. SIXTH GRADE The teacher should read carefully the general introduction to the syllabus. She should be familiar with the contents of the syllabus for the fifth year, and should fully understand the aims and objectives of the work for the end of the six-year period. Condensed Outline First Half A Review of integers B Common fractions reviewed with emphasis on fractional relation- ships C Decimals. More difficult work. Short processes Second Half A Bills with special emphasis on correct form B Personal accounts C Denominate numbers D Percentage I Finding the percentage II Finding what per cent one number is of another III Practical applications. Use first case of percentage 1 Commission 2 Profit and loss 3 Simple interest (1 year only) 4 Simple discount Expanded Outline Sixth Grade (First Half) A Integers I Drill for speed and accuracy 1 Addition with a review of combinations 2 Subtraction with attention to all difficulties ELEMENTARY ARITHMETIC SYLLABUS 83 3 Multiplication with review of tables if needed 4 Division with division tables including the forms 1 12 — of 12=—6 —=6 2 2 1 14 —of 14=7 ——f Z 2 1 16 — of 16 = 38 a 2 2 5 Two and three-step problems with integers. Teach children to think out the more difficult problems in the steps of the solution, thus: a Given b To find c Estimate result (make an approximation) d Solution: Process e Proof: Verifying or checking result 6 Reading and writing numbers to billions Relate to Geography in reading population, amount of national debts of various countries, values of exports and imports, etc. Note. For corrective work on the fundamentals use the Courtis Practice Drills, the Studebaker Drills, the Compact Efficiency Drills if available, or the Fassett Number Cards. They save time. Also consult second, third and fourth grade outlines for suggestions in grading the difficulties. For measuring the efficiency, accuracy and speed of a grade some standard test should be given 2 or 3 times a year. For reasoning, the Stone tests are good. Aim to keep the class up to standard. B Common fractions I Drill for speed and accuracy. It is suggested that each school system set its own standards for speed and accuracy in fractions. II It is also urged that some project or activity be undertaken by the class which will involve fractions. As each diffi- culty is encountered, stop for drill on the particular phase of the work. If the teaching of the fractions is done when some real need is felt for that specific piece of work, 84 Ww De A THE UNIVERSITY OF THE STATE OF NEW YORK the class will retain the acquired knowledge more clearly and will be much more interested in the work. A few suggestions for creating such arithmetical situations are: The cost of installing the radio Bringing in home accounts Making a collection of fractions from home problems, father’s business problems, cookbook problems etc. Imaginary or real store accounts III Place emphasis on 1 Sap fen: Wal Ss. fesy, (85) 8 9 Addition Subtraction Multiplication Division | Reduction, using factoring method of L. C. M. Principles of fractions Fractional relations. The study of the whole and part should receive emphasis. This is the foundation of the work in percentage. (Review just before taking up percentage ) a Fractional part of a number, as 4% of 21 b Fractional part one number is of another, as 2 is what part of 4° Make application here to school credits as 3 words missed out of 10 in spelling means what part missed ? c Find the whole when fractional part is given. Use small numbers and illustrate with diagram and otherwise. Problems from the text Problems from life C Decimals I Refer constantly to fifth grade outline, page 75. II Practice required in manipulating decimals up to and includ- ing six places. IIf The decimal is only a new form of writing the fraction (see 1 4 3 fifth grade course) Decimal notation a Much practice in reading as far as millionths b Drill in writing numbers Addition Subtraction ELEMENTARY ARITHMETIC SYLLABUS 85 4 Reduction of decimals to fractions with a study of aliquot parts (see page 77) 5 Multiplication For developing rule for pointing off see fifth grade outline. Short processes. 6 Division of decimals Make a point in sixth grade of teaching the under- lying principles of pointing off. In the fifth grade the pointing off was taught as a mere mechanical process. 7 Problems 8 Short processes in division 86 THE UNIVERSITY OF THE STATE OF NEW YORK Sixth Grade (Second Half) A Bills I Correct form 1 Heading 2 Ruling for purchases only 3 Ruling for purchases and credit 4 Dates 5 Receipting Note. Bring to the class some bills from local busi- ness concerns. Study different forms accepted by busi- ness houses. Headings Harris Anderson & Co. Des Moines, lowa Sold to Mrs Raymond Holmes 67 Avenue A Des Moines September 5, 1922 Or New York City, June 12, 1921 Messrs. Raymond & Co. Syracuse, N. Y. Bought of Jones, Tompson & Co. Ruling: Purchases Only 3 Ibs. coffee @ $.54 $1.62 etc, Received payment Signed by firm name Ruling: Purchases and Credits 10 lbs. coffee @ $.54 $5.40 20 lbs. tea @ $.60 $12.00 Cash $7 .00 $17.40 Balance $10.40 Paid Signed by firm ELEMENTARY ARITHMETIC SYLLABUS 87 II A simple form for receipt by principal and by agent. Bring printed receipts to class. B Keep a simple personal expense account I Debit, credit and balance Assume you have an allowance of $5 a month; required to buy school supplies, personal supplies, etc. II Teach budget idea. Figure out a budget for a school account or a boy’s or girl’s personal account. Show how the budget may be used as an aid to thrift. C Denominate numbers I Complete all denominate number tables that have not been completed. ie winear 2 Avoirdupois 3 Liquid 4 Dry 5 Time 6 (Optional) Square measure a Square inch, square foot, square rod, acre b Areas of rectangles Note... Square measure may be taught if a need is felt for it. Possibly some project may call for it, otherwise it may be left until later. Possible Method of Procedure Square measure (if taught) 1 Pupils bring to class actual measurements used in their experiences of buying or making articles. Domestic science and manual training will give practical measurements. 2 Observe nearby lots; tell why they are attractive. Bring in a point of civic pride. 3 Study a particular lot, possibly the school lot. a Measure the lot b Measure the building Note 1. Show a need for changing from one denom- ination to another. Note 2. Draw the lot to a scale. Show how to find the surface expressed in square units by so many rows of a certain number of square units as 6 rows of 5 square feet. 88 THE UNIVERSITY OF THE STATE OF NEW YORK Note 3. We may find the proportional relationship of building lot. The walk may be laid about the part of the lot where it is needed. II Reduction of denominate numbers 1 Change from lower denominations by division. Two-step change only — 3 denominations 2 Change to lower denominations by multiplication. Two- step change — 3 denominations III Addition and subtraction of denominate numbers 1 Two denominations only IV (Optional) Multiplication and division of denominate numbers Note. If deemed necessary, this may be taught. The committee recommends omission. V Problems: original problems by pupils For suggestion activities see end of sixth grade outline. D Percentage I Review fractional relationships and aliquot parts 1 Fraction decimal and per cent equivalent Fraction Decimal Rene cent 4 33 75% II Finding percentage 1 Have pupils find the 1/100 part of numbers 2 Lead pupils to see that we find 1/100 part by multiplication 3 Teach that to find the percentage of a number you multiply the number by per cent 4 Many practical problems III Finding one number is what per cent of another. Use simple numbers, relate to fraction parts and apply to problems that are practical as: A child has 8 words correct out of 10 words, what per cent are correct? 100 IV The whole amount of anything is 100 per cent of it or — 100 V Applications of percentage 1 Commission a Commodities bought or sold for another at a given rate per cent ELEMENTARY ARITHMETIC SYLLABUS 89 b ‘Terms that are necessary, as agent, commission, amount of sales etc. Find commission. 2 Profit and loss a Gain or loss and selling price at a given rate of gain or loss b ‘Terms that are commonly used. Find the gain or loss when the cost and rate of gain or loss are known. 3 Simple interest Use time exactly, 1 year 4 Simple discount 10 per cent off etc. Find discount Arithmetical Activities The following arithmetic problem work is suggestive of what can be done with a schoolroom situation. It is furnished by Miss Hession of Dunkirk and was worked out in a sixth grade. It proved much more interesting to the class than the ordinary text problem besides furnishing practical work in liquid measure, in reduction from half pints to quarts and gallons, decimals, fractions, and United States money simple account keeping etc. Milk was furnished to the children of the school and the following data was used for arithmetic. Monday Kindergarten ordered 14 half pint bottles Monday Grade I ordered 25 half pint bottles Monday Grade III ordered 13 half pint bottles _ Monday Grade IV ordered 15 half pint bottles Monday Grade V ordered 10 half pint bottles Monday Grade VI ordered 5 half pint bottles Question: 1 How many bottles? 2 How many quarts? 3 How many gallons? 4 Tf children pay 3 cents per bottle and the milkman gets 10 cents per quart, how much money is left? 5 How many pounds of graham wafers can be bought at 12% cents per pound with the money left? 6 How many boxes of straws at 23%4 cents per box? 7 How. long will it take to pay for a 10-pound box of graham wafers at 121% cents per pound? 90 THE UNIVERSITY OF THE STATE OF NEW YORK 8 How long for 3 boxes of straws at 231% cents per box? 9 How long for one 10-pound box of wafers at 12% cents per pound and 1 box of straws at 2314 cents per box? Note. These problems are not all given at one lesson. Problem 1 may be dealt upon for at least a week as the supply changes each day. Then problem 2 may be taken up etc. An account of the total supply for the week, month etc. may also be kept. Other Activities Possible 1 Collecting and making problems from business, from home etc. 2 Keeping accounts Garden accounts, see fifth grade Earning and saving money 3 Drawing to a scale 4 Playing bank for making out check, deposit slip, saving account and simple interest Information Pertinent to Sixth Grade 1 Knowledge of business forms: bills, receipts 2 How to keep a simple personal expense account, debit, credit and balance 3 Idea of a budget 4 Information pertaining to household costs, especially of food, shelter, fuel, clothing 5 Farm arithmetic ELEMENTARY ARITHMETIC SYLLABUS 91 INTRODUCTION TO ARITHMETIC IN UPPER GRADES In the work in arithmetic during the first 6 years, stress has been laid on the development of skill in the fundamental operations. There has been considerable emphasis on the mechanical manipula- tion of numbers. The child should have become proficient in handling whole and mixed numbers in fundamental processes. He should have gained a knowledge of simple denominate numbers. He should have a working knowledge of the decimal system and working ability in the simple relationships involving arithmetical processes. Problems have been given from time to time so that he can grasp a simple situation and from it lead to a conclusion. The point of view changes somewhat with the beginning of the work in the seventh year. Pupils are entering upon the early adolescent period and are gradually developing a social consciousness. The “why” of things begins to mean more to them. They begin, although dimly, to sense something of the great social organism of which they are a part. The time is coming earlier or later, very early with a large majority, when they will be compelled to take their places in the world. They begin to develop an interest in what it is all about. The school must recognize this psychological change in the pupils. The work in arithmetic, therefore, as in other sub- jects, in the higher grades must be approached from a different point of view, and it has at the same time a somewhat different aim. While the work is organized by half years for both seventh and eighth grades, it is so organized merely for grouping purposes. It is appreciated that the subject must now be handled in larger units of study and that the particular distribution of topics and arrange- ment of material will depend in large measure on the school organiza- tion of the individual community. The work for these years should therefore be interpreted largely as a topical arrangement rather than as necessary to be included in half years. The grouping should be determined in terms of local needs. In the seventh year the solution of problems should be greatly emphasized. Speed and accuracy are still important factors in the work, but the development of the power to reason, at the period of life when this phase of mental growth begins to manifest itself prominently, is the great aim. At this period of his school life the pupil’s ability will be determined in part by the special courses which he may be pursuing in the intermediate school or in the junior high sckocl where such an organization obtains. 92 THE UNIVERSITY OF THE STATE OF NEW YORK It is fundamentally important that the teacher fully appreciate the purpose in curriculum modifications at this point. Problems must be interesting. They must bear on the pupil’s experiences in the shop or the kitchen, on the playground or elsewhere. They must be of a variety that will lead the pupil to reflect, to investigate and to feel that he himself is concerned in the social and economic environ- ment of which he is the center, and further that the problems that _are presented to him come out of a living world in which he is to take his place. Here are to be emphasized the principles of business, of invest- ment and of thrift, and the situations which are to make for honesty and good citizenship. The character of the work in this grade may determine largely the strength or weakness of the pupil in his future work in mathematics. ELEMENTARY ARITHMETIC SYLLABUS 93 SEVENTH GRADE The teacher should read carefully the general introduction to the syllabus and should be familiar with the development of the subject as given in the syllabus and especially with the work of the sixth year, the aims attained at that point, and the new approach to the subject in the higher grades. Condensed Outline First Half Drill on fundamentals Review of fractions and decimals Percentage Applications of percentage I Profit and loss II Commission III Commercial discount IV Insurance V Taxes GUO pS Second Half Interest Notes Bank discount Banking Measurements ,HOOWS I Review of tables II Longitude and time III Linear measure IV Areas — square, triangle, rectangle, parallelogram, circle V Board measure F Construction to scale. Map scales G Miscellaneous work on problems in percentage and its applica-. tions, also on measurements Expanded Outline Seventh Grade (First Half) A Keep up drill on fundamentals with’ speed tests. Special attention should be devoted to pupils who are below standard. Graphs drawn by the pupil may be used to keep his scores, either for. purposes of comparisons of medians with those of other classes or with those of other individuals. 04 THE UNIVERSITY OF THE STATE OF NEW YORK B Review work in fractions, including decimals. Drill on expressing numbers as fractions, decimals, and per cents. C Reteach finding per cent of a number. Lead pupils to see that since in finding per cent of a number, the number is multiplied by the per cent, therefore the per cent of a number or per- centage is the product of that number and the rate, another name for per cent. Number times rate per cent = percentage If the per cent of a number or percentage is a product of said number and the rate, either factor can be found by dividing the product by the other factor. I Examples 1 Bie of 400=? — of 400 = 24 100 2 24 is 6% of what number? 24 — .06 = 400 or 6 100 24 — —_— = 24 x —-= 400 100 6 3 24 is what per cent of 400 24 6 24 — 400 = .06 or —-=—-=6% 100 100 Note. Many prefer the following presentation: 1 To find the per cent one number is of another: In a certain school of 600 pupils, 240 are boys, what per cent of the pupils are boys? 240 240 boys are —— of 600 pupils 600 240 2 — =- or .40 or 40 per cent 600 2 A given number is a certain per cent of a number, what is the entire number? For example, in a school where 40 per cent are boys, there are 240 boys, how many pupils are there in the school? 40 2 40% = — or — 100 5 2/5 of the pupils enrolled = 240 pupils ELEMENTARY ARITHMETIC SYLLABUS 95 1/5 of the pupils enrolled = 1/2 of 240 pupils or 120 pupils 5/5 of the pupils = 5 & 120 pupils or 600 pupils Note. Some teachers will prefer not to change their per cents to fractions. In that case the last solution might be given as follows: 40% of the pupils = 240 pupils 1% of the pupils = 1/40 of 240 pupils = 6 pupils 100% of the pupils = 100 X 6 pupils = 600 pupils Note. The pupil should know thoroughly the fol- lowing cases of percentage: 1 How to find a per cent of a number 2 How to find of what number a given number is a certain per cent 3 How to find what per cent one number is of another D Much drill on these processes as abstract work FE Drill in problem work involving these three cases: 1 Use problems that come in the child’s life as: Attendance at school Amount of work finished Per cents earned Games F Complete profit and loss Cost is sum paid for article Per cent of cost is gain or loss when article is sold Note. Here discuss why a merchant should receive a gain; what is considered a legitimate profit; why this profit may be greater in one place than in another. Show the relation of turnover to profit; the meaning of overhead; the reason for marked-down sales; the value of good salesmanship. I Problems 1 To find gain or loss: I buy a house for $8000 at a gain of 20%. What is the gain? $8000 .20 or 1/5 of $8000 = $1600 $1600.00 2 To find selling price in above problem: 96 THE UNIVERSITY OF THE STATE OF NEW YORK Note. Either of the following solutions is good: 100% represents the cost or 5/5 20% represents the gain or 1/5 120% represents the selling price or 6/5 or 6/5 of $8000 = $9600 $9600.00 selling price or $8000 cost 1600 gain $9600 selling price 3 If the above house were sold at a loss of 20% what was the selling price? 100% represents the cost or 5/5 20% of the cost represents the loss or 1/5 80% of the cost represents the selling price or 4/5 $8000 .80 $6400 selling price or 4/5 of $8000 = $6400 or $8000 cost 1600 loss $6400 selling price 4 If the sale of the above house at a loss of 20% involved a loss of $1600, what was the cost? 20% of the cost = $1600, the loss .20) $1600 1/5 of the cost = $1600 $8000 20) $160000 5/5 of the cost = 5 & $1600 = $8000 5 If a man gained $1600 on a house that cost $8000, what per cent did he gain? 20 1600 8000)1600 or $1600 is 8000 of $8000 1600 -20 = 20% — = 1/5 or 20% 8000 ELEMENTARY ARITHMETIC SYLLABUS 97 G Commission reviewed and new cases added I Definition: a term used in the commercial world to designate the name of money paid to an individual for transacting business for another. Review of definitions of terms taught in sixth year. II Reteach amount of sales & rate of commission = commission 1 When total cost is required, the commission and any other expenses connected with the purchase must be added to sales price. This is when buying. Z When profit from sale by a commission merchant is required The commission and any other expenses incident to the sale are added. This sum subtracted from sales price = proceeds III Facts about which the pupil should be positive 1 Amount of sales times rate of commission equals com- mission 2 Amount of sales plus commission equals total cost 3 Amount of sales minus commission equals sum remitted to principal 4 Amount of sales equals commission divided by rate of commission 5 Rate of commission equals commission divided by sales Notre. Any good form should be accepted. See types under profit and loss. Label work so thought is clearly expressed. The multiplier should be an abstract number. Much drill in real problems which are furnished by teacher, also by pupils. ‘H Commercial discount I Terms used: list price, rate of discount, net price II Why discounts are given and that more than one discount is frequently given III List price times rate equals discount IV Discount is a per cent of a number An article listed at $250 is sold at a discount of 25 per cent. How much is paid for this? 98 THE UNIVERSITY OF THE STATE OF NEW YORK 100% = marked (list) 25% = rate of discount 75% (net) $250 list price 0, woos $750 1250 or 3/4 of $250 = —— = $187.50 net price 1750 . + $187.50 net price V List price times (100% minus rate of discount) equals net price VI $250.00 marked price or list price Ray Arte) 6, $62.50 discount Therefore, list price minus net price equals discount An article for which I paid $187.50 was sold at a discount ~ of 25 per cent. What was the original price? If at a dis- count of 25 per cent, it was sold for 3/4 of original price, $187.50 3 , 4 $187.50 — — = 62.50 * — = $250 4 * For other forms of solution see pages 93-94. If an article sells for $187.50 at a discount of $62.50, what is the rate of discount? $62 250 = $187750°)25%en 1/4" 0r8Z5% VII To find the net price when a series of discounts Goods marked at $125 have discounts of 10%, 5% and 2%. Find the price received by merchant. 100% 100% 100% 10% 5% 2% .90 OS .98 $125 list price $112.50 new list price $106.88 new list price .90 95 .98 _—_ $112.50 net price 56250 85504 101250 96192 $106 .8750 $104.74 net price $125.00 104.74 $20.26 discount ELEMENTARY ARITHMETIC SYLLABUS 99 VIII Bills when discounts are given Form Schenectady, N. Y., January 3, 1923 Teed, ormtith or CO. Sold to D. S. Black Terms 10%, 5% ~ 500 yds. silk @ $2.50 $1250.00 Less 10%, 5% 181.25 $1068.75 Received payment Jat. Siiths we Co. I Insurance I Definition of terms used II Kinds of insurance. Why a protection? III Growth of insurance Is it profitable? For whom? Why? IV Essentials to be found in policy 1 Study of a real policy a The sum for which insured b The agreement clearly stated of company to insured party c Term of premium Note. Investigation of rate of insurance compared with neighboring places, also as compared with rural sections. What factors determine the rate of insurance? V Lead pupils to see this is an application of percentage and solved in same way, therefore 1 Face of policy times rate equals premium 2 Premium divided by rate equals face of policy 3 Premium divided by face of policy equals rate J Taxes I Show why taxes are necessary, how all must share. Lead pupils to feel that taxes should not be avoided or neces- sarily lowered, but that the public should receive adequate returns for these taxes. II Talk about budget. Why is it good? III Kinds of taxes: city, county, state, federal; also personal on property as well as income. When levied on imported commodities a tax is known as duty. IV Bonding individuals for tax collectors. Why? Does this build character? Study the form. 100 THE UNIVERSITY OF THE STATE OF NEW YORK V_ Assessors VI Facts to be taught. How to find. 1 Rate of tax. Tax to be raised divided by total valuation 2 Individual tax on certain piece of property. Valuation of property times rate of tax 3 Total tax to individual when collector’s fee is included a Tax times rate of fee equals collector’s fee b Tax plus collector’s fee equals total tax Note. Use the common tax of your own community. Teach common method of expressing rate; dollars on a thousand. Compare this rate with neighboring communi- ties and see why they differ.. Decide whether the cause justifies this difference. Seventh Grade (Second Half) A Interest I Terms used: principal, or 1f note face of note; rate of interest; interest; and amount Interest is very similar to rent. Rent is pay for the use of property at given rate for certain time, while interest is pay for the use of money at a given rate for a certain time. II Therefore, principal times rate times time equals interest III Time is the troublesome factor because it may be expressed in three units of measure Have a talk with the pupils about the various ways which people have used to compute interest but lead them always to see it is the product of the three factors above mentioned. Allow any standard way of computing interest Cancellation method and 6 per cent method are recom- mended For short time loans the bankers’ method is desirable IV (Optional) Show how the converses may be solved. Interest divided by principal times rate equals time Interest divided by time times rate equals principal Interest divided by principal times time equals rate Much time is spent in finding interest. Accuracy and speed is important in this. Problems in which interest is employed should be given. Little time is spent with the con- verses and their application. ELEMENTARY ARITHMETIC SYLLABUS 101 B Notes I Definition: a written promise to pay a certain sum of money to a certain party at a certain time with certain rate of interest or without interest II Essentials of a note 1 Parties 2 Negotiable or nonnegotiable 3 Indorsements. Selling of notes The pupils should be able to write any form of note and to know the names of the parties to a note. C Bank discount I A note is like a commodity for sale II Banks being dealers in money are the places to sell or buy notes III Term of discount. Pupil should see why there are two dates IV How the interest bearing note is affected V Discount on an interest bearing note. Pupil should under- stand why the new face amount is sum discounted VI Terms the pupil should be taught: bank discount, proceeds, new face, term of discount, date of maturity D Banking (Jn this grade only slightly studied) I Different kinds of banks 1 Savings 2 National 3 Private II How to deposit money III How to withdraw funds IV Checks 1 Why good 2 How to write 3 Certified V Drafts 1 How they differ from checks 2 Why used VI Other good ways of sending money E Measurements I Review all tables already learned and teach those your school requires and not already taught II Review all work in denominate numbers so pupils will reduce answers to the terms in general use 102 THE UNIVERSITY OF THE STATE OF NEW YORK III Longitude and time 1 Teach only enough to find practical change in time and how the longitude of the places causes this change. 2 Many schools may feel that this is a nonessential in arithmetic, a sufficient amount of this topic being taught in geography. 3 When it is taught use following steps: a To find the difference of longitude. If places are in same direction of longitude from the prime meridian, subtract; if in different, add. This gives difference of longitude b Divide difference of longitude by 15 to obtain difference of time Note. Show how this 15 is obtained. c Having difference of time, show how to get the time of a place (1) If traveling eastward, add (2) If traveling westward, subtract IV Review linear measure. Teach meaning of construction of straight line, circumference and perpendicular V Teach names of type forms of following surfaces: square, triangle, rectangle, parallelogram, circle If table of square measure has not been taught, introduce it here. | | Show when areas are used. Be sure pupils see the surfaces of various solids. Spend much time that pupil may see that circumference of circle is linear measure while the surface is square measure. VI How to find areas of surfaces named above. Make study in and about building to find these VII Board measure 1 A board foot is a piece of timber 1 foot long, 1 foot wide and 1 inch or less thick. Use cancellation method 2 Number of pieces times length in feet times width in feet times thickness in inches equals number of board feet 3 When price is required, include it in above formula. For example, find the cost of 18 pieces of timber 16 feet long 18 inches wide and 1%4 inches thick at $30 per M. Lae) 30 1944 18 xX 16&*& — xX — X —= LAZY 1000"? TOO or $19.44 ELEMENTARY ARITHMETIC SYLLABUS 103 For schools that take up the junior high school idea or the 4-3-3 plan instead of the older 8-4 plan, the following is suggested. It may be introduced in other places during the year rather than placed at any one step. F Geometry I Approach this subject by teaching pupil by means of well- planned observation and experiment the most important geometric forms which are found in our everyday life. II Teach how to construct these geometrical figures 1 Use T-square, triangle, protractor and compasses 2 Construction of a perpendicular bisector of lines, polygons which the pupil has found in practical life and forms found in nature 3 Direct measurements drawn to a scale From these compute some capacities and thus lead pupils to see when we demand absolute and very accurate measurements; when we may use approximations. 4 Indirect measurements as measuring heights etc. G Review all applications of percentage by means of miscellaneous problems. Use problems that are real but are sufficiently simple to enable pupils to feel they are masters of the situations and thus keep up their interest. Lead pupils to bring in problems and stimulate them to give problems which will test the ability of their classmates. Fre- quently take some of their problems for your tests. Lead pupils to see the three forms of percentage. Let the problems also include banking, bank discount and measure- ments. H (Optional) Situations for seventh grade work I A boy is given $100. How shall he invest it so that he may realize the more for his use when ready to enter college? II Study of wills. How gifts may be made to mean the most to the beneficiary III Build a radio set IV The parcel post 104 THE UNIVERSITY OF THE STATE OF NEW YORK EIGHTH GRADE The teacher should be familiar with the development of the work during the previous years, should study very carefully the syllabus for the seventh year, and should keep clearly in mind the aims of the course in terms of the larger social interests of the pupils of early adolescent age. Condensed Outline First Half A. Review fundamentals. Check proficiency of the class by means of standard tests B Measurements I Reviewed II Cubic measure, volumes III (Optional) Metric system C Ratio and proportion D Powers and roots; square root and its applications E General review to percentage F Review percentage and its applications — studied more extensively ‘than in the seventh year G Banking practice H Stocks and bonds I Investments J Review completed including new work of the grade Second Half A Drawing to scale B Lines and angles C Construction D Graphs Lebar II Circle III Broken line IV (Optional) Pictographs E Introduction to simplest literal symbols and expressions I Meaning II Definitions III Reading and writing from dictation simple literal expressions Note. It is recommended that the consideration of nega- tive quantities and the work in subtraction be deferred until ‘the ninth year. ELEMENTARY ARITHMETIC SYLLABUS 105 F Addition of literal terms (positive results only) G Numerical substitution H Certain axioms I The equation J The problem K The formula Expanded Outline HKighth Grade (First Half) A Review fundamentals Tests for speed and accuracy, include work in fractions, decimals, percentage and interest. Keep this work simple enough to retain interest because of success gained but sufficiently advanced to produce a real development of power. Use aliquot parts in the above whenever an advantage. Gain skill in multiplication by extending their use, as: 125 & 7247 (consider 125 as ¥% of 1000) add three ciphers to 7247 and divide by 8. Emphasize short processes of multiplication and division. Demand accuracy and speed. B Measurements I Work of previous grades II Where volume is required III Table of cubic measure. Teach 231 cubic inches in gallon (2150.42 cubic inches in bushel rather than 1%4 cubic feet) IV How to find volume of parallelopiped, prisms, cylinder Lead pupils to see that they must think of the shape of the base before they can compute volume. Be sure pupils know when to find surface, when volume ; also that surface means square units, two dimensions — volume means cubic units, three dimensions. Spend considerable time on circles and cylindrical type forms. V Give time for some real problem of your community in measure work as: constructing a road, laying out a park, constructing a public building, decorating a set of rooms There may not be time to do all of this but in a general way pupils should be encouraged to familiarize themselves with costs of these things. VI (Optional) Metric system 106 THE UNIVERSITY OF THE STATE OF NEW YORK C Ratio and proportion I Lead pupils to see that we often express measures by com- parison as size of buildings, fields, height of hills, trees etc. Then show them that we can also express a com- parison of number. Division is really such a comparison. Fraction is an expressed division. It is a ratio. An equality of ratios gives a proportion. II The pupils now know definition of ratio and proportion. Teach other terms used: means, extremes, antecedent, consequent, also symbols used, : and :: or = III Product of extremes equals product of means IV How to solve by cancellation so as to find the missing term V How to state a proportion VI How to divide a number according to given ratios 1 Add the terms of the ratio 2 Form fractions, using the sum of the terms as denom- inator and each term in order as numerator 3. Multiply the number by these fractions Show pupils how proportion may be employed in many problems, always when a comparison of similar conditions exists. Give real problems for drill in ratio and proportion. D Powers and roots I To find the area of a circle, the radius squared was used. From this teach the pupil that he used the second power which is a product obtained by multiplying a number by itself. The third power may be found by using the same factor 3 times and the fourth power by using a number as a factor 4 times, etc. The power may be known from the number of times the number is used as a factor. II The power is indicated by a little figure written above and to'the*right*of'a’ nuniber*asi2> ee Zee cee Cle. III This little figure is called the exponent and always shows how many times the number is taken as a factor. 1 Give drill in raising numbers to different powers. Use whole numbers, fractions and decimals. This gives opportunity to acquire speed and accuracy. 2 Be sure that the pupil knows that the second power of a number is the square. (He should also know that the third power is the cube.) To find the area of a square, square the length of the side. ELEMENTARY ARITHMETIC SYLLABUS 107 IV The number repeated as a factor to produce the power is a root. A root of a number, then, is one of the equal factors used to produce the number. The square root of a number is one of its two equal factors. V Practice in finding roots by means of factoring VI Square root The explanation of square root may be based on a geometrical figure. Some teachers may prefer to make the teaching of square root a purely mechanical process, deferring the explanation until the topic is studied in algebra. VII How to find hypotenus of a right triangle when the legs are known and the length of one leg when the hypotenuse and other leg are known. The diagonal of a rectangle; solution of problems involving square root. E General review as far as percentage I Notation and numeration TT Fundamental operations III Fractions 1 Definitions and principles 2 Operations 3 Fractional relationships 4 Miscellaneous problems IV Decimals I Reading and writing 2 Operations — special emphasis on division 3 Aliquot parts 4 Short processes (much drill) V Denominate numbers F Review of percentage I The three relationships, problems, applications of percentage In the review of applications of percentage it is not meant that the problems shall be difficult or that the. study of the various topics shall be exhaustive. It is, however, advised that the teacher shall send his pupils afield for material that not only will motivate the work but will furnish a fund of useful and interesting information. This will be valuable in later life and may not be made available to the pupil in any other wav. It is suggested that interest and enthusiasm may be stimu- lated by the organization of members of the class into a 108 ' THE UNIVERSITY OF THE STATE OF NEW YORK Wide World Club. Regular meetings may be held and local business and professional men may be invited in to speak. Suggestions for talks: Banker, banking practice, bank loans and discounts, investing money Village, city or other official, taxes and the tax budget Insurance agent, fire insurance, life insurance Merchant, the overhead, the turnover, merchandising Traveling salesman, salesmanship, different methods of selling: on salary, commission, bonus; prevailing commissions allowed County, village or city treasurer, the work of the treas- urer and the collection of taxes President of the board of education, financing the school Street commissioner or superintendent of highways, road construction, kinds and cost; concrete as a building material; its value and cost; keeping the streets clean Postmaster, the business of the local post office Contractor, building construction, steel, brick, stone Water commissioner or superintendent, the local service; its problems and cost; water bonds as an investment; municipal ownership Advertising agent, cost and value of advertising, rates, keyed ads Many other classes of speakers will occur to the wide- awake teacher. Speakers should be urged to tell about the arithmetic of their business, but this should not drift into dry statistics. Pupils should be encouraged to take notes and from notes construct problems based on conditions they have heard described. They should be encouraged to ask questions and discuss the talks they have heard. 1 Commercial discount a Same as in seventh year b Emphasize bills, collect all forms that pupils can find. Study advantages c Overhead charges d Advertising. Have pupils collect and bring to class adver- tisements announcing discounts and marked-down sales e Salesmanship. Study factors involved in good salesman- ship. The commercial traveler. Why necessary ELEMENTARY ARITHMETIC SYLLABUS 109 2 Profit and loss a Marking goods so as to allow discounts and still make a profit b Purchasing goods at a discount for cash; advantages c Legitimate profits, variations in profits, in different lines of business d The turnover 1 Its meaning 2 Its desirability and advantage e Including interest in sale price when money is borrowed for purchasing goods f Multiple discounts 3 Insurance a Kinds: fire, life, accident, health, burglary, liability etc. b Principle involved: Many parties cooperating by paying comparatively small sums for a longer or shorter period can make certain the payment of losses in the relatively few cases when they occur. The law of average. 'c¢ A protection against loss ¢e@ d Investment of insurance funds. State supervision of in- surance companies e Have pupils discover names of prominent insurance com- panies. Encourage them to discover local rates of fire insurance and find out factors that may reduce those rates. Let the class examine policies covering fire, accident or life insurance f Discuss the advantages of carrying insurance and the danger of neglecting to do so g Have a local fire insurance agent talk on fire hazards, board of underwriters, local fire insurance rates for various kinds of risks, (whether high or low and why) the value of a good fire department, etc. h Have a life insurance agent give a simple talk on the value of life insurance, two or three common forms of policy and the cost 4 Commission a Same as in the seventh year but more difficult problems. Combine it with other items of expense to arrive at total cost 110 THE UNIVERSITY OF THE STATE OF NEW YORK b Encourage pupils to discover common articles of mer- chandise handled by commission agents and varying rates of commission charged. Factors that make these differences Saaaces a Take up as in seventh grade b Reason for taxes c Secure a local budget. Find the total assessment. Make problems from the local conditions d Effect of taxes on rent. Who really pays the taxes? The effect of high taxes on industry and the location of industry e General provisions of the income tax, exemptions, state and national. Simple problems f Tax-paying and good citizenship g Kinds of property exempt from taxes. Why? h Have a talk from some village, city or county officer on the tax question and the making up of the budget 6 Interest and bank discount a Treat the subject as in the seventh year. Introduce partial payments on a note. Use not more tHe two simple payments b (Optional) The Federal Reserve Bank and the relation of the members of the system to the Federal Reserve Bank. Rediscount c Simple savings bank problems involving compound interest. Use 4 per cent only. d See if pupils can find out how interest varies in different banks e Secure forms of note. Discover the method of discounting at. a local bank. The meaning of protest. The fee charged G Banking practice I Kinds of banks (see page 100). Trust companies. The broader powers of the latter. Bank examiners II Two reasons for banks 1 To keep money safely 2 To lend money needed for business and industry III Security for loans. Collateral. Find out different kinds ~IV Checking accounts ELEMENTARY ARITHMETIC SYLLABUS Lut V Interest accounts VI Checks, certified checks, drafts, indorsements VII Certificates of deposit. Bank book. Monthly bank state- ments to depositors VIII Penalty for writing checks on bank not containing the drawer’s account. Seriousness of drawing a check for more than the amount of the deposit. Reconciling balance each month IX. Safe deposit boxes X The class should visit a bank and be permitted to see the vault, safe deposit boxes, etc. and have as much explained to them as they are able to comprehend H_ Stocks and bonds I Terms and definitions: Stocks, bonds, par, premium, quoted price, brokerage, discount, preferred, common etc. Note. Help pupil to see difference between a share of stock and a bond. If possible show class a stock certifi- cate or a liberty or other bond. Comparative advantages of owning stocks or bonds. II (Quoted price plus brokerage) times number of bonds equals cost III (Quoted price plus brokerage) times number of shares equals proceeds IV Difference between cost and net selling price equals gain or loss. Periods of depression (Optional) V Dividend. It is always the par value (usually $100) times rate of dividend. This is often expressed thus: U. S. 4s, meaning 4 per cent dividend. VI Rate of income. Divide the dividend as computed above by the cost of share VII Have pupils trace fluctuations of some bond for a week or more and report VIII Meaning of buying on margin; its danger Note. The pupil should be made to see that it is not easy for an inexperienced person to know when to invest to the best advantage. Competent advice should always be sought. Emphasize. I Investments I Kinds 1 Speculative or uncertain 2 Conservative, usually safe Lis THE UNIVERSITY OF THE STATE OF NEW YORK II Principles 1 Absolutely safe investments bring lower rate of interest 2 Long term investments usually bring lower rate of income 3 Promises of high rates of interest usually signify doubtful security 4 Danger of investing money without obtaining information from some disinterested authority, one who is in a posi- tion to know fully the safety of the investment Note. Pupils should be encouraged to find from banks or other sources information concerning the reliability of investments; also service rendered by reports of Dunn and Bradstreet. Arrange with a local banker to talk to the pupils about investments. Arrange with him to permit a group of chil- dren to have access to Moody’s ratings. Let them report the rating of some well-known and highly esteemed railroad or other good bond. Send another group to the bank to learn the rating on a bond of uncertain value. A written report should be required. 5 Secure from newspapers or elsewhere accounts of dishonest investment schemes Situations for first half of eighth grade (optional) A Building a house I Figure all expenses: both interior and yard accessories, walks, streets etc: II Cost to include insurances of various kinds, as fire, walk, windows III Rents obtained 1 If two flats unfurnished, with heating systems but no heat furnished 2 If four small apartments furnished and heated etc. 3 Whick is the better investment B Investing a gift of $500 in any way to secure greater income for — next 10 years I Income when used will furnish what part of living expenses ? | | II What would be value of same if income is invested each year? III How to invest that it shall be profitable IV Study of bonds and stocks; how to know whether reliable V When to buy? ELEMENTARY ARITHMETIC SYLLABUS 113 VI When to sell? VII Why issued? VIII Why they fluctuate? IX Foreign bonds X Study of banks XI How banks invest? XII Mortgages XIII Real estate transfers XIV Savings and loan associations C Making an interest table D Making a tax table, using figures given by tax collector E Organization of a stock company — paying dividends, selling stock, etc. F Working problems in stocks and bonds, taking figures from stock reports in newspapers G Reading of gas and electric meters and bringing problems to school H Sending money by post office money order; filling out real blanks I Paying imaginary debts to one another by check J Making out real household accounts from bills brought by the children from home Eighth Grade (Second Half) A Drawing to scale I Why used? II By whom? III Accuracy in drawing absolutely essential IV Instruments used 1 Real measurements and plans drawn. Measure school- room. Draw plan to show exact size of room, location of windows, blackboards and furniture 2 Study map scales (more difficult work) 3 Study simple plans of articles constructed in the shop. Close cooperation with teachers of manual training and sewing B The line. The path of a point in motion I Kinds 1 Straight 2 Curved 3. Broken 114 THE UNIVERSITY OF THE STATE OF NEW YORK The angle I Kinds 1 Right 2 Acute 3 Obtuse C Construction I To bisect a straight line II To construct a perpendicular (1) at any point in a line, (2) at end of a line III To bisect an angle IV To construct equal angles V To construct a triangle equal to a given triangle VI To enlarge or reduce a rectangle or parallelogram D Graphs Gradually it has come about that people feel that a repre- sentation of facts by numbers conveys very little to the human mind. Pictures will often accomplish much more. Comparison is also a help. Hence has come the method of representing values by graphs. There are many ways of doing this. Any modern text will give some plan. Any study which gives the pupil ability to express values accurately and fully is good. There should also be a study to enable pupils to read graphs in order that they can gain from printed graphs the entire story they should tell. I Definition: the pictured form of presenting facts II Kinds iieine 2 Bar 3: Circular 4 Pictographs III Essentials 1 Accuracy 2 Clearness 3 Neatness 4 Appealing to the eye and interest IV Things to note 1 Any printing on the graph itself should be avoided, wherever possible 2 All tabulations should be at the right side or below ELEMENTARY ARITHMETIC SYLLABUS 115 3 Facts should be printed on another paper or far removed from the graph ; 4 You should show clearly upon what basis you are repre- senting your facts V For illustrations of graphs see 1 Line graph, plate 1, page 119 2 Bar graph, plate 2, page 120 3 Circular graph, plate 3, page 121 Pictographs are really pictures used frequently by magazines. They should be of exact scale to tell the story truthfully. V1 There should be discussions to see what sort of graph will best illustrate the truth to be conveyed After a graph has been made the children should feel that it is really helpful in conveying the facts or the graph should be rejected. Practice on interpretation and tabula- tion from graphs. Coloring may be used to increase clearness. VII Much practice in constructing graphs. Abundant material for problems can be found in the World Almanac, census reports or statistical tables. Plot graphs of temperature and rainfall, school attendance. Two weeks or more may be profitably devoted to graphs. E The use of literal terms The work is meant to be in the nature of an easy approach to the subject. Only enough is given to awaken an interest in something quite new to the pupil. Too often the work in the eighth year proves an actual detri- ment to the pupil’s future work in algebra and many teach- ers of the algebra of the ninth year would much prefer that the pupils come to them entirely new to the subject. This work should be taken up slowly and each step should be thoroughly understood. Pupils who have done the work as outlined should be well prepared to undertake the diffi- culties of the fundamental operations. I Introduction to the use of symbols 1 This work should be very simple in nature and should be designed to make the pupil familiar with the fact that quantities may be represented by letters or by letters combined with figures. Such quantities may be added, subtracted, multiplied and divided as with num- bers. The expression a@ means something mathemat- 116 THE UNIVERSITY OF THE STATE OF NEW YORK ically exact when it appears in an algebraic combina- tion. Show that 3a is as easy a thing to consider as 3 bushels, or 3 feet. The addition 3a-+ 2a = 5a is performed as easily as $3 + $2 = $5. Show also that a in the above expression may have any value that we may choose to give it 2 The pupils should become familiar with the following: Coefficient, term (known and unknown), factor, ex- ponent, power, root 3 Give pupils much practice in reading and writing very simple literal expressions as the work advances 4 Simple literal terms. Let them not contain more than three letters. Deal only with small coefficients. It is recommended that there be no elaborate discussion of negative quantities. It is also recommended that the teacher present to the class only such combinations as will give positive results. F Addition of like terms. Practice in uniting terms so that the pupil shall not be confused with negative results. In other words, let the succession of terms be exactly the same as the pupil has had in numbers in arithmetic, for instance 8a — 4a — 2a + 5a-+ a, ete. Note. If the teacher prefers, the negative quantity may be introduced and the regular work in algebraic addition may be taken up. The recommended procedure, however, has the advantage that addition in the ninth year will still be new to the pupil and the teacher will not therefore experience a lack of interest on the part of his pupils in that year. Show that 3a7b means 3 X a? & b and may be written 3.a?.b. G Simple numerical substitution. Secure simple material from any good text -H These axioms illustrated: addition, subtraction, multiplication, division. (Optional) The same roots of equals are equal. The same powers of equals are equal I The equation I Members of the equation II Transposition of terms and the axioms involved III Clearing of fractions with not more than two or three small denominators Note. The teacher may well approach the equation by the statement and solution of problems involving the use of X. Some teachers may prefer to give some simple ELEMENTARY ARITHMETIC SYLLABUS 117 equations to perfect the mechanical manipulation before undertaking problem solution. J The solution of the problem I Read the problem carefully for the entire thought II Read the problem and determine what value is to be found first, or what value will make it possible to find all other values that may be required III Let X represent (equal) this value IV Read the problem again and with X as this value satisfy all other conditions in the problem V State the equation VI Solve the equation 1 Clear of fractions if any are involved 2 Transpose, if necessary, known terms to right side of equation and unknown to left 3 Unite the terms. Equations involving negative members when simplified are to be avoided 4 Divide both members of the equation by the coefficient of X 5 Using the value of X find any other values required by the problem VII Check the work by substituting the value of X and see whether the answer is correct. It is also advisable to read the problem to see whether the answer satisfies all the conditions stated Notr. The teacher should make many problems for pupils to solve. Any easy text in algebra will furnish good material, and pupils may be encouraged to con- tribute original problems for class use. K The formula I The formation II The interpretation III The evaluation The teacher should in the beginning seek the material for the formula from the work in arithmetic previously studied. The formulas should be developed by the pupils themselves from what they already know. To illustrate, interest is found by multiplying what together? Principal, rate and time. Supposing that stands for principal, r for rate and ¢ for time, what will stand for interest? Interest (Int.) = prt. From what 118 THE UNIVERSITY OF THE STATE OF NEW YORK the pupils already know about equations, they can develop from prt = Int. formulas as follows: Int. Int. Int. rt pt pr In commission if s = sales, y = rate of commission and c = com- mission, the class should develop c = rs. | Likewise develop formulas in profit and loss, insurance, taxes. Since the pupils know rules for areas of triangles, rectangles, parallelograms, trapezoids, let them derive their formulas, giving them letters to represent dimensions. Give practice on such formulas as D = rt. (Where D = distance, ry = rate of travel and ¢ the time traveled.) C = pn where C = cost, p = unit price and m = number purchased. Pupils should be given much practice in working out examples by use of formulas. Several of the more elementary books in algebra contain many simple formulas found in the lists of literal equations. Only the simplest of these should be used, but. much practice will not only interest the pupils but will strengthen them greatly for the work in algebra in the ninth year. Occasionally the teacher may well give a formula which is a state- ment of an arithmetical principle that the pupils know to test whether they can interpret the formula in terms of what they already know. Give considerable practice in the evaluation of simple formulas. ELEMENTARY ARITHMETIC SYLLABUS ~- 119 PCATESL Deaths from Diphtheria in New York City 1906 1907 1908 1909 1914 19is 1916 {917 1913 i919 1920 © iD2ZI [222 , 1903 , 1904 1905 1910 Dit 1912, 1913 2200 2100 2000 i900 IG0O I17OO [600 1500 1400 1300 1200 1100 1000 960 300 From the above make a table showing yearly the number of deaths from diphtheria from 1903 to 1922 inclusive. What has been the general direction taken by the line? . Can a reason be given for the decrease shown by the graph? Find out what you can about the present treatment for diphtheria. 120 THE UNIVERSITY OF THE STATE OF NEW YORK PruATEZ Cost of Public School Education in 1919-20 238328333888 gS Cee i Ge) esgssssssaas SssgseeRgsessRaggeg NEW YORK 106,045,319 ono Ee pues Sees sat om 67,426,541 CALIFORMIA ee cee 48,980,298 NEW JERSEY 40,909,827 MASSACHUSETTS GED 40,908,940 COLORADO ae 13, 200,165 VIRGINIA Pa 12,975,089 ALABAMA me 9,118,691 Compare the length of the bar representing Alabama’s school expenditure with that representing New York’s. Compare the population of the two states and determine which population is spending more for schools per capita. Determine the same for the other states. Which is spending more per capita of population, New Jersey or Massachusetts ? ELEMENTARY ARITHMETIC SYLLABUS yal RLATECO Distribution of Population in New York State in 1920 25,000 — 500,000 Si hesS ABOVE 500,000 47-3 % K> % GI % * 5,000 RURAL AND VILLAGES LESS THAN 2,500 a7: 2%o Encourage pupils to tell in their own words the meaning of the above graph. How might the same graph drawn fifty years ago appear? What variation perhaps will be shown in the next twenty-five years? Give some reason for this.