AmTHMETIC 
 
 » 
 
 I9I9 
 
 REPORTor PROGRESS 
 
 DULUTH PUBLIC SCHOOL.^ 
 
 Y ^ V • .. :i^.~
 
 This book is DUE on the last date stamped below 
 
 .;^2» 
 
 DEC l3 Idtf 
 
 JUL Jti i 
 
 '^'=^'0 1 4 1931 
 1941 
 
 Southern Branch 
 of the 
 
 University of California 
 
 Los Angeles 
 
 Form i, 1 
 
 J s 7 
 B ^% 
 
 v.\ 
 
 ;-"^->^- 
 
 'N
 
 VERSITY OF CALIFORNIA, 
 
 LIBRARY, 
 
 ARITHMETIC 
 
 1919 
 REPORT OF PROGRESS 
 
 DULUTH PUBLIC SCHOOLS. 
 
 82390
 
 ■ V. I 
 
 CONTENTS. 
 
 AIMS 1 
 
 BRIEF SURVEY 1 
 
 GENERAL DIRECTIONS 6 
 
 Identical Forms for all Grades 6 
 
 Checks 8 
 
 Principles of Method 11 
 
 Provision for Individual Differences 13 
 
 Standard Practice Material 14 
 
 Standard Tests in Arithmetic 15 
 
 GRADE I-B and I-A 18 
 
 GRADE II-B : 20 
 
 GRADE II-A 2(5 
 
 GRADE III-B 34 
 
 GRADE III-A \ . 40 
 
 (iRADE IV-B 47 
 
 GRADE IV-A 54 
 
 (tRADE V-B 61 
 
 (4RADE V-A 66 
 
 (iRADE VI-B 71 
 
 GRADE VI-A 76 
 
 (iRADE VII-B 82 
 
 GRADE VII-A 87 
 
 GRADE VIII-B 91 
 
 GRADE VIII-A 96 
 
 GRADE IX-B 101 
 
 (;RADE IX-A ! 105 
 
 GENERAL BIBLIOGRAPHY 108
 
 Will each teacher i)lease make the following correctiou: 
 
 Page 87, line 3 should read, "This will involve column addition."
 
 PREFACE 
 
 This ])anii)hlet ou .\rithnietic is oue ot" a series of five which 
 constitutes the Course of Study so far available in jjrinted form tor 
 use in the Public Schools of the City of Duluth. The other panijth- 
 lets are as follows: one on ]\Iusic and Physical Education; one on 
 Geography, History and Nature Study; one on Drawing- and In- 
 dustrial Art; one ou English including lieadiug, Phonics, Spelling, 
 PennianshiiJ, Language, Composition, Grammar, and Literature. 
 
 This Course of Study was constructed during the school term of 
 1918-1919 and during the summer of 1919. It was introduced in 
 September 1919. It is the product of the combined effort of the 
 teachers, principals, and supervisors in the Public Schools and the 
 State Normal School of Duluth. 
 
 The general supervision of the entire course was under an execu- 
 tive committee consisting of a principal, a supervisor, and a superin- 
 dent of the training department of a normal school. Each subject 
 was in charge of a special committee consisting of teachers, principals 
 and supervisors with the teachers largely predominating. While the 
 number of teachers on these committees was made as large as possible 
 in order to secure the benefit of class room experience, not all were 
 able to participate in the work on account of the lack of time and faci- 
 lities for reaching them. Much credit is due all who have so will- 
 ingly and efliciently assisted in bringing this course of study to its 
 present standard. The fact that it is an outgrowth of the best class 
 room practice in the city is due largely, however, to the teachers who 
 helped in its construction. 
 
 The general i)lan for each subject in the Course, the i)rinciples 
 for the selection of subject matter, and the organization of subject 
 matter were agreed upon by the executive coiniiiittee and the chair- 
 man of each special committee after much study and careful delibera- 
 tion. Each special committee observed these principles of selection 
 and plan of organization in preitaring the subject assigned. Sugges- 
 tions on the Course in English were received from a grouj) of business 
 men in order to secure the i)oiut of view of those outside the schools. 
 Similar help was received from a group of musicians on the Course 
 in !Music. 
 
 The general jjlan adopted for each course is as follows: 
 
 I. Table of Contents. 
 
 II. Aiius and purposes for all grades. 
 
 A statement of the pur])oses of the subject as a whole.
 
 III. Outline oi 8uV>je(.'t matter. 
 
 IJrief survey of subject matter throughout the Elemen- 
 tary and Junior High Schools. 
 
 IV. General Directions. 
 
 V. Detailed outline of subject matter. 
 
 VI. General Bibliography. 
 
 As a basis for the selection of subject matter for this Course of 
 Study the following social values were used: 
 
 I. That subject matter was selected which is most frequently 
 used by the greatest number of people in life situations. The term 
 "use" is not restricted to the mere economic sense but includes all 
 those matters which society has learned to value and desires to pass 
 on to the next generation. 
 
 II. That subject matter was selected which is not only most 
 fretjuently used but is most significant when used, e.g. we teach how 
 to save life from drowning not because of the number of times it would 
 be used but because of its great significance when used. These 
 method.'* of choosing subject matter while they have l)een a guiding 
 principle have been necessarily limited by such considerations as, ex- 
 pense of teaching, time of i)uijils, ability of teachers and pupils and 
 organization and availability of material. 
 
 In the organization of subject matter an attempt has been made 
 to arrange it around projects suited to the abilities and interests of 
 the pupils for whom it is intended and adapted to the successful use 
 of well recognized methods of teaching and to the needs of the state 
 and community. These projects, according to the nature of the sub- 
 ject matter, lend themselves to one of the following types: 
 
 Type 1. 
 
 "lu which the purpose is to embody some idea or plan in exter- 
 nal form, as building a boat, writing a letter, presenting a play : 
 
 Type 2. 
 
 "In which the i^urpose is to enjoy some aesthetic experience, as 
 listening to a .story, hearing a symjjhouy, appreciating a picture: 
 
 TvPK 8. 
 
 In which the purpose is to straighten out some intellectual dif- 
 ficulty, to Holve some problem, as to find out whether or not dew falls, 
 to ascertain howNew York outgrew Philadelphia. 
 
 TvPK 4. 
 
 In which the jiurjid.'^e is to obtain some item or degree of skill 
 or knowledge, as learning to write graile 14 on the Thorndike Scale, 
 learning the irregular verbs in French. . . . Some teachers in-
 
 Ill 
 
 deed may not closely discritniuate between drill as a project and drill 
 as a set task, althouffh the results will be markedly different." 
 
 "It is at once evident that these jjroupinffs more or less overlap 
 and that one type may be used as means to another as end. It may 
 be of interest to note that with these definitions the project method 
 logically includes the problem method as a special case. The value 
 of such a classiiicatiou as that here given seems to me to lie in the 
 ligfht it should throw on the kind of projects teachers may expect and 
 on the procedure that normally prevails in the several types." — 
 Kilpatriek. Teachers College Record, September, 1918. 
 
 This Course of Study is in no sense a finished product. It is a 
 record of past achievement and a standard of present attainment. It 
 is intended also to be a guide post for further progress. As the 
 quality of the class room instruction improves by means of this course, 
 the course should likewise be improved in the nature of the subject 
 matter and in the effectiveness of the teaching method. For this 
 purpose the suggestions and criticisms of teachers, principals, and 
 supervisors will be requested from time to time.
 
 ARITHMETIC. 
 
 AIMS. 
 
 To euable one to do llie onliiiary coiiiputiiiy re(iuii-eil in cormnoii 
 
 busiuess. 
 To give arithmetical kiiowledye that tit.s into the real situations 
 
 of home, shop, farm, or business. 
 To acquaint pupil with the simplest and shortest methods used iu 
 
 busiuess. 
 
 ATTITUDES TO BE DEVELOPED. 
 
 "a tendency not to be satisfied with guessing or approximation, 
 but to insist on finding out through the iise of figures all essential 
 matters involving numerical values. 
 
 "Standards of busiuess accuracy that will result in the keeping of 
 an accurate account of all personal or household receipts and expendi- 
 tures. This will make possible a proper adjustment of expenditure 
 to income and also a right balance among the different objects for 
 which money is spent. 
 
 "Unwillingness to rely on general estimates or rough approxima- 
 tions with reference to projects planned, as. improving a home or a 
 farm, taking a trip, investing in an automobile, etc. 
 
 "Insistence on detailed and accurately kept records of profits or 
 losses from the different enterprises of farm, shop, or busiuess. 
 
 "The development of such a sense of values aud the inevitaVde 
 logic of figures as will render one proof against the get-rich-quick 
 schemes ]»lanned by unscrupulous i)romoters to catch those who 
 tlirough ignorance of Vjusiness believe wealth to be attained V)y some 
 kind of magic. 
 
 'a sense of pleasure and satisfaction in the use of figures and in 
 the certainty which comes from their wise application to one's affairs." 
 Hetts: Classroom ^Methods and Management. 
 
 BRIEF SURVEY THROUGHOUT THE GRADES. 
 
 (rKAI)K I. 
 
 Xum])er work in this grade is taught inciueutally wliere need 
 arises in games or activities. 
 
 GRADE II. 
 
 NOTATION AND NUMERATION. Counting, rea<liug aud writing 
 numbers to 100. 
 
 ADDITION. The ad<lition combinations iu which the sum is less 
 than 10: svmV)ols + aud ^=.
 
 SUBTRACTION. The reverse of the adilition combinations: the 
 
 sytiihdl — . 
 
 MEASURES. The (juarter. half-(luHar, yallou, ijouiid, dozen, half- 
 • lozen. huuse miiiiln'r. the date. 
 
 (4KADE II— A. 
 
 NOTATION AND NUMERATION. Counting, reading and writing 
 mimV>ers to 200. IJoman numerals to XII. Counting by 3's, 4's, and ti's. 
 
 ADDITION. The average and hard combinations. Column ad- 
 dition. 
 
 SUBTRACTION. The reverse of the addition combinations. 
 
 MULTIPLICATION. The tables of 2, 5, and 10. The symbol X as 
 "times."' 
 
 DIXISION. The reverse of the multiplication tables. The 
 symbol-^. 
 
 MEASURES. The names and number of the days of the week. 
 The names and number of the months in the year. The hours on the 
 clock face. The signs § and c. 
 
 FRACTIONS. 2, 3 and 5 of numbers exactly divisible. 
 
 GRADE III— B. 
 
 NOTATION AND NUMERATION. Reading and writing numbers 
 to 5000. Roman numerals to XX. Counting by 2 and 4 to 100. 
 
 ADDITION. The very hard combinations. Two and three place 
 column.s iuvidving carrying; not over tive addends. Mixture of one, 
 two, and three jdace columns. Drill on endings. 
 
 SUBTR.^CTION. Easy subtraction and one step borrowing. 
 Work on entlings. 
 
 MULTIPLICATION. The tables of 3 and 4. Multiplication by 
 one-place multipliers. 
 
 DUISION. Division as the reverse of multiplication. No carry- 
 ing. The long division brace. 
 
 MEASURES. The ounce and yard. Time; the hour, half-hour, and 
 quarter-hour. 
 
 FRACTIONS. 3 and \ of numbers which are exactly divisible. 
 Addition and subtraction of very simple fractions, involving no 
 reductions. 
 
 (iRADE III— A. 
 
 NOTATION AND NUMERATION. Counting by 2's to 84; 8's to 96; 
 9'8 to lOS. Reading and writing numbers to 100,000. The use of the 
 comma as in 14, 5U7. 
 
 ADDITION. Adding four and five place columns of six addends. 
 Rapid drill. Speed and time tests. 
 
 SUBTRACTION. Two-step borrowing. 
 
 MULTIPLICATION. The tables of 6 and 7. Multiplication by 1 
 and 2 place niultii)lierH, the jjroiluct to contain less than six digits. 
 
 DIV'LSIO.N. Short <livision with borrowing. Numbers not 
 exactly divisil)le.
 
 DENOMINATE NUMBERS. Oue-half and one-quarter iuch. The 
 thermometer. 
 
 FRACTIONS. I and ] of numbers to 100, iiicludiiiti' numbers not 
 exactly divisible. Adding and subtracting simple fractions and mixed 
 numbers. 
 
 GARDE IV— B. 
 
 NOTATION AND NUMERATION. Reading and writing numbers 
 to 1,000,000. Counting to 100 by ll's and 12's. 
 
 ADDITION. Drill. Addition of columns containing 7 addends. 
 
 SUBTRACTION. Three-step borrowing. Rapi<l drill. Speed and 
 time tests. 
 
 MULTIPLICATION. Tables of S and 9. Multiplication by two and 
 three digits. Zero in the multiplier. 
 
 DIVISION. As the reverse of multiplication. Rapid drill. ISpeed 
 and time tests. 
 
 MEASURES. The ton, peck, and bushel. 
 
 FRACTIONS. I and g of any two place numbers. Addition and 
 subtraction of simple fractions and mixed numbers with no reductions. 
 
 DECIMALS. U. S. money. The correct use of "and." 
 
 GRADE IV— A. 
 
 NOTATION AND NUMERATION. Counting to 100 by 15, 20 and 
 25. Roman numerals L, C and D. 
 
 ADDITION AND SUBTRACTION. Time tests. 
 
 MULTIPLICATION. Tables of 11 and 12. No multipliers over 
 four places. Drill for speed. 
 
 DIVISION. Long division, using a two place divisor. Zero iu 
 the divisor. Drill for speed. 
 
 MEASURES. Recognition of the square inch, square foot, and 
 square yard. 
 
 FRACTIONS. i\ and i\ of any two place numbers. Addition and 
 subtraction of fractions whose denominators may be seen by in- 
 spection. 
 
 DECIMALS. Addition and subtraction of U. S. money. Multipli- 
 cation and division of U. S. money by integers only. 
 
 GRADE V— H. 
 
 FUNDAMENTAL PROCESSES. Drills for speed and accuracy. 
 Division by three i)lace <li visors. 
 FRACTIONS. Reduction to— 
 
 a. Higher terms 
 
 b. Lower terms 
 
 c. An integer or mixed number 
 
 d. Like denominators.
 
 4 
 
 ADDITIUN AND SUBTRACTION OF— 
 
 a. Similar fractious 
 
 1). Dissimilar fractions 
 
 c. Mixeil numbers 
 
 <1. Applicatiou to problems of the chiliVs exi)erieuce. 
 MEASURES. .Miuutes iu the hour aud seconds in the minute. 
 DECIMALS. Expression as decimals of fractions having denom- 
 inators of 10, 100. or 1.000. The decimal point in U. S. money. 
 
 GRADE V— A. 
 
 FUNDAMENTAL PROCESSES. Drills for speed aud accuracy. 
 FRACTIONS. Multiplication aud Division of: 
 
 Fraction by a whole uumber. 
 
 Whole numlter by a fraction. 
 
 Fraction by a fraction. 
 
 Mixed uumber by a whole uumber. 
 
 3Iixed uumber by a mixed number. 
 
 Application to problems of the child's experience. 
 MEASURES. Tables of linear, dry aud liquid measure, avoirdu- 
 pois, time, aud V. S. money. 
 
 DECIALALS. Drill on U. 8. money. 
 
 PERCENTAGE. Aliquot parts of a dollar, S .25, § .50, 8 .75, S .10, 
 8 .20. 8 .1*2 2, aud their fractional equivalents. 
 
 GRADE VI— B. 
 
 DENOMINATE NUMBERS. Addition, subtraction, multiplica- 
 tion, division, aud oue step reductions. 
 
 DECIMALS. Addition aud subtraction, multiplication, aud 
 division, not over four places. Notation aud numeration to .000. 
 
 PERCENTAGE. Very simple table. 
 
 (iRADE VI— A. 
 
 DENUMINATE NUMBERS. 
 
 Two step reductions. 
 
 Tables of Square and Cubic Measure. 
 
 Area aud perimeter of rectangle. 
 
 Area of triangle. 
 
 Terms: Rase, altitude, perimeter, aud area. 
 
 lioard Measure. 
 
 C'ajiacity of bins, coal hauling, and wood selling. 
 
 IJegin simjde three step problems. 
 
 DECI.M.ALS. Notation and numeration to six places. 
 
 PERCE.N'T.AGE. Change decimals to percents. Contiuuation of 
 percentage tables from VI— R.
 
 5 
 GRADE VII— B. 
 PERCENTAGE. 
 
 Commercial Discount. 
 Profit and loss. 
 Commission. 
 Budgets. 
 Equations. 
 
 GRADE VII— A. 
 PERCENTAGE. 
 
 Taxes. 
 
 Duties or Customs. 
 
 Insurance: Fire, Marine, Life. 
 
 Interest: Simple, Compound. 
 GRAPHING. 
 
 GRADE VIII— B. 
 
 INVOLUTION. Squares of numbers to twenty-five, cubes of 
 numbers to twelve. 
 
 EVOLUTION. Square root. Meaning of power, exponent, square, 
 square root, and radical sign. 
 
 MENSURATION. Areas of rectangle, rhomboid, trapezoid, and 
 triangle. Circle, circumference, radius, diameter, arc, and sector 
 defined. Area. 
 
 GRADE VIII— A. 
 
 MENSURATION. Lateral surfaces as applied to prisms, cylind- 
 ers, pyramids and cones. Volumes of prisms, cylinders, pyramids 
 and cones. Excavations, brick, stone and concrete work. 
 
 BANK DISCOUNT. 
 
 BUSINESS ARITHMETIC. Common business forms, bills, receipts, 
 statements, bills of lading; itersoual accounts, debit, credit and bal- 
 ance, inventories; sending and investing money. 
 
 GRADE IX— B. 
 
 COMMERCIAL ARITHMETIC. Addition, subtraction, multipli- 
 cation and division of whole numbers, fractious and decimals. 
 PERCENTAGE. 
 
 GRADE IX— A. 
 
 COMMERCIAL ARITHMETIC. Banking, Iu.-urance, Commission, 
 Customs, Duties, Taxes, Stocks and Bonds, Business Forms.
 
 6 
 GENERAL DIRECTIONS AND DISCUSSION OF PRINCIPLES. 
 
 Identical Forms for all Grades and Buildings. 
 
 Ecouomy requires that in the rnechanies of numbers there should 
 he uniformity of forms and processes throughout the school system so 
 that individual children or classes upon being transferred or promo- 
 ted may not lose time or energy in changing methods of manipulation 
 to ctmform to different practices. The following processes suggest- 
 ed in |)art by the C'onnersville Mathematics Course; Brown and Coff- 
 man: How to Teach Arithmetic, and recent arithmetics, are recom- 
 mende<l for use by all pui)ils ami teachers in the grades in which they 
 occur. If a child already handles other correct methods easily, it is a 
 doubtful jiolicy to require him to change. 
 
 ISE VERY SIMPLE FORMS forthe statement of written problems. 
 
 55 books 
 25 books^ 
 ~80 books 
 
 IX .ADDITION always begin at the bottom of the coUmm in add- 
 ing, adding in what is carried. The child says, "Three, seven, twelve. 
 Write two and carry one. One, five, seven, eight. Write eight. Six^ 
 ten, thirteen. Write thirteen. The sura is 1382." Check by adding 
 down. This is advised because many examples in arithmetics are ar- 
 range<l to jtrovide for the recurrence of certain combinations if added 
 from the bottom uj). 
 
 315 
 424 
 643 
 
 1382 
 
 I.\ SUBTRACTION the child says, "Seven hundred eighty-two less 
 hree hun<lred sixty-seven. Take one from eight to put with two and 
 make it twelve. Seven ami five are twelve. Write five. Six and one 
 are seven. Write one. Three and four are seven. Write four. 
 The remainder is 415." 
 
 7S2 
 367 
 415 
 
 IN MILTIPLICATION. the child says "Five 6's are thirty. Write 
 zero. Five two's are ten and three are thirteen. Write three. Five 
 5'8 are twenty-five and one is twenty-six. The product is 2630. 
 
 526 
 5 
 
 2<i3()
 
 IN DIVISION, the child says,"Six's iu fourteen, two. Two G'sare 
 12. Six's iu 28, three. Three tl's are IS, etc." Use the same form 
 for both long or short division. 
 
 239 (>218 
 
 6)1484 2)1248(5 
 
 12 . 12 
 
 23 ~~4 
 
 18 4 
 
 54 ~~8 
 
 54 2 
 
 16 
 16 
 
 IN ADDITION of fractions proceed as follows: 
 
 1 4 o 1 o 4 
 
 i=A 5i=5A 
 
 l=r\ 2|-2t\ 
 
 10. 1 7 03 5 - 1 1 
 
 12— lo 6ii— <ii 
 
 IN SUBTRACTION of fractions proceed as follows: 
 
 3 — n 
 T — T"2 
 1 — 4 
 ^ — T2 
 
 IN MULTIPLICATION of fractions proceed as follows: 
 2 7 7 
 
 38 12 
 4 
 
 IN MULTIPLICATION of mixed numbers, proceed as follows: 
 
 8 
 
 15 13 89 
 7i X 2 1=— X — =— = 1 9i 
 2 -5 2 
 
 IN DIVISION of fractions procee<l as follows: 
 
 2 
 
 5 4 10 
 (a) t-|=-X-=-=U 
 
 639 
 3 
 
 11 
 
 77 4 44 
 
 (b) 25|-1|=-X— =-=14f 
 3 7 3
 
 8 
 11. Ill MLLTIPLICATIOX of (leciinal fractious proceed as follows: 
 
 2.2 5 
 2.5 
 
 1125 
 450 
 
 5.6 25 
 
 12. In DIVISION of decimal fractions proceed as follows: 
 
 85U.2 
 .25). ST. 55 
 
 18. In simple interest proceed as follows: 
 
 (a) Interest on 810U at 6°= for 4 years. 
 
 m^ 6 4 
 
 X — X— =824 interest. 
 
 1 100 1 
 
 (1») Interest on 8100 at 5 ° for 2 years, 6 mouths. 
 
 8^00 5 30 
 
 X — X— = 812.50 iuterest. 
 
 1 ;00 12 
 
 (c) Interest on 8100 at 6 ° for 1 year, 2 mouths, 15 days. 
 
 87 
 8,100 () ^35 
 
 X — X — =87.25 interest. 
 
 1 ;00 ^60 
 
 (30 
 
 12 
 
 Checks. 
 
 Although it is true that accuracy in arithmetical processes is 
 larfrely the result of loug, continued practice, yet eveu accountants of 
 great e.xiterience admit a liability to error and seek to verify their 
 computations hy some check. Children, too, feel a strong- desire to 
 know whether or not their solutsons are correct and if no other means 
 than the <lecision of the teacher he provided, show such a desire by 
 <|uesti()ns, a« "Is my answer right?" "What part is wrong?" 
 
 It is, then, to give children the satisfaction and self confidence 
 which comes from knowing how to determine whether or not that 
 which one does is correct, and to satisfy the demand of the business 
 world for accuracy, that a practical metliod of verification should be 
 taught. After such a check is once taught, the child should use it 
 until it becomes entiielv habitual f)r automatic.
 
 9 
 
 Some of the checks most commouly used au<l believed to he of 
 greatest heuetit in the elimination of errors are listed helow: 
 Estimatin<)- results to avoid absurd errors. 
 Estimating' the ai)j)roximate field in which the answer should 
 lie before the problem is attacked is very helpful in doinj; away 
 with ludicrous errors and establishing the sense of numerical 
 values. 
 
 Ex. A man earns §2.50 a day. What are his earning-s in a 
 week? The child at once sees that the amount of his 
 earnings must be in the neighborhood of §12. This es- 
 timation precludes error in placing decimal i)oint. 
 
 What is the length and width of our schoolroom? 
 Children should judge the length and width first with 
 the eye, and then compare their estimation with actual 
 results. 
 
 How many rods long is the walk in front of the 
 school? Children estimate the distance by walking, veri- 
 fying later by actual measurement. 
 
 Checks for the four fundamental operations. 
 
 ADDITION: Checked by adding digits in the reverse 
 order. Be careful, however, in adding down, that new and 
 strange combinations are not encountered. It is V)etter to 
 delay the iuti'oductiou of this check until all the combina- 
 tions have been learned. 
 
 65 65 
 
 24 24 
 
 Add uj) iq Add down ig 
 
 13 13 
 
 121 121 
 
 SUBTRACTION: Check (l) by adding difference to the 
 
 ■ subtrahend to form the minuend or (2) subtracting result 
 from minuend to obtain subtrahend. 
 
 Ex, Check 1 Check 2 
 
 924 689 924 
 
 Subtract_235 Add_235 Subtract 6S9 
 
 689 924 235 
 
 MULTIPLICATION: Checked (l) by dividing product 
 by multiplicand to form multii)lier or (2)l)y <lividing i)roduct 
 by multiplier to form multiplicand. 
 
 Check 1 Check 2 
 
 Ex. 23 656 
 
 656 
 
 Multiply 23 
 1968 
 1312 
 15U88 
 
 65(5) I •'><»•'*>' 
 
 23)15(IS,S 
 
 1312 
 
 138 
 
 l!Mi8 
 
 128 
 
 19(iS 
 
 115 
 
 
 138 
 
 
 138
 
 1(1 
 
 DIVISION: (iK-cki'il li\- inultiplyiua- quotieut and divisor 
 to furiii <livi<leii<l. 
 
 4:{ Check 
 
 1 :))()4.') 48 
 
 )i() ^Iultii»ly !•") 
 
 45 •il-'^ 
 
 45 48 
 
 645 
 
 Other Checks. 
 
 Kevifwiuy: .stei).>< in prolileni!5 iuvulviny: small uuml»ei's. 
 For reasuiiiiiff problems: 
 
 IJewordiug to clarify purpose of problem. 
 Ex. (l) How much more does coifee at 25 lbs. for 
 >!9.50 cost thau coffee at 33c a lb. 
 (2) IJewordiutj: 
 
 One grade of coffee sells at §9.50 for 25 
 lbs. Another grade sells for 83c. Which 
 costs the more? 
 Statiug the same type of problem iu simpler nu- 
 merical terms. 
 Ex. (l) ^Ir. Browu wishes to buy a house. He 
 must give his agent 10 per cent com- 
 mission to sell a home worth 824,000. How 
 much money will Mr.' Browu get from the 
 sale? 
 (2) Reducing to simpler numerical terms: 
 
 You sell strawberries for Mrs. Brown. She 
 tells you, you can have 10 per cent of the 
 money received from selling the berries. 
 If you sell S8.00 worth one morning, how 
 much money will you give Mrs. Brown at 
 the close of the morning? 
 Solving by another method: 
 
 A certain store takes in 8100,000 <luring a year. Its ex- 
 penses were 812,000 for rent, 82.S,(JUU for salaries. 850,000 for 
 stock and 82,000 for heat and light. What were the profits 
 for the vear? 
 
 Method (1) 
 
 (2) 
 
 812,000 
 
 8100.000 
 
 50,000 
 
 Subtract 12.000 
 
 2.000 
 2.s.0ll(i 
 
 8 8S.00O 
 Subtract 50,000 
 
 892,000 
 
 8 8S,000 
 
 81(10.(101) 
 
 Suljtract 2.S.0UO 
 
 1)1!. (MMI 
 
 8 10,000 
 
 8 .S,00() Piotit 
 
 Subtract 2,000 
 
 8 S,000 Profit
 
 11 
 
 Inaccuries to Be Avoided: 
 
 See JJiowTi and ('oftiiian: IIow to Teaeli ,\ritliMietic-. 
 Inaccurate: 3+4=7+5=12 X2=24 
 Accurate: 3+4=7; 7+5=12; 12 --2=24 
 
 Inaccurate: 4| = i^.2 Accurate: 4|=4/2 
 
 0-2 12 O ., — 6 1 2 
 
 _L2_ 
 
 ll 1 
 
 9i^=10}i lUl' 
 
 Inaccurate: 2 >:$50=100 Accurate: 2X$50=$100 
 2x50=8100 2-50=100 
 
 8100^4=25 §100 : 4=!!i25 
 
 100-^4=825 100 : 4=25 
 
 8100-^84=825 8100 : 84=25 
 
 Inaccurate: 4 ft.X5 ft. =20 sq. ft. 
 Accurate: 4/5 A 1 sq. ft. =20 sq. ft. 
 4X5 sq. ft.=20 sq. ft. 
 
 Inaccurate; 27 cu. ft. ^9 sq. ft. =3 ft. 
 Accurate: 27 cu. ft. -^9 cu. ft. =3 
 
 Problem: Two-fifths of a number etiuals 12. Find the 
 
 number. 
 Inaccurate: 5 =the number Accurate: 5 of the nuiiiber=the 
 
 number 
 g=12 I of the number =12 
 
 5=i of 12=6 I of the number=.', 
 
 of 12=6 
 |=.5X6=30 i of the number= 
 
 • 5X6=30 
 Inaccurate: 100 per cent= Accurate: 100 per cent of the 
 
 100 number=the number 
 
 Inaccurate: 15° =1 hour of Accurate: 15° correspond to 1 
 time. hour of time. 
 
 Principles of method in teaching Arithmetic as derived form scientific in- 
 vestigation from the 18th Year Book of the National Society for the 
 Study of Education. 
 
 In teachinfj the numi)er concept, which is fundamental in arith- 
 metic, purposeful experience with concrete objects should be i)rovided. 
 Counting and measuriuy are two of the most fruitful forms of experi- 
 ence. 
 
 In jreneral, the meaning should be taught before the word or other 
 
 symbol is given to the child. This aj)plies to the number symbols 1. 
 
 2, 3, 4, etc., and to the technical words of arithmetic such as, "athl." 
 
 'subtract," "foot,'" 'V«ii"'li' l'**"i"li' J^^i"-' "how inucli."' and the like. 
 
 Learning the tables is a matter of memorizing and the rules for 
 memorizing ajiply. 
 
 The child should understand the meaning of the comliina- 
 
 tious which he is memorizing.
 
 12 
 
 Attentive rei>etitious are ueoessary to fix the associations iu 
 
 learniiifr the tables. These repititious may be given either in 
 
 ischitetl ilrill or in the use of these number facts in the doing of 
 
 exainjiles. 
 
 For jiermanent memorizing: [whit-li i.s desired in tliis case} 
 
 tlie repetitions of drill must be carrie<l beyond the i)()int where 
 
 iminetiiate recall is just barely jjossible. 
 
 The learning should be done under some pressure or con- 
 centration. 
 
 Memorize the number facts in t^roups, not one fact at a time. 
 
 In learning the tables, the different combinations are not equally 
 ditHcult and the number of repetitions of the several combinations 
 should correspond to the degree of difficulty, the most difficult receiv- 
 ing the largest number of repetitions. 
 
 Tlie actual degree of difficulty of a combination to any one child 
 is an "individual iieculiarity". This condition makes it necessary to 
 suitjilement class drills by i)rovisions for individual practice. 
 
 In adilitioii and niultiplieatiou l)oth forms of each combination 
 should be taught. 
 
 In column addition, grouping digits to make 10 or some other 
 convenient number is not helpful. A variety of procedure is to be 
 expected, and the best results are obtained when pupils are urged to 
 work rajiidly but are allowed to choose their own methods. 
 
 The Austrian, or additive, niethod of subtraction is not superior 
 to the "take-away" method. 
 
 In "borrowing" it is better to ijicrease the subtrahend by one 
 than to decrease the minuend. 
 
 The Austrian, or multiplicative method of division is superior to 
 the direct association method in the initial stages of learning. 
 
 The Austrian method of placing the decimal point in the division 
 of decimals is more efficient than the traditional method. 
 
 Most errors belong to recurring types. The most frequent of 
 these types should receive special em])hasi8 so that they can be elim- 
 inated. 
 
 Pupils should be taught to use an abbreviated phraseology. 
 
 After the initial stage of practice, drill upon the fundamental 
 combinations should be given by means of examples. 
 
 The ijeriod of practice should be from 10 to 15 minutes. 
 
 Children's knowledge of their previous performances, combined 
 with the desire to surpass those records, is the greatest factor con- 
 tributing to imj>rovement. 
 
 A preliminary practice of five minutes at the beginning of the 
 recitation serves as a "mental tonic." 
 
 In the ojierations of arithmetic, emphasis should be i)laced upon 
 rapi<l work rather than ui)on accuracy. 
 
 I'upils Itelonging to the same class have been shown to differ 
 widely in achievement. This condition makes it necessary to provide 
 f<ir individual instructions, 
 
 'Fhe Courtis Standard Practice Tests are superior to Thompson's, 
 miiiiinum essentials.
 
 13 
 
 Arithmetical abilities are specific, and explicil traiiiiiiir iimst be 
 provided tor each one. 
 
 The development of the abilities of a pupil to do different types 
 of examples is frequently not uniform. To meet this situation, dia<r- 
 uosis and corrective instructions are required. 
 
 Arithmetical study and ])ractice which is motivated by "itractical" 
 !)roblems produces results sui)erior to those secured by usiii','' tlie 
 problems in the textbook. 
 
 [Systematic measurement of the results of teaching l)y standard- 
 ized tests produces a hijiher degree of efficiency. 
 
 Pupils need to be taught the meaning of technical terms that are 
 used in the statement of problems, as a prerequisite for reasoning in 
 solving problems." 
 
 Provision for Individual Differences. 
 
 The usual class instruction in arithmetic does not meet the needs 
 of individuals. Classes conducted in a manner similar to those de- 
 scribed by Monroe are all too common: "Frequently the writer has 
 visited classes in arithmetic which were being drilled upon the fund- 
 amental operations. A fairly uniform procedure was followed. The 
 same example was dictated to all of the pupils, regardless of whether 
 they needed drill upon this particular type of example or not. Nat- 
 urally some pupils finished very quickly, and, as they waited for their 
 classmates to finish, there was a tendency for them to become dis- 
 orderly—a perfectly natural tendency. When a majority of the class 
 had finished the example the teacher stopped the work and read the 
 correct answer. The process was then repeated. The result was 
 that those pupils who worked slowly completed few, if any, examples 
 during the entire period, and, therefore received little satisfactory 
 drill. The bright pupils spent a considerable proportion of their 
 time waiting on the other members of the class, and i)robal)ly did not 
 need the particular kind of drill which they received." 
 
 lie gives the following suggestions as to how this condition may 
 be remedied: "instead of dictating only one example at a time, the 
 teacher can dictate several, and stojj the work as soon as a few of the 
 faster workers have finished. The slow-working pupils will have 
 some examples finisned. 
 
 "The teacher must recognize that the rate at which the jMipil 
 performs the o])erations is important, as well as the accuracy. This 
 means that in teaching, the teacher must obtain a measure of the 
 pupil's speed, as well as a measure of his accuracy. If examjtles are 
 dictated in groui>s, and the work stoi)ped as suggested in the above 
 paragraph, the number of examples which the \ni\)i\ does during llu* 
 class period is a measure of his rate of working. The i>er cent cor- 
 rect is a measure of his accuracy. 
 
 "The instruction can be made still more eftecti\c it the teacher 
 will prei)are a numl)er of sets of exami)les, each set being confined to 
 examples of the same type. These sets of exaniiiles should be written
 
 u 
 
 oil t-anls. Then, instead of dictatiuff examples, the teacher can dis- 
 tril»ute tlie cards and have the pupils copy the examples from the 
 car.ls. If the teacher studies the needs of her pupils, it will be pos- 
 sihle fi>r her to distribute the cards so that each pupil will have the 
 type of example ujiou which he needs practice. The pupil is probably 
 injured liy beini^ i-e(|uire<l to practice upon the wrong type of example 
 and. hence, it is very imjiortant that each i)ui)il be gfiven the type of 
 examiile upon which he needs practice." Educational Tests and 
 ^leasurements. ]». .t.i. 
 
 Although problem work cannot be made as definitely individual, 
 still, a careful classitication should l)e made of failures: children with 
 a lack of knowledge of the processes; those who need drill; those who 
 lack ability in reasoning or computation. The proper remedial mea- 
 sures may then be taken by devoting some of the class time for special 
 \vi>rk with the above mentioned groui)S. 
 
 "In aiming to secure good quality, the teacher will not allow the 
 subject to take i)recedence in his mind over the jtupils. A true teacher 
 is never merely teaching a subject. lie is always assisting a human 
 being by means of a subject to grow and adapt himself to 
 his surroundings. It is then, after all, the pupils' interest, success, 
 growth, improvement that the teacher has in mind when he sets up 
 high stan<lards and puts into operation stimulating incentives. 
 
 If, however, these produce failure and discouragement in any 
 in<Iividual. the standards and incentives are surely not accomplishing 
 the most useful results. Indivi<luals and classes differ in their mathe- 
 matical aptitude. They come to a teacher with different kinds of 
 jireparation. If instruction is to be given from the standpoint of the 
 l»upils, it must first be <liscovered what each individual knows and 
 also how he knows it. Therefore, standards should be adjusted to 
 each class and to each individual. For the more able pupils special 
 advanced problems should be assigned or offered as optional work for 
 which special credit is given. For the less able, special easier work, 
 should ]je arranged to secure the needed review and drill." Ken<lall 
 an<l .Mirick: "How to Teach the Fundamental Subjects." 
 
 ( )nly l)y eradicating individual weaknesses, will the class show 
 the greatest amount of progress, for it is usually only a few members 
 of the gr()U]> who ])ull down the whole class standard. (Graphing and 
 explaining the individual and class results in standard tests is a won- 
 derfid stimulant of individual effort. The degree to which personal 
 needs are met is tlie measure of educational efficiency. 
 
 Standard Practice Material. 
 
 Speed and accuracy in the funihunental ojierations may be ac- 
 «iuire<l oidy by means of fre<iuent drill projects. The fundamentals 
 should have the right of way in any grade. In the upper grades the 
 abilities of the pui>ils to <lo different types of exami)les are frecpiently 
 not uniform. To meet this sitiiation every teacher of arithmetic in 
 grades five to eight should check the abilities of incoming groups in 
 tlie four fuiidaiiiental oi>erations and diagnose the deficiencies of each 
 chilrl by the use of the standard tests of Woody or Courtis. Definite 
 practice material shouM be use<l to correct conditions discovered.
 
 15 
 
 Two sets of praftice material which serve the imrixise of testiug 
 and ])ractit'iut): in l»oth speed and accuracy are: 
 
 Courtis Standai-d l*ractice Tests, C'anI Cabinet FMition, Cabinet 1, 
 World IJook Company. 2126 Prairie Avenue, Chicafro. 
 
 Studebaker Economy Practice Exercises, Set B-One, Scott- Fors- 
 niau and Company, 528 South Wabash Avenue, Chicatjo. 
 
 These are particularly valuable in grades five and six. A few 
 minutes should be yiven to this work every day, each child at his own 
 rate. An individual may be excused for other activities when he 
 reaches the grade standard. These i)ractice tests in themselves con- 
 stitute the true drill project material with all the characteristics of 
 ' puri)osefur' activities when correctly used. 
 
 Children should be lead to note their progress during a given in- 
 terval of time between two tests rather than any one unrelated score. 
 Team work and graphing of scores will emphasize this. 
 
 At the cdose af each semester, standard tests in the fundamental 
 operations should be given. 
 
 Standard Tests in Arithmetic. 
 
 THE WOODY ARITHMETIC SCALES. 
 
 Woody has organized a set of four scales, one for each of the 
 fundamental processes. His ''fun<lamental idea was to derive a series 
 of scales which would indicate the type of problems (exam])les) and 
 the ditticulty of the ]»rol)lems (examples) that a class can solve cor- 
 rectly The ditticulty of each exami)le has been determined 
 
 and the examples of each scale are arranged in oi'der of increasing 
 
 difficulty The score of a pupil is a statement of the particular 
 
 examples which he has done correctly. The score of a class is the 
 degree of difficulty of the example which was done correctly by just 
 50 per cent of the class. Ability as measured by these scales means 
 simply that certain types of examples can be done correctly and that 
 certain other types cannot be done correctly. The speed at which the 
 examjdes can be done is not includeil in the meaning of ability." 
 Monroe: Educational Tests an<l Measurements, p. 29. 
 
 Following are the medians as given by Wuddy foi' these scales: 
 
 SEIMKS A — ACCrUACV. 
 
 GRADE ADDITION SUBTRACTION MULTIPLICATION DIVISION 
 
 Second 6.S 5.1 
 
 Third 14.5 11.2 4.7 5.S 
 
 Fourth IS.B 15.() 11.1 9.S 
 
 Fifth 28.1 2U.4 1S.8 16.4 
 
 Sixth 29.7 24.9 26.1 28.7 
 
 Seventh 82.4 2S.5 8U.0 27.4 
 
 Eighth 88.9 81.7 82.9 :!(».! 
 
 COURTIS STANDARD RESEARCH TEST. 
 
 This series consists of four tests, printed on tour consecutive 
 liages. They are suitable for a general survey of the abilitii-s of pujtils 
 to perform the operations with integers. 'I'he following are sami>les:
 
 16 
 
 TEST NO. TWO SUBTRACTION. 
 
 "This test cousists of tweuty-four examples, each iuvolviuo- the 
 same miniber of subtractions. The followiug are samples. Time al- 
 lowed. 4 minutes. 
 
 10779.1491 750SSS24 91500053 S7939983 
 
 77197»i-i9 ■')74(M>394 1!)9(il5(>3 7-22U731H 
 
 TKST No. THKEE — MILTII'LICATION. 
 
 This test consists of twenty-four examples of this type. Time al- 
 lowed, 6 minutes. 
 
 S246 3597 5739 2648 9537 
 
 29 73 85 46 92 
 
 In marking the test papers, wliich is done by the use of a printed 
 answer card wliich is run along across the page, no credit is given for 
 examjdes i)artly right nor for examples partly completed. A pupil's 
 score is the numljer of examples attempted and the uumlier right. 
 This simple plan of marking the papers insures uniformity and ac- 
 curacy. 
 
 Each of the examples of a test calls for the same number of 
 operations under approximately the same conditions. This makes 
 the examples of each test approximately eq\ial in difficulty. Any ex- 
 ample of the addition test, say the seventh, is just as difficult as any 
 other, say the second. Thus, the tests consist of twenty-four equal 
 units, just as a yardstick consists of thirty-six equal units (inches). 
 The measure of a pupil's ability is represented by the distance he ad- 
 vances along the scale in the given time, i. e., by the number of ex- 
 amples done and by the per cent of these examples which have been 
 done correctly. 
 
 Since an example of one of these tests is defined as so many 
 operations under certain conditions, it is possible to construct other 
 tests equal in difficulty. Four forms have been constructed. This 
 makes it possible to use a different form when the tests are given a 
 secontl time.'" Monroe: Educational Tests and Measurements, p. 23-25. 
 
 These are the medians obtained by Courtis: 
 
 SEKIES B — SPEED. 
 GRADE ADDITION SUBTRACTION MULTIPLICATION DIVISION 
 
 Third 4 5 
 
 Fourth 6 7 6 4 
 
 Fifth 8 9 8 6 
 
 Sixth 10 11 9 8 
 
 Seventh 11 12 10 10 
 
 Eighth 12 13 11 11 
 
 SERIES B — ACCUEACV. 
 <5RADE ADDITION SUBTRACTION MULTIPLICATION DIVISION 
 
 Third 41 49 
 
 Fourth 64 80 67 57 
 
 Fifth 70 83 75 77 
 
 .Sixth 73 85 78 87 
 
 .Seventh 75 86 80 90 
 
 Eighth _. 76 87 81 91
 
 17 
 'THE STONE REASONING TEST. 
 
 "Stoue has worked out a reasoniufj;' test which has heen used in 
 several cities, and in a number of city school surveys, so that we have 
 rather definite standards as to Avhat may l)e expected from its use." 
 
 ACTUAL MEDIANS OBTAINED. 
 
 GRADE Butte, Bridgeport, Salt Lake Nassau, Lead, 
 
 1914. Conn., 1913 City, 1915. Co., NY, 1918. S. Dak., 1916 
 
 Fifth 2.2 t).l 3.7 ___ 4.5 
 
 Sixth 3.9 5.2 (j.4 4.5 6.1 
 
 Seventh 5.8 6.8 8.6 __. 9.3 
 
 Eighth 7.7 4.5 10.5 8.2 11.4 
 
 Tentative Standards suggested by Stone (1916) for his Reason- 
 ing Tests: 
 
 That 80 per cent or more of 5th grade pupils reach or exceed a 
 score of 5.5 with at least 75 per cent accuracy. 
 
 That 80 per cent or more of 6th grade pupils reach or exceed 
 a score of 6.5 with at least 80 per cent accuracy. 
 
 That 80 per cent or more of 7th grade pupils reach or exceed 7.5 
 with at least 85 per cent accuracy. 
 
 That 80 per cent or more of the 8th grade pupils reach or exceed 
 a score of 8.75 wnth at least 90 per cent accuracy. 
 
 The time allowance for the test is fifteen minutes. Stone's plan 
 for marking the test papers allows credit for examples partly right 
 and for examples which are not finished. The problem values have 
 been determined upon the basis of difficulty. It should be noted that 
 this plan for marking the test papers is not as simple as that em- 
 ployed for marking the test papers on the operations of arithmetic." 
 3Ionroe: Educational Tests and Measurements, p. 36, 37.
 
 IS 
 
 (iKADK I-J} au.l I-A 
 DIRECTIONS. 
 
 DIVISION OF TIME— All the arithmetic work of the t^-rade will 
 l>e oral. Only such writin<r of luunbers as is required ineideiitally in 
 the seiirintr of y:aines neeil l)e done. All the work of the {ji'ade will 
 lie eonerete. 
 
 The I-B grade will have no arithmetic work except iuei<lentally. 
 Do not give a separate period to the teaching of numbers in the I-A 
 grade hut teach the numbers in connection with the other work. 
 
 The following ways of teaching the number concept are recog- 
 nized as best: 
 
 Relating the number work to the chihrs experience thru the use 
 of objects. Dr. Suzzalo in his Teaching of Primary Arithmetic, pp. 
 49-5U. makes a plea against the very narrow range of objects used in 
 this work, or to (piote exactly: "It is too frequently the case that teach- 
 ers will treat the fundamental combinations with one set of objects, 
 e. g. lentils. In all the objective experience within that field there 
 are two persistent associations, 'lentils,' and the relation of addition. 
 It follows then, that in the effective use of this method the range of 
 material should be as wide as the teacher's ingenuity can make it. 
 
 Through rhythm. Songs which have marked rhythm may be 
 claitped. tajtited. or counted. 
 
 Through games, drawings, and construction work. 
 
 SUBJECT MATTER AND PROJECTS. 
 
 COUNTING: l>y I's incidentally where need may arise in games 
 or activities. 
 
 Projects: To count pencils or crayons for the class; erasers, seats, 
 chairs, papers needed for each row; pictures on the wall; beau bags; 
 boys in the room; girls in the room; boys and girls absent; etc. 
 
 To count ai)i)les, beads, objects of different colors and shapes, 
 inch Sfpiares or heavy cardl)oard strips cut from bristol board, toy 
 money, jjinl and quart measures, circles, and speer blocks of various 
 forms and sizes. 
 
 To jilay Hop Scotch, Dominoes; IJeast, Iiird, and Fish, etc. 
 
 To teach the sense of number thru the medium of play. 
 
 To tell the pages and lessons in the readers. 
 
 MEASURES: Cent, nickel, dime, dollai-, inch, foot, pint, (piart., 
 and recognition of circle and square. 
 
 Projects: liec(»gnize these measures in games: 
 
 Play store. 
 
 Measure the sand table and the sand in it. 
 
 Construct siinjile objects suggested by the industrial arts outline. 
 
 Optional Work. 
 
 Hea<ling and u riling numbers to 50. 
 (Vninting biicku ai-ds by I's from 50.
 
 19 
 
 Standards of Attainment. 
 
 See sul»jeft matter. 
 
 Bibliography. 
 
 J>r(i\vii and ("offinau: ITow to Teach Aiitliiiietic. C'bai). X. Pri- 
 mary Arithmetic. 
 
 Johusou: Education l)y Flays and (iames. 
 
 Klapper: The Teaching of Arithmetic, ('liaji. \'I. Teaching- tlie 
 Number Concept. 
 
 McMurray: Special Metho<l in Arithii.etic. C'hai*. III. ^letliod 
 in Primary Grades. 
 
 Suzzalo: The Teacliinjg: of Primary Arithmetic. C'lia]). \. ami \]. 
 Objective Work.
 
 (;ka1)K ii-r. 
 
 DIRECTIONS. 
 
 DIVISION OF TIME: Nearly all work should be oral. Three- 
 tourtlis of the work in this trraile shoiiM Ite coiu-rete. 
 
 REVIEW. Measures aii<l c•ouIltillL^ 
 
 MAIN TOPICS: Counting u]. to 100. addition and subtraction 
 eoinbinations, niea.sures. 
 
 NOTATION AND NUMERATION. In reading numbers, do not al- 
 low the ehiltlren to misuse and. '245 should be read, "two hundred 
 t\>rty-tive" not "two liuiidred and torty-tive." And indicates a deci- 
 mal point. This caution apidies also to the reailiufj: of the date. See 
 IV-I> Directions. 
 
 ADDITION AND SUBTRACTION. In all types of work, omit tlrill 
 ou facts and processes fully known, and stress those coml)inations or 
 operations of which the class is not sure. Provide special help for 
 individuals to overcome their ]»articular weaknesses. This involves 
 frequent testing to discover both class and indiviilual shortcomings. 
 
 Make the combinations automatic, thru memory work, permitting 
 no counting after the lirst experience. 
 
 Test the pupils frequently on the combinations committed to 
 memory. Do not permit the child to hesitate iu giving these combin- 
 ations. If he does so, it is better to state the result or call at once on 
 another ])upil. 
 
 Use {\)e column form only iu teaching addition and subtraction, 
 avoiding the use of plus and minus signs. Teach the recognition of 
 these signs only. 
 
 Teach the addition and subtraction combinations simultaneously 
 using the Austrian or "change making" methoil iu subtraction for the 
 sake of uniformity over the city. See page 21. 
 
 Smith iu the Teachers' College Record 1909, p. 46 says: "The 
 addition consists iu finding what number must be added to the subtra- 
 hend to make the minuend. Thus in thinking of 17 — 8 we think, 8 
 
 ami 9 are 17," writing down 9 Is the general plan the best 
 
 one? On the side of advantages we have: (l) It is the common meth- 
 od of making change (2) It avoids the necessity of making a 
 
 separate subtraction table. There is, therefore, an economy of time 
 and an increased efficiency in the very important subject of aildition. 
 (y) The facts of addition being used so much md^ often than those of 
 subtraction there is naturally an increase in speed and certainty when 
 we emjiloy the addition instead of the subtraction tal)le." 
 
 Avoid study jicriods which fix l)ad habits. 
 
 Play games for the game's sake, V)ut avoiil games which are too 
 largely jdiysical exercise and too little arithmetic. In playing games 
 keej) the groups small or the attention is scattered, and there is lack 
 of concentration. The rest of the pupils may be given seat work re- 
 lating to some other subject or to the number games they have just 
 jilayed.
 
 21 
 
 The child ren should keep their own scores, uo matter how long- 
 they take to do it. (4ive the necessary help where it is neede«l. 
 Later, a child's score may l>e disrej^arded if he cannot yet it in- 
 dependently. 
 
 SUBJECT MATTER AND PROJECTS. 
 
 NOTATION AND NUMERATION. Counting hy I's hack from 
 fifty; by lU's to lUU, beginning at U, 1, 2, 8, etc; by o's to lUO; by 2\s 
 to50; by Vs to 100. lieading and writing numbers to 100. 
 
 ADDITION. The addition combinations to be taught are as follows: 
 Group I 000 000 000 
 Very Easy 1 2 8 4 5 6 7 S 9 
 
 and their reverses. 
 Group'TI 111111112 2 8 4 5 
 Easy 2 8 4 5 6 7 8 9 3 2 8 4 5 
 
 and their reverses. 
 Arranged by Courtis according to dificulty. 
 Use these combinations in columns which involve no carry- 
 ing. 
 Teach the recognition of + and =. 
 
 SUBTRACTION. Involving the addition combinations taught in 
 this gra<le. Teach the use of the sign — . 
 
 MEASURES. Quarter, half-dollar, gallon, pound, dozen, half- 
 dozen, house" number, the date. 
 
 Optional Work. 
 
 lieading and writing numbers to 150, 
 Heading and writing Roman numerals to XII. 
 Counting by 2's to 100. 
 Table of 2's. 
 
 GENERAL PROJECT. 
 
 This is a suggested project which will involve much of the sub- 
 ject matter of the grade. Lest it should not provide sufficient drill 
 to fix the processes, minor projects are given as details un<ler each 
 topic of sul)ject matter. The latter are unrelated but might be an 
 outgrowth of a larger project similar to the one here given. 
 
 PLAYING STORE. For this project each child may make liis own 
 store as an Industrial Art problem. Interest in the project may be 
 increased by having different kinds of stores such as, a grocery, dairy, 
 toy shop, bakery, etc^ Toy money, articles of nierchaudi.»<e, casji reg- 
 isters and safes for these stores may be made during the Industrial 
 Art periods and seat i)eriods. 
 
 I>y utilizing this ])roject. direct apjilication of the various topics 
 in the II-I> outline may be made as follows: 
 
 COUNTING TO 100. ChiMren may count pennies in tlic cash 
 register. See that each store keei>er is provided with at least lUd. 
 
 Children may count eggs, balls, marbles, cookies and otlier arti- 
 cles of merchandise in stock. Counting should be done by I's, 2's, 
 aud 5's. Storekeeper may have a clerk to check up with him.
 
 READING AND WRITING NUMBERS TO 100. Each child mayuum- 
 l»er his store, chtiusiuy: a munl'er between oU to lUU. No two iiuinbers 
 i<houM he alike. These uuinV)ers may be written on cards and placed 
 4>n front of stores. The children will like to take turns in reading 
 the store numbers, usinj; comidete sentences as: ''The number of 
 .lohn's store is 9'i." 
 
 Price taifs may be iiiaile and attached to artit-les of merchandise. 
 Children shouhl be eucourageil to tind out correct jtrices of articles in 
 their stores. 
 
 Keadin": price tajjs will be of far more interest than just reading 
 numbers written on the board. Complete sentences should be used, 
 e. jr., "One poun<l of butter costs fifty-five cents," 
 
 MEASURES. Quarter, half-dollar. During- the process of making 
 his toy money, the child rai)idly learns to recognize the fliffereut 
 pieces. Customers may ask storekeepers to change a dime into 
 nickels or pennies, or a half-dollar into (piarters. 
 
 l*t>und. To develoi) the concejtt of a pound, butter cartons filled 
 with sand or pound candy boxes may be used. Children must be 
 taught what objects are sold by the pound. Encourage criticism of 
 such a statement as, "I want a pound of vinegar." 
 
 (Gallon. A gallon oil can in the grocery store and a milk can in 
 the dairy will give a correct concept. Children should learn what 
 l»roducts are sold Ijy the gallon. 
 
 Dozen and a half-dozen. After the class has been taught that 
 any twelve articles makes a dozen, a "dozen lesson" at the different 
 stores will be greatly enjoyed. Everything must be purchased by 
 the dozen or half-dozen. Cutomers must count over the articles to 
 make sure that the dealers have given the correct number. If a cus- 
 tomer should ask for one dozen sugar or tea the storekeeper must 
 correct the error or forfeit his position. 
 
 HOUSE NUMBER AND DATE. Each storekeeper may keep order 
 books. On each order slij) he may write the name and address of 
 each customer and the date. 
 
 Combinations. 
 
 ADDITION. Addition combinations will necessarily be used when 
 two articles are purchased e. g., a pencil for Ic and a ruler for 5c. 
 (5 + 1). When drill on a i)articular combination is required the chil- 
 dren may use the iiumbers of that combination in making their pur- 
 chases. The storekeeper may use the combinations in checking up 
 his stock, e.g., 5 cans of corn on one shelf plus 5 cans on another=10. 
 Two white V)alls +8 red balls = 5 balls. 
 
 SUBTRACTION. Since the Austrian Method is to be taught, the 
 store aftords splendid opi>ortunity for making jiractical apjilication. 
 Making «diange will fix su])tracti(m combinations both in the mind of 
 the storekcepei- and the customer. i)rovideil the same numbers are 
 used frei)uently. 
 
 Storekeepers should be encouraged to check up sales from time 
 to time as, "There were ten loaves of bread on the counter. Five 
 loaves are left. Tlit re were five loaves soM. (Five and what number 
 makes ten ?)
 
 •28 
 Correlation With Other Subjects. 
 
 LANGUAGE. Insist upon fiooil Kn<ili,sli from the rliiMii-n who are 
 l)layint>- store. Encourafje use of conijilete sentences. IJoth ciistoiiier 
 and dealer should criticise freely any errors in Enyflisli. 
 
 Occasionally let the children tell about i>layin{j st<jre for a lan- 
 iiuaye lesson. This may be in the form of a conversation lesson. 
 
 SPELLING. Storekeepers may wish to write labels for some of 
 their merchandise. A few simple words selected from the store vo- 
 cabulary will motivate the spelling lessons. The words learned (lur- 
 ing the spelling period may then be written on labels and pasted on 
 to the wares. 
 
 REMARKS. Occasionally some store may liold a sale. Ked letter 
 price tags may be attached to articles on sale. 
 
 As a reward for good work, some child may be made an inspector 
 whose duty it is to see that all stores are in proper order and that 
 price tags are correct. 
 
 Ordering by telephone will add interest to the store ami keep 
 the play from becoming monotonous. 
 
 MINOR PROJECTS. 
 
 NOTATION. To climb up and down the ladder. The ladder may 
 reach from 1 to 50 or 100 and steps may be Ts, '2's, o's and lO's. 
 
 To write the names of the steps on the ladder and then read tlie 
 names. 
 
 To guess who is counting. A child comes to the front of the 
 room and hides his face. A child i)ointed out by the teacher counts 
 by 2's to 24. The one in front guesses the narae.x 
 
 ADDITION. To play the Family Game. Each child is given any 
 niimber from I-IO. A number such as 18, 14, lo, 16, 17, IS is writ- 
 ten on the board. If 14 is chosen the child having number <> finds 
 number 9 and stands by him. All the combinations are thus paire<l 
 off. The cards are redistributed and the same or a different number 
 is jdaced on the board. x 
 
 To catch the most fish. Combination cards, each representing a 
 fish, are on the fioor or on benches. The child catches one each time 
 he calls the combination correctly. If his answer is wrong, the fish 
 gets back into the water. 
 
 To get an ajjple. An apjtle i^; drawn on the board. < )n each side 
 of it are written combinations. Two chiltlren now lace in calling 
 these combinations, the first one through, getting the apple. 
 
 To play bean bags. Three circles of different sizes, each liaving 
 a different value as 2, 3, or 4, are drawn on the floor. Two leadeis 
 choose the people for their sides. Each cliild throws two bags. The 
 score for each side is determined when all have had a turn. Adaptt-d 
 from the Baltimore Course of Study. 
 
 To climb a ladder. Combinations are written on eacii rung. The 
 game is to see who can climb the highest. If a second trial is given, 
 the child i-an see how many rungs higher he climlu-d than in the tirst 
 trial.
 
 
 
 
 24 
 
 tin. 
 
 1 how iiuK-h thinjr!* 
 
 w 
 
 ill cost 
 
 1. 
 
 A picture jtostcar 
 
 .1 
 
 Ic 
 
 
 A ruler 
 
 
 3c 
 
 • ) 
 
 A liox ot" I'lialk 
 
 
 Ic 
 
 
 A penlwtMer 
 
 
 «c 
 
 3. 
 
 A |iai)er doll 
 
 
 'Ic 
 
 
 A l.all 
 
 
 t c 
 
 4. 
 
 A jitick of caudy 
 
 
 8 c 
 
 
 A peueil 
 
 
 8c 
 
 Oral work oulv. 
 
 The child answers in his turn as follows:"! want to buy a picture 
 postcanl f«>r Ic and a ruler for 3c. I shall pay 4c for both of them." 
 X Adapted from Decatur Course of Study, 
 
 SUBTRACTION. The first four projects above may be used for 
 subtraction also. 
 
 To play ]\Iore or Less. .V child has a certain number of objects 
 concealed in his hand. He asks, "How mauy?" Ans. Five. If the 
 answer is wrong- he says,"No, it is two more than five," or No, it is 
 three less than five." Adapted from the Baltimore Course of Study. 
 To <letermine the differences in prices: How much more does a 
 penholder at t)c cost than a l)ox of chalk at Ic? A ball than a paper 
 doll? 
 
 To find out how much more I must save to buy a tablet, a pencil, 
 crayons, and other school necessities. 
 
 The child pretends he has saved Ic. How much more must he 
 save to buy these articles: 
 A paper doll at 2c 
 A i>encil at 4c 
 A box of crayons at 8c 
 A tablet at 7c 
 Child answers: '"I have saved Ic, and a tablet costs 7c, so I must save 
 ♦ic more to l)uy it," etc. 
 
 MEASURES. To play store. See the general project under ad- 
 dition. 
 
 To i>lay milkman. Let one child be milkman and come to each 
 child's house (a corner) and sell him a i>int, quart, or gallon of milk. 
 This may involve money also. 
 
 To order groceries. Let one child play telephoning to the store 
 t<j order a pound of butter or a dozen or half-dozen eggs, giving the 
 house numlter to the clerk. 
 
 To invite some friends to a Christmas tree party. The child who 
 is inviting must give the house number or the guests will be lost. He 
 may also give the date. Note: Do not ask for both of these facts at 
 the same time. 
 
 To )»uy thrift stami)s. Children jday at buying and actually buy 
 one, two or more thrift stami)S, naming the coins they use. 
 
 To give and receive correct measure. One child is the grocer, 
 while others are customers leaving their orders. Pint and quart milk 
 bottles are used to show liquid measures, while cardboard objects 
 serve for use in counting out the dozen and half-dozen. The child
 
 25 
 
 should he eucouratred to ask the yrocer for many different thinjjf* s<»l<I 
 hy each Tneasure, thus faniiliarizinfj himself with the use of tliese 
 measures. 
 
 Standards of Attainment. 
 
 See Standards of (yrade II-A. 
 
 Bibliography. 
 
 See (irade II-A.
 
 •2(3 
 
 (iKAl)K II-A. 
 
 DIRECTIONS. 
 
 DUISION OF TIME. Oue-half of the iirithmetic time allotment 
 in this trraile should be giveu to oral work. 
 
 One-half of the work should relate to concrete problems. 
 
 RE\1H\V. Review measures taught iu Grade II-B, counting to 
 100 by Ts and ")\s, to 50 by 2's; backward from 50 by Ts. Column 
 addition involving the very easy, easy and medium difficult combina- 
 tions. 
 
 MAIN TOPICS. Addition and subtraction. 
 
 Head the general directions for (4rade II-B. Apply the sug- 
 gestions given there for counting, reading and writing numbers and 
 work on combinations in this grade also. 
 
 ADDITION. ''Before extended drill in addition is begun, entire 
 familiarity with the number scale to 200 must be obtained as shown 
 }»y ability: 
 
 To read the numl)ers. 
 
 To write the numbers. 
 
 To visualize the number symbols. 
 
 To know that any number and 1 more give the next number in 
 the scale, and that the number that immediately precedes any number 
 iu the scale is one less than the number, and that one less than any 
 number is the number preceding it. 
 
 To count by tens, beginning with any number. 
 
 To know the sum of a given decade and any number less than 
 10, as: 20+5=25; S0+6=86; 90+3=93, etc. 
 
 'Use the combinations mastered in building columns for adding. 
 2 3 3 An examination of these illustrative columns will show how 
 6 5 4 columns can be constructed and extended upward as far as 
 
 2 2 3 the teacher likes. 
 
 3 9 3 
 
 4 3 4 
 3 6 5 
 
 3 2 6 
 
 4 3 2 
 3 4 9 
 
 2 5 3 
 
 In l>eginning column addition with children in the primary 
 
 3 grades. ]tlace the following column on the l)oard. Take the 
 
 4 clialk, and, beginning at the foot of the column, say: "Two, 
 3 three, five," ijointing to the numbers as named, and write the 
 9 32 5 to the right of the 3. Then say, "Five, four,-nine." Write 
 3 23 the 9 to the right of 4. Then say, "Nine, three.-twelve," and 
 6 20 write 12 to the right of the 3. Then continue, ' Twelve, two,- 
 
 2 14 fourteen, writing the 14 to the right of the 2, and so on until 
 
 3 12 the column is added. At each step have the ehihlren, collect- 
 
 4 9 ively or imlividually. repeat after you each statement. Drill 
 3 5 the pupils until they can go through this without error. If 
 2 there is any hesitancy about the combinations, point to the
 
 27 
 
 combination above, so that they may learn where to find the 
 correct form if they shouM fory-et. 
 
 After this process and language form is established, write similar 
 columns on the board for each pupil, with instructions for him to do 
 the exercise himself. The teacher should pass from one to anotlier, 
 hearing each give the form. As a pu])il tinishes, let liim exchange 
 examples with another pupil, first erasing the side columns. To a- 
 void confusion, it is well to write two or three examples in excess of 
 the number in the class, so that no pupil need wait. As a furtlier 
 convenience, it may be helpful for the pupil who iinislies a column to 
 write his name underneath it. The teacher, passing around, later 
 erases the answer and the side columns, and writes "C" (correct) or 
 "X" (wrong) after his name. The place is then ready for another 
 pupil. 
 
 With a few pupils, there will be a continual tendency to make 
 mistakes in the left-hand figure, to write 42 instead of 32, etc. This 
 means that insufficient work has been done on the number scale. 
 Suppose, as in the illustration given, the pupil writes 42 instead of 
 32. To correct this, several methods are at the option of the teacher, 
 (l) She can go back for more drill in the decades, then make 
 the application to the difficulty in hand" (2) 8he may have 
 him write, in ascending column, the number beginning with 
 23, until the next 2 is reached. (3) She may draw a line un- 
 der 23, and ask, "What 2 next above 23?" (Answer, "32.") 
 After the combinations already mentioned have been 
 mastered, and every child can work out the side columns of 
 any column of figures built up out of these combinations, 
 readily and without mistake, the same combinations, in their 
 reverse form, should be treated in like manner. 
 
 The purpose of this is to drill the pupils in learning new com- 
 binations and in visualizing the end figures of the successive partial 
 sums. After this form has been mastered, the teacher should con- 
 tinue ad<lition without writing the sums at the side, and train the 
 pujiil to add without this help. In starting this, it is well to re<iuire 
 the pupil to add directly, thus: five, nine, twelve, fourteen, twenty, 
 twenty-three, etc. If he makes mistakes, have the pupil, in imagina- 
 tion, go through the form of the partial sums in the side column, 
 without actually writing them. First attemi>ts will be slow, but a 
 few exercises will cause him to depend ui)on his own visual imagin- 
 ing. Proceed in the same way to add other columns in review. 
 
 In all this early work, the child should never be permitted to 
 perform any work in addition at his seat, V)ut always at the board, in 
 full view of the teacher. Children, if allowed the time, will fall back 
 into the habit of counting up the sums serially. It is a mistake to 
 think that chihlren will outgrow this habit, once it is formed. Chang- 
 ing one's habits is not so simple a matter as this. To jirevent this 
 habit from being formed, the teacher must first give in columns only 
 those combinations which the children have first learned thorougiily. 
 
 3 
 
 
 4 
 
 
 3 
 
 
 9 
 
 42 
 
 3 
 
 23 
 
 6 
 
 20 
 
 2 
 
 14 
 
 3 
 
 12 
 
 4 
 
 9 
 
 3 
 
 5 
 
 2 

 
 28 
 
 au«l, sec'oud, always iusist that the work he performed at the V^oard 
 ami ill full view of the teacher. Do uot permit the child to stop and 
 think. He either knows the sum or uot. If he shows the least hesi- 
 tancy he must either be told the answer or l)e permitted to look at 
 the i-omhiuation involved in the answer. For this purpose the com- 
 hinatioiis should, with their sums, always lie written on the hoard in 
 full view of the child. 
 
 Concert work is y:ood, hut it should not be eiiii)l()yed exclusively, 
 for many children are thereby made dependent in their work. Again, 
 if a teacher uses it too generously, she cannot know what the indi- 
 vi<luals are capable of doing. In addition work, the teacher must 
 keep in mind the fact that her class will not pi'oeeed uniformly in 
 the acquisition of the work, and that in consequence she must provide 
 some way to give much individual instruction." Los Angeles Course 
 of Study. 
 
 SUBTRACTION. See Grade II-B which gives suggestions relating 
 to the Austrian Method of Subtraction. 
 
 MULTIPLICATION, DIVISION AND FRACTIONS. Introduce 
 multijdication as a short method of addition. If a child through fre- 
 (juent repetitions knows that 8+8+8 ai'e always 9, the transition to 
 three 8"'s are 9 will V)e easy. Teach a few multiplication com))inations 
 at a time rather than the whole table. Each combination and its reverse 
 should be taught together, as two S's are 10 and five 'i's are 10. The 
 f(»rm for the tables 'i-'i's are 4, 8-'2's are 6, 4-2\s are S leads the child . 
 easily toward division. If he knows that 8-2's are 6, he can readily 
 answer the question, "How many 2''s are there in 6?" orally. Division 
 and fractions should be used with multiplication. Thus one combina- 
 tion may take six different forms: 
 
 Five 2's=10. 
 
 Two 5^8=10. 
 
 10--2=5. 
 
 10-5=2. 
 
 \ of 10=5. 
 
 I of 10=2. 
 
 Variety in expression deepens the general impression of the rela- 
 tive values of 2, 5 and 10. 
 
 PROBLEM WORK. All problems are to be oral and based on the 
 child's experience. Present only one step problems with simple 
 numl>er8. 
 
 SUBJECT MATTER AND PROJECTS. 
 
 NOTATION AND NUMERATION. 
 
 Iteading and writing numbers to 200. 
 Rea<ling and writing Roman numerals to XII. 
 Counting to 4S by 4^8 and 6'8, 
 Counting to 80 bv 8'8. 
 
 I
 
 '2 
 
 3 
 
 8 
 
 :} 
 
 8 
 
 8 
 
 4 
 
 2 
 
 (> 
 
 2 
 
 4 
 
 5 
 
 () 
 
 7 
 
 5 
 
 7 
 
 8 
 
 4 
 
 4 
 
 4 
 
 ♦) 
 
 1 
 
 S 
 
 \) 
 
 7 
 
 1 
 
 S 
 
 9 
 
 «) 
 
 < 
 
 S 
 
 » 
 
 29 
 
 ADDITION. 
 
 Addition ( 'ombinatious. 
 
 Group III 2 2 2 
 
 Average 8' 4 o 
 
 Group IV 2 2 8 
 
 Hard S 9 8 
 
 Arrauged by Gourtis. 
 
 SUBTRACTION. Subtraction combinations, the reverse of the 
 addition combinations, 
 
 MULTIPLICATION. Construct and learn tables of 2\s, TVs and lO's. 
 Teach the symbol X as "times." 
 
 1X5=5 
 Use this form for the tables: 2X5=10 
 
 3X5=15 
 
 DIVISION. Teach the reverse of the multiplication tables. 
 Teach the symbol ~^, 
 
 DENOMINATE NUMBERS. Names and number of the days of the 
 week. 
 
 Names and number of the months in the year. 
 The hours on the clockface. 
 The signs § and c. 
 
 CONCRETE PROBLEMS. Problems relating to the following: 
 The Milkman. 
 Carfare. 
 Marketing. 
 A Bird Calender. 
 The School Week and Month. 
 
 FRACTIONS. Fart-taking: 
 
 2 of the numl)ers to 24 which give an integer as a result. 
 .' of the numbers to 50 which give an integer as a result. 
 u) of the numbers to 50, which are exactly divisible. 
 
 OPTIONAL WORK. 
 
 Extensive work in addition an<l subtraction. Caution: Keep tin- 
 work simple enough to prevent counting. 
 TaV)le of 8\s. 
 
 GENERAL PROJECT. 
 
 This is a suggested project which will invi»lve nnu-li of tiu- sub- 
 ject matter of the gra<le. Lest it should not provide sutlicietit drill 
 to ftix the processes, minor projects are given as tletails under each 
 topic of subject matter. The latter are unrelate<l but might be an 
 outgrowth of a larger jtroject similar to the one here given. 
 
 Playing Cafeteria. Individual cafeterias for this proj«'<-t may 
 be constructed out of wooden or cardboard boxes, during the Industrial 
 Arts period. If desired, one large cafeteria maybe rniistni.t..,) .••ich
 
 30 - 
 
 child haviutr some part in the txmstnu-tion or furuishing. Toy money, 
 toy foods, cash retristers and <li.slies may l)e maile during- Industrial 
 Arts or seat i)eri(.)ds. 
 
 The various topics ou II- A outline may be introduced into this 
 project as follows: 
 
 READING AND WRITING TO NUMBERS 200. The cafeterias may 
 l)e numbered, only uuml)ers between lUO and '200 being- used. The 
 children may then take turns in readin": the numbers on the different 
 cafeterias. 
 
 READING AND WRITING ROMAN NUMERALS TO XII. Each 
 cafeteria will need a clock. The proprietors may make these clocks 
 and set them. Suggest to the children that they talk about the time 
 occasionally, e. g.. "Why, it's nine o'clock already." 
 
 COUNTING. By frequently counting the money in their registers 
 ])y B's, 4's, and 6's the children may obtain skill in counting. 
 
 MULTIPLICATION. Application of tables may be made in figur- 
 ing costs; e. g., If one cookie costs B cents what must a customer pay 
 for five? If one potato costs 4 cents what must a customer pay for 
 t wo? 
 
 DIXISION. Playing cafeteria also involves knowledge of the use 
 of division, e. g., a proprietor may have 20 sandwiches ready to 
 serve. If each customer buys two, how many may be served? 
 
 A patron may have 12c left for cakes. How many cakes can he 
 purchase if they are priced 3 cents? 
 
 ADDITION. The customer need not pay for his order until he 
 receives a check on which all the items have been adde<I. 
 
 At the end of the day the proprietor may add up all his sales 
 and place the sum on his file. 
 
 SUBTRACTION. Drill in suljtractiou combinations may be ob- 
 tained by making change. 
 
 DENOMINATE NUMBERS. In making bills of fare the children 
 will soon become familiar with the 8 and c. 
 
 If each cafeteria will have a day calendar there will be little 
 difficulty in learning the names of the months an<l days. 
 
 CONCRETE PROBLE.MS. Following a lesson at the cafeteria, the 
 children should ]>e encouraged to make up problems about their pur-j 
 chases. Complete statements should be required in the solutions. 
 
 Correlation. 
 
 L.\NGl .\( .E. C'omijlete statements shouhl be required at all times. 
 Any errors in grammar should be corrected immediately by the chil- 
 dren themselves. 
 
 Occasionally the class may write short sentences about their cafe- 
 teria. 
 
 A .><tu<ly of the different foods served will jtrovide language les- 
 sons that are well woith while.
 
 81 
 
 SPELLING. The chiMren may want to know how to spell differ- 
 ent words for their l)ills of fare. A few of these words may he intro- 
 duced into the spelling lessons. 
 
 IXDUSTRL\L ARTS. Bills of fare, i)osters, toy food, money, rejjiw- 
 ters, and dishes may be made in the Industrial Arts class. Different 
 mediums may be emj)loyed, such as clay ni(idellin<r. i)aper cuttiutr, 
 construction work, etc. 
 
 INSTRUCTIONS. 
 
 Prices on bills of fare must be correct. Chihlreu should find out 
 the prices for themselves. 
 
 Bills of fare should be changed frequently. 
 
 A child who makes au eror in Arithmetic while acting as cashier 
 or proi)rietor must forfeit his position. 
 
 MINOR PROJECTS. 
 
 NOTATION. 
 
 To play "Buzz." The children count by 3's, 4's, 6's, substituting 
 the word "Buzz" for the number chosen as 1, 2 Buzz -4, 5 Buzz, etc. x 
 
 ADDITION AND SUBTRACTION. 
 
 To guess numbers. A child stands before the class saying: 
 'T am thinking of two numbers whose sum is I'i." 
 "Are they 6 and 6 ?" 
 ||Xo." 
 
 "Are they 5 and 7 V" 
 les. 
 
 The pupil guessing correctly goes to the front, x 
 
 To match sums. The teacher or a child holds a card with a num- 
 ber combination as 6+6. The one who can think of a number com- 
 bination whose sum is the same as the teacher's stands. If wrong, he 
 must sit. 
 
 To play Tenpins. Each teni)in is given a detinite value. A child's 
 score is the total value of all the teni)ius he knocks down in one 
 turn. X 
 
 To play Cat and ,AIouse. A circle is drawn on the hoard, num- 
 bers being written on its circumference. A number is placed in the 
 circle to be combined with the numbers around it. One child is the 
 cat. He points quickly to a number and calls on a classmate. The 
 I)upil who fail is caught and put out of the game. If the cat does 
 not notice a mistake, one of the mice takes his place, x 
 
 To play Dog and Lost Sheep. Each child has a combination of 
 any one number as 10. One, the dog has nothing. The ilog says. I 
 am looking for a lost sheep. Are you 5-|-5 V" "No, I am not 10." 
 The dog goes to another sheep saying "Are you 4-+-6':'" and so on 
 until he guesses the right one who answers, "Ves, I am 'i-fS." The 
 dog then chases the shec]) until he is caught, and the sheep becomes 
 the tlog. X 
 
 To avoid being put oft a street car. The children are in two rows 
 facing each other, each having a combination card. One dnld is c«)n-
 
 32 
 
 .liK-tor; child reu pay him their fare, which is the answer to the com- 
 t)iuatiou. If the coinbiuatiou is uot correctly called, the child is not 
 aide to pay his fare aud is put off the car. The children able to re- 
 main on the car the longest are the winners. 
 
 To tret the most valentines. Cards bearinjj combinations (valen- 
 tines) are put int<» valentine box havinjr a bigoi)ening in the cover. 
 The children then pull out valentines, keeping- them if the answers 
 to their coinbinations are correctly calle<l. 
 
 To tin<i from the jtrice list how much some candy and a tup will 
 cost; a whistle and a toy pistol: dominoes and an orange; a ball an<l 
 a l>ook. 
 
 To tind how much more a pistol costs than an orange; a paper pad 
 than a top: a truin]n't than a whistle. 
 
 rinOK list: 
 
 Dominoes 9c Top 4c 
 
 Whistle tic ]>ook 9c 
 
 Trumpet 8c Ball 7c 
 
 Candy 7c Orange 5c 
 
 Toy pistol 8c Apple 6c 
 
 I'ajter pad 5c 
 
 X Adajiteil from the Decatur Course of Study. 
 
 MULTIPLICATION AND DIVISION. To climb a ladder. Two 
 ladders each containing combinations are placed on the board. Pupils 
 race up and down the rows, or see how high they can climb. 
 
 To sjtin the arrow. A circle of cardboard in the center of which 
 is fasteneil an arrow, has numbers on the circumference. A child 
 spins the arrow, an<l the class gives the i)ro<luct of the number on the 
 arrow and that on the circumference. The pupil who gives the an- 
 swer first may spin the arrow in the next game. 
 
 To race. Products are written upon the board. Two chiltlren 
 are chosen to see who can most quickly point to the answer to a com- 
 bination given by the other pupils. Scores may be kept. 
 
 To see who can get the most cards. Cards containing combina- 
 tions are jtlaced along the board. Two children, starting at either end 
 race to see who can reach the center first. 
 
 To i»lay (irab. P^ach child in turn grabs a card from the (rrab 
 I5ag. If he gives the correct answer, he may keep the card; if he gives 
 the wrong answer he must i)ut the card back. The one holding the 
 most cards at the end is the winner. The Detroit Course of Study. 
 
 To plan for a party. Mother has 15 cakes, 20 sandwiches, and 
 10 pieces of candy. There will be five girls at the party. 
 
 How many cakes will each get? How many sandwiches? Pieces 
 of caiiily? 
 
 I'se this same idea for multiplication. 
 
 To find out how much it will cost to go downtown. If there are 
 five boys, what will it cost to go downtown? Three boys? Seven girls? 
 
 DENOMINATK NCMBERS. 
 
 (Merely SugK-eHted) 
 
 To make a monthly bird calendar, putting in the names of the 
 days of the week.
 
 33 
 
 To find various thin<>-s on the calendar. The number of days in 
 a week. The number of Sundays or of any other day. Tlie date on 
 which the second Monday occurred, etc. The day on wliii li tin- four- 
 teenth, tenth, etc., occurred. 
 
 To make a clockface, usiny Koinan numerals. I'laciniif the hands 
 of the clock at risinjif time, bey-i lining school time, the various hours, 
 Curfew time. Findint>- how many hours it takes the hand to iro from 
 9 in the morning: to twelve in the morniny; from 2 in the afleriioon to 
 5 in the afternoon, etc. 
 
 To write the number of <lollars investe<l in the school Saviii<;s 
 Bank. To find the total number of dollars saved in the room. 
 
 To write the number of cents spent for Thrift 8tami)s, individu- 
 ally and as a class. If the sum is larg^e, the teacher may <lo the add- 
 intj, beinfj assisted by the class. 
 
 To go marketinjjf usiny toy money. Local conditions will deter- 
 mine the j)articular forms this may take. 
 
 Standards of Attainment. 
 
 Woody Test: Series A. 
 Addition 6.8 
 
 Sul)traction 5.1 
 
 Bibliography. 
 
 Brown and C'offman: How to Teach Arithmetic. 
 
 Chap. X Primary Arithmetic. 
 
 Chap. XI Teachiu{>- the Fundamentals. 
 
 Pp. 182-186 Part Taking. 
 Iloyt and Peet: Everyday Arithmetic, JJook I. 
 Kendall and Mirick: Teaching the Fundamental Subjects. 
 
 Pp. 170-190. Skill in Calculation. 
 
 Skill in Api)lication. 
 
 Inductive Teaching. 
 
 Mental and Oral Lessons. 
 
 Klapper: The Teaching of Arithmetic. 
 
 Pp. 49-51. Motivation in Arithmetic. 
 
 Pp. 70-75. ()V)jective Teaching. 
 
 Pi*. 88-91. The Drill. 
 
 Pp. 100-108. Rationalization of Processes, 
 Testing Ability. 
 Fundamental ( )perations. 
 
 Pj). 158-1(S2. Addition and Subtraction. 
 
 Pp. 182-208. Multiplication and Division. 
 Strayer and X'orsworthy: How to teach. 
 
 Pp. 204-205. The Drill Lesson. 
 Suzzalo: The Teaching of Primary Aritliiiu't ir. 
 
 Chap. IX Si)ecial Methods for()l>taining .Xcfuracy and Speed. 
 Wilson: Motivation of School Work. 
 
 Pp. 158-165. The Motivation of Aritliinetic.
 
 34 
 
 GRADE III-B 
 DIRECTIONS. 
 
 L)I\ ISION OF TIME. Three-fourths of the arithmetic time allot- 
 meut ill this trra<le shouhl be fjiveu to oral work. Oue-fourth of the 
 work should relate to concrete problems. 
 
 RE\IE\V. Keadiiijr and writing- numbers to 200. 
 The addition and subtraction combinations. 
 Tables of ^'s, S's, and lU's both in multiplication and divis- 
 ion. 
 
 MAIN' TOPICS. Addition, subtraction, and multiidication. 
 
 Read the directions for Grade II-A relatiufj- to teaching of addi- 
 tion. multii»licatiou, and division. 
 
 .\DDITIO.\'. Do not drill on combinations already mastered. 
 Concentrate attention only on those number facts and processes of 
 which the children are not sure. This will require that the teacher 
 does preliminary and constant testing to discover class and individual 
 weaknesses. 
 
 Hesitancy in giving combinations should not be permitted; for 
 nothing so directly promotes guessing or counting the figures. Instead, 
 tell the child the combination until he has memorized it. A combina- 
 tion that is not entirely automatic with the child should never be used 
 in a concrete problem. Both forms of combinations in addition and 
 multiplication should be given. Knowledge of 3+9=12 does not im- 
 l)ly the knowledge of 9+3=12. The same applies to 3 -5 and 5X3. 
 Provide variety in work on the combinations and tables. 
 
 Teach slowly and carefully in this order (l) the ])rocess, (2) drill, 
 (3) application. The more eftective the teaching the less need of 
 drill. Avoid teaching by drill. Grouj^ work for those needing spec- 
 ial help should be organized, but ordinarily the teacher shouhl aim to 
 hold the attention of all. The problems should be made to increase 
 gra<lually in <lifticulty by increasing the number of orders and the 
 uumVjer of addends. 
 
 SUBTRACTION. Use the Austrian or making change method of 
 subtraction. For a discussion of this method, see grade II-B. 
 
 MILTIPLICATIOX AND DIVISION. The child should not be 
 conij)elleil to give a reason for every process, for he learns to do by 
 doing. Make explanations short and to the point, giving reasons for 
 the processes only when an especial value in doing so is apparent. 
 
 Addition, subtraction, multiplication and division should be drill- 
 ed ujMtn simultaneously. 
 
 liefjuire the children to say ''the difference is"; "tlie subtrahend 
 is"; "the product is"; thus familiarizing them with the language in 
 arithmetic through its correct use. Be sure that the terms multipli- 
 cand, rnultiidier, jiroduct, divisor, dividend, and quotient are under- 
 8to(Ml an<l correctly use<l. 
 
 CONCRETE PROBLEMS. Pupils work on concrete problems | 
 should in this grade l)e oral. In beginning work with concrete pro- 
 blems, have all i)roblems in one lesson relate to the same process.. 
 At another tiirie. jirobJems relating to another i>rocess may be used. 
 
 I
 
 35 
 SUBJECT MATTER AND PROJECTS. 
 
 NOTATION AND NUMERATION. Kea-liiiy- an-l u litii.n iiunilMTs 
 to 5000. 
 
 Komau numerals to XX. 
 Counting by 2 and 4 to 100. 
 
 ADDITION. Combinations. 
 
 Group V 5 5 5 6 6 7 7 S 
 
 Very hard 6 7 8 9 7 8 9 8 
 
 Two and three place columns involving carryiny-; not over five 
 addends, as 34 345 
 
 67 672 
 82 823 
 25 246 
 14 325 
 
 Also mixtures of one, 
 
 two an 
 
 3 
 
 35 
 
 372 
 
 4 
 
 316 
 
 id three place columns, as 
 
 Drill on endings, as 
 
 
 
 6 16 
 
 26 
 
 36 7 17 
 
 9 9 
 
 9 
 
 9 9 9 
 
 SUBTRACTION. Easy subtraction, as 
 
 146 235 397 295 
 
 24 21 72 53 
 
 Subtraction, with one ste]) borrowing, as 
 
 82 54 9.S 192 
 
 78 46 69 56 
 
 Subtraction of two i)lace numbers from three and four place 
 numbers, as 
 
 4,S23 .").3iMi 
 612 22 
 
 Work on endings, as 
 
 15 25 35 45 1(5 26 
 
 _6 _6 _6 J> __I _i 
 
 MULTIPLICATION. Tables of 3's and 4's. 
 
 Written i)r(»blems involving the one-jdace multiplier only.
 
 86 
 
 DIXISloN. Divisiuu only as the reverse of multiplicatiou with 
 no carryiuti. I'se the long divisiuu brac^. 
 
 DENOMINATE NL'MBERS. The ounce and yard. 
 Time: the hour, half-hour an<l quarter hour. 
 
 CONCRETE PROBLEMS. Problems relating to these subjects. 
 Travel and In<lustry. 
 Weitjhiug' and measuring pupils, 
 liuyinij from a cataloicue. 
 Siiarinji: a i)urchase. 
 Time tables. 
 
 FR.ACTIONS. Part-takinir of I of the numbers to UU and 4 of the 
 numV»er to SO which are exactly divisible. 
 
 Addition and subtraction of very simjde fraction — no reduction. 
 
 OPTIO.VAL WORK. Practice in addition and subtraction extend- 
 ed as far as possible until so ditficult that counting is necessary. 
 Multiplication by two-place multipliers. 
 Addition and subtraction of simple fractions. 
 
 GENERAL PROJECT. 
 
 This is a suggested project which will involve much of the subject 
 matter of the grade. Lest it should not provide sufficient drill to fix 
 the processes, minor projects are given. The latter are unrelated but 
 migiit be an outgrowth of a larger project similar to the one here 
 given. 
 
 Playing Post Office. As an Industrial Arts problem, the 
 children may construct a toy post office with two different windows, 
 one for stamps and another for parcels post. They may also construct 
 two large boxes, one for mailing letters, the other for packages. 
 
 One child may sell stamps, another weigh packages, a third act 
 as sorter and a fourth as postmaster. The post office should contain 
 as many small letter boxes as there are children in the class. 
 
 Either toy stamjis or cancelled stamps may be used for postage. 
 The children may make their own envelopes. If a pattern is used, 
 enough envelojtes may be made during one seat period for several 
 po.st office lessons. Each envelope need not necessarily contain a 
 letter. JJooks or ])oxes filled with sajid may be used for packages. A 
 seat j»eriod may be utilized for coining toy money. 
 
 The following are suggestions showing how the subject matter of 
 the III-]> Arithmetic outline may l)e applied in this i)roject: 
 
 READINC, AND WRITING NUMBERS TO 5000. Each post box 
 should be numliered. If the children will use only numbers between 
 lOUU and ')(!(»() they will soon become familiar with'these. 
 
 COUNTIN(; HV 2 AND 4 TO 100. The stamp seller may count his 
 pennies by *2'm and 4's. 
 
 The sorter may count letters in the same manner. Daily practice 
 of this nature tends to develop swift counting.
 
 37 
 
 ADDITION. The staniii seller may keejt (•ouiit of the iminl)er of 
 stanips sold. At the eii<J </^^he day he may acM the total nuiiil)er 
 sold. This will involve v olimi e ad<litioii. 
 
 The sorter may keep a daily record of the iiuiiiher of letters de- 
 livered. At the end of the week he may theii find the sum of all 
 letters handled. 
 
 The child who Imys stamps must also ajtply combinations e. ^., in 
 purchasing nine Ic stamps and four '2c stamps, 9+^ should be imme- 
 diately recalled. A jturchase of 16c and one of 9c means drill on 
 endinifs. 
 
 SUBTRACTION. Makinir change at the stamp ami jiackaire win- 
 dows jjfives practice in subtraction. 
 
 MULTIPLICATION. Practical aj^plication (.f tables is ma-le when 
 the child purchases four 2c stamps, eight 8c stam]is. etc. 
 
 The package weigher, who uses a very simple table of postage 
 rates, may be asked the amount of postage required ou a 8-pound 
 package, the rate being 5 cents a pound. 
 
 DIVISION. Division comes into practical use when a child con- 
 fronts a problem of this nature: I have 18 cents; how many 2 cent 
 stamps can I buy? 
 
 DENOM I NATE NUMBERS. 
 
 Ounce. If the class is particularly advanced, a scale showings 
 ounces as well as i)ounds may be used for weighing packages. 
 
 Time. I'ost office employees may be asked questions concerning 
 the time, e. g.. When will my mail l)e ready? When does this office 
 close? The time may be shown by using a pasteboard clock with 
 movable hands. 
 
 Suggested Correlations. 
 
 LANGUAGE. Learning to write letters and address them prop- 
 erly is a Language project. The Post Office project will motivate the 
 Language lessons. 
 
 Occasionally the children may write a short paragraph on the 
 Post Office (iarae. 
 
 INDUSTRIAL ARTS. Most of the materials nee.ied for the P(.si 
 Office may be made during the Industrial Arts period. 
 
 REMARKS. Allow children to take turns in playing s(.rter. imr- 
 cel clerk, stamj) seller, etc. 
 
 The position of pc)Stmaster may be njade an lionorary one. as a 
 reward for good work. The postmaster's duty may be to clicrk u|. 
 errors and sui)ervise the office in general. 
 
 When real letters are sent, allow the children to get the mail 
 from their boxes. 
 
 Letters should be addressed only to members of the class. 
 
 A cross ma<le with colored crayon may be used for cancelling. 
 
 S2390
 
 38 
 MINOR PROJECTS. 
 
 ADDITION AND SLBTRACTION. To i>lay trames learned iu a 
 lower yrade. 
 
 To play hop-scotch. A iliairiain coutaininff the numbers from 1 
 to lU is drawn on the floor. A cliild lioi»8 in any direction; his score 
 lieinir the sum of the numbers hoi»i)ed ui)on. If the wronjj sum is 
 •riven, his score is rejectetl. Sides may be chosen. This may also l)e 
 played, scoring the difference of the numV)ers. 
 
 To deliver letters to the right house. One half of the class has 
 cards with numbers as 5, 7, S, etc., while the other half has cards with 
 combinations as 9—4, 14—7, etc. The children having- answer cards 
 are at places around the room. Others, the postmen, luring their let- 
 ters to the correct houses. The combinations and answers are read 
 as delivered. The children may be score<l and timed. 
 
 To see who can get rid of his cards first. Each child is given 
 several cards, each containing a number. The teacher calls out 3+3 
 and all those holding the card 6 give it to the teacher. The child 
 giving away all his cards first is the winner. 
 
 To play "Tit-tat-toe". Use the well-known diagram. The child- 
 ren try to find the three numbers in a row, vertical, horizontal, or 
 oblique, which give the largest sum. One child says when it is his 
 turn. 6, 4, 2, equals 12. The number 12, with the childs initials, is 
 written after the row containing these numbers. The other child 
 says 4, 7, 3, equals 14. This is recorded, and so on until every one 
 is satisfied that the combination giving the largest sum has been 
 found. X 
 
 X Adapted from The Decatur Course of Study. 
 
 MULTIPLICATION AND DIVISION. To capture the giant who 
 lives on the hill. A hill with a house at the top is on the board. 
 Comliinations are on both sides of the hill. Two children, one on 
 each side, force their way to the top by calling the combinations cor- 
 rectly. The child who reaches the top first captures the giant. 
 
 To get through the lines without being stopped. Children hav- 
 ing c(jmbination cards are guards on duty, their posts being a good 
 distance ai>art. Other children are messengers, attempting to get 
 through the lines, but must first give the i)assword, i. e., the correct 
 answer to the guard's combination card. If he is not able to give it, 
 he is detained until he can. 
 
 To see who can get the most cards. The combinations are written 
 on cards, which are shown by the teacher. The first child telling the 
 sum correctly receives the card. The one who receives the most 
 cards may be the teacher for the next game. 
 
 To keep out of the center of the circle. The teacher gives each 
 child a number combination. The one who fails iu giving his answer 
 goes to the center, but can regain his place if he succeeds in answer- 
 ing nK)re quickly than the one being asked.
 
 39 
 
 To climb a ladiler. (\)iiil)iiiatit)ii.s are written on cacli ruiii,r. 
 The {fame is to see who can climb hii^hest. If a sec<.ii<l trial is ^ivcn. 
 the child can see how many ruu^a higher he can clinil». 
 
 To read the letters which the postman brings. Aisles are the 
 streets; seats are the houses. The i)ostman delivers to each house a 
 letter, combination card face down. When all are delivered, the let- 
 ters are read as: 2 • 10=-20, 5 -4—20, 8 10^30, 40 • lo' 4, etc. 
 The child, not able to read his letter, fails to gfet another until he can 
 and the child who reads the most letters in a given time wins. De- 
 livery is made several times. 
 
 DENOMINATE NUMBERS. 
 
 These are merely suggestive. Any others which are related to 
 the child's experiences and which involve the required i)roce8ses are 
 valuable. 
 
 To plan for a i)icnic or j)arty: the food necessary; the cost; the 
 amount an<l cost of the decorations. 
 
 To record the weight and measurement of pui)ils, for comparison 
 later in the year, noting the increases made. 
 
 To buy articles. Pictures of various objects with their prices are 
 mounted. Children make purchases, giving their sum. Toy money 
 may be used and change given. 
 
 To show time correctly on a i>asteboard clock with nidvable hands 
 such as, the time for the first bell; for Ijeginning school; for reciting 
 the various suV>jects; for recess; for dismissal. 
 
 To <letermine the number of blocks walked in coming to school; 
 differences in j>upirs distances; number of blocks walked the entire 
 daj'; in the five school daj's; by the whole room. Finding the tin)e 
 taken by various pupils in coming to school; at what time they should 
 start to prevent tardiness, etc. 
 
 To determine the earnings and savings of the whole class. 
 
 To make an original i)roblem from a set of figures, as, 8 ■ 5, Jojiti 
 has 5 marbles and James has 3 times as many. How many has James? 
 
 Standards of Attainment. 
 
 See (irade IIT-A. 
 
 Bibliography. 
 
 See (irade II I -A.
 
 40 
 (;KA1)K III-A 
 
 DIRECTIONS. 
 
 U1\1S().\ OK TI.MH:. Three-tuurths (if the aritluiieiic time allot- 
 ment in thin irrm\e should be giveu to oral work. Three-eighths of 
 the work should relate to coucrete problems. 
 
 REXIEW. The addition ami subtraction combinations. 
 The tables of 2, .5, 10, 3, and 4. 
 IJorrowiufif in suV)traction. 
 C'arryiuff in multiplication, 
 C'ouutiny. 
 
 MAIN TOPICS. The fundamental process. Emphasis on addi- 
 tion, subtraction, and multiplication. 
 
 See the (General directions for (trades IIB, IIA, and IIIB relating 
 to the fundamental process. 
 
 MULTIPLICATION. Two-place nuiltipliers jireseut two difficult- 
 ties. 
 
 1 . Placing the second partial jiroduct. 
 *i. Adding the partial products. 
 E. g. 2654 
 
 25 
 
 1B270 
 5308 
 
 t)6350 
 The first difficulty is most readily overcome by attempting little 
 or no explanation. The first figure S is placed under the multipier 2 
 so therefore the S is placed under the 7. Place the emphasis on the 
 mechanical process, not on the theoretical explanation. Require no 
 theoretical explanation of processes from the child. It is enough to 
 expect him to use them. 
 
 DIVISION. UsQ the long division brace as 5)230 . This does 
 not mean, however, to use the long process for division by one place 
 numl)ers. 
 
 CONCRETE PROBLEMS. All problem work in this grade is 
 oral. Only one-steji simple problems should be used. 
 
 SUBJECT MATTER AND PROJECTS. 
 
 NOTATION AND NUMER.VriON. 
 
 Counting by 2'8 to 84. 
 
 Counting bj- .S's to 96. 
 
 Counting l)y 9'8 to 108. 
 
 Kea<ling and writing numbers to 100,000. 
 
 The use of the comma as in 14,567.
 
 41 
 
 ADDITION. 
 
 Rapid drill. 
 
 Speed and time tests. 
 
 Four ami five jdaee coluTiiiis of six a<ld('iids. 
 
 ^Mixtures of four and five place numbers as: 
 4o(i7 \'1^))\1 84r)(>7 
 
 1.S35 2S17«) *21.S1> 
 
 1687 19.S84 3r)4()2 
 
 3456 27.S53 :ilS9 
 
 54:}2 24176 4276 
 
 14S() 28.>()4 8.S24(> 
 
 SUBTRACTION. 
 
 Mixed Problems: Easy au<l Hard Subtraction. Trainintr in .judt»-- 
 ineut as whether to borrow or to subtract the numbers as they staml. 
 562 157 42 975 
 
 J.31_ _8S_ 1S_ 14 
 
 Hard SuV)tractiou with two stej) borrowing, as: 
 
 832 501 30U 9.S21 1870 
 
 148 132 167 1748 483 
 
 Hard Subtraction, Mixed, one step and two steps, as: 
 721 200 465 619 706 
 
 418 103 158 :US 5S6 
 
 Classification gfiveu in School Education. 
 MULTIPLICATION. 
 
 The tables of 6's and 7"'s. 
 
 Multiplication by one and two place numbers, the multiplicand 
 to contain two or three figures, and the i)roduct to contain not more 
 than six digits. 
 DIVISION. 
 
 Written examples in short division with borrowing. I'se the 
 long division brace. The dividend should contain not more than five 
 digits and the divisor only one digit. Inclu<le numbers not exactly 
 divisible. 
 
 FRACTIONS. ,'; and \ of numbers to 100, including numbers not 
 exactly divisible. 
 
 Adding and subtracting simple fractions and mixed numbers. 
 (No reductions). 
 
 DENOMINATE NUMBERS. One-half and one-(juarter inch. 
 The thermometer. Small sub-divisions need not be used. 
 
 CONCRETE PROBLEMS. Addition of bills. 
 Postage, 
 (rarden jdans. 
 Temperature, 
 Temj^erature records. 
 OPTIONAL WORK. Table of 8's and 9's. 
 Multiplication by three-idace digits. 
 Work for excei)tional sjjeed. 
 More extensive problem work.
 
 42 
 
 GENERAL PROJECT. 
 
 >This is a sujrtiefsteil jirojeot which will involve much of the sub- 
 ject matter of the yfiJi'le. Lest it should uot provide sufficient drill 
 to fix the processes, minor projects are given. The latter are unre- 
 lated, but miifht be an outgrowth of a larger project similar to the 
 one here given. 
 
 rianning and making a class garden. This project is approi*- 
 riate for the second semester of the school year. The planning 
 ef the garden maj- be begun as early as February. The early weeks 
 of the second term may also be utilized for the planting of window 
 boxes so that, as soon as the weather permits, plants may be set out. 
 
 In the suburbs, it will not be difficult to secure a lot large enough 
 for forty miniature ganlens. In a thickly populated district, more 
 <lifficulty may be experience<L Very often, however, a patch of school 
 property adjt>ining the building may V)e available for a garden. 
 
 There are certain exjieuses connected with a problem of this 
 nature. In some cases the land must be fertilized and plowed. Then, 
 too, seeds must be purchased. It may even be necessary to ijurchase 
 a few garden implements, although most of the children will be able 
 to bring these from home. How shall these expenses be paid ? 
 
 hleveral suggestions present themselves, (l) The class may earn 
 the money by collecting and selling rags, newspapers, magazines and 
 tin foil. (2) In some schools it may be possible to hold a candy sale. 
 (3) The class may give an entertainment, charging a small admission 
 fee. (4) Pennies earned by running errands or selling papers may 
 be added to the garden fund. 
 
 This garden project facilitates a correlation of Arithmetic with 
 Nature Study and Industrial Arts. 
 
 The following are merely suggestions showing how a practical 
 application of 1 1 1- A Arithmetic processes may be made in planning 
 the garden: 
 
 If each of the forty members of the class should earn 25 cents 
 for the garden fund, how much money would we have V 
 
 Tin foil sells for 40c a pound. We have eleven pounds, IIow 
 much will that sell for? If we need 85.00 to pay for fertilizing and 
 plowing the garden, how much more money must we earn ? 
 
 Newspai)t-rs sell at the rate of 100 pounds for 25c. How many 
 pounds would we have to sell to secure 81.00 ? 
 
 Small garden hoes cost 35c a piece. IIow much will four cost ? 
 
 The garden lot affords splendid opportunity for making measure- 
 ments. The following problems in mensuration suggest themselves: 
 
 Find the length of the lot in yards. IIow many feet is that? 
 
 L'^sing a ruler, now measure the length of the ground in feet. 
 How does this numl)er comi)are with your first? 
 
 Find the width of the lan<l in yards and feet. How much greater 
 is the length tiian the width ? 
 
 What is the distance around the entire lot in feet? What would 
 be the distance around one-sixth of this lot? One-seventh? 
 
 Assuming that the lot may be 50 feet wide and <S0 feet long, each 
 member of the class of 40 might then be given a small plot ten feet
 
 43 
 
 S(iuare. Childreu in the A Tliird (irade could not work this out for 
 themselves as scjuare root is involved. However, after the class has 
 been told that each meinher may have an individual {rarden ten feet 
 on a sitle, they can easily measure off the ground themselves. 
 
 Letting one-inch stand for two feet, draw a plan of your jjanlen. 
 What measure on your ruler will you use to show a foot? a half-foot? 
 Paths should be one foot wide. In makiny your yfarden jdan, show 
 what you intend to plant in each bed. 
 
 In regard to seeds, the childreji may j^rice these at different 
 places, viz.: the grocery store, florist shoj) and seed store. They may 
 then compare costs. If all seeds are purchased at one time a re- 
 duction may be obtained. The purchasing of seeds may involve such 
 problems as these: 
 
 If Mr. A. sells lettuce seeds at <S cents a i)ackage, and Mr. ]>. the 
 same at (J cents a package, what will be the difference in price if 
 seven packages are purchased? 
 
 If one package of seeds will produce fifty head of lettuce, how 
 many heads will twelve packages produce? What will twelve jiack- 
 ages cost at six cents a package? 
 
 Beans, and peas are sold at loc a half pint. Find the cost of a 
 quart. 
 
 §!4.50 have been earned for seeds. How many packages will this 
 buy at eight cents a package? 
 
 Account must be kept of all seeds pur'fhased. How much more 
 money was spent for peas than for radishes? How much less was 
 spent for carrots than for beans? 
 
 Temperature problems may also form part of this project, e. g., 
 
 What is the freezing point on the themometer? How shall we 
 protect our plants when thermometer reaches 82 degrees above zero? 
 
 At what time of day is the mercury low? When is it highest? 
 Why should we not work in the garden at noontime? 
 
 How much below the freezing point is 12 degrees? What is the 
 difference between 6S degrees and 98 degrees? 
 
 The children should keep account of all time spent working in 
 their gardens. 
 
 If it takes a child 90 minutes to hoe 6 rows of beans, how long 
 will it take him to hoe one row? 
 
 If one child destroys 47 weeds in a niorniiig. how many wee<ls 
 will 40 children destroy? 
 
 If each child spends 20 minutes a day in the garden, how much 
 time will 40 children sjtend? How many minutes will this amount to 
 in a week? In a month? 
 
 Close account should also be kejjt of all i)rodiu-ts sold, 'i'liis 
 phase of the projects points toward such jn-oblems as these: 
 
 If cabbage sells for fifteen cents a head, how much will be .leriv- 
 ed from the sale of 86 heads? 
 
 Tomatoes sell for Kic a i)ound. How much is that an ounce? 
 How much would seven ounces sell for? Seven pounds? 
 
 r)4c was spent for lettuce seeds. The lettuce sold for n2.). 1- ind 
 the profit.
 
 44 
 
 If the pi-Ktit ou peas anmuiitt* to *l«).4:i wliile that ou carrots is 
 *19.5t). tin«l the differeuce. 
 
 If peas sell for 6f a (luart; liow many (luarts must be sohl for !i!l.50. 
 
 Frai-tioiis. too, may euter the class j^ardeii. 
 
 (.'utworms destroyeil 7 of the tomato i)laiits in the garden. If 
 there were 4'i jilaiits set out. how many were destroyed. 
 
 If jieas sell for x*2.00 a bushel, how much should be chartjed for 
 \ bushel? 
 
 REMARKS: 
 
 This i»roject may be introduced into the Nature Study, Industrial 
 Art.s ami I^aniruaire lessons as well as in the Arithmetic work. 
 
 In the Nature study class the children may exi»eriment with diff- 
 erent kinds of soil to find out which is the most suitable for cultiva- 
 tion of certain vegetables. In this class, too, the time of plantiufj: 
 should be discussed, for the i^rospective yardeners must know which 
 seeds to plant first. Account should be kept of the number of days 
 that pass before each vegetable is ready for gathering. 
 
 In the Language class the children may write compositions about 
 different phases of gardening. A study of such masterpieces as The 
 Sower or Plowing will be enjoyed at this time. 
 
 This project may also be used to motivate Industrial Arts lessons. 
 Appropriate garden booklets may be made containing crayon draw- 
 ings, water color sketches, and paper cuttings. C'omjjositions written 
 in the Language class may have a place in this booklet. Attractive 
 garden posters may also be made. 
 
 Each chihl shouhl keep strict account of money expended for 
 seeds, time spent in the garden daily, and amount realized from sale 
 of products. Any products that are used in the home should be list- 
 ed as sold in the account book. These rei)orts may be used in the 
 Sejitember work. 
 
 During the summer months, garden supervisors are appointed by 
 the IJoard of Education to visit school gardens and give the children 
 advice as to the care of their gardens and the disposal of their 
 products. 
 
 MINOR PROJECTS. 
 
 ADDITION AND SUBTRACTION. To play any of the games 
 learned in a lower grade. 
 
 To see which row can score highest. The numbers from 1-20 
 are written on the board. One score-keeper from each row is ap- 
 pointed. The leader iM)int8 to a number on the board. The first 
 cliild in the first row gives the number which, added to the number 
 pointe<l to, will make 20. Each child may have 8 or 4 chances. The 
 game is continued until all have had a chance. The row having the 
 liighest total score wins. 
 
 To ad<l iiuml>ers correctly, as fast as the teacher gives them. 
 Children are at their seats; the teacher tells them to begin with a 
 certain nunil»er as "2," then the teacher says "add 3," the chihlren 
 writing .o, then "add (3," the children writing 11,
 
 45 
 
 To i)lay JJasket IJall". Sides are t-liosen. A line upon whicli llic 
 players stand is drawn seven feet from the \vaste-l)asket. TLey throw 
 beau batrs into the basket, each bag counting- a nuinl)er of j)oints. 
 Several turns may be given. The score is the sum of all bags thrown. 
 Two score-keepers score for their sides. Adapted from the Decatur 
 Course of Study. 
 
 To keep out of the tisli pond. The class forms a cii'clc. Kach 
 child is given a combination. If he cannot give the answer correctly, 
 he becomes a fish and must stand in the center. He can get out by 
 giving some one else's combination l)efore they do, thus exchanging 
 places. 
 
 MULTIPLICATION AND DIVISION. To avoid l)eing caught by 
 the fox. Children form a big circle. The fox calls a combination 
 asking a goose to give the answer. I'nless he answers correctly, he 
 is caught and becomes a fox. The geese may regain their places by 
 giving the combiuatious more (juickly than the geese in the circle. 
 Foxes permitting a wrong answer to stand as correct, may be caught 
 by the geese. 
 
 To play Snow-man. A snow-man is drawn upon the board and 
 snow-balls each containing a combination are placed beside him. ^V 
 child pretends to pick up a snow-ball, at the same time giving an 
 answer. If the answer is correct, part of the snow-man is erase<l. 
 
 To play the game of I>ean Bags. Three circles of different sizes, 
 each having a different value as 7, <S, 9 are drawn on the floor. Two 
 leaders choose people for their sides. Each child throws two bags, 
 the score being their sum or product. The score for each side is de- 
 termined Avhen all have had a turn. Adapted from the Baltimore 
 Course of Study. 
 
 To see who can get rid of his cards first. Cards containing the 
 combinations are <listributed evenly to the class which sits in a circle. 
 The first child holds up a card, calling on another child to give the 
 answer. If that chihl does not know, he must take the card, but if 
 he does, the first child keeps the card. The next chihl then has a 
 turn, and so on until one child is out of cards. That person is winner. 
 
 To gain a greater number of cards than an opponent. Comldna- 
 tion cards are placed along the blackboard. Two children starting 
 at opposite ends, capture combination cards by railing them correctly. 
 The child, having the most at the end of one minute, wins. 
 
 DENOMINATE NUMBERS. 
 
 Suggestive Only 
 
 To keep accounts including amounts ami costs for a dctinite 
 length of time of the following: 
 Selling jiajiers. 
 Delivering milk. 
 Collecting scrap. 
 Kuniiing errands. 
 Ibiying '^rhrift Stamjts. 
 Recording the cost of pencils, jtajiers, iiem-il boxes, erasers, 
 etc., bought for school use.
 
 46 
 
 Keei»intr a record of the outside temperatures for a 
 week or month. 
 
 Estimating the amount ami cost of seeds needed in mak- 
 intr a trarden. Checking the selling-price of the vegetables 
 in the fall. 
 
 Finding the postage necessary for a number of letters. 
 Five letters and three postcards. 
 
 Standards of Attainment. 
 
 Woody test; Series A. 
 
 
 Addititm 
 
 14.5 
 
 Subtraction 
 
 11.2 
 
 .Multii»icatiou 
 
 4.7 
 
 Division 
 
 5.8 
 
 Courtis Test; Series B, Speed. 
 A<l<lition 4 
 
 Subtraction 5 
 
 Courtis Test; Series B, Accuracy. 
 Addition 41 
 
 Subtraction 49 
 
 Bibliography. 
 
 TEACHER'S READING: 
 
 Brown and Coffman: IIow to Teach Arithmetic. 
 
 Chap. VIII The Value of Drill. 
 
 Chap. X Primary Arithmetic. 
 
 Cliap. XI Teaching the Fundamentals. 
 Kendall and Mirick: IIow to Teach the Fundamental Subjects. 
 
 Pp. 10.S-lS(j Mathematical Skill. 
 
 Pj.. 195-200 Drill-Tests. 
 Klapper: The Teaching of Arithmetic. 
 
 Chajt. Ill and IV General Principles. 
 
 Chap. VII The Fundamental Oi)erations. 
 Sn)ith: The Teaching of Arithmetic. 
 
 Chaii. Xn' Details for Exi)eriment. 
 
 Chaj). XVII Work of the Tliird School Year. 
 Strayer and Xorsworthy: How to Teach. 
 
 "pp. 204-205 The Drill Lesson. 
 Suzzalo: Tlie "Teaching of Primary Arithmetic. 
 
 Chap. IX Sjtecial Methods for Obtaining Accuracy and Speed, 
 Wilson and Wilson: Motivation of School Work. 
 
 Chap. IX Motivation of Aritlnnetic. 
 
 SUPPI.I:M KNTARY BOOKS: 
 
 II<>\t and Peet: Everyday Arithmetic, Book I. 
 
 Stunt- Millis: Primary Arithmetic. 
 
 Thoriidike: .\rithiiietic. Book One. 
 
 Weill wortli and Smith: Essentials of Arithmetic, Primary Book,
 
 DIRECTIONS. 
 
 47 
 GKADK IV-i;. 
 
 DIVISION OF TIME. Thrt'e-f..urtlj.s ,.f all the aiitliiiR-tir time 
 allotnieut sliDuld l)e given to oral work. Most of tlie work should he 
 abstract. 
 
 REVIEW. Multiplieatiou by 8, 4, 5, G an.l 7. 
 Part takiug-. 
 
 MAIN TOPICS. The four fundamental i)rocesses. Emi)hasis on 
 niultii>lieati()n. 
 
 This grade is especially suited to mechanical drill. Abstract work 
 is performed with interest, without regard to its concrete api»licatioii. 
 The desired aim is speed and accuracy in all the fundamental opera- 
 tions. The use of standard tests provides an incentive to each indi- 
 vidual to increase efficiency in the fundamentals. Only such work 
 as cannot be done mentally should be recorded on paper. Various 
 types of oral work may be given as follows: 
 
 Projects involving work on the combinations in aildition, 
 
 subtraction, multiplication and division. 
 
 Projects involving measures and their ajjplication. 
 
 Problems without figures. 
 
 Problems with small numbers, emphasizing one process in 
 
 the solving of one-step concrete problems. The amount of 
 
 work done on concrete problems need not be large. 
 Make the children independent by teaching them to check and 
 verify their ow^n results as follows: 
 
 Check addition by aibling the columns in reverse order. 
 
 Check subtraction by adding subtrahend and difference to 
 
 give the minuend. 
 
 Check multiplication by multiplying a second time until di- 
 vision is understood. 
 
 Check division by multiplication. 
 The review work need not be disheartening for the process of 
 re-learning is much easier than the original process. Avoitl going 
 into the advanced work before the previous work is sufficiently at 
 command. If it is assumed that the pupils have forgotten much of 
 the previous work, discouragement on both the teacher's and pupil's 
 ]»art will be avoided. 
 
 ADDITION. Permit no eouutiiig or hesitancy in giving combin- 
 ations. If the child hesitates, tell him the answer. Center attentit)n 
 on the combinations which present difficulty to your class. The fol- 
 lowing columns contain all the combinations. They may be placed 
 on the board ])ermanently and used for time tests. 
 
 
 
 o 
 
 1 
 
 
 4 
 
 !) 
 
 !» 
 
 s 
 
 2 
 
 2 
 
 5 
 
 7 
 
 9 
 
 !» 
 
 7 
 
 • > 
 
 •6 
 
 8 
 
 4 
 
 r. 
 
 1 
 
 ■") 
 
 () 
 
 1 
 
 4 
 
 4 
 
 S 
 
 () 
 
 1) 
 
 8 
 
 •I 
 
 ."> 
 
 8 
 
 8 
 
 o 
 
 2 
 
 ( 
 
 7 
 
 s 
 
 1 
 
 2 
 
 <) 
 
 4 
 
 5 
 
 (> 
 
 1 
 
 s 
 
 H
 
 48 
 
 MULTIPLICATION. Clear n]> dittifulties an<l strentftheu individ- 
 ual \veaknes.>*ej? iu multiiilic-atioii !>> two tiuuifs before iiiultii)licatiou 
 l»y tliree ti«:ures is atteinjiteil. 
 
 DIVISION. I'se Itiiiir ilivi.xiou brace for the short division 
 process. Ill 
 
 (j) <)<)() 
 
 FRACTIONS. The subject of fractions as a definite topic is 
 taken u\> in tlie Fifth (iraile. In the Fourtli (rrade, keej) the work 
 simple and as much as i)ossible. ol)jective. 
 
 CONCRETE PROBLEMS. Aim for thorough mastery of the one- 
 step problems in this yfrade so that the child will know exactly when 
 to atld, subtract, multii»ly or divide. For this much practice must be 
 triven. If lartje numbers seem to confuse the child, f>ive the same 
 tyi»e of ]>roblems usiny smaller numbers. 
 
 DECIMALS. ]i, reading: U. S. money read the decimal point as 
 and. -s24..')0 is to be read. "Twenty-four dollars and fifty cents. 
 
 SUBJECT MATTER. 
 
 NOTATION AND NUMERATION. 
 
 Iieadin<>- and writing- numbers to 1,000,000. 
 
 Counting' to 100 V>y ll\s and 12's. 
 ADDITION. 
 
 Furthei- drill on addition. 
 
 Increased length of columns-seven addends. 
 SUBTRACTION. 
 
 Hard subtraction, three steps, as — 
 
 «001 742S 40009 • 400i»l 
 
 5783 6679 12078 14829 
 
 liapid drill. 
 
 Si)eed ami time tests. 
 MULTIPLICATION. 
 
 Tables of 8''s and 9's. 
 
 .Multii»liers of two and three digets. 
 
 Zero in the multiplier. 
 DIVISION. 
 
 Reverse of multiplcation. 
 
 Rapid <lrill. 
 
 Si)ee<l and time tests. 
 DENOMINATE NUMBERS. 
 
 'I'on. peck and bushel 
 FRACTIONS. 
 
 I'art taking-. 
 
 K and 9 of any two place numbers. 
 
 Addition and subtraction of simi>le fractions and mixed 
 
 numbers. NO leductions. 
 DECIMALS. 
 
 r. S. money correctly uiitten and read. 
 
 Correct use of "'ainl.'"
 
 49 
 GENERAL PROJECT. 
 
 This is a su^yested jji-oject whicli will involve niucli of tlie suli- 
 ject matter of the yrade. Lest it should not provide sutticieiit drill 
 to fix the processes, minor projects arc yiveu as details under each 
 topic ot" subject matter. The latter are unrelated but niitjht be an 
 outgrowth of a largfer project similar to the one here given. 
 
 Uuying Coal. This project necessarily implies a correlation (»f 
 Arithmetic and (geography. In the Geography class the children 
 may trace the route followed by vessels bringing coal from Erie to 
 Duluth. ()l)servatiou trips to the <locks during the uidoadiiig of coal 
 barges will vitalize the geography lessons. 
 
 In developing this project, the following idoblems eiiilMnlying 
 Arithmetic processes to be emphasized in IV-IJ are suggested: 
 
 1. N'essels bringing coal from Erie to Duluth consume 30 tons of 
 vval a day. How many pounds is this? It takes ten days to make 
 the rouml trip. How many tons are consumed during the trip? At 
 §3.00 a ton what would this amount to? 
 
 2. A freight boat travels at the rate of nine miles an Ikmii-. jjuw 
 far will it travel in twenty-four hours ? in five days. 
 
 3. I)oats which bring us our coal travel 1,02.5 miles. Find one- 
 eighth of this distance; one-ninth. 
 
 4. It takes one hour to unload 1.000 tons of coal. How many 
 tons can be unloaded in ten hours? 
 
 5. A vessel brings 7,000 tons of coal. If the coal sells at -sit. (Hi :i 
 ton, how much is the cargo worth? 
 
 The children may bring to school samjdes of various kinds of 
 coal. To each sainple a tag might be attached giving the price per 
 ton. This will suggest problems of the following nature: 
 
 Which kind of coal is most expensive? 
 
 How much more does a ton of nut coal cost than pea? What 
 would be the difference in price in buying ten tons of either ? 
 
 If a ton of stove coal costs nine dollars, find the cost of 4S tons. 
 
 Find the total cost of a ton of bii(iuets, a ton of screenings, a ton 
 of nut coal and a ton of i>ea. 
 
 The pupils may find out from their jtarents the cost of the tuel 
 burned in their homes during the winter. Encourage tlic childien to 
 think of ways of saving fuel. e.g. (1) by using storm windows, (2) 
 by closing all cracks and crevices, (3) by keeping outer doors locked. 
 (4) by cleaning out the furnace or stove at frequent intervals. (•')) by 
 shutting off the heat in rooms which are not in use. 
 
 Comparisons may be made as--- 
 
 At Mary's house'the fuel bill was .sOO.()S; at .l..lni"> ii was >;lii().4:{. 
 Fin<l the difference. 
 
 Mr. A bought 19 tons of coal an. I Mr. T. 14. Find tlic dift.Tcnc*' 
 in tons; in i)ounds. 
 
 In an eight-room house 19.14;') jiounds of coal wert- (•oiisuim-d. 
 How many i>ounds were required to heat one room ? 
 
 From an observation lesson the childien ina\- h-arn li<>w c<>al is 
 weighed.
 
 50 
 
 A loa«le«l coal watrou weighs 10,900 pouudf*. If the team alone 
 weiifhs 5,900 iioumls, how much does the coal weigh 'i 
 
 A coal team weighs (>,'200 pounds. Two tons of coal are loaded 
 onto it. Fiinl the weight of the team and coal together in pounds. 
 
 A man can loa<l ti.OOO pounds of coal in an hour. How many 
 piMunIs can he load in S hours? in 9 hours? 
 
 I4S.7tjO ]iouuils of coal are to be placed on 8 equal loads. How 
 many iiouuds in each load? 
 
 Following are problems V)ased on tlie chief engineer's rei)ort of 
 coal consumed in Duluth schools during 1917-18: 
 
 The fuel l»ill for the Bryant School was §1,549.51; for the Fair- 
 mount *908.88; for the Franklin xl, 147.49; for the Lowell 8650.39. 
 Find the total amount for these four schools. 
 
 1GO,U50 jtounds of screenings and '286,500 pounds of soft coal 
 <lust were purchase<l for the Jackson school. How many more pounds 
 of dust were useil than screenings? 
 
 The fuel bill for the Madison School amounted to 8788.77. This 
 was 8'i9.S.18 less than the bill for the Longfellow. What was the bill 
 for the Longfellow School ? 
 
 Fuel purchased for the Monroe School amounted to 8712.80, for 
 the Salter 8881.76, for the Webster 8408.78, for the Lester Park 
 ^875.05 and for the Central High, 810,551.07. How much more was 
 spent for the Central High School than for the other four together? 
 How much less was expended for the Monroe School than for the 
 Lester Park ? 
 
 REMARKS: This project may be correlated with the Language 
 lessons. After a visit to the coal docks, the class may write original 
 compositions on the unloading of coal, number and position of docks 
 and flistribution of the coal to points in the Northwest. These com- 
 positions may be put in booklets made during the Industrial Arts 
 jteriod. Such booklets may contain appropriate drawings and paper 
 cuttings, e.g. a coal barge entering the canal. 
 
 MINOR PROJECTS. 
 
 ADDITION AND SUBTRACTION. To solve live problems cor- 
 rectly within a given amount of time. 
 
 Problems in aiMition and subtraction are given, the child start- 
 ing and stopi)iiig on signal. Individual scores are kept on the boanl 
 from ilay to day. the child noting his imi)rovement. 
 
 To add and subtract numbers as fast as the teacher can give them. 
 The riietho<l is a follows: 
 
 Tkachkk: liegin with 8, add 6; (children write 14); sub- 
 tract 5; (children write 9) etc., using the difficult 
 combinations. 
 
 To run a race.T he combinations in addition, subtraction, multi- 
 plication, or division are printed on cards, sufficiently large so that all 
 the class can see. A correct answer counts 1; an incorrect answer 0. 
 A score-keeper is aj)pointed. The game is to see which row scores 
 highest.
 
 51 
 
 To write the correct iiumlters. A cliilil «joes to tlie teadier's ilesk 
 selectiujj three tickets, each coutaiuiuu" a number from 0-10. Wliile 
 he adds the class listens aud writes the numbers which must have been 
 taken. For instance, if the child chooses 5, 6, 7, he says 5, 11, IS. and 
 the class writes 5, 6, 7. Those at their seats cheek to see how many 
 combinations they had were correct. After a little jtractice, more 
 than three numbers may be chosen. 
 
 MULTIPLICATION AND DIVISION. 
 
 To guess numbers which make a certain product. 
 
 "I am thinkinfj' of a number between 30 and 40." 
 
 "Is it 6 6?" 
 
 ''No, it is not 36." 
 
 "Is it 7-5?" 
 
 "Yes, it is 35." 
 
 Adapted from the Decatur Course of Study. 
 
 To race. A list of combinations is written upon the V)oard. One 
 child starts aud names the results quickly. If an error is made, the 
 child's name is written at that place and another child begins at the 
 beginning. He goes as far as he can and his name is written where 
 he stops. If a child finishes, his name is written at the side as a 
 winner. A child who missed may have another chance, beginning at 
 the place where he made a mistake. 
 
 To see which leader can win. Three leaders are chosen each 
 being given five cards containing combinations. Each in turn hoMs 
 up a card aud calls on some one for a (juick answer. The child an- 
 swering correctly joins that leaders' group. The leader who gels the 
 largest number iu his group wius. 
 
 To race. The teacher writee an uneven number of examples 
 across the board. One child begins to work at the right, the other at 
 the left. The one who reaches the middle example first has the priv- 
 ilege of working it and wins the race. 
 
 To race. The pui)ils draw circles on the Ijlackboard. entering 
 the numbers from 1 to 9 inside the circumference. Each child is 
 given a number to i)lace iu the center of his circle. The game is to 
 see who can first write the products of the inner number and the out- 
 er numl)ers on the outside of the circle. 
 
 To i)lay Bean Bag. Two concentric circles are drawn on the 
 floor. The outer circle is divided into quarters. Write a figure in 
 the inner circle and in each (juarter of the outer circle. The child 
 scores the sum of the number hit in the outer circle times the inner 
 number. The scoring is done by groups. 
 
 Adaptetl from the Decatur Course of Stuily. 
 
 To play Heturn Ball. Draw a large oblong on the blackboard. 
 Mark it off into a number of squares and triangles, writing a number 
 iu each si)ace. The children use a return ball and try to strike the 
 largest number in a si)ace. If the jdayer strikes a number, lie is al- 
 lowed to multiply b\- o. or any number. 
 
 Adapte<l from the Decatur Course of Study.
 
 52 
 
 DENOMINATE NUMBERS. FRACTIONS AND DECIMALS. To 
 fiii.l the relatiou ut" the pec-k t.. the bushel tlirouirli experimeutation 
 with linth. 
 
 To riii<l out the i.ro.lufts wliich are sold by the peck or bushel. 
 
 To obtain the jiriees for such i»ro<luee. 
 
 To make oriyiual jtrobleius relatiujj to the above, using %ures 
 whii-h are accurate. 
 
 To fiud the cost iier ton of various kiuds of coal. 
 
 To tiii.l the amount of coal needed for family use during the 
 winter. 
 
 To find the cost of all the coal needed during the year. 
 
 To find the cost of clothing needed by boy or girl during the year. 
 
 T<t solve jiroblems relating to the rainfall and temperature of 
 Duluth: 
 
 Kainfall Temperature 
 
 (by inches) C^Fahreuheit) 
 
 Dec. 1 IT 
 
 Jan. 1 10 
 
 Feb. 1 13 
 
 Mar. 1', 24 
 
 Apr. 2 38 
 
 May 3 48 
 
 June 4 58 
 
 July 4 65 
 
 Aug. 3 64 
 
 Sept. 3.^ 56 
 
 Oct. 3i 45 
 
 Nov. V, 29 
 
 Finding the total rainfall for Duhith; amount of rainfall in win- 
 ter, spring, summer, fall; diftereuce in rainfall V)etween any two 
 months. 
 
 Finding the hottest month: the coldest: the difference between 
 the two; difference between any two months. 
 
 To solve problems relating to industries of Duluth. 
 
 There are four ore-docks. Each has 384 ore pockets. 
 
 I low many pockets are there in allV The ore is brought 
 down from the mines in trains of 50 cars, each car holding 
 fifty tons. How many tons of ore are brought down in a 
 trip? etc. For more statistics, see the pamphlet on Duluth 
 issued by tlie Commercial Club. \t. 46.
 
 58 
 
 Other subjects which are suffjfestive: 
 
 P(>l»ulati<)U in various years. 
 
 Public Schools. 
 
 Number of children in school. 
 
 Miles of i»aveil street. 
 
 Parks. 
 
 Iron Ore. 
 
 Coal 
 
 Number of churches. 
 
 Books in the library, etc. 
 
 To solve problems relating to travel by airjtlaue. 
 
 Distance across ocean. 
 
 Time taken to ascend; to descend. 
 
 Temperature at various levels. 
 
 Speed of aeroplanes, etc. 
 
 Time taken by lieid to cross in a hydroplane; by IJrown 
 
 and others in a non-stop flight. 
 
 To solve problems relating- to travel by water. 
 
 Length of boats. 
 
 Speed. 
 
 Number of passengers. 
 
 Cost per trij) and jter mile. 
 
 Numlier of deck-hands, etc. 
 
 Standards of Attainment. 
 
 See (irade IV- A. 
 
 Biblio^aphy. 
 
 See Grade IV- A
 
 54 
 
 (;kade iv-a. 
 
 DIRECTIONS. 
 
 DIVISION OF TIME. Three-fourths of the arithmetic time allot- 
 ment ill this irrade should l»e giveu to oral work. Most of the work 
 should l»e with aV)Stract iiumhers. 
 
 REVIEW. Part takiuj?. Multiplicatiou l>y (j, 7. S aud 9. 
 
 LEADING TOPIC. Loutr Divisiou. 
 
 IJead the <lireetioiis giveu for preeediuy grades, especially (xrade 
 4-B. The poiut of emphasis iu this half jjrade is long- divisiou. "This 
 necessitates daily drill on the multiplication tables and upon such 
 questions as, How many nines iu 67 ? in 75 ? in 71? — as a preparation 
 for division. Long- division should be approached through analyzing 
 what has been done by pupils in short division, and finding that four 
 steps, repeate<l over an<l over again, are all that must be known. 
 These steps are: 
 
 ESTIMATE how many times the divisor can be fouml in the par- 
 tial dividend necessary to use. 
 
 MULTIPLY the divisor by the estimate made. 
 
 SUBTRACT the pro<luct of the divisor and quotient figure, or es- 
 timate, from the i)artial dividend used. 
 
 BRING DOWN the next figure of the dividend, to be placed at the 
 right of the remainder, all of which now becomes a new partial divi- 
 dend with which the steps, estimate, multiply, subtract, and bring- 
 down, are again used. 
 
 When the steps are once secured from an analysis of a short di- 
 vision example, pupils should test their use of the steps Avith divisors 
 like 21, 71, 31, 51, 61, so that the ones' digit is sure to change the 
 quotient estimate, even to the extent of needing to think 40 for 39, 
 30 for i!9, 70 for 69, etc., in making the estimate. After this the di- 
 visors should be 32, 52, 92, and so on; 78, 68, 28, etc.; 97, 47, etc.; un- 
 til 25, 35, 45, etc., are reached. The general order of procedure will 
 by this time be well fixed, aud teachers need only to look out for iudi- 
 viilual errors such as putting too many naughts iu the quotient when 
 the child has once learned to jjut any there; having a remainder large 
 enough to contain the divisor once more aud not recognizing the fact, 
 but writing a one in the quotient before continuing, and similar errors. 
 As soon as the process is mastered so that accuracy is assured, pupils 
 may work for speed, both in diviiling and iu checking results." 
 Minnesota Course of Study p. 112. 
 
 SUBJECT MATTER. 
 
 NOTATION AND NUMERATION: 
 
 Counting to 100 by 15. 20. and 25. 
 
 ■^riie Koman Numerals L, C. au<l D. 
 ADDITION. 
 
 Time tests.
 
 55 
 
 MULTIPLICATION. 
 
 Tables of 11 and 12. 
 
 No multipliers over four places. 
 
 Drill for speed. 
 
 DIVISION. 
 
 Loiiff division, using a two place divisor. Zero in tin- divisor. 
 Drill for speed. 
 
 DENOMINATE NUMBERS. 
 
 Kecognitiou of the square inch, square foot and scpiarc yard. 
 FRACTIONS. 
 
 Part taking, 
 
 VI and X2 of any two place numbers. 
 
 Addition and subtraction of fractions wliose denominators may 
 
 be seen by inspection. 
 
 DECIMALS. 
 
 Addition and subtraction of V. S. money. 
 
 Multplicatiou and division of U. S. money by integers only. 
 
 GENERAL PROJECT. 
 
 This is a suggested project which will involve much of the sub- 
 ject matter of the grade. Lest it shouhl not provide sufficient drill 
 to fix the processes, minor projects are given as details under each 
 topic of subject matter. The latter are unrelateil but might be an 
 outgrowth of a larger project similar to the one here given. 
 
 To save school supplies. This project may be used to involve 
 arithmetic, language, spelling and industrial arts. The jjroject can 
 be followed as outlined. Then, when need arises, the concrete work 
 may be discontinued for a week or so to give necessary drill on those 
 processes which require special emi)hasis. Since thrift in these t\i\ys 
 of high prices is such a vital problem, and since directions given to 
 the whole class to "Do not waste your i)aper," or 'Take care of your 
 pen-points," do not produce the desired result, graphic representation 
 of the expense incurred for two or more successive months is likely 
 to decrease waste, since it is a concrete problem and will apj»eal to 
 most of the class, (rroup infiuence will compel those who are not 
 directly reached, to l)e more careful. Perhai)S not all of the subject 
 matter for the IV-A Grade will be covered by this i)roject, but most 
 of it can be brought in. The Roman Numeral 1) can be included in 
 connection with a ream of pajicr or ')(»() sheets; C as a number of 
 sheets in a pad of drawing i)ai)er. The table of I'i's is involved in 
 finding the number of articles in any number of dozen or in the gross. 
 Multiplication and long division can easily be included. The pro]»- 
 lems given are merely a suggestion as to how the minor jiroject may 
 be worked out more definitely. 
 
 LANGUAGE: To discover who pays for the sui»plies used. (The 
 School IJoard Init indirectly the i)areuts.)
 
 56 
 
 To think of \vay.< in wliich the increased cost of supplies may be 
 
 tiiet. 
 
 |>y having the parent pay more tax. 
 
 liy l>ein{: more careful in usinfr supplies. 
 
 Wliioh way is the best ? 
 The foUowinir i** a l»rire list of the articles used most frequently 
 in the schoolroom. These are real i)rices, obtained from the Account- 
 in-: Dej-artnieiit of the Tx-anl of Education: 
 
 ARTICLE 
 
 
 PRICE 
 
 
 Chalk 
 
 
 7oc per gross 
 
 2 pes. for Ic. 
 
 Drawintr Pajter 
 
 
 4c per humlretl 
 
 Ic for 25 sheets. 
 
 Desks 
 
 
 
 S5.25 a piece 
 
 Erasers 
 
 
 89c per dozen 
 
 4c a piece. 
 
 Tennianship Hooks 
 
 
 4c a piece. 
 
 ■/. 
 
 
 18c per dozen 
 
 2 cakes for 3c. 
 
 Ink Wells 
 
 
 SI. 60 per dozen 
 
 14c a piece 
 
 Paper Towelint: 
 
 
 23c a roll 
 
 5 sheets for Ic. 
 
 Pen Points 
 
 
 46c a gross 
 
 3 for Ic, 
 
 Pencils 
 
 
 $2.25 per gross 
 
 2c a piece. 
 
 Pencils (drawin 
 
 g) 
 
 S3. 12 per gross 
 
 6 for 13c. 
 
 Rulers 
 
 
 lie j)er dozen 
 
 lea piece. 
 
 Writing Paper 
 
 
 48c a ream 
 
 10 sheets for Ic. 
 
 Peaders: 
 
 
 
 
 Eldsou IV. 
 
 
 45c.. 
 
 
 Merrill IV. 
 
 
 56c. 
 
 
 Natural IV. 
 
 
 56c. 
 
 
 Reading — Literature 56c. 
 
 ARITHMETIC: To ligure the cost of the writing paper used in 
 the whole school for month. 
 
 How many sheets of writing paper were used l)y the whole class 
 in penmanship, this week? 
 
 How many weeks are there in a school mouth? 
 
 How many sheets wouhl be used in the entire month? 
 
 How many children are in the room? 43. 
 
 The whole room uses 1324 sheets in a month. How much does 
 one child use ? 30. 
 
 How many iiujjils are in the school? 
 
 The pui»ils may write a letter to each room, asking for the en- 
 r(dlment. 
 
 If each child in the school uses 30 sheets in a mouth, how much 
 is used by the entire school? 
 
 4.395 boys in the first six grades of the Duluth schools use iuk. 
 How many sheets of pa])er will they use in a month? 
 
 4,275 girls will use how many sheets. 
 
 How many sheets <lo both the l)oys and girls use? 
 
 To find the total value of the supplies, including books which 
 each child has in its desk. 
 
 To find the value of all the basal reading books in the room; all 
 the penmanshij) >»ooklets; penholders; erasers; rulers; ink wells; desks.
 
 57 
 
 etc. From these results, to detertiiiiie the total value of the ix'nna- 
 iieut supjtlies. 
 
 To tififure the cost of sui)i)lie8 (coii3uiijal)le,such as i)eiicils, pajier, 
 peu-i)oiut) ueeded by the whole I'ooni iu a month. 
 
 To help the teacher trraph ou the blackboard the cost of the 
 supplies. 
 
 
 10'^ 
 
 zo' 
 
 3(f 
 
 so' 
 
 50* 
 
 fed" 
 
 70' 
 
 80'- 
 
 :?o" 
 
 r 
 
 X' 
 
 2>^ 
 
 i^ 
 
 
 CHALK 
 
 
 
 — ' 
 
 
 — - 
 
 -—^ , 
 
 
 
 
 
 
 
 
 
 DRAWING-PAPER 
 
 
 
 
 
 _^ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 WRITING-PAPER 
 
 
 
 
 
 ' 
 
 
 
 
 — 
 
 - . 
 
 
 
 
 
 
 PENCILS 
 
 
 
 
 
 
 __ 
 
 
 — 
 
 — 
 
 — ' 
 
 
 
 
 
 PEN-POINTS 
 
 
 
 
 
 S 
 
 
 
 
 
 
 
 
 
 
 PAPER TOWELING 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 ETC. 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 LANGUAGE. To suyyest methods of elimiijatiiif>- waste; i.e., 
 writiug" ou both sides of the paper; care in usin^ pen-])()iuts and iu 
 dilipin<r ink; having^ clean hands when usiujiif books; usiny only one 
 pa])er towel at a time; picking up chalk instead of stepi)injj upon it, 
 etc. One child rniyht be made responsible for each of the sui)plies. 
 
 To take part in a campaign to see how much less the expenses 
 can be made the following month. 
 
 ARITHMETIC. To determine the saving made by indivuluals 
 and by the whole room the second month. 
 
 To show by additions to the graph, the saving which has ensued. 
 
 LANGUAGE. To tell another room how they can eliminate waste. 
 The pupils who can do this most effectively may be chosen. 
 
 MINOR PROJECTS. 
 
 FUNDAMENTAL PROCESSES. 
 
 To see how majiy problems can be done in one minute, A num- 
 ber of addition or subtraction problems are i)laced on the board. The 
 children test their rapitlity and accuracy by seeing how many of 
 these they can solve correctly in one minute, the teacher giving the 
 signals to begin and stop. The individual scores are kept on the 
 board and these should serve as a stimulus toward improvement. 
 
 To see how long it takes to do five a<ldition and subtraction 
 problems. 
 
 This test is carrietl on in the same way, the teailicr timing the 
 children as to ra])idity. Later they are scored as to accuracy ol their 
 Work, these scores also are kejtt on the board. 
 
 To play Hlackboard Uelay. Two sets of exami)les arc place.l 
 on the board. The class is di-vi<led. The leader iu each ilivision writes
 
 58 
 
 the answer to the first examples, returns tlie c-lialk to the next impil 
 who works the next example aud so on until all have had a chance. 
 There shouhl be as many examj^les as children. The wiuning- side is 
 the one whicli first conijiletes the examples correctly. A child may 
 <-orrect any errors in work done i)reviouslj'. Decatur Course of Study. 
 T.i see who can first fill out his card. ]Make cards with fiijures 
 from 1 to 12 alony the to]) and left liand maryin; the remainintj 
 si)aces. makintr up the twelve tables, heiny left blank. Proviile clieck- 
 ei"s with tlie necessary multiples irom 1 to 144. Eacii checker has a 
 ])lace in one of the blank si>ace.s on the card. When i)roperly placed 
 tlie tables froiif 1 to \'2 are complete. The one rilliny liis card first is 
 winner. Mankato Course of Study. 
 
 DENOMINATE NUMBERS. 
 
 To cut sipiare inches and square feet out of paper and use them 
 in covering the surfaces of books, desks and i)apers. 
 
 To estimate the area of books, desks, window panes, etc., verify- 
 ing- tlie same by actual measure, seeini>- who can estimate most accur- 
 ately. 
 
 To find out which blackboard is longest; how long: the sand table 
 is; how wide it is; how many square feet it contains. How long and 
 wide the teacher's desk is. How long and wide the i)upirs desk is. 
 Ildw long and wide the school room is. 
 
 To solve problems relating- to the lumber industry. Facts which 
 might be used: 
 
 UNITED STATES Millions of M Feet 
 
 Washington 4,592,058 
 
 Louisiana 4,1()1,56U 
 
 Oregon 2,098,4G7 
 
 Mississippi 2,010,581 
 
 Texas 2,081,471 
 
 No. Carolina 1,957,258 
 
 Arkansas 1,911,647 
 
 Alabama 1,528.986 
 
 Wisconsin 1,498,858 
 
 \'irginia 1,278,958 
 
 AV. Virginia 1,249,559 
 
 Michigan 1,222,988 
 
 California 1,188,880 
 
 Minnesota 1,149,704 
 
 Florida 1,055,047 
 
 All others 8,822,021 
 
 None: These numbers may be i)ut in more sim])le form if their 
 length confuses the class. 
 
 To find the total lumber pro<luctioii of the Cnited States; differ- 
 t'lice in jtroduction lietween states. 
 
 To find how much lumber, iron, grain, is produced by the lead- 
 ing countries; difference in i)roduction between countries or states.
 
 59 
 
 WHEAT (In Bushels) 
 
 COUNTRIES. 
 
 Russia (J99,4I8,()()() 
 
 I'^uited States. __ _ t)9;'),44:{,0(»(l 
 
 ludia 857,941,0(10 
 
 France 259,1X0.000 
 
 Austria-Hung-ary 241,:}i»4.000 
 
 Italy 158,:i:{7,000 
 
 Canada 149,990,000 
 
 Spain '__ ^^___1B7.44S,000 
 
 STATES. 
 
 Kansas 177,200.000 
 
 No. Dakota SI, 592, 000 
 
 Nel)raska 6S,1 ] (5,000 
 
 Oklahoma ._ 47,975,000 
 
 Missouri 43,888,000 
 
 Indiana_______ ___':____ 48,289,000 
 
 Minnesota 42,975,000 
 
 Washiufiton 41,840,000 
 
 Ohio _. 36,58S.O0O 
 
 South Dakota. 31,5()().000 
 
 All others 276,(548,000 
 
 Finding the total production of the V. 8.; difference in produc 
 tion of various states. 
 
 IKON ORE (By Tons-1910) 
 
 COUNTRIES. 
 
 United States 26,10S,199 
 
 (Terraany . 1 2,91 7, (J58 
 
 United i{inffdom___ ___. 9.81S,91G 
 
 Fran ce 3,682. 1 05 
 
 Russia 2,871.882 
 
 Austria-Hung-ary 1,958.786 
 
 Belgium 1,682,850 
 
 Canada 687,928 
 
 STATES (In millk ns of dollars) 
 
 Pennsylvania 688.922.000 
 
 Ohio 281.479,000 
 
 Illinois 124,908,000 
 
 New York 66,158,000 
 
 Alabama 21,286,000 
 
 All others 214,454,000 
 
 Standards of Attainment. 
 
 WOODY TEST. Series A. 
 
 Addition ^^-'^ 
 
 Suhtraction l-i-'> 
 
 Multiplication •'•' 
 
 Division 9.8 
 
 COURTIS TEST: Series B -Speed 
 
 Addition |^-'* 
 
 Subtraction ' •" 
 
 .Multiplication '»•" 
 
 Division "*•"
 
 (30 
 
 COURTIS TEST Series B. -Accuracy. 
 
 AtUlitiou 64. 
 
 Siilitnu-tioii ^^0. 
 
 M ult i plicat iou 67. 
 
 Division 57. 
 
 Bibliography. 
 
 TEACHER'S READING. 1)1h>\\ u aud Coftniau: How to Teach 
 
 Arithmetic. 
 
 C'hai). III. Accuracy. 
 
 Chap. V. Markiuy papers. 
 
 Pp. cS,S-91. Analysis. 
 
 Chap. VIII. Value of drill. 
 
 Chap. XI. Teaching the fuudanieutals. 
 Kendall and Mirick: How to Teach the P^undameutal Subjects. 
 
 Pp.l7U-19U. Skill in Calculation. 
 Skill in Application. 
 Inductive Teaching. 
 Mental and Oral Lessons. 
 
 Pp. 195-200. Drills. 
 
 Tests and Katings. 
 Klapper: The Teaching of Arithmetic. 
 
 Pp. 48-51. Arithmetic Must be Humanized. 
 Motivation in Arithmetic. 
 
 Chai). IV. General Principles. 
 
 Chap. VIII, Multi])lication and Division. 
 McMurry: Special Method in Arithmetic. 
 
 Chap. IV. Method for Intermediate Classes. 
 Smith: The Teaching of Arithmetic. 
 
 Chap. IV. The Nature of Problems. 
 
 Chap. IX. Children's Analyses. 
 
 Chap. X. Interest and P]ffort. 
 
 Chajj. XII. Principles of Teaching Arithmetic. 
 
 Cha}). XVIII. The Fourth School Year. 
 Strayer and Norswortliy: How to Teach, 
 "p. 204. The Drill Lesson. 
 
 l*p. 2H4-248. Measuring Achievement. 
 Suzzallo: The Teaching of Primary Arithmetic. 
 
 Chap. \'III. Methods of Pationalization. 
 
 Chap. IX. Special Methods for Obtaining Accuracy and 
 Speed. 
 Wilson: The Motivation of School Work. 
 
 Chap. IX. Motivation of Arithmetic. 
 
 SUPPLL.ME.NTARY BOOKS. 
 
 Hoyt and Peet: Everyday Arithmetic, Book 1. 
 
 Stone and Millis: Primary Arithmetic. 
 
 Thorndike: Arithmetic, IJook 1. 
 
 Wentworth and Smith: Essentials of Arithmetic, Primary Book. 
 
 i
 
 61 
 GRADK V-i;. 
 
 DIRECTIONS. 
 
 DIVISION OF TIME. Oiie-half of the work of tliis <rraile should 
 he oral and one-fourth should be concrete. 
 
 REVIEW. Fundamentals, showing- the relation to fractions, 
 tables of ll's and 12's, lony divison with two j)lace divisor. 
 
 MAIN TOPICS: Lon}; division with three i)lace divisors, re<luc- 
 tion, addition, and subtraction of fractions, and retluction of a few 
 fractions to decimals. 
 
 ADDITION, SUBTRACTION, AND MULTIPLICATION. The drills 
 jfiven on these topics should be for speed as well as accuracy. 
 
 DIVISION. Increase the divisor to three places but no divisor 
 should exceed three places. 
 
 The special work of this {>rade is ailditioii and subtraction of 
 fractions. This involves a study of the followinfi- whicdi must be un- 
 derstood before certain phases of the work can be taken up: 
 Factoring. 
 Multiples. 
 
 Tests for divisibilities. 
 
 Least common multiple of numbers commonly found as de- 
 nominators. 
 
 Reduction of improper fractions to mixed numbers ami the 
 
 reverse. 
 
 Use such fractions as conform to business practices and show 
 that the denominator is only the name of the i)art taken. 
 
 LEAST COMMON MULTIPLE. "Develop the L.C. M. only as much 
 as required in tindinfj the least common denominator. It is really doubt- 
 ful if addition and subtraction of fractions shouM extend to fractions 
 too largfe to i>ermit pupils to determine the common denominator, by 
 inspection'". Connersville Course of Stu<ly in Mathematics. 
 
 "Much valual)le time is often lost because teachers interpret the 
 expression "by insi)ection" to mean "without any computation what- 
 ever", or "by guessing". "By inspection" must be interpreted to 
 mean "without written computation"; hence, children must be taught 
 a simi>le means of determining- the least common denominator". 
 Klapi)er. 
 
 The following is suggested as jjerhaps the easiest for the children: 
 
 "Take the first two numbers and find L. C. ]M., then tind the L. 
 C. M. of that and the next figure 4, ti, 9; L. ('. M. of 4 an.l (i is \'2: L. 
 C. M. of 12 and 9 is 3(5." Klapper. 
 
 ADDITION AND SUBTRACTION OF FRACTIONS. These may \>v 
 taught simultaneously, the order of difficulty being the same: 
 
 Similar fractions. 
 
 Dissimilar fractions. 
 
 Mixed numbers. 
 
 Simple two-ste]) problems may now be useil but recpiire no more 
 than two daily. Lead the children to decide upon the processes nec- 
 essary before beginning to work.
 
 62 
 SUBJECT MATTER. 
 
 DIVISION. Three place divisors. 
 
 FRACTIONS. Need of fractions. 
 
 Keiiiu-tioii to hifj-her terms; to lower terms, to au integer or a 
 inixe<l number; to a common denominator; to like denominators. 
 
 Addition and sul»trac-tion of similar fractions; dissimilar frac- 
 tions; mixeil nurn1»ers; application, ad<lition and subtraction problems 
 of child experience. 
 
 MEASURES. !\linutes in the hour and seconds in the minute. 
 
 DECIMALS. In the reduction of fractions all fractions with 10, 
 lUO, or lUUO as denominators may be expressed as decimals. Deci- 
 mal i)oint in V. S. money. 
 
 OPTIONAL WORK. Divisors of more than three places. 
 Adilition and subtraction of larger fractions and mixed numbers. 
 Extcnde<l drill on V. S. money. 
 
 GENERAL PROJECT. 
 
 This is a suyg'estefl project which will involve much of tlie sub- 
 ject matter of the tjrade. Lest it should not provide sufficient drill 
 to tix the i)rocesse8, minor projects are given. Tlie latter are unre- 
 lated, but might be an outgrt)wth of a larger project similar to the 
 one here given. 
 
 To Plan a Camping Trip. The class should deci<le when and 
 where to go and make the general outline of their plans. 
 
 FRACTIONS AND U. S. MONEY AND MEASUREMENTS. What 
 will be the best way to go, by train, by automobile or on foot? 
 
 If an automobile travels 1 mile in Bi min., -what would it cost to 
 make the trij) by automobile at 60c per hour for a five passenger ear? 
 
 If you are delayed by a i)unctured tire 1 1 hours, by a broken 
 axle 2} hours, by a blow-out l^'s of an hour, and by minor accidents 
 ^'s of an hour, «hat is the entire delay on yotir trip? How long, 
 then, would the trij) take? 
 
 Figure the cost of gasoline at 24c per gallon if some member of 
 the class has an automobile that could be used. 
 
 What would the railroad fare be at 8c per mile with a war tax 
 of Ic? 
 
 How long would it take to walk if the jiarty walked seven hours 
 daily at tlie rate of ] of a mile in ten minutes? 
 
 If you walk 6^ miles the first day, o!,* the second, 7] the third, 
 6| the fourth, how far have you walked in 4 days? How far are you 
 still from your destination? What was your average rate i)er day? 
 At that rate how long woidd it take you to complete the trij) ? 
 
 What car must you take to get an N. P. train that leaves the 
 rniiin Station at 9:10 a.m.? 
 
 \i the train travels 2.S miles per hour, at what time will you reach 
 your destination?
 
 63 
 
 ]\[ake a list of piolialilc ueeessities in tlic way of food and nilier 
 supplies. What food sui)plies must be taken in order to have fii(Mi<rl, 
 for 1 week':' For the entire trii)? What will l)e the (-(.st ? 
 
 If eaeh i)ers()n uses 4 lb. of butter <laily. Iiow inudi niiist lie taken 
 to supply the entire class for one week? 
 
 A soldier was o-iveu 2 oz. of butter daily. On arni\- rations, how 
 niueli butter would be needed for one week? For the entire trip? 
 
 If 2 lbs. of butter are used daily for cookijiy:, how rnueh money 
 would be saved in a week by usiiiy- butter substitutes if 1 lb of olco- 
 inar-iarine, \ lb. of lard, and l lb. of suet take the place of 1 11>. of 
 butter? lIow niiudi would be saved duriiify the entire ti-ip? 
 
 What will be the cost of 3 hams weiyhinji- respectively \'1\ lbs., 
 lOs lbs., and II f lbs., at 25c ])er pound. 
 
 If 4 cupful of Hour makes 4 <iriddle cakes, how many cui)S of 
 Hour will be retpiired to nuike enoujih cakes for all, allowing (5 cakes 
 for each person. 
 
 The following recipes are intended for 6 persons: 
 To make apple sauce, 'l cupful of sugar to (j or 8 apples." 
 Lemouade, "Lemon .juice, l cupful; water, 1 qt.; sugar, 1 cui>ful. 
 Adapt these recipes to make enough for one serving for all an(i 
 estimate (piantity of each ingredient needed and cost. 
 
 LIQUID MEASURE. If each person uses I pt. of milk daily, how 
 much will be needed each day? What will it cost at 17c a (juart? 
 
 The food value of one quart of milk is about the same as that of 
 9 ounces of round steak, or (S eggs. Look up prices and see which is 
 the most econoTuical and how" much difterence there is. 
 
 Let class choose a committee to plan some athletic games and 
 contests for the trip which may be practiced in the school room or on 
 school ground such as the hammer throw: 
 
 Use cardboai'd as paper hammer. ^Measure accurately eacli con- 
 testant's distance to fraction of an inch. Determine the wiinur by 
 fraction of an inch. 
 
 IIow' long would it take >"ou to earn and save enough money to 
 go on this tri])? 
 
 If you earn 4 of a dollar every week and si)end \ of a dollar how 
 long would it take you to save one dollai? How long to earn suHi- 
 cient money for the trip? 
 
 Make out a bill for the necessary food supplies for 1 day. 
 
 Let pupils keep itemized account of pe.'sonal exi)enditures and 
 tinally include in this their share of total expenses. 
 
 MINOR PROJECTS. 
 
 DIVISION. To tinil the amount of land each child will have tor 
 a school garden. There are 5400 S(i. ft. of land besidi's the land for 
 paths. There 254 pupils in the school but 45 have gardens at home. 
 IIow many sq. ft. will each i)Upil have, w ho makes his garden at 
 school?
 
 64 
 
 FRACTIONS. To fiinl out how imu-h siigrar I shall nee<l. 
 
 I j>laii to make a cake retiuiriiiii- \ lb., another reciuirinjuf 5 1V».. a 
 pie re(juiriujf 5 11>.. aii<l .>Juiiie cookies requiriug 5 lb. How niuchsutfar 
 must I have? 
 
 ON TMK ATHLKTIC FIELD. 
 
 To timl liow many seconds it takes a ball player to make a home 
 run, it* it takes him 4 5 seconds to set to first base, A\ seconds to get to 
 second ])ase. 4| seconds to get to third base, and 45 seconds to get to 
 the home plate. 
 
 To find how many seconds over a quarter of a minute is the 
 time taken for the home run in problem 1. 
 
 To find the difference in the length of the running broad jumps 
 of Albert whose record is 12 1 feet and Charles whose record is 14 1% 
 feet. 
 
 To timl the <litlereuce in time between a record of 4f minutes 
 for a mile run and one of bj minutes. 
 
 To fin<l the difference in length of a running track i\ of a mile 
 long and J of a mile long. 
 
 IN CAMP. 
 
 To find the increase in a boy scout's walking record if on one 
 tramp he extends his record of H\ miles to one of 14 miles. 
 
 To find the distance covered in a day by a group of Camp Fire 
 girls who walk 0} miles in the morning and 4^ miles in the afternoon. 
 
 To find how far two boys are from the end of a trail that is 6^ 
 miles long, after having climbed 4jmiles. 
 
 To find how far from the farther shore is a canoe which has been 
 paddled Ij miles, if the lake is 'If miles across. 
 
 To find the difference in time in a hundred yard swimming con- 
 test, between a record of 22| seconds and a record of 1 minute. 
 
 To find the difference in a walking contest, the winner having 
 walked a mile in S^^ minutes, the next boy in the contest took 10] min- 
 utes. 
 
 To find the number of cups of sugar Dora must borrow. Dora 
 is making jelly and needs BO cup of sugar but she has only 24^^ cups. 
 How many cups must she borrow from a neighbor to finish her jelly':' 
 
 MEASURES. See second iirojcct under Fractions. 
 
 To find out what time the street car leaves. Beginning at S:.50 a 
 car passes the house every 20 minutes. At what time after 9:00a. m. 
 can 1 catch a car? 
 
 To finil out what tiiiu' I must leave the house. If it takes 5 min- 
 utes to walk to the corner, wlicn must I leave the house to catch a 
 car about 9:30? 
 
 To find out when I shall arrive down town. It takes 23 minutes 
 to go down town. H I take the 9:50 car, when will I arrive down 
 town? 
 
 i
 
 65 
 
 DECIMALS. To order some l)()oks. (Tive the eliiM a prit-e list of 
 some child reu's books as: 
 
 The Juuffle Book §1.50. 
 Niffhts with Uncle Keinus >?1.^0. 
 Kobiuson Crusoe $.75. 
 Alice ill Wonderland n.45. 
 Grimm's Fairy Tales >;.45. 
 Arabian NiiJ^lits >!.50. 
 Hans Brinker -s.50. 
 The Dutch Twins ^M. 
 
 Fees for money orders. For orders from: 
 
 §.01 to §2.50 3 cents 
 
 S2.51 to §5.00 5 cents 
 
 §5.01 to §10.00 S cents 
 
 §10.01 to §20.00 10 cents 
 
 Select several books you would like to have and find the cost of 
 the order allowinof for the cost of the money order. Vary this by 
 ordering seeds, toys, plants, tools, etc. 
 
 Standards of Attainment. 
 
 Fundamentals, see V-A. 
 
 Reasoning-: The child should have ability to use each new acquisi- 
 tion until he has control of it; he should have a working knowledge 
 of the needed processes. 
 
 OBJECTIVE. See Grade V-A. 
 
 Bibliography. 
 
 METHOD. Brown and Coffmau: ITow to Teach Arithmetic. 
 
 Ghap. XIII, Common Fractions. 
 Kla])i)er: The Teaching of Arithmetic. 
 
 Chap. X, Pp. 21S-242. 
 Wilson and Wilson: The .Motivation of school work. 
 
 Chap. IX, The Motivation of Arithmetic. 
 
 SUPPLEMENTARY BOOKS. IToyt and Peet: Fveryday Arith- 
 metic. Book Two. 
 
 Part III, Chap. Ill, Addition and Subtraction of Fractions. 
 Part IV, Chap. IV, Addition and Subtraction of Fractions. 
 
 Stone & Millis: The Stone-Millis Intermediate Arithmetic. 
 Chap. Ill, Pp. 79-97. Fractions. 
 
 Thorndike: The Thormlike Arithmetic, Book Two. 
 
 Wentworth and Smith: Essentials of Arithmetic. Intermediate 
 Book, Chap. II. Pp. 47-7(1. Addition and Subtraction of 
 Fractions.
 
 (;1{A1)K V-A 
 
 DIRECTIONS. 
 
 I)l\ ISION (JF TIMi:. Oue-balf of tlu' aritliinetic tijiie allotineiit 
 of this yraile should l»e devoted to oral work. 
 
 One-fourth of the time should he devoted to concrete ])rol)lems. 
 
 KE\'IE\V. Keductiou, addition and subtraction of fractions; U. 
 S. money; drill on the fundamentals for speed and accuracy; the 
 tables of denominate numbers i)revii>usly learned; decimals. 
 
 MAIX TOPICS. .Alultiplication and division of fractions; tables 
 of linear, dry, and liquiii measure, weij^ht, time and U. S. money. 
 
 FRACTIONS. In teaching multiplication and its reverse pro- 
 cess, division, the procedure should be (l) multiplyinjr or dividing- a 
 fraction by a whole nucnber, 2 (a) whole number by a fraction, (8) a 
 fraction by a fraction, (4) a mixed number by a whole number, (5) 
 a mixe 1 numVjer by a mixed number. 
 
 There is no good reason why a child should remember any of the 
 explanations of the processes in fractious; it is sutticient that he learn 
 tlie operation as a rational one, and that .he can perform it quickly 
 and accurately. What we want is control of fractions, power to 
 work with them, whether with or without analytic understanding. 
 Use cancellation wherever possible after the processes have been 
 mastered. Mixed numbers should conform to business ])ractice, 64, 
 12.';, (S;^, etc. There should be much mechauical drill. 
 
 DENOMINATE NUMBERS. Much of the work in denominate 
 numbers should be oral and should be related to the daily transac- 
 tions of business life. Problems involving reductions through more 
 than three denominations are seldom used in the business world. As 
 far as possible the children should have the actual measures present- 
 eil to their senses. It is easy to make tlie mistake of talking abouJ 
 these things without children actually having a kjiowledge of them. 
 Thor«tughly memorize the tables. 
 
 -AI.KJUOT PARTS. Teach the common aliquot parts of a hun- 
 dred with their fractional equivalents. These may be used as time 
 savers in multiplication. 
 
 SUBJECT MATTER. 
 
 FRACTIONS. Multiplication, division, ( anrcllation. 
 
 DENOMINATE NUMBERS, involving tables of linear, dry and 
 liquid measure, weight, tiine and U. S. money, 
 
 ALIQUOT PARTS. Those most commonly used .12.^, .25, .50, .75, 
 .20, .10, with their fractional c(iuivalents. 
 
 OPTIONAL WORK. Work for liigher degree of speed and acc-ur- 
 acy. Solve more dithcult iimblcms. 
 
 GENERAL PROJECT. 
 
 This is a suggested project which w ill involve much of the sub- 
 j<'ct matter of tiie giade. Lest it sliould not proviile sufficient drill
 
 67 
 
 to fix the i)rocesseH, minor iirojects are tfiveii. The hitlci- ui-c unrc- 
 hiled but Miitiht be an oiittiTowtli of a lary-er jiroject similar to (lie one 
 here given. 
 
 To Plan a Recreation Park on a City Block in Duluth. 
 
 LINEAR MEASURE AND FRACTIONS. 
 
 Make a scaled (lra\vin<4- of tlie lilock. 
 
 I'ut in the drawinii' of the block a ball diamond, a tennis court 
 or anything upon which the class may decide. Make measurements 
 and calculations accurately to the fraction of an inch. 
 
 In case the class decides ui)on a base ball diamond ami a iciuiis 
 court, the following problems may grow out of the project: 
 
 If 9 board feet will make 1 yard of fence 1 board high, iiow 
 many board feet of lumber will be reiiuired to put a fence 4 boards 
 high around the entire block";' 
 
 How much will the fence cost at §B0.00 per thousand board feet? 
 
 How long a pole will be required to cut 8 posts if each jiost is 
 4s feet long':* 
 
 How many such i)oles will be required for the i)osts if the posts 
 are set 6 feet apart? 
 
 If a club of 12 meml)ers is paying the exi)en8e what is each mem- 
 ber's share of the expense of the fence around the block? 
 
 Find the cost of putting a high wire fence around the tenuis 
 court at 40c per yd. What is each member's share of tins expense. 
 
 \^'EIGHT, TIME, U.S. MONEY, LIQUID MEASURE. 
 
 Find the number of pounds of rock necessary for putting '1 two- 
 inch layers of rock on the base ball diamond if it takes 275 tons for 1 
 layer. 
 
 How much will be used in the tennis court for 2 layi'rs if the 
 court is i as large as the base ball diamond. 
 
 How many trips will the truck driver need to make to deliver 
 the rock from the nearest crusher if he can take 3 tons per load? 
 
 How many hours will he work if he travels at the rate of 10 
 miles ])vv hour, allowing 15 minutes foi- loading and unloading catdi 
 loa<l V 
 
 What will it cost to have the rock delivered if the driver charges 
 >;8.;")0 per hour? 
 
 What will be the cost of the rock for base ball diam..nd ancl ten- 
 nis c(jurt if the first layer costs '^1.2o i)er ton. the second -sl.oO per 
 ton, and a third layer consisting of 1(10 tons of tine rock costing oOc 
 l)er ton? What will be the cost of spreading if labor costs .')0c per 
 hour and oTie man can spread 2 tons in 1 hour? 
 
 In making the court and diamond, water is nee<led in i>utting on 
 each layer of rock. If it requires 133;' gallons for each layer for the 
 diamond, how much will be re(iuire<l for tlie three layers? 
 
 How much will lie refpiired for the tennis court if ii '- ' :•> 
 large ? 
 
 How iiiaiiN liairels of water will be necessary for both?
 
 68 
 
 After the water is ]»ut on. each layer of rock is rolled several 
 times hy a roller drawn liy horses. 
 
 How lonjj will it take a man to roll the three layers of the tenuis 
 court if he can rt)ll 1 layer iu 40 min? 
 
 How lontr will it take to roll the base ball diamond which is twice 
 as lar^re ? 
 
 What will be the entire cost of rollinfj for both if he charjo^es 
 >;1.1*2 i)er hour? 
 
 Make a pay-roll, keepinjf the time of the labor necessary for get- 
 ting: the rock from the crusher, spreading and rolling both the base 
 l»all diamond and tennis court. 
 
 What will be the entire expense of putting rock on both? 
 
 What is each club member's share of this expense ? 
 
 The above i)roblems are suggestive only. Many others will grow 
 out of the project and l)oth the project and figures may be changed to 
 suit the needs of the class. 
 
 MINOR PROJECTS. 
 
 FRACTIONS- To find the number of yards of matting required 
 for a ]>edroom fioor, if it requires 3 strips 4^ yds. long. 
 
 To find the number of strawberry plants in a row and the total 
 number of plants required if the bed is 1H\ ft. long and 8^ ft. wide, 
 and l-l ft. is to be allowed for each plant. Rows run lengthwise. 
 
 To draw a plan of the strawberry bed to scale. 
 
 To draw to scale of 1 ft. to | of an inch, a rectangle to represent 
 a fiower betl 8 ft. wide and 16 ft. long. 
 
 To find the value of 24 yards of dish toweling. The sewing 
 teacher charged a girl 24c for a dish towel 4 of a yard long. How 
 much did the sewing teacher pay for the 24 yds. of the cloth? 
 
 DP:.\0.M1XATE numbers. To fiu<l what commodities are sold 
 Ijy the pound, the dozen, the pint, the (luart, the gallon, the peck, the 
 bushel and the ton. 
 
 To find the cost of the number of feet of wire required to fence 
 a jjlayground. 
 
 To find the difference in paying room rent by the week or by 
 the month. If you had a room to rent for a year which would you 
 rather have §2.r)0 i>er week or §10.00 a month? 
 
 To find the number of number of half-pint bottles required to 
 bottle 3' gallons of cream. 
 
 To find how wide ycm would you have to cut a ruffle iu order to 
 have it 84 iu. wide when finished, if I ' in. is turned under on one 
 side and 4 iu. on the other side. 
 
 To find the cost of 4 oz. of ciuuamom if 1 lb. costs 40c. 
 
 To find how much money Henry had left out of a dollar if he 
 bought 6 pencils at 2 for .5c, a pencil box for 25c and a set of draw- 
 ing tools for 40c. 
 
 To make an itemized acccniut of your mother's expenses for a 
 week. 
 
 To keep an account of the time spent at some useful work dur- 
 ing the week.
 
 69 
 Standards of Attainment: 
 
 The child should he able to fultill the re(iuireiiieiits indicated ])e- 
 low: 
 
 Deterniiue all ])roces8es before proceediuy to solve hi.s prohlein.s. 
 
 Know how to check liis results. 
 
 Ilantlle liis processes iu whole numbers and fractions accurati-ly 
 and with reasonable speed. 
 
 Have memorized his denominate number tables thorou<ihly. 
 
 Read and write decimals through thousandths rapidly. 
 
 Express himself freely in arithmetical lauy-uage. 
 
 Objective. 
 
 Woody Test: Series A. 
 
 Addition 23.1 
 
 Subtraction 20.4 
 
 ]\1 ultiplication 18.3 
 
 Division 16.4 
 
 Courtis Test. Series B. Speed. 
 
 Addition 8.0 
 
 Subtraction 9.0 
 
 Multiplication 8,0 
 
 Division 6.0 
 
 Courtis Test. Series B. Accuracy. 
 
 Addition 70 
 
 Subtraction S3 
 
 Multiplication 75 
 
 Division 77 
 
 Stone Ueasouing Test: 
 
 Actual Medians Obtained. 
 
 Butte, 1914 2.2 
 
 Bridgeport, Conn., 1913 6.1 
 
 Salt Lake City, 1915 3.7 
 
 Lead, S. Dak., 1916 4.5 
 
 Tentative standards suggested by Stone (1916) for his reason- 
 ing test — 
 
 That 80 per cent or more of 5th grade pupils reach or exceed a 
 score of 5.5 with at least 75 per cent accuracy.
 
 Bibliography: 
 
 TEACHER'S READING. Drown and Coftnian: How to Teach 
 .Vrithnietif. 
 
 C'haii. XII. Denominate Numbers, 
 ("liaji. XIII, Common Fractions. 
 
 Klai>iier: The Teaching of Arithmetic. 
 
 Pp. '2o2-241. Multiplication and Division of Fractions. 
 
 Wilson: Tlie Motivation of School \V(»rk. 
 Chap. IX. [Motivation of Arithmetic. 
 
 SUPPLEMENTARY BOOKS. Iloyt and Peet: Everyday Arith- 
 metic, Book II. 
 
 Part Three, Chap. IV, Multiplication and Division of Frac- 
 tious. 
 
 Part Four, Pp. 55-()3. Multi])lication and Division of Frac- 
 tions. 
 
 Part Three, Chajt. VIII, Denominate Numbers. 
 Part Four, Chap. VII, Denominate Numbers. 
 
 Stone and Millis: Intermediate Arithmetic. 
 Chap, y, Fractions. 
 Pp. 97-125, Multiplication and Divisou of Fractions. 
 
 Thorudike Arithmetic, Book Two. 
 
 Pp. 49-55, Multiplication and Division of Fractious. 
 Pp. 90-122, Denominate Numbers. 
 
 Wentworth and Siuith: Essentials of Arithmetic. 
 Pi). ()S-97. Fractious. 
 Chap. III. Denominate Numbers.
 
 71 
 
 GIJADK Vl-n. 
 
 DIRECTIONS. 
 
 DIVISOX OF TI.MK. One lliinl of tin- aritliiiu-tic time alloliiH-iit 
 of tliis yraile .sliouM l»e devoted to oral work. One llnnl of the time 
 should be devoted to couerete problems. 
 
 REVIEW. Tal)les of linear, dry and licjiiid measure; avoinlupois 
 weight; tiine, and U. S. money; fractious; aliijuot jtarts of one-iunid- 
 red. aud the reading atid writiug- of deeimals through thousandths. 
 Keej) up drill on the fun laneutal oi)eratious for speel and aeeuracy. 
 
 MAIN TOPICS. Denominate numbers; one stej) reiluetion. aildi- 
 tion, subtraetiou, multiplication, and division. 
 
 Decimals; notation aud numeration to ten-thousandths, addition, 
 subtraction, multii)lication and division. 
 
 DEXOMLXATE NUMBERS. The work should be made objective. 
 Show that reduction asceudiug aud deceuding is similar to the reiluct- 
 ion of fractions to lower aud higher terms or to the reduction of 
 whole numbers. The reduction of 2 bu. 1 pk. to quarts is similar to 
 changing I, I to l'2ths in order that they may be combined. "Long 
 written problems in addition, subtraction, multiplication and division 
 of denominate numljers should be omitted. Problems involving re- 
 duction through more than three denominations are seldom used in 
 tlie business world". How to Teach Arithmetic: Brown an<l Coffman. 
 
 DECIMALS. Adilition and subtraction of decimals offer no diHi- 
 culties exce])t those previously met in the same process with integers. 
 In multiplication, teach that the number of places in the i»roduct is 
 the sum of the places in the multii»lier and the multiplicand, and that 
 division is reverse of this. However, see Forms for all (trades, for 
 division of decimals. The clianging of fractions into decimals should 
 be emphasized, as this is a common and useful jiractice. 
 
 SUBJECT MATTER. 
 
 DEXOMIXATE Xl'MBERS. Ad<lition, Subtraction. .Multi|.lica- 
 tion and Division. 
 
 DECIMALS. A(lditi(-ii. Subtraction. .M iillijilicat ion and Divi- 
 sion. 
 
 PERCEXTACE. Construct witli the children ami teach the fol- 
 lowing table:
 
 Common Fractions 
 
 Decimal Fractions 
 
 Parts of $1.00 
 
 Per Cents 
 
 
 50 
 
 
 
 
 
 1 
 
 100 
 25 
 
 .5 
 
 >! 
 
 .50 
 
 50 
 
 1 
 4 
 
 100 
 331 
 
 .25 
 
 -S 
 
 .25 
 
 25 
 
 1 
 S 
 
 100 
 20 
 
 .33i 
 
 >! 
 
 .331 
 
 331 
 
 I 
 
 100 
 16j 
 
 .2 
 
 X 
 
 .20 
 
 20 
 
 1 
 6 
 
 100 
 12.1 
 
 .161 
 
 8 
 
 .16^ 
 
 11)^ 
 
 1 
 if 
 
 100 
 10 
 
 .121 
 
 8 
 
 .12i 
 
 12.^ 
 
 iV 
 
 100 
 75 
 
 .1 
 
 8 
 
 .10 
 
 10 
 
 3 
 
 4 
 
 100 
 
 .75 
 
 ¥» 
 
 .75 
 
 75 
 
 OPTIONAL WORK. Work for higher degree of accuracy. 
 Solution of simple three step problems. 
 
 GENERAL PROJECT. 
 
 This is a suggested project which will involve much of the sub- 
 ject matter of the grade. Lest it should not provide sufficient drill 
 to fix the processes, minor projects are given as details under each 
 topic of subject matter. The latter are unrelated, but might be an 
 outgrowth of a larger project similar to the one here given. 
 
 What profit could be made if twenty vacant lots in the vicinity of the 
 school house were cultivated as gardens ? 
 
 FRACTIONS, DENOMINATE NUMBERS, DECIMALS. There are 
 16 lots in 1 city block. How many blocks in 20 lots ? Find answer 
 to 3 decimal places. 
 
 If there are 2.75 acres in 1 block, how many acres in the 20 lots? 
 
 If .25 of a lot makes one garden, how many gardens can be made 
 on the twenty lots? 
 
 .Make a drawing of the gardens on 1 lot, 50 by 150 ft., using a 
 scale of 1 inch =^ 12.5 ft. and leaving walks 3 ft. wide around each 
 garden. 
 
 If fertilizer is used at the rate of 250 lbs. per lot, how much will 
 l>e re<iuired for the twenty lots? 
 
 What will it cost at §30 per ton? 
 
 How much will it cost to get it from the depot if the truck driver 
 charges !?:i.50 a load and carries 3 tons per load? 
 
 If labor costs 55c i)er hour and an ordinary workman spreads at 
 the rate of 2 tons an hour, what will it cost to i)ut the fertilizer on ? 
 
 What will be the entire cost of putting on the fertilizer ? 
 
 i
 
 73 
 
 How lon<>' will it take a man to jilow tlu' ciitii't' area it" lie |)l(>ws 
 2.5 acres ])er day? 
 
 What will l>e the cost of iilowiiitr it" he ehartres >!l.l'2 jier hour":' 
 
 Wliat is the entire cost of t"ertili/in<i" and iilowinu' tin- entire area? 
 
 What is the cost per garden ? 
 
 What is the cost per lot ? 
 
 If you should decide to plant o lots to ])otatoes, how many will 
 he required if each lot takes 8 bu. 2 pk. 18 (jt. of potatoes? 
 
 What will he the cost of the potatoes at -^1.15 per bushel? 
 
 Mow many hills of potatoes can be ])laiiteil on I lot. 7") by loO 
 feet, if they are planted in rows 3.5 ft. apart ? 
 
 What will be your yield of potatoes if your lots averajj^e 45 bu. 
 8 pk. ? " ' 
 
 What are these worth at x2.25 i)er bushel? 
 
 What will be your net protit after de<lucting- cost of fertilizinjc. 
 plowing- and planting? 
 
 If you should decide to i)lant 3 lots to strawberries, how many 
 plants can you put on each lot if they are placed 1.5 ft. apart? How 
 many can be put on three lots? 
 
 What will they cost at 15c per plant? 
 
 How many lots could you plant to strawberries using your net 
 protit from the potatoes? 
 
 A strawberry bed will yield for 2 years. If the plants cost !!<2.00 
 an<l the fertilizer §1.00, how much is made from the bed in 3 years 
 (1st year it does not produce any berries) if it yiehls (50 quarts per 
 year at an average price of 10c per (juart ? 
 
 Find the cost of fencing a 20 ft. sq. strawberry jiatch at -s4.5() 
 per nl. 
 
 From 2 (jts of pea seed, 2 bushels of peas were obtained. 'J'he 
 seed cost 35c per quart and the fertilizer about 20c. How much was 
 gained when green peas were selling at an average jirice of 25c per 
 (piart ? 
 
 A package of carrot seed costs 10c. The crop was al)out 3 pecks. 
 What was saved on carrots if they were selling at an average ])rice of 
 5c per (juart ? 
 
 The asparagus bed cost -si. 50 for the roots and about -si. 0(1 for 
 ihe fertilizer. Each year about <S bunches were cut from it at an 
 average price of 25c per bunch. What would be the gain in 5 years, 
 allowing 25c per year for fertilizer? The tirst year it was jilanteil m> 
 asparagus was cut. 
 
 A dozen cabbage plants cost 20c. They re(|uired about 15c 
 worth of fertilizer. 10 heads were obtained that were worth an 
 average price of lOc each. Find the gain. 
 
 From 2 (luarts of onion sets that cost 15c jier quart, alioul >^l.2.'> 
 worth of onions was obtained. Find the gain. 
 
 The corn yielded about 20 dozen ears at an average price of 20c 
 per dozen. The seed cost 15c ami the fertilizer about 50c. What 
 amount was saved on corn? 
 
 Find the total amount savetl ?
 
 74 
 
 lli.w many busliels of ouioiis can lie l»(iu<:ht for ><'20.S0 at $ .'20 a 
 lialf pei-k. 
 
 Fiu«l the aiuouut of the followiuir Vtill: 
 
 1 pk. of potatoes at 81.60 per l>u. 
 
 2 pt. cream at 8.S0 per <it. 
 
 1 qt. ci<ler viuegar at 8.60 per yal. 
 4 o/. mustard seed at 8.40 jier 11). 
 
 The above are ouly suggestive minor problems. The class 
 should decide upou what they wish to plant in the various gardens 
 and figure accurately the cost and amount which they might reason- 
 ably expect from an average yield. 
 
 MINOR PROJECTS. 
 
 DENOMINATE NUMBERS. 
 
 To find how many years, months, and days old I am to-tlay. 
 To find the difference in the length of time it takes two steamers 
 to cross the ocean. One makes the trip in 6 days. 20 hrs. 15 min. and 
 the other in 5 days, 15 hrs. 20 min. 
 
 To find the length of fence needed for my garden. Estimate, 
 then verify. 
 
 To find how much milk is used by the family for one week. 
 To find the amount of the following bill: 
 1 pk. of potatoes at 81.60 per bu. 
 I pt. cream at 8. SO per qt. 
 1 qt. cider vinegar at 8.60 per gal. 
 4 oz. mustard seed at 8.40 per lb. 
 To find the number of tons of coal in the following deliveries: 
 
 62,500 lbs., 74,560 lbs., 84,240 lbs., 66,720 lbs. 
 To find the cost of a city lot having a frontage of 2S ft. 8 in. at 
 832.50 per running foot. 
 
 To find the cost of fencing my yard at 84.50 ]jer rd. 
 
 DECIMALS. 'i'o count by 6 tenths. Begin with 8 tenths deci- 
 mall.r and coimt by (5 tenths until told to stop. Write the numbers 
 while counting. 
 
 To count by 300 thousandths. Begin with 8 thousandths and 
 count by 800 thousandths until tohl to stoj). Write numbers while 
 counting. 
 
 To find the distance through the Panama Canal. A ship enter- 
 ing from the iVtlantic Ocean travels 7.8 miles an<l is then raised to the 
 high part of the canal, Thru this i)art the shij) goes 82.0(5 miles, is 
 lowered, and then goes 10.6 miles farther to Panama City. 
 
 To find the difference in length of the Suez ami Panama canals 
 if the Suez canal is 00 miles long.
 
 To tiinl the weifflit of tlie packsack yoii take on your cam pint; 
 trip. Vol! take the foHowiuy: 
 
 5 pktjs. of choeohite eacli \vei«jliiiii>- .'4'! lli. 
 
 6 cans of coudeused milk, each wei'rliin},'- .*2S 11). 
 B pkffs. of tea each weifrhiufr .09 11>. 
 
 4 pkgfs. of canned meat, each weiffhiny 1.1 :{ lb. 
 30 yds. of rope weig-hinfj .17 pounds i)er yard. 
 4 cans of oil, weighing 1.07 i>ound8 i)er can. 
 To find the number of bushels of onions that can be bought. At 
 >^.'20 a half peck, how many bushels of onions can be bought for -s'iO.SO':' 
 To find the length of fence around my garden in feet and ro<ls. 
 The length of fence around my garden is .25 of a mile. 
 
 To find the cost of 887 lb. of sugar at >!ll.45 per C'wt. 
 To find the average rate of speed per hour of a warship. A war- 
 shi]) makes the following record in four hours: in the first hour, 19.5 
 mi., in the second hour. 21.75 mi., in the third hour. 22.2 mi., in 
 the fourth hour, 22.9 mi. 
 
 Standards of Attainment. 
 
 The child should have the habit of estimating results well fixed. 
 
 He should have acquired reasonable speed and accuracy in the 
 fundamental operations of fractions and decimals. 
 
 He should know thoroughly all denominate number tallies taught 
 thus far. 
 
 He should be able to read and write decimals rajtiilly and with 
 assurance. 
 
 Objective Standards. 
 
 See Grade tJA. 
 
 Bibliography. 
 
 MKTHOD. 
 
 J>rown and Coffman: How to Teach Arithmetic; 
 
 Denominate Numbers, Chap. XII; 
 
 Decimals, Chap. XIV. 
 Klapjier: The Teaching of .\i-ithmetic; 
 
 Decimals, ('ha]). IV., p. 7S. ]»i). 242-258. 
 Wilson and Wilson: Motivation of School Work; 
 
 Chap. IX, Motivation of Arithmeiic. 
 
 SIPI'LEMENTARV BOOKS. 
 
 Iloyt and Peet: Everyday Aiithmctic-. Txink II: 
 
 Denominate Xumliers Part III. Chap. \'II: 
 
 Decimals. Part II. Chap. \'I: 
 
 Part IV. Chaj.. III. 
 Stone-Millis: Intermediate Arithmetic-; 
 
 Decimals, pp. 1(58-1X5. 
 Wentworth-Smith: Essentials of Aritlimetic. Intt-rme liate ISouk; 
 
 Denominate Xumbers Chap. Ill: 
 
 Decimal Fractions. Chaii. I\'.
 
 GKADE VI- A. 
 
 DIRECTIONS. 
 
 L)l\"lSION OP' TIME. Due-thinl uf the arithmetic time aUotmeut 
 of this jjrade shouhl be devoted to oral work. 
 
 ( )iie-half of the time should be given to concrete problems. 
 
 REXIKW. The fundamental operations in fractions and decimals; 
 all denominate number tables; one step reductions; simple percentage 
 tables. Many oral concrete problems should be used in the review. 
 MAIN TOPICS. 
 
 SQUARE MEASURE AND CUBIC MEASURE. 
 
 Square measure: Develop and teach the table of square measure. 
 The idea of squares should be developed by drawings, by foMiug 
 paper, and with mathematical blocks. The child should have a clear 
 mental picture of a square inch, a square foot, a square yard, a square 
 rod and an acre. Accurate diagrams drawn to a scale should be 
 made. Teach terms rectangle, quadilateral triangle, base, altitude, 
 perimeter and area. The area of a triangle equals one-half the area 
 of a rectangle having the same base and altitude. 
 
 Cubic measure: Develop and teach the table of cubic measure. 
 Develop finding the volume of rectangular solids with inch cubes and 
 other mathematical blocks. Make practical applications of volume; 
 hauling coal, selling wood, capacity of bins. Solution of prol)lems 
 shoulil be deferred until all steps liave been indicated. The children 
 should be taught to estimate results and be able to tell when results 
 are impossiltle. Make clear concepts in order that insight into fu- 
 ture work will be accurate. The clear concept is the important thing 
 in this elementary beginning of mensuration. 
 
 Hoard measure: Pieces of lumber should be brought to class 
 from the shops, measured, and the number of board feet computed. 
 The children should visit a lumber yard or a house and note the 
 kinds of lumber used. Use the process that the lumber dealer uses 
 in calculating materials required in building a house. 
 
 DECIMALS. Simple three step problems may now l)e intro- 
 duced. 
 
 PERCENTAGE. Changing decimals to i)erceuts; a continuation 
 of tlie jtercentage tables taught in VI-B. 
 
 SUBJECT MATTER. 
 
 S(|uare Measure. 
 Cubic Measure. 
 IJoard Measure. 
 
 DECIMALS. Notation and numeration to six places should be 
 developed and used.
 
 PERCENTAGE. Develo]) and teacli the t'()llowiii<r as a contiima- 
 tiou of the tables begun in the N'l-ll. 
 
 Common Fractions Decimal Fractions Part of ^1,00 Percent 
 
 
 100 
 
 
 
 
 1 
 
 66:1 
 100 
 
 .661 
 
 S.66g 
 
 66^ 
 
 
 
 40 
 lOU 
 
 .40 
 
 8.40 
 
 40 
 
 3 
 5 
 
 GO 
 100 
 
 .60 
 
 >^.60 
 
 60 
 
 4 
 
 5 
 
 SO 
 100 
 
 .80* 
 
 8.80 
 
 80 
 
 ?, 
 
 100 
 
 .m 
 
 8.831 
 
 m 
 
 3 
 
 ft 
 
 100 
 
 .87i 
 
 8.87^ 
 
 87i 
 
 5 
 
 100 
 
 .62i 
 
 8.62i 
 
 62i 
 
 8 
 
 87i 
 
 .87^ 
 
 8.87i 
 
 87 i 
 
 100 
 
 OPTIONAL WORK. Solution of more difHcult problems. Drill 
 tor more speed and accuracy in fractions and decimals. 
 
 GENERAL PROJECT. 
 
 This is a sujjfoesteil project which will involve much of the sub- 
 ject matter of the jjratle. Lest it should not provide sufficient drill 
 to fix the processes, minor projects are given. The latter are un- 
 related, but might be an outgrowth of a larger project similar to 
 the one here given. 
 
 TO FIND THE COST OF IMPROVING A NEARBY VACANT BLOCK. 
 
 It will first be necessary for the class to decide what sort of im- 
 provement they prefer to make. This will depend, of course, some- 
 what upon the character and location of the block. There may be a 
 block which is wooded and could easily be made a park, or it may be 
 ojten and suitable for skating rink. Whatever the class may decide 
 upon may be made tlie basis of the ])roject and the teacher may di- 
 rect the develoi)ment of minor i)rojects and problems to suit (he 
 mathematical reijuirements of the grade. 
 
 In the vicinity of one of our schools is a bloek which has already 
 been set aside by the city as a park. It already has a skating rink 
 and some benches and tables and two or three buildings. It would 
 be an excellent community project, however, to imjtrove this further. 
 In case a class in that school should decide to take (he further im- 
 provement of that park as a project, the following minor projects 
 might naturally come out of it.
 
 78 
 
 SQUARE MEASl'RE, BOARD MEASURE. FRACTION'S, DECIMALS. 
 Make a Vml for the contract to put a cenieut sidewalk arouud the 
 ])lock. Let jHipils do this iudepeudeutly at tirst, theu work it out as 
 a class. The habit of approxirnatiuj? auswers may l)e greatly streujjth- 
 eiieil throuyhout this project. 
 
 Find the cost of fencing. Decide on kind of fence and figure 
 lumber and labor cost; kind of fence and figure painting fence, V)uild- 
 iugs, l)enches and tables, if it requires 1 gallon to give 250 sq. ft. one 
 coat, and paint costs §2.50 per gal. Which would cost less, the fence 
 or a hedge if the hedge plants cost *6 per 100 and are set 10 inches 
 apart ? 
 
 Find the amount of grass seed necessary to seed it, deducting the 
 area of the rink, if 1 qt. of seed is sufficient for 300 sq. ft. Find the 
 cost of seeding. Let pupils find out the cost of seed, labor, the 
 length of time which it would be likely to take. 
 
 Find the amount of lumber necessary for making more tables 
 and chairs. Decide on the kind of lumber to be used and let pupils 
 find out the prices. 
 
 It may be decided to put in posts for electric lights. If they are 
 9 ft. high and 4 inches S(iure, how much lumber will it lake for 1 
 dozen jxjsts? 
 
 What will be the cost of giving them 2 coats of paint? 
 
 How much earth will be dug out for setting them if they are set 
 14 in dies deep '? 
 
 What will be the cost of putting arc lights on these posts at 
 ^4.70 per light? If each light burns 500 watts per hour, how many 
 watts will all consume in one evening? 
 
 What will the cost of lighting per evening if electricity costs 10c 
 per 1000 watts ? What will be the yearly cost? 
 
 Figure the cost of making 3 or 4 flower beds. How many canna 
 plants will be necessary for a rectangular bed 6 ft. x 30 ft. if they are 
 set 2 ft. apart in rows? What would they cost at §8 per 100 plants? 
 What would they cost at §1.25 per dozen plants? 
 
 Find the number of feet of curbing around the entire block if it 
 is 400 ft. X 300 ft. ? What would be the expense of curbing at 65c 
 per running foot ? Suppose this expense were shared by the property 
 owners across the street acconling to the number of feet frontage 
 they secured. Find the share of each of the following: A owned 198 
 feet, K 5S ft., C 6<S ft., D 150 ft., E 75 ft. and F 72 ft. 6 inches. 
 
 Estimate the amount of coal necessary for heating the warming 
 house during the months of December, January and February. 
 
 How large a bin must be built to hold sufficient coal for the three 
 months if 1 ton contains about 36 cu. ft. ? How much would be 
 saved by filling the bin in April when coal is 50c cheaper than in 
 winter? 
 
 If the care taker oversees the skating rink from 7:15 until 9:30 
 six nights in the week, what is his entire time for the week. What is 
 his time for the three months ? How much money will he draw if 
 he gets 45c per hour? 
 
 If a party of 20 people engage the rink for an evening, what 
 would be each person's share of the caretaker's wage ? What would
 
 .79 
 
 l)e a fair amount for the city to charg-e for the heatiiitr and liirhtiiit,'":' 
 A part of this hlock is low and needs to he raised on theaverHfje 
 of 6 ft. At 50c per cu. yd., what will it cost to raise this area which 
 is 40 ft. by 20 ft. ? 
 
 MINOR PROJECTS. 
 
 SQUARE MEASURE. 
 
 To find the cost of varuishingr your desk at so much per scpiare 
 yard. 
 
 To find the numher of square feet of blackboard in the room, 
 
 To find how much it would cost to plaster the walls and ceilinfj 
 of a room. 
 
 To find the area of a city lot if it has a frontage of 50 feet and a 
 depth of 100 feet. 
 
 To find the length of an acre of land if it is 10 rds. wide. 
 
 To find the cost of fencing the above field with wire fencing at 
 10c a foot. 
 
 To find the number of square yards of material it would take to 
 make a triangular pennant whose base is one yard and whose altitude 
 is seven feet. 
 
 To find the cost of sodding a lawn 35 feet by 45 feet, if the soils 
 are 1<S in. by 80 in. and cost §.08 a piece. 
 
 CUBIC MEASURE. 
 
 To find number of cubic feet of air in the room. 
 
 To find the number of tons of hard coal that can be put into a 
 bin 7 ft. by 5 ft. by 3 ft. 
 
 To find the number of cords of wood in a pile. 
 
 To find the number of bricks that were needed to build one side 
 of the school. 
 
 To find the number of loads of earth nee<led to raise the school 
 yard a foot. 
 
 To find the number of gallons of oil in a tank 4 ft. wide, ft. 3 
 in. long if the oil is 6 ft. deep. 
 
 To find the number of barrels of oil in the above tank. 
 
 To be able to read and understand the rainfall table. 
 
 To make a graph showing the rainfall in Duluth by seasons, dur- 
 ing the past year. 
 
 To find the cost of putting cement sidewalks in front of a lot. 
 The lot is 50 by 140 feet and the walk is to be ii ft wide. The best 
 walk costs §1.65 a square yard; a good ([uality costs «!. 50, and «-enieut 
 blocks can be laid for §1.25 a stjuare yard. Which kind shall be put in? 
 
 BOARD MEASURE. 
 
 To find the cost of lumber needed for a fence, a sled, a window 
 box. 
 
 To find the cost of 40 pieces of lumber, 2"'x4'' and 14 ft. lout: at 
 §35 per M.
 
 To fiu<l the cost of timber for the fouudation of your house if you 
 uee<l 10 timbers 8"xl'2" aud 16 ft. loug at x4U.OO per M. 
 To tiu<l the cost of a footstool for mother: 
 4pc8. Oakl.rxl|"x8" ) 
 
 2 " " 2l"xl"xl0" ) Oak costs 10c a board foot. 
 •2 " " '2V'xr\\7" ) 
 
 1 " Piue 7"xlO'\\r* — costiug 7c a Bd. ft. 
 
 Hi screws at lOc i)er doz. 
 lOc for staiu. 
 
 Standards of Attainment. 
 
 The child should have the power to see relationships. 
 
 lie should be able to express accejitably the solutions of his 
 l»roblems orally and in writing:. 
 
 The child should have the assurance of whether he is rifjht or 
 wrong because of well controlled means of checking his work. 
 
 He should have a well-defined concei)tion of square and cubic 
 measures. 
 
 WOODY TEST: SERIES A 
 
 Addition 29.7 
 
 Subtracticm 24.9 
 
 Multiplication 26.1 
 
 Division 23.7 
 
 COURTIS TEST Series B-Speed 
 
 Addition 10 
 
 Subtraction 11 
 
 Multiplication 9 
 
 Division 8 
 
 COURTIS TEST Series A-Accuraey 
 
 Addition 73 
 
 Subtraction 85 
 
 M ultiplication 78 
 
 Division 87 
 
 STONE REASONING TESTS Actual Medians Obtained 
 
 Butte, 1914 ^_ 3.9 
 
 Bridgeport, Conn., 1913 5.2 
 
 Salt Lake City, 1915 6.4 
 
 Nassau Co. N. Y., 1918 4.5 
 
 Lead, 8. Dak., 1916 6.1 
 
 Tentative Standards (1916). 
 
 That 80 i)er cent or more of the Sixth grade pupils reach or ex- 
 ceed a score of 6.5 with at least 80 i)er cent accuracy.
 
 81 
 Bibliography. 
 
 MICTllOl). 
 
 I)i-()\vii and C'offinaii: How to Teat-h Arithmetic; 
 
 C'liaj). XXI, Mensuration. 
 Klai)i)er: The Teacliinii of Aritlinietie; 
 
 pp. 309-820. 
 McMurray: ISpecial .Method in Aritlinietic; 
 
 Chap. IV, Method in Intermediate Classes. 
 Wilsou aud Wilsoja: Tlie Motivation of School Work; 
 
 Chap. IX, Motivation of Aritlinietic. 
 
 SUPPLEMENTARY BOOKS. 
 
 Hoyt and Feet: Everyday Arithmetic, IJook Two; 
 
 Part Four, Chap. VII, Measurements; 
 Chap. VI, Perceutag-e; 
 Everyday Arithmetic, Book Three; 
 
 Part Five, Chaj). VIT, Measurements; 
 Chaj). Ill, Percentage. 
 Stone-Millis: Intermediate Arithmetic; 
 
 Chap. VIII, Mensuration; 
 
 Chap. VII, Percentage 
 Thorudike: Arithmetic, Book Two; 
 
 Chap. IV, Measurements; 
 
 Chap. Ill and V, Percentage; 
 
 Book Three; pp. 108-110, Board Feet; 
 
 Chap. V, Measurements. 
 Wentworth-Smith: Essentials of Arithmetic; 
 
 pp. 188-209, Measuring-; 
 
 Chap. VII, Percentage.
 
 82 
 GRADE VIM}. 
 
 DIRECTIONS. 
 
 I)I\ ISION OF TIME. One-fourtli of tl)e time of tliis tirade should 
 be devoted to oral work. Seveii-eio^hths of the time should he de- 
 v<)te<l to eoucrete problems, 
 
 RKXIEW. Decimals, fractious, aliquot parts of a dollar with 
 fractional equivaleuts. chantriutj fractious to decimals and decimals to 
 fractious, aud writiug decimals as per ceuts. 
 
 Drill for accuracy aud speed in uotatiou, uuiueratiou, addition, 
 subtraction, multiplication aud division. 
 
 Denominate numbers — Oral problems for accuracy and speed in 
 solving problems related to the daily transactions of business life. 
 
 LE.ADING TOPIC. Percentage. 
 
 PERCENTAGE. One of the most frequent sources of error in per- 
 centage is due to the failure of the pupil to keep clearly in mind 
 per cent of what. In all early work in percentage the teacher should 
 re(|uire the pupil to name the quantity on which the per cent is based. 
 Tliere are no new mathematical principles involved in percentage. It 
 is only the terminology that makes it hard for the child. This subject 
 should be taught as a continuation of fractions. Commercial dis- 
 count, profit and loss, aud commission should be introduced as appli- 
 cations of percentage. The only difference between profit and loss 
 and commercial discount is, that one is based on the cost, the other on 
 the list or marked price. Teaching terms other than 'rate' is confusing 
 and should not be done. In connection with percentage, teach the 
 algel)raic equation, using the term "X", in place of the unknown 
 (piantity. After the equation has been introduced it should be used 
 whenever it will aid in the solution of arithmetical problems. All re- 
 sults should lie checked by the child. This gives him confidence in 
 his own al)ility and develops habits of accuracy. Budgets of house- 
 hold expenditures and savings should be made in connection with 
 I»ercentage. 
 
 SUBJECT MATTER. 
 
 PERCENTAGE. As pertainmg to commercial discount, profit 
 aud loss, and commission. The two important processes in the sub- 
 ject of percentage are: 1. Finding some per cent of a given num- 
 ber. 2. Findiug what per cent one number is of another. All else is 
 relatively unimportant aud on these two the emphasis should accord- 
 ingly be place<l. 
 
 OPTIONAL WORK. Solution of more difticult problems. Some 
 very simple work in grai)hing. 
 
 GENERAL PROJECT. 
 
 These are suggested projects which will involve much of the sub- 
 ject matter of the grade. Lest they should not provi<le sufficient drill 
 to fix the processes, minor projects are given. The latter are unrelated
 
 S3 
 
 but might be au uuttirowlli of larger projects similar to tlie ones here 
 giveu. 
 
 TO ESTIMATE THE COST TO THE CITY OF TAUDINESS AND ABSENCE. 
 
 PERCENTAGE. The cost of education in the U.S. for 191G was 
 §914,<S04,171 of which amount ^S.'jS, 891, 3(54 was spent for public ele- 
 mentary schools. What percent was that of the entire amount':' 
 
 The cost for public high schools was !?!<S2,3"2.'),()S9. How much 
 greater percent was spent for elementary schools than for high scho(jls? 
 
 The total number of elementary and high school children in the 
 U. S. in 191(5 was estimated to be 19,704,209. What was the cost 
 per cai)ita":' 
 
 If there were that year 14, 90S i)ui)ils in the elementary ami high 
 schools of Duluth, what percent of this entire amount was spent in 
 Duluth? 
 
 In 1917 the cost of text books in your school was S2S6.9(i. What 
 percent of the entire amount was expended for each pujiil if there 
 were 53S pupils? 
 
 The State Api)ortionmeut was §79,620. SO. What percent of this 
 did each pupil draw for attending school? 
 
 In 1917 the enrollment in Duluth was 15,725. The cost of main- 
 tenance was as follows: 
 
 Wages of janitors and helpers §44,970.29 
 
 Fuel 29,S30.4(3 
 
 Water 3,57 1 .45 
 
 Light and power 2,957.3S 
 
 Janitors' supplies 3,245.42 
 
 Services other than personal 1,107.22 
 
 Kepairs and Upkeep 1S.501.SS 
 
 Instruction 475,9t)0.93 
 
 Find total cost per capita. 
 
 Find cost of instruction per capita. 
 
 What percent of entire amount was si)ent for instruction? 
 
 What is the money loss to the district if 1 puinl is absent 12 days 
 during a semester? 
 
 Reckon the amount of money you have lost to the district by 
 absence and tardiness. 
 
 What percent is it of the amount the district expended for you? 
 
 What is the percent loss for the entire room? 
 
 What does it cost the district if you fail your grade and it is 
 necessary for you to repeat? 
 
 Find out how great a per cent of the entire 1917 cost was lost to 
 the district in your school if there was a jiercentage of failure of 3 
 per cent? 
 
 2. To tind the jtrotits a boy would make out of a garden iilot if his 
 father gave him a certain per cent of the crop for heli)ing to culti- 
 vate it. If your father gave you 15 per cent of his crop of potatoes 
 for helping to cultivate it, how many bushels would you receive, if 
 he raised 40 bushels? 
 
 If your father sold the jxitatoes at §2 a bushel what was your 
 share of the money?
 
 84 
 
 What was your share of the i)rolits if 5 I'li- (>f potatoes at -^1.00 
 lier liu. had been useil iu ithiutiiitjy 
 
 If from the frardeu you used euougli jjroduce to reduce your 
 weekly exjjeuse account from §14.50 to 811.25, what per ceut was 
 saveil':' 
 
 I low much mouey did you contribute to the family table if 15 
 l»er cent t»fthe food used belonfjed to you? 
 
 If you sold yarden jiroduce for your father on a commission of 
 80 per ceut, what would you earn in 1 day from the following sales: 
 
 Radishes, 10 bunches at 5c 
 
 Onions, 20 bunches at 7c 
 
 Peas, 6 lbs at 15c 
 
 Beans. 10 lbs at 15c 
 
 8. To tind what per ceut of the games played by the Junior High 
 School base ball leag'ue our team won. The Junior Ilig'h School 
 base ball league played 20 games in all. Our team won 18 games. 
 What per ceut of the 20 games did they win? Use as a cheek to 
 the above problem. Find how many g-ames were played by the 
 leag'ue. Our baseball team won 18 games which was 90 per ceut of 
 the whole uuml)er played. How many games were played? 
 
 To rearrange the following tables according to the excellency of 
 the teams, or, To tiud how our team stands in reg^ard to the other 
 teams. 
 
 WEST END GRADE SCHOOL 
 
 WON LOST 
 
 Lincoln _7 2 
 
 Lincoln Cubs 8 
 
 Bryants 8 2 
 
 Moil roe 3 6 
 
 EAST END GRADE SCHOOL 
 
 WON LOST 
 
 Washburn 4 5 
 
 Eudion 6 8 
 
 t'ran klin 5 4 
 
 •Jefferson 3 6 
 
 To tiud ])ei ceut of games won. 
 
 To grai)h number of games won and lost by schools. 
 
 MINOR PROJECTS. 
 
 T(j find how much Current Events will cost a class of forty if a 
 discount is allowed for thirty or more. 
 
 To tind liow much profit has been made from the garden. 
 
 To tiud li(jw much is made or lost each day froTu the sale of news- 
 l)ai)erH. Find the jter cent of gain or loss. 
 
 'I'o find what your father gained when he sold your house or 
 farm. 
 
 'i'o Hud the jicr cent of your class wiio are buying thrift stamps.
 
 So 
 
 To fiud the cost of a bicycle listed at a certain piirc, uii w liicli 
 two successive discounts are allowed. 
 
 To find the value of a book if it decreases in value "i') per cent 
 each year and has been use<l three years. 
 
 To find per cent of discount on goods daniage<l by tire oi- water. 
 
 To lind which you would rather have, if you were a salesman, a 
 stated salary or a wage plus commission on your sales. 
 
 To investigate fees for collecting debts. I employeil a lawytT to 
 collect a debt of »i!3,(iOO.()0. He succee<led in getting SU per cent of 
 it, on which he charged a commission of 5 per cent. How much did 
 I receive V 
 
 To find how much is saved by buying in large quantities. How 
 much does your mother save by buying canned goods at «1.15 per 
 dozen, instead of one at a time for l*ic. 
 
 To find how much a family will have for special vacation ex- 
 penses. A family earn §1,420.00 this year. If they spend 1(3 iier 
 cent for car fare and rent, 32 per cent for food, 18 per cent for fuel 
 and other running expenses, 10 per cent for books, music, church, 
 savings and insurance 10 per cent, how much will they have for 
 special vacation expenses? (Problems like this may be graphed.) 
 
 Standard of Attainment. 
 
 The child should have the mastery of the two principles of jier- 
 centage as applied to commercial discount, i)rotit and loss, ami com- 
 mission. 
 
 The habit of checking his own results should be firmly estab- 
 lished. 
 
 The child should have a good working knowledge of all terms 
 used thus far and shouhl be able to express himself accurately. 
 
 OBJECTIVE. See dirade 7-A. 
 
 Bibliography. 
 
 TEACHER'S READING. Hrown and (otfman: liow I- Te.i.li 
 Arithmetic. 
 Chap. XV. Percentage. 
 
 Klapper: The Teaching of Arithmetic. 
 
 P]). 2o4-2GS. Percentage. 
 
 Pp. 287-290. The Equation. 
 Wilson and Wilson: Motivation of School Work. 
 
 Chap. TX. ^lotivatioii of Arithmetic.
 
 86 
 SITPI.KMENTARV BOOKS. 
 
 Iloyt and Peet: Kveryilay Aritlimetic. l>ouk III. 
 
 I'nrt Five. CMuip. IX. Town and City IniprDveineut. 
 
 C'liait. VI, J>aiikiii«i. 
 
 Part Six, C'liaj). XI. The Equation ot" Percentage. 
 Stone and Millis: A<]vanced Aritlinietie. 
 
 Chap. IV. and VT. Percentage. 
 Thorndike: Arithmetic. Book III., Chap. II and III. .Vjjplica- 
 
 tions of Percentage. 
 Wentworth-Smith: Essentials of Arithmetic. 
 
 Intermediate Book. Chai». VII. Applications of Percentage. 
 
 Advanced Book: Chai).III. Percentage and Its Applications. 
 
 Chap. V. Banks and Banking. 
 Wentworth. Smith and Brown: Junior High School ^Mathematics. 
 
 Part 1.
 
 87 
 (tKADE VII-A. 
 
 DIRECTIONS. 
 
 DIVISION OF TIME. One-fourth of the aritliiiietie time allot- 
 meut of tliis tirade should l»e devoted to oral work. Seveii-eijrhtlis of 
 the time should l)e devoted to concrete problems. 
 
 REVTPLW. The same as for VII-]>, usiiiy more ditticult woik. 
 Review the principles of i)ercentai>e and its ajjplications. 
 Use equations wherever they will aiil in the solution of the pro- 
 blems. 
 
 LEADING TOPICS. Percentage. Taxes, duties or custi^ms. in- 
 surance, simple and compound interest should be introduced as further 
 application of percentage. The informational value of taxes, duties or 
 customs, and insurance should be emphasize*! rather than the math- 
 ematical content. 
 
 The features of taxes shoulil be presented from the stand jioint of 
 civics rather than that of arithmetic. The problems should be based 
 on local couditi(ms. Make the work simple. The pupils should be 
 familar with the terms; valuation, assessment, assessors, delinquent 
 taxes, levy, rate of taxation, and board of equalization. 
 
 The subject of national duties and revenues is closely related to 
 that of taxes and may be presented in a similar manner. The math- 
 ematical problems should be brief. 
 
 Only the common types of property and i)ersonal insurance 
 should be considered. The terms, ])remium, policy, endowment, mat- 
 urity, face of policy, and adjuster should be taught. The forms for 
 l)olicies should be obtained in order that the work may be ma le real 
 and practical. Children shouhl be shown that insurance is a matter 
 of i)rotection rather than of money making for the policy holder. 
 
 "Pupils sometimes fail to master the subject of interest, but the 
 failure maj' usually be traced to the lack of an accurate understand- 
 ing: of the terms used and of an acquaintance with business jiroceeil- 
 ure rather than to any mathematical difficulties involved. There is 
 no jjroblem in simple interest, the solution of which requires a degree 
 of mathematical knowletlge not in the i)ossession of tlie jmitil who is 
 Ijrepared to begin a formal study of the suV)ject. Simjile interest is 
 an easy application of percentage with the time element as an im- 
 l>ortant factor. 
 
 "Numerous definitions have been suggested for the term interest. 
 The statement that "interest is money rent" is i)robably as good as 
 any that have been proposed. 
 
 "The pupils shouhl understand how men. when leniling money, 
 require a certain i»ayment for the use of the money. The teacher 
 should make clear to the pui)ils how a man can aflord to borrow a 
 given sum for a year, be security for the amount liorrowed. and at 
 the end of year pay back to the lender not only the amount originally 
 borrowed, but an additional amount, which is called interest. Tin- 
 sum ui)on which the interest is l>ased is called the i)rincip:il. t.. dist- 
 tinguish it from interest.
 
 88 
 
 "The impilfi slumlil consider the various factors wliich (leterniiue 
 interest rates. Tlie rehition of interest rates to the hnv of supi)ly and 
 demand shoiihl l)e pointed out. Especial attention sliouhlbe directed 
 to the security of the h)au as a factor in deterniinintj interest rates. 
 The I'niteil States can l)orrow hirfje sums of money at a low rate of 
 interest. The teacher shouhl impress upon the pupils the fact that 
 very hiyh rates of interest are often synonymous with poor security. 
 It would not be correct to say that high rate of interest are always 
 directly associated with poor security, for where profits upon capital 
 are large, the rates of interest are high as a result of the law of supply 
 and demand. However, the teacher should caution the pui)il to investi- 
 gate with more than usual care the security of any loan when very 
 high returns are promised. In this connection some consideration 
 shouhl he given to loan sharks. 
 
 "A third factor that determines interestrates is the time for which 
 the loau is made. The rate is usually lower upon a loan for a long 
 period than for a short period. The amount of the loan is also a 
 factor in determining interest rates. The rate for a small loan is 
 often higher than for a large one." From '"How to Teach Arith- 
 metic," Brown and Coffmau. 
 
 Compound interest should be taught only in connection with 
 savings accounts and War Saving Stamps. 
 
 Solve many interest problems. Examine interest tables and 
 solve a few problems to show the use of them. 
 
 SUBJECT MATTER. 
 
 PERCENTAGE. As applied to taxes, duties, or customs, insurance, 
 simple and compound interest. 
 
 This is a suggested project which will involve much of the sub- 
 ject matter of the grade. Lest it should not provide sutticient drill to 
 fix the processes, minor projects are given. The latter are unrelated 
 but might l)e an outgrowth of a larger project similar to the one here 
 given. 
 
 GENERAL PROJECT. 
 
 To determine the cost of owning property in Duluth and vicini- 
 ty. 
 
 Let class decide upon price of lot and house. Find out different 
 kinds of taxes paid in Duluth. 
 
 Find out how the tax rate is determined and how much it is. 
 
 Fiml out how it is levied. 
 
 What will be the taxes on the house aiad lot for one year at that 
 rate? 
 
 Compare tax rate with that in Minneapolis. 
 
 Compare gas and lighting rate? 
 
 In which place can one own property at less cost? 
 
 What is the yearly interest on the amount invested in the house 
 and lot at 6 per cent? 
 
 Find out the average rental price in that i)art of the city and 
 determine wliether or not it i)ays to own property.
 
 sn 
 
 Fiud the premium ou the tire iiisuraiu-e that shouM he ithiced ou 
 the property. 
 
 Do rates vary iu different i)arts of the city? Wliy? 
 
 Are there other kinds of insurance than tire insurance? Sliould 
 there be any other insurance on the i)roi)erty':' If so find tlie cost. 
 
 If you owned property at Pike Lake wouhl you pay the same 
 rate of taxes? 
 
 Find the amount of taxes ou a cottaj^e there, after decidinj^ ou 
 cost of property. 
 
 Determine also the amount of insurance consideriutr hjcation and 
 value of property. 
 
 Suppose your father is one of live men who own cotta«-es at Pike 
 Lake. The men decide to tax themselves 8.(i per cent of the value of 
 their cottages, each one to pay according to the value of tlieir jtrop- 
 erty, to hire a man to protect them from tire and thieves. What is 
 each man's tax if the properties are valued at §2,000, §2,500, §1,800, 
 §8,200, and §2,400 respectively. 
 
 If you reckoned the interest ou the money invested as rent, how 
 large a rent does each man pay for his lake cottage if the rate of in- 
 terest is 6 per cent. 
 
 MINOR PROJECTS. 
 
 To find how moving grains, lumber, coal and other commodities, 
 are protected. 
 
 To find out w^hat the principal kinds of life insurance are. 
 
 To figure the premium on a §500 straight life insurance policy 
 for yourself. The teacher will give rates of the different insurance 
 companies. 
 
 To find the interest on the pupil's Victory Bond for one year. 
 Find the interest for five years. 
 
 To fiud what one would have to pay for the use of §20 for two 
 years at 8 per cent. 
 
 To find what the money the pupil now has in the bank will l»e 
 worth in three years, eight months. He has §85 in the bank and the 
 bank pays 3 per cent interest. 
 
 To find the interest at the date of maturity on five War Savings 
 Stamps bought iu January, 1918. 
 
 To find how the expenses of the national government are met. 
 
 To find the chief items of national cxjieuse. 
 
 To grajih the military exi)enses of the Ignited States for tlie last 
 five years. 
 
 To compare resources of Minnesota with those of Wisconsin. 
 
 Standards of Attainment. 
 
 The child should have a jtractical working knowledge of the 
 principles of percentage as ai»i)lied to commercial discount, profit and 
 loss, commission, taxes, duties or customs, insurance or interest.
 
 90 
 OBJECTIVE STANDARDS. 
 
 WOODY TEST Series A 
 
 Additiou 82.4 
 
 Subt raft iou 2S.5 
 
 M a\ t ij tlicat ion 30.6 
 
 Division 27.4 
 
 COURTIS TEST Series B-Speed 
 
 Addition 11 
 
 Sal)t ractiou 12 
 
 Multiplication 10 
 
 Divisi(:)U 10 
 
 COURTIS TEST Series B-Accuracy 
 
 Addition 75 
 
 Sul)t raction 8(3 
 
 ^lultiitlication 81 
 
 Division 90 
 
 STONE REASONING TEST Actual Medians Obtained . 
 
 r.utte, 1914 5.8 
 
 Bridgeport, Conn., 1918 6.8 
 
 Salt Lake City, 1915 8.6 
 
 Lead, S. Dak., 1916 9.8 
 
 Tentative Standard suggested by Stone (1916) for bis lieasoning 
 Tests— 
 
 Tbat 80 per cent or more of Seventb (4rade pui»ils reacdi or 
 exceed 7.5 with at least 85 per cent accuracy. 
 
 Bibliography. 
 
 TEACHER'S READING. Brown and Coffmau: How to Teach 
 
 Arithmetic. 
 
 Pp. 228-270. Percentage. 
 Klapi)er: The Teaching of Arithmetic. 
 
 Pp. 829-884. The Graph. 
 Wilson and Wilson: Motivation of School Work. 
 
 Chap. IX, Motivation of Arithmetic. 
 
 SUPPLEMENTARY BOOKS. 
 
 Iloyt and Peet: Everyday Arithmetic. Book TIT. 
 
 T^art VT. pp. 85 and 148. Taxes. 
 
 J*art V. pp. 74) 
 
 I*art VT. pj). 144) Insurance. 
 
 Part VT. pp. 78, Duties and Customs. 
 Stone and Millis: Advanced Arithmetic. ,. 
 
 Cliaj). yi. Ai)plications of Percentage. |s 
 
 'J'horndike: Arithmetic. Book III. /> 
 
 Wentworth and Smith: Ailvanced Arithmetic. 
 Wentworth, Smith and Brown: .luiiior High School Matbemetics. 
 
 Book 1.
 
 91 
 GRADE VIII-I}. 
 
 DIRECTIONS: 
 
 DIVISION OF TIME. One-fourth of the arithmetic time alh.t- 
 nieut of this {jrade sliould be devoted to oral work. Seven-eiyhths of 
 the time should be f>iveu to coux?rete work. 
 
 REVIEW. Tables of linear, square aud cubic measure. Keej) up 
 oral drill on fuudameutal ])rocesses in whole numbers, decimals, and 
 fractions. Use equations whenever they will aid in ihc solution <>f 
 problems. 
 
 MAIN TOPICS. Involution; Evolution; Mensuration. 
 
 INVOLUTION AND EVOLUTION. All that is necessary to teach 
 under involution in thisyrade is the squares of the numl»ersto twenty- 
 live. The children should be taujiht a quick method of squarin}»- any 
 number and then should scpiare and memorize the numbers to twenty- 
 live. Knowing these squares will save much time in comi)utation of 
 problems later on. 
 
 Square root is needed in much of the work in mensuration. Jf it 
 is applied to the solution of actual problems, the cliildren will like it. 
 Many oral problems should be given, usino- the numbers to tweuty- 
 Hve, and numbers the square root of which may be easily found by 
 inspection. Square root should be taught as a process of finding tlie 
 side of a square whose area is knoAvn. After that idea is tlioroughly 
 fixed the mode of finding roots by grouping the factors of the number 
 should be taught. The meaning of jtower, exponent, scpiare, the si(uare 
 root, and radical sign should be taught. Children may be asked to 
 raise numbers to the third, fourth, or fifth powers. Cubes to twelve 
 should be found and learned. The pupils should work enough ex- 
 amples to acquire facility in finding the square root. The most im- 
 portant thing in this work is that the pupils get a clear concept of a 
 square. It should also be impressed upon them that only abstract 
 numbers can be squared, because a scpiare is the product of two ecpial 
 factors. The abstract number 9 is a perfect Sfjuare. but -^9.00 is not. 
 
 MENSURATION. The teacher should be sure that she kn(»ws 
 why mensuration should be taught before she attempts to teacli it. 
 ^Mensuration should be taught partly because of its constructive 
 thought value, pai'lly because of its immediate usefulness; but more 
 especially V)ecause of its value in intei'preting the "natural features of 
 
 tthe world.'' "It is not a habit subject to be acquired through drill, 
 but a thought subject to be developed."— IJrown aii<l Coffman. The 
 children must thoroughly understand all terms an<l measurements 
 used. They should be drilleil in estiniating distances and areas until 
 they can do it with a high degree of accuracy. .Ml measurements 
 shouhl l)e accurate and all drawings shouM be ma.Je to an accurate 
 scale. Inaccurate measurements an<l drawings are worse than none 
 at all. The i)rinciples of mensuration should be demonstrated and 
 illustrated. Formulas should Ite developed by the teaclier witli the 
 pupils. The children should never be asked to use a formula winch
 
 92 
 
 they have mechanically memorized. The formula is au outgrowth of 
 the solution oi the problem. After a i)riiu-ii»le has heeu mastered hy 
 the pupils, a formula shou d he developed aud used. This work is 
 «lesiiruetl to illustrate very clearly the value of indicatiufj all of the 
 steps iu a problem before auy solution of it is attempted. Here also 
 is au opportunity for more extensive use of the equation. In apply- 
 iu}^ the principles of meusuratiou to. practical problems, only the 
 modes of computatiou of actual business life should be used, Use as 
 many short cuts as possible. 
 
 SUBJECT MATTER. 
 
 AREAS, I.WOLITIOX, E\'OLUTIOX. 
 
 To estimate the cost of papering, painting and carpeting a 
 five-room house. 
 
 GENERAL PROJECT. 
 
 How many sq. y<ls. of linoleum will it take to cover the kitchen 
 if it is 12 ft. sq., allowing 15 sq. ft. for Hoor space, occupied by built- 
 in cupboards. What will the linoleum cost if it is 2 yds. wide, and 
 costs xl.75 per running; yard? 
 
 Find the cost of painting- the floor if a pint of i)aiut will cover 
 about 9 sq. yd. of floor space in painting- the first coat and about 12 
 sq. yd. in painting the second coat, the paint costing 60c a quart. 
 
 Find the cost of painting the kitchen walls and ceiling using 
 paint as in the preceding problem, if the walls are 7 ft. high. 
 
 If the floor area of a square kitchen is 576 sq. ft. how long a piece 
 of carpet would it take to reach from one corner to the corner diagon- 
 ally opposite ? 
 
 If the floor area of a square kitchen is 169 sq. ft. how many feet 
 of moulding must l)e bought';' 
 
 How much would it take if the floor area is 108 sq. ft. and the 
 kitchen is 12 ft. long? 
 
 Upon which floor could you lay linoleum which is 2 yds. wude» 
 with greater economy? 
 
 The living room is 22 ft. by IS ft. It has a fireplace across one 
 corner which cuts off 4 ft. from the width and 5 ft. from the length 
 of the room. Find the length of the tiled area in front of the fire 
 jdace. Find the area of the floor de«lucting the area cut off by the 
 fire i>lace. 
 
 What is the largest size square rug which could be used in the 
 room if it lacked at least 1 ft. of touching the base board all around ? 
 
 Find the area of the room if a book case were put across the 
 corner a<ljacent to the fireplace, cutting off the same distances from 
 the width and length as the fireplace does. Make a drawing of the 
 room using the scale ■' in. to 1 ft. 
 
 Find the cost of carj^eting the room with cari>et 27 in. wide at 
 %1.75 per yard, if the tiled area in front of fireplace is 18 in. wide. 
 
 Find the cost of jjapering the room without the book case, if a 
 douljle roll of paper costing §1.25 is 16 yds. long aud 18 in. wide, 
 anrl the paper hanger charges >»1.00 per double roll. Allow for no 
 waste in matching paper.
 
 98 
 
 Fiud the uuinber of hoard feet of lunilicr in a square i)iece of 
 wood larg-e enoutfli to make tlie t(jp of a circular taldc :{() in. in 
 diameter and I of an inch tliick. 
 
 Fiud tlie dimensions of tlie top of a square table with the same 
 area as the circular table in the preeedinfj problenj. 
 
 The living room has a rectanijular window 4 ft. ])y <i ft. Fiu<l 
 to three decimal ])laces the side of a square window which wouM ad- 
 mit the same amount of light. 
 
 Tn a well lifflited room the window space should be at least ,', of 
 the floor space. How many windows like that in the i)recedin<,'- prol)lem 
 shouhl the living room haveV 
 
 Each i)erson breathes on an average of 85 cu.ft. of air in an hour. 
 If the sleeping room is 14 ft. long, 12 ft. wideand 7 ft. high, how long 
 will the air in it supply 2 persons. 
 
 The hallway is 3 times as long as it is wide and has an area of 
 108 sq. ft. What are its demensions? 
 
 IIow many tiles 6 in. by 4 in. will it take to cover the hall flour? 
 
 A tile floor according to this design is put on the floor. Find 
 the area of the white tile; of the black tile. 
 
 I 
 
 Z Z" X Z' 
 
 1^ ^^^^^ ^^^^P 
 
 In the dining room there is a r«ig 12 ft. by 10 ft. It leaves a 
 margin 2 ft. in width on all sides. What are the dimensions of the 
 floor? Make a drawing on the scale of \ in. to 1 ft. 
 
 What is the area of the floor? 
 
 That part of the floor outside the rug and that covered by it to a 
 width of 1 ft. on all sides is to be varnished. What is the area of 
 this portion? 
 
 What will the varnish cost if it takes 1 gal. f..r every 2(MI s.j. tt. 
 and the price of varnish is *4.7o per gal. 
 
 There is a wainscot in this room which is 8 ft. high. Find its
 
 94 
 
 area. It the ruuiu i;; 7 ft. ljiy:li, ^ud the remaiuiu<; portion of the 
 walls. 
 
 MEASUREMENT. 
 
 Define terms of a circle; circumference; radius; diameter; arc; 
 sector. 
 
 Find the ratio of circumference to the diameter by measuriujj the 
 circumference and diameter of cylindrical bodies, dividing and aver- 
 aging quotients. Call this ratio pi ("n"). Use 31 as its value in 
 rough calculation, and 3.1416 for accurate calculation. Teach the 
 l)ui>ils to use judgment as to which to use. 
 
 Develop the formula for finding the circumference when diameter 
 is given, to fiml the diameter or radius when the circumference is 
 given. 
 
 Develop the formula for area of circle by dividing a circle into 
 twelve or more equal sectors and fitting them together to form a 
 rhomboid. 
 
 Projects. 
 
 To find the ratio of the circumference to the diameter by divid- 
 ing the circumference of a dollar, tin cup, or face of the clock, by its 
 diameter. 
 
 To average the results obtained by the entire class and compare 
 that average with 8.1416. 
 
 To work out formulas for finding the circumference, diameter, 
 radius and area of the circle. 
 
 Develop concrete problems from the general project or decide 
 upon another project to use. 
 
 BOARD MEASURE. 
 
 A continuation of the board measure begun in Grade VI- A. 
 Develop formula for finding board feet. 
 
 Project. 
 
 To estimate the numljer of board feet of lumber required to 
 build the framework of a two room coom cottage at Sunset Lake. 
 Problems of Industry secured from the Shop Teachers. 
 
 Standards of Attainment. 
 
 Through the long continued use of checks the pupil should by 
 this time be wholly reliant ujjon himself for accuracy and results. 
 
 The jtupils should have acciuired the habit of analyzing his 
 proVjlems thus: What is known; what is wanted; what is the best 
 method of jirricedure. 
 
 He should be able to work quickly, accurately and economically.
 
 95 
 Bibliography. 
 
 TEACHER'S READING. 
 
 TJrown and Coffnian: How to Teach Aritliiiit'tic. 
 
 Chap. XIX. Involution and Evolution. 
 
 Chap. XXI. Mensuration. 
 
 Chap. XXIII. Short Cuts. 
 Klapper: The Teaching of Arithmetic. 
 
 Pp. 309-820 Measurement. 
 Wilson and Wilson: Motivation of School Work. 
 
 Chap. IX. Motivation of Arithmetic. 
 
 SUPPLEMENTARY BOOKS: 
 
 Hoyt and Peet: Everday Arithmetic, Book III. 
 
 Chap. IX Powers and Poots. 
 Stone and Millis: Advanced Arithmetic. 
 
 Pp. 178-l.S(j Square Poot. 
 Thorndike: Arithmetic. Book III, 
 
 Pp. 19<S-204 Squares and Cubes. 
 Wenworth and Smith: Essentials of Arithmetic. 
 
 Advanced Book, Chap. VII. Square Poot and Mensuraiotn. 
 Wentworth, Smith and Brown: Junior IIi<jh School ^Mathematics.
 
 96 
 GRADE VI 1 1- A 
 
 DIRECTIONS. 
 
 DIVISION OF TIME. Oue third of the Arithinetie time allot- 
 ment of this grade should be devoted to oral work. 
 
 Seven eighths of this time should be given to concrete problems. 
 
 REX'IEW. Tables of linear, square and cubic measure; s<iuares 
 to tweuty-tive, cuV)es to twelve; square root, area of triangle; area, cir- 
 cumference, diameter, ami radius of a circle. 
 
 Keei» u]) <lrill on all fundamental operations for rapidity and 
 accuracy. 
 
 MAIN TOPIC. Mensuration of Solids. 
 
 MENSURATION. 
 
 Lateral Surface. In teaching the lateral surface of prisms, cylin- 
 deas, pyramids and cones, make forms of stiff paper which, when 
 flattened out, will be easily recognized by the pupils as rectangles, 
 triangles, or sections of a circle. They will readily see how the 
 formulas i)reviously learned can here be applied. Teach terms base, 
 slant height, lateral edge and lateral surface. 
 
 Volumes. Rectangular solids must be imaged as made up of 
 equal layers, composed of equal rows of unit cubes. The teacher 
 should build various rectangular solids of small cubes until the child- 
 ren see that the volume of a rectangular solid is equal to the pi'oduct 
 of the number of units of its three dimensions. Another way to get 
 them to see it is by placing a number of these cubes in the form of a 
 rectangle so many cuVjes long, so many cubes wide, and one cube high. 
 They will easily see that the number of cubes here is identical with 
 the number of squares in the area of the base. Then build on three 
 more layers and the solid will then contain three times the number of 
 cubic units on the base. Thus they have found out that the number 
 of square units in the area of the base times the number of linear 
 units in the height equals the number of cubic units in the volume of 
 a rectangular solid. From this the formula for finding the volume 
 of rectangular solids and cylinders may be deduced. 
 
 To find the volume of a pyramid, compare it with a prism of the 
 same base and altitude, by filling the pyramid with sand and pouring 
 it into the prism. The volume of a cone may be compared with that 
 of a cylinder in the same way. In mensuration, models should not 
 be used any longer than is absolutely necessary- As soon as the 
 children can projicrly image the figure under discussion the object 
 should be dispensed with. It is not imjiortant that the children re- 
 member the formulas for finding areas and volumes of prisms, cylin- 
 ders, pyramids and cones, but it is important that they learn to in- 
 terpret and use formulas. Much may be done with the equation in 
 this work.
 
 SUBJECT MATTER. 
 
 J^ateral area, voliinie and cajiacity of in-isms, cyliniltTs. iiyraniiils, 
 and coues. 
 
 GENERAL PROJECT. 
 
 To estimate material used in constructing a school building. 
 
 Make measuremeuts of school lirounds and Imildiny- and draw a 
 plan to scale. 
 
 How many loads of earth were excavated in makini; the base- 
 ment, if the loads averajjed 86 cu. ft.? If this earth was spread even- 
 ly over the school grounds, how deep was it covered? 
 
 Find the number of bricks required for the walls, if it takes 14 
 bricks per sq. ft. Deduct for windows. 
 
 Find out as accurately as possible the height and width of wall. 
 
 Find the amount of cement used in the foundation. 
 
 How many sq. ft. of glass in windows of the building? 
 
 Find the radiating surface of the steam pipes in one room. 
 
 How much radiating surface is there for every cu. ft. of space in 
 the room? 
 
 From this result estimate the radiating surface required for the 
 entire building. 
 
 If each pupil requires 35 cu. ft. of air per hour, how many ])U]»ils 
 should be seated in your room if the air is changed every houry 
 
 How much air must be brought in every minute to supply the 
 pupils in your room? 
 
 Find out the following: 
 
 How many cu. ft. of space per pupil there is in your room? 
 
 How much paint was used in painting the lower part of the hall 
 walls if 1 gal. of paint covers 200 sq. ft.? 
 
 How many tons of coal are used during 1 year in your school 
 building. 
 
 How large a bin had to be made to contain sufficient coal for the 
 entire year. 
 
 If the price of coal next year is 8 per cent more than last year, 
 how many more dollars will the coal supply cost? 
 
 Make out a thirty day note for the price of the coal, payable to 
 the nearest coal dealer at the Western State Bank. Discount it at (i 
 per cent. 
 
 How many tons of hard coal can l>e put into an inverted i»yra- 
 midal hopper, which is used to feed a furnace, if the hopper is (3 ft. 
 on a side, and its altitude is 12 ft.? (35 cu. ft. of hard coal e<iuals a 
 ton.) 
 
 What is the cost at ?!l.l5 per s(j. yd. of painting the pyramidal 
 tower if the base is a s<iuare 9 ft. on a side, and the slant height is 1.) 
 ft.? 
 
 How many sq. ft. of plastering are there in your room? 
 
 If your room represents the average, how many are there in all 
 the rooms in your building. 
 
 How many loads of gravel were used on the grounds it 1 load 
 equals 86 cu. ft.?
 
 98 
 
 How nuK-h ceineut was used for the sidewalks if the cement was 
 H in. thick? 
 
 What did they cost at (j'2c per S(i. yard? 
 
 How many lots 50 ft. by 150 ft. could lie made of the school 
 grounds? 
 
 Find out the average value of lots in your vicinity and determine 
 the value of the school grounds. 
 
 What tli<l the curb around tlie grounds cost at *i7c a foot? 
 
 If the heating jtlant in your buildings is large enough for 20 
 rooms and there are only 14 rooms in the building, what percent of 
 its cai>acity is unused? 
 
 What did the i)lant cost if it was bought for §1*250 with 80 per- 
 cent and 10 percent off? 
 
 At another firm it could have been bought for §1150 with 20 per- 
 cent and 5 percent off. Which was the lower price? 
 
 Find out the dimensions of the steam boiler and determine how 
 many gallon it contains? 
 
 Determine the number of sq. ft. in the lateral surface of the 
 V)oiler? 
 
 BANK DISCOUNT. The children should see the only difference be- 
 tween bank discount and simple interest is that in one case the inter- 
 est is paid when the money is borrowed, in the other the interest is 
 paid when the money is paid back. 
 
 Interest paid in advance on a note is called discount. The face 
 of the note less the discount is called the proceeds. 
 
 Project. 
 
 To make out a 30 day note for S350, dated today, payable to 
 Frank Lee's order at some bank. Discount it at 5 percent. 
 
 BUSINESS ARITHMETIC. Common businessforms, bills, receipts, 
 statements, bills of lading. Keeping personal accounts, debit, credit, 
 and balance. Inventories, purpose of inventories, when and how 
 taken. Sendingmoney. Investing money. 
 
 The work in business arithmetic should be carried on largely 
 from the informational side. The children should know the forms 
 and jn-actices in daily use. However, only the simplest and most com- 
 mon forms should be considered. If possible, representative business 
 men should be secured to talk to the children, especially on banking 
 and investing money. The children should be taught to beware of 
 "(Tct-rich-quick'" jjropositions. The element of safety is the first con- 
 sideration of any investment. The security of a loan, for instance, is 
 to be considered before the rate of interest. It is always better to 
 take a lower rate of interest than to take a risk. In general the higher 
 the rate of interest the greater risk. Security is required by those 
 loaning money. Teach the meaning of the terms real, collateral, and 
 l>ersonal. 
 
 A few realistic problems should be solved to illustrate the differ- 
 ent topics as they are taken up.
 
 99 
 MINOR PROJECTS. 
 
 Usiug the foUowiiiff sample price list, estimate the cost of supplies 
 in j'our room for one year. 
 
 Composition books *4.S0 jjer ^ross less 5 percent. 
 
 Drawing paper 1.80 per doz. pkgs. less 10 i)er cent. 
 
 Drawinfj pencils 4.80 per gross less '20 per cent 
 
 Penholder 8.40 per gross less 12 per cent. 
 
 Pens 6.TC per gross less 2.5 per cent. 
 
 Rulers 40c per doz. net. 
 
 Estimate the cost in the entire building for one year. 
 
 To write a receipt for Junior lied Cross dues. 
 
 To keep personal accounts. 
 
 To make an inventory of contents of the room. 
 
 To write a check to pay for a lost book. 
 
 To make out a promissory note payable to the Athletic Associa- 
 tion or Manual Training de])artment. 
 
 To graph certain stock quotations for a week. 
 
 To find out what causes the fluctuations which your graph shows. 
 
 To find how" much it would cost to insure a bicycle for a year. 
 
 To borrow money from the school bank, give a note and find the 
 proceeds. 
 
 OPTIONAL WORK. 
 
 Solution of algebraic problems of one unknown quantity. 
 Drill for a higher rate of si)eed and accuracy. 
 
 Standards of Attainment. 
 
 The pupils should have the habit of checking by some certain 
 means fully established. 
 
 They should use common sense rather than recall some mean- 
 ingless rule or formula. 
 
 They should have the ability to interpret and use formulas. 
 
 They should be able to use the laws of equations in solving 
 problems according to a good form developed from accurate thinking. 
 
 The habit of looking for short cuts should be well established. 
 
 The pupils should by this time be wholly reliant upon themselves 
 for accuracy of results. 
 
 OBJECTIVE STANDARDS. 
 
 WOODY TEST. Series A 
 
 Addition 33.9 
 
 Subtraction 31.7 
 
 Multiplication 32.9 
 
 Division 30.9 
 
 COURTIS TEST: Series B -Speed 
 
 Addition 1- 
 
 Subtraction 13 
 
 Multiplication H 
 
 Division H
 
 100 
 
 COURTIS TEST Series B— Accuracy. 
 
 A<l<litiun Tt). 
 
 Sulitractiou !^7. 
 
 Multiplication <"^1. 
 
 Division 91. 
 
 STONE REASONING TESTS Actual Medians Obtained 
 
 r.utte, 1914 7.7 
 
 r>ri(lg-eport, Couu., 1918 4.5 
 
 Salt Lake City, 1915 10.6 
 
 Lead, S. Dak., 1916 11.4 
 
 Nassau Co., N. \\, 1918 8.2 
 
 Tentative Stamlards sug-ffeste<l hy Stone (1916) for his reasoning 
 tests: That ^0 per eent or more of the Eijj-hth grade pupils reach or 
 exceed a score of 8.75 with at least 90 per cent accuracy. 
 
 Bibliography. 
 
 METHOD. TEACHER'S READING. 
 
 Brown and Coffman; How to Teach Arithmetic 
 
 Chap. XXI Mensuration 
 
 Chap. XIX Involution and Evolution 
 
 Chap. XVII Banking 
 
 Chap. XXIII Short Cuts 
 
 Chap. IX Waste in Arithmetic 
 Judd: Socializing Arithmetic 
 Klapi)er: The Teaching of Arithmetic 
 
 pp. 299-820 Mensuration 
 
 pp. 821-826 Business Forms 
 
 l»p. 886-877 Need of Scientific Standards 
 Wilson and Wilson: Motivation of School Work 
 
 Chap. IX Motivation of Arithmetic 
 
 SUPPLEMENTARY BOOKS. 
 
 Alexander and Dewey: Arithmetic 
 
 Durrell: Arithmetic, Book III 
 
 Harvey: Practical Arithmetic 
 
 Iloyt and Beet: Everyday Arithmetic, Books Two and Three 
 
 Iloyt and Peet: Business Arithmetic 
 
 Thorndike: Arithmetic, Book III 
 
 Vosburgh and (rentlemen: Junior High School Mathematics: 
 
 Business Arithmetic 
 Weutworth, Smith & Brown: Junior High School Mathematics.
 
 DIRECTIONS. 
 
 Kil 
 (JKADK IX-i;. 
 
 I 
 
 DIVISION OF TIME. Oiu'-tliinl of tlu' time alL.tiiiciit f-.r aritli- 
 nietic ill this yrade sliouM l)e uiveii to intensive <liill woik in lapiM 
 eak'ulatiou. 
 
 Mueli of the remainder of tlie time .should l»e t,'-iven to .solvin<r 
 practical problems without a pencil. 
 
 MAIN TOPICS. Additiou, subtraction, multiplication, ami ili\ i- 
 sion of whole numbers, fractions and decimals. 
 
 Percentage. 
 
 FUNDAMENTAL OPERATIONS IN WHOLE NUMBERS. 
 
 This course is designed primarily for pupils takiuff the commer- 
 cial courses, who, therefore, need to be able to solve accurately and 
 quickly the problems arising in ordinary business transactions. The 
 work is not to be considered in the nature of a review of the work (»f 
 the grades. It should be taken up and carried on as an advanced 
 subject. Something of the history of the origin and development of 
 numbers and the processes used in arithmetic should be given. Un- 
 usual care should be taken to develop skill in the use of the tools of 
 arithmetic. Special attention is called to the constant appeal to 
 the pupils to become independent of the pencil in solving 
 problems, hence due amount of practice should be given in oral work. 
 Short methods which are practical should be used. One good short 
 method should be found and that only should be taught. Since abso- 
 lute accuracy is a prime essential in business transactions verification 
 and checking should be practiced whenever jK)Ssible. 
 
 The classes in commercial arithmetic should be tested at the Ije- 
 ginniug of the year, so that the needs of each individual may be 
 known to himself and to the teacher at once. The pupils should then 
 be grouped and work assigned to suit the needs of each group. There 
 is always work for those who excel as well as for those who <lo not. 
 
 Waste no time. Go to work the first day. Plan each day's work 
 so that the pui»ils will feel when the lesson is over that it has been 
 worth while. Because this work is mechanical is no excuse for its 
 l)eing dull and uninteresting. Even the oMer i)uitils enjoy the game 
 an<l play idea. The commercial arithmetic teacher must keeji ahead 
 of her pupils in skill and accuracy. 
 
 ADDITION. Master the forty-five combinations. Find groui»s of 
 figures aggregating ten and twenty in the body of a column as in the 
 following: 
 
 See Moore and ^Miner fori»racticei>roblenis 
 of this kind. The children should be taught 
 to look for easy ways of doing things. The 
 work should be varied to keej) up interest. 
 
 In all written work make jilain, legible 
 
 figures of a uniform size, write them equal 
 
 distances from each other, ami be sure that the 
 
 units of the same onler stand in the same ver- 
 
 6 tical column. 
 
 4 
 •> 
 
 2 
 
 17 
 
 o 
 8 
 
 2 
 
 4 
 
 
 9 
 
 8 
 
 7 
 
 9 
 
 8 
 
 2 
 
 
 
 1 
 
 
 5
 
 102 
 
 "In Imsiuess it is imi)ortant that tig-ures be made rapidly; 
 l»ut raj»i(lity should never be secure<l at the exi)eiise of leg:ibility." 
 ---More aud 3Jiner. 
 
 Casting- out 9's is a coniinou cheek tor addition. But the sim- 
 plest way of checking- a<Mition is to add the columns in reverse order. 
 If the results obtained by both processes agree, the work may be as- 
 sumeil to l>e correct. The following is an illustration of this method: 
 
 34 54966 40 
 
 40 78728 18 
 
 87 47929 37 
 
 18 78425 40 
 
 40 45623 34 
 
 383920 78249 383920 
 
 Sl'BTRACTIOX. Use the "making change" method for the sake 
 of uniformity. If a pui)il uses the "take away" method skillfully do 
 not require him to chang-e. 
 
 MULTIPLICATION'. Multiplication should be presented as a 
 short method of aildition. Use some practical short methods and ali- 
 quots. Learn some good method of checking the work. Casting- out 
 9's is sugg-ested. 
 
 DIVISION. Teach the meaning- of division, as: 
 
 X24.00 divided by 'S2.00 equals 12, the number of times $2.00 is 
 contained into §24.00, 
 
 824.00 divided by 2 equals 812.00. If 824.00 is divided into two 
 jiarts. each jiart contains 812.00. 
 
 The work in the four fundamental operations given in Moore 
 and Miner's Practical Business Arithmetic is good and may well l)e 
 followed. 
 
 FRACTIONS AND DECIMALS. Combine fractions aud decimals 
 and show that decimals is but another way of writing fractious. 
 
 In addition and suVjtractiou of fractious emphasize that only like 
 numbers are added and subtracted. Special attention should be given 
 to cancellation. 
 
 In all the operations iu decimals make sure that the pupils know 
 exactly where to place the decimal point. Fractions with impossible 
 denominators and com]»lex fractions should be omitted. Only such 
 fracti(tns and decimals as are actually used in business transactions 
 shduM be used. 
 
 PERCENTAGE. Teach the subject as outlined in ]Moore and 
 Miner's Practical Business Arithmetic. Be sure that the pupil under- 
 stands that a jjercent of a uumljer is so many hundreilths of it. To 
 illu.strate. 6 i>er cent eiiuals iJio, or. ,06, wiitteu decimally. Under 
 percentage take uj) loss and gain, marking goods, commercial discount 
 and interest. In commercial discount use the sum an<l difference 
 methoil for oral prolilems and cancellation for written problems. In 
 interest use the 60 <lay method and develop by the cancellation 
 method.
 
 103 
 
 lu connection with the work in i)ercentatre use {frai»hin<r. Tlie 
 pupils shouhl make <irai)hs until they are aide to understand an<l np- 
 ]»reciate the graphs so commonly used now in niatfazines and the 
 daily papers. 
 
 SUBJECT MATTER. 
 
 ADDITION'. l{ai)id adilition depends entirely ujton the aliility to 
 group. The same principle which is employed in reading a sentence 
 applies to aihling a single cohunn. ''From two to four figures should 
 be read at sight. as a single number, and the group so t'orme«l should 
 be rapidly combined with other groups until the result of any given 
 column is determined." — Moore and Miner. 
 
 Only correct results can be accepted, of course. Accuracy must 
 never be sacrificed for rapidity. Add horizontally as well as verti- 
 cally. 
 
 PROJECTS. To find how I rank in addition if the following 
 groups should be read at the rate of 150 per minute. 
 
 1 1 2 2 4 1 3 3 4 3 1 4 2 4 7 S 9 S .5 G 4 5 5 7 
 1 3 1 2 1 5 2 3 2 6 7 3 5 6 7 9 9 <S 5 1 4 3 4 2 
 
 1 
 
 5 
 
 6 
 
 6 
 
 8 
 
 9 
 
 8 
 
 7 
 
 7 
 
 4 
 
 9 
 
 ( 
 
 6 
 
 7 
 
 5 
 
 3 
 
 o 
 
 4 
 
 5 
 
 7 
 
 
 
 8 
 
 6 
 
 (5 
 
 9 
 
 6 
 
 1 
 
 2 
 
 3 
 
 5 
 
 8 
 
 3 
 
 8 
 
 7 
 
 9 
 
 9 
 
 8 
 
 9 
 
 9 
 
 8 
 
 4 
 
 2 
 
 All the possil)le groujis of three figures each are given on pages 
 14 and 15 of Moore and Miner. 
 
 To determine my ability to add if the sums of the grouits of 
 three figures each should be name<l at the rate of 120 per minute. 
 
 To graph my ability to add as compared with what my ability 
 should be. 
 
 To determine who in our group can first get to the to]* <>f a 
 column of figures with the correct result. 
 
 To see how fast I can name the results in the following l)y taking 
 advantage of the lO's and 20's wherever possible: 
 
 1 2 7 () 7 5 2 3 2 4 1 9 1 2 3 
 9 8 3 4 3 4 8 5 7 3 1 2 (J 2 S 
 7 4 5 2 9 9 7 5 3 3 S 9 3 9 9 
 3 () 5 8 1 1 3 4 5 7 1 4 (5 9 7 
 
 2 7 7 3 8 tj 4 6 5^2 (j 1 2 
 
 2 1 3 5 2 
 
 To see which group can obtain the highest per cent f«>r a-lding a 
 certain number of groups of a given length in a given time 
 
 To see how well and how rapi<lly I can write the t<. 11. .wing from 
 rapid dictation: 
 
 ?!75.18. .S123.95. !sl47.25. S9.50. ?;1S1.5(), .sl72.2»i, x,s.-).!»s. .s:{24.!t7 
 ?i48.lU, x(j9.20, x312.«i0, x415.90. 
 
 9 3S 
 
 25 
 
 4 S 
 
 4 
 
 7 ] 
 
 1 
 
 U 1 
 
 3
 
 104 
 
 To prove that I have writteu tlie sums c-orrectly l)y addiiiy aud 
 compariufjf with the correct result: 
 
 To <leterraiue by casting out O's whether tlie result iu the 
 t'ollowius: exaini»le is correct: 
 
 54,(j(jy 
 15,218 
 36,425 
 45,325 
 6S,619 
 
 220,256 
 SUBTRACTION. 
 
 Projects: 
 
 To tiud how much change I should receive from §1.00. if I spent 
 §.74. 
 
 To tind the coins which I must give in exchange for a S2.00 bill, 
 if some one has jiurchased §1.19 worth of flowers. 
 
 To find how much money I have left iu the school bank if I had 
 §200.00 on deposit and draw §157.00. 
 
 To see how quickly I can find the missing numbers in the fol- 
 lowing: 
 
 50 50 50 50 50 50 50 50 50 
 __ 47 __ 34 24 __ __ 19 __ 
 
 23 __ 24 __ __ 7 8 __ 21 
 
 FRACTIONS, DECIMALS, PERCENTAGE AND INTEREST. What 
 has been said in regard to whole numbers is also apjilicable to frac- 
 tions, decimals, percentage, and interest. Before taking uj) the study 
 of percentage, look this subject up carefully from the outlines of the 
 grades below. In connection with the work in percentage continue 
 the use of the graph. Moore and Miner has some good suggestions 
 for this work. Use material from magazines and daily pai)ers. 
 
 In all of this work use the methods taught in lower grades wher- 
 ever at all practical.
 
 105 
 
 GRADE IX- A. 
 DIRECTIONS. 
 
 DIVISION OF TIME. At least fifteen minutes n( the allotiiieiit of 
 aritlimetic time in tliis grade should he yiveu to rapid ealeulation. 
 
 The remainder of the time should be given to the application of 
 the knowledge gained during the iirst semester and to store the impil's 
 mind with business information. 
 
 MULTIPLICATION AND DIVISION. Moore and .Miner's Pract- 
 ical Business Arithmetic is full of suggestive material for interesting 
 projects. Projects similar to those suggested in a<lditi()ii ami subtract- 
 ion should l)e given. 
 
 All of tliis work should be put before the pui)il in the form of 
 interesting projects in order to keep him interested an<l enthusiastic. 
 
 MAIN TOPICS. Banking, insurance, commission, customs, duties 
 and taxes, stocks and bonds, and business forms. 
 
 BANKING. Teach bank discount. Partial payments should be 
 computed according to U.S. rule, since most installment plans are based 
 on this method of payment. Teach the methods of balancing a 
 V)auk pass book and making the banker's daily balance. 
 
 Teach the following methods of sending money: the l)ank di-aft, 
 I)OStal money order, and the express money order. Discuss savings 
 bank accounts, and compound interest as ai)plied to savings accounts, 
 also clearing house problems. Show how building and loan associa- 
 tions differ from banks. Give a few problems to be solved under 
 each topic discussed. The problems in Moore and .Aliner are good. 
 
 INSURANCE. P^xplain the following kinds of policies: ordinary 
 life, twenty-payment life, and endowment. 
 
 Careful attentioJi should be given to questions of extension of 
 time, cash surender values, and dividends which ai'e of imixirtance to 
 every policy holder. 
 
 Local insurance agents will upon application, sui)iily sami)le pol- 
 icies and rate books free of cost. 
 
 This adds much interest to the work and sliouM be used. It is 
 the l)est kind of material in teaching insurance. 
 
 COMMISSION, CUSTOMS AND DUTIES, AND TANES. Discuss 
 commission, customs and duties, and taxes. See outlines for the Sev- 
 enth Grade for suggestions for this work. Under ta.xes take up tlie 
 Federal Income Tax. Work enough practical problems to make clear 
 the information given upon each tojiic. 
 
 STOCKS AND BONDS. Distinguish bt-twccn tlu-in. K.xplain 
 common and preferred stock. Discuss reason why stocks tluctuatc in 
 l»rice, and the meaning of assessments and dividends. The prolileiiis 
 from the text should be supiilemented by i)robleiiis i.repared by \isiiig 
 stock ([notations from the daily papers. 
 
 Teach bonds simply as notes issued by cori»oratioiis. munciiiali- 
 ties, states, and nations. Uomiiare a bond with a promissory note and 
 point out the similarities. Discuss registere.l bonds and Liberty 
 Bonds. Study Ijond (juotationsin the pai>ers and figure the ratesof in-
 
 1U(3 
 
 terest they payou iuvestiueutsat various quotations. Interest the pupils 
 in hrino-iuof in all of the outside information they can obtain. Encoui-agre 
 tliem t*t lie t)n the look out for valuable information along commercial 
 lines. Teach them to be able to adapt themselves to new conditions. 
 BUSINESS FORMS. Blank forms of checks, notes, commercial 
 and bank drafts, and receipts should be prepared in the priutiugdei)art- 
 ment and given pupils for practice in filling them out under the di- 
 rection of the teacher. Blank, full, special and restrictive indorse- 
 ments should be made familiar to the pupils through practice in in- 
 dorsing. Use available pay-roll forms from business houses. Spend 
 some time in figuring payrolls, billing, and making out cost sheets. 
 
 OPTIONAL WORK. 
 
 Use more difficult jiroblems pertaining to the different topics. 
 
 Do some work in accounting to emphasize form and neatness.. 
 
 Make graphs representing war expenditures, increase or decrease 
 in i>rice of certain commodities during a certain period, or fluctuation 
 of stock. These are interesting and instructive. 
 
 Standards of Attainment. 
 
 The pupil must have acquired a high degree of accuracy and 
 speed in haiulling the fundamental process; through checking his own 
 work he must have become self-reliant; through the study of the 
 practical and everyday features of arithmetic he must have a working 
 knowledge of the usages and the phraseology of business and com- 
 merce. 
 
 See Objective Standards for (xrade VIII. 
 
 Bibliography. 
 
 Moore and Miner: Practical Business Arithmetic. 
 
 Power and Locker: Practical Exercises in Rapid Calculation, 
 
 (1917). 
 Mcintosh: Exercises in Papid Calculation. 
 See bil)liography for Seventh and Eighth Grades.
 
 107 
 
 GRADES IX-i; aii.l A. 
 
 The foUowiny is a statenieut of the subject matter li> Vm- covered 
 by the Ninth Grade in Algebra: 
 
 TEXT 
 
 Ilawkes, Luby and Touton: First Course in Altjfebra, 
 Nine-B: pages 1 to 121. 
 Nine- A: pages 122 to 289. 
 
 At least six weeks of the Niue-1> (irade should be given to 
 factoring. The success of the Niiie-A Grade work depends hirgely 
 upon the pupils' ability to factor. 
 
 It is not expected that all of the exercises under eacli to])ic shall 
 be solved. It is better to solve a few and check them than to solve 
 many without checking. Long involved solutions and i»r(d)lems 
 which are of no vital interest to the pujjils should be omitted. 
 Teachers may substitute other problems within the comprehension 
 and experience of the children. Dividing and multii)lying by ex- 
 pressions of more than two terms is not practical. 
 
 The work of the entire course should center about the equation, 
 taking up other topics only as there is need for them in the solution 
 of equations. 
 
 There is no need for many formal definitions. Most of the 
 terms in algebra may be learned from intelligent use. A <lefinition 
 should be taught only when it is necessary in order to get a clear 
 concept.
 
 lOS 
 
 General Bibliography: 
 
 METHOD. Browu aud Cotfmiiu: How to Teach Arithmetic: Row, 
 Peterson and Co. 
 
 Courtis, S. A.: Measurement of Efficiency and (Growth in Arith- 
 metic: The Elementary Scliool Teacher, Chicago; Vol. X, 
 1909, pp. 55-74, pp. 177-199; Vol. XI 1910, pp. 171-185. 
 pp. 8U0-370, pp. 5'2S-539; Vol. XII, 1911, pp. 127-137. 
 
 Freeman, F. X'.: The Psychology of the Common IJrauches: 
 IIoughton-Mifflin Co. 
 
 Jessup, W. A.: Economy of Time in Arithmetic: tState I^niversity 
 of Iowa, Iowa City, Iowa. 
 
 Jessup aud Coffman: The Supervision of Arithmetic: The Mac- 
 millan company. 
 
 Kendall and Mirick: How to Teach the Fundamental Subjects: 
 Chap. Ill Mathematics: Houghton-Mifflin Co. 
 
 Klapper: The Teaching of Arithmetic: D. Appletou and Co. 
 
 McMurry: Special Method in Arithmetic: The Macmillan Co. 
 
 Smith, David E.: The Teaching of Arithmetic: Ginn and Co. 
 
 Stone, C. W.: Arithmetical Abilities aud Some Factors which 
 Determine Them: Teachers College Bureau of Publication. 
 X. V. 
 
 Suzzalo, Henry: The Teaching of Primary Arithmetic: Houghton 
 Mifflin Co. 
 
 U. S. Bureau of Education: A Course of Study in .\rithmetic: Bos- 
 ton, Mass. 
 
 Wilson, G. M.: A Course of Study in Elementary Mathmetics, 
 Connersville, Indiana Schools. 
 
 Wilson and Wilson: The Motivation of School Work; Chap. IX: 
 Houghton Miffliu Co. 
 
 DISCUSSIONS OX THE USE OF STANDARD ARITHMETIC TESTS. 
 
 Ashbaugh, E. J.: The Arithmetical Skill of Iowa School Child- 
 ren; Bulletin X'o. 24, Extension Division, University of 
 Iowa. 
 
 Ballon, Frank W.: Educational Standards and Educational 
 Measurements, with Particular Reference to Standards in 
 the Four Fundamentals in Arithmetic; Bulletin 10, 1914: 
 Boston Public Schools Bulletins 3 and 7, Department of 
 Educational Investigation and Measurement, Boston. 
 
 Buckingham, B. R.: Xotes on the derivation of Scales, with 
 special reference to Arithmetic: Fifteenth Yearbook of the 
 National Society for the Study of Education, Part 1. 
 
 Counts, (4. S.: Arithmetic Tests and Studies in the Psychology 
 of Arithmetic; University of Chicago Supplemeutaj-y Edu- 
 cational Monographs, Vol. I, No. 4_ 
 
 1
 
 109 
 
 Courtis, S. A.: Third, Fourth aud Fifth Annual Accountiuffs, 
 1918-191G: Dei)artnK'ut of Co-operative Research. .S2 Eliot 
 Street, Detroit. The llelial)ility of Siiiyle Measurements 
 with Standard Tests: Klenientary School .Journal, March ami 
 June, 1918. 
 
 Haffgerty, M. K.: Arithmetic: A Co-operative Study in Educa- 
 tional Measurements: Indiana Cniversity Studies, No. 27. 
 Studies in Arithmetic: Indiana Cniversity Studies, No. H'l, 
 Vol. 8. Sept., 1916. 
 
 Judd, Chas. IL: Measuring the Work of tiie Puhlic Schools: 
 Cleveland Foundation, Survey Ke])ort, Clevehunl, Ohio. 
 
 Monroe, W. S.: Educational Tests and Measurements, Chaj). II, 
 Arithmetic: lIoughton-MitHin Co. 
 
 Report of the Use of the Courtis Standard liesearch Tests: 
 Kansas State Normal School, Emporia; Bulletin, new series, 
 Vol. 4, No. 8. 
 
 Smith, J. 11.: Individual Variations in Arithmetic: Elementary 
 School Journal, Vol. 17, pj). 195-'2(J0. 
 
 Stone, C. W.: Arithmetical Abilities and Some Factors Deter- 
 mining Them: Teachers' College Contributions to Educa- 
 tion, No. 19. 
 
 Standardized Ueasoning Tests in Arithmetic and How to Utilize 
 Them: Teachers' College Contributions to Education, No. S^: 
 Teachers' College, Columbia University, N. Y. 
 
 Theisen, W. W.: The Use of Some Standard Tests; Chap. II. 
 Arithmetic: State Department of Public Instruction, Madi- 
 son. Wisconsin. 
 
 AVoody, C: Measurements of Some Achievements in Arithmetic: 
 Bureau of Publications, Teachers' College. 
 
 OTHER DISCUSSIONS OF ARITHMETIC. School Survey Reports: 
 Butte, Cleveland, Denver, (irand Rapids, Janesville, Salt 
 Lake City, St. Paul, St. Louis, Nassau Co., N. V. 
 
 Second Report of the Committee on Elimination of Subject .Mat- 
 ter: Chap. V Arithmetic: Iowa State Teachers' Asst)ciation. 
 
 Bush, M. G.: The Fundamental Number Facts: School and So- 
 ciety, Sept. 1, 1917. 
 
 Chase, S. E.: Waste in Arithmetic: ''I'eaclK'rs' College Record, 
 Sept., 1917. 
 
 Oist, S. A.: Errors in the Fundamentals of Arithmetic: School 
 and Society, August 11, 1917. 
 
 Holloway, II. U.: The Relative DitHculty of the KlciiKMitary 
 Processes State (Gazette Pub. Co., Trenton. N. ^. 
 
 Kirby, T. J.: Practice in the Case of School Children: Rureau of 
 Publications, Teachers' College, N. V.
 
 110 
 
 standard Tests. 
 
 FUNDAMENTAL OPERATIONS. 
 
 Courtis Standard Research Tests, Series I>. The tests may be 
 secured from S. A, Courtis, 82 Eliot Street, Detroit Micliigau. 
 
 Research Tests in Arithmetic, Addition of fractions, desiy^ned hy 
 F. W. IJalloii. ("oities of these tests are not obtainable. 
 
 The Cleveland Survey Arithmetic Tests. Copies of the test 
 paj^ers may be obtained from Charles II, Judd, School of 
 Education, University of Chicago, Chicago, Illinois. 
 
 Stone's Arithmetic Test for fundamental Operations. Designed 
 as a general test. Copies may be obtained from the Rureau 
 of Publications, Teachers College, Columbia University. 
 New York City. 
 
 Arithmetic Scales devised by Clifford Woody. Copies may be 
 obtained from the Bureau o:^ Publications, Teachers College, 
 Columbia University, New York City. 
 
 REASONING TESTS. 
 
 Stone's Reasoning Test. For copies of the test, address Bureau 
 of Publicatious,Teachers College, Columbia University, New 
 York City. 
 
 Starch's Arithmetical Scale A. Copies may be obtained from 
 Daniel Starch, University of AVisconsiu, Madison, ^Visconsin. 
 
 Buckingham's Reasoning Test i n Arithmetic. Used by Buck- 
 ingham in the Survey of t he Gary, and the Prevocational 
 Schools of New York City.
 
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