|sAN FRANCISCO STATE NORMAL SCHOOL 
 
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 I^9Q I UC-NRLF 
 
 ^^^ B ^ SIS ^si 
 
 ^H D 
 
 How to Teach the Critical 
 Difficulties of Arithmetic 
 
 to accompany the 
 
 CALIFORNIA STATE SERIES ADVANCED 
 ARITHMETIC 
 
 • / r.i 
 
 For Normal Students and Experienced Teachers 
 
 By MARY A. WARD 
 
 Supervisor of the Teaching of Arithmetic 
 
 California 
 1 State Printing Office 
 
 , 15484 1915 
 
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 TABLE OF CONTENTS. 
 
 Page. 
 
 INTRODUCTION 1 
 
 CRITICAL, DIFFICULTIES IN THE ADDITION AND SUBTRACTION OF 
 
 FRACTIONS AND HOW TO TEACH THEM 2 
 
 CRITICAL DIFFICULTIES IN THE MULTIPLICATION AND DIVISION OF FRAC- 
 TIONS AND HOW TO TEACH THEM 9 
 
 MISCELLANEOUS QUESTIONS ON FRACTIONS 11 
 
 TESTS IN ADDITION, SUBTRACTION, MULTIPLICATION AND DIVISION OF 
 
 FRACTIONS 12 
 
 CRITICAL DIFFICULTIES IN THE ADDITION AND SUBTRACTION OF DECI- 
 MALS AND HOW TO TEACH THEM 13 
 
 CRITICAL DIFFICULTIES IN THE MULTIPLICATION AND DIVISION OF DECI- 
 MALS AND HOW TO TEACH THEM 15 
 
 MISCELLANEOUS QUESTIONS ON DECIMALS 19 
 
 TESTS IN THE ADDITION, SUBTRACTION, MULTIPLICATION AND DIVISION 
 
 OF DECIMALS 21 
 
 TEST QUESTIONS ON AVERAGE, RATIO, TABLES, BILLS AND RECEIPTS 21 
 
 SQUARE MEASURE 22 
 
 CUBIC MEASURE 25 
 
 SUGGESTIONS FOR TEACHING EACH CASE IN PERCENTAGE 26 
 
 TEST ON FORMAL WORK OF THE THREE CASES OF PERCENTAGE 2S 
 
 TEST ON PROBLEM WORK OF THE THREE CASES OF PERCENTAGE 29 
 
 TEST ON COMMISSION, INSURANCE, TAXES, DUTIES, TRADE DISCOUNT AND 
 
 INTEREST 31 
 
 COMPLETE REVIEW OF MENSURATION 33 
 
 CLASSIFICATION OF PROBLEMS OF PAGES 18-160 OF TEXT 37 
 
 306779 
 
I 
 
HOW TO TEACH THE CRITICAL DIFFICULTIES OF 
 
 ARITHMETIC. 
 
 INTRODUCTION. 
 
 The helps and suggestions given in this bulletin for teaching each of the 
 topics of the State Series Advanced Arithmetic may be applied to the work 
 of any arithmetic text. Teachers will find the miscellaneous questions and 
 tests given at the end of each topic useful in many ways. They may be used 
 as thorough reviews at the beginning of a new term or as tests for new 
 pupils who enter late in the term. Such tests should help to determine the 
 points in which a pupil's knowledge is deficient. The classification of the 
 problems of pages 18 through 165 of the State text will be of much service 
 to the teacher who is using this portion of the textbook. 
 
 High school students desiring to enter the Normal School, will find that 
 the bulletin outlines the scope and type of the arithmetic work with which 
 students must be familiar before they can be entrusted with the teaching of 
 this subject in the classes of the training school. Students will be expected 
 to be familiar with the tests, suggestions and methods of work given in this 
 book. 
 
 Besides the subjects treated in this outline, normal students are expected 
 to be familiar with the following: 
 
 1. The system of addition combinations as given in chapter three of the 
 
 State primary text. 
 
 2. The method of finding the quotient figures in long division. (Advanced 
 
 State Text, pages 63-67.) 
 
 3. Additive method of subtraction. (Advanced Text, pages 21-22.) 
 
 MARY A. WARD. 
 
 State Normal School, San Francisco, 
 February, 1915. 
 
HOW TO TEACH THE CRITICAL DIFFICULTIES OF 
 
 ARITHMETIC. 
 
 COMMON FRACTIONS. 
 
 The following sections summarize the various types of problems which 
 should be taught in the addition, subtraction, multiplication and division of 
 fractions. The suggestions given here for the teaching of fractions are 
 intended to be used in connection with the indicated sections of the Cali- 
 fornia State Series Advanced Arithmetic, but may be used in connection 
 with the fraction work of any arithmetic text. Normal students will be 
 required to work problem^s similar to the types given here and to use the 
 form and methods of work indicated in this outline. In order to follow 
 the references, students should if. possible provide themselves with a (.-opy 
 of the California State Series Advanced Arithmetic. Students should be 
 especially familiar with the work of Sections 26 and 27. 
 
 1. Fractions: 
 
 1. Teach pupils that a piece and a fraction of an object are one and 
 the same. 
 
 2. That the number of equal pieces into which the whole quantity 
 is divided determines the name of the piece or fraction. Ask — "If a 
 line is divided into 4 equal parts, what is each part called ? If divided 
 into 2 equal parts? 5 equal parts? etc." (Text page 90.) 
 
 3. Ask — -"How many fourths, thirds, tifths, etc. are there in one 
 whole?" 
 
 4. Teach pupils the names of the parts of a fraction — numerator and 
 denominator. (Text page 93.) 
 
 5. Suggest that the first letter of the word down is d. The name 
 of the term which is down or below^ the line, denominator, also begins 
 with a d. 
 
 6. Have pupils show ^, f, f, |-, etc., of lines, rectangles, circles. Be 
 careful to see that pupils point not to the line of division, but to the 
 portion of the line or surface when showing ^, f , ^, etc., of a line, rec- 
 tangle, or circle. (Text page 93, numbers 6-7.) 
 
 7. Teach pupils to distinguish between proper and improper frac- 
 tions and mixed numbers. Do not teach definitions of these terms. 
 (Text page 94.) 
 
 8. Have pupils classify each of the following as proper fractions, im- 
 proper fractions or as mixed numbers: If, |, f, |, V, If? V- Pupils 
 often fail to regard ^, |, etc., as improper fractions. 
 
 Write a miscellaneous list of proper, improper fractions and mixed 
 numbers on the board. Have pupils tell whether each is I (improper), 
 M (mixed number) or P (proper). This form of drill may be either 
 oral or written. 
 
 (2) 
 
2. Changing Improper Fractions to Mixed Numbers. 
 
 Before beginning the addition of fractions of like denominators, it is 
 necessary that pupils know how to change improper fractions to mixed 
 numbers. There is no necessity for teaching- pupils to change mixed 
 numbers to improper fractions at this time. The application of the 
 drill upon changing mixed numbers to improper fractions can not be 
 made until pupils are ready for work in the multiplication of frac- 
 tions. For this reason the work of Sections 119-120 of text should be 
 delayed until pupils are ready to begin the multiplication of fractions. 
 
 3. Change Y- to a Mixed Number: (Text page 96.) 
 
 a. Teach the pupils to read V as 19 divided by 3. Emphasize the 
 fact that the horizontal line is the sign of division. Pupils should 
 read y in two ways— as 19 thirds or as 19 divided by 3. 
 
 1). Ask: What is the answer when 19 is divided by 3? Nineteen 
 thirds equal what? Proceed in the same manner with other improper 
 fractions. Do not teach by rule the changing of improper fractions 
 to mixed numbers. 
 
 c. When pupils can readily change improper fractions to mixed 
 numbers, show the reason the numerator is divided by the denominator. 
 
 Ask : How many thirds are there in one whole "? How many Avhole 
 numbers are there inf, f, f, -y^, i/, etc., |, |, |, V, ¥, ¥? 
 
 d. Require pupils to change mentally all small improper fractious 
 to mixed numbers. 
 
 4. Reduction to Lower Terms. 
 
 Much time is wasted in attempting to teach pupils to find the 
 greatest common divisor of two or more numbers. That the reduc- 
 tion of fractions to lower terms may be taught without the use of 
 such terms is shown by the following suggestions (Text 87-89) : 
 
 a. Teach that the even numbers are 0, 2, 4, 6, 8. 
 
 &. Ask : Which of the following end in even numbers 234, 40, 43, 
 27, 22, 35, 19, 36? 
 
 c. Teach pupils to recognize which of the following fractions have 
 numerators and denominators that end in even numbers. 
 
 2.4 16 10 3 2. .'5 7 pf^ 
 
 d. Teach that when both the numerator and the denominator of a 
 fraction end in even numbers, the fraction may be reduced by dividing 
 both numerator and denominator by 2. 
 
 Reduction by 5: Teach that when the numerator and the denom- 
 inator of a fraction end in 5, or and 5, the fraction may be reduced 
 by dividing both numerator and denominator of the fraction by 5. 
 
 Reduction by 10: Teach that when both the numerator and the 
 denominator of a fraction end in 0, the fraction may be reduced by 
 dividing both the numerator and the denominator of the fraction bv 
 
 ■ 10. Thus, &{J=^. 
 
 (3) 
 
7. Reduction by 3 : Reduce ^. 
 
 a. In the numerator, 12, add together the 1 and 2. The answer is 3. 
 May 3 be divided by 3 ? 
 
 &. Add together the 3 and 9 in the denominator, 39. The answer 
 is 12. 
 
 c. May 12 be divided by 3 ? 
 
 d. May the sum of the numbers in the numerator of ^| be divided 
 by,3? 
 
 e. May the sum of the numbers in the denominator of ^| be divided 
 by 3? 
 
 /. When the sum of the numbers in the numerator and the sum of 
 the numbers in the denominator of a fraction are divisible by 3, the 
 original fraction is divisible by 3, thus, || = j%. 
 
 g. Reduce each of the following fractions by dividing both the 
 numerator and the denominator of the fraction by 3 : 
 
 iiJLL 4 8. IL 4 5. 
 1728>^3' 7 25 72- 
 
 li. Give much drill upon the reduction of small fractions such as 
 f > ¥j I) -Ni li 6tc. Pupils should be able to reduce such types men- 
 tally without any hesitation. 
 
 8. Summary of Types in Reduction to Lower Terms: (Text pages 101, 
 
 104, 105.) 
 <*• Iff J ctI- Divide by 2 as the numerator and the denominator end 
 
 in even numbers. 
 &. /oWj f M- Divide by 5 as the numerator and the denominator end 
 in 5, or and 5. 
 
 c. If {], -}f Q. Divide by 10 as the numerator and the denominator end 
 
 in 0. 
 
 d. Ill, |gf . Divide by 3 as the sum of digits of numerator and the 
 
 sum of digits of denominator are both divisible by 3. 
 
 9. Difficulties: 
 
 1. Besides the above types, pupils often fail to see the number by 
 which fractions, such as the following, may be reduced: ^, ^|, \\, ||, 
 
 3 8- 
 
 2. Pupils often reduce a fraction once but fail to reduce it to its 
 lowest terms, thus, ^l = f = f . 
 
 3. Train pupils to divide both numerator and denominator of frac- 
 tions by largest number possible. 
 
 10. Types in Addition of Fractions of Like Denominations : 
 
 For drill material see text page 97. 
 
 a. h. c. d. e, 
 
 J- ^ - 24-1- 6^ 
 
 V s: a "^X 'Jit 
 
 2 1 X A 6 A 
 
 8 S S ^8 ^ 
 
 41 141 
 
 i f t 9 25 
 
 11 2 42| 51 
 
 (4) 
 
Teach each of the above types in the given order. Do not pass 
 from type a to 6 until pupils understand type a perfectly. Only a 
 few problems are needed to teach the principles of each type. Follow 
 closely the suggestions given under the following sections : 
 
 1. Notice that in type a the answer is a fraction only. Construct 
 problems to fulfill this condition. 
 
 2. In type & the answer ^-^ is to be changed mentally to the mixed 
 number, 1^ or 1^. Nothing is put upon the paper except the answer 
 in its simplest form. Notice the j^osition of the | in the answ^er. 
 Where is the whole number, 2, placed! From the beginning of work 
 with this type, insist upon pupils making all reductions mentally. 
 
 3. In type c pupils must be shown where to place the whole 
 number 2. 
 
 4. In type d, note the irregularity of the column. All changing 
 and carrying is done mentally. 
 
 5. In type e, note the irregularity of the column and the position 
 of the answer. 
 
 6. While teaching addition of fractions of like denominators, con- 
 tinue drill work upon the reduction of fractions to lower terms and 
 the changing of improper fractions to mixed numbers. 
 
 11. Language Forms for Addition of Fractions : 
 
 Use the following language forms in dictating exercises in addition : 
 a. Find the sum of 4f , 6^, etc. 
 h. Add the following: 9|, 6^, etc. 
 c. Add : 6f , 81, etc. 
 
 12. Addition of Fractions of Unlike Denominators: 
 
 Before introducing work in the addition of fractions of unlike 
 denominators, pupils should be able — n. to change fractions to higher 
 terms; &. to find the smallest common denominator for several unlike 
 fractions. 
 
 13. Changing Fractions to Higher Terms : (Text pages 102-103.) 
 
 a. Teach pupils to change all possible fractions to 6ths, 8ths, 12ths, 
 ISths, 20ths, 24ths, 30ths, 36ths, 48ths, 60ths. 
 
 h. For purposes of illustrating the changing of fractions to higher 
 terms, use fractions such as f, f, | rather than \, i, or ^. Since the 
 numerators of the latter are 1, they are exceptions to the general rule. 
 
 14. The Smallest Common Denominator: (Text page 112, col. 3.) 
 
 fl. Teach pupils that 1, 2, 3, 5, 7 are prime numbers. That is, such 
 numbers are divisible only by themselves and one. 
 
 6. Teach pupils to find the smallest common denominator for unlike 
 fractions by the following method : 
 Find the smallest common denominator for f, §, f, |, ^\^ f, |. 
 c. Write the denominators in a horizontal line. Thus, 
 2 ) ^ — ^— jj— 8 — 12 — 7 — ^ 
 
 2)] 4— 6 — 7 
 
 2 3 7 
 
 2X2X2X3X7 = 168. 
 
 2—15484 
 
d. Have pupils cross out all the small denominators which are con- 
 tained an even number of times in the larger denominators. Since 
 4 — 3 — 6 — 2 are contained in the larger number, 12, they are crossed out. 
 
 e. Have pupils divide the remaining numbers by the smallest prime 
 number which is contained "in at least two of the numbers that are 
 left. Continue as shoMTi in the model. Emphasize the fact that the 
 divisor must be contained in at least two of the numbers to be divided. 
 
 f. Pupils should be perfectly familiar with the method of finding 
 the smallest common denominator before be^ning the addition of 
 fractions of unlike denominators. The above method of finding the 
 common denominator wdll be more successful with the ordinary pupil 
 than the inspection method given on pages 106 and 107 of the 
 Advanced Text. 
 
 15. Difficulties in Teaching Pupils to Find the Common Denominator: 
 
 a. Pupils sometimes cross out both the large and small numbers. 
 
 Teach that the larger numbers must be left to take care of the smaller 
 
 numbers. 
 
 &. Pupils often fail to cross out small numbers when it is possible 
 
 to do so in the second or third line of the work thus : Find the smallest 
 
 common denominator for #, vV, 1, 4 
 
 16. 
 
 2) 
 
 T2") 
 
 .4- 
 
 12- 
 
 -8- 
 
 -9 
 
 2) 
 
 
 6- 
 
 -4- 
 
 -9 
 
 
 
 :1 — 
 
 -2- 
 
 -9 
 
 2 X 2 X 2 X 9 = 72. 
 
 c. Pupils often waste time dividing by a number which is con- 
 tained in only one of the denominators. In the above model, pupils 
 might divide the last line by 2 and then the next line by 9, which 
 would be a great waste of time. Insist upon pupils dividing by the 
 smallest number which is contained in at least two of the denomi- 
 nators. 
 
 d. In teaching the above work, always dictate the unlike fractions 
 which are to be added. Do this so that pupils may see that they are 
 learning to find the smallest common denominator of several numbers 
 solelj^ for the purpose of adding unlike fractions. 
 
 e. Be certain that the fractions which you dictate for the above work 
 have denominators which illustrate the several steps of finding the 
 smallest common denominator. 
 
 Forms for Addition of Fractions of Unlike Denominators. 
 
 All that remains to be taught in the addition of fractions of unlike 
 denominators is the form of the work. 
 a. 
 
 J.J — ,,^^ j_j 
 
 81 = 84^ 8| 
 
 AS. — 4. 9 4.3 
 
 1). 
 
 ^4 
 
 2T 
 
 A 
 
 29^'^ 
 -3 — 2 
 
 29 y^ 
 
 -8 = 24 S. C. D. 
 (6) 
 
a. Pupils who are learning the addition of fractions should use the 
 form of model a. 
 
 h. Model 1) is shorter and may be used after the pupils have passed 
 the work of multiplication of fractions, or sooner if the teacher feels 
 that the pupils know what they are doing. 
 
 c. The M'ork of finding the new denominator should always appear on 
 the pupil's paper. In this way a pupil's knowledge of this principle 
 vciSiY be tested. 
 
 d. Notice that the answer to the addition problem of the above model 
 is written on the paper only in its simplest form. Train pupils to do as 
 much as possible of the reduction mentally. 
 
 e. Train pupils to reduce fractions of the given problem to lowest 
 terms before attempting to find the common denominator of the frac- 
 tions. In this way much useless work is saved. 
 
 Find the sum of -/-j, -{^i^ i> li f) iV- First reduce the fractious to 
 lower terms. The problem then becomes : Find the sum of ^, f , 
 
 2^11 
 3> 4> 4) 4- 
 
 17. Difficulties Pound in the Teaching of Addition of Fractions. 
 
 a. The changing of fractions to higher terms. 
 
 b. Finding the smallest common denominator for several unlike 
 fractions. 
 
 c. Changing mentally improper fractions to mixed numbers and 
 reducing fractions to lower terms. 
 
 d. Failure of pupils to use the correct form when adding fractions of 
 unlike denominators. Teach pupils why the following form is incor- 
 rect : 2|=^V 
 
 e. Be careful to see that the problems you use for drill work illustrate 
 the principle upon which you are drilling. In teaching the method of 
 finding smallest common denominator, do not use a problem such as 
 
 /. Do not introduce the subtraction of fractions until pupils have a 
 thorough knowledge of each of the above difficulties. 
 
 SUBTRACTION OF FRACTIONS. 
 
 18. Types in Subtraction of Fractions Which Involve No Carrying: 
 
 For review work after types a, h, c, and d have been taught, use the 
 material on pages 98-99 of text. 
 
 Type a. Larger Fraction in Minuend : 295f ^ 
 
 - m -I 
 
 Type 1). Fractions in Minuend Only: 249f 
 
 — 18 
 
 Type c. Fraction Only Subtracted From Mixed Number: 
 
 832X 
 a 
 
 Type d. Answer a Whole Number Only : 2H 
 
 18 4 4 
 
 -■"O 9 9 
 
 (7) 
 
Suggestions : 
 
 1. Teach each of the above types one at a time. Do cot attempt to 
 make pupils conscious of which types they are learning. 
 
 2. Insist upon pupils making all reductions mentally. 
 
 3. Teach pupils to prove their answers to all problems in the sub- 
 traction of fractions. 
 
 4. Be certain that pupils can work problems of the above types when 
 arranged in a miscellaneous order. 
 
 5. For extra drill work, have pupils make up and solve problems of 
 their own which involve the different tj^pes. Such problems are a good 
 test of pupil's knowledge of the work covered. 
 
 19. Types in Subtraction of Fractions That Involve Carrying: 
 
 a. Fraction in Subtrahend Only: 1018 1018 
 
 — 27# — # 
 
 1. Teach the above type b.y borrowing method. 
 
 2. Do uot permit pupil to put changes on paper. 
 h. Smaller Fraction in Minuend : 
 
 32i 321 
 
 -19| - f 
 
 1. Teach the above type by the borrowing method. 
 
 2. Train pupils to make all changes mentally. 
 
 3. As this is the most difficult type in the subtraction of fractions, 
 pupils should receive a great deal of drill upon it. 
 
 4. Teach pupils to prove all answers to subtraction problems. 
 
 20. Language Forms: 
 
 a. When pupils can work readily any of the above types in the sub- 
 traction of fractions, the following language forms should be taught 
 and applied : 
 
 a. Find the difference betw^een 18| and 44^. The difficulty lies in 
 the fact that the smaller number is dictated first. Pupils are inclined 
 to place as the minuend the first number stated in the problem. 
 
 1). Find the difference between 32f and 14^. 
 
 c. Subtract 9J from 46^. 
 
 d. What must be added to 24^ to give 32f ? 
 
 21. Subtraction of Fractions of Unlike Denominators : 
 
 For drill material see text page 110. 
 
 3 i 3 .5 
 
 f 8 'i'6 
 
 1 8 
 
 241 = 24^ 
 - 6|= f:-8_ _4 
 
 17 ~ 11 
 
 a. Notice tlie form of the work. 
 
 h. Use the language forms given under the subtraction of fractions of 
 like denominators. See section 20. 
 
 c. Teach the reason this form of work is incorrect : 2'4 = .%. 
 
 (8) 
 
a. 
 
 1X4- ? 
 
 h. 
 
 f X 1 = ? 
 
 c. 
 
 ^XA=? 
 
 MULTIPLICATION OF FRACTIONS. 
 
 The nmltiplieation of fractions is to be taught by the cancellation 
 method. The following is the order in which the ditferent types are to 
 be taught : For drill material see advanced text page 114, section 152. 
 (No cancellation.) 
 (Cancellation cione diagonally.) 
 (Cancellation done vertically. Emphasize the fact 
 that cancelling is simply dividing a number above 
 the line and a number below the line by the same 
 number. Show that for this reason we never cancel 
 horizontally.) Text pages 119, 120, 121. 
 
 d. 24 X ^= ? (Teach that 24 is to be considered as the fraction \*-. 
 
 Pupils often fail upon this type through regarding 
 24 as the denominator of a fraction.) Text pages 
 115-116. 
 
 e. f X 12 = ? (The same difficulty appears in this type as in Type d.) 
 
 22. Changing Mixed Numbers to Improper Fractions: 
 
 It is at this point that pupils should be taught to change mixed num- 
 bers to improper fractions. (See text page 95.) 
 Change 2f to an improper fraction. 
 
 a. First, teach pupils to change 2f to an improper fraction simply 
 as a mechanical operation : 4X2^8 + 3 = \i-. 
 
 J). Later show pupils why this is done. Ask. "How many fourths 
 are there in one whole" ( f ) Two wholes? Three wholes? etc." 
 
 c. In 2f we have the whole number 2 or f and f more or y . 
 
 d. Pupils should be able to readily change any mixed number to an 
 improper fraction before the following types in the multiplication of 
 fractions are introduced. 
 
 /. 2-j X f = ? (Pupils often attempt to cancel before changing the 
 mixed number to an improper fraction. Use small 
 numbers.) 
 g. 2| X 24= ? (There are two difficulties to be noted in this type. 
 1. Changing the mixed number to an improper 
 fraction. 2. Regarding 24 as a correct fraction.) 
 h. I X 2f = ? (The same difficulty appears in this type as in Type g.) 
 i. 5§ X 4f = ? (Be careful while teaching the principle of this type 
 to use small numbers.) 
 Suggestions : 
 
 1. Drill upon each of the above types until it has been mastered. Do 
 not be in too great a hurry to introduce a new type. As each new type 
 is mastered, review in a miscellaneous order all previous types. 
 
 2. There is no necessity for using large numbers when teaching the 
 multiplication of fractions. No number larger than 25 should be 
 used. 
 
 3. Insist upon all answers being reduced to lowest terms. 
 
 (9) 
 
23. Language Forms for Multiplication of Fractions: 
 
 a. Find the product of | and ^. 
 l. Multiply 12f by |. 
 
 c. Find y of 21. (Pupils should be able to illustrate this type with 
 
 the aid of a diagram.) Text page 138, Sections 179 and 181. 
 
 d. Find f of f. 
 
 e. What is | of 6f ? 
 
 DIVISION OF FEACTIONS. 
 
 1. Have pupils read problems in the division of fractions and tell 
 which fraction is the divisor and which the dividend. 
 
 2. Emphasize the fact that it is the divisor which is inverted. 
 
 3. Emphasize the fact that fractions may be cancelled only when 
 the}' appear with multiplication sign between them. 
 
 4. Pupils should be taught to prove their answers to division 
 problems. 
 
 5. The following is a list of the types of problems in the division of 
 fractions. The types are listed in the order in which they are to be 
 taught. 
 
 a. f--f=? 
 c. 24 ^ f = ? 
 e. 6 -f- 4§ = ? 
 
 g. 2f-|=? 
 
 After each of the above types has been taught, the work of sections 
 158, 168, 180 of text may be used as review work. 
 
 24. Language Forms in Division of Fractions: 
 
 The following are the language forms to be used in the work of the 
 division of fractions : 
 
 a. Divide f by |. 
 
 Z). How many times is 1§ contained in 20? 
 
 Pupils often fail upon this type because they do not realize which 
 number is the dividend and which is the divisor. First teach the 
 principle of the work by using integers. 
 
 25. Given Part to Find Whole: 
 
 Perhaps the most difficult of all problems in fractions is the following : 
 18 is f of what number? (Text pages 59, 139, 140, 141, 142, 143, 144.) 
 The solution should be as follows: 
 f of the number = 18. 
 I of the number =6. 
 The number is 24. 
 Pupils should be able to show the meaning of the above statement by 
 means of a diagram, also be able to tell why 18 is divided by 3, not 4, 
 to find \ of the unknown number. Show pupils why the following form 
 of work is incorrect : f = 18, etc. 
 
 (10) 
 
 &. i--9^?4 
 
 
 8 1 
 d2|-.-6=3X^ = 
 
 4 
 '9 
 
 /■ J- ^ 24 = '? ^ 
 
 
 /i. 4a -^ 6f = ? 
 
 
b. Simplify : ^— — - Pupils should be first taught to read ~~ as 137^ 
 
 divided by 200. As a second step have pupiLs write the problem 
 thus : 1371 -f- 200 = ? 
 
 c. What fraction of f is -J? (Text pages 131, 145, 146.) 
 f is what fraction of ^ ? 
 
 22 is what fraction of 22 ? 
 
 d. What is the ratio of 7 to 15 ? Of | to f ? Of 16 to 4 ? (Text pages 
 
 54, 55. 56, 91-92. Pages 130, 19-20. Pages 145-146.) 
 
 QUESTIONS ON FRACTIONS. 
 
 26. The following questions and the tests of the next section will show 
 in a general way the knowledge of fractions wiiich a child should 
 have upon completing the w^ork of this topic. These questions 
 will also serve as a guide for Normal students desiring to prepare 
 themselves for a test in the subject of fractions. Questions marked 
 wdth a star are for teachers only. 
 
 1. Illustrate two different types of improper fractions. 
 
 2. Show with aid of a diagram that | or | of a line is the same as 
 
 "I of the line. 
 
 3. May J or | be changed to 12ths ? Give a reason for your answer. 
 
 4. Illustrate with diagrams what is meant by each of the following 
 
 statements : 
 a. Find f of $12. h. $12 is § of what amount? c. i is what 
 fraction of 4? d. Find f of f of a line. 
 
 5. Which of the following fractions may be reduced to lower terms : 
 
 i, rs, i> ^2? How can you tell? 
 
 6. When 2 is divided by f, why is the answer greater than 2? 
 *7. In changing -y- to a mixed number, why is 27 divided by 5? 
 *8. In changing 3f to an improper fraction, why is the whole number, 
 
 3, first multiplied by the denominator of the fraction, f ? 
 *9. Illustrate the difference between a prime number and an odd 
 number. 
 
 10. Why must unlike fractions be changed to fractions of the same 
 
 denominator before they may be added? 
 
 11. Read Y- in two ways. 
 
 12. Which of the following are in a correct form to be cancelled? 
 
 Change those that are not, so that they may be cancelled, 
 a. 24X1= ? d f Xi=? 
 
 h. 22.Xt=? e. f X2i=? 
 
 c. 8f X9i=? /. f X14=? 
 
 13. How many thirds, fourths, sixths, are needed to make a whole? 
 
 14. $18 is f of what amount? Show with the aid of a diagram why 
 
 $18 is divided by 3 to find i of the unknown amount. 
 
 15. State and solve a problem of your own in which it is necessary 
 
 to invert one of the fractions. 
 
 (11) 
 
3i = 
 
 if 
 
 2i = 
 
 A 
 
 6| = 
 
 A 
 
 12 
 
 If 
 
 16. Tell what is wrong with each of the following problems and 
 rewrite each problem in its correct form: 
 
 a. J). c. 
 
 6f 6| 
 
 Q2 3 
 
 ^4 8 
 
 j6i_ 2f 
 
 15 Ol 6 U2 
 
 ■'■"2T ^^¥ ^8' 
 
 17. In the multiplication of fractions, tell in what directions numbers 
 
 may be cancelled. Is it possible to cancel horizontally? For 
 instance, can you cancel the 4 and 8 in | X -i\ = ? Give a 
 reason for your answer. 
 
 18. Write two mixed numbers and change them to improper fractions. 
 
 19. Find the smallest common denominator for the following frac- 
 
 tions and show all of your work : 
 
 // 2 1 ^ 2 
 
 7) ^ 3 1 6_ 3 
 
 20. Correct the following statement: 
 
 42 — 4 
 
 21. In the fraction ^2^ which term is the numerator and Avhich term 
 
 the denominator of the fraction? What does each term tell? 
 
 22. Change the following fractions to 36ths : -j^, |, |. 
 
 23. Does the value of a fraction increase as its denominator increases? 
 Is ^i of a line greater or less than -j% of a line ? 
 
 SUMMARY OF THE FORMAL WORK IN FRACTIONS. 
 
 27. The following tests in the formal work of fractions are intended for 
 pupils of the grammar grades who have completed the work in 
 fractions. Inexj^erienced teachers will find these tests useful as 
 guides when planning reviews. The tests will also serve to indicate 
 to Normal students some of the fundamental facts in fractions with 
 which they must be familiar. 
 
 Test in Addition and Subtraction of Fractions. 
 
 1. Reduce to lowest terms : ^f , ff^, |H, ^S\, if • 
 
 What are the tests as to whether or not fractions may be reduced 
 by dividing by 2, 3, 5, or 10? 
 
 2. Classify each of the following as proper or improper fractions 
 
 or mixed numbers : f, If, |, -V , f- 
 
 3. Write two proper fractions which may be reduced to lower terms. 
 
 Reduce your fractions. 
 
 4. a. Write two improper fractions. Change them to mixed numbers. 
 h. Write two mixed numbers and change them to improper frac- 
 tions. 
 
 5. Find the sum of: S,%-, 8f, 2^, 3i, |, 2-,%^, 2\, S^. Show all of 
 
 your work for finding the common denominator. 
 
 6. Show by a diagram that f , | and | of a line are equal. 
 
 7. Change f, /a to 24ths. 
 
 (12) 
 
8. Find the difference between 20f and 108f 
 
 9. Subtract 12f from 25. 
 
 10. What must be added to 206$ to make 1000^? 
 
 11. 206f - ISf . 
 
 Test in Multiplication and Division of Fractions. 
 
 1. Find f of 20f. 
 
 2. What is i of t? 
 
 3. How many times is | contained in 2|? 
 
 4. Divide f by If. 
 
 5. $2f is I of what amount? 
 
 6. Divide 1 ft. 9 in. by 7 inches. 
 
 '• 300 ~ ■ 
 
 8. 17 is what fraction of 29? 
 
 9. What part of | is | ? 
 
 10. f are what part of | ? 
 
 11. What is the ratio of 7 to 11 ? Of 12 to 4? Of t to 4? 
 
 12. If i of a yard of satin is worth $1.60, what is the price of satin per 
 
 yard ? 
 
 13. What fraction of $2.10 is $1.05? 
 
 14. (/. 12f X7f=? &. 3i--24=? c. |-f-t=? 
 
 HOW TO TEACH THE CRITICAL DIFFICULTIES OF DECIMALS. 
 
 This summary of type problems in decimals and the suggestions made 
 here in regard to the form and the method of work to be employed in 
 the teaching of this subject may be used in connection with the indicated 
 sections of the California State Series Advanced Arithmetic or any 
 other arithmetic text. 
 
 Normal students will be expected to be familiar with the difficulties, 
 form of work and methods of teaching suggested in the following 
 sections. 
 
 28. Reading and Writing Decimals : 
 
 1. Teach pupils to read and write decimals through millionths' place. 
 
 2. Be careful to see that pupils pronounce and spell correctly the 
 names of the decimal places. In writing the word hundredths, pupils 
 often omit the second d. In pronouncing the word hundredths, pupils 
 often neglect to sound the th. Point out that 100 and one hundredth 
 are not the same. The fourth and fifth decimal places are "ten- 
 thousandths ; hundred-thousandths, ' ' not tens of thousandths or hun- 
 dreds of thousandths. 
 
 3. Teach that the hyphen between the ten and thousandths in the 
 word three ten-thousandths, makes the two words one. 3 ten-thou- 
 sandths is written decimally as .0003. The name of the decimal place 
 is ten-thousandths. The problem requires us to write three of them. 
 
 When there is no hyphen between the ten and thousandths, they are 
 to be regarded as separate words. Thus, ten thousandths would be 
 
 3—1.5484 (13) 
 
written decimally .010. The name of the decimal place is thousandths. 
 The problem requires us to write ten of them. 
 
 4. In reading a number such as 273.003, there is only one and. 
 This is the decimal point. The above number should be read ''tw^o 
 hundred seventy-three and three thousandths." Have pupils read 
 $20.85, 1.04, etc. 
 
 5. Ask : What is the name of the third place after the decimal point ? 
 The fourth place? The second place, etc. How many decimal places 
 are needed to express ten-thousandths? Hundredths? etc. 
 
 6. When w^riting decimals from dictation, insist upon pupils placing 
 the decimal point as soon as the "and" is mentioned. This is the only 
 way in which pupils will acquire the habit of always inserting the 
 decimal point. 
 
 7. Teach which is of the greater value and why — 3 tenths. 3 hun- 
 dredths or 3 thousandths, etc. Pupils often think that three hundredths 
 or three thousandths of a quantity is greater than three tenths of the 
 same quantity. 
 
 8. In a number such as 2.2222. have a pupil tell the value of each 2 
 as compared with the 2 to its right or left. 
 
 9. Show pupils that .4, .40, .400 of a quantity are all of equal value. 
 Point out the value of the naught in numbers such as the above. 
 
 10. Show pupils how the naughts change the value of each of the 
 following numbers : .002, .02, 1.02, 1.10. 
 
 11. Explain that our system of money is a decimal system. (Text, 
 pages 9, 14.) 
 
 12. Have pupils see that .6 is onlj* another way of writing the fraction 
 1%. Emphasize the fact that a decimal is a fraction of a quantity. 
 
 ADDITION OF DECIMALS. 
 
 The addition of decimals is merely an application of the reading and 
 W'riting of decimals. 
 
 Dictate exercises in which some of the numbers are liuudredths, 
 others thousandths, integers, etc. Do not permit pupils to fill out 
 vacant places with naughts. Occasionally write the decimals out in 
 words and have pupils add the numbers. 
 
 (For drill work, see Text page 18. Teach also the Avork of page 14.) 
 
 29. Language Forms : 
 
 The language forms to be used in the addition of decimals are: 
 
 a. Add the following : .003, .2, 1.46, 83, .004. 
 
 &. Find the sum of: 1.14, 89.2, 10006, 1.0012. 
 
 c. Work in the addition of decimals may be varied by using the 
 following form : 
 
 Write in the order of their value and add : Sixteen ; thirty-two and 
 four hundred-thousandths ; four hundred-thousandths ; one and three 
 millionths; eight hundredths. The above exercise is a good test a. of 
 the pupil's ability to write decimals; h. to ascertain their relative value. 
 
 c. Emphasize the fact that in the addition of decimals, the decimal 
 points must be kept under each other. 
 
 (14) 
 
SUBTRACTION OF DECIMALS. 
 
 28. Types in Subtraction of Decimals: 
 
 a. h. c. d. e. f. 
 
 275.85 275.85 275.85 .7 295 296 
 
 -14.25 -14. - .974 -.0024 - 14.006 - .009 
 
 Little or no difficulty will be experienced in teaching the subtraction 
 of decimals provided pupils are proficient in the reading and writing 
 of decimals. Emphasize the fact that decimal points must be kept 
 under each other. 
 
 29. Language Forms for Subtraction of Decimals. 
 
 Often pupils, who experience little difficulty in solving subtraction 
 exercises in which the minuend and subtrahend are correctly placed, 
 fail when confronted with a problem requiring discrimination between 
 the two terms. As this is the form in which subtraction is usually 
 encountered in life and even in the problems of a textbook, it is neces- 
 sary to teach the various language forms which serve to indicate that 
 the process of subtraction is to be employed. Only the more common 
 forms are enumerated below in the order of their difficulty : 
 
 1. Take 4.06 from 19.65. 
 
 2. Subtract 82.3 from. 963.64. 
 
 3. Find the difference between 444.85 and 200. 
 
 4. Find the difference between .245 and 48.65. 
 
 (This last type is especially difficult because the smaller num- 
 ber is given first. Pupils often attempt to subtract the numbers 
 in the order in which they appear in the problem.) 
 
 5. What must be added to 8.56 to make 27.2? 
 
 6. Teach Text page 25. 
 
 7. The addition and subtraction of decimals may be taught in a 
 very short time provided pupils can read and write decimals with- 
 out hesitation. 
 
 MULTIPLICATION OF DECIMALS. 
 
 30. Two Principles to Be Taught : 
 
 There are two things to be emphasized in teaching the multiplication 
 of decimals — (1) the correct placing of the decimal point in the 
 answer. (2) The correct placing of partial products with the different 
 types of multipliers. 
 
 31. The Placing of the Decimal Point: 
 
 1. Do not insist upon pupils writing the multiplier so that its decimal 
 point is under the decimal point of the multiplicand. This form often 
 causes pupils to subtract instead of multiplying the exercise. Use this 
 form : 
 
 2.4 2 9 6 ^^. 2.4 2 9 6 
 
 X .9 6 "'^ X.9 6 
 
 do) 
 
2. Do not use multiplicands or multipliers which have a great many 
 figures. The fatigue induced by the working of such long problems 
 leads tf> many mechanical errors. Two short problems are better than 
 one long one for teaching a new principle. 
 
 TYPES OF MULTIPLIERS AND MULTIPLICANDS. 
 
 The tj'pe problems of multiplication are enumerated below in the 
 order of their difficulty. 
 
 The same general form and method of work should be employed in 
 teaching pupils to place their partial products when using each and 
 every type of multiplier. The one rule, "Place your first answer 
 directly under the number you have multiplied by" gives excellent 
 results. 
 
 32. Types of Multipliers : 
 
 a. b. c. d. e. f. 
 
 24.82 900.2 427.08 6.347 14.2 .007 
 
 X.422 X 2.5 X 6.009 X 4.80 X-06| X .003 
 
 1. In type a, the emphasis is only upon the correct placing of the 
 decimal point in the answer. 
 
 2. In type &, the naughts of the multiplicand offer a difficulty to 
 many pupils. 
 
 3. In types c and d, pupils often fail to place the partial products in 
 their correct place. Use the suggestion of "Place your first answer 
 directly under the number you have multiplied by" and the difficulty 
 wall disappear. 
 
 4. In type e, be careful to see that pupils do not consider the fraction 
 as occupying a decimal place and also that the product obtained by 
 multiplying by 6 is correctly placed. 
 
 5. In type f, teach pupils that naughts must be added in the product 
 to the left of the integer. Show why naughts must be added to the left 
 and not to the right of the integer. Pupils often subtract instead of 
 multiplying this tj^pe of problem as its appearance is very much that of 
 a subtraction exercise. 
 
 33. Language Forms To Be Used In Connection With the Multiplica- 
 tion of Decimals: 
 
 1. Find the product of 24.84 and .32. 
 
 2. What is six hundredths of 24.84? 
 
 3. Find decimally /v of iVoV- 
 
 4. Multiply .002 by .003. 
 
 DIVISION OF DECIMALS. 
 
 In teaching the division of decimals, follow closely each of the fol- 
 lowing steps : 
 
 1. Be certain that pupils know the difference between the terms 
 dividend and divisor. 
 
 (16) 
 
2. Write on the board division problems in which there are more 
 decimal places in the dividend than in the divisor. Write other 
 problems in which there are less decimal places in the dividend than 
 in the divisor. 
 
 3. Teach that there must always be at least as manj- decimal places 
 in the dividend as in the divisor. Emphasize the fact that the dividend 
 may have more decimal places than the divisor. 
 
 From the problems on the board, have pupils select those in which 
 the relation between the number of decimal places in the dividend and 
 the divisor is correct. 
 
 4. Teach pupils that naughts are added to dividends that need more 
 decimal places. Show why these naughts do not change tlie value of 
 the decimal. 
 
 5. Dictate many problems to pupils and have them fix the dividends 
 that need attention. Do not leave this step of the w^ork until it is 
 thoroughly mastered. 
 
 6. Placing Decimal Point in Answer : 
 
 Use the following method to teach pupils to place decimal points in 
 the answer. Insist upon pupils placing the deciiual point in the answer 
 before the problem is solved. 
 
 a. Write the following problem on the board : 
 
 .0 2 ) 4 1 8.2 8 4 9 
 Place your chalk on the decimal point iu divisor and count the 
 number of decimal places, saying "decimal point— one, two, 
 three." Place your chalk on the decimal point in the dividend 
 and count off the same number of decimal places, saying 
 "decimal point — one, two, three." Place decimal point in 
 quotient above and after the last number you counted. Try 
 other types and iLse the same language form. 
 
 .4 ) 8 2.4 2 9 
 
 c. Dictate problem after problem and have pupils count off the 
 
 decimal places and fix the decimal point in the cpiotient. At 
 this stage of the work, do not permit pupils to solve the 
 problems. All emphasis should be upon the correct placing of 
 the decimal points. 
 
 d. The following are the advantages of the above method : (l)It is 
 
 not necessary for pupils to memorize any rule for pointing off 
 the answer in division of decimals. Memorized rules are soon 
 forgotten. (2) The decimal is placed before beginning any of 
 the work of division; (3) when the decimal point is placed in 
 the answer, pupils can tell at a glance the number of places to 
 which the answer has been carried. 
 34. Carrying- Answers Out to Three Decimal Places : 
 
 1. When pupils can correctly place the decimal point in ;uiy type 
 problem, show them that the problem is really one in the division of 
 integers. 
 
 (IT) 
 
2. Teaeli pupils that to carry an answer out three places decimally, 
 means to have three figures in the answer after the decimal point. 
 Pupils sometimes think that adding three naughts to the dividend has 
 carried the answer out decimally three places. 
 
 3. After carrying the answer out to three decimal places, a plus sign 
 should he used, not a common fraction remainder. 
 
 Suggestions : 
 
 1. Insist upon pupils alwaj^s placing the decimal point in the answer 
 before attempting to perform the division of the problem. This is the 
 onl}^ way in which pupils will ever acquire the habit of always placing 
 the decimal point in the quotient. 
 
 2. Have all answers carried out to three decimal places. 
 
 3. Do not place the emphasis upon the length of the problem, but 
 upon the correct placing of the decimal point. 
 
 4. Occasionally make use of the divisor 3.1416 which will be used 
 later in the work of mensuration. 
 
 5. Use the following language forms when dictating problems: 
 
 35. Language Forms for Division of Decimals. 
 
 a. How many times are .2 contained in 42.006 ? 
 h. Divide 4.09 by .6. 
 
 c. Divide 2 by 11. 
 
 d. $20.45 is .006 of what amount? 
 
 36. Types to be Found in the Division of Decimals. 
 
 .004 
 
 9- 
 
 Divide 2 by 7. 
 
 i. 
 
 
 6.5 
 
 ) .0042 
 
 
 .07 
 
 d. 
 
 
 ) 63.2 
 
 
 
 /. 
 
 
 27 
 
 ) 1.0246 
 h. 
 
 Chi 
 
 mge I'l' to a decimal. 
 
 
 
 J- 
 
 
 3.1416 )~22' 
 
 $1.05 ) $728.45 
 
 For drill work after all the above principles have been taught see 
 Text, pages 68-69-70. 
 
 FRACTIONAL EQUIVALENTS. 
 
 Have the pupils memorize the following : 
 
 .50 =i .40 =f .m = i 
 
 .33i = i .60 =f .37i = f 
 
 .66f = | .80 =f .62| = f 
 
 .25 =i • .161 = i .S7i = x 
 
 .75 =1 .83^ = 1 .10 =Jo 
 
 .20 =i .141 = ^ .30 =xV 
 In connection with this section teach the material on pages 156, 157 
 and 158 of text. 
 
 (IS) 
 
37. To Divide Integers and Decimals by 10, 100, 1000. 
 
 The following are the types of numbers which pupils should be taught 
 to divide first by 10, then 100, then 1000 (text pages 37, 40) : 135, 40, 
 400, 6, .2, .06, 1.3, $4, $200, $1.60, $.85, $.40. 
 a. Write 4.6 on the board. Erase the 6. This leaves 4 with a decimal 
 
 point after it. Have pupils read this number. 
 h. Write 16, 18, 32 on the board. Have pupils locate the decimal point 
 in each of these numbers. 
 
 c. Teach that to divide 16, 18, 32, etc., by 10, the decimal point is moved 
 
 one place to the left. 
 
 d. Teach that numbers, such as 50, 60, 40, etc., are divided by 10 by 
 
 moving the decimal point one place to the left. Emphasize the fact 
 that such numbers are divided by 10, not by crossing off the 
 naughts but by moving the decimal point one place to the left. 
 
 c. Care must be taken to see that pupils do not make mistakes in 
 dividing numbers such as $.40, $1.40, etc., by 10. Ask: ''Does 
 crossing off the naught divide the above numbers by 10?" "How 
 are they divided by 10 ? " 
 
 /. Ask: "How is a number divided by 10?" 
 
 g. When the decimal point has been moved one place to the left, what 
 has been done to the number ? 
 
 h. Repeat the same steps in teaching pupils to divide numbers by 100 
 and 1000. Use the types listed above. 
 
 (. Ask: "How are numbers divided by 10, 100, 1000?" When deci- 
 mal points are moved one, two or three places to the left, what has 
 been done to the number ? 
 
 38. To Multiply Numbers by 10, 100, 1000. 
 
 a. Teach that numbers are multiplied by 10, 100, 1000 by moving the 
 
 decimal point one, two, or three places to the right. 
 h. Ask: "How is a number multiplied by 10? 100? 1000?" 
 
 c. When is the decimal point moved to the right ? 
 
 d. When is the decimal point moved to the left? 
 
 e. Put a column of nuinbers on the board. Mix in decimals with the 
 
 integers. Call upon pupils to give the answers when the various 
 numbers are divided by 10, 100, 1000, or when multiplied bv 10. 
 100, 1000. 
 
 MISCELLANEOUS QUESTIONS ON DECIMALS. 
 
 After all of the principles in decimals have been taught, the following 
 questions and the tests of the next section may be used to ascertain the 
 extent of the pupils' knowledge. 
 
 Normal students will find these c|uestions and tests useful in tlieir 
 preparatory work in arithmetic. 
 
 1. Write in their order the names of the decimal places through 
 millionths. 
 
 2. Explain why our system of money is a decimal system. 
 
 3. Which is of the greater value, $1.10 or 1100 mills? How much 
 greater ? 
 
 (10) 
 
4. Write in figures three hundred thousandths; three hundred- 
 thousandths ; three hundred thousand. How do you distinguish each 
 type? 
 
 5. In the expression 111.111, how does each 1 compare in value with 
 the one to its left ? 
 
 6. If a naught is added to the right of $2.10. has the number been 
 multiplied by ten? If the naught at the end of $2.10 is crossed out, 
 has the number been divided by ten ? Explain in full. 
 
 7. What is the difference in value between 100 and .01 ? 
 
 8. Where 4s the decimal point in a number such as 24 ? 
 
 9. If the decimal .9 is a fraction, where is its numerator and 
 denominator ? 
 
 10. In a decimal expression such as .665, does the fraction, f, count 
 as a decinial place ? 
 
 11. Show that .8, .80, .800 of a quantity are all of equal value. 
 
 12. Write the table of United States money. 
 
 13. In the multiplication of decimals, how do you know the number 
 of decimal places to point off in the answer ? 
 
 TEST IN ADDITION AND SUBTRACTION OF DECIMALS. 
 
 1. Express each of the following decimals as common fractions : .024, 
 .004, .8, 2.101, .400, .00265. 
 
 2. If two naughts are added to the right of the 1 in the decimal 
 expression .1, what is the effect upon the value of the decimal? If two 
 naughts are added to the left of the 1 in what way is the value of the 
 decimal .1 changed? 
 
 3. How does .4 compare in value with .44 and with .004 ? 
 
 4. Show that .1, .10, .100 are of equal value. 
 
 5. What is the name of the fourth decimal place? The third deci- 
 mal place ? The second decimal place ? 
 
 6. Write at least five aliquot parts of a dollar. 
 
 7. Arrange the following decimals in the order of their value and add : 
 Six hundred six and six hundred-thousandths ; eighteen and one mill- 
 ionth ; twelve and six tenths ; five hundred seven thousandths ; six hun- 
 dred; fifteen hundredths. 
 
 8. Find the difference between .1 and 1.426. 
 
 9. a. h. 
 296 A $110 
 
 ^ 18.008 — $ .85 
 
 10. Find out by changing to common fractions whether you would 
 rather have .5 of a pie. .50 of a pie, or .500 of the same pie. 
 
 11. What is the difference in value between (a) 1 and .1; (h) 1. and 
 .01; (c) 1. and .001; {d) 100 and .01 ? 
 
 12. Place the decimal point in each of the following numl^ers : sixteen 
 hundredths; eighty-five; five thcusandths; two hundred seventy-five; 
 sixteen and four thousandths ; twelve. 
 
 13. Write decimally and add : four dollars and sixty-five cents ; 
 twenty-five and seven tenths dollars; forty-eight dollars and nine 
 
 (20) 
 
cents; eight thousand three hundred sixteen dollars and sixteen cents 
 four mills ; four and sixteen hundredths dollars ; eight dollars and four 
 cents ; nine mills ; three and two hundred fifteen thousandtlis dollars. 
 
 14. What is .1 of a dollar? One hundredth of a dollar? One 
 thousandth of a dollar? 
 
 TEST IN MULTIPLICATION AND DIVISION OF DECIMALS. 
 
 1. Find the product of 80.04 and 40.8. 
 
 2. Divide each of the folloMing problems in the shortest way possible : 
 
 $1.10 ) $240 300 ) 150.1 400 ) .8394 
 
 $600 ) $84.80 $.10 ) $1.10 .40 ^400 
 
 3. Change -^f to a decimal. (Carry quotient to three decimal places.) 
 
 4. Multiply .004 by .003. 
 
 5. Place only the decimal point in the quotient of each of the follow- 
 ing problems : 
 
 a. 3.1416 )r259 h. .003 ) 2.429 c. .019 yii d. 34 ) 802.34 
 e. .7 y^ /. 19 Yl g. $.01 Y^ /'• $1 ) $1.01 
 
 6. Divide 2 by 9. 
 
 7. ]Multiply each of the following numbers in the shortest way pos- 
 sible, first by 10. then 100, and then 1000: 
 
 $.45, $1.40, .0008, .6, .94, $1,008, 2.75, .009, 46., $.05. 
 
 8. Find decimally fo of 42.094. 
 
 9. What is 6| hundredths of 245.6 ? 
 
 10. Write each of the following as decimals : f, f, |, f , =. |, f, ^, f, -|. 
 
 RATIO, AVERAGE, TABLES, BILLS AND RECEIPTS. 
 
 The following questions cover several miscellaneous topics. The 
 questions will indicate the principles to be taught under each topic. 
 
 39. Test on Ratio and Average. 
 
 1. What is the ratio of a pound to an ounce? 
 
 2. From the following table, find Fred's average gain in weight and 
 in height each year : 
 
 Age. Height. Weight. 
 
 11 yr. 4' 11" 98 lb. 
 
 12 " 5' 2" 104 lb. 4 oz. 
 
 13 " 5' ^" 107 lb. 9 oz. 
 
 14 " 5' 4" 110 lb. 1 oz. 
 a. Be certain that pupils first find the gain in height. 
 
 6. Since there were 3 gains, the actual gain is divided by 3 to find the 
 average gain in height. 
 
 3. The expenses of a picnic were $3.40. Divide this amount between 
 Mrs. Brown and Mrs. White in the ratio of 2 to 3. 
 
 4. ]\Iiss Saunders had 57 pupils in her class. There were 42 pupils in 
 ]\Iiss Ryan's class. Find the ratio of the number of pupils in Mi.ss 
 Saunders' class to the number in Miss Ryan's class. 
 
 5. In 1890 the population of San Francisco wa& 298,997. In 1906 
 the population of San Francisco was 450,000. Find the ratio of the 
 population in 1906 to the population in 1890. 
 
 (21) 
 
6. During the month of September Mrs. Fisher used 645 cubic feet of 
 gas; in October, 750 cubic feet; in November, 1,200 cubic feet; in 
 December, 1,950 cubic feet ; in January, 2,645 cubic feet. Find the 
 average gain in the number of cubic feet of gas used each month. 
 
 7. What is the ratio of f to f ? 
 
 8. Is the ratio between two quantities ever expressed as a concrete 
 number ? Why ? 
 
 9. Is the ratio of 7 to 9 the same as the ratio of 9 to 7 "? 
 
 10. During a certain year the rainfall for six months in San Francisco 
 was as follows : December, 5.11 inches ; January, 4.98 inches ; Feb- 
 ruary, 3.72 inches; March, 3.28 inches; April, 2.14 inches; May, 0.68 
 inches. Find the average rainfall for each month. 
 
 40. Tables: Linear, Square, Weights, Country, Time, Liquid. 
 
 a. How many pints of milk may be poured into a five-gallon can? 
 h. Mr. Bray owns a section of land. What is the value of the entire 
 farm if one acre is worth $75 f 
 
 c. What is the value of 956 eggs at $.32 a dozen ? 
 
 d. Helen practices two and one half hours a daj^ on the piano. How 
 many minutes does she practice each day? 
 
 e. At $11.50 per ton, what is the value of 16,480 pounds of hay? 
 
 /. I bought 2| pounds of sugar. How many ounces of sugar did I 
 purchase ? 
 
 g. What is the cost of a great gross of pencils if one pencil costs l^ff. 
 
 h. ]Mr. Brown owns property having a frontage of | of a mile. 
 Express the width of the lot in feet. 
 
 i. A square foot is how many times as great as a square inch ! 
 
 j. Which is the greater in value, $2.10 or 8,124 mills? How much 
 greater ? 
 
 A'. How many steps, each 12 inches long will a man take in walking 
 one mile ? 
 
 /. What is the length and wndth in feet of a square room, the floor of 
 which has an area of one square rod? 
 
 m. Find the sum of 4' 6" ; -}" ■ V i" ■ 2' IV'. 
 
 n. Find the difference between 102^' and ^". 
 
 0. How many dozen are there in a gross ? Ounces in a pound ? 
 Pounds in a hundredweight? Minutes in an hour? Units in a score? 
 Quarts in a gallon? Pints in a quart? Mills in a cent? Pounds in a 
 short ton ? Hundredweight in a long ton ? 
 
 II. 
 
 a. If a dealer has 288 pencils in stock, are there enough to fill an order 
 for a gross of pencils? 
 
 b. Which is the larger amount, 18 cents or 25 mills? 
 
 c. How many pounds are there in a long ton? How many cwt. are 
 there in 3485 pounds? 
 
 d. Alice practiced 85 minutes on the piano. How much more or less 
 than an hour did Alice practice? 
 
 e. How many pint bottles may be filled from a gallon of milk? 
 
 (22) 
 
/. Last Christmas I sent to New York a package weighing 22 ounces. 
 How much more or less than a pound did the package weigh? 
 
 g. How many cubic inches are there in a gallon? 
 
 h. How many cords of wood are there in 15,744 cubic feet of lumber? 
 
 i. How many pounds of coal are there in 2 hundredweight? 
 
 j. How many minutes are there in 2 hours 45 minutes? 
 
 A'. How many years are there in "four score and seven years"? 
 
 1. How many hundredweight are there in one ton? 
 
 m. If a ton of coal is worth $14, what is the value of three hundred- 
 weight of coal? 
 
 n. A gallon of water is the same as how manj' cubic inches of water? 
 
 41. Bills and Receipts. (Text pages 42-44.) 
 
 1. Mr. Berryman rented a house at 1207 Sacramento Street from Mr. 
 J. R. Hansen for the sum of $35 a month. On May 1st, Mr. Berryman 
 paid the rent for one month in advance. Write the receipt Mr. Hansen 
 gave to ]Mr. Berryman. 
 
 2. ]\Iake out and receipt the following bill : I\Irs. J. H. Moore, resid- 
 ing at 3746 Sacramento Street, bought of her dealer the following: 
 January 3, 1914, 4 yd. blue ribbon @ $.30; 2^ yd. lace @ $.75; 6 
 handkerchiefs @ $.25; | yd. lace yoking @ $2.25. 
 
 3. Make out a receipt of your own showing that money has been paid 
 by you to some other person for work done. 
 
 4. Make out and receipt the following bill of goods purchased from 
 your grocer by you on May 1st: 
 
 3 lb. coffee @ $.35 
 
 1 tb. tea @ .80 
 
 2 doz. eggs @ .30 
 
 1 roll of butter__@ .65 
 
 5. Mrs. Adams bought from ]\Irs. Dean a sewing machine for the sum 
 of $60. Write a receipt showing that Mrs. Dean has received the money 
 from Mrs. Adams. 
 
 AREA OF RECTANGLES. 
 
 (Text pages 80-84.) 
 The material of pages 80, 81, 82, 83 of the Advanced Arithmetic may 
 be supplemented with much additional work from Bulletin No. 31, pages 
 1-26. 
 
 42. Test on Area of Rectangles. (Text pages 80-84.) 
 
 The following questions may be used to test a pupil upon the work 
 of pages 80 to 84 of text. 
 
 1. Draw three different figures, each of which is a rectangle. Label 
 
 the base and altitude of each figure and shade its surface. 
 
 2. Illustrate with diagrams the difference between a horizontal rect- 
 
 angle and a square. 
 
 3. Illustrate with diagrams the difference between two square inches, 
 
 two inches and a two-inch square. 
 
 4. Draw on your paper four right angles, each of which is in a 
 
 different position. 
 
 (23) 
 
5. Is a very thin sheet of tissue paper a surface or a solid ? Give a 
 
 reason for your answer. 
 
 6. Illustrate by means of a diagram which is the larger, a square 
 
 mile or a section of land. 
 
 7. A lot is 125 feet deep and has a frontage of 25 feet. What is 
 
 the perimeter of the lot? 
 
 8. How many feet of picture molding are needed for a room 21 feet 
 
 long and 16 feet wide? 
 
 9. Make a list of three rectangular surfaces that appear in your 
 
 classroom. 
 
 10. Find the perimeter and the surface of this page of your book. 
 
 11. A forest extends over two sections of land. How many acres of 
 
 land are covered b}' the forest ? 
 
 12. What is the length in feet of the side of a square field, the area 
 
 of which is one acre? 
 
 13. Draw two oblique, two horizontal and two vertical parallel lines. 
 
 14. Draw a sc^uare foot on your paper. (Let a line 6 inches long 
 
 represent a foot.) Divide your square foot into the greatest 
 possible number of square inches. What is the answer ? 
 
 15. Draw a square yard on your paper. (Let 6 inches represent 
 
 1 yard.) Show" by your diagram the number of square feet 
 there are in one square yard. 
 
 16. What is the perimeter and what is the area of a field one mile 
 
 scpare? Diagram your field. 
 
 17. Tell for what, in the scpare measure or the linear measure, each 
 
 of the following numbers stands : 640, 3, 12, 9, 320, 160, 30^, 
 208f , 16|, 144, 5i, 5,280. 
 
 18. With a diagram, show the number of square inches there are on 
 
 the surface of a three-inch sc|uare. What is its area? 
 
 19. Draw, if you can, a rectangle which contains no right angles. 
 
 20. Illustrate with marked diagrams the difference between the perim- 
 
 eter and the surface of a four-inch square. 
 
 21. Express 14| square feet as scjuare inches. Change 4 acres to 
 
 square rods. 
 
 22. Draw a horizontal rectangle, a vertical rectangle and a square. 
 
 Label the base and altitude, and shade the surface of each 
 figure. 
 
 23. Draw a line which is perpendicular to another line. 
 
 24. If the area of a rectangle is given in acres and the length of one 
 
 side in rods, how is the perimeter of the rectangle found? 
 Illustrate with a problem of your own. 
 
 25. If the area of the floor of a square room is one square rod. what 
 
 is the length and width of the room expressed in feet? 
 
 26. At $175 an acre, what is the value of a field which is 400 rods 
 
 long and ^ of a mile wide? 
 
 27. How many rails, each 16| feet long, are needed to enclose a square 
 
 field, the area of which is one sc^uare mile? 
 
 (20 
 
28. A field contains 3 acres of land and is ^ of a mile long. What is 
 
 the Avidth of the field? 
 
 29. If the area of a field is 10 acres and one side is 20 rods long, 
 
 what is the perimeter of the field? 
 
 30. ]May an acre or any part of an acre ever be used to measure the 
 
 length of a line ? Give a reason for your answer. 
 
 31. j\Ir. Jones paid $550 for his farm. Each acre cost him $55. If 
 
 the length of the farm is 180 rods, what is its perimeter ? 
 
 CUBIC MEASURE. 
 
 (Text pages 84-86.) 
 The material of pages 84, 85, 86 of the text may be supplemented with 
 much additional work from pages 55 through 68 of Bulletin No. 32. 
 
 43. Test on Cubic Measure. (Text pages 84^86.) 
 
 The following questions will serve as a test of a pupil's knowledge 
 after he has completed pages 84 through 86 of text. 
 
 1. How many cords of wood are there in a pile which is 12 feet long, 
 
 8 feet wide and 4 feet high? 
 
 2. How many cubic yards of dirt were removed from a lot 114' X 50' 
 
 if the earth was removed to a depth of 6 feet ? 
 
 3. A rectangular tin can is 8" X 10" and 16 inches high. How 
 
 many gallons of water will the can hold? 
 
 4. A swimming tank is 110 feet long, 75 feet wide and has a depth 
 
 of 5 feet. How many gallons of water are there in the tank 
 when it is one half full? (7^ gal. = 1 cu. ft.) 
 
 5. Illustrate with a drawing the difference between a two-inch square 
 
 and a two-inch cube. 
 
 6. How many faces has a cube ? "What is the volume of a three-inch 
 
 cube? 
 
 7. Is a sheet of paper a surface or a solid ? Why ? 
 
 8. In order to find the volume of a prism, how^ many dimensions must 
 
 be known? State and solve a problem of your own in which 
 you are required to find the volume of a prism. 
 
 9. A gallon of water is the same as how many cubic inches of water? 
 
 10. If the length and width of a room are given, tell whether or not 
 
 you can find (a) the area of the ceiling, (6) the volume of air 
 in the room, or (c) the number of feet of picture molding 
 needed for the room. 
 
 11. ]\Ir. Harris removed one foot of dirt from a lot having a depth of 
 
 114 feet and a frontage of 75 feet. How many cubic yards of 
 earth were removed from the lot ? 
 
 12. What is the volume of a four- inch cube ? 
 
 13. A rectangular vessel is 26 inches long, 18 inches wide and 9 inches 
 
 deep. What is its capacity expressed in gallons? 
 
 14. Write the table of cubic measure. 
 
 (2.5) 
 
PERCENTAGE. 
 
 (Text pages 166-190.) 
 
 The work of each of the three cases in percentage may be divided into 
 two parts — formal work and problem work. Bulletin No. 29 presents. 
 in the order of their difficulty, each of the points to be taught under 
 the formal and problem work of each of the cases in percentage. As 
 this bulletin is for self -instruction, it contains all necessary explanations 
 of difficulties. Normal students are expected to be familiar with all 
 of the problems of this bulletin as well as the problems in the State Text 
 on pages 166-190. 
 
 The mechanical or formal work of each case in percentage should be 
 mastered before the problem work is introduced. For this reason, the 
 formal work of each case of percentage is taught before pupils are 
 required to center their attention upon the problem work of a given 
 case. Most of the complicated and impractical problems which fall 
 under the second case in percentage have been omitted from Bulletin 29. 
 
 43. Formal Work of Case One in Percentage. 
 
 a. The most difficult piece of formal Avork in the first case in per- 
 centage is the changing of the various types of per cents to correct 
 decimals. 
 
 &. Pupils should be taught to change each type of per cent as a 
 separate step of work. As soon as possible, work should be given which 
 will be an application of the type taught. Unless this is done, pupils 
 will think of the changing of per cents to decimals as a line of work 
 separate from the rest of percentage. 
 
 c. Bulletin No. 29, pages 3 through 23, presents in the order of their 
 difficulty, each of the difficulties that occur in the formal work of 
 Case One. 
 
 d. The following are the most difficult sections: 7, 8, 11, 12, 16, IS, 
 22, 25, 26, 28, 29, 30. Teachers should question pupils closely upon 
 each of these sections before permitting them to take up any new work. 
 
 e. The tests of sections 32 and 33 may be used as guides Avhen making 
 out tests upon the formal work of Case One. 
 
 Problem Work of Case One in Percentage. 
 The failure of many pupils in the problem work of arithmetic may 
 be traced (1) to their inability to read a problem and ascertain what 
 is given in the problem and what is recpiired to be found; (2) to lack 
 of ability to understand the situation, words and phrases of the prob- 
 lem; (3) helplessness in attempting the solution of the problem; (4) 
 failure to test the answers to problems. 
 
 44. Statement of Problems : 
 
 Pupils should be required to write a statement of every problem 
 before attempting its solution. 
 
 When a pupil can write a correct statement of what is given in 
 a problem and what is to be found, it is an indication to the teacher 
 that the pupil has at least read the problem. The method of teaching 
 
 (26) 
 
pupils to write the statement of a problem is to be found on page 28 of 
 Bnlletin No. 29. Pnpils should master the writing of statements before 
 being permitted to proceed to the next step of work. 
 
 45. Wording of Problems: 
 
 As far as possible, the situations in the problems of the bulletin are 
 those with which pupils are familiar or which can readily be explained 
 to them. 
 
 46. Solution of Problems and Proof : 
 
 Under each of the tj'pe problems, questions are asked to help pupils 
 in their method of attacking the problem. In each case in percentage, 
 emphasis is placed upon having the pupil first w^ork out the meaning 
 of the per cent in the problem. As part of the regular problem work, 
 pupils are required to prove the answers to their problems whenever 
 possible. 
 
 The problems of Case One on pages 52 through page 58 should receive 
 special attention. 
 
 The tests on pages 60, 61 may be used as guides when planning 
 tests on the problem work of Case One in percentage. 
 
 Formal Work Case Two in Percentage. 
 
 The second case in percentage is usually more difficult for pupils to 
 master than either of the other cases. Experience has shown that pupils 
 gain more power is attacking problems of this case Avlien taught the 
 method of unitary analysis rather than the method of the text. All 
 explanations of difficulties of the formal work of Case II of percentage 
 will be found on pages 62, through 72 of the bulletin. Care must be 
 taken to see that pupils follow closely the work of the models. 
 
 The suggestions given on pages 69-70 are important. 
 
 The tests on pages 74-75 may be used as guides, when planning tests, 
 on the formal work of Case II of percentage. 
 
 The Problem Work of Case Two in Percentage. 
 
 Problems similar to those on page 178 of text are very difficult for 
 pupils to understand and also very impractical. Problems of this 
 type have been entirely omitted from the Avork of the bulletin. Normal 
 students are however required to be familiar with such problems. 
 
 Review of Cases One and Two. 
 
 The test of a pupil's knowledge of the first or second cases in per- 
 centage comes when the problems of these two cases are mixed. Such 
 a review may be found on pages 90-95 of the bulletin. 
 
 Case Three Formal. 
 
 Pupils must be taught to change decimals to per cents before begin- 
 ning the real work of Case Three. Each type of decimal is taught and 
 applied in the various sections of pages 99 through 110 of the bulletin 
 (Text 179-182). A test of the formal work of Case Three of percentage 
 may be found on page 107. 
 
 (27) 
 
Problem Work of Case Three. 
 
 The difficulties met in teaching the problems of Case III in percentage 
 are explained and illustrated on pages 114 through 122 of the bulletin. 
 
 Review of Cases One, Two, Three. 
 
 The test of the pupil's knowledge of percentage comes when problems 
 of the three eases are presented in a miscellaneous order. Such a review 
 may be found on pages 184 through 191 of text or pages 122 through 
 126 of bulletin. Normal students are expected to be familiar with all 
 of the above mentioned problems. 
 
 TEST ON FORMAL WORK OF PERCENTAGE. 
 
 The following miscellaneous review of the three cases of percentage 
 may be used as a guide when it is necessary to test a pupil 's knowledge 
 of the three cases of percentage. At the beginning of a term or when 
 a new pupil enters a class, such a test is useful. Pupils can be then 
 drilled upon the points in which their knowledge is deficient. 
 
 1. Illustrate with a diagram and a problem the difference between 
 
 i of $120 and ^% of $120. 
 
 2. Write each of the following expressions in two other ways : 
 
 lOlf %, f %, 200%, 104%, 45%, 66§%, f %, 1^%, 1^ i%, 1^%, 
 t%, 12%, 110%, Jo, liV%, 1%, 133^%, 8-^%, 250%, 6, 103%, 
 82%, 101f%, 85%, 111%, 2. 
 
 3. $102 is 31% of what amount? 
 
 4. Find the entire cost of a book when 83-?,% of the cost is $2.50. 
 
 5. AVrite each of the following as per cents : ^, f , f , f , f , i, ^, fo, 
 
 1 1 
 
 6. Change ^| to a per cent. 
 
 7. What per cent of 2 is 1.75 ? 
 
 8. What per cent of $810.20 is $350 ? 
 
 9. Find in the shortest way— a. 166|% of 270. b. 8^% less than 
 
 180. 
 
 10. Express each of the following as per cents : 
 
 a. 3.1, .0401, 1.1, 2.3, 2|, .0075, .00125. 
 h. .8906, .241, .161, 1.001, 1.013, 2.5, .024. 
 
 c. li .0075, .75, .010, 1.011, 1.111, 3. 
 
 d. 2i, .022, 1.4, 1.067, 2|, 4.25, 3.2. 
 
 11. Find the number that is 17-|% less than 1,000. 
 
 12. $850 is 1|% of what amount? 
 
 13. Find 100% more than $600. 
 
 14. Find Y/c less than 210. 
 
 15. AVhat per cent of a mile is a rod? 
 
 16. Which is the largest amount, -jVir 1'^% o^" -^6 of $250 ? 
 
 17. What is the difference between 130% of $210 and 30%^ more than 
 
 $210 ? 
 
 18. What is the difference between 5% less than $115 and 95% of 
 
 $115? 
 
 19. Write a per cent of a quantity that is equal to less than the 
 
 whole of a quantity. To more than the whole of the quantity ? 
 (28) 
 
20. Why is it incorrect to say that $400 = 125 7o ? 
 
 Correct the statement. 
 
 21. Tell two ways of finding 331%, 40% or 25% of a quantity. 
 
 22. Six out of every 100 is the same as how many per cent! 
 
 23. Is there any difference between ^, 12^%o and i% of 64? Illus- 
 
 trate with work and a diagram. 
 
 24. What per cent of I is f? 
 
 25. Tell without working the following problems in which cases the 
 
 answers will be larger and in which cases smaller than the given 
 
 cjuantity. 
 
 a. Find 105%o of 650. h. What is 4%o less than 220? c. Find 
 
 i% of 125. d. What is 2^% more than 175? e. Find 200%o 
 
 of $150. 
 
 26. What per cent of a quantity is always equal to a quantity ? 
 
 TEST ON PROBLEM WORK OF PERCENTAGE. 
 
 (Text pages 171-190.) 
 The following test on the problem work of the three cases of per- 
 centage may be used as a guide when planning review work or as a 
 test for new pupils who enter the class late in the term. Problems 
 at the end marked with stars are the additional types with which 
 Normal students are required to be familiar. 
 
 1. Mr. Hansen paid $6,000 for a piece of property. He wishes to 
 
 rent the property so that he may have a net income of 6% 
 on his investment. His yearly expenses average $140. Find 
 the monthly rent Mr. Hansen must receive from his property 
 in order to have a net income of 6%. (Difficult for pupils.) 
 
 2. Find the selling price of a horse sold at a loss of 15%o which 
 
 amounted to $42. 
 
 3. Given the gain and the selling price, tell how to find the gain 
 
 per cent and illustrate with a problem of your own. 
 
 4. At the close of the year 1913, there were 640 pupils in a certain 
 
 school. How many more pupils must be enrolled during 1914 
 to have the enrollment at the end of the year 1914 25%o greater 
 than at the close of the year 1913? 
 
 5. When the gain and the gain per cent are given in a problem, 
 
 how is the selling price obtained? Illustrate with a problem 
 of your own. 
 
 6. If the increase per cent is required, what two things must be 
 
 known? Illustrate with a problem of your own. 
 
 7. Mr. Saunders paid $10,000 for a piece of property. The monthly 
 
 rent from the property amounts to $125. If the annual expenses 
 average $280, what is Mr. Saunders' net rate of income from 
 the property? 
 
 8. State and solve a problem in which you are required to find the 
 
 interest due on a sum of monej^ borrowed for one year at 6% 
 interest. 
 9o What per cent of f of a yard is f of a yard? 
 
 (29) 
 
10. There were 285 words in a spelling test. George spelled 265 
 
 words correctly. Find his per cent of error. 
 
 11. There were 500 words in a spelling test. Albert missed 82 words. 
 
 What was Albert's per cent of correct work? 
 
 12. The population of a certain town was 25,000 in 1900. In 1910, 
 
 the population had increased 5,500. The population in 1910 
 M-as what per cent of the population in 1900? What was the 
 per cent of increase? 
 
 13. By selling a horse for $30 more than it cost, Mr. Tompkins 
 
 gained 20%. Find the cost and the selling price of the horse. 
 
 14. Mr. Hamilton bought tomatoes at the rate of three cans for $.25 
 
 and sold the tomatoes at the rate of two cans for $.25. Find 
 the per cent of gain or loss. 
 
 15. There are 215 boys and 289 girls in a certain school. What per 
 
 cent of the pupils are boys? 
 
 16. Mr. Hooj^er bought a lot for $3,000, upon which he built a house 
 
 that cost $4,500. The yearly expenses of the property average 
 $320. Find the montlily rent Mr. Hooper must receive from 
 his property in order to have a net income of 6%. 
 
 17. Louise saves 20% of her yearly salary and spends $960. What 
 
 is Louise's yearly income? 
 
 18. A public library had 7,500 books in circulation during the month 
 
 of July. This number represented 75% of all the books in the 
 library. Find the number of books not in circulation during 
 the month of July. 
 
 19. Mr. Green receives an annual income of $1,200 from prop- 
 
 erty which originally cost him $8,500. If the yearly expenses 
 average $240, find Mr. Green's net rate of income. 
 
 20. There are 520 pupils in a certain school. 210 of the pupils are 
 
 boys. What per cent of the school are girls? 
 
 21.*After losing 10% of its weight, a piece of ice weighed 20 pounds. 
 What was its original weight? 
 
 22. -Mr. Harris drew $350 out of the bank. This amount was 25% 
 more than the amount he had left in the bank. How much 
 had he originally in the bank? 
 
 23.*Two men invested $10,000 in a grocery business. One man put 
 in 20% more than the other man. Find the amount of capital 
 invested by each of the partners. 
 
 24.*A ring marked at 20% above cost is sold at a discount of 5%. 
 Find the gain or loss per cent. 
 
 25.*Go6ds marked 15% above cost are sold at a discount of 15%. 
 Find the gain or loss per cent. 
 
 26.*If a merchant sells | of a yard of silk for what ^ of a yard cost 
 him, find his gain or loss per cent. 
 
 27.*In 1911, there were 24,072 girls attending high school in Cali- 
 fornia. This was 18% more than the number of boys attend- 
 ing. Find the number of pupils attending high schools in 
 California during the year 1911. 
 (30) 
 
28.*By selling a horse at a profit of $25, Mr. Lucas received 115% 
 of the cost of the horse. What was the selling price of the 
 horse ? 
 
 29.*If four tablets are sold for the cost of five tablets, find the gain 
 or loss per cent. 
 
 COMMISSION, INSURANCE, TAXES, DISCOUNT, INTEREST. 
 
 The material of pages 192 through 217 of the text may be supple- 
 mented by the explanations and drill work of Bulletin No. 30. The 
 topics of commission, insurance, taxes, etc., are merely an application 
 of the work of percentage to business transactions. Pupils usually 
 fail to master such topics because they have no understanding of 
 business situations or of the terms used in the problems. Normal 
 students will be expected to be familiar with any of the problems of 
 the text on pages 192 through 277 as well as all of the work in Bulletin 
 No. 30. Students should strive to obtain as full a knowledge as possible 
 of these business transactions. 
 
 Test on Commission, Insurance, Taxes, Interest. 
 
 The following miscellaneous review problems may be used as a guide 
 when it is necessary to test a pupil's knowledge of commission, insur- 
 ance, taxes, interest or discount. At the beginning of a term or when 
 a new pupil enters a class, such a test is useful. Pupils can then be 
 drilled upon the points in which their knowledge is deficient. 
 
 1. What must be given in a problem in order that you may find 
 
 the rate of an agent's commission? Illustrate by stating and 
 solving a problem of your own. 
 
 2. What is meant by the net proceeds of a sale? Illustrate by a 
 
 problem of your own. 
 
 3. When cartage and other expenses are to be paid, is the agent's 
 
 commission figured on the amount left after deducting expenses ? 
 Give a reason for your answer. Illustrate by stating and solv- 
 ing a problem of your own. 
 
 4. Name as many different occupations or situations as you can in 
 
 which a man receives a commission as all or part of his pay 
 for services rendered to others. 
 
 5. What must be kno^vn in a problem in order that you may find 
 
 the amount of an agent's commission? Illustrate by stating 
 and solving a problem of your own. 
 
 6. If the selling price and the net proceeds are given in a problem, 
 
 tell how you may find the rate of the agent's commission. 
 Illustrate by stating and solving a problem of your own. 
 
 7. What is the difference between the assessed value and the real 
 
 value of a piece of property? Illustrate with a problem of 
 your own. 
 
 8. What must you know in order to find the rate of taxation which 
 
 will yield a certain amount of revenue? Illustrate by stating 
 and solving a problem of your ow^n. 
 
 (31) 
 
9. If the amount of a man's tax bill and the rate of taxation are 
 knoAvn. what amounts may be found? Illustrate by stating 
 and solving problems of your own. 
 
 10. Is every property owner compelled to insure his property against 
 
 loss by fire? 
 
 11. What must you know in order to find the amount of a man's 
 
 tax bill ? Illustrate by stating and solving a problem of your 
 own. 
 
 12. In what three ways may a rate of taxation be expressed ? Illus- 
 
 trate by stating and solving a problem of your own. 
 
 13. Name at least five things for which the monej'' derived from taxes 
 
 is used. 
 
 14. a. Name some of the items of expense of the national government. 
 h. Name some of the sources of income of the national gov- 
 ernment. 
 
 c. What is the difference betw^een a specific duty and an ad 
 valorem duty? Illustrate by stating and solving a problem 
 of your owm. 
 
 15. Is the rate of taxation the same for all taxpayers who live in a 
 
 certain citj^, town or district? 
 
 16. Give some reasons why two men living in the same block may 
 
 pay different rates of insurance. 
 
 17. Name at least five things which would raise the rate of a man's 
 
 insurance. 
 
 18. Illustrate what is meant by the face of a policy. What is the 
 
 premium ? 
 
 19. How is it possible for an insurance company to make the premium 
 
 on a three-year policy or a five-year policy cheaper than that 
 of an annual policy renewed each year for the same length of 
 time? 
 
 20. What must you have given in a problem in order to find the rate 
 
 of insurance? Illustrate by stating and solving a problem of 
 your own. 
 
 21. What must you have given in a problem in order to find the 
 
 amount of the premium due on a three-year policy ? Illustrate 
 by stating and solving a problem of your own. 
 
 22. Name at least four important things which should appear in a 
 
 policy. 
 
 23. If the amount of premium and the rate of insurance are given 
 
 in a problem, what unknown amount may you find ? Illustrate 
 by stating and solving a problem of your own. 
 
 24. Mr. Wilkins' property valued at $8,000 was assessed for 60% of 
 
 its value at the rate of $2.27 per $100. Find the amount of 
 Mr. Wilkins' tax bill. 
 
 25. Mr. Ruskin received $45 for selling some fruit on a commission 
 
 of 3%. Find the net proceeds of the sale. 
 
 26. What is the compound interest on $500 borrowed for 1^ years 
 
 at the rate of 6% if the interest is compounded semi-annually? 
 (32) 
 
r 
 
 27. What was the total duty on six dozen table knives worth $-1.50 
 
 per dozen if the specific duty is 16^ on each knife and the ad 
 valorem duty 15% ? 
 
 28. A watch marked $45 was sold by a wholesale dealer to a retail 
 
 merchant at a discount of 20%. The retail dealer sold the 
 watch at its original marked price. Find the retailer's gain 
 or loss per cent. 
 
 29. A hotel worth $50,000 was insured in one company for $12,000, 
 
 in another company for $15,000 and in a third company for 
 $1jO,000. If the hotel was damaged to the extent of $20,000, 
 how much of the damage would each company pay? 
 
 If the hotel were damaged to the extent of $40,000, what 
 amount of the damage would each company pay? 
 
 30. What is the interest on $350 borrowed October 8, 1912, and 
 
 returned with interest at 5% on November 6, 1913? (Use 
 exact time.) 
 
 31. Mr. White's taxes amounted to $75. What was the actual value 
 
 of his property if the tax rate was $2.04 per $100 and the 
 property was assessed for 60% of its real value ? 
 
 32. Mr. Roberts insured his house worth $6,000 for f of its value 
 
 for three years at $1.10 per $100. What was the amount of 
 his premium? 
 
 33. A real estate agent sold two lots for $12,500. He returned $8,500 
 
 to the owner of the lots. What rate of commission did the 
 agent charge? 
 
 MENSURATION. 
 
 The work in mensuration on pages 221 through 252 of text may 
 be supplemented by the work of Bulletins Nos. 31 and 32. Normal 
 students are expected to be familiar with the work of these bulletins 
 as well as the work of above pages of the text. 
 
 Test Questions on Mensuration. 
 
 The following questions and problems are a test of the work of the 
 text, pages 221 through 252 and of Bulletins Nos. 31 and 32. Normal 
 .students should be familiar with the answers to these questions. 
 
 1. Explain with the aid of a diagram why the area of an obtuse- 
 
 , , , . , . b X a 
 angled triangle is — - 
 
 2. Write the formula for the area of a circle and for the circum- 
 
 ference of a circle. State and solve a problem to illustrate 
 each. 
 
 3. Illustrate with a drawing the difference, if any. between three 
 
 square inches and a three-inch square. 
 
 4. State and solve a problem of your own in which you are required 
 
 to find the area of the entire surface of a cylindrical log. 
 
 5. How do you know when to use the square measure and when to 
 
 use the cubic measure in working with cylinders ? 
 
 (33) 
 
6. If the area of a rectangle is given in acres and the length of 
 
 one side in rods, how is the perimeter of the rectangle found? 
 Illustrate with a problem of your own. 
 
 7. How many acres are there in a section and one half of land ? 
 
 8. Express in feet the length and the width of a square field the 
 
 area of which is one acre. 
 
 9. Write the table of square measure. 
 
 10. What is the perimeter of a field one mile square? What is its 
 
 area? 
 
 11. If the circumference of a circle is divided by 3f and that answer 
 
 by 2, what is the result called? 
 
 12. If the area of the floor of a square room is one square rod. what 
 
 is the length and width of the room expressed in feet ? 
 
 13. With the aid of a diagram find the area of the largest circle 
 
 which may be cut from a three-inch sc|uare. 
 
 14. Why is it that the longest oblique line of an obtuse-angled triangle 
 
 is not the altitude of the triangle? 
 
 15. If the area of a right-angled triangle is 270 square rods, what is 
 
 the area of the rectangle which has the same base and altitude 
 as the triangle? 
 
 16. Write three formulas for the area of a triangle. State and solve 
 
 a problem to illustrate the advantage of each formula. 
 
 17. Given the circumference of a circle, how may its radius be found ? 
 
 Illustrate by stating and solving a problem of your own. 
 
 18. Show by a diagram the relation between the diagonal of a 
 
 rectangle and the longest side of a right triangle. 
 
 19. Draw a trapezoid. Assign dimensions to it and find its area in 
 
 acres. 
 
 20. Is the altitude of a figure alwaj's drawn within tlie figure? 
 
 Illustrate. 
 
 21. What is the name of the solid, all dimensions of which are equal ? 
 
 22. Draw three different kinds of triangles. With dotted lines show 
 
 the parallelogram of which each triangle is one half. Label 
 the base and altitude of both the parallelograms and triangles. 
 Assign dimensions to each triangle and find its area in acres. 
 
 23. What is the formula for the volume of a cylinder ? Illustrate 
 
 with a problem of your own. 
 
 24. Is the perimeter of a triangle, square, rectangle, or circle a line 
 
 or a surface? What measure is used in finding the perimeter 
 of a figure? 
 
 25. Illustrate what is meant by one line being perpendicular to 
 
 another line. 
 
 26. Given the circumference of a circle, how may its area be found? 
 
 Illustrate by stating and solving a problem of your own. 
 
 27. Illustrate with drawings the difference, if any, between two 
 
 square inches, two inches, a two-inch cube, and a two-inch 
 square. 
 
 (34) 
 
28. What is the ratio of the diameter of a circle to its circum- 
 
 ference? What is the ratio of the circumference of a circle 
 to its diameter? 
 
 29. Draw as many different parallelograms as possible and label the 
 
 base and altitude of each. Explain with the aid of a diagram 
 why the formula for the area of a parallelogram with no right 
 angles is the same as the formula for the area of a rectangle. 
 
 30. What is the perimeter and the area of a six-inch square ? 
 
 31. Tell for what each of the following numbers stands in the square 
 
 table : 640, 30^ 9, 144, 160, 208i 
 
 32. Is it entirely correct to say that a triangle is one half of a paral- 
 
 lelogram? If not, make a correct statement. 
 
 33. Write the table of Linear Measure. 
 
 34. Is the circumference of a circle a surface or a line ? 
 
 35. With a problem, illustrate the difference between finding the 
 
 surface and the volume of a cylinder. 
 
 36. Draw a cir'^le. Shade its surface and label its center, radius and 
 
 circumference. 
 
 37. Given the circumference of the base of a cylinder, how is the area 
 
 of the base of the cylinder found? Illustrate by stating and 
 solving a problem of your own. 
 
 38. How do you find the area of the convex surface of a cylinder 
 
 when the circumference of the base of the cylinder and the 
 altitude of the cylinder are given? Illustrate by stating and 
 solving a problem of your own. 
 
 39. A tree was broken by the wind in such a way that the top of the 
 
 tree struck the ground at a distance of 45 feet from the foot 
 of the tree. If the broken part of the tree was 65 feet long, 
 find the length of the part left standing. 
 
 40. A lot is 125' X 340'. How much shorter is it to walk diagonally 
 
 through the lot than around the two adjoining sides of the lot ? 
 
 41. Which field has the greater perimeter and how much greater, a 
 
 rectangular field 210 feet by 350 feet or a square field of equal 
 area? 
 
 42. The area of a square field is 8 acres. What is the perimeter of 
 
 the field expressed in feet? 
 
 43. Which has the greater area, a rectangular table top 4' 3" x 3' 
 
 or a circular table top which has a diameter of 4' 3"? How 
 much larger or smaller is the rectangular table top than the 
 circular table top? 
 
 44. The perimeter of a square field is 320 rods. At $6o an acre, 
 
 what is the value of a triangular field which has the same base 
 and altitude as the square? 
 
 45. Mr. Hamilton owns a" farm one eighth of a mile .square. IMr. 
 
 White's farm has an area of one eighth of a square mile. What 
 is the value of each farm at $125 an acre ? 
 
 46. If a horse is tied to the bottom of a stake by a rope 20 feet long, 
 
 what is the area of the surface over which the horse may graze ? 
 (35) 
 
47. How mauj' feet of picture molding are needed for a room the 
 
 length of which is 18 feet and the area of the floor 378 square 
 feet? 
 
 48. Ethel is embroidering a round centerpiece for her mother for 
 
 Christmas. If the greatest width of the centerpiece is 36 
 inches, how" many inches of the outer edge of the centerpiece 
 will Ethel have to embroider each day in order to finish the 
 outer edge of the centerpiece in 8 days? How many square 
 feet of the surface of a table will be covered by the centerpiece 
 when finished? 
 
 49. A rectangular field 114 rods long has an area of 1^ acres. ' If 
 
 land is worth $250 an acre, what is the value of a triangular 
 field which has the same base and altitude as the rectangle? 
 
 50. The length of the upper base of a trapezoid is 12 feet and the 
 
 lower base is 8 feet. If the perpendicular distance between the 
 sides is 4 feet 6 inches, find the number of square feet on the 
 surface of the trapezoid. 
 
 51. How many revolutions will a 28-inch bicycle wheel make in trav- 
 
 eling a mile? 
 
 52. How many cubic feet of marble are there in a column of marble 
 
 20 feet long and 22 inches in diameter? 
 
 53. How man}' rods of fence are needed to enclose a square field the 
 
 area of which is 8 acres? 
 
 54. At $.12| a scpiare foot, find the cost of laying a three-foot cement 
 
 walk around a circular flower bed 30 feet in circumference. 
 
 55. What is the diameter of a circle which has an area equal to that 
 
 of a seven-foot square? 
 
 56. The hypotenuse of a right triangle is 70 feet. The altitude of 
 
 the triangle is 30 feet. What is the perimeter of the triangle .' 
 
 57. What is the length of the longest stick which may be placed within 
 
 a box cubical in shape, each edge of which measures 2 feet ? 
 
 58. Express in gallons the capacity of a covered cylindrical tank 
 
 which is 20 feet long and 3 feet in diameter. 
 
 59. What is the area of the entire outer surface of the above tank? 
 
 (Include cover.) 
 
 60. Tell some method of proving that the ratio of the circumference 
 
 of a circle to its diameter is 3i. 
 
 61. A cistern filled with water is 6 feet in diameter and 40 feet deep. 
 
 How many gallons of water does it contain? 
 
 62. If there are 31^ gallons in a barrel, how many barrels of water 
 
 may be filled from this cistern? 
 
 63. If the bottom and sides of the interior of the tank are cemented. 
 
 find the cost at $.12 a square foot. 
 
 64. A marble column is 12 feet high and 38 inches in circumference. 
 
 What is the area of the convex surface of the column? 
 
 65. What is its volume expressed in cubic yards ? 
 
 66. What is the length of the diagonal of a square, each side of 
 
 which is 10 feet? 
 
 (36) 
 
CLASSIFICATION OF PROBLEMS OF PAGES 18-160 OF 
 ADVANCED TEXT. 
 
 By grouping under their proper headings all problems of like types 
 which are scattered throughout the various pages of the text, there will 
 be found to be in most instances, sufficient problems to teach a given 
 type. 
 
 Pupils who are accurate and understand their work quickly should 
 not be required to work as many problems of a given type as pupils 
 who are slow and inaccurate. The problems under each type have been 
 divided into three groups— a, problems not marked with stars; &, 
 problems marked with one star; c, problems marked with two stars. 
 Pupils who work the non-starred problems correctly should be per- 
 mitted to skip the single and double starred exercises. 
 
 Teach pupils to write a statement telling what is given and what is 
 required to be found in each of the following problems : 
 
 Addition and Subtraction of Integers and Decimals. 
 
 Page 19—5-6 ; *7-8 ; **9-10-ll. Page 23—5-7-9 ; *6-8. Page 24— 
 2-3-7; *4-5. Page 25—6; *5-7. Page 28— 1-2-3-4-5-6-7. Page 30— 
 4-5-6. Page 73—5-6-7. Page 36—1 through 8. 
 
 Multiplication and Division of Integers and Decimals. 
 Page 35— Sec. 37, 2-3-4. Page 41— Sec. 45, 3. *Page 159—7-8-9-10. 
 Page 130—12. Page 50— Sec. 58, 2-3 ; *4-5 ; **6. Page 50— Sec. 59, 
 1-3; *2-4. Page 53—3-5-8; *4-6-7-9. Page 56— Sec. 67, 1-2; *3- 
 4-5. Page 56— Sec. 68, 1-4-5 ; *2-3. Page 67— Sec. 86, 1-2-4^7. Page 
 72—9. *Page 67—3-5-6-8. **Page 129—10-11. Page 159—1-2-4- 
 12-13; *3-5-14. Page 56— Sees. 67-68 (entire). Page 164—6-7-8-9- 
 10-11-12-13-14-15-16-23-24. 
 
 Addition and Subtraction of Fractions. 
 Page 98—4-8-9-10. Page 130—22-23. Page 110—2-3. *Primary 
 Text page 188—1-2. *Page 190—5-6-7-8. *Page 191—5-6-7. *Page 
 .192—5-6. **Page 203—1-2-4-5-6. Page 204—1-2-3. 
 
 Multiplication and Division of Fractions. 
 
 Page 118—4. Page 144— Sec. 187, 4-5. Page 130—25. Page 127— 
 6-7-8-10-11. Page 127— Sec. 171, 2. Page 128—8-9-10-11. Page 
 144— Sec. 187, 3-6. Page 118— Sec. 160, 2-3-5. 
 
 Given Whole to Find Part. 
 Page 57—2-3-4-6-9-11; *5-7-10-12; **8-13-14. Page 58—1-4; 
 *2-3 ; **5-6. Page 129—6-7. Page 144— Sec. 187, 1. 
 
 Given Part to Find Whole. 
 
 Page 59— Sec. 72, No. 1 through 12. Page 139—2-^5-7-9-10. *Page 
 129—8-9. Page 139—3-6-8-11-12-14-15-17-23; **13-16-18-19-20- 
 21-22-24-25-26-27. Page 141— See. 183, entire. Page 143—3-5-6-7- 
 8-10; *l-2-4-9-ll. Page 144— Sec. 187, 2. 
 
 (37) 
 
Profit and Loss. 
 
 Page 142—14-15-16. Page 152— Sec. 199. 
 
 Bills and Receipts. 
 Pages 42-43. Page 61—1-2. Bull. No. 29— Pages 41-42-43-97-98. 
 See Test Sec. 39 of this book. 
 
 Average and Ratio. 
 
 Page 41— Sec. 45, 1. Page 118— Sec. 160, 6. Page 54—55-145-146. 
 Bull. 29— Pages 73-74. 
 
 Area of Rectangles. 
 
 The following problems of the text maj^ also be used as additional 
 drill work while teaching pages 80 to 84: 
 Page 130— No. 15-16-17-18. 
 Page 128— No. 7-15-17. 
 Page 129— No. 1-2-3-4-5. 
 Page 165— No. 24. 
 See Test See. 42 of this book. 
 
 Cubic Measure. 
 
 Pages 8^85-86. 
 . Page 236—6-7. 
 
 (38) 
 
SAN FRANCISCO STATE NORMAL SCHOOL PUBLICATIONS. 
 
 The Teachers' Manuals. 
 
 Some years ago the San Francisco State Normal School undertook the publication 
 of courses of study for teaching the various subjects of the curriculum of the elementary 
 school. These courses were prepared by members of the faculty and were the outgrowths 
 of daily experience in directing and supervising the teaching by student teachers in the 
 Normal Elementary School. The original purpose of these publications was to furnish 
 to these student teachers directions for teaching each of the subjects. An essential 
 necessity in their construction was that they should be very explicit, specific and 
 practicable in use. Gradually there grew a demand for them by teachers in the public 
 schools, and the Normal School began to print larger editions in order to fill this new 
 need. The demand from the public school sources has now grown to such proportions 
 that one chief service of the institution is that of its publications. 
 
 Pupils' Exercise Books. 
 
 Up to 1912 the publications had been confined to courses of study for the assistance 
 of teachers. During 1912 the publication of pupils' exercise books, accompanying the 
 teachers' bulletins, was commenced. In one type of these exercise books the pupils 
 write directly in printed lessons. This device saves a large amount of labor and time 
 of the teacher in copying upon the board and in oral instructions. Further, it saves the 
 pupils' time in copying from the board. But pupils can make progress two or three times 
 faster than by the usual method, and the work is done much more effectively and without 
 the sense of drudgery either to pupil or teacher. The exercise books are printed upon 
 paper that will take ink. They cost little or no more than the common blank books 
 of the same quality of paper. 
 
 Monographs. 
 
 There is now commenced a series of monographs of a practical nature, aimed to 
 assist or suggest further development of a greater efficiency of school instruction. 
 
 Three Series. 
 
 There have been three series of publications in time — one issued prior to the great 
 fire of 1906, of which no numbers now remain; a series begun in 1907 and continued 
 to 1914, and, finally, the Pupil's Self-Instruction Series, begun in 1914. The latter two 
 will be found listed below. 
 
 System of Publication. 
 
 The expense of these publications is borne chiefly by a revolving fund obtained by 
 their sale. They are printed in the State Printing- Office and sold practically at manu- 
 facturing cost. They are issued merely upon the autjriority of the individual authors and 
 the editor of the series, and do not represent a general or necessarily permanent policy 
 of the school, nor a consensus of its faculty or trustees. 
 
 How to Order. 
 
 All orders must be accompanied by school district warrant check, money order or 
 stamps. We cannot fill orders which require keeping of accounts. As most of the 
 purchases of bulletins and pupils' exercise books are now made by the school districts, 
 teacliers who send orders should be careful to secure the signature of trustees to warrants 
 in payment for orders, so that delays may be avoided. Be careful, also, in filling out 
 orders that the bulletins are listed and are not out of print. We cannot exchange 
 publications once purchased unless error has been made and the request is made within 
 three days. Kindly avoid, so far as possible, conditions which require special corre- 
 spondence in the business department. 
 
 MONOGRAPHS. 
 
 Monograph A. A remedy for Lock-Step Schooling; a preliminary report upon the 
 weakness and impossibilities of the class system of instruction, and progress to date in 
 substituting therefor an individual system of teaching. By Frederic Burk. Issued free 
 upon application. 
 
 Monograph B. Outline courses in general information and general intelligence. This 
 monograph undertakes to map out the beginning of a reorganization of the high school 
 course of study, not only for the better preparation of those intending to become teachers, 
 but also in the general cause of wider preparation of all students in industrial, civic and 
 social intelligence. To students intending to enter the San Francisco State Normal School 
 
 (39) 
 
the monograph will be sent free. Section I printed as separate pamphlet; American 
 History and Civics, by P. F. Valentine. Section II, pampltlet for General History, Science, 
 and Literature, Arithmetic, Geography, and Music. Section III, Spelling, Language, and 
 Grammar. Price — the three sections will be sent for 25 cents, postpaid; in lots of 25 or 
 more, expressage or freight paid by purchaser, 7% cents per section. 
 
 Monograph C. (In preparation.) This will be a sequel to Monograph A, giving the 
 data of results of one year's experience in operating an individual system of instruction 
 in the Normal Elementary Department. It will show the records of about 500 pupils, 
 their variations in rates of progress, etc. 
 
 Monograph D. Critical Difficulties in the Teaching of Arithmetic. For teachers, 
 and for students preparing for admission to the Normal School. By Mary A. Ward. Price 
 15 cents, 3 cents added for postage. 
 
 PUPIL'S SELF-INSTRUCTION SERIES. 
 
 (Adapted to an Individual Method of Teaching.) 
 
 The occasion and general plan for this series is set forth in Monograph A. We 
 are at this date (August, 1914) beginning the publication of a series of pupils' exercise 
 books and teachers' manuals adapted to use under the individual system of instruction. 
 They, of course, may also be used under the class systein and will assist in teaching 
 by the state series texts. Their plan of construction embodies the features outlined in 
 Monograph A — the "elastic" lesson, by which the number of exercises to secure compre- 
 hension or accuracy varies according to individual need; the adaptation to simplicity of 
 language; the lesson directions whereby the pupil can make his own rate of progress and, 
 to a large extent, independently of prescribed lessons or help from the teacher; the 
 cumulative reviews by which all principles once learned are carried forward automatically. 
 
 This series will be found invaluable, especially for the rural school where pupils must 
 depend largely upon their own resources. The teachers' manuals will give full directions 
 for operation of the system. 
 
 Prices. 
 
 Except where specially stated, the price of the numbers of the Self-Instruction Series 
 will be as follows: 
 
 At the Normal School, 10 cents each; 
 
 By mail, 12 cents each, postpaid; 
 
 By express or freight, $7.50 per hundred, transportation charges paid by buyer. 
 
 Arithmetic. By Frederic Burk and Mary A. Ward. 
 
 No. 20 — Teachers' Manual to accompany pupils' books, Nos. 21, 22, and 23, giving 
 directions, answers to examples and supplementary examples. Price — 25 cents each, post- 
 paid. 
 
 No. 21 — Pupils' Exercise Tablet in addition and subtraction. 
 
 No. 22 — Pupils' Exercise Tablet in multiplication and short division. 
 
 No. 23 — Pupils' Exercise Tablet in compound multiplication and long division. 
 
 No. 29 — Problems in Percentage. Double number. Price — 20 cents each at Normal 
 School; 24 cents by mail; $15.00 per hundred by express. 
 
 No. 30 — Applications of Percentage. 
 
 No. 31 — Problems in Mensuration, Part I. 
 
 No. 32 — Problems in Mensuration, Part II. 
 
 Language. By A. S. Boulware and Ethel G. Smith. 
 
 No. 42 — Pupils' Exercise Tablet in Language, Part I (for fifth grades). 
 
 No. 43 — Pupils' Exercise Tablet in Language, Part II. 
 
 No. 44 — Pupils' Exercise Tablet in Language, Part III. 
 Grammar. By Ethel G. Smith and Frederic Burk. 
 
 No. 51— Part I. 
 
 No. 52— Part II. 
 
 No. 53— Part III. 
 
 No. 54— Part IV. 
 
 No. 55 — Part V. (In preparation.) 
 
 No. 56 — Part VI. (In preparation.) 
 History. By P. F. Valentine. 
 
 No. 80 — Pupils' Exercise Book, Part I, to accompany advanced state text. (Columbus 
 through Jefferson.) 
 
 No. 81 — Pupils' Exercise Book, Part II, to accompany advanced state text. (Madi- 
 son through Civil War.) 
 
 (40) 
 
No. 82 — Pupils' Exercise Book, Part III, to accompany advanced state text. (Civil 
 War to present.) 
 
 No. 83— Difficulties of liistory texts simply explained. (Democracy, the Constitution, 
 Centralized Government, Religious Toleration, Monroe Doctrine, Spoils System, Civil 
 Service Reform, the Tariff, etc.) 
 
 Geography. By F. W. Hoffmann. 
 
 Bulletin No. 18, Teachers' Manual, with two pupils' exercise books in Map Geography, 
 is partly constructed upon the individual plan. It is already published and may be 
 obtained upon application. (See next hst.) 
 
 IN PREPARATION. 
 
 There are in preparation, for publication during the year, the following: 
 
 Arithmetic. Two tablets in formal arithmetic for fractions and decimals; one exer- 
 cise book in arithmetic problems in integers; one exercise book in denominate numbers; 
 one exercise book in application of percentage. 
 
 Language. Two exercise tablets for third and fourth grades, respectively. 
 
 Phonics. A series of exercise books. 
 
 Writing. A series of exercise books. 
 
 Drawing. A series of exercise books. 
 
 Music. A series of exercise books in formal note work. 
 
 TEACHERS' MANUALS AND PUPILS' EXERCISE|BOOKS. 
 
 (Series Published 1907 to 1914.) 
 
 (Out of print: Nos. 1. 2, 3, 5, 6, 7, 8 and 13. The materials of these have largely been 
 absorbed in revised editions represented by the later numbers.) 
 
 No. 4 — A Course of Study in IVIap Geography; paper bound, 52 pages. By Allison 
 Ware. Price — by mail, postpaid, 30 cents. 
 
 Outline Maps — In connection with Bulletin No. 4, the school publishes a series of nine 
 outline maps from which pupils may trace outlines for use in location. These maps are 
 9 by 12 inches in size. They represent the following areas: North America, South 
 America, Europe, Asia, Africa, Australia, United States, California, and the hemispheres. 
 Price — by mail, postpaid, for set of nine, 15 cents. (See also Bulletin No. 18.) 
 
 No. 9 — A Course of Study in Language; 174 pages. By Effie Belle McFadden. Price — 
 by mail, postpaid, 30 cents. 
 
 No. 10 — A Course of Study and Teachers' Handbook in the Common Literature of 
 Life; 207 pages, paper bound. By Allison Ware. Price — postpaid, 40 cents. 
 
 No. 11 — A Course of Study In Formal Arithmetic and Teachers' Handbook. By David 
 Rhys Jones. This bulletin is published in various parts as follows: 
 
 Part I. Teachers' Handbook and exercises for integers; 109 pages, paper bound. 
 Price — by mail, postpaid, 30 cents. 
 
 Part II. Teachers' Handbook and exercises for common fractions, decimals, per- 
 centage, denominate numbers and mensuration; 84 pages, paper bound. Price — by mail, 
 postpaid, 30 f^ents. 
 
 The Pupils' Exercise Books, Nos. 1, 2, and 4, accompanying the Handbooks, are out 
 of print; a limited stock of No. 3 (fractions and decimals) is still on hand. Price — 10 cents 
 by mail; the pupils' work, however, is included in the Teachers' Handbook. Individual 
 Series No. 21 takes the place of No. 1; No. 22 and No. 23 take the place of No. 2. 
 
 No. 12 — Review Courses of American History by means of Composition Topics, and 
 Teachers' Handbook to the use of the California State Series Texts. By P. F. Valentine. 
 
 Part I. Teachers' edition containing introduction and directions for use of the com- 
 position method, the pupils' topics of the primary text, the pupils' topics of the advanced 
 text, a paragraph directory to the text, and a cumulative fact review of the advanced 
 text; 73 pages. Price — postpaid, 25 cents. 
 
 Part II. Pupils' edition containing the composition outlines which follow the state 
 primary text in history; 10 pages. Price — postpaid, 5 cents; in lots of 25 or more, freight 
 or expressage paid by purchaser, 4 cents per copy. 
 
 Part III. Pupils' edition containing the composition outlines which follow the state 
 series advanced text in history, the paragraph directory to the text, and the cumulative 
 fact review for the same; 48 pages. Price — postpaid, 10 cents; in lots of 25 or more, 
 freight or expressage paid by purchaser, 8 cents per copy. 
 
 No. 13 — A Course of Study In Applied Problems In Arithmetic, for Supplementary Use. 
 By Mary A. Ward. 
 
 Out of print — order No. 29, Self-Instruction Series. 
 
 (41) 
 
No. 14 — A Course of Study in the Teaching of Composition, Language and Spelling; 
 paper bound. By Effle B. McFadden, assisted by Ethel G. Smith. Teachers' edition for 
 first three years. Price — postpaid, 25 cents. Nos. 1 and 2 of the pupils' exercise books 
 heretofore accompanying this handbook are out of print; they are replaced by Nos. 40 
 and 41, Self-Instruction Series. A small stock of Nos. 3 and 4 still remains. Price — 10 
 cents, postpaid. 
 
 No. 15 — A Simplified Course of Study in the Teaching of Grammar; teachers' edition, 
 paper bound, 187 pages. By Frederic Burk, Effie B. McFadden and Irving Brazier. Price 
 
 postpaid, 40 cents. The pupils' exercise books accompanying this handbook are all out 
 
 of print and are being replaced by Nos. 51, 52, 53, and 54 of the Self-Instruction Series. 
 
 No. 16 — A Course of Study in Phonics. By Corrine H. Johnstone and Frederic Burk. 
 Teachers' Edition, Part I, 90 pages. Price— postpaid, 20 cents. 
 
 Pupils' Phonic Exercise Book No. 1 (containing exercises reprinted from teachers' 
 edition). Price— postpaid, 10 cents; in lots of 25 or more, freight or expressage paid by 
 purchaser, TVz cents. 
 
 No. 17 — A Composition Course in American Government and Pupils' Handbook to the 
 State Series Text (Dunn's Community and Citizen) with Supplement containing revised 
 or additional paragraphs upon conservation, some California laws. Interstate Commerce 
 Commission, California's compulsory education, direct election of United States senators, 
 direct primary, township and county, the commission form of city government, the 
 initiative, referendum and recall, the cabinet; paper bound, about 40 pages. By P. F. 
 Valentine. Price — postpaid, 15 cents; in lots of 25 or more, expressage or freight paid 
 by purchaser, 10 cents per copy. 
 
 No. 18 — A Course of Study in Map Geography; can be used in trades as low as fourth. 
 By F. W. Hoffman. 
 
 Teacl 
 postpaid. 
 
 Pupil, 
 Price — 12 ■ 
 
 Pupilf 
 maps and 
 more, freig 
 
 Outlin 
 
 No. 1, wor 
 postpaid. 
 
 Pupils 
 Price — 10 c 
 
 THIS BOOK IS DUE ON THE LAST DATE 
 THIS BO"^sT^j^ipED BELOW 
 
 Doks. Price — 10 cents. 
 
 AN INITIAL FINE OF 25 CENTS 
 
 AJM 1J»J-XX«." RETURN 
 
 W.UU BE ASSESSED FOR Fe^T^ ^^^^^ 
 
 THIS BOOK ON ^HE DATE D pQURTH 
 
 ^^V^;r T^^^-l°ON -% SEVENTH DAV 
 
 pupils write directly. 
 
 dual instruction, with 
 stpaid; in lots of 25 or 
 r both exercise books. 
 ?iven for mounting) ; 
 Price — 10 cents each, 
 
 and California). 
 
AriORD BROS. 
 
 )D 2257 
 
', ^^ 
 
 
 i