ii^ :-'x>v^yryy>y>>!>.#>x/ . y-^/'jd'J^JC^Z^j'J'y^jrAyj'y-yy, DEPA No. foe c c cc PEIMARY ARITHMETIC NUMBER STUDIES '-' ' ' -J :>•>-,■' ' -' ' " > ) ' '»',',''''>'' ' > ' J ' 5 ) 1 > F0i2 THE SECOND, THIRD, AND FOURTH GRADES BY A. R. HORNBROOK, A.M. TEACHER IN THE PUBLIC SCHOOLS OF EVANSVILLE, IND. -oo'ioioo- NEW YORK •:• CINCINNATI :• CHICAGO AMERICAN BOOK COMPANY *• • » • ' • • ei : : ^ r « . r ' . • ' • • • * • * * ' ' • • .i-^. » • « »,• Copyright, 1898, by A. E. HORNBROOK. PBIM. ABITH. — HORNBROOK. VafiFV. PREFACE The progress of a beginner in arithmetic is of a desirable kind when it involves a snccession of insights into the relations of numbers and an increase of expert- ness in dealing with them. It is the aim of this first book to secure these ends. Its material has been chosen with careful reference to the development of the number sense of little children as observed by the author and as reported by many other observers. It is believed that when a child realizes the meanings of the first ten number names, has learned to make com- binations within 10, and is able to count to 100, he is ready to take up the first hundred as an aggregation of tens, to consider other numbers as aggregations, and to discover their relations. At that point this book begins. The use of diagrams called " number tables " as a con- crete basis for the child's thinking while he is getting his first ideas of the facts of the addition and multiplication tables is a distinctive feature of the work. Children readily learn from a number table like that on page 14 such facts as "5 tens = 50." The five columns of num- bers are as concrete to them as five sticks, and the figures "•50 " at the end of the fifth column make them much more suggestive. Much of the work given in this book would be entirely too difficult for the children for whom it is intended if it lacked the basis of the measurements of the number tables. - . . - i /« 54 *^'*}44 4 PREFACE The treatment of numbers used in this book leads to the presentation of the multiplication tables in an order different from that usually followed, and more economical of children's time and effort. 10, "the master key of number," under the decimal system, is presented first with its multiples. The child's instinct for grouping by pairs is next utilized by giving the table of twos. Work in addition and subtraction follows in which the relations of numbers to 10 and to 2 are frequently brought to mind. By objective work in feet and yards illustrat- ing_ combinations in addition, the pupil gains a knowledge of multiples of 3. The smaller multiples of 4 are learned by similar work upon quarts and gallons, pecks and bushels. The fives as a subdivision of the tens are pre- sented in the next chapter, and in order that the child may have time to become familiar with the multiples of 5, most of the work of that chapter relates to them. The child has been dealing with 10 and its divisions, and has had much practice in combining 10 with other num- bers. To learn the table of elevens is an easy task for him. One little fellow remarked, "Learning the table of elevens is just like going down stairs, and you can always tell what step you are on. The first step is made of I's and the second step is made of 2's, and it is that way all the way down." A glance at the oblique line made by the multiples of 11 in the number table on page 114 will explain his remark. 9, as a departure from 10 on the other side, is next given. The table of nines is reenforced by that of the threes, which receives formal treatment in the next chapter. The treatment of 8 is followed by that of its subdivision 4. Work in fractions, which is generally so successful in first grades, is continued throughout the book in connec- PREFACE 5 tion with simple geometric forms, and leads naturally to the recognition of ratios. Only the rare, precocious child is able to found a process upon a course of reasoning, however clearly it may be presented. For that reason, only those processes of written work that can be based upon the child's intui- tions of number are accounted for ; others are given simply as processes leading to desired results, without any attempt at forcing a knowledge of the underlying principles into the immature mind. The child is led to construct, to observe, to report, and to remember, but the reasoning required of him in the first book is limited to simple inferences. Formal analysis, that most effective deadener of the mathematical sense of little children, has been omitted. The successful teacher knows how to stimulate the ex- pression of the child's own insights into iiumber by light, skillful touches upon his mind in easy conversational exercises. The development of the plan of the work is indicated by many notes to the teacher. To the many primary teachers who have kindly con- tributed the results of their schoolroom experiences, the author offers grateful acknowledgments. CONTENTS CHAPTER I. Squares — Counting II. Tens Cents and Dimes Written Addition Written Subtraction Tens and Units Roman Numeral X III. Twos . Even Numbers . Foot and Inch . Halves Quart and Pint Horizontal Line Triangle . Thirds Fourths Vertical Line . Roman Numeral I IV. Addition Sum . Yard Rectangle Thousands Gallon Perimeter Decimal Point . Roman Numerals V, L, and C Peck V. Subtraction Difference Minuend . Subtrahend Pound and Ounce VI. Applications of Addition and Industrial Problems Days in Months Odd Numbers . VII. Fives . Equilateral Triangles Roman Numerals D and M Quotient .... Subtraction PAGE 9 14 17 21 26 32 34 35 36 39 40 41 42 44 46 48 48 49 51 53 56 64 64 66 71 74 81, 82 79 83 84 89 92 93 95 96 98 100 104 108 108, 113 . Ill 75, The table of contents shows the chapter in which a subject first appears. Each subject reappears in succeeding chapters. 7 8 CONTENTS CHAPTER PAGE VIII. Elevens 114 Written Multiplication 120 Product 120 IX. Nines 122 Multiplier 123 Square Yard 124 Square of a Number 125 Divisor 130 X. Threes 132 Multiplicand 133 Parallel Lines 135 Trapezoid 136 Khombus . 137 Ratio 139 XI. Eights 142 Denominator 147 Quart and Peck 148 Short Division 148 Dividend . 149 Perpendiculars 149 Area of Eight Triangle 151 XII. Fours 154 Numerator 159 Square Prism 164 Partial Products 165 Ton 166 XIII. Sevens 167 Factors 169 Compound Fractions 173 XIV. Sixes 180 Rod 186 Hexagon 188 Interest 190 XV. Twelves 192 Square Foot 195 Long Division 199 Cubic Foot 202 Common Multiple . 203 XVI. Review 205 Average 206 Common Divisor 211 Addition of Compound Denominate Numbers . . .221 Subtraction of Compound Denominate Numbers . . 222 Multiplication of Compound Denominate Numbers . 223 Per cent 227 Bills 230 XVIL Fractions 232 1 5 1 1 i > ■) > ■> 5 1 ■> ■> ■) T 5 1 1 1 J J ELEMENTARY ARITHMETIC -O-OJ^OO- CHAPTER I SQUARES — COUNTING Inch squares cut fi'om white paper should be prepared in such abundance that each child may have enough to make the figures given in this chapter. Draw the figures on the blackboard and give the work orally at first. 1. How many squares in Fig. 1 ? 2. Place squares in a column like Fig. 1. 3. If we call the square at the top the first square, and the next one the second, and so on, how shall we number the last square? How shall we number the next to the last square? 4. Show the fourth square in your column. Show the sixth square, the ninth square, the third square, the seventh square, the fifth square, the eighth square. 5. Push the lowest two squares away. How many squares are left? 6. Make the column whole again. Take away four squares at the lower end of the column. How many squares are left? 7. Divide the column into two equal parts. How many squares in each part? 8. Take away from the whole column six squares, and tell how many are left. 9 Fig. 1 < ( i 10 SQUARES — COUNTING 9. Take away from tlie whole column three squares, and tell how many are left. Give exercises in parting and wholing the column of squares until the combinations up to 10 are thoroughly reviewed. 10. Place squares as in Fig. 2. How is it different from Fig. 1? How many squares in Fig. 2? 10 and 2 are how many? 11. Add 2 more squares to the short col- umn. Tell how many squares there are now in tlie short column. How many in the whole figure? 10 and 4 are how many? 12. Add 2 more squares. How many squares in the short column now? How many in tlie whole figure? 10 and 6 are how many? 13. Add 2 more squares. How many squares in the short column? How many in tlie whole figure? 10 and 8 are how many? 14. Add squares to the short column until the columns are equal. How many did you add? How many squares in each column? How many in the whole figure? 10 and 10 Pig. 2 are how many ? 15. Take away the last two squares from the figure you have. How many are left? 2 from 20 leave liow many ? Take away 2 more and tell how many are left. 2 from 18 leave how many ? 16. Keep taking away two more and telling how many are left until the right-hand column is all gone. 2 from 16 leave how many? 2 from 14 leave how many? 2 from 12 leave how many? SQUAFiES — COUNTING 11 Give similar exercises on successive days until these facts of number measurement have been called into the consciousness of the children so often and so clearly that they have become a part of their mental property. Do not let them memorize number statements such as " 10 and 2 are 12 " until it is evident that their statements are sup- ported by their own perceptions of number truths. 17. Put the 20 squares back into 2 equal columns. Number them as in Fig. 3. 18. Find the 17th square. How many squares in this figure come after the 17th? How many squares before the 17th square are numbered in this figure ? 19. Find the 15th square and show how many squares come after it. How many squares come before the 15th square in this figure ? Show how many squares, in this figure, come after the 13th. After the 16th. After the 14th. After the 11th. After the 18th. After the 12th. 20. 3 from 20 = how many ? 5 from 20 = how many ? 8 from 20 = how many ? 4 from 20 = how many ? 6 from 20 = how many 9 from 20 2 from 20 7 from 20 = how many = how many ? = how many ? 1 n ■> 12 ■> o 113 4 U 5 15 G 16 7 17 8 18 9 19 10 20 Fig. 3 Let the children separate the figure before them into unequal parts of their own choosing, telling how many squares they take away, and how many are left. 12 SQUARES — COUNTING 21. Place squares as in Fig. 4. How many squares in it ? How many more squares than in Fig. 3 ? 22. Show the 20th square in your figure. Let the squares be counted in the order indi- cated in Fig. 3. 23. Show the 21st square and tell how many squares come after it. 24. 21 + 2 = how many ? 20 + 3 = how many ? 25. Add 2 squares to the short column. How many are there now in the short column ? How many in the whole fig- ure ? 23 -f- 2 = how many ? 26. Add 2 more squares and tell how many are in the short column. How many are in the whole figure ? 25 4- 2 = how many ? 27. Add 2 more squares and tell how many are in the short column. Hoav many are in the whole figure ? 27 + 2 = how many ? 28. Add squares enough to make the short column as long as the others. How many did you add ? How many squares in your whole figure ? 29. Divide the whole figure yoc have made into three equal parts. How many squares in each part ? 30. 10 4- 10 + 10 = how many ? 31. 10 from 30 = how many ? 4 from 30 = how many ? 6 from 30 = how many ? 3 from 30 = how many ? 5 from 30 = how many ? 7 from 30 = how many ? Fig. 4 SQUARES — COUNTING 13 32. Place squares to make Fig. 5. How many squares does it take ? 30 + 4 = how many ? 33. Add 2 squares to the short column and tell how many squares in it. How many in the Avhole figure now ? 34 -}- 2 = how many ? 34. Add 2 more squares to the short column. How many in it now ? How many in the whole figure ? 36 + 2 = ? 35. Add squares enough to make the short column equal to the others. How many did 3'ou add ? How many are there in the whole figure ? 38 + 2 = ? 36. How man}^ columns in the whole figure ? Separate the figure into 4 equal parts. How many squares in each part ? 37. Put the parts together again. Show the 31st square. How many squares come after it in the tigure ? How many are before it ? 38. Show the 33d square and tell how many squares follow it ; the 35th ; the 32d ; the 36th ; the 38th. 39. 2 from 40 = ? 4 from 40 = ? 6 from 40 = ? 7 from 40 = ? 5 from 40 = '.^ 9 from 40 = ? 8 from 40 = ? 3 from 40 = ? 10 from 40 = ? Add squares, a few at a time, to the figure on the board and let the children count and combine them. As each column is completed, number the last square. Continue this work from day to day until the figure of 100 squares is completed. Fig. 5 CHAPTER II TENS Cents and Dimes, Addition and Subtraction, Tens AND Units, Roman Numeral X The questions upon the table which immediately follow it are designed to lead the children to analyze it as an object of vision with- out reference to its symbolism. Similar questions should be given a few minutes every day until the children are familiar with the relative positions of the numbers. NUMBER TABLE* 1 11 21 31 41 51 61 71 81 91 2 12 22 32 42 52 62 72 82 92 3 13 23 33 43 53 63 73 83 93 4 14 24 34 44 54 64 74 84 94 6 15 25 35 45 55 65 75 85 95 6 16 26 36 46 5Q 6Q 76 86 96 7 17 27 37 47 57 67 77 87 97 8 18 28 38 48 58 68 78 88 98 9 19 29 39 49 59 69 79 89 99 10 20 30 40 50 60 70 80 90 100 * The number table should be written in large figures upon the board, or a chart should be made of it. The figures may be drawn with char- coal upon manila paper, or paiuted upon shade cloth. 14 TENS 15 1. How many columns of numbers in this table? 2. How many numbers in each column? 3. Point out and name the first ten numbers. 4. Point out and name the second ten numbers. The fourth ten. 5. Show the tenth (or last) ten numbers. The ninth ten. The third ten. 6. What is the first number of the second ten? Of the third ten? Of the tenth ten? Of the fifth ten? 7. What is the last number of the first ten? Of the third ten ? Of the fourth ten ? Of the second ten ? Of the ninth ten ? 8. What is the last number of the tenth ten? 9. How many numbers are there in the whole table ? 10. Point out and name the second number of the first ten. 11. What is the second number of the second ten ? Of the fourth ten ? Of the tenth ten ? 12. Show the second number in each ten, and tell what figure it ends with. 13. Point out and name the third number of each ten, and tell what figure it ends with. Let children point out the corresponding numbers in each ten until they see their regular decimal succession. 14. In the table of numbers, which number is written just above the number 4 ? Just above the number 14 ? 24? 44? 15. Which one is written just below the number 14 ? Just below 24 ? 34 ? 16. Find 35 and show what number is written just at the right of it. At the left of it. 16 TENS 17. What number is written just at the right of 41 ? At the left of it ? 18. Begin at the number 8 and read toward the right, naming every number. 19. Begin with number 3, and read until you reach 93. 20. Begin with number 97, and read to the left until you reach 7. 21. In which column do you find the number 25 ? 48 ? 67 ? 94 ? 79 ? 22. 40 is at the end of which column ? Where is 80 ? 23. Look at the last number of all the columns and tell what figure each number ends with. 24. Name the next to the last number of each column. What figure is the same in each ? 25. Name all the numbers in the table that end in 7. In 5. In 3. In 8. 26. 21 is at the beginning of which column ? Where is 51 ? 71 ? 91 ? 27. Name all the numbers in the table that end in 4. 28. Point out the 12th number. The 22d. The 32d. The 42d. The 16th. The 26th. The 36th. The 46th. The 56th. 29. How many numbers in the first two tens ? 30. 10 numbers and 10 numbers are how many num- bers? 31. 20 and 10 are how many ? 32. 30 and 10 are how many ? 33. 30 and 10 = how many ? 50 and 10 = how many ? 60 and 10 = h()^y many ? 70 + 10 = ? 90 + 10 = ? TEN8 17 The teacher should provide herself with actual money, consisting of dimes and cents, with which to illustrate the following work. Give much oral work. Let the children make problems for the class to solve. It will be seen that in this work the child's attention is drawn to the facts of number, and not yet to the processes of addition and subtraction. 34. How many cents eqnal a dime ? What else are dimes called ? Ans. Ten-cent pieces. 35. How many cents eqnal two dimes ? Tliree dimes ? Four dimes ? 36. If you had 20 cents, how many dimes would you have ? 37. If you had 30 cents, how many dimes would you have ? 38. 40 cents equal how many dimes ? 39. Which is the more money, 31 cents or three dimes ? How much more? 40. If you had 10 cents and your father gave you 10 cents more, liow many cents would you have ? How many dimes would they equal ? 41. If 3'ou had 20 cents and your father gave you 10 cents more, how much money would you have ? 42. If you had 30 cents and your brother gave you 10 cents more, how much money would you have ? 43. If you had 20 cents and gave away 10 cents, how many cents would you have left ? 44. If you had 30 cents and lost 10 cents, how mucli money would you have ? 45. If you had 10 cents and your mother gave you a ten-cent piece, how much money would you have ? HORN. ARITH. 2 18 TENS 46. If you liad a dime and your mother gave you 10 cents, liow many cents' worth of apples could you buy with your money ? 47. How many tens make 20 ? Point them out in the table. 48. Show how many tens make 30. 40. 60. 80. 50. 90. 49. Sixty means six tens ; what does seventy mean ? Eighty ? Ninety ? 50. When we mean three tens, we do not say threety ; wliat do we say ? 51. How do we express two tens? Four tens ? Five tens ? Ten tens ? 52. Can }Vou find out how many tens make 50 without counting them ? 53. If you have 10 cents and your brother has 11 cents, how many more has he than you have ? 54. If you have 10 cents and your sister has 2 cents more than you, how many cents has she ? 55. 10 cents and 8 cents equal how many cents ? 56. One dime and 5 cents equal how many cents ? 57. One dime and 7 cents equal how many cents ? 58. One dime and 6 cents equal how many cents ? 59. One dime and 8 cents equal how many cents ? 60. Find 10 in the table, add 4, and point out the num- ber which is the answer. In the same way add 9 to 10. 61. Add 7 to 10. Add 6 to 10. 5 to 10. 3 to 10. 8 to 10. 62. 10 + 2 = ? 10 + 9 = ? 10 + 4 = ? 63. Find 20 in the table, add 3, and point out the num- ber which is the answer. TENS 19 64. In the same way add 6 to 20. Add 4 to 20. Add 7 to 20. Add 9 to 20. 65. 20 + 5 = ? 20 + 2 = ? 20 + 8 ? 20 + 4 = ? 66. If you have two dimes and one cent, how many cents in money have you ? 67. If you have 2 dimes and three cents, how much money have you ? How many cents are equal to : 68. 2 dimes and o cents ? 69. 2 dimes and 4 cents ? 70. 20 cents hicking 1 cent ? 71. 20 cents lacking 2 cents ? 72. 2 dimes and 7 cents ? 73. 2 dimes and 9 cents ? 74. 20 cents lacking 3 cents ? 75. 20 cents lacking 4 cents ? 76. Find 30, add 4, and point out the answer. Add 3 to 30 in the same way. 77. Add 5 to 30. 7 to 30. 2 to 30. 8 to 30. 6 to 30. 9 to 30. 78. Three dimes = how many cents ? 79. If you have 3 dimes and 2 cents, how many cents in money have you ? 80. If you have 3 dimes and 4 cents, how much money have you ? How many cents are equal to: 81. 3 dimes and 3 cents ? 85. 3 dimes and 5 cents ? 82. 3 dimes and 7 cents ? 86. 3 dimes and 9 cents ? 83. 30 cents less 1 cent = ? 87. 30 cents less 4 cents = ? 84. 30 cents less 3 cents = ? 88. 30 cents less 2 cents = V 20 TENS 89. 11 cents equal how many dimes and how many cents over ? 90. How much more than a dime are 13 cents ? 15 cents ? 17 cents? 14 cents? 19 cents? 91. How mucli more than 2 dimes are 21 cents ? 24 cents ? 22 cents ? 92. How much more than 3 dimes are 33 cents? 31 cents? 37 cents? 93. If you buy something for 9 cents and give the clerk a dime, how much change ought you to have? 94. If you buy something which costs 29 cents and give the clerk 3 dimes, how much change should you have ? 95. If you buy something that costs 18 cents and give the clerk 2 dimes, how much change ought you to have ? 96. If James had 3 cents more he would have 10 cents. How much money has he ? 97. John has a dime, a nickel, and 2 cents. How much money has he ? Walter has 2 dimes. How much more has he than John ? 98. Henry has 2 dimes, a nickel, and 3 cents. How much more money must he get to have 30 cents ? 99. Mary has 3 dimes. If her mother should give her 9 cents, how much money would she have? How much more must she get to buy something worth 41 cents ? 100. John has 3 dimes, and James has 7 cents. How many cents liave they both ? 101. Find 40 in the table, add 5, and point out the answer. 102. Add 8 to 20. Ada 4 to 20. Add 3 to 30. Add 7 to 30. Add 2 to 40. TENS 21 103. Add 8 to 40. Add 8 to 50. Add 8 to 50. Add 4 to 60. Add 6 to 60. 104. Add 3 to 70. Add 9 to 70. Add 5 to 80. Add 7 to 80. Add 1 to 90. Show the children how older people write numbers when they add them, and let them furnish numbers for many exatnples similar to the following. It will be observed that of the numbers combined, one is a multiple of 10, and the other a number less than 10. 105. A7IS. To 10 To 10 To 10 To 20 add 3 add 5 add 7 add ■:: 18 A71S. A71S. Ans. ^ To 20 To 80 To 40 To 50 add 6 add 3 add 2 add 4 Ans. Ans. Ans. - To 60 To 70 To 80 To 90 add 3 add 6 add 2 add 5 Ans. Ans. Ans. Ans. Ans. 106. If you divide 20 cents between two girls so that each gets the same, how much will each get ? How much is one half of 20? 107. How many dimes are one half of 4 dimes ? 108. If you divide 40 cents equally between two boys, how much will each get? 109. Three girls have 10 cents apiece. How much have they all together ? 110. Mary had 20 cents, and Kate had 10 cents. How much did they both have ? 111. Anna had 3 dimes, and Lucy had 10 cents. How many cents' worth of oranges could they buy with all their money ? 22 TENS 112. Helen had 40 cents, and her mother gave her 10 cents. How much money had she then ? She bought a doll for 49 cents. How much money had she left ? 113. How many cents make a dollar? How many cents make half a dollar? 114. Which is the more money, 53 cents or half a dol- lar ? How much more ? 115. Find 30 in the table, add 10, and show the answer. 116. Add 10 to 50. Add 10 to 70. Add 10 to 40. Add 10 to 80. Add 10 to 90. 117. To 20 To 40 To 80 To 60 To 50 add 10 add 10 add 10 add 10 add 10 118. Begin at 10 and count by tens to 100. 119. Fill out and learn the following : 1 ten = 6 tens = 2 tens = 7 tens = 3 tens = 8 tens = 4 tens = 9 tens = 5 tens = 10 tens = 120. 10, 20, 30, etc., are called multiples of 10. Begin with 10 and name the multiples of 10 as far as you can. " Multiple " is not a difficult word for children when it is used in its objective sense, as in this case. The author's pupils used to call the multiples "bright numbers" until experience showed the advan- tage of giving them the true name. 121. What is the first multiple of 10 ? Ans. 10. 122. What is the second multiple of 10 ? 123. Point out the third multiple of 10. 124. What is the fourth multiple of 10 ? The seventh multiple of 10 ? The tenth multiple of 10 ? TENS 23 125. What figure does each of the multiples of 10 end witii ? 126. 30 is which multiple of 10 ? 50 is which multiple of 10 ? 70 is which multiple of 10 ? 127. How many tens in 70 ? In 60 ? In 40 ? In 90 ? 128. 80 equals how many tens ? 50 equals how many tens ? 70 equals how many tens ? 100 equals how many tens ? 3.29. Write the table of numbers, setting them in straight columns, and making the multiples of 10 larger and brighter than the other numbers. Let the children use colored crayons to make the multiples distinct. Give a few of the most expert pupils a certain space at the board where they can spend some time each day making their tables, until they consider them fit to be presented as their completed work. As each one finishes his table, give the space to another child until all have written it. The necessities of the acts of construction make the mental picture distinct. 130. 3 tens + 4 = ? 7 tens + 7 = ? 8 tens + 6 = ? 9 tens + 5 = ? 3 tens + 2 = ? 4 tens + 8 = ? Give occasionally chart exercises similar to the following : Point out 5, add 4, add 2, take away 1, add 7, take away 3, add 5, subtract 1, etc. Have the children make the combinations without counting as soon as possible. Encourage them to recite without look- ing at the number table, but do not allow guessing. Require them to go back to the number table whenever they show indefinite ideas of numerical distances. 131. Cover up the last two numbers of the first ten in your number table. How many numbers of that column are left in sight ? 132. If we cover the last two numbers of the first twenty, how many of the twenty are left? 24 TENS 133. Take 4 from 30 in the same way and show how many are left. 134. Take 2 from 70. 3 from 80. 4 from 90. 3 from 100. 135. Which is more, 2 tens or 19 ? How much more ? 136. Which is more, 3 tens or 28 ? How much more? 137. 4 tens are how many more than 39 ? Than 37 ? Than 33 ? Than 31 ? 138. 62 is how many more than 6 tens? 139. 83 is how many more than 8 tens ? 140. What number is 2 more than 28? 2 more than 38 ? Than 48 ? Than 78 ? 141. What number is 3 more than 10? 3 more than 20 ? Than 30 ? Than 80 ? 142. What number is 3 less than 10 ? 3 less than 40 ? Than 50? Than 60? Than 80? 143. AVhat number is 4 more than 3 tens? 4 more than 6 tens ? Than 5 tens ? Than 7 tens ? Than 9 tens ? 144. What must be added to 79 to equal 8 tens? 145. How many must be added to 48 to equal 5 tens ? 146. How many must be added to 7 tens to make 72 ? 147. How many must be added to 3 tens to make 35 ? 148. How many must be taken from 4 tens to leave 38 ? 149. How many must be taken from 5 tens to leave 47 ? 150. How many must be subtracted from 95 to leave 9 tens? 151. How many must be subtracted from 28 to leave 2 tens ? TENS 25 152. John had 40 cents and lost 2 cents. How many had he left ? 153. Mary has 10 cents. Anna has 3 times as many. How man}^ has Anna ? Let the children make story problems like the preceding. 154. 5 + 4 = ? 15 + 4 = ! 25 + 4 = -^ 35 + 4 = ? ^^ + 4 = •/ ()5 + 4 = ? 75 + 4 = ? 85 + 4 = ? 155. Add 3 to several numbers that end in 5, and point out the answers. 156. Add: 16 26 36 46 bQ m 76 86 96 33333 3 '333 Will not some of the pupils find out for themselves that the addi- tions can be performed easily by adding the units and bringing down the tens? This should be shown to all. 157. Add 4 to several numbers that end in 2. Write the numbers as grown people write large numbers when they add them. 158. Add 5 to some numbers that end in 3. 159. Add 2 to numbers that end in 7. Add 4 to num- bers that end in 1. Give oral as well as written work on these combinations until they are mastered and the cliildren no longer count. Observe that the sum of the units given in this work is less than 10. 160. Add: 46 33 23 51 71 84 62 m 41 161. If 3^ou had a figure made of 13 squares and should add 5 squares, how many squares would there be in the figure ? If the children are uncertain, let the squares be placed, not otherwise. 162. li you had 36 cents and earned 3 cents, how many would vou have ? 26 TENS 163. Mary had 23 nuts and picked up 4 nuts; how many had slie then ? Let the children make story problems. 164. Take 2 from 7. 2 from 17. 2 from 27. 2 from 37. 2 from 47. 2 from 57. 165. Take 2 from all the numbers on the number table that end in 8. Point out the answers. 166. In the same way take 2 from all the numbers on the number table that end in 5. 167 From 39 Show that instead of thinking of the whole , ^y ^9 we can take 2 from 9 separately, and bring LaKe —i 1 ii .1 i down the o tens. 59 29 49 79 99 69 89 2 9 2 2 2 2 2 168. Write some numbers that end in 6, and take 3 from each of them. 169. Take 3 from numbers that end in 9. 170. Take 4 from numbers that end in 8. 171. Take 4 from numbers that end in 6. 172. Take 5 from numbers that end in 9. 173. Take 5 from numbers that end in 7. 174. Take 8 from numbers that end in 9. 175. From 27 38 46 88 96 84 take 2 3 4 3 5 3 67 5 78 4 89 3 176. If you had })laced 27 squares in a figure and should take away 5 squares, how many would be left ? 177. If there were 38 squares in a figure and you took away 4 squares, how many would be left ? 178. John had 17 marbles and lost 4 marbles. How many were left ? TENS 27 179. Mr. Smith earns 28 dollars a week and spends 7 dollars for board. How much has he left ? 180. Mary had 89 cents and spent 6 cents. How many had she left ? Call for similar number stories. 181. Point out G, add 10, and point out the answer. Add 10 to 26. To 36. Go on adding tens until you reach 96. 182. In the same way add tens to 5 until you reach 95. 183. Mary may name a small number and the others may add tens to it. It is to be hoped that some of the children will go beyond the limits of the 100, and that the others will readily follow. 184. To 35 64 86 48 57 33 72 add 10 10 10 10 10 10 TO Call attention to the convenience of the plan of adding the columns of units and tens separately, and let the children prove by trial that it gives the same result as reckoning on the number table. 185. If you had 13 squares in a figure and added 10 squares, how many squares would there be in the figure ? 186. 27 squares and 10 squares = how many squares ? 187. If you had 24 cents and earned 10 cents, how much money would you have ? 188. 10 cents added to a nickel = how many cents ? 189. 10 cents added to a (juarter of a dollar = how many cents ? 190. Point out 48, and subtract 10. Take 10 from 38. From 28. From 18. 191. Subtract all the tens you can from 93. From 97. From 95. From 91. From 94. 28 . TENS 192. John may name a number, and the others may subtract from it as many tens as they can. 193. From 87 79 64 21 47 86 73 take 10 10 10 10 10 10 10 194. If there were 19 squares in a figure, and we took away 10 squares, hoAV many Avould be left ? 195. 38 squares lacking 10 squares = how many squares ? 196. If Anna had 25 cents, and h)st 10 cents, how many would she have left ? 197. If William's father had 35 dollars, and spent 10 dollars for William's suit of clothes, how many dollars would he have left? 198. 48 chickens were in a coop, and 10 chickens were sold. How many were left ? 199. Add 2 tens to 21. 3 tens to 42. 4 tens to 54. 3 tens to 26. 4 tens to 38. 200. Name a number smaller than 50, and add 5 tens to it. 201. Name a number smaller than 30, and add 7 tens to it. 202. Name a number smaller than 40, and add 4 tens to it. 203. Name a number smaller than 20, and add 5 tens to it. 204. Add: 68 43 57 29 19 36 30 40 30 40 80 50 205. Subtract 2 tens from 47. 3 tens from 83. 5 tens from 79. 206. Name a number larger than 50, and subtract 4 tens from it. TENS 29 207. Name a number larger than 80, and subtract 7 tens from it. 208 Name a number larger than 40, and subtract 2 tens from it. 209. Name a number larger than 70, and subtract 5 tens from it. 210. From 79 66 85 73 89 98 54 take 30 20 50 40 60 80 30 211. Find 17, add 10, add 20 to the result, subtract 10, add 30, add 20, subtract 30, add 10, subtract 20. 212. Find 93, subtract 10, subtract 20, subtract 10, add 20, subtract 20, add 10, subtract 20. Give similar chart exercises frequently. 213. Find 25 and add 10, and then 1. 214. Find 23 and add 11. Add 11 to 46. Add 11 to 75. To 81. To 58. Call the children's attention to the relative position of numbers in the diagram whose difference is 11. Do not let them count 11, but add 10 and then 1. 215. Begin with 11 and add elevens until you reach 99. 216. Find a number greater than 12 and less than 16, and add 2 elevens to it. 217. Find a number greater than 23 and less than 26, and add 2 elevens to it. 218. Add 2 elevens to each of the numbers that are between 34 and 37. 219. Think of 12, and without counting, add 11 to it. Keep on adding elevens until you reach 100. 220. Think of a number greater than 21 and less than 25, and add 2 elevens to it, 30 TENS 221. Add 23 75 68 74 26 33 48 87 11 11 11 11 11 11 11 11 222. Point out on the number table the number that means 2 elevens, and add elevens to it until you reach 41. How many elevens did you add? 223. Add 36 74 37 55 41 34 22 22 22 22 22 22 224. If James had 11 cents and John had as many more, how many did they both have? How many dimes, and how many cents over would their money equal? 225. If James had 24 cents and John had 11 cents more than James, how many did John liave ? How many dimes and how many cents? 226. Mr. Smith paid 75 dollars for a horse and 11 dol- lars for a harness. How much did they both cost ? Call for story problems, using the number 11. 227. Find 59 and take away 11. Take 11 from 83. From 75. From 84. From 97. From 48. 228. From 68 43 84 65 48 76 take 11 11 11 11 11 11 From 35 66 84 29 77 98 take 22 22 22 22 22 22 229. A man had 33 dollars and lost 11 dollars. How much had he left? 230. 24 apples less 11 apples = how many apples ? 231. 36 squares less 11 squares = how many squares ? 232. Find 25 on the chart, add 10, and then 2. 233. Add 12 to 24. Keep on adding twelves until you reach 96. Remember that 12 means 10 and 2. TENS 31 234. Think of a number greater than 40 and less than 48, and add 12 to it. 235. Add 12 to each of the numbers greater than 20 and less than 26. 236. Think of a number between 31 and 34, and add 3 twelves to it. 237. Think of a number l)etween 63 and 66, and add 2 twelves to it. 238. Add: 82 47 63 75 84 5() 24 66 87 12 12 12 12 12 12 12 12 12 239. Write 12 under 76 and add them. Write 12 under 44 and add them. 240. If you had 12 cents and received 12 cents, how many cents would you have ? If you gained another 12 cents, how many cents would you have ? How many dimes, and how many cents over ? 241. If you had 15 cents and gained 12 cents, how many cents would you have ? 242. Point out on the number table the numbers that mean 2 twelves. 3 twelves. 4 twelves. 5 twelves. 6 twelves. 7 twelves. 243. Find the number that means 3 twelves ; write it under 42 and add. 244. Write under 21 the number that means 4 twelves, and add. 245. Add 15 to the number that means 2 twelves. 246. Find 48, take 12 from it, and point out the number that is left. Keep on taking twelves until nothing is left. 247. Find 59 and take twelves from it until 11 remains. 32 TENS 248. From 88 96 34 78 25 57 take 12 12 12 12 12 12 249. Find 23 on the chart and add 13 to it. Add 13 to 13. To 53. 250. Add 14 to 23. Add 14 to 33. To 53. To 73. 251. Add 15 to 24. To 44. To 74. To 34. 252. Add 16 to 21. To 31. To 51. To 81. To 61. 253. Add 17 to 22. To 82. To 32. To 52. . 254. Add 22 31 84 24 25 85 81 33 24 51 1717131313131414 15 15 255. From 77 86 47 62 48 55 78 69 58 53 take 24 35 33 30 26 43 24 37 27 21 256. How many chiklren are there in your class? If they were pUiced in groups of 10, how many groups would there l)e, and how many children over ? 257. 12 cents = how many dimes and cents ? 258. 18 cents = how many dimes and cents ? 259. 33 cents = liow many dimes and cents ? 260. Write 32 cents in dimes and cents. 261. Write in dimes and cents 45 cents. 75 cents. 24 cents. 38 cents. 262. How many are 3 tens and 5 ? 7 tens and 6 ? 2 tens and 7 ? 4 tens and 3 263. How many ones in 4 ? How many ones in 7 ? 264. Sometimes the word "unit" is used to mean one. How many units in 6? In 9? 265. 11 means 1 ten and 1 unit. What does 12 mean? Ani<. 1 ten and 2 units. TENS 33 Take numbers consisting of tens and units, as 42, and lead the children to see that the figure 4 stands for 4 tens (which they may show by the number table, or by columns of squares), and that the figure 2 stands for the 2 remaining units. 266. What does 17 mean? 21? 32? 64? 57? 63? 89? 267. Separate 37 into tens and units. 268. Separate into tens and units 48, 57, 65, 39, 82, 68, 95, 24. 269. How do you write 3 tens and 7 units together? 5 tens and 2 units ? 7 tens and 8 units ? 270. In the number 25, which figure stands for tens ? Which figure stands for units? 271. In the number 68, which figure is in the tens' place, and which is in tlie units' phice ? 272. Write a number whicli has 4 in the tens' phice and 7 in the units' place. 273. What number has 5 in the tens' place, and 3 in the units' place ? 274. Write some numbers of two places, and tell what figures you put in the tens' places, and what figure in the units' places. 275. Write a number which has 9 in the units' place. Can you write a number which has 10 in the units' place? 276. When we mean 20 and 9, we write 29. What do we write when we mean 20 and 10 ? 20 and 11 ? 20 and 12 ? 4 units and 2 units = liow many units ? 277. To 3-1 (3 tens and 3 tens = how many tens ? add 64 How many tens and how many units in the answ^er ? HORN. ARITH. 3 34 i^^^!^ 278. Write 52 under^ 46 and add them. Why is it best to write units under units, and tens under tens, when we add numbers ? 279. Add 76 and 22. Add 23 and 34. Add 25 and 42. Add 31 and 63. Add 34 and 25. Add 11 and 78. Show how many tens and how many units in each answer. 280. Write 22 under 48, and subtract the smaller num- ber from the greater. 281. From 39 take 11. From 27 take 15. From 78 take 22. From 66 take 21. From 83 take 62. 282. What is the largest number that is written with one figure ? 283. What is the smallest number that is written with two figures? 284. What is the largest number that can be expressed by two figures ? 285. What is the smallest number that can be expressed by three figures ? It should be explained that there were people living long ago, called Romans, who expressed numbers by letters instead of figures, and that sometimes we still use then' notation. 286. X stands for 10 in Roman notation. Find the lOtli chapter in this book, and tell on what page it begins. Make a 10 like that on the clock. 28V. Since X stands for 10, what do two X's stand for ? What do three X's stand for ? 288. Write in Roman notation, 10, 20, 30. CHAPTER III TWOS Even Numbers, Foot and Inch, Halves, Quart and Pint, Horizontal Line, Triangle, Thirds, Fourths, Verti- cal Line, Roman Numeral I NUMBER TABLE 1 11 21 31 41 51 61 71 81 91 2 12 22 32 42 52 62 72 82 92 3 13 23 33 43 53 63 73 83 93 4 14 24 34 44 54 64 74 84 94 5 15 25 35 45 55 6,"^ 15 85 95 6 16 26 36 46 56 66 16 86 96 1 17 27 37 47 57 67 77 87 97 8 18 28 38 48 58 68 78 88 98 9 19 29 39 49 59 69 79 89 99 10 20 30 40 50 60 70 80 90 100 1. Begin with 2, and connt by twos to 40, pointing ont the nnmbers on the number table. Count without the number table. 35 36 TWOS 2. These numbers that you have been giving, 2, 4, 6, 8, and so on, are called Even Numbers. Name all the even numbers in the 1st ten. In the 2d ten. In the 3d ten. Call attention to the endings of the even numbers. 3. Name all the even numbers in the 10th ten. In the 9th ten. Tell how many tens and how many units in each even number in the 9th ten. 4. What is the smallest even number that you can think of? 5. Name all the even numbers that end in 2. In 4. In 8. In 0. What other figure may an even number have in its units' place besides 2, 4, 8 or ? 6. What is the smallest even number in the 3d ten ? In the 7th ten? In the 10th ten? 7. What is the largest even number in the 4th ten? In the 6th ten? In the 8th ten ? 8. Think of the largest even number that is less than 29 and write it. 9. What is the next even number after 34? Before 34? 10. Find the third even number. The fifth even number. 11. What even number comes just before 49? How man}^ tens and how many units in it? A device for leading children to recognize even numbers is to write an even number out of the children's sight, and then let them guess what it is, giving them a clew^ as " It is in the 3d ten " or " It ends with 4," or " It is between 91 and 90," •' It is larger than 26, but not so large as 38," Sontetimes allow the children to write the hidden number. TWOS 37 12. Write the tirst 30 numbers, marking them off into groups of two, as follows : 13. How many groups of 2 equal 8? How many equal 12? 18? 14? 20? 16? 24? 30? 14. How many ones in 3 twos? In 5 twos? In 8 twos? In 6 twos? In 9 twos? In 12 twos? In 7 twos? 15. Name some even numbers, and tell how many twos they equal. 1 11 21 2 12 22 3 13 23 4 14 24 5 15 25 6 16 26 7 17 27 8 18 28 9 19 29 10 20 30 16. Fill out ai 1 two = 2 twos :^ 3 twos 4 twos = 17. 54 -h 9 — > 5 twos = 9 twos = 6 twos = 10 twos = 7 twos = 11 twos = 8 twos = 12 twos = 84 + 2 = ? 94 + 2 = ? 74 4- 2 = ? 18. Find 42 in the number table, add 2 twos, and point out the answer. 19. Find 34, and add 3 twos. Add 4 twos to 50. Add 5 twos to 20. 20. Add 2 twos to 18. Add 3 twos to 28. Add 4 twos to 36. 21. Find 76, add 6 twos, and show the answer. Add 7 twos to 64. Add 8 twos to 24. Add 9 twos to 20. 22. Point out even numbers and add some twos to them. 23. How many twos must be added to 12 to equal 18? To equal 22? To equal 16? 38 TWOS 24. How many twos must be added to 14 to equal 20? To equal 24? To equal 18? 25. If you had 18 cents and gained 2 cents, how many would you have? How many dimes Avould it equal? 26. If you had 16 cents, and your mother gave you 2 cents, and your father gave you 2 cents, how many cents would you have? 27. If you had 18 cents, and 3 people each gave you 2 cents, how many cents would you have? 28. If you had 10 cents, and 5 people each gave you 2 cents, how many cents would you have? 29. Add : 54 66 36 32 86 64 56 12 22 42 54 12 32 22 42 52 74 22 34 72 84 34 26 14 46 24 22 12 30. Beginning at 20, count backwards by twos. 31. 2 from 36 leaves how many? 2 from 48 leaves how many? 2 from 96? 18-2:=? 28-2=? 78-2 = ? Give questions in subtraction similar to the addition drill in Ex. 18-24. 32. If you had 36 cents and lost 2 cents, how many would you have left? 33. If you liad 24 cents and lost 2 cents, how many would you have left? 34. If you had 2 dimes and bought something for 2 cents, how much money would you have left? 35. If you had a dollar and lost 2 cents, how much would you have left? TWOS 39 36. If 20 children were at a party and 2 went home, how many would be left? How many would be left when 2 more went home ? When 2 more went home ? 37. If there were 22 children belonging in your class and 2 were absent, how many would be present? If 2 of those were dismissed, how many would be left ? 38. Find 20 and show how many twos must be sub- tracted from it that l-i may be left. 39. How many twos must be subtracted from 22 to leave 16 ? To leave 12 ? 40. Beginning at 30, count backwards 2 twos. What number did you reach ? 41. Beginning at 40, count backwards by twos until you reach 32. How many twos did you count off? 42. How many twos must be taken from 36 to leave 28 ? 24 ? 20 ? 43. If you had 40 cents and gave 2 cents to each of 4 boys, how many cents would you have left ? 44. From 68 76 86 38 44 56 32 6Q 54 86 take 26 34 42 12 22 34 22 42 24 36 45. If you had 40 cents and each of 5 boys gave you 2 cents, how many cents would you have ? How many dimes would they equal ? 46. Draw with a foot rule a line a foot long, and mark off the inches. How many inches make a foot ? 47. Draw a line 10 inches long, marking tlie inches. A 10-inch line lacks how many inches of being as long as a line a foot long ? Show how many times a 2-inch line can be measured off on a 10-inch line. 40 TWOS 48. Make the line that is a foot long, 2 inches longer. How long is it now ? How much longer is it than the 10 -inch line ? 49. How many times can a 2-inch line be measured off on a 14-inch line ? 50. Lengthen the 14-inch line 2 inches, and tell how long it is. How many times will it contain a 2-inch line? 51. Lengthen the 16-inch line 2 inches and tell how long it is, and how many times it contains a 2-inch line. 52. Lengthen the 18-inch line 2 inches and tell how long it is and how many times it contains a 2-inch line. 53. 16 divided into groups of 2 = how many groups ? 54. 24 divided into groups of 2 = how many groups? 55. 20 - 2 = ? Show division as a process of separating tlie larger number into groups of the less. Illustrate by grouping objects, marks, or the num- bers of the number table. 56. 14 -f- 2=? 22- 2='> 12- 2=*^ 18-- 2=*;^ 20-1-10=? 40-10=? 70-10=? 30-10=? 57. How many times can a 10-inch line be measured off on a 20-inch line ? 58. How long is a line which is one half as long as a 20-inch line ? 59. Cut a slip of paper 12 inches long, double it to find the middle, and tell how many inches long one half of the strip is. 60. Measure off a 14-inch line and find how many inches one half of it measures. TWOS 41 61. Fill out and learn the following : One half of 2 = ? i of 12 = ? One half of 4 = ? i of U = ? i of 6 = ? -1- of IG = ? 1 of 8 = ? 1 of 18 = ■? -1- of 10 = ? 1 of 20 = ? 62. Find 4 in the table, and point out the number which means one half of it. 63. Show 10 and the number which is -J of it. Show J of 12. Of 16. Of 20. Of 14. Of 18. ^ 64. Mary has 14 cents, and Julia has ^ as many. How many has Julia ? 65. How many eggs in ^ a dozen ? 66. When candy is 10 cents a pound, how many cents will |- a pound cost ? 67. AVhen candy is 20 cents a pound, how much will I a pound cost ? 68. How many pints make a quart ? Let the children pour the water from pint measure to quart meas- ure, and find out the fact for themselves. 69. How many pints in 2 quarts? In 4 quarts? In li quarts ? In 7 quarts ? In 9 quarts ? In 3 quarts ? In 8 quarts? In 12 quarts? In 5 quarts? 70. If a bucket has 12 joints of water in it, how many quarts of Avater can be taken out of it ? The child who cannot imagine the process should be allowed to work it out practically. 71. How many quarts in 12 pints? In 18 pints? In 14 pints? In 10 pints? 72. A pint of milk equals what part of a quart? 42 i'Wos 73. If a quart of milk costs 6 cents, how much does a pint cost ? 74. If a quart of vinegar costs 8 cents, how much does a pint cost ? 75. If a quart of molasses costs 18 cents, how much does a pint cost ? Teach the horizontal lines. 76. Draw a horizontal line 20 inches long. Into how many 2-inch lines can it be divided ? A 12-inch line = how many 2-inch lines ? A 16-inch line = how many 2-inch lines? An 18-inch line = how many 2-inch lines? 77. Seven 2-inch lines = how lono" a line ? P^leven 2-inch lines ? Twelve 2-inch lines ? This should be read: ''2 inches mul- 78. 2 inches x (> = ? ^- ,• i i <> o tiplied l)y (j = r ?> 79. 2 inches x 7 = ? 2 inches x 9 — ? 2 inches x 12 = ? 2 inches x 10 = ? 2 inches x 4 = ? 2 inches x 8 = ? 80. What name is given to numbers which equal any number of twos ? 81. Name all the even numbers between 11 and 19. Between 21 and 31. 82. Begin at 18 and name all the even numbers to 84. 83. How many even numl)ers are tliere less tlian 20? How many less than 26 ? 84. What even number is 3 more than 31 ? How many tens and how many units in it? 85. How many gloves in 7 pairs of gloves ? 86. How many shoes in 9 pairs of shoes ? 87. How many pints of milk in 8 quarts ? In 10 quarts ? lull quarts ? TWOS 43 88. How many slices can 5 boj^s wear at the same time ? 89. 12 men paid 2 dollars apiece to lure a sailboat. How much did they all pay ? 90. 11 boys put in 2 cents apiece to buy a ball. How much did they all put in ? 91. How many wheels have 10 bicycles ? 92. T chickens have how many feet ? 93. How much will 11 two-cent postage stamps cost? 94. 8 ])airs of horses = how many horses? 95. How many wings have 12 Ijirds ? 96. If you had 18 a|)plcs, to how many cliildren couhl you give 2 apples apiece ? 97. 20 children are going to march by couples. How many couples will there be? If there were 1(3 children, into how many couples could they be formed ? 1(3 h- 2 = ? 14 -^ 2 = *? 98. If there were 13 cliildren, into how many couples could they be formed ? (Illustrate if necessary.) 99. Place inch-squares Fig. 1 as in Fig. 1. Push the squares apart so as to divide your fig- ure into halves. How many square inches in the whole figure? How many in each half? After the children have made the figures, it would be well for the teacher to have them close their books and to give them the questions orally, develoi^ing ideas of forms and of the ratio of their parts as far as the abilities of each class allow. 44 TWOS Fig. a 100. Place inch -squares as in Fig. 2. Divide the figure .into halves. How many squares in each half? Can you put the halves of Fig. 2 together so as to make Fig. i ? 101. Which is the larger, Fig. 1 or Fig. 2 ? 102. Place inch-squares as in Fig. 3. Divide the ligure into halves, and tell how many square inches in each half. Can you put the halves of Fief. 3 toofether so as to make Fig. 2 ? 103. Which is the largest. Fig. 1, Fig. 2, or Fig. 8 ? Show some horizontal lines in Fig. 3. Let the children combine squares into figures of their own devis- ing, divide the figures into halves, and report upon them. Encourage symmetrical forms. 104. Cut an inch-square in two so as to make triangles like these. Fig. 105. Place the triangles as in Fig. 4. How many triangles would it take to make 6 such figures. 106. Place other triangles as in Fig. 5. How many triangles would it take to make 9 figures like Fig. 5 ? Fig. 4 Fig. 5 TWOS 45 107. Place others as in Fig. 6. How many triangles would you need to make 7 figures like Fig. 6 ? 108. Place others as in Fig. 7. Wliicli is the largfest of these fio-ures ? Fig. 7 Let the children show the equality of the figures by rearranging each of them into a square. 109. Copy Fig. 8 by placing triangles. How many triangles does it take? How many inch squares does it take to make the triangles used in copying Fig. 8? 110. Make Fig. 9. Which is greater, Fig. 8 or Fig. 9? How many square inches in each? How many square inches in both? Fig. 8 Fig. y 111. Place three triangles so as to make other figures, and show how many square inches in each of the figures. 112. How many such triangles can you make from one inch square? From 3 squares? From 7 squares? From 8 squares? From 9 squares? From 10 squares? 113. 12 such triangles = how many inch squares? If the children are uncertain, let them work out the problem by arranging the 12 triangles into inch squares. 114. 14 such triangles = how many inch squares? 18 triangles = ? 20 triangles = ? 22 triangles = ? 24 tri- angles = ? 115. 10 = how many twos? 2 tens = how many twos? 2 tens = how many units? 10 twos = how many units? 46 TWOS 116. 5 tens + 2 units = how many units? 6 tens +4 units = how many units? 117. If Mary had 3 dimes and earned 8 cents, how much money woukl she have? 118. If John had 7 dimes and a nickel, how much money wouhl he have? Call for story problems. 119. What is an even number ? 120. Name an even number smaller than 20, and tell how mau}^ twos it equals. 121. Into how many parts is this circle divided? 122. When anything is divided into three equal parts, what is eacli part called ? Ans. One third, written i. Show -J of the circle. Show J of it. 123. Draw a line 3 inches long. Mark off -J of it. 124. Cut a strip of paper 3 inches long and fold it into thirds. Show -^ of it. Show J of it. 125. Find Fig. 8, Ex. 109, and show how much J of it is. Show I of it. 126. Show I of Fig. 9, Ex. 110. Show i of Fig. 2, Ex. 100. Let the children estimate | of the length of a book or stick or some convenient object. Cut a strip of paper as long as the object, and fold it into thirds to test the correctness of the estimates. Provide a tumbler of cylindrical shape, and require a child to bring it to you i full of water ; | full ; | empty ; i empty. TWOS 47 127. If you jjut ^ of a stick into the ground, how much will be above ground ? 128. The snow was so deep that it covered J of a fence post. What part of the post was bare? 129. What is J of 6 squares ? ^ of 9 squares = how many ? (Illustrate.) 130. How much is ^ of 3 cents? | of 3 cents = how many ? 131. Turn to the number table and show ^ of 3. ^ of o f 30. 6. i of 9. If the children do not see these relations readily, do not let them memorize the statements of them, but postpone the subject. 132. Place triangles as in Fig. 10. How many triangles does it take ? How many square inches do they equal ? yig 10 133. Place triangles as in Fig. 11 and Fig. 12. Which is the larger figure ? How many triangles in both ? How many square inches do they both equal ? Fig. 11 Fig. 12 134. Place triangles as in Fig. 13, Fig. 14, and Fig. 15. Which is the largest figure ? How many triangles in all ? How many square inches do they all equal ? Fig. 13 Fig. 14 Fig. 15 135. Place 4 triangles so as to make a figure different from any in the book. 48 TWOS 136. Fold a strip of paper into 4 equal parts. What is each part called / Ans. One fourth, written -|. 137. Draw a picture of a pie cut into fourths. If ^ were eaten, how many fourths would be left ? 138. Take a piece of string and divide it into fourths. 139. If eight children made a class, how many children would ^ of the class be ? (Illustrate.) 140. Pour water into a glass until it is ^ full. 141. Turn to Fig. 10, Ex. 132, and show ^ of it. Show |. Show 1^. 142. Show ^ of Fig. 11. Show I of it. 143. Find Fig. 12, cover up ^ of it, and tell how many fourths are in sight. 144. Draw a horizontal line 8 inches long and show ^ of it. Show 1^ of it. Show | of it. 145. A line in this position is called a vertical line. Hold your pencil up to show a vertical line. 146. Draw a vertical line 4 inches long, divide it into fourths, and show -J of it. Show | of it. Show | of it. 147. Turn to the number table and show ^ of 4. ^ of 8. { of 40. 148. Take an inch square of paper and fold it into fourths shaped like those in Fig. 13. Fold an inch square into fourths of other shapes. 149. Draw a horizontal line 1 foot long and divide it mto fourths. How many inches long is ^ of a foot ? 150. Draw a circle (by marking around a coin, tumbler, or some circular object), cut it out and fold it into fourths. How many fourths in the whole of anything ? TWOS 49 151. Draw a horizontal line 9 inches long. Show J of it. Show I of it. 152. What is the sixth multiple of 10 ? 153. Write in order in a horizontal line all the muh tiples of 10 which you have learned from the number table. Are there any other multiples of 10 ? If so, write some. 154. Which will cost more, 6 pints of milk or 3 quarts ? Explain. 155. 6 pints + 1 quart — 1 pint = how many pints ? 156. 7 pints + 1 (|uart = how many pints ? 157. 3 quarts + 2 pints = how many quarts ? 158. 4 quarts — 1 pint = how many pints ? 159. 3 dimes + 5 cents — 1 cent = how many cents ? 160. 1 foot -h 3 inches = how many inches ? 161. Find some horizontal lines on this page. 162. What is the largest even number that can be written wdth one figure ? 163. What is the largest even number that can be written with two figures ? 164. What is the smallest even number that can be written with two figures? 165. How many thumbs do ten boys have ? 166. How many toes do two boys have ? 167. Which is more, 2 times 10 or 10 times 2 ? 168. What does X stand for in Roman notation? XX? XXX? 169. I stands for one. Find I on the clock. What does n mean on the clock ? What does III mean? HOKX. Aiam. — 4 50 TWOS 170. Which chapter of this book are you studying ? What is the number of the chapter before this ? 171. X H- I = liow many ? ' XX + III = X + II = how many ? XXX + I = X -f III = how many ? XXX +11 = XX H- I = how many ? XXX + 1 1 1 = 172. Read XI from the clock. Read XII from the clock. 173. Read XIII. XXII. XXIII. XXXI. XXXII. XXXIII. 174. Write in Roman numbers 12, 32, 22, 13, 21, 31, 23, 33. 175. On which page of this book does the 11th chapter begin ? The 13th chapter ? The 12th chapter ? 176. Add: 72 86 14 24 66 48 • 32 24 12 74 32 20 40 66 177. From 88 66 38 54 76 84 take 12 24 26 22 36 44 CHAPTER IV ADDITION Sum, Yard, Rectangle, Thousands, Gallon, Perimeter, Peck, Roman Numerals V, L, and C Do not take up a new combination of numbers until the pupils are able to give promptly those already taken. Exercises in form, in simple fractions and measurements, and in Roman Numerals are introduced between the combinations not only to illustrate them, but to extend profitably the time during which they are learned. 1. Add: 48 How many units in the answer? How Sb many tens ? 2. Add: 73 How many units in the answer ? How 35 many tens ? Wliat do 10 tens make ? 3. Add: 24 76 81 45 38 52 94 82 88 81 38 27 62 71 53 18 78 25 Tell how many hundreds, how many tens, and how many units in each of the answers. The learning of the addition combinations is a gradual process accomplished by many repeated perceptions on the part of the learner. Inexperienced teachers are cautioned not to be discouraged if the same pupil who has one day given the combinations correctly misses them at a later date. This merely shows that those paths in the undevel- oped little brain need to be traversed again. Vary the work l>y having pupils place and count squares, make and count dots, or count objects, real or imaginary. 51 2 ADDITION NUMBER TABLE 1 11 21 31 41 51 61 71 81 91 2 12 22 32 42 52 62 72 82 92 3 13 23 33 43 53 63 73 83 93 4 14 24 34 44 54 64 74 84 94 5 15 25 35 45 55 65 75 85 95 6 16 26 36 46 56 66 76 86 96 7 17 27 37 47 51 67 77 87 97 8 18 28 38 48 58 68 78 88 98 9 19 29 39 49 59 69 79 89 99 10 20 30 40 50 60 70 80 90 100 4. Find 9 on the number table and add 2. Add 2 to 19. Add 2 to : 29, 39, 49, 59, 69, 79, 89, 99. Show 9 + 2 as 9 + 1, which completes the ten, and 1 more which makes 11. Call attention to the relative position in the number table of 9 and 11, 19 and 21, etc., and train pupils to associate with them the thought of 2 as denoting the interval between them. Do this with each com- bination, noting the respective intervals. 5. 19 cents and 2 cents = how many cents ? How many dimes and how many cents over ? 6. Think of 9 little birds and of 3 more little birds coming to join them. Draw a picture of them. 7. 49 cents + 2 cents equal hoAV many dimes and cents? 8. 99 cents and 2 cents = how many cents ? How many dimes and cents ? How much more than a dollar ? 9. Add: 93 94 92 95 27 25 96 94 25 24 24 26 23 91 94 21 24 92 ADI)1TI(.)N 53 Tell how many liuiidreds, liow many tens, and how many units in each of the answers. 10. How many must be added to 9 to equal 11 ? 11. How many must be taken from 11 to leave 9? 12. Show by the number table which is greater, 9 + 2 or 2 + 9. 13. Find each number in the number table that has 9 in the units' place, add 8 to it, and remember the result. Show that in arldiug 3 to 9, 1 completes the first ten and the remaining 2 make 12. 14. Which is the greater, 9 -f 3 or 3 -f- 9 ? Show it on the number table. % As each combination of units is taken up, lead the children to ob- serve that the same result is obtained by combining the numbers in either order, and give drill upon the combinations stated in each way. 15. Add: 95 96 98 93 92 93 93 34 23 32 31 21 36 26 35 25 94 91 How many hundreds, how many tens, and how many units in the answers ? 16. When numbers are added, the result is called their Sum. What is the sum of 29 and 3? 49 and 3? 89 and 3? 17. Jane had 19 cents and gained 3 cents. How many cents had she then ? 18. Louise paid 39 cents for a doll and 2 cents for a postage stamp. How much did both cost ? 19. John had 3 cents more than Thomas. Thomas had 29 cents. How much did John liave? 20. 99 cents + 3 cents = how many cents ? How much more than a dollar ? ^4 ADDITION 21. 9 pints + 8 pints = liow many pints ? How many quarts ? 22. How many must be added to 9 to equal 12 ? 23. How many must be added to 29 to equal 82 ? 81 ? 24. 41 is how many more than 39 ? 25. How many must be taken from 52 to leave 49 ? 26. 9 cents are how many less than 11 cents ? 12 cents ? At each lesson review combinations previously learned. Lead pupils to see that as 10 + 3 = 13, 9 + 3 must equal 1 less than 13, 9 + 4 must equal 1 less than li, and so on. 27. Find 9 and add 4. Add 4 to 19. To 29. To 69. 28. What is the sum of 89 and 4 ? 69 and 2 ? 39 and 3 ? 29. Add: 93 96 98 40 87 i^ 95 97 92 97 42 82 41 99 9L^ 92 88 -U 40 22 30. Point out 9 on tlie number table, and without counting show the number that is 4 more than 9. 31. Show 29 and the nundjer that is 4 more than 29. 32. Show 89 and the number that is 3 more than 89. 33. Name the number tliat is 2 more than 49. 34. Name quickly the number that is 4 more than 59. 35. Sliow 41 and the number that is 2 less than 41. 36. Show 53 and the number that is 4 less than 53. 37. Name the number that is 4 less than 13. Drill pupils on the combinations until the mind furnishes instanth^ the correct result. Then apply the combinations to the facts of childish experience. 38. There were 29 apples in a basket and 4 apples were put in. How many apples were in the basket then? 39. Lucy had 19 squares on her desk and her teacher gave lier 3 squares more. Hoav many squares had she then? ADDITION 55 40. William had 39 cents, and Alfred had 4 more than William. How many cents did Alfred have? How many dimes and how many cents over? 41. Add 5 to each of the numbers less than 100 that have 9 in the units' place, and learn the result. 42. Add: 98 92 33 94 99 54 47 93 31 51 43 96 50 50 94 92 54 95 43. What is the sum of 29 and 5 ? 69 and 5 ? 39 and 3 ? 44. Show 79 and the number that is 5 more than 79. 45. Show 39 and the number that is 5 more than 39. 46. Name the number that is 5 more than 49. 47. Name the number that is 4 more than 89. Than 59. 48. Mary may name a number that ends in 9, and the rest may think of a number that is 5 more than hers. What number is 3 more than hers? 2 more? 4 more? Make a general exercise of the work of Ex. 48. As new numbers are taken up use similar exercises. 49. Think of 74 and give the number that is 5 less than 74. 50. may give a number ending in 4, and the rest may give the number tliat is 5 less. 51. There were 29 people in a car, and 5 got on the car at the station. How many persons were in the car then? 52. There was a school of 39 children, and 4 new pupils were brought into it. How many pupils were in the school then ? 53. Add 6 to 9, and to each of the numbers less than 100 that end in 9. 54. Add: 96 94 45 97 94 98 54 61 44 97 53 Qd 91 62 35 61 93 95 92 50 56 ADDITION 55. Show 69 and the number that is 6 more than 69. 56. Show 49 and the number that is 6 more than 49 ; 4 more than 49 ; 2 more than 49 ; 5 more than 49 ; 3 more than 49. 57. Think of 79 and tell the number that is 6 more than 79 ; 3 more ; 5 more ; 2 more ; 4 more. See Ex. 48 and the note following. 58. What number is 6 more than 59? 2 more? 5 more? 3 more? 59. What number is 6 less than 75? 6 less than 95? 60. How many must be added to 49 to equal 55? 61. Six pupils of a school stayed at noon. The 39 other pupils went home to dinner. How manj pupils were there in all? 62. What is the difference between 9 and 14 ? Between 19 and 25 ? 29 and 33 ? 69 + ? = 74 ? 69 + ? = 72 ? 69 + ? = 75 ? 63. 33 cents are how many more than 29 cents ? 64. Arthur found 19 eggs in one nest and 6 eggs in another. How many did he find in all ? 65. Lizzie has 39 cents and needs 6 cents more to buy the doll she wants. What is the price of the doll ? 66. Draw a line on the board 3 feet long. It is a yard long. We can call it either 1 yard or 3 feet. Let the boys draw lines of varions lengths on the floor, marking off the foot units and the yard units. Use them in working out the fol- lowing, and refer to them whenever the children's imaginations fail to give the correct ideas of feet and yards. 67. How many feet in 3 yards ? 4 yards ? 2 yards ? 6 yards ? 5 yards ? 8 yards ? ADDITION 57 68. I low many yiirds in O feet? 12 feet? 9 feet? 15 feet ? 18 feet ? ^ 21 feet ? 27 feet ? 30 feet ? 69. t) feet + o feet = how many feet ? How many yards ? 70. 9 feet + 3 feet = how many feet ? How many yards ? 71. How many yards in 12 feet + 3 feet ? 15 feet + 3 feet? 72. How many feet in 1 yard + 1 foot ? 2 yards + 2 feet ? 73. How many feet in 4 yards + 1 foot ? 3 yards -f- 2 feet ? 74. 2 yards + 3 feet = how many yards ? 75. 7 feet = how many yards and how many feet ov^er ? 76. How many yards and feet in 11 feet? 13 feet? 8 feet ? 77. 9 feet -f 1 feet = liow many yards and feet ? 78. How mucli do 9 feet + 5 feet hick of benig 5 yards ? 79. (9 + 5)-2 = ? (19h-5)-2 = ? (9 + 3)-r-2 = ? 80. To each of the numbers less than 100 that have 9 in the units' phice add 7. 81. What is the sum of 19 and 7 ? 39 and 7? 89 and 7 ? 82. Add: 93 98 91 45 9G 97 93 91 7461729352627438 Tell how many hundreds, how many tens, and how many units in each answer. 83. Write 91 and 73 and find their sum. 98 + 40 = ? 84. Show 9 and the number that is 7 more than 9. 85. Show 89 and the number that is 7 more than 89. 58 ADDITION 86. Name the number that is 7 more than 49. 5 more ; 3 more ; 6 more ; 2 more ; 4 more. 87. What number is 7 less than 16 ? 7 less than 46 ? 7 less than 86 ? 5 less than 76 ? 4 less than 63 ? 88. What number must be added to 39 to equal 46 ? 89. There are 9 persons at home in Mr. Smith's family. When they have 7 visitors, how many persons are in the house ? 90. 9 pints + 7 pints = how many quarts ? 91. Thomas had 19 cents and his father gave him 7 cents. How many cents had he then ? 92. 49 men were working on a building when 7 other men Avere hired to help. How many workmen were there in all ? 93. Find each of the numbers that have 9 in the units' place and are less than 100, and add 8 to it. 94. 9 squares and 8 squares = how many squares ? 95. Add: 93 95 87 86 92 95 24 96 8684919374439181 96. Show 9 and the number that is 8 more than 9. 6 more than 9. 4 more than 9. 7 more than 9. 97. Think of 29. What number is 8 more ? 5 more ? 2 more ? 7 more ? 4 more ? 98. i^i^y Ht'ime ^ number ending in 9, and otliers may add to it numbers less than 9. 99. 19 books were on a shelf and 8 others were added. How many books were there then ? 100. 89 persons were in a meeting. When 8 other persons came in, how many were present ? 101. One coAV gave 9 quarts of milk and another gave 8 quarts. How much milk did l)oth cows give ? ADDITION 59 102. 9 ft. -f- 8 ft. = liow many yd. and ft. ? Contractions of denominations will be used hereafter interchange- ably with the complete words. 103. ]\lary picked 9 pt. of berries and Anna picked 7 pt. How many pt. did both pick ? How many qt. '! 104. Add 9 to 9, and to each of the nnmbers less than 100 that end in 9. 105. Add: 95 98 9^ 97 93 49 93 92 93 91 35 92 94 90 64 87 106. Show on the number table 29 and the number that is 9 more than 29. 107. Show the number that is 9 more than 49. 9 more than 79. 9 more than 39. 8 more than 59. 7 more than 69. 6 more than 79. 5 more than 89. 4 more than 29. 108. 29 cents 4- 9 cents = how many cents ? 109. 99 cents + 9 cents = how much more than a dollar? 110. Anna and Mary each picked 9 pt. of berries. How many qt. did both pick ? 111. How many, sums did you find in working Ex. 95 ? How many in working Ex. 105 ? How many in both ? BLACKBOARD EXERCISE. Practice rapid adding of each figure on the edge of the square to the one in the middle. Then change 9 to 19, 29, 39, etc. 5 2 7 8 9 3 6 4 9 50 ADDITION 112. John has 9 marl)les and James has 7 marbles. How many have both ? 113. Make story prol)lems in addition with 9 as one of yonr nnmbers. 114. Add: 72 96 93 92 45 93 68 92 95 64 35326584947291213291 115. How many hundreds, how many tens, and how many units in the number 287? In 307? 422? 330? 976 ? 116. Write a number that has 3 hundreds, 8 tens, and 7 units. 117. Write a number made of 2 hundreds, 6 units, and 3 tens. 118. Write a number having 4 in the units' phice, 6 in the tens' place, and 7 in the hundreds' place. 119. Write a number having 9 in the tens' place, 3 in the hundreds' place, and 4 in the units' place. It should be explained that of the 16 obtained 120. To 29 by adding 9 and 7, the 6 units should be written add 7 in the units' place, and the ten should be combined with the 2 tens. The child should be led to see that the 36 obtained in this way is the same result as that which he got at first by counting. 121. Add: 49 89 69 69 39 49 29 59 69 J7jr_7272727272716 69 27 326 235 144 63 349 129 27 49 469 749 629 29 542 315 While the class are taking the form work and fractions which fol- low, keep the addition work clear in their minds by short drills and by giving examples like the preceding for seat work. 122. Fold an inch square of pa2)er into 4 equal parts. ADDITION 61 123. When anything is divided into 4 equal parts, what is each part called ? 124. When anything is divided into 5 equal parts, what is each part called ? Ans. |. 125. Place triangles as in Fig. 1. How many triangles does it take to copy Fig. 1 ? • Fig. 1 126. Take away -J of your figure. How many fifths are left ? Put back the ^ and take away |. How many fifths are left ? 127. Build Fig. 2 with triangles Show i of Fig. 2. -I of it. I of it. -p^^ 2 128. How many triangles did you use in building Fig, 2 ? How many square inches would they equal ? 129. Build Fig. 3 with triangles. Which is larger, Fig. 2 or Fig. 3 ? 130. How many triangles did you use in mak- ing Fig. 3 ? How many square inches would they equal ? Fig. 3. 131. Can you take away one triangle from Fig. 3 so as to leave a large square ? 132. Show ^ of Fig. 3. Show f of it and tell how many fifths are left. Show | and what is left. 133. Put 5 triangles into a figure different from any in the book. Show |- of it ; |- of it. Let the pupils carry on the processes of building figures and part- ing, wholiiig, and parting them again as long as the exercise is inter- esting and instructive. If the quicker ones anticipate the "work to come and show sixths, seyentbs. and eighths, so much the bettei*. The 62 ADDITION presentations of the teacher and of the book must be in an orderly progress-ion, but children should be encouraged to make their own discovei'ies freely. 134. Draw a vertical line 5 inches lono*. Mark it off into inches. Show ^ of it. -|. 5* 2 5' 1 incli 135. Draw a horizontal line 10 inches long and show -J of it. Show | of it. Show |- of it. ShoAv I of it. 136. HoAV many inches long is a line that is |- of 10 inches long ? How long is -| of a 10- inch line ? I of a 10-inch line ? -| of a 10-inch line ? 137. 10 pupils may stand in a line. 1 5 of them at this end of the line may step forward. 2 inches Take places again. | at the other end of the line may step forward. | ul 10 = how many ? I of 10 = lit)W many ? 138. Turn to the number table and show ^ of 50. I of 50. 5 f of 50. I of 50. i of 100. 3 inches Fig. 4 4 inches 139. Place triangles so as to make Fig. 4. How many square inches ' does it equal? Hoay many triangles in Fig. 4 ? Separate it into halves. How many triangles in each half? Put the halves together again. 140. When anything is divided into 6 equal parts, what is each part called ? 141. Show i of Fio-. 4. I of it. Show | of 6 1 9 it. Whicli is greater, 1 or | ? ^ or # r A or 2 9 1 6 • 2 3 '/» G • 5 inches 1 _L 2 G 1^ 6 142 many Oths ? how many Oths? 1 + I = lu.w ADDITION 63 Fig. 5 Fig. 6 Fig. 7 143. Copy Figs. 5, 6, and 7 by placing triangles. 144. Can you take away two triangles from Fig. 5 so as to leave 2 square inches ? 145. Can you take away two triangles from Fig. 6 so as to leave 2 square inches ? Notice the triangles whose square corners are at the center of the hgure. 146. Can you take away two triangles from Fig. 7 so as to leave a large square ? 147. Can you take away two triangles from Fig. 7 so as to leave 2 square inches ? 148. From Fig. 7 take away J, and show how many sixths are left. 149. How many sixths are left if you take away | ? | ? 150. I — -1 = how many sixths ? | — | — how many sixths ? f — f = how many sixths ? | — f = how many sixths ? 151. Place triangles as in Fig-. 8. Can you take away four triangles and leave 1 square inch ? 152. Separate Fig. 8 into halves by taking away the lower row of triangles. How many sixths in each half ? 153. Place trianafles as in Fig-. 9. Separate the figure into three equal figures of the same shape. How many sixths in each third? Fis. 9 Fig. 8 64 ADDITION Fig. 10 154. Place triangles as in Fig. 10. Separate the figure into halves. ^ = i'- 155. Place six triangles so as to make a different figure from any in the book, and divide it into halves. 156. Make a figure of six squares. Show 1 of it. Show I of it. Show I or ^ of it. Show | of it. 157. Draw a vertical line 6 inches long. Show 1 of it. Show I of it. Sliow I of it. What does | of it equal? 158. Figure 11 is a rectangle. How many sides has it? How many corners ? What kind of corners has it ? Let pupils find rectangular surfaces, as window panes, blackboards, etc. 159. Draw a rectangle (3 inches long and 1 inch wide, and divide it into 6ths by vertical lines. 160. Draw a rectangle whose horizontal lines are each 5 inches long and vertical lines 1 inch long, and divide it into fifths. 161. Draw a rectangle 3 inches long and 2 inches wide, and divide it into 6ths. 162. Draw a picture of a pie cut into sixths. 163. Turn to the number table and show 1 of 60. | of 60. I of 60. I of 60. 164. Add: 226 433 659 319 529 124 Fig. 11 649 377 32 519 739 249 429 535 349 6:! 8 165. Add : 437 How many hundreds, liow many tens, 641 and how many units in the answer ? 166. 10 hundreds make 1 thousand. Which place do the thousands have ? ADDITION 65 167. How many thousands, how many hundreds, how many tens, and how many units in the number 7654 ? 4326? 6304? 3829? 5340? 2002? 168. Write a number made of 5 units, 2 hundreds, 7 thousands, and 8 tens. 169. Write and read a number. which has 5 in the fourth place, 8 in the third place, 3 in the second place, and 1 in the first place. 170. Write a number of 4 places and tell what is in each place. 171. Write 4351 under 5437 and add them. Why is it best to put units under units, tens under tens, hundreds under hundreds, and thousands under thousands ? 172. Add: 3219 8647 7935 6396 3456 1639 1234 1239 1249 1272 1939 1924 173. Add 3 to 8, and add 3 to each of the numbers less than 100 that end in 8. Show that in adding 3 to 8, 2 completes the first ten, and the re- maining 1 makes 11. 174. Find the sum of 18 and 3. 48 and 3. 78 and 3. 175. Show on the number table the number that is 3 more than 8. 3 more than 28. 3 more than 88. 3 more than 58. 3 more than 38. 176. 8 ft. + 3 ft. = how many yd. and ft.? 177. 18 ft. + 3 ft. = how many yd.? 178. There were 38 sheep in a flock and 3 sheep were added to it. How many were there then ? 179. John worked 28 examples on Monday, and on Tuesday he worked 3 more than on Monday. How many did he work on Tuesday ? HORN. ARITH. 5 66 ADDITION « 180. Add: 325 1928 2878 2463 349 284 813 819 633 913 928 815 631 358 181. Find the sum of 1328 and 2843. 4928 and 2953. 182. Find the sum of 80 and 30. 90 and 30. 90 and 50. 183. Find the sum of 8 and 4. 28 and 4. 38 and 4. 68 and 4. 88 and 4. 18 and 4. 58 and 4. See suggestion after Ex. 173. 184. Show on the number table the number that is 4 more than 78. 4 more than 48. 4 more than 58. 185. Add: 827 869 1238 1835 1598 1318 823 ) 151 M^ 1^ 1429 4123 4824 448 186. Mary found 38 peaches under one tree and 4 peaches under another. How many peaches did she find in all ? 187. James went to the grocery to buy sugar. In bringing it home he spilled 4 pounds. He brought home 8 pounds. How much did he buy ? 188. There were 28 pounds of butter in a jar and 4 pounds more were put into the jar. How many were in it then ? 189. Find the sum of 80 and 40. 90 and 40. 90 and 70. 190. 8 ft. 4- 4 ft. = how many yd.? 191. 8 pt. + 4 pt. = how many qt.? 192. How many quarts make a gallon ? Let the children use quart and gallon measures and find out the fact for themselves. 193. How many quarts make 2 gallons ? 3 gallons ? 4 gallons ? 5 gallons ? 194. How many quarts in 2 gallons and 1 quart ? 3 gal. and 2 qt.? ADDITION 67 195. There were 2 gal. and 3 qt. of molasses in a jug, and Mary used a quart of it to make candy. How many qt. were left ? 196. How many gal. in 8 qt.? 12 qt.? 20 qt.? 16 qt.? 197. How many gal. and how many qt. over in 9 qt.? 11 qt.? 13 qt.? 15 qt.? 198. How many gal. in 8 qt. + 4 qt.? 9 qt. -|- 7 qt.? 199. 9 qt. less 1 qt. = how many gal.? 200. Add 5 to each of the numbers less than 100 ^vliose unit figure is 8. 201. Find the sums : 873 729 8386 628 989 8359 648 933 445 524 943 5353 843 250 1680 289 488 884 202. 8 ft. H- 5 ft. = how many yd. and ft. ? 203. 58 lemons were in a box. If 5 more lemons were placed in the box, how mau}^ would it contain? 204. Mr. Smith set out 5 new trees in his orchard, which already had 88 trees. How many trees were there then? 205. A book cost 28 cents and a tablet cost 5 cents. How much did they both cost? 206. Find the sum of 80 and 50. 80 + 30. 90 + 90. 207. 8 qt. + 5 qt. = liow many gal. and qt. ? 208. 2 gal. + 4 qt. = how many qt. ? How many gal. ? 209. If you drank a pint of milk every day, how many pints would you drink in 2 weeks? How many quarts? 210. Add 6 to some numbers that end in 8, and tell how many tens and how many units in each answer. 211. If you picked 8 apples from a tree and should pick off 6 more, how many would you have? 68 ADDITION 212. When John worked 18 problems in the morning and 6 in the afternoon, how many did he work in the whole day? r. 213. 8 ft. + 6 ft. = how much less than 5 yd. ? 214. Find the sums : 846 3858 4888 3884 4898 8888 3618 698 5596 4456 5659 9586 3456 4845 How many thousands, how many hundreds, how many tens, and how many units in each answer? 215. Write and read a number which means 1 thou- sand, 3 hundreds, 5 tens, 7 units. 216. Write a number that means 2 thousands, 5 hun- dreds, tens, 4 units. 217. Write a number that is made of 11 thousands, 5 hundreds, 3 tens, units. 218. Put 4 tens, 5 hundreds, 3 units into one number. 219. Put 6 units, 7 hundreds, 8 tens into one number. 220. Put 2 thousands, 1 unit, 6 tens, 4 hundreds into one number. 221. Put into one number 12 thousands, 2 units, 4 hundreds, 3 tens. 222. Put into one number 25 thousands, 4 tens, 1 hun- dred, 6 units. 223. 8 qt. + 6 qt. = how many gal. and qt. over? 224. Add 7 to several numbers whose unit figure is 8. 225. Harriet is 8 years old and Lucy is 7 years older. How old is Lucy? 226. A carriage cost 88 dollars, and a horse cost 7 dol- lars more than the carriage. How nuich did the horse cost ? ADDITION 69 227. 8 ft. and 7 ft. = how many yd. ? 228. Make prublenis using 7 and numbers that end in 8. 229. Add : 5898 3968 2897 5848 8989 25868 9697 1497 3958 3697 4069 48036 230. What is the sum of 80 and 70? 80 and 60? 80 and 30 ? 80 and 40 ? 231. 8 qt. + 7 qt. = how many qt. ? How much less than 4 gah? 232. Add 8 to some numbers whose unit iigure is 8. 233. Anna has 2 dolls each of which cost 8 cents. How much did they both cost ? 234. Mr. Smith has 2 horses each of which cost 80 dollars. How much did they both cost ? 235. 8 qt. + 8 qt. = how many gal.? 236. 8 ft. + 8 ft. = how many yd. and ft.? 237. Find the sums : 867 2183 8482 8623 " 8623 6683 3468 9817 832 9434 8351 8351 7986 8275 8522 6528 238. John drew a line 5 ft. long. How much over a yard was it in length ? 239. Samuel drew a line 8 ft. long and William drew one 3 yd. long. How much longer was William's line than Samuel's ? 240. Draw on the board a square 1 ft. long and 1 ft. wide. If you had a string long enough to lie all around on the line that bounds the square, how many ft. long would the string be ? How many yd. and how many ft. over ? 70 ADDITION 241. A rectangle is 2 ft. long and 1 ft. wide. Draw a small picture of the rectangle, and find how many ft. around it. How many yards ? See that the proportions of the " picture " are correct. 242. Draw a picture of a rectangle 3 ft. long and 1 ft. wide, and show how many ft. in the distance around it. 243. Draw a picture of a rectangle whose horizontal lines are 3 ft. long and whose vertical lines are 2 ft. long, and tell how far it is around it. 244. Add 9 to each number less than 101 whose unit figure is 8. 245. Show in what direction from 28 in the number table is the number that is 9 more than 28 ; 9 more than 38 ; 9 more than 48. 246. How much will a span of horses cost if one horse costs 80 dollars and the other 90 dollars ? 247. 8 qt. 4- 9 qt. = how many qt. more than 4 gal.? 248. 28 cents + 9 cents = how many dimes and cents ? 249. 8 ft. + 9 ft. = how many ft. less than 6 yd.? 250. Add: 2478 2348 8124 9825 8483 1628 8619 3798 7318 8938 9329 849 BLACKBOARD EXERCISE 6 9 4 3 8 7 5 8 2 Let each of the numbers on the edge of the square be added to the number in the middle. Then use 18, 28, etc., instead of 8. If the children cannot give any combination at sight, review it. ADDITION 71 251. Find the largest even number that is less than 29 and add 7 to it. 252. Add 8 to the largest even number that is less than 49. 253. Five boys have how many toes ? Nine boys have how many toes? 254. Give all the multiples of 10 tliat are less than 100. 255. Find the number that is 1 less than the third multiple of 10. Add 6 to it. Add 4 to it. Add 7 to it. Add 9 to it. Add 5 to it. 256. Find the number that is 2 less than the fifth mul- ti[)le of 10 and add 9 to it. Add 3 to it. Add 7 to it. Add 4 to it. Add 8 to it. 257. 10 twos = ? Find tlie number that is 1 less than 10 twos and add 4 to it. Add 7 to it. Add 5 to it. Add 8 to it. 258. Draw a rectangle 5 in. long and 4 in. wide, and tell how far it is around it. 259. The line around a figure is called its Perimeter. Draw a rectangle 4 in. long and 3 in. wide, and tell how long its perimeter is. 260. How long is the j)erimeter of a rectangle which is 3 in. lono- and 3 in. wide ? Draw. Recall fractious previously learned. 261. Copy Fig. 12 by placing triangles. How many triangles in Fig. 12 ? 262. Show -^ of the figure. Show f of the figure. Show ^ of it. How many ■^ 2. _^ |. _ \lQ^^Y many sevenths ? 9 remain r |- + I = how many sevenths ? Fig. 12 72 ADDITION 263. Copy Fig. 13. How many tri- angles in the figure ? 264. Take away ^ of the figure and show how many sevenths remain. Take away |- and show how many seventlis remain. Take away ^. _? 6_4_1 4_2_1 2_2_?_ 7" 7 T ~" 7 1 Y ~ 7 7 7 "" 7 266. Place 7 triangles so as to make a figure different from those in the book, and show sevenths of it. 267. Draw a rectangle 7 in. long and 1 in. wide, and find the length of its perimeter. 268. 7 + 4 = ? Add 4 to each of the numbers in the number table whose unit figure is 7. 269. 17 cents + 4 cents = how many dimes and cents ? 270. 70 cents + 40 cents = how many cents ? How many dollars and dimes ? 271. 7 qt. + 4qt. = how many gal. and qt. ? 272. John has 47 dollars and needs 4 dollars more to buy a bicycle. What is the price of the bicycle ? 273. Add : 7425 7378 17849 8795 2135 8123 4539 1839 29847 9249 1719 6427 274. Add 5 to each of the numbers in the number table whose unit figure is 7. 275. 7 cents + 5 cents = how many dimes and cents ? 276. 70 cents + 50 cents = how many dollars and cents ? 277. 7 pt. -f 5 pt. = how many qt. ? 7 ft. + 5 ft. = how many yd.? 278. A line 17 in. long was lengthened 5 in. How many ft. and in. long was it then ? ADDITION 73 279. 5 qt. of milk were poured into a can that already held 27 qt. of milk. How many gal. were in the can then ? 280. Add : 7642 1729 8757 9562 8967 3724 5251 8449 5435 8737 7724 8573 281. Add 6 to each number smaller than 108 whose unit figure is 7. 282. 60 + 70 = ? 90 + 40 = ? 80 + e50 = ? 80 + 70 = ? 283. Find the sum of 147 and 6. 327 and 6. 437 and 6. 284. John had 107 cents and gained 6 cents. How many dollars, dimes, and cents had he then ? 285. How many dollars, dimes, and cents in the sum of 207 cents + 6 cents ? 286. 407 cents + 6 cents = ? 217 cents + 6 cents = ? 287. 427 cents + 6 cents = ? 967 cents + 6 cents = ? 288. Add: 3754 325 9626 8768 9387 272 8927 947 947 1207 8904 9986 289. Add 7 to each number smaller than 108 whose unit figure is 7. 290. 70 + 70 = ? 80 + 80 = ? 90 + 90 = ? 291. Find the sum of 187 and 7. 157 and 7. 277 and 7. How many dollars, dimes, and cents in the sum of : 292. 314 cents + 7 cents ? 295. 227 cents + 4 cents ? 293. 287 cents + 7 cents ? 296. 357 cents + 6 cents ? 294. 537 cents + 7 cents ? 297. 947 cents + 7 cents ? 298. Name the days in the week. How many days in 2 weeks ? 3 weeks ? 74 ADDITION 299. Mary made a visit of 17 days. Then her mother allowed her to stay a week longer. How many days in all did she stay ? 300. 27 days + 1 week = hoAV many days ? 301. A line 7 ft. long was lengthened 7 ft. ITow nnich did it lack then of being 5 yd. long ? 302. How wide is a square that is 7 in. long? Draw a square 7 in. long and hnd the length of its perimeter- 303. Mr. Smith is 27 years old, and j\lr. Brown is 7 years older. How old is Mr. Brown ? 304. Add: 2736 7826 4727 7268 50627 9773 9729 7273 9737 7981 7197 6576 305. Add 8 to each number less than 118 ^vhose unit ligure is 7. 306. 70 + 80 = ? 70 + 50 = ? 70 + 60 = ? 307. Find tlie sum of 137 and 8, 267 + 8. 967 + 8. 308. 7 ft. + 8 ft. = how many yd. ? How many dollars, dimes, and cents in the sum of : 309. 547 cents + 8 cents ? 312. 187 cents + 4 cents ? 310. 627 cents + 8 cents ? 313. 189 cents + 7 cents ? 311. 917 cents + 8 cents ? 314. 218 cents + 7 cents ? 315. It is the custom to write 364 cents *^3.64 and to call it 3 dollars and 64 cents. Read: 84.84 ; $9.87. 316. Add:. $7.47 $8.27 $25.37 $48.47 $18.57 2.38 1.48 97.28 23.27 25.36 317. The point Avhich separates dollars and cents is called a Decimal Point. In writing columns of dollars and cents to be added or subtracted, why is it best to put the points in a vertical line ? 318. Add 9 to each number that is less than 118 and lias 7 in the units' place. ADDITION 75 319. What is the sum of 12T and 9 ? 23T + ? 487 + 9 ? 320. AVhat is the sum of TO and 90 ? TO and 40 ? 321. '^ 1.2T + 9 cents = ? -^ 2.3T + 9 cents = ? 322. T (|l. + 9 qt. = how many gal.? 323. T ft. + 9 ft. = how many yd. and ft.? 324. T pt. -f 9 pt. = how many qt.? How many gal.? 325. Add: .it54.5T -^BG.BT ^lo.TT i36.5T !i?18.9T 9.29 19.3T 98.2T 23.3T 18.99 Practice sight addition of tlie num- bers on the edge of the square to the number in the middle, and then sub- stitute for 7 other numbers whose unit figure is 7. 326. What time is it when the hour hand of the clock is on Y ? 327. V stands for 5 in Roman notation. On what page does the 5th chapter of this book begin? Show that two Vs placed as follows ^ make X or 10. 328. I written after V means I added to V, or 6. What does YII mean? VIII? Find the heading of the 6th chapter in this book. The 8th. The Tth. Explain that when the smaller numeral is written after the greater their sum is to be found. 329. Read XX. XVI. XVII. XVIII. XXV. XXVI. XXVII. XXVIII. XXX. XXXV. XXXVII. 330. AVrite in Roman notation 15, IT, 23, 25, 28, 30. 7(3 ADDITION 331. Write in Roman notation the first even number in the second ten. 332. Write in Roman notation the nnmber that is 7 more than the 3d multiple of 10. 333. Add 5 to several numbers whose unit figure is 6. 334. Find the sum of 186 + 5. 210 + 5. 296 + 5. 335. Add: 111.26 -i^7.36 $59.36 -^18.76 134.55 182.75 13.45 5.45 6.52 97.85 97.46 34.86 336. 6 qt. + 5 qt. = how many gal. and qt. ? 337. Write in Roman numbers the answers to the fol- lowing : Q + F^ = ? 26 + 5=? 36+5 = ? 338. 46 pupils were in a school and 5 new pupils en- tered. How many puj^ils were in the school then ? 339. Add: 8653 84676 65745 36235 50685 60606 7527 79515 96186 75486 64506 39545 How many thousands are there in each answer ? 340. Write a number that means 25 thousands, 3 hun- dreds, 5 tens, 7 units. Write numbers made of : 341. 75 thousands, 7 hundreds, tens, 4 units. 342. 131 thousands, 2 hundreds, 7 tens, 6 units. 343. 475 thousands, 3 hundreds, 8 tens, units. 344. 187 thousands, hundreds, 3 tens, 4 units. 345. Add 6 to each of ten numbers whose unit figure is 6. 346. Add: 8466 19267 83646 96356 72685 96564 7846 56762 47625 64925 66606 86165 347. 6 ft. + 6 ft. = how many yd. ? ADDITION rr 348. 6 qt. 4- 6 qt. = liow many gal. ? 349. G pt. + G pt. = how many qt. ? How many gal. ? 350. 326 + G = ? 586 + 6 = ? 916 + 6 = ? 471 + = ? 351. A man paid 8 60 for a horse and twice as much for his carriage. How much did the carriage cost? 352. Add: $13.76 824.96 818.06 828.76 8128.16 2.86 3.36 1.05 17.06 25.06 353. Copy Fig. 14 by placing inch- squares. Find the length of the perim- eter of the figure you have made. If each inch line were a foot line, how many yards long would the perimeter be? Fig. 14 354. Copy Fig. 15 by placing squares. Find the length of the perimeter. How many yards long would the perimeter be if each of the inch lines were changed to a foot line ? 355. Add 7 to each of ten num- bers that have 6 for the unit figure. 356. Find the sum of 60 and. Fig. 15 70. 60 + 50. 436 + 7. 596 + 1 357. Add: 8252.75 8187.36 8432.66 964.27 187.17 719.57 9 t 8314.86 938.97 358. 6 qt. + 7 qt. = liow many gal. and qt.? 359. 6 ft. + 7 ft. = how many yd. and ft.? 360. Write in Roman notation the sum of 15 and 7. T8 ADDITION 361. Find in the 4th ten the even nnmber that ends in G and add 7 to it. 362. In the 5th ten lind the even nnmber that ends in 8 and add 7. ^ 363. How many days in 5C) days + 1 week? 364. Add 8 to each nnmber ending in 6 and less than 108. 365. Write 22 dollars and 16 cents. Under that write 31 dollars and 18 cents. Add them. 13.46 + -^8.28 = ? 366. Write 48 dollars and 36 cents, and add to it 27 dollars and 38 cents. $5.08 + 'f 1.96 = ? 367. To 125 dollars and 58 cents add 134 dollars and IS cents. 368. 16 c[t. of milk and 8 qt. of milk = how many gal. of milk ? 369. 6 ft. + 8 ft. eqnal how mnch less than 5 yd. ? 370. Add: f 1375.58 18934.68 $2456.76 $3467.86 8277.16 7927.26 7778.18 8881.38 371. Add 9 to each number ending in 6 that is less than 109. 372. 60 + 90 = ? 60 + 80 = ? 60 + 50 = ? 60 + 70=? 373. 6 qt. + 9 qt. = how many qt. ? How many qt. must be added to eqnal 4 gal.? 374. 16 qt. + 9 qt. = how many gal. and qt.? 375. If you measure off 9 in. on a tape measure and then measure off 6 in. more, how many ft. and in. will you have measured off ? Illustrate if necessary. 376. Add: $366.66 $276.76 $346.86 $868.76 839.19 988.19 997.09 666.09 ADDITION 7om 11 take 5 9 3 7 4 8 tract quickly each of the numbers 5 11 9 on the edge of the square from 11. Then change 11 is 8 4 to 21, 31, etc. 28. A line lacks 1 inch of being 1 foot long. If 9 inches of it were rubbed out, how long would it be then ? 29. 21 qt. — 5 qt. = how many gal.? 30. 21 pk. — 9 pk. = how many bu.? 31. Write in Roman notation the number which is the difference between 21 and 4 ; the difference between 61 and 4. 32. Copy Fig. 1 by placing triangles made by cutting inch- squares in two. Show J of the figure. Show ^ of it. How many eighths in | ? 33. Divide the figure into 4 equal parts. How many eighths Fig. 1 inf? Ill i ? Ill i ? SUBTRACTION 87 Fig. 3 Show I of it. 34. Copy Fig. 1 by drawing. 35. Copy Fig. 2 by placing an inch- square and triangles made by bisecting an inch-square. Show ^ of the figure. ShoAv ^ of it. How many fourtlis equal ^ ? 36. Copy Fig. 2 by drawing. 37. Copy Fig. 3 by placing triangles. Show J of the figure. Show ^ of it. How many sixths in -J? Sliow J of it. How many sixths in J ? Show ^ of it. How many sixths in | ? 38. Copy Fig. 3 by drawing. 39. Show on the number table J of 20 ; | of 40 ; J of 60 ; 1 of 80 ; J of 100. 40. Show 1 of 30 ; f of 30 ; i of 60 ; | of 60. Pig. 3 J of 90 = ? f of 90 = ? 1 of 40 = ? f of 40 = ? 41. From numbers ending in 2 subtract 3. From the same numbers subtract 9. 42. Find differences : 82 92 72 182 13 43 29 79 562 4262 1319 8232 2329 7212 2449 43. When butter is 22 cents a pound and lard is 9 cents a pound, the price of a pound of butter is how much more than the price of a pound of lard ? 44. From numbers ending in 2 subtract 4. From the same numbers subtract 8. 45. Find differences : 272 9262 6224 8172 9182 8272 9262 134 7814 2354 2518 6814 2714 5438 88 SUBTRACTION 46. Eva spent 22 cents in one day. She spent 8 cents before dinner. How much did she spend after dinner ? 47. From numbers ending in 2 subtract 5. From the same numbers subtract 7. 48. Find differences : 131.72 1818.23 1712.92 1921.92 $681.82 1492.62 6.15 391.51 87.65 238.17 126.15 385.17 49. Henry bought a sled for $.72 and traded it for another sled and a nickel. How much was the other sled worth ? 50. From numbers ending in 2 subtract 6. 51. Find differences : 12352 81292 92342 82322 98292 22222 6146 2536 24128 37156 63526 17516 From 12 take . . 52. The city where Alfred lives is 22 miles from Bos- ton. When he has ridden 6 miles towards Boston, how far is he from it ? BOARD WORK. Practice subtracting rapidly each number on the edge of the circle from the number at the center. Then replace 32 with 52, 72, 92, etc. 53. 22 qt. — 4 qt. = how many gal. ? 54. Write in Roman numbers the difference between 22 and 7. 55. Mary bought a dozen eggs and broke 5 eggs carry- ing them home. How many were left ? 56. One dozen minus one half a dozen = how many? SUBTRACTION 89 57. If 9 eggs were taken from a nest where a dozen eggs were found, liow many would remain? 58. The number from which another number is sub- tracted is called the Minuend. Make problems, using 12 as a minuend. 59. Use 13 as a minuend, subtracting 4. Subtract 4 from eisfht other numbers that end in 3. From the same numbers subtract 9. 60. Read minuends and find differences : 8373 2383 9323 4393 6373 33333 7533 3439 924 5439 2489 2429 21919 3214 61. 13 qt. of berries — 9 qt. of berries = how many gal. ? 62. From several numbers ending in 3 subtract 5. From the same numbers subtract 8. 63. Find differences : 7343 4283 6293 8083 9639 33333 8343 985 948 1745 4728 6255 18175 2519 64. (13-5)^2 = ? (23-5)-^2 = ? (12-6)-3 = ? (22-7)^3 = ? (13-5) -4 = ? (33-6) +4 = ? 65. If a piece of ribbon 5 ft. long is cut from a piece 4 yards and 1 foot long, how much ribbon is left ? 66. Write in Roman notation the difference between 73 and 5 ; between 33 and 5. 67. From numbers endinsr in 3 subtract 6. From the same numbers subtract 7. 68. 823 - 456 = ? 733 - 276 = ? 1039 - 462 = ? 69. 23 pk. — 7 pk. = hoAv many bu.? See board work used with numbers 11 and 12. 70. Write in Roman numbers tlie difference between 7 and 33. 90 SUBTRACTION 71. 23 qt. — 7 qt. = how many gal.? 72. 33 pk. — 9 pk. = bow many bu.? 73. 23 ft. — 5 ft. = how many yd.? 74. John had 13 cents and spent 8 of them. How many were left ? 75. 13 Readers belong to the library and 6 are in use. How many are on the shelves ? 76. Make problems in which j^ou use 13 as a minuend. 77. Read XV, XIII, C, CVI, CX, CXIII, CXXII. 78. IV means 4. Read XIV, XXIV, XXXIV, LXIV. Show that when the smaller Roman numeral is written before the larger, their difference is expressed. 79. What does IX mean ? How can you tell ? Read XIX, XXIX, LIX, LXXIX, XXXIX. 80. Write in Roman numbers, 59, 89, 39, 14, 64, 74, 24. 81. From 14, and from some other numbers whose unit figure is 4, subtract 5. Subtract 9 from the same num- bers. 82. Find differences : 9484 8474 7494 2094 6434 7494 8484 6935 3529 _935 1829 1875 2965 2569 83. Of 14 horses hauling loads, 5 were white. Hoav many of them were not white liorses ? 84. Write in Roman notation the number that is the difference between 14 and 5 ; between 24 and 9. 85. 2 weeks — 5 days = how many days ? 86. From several numbers Avhose unit figure is 4 sub- tract 6. From the same numbers subtract 8. 87. Find diiferences : 2494 3484 7474 9484 11464 14846 9234 1866 1628 2628 7866 628 8265 4968 SUBTRACTION 91 88. 8 yd. — 6 ft. = how many ft.? How many yd.? 89. 6 gal. — 6 qt. = how many qt.? 90. Write the difference between XXXIV and VIII. 91. From several numbers whose unit figure is 4 take 7. 92. Find differences : 324 674 924 1246 6549 846 7434 6434 3444 207 538 619 Jl62 2275 193 2918 2726 1968 93. Edwin had 24 marbles and lost all but 7. How many did he lose ? Use board work as with numbers 11 and 12. 94. Write in Roman numbers the number that is the difference between 14 and 8 ; between 24 and 8 ; between 34 and 6. 95. A hen sat upon 1 dozen and 2 eggs. 5 eggs failed to hatch. How many chickens came out ? 96. If a room is 24 ft. long and 9 ft. wide, how many ft. greater is its length than its width ? How many yd.? 97. If you measure off a line 14 feet long on the floor, and another line 8 feet long, what is the difference be- tween them in feet ? In yards ? 98. Make problems using 14 as a minuend. 99. Read X, I, V, L, C, IV, IX, XIX, XXIX. 100. X written before L means 40. Can you tell why ? Read XL, XLI, XLII, XLIII, XLIX, CL, CXL. 101. Write in Roman notation the difference between 40 and 9 ; 4 and 40. 102. From 15, and other numbers whose luiit figure is 5, subtract 6. From the same numbers subtract 9. 92 SUBTRACTION 103. Find differences : 975 T595 6935 8357 7358 4595 256 4976 2486 2994 2466 1936 104. Joseph had Q'^ feet of kite string, and his mother used 6 feet of it to tie up a bundle. How many feet of string had he left ? 105. Express in Roman numbers the difference between 15 and 6. 106. From 15, and other numbers ending in 5, take 7. From the same numbers take 8. 107. Find differences : 175 735 8595 1125 3525 6352 8958 7257 9859 128 219 867 1016 2778 2161 6274 2164 3694 108. 15 qt. — 7 qt. = how many gal.? 109. 25 pk. — 2 bu. = how many pk.? 110. In Roman numbers write the difference between 7 and 35 ; between 45 and 8. 111. If I saw off 8 in. from a board that is 1 ft. and 3 in. long, how many in. are left ? 112. 355 is a minuend and 127 the number to be sub- tracted. What is the difference ? 113. A number which is subtracted from another num- ber is called a Subtrahend. Make a problem with 9 as a subtrahend. 114. Make problems in wliich you use 15 as a minuend, and some number less than 10 for a subtrahend. 115. Subtract 7 from 16, and from other numbers end- ing in 6. From the same numbers subtract 9. 116. Find differences : 13.76 15.76 128.26 19.46 118.68 19.36 Subtrahend 2.37 2.49 9.87 2.59 13.75 4.98 SUBTRACTION 93 117. Write in Roman notation the difference between 56 and 9. Between 66 and 7. 118. Anna is 16 years old and Mary is 9 years old. How much older is Anna than Mary ? 119. William is 16 years old and Thomas is 9 years younger. How old is Thomas ? 120. I have a string that is 1 ft. and 4 in. long. If I break off a piece 7 in. long, how much will remain ? 121. Anna expected to spend 26 days in visiting a friend, but was called home a week sooner than she expected. How long did she stay ? 122. HoAV many ounces make a pound ? Let the children weigh out sand, sawdust, coal, or some other sub- stance until they realize the meaning of the terms "pound" and " ounce." 123. How many ounces in 1 pound lacking 7 ounces ? 1 pound lacking 9 ounces ? 124. At 10 cents a pound, how many pounds of candy can be bought for 50 cents ? 125. J a pound of sand weighs how many ounces ? |^ a pound of sugar Aveighs how many ounces ? 126. Use 8 as a subtrahend with each number less than 100 whose unit figure is 6, 127. Find differences : 116.16 126.36 i278.36 8376.46 1236.56 1376.86 8.08 8.18 54.28 18.28 98.38 98.78 128. Make problems using 16 as a minuend, and a smaller number as a subtrahend. 129. Use 9 as a subtrahend with each of the numbers less than 120 whose unit figure is 7. 94 SUBTRACTION 130. What number added to 39 makes 47? What number added to 339 makes 347 ? 131. Find differences : 1377.57 1627.87 1547.27 12275.87 1648.77 88.29 38.39 68.49 999.38 294.89 132. Anna had a dime and 7 cents and bought an 8-cent doll. How much had she left ? 133. A flower bed is 27 feet long, and John has weeded 9 feet of it. How much remains to be weeded ? 134. Make problems using 17 as a minuend. 135. XC means 90. Can you tell why ? 136. Read XCI, XCV, XCIV, XCIX, XCVI, XCIH, XCVIII. 137. In Roman numbers write : All tlie numbers that end in 9 up to 99. All the num- bers that end in 4 and are less than 100. Your age. All the even numbers in the first two tens. 138. Subtract 9 from several numbers ending in 8. 139. Minuends 727 113.47 167.27 167.75 $98.75 Subtrahends 259 9.18 29.18 38.69 26.97 Differences ? CHAPTER VI APPLICATIONS OF ADDITION AND SUBTRACTION Industrial Problems, Days in Months, Odd Numbers 1. Add 729 to itself. Add 1348 to itself. 2. Find the sum of 648 and the number tliat is 1 S^ieater than 648. 3. Find the sum of 276 and the number that is 1 less tlian 276. 4. Add 7 times 2 to 45. (2 x 8) + (10 x 7)= ? 5. Add 6 times 2 to the 6th multiple of 10. 6. What must be added to 9 to equal 12? 17? 15? 7. 26-? = 13. 64-? = 58. 35-? = 27. 26-? = 17. 8. Copy Fig. 1 by drawing 3 inch-squares and bisecting them. 9. Show 1 of the figure. Show | of it. Show -^ of it. How many 6ths does ^ e(|ual ? Show ^ of it. How many 6ths in J ? Fig. 1 10. 11. 12. 13. 14. 6. _ 1 _ ? 6 6 ~" 6 5 _ 1 _ ? "6 2 "~ 6' 5. _ i — 1 2. 6 3 ~" 6* 3 1 — 1 6 " 6* Show on the number table ^ of 50. -| of 50. |- of 50. Show 1 of 60. I of 60. I of 40. f of 40. Add I of 50 to 50. To 25. To 30. To 20. Add 1 of 60 to 60. To 30. To 40. To 25. 15. Subtract i of 30 from 30. J of 30 from 75. 95 96 APPLICATIONS OF ADDITION AND SUBTRACTION 16. Subtract i of 40 from 40. ^ of 40 from 60. 17. Find the difference between 464 and 820. Which number is the minuend ? 18. Find the difference between 398 and 785. Which is the greater number, the minuend or subtrahend ? 19. Find the difference between 4246 and 3278. Where is the minuend written in subtraction ? 20. 484 is how many more than 376 ? Which number is the subtrahend ? 21. 324 is how many less than 486 ? 22. How many children are there in a ward school which has 139 children in the first grade, 137 in the 2d, 747 in the 3d, 128 in the 4th, 98 in the 5th, 83 in the 6th, 77 in the 7th, and 48 in the 8th? 23. 158 children were in the first grade of a school, and 43 were transferred to another building. How many re- mained ? 24. There were 676 children in a school building when 183 others were transferred to it. How many were there then ? 25. There were 392 books in the school library, and 219 new ones were added. How many were in the library then ? 26. 18,943 bushels of coal were dug from a mine in one week and 29,312 the next Aveek. How many in the two weeks ? Take up the subject of coal mining, showing coal and pictures of mines, and reading or telUng stories about mines and miners. Then let the children give problems about them. In the same way deal with the different industries referred to in the following problems, letting the children furnish facts when they can about industries of which they have some knowledge. APPLICAIIOXS OF ADDiriON AND SUBTRACTION 97 27. 18,946 bushels of coal were dug from a mine in one week, 29,321 the next week, and 31,457 the next week. How many were dug out in the three weeks ? 28. 7281 cattle were on a cattle ranch and 943 were killed. How many were left ? 29. An iron foundry made 875 stoves in one Aveek, 873 in another week, and 884 in another week. Hoav many in all ? 30. A cotton mill wove 10,87(3 yd. of cloth in one week, 9343 in the next week, and 11,833 in the next week. How many in the three weeks ? 31. A farmer raised 2343 bu. of corn in one year, 3124 in another year, 1957 in another year, and 2417 in another year. How many did he raise in those four years ? 32. One farmer raised 1247 bu. of wdieat, another far- mer raised 3268 bu., and another farmer raised 5324 bu. How many bushels of wheat did they all raise ? 33. A lawyer earned $ 5727 in one year, $ 2938 in the next year, and 'ff 11,536 in the third year. How many dollars did he earn in the three years? 34. Gold worth 12,342 dollars was taken from a gold mine in one month, 98,676 dollars' worth in the next month, and 321 dollars' worth in the next month. How many dollars were taken out in the three months ? 35. A farmer sent to market one year 1224 pounds of butter, 1376 pounds the next year. 1312 pounds the next year, and 1678 pounds the next year. How many pounds of butter did he send in the four years ? 36. A milkman sold 943 quarts of milk in Jan., 836 in Feb., 972 in Mar., and 937 in Apr. How man}^ quarts did he sell in all ? HORN. ARITH. 7 98 Ai'l'LlCAllOxNS OF ADDITIONS' AND iSUBTKACTlOxN 37. Copy and learn : Thirty days hath September, April, June, and November. All the rest have thirty-one, except February alone, Which has just twenty-eight in fine, till leap year gives it twenty-nine. Let children consult calendar. 38. Copy, writing the number of days in each month opposite its name : March [ June Spking April Summer] July May [ August Fall Sej^tember October November December WiNTEK ] January February 39. How many days in the spring months ? In the summer months? In the fall months? In the winter months ? 40. How many days in the last 5 months of the year ? In this month and last month ? 41. The year in which February has 29 days is called leap year, and comes once in 4 years. A boy named Walter Jones was born February 29th, 1884. In what year can he first celebrate his birthday on the 29th of February ? 42. Find the number of days in the month in which you were born, add to it the number in the montli before and the month after. 43. Find the number of days in the montli in Avhich Christmas comes, add to it the days in the month before and the month after. 44. How many days in the first ten months of a leap year ? APPLICATIONS OF ADDITION AND SUBTRACTION 99 45. Add the days in the month in which Thanksgiving comes to those in the month after and the month before. 46. A store sold 927 yards of carpet in one day, 713 the next, and 837 the next. How many in all ? 47. Susan's father earned iloOO in one year and spent 11321. How much did lie save ? 48. John gets 2 cents a quart for picking herries for a farmer and 1 cent a quart for selling them. How many cents did he earn in the day in which he picked 12 quarts and sold 10 of them ? 49. A bookkeeper earned f 1400 in one year and saved f 227. How much did he spend? 50. A farmer raised 2827 busiiels of corn ; another farmer raised 3431, another 9852, and another 6856. How many bushels did all raise ? 51. Mr. Smith's salary was 11300 last year and 11450 this 3^ear. How much more does he receive this year than last year ? 52. ]\Ir. Ward has 3 horses. Black Beauty, Whiteface, and Fleet. Black Beauty is valued at ^375, Whiteface at 1125, and Fleet at ^575. How much are they all worth ? 53. Anna worked 73 problems in addition in 1 week, her sister worked 98, and her brother 113. How many did they all work ? 54. John had a knife worth 49 cents, w^hich he traded for William's knife and 2 nickels. How much was William's knife worth ? 55. Find the sum of all the even numbers in the first ten. 100 APPLICATIONS OF ADDITION AND SUBTRACTION 56. Numbers which are not even are called Odd Num- bers. Write all the odd numbers in the first ten. Find their sum. 57. Write all the odd numbers in the second ten. Find their sum. 58. Find the sum of the odd number which comes just before 30 and the odd number which comes just after 30. 59. Draw a horizontal line 9 inches long, and divide into halves. 60. Draw a vertical line 5 inches long, and find how many inches in ^ of it. 61. Copy Fig. 2 by drawing inch- squares. Divide the figure into two equal parts by one straight line. How many inch-squares in each half ? Fig. 2 62. Can you divide a group of 9 children into 2 equal groups ? 63. Can you divide 7 apples equally between two boys without cutting any apples into halves ? 64. Can you find a whole number that is just | of 11? 65. Think of different odd numbers, and see if you can find a whole number that is just |- of any of them. 66. Think of some even numbers, and tell what -| of each of them is. 67. How are even numbers different from odd num- bers? 68. Make a list of the odd numbers in the first two tens, and find their sum. 1 1 ' ' ' ' > J 5 ^ > APPLICATIONS OF ADDITION AND tSlTP VfiAcrt;(!)N !liAl» ',' ' 69. Make a list of the even iiumljers in tlie first two tens, and find their snni.* 70. Mary was in school 20 days in the month of Jan- uary. How many days was she out of school ? 71. Use 23487 as a minuend and 14798 as a subtraliend. 72. When one number is subtracted from another, some- times the difference is called the Remainder. Find the remainder Avhen 2987 is subtracted from 8012. 73. Arthur had $38.72 and spent I29.8G. How much was the remainder ? 74. Alfred took $12.38 from $21.75. How much was the remainder ? ^ 75. Find the last remainder when from $829.75 there is subtracted first $28.93, then $478.38 from what was left, then $312.69 from what was left. 76. Subtract 9 from 50 and write the remainder in Roman numbers. ^ 77. Read C, CX, CL, CI, CTH, CIX, CXL, CXLIII, CXX, CCXXV, ( CCXV, CCCCLX, CXLIX. 78. Write in Roman notation all the numbers of two places that have 9 in the tens' place ; all the numbers of two places that have 9 in the units' place. * The game of Odd or Even is useful at this stage. Having the class at the board, the teacher, or the child leader, holds out her closed hand, containing a number of objects, — grains of corn or pieces of paper. Each child writes "odd" or "even." When the hand is opened, those who guess correctly credit themselves with the number, the others with 0. After five trials the scores are added. If instead of using objects, the leader simply writes a number on paper, large numbers can be conven- iently used, and the game thus varied. j^Oii Ai'.PiLlCA:^,tUNi»> OF ADDITION AND SUBTRACTION 79. What page of your book are you reading? The numbers which show the pages are written in Arabic nota- tion. In which kind of notation are the numbers in tlie number table written ? 80. Write in Arabic notation CXC^V, CCXCI, CCCXCVI, CCCCXCIV. 81. Write in Arabic notation, and add, CLXXV, l^IV, LXXXIV, LXTX, XLIX, CXLVJll. 82. Subtract 4 from 91 and write the remainder in Roman numbers. 83. Write in Arabic notation CCXCV and CCCLXXVI and find their difference. 84. C'o}>y Fig. o by phicing triangles made by bisecting iiicli-squares. Show ■1 of your ligure. | ~ i = • 85. Divide the liij^ure into halves. How many sixtlis in one half ? 86. Copy Fig. 8 by drawing. 87. Copy Fig. 4 by placing triangles. How many triangles does it take ? How many triangles would it take to make five such figures ? To make 7 such figures ? To make 10 such figures ? Fig. 3 88. Show Jq of your figure. 1 _ 1 _ y J) 3_ _ V _6^ _ 10 10 ~" • 10 10 "~ • lIT 4 10 Fig. 4 89. Divide the fiofure into halves. How many lOths in | ? 90. Can you take 4 triangles away from Fig. 4 and leave it just like Fig. 3 ? Copy Fig. 4 by drawing. APPLICATIONS OF ADDITION AND SUBTRACTION 103 Fig. 5 91. Copy Fig. 5 by placing triangles. How many tri- angles does it take ? How many trian- gles would it take to make 9 such fig- ures? To make 6 such figures? 92. Divide the figure into halves and show how many tenths in ^. 93. Can you take away 4 triangles from the figure so as to make a figure just like Fig. 3 ? 94. Can you show how Fig. 5 can be made just like Fig. 4 by turning four of the triangles around ? Copy Fig. 5 by drawing. 95. Subtract 327 from 982. Subtract it ao-ain from tlie remainder and again from the second remainder and see if 3"our answer is 1. 96. Keep on subtracting 224's from 1123 in the same way until the remainder is 3. 97. Subtract 123's from 369 until nothing remains. How many 123's does it take to equal 369 ? 98. Add 32's together until you get 192. 32's did you use ? 99. Add 24's together until you get 144. 24's in 144 ? How many How many 100. AVrite the names of the months that have 31 days, ^nd find how many days in all of them. 101. Five vertical lines are drawn on the board one foot apart. How far apart are the two outside lines ? Let the children try imagining before iUustratiiig. 102. If it costs 10 cents to saw a log into two pieces, how much will it cost to saw it into three pieces? CHAPTER VII FIVES Equilateral Triangles, Eoman Numerals D and M, Quotient NUMBER TABLE* 1 11 21 31 41 51 61 71 81 91 2 12 22' 32 42 52 62 72 82 92 3 13 23 33 43 53 63 73 83 93 4 14 24 34 44 54 64 74 84 94 5 15 25 35 45 65 75 85 95 6 16 26 36 46 r,(cy Gij 76 86 96 7 17 27 37 47 51 67 77 87 97 8 18 28 38 48 58 68 78 88 98 9 19 29 39 49 59 69 79 89 99 10 20 30 40 50 GO 70 80 90 100 1. Begin with 5 and count l)y fives to 100 Let pupils practice this until they can count rapidly. 2. The number table is divided into groups of five numbers. Name all the numbers in the first group of * Charts containing this and other number tables should remain upon the wall in sight of the children all the time, except when tests are given. By this means the children unconsciously become familiar with the mul- tiples and their relative positions, 104 FIVES 105 five. Name all the numbers of the second five. The third five. The fourth five. The fifth five. 3. Show the sixth five. Show the next five. Which five is it ? 4. Point out the second five and name the first and last number of it. 5. Show the third five and name the first and last number. 6. What number in the third five is next to the last ? 7. What is the last number of the fourth five ? Of the fifth five ? Of the sixth five ? Of the seventh five ? 8. In which five is 13 ? 29 ? 31 ? 46 ? 9. Name an odd number in the seventh five. Let the children select numbers and tell in which five they are found. 10. Name in order the multiples of five up to 100. l^earn to name them without looking at the number table. Fill out and learn the following table of fives : 1 five = 5 fives = 9 fives = 2 fives = 6 fives = 10 fives = 3 fives = 7 fives = 11 fives = 4 fives = 8 fives = 12 fives = Pupils must first learn the nniltiplication tables in regular order so that they may see the aggregations of which multiples are com- posed. But in later work care should be taken not to use a fixed order. The child should learn the statements of the multiplication tables as separate facts, so that eacli may spring singly into his con- sciousness when needed. 11. Name the second multiple of 5 ; the fourth multiple of 5 ; the fifth, the sixth, the tenth, the eighth, the seventh. 12. 3 fives = how many? What is ^ of 15? 5 is ^ of what number? 100 FIVES 13. 5 is i of what number ? How can you tell ? 5 is 'I of what? I of what? ^ of what? ^2" ^^ what? ^ of what? -^^ of what? ^ of what? -^j of what? 14. 50 is which multiple of 5? Name another number that 50 is a multiple of. 15. 10 is Avhich multiple of 5? Name another number that 10 is a multiple of. 16. Which multiple of 5 is 15? 25? 35? 40? 60? 17. Name multiples of 5 and tell quickly which multi- ples they are. 18. Name all the numbers in the 6th group of five and tell wiiich is the middle number. 19. 5 cents + 5 cents + 5 cents + 5 cents = liow many dimes ? 20. 5 is wliat part of 25? Of 40? 20? 60? 35? 21. Alary had 15 cents, and Anna had J as many. How many had Anna? 22. How much is | of 15 cents? |- of 30 cents? 23. How many times can a line 5 inches long be meas- ured off on a line 20 inches long? On a line 15 in. long? On a line 35 in. long? On a line 45 in. long? 24. 5 multiplied by 3 = ? F> x C> =? 5x8=? 25. What number is 3 more than 4 fives ? 2 more than 5 fives? 26. How much is 1 less than 3 fives? 2 less tlian 5 fives ? 27. Which is more, 17 or 3 fives? How much more? 28. Wliich is more and how much more, 4 fives or 18? 2 fives or 14? 4 lives or 23? 29. Wliich is more and how much more, 6 fives or 28? 4 fives or 2 tens? 10 fives or 5 tens? 5 fives or 26? FIVES 107 30. How many nickels equal 15 cents ? 20 cents ? 31. If 6 little girls have a nickel apiece, how many cents' worth of peaches can they all buy ? 32. How many cents will it cost for 7 children to ride on a street car, if they each pay 5 cents fare ? 33. Point out the last number of the lifth five, add 2 fives, and point out the result. 34. Add 2 fives to 30 and point out tlie result. How many fives does it equal ? 35. Add 2 fives to 40. To 50. 35. 45. 55. 15. 36. Add 3 fives to 10. 20. 15. 35. 45. 30. 40. 37. Mary may think of an even number and tell wliich five it is in. The class may guess the number. 38. John may think of an odd number and tell which five it is in. 39. Begin at 100 and c(Hint backwards by fives quickly. 40. Take 2 fives from 20. 45. 35. 40. 50. 30. 41. How many fives in 15? 30 ? 40 ? 55 ? 35 ? 42. How many fives in a ten and half a ten ? In 2 tens and a lialf ? In 3 tens and J a ten? 43. Place two rows of five squares each as in this figure, and tell how many squares there are. Place another row of five squares above them and tell how many squares there are. Keep on placing rows of 5 squares each until the figure is as wide as it is long or until it is square. How many little squares are there in it then? Find the middle square of all and write M in it. 44. 25^5 = ? 40-5 = ? 15^5 = ? 35^5 = ? 108 FIVES 45. Read XC, XCIV, CCLII, CCCX, CCCLXVI. 46. D. stands for 500 in Roman notation. Read DC, DL, DXC, DCCC, DXLVIII, DCLXVI, DCCIX. 47. Write in Arabic notation DCCXXV and DCCCXXXVII. Then find their sum. 48. Write in Roman notation 605, 607, 609, 611o BLACKBOARD EXERCISE Mnltij^ly 5 by each of the numbers on the edge of the triangle. Answer quickly. Fig. 1 10 5 9 49. Triangles whose sides are all equal like Fig. 1 are called Equilateral Triangles. Are those triangles equilateral that are made by cutting a square inch into halves ? 50. If each side of Fig. 1 were 5 in. long, how long would the perimeter be ? Equilateral triangles should be furnished for the following work. 51. Copy Fig. 2 by placing equilateral triangles. If each side of the triangles you used were 5 in. long, how long would the perimeter of your figure be ? 52. Copy Fig. 3 by placing equilateral triangles. If each side of the triangles were 5 in. long, Avhat would be the length of a line that would lie all around Fig. 3 Fig. 3 the figure? FIVES 109 53. Show f of Fig. 3. ShoAv -| of it. f - f = ? 54. Draw a vertical line 5 in. long. Divide into inches and mark the divisions. One in. is what part of 5 in.? 3 in. are what part of 5 in.? 4 in. are what part of 5 in.? 55. How mucli do | of the line lack of being the whole line ? Show | of the 5-inch line. 56. Draw a line | as long as the 5-inch line. How much longer is it than the 5-inch line ? 57. Draw a line -| as long as the 5-inch line. -|. ^. 58. Mary has 5 cents, and Anna has ^ as much. How many cents has Anna ? John has ^ as much money as Mary. How many cents has John ? Kate lias |^ as much as Mary. How many cents has Kate ? Thomas has | as much as Mary. How many cents has Thomas? Illustrate with actual mouey if the children fail to think out this work. 59. 1^ of anything is how much more than the whole of it? ^ is how much more than the whole? -|? J^? 60. Numbers like -i, J, ^, that show parts of anything are called Fractions. Write some other fractions. 61. DraAV a line 3 in. long and another line ^ longer. How many inches in the long line ? 62. Show on the number table 1 of 25. Show -| of 25 ; f of 25 ; I of 25. 63. If John had 25 cents and James had ^ as much, how many cents would James have ? 64. Make story problems about fifths of 25. 65. How many fives must be added to 20 to equal 35 ? 45 ? 30 ? 66. How many fives must be subtracted from 60 to leave 45 ? To leave 35 ? 50 ? 40 ? 20 ? 30 ? 55 ? no FIVES 67. 3 fives are how many more than 13 ? 6 fives — 3 = ? 68. What must be added to 8 fives to erjual 43 ? To 7 fives to equal 39 ? 69. What must be subtracted from 1(3 to leave 3 fives ? 2 fives ? 1 five ? 70. 33 is how much more than 6 fives ? Than 5 fives ? 71. 3 fives + 4 = ? 4 fives + 2 = ? 8 fives + 3 = ? 72. How much does 49 lack of being equal to 10 fives ? 47 is how many more than 9 fives ? 73. Can you bring in (or name ) a flower that has five petals ? How many petals would 7 such flowers have ? 9 such flowers ? 74. How many school days in 3 weeks ? 5 weeks ? 7 weeks ? 11 weeks ? 75. How many cents = 10 nickels and 3 cents ? 8 nick- els and 4 cents ? 4 nickels and 5 cents ? 76. (3 times 5 pk. = liow many bu. and pk. ? 77. How manv tens = 4 fives ? 6 fives ? 10 fives ? Blackboard Exercise Divide quickly each num- ber on the edge of the ti'i- angle by 5. F1VE8 111 78. The number that sliows how many times one num- ber is contained in another is called a Quotient. What is the quotient of 25 divided by 5 ? 16 divided by 2 ? 79. Give quotients of 50 -r- 10 ; 30 -f- 5 ; 24 ^ 2 ; 55 -^ 5. 80. -ig5- = ? (This is read " 15 divided by 5 " or " 15 fifths.") on 5^0_9 10. _y _2 2._? 40 — 9 60_? 70_? °-^' 10 ~" • 10 ~ • 2 ~ • 5 • 5 ~ • 10 ~ • 10x4 >, Show the process of cancellation and let v> v^ -) the pupils prove by trial with small num- bers that the same result is obtained as by dividing the i^roduct of the numbers above the line by the product of those below. Do not attempt to give the underlying principles, as the power to perceive them usually comes at a much later stage of the child's psychological development. Q3_ 10x20x11 _, Q5^ 33x5x8 ^, ^^ 0x5x11 ^, 2x5x55 ' * 11x2x15 " '2x44x25 84. 222ll^ili^=v 86. Ji^^^y 88. ^QX^ =? 2x11x20 2x3x2 2x10x11 The division of one indicated product by another by cancellation may be made an interesting class exercise, and it is very useful in helping children to Ijecome expert in recognizing ratios. As new numbers and their multiples are learned give class exercises in this work, combining the new numbers with those previously learned. 89. Show on the number table I of 35 ; f of 35 ; | of 35 ; f of 35 ; 4 of 35. 90. If Thomas had 35 cents and William had ^ as many, how many did AVilliam have? 91. Make story problems about sevenths of 35. 92. Kind the 6th multiple of 5 and add 7 to it. 93. Add 8 to the 4th multiple of 5. To tlie 7th. To tlie 9th ? To the 11th ? 94. Take 6 from the 12th multiple of 5. From the 9th. From the 7th? From the 4th ? From the 11th ? 112 FIVES 95. Add 2 tens to the 6th multiple of 5. To the 9th. 96. Add 3 twos to the 3d multiple of 5. To the 7th. Let pupils compose similar questions and briug them to the recita- tion to be solved by their classmates. 97. A child was asked, " What is a multiple of 5 ? " She answered, " The number you get when you multiply 5 by any number is a multiple of 5." Was she right? Explain. 98. What is a multiple of 10 ? A multiple of 2 ? 99. What number is the fourth multiple of 10 ? What number must 10 be multiplied b}^ to give the fourth mul- tiple of 10 ? 100. By what must 5 be multiplied to give the tliird multiple of 5 ? 101. By what must 5 be multiplied to give 45 ? Which multiple of 5 is 45 ? 102. If you Avere to spend 5 minutes a da}^ playing with a kitten, how much time would you spend in a week ? 103. A pansy has 5 petals. How many petals do 9 pansies have ? 104. At 5 dollars apiece, how much will 11 hats cost ? 7 hats ? A dozen hats ? 105. At $5 apiece, how many hats can be bought for $40? 160? 125? 106. At 5 cents apiece, how many oranges can be bought for 30 cents ? 45 cents ? 20 cents ? b5 cents ? 107. Find sums : 108. Find differences : i^3.15 111.55 111.15 129.65 $69.57 158.58 6.75 38.57 67.25 13.17 32.85 12.95 5.76 24.55 16.75 FIVES 113 109. Add 8 thousand 2 hundred 86 to 9 thousand 3 hundred 74. 110. From 5 thousand 3 hundred 24 take 2 thousand 1 hundred 95 and mark the subtrahend. 111. Anna bought some groceries for her mother. She paid ^1.15 for tea, if) 3. 37 for flour, and til.2o for sugar. How much was the whole bill ? 112. Mr. Williams paid a doctor's bill of 815.75. He gave 4 five-dollar bills. How much change should he receive ? 113. Make story problems about buying. 114. Name the multiples of 5 that are even numbers. What figure does each of the even multiples of five end in ? 115. Name the multiples of 5 that are odd numbers. What figure do they end in ? 116. Write the odd multiples of 5 in a horizontal line. Think how the number table of five looks, and write the multiples of 5 that are even numbers in a horizontal line under the line you have just written. Leave space be- tween the lines as in the number table. 117. Write in Roman notation all the multiples of 5 up to 100. 118. M stands for 1000 in Roman notation. Write in Arabic notation MC, MCCC, MD, MDC, MDCCC, MDCCCC, MDCCCLIX. 119. Write in Roman notation 1800, 1830, 1840, 1850, 1860, 1890, 1896, 1861, 1876, 1883. 120. Write in Roman notation the number of the page on which you are reading ; the number of the page on which the 7th chapter of this book begins ; the number of the page on which the 12th chapter begins. HOKX. ARITH. 8 CHAPTER VIII ELEVENS Written Multiplication, Pkoduct How many units in each answer ? How many tens ? How many units and how many tens in each answer ? 3. Write 9 elevens and find their sum. -j-j 4. Find the sum of 7 elevens. — 10 elevens. NUMBER TABLE 1. Add ; 11 11 11 11 11 2. Add : 11 11 11 11 11 11 1 11 21 31 41 51 61 71 81 91 2 12 22 32 42 52 62 72 82 92 3 13 23 33 43 53 63 73 83 93 4 14 24 34 44 54 64 74 84 94 5 15 25 35 45 55 65 75 85 95 6 16 26 36 46 5i] 66 76 86 96 7 17 27 37 47 57 67 77 87 97 8 18 28 38 48 58 68 78 88 98 9 19 29 39 49 59 69 79 89 99 10 20 30 40 50 60 70 80 90 100 114 ELEVENS 115 5. Begin at 11 and count by elevens until you reach 99. How many multiples of eleven are there in the first hundred numbers ? Learn them. 6. Begin at 99 and count backwards by elevens rapidly. 7. Fill out and learn the table beginning " Once 11 is eleven," and ending ''12 times eleven are 182." 8. AVhat is the third multiple of 11 ! Mx */ 8th '^ 6th ? Call attention to the fact that the od multiple of 11 is expressed by two 3's, the 5th multiple by two 5's, etc. 9. How many elevens in 41 ? (3(3? 77? 33? 121? 10. 11 multiplied by 5 = ? 11 x 8 = ? 11 x 4 = ? 11. Add tAvo elevens to 33. 55. 22. 77. 44. QQ. 12. How many elevens must be added to 22 to equal 55? 44? 66? 88? 13. Take 2 elevens from 77. From 44. 88. 22. m. 14. How many elevens can be taken from 99? From 132? 110? 15. How many elevens must be taken from 77 to leave 44? 22? ^b'^ 33? m^! 16. How many elevens must be taken from 55 to leave 4 elevens ? 17. Name multiples of 11, and take elevens from them. 18. If you had 22 cents and your mother gave you 11 cents, how many cents would you have? How many dimes and cents ? 19. Mary had 44 cents, Julia had 11 cents more than Mary. How many cents did Julia have? 20. Eight boys gave 11 cents apiet^e toward a picnic. How many did they all give? llg ELEVENS 21. John solved 11 problems on Monday and twice as many on Tuesday. How many on both days? 22. Make problems with the number 11. 23. Which is the, greater, 57 or 5 elevens, and how much ? 24. Which is the greater, 6 tens or 5 elevens, and how much ? 25. Which is the greater, 11 fives or 5 elevens, and how much ? 26. How much does 42 lack of being as great as 4 elevens ? 27. How much does 58 lack of being as great as 6 elevens ? 28. 26 is how many more than 2 elevens? 29. 91 is how many less than 9 elevens ? 30. 49 is how many less than 5 elevens ? 31. 6 elevens — 7 = ? 7 elevens — 8 = ? 4 elevens — 9 = ? For a class exercise let the children choose numbers and tell how much they exceed or fall short of multiples of 11. 32. Find the third multiple of 11, take 3 from it, and tell how many tens in the answer. 33. Find the 5th multiple of 11, take 5 from it, and tell how many tens in the result. 34. Name the first multiple of 11, subtract 1, and tell how many fives in the result. 35. Find the second multiple of 11, subtract 2, and tell how many fives in the result. 36. Add 5 to the 4th multiple of 11. Add 7 to the 6th multiple of 11. 37. How many elevens does it take to equal the number that is the 3d multiple of 11? The 5th multiple of U? ELEVENS 117 38. Think of different multiples of 11, and tell how many tens and how many units in each. 39. If a cow gives 11 qt. of milk each day, how many qt. will she give in a week ? In 10 days ? 40. If it takes 11 buttons for a boy's suit, how many buttons will it take for 4 suits ? 7 suits ? 9 suits ? 41. If 11 cents were given to each of 5 boys, how many cents would all get ? 42. When tops are 11 cents apiece, how much will 4 tops cost ? 8 tops ? 11 tops ? 43. If 33 cents were divided equally among 3 boys, how many cents would each receive ? 44. If 55 cents were divided equally among 5 boys, how many cents would each receive ? 45. Of what number is 11 one half? 11 is ^ of what? i of what ? I of what ? ^ of what ? ^ of what ? 46. 11 is what part of 33 ? Of 77 ? 44? 99? 55? 47. How much is ^ of 33 ? J of 33 = ? 48. iNIr. Smith had §33 and spent J of them. How many dollars did he spend ? How many had he left ? 49. How much is i of 55 ? | of 55 = ? | of 55 = ? 50. 55 cents — I of 55 cents = ? 55 cents — | of 55 cents = ? 55 cents — -| of 55 cents = ? 51. Make story problems about fifths of 55. 52. How much is 1 of 77 ? f of 77 = ? f of 77 - ? f of 77 = ? A of 77 = ? f of 77 = ? 53. Make story problems about sevenths of 77. CHART DRILLS 1st. Taking some multiple of 11 as a basis, as for instance 55, point to that and let the children give the numbers that are equal to f of it, I of it, 4 of it, f of it, etc. 118 ELEVENS Fig. 1 2d. Taking a multiple of 11, as 55, as a basis, let the children point to some other multiple, as 3:3, and tell quickly what part of 55 33 equals. Use these drills frequently until the children can give the ratios of the multiples at sight. 54. 2 times 11 qt. = how many gal. and qt.? 55. 4 times 11 pk. = liow many bu.? 56. 3 times 11 ft. = how many yd.? 57. 5 times 11 days + 1 week = how many days? 58. Take 2 elevens from each of the odd numbers in the third ten. 59. Copy Fig. 1 by placing equilateral triangles. How long would the perime- ter of the figure be if the side of eacli triangle were 11 in. long? 5 in.? 60. Copy Fig. 2 by })lacing equilateral tri- angles. Which is greater, Fig. 1 or Fig. 2? Which has the longer perimeter ? 61. How long would the perimeter of Fig. 2 be if each side of the triangle were 10 in.? Fig. 2 5 in.? 11 in.? 62. Place 7 e(pnlateral triangles, making a figure differ- ent from those in the book, and make problems about the perimeter of the figure. 63. Place equilateral triangles as in Fig. 3, and find how long the perimeter of the figure would be if each side of the triangle Avere 11 in. long. 64. Sliow J of the figure you have made. Show I of it ; f ; | ; | ; 1 8' 1 _ 5 _? 8 8 ~ • 2 — ? 8_3_? 1— how Fig. 3 many eighths ? ELEVENS 119 65. Can you separate the figure into 4 equal parts shaped just like the figure itself, only smaller? How many eighths in each of those ? ^ = how many eighths ? 66. Place 8 equilateral triangles in such a way as to make a figure different from Fig. 3, and make problems about them. 67. Copy Fig. 4 by j^tlacing equilateral triangles. How many triangles in Fig. 4? How many triangles would it take to make 9 such figures ? 7 such figures ? 5 such figures ? 8 such figures ? ^^' 68. How many such figures could be made from 44 equilateral triangles ? From 99 equilateral triangles ? From 33 equilateral triangles ? From Q6 equilateral triangles ? 69. Draw a horizontal line 11 in. long, marking the inches. 1 inch is what part of it ? 2 in. is what part of 11 in.? 3 in. is what part of 11 in.? 5 is what part of 11 ? 7 is what part of 11 ? 70. Draw a line that is ^^ as long as an 11-inch line. Draw a line that is ^| as long as an 11-inch line. 71 11 ^3_ _ _?_ 11 9_ _ _?_ __9 7_ — JL '■^* 11 11~11* 11 11 ~ 11* 11 11~11 Let these subtractions be shown objectively if necessary. 72. How many players in 4 football teams ? In 9 foot- ball teams ? 73. Write in Roman notation all the multiples of 11 that are less than 135. 74. A string 77 in. long can be cut into how many strings 11 in. long ? 75. Make a drawing of two rows of squares, 11 squares in a row, and tell how many squares in it. 120 ELEVENS 76. Make 3 rows of 11 squares each, and tell how many squares there are. Find the middle square of the middle row and write the first letter of your name in it. 77. What is the quotient of 44 divided by 11 ? 78. 55 -r- 11 = ? 11 = ? H = ? £6 ^ ? 3 3 _ 11 11 • 11 • 11 ~ • See note after Ex. 88, p. 111. 79. Show by grouping numbers on the number table how many twos equal 2 elevens ; how many fives equal 5 elevens. 80. Write the multiples of 11 in the same position that they liave in the number table. 81. Multiply 11 11 11 11 11 11 11 by^_8_6_5J7_9_8 Lead the children to see that they can get tlie same result by mul- tiplying the units and then the tens as by combining numbers on the number table, and in an easier way. 82. Multiply 111 112 113 211 131 142 111 by _3 _4 _2 _3 __3 _2 8 83. Wlien one number is nuiltiplied by another, the result is called a Product. What is the product of 111 and 4 ? 84. Find the product of 121 and 3; 221 and 4; 122 and 3 ; 512 and 3 ; 512 and 4. 85. If one side of a square were 321 ft. long, how long would the perimeter of the square be ? 86. If each side of an equilateral triangle were 133 ft. long, what would be the length of the perimeter of tlie triangle ? 87. Multiply 35 It is left for the teacher to show that the l)v 4 ^'^'^ teufi obtained by multiplying the 5 units by 4 must be added to the 12 tens obtained 140 l)y multiplying the 3 tens by 4. ELEVENS 121 88. Find products: 125 152 251 215 255 515 4 8 7 6 5 9 89. Multiply 555 by each of the numbers that are greater than 1 and less than 10. 90. Multiply 2222 by each of the numliers greater than 1 and less than 10. 91. Add 413 to itself and see if the sum is 826. 92. Add 312 to itself and 312 to their sum, and see if the answer is 936. 93. Add 121 to 121, and keep on adding 121 until you have 484. 94. Add 211 to 211, and keep adding 211 until you get 1055. How many 211's does it take to make 1055? 95. Can you hnd a better way of finding the sum of five 211's than by adding them ? If not, ask your teacher. 96. Find the sum of three 125's. Four 215's. Six 512's. Five 511's. 97. Write the 5th multiple of 11, under it the Tth mul- tiple of 5, under it the 10th multiple of 2, under that the 6th multiple of 10, and add. 98. Write the first odd number in the 4th ten, under that the last even number in the 4th ten, under that the first odd number after 30, under that the first odd number after 37, and find their sum. 99. Thomas paid #8.25 for a suit of clothes, i^l.25 for some handkerchiefs, -f .37 for a necktie, and i .25 for some collars. How much was the whole bill ? 100. He gave the clerk a ten-dollar bill and a five-dollar bill. How much change should he get ? 101. How much will 7 horses cost at #125 apiece? 102. Tell what these words mean : Sum^ Difference^ Product^ Quotle7it. CHAPTER IX NIISTES Multiplier, Square Yard, Square of a Number, Divisor NUMBER TABLE 1 11 21 31 41 51 61 71 81 91 2 12 n 32 42 52 62 72 82 92 3 13 23 33 43 53 63 73 83 93 4 14 24 34 44 54 61 74 84 94 5 15 25 35 45 ^^ Q^ 75 85 95 6 16 26 36 40 m m 76 86 96 T 17 27 37 47 57 67 77 87 97 8 18 28 38 48 58 68 78 88 98 9 19 29 39 49 59 69 79 89 99 10 20 30 40 50 60 70 80 90 100 1. Begin with 9, and learn to count quickly by nines to 99. 2. Learn the multiples of 9 that are less than 100. How many are there ? What is the next multiple of 9 ? 3. Begin with 108 and count backwards by nines to 0. 4. Fill out and learn the table of nines ending with 12 times 9 = 108. 122 NINES 123 BLACKBOARD EXERCISE Lead the children to see that as 9 falls 1 short of 10, 2 nines fall 2 short of 2 tens, ;> nines fall 3 short of 3 tens, and so on. Will not some child discover that in each of the first ten multiples of 9 the sum of the digits is 9 ? 5. What is tlie 4tli multiple of 9 ? 6th ? 8th ? 9th ? 6. How maiiv nines in 63 ? 81 ? 45 ? 99 ? 54 ? 36 ? 7. 72 + 9 = ? 54 + 2 nines = ? 81 + 2 nines = ? 8. 63 - 2 nines = ? 36 - 2 nines = ? 81-2 nines = ? 9. \\) what nuniljer nmst 9 be multiplied to give the produetSl? 36? 63? 108? 54? 72? 99? 45? 18? 10. Multiply 29 119 119 219 259 295 295 by 3 3 4 5 6 7 8 11. A number that is used to multiply another number is called a Multiplier. Name the multipliers in Ex. 10. 12. Use 4 as a multiplier of 99. 13. Use 5 as a multiplier of 999. 14. Use 6 as a multiplier of 9999. 15. How many nines must be added io 27 to equal 45? 36? 54? 72? 63? 16. How many nines must be taken from 72 to leave 54 ? 63 ? 45 ? 27 ? 17. How many nines must be taken from 54 to leave 5 nines ? 3 nines ? 18. How many nines must be taken from 45 to leave 2 nines ? 4 nines? 124 NINES 19. Write in Roman notation all the multiples of 9 that are less than 109. 20. Which multiple of 9 is 99 ? 36 ? 63 ? 27 ? 72 ? 21. Draw 3 rows of squares, 9 squares in a row. Keep on adding rows of squares until you have as many rows as there are squares in a row. How many squares in all ? 22. If a rectangle is just as long as it is wide, it is a perfect square. Is your drawing a j)erfect square ? Find the middle square and make the sign of multiplication in it. 23. Place 4 squares so as to make a perfect square. How long is one side? 24. Place 9 squares so as to make a perfect square. How long is the perimeter? 25. Place 4 rows of squares, 4 squares in each row. How many squares in all? Is the figure a perfect square? 26. If you place 4 rows of squares, 5 squares in a row, will the figure be a perfect square ? If not, what can be added to it to make it a perfect square? What can be subtracted from it to leave a perfect square? 27. How many square inches in a square whose sides are each 5 in. ? 2 in. ? 10 in. ? 9 in. ? 11 in. ? 28. John may draw on the floor a square, a side of which is 3 ft. long, and mark it off into square feet. How many square ft. in it? 29. A square measure which is 3 ft. long and 3 ft. wide is called a Square Yard. How many square feet make a square yard? 30. How many square ft. in 4 sq. yd.? In 7 sq. yd.? 3 sq. yd. ? 11 sq. yd. ? 5 sq. yd. ? 8 sq. yd. ? 2 sq. yd. ? 9 sq. yd. ? 6 sq. yd. ? 12 sq. yd. ? NINES X25 31. How many square yards in 36 sq. ft. ? In 54 sq. ft. ? 99 sq. ft. ? 45 sq. ft. ? 108 sq. ft. ? 27 sq. ft. ? 32. 1 square foot equals what part of a square yard? 33. What fraction of a sq. yd. is 2 sq. ft. ? 3 sq. ft. ? 5 sq. ft. ? 8 sq. ft. ? 6 sq. ft. ? 34. Show ^- of the square yard drawn on the floor. How many ninths does it equal? 35. Show f of the square yard and tell how many ninths it equals. 36. Multiply 9 by itself. 37. When a number is multiplied by itself, the result is called the Square of that number. What is the square of 9? 2? 10? 5? 38. Add the square of 2 to the 7th multiple of 9. Exercises like the following are useful : "Take the square of 5, add 5, take |, add 5, take ^, square, add the square of 2, subtract 10, add 1, divide by 10, add the second multiple of 5, take ^, add the square of 3," etc. 39. Copy Fig. 1 by placing equilateral triangles. If a side of each of the tri- angles you use were 9 in. long, how long would the perimeter of your figure be? 40. Find the length of the perimeter of the figure when each side of the small "fig~T triangles is 5 in. 10 in. 41. How many small triangles in the large triangle that you have made? 42. How many triangles in 6 such figures? In 8 such figures? 11 such figures? 4 such figures? 7 such fig- ures? 9 such figures? 10 such figures? 5 such figures? 126 NINES 43. How many figures like that you have made could be made from 18 small triangles? From 72 small trian- gles? 54? 99? 63? 36? 81? 45? 27? 44. Show -^ of the figure you have made. Show J of it. Show -^ of it. Show I of it. 45. 9. 9 1 — 1 9 ~" 9' 8. _ 3 9 9 6. _ 3. _ ? 9 9 ~ • 5 _? 9 • Fig. 2 46. Separate your figure into thirds as in Fig. 2. How many ninths in each third ? 47. -1 = how many ninths ? | = how many nintlis ? 48. 1 is what part of 9 ? 2 is what part of 9 ? 4 is what part of 9 ? 5 is what part of 9 ? 8 is wliat part of 9 ? 9 is how many ninths of 9 ? 3 is Avhat part of 9? 6 is what part of 9? 49. 2 times 9 ft. = liow many yd.? 50. 3 times 9 sq. ft. = how many sq. yd.? Children sometimes fail to distinguish linear yards and square yards. Whenever their imagery of these becomes confused or indefi- nite, refer them to the actual figures drawn on the floor. 51. 4 times 9 sq. ft. = how many sq. yd.? 52. 2 times 9 ph. + 6 pk. = how many bu.? 53. 3 times 9 qt. — 7 qt. = how many gal.? 54. Write all the multiples of 9 that are odd numbers less than 100. 55. Write all the even multiples of 9 that are less than 108. 56. Write the multiples of nine from 9 to 81 in a slant- ing line as they are in the number table. 57. Add 5 to tlie 4tli multiple of 9. 58. Subtract the square of 2 from tlie 2d multiple of 9. NINES 127 59. Subtract 5 from the 6tli multiple of 9. 60. Subtract 3 twos from the 3d multiple of 9. 61. Subtract 11 from the 6th multiple of 9. 62. Subtract 2 elevens from the 5th multiple of 9. Let class prepare similar questions. 63. If there are 9 desks in each row and 6 rows in a schoolroom, how many cliildren can have desks of their own ? 64. 8 children give 9 cents each to a Children's Aid Society. How manj^ are given by all ? 65. 11 children give '1.09 apiece for a trip to the country. How much do they all give ? 66. The fare to Chicago from a certain city in Wiscon- sin is 8 9. How much will it cost 8 persons to make the trip ? 67. A round trip ticket to Chicago from a town in Wis- consin costs $ 9. How much will it cost for 7 persons to go to Chicago and back ? 68. If a grown person's ticket costs twice as much as a child's, how much will it cost for little Mary and her mother to make a journey to Atlanta when Mary's fare is §9? 69. If a boy's suit cost f 9, how much will 10 such suits cost ? 70. If a dressmaker receives $ 9 for making a dress, how much will she earn by making 11 such dresses? 7? 4? 71. It a man earns 9 dollars in a week, how much will he earn in 7 weeks ? 9 weeks? 12 weeks? 6 weeks ? 72. If 18 cents are divided equally between two boys, how many cents will each boy receive ? 73. If a ball costs 9 cents, how many balls can be bought for !^. 27? 1.63? 1.36? |.81? 11.08'? 128 NINES 74. What is the quotient of 45 divided by 9? 36^9 = ? See note after Ex. 88, p. 111. 75. Which is greater and how much, 3 nines or 28 ? 34 or 4 nines ? 58 or 6 nines ? 55 or 7 nines ? 76. 8 nines are how many more tlian 70? How many less than 80 ? 77. 5 nines — 4 = ? 7 nines — 5 = ? 6 nines — 7 = ? 78. How much does 61 lack of being equal to 7 nines ? 79. How much do 5 nines lack of being equal to 48 ? 80. How much do 6 nines lack of being equal to 6 tens ? 81. 3 tens — 3 nines = ? 5 elevens — 5 nines = ? 82. How many nines and how many over in 28 ? 38 ? 83. Choose numbers less than 100 that are not multiples of 9, and tell hoAV many nines in them and how many over. 84. Turn to the number table of 9 and the number table of 5, and show which is greater, 5 times 9 or 9 times 5. 85. Compare 9 x 11 and 11 x 9. 9 x 10 and 10 x 9. 9 X 2 and 2 X 9. 86. What is the product of 90 multiplied by 4 ? By 7 ? 87. Give tlie product of 9 maltiplied by 20. 30. 40. 88. How much is ^ of 9 ? 1 nine and 1 of 9 ? 89. What is the product of 9 multiplied by 5 J ? 2 J ? 90. Show on the number table the product of 9 by 1^. By 31. 81 61. 41. 101 71 91 51 21 91. What is the product of 9 by 1|? By 6f ? 4|? 2|? 8|? lOf? 7|? 5f? 3f? 9|? Give exercises like Ex. 90 and 91 until pupils are prompt in that work. Give similar exercises on each number as it is taken up. 92. 9 is ^- of wliat number ? 9 is J of what ? ^ of what ? ^ of what ? -^ of what ? ^ of what ? ^ of what ? NINES 129 93. 9 is what part of 27 ? 54 ? 36 ? 99 ? 63 ? 81 ? 45 ? 94. What is J of 27 ? How much will ^ of a yard of ribbon cost at 27 cents a yard ? How much will | of a yard cost ? 95. ^ of 36 = ? If you have 36 marbles, and lose ^ of them, how many marbles will you lose ? How many Avili you have left ? 96. Show on the number table ^ of 45 ; |- of 45 ; | of 45 ; I of 45. 97. Tell what part of 45 is 18, 36, 9, 27. See note on Chart Drill after Ex. 53, pp. 117, 118. 98. May has 36 cents. Ann has ^ as many. How many has Ann ? Louise has | as many cents as May. How many cents has Louise ? 99. John has |- as many marbles as James, who has 45. How many marbles has John ? 100. Make story problems. 101. Fill out the following, and learn to give the state- ments in any order : 1 of 54 = I or 1 of 54 = f or i of 54 = I or I of 54 = f of 54 = 102. What part of 54 is 18 ? 45 ? 36 ? 27 ? 103. Thomas had J as much money as William, who liad 54 cents. How much had Thomas ? Train pupils to give results from their memory of the ratios of numbers. When, for instance, they can recall the fact that f of 54 is 45, do not have them go through the process of finding i of 54, and then f of it. 104. Make a table showing the sevenths of 63 from -^ to 1^, like the table that shows the 6ths of 54. HORX. ARITH. 9 130 NINES Children get interesting practice for a short time from exercises like this : " Let us play that Mary has 63 cents. Louise, how many sevenths of Mary's money will you think of ? " "I will think of f of it, or 45 cents," replies Louise. Then other pupils " think " and give their thoughts promptly. 105. What part of 63 is 27? 36? 18? 54? 45? 106. Make a table showing the eighths of 72, and study it until you can tell quickly what part of 72 is 18, 63, 45, 54, 36, 27. 107. Take 81 and find 1 of it ; |, J, |, |, |, |, |. 108. What part of 81 is 18 ? 36 ? 72 ? 27 ? 63 ? 54 ? 45 ? 109. If the whole of anything costs 81 cents, how much would I of it cost at that rate ? How much would | cost ? A? 5 ? 1? 9 • 9 • 9 • 110. If the whole of anything costs 81 cents, what part of it could be bought for 9 cents ? 18 cents ? 36 cents ? 63 cents ? 72 cents ? 54 cents ? 45 cents ? 27 cents ? 111. Make story problems about 9ths of 81. 112. What is iV of 90 ? -^^ of 90 ? j\ of 90 ? JL of 90? -^9oOf90? -/oOf90? ^Vof90? ^%oim? 113. What part of 90 is 9 ? 27 ? 18 ? 36 ? 63 ? 54 ? 45 ? 72 ? 81 ? 114. Thomas had 90 cents, James had ^^ as much money. How many cents had James ? William had -^^ as much. How many had William ? 115. If a yard of lace costs 90 cents, what part of it could be bought for 18 cents ? 63 cents ? 45 cents ? 116. Make story problems about lOths of 90. 117. What is the quotient when 63 is divided by 9 ? 118. A number that is used to divide another number is called a Divisor. Pick out the divisors : 63 -j- 9 = ? 33 - 11 = ? -^gO = ? -3_6. = 9 NINES 131 Divisor 9 )36 1 Quotient. Explain this as a new way of expressing division. 119. Divide and mark divisors and quotients : 9)72 5 )45 9)81 9 )63 5)35 9 )54 9 )99 120. In the new way set down 27 as a dividend and some number that will exactly divide it as a divisor and write the quotient. Do the same with 25, 18, 44, 55, 40, 66, 20, 50, 70, 77. Let pupils choose other numbers and their divisors and find quo- tients. 121. How much do five 259's equal ? Six 859's ? 122. If a piano costs f 295, what will 7 pianos cost at the same price ? 123. If there are 9 buttons on each shoe, liow many buttons are there on 3 pairs of shoes ? 124. Mrs. Smith has ^11.75 and wants to buy a rock- ing chair that costs $15.00. How much more money must she have ? 125. Louisa's mother had $20. She spent $4.75 for coal, $3.15 for shoes, and $8.75 for a cloak. How much had she left ? 126. Make story problems. 127. How many dollars and cents are 9 times $125.59? $212.55? $213.39? $991.95? $195.59? 128. Write in Arabic notation MDCCCLXIX and MDCCCXLIX and find their sum. 129. What is a multiplier? A divisor? The square of a number ? A square yard ? 130. Show three ways of expressing division. CHAPTER X THREES Multiplicand, Parallel Lines, Trapezoid, Ehombus, Eatio 1. Begin with three and count quickly by threes to 39. 2. Begin with 39 and count backwards by threes to 0. 3. Write the first 4 tens in columns, putting a square in the place of every third number, as below. 4. Learn the missing multiples of 3 and write them in the squares. 5. Write and learn the table of threes as far as "12 times 3 = 36."* 6. How many threes in 33? 27? 15? 12? 18? 7. Add 3 threes to 21, 27, 18, 15, 9, 30, 24, 12. 8. Subtract 3 threes from 21, 36,27,18,30,33. ♦Playing *' Numbers Out," a device for learning the multiplication table, is contributed by a very successful teacher and v^^armly indorsed by her pupils. In Numbers Out, the children stand around the room, leaving one side of the room where there is a blackboard vacant. Beginning at one end of the class, they number themselves 1, 2, etc. In playing ♦* Threes Out," when 3 is reached, or any multiple of 3, the child, 132 1 11 31 3 23 33 13 23 4 14 31 5 25 35 16 26 • 7 17 37 8 28 38 19 29 10 20 40 TIIiiEES 133 9. 4 threes + 2 threes = ? 11 threes — 2 threes = ? 10. How inaiiy threes must be added to 24 to equal 30 ? 11. How many threes must be taken from 21 to leave 18? 12? 9? 15? 12. What number will be equaled by adding 2 to 6 tln-ees ? By adding 1 to 9 threes ? By subtracting 1 from 10 threes ? By subtracting 2 from 8 threes ? Call for similar questions. 13. Multiply 13 13 13 13 13 13 13 13 by^ 3^^_6_7_8_9 14. Find products of 23 multiplied by each of the num- bers from 2 to 9. 15. Use as a multiplier of 53 each of the numbers from 2 to 9. 16. A number which is multiplied is called a Multipli- cand. Name the multiplicand in Ex. 15. In Ex. 13. 17. Use 33 as a multiplicand with each of the numbers that are greater than one and less tlian 10 as multipliers. 18. Show by grouping on the number table which is greater, 9 x 3 or 3 x 9, 11 x 3 or 3 x 11, 5 x 3 or 3 x 5. Turn to number tables in advance and let pupils show the equality of 8 X 3 and 3 x 8, 7 x 3 and 3 x 7, etc. 19. Draw on the boai'd a horizontal line 1 foot long and show how many times a 3-incli line can be measured off upon it. instead of calling the number, says "Out," goes to the blackboard, writes his number large and bold as high as he conveniently can, and takes his stand under it. When a suthcient number of children are out, the teacher calls on them to make statements about their numbers. "I stand for 27 or 9 threes," says one. "18 is my number. It equals 6 threes," says another. A child who in numbering around names a multiple of 3, or who says " Out " for any number that is not a multiple of 3, or who makes a wrong statement about his number, misses the game. 134 THREES 20. How many times can a 3-inch line be laid off upon a 9-incli line ? Upon a 15-inch line ? Upon a 27-inch line ? 21. A 3-inch line equals what part of a 12-inch line? Of a 15-inch line ? Of a 27-inch line ? 22. 3 is |- of what number ? 1 of what ? yL. of what ? J of what ? -J- of Avhat ? ^ of wliat ? -^^ of what ? 23. -V_ = 9 3^6. = '/ 24 - 3 = ? -1/ = ? 21 - 3 = ? 24. (3 X 8) - 2 = ? (3 X 6) ^ 2 = ? (3 x 10) -^ 5 = ? See note after Ex. 88, p. 111. Give quotients : 25. 3)18 3)27 3)21 3)15 3)24 3 )12 26. Write 9 as a divisor of each of the multiples of 9 that are less than 100, and give quotients. 27. 5 yards of ribbon are how many feet long? 4 yd. ? 7 yd. ? 9 yd. ? 28. A certain room is 21 feet long. How many yards of carpet must there be in each strip that runs the whole length of the room? 29. Measure the length of a room and tell how many yards of carpet it would take for each strip. 30. John lets out 36 ft. of kite string. How many yd. of strinor are let out? 'fc> 31. A rug is 12 ft. long and 9 ft. wide. How many yd. long is it? How many yd. wide? Picture it, and show how many yd. of binding it a\'ou1(1 take to go all around it. (Draw to a scale.) 32. Mary has a flower bed 3 yd. long and 2 yd. wide. Tiiiidv how it looks, and tell liow many feet of border it would take to go all around it. THREES 135 33. If you place rows of squares, each row containing three squares, until the figure is a perfect square, how many squares will there be in it? Let those who fail to image rightly do the actual placing or draw- ing of squares, but encourage imagery by excusing from objective work those who are able to give correct results without it. 34. How many square ft. in a square which is 3 ft. long? AVhat do we call such a square? 35. How many sq. ft. in 8 sq. yd. ? 4 sq. yd. ? 11 sq. yd. ? 9 sq. yd. ? 7 sq. yd. ? 12 sq. yd. ? 36. Add the square of 3, the square of 9, and the square of 10. 37. If you place 3 squares in a horizontal row and add equal rows of squares until you have 18 squares, how many rows will there be ? 38. If 24 squares are placed in the same way, how many rows will there be ? 39. What number of cents can be divided into five equal parts each of which is 3 cents? Each of which is 11 cents? 40. Draw a rectangle having horizontal lines 5 inches long and vertical lines 3 inches long. Divide it into square incJies, and find how many square inches there are. How many rows of square inches, and how many square inches in each row ? 41. When lines run in the same direction, they are said to be Parallel Lines. Draw two parallel horizontal lines. Draw three parallel horizontal lines. Three parallel ver- tical lines. 42. Draw two parallel lines slanting downwards to the left. Draw three parallel lines slanting downwards to the right. 136 THREES 43. Show parallel lines on the door ; on the window ; on your desk. 44. Can you name two streets or roads that are parallel ? 45. Think of two fences that are parallel, and tell where they are. 46. How many lines in the perimeter of this figure ? Which of the lines are parallel ? 47. If a four-sided figure has only two parallel sides, it is called a Trape- zoid. Draw a trapezoid like this. 48. Draw a trapezoid which shall be in this position. Draw another trapezoid in this position. Let pupils draw different kinds of trapezoids in different positions, and show parallel lines. 49. Copy Fig. 1. Can you separate the figure into two equal trapezoids ? Show ^ of the hgure. ^ of the figure equals how many sixths? Fig. 1 50. Copy Fig. 2 by placing equilateral triangles. How many triangles in Fig. 2? How many triangles would it take to make 7 such figures ? To make 9 such figures ? Can you divide Fig. 2 into 3 equal trapezoids? Hoav many triangles in each trapezoid ? Fig. 2 THREES 137 51. Show ^ of Fig. 2. Show ^ of it. How many ninths in i of it ? Show ^ of Fig. 2. How many ninths in I of it ? 52. How long would the perimeter of Fig. 2 be if a side of each triangle were 3 in.? How long if each side were 9 in.? If each side were 5 in.? 53. Copy Fig. 3 by placing triangles. How many tri- angles are used? How long would tlie perimeter of the figure be if each side of the triangles were 9 in. long? 3 in.? 54. Can you divide Fig. 3 into 3 equal trapezoids ? 55. Show how many ninths in ^ of Fig. 3. In f of Fig. 3. Fig. 3 56. Copy Fig. 4 by placing equilateral triangles. How many triangles does it take ? 57. HoAv long would the perimeter of the figure be if a side of each triangle were 3 in. long? 5 in. long ? 9 in.? 58. Show J of Fig. 4. Show ^V ^^ ^^* Show -^-Q of it. How many tenths equal ^ of it? 59. Separate Fig. 4 into 5 equal diamond - shaped figures. Each diamond is what fractional part of Fig. 4 ? 1 = how many tenths ? 60. A diamond-shaped figure is called a Rhombus. How many sides has a rhombus ? Has it any square corners ? 61. How long is the perimeter of a rhombus, each of whose sides is 11 in.? 5 in.? 9 in.? 3 in.? 138 THKEES 62. Copy Fig. 5 by placing equilateral triangles. How- many triangles in Fig. 5 ? Fig. 5 63. Show ^ of the figure you have made. Show -^q of it ; ^^^, -f^, -^q. 64. Separate your figure into 5 rhombuses, i = how many tenths ? f = how many tenths ? -| = how^ many tenths ? ^ = how many tenths ? 65. 3 is what part of 9 ? 27 ? 36 ? 18 ? 24 ? 33 ? 21 ? 66. What is i of 15 ? | of 15 ? |- of 15 ? | of 15 ? 67. What part of 15 is 12 ? 9 ? 3 ? 6 ? See note on Chart Drills after Ex. 53, pp. 117, 118. 68. If a yd. of cloth costs $.09, how much will -^ of a yd. cost ? I of a yd. ? 69. When nuts are | .12 a pound, what part of a pound can be bought for 1 .03 ? f .09 ? | .06 ? 70. If a yd. of ribbon costs f .15, how much Avill | of a yd. cost? I of a yd.? fyd.? f yd.? | yd.? 1 3-d.? 71. If 15 yd. of cloth cost a certain sum of money, what part of the money will 3 yd. cost ? What part will 9 yd. cost? 6 yd.? 12 yd. ? 72. How much is j of 21 ? f of 21 ? | of 21 ? f of 21? I of 21? fof21? 73. What part of 21 is 3 ? 9 ? 18 ? 6 ? 15 ? 12 ? 74. If a yd. of ribbon costs 21 cents, how much wall | of a yd. cost ? |- of a yd. ? ^ yd. ? -| yd. ? | yd. ? 75. Take 24 and show what part of it is 3, 9, 1 2, 6, 18, 21. 76. How many hours is it from 9 o'clock Monday morning till 9 o'clock Tuesday morning ? THREES 139 77. How many days from Monday morning to Wednes- day morning ? How many hours / From 9 o'clock to 12 o'clock is what fractional part of a day ? Refer to clock or watch. 78. What fractional part of a day is 6 hours ? 9 hours ? 15 hours ? 21 liours ? 79. From 6 o'clock in the morning to 6 o'clock at night equals how many hours ? What part of a day ? 80. From 11 o'clock in the morning until 2 o'clock in the afternoon equals how many hours ? What part of a day ? 81. From 10 o'clock in the morning till 4 in the after- noon equals what part of a day ? 82. Take each multiple of 3 that is less than 27 and show what part it is of 27. 83. 3 equals what part of 30, or what is the ratio of 3 to 30 ? Use these expressions interchangeably. 84. What is the ratio of 1 to 2 ? 1 to 3 ? 2 to 3 ? 1 to 5? 2 to 5? 4 to 5? 85. Give ratio of 3 to 9. 3 to 12. 3 to 24. 3 to 15. 86. What is the ratio of 9 to 27 ? To 81 ? 36 ? 72 ? 87. What number is ^^ of 30 ? j^^ of 30 ? ^V of -30 ? foOf30? ^2_of30? ^4_of30? fo-ofSO? ^s^of.SO? If the children try to memorize the statements of ratios without perceiving the relations of numbers, let them work out the ratios by dividing lines or grouping numbers on the number table. 88. What part of a yard is 3 inches? 6 in.? 18 in.? 24 in.? 33 in.? 21 in.? 15 in.? 9 in.? 27 in.? 89. What is the ratio of a 3-in. line to a line a yd. long ? 90. What is the ratio of 1 to 100 ? 6 to 100 ? 10 to 100 ? 140 THREES 91. Point out nuiiibers on the number table and tell tlieir ratio to 100. 92. What is the ratio of a foot to a yard ? 93. Think of a square foot, and with your finger out- line in the air its perimeter. 94. Outline in the air a square yard. What is the ratio of a square foot to a square yard ? 95. Show with your hands as nearly as you can the size of a pint measure ; the size of a quart measure. What is the ratio of a pint to a quart? In the same way show size of inch and foot ; peck and bushel ; quart and gallon, and give ratios. 96. How much is 30 multiplied by 4? By 6 ? 8? 3? 9? 7? 97. How much is 3 multiplied by 20 ? By 80 ? 30 ? 90? 60? 40? 98. How much is 3 multiplied by l^ ? 21 ? BJ- ? 81 ? 71? 91? 31? 51? 41? 121? 101? 111? See note after Ex. 91, p. 128. 99. 4 times ^ = how many whole ones and how many thirds over? 100. How much is 5 times ^? 8 times -^? 9 times J? 10 times -J? 12 times J? 15 times i? 18 times -J? 101. Find quotients : 3 )24 3 )18 9 )72 5)30 9 )63 9 )45 102. When 4 threes are subtracted from 14 what is the remainder ? 103. How much is the remainder when 2 threes are subtracted from 8 ? 104. Divide 10 by 3. What is the quotient and what the remainder? J THREES 141 105. jNIark divisor, quotient, and remainder in the fol- lowing examples : Divisor 5 )17 Quotient 3, Remainder 2. 5 )26 9 )38 3 )13 9 )19 9 )78 5 )39 3)19 9 )65 106. When a multiple of 3 is divided by 3, is tliere ever a remainder ? Explain. 107. Write all the numbers between 21 and 30 that are not multiples of 3, divide each of them by 3, and mark quotient and remainder. 108. Write in Roman notation the first 13 multiples of 9. 109. Write in Arabic notation MDCCCXCIX and MDCCCCV, and find their difference. 110. Find sums : 111. Find differences : 19.13 18.23 129.75 $18.29 8384.78 8.23 2.43 3.83 12.63 31.96 2.13 6.73 112. Jolm has $10.09 and wants to buy a bicycle that costs 1 25. How much more money must he get ? 113. John earns 12.07 to add to his 110.09. How much does he still lack ? 114. Some one gives him $.27. How much does he still lack ? 115. He earns $2.18 more. How much does he still lack ? Let the children make problems in which, as in the foregoing, there is a continued striving toward some desired end. 116. AVhat is meant by the words: Multiplicand^ Par- allel Lines, Trapezoid, Rhombus f CHAPTER XI EIGHTS Dexominator, Quart and Peck, Short Division, Divi- dend, Perpendicular Lines, Area of Right Tri- angle "1 NUMBER TABLE 1 11 21 31 41 51 61 71 81 91 2 12 22 32 42 52 62 72 82 92 3 13 23 33 43 53 63 7.) < 'J 83 93 4 14 24 34 44 54 64 <4 84 94 6 15 25 35 45 55 Gh 75 85 95 6 16 2(J 36 46 56 Ca) 76 86 96 7 17 27 37 47 57 67 77 87 97 8 18 28 38 48 58 68 78 88 98 9 19 29 39 49 59 69 79 89 99 10 20 30 40 50 60 70 80 90 100 1. Count by eights to 96. How many eights did yon count ? 2. Begin with 96 and count quickly hy eights bach to 0. 3. Write and learn the table that ends with "12 times 8 are 96." 142 EIGHTS 143 o 4. What is the 3d multiple of 8 ? 5th ? 9th ? 6th ? 11th? 7th? 12th? 8th? 4th? 10th? 5. Which multiple of 8 is 32 ? 48 ? 64 ? 16 ? 56 ? 6. (iive quotients : 8)16 8}48 8)64 8)80 8)56 8)32 8)72 8)24 7. Add 2 eights to 40. To 64. 48. 56. 24. 16. 40-2 eights = ? 80-2 eights = ? 56-2 eights = ? 8. How many eights must be added to 24 to equal 40 ? 56'^ 72? 48? 80? 64? 88? 96'^ 9. How many eights must be taken from 80 to leave 64? 48? 32? bi)? 24? 40? 16? 10. How many eights must be added to 32 to equal 9 eights ? 7 eiglits ? 5 eights ? 8 eights ? 6 eights ? 11. How many eights must be taken from 64 to leave 6 eights ? 3 eights ? 5 eights ? 4 eights ? 2 eights ? The game "Eights Out," like that described in the footnote on page 132, is useful. 12. 5 eights -f- 8 = ? 7 eights + 6 = ? 2 eights + 5 = ? 13. Find quotients and remainders : 8 )28 8)75 8)50 8)71 8)39 8)53 8 )47 14. How many must be added to 21 to equal 3 eights ? 15. may tliink of a number and tell how many must be added to it to equal eights. 16. may think of a number and tell how many must be subtracted from it to leave eights. 17. may think of a number less than 14, subtract it from 9 eights, and tell what is left. 18. Is the 3d multiple of 8 even or odd ? Can jon write a multiple of 8 that is an odd number ? Call attention to the fact that the endings of the multiples of 8 differ by 2 in regular order, 8, 6, 4, 2, 0. 144 EIGHTS 19. iMark products, multiplicands, and multipliers : 18 18 18 18 38 38 38 38 20. Use 28 as a multiplicand with each of the numbers that are greater than 1 and less than 10 as multipliers. 21. Use 5 as a multiplier with 119, 218, 318, 518, 918. 22. Eugene sold 3 times as many papers as his brother, who sold 28 papers. How many papers did Eugene sell ? How many more than his brother ? 23. Use 9 as a multiplier of 181, 251, 381, 581, 881, 981. 24. Use 82838 as a multij^licand with each of the odd numbers that are less than 10 and greater than 1. 25. Use 85898 as a multiplicand witii each of the even numbers that are less than 10. 26. What number must be used as a multiplier of 8 to produce 32? 24? 56? 72? 96? 64? 88? 48? 16? See note after Ex. 88, p. 111. 27. How many sheets of paper must be divided among 5 children to give each child 8 sheets ? 10 sheets ? 9 sheets ? 28. If there are 8 cherries in a bunch, how many cher- ries are there in 10 bunches ? In 12 bunches? In 6 bunches ? 29. If it takes 8 eggs for a cake, how many cakes can be made with 2 dozen eggs ? With 48 eggs ? 30. Make problems using the number 8. 31. What is the ratio of 8 to 16 ? Use chart drill as suggested in note after Ex. .53, pp. 117, 118. 32. A boy offers to trade a big apple for 16 marbles. How much of the apple ought he to give for 8 marbles ? 33. AVhat is the ratio of 8 to 24 ? Of 16 to 24 ? EIGHTS 145 Children slioiild be led gradually to see such facts as that the ratio of the first multiple of any number to its second multiple is | ; of the 2d to the 3d, f ; of the 5th to the 10th, ^. 34. Two boys receive 24 cents for cutting some wood. The big boy does f of the work. How much should he receive ? How much should the small boy get ? 35. What is ^ of 32? fof32? fori of 32? 36. What part of 32 cents do 16 cents equal ? 24 cents ? 37. Mary has ^ as many cents as Harriet, who has 32 cents. How many cents has Mary? 38. If 8 men can do a piece of work in 4 days, how long will it take 1 man to do the same work ? 2 men ? 39. What is the ratio of 8 to 40 ? How much is f of 40 ? 4 of 40? fof40? 40. What part of 40 is 16 ? 32 ? 24 ? 41. How many cents are | of 40 cents? | of $.40? 42. John had 40 cents and lost | of them. How many cents did he lose, and how many had he left ? 43. A C D E F B . The line AB repre- sents a distance of 40 miles, divided into 5 equal parts. How far is it from A to (7? ^ to .E^? J. to i) ? A to F? CtoE^i BtoB? FtoB? 44. What is the ratio to the whole distance of the dis- tance from Ato 0? AtoF? FtoB? BtoB? O to F? CtoF? AtoB? FtoB? 45. Fill out and learn the following : 1 of 48 = The ratio of 8 to 48 is •| or |- of 48 = The ratio of 16 to 48 is I or 1 of 48 = The ratio of 24 to 48 is I or f of 48 = The ratio of 32 to 48 is f of 48 - The ratio of 40 to 48 is HORN. ARITH. 10 146 EIGHTS 46. Tell quickly what is the ratio to 48 of each of the multiples of 8 less than 48. 47. Three boys caught 48 fish. John caught ^ of them, James -J of them, and Henry ^ of them. How many fish did each boy catch? 48. Two men bought 48 bu. of apples, one man paying for J of them, and the other man for the rest. How many bu. ought each man to receive? 49. If the apples cost $ 15, how much should each man pay? 50. Make a table showing sevenths of 56 from y to |^. 51. Learn to give quickly the numbers whose ratio to .^6 I'c 5 3 6 4 2 OU lb y, ■^, f, y, y. 52. If 56 marbles were divided equally among 7 chil- dren, how many marbles would 2 children receive? How many would 4 children receive ? 6 children ? 53. Albert missed 8 words in spelling 56 words. What fractional part of the words were spelled wrong ? Right ? 54. John had 56 cents and spent 16 cents. How many sevenths of his money did he spend ? How many sevenths did he keep ? How many cents ? 55. Make story problems about sevenths of 56. 56. Draw on the board an 8-inch square and divide it into inch-squares? How many rows of squares? How many squares in each row? 57. How many squares in ^ of the figure? In | of it? In f of it? In I of it ? In i of it ? In i of it? 58. What part of the whole figure are 8 squares ? 24 squares ? 56 squares ? 16 squares ? 32 squares ? 48 squares? 40 squares? 1 square? 7 squares? 13 squares? EIGHTS 14 n 59. Make a list of the numbers that are ^, |, |, |, ^, |, i I of 72. 60. Learn to give quickly the ratio to 72, of 24, 8, 48, 64, 32, ,%, 16, 40. 61. If 72 men do a piece of work in a day, how much of it is done by 8 men ? 32 men ? 40 men ? 64 men ? 16 men ? 62. 8 has the ratio ^ to what number? Find -^^ of that number -"^ -3_ _4_ __8_ _5_ _S_ _6_ 63. Give quickly the ratio to 80 of each of the multi- ples of 8 that are less than 80. 64. If a yd. of cloth costs f .80, how much of it can be bought for 1.08? 1.16? ^.40? 1.24? $M? i.64? 65. Make story problems. 66. In fractions the number that is written below the line is called the Denominator. Name the denominator of |-, ^, ^^~. 67. Write and read a fraction Avith 8 as the denomi- nator and 7 for the number above the line. 68. AVrite and read a fraction with 8 as the denomi- nator and some odd number for the other number. 69. Write and read a fraction with 9 as the denomi- nator and some even number for the other number. 70. Write several fractions with 8 for the denominator and some other multiple of 8 for the number above the line. Tell what each equals. Will not the children see that a fraction is a form of division ? 71. Write several fractions with 9 for the denominator and a multiple of 9 for the other number, and tell what each fraction equals. 148 EIGHTS 72. What is the value of the fraction that has 6 for its denominator and 3 for the number above the line ? 73. What is the product of 80 multiplied by 6? 8? 11? 9? 7? 74. What is the product of 8 multiplied by 30 ? 70 ? 50? 90? 60? 40? 80? 75. What is the product of 80 multiplied by 30 ? 40 ? 90? 60? 70? 76. How much is 6 eights and |^ of 8 ? 7 eights and 1 of 8 ? 9 eights and | of 8 ? 77. Louise has 8 cents, and Mary has 2| times as many. How many cents has Mary ? 78. Make story problems. 79. How many quarts make a peck ? Use actual measurements. 80. How many qt. equal 3 pk. ? 5 pk. ? 9 pk. ? 10 pk. ? 81. Fill out and learn the table of Dry Measure. pints (pt.) = 1 quart (qt.). quarts = 1 peck (pk.). pecks = 1 bushel (bu.). 82. How many quarts in 3 pk. -H 7 qt. ? 4 pk. + 5 qt. ? 83. What is the ratio of a qt. to a pk. ? 3 qt. to a pk. ? 84. How many pk. and qt. in 18 qt. ? 27 qt. ? 33 qt. ? 85. Find quotient : 3 )36 Show the process of dividing tens and units separately. Lead the children to see that they get the same result by this process as by grouping numbers. 86. Find quotients : 5^55 2)28 9)99 8)88 2)242 2)264 3)336 2 )226 EIGHTS 149 87. Mary is making badges 3 inches long. How many can she make out of a piece of ribbon 63 inches long ? 88. Among how many children can 88 cherries be divided, giving each child 4 cherries ? 89. Find quotients and remainders : 3 )964 2 )245 3 )865 2 )4843 9 )93 9 )185 9 )276 90. Use 8 as a divisor of 489, 645, 568, 168, 327, 720. 91. A number which is divided by another number is called a Dividend. Name the dividends in Ex. 89. 92. With 5 as divisor use as dividends 105, 255, 458. 93. AYith 9 as divisor use as dividends 279, 364, 723. 94. With 8 as divisor use as dividends 643, 167, 489. 95. If 8 squares are placed in a row, how many rows must there be to use 40 squares ? 56 squares ? 96. If 8 squares are placed in a row, how many rows must there be to make the figure a perfect square ? How many squares in the figure ? What is the square of 8 ? 97. A triangle that has a square corner is called a Right Triangle. Draw a right triangle. 98. Make right triangles by bisecting an inch-square. The lines that meet to form the square corner are perpen- dicular to each other. How long is each of the perpen- dicular sides of the triangles you have made ? 99. Place 8 triangles as in Fig. 1. If you take away the four outside triangles, what kind of a figure will be left ? 100. What is the ratio of the figure that is left to the figure as it was at first ? 101. To how many square in. is your copy ^^' of Fig. 1 equal. 150 EIGHTS 102. Show by Fig. 1 how many eighths equal ^ ; how many eighths equal ^. 103. In the fraction -^, which number is the denomi- nator ? 104. Copy Fig. 1 by drawing. Make each of the per- pendicular sides of the triangles one inch long. 105. Copy Fig. 2 by placing triangles. How many such figures could you make with 24 triangles ? 72 ? 48 ? 80 ? 96 ? 106. Copy Fig. 2 by drawing, making the perpendicular sides of each triangle 1 inch long. How many square inches in your copy ? Fig. 2 107. What is tlie ratio of one of the triangles to the whole figure ? Of three triangles to the whole ? Of 5 triangles to the wliole ? 108. How many sq. in. in a rectangle 8 in. long and 7 in. wide ? 8 in. long and 5 in. wide ? 8 in. long and 8 in. wide ? What is a rectangle called that is as long as it is wide ? 109. Draw two square inches, divide them into halves, and letter them as in Figs. 3 and 4. Which is greater, the triangle ABD, or the rectangle EFOD ? Fig. 3 Fig. 4 EIGHTS 151 Fig. 5 Besides bringing out the fact that they are each one half of a square inch, let the equality of the figures be shown by cutting off the upper part of the tri- angle and fitting it to the lower part to form a rectangle, as in these figures. 110. Copy Fig. 5 by drawing, making the horizontal lines 2 inches long and the ^ vertical lines 1 inch long. 111. How many square inches in the tri- angle 7li>C? In ABO? Let the children prove their answers by cutting and fitting surfaces. 112. How many sq. in. in a right triangle 8 in. long and 3 in. wide ? 8 in. long and 7 in. wide ? 113. How many sq. in. in a triangle, one of whose per- pendicular sides is 9 in. long and the other 7 in.? 114. What is the area of a right triangle 9 in. long and 8 in. wide ? 115. What is the area of a right triangle 9 in. long and 6 in. wide? 116. When the length of the perpendicular sides of a right triangle is given, how is the area of the triangle found ? 117. Make a trapezoid by placing 5 equilateral triangles. 118. Copy Fig. 6 by placing equi- lateral triangles. How long would the perimeter of Fig. 6 be if each side of the triangles w^ere 8 in.? If each side were 5 in.? 11 in.? 9 in.? 119. Can you take away 3 rhombuses from Fig. 6 and leave a trapezoid? 120. What is the ratio of each rhombus to the whole figure ? Fig. 6 152 EIGHTS 121. What is the ratio of the trapezoid to the whole figure ? 122. Copy Fig. 7 by placing equi- ateral triangles. Can you take away 3 trapezoids from the figure and leave one triangle ? 123. One trapezoid has what ratio to the whole figure ? Two trapezoids have what ratio to the whole figure ? 124. How long would each side of the triangles be if the perimeter of Fig. 7 were 12 in.? 60 in.? 96 in.? 125. Find quotients : 8 )1688 8 )3288 8 )4880 8 )6488 2)f| 126. Divide: 2 )356 178 Show this process of short division. 127. Divide : 3)726 3 )654 3 )9381 3 )427 5 )115 5 )275 5 )385 5 )4355 9 )828 9 )738 9)1269 9 )1648 128. Mr.HoAve divided $2268 among 4 grandchildren. How much did each receive ? 129. How many weeks are there in 1323 days ? 130. What number multiplied by 8 will give 968 ? 131. Divide 187 by each of the numbers 2, 3, 5, 8, 9. 132. Divide 437 by 2, 3, 5, 8, 9. 133. Divide 493 by 2, 3, 5, 8, 9. 134. Divide the first odd number after 209 by 2, 3, 5, 8. 135. At 8 cents a qt., how much will a pk. of berries cost ? EIGHTS 153 136. At 8 cents a qt., how much do half a pk. of berries cost ? 137. How many ounces in |^ a pound ? In 1 J pounds ? 138. At 8 cents apiece, how much will a dozen pine- apx:)les cost? How much change ought you to get from a dollar bill after paying for them ? 139. If you bought 3 yd. of ribbon at 8 cents a yd., and gave the clerk a quarter of a dollar, how much change ought he to give you ? 140. If you bought 6 yd. of calico at 8 cents a yd., and gave the clerk half a dollar, how much change should you get? 141. Anna bought 7 yd. of ribbon at 8 cents a yd., and gave the clerk a half dollar and a dime. How much change was due ? 142. Write and add the square of 8, the square of 5, and the square of 9. 143. A Sunday school wishes to buy an organ which costs 1)125. The treasurer has $75.08. How much more is needed ? 144. 114.23 more were paid in. How much was still needed? 145. After 128.13 more were paid in, how much was needed ? 146. Mr. Brown gave |10 toward the organ. How much more was raised than the organ cost ? 147. If the rent of a house is 122.50 a month, how much will the rent for the summer months be ? 148. Write in Arabic notation, and divide by 8, CXV, XCV, CLXII, CCCCLIX, MDCCCXC. 149. Explain the words : Denominator^ Dividend, Divisor, Right Triangle, Perpendicular Lines. CHAPTER XII FOURS ' Numerator, Square Prism, Partial Products, Ton 1. Begin with 4, and count by fours to 48. 2. Count by fours from 48 until nothing is left. 3. Write the first five tens in vertical columns, putting a square in the place of every fourth number. 1 11 21 31 41 33 D 43 3 D 13 14 33 33 31 43 n 6 15 D 35 36 35 45 46 17 18 07 D 37 38 47 9 10 19 29 30 154 39 D 49 50 FOURS 155 4. Learn the missing multiples of 4, and write them in the squares. 5. Write and learn the table ending " 12 times 4 are 48"." 6. Name different multiples of 4, and tell whether they are even or odd. 7. Name the 3d multiple of 4, the 7th, 9th, 12th, 2d. a -2 0.''' X§.'^ -1^'^ M'^ 40-'^ 48-''* ^-'^ -M-''' AA'^ 9. Add 2 fours to 20, to 36, to 24, to 16, to 32. 10. Take 3 fours from 48, 16, 40, 32, 44, 24, 20. 11. 40 -2 fours =? 48 -3 fours =? 32 -4 fours =? 12. 8 fours + 2 fours =? 7 fours + 5 fours =? 13. 7 fours — 3 fours =? 9 fours — 4 fours =? 14. How many fours must be added to 16 to equal 28? 15. How many fours must be taken from 32 to leave 28? 20? 12? 16. How many fours must be added to 20 to equal 6 fours ? 8 fours ? 17. How many fours must be taken from 16 to leave 3 fours ? 1 four ? 18. 3 fours + 7 = ? 9 fours — 1 = ? 19. 7 fours + 3 = ? 11 fours - 2 = ? Call for similar questions from pupils. 20. How many fours in 8 ? In 2 eights ? 3 eights ? 21. (8 X 3) - 4 = ? (8 X 5) ^ 4 = ? (8 x 6) -^ 4 = ? See note after Ex. 88, p. 111. 22. How many quarters of a dollar equal a whole dollar ? How many quarters equal 3 dollars ? 5 dollars ? 7 dollars ? 23. How many quarters of a dollar equal one half dollar ? 1 and 1- dollars 1 2^- dollars ? 156 FOURS 24. If I cut 6 apples into fourths, to how many boys can I give one fourth of an apple ? 25. To how many boys could I give ^ of an apple if I had 5 apples ? 7 apples ? 9 apples ? 10 apples ? 26. HoAV many dollars will it take to give 20 boys a quarter of a dollar apiece ? 36 boys ? 48 boys ? 24 boys ? 27. How many pounds of coffee, at a quarter of a dollar a pound, can be bought for 1 2 ? For 1 3 ? For $5 ? 28. How many fourths in the whole of anything ? 29. How many fourths in two whole things ? In 4 whole ones ? 3 ? 7 ? 30. How many fourths in 2 wliole ones and ^ ? In 3| ? 31. How many fifths in 2i ? 3f ? 5| ? 4^ ? 8^ ? 9f ? 32. Which is greater, and how much, 7 bu. or 29 pk.? 9 bu. or 34 pk.? 18 pk. or 5 bu.? 23 pk. or 6 bu.? 33. How mau}^ pk. in a bushel and a half ? In 2 bushels and a half? In 3|^ bu.? 4^ bu.? 5-| bu.? 6J bu.? 34. What is the ratio of a quarter of a dollar to 2 dollars ? A peck to 3 bushels ? A quart to 4 gallons ? 35. # = ? Ans. 2J-. 36 1—9 11—? _1J — '? 1_5 — ? i_a — ? _1^ — ? OD. 2 ~~ ' 2 ~ ' 2 ~ ' 2 * 2~' 2 5 — ? 9 — ? 11—? _1 3. _ ? J_a — ? 2JL — ? ^— • ^— • 1[— • ¥~- 4~- 4~- 37. Divide: 4 )897 4 )9893 4 )827 4 )7389 4)6253 Let the quotients be expressed in mixed numbers. 38. Divide by 4 each of the numbers between 200 and 300 Avhose unit figure is 7. 39. Use 4 as a divisor of each of the numbers between 300 and 400 whose unit figure is 5. FOURS 157 Fig. 1 Fig. 2 40. How many equilateral tri- angles in Fig. 1 ? Copy it by plac- ing triangles. Can you separate your figure into three equal large triangles ? Each large triangle is what fractional part of the whole figure ? Each small tri- angle is what part of the large triangle to which it belongs ? 41. How much is :| of J? How many twelfths in -J? In -I? 42. Copy Fig. 2 by plac- ing triangles. How many triangles in it ? 43. Each triangle is what fractional part of the whole figure ? Show J of the figure. Show J of it, and tell how many 12ths | equal. Show ^ of it, and tell how many twelfths ^ equals. 44. Separate your figure into 6 rhombuses. Each rhom- bus is what fractional part of the whole figure ? Each triangle is what fractional part of a rhombus ? 45. ^ = how many 12ths ? | = how many 12ths ? I = how many 12ths ? ^ = ho^\' many 12thg ? I = how many 12ths ? What is j of } ? If the children cannot see the facts whicli these questions are intended to bring out, do not let them memorize them. Come back to the work again with a different figure and lead them on more slowly. • 46. Copy Fig. 2 again. Separate it into 4 equal trape- zoids. Each trapezoid is what fractional part of the whole figure ? Each triangle is what fractional part of a trapezoid ? 158 FOURS 47. ^ = how many 12ths ? -| = how many 12ths ? |. = how many 12ths ? i of ^ = what ? 48. Put the figure together again. Separate it into 3 equal parts. How many triangles in each part ? 49. 1 = how many twelfths ? f = how many 12ths ? ^ of ^ = what ? 50. Draw^ a circle and draw a line across it, dividing it into halves. Show pupils how to draw a circle by the aid of dividers or a string or a slip of pasteboard turning on a pin. Fig. 3 51. Divide each half of the circle into halves. What is J of ^ ? Fig. 4 52. Divide each fourth of the circle into halves. What part of the whole is J of ^ ? Fig. 5 53. Divide each eighth of the circle into halves. How manv divisions in the whole circle ? What part of the whole is J of ^ ? Let these circles, large and bold, be drawn upon the board and left there for some time. The children should discover and report from them the facts called for in the following questions, and in many similar ones. Gradually discard this work with the concrete, and lead the pupils to the use of figures as symbols. 5*- ^-*=? J-f=? i-*=? i+i=? i+i=? i+i-? Fig. 6 FOURS 159 55. The whole circle — J-g = how many 16ths ? 56. i - Ve 3 _ _ 8 16 ! + -!- = 8 ^ 16 16 „ — '> T ■g" 16 8 ' 16 • ¥ "^ 16 • 57. Name the denominators of some of the fractions in Ex. 56. 58. In a written fraction, the number above the line is called the Numerator. Name the numerators of some of the fractions in Ex. b6. 59. Write a fraction with an odd number for the numerator and an even number for the denominator. 60. Write a fraction with 4 as the numerator and 8 as the denominator, and tell what the fraction equals. 61. Write a fraction with 4 as the numerator and 12 as the denominator, and tell what it equals. 62. Write a fraction with 5 as the numerator and 10 as the denominator, and show what it equals. 63. Write some fractions in which the luimerator is just J as large as the denominator, and show what each fraction equals. 64. Write some fractions in which the denominator is just 3 times as large as the numerator, and show what each fraction equals. 65. Write some fractions with 4 as the denominator and a multiple of 4 as the numerator, and tell what each fraction equals. 66. What is i of 12 ? I of 12 ? 67. What is* the ratio of 4 to 8 ? 4 to 12 ? 8 to 12 ? Use chart drill as suggested in note after Ex. 53, pp. 117, 118. 68. There are 12 apples in a basket and |- as many on a plate. How many apples are on the plate ? 160 FOURS 69. What is ^ of 16 ? I of 16 ? | or J of 16 ? 70. What is the ratio of 12 to 16 ? Of 8 to 16 ? 71. If John has 16 marbles and James ^ as many, how many marbles has James ? 72. The price of 4 marbles is what part of the price of 8 marbles ? How much will 4 marbles cost if 8 marbles cost 10 cents ? 14 cents ? 20 cents ? 24 cents ? 40 cents ? 73. The cost of 4 marbles is what part of the cost of 12 marbles ? How much will 4 marbles cost if 12 mar- bles cost 15 cents ? 24 cents ? 30 cents ? 18 cents ? 74. What part of a pound is 4 ounces ? 8 ounces ? 75. If a pound of candy costs 40 cents, what part of the money will 4 ounces cost ? How many cents will they cost ? 76. How much will 8 marbles cost if 16 marbles cost 10 cents ? 20 cents ? 30 cents ? 40 cents ? 50 cents ? 77. What is I of 20 ? | of 20 ? | of 20 ? | of 20 ? 78. What is the ratio of 8 to 20? 16 to 20 ? 12 to 20 ? 79. If a yard of ribbon costs 20 cents, how much will ^ of a yard cost ? f ? ^? f ? 80. If 20 men do a piece of work in a day, how much of the work is done by 4 men ? 12 men ? 8 men ? 81. If f 40 is paid for the work, how much should be paid to 4 men ? 12 men ? 8 men ? 16 men ? 82. Fill out and learn the following : i of 24 = The ratio of 4 to 24 is — I or J of 24 = The ratio of 8 to 24 is — / I or J- of 24 = The ratio of 12 to 24 is — I or I of 24 = The ratio of 16 to 24 is — I of 24 = The ratio of 20 to 24 is — When these ratios can be given instantly, give and call for many illustrative problems. FOURS 161 83. How many hours in i of a day ? In |^ of a day ? 84. What part of a day is 12 hours ? 8 hours ? 85. John bought something for 24 cents and sold it so as to gain 6 cents. What was the ratio of the gain to the cost ? 86. Take the number 28 and make a table showing ^, f, ••• -{ of it. 87. Make a table showing what ratio each of the mul- tiples of 4 less than 28 has to the number 28. 88. If a pound of candy costs 28 cents, how much will f of it cost? f? f? I? 5? 89. If 28 men do a piece of work in a week, what part of it is done by 4 men ? 12 men ? 8 men ? 16 men ? 90. If they are paid |>14 a day, how much will 4 men receive for each day's work ? 8 men ? 18 men ? 24 men ? 91. Take 32 and make a table showing i, |, ••• | of it. 92. Make a table showing the ratio to 32 of each of the multiples of 4 that are less than 33. Require these ratios in their lowest terms. 93. If a pound of candy costs 32 cents, how much will | of it cost? f? I? 1? 1? I? 94. If 32 cents are paid for some candy, how much of it can be bought for 4 cents ? 12 cents ? 28 cents ? 95. A flower bed is 8 ft. long and 4 ft. wide. Make a picture of it. How many sq. ft. in ^ of it ? In | of it ? 96. Make a table showing -J, |, ... -| of 36. 97. Make a table showing the ratio to 36 of each of the multiples of 4 that are less than 38. 98. j\[r. Smith is 36 years old. How old is his son whose age is | of Mr. Smith's age ? How old is his daughter, whose age is | or -J of her father's ? His wife's HORN. ARITH. 11 162 FOURS age is I of his, how old is she ? His brother's age is | of his, how old is his brother ? His sister's age is -J of his, how old is his sister ? 99. A bolt of cloth contains 36 yd. What part of it is 20 yd.? 24 yd.? 32 yd.? 16 yd.? 12 yd.? 28 yd.? 100. If the whole bolt is worth $ 18, how mucli will 4 yd. cost ? 8 ydo? 12 yd.? 20 yd.? 28 yd.? 32 yd.? A B 101. Fig, 7 rep- resents a rectangle 9 feet long and 4 feet wide. How many square yards <-^ are represented by ABOD? Draw Fig. •^^^- ' 7 and show how many square yards the whole figure represents. 102. What is the ratio of the square, ABOB^ to the whole figure ? 103. Make a table showing ^ ••• i^ of 40. 104. Make a table showing the ratio to 40 of each of the multiples of 4 that are less than 43. 105. A certain town is 40 miles from New Orleans. D How far from the town is a man who has traveled -^ of the distance from it to New Orleans ? How far is he from New Orleans ? 106. When he has traveled -^^ of the way to New Orleans, how far is he from the town he started from ? How far is he from New Orleans ? How far is he from each place when he has traveled -^^ or J of the way ? ^ of the way ? -^ of the way ? 107. Find, by measuring, how many gills make a pint. How many gills in 3 pints ? 5 pt. ? 9 pt. ? 6 pt. ? FOURS 163 108. Fill out and learn the table of Liquid Measure. gills (gi.) = 1 pint (pt.) pints = 1 quart (qt.) quarts = 1 gallon (gal.) 109. How many gi. in a qt.? 2 qt.? 5 qt.? 9 qt.? 110. How many gi. in IJ pt.? 3^ pt.? 5|- pt.? 6^pt.? 71 pt.? 11 qt.? -ifqt.? 111. How much is 4 multiplied by 30? 20? 60? 90? 70? 50? 80? 40? 120? 150? 112. How much is 40 multiplied by 2 ? 7 ? 8 ? 12 ? 113. How much is 40 multiplied by 30 ? 80 ? 60 ? 114. How many are 3 fours and J of 4 ? 7 fours and i of 4 ? 9 fours and | of 4 ? See note after Ex. 91, p. 128. 115. Thomas has 4 marbles, and James has 3| times as many. How many marbles has James ? 116. Make story problems. 117. 5 times ^ equals liow many wliole ones and fourths over ? 118. How many whole ones and how many fourths over in 6 times ^? 7 times ^? 9 times |^? 12 times f ? 8 times 1? 11 times f ? 16 times ^? 30 times i? 40 times ^ ? 29 times ^ ? 3 times |^ ? 5 times -|. How many square inches in : 119. A rectangle 9 in. long and 4 in. wide ? 120. A right triangle 9 in. long and 4 in. wide? 121. A rectangle 4 in. long and 2J in. wide? 122. A right triangle 4 in. long and 3| in. wide ? 123. A rectangle 8 in. long and i in. wide ? 124. A right triangle 12 in. long and ^ in. wide? 134 FOURS Give each child inch -cubes, and lead the class to find surfaces, edges, and angles. 125. Each side of a cube is called a face. How many faces has a cube ? 126. Show some parallel lines on a cube. Show per- pendicular lines. 127. If you were to paste a strip of paper along each edge of an inch-cube so as to bind the edges, how many inches long would all the strips be ? 128. How many right angles has each face ? How many right angles have all the faces? 129. Make a layer of inch -cubes 4 inches long and 2 inches wide. How many cubes in it ? 130. Cover this layer with another layer of inch-cubes. How many cubes in the whole figure ? 131. How man}^ cubes would it take to build it up 3 layers high ? To build it 5 layers high ? 7 layers ? 6 layers ? 9 layers ? 132. If it were 4 layers high, one cube Avould be what part of the whole figure ? 133. Figures like these you have built are called Square Prisms. Build with inch-cubes a prism 5 in. long, 4 in. wide, and 2 in. high, and tell how many cuV)es in it. 134. Build a prism 3 in. long, 3 in. wide, and 2 in. high. How many layers of cubes are in it? How many cubes in each layer? Let the children use the cubes to build prisms until they are able to get the required facts by means of their mental imagery, then encourage them to " think how it looks." Do not let illustrative work become formal nor take the place of thuiking. FOURS 165 135. Find the number of cubic inches in each of the fol- lowing prisms, the measurements being given in inches : Length Width Height Length Width Height 5 3 1 4 3 3 5 2 2 8 4 2 5 4 2 8 3 2 3 4 2 4 2 9 136. How many cubic inches in a cube each edge of which is 2 in. long? 3 in.? 4 in.? 5 in.? 137. How many square inches in all the faces of a 2- inch cube ? 3-inch cube ? 4-inch cube ? 5-inch cube ? 138. An inch cube equals what part of a 2-inch cube? What is the ratio of an inch cube to a 3-incli cube ? To a 4-inch cube ? To a 5-inch cube ? 139. How many inch-cubes can you put into a box that measures on the inside 5 in. long, 4 in. wide, and 3 in. deep? 140. Estimate the length, width, and depth of a box, and tell how many cubic inches it can hold. Let the children inclose portions of space with blocks and tell how many cubic inches in them. ft/ 141. How many cubic feet of air can there be in a closet that is 4 ft. long, 3 ft. wide, and 9 ft. high ? 142. Multiply 444 by each of the numbers that are greater than 2 and less than 10. Multiply 44 The process of mnltiplying by a number by 11 of more than one place should be shown "~TT simply as a process which brings the desired result. Later, when the children have become '^^ expert and are ready for the insight, show 484 them that in multiplying, for instance, 444 by 111, they are finding the sum of one hundred 444's, ten 444's, and one 444. 166 FOURS 143. Find products: 64 54 24 54 84 94 n n 11 1? 12 12 144. Multiply 14 by itself. 145. Square 15, 21, 22, 23, 35, 38, 88, 89, 48, 49, 58, 59. 146. Multiply 444 by each number between 10 and 20. Find the cost of: 147. 18 yd. of cloth at il.55 a yd. 148. 29 hats at f 1.45 apiece. 149. 35 pounds of tea at il.l5 per pound. 150. 345 tons of coal at $8.25 per ton. 151. There are 2000 pounds in a ton. How many pounds in 40 tons? In 70 tons? IJ tons? 2^ tons? 152. Add: 41 4J 3 J 51 6 J 9| 2 41 4^ 7| 8f 8| fi !i 5i ?i ^ if 153. ]\Iary wants to buy for her mother a Christmas present that costs f 3.00. She has -t 1.22. How much does she lack ? 154. She saves 1^.35 more. How much does she still lack? 155. She earns 15 cents a week for 7 weeks. How much does she still lack ? 156. She saves 5 cents a week for 9 weeks. Does she lack any then ? If so, how much ? 157. Find (luotients: 4 )3801 4 )7897 9 )8205 158. Find I of 8476, 8264, 8148, 9365. 159. One of the girls may name a number of 5 places, and the class may use it as a dividend with 4. 160. Write in Arabic notation MDCCXCIX and MDCCCXLIV, and find their difference. CHAPTER XIII SEVENS Factors, Compound Fractions 1 SIUMBER TABLE 1 11 21 31 41 51 61 71 81 91 2 12 22 32 42 52 62 72 82 92 3 13 23 33 43 53 63 73 83 93 4 14 24 34 44 54 64 74 84 94 5 15 25 35 45 55 65 75 85 95 6 16 26 36 46 56 6G 76 86 96 7 17 27 37 47 57 67 77 87 97 8 18 28 38 48 58 68 78 88 98 9 19 29 39 49 59 69 79 89 99 10 20 30 40 50 60 70 80 90 100 1. Begin at 7 and connt by sevens to 98. Practice until you can count quickly. Ho^y many multiples of 7 are less than 100 ? 2. Begin with 98 and count backAvards by 7 to 0. 3. Write and learn the table ending '•■ 12 times 7 are 84." 4. Name in order all the multiples of 7 that are less than 100 and are odd numbers. 167 168 SEVENS 5. Name in order all the even mnltiples of 7 that are less than 100. 6. Name a few multiples of 7 greater than 100. 7. How many sevens in 21 ? 84 ? 42 ? 56 ? 63 ? 28 ? 8. Give the 5th multiple of 7 ; 7th, 9th, 11th, 4th, 6th. 9. Multiply 777 by each of the numbers that are greater than 45 and less than 50. 10. If a bicycle costs 157, how much will a dozen bicycles cost ? 11. At §87 apiece, what is the cost of 24 bicycles ? 12. At i477 apiece, how much would 13 pianos cost? 13. Add 2 sevens to 42, 63, 21, 49, 70, 14, 28, 56, 35. 14. Take 2 sevens from 77, 42, 21, 56, 84, 63, 28, 49, 35. 15. How many sevens must be added to 28 to make 42 ? 5(j? 70? 84? 63? 35? 49? 77? 16. How many sevens must be subtracted from 84 to leave 70 ? 5(j ? 77 ? 63 ? 49 ? 42 ? 28 ? 14 ? 35 ? 17. How many sevens must be added to 21 to make 4 sevens ? 7 sevens ? 9 sevens ? 6 sevens ? 8 sevens ? 18. How many sevens must be taken from 77 to leave 9 sevens ? 6 sevens ? 8 sevens ? 10 sevens ? 5 sevens ? 19. 5 sevens4-3 = ? 8 sevens + 5 = ? 6 sevens + 6 = ? 20. 8 sevens — 5 = ? 11 sevens — 6 = ? 7 sevens — 5 = ? 21. Which multiple of 7 is 35? 49? 77 ? 21 ? 56? 84? 22. Add 2 to the 3d multiple of 7. To the 6th. 10th. 23. Subtract 4 from the 2d multiple of 7. From the 7th, 9th, 5th. 24. 50 is how many more than the 7th multiple of 7 ? How many less than the 8th multiple of 7 ? Call for similar questions from pupils. SEVENS 169 25. 7— • 7~- 7' 7- 7* 7 1 5 _ '> 3 8—9 6_6 _ 9 89. _ ? 3^L — '? 5_3 _ ? 7— • 7~~* 7~- 7'~- 7~' 7 26. At 7 cents a yard, how many yards of ribbon could you buy for 15 cents ? 23 cents ? 29 cents ? 36 cents ? 27. Make story problems. 28. Divide by 7 each of the numbers between 900 and 1000 whose unit figure is 4. 29. At i 7 apiece, how many music boxes could be bought for 1500? 1800? 1900? 30. Multiply 797 by each of the even numbers between 31 and 39. 31. Multiply 897 by each of the odd numbers between 40 and 50. 32. What number must 7 be multiplied by to make a product of m ? 63 ? 84 ? 49 ? 42 ? 33. Numbers that make a product are called Factors of that product. Name two factors that make 14, 35, 77, 21, 49, 63, 84, 42, 28, 56, 70. 34. 8 is a factor of 24. Name the other factor that helps 8 to make 24. 4 is also a factor of 24. Name the other factor that helps 4 to make 24. 35. 2 is one of a pair of factors that help each other to make 24. What is the other factor ? 36. Give all the pairs of factors that you can of 12, 18, 20, 30. 37. 44 = 4 X ? 25 = 5 X ? 28 = 4 x ? 42 = 7 x ? 38. Write all the multiples of 8 that are less than 48, and the pairs of factors into which they can be divided. Let pupils begin with the smallest factors and work regularly ; as, 8 = 2x4; 16 = 2 X 8 or 4 X 4 ; 24 = 2 X 12 or 3 X 8 or 4 X 6. This is an excellent review of the multiplication table. 170 SEVENS 39. Write all the multiples of 9 that are less than 50, and give factors of them. 40. How many times can a 7 -inch line be measured off upon a 21 -inch line ? Upon a line that is 2 ft. and 4 in. long ? Upon a line that lacks an inch of being equal to a yard ? Upon a line that lacks 4 in. of being 5 ft. long ? Upon a line that is 1 yd. 1 ft. and 1 in. long ? 41. How many rows of squares, 7 in a row, make 42 squares ? 28 squares ? 63 squares ? 49 squares ? 42. Place or draw 9 inch-squares so as to make a perfect square. How long is the figure ? How wide is it ? 43. Draw a square containing 49 square inches and mark them off. Give length and width of the figure. 44. Draw 16 inch-squares placed in a perfect square. How long is each side of the square ? 45. Draw 25 inch-squares arranged in a perfect square. How long is each side of the square ? 46. What is the square of 3? 4? 5? 7? 47. Find the square of 17 ; 27 ; 77 ; 97 ; 47. 48. What number, multiplied by itself, will give 9 ? 25 ? 16 ? 100 ? 49. Give other numbers that are made of two equal fac- tors. 50. A number that is made of two equal factors is called a Perfect Square, and each of the equal factors is called a Square Root of the number. What is the square root of 49 ? 25 ? 81 ? 16 ? 51. Write some other numbers that are perfect squares, and give their square roots. Oral class exercises like the following are useful : " Take the square root of 9, double it, add 4, divide by 2, square, subtract 5, take |, add," etc. SEVENS 171 52. The floor of a square room contains 49 sq. ft. How long is one side of the room ? How many feet around all the edges of the floor ? 53. How many feet around the edge of the floor of a closet if the floor is square and contains 25 sq. ft.? 54. How many feet around the edge of a square floor that contains 16 sq. yd.? 25 sq. yd.? 6-4 sq. yd.? 55. How many feet around a room 11 ft. long and 7 ft. wide ? 56. How much will a pt. of oil cost at 7 cents a gi.? At 9 cents '' 3 cents ? 8 cents ? 57. If it takes you 7 minutes to Avalk to school, how many minutes do you spend in walking to school each day ? How many in a week of 5 school days ? In a school month or 20 days ? 58. At 7 cents a qt., how much will a gal. of milk cost ? 59. How much will 3 yd. of wire cost at 7 cents a ft. ? At 7 cents an in. ? 60. 7 times 7 in. = how much more than 4 ft. ? 61. How much will 2 qt. and 1 pt. of berries cost at 7 cents a pt. ? At 14 cents a qt. ? 62. How many apples would it take to give 6 boys 7 apples apiece ? 63. How many marbles must you have to give 7 marbles to each of 8 boys? 64. How many days in 4 weeks ? 5 Aveeks ? 8 weeks ? 6 weeks ? 3 weeks and 4 days ? 5 weeks and 1 day ? 65. What part of a week is 1 day ? 3 days ? 5 days ? 66. HoAv many weeks and sevenths of a week in 11 days? 22 days? 30 days? 36 days? 44 days? 50 days ? 69 days ? 82 days ? 172 SEVENS 67. How many weeks in 583 days ? 687 days ? 599 days ? 1601 days ? 68. How many inch cubes will it take to make a square prism 7 in. long, 3 in. wide, and 2 in. high ? 69. How many cubic inches in a box 7 in. long, 4 in. wide, and 2 in. deej) ? 70. How many cubic feet of space in a closet 7 ft. long, 6 ft. wide, and 8 ft. high ? 71. How many cubic feet of space in a closet 7 ft. square and ft. high ? 72. Arrange 7 equilateral triangles as in Fig. 1. Can you take aAvay two trapezoids from Fig. 1 and leave one triangle ? 73. What is the ratio of one triangle to the whole figure ? Of one trapezoid Pj^ "I^ to the whole figure ? Of two trapezoids to the whole figure ? 74. Copy Fig. 2. Take away two trapezoids and show what is left. 75. Can you show how Fig. 1 may be changed into Fig. 2 by changing the position of one trapezoid ? 76. How long would the perimeter of Fig. 2 be if each side of a triangle were ^'^- ^ 7 in. ? 8 in. ? 9 in. ? 5 in. ? 11 in. ? 77. Take away f of Fig. 2. Tell how long the perim- eter of the figure that is left would be if each side of the triangles were 7 in. long. 78. How many sevenths make the whole of anything ? 79. How many sevenths in 2 whole ones? 9? 5? 7? 80. How many 7ths in 2| ? 3f ? 5f ? 7| ? 8 f ? 9f ? SEVENS 173 Fig. 3 Fig. 4 Fig. 5 81. Copy Fig. 3 by placing equilateral triangles. Each tri- angle is what part of Fig. 3 ? 82. Divide each equilateral tri- angle into two right triangles as in Fig. 4. Each right triangle is what part of the whole figure ? J of ^ = ? 83. Copy Fig. 5 by placing equilateral triangles. 1 triangle is what part of Fig. 5 ? 84. Separate your copy into halves. 1 triangle is what part of a half of the figure ? i of ^ = ? Show pupils that "when we wish to find the value of a compound fraction, instead of dividing and subdividing an object and counting the parts, we multiply the numerators of the fractions together for a new numerator, and the denominators for a new denominator. Let them try the plan with some small fractions and prove its correctness by building figures and separating them into parts. 85. |0fl = ? J0f3=? i0fi = ? i0ii = ? 86. A fraction of a fraction is called a Compound Frac- tion. Write some compound fractions and find their values. 87. }0ii = ? |0f J = ? |ofVo=? }0ii = ? 88. |of J = Jj. \¥hat will f of 1 equal ? How much are f of ^ ? -fofj? fofj? iof|? ioii=:? ioff=? |off = ? foff = ? I of |- = ? (Cancel when you can.) |- of ^^ = ? 92. What is the ratio of 7 to 14 ? 7 to 21 ? 14 to 21 ? 28 to 21 ? 35 to 21 ? Use chart drill. 89. 90. 91. 174 SEVENS 93. If a yard of ribbon costs 21 cents, how much ribbon can you get for 7 cents ? 14 cents ? 28 cents ? 42 cents ? 94. What is the ratio of 7 to 28 ? 21 to 28 ? 14 to 28 ? 95. If 28 ajDples cost 10 cents, how much will 14 apples cost ? 96. A bed of pinks 2 feet square is a part of a floAver bed 7 ft. long and 4 ft. wide. Make a picture of them, drawing to a scale of 1 inch to a foot. What is the ratio of the bed of pinks to the whole flower bed ? 97. How much is i of 35 ? -| ? | ? 98. 35-iof35 = ? 35-fof35 = ? 35--|of35=? 99. What part of 35 is 28 ? 14? 21? 42? 100. Mary had 35 cents and spent ^ of them. How many had she left ? 101. Mr. Baker borrowed f 35. When he had paid | of it, how many dollars did he still owe ? 102. John had 35 marbles, and Albert had ^ as many. How many had Albert ? 103. Mr. Lane's watch chain is worth -| as much as his watch, which is worth f 35. How much is the chain worth? How much are they both worth? The watch is worth how much more than the chain ? 104. Fill out and learn the following: 1 of 42 = The ratio of to 42 is f or ^ of 42 = The ratio of to 42 is f or 1 of 42 = The ratio of to 42 is f or I of 42 = The ratio of to 42 is f of 42 = The ratio of to 42 is f of 42 = The ratio of to 42 is SEVENS 175 105. Fill out the following table of Avoirdupois Weight : — — ounces (oz.) = 1 pound (lb.). pounds = 1 Ton (T.). 106. How many oz. in 2 lb. and 3 oz. ? 2 lb. 7 oz. ? 107. A jar contains 2 lb. and 10 oz. of l)utter. How many oz. in the jar ? How many oz. in ^ of it ? | of it ? lofit? fofit? lofit? 108. There are 42 gal. of oil in a barrel. When i of the oil is drawn out, how many gal. are left ? How many are left when | or ^ of the oil is drawn out ? When |- or I is drawn out ? 109. Grace wishes to buy a doll that costs 42 cents. She has 35 cents. What part of the price has she, and how many sixths of the price does she need ? 110. Make a table showing the 7ths of 49 from ^ to ^. 111. Make a table showing the ratio of each multiple of 7 that is smaller than 50, to 49. 112. Mr. Jones can do a piece of work in 49 days. How many days will it take him to do ^ of it ? How many Aveeks ? How many days will he need to do |^ of it ? 1^ ? 1^ ? ^ ? How many weeks in each case ? 113. There are 49 yards in a bolt of cloth. What frac- tional part of it remains after 7 yd. of it are sold ? After 21 yd. are sold ? After 35 yd. are sold ? 114. Gertrude's age is f of 49 years. Her father's age is ^ of 49 years, and her mother's age is |- of 49 years. How old is each one of the family ? 115. How much younger is Gertrude than her father ? Than her mother ? 116. Gertrude's age is in what ratio to her father's age? To her mother's age ? 176 SEVENS 117. Make a table showing the 8ths of 56 from ^ to |. 118. Make a table showing what ratio each multiple of 7 that is smaller than G-l has to 56. 119. Mary spent 56 days in a visit to her aunt at St. Louis. When she had been there a week, what part of her visit was past ? What part of it was past when she had been there 21 days ? 35 days ? 49 days ? 120. A farmer brought 56 lb. of butter to market and sold I of it. How many lb. had he left ? 121. He received $ 14 for his 56 lb. ; how much did he receive for 28 lb. of it ? 122. Anna has 56 cents. How much will she have when she has spent ^ of it ? | of it ? | of it ? | of it ? 123. Make a table showing the 9ths of 63 from J to ■^-. 124. Make a table showing what fractional part of 63 each multiple of 7 is that is less than 75. 125. A gentleman's house is 63 miles from Denver. When he has traveled -J of the way to Denver, how far is he from his own house ? How^ far from Denver ? Tell how far he is from each place when he has traveled -| of the way. |. f . -J. |. 126. A garden is 63 ft. long and 42 ft. wide. What is the ratio of its width to its length ? 127. Make a tal)le showing lOths of 70 from -^ to \^. 128. Make a table showing what fractional part of 70 each multiple of 7 is that is less than 75. 129. Arthur, William, and Thomas gave 70 cents in charity. Arthur gave -^^^ or ^, of the money, William gave y^^, and Thomas gave -f^, or |. How many cents did each give ? SEVENS 177 130. John had 14 cents, and wished to buy a 70-cent cap. What part of the cost of the cap had he ? What part did he lack ? Wliat part did he lack when he had gained 21 cents more ? 131. If one fan costs 7 cents, how many fans can be bought for 56 cents ? 28 cents ? 63 cents ? 84 cents ? 132. If 2 fans cost 14 cents, how much will 3 fans cost ? 5 fans ? 7 fans ? 4 fans ? 8 fans ? 133. How much will 8 fans cost if 7 fans cost 21 cents ? 63 cents ? 49 cents ? 84 cents ? 28 cents ? 56 cents ? 134. How much will 9 fans cost if 7 fans cost 21 cents ? 49 cents ? 63 cents ? 14 cents ? 28 cents ? 56 cents ? 135. What is the product of 70 multiplied by 8 ? 6 ? 11 ? 136. What is the product of 7 multiplied by 30 ? 60 ? 90 ? 137. How much are 2 sevens and | of 7 ? 3 sevens + f of 7 ? Let pupils practice giving quickly the products found by multiply- ing 7 by each of the smaller integers + i, f, f, etc. 138. How many sq. in. in a rectangle 7 in. long and 5| in. wide ? How many in a right triangle of the same length and width ? 139. Multiply 9 27 77 7 7 247 328 by 10 10 10 100 1000 20 40 140. Tell how you multiply a number by 10 or 100. 141. Show how you multiply a number by 20, 30, or any multiple of 10. 142. In a right triangle one of the sides that makes the right angle is called the Base and the -I other the Altitude. Find the area of a right tri- § angle whose base is 7 inches and altitude 10 ^ inches. Base HORK. ARITH. — 12 178 SEVENS 143. By holding a paper triangle in different positions, show that the same line may sometimes be called the base and sometimes the altitude. 144. Draw right triangles, making the bases and alti- tudes of any measurements you choose, and find the areas. 145. Add: 7I 9f 6f H ^ n ^ 4f 4f H If 3f If n 6f ^ 9f H 2^ 8f n 146. There are 4| yards of cloth in May's dress, and 2| yards in her jacket. How many yards are there in both ? 147. From 7|- 35f 16f 27f 25^ 64f 34 take 4 Hi _^ JH; ly 2f 1} 148. For five years Mr. Smith's family saved money to raise the sum of $1000, to send John to college. The first year they saved §85.75, the second year f 98.50, the third year ^f 197.50. How much more was needed to make up the 1 1000 ? 149. In the fourth year they laid aside $195.45, and John's uncle sent him a check for $ 150. How much more was needed ? 150. In the fifth year they saved $245.50, and John earned the rest in vacation. How much did John earn ? 151. If a boy earns $3.95 a week, how much will he earn in 17 weeks ? In 19 weeks ? What will be the cost of : 152. 19 tons of hay at $9.75 per ton ? 153. 23 barrels of flour at $4.75 per barrel ? 154. 24 copies of "Alice in Wonderland" at $1.27 per copy ? 155. 28 bu. of wheat at $ .87 a bu.? SEVENS 179 Find cost of each article if : 156. 3 tons of hay cost 'f 24.75. 157. 5 barrels of flour cost -i^ 26.25. 158. 7 geographies cost f 4.55. 159. 9 bushels of wheat cost 88.19. Let the children find the actual prices of various articles in common use in their locaUty, and bring in problems based upon them. 160. Name two factors of the 3d multiple of 7. Of the 5th. 9th. lltli. 8th. 6tli. 3d. 7th. 12th. 161. Of wliat number is 7 the square root ? How long is one side of a square that contains 49 square inches ? 162. What number that has 7 for a factor is nearest to 20 ? 30 ? 50 ? 163. How much greater is tlie product of 7 and 8 than their sum ? 164. Thomas went fishing at 7 o'clock in the morning, and came home 7 hours later. At what time did he come home ? 165. Write in Arabic notation CCCCXXVII and VH, and find their quotient. 166. Write in Roman notation the present year, the year in which you were born, the year in which the Declaration of Independence was signed. CHAPTER XIV SIXES Rod, Hexagon, Ixterest NUMBER TABLE 1 11 21 31 41 51 61 71 81 91 2 12 22 32 42 52 62 72 82 92 3 13 23 33 43 53 63 73 83 93 4 14 24 34 44 54 64 74 84 94 5 15 25 35 45 55 65 75 85 95 6 16 26 36 46 56 66 76 86 96 7 17 27 37 47 57 67 77 87 97 8 18 28 38 48 58 68 78 88 98 9 19 29 39 49 59 69 79 89 99 10 20 30 40 50 60 70 80 90 100 1. Add 6 to 0, and keep on adding sixes until you have 12 sixes. What number have you ? Which multiple of 6 is it ? 2. Begin at the 12th multiple of 6 and count backwards by sixes until nothing is left. 3. Learn the table of sixes as far as " 12 times 6 are 72." 180 SIXES 181 Give rapid drill on 6's by aid of this figure. 4. How nifiny sixes in 24 ? 42 ? 54 ? 66? 36 ? " 48 ? 72 ? 30 ? 5. Add 2 sixes to 18. 30. 48. \ ^ 60. 54. 24. 6. Subtract 2 sixes from 48. 60. 24. 36. 66. 11 7. How many sixes must be added to 12 to equal 30 ? 36? 24? 18? 42? 8. How many sixes must be taken from 48 to leave 5 sixes ? 7 sixes ? 4 sixes ? 6 sixes ? 9. Read and give quotients quickly: 183060364872246642 54 6666666666 10. If you put 36 square inches into a perfect square, how long would one side of the figure be ? How long would the perimeter of the figure be ? 11. Of what number is 6 the square root ? 12. Multiply 6666 by the square root of 36 ; by the square root of 49 ; by the square root of 64; by the square root of 81. 13. Use 6666 as a multiplicand with each of the numbers greater than 55 and less than 63. 14. Use 6 as a divisor with each of the numbers between 500 and 600 whose unit figure is 9. 15. What factor helps 6 to make 18 ? 30? 54? 72? 60? 16. Name two numbers, neither of which is 6, that mul- tiplied together give 18. 30. 24. 72. 12. 36. 17. Write as many sets of factors of 24 as you can. Of 26. 48. 30. 28. 40. 32. 20. 16. 182 SIXES 18. Write the multiples of 6 as far as 72, and give the sets of factors into which they can be resolved, as: 6 = 2x3; , 12 = 2x6 or 3x4; 18 = 2 X 9 or 3 X 6 ; 24 = 2 X 12 or 3 X 8 or 4 X 6. Let the children select composite numbers, and call on the class for the factors of them. Factoring is very useful in helping children to see the relations of numbers, and is not difficult for them if they know the multiplication tables. 19. Name some numbers that are made of two equal factors, and give the factors. 20. Fill out the blanks in the following sentence : " One of the two equal factors that make a number is called the of that number." 21. 15 X 15 = 225. What is the square root of 225 ? 22. Of Avhat number is 16 the square root ? 21 ? 23 ? 18? 23. How many lb. of sugar could be bouglit for 48 cents when sugar is 6 cents a lb. ? 8 cents a lb. ? 4 cents alb.? 24. To how many boys could you give 6 marbles apiece if you had 36 marbles ? 66? 42? 30? 25. If there were 6 peas in a pod, how many peas in 9 pods? 7? 12? 8? 6? 26. Mrs. Adams cut the pies at a picnic. She cut 7 pies into sixths, and they were all eaten up. Each person had one piece of pie. How many persons were at that picnic? 27. Anna has 3 times as much money as Mary, who has 6 cents. How much have both ? 28. John has 5 times as much money as James, who has 6 cents. How much more has John than James ? SIXES 183 29. Five boys started a game of marbles. Fred, one of the boys, had no marbles, and so each of the other boys lent him 6 marbles. How many had Fred to start with ? 30. Fred won 35 marbles. After he had paid his play- mates, how many did he have left ? 31. What is the ratio of 6 to 12 ? 6 to 18 ? 12 to 18 ? 32. The cost of 12 apples is what part of the cost of 18 apples ? What would 12 apples cost if 18 apples cost 30 cents ? 15 cents ? 21 cents ? 24 cents ? 9 cents ? 33. What is the ratio of 6 to 24 ? Of 12 to 24 ? Of 18 to 24 ? 34. How much is 1 of 24 ? f of 24 ? | of 24 ? 35. How much will 12 fans cost if 24 fans cost f .50? 1.60? i.80? 36. How many hours is it from Cj A.M. on Monday till 6 A.M. on Tuesday ? 37. From 6 a.m. till noon is wliat part of 24 hours? 38. From 6 a.m. till 6 p.m. is what part of 24 hours? 39. From 6 A.M. till midniglit is what part of 24 hours? 40. What is the ratio of G to 30 ? How much is f of 30? I of 30? I of 30? 41. What part of the price of 30 apples is the price of 6 apples? 24 apples? 18 apples? 12 apples? 42. If 20 cents are paid for 30 apples, how much Avill 6 apples cost? 12 apples? 24 apples? 18 apples? 43. Fill out and learn the following : i of 3G = The ratio of — to 36 is — I or J of 36 = The ratio of — -to 36 is — J or |- of 36 = The ratio of — to 36 is — J or I of 36 = The ratio of — to 36 is — 1^ of 36 = The ratio of — to 36 is — 184 SIXES 44. What part of the price of 36 hats is the price of 6 hats ? 18 hats ? 30 hats ? 24 hats ? 12 hats ? 45. How many inches in J of a yard ? In |- ? -i ? | ? 46. To a line a yard long what is the ratio of a line 6 in. long ? Of a line 1 ft. long ? Of a line a foot and a half long ? Of a line 2 ft. long ? Of a line 1^ ft. long ? 47. Make a table showing 7ths of 42 from ^ to -|. 48. Express the ratio to 42 of each of the multiples of 6 less than 50. 49. 42 gallons of oil are in a barrel. Tell what frac- tion of it is gone and what fraction of it is left when 12 gal. have been drawn out. 24 gal. 30 gal. 36 gal. 50. If the whole barrel was w^orth $ 28, how much would 6 gal. cost ? 12 gal. ? 24 gal. ? 30 gal. ? 36 gal. ? 51. Make a table of the 8ths of 48 up to |. 52. Express the ratio that each of the multiples of 6 less than 61 has to 48. 53. How many hours from 12 o'clock noon on Monday to 12 o'clock noon on Wednesda}'? 54. From 12 o'clock till 6 p.m. on Monday is what part of 48 hours ? 55. From noon till midnight is what part of 48 hours ? 56. From midnight till 6 p.m. is what part of 48 hours ? 57. From 6 a.m. Monday to 6 p.m. Tuesday is what part of 48 hours ? 58. From 6 A.M. Monday till noon on Tuesday is what part of 48 hours ? Make a table showing 9ths of 54 up to JJ- 59. Express the ratio that each of the first 10 multiples of 6 has to 54. SIXES 185 60. The price of marbles is what part of tlie price of 54 marbles? If 54 marbles cost 18 cents, how much will 6 marbles cost ? 18? 30? 12? 24? 48? 42? 36? 61. 60 seconds make a minute. How many seconds in 2 minutes ? Give the children ideas of minutes by requhing them to keep per- fectly still for one minute by the watch, and of seconds by having them beat time, one beat to a second. 62. Fill out and learn the table of Time : seconds (sec.)s= 1 minute (min.). minutes = 1 hour (hr.). hours = 1 day (da.). days = 1 week (wk.). 63. How many sec. in 3 min.? 5 min.? 9 min.? 4 min.? 8 min.? 11 min.? 7 min.? 6 min.? 12 min. 64. How many sec. in 8 min. and 2 sec? 4 min. and 7 sec? 9 min. and 3 sec? 8 min. and 10 sec? 65. Make a table showing lOths of 60 up to ^^. 66. Make a table showing the ratio to 60 of each of the first 12 multiples of 6. 67. What part of a nun. is 12 sec? 30 sec? 18 sec? 54 sec? 42 sec? 36 sec? 48 sec? 68. How many min. in 2 hr. and 8 min.? 5 hr. and 9 min.? 6 hr. and 25 min.? 7 hr. and 48 sec? 69. How many minutes in ^ an hour ? ^? ^? J-? |? i ? 2 ? _i_ ? __3_ ? JL. 9 _9_ ? 3 • 3 • 10 • 10 • 10 • 10 • 70. How many minutes in 3 J hr.? 5^ hr.? 7^ hr.? 8fhr.? 43^^hr.? Q-^hv.? 10-^% hr.? 71. What part of an hour is 15 minutes? 30? 45? 10? 5? 6? 24? 54? 36? 48? 42? 18? 186 SIXES 72. Draw a line on the floor 5i yards long. How many feet long is it ? It is 1 rod long. How many feet in 2 rods ? 3 rods ? 4 rods ? 7 rods ? 11 rods ? Let rods be measured off in the yard or upon the pavement by- means of a string 5| yards long, and give practical questions in meas- urement. 73. How many feet in the perimeter of an equilateral triangle, each side of which is 1 rod long ? 74. Make a table showing lltlis of 66 up to ^^. 75. Express the ratio of each of the first 12 multiples of 6 to 66. 76. Robert's house is 66 rods from the schoolhouse. When he has gone 6 rods on his way to school, what part of the distance has he gone, and what part has he yet to go ? 77. What part of the distance has he gone, and what part has he to go when he has gone 18 rods ? 30 ? 42 ? 48 ? 80 ? 78. Draw on the floor a square that is 1 rod long each way. How many feet in its perimeter ? 79. How many square rods in a garden that is 9 rods long and (3 rods wide ? 8 rods long and 7 rods wide ? Let the children mark off on the playground a jDieee of ground a number of rods long and a number of rods wide, and find the number of square rods. 80. 320 rods make a mile. How many rods in 4 miles ? 9 miles ? 12 J miles ? Call on pupils to mention places that are about a mile distant. 81. Fill out and learn the table of Long Measure : inches (in.) = 1 foot (ft.). feet = 1 yard (yd.). \ = 1 rod (rd.). yards j ^ rods = 1 mile (mi.). SIXES 187 82. John lived a mile south of the post office. His cousin Henry lived 2i mi. south of it. How many miles must John walk to go to the post office, then on to his cousin's house, and home again ? How many rods ? 83. ^lake a table showing the 12ths of 72 up to -^|. 84. Express the ratio of each of the first 12 multiples of 6 to 72. 85. A field is 9 rd. long and 8 rd. wide. How many so rd in -i. of if^ -5-'^ -^"^ il'^ ^-'^ 1'^ ^'^ l*^ &q. iu. ill ^2 ^^ ^^ ' 12" 12* 12" 2* 4* ?* 3* 1'? 19 5 ? 3 • 6 • 6 • 86. What part of a field 12 rd. long and 6 rd. wide is a lot 2 rd. long, and 2 rd. wide? 4 rd. long and 3 rd. wide? 9 rd. long and 2 rd. wide? 8 rd. long and 3 rd. wide? 6 rd. square? 11 rd. long and 6 rd. wide? 12 rd. long and 4 rd. wide? 15 rd. long and 4 rd. wide ? 87. Draw a diagram of a right triangle whose base is 6 rd. and altitude 3 rd., drawing to a scale of 1 inch to a rod. Find its area. 88. What is the ratio of that triangle to a rectangle 6 rd. long and J as wide as long ? Draw. 89. HoAv many inch cubes will be needed to make a square prism 6 in. long, 3 in. wide, and 2 in. high ? 90. How many cu. ft. in a tank 6 ft. long, 5 ft. wide, and 7 ft. high? 91. How many cu. iuo of air can there be in a box 6 in. long, 4 in. wide, and 3 in. high ? 92. If a solid 3 in. long, 2 in. wide, and 2 in. high were placed in the box mentioned in Ex. 91, how many cu. in. of air would be left in it ? 188 SIXES 93. How many cu. ft. of air can there be in a room 60 ft. long, 40 ft. wide, and 12 ft. high? 94. If a solid 6 ft. long, 5 ft. wide, and 4 ft. high were placed in tlie room mentioned in Ex. 93, how many cu. ft. of air would be left in the room? 95. Estimate the length, width, and height of the schoolroom, and find about how many cu. ft. of air the room will hold. Let the children suggest different rectangular solids to be placed in the room, and find the cu. ft. of air displaced by them. 96. What is meant by ^ of anything? 97. Draw a picture of a pie cut into sixths, and tell how many sixths of a pie in 2 equal pies. In 4. 7. 8. 5. 98. How many sixths in 1} ? In 2^ ? In 3f ? In 91 ? In 8| ? In 71 ? In 6| ? ^^* 6~' 6~" 6~~' 6~' 6 6 100. Co]3y Fig. 1 by placing equilteral triangles. How many sides has the figure ? A figure bounded by six straight lines is called a Hexagon. 101. How long would the perimeter of the hexagon be if each side of a triangle Avere 6 in. long ? 8 in.? 9 in.? 7 in.? 102. How many such hexagons could you make from 42 such triangles ? 36 ? 72 ? 24 ? 54 ? 66 ? 103. Divide your hexagon into trapezoids. What is the ratio of each trapezoid to the hexagon ? 104. Divide your hexagon into rhombuses. What is the ratio of each rhombus to the hexagon ? T^IG. 1 SIXES 189 105. One of the triangles is what part of the hexagon you have made ? 3 triangles ? 2 triangles ? 4 triangles ? 5 triangles ? 106. Divide each triangle that makes your hexagon into 2 right triangles, as in Fig. 2. Hold up one of the right triangles, and show its right angle. 107. Each of your right triangles is what -^^^- ^ part of an equilateral triangle ? What part of the hexa- gon ? J of 1 = ? 108. Draw several hexagons of different shapes. 109. Form 6 equilateral triangles into a figure that is not a hexagon. Show J of J of the figure. 110. What kind of a fraction is i of | ? How can the value of that kind of fractions be found ? 111. Find value : iofj Joff ^ofi fof^ iofl fofl foff foff foff fofl fofl I of A lofi foff foff I of I JjOfl ^Vofi 5 nf 1 5 nf A JL. nf 1 _9_ of 5 _8_ of ^ -9- of ^ 11 ^^6 TT '^^ 6 12 ^^ 6 12 ^^ 6 12 ^^ 6 12 ^^ 6 112. How many rods in 1 mile and 80 rods ? 113. George rode on his bicycle 1 mile lacking 10 rods. How many rods did he ride ? 114. At 6 cents a pt., what will be the cost of 3 qt. of milk ? Of a gallon ? Of a gallon and a half gallon ? 115. At 6 cents a qt., how much will a pk. of beans cost? A pk. and a half? A bu.? 116. Mary borrowed 60 cents from her brother, and paid him 6 cents of it every week, How many weeks did it take her to pay the whole ? 190 SIXES 117. Paying 6 cents a week, how many weeks would it have taken her to pay 48 cents ? 72 cents ? 78 cents ? 84 cents ? 118. Sometimes when people borrow money they pay 6 cents for every dollar that they keep for a whole year, besides paying back the dollar. If you lent i 3 for a year at that rate, how much would you receive at the end of the year besides the $ 3 you had lent ? Explain that interest is paid for the use of money just as horse hire is paid for the use of a horse, or rent for the use of a house. 119. The money which is paid for the use of money is called Interest. If 6 cents are paid for the use of a dol- lar for one year, how much interest must be paid if it is kept 4 years ? 7 years ? 9 years ? 6 years ? 8 years ? 120. If the interest of a dollar for one 3^ear is 6 cents, what is the interest of 3 dollars for a year ? Of i 5 ? f 7 ? 19? 112? 16? 18? 14? .$11? 121. John lent $S for a year at 6 per cent. (That means 6 cents for every dollar.) How much interest did he get ? 122. At 6 per cent, what is the interest of -f 1 for IJ years ? For 2 J years ? 4J years ? 6^ years ? 8^ years ? 123. James lent some money at 6 per cent, and received 42 cents interest at the end of a year. How many dollars did he lend ? How many dollars must he lend at that rate in order to get 72 cents ? 36 ? 54 ? 66? 48 ? 124. Arthur has a dollar in a bank that pays 6 per cent. How much interest will it give him in 2^ years ? In 4J years ? In 5^ years ? In 9i years ? 125. What is the interest of a dollar for 7 years at 5 percent? (5 cents for every dollar.) For 9 years? P^or 12 years ? SIXES 191 126. How much will William receive at the end of a year on each dollar that he has in a bank which pays 3 per cent ? How much would he receive for i 10 ? 127. If a bank paid 5 per cent, how much would May receive at the end of a year if she had $ 8 in it ? How much if she had 1 5 ? 17? 14? |10? |9? 128. At 6 per cent, how long must I keep a borrowed dollar to pay 2-1 cents interest ? To pay 18 cents ? 54: cents ? 72 cents ? 60 cents ? 129. Which gives the more interest at the end of a year : $ 8 loaned at 6 per cent, or f 7 at 7 per cent ? 130. Per cent is sometimes written %. What is the interest of a dollar for 3 years at 6 % ? For 3 years at 8 % ? 131. Anna may tell how much money she would like to have at interest, and how much it would bring her each year at 6%. At 5%. At 4%. Let this exercise be general. 132. If it costs §286.50 to make 6 wagons, how much will it cost to make 1 wagon ? 5 wagons ? 133. If it costs $47.75 to make one wagon, for how much must the maker sell it to gain -f 5.25? 134. At 6 cents a ft., how much will it cost to fence a lot 1 rd. square ? 2 rd. square ? 20 rd. square ? Notice that in the following examples one unit of the minuend must be reduced to fractional units. 135. From 27f 38 47 56 28 96 take lli Ji ii Ji ^ _1 136. Write in Roman notation two factors of 77. CHAPTER XV TWELVES Long Division, Square Foot, Cubic Foot, Common Multiple NUMBER TABLE 1 11 21 31 41 51 61 71 81 91 2 12 32 42 52 62 72 82 92 3 13 23 33 43 53 63 73 83 93 4 14 24 34 44 54 64 74 84 94 5 15 25 35 45 55 65 75 85 95 6 16 26 36 46 m 66 76 86 96 7 17 27 37 47 57 67 77 87 97 8 18 28 38 48 58 68 78 88 98 9 19 29 39 49 59 69 79 89 99 20 30 40 50 60 70 80 90 100 1. Count by twelves to 96. How many twelves did you count? 2. Count by twelves from 96 to 144. How many twelves from 96 to 144? 3. Count backwards by twelves from 144 to 0. 4. Write and learn the table of twelves up to 144. 192 TWELVES 193 5. How many twelves in 48 ? 72 ? 36 ? 96 ? 60 ? 6. Name the 7th multiple of 12 ; the 5th ; 9th i 6th. 7. Add 2 twelves to 24, 60, 36, 84, 12, 48, 72, 108, 96. 8. Take 2 twelves from 96, 60, 144, 108, 48, 84, 36, 72. 9. How many twelves must he added to 24 to make 60? 36? 72? 48? 96? 10. How many twelves must be subtracted from 96 to leave 72 ? 48 ? "^ 24 ? 84 ? 36 ? 60 ? 11. How many twelves must be added to 36 to equal 6 twelves ? 5 twelves ? 9 twelves ? 7 twelves ? 12 twelves ? 12. How many twelves must be taken from 108 to leave 7 twelves ? 4 twelves ? 6 twelves ? 3 twelves ? 5 twelves ? 13. 3 twelves + 5 = ? 7 twelves + 9 = ? 9 twelves + 10 = ? 14. 9 twelves — 4 = ? 7 twelves — 6 = ? 4 twelves — 9 = ? 15. How many sixes in 2 twelves? 3 twelves? 5 twelves ? 8 twelves ? 6 twelves ? 16. How many threes in 12? In 2 twelves? 3 twelves? 4 twelves ? 6 twelves ? 5 twelves? 17. How many fours in 12 ? In 2 twelves ? 3 twelves ? 4 twelves ? 5 twelves ? 7 twelves ? 18. How many twelves in 4 sixes ? In 10 sixes ? 8 sixes ? 19. How many twelves in 3 eights ? In 6 tens ? 12 sixes ? 20. 12 is ^ of what number ? i of what? J of what ? 1 of what ? -^ of what ? -i of what ? -l of wliat ? 21. Make a list of the multiples of 12 that are less than 145 and of the factors that compose them, as : 12 = 2 X 6 or 3 X 4 24 = 2 X 12 or 3 X 8 or 4 X 6. 22. There are four numbers besides 12 that are con- tained in every multiple of 12. Name them. HORN. ARITH. 13 194 TWELVES 23. Add the number whose factors are 8 and 7 to the number whose factors are 8 and 9, How many eights does their sum contain ? 24. Find the difference between the number that is composed of 8 and 6 and the number composed of 4 and 12. 25. What is the ratio of the number whose factors are 3 and 7 to the number whose factors are 5 and 7 ? 26. How many hours does it take the hour hand of a clock or watch to go once around the face ? How many hours to go one half the way around ? -J of the way ? ■| of the way ? i of the way ? -I of the way ? 27. How many hours does it take for the hour hand to go around twice ? 3 times ? 5 times ? 6 times ? 28. In 24 hours, how many times does the hour hand go around the face of a clock ? In 60 hr. ? 72 hr. ? 96 hr. ? 29. How many times does the minute hand of a clock go around the face between noon and midnight? How many times between noon to-day and noon to-morrow ? Between noon on Monday and noon on Wednesday ? 30. Name the months of the year in order, beginning with January. 31. How many months in 2 years ? In 4 years ? In 8 years ? In 10 years ? In 12 years ? In ^^ of a year ? 32. How many years in 25 months ? 39 months ? 50 months ? 67 months ? 109 months ? 88 months ? 33. If you save a dollar a month for 3 years, how much money will you save ? 34. HoAv much money will you save if you lay aside one dollar a month for 2J years ? For 5 years ? 3 J years ? 35. How many months had you lived when you were just 6 years old ? TWELVES 195 36. Margaret was 9 years old on the 1st day of last month, llow many months has she lived and how many days over? Let each child in the class reckon up the number of whole months he has lived and the days over. 37. How many eggs in 7 dozen ? 12 dozen ? 3J dozen ? 8^ dozen ? 2i dozen ? 6 J dozen ? 38. How many dozen eggs in 48 eggs? 72 eggs? 39. Gertrude is visiting her cousin Alice. She has spent 12 days at Alice's house and -i of her visit is gone. How long was the visit to be and how many days longer can she stay ? 40. How many in. long is a line that is 2 ft. long? 2i ft.? 31 ft.? 41 ft.? 6 ft.? 71 ft.? 81 ft.? 9 ft.? 41. How many ft. long is a line that is 86 in.? 60 in.? 84 in.? 108 in.? 42. Draw on the board a square 1 foot each way, and divide it off into square inches. How many rows of squares? How many squares in eacli row? How many squares in all ? 43. How many sq. in. in 2 sq. ft.? 3 sq. ft.? 5 sq. ft.? 44. How many sq. in. in a square which is 2 ft. long? 45. How many sq. in. in a sq. yd.? The square foot with its subdivisions of square inches should remain on the board where the children can see it for several davs, and occasional short drills should be given by questions similar to those relating to dozens. 46. What is the ratio of 12 to 24 ? 12 to 36 ? Use chart drill. 47. How many inches in i of a yd.? In J of a yd.? 48. What is the ratio of 12 to 48? What is | or -J of 48? I of 48? 196 TWELVES 49. A lot has a frontage of 12 rods and a depth of 48 rods. What is the ratio of the length of the front fence to the side fence ? How many rods of fencing will it take to inclose it? How much will the fence cost at 1.15 a ft.? 50. What is the ratio of 12 to 60 ? What number is I of 60 ? I of 60 ? f of 60 ? 51. A's house is 60 rd. from B's house. When A has gone |- of the Avay to B's house, how many rd. is he from his own house ? How far is he from B's house ? How far is he from each house when he has gone ^ of the way to B's house ? 52. Susan has to practice on the piano an hour every morning. How many fifths of her practice hour are past when she has practiced 24 minutes ? 48 minutes ? 53. Fill out and learn : i of 72 = The ratio of — to 72 is — f or 1 of 72 = The ratio of — to 72 is — f or ^ of 72 = The ratio of — to 72 is — I or f of 72 = The ratio of — to 72 is — I- of 72 = The ratio of — to 72 is — b 54. A farmer buys a wagon for $ 72 and pays $ 12 cash. What fractional part of the jjrice does he pay, and what part does he still owe ? 55. At the end of six months he pays 1 24 more. What part of the price does he still owe ? 56. He pays 1 24 at the end of another six months. What part of the price does he still owe ? 57. Draw a rectangle 1 ft. long and 6 in. wide. Divide it into 6ths. How many sq. in. in each sixth? How many in | of it ? In | or i of it ? In | or | of it ? TWELVES 197 58. Make a table showing sevenths of 84 from ^ to ^. 59. Make a table showing the ratio to 84 of each multi- ple of 12 less than 100. 60. A fence 84 ft. long costs $21. 12 ft. of the fence cost what part of the money ? How many dollars ? How many dollars do 36 ft. cost? 60 ft.? 72 ft.? 24 ft.? 61. Make a table of the 8ths of 96 as far as |. 62. IMake a table showing the ratio of each of the first 8 multiples of 12 to 96. 63. If an acre of land costs % 96, how much do | of an acre cost? f? |? |? i? |? 64. 160 square rods make an acre. Into how many pieces of ground 2 rods square can an acre be divided ? Let children get ideas of an acre by measuring off distances on the playground, estimating the length of a line about 290 feet long and the area of a square of these dimensions. 65. A farmer has a farm of 96 acres. 12 acres are planted with potatoes, 24 with corn, 36 with wheat, and the rest is pasture land. What part of the farm is planted with potatoes? Corn? Wheat? What part is pasture land? 66. Make problems about 8ths of 96. 67. Make a table showing 9ths of 108 from J to |-. Make a table showing the ratio of each of the first 9 mul- tiples of 12 to 108. 68. The cost of 12 pencils will be what part of the cost of 108 pencils ? Supposing that 108 pencils cost 45 cents, how much would 12 pencils cost ? Let children suppose different prices for different numbers of pencils. 69. What part of the cost of 108 pencils is the cost of 24 pencils ? Of 36 pencils ? 60 pencils ? 72 pencils ? IC)^ TWELVES 70. If 1 dozen pencils cost 5 cents, how many times as mncli will 108 pencils cost ? How many cents would that be? 71. Make problems about 9ths of 108. 72. Make a table of the lOths of 120 up to 1^. 73. Make a table showing the ratio to 120 of each of the multiples of 12 that are less than 120. 74. John had il.20 and spent 12 cents. What part of his money did he spend ? What part had he left ? 75. What part had he spent and what part was left when he had spent 24 cents? 48 cents? 1.60? |.72? 76. Make a table showing yL to i| of 132. 77. Make a table showing the ratio to 132 of each of the multiples of 12 less than 132. 78. How long is a line that is -f-^ as long as a line 132 inches long ? -/y '^^ ^^^^^ • TT ^^ ^^^^^ ^ ri ^ t\ ^ 1 1 6 '/> 12 9 _4_ V _8^ 9 11 9 TT- 11- 11- 11- 11- 79. A certain house is 132 miles from Philadelphia. What fraction of that distance has a man traveled who has gone 12 miles toward Philadelphia ? What part of the distance does he still have to travel? 80. When he has traveled 24 mi., what part of the dis- tance has he traveled and what part remains ? What part has he gone over, and what part remains, when he has traveled 48 mi. ? 60 mi. ? 84 mi. ? 96 mi. ? 120 mi. ? 81. Make a table showing 12ths of 144 from -f^ to ||. 82. Make a table showing the ratio of each of the first 12 multiples of 12 to the number 144. 83. A jar of butter is priced at 11.44. How much will -JL of it cost *? -3^ or 4 of it '^ -5- '^ -"- '^ -^- '^ -9- '^ ^1 *? 12 ^^ ^^ ^^^^ ' 12^ Ol ^ Oi lU . -j^2 • 12 • 12 • 12 • 12 • TWELVES 199 84. What part of a square foot is 48 square inches ? 24 sq. in. ? 60 sq. in. ? 96 sq. in. ? 120 sq. in. ? 85. How many inches of ribbon will it take to bind a lamp mat that is one foot square ? 86. What is the square of 12 ? 87. -J of Harry's money is 12 cents. How much has he ? How many cents had he when ^ of his money was 9 cents ? 7 cents ? 11 cents ? 13 cents ? 88. Louisa has 12 cents. John has 3 times as many cents and 5 cents more. How many cents has John ? 89. If i\lary had 3 cents more, she would have twice as much as Jennie, who has 12 cents. How many cents has Mary ? 90. Divide 276 by 12 by long division. SOLUTION 12)276(23 i)A Show the process and let the children become "■^ expert in it before giving an explanation of it. 36 91. By long division find the quotient of 288 and 12. 92. By 12 divide 1728, 3456, 432, 264, 384, 636. 93. Divide 373 by 12 and see if you get the answer 94. Divide by 12 each of the numbers between 500 and 600 whose unit figure is 6. 95. Divide by 11 each of the numbers between 600 and 700 whose unit figure is 5. ' 96. How" many feet in 189 inches ? 474 in.? 699 in.? 97. How many dozen eggs in 972 eggs ? 852 eggs ? 200 TWELVES 98. John was given #2.64 with which to buy coffee. How many lb. could he buy at 12 cents a lb.? At 11 cents a lb.? 99. How many years in 898 months ? 961 months ? 6846 months ? 7849 months ? 100. At his last birthday Mr. Smith had lived 336 months. How many years old was he ? 101. How many years old will you be when you have lived 216 months? 348 months? 396 months? 456 months ? 102. A dozen readers cost 14.20. What is the price of each ? 103. How much rent does Mr. Jones pay each month when his yearly rent is $ 288 ? How much when his yearly rent is -f 384 ? 104. William has 12.00 at interest at 6%. How much does it gain each year ? In how many years Avill it gain 11.56? 12.16? $3.72? 105. Fig. 1 represents a floor 12 ft. long and 6 feet wide. How many square ft. Q are represented by AB CD ? How many square yd. will it take to cover the floor ? Draw the figure and outline the square yd. D B Fig. 1 106. Draw a figure to represent a square floor 12 ft. long, and show how many square yd. of linoleum it will take to cover it by outlining with a heavy line each figure that represents a scpuire yard. TWELVES 201 Fig. 3 107. Arrange equilateral triangles as in Fig. 2. Show ^ of the figure and tell how many 12ths it equals. 108. Separate the figure into 4 trapezoids. ^ = how many 12ths ? |^ = how many 12ths ? 109. Separate the figure into 6 rhombuses. 1 = how many 12ths ? | = how many 12ths ? 110. How long is the perimeter of Fig. 2, if each side of the triangles is 12 in. long ? 9 in. ? 8 in.? 111. If the perimeter of Fig. 2 is 70 inches, how long is each side of the triangles ? 112. How many triangles in this six- pointed star ? How many triangles would it take to make 9 such stars ? 6 such stars ? 11 such stars ? 113. Copy Fig. 3 by placing equilat- eral triangles. Separate your copy into 6 equal rhombuses. Each rhombus is what fractional part of the figure ? Each triangle is what fractional part of a rhombus ? J of J = ? 114. Remove triangles from the figure until you have a hexagon left. What fractional part of the figure did you take away ? 115. Copy Fig. 4 by placing equilateral triangles. Separate your figure into 4 equal trapezoids. Each trapezoid is what part of the whole figure ? Each triangle is what part of a trapezoid ? -i of |^ = ? 116. If the perimeter of Fig. 4 were 396 inches, how lonof would each side of the triano^les be V Fig. 3 Fig. 4 202 TWELVES ¥m. 117. Can you make Fig. 5 by changing the place of one of the trapezoids in Fig. 4 ? 118. Show by Fig. 5 that ^ is equal to -J of i. 119. How long would each side of the triangles be if the perimeter of the figure were 72 in. ? 396 in. ? 528 in. ? 120. How many inch -cubes would it take to make a layer of them 12 in. long and 12 in. wide ? 121. How many inch -cubes would it take to make 2 such layers ? , 8 layers ? 9 layers ? 11 layers ? 12 layers? 122. A cubic foot is 12 inches long, 12 inches wide, and 12 inches liigh. How many cubic inches make a cubic foot ? Let children show their ideas of the dimensions of a cu})ic foot by outlining with their hands. 123. Find how many cu. in. make 2 cu. ft. 3 cu. ft. 5 cu. ft. 7 cu. ft. 9 cu. ft. 11 cu. ft. 124. How many cubic inches in 1 of a cubic foot ? 2^ cu. ft.? S^ cu. ft.? 41 cu. ft.? 6^ cu. ft.? 125. A line 1 in. long is what fractional part of a line 1 ft. long ? Half an inch is what part of a foot ? J of i2- = ? 126. AVhat kind of a fraction is 2 ^^ i^ ^ How do you find the value of it ? 127. 128. 129. ±S. — 9 12 "■ • 3 uj- ^ 2 " 3 ^^ 1 2 ° 5 ^^ 1 2 6 of -6_. _ ? y ^^ 12 — ' A square inch is what part of a square foot ? 4A = ? 4 8 _ ? 14 = ? 1X3. = 9 12 • 1^ • 12 • 48 _? 1^ 6 5.—? 1# — ? 12 ""• 12 ~~ • 12 1X3. 12 ^# = 12 1^ =? 12 12 • 13 —? 12 JL4^ = 9 12 130. At 12 cents a yd., how much ribbon can be bought for 29 cents? f.31? |.89? 11.29? *2.89? #1.58? TWELVES 203 131. Name the unit figure of each of the multiples of 12 given in their order up to 144. Can any of these mul- tiples of 12 be odd numbers ? 132. Which multiple of 12 is 36 ? How many times does 36 contain 12 ? 133. Think of a number and of one of its multi})les, and see if this definition is true. " A multiple of a number is a number that will contain it Avithout a remainder." 134. There is one multiple of 12 less than 100 that is also a multiple of 5. What is it ? 135. What number less than 100 is a multiple of 12 and also of 7 ? 136. When a number is a multiple of two other num- bers, it is called a Common Multiple of them. Name a common multiple of 5 and 12, 8 and 7, 11 and 5. 137. 42 is a common multiple of 6 and what other number ? 138. Turn to the number table of 9's, and see how many of the multiples of 9 in it are also multiples of 12. Make a list of these common multiples of 9 and 12. Which is the least ? 139. Make a list of the multiples of 12 and of 8 that you have learned. Show those that are common multiples of 12 and 8. Which is the least common multiple ? 140. Turn to the number table of 6's, and the number table of 8's, and you will see that the multiples of 6 meet the multiples of 8 in the numbers 24, 48, 72, 96. What is the least number that contains both 8 and 6 ? A way of showing this meeting of the multiples is to have tha pupils put the first hundred numbers on the board, writing all the multiples of one number with crayon of a certain color and those of other numbers with other colors. For instance, if the multiples of 6 204 TWELVES are red, 8 blue, and 9 yellow, 72 will show itself in its varicolored representation as a multiple of them all. 141. Make a list of the common multiples of 6 and 9 tliat are less than 100, and tell which is the least common multiple. 142. What is the least common multiple of 6 and 7 ? 6 and 11 ? 6 and 5 ? 143. Find the least common multiple of 6 and 8 without looking at the number tables. Show the following method : To find the least common multiple of two numbers, take the largest of the numbers and name its multi- ples in order until one is found that is also a multiple of the other number. Thus, to find the least common multiple of 6 and 8, name the multiples of 8 in order, 8, 16, 24, trying each one to see if it will contain 6, until one is found that will contain it. 144. Find the first number in which the multiples of 8 meet the multiples of 3. In what number do the multiples of 8 first meet the multiples of 5 ? 7 ? 10 ? 145. Find the least common multiple of 12 and 9, 12 and 8, 12 and 10, 12 and 7. 146. Find the least common multiple of 9 and 6, 9 and 4, 9 and 5, 9 and 8, 8 and 10, 10 and 6. 147. Mary may name two numbers less than 13, and the class may find the least common multiple of them. Let this exercise be general. 148. 6 is a common multiple of what two other numbers? 149. Of what two numbers is 35 the least common mul- tiple? 21? 77? 15? 150. Write in Roman notation the least common mul- tiple of 12 and 7. CHAPTER XVI REVIEW Average, Commox Divisor, Adding, Subtracting, and Multiplying Denominate Numbers, Per Cent, Bills 1. Find by long division the q^uotient of 748 and 11. Of 799 and 12. 2. Fill out this table of the products of 21 multiplied by numbers from 2 to 9. application 21 X 2=42 21)1491(71 21 X 3 = 63 147_ 21 X 4 = 21 21 3. Divide 2541 by 21. Tables of products are very helpful to pupils in beginning long division, but they should be encouraged to dispense with them as soon as possible, and to estimate their trial quotients carefully. 4. Find how many times 21 is contained in each of the following numbers ; 2352, 2583, 2982, 3003, 2835. 5. By 31 divide : 3503, 3787, 3875, 4154, 3906, 4092. 6. By 22 divide : 2706, 2662, 2816, 2984, 2882, 2728. 7. How many times is 32 contained in 3872 ? 4224 ? 8. By 41 divide : 5125, 5882, 4961, 5371, 5494, 5412. 9. Use 42 as a divisor of : 5337, 5166, 5418, 5124, 5712. 10. How many times is 25 contained in 1775 ? 8126 ? 11. How many times does 879 contain 22 ? 32 ? 62 ? 12. How many times does the square of 21 contain 49 ? 206 REVIEW 13. If 31 men own in equal shares a mine whicli pays 118,775 this year, what is each man's share of the profits ? 14. How much is /^ oi 6824 ? 8965 ? 4869 ? 12428 ? 15. If 16 lb. of beef cost $2.56, how much does 1 lb. cost ? 16. At i .33 a gallon, how many gallons of vinegar can be bought for 14.62? |4.95? 17.92? $8.91? $8.25? 17. Think of a number that is as much greater than 10 as it is less than 14. 18. Find a number that is as much less than 20 as it is greater than 12. 19. What number is halfway between 10 and 20? 24 and 30 ? 40 and 50 ? 25 and 45 ? 20. A number that is halfway between two numbers is called the average of those numbers. What is the aver- age of 16 and 20 ? 18 and 22 ? 21. How long is a line whose lengtli is the average of an 8-inch line and a 10-inch line ? 22. To find the average of two numbers, divide their sum by 2. Find the averages in the last five examples in that way. 23. Julia is 12 years old and Jennie is 16 years old. What is the average of their ages ? 24. What is the average of 146 and 178 ? 234 and 478 ? 25. Sometimes the average of numbers is a fractional number. Find the average of 7 and 8, 17 and 20, 46 and 53, 18 and 47. 26. Find the average of 19 and 49, and tell how much greater the average is than 19, and how much less than 49. 27. The averaore of 27 and 55 is how much more than 27 ? How much less than 55 ? REVIEW 207 28. To find the average of 3 numbers, divide their sum by 3. Find the average of 20, 22, and 24. 33, 36, and 39. 29. To find the average of 4 numbers, divide their sum by 4. Find the average of 21, 48, 72, and 95. 30. Alfred's % in an arithmetic test was 95, in geogra- phy 94, in spelling 90, and in writing 93. What was his average %? 31. What was the average age of the children of a family of which the youngest was 8 years old, the next 12, the next 15, and the oldest 17? Show that ill finding the average of numbers, their sum is divided by the number of them. Make class exercises by averaging the ages of different groups, or their standing in tests. 32. In 5 days Fanny worked 75 problems. How many did she average a day ? 33. The Blount Plow Works made 12,345 plow points in June, 12,675 in July, and 12,945 in August. What was the average made in the summer months ? 34. If twice as many plow points were made in Decem- ber as in June, 5555 more in January than in July, and 3345 more in February than in August, how many plow points were made in the winter ? What was the average number of plow points made in the winter months ? 35. Harold earns 810.25 per week, Fred earns '111.50 per week, and Ernest earns $12.75 per week. How much are the average wages of the boys ? 36. 14 companies of soldiers have 1372 men enrolled. What is the average number in a company? 37. A school of 43 pupils was found to weigh 3483 pounds. What was the average weight of the pupils? 38. The combined height of the pupils of the same school was 172 feet. What was the average height? 20g REVIEW 39. Use 71 as a divisor and as dividends, 994, 6045, 3903. 40. With 2982 as a dividend use as divisors, 71, 15, 28. 41. Wliat is the quotient when 884 is the dividend and 52 the divisor ? 42. What is the quotient when 1632 is the dividend and 51 the divisor? Multiply the quotient by 51 and compare it with 1632. 43. What is the quotient when 2542 is the dividend and 62 the divisor ? Multiply the quotient by the divisor and compare it with the dividend. 44. Find quotient when 3888 is the dividend and 81 the divisor. Compare the product of divisor and quotient with dividend. Give examples similar to the above until it is seen that when there is no remainder the dividend is equal to the product of the divisor and quotient. Then require examples proved. 45. 2208 -f- 92 = ? 1952-122 = ? 3025 -f- 121 = 8734 - 312 = ? 4551 -^ 123 = ? 8988 -^ 214 = 46- 13 2 — • ~\Tl~ — ^12 —• 2 42 • 13 1 47. Find quotient when 697 is the dividend and 21 is the divisor. Multiply your quotient by 21, add the re- mainder 4, and see what number the result equals. By oral work with small numbers lead the class to see the method of proof in this case. Find quotient and remainder, and prove : Dividend. Divisor. Dividend. Divisor 48. 3839 142 53. 4839 156 49. 15699 215 54. 17898 213 50. 4294 126 55. 5307 221 51. 5782 134 56. 5808 215 52. 4879 212 57. 15413 214 REVIEW 209 58. When 2 oranges can be bought for 5 cents, how much will 1 orange cost ? 3? 7? 11? 12? 20? 40? 59. How many cubic inches in a two-inch cube ? Draw a picture of it. HoAV many square inches in all its surfaces ? 60. How many lb. in a ton ? In 2 J tons ? 7|- tons ? Fig. 1 61. A farmer sold 5-J tons of hay at ^ 10 a ton. How much did he receive ? He bought a wagon for $ 48.25. How much did he have left ? 62. Find sums : 63. Fi] ad diffe rences • 91 8J Si ^ n If H 61 9^ 5f 381 29^ or 1 81* 29i 64. From 8 2 6 5 6 7 8 take 6-J 0,- ^ !i !i 5i H 65. Multiply : 3 J 7 4 If 7 21i 8 11 96^ 12 66. 1 ft. is what part of a yd.? 1 in. is what part of a ft.? ^2 of J of a yd. is what part of a yd.? 67. How many feet does a man walk who A\'alks three times around a lot 4 rods square ? Draw diagram. 68. A night watchman has the duty of walking four times each night around a lot that is 18 rd. long and 15 rd. wide. Hoav many ft. does he Avalk? HORN. ARITII. 14 210 KEVIEVV 69. Draw a picture of a 3-inch cube. How many cu. in. in a 3-inch cube ? How many sq. in. in all its faces ? 70. What is the ratio of 6 to 3, or how many times does 6 contain 3 ? 71. Draw a line 3 inches long and divide it into inch lines. Each inch line has what ratio to the 3-inch line ? A line 2 inches long has what ratio to the 3-inch line ? A line 4 inches long has what ratio to the 3-inch line? A line 6 inches long has what ratio to the 3-inch line? What does f equal? By illustration with lines of different lengths lead children to see that the ratio of a larger number to a smaller one is the quotient of the larger divided by the smaller. Later they may be shown that every ratio is a quotient. 72. What is the ratio of 12 to 3 ? 21 to 3 ? 25 to 3 ? 73. 6 apples will cost how many times as much as 3 apples ? If 3 apples cost 2 cents, how much will 6 apples cost? 9 apples? 15 apples? 74. When corn is #.42 a bushel, how many bushels canbebought for 17.56? '$ 13.02? .^14.70? #10.08? 75. How many cubic inches in a 4-incli cube ? Draw a picture of one and tell how many sq. in. in its surfaces. 76. How many 2-inch cubes can a 4-inch cube be divided into ? 77. What is the ratio of 20 to 4 ? 36 to 4 ? 48 to 4 ? 78. What is the ratio of 576 to 12? 1728 to 12? 996 to 12 ? 79. 8 apples will cost how many times as much as 4 apples sold at the same rate ? If 4 apples cost 5 cents, how much will 8 apples cost ? 12 apples ? 16 apples ? REVIEW 211 80. 5 is one of a pair of factors that make 55. What is the other factor ? Find the factor that helps 12 to make 516. 81. How many yd. in 6 rd. ? 8 rd. ? 3 rd. ? 11 rd. ? 82. How many nickels equal 35 cents ? i.75? $.93? 83. Name four numbers of which 5 is a factor. 84. When a number is a factor of two or more numbers, it is called a Common Divisor of them. Name a common divisor of 6 and 9, 12 and 8, 15 and 20, 25 and 35, 21 and 35, 18 and 20, 18 and 27, 14 and 21, 15 and 20. 85. Turn to the number table of sevens and tell what number is a common divisor of all the multiples of 7 that are in the table. 86. Name two numbers that have a common divisor, and tell what it is. 87. Name three numbers that have a common divisor, and tell what it is. 88. Turn to a number table and look at the numbers whose unit figure is 5. What number is a common divisor of them all ? 89. Give a number that is a common divisor of all the numbers whose unit figure is 0. 90. What number is a common divisor of all the even numbers ? 91. Name a common divisor of all the numbers that are printed in heavy type in the number table on p. 114. On p. 122. On p. 142. On p. 180. 92. Write out all the pairs of factors that make 20, and all those that make 45, and tell which is the greatest divisor that is common to 20 and 45. 212 REVIEW 93. Find three common divisors of 12 and 18. Which is the greatest ? 94. Find two common divisors of 12 and 20, and tell which is the greatest. 95. Make a list of common divisors of 20 and 40, and tell which is their greatest common divisor. 96. Make a list of the divisors, and pick out the great- est common divisor of 20 and 30, 15 and 24, 30 and 40, 24 and 30, 24 and 36, 25 and 30, 36 and 40, 35 and 49. 97. What is the ratio of 30 to 5 ? 40 to 5 ? 45 to 5 ? 98. 10 hats will cost how many times as much as 5 hats ? 20 hats Avill cost how many times as much as 5 hats ? If 5 hats cost $ 3, how much will 10 hats cost ? 20 hats ? 30 hats ? 35 hats ? 99. How many cubic inches in a 5-inch cube ? How many square inches in its faces ? 100. Add: If 61 If 101. From 61| 94f 27 16 7^- 2| 8f take 38J 37| JJ _2| 102. A bolt of cloth contained 37| yd. When 121 yd. were sold, how many yards remained ? 103. Multiply 751 85f 28f 27^ 4| 6f by 7' 9 15 6 7 5 104. If $1285.75 is divided among 5 men, how many dollars and cents will eacli man receive ? 105. How many sixths of a pie in 2 pies and i of a pie ? In 31 pies ? 51 ? 7f ? 81 ? lOf ? 9i ? 20i ? 30f ? 106. From 17 18 24 107. Multiply 61 81 8f take 21 41 ^ ^^ L L L 108. Find 1 of 1248.66, Of 1366.72. Of 1968.22 Of 11575.36. REVIEW 218 109. At 6%, what is the interest of a dollar for 6 months, or i a year? 1 year and 6 months? 2 years and 6 months ? 110. How many seconds in i a minute ? In ^ ? In J ? How many minutes in one quarter of an hour? In | of an hour ? In i of an hour ? 111. How many cu. in. in a 6-inch cube ? Into how many 3-inch cubes can a 6-inch cube be divided ? 112. How many square inches of paj)er would it take to cover a box in the shape of a cube, each side of which is 6 inches ? 113. Write all the pairs of factors that make 36, and those that make 48, and pick out the greatest number that is a divisor of both. 114. In the same way find the greatest common divisor of 42 and 54, 36 and 60, 48 and 72. 115. What is the ratio of 18 to 6 ? 30 to 6 ? 6 to 54 ? 116. 18 pencils will cost how many times as much as 6 pencils ? If 6 pencils cost 5 cents, how much will 18 pencils cost ? 30 pencils ? 24 pencils ? 42 pencils ? 117. How much is | oi ^? i of ^ ? | of i ? -^^ of | ? 118. HoAV many times are 4 and 6 each contained in their least common multiple ? 119. How many days in 3| weeks ? In 4| weeks ? In Sf weeks ? In 9|- weeks ? 120. 122. Add: 8|. 5f 8f 121. From 82f 92 54 fi2 76 5f take 46f 57| 49f Multiply 51 by 8 9f 7f 8| 21j 96| 6 7 8 22 12 214 REVIEW 123. 7 men have equal shares in a gokl mine. They take from it in one year f 17,355 worth of gold. How much is each man's share ? 124. At 7% how much is the interest of a dollar for 5 years ? 2i years ? 8 years and 6 months? 125. At 7% interest how long would it take a dollar to gain 28 cents ? 42 cents ? 84 cents ? 63 cents ? 126. How many sq. in. in a rectangle 7 in. long and 5^ in. wide ? 127. How many sq. in. in a right triangle whose base is 7 in. and altitude 5^ in.? 128. How many cu. in. in a 7-inch cube ? How many sq. in. in all its surfaces? 129. iNIake lists of all the factors of 42 and 56, and find the greatest number that is contained in both of them. 130. In the same way find tlie greatest common divisor of 42 and 63, 35 and 70, 21 and 42, 28 and 56. 131. Name a multiple of 7 that is a perfect square. What is its square root ? 132. How much is 1 of 4^ ? f of | ? | of f ? f of f ? 133. What is the ratio of 7 to 14 ? 7 to 42 ? 35 to 7 ? 49 to 7 ? 28 to 7 ? 63 to 7 ? 134. 14 tops will cost how many times as much as 7 tops of the same kind ? If 7 tops cost 10 cents, how much will 14 tops cost ? 21 tops ? 35 tops ? 28 tops ? 135. Name two numbers whose greatest common divisor is 8. Tell how many times 8 is contained in each of them. 136. Name three numbers whose greatest common divi- sor is 8, and tell how many times each of them contains 8. 137. Name a multiple of 8 which is a perfect square. What is its square root ? REVIEW 215 138. How many quarts in 3 pecks ? In a bushel ? In 51 pk.? lOfpk.? 41 pk.? 7|pk.? 201 pk.? 139. How many pecks in 9 quarts? 21 qt.? 37 qt.? 46 qt.? 58 qt.? 63 qt.? 77 qt.? 89 qt.? 140. How many 8tlis in 3 Avhole ones ? In 41 ? 6J ? 7|? 5J? 9|? 12|? 8f? 141. Mr. Kent works in a factory where 8 hours make a day's work. How many hours does he work in a week ? In 10 weeks ? 142. How many sq. in. in an 8-inch square ? In a rec- tangle whose base is 8 in. and altitude 6| in.? In a right triangle whose base is 8 in. and altitude 6 J in.? 143. How many cu. in. in an 8-inch cube ? 144. How many sq. in. in all the surfaces of an 8-inch cube ? 145. Add: 411- 75|- 146. From 28| 76f 9 ^ 51| or I 31| take 19^ 41f 6^ 147. Find product: 31i 8f 21| 6 J 41 6^ 9 J 5 3 4 2 12 13 14 148. Find I of 1245.76. Of $334.32. Of 1676.24. Of -1^889.60. Of 1498.48. 149. If i 7288 were divided among 8 men, how much would each man receive ? 150. At 8%, how much is the interest of a dollar for 5 years? 7 yr. and 6 mo.? 9 yr. and 6 mo.? 11 yr. and 6 mo. 151. How much is 1 of -1 ? f of f ? foff? fof|? 152. What is the ratio of 8 to 16 ? 48 ? 32 ? 96 ? 72 ? 153. What is the ratio to 8 of 48 ? 64 ? 40 ? 56? 88 ? 216 REVIEW 154. 24 pencils will cost how many times as many cents as 8 pencils ? When 8 pencils are sold for 4 cents, how many cents will 24 pencils cost ? 40 pencils ? 5Q pencils ? 155. How many sq. ft. in 3 sq. yd.? 6|- sq. yd.? 3 J sq. yd.? 41 sq. yd.? 7 J sq. yd.? 8f sq. yd.? 156. If there are 9 squares in a row, how many rows are needed to make a rectangle containing 54 squares ? 72 squares ? 96 squares ? 63 squares ? 157. What is the square of 9 ? What is the square root of 9 ? Of what number is 9 the square root ? 158. How many cu. in. in a 9-inch cube ? 159. How many sq. in. in the whole surface of a 9-inch cube ? 161. From 9^ 9f 76^ take 1| 4f 29| 162. Florence's mother bought 33 yd. of calico, and used 11^ yd. in making a dress. How many yd. were left? 163. Make problems in which you use fractions. ~ 164. Multiply: 91 9f 7f If 9| 27^ 38^ 29i _4_5_6_7 ^ 24 29 9 165. There are 18^ acres in a farmer's tiela. How many acres would there be in 27 such fields ? 166. Find quotients : 9 )1758.79 9 )1239.71 9 )1998.47 9)1621.34 167. 9 boys owned a boat worth 1218.70. How much was each boy's share worth ? 168. Write two numbers of which 9 is the greatest common divisor, and tell the number of times 9 is contained in each. 160. Add: n 2^ 8f n H 8| n 5| n REVIEW 217 169. A man bought a lot for #500, built a house for 11000, and a stable for 1 200. He sold the property for f 3560. Did he gain or lose, and how much ? 170. 27 marbles will cost how many times as much as 9 marbles at the same rate ? When 9 marbles are sold for 5 cents, how much will 27 marbles cost ? 36 ? 63 ? 81 ? 171. Name the multiples of 9 until you reach one that is also a multiple of 6. What name is given to the smallest number that exactly contains both 9 and 6 ? 172. Find the least common multiple of 9 and 5, 9 and 4, 9 and 8. 173. 174. ' 10 How much is 1 2 01 1? 9 • ^ of ^ '^ ^ of ^ ''' f of ^ '^ Find sums : 175. Find differences : 6A »^ 1t^ 2^ Q 9_ ^10 9 7 ^10 7i»o 6J^ 9^ S^^ h\ n 51 4f 9 ^10 4_1_ 9_9_ Q_9_ ^10 ^10 ^10 176. Mr. Wilson had 24^9^ acres of land, and sold 21 1 acres. How many acres had he left ? 177. Multiply 35 375 25 15 13 by _10 _10 _100 1000 10000 178. Give a short way of multiplying a number by 10. By 100. By 1000. Make some examples and explain. 179. Multiply 24 41 82 51 212 by _20 _300 ^00 5000 30000 180. Make some examples like the above, and tell how you multiply when the multiplier ends in naughts. 181. Find products : 339^ 462J^ 596J^ 463^3_ 558_7_ 661^2^ 287-^0 10 120 20 30 10 50 60 218 REVIEW 182. If it takes 5^-q yd. of bunting to drape a window, how many yd. will it take for 8 windows? 183. If it takes 1-^-q yd. of bunting to drape a door, how many yd. will it take for 4 doors ? 184. Divide: 10 )3270 . After dividing in the usual way, lead the children to see that the same result will be obtained by cutting off the naught in the units' place. 185. Divide by 10 in the shortest way : 4280, 3270, 47500. 186. Make examples and show how you divide by 10 any number that ends in naught. 187. Make examples and show how you would divide by 100 any number that ends in 2 naughts. 188. Make examples and show hoAv you can divide by 1000 any number that ends in 3 naughts. 189. At 6 cents a square foot, how much will it cost to sod a square yard of the lawn? 15 sq. yd. ? 30 sq. yd. ? 190. If the binding used costs 10 cents a foot, how much will it cost to bind a rug 1 yd. square ? 2 yd. square ? 191. If you put $10 into a bank that pays 3%, and take none out, how much will you have in the bank at the end of 6 years? 12 years? 18 years? 192. Mary bought 9 yd. of lace at 8 cents a yd., and handed the clerk 75 cents. How much change should she receive ? 193. Make problems about buying and making change. 194. A boy picked 2 gal. of berries on Saturday and 3 gab on Monday. He sold them for 10 cents a qt. How much did he get for them ? 195. Divide 12078 by { of 236. REVIEW 219 196. Name three numbers of which 10 is the greatest common divisor. 197. Find factors of 40 and 60, and pick out the greatest common factor. 198. Name in order the multiples of 9 until you find one that is also a multiple of 4. By it divide 432, 1296, and 2592. 199. What is the least common multiple of 10 and 4 ? 10 and 7 ? 10 and 6 ? 10 and 8 ? 10 and 12 ? 200. 2 ^-^ iV ~ ^ 2 ^^ "iV of a dollar = how many cents ? ^ of ^ of a dollar = how many cents ? ^ of -^^ of a dollar = how many cents ? 201. 40 oranges will cost how many times as much as 10 oranges ? If 10 oranges cost 25 cents, how much will 40 oranges cost ? 20 ? 50 ? TO ? 90 ? 202. How many cu. in. in a 10-inch cube ? How many sq. in. in its surface ? How many in. long are all its edges taken together ? 203. How many sq. in. in a rectangle 10 in. long and 9^ in. wide ? How long is its perimeter ? 204. The perimeter of a square is 40 in. How long is one side ? How many sq. in. in the square ? 205. A rectangle is 10 in. long and its perimeter is .30 in. How wide is the rectangle and what is its area? Draw diagram. 206. John may draw a rectangle, not letting any one else see it. He may give its length and the length of its perimeter to the class. They may find the width of the rectangle and its area. 220 REVIEW 207. Mary may draw a rectangle and give its width and the length of its perimeter. The class may find the length of the rectangle and its area. Let this exercise be general. Encourage the children to dispense with the drawing of the figure as soon as they are able to visualize it clearly. 208. What do you mean by -^ of anything ? Illustrate. 209. -^j of a yd. of cloth and -^j of a yd. and ^^ of a yd.= how many whole yd.? 210. How many llths in 2 whole ones ? In 3 ? 4^ ? 211. How many whole ones in |^ ? if ? || ? ^ ? ^ ? 212. Find sums : 213. Find differences : 3A h\ hS ■t"T\ 7 6 6 5A lA 8j^ 8A 3tV h\ lA 214. From a piece of goods containing S^j yd., o^j yd. were cut off. How much remained? 215. Write products : 11 11 11 11 11 11 11 11 11 81 Q 1 91 ^1 ^1 ^1 71 .'^i 4 2_ __2 ^'tt _% _% _^ _2a __1 __I ^tt 216. How many sq. in. in a rectangle 11 in. long and 7^ in. wide? How long is its perimeter? 217. Find area and perimeter of a rectangle 11 in. long and 3J in. wide. 218. How many sq. in. in a right triangle whose base is 11 in. and altitude 8 in.? 219. How many cu. in. in an 11-inch cube ? How many sq. in. in all its surfaces ? AVhat is the length of all its edges taken together? 220. Write three numbers, of which 11 is the greatest common divisor. REVIEW 221 221. How many sq. in. in a square foot ? How many sq. ft. in 1584 sq. in.? 3024 sq. in.? 4752 sq. in.? 222. How many cu. in. in a cubic foot? How many cu. ft. in 19,008 cu. in.? 20,736 cu. in.? 25,920 cu. in.? 223. What is the least multiple of 11 that will contain 3? 7? 5? 8? 224. How much is J of 3^^ ? | of -f^ ? | of y\ ? 225. Use 11 as a divisor with 462, 484, 572, 683, 782. 226. The expense of an excursion which cost S 374 was shared equally by 11 men. How much did each man pay ? 227. 33 yd. of cloth will cost how many times as much as 11 yd. at the same rate? If 11 chocolate drops cost 5 cents, how many can be bought for 10^? For 15^? 228. How much is f of 12 ? | of 12 ? | of 12 ? 229. Multiply 12 by OJ. By ^, 5f 2i. 3^. 230. How many 12ths in 3 whole ones ? In 4^2 - ^ii ^ f,_5^'? ^J^9 6_7_9 Q1JL9 231. How many whole ones in J^J ? In -H ? U? ^|^? 232. Find sums : 233. Find differences : 8tV 6^ 211 811 61 or 3^ 9^ 7i 61 or ^2 ^12 Hi 5 7 9 8 9 1_ 71 41*' IJL B-5- 4 J- 411 ""12 *^ ^1 ^12 ^12 ^12 ^12 234. A farmer had 96\^ acres of woodland and 238^2 acres of cleared land. How many acres had he in all ? 235. To 3 ft. and 4 in. add 2 ft. and 5 in., placing the work as below. ft. in. 3 4 2 5 5 9 Ans. 9 •J REVIEW ft. in. ft. in. ft. in. ft. in. 236. Add: 7 4 4 11 6 8 9 8 3 7 3 1 2 5 1 7 ft. in. ft. in. ft. in. ft. in. 21 3 16 8 8 9 10 9 8 10 5 9 2 7 4 6 ft. in. ft. in. ft. in. ft. in. 237. From 11 8 9 7 8 9 12 11 take 6 2 . 3 4 4 ^ D 2 7 238. If you have a string 4 ft. long and cut off 1 inch, how many ft. and in. long is it then? How long is it when you have cut off 2 ft. more ? fto in. ft. in. ft. in. ft. in, 239. From 40 7 80 90 take 21 23 46 38 240. A room is 12 ft. high. The border around the tojj of the wall is 1 ft. 6 in. wide. How far is the lower edge of the border from the floor ? 241. ft. in. From 3 1 ^^^ ^^^^^ ^^ illustrated by measurement if nee- take 5 ^'"''^- 242. ft. in. ft. in. ft. in. ft. in. ft. in. From 6 2 9 3 8 1 6 4 9 6 take 2 4 3 5 3 7 2 7 3 8 243. Arthur is 4 ft. 7 in. tall and Mary is 5 ft. 1 in. What is the difference in their heights ? 244. How tall were you when you were 1 ft. and 6 in. shorter than you are now ? Make a general exercise by having the children measure the lieights of their classmates or of different objects in the schoolroom, ^nd find differences. ft. in„ 12 10 3 ft. in. 9 7 4 ft. in. 10 6 6 ft. in. 5 7 5 ft. in. 9 2 8 REVIEW 223 245. ft. in. ft. in. ft. in, ft. in. ft. in. Multiply: 32 76 53 25 56 5 2 4 3 4 ft. in. 7 8 2 246. How long is the perimeter of a square, one side of wliicli is 5 ft. and 4 in.? 4 ft. 7 in.? 6 ft. 3 in.? 2 ft. 11 in.? 247. The long sides of a rectangle are each 4 ft. and 8 in. long. The short sides are 3 ft. and 4 in. How long is the perimeter ? 248. Make problems about the perimeters of figures. 249. Mr. Wilson has a flower bed in the shape of a six- pointed star. See Fig. 3, p. 201. Each side of the points is 2 ft. and 6 in. long. How long is the perimeter of the flower bed ? 250. How long is the perimeter of an equilateral tri- angle, one side of which is 5 ft. and 1 in. long.? 4 ft. 4 in.? 251. How many sides has a hexagon ? On what l^age in your book can you find one ? 252. Julia has a flower bed in the shape of an equi- lateral hexagon, bordered with pinks j)laced 1 ft. apart. Each side of the border is 2 ft. long. Draw a diagram of the flower bed, and find how many plants are in the whole border. 253. Add : yr. mo. yr. mo. yr. mo. yr. mo. yr mo yr. mo. 8 3 9 6 11 3 4 11 21 8 31 7 7 5 8 8 4 10 7 3 17 9 12 11 254. Helen is 11 yr. and 7 mo. old and Emma is 4 yr. and 8 mo. older than Helen. How old is Emma ? 224 REVIEW 255. 3 yr. and 7 mo. ago Edwin was 8 yr. and 9 mo. old. How old is he now ? 256. How old will you be in 2 yr. and 3 mo. from noAV ? 257. Make problems about ages in years and months. ^oo. yr- mo. yr. mo. yr- mo. yr. mo. yr. mo From 13 6 15 9 16 11 18 19 take 7 3 12 4 5 6 5 1 4 3 259. Albert is 13 yr. and 8 mo. old. How long before he will be 15 years old? 260. yr, mo. yr. mo. yr. mo. yr. mo. yr mo. From 36 1 48 2 37 7 24 5 31 4 take 2 3 12 5 13 9 19 8 24 9 261. How old were you 3 yr. and 2 mo. ago ? 262. is yr. and mo. old, and is — yr. and mo. old. Find difference between their ages. • For a general class exercise compare ages of different members of the class, disregarding days. 263. yr. mo. yr. mo. yr. rao. yr. mo. yr. mo. Multiply: 53 74 86 12 8 11 7 3 3 4 3 3 264. Harriet is 11 years and 6 months old. Her mother is three times as old. How old is her mother ? 265. Make problems in which years and months are multiplied. 266. Divide by 13 each of the numbers between 2000 and 3000 that end in 97. 267. Mr. Anderson owns -^^ of a mine that paid one year 187,872. How much did he receive? 268. The next year the mine yielded only 16472. How much did he receive ? REVIEW 225 269. The next year the mine lacked $765 of paying expenses. How much did Mr. Anderson have to pay out ? 270. Mr. Brown has a salary of $3500 a year. How much does he receive each month ? 271. If he saves $125 every month, how much does he save in a year ? How much will he save in 12 years ? 272. A dealer in wagons paid $564 for a dozen wagons of the same kind. How much did each wagon cost ? 273. If he gained $5.50 on each Avagon, how much did he gain on the dozen ? 274. Mr. West, who has a stationery store, bought 11 dozen tablets for $3.96. What was the price per dozen ? How much did each tablet cost ? If he sells them at $.05 each, how much does he gain on them all? The cost of one is in what ratio to the gain on one ? 275. What is the ratio of 21 to 12 ? 48 to 12 ? 72 to 12 ? 36 to 12 ? 108 to 12 ? 276. 24 bicycles will cost how many times as much as 12 bicycles of the same kind ? If a dozen bicycles cost $600, how much will 24 bicycles cost ? 36 ? 48 ? 96 ? 72 ? 277. If a dozen bicycles cost $500, how many can be bought for $1500 ? For $2500 ? For $3500 ? For $4500 ? For $2000? 278. Square 14, 98, 195, 117. 279. If 84 men form a military company, how many companies can be formed by 1092 men ? 1764 men ? Divide: 280. 46968 by 206 283. 939695 by 815 281. 88392 by 509 284. 12750 by 315 282. 634876 by 411 285. 12750 by 316 HORN. ARITH. 15 Dividends Divisors Dividends Divisors 22260 212 292. 84941 841 122811 611 293. 178488 888 48569 423 294. 48144 472 136841 671 295. 71173 691 12750 125 296. 198198 18 226 EEVIEW 286. Multiply 212 611 423 671 1228 1728 6843 by 105 201 103 204 1004 1005 2001 in the quotient is the special difficulty of the following examples. Find quotients : 287. 288. 289. 290. 291. 297. Mr. Hunt had 851^ acres of corn, 108J acres of wheat, 2J acres of cabbage, 54 acres of oats, 13 acres of potatoes, IJ acres of radishes, and 15|- acres of rye. How many acres of grain had he ? How many acres of vege- tables had he ? 298. What is the ratio of 385 to 35 ? Of 462 to 42 ? 299. How many inch- cubes would it take to cover a square foot ? How many layers of inch cubes to build a cubic foot ? How many inch cubes to build a cubic foot ? 300. How many cu. ft. in a coal bin which is 10 ft. long, 8 ft. wide, and 7 ft. high ? 301. How many cu. ft. in a cellar 24 ft. long, 20 ft. wide, and 7 ft. deep ? 302. How many cu. ft. in a ditch 40 ft. long, 5 ft. wide, and 3 ft. deep ? 303. How many cu. ft. of air in a room 30 ft. long, 20 ft. wide, and 10 ft. high ? 304. Class Exercise. may think of a room that has four smooth walls, give its probable length, breadth, and height. The class may find how many cubic feet it contains. REVIEW 227 305. 1 cent is what part of 1 dollar ? 306. "Y-QQ of anything is sometimes called 1% of it. What shall we call -^f q ? -^ ? y^o • i oo ^^* ^^^^ whole ? 307. Have you ever stood 100% on an examination or test? What does 100% mean? 308. If you lacked 2% of being perfect on an examina- tion, Avhat % would you stand ? 309. When a man loses 100 % of his money, wliat % of it has he left ? 310. 7 cents is Avliat % of a dollar? What % of a dol- lar is 9 cents? 13 cents ? 21 cents? 99 cents? 311. $ 17 is what % of # 100 ? What % of 1 100 is 1 19 ? 312. 9 inches is what % of 100 inches? AYhat % of loo inches is 31 inches? 41 inches? 1 yard and 1 incli? 313. If you get 6 cents' interest for every 100 cents you lend for a year, what %.are you getting? 314. Mr. Ta3lor gets 5 cents' interest each year for every dollar he lends. What % does he get ? How much interest does he get each year for ^$7.00 ? 315. Point out numbers on the number table and tell what % they are of 100, and what % they lack of being equal to 100. 316. 50 cents is what part of a dollar ? What % of a dollar ? 317. 25 cents is what part of a dollar ? What % of a dollar ? 318. 75 cents is what part of a dollar ? Wliat % of a dollar ? 319. I of 100% = how many % ? 320. How much is 50% or i of 18 ? 50% of 24 ? 228 REVIEW 321. Turn to Fig. 2, page 201, and show 50% of the figure. Show 50% of Fig. 3, page 201. Of Fig. 2, page 189. 322. George had f 10, and lost 50% of his money. How much did he lose, and how much had he left ? 323. Mr. Hall is 6 feet high. The height of his son Charles is 50% of Mr. Hall's height. How tall is Charles? 324. Caroline's age is 50% of that of her teacher, who is 25 years old. How old is Caroline ? 325. How much is 50% of 28? 280? 140? 360? 840? 326. 50% of a gallon = how many quarts? 50% of a pound = how many ounces ? 50% of a peck = how many quarts? 50% of a ton = how many pounds ? 50% of a foot = how many inches? 50% of a square foot = how many square inches? 50% of a cubic foot = how many cubic inches? 50% of a yard = how many feet? How many inches ? 327. 50% of a square yard = how many square feet? How many square inches ? 328. 3 is 50% of what? 7 is 50% of what? 11 is 50%, of what? 13 is 50% of what? 329. John had 8 cents, and gained as much more. How much had he then ? How much would he have had if he had gained only 50% as much more ? 330. Thomas had 12 cents and gained 50% more. How much had he then ? 331. may give a problem to the class about some one who had some money and gained 50%. 332. Find 50% of 10136 and divide it by 24. REVIEW 229 Fig. 2 333. Find 50% of 8148, and divide it by 32. 334. ^ of 100 % of anything = how many % ? 335. Draw a circle and divide it into fourths. Write in each fourth the % which it is of the whole circle. 336. Draw a square 2 inches long and show 25% of it. Show 25% of Fig. 5, page 158. Of Fig. 1, page 157. 337. Henrietta is 8 years old, and her little sister's age is 25% of hers. How old is her little sister? 338. Mr. Adams had 112 and lost 25% of it. How much did he lose, and how much had lie left ? 339. Find 25% of 40. 24. 36. 248. 432. 888. 340. 6 is 25% of what number ? 341. 25 per cent of what number is 3 ? 5 ? 7 ? 12 ? 342. Richard had 20 cents and gained 25%. How much had he then ? 343. Mr. Walker had 1400 and lost 25% of it. How much had he then ? 344. Make problems about some one who gamed or lost 25% of a sum of mone3^ 345. Find 25 % of 34272 and divide it by 43. 346. Find 25% of 8028 and divide it by 61. 347. When a man loses 25 % of his money, what % has he left ? How many fourths of his money are left ? 348. Draw a line 12 inches long and show 75% of it. 349. Show 75% of Fig. 2, page 189. Of Fig. 5, page 202. 350. Find some figures in the book that you can show 25% of. 50%. 75%. 230 REVIEW 351. 75% of a gallon = how many quarts? 75 % of a bushel = how many pecks ? 75 % of a pound = how many ounces ? 352. Draw a rectangle 8 inches long and 2 inches wide, and show 75 % of it. 353. Show 75 % of these figures : A rectangle 8 in. by 4 in. A rectangle 4 in. by 3 in. A 4-inch square. A rectangle 10 in. by 4 in. 354. Ida's age is 75% of the age of Ella, who is 12 yr. old. How old is Ida ? 355. Mr. Edwards' horse, Claybank, sold for 75 % of the price of another horse of his called Redtop. Redtop's price was $ 400. What was Claybank's price ? 356. William had 20 cents and lost 75% of it. How much did he lose ? How much had he left ? 357. Arthur had 8 cents and gained 75%. How much had he then ? 358. Thomas had 20 cents and gained 75%. How much had he then ? 359. Make problems in which 75% of a sum of money is lost or gained. 360. Find amount of the following bill : Boston, Mass., Jan. 3, 1898. Mr. Thomas Reed, Bought of James Wilson and Co., 12 1b. Coffee at $.30 ? 4 lb. Butter at .25 ? 25 lb. Sugar at .05 ? 3 lb. Starch at .15 ? REVIEW 231 Get billheads from local merchants. Let children make imaginary purchases of one another, copying the billheads, making out bills, and receipting them. 361. Lucy Wood is going to the country. Her mother bought 8 yards of gingham at ^.11 a yard to make her a dress, and 2 yards of lace for it at $.24 a yard. She paid $1.15 for the makmg of the dress. How much did the dress cost ? 362. She bought a hat for $.50, some flowers for $.35, and 3 yd. of ribbon at $.18 a yd. She paid the milliner $.50 for trimming the hat. How much did the hat cost? 363. She has bought a rain cloak for $3.75, an umbrella for $ 1.25, and a pair of rubbers for $ .35. How much has Mrs. Wood spent to keep Lucy from getting Avet ? 364. hicluding $2.50 for a pair of shoes and $2.75 for a tennis racket, ho\v much has Lucy's whole outfit cost ? For a class exercise let pupils find cost of preparing an outfit to go camping, to go to the city, to the seashore, to a picnic, etc. Let pupils suggest items and estimate cost. 365. Divide 2025 by the square root of 144. 366. Divide 89286 by the 9th multiple of 8. 367. Divide 30292 by the 8th multiple of 9. 368. Divide 1487 by the least common multiple of 9 and T6^ 9 t' 3 _ ? 8 — 32' 9 T6- 3 ? 4 — 32' 9 9 5 _ JL 8 32' ? 16* K the children have not discovered it for themselves, show them that the method of reducing a fraction to lower terms by dividing both terms of it by the same number, or raising it to higher terms by mul- tiplying both terms by the same number, gives the same result as by dividing figures and counting the parts. 39. ^ of anything equals how many 6ths of it ? How many 8ths of it? How many lOths ? How many 12ths ? How many 14ths ? How many 20ths ? How many lOOths ? Show that we may express the same fractional values by large numbers or by small numbers, provided that we do not change the ratio of the numerator and denominator. 40. Write a fraction whose denominator is 5 times its numerator and reduce it to lowest terms. 41. Write several fractions whose denominators are just twice the numerators. To what fraction is each one of those fractions equal ? 42. Change J to some equivalent fractions. Change ^ to some equivalent fractions. Change i to some equiva- lent fractions. 236 FRACTIONS 43. By what number must both terms of the fraction I be multiplied to change it to -I ? Which is the greater, 44. Change | to 15ths and tell by what number you multiplied both terms. How do you find out by what number both terms must be multiplied ? 45. Change the following fractions to higher terms and tell in each case by what number you multiplied each term : f to 21sts. 4 to 12ths. I to 20ths. f to 24ths. | to 28ths. 46. Sometimes we let x stand for a number that we are trying to find. Write out the following, putting the true number in the place of x: 6 _Jr_ 3. JL- 5. _x A ^ 7 «■ 4 r y~~2 1* 4~l0- 8 ~" 2¥* 9 ~ l¥' 11~~¥^' 5 ~~ 3T* X is no more difficult in this place than the interrogation point. 47. Class Exercise. — One of the girls may give a fraction. One of the boys may mention a higher denomi- nator that it may have, and the class may change it so that it has that denominator. 48. Which is the greater, ^ of a foot or ^ of it? ^ of an apple or y\ of it ? -I- of a dollar or -f^^ of it ? 49. y^o" of anything is what % of it ? 50. Which is greater, |- of a sum of money or 49% of it ? 51. Change \ to hundredths, and write it as %. Change f to %. \ to %. 4 to %. yV to %. yV to %. y9^ tO %. 52. Class Exercise. may give a fraction that he can change to lOOths or %, and some one else may change it. 53. Change to 18ths : |, \, -J, ^. Why can you not change these fractions to 17ths ? 54. Find a denominator to which all these fractions can be changed, and change them : \, f, f. FRACTIONS 237 55. The least common multiple of the denominators is the most convenient denominator. What is the least common multi^^le of the denominators of \, |, and J? Chansre these fractions to 12ths. 56. Find the least common multiple of tlie denomi- nators of the fractions ^ and f , and reduce them to equiva- lent fractions that have it for their denominator. 57. Give a common denominator to |, |, and | without changing their values. 58. Without changing values give common denomi- nators to -3, 4? 5"' ^^ "s? T' to^* ^-^^ 4' ¥' e^ 12* -'-^' s? 5> is* Tn 3 12 5 Tn ii 1 1 Tn 1 ^ ^L To i ^ To K +. _9_ q^o -8 J^ 1 ^Pn 5 5 2 3 10* -"-^-^ 9' 27' 3* -*-^ 8? 6' 3' 4* 59. Class Exercise. may write on the board two fractions, neither of whose denominators is greater than 12, and the class may change them to the same denominator without changing the value of either of them. 60. As we raise fractions to higher terms by multiply- ing both terms of the fraction by the same number, so we may bring them to lower terms. How ? 61. Write the true number in the place of x : 4 X 18 X J^O — £. 12 — _J^ 3_0 — X 11 =i 28:=:iL "8~2"' 2T T* 18~~9* 22~11* 40 4' 24 8* 3T 5* 62. Reduce the following to lower terms, and tell by what number you divide the terms of each fraction : 4 7 3 _3 __5_ JL JL 6. 11 2.5 3.5. _7JL _2_5_ _95_ "J' If 9' 11' 15' 10' 11' 8' 2 1' 30' 40' 100' 100' 100* 63. What number must both terms of ||- be divided by to reduce that fraction to lowest terms ? 64. What is the largest number that will divide both 22 and 33. What name do we giye to the largest number that will divide two numbers ? 238 FRACTIONS 65. Divide both terms of -f^ by their greatest common divisor. Which is the greater, the fraction you get or ^q? Wliich is in higher terms ? 66. Divide botli terms of || by their greatest common divisor. What liave you done to the fraction ||? 67. Reduce the fraction -^^ to its lowest terms. What number is the greatest common divisor of both terms ? 68. Reduce to lowest terms, and tell what common divisor you use with each fraction : |, |-|, i|, l^|, |, -^^^ _^4_ _8_ 20 10 10 2.8 16' 3 2' 32' 16' 32' 32 ' 69. If you cannot see the greatest common divisor at first, and if, after dividing, your fraction is not in its lowest terms, Avhat "can be done about it ? 70. Reduce to lowest terms : 15. 11 2 5 4 8. 3 2.1 18 21 _SJ)_ _5_5 3_5 30' 30' 30'' 60' 45' 81' 36' 54' 100' 10 0"' 100' 71. Change 50% to a fraction in its lowest terms. 72. Change to a fraction in its lowest terms: 25%, 75%, 20%, 40%, 00%, 80%, 30%, 70%, 90%, 45%, r,n%, 35%. 73. Which is more, 15% of a dollar, or ^^ of it? 23% of a dollar, or i of it? 31% of a dollar, or\3_ of it? 74. Class Exercise. may name a number of %, and the class may reduce the expression to a fraction in its lowest terms. 75. Can you reduce f to lower terms ? Explain. 76. How do you reduce a fraction to lower terms ? 77. What is the use of reducing fractions to lower terms ? 78. Write -^ in its lowest terms, and then change it to 14ths. FRACTKJNS 239 79. Put ^ into its lowest terms, and then change it to 21sts. 80. Bring | to its lowest terms, and then to 12ths. 81. Change y\ to its lowest terms, and then to lOOths. 82. Change ^^ to its lowest terms, and then to %. 83. Change to its lowest terms and then to % : |, i|, 4 21 i J. 6 6 T6> 3 6' 4 4' T8' 84. What is the ratio of 15 to 20 expressed in its lowest terms ? 85. Express in its lowest terms the ratio of 4 oz. to a lb. 8 oz. to a lb. 12 oz. to a lb. 14 oz. to a lb. 86. Give ratio in lowest terms of 18 to 20. 28 to 21. 30 to 35. 40 to 50. 50 to 40. 45 to 50. 18 to 27. 45 to 36. 72 to 84. 16 to 20. 30 to 24. 48 to 54. 87. The flag of Company E, 159th Reg. Ind. Vol., is 72 inches long and 54 inches wide. Express the ratio of its width to its length in lowest terms. Express the ratio of its length to its width. It is sometimes well to let the children take sides and see who can stand the longest without failure — giving fractions and reducing them to the lowest terms or to higher terms. 88. How many whole ones in | ? Y- ? ¥ ^ "^ ^ ¥ ^ 89. A fraction whose numerator is equal to or greater than its denominator is called an Improper Fraction. Find some improper fractions on page 156. Show pupils that an improper fraction is merely a form of division with which they have been working for a long time. 90. Write an improper fraction whose numerator is 10, and find its value. 91. Class Exeiicise. may name an improper fraction, and the class may tell its value. 240 FRACTIONS 92. What kind of a fraction is |- ? What does it equal ? 93. A number that consists of a whole number and a fraction is called a Mixed Number, as 1^. Give some other mixed numbers. 94. In a mixed number, the whole number is called the integral part, and the fraction is called the fractional part. Give the integi'al part of the mixed number 3|-. Of 7^. Of 6i. Of 144i. 95. Give some other mixed numbers and tell which is the integral and which is the fractional part of each. 96. Give a mixed number the integral part of which is 7. 97. Give a mixed number whose fractional part is f . 98. Can you see any reason why a number that is made of a whole number and a fraction is called a '' mixed " number? 99. Change the improper fraction ^ to an equivalent mixed number, or find how many whole ones in |. 100. Change to an equivalent mixed number : J, li, i^, 9 _1_1 1.5 _2_4 3_0 _2_2 _1_7_ 2. 5. 1 6 _5_5. J_3 4X _5_9 4 9 7 J 5"? 1 U 7 ' 1 1' 3"? 5 ? 12^ ~Y~J 9 ' 8 ? 5 ? 8 J 12* 101. Look at the definition of an improper fraction in Ex. 89 and tell whether or not f is an improper fraction. How many whole ones does it equal ? 102. Can you change the improper fraction J^^- to an equivalent mixed number? Explain. 103. Change to equivalent whole numbers the following imnrnnpr fraction'^ • 18 2 4 21 4 8 4 8 4 8 4 8. 7_2 _6 4_ 3_6_ _6_3. 104. Write some fractions whose denominators are each 7 and whose numerators are multiples of 7, and change them to equivalent whole numbers. 105. Ag^ of a pie are equal to how many whole pies? 18? 3Q.? 4_2 9 72 ? 6' • F^ • '6 • 6 • FRACTIONS 241 106. Write some fractions whose denominators are each 8 and whose numerators are multiples of 8, and reduce them to equivalent whole numbers. 107. Class Exercise. may give an improper fraction that can be reduced to a whole number, and the class may reduce it. 108. Give an improper fraction that can be reduced to a mixed number and reduce it. 109. Reduce to equivalent mixed numbers : ^, ^J-, -L®-, ^^, 6 4 9 9 4i)_ S2_ 5.1 4 7 _8 1 ~9 "> 1 2 J " 7 J 9 J 1 2 J o ? 1 1 • 110. Class Exercise. may give an improper fraction that can be reduced to a mixed number, and the class may reduce it. 111. Reduce to whole or mixed numbers : -y-, ^gS ^^^ 6 1 4_8 4 8 4_8_ 5_0 JJ) 3JL 6 1 12' 6 ' 12' 8 ' 9 ' 8~' "5 ' T"' 112. Tell how you reduce an improper fraction to a whole or mixed number. 113. Reduce the following to equivalent mixed numbers h\r Inner rliviQinn • -2-99 683 849 476 1000 8246 833 789 114. How many 7ths in 5 whole units ? In 9 whole units ? In 11 ? In 13 ? What kind of fractions have you been changing these whole numbers into ? 115. How many 8ths of an inch in 3 inches? 6 in.? 9 in.? 4 in.? 12 in.? Into what form have you been chano^ing- these whole numbers ? 116. Change 5 into an improper fraction whose denomi- nator is 10, or find how many lOths in 5. 117. Change 6 into an improper fraction whose denomi- nator is 8. Into one whose denominator is 5. Into one whose denominator is 7. HORN. ARITH. 16 242 FRACTIONS 118. Change 4 to 6ths. To 8ths. To lOths. To 9ths. To 12ths. 119. Class Exercise. — John may name a whole num- ber, and the class may reduce it to an improper fraction, with a denominator that Mary may choose. 120. How many 7ths in 2 1 ? In 4f ? In 5f ? In 8f ? In 6^ ? Into what form have you been changing these mixed numbers ? 121. Change into equivalent improper fractions : 5|-, 122. Write the following, putting the true numbers in the place of x: 7i = f. 3f=f. 6i = |-. 5f = f. ^=^. i 2 _J_ Q3 £. C7 X 73 X ^11 — 11' fJrj — rj. o^ — -g-. 'TT — TT* 123. Give a mixed number whose fractional part is ^, and reduce the mixed number to halves. 124. Give a mixed number whose fractional part is J, and reduce the mixed number to an equivalent improper fraction. 125. Give a mixed number Avhose integral part is 4, and reduce it to an equivalent improjjer fraction. 126. How do you reduce a mixed number to an im- proper fraction ? 127. Reduce to equivalent improper fractions tlie fol- lowing : 12i, 621 173^ 411^ lej, 33^, 871 37^, 66|, 31J. 128. Class Exercise. may put a list of mixed numbers on the board, and the class may reduce them to improper fractions. 129. yV of a foot -f- 3^ of a foot -|- |J of a foot -f y^^ of a foot equals how many feet ? 130. -J of anything + | of it + | of it = what part of it? FRACTIONS 243 131. Write four fractions whose (leuoininator is 7 imd tincl their sum. 132. Class Exercise. may give three fractions having the same denominator, and the class may find their sum. If the sum is an improper fraction, reduce it to a whole or mixed number. 133. From ^ take tV ^ - t% = ? A-t\=? 134. Recopy Fig. 2, p. 233, and find from it how many 32ds of it J + 3V equal. 135. Find from Fig. 2 the values of x in the following equations, and write them in place of it: \+^2—T2' 3_i__l — JL. 1 -U—l— — ^_ •5_1_-J_ — _=^_ 5._|_JL — _^_ I_|__l_ — ^■^_ 4 " 32~~32' 8 '32 ^2* 8 '32 32" 8^32 32* 8^32"~32- 1 _i_ 1_ — _x_ _3 I 1_ — _Jc_ _5 I 1_ — _i_ _7__ J !__ — _-^_ T6"'T""3^2~32* 1G^32~32- 1G^3 2~32* 16'32~32- 9 I 1 _^_ iO J 1_ — _x_ iJi J 1_ — jc_ JL2. I 1_ — _x_ 16~'~32 — 32" 10" ^32 32" 16'32 3 2* 16132 3 2' 136. How much does }f -f 3^2^ of Fig. 2 lack of being the whole figure ? 137. Change each of the plus signs in Ex. 135 to a minus sign, and write the equations again, putting in the true numbers in place of x. 138. Can you find a shorter way of adding or subtract- ing fractions than by dividing a figure and counting the parts ? Bring out the idea that fractions must be reduced to the same denominator before adding them or subtracting one from another. 139. To change to 32ds by what number must both terms of ^ be multiplied ? ^ ? ■^? 140. Find values of x in the following equations, first reducing all fractions to 32ds : l_Lll— ^L. JL-I-U- — ^ 3._1_L — _*_ i_Lii — _*_ 3.4_JL1. — _*_ '^T'a^ — S2* 4IR2 — ■rT* 4 S2 — S2' 8"^ 3 2 — S'2* Ri^ ■S" ' 32 — 32* 4 '^32 — ST* 4 32 — 32' 8 "^ 3 2 — 32* 8 "^ 3 2 32* 5 JJ, _x_ 7. I_L — _=?_ J 1-11 — _=L- _3 L 11 — J?_ 8 32 — 32" 8 32 — 32* 16'32 32" 16'32 32* 244 FRACTIONS 141. How many inches in 1 foot and 3 inches ? What must you do with the 1 foot before you can add 3 inches ? 142. To add ^ and ^, what must you change ^ into ? Why not change ^ to halves ? 2 + i~ -^ i ~ i~- 143. To add or subtract 4ths and 8ths, what common denominator must they have ? 4 + 8~^ i~l~^ 144. Can you see the use of learning to reduce frac- tions to higher terms ? 145. Find values of x in the following: 1 + 1 = |. 2 ^ 6 ~ 6* 3^6~6' 3^6""6- 2 3 6* 2 6 6* 146. Draw a circle, divide it into 6ths, and show that your work was right in the preceding example. 147. Divide the circle into 12ths, and prove your work after you have found the values of x in the following : 1 _i _L _iL- 1 J 1_ _^ X J 1_ _i^ 1 J L_ ^— 2 "^ 12 ~ 12' 4 "1" 12 "~ 12* 3 "1" 12 "~ 12' 6 ' 1 2 ~" 12* 3 "1" 4 ~ T2- 4. ~ 12 ~~ T2- 3 ~ 12 ~~ ^2' 6 ~ 1 2 ~" 12* 6 I _1_ _ ^l_ 2 I _1_ _ _^«;L 5 1_ — _±^ 2. 1^ — _^_ 6 ^ 12 ~~ 12' 3 ' 12 ~" 12* 6 12 ~~ 12' 3 1 2 ~~ 12* 3 ^ f 6 ■" 12- 3 4 ^ 12 ~~ 12- 3 4^6 12' 4 6 ^ 12 ~ 12* 6 4 3 ~~ 12* 6 3 ^ 12 12* 148. Draw a rectangle 5 inches long and 2 inches wide, and prove your work after finding the values of x in the following equations : 111 X 1 1 X 1 1 _x _3 1. — _x_ o" "I 2" ~ T"5'* "5 T^ ~ Tiff' 2" 10~~10" 10 5~10* 149. Divide each square of your rectangle into halves, and prove your results in the following : 1 i_ 1 — X 1 L — _*_ _1 I 1- — -J?_ 1 3L — _*- 2'^20 — 20' 2 20 — 20' 10'20 20* Z 20 20* 150. In adding | and -j^ why do you reduce ^ to 20ths ?, Lead the children to observe that in all this concrete work they have used as a common denominator the number that is th^ le^st con^uiou nmltiple of the denominators, FRACTIONS 245 151. What is the least number that will contain 8 and 3 ? Change ^ and \ to 24ths, and find their sum. Find their difference. 152. Change ^ and f to a common denominator, and add them. Find their difterence. 153. Change to a common denominator and add : 2nndl 1-1-1 1-1-2 3ii i-1-2 _3_i2 i4_3 154. Philip lost y of his money and spent i of it. What part of it had he left? If he had il4 to begin with, how much had he left ? 155. Fred took a bicycle trip from his home to Indian- apolis. In the first 5 days he rode j\ of the distance. On the 6th day he rode -^ of the distance. What part of the distance had he still to ride ? 156. Arthur spent ^ of his money at one time, and gave ^ of it at another time. What part of it did he spend ? What part of it had he left ? If f 2 was what he had left, how much had he at first ? 157. Mrs. Sampson spends | of the money she receives as interest for board and ^ of it for clothes. What part of it has she left ? 158. Mr. Perkins laid off J of an acre for turnips, ^ of an acre for tomatoes, and ^ of an acre for peas. How many acres did he lay off for all ? 159. Write two fractions that can be reduced to 20ths, and find their sum. 160. Write two fractions that can be reduced to SOths, and find their difference. 161. A lady spent ^ of her money on Monday, and i of it on Tuesday. What part of her money did she spend, and what part had she left ? If at first she had i 18, how much did she spend on Monday ? On Tuesday ? 246 FKACTIONS 162. George spent ^ of his money for a watch, and ^ of it for a coat. What part did he spend and what part had he left ? If he had $ 20 to begin with, how much had he left? 163. Out of a flock of chickens ^ died, \ were sold, and Yo were lost. What part of them were left ? If there were 20 chickens in the tirst place, how many remained ? 164. Find difference of ^ and |. Find their sum. 165. A man bought -J of an acre of land, and sold ^ of an acre to his brother. What part of an acre did he keep? 166. Mrs. Miller paid I of a dollar for some butter, i a dollar for some coffee, f of a dollar for some sugar, and had ^ a dollar left. How much had she at first ? 167. A milkman left ^ of a gallon of milk at one house, f of a gallon at another, | of a gallon at another. How many gallons of milk did he have in all ? 168. A field is J of a mile long and i of a mile wide. What fraction of a mile is the difference between its length and its width ? 169. Irene spent J of an hour in school in writing, |^ of an hour in preparing her geography lesson, and J of an hour in reciting it. How much time did she spend in all ? 170. If the session of school was 3 hours long, how much time had she left? 171. Find sums : ^ 8| ^n 8f If 2f .3f 2,V 7| 4f C)5 4f 8f 8t\ 4^ 172. Find differences : 8| 5i 9| 8| 4i Sf 2| 3f 6| If 5i If 6f 8t\ 4-7- FRACTIONS 247 173. Mr. Turner had 83-|- acres of wheat, 78^ acres of corn, and 13|- acres of oats. How many acres had he in cultivation ? 174. Mr. Green's Jersey cow Bova gave milk enough to make 17|^ lb. of butter in one week, 18| lb. the next week, 19^ lb. the next week, and 18J lb. the next week. How much was her average weekly yield of butter ? 175. Bova's price was $575. She and her calf, Good Boy, were sold for $600. What Avas the price of the calf ? Find its ratio to the price of the cow. 176. Make problems in which fractions are added. 177. Find differences : From 8iorT\ 94 H ^ 9f ^ ^ 8f 7J take 3i or A ?I?i?iilli?l!lli 178. H the bread that you eat in 1 day requires 4 oz. of flour to make it, how many oz. of flour will you eat in a year of 365 days ? How many lb.? 179. 196 lb. of flour make a barrel. If 3^ou ate 5 oz. of flour each day, how much less than a barrel would you eat in a leap year ? 180. John rode on his bicycle to a town 28 miles away. He stopped to rest and found he had traveled 9| miles. How much farther had he to go ? 181. After riding 6| miles farther, how many miles re- mained ? 182. Multiply 6 by \. To multiply 6 by \ we take \ of 6. Multiply 24 by \. 18 by i. 27 by J. 30 by yV- 183. Multiply I by |-. To multipl}^ i by |- we take ^ of \. How much is |- of ^ ? 1QJ. Iv5— ? 4v'5— ? _6_v2J.— ? 7.vi6._? 15 y 11—9 184. 8'^T—- "S-^y— • TT^~9^— • ¥^2 1 — - T^'^TZ Iv2v3_9 .5v7v6_'> 2v6v2-l— 9 5.v6v^5. — 9 248 FRACTIONS 185. Class Exercise. may give some fractions, and the class may find their product. 186. Give an improper fraction and tell what an im- proper fraction is. 187. A fraction that is not improper is, of course, proper. -| is a proper fraction. Compare its numerator and de- nominator and tell why it is a proper fraction. 188. Give a proper fraction whose denominator is 7. Give three other proper fractions that express 7ths. 189. Give four proper fractions that express llths. 190. Multiply a proper fraction by an improper fraction. 191. Find the product of two improper fractions and reduce this product to a whole or a mixed number. 192. Multiply a proper fraction whose denominator is an odd number by a proper fraction whose denominator is an even number. 193. Is 99% a proper or an improper fraction? 194. Reduce the mixed number 2^ to 9ths and tell how you reduced it. 195. I wish to multiply the mixed number 2^ by J. What must be done to the mixed number so that it may be multiplied by a fraction ? 196. Reduce the mixed number 8J to an improper frac- tion and multiply it by ^. 197. Reduce the mixed numbers 2^ and 2^ to improper fractions and find their product. 198. Reduce the following mixed numbers to improper fractions and find values of x : 2-1 x^l =x. 21 X 2| = X. 3f X 2^=x. 4| x 4^ = x. •y- X l-j\ = x. 3i X 24 = x. 8i X H=x. 5f X 3j = x. FRACTIONS 249 199. How much will 2| pounds of soap cost at 12|^ cents a pound ? Find the cost of: 200. 5^ tons of hay @ 12|- dollars a ton. 201. 41 quarts of strawberries @ 8^ cents a quart. 202. 3| acres of land @ 62|- dollars per acre. 203. IJ dozen pencils @ 33^ cents per dozen. 204. 3|- quarts of milk @ 6^ cents per quart. 205. 5| yards of carpet @ 66|^ cents per yard. 206. A whole number is called an Integer, as 4, 10, etc. Name two integers and give their product. Is the prod- uct an integer or a fraction? 207. Multiply 2V ^y ~V"- Does it make any difference in the result whether that multiplier is called 16 or ^-? 208. Multiply I by 10. j\ by 22. -f^ by 7. | by 6. ^Vbyie. A by 21. ^^hj20. ^3 by 27. if by 12. 209. Multiply 18 by -^. 24 by ^5. 42 by f 81 by -|. 210. Write a fraction and multiply it by an integer. 211. Multiply an integer by a fraction. 212. When you wish to multiply a mixed number by an integer or an integer by a mixed number, do not reduce either of them to an improper fraction. Multiply li Multiply 12 by 8_ by _3i Why is it best not to reduce either of the numbers to an improper fraction ? 213. How many rods in 2|- miles ? 3^ miles ? 7|- miles ? 214. How many minutes in 3|- hours ? 7f liours ? Ij hours ? 250 FRACTIONS 215. At 12|- cents a yard, how much will 8 yards of lace cost ? 216. At 37|- cents a yard, how much will 15 yards of sheeting cost ? 217. Tell how you multiply a mixed number by an integer. 218. ^Multiply f by itself. OIQ ^nncjTP • 3. 4 5. _3_ _7_ 3 J_0. A 1 ii 220. bquare : I3, l>2^, 2^, Ig^, l^^, ly, z^, 42^, o^. 221. How many yards long is a rod ? Tell how you find the number of square yards in a square rod. Let the square rod with its divisions be drawn on the floor of the schoolroom and remain until it is worn off. 222. How many square yd. in a sq. rd. ? In f of a sq. rd. ? In y\ of a sq. rd. ? In 3% of a sq. rd. ? In || of a sq. rd. ? 223. How many ft. long is a rod ? How many square feet in a square rod ? 224. How many sq. ft. in | of a sq. rd. ? In -^j of it ? In if of it ? In tIt of it ? 225. How many yd. in the perimeter of a sq. rd. ? How many ft. ? 226. How many feet of fence will it take to inclose a burial lot 2 rd. square ? How much will it cost at 50 cents a foot ? 227. How many yd. of fence will inclose a lot 8 rd. square ? How much will it cost at 11.75 a yd. ? 228. Write an improper fraction whose denominator is 5, and change it to a whole or a mixed number. 229. Draw a line J of an inch long, and see how many times a line i of an inch long is contained in it. ^ ^ 4 = ^ FKACTIONS 251 230. How many times is | of a pie contained in -i- of a |jiC . 2*6 ' 231. Turn to Fig. 1 and show \ of it. Show Jg- of Fig. 1. How many times is Jg contained in J ? ^ ^ Jg- = ? 232. Each triangle is what part of Fig. 3? Show \ of Fig. 3. Show | of it. How many times is |- contained in i ? i -^ i = ? Show that in dividing one fraction by another, the same result is obtained by inverting the divisor and multiplying, as by actually measuring off one part of an object upon another part, and counting the measurements. 233. Divide: fbyf. | by f . f by 4. 1^^=^^ A-^-f = ? 234. Write a fraction whose denominator is 7, and divide it by a fraction whose denominator is 14. Class drills like the following are useful : " Take |, multiply it by 4, add I, reduce, add ^, change to improper fraction, divide by 8, square, nmltiply by 5, add ^, reduce, divide by 3, subtract y etc. 235. Write an improper fraction and divide it by another improper fraction. 236. Write an improper fraction and divide it by a proper fraction. 237. Write a proper fraction and divide it by another proper fraction. 238. Write a mixed number, reduce it to an improper fraction, and divide it by some other fraction. 239. Reduce 3|^ to an improper fraction, and divide it byA- 240. Divide: 2|byf. 2f by j^. 4fby|. 9|by^. 241. Reduce to improper fractions and divide : 9J by 3i 6iby2f 16|by6J. 7| by 2|. 8| by 8J. 252 FRACTIONS 242. Reduce mixed numbers to improper fractions, and find values of x : ^^^^x, 7y\-^5f = 2;. 8|^13J=:r. 333^-21=2:. 243. Tell how you divide one fraction by another. 244. At 2|- cents apiece, how many oranges can be bought for 15 cents ? 25 cents ? 40 cents ? 50 cents ? 245. At 3|^ cents, how many balls can be bought for 13i cents ? 33|^ cents ? 36| cents ? 43J cents ? 16J cents ? 246. At 6^ cents per yard, how many yards of ribbon can be bought for 25 cents ? 50 cents ? 75 cents ? 247. At 8i cents per pound, how many pounds of rice can be bought for 25 cents ? 75 cents ? 50 cents ? 248. How much does the quotient of J ^ ^- lack of being equal to 1 ? 249. 1 is how much greater than the quotient of ^ ^ 15 ? Ti • 250. What is the sum of | and f ? What is their differ- ence ? Product ? Quotient of greater divided by less ? Quotient of less divided by greater ? 251. Abraham Lincoln w^as born in MDCCCIX. How old was he in MDCCCLXV, the year in which he died? FRACTIONS 253 DRY MEASURE 2 pints (pt.)= 1 quart (qt.). 8 quarts = 1 peck (pk.). 4 pecks = 1 bushel (bu.). LIQUID MEASURE 4 gills (gi.j) = 1 pint (pt.). 2 pints = 1 quart (qt.). 4 quarts = 1 gallon (gal.). MEASURE OF TIME 60 seconds (sec.)= 1 minute (min.). 60 minutes = 1 hour (hr.). 24 hours = 1 day (da.). 7 days = 1 week (wk.). 12 months = 1 year (yr.). 365 or 366 days = 1 year. LINEAR MEASURE 12 inches (in.)= 1 foot (ft.) . 3 feet = 1 yard (yd.). 5 J yards = 1 rod (rd.). 16|^ feet = 1 rod. 320 rods = 1 mile (mi.). AVOIRDUPOIS WEIGHT 16 ounces (oz.)= 1 pound (lb.). 2000 pounds =1 ton (T.). SQUARE MEASURE 144 square inches (sq. in.)= 1 square foot (sq. ft.). 9 square feet = 1 square yard (sq. yd.). CUBIC MEASURE 1728 cubic inches (cu. in.)=: 1 cubic foot (cu. ft.). 27 cubic feet ^ 1 cubic yard (cu. yd.). Concrete Geometry for Beginners By A. R. HORNBROOK, A.M. Teacher of Mathematics in High School, Evansville, Ind. Price, 75 cents The aim of this book is to give pupils clear ideas of mathematical terms and forms at an early age and to guide their perceptions in such a way as to lay a good foundation for further study by means of personal obser- vation and invention. Without giving rules or formal modes of reasoning to be learned, the book leads the beginner to construct, to observe, and to infer for himself, and to report the result of his work in mathematical language. Its material and methods have all been sub- jected to the test of the schoolroom and found to be practical, and easily within the comprehension of young pupils. The great number of original problems in the book will give pupils that familiarity with geometric forms and facts which is essential to logical reasoning, and which will greatly facilitate their future progress in mathematical study. Although designed especially for use in grammar grades, the book will be found useful for supplementary work for beginners in demonstrative geometry. Copies of Hornbrook' s Concrete Geometry 7vill be sent, prepaid , to any address on receipt of the price by t/ie Publishers : American Book Company New York Cincinnati Chicago (66) Mental Arithmetic BAILEY'S AMERICAN MENTAL ARITHMETIC ... 35 cents For Advanced Grammar Classes, High Schools, Academies, and Normal Schools. Though only recently published, this book has met with the highest favor, and is already in satisfactory use in the best schools. DUBBS'S COMPLETE MENTAL ARITHMETIC ... 35 cents For use in any school where Mental Arithmetic is taught. The rapid introduction of this book on its own merit is the best evidence of its sterling worth. MILNE'S MENTAL ARITHMETIC 35 cents This book follows the same inductive plan and method of develop- ment which has proved so successful in the author's other works. RAY'S NEW INTELLECTUAL ARITHMETIC . . . 25 cents The Mental Arithmetic of Ray's Series of Arithmetics. ROBINSON'S NEW INTELLECTUAL ARITHMETIC . . 35 cents The Mental Arithmetic of Robinson's Series of Arithmetics. ARITHMETIC TABLETS AND BLANKS National Number Tablets. 12 Nos. . Per doz. Piper's Graded Seat Work in Arith. 4 Nos. Each Ray's Test Example Tablets. 8 Nos. . Per doz. Silver's Primary Exercises in Arithmetic. Nos. I and 2 . . . . . . . Each Nos. 3 and 4 . . . . . . . Each Teachers will find these tablets very convenient and useful accessories in the study of arithmetic. 90 cents 8 cents $1.00 10 cents 15 cents Copies of any of the above Mental Arithmetics will be sent, prepaid, to any address on receipt of the price by the Publishers : American Book Company New York Cincinnati Chicago (49) Arithmetic Blanks Arithmetic blanks with graded examples are a most convenient, economical, and useful aid in class room work. They assist the teacher by furnishing a large number of carefully classified and graded examples which may be used for regular class drills and for examination tests. The examples, being without answers, furnish a uniform standard of comparison and a complete test of the pupil's progress. The best and cheapest arithmetic blanks are the following: NATIONAL NUMBER TABLETS Twelve numbers Per dozen 90 cents This series comprises twelve tablets or numbers and supplies suf- ficient work to cover the whole course of written arithmetic. The tablets and lessons are carefully graded and so arranged that two tablets furnish enough supplementary work for a school year. RAY'S TEST EXAMPLE TABLETS Eight numbers Per dozen $1.00 These tablets furnish in convenient form well selected and carefully graded test examples, each sheet having printed at the head from five to ten problems. The eight numbers cover a full course of arithmetical operations. SILVER'S PRIMARY EXERCISES IN ARITHMETIC Numbers i and 2 Each 10 cents Numbers 3 and 4 ...... Each 15 cents A series of graded exercises in the fundamental rules of arithmetic for beginners; one page for each school day, printed in large, bold type, giving the pupil a large amount of practice. The answers to the examples are to be recorded by the pupil on the printed page. These blanks will be found a very useful supplement to any text-book in arithmetic. Specimen copies of any of the above Arithmetic Blanks -will be sent^ prepaid, to any address on receipt of the price. 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