IP iii i 1 III i i II 1 ;!'" ! lillH 1fi'1iilfi!l!i|ip1l|ipiijp^ ^ ^-" ^-ji jiiii i^^^l^^l^l^^^l^^ ■ lijiiiiiliiil jHo DE No. vS~ N COAiPLIhdElUTS AMERICAN BOOK CO A. P. GUNN. Crea'l Ao'i, PIN E & BATTERY ■""^ FRANCISCO THE (^.;::],AV-'-=- ARITHMETIC PRIMER AN INDEPENDENT NUMBER BOOK DESIGNED TO PBECEDE ANY SERIES OF ARITHMETICS BY FRANK H. HALL ATTTHOR OP "THE ARITHMETIC READERS," "THE ARITHMETIC OF THE FARM AND ■WORK-SHOP," "THE WERNER ARITHMETICS," "THE HALL ARITHMETICS,' AND A MONOGRAPH ENTITLED, "ARITHMETIC: HOW TO TEACH IT." NEW YORK .-. CINCINNATI .-. CHICAGO AMERICAN BOOK COMPANY HALL'S MATHEMATICAL SERIES THE WERNEK ARITHMETICS A Three-Book Course for Graded Schools Book I. For third and fourth grades, cloth, 256 pages, 40c. Book II. For fifth and sixth grades, cloth, 288 pages, 40c. Book III. For seventh and eighth grades, cloth, 320 pages, 50c. TEACHERS HANDBOOK giving oral work preparatory for Book I., suggestions to teachers who are using The Werner Arithmetics, answers to problems in Books II. and III., and a large amount of supple- mentary seat-work. Cloth, 137 pages, 25c. THE HALL ARITHMETICS A Two-Book Course for Graded or Ungraded Schools Hall's Elementary Arithmetic, cloth, 248 pages, - - 35c. Hall's Complete Arithmetic, cloth, 448 pages, - - - 60c. ''H PEP'y- Copyright, 1901, by FRANK H. HALL SfjE ILakcsitiE ^ress R. R. DONNELLEY & SONS COMPANY CHICAGO SUGGESTIONS TO TEACHERS AND PARENTS. CHAPTER I. Note 1. — Most pupils on entering school are somewhat familiar with the number idea. A majority of children at six years of age can separate from a group, four, five, or six objects. Some can count, with a good degree of accm-acy, ten or twelve objects. It is therefore unnecessary, except in rare cases, for the teacher of first-grade pupils to spend time in trying to " develop the idea of five " or six. The work suggested in this chapter, then, is, (1) for mothers, (2) for teachers of backward pupils, and (3) for a review of that with which many of the pupils in all first grades are akeady familiar. 1. Train the child to distmguish one object from two objects. Bring one apple. Bring two apples. Bring two pencils. Bring one pencil. Hold up one hand. Hold up two hands. Show me one thumb. Show me two thumbs. Give Mag one cherry. Give John tivo cherries. Make one mark. Make tioo marks. How many horses? How many marbles? One apple and one apple are . One book and one book are . One boy and one boy are . With two objects in view, the attention of the child being directed to them, say: One and one are . Repeat many times, using a variety of objects. With the objects concealed from view, but presumably imaged by the child, say: One and one are . Repeat many times, taking care that at first there come into the :nuid of the child images of two certain objects suggested by the words, One and one. XoTE 2. — The care suggested in the foregoing is necessary lest the child shall simply memorize the expression. One and one are two, without thinking its meaning. The attention of the author was once called to a pupil who was able to recite a hundred number statements like. Four and four are eight, one half of four is two, four is one half of eight, and who yet had no knowledge of number whatever — could not select two or three objects from a group. It is a common experience to find pupils in the second and third grades who have memorized number facts without perceiving them. 54NI7 IV Note's.'— At this stage of the work the child should see number as mag- nitude. The one is to him one book, one apple, one marble, one boy. Later he will learn to see number as ratio. " The formulae of arithmetic and algebra are capable of double interpre- tation. For instance, such a symbol as 3 meant, in the first place, a number of letters or men or any other thing ; but afterwards was regarded as mean- ing an operation, namely, that of trebling anything." — Common Sense- of the Exact Sciences, D. Appleton & Company. 2. Make the child familiar with the number three; and with the fact that two and one are three. In doing this work, observe the same order as that suggested in the preceding section. (a) Use objects and name them in connection with the numbers employed; as, Bring two ajjples, Bring three apples. Give May two cherries. Give John three cherries, Show me two fingers, Show me three fingers, Two marUes and one marUe are three marlles, One look and two hooks are three books. (b) Use objects but do not name them; thus, with two marbles and one marble before the pupil, lead him to say. Two and one are three. One and two are three. (c) Conceal the objects and lead the pupil to image them, and say, Two and one are thixe, One and tvjo are three. Note 4. — To assist the pupil in imaging objects (as two marbles and one marble), these may be exhibited and immediately concealed from view just before the child is required to say, Tico and one are three. He makes this statement of a number fact and thinks (images) marbles. Later this will not be necessary, for he will early learn the generalization that tico and one are three, no matter what may be the objects to which these numbers are applied. But he must arrive at this conclusion, not by memorizing the words that express the fact, but by thinking it in connection with a variety of objects. Note 5. — To lead the pupil to think that two and one are three, it is not best, at first, to ask how many two and on« are. Rather, let the teacher say. Two and one are , and allow the pupil to complete the statement. This simplifies the work, since it relieves the pupil of the necessity of interprettng a question and framing an answer. It allows him to concentrate his thought upon the number fact. When he is familiar with many number facts he will be able to interpret and to answer questions concerning them. One of the most common errors in arithmetical teaching is the attempt to have a child " express in good English " and in complete sentences that which he does not perceive. For reasons that will now be apparent, the attention of young pupils may- be directed to unfamiliar number facts by means of the incomplete assertion better than by the interrogative sentence. 3. The attention of the child may now be directed to four objects. Following the plan outlined on the preceding pages, he should learn that three and one are four, that one and three are four, that two and two are four, and that two twos are four. XoTE 6. At this stage of the work, the author does not deem it advisable to employ figures or written words to represent number. For some time — many weeks, possibly many months — let the spoken word be the only number symbol employed. This will, on the one hand, minimize the danger of lead- ing the pupil into symbol juggling— the use of empty figm-e symbols; and on the other hand, will avoid the exhaustion of the child's strength (which for the moment we desire to employ in the seeing of magnitude relation and expressing it numerically) in the recognition and pronunciation of written or printed words. 4. The idea of half may now be taught. Give sister half of your candy. Give me half of your apple. Give Henry half of your clay. Continue such work until the child is familiar with the fact that the halves of anything are alike as to quantity. Give the child four toothpicks. Then say, Give me half of the toothpicks. Do the same with two toothpicks; with three toothpicks. ]>foTE 7. — If this work is properly performed, little will need to be told the child or done for the child. As soon as he understands what you mean by half, he can find half of four toothpicks, half of two, and half of three. Later, he will, without assistance, find half of six, half of five, half of eight, half of seven, half of a foot, half of six inches, half of five inches, half of a half, etc. It is the idea of half that is to be taught; then the child should be allowed to do — to find half of many things and of many groups of things. He will easily memorize some of the number facts which he perceives ; not by remembering the words that express them, but by imaging the related magni- tudes. When you say. Half of four is , he should see with the mind's eye the four (objects) in two equal groups. He then tells you what he sees ; namely, that half of four is two. It is an image that he remembers, not a plii'ase or sentence. 5. The child may learn to count to ten ; or, if this is done with little effort, to twenty, or even to a hundred. "WTien he is able to take ten objects from a group, he may be allowed to put toothpicks in bunches of ten. These may be held together by little rubber bands. He may learn that two tens are tiventy; three tens, thirty; four tens, forty, etc. He may do this even before he can count from ten to twenty. He \nl\ then be able to tell you that half of forty is twenty, and that half of twenty is ten. Note 8. — A necessary caution in connection with the foregoing is, that the child should not be allowed to think that saying the names of the num- bers in regular order is counting. Neither must he be permitted to entertain the idea, for instance, that the ninth object is /n'ne; the tenth object ten, etc. He can "count to ten" if he can take from a group of more than ten objects any number of objects not greater than ten that may be suggested. He may be able to bring twenty objects (two tens) or thirty (three tens) or forty (four tens) before he can " count to twenty." 6. Work with the number fve. Lead the child to discover that two fires are ten; that one half of five is ttco and one half; tlmt four and one are five; that three and tv:o are five; that tivo twos and one are five. As soon as the child perceives one of the above number facts, he should be encouraged to remember it. This he should do by recalling his image of the objects so grouped as to suggest the fact to be remembered. Thus, the -words, One half of five is , should bring into his consciousness an image of five objects in two equal groups. XoTE 9. — In Note 4 the use of a variety of objects for the pm-pose of leading the pupil to abstract the idea of number and to generalize num- ber facts, is recommended. But in teaching a particular number fact, imi- formity of objects is preferable to variety. Moreover, the objects employed should"^ be simple and unattractive in themselves. For this early work, nothing can be better than ordinary toothpicks. They are inexpensive, may be broken into halves or thirds or fourths, are easily grouped, and may be conveniently put up in packages of ten. " The mental comprehension of number is disturbed if things which awaken other ideas or desires are employed. The mind is capable of only a certain amount of interest, and when this interest is wholly or partly with- drawn, but little can be expected for the particular thing at hand. For this reason, while teaching the abstract number there should be but few things shown the child, and these should be simple and uniformly the same." — Levi Seeley, in Gruhe's Method of Teaching Arithmetic. 7, Work with the number six. The child should now discover and memorize the following number facts: Five and one are six. Four and two are six. Three and three are six. Ttvo threes are six. One half of six is three. Tliree ttvos are six. He may be taught to think of tivo as one third of six; and of fo2ir as two thirds of six. He may learn to count to twelve if he has not already done so; and then he may discover and memorize the fact that two sixes are twelve. 8. Teach the figures from 1 to 12. Give the child a foot rule, and help him to become familiar with the terms, inch and/oo^. Teach him to use the rule in measuring. At first let him measure distances that are an integral number of feet or an integral number of inches. Later, teach him the meaning and use of the expressions, tivo and one half feet, two and one half inches, three and one half inches, four and one fourth inches, four and three fourths inches, etc. Follow this by encouraging him to measure distances suggested by the above expressions. Help him to discover that one half is tioo fourths; that one half and one fourth are three fourths; that there are four fourths in a ivhole. With the rule concealed from view, ask him to draw a line two inches long; a line three inches long, etc. Then allow him to test his work by applymg the rule to the lines he has drawn. He may be told that a dozen is tvjelve. He may then discover that half a dozen is six; that a fourth of a dozen is three; that three fourths of a dozen are nine. Note 10. — While doing the above number work, the pupil must not be allowed to lose sight of magnitude. There must be no attempt to have him memorize the words, one half and one fourth are three fourths. He must rather discover this number fact and remember it, at first,_ by recalling his image of the rule and its divisions ; or of a pie and its divisions. Note 11. — The foregoing pages suggest what the average child may easily learn concerning number by the time he is six years of age. It is not expected that the mother will present this to the child in formal number lessons in the exact order here given, but rather that, beginning when the child is three or four years of age, she will incidentally, in connection with his play or her own work, lead the child to perceive the number facts herein given and to retain them in mind hy means, chiefly, of his memory images of the magnitudes considered. Besides the mere counting, there are less than fifty of these number facts suggested. One fact learned each week, the work will be accomplished in a year. Some children may learn much more than the amount here suggested ; but if all, before entering school, could be introduced to the subject somewhat after the manner herein suggested, the number of pupils thought to be " born short " in respect to mathematical power would be greatly diminished. Note 12. — If the first-grade teacher finds pupils in her class who are seemingly "deficient in the number sense," she should make a careful study of each case ; and, using the foregoing pages as suggestive of the order of procedure, should, in connection with the natm-e work, the drawing, the dis- tribution of material, the paper-folding, etc., help the child to see magnitude relation in a small way, and to express it numerically. CHAPTER II. THE FIRST YEAR OF SCHOOL. Note 1. — Soon after the beginning of school the teacher should carefully note the real number knowledge which each pupil possesses. It is not the knowledge and power of the class as a whole that the teacher must discover, but rather the number knowledge of the individual. In making these discov- eries the teacher should make use of the suggestions in Chapter I. Pupils who have mastered the work there outlined may safely engage in work sug- gested by the following topics. XoTE 2. — The number-work of the first school year should be mainly oral, should occupy comparatively little of the pupil's time, and should be presented incidentally in connection with (1) the administrative duties of the teacher, (2) the nature work, (3) constructive work, and (4) reading. XoTE 3. — No attempt is made to give these topics in the order in which the work should be presented. Any one of the subjects may be introduced when the pupil seems ready for the little problems given and especially when he feels the need of number knowledge in order that he may perform some task assigned. The thoughtful teacher, however, will always keep in mind the work done by each pupil, and will see that it is frequently reviewed, and that each new step is in some way related to that with which the child is already familiar. I.— ADMINISTRATION. 1. Attendance. How many girls tardy ? How many boys tardy ? How many pupils tardy ? How many girls absent ? How many boys absent ? How many pupils absent ? How many boys present in Eow 1 ? How many girls present in Row 1 ? How many pupils present in Row 1 ? ^ How many more pupils present in Row 1 than in Row 2 ? ^Miich is the larger, Class A or Class B ? How many pupils in Class A? How many pupils in Class B? How many pupils absent from Class B ? How many pupils present in Class C ? 2. Distribution of Materials. A monitor from each row goes to the teacher for pencils, sheets of paper, scissors, crayons, or books. How many do you need ? Teacher gives some to one of the moni- tors. How many have you ? Do you need as many as you have ? Do you have as many as you need ? Eeturn to me all you do not need. Monitor goes to the supply and counts for himself under the eye of the teacher. Count by ones. Count by twos. Count by threes. Carry books to Room 2. How many can Mary carry ? How many can John carry ? How many did Mary carry ? How many did John carry ? How many more did John carry than Mary ? 3. Time. Observe the clock face for one minute ; for five minutes. Tell the pupils how far the long hand moves in one hour. Observ^e how far the short hand moves in an hour. Teach the figures on the dial. How far does the long hand move in half an hour ? In a quarter of an hour ? A quarter of an hour is fifteen minutes. A half hour is thirty minutes. An hour is sixty minutes. Count by fives to fifteen ; to thirty ; to sixty. Ten is how many fives ? Fifteen is how many fives ? In five minutes you may have a recess ; watch the clock. Observe the position of the hands when school begins ; at the beginning of the recess ; at the end of the recess ; at the close of school ; do this for both forenoon and after- noon. Five minutes and one minute are . Ten minutes and one minute are . Fifteen minutes and one minute are . Five minutes and two minutes are . Five and three? Five and four ? Ten and two ? Ten and three ? Ten and four ? Ten and five ? Fifteen and two ? Fifteen and three ? Fifteen and four ? Fifteen and five ? John is how many minutes tardy ? What time do you start for school ? In how many minutes can you come to school ? How many hours of school each day ? At what time are you dismissed ? How many days of school each week ? How many days of each week do you remam at home ? A week is how many ? Note 4. — Do not attempt to give all of the foregoing regularly and on consecutive days, before attempting much that follows this topic. Rather, introduce portions of the suggested matter incidentally in connection with and as a part of the administrative work. But whenever a number fact that should be memorized has been given, the teacher should hold it in her own mind for fi-equent review. So far as possible she should know the memory possessions of each pupil ; and, taking care that he loses nothing of what he has already acquired, she shoiUd see to it that a little, from time to time, is added thereto. The careless teacher allows a pui>il to forget almost as much as he memorizes dm-iug a term. This is usually the case with some pupils when the teacher presents the memory work to the class as a whole and loses sight of the individual pupil. The teacher should constantly ask herself : How much more does Mary^ know to-day than slie knew yesterday? "What does William need to review in order that he may be ready for the new number fact? "What has John per- ceived and memorized? II.— NATURE STUDY. 1. Plants. Count buds, leaves, petals, etc. Measure growth of twigs. Measure length of needles from pine trees; from spruce trees; from fir trees. Find circumference of trees in school yard. Find growth of corn or of a vine for one day ; for one week. 2. Animals. Count legs of fly, spider, cray-fish, beetle. How many legs have two toads ? Two butterflies ? Two spiders ? Three cray-fish ? Three frogs ? Three beetles ? How many toes on a cat's fore-foot? On hmd-foot ? On two fore-feet? On two hind-feet? How many shoes does a horse need ? A pair of horses ? How many shoes does an ox need ? A yoke of oxen ? How tall is your pony ? (To be measured over the front feet to top of shoulder.) How tall are you ? 3. Weather Record. Number of clear, cloudy, and rainy days in the week ; in the month. Prevailing winds. How many days was there a west wind? An east wind? A north wind ? A south wind? A northeast wind ? A northwest wind ? A southeast wind ? A southwest wind ? How many days did it rain last month ? How many this month ? How many more last month than this month? To-day is Monday; last Friday was May sixth; what month and day of the month to-day ? Teach pupils to read the thermometer. Eepresent the thermometer scale on blackboard, one inch equal two degrees. Mark daily readings. Piepresent five thermometer scales side by side on blackboard. Mark the reading Monday at twelve o'clock on first scale ; Tuesday at same hour on second scale, etc. At end of week connect the points of daily mark- ing, thus gi^"ing a graphic representation of variation. IN'oTE 5. — The pupils will be interested in noting the variation of the outdoor temperatm-e during a school day. Put eight scales side by side on Xll blackboard. On the first, mark the standing at nine o'clock ; on the second, at ten o'clock, etc. Connect the points of hourly marking. Then ask : At ■what time was it -warmest? How many degrees warmer was it at twelve o'clock than at nine o'clock? At three o'clock than at nine o'clock? Ill — CONSTRUCTIVE AVORK. 1. Drawing. Draw a horizontal line one inch long; two inches long ; three inches long. Draw a vertical line one inch long ; three inches long. Draw two parallel horizontal lines three inches long and one inch apart. Draw several parallel vertical lines one half inch apart and two inches long. Draw a four-inch line; bisect it. One half of four inches is . Draw a five-inch line; bisect it. One half of five inches is . Draw a one-inch square. Draw a two-inch square and divide it into one-inch squares. A two-inch square is equal to one-inch squares. Draw a three-inch square; di\dde it into one-inch squares. A three-inch square is equal to one-inch squares. Draw an oblong one inch by three inches ; divide it into one-iach squares. Draw an oblong two inches by three inches; divide it into one-inch squares. Divide a one-inch square into half-inch squares. How many half-inch squares in a one-inch square ? Note 6. — In doing the work suggested in the foregoing paragraph at first allow the pupil to use a ruler; then require him to draw without a guide and to test his work with the ruler or with paper of known dimensions. Note 7. — After the pupil has drawn a figure several times he should be able easily to bring the image of the figm-e into consciousness when its description is given by the teacher. Thus, when the teacher says, "Imagine a two-inch square ; divide it into one-inch squares," the pupil should see all the lines with the mind's eye almost as clearly as they would be seen were the figure on the blackboard in plain view. If he images well he will be able to tell, without drawing, the number of one-inch squares in a two-inch square ; the number of two-inch horizontal lines, and the number of two-inch vertical lines, in the figure. Cultivate from the first the imaging power. To answer the question, " A three-inch square equals how many one-inch squares ? " by word memory is valueless ; to answer by drawing the figure and counting the squares is better ; to answer by imaging the figure and counting the squares is best. First-grade pupils can be taught to image such figm-e& 2. Paper Folding. paper. How wide Give to each pupil a four-inch square of How long is the paper? How is the paper? many corners ? How many edges ? Fold the right edge upon the left edge. Crease. Unfold. How many oblongs? How wide is each oblong? How long is each oblong? Each oblong is what part of the four-inch square? Fold the right edge of the paper to the crease; the left edge. Unfold. How many oblongs? How wide is each oblong? How long is each oblong? Each oblong is what part of the four-inch square ? ) Fold the lower edge upon the upper edge. Unfold. How many oblongs? How wide is each oblong? How long is each oblong? Each oblong is what part of the four-inch square ? One half is how many eighths? A fourth is how many eighths? Two fours are . Four twos are . Fold the lower edge of the paper to the crease; the upper edge. Unfold. What is the size of each httle square ? How many one- inch squares in one row? How many rows? How many one-inch in all? How many one-inch squares equal a four-mch squares square? Four fours are , fourth of sixteen is four and four are — One half of sixteen is . One Two eifrhts are . Four and four and Fif . 3. ' J Note 8. — By cutting the paper as indicated by the heavy lines in Fig. 1, and folding and pasting the corner squares, an open box two inches by two inches by one inch may be made. Note 9. — By cutting tlie paper as indicated by the heavy lines in Fig. 2, and folding and pasting, a square prism one inch by one inch by two inches may be made. Note 10. — By cutting the paper as indicated by the heavy lines in Fig. 3, discarding the four ujDper squares, and folding and pasting the lower tliree fourths of the sheet, a one-inch cube may be made. Note 11 — By using an eight-inch square of paper, folding and cutting as indicated by Fig, 1, an open box four inches by four inches by two inches may be made. Note 12.— By using an eight-inch square of paper, folding and cutting as indicated by Fig. 2, a square prism two inches by two inches by four inches may be made. Note 13. — By using an eight-inch square of paper, folding and cutting as indicated by Fig. 3, a two-inch cube may be made. Note 14. — By using a twelve-inch square of paper, folding and cutting as indicated by Fig. 3, a three-inch cube may be made. 3. Building with One-inch Cubes. Note 15. — Provide at least twenty-seven one-inch cubes, a two-inch cube, a three-inch cube, a square prism one inch by one inch by two inches, a square prism one inch by one inch by three inches, a square prism two inches by two inches by three inches, and a rectangular prism one inch by two inches by three inches. These should be made of wood. A cube has faces. Each face of a cube is a square. A cube has corners. A cube has edges. Three edges meet at each corner. A square prism has faces. Two of the faces are squares. Four of the faces may be oblongs. Some prisms have no square faces. A face of a two-inch cube is a two-inch square. A two-inch square equals one-inch squares. A face of a three-inch cube is a three-inch square. A three-inch square equals one-inch squares. A two-inch cube equals one-inch cubes. A three-inch cube equals one-inch cubes. One half of a two-inch cube equals one-inch cvibes. One fourth of a two-inch cube equals one-inch cubes. One eighth of a two-inch cube equals a cube. Note 16. — In the beginning the child must work with sense magnitudes; but very early he can be taught to build with, separate into parts, and compare imaged magnitudes. To this end the teacher and pupil must frequently talk about seuse magnitudes that are present to the mind's 'eye only. The pupil must be led to see the four sides of a one-inch square, or of a two-inch square ; the six square faces of a one-inch cube, or of a two-inch cube ; the four one- inch squares in the face of a two-inch cube ; tlie eight one-inch cubes in a two-incli cube ; and the twenty-seven one-inch cubes in a three-inch cube, tvliile all these objects are concealed from view. At this stage of the work the niatlieniatical skill of the pupil depends mainly upon his ability to call the images of magnitudes into consciousness when the symbols of these (their names or descriptions) are spoken by the teacher. lY.— READING. 1. Pages. Teach the child to find the page by number. How many pages did you read yesterday? Upon what page do you begin to-day ? If you read two pages to-day, upon what page will you begin to-morrow? Henry is reading on page 28. Mary is reading on page 32. Mary is how many pages in advance of Henry? How many pages must you read before you reach page 40 ? How many pages in your book ? Twenty pages and twenty pages are . 30 and 20? 50 and 50? 50 and 20? 29 and 1? 29 and 2 ? 39 and 1 ? 39 and 2 ? 2. Lines. Find the first line on the page; the second line; the third line, etc. Eead the fourth line. Kead the second line. Find the third line on page 21. Find the fourth line from the bottom of 'the page. How many lines on page 15 ? 3. Words. How many words in the word list at the beginning of lesson ? How many words in the first line ? In the second line ? Find the third word in the second line. Can you find the word tu-o? How many times does the word two occur on page 10? 4. Letters. How many letters in the word John ? In the word aijple ? Fmd the longest word on page 10. How many letters in the longest word on page 12 ? How many words can you think of, each of which is made of two letters ? Of three letters ? Of four letters ? Note 17. — The Pedagogical Pause. We have grammatical pauses and rhetorical pauses. There is yet another pause that may very properly be called the pause pedagogical. It is the pause that the teacher should make as a pedagogical necessity in questions and statements to pupils. This pause should follow a word or phrase which sj-mbolizes something that maybe diffi- cidt for the hearer to image. The length of the pause should be measured by the time required by the pupil to image that for which the word or phrase stands. This pause is indicated in the following sentences by the double asterisk : The written word hoy * * contains how many letters ? One half of a foot * * is how many inches? How many half inches * * in three inches ? One half oifour toothpicks * * is One half of three toothpicks * * is Think of a two-inch square * * Divide it into one-inch squares * * How many one-inch squares? How many one-inch squares of paper * * do you need to make a three-inch square ? How many one-inch cubes * * do you need to build a two-inch cube? A four-inch cube ** equals how many two-inch cubes? A four-inch square ** equals how many two-inch squares? A ttoo-inch cube * * equals how many one-inch cubes? A two-inch square ** equals how many one-inch squares? With the bright pupil this pause may be made very short. With the dull pupil it should be made very, very long. With any pupil it must be made long enough for him to image that for which the icord symbols stand. To proceed without the imaging on the part of the pupil, is to tempt him into word memory work. A proper observance of this pause and special attention to the imaging power of individual pupils can not fail to promote better teaching in elementary number-work. Note 18. — It will be observed that the child has already reached a point at which a kind of double imaging is necessary. The teacher may speak the word " cube " when she desires to bring into the consciousness of the child an image of the written word cube. Again, she may speak the word "cube " and desire to bring into the consciousness of the child an image of that for which the written word stands. The teacher must now strive to see into the child mind and to note the character of the imaging that is lieing done. If the child images the icord cube when the teacher expects him to image that for which the word stands— the real cube — and if the teacher proceeds on the supposition that the child is imaging the one when he is really imaging the other, the results must be unsatisfactory and often positively harmful to the pupil. The most practical and profitable child-study is that which enables the teacher to read the child mind, and to note correctly the kind and amount of imaging activity. The difficult thing in the early work in mathematics is not, to see relation, but to bring into consciousness images of the things related. CHAPTER III. THE SECOND YEAR OF SCHOOL. The number-work of the second school year must be done mainly orally. A book must not be put mto the hands of the second grade pupil for the jiurpose, of teacJmig Imn numhei' facts. If a number book is used at all in this grade, it should be made up of reading lessons in which the child simply reviews number facts with which he is already famiUar. A well-arranged book used as here suggested will not only review number facts, but will furnish reading matter in which the pupil wiU be greatly interested. The Primer of Arithmetic is especially designed to show the order in which the number facts may be presented orally, and to furnish reading lessons in which the pupil will be led to review these facts. The teacher should take great care to prepare the pupil, by oral instruction, for the page he is to read. The oral work should always be considerably in advance of the reading lesson. In fact, the second- grade pupil should not be asked to read statements expressing number knowledge until he thoroughly understands the numerical relations involved. When he reads, his task should be to recognize through tlie eye and to express that with which his ear is quite famihar. A good book used in the second grade for this purpose will be helpful; used (in the hands of the pupils) to teach number facts, it will be -positively harmful. If the lessons herein suggested for the home and for the first school year are properly taught, but little time will be required to master the lessons suggested for the second school year. No formal number-work at all, in the first two years of school, is better than too much work, or work improperly done. Too much number work will give to the pupil a bias, on account of which magnitude relation will, ever after, occupy too prominent a place in his thought. Num- ber-work improperly done will create a distaste for such effort, and result in arrested development and possible permanent injury to the pupil. " It is perhaps not too much to say that nine tenths of those who dislike arithmetic, or wlio at least feel that they have no aptitude for mathematics, owe this misfortune to wrong teaching at first." — McLellan and Dewey, in Psychulo(iy of Number, page 146. " There is no subject taught in the elementary schools that taxes the teacher's resources as to methods and devices to a greater extent than arith- metic. There is no subject taught that is more dangerous to the pupil in the way of deadening his mind and arresting its develoi^ment, if bad methods are iised." — Dr. Wm. T. Harris, in Editor's Preface to Psychology of Number, D. Appleton & Co. THE PLAN OF THE BOOK. The Primer of Arithmetic is not made up mainly of graded miscellaneous problems. It is built on a plan. Each four-page group constitutes the larger unit of the plan. Twenty-seven presen- tations of this gradually changing unit make up the book. In the first eleven groups (44 pages) the page units are as follows: 1st page — New number facts. 2d page — Number facts applied to linear measurements. 3d page — Number facts applied to surface measurements. 4th page — The " Elementary Spiral " (a problem in each of the five fundamental processes), followed by miscellaneous problems. In the remainder of the book, the 1st and 4th pages of each group are similar to the corresponding pages of the first eleven groups. The 2d page teaches the application of primary facts to larger numbers. The 3d page deals with sohd, surface, linear, and capacity measurements. The teacher and pupil soon discover that the mastery of the first page of a group (and all preceding it) makes the work of the other three pages easy and interesting. The child quickly learns that it is necessary that certain number facts should be memorized. The teacher knows exactly where to find these facts, and is given definite direction in regard to the order in which they should be presented. The last three pages of each group, particularly in the first 44 pages, furnish at ouce a review of the newly learned number facts. and a reward for doing the work presented on the 1st page of each group. It will be observed that the plan of the book is such that each page, between pages 4 and 100, sustains a kind of double (or quadru- ple) relation to other parts of the book. It is, of course, related to the page preceding it and to the page following it. It is also closely related to the 4th page before it and to the 4th page after it. Com- pare page 23 with pages 19 and 27; page 30 with pages 26 and 34; page 62 with pages 58 and 66; page 52 with pages 48 and 56. Such an arrangement as this greatly aids in gi^^ng to the child a lively sense of the relation of the new to the old. Moreover, it does not allow him to forget the fundamental number facts that are so essential to his progress. The one direction to the teacher, more important than all others, and hence here repeated, is : By means of oral instruction, teach thoroughly every fact given in a 4-page group, before asking or allowing the pupil to attempt to read any page of the group. The reading thus becomes a pleasant review (a seeing again), by means. of ^j/'in^ftrZ language, of that with wdiich the child is already familiar when presented to him in spoken language. It is the hope of the author that this little book will help to diminish, rather than increase, the amount of time devoted to number-work in the first two grades of the school. If the work is properly done, but little time will be required in its performance. Mere " figure processes " are designedly omitted from most of the pages of the book. The child should be well grounded in elementary number processes before he begins to practice figure manipulation. Such nuriiber %vork furnishes the only proper basis for the figiire work that will follow. To teachers who may find difficulty in supplying profitable " seat- work " for their pupils, and who have heretofore kept them busy with long figure-problems, the following is suggested : In place of mere figure work, require the pupils to copy certain pages from this book, carefully filhng all the blanks. Select pages with which the child is quite familiar, and put the main emphasis upon accuracy. Make the lesson simple enough, and so impress the pupils with the importance of absolute accuracy that at least 75% of all the papers (or slates) examined will be free from errors — in spelling, in use of capital letters, in punctuation, or in figures. Commend only those who have "perfect papers" — perfect in respect to the four points named above. The educative value of such an exercise as this is not to be com- pared with mere figure manipulation in which "90% of accuracy" is accounted good, 80% fair, and 70% good enough to "pass." Let the pupils early learn that in arithmetic, at least, nothing short of almost perfect accuracy is of any value whatever. Measure the pupils in all the grades, not so much by the ratio of the number of accurate answers to the number of problems given, but rather by the amount of work they are able to do ivithout snaking any errors. Allow no pupil to leave any grade of the school with the impression that in number-work "90% of accuracy " is good. r. H. H. Jacksoxville, III., Jan. 1, 1901. THE PRIMER OF ARITHMETIC. one two three four four three I I n III II 11 2 • • • 3 4 inch inches 1 and 1 are 2 and 1 are 2 and 2 are 3 and 1 are 2 inches and 1 incli are inches. — inches. 2 inches and 2 inches are - 4 inches less 1 inch are inches. 4 inches less 2 inches are inches. 3 inches less 1 inch are inches. 1. One half of 4 inches is inclies. 2. One half of 3 inches is 3. One half of 2 inches is inch. 4. John had a pencil exactly 4 inches long. He broke it into two equal pieces. Each piece was inches long. A 1-incli square. A 2-incli fiqnaro. 1 in. by 1 in. 1. A square has - corners. 2. A square has square comers. 3. A square has 2 i'-iclies by 2 iucLtiB. - equal sides. 4. The red s(|uare is a square. 5. The yellow square is a square. times as large as — as large as the 6. The yellow square is - the red square. 7. The red square is yellow square. 8. A 2-inch square equals 1-inch squares. 9. A 1-inch square equals what part of a 2-inch square ? 10. Two 1-inch squares equal what part of a 2- inch square? 1. John had 3 cents. His father gave him 1 cent more. He then had cents. 2. Mary had 3 cents. She spent 1 cent. She then had cents. 3. At 2 cents each, two lemons cost 2 times 2 cents are cents. cents. 4. How many 2-cent stamps can you buy with 4 cents ? 5. One half of 4 cents* is cents. 6. One half of 4 apples is - 7. One half of 3 apples is - 8. One half of 2 apples is - 9. Two apples are 1 half of 10. One api:)le is 1 half of — apples. - apple. — apples, apples. 11. One and one half apples are one half of apples. 12. Henry sold apples at the rate of 2 apples for 1 cent. For 4 apples he should receive cents. 3 and 1 = 4 less 2 = 2 and 1 = 4 less 3 = 2 and 2 = 4 less 1 = -11!! I II five live 4 and 1 are . 3 and are 5. 5 less 1 = 5 less 2 = 1 half of 5 is 5 less 4 = 5 less 3 = 2 twos and 1 are 2 books and 2 books are 3 boys and 2 boys are — 2 men and 1 man are — 1 girl and 4 girls are — books. - boys, men. girls. 2 apples and 2 apples are apples. 4 marbles and 1 marble are marbles. 4 inches and 1 inch = 5 inches less 2 inches = 3 inches less 2 inches = 2 inches and 3 inches = 5 inches less 3 inches = 3 inches and 2 inches = 5 inches less 1 inch = 1. One half of 5 inches is 2. Two inches are 1 half of inches. 3. One and 1 half inches are 1 half of inches. 4. Two and 1 half inches are 1 half of inches. 5. One inch is 1 half of inches. 6. In 1 inch there are half-inches. 7. In 2 inches there are half-inches. 8. In 2 and I half inches there are half-inches. 9. Henry had a pencil exactly 5 inches long. He broke it into two equal pieces. Each piece was inches long. An oblong. 7 An oblong. An oblong. 1 in. by 2 in. 1. has An oblong — corners. 2. An oblong lias sides. 1 in. by 3 in. 3. Two of the sides of an ob- long are longer than the other two sides. 4. The red oblong is inch wide and inches long. 5. The yellow oblong is ^ incli wide and ■ inches long. 6. How long is the bine oblong? 1 in. by 5 in. 8 1. Hemy had 3 cents. He earned 2 cents more. He then had cents. 2. Helen had 5 cents. She spent 3 cents. She then had cents. 3. At 1 cent each, 5 apples cost cents. 5 times 1 cent are cents. 4. How many 2-cent stamps can yon buy with 5 cents? 5. One half of 5 pies is pies. 6. An oblong 1 inch wide and 5 inches long is equal to 1-incli squares. 7. An oblong 1 inch by 5 inches is times as large as a 1-inch square. 8. Cut from paper a 2-incli square. (a) Place it upon your desk with one edge par- allel with the front edge of your desk. (b) Fold the front edge upon the back edge. Crease. Unfold. (c) Fold the right edge upon the left edge. Crease. Unfold. (d) A 2-inch square is equal to 1-inch squares. SIX SIX SIX III III II 11 I I II 5 and 1 are . 4 and are 6. 3 and are 6. 6 less 5 = 6 less 4 = 6 less 2 6 less 3 4 books and 2 books are 2 hats and 3 liats are — 3 cows and 3 cows are — 5 birds and 1 bird are — 2 flaos and 3 fla2:s are — book^ hats. - cows. - birds. ^ . ^.. flass. 4 sleds and 2 sleds are sleds. 3 twos = C-l 2 threes = 10 V 1 M M ' \,M ' \-n:y i.i.l.Tii,i,i,T,\,\,\ 3 inches and 3 inches = 6 inches less 3 inches = 4 inches and 2 inches = ^ 6 inches less 2 inches = 5 inches and 1 inch = 1. A foot is twelve inches. 2. One half of a foot is inches. 3. One half of six inches is inches. 4. Six inches are one half of . 5. Three inches are one half of . 6. In three inches there are half-inches. 7. My pencil is about inches long. 8. My slate is about inches wide. 9. Draw a horizontal line 3 inches long. 10. Draw a vertical line 3 inches long. 11 1. The red oblong is 2. The red oblon.s; is inches wide, inches hins;. 3. An oblong two inches wide and 3 inches long is equal to 1-inch squares. 4. Two threes are -. Three twos are . 5. An oblong 1 inch wide and 6 inches long is equal to 1-inch squares. 6. The red oblong is how many 1-inch squares larger than a 2-inch square? 7. The red oblong is how many times as large as a 1-inch square? 8. One half of the red oblong equals how many 1-inch squares? 12 1. James had 4 arrows. He made 2 more. He then had arrows. 2. Kichard had 6 arrows. He lost 4 of them. He then had arrows. 3. At 2 cents each, 3 oranges cost cents. 3 times 2 cents are cents. 4. How many 2-cent stamps can you buy with 6 cents? 2 cents are contained in 6 cents times. 5. One half of 6 days is days. 6. Cut from paper an oblong 2 inches by 3 inches. Cut from paper a 2-inch square. The oblong is times as large as the square. 7. Cut from paper 6 1-inch squares. (a) Arrange these into an oblong 1 inch by 6 inches. (b) Arrange them into an oblong 2 inches by 3 inches. (c) Make with them a 2-incli square. What part of a 2-inch square can you make with what remain? 8. Two 1-inch squares are of a 2-inch square. C-4 13 seven seven seven seven o • G ® • • 9 • • 6 and 1 are . 5 and are 7. 4 and are 7. 7 less 1 = 7 less 2 = 7 less 3 = 1 half of 7 is 5 pins and 2 pins are 4 caps and 2 caps are 4 hens and 3 hens are 3 eggs and 3 eggs are D-l 7 less 6 = 7 less 5 = 7 less 4 = pins, caps, hens. eggs. 14 4 inches and 3 inches = 7 inches less 3 inches = 5 inches and 2 inches = 7 inches less 2 inches = 4 inches and 3 inches = 7 inches less 4 inches = 6 inches and 1 inch = 7 inches less 1 inch = 1. One half of a foot and 1 inch are inches, 2. One half of a foot less 1 inch equals inches. 3. One half of 7 inches is inches. 4. In three and 1 half inches there are half- inches. 5. Three and 1 half inches are 1 half of inches. 6. Draw a horizontal line 6 inches long. With- out measuring, divide it into two equal parts. Each part should be inches long. 7. Draw a horizontal line 5 inches long. With- out measuring, divide it into two equal parts. Each part should be inches long. 15 1. The yellow oblong is inches Avide and inches long. It contains 1-inch sqnares and 2 halves of a 1-inch square. We say that it contains 7 square inches; that is, it is 7 times as large as a 1-inch square. 2. An oblong 2 inches by 3 inches contains square inches. 3. A 2-inch square contains square inches. 4. Think of an oblong 2 inches wide and 2 and 1 half inches long. How many square inches does it contain? 5. Two times 3 and 1 half are . 6. Two times 2 and 1 half are . 16 1. Mr. Smith had 4 cows. He bonglit 3 more. He then had cows. 2. Mr. Harris had 7 cows. He sold 2 of them. He then had cows. 3. At 2 dollars each, 3 sheep cost dollars. 3 times 2 dollars are — - dollars. 4. How many 2-cent stamps can you buy with 7 cents? 5. One half of 7 dollars is dollars. 6. An oblong 1 inch by 7 inches contains square inches ; that is, it is times as large as a 1-inch square. 7. In a week there are days. 8. The first day of the week is . 9. The 7th dav of the week is . 10. The second day of the week is . 11. George goes to school 5 days of each week, and is at home the other days of the week. He is at home days. In 2 weeks he is in school days. IM 17 eight • • • • • • • • 8 eight • • • ® • • • • 8 eight o • • • 8 eight • • • • 8 7 and 1 are . 6 and — — are 8. 5 and — 4 and — — are 8. — are 8. 8 less 4 are . 8 less 1 = 8 less 7 = 8 less 2 = 8 less 6 = 8 less 3 = 8 less 5 = 1 half of 8 is . 5 cents and 3 cents are cents. 4 days and 3 days are days. 6 nails and 2 nails are nails. 4 twos 9 • a • • • • • E-l 2 fours = 18 4 inches and 4 inches = 8 inches less 4 inches = 5 inches and 3 inches = 8 inches less 3 inches = 6 inches and 2 inches = 8 inches less 2 inches = 8 inches less 6 inches = 1. One half of a foot and 2 inches are inches. 2. One half of a foot less 2 inches equals inches. 3. One half of 8 inches is inches. 4. In 4 inches there are half-inches. 5. Four inches are 1 half of inches. 6. Draw a horizontal line 7 inches long. With- out measuring, divide it into two equal parts. Each part should be inches long. 7. Draw a horizontal line 8 inches long. With- out measuring, divide it into two equal parts. Each part should be inches long. 8. Draw a vertical line 4 inches long. Without measuring, divide it into two equal parts. Each part should be inches long. 19 1. The green oblong is — inches long. In it there are inches wide and 1-inch squares. 2. An oblong 2 inches by 4 inches contains square inches; that is, it is times as large as a 1-inch square. 3. Two times 4 are . 4. Four times 2 are . 5. Three times 2 are . 6. Two times 2 are . 7. Two times 2 cents are cents. 8. Three times 2 cents are cents. 9. Four times 2 cents are cents. 10. Two times 3 cents are cents. 20 1. Mrs. Smith had 5 boxes of berries. She picked 3 boxes more. She then had boxes. 2. Mrs. Harris had 8 boxes of berries. She sohl 6 box-es. She then had boxes. 3. Mary gets 2 cents a box for picking berries. For picking 4 boxes she gets cents. 4 times 2 cents are cents. 4. How many 2-cent stamps can you buy for 8 cents? 2 cents are contained in 8 cents times. 5. One half of 8 boxes of berries is boxes of berries. 6. An oblons; 1 inch by 8 inches contains square inches ; that is, it is times as large as a 1-inch square. 7. Cut from paper an oblong 2 inches by 4 inches. (a) Place it on your desk with the long edge par- allel with the front edge of your desk. (b) Fold the right edge upon the left edge. Crease. Unfold. (c) Observe, that an oblong 2 inches by 4 inches is equal to two squares. 21 nine nine nine nine • • • 8 and 1 are 7 and 6 and 5 and 9 less 2 = 9 less 3 = 9 less 4 = are 9. are 9. are 9. 9 less 7 = 9 less 6 = 9 less 5 = 1 half of 9 is 3 threes = • • e • F-l 1 third of 9 = 22 5 inches and 4 inches = 9 inches less 4 inches = 6 inches and 3 inches = 6 inches less 3 inches = 7 inches and 2 inches = 7 inches less 2 inches = 1. One half of a foot and 3 inches are inches. 2. One half of a foot less 3 inches equals '■ inches. 3. One half of 9 inches is inches. 4. One third of 9 inches is inches. 5. Two thirds of 9 inches are inches. 6. Draw a horizontal line 9 inches long. Divide it into two equal parts. Each part should be inches long. 7. Draw a horizontal line 9 inches long. Divide it into three equal parts. Each part should be inches long. 8. Draw a vertical line 6 inches long. Divide it into three equal parts. Each part should be inches long. 23 1. The blue ligure is a square. In it there are 1-inch squares. .2. A 3-inch square contains square inches; that is, it is ■ times as large as a 1-inch square. 3. Three times 2 are . 4. Three times 3 are . 5. Two times 3 marks are - 6. Three times 3 marks are 7. Two times 4 marks are - 8. Four times 2 marks are - F-a marks. - marks. marks, marks. 24 1. Jane paid 5 cents for a tablet and 4 cents for a pencil. For both she paid cents. 2. Lily is 9 years old and her sister is 6 years old. Lily is years older than her sister. 3. At 3 cents each, 2 oranges cost cents; 3 oranges cost cents. 2 times 3 cents are cents. 3 times 3 cents are cents. 4. How many 3-cent stamps can you buy for 9 cents? 3 cents are contained in 9 cents times. 5. One third of 9 days is days. 6. Two thirds of 9 days are days. 7. Look at the figure on page 19. If this figure were 1 inch longer than it is, it would contain square inches. An oblong 2 inches by 4 and 1 half inches contains square inches. 4i and 4i are — 2 times 4i are — 3i and 3^ are — 2 times 3i are — 2 times 2i are — 25 ten ten ten ten ten • • • • • 9 and 1 are 8 and 7 and 6 and 5 and 10 less 2 = 10 less 3 = 10 less 6 = ten 10 10 10 10 10 are 10. are 10. are 10. are 10. 10 less 8 10 less 7 10 less 4 10 26 5 inches and 5 inches = 10 inches less 5 inches = 6 inches and 4 inches = 6 inches less 4 inches = 7 inches and 3 inches = 7 inches less 3 inches = 1. One half of a foot and 4 inches are inches. 2. One half of a foot less 4 inches equals inches. 3. One half of 10 inches is inches. 4. In 5 inches there are half-inches. 5. Five inches are 1 half of inches. j One half of 4 inches is — ( Four inches are 1 half of ( One half of 3 inches is — 8. 9. nches. - inches. ( Three inches are 1 half of inches. One half of 2 inches is inch. Two inches are 1 half of inches. . One half of 5 inches is . Five inches are 1 half of inches. 27 1. The yellow figure is an oblong. It is inches wide and inches long. In it are I-inch squares. 2. An oblong 2 inches by 5 inches contains square inches; that is, it is times as large as a 1-inch square. 3. Two times five 1- inch squares are 1-inch squares. 4. Five times two 1- inch squares are 1-inch squares. 5. Two times five books are books. 6. Five times two books are books. 7. Two times 5 = 8. Five times 2 = 28 1. Alice is 8 years old. In two years more she will be years old. 2. Susie is 10 years old. She began to go to school 4 years ago. Then she was years old. 3. At 5 cents each, 2 pencils cost cents. 2 times 5 cents are cents. 4. How many 2-cent stamps can you buy for 10 cents? 2 cents are contained in 10 cents times. 5. One third of 9 years is years. ^ (One third of 3 inches is inch. 6. < ( Three inches are 1 third of inches. 7. Look at the figure on page 19. If this figure were 1 inch shorter than it is, it would contain square inches. An oblong 2 inches by 2A inches contains square inches. 2i and 2h are . 2 times 2h are . 2 times 3i are . 2 times li are . 2h and 2h and 2i = 3 times 2h are . G-4 29 eleven 10 and 1 are 9 and - 8 and - 7 and - 6 and - 11 less 1 = 11 less 2 = 11 less 3 = 11 less 4 = 11 less 5 = One half of 11 is 11 are 11. are 11. are 11. are 11. 1 less 10 = 1 less 1 less 1 less 1 less 9 = 8 = 7 = G = 8 chairs and 3 chairs are 5 chairs and 1 chairs are 7 chairs and 4 chairs are chairs, chairs, chairs. 4 flags and 6 flags are 6 flags and 5 flags are 8 flags and 2 flags are flags, flags, flags. 30 6 inches and 5 inches = 11 inches less 6 inches = 7 inches and 4 inches = 7 inches less 4 inches = 8 inches and 3 inches — 8 inches less 3 inches = 1. One half of a foot and 5 inches are inches. 2. One half of a foot less 5 inches eqnals inch. 3. One half of 11 inches is inches. 4. In 51 inches there are half-inches. 5. Five and 1 half inches are 1 half of inches. 6. My book is abont inches wide. 7. My book is about inches long. 8. My pencil is about inches long. 9. Draw two parallel vertical lines one half-inch apart and 4 inches long. 10. Draw two parallel horizontal lines one half- inch apart and 4 inches long. 31 1. An oblong 2 inclies wide and 5h inches long contains square inches. 5h and 5h are . Two times 5h are . 2. Which is the larger, a 3-inch s(|Liare or an oblong 2 inches by 5 inches? The square contains sc^uare inches. The oblong contains square inches. 3. Which is the larger, a 2-inch square or an oblong 1 inch by 5 inches? The square contains square inches. The oblong contains square inches. 4. How many square inches in an oblong that is half a foot long and 1 inch wide ? 5. I am thinking of an oblong that is 2 inches wide. It contains 6 square inches. How long is it? 2 times are 6. 6. I am thinking of an oblong that is 2 inches wide. It contains 8 square inches. How long is it? 2 times are 8. 2 times are 5. 2 times are 7. 32 1. Jason rode on his wheel 6 miles Monday and 5 miles Tuesday. In all he rode miles. 2. Little Joe started in the morning to ride to his uncle's home, 11 miles away. By noon he had ridden 7 miles. He had miles farther to go. 3. Bennie can easily ride his wheel at the rate of 5 miles an hour. In 2 hours he can ride miles. 2 times 5 miles are miles. 4. How many 5-cent stamps can you buy with 11 cents? 5. One third of 9 miles is miles. 6. There are 5 school days in a week. In two weeks there are school days. 7. Frank had 6 cents and his brother had half as many. Together they had cents. 8. Two yards of ribbon at 4 cents a yard will cost cents. 9. A nickel is cents. 10. A dime is cents. 11. A nickel and a dime are cents. 12. A "quarter" is cents. 33 twelve 12 1^^ 11 and 1 are 10 and 9 and 8 and 7 and 12 less 2 = 12 less 3 = 12 less 4 = 12 less 5 = 12 less 6 = are 12. are 12. are 12. are 12. 12 less 10 12 less 9 12 less 8 12 less 7 2 sixes = One lialf of 12 is 9 books and 3 books are books. 8 books and 2 books are books. 7 books and 5 books are books. 5 girls and 6 girls are 4 girls and 8 girls are 3 girls and 7 girls are girls, girls, girls. 34 6 inches and 6 inches = 12 inches are 1 . 1 foot less 5 inches = 1 foot less 4 inches = 7 inches and 5 inches = 1 foot less 3 inches = 8 inches and 4 inches = 8 inches are 1 fourth of a foot. 6 inches are of a foot. 9 inches are of a foot. i of a foot and 1 inch are inches. i of a foot and 1 inch are inches. f of a foot and 1 inch are inches. f of a foot less 1 inch are inches. 1. Draw on the blackboard two parallel horizon- tal lines 1 foot long and 2 inches apart. 2. Draw a 1-foot square. Draw a 2-foot square. The 2-foot square is how many times as large as the 1-foot square? 3. Draw an oblong 1 foot wide and 3 feet long. The oblong is equal to 1-foot squares. 35 1. An oblong 2 inches wide and 6 inches long contains square inches. 6 and 6 are . Two times 6 are . Six times 2 are . 2. Which is the larger, a 2-foot square or an oblong that is 1 foot by 5 feet? The square contains square feet. The oblong contains square feet. 3. Wliicli is the larger, a 3-foot square or an oblong 2 feet by 5 feet? The square contains square feet. The oblong contains square feet. 4. How many square inches in an oblong that is half a foot long and 2 inches wide? 5. I am thinking of an oblong that is 2 inches wide. It contains 10 square inches. How long is it? 2 times are 10. 6. 1 am thinking of an oblong that is 2 inches wide. It contains 12 square inches. How long is it? 2 times are 12. 6 times are 12. 36 1. Herbert worked 7 hours Monday and 5 liours Tuesday. In both days he Avorked hours. 2. Mr. Jones works 12 hours each day. His son works 8 hours. The father works hours more than the son. 3. Alfred worked 2 hours a day for all the work- ing days of a week. In all he worked hours. 6 times 2 hours are hours. 4. Wilbur can cane a chair in 2 hours. In 10 hours he can cane chairs. 2 hours are contained in 10 hours times. 5. One third of 6 hours is hours. 6. There are 6 working days in a week. In 2 weeks there are working days. 7. Peter had 8 cents and his brother had half as many. Together they had cents. 8. Two yards of ribbon at 6 cents a yard will cost cents. 9. A year is 12 months. One fourth of a year is months. Three fourths of a year are months. 37 3 fours = 4 threes = 2 fours = 3 threes = 1. Four times 3 stars are stars. 2. Three times 4 stars are stars. 3. One third of 12 stars is stars. 4. Two thirds of 12 stars are stars. 5. Four stars are of 12 stars. 6. Six stars are — of 12 stars. 7. Three stars are of 12 stars. 9 stars and 3 stars are stars. 8 stars and 4 stars are stars. 12 stars less 3 stars are stars. 12 stars less 4 stars are stars. 6 stars and 6 stars are stars. 12 stars less 5 stars are stars. 12 stars less 2 stars are stars. 12 stars less 6 stars are stars. 38 1. Draw a line on the blackboard 3 feet long. Three feet are 1 yard. 2. Draw a line on the blackboard 2 yards long. Two yards are feet. 3. Measure off three yards of string. , Three yards are feet. 4. Measure off four yards of string. Four yards are feet. 5. One yard is feet. One half of a yard is' feet. 6. One third of a yard is foot. 7. Two thirds of a yard are feet. 8. One and 1 third yards are feet. 9. One and 2 thirds yards are feet. 10. One and 1 half yards are feet. 11. Draw a 1-foot square. Draw a 3-foot square. The 3-foot square is how many times as large as the 1-foot square? 12. Which is the larger, a 1-yard square or a 3-foot square? 13. Draw a line one yard long. Divide it into 2 equal parts. Each part should be long. 39 1. This oblong is long. It contains - - inches wide and square inches. inches 4 threes are 3 fours are - 2 sixes are - 2. I am thinking of a 1-yard square divided into 1-foot squares. How many rows of 1-foot squares? How many 1-foot squares in each row? 3 threes are . 40 1. Albert had 6 liens. He bought 5 more. He then had hens. 2. Hattie had 10 hens. She sold 6 of them. She then had hens. 3. A horse needs 4 shoes. Three horses need . shoes. 3 times 4 shoes are shoes. 4. Hiram earns 4 cents an hour. In how many hours can he earn 12 cents? 4 cents are contained in 12 cents times. 5. One third of 12 cents is cents. 6. Buttons, eggs, oranges, lemons, etc., are often bought by the dozen. A dozen is 12. 7. Men sometimes count eggs by taking 3 eggs in each hand. 3 eggs and 3 eggs are eggs. 6 eggs and 6 eggs are eggs. One six is half a dozen. Two sixes are dozen. Four sixes are dozen. 3. Two dozen eggs are sixes of eggs. 41 EEYIEW OF Xr:\IBER FACTS. Read first by column ; tlieii by line. 3 + 2 = 2 twos = 5-2 = 5-3: 4 + 2 = 3 twos = 6-2 = 6-4: 4 + 3 = 4 twos = 7-3 = 7-4: 5 + 2 = 5 twos = 7-2 = 7-5: 5 + 3 = 6 twos = 8-3 = 8-5: 6 + 2 = 2 threes = 8-2 = 8-6: 5 + 4 = 3 threes = 9-4 = 9-5 6 + 3 = 4 threes = 9-3 = 9-6 7 + 2 = 2 fours = 9-2 = 9-7 6 + 4 = 3 fours = 10-4 = 10-6 7 + 3 = 2 fives = 10 - 3 = 10-7 8 + 2 = 2 sixes = 10-2 = 10-8 6 + 5 = i of 2 = 11-5 = 11-6 7 + 4 = i of 3 = 11-4 = 11-7 8 + 3 = i of 4 = 11-3 = 11-8 9 + 2 = i of 5 = 11 - 2 = . 11 - 9 7 + 5 = i of 6 = 12-5 = 12-7 8 + 4 = ^ of 7 = 12-4 = 12-8 9 + 3 = i of 8 = 12-3 = 12-9 i of 9 = i of 10 = i of 11 = i of 12 ^ of 6 = i of 9 = f of 9 = i of 12 1 of 12 = i of 8 = i of 12 = 1 of 12 12 inclies are 1 foot. 3 feet are 1 yard. 1 week is 7 days. 1 year is 12 months. 1 dozen is 12 ones. 42 1. This room is about yards wide and about ■ yards long. 2. This room is about yards high. 3. The door is more than yards high. Is it more or less than 1 yard wide ? 4. The teacher's desk is feet inches long. It is feet inches high. IIow wide is it? 5. Kuby is feet — - inches tall. 6. One half of 2 feet is foot. 7. Two feet are 1 half of feet. 8. One half of 4 feet is feet. 9. Fouf feet are 1 half of feet. 10. One half of 6 feet is feet. 11. Six feet are 1 half of feet. 12. Adam is 3 feet 8 inches tall. Oscar is 3 feet 4 inches tall. Adam is inches taller than Oscar. 13. Abbie is 3 feet 9 inches tall. Sarah is 3 inches taller than Abbie. Sarah is tall. 14. Four feet and 1 foot 2 inches are . 43 1. The above figure is inclies long. It is — inches wide. 2. The perimeter of a figure is the distance around it. The perimeter of the above .oblong is inches. 3. The area of a figure is the amount of its surface. The area of the above oblong is square inches. 4. The perimeter of a 2-inch square is inches. 5. The area of a 2-inch square is square inches. 6. Draw an oblong and tell its perimeter and its area. 44 1. William had 7 slieep. He bought 5 more. He then had sheep. 2. Elsie had 12 chickens. The rain killed 5 of them. She then had chickens. 3. At 6 cents each, 2 melons cost cents. 2 times 6 cents are cents. 4. Ernest earns 6 cents an liom\ In how many hours can he earn 12 cents? 6j^ are contained in 12^ times. 5. Dora paid 12 cents for three lemons. One lemon cost cents. One third of 12 cents is cents. 6. At 6^ a dozen, 2 dozen buttons cost cents. Half a dozen cost cents. 7. Six eggs and 6 eggs and 6 eggs and 6 eggs make dozen eggs. 8. Six inches and 6 inches and 6 inches and 6 inches make feet. 9. Six months and 6 months and 6 months and 6 months make years. 10. Three eggs are of a dozen. 45 twenty 20 thirty 30 2 tens and 1 are . 3 tens and 2 are . 4: tens and 3 are — — . 5 tens and 5 are . 12 3 4 5 11 12 13 U 15 21 22 23 24 25 31 32 33 34 35 34 is 3 tens and . 53 is 5 tens and . 2 tens are . 3 tens are — . 4 tens are forty. 5 tens are fifty. . 2 tens and 5 are . 3 tens and 6 are - — . 4 tens and 4 are . 5 tens and 9 are 6 7 8 9 10 16 17 18 19 20 26 27 28 29 30 36 37 38 39 40 46 is 4 tens and . 27 is 2 tens and . 46 10 and 1 are 10 and 2 are 10 and 3 are 10 and 4 are 10 and 5 are 10 and 6 are 10 and 7 are 10 and 8 are 10 and 9 are 19 and 1 are 20 and 1 are 20 and 2 are 20 and 3 are 20 and 4: are 20 and 5 are 20 and 6 are 20 and 7 are 20 and 8 are 20 and 9 are 29 and 1 are 1. Charlie's book cost 25 cents and liis pencil cost 2 cents. Together they cost cents. 25 is tens and . 27 is tens and — -. 2. Thomas had 34 marbles, him 2 more. He then had His brother gave marbles. 12 + 2 = 14 + 2 = 16 + 2 = 18 + 2 = 22 + 2 24 + 2 26 + 2 28 + 2 32 + 2 34 + 2 36 + 2 38 + 2 47 A 1-iuch cube. A 2-inch cube. 1. With 1-incli cubes, build a 2-inch cube. It takes 1-inch cubes to make a 2-inch cube. 2. A 2-inch cube contains cubic inches ; that is, it is times as large as a 1-inch cube. 2 times 2 cubic inches are cubic inches. 2 times 4 cubic inches are cubic inches. 48 1. Aaron paid 35 cents for a book and 3 cents for paper. For both he paid cents. 2. Maggie had 35 chickens. A rat killed 3 of them. She then had chickens. 3. A bntterfly has wings. Three butterflies have wings. 3 times 4 wings are — — wings. 4. At 2 cents each, how many oranges can you buy for 10 cents? 2^ are contained in 10^ times. 5. Willis paid 8 cents for 4 lemons. One lemon cost cents. One fourth of 8 cents is cents. 6. Thirteen eggs are 1 dozen and . 7. Thirteen cents are 1 dime and cents. 8. Thirteen inches are 1 foot and inch. 9. Thirteen months are 1 year and month. 10. Thirteen feet are 4 yards and foot. 11. Three buttons are 1 fourth of a . 12. Three inches are 1 fourth of a . 13. Three months are 1 fourth of a — — . 49 7 stars and 7 stars are stars. 2 sevens are . 7 twos are — 2 times 7 stars are stars. 7 times 2 stars are stars. 1. One week is days, and two weeks are — days. 7 days and 7 days are days. 2 times 7 days are days. Six tens are sixty. Seven tens are seventy. Eight tens are eighty. Nine tens are ninety Ten tens are one hundred. 2 tens and 2 tens are . One half of 40 is . 3 tens and 3 tens are . One half of 60 is . 4 tens and 4 tens are . One half of 80 is . 50 12 and 2 are 32 and 2 are 52 and 2 are 72 and 2 are 94 less 2 are 74 less 2 are 54 less 2 are 34 less 2 are 14 less 2 are 22 and 2 are 42 and 2 are 62 and 2 are 82 and 2 are 84 less 2 are 64 less 2 are 44 less 2 are 24 less 2 are 14 less 1 are 1. One half of 40 = i of 6 = One half of 46 is . 2. One half of 60 = i of 8 = One half of 68 is . 3. One half of 80 = i of 4 = One half of 84 is . 4. Nancy's book cost 40 cents. Her sister's book cost half as much. Together they cost cents. 5. Julian's book cost 30 cents. His brother's book cost twice as much. His brother's book cost cents. 51 A 1-iucli cube. 1. This prism is 1 inch wide, 1 inch thick, and 4 inches long. 2. It is equal to 1- inch cubes. 3. It contains cubic inches; that is, it is times as large as a 1-inch cube. 4. The prism is equal to what part of a 2-inch cube? 5. One half of a 2-inch cube contains cubic inches. 6. A prism I inch by 2 inches by 4 inches con- tains cubic inches. M-3 iiiii'i' iii'iinHHiii niiiHniiiiiii A square prism. 52 1. Mr. Davis had 42 slieep. He bought 6 more. He then had sheep. 2. Mr. Evans had 59 sheep. Six of them died. He then had sheep. 3. At 7 cents a yard, 2 yards of ribbon cost cents. 2 times 7 cents are cents. 4. HoAv many 2-cent stamps can you buy for 1-1 cents? 2^ are contained in 14^ times. 5. Albert paid 14 cents for 2 melons. One melon cost cents. One half of 14 cents is cents. 6. Fourteen eggs are 1 dozen and - 7. Fourteen cents are 1 dime and - 8. Fourteen inches are 1 foot and - 9. Fourteen days are weeks. 10. Fourteen months are 1 year and 11. Fourteen feet are 4 yards and — 12. Nine buttons are 3 fourths of a 13. Nine inches are 3 fourths of a - - cents. - inches. - months, feet. 53 5 stars and 5 stars and 5 stars : 3 and 3 and 3 and 3 and 3 = 3 times 5 stars are stars. 5 times 3 stars are stars. 1. Two nickels equal cents. 2. Three nickels equal cents. 3. One dime and 1 nickel ecjual cents. 4. Two dimes equal cents. 5. Two dimes and 1 nickel equal cents. 6. Three dimes equal cents. 7. Three dimes and 1 nickel equal cents. 8. Four dimes and 1 nickel equal cents. 9. Five dimes and 1 nickel equal cents. 10. A quarter and a nickel equal cents. 11. One yard is feet. Two yards are — feet. Three yards are feet. Four yards are feet. Five vards are feet. 54 18 and 2 = 28 and 2 = 38 and 2 = 48 and 2 = 58 and 2 = 68 and 2 = 78 and 2 = 88 and 2 = 21 less 2 are . 31 less 2 are 41 less 2 are - — . 51 less 2 are 61 less 2 are . 71 less 2 are 81 less 2 are . 91 less 2 are 101 less 2 are . 151 less 2 are 30 = 20 and 10. 1. One half of 30 is (1 ten and 5) . 50 = 40 and 10. 2. One half of 50 is (2 tens and 5) . 70 = 60 and 10. 3. One half of 70 is (3 tens and 5) . 90 = 80 and 10. 4. One half of 90 is (4 tens and 5) . 5. Jnlia's book cost 50 cents. Her brother's book cost half as much. Her brother's book cost cents. 55 1. A cube has equal faces. In tliis picture of a cube, three of the faces can be seen and are hicklen from view. 2. Each face of a 1-inch cube is a inch square. 3. Each face of a 2-inch cube is a inch square. 4. The area of one face of a 2-inch cube is s(|uare inches. 5. Each face of a 3-inch cube is a . 6. A cube has corners. In this picture of a cube, 7 of the corners are in sight and is hidden from view. 7. A cube has edges. In this picture of a cube, 9 of tlie edges are in sight and are hidden from view. N-3 56 1. There were 36 pupils in a certain room. Four more Avere sent in. Then there were pupils in the room. 2. There Avere 50 pupils in a certain room. Four were sent out. Then there were pupils in the room. 3. At 5 cents a yard, 3 yards of ribbon cost cents. 3 times 5 cents are cents. 4. How many 3-cent stamps can you buy with 15 cents? 3^ are contained in 15^ times. 5. Oscar paid 15 cents for 3 tablets. One tab- let cost cents. One third of 15 cents is cents. 6. Fifteen eggs are 1 dozen and . 7. Fifteen cents are 1 dime and cents. 8. Fifteen inches are 1 foot and inches. 9. Fifteen days are 2 Aveeks and day. 10. Fifteen months are 1 year and months. 11. Fifteen feet are yards. 57 8 stars and 8 stars are stars. 4: and 4 and 4 and 4 = 2 times 8 stars are stars. 8 times 2 stars are stars. 4 times 4 stars are stars. 1. A spider has legs. Two spiders have legs. 2 times 8 legs are legs. 2. A dog has legs. Four dogs have legs. 4 times 4 legs are legs. 3. A bird has 2 legs. Eight birds have legs. 8 times 2 legs are legs. 4. An ox needs - — shoes. Two oxen need shoes. 2 times 8 shoes are shoes. 5. A horse needs shoes. Four horses need shoes. 4 times 4 shoes are shoes. 58 19 and 2 = 39 and 2 = 59 and 2 = 79 and 2 = 29 and 2 49 and 2 69 and 2 89 and 2 22 less 2 are 45 less 2 are 67 less 2 are 89 less 2 are 102 less 2 are 33 less 2 are 54 less 2 are 73 less 2 are 98 less 2 are 152 less 2 are 26 = 20 and 6. 1. One half of 26 is (1 ten and 3) . 48 = 40 and 8. 2. One half of 48 is (2 tens and 4) . 64 = 60 and 4. 3. One half of 64 is (3 tens and 2) . 82 = 80 and 2. 4. One half of 82 is (4 tens and 1) . 5. Jennie's book cost 46 cents. Her sister's book cost half as much. Her sister's book cost cents. Take a 4-inch square of paper. Fold lower edge upon upper edge. Crease. Unfold. Fold lower edge to the crease. Crease. Unfold. Fold upper edge to the middle crease. Crease. Unfold. Fold right edge upon left edge. Crease. Unfold. Fold right edge to the middle crease. Crease. Unfold. Fold left edge to the middle crease. Crease. Unfold. Cut as indicated by the heavy lines. Then fold into a box 2 inches square, having the four squares in the middle for the bottom of the box. Paste the corner squares upon those next to them. The box will contain cubic inches. 0-3 60 1. There were 28 pupils in a certain room. Four more were sent in. Then there were pupils in the room. 2. There were 42 pupils in a certain room. Four were sent out. Then there were pupils in the room. 3. There were 8 pupils in each row of seats. How many in 2 rows? 2 times 8 pupils are pupils. 4. How many 4-cent stamps can you buy with 16 cents? 4^ are contained in 16^ times. 5. Bernie paid 16 cents for 2 boxes of berries. One box of berries cost cents. One half of 16 cents is cents. 6. Sixteen eggs are 1 dozen and . 7. Sixteen cents are 1 dime and cents. 8. Sixteen inches are 1 foot and inches. 9. Sixteen days are 2 weeks and days. 10. Sixteen months are 1 year and months. 11. Sixteen feet are 5 yards and foot. 61 i^^^i^ ^^-i^ iViV^ lin^-i^ ^^t^ i:xiki^ 9 stars and 9 stars are stars. 6 and 6 and 6 are 3+3+3+3+3+3= 9 twos are . 2 times 9 stars are stars. 9 times 2 stars are stars. 3 times 6 stars are stars. 6 times 3 stars are stars. 1. A butterfly has legs. Two butterflies have legs. Three butterflies have legs. 2 times 6 legs are legs. 3 times 6 legs are legs. 2. On one front foot and one hind foot a cat has toes. On both front feet and both hind feet a cat has toes. 2 times 9 toes are toes. 3. Six teaspoons make a set. In 2 sets there are teaspoons. In 3 sets there are tea- spoons. 62 19 and 3 = 39 and 3 = 59 and 3 = 79 and 3 = 99 and 3 = 29 and 3 49 and 3 69 and 3 89 and 3 98 and 3 23 less 3 are 46 less 3 are 68 less 3 are 89 less 3 are 103 less 3 are 34 less 3 are 53 less 3 are 75 less 3 are 92 less 3 are 153 less 3 are 1. One half of 22 = i of 24 = 2. One half of 26 = i of 28 = 3. One half of 42 = i of 44 = 4. One half of 46 = i of 48 = 5. One half of 62 = i of 64 = 6. One half of 66 = h of 68 = 7. One half of 82 = * of 84 = 8. One half of 86 = I of 88 = Add by lO's to 100; thus, 10, 20, 30, 40, etc. Add by lO's from 2 to 102: thus, 2, 12, 22, etc. 63 2 4 5 (5 Take a 4-inch sqiiare of pajier. Fold as suggested on page 59. Cut off a row of 1-inch squares fi-om one side of the paper, and a row from one end of what remains. You will then have a 3-inch square. Cut this as indicated by the heavy lines in the diagTam on this page ; then fold into a box 1 inch square and 1 inch deep, making the center square the bottom of the box. Paste Xo. 1 upon Xo. 2, and Xo. 3 upon Xo. 1. Paste Xo. 4 upon Xo. 5, and Xo. 6 upon Xo. 4. The box will contain cubic inch. Observe that this box will contain exactly one fourth as much as the box described on page 59. ]\Ieasure 1 cubic inch of sand ; 2 cubic inches ; 3 cubic inches ; 4 cubic inches. To THE Teacher. — Provide a few small boxes or bottles and require the pupils to estimate in cubic inches the capacity of each. Then require the pupils to measure each, using sand and the box described above. P-3 64 1. The mercuiy in the thermometer stood at 68. It went up 10 degrees. It then stood at . 2. The mercury in the thermometer stood at 74. It went down 10 degrees. It then stood at . 3. At one dime a yard, 5 yards of ribbon cost cents. 5 times 10 cents are cents. 4. Hiram paid 40 cents for tickets to the mati- nee. The tickets cost 10 cents each. There were tickets. 10^ are contained in 40^ times. 5. Ada bought 3 ferry tickets for 18 cents. One ticket cost cents. One third of 18 cents is cents. 6. Seventeen eggs are 1 dozen and . 7. Seventeen cents are 1 dime and cents. 8. Seventeen inches are 1 foot and inches. 9. Seventeen days are 2 weeks and days. 10. Eighteen eggs are 1 and dozen. 11. Eighteen inches are 1 and feet. 12. Eighteen days are 2 weeks and days. 10 stars and 10 stars are — — stars. 5 and 5 and 5 and 5 = 4 + 4 + 4 + 4 + 4 = 5 times 4 stars are stars. 4 times 5 stars are stars. 2 times 10 stars are stars. 10 times 2 stars are stars. 2 nickels are equal to dime. 3 nickels are equal to cents. 1. A horse needs shoes. Two horses need shoes. Three horses need shoes. Four horses need shoes. Five horses need shoes. 2. The area of an oblong 4 inches by 5 inches is square inches. 4 times 5 sq. in. are 5 times 4 sq. in. are Q-l sq. m. sq. in. 66 18 and 4 = 28 and 4 38 and 4 = 48 and 4 58 and 4 = 68 and 4 78 and 4 = 88 and 4 22 less 4 = 32 less 4 42 less 4 = 52 less 4 68 less 4 = 78 less 4 89 less 4 = 97 less 4 102 less 4 = 152 less 4 3G = 30 and 6. 1. One third of 36 is (1 ten and 2) — . 39 = 30 and 9. 2. One third of 39 is (1 ten and 3) . 63 = 60 and 3. 3. One third of 63 is (2 tens and 1) . 4. Bessie's book cost 39 cents. Her sister's book cost one third as mnch. Her sister's book cost cents. Subtract by lO's from 100 to 10. Subtract by lO's from 102 to 2. Q-2 67 Fold as suggested on page 59. Cut off a row of l-inch squares from one side of the paper. Cut also as indicated by the heavy lines in the diagram on this page. Fold and paste so as to make a 1-inch cube. Observe the number of its corners, edges, and faces. Each face is what kind of a square? How many such cubes can be put into the box described on page 59? How many such cubes are necessary to build a 2-inch cube? See page 47. 1. A cube lias faces. 2. A cube lias corners. 3. A cube lias edges. 4. Each face of a cube is a square. Q-3 68 1. The mercury in the thermometer stood at 78. It went lip 4 degrees. It then stood at . 2. The mercury in the thermometer stood at 92. It went down 4 degrees. It then stood at . 3. At 20 cents a yard, 2 yards of lace cost — — cents. 2 times 20 cents are cents. 4. Miriam earns 4 cents an hour. To earn 20 cents, she must work hours. 4 cents are contained in 20 cents — — times. 5. Belle paid 20 cents for 4 tickets. One ticket cost cents. One fourth of 20 cents is cents. 6. Nineteen buttons are I dozen and . 7. Nineteen cents are I dime and — - cents. 8. Nineteen days are 2 weeks and days. 9. Nineteen inches are I foot and inches. 10. Nineteen months are 1 year and months. 11. Twenty buttons are 1 dozen and . 12. Twenty inches are 1 foot and inches. 13. Twenty months are 1 year and months. Q-4 69 7 and 7 and 7 = 7 threes = 3 times 7 stars are stars. 7 times 3 stars are stars. 2 3-cent stamps cost cents. 3 3-cent stamps cost cents. 4 3-cent stamps cost cents. 5 3-cent stamps cost cents. 6 3-cent stamps cost cents. 7 3-cent stamps cost cents. 1. There are days in one week. 2. There are days in two weeks. 3. There are days in three Aveeks. 4. Two dimes and cent equal 21 cents. 5. The area of an oblong 3 inches by 7 inches is — square inches. 3 times 7 sq. in. are sq. in. 7 times 3 sq. in. are sq. in. 70 17 and 4 = 27 and 4: = 37 and 4:= 47 and 4 = 57 and 4 = 67 and 4 = 77 and 4 = 87 and 4 = 97 and 4 = 96 and 4 = 21 less 4 = 31 less 4 41 less 4 = 51 less 4 65 less 4 = 74 less 4 87 less 4 = 94 less 4 101 less 4 = 151 less 4 m = 60 and 6. 1. One third of 66 is (2 tens and 2) — 69 = 60 and 9. 2. One third of 69 is (2 tens and 3) — 93 = 90 and 3. 3. One third of 93 is (3 tens and 1) — Add by 2's from 2 to 50. Snbtract by 2's from 50 to 2. Add by lO's from 4 to 104. Subtract by 10' s from 104 to 4. 71 Take an 8-inch square of paper. Fold as directed on page 59. This will make 2-inch squares instead of 1-incli sciuares. Cut as directed on page 07. Fold and paste so as to make a 2-inch cube. Observe the number of its corners, edges, and faces. Each face is what kind of a square? The area of each face is liow many square inches? Each face is how many times as large as the face of a 1-inch cube. EEVIEW PAGE 43. Think of a ?>-inch square. 1. The perimeter of a 3-incli square is inches. 4 times 3 inches are inches. 2. The area of a 3-inch square is square inches. 3 times 3 sq. in. are sq. in. inches. Think of a 4-inch square. 3. The perimeter of a 4-inch square is - 4 times 4 inches are inches. 4. Tlie area of a 4-inch square is square inches. 4 times 4 sq. in. are S(|. in. 5. Tell the perimeter and the area of a 2-inch square. Of a 1-inch square. 72 1. The mercury in the thermometer stood at 64. It went up 20 degrees. It then stood at . 2. The mercury in the thermometer stood at 46. It went down 20 degrees. It then stood at . 3. In an orchard there were 3 rows of trees, and 7 trees in each row. In all there were trees. 3 times 7 trees are trees. 4. A lady divided 18 apples among some boys, giving to each boy 3 apples. There were boys. 3 apples are contained in 18 apples times. 5. Lilian divided a piece of ribbon 20 inches long into 4 equal pieces. Each piece was inches long. One fourth of twenty inches is inches. 6. Twenty-one buttons are 1 dozen and 7. Twenty-one cents are 2 dimes and - 8. Twenty-one inches are 1 foot and inches. 9. Twenty-one months are 1 year and months. 10. Twenty-one days are weeks. 11. Twenty-one feet are yards. 73 8 and 8 and 8 = 3 times 8 stars are stars. 8 times 3 stars are stars. 1. An ox needs shoes. Three oxen need shoes. 3 times 8 shoes are shoes. 2. A tripod has 3 legs. Eight tripods have legs. 8 times 3 legis are lees. 'b'- 3. Three weeks and days are 24 days. 4. Two dimes and — - cents are 24 cents. 5. The area of an oblong 3 inches by 8 inches is — square inches. 3 times 8 sq. in. are sq. in. 8 times 3 sq. in. are sq. in. 6. Two eights are . Three eights are . 7. Two sevens are . Three sevens are . 8. Two sixes are . Three sixes are . 9. Two fives are . Three fives are . 74 17 and 5 = 27 and 5 37 and 5 = 47 and 5 57 and 5 = 67 and 5 77 and 5= 87 and 5 21 less 5 = 31 less 5 41 less 5 = 51 less 5 69 less 5 = 78 less 5 86 less 5 = 95 less 5 101 less 5 = 151 less 5 1. One half of 24 = i of 25 is 2. One half of 26 = i of 27 is 3. One half of 22 = i of 23 is 4. One half of 44 = h of 45 is 5. One half of 46 = i of 47 is 6. One half of 42 = i of 43 is 7. One half of 48 = i of 49 is i of 4 inches = 4 inches are i of ^ of 5 inches = 5 inches are i of i of 6 inches = 6 inches are i of 75 Take a 12-inch square of paper. Fold as directed on page 59. This will divide the paper into 3-inch squares. Cut as dii'ected on page 67. Fold and paste so as to make a 3-inch cube. Observe the number of its corners, edges, and faces. Each face is what kind of a square? The area of each face is how many square inches? Each face is how many times as large as the face of a 1-inch cube? TJdnk of an ohlong 2 i7iches hy 3 inches. 1. The perimeter of an oblong 2 inches by 3 inches is . 3 in. + 3 in. + 2 in. + 2 in. = 2. The area of an oblong 2 inches by 3 inches is square inches. 2 times 3 sq. in. are sq. in. 3 times 2 sq. in. are sq. in. Think of an oblong 2 inches hy 4 inches, 3. The perimeter of an oblong 2 inches by 4 inches is inches. 4 in. + 4 in. + 2 in. + 2 in. = 4. The area of an oblong 2 inches by 4 inches is square inches. 2 times 4 sq. in. are sq. in. 4 times 2 sq. in. are sq. in. 76 1. A farmer had 35 slieep in one pen and 6 in another. In both pens there were sheep. 2. Edwin drew a line 32 inches long. He erased 6 inches of the line. What remained was inches long. 3. An electric car goes at the rate of 8 miles an hour. In three hours it goes miles. 3 times 8 miles are miles. 4. How many 2-quart jars can be filled from a pail containing 14 quarts? 2 quarts are contained in 14 quarts times. 5. Mr. Nelson paid 48 dollars for the rent of a house for 2 months. This was at the rate of dollars a month. One half of 48 dollars is dollars. 6. Twenty-two days are 3 weeks and day. 7. Twenty-three days are 3 weeks and days. 8. Twenty-four days are 3 weeks and days. 9; Twenty-two cents are 2 dimes and cents. 10. Twenty-four cents are 2 dimes and — cents. 11. Twenty-four feet are yards. 77 i^i^i^ ^^^ 6 and 6 and G and 6 = 4 times 6 stars are stars. 6 times 4 stars are stars. 2 times 12 stars are stars. 2 dozen are . 2 years are months. 2 feet are inches. 1. One and 1 half feet are inches. 2. One and 1 half years are months. 3. One and 1 half dozen are . 4. One and 1 fonrth feet are inches. 5. One and 1 fourth years are months. 6. One and 1 fourth dozen are . 7. The area of an oblong 4 inches by 6 inches is — square inches. 4 times 6 sq. in. are sq. in. 6 times 4 sq. in. are sq. in. 78 16 and 5 = 36 and 5 = 56 and 5 = 76 and 5 = 26 and 5 46 and 5 66 and 5 86 and 5 22 less 5 = 42 less 5 = 68 less 5 = 85 less 5 = 32 less 5 52 less 5 79 less 5 96 less 5 64 = 60 and 4. 1. One half of 64 = i of 65 is 2. One half of 66 = i of 67 is 3. One half of 62 = i of 63 is 84 = 80 and 4. 4. One half of 84 = i of 85 is 5. One half of 86 = ^ of 87 is 6. One half of 82 = i of 83 is i of 7 inches = i of 8 inches = i of 9 inches = 7 inches are i of 8 inches are i of 9 inches are i of 79 Take an 8-inch square of paper. Fold as directed on page 59. This will divide the paper into 2-inch squares. Cut as directed on page 63. Fold and ]iaste so as to make a box 2 inches by 2 inches by 2 inches. Fill the box with sand, using the box described on page Go as a measure. Observe that the box will hold 8 cubic inches of sand; that 4 cubic inches fill the box "half fulL" Observe that the solid content of the box is the same as the solid content of a 2-inch cube. See page 47. Think of an ohlon