. ha ieee ore sm - . . sgoort nen ' Tem - ? “os ee MATHEMATICS LIBRARY THE UNIVERSITY OF ILLINOIS LIBRARY The Frank Hall collection of arithmetics, presented by Professor H. L. Rietz of the University of Iowa. AOA SIF RAA hu Return this book on or before the Latest Date stamped below. University of Illinois Library L161—H41 /. oe ® Yl eitlid Bee i Dd “4 Pity vue ia aX: PUIRIT) te eer Peed ak = APPLETONS’ STANDARD ARITHMETICS NUMBERS ILLUSTRATED AND APPLIED IN LANGUAGE, DRAWING, AND READING LESSONS AN ARITHMETIC FOR PRIMARY SCHOOLS tof) Anprew fF. * RICKOFF AND @ IS AOR MOP STE NEW YORK, BOSTON, AND CHICAGO D. APPLETON AND COMPANY 1886 a ¥ 7 ee f * , rid O bas : ._/ Jie . Ym fo i Pe i . , 1 * ca) Ft a ER be 2 - in af iis - CopyricHt, 1886, M “he HARE By’ D. APPLETON AND COMPANY. a - ‘ - hie ‘ ] t ie \ \ ate { : t . . a . ¢* A ' ys * f » | > Tis = * ; . + 3 P a hd : >| ; a eo52S + , we = ; 7 ow i |e ; 1s) i , Ls : Lad A : CLT Re ey; AUT Was 2G oh Nit) 7 a 6 can Pa XK A400 Oye 2a eee > : é . ne t < Do ; > ae ’ : Dit I a ai ee Mu dove 2] R4e2r~~ — weuenenes upraa PREFACE. Ir is the design of this book, in the first place, to familiarize the chiid with numbers and their combinations, not by means of repeating such for- mule as 4 and 3 are 7, but by provoking observation to lead him to the adop- tion of the formula as a statement of his own experience. In this way an intimate and spontaneous association of thought and expression will de in- duced, and that listlessness avoided which makes it possible for many chil- dren to repeat 4 and 3 are 7, without a thought of 4 or 3, or of the com- bination of 4 and 3. It is not difficult to understand how it is that so much effort is wasted in teaching a child to answer always with readiness and confidence the question, “‘ How many are 4 and 3?” when it is recollected that the custom has too often been, not to lead him to tell us what he has observed and knows, but to repeat a form of words which he had committed to memory as mere words. The methods of onr schools are happily greatly improved in this respect, and it is the purpose of this book, especially of Part I, to afford a great variety of exercises in which the pupil may gain a wide ex- perience in the application of number to objects, and a ready perception of their relation to each other. The pictorial illustrations at the head of the pages, entitled ‘* The Conversation,” are designed for language-lessons in which the immediate - design is to excite thought and cultivate expression, but their adaptation to the ultimate purpose of the book will be readily understood. In the smaller illustrations, under the heading “‘ What can you tell?” the imagination is called into more active play, and the child is led to give more independent and original expression to the ideas gained from the pictures than the purpose of the foregoing exercises would permit, which was to follow out a logical and consistent plan of development. Here he is to find for himself the thread of the “story” hinted at in the picture, and weave it into a connected form for himself, the basis being the special combina- tions suggested in the picture. In these illustrations, the operations in numbers are more definitely brought to notice than previously, but it is especially desirable that po form of words or even process of objective illustration be repeated with such a 4.63940 4. 7E PREFACE. degree of uniformity that the mere form may assume undue importance, or supersede the exercise of intelligence. The Slates are to supplement and carry on the object-work, suggestions for which will be found in the general notes. They serve the twofold pur- pose: first, of teaching the child the use of the slate, at the same time familiarizing him with the language conveying the ideas, position, direction, ete.; and, second, of picturing out the various combinations with more dis- tinctness and freedom from distracting surroundings than can be attained in any other mode of illustration. The Diagrams, which, after Lesson V, take the place of the slates, are designed for more extended slate exercises, still involving the use of num- bers. Their usefulness as primary drawing-lessons can not fail to be rec- ognized. The children are not only to discover and describe the various combina- tions depicted, but, taking them as models, are to exercise their ingenuity in making as many others as possible with marks, dots, etc., upon their slates, or with sticks and other counters. ‘This is a training in form, num- ber, and arrangement, and, if counters of different colors can be obtained, admirable lessons in color may also be given, Thus is the inventive faculty called into play, imagination exercised, and taste cultivated, while the child is becoming accustomed to the number in all its combinations. The con- stant handling of the number, in this and the other exercises, affords the child experience, and, as far as the purely arithmetical aim of these first les- sons is concerned, this is the sole object, not to teach him to say that 2+ 1+2=5, but to lead him to know it by experience. Figures and arithmetical signs have, of course, no place in Part I. Normal or schematic representations and names only are given. These are to be learned by sight, as suggested in the general notes. In the Seript Lesson at the foot of each page will be found the name of each number in script form. This is given for copy-work. Hints.—The suggestions and questions to be found under this heading on the teacher’s page are merely intended as hints of an extended work and great variety of exercises, to be supplied at the discretion of the teacher. The Parts from I to V indicate divisions of the subject; they do not correspond with successive grades in school classification. P fs gad b fie Suggestions for exercises to precede and supplement the /essons an the i/lustrated pages. SLATE EXERCISES. Object Lesson on S/ate,—As the slate and pencil are most important implements of study in this branch, it is well to make them first odjects of study—for one reason, that the child may become so well acquainted with them and their use that his ignorance of the tools may not interfere with the efficiency of the work to be done with them, and, also, because the study of them affords excellent opportunity for certain preparatory work necessary to the introduction of written arithmetic. Although every teacher has, doubtless, her form for object lessons on the slate, yet sugges- tions are here given for such lessons, that certain points having a more es- pecial bearing upon the subject under consideration may not be overlooked. The children are first encouraged to tell, in their own way, all they can see or be led to observe about their slates; then, by skillful questioning, not hinting too much as to answers expected, the teacher draws out the following, in consecutive form. /rame.—Use of frame; what made of; if covered, with what, and why [a little lesson on quietness]; if rubber corners, why. Jor slate part.—Of what made; hard; breakable; color. Parts.—Sides, how many; corners, how many; faces, one looking up at you, upper face; one looking at desk, lower face; how many faces has a slate? How many faces have you? Care of slate-—Breaking, scratch- ing, cleanliness, etc. An Object Lesson on the Pencil has its place here, and the following points to be made are suggested: A conversation on pencils in general; use lead-pencils for white paper, black on white, stone-pencils for slate, white on black; compare crayon and blackboard. Shape.—Long, round, pretty ; easy to hold. Hnds.—How many; one blunt, one sharp; why sharp. How to hold pencils.—Position of fingers; of wrist [writing position]. Use.— Make firm, light lines; heavy lines scratch and are not easily erased. Care of pencils.—Easily broken; carefully handled; kept sharp for neat work and light lines. 6 SUGGESTIONS FOR EXERCISES. Lessons on Position and Direction.—It is an accepted theory that the child must go from the known to the unknown, and that each newly-ac- quired experience be made a stepping-stone to the next. Few children on first entering school, without previous training, would be able to obey the direction, ‘‘ Make a ring in the left upper corner of your slate.”’ On the other hand, few, if any, would be found that did not know which is the right band. With this bit of “terra firma” to stand upon, the teacher, after giving a short and entertaining exercise upon ‘‘right hand and left hand,” begins the lessons on the use of the slate by the following exercise, which has for its purpose the learning of Position.—The children being first directed to place their slates lengthwise on their desks {the word will be readily learned by ‘‘telling and doing ”’], the teacher begins by having them point out the right side of the slate; left side ; the lower side [side lower down on desk]; upper side [side farther wp on the desk]. ‘‘ How many corners on left side? Where are they? Who can point to a left corner? To the left wpper corner? To the left lower corner? Who can tell me the name of this corner [pointing to corner of slate pictured on blackboard]? Who can find the right upper corner? Who ean find the right dower corner? Jennie may point to any one corner of her slate and tell me its name. Johnnie may show and tell another, etc., etc.” After this follows an exercise on Direction.—‘‘ Put your finger on the left side of your slate, in the upper corner. Now slowly move it downward as far as you can. Where does it stop [left lower corner]? Where did you start? Now start at the left lower corner, and move your finger upward. Where does it stop?” Re- peat with the right side. ‘‘ Put your finger in the left upper corner again. Now move it slowly across the top of your slate, toward the right side. Where does it stop [right upper corner]? Where did you start? Now start at the right upper corner, and move it back toward the left.” Ques- tion as before, and then repeat with the lower side of the slate. CLASS EXERCISES, Tue Class Exercises outlined here are given only suggestively, to be amplified or condensed, omitted or repeated, at the discretion of the teacher ; their purpose being, mainly, to show what kind of oral work may advan- tageously supplement the work for the children’s pages. ~ By means of such exercises as the following, each of which has its definite purpose, it will be found that memory is cultivated, imagination stimulated, SUGGESTIONS FOR EXERCISES, — 7 and observation trained; sight, hearing, and touch are exercised, and thus a harmonious development of the child’s powers is attained; the habit of attention, more or less sustained, is formed, and prompt action in obedience to command is acquired, as also a ready expression of thought. 1. The teacher calls upon Johnnie to come and find out what she has in her right hand. Johnnie finds ‘‘a marble.” Jennie finds ‘‘two marbles” in her left hand. ‘‘ Would you rather have Jennie’s marbles or Johnnie’s? Why? Who would rather have mine [showing a handful]? Why?” 2. ‘*How many more marbles has Jennie than Johnnie? What can I do so that Johnnie and Jennie will each have the same number of marbles? 3. ‘Ned, bring me one little girl; now bring me one little girl again. How many times did Ned bring me one girl? [Putting arms around them and bringing them close together.] How many are there? [Sends one to her seat.| How many did I send back? How many are left here? 4. “Hold up as many hands asI do. Hold up twice as many. Who can show me this many [two] pencils?) Show me half as many.” 5. ** Now, let me see all the little heads bowed down upon the desks. Shut your eyes tight. Listen! [Claps twice.} Wakeup! Who can tell me what he heard? How many claps? 6. ‘* Who would like to play blindman? Well, blindman, feel these pebbles and tell me how many there are.” She tests him with numbers, from one to three, and then with a large number, calling out the expression “many pebbles.” 7. “Clap your hands this many times. [Makes two rings.] Clap once for each star I make. [Makes ten stars, and covers them quickly.| Tell me, without seeing, how many stars I made. How many times you clapped.” 8. “Show me as many counters as you have heads; arms; chins; cheeks.” 9. ““ Who knows of something that has one wheel? Two wheels? Three wheels? Two feet? Four feet? More than four feet?” 10. ‘‘ How many eyes has a cat? What has one eye? [Needle.] How many ends has a pin? Namethem. How many wings has a bird? A fly? 11. If Nellie earns one penny making lamp-lighters to-day and one penny to-morrow, how many will she have? If you had two pencils, and lost one yesterday and one to-day, how many would you have left?” 12. ‘* Nellie, find two blue stars [paper]. Jennie, find one red, one blue, and one yellow star. Walter, find three different colored stars.” 13. Who can touch two different things? Three different things? 14. ‘tI hear the clock ticking, a bell ringing, and Will writing on his slate. How many soundsdol hear? Who can tell of two different sounds?” 8 LESSONS ON ONE AND TWO. /, Outline of Lesson on the Words and the Schemas,— What is this, children [e] [pointing to it on the blackboard]?” ‘‘ A dot.” ‘‘ How many dots?” ‘ Onedot.” ‘‘Say again the word which tells how many dots; say it slowly. It sounds like the tone of a great bell. Would you like to see how it looks?” The teacher prints slowly, and in large letters, the word one, and then calls upon individuals to.point to it and pronounce it. Then, ‘“Show me one book; one slate,” etc. ‘Show me [pointing to the word but not pronouncing it] [one] pencil; [one] desk,” ete. In the same way, by showing [eo el, the word “two” is introduced. ‘‘Sounds like the cry of an owl.” The teacher then prints the word, as before, and lets the printed form ‘‘ tell” how many fingers, hands, books, etc., to be shown or pointed to. Changing the questioning from one thing to two things, she tests their conception of the meaning of the two words, Then, showing one and two of various articles. she calls upon first one and then another to point to the word which tells how many things she is show- ing. The class is then permitted to find these words (and show how many) in chart or readers, or any text at hand, and finally to discover both the words and the schemas in their arithmetics. /1, Hints for Language Lesson on“ The Conversation,’’—The general aim of this exercise is to give a liberal training in reading, and expressing the thought contained in pictures, while the special points to be brought out through this medium are a recognition of the number illustrated, first as a whole and then as to its groupings. This is to be attained by means of a familiar conversation in which the children are encouraged to give full and free expression to the ideas they gain from the picture, their attention being directed to special points by the questions of the teacher. Since the principal aim of this lesson is to stimu- late thought and cultivate the imagination, it would not be advisable to throw the child off the track of the thought by insisting upon accurate description, nor by too frequent corrections of language. The teacher would do well to first bring the children into sympathy with the beauty and spirit of the scene in this first illustration, by noting the softened light of early dawn, the fleecy clouds, the rising sun, the long shadows, and the signs of awakening life. ‘‘ How many of you ever saw the sun rise? Who ever saw the moon? The stars? How many stars? How many suns? What are the children doing? Do blackberries grow on trees? What is the man doing? Is this place in the city or country ? What else do you see in the picture?” This last will call out an enumeration LESSONS ON ONE AND TWO (continued). 9 of the objects in the picture, from which, together with the above questions, the teacher may skillfully draw out a more or less connected narrative, which will be “the story the picture tells.’ The special point, number, may be further developed by questions such as the following: ‘* How many hands has the boy? How many has the girl? How many eyes has the boy? Do you see the girl’s eyes? . How many feet has the boy? How do you know? Choose the colors you would have for this little girl’s dress ; her apron, etc. How many hats has the girl? The boy? How many hats in the picture? Count the sheep. Count the birds. How many more doors than windows has the barn?” ete. //1, Slate-work,—In all Recitation Exercises upon the slate, let the con- scious effort and ambition of the children be “to tell a nice long story,” i. e., to give a full and exact description. The degree of accuracy with which the child describes the slate, and the location and arrangement of the objects thereon, will be a test of the exactness of his observation. With this in view, corrections of language can advisedly be made here, and the exercise may thus serve as a training, both in keenness of perception and accuracy of expression. In beginning the lesson on the slate in the book, the teacher should first call the attention of the children to the resemblance of this pictured slate to their own slates; also to the differences. ‘‘ How many corners has your slate? How many has this slate? Are they sharp corners or round corners? How many sides has this slate? Has yours? How many faces? And yours? Point to the left side of this slate; upper side; right side; lower side. Who can tell me what he sees in the middle of this slate?” Require a full and correct statement, first as to ‘‘ what” and ‘ where,” and, after arrange- ment has been discovered, as to ‘‘how” arranged. For example: ‘I see [or there are] two stars in the upper right corner of the slate. The stars are one under the other.” Or “I see two flags, side by side, in the lower left corner of the slate,” etc., etc. Every such statement will, of course, have to be at first built up, point by point, because the child is as yet un- trained in observation, and does not see it all atonce; and, also, because he has not learned how to express what he does see, After all the objects on the slate have been thus ‘‘ located,” comparison as to number is next noted. “How many flags are there? How many more flags than rings? How many stars? Are there more stars than rings?” etc., ete. A Drawing Exercise may follow, in which the children are directed to either copy on their own slates the objects as they are on the pictured slate, 10 LESSONS ON ONE AND TWO (continued). in their books, or on a similar one which the teacher shall have drawn upon the blackboard. The Dictation Exercises should be introduced by a more or less rapid review of the object-lesson upon the slate. The little workers being ready, with slates in proper position, the teacher directs them to make “‘ a row of stars down the left side of the slate; across the upper side; down right side; across lower side.” Then, on the other face, to ‘‘ make a star in the upper left corner; two rings, side by side, in the lower right corner ; asquare in the center of the slate,” etc., till each place is located and filled. /V, What can you te// ?7—In these exercises the children are to be left, as much as possible, first, to give spontaneous and unaided expression to ‘the story the picture tells’; and, second, to observe the detail with special reference to number. The following are some of the points that may be made in these pictures : First picture.—A rabbit in field; standing on two hind-legs; holding up two fore-legs; has two long ears; two eyes; we see only one eye, etc. The lily has one open blossom, two buds; half-way up there is only one stem; above, two stems; two leaves on stem with one blossom on it; one leaf on the stem that has two buds, etc. By counting, we find there are ten leaves in all. Second picture.—A garden; wall, with bicycle against it; and a wheelbarrow; vine on wall, etc. Bicycle has two wheels; the wheel- barrow has one wheel (compare); bicycle has two pedals, two handles, one saddle, etc.; wheelbarrow has two handles, two legs, ete. Third picture.— Looking out of the window; a bird-cage; bird flying away; no bird in cage now. ‘How do you think it happened that the bird got out?” Out- side the window a house can be seen, etc. Fourth picture.-—Boys flying kites in a field or vacant lot. Two boys, each boy one kite; so two kites— twisted together; each kite one tail; two kites, two tails; same with strings. If one boy runs off with his kite, one boy and one kite left, etc. V, The Seript.—As each school has its established system of writing-les- sons, the following brief suggestions are only offered: That the recognition of the written form of the word may be given in the same way as the printed form; that the written and printed forms be compared; that the written word on the blackboard be traced with the pointer, individually, and traced in the air with pencils by the class; and, finally, that it be copied, first as a whole, and then practice given on the accompanying letters. V/, Hints,—The oral work on One and Two will be found on page 7. For ‘‘ busy work ” hints, see lessons on Three. The Conversation, 12 LESSONS ON THREE. /, The Word and Schema.—As in Lessons on One and Two. /!, The Conversation.—After the general conversation the following special points to be made: ‘‘How many little girls are there? Names? Each girl has one apple; doth want the other one. If (May) takes it, how many will she have? How many more than (Nita)? Which will have twice as many as the other? Which half as many? What can be done so that each will have as much as the other? Look at the picture below and tell.” In the second picture the story will be readily grasped by the children. A few questions only will be needed to give direction to the thought, and bring out the facts in the number. //!, Slate-Work,—Recitation: The pupils to give a full and exact de- scription of each group, as ‘‘ what,” ‘‘ where,” ‘how many,” and “how arranged,” as ‘In the middle of the lower side of this slate there are three flags, two of the flags are side by side, and the other flag is below them.” Also compare with slate in Lessons on One and Two. Drawing: The children may be directed to copy exactly what is on the pictured slate, or, to devise original arrangements of these same objects. Dictation: Review as in the first lesson. Then dictate from the pictured slate; second, have the groups placed in the corners; and; third, dictate ones and twos of objects to be drawn, as well-as threes. In each case have the written work described by individuals. /V. What can you te/l/?—The story of the hen and her three ducklings will be easily gathered by the children from the two pictures. The points in number to be noted are: There are three ducklings and two water-rats. If two rats get one duck each, then two will be taken and one left, etc., etc. V. The Script.—To be taught as directed in Lessons on One and Two. V/, Hints for busy work, to be copied from the black- [e e| [ol © © © Ad board by the pupils. [e @ e| Rea A A A Have the children write or tell number stories about e e eee ies e ee 24 objects in view, as, ‘‘There are (2 books) on the (table) and (1 book) on the (chair). Objects to be represented in outline drawings. “Draw a picture of three things, and write or tell a number story about it; mention three red things seen on the way to school (other colors); three things with wheels; three like things; three different things.” Oo”, ? The Conversati What can you te// 14. LESSONS ON FOUR. /, The Word and Schema.—To be given as suggested in preceding lessons. /1, The Conversation.—After the general conversation upon ‘‘ The Nut- ting Party,” the following points in number may be noted: 1 child in a . tree, 3 on the ground—4 in all. 2 children with hats off, 2 with hats on— 4 hats in all. 1 child kneeling, 2 standing, 1 sitting in tree—4 in all, ete. ‘‘What do you think the boy in the center picture is going to do? What hashein hishand?” Point to 2 things alike in this picture. Find 3 different things. How many pints of nuts were there in the basket? If he sells 2 pints at 2 cents each, how much money will he get for them? ///, Slate-Work.—fecitation: As in the preceding lessons. Special questions as to number may be, ‘‘ How many dots in the center? If you erased 1 star, how many would be left? How many times could you erase 2 squares? To how many boys could you give 1 flag each?” ete. Drawing: Besides the exercises in the preceding lessons, the class may draw picture-slates of the size of that in the book, and exercise their ingenu- ity in varying the location and arrangement of the pictured objects. A training-lesson in language may be founded upon this exercise, by having each pupil describe in precise terms one of his picture-slates. Dictation: These lessons would differ from the preceding ones only in the greater variety of exercises possible from the material suggested. lV, What can you te// ?—The exciting incident pictured in this series of three pictures can not but unloose the tongue of even the shyest child, and ‘“‘ what they can tell” will find spontaneous expression, which need only to be directed. Compare 3 cats, 4 mice. ‘‘How many mice get away? Are caught? How many does each cat get? How many fore-feet has a cat? Hind-feet? How many more tails have 4 mice than 3 cats?” ete. In the picture, ‘‘ The Horse at the Blacksmith’s Shop,” the points in question are: ‘‘How many legs has a horse? A boy? How many more has the horse than the boy? How many times 2 shoes does a horse wear? A boy? Which has twice as many feet as the other? Half as many?” ete. V, The Seript-Lessons.—To be given as suggested in lesson on One and Two. V/, Hints for busy work, see ‘‘Three.” Have the class o 2 2 ®Q make a picture-story of a boy who had 4 cents, and bought 3 sticks of candy at a cent apiece, as shown here. “If I | | | give Carl one fourth of these four pencils, how many will he get? How many shall I have left? How can I give these 2 apples to four boys? Would you rather have these (4 pennies) or these (2 two-cent pieces)?” The Conversation, - ———— SES = Se SE 16 LESSONS ON FIVE. /, The Word and Schema.—To be given as suggested in Lessons on Three. /1, The Conversation.—‘ Tell me a story of your own about this tea- party ’’ will call out the individual ideas gained from the picture. ‘ How many children are talking? Listening? At the sides of the table? How many cups and saucers? Blocks? If each child takes a block, how many will be taken? How many left? etc., ete. ‘‘ How many leaves on each stem of the ivy? How many ‘fives’? Count the ivy-leaves by ones. How many leaves on each stem in the rose- vine? How many blossoms? How many petals in each blossom? I see some rose-petals falling. Who can tell a number-story about this?” ///, Slate-Work.—Recitation: After each group of pictured objects has been fully discussed and all the combinations within five found, com- pare with the slate for Four to find resemblances; and with that for Three to find differences. Then compare the groups, as the flags with the flags on all the preceding slates, as to number, location, and arrangement. Drawing: The suggestions given on all the preceding lessons may be put into use here, especially the last one on Four. Dictation: ** Quick-work” exercises in locating and arranging at once and then describing exactly and fully the matter dictated may serve both as a review and as a test of the child’s understanding of the terms he has been accumulating in the preceding lessons. /V. What can you tel/ ?7—“ Study the picture, ‘The Dog-Show,’ and tell me a story about it.” Direct the thought by questioning as to the kinds of dogs, the number, etce.: “If the Newfoundland dog should pick up the poodle and carry it off, how many dogs would be left? Suppose a rat should go scampering past, how many dogs would be left? How many tails do you see? How many don’t you see? Tell a number-story about each dog.” The next illustration was designed for the comparison of “ five” with numbers under five, and also for a lesson in comparing the feet of these animals with the human hand. The toads and toadstools illustrate two twos in five, and one left over, etc. “If that little toad goes off, how many will be left? If the wind blows down 1 toadstool, what then?” etc. V. The Script.—To be given as heretofore directed, and a comparison made between this word and the preceding ones. V/, Hints;—Draw your hand. “If I had 5 cakes, to how many little girls could I give 2 cakes each? John has (showing 5 pennies), and Carl has (showing a nickel), which can buy the most candy ?” 18 LESSONS ON SIX. /, The Word and Schema.—To be given as heretofore suggested. //, The Conversation.—“ Coasting” and ‘‘ The Snow Fort” will require but little, if any, questioning to draw out “the story the picture tells” of these the favorite winter sports of childhood. The topic, ‘ Seasons,” might also be introduced here. Special points: First picture—‘* How many sleds? How many boys on the first sled going down? On the second? Altogether? How many girls going down?” ete., etc. Second picture—‘‘ How many boys in the fort? In front of the fort?” Other things to be examined as to number: snow- balls, trees, children, birds, and branches of trees outside, etc. ///, The Diagrams are, first, to be examined by the pupils to find the combinations in six, and read in class; second, to be reproduced at the desks, either in blocks or in squares of various colored pasteboard, or copied ontheslate; third, to serve as models from which the children are to make other and original designs. The inventive faculty will thus be exercised, and training almost without limit, in form, arrangement, and combinations of the number afforded, as also invaluable lessons in color. lV, What can you te// 7—First picture: ‘‘The mother has four children and six cakes. How can she divide the cakes equally, and how much will each child receive?”? Other points are, the comparison of other numbers under six. Second picture, the story of ‘‘ The donkey that wouldn’t go”; time of day and season. Special points, ‘‘ How many children in the cart? Out of the cart? Altogether? On front seat? On back seat? How many feet has the donkey? The boy? Double as many? Half as many? Both together?” ete. Third picture, ‘‘The Rainy Day.” Make up a little story of your own about this picture. Special points, ‘‘How many little girls are there under this umbrella? How do you know? How many pairs of arms? Eyes? Heads?” ete. V, The Sor/pt.—Lessons to be given on the plan of the others. V/, Hints,— Draw as many rubber boots as three boys would wear. As many mittens. How many more legs has a spider than an elephant? Than a fish?” ete. A mother said, “I will give you one half of these six cakes if you tell me how many that will be?” ‘Draw six cents on your slate. Draw as many pencils as you can buy at two cents each.” Have the children make six-inch rules—of paper, pasteboard, or wood—for them- selves, and encourage them to measure their desks, slates, books, etc. ‘sation ? The Conver Bee ~ ® ~ >] S | o | $5.0) ae | ww F- % = 20 LESSONS ON SEVEN; /, The Word and Schema.—As previously suggested. //, The Conversation.—Upon wild rabbits, pet rabbits, pets in general. Characteristics of rabbits. ‘‘Cousin Hare”; fable of ‘‘Hare and Tor- toise,”’ story of ‘‘Brer Rabbit and the Tar-Baby,” from “‘ Uncle Remus,” adapted by the teacher, are suggested as topics for the general language- lesson. Special points in number upon all the combinations in seven may be made in the groupings of the rabbits. ///, The Diagrams may be copied in lines on the slate or represented in objects by match-sticks, splints, etc., and, besides the simple arrangements showing the combinations of number, may represent real objects, such as houses, fences, chairs, tables, flower-pot with plant, and an endless number of like things, and also fancy geometrical designs and figures. lV, What can you te//?—The merry circle around the Christmas-tree will arouse the pleasantest recollections, and may be turned to great ad- vantage in inducing the children to relate their own experiences. Hereto- fore the work in language has been either to describe or narrate. After the general conversation, each child may be called upon to relate an experience ~ which shall have in its plot some combination of seven. The second pict- ure was specially designed to give occasion for a language-object upon “the table,” manners, setting a table, etc., while affording excellent material for the study of seven. ‘Tell all you can about the family who are going to have tea. How many grown people? Children? What ages? Where going to sit?’ etc. Special points also on the number of things on the table. The third picture shows an incident familiar and easily described. All the combinations of seven may be noted in the balloons. V, The Script.—As in previous lessons. V/, Hints.— Put seven blocks on your desk. Find another number in these blocks—another.” (Six, five, four, three, two two’s, and two three’s will be found.) Have a pupil tell a number-story, and the class picture it. As “Seven birds were on a fence, and three flew away.” Write ‘“‘ Four and three are ——,”’ and have the children copy, fill blanks, and complete. ‘‘ Write a number-story of your own about the Christmas- tree.” ‘‘How can I divide seven oranges equally between two children? Equally among three? Who can picture it? How many 2-cent stamps can I buy for seven cents?’ The children should have breakable objects, and be directed to find one half and one third, also to put together and compare with an undivided whole. a, Se as ®D “~~ S 1) se & D © ~ D = 22, LESSONS ON EIGHT. /, The Word and Schema.—As heretofore, the word as the spoken or written, and the schema as the concrete expression of the number. /1, The Conversation.—“ The Apple-Gathering ” affords an effective lesson on country life and occupations. The topic of seasons may be re- sumed, the teacher taking the children back in memory to the spring-time when these trees were white with fragrant blossoms that spring rains and summer suns have ripened to the round red fruit. Now, in the autumn, the father and his children gather the apples, and early in the morning he goes off to the city and sells them. Here, also, we see the fruit- store of the man who bought the apples. ‘What is he doing? What measure has he in his hand?” Special points, the grouping of the barrels, the persons, baskets, trees, horses, ete., and the articles in the store. ///, The Diagrams are to be used in the same way as those for six. They may be represented in objects by pennies, button-molds, etc., and by variously colored pasteboard disks. The forms may each be read in several different ways, as for instance: No. 4 may be read as either 3 and 2 and 3, or as 1 and 8 and 38 and 1. ‘The class may also be asked to ‘find other num- bers” in the eight. One may find a three, pointing to the upper row in No.1. The question, ‘‘ What eise is there besides the three?” will call out the fact that 3+?=8. In No. 2, one may find “a four,” and immediately discovers that there are ‘‘two fours” in the eight spots, ete. /V, What can you te// ?—The children may “imagine” or ‘ dream” stories about the people to whom this room belongs, two number combina- tions of eight being required as the basis of each story. Special points, the panes of glass, legs of tables, chair, etc. In the second picture there will be no lack of “stories,” which may be given direction by requiring, as before, number to be the basis of them. The third picture is presented for the comparison of the even numbers under eight with eight. ‘‘ How many more blossoms would you have to add to the first plant to have eight? Which plant has twice as many flowers as another? Three times as many? How many times could you pick two of the lilies?” etc. The comparisons are numerous; only a few types have been given. V. The Script.—To be given as previously suggested. V/, Hints.—Name eight different things that can run, hop—that have eyes, ears, hair, fur, wings. Eight kinds of fruit, vegetables, nuts, games. Before the eyes of the children, the teacher folds a paper in four folds, and then cuts out two paper dolls, and asks the children to guess how many. es SS The Conversation, 24: LESSONS ON NINE. /, The Word and Schema,.—To be given as heretofore suggested. /1, The Conversation,—This beautiful sea-shore scene will afford so many and varied object language-lessons that only a few hints on each of the different lines of thought suggested by it can be given: Land and water —the sea-shore—rocky coast—sandy beach—calm and storm—light-houses —pleasures of the sea—toils of the sea (fishing, whaling, transportation)— ships and boats—the different kinds of motive power—the vessels in the picture—what the sea yields for our use—things under the water—at the bottom of the sea. These last two are illustrated by the frame of the pict- ure, and give opportunity for opening a new world to many of the children. Interesting facts about the wonders and beauties of submarine life may be gathered from almost any natural history. The special points in num- ber may be brought out by questioning on the ship-sails; the legs, ete., of the crab; the points of the star-fish, the sea-anemone, etc., ete. //!, The Diagrams,—Too much stress can not be laid upon the value and variety of the exercises which may be founded upon these diagrams. The designs which may be formed, either with long counters or by lines on the slate, are almost without limit, and, if the pupils are required to pay attention to the combinations of number in making their designs, these ex- ercises can not fail to familiarize them with the number in its every aspect- lV, What can you te//?—After the story of Little Bo-peep has beer rehearsed, require each pupil to tell in what groups he thinks the sheep ran off: one will say, ‘‘ First three, and then three more, and then three more,” and so on, with all the combinations in the number. Again, looking at the illustrations from different points of view, many of the combinations can be seen, as, one and eight, or one and three and five, etc. The story and illustration of ‘‘ Little Boy Blue” may be treated in a similar way. V, The Script.—This lesson introduces a new feature, that of the slant- ing lines, which serve for practice in slant and distance. V/, Hints,—Draw a number-story picture of nine boys and six sleds. Draw another of three rabbits and nine carrots. Draw nine oblique lines (in groups of three, or five and four, etc.). Draw nine horizontal lines; nine vertical lines. Have exercises in making up nine with one, two, three, and five cent pieces. Conduct sales of slate, pencils, sponges, paper-dolls, tops, etc., and have the children picture the operations (as in ‘ Hints,” Lessons on Four). Have the children each bring a flower, and find and write number-stories about the parts of the flower. . The Conversation, a ane 4 26 LESSONS ON TEN. !, The Word and Schema.—As heretofore. /1, The Conversation.—The game ‘ Keeping Store,”’ besides being of un- failing interest to the children, gives special advantages for the study of the combinations within ten, as in buying and selling, exchanging one number of things (pennies) for another, and ‘‘ making change,” etc., both customer and merchant must be ready with his calculations. A study of the articles for sale will also yield most of the combinations. Each pupil may be allowed to make an imaginary sale or purchase of the things pictured here, and tell of the transaction. //1, The Diagrams in this lesson are like those in six, and may be used in the same ways. With space less limited far more beautiful designs may be made. It is specially desirable that the pupil should not, in these exercises, be permitted to overlook the element of number, -but should note the combinations he forms. In this, and in the other exercises founded upon the diagrams, may be produced what Froebel calls ‘forms of life— such as actually exist, and come under our observation as works of art and industry ; forms of knowledge—such as relate to number, order, proportion, etc.; and forms of beauty—representing ideal forms, models of symmetry and order.” /V. What can you te//?—If the children have been duly encouraged in their previous work to give a numerical turn to the stories they tell, the teacher will find no difficulty in gathering from the various versions which they will give of ‘The old woman who had so many children,” etc., and ‘St. Nicholas,” the many combinations within ten—combinations which will cover the ground of the four rules of addition, multiplication, subtrac- tion, and division, without, however, having these recognized by the chil- dren as distinct operations. V, The Script consists of a lesson on “ ten,” with a review of ‘‘eight” and “nine.’’ A review should also be made of all the previous script-lessons. V/, Hints.—Make a paper disk and perform operations in fractions be- fore the class, and have them tell what you have done. Then let each child make at home a “ paper cake,” and then have them divide their “ cakes” in halves, fourths, eighths. This will afford many interesting exercises. At another time the cakes may be divided into thirds and sixths. By compari- son, and by putting parts together, without formulating what they do, they will be unconsciously learning to perform with parts all the operations that they do with wholes. ot fo ce Fog Ae 1 Ws N. What can you te//? a NG. 4 M1 " ee th Bnd ue tn NE; he. At 2 Ww bSualths ’. 28 NOTES ON LESSONS IN PARTS II, Ill, AND IV. Part //,—* Lessons I to X. The main purpose of these lessons is to introduce the child to the printed forms of arithmetical expressions. They do not embrace all, nor the majority, of the combinations within ten, but may be taken as types for blackboard exercises, in which are introduced the other combinations. They constitute a systematic series of reading- lessons, and may besides be copied on the slate, the children filling the blanks with pictures of the objects, and the names of the numbers, thus serving as writing and drawing lessons also. * Lessons VI, VII, and VIIJ.—‘“ Making up and writing original prob- lems.” These illustrations are given as themes for oral, blackboard, and slate exercises, in making original number-stories like those in the text. * Lessons XI and XII. Ten is here dealt with as a unit, that the child may get, at the outset, a correct view of our decimal notation. * Lesson XIII. The first of a séries of test-lessons, and which are, there- fore, purposely miscellaneous in character. * Lesson XV. The purpose of this lesson is to picture out the real mean- ing of “eleven, twelve,” etc., and make the child realize it when counting. * Lessons XX to XXIII, and also Lesson XXXV. A summary and re- view of the numbers within ten, analyzed according to the Grube method. * Lessons XXXII and XXXIII may be either dictated by the teacher, or used as a silent desk exercise. Part ///,—* Lessons IV, VIII, XV, XIX, XXII, and XXV, constitute the groundwork of all operations within the hundred; each one introduces a new step, and should be thoroughly practiced with objects. *Lessons XXIV to XXXII embrace a complete review of numbers from ten to twenty, which should now be thoroughly practiced. Part [V.—* Lesson IV. In all cases of successive additions, subtrac- tions, etc., state results, not the operations that produce them. *Lessons V, VII, 1X, XIJ, XVI, XXII, XXIX, and XXX, each intro- duces a new step, and it is especially recommended that objects be used for illustration until the pupils have thoroughly mastered the subject. *Lesson XV. In this, as in all the lessons on fractions, pupils should be required to draw squares, and actually perform the operations with them before expressing in figures. . * Lessons XX and XXIV. The large numbers given here are for prac- - tice exercises only—concrete applications in them would be inadvisable. * Lesson XXV. It is recommended that the teacher herself illustrate each step of this lesson with objects. PAR TELE: READING LESSONS IN NUMBER, AND DICTATION EXERCISES. Combinations from One to Ten, LESSON I. ADDITION, Phrase and words to be /earned by sight. | How many | and = are One —— and one 2. How many, are aye) & Two ——— and one . How many , are One —— and ‘one ———. and one 4. , How many, are eB ON ee BN ONE cee cS 0 eee 380 LESSON II. Phrases and words to be learned by sight, EL See | a in on There are | Two. and two 2g Pp ae SE ZF ope many in all)? Two —— = 4nd one Send two 2 Sere LESSON III. 31 Words to be /earned by sight and sound. by tin’der ind arg How many In all? the 4. How many ,in all? Three ied PAHs reS: a GPAh As seat 3. ,L see, eure’ in the <2 TEAS the By and OL, under the J How many in all e Two and two. end two are —+ => — 32 LESSON IV. SUBTRACTION, Phrases to be learned by sight and words by sound. are left i 7 léss will be left , thése take away | fly away , hop away , 1. If I take away, two of these SQ igs _ how many , will be left ? [eS ae es ee 2. If two of these My f; a will be left? are fly away, how many Four less two are = — of these. ea. How many ,are left, ? Segoe 4. If two of these hop away, and two | fly away, how many , will be left? is SS ae eT eee Five __— less two -_— less two LESSON. V. 33 Phrases to be learned by sight and words by sound, , were there | left ran ran away | lambs on’ly ran away, and_ these 4 are left. How many , were there ts gall in all ? —__— lambs less three lambs are two lambs. ix There were, Sess Now there are | only two. How many ,ran away |? Six. /ess ares tWost- Mies. If three fly away ,, es A Wii how many will be left |? Six less three ——— are : 4. If I take away, two of these and you take away, two, how many : will be left : ? ore 2227/009 Lwon%e22less tWoeteere =n 34 LESSON VI. Phrases to be learned by sight and words by sound, Here are, roll away | frogs of them, | swim away , bees out of runs away | goat 1, | Here are, If one goat Tuns away ), oer many , will be left, ? Six goats less one goat are —— goats. If two of these run away, and you take away one, how many will be left, ? Six ——— less two ——— /ess one gre Lad be If four of them ‘There are, <4 hop away ,, how many | will be left ? Six frogs /ess ee frogs are 2 frozs. Here are, g If two ,run away , and three , swim away ,, how many will be left ? Seven less twoe.2-Yess thréess 229 ere ae, LESSON VI (continued). 35 G25. If two bees See fly away ,, how many will be left |? Or . Eight bees are on this @ Eight bees less two bees are —— bees. . If you take ALLEL out of, this how many will be left? Pr) EI Rat oer Oe8 Ele (eae OSG ae, val —<— eee, )SWIM away |, how many will be left? ~I are ——— ——, Nine ——— less six If five of them roll away |, how many will be left ? iepes = (ess five #3 are __—, * Making up and writing out original problems, TTT T heat m Wit Bs ' ; y cog yh Hh i he} | || Fe ; J 4 é y ne heeds 7 ek z 7 f a se hag f | he a : ¥ é 7 : Fg VRE R an cats Pardini epitty ; Haiysatieele 24) YU - iy . 7 " om > 4 i \) Heerrinesary: 35 LESSON VII. MULTIPLICATION AND DIVISION. Phrases to be learned by sight and words by sound, How many times , légs this Are any left, fone DOX@ win ke can you find | tops kites 1. How many legs have these ..44—f}—_: | Two times two legs are —— legs. : a, ts i He f 2. | How many times ye mi have these bad tame How many in all? Three times !¥v ie OY Aoi tii a Meat 3. , How many times, can you ,take away, two tops from these PM? How many = A At will be left? From five tops you can take, 2) op el be Er 4. In these OeKrgrtmmre times, Megas [A two ie "ca you at [n_ six. kites there are ——— times LESSON VII (continued). 37 5. From these ¢ bow many times, can you take two pinks ! ? Are any left? From seven pinks \ you can take, 7 ist WANES, and —— pink will be left. 6. How many times four legs have }y How many legs in all? Two times —— legs are —— legs. 7. How many times [LES in this How many in all? There are times Bis inthe’ Dox, TINGR LG tee Oe ae ae, Making up and writing out original problems. 38 LESSON’ VITE Words to be learned by sight and by sound, her apples from bids toes _ fingers éges nuts 1. How many times can youtake TELE out of : this box ee Are any left? = ( - =e From six —— you can take Ga Ay A times. a Nis 2. |How many times | five fingers on these a at How many fingers in all? [here sare == Vingers@ iN alle Ge) free a ers there are —— times five fingers. 3. How many toes has a cat on her ¢@¥ | Two times five toes are ——— toes. 4. How many toes has a cat on her Two. times —— toes are ——— toes. From eight buds you can take Ce all _ me | LESSON VIII (continued). 39 How many times, three nuts are there on - How many in all? There are —_— times three nuts. Three times 2s are ——— «nuts. How many times | can you take four apples = 4O *LESSON IX. Phrases to be learned. by sight and words by sound. one half, one fourth | mine one third, | one fifth, shélls as many | she gave | eatch 1. Mark has ae®, but I have only , one half, as many, How many have I? One half of two — Is Vi. thoi! +n ¥Z y | OO pa but Carl has only ; one third | as many, How many has Carl? Onesthirdrofe tire? === 5a Liylhy, Wey — 4 PP ai 3. What is one fourth , of IS One fourth of four apples 1s —— . 4. If you catch one fifth of these Jy many | will you have ;? One fifth of five —— is 5. What is | one half, half, of eo.cy 9 Hip D One half of six is —— — reat te ee vey a ee a —- LESSON 1X (continued). 41 6. May had ae e@ey She gave, one third, of them to Ned. a oh did he get)? One third of six —— are —— —— 7. If I give one half, of these Reo you, how many , will you ee e)? One half of four —— SIE ee 8. If I take away one half, of these Ee how many , will be left, ? One half of eight are so 9. One fourth , of these Geax SS How many shells have |? One fourth ill eight shells are —— shells, 10. Ann had ¢@ Nt ig ui She gave, one third of paid Tian me. How many ,did I get,? One third-of, nine: 2+ —. ace. = ex 11. If you take , one half, of these < - “ie how many eggs , will you have ;? . One half of ten eggs are —— eggs. ee ~ X x 42 LESSON X. Phrases to be /earned by sight and words by sound. more than, have l, log pond 1. Carl has and I have 2. How many more has Carl than |? lhree —— are. —— more thanz one == 2. Nellie has 4@ many more have | than Nellie? our eae hg —— ,;more than, one ——. 3. May has ¥) and Ned has Ge. How many more has May than Ned? Five are 232 = 7 more: Wan oe 4. Here are GED “2 In the pond there are five frogs. How many more frogs are there on the log, than in the pond ? Six frogs LE Sa more than five frogs. 5. 1 have mye How many Ari Hb! have [, Te May? Eight ——*are —— ,;more*than, two ——. *LESSON XI. 43 COUNTING BY TENS. One ten Two tens Three tens Four tens a & \@ \@ e |e ® a |\@ ten. twenty. thirty. Five tens Six tens seventy. Ten tens eighty. aero one hundred. * LESSONCAIE DICTATION EXERCISES, WORKING WITH TENS. Phrases to be learned by sight and words by sound. | Give the name, like these , add make /have you made, ;enough more, e-rase’ made . Make two tens like these and name them. . Make four tens. Give the name : . Make three tens. Name them. Add enough more tens , to make, five tens. . Make six tens. Name them. Erase three tens. ‘Name the tens , you have left. . Make three tens and four more tens. How many tens in all? Give the name. . Make eight tens. Name them. Erase half of the eight tens. How many are left? . Make five tens. Namethem. Add enough more tens to have nine tens. . Make nine tens. Name them. How many times can you take, three tens from ninety? LESSON XII (continued). AS 9. Make five tens. Name them. Add five more tens. How many tens in all have you? How many times, have you made five tens? : Give the name : of two times five tens. 10. Make three tens. Name them. Add three more tens. How many tens have you now? Name? Add three more tens. How many times ; have you made, three tens (thirty)? How many tens have you in all? Name? | 11. Make two tens. Name them. Add two more tens (twenty). How many tens have you? Name? Add two more tens. How many tens have you now? Name? Add two more tens. How many now? Name? Add two more tens. How many times ,have you made, two tens? How many tens in all? Name? A6 *LESSON XIII. MISCELLANEOUS EXAMPLES. Phrases to be learned by sight and words by sound, Picture it has he left , shall — séll with counters, on your slate | slate dots 1. How many are WA and Ave and we ? 2. Ned had GQ : Fae and EHP m&. Four of them ran away. How many ,has he left |? ; rs Axo, 3. I have two times #4 and ade more. How many, have I in all? | Picture it, with counters, or with dots ,on your slate . €@)K& and two times Ber How many in all? | Picture it. 2 . How many times >< in this Oy Haye ab Rs 4. Ihave three times _ 8. How many are three times three kites? LESSON XIV. A'7 Phrases to be learned by sight and words by sound, the more, to which | much cénts the more money , more apples bo . Carl has _ Nellie found @ How many more legs has a ee. than an and May has G, : 223. Which has the more money ,? Aa a eee ee ee . Nita has (8). she than Carl? How much more than May? How many more legs has a “than a SS ? 4 % Tey a Cot 16+1= 144+2= 1+4= 16-3'= l1l+6= 25 4 Copy, complete, and afterward prove with counters. Try to do them in five minutes. 1+-4= 4-5 = ede d+9= 11+4= 14+9= 1h Ge ee i ioe ts) = Ap age bis o+4= 7+1= oS 14+1= 1l+4= IB fos pet ce: 1+3= * ORIGINAL PROBLEMS, | Write or tell number stories about anything you like, using: 3+3 12+5 13 +6 ie ayellak yy ei) LO 12+5 14+3 beyond RAPID ORAL CALCULATIONS, Read off quickly, giving the answers at once, aL 2 ela 13+3 1444+4 15+5 1Atte Td 8+16 44+11 5+12 174+3 1941 10+4 5+13 16+4 18+2 1347 5+14 15+4 8+11 134+6 1742 LESSON VII. 83 LEARNING THE SIGN —. Words to be /earned by sound, more than léss ex-préss’ dash’es ew) ~] . How many are 14 less 3? 16-—4=? 17-5= . How many more are 13 than 10? 15—-3= e@ 98 e@ ..°¢ . How many are less three dots ? Eight less three are five. . 8—3=5 is a shorter way of writing this. . What word is the sign — used for? . May has this many pencils, |[|]], and Carl has this many, |]. How many more pen- cils has May than Carl? Five 1s three more than two. . 5—2=8 is the way we express this in figures and signs. . Write these sentences with figures and signs in place of the words and dashes: Seven less five are ——. Ten is —— more than six. Sixteen less six are ——. Nine is —— more than seven. Twelve less two are ——. Seventeen is —— more than seven. Eighteen less nine are ——. Nine- teen is —— more than one. 9 Pha Toso? 190 =P 16-3 =? 84 *LESSON VIII. SUBTRACTIONS BETWEEN 0 AND 20. Phrase to be /earned by sight and words by sound, ,all the subtractions , un-tying _fig’ures ie 2. with-out’. eount’ers Put one bundle of ten and ten single counters on your desk. How many in all are there? From your 20 counters take 8 counters, and tell by figures and signs how many are left. Thus: 20 c.-- 8. = 12ie. . In the same way find and tell how many are 20 c. less 6c.; 20c. less 3c¢.; 20c. less 5c. ; 20 c. less 9¢.; 20c¢. less 7c.; 20c. less 2c. ; 20 c. less 4¢.; 20c. less Le. . Have 18 counters on your desk. Now find and tell how many are 18 counters less 8 counters; 18 less 6; 18 less 4; 18 less 2; 18 less 7; 18 less 5; 18 less 3. . Have 17 counters on your desk. Find and tell all the subtractions you can make without untying the ten bundle. . Have only 16 counters on your desk. In the same way as before, find and tell all the subtractions you can make. Do the same thing with 15 counters; with 14; with 13; with 12; with 11. LESSON IX. 85 CONCRETE APPLICATIONS, Phrases to be /earned by sight and words by sound, , this morning | ate broke sehool , this afternoon , éat brok’en _ stable 1. May had two dimes. She spent one cent for a paper doll. How much money had she left ? 2. Carl, too, had two dimes, and he spent 5 cents fora top. How much had he left? 3. After Carl had bought his top he lost 4 of his cents, and then he put what he had left into his bank. How much did he put into his bank ? 4. May gave the rest of her money, all but 6 cents, to Will and Nita. How much did she give away? 5. Nell had a dozen and a half of eggs in a basket. She let it fall and broke 7 of the egos. How many were not broken? 6. 17 goats less 6 goats are how many ? 7. On his way to school Tom found 14 nuts. He ate all but three. How many did he eat? 8. This morning there were 19 cows in the stable; now there are only 15. How many have been taken out? 86 oO 10. 11. 14, 15. 16. LESSON IX (continued). . This afternoon Jack had 16 bags of pop-corn to sell. He sold only five. How many has he left? Jennie’s little white hen sat upon 15 eggs. All hatched out but 2. How. many little chickens has she ? | This afternoon Will and Walter found 19 pond-lilies. Will found 8 of them. How many did Walter find ? . There are 18 trees in a field. 5 are apple- trees and the rest are plum-trees. How many plum-trees are there? . John had 17 marbles and lost 4. How many has he left? Nat is 13 years old and Will is 11. How much older is Nat than Will? May has 16 buttons and Nita has 13. How many more has May than Nita? Picture it, thus: May’s buttons SPETEETTEE oe. ee0 Nita’s buttons [ee ee0e-ceeeeolece /6 is 8 more than (8. Carl had 19 chestnuts. He gave his squirrel 7 of them. How many has he left? Pict- ure it, thus: [ee eee-coccelececccces 19 less 7 are (2, LESSON X. 87 OBJECT AND SLATE WORK. Work out with your counters, copy and complete, 20 -—-7= 19-—9= L3 — lis l1l-l= Lin3 = 14—2= 12-2= 18—-4= Picture these with dots (as in example /5, page 86) and copy, giv- ; ing the answers, How many more are io than*s ? 15° than’3 ? 18 than 6? 14 than 1? 16 than 2? 17 than 3? * Copy, complete, and prove with counters, i—?=6 20-—?=7 14-3=? 15—?=12 17—6=? ?-17=2 ?—-T=11 13-3=? RAPID ORAL CALCULATIONS. Read and give the answers at once. 10 less4=? 9less3=? 6 less 4=? 13 less 2=? ZO AK? 19 Hs B=? 16)-— 457: 18':- 8=? eet a 1 20) = 8S Pld = oS? What is the difference between 19 and 13? 15 and 12? 14 and 20? 14 and 17? 2 and 10? 16 and 1? How many more are Piethan elo? 16ithan 313.2, 19;than 12? 12 than 10? 18 than 5? .20 than 17? 88 LESSON XI. MAKING UP TENS. . Group your counters like these marks: |]]][]], Hi]|. How many counters in the first eroup? How many in the second group ? . Make a ten of the first group by adding counters from the second group; thus: WME ow many counters did you put to the seven to make ten? Then how many of the 9 are left? . Show BOY figures what you have done; thus: (+342=12. Then i-9 = . Place your counters in groups of 6 and 9; thus: |FULL, TIPLEE, |. Make a ten of the six with counters taken from the nine; thus: TIETE TTD TE . How many counters did you add to the 6 to make a 10? How many of the 9 are left? Then 6+44+5=? 64+9=? . Place your counters in groups as these figures tell you, then add, making a ten of each first group. Write every example both ways; thus: 7+34+3=18. TOS 133 Tet O44S-+D! 9b 4S 9) 54 Oa Ge ae CTT UA Be 6 20 8 Be 9 SEO BESET ay LESSON XII. 89 CONCRETE APPLICATIONS. Words to be learned by sound. both pair oar’den write used a-20' shoes rose’bush wrote doz’en 1. In my garden there are 6 roses on one rose- bush and 5 roses on another. How many roses are there on both bushes? 2. Carl wrote 8 words on his slate this morning and 4 this afternoon. How many in all did he write? 3. If there are 7 buttons on one of little Nell’s shoes, how many are on the pair? Nine roses and 5 roses are how many? In one box there are 9 pencils and in the other box there are only 2. How many pencils in both boxes? 6. Jane used a half a dozen eggs this morning and a half a dozen this afternoon. How many in all has she used to-day ? 7. There are 8 bees on the bee-hive and 7 bees on the stand. How many altogether ? 8. Four years ago Ned was 9 years old. How old is he now? 9. If | buy 7 marbles and you buy 5, how many shall we both have? ie LESSON XIII. 90 Words to be learned by sound, spider fiéld an-oth’er al-to-géth’er 1. Will gave me 8 daisies this morning and 5 10. 1a: this afternoon. How many daisies has he given me to-day? . How many are 9 cents and 6 cents? . Carl has drawn a line 7 inches long. How many inches must he add to make it 11 inches long ? . Carl has drawn another line 9 inches long. How many inches must he add to make it a foot long? . How many legs has a fly? How many legs has a spider? How many legs have a spider and a fly together ? . How many are 7 frogs and 6 frogs? . There are 9 lambs in one field and 8 lambs in another. How many in both? . If you have three cents and I give you 8 more, how many will you have? . Nine blocks and how many more are 16 blocks ? . Hight dots and how many more are 16 dots. If there are 9 buttons on one of your shoes, how many buttons are there on the pair? LESSON XIV. 91 OBJECT AND SLATE WORK. Work out with counters, copy and complete. 7+5=? 8+5=? 9+9=? ?+6=11 9+ ?=11 r+7=14 8+ ?=14 ?+5= 14 8+8=? 9+7=? 8+ ?=15 9+ ?=16 Copy, complete, and prove with counters. 6+6=? °+7=15 7+?=18 7+9=11 9+4+8=? 8+4=? 4+9=? 6+ ?=18 ?+6=15 ?+6=15 8+3=? 6+5=? ORIGINAL PROBLEMS. Make as many problems as you can of these, 249=16 ?+?=18 2+?=13 40 =17 ?+?=15 P+?=12 o4+9=14 2+?=19 2+? =20 RAPID ORAL CALCULATIONS, Read and give the answers at once. 10+10= 84+5= 4+9= 8+ 8 = 6+9= 7+9= T+7 = 7+6= 8+6= I+ 9 = 8+7= 9+95= 92 1. Place 13 counters on your desk, the LESSON XV. BREAKING UP TENS. 10 tied in a bundle and the 3 single {J | sticks beside it. Take 9 counters from these. Can you take 9 counters from the 3 counters? How many of the 9 can you take? What must Bek cee do to get the rest of the 9? Mien Ime 3—6=—7 [ose — . Place your counters as these figures tell you (the ten tied always in a_ bundle), From each number take 6. Write your work both ways; thus: 15-5—-1=9. 15 — 6 Dagel2ad Its: . Place your counters as these figures tell you, and from each number take 7. Write your work both ways. 16, 13, 12, 15, 11, Zs . From each of these numbers take 9. Write your work both ways. 17, 138, 16, 11, 16, 12, 15, 18, 14. . From each of these numbers take 8. Write your work both ways. 11, 12, 18, 14, 15, Noy y IEG. . From each of these numbers take 5. Write your work both ways. 15, 14, 13, 12, 11. LESSON XVI. 93 CONCRETE APPLICATIONS. Words to be learned by sound, léarn béans péas’ length dif’fer-ence Hh 2. May had 13 buttons on a string. She lost off 7. How many has she left ? Fred is 16 years old? How old was he 8 years ago? . My slate is 11 inches long and May’s is only 8 inches long. What is the difference in length between them ? . Carl drew 17 birds on his slate, but erased 9 of them. How many has he left? Picture it, thus: Peeeeesedasossees {7 lessG4 aré 22e., . Nita has a dozen blocks and Will has half a dozen. How many more blocks has Nita than Will? Picture it, thus: Nita’s blocks [ee eeeeeee cleo Will’s blocks eccooeocee [2 are =2.a\more than 6: . Fred planted 11 beans in his garden, and 9 peas. How many more beans than peas did he plant? Picture it. . There are 15 birds in a tree by my window. If 8 fly away, how many will be left ? 94 LESSON XVI (continued). . In the pencil-box there are 14 sharp pencils and 8 dull ones. How many more sharp pencils than dull ones? . I see 11 frogs on a log. If 6 hop away, how many will be left ? . Carl had 13 words to learn. He has learned 8, how many has he yet to learn? . How many more are 15 rabbits than 9 rab- bits ? . Tam 12 years old and Carl is 7. How much older am I than Carl? . May made 16 paper-dolls and gave 9 of them to Nita. How many did she keep? . Carl had 14 doves. 7 of them flew away. How many has he left ? . Four months of this year are past. How many are still to come? . There were 11 ducks in the pond. 7 have come out. How many are still in the pond ? . Fred made 14 kites. He sold 9 of them. How many has he left ? . If Jane buys a dozen oranges and gives away 8, how many will she have left? . John had a dozen and a half pop-corn-balls. He sold 9. How many has he yet to sell? . How many are 13 apples less 9 apples ? LESSON XVII. 95 OBJECT AND SLATE WORK, Work out with counters, copy and complete, 1l—6=? ?—6=6 I17-9=? ?—-8=8 13-—?=6 16-9=? ?-9=5 12-9=? Picture with dots, as in Lesson XVI, How many more are 11 than 7? 16 than 7? 15, than 6? 17 than 8? Teese 12 than 5? Copy, complete, and afterward prove with counters, 15— ?=6 12—-—9=? °—9=9 11-—8=? ?—d9=18 dE Deere Breese a 16—/%=—8 fo (at RAPID ORAL CALCULATIONS, Read and give the answers at once. How many are 10 less 9 ? LOai6 2 lou ier iL sehes Crees SS VEPs Ding Leo? 13s 97 What is the difference between 9 and 13? 3 and 12? 14 and 6? 16 and 17? 17 and 8? 18 and 9? How many more are | 15 than 8? 18 than 9? 16 than 9? 17 than 9? 13 than 6? 17 than 8? 96 LESSON XVIII. MULTIPLICATION. LEARNING THE SIGN X, Words to be learned by sight and sound, times place an-oth’er showing 1. May wrote two words this morning and two words this afternoon. How many times did she write two words? How many in all? Two times two are four. 2, 2x 2=4 1s the shorter way of writing this. . What word is this sign X used for? 4. How many mittens are there in a pair? How many in four pairs? Four twos are eight. . Four 2’s = 8 is the shorter way of writing this. 6. Write these sentences with figures and signs in place of the words and dashes: Co Or Three times three are ——. Six twos are ——. Four times five are ——. Three fours are ——. Two times four are ——. Five threes are ——. Nine times two are ——. Two eights are —_. Six times three are ——. Four fives are ——. 7. How many are 2X5? 6X2=? 10X1=? 2X8=? 9X2=? 6X3=? 8. How many are two fives? Three fours =? Two 8's =? Nine l’s =? Seven 2’s=? dees al, L2. ~) LESSON XIX. 97 EXERCISES IN MULTIPLYING. Group 20 counters in 2’s, thus: |] |] qT] UT] UI HUE UT Up dy.) Plow many 2’s are there? Find with your counters and tell by figures and signs how many are ten 2’s. Three 2’3. Five 2’s. Seven 2’s. Nine 2’s. Two 2’s. Four 2’s. Six 2’s. Hight 2’s. Group your 20 counters by 4’s, thus: |] ]] {II LEE VUTL UIE. How many 4’s are there? How many areo x4? 2x4? 4x4? 3X4? Group your counters in 5’s, thus: ||]]] Tf] PUTT UEP]. How many 5’s are there ? How many are four 5’s? Two's? Three 5’s? Group your 20 counters in 3’s, thus: |]] [jf] HE LVL VTE ULE df.) low many 3’s are there and what else ? Put aside the two counters left over. How many counters have you now? 6X3=? How many are 4X3? 3X3? 9X3? 2X8? Group your 20 counters by 6’s. How many of them do you need to make even 6's? Put the others aside. How many 6’s are there? How many are two 6’s? Three 6’s? Group your counters by 9’s. How many of them do you need to make even 9’s? How many are two 9's? 0 98 LESSON XX. CONCRETE APPLICATIONS. Words to be /earned by sound, prompt éarn old least réal-ly léarn old’er most bréak’fast vérse dld’est month 1. Every day for one week May was given 2 cents for being prompt at breakfast. How many times two cents had she at the end of the week. How many cents in all? . Carl was given 3 cents for every basketful of weeds he picked out of the garden. How much did he get for 5 basketfuls ? . When eges are 3 cents each how much will a half dozen cost ? May was to get 4 cents for every verse she learned. She learned one verse every week for a month. How much did she earn? Carl, who is older than May, was to get 2 cents for every verse he learned. He learned 2 verses every week for a month. How much money did Carl earn? Nell, who is the oldest, was to get only 1 cent for each verse she learned. She learned 3 verses each week for a month. How much did Nell earn ? 10. a), 12. 15. 14. 15. 16. LESSON XX (continued). 99 Which of these children earned the least money? Why? Which one learned the most verses ? . Jack made 2 large kites. Fred gave him 2 new lead-pencils for each kite. How many pencils did Jack get ? . Fred had just paid 5 cents each for the lead- pencils. How much money did the kites really cost him ? Nellie bought 3 apples at 2 cents each, and 2 lemons at 4 cents each. How much did she pay for the apples? How much for the lemons? How much for both lemons and apples ? Nita wrote 2 rows of words on her slate, and there were 5 words in each row. How many words in all did she write ? May wrote 2 rows of words of 10 words each. How many words did May write ? In my spool-box there are 3 rows of spools of 3 spools in each row. How many in all? Annie has 2 such boxes of spools. How many spools has Annie ? How many legs have half-a-dozen sparrows ? My desk is 2 times 8 inches wide. How wide is it? 100 LESSON XXI. OBJECT AND SLATE WORK. Work out with counters, copy and complete, 2 threes = ? y Aemaesitee ie ean | by 3 fours=? ? twos = 14 ? threes = 15 ? sixes = 18 4. PWas 20 5 twos =? 8 ? =16 Copy, complete, and afterward prove with counters. 2xX2= 2xX9= 2x 10= 3X3= aX3= 5X 4= TX2= 4x4= 4x 2= aX6= 2xX8= 2X 6 = ORIGINAL PROBLEMS. Make as many problems as you can from these. ?x?=12 Perea oa LS rx?=16 RAPID ORAL CALCULATIONS, Read and give the answers at once. 3 twos are 7 twos are 4 fours are 5 fours are 3 fives are 5 twos are 2 fives are 6 twos are 2 eights are Dex 4 10.42) = 5X Qier bo of the rest to Jane. How many did she give to Jane? . Will wrote 6 words on the blackboard, which was !/3 of the lesson. How many words in all the lesson ? 10. 14, 15. 21. 22. LESSON XXVII (continued). 109 IT had 20%. I spent !/ of it for car-fare, and gave '/; of the rest to May. What did she get? . [had 16 dimes. I gave !/s of them to May. How many dimes had I left ? . I spent 1/2 of the dimes I had left for a sled. How much did my sled cost ? Carl has $18. His father gave him !/s of it, and his mother gave him !/9 of it. How many dollars did each give him? (The sion 5 stands for dollars.) His grandmother gave him !/, of it, and Carl earned the rest. How much did he earn? There are 8 quarts in 1 peck of peas. How many quarts in a half-peck ? How many quarts in a peck and a half? . Carl sold 6 quarts of nuts last fall. What part of a peck was that ? There are 12 inches in a foot. My reader is '/, of a foot long. How many inches long? My copy-book is */, of a foot wide. How many inches wide ? Carl’s tool-box is 11/, of a foot long. How many inches long is it ? What part of a foot are 9 inches? 6 inches? My room is 5 yards long. How many feet? EO LESSON XXVIII. OBJECT AND SLATE WORK, Work out with counters, copy and complete. 1/4, of 20 =? Ot leas V4 of 12 =? erOt 2 8G Ue oft) = 3H ? of 16=4 '/g of a= 2 1/9 of 18 =? M/s of 2? =3 eotalye= 2 Lobe 1/5 of 16 =? ALIS) 1/o Of 2 YO 1/4 184 /g Of 2 O24 6s Aye One Copy, complete, and prove with counters, 1/5 of 11 =. 1/7 of Ideal 4/5 of LOS 1 /sc0t 13 We of 12 = 41/g of LT = et/eiof 1d maths oF 19 ORIGINAL PROBLEMS. Make as many different problems as you can of these, aay e. Wh eset Cit eye > 1-Olsba Ov We Sy VO e—er Mn) bite = RAPID ORAL CALCULATIONS, Read and answer at once. Vsof20= 'Yso0f12= Yeof 9= 3/5 of 15= Wg of 19= Ysofll= ',0f 1I8= 1/4 of l2= oust Y/gio£,.: Bis) vote ids Ota aust. or Dist/,of 418 1/3 of 9is 1/5 of 10 is 1/, of LESSON XXIX. pis bp | REVIEW. Analysis and synthesis of each number from /0 to 20, Bt meee Steed W-38=? | 84+27=11 ?+5=11 64+5=? ?—2=9 11-4=? How many more are ll than8? Than2? 5? 7? How many less than ll are 4? Are 3? 9? 6? How many must you add to 7 to make 11? To 6? To4? To 2? Tie your counters together by 2’s, and see if you can make 11 with 2’s. Try the same thing with 3’s; with 4’s; with 5’s. Take 11 counters and see if you can divide them into groups of 2’s; of 3’s; of 7’s; of 8's. eos). U2? lle rs otf, Lr 2b 12 2+10=? 12-4=? 34+?=12 74+5=? Peer = 12-2 —? ey12+ 6 =? 12+ 4% Bee teat 2S S19) 4 xs? 12 12-89 WP H4 WfokF 8 oP 66 6 2x GR? Ones? GHt = SP0f12S6 “tyor lar What is the difference between 12 and 8? Be- tween 12 and 7? 6and12? 9 and 12? What must you add to 10 to make 12? To 8? Tote 16.8? i sep LESSON XXX. 13 18-37 OP PES Sot UO caeae Org Ooh =o 13 4 ee e— 6. a ee Poh) Lo 2 3 a ar eee Moe lol 42-7213) 2a OS oe acne How many less than 13 are ll? Are 12? 9? 6? How many more are 13 than4? Than 8? 12? 7? Tie your counters together by 2’s. Try to make 13 with 2’s. Try the same thing with 4’s; with 5’s; 6’s; 8's; T’s. 14 11+ ? =14 !/,of ? =2 210 — 5 a dae ee OD As RM erat Yenaets nat pa VA meteor ean te es Wa 11 10 4 A eee ]4-- 2? =7 2Aet = 149) LIS a 14d ie Tsien Party O14 14— ? = 100 a 0 ee PR Set ey (ait ee © LO A ee 15 The S15 215-6 =? 4 SP te Meorlos=? «4-2 =156 3x? =15' 1946 =f I-Fl4—? Jot? i=3 ?-I18=2 “o9X32 Li 11. /5 of lO 2, A of LO Se oak se Mit Galo ?oSele los 8 ep Terie 1S S224 3 SHA One Sl db tee I3+.%2=15 15-? =13. 8+7 +? Sas unde bara tle shane ap ore S= lime iarellontGar barthieibariin tear is = 1 fer ttt = su) 8 Yor ne oe Peo Ie 9+18= 14 2 4+19= Gt Lis “Utd oe Me OSD eo Se eS oe ene O+17= Begin with 1 and add 4’s to 41. Begin with 3 and add 4’s to 43. Begin with 2 and add 5’s to 52. Begin with 2 and add 7’s to 72. LESSON V. 125 SUBTRACTION, IN NUMBERS BELOW ONE HUNDRED, . Place 9 counters on your desk; take away 4 of them. What have you left? 9 less 4 are 5, . Work out these examples with your counters, copy them in the form of a table, and com- plete: 19-4; 29-4; 39—4; 49—4; 59 oe ee Oeming: 1. Oh 489) 2 A. GO aA . In the same way, work out and write a table of subtractions of 8 — 6, etc., to 98 — 6. . In the same way, work out and write a table of subtractions of 7 — 3, etc., to 97 — 3. . In the same way, work out and write a table Or subtractions’ of 6°-'D, etc!, to 96— 5: also, one of 5—3, etc., to 95 —3; also, of 4—2 etc. to 94— 2:"of3'— 3, ete, to 93 —s: . How many more are 9 counters than 4 count- ers? Place 9 counters on your desk ; under these place 4 counters. . Now, to find the difference between these two numbers, we will take as many {|| || counters from the greater num- ber (9) as there are counters in the smaller number (4), and what is left will be the difference between them. Then 9 are 5 more than 4. 9—4=85, 126 LESSON V_ (continued). 8. Notice that we express these two questions, “9 less 4 are how many?” and ‘9 is how many more than 4?” by figures and signs, in exactly the same form, thus: “9—4=?” 9. In the same way find how many more are 9,than 6) jev thangdi? GithannZan) saganel 19 than!6. -27-than.5-- t6 than -2Pisithanal. 39 than 6 d7than5 S56than2 23 than 1 69 than6 77 than5 86than2 73 than 1 CONCRETE APPLICATIONS. 1. How much older is Anna, who is 19 years old, than Frank, who is only 6 years old? 2. John had 48 marbles, but he has lost half a dozen. How many has he left? 3. There were 97 trees in the park. - The wind blew down 6 of them. How many are left ? 4. I have 32 buttons on a string, but Anna has 38. How many less have I than Anna? 9. Our hall is 40 inches wide. Will yard-wide carpet cover it from side to side? How much wider is the hall than the carpet ? 6. Nellie found 4 dozen eggs in the barn last week, and only 45 this week. How many less this week than last ? LESSON VI. 127 OBJECT AND SLATE WORK, Work out with counters, copy, and complete these examples, writing the sign — in place of the word “ less,” How many are lv less4 16less3 i18less5 19 less 7 27 less4 26less3 28less5 £99 le&S 7 D7 less4 S8361less3 ‘58 less 5 . 29 less 7 Wles4 14less3 15les4 #4217 less 6 Soles 4 7 24 "ess 3°") 25 lesa 4") VT Tess 6 Sless4 34]less3 35 ]less4 77 less 6 Copy and complete, writing the sign — in place of the words “than” and “and,” How many more are 13than3 18than6 59than5 26 than 4 33 than3 28 than6 69 than5 46 than 4 73 than3 98than6 79thand5 86 than 4 What is the difference between 9and8 45 and2 38and7 £469 and3 39 and8 85and2 18and7 79 and3 5Dand8 Q95and2 58and7 # £89 and 3 RAPID ORAL CALCULATIONS. oT hss 89-3 = 1 26 -— 5 = 9 -— 8 = A514) 6 —4— 0 63-328. -98=7= a2 —-D =i (Sr ie 9b-1= 67-6= 128 LESSON VII. 1. 23 counters less 7 counters are how many ? Arranging the counters thus, ff] we see at once that we can not [ff a take 7 ones from 3 ones, so we untie one tl of the tens and arrange | | | the counters thus, having 1 ten and 13 single counters. Now, taking the 7 ones from the 13 ones, we find that we have six ones and one ten (16) left. The work we have done with our counters is expressed by figures and signs thus: 23 —7 = 16. 2. In the same way, find and express how many Ale. 49 i 50 Ol ae One) Ome a cee ie IN He Reoee eS LHR 6 teeta Mp ser. 3. Make with counters and write a table of sub- tractions, by taking 7 counters from each of these numbers: 15, 25, 35, 45, 55, 68, FAD eeCo) eo 4. In the same way, work out and write tables of subtractions for 17 — 9, etc, to°9T — oe 13-5), 6teeo lai Oy ete epee wee Ghee Tl 3, ete. 18h 9) GlCreLo. vee cen 12 = 4, ete. ; 19 Ll eter eects 136, etc.; “14 '3) etc, slo ad eo reme 1673, etc.; “15 9) "etc. ieee re LESSON VII (continued). 129 ). How many more are 23 than 7? Place 23 6. In Peake counters on your desk; place 7 counters under them. To find the difference between these two numbers, we must take as many counters from the larger number (23) as there are counters in the smaller (7). Can you take 7 ones from 3 ones? What, then, must you first do? Work out the answer to this question with counters, and express it in figures and signs, thus: 25 — 7 = 16. the same way, find and express the differ- ence between 24 and 9; 27 and 8; 51 and 6; 43 and 9; 52 and 5; 66 and 9; 81 and 8. CONCRETE APPLICATIONS, seven years John will be 34 years old. How old is he now? 2. Carl’s kite-tail was 27 feet long, but 8 feet of it were torn off. How much is left ? . Nellie had 42 buttons on her button-string ; now she has 34. How many has she lost? A table is 3 feet wide, and is 7 inches longer than it is wide. How many inches long is it? . Walter is 8 and his brother is 7 years old. What is the difference in their ages? 130 LESSON VII (continued). 10. . Carl had 55 cents; he spent 9 cents for slate- pencils and candy. How much had he left? . This afternoon he has spent 8 cents more, for writing-paper. How much remains now? . Harry and Ned have, together, 43 marbles; 9 of them are Ned’s. How many are Harry’s? . Tom had 385 pop-corn balls to sell on the trains. On the 4 o'clock train he sold 8. How many had he left? On the 4.15 train he sold 9 more. How many had he then left? On the 4.30 train Tom - | sold all the rest but 9. How many did he sell ? . Will’s top-cord was | yard, 1 foot long. He broke off9 inches. How long was it then? . Tom had a quarter of a dollar, and spent 6 cents. How much was left ? . Nell had a basket of 2 dozen eggs; she let the basket tip over, and 8 eggs fell out and were broken. How many were left? . If you sleep 7 hours, how many hours of the day are you awake? . 9 of the 52 weeks of this year are past; how many are still to come? . In my garden to-day there were 33 carna- tions; I picked 7. How many did I leave? butt (Dn) al al LESSON VIII. 131 OBJECT AND SLATE WORK, Work out with counters, copy, and complete. S07 te 6 Rb 8H lao Sur MeO Minh Ose ~~ Sr675 = 0S OsThine + om 2S 6S = D Speas or 4570-657. 85977 b= Copy, complete, and prove with counters. 30 —-8= 24 —-8= 33 — T= 45 -6= 72-9= 83 —9= 23 —4= 62—6= gl —9= 92 —-3= 91 —-—2= Day Dae Do Oi ar 60 10s 6570. nn6 ib = iia Ore © Ol 76s 7Ocpbnnb bia = Way; O26 aA 2517 8K ee 1-9= RAPID ORAL WORK, Read, stating remainders only, thus: “80, 27, 24," ete, YB PSE og hates Tae Be Soe im ts One 8 ends Manet We Bees ee eg a(S} 0) LC) SLES a RAE OM Rea Some RU Rae Ge P BoPe STP VAP Bie eras reales Meare 0 Woe Begin with 40, and take as many 4’s as you can. Begin with 38, and take 3’s; 5’s; 7’s; st Begin with 54, and take 8’s; 9’s; 2’s; 7’s; 6's. Begin with 67, and take 6’s; 8’s; 4’s; Os. rats 2 132 *LESSON IX. MULTIPLICATION. Making and Learning the Tables. ssi . Group your counters by threes until you have 10 threes. How many counters in all is this? Copy and complete each of these examples as you find the answer, and write in the form of a table. How many are three 3’s? Five 3’s? Seven 3’s? Nine 3's? Ten 3’s? Hight 3’s? Six 3’s? Four 3s? Two 3's? . Arrange your counters in 10 groups of 4 each. Find answers, copy, complete, and write as a table, how many are two 4's? Four 4’s? Six 4’s? Hight 4’s? Ten 4’s? Nine 4’s? Seven 4's? Five 4’s? Three 4’s? . Arrange your counters in 10 groups of 5 each. Find answers, copy, and complete. How many are "3 *D2 oO KD? 7 Xora DEX DP MASK 9 1) MG) Pie eee Oe 4, Arrange your counters in 10 groups of 6 each. How many are one 6? Two 6’s? Three 6’s? Four 6's? Five 6’s? Six 6’s? Seven 6's? Hight 6’s? Nine 6’s? Ten 6’s? . Arrange your counters in 10 groups of 7 each. How many are7X7? 9X7? 3X7? 6X7? DIANE 2 PAIX T 2) BOX TOE D OT I TOUR ee bo ey Or LESSON IX (continued). 133 6. Arrange your counters in 10 groups of 8 each. How many are two 8's? Four 8's? Hight 8’s? Three 8’s? Six 8’s? Nine 8's? Five 8’s? Ten 8’s? Seven 8's? 7. Arrange your counters in 10 groups of 9 each. How many are 10X9? 9x9? 8x9? 1X9? 6X92 OK 924%97 3X92.2XK 9? 8. Arrange your counters in 10 groups of 10 each. How many are2X10? 3x10? 4X10? 9X10? 6X10? 7x10? 8x10? 9x10? 10 x 10? DICTATION EXERCISE, e@e$88 86 1. How many dots in this square of °¢e¢¢e 5 dots each way? 5X5=? HESS Beek x 2. Make a square of dots, four dots **¢ee¢e 9 each way. How many dots inall? 4x4= 3. Make a square of 7 dots each way. 7 X 7 =? 4. Make a square of 8 dots each way. 8X 8=? 5. Make a square of 6 dots each way. 6x 6=? 6. Make a square of 9 dots each way. 9X 9=? 7. Make a square of 3 dots each way. 3X3=? 8. Make a square of 2 dots each way. 2x 2=? 9. Make asquare of 10 dots each way. 10 x10=? 10. Complete, and learn this table of squares. 134 ee yn pee ed LESSON vx: CONCRETE APPLICATIONS. . John earns $3 a week. How much does he earn in 2 months? ($ stands for dollars.) How many shoes will it take to shoe 7 horses? How many days are there in 8 weeks ? How many school-days in 9 weeks ? . Ned has 6 quarts of strawberries. How many pint-baskets can he fill with them? (There are 2 pints in 1 quart.) . How many quart-baskets could you fill from 1 peck of plums? (There are 8 quarts ina peck.) How many from 2 pecks? 3 pecks? . How many sides have six i Picture this and the next 7 examples. . How many sides have 9 triangles ?. . How many faces have 7 cubes? . How many sides have 8 pentagons ? . Four pentagons have as many sides as how many squares ? . Three pentagons have as many sides as how many triangles ? Three squares have as many sides as how many triangles ? . Make 12 pentagons, 20 triangles, and 15 squares, and count the sides in each set of figures. a". ap 1X2= 6xX%2= 1X3= 6X38= 1x4= Pee A OR EXIS — * Bde SBA2Z= 8X2= 38X38=°8x3= 38x4= 4X2= 9xXx2= 4x3= 9x38= 4x4= LESSON XI. OBJECT AND SLATE WORK. Copy, complete, and /earn, 135 5 X4'= yee. 8x4= 9x4= OX2=10K2= 5X3=10X3= 5x4=10x4= Work out with counters, copy, complete, and /earn., 1X5= 6X5= 1X6= 6X6= 1x7T= 2X5= (X5= 2X6= TX6= 2XT= peo SOND os6=> 8X6= 3XT= aoe ey AK oO 9X OS AKT SH 6xX7T= TX7T= Sxi= IX7T= en tee tae Os LOG oe = AO Xe 1X8= 6x8= 1x9= 6X9= 1X10= 6X10= ZxO= (X8= 2x9= TX9= 2X10= TX10= BRK ioe io eeo BUS 69 SLO BxahoS 4X8= 9xX8= 4x9= 9xXx9= 4x10= 9X10= 5X8—10X8=.5x*9=10:X%9=.-59%10=10X10= six 6’s nine 9’s eight 8’s RAPID ORAL CALCULATIONS. 9X7T= 7X8= 8xX4= 6X9= >X8= OX T= eight 6’s nine 8's six 7’s | | 136 *LESSON XII. DIVISION—CASE /, 1. Arrange 24 counters on your desk; separate them into groups of six. How many 6's are there? 24+6=? 2. In the same way, work out with your count- ers, copy, and complete these examples: Did — "00 4" 20rd OU Danae ee 96+7= 16+4= 25-5= 3676= 49-7= 64°38 — Sl=9=' 48-0 =) 04-9 — 606 3. Also work out with your counters, copy, and complete these : 24 A= BD> Da gde Say 0 = 1. pieces 28-7= 4075= 5426= 21+7= 32-8= 63=-T= 49+9= 30+6= 24+8= 28+4= CASE //, 4, Separate 32 counters into 8 equal groups, and find and tell how many counters in one of these groups. 1/s of 32 =? 0. Work out with counters in the same way, and copy and complete these examples: 1/5 Cie 1/6 of 138 = =p of 24 = 1 of 27 = V/gof 28 = Myof386= '/,of21= 1/,0f 42= Vgof56= Yrof68= 1/,of54= 1/)0f 81= V,of16= YWoofT2= Ygof35= ',of 72= bo On LESSON XIII. 137 DICTATION EXERCI/SES, . Make 36 dots on your slate, so that there will be the same number of dots each way. The square of what number is 36? Arrange 64 dots in a square. The square of what number is 64? . Arrange 25 dots in a square. 25 is the square of what number ? . Arrange 49 dots ina square. 49 is the square of what number ? . Arrange 81 dots in a square. 81 is the square of what number ? . Find, by trying, which of these numbers can be arranged in squares and which can not: 39, 40, 16, 22, 12, 9, 24, 48, 49, 88, 72, 64. CONCRETE APPLICATIONS. Our house is 27 feet wide. How many yards wide is that? (3 feet in 1 yard.) How many hours do I sleep if I sleep 1/4 of the day? (A day is 24 hours.) I have studied !/, of an hour. How many minutes is that? (An hour is 60 minutes.) Carl has 36 marbles, 9 in each of his pockets. How many pockets has he ? 1388 D. 6. 7. eh 2 10. iN 12. 13. 14. 15. 16. LESSON XIII (continued). Nita has set up 45 blocks in 5 rows. How many in each row? If you make 72 dots in 8 rows, how many dots will there be in each row ? How many rows would there be if you make them rows of 12 each? How many rows if in rows of 24 each? Carl ate 8 plums, which were !/; of all he had. How many had he at first. Nita is 6 years old, which is !/; of her moth- er’s age. How old is her mother ? How many quart bottles can be filled from 24 pints of milk? (2 pints in 1 quart.) How many gallon jugs would be filled from 32 quarts? (4 quarts in | gallon.) John made 40 cents to-day selling pop-corn balls at 2 for 5 cents. How many did he sell ? Thomas earns !/g as much money a month as his father does, whose wages are $48 a month. How much does Tom earn? Aunt Sarah says that when I have earned 40 cents she will give me !/, as much. How much shall I then have? I have !/, as many marbles as Tom, who has 42. How many have I? 139 LESSON XIV. OBJECT AND SLATE WORK. Write the complete answers. How many times 3 in 4 in 5 in 6 in 7 in 27 36 35 36 49 30 24. 30 42 42 21 32 45 54. 56 How many 8’s in 9’s in 10’s in 24: F2 yA ida 43 AO7E5'30 o2t 00 a LL Bers. 40 60 rh 64- H4e— 281 80 3690 40 48 a6+0 48 10 +100 Work out with counters, copy and complete. 39-7 = 36+ 7= 28=-6= Vsof28= 1/, of 44= 4] +5 = 38 +4 = 2/3 of 33 = = 8/g Of 832 = RAPID ORAL CALCULATIONS. 36 +9 = 32+ 8= 63-9 = 54 +6 = 49=+-7T= (2a ot Pao Perorie */,orae =m ty oh Gee '/pof 56 1, of 2B= 1, 0f382= %,o0f64= 5/, of 40= Ys of 35= %so0f45= Yyof54= 3/5 of 42= 140 LESSON XV. if 10. FRACTIONS. Draw a square.inch. Draw a line down the middle. Into how many equal parts does this divide it? Express this in figures. . Divide each half into two equal parts. How many parts in all, now? 1/2, 0f !/,=? . Now, divide your square as this one is divided, by drawing a line through the middle of each fourth. Into how many parts does this divide it? Then 1/5 of 1/4 =? . How many 8ths in 1/4? 3/4 = how many 8ths? 1/, = how many 8ths? . How many times ?/s are there in the square ? Then 8/g “3 2/s 40 ve of 8/g =? . How many times */s are there? 1/, of 8/g =? . How many more are ®/g than 3/g? %/g+?= e/a: ALT mee 3/9 7/3 =? 2/3 +? = 7). Sig Vg =? ORES Acerca hy . How many 8ths are 8/, less 2/g? 8/g—®/g=? 8/,— 7/3 =? . How many 8ths are 2 times 2/g? 3X 2/g=? 2X 3/g =? How many times ?/g in 8g? 4/,=%/g=? Vp of 8/g =? a of 8/g =? 1/5 of 6/g =? .- _ "LESSON AVI. 141 ADDING BY TENS. . Do you think it is any easier to add together 3 single counters and 4 single counters, than it is to add 4 bundles of ten and 3 bundles of ten? ‘Try it and see. . How many tens are 4 tens and 5 tens? How many single counters in 4 tens? How many single counters in 5 tens? Then 40 and 50 are how many ? . How many are 6 tens and 2 tens? 60+20=? LO 20 =?) 40-7 30:12 10H 00S? . How many are 3 tens and 4 tens and 2 tens? 30+ 40+ 20=? 20+10+30=? . How many are 30+ 20+8? 50+40+9=? 380+15790=? 401+15740=? . How many are 10+ 20+30+ 20+10+7? . Place 32 counters on your desk. Under them place 26 counters, so that the 2 tens will be under the 3 tens, and the 6 singles will be under the 2 singles, thus : Now add these two numbers i 4 A || by putting the units with the i units, and the tens with the H) al ] tens. What is the new number pee you have made? This is called The sum of the two numbers. 142 LESSON XVI (continued). 8. Express in figures what you had done with your counters, by writing the numbers ;. », to be added one under the other, and 32 then draw a line under them, and 26 write the sum of the numbers below 58 the line, thus: 9. Work out with your counters, and express in. figures, as above, these additions: 42 and 25; 55 and 14; 62 and 27; 81 and 15; 75 and 22; 33 and 44; 51 and 36; 23 and 42 and 14; 21 and 34 and 23 and 15. 10. Place on your desk 47 counters, and under them 38 counters, and under these 14 counters. When you add the 8 units (single sticks) to the 7 and the 4 units, what do you find? What will you do with the ;,, new ten thus made? (Add it in with 47 the 1 and 3 and 4 tens.) What you 38 have done with your counters is here 14 expressed in figures. 99 11. Add 1 more counter to your 9 single counters. You can now make another ten. How many tens in all have you? 12. There is another name for ten tens; do you know it? Write Ten tens make one hundred. LESSON XVII. 143 ADDITION, NOTATION, AND NUMERATION, IN HUNDREDS, 1. How many. are 78 and 47? Let us picture this example, with dots for the ales units and rings for the tens, thus: |o_o 0}. 2. We add first the 8 units and the 7 [° ° °°" units, and find that we have 15 | © ° units, or 1 ten and 5 units. Un- der the line, in the units’ place, we will put the 5 units (dots), and add the | ten (ring) to the tens (rings). This 1 ten added to the 4 and ,,. t. u. the 7 tens makes 12 tens, or O90 4 1 hundred, and 2 tens. aS 3. We put the 2 tens (rings) in the 0 0 hens} plaice, and*express the Leys °° be hundred by a large ring, which “at we put in the hundreds’ place. 4. The work, when completed, will look like this, and it may be expressed in figuresthus: ,,,, Do. In this number we can tell which rings 78 O stand for hundreds and 47 O which for tens, first by their 555 O Doh: size, and second by their place, for tens are always in the second place to the left, and hundreds are always in the third place to the left. 144 LESSON XVII (continued). 6. After this we will use only the second of these two ways of showing which are units, which are tens, and which are hundreds. That is, instead of using dots and different sized rings, we will use dots only, and let their places show whether the dots stand for units, or tens, or hundreds. Bhai 7. So, instead of writing it thus, with |Ol? ie dots and rings, we will express |. this number by dots only, thus, | | | and it may be expressed in figures thus, 235. 8. Copy and complete these examples in addi- tion, and then express them in figures: e@e | eee 9. Express these numbers in words: 125, (28,809 (367,.590, 982, 260 7541 315 619 701 400 600 780 708 611 10. Express in figures: one hundred, twenty-one; six hundred, forty-two; eight hundred, eighteen; seven hundred, seventy; five hun- dred, five; four hundred; three hundred, thirty-three ; nine hundred, nineteen. LESSON XVIII. 145 ADDITION, NOTATION, AND NUMERATION, IN THOUSANDS, 1. You have learned that ten units make one ten, and that ten tens make one hundred. Now, what do you think ten hundreds make? Ten hundreds make one thousand. 2. Add 5 hundreds, 7 tens, 6 units, and 7 hundreds, 4 tens, 8 units. Express these numbers by dots, thus. Now, adding the 6 and the 8 units, you find that you have 14 units, that is, 1 ten and 4 units; you put the units under the line in the units’ place, and, adding this 1 ten to the 4 tens and 7 tens, you obtain 12 tens, which are equal to 1 hundred, and 2 tens. Putting the tens in the tens’ place, and adding this 1 hundred to the 7 hun- dreds and 5 hundreds, you find that you have 13 hundreds, that is, 1 thousand and 3 hundreds. Now, put the 3 hundreds in the hundreds’ place and the 1 thousand in the fourth place to the left, the thousands’ place, and the answer—that is, the sum of these two numbers—will be, O76 1 thousand, 3 hundreds, 2 tens, and 148 4 units, expressed in figures thus: 1,324 146 LESSON XVIII (continued). 3. Copy and complete these examples in addi- tion, and then express in figures: ee |eees)/ eee ee | eeeleeeor 4. Express these numbers in figures: one thou- sand, six hundred, fifty-four; five thousand, three hundred, ten; four thousand, twenty- seven; eight thousand, eight hundred, eighty-six ; two thousand, two. Oo. Express these numbers in words: L276 4.1; 5,720, \uq,6)8026514 %000n: bis, 001 10,60 4440 1,001 2,220 1,100 3,030 5,005 1,010 8020 9,001 6. Ten thousands make “a ten” of thousands, just as ten ones make “a ten.” This num- ber, 10,856, is read, ten thousand, eight hundred, fifty-six. 7. Read these numbers: 10,000; 20,000; 30,000; 15,000; 17,000; 12,856; 14,805; 10,010; 13,927; 15,005; 16,600. LESSON XIX. 147 CONCRETE APPLICATIONS. . There are in my garden 35 roses, 27 carna- tions, and 42 sweet-peas. How many in all? . May has 125 buttons on her button-string, and Nita 108. How many have they both? . Twelve dozen of any thing are called a gross. 12x 12=? How many buttons in a gross and a half'a gross ? my reader there are, on the first page, 123 words, the same number on the next page, and also on the next. How many on these 3 pages? There are two ways of doing this example. See if you can find out for yourself what these ways are. . In another book there are on one page 237 words, on the next, 209, on the next, 223, on the next, 207, and on the next,. 252 words. How many words on these 5 pages? Can you do this example by both addition and multiplication? Why not ? . Last summer Mr. Jones raised 380 bushels of wheat, 245 bushels of oats, and 897 bush- els of corn. How much grain in all? . In 1 mile there are 5,280 feet, and in a !/, of a mile 1,320 feet. How many feet in a mile and a quarter ? a melt 148 *LESSON XX. OBJECT AND SLATE WORK. Work out with counters, and copy and complete. 45 and 36 = 32, 26, 18, and 30 = 27 and 39 = 16, 26, 36, and 6 = 48 and 56 = 12,42, 8 and 36= Write in columns and add. | 19, 9, 13, and 24 13, 15, 16, 17, 18, and 19 16, 28, 32,and4 12,19, 27, 18, 20, 12, and 8 Copy and add. 14h 19) cule 26 16 Av 6 one Sees 15 Ris 62 JG 8: VES 7am 195 9 OW Bil 136s 19% 4200 9 32-32 12 29 6 18 16 327 693 427 582 1,069 3,686 502 58 470 399 1,672 4,864 1,883 12,861 25,005 120,120 5,972 8542 19,699 22,903 3,645 7,207 4,805 37,456 808 20,101 15,062 18.379 RAPID ORAL CALCULATIONS. 30 60 20 40 20 47 29 40 30 50 Li 30 33 29 20) 9 18 23 38 15 25 LESSON XXI. 149 . Draw a square and divide it into 6ths. Then divide each 6th in half by , drawing a line down the middle of it. Into how many parts does this divide the square? Express this in fig- a ures. Then !/, of 1/6 =? . How many 12ths in half the square? Then peat /iat Titer yh rete = i ays 9 it’s = "ap: . How many more are !2/19 than 6/19? "/ig +? = ayAL GH bea Atay Rel at ba mil gaa tcf 2 CP . How many are !2/19 less 5/2? ~ Less 8/12? . How many are 3/1, and 5/19? = "/ig $2? = %/ta. 4/19 Bila T/h9. 9/19 ttm 4/19. . How many 12ths are 3 times 4/12? 2 X 6/19 =? 4X 3/19 =? OXF /19 GR 3 X 2/p= ? . In /;2 how many times 2/12? 4/19? %/19? 3/19? 1 /o of Bhe=? 1/s of hig =? 1/6 of 2he=? A Ad of Lah =? 1/3 of S/ig=? — 1/o of S/n =? . In how many different ways can you express this much of the square-—°/19 ? . Answer the same question about 8/12; about 12 fis; fins 1 /iehoesies) aes 7/1! /aey 4/19. 150 * LESSON’ CEA. SUBTRACTION AND COMPARISON. 1. Is it any easier to take 4 single counters from 9 single counters than it is to take 4 tens from 9 tens? Try both, and see. 2. 9 tens less 4 tens =? Then 90 less 40 =? 3. 8 tens less 5 tens are? 80—90=? 60—20=? 4. 57 less 34 are how many ? aaa | Place 57 counters on | ! your desk; from this group of 57 take out 34, What is left ? 5). To express this in figures, you write first the number of counters you have (57); 4.x. then under this write the number (384) 57 you wish to take away; then, drawing 34 a line, you write under it the figures 93 which tell how many of the 57 would remain. This is called Zhe Remainder. 6. 32 less 15 are how many? Arrang- ing your counters thus, we see i i al at once that we can not take 5 ones from i) HII] the 2 ones, so, untying i | one of the tens, we ar- . range the counters thus, having 2 tens and 12 ones. Then, taking from these the 1 ten and 5 ones (15), we find that we have H ten and 7 ones (17) left. 10. ie 12. . In the same way, find and express in 72 LESSON XXII (continued). 151 . The work we have done with counters 99 may be expressed in figures thus: ~ 5 15 17 figures and signs how many are 41 less 14; 37 less 19; 53 less 25. . How many more are 637 than 425? Let us use dots to find the difference between these two numbers. leet. Taking as many units, tens, and hundreds from the greater num- |). ber as there are units, tens, and hundreds in the less, we find that we have left 2 units, 1 ten, and 2 hun- dreds, or 212, which is the difference 495 between them. ae 212 This may be expressed in figures thus: Copy, complete, and then express in figures : 13. Picture with dots, copy, and complete : 1234 3582 5,796 8092 5,608 123 401 4563 6,000 3,006 152 LESSON XXIII. CONCRETE APPLICATIONS. 1. If there are 675 daisies in a field, and Jane picks 325, how many will be left ? 2. Mr. Hudson raised 857 bushels of corn on his farm last summer, and has sold 569 bushels. How many bushels has he left ? 3. He also raised 388 bushels of wheat. How much more wheat than corn did he raise ? 4. In the flower-show there were 1,587 yellow flowers; 407 were roses. How many of other kinds of flowers ? 9. There were 500 white flowers. How many more yellow than white flowers ? 6. There were 859 red flowers. How many more yellow than red flowers? How many more red than white flowers ? 7. I have walked 3,160 feet. How many more feet must I walk to make a mile? How many feet in a mile. 8. Will has walked 4,085 feet. How much farther has he gone than I? How many more feet must he walk to make a mile? . 9. Frank’s father made $9,875 last year, and spent $6,750. How much did he save ? 10. He made $1,790 less this year than last. What did he make this year ? *LESSON XXIV. OBJECT AND SLATE WORK, 153 Work out with counters, and copy and complete these Subtractions. 47 D4 62 48 36 33 37 20 29 16 _Comparisons. 40 65 79 86 49 25 40) 37 68 37 Picture with dots, and copy and complete, 146 269 30D 627 34 2 «136 = 1038-81982 Copy and complete, 427 862 948 1,327 116 502 908 427 12,678 15,654. 28,028 2.564 5,432 8,020 89,956 65,902 99,090 9,056 60,092 9.090 RAPID ORAL CALCULATIONS, Read thus: “twenty from forty, twenty,” ete, 40 60 08 86 330 2 40 «50 40120 2,849 1,627 39,267 7,267 127,342 79,563 DD6 246 154 *LESSON XXV. MULTIPLICATION—A SHORT METHOD OF ADDITION. 1. Suppose we are asked to find out how many trees there would be in an orchard of 6 rows of trees with 28 trees in each row? One way to find out is by Addition; that is, to write down the number of trees in each row (28) as many times as there 28 are rows of trees (6). We would 28 then add, first, the units, thus: “8,16, 28 24, 32, 40, 48 units—4 tens and 8 28 units’; putting down the 8 units, we 28 would then add the 4 tens to the col- 28 umn of tens and add thus: “4, 6,8, 168 10, 12, 14, 16 tens—1 hundred and 6 tens”’—which we write in their proper places, having for our answer 168 trees. 2. Another and a much shorter way to find out the same thing is by Multiplication; that is, instead of writing down the 28 six times, to write it only one time, and under it to write the figure 6, to show how many , times 28 trees there are; then 28 trees we multiply 28 by 6, thus, saying — 6 “6 times 8 (or 6 8’s) are 48—4 168 trees tens and 8 units”—we write the 8 units in their proper place; then, over LESSON XXV (continued). 155 the 28 we place a small figure 4, to remind ourselves that we have these 4 tens to add to the other tens; then, after multiplying the tens, saying “6 times 2 tens are 12 tens,” we add in the 4 tens, which gives us 16 tens, that is, 1 hundred, 6 tens, which we write in their proper places, and thus show that 6 X 28 trees = 168 trees. 3. Do this example both by addition and by mul- tiplication. There are 8 rows of leaves in our hall carpet, and 24 leaves in each row; how many leaves in all? 4. Write 43 seven times and add; multiply 43 by 7. Add 72 five times; multiply 72 by 5. ). Do these examples both ways: 38 39 27 124 672 987 — + -_———---—- —--— ———_—_____— pee 9 tr Shad Fad oiling 536 881 269 380 1,246 eri ack nctct 6. Copy and complete these examples: 2,579 3,520 4,408 5,045 mies usd Ue bas ad 15,232 25,550 32,003 51,020 6 7 3 9 — 156 LESSON XXVI. 10. CONCRETE APPLICATIONS. John’s grandfather is 3 score and 10 years old. What is hisage? (A score is 20.) . John’s father is 2 score and 5 years old, and John is 5 years less than a score. What are their ages ? . In a pound of sugar, or of flour, there are 16 ounces. How many ounces in 8 pounds? . There are 100 pounds in a hundred-weight. How many hundred-weight and how many pounds over in a barrel of flour (which weighs 196 pounds) ? . Our horse is 15 hands high. How many inches is that (the hand-measure is 4 inches)? . There are 2 pints in a quart, 8 quarts ina peck, and 4 pecks in a bushel. How many pints in 1 bushel? . How many pint boxes could be filled from 2 bushels of strawberries ? . There are 3651/, days in a year. How many days in 4 years? In 6 years? In 8 years? . If there are 4 gills in a pint of vinegar, and 2 pints in a quart, and 4 quarts in a gallon, how many gi//s in a gallon? How many gill bottles could be filled from 3 gallons of essence of lemon? LESSON XXVII. 157 DIVISION. 1. Divide 6 units by 3; divide 6 tens by 3; divide 6 hundreds by 3. Thus: 6+3=2 60 + 3= 20 600 + 3 = 200 2. Here is another form of expressing this same thing. It is a better 3)6, 3)60, 3)600. TOCRLOSUSE Ie CLYICINID a 6 Sa yo 1 2 ROOT S200 arge numbers. 3. Copy and complete 3)9, 3)90, 3)900 ; 2)8, 2)80, 2)800; 4)16, 4)160, 4)1,600. 4, 888 divided by 2 equals what ? 888 is equal to 5. Here isa short- 8 hundreds, 2)800 er way of 8 tens, 80 showing SUITS, St ey-ihtF 8 the same 800 divided by2= 400 thing : 80 divided by2= 40 8 divided by 2 = a eee me a +44 888 divided by 2= 444 6. Do each of these examples in both ways: 999 +3; 666+2; 444+2; 88874; 555-5; 848 +4; 426+ 2; 986+3; 248+ 2. 7. Do these examples in the shorter way only: 333 +3; 888+8; 666+3; T77+7; 444-1; 224+2; 363+3; 624 + 2; 663 + 3; 882 = 2. 158 a: 10. 11. LESSON XXVIII. CONCRETE APPLICATIONS, Carl has been writing for 80 minutes, half the time on his slate and half in his copy-book. How many minutes has he been writing in his copy-book ? . There are 600 minutes in ten hours. How many minutes in !/; that time? . A ranchman has 888 sheep divided into 8 flocks. How many sheep in each flock. . In a tulip-garden of 7 beds there are 777 plants, an equal number in each bed. How many is that? . Inarow of ten houses there are 90 windows. How many windows to each house ? . In 666 lead-pencils how many packages of a half dozen ? . In 464 shoes how many pairs ? . How many horses can be shod all round with 848 shoes ? . | have made a certain number of triangles, and find that there are in all 393 sides. How many triangles have I made? I have 550 ¢ in 5-cent pieces. How many 5 ¢ pleces ? There are 488 carriage-wheels in a shop. How many carriages will they supply. LESSON XXIX. 159 . Whatis!/, of 90? 9 bundles of ten divided 1 in- to 2 equal groups i i) i ii gives 4 tens in |] ff if each group and 1 ten over. Unig the ten left over, we divide it in half and find that there are five sticks in each half, i Hi | | which, added to the 2)90 ii. tens, gives 45 sticks ~““— in each group. Then 1/2 of 90 = 45. 5 . Suppose we wanted to find 1/2 of 94. One way would be to find, first, 2)90 the half of 90, just as 4 we did before, and petit : then find the half of " oe ee the 4 and add it to the ; : nit es half of the 90, thus: /2°° 94 sives 47 . But a shorter way would be to find the half of the 9 tens, and, then adding the sticks of the 1 bundle of tens left over to the 204 4 sticks, find at once the half of all ~—— the single sticks, thus: #1 . Do each of these examples both ways: 70 + 2; nee 24, Use oO 2A SO O85 +O. . Do these the shorter way only: 45+3; 72 ~63 S13; 95+5; 68-43) 844.6; 56 +2; 64+4; 92-4. 160 LESSON XXX. 1. What is '/3 of 729? We will take 7 bundles of one hundred, 2 bundles of ten, and 9 single sticks. 2. Dividing the 7 bundles of a hundred into 3 equal groups or thirds, we find 3)700 that we have 2 bundles in each group and | hundred over. In this 9 bundle of a hundred there are 10 tens; adding these to the 2 tens we or find that we have 12 tens. Wethen pe 3" divide these into three equal groups, 943 and find that there are 4 tens in each group, and the 9 single sticks divided into thirds gives 3 sticks to each third. Now, adding together the 2 hundreds, 4 tens, and 3 units of each third, we find that !/, of 729 is 243. The work we have done with counters is here expressed in figures. ’ 3. Here is the shorter way of expressing 3)/29 this: 943 4, Work out with counters and express in both forms these examples: 746 +3; D4 4°) 546 D2 028 = 4a (O60 ae 906+2; T65+3; 948+4; T6595; 864 + 6. APPLETONS’ US Bile SVS ea lho pn YE a Dd nS Magnificently Illustrated. Philosophically Treated, THE SERIES: J. NUMBERS ILLUSTRATED And applied in Language, Drawing, and Reading Lessons. An Arithmetic for Primary Schools. By ANDREW J. RICKOFF and E. C. DAVIS. Il. NUMBERS APPLIED. A Complete Arithmetic for Intermediate and Gram- mar Schools. Prepared on the Inductive Method, with many new and especially practical features. By ANDREW J. RICKOFF. i-@ This series is the result of extended research as to the best methods now in use, and many years’ practical experience in class-room work and school supervision. The appearance of this series has been awaited with great interest by leading educators, as it is intended to give all that has proved most successful in arithmetical work, while it presents some new methods of illustration, pictorially and otherwise, that will make the introduction to the study especially interesting and instructive. Send for full particulars at once. A glance, even, through these books will be instructive to any teacher. D. APPLETON & CO., PUBLISHERS, NEW YORK, BOSTON, CHICAGO, ATLANTA, SAN FRANCISCO. APPLETONS’ INSTRUCTIVE READING-BOOKS FOR SUPPLEMENTARY READING. THE NATURAL HISTORY SERIES. By James Jononnor, author of “Principles and Practice of Teaching,” ‘ Geographical Reader,” “How We Live,” ete. ‘No. 1. Book of Cats and Dogs, and other Friends. For Little Folks. 12mo. 96 pages. Deals with the familiar animals of the house and farm-yard. No. 2. Friends in Feathers and Fur and other Neighbors. For Young Folks. 12mo. 140 pages. Gives an account of the chickens, ducks, and geese about home, and of the birds, squirrels, rabbits, and other animals found near home. No. 8. Neighbors with Wings and Fins and some others. For Boys and Girls. Interspersed with interesting stories, it gives descriptions of birds, reptiles, and insects, in such a way as to lead to scientific classification. No. 4. Neighbors with Claws and Hoofs and their Kin. For Young People. Begins with the familiar animals of house and field, and reaches out to a general description and classification of mammals, No. 5. Glimpses of the Animate World: Science and Literature of Natural History. For SchoolorHome. 12mo. 414 pages. Treats of special topics, and is made up of the literature of natural history. The articles are from the pens of some of our most distinguished scientists and literary writers. The publication of this series marks a distinct and important advance in the adaptation of special knowledge and general literature to the intelligent comprehension of pupils of all grades of attainment. The importance of this movement and its value to the present generation of school-children can not be overestimated. While in no wise tending to do away with the regular school-readers, philosophically constructed in accordance with correct educational principles, the “ Instructive Read- ing-Books” introduce suggestive and valuable information and specific knowledge, covering many of the subjects which will eventually be more minutely investigated by the maturing of the pupil’s mind. Natural History in the Instructive Reading Course is to be followed by History, Geography, Science, and the Industries—all topics to arouse attention in their turn, and to fill the mind with knowledge of the great- est use. These are now in preparation, and will be announced in detail from time to time. New York: D. APPLETON & CO., 1, 3, & & Bond Street. THE PHYSIOLOGY FOR THE TIME. How we Live; or, the Human Body, and How to take Care of it. An Elementary Course in Anatomy, Physiology, and Hygiene. By James JoHONNOT, author of “ Principles and Practice of Teaching,” “Geographical Reader,” ‘‘ Natural History Reader,” etc., EugEens Bouton, Ph. D., and H. D. Dipama, M.D. Thoroughly adapted to elementary instruction in the public schools ; giving special attention to the laws of Hygiene (including the effects of alcohol and narcotics on the human system) as ascertained from a care- ful study of Anatomy and Physiology; containing also a full Glossary of Terms, complete Index, etc. FOR HIGH SCHOOLS, ACADEMIES, AND ALL SCHOOLS OF S/M/LAR GRADE. The Essentials of Anatomy, Physiology, and Hygiene. By Roger S. Tracy, M.D., Sanitary Inspector of the New York City Health Department. This work has been prepared in response to the demand for a thoroughly scientific and yet practical text-book for schools and academies, which shall afford an accurate knowledge of the essential facts of Anat- omy and Physiology, as a scientific basis for the study of Hygiene and the Laws of Health, the applications of which are clearly and carefully stated throughout. It also treats of the physiological effects of alcohol and other narcotics, fulfilling all the requirements of recent legislative enactments upon this subject. Teachers and School-Officers should correspond with us before intro- ducing a new work upon this subject. D, APPLETON & @0.,, Publishers, NEW YORK, BOSTON, CHICAGO, SAN FRANCISCO. APPLETONSDS’ AMERICAN STANDARD GEOGRAPHIES. BASED ON THE PRINCIPLES OF THE SCIENCE OF EDUCATION, And giving Special Prominence to the Industrial, Commercial, and Practical Features, The remarkable success which Appletons’ Readers have attained is due to the fact that no effort or expense was spared to make them not only mechanically superior, but practically and distinctively superior, in their embodiment of the best results of modern cxperience in teaching, and of the methods followed by the most successful and intelligent educators. In the same spirit, and with the same high aim, this new series cf Geog- raphies has been prepared, and it is in harmony, therefore, with the active educational thought of the times. The series comprises two books for graded schools. I. Appletons’ Elementary Geography. Small 4to. 108 pages. In this book the aim is to develop and present the subject in accord- ance with the views of advanced teachers, and to embody the most natural and philosophical system. It treats the subject objectively, makes knowl- edge precede definitions, and presents facts in their logical connections, taking gradual steps from the known to the unknown. II. Appletons’ Higher Geography. Large 4to. 129 pages. In this volume, the aim has been to combine beauty of typography, usefulness of illustration, attractive maps, and every element of mechan- ical superiority, with a variety of original features, and the improved methods followed by the most successful teachers of the day. Prominence is given to a consideration of the leading Industries, as the results of certain physical conditions, and especially to Commerce, a feature which will not fail to be acceptable in this practical age. The pupil is taught to what the great cities owe their growth, the main routes of travel and traffic, where and how our surplus products find a market, whence we obtain the chief articles of daily use, and the exports which the leading commercial cities contribute to the world’s supply. The Maps, both Political and Physical, challenge comparison in point of correctness, distinctness, and artistic finish. Special editions for the various States have been prepared, giving extra maps and descriptive matter. D. APPLETON & CO., Publishers, NEW YORK, BOSTON, CHICAGO, ATLANTA, SAN FRANCISCO. APY LETONS “READERS. SOME DISTINGUISHING FEATURES. Modern Methods made easy.—Education is a progressive science. Meth. ods of the last century must be discarded. The question ‘* How shall we teach reading?” is fully answered in these books, and teachers who have adopted and followed this method have greatly improved their schools, Word and Phonic Method.—By taking at first words with which the child is quite familiar, and which contain sounds easily distinguished and continu- ally recurring, both teacher and pupil will find the sounds a great help in reading new words as well as in acquiring a distinct articulation. Spelling.—Words selected from the lessons are given for spelling with each piece, thus affording the best opportunity for oral and written spelling- lessons as well as for definitions. In the Third, Fourth, and Fifth Readers, graded exercises in spelling analysis, together with daily lessons of words often misspelled or mispronounced, are placed in the Appendix for constant study. With these Readers no “ Speller” will be needed. 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Selections,—The selections embrace gems of literature from Jeading authors, No other readers include such a wide range of thought. showing from the sim- ple stories for children in the earlier books, to the extracts from the best authors in the Fourth and Fifth, unity of design and a just appreciation of the needs of our schools. Great Success,—Since the publication of these Readers, their sale has aver- aged nearly a million copies a year. Indorsements.—These Readers have received the indorsement of nearly every educator of note in the United States, but the best proof of their merita is found in the great improvement manifested everywhere they are used. D. APPLETON & CO., Publishers, NEW YORK, BOSTON, CHICAGO, ATLANTA, SAN FRANCISCO. JAMES JOHONNOT’S EDUCATIONAL WORKS. The Sentence and Word Book; a Guide to Writing, Spelling, and Composition by the Word and Sentence Method. 12mo. Cloth. A Geographical Reader. A Collection of Geographical Descrip- tions and Narrations, from the best Writers in English Literature. Classified and arranged to meet the wants of Geographical Students, and the higher grades of reading classes. 12mo. Cloth ‘““Mr. Johonnot has made a good book, which, if judiciously used, will stop the immense waste of time now spent in most schools in the study of geography to little purpose. The volume has a good number of appropriate illustrations, and is printed and bound in almost faultless style and taste.”"—A ational Journal of Education Appletons’ Instructive Reading-Books. Natural History Scries. No. I. Boox or Cats anp Docs, AND OTHER FrRienDs. For Little Folks. No. IJ. Frrenps 1n FeatHers aNnD Fur, AND OTHER NEIGHBORS. For Young Folks. No. Il]. Neiausors wits Wines anp Fins, AND SOME OTHERS. For Boys and Girls. No. IV. NeigHsors witH CLAws AND Hoors AND THEIR Kin. For Young People. No. V. GLIMPSES OF THE ANIMATE WorLp: SCIENCE AND LITERATURE oF NaturaL History. For School and Home. Supplementary reading-matter has come to be recognized as indispensable to rapid progress and the attainment of right methods in the art of reading. Nothing is more delightful to children of all ages than studies of animate nature. Not only are perception and memory appealed to, but the imaginative and comparative faculties are also brought into healthful play; such subjects are in a peculiar sense fitting embodiments of reading exercises. When in- troduced, monotone and sing-song will at once disappear, and natural reading take their place. The subject-matter of these books, the fascinating style in which it is presented, and, withal, the beauty and accuracy of the illustrations, make the series incom- parably superior to anything yet issued for supplementary reading. Principles and Practice of Teaching. 12mo. Cloth. Contenrs: I. What is Education? IT. The Mental Powers: their Order of Development, and the Methods most conducive to Normal Growth. III. Object- ive Teaching: its Methods, Aims, and Principles. IV. Subjective Teaching: its Aims and Place in the Course of Instruction. VY. Object-Lessons: their Value and Limitations. VI. Relative Value of the Different Studies in a Course of In- struction. VII. Pestalozzi, and his Contributions to Educational Science. VIII. Froebel and the Kindergarten. IX. Agassiz, and Science in its Relation to Teaching. X. Contrasted Systems of Edacation. XI. Physical Culture. XIL. reste es Culture. XIII. Moral Culture. XIV. A Course ofStudy. XV. Country chools. How we Live: or, the Human Body, and how to take Care of it. An Elementary Course in Anatomy, Physiology, and Hygiene. By James Jouonnot, Eucene Bourton, Ph. D., and Henry D. Dipama, M. D. 12mo, Cloth. D, APPLETON & CO., Publishers, NEW YORK, BOSTON, CHICAGO, ATLANTA, SAN FRANCISCO. APPLETONS’ Elementary Reading CHARTS | CHARTS CHARTS CHARTS. CHARTS CHARTS. CHARTS CHARTS CHARTS. CHARTS. CHARTS. CHARTS. CHARTS. CHARTS CHARTS. CHARTS. CHARTS. CHARTS CHARTS. CHARTS CHARTS. Forty-seven Numbers. Prepared by REBECCA D. RICKOFF. With Patent Revolving Supporter. Designed to make learning to read a pleasant pastime. Designed to cultivate the observing powers of children. | Designed to teach the first steps of reading in the ght way. Designed to train the mind of the child by philosophical methods. Designed to furnish the primary classes with a variety of inter- esting occupations in school-hours. Every step in advance is in a logical order of progression and development. Pictures, objects, and things are employed, rather than abstract rules and naked type. The beautiful and significant illustrations are an especially notice- able and attractive feature of these charts. Every chart in the series has in view a definite object, which is thoroughly and systematically developed. They are in accord with the educational spirit of the day, and with the methods followed by the best instructors. 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