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Tous les autres exemplaires originaux sont filmds en commenpant par la premidre page qui comports une empreinte d'impression ou d'lllustration et en terminant par la dernidre page qui comporte une telle empreinte. Un des symboles suivants apparaitra sur la dernidre image de cheque microfiche, selon le cas: Ir symbole — ^ signifie "A 3UIVRE", le symbole V signifie "FIN". Les cartes, planches, tableaux, etc, peuvent dtre filmds d das taux de reduction diff^rents. Lorsque le document est trop grand pour dtre reproduit en un seul clich6, il est film6 d partir de I'angle supdrieur gauche, de gauche d droite, et de haut en bas, en prenant le nombre d'images ndcessaire. Les diagrammes suivants illustrent la mdthode. 1 2 3 4 5 6 ^. 1- 1^ / ■^ . National Library Bibliotheque nationale of Canada du Canada THE BEST VERTICAL SYSTEM PUBLISHED. Authorized in the Province of Quebec. Grafton's Vertieal Penmanship. Compiled by Competent Teachers. Easy to L^arn. Practical. Easy to Write. Graceful. Thoroughly Graded. Printed and ruled in the best style on a very fine quality of paper specially made for this series. Duplicate head-lines are repeated in the centre of each page, and ruled verticpl lines assist the pupil in writing perfectly upright. Price, 8 cents each. No expense has been spared to make these books superior in every respect. The system is simplicity itself. The writing staff is divided into only three spaces, and all letters, both small and capital, are brought within these three spaces. The advantages of this must be apparent to every practical teacher. It makes the systeir simpler and easier to teach, it follows more closely the proportici^s of ordinary type, and produces a style of penmanship having more character "and dignity than any other scale. Every single letter form has been carefully considered, so that it might combine in the highest ^degree the qualities of grace, legibility, and ease of execution. After much deliberation, the proportion of 3 in width to 4 in height has been adopted as the basis of the capitals and small letters ; the capitals and loops of small letters are made just twice the height of the small letters «, c, tn, n, etc. ; and the initial and terminal strokes are made to extend one half-space vertically and hori- zontally — a unique feature, simplifying the letters considerably. An examination of Grafton's Vertical Copy Books will convince anyone that they are not books hastily prepared in response to a sudden demand, but that they are the outcome of long-continued study and thought on the part of both the authors and the publishers. In fifteen months the edition of these Copy Books published in the United States has been adopted by .School Poards, over all competitors, in cities and towns having an aggregate population of 5,000,000. Descriptive Circular, showing specimen of writing in the different numbers, upon application. F. E. GRAFTON & SONS, Publishers, MONTREAL. 1 r.^ A NEW BOOK ON A NEW PLAN, wliicli sliould be in every school in the covmtry. "ir INTERESTS I'Ul'II.S AM) MAKES THEM THINK." GFafton's Word and Sentence Book. ATTRACTIVELY AND STRONGLY BOUND. PRICE, 30 CENTS. While the ai)ilily to spell correctly is justly regarded as one ot the most important oi)jects of education, as it is also one of its most distin- guishing marks, the old style of " Speller," with its lists of words in parallel columns, classified only with reference to their length, is, and deserves to he, in disrepute amon<; intelligent teachers. The Word and Sentence Book recognizes the truth of the educa- tional maxim, "We learn by doing." While it contains an unusually large list of words, classified with respect to their meaning, it provides also, in great numl)er and variety, carefully prepared dictation exercises requiring the pupil to write the words in different combinations. It will be found that by the use of these exercises the primary object of a spelling-book, to enable pupils to spel' correctly in written composition, can be most successfully accomplished. In order to get the best results I he pupil must be interested in the book, and it is l)elieved that Grafton's Word and Sentence Book, containing, as it does, not only the best selected and best classified list of words, but a far greater amount of useful and interesting information than can be found in any other book of its kind, will insure his interest. What they say of the New Speller : " It is the best." " Is delighted with it." '■ Selections for copying and memorizing are in good taste." " Has found it perfectly satisfactory." " I would like to |)Ut every class in school through it." '* The best he has; ever seen." " It heads the list." *' It covers the whole ground.' " Tiie most practical and interesting he has ever seen. He predicts for it great success." Anthoyizcd for use in the Province of Quebec, Adopted for use in the Public and llii^h Schools of Montre>u. F. E. GRAFTON & SONS, Publishers, rvlONXRKAL, /■ GRAFTON'S G RADED ARITHMETIC BOOK I, TEACHERS" MAXIJAI -J WITH a:s^weks BY K w. Airrnv, MI'KKINTKNDKNT OK CITV .SUIIOOLS, MO.NTKKAl.. MOiN TH KAL: F. K. (JUAFTUN .V SONS, IMJlILlSllFUS. ksitO. QA Z:^^ Entered accordijig to Act of Parliament of Canada, in the year one thousand eight liundred and ninety-six, by F. E. Graftox & Sons, in the Office of the Minister of Agriculture. e year Sons, GENERAL SUGGESTIONS. 1. In teaching follow the method and instructions given in this manual. Little time, however, is needed for teaching compared with the time needed for practice. A sufficient number of carefully graded examples will be found in the pupils' book. The best pupils may work them all ; slower pupils should work part only. Do not make quick pupils keep pace with slow ones. Both must be taught together, but in busy work the one should do mucli more than tlie other. Quick pupils may be asked to prove their work. Avoid luwj examples, which dis- courage and disgust little cliildren. 2. Sight Exercises, when oral, should be conducted in a spirited manner. When results are irriUen, an exercise should be assigned, and pupils allowed to perform the mental work and record the answers at their own speed. :>. Endeavour first to make pupils undcrdand the process of a rule; then train fhe„i to he accurate; and finally drill in rapiditif. Xever attempt to gain rapidity to the neglect of accuracy. 4. do slowly, especially at first ; do not measure the ability of the child by yoin- ability. Bring yourself down to the level of tlie child's mind: be ])atient : repeat everything many times ; i-eview daily. .J. Problems {oral and irritten). It should not be forgotten that the number-lesson may be nuide an excellent language-lesson. It is of the highest importance that the child give his answers in I (iKADEI) AHITil.METIC. complete .seiitencus, plainly s])()keii, with clear accent. Explanatory statements made by pupils should be simple but clear. Tliey should represent the pupil's thought, and be clothed in language of his own choice. A fonnula or form of analysis may be given to the class later. A problem is not finished vvhen the answer is found, but when it has been analysed. The language may be taken as a safe test that a pupil has completely mastered a ste}), though it does not follow from a ])U[)irs inability io make an oral statement that he has failed to understand the process. Lead chUdrcn to malr orifjinal prohhui^. 0. Slates, scribblers and pencils should be kept in good condition. Figures should be large and distinctly made and written in lines parallel to the upper edi^e of the slate or book. With beginners it is of prime impor- tance that all lines or columns of figures should be large, even, distinct. NUMBERS 1 TO 10. A knowledge of nund)ers up to 10 is presupposed in the exercises of this book. It is presumed that a child not only knows them, but can use his knowledge. Of what use to the child if he can count to 100, but is unable to separate the number 9 into its elements and use them i By means of the eye and by handling objects (sticks, blocks and other counters) he has mastered the first ten numbers and their combinations. At the 10, if not before, the use of objects should be abandoned. The child should now be able to gain the al>stract idea without the help of objects. Objects l)ecome a cuml>rance as soon as the child can do without them, as they withdraw the attention from the abstract numlxn'. TEACH EKS' MAMAL. o The moi'e tlioroughly the iiumburs fromOiu* to ten are known, the siii'er and more rapid will he all later work in arithnielic. They are the fonndatioii of the wliole number system. A ]-i«^dit eoneeption of the first ten numbers will be much facilitated by ai-rangino- them in geometrical patterns. *" With a small nunil)er of objects a random groupin*;' is instantaneously recognised; but not with many objects. Careful observation has shown that with most of us the highest number instantly rec(Kniised in a promiscuous asseml)lage of things is live. Higher numbei's tlian five are subdivided l>y the eye into more easily recognised small groups. If nine pebbles be thrown upon a table Ijcfore us, most of us will sav nientallv, here are three and three and three, nine. A few of the more expert will say, here are five and four, nine, on the table. Scarcely one will say at (nice nine, as we slundd all say three, if but three were thrown down before us. What is difficult or imposible for us to do, when o])jects ai'e promiscuously presented, Ijecomes easy in a definite arrangement. This • • • is at once recognised as nme, and that without « « « exi)licit l)reaknig upinto three and three and three, although that subdivision is implicit in the conception. The formation of such con- ceptions of the first ten numbers should l)e regarded as an essential preliminary to arithmetical rules, should be begun at home or in the kindergarten and completed in the first year of the jnimary school." The ])attern which is presented to the class as the ////>t'- form of aiiumber should be carefully chos -n. It ought This (|Uotatio]i is from a treatise on the four simple rules of arithmetic, hy Dr. Robins, i'rineipal of the Me(Jill Normal School, which was lent to the author in manuscript, and to which he desires to acknowledge liis indebtedness for some valuable suggestions. n GRADED ARITHMETIC. # • • • • • • • • • 8 #.• • ••• 9 • • • • to • ••• • • • • • • • • • • JEACHERS MANUAL, 5 to be (1) n-ell biilanced, (2) easily derived from preced- ing patterns. If the triangle is accepted as the typical three, and th(i square as tlie typical four, good patterns may readily be constructed from these two forms with the aid, perliaps, of some linear arrangement. Special care should be taken in the selection of patterns for numbers al)ove five. The patterns placed first in the series on page 4 can be recommended, and analysis will show how they are related to each other, e.g., the patterns for the seven, the eight, the nine and the ten are all developed from the six, the two triangles of the six beintr separated by a linear arrangement of 4 to make 10, 3 to make 9, etc. « When a number is being decomposed into its elements for the purpose of comparing and measuring it with other numbers, the remaining patterns will be found useful as suggesting new combinations. If some desired combina- tion is not leadily seen, it will be made plain by the use of coloured chalks. Pupils must be trained to make patterns for tliemselves and to discover in them fresh combinations. Notes for Book I. Figures in heavy type at the top of each page indicate corresponding pages in the Pupils' Book. Roman numerals and capital letters on a page indicate corresponding exercises in the Pupils' Book. I. Numbers 10 to 20. 1. The Ten. We have now reached the first number that must be considered as another kind of one the ten. We write the figure 1 as before, but to show that this 1 contains ten times as much as the siniple 1, we move it one place to the left, and say this 1 is a ten. The vacant 2 I* •.•*' 6 GlUltED AUITllMKTIC 1-2 place of the simple one will be indicated by a cipher, so — 10. Teach the tt'n as a group; ten dots joined together, ten sticks bound together, etc. 2. The teaching of each number al)ove ten nuist precede the working of the exercises in the hook. Show that the numbers from ten to twenty are formed by adding the first nine numbers to teiK Their names, from I.') to 19, indicate this, e.g., fuitrkeii means four and ten : Jifteeu, five and ten, etc. In writing these numbers the ten is expressed by a 1 in the seeond place and the figure expressing the one.^ is put in the first place. I, A. This exercise must be worked across the page, not in columns downwards. It involves one new step, viz., the coinh'iuation of the ten n-ifh the nine digits;. I. B. This exercise introduces the key to addition through the ten. When the sum of two numbers exceeds 10, one (usually the upper in the column) is broken into two parts, the first of which is sufticiont to raise tiie lower number to 10. The remaining part is then added to the 10 thus formed, e.g., the sum of G and 5 is found bv breaking the ;" into 4 -f 1 , and the operation i.'ecomes and 4 are 10 and 1 more makes 11. This Hiental rciirningcnient of (5 and o into (I and 4 ami 1, for the [)urpose of addition, is fundantenttd, and must be thoroughly taught. It removes the necessity of committing to memory an addition table, and enables a pupil, who knows the elementary sums up to 10, to add at once more dillicult numbers through the ten. Practice will soon enable pu[)ils to '.idd l>y this method rapidly and acciir- alelv : when thev no lonsrer need the intiu-mediate step, they must be encouraged to do without it, and in most cases they will tliemsclves dispense witli it. The process is purely nilaining it, oral 1-2 2-6 teachers' manual. expression of it must not be permitted : in adding 9 and 8 the thouglit is 9, 10, 17, nothing more, instead^'of 9, l?] without the intermediate 10. The fourtli line of exercise B must be worked exactly as the fourth column of example A. ^ I. C. Addition and su])traction sliould be tauglit together. The latter process is the opposite of the former, and should be derived from it. As soon as a child sees that 7 and 5 are 12, he is ready to see that 12 less 7 is 5, and 12 less 5 is 7. The difference between 12 and 7 nnist be inferred from the knowledge that 7 requires 5 to make 12 and not by counting 7 off' 12. Examples must be worked through the tni as in addition, e.g., 12 - 7 presents itself m this form : " What number must be added to 7 (the lower number) to make 12 (the upper) ? " The result (5) is foundjnentally by raising 7 to 12 in two steps, thus, 7, 10, 12 (7 and o are 10 and 2 more are 12). I. D. Such exercises in computation as are here given must.be practise.l fre(iuently not for any great length at one time, but in a sj.irited manner at frequent intervals. Successive results only must be named as rai)idly as they can be given, e.g., adding by threes from 1 would rc.piire pupils to say, 4, 7, 10, 13, 10, etc. The spelling jyrocess, 1 and 3 are 4, and :{ are 7, and 3 are 10, etc., cannot be allowed. If. A. The teaching of numbers from 20 to 100 sliould precede and accompany these exercises. Tlie exercises, if studied by iIr, teacher in advance, will tiiemselves indicate the meihod of teacinng. As before, every ten should lie regarded as a groujt or bundle, and tli-' number of such grouj)s or bundles should be called so many tens, tlie surplus left over being called ones or units. 8 GRADED ARITHMETIC. 6-10 A child must be taught to give clearly and exactly an analysis of numbers, written or spoken. Upon his ability to do this rapidly will depend his power to compute. E.rj., in 49 he must see at once 4 tens and an added 9. Concerning such a number he must be able to tell (a) that it consists of 4 tens and 9 units, (6) that it requires 1 unit more to make it 5 tens. In counting, the following device may be tried with advantage : — Count by ones (say) from 30 to 50: 31, 32, 33, 34, 35, 36 "Stop!" the teacher says, "Where are we?" Ans. — "We have passed the third ten by 6 ; we require 4 more to make 4 tens, and still another tea to make 50. " II, B. All these exercises • ) 4 4 5 5 G 6 8 8 9 9 16-26 TEArilKHS' MANIAL. 11 6 (rt) Let the results that are to be memorized in the table first be proved l)y rows of dots as sliowii in the pupils' book, p. 18. (b) Let pupils make the table by counting (adding) by equal iiicremeuLs, c.,;., ;f the table of 6's is to be learnt, this step calls for adding by 6s up to 60 : -6, 12, 18, 24, 30, 36, 42, 48, 54, 60. The rate at which these successive numbers are named will at first be slow, but will ])ecome gradually quicker as facility is attained. {(') The table must then be read and recited as follows : — 1 six, 6 ; 2 sixes, 12 ; 3 sixes, 18 ; 4 sixes, 24 ; etc. No other words should be spoken or thought of. {(l) Study, i< citation and frequent repetition are still required to fix the table in the memory. Division Table. Tlie division table, tliough given in full, must not be eoniniitted to memory. Division must be taught as the reverHC of nudtipHcatlon ; and the elementary qifoilenfs must be derived from the elementary |>/W/ is contained in MO how many times ? etc. Ihe boo!' pxercises, pp. 18-2(i, on each table sliouhl accompany the learning of the table. 'I'lie problems at this stage both in multi- plication and division ought to be illustra^^ed l)y drawings, rows of dots, etc. /v.//.. How many cents must I pay for 4 tliree-cent stamps? This problem may be illi:strated by drawing four oblongs represent- ing the stamps, and placiir, dots above each to repres'^nt the cents. /{'tifrex, (hiidx, Juui/h>^, etc., should l)e taught frcun fraction disks, and problems involving their use illustrated afterwards, *'.r/., 16 ounces in a lb. How many ounces in | lb. '/ 16 dots (4 rows of 4 12 (MiADEI) AIUTHMETIC. 18-26 each) may be drawn, representing 16 ounces, and one-quarter of them then marked off. Tlie task of committing to memory tlie multiplication table may he shortened by several devices which are educationally sound. Tlie following are recommended : — 1. The table of tuos should be derived from addition as sometiiing already known ; f\nd that of (eus from numeration and notation. These tables may be taken first. 2. Prove (but do not enunciate) the principle that tlie product of two factors is the same in whichever order they are taken, c.j/. , 3 x 5 = 5x3. This may be shown by rows of dots which, read horizontally, make 3 fives; read vertically, nuike 5 threes; or by adding a oolunm of 3 fives and then one of "> threes. This principle, which proves that 3 fives and 5 threes give tiie same result, aiul need not be memorised as independent facts, reduces the number of products to be memorised in each table progressively by one. In this way, when nine times is reached, 9 nines is the only result not already learned. 3. Five time.s is easy to remember, but, for the sake of the training involved, is l)est taught in connection with ten timej<, creri/ two fives makinij a ten. J fiveA- 1 ten ; 3 fives = 1 ten and five or 15 ; 4 fives = 2 tens or 20 ; T) fives = 2 tens and five or 25, etc. 5. The nines, which are difiiicult to memorise, may be thus oimpli- fied : Nine is one less than tt-n ; therefore, 2 nines are 2 less than 2 tens or IS ; 3 nines are 3 less than 3 tens or 27 ; etc. These processes, once understood, must be worked mentally, not repeated aloud. Dull pupils will still use their unreasoning meniory. VI. Practical Questions. A contains easy mental ([uestions in multiplication. ?> and contain similar 'i»estions in division. In both cases the ([uestions should be considered as sam[)les, aiul their number increased. This exercise is intended to devel()[) aral rxpiraslon and anulf/si.r The analysis given in the Ai-ilhmetic must be followed. Each exam])le sliould be analysed dearbf and i'onvml II, iiniX recited with (h'l^fincfHi'.^x and prompt If it o 3 3 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7 8 8 8 8 9 Before passing to written work, let the teacher write four columns of figures on the blackboard, as represented in the margin. This is a practical exercise to make pupils rapid and accurate in the mechanical processes of add- ing, subtracting, multiplying and dividing. For the method of con- ducting this exercise see directions on p. 10 of this Manual. VII. Numbers to lOOO. 1. Teach the place-value of a figure, ie., that the value of a figure in i\\Q first place is so many units; in the second place, so many tens; in the third place, so many hundreds. 2. Require pupils to write numbers from dictation, their component parts being given in order and out of order, e.g., 1 unit, 5 tens, 6 hundreds; and to name at sight the component parts of any number of 3 figures. 3. The addition and nniltiplication of hundreds will result in giving a figure in the fourth place, viz., units of thousands. Give the name thousands to figures in the fourth place, but do not teach the thousand period. 4. In writing numl)ers from dictation give special drill on numbers that present difficulties, e.g., 101, 106, 413, 210. Such numbers as 756 are seldom read or written incorrectly. VIII. The result found by addition is called the sum. The numbers to 1)0 added are called addends. Th.ese terms nuist be now explained, and afterwards used constantly. No deliuitions winuX ha meiKorised. 3 14 GRADED ARITHMETIC. 31-33 Addition. — Two cases, (n) v'itlfou.t carrj/ing, (h) irit/i carrying. A. :Ul + 415 + 1-2 (no carryiiio) :— ^Ml = o hundreds + 4 tens + 1 unit. 415 = 4 liundreds + 1 ten + 5 units. 122 = 1 hundred + 2 tens + 2 units. 87ook without copying, results only being recorded on their slates. They will get plenty of practice in copying other examples not arranged in columns. 34-36 TEACHEHS' MANUAL. 15 NoTK 4.— Book exercises must be commenced as soon as pupils nnderstaml the. 2>rocens of the rule. Do not put otF the meclianical drill until they aie able to reproduce the extended work. In teaching the four simple rules observe the following steps : -(1) ^fake pupils under- stand the process of :ke ride. (2) Train them to be accurate by mechani- cal practice. (3) Drill them in rapidity. (4) Make them show the process by reprodncimj the extended work. IX. The inetliod of finding the j^art that remains, when a smaller number is taken from a larger, is called subtraction. The number that remains after subtractinfj" IS called the difTerenoe. Subtraction is very closely connected with addition (being its ojyposite), and must be tauglit by means of addition. The larger nund)er is tlie sum, tlie smaller number is our of tiro addends. The remainder is the other addend. To show tliis relation introduce subtraction as a variation of addition in the following way: — 432 546 Suppose this example in addition has just been worked. Eri«,se one of the addends, say r>4G, and then ask the class to 978 ^^'«^t>ver what it was. Lead them by judicious (inestioniiig ; to reproduce the missing addend, ('.(/., the sum is 8 and one of the two addends is 2 ; therefore, the other nmst be (». The sum is 7 and one of the two is 3 ; therefore, the other must be 4, and so on. When the nussing addend is thus reproduced, prove its correctness by addition. The other addend may next be erased and reproduced in the same way. The next step will be to rearrange the numbers 978 978 after the manner usual in subtraction, and agu,iu 540 432 find the missing addends. This method has the following a«lvantages to recommend it : — (a) It shows clearly tlie relation between addition and subtraction. (h) We easily derive from it a rule for subtruition, viz., to raise, the toicer line so as to equal the top line. (c) We easily derive the pntof af R..htr,vr?inn, vi/.., that (he remainder and lower line (the two addends) will yire the top line (the sum). ((/) The borrowinii becomes the carrying of addition. 16 GRADED ARITHMETIC. 34-37 Note. — Do not mention or teach the ternia minuend or nubtrahend. They will come in later. Pupils must be able to show the work extended as before : — 623 = 6 hundreds -f 2 tens + o units. 237 = 2 hundreds + 3 tens + 7 units. 386 = 3 hundreds + 8 tens 4- 6 units. 7. units require 6 units to make 13 units. Write down the 6 units and carry 1 ten. 4 tens (3 + 1) require 8 tens to make 12 tens. Write down 8 tens and carry 1 hundred. 3 hundreds (2 + 1) require 3 '. hundreds to make 6 hundreds. Note 1. — Carrying 1 ten means that having raised 7 up to 13 (instead of 3), we are 1 ten to the good in raising the next figure of the lower line up to the next figure of the upper line. Do not waste time in explaining this. It will be evident when tlie subtraction is proved by adding 386 and 237. Note 2. — Though the exercises in the text-book give 4 figures (thousands), extended work need not go beyond 3 figures (hundreds). X. Multiplication is a short method of adding a Humbcr to itself. Prove this by working a few examples (as 3 times 413) both by addition and multiplication. Pupils should occasionally be reipured to prove an example by addition. The number to be multiplied is called the multipUoand. The immber by which we multiply is called the multi- plier. The res\dt is called the product. Explain and afterwards constantly employ these terms. A complete analysis of the steps used in multiplication is too difficult for pupils at this stage ; but they must be taught to show extended work, as before, up to hundreds. 495 = 4 hundreds -f 9 tens + 5 units. I I, 1980 = 19 hundreds + 8 tens + units. 38-40 teachers' manual. 17 as t I 4 times 5 units (4 fives) are 20 units, 20 units are 2 tens and units over. Write down and carry 2 tens. 4 times 9 tens are 36 tens, which with 2 tens carried make 38 tens. 38 tens are 3 hundreds and 8 tens over. Write down 8 tens and carry 3 lumdreds. 4 times 4 hundreds are 16 hundreds, which with 3 Imndreds carried make 19 hundreds. XL Division is finding how many times one number is contained in another {e.g., how often 5 days are contained in 20 days), or separatiuj^' a number into eqaal parts {e.g., distributing 20 apples equally among 5 boys). In division there are two numbers, called dividend and divisor. The dividend is the number to be divided. The divisor is the number by which we divide. The result or answer found by division is called the quotient. The part of the dividend left after dividing is called the remainder. Explain and afterwards employ these terms. A complete analysis of division need not be attempted, but pupils must be taught to show extended work up to hundreds. 5 ) 732 = 5)7 hundreds -|-'> tens -f 2 units. Iri6 laith 2 rem. 1 hundred + 4 tens + units ivith 2 rein. 5 is contained in the liundreds, wluch are 7, once with 2 remainder. Write down 1 under the lumdreds. The 2 hundreds remaining are , equal to 20 tens, to wluch are added the 3 tens in tiie dividend, making 23 tens. 5 is contained in 23 4 times with 3 remainder. Write 4 under the tens. The 3 tens remaining are equal to .30 units, to which are added the 2 units of the dividend, making 32 units. ') is contained in 3.1 6 times with 2 remainder. Write the 6 under the units and tlie 2 as a remainder. 4 ) 1.36 -- 4 ) 1 hundred + 3 tens + units. 34 ^ hundred + 3 tens I- 4 units. 4 is not contained in the hundreds, wliich are I. We show this (in the exten. and X ten should be taught first. Ihe others are combinations of these three. When I lyreeedes V or X, it must be subtracted from it : when I iollows V or X, it must be added. XIII. Oapacity.-Bj way of introducing a table, have a talk with the children to find out what they know about It. Even httle children will know son/ething about buymg milk, coal-oil, etc., by the pint, quart and gallon Have these measures in your class-room, and prove the reahty of your table, using water or dry sand. The table must then be copied and learnt, and .Irill given upon it. 1 ^fT; ^^''f*^--^-'^''^^ measuring strip here suo-gested should be o thick paper or cardboard. It may be prepared by the pupil at home, if it cannot be conveniently done in school. No work is to be done by means of this measure : 1 is to be used as a test of correetness out,,. The len.rth of all hues or objects must be guessed, then measured, the ditterence between the estimate and true measure beinc. found and recorded. " XV. Mental Examples. In these examples, when the problems are cdven in concrete form, an oral statement should be called'for; E.o.,Mi roses; 9 faded; liovv .nauy left ? A»,. 40 roses lesJo or 31 roses ar. left ; or 31 roses are left, the difference het.ee.: 40 u'l I ji 46-48 teachers' manual. 19 For oral statements of examples in nmltiplication and division see p].. 27 and 28 l»upils' IJook. XVI. When problems involvinf,^ concrete numbers are given in the Test Exercises, the following [dan is recom- mended : — 1. Let pupils first endeavour to solve such examples without jiid and without giving any written statement of the process. I 2. Let the teacher draw from the class \)y (piestioning an oral analysis of these examples. 3. Let pupils now work the examples a second time, giving a written statement suliicient to indicate the l)rocess. B.i/., If 4 rishermen catch 920 fish and divide them equally, how many fish will each have ? (a) What is given in this example? Ans. (drawn from several pupils). Tliat there are 4 fishermen ; that they catch 920 fish ; that they divide them equally, (/') Wliat is required ? Ans. To find how many fish each fisherman will iiave. (c) Wliich of the four methods or rules shall we use to fin 1 this? Ans. Division. ((/) Employing division, what shall we do ? Aiis. Divide 920 by 4, or 920 fish into four ecpial parts. (e) Why ? Ans. Hecause each fisherman ought to get .| of the whole. The pupils, after this oral analysis, proceed to work the example a second time, and are now expected to give some written statement of the process. Something Hke the following will be sufficient : — Each fisherman will have | of 920 fish. 4 )_920 fish- I of 920 fish is 280 fish. ^"oliSi^ XVII. Before teaching numbers to 1,000,000, study the questions in this exercise and the test questions that 20 GRADED ARITHMETIC. 48-52 follow. The division of uurnk^rs into periods cf .3 fi^^ures each m,ist now be taught. Kevicw the place-valuc^of a hgure. 1 he three places of the units' period have already been taught, viz., the units' place, the Jirsi ; the tens' place, the ,cco>id ; the hundreds' place, the third On connnenc.ng instruction on the tkomamis' period, teach the names of the 3 nlaces, thousands (or units of thou- sands), tens of thoMSnnds, hundreds of thousands; also the corresponding numbers of the places, fncrth, fifth and NotaUon, numeration, decimal scale must be explained and the definitions given in the book connnitted to memory. XVIII. The object of this exercise is not to teach Canadian money (which is dealt with in Book U (^h f) but only how to rea) The cents occupy t,co plaees, the first being sin-de cents (units), the second ten-cent pieces (tens). " ' {c^ AVhen the number of cents is less than 10 a nought must be put i.i the second place to indicate the absence of tens. , ^}^' 'f ■«i"'ti'>iis relntinK t., n.l.lition here „iven (an, a 1 .lefinitions hereafter given i„ the |,n,,il.,. \„^,) are to be connnitte,. to nuMnory. liefore tins is done, not only the terms „se.l, 1ml the aritlnnetieal rnle to whieh hey apply shonhl ),e fnlly nn.lerstoocl, ... The ,,„estion, How sl,all pnp,ls learn to express eaeh rnle an,! it, reasons ? does not so n.neh ,.o„eern „rith,„etic as co.n- ♦ 'I' This extract is from Dr. Robins' treatise. 62-53 teachers' manual. 21 i position. It is not a question of calculation, it is a ^piestion of tlie right use of language. Nothing must be 'lone that the child does not understand as he does it. It IS not good training to say to a child, M)o or learn this Uint or the other thing. You will understand it later' He may understand it better later, hut even as he does it or learns it, he should have vvitliin himself a sufHcient reason for the doing or learning. Tiierefore, not so much in the interest of the i^upils' arithmetical instruction and training, as of his general mental culture, let him attempt the enunciation of the rule, and let him arcme the correctness and convenience of the rule. Let this'be done at first in answer to questions given by the teacher ' W hat do you do first in adding several numbers (or in subtracting, multiplying or dividing one number from or l)y another)? Wliat then ? And then r The answeivs will very likely be imperfect. If erroneous, the error will probably be an error cf omission, and the imperfection or omission may be l,rought to light by further .luestionin.r or, better still, by the teacher's attempting to do exactly what the pupil has said shouM be first done. So step by step all the elements of the rrde are i)rominentlv and distinctly presented." XIX. A. Drill in (nl,/i,i(/ h,/ rqnal hirremcnfs must be given ut frecpienl intervals. (For method see p. 7, I. D.) XIX. ('. To the first number add the uniU of the s.voiul number, and to the result add the tain of the second number: o/., (]:.4-:;o. (J5 + (; = 71 : 71 -f ;10= 101. Tl... fuUowiug n.eth.Hl ,uay lu- „s,m1, if preferre.l AcM the tens frst : then the units. Combine the two, ..,,., O.'. f30. ten, ,3 Wn, ^ 9 tens or 00 ; .-> t «i . ! 1 : 90 I 11 = 101. In acKl.ng d^^lkrs and cents (involving fonr figures) pupils may be Hllowea at u'st trial to recor.l th. su.n of th. oents before adclin/the cioiiars. Ihe exercise n.ay b. repeated later without such indulgence. 22 CiiADED AIUTHMETIC. 64-61 XIX. I) and E. The suggestions, including the notes niade tor the teaching of written addition on p. 14 still hole good. Kead and follow them. Extennust be .vorked f.y a.i,i.y m ,o lU.lf nine Ls. They n^ "^ proved by nniltiplication. I'Ky niaj ho NOTK.-A figure in the ..venth place will son.etin.es oceur in the place. Gn e the nan.e milUon. to a figure in the seventh place but do not teach the inillions' period. ^ ' XIX. F. All problems ought to be analysed orallv bv asking (a) what is given, (/>) what is required, (.) xW.^t steps and methods must be employed. For method of analysing see \\ 1 1), XVL ^ The dcnomwatUm must be marked in all answers involving concrete quantities. XX. A and K JJaise the smaller number so as to equal the greater. This may conveniently be done in two steps- (1) raise the >n>ifs of ihe smaller to tlie units of the larger; (2) raise the tens. fO;79-l9. 19 requires 60 to make 79. (Second step only the units being already the sanie.) * ^ «« :]o It '''"'"''■'' * '" '""''' •'^^' ■""•' *^ "''''' t" •"•■^ke 78. .1„.. 44. 8b 29. 29 reqiures 7 to n.ake .%. a..,l 50 more to make 86. Aus 58 1 he fi,st step gives the ,n,its ; the second the ten.. XX. ('. Draw attention to the dillcirnt ways ,,f y-onhnt/ a (,uestion in subtraction. The method of workin- remains the same. " XXI. ('. I)(, not forget the oral analvsis of tlio ])roblems. XXII. This e.xerci.se tukes up examples f„r the solution Ol which both >»Ui/iot, iuul snh/nn.fi,,.. .,,... n .nir-. 1 feee that eadi step is fully understood. The examples 54-61 le notes, 14, still methods addition. y may Ik; iir in the the sixth e, but do 'clllv 1)V ') what hod of Liiswers equal ► steps: of the Illy, the Ans. 44. ins. 58. ys of -H'k'in^' ( the lution iiirnd. nples 61-66 teachers' manual. 23 increase in difficulty so that some of those given under 1> and E may require explanation. Troblenis must be analysed orally. The short method of addimr by multi- plication must not be permitted. XXII. 15. Brackets, inclosing numbers, indicate that whatever is contained within thelmickets is to be treated as a single number. Therefore, if within brackets there are several muubers connecte.l by Mgns, the operations denoted by these signs niu.t be performed before any operation denoted by a sign outside tlie brackets. E.<,.] 31 - (8 + 6). The numbers 8 + G inside the l)rackets are to be treated lis a single number, and our first step is to fnid their sum. 31 - (8 + 6^ -SI - 14 (hrst step) = 17 (second step). This example should be read rmm 31 fnke the sum of 8 aiuf 6." XXIII. The millions period is to be taught like the thousands' period. Ft occupies the sercnfh^ eighth and niNth places, consisting respectively of millions (units of millions), ti'nsof ui ill ions and hiimlreds of millions. Dictate numbers as follows: 801 million, GOT thousand, 7G0 ; 10 million and 1 ; etc. XXIV. The tables of elvrens and tnrhrs are best learned from tens. Eleven is 1 ten and 1 : i' elevens are 2 tens and 2 or 22 ; :! elevens are 3 tens and ;! or 3.*!, etc. Twelve is 1 ten and 2 : 2 twelves are 2 tens and 4 or 24 ; M twelves are ."» tens ami 6 or .".0, etc. 11 X 11 and 11 X 12 must be memorised and should receive special attention on account of their difficultv. XXV. A. In multiplying at sight I>cgin with the t>>is. K.ij.. 40 X = 4 tens X t) = ;;« tens = ;!|J0. «4x7: 7 times .SO = :)GO; 7 times 4 = 28: :>G0 and "8 = 588. I ^'^ GRADED AKITIIMETIC. 67-68 XXV. K 22. Division by factors. Row to find the remainder when one exists, e.g., 670408-^85 (5 x 7). 5 )_070408 ^ ) ^''-^Q'^l — o units over"! ^ 1Q1-.1 "^ y?, ['^ fives + :» units = 18rem. 1 Vi 1 1)4 — . ) fives ov er ! The first division hy 5 distributes the units into groups of fire each Mith 3 units over. When we agcain divide the^e groups of five l)y 7 we get .S over ; but these are 3 M-% not 3 units. The true remainder is then found by adding the two. From this ,lerive the rule, " yfumply the sfcond remainder by the^p'rst divi.or and add thejrst remainder." XXVI. A. Multiplyi'ig a nunil)er by 10 is to raise it from units up to tens, e.g., 2:1 x 10 L 2:5 tens = 2:J0. Examining the result we find that the fi,crures composing the number remain unaltered, but each has been moved to the next higher place, and a cipher has been inserted in tlie units' place. Hence a number is multiplied by 10 by annexing a cipher. For the same reason annexing tu^o ciphers increases the value of a number 100 times (raising each digit two places), and tiierefore multiplies the number by 1 00. To multiply a number by 20, nniltiply by 10 and 2, i.e., annex a cipher and multiply by 2 ; to^nultiplv bv Voo! multiply by 100 and 7, i.e., annex two ciphers and mnltiplv bv 7, etc. In the same way and for the same reason, a number is (Uvidvil by 10 by moviug (^ach figure to the next /oicer ].lace: or by 100 by moving each figure fmj places down. The last figure or ligures of the number, which iire thus cut otr, become the remainder. XXVL r.. \'\. Vou are now ready to explain multipii- cation where the multiplier consists of more than one i 68-69 TEACHERS MANUAL. 25 700, and figure. Suppose the product of 571 and 23 is required. The multiplier 23 = 20 (2 tens) 4- 3, and the product is found by multiplying 571 first by the 3 units and then by the 2 tens. The partial products are then added. 571 571 23 23 1713 no = 571 X 3 or 1713 11420 = 571 X 20 more briefly 1142 13133 = 571 X 23 33133 The cipher at the right of the second partial product does not affect the result of the addition, and may be omitted (as in shorter method), if care is taken in writing down the partial results, so that the ^first ^figure of each shall be directly under that figure of the multiplier wiiich was used to obtain this result. Note 1. — Do not wait for a complete understanding of full method before giving practice by shorter method. NoTK 2. — It is essential to good work in multiplication that figures should be large and plain, and columns and lines even an product of the signijicant fynreii, and to the result annex as mautj ciphers as are on the right of both /actors. (live tost (questions like the (oWowiu^^: " How mang ciphers will there he in the product of 070 and HOO i " The above rule is not to be memoriseil. If a multi])lier, e.g., 4007, contains cipiicrs, not to tlie riglit of, but between, its significant Hgures, two points 26 GRADED AltlTIIMRTrC. 69-73 2341 2007 16387 4698387 need careful attention. (1) The products that correspond to these ciphers will consist qt ciphers and need not be written. (2) Tho first figure of each partial result >nust he written under the figure used as a multiplier. XXVII. C. For method see Manual, XV. p. 18. XXVII. 1). For method see Manual, XVI. p. 19. XXVIII. Long Division. Tlie difficulty of teaching, long division will be nu.ch lessened b, a cLful 3 f of the exercises. The book exercises have been, tl^e o ^' arefull, graded, and teachers are advised not to in ' h or nicrease then, without due precaution, lest son^ te -ay be unwittingly on.itted or prematurely introduced' A. Short Method. Examples in A, 1 to 17. •H is not contained in the first figure of the dividend. ° _ ol is contained in 40 (the first two hgnres) I time with 9 over, (or if preferred, 3 is contained in 4', takincr — the first figure only of the divisor). ° J\nte the 1 over the second figure of the dividend (ine fiist figure taken). Annex 3 the next figure „f the dividend, to the —der 9. 31 . eontamed in 93 3 tunes wUh n: Write the 3 over the third figure of the divi.lcnd. Note l.-I'upib ...ust be n.a.le t<. olKserve that there ire fan. •/ , in each partial npp,.aM(>n ■ i\\ /)!>■/ .> '"^V^"*-'^ '"« ./o«/- .s^^w 13 31 ) 403 31 93 9;i 73 teachers' manual. 27 Note 2.— If, when we multiply, the product is greater than the partial dividend, the quotient figure is too large and must be diminished. Note 3.— If, wiien we subtract, the remainder is equal to, or greater tlian, the divisor, the quotient figure is too small and must be increased. Note 4.— The method of writing the figures of the quotient above tlie correspondinij figures of the dividend is recommended for two reasons ; (a) it closely resembles the method of short division, the quotient figures being now written immediately above instead of below; {!>) it prevents tlie omission of figures (especially O'a) from the quotient, for (the place of the first quotient figure being determined) there will be a quotient figure over each succeeding figure of the dividend. C. Extended Method. 1 ten +.•) units = 1.'^. Aus. 40:^-f-r51 403 = 40 tens + ;"! units. — ol ) 40 tens-f o units 9 tens = 90 units As 31 is not contained in the hun- dreds, which are 4, we arrange the dividend into 40 tens and 3 units. 31 is contained in 40 (/e«.s) 1 {feu) witli 9 {fi'us) over. Write the 1 (en abo\e tiu; teH.< of the dividend. Add the 3 units of the dividend to the 90 (9 teny) of the remainder. 31 is contained in 93 (units) 3 (units) with no remainder. Write 93 93 Quotient = 1 ten f 3 the 3 units al)ove the units of the dividend, units = 13. Note.— Teach tlie short method first and give practice upon it. Introduce the extended method later. XXVIII. A. Examples 18 to 51 -'494 ^ 2!) 29 )~2494 How to flad the quotient figures. AVlien the seeund figure of the divisor is greater than a, use tlie iirst figure only as ii trial divisor, ^(j Ans. 232 174 174 and inerea.se it by 1. Also increase the trial dividend bv 28 fiUAUEU AlilTHMETjc. 74-80 L ?"'; ""'?' "' '"^'"'S ^"'^ olteu is 20 contained in omtled. The.r omission will be best prevented by elose attention to Note 4, p. 27. '' ThfS;^' '""f°" '' "'' '■"•''■'^ °'' '""'"plication. The forner separates a number into equal parts; the corresponds to the product, and the Jifisor and ««<,<,Vv< to tho^.««„/^.aud ,nuU^,can,. Explain and 'ilil: XXX. A. See instructions, p. 18. XV. XXX. B. See instructions, p. 19. XVI. book, p. 41). Now teach secomk, using the small dial and second-hand of the clock-face. Show' how IS t lime of day. explaining a..vi., ,,., and imi. and the ecmon. w,th their month.,. Teach the immber of ■ays 1,1 each month. Kebniary 28 or 29 ; those witl .-"o others in. ^^^^' '^■' " ''''"«"''^«'-' «'c; «« the Oapacity.-Jteview previous work p. 42 pupils' book and see suggestions p. 1.S of manual. Measurements. — J'eview previous work p. 4:; pupils' book, and see suggestions p. 18 of tbis manual. 81^89 teachers' manual. 29 Weight. — For this lesson you need scales and the following weights : 1 lb., 8 oz. Q lb.), 4 oz. (J lb.), 1 oz. Introduce by a talk about these weiglits. Roman Notation. — Tlie following ket/ to Eoman nota- tion must be explained : — ^ 1. When letters representing equal values are placed side by side, their vtilues are to be added, e.g., XX. 2. AVhen a smaller number {i.e., a letter representing that number) is placed on the right of a larger number, it must be added to the larger, e.g., VI, LX, xV. 3. When a smaller mimljer is placed at the left of a larger, it must l)e subtracted from the larger, e.g., IV, XL. Mental Problems, p. 84.— An oral analysis of these problems is rc(|uirc(l The answer may be recorded by all pupils and the oral analysis given by one. Review Examples.— For suggestions see p. 19, XVI. of manual. ANSWERS. V^ HI. 1'ages i 31, 32, 33. ■ A. , 1 . 10^6. 2. 77.') . 3. 889. 4. 898. 6. 965. 6. 1 7 . 798. 8. 889. 9. 797. lO. 708. 11. 789. 12. 988. Hj 1. 781. 2. 1117. 3. 1331. 4. 1714. 5. 2058. 6. 2301 . H 7. 2.j68. 8. 29.39. 9. 2043. 10. 2143. 11. 2080. 12. 2094. H 13. 24.%. 14. 2720. 15. 2!0»). 16. 2a:).-) . 17. 2104. 18. H 19. 1872. 20. 22S-). 21. 2082. 22. 2791 . 23. 2359. 24. 2740. H C. 1. :i'2SC}. 2 3012. 3. .3081. 4. 2474. 5. 2096. 6. 2236. H 7. 2;wi . 8 170,-). 9. 37!;"). 10. .3.-)43. 11. .3.J40. 12. 2197. ■ 13. 2197. 14. 23)1. 15. 2204. 16. 170:). 17. 3.393. 18. 3347. I D. 1. 9i:{. 2. 910. 3. .->o.-). 4. 807. 5. 805. 6. 1055. 1 7. iG9:i. 8. 1747. 9. 1092. lO. 1 •")78 . 11. 1397. 12. 1025. ■ E. 1. 2.'{0. 2. 1207. 3. .378. 4. 9r)8. 5. 965. 6. 768. ■ 7. (58.3. 8 331 . 9. Oi>7. 10. 1039. 11. 001. 12. 231. ■ 30 F. 1 . 825. 7. 953. a 1. 711. 7. 18S7 GKADED AKITIIMETIO. 2.658. 3.678. 4.908. 6.864. 6 098 8. 1498. 9. 1604. 10. 341. 11. 1723. 12. 1681, 2. 103. 8. 1442. 3.210. 4.407. 5.408. 6.1275. 9. 1564. 10. 2168. 11.2650. IX. Pages 34, 35, 36. B. C. I). 1 . .3243. 7. 5022. 1 . 2728. 7. 1825. 13. 2462. 19. 1881. 1. 4623. 7, 3754. 13. 2281. 10. 64.32. 1. 6737. 7. 9.397. E. 1. ].-,46. 7. 6382. 13. ,3212. F. G. 1 . 7560. 7. 4217. 13. 11. 1. 2.36. 7. 4299. 2. .32.32. 3. .3244. 8.5024. 9.6257. 2.2710. 3.2780. 8. .3085. 9. 1306. 14. 1250. 15. 643. 20. 5760. 21. 5365. 2.2780. 3.4790. 8. ;r)53. 9. 2432. 14. 2288. 15. 3258. 20. 3653. 21. 4499. 2.4927. 3.6834. 8. 85.39. 9. 944. 2. 2717. 8. 2861. 3. 4429. 9. 6896. 14. 7179. 15. 5026. 2. 4,j87 8. 5170. 14. 228. 2. 253. 8. 431. 3 . 4887 9. 3J31. 15. 813. 3. 36. 9. 81. 4. .3424. 10. l.'JOl. 4. 2419. 10. 4093. 16. 6.3.32. 22. .3016. 4, 1147. 10. 7.37. 16. 3154. 22. 6,3.35. 4. 2586. 10. 4636. 4. 4622. 10. .3552. 16. 8.383. 4. 5841 . 10. .3:)55. 16. 408. 4. 27. 10. 4135. 5. .3,>25. 11. 2001. 5. 1449. 11. HI. 17. .3.145. 23. 61,j4. 5. 4.372. 1 1. i;]S. 17. 2184. 23. 4930. 5. 5462. 1 1 . 3387 . 5. 4797. 1 1. 7744. 17. 8778. 5. 4763. 11. 907. 17. 124. 6. .3213. 12. 2170. 6. 2.324. 12. 4.')3. 18. .3,353. 24. 1844. 6. 3144. 12. 2901. 18. 2017. 24. 7800. 6. 5027. 12. 5493. 6. .3726. 12. .5879. 18. 619. 6. 8677. 12. 4079. 18. 233. 6. 461. 6. 578. A. 1 7 13, 19, 25. 31. 37. 110. 696. 371. 291, 302. 510. 2. 52. 8. .340. 14. 768. 20. 702. 26. l.")9. 33. 504. 38. 469. X. 1»A(;k.s 37, 38. 171 3. 123. 9. 192. 15. 4.10. 2 1 . 688 . 27. 32!). 33. 1.35. 39. 616. 4 10 1 6. 666 22. 23. 3^. .1-1) 470. 316. 5. 208. 1 1. .329. 17. 6.18. 23. .3.32. 29. 280. 35. 276. 6. .300. 12. .371. 18. 4.30. 24. 861. 30. 172. 36. 2[):}. 40. 891 ANSWERS. 31 .3144. 2901. 2017. 7S99. 5027 . 549;}. B. 1 7 13 19, C. ]). 1 5 10 942. 2 4(556. 8, 4550. 14. 1290. 20. 1 . 13446 20109 20392 33615 40338 9144, J 1 7806, 12374, . 976. 2 . !i?776. 6 . 2456 lbs. 1146. 3.2704. 4.2860. 5.3822. 6. 58S7 6S49. 9. 2748. 10. 3220. 1 1. 14.35. 12. 3822 6204. 15.3411. 16.412. 17. 14(' 18.3744. 507G. 21 . 4956. 22. 6928. 23. 6006. 24. 5157. 6 6 7 9152 13728 18304 22SS0 27456 26663 30472 34281 4. 53424 61056 6S6SS 3716, 18288, 22860, 27432, 32004, 30576, 41143. 26709, 35612, 44515, 53418, 62321, 7l2i4, 80127. 18561, 24748, 30935, 37122, 43309, 49496, 55683. . 679. 3. 11.34. 4. 445, 534, 623, 712, 801. . 978 1U. 7. 243 miles. 8.784 11.8. 9.1GShr.s. 1 1 . 8600 . 12. §475 . 1 3 . 522 cents . B. C. 1). 1, 7. 13. 19. 25. 31. 37. 43. 1. 7. 13. 19, 25, 1, 2. 3. 4. 9. 214. 86. 119. 155-.3. . GO-3. . 77-6. . 7-i. , 52-1. 839. 839. 837. 579. 836. 2. 1.72. 8.174-1. 14. 103. XI. P.uJKs 39, 40. 4. 488. 3.212. 9.2.39. 10.36. IS.IOS. 16.118. 20. U;)-4. 21.145-5. 22.91-6. 23.91-1. 27.147. 23.201-2 32. 90-4. S3. 95-,-i. 34. 124,3. 35. M7-1 33. 4S-5. 39.57. 40.41. 41.. 39. 44.21-4. 45.209. 46.269-1.47.5.3. 5. 2:4. 11. Gl. 17.2:.0-1. 23. 11. -,-7. 6. 12. 18. 24. 30. 33, 42. 48, l.'i.3. 72. 2.39-2. 78-3. 3;31-1. 135,3. 17-4. 187-1. 2.789. 3.345. 4.7.38. 5.584. 6.043. 8.389. 9.738. 10.647. 11.583. 12.7.39. 1 4 , 4S5 15. 5.37 . 1 6. 803. 1 7. 749. 1 3. 8,37. 20.496. 21.. 371. 22.695. 23.7.38. 24. .597. 26.948. 27.. 379. 28.9,17. 29.6.-7. 30.598. 1260, 840, 630, 504, 420, 360, 315, 280. 1980, 1.320, 990, 792, OiiO, 5()5-5, 495, 440. 2.376, 1584, 1188, 9,-,0-2, 792, 678-6, 594, 528. 385. 5.. 324. 6,81plmu.s. 7 . 894 times. 8 . 106 .slates. 31 childiL'ii. 10. 97 y>ls. 1 1 . 126 qts. 12. 1123. 13. 873. XVI. rAOi;.s 46, 47. A. 1. 7.36. 2. 219. 3. 47 counters. 4. 12627 people 5 2.30 fish. B. 1.1313. 2.899. 3.8SX)y(l.s. 4. 54 cows. 5 . .3600 time.s. 32 GRADED ARITHMETIC. C. D. E. F. G. 1 1 1 1 1 881. 2. 11 marbles. 3.53. 4.725-5. 5 . 2920 days 151 days. 2. 137 children. 3. 141. 4, 41909. 6. 107 cents. 850. 2.88slieop. 3.574pts. 4.133. 6.2384 299 pupils. 2.99. 3.181. 4.6867. 5.90 2147 bushels. 2.343 kittens. 3. 1188 panes. 4.52yds. 5.317 eggs XIX. PAGE.S 54, 55, 56. D. 1 . 2299S. 6. 122904. 9. •>44372. 13. 1124838. 17. 401968. E. 1 . 202821 . 5. 2009637. 9. 2498457. 13. 8883. 17. 20797. F. 1 . 71 panes. 5. 1984 pages. P . 78 strokes . 2. 19983. 6. 267974. 10. 146506. 14. 1070465. 18. 323049. 2. 154810. 6. 1,333435. 10. 931108. 14. 68331. 18. 8704. 2 . 5665 pupils . 6. 1008 pens. 10. 41015. 3. 28310. 7. 390370. 1 1 . 409448. 15. 2250524 19. 5133357. 3. 3140069. 7. 1034683. 1 1 . 72084. 15. 62378. 19. 4956. 4. 242724. 8. 262412. 12. 205782. 16. 271.305. 20. 1469297. 4. 779053. 8. 1126047. 12. 391385, 16. 1406361. 20. 578312. 3 . 2 1 92 apples . 4 . 7642 trees . 7. 2.309 sheep. 8.365d;ys. 11. 1773 potatoes. 12. 171979. XXI. Pages 59, 60. B. L 1 . 61.333. 2 . 66641. 3. 21107. 6 . 1618. 7 13031 . 8. 15708. 11 . 161376 12 4 11.389. 13. 492064. 16 . 237707 17 168299. 18. 162798. .. 1 39672. 2 730926. 3. 80857. 6 92514. 7. 96444. 8. 662733. 11. 877507. 12. 291111. 13. 111109. 16. 20000. 17. 6778. 18. 1. 21. 730926. 22. 78.552 . 23. 80857. 26 904(). 27. 6.3612. 28. 594044. 1. 07805. 2 103875. 3. .$242.19 6. 42 yrs . 7. 1815. 8. 969370. 11. 749 yrs. 12 1. 152367 males . 1 3 5. 47693. 1 5 . 44769 grain 16. 13269 ft. 4. 2165. 9. 17368. 10. 321.S!. 14. 370504. 15. o2U:iS. 19. 11999. 20. 253676. 4. 170595. 5. 599071. 9. 54322. lO. 779044. 14. 260679. 15. 299999. 19. 39672. 20. 699731. 24. 16.3896. 25. 22468. 29. 212503. 30. 91002. 4. $126.17. 6. 42.3570. 9. 18,394. 10. .ii319.'595. 586 trees. 14. 339752. ANSWEKS. 33 XXII. Pages 62,63. D. 1.24r)6. 2.6H98. 3. 096. 4.1436. 6.201. 6.676. 7.10388. 8.426. 9.90408. 10.92. 11.1. 12.28177. 13.605483. 14.4098. E. 1.988. 2.177. 3.200. 4.10. 6 19013. 6.827. 7.65454. 8.710. 9.2397. 10.5329. 11.^5072.43. 1 2 . 7065 . 13. 985 . 1 4 . 498 pages . 15. $4405 . 1 6 . 382 pupils . 1 7 . 44 yds . 1 8 . M ary 786, John 1069 . XXIV. Page 65. 3. 664279, 724668 ; 779845, 850740 ; 701899, 765708 ; 963952, 1051584- 9999099, 10908108. 4.66555-2, 61008-11; 1089905-4, 999079-11; 1000999, 917582-5- 6910 J 00-1, 6334258-5 ; 9829199-9, 9010099-10. B. 1. 5. 9. 13. 17. 20 24. 27. 28 29 30. 31. XXV. Pack 67. 12510. 2.12892. 3.224608. 4.162315. 206952. 6.368109. 7.438480. 8.2271528. 6364248 . 1 C 3905208 . 11.1 026480 . 12. 7827280 . 854904 . 14. 2823408 . 15. 846530 . 56 . 1 6 . S 1 75 1 79 . 24 . $200408 . 88 . 18. $201 150 . 60 . 19. $145952 . 96 . $571069.80. 21.8275096.88. 22.4419-4. 23.18119-8. 19154-18. 25. $3.24. 26. $5.63. 244779-12, 156659-1, 111899-11,48351-45,69937-4. 547597-9, 456331-6, 342248-12, 586711-10, 228165-24. 44018-21, 29956-71, 25677-35, 19608-23, 43138-3. 272088-12, 105812 12, 57715-75, 90696-12, 52906-12. 200355-38, 127499-14, 103888-20, 58437-20. XXVI. PAGE.S 68, 69. 14.483. 15.1426. 16.25228. 17.25116. B. 13. 182. 18. 10952 C. 1 . 1760. 6. 6084. 11. 68026. 12 16. 199617. 20. 82052571. 24. 6754755. 28. 76661002. 19. 27306. 2. 2625. 3. 7. 14450. 8. 34466. 13. 17. 7278538 21. 760852. 20 169. 15215. 1 10768, 18 22 114885. 21. 145314. 4.625. 5.2116. 9. 11414. 10. 4250. 14. 254736. 15. 486726. 3592212. 19. 3131672. 9;}9015 25. 68604840. 26. 54949721, 29. 64240 198. 30. 16978476. 23 27 2408217. 9320556. 34 GRADED AIUTHMETIC. D. 17. 372000. 18. 21. 207.-)9<)-iC00.22. 25. 800000. 29. 008 i GOO. 33. 89044708. 37. 407 11)0047. 41 . ;{07'2'J9800. 45. 9104088. 26. 30. 34. 38. 42. 46. 118400. 19. 300.3200. 23. 950()0000. 27. 1594700. 31. 87701204. 35. 301277500.39. 3897 108S 1.43. 15087224. 47. 373.520. 20. 5544700. 24. 3800040. 28. 17559.500. 32. 278097188 36. 421001.350.40. 4029.')9,52. 44. 19704888. 48. 10940000. 40593000. 8271900. 70783.300. 190151117. 280.507995. 70370700. 1057904889. XXVII. Page 72. 2700 teet . 6 . ."j^ri . 15 . 7 . Soii", IX 1. 82280 yds. 2 . .300 miks. 3 317 t 6 9. 12. 15. / tons. 8. 27 4. SM lO. ,000,000 seeds. A. 1 9 14 19 24 30 35 40 45 50 B. 1 7 13 18. 23. 28. C. 1. 5. 9. 13. 17. 21. 26. 29. 33. 37. ;:!OJ,000,000 eggs. lO. 149170 people. 1 1 , 122896 fish 174 mail.!. -.s. 13. 2,895,588 oiuioes. 14. 817200 pa-^es. 402000 bricks. XXVITI. PA(ih>; 73, 74. . 13. 2. 15. 3. 10. 4. 21. 5. 10. 6. 23. 7. 22. 8. 1.3. . 47 . 10. 23 11. 11. 341 22. 1 2 . l;;0-21 . 13. .39 . 192-1. 15. 222-21. 16. 545-10. 17. 50-11. 18. 86 .98. 20. 175-28. 21.84 5. 22.127-16. 23.259-11. . 531-7. 25. 198-12. 26. 27. 27. 40. 28. .357. 29.6.347, .61. 31.494-13. 32.90 29. 33.53-05. 34.93-58. . .3217-12. 36.2773-27. 37. 1];:91. 38. 1443 7. 39. 1489-.33. . 428-.']4. 41.. 3.3.38. 42.2010 10. 43.42312. 44.5447. .527 31. 46. 4;:9. 47.902. 48.472. 49.615. 1231-20. 51 . 8.37-20. 205. 2. 3207 12. 3. 104-3. 4. 1055-5. 5. 60-18. 6 .3054. 403. 8.807-5. 9.704-7. 10. .504. 11.400 12 70. 500-21. 14.8000-8. 15..%\50. 16.10070. 17.6410.30. .3280-15. 19.1608.20.1020. 21 .slO.OO. 22..><20.04. $!4.70. 24.8.30.05. 25..S56.2). 26.f<70.03. 27..•^70.U3. 30000-56. 29. 10010-6. 30. 2010-23. .35-17. 2.. 3.59 78. 3.1099 1'J. 4.9013.59. 861-64. 6.49-455. 7.113.588. 8.96.385. .39:!8-373. 10. 57 64. 1 1 . -181 S5. 12.4512-199. 813-16. 14. 508.30 1 -.32 .15,1 7827 1 1 -401 . 16. 08.3247 - 1 00 . 245-116. 18. 1-33- 1.37. 19.26.3SS. 591-27. 22.697 35. 23 720 306. 1010 490. 26.19 948. 27.3751.372. 2-11.30. 30. 179;il85. 31.7 3t!l9. 3.32 1676. 34. 6.3 2617. 35. 429.3.380. 413 1.366. 38. 114 728. 20. IS.-)0 83. 24. 99S 847. 28. 5400 5. 32. 31 1640. 36. 716-2.387, I AXSWEl^S. 35 v>. 1 6 9. 12 16 1 2 3 4 5 6 7 8 1 3 1 1 1 1, 1 . 1. 1. 3. 1. 6. XXX, P.v(iE 77. .'U yrs. 28 weeks over. 2 . 75 dresses . 3 . $ryO . 4 . 27 »Iws 120 1.OX..S 6.2SNc>ggs. 7.7;)lmns. 8. 17 ears. 106daily, 742 v.uddy. 10. 40 loius. 11. 7010 boxes. .^o.-U . 13. (J.S i ,S2 acres, .$28 oNir . 14. 9080 . 16. S804 7:^298 .lo/cn. 17. 1908. lii;viK\v KV..VMIH.K.S. Pa,;ks 80, 87, 88, . 5023 soldiers. 2. lOOO.S plums . .'i9 payiiicnl.s, .*] yrs. A mos. 4. U1816. 2,-),-..*} . 2 . 880 counters . 3.8 vvks . days . 4 ] 95049 . ^40 apples. 2.770. 3.201-8. 4 40»<)9 ^.'{149. 2.77054. 3.4470. 4. 150 strokes * •-'45 days. 2.10119 men. 3. 85. 4. 30 marbles. 2.' "7(10 . 2 . .S.3 . NO . 3 . S227 . 4 . .$ 1 08 . ;-»2, 2. 174;iO. 3. 80 counters eaeh, 10 over. 4. 4840 trees. 40 apples eaeli, 1 1 ai)ples over. 2. 40 l)oys, 1 1 apples over. 2(l9-.-)84. 4 12.-,9. .*2:!4. 2. .>!1 , 00. 4. 3 ft. ill. 6. 300 laiiiutes $000, $900, 5;2100. H few xaseful Boohs FOR TEACHERS. 50 00 75 BROOKS' NORMAL METHOD OF TEACHING $1 BALDWIN'S ART OF SCHOOL MANAGEMENT FITCH'S LECTURES ON TEACHING i GL ADMAN'S SCHOOL METHOD, net to Teachers HACKVVOOD'S NOTES OF LESSONS ON MORAL SUBJECTS 70 SINCLAIR'S FIRST YEAR AT SCHOOI 50 SPENCER'S EDUCATION 50 THE STUDENT'S FRCEBEL.— Pr.rt I : Theory of Education . 90 Part II: Practice of " .90 OBJECT LESSONS FROM NATURE.— Part I 70 Part II 70 These are two capital books. Each part is complete in itself, and may be purchased separately. LONGMANS' OBJECT LESSONS i .00 Hints on preparing and givinjj; them. CUSACK'S MODEL DRAWING 1.25 For Teachers and Students of Public and Elementary Schools. HOW TO SHADE FROM MODELS i.co A Practical Manual. THE NEWCALISTIHCNICS 1.25 A Manual of Health and Beauty. METHODS OF MIND TRAINING, CONCENTRATED ATTENTION AND MEMORY, by Cathkrink Aikin. r .00 One of the most helpful and fascinating books for Teachers ever published. F. E. GRAFTON & SONS, Publishers, Booksellers and Stationers, 250 ST. JAMES STREET, niontpeal.