M«fflrfe4 - 
 
 
THE GIFT OF 
 
 M&b&G. ^ijijaa.. 
 
olin.anx 
 
 3 1924 031 296 910 
 
Cornell University 
 Library 
 
 The original of this book is in 
 the Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924031296910 
 
Quincy Course 
 
 —IN— 
 
 Arithmetic. 
 
 BY 
 COL. FRANCIS W. JARKER, 
 
 Cook Co. (III.) Normal School. 
 
 CHICAGO: 
 A. FLANAGAN, Pubi,ish«r. 
 
A -4 5*f £ 
 
 <h^r3r-g-l-g 
 
 I<!hmJ 
 
 CORNEL 
 UNIVERSITY 
 V LIBRARY A 
 
 ^ ,1 i li'i'I ■ ■! I'tjTTi l .'Vt'^ 
 
 Entered according to Act of Congress, in the year 1879, 
 
 By Francis W. Parker, 
 
 In the office of the Librarian of Congress, at Washington. 
 
SUGGESTIONS AND DIRECTIONS. 
 
 1. A knowledge of numbers and their relations is gained' in 
 the same way as a knowledge of color, form, size, or weight. The 
 ideas of which red, square, large, and six are the signs, are taken 
 (abstracted) from objects. What these words (red, etc.) are to the 
 mind depends entirely upon the products of sense-perception, and 
 no amount of study of signs, alone, will ever bring the slightest 
 knowledge of the things they represent. Signs may be learned with- 
 out a definite association with the ideas that they should recall ; they 
 are learned, in fact, and the learning of figures, and not numbers, — 
 signs, not things, — is the fundamental mistake in teaching arith- 
 metic. _ 
 
 2. a number can be separated into equal or unequal 
 parts. Equal or unequal numbers may be combined. Noth- 
 ing MORE CAN BE DONE WITH A NUMBER OR NUMBERS. 
 
 To illustrate: Present a number of marks, thus, Mil — the sign 
 is 4. Present the same objects thus, — II II The relations seen may 
 be expressed : " In four I see two and two," " I see two twos," 
 " Two marks and two marks are four marks," or " Four marks less 
 two marks are two marks." The two relations of separation and com- 
 bination are seen together; they are reciprocal ; one suggests the 
 other; one cannot be thought without the other is known, either con- 
 sciously or unconsciously. Therefore the teaching of one relation is 
 greatly aided by the teaching of the other at the same time. 
 
 3. The combination of numbers is Addition ; of equal num- 
 bers is Multiplication. The. separation of a number is Subtrac- 
 tion ; the separation of a number into equal parts is Division ; the 
 separation of a unit into equal parts gives fractional units. All 
 .that belongs to pure arithmetic has for its foundation, root, and 
 source, the simple putting together and putting apart numbers of 
 things. 
 
 4. Plants' and numbers should be taught by precisely the 
 same method. First, the whole plant, or number, is observed ; then 
 the parts on the plant or number, or severed from the whole ; afterward 
 the plant or number is to be compared with all o.ther known plants or 
 numbers. (3) 
 
Grube simply extended the method of object-teaching to the teach- 
 ing of numbers of objects. 
 
 5. The proper way of teaching language in number pre- 
 sents A COMPLETE and logical illustration of the manner 
 in which all language should be taught. 
 
 Ideas are gained from objects. When an idea becomes by sense- 
 perception a clear idea, it demands expression. From repeatedly 
 seeing and handling three blocks, three marbles, three sticks, etc., 
 there comes into the mind the idea of which three, 3, or III. is the 
 sign. 
 
 The relation of two or more ideas — a thought — is seen again and 
 again. When this thought is clear, it of itself demands expression by a 
 sentence. To illustrate : supposing the ideas and signs, 3, 2, 5, to be 
 already known, then the repeated observation of these objects, III, II, 
 □ □□,□□, T T T> T T> * * *> * *> ^us arranged, awakens 
 the thought expressed by the sentence, "Three and two are five." 
 The very important law is, teach clear ideas first, then their signs ; 
 teach relations of ideas and then the sentences which express them. 
 
 We can express that only which is clear in the mind ; it is danger- 
 ous to force expression of that which is dim ; lead to clearness before 
 any attempt at expression is demanded. Ideas grow, very slowly, and 
 the most important part of a teacher's duty, after presenting the 
 proper opportunities for the growth of ideas, is to wait and watch. 
 
 Pupils should be allowed to express their thoughts in the idioms 
 which they have been using all their lives. The early introduction of 
 terms, phrases, and sentences entirely foreign to their minds, such as 
 — "divided by," "multiply," "subtract," "taking one number from 
 another," " taking one number so many times," is disastrous ; for 
 such forms "cannot, for a time, contain a child's thoughts. The child 
 should be slowly led to these expressions by permitting it at first to 
 use its own words and slowly make the new forms known by associa- 
 tion and repetition. 
 
 The teaching of each number, definition, principle, rule, process, or 
 problem, is a language, as well as an object lesson ; in other words, 
 'aach number, 'definition, etc., presents an opportunity to teach lan- 
 guage. The learning of sentences before the thoughts they express 
 are known robs the child of power to express thought. Great free- 
 dom of expression should be allowed, stiff formulas avoided, and the 
 same thought expressed in as many different ways as possible, 
 always, however, leading up to the best possible form of expression. 
 
 6. Most children know very little of number when they 
 enter school. So far as the writer has investigated this important 
 matter, a large majority of little ones of five years know less than three. 
 
The words they learn in reading are signs of ideas made clear by the 
 incessant sense-activity of five years. They can be readily associated 
 with written words ; but fhe^ase is entirely different with ideas of num- 
 ber not yet acquired, and that must be learned by the exceedingly 
 slow process of observation. The great,evil of learning the signs with- 
 out the ideas must be avoided, or the result will be ruinous. To learn 
 the ideas and two sets of signs, oral and written, at the same time, 
 seems to be also unscientific, for the multiplicity of dead forms, in- 
 stead of the ideas of number which make the forms a necessity and 
 give them life, will absorb the child's attention. Following nature's 
 order, oral signs (words) are thoroughly learned first ; therefore it is 
 thought best in this course to teach the ideas of number and associ- 
 ate the oral signs alone with them, during the first year. Then, too, 
 when the ideas and their oral signs are firmly fixed in the mind they 
 will give zest and impulse to the teaching of the new (written) forms 
 of expression. Besides, teaching the figures furnishes a capital 
 means of thoroughly reviewing the former work. 
 
 7. Cultivation of the power to reason, and the forma- 
 tion of the habit of accurate and rapid calculation, are 
 the two great motives in teaching arithmetic. 
 
 The first and greater aim is reached by leading to distinct ideas of 
 numbers and their relations ; the second by slow, graduated exercises 
 in combination and separation of numbers, adapting each step to the 
 easy grasp of the pupil, and never taking an advance step until all 
 previous ones are mastered ; and, above all, never permitting, if pos- 1 
 sidle, a pup'ii to make a mistake. 
 
 8. The first steps in number should be taken with great 
 care. After the child has been made thoroughly at home in the 
 school-room, the teacher should ascertain by careful and, repeated 
 tests just what it knows of numbers. -This examination should be 
 made under the most favorable circumstances, and extend over a 
 period of not less than two we"eks. 
 
 " Bring me so many blocks." The teacher holds up each time the 
 number. "Show me so many.'' "Touch so many." "Make so 
 many marks upon the blackboard." " Take some blocks in your 
 hand." " How many have you ? " This question is the first request 
 for a sign of number. Then may follow the directions, "bring," 
 "show," "touch," "make," three blocks, three marks, etc. "How 
 many hands have you ? arms ? legs ? feet ? noses ? eyes ? ears ? 
 
 mouths ? chins ? " " How many have I in my hand ? " " Now 
 
 liow many ? " " Clap your hands three times." " Stamp three times. 
 " Open your mouth three times." " Shut your eyes three times." 
 
These questions indicate something of the way a child's knowledge 
 of number should be tested. The exercises for a time should not be 
 continued more than three minutes. 
 
 9. When a child's knowledge of number is ascertained, 
 begin to build there. Lose no time in trying to teach children 
 what they already know, and never take an advance step until the 
 preceding ones are thoroughly mastered. 
 
 10. A great variety of objects should be used in teach- 
 ing number. For the use of one kind of objects alone leads children 
 to associate their ideas of number with them, while clear ideas are 
 more easily taken (abstracted) from a great many different kinds of 
 objects. Besides, changing from one class of objects to another sus- 
 tains the interest. It is plainly evident, too, that seeing and handling 
 many classes of objects trains the observing powers to make distinc- 
 tions and to classify things. See S. 1, Course of Study. 
 
 11. Until ten is taught, pupils should handle the objects 
 themselves. Two senses are thus trained, — sight and touch. The 
 plan of causing children to do everything for themselves is thus put 
 in practice. " Take two blocks in one hand and three in the other." 
 " Put them together." '■ How many have you." '' Two blocks and 
 
 three blocks are • ? " " Make three marks, a dot, now make 
 
 two more marks." "How many marks have you made?" "Five 
 less two are how many ?" Let pupils in answering this question take 
 five objects and separate them into two parts, one of which is two. 
 
 For tests and reviews *the teacher should handle the blocks, — 
 alone, suggesting the questions by different movements with the 
 blocks, thus, — Mill. "I see five marks" (or five marks simply), 
 III, II. "In five marks I see three marks and two marks." "Five 
 marks less two marks are three marks." Erase the dot, 1 1 1 1 1. 
 "Three marks and two marks are five marks." Show II, II, thus, 
 then put them together, llll. "Two ma,rks and two marks are four 
 marks." " Say it another way." Ans. — Two twos are four. 
 
 Marks on the blackboard should be made large and distinct, gener- 
 ally with colored crayon. 
 
 12. The teacher should thoroughly learn to make all 
 the combinations indicated by table (see page 11), and in the order 
 there laid down. Three ways of giving a lesson are suggested. 
 1st. Pupils standing around a long table plentifully supplied with 
 objects. Let each pupil have a definite number of objects, — selected 
 by the pupils themselves. (It may be well at times for one or two 
 pupils- to arrange the blocks for a lesson.) The teacher at the head 
 
of the table rapidly and skilfully directs the movements of the class. 
 2nd. Class at blackboard, —directed by the teacher, — at blackboard 
 on the opposite side of the room. The ways of using the blackboard 
 in number are innumerable. 3rd. Class in their seats with objects 
 on the desks before them. The lessons must be short, never exceeding 
 ten minutes during the first year. 
 
 13. The written signs of number are to be learned the 
 second year. The process of teaching figures is precisely the same 
 as. in teaching written words. Show a number of objects, and then 
 write (on blackboard) the sign. Write the sign and ask pupils to show 
 that number of objects. Show a number of objects, — and have pupils 
 write the sign. Class at the board. Show numbers of objects one 
 after the other, and have pupils write the signs. Show III, II, thus, 
 then change, Mill. "Write that." "Three and two are five." Teacher 
 erases and and writes +, are and writes =. '■ Now read it the same 
 way as before." Show objects as in oral teaching and have pupils 
 write the answers. 
 
 7 — 4 = 3. 
 
 " Erase answers." 
 
 7 — 4 = 
 
 6 + 2 = 8, 
 
 
 6 + 2 = 
 
 9-^-3 = 3. 
 
 
 9-^3 = 
 
 3 3's = 9. 
 
 
 3 3's = 
 
 3X3 = 9. 
 
 
 3X3 = 
 
 "Write answers rapidly." "Erase answers again." "Read col- 
 umns." " Erase second line.'' 
 
 7 — = 3. " Fill up the columns." 
 
 6 + = 8. " Erase again." 
 
 9 S- = 3. " Read." 
 
 3 's =9. " Erase first line. 
 
 3X =9. 
 
 Use 4n these exercises all the forms of stating processes to be 
 found in arithmetical calculation, the pupils learning them by seeing 
 the relations which they express. In division, for example, 6 -=-2 = 3, 
 
 216 4 
 
 2)6(3, ; - ; in multiplication 2 4's = 8, 2 X =8, 2. when these 
 
 3 8 
 
 forms are firmly fixed in the mind, give the same exercises without 
 using objects. From 10 proceed, number by number, to the develop- 
 ment of 20, using both oral and written work. For reviews give an 
 exercise like this orally. Have pupils write out answers upon slates 
 or board, in columns, without hesitation. 9 + 3 ; 6 + 4 ; 3's in 
 12 ; 12 — 6 ; \ of 8 ; 7X2. Let pupils change slates and correct, 
 
8 
 
 the teacher reading the answers. All means of training children to 
 read numbers at sight — i.e., add, subtract — should be used. Avoid 
 forming the habit of hesitating. 
 
 14. Reasoning is seeing the relation of things. A prob- 
 lem represents things as being in certain relations to each other ; the 
 solution of the problem depends primarily upon the mind's compre- 
 hension of the things represented and their relations. In other words, 
 the mind must grasp the exact conditions stated in the problem, as 
 the first step to its solution. 
 
 15. The habit of trying to perform the problems with 
 the figures alone, without attempting to understand the 
 conditions, is fatal to all progress and completely barren 
 of good results. 
 
 The proper use of objects is the first step in forming this all-im- 
 portant habit of thinking (seeing relations of things), but at every 
 step during the whole course, when pupils cannot mentally picture out 
 the things represented, recourse should be had to objects. See S. 14. 
 
 16. Pupils should be trained to illustrate every prob- 
 lem WHEN POSSIBLE BY DRAWING AND OBJECTS. It is better tO 
 
 perform one problem understandingly than a hundred that are only 
 partially, understood. Quality, not quantity. See S. 17. Explana- 
 tions by the teacher are entirely unnecessary, if the proper occasions 
 are presented for the mind to act. 
 
 It is a good plan to write a problem upon the blackboard, have a 
 pupil read it, then, with the close attention of the entire class, ascer- 
 tain, by sharp questioning, if they, understand it. Follow this by 
 asking a pupil (generally the dullest) to take the first step in per- 
 forming the problem; ask another, and another, to take the first step ; 
 when satisfactory, have the work written upon the board, and then 
 take the next step in the same way, and so on to the end. 
 
 It is also a good plan to cause the work of a problem to be written 
 out upon the board before any calculation is made, thus : — 
 
 " How many apples at 2 cts. apiece can you buy for 8 oranges at 4. 
 cts. apiece ? " 
 
 8x4 cts. 
 
 Work written : 
 
 2 cts. 
 
 17. See 12, a. Each definition, rule, process and principle, 
 
 AS IT OCCURS IN TEACHING, PRESENTS AN EXCELLENT OPPORTUNITY 
 
 for an object and language lesson. A definition is a descrip- 
 tion of a thing, or things in their relations, that can be seen. A rule 
 is a description of a piocess that can be placed before the pupil's 
 
eyes J a principle can be discovered in a process. When the pt6ptt 
 time comes (beginning of the fifth year in the course) this interesting 
 series of object and language lessons should begin and be kept up 
 through the entire course. 
 
 Subjects for Object and Language Lessons. 
 
 DEFINITIONS. RULES FOR 
 
 A Number. Addition. 
 
 Addition. Multiplication. 
 
 Multiplication, Subtraction. 
 
 Subtraction. Division. 
 
 Divison. 
 
 Notation. Principles of Notation. 
 
 Numeration. 
 
 Fractions. 
 
 These are a few of the subjects that should be used to test a 
 pupil's power of seeing and describing things and processes. 
 
 Lesson upon the Definition of a Number. 
 
 Hold up a number of objects, another and another of different 
 objects. "What is this?" "What are these?" "It is a lot of blocks." 
 "Five shells." "A number of things." "Is this a number?" "Yes." 
 If this is a number, what is a number?" (showing a number of ob- 
 jects.) " It is a lot of things together." " It is several things." "It 
 is many things:" "Do not say 'lot' or 'several.'" "I am putting 
 down a number of things upon the desk, now I take them up one by 
 one. What did I do?" "You took them up one by one." " Do not 
 
 say, 'you took them up.' Say you ' collected ' them." " What 
 
 are they together after I have collected them." "A collection." "A 
 number." "Take your slates and write what a number is." Each 
 pupil should read his definition; the best (if correct) should be taken 
 as a model and written by all. 
 
 Lesson upon the Rule for Addition. 
 
 " We will try to find the best way to add numbers upon the slate 
 or blackboard." Teacher writes numbers on different parts of the 
 board. "Can we add these figures as they are?" "No." "Yes." 
 "No, we cannot." " Try it." "Yes, we can add them, but it is not 
 the best way." " What is the best way ? " " Write them together." 
 Teacher writes the numbers together, but does not put units under 
 units, etc. " That is not the best way." " Please take your slates 
 and write the best way of writing numbers to add." Thus each step 
 in the process can be found out and written by the pupils. 
 
10 
 
 In " carrying," "borrowing," the processes of long multiplication 
 and division, "the best way" should be discovered by the pupils. 
 See Fifth Year, T. R. a. 
 
 18. After the fundamental work has been mastered, noth- 
 ing NEW CAN be introduced ; everything can and should be 
 referred directly back to the simple acts of combination 
 and separation. 
 
 Terms "factor," "principal," "interest," "base," etc., are simply 
 old, well-known friends under new names. 
 
11 
 
 ro CO" ■* lO •* O r-T t-T CO~ 00* 05 ID o" rT ^l" O* rt" 3 
 
 ■I'+'S'S+'S l , S+'o'o+^+ + C-g I 
 
 £ f ^ «f 1 «f f f" f <f oo £ •" -• f o 8 I 
 
 03^0 l.^o H oJI , S o.|-T« g 
 
 V, >-| . - th . r-T - - ,<r of „ „ He, ~ S 
 
 X .1. "♦ ^ I s i »> ri , - I ° - - II * 
 
 rf'i-l 1^4".^ l.ao^^H, .-II a 
 
 S ■*" - *>" j -r °*" *-" S ^ o « « 
 
 C -1 <" . MM .CO o **» ., ' -Jo m 
 
 
 £» * * * + X • -|- *-" ^ -I- - + 2 ; 2" 5 
 
 £i S ° ^ M -f- " .,, - O ' II 
 
 W | -*" J+H -X^N.|. + ® S " 
 
 WS l!' V .«- H « «? ? S + + rH 
 
 £ H.J, 0*<f| »• „- ° I .Ctlxj^ 
 
 C fc. "I" I W - X «r 00" ?, -*" ' m e 
 
 HI 
 
 zrMc±i*zY*?.11t i 
 
 i- - - - "3 I „- "S o « 
 
 O ° ^ °?" w - ™ T 7. ^ °*° ° X """ ... 5 ^ 
 
 S I "h Xo .^ H co , af m .1. ^ 5 
 
 O < ^ J, _ + • . M ^ ,o- «g X «f -o + o ■** I. 
 
 00 (M<mX„^.I Joooo h s . ih S 
 
 < X " rf ms ms «f - M=-g- ^S • 7 >S +J> ^ 5 
 
 -S XN oi ■= «-> CN O ^ ^> _L 
 
 7 "I" ,„- O A 'f .' LO OS -• 
 
 CS|0» 
 
 ^I-S-- ^ /\^-"°.^",-S, s l 
 
 <N 
 
 C3 "I" CO" N . <* ' . O V '=" ^ " : 
 
 " - 7 u- v- O. I O + ° „4-0 «5»l--$ 
 
 a + ^y^^" 5 X . ^ g*"2 ! 
 
 » co- =d ^ or 7 | Zj - | 
 
 ms w - co - t- co" " ro co" o 5 7 I -s „ 1. 
 
 7+4 v*^cV^oo*x°^ S "8-15 
 
 I ^ ~ cf ° «| 
 
 1-1 _r _," "^ of "^ co" *- cff co co" os" cf °? r -' 5- || 
 
 a ■i-+s , s i "^ i o i'ssi o + i'^-irs! 
 
 v w ^ CO CO ' v — ^ cv° CD q»N «*- 00 -«*» ^-^ OS w|o» iH rH H^ OS . K 
 
12 
 
 QUESTIONS SUGGESTING THE METHOD OF DE- 
 VELOPING NUMBER. 
 
 Taken largely from Horace Grant's Arithmetic for Young Children. 
 
 Operations with Objects to 4. 
 
 Let the teacher show three blocks, saying to the child, " Bring me 
 as many blocks from the table ; as many pebbles ; make as many 
 marks upon the blackboard," etc. Let the teacher take up a block, 
 saying, "This is one block." Let the child be told to put one block 
 on the table ; one pebble, one bean, etc. The teacher should then 
 speak to the pupil as follows : — 
 
 " Show me one finger. Show me one chair. How many heads 
 have you ? How many noses have you ? I have placed two blocks 
 on the table ; now I have taken them off. Try if you can put two 
 blocks on the table, two beans ; etc. Put away the two blocks, and 
 put out two shells. Take away one shell, and how many are left? 
 Take away the other shell ; how many are left ? Put out two horse- 
 chestnuts; say 'one' for every horse-chestnut that you have put out. 
 Clap your hands once. Show me two fingers ; show me two thumbs. 
 Take two steps on the floor ; go backwards one step. How many 
 mouths have you ? You have one head what else have you one of? 
 Tell me all the things you have two of. Here is one block and one 
 block ; what are one block and one block called ? Hold up one hand, 
 and also the other hand ; how many have you ? Put down one ; how 
 many are up now? Clap your hands twice or two times." ('Twice' 
 and 'two times' should be used alternately, and in such a manner that 
 the child shall perceive that they mean the same things.) 
 
 "Look at what I am doing: I have put out three pebbles, and now 
 I have put three back. Put out two sticks, now put out another ; how 
 many have you put out ? Say 'one' for each stick you have put out ; 
 how many times have you said 'one' ? Hold out three fingers. Point 
 to three children ; to three boys ; to three girls ; to three teachers. 
 Show me three legs of a chair. Strike the table once and once ; how 
 many times have you struck it? Shut your hand ; open one finger ; 
 open another finger ; how many are open ? / Open another finger ; 
 how many are open now ? Lay down one splint and one bead ; how 
 many things have you laid down? Lift up your foot three times. 
 How many plates at dinner must be put out for you and me ? Take 
 three steps forward and two steps backward. How many joints has 
 your thumb? How many joints has your forefinger? How many 
 joints has your forefinger more than your thumb ? I have put one 
 cent upon the table ; put as many beside as shall make three alto- 
 gether? Nod your head twice and once ; how many times have you 
 nodded? Stamp your foot twice and once ; jump twice and once ; 
 take two and- one steps. Make a mark on the blackboard for every 
 window in this room." (Pupils may be allowed to draw all the 
 windows.) 
 
 "Draw a square and put three birds in it. Draw an apple and put 
 three seeds in it. Make a mark for each door in the room. Draw 
 
IS 
 
 each door in the room. Draw a boy going out of each door. Make 
 three squares on the blackboard ; three boxes ; three lines, etc. 
 Clap your hands once and once and once ; how many times have you 
 clapped them. Put three slates on the table ; now put another ; three 
 slates and one slate are four slates. Put three nuts and one nut on 
 the table ; how many nuts have you placed there ? Put out two peb- 
 bles ; now two more ; how many have you put out altogether ? Say 
 ' one ' for every pebble you have put out. Make four marks' on the 
 blackboard. Bring me four blocks. How many blocks have you 
 brought: me ? How many ones ? how many twos ? how many fours ? 
 Carry one back ; howmany have you left. Shut your hand ; open 
 one finger ; open two other fingers ; how many are open ? Put down 
 as many splints as you have hands and mouths. Show me one chair 
 and one table ? How many things have you shown me ? how many 
 tables ? how many chairs ? Show four fingers ; touch four of my 
 fingers. How many are you and I ? How many are we if we count 
 John and Mary? Put on the table two blocks ; make a mark for this 
 one, now make a mark for the other ; How many marks have you 
 made ? Rub out one of the marks ; how many marks are there now ?- 
 Rub out another ; how many have- you now ? How many have you 
 rubbed out altogether ? Here are four pictures ; how, many ones do 
 you see ? 
 
 "Walk four steps ; jump four times ; hop on one foot four times ; tap 
 the table four times. Tell some little stories about four things. Who 
 can see four things in this room ? Now, John, tell me the four things 
 you see. James ; Sarah. How many things have I altogether in my 
 pocket, if I have a pencil, a knife, and a key ? Put out two blocks ; 
 now put out two more ; how many twos have you put out? Five twos 
 are called what ? Make a mark for each leg of this chair ; how many 
 marks have you made ? Put four pebbles in a row. Put four pebbles 
 in two rows ; how many are in each of these rows ? 
 
 " How many fingers have you on your right hand without counting 
 your thumb and little finger. How many have you if you count your 
 thumb ? Say your own' name three times. Put down two blocks ; 
 put down two more ; now take away one block ; how many are left ? 
 If John and you and I had one cent each, how many cents should we 
 all have ? How many legs has this cat ? (Showing picture.) rfow 
 many legs has the cat more than you have ? If you had one leg more 
 should you have as many legs as the cat ? How many legs have you 
 less than the cat ? Take out three beans. Take one of three beans, 
 how many are left ? Put out one block ; put two in a row below that; 
 put three in a row below that ; put four in a row below that ; make as 
 many marks, as three ones in three ; rub out one mark ; how many 
 marks are left ? Rub out another mark ; how many are now left ? 
 How many handles have* three knives ? Take two blocks yourself, 
 and give as many to me as will make four altogether. Hand me one 
 block with your eyes shut ; three blocks ; four blocks." 
 
 Operations with Objects to 6. 
 
 " Bring me four pebbles ; now bring me one more ; four pebbles and 
 one pebble are called five pebbles. Put out five blocks ; five shells ; 
 five beans ; make five marks ; stamp five times ; show me five boys ; 
 five girls ; five children. Open your hand wide ; how many fingers 
 
14 
 
 are open (with the thumb) ? Show your hand ; how many fingers are 
 open ? how many are shut ? Put out five blocks ; put them in two 
 rows ; how many are in each row. Put them in three rows ; how many- 
 are in each row. Open your hand ; then shut one finger ; how many 
 fingers are opened ? shut three fingers ; how many are left open ? How 
 many are left open when you shut two fingers ? when you shut four 
 fingers. 
 
 " Try how many ways you can arrange three blocks. ... j • ■• -•" etc. 
 
 " If I buy two cakes at one cent each, how much shall I pay the 
 baker ? Here is a five cent piece, and there are some cents ; put down 
 as many cents as would buy as much as fhiS'five cent piece. Show 
 me a couple of fingers ; show me two couple of fingers. ('Couple ' 
 should be explained if not understood by the child.) How many 
 fingers have you altogether on your right hand ? on your left hand ? 
 W at else have you five of? How many panes of glass are there in the 
 lowest row of panes in the window ? How many more do 1 need to 
 make up five panes ? How many ducks are a couple of ducks ? Has 
 this chair more legs or fewer legs than five ? How many fewer ? 
 Are your arms and mine five ? How many arms have you less than 
 -four ? How many less than five ? Take two blocks for yourself, 
 take one for John, and give me as many as will make us have five 
 among us. If -I were to give two apples for you and John, how many 
 should you have and how many should he have ? 
 
 " If John had two cents and you and I had a cent each, how much 
 would all of us have. If you have three cakes to divide between 
 yourself and your sister and me, how many should each of us have ? 
 Shut your eyes and take up three from these five counters. If you 
 had one and two nuts in your right hand, and two and one nuts in 
 your left hand, which hand would hold most nuts ? 
 
 " This stick, or measure, is one foot long ; this other stick, or meas- 
 ure, is one yard long ; show me how far a foot goes on the yard 
 measure. Try if you can find how many feet are as long as a yard. 
 Measure this chair. Is it a foot broad ? How many feet broad is it ? 
 Measure this table. Is it a yard high ? Is it yard across ? Meas- 
 sure the door with the yard measure. Is it a yard ? Measure it with 
 a foot measure. How many feet wide, is it? Measure two yards 
 along the floor beginning at the door. Measure three feet in the same 
 manner. Try if you can measure the length of a yard on the floor 
 with a foot measure. Draw a line a foot long on the blackboard. 
 Now take your measure and see if you are right. John, James and 
 Abby may draw a line one yard long on the blackboard. Frank Gilbert 
 may take the measure and see who has drawn the best. Guess how 
 long your desk is ; take the measure and see if you guessed right. 
 Draw a triangle; how many lines or sides are there ? If this side 
 were a foot long, and the other sides were^each of the same length, 
 how many feet would all the sides together measure ? When I say 
 ' one, one,' how many words do I say ? when I say ' two,' how many 
 words do I say? 'Two' is a short way of saying what? What 
 would two things be called if one were taken away,. What would 
 ' one ' be called if another were added or put to it. Take up five 
 blocks; now take up another block; five blocks and one block. are 
 called six blocks. Show me six shells ; six beans ; six slates ; make 
 six marks, — say six words. Take six beans and put them in twos ; 
 how many twos are there ? Put them in threes ; how many threes are 
 
15 
 
 there ? Put them into ones ; how many ones are there ? How many 
 fours can you find in six shoe pegs ? How many fives ? In six shoe 
 ' pegs how many threes do we find ? How many twos ? 
 
 " Write six words on your elates. Write six rows of i's — each row 
 containing six i's. 
 
 " If I cut this square piece of paper into two parts of the same 
 size, what is each part called ? How many halves is it cut into? Here 
 is a splint which is cut into two parts of the same size ; what is each 
 one of these parts called ? How many halves are there in the whole 
 splint ? Here is an apple that is not cut ; how many halves could it 
 be cut into. Draw a line ; divide it into two parts of the same length ; 
 what is each part called ? Half of what ? The whole line is made up 
 of how many halves ? If I gave you, one apple between yourself and 
 Philip, how much of it ought you to keep, and how much of it should 
 you give to him. 
 
 "This weight is called a pound weight. Take it in your hand. A 
 piece of bread or a stone, or anything just as heavy as this, would 
 weigh a pound, or be a pound weight. This weight is half a pound. 
 Take it in your hand. How many of those weights do you think 
 would weigh as much as a pound weight. (A number of very enter- 
 taining exercises, similar to those with the yard measure, may be per- 
 formed by the child with a small pair of scales and a few weights.) A 
 girl carried a pound of sugar in one hand, and two pounds of sugar in 
 the other ; how many pounds of sugar did she carry. If a package 
 weighs twice as much as the pound weight, how much would it weigh ? 
 Suppose that you picked up a stone that is half as heavy as this 
 pound weight, how much would it weigh. How many such stones 
 would weigh a pound ? A man went to market and bought a pound 
 of meat, two pounds of bread, and a pound of butter ; how many 
 pounds had he to carry home in his basket ? 
 
 " A farmer had two sheep, each of which had two little lambs ;' how 
 many lambs were there ? Another farmer had two sheep and three 
 lambs ; one of the sheep had one lamb only ; how many lambs must 
 the other sheep have had ? 
 
 " Try in how many ways you can arrange four blocks. Put out. two 
 shoe pegs; take away half. Put out twice as many pebbles as one. 
 Put down four blocks ; take take away half. Put out half as many 
 pebbles as four. Arrange six blocks in pairs ; how many pairs do 
 you find. Put down six shells.. If one shell be taken out of six, how 
 many remain ? if two be taken ? how many are left if three are taken ? 
 If once two be taken from six shells, how many are left ? if two twos or 
 twice two ? if three twos or three times two ? How many things are 
 three chairs, two lamps and a fiddle ? Try if you can find out without 
 looking, how many horse-chestnuts there are in each of my hands. A 
 hen had six chickens, but some rats killed two of them ; how many 
 chickens had she then left of six. Another hen had also six chick- 
 ens, and some rats ate one, and two fell into a ditch and were drowned ; 
 how many chickens had this hen left." 
 
 Operations with Objects to 7. 
 
 " Tell me another name for one and one. Tell me another name 
 for one, one and one. Twice one, or two times one, is called what ? 
 " Make a square ; how many sides has this square ? how many 
 
16 
 
 lines ? hov; many corners ? If the top line is a yard long, how many 
 yards long would all the sides measure ? Divide the square into two 
 equal parts ; how many parts did one-half make ? two halves ? Some 
 men put a post into the water to tie boats to ; the post was three feet 
 under water, and two feet above water ; how long was the post? If 
 two feet were under water and two feet above water, how long would 
 it be? 
 
 " How many legs must be put on a dog for him to have five legs ? 
 
 "A little insect, that had six legs, met with an accident and had two 
 broken off; how many legs was it obliged to walk with afterwards ? How 
 many feet have you and Willie Worster and I ? How many hands ? 
 how many noses ? Three old soldiers were walking along the road ; 
 William had but one leg, Charles had two legs, and John only one ; 
 How many legs had these three soldiers to walk with ? How many 
 wooden legs would they need, so that each soldier should have two 
 legs ? Make six marks of any kind that you like. How many are 
 left if you rub out two and one ? Are any left if you rub out three 
 and two ? What number is the half of two ? What number is twice 
 two or two twos ? Cut this bit of paper into two parts of the same 
 size ; what is each of these parts called ? If an orange were divided 
 equally between you and George, what part would each of you have ? 
 If a sailor had half as many hands as you, how many would lie 
 have ? 
 
 " This can holds a quart and this dish holds a pint ; now if I fill the 
 dish with water and pour it into the can, the can would be half full ; 
 what must I do to fill the can. How many pints in a quart ? In a 
 quart how many pints ? What part of a quart is a pint ? One is what 
 part of two ? A spider caught in his web, one fine day, two flies and 
 a bug and a bee ; how many insects did he catch that Say ? how many 
 animals ? How many did he eat for dinner if he ate all but the bee ? 
 John took two buttons off his jacket, and Harry lost twice- as many ; 
 how many did John and Harry lose together? If your cat had half as 
 many legs as she has now, how many would she have ? 
 
 " Make six marks ; make five under them ; four under them ; three 
 under these ; two under these ; one under these. How long must I 
 keep a puppy that is a month old now before he is six months old ? If 
 I were to place six nuts on the table and tell you to take one-third of 
 them, how many would you take ? 
 
 " I have five hens and a duck ; how many hens have I ? how many 
 ducks ? how many fowls ? If I were to divide a quart of cherries 
 into two equal parts, what would each of those parts be called? 
 What part of a quart is a pint? Two pints are how many halves 
 of a quart ? How much is two added to one more than one added 
 to two ? Here are five pebbles and there are six pebbles ; which is 
 the larger number. How much larger ? 
 
 " Put out six splints ; now add another to them ; six splints and one 
 make seven splints. Show me seven shells ; seven shoe pegs ; seven 
 boys ; seven girls ; seven fingers ; make seven marks. 
 
 " Write seven words ; write i seven times ; take seven steps ; clap 
 your hand seven times ; go into the entry and rap on the door seven 
 times ; put blocks on seven desks ; shake hands with seven children. 
 How many days are there in a week ? Put out seven blocks ; how 
 many twos can you find in them ? how many threes ; how many fours ? 
 how many fives ? how many sixes ? how many sevens ? Show me 
 
17 
 
 six pebbles ; how many shall you add to them to make seven ? Show 
 me four blocks ; how many more will make seven. Five splints and 
 how many are seven splints ? Three boys and how many are seven 
 boys ? Have two chairs seven legs altogether or more than seven. 
 Here are a five cent piece and a two cent piece ; how many cents in 
 all? How many single stockings are there in. a pair ? in two pairs ? 
 A man had six chickens, sold two, lost one and ate two ; had he any 
 left? A house had three windows in front, and three behind ; had it 
 more or less than seven windows altogether ? 
 
 " A man had four books, and afterwards bought as many as with 
 the four made seven ; how many did he-buy ? How many times must 
 I empty a quart jug into a tub to have seven quarts of water ? A 
 gallon is four quarts ; how many quarts is half a gallon ? Draw a tri- 
 angle ; how many sides has one triangle ? how many corners ? Ask 
 the same of two triangles. Make the number of marks which are 
 one less than seven ; two less than seven ; four less than seven. 
 How many eyes must you have to have seven. A hen laid seven 
 eggs in two months ; the first month she laid three eggs ; how many 
 did she lay the second month ? Another hen laid seven eggs in three 
 weeks ; one week she laid three, the next she laid two ; how many 
 did she lay the third week ? There are three windows in a room, and 
 each required two yards of holland for a curtain ; how much holland 
 must be bought for all the curtains ? Put all the numbers you know 
 upon the table. 
 
 "Name the days of the week. Write their names. How many are 
 there ? Four weeks are as much time as one month ; how many 
 weeks are there in half a month ? How many weeks are there in a 
 month and a half ? 
 
 " I have cut this square piece of paper info four equal parts ; what 
 is one of the parts called ? Look at this splint ; it has been cut into 
 four ; what is each part called ? how many fourths or quarters are 
 there in the whole splint ? Here is a foot measure ; show me half a 
 foot; show me a quarter of a foot. Draw a square, each side of which 
 is a foot long ; now divide the square into four equal parts ; how many 
 squares can you see ? how long are the sides of the little squares ? 
 Draw a line ; divide it into halves; divide it now into quarters; how 
 many quarters did you divide each half into ? 
 
 "Draw four horizontal lines, and three vertical lines; how many 
 lines have you drawn ? Draw three slanting lines, two vertical lines, 
 and two horizontal lines ; how many lines have you drawn ? Draw 
 a tree ; put five birds on one limb, and two birds on the ground close 
 to the tree ; how many birds on the tree ? Each day I read a page 
 in my new book ; how many pages do I read in a week ? 
 
 "Here are five blocks and seven blocks ; which is the larger num- 
 ber? Write seven rows of i's on your slate, with seven i's in each 
 row." 
 
 Operations with Objects to 8. 
 
 "A quart is two pints ; how many quarts are there in four pints ? 
 in three. pints? in seven pints? Do you recollect how many feet 
 there are in a yard ? how many feet are there in two yards ? in a yard 
 and one-third? How many yards are there in six feet? in seven feet? 
 in four feet ? Draw a square one-half yard long on each side ; how 
 long are the four lines ? Put down four blocks ; take away one-quar- 
 
18 
 
 ter of them ; how many are left ? Four times what are equal to four ? 
 What is the half of four ? what is the quarter of four ? Mr. Spear 
 walked two miles in an hour, and his son walked four miles in an 
 hour ; how much faster did the son walk than his father ? It would 
 take Mr. Spear three hours to walk from here to Boston ; how many 
 miles is it to Boston? How long would it take Mr. Spear's son to 
 walk to Boston ? Seven cubes and one cube are called eight cubes. 
 Show me eight shells ; how many fours can you find in them ? how 
 many twos ? how many eights ? 
 
 " Take eight steps ; show me eight fingers, eight boys, eight girls ; 
 make eight squares, eight triangles ; write eight words ; write the 
 letter i eight times ; draw a line, and divide it into eight parts. Give 
 each one of these children eight beans ; now take away half of eight 
 beans from each one ; how many has each left? Eight pebbles mean 
 the same thing as twice how many ? 
 
 " How many legs have a man and a horse ? how many legs have 
 two horses ? A careless little girl lost a needle every day for a week, 
 but found two of the lost needles ; how many were not found ? Four 
 women bought a pound of tea ; how much must each have for her 
 share when the tea is divided ? Take eight marbles ; can you sep- 
 arate them into two equal parts ? into three equal parts ? into four 
 equal parts? into five? into six? into seven? into eight? Eight is 
 another way of saying four times how many ? I will cut this square 
 of paper into three equal parts ; what is each part called ? How 
 many thirds are there in the whole square ? What part of a yard is a 
 foot? What part of a yard is two feet? How many lame legs must 
 two dogs have between them if they have seven sound legs ? A dOg 
 was run over by a wagon, and one leg was hurt ; how many legs 
 had the dog then ? Can you tell me two numbers that are equal to 
 eight ? 
 
 "A young apple-tree had upon it the first year one apple ; next year 
 twice as many ; in the third year twice as many as on the second ; 
 how many apples did the tree produce in three years ? How many 
 thirds are there in one apple ? in two apples ? in one apple and a 
 third ? How many sides (faces) has this cube ? Take eight shells ; 
 separate them into four equal parts ; what number will each of these 
 parts be ? what part of eight will it be ? 
 
 " If three boys had seven apples to divide among them, and one 
 boy took three, how many apples would be left for each of the two 
 other boys ? One hen had eight chickens ; another hen had half that 
 number ; how many had she ? 
 
 "A snail climbed up two feet of a wall every day, but slipped back 
 one foot every night ; how many feet did he get up in three days ? 
 Take these blocks, and show me all the numbers that you know. 
 Two and what are equal to five ? Can you find two equal numbers 
 that are equal to five ? " 
 
 Operations with Objects to 9. 
 
 " Put down eight blocks and one block ; eight blocks and one are 
 called nine. Put these nine blocks back, and put down nine pebbles. 
 Seven pebbles and how many are nine pebbles ? Into how many 
 threes can you separate these nine pebbles ? Try how many lots of 
 four cubes there are in nine cubes ; how many lots of two cubes, five 
 
19 
 
 cubes, eight cubes, three cubes, seven cubes. Draw a si^iare ; how 
 many sides has this square ? how many sides have two squares ? 
 Now divide the square into three equal parts, with horizontal lines ; 
 divide each equal partMnto three equal parts, with vertical lines; what 
 is the shape of each one of the parts ? how many little squares are 
 there on one side of the large square ? how many squares are there 
 on four sides ? how many squares are there that do not touch a side 
 of the large square ? 
 
 "A man had eight geese ; one day he sold a quarter of them ; how 
 many did he sell ? how many had he left ? Next day he sold another 
 quarter of his eight geese ; how many had he then left? how many 
 had he sold altogether ? Two quarters of the geese are the same as 
 what part? What part of his whole flock then had he sold? Take 
 as many pebbles as you have heads, ears, eyes, hands, and noses. 
 There is a story of a giant who had two-mile boots (or boots in which 
 each step he took was two miles long); how many steps must this 
 giant take to go eight miles ? How many cakes must- 1 make so that 
 you and John and Anne shall have half a cake each? How many 
 cakes must I make so that six persons shall have half a cake each ? 
 that seven persons shall have the same quantity ? How many threes 
 are there iri nine ? fours ? fives ? etc. 
 
 "One loaf weighs half a pound, another weighs a quarter of a 
 pound; how much do both weigh together? Little Philip is eight 
 years old, his sister Mary is five ; how much older is he ? What is 
 half of two apples? of three apples ? four apples? five apples? six 
 apples ? 
 
 . "A pound of new potatoes sometimes costs four cents ; how much 
 do two pounds then cost? how much does half a pound cost? how 
 much does a pound and a half cost ? Put nine beans in a row; seven 
 beans and how many are nine ? five beans and how many are nine ? 
 
 "A lame horse went along the road two miles an hour; another 
 horse went six miles an hour ; how much faster did one go than the 
 other? Draw a picture of a road on the blackboard (a broad line); 
 make a mark across the middle of the line. 
 
 Here is the stable. I start with my horse from the stable in one 
 direction and go five miles ; you start and go /our miles in the oppo- 
 site direction ; how far should we then be from each other? If I 
 travel six miles an hour and you travel four, in opposite directions, 
 (show on board,) how far would we be apart in half an hour ? How 
 far should we be apart if we walked an hour in the same direction ? 
 
 " How much does a quart of pop-corn cost at three cents a pint ? 
 A cat caught a mouse every other day for four days ; how many did 
 she catch? If eight eggs cost four cents, how much do two eggs 
 cost? Make nine crosses in one row, nine more in two rows, nine 
 more in three rows ; how many threes do you find in nine ? How 
 much do I pay for a yard 'and a half of calico at six cents a yard ? 
 Tell me all the sets of two numbers that are equal to nine ; to seven ; 
 to .nine. Tell me what numbers you can separate six into. Tom- 
 mie Lennen bought four cents' worth of ginger-bread and sold it for 
 one-half more ; how much did he sell it for ? How many equal num- 
 bers are there in nine, two, three, etc. ?' A boy drove a calf to mar- 
 ket at the rate of two miles an hour ; how many hours did he spend 
 
20 
 
 in going to market, which was eight miles off? If six be divided by 
 two, or divided into twos, how many such parts will be found in it ? 
 Two, then, is contained in six how many times ? For shortness we 
 say, 'Wo's in six three times;' or, 'two's in six three;' or, 'six 
 divided by two is three.' Twice what is six ? Three times what is 
 six ? Half six is what ? What should I pay the butcher for half a 
 pound of meat, if a pound cost seven cents? I go to market with 
 nine cents in my pocket ; I buy two apples at a cent apiece, four eggs 
 at half a cent apiece, and with what is left of my money I buy three 
 peaches ; how much do I pay apiece for the peaches ? " 
 
 Operations with Objects to 10. 
 
 "Show me nine marks ; add one to them ; nine and one are called 
 ten. Show me ten blocks ; ten shells ; ten shoe-pegs ; ten horse- 
 chestnuts ; ten pebbles ; ten splints ; ten boys ; ten girls ; ten chil- 
 dren ; ten slates ; ten books ; ten panes of glass. Make ten trian- 
 gles ; ten squares; ten crosses; ten vertical lines; ten slanting 
 lines ; ten horizontal lines. Draw an apple-tree and put ten apples 
 upon it ; now put ten birds in the branches. Draw a house with ten 
 windows and half as many chimneys. Draw ten pens, and put ten 
 pigs in each pen. Make a stable with blocks and put ten horses 
 (shells, etc.) in it. Take ten steps ; shake hands with ten persons. 
 Draw a fence with ten posts, and ten slats (picket) between each two 
 posts. | ) ) I f 
 
 " Separate ten cubes into two equal parts ; how many are there in 
 each part ? How many ringers have you on both hands ? on each 
 hand ? How many fives of fingers does it take to make ten fingers ? 
 Separate ten pebbles into lots of two each ; how many such lots can 
 you find ? Put out ten shells, — take away three ; how many are 
 left ? Take away eight ; how many are "left ? I have put out six 
 blocks ; give me as many as will make ten blocks." 
 
 Give a large number of examples of this kind. First have pupils 
 handle counters, then have them answer by simply seeing. 
 
 "Here are how many blocks ? " '"Ten." "Turn around (teacher 
 takes away four) ; turn around ; how many now ? How many did I 
 take away ? Shut your eyes (takes away six) ; open your eyes ; how 
 many have I taken away ? how many are left ? I put down three 
 counters ; give me as many as will make ten counters. Shut your 
 eyes ; give me ten blocks. Write ten i's ; write ten words ; ten sen- 
 tences. Write ten i's in five rows ; in two rows. How many weeks 
 are there in two months? How many months are there in six weeks ? 
 How many weeks are there in a month and a half? How many weeks 
 are there in three-quarters of a month ? How many weeks are there 
 in a month and a quarter ? What part of a month is a week ? What 
 part of a month is a fortnight ? Take ten shells ; now find how many 
 twos you can divide them into; how many fives; threes; eights; 
 fours ; sixes ; etc. 
 
 "A woman bought eight cups and saucers ; her servant broke two, 
 and her little boy broke three ; had she any left ? A mother gave 
 eight apples to her eldest daughter, to divide with her two brothers 
 and one sister; how many should each of the four children have? 
 
21 
 
 If only four apples had been given how many should each child have 
 had ? how many if two apples had been given ? if one had been 
 given? If I were to divide an apple into eight parts, What would 
 each part be called ? 
 
 " How many pages in a book should you read in a week if you read 
 one in a day ? How many if you read half a page a day ? Seven 
 pebbles and what are ten ? three and what are ten ? etc. A farmer 
 had nine acres of wheat, and at harvest-time he reaped three acres a 
 day; how many days did it require to reap all his wheat? -How many 
 weeks are there in a month and a week? Takeout ten cubes and 
 put them in a row ; five cubes and four cub^s and what are ten ? 
 three cubes and five cubes and what are ten ? A wooden pile, or 
 post, was driven three feet into the bottom of a river, two feet more 
 were covered with water, and three feet more were above water ; how 
 long was the post from top to bottom ? what part of the post was cov- 
 ered with water? If you are five years old now, in how many years 
 will you be eight ? Who will be eight in two years ? why ? 
 
 " Put ten counters in a row ; put other ten counters in two rows ; 
 put other ten counters in three rows ; four rows ; five rows. Here 
 are four little boxes ; now put these ten beans into the boxes in as 
 many ways as you can. Take five splints ; if each of these splints 
 were divided into two parts, how many half cubes should you have ? 
 A tall man's steps were each a yard long, his little boy's steps were 
 a foot long ; how much longer were the father's steps than the son's ? 
 If you put two cents into your savings every day, in how many days 
 would you have ten cents in the bank ? Supposing that you put in a 
 five cent piece every week, in how many months^ would you save ten 
 five cent pieces ? What must I pay the milkman in a week for a pint 
 of milk every day, except Sunday, when milk is three cents a quart ? 
 Tell me all the numbers that are the same as seven. A bureau has 
 five drawers; each drawer has two handles ; how many handles are 
 there in the bureau ? Find all the sets of two numbers that will make 
 ten. Show the twos of ten ; the threes ; fours ; fives. Find how 
 many ways you can make ten with the blocks. Show me three num- 
 bers that are ten ; four numbers ; five ; six ; seven ; eight. I build 
 here with the blocks five pig-pens; no, you may build them; now 
 how many pigs have I in each pen? None: then I will put three 
 pigs into each of four pens, and one pig into this last pen ; how many 
 pigs In all ?" 
 
22 
 
 .Ji jJ-6 
 
 5 s S» 
 
 u 5 « 3 
 
 o o"i > 
 
 - S g p - 
 
 o5.g Be 
 
 •- o S 
 
 |es£ 
 
 o 
 
 CD u C 
 
 ""§5. 
 
 !«.a 
 
 I Hi 
 
 a=£° 
 
 « . & = 
 
 "2 w ^ <u 
 
 5 aff&a 1 
 
 K 1 R p- 
 
 s? g I o I '€ 
 I S ? - S-.S 
 
 S.S 
 
 
 M.5 
 
 2 
 
 "rt S" rt S w 
 
 -e „S,m-9 
 
 
 ~i 
 
 s s 
 
 s 
 
 ■a 
 
 o 
 
 (C 
 
 .13 
 
 
 
 S 
 
 
 
 ** 
 
 fl*« 
 
 <H 
 
 P 
 
 
 J»l 
 
 • a , 
 
 v, 5 
 
 
 
 OJ= 
 
 <•& 
 
 HH 
 
 1 s 
 
 3 .1 
 
 MS- 
 
 sis'. 
 
 i ffq CO 
 
 CO 
 
 in 
 
 c4 
 
 I 
 
 lis « 
 
 Sol 
 
 
 5 » 
 
 
 
 'H 
 
 u5 
 
 Uh.C 
 
 «* 
 
23 
 
 II 
 
 lit 
 
 4-. £-> 
 
 6a 
 
 s 
 
 of I 
 sely 
 sign 
 
 3 
 
 2 "l? 
 |B.8 
 
 
 «-• c-fa 
 
 
 
 
 *■# 3 ** 
 
 p^ 
 
 (A 3 5 
 
 
 
 
 M3 £ 
 
 ■Sfe 
 
 ■So 
 I • 
 a* 
 
 Q <U 
 
 J>§ S 
 
 .Bam S 
 
 It"- s 
 
 .6°. .§ 
 
 "°. «*i tD . | 3- 
 
 
 4 S £ i i J 
 „-X2 hog** 
 
 is" 8 !'-! 
 
 •a » ! |-E.§-2 
 
 «*« _ b* o ® 
 3 O m B-c"" 
 
 is s s s & 2 
 
 3"-a •-•S"S«? 
 
 I 111 
 
 rt 5 
 
 *S 
 
 si 
 Is 
 
 S e 
 c o 
 
 ^2 rt 
 
 . « » 
 '■5 K 
 
 g-o S 
 S « 
 
 s-3 r 
 
 '* S 
 
 «• a 
 
 B"S;S 
 
 ah « 
 
 ■ H Stc II 
 
 
 ■2 bfl 
 rtu* 
 
 to *^ 
 
 rt o rt 
 
 9 « c 
 g S « 
 
 £ ° a « 
 
 "Ortl » 
 
 ■a m g s a 
 g.| s H-a 
 
 -£ S « S "c 
 
 I -" — u 
 
 *-5 C* CO vj IH « 
 
 ■S'S.s ~js 
 
 ■cSSg-g 
 
 X 
 2* 
 
 s a 
 
 rt» JS 
 
 °72 
 s+ = 
 
 rt -}-<o 
 
 s»s 
 
 ■o S- « 
 « s 
 
 1 
 
 5:2 o 
 
 - ■■» 
 
 .= ><■§ 
 
 ^t: 
 
 « -si 
 
 ■- CO 
 
 1-1 
 
 o 
 
 rt-.e5 
 r- U S 
 
 COS 
 
 ■(j m 
 
 
 •21 
 
 o 
 
 o 
 
24 
 
 CfL, 
 
 •■fl -3 
 
 
 
 .y CO +j 
 
 P- - 5 E « 
 _; io >> • C 
 
 5B 5 
 
 •a+«8 
 
 tt) CO n 
 
 en w 3, 
 u M _ 
 
 Su OS 
 y CJ fc 
 
 ~ * 
 
 y s ii ■£ "^ 
 
 ^ o-h 3 S S « 
 8 S 1 
 
 « o S c 5. - 
 
 - fi»fi-C ^ k V 
 
 : €S - * g-* S 
 * »-2 S Z Sf* y, 
 . 8g£^ °_ «J 
 
 ij* h 1S:i 
 
 ; <-> if 4 " " h fe-o 
 
 SCO no ^ fl 
 " I P. 
 
 S"U3 -* 
 
 o S rt,H 
 
 fe.1 «TS 
 
 §"8-2 + 
 
 2; u 3 ■* 
 o.x c 
 
 " o S 
 
 ^i 
 
 ^^3 + 
 £ -* S 
 
 « 2 8 
 
 O in 
 
 J2» 
 E c - - rt 
 
 o M„, 4- »ia 
 r»dH «" gsStn 
 + 05SH++* 
 
 Ph +3 eq c*l 
 
 S S-3-e 
 "s s JO- 
 'S* 3 " 
 
 Is- 
 
 ^4 
 
 ■a 
 
 SsO* 
 
 £« -, » 
 
 .6 « e h S C 
 
 ."8«5 . 
 H H 
 
 Eh 
 & 
 
 a 
 
 o 
 in 
 
25 
 
 «? 
 
 ■ggs 
 
 fsT 
 
 W I. 
 
 ~ 3 L 
 
 > tn c 
 
 Pi ^3 
 
 «' s c 
 
 §■§ 
 8 1 
 
 G o 
 
 .1 
 
 = h .5 - ^, 
 
 fijS bo ' 
 
 §2J? 
 
 S »? o'o ■ o =. 
 
 = J= 
 
 tfl 
 
 <u — I 
 
 *73 JC ^ 
 &. *3 N 
 
 .•||l|-g.|g 
 
 v. . B u o o . 
 
 2 S am & 
 
 S 
 
 a J 
 
 -s 
 ■>»a 
 
 c a 
 
 U C 
 (C ■ — 
 
 1-4 
 
 i «T3 « ** O 
 
 S3 S « 2 oj 
 _e 8 s w c bo 
 
 d H CJ cs 
 
 ffi » iS".9 
 
 *- l. ** M S 
 
 e*.2„ **.,* 
 
 **5 to S* m rt w 
 
26 
 
 Id 
 u 
 
 U 
 Si 
 
 w 
 
 W 
 
 -Sjs 
 
 Is 
 
 c 
 
 S * 
 * 2 
 
 measures with objects. 
 
 taken in leading pupils 
 es. Train them to 
 Highly understand- 
 hey attempt to. per- 
 
 ch tables of weights and 
 ith objects, to write them 
 e them in performing a 
 hen erase. 
 
 c 
 
 £ ® 
 
 ■o .a p,d* 
 
 rf > " *- 
 <u ? 5 .„ 
 
 < 
 
 y. 
 
 o 
 
 IF** 
 «3 
 
 S - S ° £ 
 
 ■" 2 51 ~ O 
 
 Mil 
 
 H 
 
 in 
 
 « ^ cn 3 g S 
 
 mm l$U 
 
 _ "- 2 u 
 
 u 
 u 
 o 
 s 
 (A 
 
 *lf 
 
 w £ S 
 
 ■ «j* a 
 lO J. ** g 
 
 . to e <u 
 S 3.3 
 
 
 (fl 
 
 -co «r 
 
 >L u." x 
 
 
 *r 
 
 
 «S| is 
 
 
 To 
 
 wo" 
 
 s - M 2 
 
 
 ** 
 
 O b- « 
 
 xS "2 <h u 
 
 o 
 
 H 
 
 u 
 u 
 
 X 
 
 3 
 
 p 
 ■f. 
 
 < 
 
 < 
 
 H 
 7. 
 
 W 
 Q 
 
 G 
 
 s 
 o 
 '> 
 
 V 
 U 
 
 P, 
 
 bO 
 
 C 
 
 ~ (A 
 
 •- ... 
 
 > 5 
 
 ,S§ 
 
 « rt « 
 Com 
 
 "" a S 
 
 ■&'■§« 5 
 
 IS 1 " gHO 
 
 = s3 ;'• 
 
 a m t.Z, 
 
 m g g a *= 
 •JjS | «| 
 
 t| all 
 
 'o rt "3 "w * 
 & • ^ S3 
 
 en 
 ■ <u 
 
 5 2 H» 
 
 §3° + 
 
 "■of-** 
 
 'o a 2 ° m « 
 
 ■"• » -o s •£ a 
 »8b . .8 
 
 , rt.t! « •cr) 
 
 ** r-l r-l 
 
 
 H 
 
 C9 
 
 CI r-l i-t 
 
 
 03 
 
 
 
 
 H 
 
 
 
 i-l 
 < 
 
 2 8 
 
 o 
 o 
 
 O 
 O 
 O 
 
 P 
 
 iH 
 
 cT 
 
 Z ■ 
 
 
 r-l 
 
 in 
 
 H oi 
 
 d 
 
 . 
 
 w 
 
 cc 
 
 i— i 
 
 t— 1 
 
 
 H 
 
 
 r— ( 
 
 
 H 
 
 
 
 S 
 
 Hi 
 W 
 
 ►—5 
 
 
 H-4 
 
 t< 
 
 
 
 HH 
 
27 
 
 < 
 
 S 
 < 
 
 O 
 
 Pi 
 < 
 
 >< 
 a 
 
 r- 
 
 w 
 u 
 z 
 w 
 
 « 
 
 a 
 ' a 
 
 K 
 
 i < 
 
 Z 
 
 o 
 p 
 
 u 
 
 a 
 •j 
 
 9 
 (A 
 
 R. Grant's Arith., Part II.; McVicar's Hand-book 
 of Arith. ; White's Manual of Arith. ; How to Teach, 
 pp. 67, 88. 
 
 S. 16. See T. R. a, 1! Prim. If pupils are weak, 
 and full of faults, begin the work again, and take 
 time to repair the poor instruction. 
 
 R. See S. 8. 
 
 S. 17. The main purpose of this teaching is the 
 development of mental power. The learning of 
 each problem, definition, rule, or principle, presents an 
 opportunity to increase the power of seeing and 
 thinking. The explanation or learning by heart of one 
 of these, deprives the pupil of one important 
 means of growth; as if the trainer of gymnasts lifted 
 
 all the heavy weights for (?) his pupils. Never 
 tell a pupil anything that he can be led 
 to discover for himself. 
 
 S. 18. U. S. money, compound numbers, decimals, 
 weights and measures can be made important aids 
 if properly used, in teaching and applying the funda- 
 mental rules. Simple examples only should be used. 
 
 S. 19. Have pupils work together at blackboard 
 quite often. 
 
 S. 20. Walton's Tables are recommended for drills 
 in calculation. 
 
 R. See 13 a, D Gram. ; How to Teach, p. 113. 
 
 i 
 
 tn 
 
 z 
 o 
 
 p 
 
 o 
 w 
 
 « 
 
 3 
 
 a 
 z 
 
 ■< 
 •j 
 
 < 
 
 z 
 
 a 
 a 
 
 z 
 
 T. R. a. Mental activity, delight in the work,* power 
 to cornprehend new subjects, and the habit of trying 
 to understand examples, are the best proofs' that 
 the previous work has been well done. 
 
 12 a. Up to this grade the teaching has been purely 
 elementary. Pupils have been seeing and 
 doing; they have acquired clear, growing con- 
 ceptions of numbers and operations with them by 
 constant and systematic exercise. Now they are 
 required to look closer at what they see and do ; to 
 describe things and processes (definitions and rules), 
 to discover reasons that make certain processes nec- 
 essary (principles). This teaching must be purely 
 objective. 
 
 13 a. Use V. 8. money, decimals and com- 
 pound numbers in teaching fundamen- 
 tal rules. Teach meter, decimeter, dekameter, li- 
 ter, deciliter, dekaliter, gram, decigram, dekagram. 
 
 13 b. Addition and subtraction of fractions, using sim- 
 ple examples, with objects (elementary work). 
 
 13 c. Writing bills, and keepjng ordinary simple ac- 
 counts, measuring wood, carpets, room walls, &c. 
 
 13 d. Use very few examples exceeding millions. 
 
 16 a. Multiplication of fractions (elementary). 
 16 b. Simple examples in.percentage and interest. 
 16 c. Cubic measure, with objects, 
 16 a. Division of fractions (elementary). 
 16 b. Shorten process, by casting equal factors out of 
 dividend and divisor. 
 
 w 
 
 i-l 
 < 
 P 
 
 £ 
 
 W 
 
 Cfl 
 H 
 
 Test Results. 
 
 12. Numeration and 
 Notation. 
 
 13. Addition. 
 14. Subtraction. 
 
 1.1 Multiplication. 
 16. Division. 
 
 
 i— ! 
 
 1— 4 
 
 1— I 
 
 
28 
 
 
 S 5? 
 
 - C " . c 
 
 ■5.5 fl s 
 
 s s« E 
 
 £ "0 ft • ,- 
 
 .E 2 da 
 
 t3 " h ct 
 -Bis 3* =• 
 
 -3 3.5H 
 g C g O 
 
 hS* - 
 
 OJ U c .h 
 
 •* 2 5 
 
 15 3,6 
 
 
 c 
 
 o 
 
 c 
 
 T3 
 
 +j .s 
 I . i-* t3 
 
 3 +* 5 ns 
 
 . «c|i 
 J 2 « c ° 
 
 C ,„ rt ri O 
 
 c $ «-*.2 £ 
 
 ■«h -2 2 § 
 
 S-S Si-a 
 
 B fi i. j e 
 
 je.S'Sa 
 
 C g 
 "1 * ST 
 
 "43 p rt 
 
 « =5 
 £ G^ 
 
 sis 
 
 ^'43 .2 
 
 a 
 
 H 
 
 
 . S o • 
 ■«■< ph c H i 
 
 S rH IN CO 
 
 "5 S 
 
 =3 3 
 
 5 6 
 
 CO 
 
 
 c 
 
 
 
 c 
 
 o 
 
 s 
 
 
 
 f 1 
 
 
 fa 
 
 x: 
 
 h 
 
 * •§ 
 
 ■43 
 
 .2 
 
 5 .-S 3 i 
 
 - S3 I 5" ■! 
 
 5 SS'tSE"! 
 
 a) O.K. o 
 
 3 o ts a « 
 
 ,. x S r" s c 
 
 5 . **•&"§ J 
 
 . s 
 
 1 s g g ag 
 
 •E'S I 2 
 
 .n C-43-43 
 
 S,-, u o 
 o rt 2 2 
 
 'H'ta **- 
 
 PI 
 
 ■ .— w 3 *■■ « pi_j ^ qj C fli il 
 r G <V G H ft, 
 
 ■ S i 
 
 a 
 o 
 
 •43 
 
29 
 
 > a.u a 
 
 — C s, u 
 
 SB1«S 
 
 ° ~ ^ j: -^ 
 
 *^ c JS "o ^ 
 » = u ga 
 
 S r S ^ 
 
30 
 
 « 
 
 w 
 u 
 
 Id 
 a: 
 
 w 
 b 
 a 
 
 C 
 Z 
 
 O 
 
 H 
 « 
 M 
 
 (J 
 D 
 
 en 
 
 R. Felter's Arith. ; White's Arith.; How to Teach, 
 p. 180. Understand thoroughly the whole of this 
 ayurse of study before attempting to test results. S. 7. 
 
 S. 25. All commonly used tables should be thor- 
 oughly known. Not used, dram (av. weight), roods j 
 barrel, hogshead (not often). 3. lu. 
 
 S. 26. A large number of examples in measure- 
 ments of surfaces and solids, as floors, carpets, ceil- 
 ings, wood, stone, timber, &c. Second and third pow- 
 ers of numbers. Roots that are easily found. 
 
 S. 27. The metric system should be compared with 
 the cumbrous system, or lack of system, now in use. 
 The miter, liter and gram should be made familiar 
 objects to pupils. The simple principles should be 
 learned. 
 
 
 O 
 
 H 
 (J 
 
 w 
 a 
 
 < 
 
 < 
 
 H 
 
 HI 
 Q 
 U 
 
 19 a. All the work in compound numbers is limited 
 to that wliich is practiced in common life. Weights 
 and measures not commonly, used are not to be 
 learned. Very little of addition, subtraction, mul- 
 tiplication and division is to be taught. 
 
 19 b. 1. Reduction Descending. 
 
 2. Redtiction Ascending. 
 
 3. Weights. 
 
 4. Linear measure. 
 
 5. Surface measure. 
 
 6. Solid measure. 
 
 7. Dry and liquid measure. (Omit beer measure.) 
 
 8. Circular measure. Questions in the difference 
 of latitude and longitude. 
 
 9. Time. Preparation for the study of interest. 
 
 10. Miscellaneous table. 
 
 11. Money. Few examples, showing difference be- 
 tween U. S. money and English, French, and 
 German moneys. 
 
 12. Fractional compound numbers, limited to — 
 
 a. Common fractions to integers of lower denom- 
 inations. 
 
 b. Decimals to integers of lower denominations. 
 
 c. Compound numbers to fractional numbers of 
 a higher denomination. 
 
 d. Compound numbers to decimals of a higher 
 denomination. 
 
 e. Metric system. Teach with objects. 
 
 
 < 
 
 p 
 
 z 
 
 W 
 
 Tkst Results. 
 
 19. Compound Num- 
 bers. 
 
 • 
 
 19. Compound Num- 
 bers. 
 
 
 Id 
 H 
 
 H-4 
 
 t— H 
 1— \ 
 
 
31 
 
 
 BOS •"! 
 
 , 3; '/: v d . 
 
 _C Sfi * l- c . > 
 
 3 W ® rt 5 ■„- J3 
 
 *a 
 
 S-9 
 
 •S g,a 
 
 -T 5 s c +j 
 ■5 So£i^ 
 
 O — -t-i 0*0, (fl 
 
 -8 8 s 
 
 2 B 
 
 a-d 
 
 " en 
 
 23 
 1j 
 
 JgJS 
 
 
 §1 
 
 S Mod 
 
 86" 
 
 
 » ' j o j 
 in >* (J, w g . 
 
 to to 
 
 8" 
 
 •° oT 
 
 c 2 
 <c S3 
 
 a 
 
 J2 
 
 s"M 
 
 u S 
 
 S S 
 o 
 
 bn> (g 
 .S'ScJ 
 
 « c 
 
 a- 
 
 a, 0. 
 
 
 s o 
 
 SB SI 
 
 s s 
 
 a a 
 
 «&E £ . 
 
 _? M CO 
 
 o 
 ©■1 
 
 +s > h4 
 
 BJ-g 
 
 n J=lC 
 
 G . In 
 
 <:«:.s 
 
 S.Mf 
 
 "S !< h u 
 u ** o _« 
 
 to aT+j Q 
 ■»= tn in f 
 
 e rt o _ 
 
 ** r-H 
 
 OJ4-> c 
 
 S-S § 
 
 °. s 3 
 
 tn' .. <u 
 
 & aJ <-> 
 
 £ >H 
 
 .5 'm-t 
 
 HS> 
 < &> 
 
 ct 
 
 S 
 
 © 
 
 < 
 
 o 
 
 < 
 
 w 
 
 X 
 H 
 W 
 
 o 
 3 
 
 w 
 
 O 
 
 z 
 
 « 
 « 
 
 « 
 
 a 
 z 
 < 
 
 [A 
 Z 
 
 o 
 
 H 
 [n 
 Id 
 
 
 ■ 
 D 
 en 
 
 K. S 16, 17. 
 
 1 
 
 S 
 
 35 ^ 
 
 '(5 
 
 u 
 
 V 
 
 s* 
 
 o 
 
 "be 
 rt 
 
 aj 
 ^3 
 O 
 
 H 
 
 cr u 
 
 .crSf 
 
 rt 
 
 tn 
 Z 
 
 •o 
 
 H 
 U 
 
 W 
 OS 
 
 5- 
 
 
 
 z 
 
 •< 
 
 < 
 
 H 
 
 z 
 
 w 
 
 D 
 
 u 
 z 
 
 3. The base, interest and time being given, to find 
 the rate. 
 
 4. The base, rate and interest being given, to find 
 the time. * 
 
 20 c. Compound Interest. Limited to a few exam- 
 ples. 
 
 C 
 
 V 
 
 ■ S 
 
 c & 
 
 §" 
 
 s "s 
 
 a, e 
 ^ o 
 
 'v. 3 
 
 *< W 
 •« « 
 
 o o 
 
 CI w 
 
 O 4J 
 
 « § 
 
 II 
 
 £/2 (J 
 
 e •«■ 
 
 t/J 
 
 < 
 H 
 
 z 
 
 u 
 
 « 
 
 Tk.st Results. 
 20. Percentage. 
 
 ai 
 brj 
 
 B 
 o 
 
 o 
 
 .2 
 
 S 
 
 >■ 
 W 
 
 
 )-H 
 
 ►— 1 
 
 t— 1 
 
 t— 4 
 1— 1 
 
32 
 
 £ Ml 
 
 *3 "S 
 
 CO 
 W 
 
 Q 
 
 O 
 ffl 
 
 Pk 
 
 o 
 
 S 
 
 p4 
 
 o 
 w 
 
 ffl 
 
 H 
 
 & 
 O 
 
 55 
 O 
 to 
 co 
 
 W 
 
 Pi 
 
 < 
 
 W 
 
 S 
 w 
 
 w 
 
 o 
 w 
 
 CO 
 
 o 
 u 
 
 d 
 
 s 
 
 ■a, 
 
 > ^ 
 
 o o 
 
 Z -WW 
 
 a a ° 
 
 ™ S..2 
 
 .2 SS 
 
 ° = S 
 
 1 '5 x 
 
 ag 
 a" 
 
 I 
 
 a 2 
 •-« s 
 
 8 Is. 
 
 <u fl o 
 £ 0« 
 
 (SSfe 
 
 "5 c •-' 
 
 ■» fl s *d ° 
 
 «Sl |.I 
 ■fSe ?2 
 
 ■a t> a, o a. 
 
 B QCJ 
 
 5 60 
 
 S.E 
 ■0-C 
 >»* 
 
 iffl-g 
 
 If 
 
 "^53 
 
 Pg O 
 
 a- 2 
 
 J3 
 
 ■a 
 3 
 
 oS 
 
 c c 
 
 >.■& 
 
 .e-d" 
 
 - 5 
 o 
 
 <H n 
 >, 
 
 si 
 
 0> 
 
 <h a 
 
 aj • , 
 
 6* 
 
 in S 
 
 'S..2 
 
 a> o ' 
 
 — A 
 
 en rt 
 
 8"! 
 
 
 
 E0 S 
 
 «^ 
 .a— 
 
 ■S3 
 
 •-PS 
 
 M S 2 ^ & 
 
 ■h rt o ^ Si 
 
 ••2 S g . 
 
 m^ 
 
 ** -Zs 
 
 0.-= 0« 
 
 S e-S = £ 
 &> s 2 § 
 
 a© h *h 
 
 (-1 to-. 
 
 ■a - 
 
 rt f_, « 
 
 I 1 ! 
 
 °ts 
 ■3 2 
 oj au 
 « m S 
 
 
 
 
 
 
 
 > 
 ttJ 
 
 0) 
 
 O 
 
 
 
 
 o 
 
 
 o 
 
 en 
 © 
 
 •A 
 
 u 
 
 <u 
 
 o 
 
 
 & 
 
 sj 
 
 So 14 Hi 
 
 H 4j irt 
 .IBbOH 
 
 "SB 
 
 •s 5 
 
 ^3 o cc 
 
 TOO) 
 
 SO-i2 
 
 § -2 
 
 .Scft 
 
 S-'a 
 
 td o rfi 
 
 OJ ^ 4) -^ X ' 
 
 ^^ a=j g-s 
 
 H'C-3 ° e a 
 
 " s ss^ I 
 
 ** tn Ti ^ -C 
 
 c qj rS a) ?? <u 
 
 ^«3 <D « ^, . 
 
 stalk?! 
 p-« o s'e-g-5 
 
 ex c S i- y 
 
 rt in OJ & br.i-, 
 
 s^s's 
 
 « W - 
 
 •■5 2° 
 
 41 <U -/) 
 
 _C be aj 
 53 d <-» 
 O rt 4> 
 
 S.S ^ 
 
 o en p b) 
 
 " I ,,-S 
 « cl " 
 
 ^ S 
 
 
 4) 
 
 
 rt 
 
 S 
 
 ■n 
 
 T) 
 
 g 
 
 s 
 
 
 
 
 
 fc/) 1 ^ 
 
 4) 
 
 ■3 
 
 4> 
 
 O 
 
 0) 
 
 a, 
 
 a 
 
 o 
 
 u 
 
 4) 
 
 o 
 i- 
 
 a 
 
 C 
 
 ;r 
 
 4) 
 
 
 
 
 .8 
 
 X 
 
 4) 
 
 H 
 
 
 
 -^ 
 
 
 
 
 4) 
 
 
 
 
 
 
 
 
 
 ii) 
 
 J 
 
 
 -^ 
 
 
 
 
 
 
 r^ 
 
 a 
 
 4) 
 
 •2 1 
 
 to __• 
 
 6£ 
 
 
 — fa Cl. 
 >• S w 
 
 D k o 
 
 u &,y o P " 
 9 aii S3 a „ 
 
 .- __. te ^ T3 ^3 
 « rt A ■*-••*. « 
 
 ■S.- : g,s s^s 
 
 £ e g" >, - " g 
 
 u 3 S-OT5 Ofi 
 
 S § *Sh 
 
 •S3 ? O >i 
 
 o ■S5-2 
 H aao» 
 
 t tso-d I 
 
 .2«.y "-o 
 
 2-cd c « 
 
 3 fen S.y 
 
8ir 
 
 ■ss 
 
 . "a 
 
 a o 
 
 3 * 
 "^ a) 
 *• J3 
 rt ^ 
 
 1.s 
 
 en 
 
 £* > 
 
 "fee 
 ■a IB 
 
 1 81 
 
 "So 
 v o 
 
 ■a" 5 
 a a ™ 
 
 S-d o 
 a « .a 
 
 a ,„-■= 
 
 ~ *■< 
 a) o a) 
 
 JS'S 
 
 a) a w> 
 > o a> a 
 
 bo 
 
 STa % a 
 
 -. rt " -C 
 
 '=§§'§ 
 s °t) a g 
 
 3-a3,Ji £° 
 X! — ' u ■*-» 
 
 ■" rt « « £ 
 «j o W » a 
 
 a g lit!* 
 S g-S.S S 
 
 4i c«.S 3 
 
 iS 
 
 „^t8la 
 
 , g :£ -.a as- a) M a) 
 > •« xi-x: x! _u 
 
 b ai- g»a*« 
 
 S gS-o,H g- ft 
 
 " yc El... i, (o -m 
 
 ^■og'^aS" 
 o a -H ^ **-. ° a a 
 xi Sac OB § o 
 
 13 
 
 
 ■a 
 
 = 
 
 a c 
 
 rt 
 ■&>> 
 
 «rft 
 8 & 
 
 rt rt 
 
 "SH 
 
 3 S 
 
 3 8s 
 
 .2 rt 
 a; C ! 
 
 +J ig «■ rQ 
 
 5 a U O W 
 
 
 ttJ 43 « _ OJ 
 
 6. Stg 
 
 in B 53 
 
 ."£ -^ XI 
 
 I w; ^ 3 
 ■""■>- S a 
 
 ,££|£ 
 "5«'S 
 
 « £ 
 
 a g 
 « £ 
 
 '5 -a 
 
 a = 
 
 oj^ 
 
 1 ^1 '• 
 
 »>2xi ™ o 
 
 St: MSp xi 
 Sgg^ s 
 
 -mXI- +3 
 
 2 ." " -a o 
 
 • ;S| a- 
 
 <H a^ ■S 
 
 '"« ^S-iJ 
 
 xj hjtg .U 
 
 •«•£ JS >> ft 
 
 " - ° x: xi a. 
 
 rt Si rt -tJ g 
 
 HsH a uH 
 ^> ** p— 
 
 .S aj-.a-S g 
 u.SgS* 
 
 a~.S— a 
 
 •g ft o ai aT 
 
 o — a o S 
 
 & aT— ■£ " 
 ^ d jS rt 
 
 ■fif-Sg-g 
 
 «•-* aj 
 — iS a 
 u a = 
 
 -1-8 
 
 o rt-g 
 
 .2 a a a" 
 _-.-S «- o 
 
 R o « gS 
 ft ft *J 2 w 
 
 . . •.-! "Xi O 
 
 ft 
 
 aUtJS 
 
 «■■= « 
 
 5/3 
 
 ■" O *■ °X! 
 
 = 1*1 g 
 
 rt O rt O ° 
 aj cu a* ft ^ 
 f -1 aj t"" 1 v o 
 
 _^- r» O 
 
 73 *q3 
 
 £ a C^ S 
 rt ii £.3 2 
 
 f ^l.-Srt 
 5« Bl .TJ 
 
 !^ = .2 ° « 
 
 ° x: — _- ^ 
 aj o « 6 2 
 rt ai ftg in 
 
 b a c 
 
 R to rt 
 
 rt s S 4) 
 
 £,"[ rt • S3 
 rt "T I rt 
 
 fci S e.S.g 
 
 . a ■ 
 "II 
 
 O.JJ „- 
 
 o 5 a 
 a mj5 
 
 » g « IB 
 
 ■3 ° > » 
 xi <J ^ S 
 
 e 
 ..fe 
 
 ^ en 40 • 
 
 l sis 
 
 > -£S "^3 O 
 
 u C rt 0-- rt S 
 ■s rt +i - O 
 
 «>•&« ■ gSqs 
 "ch » o a 
 
 i 
 
 s 
 
 rt 
 
 +3 
 
 ft 
 
 > 
 
 
 'c8 
 
 
 
 
 v 
 
 
 4-> 
 
 U'i 
 
 j 4) 
 
 
 1 *■* s* 
 
 ft.g 
 
 (U aj 
 
 Hft 
 
 W3 
 
 A 
 
 -S 
 
 | 
 
 
 SI 
 
 1 
 
 >> 
 
 
 Si 
 
 
 o 
 
 ^3 
 
 
 tee 
 
 4rf O 
 
 
 0(2X1 
 
 
 
 
 rt 
 
 U 
 
 
 « 
 
 
 •fi 
 
 
 a 
 
 h 
 
 C T3 
 
 
 ^2 
 
 
 
 -J 
 
 Oh>» 
 
 
 WU 
 
 
 o o . 
 m- a"*; a 
 o'.o 55.0 
 
 g-a ft-a 
 
 a5 ft 
 ■«3-3 S* 
 B.H.a a 
 
 mi 
 
u 
 
 Use the opposite sides of the face of the cu- 
 bical body to teach that two lines are parallel. 
 
 Use the two adjacent sides of the face to teach 
 that two lines are perpendicular to each other. 
 
 Use the adjacent sides of a face which is an 
 oblique parallelogram to teach that two lines are 
 inclined. 
 
 Speak of two lines as being parallel, not of 
 two parallel lines. 
 
 Use the two sides of a face which is an oblique 
 angled parallelogram to teach tlie oblique angle' 
 and its varieties. 
 
 The angle comes as one part of the face of 
 the cubical body. 
 
 Use a right angle cut "from a piece of card in 
 measuring the angles. 
 
 Let the pupils make the faces of the cyl'mdri- 
 body and the cubical body on the slate. 
 
 Use a number of rectangular blocks of differ- 
 ent sizes. 
 
 Have pupils find other rectangulaf bodies and 
 note their parts. 
 
 Explain the word " rectangular " as applied to 
 the face, — as a face having all its angles rigbt 
 angles. 
 
 Teach that two lines are parallel to each other 
 when they have the same direction. 
 
 Teach that two lines are inclined to each other 
 when one leans either toward or from the other. 
 
 Teach that two lines are perpendicular to each 
 other when they differ in direction and one nei- 
 ther leans toward nor from the other. 
 
 Teach that the two sides of the face of the 
 cubical body extending from the -same point in 
 different directions form an angle. 
 
 Teach that when the sides of the angle are 
 perpendicular to each other the alible is a right 
 angle, when they are* inclined to each other the 
 angle is oblique. 
 
 Teach the varieties of oblique angles. 
 
 Measure the faces of cylindrical body by a 
 piece of card of size of one face to show .that 
 they are equal. 
 
 Measure the sides and angles of the faces of 
 the cubical body to show that they are equal. 
 
 1 
 
 The form of the body is determined by the 
 faces which bound or limit it. Hence in teach- 
 ing the form of a new class of bodies observe 
 the body first in its faces as a whole, then its 
 parts, then the number and qualities of the faces. 
 
 7. KelatWe position of two lines. 
 Two lines parallel to each other. 
 Two lines inclined to each other. 
 Two lines perpendicular to each other. 
 
 •SJS 
 
 C bo 
 
 S — — - 
 
 s I 
 
 »■§. 
 
 . « 
 e s 
 
 < o 
 
 9. The Faces of these bodies. 
 
 Their number and the qualities of their 
 parts. 
 
 Spherical body — .surface one, no faces. 
 
 Cylindrical body — two equal plane faces, per- 
 fectly round or circidar, one curved face. 
 
 Cubical body — faces have four straight, equal 
 sides, four right angles ; faces are square. 
 
 10. Rectangular bodies. All plane faces as 
 -wholes. 
 Parts — faces, edges, corners. 
 Faces — have four straight sides, four right- 
 angles, opposite sides parallel ; all the faces 
 are rectangular. 
 
35 
 
 c 
 
 E 
 
 -a 
 
 o i 
 
 «r .2 4 
 
 =3 
 
 o 
 
 o 
 .a 
 
 
 (A 
 
 O 
 O 
 
 to 
 
 in J- 
 
 rt — ■ 
 
 • u 
 
 3 
 
 .1 
 
 "3 «J 
 
 rt 
 
 
 t>c£ 
 
 
 
 
 
 
 
 5j 
 5 
 
 o 
 
 
 <*■» 
 
 TJ 
 
 TJ g> 
 
 o 
 
 * j5 
 
 li 
 
 
 J2 « 
 
 .a fi 
 
 s 
 
 s >- 
 
 _ « 
 
 c 
 
 ■ P/<U 
 
 £ rt 
 
 (4 
 
 
 f 1 . 
 
 P 
 
 0) X +3 
 
 OT C 
 
 
 T3 
 
 <u 
 
 
 C 
 
 £ 
 
 
 £ 
 
 <+■! 
 
 
 CD 
 
 O 
 
 
 
 a 
 
 
 
 .2 j2 
 
 
 o 
 
 
 
 O Oh 
 
 
 -4-« 
 
 *o 
 
 
 o 
 
 <u ji 
 
 
 
 _> +3 
 
 
 a 
 
 
 
 *o 
 
 rt o 
 
 
 X 
 
 £ w 
 
 
 
 >-■ aj 
 
 
 "" 
 
 , Td'jS 
 
 
 
 t; ca 
 
 
 tn 
 
 " rt rt 
 
 
 rt 
 
 Sfeg' 
 
 
 
 
 ■ T3 
 O 
 
 ■g « 
 
 
 > 
 
 S-^H 
 
 
 " 3 
 
 <y *U 
 
 
 co 
 
 (j rt' . 
 
 
 ,0 
 
 rt « tn 
 
 
 o 
 
 ^Ht 
 
 
 
 
 
 o c 
 
 
 1 
 
 
 Ui — 
 
 a on . 
 
 
 t/i in 
 
 
 8 8 
 
 S5 en a 
 
 o 
 
 1! 8 
 
 -< to 
 ^5 « 
 
 ift v -~ 
 4) in M ^ 
 
 a 
 
 ill 
 
 
 
 a 
 
 ^ 
 
 L rt « 
 
 _s 
 
 ffl (/^ 
 
 tn ,,u '^ 
 
 'S 
 
 tz 
 
 ^ u « 
 
 JH 
 
 [fl ^ 
 
 _«^H 
 
 
 •< An ft 
 
 t-J 
 
 
 
 fH 
 
 
 
88 
 
 I .-a. 
 
 G 0) (A 
 
 ' S3 1 
 "■8.8 
 
 01JS -a 
 .2 g "8 
 
 o^S 
 •a s.-a. 
 
 4J •*- 
 
 O i> 
 
 « E 
 
 •Be* = f « 
 
 ■5:5 JS"S 
 
 "_ so, 
 
 ^h o id 
 
 J" c 
 
 l«p 
 
 o O. u 
 u rt S 
 
 t, 4 u 
 
 ° rt « 
 
 £ s "-a 
 ! •£ 3 g 
 
 H. — £ 
 : ?« S 
 ! is. „- & 
 
 : «s «■ 
 
 ' a. > -2 u 
 i 5 K & p 
 
 "rt »■ 
 
 ! •■ S 
 
 •rf13 cfl 
 
 i°'E 
 
 S-B g 
 
 -O <o 
 
 § S -a 
 
 ■a* ^ 
 
 2= -a 
 
 •b S 
 
 t: 
 
 1 5? 
 
 ~ ~ 
 
 •3 == 
 
 
 ■ U-g 5.8 g 
 
 o fc£ 
 
 b « ,«g « 
 
 rt o " 
 
 rt rt rt 
 P >»R 
 
 o o : 
 
 S « to 
 
 ^3 C 
 
 .8 
 
 j= o x B j. 
 
 LI U "tJ ° U 
 
 tf « 
 
 
 O -.5 
 
 " tp 5 j s 
 
 «J - 0) C^j 
 1 " *■ to o 
 
 
 ■" « £ £ 
 
 *3 o 
 
 f & 
 
 
 i 1 
 
 
 _ a) •— i "O 
 *S &*^> c! 
 c P j; 
 
 & I.S- I 
 
 « o « o 
 •^ •? * -5 
 
 1| 
 
 A3 
 
 S - 
 
 3 ^ « re 
 
 2 -g - 
 
 ?S (/I 
 
 « "3 -° .8 
 
 O u ^-g 
 
 t. S -3 J 
 
 2 -5 C _ 
 
 •3 a »h ^ 
 
 S 1 "" m 
 '3 
 
 J X a." 3 
 
 8 u 
 
 •a -='5 
 
 o 
 
 l-s s 
 
 ago? 
 
 ■s-S-a 
 
 - " « 
 
 •?■=•" 
 
 rt ^ I 
 ^ g in A ? 
 
 « c « a if t « 
 
 » ~ "> * ■£ in ». 
 
 < s »■< 
 
 
 so e ; 
 
ST 
 
88 
 
 Let the pupils draw and name as directed 
 above. 
 
 Use the gonigraph. 
 
 Have many examples of each kind so as to 
 give the pupil an idea of the extent «f the 
 class, as well as of the form. 
 
 Let the pupils draw and name the polygons. 
 Use the gonigraph. 
 
 Other polygons may be taught if needed. 
 Polyedral bodies cut from thick pasteboard 
 make good illustrations. 
 
 Teach the quadrilaterals by using such bodies 
 as will show the different varieties of quadrilat- 
 erals in their faces. The faces of cubical and 
 rectangular bodies will illustrate the square and 
 rectangle. 
 
 Teach rhomb and rhomboid by using bodies 
 which have bases of this form. 
 
 Teach parallelogram by using all the bodies 
 in which the opposite sides of the faces are par- 
 allel. 
 
 Teach trapezoid and trapezium by using bod- 
 ies which have bases of this form. 
 
 Teach the different polygons by using the 
 faces of the polyedraL bodies which show them. 
 
 Teach' in such a way as to explain the mean- 
 ing of the name applied to each form. 1 
 
 (3) Quadrilaterals. 
 
 As J a whole— any face which has four straight 
 
 sides ^nid four angles. 
 Parts — sides and angles. 
 A Square — a quadrilateral which has four 
 
 right angles and four equal sides. 
 A Rectangle — a quadrilateral which has 
 
 four right angles, two long sides and two 
 
 short sides. 
 A Rhomb — a quadrilateral which has four 
 
 oblique angles and four equal sides. 
 A Rhomboid — a quadrilateral which has 
 
 four oblique angles, two long sides, and two 
 short sides. 
 A Parallelogram — a quadrilateral wliich 
 
 -has its opposite sides parallel. 
 A Trapezoid — a quadrilateral which has 
 
 only two sides parallel. 
 A Trapezium — a quadrilateral which has 
 
 no two sides parallel. 
 
 (4) Polygons. 
 
 As a whole — any face which has more than 
 four straight sides and more than four an- 
 gles. 
 
 Parts — sides and angles. 
 
 A Pentagon — a polygon which has five 
 sides and five angles. 
 
 A Hexagon — a polygon which has six . 
 sides and six angles. 
 
 An Octagon — a polygon which has eight 
 sides and eight angles. 
 

 
 s