LIBRARY
OF THE
UNIVERSITY OF CALIFORNIA. •^Received (AN 6 1893 • l89
Accessions No. ^-f^QO^t. Class No. \
GRUBE'S METHOD OF TEACHING
ARITHMETIC
Explained and Illustrated. Also the im- provements upon the method made by the followers of Grube in Germany.
LEVI SEELEY, A.M., PH.D.
*' THAT MAN WILL BE A BENEFACTOR OF HIS RACE WHO SHALL TEACH us
HOW TO MANAGE RIGHTLY THE FIRST YEARS OF A CHILD'S EDUCATION.'*
Garfiefd.
NEW YORK AND CHICAGO:
E. L KELLOGG & CO.
[US 17 BE SIT 7)
L_-
COPYRIGHT BY
E. L. KELLOGG & CO.,
1888.
PREFACE.
THERE is a widespread feeling among American teachers that there is need of better methods of teaching Number, especially in the primary classes. During the last few years, the Grube System, having been introduced into a few schools and discussed at teachers' institutes and in educational journals, has attracted the attention of thoughtful educators in various parts of our land. Many of the later Arithmetics have devoted a few pages in outlining this system or in giving a few hints in regard to it. The excellent results apparent in those schools that have tried the system, the enthusiasm of its ad- herents, and the belief that it is based on sound philo- sophical principles, have created a desire to a better understanding of it. The design of this little book is to give a plain, concise exposition of the Grube theory, and, at the same time, to illustrate the method of teaching Number in accordance with it. It is intended to be a helpful book for the primary teacher.
This book is not simply a translation of Grube's treatise, nor is it the Grube system exclusively; it includes all of that system, and in addition the latest and best ideas of the disciples of Grube in Germany, whose works were studied
4 PREFACE.
and whose personal acquaintance was made by the author in their school-rooms and in their educational as- sociations during a three years' study of the German schools.
In the preparation of this book I have examined with care the following works : Grube's ' ' Leitfaden fur das Rechnen in der Elementarschule" (the original ex- position of Grube's system), German works on Arith- metic by Brautigam, Gopfert, Lincke, Schellen, Bert- helt und Petermann, Rein's "Theorie und Praxis," Sol- dan's " Grube's Method," Indianapolis School Manual of 1876, Quincy (Mass.) Course of Study of 1879, anc* other treatises.
LEVI SEELEY.
LAKE FOREST, ILL., 1888.
SKETCH OF GRUBE.
AUGUST WILHELM GRUBE was born in Wernigerode at the foot of the Harz Mountains, Germany, on the i6th of December, 1816. His father was a tailor, and August was his only child. He commenced school when four years of age, and very early decided to devote himself to teach- ing. Grube often said in later years that it was his love for his teacher that awakened in him the wish to become a teacher.
When eight years of age he entered the Lyceum of his native city, where he remained till his fifteenth year, after which he entered the Teachers' Seminary at Weis- senfels near Leipzig. In 1836, when twenty years of age, he completed his work here, and obtained a testimonial which stated that Grube was well fitted to teach in the best grade of schools. After teaching a short time in a public school, he took the position of family teacher (Hauslehrer) in the family of a count in the province of Posen. A like position in the family of a wealthy manu- facturer near Bregenz occupied his time until he gave up teaching and devoted himself to authorship. He died January 27, 1884.
" Grube was one of the most fruitful and, at the same time, most important pedagogical authors of the present time; a man endowed with philosophical penetration and sound knowledge, great from inclination and char-
6 SKETCH OF GRUBE.
acter, likewise rich in the experiences of life and of the schoolroom. He has by means of his writings exercised an extensive, blessed influence upon the educators of our time."
His works cover all departments of pedagogics. From the many we name " Pedagogical Studies and Criticisms" (Padagogische Studien und Kritiken), in one part of which he discusses " Darwinism and its Consequences," taking a stand against Darwin. Especially has Grube rendered great service to the young by his " Geographical Char- acter-pictures" (Geographischen Charakterbilder), " Bi- ographies from Natural Science" (Biographien aus der Naturkunde), "Character-pictures from History and Tra- dition" (Charakterbilder aus der Geschichte und Sage). Grube was the forerunner of new methods of teaching geography in Germany. He opposed the practice of making the study of geography a memorizing of num- bers and facts, and arranged the material to be taught so that it could be used to advantage. He connected the teaching of geography with pictures of the landscape, with productions, temperature of the country, and would show how the customs, religion, government, history, and hap- piness of the people are related to and dependent upon the country.
But of particular interest to us in connection with this work is Grube's " Guide for Reckoning in the Elementary School, according to the Principles of an Inventive Method" (Leitfaden-fiir das Rechnen in der Elementar- schule, nach den Grundsatzen einer heuristischen Methode). This book marked an epoch in the teaching of Number in Germany and has exerted a wide influence on American teaching.
CONTENTS.
PAGE
I. SKETCH OF GRUBE ...... 9
II. INTRODUCTION . . . . . .11
III. ADVANTAGES OF GRUBE'S METHOD . . .13
IV. DIRECTIONS TO TEACHERS. . . . .17 V. THE FIRST YEAR ...... 19
First Step — The One . . . . .19
Second Step — The Two . . . .22
Third Step— The Three . . . 26
Fourth Step— The Four . . . .30
Fifth Step— The Five . . . . • 34
Sixth Step— The Six . . . . -37
Seventh Step — The Seven . . . .41
Eighth Step— The Eight . . . • ' 44
Ninth Step— The Nine . . . . -47
Tenth Step— The Ten . . . . .50
VI. THE SECOND YEAR. OBSERVATIONS . . .54
Eleventh Step— The Eleven . . . • 5$*
Twelfth Step— The Twelve . . . .58
Thirteenth Step— The Thirteen . . .62
Fourteenth Step — The Fourteen . . .63
Fifteenth Step— The Fifteen . . . .64
Sixteenth Step — The Sixteen . . . .66
Seventeenth Step — The Seventeen . . .68
Eighteenth Step — The Eighteen . . .70
Nineteenth Step — The Nineteen . . .71
Twentieth Step — The Twenty . . -73
CONTENTS.
YI. THE SECOND YEAR. OBSERVATIONS.— Continued.
Thirtieth Step — The Thirty . . . -77
Fiftieth Step— The Fifty . . . .80
Hundredth Step — The Hundred . . .82
VII. APPENDIX TO SECOND YEAR'S WORK . . .90
Section I. — First Unity . . . .90
Second Unity . . . • 94
Section II.— Third Unity . . . .96
Fourth Unity . . . .98
Fifth Unity . . . -99
Section III.— Tenth Unity . . . . 101
Eleventh Unity .... 104
Twelfth Unity . . . .105
Thirteenth Unity . . . .106
VIII. SECOND COURSE. THE THIRD YEAR . . .107
I. First Half of the Third Year . . . .107
A. The Pure Number — First Step . . .108
Second Step . , . 109
Third Step . . . in
Fourth Step . . .113
Fifth Step . , .115
Sixth Step . . .118
B. The Applied Number . . . .118
II. Second Half of the Third Year . . .122 " A. With Abstract Numbers— First Step : Numeration . 123
Second Step : Addition . 127
Third Step : Subtraction . 130 Fourth Step : Multiplication 133
Method for School Work . . . 136
Fifth Step — Division . 140
Oral Method— A. Without Remainder . . 140
— B. With Remainder . . 142
Written Method— A. Without Remainder . . 143
B. With Remainder . . 146
Concrete Numbers .... 150
IX. THIRD COURSE — FRACTIONS. THE FOURTH YEAR . 155
I. First Half of the Fourth Year . . . 155
First Step : Halves . . . .156
CONTENTS.
IX. THIRD COURSE. THE FOURTH YEAR.— Continued.
Second Step : Thirds .... 159
Third Step : Fourths . . . .162
Fourth Step: Fifths . . . .166
Fifth Step : Sixths . . . .169
II. Second Half of the Fourth Year . . .170
1. Unity1 ...... 171
2. Expansion and Reduction . . .172
3. Common Denominator . . .174
4. Number relations in fractional form . -175
INTRODUCTION.
PESTALOZZI was the pioneer who broke new ground in elementary instruction and led the way from mechanical, abstract methods to those which are more natural and psychological. He laid down the principle that all mathe- matical-knowledge is founded upon immediate observation, and therefore must proceed from the concrete to the general or abstract by means of innumerable examples. This dis- covery was not only of vast importance to pupils in the schools, but it opened up to teachers the psychological principles of all pedagogics.
In 1842, only fifteen years after Pestalozzi's death, ip- peared Grube's " Leitfaden fur das Rechnen in der Ele- mentarschule." In this work Grube gives a thoroughly developed system of teaching number. Pestalozzi was un- fortunately lacking in system. While he brought to light pedagogical principles, he developed no system of peda- gogics. He taught the world that the proper way to teach the child is to go directly to Nature, let her operate on the mind and follow her harmonious development. Grube found the germ of his system in Pestalozzi's teachings, but went farther than his master in that he broke away from the idea of teaching the four processes, addition, sub- traction, multiplication, and division, separately and in the order named. This is one of the great and most impor- tant features of the Grube system. Grube held that the
1 2 IN TROD UCTION.
four simple processes of arithmetic should go together in the small numbers, believing it to be the natural process of the mind.
By the use of objects the child is brought to see the relations of numbers until he is able to reproduce the re- lations without the objects. As the elementary work properly covers the period from the sixth to the tenth year, the period of observation, and as the method is purely elementary, Grube discusses only the first four years' work. His plan provides for three hours (full hours) per week. The end to be reached is a thorough knowledge of the fundamental rules and common frac- tions. His work is divided into three parts or courses : I. Whole numbers from i to 100, employing the first
two years.
II. Whole numbers above 100, employing the third school year.
III. Fractions, employing the fourth year.
He lays down the work definitely for each half-year, which we shall fully discuss later.
ADVANTAGES OF GRLJBE'S METHOD.
I. It recognizes the psychological fact that nearly all the knowledge obtained by the child in its earlier years is by means of the senses.
By observation and not by reasoning does the child gain his first knowledge of number. In the earlier years the child's reasoning powers must be brought very little into play. He is not yet especially ready for reasoning, and Nature did not intend that he shall gain knowledge at this early period through the reason. He is eager for knowl- edge, but such knowledge as is obtainable by the senses. He learns mechanically. He comes to know all the com- binations and manipulations taught him so as to give them with absolute accuracy and great rapidity without stopping to think.
II. As it makes the first year's work a study of the numbers 1 to 1 0 it lays a solid foun- dation.
The knowledge thus obtained becomes an acquirement which will be a methodical, substantial product. Accord- ing to our usual crude methods this may seem but little work for an entire year, yet by this scientific study the first ten numbers will be found to furnish ample work. They are the foundation of the whole number system ; all larger numbers are only a repetition, in a sense, of the first orders. The more thoroughly the numbers from one to ten are known, the surer and more rapid will be all later work in arithmetic. Let this foundation be well
14 ADVANTAGES OF GRUBE'S METHOD.
laid and the structure is well begun. Within these limits there is so much rich material for all-sided practical ap- plications that the teacher will find plenty to do to accom- plish the teaching of the first ten numbers in one year.
III. The Grube method progresses gradually and naturally according to the ability of the child.
It proceeds from the knowledge already gained to new knowledge by a very easy step. The knowledge possessed is utilized in mastering new knowledge. The child must not be subjected to mental over-exertion at any penod. This is especially dangerous during the first years. The Grube method does not require too much, and yet it gives endless and suitable variety so that the child does not tire of number.
IV. It develops the mental powers evenly and in all directions.
One-sided teaching should always be avoided. All de- velopment should be harmonious and natural. Grube considers each number by itself as an entirety and teaches all about it completely, thus building the mathematical structure step by step.
V. Elementary teaching of number should proceed from observation, or, better, it should proceed from things.
Grube's system calls for the use of things — balls, marbles, cubes, blocks. It uses objects repeatedly until the child is thorough master of the number and can make the com- binations abstractly. After a time the simple remem- brance of the objects used will always be sufficient to re- call to the consciousness the number until no object is longer necessary and the pure relations of numbers are fixed. Then the child needs no object, or intermediate process, to help him to know the number, but he knows
ADVANTAGES OF GRUBE' $ METHOD. Ij
it instantly, and the simple, fundamental processes are mechanical. So we pass from the object to the symbol, from this to the comprehension of number, and lead in this way the interest from the empire of objects over to the empire of the number forms.
VI. The Grube method makes the teaching of number an excellent language-lesson.
The answers and statements are to be complete sen- tences ; and as the subject is always kept within the ability of the child to comprehend, the number-lesson becomes one of the most valuable means of teaching language.
VII. The child acquires the habit of close observation.
As only that which is within the child's comprehension is brought before him, and as familiar objects are placed before him so frequently and so systematically, he acquires the habit of accurate and close observation. He learns to see what is brought to his notice, and to see all about it. This is one of the most important features of the Grube method, in that it is thoroughly psychological.
VIII. It develops and trains the attention.
As the child can understand the matter, it interests him, and interest is the first factor of attention. Because of its harmonious, all-sided development it cultivates the power of attention and leads the child to the habit of commanding and fixing the attention at will.
IX. It forms the habit of thoroughness in the child.
Mastering each number in all its details and possible combinations, it becomes, like a habit, a part of the very being of the child, until he is able to use it exactly as he uses the eye or the hand, without conscious thought.
l6 ADVANTAGES OF GRUBE S METHOD.
Beginning thus early in his school life to gain a thorough mastery of each step, he is led to desire mastery in other departments of learning and of practical life.
X. The Grube method gives pleasure and awakens a love for the study of number.
If the pleasure of work is not found in the work itself, all incentives and threatenings will be in vain. The desire to know a thing must be produced in the child himself. The impulse can begin in the young mind only when there is the consciousness of continual unity in the de- velopment of his powers, and then he will be driven by this impulse to further development by his self-activity. This method contains such unity and thus awakens in the child a love for the subject.
XI. It makes the child self-active in a proper manner.
Becoming complete master of a number, he is able to combine and operate with it by making original examples. Thus number becomes to him from the first a living, practical reality.
XII. The Grube method is a logical one.
It proceeds systematically and according to an order of sequence ; it is psychological in that it teaches the use of the senses, in that it proceeds from the simple to the more difficult, and in that it goes out from the known to unknown and makes constant use of the known ; it is practical because it gives a sure foundation for all future work in arithmetic, and because it brings the child im- mediately to measure and compare numbers and to make use of the relations of the same.
DIRECTIONS TO' TEACHERS.
THE first course includes whole numbers I to 100. Two years are required for this work, the first year being spent on i to 10 and the second on the rest, 10 to 100.
It must not be forgotten that the number-lesson must at the same time be a language-lesson. It is of the highest importance that the child give his answers in complete sentences, plainly spoken, with clear accent. Great importance must be attached to the explanation of every example from the outset. So long as the child is not master of the language necessary to express the operations performed with the number, he is not master of the representation or idea of the number itself, he does not know the number.
An example is not finished when the answer is found, but when it has been analyzed. The language may be taken as a safe test that a pupil has completely mastered a step.
So far as possible the pupil must be led to speak for himself and not to depend upon half the answer being put into his mouth by the teacher. Concert and individual answers must interchange in order that the interest of the class be maintained.
The uniform objects to be used are tbi fingers and blocks; for blackboard or slate use simple straight lines. Too many kinds of objects must not be used. The child has only a certain amount of strength and mental power which he can apply, and his interest must not be divided.
The mental comprehension of number is disturbed if things which awaken other ideas or desires are employed. The mind is capable of only a certain amount of interest, and when this interest is wholly or partly withdrawn but little can be expected for the particular thing at hand.
l8 DIRECTIONS TO TEACHERS.
For this reason, while teaching the abstract number, there should be but few things shown the child, and these should be simple and uniformly the same. The best things are blocks, which awaken little interest in them- selves, and these must be the chief objects used through- out. Other objects should be referred to after the child has a number well fixed, byway of application, but should seldom be shown him, at least during the study of the first numbers. Thus apples, nuts, etc., which awaken desire, stimulate the appetite, and thus divide the atten- tion, must not be used as objects in teaching number. All the interest which the child gives to the color, taste, etc., of objects is just so much lost to number.
The operations in any step consist simply of compar- ing and measuring what has been gained in the preced- ing steps with that which is new. The child proceeds from the known to the unknown, from the easy to the difficult ; hence the method must follow a psychological law. Th3 pure number is first learned, and then it is ap- plied to things in order to fix it and make its practical use apparent.
The work of teaching a number is not complete until the child has been taught to make neatly and with dis- patch the figure which stands for the number. This makes excellent employment for the children at their seats, and is a good preparation for written arithmetic, for which the foundation is being laid.
Go slowly — clo not measure the ability of the child by your ability; bring yourself down to the level of the child's mind ; be patient ; repeat everything many times ; review daily; use many examples and lead the children to make original problems.
Lastly, do not expect too much of the children ; give them the kind and quantity of food that they can digest, remembering that real, sound, intellectual growth is slow, especially at the beginning.
THE FIRST YEAR: 1-10.
THE work of the first year embraces a study of the num- bers, i, 2, 3,4, 5, 6, 7, 8, 9, 10. Each number is taken by itself, measured by those that precede it, compared and studied in all its possible operations.
FIRST STEP.
THE ONE.*
The one can only be measured by itself. The child has only to learn the idea of unity.
* Many insist that the child already knows the one, and that it is folly to spend time in teaching it. But it must not be forgotten that the child when he enters school must begin the study of number, must begin to form habits of correct speaking and thinking, must learn to observe carefully what is done, to tell what he sees done, and to answer in complete sentences. Then, too, the Grube system builds step by step, always making use of the knowledge possessed. The one is the first step. For these reasons and for these purposes a short time can be profitably spent with the one. It may be further added that the fact of the child's knowing the one will make it especially valuable in getting him to feel confidence in himself, an important matter for the child when he begins school.
2O THE FIRST YEAR: i-io.
I. The Pure (Abstract) Number.*
(The teacher shows the block) How many have I ?
P. You have one.
T. Charles, take this block. Now how many has Charles ?
P. Charles has one.
T. Charles, give it to me. Now how many has Charles?
P. Charles has none.
T. How many have I ?
P. You have one.
(Lead the children to watch the movements of the teacher, to describe accurately what he does.
Many other questions similar to the above can be given. Great care must be given to the language. Cor-
* The idea of a number is given the child by the use of an object or objects which are placed before him to see. He sees the concrete form, but does not name it. It suggests to him the abstract number. Thus when the child sees one block he is led to think of it as a one, two blocks as a two, three blocks as a three, etc. Do not let the child say " one block," " two blocks," but simply u one," u two," etc. Requiring the child to name the object withdraws his attention from the number itself. It is much easier for him to speak of the abstract number, and it is also easier for him to read and write it. The block gives him the picture of the number, and that is all that is wanted of it ; after getting the picture of it he only needs to use the name that stands for the num- ber. After becoming thoroughly familiar with the number, the child must go a step farther and use it with various objects and make ap- plication of it. This procedure is entirely psychological in that it pro- ceeds from the simple to the complex, from the known to the unknown.
FIXST STEP. 21
rect statements must be given. Never accept anything else.)
T. Now let us put this one on the board (makes a draw- ing of the block). How many have I ?
P. You have one.
T. But it takes too long to make this kind of a one ; we will let this I stand for our one. How many have I ?
P. You have I.
(The use of the mark is simply to save time. The child readily proceeds from the object [the block] to its pic- ture, and from that to the straight mark [ I ] as standing for one.)
T. Now we will learn to make something else that stands for one. Watch me and see what it is. It is this /. What does that stand for ?
P. That stands for one.
T. Now you may make that on your slates.
(By this method the pupils are taught the figure. The block, its picture, and the mark [I] must be taught as the number one. The child thinks of them only as the one. But he must learn to make the character which stands for the one ; namely, the figure [/]. No number must be left and no step be considered complete until the pupils have learned to make the figure which stands for the number. Great care must be taken by the teacher not to confuse the number with the figure.)
II. The Applied Number,
T. What thing do you find but once in the school- room?
P. I find one stove (desk, clock, etc.).
T. What have you one of at home ?
P. I have one dog (cat, sled, etc.).
(But little time need be given to the one, the aim being chiefly to get good expressions from the children, and to start them in the mode of thought to be pursued here- after with the other numbers.)
22 THE FIRST YEAR: i-IO.
SECOND STEP.
THE TWO. I. The Pure Number. — Measuring and Comparing.
T. How many have I ?
P. You have two.
T. ( Takes a block in each hand and holds up one.) What have I here ?
P. You have one.
T. And what have I here ? (the other hand.)
P. You have also one.
T. Now watch and see what I do. (Moves the hands holding the blocks slowly together?) What did I do ?
P. You put one and one together.
(Continue this operation until they can make a good statement of the fact.)
T. Very well. Now we will put this little story on the board. [] and Q make p Q That is good ; but I can make these marks [II] instead of the pictures [Q O], and that will be easier. What have I here [I I] ?
P. You have two.
T. But I can make something else which means two. It is this : 2. What have I now ?
P. You have two.
(The teacher should in this manner teach each figure. When the pupils have learned the number, and have seen it expressed by pictures of blocks and by marks, they are then ready to learn the simplest way of expressing it, that is, by the figure. Make the figure slowly before the class, so that they can imitate your manner of making it.)
SECOND STEP. 23
T. Now what have I ? (Two blocks shown together.)
P. You have a two.
(It is most important here that the child be taught that the two cubes is a 2 and not two blocks. The child must think of it in its entirety — a two.)
T. How many 2's have I ?
P. You have one 2.
T. How many times have I 2 ?
P. You have 2 one time or once.
T. How many does one 2 make ?
P. One 2 makes 2.
T. Now let me write — One 2 is 2.
(This will be reached only after repeated and patient efforts, but it pays to give the necessary time, because it is fundamental and all later work will be better and more easily done. Show the blocks and lead the child to see for himself. Always go back to the blocks when the child is in doubt. After the child has learned the figure it may be used to indicate the number. Until then use objects, pictures, or marks to stand for the number.)
T. Again, notice what I have. (The two blocks in one hand.)
P. You have 2.
T. Now see what I do. ( The teacher slowly takes i from
2.)
P. You took i away from 2.
T. And what does that leave ?
P. It leaves i.
T. Who will tell me the whole story about what I did ?
P. You took i away from 2 and it left i.
T. Good ; we will write that also.
D D less n leaves Q
Very well ; now we will write that another way. 2 less i leaves i.
Let us now see what else we can do. (The teacher holds the 2 blocks before the children and takes i away) What have I done?
P. You have taken i away.
( Teacher then takes the other away) Now what have I done ?
24 THE FIRST YEAR: i-io.
P. You have taken i away again.
T. How many times have I taken i away from 2 ?
P. You have taken i away two times.
T. Then how many I's are there in 2?
P. There are two I's in 2.
T. Now we will write that on the board. In 2 there are two i's.
Or we may say 2 divided by i makes 2.
(This last statement will be somewhat difficult for the child, and should not be attempted until by the method above given he has mastered the idea. Then we may teach this as another way of expressing the same thing. It is necessary that the child learn this way of expressing division, as it is the simplest expression and is the form that he will most often meet with. After mastering the idea it will not take long for the child to learn the form in common use.)
T. Now we will write what we have learned about 2.
1 and i make 2.
One 2 is a 2 or makes a 2. Two I's are a 2 or make a 2.
2 less i is i.
In a 2 there are two i's.* In a 2 there is one 2.
(At this time the signs can be taught. Write each ex- pression using the words, and underneath write the same expression using the signs.) For example : i and i make 2,
1 + 1 = 2.
2 divided by i makes 2. 2 -*- 1 = 2.
(Require the child to read the expression containing the words, and then that containing the signs. He must read both exactly alike. Hereafter use only signs, and require the pupils to make and use them. Go slowly, repeat many times, seek to get correct expressions, teach the
* I can take i from 2 twice.
SECOND STEP. 2$
children to observe accurately what you do and to de- scribe it.)
The four processes must be repeated until the pupils can give all operations with great rapidity. They must also be able to make their statements fluently, to read readily from the board, and to write exercises from dicta- tion, all to be included within the 2.
The pupil must be able to answer such combinations as follows, the teacher developing them by use of the blocks :
What number is found twice in 2 ?
Of what number is 2 the double ?
Of what number is i the half?
What number must I double in order to get 2 ?
I know a number which has i more than i. What is it?
What number must I add to i in order to get 2 ?
All possible combinations of the 2 should thus be given.
II. The Applied Number.
(The pupils are now prepared to apply their knowledge in practical examples embracing other objects than the blocks. It is no longer necessary to show them the ob- jects. Let them also make examples.)
Fred had 2 cents and spent i cent for cherries. How much had he left ?
A slate pencil cost one cent. How much will 2 pencils cost ?
Charles had I dime in his savings bank ; his sister had twice as many. How many had his sister ?
If a cake cost i cent, how many cakes can you buy for 2 cents?
James had 2 apples and Frank had half as many. How many had Frank ?
George had i marble and John twice as many. How many had John ?
26 THE FIRST YEAR: i-io.
THIRD STEP.
THE THREE.
I. The Pure Number. — Measuring and Comparing, (a)— With one.
(The cubes should be placed on a table where all the class can see what is done with them. It is inconvenient to operate with three or more blocks in the hands. There- fore, the blocks must be shown on the table.)
T. How many have I ? (Shows the 3 blocks separately.}
P. You have i + i + i.
T. How many is that?
P. i+i + i = lll.
T. Good ; I will print this little story on the board :
p + D + Q = D Q O
then
i + i + i = lll.
(When sure that the pupils know the 3, he writes 1 + 1 + 1 = 3, teaching the figure 3 by the same method as the 2 was taught. The symbol representing the number must be taught as soon as the pupils have ob- tained the idea of the number, and not before. After they have learned the figure which stands for a number, no other characters need be used to represent the number.)
T. Now tell me how many times I have i. (Picks uptke blocks one after another, the children counting?)
P, You have one 3 times.
THIRD STEP. 27
T. One 3 times makes how many? P. One 3 times makes 3. T. Then how many are 3 times one ? P. 3 times i are 3, or
3x1=3.
T. Now, what have I done ? (Holds up the 3 blocks and takes i away.)
P. You have taken i away from 3.
T. How many are left ?
P. There are 2 left.
T. Who will write that for me on the board ?
P. (Child writes 3—1 = 2.)
T. What else have I done ? (Takes another block away from the remaining 2.)
P. You have taken i away from the 2.
T. How many does that leave ?
P. It leaves i.
T. Now read 3 — 1 — 1 = 1.
Still, again, how many times can I take I away from 3 ? ( Takes i away 3 times, children counting?)
P. You can take i away 3 times.
T. Then how many ones are there in 3 ?
P. There are 3 ones in 3.
T. Then I will write i in 3 = 3 or 3-7-1 = 3. Now see what we have learned.
, fi + i + i=3 3Xi =3
3-1-1=1
U+;i =3
(b)— Measuring with 2.
(In the same manner the following tables will be devel- oped, the children always seeing the manipulations with the blocks and acquiring the statements of themselves. If a child hesitates and does not yet comprehend, go over the operations again. Call him up and let him handle the blocks until he has mastered the process and understands the relations.)
28 THE FIRST YEAR: i-io.
'2+1= 3, 1+2 = 3 I I 1 I x 2 + i =3
I |3 — 2 = i, 3—1=2
3-7-2 = 1, and i remainder.* 3 is I more than what ? 3 is i more than 2. 3 is 2 more than what ? 3 is 2 more than I. 2 is i less than what ?
2 is i less than 3.
In the same manner find that — I is 2 less than 3.
3 is 3 x i. i is i of 3.1
What 3 equal numbers make 3 ?
What numbers with i make 3 ?
The 3 contains a 2 and a i.
(All of these combinations have been wrought out using the blocks. Use only the blocks, the fingers and marks. Do not divide the attention by the introduction of new objects or things that awaken a desire of possession in
* This should be developed as follows :
T. How many have I here ? (Holding up the 3 blocks.)
P. You have 3.
T. How many ones in it ?
P. There are 3 ones.
T. Now see how many twos you can find. (Children take the blocks and find out for themselves.) How many are there ?
P. There is one 2.
T. Who will find another 2 ?
P. There is no other 2, there is only i left.
T. Then how many twos in 3 ?
P. There is one 2.
T. And what is there left ?
P. There is i left.
T. Now tell me the whole story.
P. In 3 there is one 2 and i left.
T. Very well, but I like the word remainder instead of left. Now try again.
P. In 3 there is one 2 and i remainder.
t To teach this I would take 3 blocks and place i block near them. Then ask which is the larger and how many times. Reverse the pro- cess, and ask which is the smaller, how many of the ones it takes to make a three, and finally, what part of the three the one is.
'THIRD STEP. 29
the child. Require the pupils to make the figures I, 2, 3 forwards and backwards, thus, i, 2, 3, 3, 2, i.)
Rapid Work.
Teacher gives examples orally as rapidly as possible, the children giving only the answer. How many are 3—1 — 1 + 1? 3x1—2x1—1=? 1+1x1+1—3=?
3—2+1+1—2=?
2 + 1+1—2 + 1 — 1=?
(Many examples of this kind should be given until the pupils are able to give the answer instantly. This is largely oral work.)
Combining.
From what number can you take the double of i and still have i remaining ?
What number is 3 times i ?
I use a number once, and then once again, and then once again, and obtain 3. What is the number?
n. The Applied Number.
If you would buy a 3-cent stamp, how many cents must you have?
Anna had a 3-cent piece, and bought 2 cents' worth of candy. How much change should she get ?
If a pencil cost i cent, how much will 3 pencils cost?
Charles has 2 apples and i apple. How many apples has he ?
Mary divided 3 flowers among her father, mother, and brother. How did she divide them ?
Martha, Fanny, and William have each i book. How many books have they ?
A boy had 3 apples and ate i apple. How many had he left?
David has i dollar. How much more must he earn in order to have 3 dollars ?
An uncle divided 3 dimes equally among his 3 nieces. How much did each receive?
My father gave each of his boys i dollar, and it took 3 dollars. How many boys had he ?
3O THE FIRST YEAR: i-io.
FOURTH STEP.
THE FOUR.
(It will not be necessary longer to pursue the question and answer method, as it has been sufficiently illustrated. The course indicated should be pursued for all subsequent numbers. Develop all the relations which come under " Measuring and Comparing" by use of the blocks. Lead the children to the statement of what you develop. The further you proceed the greater opportunity for variety, but limit that to such as may be obtained by use of such objects as have been heretofore specified, namely, blocks, fingers, marks.)
I. The Pure Number. — Measuring and Comparing.
(a) With i.
i i i i 4
i / r 1 + 1 + 1 + 1=4, 1 + 1 = 2 + 2 = 4. i t \ 4x1=4.
| ; 4_! _i-i = i.
I / [ 4 + i = 4.
(b) With 2.
r 2 + 2 = 4.
I I 2 J 2x2=4. I I 2 1 4—2 = 2.
[ 4+2 = 2.
FOURTH STEP. 31
(c) With 3.
f 3 + i = 4, i + 3 = 4- Ill 1x3 + 1=4.
I 4 — 3 = i, 4— 1=3.
4-5-3=1 and i remainder (3 in 4 once,
and i remainder.)
How many more legs has a horse than a man ? How many times the number of wheels of a bicycle has a wagon ?
How many more legs has a chair than a piano-stool ? 4 is how many more than 3 ? (Bring out the following facts with the blocks): 4 is i more than 3, 2 more than 2, 3 more than I.
3 is i less than 4, i more than 2, 2 more than i. 2 is 2 less than 4, i less than 3, i more than i.
i is 3 less than 4, 2 less than 3, i less than 2.
4 is 4 times i, 2 times (or double) 2.
i is one fourth of 4, 2 is one half of 4.
Of what equal and what unequal numbers is 4 made up ?
Rapid Work*
2x2 — 3 + 2X i + i — 2 doubled = ? 4— i — i — i + i — 3 is how much less than 4 ? 3—1 X2 — 3 + 2+1-1-2=?
2X2X I— 3 + 2— I X 2 -^ 3 = ?
1+1+1—2x4—3x2=? 4-^-2 + 1 + 1 — 3 — 1=?
(Continue the work as described before, always keeping within the combinations of the 4.)
Combining.
What number must I take 2 times in order to get 4 ? Of what number is 4 the double ?
* These expressions are intended only for dictation, the pupil work- ing as fast as they are dictated, and obtaining the result of each step with no reference to what is to follow. Thus 3 — 1x2 — 3 + 2+1-*-
2 would be when worked out from dictation 3— 1 = 2, 2x2 = 4, 4 —
3 = 1, 1+2 = 3, 3 + 1=4, 4-+-2 = 2. Of course the pupils do not re- peat the numbers as in the exercises here given, but obtain each result mentally as soon as the teacher dictates. The end sought is rapidity as well as accuracy. When expfessions are" written for the pupils,
32 THE FIRST YEAR: i-io.
Of what number is 2 the half ?
Of what number is i the fourth ?
What number can be taken twice away from 4 ?
What number is 3 greater than i ?
How much must I add to the half of 4 in order to get 4 ?
How many times i is the half of 4 less than 3 ?
If I take i from 4, how many times i have I left?
If I add i to i, what part of 4 have I ?
If I take 3 from 4, what part of 3 have I left ?
How much is the half of 4 more than the third of 3 ?
How much is the fourth of 4 less than the half of 4?
II. The Applied Number.
Caroline hajd 4 tulips in her vase which she neglected to water. One wilted, then another, then another. How many had she left ?
How many cents in 2 two-cent pieces ?
In a one-cent and a three-cent piece ?
How many cakes can you buy for 4 cents if each costs i cent?
When each costs 2 cents ?
If a top costs 2 cents, how much will 2 tops cost?
John paid for 2 cakes a three-cent and a one cent piece. What was the cost of each ?
One quart has 2 pints. How many pints in 2 quarts ?
Charles had 4 chestnuts, and gave Frank i and Henry i. How many had he left ?
William had 3 peaches and ate 2. How many had he left?
Anna received an orange on Monday, one on Tuesday, one on Wednesday, and one on Thursday. How many did she receive in all ?
care must be taken to have them mathematically correct and not to in- clude combinations beyond the 4, or the number which is being taught. The above expressions, as well as those which follow later under the head of "Rapid Work," may not always be correct when taken as a whole and considered as a mathematical expression; but they are cor- rect as dictation exercises, and a few are given for the purpose of sug gesting the method to the teacher. Such exercises will be found very valuable.
FOURTH STEP. 33
George had 4 apples and ate i each day. How many days did they last ?
If a pint of milk costs 2 cents, how many pints will 4 cents buy ?
What part of 4 cents is i cent ?
What part of 4 cents is 2 cents ?
What part of i quart is i pint ?
What part of i gallon is i quart ?
My father had 3 cows and bought i more. How many has he now?
If he sells 2 cows how many will he have left ?
Mary had i pin and found 3 more. How many had she then ?
If a letter requires 2 two-cent stamps, what will it cost to mail it?
I had 4 quarts of milk, and sold 3 quarts. How many quarts had I left ?
A mother has 2 sons and 2 daughters. How many children has she?
I have 4 pears which I wish to divide equally between my 2 sisters. How many can I give each ?
(Give many examples from the life of the children until they can make application of all the relations contained within the 4. Let them also make examples. Do not let them leave a number for a new step until they can make the figure, perform rapidly all the combinations whether given to them orally, or written on the board, make complete statements of such combinations, and apply them in their own surroundings. The great suc- cess of this method depends upon thoroughness in these particulars.)
34
THE FIRST YEAR: i-io.
FIFTH STEP.
THE FIVE.
(As it is difficult to work with five blocks in the hands, they should be manipulated on a table before the children so that they can be plainly seen. It would be better still if the children could be gathered around a large table and each have the same number of blocks, so as to perform the same work as the teacher does with hers. The blocks can very well be dropped and only the fingers used after this.)
I. The Pure Number. — Measuring.
(a) With i.
' + 1 + 1 + 1 + 1=5 5 x i = 5, i x 5 = 5 5 — i — i — i — i = i
FIFTH STEP. 35
(b) With 2.
oo 2 f2 + 2 + i =5 oo 2 \ 2 X 2 + ' = 5
0/1 5-2-2 = 1 [5 + 2 = 2(1)
(<:) With 3.
r3 + 2 = 5, 2 + 3 = 5 ooo 311x3+2=5 oo 215-3 = 2, 5-2 = 3
I S •*• 3 = i C2) (<0 With 4.
f4 + i = 5, i +4 = 5 oooo 4 J i x 4 + i = 5 o / ] 5-4= i, 5 - i =4
[5+4=1 (i)
After the pupils are familiar with all the combinations they may be required to fill out missing numbers ; for ex- ample :3 + =5, 4x1 + =5, 4 x +1 = 5.
5 is i more than 4, 2 more than 3, 3 more than 2, 2 more than i.
4 is i less than 5, i more than 3, 2 more than 2, i more than 3.
3 is 2 less than 5, i more than 2, 2 more than I. 2 is 3 less than 5, i more than i.
5 = 5x1.
i = | x 5 (i is the fifth part of 5.) The five consists of unlike numbers, 3 + 2, and of 2 like numbers and i unlike number, 2 + 2 + 1.
Rapid Work. 5 — 2 — 2 + 2—1 X2, the half, less i =
2X2 + 1— 3 X I X2 — 3— 4 = ?
Combining.
How much must I add to 2 in order to get 5 ?
How much must I take away from 5 in order to get 2 ?
What number is the fifth part of 5 ?
How many times 2 have I added to i to get 5 ?
36 THE FIRST YEAR: i-io.
I have taken from a number twice 2 and have i left What is the number ?
If to 2 times 2 I add i, what do I get ?
I take 3 times i from a number and have 2 left. What is the number?
What number shall I add to 2 to get 5 ?
If I add i to a number I get 5. What is the number?
What must I add to the half of 4 to get 5 ?
Take 2 times i, then add it to 2 ; this lacks how many of 5?
4 times one third of 3 added to i makes what ?
Take 3 from 5 and how many will 2 times this lack of 5 ?
(These combinations must be multiplied to insure readi- ness and accuracy. While in the examples above only those involving 5 are given, review of the numbers already learned must never be forgotten.)
II. The Applied Number.
How many 3-cent and 2-cent loaves of bread can you buy for 5 cents ?
John received from his father a 3-cent and a 2-cent piece. He bought 2 sheets of paper at 2 cents each. How much change did he get?
Bertha knit 3 times around, and her sister 2 times more than she. How many times around did the sister knit?
A father divided 5 cherries among his 3 children. The youngest got only i, and the other two each the same number. How many did the others get?
Charles gave his 2 sisters each i apple, his brother 2 apples, and had i for himself. How many did he have at first?
The milkman had 5 quarts of milk, and sold 2 quarts to Mrs. Wilson, i quart to Mrs. Rand, and I quart to me. How many quarts did he have left?
John has 2 marbles, David has 3. How many have both ?
John wins 2 from David. How many has David left ? How many has John now ?
How many apples can you buy for 5 cents if i apple costs you 2 cents ? How many cents will you have left ?
SIXTH STEP,
37
(These examples must be multiplied, making use of things with which the children are familiar. Continue this practice until they can perform all the operations with absolute accuracy and great rapidity. Whenever a child is in doubt, take him to the blocks and make it clear to him. Or, better, lead him to find out the truth himself by use of the blocks.)
SIXTH STEP.
THE SIX.
(The pupil is now able to fill out the operations himself according to the method already pursued. If he can do that readily, it may not be necessary to use the objects longer, except when the child is puzzled. Many Germans use no objects in teaching number after the four. Just as soon as the child is familiar with the method, and can grasp the idea of the number without the visible object before him, the objects should be abandoned. But
THE FIRST YEAR: i-io.
blocks should always be at hand to be used in removing doubt in the child's mind, when a point is not perfectly clear to him.)
I. The Pure Number.— Measuring.
1+1+1+1+1+1=6 6 x 1=6, i x6 = 6 6— i— i— i— 1—1=1 6-i-i =6
(Teach each figure as soon as the child has use for it.)
|
oo 2 o o 2 oo 2 |
1Z, -f «5 f -^ = <->. ^ f >! 3x2 = 6 6—2—2=2 f |
= 4» 4 1 |
|
6 + 2 = 3 |
||
|
ooo 3 |
[3 + 3 = 6 1 2 x 3 = 6 |
|
|
ooo 3 |
|673=3 |
|
|
[4 + 2 = 6, 2 + 4 = 6 |
||
|
oooo 4 |
1 i x 4 + 2 = 6 |
|
|
oo 2 |
16-4=2 |
|
|
[6^-4=1 (2) |
||
|
[5 + 1=6, 1 + 5=6 |
||
|
ooooo 5 |
I i x 5 + i =6 |
|
|
0 / |
l6-5=i |
|
|
U-5 = i(i) |
||
|
6 = |
5 + i, 4 + 2, 3 + 3, 2 + |
4, + 5 |
|
5 = |
6— i, 4 + i, 3 + 2, 2 + |
3. +4 |
|
4 = |
6 — 2, 5 — i, 3 + i, 2 + |
2, +3 |
|
3 = |
6 — 3, 5 — 2, 4 — i, 2 + |
I, +2 |
|
2 = |
6 - 4, 5 - 3, 4 - 2, 3 - |
I, + I |
|
I = |
6 - 5» 5 ~ 4, 4 - 3> 3 - |
2, 2 — I |
|
6 = |
6xi, 3x2, 2x3 |
|
|
3 = |
i x 6 (is half of 6) |
|
|
i x 6 |
||
|
i = |
i x 6. |
SIXTH STEP.
39
Of what 3 like numbers is 6 composed ? Of what 3 unlike ? The following tables can be profitably given :
(*)
6 - i = 5 6 — 2 = 4
6~3 = 3 6 — 4 = 2 6-5 = 1 6-6 = 0
5 - 2 = 3 3-2 = 1
4 = 6
3 + 3 = 6
5 + i=6 (c) 1+2 = 3
Odd numbers : i, 3 5 ; 5, 3, 2 + 2 = 4
4 + 2 = 6
Even numbers : 2, 4, 6 ; 6, 4, 2. + 1=2 + 2 = 3 + 3 = 4 + 4 = 5 + 5 = 6
(g) 6 = 6
6 = 5 + 1 6 = 4 + 2
6 = 3 + 3 6 = 2 + 4 6=1+5
(Many other tables can be made embracing multiplica- tion and division as far as the 6. The pupils can easily be led to make these themselves. Dictate numbers, and require the pupils to name or write them promptly, so as to test their knowledge of the order of the numbers, and their ability to make the figures. For example : Write the numbers from j to 6. What number comes after 4?
|
6-: 4~- |
2 = 4 I = 2 |
|
|
2 — : |
I = 0 |
|
|
2. |
||
|
(/) |
6- |
= 5 |
|
5- |
= 4 |
|
|
4 — |
= 3 |
|
|
3- |
— 2 |
|
|
2 — |
= I |
|
|
I — |
= 0. |
|
|
w |
6- |
i = 5 |
|
6-: |
2 = 4 |
|
|
6- |
5 = 3 |
40 THE FIRST YEAR: i-io.
What comes before 4? What comes between 2 and 4? Express the number that comes after 5. Express the number between 3 and 5.)
Rapid Work.
i x 2 + i x 2 — i x 1 — 5 + 5 = ?
4 + 2 — 3 is how much less than 6 ?
3 — 2 x 5 + i — 4 x 2-7-2 + 4—5 = ?
5 — 4 + 3-5- 2 + i x 2-*-3 = ? «
These should be given orally to the pupils, or written on the board as rapidly as they are able to work them. Allow no counting of the fingers or use of objects in this operation. The pupils must know eveiy operation, and be able to perform rapidly and accurately without any hesitation. This is a test of the thoroughness of the work.
Combining.
What number can you take 3 times from 6 and twice from 4?
How many times i has half of 6 more than half of 4, and how much less than 5 ?
I have taken a number twice away from 6 and have 2 left. What is the number?
How many times is i of 6 contained in 4 ? The half of 4 = what part of 6 ? What number is 3 times 2 ?
II. The Applied Number.
How many times i cent, 2 cents, and 3 cents in 6 cents ?
How many quarts in 6 pints ?
What will 3 liters of milk cost at 2 cents a liter ?
William got 3 tops for 6 cents. What did i cost?
I have 6 apples in 3 pockets. How many apples in each pocket ?
How many lead-pencils at 2 cents each can I buy for 6 cents ?
I gave each of my 3 sisters 2 oranges. How many oranges did I give away ?
A father divided 6 dollars equally among his 3 children. How much did each get ?
SEVENTH STEP. 4!
Fanny took 6 cents to the store and bought 2 candies at 2 cents each. How much money did she have left ?
Joseph gave Charles 2 marbles, Henry i, and had 3 left. How many had he at first ?
I have 5 dollars and borrow i more, and lend 2. How much have I left ?
If David earns 2 dimes a day, how much will he earn in 3 days ?
Mary gave each of her 5 friends a candy, and had i for herself. How much did she have at first ?
I have 2 books on the table, i on the chair, and 3 in my book-case. How many have I in all ?
Jessie buys 3 pints of milk at 2 cents a pint. How much is the cost ?
SEVENTH STEP.
THE SEVEN.
1+1+1+1+1+1+1=7
7x1=7
7— i— i— i— i— i— 1=1
7-7-1=7
42 THE FIRST YEAR: i-io.
00 2 f 2+2+2+1=7
oo 2 I 3x2+1=7
OO 2 ] 7—2—2—2=1 O 1 ( 7-H2 = 3(l)
o o o 3 f 3 + 3+ 1=7 ooo I ^3 + r=7 oo.o ,\ >+l-lul
f 4 + 3 = 71 3 + 4 = 7
oooo 4 1x4+3=7
ooo 3 ' 7 — 4 = 3
[ 7 + 4 = i (3)
(5 + 2 = 7; 2 + 5 = 7 i x 5 + 2 = 7 7 — 5 = 2 7 - 5 = i (2) *
ooooootf f 6 + 1=7; 1+6 = 7 o/ 1x6+1=7 1 7-6=1 ( 7 -v- 6 = i (i)
In what ways can a father divide 7 apples among 2, 3, 4 children ?
7 = 6 + i, 5 + 2, 4 + 3, 3 + 4, 2 + 5, i + 6.
6 = 7 - i, 5 + i, 4 + 2, 3 + 3, 2 + 4, i + 5. 5 = 7 — 2, 6 — i, 4 + i, 3 + 2, 2 + 3, i + 4. 4 = 7 — 3, 6 — 2, 5 — i, 3 + i, 2 + 2, i +3.
7 = 7 x i, i = | of 7.
What like numbers does 7 contain ?
(Make tables like those on page 39, and require the pupils to do the same. Follow this course with all suc- ceeding steps.)
* I would have the pupils find out how many fives in 7 by use of blocks. Then express 7-1-5 = 1 with a remainder of 2. Then write the remainder in parentheses, as : 7 •*- 5 = i (2). The children will very easily learn that the number in parentheses is the remainder.
SEVENTH STEP. 43
Rapid Work. 3x2 + 1 — 2X 1 — 3x3 + 1?
2+I+2+I + I? 1+2+4 — 3 — 2X3?
Combining.
From what number can you take i seven times ? What number contains 7 seven times ? To what number must I add 3 x 2 to get 7 ? I take a number 3 times and get i less than 7. What is the number? How many times i is 7 greater than the double of 2 ?
(The double of 2 is 4. 7 is 3 more than 4, and has therefore 3x1 more than 4. Therefore 7 is 3 x i greater than the double of 2.)
II. The Applied Number.
A week has seven days. What is the name of the first, the second, the fifth, the third, the seventh day?
I took a trip lately that lasted just a week ; how many days was I on the journey ?
How much money did I need for the journey, if I used one dollar a day ?
If you put i cent in your savings bank each day, how much will that make in a week ?
How many threes would that make?
How many quarts in 7 pints ?
George was sent by his mother to fetch 2 3-cent loaves of bread. She gave him 7 cents. Was that enough ? How much did he have left ?
Henry took a 5-cent and a 2-cent piece and bought 3 candles at 2 cents each. How much money should he bring back ?
(The teacher must multiply examples of each kind until the pupils have mastered the number. Never leave a number to take up a new one until the former is thoroughly learned.)
44
THE FIRST YEAR: i-io.
EIGHTH STEP.
THE EIGHT.
I. The Pure Number. — Measuring.
O 7
O /
O /
O /
o /
o /
o /
o /
o o 2
o o 2
o o 2
o o 2
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1=8
8xi=8
8—1—1—1—1—1—1—1=1
2 + 2 + 2 + 2 = 8
4X2 = 8
8 — 2 — 2 — 2 = 2
8 -f- 2 = 4
|
0 0 |
o o 0 |
0 o 0 |
3 f 3 M |
3 2 8 8 |
x |
3 3 3 3 |
+ 2 + 2 -3 = 2 |
= 8 = 8 = 2 (2) |
|
4 |
+ |
4 |
= 8 |
|||||
|
0 |
o |
o |
2 |
X |
4 |
= 8 |
||
|
o |
o |
0 |
8 |
— |
4 |
= 4 |
||
|
8 |
-r- |
4 |
= 2 |
EIGHTH STEP. 45
|
00000 o o o |
.1 3i |
5 + 3 = 8, 3 + 1x5+3=8 8-5 = 3 8 -*- 5 = i (3) |
5=8 |
|
( |
6 + 2 = 8, 2 + |
6 = 8 |
|
|
000000 |
6 |
i x 6 + 2 = 8 |
|
|
O 0 |
2 |
8-6 = 2 |
|
|
( |
8 •*- 6 = i (2) |
||
|
( |
7 + i = 8, i + |
7 = 8 |
|
|
O 0 0 0 0 0 0 |
7 |
1x7+1=8 |
|
|
0 |
/ |
8-7 = 1 |
|
|
8 •+- 7 = i (0 |
8 = 7 + i, 6 + 2, 5 + 3, 4 + 4, etc.
7 = 8 — i, 6 + i, 5 + 2, 4 + 3, etc. 6 = 8 — 2, 5 + i, 4 + 2, 3 + 3, etc.
5 = 8 — 3, etc., completing the table.
8 = 2 x 4, 4 x 2, 8 x i. i=ix8, 2=Jx8, 4 = ^x8.
The 8 consists of 4 equal numbers, for it equals 4x2, and of 2 equal numbers, for it equals 2x4, also of 2 equal and one unequal number, namely, 2x3 + 2.
Rapid Work. 8 — i — 2 — i — 2 ?
I+2 + I+2 + 2 — 5? 2X2 + 3 + 1-^-4 + 2?
4 + 3-5 x 4-5-2-3 x 7? Combining.
What number contains the fourth of 8 three times ?
What is the difference between a half of 8 and a half of 6?
What number must I double in order to get 8?
What number must I take 4 times in order to get 8 ?
What number has 5x1 more than 3 ?
(The number which has 5 x i, or 5, more than 3, is 5 + 3 = 80
Take the third of six 4 times.
46 THE FIRST YEAR: i-io.
II. The Applied Number,
How many twos, threes, and fours are found in 8 ?
How many gallons in 8 quarts ?
How many quarts in 8 pints ?
How many weeks in 8 days ?
William wanted to buy 4 spools of thread at 2 cents each. How much money must he have? He paid the sum in 2-cent pieces. How many did it take ?
If 2 gallons of molasses cost 8 dimes, what will i gallon cost ?
If i bushel of corn costs 8 dimes, what will i peck cost?
(i bushel contains 4 pecks. If 4 pecks cost 8 dimes, i peck will cost 2 dimes.)
John has a 5-cent and a 3-cent piece. If he buys 3 tops at 2 cents each, how much money will he have left ?
A merchant has a piece of cloth 8 feet long from which he cuts off 2 yards. How much remains ?
A yard = 3 feet, 2 yards will = 6 feet. If he cuts off 6 feet there will remain 2 feet.
(In this way, as soon as possible, parts of the tables of compound numbers should be introduced, and the chil- dren will thus gradually become familiar with the entire tables. Make examples from every-day life which will be suggested by the errands the children must do or by the employment of their parents. Bring their knowledge of number into immediate, practical use.)
NINTH STEP
47
NINTH STEP.
THE NINE.
The Pure Number. — Measuring.
i+i+i+i+i+i+i+i+i=9 9x1=9
9—i—i—i—i—i—i—i—i=i 9-*- i =9
2+2+2+2+1=9
4X2+1=9
9 — 2 — 2 — 2 — 2 = 1
9H- 2=4(1)
48 THE FIRST YEAR: i-io.
:::j{ Xi-*='
«°-M s^s-r1 :::: if «i±|=f
1 9-4-4=1 [ 9 -i- 4 = 2 (i)
(Measure 9 with all the other numbers according to the plan followed with the preceding numbers.)
9 = 8+ i, 7 + 2, 6 + 3, etc. 8 = 9 — i, 7 + i, 6 + 2, etc. 7 = 9 — 2, 6 + i, 5 + 2, etc. 6 = 9 — 3, 5 + i, 4 + 2, etc.
(Continue in the same way ; also construct tables as on
P- 39-)
9 = 9 x i, 3 x 3.
9 can be separated into —
3 equal numbers, 3 + 3 + 3 ;
4 equal and i unequal numbers, 2 + 2 + 2 + 2+1;
2 equal and i unequal numbers, 4 + 4 + i ;
3 unequal numbers, 2 + 3 + 4 ; 2 unequal numbers, 5+4.
Rapid Work.
3X3— 3— 2x2— 5— 1x4? i +2+3-3—2+3—4? 9-j- 3 + 4 + 2 — 1 -r-4 x 3?
(These should be given very rapidly, and should involve all kinds of operations which the pupils have already had. The pupils should be able to follow as rapidly as the teacher can dictate the combinations, and be ready with the answer as soon as the teacher ceases.)
NINTH STEP. 49
Combining.
How many times i is 4 x 2 less than 3x3?
What number can I take four times away from 9 and have i left?
What part of the 6 is the third part of 9 ?
Separate 9 into two unlike numbers, one of which is i greater than the other.
II.— The Applied Number.
How many gallons in 9 quarts ?
How many weeks in 9 days ?
How many 2's, 3's, 4's, and 6's in 9 cents ?
Mary had 9 verses to learn. She learned 3 verses each day. How many days did it take her ?
Her brother wrote 9 pages in 3 days. How many each day?
What cost 3 sheets of paper if i sheet cost 3 cents ?
William was to fetch his father 4 sheets of paper, each sheet costing 2 cents. He had 6 cents and 3 cents. How much money must he bring back ?
The milkman asks 3 cents a pint for milk. How many pints can I get for 9 cents ?
I give Fannie 4 2-cent pieces and a i-cent piece. She gave her sister 3 cents and her brother 4 cents. How much had she left ?
A boy buys 2 lemons at 2 cents each, and 1 orange for 4 cents. He has i cent left. How much had he at first?
SO THE FIRST YEAR: i-io.
TENTH STEP.
THE TEN.
(We have now reached the first number which must be considered as another kind of unity, or another kind of One,— the Ten. So we write again the figure i, but to show that this i contains ten times as much as the first i, we move it one place to the left, and say, this i is a ten. The vacant place of the simple i will be indicated with the cipher, so — 10. The pupils should be taught as follows :)
Show me 10 fingers. Now i finger. Indicate the i finger with a figure. Indicate the fingers of both hands with a figure.
(The children are shown how they must write i in the ten's place, and the cipher at the right for unit's place, place.)
(Ten splints also may be used, and then bound into a bundle to represent i ten. This must be continued'until the pupils comprehend the 10 and its relation to the unit.)
TENTH STEP. 5 1
Measuring*
(Grube now abandons the writing out of the tables, such as 10 with i, 10 with 2, 10 with 3, etc., as practised with all preceding numbers. The teacher can require the pupils to do so if he deems it desirable ; but it will prob- ably be found unnecessary to go farther in this direction.)
The 10 consists of two equal numbers, 5 + 5 ; of 5 equal numbers, 2 + 2 + 2 + 2 + 2; of 2 equal and i unequal numbers, 3 x 3 + i ; of 4 unequal numbers, 1+2 + 3 + 4.
1 is the half of 2 6 is 3 times 2
the third of 3 2 times 3
the fourth of 4, etc. 9 is 9 times i
2 is the half of 4 3 times 3
the third of 6 8 is 8 times i
the fourth of 8, etc. 4 times 2
10 is 10 times i 2 times 4
5 times 2 7 is 7 times
2 times 5 5 is 5 times
3 is the half of 6 4 is 4 times
the third of 9 2 times
4 is the half of 8 3 is 3 times
5 is the half of 10 2 is 2 times
6 is 6 times i i is i times
What numbers go without remainder into 10, 9, 8, 6, 4? What are only divided by i and themselves ? (The prime numbers, i, 3, 5, 7.)
Rapid Work.
2x3 + 2 + 1—6 + 5 — 3x2-7-5?
10 — 7x3 + 1-^-5x4+14-3 + 6?
2x2 + 2 + 3 — 7x5-5-2 + 4-4-3?
10— 2— I— 2— I— 2— I?
i + 3 + 3 + 4?
Combining.
What number has i more than the double of 3 ? How much is 2 x 5 greater than the difference between 3x3 and the double of 4?
52 THE FIRST YEAR: i-io.
A father divided 10 apples among his 4 children so that each older received i more than the next younger. How many did each receive ?
(The 10 consists of the 4 unequal numbers, i + 2 + 3 + 4; each is i larger than the next below. Therefore the father could give the youngest i apple, the next 2, etc.)
N. had learned 4 proverbs. His brother said to him, " I know twice as many again as you and 2 more." How many did he know?
Herman said, "I am 5 times as old as my brother." The brother was 2 years old. How old was Herman ?
II. — The Applied Number.
In 10 pints how many quarts ?
In 10 days how many weeks and days ?
In 10 cents how many 2-cent pieces? 3-cent pieces? 5-cent pieces ?
Fred had 6 cents, 3 cents, and i cent. He went to a stationer and bought 4 sheets of paper at 2 cents a sheet, and 2 sheets at i cent a sheet. Did he have money enough ?
Karl had the same amount of money, and bought 3 sheets at 3 cents each. How much did he have left ?
How many pints of milk can be bought for 10 cents if i pint costs 2 cents? 5 cents?
How many biscuits can I buy for 10 cents at 2 cents each ?
A dime has 10 cents. How many 2-cent pieces equal a dime ? How many 5-cent pieces ?
Ten dimes make a dollar. How many 2-dime pieces in a dollar ? 5-dime pieces ?
This completes the first school year, and the most im- portant steps in number have been mastered. One year is not too long if the work has been thoroughly done. Of course the child knows only the numbers i to 10 ; but he knows them and can use his knowledge ; therefore they are of some value to him. Of what use to the child if he
TENTH STEP. 53
count to 100, but could not separate the number 9 into its elements and use them ? The process of " measuring " must be thoroughly mastered by the child. He must be- come so thoroughly acquainted with all of the operations of each step that without hesitation he can perform them. The eye becomes trained by use of the blocks to habits of accuracy, and the child learns to be attentive. By means of the eye and by handling, he gains an idea of the num- ber and its combinations. Thus the objects appeal to his senses, and he is soon able to pass over from the number obtained from concrete things to the abstract. When objects no longer are necessary to give the idea, the concrete objects must not be named, as that withdraws the attention from the abstract number itself. Having mastered the abstract number, the child is able to apply it with concrete examples. This will not be difficult, as little exam pies are made from the every-day relations of life.
At the 10, if not before, the use of objects should be aban- doned entirely. The child is now able to gain the abstract idea without the help of objects. Objects become a cum- brance as soon as the child can do without them. Some think they can be abandoned after the 4 or 5. Each teacher must settle the question with the class he may have. When the class are able to get the idea without the objects, then is the time to give them up. Certainly that end will be reached when the 10 is completed.
Reviews must be frequent, and every step must be mas- tered before proceeding to the next. After knowledge has been obtained by illustration and observation it must be thoroughly memorized.
54 THE SECOND YEAR: 10-100.
THE SECOND YEAR: 10-1.00.
(See Appendix, p. 90.)
OBSERVATIONS.
1. Grube says: " Fingers and lines continue to be used for illustration. One can well say that Nature has given to man the decimal system of number in the hand."*
2. The procedure in the following steps is the same as that given for the smaller numbers. Multiplication and division should be given both as written and oral work, while addition and subtraction need only be oral. The pupils must continue the " measuring" of each new num- ber by the numbers from i to 10 until the greatest mechan- ical skill is reached. This mechanical skill is connected with the greatest self-activity on the part of the pupil.
3. For the operations with the pure as well as the ap- plied numbers, a greater diversity in the manner of expres- sion in the examples can be employed, in order that the pupil may become more and more free from the formulas of the earlier work. Applied examples should be gathered from the pupil's surroundings, from material with which he is familiar. Here is an excellent opportunity to lead the pupil to invent examples, and the privilege of giving an example to the class may be accorded to the pupil first solving a given example. This originating of examples will not be difficult, because the pupil always proceeds from the preceding step, and only adds to the already known.
* While Grube continued to use objects to illustrate the number, his modern followers abandon them, as we have already shown. Sim- ple lines may be used profitably, as the following pages illustrate, to show the relations of units to tens, etc. But the numbers do not need to be longer taught by the use of objects, as heretofore.
ELEVENTH STEP. $5
ELEVENTH STEP.
THE ELEVEN.
I.— The Pure Number.
10 times one or 10 ones make i ten. If I have 10 ones taken together, I have i ten and no (o) ones or units besides.
llllllllll = i ten and o ones = 10.
If another one is added, it belongs to the second ten.
I I I I I I I I I I 10+ i = 11.
What is the i at the right ? What the i at the left ? Where does the one (unit) belong ? How many ones must be added in order to make the second ten full ? What do we call i ten and i one in one word ? What is the 1 1 ?
50 THE SECOND YEAR: 10-100.
Oral. Measure with i.
i + i + i + etc. =11. (i + i = 2, 2 + i = 3, 3 + i = 4, etc.)
II X I = II.
1 1 — i — i — i, etc. = i. (i i — i = 10, 10 — i = 9, 9 T- i = 8, etc.) ii -j- i = ii.
Measure with 2.
2 + 2 + 2 + 2+2+1=11.
5X2 + 1 = 11.
11—2 — 2 — 2—2 — 2 = 1.
Measure with 10.
10 + i = ii.
i x 10 + i = ii (i ten + i one = n).
11 — 10 = i.
ii -5- 10 = i (i) (In ii is i ten + i one).
(Each pupil gets by this means a principle, and as he knows the course to follow, all assistance from the teacher must cease.)
All numbers hereafter are measured only by the num- bers from i to 10.
|
Written. |
|
|
\\ \\ \\ \\ 11 = 10+ i |
= ii |
|
1 |
|
|
II = II X I II |
-*- i = ii |
|
5x2 + 1 |
' 2 = 5(1) |
|
3x3 + 2 |
3 = 3 (2) |
|
2x4 + 3 |
4 - 2 (3) |
|
2x5 + 1 |
5 = 2 (i) |
|
x 6 + 5 |
6 = i (5) |
|
x 7 + 4 |
7 = i (4) |
|
x 8 + 3 |
8 - i (3) |
|
x 9 + 2 |
9 = i (2) |
|
X 10+ I |
10 = I (I) |
ELEVENTH STEP. 57
Comparison (oral). 1 1 = 10 + i, 9 + 2, 8 + 3, etc. 1 1 = 1 1 x i, i = ^ x 1 1 (i is ^ of 1 1). Form 1 1 from 3 equal and i unequal number,
4 equal and 2 unequal numbers.
5 equal and i unequal number. 4 unequal numbers.
Rapid Work.
I have 6 cents, 3 cents, i cent, and i cent, and give away 4 cents, 2 cents, and 3 cents. How much have I left ?
ii — 2 — 3 — 4 + 3 — i-*-2 x 5?
II— 2 — I— 2 — I— 2— I? 2X 5 + 1—9x4 + 3 — 7+4?
ii — 5 + 3 + % + 5 X4 — 3x2 + 1? (Let all the possible combinations within the 11 be given orally until the pupil can reckon as rapidly as the teacher can give them.)
Combining.
How many I's must I add to 5 x 2 to get 11 ?
From what number must I take 3 x 3 to get 2 ?
How often can I take the fourth part of 8 away from n?
What number is i ten greater than i ?
What is the difference between 4x2 and 1 1 ? «
II. The Applied Number.
n cents = 3 three-cent pieces and i two-cent piece.
I 1 pints contain 5 quarts and i pint. 1 1 days = i week and 4 days.
N. made a journey of 11 days and used just II dollars. How much was that per day ?
B. used on a journey 1 1 dollars. If he used i dollar a day, how many days was he on the way?
Fanny had 2 five-cent pieces and i cent. She bought 2 lead pencils at 3 cents each and 2 at 2 cents each. How much money had she left?
A boy was given 2x4 + 3 cents for doing errands. He put 5 cents + 3 cents in his bank, spent 2 cents for candy
58 THE SECOND YEAR: 10-100.
and gave the rest to his sister. How much did he give her?
If milk costs 4 cents a quart, how many quarts can I buy for ii cents and how many cents would I have left?
A mother gave one son 3x2 cents, and another 2x2 4- i cent. How much did she give both ?
Henry had 5 apples, John gave him 2 and William gave him 2x2. How many had he then ?
Mr A. had n nuts which he divided among four boys. To the first he gave two and to the others each an equal number. How many did each of the others receive ?
TWELFTH STEP.
THE TWELVE.
I. The Pure Number.
I I I I I I I II 1 = 10 + 2=12
1 1
Oral.
i + i + i + i, etc., = 12
12 x i = 12
12 — i — i — i, etc., = i
12-*- I = 12
TWELFTH STEP. 59
2 + 24-2 + 2-1-24-2 = 12
6X2=12
12 — 2 — 2 — 2 — 2 — 2 = 2 12 -T- 2 = 6
3 + 3 + 3 + 3 = 12
4 x 3= 12
12-3-3-3 = 3 12-5-3 = 4
(Measure in the same manner by all the numbers as far as 10.)
Written.
12= 12 X I 12 -5- I = 12
6X2 2=6
4x3 3=4
3><4 4 = 3
2X542 5 = 2 (2)
2x6 6=2
1x74-5 7 = i (5)
1x84-4 8 = i (4)
1x9 + 3 9 = i (3)
I X 10+ 2 10 = I (2)
(i ten + 2 units.) (In 12 are i ten and 2 units.)
12 = 11 + i, 10 + 2, 9 + 3, etc.
12 is i more than n, 2 more than 10, etc.
1 is the twelfth part of 12.
2 is the sixth part of 12.
3 is the fourth part of 12.
4 is the third part of 12. 6 is half of 12.
From what equal numbers can 12 be formed? From what unequal numbers ?
Form 12 from 3 numbers, the first of which is 2 and the following always increasing by 2.
60 THE SECOND YEAR: 10-100.
Rapid Work*
2x2 + 2x2-5-4—1? 2+3+3+2+2—4+4+4x2?
From 12 apples, one half are eaten, then half the re- mainder, then i. How many remained?
From 12 cents take away i three-cent and i two-cent piece, then again i three-cent and i two-cent piece. How much remains.
12 — 6 + 3 — 5 + 7 — 1-5-2 — 3? IO + 2-T-4X2 + 3-5-3 + 9?
8 + 2 + 2-5-3-7-2x6 — 11 + 7?
5x2 + 2-^-6 + 7-5-3x4 — 7?
6-5x12 — 9x3 — i-v-4?
Combining.
The third part of 12 is what part of 8 ? The half of 12 is how many times 3 ?
What is the difference between \ of 12 and £ of 10? 12 is 3 times what number? What number must I subtract from 12 to get 9?
(As 9 + 3 = 12, 3 must be taken from 12 to get 9.) What number subtracted from 12 leaves 4?
n. The Applied Number.
12 things make a dozen.
12 months make a year.
What part of a dozen are 6 things ?
What part of a year are 6 months ?
3 months = a quarter of a year.
4 months = a third of a year, How many gallons in 12 quarts? How many quarts in 12 pints?
How many sixes, fours, threes, twos in 12?
* It must not be forgotten that the exercises of " Rapid Work" are to be worked as fast as the teacher dictates, and that it is oral work only. The object is to gain facility in head-work. All idea of paren- thetical expressions is excluded here.
TWELFTH STEP. 6 1
In a month there are four weeks. If a boy earns 12 dollars a month, how much does he earn per week ?
A father pays 2 dollars a month for his son's lessons. How much is that for 3 months? How much for 6 months ?
Charles divided 12 cents equally among 4 poor boys. How much did each receive ?
How many sheets of paper at 3 cents each can you buy for 12 cents?
(As many times as I have 3 cents I get a sheet of paper. 12 cents =4x3 cents, so I get 4x1 sheet, or 4 sheets.)
Illustrated.
Ill o
III o
III o
III o
A foot contains 12 inches. How many inches in •£ of afoot? t? $?
What part of a foot is 4 inches ? 6 inches ?
A troy pound contains 12 ounces. What part of a pound is 3 ounces ? 4 ounces ?
If a top costs 3 cents, a whistle 2 cents, and a ball 7 cents, how much do all cost ?
John takes a dime and a 2-cent piece to the store and buys 4 lead pencils at 2 cents each, and a sponge for 3 cents. How much money had he left ?
John has 8 cents ; how much more must he earn to have a dozen cents ?
62 TtiE SECOND y&A&: 16-106.
THIRTEENTH STEP.
THE THIRTEEN.
I Illlll I I 1= 10 + 3= 13 III
(Measuring and rapid work the same as before.)
Combining. Make 13 by multiplying 3*3 and 2's.
(3x3 + 2x2 = 9 + 4= 13.)
How does the difference between 13 and 9 compare with the difference between 12 and 8? Subtract 6 from 13.
(13 — 6 = 7, for 13 — 3 = 10 and 10 — 3 = 7.) What number = 7 + 6 ? What number = 8 + 5 ? What number must I add to 4 x 3 to get 13 ?
n. The Applied Number.
Mary has a ro-cent and a 3-cent piece, with which she buys 4 oranges at 3 cents each. How much has she left ?
A gentleman divides 13 apples among some children, giving the first child 3 apples, and the others 2 apples each. How many children were there ?
FOURTEENTH STEP.
FOURTEENTH STEP.
THE FOURTEEN.
I I I I I I I I I I = 10 + 4 = 14. I I I I
What number must I double to get 7x2?
2 sparrows lit upon a tree, and then 2 more, and 3, and 3 and 2x2; 3 + 4 + 5 soon flew away. How many re- mained ?
I have taken a certain number 3 times away from 14, and have 2 left. What is the number ?
(If I have 2 left, I must have taken 12 away ; 12 is 3 X4, so I have taken 4 three times away from 14 if I have 2 remainder.)
H.— The Applied Number.
14 days = 2 weeks.
14 cents = 7 two-cent pieces, 4 three-cent and I two- cent piece, etc.
14 things = i dozen and 2 things.
14 months = i year and 2 months.
If I travel for 2 weeks, and spend i dollar each day, how much will my journey cost ?
64 THE SECOND YEAR: 10-100.
What will i dozen and 2 pencils cost at i cent each ?
How many sheets of paper can I buy for i dime and 4 cents, if a sheet costs 2 cents ?
Marie knit in i year 7 pair of stockings. How many dozen did she knit, and how many stockings over?
If I sell 7 pigs at 2 dollars each, how much do I receive ?
A man buys 14 pounds of sugar for i dollar and 4 dimes. What does i pound cost ?
FIFTEENTH STEP.
THE FIFTEEN. I.— The Pure Number.
II I I I I I I I I 10 + 5 = 15. I I I I I
How many units belong to the second 10? How many are still lacking to make the second 10 full ? Write 15 in dots so that 5 always stand together.
Write these fives under each other. • • • • • j f 3 x 5
Write them so that 3 stand in a row.
:f5X
3
Of what unequal numbers does 1 5 consist ? Of what 3 unequal numbers ? What is the difference between 15 and 8?
FIFTEENTH STEP. 65
(a. I observe how much I must add to 8 to get 15. 8 -f 2 = 10, 10 + 5 = 15, 8 + 7 = 15. The difference be- tween 8 and 15 is 7.
b. I take 8 from 15; 15 — 5 = 10, 10 — 3 = 7, 15—8
= 70*
In what number is found an entire and a half ten ?
What 5 numbers give the sum of 15 ?
Each following of these 5 numbers is to be i larger than the preceding. 1+2 + 3 + 4 + 5.
At Easter a mother divided among her 5 children boiled eggs according to age, so that each older got i egg more than the next younger. The middle child in age got. 3 eggs. How many did each of the others get? How many eggs were distributed ?
II. — The Applied Number.
Compare i doz. with 15. i doz. =4x3 things.
15 = 5 x 3 things = i doz. + 3 things. 15 cents= i dime + 5 cents. 1 5 days = 2 weeks + i day.
Mary buys 15 pints of milk; how many quarts does she buy ?
I sent a friend 3 five-cent postage stamps. How much are they worth ? I bought 4 two-cent, 2 one-cent, and i five-cent stamp. How much did all cost ?
* A strong point practised by the Germans in their arithmetical cal- culations is here illustrated. They make the 10 an important factor in all operations. This will be more fully discussed later.
66 THE SECOND YEAR: 10-100.
SIXTEENTH STEP.
THE SIXTEEN. I.— The Pure Number.
I I I I I I I I 1 I 10 + 6 = 16. Mill!
How many ones, twos, threes in the second 10 > Divide the 16 lines into twos, fours, eights.
I I I I
Write 16 in twos perpendicularly.
In fours. In eights.
4x4 2x8
Where do we find equal numbers of points horizontally and perpendicularly ?
(Many pupils will now be able to construct a square and divide it into equal parts.)
Rapid Work.
How many are 2 + 2 + 2+2 + 2 + 2 + 2 + 2? How many are 1+2+3 + 1 + 2 + 3 + 1+2 less than 16?
How many are 16 — 3 — 3 + 2 —3 ? 16^4 + 3 + 2-5-3 x 5 — 9 + 10?
Combining.
How do you find the half of 16? (16 = i ten + 6 units. \ of i ten is 5 units, \ of 6 units
SIXTEENTH STEP. 67
is 3 units. 5 units + 3 units = 8 units. Therefore i of 16 = 8.)
A had 6 dimes. B said, " If you take 4- of your money
8 times, you will have as much as I." How much had B ?
(B's amount was 8 x £ of A's ; i x 6 dimes [the third
part] = 2 dimes. Therefore B had 8x2 dimes = 16
dimes.)
A. B. I I I I I I I I
What number =2 x J of 16 ? What part of 16 is that?
(J of 16= 4, 2 x 4 = 8. Therefore 8 is 2 x J of 16. 16 = 2 x 8. Therefore 8 = J of 16.)
•I:
II. — The Applied Number.
(Applications of the denominations of compound num- bers should be made as soon as a number embraced in any table is reached. If possible, place the various measures in the hands of the pupils and let them measure out the various denominations. Dots may be used to give a pic- ture of the relations of the various denominations to each other. In this way all the tables of compound numbers will be gradually and intelligently learned.)
In i bushel there are 4 pecks. How many bushels in 1 6 pecks ?
A pound avoirdupois contains 16 ounces. How many ounces in half a pound? What part of a pound is 4 ounces ?
If a pail holds 2 gallons, how many, pailfuls will 16 gal- lons make ?
A farmer sold 16 bushels of potatoes at half a dollar a bushel. How much does he get for all ?
68 THE SECOND YEAR: 10-100.
SEVENTEENTH STEP.
THE SEVENTEEN.
I. — The Pure Number.
MINIMI! 10 + 7 = 17. II I I I I I
How many ones have we now ? How many are lacking of 2 full tens ? How many more has the first row than the second ?
With what numbers can we measure the 17? Begin with 1 6.
17 = 16 + i 15 + 2 14 + 3, etc.
Of how many ones, twos, etc., does 17 consist?
17 = 17 x i
8x2 + 1 5x3 + 2 4x4 + 1, etc.
Make 17 from 3 equal and i unequal number.
17= 3x5 + 2 3X4 + 5 3x3 + 8, etc.
Make 17 from 4 equal and unequal numbers, also from 5 equal and i unequal number.
Rapid Work.
17 — 2— 2— 2 — 2 — 2 — 2 — 2?
17-3-3-3- etc. 17 — 4 — 4 — 4, etc. 17-5-5. etc.
SEVENTEENTH STEP. 69
1 + 2 + 3+4+5 lacks how many of 17 ?
2 threes, i five, and i three lacks how many of 17 ? 17 — 7-^-5 x 8-7-4—1 x 5?
3x5 + 2—1-5-4 + 8 + 5—1?
Combining.
How many ones must I add to 3 x 5 to get 17 ?
How many ones must I add to 5 x 3 to get 17 ? (An- swer: The same.)
What relation has 4x4 and 3 x 5 to 17 ?
I have taken 4x4 from 17 and obtained just the same as if I had taken double another number from 17. Of what number must I have taken the double ?
II. — The Applied Number,
How many pounds in 17 ounces ?
Four brothers divided 17 cents so that the oldest had i one more than the others. How many cents did he get?
In 17 quarts how many gallons ?
Charles had a dime, a five-cent piece, and a cent. How much did he lack of 17 cents ?
A milkman had 2 cans of milk, each holding 10 gallons. One was full, and the other had 7 gallons in it. How many gallons had he in both ? How many gallons did the second can lack of being full ?
Henry divided 17 cents equally among 5 poor children. How many cents did each get, and how many had he left?
If Charles can walk 3 miles in an hour, how long will it take him to walk 17 less 2 miles?
7O THE SECOND YEAR: 10-100.
EIGHTEENTH STEP.
THE EIGHTEEN. I.— The Pure Number.
I I I I I I I I I I 10 + 8 = 18. 1 I I I I I I I
Write the number 18 in dots so that 2 always come to- gether.
How many pairs has the first ten ? How many pairs are lacking in the second ten ? Write the number 18 so that 3 lines come together.
Ill III III III III III
How many threes and how many sixes has the 18 ? Write the sixes in horizontal lines.
I I I I I I
I I I I I I
I I I I I I Write the 18 in fives.
Of what 2 equal numbers does 18 consist? Of what 3 6, 9?
Of what 3 equal together with i unequal number? Of what 4 equal together with i unequal number?
Rapid Work. Add rapidly 2 twos, i three, 2 twos, i five and i two.
2 + 2 + 2 + 3 + 3 + 3 +3?
Count upwards by twos, commencing with 2 (2, 4, 6, etc.) to 1 8.
The same backwards. The same commencing with 3. Also backwards,
NINETEENTH STEP. 7 1
Combining.
Of what number is 18 sixfold ?
What is the number of which 12 is twofold and 18 threefold ?
What part of 12 is this number? Of 18 ?
What part of 1 2 is 1 8 greater than 12?
What number must I multiply by 3 to get 18 ?
How much greater is the double of 9 than the clouble of 8, 7,6?
II. — The Applied Number.
If a pound of meat costs 9 cents, how many pounds can be had for 18 cents ?
How many weeks are there in 18 days ?
If Mr. A works 3 weeks at i dollar a day, how much does he earn ?
Fred was sent to market with 18 dimes. He bought 4 pounds of veal at i dime a pound ; 6 pounds of beefsteak at 2 dimes a pound ; cabbage for i dime. How much money had he left ?
If a child is 1 8 months old, how many years old is it?
A farmer had 18 pecks of clover seed. How many bushels had he?
NINETEENTH STEP.
THE NINETEEN.
I.— The Pure Number.
1.
II I I I I I I I I 10 + 9 = 19. I I I I I I I I I
What have we now ?
We have i ten and 9.
How many does this lack of 2 full tens?
Jt lacks but i and then the second ten is complete.
2 THE SECOND YEAR: 10-100.
Write 19 in lines of 2 each.
II II 11 II II II II II II I Write the ten in twos and the nine in threes.
Write in lines of 5 each.
I I I I I I
I I I I I I I I I I I
How many fives in 19? sixes? sevens? eights? Il- lustrate these by lines.
I I I I I I I I I I I I I II
II I I I I I I I I I I I I I I I I I
etc.
Rapid Work.
19—1 — 2 —2, etc.
(An excellent practice is found in starting from a given number and counting upward as far as 19 and backward by twos, threes, fours, etc. For example : by twos from 3; as, 3, 5, 7, 9, u, 13, 15, 17, 19; 19, 17, 15,' etc.: or by threes from 4; as, 4, 7, 10, 13, 16, 19; 19, 16, 13, etc. This must be done with greatest rapidity and without hesita- tion.)
Combining.
5 times what number + 4 times what number make to- gether 19?
3 times a number +1 = 19. What is the number ?
6 times a number +1 = 19. What is the number ? How can I divide 19 apples among 6 children so that
at least 5 get the same number ? How many would the sixth get ?
TWENTIETH STEP. 73
II. — The Applied Number.
Gussie had 2 dollars, or 20 dimes. George had i dollar and 9 dimes, or 19 dimes. How much does George lack of having as much as Gussie ?
" My little brother," said Anna, " is \\ years old ;" " And mine," said Bertha, " is just i month older." How many months old was the latter? How much over i year?
A cloak requires 3 yards of cloth, each yard costing 6 dollars. Reckon also i dollar for the velvet collar. What will the cloak cost ?
TWENTIETH STEP.
THE TWENTY. I.
I I I I I I I I I I 10 + 10 = 20. I i I I i I I I I I
Now how many tens have we ?
Show me 20 fingers. (Let two children hold up both hands.)
Here are twenty sticks. How many bundles of tens can we make from them ? We bind them together and have what number?
Write 20 so that the lines fall in twos.
II II II II II II II II II II
How many fours are there ?
The 10 can be separated in two equal parts ; how many the 20 ?
1 4 x 5=20
74 THE SECOND YEAR: 10-100.
How many points in one of these horizontal lines ? In one of the perpendicular lines ?
Now write 4 dots in a horizontal line ; how many rows are there ?
\ 5 x 4 = 20
J Of what equal numbers does 20 consist?
20 x i, 10 x 2, 5 x 4, 4 x 5, 2 x 10.
Of what number is 20 the double ? The fourfold ? The fivefold? The tenfold? What part of 20 is 2, 4, 5, 10?
Measure with i.
1, 2, 3, 4, 5, 6, (In which ten are we? How many does it lack of being full?), 7, 8, 9, 10, n, 12, 13, 14, 15.
(How many have we in the second ten ? How many in all? How many ones must we take to fill the second ten?)
20 X I = 20.
20, 19, 1 8, 17, (Stop! how many ones must we still take away before we get to the first ten ?) etc.
20 -*• i = 20.
I can take i away from 20 twenty times, or i is con- tained in 20 twenty times.
Measure with 2.
2, 4, 6, 8, 10, 12, (How many times 2 have we now?) 14 16, (How many now?) 18, 20.
10 x 2 = 20. 20, 1 8, 1 6, 14, etc.
?0 -T- 2 = 10,
TWENTIETH STEP. 75
Measure with 3.
3, 6, 9, 12, 15, 1 8, (How many must we still add to get 20?) 20.
6 x 3 + 2 = 20.
How many times can I take 3 from 20 ?
(As often as I take 3 away, we will count a finger.)
20, 17, 14, n, 8, 5, 2.
20 -5- 3 = 6 (2), etc.
20 = 20 X I 20 -*- I = 20
IO X 2 2 = IO
6 x 3 (2) 3 = 6 (2)
5 ><4 4 = 5
4x5 5=4
3 x 6 (2) 6 - 3 (2)
2 x 7 (6) 7 = 2 (6)
2 x 8 (4) 8 = 2 (4)
2 X 9 (2) 9 = 2 (2)
2 X 10 10 =2
Rapid Work. How much is (2 x 2) + (2 x 2) + (2 x 2) + (2 x 2)
+ (2X2)?
How much is (3 x 2) -f (3 x 2) + (3 x 2) less than 20 ? Subtract 4, 3, 2, i from 20 and again 4, 3, 2, i, and how many remain ? 20 — 13 — 6? 20— ii ? 9? 8? 6? 4? How many dozen and how many units in 20?
20 -7-2 + 5-^-5x3 + 1 X2 = 20 — 4-7-4 + 6 X2 — 12 -*• 8 ?
Combining.
4 times 5 = 2 times what number ?
5 times 4 = 4 times what number ?
What is the difference between 4x5 and 5x4? Be- tween 4x5 and 4x4?
? THE SECOND YEAR: 10-100.
A gardener divided 20 apples among children so that he gave each child the same number and still had 2 to save for Fred. How many did each child get ?
(If Fred received 2, there remained 18 to divide. 18 can be divided into 3 equal parts, etc.)
II.— The Applied Number.
20 things = a score.
20 cents = 2 dimes.
How many weeks in 20 days ?
How many pounds in 20 ounces ?
Charles had 20 cents to spend. He bought 3 apples at 2 cents each, a ball at 10 cents, gave his sister 2 cents, and kept the balance. How much did he keep ?
James is 20 years old and his brother Chester is 3 years younger than he, while his cousin John is 6 years younger than Chester. How old is John ?
2 dimes is how many times 5 cents ?
20 cents equal how many 2- cent pieces ?
A farmer had 20 pecks of corn. How many bushels had he ?
A board is 20 inches long. How much of it must I saw off in order to leave j ust a foot ?
A merchant found just 20 feet in a piece of cloth. How many yards did the piece contain ?
A milkman had 2 ten-gallon cans full of milk. He sold ii gallons in the morning, and the rest in the evening. How much did he sell in the evening?
If I can walk 4 miles in an hour, how long will it take me to walk 20 miles ?
How long if I walk 5 miles an hour ?
In the morning there were 9 boys and 11 girls at school. In the afternoon 3 pupils stayed out. How many were there in the afternoon ?
If I pay i dollar for 2 bushels of potatoes, how much must I pay for 20 bushels?
In a score of years how many birthdays will you have ?
(In this manner all the foHowing numbers are treated, and the teacher will now be prepared to continue the
THIRTIETH STEP. 77
course himself. A written preparation should be made in order that nothing be omitted, and that the pupil be induced in the best manner to prepare the exercises him- self. Especial attention should be given to such numbers as 24, 30, 50, 60, etc., which are more often applied. Such numbers as 23, 29, etc., need but little attention. Two or three steps more will suffice.)
THIRTIETH STEP.
THE THIRTY. I.— The Pure Number.
I I I I I I I I I I 10 -f 10 + 10 = 30.
I I I I I I I I II
I I I I I I I I I I
(3 times the fingers of 2 hands.) If I add i to the 29 the third ten will be full. 3 tens taken together is called 30.
Measure with i.
Count by ones upwards: i, 2, 3, 4, 5, 6, 7, (Stop! a pupil says : " We are in the first ten ; it lacks 3 of being full, and 20 more of 30.") 8, 9, 10, 11, 12, 13, (We are in the second ten, etc.) 14, 15, etc.
30 x i = 30.
Count downwards : 30, 29, 28, (We are in the third ten ; from the third ten 2 have been taken away.) 27, 26, etc.
30 -j- i = 30.
Measure with 2. 2, 4, 6, 8, 10, 12, 14, (We are in the second ten ; it lacks
THE SECOND YEAR: 10-100.
3 twos of being full, and 5 twos more of completing the 3 tens or thirty.) etc.
15 x 2 = 30.
Count downwards with 2 : 30, 28, 26, 24, (We are in the third ten, have taken 3 twos from the third ten and there remain 2 twos in the third ten.)
30 +-2= 15. Continue with the other numbers.
Measure with 10.
10 + 10 = 20, 20 + 10 = 30.
3 x 10 = 30. 30 — 10 = 20, 20 — 10 = 10.
30 -f- 10 = 3.
|
3o = |
30 x i |
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15x2 |
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10 x 3 |
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7 x 4 (2) |
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6x5 |
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5x6 |
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4 x 7 (2) |
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3 x 8 (6) |
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3 x 9 (3) |
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3 x 10 |
Divide the dots into twos :
Into threes :
3 x 10 = 30.
30 -h i = 30
2 = 15
3 = 10
4 = 7(2)
I = 6
6 = 5
7 = 4(2)
8 = 3 (6)
9 = 3 (3) 10 = 3
(• 30= 15 X 2.
>• 30 = 10 x 3.
THIRTIETH STEP. 79
Oral :
30 = 29 + i, 28 + 2, 27 + 3, etc.
30 is 3o-fold of i i is ^ of 30
15-fold of 2 2 is TV of 30
lo-fold of 3 3 is TV of 30
6-fold of 5 5 is £ of 30
5-fold of 6 6 is ^ of 30
3-fold of 10 10 is £ of 30
2-fold of 1 5 1 5 is i of 30
Of what equal numbers does 30 consist ? Of what 2, 3, 4, 5, 6 unequal numbers ?
Rapid Work.
30-4-15 +3x5+ 5-5-10?
(3 x 5) + (2 x 4) + 7 •*• 10 x 3 ?
4x6, the half, again the half, x 5 ?
Combining. 30 - 19.
(19 = 10 + 9, 30 — 10 = 20, 20 — 9 = II, 30— 19= II.)
How do you get the double of 15 ?
(15 = i ten and 5 units ; 2 x i ten = 2 tens 52x5 units = 10 units = i ten ; 2 tens + i ten = 3 tens = 30.)
Compare 30 with 16.
(30 = 3 tens ; 16 = i ten + 6 units. I must add to the 6 units 4 units to complete the second ten, and still i ten to get 3 tens. Therefore 30 has i ten + 4 units = 14 more than 16, and 16 is i ten + 4 units, or 14 less than 30.)*
10 x 3 = 6 times what number?
If I take 3x5 from a number, I get 5 x 3 as remain- der. What is the number ?
* This method of adding1 and subtracting is universally practised at pres- ent in the German schools, with most excellent results. The tens are the stepping-stones of the method.
SO THE SECOND YEAR: 10-100.
II. — The Applied Number.
In 30 days how many weeks ? How many dozen ? How many score ?
If it costs me 2 dollars a day when I travel, how many days can I travel and spend 30 dollars ?
A dollar contains 10 dimes. How many dollars in 30 dimes?
A workman received 6 dollars a week. How much will he receive in 5 weeks ?
A piece of linen was 10 yards long. How many feet long was it ?
Charles had a cane 2 feet 6 inches long. How many inches long was it ?
William had 7 gallons and 2 quarts of water in a tub. How many quarts did he have ?
If a shirt require 3 yards of cloth, how many shirts can be made from a piece containing 30 yards ?
A golden eagle equals 10 dollars. If Joseph had 3 eagles, how many dollars had he ?
Mr. A. sold 3 sheep at 5 dollars each, and 4 pigs at 3 dollars each. How much did he lack of getting 30 dol- lars?
Five francs make i dollar. How many dollars are equal to 30 francs ?
A teacher divided 30 apples among his pupils, giving the boys half and the girls half. The boys received each 3 apples, and the girls each 5. How many pupils had he ?
FIFTIETH STEP.
THE FIFTY.
X 10 = 50
FIFTIETH STEP. 8 1
How many ones stand perpendicularly under each other ?
Write these fives in horizontal lines. Write 50 in twos. How many stand in a line ? (Continue as in preceding numbers.)
Oral and Written.
50 = 50 x i 50 -*- i = 50
25 x 2 2 = 25
16 x 3 (2) 3 = 16 (2)
12x4(2) 4=12(2)
10 x 5 5 = 10
8 x 6 (2) 6 = 8 (2)
7 x 7 (i) 7 = 7 (i)
6 x *8 (2) 8 = 6 (2)
5 x 9 (5) 9 = 5 (5)
5 x 10 10 = 5
Rapid Work.
50 -i- 2 -f- 5 x 6 — 1 5 -*- 3 ? 10 x 5 — 10 — 10 — 10 — 10 —5 ? 25 x 2 -T- 5 + 15 — 5 + 20 + 10 ? 30 -r- 2 x 3 + 5 -i- 5 + 40 -*- 50 ? 50 -2-4-4-^-4 + 5x3- 15?
Combining.
The nfth part of 50 is double what number ? The half of 50 is 5 times what number ? How does \ of 50 compare with \ of 25 ? £ of 25 = TV of what number ?
II.— The Applied Number.
How many weeks in 50 days ? How many pounds avoirdupois in 50 ounces ? If I have 50 cents, how many lo-cent pieces does it equal ? How many 5-cent pieces ? In 50 inches how many feet ?
82 THE SECOND YEAR: 10-100.
Charles receives from his aunt a 2$-cent piece; from his father he received another. How many cents did he get?
A milkman shipped to New York 5 cans of milk, each containing 10 gallons. How many gallons did he ship?
Mary buys 50 cents' worth of muslin, paying 10 cents a yard. How many yards does she get ?
A coal-dealer invested 50 dollars in coal at 4 dollars a ton. How many tons did he buy ?
HUNDREDTH STEP. (The Last Step of the First Course.)
THE HUNDRED.
The counting upwards and downwards with numbers from i to 10, beginning with both i and 2, must be done without hesitation, rapidly and accurately. In counting in concert, the teacher should frequently stop the pupils with questions as before. To illustrate :
Counting with 2, beginning with i : i, 3, 5, 7, 9, 11, 13, (Stop ! In what ten are we? How many units does it lack of being complete ? How many tens still remain to complete 100? How many units ?) 15, 17, 19,21, etc.
Counting with 6, beginning with 2 : 2, 8, 14, 20, 26, 32, (Stop! In what ten are we? How many units are lacking to complete the ten? How many tens to com- plete ico ?) 38, 44, 50, etc.
Counting with 6, beginning with 99 : 99, 93, 87, 81, (Stop ! Ask questions as above.)
(Count both upwards and downwards with all numbers from i to 10 in this manner. This will be found a most valuable exercise. If the drill has been faithfully kept up with all preceding numbers, it will now be very easy and satisfactory.)
HUNDREDTH STEP.
Written. Pupils should write the above tables as follows :
1+2 = 3 or 1+7=8 or ioo — 8 = 92
3 + 2 = 5 8 + 7 = 15 92 — 8 = 84
5 + 2 = 7 15+7 = 22 84 — 8 = 76
7 + 2 = 9 22 + 7 = 29 etc.
etc. ' etc.
10 x 10 = 100
100 = loo x i 50 x 2 33 x 3 (i) 25 x 4 20 x 5 16 x 6 (4) 14 x 7 (2) 12 x 8 (4) ii x 9 (i)
IO X IO
100 •*• I = 100
2 = 50
3 = 33 CO
4 = 25
5 = 20
6 = 16 (4)
7 = 14 (2)
8 = 12 (4) 9=11 (i)
10 = 10
ioo = 99 + i, 98 + 2, 97 + 3, etc.
100 is loo-fold of i 5o-fold of 2 25-fold of 4 2o-fold of 5 5-fold of 20 2-fold of 50
1 is TJ-g- of ioo
2 is TV
4 is A-
5 is A 20 is \ 50 is i
84
THE SECOND YEAR: 10-100.
Rapid Work.
IOO -*- 2 + IO -4- 30 X 25 X 2 ?
4 x 25 - 50 -*- 5 x 10 - 75 x 3 + 25 ?
All possible combinations in multiplication have al- ready been learned in the course of the various exercises, but may now be arranged in the multiplication table and practised.
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
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2 |
4 |
6 |
8 |
10 |
12 |
H |
16 |
18 |
20 |
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3 |
6 |
9 |
12 |
15 |
18 |
21 |
24 |
27 |
30 |
|
4 |
8 |
12 |
16 |
20 |
24 |
28 |
32 |
36 |
40 |
|
5 |
IO |
15 |
20 |
25 |
30 |
35 |
40 |
45 |
50 |
|
6 |
12 |
18 |
24 |
30 |
36 |
42 |
48 |
54 |
60 |
|
7 |
14 |
21 |
28 |
35 |
42 |
49 |
56 |
63 |
70 |
|
8 |
16 |
24 |
32 |
40 |
48 |
56 |
64 |
72 |
80 |
|
9 |
18 |
27 |
36 |
45 |
54 |
63 |
72 |
81 |
90 |
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10 |
20 |
30 |
40 |
5o |
60 |
70 |
80 |
90 |
IOO |
HUNDREDTH STEP.
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[2 XIO |
X *O |
X 00 |
SI |
X at |
X Ol |
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CO X 10 II |
CO X 00 II |
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86 THE SECOND YEAR: 10-100.
The multiplication table can also be written in rows as follows :
2 x 1=2; 2 X2 = 4; 2x3 = 6, etc.; 3Xi=3;3X2 = 6;3X3 = 9, etc.; 4 x i=4;4X2 = 8;4X3 = i2, etc.;
and so forth until
10 x i = 10; 10 x 2 = 20; 10 x 3 = 30, etc.
The pupil must learn that the product is the same when the factors are alike, no matter in what order taken.
7x6 = 42, 6x7 = 42.
Besides what has already been given, the four funda- mental rules must be practised with numbers above 10 until the pupils are brought to complete mastery of them, and are able to make various combinations with great rapidity and accuracy. A few examples follow which must be worked only orally :
a. (Addition) —
14 + 13 + 12 + ii
15 + 17 + 19 + 18 25 + 37 + 39 + 17 42 + 15 + 26 + 37
etc.
(The method of working is as follows : 42 + 10 = 52, S2 + 5 = 57, 57 + 20 = 77, 77 + 6 = 83, 83 + 30 = 113, 113 +7 = 120.)
b. (Subtraction) —
90—19—12 — 11 98 - 32 - 41 - 24 90 — 16 — 17 — 28 — 29 97 — 12 — 34— 16 — 27 etc.
(To be worked as follows : 97 — 10= 87, 87 — 2 = 85, 85-30 = 55, 55-4 = 5i, 51 -10 = 41, 41-6 = 35, 35-20= 15, 15-7=8.)
HUNDREDTH STEP. 87
c. (Multiplication) —
3 x 30, 4 x 22, 2 x 44, 2 x 27, 3 x 25, 4 x 18, 12 x 5, 33 x 2,
35 x 2, 45 x 3, 15 x 7, 19 x 4,
42 x 6, 17 x 9,
etc.
(Employ the following method of solution: (17 x 9) 10 x 9 = 90, 7 x 9 = 63 ; 90 + 60 = 150, 150 + 3 = 153. (Likewise 15 x 37) 10 x 37 = 370, 5 x 30 = 150, 5x7 = 35 ; 37o + ioo = 470, 470 + 5° = 52o> 52o + 35 = 555-)
d. (Division) —
60-5-3, 69-5-3, 96-5-4, 72 -5-4, 84 •*- 4, 84-5- 12, 68 -*- 13, 72 -*- 18, 53 -1-4, 62 -i- 5, 72 -*- 3.
(Method as follows : 72 -f- 3 is 60 -5- 3 = 20 and 12 -*- 3 = 4. Thus 72 [60 + 12] -j- 3 = 24 [20 + 4]. Another : 84 -*- 12 is 60 -5- 12 = 5 and 24 •*- 12 = 2. Thus
84 [60 -4- 24] -*- 12 = 7 [5+2].
Another : 40 -s- 18 is 36 -5- 18 = 2. Thus 40 •*• 18 = 2 (4).)
Combining.
The teacher will write a row of figures on the board to be added, subtracted, multiplied, and divided, placing the signs between. He then points to the exercises, and the pupils solve them rapidly. The teacher must always solve the examples himself. This holds good for all work in arithmetic, and is very important.
Such examples as the following must be rapidly worked by the pupils :
(3 x 29) — (4 x 1 6) + 7 -s- 10 x 9 x 3 ? (15 x 6) -(45 -r-9) x 37? (4 x 20) — 1 5 — 5 -*- 20 x 45 ?
90 — 45 — 5 X 2 + 20 -T- 10 + 6 X 6?
As a test whether every number is comprehended by the pupil, let him name the factors of numbers from i to ioo without naming the numbers. Prime numbers can only be measured with i. For example: i x i, 2 x i, 3 x i, 2 x 2, 5 x i, 2 x 3, 7 x i, 2 x 4, or 2 x 2 x 2, 3 x 3, etc.
88 THE SECOND YEAR: 10-100.
It is important that the children know readily that 52 = 4 x 13, 68 = 4 x 17, 95 = 5 x 19, and so on to 100.
Such exercises as the following should be thoroughly drilled :
2X11 = 22 2X12 = 24 2X13 = 26
3X11=33 3x12 = 36 3X13 = 39
4 x ii = 44 4 x 12 = 48 4 x 13 ='52
5 x ii = 55 5 x 12 = 60 5 x 13 = 65
etc. etc. etc.
2 x 14 = 28
3 x 14 = 42, and so on till 5x19 = 95.
The prime and composite numbers can be arranged as follows :
1 prime number
2 "
3 "
4 = 2X2
5 prime number
6 = 2x3
7 prime number
8 = 2X2X2
9 = 3X3
10 = 2 x 5
11 prime number
12 = 2x2x3
13 prime number
14 = 2 x 7
15 = 3 x 5
16 = 2X2X2X2
17 prime number
18 = 3 x 3 x 2
and so forth till 100. This must be thoroughly drilled.
Combining.
A man gave away J of 100 dollars, and then \ of the rest. What part of the whole amount had he still ?
I have taken a number 3 -times and obtained 4 more than \ of loo. What was the number ?
5 times what number is 5 less than 100 ?
HUNDREDTH STEP. 89
(The number which is 5 less than 100 is 95. If 95 is 5 times a number, that number is contained 5 times in 95. £ of 95 = 19.)
75 is 3 times J of what number ?
50 is 4 of 4 times what number ?
60 is 6 times T^ of what number ?
II.— The Applied Number.
100 cents = how many dimes ? quarters ? halves ?
loo feet = how many yards ?
100 days = how many months (of 30 days)?
loo months = how many years ?
loo quarts = how many gallons ?
100 pints = how many quarts ?
loo things = how many dozen ? score ?
loo ounces^ how many pounds?
100 days = how many weeks?
Charles received 4 quarters as a birthday present. He spent 50 cents for fruit and nuts, gave 20 cents away, and lost 10 cents. How much had he left ?
How many days in 100 hours ?
How many hours in 100 minutes ?
A farmer had 100 pecks of clover seed. How many bushels had he ?
Mary had a cord 100 inches long. How many feet is that ?
Mrs. A. took 8 dozen eggs to the store. How many eggs did she take ?
My grandfather is fourscore and four years old. How many years is that ?
January has 31 days, February 28, and March 31. How many days do they lack of 100 ?
Mr. Thomas sold 4 cows at $25 each, and laid out the money in sheep at $5 a head. How many sheep did he buy?
How many years in \ a century? \ ? -j^-?
If a train of cars goes 20 miles an hour, how long will it take to go 100 miles ?
A ship sailed 2 leagues an hour. How long did it take her to sail 96 miles ?
9° THE SECOND YEAR: 10-100.
If I saw a board 100 inches long in 4 equal pieces, how long will each piece be ?
(Examples of this kind should be made by both teacher and pupils until the pupils are able to thoroughly apply the number 100.)
(This completes the work of the first two years accord- ing to Grube. Later educators in Germany accept the first year's work, but modify the second very materially. We append herewith a method which meets with very general favor in Germany. The two methods must not be mixed, and no attempt should be made to combine them. Accept one, reject the other.)
APPENDIX 70 THE SECOND YEAR'S WORK,
The decimal system is the basis of all arithmetical operations after the 10 (the first year's work). The chil- dren should be introduced to this system the second year. They must not longer consider each number as an indi- vidual, but as a part of a system. Accordingly, the fol- lowing should be the plan of procedure.
SECTION L
^Development of the numbers from 10-100 in the pure tens, and practice of the four fundamental operations within this range.
FIRST UNITY. (Addition and Subtraction.)
First Step.
a. i + 2 = 3 10 — 2 = 8
3+2=5 8-2=6
5+2=7 6—2=4
7+2=9 4-2=2
APPENDIX. QI
1+3=4 10 — 3 = 7
4 + 3=7 7-3 = 4
7+3 = 10 4-3 = 1
The same with 4 and 5.
b. We wish to count 100 miles. Children count 10 on the numeral frame. That we will call 10 miles. Again 10 miles must be counted.
10 miles + 10 miles = 20 miles.
We must count very much farther before we get 100 miles. Count another 10 on the frame :
20 miles + 10 miles = 30 miles. Continue in this way until we get
90 miles + 10 miles = 100 miles.
In the same manner reverse the process, commencing at 100 and subtracting 10 each time :
10 m. + 10 m. = 20 m. 100 m. — lorn. = 90 m.
20 " + 10 " = 30 " 90 " — 10 " = 80 "
30 " + 10 " = 40 " 80 " — 10 " = 70 "
40 " + 10 " = 50 " 70 " — 10 " = 60 "
till till
90 m. + 10 m. = 100 m. 20 m. — 10 m. = 10 m.
c. Run rapidly through the tens forwards and back- wards, using the numeral frame or fingers:
10 100
20 90
30 80
40 70
till till
100 10
d. With the frame develop the ordinals. (Children say as the teacher indicates.)
That is the first 10 miles;
" second 10 "
" third 10 " till
That is the tenth 10 miles.
92 THE SECOND YEAR: 10-100.
Or,
That is the tenth 10 miles ; " ninth 10 " etc.
e. Writing the tens forwards and backwards from dic- tation:
10 100
20 90
30 80
etc. etc.
Second Step.
a. Oral practice in intervals of 20, 30, 40, forwards and backwards :
20 + 20 = 40 loo — 20 = 80
40 + 20 = 60 80 — 20 = 60
60 + 20 = 80 60 — 20 = 40
80 + 20 = 100 40 — 20 = 20
10 + 20 = 30 90 — 20 = 70
3O + 2O == 50 70 — 2O = 50
50 + 20 = 70 50 — 20 = 30
70 + 20 = 90 30 — 20 = 10
2O + 30 = 50 80 — 30 = 5O
50 + 30 = §0 50 — 30 = 20
10 -f 30 = 40 ioo — 30 = 70
40 -f 30 = 70 70 — 30 = 40
70 + 30 = ioo 40 — 30 = 10
In this same manner with the other numbers.
b. Practise writing from dictation after each group.
c. Exercises like the following :
1 + 2=3 10 — 2 =8
10 + 20 = 30 IOO — 20 = 80
2 + 3 =5 7-4 =3 20 + 30 = 50 70 — 40 = 30
4 + 3 =7 8-5 =3
40 + 30 = 70 80 — 50 = 30
APPENDIX. 93
d. Concrete examples :
A man travelled 30 miles, and afterwards 20 miles more. How far did he go ?
John had 70 miles to go. After he had gone 40 miles, how much farther had he still to go ?
(Other examples of this kind.)
Third Step.
The pupils have now learned that the tens progress the same as the units, that they can be added and subtracted the same as the units, and they have also learned how to write the tens.
Fourth Step.
a. 10 miles + 20 miles + 30 miles = ? 30 " +40 " +20 " = ? 50 " +30 " +20 " = ?
etc.
b. 90 miles — 30 miles — 20 miles = ? loo " — 40 " — 30 " = ?
80 " — 10 " — 50 " = ? etc.
C. 30 + 40 — 50 + 10 — 20 = ? 70 — 30 — 20 + 60 + 10 = ?
60 + 40 — 50 — 30 + 40 = ? etc.
d. 40 = 10 + 30 40 == 60 — 20
40 = 20 + 20 40 = 50 — 10
50 = 10 + 40 50 = 80 — 30
50 = 20 + 30 50 = 90 — 40
60 = 10 + 50 30 = 70 — 40
60 = 20 + 40 30 = 90 — 60
60 = 30 + 30 60 = loo — 40 etc.
94 THE SECOND YEAR: 10-100.
60 = 20 -f 20 + ? .70 = 10 + 30 + ? 50 = 30 -f 10 + ? 80 = 40 + 10 + ? 90 = 70 -I- 30 — ? 60 = 80 + 10 — ?
40 = 50 + 20 — ?
Each group to be written from dictation.
/. Concrete examples :
If a man travels 20 and 30 and 40 miles, how far has he gone ?
From A to B is 30 miles, from B to C is 50 miles, from C to D is 20 miles. How far is A from D ?
If I had 90 miles to travel, and have already gone 30 miles, how many miles remain to travel?
(Make more examples of this kind.)
SECOND UNITY.
(Multiplication and Division.) First Step.
Review first a few numbers :
1x1 = 1 i in 1 = 1
2x1 = 2 i in 2 = 2
3x1=3 i in 3 = 3
4x1=4 I in 4 = 4
till till
10 x i = 10 i in 10 = 10
Second Step.
a. Count off the first 10 miles on the numeral frame, also the 2d, 3d, etc., till the loth.
b. Name the ordinals forwards and backwards from the frame as follows :
That is the first 10 miles ;
" second 10 " till
That is the tenth 10 miles.
APPENDIX. 95
Also,
That is the tenth 10 miles ; " ninth 10 " etc.
c. i x 10 miles = 10 miles ;
2 x 10 " =20 "
3 x 10 " =30 "
4 x 10 " =40 "
till 10 x 10 miles = 100 miles.
The same backwards. d. Then briefly :
1 x 10 = 10 10 x 10 = TOO
2 x 10 = 20 9 x 10 = 90
3 x 10 = 30 8 x 10 = 80
4 x 10 = 40 7 x 10 = 70
till till
10 X 10 = IOO I X 10 = 10
Third Step.
#. Practise with concrete and abstract numbers, oral and written, without the numeral frame :
10 miles in 10 miles once ; 10 " 20 " twice; io = ^of2o
10 " 30 " 3 times; io = iof3O till
10 " 100 " lotimes. io=TVof 100, etc.
b. Multiplications and divisions of abstract and con- crete numbers not following regular order :
If a boy walks 10 miles a day, how far will he go in 2 days? In 3, 5, 7, 6, 9, 8, 10?
How much is i of 20 miles? £ of 30 miles? f of 30 miles? £ of 50 miles ? f of 50 miles ? etc.
Fourth Step.
Writing and reading of tens in both the form of mul- tiplication and division.
96 THE SECOND YEAR: 10-100.
Fifth Step.
GENERAL APPLICATION. 7O -r- IO = ? 60 -5- IO = ? 40 -f- IO = ?
How many times 10 in 30? 90? 50? 100?
(6 x 10) + (4 x 10) = ?
(7 x 10) - (5 x 10) = ?
4 x 10 miles are how many more than 2 x 10 miles? How many less than 7 x 10 miles ?
How much is J of 20? f of 30 ? f of 40? | of 50 ?
How much is \ of 20 -f \ of 30 + % of 30 ?
How much is f of 50 — f of 30?
What number is 3 x 10 more than 5 x 10 ?
A man has 70 miles to travel. How long will it take him if he travels 10 miles a day ?
How many miles has he still to go after he has travelled 4 days ?
If a boy earns 10 cents a day for 7 days, how much does he lack of i dollar ?
SECTION II.
Development of tbe whole range from i to 100 with all numbers between as sums of tens and units, in sec- tions from i to 20, 20 to 30, 30 to 40, 40 to 50, 50 to 60, 60 to 80, 80 to 100. This includes addition and subtraction within these limits.
THIRD UNITY. 1 to 2O.
(Addition and Subtraction of Pure Units and Tens. 1O + 6J 2O-5.)
First Step.
Count the numbers from i to 20 on the frame forwards and backwards. Write the numbers 1 1 to 20. Children should be taught —
1. To distinguish tens and units.
2. Different values of figures according to the position or order in which they are found.
3. That the units progress just the same in connection with the tens as when alone.
APPENDIX. 97
Second Step.
10 years 4- i year == 1 1 years ; 10 " +2 years =12 " 10 " + 3 " = 13 " till
10 years 4- 10 years = 20 years.
20 years — 10 years = 10 years ; 19 " — 9 " = 10 18 " - 8 " = 10 " till
11 years — i year = 10 years.
i ten 4- i unit = 11 i " 4- 2 units = 12 i " 4- 3 " =13
till i ten + 10 units = 20
11 = i ten 4- i unit.
12 = 1 " +2 units.
13 = i " 4-3 units, etc.
10 4- i = ii 2d— i = 19
10 4- 2 = 12 20 — 2 = IS
10 + 3=13 20 — 3=17
till till
10 4- 10 = 20 20 — 10 = 10
Dictate the following for the pupils to write rapidly : 14, ii. 17, 13, 19, 12, etc.
The number consisting of i ten and 5 units, i ten and 7 units, i ten and 6 units, etc.
A child is 10 years old. How old will he be in 3 years? In 5, 6, 9, 7, 2?
(Other practical questions.)
Third Step.
Separating the numbers from ii to 20 into tens and units, as well as forming these numbers from this and units. This must be oral and written.
98 THE SECOND YEAR: 10-100.
15 == i ten -f 5 units. 18 = i ten + 8 units.
i ten -f 9 units = 19, etc.
FOURTH UNITY. 1 to SO.
(Addition and Subtraction of Pure Tens in con- nection with Mixed Tens : 1 4 -f 1 O ; 28 - 1 O-
First Step.
Count the numbers from 20 to 30 forwards and back- wards on the frame. Write the same.
Separate into tens and units. Form the new numbers from tens and units.
Second Step.
a. Review the former exercises oral and written, and extend them.
20 miles + i mile = 21 miles; 20 " +2 miles = 22 " 20 " +3 " = 23 etc.
30 miles — i mile = 29 miles ; 30 " — 2 miles = 28 " 30 " — 3 " =27 " etc.
6. 10 miles 4- i mile = n miles;
20 " + I " = 21 "
10 " -f 2 miles = 12 "
20 " + 2 " = 22 "
10 " + 3 " = J3 " 20 '• + 3 " = 23 etc.
10 miles — i mile = 9 miles; 20 " — i " = 19 " 30 " — i " = 29 10 " — 2 miles =8 " 20 " — 2 4< = 18 "
30 " —2 " =28 '«
etc.
APPENDIX, 99
Give also abstract numbers in irregular order :
1 + 10 = ii 11 + 10 = 21
2 + 10=12 12+10=22
3+ 10= 13 13 + 10 = 23
2 + 10= 12 21 — 10 = II
12— 10= 2 22—10=12
13— 10= 3 23- 10= 13
etc.
Practise these exercises and others until the pupils are able to give all possible combinations below 30 in adding and subtracting.
Third Step.
a. Explain the numbers 12 and 21. (12 = i ten and 2 units; 21 = 2 tens and i unit.) Other numbers in the same manner.
b. 14 hours + 10 hours = 24 hours;
18 " +10 " = 28 " 7 " + 10 " = 17 "
etc.
29 hours — 10 hours = 19 hours ;
19 " — 10 " =9
etc.
20 — 8 + 10 — 20 = 26 — 20+10+10 =
15 + 10— 20 + 25 =
FIFTH UNITY. First Step.
a. Length of the schoolhouse • 30 paces (children know as far as that), and still 8 paces.
Width of the schoolhouse : 16 paces. Other exercises that require counting.
b. Review the numbers i to 30.
100 THE SECOND YEAR: 10-100.
Second Step.
a. Extend the counting to 40 : 30 paces + 8 paces, etc.
b. Count on the numeral frame from 30 to 40, i to 40, forwards and backwards.
c. Write the new numbers from dictation, and separate them into tens and units.
Third Step.
Exercises like the previous ones extended to 40, a. 10 + i = ii 20 + i = 21 30 + i = 31
10 + 2 = 12 20 + 2 = 22 30 + 2 = 32
etc.
10 — i = 9 20 — i = 19 30 — i = 29 40 •— i = 39
10 — 2 = 8 20 — 2 = 18 30 — 2 = 28 40 — 2 = 38
etc.
£.10+1 10 + 2 10 + 3
20+1 20 + 2 20 + 3 30+1 30 + 2 30 + 3
etc.
10 — I 10 — 2 10—3 2O — I 2O — 2 2O—3
30—1 30-2 3° - 3 etc.
C. II + 10 21+10 11+20 12 + 10 22 + 10 12 + 20
11 — 10 21 — 10
12 — IO 22 — IO
etc.
d. Mixed exercises :
10 + ii + 10 =
20 + ii — i + 10 =
40 — 1 1 + 6 =
40 — 21 + 6 =
30 — 22 + 10 = etc.
APPENDIX. 10 1
Fourth Step.
a. Write 12, 21, 13, 31, 23, 32. Separate into tens and units.
b. 24 paces + 10 paces — 20 paces =
20 years + 17 years + 2 years —15 years = etc.
Continue until the new numbers are thoroughly mas- tered.
SIXTH UNITY. 1 to 5O. SEVENTH UNITY. 1 to 6O. EIGHTH UNITY. 1 to SO. NINTH 'UNITY. 1 to 1 OO.
The method pursued in these will be the same as previ- ously given, and therefore needs no further explanation. Dollars and cents can be taught in the last " unity."
SECTION m.
^Development of the whole range from i to wo by building each number from products of small num- bers. This includes multiplication and division within these limits.
TENTH UNITY.
(Multiplying and dividing by 2.)
First Step.
Children should be drilled in the kinds of coins from i dollar down, the coins being shown them. They should be able to tell quickly how many cents each coin equals in value. Attention must be called to the differences be- tween the coins, so that the children can readily dis- tinguish them. Little examples should be made, the children working them out with actual or toy money in their hands. Let the children make examples.
Second Step.
a. Count the cents which make 10 two-cent pieces. Count the same number on the numeral frame. The
102 THE SECOND YEAR: 10-100.
children arrange the balls on the frame in 10 twos per- pendicularly under each other, and say as the teacher points :
That is the first two cents ;
" second "
third
till
That is the tenth two cents.
b. Count 2 cents, 4 cents, 6 cents to 20 cents.
c. Give the exercises that follow :
That is i x 2 cents ; " 2x2 " 3x2 "
till
That is 10 x 2 cents. d. 1x2 cents = 2 cents;
2X2 " = 4 "
3x2 " = 6 "
etc.
2 cents = 1x2 cents ; 4 " =2x2" 6 " =3x2"
etc. e. Abstract :
1X2=2 2=1X2
2X2=4 4=2X2
3 x 2 = 6 6 = 3x2
till till
IO X 2 = 2O 2O = IO X 2
Practise and write forwards and backwards.
Third Step.
Complete these numbers with abstract and concrete oral and written examples.
a. 1x2 cents = 2 cents ; 3x2 " =6 " 5x2 " =10 " 7x2 " =14 " 9x2 " = 18
APPENDIX. 103
4 cents =2x2 cents ; 8 " = 4 x 2 "
12 " =6X2"
16 " =8x2"
2O " = IO X 2 "
b. 2 in 2 = once ; 2 in 4 = twice ; 2 in 6 = three times ;
till 2 in 20 = ten times.
i of 4=2 2 = £ of 4
i of 6 = 2 2 = | of 6
t of 8 = 2 2 = i of 8 till till
TV Of 20 = 2 2 = TV Of 20
c. Concrete and abstract exercises in multiplying and dividing with 2 taken in irregular order.
Fourth Step.
a. An orange costs 2 cents. What will 4 oranges cost ? 8? 10?
If you practise music 2 hours a day, how many hours will you practise in 4 days ? In a week ? In 10 days ?
A boy steps 2 feet each time he steps ; how many feet will he go in 7 steps ? In 5 ? In 9 ?
(Other examples.)
b. 6x2 years + 2x2 years ; 5x2 " +4x2 " 4x2 " 4-3x2 "
etc.
10 x 2 years — 3x2 years ; 7x2 " — 5x2 8x2 " — 2x2 etc.
c. 4x2 hours + 15 hours = 9x2" +12 " 7x2" 4-10 "
etc.
104 THE SECOND YEAR: 10-100.
8x2 hours — 10 hours ;
10 X 2 " — l6
12 X 2 " — l8 "
etc.
d. 10 = I X 10 12 = I X 10 -f 2 14 = I X 10 + 4
10 — 5X2 12 = 6X2 14 =6X2 + 2
11 = i x 10 -f i 13 = i x 10 + 3 15 = i x 10 -f 5 11 = 5x24-1 13 = 6x2 + 1 15 = 2x6 + 3
In the same manner to 20.
e. 1x2 tens = 2 tens = 20 i x 20 = 20
2x2 " =4 " =40 2 x 20 = 40
3x2 " =6 " =60 3 x 20 = 60
4* x 2 " =8 " =80 4 x 20 = 80
5x2 " = 10 " = loo 5 x 20 = loo
20 in 20 = once. 20 is i of 40
20 in 40 = twice. 20 is \ of 60
20 in 60 = 3 times. 20 is J of 80
20 in 80 = 4 " 20 is \ of 100 20 in 100 =5 "
/. 2 X II = 22 ^ Of 22 = II
2 X 1 2 = 24 \ Of 24 = 1 2
2 X 21 = 42 \ Of 42 = 21
2 x 22 = 44 £ of 44 = 22
(Other numbers in irregular order.)
ELEVENTH UNITY. (Multiplication and Division by 3.) First Step.
a. Count off threes on the numeral frame and arrange them perpendicularly under each other. > Name them as follows, the teacher pointing as the pupil names :
That is the first 3 ;
second 3 ;
third 3 ;
till
That is the tenth 3.
APPENDIX. 105
b. Count in intervals of 3: 3, 6, 9, 12, 15, to 30, for- wards and backwards.
c. 1x3 dollars = 3 dollars ; 3 dollars = 1x3 dollars ; 2x3" = 6 6 =2x3
3x3" = 9 " 9 " =3x3
till till
10 x 3 dollars = 30 dollars. 30 dollars = 10 x 3 dollars.
Practise forwards and backwards with concrete and ab- stract numbers.
Second and Third Steps by the same plan as previously given.
Fourth Step.
a. i x 3 = 3 i x 30 = 30
2x3 = 6 2x30=60
3x3 = 9 3x30=90
b. 3x1=3 3 x ii = 33 3 x 21 = 63 3 x 31 = 93 3x2 = 6 3x12 = 36 3x22 = 66 3x32 = 96 3x3 = 9 3x13 = 39 3x23 = 69 3x33 = 99
'• 3-3=1 33-3=11 634-3 = 21 93-3 = 31 6-5-3 = 2 36-^-3 = 12 66-5-3 = 22 96-5-3 = 32 9-3 = 3 39-*- 3 = 13 69-5-3 = 23 99-5-3 = 33
TWELFTH UNITY. (Multiplication and Division by 4.)
Follow the same plan as before. In the last step in- troduce also :
b.
|
4 4 4 4 |
a x X X X |
I 2 3 4 |
I X 2 X 3 x 4 x = 4 = 8 = 12 = 16 |
4 4 4 4 |
4 4 4 4 |
4 8 12 16 X X x X |
II 12 13 H |
= 44 = 48 -$ |
i 2 3 4 |
x 40 x 40 x 40 x 40 4 x 4 x 4 x 4 x |
= 40 = 80 = 120 = 160 21 =84 22 = 88 23 = 92 24 = 96 |
J06 THE SECOND YEAR: 10-100.
4-5-4=1 44-7-4=11 84-7-4 = 21
8-7-4 = 2 48-1-4=12 88-7-4 = 22
12-4-4 = 3 52-7-4=13 92-7-4 = 23
16 -j- 4 = 4 56 -T- 4 = 14 96 -T- 4 = 24
THIRTEENTH UNITY. (Multiplication and Division by 5.)
Add to the plan of the previous steps :
#.1x5 = 5 i x 50 = 50
2x5= 10 2x50= loo
3x5 = 15 3x50=150
4x5 = 20 4x50 = 200
5X5 = 25 5X50 = 250
b. 5x1 = 5 5 + 11 = 55
5x2 = 10 5 x 12 = 60
5x3 = 15 5x13 = 65
5 x 4 = 20 5 + 14 = 70
5x5 = 25 5 + 15 = 75
*. 5-4-5 = 1 55-1-5 = 11
10-7-5 = 2 60-7-5 = 12
15^-5 = 3 65-7-5 = 13
20-7-5=4 70-7-5 = 14
25-7-5 = 5 75-5 = 15
FOURTEENTH UNITY.
(With the 6.)
FIFTEENTH UNITY.
(With the 7.)
SIXTEENTH UNITY. (With the 8.)
SEVENTEENTH UNITY. (With the 9.)
Nothing further need be given to illustrate this system. The remaining steps are carried out in the same manner as those which are given in full,
SECOND COURSE
i.
NUMBERS ABOVE 1OO.
THE THIRD YEAR.
FIRST HALF OF THE YEAR.
1 OO to 1 OOO.
1. As the numbers between 100 and 1000 are combina- tions of the numbers within the first hundred, the only purpose of this course is to reduce them to their ele- ments.
2. Thereby the pupil comes into possession of the secret of all accurate and rapid mental work in arithmetic, namely, always to operate with the smallest possible numbers; hence he needs none of the so-called "arith- metical knack."
3. In order to lead to an allsided representation of the number, it is impossible to consider all of the fundamen- tal rules at once as heretofore ; this will receive wider at- tention in the second half of the year. Mental and written arithmetic are now united at every step.
4. As the necessity of isolating each number now dis- appears, and because the allsided penetration and com- prehension of each number must take place, the material will be divided in two parts only :
A. The pure number : measuring, separating, compar- ing, and combining.
B. The applied Number.
IOS THE THIRD YEAR: IOCKIOOO.
The child is now sufficiently mature gradually to leave simple mechanical processes and make more use of the understanding and reason. He must, however, master the processes, so as to be able to give them rapidly and almost mechanically. This side of the work is not to be neglected. Illustrations should still be used where needed, but he will learn chiefly by analogy from smaller numbers.
A. — Allsided Contemplation of the Pure Number.
(First Quarter.)
FIRST STEP.
Measuring of the numbers by the units of the Decimal
System, by units, tens, and hundreds. * a. (Oral.)
Count upwards and downwards from 100 to 1000. 10 splints can be bound together, and that is i ten. Around 10 of these bundles (100) a wide ribbon may be tied ; 10 of these bundles make 1000. In this way the pupil gains a comprehension of 1000. Solid blocks divided by lines into 10 and 100 units can also be used.
During the counting the pupils must often be stopped and questioned as to which hundred and which ten they are in : How many units and tens are lacking in the tens and hundreds respectingly, and in the hundreds how many are lacking from a thousand ?
For example : The teacher gives 768. The pupil will explain :
768 = 7 hundreds, 6 tens, 8 units. It lacks 2 units of completing the 7th ten, then 3 tens of completing the 8th hundred, and finally 2 hundreds of completing 1000.
829 == 8 hundreds, 2 tens, 9 units. Complete the analysis of 829.
Analyze in the same way 999, 500, 463, 271, 604.
What number is composed of 3 hundreds, 6 tens, and 5 units?
How many units in 7 hundreds, 8 tens, and 9 units ?
How many units in one thousand ? Tens ? Hundreds ?
Of what does 669 consist ?
(6 x 100) + (6 x 10) + (9x1)
FIRST HALF OF THE YEAR. IOQ
b. (Written.)
To make it easier for beginners in writing, use the fol- lowing plan :
h. t. u.
i o i = 101
4 8 o = 480
10 o o= 1000
Numbers must be dictated for the pupils to write. When written on the blackboard, the figures should be named and the numbers read, etc., in order to acquire perfect mastery of the subject.
Finally, analogous to the oral work, dictated numbers should be written out as follows :
615 = 6 x 100 + i x 10 + 5 x i
2O4 = 2 X ICO -fOX IO + 4 X I
390 = 3 x loo + 9 x 10 + o x i
1000 = 10 X IOO + 0X10 + 0X1
or 615 = 600 +10 + 5
204 = 200 + 4 390 = 300 + 90
IOOO = IOOO
In the following steps we shall give only one form, which will answer for both oral and written work. Neither is to be omitted, but they must be united, the oral taking precedence in order of time. Follow the plan suggested in the First Step.
SECOND STEP. The pure hundreds measured with hundreds.
Measuring, comparing, rapid work, combining, are to be the same as in the First Course. With the number 2 in the First Course we obtained the following scheme :
1 + 1=2
2X1=2
2—1 = 1
2 -f- I = 2
HO THE THIRD YEAR: 100-1000.
Conformably to this the pupil now learns : 200.
100 + 100 = 200 2 X IOO = 2OO
2OO — IOO = IOO 200 -T- IOO = 2
What number is contained twice in 200? Of what number is 200 the double ? Of what number is 100 the half? What number must I double in order to get 200 ? (Give many other examples, following the method em- ployed in teaching the 2. See Second Step in the First Course.)
300.
100 -f loo + ioo = 300 3 x ioo =300 300 -i- ioo = 3
200 + IOO = 300 3OO — IOO = 2OO 3OO — 2OO = IOO
300 -f- 200 = i, with remainder of ioo.
300 is ioo more than 200, 200 more than ioo.
200 is ioo less than 300, ioo more than a hundred.
ioo is 200 less than 300, ioo less than 200.
300 is 3 times ioo.
ioo is \ of 300.
Of what equal and what unequal numbers does 300 consist ?
How much is 300 — ioo — ioo + 200 ?
300 •«- 3 — ioo -f 200 -f- ioo x ioo? 300 — 200 + ioo + ioo -5- 3 — ioo ?
From what number can you take away 2 x ioo and have ioo left ?
£ of 300 is how much less than \ of 300 ? Which is greater, \ of 300 or \ of 200 ?
FIRST HALF OF THE YEAR. 1 1 1 400.
I. Measure with 100 :
100 + 100 + 100 -f 100 = 400 4 x 100 = 400
400 — 100 — 100 — 100 = 100 400 -f- 100 = 4
2. Measure with 200 :
200 4- 200 = 400 2 x 200 = 400 400 — 200 = 200
400 -5- 200 = 2
3. Measure with 300 :
300 + 100 = 400 100 + 300 = 400 i x 300 + 100 = 400 400 — 200 = 200 400 — 300 = 100 400 -5- 300 = i (100)
400 is 100 more than 300 200 more than 200 300 more than 100
300 is 100 less than 400 100 more than 200 200 more than 100, etc.
THIRD STEP. Mixed hundreds measured mith mixed hundreds.
220 = 2 x no, also i x 220 440 — 4 x no, also 2 x 220 660 = 6 x no, also 3 x 220
880 = 8 x 1 10, also 4 x 220, 2 x 440 990 = 9 x 1 10, also 3 x 330
112 THE THIRD YEAR: 100-1000.
How may 888, 999, be considered ?
888 = 8 x in, also 4 x 222, 2 x 444 999 = 9 x in, also 3 x 333
999 •*• 333 = 888 +• 222 =
999 -*- 3 = 888 -5- 4 =
999 -f- 1 1 1 = 888 -H 8 = etc. etc.
Of what number is 120 the J, £, i? What is i of 844 ? Of what number is 844 fourfold? What number can I take 4 times from 844 ? What number is contained 4 times in 844? \ of 844 is how much greater than J ? i of 333 is -J- of what number?
Of what number is \ of 333 the ninth? (\ of 333 = in. in is \ of 9 times 1 1 1 = 999.) Compare 365 with 244.
365 = 3 hundreds 4- 6 tens + 5 units ; 244 = 2 hundreds + 4 tens + 4 units.
3 hundreds — 2 hundreds = i hundred ; 6 tens — 4 tens = 2 tens ; 5 units — 4 units = i unit. Therefore 365 — 244 = i h.+ 2 t.-f i u.= 121, or 365 is 121 greater than 244, and 244 is 121 less than 365.
What is the difference between 743 and 120?
743 = 7 h.-f 4t.+ 3u.; 120= i h.+ 2 t. + ou.; 7 h. - i h. = 6h. 4t. — 2t. = 2 t.; 3u.— ou. = 3 u. Therefore 743 — 120 = 6 h. + 2 t. + 3 u. = 623.
What number = 743 + 221 ?
743 = 7 h.+ 4t.+ 3 u.; 221 =2 h.+ 2 t.+ i u.; 7 h. + 2 = 9 h.; 4 t. + 2 t. = 6 t.; 3 u. + i = 4 u. Therefore 743 + 221 = 9 h. -I- 6 t. + 4 u. or 964.
How much is in -f 212 + 313?
How much is 112 + 113 -f 114?
Subtract 322 and 124 from 659.
Continue these exercises until the subject is undei- stood.
FIRST HALF OF THE YEAR. 1 13
FOURTH STEP. Measuring of hundreds with tens.
I.
a. The pure hundreds. Since 100 = 10 x 10,
2 x 100 or 200 = 2 x 10 x 10 = 20 x 10
3 x 100 or 300 = 3 x 10 x 10 = 30 x 10
4 x loo or 400 = 4 x 10 x 10 = 40 x 10
10 x loo or looo = 10 x 10 x 10 = 100 x 10 =1000.
b. Hundreds with tens. Since 100 = 10 x 10,
no = (10 x 10) + (i x 10) = ii x 10
120 = (10 X 10) + (2 X 10) = 12 X 10
130 = (to x 10) + (3 x 10) = 13 x 10
140 = (10 x 10) + (4 x 10) = 14 x 10
150 = (10 x 10) + (5 x 10) = 15 x 10
190 = (10 x 10) + (9 x 10) = 19 x 10
240 = (20 x ip) + (4 x 10) = 24 x 10
990 = (90 x 10) + (9 x 10) = 99 x 10
c. Hundreds with tens. Since 100 = 10 x 10,
ioi = (10 x 10) + i 109 = (10 x 10) + 9 906 = (90 x 10) -f 6 814 = (81 x 10) + 4
How many tens in 500, 900, 1000? What number = 53 tens ? What number = 9 units more than 53 tens ? How many times 10 is 660, 420, 870 ? Of what number is 10 the 42d part ? The 66th ? The 84th ? The ;oth ?
How many tens in 879?
114 THE THIRD YEAR: 100-1000.
II.
Comparison.
Compare 400 with 900.
(400 = 40 tens ; 900 = 90 tens ; 90 tens — 40 tens = 50 tens. Therefore 900 has 50 tens more than 400, and 400 has 50 tens less than 900.)
How many are 55 tens less than 600? Than 660? Than 990?
(As 600 = 60 tens, 55 tens are 5 tens, or 50 less than 600.)
Of what 4 equal tens does 880 consist ?
(880 = 88 tens, and as 88 tens = 4 x 22 tens, 880 is composed of 4 x 22 tens.)
What is the sum of 800, 180 and 20?
(800 + 180 -f 20 = 80 + 18 + 2 tens = 100 tens = 1000.)
What is the difference between 160 and 210?
(210 or 21 tens — 160 or i6tens = 5 tens = 50.)
60 x 10 = how many times 100 ?
What number has 8 tens and 9 units more than 490?
(The number which has 8 tens and 9 units more than 490 must equal 490 + 8 tens + 9 units. 490 or 49 tens + 8 tens = 57 tens = 570. 570 -f 9 = 579.)
I have taken a number 87 times, added 9 to it and ob- tain 879. What is the number ?
(879 = 87 tens + 9 units. Therefore I must have taken ten 87 times.)
How many more tens has 73 x 10 than the double of 240?
How many times is yj^ of 1000 contained in 500?
(looo = loo tens ; yj-g- of 100 tens = i ten ; 500 = 50 tens ; 50 tens contains i ten 50 times.)
\ of 630 = J of what number ?
(630 — 63 tens ; i of 63 tens = 21 tens ; 21 tens is i of 4 x 21 tens = 84 tens = 840.)
-5*5- of 680 + ^ of 240 is how much less than 10 x 36 ?
(As 680 = 68 tens, ¥V of 680 = i ten ; and -fa of 240 = I ten ; both together = 2 tens. 10 x 36 = 36 tens, and 36 tens — 2 tens = 34 tens.)
The factors must also be changed in these exercises.
FIRST HALF OF THE YEAR. 11$
110= II X 10= 10 X II 22O = 22 X IO = IO X 22
680 = 68 x 10 = 10 x 68 990 = 99 x 10 = 10 x 99
Of what is 990 composed ? How can 130 be composed from 13 ? 280 from 28 ? 560 from 56 ? What number must I take 10 times to get 670? Of what number is 67 one tenth ? What is -gV of 670 ?
How many times is 79 contained in 790 ? What number can I take 10 times from 790 ? What 79 times ? 79 x 10 =10 times what number ?
These exercises lead us naturally to the next step.
FIFTH STEP. Measuring a number by its factors.
I.
a. The pure hundreds.
100 = 2 x 50, 4 x 25, 5 x 20, 10 x 10
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Therefore, |
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200 = |
2 |
x |
2 |
x |
50 |
= |
4 |
X |
50 |
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2 |
x |
4 |
X |
25 |
= |
8 |
x |
25 |
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2 |
X |
5 |
X |
20 |
= |
IO |
X |
20 |
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2 |
X |
10 |
X |
10 |
— |
20 |
x |
10 |
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300 = |
3 |
X |
2 |
X |
50 |
= |
6 |
X |
50 |
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3 |
x |
4 |
X |
25 |
= |
12 |
X |
25 |
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3 |
X |
5 |
X |
20 |
= |
15 |
X |
20 |
3 x 10 x 10 = 30 x 10 = 20 x 15 b. Hundreds with tens.
220 = 10 X 22. As 10 = 2 X 5, 22O = 2X5X22 = 2
x 1 10 ; and as 22 = 2 x 1 1, 10 x 2 x 1 1 = 10 x 22 ; and as 10 = 5 x 2, 5 x 2 x 22 = 5 x 44.
Il6 THE THIRD YEAR: 100-1000.
960 = 10 x 96.
10 X 2 X 48 == 20 X 48 = 48 X 20
10 x 3 x 32 = 30 x 32 = 32 x 30 10 x 4 x 24 = 40 x 24 = 24 x 40 10 x 6 x 16 = 60 x 16 = 16 x 60
10 X 8 X 12 = 80 X 12 = 12 X 80
Or leaving the second factor unchanged :
iox96 = 2X5X96 = 2x 480 $X2X96 = 5x 192
c. Hundreds with tens and units.
426 = (10 x 42) + 6 or (4 x 100) -f 26
(10 x 2 x 21) + 6 or (20 x 21) + 6
(10 x 3 x 14) + 6 or (30 x 14) + 6 = (i4x 30)4-6
(2 x 5 x 42) -f 6 = (2 x 210) + 6
(5 x 2 x 42) + 6 = (5 x 84) + 6
896 = (8 x 100) + (8 x 12) = 8 x 112
(8 XII 2) = 2X4X1 12 = 2X 448
4x2 x ii2=4x 224
(10 x 89) + 6 = (2 x 5 x 89) + 6 = (2 X445) + 6 (5 x 2 x 89) + 6 = (5 x 178) + 6
489=3(10 X48) + 9 (2 x 240) + 9
(5 x 96) + 9 Or:
(10 x 4 x 12) + 9 = (40 x 12) + 9 = (12 x 40) + 9 (10 x 3 x 16) + 9 = (30 x 16) + 9 = (16 x 30) + 9 (10 x 2 x 24) + 9 = (20 x 24) + 9
300 is composed of how many twos ? Threes ? Fives ?
How do you find the 25th part of 300 ?
(^5 of 100 = 4, so ^V of 300 = 3 x 4 = 12.)
How does 300 arise out of 15 ?
(As 300 = 10 x 30, and 30 = 2 x 15, 300 = 10 x 2 x 15 = 20 x 15. I have taken 15 twenty times and obtained 300. Or: as 300 = 2 x 150, and 150= 10 x 15, 300 = 2 x 10 x 15 = 30 x 15.)
How many times must I take 44 to get 220 ?
(220 =IOX22=5X2X22=5X 44.)
FIRST HALF OF THE YEAR. 1 1/
What number must I take 5 times from 426 in order to have 6 remainder ?
(426 = 10 x 42) + 6 = (5 x 2 x 42) -f 6 = (5 x 84) + 6. So I must subtract 84 five times from 426, and have 6 remainder.)
How many times is 24 contained in 489?
(489 = [10 x 48] + 9 = [10 x 2 x 24] + 9 = [20 x 24] + 9. 24 in 489 20 times with 9 remainder.)
II.
Comparison.
What is the difference between 980 and 377?
(980 = 98 tens, and 377 = 37 tens and 7 units. 98 tens — 37 tens = 61 tens — 7 units = 60 tens + 3 units = 603. Or : 900 — 300 = 600 ; 80 — 77 = 3 ; 980 — 377 = 603.)
The difference between 980 and 377 is 3 times what number ?
(The difference between 980 and 377 is 603. 603 is 3 times i of 603. \ of 600 = 200, \ of 3 = i, 200 + i = 201. Therefore, etc.)
•J- and £ of 480 taken together is how many less than twice 480 ?
By what number must I divide 365 to get 5 ?
(If I divide 365 by a number and get 5, that number is contained 5 times in 365, or is \ of 365. \ of 300 = 60 ; \ of 65 = 13 ; 60 -f 13 = 73. Therefore, etc.)
What is the difference between ^ and ^ of 660 ?
The sum of 326 and 418 is how much greater than the sum of their halves ?
I take 4 units from a number, and then divide the re- mainder by 1 6, and obtain a quotient of 60. What is the number ? %
(The unknown number is 16 x 60 + 4. 60 = 6 tens; 1 6 x 6 tens = 96 tens = 960; 960 + 4 = 964. Therefore, etc.)
What number is 10 more than the double of 5 x 99?
Il8 THE THIRD YEAR: 100-1000.
SIXTH STEP. Reduction of numbers from i to 1000 to their elements.
It does not matter in what order the numbers are taken, the chief object being practice in rapidly and accurately separating the numbers into their elements. The pupil is now able to tell at a glance into what parts the number must be separated. The teacher should make the work partly oral and partly written.
3GO.
300 + 60 3 x loo + 3 x 20 i So -f 1 80 3 x 1 20
200 4- 160 10 x 36 320 + 40 5 x 72
336 + 24 20 x 18 = 18 x 20
etc. 9 x 40 etc.
320 + 45, 2 x 150 -f 65, 2 x 182 + i, 7 x 50 + 15, 14 x 25 + 15, 18 x 20 -f 5, etc.
These six steps complete the work of the first quarter. The work now changes from the pure to the applied number. At this point especial attention should be given to compound numbers, weights, measures, money, etc. This will employ the second quarter. The teacher must supply a great many more examples than are here given, as the " applied number" is of great importance.
B. — All-sided Contemplation of the Applied Number. (Second Quarter.)
a. The tens.
10 cents = i dime. 10 dimes = i dollar. 10 dollars = i eagle.
3 dimes = 30 cents.
5 dimes = 50 cents or \ dollar. 50 " =5 dollars or \ eagle.
FIRST HALF OF THE YEAR. I 1 9
1 dollar — 10 dimes.
2 dollars = 20
3 " =30 5 " =50
10 " = 100
100 cents = 10
or i dollar.
loo dollars = 1000 dimes. 900 dimes = 30 x 30 dimes. 870 " = 29 x 30 " 840 " = 28 x 30 810 " = 27 x 30 " etc.
9, n, 17, 28 dollars = how many dimes?
9 dollars, 4 dimes, 24 cents = how many cents ?
314, 365, 720, 799 cents, how many dimes? How many dollars ?
25 dimes + 9 dimes -f 17 dimes + 15 dimes = how many dimes ? How many dollars ?
From 2 dollars and 6 dimes take 15 dimes. 17 dimes. 25 dimes.
Divide the class into two divisions and let the first division give 3 times, and the second division 4 times, the numbers as the teacher names them.
|
Teacher. |
ist Division. |
2d Division. |
|
25 dimes. |
75 dimes. |
100 dimes. |
|
9 " |
27 " |
36 « |
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15 « |
45 |
60 " |
|
17 " |
5i " |
68 •« |
Again, the first division can give 3 times the number given, and the other division add the results.
Teacher. ist Division.
19 dimes. 5 dollars and 70 cents. 12 " 3 " " 60 " 22 " 6 " " 60
2d Division 15 " " 90 "
In long examples the results of multiplying can be writ- ten in order to get the total correctly.
120
THE THIRD YEAR: 100-1000.
14 dollars 9 dimes.
3
17 18
19
102
7 4 8 6 5
9
This will be added as follows, the operations being mental, only the above being set down :
14 dollars 9 dimes + 15 dollars = 29 dollars 9 dimes, -f 7 dimes = 30 dollars 6 dimes; 30 dollars 6 dimes + 16 dollars 4 dimes = 47 dollars; 47 dollars -f 17 dollars 8 dimes = 64 dollars 8 dimes; 64 dollars 8 dimes 4- 18 dol- lars 6 dimes = 83 dollars 4 dimes ; 83 dollars 4 dimes + 19 dollars 5 dimes = 102 dollars 9 dimes.
2. Reverse the process, subtracting 19 dollars 5 dimes from 102 dollars 9 dimes, and so on until the first number, 14 dollars 9 dimes, is reached.
3. Multiply each of these numbers by 2, 3, 4, 5, and add the products, and divide the sum by 2, 3, 4, 5, and see if the sum of the original numbers is obtained.
14 dollars 9 dimes x 3 = 44 dollars 7 dimes.
II
17 18
12-
102
7 4 8 6 5
47 49 53 55 J8
308
3 in 308 dollars 7 dimes = 102 dollars 9 dimes.
b. Tens and units.
24 dimes = how many cents ? 24 dozen = " " things ? 24 dollars = " " dimes ?
FIRST HALF OF THE YEAR.
121
Pupils must be drilled in tables from
12 x i to 24 x 12,
15 x i to 15 x 15,
16 x i to 16 x 16, 24 x i to 24 x 24.
One dollar = 10 dimes = 100 cents. One half " = 5 One fourth " = 2^
One fifth " = 2 =20
One tenth " = i = 10
One twentieth = £ =5
One fiftieth " = | =2
One hundredth " = -^ = i cent.
100 cents = 10 = i dollar. So " = 5
20 = 2£
etc.
The English weights and measures cannot well be ap- plied to the Grube system, as the scales, instead of being decimal, are varying. And yet, with their present knowl- edge of the numbers as high as 1000, and with the appli- cations of compound numbers made in the First Course during the first two years, the pupils should be able to master the subject of compound numbers in connection with the above work during the second quarter.
II.
THE FOUR FUNDAMENTAL RULES IN ABSTRACT AND CONCRETE NUMBERS -UNLIMITED RANGE.
THE THIRD YEAR.
SECOND HALF OF THE YEAR.
The division of the work is as follows : A. — With abstract numbers.
!i. Numeration.
2. Addition.
3. Subtraction.
4. Multiplication.
[ 5. Division.
B. — With concrete numbers.
f i. Numeration.
12. Addition.
3. Subtraction.
4. Multiplication.
5. Division.
Mental and written arithmetic must not be separated, as though they were different studies ; mental arithmetic is the foundation ; here the idea has its beginning; on this the written statement depends.
It becomes necessary in large numbers, and where several numbers are involved, to write them down in
SECOND HALF OF THE YEAR. 123
figures, so that they can be seen by the eye. The writ- ten work may thus be considered as only an assistant to the mental work.
It must not be thought that in addition there shall be only adding, and in subtraction only subtracting ; but, as in the preceding steps, all the operations must be united as far as may be. Many examples must be given. The exercises in " rapid work" must not be neglected, and emulation among the pupils must be aroused to secure rapid and accurate work.
Only the important points of each step will be given, the filling in with material being left to the skill and tact of the teacher.
A. — With Abstract Numbers.
FIRST STEP.
Numeration.
a. (Oral.) Thousands and millions. If we have the 10 hundreds we have a new unity — the thousand. We can combine thousands into tens of thousands ; tens of thou- sands into hundreds of thousands, etc., exactly as we combined units into tens, tens into hundreds, etc.
unit = i unit = i.
unit = 10 units = 10.
ten = 10 tens = 100.
hundred = 10 hundred = i thous. = 1000.
thousand = 10 thousand = i tenth. = 10000.
ten thous. = 10 ten thous. = i h'd th. = 100000.
hund. th. = 10 hund. thous. =i million = loooooo.
i x 10 x 10 x
JO X
10 x 10 x 10 x
In the same way :
i x 2 units = 2 units — 2
10 X 2 " = 20 " = 20
10 x 2 tens = 20 tens = 200 etc.
i x 3 units = 3 units = 3 10 x 3 " = 30 " = 30 10 x 3 tens = 30 " = 300 etc.
I24
THE THIRD YEAR: 100-1000.
i unit — i x I unit
i ten = 10 x i unit = 10 units.
i hundred = 10 x i ten = 100 units. i thousand — 10 x i hundred = 100 x i ten = 1000 units. etc.
The units are units of the first order, the tens .are units of the second order, the hundreds are units of the third order, etc. The units of each order are 10 times more than the units of the preceding order. In each order there can be only 10 units, and the tenth unit is the first of the following order.
b. (Written.)
We will write the number 1852, placing the units, tens, hundreds, and thousands, each in a separate box.
Counting from the right towards the left, we have units in the first box, tens in the second, hundreds in the third and thousands in the fourth, tens of thousands in the fifth, hundreds of thousands in the sixth, millions in the seventh.
Or,
Read'. 6 millions.
0 hundred thousands. 9 ten thousands.
2 thousands.
1 hundred. 6 tens.
o units.
6 million 92 thousand i hundred and 60.
Give the name of the units of the ist, 3d, 5th, 4th, 6th order. What are units of the 8th order called ? etc., etc.
SECOND HALF OF THE YEAR.
12$
Then the teacher writes 9 in each box successively, and the children read it.
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8 |
7 |
6 |
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2 |
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9 |
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9 |
o |
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9 |
o |
o |
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9 |
o |
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0 |
o |
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9 |
o |
o |
o |
o |
0 |
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9 |
o |
0 |
o |
o |
o |
o |
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9 |
o |
o |
0 |
0 |
0 |
o |
o |
Again the teacher writes and the children read :
90,000,000
9,000,000
900,000
90,000
9,000
900
90
9
Instead of boxes columns may be used as follows. The teacher writes, and the children rea
126
THE THIRD YEAR: loo-iooo.
|
a. Mil. |
8 T. Millions |
7 Millions |
6 H. Thous. |
4. Thous. |
4 Thous. |
3 Hund. |
2 Tens. |
X Units. |
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3 |
O |
S |
6 |
4 |
8 |
7 |
0 |
O |
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2 |
6 |
5 |
9 |
O |
8 |
4 |
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7 |
0 |
o |
9 |
3 |
6 |
4 |
2 |
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9 |
8 |
6 |
O |
The pupils should arrange such columns on their slates, and on the blackboard then write numbers in them from dictation. They must also be drilled in such exercises as :
What is the 3d order, the 5th, the 8th, the 6th, the 4th, etc. ?
The hundreds are what order ? The millions ? The tens of thousands ?
The orders must be known from right to left and left to right, so as to be given without hesitation with great facility.
Then the numbers may be divided into periods (threes), attention being called to the fact that there are "units," " tens," and " hundreds" of units ; also units, tens, and hundreds of thousands, millions, etc.
The periods may be shown as follows :
Million. Thousand.
Units.
|
h. 3 |
t. 2 |
u. 5 |
h. I |
t. 7 |
u. 6 |
h. I |
t. 8 |
u. 0 |
Then all columns, etc., may be abandoned, and the numbers written in periods. Always have the commas, which separate the periods, placed in at the time of writ- ing the number, and not after all the figures have been written. Thus: 325 million (comma), 176 thousand (comma), 180. Thus the pupil becomes perfectly sure in both writing and reading numbers.
SECOND HALF OF THE YEAR. 127
SECOND STEP.
Addition. ORAL AND WRITTEN.
a. Numbers of one figure :
4 units -I- 5 units = 9 units (4 + 5 = 9). 4 tens + 5 tens — 9 tens = 90 (40 -f 50 = 90). 4 hund. + 5 hund. = 9 hund. = 900 (400 4- 500 = 900)0 4 th. + 5 th = 9 th. = 9,000 (4,000 + 5,000 = 9,000). 4 ten th. + 5 ten th. = 9 ten th. = 90,000 (40,000 + 50,000 = 90,000).
etc.
b. Numbers of two figures and one figure :
43 units -f 5 units = 48 units (43 + 5 = 48). 43 tens + 5 tens = 48 tens = 480 (430 + 50 = 480). etc.
c. Numbers of two figures and two figures :
43 units -f 28 units = 71 units (43 + 28 = 71). 43 tens + 28 tens = 71 tens = 710 (430 -f- 280= 710). etc.
d. Numbers of three figures and one figure :
416 units + 8 units = 424 units (416 + 8 = 424). 416 tens + 8 tens = 424 tens = 4,240 (4,160 + 80 = 4,240). etc.
e. Numbers of three figures and two figures :
416 units + 23 units = 439 units (416 + 23 = 439).
416 tens + 23 tens = 439 tens = 4,390 (4,160 + 230 = 4,390)-
416 hund. + 23 hund. = 439 hund. = 43,900 (41,600 -f 2,300 = 43,900).
416 th. + 23 th. = 439 tn- = 439,000 (416,000 + 23,000 = 439,000).
etc.
128 THE THIRD YEAR: 100-1000.
/. Numbers of three figures and three figures :
416 units 4- 123 units = 539 units (416 + 123 x 539). 416 tens + 123 tens = 539 tens = 5,390 (4,160 + 1,230
= 5»39°)-
416 hund. 4- 123 hund. = 539 hund. = 53,900 (41,600 4- 12,300=53,900).
etc.
The corresponding method for operations to acquire rapidity. For examples :
a. 7 4-8 +9 +6 70 +80 +90 +60 700 4- 800 4- 900 4- 600 7000 4- 8000 4- 9000 4- 6000 70000 4- 80000 4- 90000 + 60000
etc.
b. 25 4- 9 + 3 + 8 250 4- 90 4- 3° + 80 2500 4- 900 4- 300 + 800
etc.
c. 25 +36 4-47 + 58 250 4- 360 4- 470 + 580 2500 4- 3600 4- 4700 4- 5800
etc.
d. 254 + 6 4- 8 4- 9 2540 4-60 4-80 4-90 25400 4- 600 4- 800 4- 900
etc.
e. 254 4-27 4-38 +49 2540 4- 270 4- 380 4- 490 25400 + 2700 4- 3800 4- 4900
etc.
/. 254 4- 316 + 449 2540 4-3160 4- 449° 25400 4- 31600 4- 44900 etc.
SECOND HALF OF THE YEAK.
129
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The pupils must be led step by |
step in written addition |
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as well as oral. |
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Let them write in |
columns : |
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(i) 36 |
Shorter : 36 |
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24 |
24 |
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15 |
15 |
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23 |
23 |
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50 |
50 |
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18 |
units |
148 |
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13 |
tens |
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148 |
units |
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(2) 365 |
or 365 |
Shorter : 365 |
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21 |
21 |
21 |
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1430 |
1430 |
H30 |
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2045 |
2045 |
2045 |
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320 |
320 |
320 |
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3000 |
II |
4181 |
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1000 |
170 |
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170 |
1000 |
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II |
3000 |
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4181 |
4181 |
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(3) 5946 |
5946 |
5946 |
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847 |
847 |
847 |
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239 |
239 |
239 |
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6320 |
6320 |
6320 |
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2200 |
130 |
PX*)W |
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13352 |
||
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130 |
22OO |
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22 |
1 1 000 |
13352 13352
(Place the number to be "carried" under the column to which it belongs, in parenthesis, using smaller figure.)
Pupils should be able to answer promptly questions on the addition processes.
I3O THE THIRD YEAR: 100-1000.
THIRD STEP. Subtraction.
ORAL AND WRITTEN.
a. Numbers of one figure :
9 units — 5 units = 4 units (9 — 5 = 4). 9 tens — 5 tens = 4 tens (90 — 50 = 40). 9 hund. — 5 hund. = 4 hund. (900 — 500 = 400). etc.
b. Numbers of two figures with one figure :
12 units — 5 units = 7 units (12 — 5 = 7). 12 tens — 5 tens = 7 tens (120 = 50 = 70). 12 hund. — 5 hund. = 7 hund. (1200 — 500 = 700). etc.
Continue in the same manner as in addition. From 9456 take 7321.
a. 9456 = 9000 + 400 + 50 + 6 = 9456 7321 = 7000 + 300 + 20 + i = 7321
2000 + 100 -f 30 + 5 = 2135
b. 4325 — 1123 == 4325 Minuend.
1123 Subtrahend.
3202 Remainder. c. (Borrowing without ciphers.)
16 15 18
(I) 17-6-8* 1768
679 679
1089 1089
(2) 4 5-4-2 45'4'2
4159 4159
383 383
* Grube" placed dots at the bottom where " borrowing" was employed ; but in order to avoid confusing them with the later use of the decimal point, we place them at the top. The Germans use the comma for the deci- mal point, therefore this danger does not arise with them.
SECOND HALF OF THE YEAR. 13!
Where "borrowing" is necessary, make the necessary changes in the minuend, placing smaller figures above to indicate the changes. Then work the same example without making the changes.
d. (Borrowing with ciphers.)
(1) 7 4- 6-0 74-6-0 3269^ 3269
4191 4I91
(9 units cannot be taken from o units, so I borrow from the tens ; that is, take one from the 6 tens, which I indicate by a point after the 6. The ten taken away = 10 units. 9 units from 10 units = i unit ; etc.)
10 16
(2) 74*0-6 74-06 3269 3269
4137 4137
10 13 16
(3) 7'04-6 7-04-6 3269 3269
3777 3777
9 10 10
(4) 74-00 74-00
3269 3269
4i 31 4i 31
* 9 9 6101014
(5) 7-004 7-004
3269 3269
3735 3735
132 THE THIRD YEAR: 100-1000.
9 9
10 10 10
(6) 7*0 o o __ 7-0 o o
3269-3269
3731 3731
999
10 lo 10 16
(7) 7*0 o o 6 __ 7*0 006
32697-32697
37309 37309
This kind of work must not be left until the pupil is able to perform it rapidly. He must understand that as 7 units cannot be taken from 6 units, he must borrow of the tens. As there are no tens, hundreds, or thousands, he must go to the tens of thousands to borrow, and reduc- ing one of this to the next lower denomination, then bor- rowing from that, etc., until we come to the order where we need to increase the minuend figure.
The explanation of subtraction is based on addition. Take two numbers :
a. 3480
b. 2375 Adding.
c. 5855
How large is a ? (a = 5855 — 2375.)
How large is bt \b = 5855 — 3480.)
How was c found ? (By adding a and b.)
How were a or b found ? (By subtracting the known a
or b from their sum c.)
What do we call c in this subtraction ? (The minuend.) What do we call the a or b ? (The subtrahend.) What do we seek ? (The remainder or difference.) If the minuend were unknown how would you find it ?
(By adding the subtrahend and difference.)
5855 5855
- 3480 - 2375
\ Proof.
SECOND HALF OF THE YEAR. 133
How can I prove subtraction ?
What is the whole in an example in subtraction called ? (The minuend.)
In addition? (The sum.)
What is the difference between addition and subtrac- tion ? (In addition the parts are given from which the whole is to be found ; in subtraction the whole and one of the parts are given from which the other part is to be found.)
The sum 5855 is composed of three numbers ; the first = 1320, the second = 1427 ; what is the third ?
5855 r 132° - H27 = 5855 - (1320 + 1427.) 5855 1320 Proof: 3108
1320 + 1427 , \ 1320
* \ 1427
4535 2747 1427 5855
5855 3108 - 2747
3108
FOURTH STEP. Multiplication.
ORAL AND WRITTEN. 1. Multiplier One Figure.
a. Multiplicand One Figure.
3x9 units = 27 units (3x9 — 27). 3x9 tens = 27 tens (3 x 90 = 270). 3x9 hund. = 27 hund. (3 x 900 = 2700). etc.
b. Multiplicand Two Figures.
3 x 29 units = 87 units (3 x 29 = 3 x 20 = 60
3 x 9 = 27
87).
3 x 29 tens = 87 tens (3 x 290 = 870). 3 x 29 hund. = 87 hund. (3 x 2900 = 8700).
134 THE THIRD YEAR: 100-1000.
c. Multiplicand Three Figures.
3 x 529 units = 1587 units (3 x 500 = 1500 3 x 20 = 60 3 x 9 = 27
1587).
3 x 529 tens = 1587 tens (3 x 5290 = 15870). etc.
d. Multiplicand Four Figures.
3 x 5293 units = 15879 units (3 x 5000 = 15000
3 x 200 = 600 3 x 90 = 270 3 x 3=9
15879).
3 x 5293 tens = 15879 tens (3 x 52930 = 158790). 3 x 5293 hund. = 15879 hund. (3 x 529300 = 1587900).
2. Multiplier Two Figures.
a. THE PURE TENS. i. Multiplicand One Figure.
60 x 5 units = 300 units (60 x 5 = 300).
60 x 5 tens = 300 tens (60 x 50 = 3000).
60 x 5 hund. = 300 hund. (60 x 500 = 30000).
60 x 5 thous. = 300 thous. (60 x 5000 = 300000).
2. Multiplicand Two Figures.
60 x 56 units = 3360 units (60 x 50 = 3000
60 x 6 = 360
60 x 56 tens = 3360 tens (60 x 560 = 33600). etc.
SECOND HALF OF THE YEAR. 135
3. Multiplicand Three Figures.
60 x 562 units = 33720 units (60 x 500 = 30000
60 x 60 = 3600
60 X 2 = I2O
33720).
60 x 562 tens = 33720 tens (60 x 5620 = 337200).
b. MIXED TENS.
(1) 25 x 9 units = 225 units (20 x 9 = 180
5x9= 45
225). - 25 x 9 tens = 225 tens (25 x 90 = 2250).
(2) 25 x 96 units = 2400 units (20 x 96 = 1920
5 x 96 = 480
2400). 25 x 96 tens = 2400 tens (25 x 960 = 24000).
3. Multiplier Three Figures.
a. PURE HUNDREDS. i. Multiplicand One Figure.
300 x 9 units = 2700 units (300 x 9 = 2700). 300 x 9 tens = 2700 tens (300 x 90 — 27000). 300 x 9 hund. = 2700 hund. (300 x 900 = 270000).
2. Multiplicand Two Figures.
300 x 91 units = 27300 units (300 x 90 = 27000
300 x i = 300
27300 300 x 91 tens = 27300 tens (300 x 910 = 273000).
136 THE THIRD YEAR: 100-1000.
3. Multiplicand Three Figures.
300 x 914 units = 274200 units (300 x 900 = 270000
300 x 10 = 3000 300 x 4 = 1 200
274200). 300 x 914 tens = 274200 tens (300 x 9140 = 2742000).
b. MIXED HUNDREDS.
304 x 9 units = 2736 units (300 x 9 = 2700