James B Thomson.

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Qri)oms0n*s lixQi Ccssons.

MENTAL

ARITHMETIC,

OB

FIRST LESSONS IN NUMBERS

FOR CHILDREN.
By JAMES B. THOMSON, LL.D.,^

AUTOOU Ol» PaAOTlOAI. ARITUMBTIO, BDITOB OP DAY'h SOHOOl AMHBOA
I.BOBNDRli'8 QBOMSTRV, BTO.

ONI HUNDRED AND FIFTIETD EDITION, REVISED AND BNLABOBD.



NEW YORK:
CLARK & MAYNARD, 6 BARCLAY ST.

CHICAGO : a C. GRIGGS & CO.

* * * *



1



BiiUjkxI oooordlDg to Act of OongrcBe, In Uie ytor iy3€,

By JAMES B. THOMSON,

in the Clerk'u OflJoo of the Northern Dlstrlot of Now York.



PREFACE



At what precise age a child should begin to gc to schoal,

or commence the different studies, it is not our province to

I decide. Whatever may be the diversity of opinion on

I this point, all practical teachers seem to agree, that Men-

\ id A^thmetic is among the first exercises which should

j be presented to the youthful mind. The correctness of

I this sentiment is corroborated by the ease with which

I children understand simple combinations of numbers, their

! fondness for the!>e exercises, and the obvious advantages

I which may be derived from them. But in order to be-

I come interesting or profitable, it is manifest, this branch,

j as well as others, must be taught in such a manner that

the pupil shall understand it. The examples, therefore,

j must, at first, be simple, containing small numbers, and

have reference to sensible objects with which the learner

is accjuainted ; the transition from easy to more difficult

(questions must be gradual; and the reason for every

step in the solution distinctly seen.

It is believed that much dislike for the study of Arith-
metic, and much unnecessary discourage7nent, have been
occasioned by the abruptness of the transitions from easy
to difficult questions. It is too often forgotten that the
powers of the child's mind, Lke those of his body, are
feeble ; that while familiar mental exercises which he can
comprehend, afford him the highest delight, he turns from
intricate questions, which he does not and cannot under-
itand, with indifference and disgust.

It is the design of this little work to furnish a ser^^ of
mental exercises in numbers, adapted to the wants and
cnparities^of children. It commences with practical ex-
amples, which relate to familiar objects and require the



PREFACE.



simplest combinations. The pupil is then introduced to
others involving the same principle but somewhat harder,
special care being taken to make the transition very grad-
ual, so that instead of being disheartened at the rugged-
ness of the way, he shall be stimulated to take the next
step by the hope of victory.

From the fact that children comprehend and remember
words more easily than figures, and reason upon them
with so much greater facility, the numbers and Tables
in the first part of the book are expressed in words.

After the pupil has become practically acquainted with
the principles of a rule, and is able to solve questions un-
der it with facility, the operation is then defined, and the
more prominent terms are briefly explained. This, it is
believed, teachers will be glad to see. There is no rea-
son why a child should not be informed, that a certain
operation upon numbers is called Addition ; another Sub-
traction ; &c., as well as to be told that a certain opera-
tion of his vocal organs in connection with those of his
mind, is called reading ; another singing ; &c.

With this brief explanation of the object and plan of
the work, the author commends it to the friends of edu-
cation, by whom his former efforts to subserve this noble
cause, have been so favorably received.

Although designed particularly as an introduction to
the *' Practical Arithmetic," it may be used as a prepar-
atory work to any of the larger systems of Arithmetic
now aefore the public.



MENTAL ARITHMETIC.



SECTION I



INTRODUCTION,



The first step in acquiring a knowledge of numbers is
to learn to count. Most children are able to repeat the
na7}ies of numbers, one, two, three, (Sec, before they begin
to go to school ; but there are fewer who fully comprehend
the meaning of these terms; who perceive, for example,
that eleven expresses more things than seven, or fewer
than thirteen. While such is the case, no substantial
progress can be made in Arithmetic.

Great pains should therefore be taken to show young
pupils, in the outset, how many things thf» name of each
number denotes, and to establish in their minds a correct
idea of more and less. Counters, made of round pieces
of wood or leather, also beans, kernels of com, &c., may
be used for this purpose ; but the most convenient appa-
ratus is the Numerical Frame.* The bails upon the
wires are more easily arranged and are seen at once by
every member of the class, while the liability of falling
upon the floor and getting lost, is entirely avoided.

LESSON I.

Having slipped all the balls to the left side, the
teacher holds up the Numerical Frame before the
class and requests their particular attention.

• Every instniclor who is called upon to tearh the nidimfntfl of
Arithmetic, should be furniBhed wiih this useful instrument, it



INTRODUCTION.



With his pointer he now moves the first ball on
the bottom wire to the right side and says, this is
oiie, the class repeating it with him. Moving across
another on the same wire, he says, this makes two,
the class repeating it with him as before ; moving
an )ther, this makes three ; another still, this makes
' four ; and yet another, this makes Jive ; and so on
up to ten.

Tliis process should be repeated and varied ac-
cording to circumstances, u.itil the class can count
ten in concert and individually with readiness. If
this cannot be accomplished in one exercise, another
should be devoted to it.

Note.—U the (hil<lr»»n are young, or have never learned the
names of any of the numbers, when they get to three or Jive h will
l)e ^xp«?nient to stop and review as far as they have been. Care
Hhould be taken not to present too many new ideas to the young
mind at once, lest it become bewildered ; nor should the exercise
be continued so long as to weary it, and thus create a lasting dis-
relish for the study.

LESSON II.

Slippine a»: the balls to the left side of the frame
as before, move the first on the lower wire to ihe
right side, and ask the class to count it.

Now move out two on the second wire, taking
one at a time, and let the class count as you niuve
them, one, tioo. Then pass across three on the third
wire, taking one at a time, while the class coimt
one, two, three.



costs but a trifie, and, with proper care, will last an age. Us
raore important uses will be pointed out in their proper place.

The lessons in this Section are designed for pupils who have
not learned to count, or may not comprehend how many things
are dr«noted by the names of numbers. I'hose who thoroughly
undeti-tand tliese points, cen begin with Addition.



INTRODUCTION. 7



Proceed in this manner to the iejith or Icist, in-
creasing one ball on each successive wire.

Again, beginning at the bottom, let the class count
he balls moved out on each Avire, and observe that
two is one more than one; that three is one more
than two ; that four is one more than three, &c.

Next, let the class retrace this process ; that isj |
beginning at the top, let them count the balls moved
out on each wire till they arrive at the bottom one.

Let them also begin at ten and count backwards
to one, several times in quick succession. Thus,
teji, nine, eight, seven, six, &c.

Finally, move out any number of balls under ten
promiscuously, and call upon some one to count
them ; then move out a different number, and let
another count them ; and thus continue to vary the
exercise, till every one in the class can count ten
understandingly.

LESSON III.

Note. — As soon as the class clearly comprehend how many thinffs
are expressed by the name of each of the numbers up to ten, they
are then prepared to learn to count from ten to twenty, &,c.

Flaving counted out ten balls on the lower wire,
move across one on the second wire saying, this
makes eleven, the class repeating it with you. Pass-
ing across another, this makes twelve ; another, this
makes thirteen ; another still, this makes fourteen ;
and so on up to twenty.

Repeat this process, at the same time explaining
to the class that the term thirteen, is composed of
the words three and ten, and means the same as ihvf'e
counted on to ten, or three and ten put together.
Also, that the term fourteen is con.posed of the



INTRODUCTION.



words four and ten, and means the same as four and
ten put together; that fifteen means five and ten;
sixteen^ six and ten ; seventeen, seven and ten; eiyh-
teen, eight and ten; nineteen, nine and ten; and
twenty means two tens.

Next, having counted off twenty balls ; that is,
ten apiece on each of the two lower wiies, pass
across one on the next wire, saying this makes twen-
ty-one, the class repeating it with you as before.
Passing across another, this makes tioenty-two ; an-
other, this makes tioenty-three ; another still, this
makes twenty-four ; and so on to thirty.

Here again tlie teacher should be careful to ex-
plain that the term twenty-one, is composed of the
words iiventy and one, and means the same as one
counted on to twenty, or twenty and one put together.
Also, that the term twenty-two, is composed of the
words twenty and two, and means the same as twenty
and two put together; that tzoenty- three mea.ns twen-
ty and three ; «fec.

In a similar manner children may be easily taught
to count from thirty to forty, fifty, sixty, &c., to a
hundred, and to comprehend Iiow 7nany tilings are
expressed by the name of each number.

Nott to tht Teacher. — It is advisable to exercise the pupil
in writing the Jigures upon his slate, or the black-board, in
the early part of his course. This will afford him pleasing
occupation, and at the same time, will be of great assistance
in enabling him to understand and apply them when he
shall have occasion for their use



SECTIOJN II.
ADDITION.

LESSON I.

1. How many little hoys are there in this class?

2. How many little girh are there ?

3. How many little girls and boys are there,
counted together ?

4. Will each of you show n e your right hand ?
Now if you put your left hand by the side of it, how
many hands will each one show me ? Why ?

Ans. Because one and one more make tv:o.

5. If you show me two fingers on your left hand
and one on your right, how many fingers will you
show me ? Why ?

Ans. Because two and mie more make three.

6. If you open three fingers on your left hand and
one on your right, how many fingers will you have
open ? Why ?

7. How many fingers have you on your right
hand ? How many thumbs ? How many fingers
and thumbs counted together, are there on your right
hand?

8. If I move out five balls on the lower wire of
Tfiy Numerical Frame and one on the next, how
many will they all make ? Why ?

Note. — When a general question ia asked whicl is not desi^Hd
to be answered in concert, it is an excellent way to have afi who j
think they know, raise a hand ; the i call upon some one promifl- j
cuously to answer it. This methcxJ prevents much confusion and |
secures the attention, wliile it effectually avoids the temp'ation ,
to learn the answers to certain questions which will fall to »>ach,
if the teacher always begins at the same end of the cla^^ and
proceeds through it iii regular rotation.



10



ADDITION,



9. I^ I move out six on the lower wire and one
on the next, how many will they make ? Why ?

10. If I move out seven on the lower and one on
the next, how many will they make ? Why ?

11. If I move out eight on the lower and one on
the next, how many will they make ?

12. If I move out wne on the lower and one on
the next, how many wi 1 they make ?

Note. — A« loon m a child learns a principle in Arithmetic, it i«
iniportajit for him to see ita application, ana begin to practice it.
For this reason, it js recommended to let the class iludy the next
eight lessons in connection with the corresponding paru of the Ta-
ble be.ow. Thus, as soon as they learn to add twos, let them take
lesson second yvhich contains exercises in adding two ; as soon as
they learn to add threes, let them take lesson third, &c. In this
way the Table is kept together, which is essential for reference
ana review, ar.d at the same time we secure all the advantages
of studying it jx connection with examples which put it into prac-
tice, as fast as it is leamexi.

ADDITION TABLE.



One


•nd


oae wr.


twa


One


•nd six V seven.


One


u


two "


threek


One


" seven « eight.


One


u


three "


four.


One


" eight " nine.


One


U


four "


five.


One


" nine " ten.


One


u


five "


six.


One


« ten " eleven.


Two


and


OJtB •«


three.


Two


u»d six Ml eight.


Two


u


two "


four.


Two


" seven " nine.


Two


(1


three "


five.


Two


" eight " ten.


Two


H


four "


six.


Two


•' nine " eleven.


XW)


n


five '*


seven.


Two


" ten " twelve.


Ihree


•Ad


on«J «•


four.


Three


Hid six u* nine.


Three


u


two "


five.


Three


" seven " ten.


Three


u


three "


six.


Thr«e


" eight " eleven


Three


II


four "


seven.


Three


" nine " tweha


Three


II


five . "


eight.


Three


" ten " thirteen.



1

1




ADDITION.


11


Four


and one


art five.


Four and six


are ten.


1 Four


" two


» «x.


Four " seven


" eleven. (


1 Four


' three


" seven.


Four " .eight


" twelve.


! Four


' four


" eight.


Four " nine


«♦ thirf.'en. |


i Four


« five


" nme.


Four " ten


** fourteen.


; Five


and one


are six.


Five and six


are eleven.


Five


" two


" seven.


Five " seven


" twelve.


Five


" three


" eight.


Five " eight


" thrrieen.


Five


« four


" nine.


Five " nine


" fourteen


Five


" five


" ten.


Five " ten


" fifteen.


Six


ana one


ar« seven.


Six and six


are tWclve.


Six


» two


" eight.


Six '" seven


" thirteen.


1 Six


" three


" nine.


Six " eight


" fourU^en.


Six


" four


" ten.


Six '< nine


« fifteen.


Six


" five


*' eleven.


Six " ten


" sixteen.


Seven


aud one


are eight.


Seven and six


are thirteen.


Seven


'» two


" nine.


Seven " seven


" fourteen.


Seven


" three


" ten.


Seven " eight


" fifteen.


Seven


" four


" eleven.


Seven " nine


" sixteen.


Seven


" five


" twelve.


Seven " ten


" seventeen.


Eight


and one


are nine.


Eight and six


are fourteen. I


Eight


" twn


" ten.


Eight " seven


" fifteen. |


Eight


" thific


" eleven.


Eight " eight


" sixteen.


Eight


« fom


'* twelve.


Eight " nine


" seventeen.


Eight


" five


** mirteen.


Eight " ten


*• eighteen, j


! Nine


and one


are ten.


Nine anu six


are fifteen. -
" sixteen.


Nine


" two


" eleven.


Nine " seven


Nine


" three


" twelve.


Nine " eight


" seventeen.


Nine


« four


" thirteen.


Nine " nine


'* eighteen, j


j Nine

1
1


« five


" fourteen.


Nine " ten


" nineteen.



12 ADD7TI0N.



LESSON II.

1. If a pear costs two cents, and an apple costs one
cent, ho^ many cents will they both cost ? Why ?

2. Charles had ttao books, and his father gave him
two more: how many books did he then have? —
Why?

3. Robert brought in three sticks of wood, and
George two sticks : how many sticks did both boys
bring in ? Why ?

4. Let each one in the class open all his. fingera
on his left hand, and two on his right : how many
fingers has each one open ? How many are four
and two ?

5. Sarah had Jive dresses, and on her birthday
her aunt gave her two more : how many dresses had
she then ? Why ?

6. In this class there are two girls, and six boys :
how many scholars does the class contain ?

7. John has seven brothers, and two sisters : how
many brothers and sisters has he together ?

d. A little boy gave a blind man eight cents, and
his sister gave him two cents more : how many cents
did they both give him ?

9. William has two marbles in his hand, and nint
in his pocket : how many marbles has he in all ?

10. Mary recited two perfect lessons last week and

^1 Nole. — llje process of addins and the structure of the Table
raay be easily illustrated by the Numerical Frame. Thus, to show
that five and four make nine, mi-ve out five balls on one wire and
four on another, and let the pupil count them together. But as
I soon as the pupil understands the principle of the oiHiration, he
should tlien learn to i»crforra it without the assistance of visible
objects. Nothing is more deleterious to mental growth and dis-
cipline, than the habit of solving questions by counters, by the
fingers, marks upon a slate, &c. (S^n.



ADDITION. 13



tfifi this wee.i : how many perfect lessons has sK-e re-
cited in two weeks ?

LESSON III.

1 If you pay three cents for a pint of nuts, and I
one cent for an apple, how many cents do you pay
for boili ? Why ?

2. Henry has tkree young doves, and two old j
ones : how many doves has he ? Why ?

8. A begjxar met some generous little boys, one
of whom gave him four cents, and another gave him
three cents: how many cents did they both give
him ? Why ?

4. Susan bought three yards of blue ribbon, and
three yards of white : how many yards of ribbon did

I she buy in all ? Why ?

5. Harriet gave her teacher Jive pinks, and three
roses : how many flowers did she give her teacher ?

6. Matthew sold a quart of chestnuts for 6ix cents,
and a pint of beechnuts for three cents: how much
did his nuts come to ? i

7. flenry picked three ripe peaches from one lree> I
and s€i'>en from another : how many peaches did lie '
pick from both trees ? I

8. Frank has nme walnuts in his pocKet, and he j
lost three coming to school : how many had he when
he started from home ?

9. How many are eight and three ? I

10. Jane read ten pages of hislory in the morning, '
and three in the afternoon : how many pages did slm i
read during the day ? I

Ncte. — It is ad\l8able lo have the class review iIjc prerrdinjr i
part of the Table, and thus continue to do at every re-itati(jn, !
till they get through it. {



14 ADDITION.



LESSON IV.

1. Joseph received four peaches from his couiiin.
and one from his sister : how many peaches had he ?

2. A boy paid four cents for an orange and two
cents for a pear : how much did he pay for both ?

3. Emily bought a yard of silk for five shillings,
and a pair of gloves for four shillings : how many
shillings did she pay for both ?

4. Dick's father gave him six marbles, and he
gained four more : how many had he then ?

5. Henry gave four cents for a lemon, and four
cents for an orange : how many cents ^lid he give
for both ?

6. A farmei g^yefour dollars for a hog, and eight
dollars for a cow : how much did he pay for both ?

7. How many are four and three ?

8. A market boy soldybwr shillings worth of milk,
and vegetables to the amount of ^even shillings: how
much money did he have to carry home ?

9. James g&yefour shillings for a knife, and nine
shillings for a pair of skates: what did he pay for
both?

10. Henrietta bought a slate for ten cents, and a
sponge for four cents : how much did she pay for
both?

LESSON V.

1. A man bought a plough for fve dollars; and a
•hovel for two dollars : how many dollars did he pay \
for both ? j

2. A farmer raised three bushels of plums n one |
tree, and j?re bushels on another : how many bushels
did he raise on both trees ?



ADDITION. 15



I 3. A man sold a bushel of plums fear five dollars,
and a bushel of peaches (or four dollais: how much
did both come lo ?

4. There are six sweet apples, said five sour ones ,
I in the fruit dish: how many apples does the dish

contain ? j

5. A man paid^t'e shillings for a turkey, &nd five
shillings for a peck of peas: how much did he pay
for both ?

6. A man caught eight red squirrels, erndfive grey
ones : how many did he catch in all ?

7. A cabinet-maker asks five dollars for a rock-
ing chair, and seven dollars for a table : what is the
price of both ?

8. If I pay five shillings postage on letters, and
nine shillings on pamphlets, what is the amount of
my postage ?

9. Harriet bought a comb for ten cents, and a pa-
per of needles ioT five cents : how much did she pay
for both ?

LESSON VI.

1. Charlotte picked six white roses, and two red
ones : how many roses did she pick ?

2. Harriet gave three cents for a sponge, and six
cents for a slate : how many cents did she pay for
both?

3. IIow many are six and six ?

4. How many are six ^nd five ? Y

5. How many are six and ybwr?

6. If you read six pages in the morning, and seven
in the evening, how many pages will vou read in a
day 1



1 8 ADDITION.



7. John hoed ei^ht rows of cca*n in one day, and
s,x in another day: aow many rows did he hoe in
bolh days ?

8. His employer gave him nine pence for liis first
day's work, and six pence for the second day 's work :
how many pence did John receive ?

9. A lady bought a muff for J^n dollars, and a hat
for six dollars : what was the amount of her bill ?

LESSON VII.

1. Sarah had seven pins on her cushion, and she
afterwards found two more : how many pins had she
then?

2. How many are seven and three ?

3. If seven flower pots stand in one window, and
foiir in another, how many are there in both win-
dows ?

4. How many are seven and six?

5. How many are seven snid Jive?

6. How many are seveii and seven?

7. Oliver paid seven cents for a writing-book, and
eig/it cents for a slate: how much did he pay for
both?

8. How many are seven and ten ?

9. How many are seven and nine ? ft

LESS jN VIII.

1. If a barrel of flour cost eight doLars, and a keg
cf lard 'wo dollars, how much will Wh cost?

*2. A merchant tailor asks eight liars for a pair
of pants, and three dollars for a vest: what is the
price of both ?



ADDITION. 17



3. If you pay eight cents a mile for the use of a
horse, and Jive cents for a buggy, how much will you j
pay a mile for both ? i

4. How many are 6(i0'/i/ and yb?/r? !
f>. Julius gave one of his companions six ap* '*3, i

and had eight left : how many had he at first ? |

6. Henry had eight marbles, and his brother gave
him eight more : how many did he then have ?

7. A shopkeeper sold eight yards of satin to one
lady, and seven to another: how many yards did he
sell to both ?

8. Catharine's book has eight pictures in it, and
iMary's has 7une : how many pictures do both books
contain ?

9. If you pay eight cents for a pound of sugar,
and ten cents for a pound of figs, what must you
pay for both ?



LESSON IX.

1. There are nine shade trees standing in front of
the school house, and two have been cut down : how
many trees were set out ?

2. Alexander's kite line is now nine yards long,
i by accident he broke off three yards: how long was

his line at first ?

3. flow many are ni?ie and Jive?

4. IIow many are iwie smd four?

5. If a ton of coal costs nine dollars, and a co.-d
of wood costs s^ix dollars, how much will they both

jCOSt?

I 6. A hunter gave nine dollars for a gun, and
I seven dol ars for a dog : how much did he pay for

\ both ?

I



18



ADDITION,



7. If you have nine chestnuts in cne pile and nine
in another, how many will they make i( you put
thtm all into one pile ?

8. In Margaret's flower garden there are nine
lady's slippers, *nd eight bachelor's buttons: how
many flowers has she ?

9. Sophia wrote nine lines at the last exercise,
and Henrietta wrote ten : how many lines did both
write ?



LESSON X*

1 . Ilow many are one and ten ? Three and ten ?
Six and ten 1 Four and ten ? Seven and ten. Five
and ten ? Eight and ten ? Nine and ten ?

2. How many are eleven and ten ? Twenty-one
and ten ? Forty-one anjd ten ? Thirty-one and ten ?
Fifty-one and ten ? Seventy-one and ten ? Sixty -one


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Online LibraryJames B ThomsonMental arith → online text (page 1 of 7)