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ARITHMETIC.

PART FIRST.

Arithmetic is the science of numbers.
A unit is a whole thing of any kind.
A fraction is a part of a thing.

Thus a dollar is a unit; a man is a unit; a picture is a unit; a bushel of apples is a unit, &c.

A half of an apple is a fraction; a quarter of a dollar is a fraction; a third of a loaf of bread is a fraction, &c. Let the pupil mention other units and fractions.

If an apple is cut into two equal parts, each part is called one half of the apple. If it is cut into three equal parts, each part is called one third. If it is divided into four equal parts, each part is called one fourth. If it is divided into five equal parts, each part is called one fifth, fc.

If a unit is divided into six equal parts, what is one of those parts called? If a unit is divided into seven equal parts, what is one of those parts called? If a unit is divi. ded into eight equal parts, what is one of those parts called ? Into nine? Into twenty? Into an hundred ? Into fifteen ? Into twenty-two? How many halves make one unit? How

many

thirds make one unit? How many fourths ? How many fifths ? How many sixths ? How many sevenths ? How many eighths ? How many ninths ? How many tenths? How many twentieths? How many hundredths ?

For illustrating the exercises which immediately follow, the teacher should be provided with a proper number of the several coins of the U. S. viz : eagles, dollars, dimes, cents and mills. As mills have never been coined, round bits of stiff paper may be employed to represent them. The pupil should first see the several coins and learn the value of them.

What is arithmetic? What is a unit ? What is a fraction? Mention & unit and a fraction.

one mill?

Ten mills are one cent.
Ten cents are one dime.
Ten dimes are one dollar.

Ten dollars are one eagle.
How many mills make a cent? What part of a cent is

What part of a cent is two mills? What part of a cent is three mills ? What part of a cent is four mills ? What part of a cent is five mills ? Six mills ? Seven mills ? Eight mills ? Nine mills ?

How many cents make a dime ? One cent is what part of a dime ?

Two cents is what part of a dime ? Three cents is what part of a dime? Four cents is what part of a dime? Five cents ? Six cents ? Seven cents ? Eight cents ? Nine cents ?

How many dimes make a dollar ? What part of a dol. lar is one dime? What part of a dollar is two dimes ? What part of a dollar is three dimes ? Four dimes? Five dimes ? Six dimes ? Seven dimes ? Eight dimes ? Nine dimes ? How

many dollars make an eagle? One dollar is what part of an eagle? Two dollars is what part of an eagle ? Three dollars? Four dollars ? What part of an eagle is five dollars ? Six dollars ? Seven dollars ? Eight dol. lars ? Nine dollars ?

The same thing may be considered sometimes as unit and sometimes as a fraction—thus, one dollar is a unit or whole thing of the kind or order called dollars, and one dollar is also the tenth part of an eagle, or the fraction of an eagle. One cent is a unit or whole thing, of the or. der of cents, and one cent is also the tenth part of a dime, or the fraction of a dime. One mill is a unit of the order of mills, and one mill is the tenth part of a cent, or the fraction of a cent. One day is a unit or whole thing of the order of days, and one day is also the seventh part of a week, or the fraction of a week. One week is a unit or whole thing of the order of weeks, and one week is the fourth part of a month, or the fraction of a month. One month is a unit or whole thing of the order of months, and one month is also the twelfth part of a year, or the fraction

a

of a year.

Of what order is one dollar a unit? Of what order is it

a fraction? Of what order is one cent a unit ? Of what order is it a fraction? Of what order is one week a unit ? Of what order is it a fraction? Of what order is one foot a unit? Of what order is it a fraction ? One day is a unit, of what order, and a fraction of what order, &c. ?

What is half of four cents ? What is a third of six cents ?

Let the pupil take six cents, and divide them into three equal portions, and then tell what is one of these parts ?

What is a fourth of eight cents ? Let the pupil divide eight cents into four equal portions, and tell how many in each portion. There are twice six cents in twelve cents, what

part

of twelve is six cents ?

There are three times four cents in twelve cents, what part of twelve is four cents ?

There are three times five cents in fifteen cents, what part of fifteen is five cents ?

There are three times three in nine, what part of nine is three?

The are two times three in six, what part of six is three ?

There are four times two in eight, what part of eight is two ?

There are four times three in twelve, what part of twelve is three ?

There are five times six in thirty, what part of thirty is six ?

There are three times seven in twenty-one, what part of twenty-one is seven ?

There are four times six in twenty-four, what part of twenty-four is six ?

There are six times seven in forty-two, what part of forty-two is seven?

What part of twelve is three ? Is four ?
What part of nine is three ?
What part of fifteen is three? Is five ?
What part of sixteen is four ?
What part of eighteen is three ? Is six ?
What part of twenty-one is three ? Is seven ?
What part of twenty-four is six? Is four ?

What part of twenty-eight is seven ? Is four?
What part of thirty-two is eight? "Is four ?
What part of thirty-six is nine ? Is four ?

If an apple is cut into two equal parts, what is each part called? If it is cut into three equal parts, what is each part called ?

The more parts a thing is divided into, the smaller these parts must be.

If one thing is divided into twice as many parts as another thing, each part is twice as small.

If one apple is cut into twice as many pieces as another, how much smaller is each piece? How much larger is a half than a fourth? Ans. There are twice as many fourths as halves in a thing, therefore a half is twice as large as a fourth.

If one apple is cut into four pieces, and another into eight pieces, how much larger are the fourths than the eighths ? Ans. As there are twice as many pieces when there are eighths, as when there are fourths, an eighth is twice as small as a fourth.

If one apple is cut into twelve parts and another into six parts, which has the most parts and which has the largest parts? How much larger is a sixth than a twelfth ? Ans. Twelve is twice as many as six, therefore a sixth is twice as large as a twelfth.

Which is the largest, a fifth or a tenth? How much larger is a fifth than a tenth ?

Which is the largest a seventh or a fourteenth?
How much smaller is a fourteenth than a seventh?
Which is the largest a third or a fifth ?
Which is the smallest a half or a fourth ?

Which is the smallest a third or a half? Ans. The more pieces there are, the smaller they must be, therefore a third must be smaller than a half.

If one apple was cut into four pieces, and another into six pieces, which would be the largest a fourth or a sixth ?

Which is the largest a sixth or a ninth?
Which is the largest a fifth or a fourth ?
Which is the smallest a twelfth or a tenth?
Which is the smallest a seventh or a ninth ?
Which is the smallest an eighth or a seventh ?
Which is the smallest a fifteenth or a fifth ?

Which is the largest an eighth or a sixteenth ?
Which is the largest a fifth or a half?

If an apple is divided into four pieces, what is each piece? If it is divided into twice as many and twice as small pieces, how many are there, and what are they called ?

If an apple is divided into thirds, what would you change them to, to make them twice as many and twice as small ?

Make two fourths twice as small and twice as many pieces, and what is the answer?

What part of a thing is twice as small as a half? As a third ? As a fourth ? As a fifth ? As a sixth ? seventh? As an eighth ? As a ninth? As a tenth ? As an eleventh ? As a twelfth ?

What part of a thing is twice as large as a fourth ? As a sixth ? As an eighth ? As a tenth? As a twelfth ? As a fourteenth? As a sixteenth ? As an eighteenth ? As a twentieth ?

As a

ADDITION.

Two cents, and four cents, and six cents, and nine cents, are how many ? Sixteen cents, and twelve cents, are how many ?

Five dollars, and four dollars, and nine dollars, are how many ?

Four halves of an apple, and six halves, and nine halves, are how many halves ?

Five sixths of an apple, and four sixths, and nine sixths, are how many sixths ?

Three fifths of an orange, and four fifths, and pine fifths, and twelve fifths, are how many fifths ?

Addition is uniting several numbers in one.

When whole numbers are added, it is Simple Addition. When fractions are added, it is Fractional Addition.

Six dimes, five dimes, and four dimes, are how many?

What is addition? What is simple addition? What is fractional addition ?

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