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FACTORING.

PAGE.

Definitions......

120

Six Propositions,......

122

Examples for Practice......

125

GREATEST COMMON Divisor.

First Method, by Factoring,...

127

Second Method, by Division,..

128

LEAST COMMON MULTIPLE.

First Method, by Factoring,...

131

Second Method, by Division,.....

132

COMMON FRACTIONS.

Origin and Nature of Fractions,..

134

Definitions..........

138

General Principles,

139

Reduction of Fractions,....

142

To reduce a fraction to its lowest terms,

142

To reduce an improper fraction to a whole or mixed number ......... 143

To reduce a whole or mixed number to an improper fraction,......... 144

To reduce compound to simple fractions.........

146

To reduce compound to simple fractions by Cancellation............ 147

To reduce fractions to a common denominator,.....

148

To reduce fractions to the least common denominator,...

150

Addition of Fractions.......

152.

Subtraction of Fractions,....

154

Multiplication of Fractions,.

156

Division of Fractions........

162

Fractional Exercises in Compound Numbers..........

169

Reduction of Fractional Compound Numbers............

171

Addition and Subtraction of Fractional Compound Numbers......... 174

Promiscuous Examples,....

175

DECIMAL FRACTIONS.

Origin and nature of Decimals..........

177

Decimal Numeration and Notation..............

180

Addition of Decimals.......

183

Subtraction of Decimals..........

184

Multiplication of Decimals,..

185

Division of Decimals,.......

187

Reduction of Decimals.....

189

Promiscuous Examples,.....

193

RATIO.

The nature of Ratio,.......

195

Méthod of expressing Ratio,.........

196

PROPORTION.

The nature of Proportion............

197

Simple Proportion...........

199

INTELLECTUAL ARITHMETIC should be thoroughly studied by all, and especially by the young, before commencing PRACTICAL. For this purpose, attention is called to “Ray's Arithmetic, Second Book," which has been carefully prepared, and is now published with important improvements.

When admissible, pupils studying Arithmetic should be taught in classes; the presence of the class being a stimulus to both teacher and pupil. This arrangement also economizes time, since the same oral illustrations, necessary for the instruction of a single pupil, serve for a class.

The time occupied at each recitation ought not to be less than thirty minutes, nor more than one hour. The class should not be too large; and, if possible, the attainments of its members equal.

Every school should have a blackboard, on which pupils can solve the questions and explain the method of solution.

A prime object in recitations is to secure attention ; to do this, the exercises must be interesting, and all must be kept employed. Let as many be called out as can obtain positions at the blackboard, and let all solve the same question at once.

When the solutions are completed, let some one be called on to explain the process, giving the reason for each step of the operation. Exercises thus conducted animate the class; and by requiring the learner to explain every process, and assign a reason for every step, he learns to rely on his own reasoning powers.

In assisting pupils to overcome difficulties, it is preferable to do it indirectly, by making such suggestions, or asking such questions, as will enable the learner to accomplish the object.

Frequent Reviews will be found of great benefit.

The pupil should be rendered familiar with the answers to the questions in the REVIEW at the foot of the page. This review is intended to aid the teacher, but not to prevent his asking other questions, or presenting different illustrations.

A RITHMETIC.

1. A Unit, or one, is a single thing of any kind; as, one apple, one dollar, one pound.

2. Number is a term signifying one or more units; as, one, five, seven cents, nine men.

3. Numbers are expressed by ten characters, called Figures; as, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.

4. Arithmetic treats of numbers, and is the art of computing by them. The fundamental rules are five; Notation and Numeration, Addition, Subtraction, Multiplication, and Division.

two; three;

..

I. NOTATION AND NUMERATION. ART. 1. A single thing is a Unit, or one; written, 1 One unit and one more, are

2 Two units and one more, are

3 Three units and one more, are four; 4 Four units and one more, are five;

5 Five units and one more, are six ;

6 Six

units and one more, are seven ; 7 Seven units and one more, are eight;

8 Eight units and one more, are

9 ART. 2. These nine characters are called digits, or significant figures, because they denote something.

The character, 0, called cipher, naught, or zero, is employed to denote nothing: thus, to show that there are no cents in a purse, write, the number of cents is 0.

REMARK.—The cipher is sometimes termed an auxiliary digit, because it helps the other digits in expressing numbers.

nine;

REVIEW.--ART. 1. What is a single thing? What are one unit and one more of the same kind? Two units and one more, &c. ? 2. What are these nine characters Called? Why? What does naught, or zero denote? Rem. What is the cipher termed? Why?

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