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SUGGESTIONS TO TEACHERS.

The progressive drills, the clear, rigid analyses and logical methods insisted upon in the Elementary Arithmetic should be continued through the Complete Arithmetic.

1. Never pass a lesson until it is fully understood. Encourage pupils to ask questions, and thoroughly test their knowledge of each principle.

2. Whenever practicable, require pupils to illustrate each definition and principle by original examples. You will observe that such requirements are an original feature of the Standard Arithmetics. In some cases, only partial answers are given-enough to afford the pupil all the guidance necessary.

3. Too great weight cannot be attached to the oral exercises. In every instance state the problem clearly and but once.

Then call upon some pupil to solve it, who should immediately repeat the example, and reason it out to a correct result. If he is unable to state it correctly, he should be charged with a failure. Attention will thus be fostered, memory strengthened, a rigidly logical method of solution acquired, and pupils will soon do creditable and even brilliant work.

4. In solving problems on the blackboard, the same method should be insisted upon, until the analysis is thoroughly understood. He who formulated the first rule for the addition of numbers must have done so from investigation, and such should be the course of every student of arithmetic.

5. Frequent reviews are more profitable to the pupil than additional drill in new subjects, since by the former that which is imperfectly understood becomes clear and is permanently fixed in the mind.

6. In the preparation of original problems, require pupils to make their examples consistent. Thus a dollar should not be divided into ninths or elevenths, nor a pound into thirds, nor a yard into fifths, etc.

THE COMPLETE ARITHMETIC.

DEFINITIONS.

ART. 1.—A Unit is one, or a single thing; as one, one cent, one bushel.

ART. 2.-A Number is a unit, or a collection of units; as one, five, seven boys, nine girls.

ART. 3.–Arithmetic treats of numbers and their uses.

ART. 4.-A Concrete Number is a number applied to one or more objects; as three pounds, five eggs.

ART. 5.—An Abstract Number is a number not applied to any object; as three, four, five, nine.

ART. 6.-Like or Similar Numbers are those whose units are of the same kind; as six girls, three girls; eleven miles, four miles.

ART. 7.-Unlike or Dissimilar. Numbers are those whose units are of different kinds; as six girls, three miles; five pounds, three boys.

ART. 8.—An Integer or Whole Number is composed of whole units; as eight, ten, fourteen.

ART. 9.—A Problem is a question offered for solution.

ART. 10.—A Solution in arithmetic is the process of obtaining the required result.

ART. 11.—The Proof of a solution is the process by which its correctness is tested,

ART. 12.-A Rule is a statement of the process by which a problem is solved.

ART. 13.-A Principle is a general truth.

Notation and Numeration.

ART. 14.-Notation is the art of writing numbers.

ART. 15.-Numeration is the art of reading numbers. ART. 16.—Numbers may be expressed in three ways : 1. By words; as four, five, eight, etc. 2. By figures, called the Arabic Method. 3. By letters, called the Roman Method.

ART. 17.—The Arabic Notation requires ten characters, called figures, to express numbers.

They are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

ART. 18.-The first, called naught or cipher, when standing alone, expresses no value.

ART. 19.—The other nine characters are called sig. nificant figures, because they always express value.

ART. 20.—By combining these figures, any number can be expressed.

ART. 21.—The law governing the combination of figures is-In any whole number the first figure on the right expresses units; the second, tens ; the third, hundreds ; the fourth, thousands, etc.

EXERCISES.

ART. 22.–1. How many tens and units in 37 ? 73 ? 65 ? 84 ? 91 ? 83 ? 177 ? 88 ? 90 ?

2. How many hundreds, tens and units in 329 ? 763 ? 907 ? 378 ? 593 ? 105 ? 810 ? 503 ?

3. How many thousands, hundreds, tens and units in 1886 ? 9230 ? 4376 ? 8295 ? 7034 ? 2010 ? 6352 ? 1101 ? 7839 ? 6354 ? 1878 ?

ART. 23.-For convenience in writing and reading numbers, they are separated into groups of three figures each, called periods, as shown below :

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} Units' Period.

spo puno •с

Tens. 00 Os Units.

Hundreds of Millions.

Tens of Millions.
C7 Millions.

74,

o Hundreds of Billions.

Tens of Billions.
CO Billions.

429,

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420, 341, 214, 412, 513, 524, 417,

38, 315, 219, 305, 400, 815, 921, 325,

327 232 549 321 418 325 328 415 318 618 417 898 319

152, 225, 412, 548, 462, 532, 654, 543,

681, 186, 514, 329, 865,

321,

937,

625, 415,

517, 416, 928,

ART. 24.-Write the following in figures :
1. 5 tens 7 units; 4 tens 3 units.
2. 9 tens 0 units; 6 tens 4 units.

3. 8 tens y units; 1 ten 1 unit. 4. 6 tens 6 units ; ry tens 9 units. 5. 2 hundred 5 tens 8 units. 6. 4 hundred 1 ten 0 units. 7. One thousand one hundred seventy-four. 8. One thousand eight hundred eighty-five. 9. One thousand seven hundred jeventy-six. 10. Five thousand three hundred nineteen. 11. Twelve thousand three hundred twenty-one. 12. One hundred thousand one hundred one. 13. Seven thousand seven. 14. Seventy thousand seven. 15. Seven hundred thousand seven. 16. One million. 17. One hundred million one hundred one. 18. Five quadrillion five billion five million five. 19. One million ten. 20. Six hundred twenty-eight thousand sixty.

21. Write on the blackboard and read, a number consisting of three periods ; of four periods.

22. Write on the blackboard and read, a number consisting of five periods.

ART. 25.-Rule for Notation.-Begin at the left, write the figures of each period in their proper order, filling all vacant periods and places with ciphers.

ART. 26.—Rule for Numeration.—Begin at the right and point off the number into periods of three figures each.

Begin at the left and read each successive period containing one or more significant figures as though it stood alone, giving the name to each period except that of units.

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