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2.2.2.61. Observe, however, that nearly all these steps will become superfluous by practice, and should be dispensed with as soon as possible.

Specimen of the FORM of the Table of Prime Factors.

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As soon as the members of a class have each formed a table from 1 to at least 100, it may be well to direct their attention to the fact that every prime number greater than 3 is either 1 more or 1 less than 6, or one of its multiples. This singular property of numbers may be thus accounted for. Neither 6 nor either of its multiples can be prime, since they must necessarily be even; now, if the numbers between any two of its adjacent multiples be examined, take, for instance, 13, 14, 15, 16, 17, it will be obvious that the middle number cannot be prime, since every multiple of 6 must also be a multiple of 3, and the middle number contains exactly one 3 more; and as the even numbers on each side of the middle one cannot of course be prime, it follows that no number greater than 3 can be prime, unless it be 1 greater or 1 less than 6, or than one of its multiples. Observe, however, that, although primes can be found in no other situation, it does not necessarily follow that the converse is true, namely, that every number in that situation must be prime, as an inspection of the table will show.

15. Write in a column on the slate the prime numbers between 5 and 97 inclusive, and repeat the exercise daily till they become familiar.

16. What are the prime factors of 14 and 35 ? What is their greatest common divisor ; that is, what prime factors are common to both numbers ?

What are the prime factors of 16 and 12? What is their greatest common divisor? Mention the prime factors of 16 and 18, and say which are common. What is the greatest common divisor of 28 and 42 ? Of 16 and 36 ? 18 and 42 ? 19 and 57 ? 72 and 30? 20 and 45 ? 51 and 17 ? 75 and 125 ? 39 and 26 ? 36 and 54 ? 14 and 63 ? 15 and 125 ? 27 and 84 ? 30 and 81 ?

17. What is the greatest common divisor of 12, 27, and 51 ? Of 9, 45, and 54 ? 63, 18, and 36 ? 15, 39, and 27 ? 4, 26, and 38 ? 14,49, and 63 ? 15, 105, and 75 ? 24, 78, and 42 ? 85, 34, and 51 ?

18. What is the greatest common divisor of 4, 32, 12? Of 45, 75, 60 ? 33, 77, 22 ? 39, 91, 78 ? 46, 69, 92 ? 34, 85, 102? 16, 128, 64 ? 116, 29, 87 ?

19. Mention two numbers of which 10 is the common multiple ? Of which 15 ? 22 ? 26 ? 34 ? 35 ? 42 ? 46 ? 51 ? 52 ? 106 ? 112 ?

20. What is the least common multiple of 2 and 3 ? Of 3 and 5 ? 7 and 3? 5 and 11 ?

21. Is 18 a common multiple of 3 and 2 ? Its least common multiple ? Is 260 a common multiple of 5 and 13? Its least common multiple ?

SECTION III. Fractional Change of Form. 1. WHEN a bushel of wheat is divided into 4 equal parts, what is one of them called ? How many of these fourths make half a bushel ? Is =ì, then? If a bushel was divided into 6 equal parts, how many of them would make half a bushel ? Is=i=i? [Show these fractions on blackboard.] If divided into 8 parts, how many would make half a bushel ? If divided into 10, 12, 16, 18, &c., parts ? [Show a number of these fractions, and let the class observe the relation between the two terms of each fraction.] In how many ways could be represented ? [Infinite.] Which is easiest understood, the smallest fraction, or one of the larger ones?

2. When an article is divided into 3 equal parts, what is one of them called? If divided into 6 equal parts, what

would one of them be called ? How many of these last would make of the article? Is =}, then? (Black-board.] Express { in as many different forms as you can. Which is easiest understood, the smallest or one of the larger ? Into how many different forms can a fraction be changed without altering its value? [See p. 82, 1. 5.]

3. What is the least common multiple of 2 and 3? How, then, may į and } be changed to equivalent fractions with a least common denominator ? How may s and be changed to equivalent fractions with a least common denominator? 1 and ? and ? and ? and ?

4. Change to an equivalent fraction of the same denomination as 3. to same denomination as 15. Make ), į and {, of same denomination, by a change in the two former ? What is the least common multiple of 3, 5, and 25 ? Change ģ, and 25, then, to equivalent fractions with the least common denominator. [Use black-board.] Change , à, and is, to fractions with least common denominator, commencing by a change in the last fraction. Change 4, ş, }, \, and 2%, to fractions with least common denominator, commencing by a change in the second and fifth.

5. Change the following sets of fractions to equivalent ones, with the same lowest denominator by division : $ and ; & and 16; 2o and it, and 3 ; , }, and 16; to, }, and

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6. What is the least common denominator of f, g, 4, , 2,

? [Black-board.] How shall be changed to an equivalent fraction of that denomination? How shall ? 1? f? ? ? What is the least common denominator of , at ? How shall be brought to that denomination ? ? ?

? What is the least common denominator of , 3; %; o's, *. 14, 28

er? How shall 4 be brought to that denomination ? How shall ? ? ? ? ? ?

1 The above exercises should be studied without the aid of the slate. In reciting, the teacher should write the given fractions on the black-board, and call on the pupils to work out the answers mentally.

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SECTION IV. - Addition and Subtraction of Common

Fractions. Explanation. — Numbers of different denominations can neither be added together nor subtracted from each other. Thus, 6 chairs and 3 tables make neither 9 tables nor 9 chairs ; and 3 tables cannot be taken from 6 chairs, nor 3 chairs from 6 tables. Their denomination, however, can be made the same, and then they can be either added or subtracted. Thus, by calling both chairs and tables pieces of furniture, they have the same denomination; and, when added, make 9 pieces of furniture; and the 6 or the 3 may be subtracted from the 9.

The same remark holds good with respect to abstract numbers (that is, numbers used without being applied to things), whether they are whole or fractional numbers. Thus 6ty cannot be added to 3 hundred or to 3 units, because they are of different denominations, and would make neither 9 hundred, 9ty, nor 9 units. Neither can 3ty be subtracted from 6 hundred without changing one of the hundred (mentally) to 10ty, and thus we should have 570. This becomes more evident when applied to fractions. À can neither be added to, nor subtracted from, 5, while both retain their present forin. can be changed to is, and to 19, and, as they have now the same denomination (twelfths), they can either be added or subtracted.

But the case is entirely different in regard to multiplication and division. In multiplication, as has already been observed, although it is in reality nothing more than addition, yet one of the factors is not to be added. It merely points out the number of times that the other factor is to be taken. Thus, the 6 chairs and 3 tables may be multiplied by 3, giving 18 chairs and 9 tables, because 3 times 6 chairs make 18 chairs, and 3 times 3 tables make 9 tables; and so with any other number whatever. Abstract integers may also be multiplied or divided by numbers of different denominations, since it is evident that 6 hundred or 6ty can be taken three times as well as 6 units can; and it is equally evident that either of these digits can be divided by 3 of any denomination whatever. In like manner,

and , or any other fraction, though of different denominations, may be multiplied or divided by 2, 3, 4, or any other number, since the one is only taking each of these fractions so

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many times, and the other is only finding how many times they contain the divisor.

1. Can numbers of different denominations be added together or subtracted from each other? Give me examples to show why. What should be done, then, when it is necessary they should be added or subtracted? Can factors of different denominations be used in multiplying? Give an example to show why. May the divisor and dividend be of different denominations? Give an example to show why.

2. By how many methods can a fraction be changed without altering its value ? [See p. 83, 1. 36.] Which is more intelligible, a fraction with large or with small terms ? Which, then, is the preferable mode of changing the form of a fraction? Can the form of a fraction always be changed by division? When can division be used ? Ans. When its terms have a common divisor. Can jy be changed by division ? Can

? ? ? ? Why? Can 1 ? ? ? Can every fraction be changed in form by multiplication? Why? Because, though every two numbers may not have a common divisor, yet any two, &c.

3. What is the least common multiple of 3 and 15? What, then, is the sum of sand ş? What is the least common multiple of 4 and 8? What, then, is the sum of sand ? What is the sum of and f? Of , 4, and g? Of 4, zs, and ? Of j, k, and ? Of , , and ? (By division and multiplication the least common denominator becomes 20.] Of \, 6, and ? Of is, , and 16? [Least common denominator 4.]

4. What is the least common denominator of and ž? What is the difference of these fractions, then ? The difference of g and zo? Of and ? Of and 1 ?

5. How many fourths are there in 1 ?" In 2? In 5? In 9 ? How many fourths in 14? In 5%? What, then, is the difference between 6 and 11? How many fifths in 1? Eighths in 1? Sixths? Ninths? What, then, is the difference between 4 and 24 ?

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