LESSON XXIII.

Exercises on the Fundamental Operations.

PHILO caught 15 fish and gave John 6 of them; how many did Philo keep?

2. Jacob gave 10 cents for crackers and 6 cents for cheese; what did his lunch cost?

3. Janson walked 18 miles in 2 days, and walked 12 miles the first day; how far the second day?

4. Mary's book contained 20 leaves, and a dog tore ou 11 leaves; how many leaves remained?

5. Fanny solved 22 problems, and Alice solved 10 problems; how many did Fanny solve more than Alice?

6. Maria read 23 verses to-day and yesterday, but only 11 to-day; how many did she read yesterday?

7. How many are 6+6? 6+16? 6+26? 6+36? 6 +46? 6+56? 6+66? 6+76?

8. How many are 6—6? 16 — 6? 26 — 6? 36—6? 46 -6? 56-6? 66-6? 76 — 6?

9. How many are 7+7? 7+17? 7+27? 7+37? 7+ 47? 7+57? 7+67? 7+77?

10. How many are 7-7? 16-7? 25-7? 35-7? 45 -7? 55-7? 65-7? 75 — 7?

LESSON XXIV.

Exercises on the Fundamental Operations.

HOW many words will 12 boys spell, if each boy spells

8 words?

2. How many dimes will 11 books cost, if one book costs 10 dimes?

3. There are 110 birds in flocks of 10 birds each; how many flocks are there?

4. There are 132 pupils in a school, and 12 pupils in each class; how many classes are there?

5. Fill out the following exercise on the number 3:

6. Write a similar exercise on the number 4; thus,

NOTE.-The teacher may require the pupils to write similar exercises on all the numbers from 5 to 24.

7. Make a division table from the multiplication table of 2 times; thus,

8. Write, in a similar manner, the division table of 3 times; Of 4 times; Of 5 times; Of 6 times, etc.

9. Name all numbers in combination of two each that make 4; 5; 6; 7; 8; 9; 10; 11; 12.

10. Write the numbers which in combination of three each form the numbers 4; 5; 6; 7, etc., to 20.

11. Write a table of each number from 6 to 24; thus, 5+1=6; 4+2=6; 3+3=6; 2+4-6; 1+5=6. 6—1—5; 6—2=4; 6—3—3; 6—4—2; 6—1—5. NOTE.-Bequire a similar drill in the same numbers with three parts four parts, etc.; involve also multiplication and division.

Terms and Principles in Multiplication and Division.

THE teacher will present the following lesson inductively, as suggested on pages 26 and 27.

1. When we find the result of a number taken any number of times, the process is called MULTIPLICATION.

2. The number taken a certain number of times is called the MULTIPLICAND.

3. The number which denotes how many times the multiplicand is taken is called the MULTIPLIER.

4. The result obtained is called the PRODUCT. Each of these three is called a TERM.

5. What is the product of 8 apples multiplied by 4?

6. In this problem which is the multiplicand, which the mul tiplier, which the product?

7. When we take 8 apples 4 times, is the result apples or something else?

8. Can the product be anything else than apples?

9. The product then is of the same denomination as which term?

10. Can we take 8 apples 4 peaches times, or simply 4 times? 11. Is 4 an abstract or a concrete number? What kind of a number then must the multiplier be?

12. When we find how many times one number is contained in another, the process is called DIVISION.

13. The number which contains the other is called the DIVIDEND, the number contained is called the DIVISOR, and the number denoting how many times the divisor is contained is called the QUOTIENT.

14. If we divide 8 apples by 2 apples, is the result apples? If not, what is it?

15. Are 2 apples contained in 8 apples 4 peaches times, or 4 apples times, or simply 4 times?

16. What kind of a number is 4, and what kind of a number then must the quotient always be?

17. How many times 2 equals 8 apples? Are 2 pears contained any number of times in 8 apples?

18. What 2 are contained a number of times in 8 apples? 19. The divisor then must be of the same denomination as which term?

THE

INTRODUCTION.

Suggestions to the Teacher.

HE first lessons in fractions should be given in accordance with the following principles:

1. The first lessons in fractions should be given orally. No textbook is needed in teaching the primary ideas of the subject. The teacher should drill the pupils for several days before taking the subject up in the book.

sons.

2. Mental and written exercises should be combined in the first lesThe order is first the idea, then the oral expression of it, and then the written expression of it. As soon as a pupil has an idea of an operation, he should be taught to express it in written characters. The seeing of the operation will help to make it clear to the understanding and to fix it in the memory.

3. The elements of fractions should be taught by means of visible objects. The pupil should be led to see the fractional idea and relation in the concrete, before he is required to conceive it abstractly. The objects to be employed are apples, lines or circles on the blackboard, etc. An arithmetical frame with long rods cut in sections, is used in the schools of Sweden, Prussia, etc.

4. The operations in written fractions should be taught to young pupils mechanically. They should be drilled upon the operations until they are thoroughly familiar with them, even before they understand fully the reasons for these operations. This is in accordance with the principle that with young pupils practice should precede theory.

5. The several things to be taught in fractions are as follows: 1. The idea of each fraction and the expression of it; 2. The fractional parts of numbers; 3. Solving problems requiring the fractional parts of numbers; 4. A few of the simpler cases of reduction. After the learner is familiar with these introductory exerises, he is prepared to take up the subject more thoroughly, and earn to dispose of all the ordinary cases.

6. The simple cases should be solved by analysis and induction. Special problems should be given for solution and the methods be inferred from the analysis of these problems. The principles of fractions should be illustrated rather than demonstrated. The pupils should commit the principles and learn to apply them readily.

LESSON I.

One Half.

IF I divide an apple into two equal parts, what is one of these parts called? Ans. One-half.

2. How many halves of an apple in one apple?

3. If I divide anything into two equal parts, what is each part called? How many halves of anything equal the whole?

4. What do we understand by one-half of anything? Ans. One-half is one of the two equal parts into which anything may be divided.

5. What is one-half of six?

SOLUTION.-One-half of 6 is three, since 6 divided by 2 is 3. 6. What is 1 half of 4? Of 8? Of 10? Of 12?

Of 14?

Of 20? Of 18?

Of 22?

Of 26? Of 28?

7. What is 1 half of 16? 8. What is 1 half of 24? 9. If a yard of tape costs 4 cents, what will 1 half of a yard cost?

SOLUTION.-If 1 yard of tape costs 4 cents, 1 half of a yard will cost 1 half of 4 cents, which is 2 cents.

10. Mary had 6 oranges, and gave 1 half of them to her brother; how many did she give away?

11. A farmer sold 1 half of 12 sheep to his neighbor; how many sheep did he sell?

12. Sarah had 14 apples, and gave 1 half of them to Jane; how many apples did Jane receive?

13. William sold 1 half of 20 oranges; how many oranges did he sell?

WRITTEN EXERCISES.

1. One-half is written thus, 2. What is of 24?of 46? 3. Find of 96; of 284;

; two-halves is written 4. of 84?of 102? of 112! of 396; of 274; of 652.

4. A farmer raised 680 bushels of apples, and sold of them; how many did he sell?

5. A had 250 dollars, and spent of it; how much did he spend? How much remained?