Grade Pupils. TO THE TEACHER.-Pages 6 and 7 of this book are not for the pupil's use. They contain the number facts and terms that should be taught orally to Second Thirty-Three Facts of Addition. 9 9 9 9 10 10 10 10 10 11 11 11 11 12 12 12 12. If the thirty-three facts of addition given above are taught prop erly, the pupil will, at the same time, acquire a knowledge of the corresponding facts of subtraction; thus, if it is clear to the pupil that 7 and 5 are 12, he will also know that 12 less 7 are 5, and that 12 less 5 are 7. Twelve Facts of Multiplication. = 2 twos = = 4. 3 twos 6. 2 threes = 6. = 9. 4 threes=12. 2 sixes = 12. 2 fives = 10. 3 fours = 12. If these twelve facts of multiplication are taught properly, the pupil will, at the same time, acquire a knowledge of twelve facts of division; thus, if it is clear to him that 3 twos are 6, it must be equally clear that 6 is 3 twos; or, to use the ordinary mathematical expression of this fact, that 2 is contained in 6 three times. These twelve facts of multiplication will also give to the thoughtful pupil twelve other facts of division, often called 10 dimes = 1 dollar. 2 pints = 1 quart. 4 quarts = 1 gallon. Test and cultivate the conceptive power of the pupil by questions in pairs like the following: It will be observed that these facts are combinations of facts given before. The work should be performed first with objects. It is very desirable that the pupil shall give the answer to such questions as 6 is of how many, not from memory, but by thinking 6 objects divided into 2 groups and by adding to the 2 groups another similar group. (111 111) 111. To aid in securing correct mental processes it will be found profitable to give questions in pairs like the following: If this work is done properly, the pupil will be able to answer questions outside the "beaten track;" thus, he will readily answer the following: Ten is two thirds of how many? Two thirds of eighteen are how many? The real measure of the value of arithmetical training is the ability of the pupil in solving problems that are in some respect unlike those he has had before. One half of three is how many? Three is one half of how many? All the facts of combination and separation should be first worked out with objects and may be applied to measurements. The pupil should be able to fill the blanks in such statements as the following: Not by definition but by frequent use, the pupil should become familiar with the following terms and expressions: add, subtract, multiply, divide, sum, difference, product, quotient, square, oblong, triangle, square inch, 2-inch square, 3-inch square, is contained in. Do not use the sign (X) for the word times. Whenever this sign is employed in the lower grades it should be used for the words multiplied by. Teach the pupil to count objects to 100, and to count (add) by twos and by tens. SUGGESTIONS TO TEACHERS. 1. Do not permit the pupil to attempt the work of any page of this book until he has been properly prepared for it. This preparation will consist of either (a) an oral presentation of work similar to that found on the page or (b) a review of work already done. The amount of oral work may be, often should be, many times as much as is given in the book. Be sure that the oral work precedes the book work. As often as the pupil is confronted with a seemingly insurmountable difficulty, have recourse, either to review, or to an oral presentation of the subject. If the child cannot solve the problem presented, do notexplain; but give him problems that he can solve and so lead up to and over the difficulty. 2. Often ask-"What is the meaning?" If the pupil knows the meaning of the following expressions, he finds little difficulty in answering the questions correctly: Three fourths of the difficulty usually found in common and decimal fractions vanishes, if the teacher selects problems wisely (that is, considers them in their proper order) and habitually requires the pupil to tell the meaning of the expression before he attempts the solution of the problem. Too often the pupil is required to "juggle with the figures" and then explain the juggling. 3. The order of procedure in the lower grades should usually be as follows: (a) Work with objects. (b) Work without objects, but with concrete problems. (c) Work with abstract problems. (d) Require the pupil to convert the abstract problems into concrete problems. 4. Do not waste time in "pretty," useless, namby-pamby number stories in connection with splints or toothpicks. The time for the number story is immediately after the solution of an abstract problem; the purpose, to lead the pupil to keep in view the possible application of his abstract work. Require the number story (the concrete problem) as often as there seems to be danger that the pupil does not see the application of an abstract problem. This danger arises mainly in connection with multiplication and the two cases of division, especially in common and in decimal fractions. 8 11. William earned 5 cents and his brother earned 6 cents; together they earned cents. 12. James had 12 cents; he gave his sister 4 cents; he then had cents. 13. John paid 2 cents each for 5 pencils; for all he paid cents. 14. Harry has 10 cents; oranges cost 5 cents each; he can buy oranges. 15. Richard paid 8 cents for 2 lemons; one lemon cost cents. *Pages 9 to 22 inclusive, are at once a review and a test. If the pupil is prepared to read in a "third reader," and if the second grade number work outlined on pages 6 and 7 of this book, has been mastered, he will read these lessons without much hesitation. If he finds difficulty in calling the words, lay the book aside and teach him to read. If he cannot readily fill the blanks, teach him orally the second grade number work outlined on the preceding pages. 12. Mary's pencil is 3 inches long; Alice's pencil is 4 inches long; together they are inches long. 13. Jane had a pencil 8 inches long; she broke off a piece 2 inches long; what remained was inches long. A square has sides. 14. Think of a square. side of a 2-inch square measures of a 2-inch square together measure One inches. All the sides inches. 15. Ann had a piece of ribbon 10 inches long; she cut it into 2-inch pieces; there were pieces. 16. I am thinking of a square. It has All its sides together measure 12 inches. square. 17. Three times 2 are 18. Three times 4 are 19. Four times 3 are 20. Two times 6 are 21. Two times 4 are 22. Two times 2 are Three times 3 are |