11 41 By the second method, the child is instructed that whenever he increases by ten any figure of the minuend, he must add 1 to the next figure of the subtrahend. Seeing that he cannot take 9 from 1, he says 9 from 11 leaves 2; 1 (to carry) and 2 are 3, 3 from 4 leaves 1. 29 While this method is just as logical as the next, it is not so easily "explained," and, for this reason, is not so much favored by many teachers of the present day. 29 The third method consists of diminishing the next left- 311 hand figure of the minuend after “borrowing." Where 41 the minuend contains ciphers, this method is particularly confusing to beginners, especially where they are forbidden, as should be the case, to write the changes that are made in the figures of the minuend. Except in the addition of long columns, children should be required from the beginning of slate work to abstain from counting, writing "carrying" figures, and the like. The guide figures introduced into the foregoing explanations of methods of subtracting should not be used by pupils. 61. As a change from sight work, and to increase the pupils' readiness in the solution of mental examples, these drills are useful. Not requiring any preliminary writing on the board, they can be taken up at any time the class is unoccupied for a few minutes waiting for the signal to go home, for example. The pupils all stand; the teacher announces the number to be added, 2 for instance, and begins by saying 1 herself. The first pupil says 3, then sits; the next, 5; and so on. After 39, or some other convenient number, is reached, the teacher begins by saying 2, and the pupils, in order, give 4, 6, 8, etc., to 40. The intelligent teacher will be careful to suit these drills to the capacity of her pupils. She will not weary beginners by spending too much time on the more difficult drills with 7, 8, and 9; nor will she waste the time of older scholars by dwelling on the addition by twos. V NOTES ON CHAPTER TWO 74. Slate multiplication is commenced as soon as the table of 2 times is learned. The first examples contain no carrying. 76. Division tables should not be memorized. 81. Do not permit children to prefix an unnecessary cipher in the quotient of 100÷2; that is, do not have the answer written 050. 84. Many scholars think that when a slate problem contains a very small number and a large one, they must either multiply or divide. Examples 1-4 are given with simple numbers to show them that the nature of the operation depends entirely upon the conditions of a problem. While pupils should not be required to use a pencil to solve a problem that can be solved mentally, it would help the class to have these four examples worked on the board as an indication that in the subsequent examples there may be needed any one of the four operations learned thus far, and to serve as a model in their arrangement of the other problems. While many teachers require the pupils to write the denomination of each addend, of the subtrahend and the minuend, of the multiplicand, and of the dividend, it is scarcely necessary. In later life it is not done; and confusion is sometimes produced in the minds of young scholars by attempting to make them understand why, for example, 60 pints divided by 2 will sometimes give a quotient of 30 pints, and at other times, as in the 6th 22 problem, an apparent quotient of 30 quarts. It will be found. more satisfactory, even if less scientific, to have the denomination written only with the result. Although no formal instruction in finding halves and thirds of numbers has as yet been given, the average pupil will be able to solve problems 10, 11, and 14. 85. Lay no stress on the local value of the figures. Practice will enable the children to read and write correctly numbers of four figures. Teach the pupil to write the comma when the word "thousand" is said and after the number of thousands, the comma to be followed always by three figures. 97. Children should be led to see that 12 x 2 is the same as 12+12; so that when they come to 15 × 2, they will have no difficulty in deducing the rule for writing 0 and carrying 1 when they multiply the 5 by 2. 98. Give the pupils time to find for themselves the quotient of 30 ÷ 2. If it becomes necessary to show some of them how to work the example, do not elaborate the meaning of the 1 (ten) remainder when the tens' figure, 3, is divided by 2. An experienced mathematician, in dividing 9752 by 2, does not say 2 into 9 thousand 4 thousand times with a remainder of 1 thousand, 2 into 1700 8 hundred times with a remainder of 1 hundred, etc. In dividing 30 by 2, children should not be permitted to write the first remainder, 1, before the 0, to indicate that 2 is to be divided into 10 for the second quotient figure. Children learn to work just as well without these unnecessary scaffolds. 104. While these drill exercises introduce a multiplier greater than 2, they contain no combinations, except 3x 3, other than those found in the preceding work. After working these examples, the pupils will have learned that twice 9 is equal to 9 twos, that when he knows the table of 2's, he knows a portion at least of the table of 3's, 4's, etc., to 9's. 111. When the teacher places the pointer on a number in one of the two outer spaces of the first circle, the pupil promptly gives the result obtained by adding to it the number contained in the inmost space. When this last number has been combined with all the others, it is replaced by a different number. 112. These drills are useful to impress upon a child the fact that when he knows, for instance, that 6 and 5 are 11, he should also know that 6 and 15 are 21, that 6 and 25 are 31, etc. They may also be employed as subtraction drills. 115. Division drills are necessary to enable pupils to acquire facility in obtaining quotients and remainders. When pupils are dividing by 2, the numbers from, say, 9 to 19 are written on the board with 2 underneath. 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 ÷ 2. When the pointer is placed at the 9, the pupil answers 4 and 1; when placed at 14, he answers 7; at 17, 8 and 1; etc. Other divisors may be employed, but care should be taken not to have any quotient figure but 1 or 2 at this time, as pupils have not yet learned the table of 3's. Thus, when 6 is used as a divisor, the teacher should not use a dividend greater than 17. When the three-times table is known, numbers from 12 to 29 may be written. Facility in division will come only by practice, and it may be necessary for the teacher to supplement the examples of the book by others of her own. 118. Do not fail to keep up practice in addition and subtraction. 119. Subtraction examples in which the subtrahend is given before the minuend should occasionally be used. |