Addition of Higher Orders. 24. Note.-The illustrations of this book are not intended to be merely observed and read about, but they are designed to picture to the eye, as far as possible, the actual work which it is intended shall be done by the pupils with objects. These objects should be supplied by the school authorities, or, with slight suggestions by the teacher as to what is best or most available, according to the circumstances of the school, they may be brought in by the pupils. They should be as large as possible, and yet not inconvenient to handle in great numbers. SLATE WORK. Example. Find the sum of 738, 236 and 573. One who knows nothing more of arithmetic than how to count to ten might find the sum of these numbers by some such means as the following: Suppose that he has a large number of sticks, some of them tied up in bundles of ten, and some in bundles of a hundred each, and that he has, besides, some single sticks. these were placed in rows or shelves, as in the picture at the right, he might count first the single sticks, taking them in his hand as he does so, and when he has reached ten, tie them in a small bundle, leaving the remaining single sticks at the right on the shelf below. He could then count the bundle which he had just made, with the tens' bundles on the shelves, and tie each ten of these bundles into larger bundles of a hundred each, and leaving the odd bun dles of tens on the shelf below where they formerly were, he could count all the bundles of hundreds together. And, again, tying ten of these larger bun dles into one, he would have the sticks arranged as here represented; that is, one bundle containing a thousand, five of a hundred each, four of ten each, and seven single sticks. 25. The foregoing method is the same as that which is indicated in the following arithmetical process: Thous Ohind. Tens. Units. Thous. Hund. Tens. Units 8 6 ori 3 6 5 7 3 Having seen how it is, that this process really produces a number equal to the sum of the numbers added, the pupil is prepared for the rule for addition. 26. Rule.-1. Arrange the numbers to be added so that the figures of the same order shall stand in the same column, units under units, tens under tens, and so on. 2. Begin at the lowest order, and add each column separately. If the sum of any column is less than 10, write it underneath. If it is equal to or greater than 10, place the right-hand figure of the sum under the column added, and unite the left-hand term or terms with the next column. Proof. In order to be quite sure that the addition is correct, add each column both upward and downward. If the two results are the same, there is little danger of error, 78-88. Add together the numbers in each column. Add the following numbers, first arranging them in columns : 115-124 3254-4015-7845–1572 - 433-7958= 968 — 916-3417 + 468) — 14 The following may be solved first without the use of the slate. Only results should be pronounced. (In the last line, Ex. 127, for instance, Bay 892, 952, 956.) 153. Add 768, 5,643, 12,354, 678,901, 5,847, 2,146,353, 975,321, 64,387,510. 154. Add nineteen, ninety, seventy thousand four hundred eight, 87, 1,625,847, 269,751, 3,894, twelve hundred sixty-one, 5,050,050, six hundred thousand six. 155. Add 198,725, 918,273, 1,928,370, 4,354,651, 34,234,534, 6,712,893, 647, 19, 1,345, 67,351. 156. Add 283,857, two thousand twenty, 998,722, five millions fifty thousand fifty, eight hundred thousand seven hundred twelve, 27, 192,875, 909,090, six hundred eight thousand four hundred ten, 34,827, fourteen hundred fourteen. Direction. In adding these columns, do not say (see Ex. 157) 4 and 3 are 7, and 8 are 15, and 5 are 20, and 4 are 24, etc., but simply speak results; thus: 4, 7, 15, 20, 24, 32, 38, 41. The repetition of the numbers to be added increases liability to error. Some can learn to add mentally numbers of even three places. Treating the tens and hundreds as units and tens (see note, p. 25), they would say, in Ex. 161, 27, 57, 65 tens : 650, 655, 661, and set down the answer at once. 27. Adding two or more columns of figures at once is valuable practice in "mental arithmetic." It should be carried as far as time and the ability of the pupil will permit. 161-169. 275 369 876 629 483 987 797 519 625 529 875 759 654 686 386 Examples for Practice and Review. 357 982 678 Applications.-1. I gave 83 marbles to Lewis, 34 to William, and 97 to Charles. How many did I give away? 2. In one book there are 89 pages, in another 246, and in a third 387. How many pages in all ? 3. A certain tract of land was divided into four farms, one containing 113 acres, another 237, a third 180, and the fourth 320 acres. How many acres did the original tract contain ? |