Note. The examples in 235 and 236 require only one oral step, that is, the direct announcement of the result; as, for instance, in adding 80 and 38, think 80 and 30 (=110) and 8, but say at once 118. In examples 237 to 241, two steps are enough; thus, in adding 59 and 32, first think 59 and 30, and say 89, then 89 and 2, and say 91. In examples 242 to 244, four steps may be necessary for the learner; thus, in adding 25, 38, and 49, say 55, 63, 103, 112. LEFT Suggestions for Blackboard Exercises. 6 RIGHT 22. In drill exercises, the double star affords some advantages over the circle, and at the same time facilitates the learning of the several series arising from successive additions of 2's, 3's, 4's, etc. Direction. Beginning at the unit figure of any given number, the unit figures of the successive sums will be found as follows: 1. In adding 3's, at the next point to the right, and so on; in adding 7's, at the next to the left. 2. In adding 6's, at the second point to the right, and thus on, from point to point, of the same star. In adding 4's, at the second point to the left, and so on. 3. In adding 9's, at the third point to the right. 4. In adding 2's, at the fourth point to the right, and thus on (following the line at the right of the last unit figure). In adding 8's, at the fourth point to the left (following the line at the left of the last unit figure). 5. In adding 5's, at the point directly opposite the unit figure of the given number, and thus to and fro. 23. Other Uses of the Figure.-A suitable number being written at the center, the numbers at the points can be combined with it, in addition, subtraction, multiplication, or division, as may be desired. The number at the center being changed from time to time, there is no end to the variety of exercises that may thus be had at little expense of time or labor on the part of the teacher. Exercises in common and decimal fractions may be given in the same way. 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. If 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 method is the same as that which is indicated in the following arithmetical process: Thous Ohind. Tens. Units. Thous. Hund Tens. Units 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. 877 213 361 425 357 862. 135 779 387 600 731 78-88. Add together the numbers in each column. Add the following numbers, first arranging them in columns: 101. 125, 126, 138, 139, 140. 102. 87, 9, 55, 394, 225, 194. 103. 83, 194, 56, 168, 473. 104. 336, 195, 987, 9, 11. 20+2103= 105–114. Add by columns. Also by lines. 2123+2364+7025+ 428+ 6354+2559+ 843+1125 + 359+ 23= 698+ 1994+1427 +2496 +2478+ 437= |