15. Multiply by 199: 3; 9; 52; 69; 81; 46. 16. Find the cost of 42 barrels of flour at $4.99 a barrel. 17. State a short way to multiply by 198. 21. Find the cost of 16 doz. handkerchiefs at $2.98 a dozen. 22. A milliner sold 24 hats at $3.98 each; how much did she receive for the hats? SEVENTH Year. Pupils in this grade are able to work out with some suggestions from the teacher many short processes for themselves. Such work is more valuable than the mere application of methods already learned. I. The following developments with exercises will suggest work of this nature in multiplication: 1. What is the cost 468 books at 15 cents each? Plan. 1. 15c. 11⁄2 X 10c. = 2. 468 X 12 X 10c. =? 2. What part of a number is added to the number to multiply it by 11⁄2? How is a number multiplied by 10? 3. What is the cost of 450 handkerchiefs at 133 cents each. Plan. 1. 133c. 13 X 10c. = 2. 450 X 13 X 10c. =? 4. What part of a number is added to the number to multiply it by 1? Tell why the following are true, with any number as multiplicand: 12. 56 yd. of carpet at 871⁄2 cents per yard? 19. 25 rolls of wall paper at 871⁄2 cents per roll? II. The following developments with exercises will suggest work of this nature in division: 1. How many sewing machines at $15 each can be bought for $285? 2. What part of a number taken from it leaves 3 of the number? How is a number divided by 10? = 3. How many yards of silk at $1.121⁄2 a yard can be bought for $900? How many of each article can be bought for the amount given? 11. Yards of carpet at $.871⁄2 per yard, $70. 12. Atomizers at 33 cents each, $40. 13. Concert phonographs at $66% each, $600. 14. Parlor phonographs at $121⁄2 each, $350. 15. Tea kettles at $1.333 each, $60. 16. Bookcases at $15 each, $855. 17. Pairs of dumb-bells at $1.121⁄2 per pair, $20.25. 18. Tennis rackets at $1.33% each, $20.40. 19. Tents at $13% each, $120. EIGHTH YEAR. All the practical "short cuts" of general use in business arithmetic should have been thoroughly taught when this stage of the work is reached. Eighth grade pupils, if taught from a text-book of the better class, have had some work in literal notation and are interested in the more complex abridgements of processes. For these pupils cases like the following may be presented: I. To multiply two numbers between 10 and 20: Add the units' digit of one number to the other number, multiply by ten and add the product of the units' digits. Thus, 15 X 17 10 (15+7) + 5 × 7 = 225. = The formula which shows this to be generally true is 100+10a10b+ ab 10 (10+ a + b) + ab. =: II. To multiply two numbers, one as much greater than a multiple of ten as the other is less. Thus, 27 X 33 (30—3) (30+3)= 30% — 39009891. = The formula which shows this to be generally true is: (a+b) (a - b) a2b2. - = Judicious Use of Arithmetic Drills in the First Four Years BY CATHARINE L. O'BRIEN, Head of Department, P. S. 10, Brooklyn. The introductory note of the "New York Course of Study" in mathematics states that counting should aid in column addition and in a mastery of the multiplication tables. How shall we secure accuracy and rapidity in this work? After objective presentation has demonstrated to the eye and understanding the number of relations, the number facts must be learned. What motive shall we give the children for this laborious work? The important goal to be reached is the excitement of a deep interest. Those topics must be chosen which are capable of arousing interest. We must search out materials which correspond with child nature and identify the child with life. We will thus get genuine interest, concentration of attention, and good mental discipline. In early number work the need for counting immediately appeals to the child. He sees his parents count money. In his own little world he counts in games. The teacher wants to know how many are present. She finds out by asking the children to count. The game is varied by having a different child begin each day. Toy money is arranged by the children in piles of 10 cents each and a child quickly counts from 10 to 100. If he does not know some one volunteers to assist and the difficulty is soon gotten over. Counting leads over to putting 1 with 1, with 2, etc., up to 9; then 2 with 1, etc., up to 9. A quick game of showing cards and getting answers pleases the little folks; when 3 is presented 2 is shown, the numbers become associated and 5 as the answer is given at sight. The form of ordinary addition is used and in the first half year sums with no carrying (no answer to be higher than 9) are put upon the board for oral drill, as: These drills are kept up through the first four years and increase in difficulty as the child progresses. To illustrate: Oral blackboard work: column addition within the combinations. 1 11 21 etc. Oral blackboard work: column addition, begin the column sometimes with a large number to give practice in higher combinations, as: Oral blackboard work: column addition, sometimes begin the column with a large number to give practice in higher combinations, as: Pupils learn the facts of addition in the first three grades. 2B. Oral blackboard work: column addition, no column to foot more than 50. Begin column with a large number to give practice in higher combinations, as: Beginning with the 2A grade the column is added from the bottom to the top 13, 22, 24, 28, 34, 39, 40, 48 and from the top to the bottom 9, 14, 20, 24, 26, 35, 39, 48. In this and the higher primary grades the teacher's drill example may contain more or less numbers from left to right than I have indicated. 3A. Oral blackboard work: column addition, no column to foot more than 60. See note 2B grade. Aim for rapidity as well as for accuracy. Have at least one 7, 8, and 9 in column. 3B. Oral blackboard work: column addition use 1 to 9 inclusive, have at least one 7, 8, and 9 in column, no column to foot more than 70. See note 2B grade. Aim for rapidity as well as for accuracy. 4A. Oral blackboard work: column addition, no column to foot more than 80. See note 3B grade. 4B. Oral blackboard work: column addition, no column to foot more than 100. See note 3B grade. If the blackboard be ruled off into square spaces the numbers may be put in readily. By changing the numbers of any one line you have a different example. By erasing every third line the numbers by slight changes are suitable for subtraction. Any one line of them may be used for drill in multiplication or division. The numbers should be put upon the board before nine o'clock and the drill given in the first recitation period. The secret of success is to make the drill bear upon the previous day's work as necessary repetition to fix needed combinations. Pupils can be made to feel a responsibility for every answer if the lesson is conducted so that no one knows who is to be called next. If the teacher calls from cards, one pupil can point and another put down the answer. If a combination is failed upon the teacher makes a note of it and the pupil who fails is called upon before the end of the recitation period to add that column again and another column. After the lesson the combinations failed upon are put upon the blackboard for study. During the lesson, the skilful teacher exercises judgment and calls upon the poorer scholars more frequently than upon the others. The exercise stands in the same relation to arithmetic as the "Setting-Up" drill does to the physical culture lesson. Classes of the same grade can be matched against each other and the results compared by taking account of the register, the number present, the number who failed, and the time taken in each class for the drill. Pupils of different sections in the same class may vie with each other |