MULTIPLICATION. INDUCTIVE EXERCISES. 46. 1. At 6 cents each, what is the cost of 4 oranges? 2. How many cents are 6 cents +6 cents+6 cents +6 cents? 3. What is the sum of 6 cents taken four times? 4. Add by 6's to 24; by 4's to 24; by 3's to 24; 8's to 24. 5. How many are four 6's, or 4 times 6? Six 4's, or 6 times 4? 6. Do the results differ? Why not? 7. James bought 5 pencils, at 7 cents each; how many times 7 cents did he pay? What is the sum of 7 cents repeated 5 times? 5 times 7 = ? 8. Add by 7's to 35; by 5's to 35. 9. What is the difference between five 7's and seven 5's? 10. What is the sum of 8 repeated 3 times? Of 3 repeated 8 times? How many are 8+8+8? 3 times 8 = ? 11. What sum is produced by taking $8 five times? 12. What is produced by taking 9 as many times as there are units in 3? In 4? In 5? 13. At $6 a ton, what is the cost of 5 tons of coal? ANALYSIS.-Since 1 ton of coal costs $6, 5 tons cost 5 times $6, or $30. Hence, 5 tons of coal cost $30. Of 5? Of the number pro 14. What is the unit of $6? duced by taking $6 five times? 15. Should we repeat $6 five times, or 5 six times? Why? 16. What will the unit of the number produced always be Ans. Like the unit of the number repeated. like? 1 23 4 23 4 6 8 10 6 912 15 18 4 812 16 20 24 5 10 15 20 25 30 612 18 24 30 36 7 14 21 28 35 8 16 24 32 40 9 18 27 36 45 10 20 30 40 50 11 22 33 44 55 12 24 36 48 60 MULTIPLICATION TABLE. 6 қ 8 9 10 11 12 12 14 16 18 20 22 24 21 24 27 30 33 36 28 32 36 40 44 48 35 40 45 50 55 60 42 48 54 60 66 72 42 49 56 63 70 77 84 48 56 64 72 80 88 96 54 63 72 81 90 99 108 60 70 80 90 100 110 120 66 77 88 99 110 121 132 84 96 108 120 132 144 72 DEFINITIONS. 47. Multiplication is the process of taking one of two numbers as many times as there are units in the other. Or, It is a short method of adding equal numbers. 1. The multiplicand is the number taken or multiplied. 2. The multiplier is the number by which to multiply. It shows how many times the multiplicand is taken. 3. The product is the result found by multiplication. 4. The multiplicand and multiplier are called factors, because they produce the product. 48. The Sign of multiplication is x. It is read times, or multiplied by. When placed between two numbers, it shows that they are to be multiplied together. Thus, 9 x 7 is read, 9 multiplied by 7, or 7 times 9. 49. Brief and rapid oral exercises should frequently be given the pupil upon the tables, and in the fundamental operations of numbers. Pupils may be exercised in multiplication upon Tables 1 and 2, as follows: 1 First, by columns, at sight, give the product of each number and the one below it. Then taking two columns, give the product of each pair of numbers. 2. Then practice on each line from left to right, and from right to left. Then, taking two lines, give the successive products at sight. 3. Next, multiply the numbers in each line and column by 2, 3, 4, 5, etc., to 12, in every case naming only results. The following device, or the Arithmetical Chart, which is constructed on the same plan, is peculiarly adapted to facilitate the memorizing of the tables. EXPLANATION.-1. Draw on the slate or board a small square, and around the inside write the numbers from 1 to 12 inclusive, in regular order, as shown in the diagram. 2. In the centre write one of the numbers, as 2, for a multiplier. Commencing with 1, read rapidly the product of the central figure and each figure in the margin, first, to the right, then to the left, naming only results; thus, 2, 4, 6, etc. Or read from the open book. 987 3. Next practice, in the same manner, with the numbers in the top and the bottom lines, and in the right-hand and left-hand columns. 1 12 11 10 2 3 4 5 6 2 4. Then use successively the figures in the margin as multipliers, and the central figure as the multiplicand. 5. Now, to vary the exercise and test the pupil, the teacher may point in rapid succession to the numbers in the margin, without regard to the order of arrangement, and the pupil, as promptly, give the product. 6. By erasing the central number, and inserting another, a new table and set of products are provided. 7. In like manner, continue this exercise until all the numbers in the margin have been used as central numbers. 8. A few minutes, at each recitation, devoted to this drill exercise, on the board, or chart, will soon make the pupils familiar with the tables. 50. An abstract number is a number in which the kind of unit is not named. Thus, 4, 7, 9, 15, 40, are abstract numbers. 51. A concrete number is a number in which the kind of unit is named. Thus, 3 men, 7 hours, 9 tons are concrete numbers. 52. The parenthesis, ( ), and vinculum, con " nect terms, and show that the numbers included by them are to be treated as one number. The signs + and separate terms. Thus, 24 × (12-7) signifies that 24 is to be multiplied by the difference between 12 and 7. INDUCTIVE EXERCISES. 53. 1. What is the product of 8 cents multiplied by 5? 2. What kind of number is the multiplicand? 3. What kind of number is the multiplier? 4. What kind of number is the product? 5. Name two numbers that are like and abstract. 6. Name two numbers that are like and concrete. 7. What is the product of 12 feet multiplied by 7? 9. What kind of numbers are the factors? The product? 10. What two parts in multiplication are always like numbers? 11. Can you multiply 8 miles by 7 days? Why not? PRINCIPLES.-I. The multiplier is always regarded as an abstract number. II. The multiplicand and product are like numbers, and may be either concrete or abstract. In examples containing concrete numbers, the concrete number is the true multiplicand, but when it is the smaller, it is often, for convenience. used abstractly as the multiplier. 54. The pupil may read the following rapidly, from the open book, thus: 64, 74; 36, 26; 15, etc.* 6. 7. Find the product of 8 by 5. 9 by 6. 8. 7 by 8. 9. 12 by 4. 9 × 12 = 11 x 10 = 10 × 12 = Thus for 8 × 4; The pupil may make applied examples for the above. “What will 4 pounds of sugar cost. at 8 cents a pound? ' may solve it thus: " 4 pounds of sugar will cost 4 times 8 cents, or 32 cents." For 3 × 9: "If a bag holds 3 bushels of wheat, how many bushels will 9 such bags hold?" etc. ORAL 10. 11. 12. 13. EXERCISES. Find the result: Of 8 × 6? 15 x 2+15 = 11 x 11- 9 = 12 × 0+25 = 10 × 12-16 12 x 11-12 = Of 6 × 5? It will be observed that the product of 6 by 5 is the same, whatever order of units 6 may represent. 10 x 8 = 9x7= 9 × 11 = X 14. 15. 16. 17. 400 by 6. 300 by 9. 800 by 7. 900 by 8. Of 800 x 6? 60 by 4. 80 by 6. 40 by 8. 50 by 9. Table No. 2 will furnish additional exercises of this kind. 18. 3000 by 7. 19. 5000 by 6. 20. 2000 by 9. 21. 7000 by 8. * For suggestions on the right use of signs, see notes on page 55. + Multiplying any number by 0 produces 0 and multiplying 0 by any number produces 0. Thus, 9 × 0, or 0 × 9 = 0. |