45. 1. How many cents are 12 cents and 8 cents ? 2. Twenty cents are how many more than 12 cents? Than 9 cents? Than 8 cents? Than 11 cents? 3. How many less are 12 cents than 20 cents? 8 cents than 20 cents? 11 cents than 20 cents? 4. If Mary has 18 cents, how many more must her father give her, that she may have 25 cents? 5. James had 25 cents and spent 7; how many cents had he left? 6. 25 is how many units greater than 18? 18 is how many units less than 25? 7. From a piece of cloth containing 21 yards, 10 yards were cut; how many yards remained? 8. How many more are 20 bushels than 16 bushels? 9. How many tens are 9 tens less 6 tens ? 10. How many hundreds greater are 7 hundreds than 4 hundreds? 10 hundreds than 2 hundreds ? 11. How many thousands less are 5 thousands than 9 thousands? 12. What is the unit of 16 rods and 12 rods? How many rods less are 12 rods than 16 rods? The unit of 4 rods? 13. What is the unit of 26 days? units like, or unlike? Of 15 men? Are the 14. Can 15 men be taken from 26 days? Why not? 15. Only what kind of a number can be taken from another? 46. Comparing two numbers by finding how many units the one is greater or less than the other is called Subtraction, and the result obtained is called the Difference. 47. The Sign of Subtraction is which signifies less. It is read minus, Thus, 14-95 is read 14 minus 9 equals 5, or 14 less 9 is 5. When a part is taken from the whole, the difference is sometimes called the Remainder. Thus, the difference between 14 and 5 is 9, or if 5 yards of cloth are cut from 14 yards, the remainder is 9 yards. 48. The Parenthesis, (), denotes that the enclosed expression is to be considered as one number. The Vinculum, has the same signification. , Thus, 25 — (9+6), or 25 −9+6, shows that the sum of 9 and 6 is to be taken from 25. 49. 1. Write on the blackboard any number, as 5. The teacher may name one part, and the pupil the other, until all its parts are named. Thus, the teacher says "1 and," the pupil adds "4 are 5;" again, "2 and," the pupil adds "3 are 5," etc. 2. In like manner, name the parts of each number from 6 to 25. 3. Again write on the slate or blackboard a line of figures, thus, 2, 5, 3, 7, 6, 8, 4, 9. Give promptly at sight the difference between each two consecutive numbers; as, 5 less 2, 5 less 3, 7 less 3, etc. 4. Next, the difference between the sum of each two consecutive numbers and the next one on the right; thus, 2 and 5 less 3, 5 and 3 less 7, etc. 5. Subtract each number in succession from 10; then from 11, 12, 13, etc., to 25. 6. Subtract by 2's from 24 to 0. Written. 24-2-2-2-2- -2-2-2-2-2-2-2. 24, 22, 20, 18, 16, 14, 12, 10, 8, 6, 4, 2, 0. Read. 7. Add by 3's from 0 to 48, and subtract by 3's back to 0. 8. From 1 to 49, and back. 9. In like manner, begin at From 4 to 52, and back. different numbers, and add forward and subtract back by 4's, 5's, 6's, 7's, 8's, 9's, and 10's. 10. Dictation exercises, to be treated as on page 20: 19-6=? 7+12-9-? 13+ 5-7-? 19-4+7=? 21-10 ? 16+ 6-8-? 20+ 7-8=? 248 ? 12+ 9-7=? 12-5+13=? 9+11-6 ? 21-8+12=? 20-12 ? 15+10-8-? 14+ 5-9=-? 25-9-6=? 12+6-5+7-3-8+10+6-7+9+3, etc. Write. Read. 12, 18, 13, 20, 17, 9, 19, 25, 18, 27, 30, etc. 9+10+6—7+2—8=? 25-9+8+6—5—10+7= ? 16-12-20-4-3+9=? 6+30-10-5+12-3+4= ? 13. Again, write on the board combinations as follows: 25-9, 206, 21-12, 14+10, 9+7-3, etc. Then require the pupil to make applied examples for the same. Thus, for 25-9: "I had 25 cents, and gave 9 cents for a slate; how many cents had I left?" Another pupil answers: The difference between 25 cents and 9 cents, which is 16 cents." Again, for 9+7—3, "James had 9 marbles he bought 7 more and lost 3; how many had he then?" etc. Exercises similar to the above should be used for occasional drill, and may be extended and varied, at the option of the teacher. 14. Any number above 20 and less than 100 can be separated at sight into two parts, one of which shall contain 1 ten and the units, if any. Thus, 56 = 40+16, 60 = 50+10, 4730+17, 8870+18, 95 80+15, etc. = 15. Let the pupil practice upon all the numbers from 20 to 99, separating each into parts, as in example 14. 16. Then find the difference between any digit and a number composed of two figures, when the unit figure is less than the digit, as 8 and 43. Thus, Separate 43 into the parts 30+13, and at once we know that the difference between 13 and 8 is 5, which united with 30 makes 35, the difference between 43 and 8. 17. Write on the slate or board a series of numbers between 20 and 99, arranged in any order, and under each number write the same digit. Thus, 21 27 30 24 36 32 23 25 33 41 etc. 18. After subtracting 3 from each number and writing the difference, erase the same and repeat, until the difference can be given at sight. 19. In the same manner practice with 4, 5, 6, 7, 8, and 9. 20. Then write a new set of numbers, and practice upon them in the same manner, and so continue until the difference between any digit and any number composed of but two figures can promptly be given at sight. 21. What is the difference between 22. Find the difference between 73 and 48? 73 48 60+ 13 40 8 25 20+ 5 Since the 8 units in the smaller number cannot be taken from the 3 units in the larger number, separate the larger number into 60+13, and take the 8 units from 13 units, and the 4 tens or 40 from the 6 tens or 60, and the difference between 73 and 48 is 2 tens and 5 units, or 25. 31? 25? 34? 7? 6? 8? 10? 24. What number added to 7 makes 23? 25. What number taken from 32 leaves 9? 26. What number added to 26 makes 31? 42? 54? 60 ? 27. What number taken from 54 leaves 15? 26? 45? 37? 28. How many units are 8 units less 5 units? 8-5 = ? 29. How many tens are 8 tens less 5 tens? 80-50=? 30. How many hundreds are 8 hundreds less 5 hundreds? 800-500 ? 31. How many thousands are 8 thousands less 5 thousands? 8000 - 5000 = ? Observe that the difference, abstractly considered, between 8 and 5 is the same, whether it represents units, tens, hundreds, or thousands, etc. 32. Give the difference of each of the following at sight: |