how many?” Then require them to count four twos, five twos, &c. This method may be brought in aid of all their earlier attempts in adding and multiplying. Care should be taken in this exercise that the words be pronounced in a quick and neat manner, with a distinct pause between the groups. If rightly conducted, this exercise furnishes one of the best aids for overcoming the habit of counting, instead of adding, — that fatal clog which keeps back many scholars in arithmetic from making any rapid progress. II eleven The successive additions of two, beginning with one, | nine furnish the series of odd numbers. This and the pre- il seven. ceding exercise exhibit all the results made by the ad- II five ditions of the number two. || three The pupil should, in connection with this exercise, be taught to group the numerical names by twos, beginning from one, so as to furnish the series of odd numbers, thus: one — two, three — four, five, &c. Finally, let him count the even series, up to twenty, and then down, till he can do it readily. Let him count the odd series up to twenty-one, up and down. * * To teach the pupils to add the number three, let the teacher make three marks on the board, and require the class to name the | | | fifteen number; then three more, and so on. Then, covering III twelve all but the lower three, let the class name the number in ||| nine view, as the teacher exhibits successively three more at III six each remove, or three-less. III three The pupils may then be led to count in groups of three; thus, one, two, three — four, five, siz — seven, eight, nine, &c. From this they may be readily led to name the multiples of three; three, siz, nine, twelve, &c. The first variation from the above method in the addition of three may be made by beginning with two, which gives the follow- | | | fourteen ing numbers. In order to secure the necessary amount III eleven of repetition in these exercises, and yet to avoid weari- | | | eight ness, it will be well often to change the manner of add- III five ing. Thus, in the accompanying example, the pupils II two may simply name the series made by the successive additions; as, two, five, eight, eleven, fourteen ; or, they may state the process more fully, thus: two and three are five, and three are eight, and three are eleven, &c. A still fuller statement of the operation would be thus: two and three are five, five and three are eight, eight and three are eleven, &c. The only remaining variety in adding three will be exhibited by beginning with one, which may be illustrated in the same way as the others, giving the numbers, one, four, setén, ten, &c. o LESSON VII. After the numbers have been mastered in the manner above suggested they should be combined in a variety of ways by means of marks on the board. The columns of marks subjoined are given merely as examples and hints on this point. III II III i il II III II II III III III - II I II III II III III i I II I III III II I III I II III III I II III III II II I II I II III Ii II II III ! II A single column may be made to furnish exercise sufficient for a recita tion; for, by adding or erasing a single mark at the beginning, a change is made through theohol. Y y LESSON VIII. The addition of the number four may be illustrated in the same way as the preceding numbers. It presents the following variations: 11 II sixteen, &c. 11 II fifteen. || || fourteen. ! III thirteens. The number four may then be combined with the preceding numbers in a great variety of ways. This will serve as a review of what has gone before, and at the same time will extend the discipline on the new element which the lesson contains. The same mode of illustration may be pursued in relation to the larger tumbers, five, six, &c., if the teacher shall find it necessary. A.* 1. How many thumbs have you on your right hand? how many on your left? how many on both together ? 2. How many hands have you? 3. If you have two nuts in one hand, and one in the other, how many have you in both 2 4. How many fingers have you on one hand 2 5. If you count the thumb with the fingers, how many will it make 2 6. If you shut your thumb and one finger, and leave the rest open, how many will be open 2 7. If you have two cents in one hand, and two in the other, how many have you in both 2 8. James has two apples, and William has three; if James gives his apples to William, how many will William have 2 9. If you count all the fingers on one hand, and two on the other, how many will there be 2 10. George has three cents, and Joseph has four: how many have they both together? * The first questions in this section are intended for very young children. It will be well for the instructer to give a great many more of this kind. Older pupils may omit these 11. Robert gave five cents for an orange, and two for an apple; how many did he give for both 2 12. If a custard cost six cents, and an apple two : cents, how many cents will it take to buy an apple and a custardo 13. If you buy a pint of nuts for five cents, and an orange for three cents, how many cents would you give for both 2 how many more for the nuts than for the orange 2 14. If an ounce of figs is worth six cents, and a half a pint of cherries is worth three cents, how much are they both worth 2. 15. Dick had five plums, and John gave him four more; how many had he then 2 16. How many fingers have you on both hands? 17. How many fingers and thumbs have you on both hands 2 18. If you had six marbles in one hand, and four in the other, how many would you have in the one more than in the other 2 how many would yo! bave in both hands P 19. David had seven nuts, and gave three of them .0 George; how many had he left? 20. Two boys, James and Robert, played at mar, bles; when they began, they had seven apiece, and when they had done, James had won four; how many had each then 2 21. A boy, having eleven nuts, gave away three of them; how many had he left? 22. If you had eight cents, and your papa should give you five more, how many would you have 2 23. A man bought a sheep for eight dollars, and a calf for seven dollars ; what did he give 'for both 2 24. A man bought a barrel of flour for eight dollars, and sold it for four dollars more than he gave for it; how much did he sell it for 2 25. A man bought a hundred weight of sugar for nine dollars, and a barrel of flour for seven dollars; how much did he give for the whole 2 26. A man bought three barrels of cider for eight dollars, and ten bushels of apples for nine dollars; how much did he give for the whole 2 27. A man bought a firkin of butter for twelve dollars, but, it being damaged, he sold it again for eight dollars; how much did he lose 28. A man bought three sheep for fifteen dollars, out could not sell them again for so much by eight dollars; how much did he sell them for 2 29. A man bought sixteen pounds of coffee, and lost seven pounds of it as he was carrying it home; how much had he left 2 o 30. A man bought nineteen pounds of sugar, and, having lost a part of it, he found he had nine pounds left; how much had he lost? 31. A man, owing fifteen dollars, paid nine dollars of it; how much did he then owe ? 32. A man, owing seventeen dollars, paid all but seven dollars; how much did he pay ? \ B. 1. Two and one are how many 2. Two and two are how many ? 3. Three and two are how many 2 4. Four and two are how many ” 5. Five and two are how many 6. Six and two are how many P 7. Seven and two are how many P. 8. Eight and two are how many P 9. Nine and two are how many 10. Ten and two are how many P 11. Two and three are how many 2 12. Three and three are how many 2 13. Four and three are how many ? 14 Five and three are how many P |