Sold it at a loss of 5 p. c., .'. 25 of 25=25-14=23% what he sold it at, and 25+=251 what he would have sold it at to gain 1 p. c.,,. in this case 251-233-14 that is to £3 in the original,,. as 1 : 20 :: £3 ; sum given for watch, i.e., As 3: 40: 3:40 = sum given to French maker for the watch.-J. G.; W. Thorneycroft; H. F.; Geo. Davies, 84. I extract the following from the Student's Hume.-Charles (the Pretender) arrived safely in the W. Isles of Scotland, and landed at Moidart in Inverness-shire (1745). Several of the Highland chieftains remonstrated against his enterprise as impracticable and insane: for his arms he had lost, and the only adherents who landed with him were his tutor, Sir Thomas Sheridan; the Marquess of Tullibardine; Sir John Macdonald, an officer in the Spanish service; Kelly, a nonjuring clergyman; Francis Strickland, an English gentleman; Æneas Macdonald, a banker in Paris; and Buchanan, who had been sent messenger to Rome by Cardinal Tencin. These were afterwards called "the seven men of Moidart."-Unit. Replies also from N.F.; and Rana. 86. (a) I true who that may selves ELE (g) to one clear harp (j) men (man) Connective (7) on stepping stones Adj. phrase to stones Ext. (dir.) (n) to higher things Adj. forming part of Infin. Comp. "(to be true") to Adj. to him. Sub. Sent. Noun. Obj. to held. Rel. Pro. agreeing with ante. "him," nom. to "sing." Def. Verb Ind. pre. 3rd plu. agr. with "men," or, Aux. Verb Com. noun plu. masc. obj. gov. by "of." -W. Carter. Also by H. F.; W H. Mitchell; and "V.” For want of space we have been obliged to omit replies to several of the queries in the August number. We have omitted those to which least answers were sent. The numbers are 60, 63, 64, 66, 68, 72, 74, 75, 79, 81, 85 (for "yards" read "paces,") and 87 to the end. Replies to the whole of these will be inserted in our next number; they should be sent as early as possible. Books for Review and all other communications for the Editor to be addressed, To the Editor of the Teachers' Assistant, at W. Stewart & Co.'s, Holborn Viaduct Steps, E.C. When the Magazine is ordered direct from the office the order should be sent to the Publishers. Stewart's TEACHERS' ASSISTANT. Practical Papers on Teaching. By T. W. PIPER. Introductory Lessons in Simple Subtraction. IN teaching Subtraction to a class of children previously unacquainted with it, we should prefer NOT to begin by informing the children that they were about to learn something new; for although by making this announcement we should, perhaps, excite the interest of the children, we should also very probably discourage others, especially the weaker ones. We prefer rather to base our teaching of Subtraction upon what we have already taught the children of Addition, and so very gradually to introduce Subtraction, that the pupils shall scarcely perceive that they have passed on to a new rule. By so doing, we act upon that almost universal rule in teaching of "proceeding from the known to the unknown;" thus teach the pupils so to use the knowledge they already possess as to obtain new knowledge. we Without any mention, then, of what we were about to do, we should one morning in our Mental Arithmetic Lesson say, "If I had three marbles in my pocket, and a boy gave me six more, how many should I then have ?" Answer, NINE. "Yes; and if a man had 6 cows and bought 3 more, how many would he then have?" Answer, NINE. We should ask ten or a dozen such questions as this, all involving the process of adding together six and three; we should, in fact, repeat the same sum in different forms (as in the two examples given above) until the pupils know thoroughly that R To make sure that each child understood this, we should provide ourselves with a piece of cardboard, similar in all respects to AB, and another piece similar to BC, and putting these together we should produce the figure AD. .6 66 66 Now comes the important point, where the idea of Subtraction is introduced. The teaching would proceed thus: "If I take from AD the piece of card called BC, what will remain ?" Answer, The piece AB." "Yes; but if I do not take away BC, but take away AB instead, what will be left ?" Answer, "BC." "Yes ; but how many squares are there in AD?" (Instead of saying 'AD," the teacher may point out with his hand all he means by AD"). Answer, "NINE." "Quite right! and how many squares are there in AB?" Answer, "Six." "And how many in BC?1 Answer, "THREE." "Now here (placing the hand over AD) are NINE squares; if I take away all this (placing the hand over AB), will there be as many there as there was at first (i.e., before AB was removed)?" [The object of this question is to open the minds of the children still further to the fact that numbers may be diminished in learning Addition, they have already learnt that numbers may be increased.] Answer, "No!" "How many will be gone?" Answer, "SIX." "And how many will be left?" Answer, "THREE." "Then when I take six squares from nine squares, how many are left ?” Answer, "THREE." In this way the children get some idea of the fact that This knowledge will be confirmed in two ways, as follows: "You told me that if a man had 6 cows and bought 3 more, he would then have -?" Answer, "NINE." "Very well; then if a man had 6 cows and wanted enough to buy to make up 9, how many would he have to buy?" Answer, "THREE." "Yes; but when he had 9, how many would be left if 6 were to die?" Answer, "THREE.". This is one way of confirming the knowledge referred to; another is as follows: Starting from the most simple case of Substraction, viz., "counting backwards," the children should be led from their knowledge of the fact that 8 is 1 less than 9, to see that (1.) 7 is 2 less than 9, and that therefore 9 - 2=7 Having thus confirmed the children in their knowledge that 9-6=3, viz., by arriving at that knowledge two different ways, i.e., by Addition and Subtraction, we shall once more propose a few such questions as the following: "A girl bought 9 yds. of ribbon; she wanted 6 yds. for her bonnet, and the rest for her hat; how much did she use for her hat?" Again, "A man sets out to walk 9 miles; when he has gone 6 miles, how far has he still to travel?" A few more such questions, and then comes this one: "Nine hundred men went to battle, and six hundred came back; how many were killed?" The use of this question is to lead the children to reason about 9 hundreds as they would about 9 cows. It is the intermediate step between the study of concrete numbers and abstract numbers; as such it is most important, and similar questions should be set upon Thousands, Millions, &c. This will be quite enough for one lesson; before taking up the subject again in a new lesson, the teacher will allow two or three days to elapse, and then he will, by a carefully-prepared course of questioning, lead his pupils to see that whenever two numbers have been added together, if one of them be taken from this SUM, the other will be left as a remainder. The course of questioning necessary to establish this proposition will be sufficient for the second lesson in Subtraction. After a due interval, at least a day or two, the subject will be again renewed, and a great number of Exercises in Mental Arithmetic will then be given, in order to make the children practically acquainted with the proposition discovered in the last lesson. As an example of the way in which this is done, take the following: "Nine and seven make how many?" "Nine from sixteen leave how many?" "How many must be taken from sixteen to leave seven?" "How many must be added to seven to make six teen?" "By how many is sixteen more than seven ?" Here it will be noticed that the question is varied as much as possible; this is done to lead the children to consider the subject from several different points of view. Other lessons will be given like this last one, and no attempt will be made to proceed any further till the children are perfectly familiar with the truth contained in the proposition above referred to. We shall use this truth as an instrument for discovering the Rule for Simple Subtraction, and we therefore cannot do anything towards finding that rule till the children see most clearly that it is a truth, and are so familiar with it that they can use it as we desire they should. When, however, they have attained to this knowledge, our next lesson will proceed to use it, thus: -?" Answer, 41 35 76 We shall write 41 on the blackboard, and 35 beneath it, and get a boy to add the numbers together, thus Then will come the questioning: "Forty-one and Thirty-five added together make"SEVENTY-SIX.' 97 "Then if 41 be taken from 76, how many will remain ?" Answer, "THIRTY-FIVE.” "Yes; and if 35 be taken from 76, how many will be left?" Answer, "FORTY-ONE." After some similar exercises will come, perhaps, this one: "If 52 be added to 27 what will be their sum? Answer, "SEVENTY-NINE.' "Then if 52 be taken from 79, how many will remain ?" Answer, "TWENTY-SEVEN." Yes; now when we mean to show that one number is to be taken from an79 52 other, we put the former number beneath the latter, thus: and we place the remainder beneath the lower number, thus, 27 NOTES.-1. Our next number will show how the Rule for Simple Subtraction is obtained from a consideration of the last Example mentioned in this paper. 2. In this paper we have used some technical words, such as Concrete, Abstract, &c. As every teacher, however young, ought to be familiar with these words, we refer those who are ignorant of their meaning to any good text-book of Arithmetic. |