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from the examples given ; also what is meant by so much in the shilling, so much in the pound, etc.,—that if a person spends twopence in the shilling in a particular way, and lays out two, three, ten shillings, he spends 4d., 6d., 20d., etc., in that particular thing.
A penny in the shilling is twenty-pence in the pound,
The whole expenditure of a family in a year is £A,' of which a per cent. is spent in bread, b in tea, c in clothes, d in house rent, e in taxes, etc., what part of the whole income is spent in each of these articles, and give an expression for the whole.
--= part of every £ spent in bread, and ---=what is spent in a year.
100 -- = the part of each £ in tea, and —- of the whole income. 100
100 In the same way — = that in taxes in the year.
Аа AB Ac
100 100 100
100 * And if the annual income of a family is £P per annum, P 700 (a + b +c+ etc.) will be the state of the pocket at the end of the year. When this expression is negative, it means they have exceeded their income. When it is = 0, they have just spent their income; and when it is positive, they have saved money.
A mass M of three metals, of which c per cent. is copper, s per cent. silver, and g per cent. gold; how much of each.
-- (c + s). 100
twenty pence in one pound is a hundred times that in a hundred pounds, and would be called so much per cent. The same in the common rule of three; they get into the way of stating their questions mechanically ; but what the
Suppose the mass 1000lbs., of which 25 per cent. copper, 40% per cent. silver, and the rest 342 gold: how much of each?" Here
M = 1000, c = 25, s = 40.5, etc.
-= 250lbs. of copper,
- =405 lbs. of silver.
-= 345 lbs. of gold.
100 The skilful teacher, who knows a little algebra may see a very extensive application of it in this way, and the satisfaction and instruction to a boy in being able to work out easy formulæ of this kind, and adapt them to particular cases, is beyond comparison greater than being taught by rules.
This makes it highly desirable that all our schoolmasters should be able to teach so much of the rudiments of algebra as to apply it to simple calculations of this kind. The merely being able to substitute numerical values for the different letters in an algebraical formula is of service.
For instance, that
(1.) (a + b)(a - b) = q?_62: that this means that the sum of two quantities multiplied by their difference is equal to the difference of their squares.
(2.) That (a + b)2=a+2ab +62, or that the square of the sum of two numbers is equal to the sum of their squares, increased by twice their product.
(3.) That (a−b)2=a? — 2ab + b2 = a” +62— 2ab, and the square of the difference of the numbers is equal to the sum of the squares of the two numbers diminished by twice their product. .
In each of these cases, let a = 6, and b=4; then (a+b) (a−b) would become (6 + 4) X (6—4), or 10 X 2 = the square of 6 or 36, diminished by the square of 4 or 16, or (62-4) = 20. (2.) (6 + 4) or 10o=62+42 +2 X 6 X 4.
or 36 + 16 + 48 = 100. That is, it is the same thing if you add the two numbers together, and square the sum, or square each number separately, add them, and to this add twice their product. (3.) (6—4)= 22 or 4 = 62 +42 -2 X 6 X 4.
or 36 + 16-48 = 4.
teacher should do, is, instead of saying as 1 yard: 2s., 6d., : : 50 yards to the answer; he should say, if one yard cost 2s. 6d. two yards will cost twice as much ; three yards three times ; 50 yards 50 times as much, having recourse to the commor-sense principle as much as possible.
The following questions, with those at the end of this section, may be useful to the teacher, as bearing upon the economic purposes of life, and will suggest others of a like kind:
The population of the parish in 1831 was 1,040 at the census of 1841 it had increased 7 per cent., what is it at present ?
In the population of the parish, 20 per cent. of them ought to be at school; in this parish, containing 1,040, only 12 per cent. are at school ; how many are at school? and how many absent who ought to be there?
The population of the county in 1841 was 355,004;82.8 per cent, were born in the county, 14.2 in other parts of England, 0:5 in Scotland, and 0.9 in Ireland; what number were born in each country ?-how many in number, and what per cent. are unaccounted for?
Give the average of the parish, how many to the square acre ; number of the houses, how many to a house, etc. These questions ought also to be the vehicle of a good deal of instruction on the part of the teacher.
A sheet containing the names of the towns in each county, arranged by counties, and giving in a tabular form the population in adjoining column, according to the census of 1831 and of 1841, is to be had for a shilling, and offers great facility to a master for making questions of this kind; as well as affording useful statistical information.*
* I have not seen a table of this kind made from the last census of 1851; but it is an educational want which ought to be supplied. In general, children, even of advanced ages, have no definite idea of the amount of population in the towns or villages which surround them, and will speak of hundreds where there are thousands, and of thousands where are hundreds. In the same way, they have no idea of their different bearings, and will tell you in set phrases, what is meant by latitude and longitude, who have no idea whether any particular town near which they live, is East, West, North or South of their own homes.
In teaching them superficial and solid measure, the fol. lowing mode is adopted:
They are first shown, by means of the black board, what a square inch, foot, yard, etc., is, by proofs which meet the eye; that a square of two inches on a side contains four square inches; of three inches on a side, nine square inches, and so on; or, in other words, that a square of one inch on a side, could be so placed on a square of two inches, as to occupy different ground four times, and in doing this it would have occupied the whole square, one of three inches, nine times : thus showing clearly what is meant by a surface containing a certain number of square feet, etc.
The same illustration with an oblong, say nine inches by two, three, etc., two or three drawings or diagrams of figures so divided are painted on the walls.
Solid Measure, The teacher takes a cube of four inches on a side, divided into four slices of one inch teick, and one of the surfaces divided into sixteen superficial inches; to this slice of one inch thick, containing sixteen solid inches, add a second, that will make 32, and so to the fourth, making 64; so that they now have ocular proof so simple, that they must understand ; that the superficial inches in a square, or rectangle, is found by multiplying together the number in each side ; the contents of a regular cube by multiplying the number of superficial inches on one side by the number of slices.
To apply this :
The master tells one of the boys to take the two-foot rule (a necessary thing in a village school), measure the length and breadth of the school-room. Yes, sir,
Length 26 feet, breadth 16 feet. What is the figure ? An oblong-sides at right angles to each other. Multiply length and breadth — what is the area ?
To another -- Look at the boards of the floor; are they uniform in width ? How are they laid ? Parallel to each other. The breadth of the room you have got, and, as the boards are laid that way, you have the length of each board; measure the width of a board. Nine inches, Reckon the number of boards. What is the area of the room? Does
it agree with your first measurement ? If not, what is the source of error; the boards will turn out to be unequal in width,
The door — what is the shape of the opening ? An oblong, with one side a good deal longer than the other. Measure the height — the width : now what number of inches of surface on the door ?
The rule again. Measure the thickness. Now how many solid inches ?
The door-posts. Measure the height, width ; now the depth. How many solid inches of wood in one post ? How many in the whole door-posts? How many solid inches in a foot ? Turn it into feet.
In the same way they may apply the rule to find out the surface of a table, a sheet of paper, surface of a map, a page of a book, etc., but always making them do the actual measurement, first taking one child, then another. 1
Again the room — we have got the area — tell us how much water it would hold, if we could fill it as high as the walls; we have got two dimensions, what is wanting? The height. We cannot reach up, sir. — Take your rule. Measure the thickness of a brick with the mortar.— About four inches..., Measure the first three courses.—A foot, sir. -Reckon the courses of the wall.-Thirty-six.—Then the height is what?-- Twelve feet. Now find out the solid contents of the room.
Find the surface and solid contents of a brick.
In fact, the two-foot rule is to the village school what Liebig says the balance is to the chemist.
Another practical application, which works well in giving fixed ideas of linear measure, is the following:
Take a hoop, say of two feet diameter; apply a string to the circumference; measure it.-Rather over six feet. Another of three will be found to be nine, and by a sort of inductive process, you prove that the circumference is three times the diameter; when farther advanced, give them the exact ratio, 3•14159, which they will work from with great facility. That a child should feel and understand this mode of inductive reasoning is very important, and is one of the most useful school-lessons he can have.