Take a rectangular sheet and rule it out as far as you can into inch squares. Cut off the strips left over, and state the relation between the area, the length, and the breadth of the rectangle you now have. The area in square inches is the length in inches multiplied by the breadth in inches. Does this result depend on the particular length and breadth, or would it be true for others? And if the length is 7 inches and the breadth b inches, what is the area? It is lxb square inches, provided 7 and b are whole numbers. 3. Let us now replace the strips that were cut away from the rectangle and try to find the area of the original sheet of paper. No inch-squares can be marked out on these strips, and we must use a smaller unit. Let us take as smaller unit a strip such that ten of them placed side by side form an inch square. Mark out this strip-unit as often as you can on the strips that were cut away. If any area still remains that has not been included take a still smaller unit, say a little square, ten of which will make a stripunit. Thus, for instance, if the sheet of paper measures 12.7 inches by 8.2 inches, the area consists of 12 × 8 inch squares, 12 x 28 x 7 strip-units, 7 × 2 of the smaller squares. How many strip-units are there?-80. were given the expression 12x2+8x7 If you without knowing how it was arrived at, would its meaning be clear?—No, unless there is an agreement about the order in which the addition and multiplication are to be performed; if we began by adding the 2 and the 8, the expression would mean 840. The expression then may mean the sum of the two products 12 × 2 and 8x7, or it may mean the product of the three quantities 12, 2+8, and 7. Brackets may be used to avoid ambiguity; thus (12×2)+(8×7) means the sum of the two products 12 × 2 and 8×7, while 12x(2+8)x7 means the product of the three numbers 12, 2+8, and 7. There can be no doubt about the meaning of either of these, but we can save ourselves some labour in writing if we agree to confine the brackets to one of the two cases. Let us agree that in the absence of brackets or other indication multiplication and division are to precede addition and subtraction. Then when we meet 12x2+8x7 we know that we must first multiply 12 by 2 and 8 by 7 and then add the results. So that with this agreement the expression as it originally stood means 80 without any ambiguity, and if we want to make it mean 840 we must use a bracket. 4. Now express in square inches and fractions the area of the sheet of paper. It is 12 × 8 + (12 × 2 + 8×7) × 10 + 7×2×180 square inches. (Note the use of the bracket; what would this expression with the bracket left out mean?) This expression, which is now recognizable as 12.7 × 8.2, has the value 104.14. For most purposes it is sufficient to give the value as 104, and if the length and breadth of the sheet are only approximate, being measured to the nearest tenth of an inch, the form 104-14 has a misleading appearance of accuracy, and even 104 is not reliable in the unit figure. If you adopted as unit from the beginning the small square 01 of an inch in the side, how many of these units would you have in the rectangle? And what is their equivalent in square inches? Ex. 1. Make a loop of string, lay it on squared paper, and find the area it encloses by counting squares. Move the loop about and pin it out, to make it enclose as great an area as possible. What does the shape of the loop then look like ? 5. Now generalize the formula already obtained for the area of a rectangle in terms of the length and breadth.The length being 7 inches and the breadth b inches the area is lxb square inches, where l and b may contain fractions. Are these fractions limited to be tenths? Or to be decimals? And what is the formula if the dimensions are given in centimetres ? 6. What is the area of a rectangle measuring a inches and b tenths by c inches and d tenths?—It consists of inch squares, that is, ахс axc + axdx 10 + bxcx 10 + bx dx 100 square inches. This form is clumsy, and a shorter way of writing it is desirable. One a little better is but even this may be improved on. Would it do to drop the sign of multiplication and write ac to mean a multiplied by c? It would not do for numerals; we cannot drop the sign in 12 x 8, for it would then be 128, that is, one hundred and twenty-eight. May we drop it except between numerals?—Yes. A system is a little illogical that adopts different conventions for letters and numerals, and this difference of convention may cause difficulty. However, these conventions are universal and very useful, and pupils that realize the difference, and realize that both conventions are no more than conventions, will find no difficulty. We agree, then, that the sign x may be dropped when no ambiguity results from dropping it, and we write the number of square inches in the rectangle as ac + ad + bc10 + bdito, Area of a right-angled triangle. 7. Suppose that the playground that we want to measure is shown in Fig. 1, on a scale of 1 to 1000. We saw one way of measuring its area, by marking out a foot square as often as possible. Can we now shorten this work?—We could mark out a few big rectangles on the playground, measure their length and breadth, and so find their But some pieces area. will remain over, and what FIG. 1. Take a rectangular sheet of paper and cut it along a diagonal, that is, along the line joining two opposite corners. What kind of figures have you now?-Two rightangled triangles. Which of them is greater?-Fitting together shows them equal experimentally, that is, equal as far as observation will tell us. We will discuss later whether two triangles obtained by a cut accurately along a diagonal of an accurate rectangle will fit accurately, that is, whether the triangles are mathematically equal. What do you know of the area of each triangle ?-It is half that of the rectangle. Now draw any right-angled triangle. Is there any rectangle of which this triangle is half? Draw this rectangle. Express the area of the rectangle in terms of the sides of the triangle. If the two sides of the triangle that meet at the right angle are p cm. and q cm, long, the area of the rectangle is pxq sq. cm. And the area of the triangle itself? -It is px q÷2 sq. cm. 8. Take again one of the triangles into which you cut the sheet of paper. Call the corner at the right angle A and the others B and C(Fig. 2). Fold B on to A and fold Con B FIG. 2. A F to A, forming creases DE and FE. Measure the sides and angles of the figure ADEF. What is the figure ADEF? And As far as observation tells it is a rectangle. mathematically what do you know of the figure ADEF?— That the folding makes the angles at D and F right angles, and makes AD half of AB and AF half of AC, while A was supposed a right angle from the beginning. Later we will discuss whether this is enough to show ADEF a rectangle. Meantime, the sides AB and AC being p and q cm. long, what is the area of the rectangle ADEF? It is When the corner B of the triangle ABC is folded in on A, is the piece of paper still a triangle ?—No. Why? -Because the three boundaries are no longer straight, and a triangle is bounded by three straight lines. When B and C are folded in on A, how does the piece of paper lie?—By observation it is made up of the rectangle ADEF and the folded-in parts which just cover the rectangle. Whether this is so mathematically we shall see later. Then what is the area of the triangle ?—It is twice the rectangle, or pq sq. cm. Thus our previous result is confirmed. 9. How much of the playground in Fig. 1 are we now in a position to measure ?— All but the triangle Q. Can you suggest a method of dealing with the triangle Q? Can you divide it into right-angled triangles ?—Yes, by a perpendicular from a corner on the opposite side. You are now in a position to find the area of the playground of Fig. 1, Make the necessary measurements of the figure, and, remembering that every length on the figure is 0.001 of the true length, find its area in square feet or in square metres. FIG. 3. If your own playground is simple enough to treat similarly measure it up and find its area. 10. Could you now find the area of a playground such as Fig. 3 shows? Could you divide it into triangles? |