State the missing numbers:—
| 8 = .... 4s | 12 = .... 6s | 9 = .... 9s |
| 8 = .... 2s | 12 = .... 4s | 9 = .... 3s |
| 8 = .... 1s | 12 = .... 3s | 9 = .... 1s |
| 8 = .... 1⁄2s | 12 = .... 2s | 9 = .... 1⁄3s |
| 8 = .... 1⁄4s | 12 = .... 1s | 9 = .... 1⁄9s |
| 8 = .... 1⁄8s | 12 = .... 1⁄2s | |
| 12 = .... 1⁄3s | ||
| 12 = .... 1⁄4s |
| 16 ÷ 16 = | 9 ÷ 9 = | 10 ÷ 10 = | 12 ÷ 6 = |
| 16 ÷ 8 = | 9 ÷ 3 = | 10 ÷ 5 = | 12 ÷ 4 = |
| 16 ÷ 4 = | 9 ÷ 1 = | 10 ÷ 1 = | 12 ÷ 3 = |
| 16 ÷ 2 = | 9 ÷ 1⁄3 = | 10 ÷ 1⁄5 = | 12 ÷ 2 = |
| 16 ÷ 1 = | 9 ÷ 1⁄9 = | 10 ÷ 1⁄10 = | 12 ÷ 1 = |
| 16 ÷ 1⁄2 = | 12 ÷ 1⁄2 = | ||
| 16 ÷ 1⁄4 = | 12 ÷ 1⁄3 = | ||
| 16 ÷ 1⁄8 = | 12 ÷ 1⁄4 = | ||
| 12 ÷ 1⁄6 = |
(8) '% of' means 'hundredths times.'—In the case of percentage a series of bonds like the following should be formed:—
| 5 | percent | of | = .05 times | |
| 20 | " " | " | = .20 " | |
| 6 | " " | " | = .06 " | |
| 25 | % | " | = .25 × | |
| 12 | % | " | = .12 × | |
| 3 | % | " | = .03 × |
Four five-minute drills on such connections between 'x percent of' and 'its decimal equivalent times' are worth an hour's study of verbal definitions of the meaning of percent as per hundred or the like. The only use of the study of such definitions is to facilitate the later formation of the bonds, and, with all save the brighter pupils, the bonds are more needed for an understanding of the definitions than the definitions are needed for the formation of the bonds.
(9) Habits of verifying results.—Bonds should early be formed between certain manipulations of numbers and certain means of checking, or verifying the correctness of, the manipulation in question. The additions to 9 + 9 and the subtractions to 18 − 9 should be verified by objective addition and subtraction and counting until the pupil has sure command; the multiplications to 9 × 9 should be verified by objective multiplication and counting of the result (in piles of tens and a pile of ones) eight or ten times,[4] and by addition eight or ten times;[4] the divisions to 81 ÷ 9 should be verified by multiplication and occasionally objectively until the pupil has sure command; column addition should be checked by adding the columns separately and adding the sums so obtained, and by making two shorter tasks of the given task and adding the two sums; 'short' multiplication should be verified eight or ten times by addition; 'long' multiplication should be checked by reversing multiplier and multiplicand and in other ways; 'short' and 'long' division should be verified by multiplication.
These habits of testing an obtained result are of threefold value. They enable the pupil to find his own errors, and to maintain a standard of accuracy by himself. They give him a sense of the relations of the processes and the reasons why the right ways of adding, subtracting, multiplying, and dividing are right, such as only the very bright pupils can get from verbal explanations. They put his acquisition of a certain power, say multiplication, to a real and intelligible use, in checking the results of his practice of a new power, and so instill a respect for arithmetical power and skill in general. The time spent in such verification produces these results at little cost; for the practice in adding to verify multiplications, in multiplying to verify divisions, and the like is nearly as good for general drill and review of the addition and multiplication themselves as practice devised for that special purpose.
Early work in adding, subtracting, and reducing fractions should be verified by objective aids in the shape of lines and areas divided in suitable fractional parts. Early work with decimal fractions should be verified by the use of the equivalent common fractions for .25, .75, .125, .375, and the like. Multiplication and division with fractions, both common and decimal, should in the early stages be verified by objective aids. The placing of the decimal point in multiplication and division with decimal fractions should be verified by such exercises as:—
|
20 1.23 ) 24.60 246 | It cannot be 200; for 200 × 1.23 is much more than 24.6. It cannot be 2; for 2 × 1.23 is much less than 24.6. |