JAN. 21.] THE NEW ZEALAND GAZETTE. 279

thousands. Simple and compound rules (money), multipliers and divisors not to exceed 99, multipliers if over 12 to be reducible to factors not over 12; sums of money in the questions and answers not to exceed £200. It is recommended that the principle of complementary addition should be extended to long division

Work of Standard II. applied to higher numbers.

In teaching simple multiplication by higher multipliers than 12, the first exercise should involve multiplication by 20, 30 .... 90, and the difference explained (in a concrete manner at first) between the results thus obtained and those obtained by multiplying by 2, 3 .... 9. Even after this point is understood, frequent reference to concrete illustrations will be desirable until the pupil begins habitually to visualise the process and the result. For this reason also the numbers should be as small as can be employed to illustrate the process. Then teach multiplication by numbers 13 .... 99: e.g., by 86, i.e., 80 times + 6 times.

The first exercises in long division should be as simple as possible: e.g., 26 ÷ 13, 260 ÷ 13, 2,600 ÷ 13; and so on.

Simple multiplication by factors should precede compound multiplication by factors.

STANDARD IV.

Numeration and notation up to 1,000 millions. Long multiplication of money; reduction of money and of the weights and measures named below; simple practice, and the making out of easy bills of accounts and receipts such as occur in ordinary retail transactions. Tables of money, avoirdupois weight, long measure (excluding poles or perches), square measure (excluding square poles or perches and roods), capacity (pint, quart, gallon, peck, bushel, quarter), time, angular measure. Mensuration—to find the area of a square and of a rectangle with given sides, expressed in one denomination only (as in inches, feet, or yards, but not in feet and inches, &c.): this should be demonstrated by making each child draw and cut out a square and a rectangle with a given integral number of inches in each side, and then fold or rule the paper so as to show the number of square inches; the principle may be extended to square feet on the floor of the class-room and to square yards in the playground. (See also clause 56.) The meaning of proper fractions, with denominator not greater than 20, is to be known, and applied to concrete examples in a simple manner—e.g., ⅔ of £4 10s.: ⅓ to be found first, and ⅔ to be shown to be three times the result.

Mental arithmetic and problems to be adapted to this stage of progress.

STANDARD V.

Simple proportion by the unitary method; practice and harder bills of accounts; the easier cases of vulgar fractions (excluding complex fractions); shorter methods in working sums required in earlier standards; easy direct percentages (e.g., to find the simple interest for a year or for a given number of months, but not years and months or days). The meaning of ·1, ·2, &c., is to be known as one-tenth, two-tenths, &c.; that of ·01, ·02, &c., as one-hundredth, two-hundredths, &c.; of ·11, ·12 .... ·49 .... ·95 .... ·99, to be known as one-tenth and one-hundredth, or eleven-hundredths, &c.; that of ·001, ·002, &c., as one-thousandth, two-thousandths, &c., and applied to concrete examples in a simple manner—e.g., the value of ·1, ·15, ·2, ·25 .... ·999 of a pound sterling or of a ton should be understood. The pupils should be able to express money and common weights and measures in decimal forms (to three or four places of decimals), and to work very easy sums thereby. Mensuration of walls and floors, and other simple rectangular areas, as far as possible from actual measurements. The relative values of the cubic foot and cubic inch, and of the cubic yard and cubic foot: these should be actually demonstrated by models, which in the former case may be made of cardboard, carton, or cartridge paper, and in the case of the latter of wood or tin. Relative values of the kilometer, meter, decimeter, centimeter, and approximate equivalents in yards and inches. Value of kilogram and gram.

STANDARD VI. (See also Standard VII.)

Vulgar and decimal fractions (excluding sums in recurring decimals, but the meaning of simple recurring decimals should be known); percentages applied to simple examples—e.g., to find simple interest for a year, months, or days; to find rate or time when one of them and principal and interest or principal and amount are given, but not to find principal when rate, time, and amount are given; to find profit or loss when cost price and rate per cent. are given, or to find rate when