(a) Very large numbers, such as 1000, 10,000, 100,000, and 1,000,000, need more concrete aids than are commonly given. Guessing contests about the value in dollars of the school building and other buildings, the area of the schoolroom floor and other surfaces in square inches, the number of minutes in a week, and year, and the like, together with proper computations and measurements, are very useful to reënforce the concrete presentations and supply genuine problems in multiplication and subtraction with large numbers.
(b) Numbers very much smaller than one, such as 1⁄32, 1⁄64, .04, and .002, also need some concrete aids. A diagram like that of Fig. 57 is useful.
(c) Majority and plurality should be understood by every citizen. They can be understood without concrete aid, but an actual vote is well worth while for the gain in vividness and surety.
Fig. 57.—Concrete aid to understanding fractions with large denominators.
A = 1⁄1000 sq. ft.; B = 1⁄100 sq. ft.; C = 1⁄50 sq. ft.; D = 1⁄10 sq. ft.
(d) Insurance against loss by fire can be taught by explanation and analogy alone, but it will be economical to have some actual insuring and payment of premiums and a genuine loss which is reimbursed.
(e) Four play banks in the corners of the room, receiving deposits, cashing checks, and later discounting notes will give good educational value for the time spent.
(f) Trade discount, on the contrary, hardly requires more concrete illustration than is found in the very problems to which it is applied.
(g) The process of finding the number of square units in a rectangle by multiplying with the appropriate numbers representing length and width is probably rather hindered than helped by the ordinary objective presentation as an introduction. The usual form of objective introduction is as follows:—
