(_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. [Illustration: 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:-- [Illustration: FIG. 58.] How long is this rectangle? How large is each square? How many square inches are there in the top row? How many rows are there? How many square inches are there in the whole rectangle? Since there are three rows each containing 4 square inches, we have 3 × 4 square inches = 12 square inches. Draw a rectangle 7 inches long and 2 inches wide. If you divide it into inch squares how many rows will there be? How many inch squares will there be in each row? How many square inches are there in the rectangle? [Illustration: FIG. 59.] It is better actually to hide the individual square units as in Fig. 59. There are four reasons: (1) The concrete rows and columns rather distract attention from the essential thing to be learned. This is not that "_x_ rows one square wide, _y_ squares in a row will make _xy_ squares in all," but that "by using proper units and the proper operation the area of any rectangle can be found from its length and width." (2) Children have little difficulty in learning to multiply rather than add, subtract, or divide when computing area. (3) The habit so formed holds good for areas like 1-2/3 by 4-1/2, with fractional dimensions, in which any effort to count up the areas of rows is very troublesome and confusing. (4) The notion that a square inch is an area 1' by 1' rather than 1/2' by 2' or 1/3 in. by 3 in. or 1-1/2 in. by 2/3 in. is likely to be formed too emphatically if much time is spent upon the sort of concrete presentation shown above. It is then better to use concrete counting of rows of small areas as a means of _verification after_ the procedure is learned, than as a means of deriving it.