Suppose that it is the 35 square feet. Solutions by (5 × 144) + (30 × 144), however arranged, or by (10 × 144) + (10 × 144) + (10 × 144) + (5 × 144), or by 3500 + (35 × 40) + (35 × 4), or by 7 × (5 × 144), however arranged, should all be listed for verification or rejection. The pupils need not be required to justify their procedures by a verbal statement. Answers like 432,720, or 720,432, or 1152, or 4220, or 3220 should be listed for verification or rejection. Verification may be by a mixture of short multiplication and objective work, or by a mixture of short multiplication and addition, or by addition abbreviated by taking ten 144s as 1440, or even (for very stupid pupils) by the authority of the teacher. Or the manipulation in cases like 53 × 9 or 84 × 7 may be verified by the reverse short multiplication. The deductive proof of the correctness of the manipulation may be given in whole or in part in connection with exercises like

10 × 2 =30 × 14 =
10 × 3 = 3 × 44 =
10 × 4 =30 × 44 =
10 × 14 = 3 × 144 =
10 × 44 =20 × 144 =
10 × 144 =40 × 144 =
20 × 2 =30 × 144 =
20 × 3 = 5 × 144 =
30 × 3 =35 = 30 + ....
30 × 4 =30 × 144 added to 5 × 144 =
3 × 14 =

Certain wrong answers may be shown to be wrong in many ways; e.g., 432,720 is too big, for 35 times a thousand square inches is only 35,000; 1152 is too small, for 35 times a hundred square inches would be 3500, or more than 1152.

The time spent in realizing the problem here is fairly well spent because (1) any successful original manipulation in this case represents an excellent exercise of thought, because (2) failures show that it is useless to juggle the figures at random, and because (3) the previous experience with short multiplication makes it possible for the pupils to realize the problem in a very few minutes. It may, however, be still better to give the pupils the right method just as soon as the problem is realized, without having them spend more time in trying to solve it. Thus:—

1 square foot has 144 square inches. How many square inches are there in 35 square feet (marked out in chalk on the floor as a piece 10 ft. × 3 ft. plus a piece 5 ft. × 1 ft.)?

1 yard = 36 inches. How many inches long is this wall (found by measure to be 13 yards)?

Here is a quick way to find the answers:—

144
35
——
720
432
——
5040 sq. inches in 35 sq. ft.
36
13
——
108
36
——
468 inches in 13 yd.

Consider now the following introduction to dividing by a decimal:—

Dividing by a Decimal