DIVISION 1819 



The child may separate it thus: 



411732 = 1600 + 100 + 32 



400+ 25+ 8 = 433 



By separating it in several ways, each of which 

 brings the correct number of 4's in .1732, he 

 will come to see that there is one way that 

 follows the lines of construction, and that this 

 particular method serves his purpose best. He 

 learns in this to look for a multiple of the 

 divisor in the first part of the dividend and to 

 look intelligently, a point that is of vital im- 

 portance in division, and does away with the 

 "wild guessing" so familiar to teachers. 



After much of this work, the child becomes 

 independent in both short or long division, 

 and sees clearly what he is doing. He sees 

 that he is finding how many times a certain 

 measure (the divisor) is contained in the un- 

 measured whole (the dividend), and he finds 

 it by thinking of the dividend as a product. 

 This method of separating the dividend into 

 its parts is much more effective than the usual 

 method of placing the remainder after each 

 division as a small figure in front of the next 



4I74312 



digit, and reducing, as J , because the eye 



183 



helps and the child sees that 732=400+320+12. 

 The latter meaning is not clear in the other 

 method. The concise method comes later, and 

 the child divides without any devices, in the 



desired form, as 



41732 



183 



He sees the 400, the 



32 (or 320) and the 12, while in the quotient 

 he sees 100, 80 and 3. 



Long Division. We need this process when 

 a problem in division becomes too complex 

 to carry in the mind. The greater the power 

 of registration of the mind, the less often we 

 have recourse to long division. The mind 

 should be led to solve problems by short divi- 

 sion beyond the divisor 9 or 12; for example, 

 7525-^25, 6500+13, 62500+25, 950-M9, 72480+ 

 24, 850+170, 840+21, 6946-^23. 



A divisor consisting of tens and units is 

 simpler when the units digit is small; for ex- 

 ample, 91 is a simpler divisor than 38; 72 than 

 49 and so on. This should be considered care- 

 fully in approaching long division. While the 

 units digit remains small, the quotient figure 

 is determined largely from the tens digit, and 

 the eye helps more in the process. 



After the preceding work in short division 

 the child is ready to look at the more difficult 

 numbers along the lines of construction. He 

 readily thinks of the parts of the dividend, and 



DIVISION 



the parts in the divisor; that is, he realizes 

 the units, tens, hundreds, etc., of each number, 

 as follows: 



64x21=n 

 21 



1200 = 20X60 



80 = 20X 4 



60= 1X60 



_ 4= IX 4 



1344 



1344-=-21=n 



60+ 4 

 11344 = 1200 + 140 + 4 



1200+_60= ( 60 X 20) + ( 60 X 1 ) 

 80 + 4 



=(4X20) + (4X1) 



Here he notes the corresponding parts in the 

 division and the multiplication. Another illus- 

 tration : 



523X42 



42 



20000 



800 



120 



1000 



40 



6 



21966 



21966-M2 = n 



500+ 20+ 3=523 

 40+2120000 + 1000+800 + 120+46 

 20000 + 1000 



(500X40) +(500X2) 



800 + 120+46 



800 + 40 = (20 X40) + (20 X2) 



120+ 6 



120+ 6= (3X40) +(3X2) 



57188-=- 58 = n (Note the units digit is large.) 



986 

 X58 



45000 



4000 



300 



7200 



640 



48_ 



57188 



900+ 80+ 6 = 986 

 50 + 8145000 + 7200 + 4000 + 940 + 48 



45000 + 7200 



4000 + 940 + 48 

 4000 + 640 



300 + 48 

 300 + 48 



Here the child sees clearly that he must con- 

 sider the units digit of the divisor, and sees 

 it is an element to be reckoned with in finding 

 each quotient figure. 



Now we bring the process gradually to a 

 more and more concise form, taking care, how- 

 ever, that the pupil understands the reason for 

 the omission of each step. 



