DIVISION 



1817 



DIVISION 



Illustration of second case : 



1677 



274(459636 

 274 

 1856 ' 

 1644 

 2123 

 1918 



4X3 = 



2056 



1918 



138 



Excess is 3. (1+3+8) =9+3; excess of re- 

 mainder is 3. 3+3=6. Excess is 6. 4+5+9+ 

 6+3+6=3X9+6. Excess is 6. Excesses agree; 

 the quotient is correct. 



The 9's are cast out in a shorter way than 

 the work above indicates; for example, in the 

 number 459636, the dividend in the last prob- 

 lem, one sees 9 and cancels it; then sees 4 and 

 5, then 6 and 3, and cancels them thus 4"$$0$6, 

 leaving the excess 6. A.H. 



DIVISION, divizh'un. Division comes eas- 

 ily out of multiplication, as shown below: 



9X8 = 72 

 4X9 = 36 

 7X4 = 28 



In the group at the left, we use 8 nine times 

 and have 72; we use 9 four times and have 36; 

 we use 4 seven times and have 28. In the 

 group at the right, someone uses a certain 

 number of 9's and has 54; uses a certain num- 

 ber of 3's and has 27; uses a certain number 

 of 6's and has 48. 



Let it be the teacher who does this. She 

 says, "How many 9's did I use?" or "By what 

 did I multiply 9?" and the child, seeing it from 

 the multiplication view, answers "6." The 

 problem appears: 



6x9 = 54 



The same method brings answers to the oth- 

 ers and they appear: 



9X3 = 27 8X6 = 48 



The question "How many 9's did I use to 

 make 54?" easily becomes, "How many 9's are 

 in 54?" and we have division in form as well 

 as in thought. The problems now are: 



54-=-9=n 

 27-f-3 = n 

 48^-6 = n 



They should be read "How many 9's in 54?" 

 "How many 3's in 27?" and so on, just as the 

 multiplication form, nX9=54, is read. The 

 very technical expression, "54 divided by 9= 

 what?" should not be used too early, and even 

 when used later should alternate with the 



clearer, more suggestive question, "How many 

 9's in 54?" This should be used far along 

 into the middle grades. 



Much work of this kind will carry multipli- 

 cation right into division. We may write the 

 two views side by side: 



Multiplication 

 nx8 = 32 

 nx9 = 27 

 nx7 = 63 



7x8 = 56 

 4xn = 24 

 4X6 = 24 



Division 

 32 8=n 

 27 9 = n 

 63 7=n 

 35 5 = n 

 56-8 = n 

 56-8 = 7 

 24 4=n 

 24 4 = 6 



35,84.77.14.49.63,56.28.42.70 



With this the transition is made from multipli- 

 cation to division. It should be accompanied 

 by concrete illustrations; use should be made 

 of blocks, squared paper, etc. 



The use of areas 

 on squared paper 

 helps greatly at 

 this place, a s 

 shown in the il- 

 lustration which 

 appears herewith. 



Teachers should 

 give opportunity 

 for concrete divi- 

 sion to such an 

 extent that the 

 child realizes that 

 he is measuring a 

 quantity of length 

 or space or 

 weight, etc., by finding how many of a definite 

 measure or unit there are in the quantity 

 measured. When the child goes from the con- 

 crete problem to figures, he must see that the 

 dividend represents the quantity he is measur- 

 ing, the divisor represents the measuring unit, 

 and the quotient tells how many of these units 

 there are in the quantity he is measuring. 

 (Quotient means how many.) 



Home Helps. The parent has excellent op- 

 portunity to help the child at home in division. 

 Here there are many quantities to be measured 

 and the child may move about freely and 

 choose his material and discuss the measuring 

 process, and have father or mother help him 

 carry out his work. Some suggestions for home 

 work: 



(a) Room 19 feet by 18 feet. Measure with 

 a yard stick. The problem becomes 19-^-3= 

 6^; 18+3=6. 



(b) Mother buys 2 dozen eggs. She uses 6 

 for each day's breakfast. Child finds, by count- 



