MULTIPLICATION 



4000 



MULTIPLICATION 



(4) 456 = 400 + 50 + 6 456 



X7 7 



2800 42 



350 350 



4j2 2800 



3l?2 3192 



Lastly, carry in mind the numbers to be 

 added to tens and hundreds, and work in the 

 most concise form; but do not, in teaching 

 young children, go too rapidly to the carrying 

 or reducing process. Let there be much of the 

 work written out in full in the early lessons, 

 as shown above. The final form is : 



456 



X7 



3192 



By placing a zero at the right, of a digit, the 

 digit is moved one place to the left, and the 

 number it symbolizes is multiplied by 10, as 

 9, 90; 6, 60. The 9 and the 6 are moved from 

 units place to tens place. So to multiply an 

 integer by 10, 100, etc., annex one, two, or more 

 zeros (see DECIMAL FRACTIONS). 



756X10=7560. Before the zero was annexed, 

 6 was 6 ones, 5 was 5 tens or 50 ones, 7 was 7 

 hundreds; now 6 is 6 tens or 60; 5 is 5 hun- 

 dreds, and 7 is 7 thousands. The number was 

 700+50+6; it is now 7000+500+60. See that 

 annexing a zero changes the place of each digit, 

 and causes it to signify 10 times as much as it 

 did before. Show what happens in each of the 

 following: 



75X100 = 7500; 846X1000 = 846000 



Multiplying when there nre zeros In the 

 multiplicand 



(1) (a) 2X304=(2X300) + (2X4)=600 + 8 = 608 

 (b) 304 (c) 304 (d) 304 



2 2 2 



600 8 ' 608 



8 600 



608 608 



Below, in (a) and (b), are shown all partial 

 products. 



(a) 3X4060=(3X4000) 

 180 = 12180 



(b) 4060 



3 



12000 



180 



12180 



(c) 4060 



3 



180 

 12000 

 12180 



(3X60)=12000 + 



(d) 4060 

 3 



121SU 



Multiplying by a number of two digit* 



23X!23=n 

 123 = 100 + 20 + 3 

 23= 20 + 3 



In (c) three partial products are combined in 

 369, and three in 2460. In (d) the zero is 

 dropped from 2460, and we have the usual form 

 of multiplication. The full expression of the 

 products, as shown in (a) and (b), helps the 

 young student to get the meaning of multipli- 

 cation, and should precede the more concise 

 form, which tends to disguise the meaning. 



3008X9863 =(3000X9863) + (8X9863) 



(a) 9863 

 3008 



29589000 = 3000X9863 

 78904 8X9863 



29667904 



(b) 9863 

 3008 



78904= 8X9863 

 29589000 = 3000X9863 

 29667904 



In (b) we see that the three zeros at the end 

 of the second partial product need not be put 

 down if the digits 9, 8, 5, 9 and 2 are put in 

 their proper places. So we may have the prob- 

 lem in the usual form: 



29667904 



In teaching the child multiplication, there 

 should be much work done in which the zeros 

 are set down, showing what the real problem 

 is. Through this he will the more quickly see 

 why they need not be expressed, and find the 

 proper place for the first digit. 



Testing Multiplication. The child may use 

 the two methods, one as a test for the other. 

 He may multiply by the usual method, and test 

 the work by employing the method in which 

 he writes partial products in full, or he may 

 work by the "long" method and test by the 

 usual method. It is a safer method of testing 



