Extended Multiplication and Division

Standard Algorithm

     65
  X34
   260    4 X 65 ( 4 groups of 65 )
 1950    30 X 65 ( 30 groups of 65 )
 2212

 

 Partial Product Algorithm

     197   (100 + 90 + 7))
    X68   (60 + 8)
       56    8 X 7    Multiply:
     720    8 X 90 
     800    8 X 100
     420    60 X 7
   5400    60 X 90
   6000    60  X 100  
 13396                Add:

 

 

Long Division Algorithm

       316R1
7 ) 2213
     21
       11
        7
         43
         42
           1

       316R1
7 ) 2213                                                        Or Shorten:
       700       100      ( 100 groups of  7 )
     1513
       700       100      ( 100 groups of  7 )
       813
       700       100      ( 100 groups of  7 )        or (3 X 100)
       113
        70         10      ( 10  groups of  7 )
         43
           7            1      ( 1 group of  7 )  From here to
         36                                              the bottom
           7            1      ( 1 group of  7 )   can be done
         29                                               in one
           7            1      ( 1 group of  7 )   subtraction
         22                                               step if it
           7            1      ( 1 group of  7 )   is a known
         15                                               fact.
           7            1      ( 1 group of  7 )  
           8
           7      +    1      ( 1 group of  7 )           or (6 X 7)
           1        316		 Interim Estimate Algorithm

The Interim Estimate Algorithm uses estimation to find how many sets of a number can be found in another number.

The assumption is that students understand division as a splitting up into equal groups action, that they can do several-digit subtraction arithmetic, and that they can multiply by a 1-digit number by multiples of 10s.

Start by removing the largest amount possible by multiplying the divisor (7) by a multiple of 10 (100 works in this case) and subtracting this (700) from the dividend (2213) which creates a new dividend (1513) and the process continues until the multiple of 100 (700) can no longer be subtracted. Then the next largest multiple of 10 is used ( in this case 10 ) is multiplied by the divisor giving us (70) to subtract and finally 1 times the divisor (7). When this is completed simply add up all the multiples of 10 to the side and place the answer in the quotient.

Terms:                     quotient
                    divisor ) dividend