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Lesson I

Multiplication by One
The Rule:
write the number

Simple enough, 25 times 1 is 25. Below are the steps toward achieving the solution to the problem; not to insult your intelligence but to illustrate the format used throughout this series. 

0 2 5  x  1
-----
    5
write the number. You should always write your answers thusly so that errors which may occur in your work may be quickly found. If you are writing on paper it is preferable that you use a pencil and graph paper, writing one digit in every-other square. 
0 2 5  x  1
-----
  2 5
write the number. 
0 2 5  x  1
-----
0 2 5
writing the number from under the zero position will keep you in good practice for later lessons.

Multiplication by Ten
The Rule:
write the neighbour

This is another way of looking at the commonly taught 'add a zero to the multiplicand,' which better fits in with the simplified rules of this multiplication system. Take a look at the simple example of 425 times 10. 

0 4 2 5  x  10
-------
      0
remember that the "neighbour" when one is looking at far right digit of the multiplicand is not "nothing", but rather is considered to be an unwritten zero. 

0 4 2 5  x  10
-------
    5 0
write the neighbour. 
0 4 2 5  x  10
-------
  2 5 0
write the neighbour. 
0 4 2 5  x  10
-------
4 2 5 0
write the neighbour.

Multiplication by Eleven
The Rule:
add the neighbour

Notice that this rule says 'add' instead of 'write' meaning 'add the neighbour to the number. A simple combination of the rules used for multiplication by one and by ten. Once again this is a simplified restatement of a method you may have already learned which is to 'multiply the multiplicand by ten and add the multiplicand.' So now let's try the simple equstion 75632 times 11. 


0 7 5 6 3 2  x  11
-----------
          2
see the 2, say [ 2 ], add the neighbour (zero).
Your mental stops should have been: [ 2 ] answer: 2. 

0 7 5 6 3 2  x  11
-----------
        5 2
see the 3, say [ 3 ], add the neighbour (2) and say [ 5 ].
Your mental stops should have been: [ 2, 5 ] answer: 5. 
0 7 5 6 3 2  x  11
-----------
      9 5 2
see the 6, say [ 6 ], add the neighbour (3) and say [ 9 ].
Your mental stops should have been: [ 6, 9 ] answer: 9. 
0 7 5 6 3 2  x  11
-----------
   `1 9 5 2
see and say [ 5 ], add the neighbour (6) and say [ 11 ]. Carry the one.
Your mental stops should have been: [ 5, 11 ] answer: 11.
Notice how easy it is to not only write but also to remember the carry when using this style of writing out the problem with spaces between the numbers. 
0 7 5 6 3 2  x  11
-----------
 `3`1 9 5 2
see the carry and the 7, and say [ 8 ], add the neighbour (5) and say [ 13 ]. Carry the one.
Your mental stops should have been: [ 8, 13 ] answer: 13. 

0 7 5 6 3 2  x  11
-----------
8`3`1 9 5 2
see the carry and the zero, and say [ 1 ], add the neighbour (7) and say [ 8 ].
Your mental stops should have been: [ 1, 8 ] answer: 8.
Thus each figure of the multiplicand is used twice; once as a number and once as a neighbour. There is a special case when multiplicands beginning with 9 followed by another large figure, the product may have a 10 in the last step. Try the next one yourself; multiply 98,325 times 11. The answer is 1,081,575.

Multiplication by Two
The Rule:
double the number

Another simple concept, but one well used in this system and worth your attention. Whenever the rule says 'double' for a step, you should try to look at the value represented and double it instantaneously (ex. if you are to double 6, do not say to yourself 'six times two is twelve' or 'six and six is twelve,' instead say 'six, twelve.') at first you may wish to practice by saying the value and then its double, but your goal should be to reduce even that to the single step of saying the double (look at the 6 and say 'twelve').

Once again I have provided a sample problem, this time to illustrate the proper mental method with regards to the carry. Try 984 times 2.


0 9 8 4  x  2
-------
      8
double the number. 
0 9 8 4  x  2
-------
   `6 8
double the number. 
0 9 8 4  x  2
-------
 `9`6 8
double the number and add the carry. Do not say [ 9, 18, 19 ]; see the carry mark and say [ 9, 19 ], or as you become better at this form of doubling, simply [ 19 ]. 
0 9 8 4  x  2
-------
1`9`6 8
double the number (zero) and add the carry.

Multiplication by Twelve
The Rule:
Double the number and add the neighbour

This is the same as multiplying by 11 except that now you use the new rule from multiplication by two and double the number before adding its neighbour. Take an easy example 252 times 12: 

0 2 5 2  x  12
-------
      4
see the number (2) and say its double [ 4 ], and add the neighbour (zero).
Your mental stops should have been: [ 4 ] answer: 4. 
0 2 5 2  x  12
-------
   `2 4
see the 5 and say its double [ 10 ], then add the 2 and say [ 12 ]. Carry the one.
Your mental stops should have been: [ 10, 12 ] answer: 12. 
0 2 5 2  x  12
-------
 `0`2 4
see the 2 and the carry, say [ 5 ], then add the 5 and say [ 10 ]. Carry the one.
Your mental stops should have been: [ 5, 10 ] answer: 10. 
0 2 5 2  x  12
-------
3`0`2 4
see the zero and the carry, say [ 1 ], then add the 2 and say [ 3 ].
Your mental stops should have been: [ 1, 3 ] answer: 3.
The product of 252 times 12 is 3,024. Now try 65535 x 12 on your own. The answer is 786,420.

Some practice in proper mental methods

Practice looking at a number (8 for example) and saying its double... try that with each of the numbers in the following list:

2, 5, 3, 8, 5, 3, 6, 9, 5, 3, 6, 8, 9, 6, 4, 2,
9, 8, 1, 0, 8, 6, 5, 4, 2, 6, 7, 4, 3

Great! Now practice looking at the number, and adding the neighbour... this is how you multiply by eleven.

2 5 3 8 5 3 6 9 5 1 4 8
9 6 1 2 4 0 9 5 8 6 5 4
2 6 8 9 7 4 3 1 6 3 5 8
3 5 1 6 6 5 9 4 5 2 3 6

Finally, using the same set of numbers, practice seeing and doubling the number, and adding the neighbour... which is, as you know, how you multiply by twelve.

enjoy...

Lesson 2