This is Math Newsletter number 1; Wednesday, July 28, 2010.
If you send any part of this newsletter to a friend, please
also include this paragraph. To subscribe or unsubscribe,
send a personal email request to kermit@polaris.net
Archive is at
http://www.kermitrose.com/math/NewsLetter/LetterList.html


Multiplication Table


Along about third grade, some kids are turned away
from mathematics by their teacher insisting that
they "memorize" the multiplication table.

Some teachers are wise enough to teach the kids to
"count by two's", "count by three's", etc,
rather than insist on their memorizing the table.

Intentional memorization is an inferior method of
accumulating knowledge. It is much better to involve more
parts of your brain by pursuing a worthy goal. Develop skill
in arithmetic by calculating and re-calculating the products
that you need to know. This will enable a stronger more
reliable memory than will intentional memorization.

I strongly suggest that you make it a prime directive for the
study of math: Never intentionally memorize any math
formula.

Here are some simple rules for knowing, without intentional
memorization, the multiplication table up to 12 times 12.

It is extremely important that you figure out why the
following rules work, and do not take them only as "rules to
memorize".

The more experience you have with arithmetic, the easier it
will be to figure out why these rules work.

The times 1 column is the easiest.
1 times anything is that anything.

1 * 1 = 1
1 * 2 = 2
1 * 3 = 3
etc


The times 2 column is a simple doubling,
adding a number to itself.

2 * 1 = 1 + 1 = 2
2 * 2 = 2 + 2 = 4
2 * 3 = 3 + 3 = 6
etc

For the 10 times column, simply append a 0, to the number you
are to multiply.

10 * 1 = 10
10 * 2 = 20
10 * 3 = 30
etc


For the 11 times column, up to 11 * 9,
simply repeat the digit.

11 * 1 = 11
11 * 2 = 22
11 * 3 = 33
etc

This rule comes about because

11 * 1 = (10 + 1) * 1 = 10 * 1 + 1 * 1 = 10 + 1 = 11
11 * 2 = (10 + 1) * 2 = 10 * 2 + 1 * 2 = 20 + 2 = 22
11 * 3 = (10 + 1) * 3 = 10 * 3 + 1 * 3 = 30 + 3 = 33
etc

11 * 10 = 10 * 11

11 * 11 = 121

Note that the middle digit of the product is the sum of the
two digits of the number to multiply by 11.

11 * 12 = 132.

Note that the middle digit of the product is the sum of the
two digits of the number to multiple by 11.


The times 5 column is easy because the last digit of the
product is always either 0 or 5.

5 times even ends in 0.
5 times odd ends in 5.

5 * 1 = 5

5 * 2 = 10.
5 * 3 = 15

5 * 4 = 20; Note that the first digit 2, is half of 4.
5 * 5 = 25

5 * 6 = 30; Note that the first digit 3, is half of 6.
5 * 7 = 35;

etc

The times 9 column also has an easy pattern.

9 * 1 = 9
9 * 2 = 18
9 * 3 = 27
9 * 4 = 36
9 * 5 = 45
9 * 6 = 54
9 * 7 = 63
9 * 8 = 72
9 * 9 = 81
9 * 10 = 90.


Note that the digits add to 9, and that the leftmost digit
is one less than the number you are multiplying.

There is a "finger" calculator that makes use of this
pattern.

Example:

For 9 times 3, hold up all fingers, put the third one down,
and read the answer, 2 fingers and 7 fingers is 27.



For 9 * 11, use 9 * 11 = 11 * 9 = 99.

There are several ways to calculate the times 6 column.

6 = 5 + 1,
so

6 * 1 = (5 + 1) * 1 = (5*1) + (1 * 1) = 5 + 1 = 6
6 * 2 = (5 + 1) * 2 = (5*2) + (1 * 2) = 10 + 2 = 12
6 * 3 = (5 + 1) * 3 = (5*3) + (1 * 3) = 15 + 3 = 18
etc

Also, 6 = 2*3,
so

6 * 1 = (2 * 3) * 1 = 2 * (3 * 1) = 2 * 3 = 6
6 * 2 = (2 * 3) * 2 = 2 * (3 * 2) = 2 * 6 = 12
6 * 3 = (2 * 3) * 3 = 2 * (3 * 3) = 2 * 9 = 18
6 * 4 = (2 * 3) * 4 = 2 * (4 * 3) = 2 * 12 = 24
etc





The times 12 column is easier than most people expect.

12 * 1 = 1 * 12 = 12
12 * 2 = (10 + 2) * 2 = 10 * 2 + 2 * 2 = 20 + 4
12 * 3 = (10 + 2) * 3 = 10 * 3 + 2 * 3 = 30 + 6
etc

Also, 12 = 2 * 2 * 3

12 * 1 = (2 * 2 * 3) * 1 = (2 * 2) * (3 * 1)
= 2 * (2 * 3) = 2 * 6 = 12

12 * 2 = (2 * 2 * 3) * 2 = (2 * 2) * (2 * 3) = (2 * 2) * 6
= 2 * (2 * 6) = 2 * 12 = 24

12 * 3 = (2 * 2 * 3) * 3 = (2 * 2) * (3 * 3) = (2 * 2 ) * 9
= 2 * (2 * 9) = 2 * 18 = 36
etc


For the times 4 column,


4 = 2 * 2,
so

4 * 1 = (2 * 2) * 1 = 2 * (2 * 1) = 2 * 2 = 4
4 * 2 = (2 * 2) * 2 = 2 * (2 * 2) = 2 * 4 = 8
4 * 3 = (2 * 2) * 3 = 2 * (2 * 3) = 2 * 6 = 12
4 * 4 = (2 * 2) * 4 = 2 * (2 * 4) = 2 * 8 = 16
etc

For the times 8 columns,

8 = 2 * 2 * 2,
so

8*1 = (2*2*2)*1 = (2*2) * (2*1) = (2*2)*2 = 2*(2*2)=2*4 = 8
8*2 = (2*2*2)*2 = (2*2)*(2*2) = (2*2)*4 = 2*(2*4)=2*8=16
8*3 = (2*2*2)*3 = (2*2)*(2*3) = (2*2)*6=2*(2*6) = 2*12 = 24
etc





For the times 3 column,

3 * 1 = 1 * 3 = 3
3 * 2 = 2 * 3 = 3 + 3 = 6
3 * 3 = 3 * 2 + 3 = 6 + 3 = 9
3 * 4 = 3 * (2 + 2) = 3 * 2 + 3 * 2 = 6 + 6 = 12
3 * 5 = 5 * 3 = 15
3 * 6 = 3 * (3 * 2) = (3 * 3) * 2 = 9 * 2 = 18
3 * 7 = 3 * (5 + 2) = 3 * 5 + 3 * 2 = 15 + 6 = 21
3 * 8 = 3 * (10 - 2 ) = 3 * 10 - 3 * 2 = 30 - 6 = 24
3 * 9 = 9 * 3 = 27
3 * 10 = 10 * 3 = 30
3 * 11 = 11 * 3 = 33
3 * 12 = 12 * 3 = (10 + 2)*3 = 10 * 3 + 2 * 3 = 30 + 6 = 36

Note that:
For 3 * 1, 3 * 4, 3 * 7, and 3 * 10,
the sum of the digits is 3.
For 3 * 2, 3 * 5, 3 * 8, and 3 * 11,
the sum of the digits is 6.
For 3 * 3, 3 * 6, 3 * 9, and 3 * 12,
the sum of the digits is 9.


The one column remaining to give examples for calculating is
times 7.

7 * 1 = 1 * 7 = 7
7 * 2 = 2 * 7 = 7 + 7 = 14
7 * 3 = 7 * (2 + 1) = 7 * 2 + 7 * 1 = 14 + 7 = 21
7 * 4 = 7 * (2 + 2) = 7 * 2 + 7 * 2 = 14 + 14 = 28
7 * 5 = 5 * 7 = 35
7 * 6 = 7 * (3 + 3) = 7 * 3 + 7 * 3 = 21 + 21 = 42
7 * 7 = 7 * (3 + 4) = 7 * 3 + 7 * 4 = 21 + 28 = 49
7 * 8 = 7 * 4 * 2 = 28 * 2 = 56
7 * 9 = 9 * 7 = 63
7 * 10 = 10 * 7 = 70
7 * 11 = 11 * 7 = 77
7 * 12 = 7 * (10 + 2) = 7 * 10 + 7 * 2 = 70 + 14 = 84



There are other tricks for calculating the products in the 12
by 12 multiplication table.

Here is one for multiplying a number 6 through 9 by another
number 6 through 9.

On each hand, Name the thumb "6", and name the other fingers,
in order, "7","8","9","10".

Here is an example of how to use this to multiply 6 * 7.

On the left hand, bend down finger 6, which is the thumb.
On the right hand, bend down fingers 6 and 7.

On the left hand, you have 1 finger bent down, and 4 fingers
still raised.

On the right hand, you have 2 fingers bent down, and 3
fingers
still raised.

The product 6 * 7 will be 10 times the sum of fingers bent
down + the product of fingers still raised.
That is,

6 * 7 = 10 * (1 + 2) + 3 * 4 = 10 * 3 + 12 = 30 + 12 = 42.




The ladder of squares:

1 * 1 + 1 + 2 = 2 * 2
2 * 2 + 2 + 3 = 3 * 3
3 * 3 + 3 + 4 = 4 * 4
4 * 4 + 4 + 5 = 5 * 5
5 * 5 + 5 + 6 = 6 * 6
6 * 6 + 6 + 7 = 7 * 7
7 * 7 + 7 + 8 = 8 * 8
8 * 8 + 8 + 9 = 9 * 9
9 * 9 + 9 + 10 = 10 * 10
10 * 10 + 10 + 11 = 11 * 11
11 * 11 + 11 + 12 = 12 * 12



Differences of squares:

1 * 3 = 2 * 2 - 1 * 1

2 * 4 = 3 * 3 - 1 * 1
1 * 5 = 3 * 3 - 2 * 2

3 * 5 = 4 * 4 - 1 * 1
2 * 6 = 4 * 4 - 2 * 2
1 * 7 = 4 * 4 - 3 * 3

4 * 6 = 5 * 5 - 1 * 1
3 * 7 = 5 * 5 - 2 * 2
2 * 8 = 5 * 5 - 3 * 3
1 * 9 = 5 * 5 - 4 * 4


5 * 7 = 6 * 6 - 1 * 1
4 * 8 = 6 * 6 - 2 * 2
3 * 9 = 6 * 6 - 3 * 3
2 * 10 = 6 * 6 - 4 * 4
1 * 11 = 6 * 6 - 5 * 5


6 * 8 = 7 * 7 - 1 * 1
5 * 9 = 7 * 7 - 2 * 2
4 * 10 = 7 * 7 - 3 * 3
3 * 11 = 7 * 7 - 4 * 4
2 * 12 = 7 * 7 - 5 * 5

7 * 9 = 8 * 8 - 1 * 1
6 * 10 = 8 * 8 - 2 * 2
5 * 11 = 8 * 8 - 3 * 3
4 * 12 = 8 * 8 - 4 * 4

8 * 10 = 9 * 9 - 1 * 1
7 * 11 = 9 * 9 - 2 * 2
6 * 12 = 9 * 9 - 3 * 3

9 * 11 = 10 * 10 - 1 * 1
8 * 12 = 10 * 10 - 2 * 2

10 * 12 = 11 * 11 - 1 * 1


Here is another method for reducing multiplication to
addition:

For example, to multiply 9 * 8, Make two columns, with the 9
at the top of one column and the 8 at the top of the other
column.
In the first column, successive have the number. In the
other column successive double the column. Discard
remainders.

9 8
4 16
2 32
1 64

Then add the numbers in the doubling column that correspond
to odd numbers in the halving column.




9 8
1 64
-----
72

9 * 8 = 72


Now you have examples of a few ways to re-calculate
the multiplication table, and never need to have
"intentionally memorized" it.

Understand why these methods are valid. It is not useful to
simply memorize these methods.

Kermit Rose