This is Math Newsletter number 1; Wednesday, July 28, 2010.

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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