How to Learn the Multiplication Table

Date: 09/01/2005 at 18:21:59
From: Erika
Subject: Multiplication

I have great difficulty remembering my multiplication tables, I guess 
it has too many numbers.  I know them if I write them down on paper, 
but I can't just answer them in my mind or if someone asks me.  I need 
help.

Like if someone asks 6x6 = ? I wouldn't know it.

Date: 09/02/2005 at 17:11:49
From: Doctor Achilles
Subject: Re: Multiplication

Hi Erika,

Thanks for writing to Dr. Math.

Multiplication tables used to give me headaches, too.  I remember when
I was in 5th or 6th grade, I had a test where I had to fill in the
multiplication table as quickly as possible and I was honestly in
tears by the end.

I think the problem is that students are told just to memorize the
table, and there's just too much there.  Plus, that doesn't really
help students anyways because you won't remember it a couple weeks later.

Instead, I think it's helpful to learn a few rules for doing
multiplication.  With a list of just 6 rules, you can fill out the
entire multiplication table, and do any multiplication problem you need.

I should warn you, the rules can take a little while to learn.  This
is a very long answer.  I suggest that you spend a few days studying
this answer.  Try to learn one rule every day for a week, and then go
back and review the rules.  Once you know these rules, multiplication
will be easy, but don't try to learn too much at once.

If you can't do some of the bigger numbers, then wait until the end. 
When you get to the last rule, it will help you solve all the
difficult problems along the way that were confusing.  Rules 4 and 5
are especially tough, but once you understand them, Rule 6 should make
it easy to do the harder problems.

Also, remember that it doesn't matter the order you do multiplication
in.  So 6x12 is equal to 12x6.  So if you know how to do 6x12, you can
do 12x6.

Here's how I do it.  I will show you the multiplication table for
numbers from 1 to 12 and fill it in in the order I do it.

I hope that the way I'm drawing the table shows up correctly for you.
If not, I think it should work if you cut and paste this e-mail into
the "notepad" program on your computer.

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

2

3

4

5

6

7

8

9

10

11

12

-------------------------------------------------

RULE #1: MULTIPLYING BY 2, 4, AND 8 IS ALL ABOUT DOUBLING:

Multiplying by 2 is easy, you just double the number:

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

2   4   6   8  10  12  14  16  18  20  22  24

3   6

4   8

5  10

6  12

7  14

8  16

9  18

10 20

11 22

12 24

Now you can fill in the 4's, they're just twice as big as the 2's.

If you get stuck, just remember: 8x4 is twice as big as 8x2.  And
8x2x2 is going to be 16x2, which is 6x2 plus 10x2, which is 12 plus 20.

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

2   4   6   8  10  12  14  16  18  20  22  24

3   6      12

4   8  12  16  20  24  28  32  36  40  44  48

5  10      20

6  12      24

7  14      28

8  16      32

9  18      36

10 20      40

11 22      44

12 24      48

And using the same logic, you know that multiplying by 8 will be twice
as big as multiplying by 4.

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

2   4   6   8  10  12  14  16  18  20  22  24

3   6      12              24

4   8  12  16  20  24  28  32  36  40  44  48

5  10      20              40

6  12      24              48

7  14      28              56

8  16  24  32  40  48  56  64  72  80  88  96

9  18      36              72

10 20      40              80

11 22      44              88

12 24      48              96

Ok, so to review: multiplying by 2 means you just double the number. 
If you have to double a difficult number like 36, just break the
number up into 30x2 plus 6x2, which is 60 plus 12 or 72.

If you have to multiply by 4, then just double twice.  So 8x4 = (8x2)x2.

If you have to multiply by 8, then just double three times.  So 9x8 =
((9x2)x2)x2.

You can also use the rule for multiplying by 9 that I'll teach you later.


You should practice finding the double of a number until you get
comfortable multiplying numbers by 2.

Ok, now that we've covered 2, 4, and 8, let's do 5 and 10.

------------------------------------------------------

RULE #2: MULTIPLYING BY 10 IS ADDING A 0.  MULTIPLYING BY 5 IS HALF OF
MULTIPLYING BY 10.

Multiplying by 10 is pretty easy, just add a 0.

6x10 = 60, 7x10 = 70.

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

2   4   6   8  10  12  14  16  18  20  22  24

3   6      12              24      30

4   8  12  16  20  24  28  32  36  40  44  48

5  10      20              40      50

6  12      24              48      60

7  14      28              56      70

8  16  24  32  40  48  56  64  72  80  88  96

9  18      36              72      90

10 20  30  40  50  60  70  80  90 100 110 120

11 22      44              88     110

12 24      48              96     120

Multiplying by 5 is a little tougher than 10, but it's not too bad.

For an even number, like 6, to multiply by 5 you first divide by 2 and
then multiply by 10.  So half of 6 is 3, so 6x5 = 30.

Odd numbers are a little tougher, but not too bad.  Just do it like this:

3x5 = (2x5) + 5 = 10 + 5 = 15

5x5 = (4x5) + 5 = 20 + 5 = 25

7x5 = (6x5) + 5 = 30 + 5 = 35

9x5 = (8x5) + 5 = 40 + 5 = 45

11x5 = (10x5) + 5 = 50 + 5 = 55

Does that make sense?

Or, if you prefer, you can just multiply by 10 and then take half of
the number:

3x5 = half of 3x10 = half of 30 = 15

5x5 = half of 5x10 = half of 50 = 25

7x5 = half of 7x10 = half of 70 = 35

9x5 = half of 9x10 = half of 90 = 45

11x5 = half of 11x10 = half of 110 = 55

So now we have:

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

2   4   6   8  10  12  14  16  18  20  22  24

3   6      12  15          24      30

4   8  12  16  20  24  28  32  36  40  44  48

5  10  15  20  25  30  35  40  45  50  55  60

6  12      24  30          48      60

7  14      28  35          56      70

8  16  24  32  40  48  56  64  72  80  88  96

9  18      36  45          72      90

10 20  30  40  50  60  70  80  90 100 110 120

11 22      44  55          88     110

12 24      48  60          96     120

Ok, so to review: multiplying by 10 means adding a zero.  Multiplying
by 5 is half of multiplying by 10.


Now that we've done 5 and 10, we have to do a few of the harder ones.
Most of these have easy rules if you are multiplying by a number
smaller than 10.


RULE #3: TO MULTIPLY BY 11, JUST DOUBLE THE DIGITS.  (This only works
up to 9x11.)

The first 9 multiples of 11 are:

11

22

33

44

55

66

77

88

99

The word "multiple" just means "answer to a multiplication problem".
So a "multiple of 11" is the answer to a multiplication problem with
11 in it.

Which are pretty easy to remember.  7x11 = 77, not too bad, right?

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

2   4   6   8  10  12  14  16  18  20  22  24

3   6      12  15          24      30  33

4   8  12  16  20  24  28  32  36  40  44  48

5  10  15  20  25  30  35  40  45  50  55  60

6  12      24  30          48      60  66

7  14      28  35          56      70  77

8  16  24  32  40  48  56  64  72  80  88  96

9  18      36  45          72      90  99

10 20  30  40  50  60  70  80  90 100 110 120

11 22  33  44  55  66  77  88  99 110

12 24      48  60          96     120

We'll leave the last couple multiples of 11 until I get to the last rule.

----------------------------------------------

RULE #4: TO MULTIPLY BY 9, FIND THE 10'S DIGIT BY SUBTRACTING 1, THEN
FIND THE ONE'S DIGIT BY SUBTRACTING 9.  (This only works up to 9x9.)

This is really hard to understand.  Once you get it, it makes
multiplying by 9 a snap.  So take your time and read through the
explanation.

Let me start by listing the first 9 multiples of 9.  Don't memorize
these, I just want to talk about them.

1x9 =  9

2x9 = 18

3x9 = 27

4x9 = 36

5x9 = 45

6x9 = 54

7x9 = 63

8x9 = 72

9x9 = 81

Notice that if you add up the digits for each of these, the answer is
always 9:

0+9 = 9

1+8 = 9

2+7 = 9

3+6 = 9

4+5 = 9

5+4 = 9

6+3 = 9

7+2 = 9

8+1 = 9

That's a really useful fact about multiplying by 9.

When you ask me to multiply by 9, I do two steps:

Step 1: find the ten's digit by subtracting 1.

Step 2: find the one's digit by using the rule that the digits have to
add up to 9.


Let's say you ask me to do 6x9.  The first thing I do is figure out
the ten's digit.  That is going to be 6-1.  So the ten's digit is "5".

Now, I find the one's digit.  I know that the digits have to add up to
9 total.  So if the ten's digit is "5" then the one's digit is "4". 
So 6x9 = 54.


Let's say you ask me to do 7x9.  The first step is to find the ten's
digit.  That is 7-1.  So the ten's digit is "6".

Now, I find the one's digit.  The digits have to add up to 9.  So the
one's digit must be "3".  So 7x9 = 63.


What if you ask me to do 3x9.  The ten's digit is 3-1 or "2".  So the
one's digit has to be "7", so the answer is 27.


What about 9x9?  The ten's digit is 9-1 or "8".  So the one's digit
has to be "1", so the answer is 81.


Ok, now that you know how to multiply by 9, let's fill in our table.

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

2   4   6   8  10  12  14  16  18  20  22  24

3   6      12  15          24  27  30  33

4   8  12  16  20  24  28  32  36  40  44  48

5  10  15  20  25  30  35  40  45  50  55  60

6  12      24  30          48  54  60  66

7  14      28  35          56  63  70  77

8  16  24  32  40  48  56  64  72  80  88  96

9  18  27  36  45  54  63  72  81  90  99

10 20  30  40  50  60  70  80  90 100 110 120

11 22  33  44  55  66  77  88  99 110

12 24      48  60          96     120

We already found 11x9 using the rule for 11.  12x9 is a little
tougher, so we'll save it until I get to the last rule.

-------------------------------------------------

RULE #5: TO MULTIPLY BY 3, YOU HAVE TO MEMORIZE 3x3 AND 3x7. 
MULTIPLYING BY 6 IS TWICE 3 AND MULTIPLYING BY 12 IS TWICE 6.

Unfortunately, multiplying by 3 is hard to do.  You have to memorize a
few things to do it.

Let me list the first 11 multiples of 3.  You don't need to memorize
all of these, I just want to talk about them.

3x1 =  3

3x2 =  6

3x3 =  9

3x4 = 12

3x5 = 15

3x6 = 18

3x7 = 21

3x8 = 24

3x9 = 27

3x10 = 30

3x11 = 33

We already know how to do 3x2, 3x4, 3x5, 3x8, 3x9, 3x10, and 3x11. 
You ONLY have to memorize these two:

3x3 = 9

3x7 = 21

To get 3x6 you can use the rule for 6's which is: Multiplying by 6 is
just doubling 3.

We know 3x3 = 9, so 3x6 is double that.  3x6 = (3x3)x2 = 9x2 = 18.

Ok, so now let's fill in the table for 3's:

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

2   4   6   8  10  12  14  16  18  20  22  24

3   6   9  12  15  18  21  24  27  30  33

4   8  12  16  20  24  28  32  36  40  44  48

5  10  15  20  25  30  35  40  45  50  55  60

6  12  18  24  30          48  54  60  66

7  14  21  28  35          56  63  70  77

8  16  24  32  40  48  56  64  72  80  88  96

9  18  27  36  45  54  63  72  81  90  99

10 20  30  40  50  60  70  80  90 100 110 120

11 22  33  44  55  66  77  88  99 110

12 24      48  60          96     120

And while we're doing that, we can fill in the table for 6's and 12's.
Don't worry right now if some of these are hard, the last rule will
show you another way.

Here are the steps to do that:

6x6 = (6x3)x2 = 18x2 = 36

7x6 = (7x3)x2 = 21x2 = 42

12x6 = (12x3)x2 = 36x2 = 72

3x12 = (3x6)x2 = 18x2 = 36

7x12 = (7x6)x2 = 42x2 = 84

9x12 = (9x6)x2 = 54x2 = 108

11x12 = (11x6)x2 = 66x2 = 132

12x12 = (12x6)x2 = 72x2 = 144

(Don't worry if the last two seem difficult, the last rule will show
you an easier way to do 11x12 and 12x12.)

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

2   4   6   8  10  12  14  16  18  20  22  24

3   6   9  12  15  18  21  24  27  30  33  36

4   8  12  16  20  24  28  32  36  40  44  48

5  10  15  20  25  30  35  40  45  50  55  60

6  12  18  24  30  36  42  48  54  60  66  72

7  14  21  28  35  42      56  63  70  77  84

8  16  24  32  40  48  56  64  72  80  88  96

9  18  27  36  45  54  63  72  81  90  99 108

10 20  30  40  50  60  70  80  90 100 110 120

11 22  33  44  55  66  77  88  99 110     132

12 24  36  48  60  72  84  96 108 120 132 144

Ok, so to summarize.  We already learned by the earlier rules how to
do all the 3's except 3x3 and 3x7.  We memorized those.

To multiply by 6, just multiply by 3 and then double the answer.

To multiply by 12, multiply by 6 and then double the answer.


Here's a little note that sometimes helps me double-check my 3's.  The 
multiples of 3 are:

3x1 =  3

3x2 =  6

3x3 =  9

3x4 = 12

3x5 = 15

3x6 = 18

3x7 = 21

3x8 = 24

3x9 = 27

3x10 = 30

3x11 = 33

Notice:

0+3 = 3

0+6 = 6

0+9 = 9

1+2 = 3

1+5 = 6

1+8 = 9

2+1 = 3

2+4 = 6

2+7 = 9

3+0 = 3

3+3 = 6

Notice that when you add the digits up for a multiple of 3, you end up
with 3, 6, or 9 (sometimes for some of the bigger numbers, you might
end up with 12, 15, 18, etc, too).

The same thing is true for multiples of 6.  The digits always add up
to 3, 6, 9, 12, 15, 18, 21, and so on.  It also works for multiples of 12.

-------------------------------------------

RULE #6: WHEN YOU GET STUCK, JUST DO AN EASIER PROBLEM.

What if someone asks you to do 6x6 and you forget how.  DON'T PANIC,
there is an easy way to do it!

You can do 6x5 without a problem, right?  So just remember that 6x6 is
6x5 + 6.

Remember, multiplication is just a short-cut for doing addition.  So
if you can do 8x5, then you just add 8 more and you have 8x6.

You can use this to solve some tough problems like 6x12 too.  It's
just 6 more than 6x11.  66+6 = 72.  So 6x12 = 72.

Or you can solve a problem like 12x11.  To do that one, just take
12x10 and then add 12 more.  12x10 = 120.  So 12x11 = 120 + 12 = 132.
 
You can also subtract if you're careful about it.

What is another way to do 8x4?  Well, 8x5 is pretty easy, but that
will be too big.  But if we do (8x5)-8, that will be exactly right. 
(8x5)-8 = 40-8 = 32.  No problem.

We can use this for problems like 6x4.  We know that 6x5 = 30.  So 6x4
must be 6 less than that.  So 6x4 = 24.

Or 12x4.  12x5 = 60.  So 12x4 must be 60-12 or 48.  Does that make sense?

What about 7x9?  To do that, we can take 7x10 and subtract 7.  7x10 is
70, so 7x9 must be 70-7 or 63!

What about 12x9?  We can just take (12x10)-12.  That's 120-12 or 108.
It may take a few extra seconds, but you can get there.


How do we deal with difficult two-digit multiplication problems?

What about something like 66x2?  The answer to that is going to be
(60x2)+(6x2).  60x2 = 120.  6x2 = 12.  So 66x2 = 120+12 = 132.

Can you solve 84x2?  How about 48x2?

What about 12x12 (the biggest number on our table)?  To solve that, we
can just take (10x12)+(2x12) = 120 + 24 = 144.


So when you get stuck, just find an easier way to divide up the problem.


As you've probably noticed, there are some problems that can be done a
few different ways.

To solve something like 6x6 we can do one of two things:

6x6 = (6x3)x2 = ((3x3)x2)x2 = (9x2)x2 = 18x2 = 36

Or:

6x6 = (6x5)+6 = (3x10)+6 = 30+6 = 36

Let's take one last look at our table:

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

2   4   6   8  10  12  14  16  18  20  22  24

3   6   9  12  15  18  21  24  27  30  33  36

4   8  12  16  20  24  28  32  36  40  44  48

5  10  15  20  25  30  35  40  45  50  55  60

6  12  18  24  30  36  42  48  54  60  66  72

7  14  21  28  35  42      56  63  70  77  84

8  16  24  32  40  48  56  64  72  80  88  96

9  18  27  36  45  54  63  72  81  90  99 108

10 20  30  40  50  60  70  80  90 100 110 120

11 22  33  44  55  66  77  88  99 110     132

12 24  36  48  60  72  84  96 108 120 132 144

We still haven't filled in 7x7 or 11x11.  How can we do those?

Those are two of the hardest ones to do.

11x11 = (11x10)+11 = 110 + 11 = 121

Let's add that:

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

2   4   6   8  10  12  14  16  18  20  22  24

3   6   9  12  15  18  21  24  27  30  33  36

4   8  12  16  20  24  28  32  36  40  44  48

5  10  15  20  25  30  35  40  45  50  55  60

6  12  18  24  30  36  42  48  54  60  66  72

7  14  21  28  35  42      56  63  70  77  84

8  16  24  32  40  48  56  64  72  80  88  96

9  18  27  36  45  54  63  72  81  90  99 108

10 20  30  40  50  60  70  80  90 100 110 120

11 22  33  44  55  66  77  88  99 110 121 132

12 24  36  48  60  72  84  96 108 120 132 144

For 7x7 we need to take:

  (7x6)+7

Which we can change into:

  ((7x3)x2)+7

We memorized that 7x3 = 21 (it was one of the two that we had to
memorize).  So we have:

  (21x2)+7

Which becomes:

  42+7

Which is:

  49

So 7x7 is 49, and we're done with our table.

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

2   4   6   8  10  12  14  16  18  20  22  24

3   6   9  12  15  18  21  24  27  30  33  36

4   8  12  16  20  24  28  32  36  40  44  48

5  10  15  20  25  30  35  40  45  50  55  60

6  12  18  24  30  36  42  48  54  60  66  72

7  14  21  28  35  42  49  56  63  70  77  84

8  16  24  32  40  48  56  64  72  80  88  96

9  18  27  36  45  54  63  72  81  90  99 108

10 20  30  40  50  60  70  80  90 100 110 120

11 22  33  44  55  66  77  88  99 110 121 132

12 24  36  48  60  72  84  96 108 120 132 144

-----------------------------------------------------

I know this seems like a lot, but with a little practice the rules
will become easy and you can do multiplication of any numbers you
need.  Take your time and study these rules and it will become easier.

I hope this helps.  If you have other questions or you'd like to talk
about this some more, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/