# Topic: Multiplication

## Summary

Multiplication is a Mathematical operation of adding together multiple copies of the same thing.

#### Intro

For example, the multiplication of Whole Numbers can be illustrated with this example....

``` 5 + 5 + 5 = 15 ```

This example above has 5 multiplied 3 times. Or, said another way, this is: 3 times 5.

This is often written as the short hand....

``` 3 × 5 = 15 ```

That symbol that looks similar to an "x", is actually a different symbol. The "×" is called the "Multiplication Sign".

When you see the "×", it is read as "times". So "3 × 5" would be read as "three times five".

Some people don't use the multiplication sign -- "×" -- to denote multiplication. Some people write it like this instead, with a Middle Dot (which is also called a middot)....

``` 3 · 5 = 15 ```

And still some put the dot lower, and write....

``` 3 . 5 = 15 ```

In many Programming Languages, multiplication is done using the Asterisk. And although the notation below would likely not be seen in any of these computer programming languages, it is suggestive of it....

``` 3 * 5 = 15 ```

All these mean the same thing though. I tend to normally use the Multiplication Sign -- "×" -- if given the choice.

#### My History

Most of the credit for teaching me how to multiply goes to my mother. She started teaching me Mathematics before I could even talk. And once some of my Elementary Schools teachers saw how advanced my brother and I were in many many of our studies (including math), they asked my mother to volunteer at the school and teach. Which she did.

I also learned alot about teaching Math, including multiplication, from when I was a Math Tutor. I figured out that one of the biggest problems people have with math, in general, is not being able to multiply in their heads.

It was one of the things I checked for first, when a student had trouble in math. It's never too late to learn to multiply in your head. (But it's best to learn it as young as possible!)

#### Learning and Teaching HOWTO Multiply

Being able to multiply in you head is an important skill! And something you should teach your kids to do while they are still young!

(Start this education before they are able to talk.)

Even in this age, with electronic calculators and computers, it's still better to be able to multiply in your head. I can multiply numbers faster in my head, than the time it would take to enter them into an electronic calculator or a computer.

Make sure your kids learn how to multiply in their heads!!!

Do not let them become dependent on machines. (It is a weakness. And they could suffer because of it. And could be put at a disadvantage.)

Kids who are not able to multiply in their head often have alot of trouble learning new or more advanced Math topics, because they are still struggling with the multiplication part of it, and not able to get to the "real" problem.

##### Counting By Multiples

One skill that's useful in learning how to and being able to multiply is being able to count by multiples.

For example....

• being able to count by 1s
• being able to count by 2s
• being able to count by 3s
• being able to count by 4s
• being able to count by 5s
• being able to count by 6s
• being able to count by 7s
• being able to count by 8s
• being able to count by 9s
• being able to count by 10s
• being able to count by 11s
• being able to count by 12s
• being able to count by 13s
• being able to count by 14s
• ...
###### Counting By 1s

This is a very very basic skill, and is the same as justing being able to count. You should start teaching your kids this before they can talk.

###### Counting By 2s

Once you child can count by 1s -- can count -- then they know the numbers (up to whatever limit you taught them up to). The next step is to teach them to count by 2s.

Just help them memorize the sequence...

2, 4, 6, 8, 10, 12, 14, 16, 18, 20,...

... and make sure you associate this with numbers of physical objects. Like use blocks, legos, pieces of food, toys, coins, or whatever, and count the objects by twos with them. And get them to do the same!

###### Counting By 3s, 4s, 5s,...

After doing 2s, do the same for 3s, 4s, 5s, etc....

First make sure they know the sequence. And then associate this with counting objects.

Note, make sure you they don't forget what they have already learned. And keep on going over the old stuff too.

##### Multiplication Table

After they can count by 1s, 2s, 3s,... the next thing to do it to get them to memorize the Multiplication Table.

The memorization of this can be aided with some "tricks" and some Mnemonics.

You can also start them off with a smaller multiplication table first, and then expand it.

1 2 3 4 5 6 7 8 9 10
1 1 2 3 4 5 6 7 8 9 10
2 2 4 6 8 10 12 14 16 18 20
3 3 6 9 12 15 18 21 24 27 30
4 4 8 12 16 20 24 28 32 36 40
5 5 10 15 20 25 30 35 40 45 50
6 6 12 18 24 30 36 42 48 54 60
7 7 14 21 28 35 42 49 56 63 70
8 8 16 24 32 40 48 56 64 72 80
9 9 18 27 36 45 54 63 72 81 90
10 10 20 30 40 50 60 70 80 90 100

And here's one you can do next....

1 2 3 4 5 6 7 8 9 10 11 12
1 1 2 3 4 5 6 7 8 9 10 11 12
2 2 4 6 8 10 12 14 16 18 20 22 24
3 3 6 9 12 15 18 21 24 27 30 33 36
4 4 8 12 16 20 24 28 32 36 40 44 48
5 5 10 15 20 25 30 35 40 45 50 55 60
6 6 12 18 24 30 36 42 48 54 60 66 72
7 7 14 21 28 35 42 49 56 63 70 77 84
8 8 16 24 32 40 48 56 64 72 80 88 96
9 9 18 27 36 45 54 63 72 81 90 99 108
10 10 20 30 40 50 60 70 80 90 100 110 120
11 11 22 33 44 55 66 77 88 99 110 121 132
12 12 24 36 48 60 72 84 96 108 120 132 144

And this one after that....

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28
3 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42
4 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56
5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
6 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84
7 0 7 14 21 28 35 42 49 56 63 70 77 84 91 98
8 0 8 16 24 32 40 48 56 64 72 80 88 96 104 112
9 0 9 18 27 36 45 54 63 72 81 90 99 108 117 126
10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140
11 0 11 22 33 44 55 66 77 88 99 110 121 132 143 154
12 0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
13 0 13 26 39 52 65 78 91 104 117 130 143 156 169 182
14 0 14 28 42 56 70 84 98 112 126 140 154 168 182 196

Note, thes multiplication tables shown above are meant to be illustrative of what needs to be learned and memorized. You and your kids should be able to multiply any and all these values, in your head, without lookig at this.

##### Error Detection Tricks
TODO
###### Error Detection For Multiplying By 3
TODO

The answer to anything multiplied by 3 should end in a: 3, 6, or 9.

TODO
###### Error Detection For Multiplying By 5
TODO

The answer to anything multiplied by 5 should end in a: 0 or a 5.

TODO
###### Error Detection For Multiplying By 10
TODO

The answer to anything multiplied by 10 should end in a: 0.

TODO
TODO
##### Mutlply Multi-Digit Numbers

For multiplying multi-digit numbers that you don't have memorized, there are Algorithms you can use to perform the multiplication.

TODO
##### Things To Know
TODO
• do you have this sequence memorized: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20,...
• do you have this sequence memorized: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30,...
• do you have this sequence memorized: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40,...
• do you have this sequence memorized: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50,...
• do you have this sequence memorized: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60,...
• do you have this sequence memorized: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70,...
• do you have this sequence memorized: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80,...
• do you have this sequence memorized: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90,...
• do you have this sequence memorized: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100,...
• do you have this sequence memorized: 11, 22, 33, 44, 55, 66, 77, 88, 99, 121,...
• do you have this sequence memorized: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120,...
• do you know the "trick" for multiplying 0 with anything else?
• do you know the "trick" for multiplying 1 with anything else?
• do you know the "trick" for multiplying 2 with anything else?
• do you know the "trick" for multiplying 5 with anything else?
• do you know the "trick" for multiplying 10 with anything else?
• do you know the "trick" for multiplying numbers that end in 0's?
• do you know the "trick" for multiplying anything that end in a 5 with itself? (ex: 15 × 15 or 25 × 25 or 35 × 35 or ... or 1125 × 1125 or ...)
TODO

-- Mirza Charles Iliya Krempeaux