The Multiplication Tables

Multiplication is a repeating sum.

For example:

The operation $3\times4$ can be changed to $3+3+3+3$, i.e. $4$ times $3$. The result is $12$.

Another example:

The operation $7\times2$ can easily be changed by $7+7$, i.e. $2$ times $7$. The result is $14$.

It should be noted that the order of the numbers is of no importance.

For example:

in the case of $2\times4$, if, instead of doing the search explained in the previous paragraph, we search the column corresponding to the number $4$ and the row corresponding to the number $2$, the result will still be $8$.
In other words: $2\times4=4\times2=8$.

Let's look at another example:

$9\times7$. We look for the column corresponding to the number $9$ and the row corresponding to the number $7$. The point where both meet shows us the number $63$, which is the answer to the operation.

Also here we can see that, if we do it the other way around, i.e. $9\times7$, the result will still be $63$.

Tricks for remembering the multiplication tables

There are many children and even adults who find it difficult to memorize the multiplication tables, but luckily there are many tricks that make this task easier.

When we multiply any number by $1$ it stays the same.

For example, when we multiply any number by :

$1\times6=6\times1=6$.

When we multiply any number by $2$, what we are actually doing is adding it to itself, i.e., $6\times2$ equals $6+6$, which equals $12$.

First, we write the numbers $0$, $1$ and $2$ as follows:

0......0......0 ......0
1........1.......1
2......2......2

Secondly, we add the digits $1$ - $9$ as follows: ascending from the bottom row of the right column to the top row of the same column. Following the order, we do the same in the middle column and in the left column up to the top row of the left column. We would obtain the following table:

03......06......09

12.......15.......18

21......24......27

We thus obtain the multiples of $3$:

$3\times1=3$; $3\times2=6$; $3\times3=9$, etc.

This time we must use our hands. We divide each of the fingers into three parts, the fourth part being the area where the finger meets the palm of the hand. At first it helps to put the numbers in each part, from bottom to top, but with practice we will learn it by heart. When we want to calculate, for example, how much is $4\times3$, what we have to do is to advance to the fourth part of the third finger, where we will have the number $12$.

This is a fairly easy table, as we can observe a clear pattern in it. If we write the results of the table of 5, we can see how there is a recurring rule:

$1\times5=5$

$2\times5=10$

$3\times5=15$

$4\times5=20$

$5\times5=25$

$6\times5=30$

$7\times5=35$

$8\times5=40$

$9\times5=45$

$10\times5=50$

If we look at the first digit of each result, we will see that, except for the first $0$ and the $5$ that appears at the end, the rest of the digits always appear twice. In addition, we can see that the results end alternately in $0$ and $5$.

The simplest method to learn this table is to resort to the table of $5$ and add to it the number we have multiplied, that is, to obtain the answer to $6\times8$, we first calculate how much is $5\times8$, which will give us $40$ as a result; secondly, to this number we add $8$ and we will obtain the result corresponding to $6\times8$; that is, $48$.

This time we use a new table and write down the numbers from $1$ to $9$ from right to left as follows:

1........4.......7

2.......5.......8

3.......6.......9

Now we add, from left to right, the digits from the $1$ to $9$. In the left column, we write the figures $1$, $2$ y $3$ from top to bottom; in the middle column, we write the digits $4$, $5$ y $6$ from top to bottom, and, in the right column, we write the figures $7$, $8$ y $9$ from top to bottom as well. We obtain the following:

21........14........07

42.......35.......28

63.......56.......49

Now we have in front of us the results of the table 7 from 1 to $9$:

$7\times1=7$, $7\times2=14$, $7\times3=21$, etc.

There is a very simple trick that allows us to find out which are the multiples of $8$. First, we arrange the numbers in two rows as follows:

0.......1.......2.......3.......4

5.......6.......7.......8.......9

Then, in each column we add from right to left the even numbers, i.e.: $0$, $2$, $4$, $6$ and $8$.

08.......16......24......32......40

48......56......64......72......80

Thus we get the multiples of $8$:

$8\times1=8$, $8\times2=16$, $8\times3=24$, etc.

To use this trick we must resort to our hands. An important detail that we must take into account is that to the big finger of the left hand we will assign the number $1$ and to the one of the right hand the number $10$. Here we illustrate this statement with a simple example: $9\times5$.

We count from left to right up to the fifth finger and bend it. This finger will act as a guide.

Then we have to ask ourselves two questions:

How many fingers are in front of the finger we have bent? $4$

How many fingers are behind the finger we have bent? $5$

In this simple way we have obtained the answer corresponding to $9\times5$, i.e. $45$.

This time no calculation is needed, just add a 0 to the number we have multiplied by $10$.

For example: $8\times10=80$, $6\times10=60$.

Phases in the memorization of the multiplication tables

The tricks that we have shown above are especially important in the initial phase to make easier the task of memorizing the multiplication tables, although the main objective is that we know them by heart and we do not have to be calculating the multiples all the time.

The method we recommend is to focus, at first, on the simplest multiples, that is, those of $1$, $2$, $5$ and $10$. Then, we recommend to include every couple of days a new table to the ones we have already memorized in order to learn them in an easy and progressive way.

$3\times 3=9$

$7\times 8=56$

$9\times 6=54$

$5\times 8=40$

$4\times 9=36$

$3\times 9=27$

$9\times 5=45$

$6\times 6=36$

$8\times 3=24$

$8\times 8=64$

$6\times 8=48$

$9\times 7=63$

$6\times 2=12$

$3\times 7=21$

If you are interested in other areas of multiplication you may be interested in the following articles:

• Solving equations by multiplying or dividing both members by the same number.
• Multiplication of powers of equal base
• Power of a multiplication
• Decomposing natural numbers as a product of powers
• Multiplication of algebraic expressions

In Tutorela you will find a variety of articles about mathematics.