Vertical Multiplication

Vertical multiplication is a basic topic in mathematics that every student must know and be able to solve.
To solve vertical multiplication exercises, in a simple and practical way, you must master the multiplication tables and follow the rules meticulously.

Correct notation: Ones under ones, tens under tens, and hundreds under hundreds.

Observe the exercise $4\times 34=$
To convert it into a vertical multiplication, we must write the numbers one under the other, ensuring that the ones are under the ones, the tens under the tens, and the hundreds under the hundreds.
Moreover, the longer number, the one that contains more digits, should be written at the top.

Solution:

Now we will multiply the ones digit $2$ by the ones digit $4$. We will write the result and continue.

Now we will multiply the ones digit $2$ by the tens digit $3$ and write the result as follows:

When the result is greater than $9$ it is stored at the top left and must be remembered to add it to the next result. In the result row, only the ones digit is noted.

Let's move on to a more complex exercise.
$36\times 8=$

Solution:
Let's write it in vertical form:

Let's multiply the ones digit $8$
by the ones digit $6$
We will get $48$
$48$ is greater than $9$.
Therefore, we will apply the second rule and note in the result row only the ones digit $8$.
The $4$ will be written at the top left and remembered to add it to the result of the next multiplication.
We store it above the $3$.
Now let's multiply the ones digit $8$ by the tens digit $3$ and let's not forget to add $4$ to the result.
$3\times 8=24$
$24+4=28$
We will note $28$.

Erase "the carried numbers" at the top left before moving on to multiply the next digit, this prevents confusion.

Add $0$ below the result to indicate that you move to the next digit, each row of results starts one place to the left in relation to the previous row.

Now we will see the multiplication of a two-digit number by another two or three-digit number, so we can apply the third and fourth rules.
Observe the exercise:  $358\times 38=$
Solution:

Let's write it in vertical form according to rule $1$.
Multiply the ones digit $8$,
by each of the digits according to rule $2$.

$8\times 8=64$
$8\times 5=40$
$40+6=46$
$8\times 3=24$
$24+4=28$
Now, according to rule $3$ let's erase the "carried numbers" on the top left to avoid confusion.
Furthermore, according to rule $4$ we will add $0$ below the answer to indicate that we have moved to the next digit and we will start writing the row of results one step to the left from the previous row.

That is:

After erasing and moving one step to the left, we can move to the tens digit $3$ and continue multiplying it with the ones, tens, and hundreds, just as we have done so far.
Notice that, the result will be written to the left of the $0$ we added in the following way: 
Make sure to write the digits correctly, each digit below the corresponding one.

$3\times 8=34$
We will keep the $2$ and continue.
$3\times 5=15$
$15+2=17$
We will keep the $1$ and continue.
$3\times 3=9$
$9+1=10$

At this point, all we have left to do is, add all the solutions obtained, in the same way we solve a common addition exercise in vertical form.

Attention: if it were a multiplication of a three-digit number by another three-digit number, when moving to the third digit, we should reserve another place with the $0$. That is, $2$ places and then the answer would be written $2$ steps to the left.