Multiplying and dividing whole numbers
In a nutshell
You can use both long and short division and multiplication to solve multiplying and dividing calculations when a calculator is not available.
Multiplication
Definitions
Factor  The number(s) that are being multiplied.

Product
 The result of the multiplication. 
Procedure
1.
 Set out the calculation: Write the factors in place value columns, with the larger number going on top of the smaller one. 
2.
 Multiply the units column of the smaller number by each digit of the larger number and write down the results in the answer space below, carrying where required. 
3.  Repeat this for each column of the smaller number, adjusting the zeros accordingly.

4.  Add the results.

Example 1
Calculate $387 \times 56$ using long multiplication.
Set out the calculation.
$\begin{array}{ccccc}&\text{TTh}&\text{Th} & \text{H} & \text{T} & \text{O} \\& & & 3 &8 & 7 \\\times & & & &5& 6 \\ \hline \\+ \\ \hline \end{array}$
Multiply each digit of $387$ by $6$.
$\begin{array}{ccccc}&\text{TTh}&\text{Th} & \text{H} & \text{T} & \text{O} \\& & & 3 &8 & 7 \\\times & & & &5& 6 \\ \hline& &^22&^53&^42&2 \\+ \\ \hline& \end{array}$
Then by $5$, adding a $0$ in the units column.
$\begin{array}{ccccc}&\text{TTh}&\text{Th} & \text{H} & \text{T} & \text{O} \\& & & 3 &8 & 7 \\\times & & & &5& 6 \\ \hline& &^22&^53&^42&2 \\+ &^11&^49&^33&5&0\\ \hline&&&&& \end{array}$
Add the results.
$\begin{array}{ccccc}&\text{TTh}&\text{Th} & \text{H} & \text{T} & \text{O} \\& & & 3 &8 & 7 \\\times & & & &5& 6 \\ \hline& &^22&^53&^42&2 \\+& ^11&^49&^33&5&0 \\ \hline& 2&1&6&7&2 \end{array}$
$387 \times 56=\underline{21\space672}$
Note: Multiplying by a singledigit number in the same way is known as short multiplication. Long multiplication involves adding two or more result of calculations and is used for largerdigit numbers, like you can see used above.
Division
Long division and short division are both methods of dividing one number by another. Long division involves writing down each step of the calculation, whereas short division is done mentally and each step is not written down.
Definitions
Dividend
 The number that is being divided by another.

Divisor
 The number that is dividing another.

Quotient
 The result of the division.

Short division
To use short division, you must be able to divide each column by the divisor. Either you need to know the times tables for it, or you can write out a list of its multiples.
Procedure
1.
 Set out the calculation: Write the dividend in the box and the divisor outside.

2.
 Divide the first digit of the dividend by the divisor, and write the quotient at the top of the box.

3.
 Carry the remainder over to the next column of the dividend to form a new number in that column.

4.
 Repeat the division process with the new number formed, writing the quotient at the top of the box and carrying the remainder.

5.
 Repeat the process until all columns of the dividend have been divided.

Note: If you still have a remainder when you divide the units of the dividend, add a decimal place and a zero to the next column and carry the remainder. Repeat until there is no remainder left and add a decimal place in the correct column to your final answer.
Example 2
Calculate $1020\div8$ using short division.
Set out the calculation.
$8\overset{}{\overline{\smash{)}1\,0\,2\,0}} \\$
Since $1$ cannot divide $8$, carry the $1$ to the next column and divide.
$8\overset{\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,}{\overline{\smash{)}\cancel1^1020}} \\$
Continue carrying and dividing each column.
$8\overset{\,\,\,\,\,\,\,\,\,\,1\,\,\,\,2\,\,\,\,7.\,\,\,5}{\overline{\smash{)}\cancel1^10^22^60.^40}}$
$1020\div8=\underline{127.5}$