Formal Algorithms for
Multiplication
Contents
The formal algorithm for multiplication (Long multiplication)
Multiplication by a single digit
Multiplication by a multiple of ten
The formal algorithm for multiplication
In the primary school, children are taught multiplication using a formal written method that is based on
the place value system
multiplication tables up to 10 by 10
the distributive property of multiplication over addition.
This algorithm (often called "long multiplication") and its teaching is illustrated below in several steps. Teaching the algorithm proceeds in three steps: multiplication by a single digit, then multiplication by a multiple of ten and then to multiplication by numbers with two or more digits. Now calculators are widespread, not all children need the third stage.
Example 1: Multiplication by a single digit
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2 3 x 4 |
23 is 2 tens and 3 ones.
3 ones multiplied by 4 gives 12 ones and
2 tens multiplied by 4 gives 8 tens (that is 80).
80 and 12 are added to give the final product 92.
Children should write multiplication in this form for some time, until the procedure is familiar and the concepts (especially the distributive law) is well understood. Ruling up and labelling columns for tens and ones is recommended in the early stages. Later it can be reduced to a more compact form:
The 3 ones are first multiplied by 4 giving the product 12, which is 1 ten and 2 ones. 2 is written in the ones column and the 1 is recorded in the tens column. Now the 2 tens are multiplied by 4 to give 8 tens. The 1 ten recorded before is added on, so the product has 9 tens.
Click here to see how multiplication is explained with Multi-Base Arithmetic Blocks (MAB).
Example 2 Multiplication by a multiple of ten
Children must learn how to multiply by multiples of ten. It is very important that they know that to multiply a whole number by ten a zero can be added to the number. It is better to say that the digits move into the next larger place value column.
10 x 2 = 10 x 2 ones = 2 tens = 20
10 x 152 = 10 x (1 hundred + 5 tens + 2 ones)
= 10 hundreds + 50 tens + 2 tens
= 1 thousand + 5 hundreds + 2 tens
= 1520
(see also Multi-Base Arithmetic Blocks)
After learning how to multiply by ten, children can see how to multiply by multiples of ten.
To multiply by 30, first multiply by ten (by putting down the zero) and then by 3
To multiply by 300, first multiply by one hundred (by multiplying by ten and then by ten again i.e. putting down two zeros) and then by 3
Example 3. Multiplication by a number with two or more digits
These multiplications require understanding of all that has come before. They are less important now that calculators are common so not all children need to master the algorithm. Extensive practice is no longer a high priority.
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5 7 x 4 6 |
To compute this product, 57 is first multiplied by 6 ones and then by 4 tens. The two results are then added to get the final result. It will be written down as follows.
Next, 57 is multiplied by 40 (this is done by multiplying by 10 &endash; putting down the zero- and then by 4)
Other ways of setting out the algorithm
There are a variety of slightly different ways of setting out the algorithm. The choice is unimportant, except that omitting zeros (as in the final example) is inadvisable. Children are more likely to keep columns aligned if they put in the zeros.
Move carry digits somehwere else, move multiplication sign to the other side
Move carry digits somewhere else and mult sign on top other side
Not advised
Teaching the formal algorithm in the calculator era.
Children should be able to do Multiplication by a single digit on paper,
Multiplication by a multiple of ten in their heads and it doesn't matter if not all children can do Multiplication by numbers with two or more digits &endash; except for special numbers done mentally.
Click here to see
Other algorithms for multiplication