Teaching
algorithms for multiplication
In
the primary school, children are taught multiplication using a formal
written method that is based on:
Understanding
the formal written algorithm for multiplication depends on assembling
together understanding of several separate steps. Therefore the
ideas must be introduced through a number of stages. Students need
to be competent and comfortable with each stage prior to moving
onto the next stage. Ample experience with place value materials
prior to the introduction of symbolic notation will assist children
consolidate knowledge at each stage.
The
sections below give the reasoning behind the steps of the formal
written algorithm, the intermediate forms which teachers use before
students learn the most efficient procedure and link to explanations
using place value material.
Stage
2: Multiplication by a single digit
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 property) is well understood. Ruling up (or using squared paper) and
labeling 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 this aspect of multiplication
is explained with place value material.
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Stage
3: Multiplication by 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", but this is dangerous terminology.
Teachers should be aware that many children may fall victim to simply
knowing the rule of "adding zeros". It is important that
children understand that the effect of the zero is to move digits
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
Click
here to see how this is explained with
place value material.
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Stage
4. Multiplication by a multiple of ten and power of ten
After
learning how to multiply by ten, children can see how to multiply
by multiples of ten. This step relies on understanding the associative
property of multiplication and understanding. This step is NOT assisted
by place value or other concrete materials.
Multiplication
by 30 is done by multiplying by 3 and then by 10 (or vice versa).
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 so moving
the digits by two place value columns) and then by 3
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Stage
5: 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 practice sufficiently to be able to accurately
calculate with very large numbers. Extensive practice is no longer
a high priority.
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, putting
down the zero, and then multiplying by 4)
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Other
ways of setting out the algorithm
There
are many 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. Using squared paper for mathematics
is a great idea to help keep digits aligned  in many countries
normal writing paper is already squared.
Move
carry digits somewhere else and move the multiplication sign
to the other side. 

Move
carry digits somewhere else and place the multiplication sign
on top other side. 

Not
advised. 

