|
|
Survey
of mental methods for multiplication
This
section shows some of the informal methods used by children to carry
out multiplication, loosely classified according to the mathematical
principles used. In practice, some mental methods used by children
will have characteristics of more than one classification.
These
methods are presented in a written form to make then easier to follow,
but in practice they would be done mentally, or with some jottings
on paper, organised in an idiosyncratic way.
Front
end multiplication
When
multiplying mentally, people generally like to do the bigger or
important quantities first. This gives an estimate of the value
expected. Such an approach is front end or left to right multiplication.
Students using this method multiply the tens before the ones and
hundreds before the tens and so on. They then add up these partial
products to get the final result. This method is based on the distributive property of multiplication over addition.
Given
below are some examples of front end multiplication. Note that there
will be more than one possible way of presenting the written form
as this is mainly used for mental computations.
|
Mike
|
Question:
23 x 8
|
|
I
know that 23 is 20 + 3.
So
I multiply 20 by 8, and then 3 by 8.
I
now add 160 to 24 to get my answer of 184.
|
|
|
Sam
|
Question:
535 x 3
|
|
I
know that with large numbers I can think of them in separate
parts so 500 = 500 + 30 + 5
I
like to start with the biggest number first.
I
can now easily add the parts together to get the answer.
|
|
|
Chris
|
Question:
24 x 15
|
|
I
am going to multiply 24 by 10 by first multiplying 20 by 10
and then 4 by 10.
I
am then multiply 24 by 5, by first multiplying 20 by 5 and
then 4 by 5.
I
then add each of these results to get my final answer.
|
|
Multiplication
in stages
Some
students simplify multiplication by multiplying by the factors one
by one instead of the number itself. This method could help children
learn multiplication tables. For instance, the 4xtables would be
much easier if thought of as double the 2xtable, or that x6 is double
of x3. Here are some examples to illustrate some of the procedures
using this principle.
|
Sarah
|
Question:
25 x 6
|
I
know that the factors of 6 are 2 and 3 which are both numbers
I can easily multiply by.
So
I multiply the 25 by 2 first and then the answer by 3.
|
 |
|
Nick
|
Question:
25 x 6 |
|
The
factors of 25 are 5 and 5. If I multiply 5 by 6 I get 30.
To multiply by 30 I can multiply by 3 and then 10.
|
 |
|
David
|
Question:
12 x 40
|
|
This
method is often used when multiplication involves multiples
of 10 as illustrated by David's method.
|
 |
Multiplication
by rounding
Children
usually use this method in multiplication problems where one of
the numbers is close to a multiple of 10. Sometimes they use it
for numbers nearby a known fact. Such a number is then rounded to
the nearest ten or the known fact and then adjusted by addition
or subtraction. Thus this method again uses the distributive property.
|
Jason
|
Question:
99 x 5 |
|
I
notice that 99 is close to 100 which is number I can easily
multiply by.
I
times 100 by 5 and 1 by five.
Since
100 fives is 500, 99 fives will be 5 less.
|
 |
|
Julia
|
Question:
25 x 7 |
|
Four
25's make 100, so seven 25's is 100 + 25+25 + 25, total 175.
|
|
Julia
|
Question:
26 x 7 |
|
I
already know 25 sevens, so I can use this.
|
 |
| Indira
|
Question:
198 x 3 |
|
This
is a difficult sum but I notice that 198 is close to 200.
200
times 3 is easy for me to solve.
Since
200 threes is 600, 198 threes must be 2 threes less, so 600
minus 6.
|
 |
Multiplication
by halving and doubling
Another
method adopted by students is to double one factor and halve the
other which leaves the product unaltered. This may be extended to
four times one factor and quarter the other, or eight times one
factor and eighth the other depending on the question. This method
is actually a special and very common case of multiplication in
stages.
|
Andrew
|
Question:
8 x 15
|
|
|
|
Luke
|
Question:
16 x 25
|
|
|
|
Alan
|
Question:
51 x 16
|
|
51
x 16 is difficult to solve,but I notice that I have two choices.
51 is close to 50 and 16 is easily halved to give 8. Halving
is probably easier.
I double 51 and halve 8 giving 102 x 8
This is now easily solved by adding 8 x 100 and 8 x 2.
|
|
This
gives a quick method for multiplying by 25. Divide by 4 and add
two zeroes.
Similarly
to multiply by 50, divide by 2 and add two zeroes, to multiply by
125, divide by 8 and add three zeroes.
Encouraging
students to look for patterns in multiplication and finding quick
methods such as those above will help develop their mental multiplication
skills and number sense.
Multiplying
by special numbers
Many
students will enjoy working out special methods for special numbers.
Here are some examples worth remembering. Classes should talk about
WHEN and WHY these shortcuts work.
|
Shortcuts
|
Examples
|
|
To
multiply by 25, multiply by 100 and divide by 4 (because 25
is 100 divided by 4)
|
36
x 25 is 3600 divided by 4, so the answer is 900.
|
|
To
multiply by 125, multiply by 1000 and divide by 8 (because
125 is 1000 divided by 8).
|
56
x 125 is 56000 divided by 8, so the answer is 7000.
|
|
To
multiply by 9, multiply by 10 and subtract the number (because
9 is 10 minus 1).
|
123
x 9 is 1230 subtract 123, which is 1230 subtract 100 (so 1130),
then subtract 20 (so 1110) then subtract 3 (so 1107)
|
|
To
multiply by 4, double twice (because 4 is 2 by 2).
|
65
x 4 is done by doubling 65 (get 130) and doubling that (get
260)
|
|
To
multiply by 11, add adjacent digits, because 11 is 10 + 1
|
62
multiplied by 11 is 682 (write down the 6, add the 6 and 2
and write down the 8, write down the 2)
6352
x 11 is 69872 (write down the 6, add 6 &3, write down
9, add 3 & 5, write down 8, add 5 & 2, write down
7, write down the 2)
(Be
careful: check what happens when the sums go over 10)
|
|
Make
up some shortcuts of your own
|
|
Other
methods to suit specific questions
Apart
from the methods classified above, children devise short cuts to
suit the question at hand. Such methods often combine more than
one of the above methods.
|
Brendan
Brendan uses repeated addition for 35 x 3. However for
the 32 x 45 he uses halving and doubling followed by front
end multiplication. Good mental calculators use a variety
of methods, depending on the actual numbers in the question.
|
|
The
multiplier (3) is quite small so repeated addition can be
easily used.
|
Question:
35 x 3
|
|
Question:
32 x 45
|
Anna
Anna
uses her knowledge of algebra: she knows the "difference
of two squares", a number pattern which she can express
as follows. |
|
Question:
8 x 12
|
|
More
examples of this method:
7
x 13 = (10 - 3) (10 + 3) = 10 x 10 - 3x3 = 100 - 9 = 91
24
x 16 = (20 - 4) (20 + 4) = 20 x 20 - 4x4 = 400 - 16 = 384
|
|
Monica
Monica
has combined halving and doubling and front end multiplication.
It is clear that she knows her number principles and multiplication
facts well but goes to great lengths to avoid the actual written
algorithm. Although students' own methods should be encouraged,
it is also important to teach students an efficient algorithm.
As Monica's example show, although invented methods can give
the correct answer, they may be more time consuming.
|
|
26
x 84 is difficult to solve. If I halve 26 to get a smaller
number this will help. To multiply by 13, I will think of
it as 10 plus 3.
Using front end multiplication the first part is solved 10
x 168 =1680
168 is then seen as 100 + 60 + 8, which are each easily multiplied
by 3.
The separate products can now be added.
|
Question:
26 x 84
|
|
