We have often been asked whether chunking, which seems so popular with
children learning addition and subtraction methods, can be extended to
apply to multiplication and division. The answer is 'yes' for
multiplication, in as much as a very similar approach to the one used for
+ and - carries over into ×. However, for division, the process changes
somewhat, but nonetheless a process in which we chunk, or break a
calculation down into manageable parts still applies.
Multiplication
We wouldn't normally use the chunking method with multiplications that
involve numbers greater than 100 and we would introduce it with a
single-digit by double-digit example and plenty of practice. However,
chunking does extend to
higher order multiplications and the way in which the method emphasizes place
value is well-illustrated by the example below.
Here we are multiplying 42 by 317 and we chunk both numbers into their
single-digit components. Now, bearing in mind that the first 4 isn't a 4,
but a 40, we can produce the first three multiplications, 40 × 300, 40 ×
10 and 40 × 7. Moving on the the second digit in 42, we produce the
second three results from 2 × 300, 2 × 10 and 2 × 7.
That's all the multiplication that we need to do, as the remaining task
is to add the numbers that we have created in the second row by the normal
method of chunking for addition. The final outcome, therefore, looks like:
God forbid that we ever have to carry out a multiplication such as this
in our heads! However, we would like to say that the single-digit by
double-digit multiplication, when done in the chunking style, very much
mirrors mental methods.
Division
Suppose that we now want to divide 317 by 42. The chunking approach is
to use our key addition strategies, such as double or multiply by 10, to
find the answer in easy stages. This turns division into a repeated
subtraction process, subtracting only numbers that we feel comfortable
with.
We would encourage the children to try times 10 first, and would expect
them to notice that 420 is more than 317, so 10 is too large. Then we
might fall back on doubling to get:
Here, we have doubled 42 and subtracted the result from 317 to get 233.
The process can continue in this way. In the diagram below, we noticed that
42 times 10 is 420 and we can halve that to get 210 as our subtrahend
(that's the posh word for the take-away number). The process is completed
as:
The layout shows that we have multiplied 42 by 2 and by 5, making 7
times in all, and that there is a remainder of 23.
At no point do we ever do anything more than apply addition processes
that the children feel comfortable with and do subtractions that they can manage.
Contrast this approach with the traditional method in which we would need
to search for the number by which to multiply 42 and then carry out the
multiplication in order to follow the steps of the traditional algorithm
correctly. Yes, it's a good thing that we do have calculators to handle
this kind of problem but if we want to know if children understand the process,
then chunking makes it clear at each stage what is going on ... and builds
on skills that the children are confident with, rather than expecting them
to learn a new recipe ... would that be for disaster?
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