Mental multiplication - via Chunking, Rescaling and Spitting
Chunking
In multiplication, the Distributive law states that: ( a + b) × c equals ( a × c ) + ( b × c )
'Distribution' is the American term for what is called 'expanding brackets' in the U.K. However the rule above helps with general multiplication, so it's not just for brackets. In essence:
Chunking is where a difficult multiplication is split up into two or more easier multiplications which are all added at the end to give the final answer. It works because of the rule above.
There are often many ways to 'chunk' a multiplication, and it takes a bit of practice to 'recognise' which is going to be the easiest option. For example:
16 x 14
is the same as ( 10 x 14 ) added to ( 16 x 4 ) . All we have done is split the 16 into 10 and 6.
16 x 10 is easy: 160. Now we just need to do 16 x 4. You can chunk this as well - it is the same as 10 x 4 added to 6 x 4.
10 x 4 is also easy, 40. Add the 40 to the 160 above to give 200. Now all that remains is 6 x 4, which we know from our times tables is 24.
Add the 24 to 200 to give 224, the final result.
Rescaling
Rule: when multiplying two numbers, if you multiply one side by a number and divide the other side by the same number, you get the same result.
The trick is to spot which multiplications are suitable for rescaling. Most often, rescaling works best when you spot where a times and divide by either a 2 or a 3 would make the whole thing easier. For example, let's do 16 x 14 again , as in the previous example:
16 x 14 is the same as 8 x 28. All we have done is halve the 16 and double the 14. Now keep going:
8 x 28 is the same as 4 x 56 . Keep going, and we get 2 x 112.
112 is reasonably easy to double mentally, giving 224 as a final answer.
The general principle is that it is usually easier to multiply a long number by a single digit than it is to multiply two medium-sized numbers together.
Another example:
18 x 4 is the same as 9 x 8 - all we have done is halve the 18 and double the 4. Now we know the answer from our times tables: 72.
Combining chunking and rescaling
Sometimes it's easier to use both techniques to arrive at answer. For example:
124 x 12. first let's rescale this by halving/doubling to 248 x 6 and continue halving and doubling to 496 x 3 .
Now we can chunk this: 496 x 3 is the same as ( 400 x 3 ) plus ( 90 x 3 ) plus ( 6 x 3 ) .
This is:
1200 + 270 + 18 = 1470 + 18 = 1468 + 20 = 1488.
We could have done this differently, like this:
First chunk the 124 x 12 to 124 x 10 and 124 x 2 . This looks easier:
124 x 10 is just 1240 ... double 124 is 248. Add them: 1240 + 248 = 1238 + 250 = 1438 + 50 = 1488.
So in other words, do you chunk first or second, or rescale first or second? The best way depends on the multiplication involved, and takes a bit of practice to get right. Both will give the right answer.
Splitting
Another technique is splitting: If a multiplication involves numbers that are easily factorised, then the calculation can be 'spread out' over a series of smaller multiplications. For example
Multiplication by 8: 8 is the same as 2 x 2 x 2 , so doubling a number three times is the same as multiplying by 8.
Multiplication by 9: 9 is the same as 3 x 3 , so multiplying a number by three twice is the same as multiplying by 9.
Example:
24 × 8 is the same as 24 × 2 × 2 × 2 , so in other words, double it three times:
24 × 2 = 48
48 × 2 = 96
96 × 2 = 192
Multiplying higher, then subtracting.
This is a bit like chunking but with a subtraction. Example:
78 x 12 .
Step 1) Multiply a higher value of 80 : 80 x 12. We can chunk this as 80 x 10 and 80 x 2 ,
which gives 800 and 160, which total 960
step 2) Multiply 2 by 12 giving 24. Now just subtract this from the total: 960 - 24 which is 966 - 30
which comes to: 936 .
Difficulty Levels - typical questions:
1: 15 x 7
2: 124 x 16
3: 362 x 58