This neat little trick shortcuts the multiplication process by breaking it into easy chunks that your brain can handle. The first thing you need to do is multiply the digits together, then double that result and add a zero, and then square each digit separately, and finally add up the results.
Slightly confused? Great – we made a video that outlines each step. There’s a definite pattern and flow to it. With practice, you will be able to do this one in your head within a very short time. Have fun!
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Here are some fun related questions that lead to something bigger later…
Let’s look at the squares of 2-digit numbers and see if there’s a pattern.
Here are a few to look at:
10×10 = 100
11×11 = 121
12×12 = 144
13×13 = 169
14×14 = 196
Just looking at these, you can see:
121-100=21
144-121=23
169-144=25
196-169=27
try a few more. Does this pattern continue?
Using this pattern, if you know the square of a number, can you find the square of the NEXT number?
For example, suppose you know that 25 squared is 625. Can you very quickly find the square of 26, using the pattern above? Now mutliply 26 x 26 using any method you like.
Do you get the same answer that you predicted using the pattern?
Does this ALWAYS work?
If you know the square of a number, can you find the square of a number that is, say, three larger?
For example, if you know that 10×10=100, can you then find the square of 13? (It’s 169, but how did that happen?)
Hint: look at this again:
100
121
144
169
Can you see this?
169 = 100 + 21 + 23 + 25
Another way to say it is:
169 = 100 + 23 + 23 + 23
169 = 100 + 3×23
Again: If you know that 10 x 10 = 100, and you want to find out what the square is for a number that is 10 + 3, which is 13,
that is, you want to find 13 squared,
can you figure out a rule that will give you the magic number 23?
Now try it with other two-digit numbers that are 3 greater than numbers with easy squares. Does your rule work for finding their squares as well?
Try other numbers. Does this pattern only hold for two-digit numbers?
Have fun!
Basically, using the first example:
23 squared is (20+3) squared
Using “FOIL” (you learn this in algebra for squaring a binomial)
you get: 23 squared = 20×20 + 20×3 + 20×3 + 3×3
THE TWO MIDDLE TERMS COME OUT LIKE THIS:
20×3 + 20×3
is the same as 2x(20×3)
which is 2×2×10×3
which is (2×3) x 2 x 10
You can get this by multiplying 2×3 and doubling it,
then tacking a zero to the end of the result,
and that’s why you get 120
THE FOUR – DIGIT NUMBER COMES FROM THE FIRST AND LAST TERMS:
Remember, 23 squared = (20×20) + (20×3)x2 + (3×3)
We already figured out that (20×3)x2 is 120 (above)
So we just need to figure out (20×20) and (3×3) and then add that to our 120 to get the final answer.
Here goes:
20×20 is (2×10) x (2×10)
which is 2×2 x 10×10 (rearranging the numbers you get the same answer when multiplying)
which is just 2×2 and then 2 zeroes added to the end of it
(that’s how you get the first 2 digits of your initial 4-digit number, 0400)
and then 3×3, the final term, added to 0400 gives 0409
and finally, adding back the 120 from the first step gives 0529
so your answer is 529.
ta-da!
(or some people like to say “QED” which is a fancier way of saying “ta-da!”)
Here’s another way of looking at it:
23 x 23 is the same thing as 23 + 23 + 23…. twenty-three of these.
———————————
But that’s a bit hard to do in your head, right?
So instead, you decide to try this:
20 + 20 + 20 …. twenty-three of these
and then
3 + 3 + 3 … twenty-three of these.
———————————-
Still too hard?
ok, how about this:
We let our friends Adam, Bob, Charles, and David help us out.
We tell Adam to add up 20 + 20 + 20…. twenty of these,
and
Bob gets to add up 3 + 3 + 3 …. twenty of these.
but then of course you still need two more friends:
Charles adds together 3 more 20’s
and
David adds up 3 more 3’s.
Then all our friends will add their results together to get the final answer.
————————————–
Now. Adam looks at his 20 20’s and says, hey that’s just 20×20,
which is the same as 4 x 100. So he writes down 0400.
David looks at his 3×3 and says, hey that’s just 9. Then Adam and
David put their answers together to get 0409.
Meanwhile Charles and Bob have looked at their problems.
Bob has 3×20, and Charles has 20×3. Both of them decide to just
look at 3×2 first, which is 6, and multiply by the 10 later. But wait!
They both got 6 as their answer. So there are two sixes! So they
double the six to get 12, and then they remember to stick the zero
on at the end to get 120.
Now, David and Adam got 0409, and Bob and Charles got 120.
Adding all of their answers together to get the total, we get 0529,
or just 529.
The end.
The real mystery, of course, is why 2-digit multiplication works the way we’re taught in elementary school.
23
x23
= 69
+460
=529
Really, why doeesn’t anyone tell you:
23 x 23 is the same as 23 x 3 plus 23 x 20.
23 x 3 is the same as (3×3) plus (3×20) [which is 69]
and
23 x 20 is the same as putting a zero at the end first and then figuring out 23 x 2
(23 x 2 is then the same as 2×3 plus 2×20) [so that’s where you get 460]
So you end up with 3×3 plus 3×20 plus (2×3) x 10 plus (2 x 20) x 10.
Which is 9 + 60 + 60 + 400
Which is 409 + 120 (adding the numbers in a different order)
Which is 529.
The end.