Great that this thread has been revived as I’ve been thinking about some of its content. With 79 x 69 I would use difference of 2 squares as per Kinma’s post on this subject, so it becomes 64^2 - 5^2.
In your opinion what is the most difficult type of 2 x 2 calculation ie when they don’t factorise eg 89 x 47, or when they end with high unit digits such as 87 x 39?
Criss cross using negative values:
90 -1
50 -3
_______ X
90X50 = 4500
(-1 X 50) + (-3 X 90) = -50 - 270 = -320
-1 X -3 = 3
__________________ +
4500
-320 (=4180)
-+3 (=4183)
Anchor method using 50 as anchor:
1st step. We add 3 to 47 to get 50.
Therefore we subtract the same 3 from 89 to get 86.
Now multiply them:
50 X 86 = 4300
Now subtract the difference from the base numbers (we changed 47 into 50 and 89 into 86. So difference is 3) and multiply by the distance of the anchor and the biggest number:
-3 X (89 - 50) = -3 X 39 = -117
4300 -117 = 4183
Btw this is based on: (a+c)(a+d) = a(a+c+d) + cd
Criss cross (using negative values):
90 -3
40 -1
__________ X
3600
-90-120 = -210 (total is 3390)
–1X-3 = 3 (total is 3393)
Anchor using 40:
40 X 86 = 3200 + 240 = 3440
-1 X (87-40) = -1 X (47) = -47
3440 -47 = 3393
You privately asked me:
Anchor method seems quick:
50 X 77 = 3850
3 x 24 = 72
3850 + 72 = 3922
Either do:
70 -3
90 -3
Or anchor:
60X94 + 7X27
Either do:
70 -3
80 -1
Or anchor:
60X86 + 7X19
Thanks Kinma, it would appear that you can convert any 2 x 2 multiplication no matter how unfriendly to something very manageable. I do like the criss-cross method because of its universal application but for me I need to strengthen my memory so I can remember all the partial products.
What helps for me is to repeat the numbers I am working with and let my mind listen to it.
So in the case of 89 x 47 when using the Anchor method I first mentally change the numbers so the lowest comes first:
47 X 89. Then I change add and subtract 3: 50 x 86.
Then do the multiplication: 4,300. I then try to see the 4,300 as well as pronounce it in my head. A couple of times until it sticks.
Then 89 - 50 = 39. Again; repeat the 39 AND try to see the 4,300 as well.
Then multiply the 39 by 3: 117. First stick with the 117 in your mind, repeat it a couple of times until you firmly ‘see’ it in your mind.
Only then do the subtraction. Done.
Thanks Kinma, always helpful and educational in your replies.
I don’t understand how you get the 3 X 37 bit. Could you please detail? Thank you very much!
I wrote that here. Let’s repeat it:
Let’s say you need to calculate 43 X 77.
This can be rewritten as 40 X 80 + 3 X 37.
40 X 80 is easy: 3200.
3 X 37 is 111.
3200 + 111 =
3311
This is called the anchor method. Do a search on this forum for details.
In the anchor method, when we have a multiplication like 43 X 77, it is much easier to begin with 40 X 80. We subtract 3 from the first number (43). We call this new number the anchor (40). Then we add 3 to the last number (80).
We multiply them (3200) and then we take the number we changed the multiplication numbers with (3) and multiply this with the difference of the highest number with the anchor. 77-40 = 37.
We multiply these two numbers. Therefore 3 X 37 (111).
Lastly we add them. 3200 + 111 = 3311.
I have written about the anchor method a lot in the past.
But here is a new way to look at this.
Again we want to find 43 X 77.
1: We could start with 40 X 77 and then add 3 X 77.
2: In order to find the first part or 40 X 77, we can start with 40 X 80 and subtract 3 X 40.
Combining the two, we can start with 40 X 80, add 3 X 77 and subtract 3 X 40.
However; if we first add 3 X 77 and then subtract 3 X 40, we can do this in one go by just adding 3 X 37.
Let’s do another one: 43 X 82.
We add 2 to the left part and subtract 2 to the right part and get: 45 X 80 = 90 X 40 = 3600.
The amount we changed the numbers with is 2.
The anchor is 45
82 - 45 = 37.
2 X 37 = 74.
Keep in mind that in 43 X 77 we subtracted 3 from 43.
Now we go in the other direction: from 43 to 45.
The 74 therefore needs to be subtracted.
3600 - 74 is 3526.
So 43 X 82 = 3526