In a different thread we talked about extending the multiplication tables from 12X12 to 20X20.
A lot of the time you will be multiplying a number between 10 and 20 with another one.
These can be calculated extremely quickly.
Here is an example:
12X13
Start with 1 for the first digit (or 1X1).
Add the 2 and the 3 to get 5.
Multiply the 2 by the 3 to get 6.
1 5 6
If the numbers get bigger you need to take care of the carry:
18X19
Start with 1 for the first digit.
8+9=17
8X9=72
1|17|72 = 342
Here you have to carry 2 times.
It might help in these cases to see the first digit as the hundreds, the second as the tens, etc.
Then the steps are:
1: The hundreds. Start with 100
2: The tens. 8+9=17 (tens) so 170, add to previous to get 270
3 8X9=72. Add to previous. 270+72 = 342.
The 20x20 is just 200 different results (since half of them are symmetrical.)
But for people who know the 12x12, the 72 are already known and trivial.
Others like the multiples of 15 for 11 up to 20, are also very easy
…165, 180, 195, 210, 225, 240, 255, 270, 285…300
But even for stuff like 19x18, I think memory recall should be needed, and not calculation. Because carrying digits is considered calculation.
But even If you forget the result of that grid, then the distributive property is our friend. Just do 19x18=(18+1)x18=18^2+18=324+18.
So, let’s say if I forget the result of 24x23 or 18x19, I just do 529+23 and 324+18 respectively, since the 2-digit squares are memory recall for me, not calculation.So, 18x19 is basically converted to a simple addition, 324+18
P.S - For that specific example you mentioned,my mnemonic rule is that 18x19 is the same as 18^2, but with the last 2 digits inverted, e.g. 342 instead of 324.
If you invert this number further: 234 = 18 x 13, again a multiple of 18.
and if you invert it further: 432 = 18 x 24, again a multiple of 18. And of course 4+3+2=1+8.
There are such patterns everywhere.
As a data point the places where I stumble as I am working beyond 12 (rote) to 20+ is not in the 1a * 1a but rather in the 6,7,8, * 1a where the step requires a subtotal and a carry.
1214 or 1317 are easy to simply read/think from left to right. It is the simpler 14*8 that make me stop and think 80+32 when it would be a lot more pleasant just to be able to read it as 132.
Another example I can enumerate by 2’s, 5’s, 10’s without pause but if I mentally try to count by 17’s I cannot sustain the same mental pace.
Where I have familiarity with these very simple relationships I don’t make errors.
You make an excellent point that memorization isn’t terribly important at this level but the challenge I find as one of the great unwashed is that I have gaps in learning of basic addition and subtraction that are being exercised by learning new multiplication tables.
This is normal. You need to stop, take care of the carry and continue. this interrupts your thought process and slows you down.
Here is how I deal with this and would love to hear how Nodas does it.
The moment I see ‘80’ in my mind, I also see ‘100-20’.
This stays in the back of my mind. It is like working with 80 and 100-20 at the same time.
Then when I see 8X4, I first see 32, then 20+12.
The ‘20’ comes up because of the previous step.
This quickly resolves to 112 when I add them.
Again, this is normal. If you recite the table of 17: 17, 34, 51, etc., when you get to 34, train yourself to also see the distance to 50. Why 50? Well, adding 17 will get you close to that number. You probably immediately see ‘16’ if you think of the distance.
If you do this your mind will jump to 51 as the next number. Try it.
This is similar to my advice with the previous step.
So your mind will come up with ‘51’. This is the low 50’s. No chance on a carry, so the next step will be easy: 68.
68 is in the high 60’s. So there will be a carry. 68 is 12 removed from 80 so your mind will jump to 85 for the next step.
Try this. It will take some getting used to, but this will save a lot of time in working with carries.