# what is teaching kids the rules and exceptions in multiplication called?

I recall reading a website quite some time ago about the rules and exceptions of multiplication with regards to teaching children. For instance: The result of multiplying any number times 9 will have a result where the digits add to 9 (1x9=9, 2x9=18 so on, but breaks at 13). It was something like that -- there was a grid and there were about 8 'rules' and 5 'exceptions' that would let you multiply any integer under 13 by any other integer under 13 -- I think.

What's this technique called? Is it a good idea to teach kids multiplication with this technique?

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These could be "divisibility tests" or "sanity checks". The one for $9$ in particular is a special case of "casting out $9$s".
I don't see why you say that it "breaks at 13": $9\times 13 = 117$, and $1+1+7=9$. What is true is that eventually you get numbers that add up to more than $9$, but if you repeat the process you will eventually get to $9$. For instance, $11\times 9 = 99$, and $9+9=18$, not $9$; however, if you repeat the process you get $1+8 = 9$.
Note that these rules don't actually teach multiplication, they just provide methods to check multiplication; they are also not definite tests: casting out nines does not detect all errors. For example, if you make the mistake of thinking that $9\times 9 = 18$, adding the digits to get $9$ will not disclose the error.