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?


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.

  • $\begingroup$ After 10 the digits add up to 18 if you look at them right; here's another for-instance: when multiplying by 8, the first digit always increases by 1 and the 2nd digit always decreases by 2 (8, 16, 24, 32, 40, 48, 56...) except at 7 and 11. So there's an exception and rule. $\endgroup$ – jcollum Nov 15 '11 at 17:35
  • $\begingroup$ I'm looking for a technique to teach my kid the patterns in numbers, hoping that it will get him thinking about math instead of just memorizing the numbers. $\endgroup$ – jcollum Nov 15 '11 at 17:39
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    $\begingroup$ @jcollum: In my experience, that sort of things is better discovered by one-self than taught; otherwise, you are teaching him to memorize the pattern instead of memorizing the number, and you aren't really encouraging much, in my humble opinion. $\endgroup$ – Arturo Magidin Nov 15 '11 at 17:48
  • $\begingroup$ you were right about the 9's tho: 9 x 235 = 2115 which becomes 2 + 1 + 1 + 5 = 9. Wow. $\endgroup$ – jcollum Nov 15 '11 at 17:54
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    $\begingroup$ "Note that these rules don't actually teach multiplication" I couldn't possibly agree more. When teaching elementary mathematics, one should be seeking unity as much as possible. Taking the single concept of multiplication and breaking it into a dozen rules intended for memorization is about the worst thing you can do for a child. $\endgroup$ – Austin Mohr Nov 15 '11 at 19:54

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