Take whatever questions the child has found interesting to study, and expand upon them. Some directions will be more fun than others: pay attention to that.

For the questions you've mentioned:

  * What happens to divisibility tests in other numerical bases? (That's where we use a different number of digits: we count up to a different number before deciding we've reached one-zero.)
      * Can you find one? (That digit sum insight might help!)
      * Why does it work? Can you prove to yourself that you won't ever find any counterexamples?
          * Can you convince somebody else? (Even somebody who's being totally unreasonable about what they'll count as proof?)
      * (If it doesn't work:) Why did it look like it worked? What's actually going on here?
      * Can we find base-10 divisibility tests for the _hard_ numbers, like 7?
      * Why the similarity between base-9's 4-test and base-10's 3-test?
  * What happens if we only use the last digit of calculations (like addition, multiplication, subtraction)?
      * This is the same as using remainder-10: what if we use remainder other things? (5, 7 and 8 are good numbers to look at.)
      * If division is the opposite of multiplication, can we do remainder-5 division?
      * What about remainder-10 division? Which numbers don't work?
      * For which remainder is 0 the only number you can't divide by?

Using techniques you'll already have studied, like “fill in the blank equations” (algebra), can really help with these problems! (No need to use letters, if squares and circles and different colours do the trick – but equally, no need to use those if letters work fine.)

If you want to make sure you're not making future school-maths boring, try university-level mathematics. (No, I'm not joking: the only reason half that stuff is “university-level” is because it's currently not in the curriculum for younger people, often because it's hard to write exam questions for.) Those questions you asked about operations lead nicely on to the study of [binars of various kinds](https://en.wikipedia.org/wiki/Magma_(algebra)#Types_of_magma). Yes, this is _incredibly_ dull – but the maths isn't! [Clock addition](https://en.wikipedia.org/wiki/Cyclic_group#Additional_properties), [flipping shapes](https://en.wikipedia.org/wiki/Dihedral_group), and [rules for transforming strings](https://en.wikipedia.org/wiki/Abstract_rewriting_system) can all be used as examples to make [Cayley tables](https://en.wikipedia.org/wiki/Cayley_table) from, for asking questions like:

  * What if we know that $a+b=b+a$, but [nothing else](https://en.wikipedia.org/wiki/Commutative_magma)?
      * Apparently rock-paper-scissors looks like that – can you make rock-paper-scissors [bigger](https://en.wikipedia.org/wiki/Rock_paper_scissors#Additional_weapons)?
          * What makes rock-paper-scissors, or its variants, fair? (Yes, this is no longer abstract algebra: sue me, it's interesting.)
      * What if we also have a value that behaves like $0$: $0+a=a+0=a$?
  * What's the fewest different values where we can have a $0$, but we can define $+$ such that $a+b$ doesn't always equal $b+a$?

Alternatively, you could go for number theory along those lines:

  * We have $+$ and $\times$ – can we go bigger? Do those operations have any surprising properties?

Don't invest too much in any one idea, though: it's no use putting together a six month curriculum if the child isn't actually interested in this area of mathematics right now. (You wouldn't want to just re-create school at home.)