My advice:
you have covered arithmetic. the next topic is algebra. the best book i know of is by harold jacobs.
of course calculus would be absurd, althouhg if he learns graphing, the idea of approximating a polynomial by its linear term conveys the key idea of differential calculus as practiced by fermat.
after algebra try geometry, again by harold jacobs.
please do not use any books by john saxon, or by anyone else who dislikes math and considers it a skill to be acquired without thinking or understanding.
do not force him/her. let the child choose what to pursue from interest.
or not to do it at all.
it is essential that the pupil enjoy what is going on, or it will be dropped like a hot rock when the pressure is absent.
This is based on doing it somewhat wrong with my own kids. after being obliged to read thorugh about half of jacobs algebra book (one afternoon a week), our older son easily won the state title in a math contest, then refused to participate any more.
Unfortunately the standards of our schools are so low, the child may soon begin to wonder why no one else does anything remotely as challenging and may begin to lose motivation.
a wonderful program for kids of that age (maybe 12 or older) is the TIP summer program at Duke, but pricy. The good thing is there are other kids there who are all bright, and all interested in learning.
the entrance requirement is to take the SAT test: yes many 12 year olds take it, and outscore many college bound seniors.
the hard part of teaching bright kids is that much of the company they keep at school in the US, including the official curriculum, teaches them to want to do nothing essentially.
the late paul torrance wrote many books on educating gifted kids, especially creative ones. some of these books have collections of challenging problems to foster creativity as opposed to rote learning. example: how many uses can you think of for a wrecked car?
remember success in scientific research requires creativity as much as mastery of classical mathematics.
we took another quiet laid back 7 year old child to the torrance center to be tested. the tester asked him to say as many words as he could think of in 30 seconds as a measure of verbal facility and imagination i suppose. the child immediately began counting: "1,2,3,4,5,6,7,8,9, 10,..." then stopped suddenly and began laughing at outwitting the question so easily. he continued slowly in a desultory tone, "pig, dog,..." as if it were no longer any fun to play such a stupid game.
his school considered this child relatively ungifted in math, but on his math sat he outscored me, a full professor of math, and high school state math champ.
so my experience is that the early technical preparation of a child is relatively unimportant, compared to the love and interest they develop for a subject. so try not to kill that love with over emphasis on advanced training.
