What can I teach a talented student who is enthusiastic about math?

I have a very clever student in Grade 7. Due to carelessness and English (which is not his native language), he failed the placement test and can't learn Grade 8 math (algebra 1) in school. School math is easy and bores him, so he chooses me to teach him in advance. He is quick in thought, but a little careless (which is not a problem here since we find a way to fix that).

He attends the math club, and tries to understand problems designed for students in Grade 10 or above. And indeed, he can understand the problem well and ask me the right questions. I'm willing to teach him for he trusts me and follows me well.

My thought is to teach what is related to his school math, but deeper and more challenging. Now he's studying algebraic expressions in school, so I start with polynomials, which works well. Now I've just finished teaching him multiplying polynomials (Pascal's triangle was also introduced and he really likes it) and plan to teach him factorization in the next lesson.

There are many topics in polynomials (division, rational root theorem, GCF/LCM), and many more are related to it (polynomial functions, solve polynomial equations, rational expressions and functions), from algebra 1 all the way to precalculus, even higher algebra (Eisenstein's criterion). So I have to make a choice. Since he also wants to participate in AMC, there are more things to learn, number theory, combinatorics, Euclidean geometry (maybe not now), etc.

The problem I face is, I now have two goals: one is to teach him something new (from higher grade math), the other one is to make it interesting and challenging, by introducing him to more advanced topics and more difficult problems. I can't merely follow the easy textbooks.

Jumping from a topic to another is not a good idea, and I want to design a curriculum for him: what to teach and in what order, and to what degree, and what problems are suitable to assign to him so that he can try to solve them by himself and we can then discuss in the next class.

Any suggestion on the course design and reference books is welcome. And we have 3 hours on Friday each week, how can I structure the class?

It may not be a good idea to let him read some books by himself, and discuss with me what he read, since he prefers listening to reading (and his English level is not good enough).

• So you have a talented student and a very weak student. – user5402 Dec 27 '17 at 10:21
• @inéquation That's true. And I have several average students. Normal distribution... – Mathis Dec 27 '17 at 10:27
• This is probably not exactly what you are looking for, but pauls math notes may have some things you can adapt into a curriculum. – user7915 Dec 27 '17 at 11:09
• If I were you and I couldn't find any good book I'd follow the textbooks and extend every chapter by introducing some new ideas. For example, say you're explaining factorization then after you've done with the basic examples of the textbook ($x^2-4+2(x-1)(x-2)$), I'll add more and more challenging exercises like $\displaystyle (5x+1)^2+(4x+2)^2-(3x+5)^2-(2x+6)^2-2(5x+1)(4x+2)+4(-x-3)(x+5)$ or $\displaystyle x^3-(2a+1)x^2+(a^2+2a-m)x-(a^2-m)$. If he managed to do these exercises easily, you can use some contest exercises for middle or high school. – user5402 Dec 27 '17 at 11:10
• My advice is to find a way to accelerate him, not enrich him. Maybe you can get through both pre-algebra and algebra this year. – guest Dec 27 '17 at 13:38