Extracurricular ideas for UK GCSE level maths student My daughter is 15 years old and enjoys her maths classes (perhaps only because her maths homework takes her the least amount of time).
Until now I have managed to introduce her to subject matter before she comes across it at school. Recently she was asking me to tell her about some new things..
Some obvious things spring to mind such as complex numbers, series, matrices, calculus etc.
But I know just enough about maths to not be entirely comfortable with the UK GCSE syllabus. For example, if I were to ask her what she thought an axiom was, she would be clueless.
I want to start teaching her some 'ground-level' stuff. Things like:


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*The meaning of axioms, statements, theorems & proofs; (classical origins & alternatives ?!) 

*The formal definition of a function; limits

*Sets of numbers (eg: just what is Natural number ?).

*Operations: associativity; commutativity etc.

*(Or even some historical background; eg: she knows of Newton & Fermat but has not even heard of the likes of Gauss or Fourier; knows how to use a blur-tool though)


But I am not a maths teacher; just a parent with a vague frustrated feeling that the above mentioned concepts are important & better off understood sooner rather than later by someone interested in learning about mathematics. Am I kidding myself?
If anyone has any additions to the above and better still suggestions of where and how to start, then I would be delighted! I realise (hope even) that this may be highly subjective.
I was thinking of sitting down with her for just an hour or two of a weekend. I don't mind spending a bit longer than that sorting out a plan before-hand.  
 A: Read through (and work the problems in) "What is Mathematics" by Courant and Robbins. It won't detract/interfere with her studies, it is quite well self-contained, it is at once rigorous, deep, and broad. It is the book I wish I had read when I was 15.
Seeing some of the other answers, I thought perhaps I might expound a bit about why I recommend this text over others. I don't mean to insult you, and please correct me if this is not the case, but I got the impression from your question that these are not topics which you have a confident footing. One concern with approaching a book such as Fraleigh's Abstract Algebra is that there is the possibility of misinterpreting some of the material. This is still possible with the book I recommend, but Courant and Robbins' book does not really cover a particular subject and is not meant to be a primer in, for instance, abstract algebra or some other course that your daughter may take later in her studies. A misinterpretation in the material covered in "What Is Mathematics" is, in my opinion, more likely to be corrected later by a rigorous course in college than a misinterpretation in one of the building blocks of Abstract Algebra.
Courant and Robbins does an excellent job of encouraging intuition, and will definitely hit that "wow" spark for a young mind. I worry that your daughter's interest in mathematics might be squashed like a book like Rosenlicht's. Better, in my opinion, to build the motivation with interesting and amazing proofs and results - that way, the desire to struggle through some of the tedium of learning the basics will be all the more worthwhile.
The book includes an excellent section on number theory in the beginning, including covering complex numbers and de Moive's formula. This is followed by an excellent geometry section covering projective and hyperbolic geometry. Topology is then covered (when I was a senior in college, I still thought the "topology" course would be about mapping, so to expose a 15 year old to this would be amazing). This also includes a proof of Brouwer's fixed-point theorem (or one of them, rather). It then moves onto limits and calculus concepts, including calculus of variations, as it applies to minimal surfaces!
The main reason I would recommend this book is that she will be excited while reading it. She will get an excellent selection of some of the most interesting and amazing results in mathematics. If she likes what she reads then she will be more interested in mathematics than before. If, on the other hand, she doesn't like it, at least she is making a very well informed opinion about it.
A: Your own suggestions are what I think of as getting ahead of the curriculum. I think that's worth doing for topics that you or your daughter consider particularly interesting. However, another approach is to go totally off curriculum. Here my suggestions would be:


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*Recreational mathematics, such as the books of Martin Gardner or Raymond Smullyan.

*"Thinking Mathematically", by John Mason et al, which is about how to attack investigative problems.

*Learn programming (I consider Python the best language for beginners).
In any case, as it's for fun, I would suggest following what most interests you or your daughter.
A: I recommend the following four books I have.


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*Maxwell Rosenlicht's Introduction to Analysis, Dover Books on Mathematics, 1968. (See list of chapters below).

*Martin Aigner and Günter M. Ziegler's Proofs from THE BOOK, Springer, 2010. (List of contents available online).

*104 Number Theory Problems: From the Training of the USA IMO Team by Andreescu, Andriga and Feng, Birkhäuser, 2007 (List of contents available online).

*Terence Tao's Solving Mathematical Problems: A Personal Perspective (I have the Portuguese translation only).


List of chapters of Maxwell Rosenlicht's Introduction to Analysis
I. Notions From Set Theory, 1; II. The Real Number System, 15; 
III. Metric Spaces, 33; IV. Continuous Functions, 67; V. Differentiation, 97; VI. Riemann Integration, 111; VII. Interchange of limit Operations, 137; VIII. The Method Of Successive Approximations, 169; IX. Partial Differentiation, 193; X. Multiple Integrals, 244.
All chapters include Problems.
A: Mathematical Proofs: A transition to advanced mathematics is an ideal book to gift your daughter. It is a really really well-written book and is very enlightening. I can vouch that it is a pleasure to read and I only wish I had  possessed this book earlier.
