Materials for teaching the axioms of the real numbers to high school students
This sounds like one of those things easiest to accomplish by visiting a university library and browsing the shelves at the locations of real analysis books, transition to advanced mathematics type books, and books dealing with constructions and axiomatizations of the real numbers. I say this because you and your student's background and interests will play a huge role in what is appropriate, and the quickest way of weeding through the large amount of material available is to be in a position where you can immediately look at something for consideration.
Off the top of my head, the following book (which fits into the 3rd category I listed) seems like it might be a good fit:
Stefan Drobot, Real Numbers, Prentice-Hall, 1964, x + 102 pages.
(My comments about the book) This book begins with an axiomatic treatment of the real numbers as a complete ordered field and then discusses various expansions of real numbers (decimal, Cantor, continued fraction), approximation of irrationals by rationals, and (briefly) cardinality and measure of sets of real numbers.
Below are some other possible books. Incidentally, there were a lot of books published in the U.S. during the 1960s having to do with constructions and axiomatizations of the real numbers.
Leon Warren Cohen and Gertrude Ehrlich, The Structure of the Real Number System, The University Series in Undergraduate Mathematics, D. Van Nostrand Company, 1963, viii + 116 pages.
Solomon Feferman, The Number Systems. Foundations of Algebra and Analysis, Addison-Wesley Publishing Company, 1964, xii + 418 pages.
Norman Tyson Hamilton and Joseph Landin, Set Theory and the Structure of Arithmetic, Allyn and Bacon, 1961, xii + 264 pages.
Edmund Jecheksel Landau, Foundations of Analysis, 1951, Chelsea Publishing Company, 1951, xiv + 134 pages.
Elliott Mendelson, Number Systems and the Foundations of Analysis, Academic Press, 1973, xii + 358 pages.
John Meigs Hubbell Olmsted, The Real Number System, Appleton-Century Monographs in Mathematics, Appleton-Century-Crofts, 1962, xii + 216 pages.
Francis Dunbar Parker, The Structure of Number Systems, Teachers' Mathematics Reference Series, Prentice-Hall, 1966, xiv + 137 pages.
Joseph [Joe] Buffington Roberts, The Real Number System in an Algebraic Setting, A Series of Undergraduate Books in Mathematics, W. H. Freeman and Company, 1962, x + 145 pages.