The book “Mathematics Mathematics for high-school teachers” by Usiskin et al. has what I find a hideously over-complicated approach, at the nutshell of which is a nice idea: How does the notion of the collection of real numbers as a geometric object (a line) mesh up with its realisation as an algebraic object (at the middle-school level, decimal expansions)?

Some things to think about in this connection: If real numbers are directed lengths, how do we add and subtract them? OK, that's not so bad; how about multiplication and division? Now how about square roots? One can draw a nice picture to show $\sqrt2$ as the length of the diagonal of a unit square (constructed on a unit length by ruler and compass) being ‘unfolded’ (by a compass) down to the real line; now we've taken square roots geometrically. Can we do the same thing with the circumference of a circle?

If one wants to think purely in terms of decimals, there's the age-old “Are $1$ and $0.\overline9$ really the same number (and why)?”. It's easy for middle schoolers, and good practice in long division, to find decimal expansions of some unfamiliar (for them) fractions, like $1/7$ and $1/13$; which ones terminate? Which ones repeat? Will one of these always happen? Can you predict in advance which is which, and how long it'll take before it terminates or repeats?

UPDATE: On further reflection, I suppose that the latter is “hands on only in a very rough sense. Perhaps one could consider the analogous sum $\sum_{i = 1}^\infty \frac1{2^i}$ by (‘building up’) laying down a strip, then another strip of half the length, then another of half that length, and so on; or (‘tearing down’) by cutting off from a long strip half its width, then half of the remaining width, and so forth.

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The book “Mathematics for high-school teachers” by Usiskin et al. has what I find a hideously over-complicated approach, at the nutshell of which is a nice idea: How does the notion of the collection of real numbers as a geometric object (a line) mesh up with its realisation as an algebraic object (at the middle-school level, decimal expansions)?

Some things to think about in this connection: If real numbers are directed lengths, how do we add and subtract them? OK, that's not so bad; how about multiplication and division? Now how about square roots? One can draw a nice picture to show $\sqrt2$ as the length of the diagonal of a unit square (constructed on a unit length by ruler and compass) being ‘unfolded’ (by a compass) down to the real line; now we've taken square roots geometrically. Can we do the same thing with the circumference of a circle?

If one wants to think purely in terms of decimals, there's the age-old “Are $1$ and $0.\overline9$ really the same number (and why)?”. It's easy for middle schoolers, and good practice in long division, to find decimal expansions of some unfamiliar (for them) fractions, like $1/7$ and $1/13$; which ones terminate? Which ones repeat? Will one of these always happen? Can you predict in advance which is which, and how long it'll take before it terminates or repeats?