Best way to teach concept of real numbers using a hands-on activity? I know a middle school math teacher looking for some suggestions for hands-on activities to teach the concept of real numbers.  I'm new to this site, so this may be a little off topic.
 A: The questioner's user page identifies him as an American, so he is presumably asking about the American middle school (grades 6-8) system.  (In other words, most entering middle school students will be 11 years old, and most departing middle school students will be either 13 or 14.)  Some of the responses so far seem to me to be pretty far away from this, so I am enclosing a link to the NCTM standards for grades 6-8:
http://standards.nctm.org/document/chapter6/index.htm
Note: I didn't see anything about "real numbers" in the above link, although that doesn't mean that a discussion of real numbers would necessarily be inappropriate.  These are intended to be minimal standards.  (Still, realistically speaking, even these minimal standards are, on average, a long way from being met.)  
A: My mind immediately jumps to grabbing a ruler and some string and trying to determine the ratio of the circumference of a circle to its diameter, or the hypotenuse of the right triangle with unit legs.  In this way, the most accessible irrational numbers to most non-math majors (i.e. up to college even for most freshman) can be explored.
It should be easy to see that numbers can be built up from the humble unit 1, and quickly overtake the natural numbers with addition, the integers with subtraction, and the fractions with division.  Informing someone that there are numbers outside these operations should be the first step.
I would then construct some irrational numbers and argue (likely without proof) that they are not rational.  0.1234567891011121314... seems like a good candidate for this.  This also gives a powerful technique to the students to construct other irrational numbers and notice how many there actually are.
We can also explore the number Pi.  Cut out in paper a square, a pentagon, a hexagon, etc, to have several shapes of about the same area.  In some sense you can consider the ratio of a length to the perimeter in relation to the number of sides each shape has (and assign it as homework).  As the number of sides goes up, you can argue that this number is approaching Pi.
I am of the impression that this question counts as community wiki since the question is subjective.
A: There is an argument to be made that the real numbers, by which I mean the completed reals, does not belong in an algebra course.  In the answer above, all of the motivation comes from the real-algebraic-closure of $\mathbb{Q}$ (the largest algebraic field extension not containing $\sqrt{-1}$).  
The reason one might want to introduce the whole continuum comes from numbers like $e$ and $\pi$, which are transcendental (the fact that these numbers are transcendental is not immediate and requires a proof that I would consider past middle-school level).  If you're willing to state those facts without proof, you can give a moral argument for $e$ by showing that it is the limit of the sequence $((1+1/n)^n)_{n\in \mathbb{N}}$, which is Cauchy, and its inclusion in the real numbers follows from the completeness of $\mathbb{R}$.  However, this argument may still be somewhat sophisticated for a middle-school algebra course.  
Edit: On Prof. Clark's suggestion, I've copied my comments into the body text (with an additional section as well):
The notion of a sequence converging to a limiting value has a very intuitive geometric interpretation, so it wouldn't be hard to give a geometric argument (say on a graph, for example) that $e$ is a real number, since after relatively few iterations, the graph does level out. Showing that it is not the solution to a polynomial is effectively proving that it is transcendental, and I can't think of an informal argument showing this, but since this answer is community wiki, if someone has an idea, this would give a "moral" argument for the study of the "whole" continuum.  
I find this approach useful because it can be introduced using the compound interest formula, which is often taught in an introductory algebra class.
We can see this as follows $$A=P(1+r/n)^{nt}.$$
Let $N:=n/r$.  Then we have $$A=P(1+1/N)^{Nrt},$$
which we can rewrite as $$A=P((1+1/N)^N)^{rt}.$$
If we increase $N$ and leave $r$ fixed, this is equivalent to increasing $n$ (this is obvious because $r>0$), which amounts to increasing the number of compounding periods per unit time.  Taking the limit (in some informal geometric sense), we can see that as we increase the number of compounding periods, we approach the continuously compounded interest formula $$A=Pe^{rt}.$$
A: What do you mean exactly?  Show them that not all numbers are rational?  That not all numbers are algebraic?  I think that you could explain historically how the Pythagoreans thought all numbers were rational.  Then challenge them to find a number which is not rational.  You might be surprised with the creativity you get.  Maybe they know rational numbers have repeating decimal expansions and so they will invent one which does not repeat.  Challenge them to explain why rational numbers have repeating decimal expansions.  If you like geometry, you could lead them to the essential step in the geometric argument here:
http://blog.plover.com/math/sqrt-2-new.html
that the square root of two is irrational.
A: Thanks to Pete for the link on what the question means. I was misled by the other responses.
Here are two thoughts:
i. What is the square root of 14? Well, it's between 3 and 4. Then we can do better by taking the average of 3 and 14/3. Then we can repeat this and do even better.
ii. Take the Fibonnaci sequence 1,1,2,3,5,8,13,... Take ratios of successive terms 1/1,1/2,2/3,3/5,5/8,8/13,... then these are approximations to the "golden ratio".
A personal anecdote: My daughter is 11 yrs old, bright and interested in maths. I tried i. on her and drew a blank look. I conclude from this that either it is a mistake to teach your own children or that she was not ready for this.
A: I guess the "hands-on" version of the real numbers is the Euclidean line.
A: 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?
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.
