TITLE: Best way to teach concept of real numbers using a hands-on activity?
QUESTION [6 upvotes]: 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.

REPLY [3 votes]: 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.