Proving that $\sqrt{2}$ is irrational with a math level of a middle school student? So I have a friend, whose professor challenged the class to prove that $\sqrt{2}$ is irrational by using only middle school math level.
No one managed to do it, and neither did I (although I didn't give it much thought, as I have other stuff to do for the moment), but I am still curious as to how to do it.
Apparently, most of the class thought about using the Pythagorean theorem: $$ a^2 + b^2 = c^2$$ And taking $a=b=1$, but it didn't help them at all.
So any ideas? I know that the middle school levels vary from country to country, but let's just suppose that we are talking about YOUR country's middle school.

Geometry was taught after elementary algebra at my middle school, and focused heavily on construction with a compass and straight edge, and so we would all have been able to understand this proof by Apostol (although perhaps only after a couple readings of it), which I will quote here:

This note presents a remarkably simple proof of the irrationality of
  $\sqrt{2}$ that is a variation of the classical Greek geometric proof.
  By the Pythagorean theorem, an isosceles right triangle of edge-length
  1 has hypotenuse of length $\sqrt{2}$. If $\sqrt{2}$ is rational, some
  positive integer multiple of this triangle must have three sides with
  integer lengths, and hence there must be a smallest isosceles right
  triangle with this property. But inside any isosceles right triangle
  whose three sides have integer lengths we can always construct a
  smaller one with the same property, as shown below. Therefore
  $\sqrt{2}$ cannot be rational.
Construction. A circular arc with center at the uppermost vertex and
  radius equal to the vertical leg of the triangle intersects the
  hypotenuse at a point, from which a perpendicular to the hypotenuse is
  drawn to the horizontal leg. Each line segment in the diagram has
  integer length, and the three segments with double tick marks have
  equal lengths. (Two of them are tangents to the circle from the same
  point.) Therefore the smaller isosceles right triangle with hypotenuse
  on the horizontal base also has integer sides.


-Tom M. Apostol, Irrationality of The Square Root of Two -- A Geometric Proof, American Mathematical Monthly 107, No. 9 (Nov., 2000), pp. 841-842.
