Paper Folding: Locating the square root of a number on the number line

Paper folding (origami) has been used by many teachers, particularly in Japan, to teach mathematics.   In this post, we are going to use paper folding to locate the square root of any number on the number line. A GeoGebra simulation of the paper folding is shown below.

Note (July 17, 2014): Applet had an error. I will update it within 24 hours.

Paper Folding Instructions 

  1. Get a rectangular piece of paper and fold it in the middle, horizontally and vertically, and let the creases (see green segments in the applet) represent the coordinate axes.
  2. Construct point M at (0,1), and fold and make a crease at y = -1 (see blue segment above).
  3. Let P denote a point with coordinates (0,n) where n < 0 (drag P along the negative y-axis).
  4. Make a fold through P (by dragging Q in the applet) that places M(0,1) on the line y = – 1.
  5. The x intercept of the fold is \sqrt{n}.
The Mathematics Behind

Theorem:  The x intercept of the fold above is \sqrt{n}.
Before we proceed with the proof, in the applet check the Show/Hide Triangles check box to display the triangles MNO and OPN. You can also uncheck the Show/Hide Fold check box to get a better view of the triangles.

Let be x-intercept of the fold.  It is clear that \Delta NOM \sim \Delta PON. This implies that
\displaystyle\frac{ON}{OM} = \frac{OP}{ON}.
But ON=m, OM=1, and OP = n.  Subsituting the values, we have
\displaystyle\frac{m}{1} = \frac{n}{m}.
Therefore m^2 = n and  m = \sqrt{n}.
This means that for every value for n < 0,  the intersection of the fold is always \sqrt{n}.  In effect,  using this method, we can locate the number or the length  \sqrt{n} on the number line.
Reference:  My old dilapidated geometry notebook, so I don’t really know to whom I shall attribute this. Our teacher in Geometry never used any reference. He could talk all day without looking at any book.

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