Two years ago, I worked as part-time geometry instructor in a technical school near our university. Most of the students in that school were quite clueless about the notion of proofs, so I tried to find ways to introduce proofs in an intuitive manner.

One lesson I developed was on proving that a quadrilateral formed from paper folding is a square. I let the students create a square from a piece of bond paper without using any measuring instrument; only folding and cutting were allowed.

As expected, most of the students used the method shown in the figure above. For the sake of discussion, we label the corners and critical points of the bond paper. Most of the students constructed the square using the following steps (see figure):

- Fold the paper such that point
*A*coincides with point*C*, and then crease well. - Fold along
*BC*by placing*F*on*DC*(represented by*F’*in the figure). - Unfold all.
- Cut along
*BC*with a scissors.

Although the students had created a square, they had no idea how to explain why the final figure was a square. I asked them “What do we need to show to prove that this is a square?” After a few minutes of discussion, they had agreed that to show that the final figure is a square, they must show that (1) the interior angles measure 90˚ and (2) the four sides are congruent.

Through a series of questions I asked, the students realized the following:

- Angle
*A*and angle*ADC*measure 90˚ since they are corners of the bond paper (the bond paper is a rectangle). - Angle
*BCD*is also 90˚, since when fold 1 was made, angle*A*coincided with angle*BCD*. - Angle
*ABC*measures 90˚ since when fold 1 was made,*ADB*coincided with angle*BDC*and each of which measure 45˚ (since segment*DB*divides angle*ADC*).

From these realizations, the students knew that it remained to be shown that the four sides of *ABCD* are congruent. Again, through a series of questions, they realized the following:

*AD*is congruent to*CD*since they coincide when fold 1 was made. For the same reason,*AB*is congruent to*BC*.*BC*is congruent to*CD*since*BCD*is a 45˚-45˚-90˚ triangle.*AD*is congruent to*AB*is congruent because*ABD*is 45˚-45˚-90˚ triangle.

Although some properties discussed above implied other properties, they were not seen by most students. We had to prove them one by one. The last part of our lesson was to write the proof in paragraph form concisely.

I think those who are teaching geometry and introducing the concept of proofs should try out this lesson. I would love to hear what happened in your class.

It feels a little bit like cheating to me, to start with assuming the corners of the bond paper are really 90 degrees. A better high school level project would be to start with a random scrap of ragged-edge paper and fold a square. It only adds a couple of steps, but I think they are conceptually important.

Hi Denise. Thank you for your comment. Although, I did not mention in my post regarding that assumption, we actually had some preliminaries. We assumed that the bond paper is a rectangle and I asked them to recall the properties of rectangles.

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