Tangram Exercise Revisited: A Discussion

In this sheet, we break out some of the details of the Tangram class activity.

Tangram Step 1. How to fold a square with a side of 6" (or some other measurement d).

Begin with a paper rectangle like PQRS; each side should be greater than 6". First fold the paper so that the fold passes through P and that side PQ is folded to lie on side PS. Point Q is folded on top of a point on side PS.

Mark the point T on side PS that is 6" from P. Then fold a second time with a fold passing through T so that the edge PS is folded onto itself; in other words, fold so that segment PT is folded onto a part of line TS. Cut on this fold.

Discard the piece of paper containing corners R and S. When you unfold the remaining piece of paper containing the corner P, the paper should form a square of side 6" with a diagonal fold. Relabel point P as point A and label the other corners of the square ABCD.

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Questions:

The big question is this. Why is ABCD a square? A second question is to tell what are the measurements of the angles and the length of the diagonal.

Why is this a square? First of all, what do you have to know about the lengths and angles of this figure from the folding and cutting that you did. Second, what else do you have to know and for what reasons do you know it.

To Answer. Lengths you know from the construction?

Angles you know from the construction?

Lengths and angles you know from reasoning and geometrical facts. What facts?