I am beginning to run out of ideas about how to explain this to you. You have actually (and more praise to you for doing so!) taken the trouble to model the situation and seen it happen. Let's see whether we can put it in terms that make sense to you.
You have several pieces of paper. From our point of view they represent several fixed areas. No matter how you move them they still represent the same total area.
Right?
However, if you put them together in different ways you do not get the same shape. You could, if you felt like it, put them together in such a way that there is a big gap in the middle. This would not surprise you and it also would not surprise you to find that the area of the paper itself (not counting any gaps) still has not changed, right?
Now, have you bothered to calculate the areas of each of your pieces of paper and add them up? (Half base times height for the triangles and base times height for the rectangles, right?) I don't have the image in front of me, so I cannot do it for you, but they all have whole numbers of squares along the sides that matter, so the calculation is not difficult.
Then see whether the two apparently triangular figures that you construct when you fit them together, apparently have the same area as the sum of all the pieces of paper. You will find that they do not. The discrepancy is small, but real.
Now, here comes the tricky bit. Remember that I said that the whole apparent triangle is not really a triangle. If you have put your pieces of paper together correctly, and you have a good straight ruler handy, you will find that the horizontal and vertical sides of the apparent triangle are nice and straight but if you very cautiously lay your ruler along the diagonal side of the apparent triangle (the apparent hypotenuse) you will find that it is not a straight line. The "solid" apparent triangle, instead of a straight-line "hypotenuse", has two straight lines meeting at an external angle slightly less than 180°, whereas the triangle with the gap has a hypotenuse with two straight lines meeting at an external angle slightly more than 180°. Now, that requires more area, and the area of your paper is not flexible; it cannot stretch. So instead it leaves a gap equal to 1 unit square. You have put the paper together in a different shape, and the new shape has a gap in it. If you calculate the shapes' areas, you will find that the discrepancy is one unit square.
Am I getting warm?
If not, then do you think you might explain why it seems strange to you that rearranging the pieces could give you a different shape?
