Pattern Blocks (mini follow-up)

Who: J1

When: 5 minutes in between other play

What: finding an area relationship and learning a surprising technique

How: I made a design and then we admired it together

A pretty little picture, no?

Last month (specifically, here: http://3jlearning.blogspot.com/2014/06/tease.html) I asked you to calculate the areas of the different pattern blocks. One of the trickier ones is the thin, tan parallelogram. This arrangement gives you one way to see a relationship that might be hard to spot.

I set out the blocks and arranged them as in the picture.  At some point in his play, J1 paused and started looking at the blocks.  I moved the top two (orange square and green triangle) away from the others and said "hmm." That was enough to launch us into a discussion of
(a) are the two clusters the same shape
(b) are they the same area
(c) are the green triangles the same
(d) can we draw any conclusions?
(e) if they are the same area, why can't we make a square from the two parallelograms?
(f) what equations can we write that describe our picture?

Do you know other examples that are like this (you add something to a problem/picture/system that suddenly makes it more clear rather than more complex)?

One example that recently came to my mind was completing the square for a quadratic equation. A more sophisticated example is moving from affine to projective space to look at curve intersections in algebraic geometry.