Veneer Square

The following represents a piece of wood in my possession, 5 in. square. 
By markings on the surface it is divided into twenty-five square inches.
I want to discover a way of cutting this piece of wood into the fewest possible pieces that will fit together and form two perfect squares of different sizes and of known dimensions. 
But, unfortunately, at every one of the sixteen intersections of the cross lines a small nail has been driven in at some time or other, and my fret-saw will be injured if it comes in contact with any of these.
I have therefore to find a method of doing the work that will not necessitate my cutting through any of those sixteen points. 

How is it to be done? 
Remember, the exact dimensions of the two squares must be given.

See answer

Math Genius