I have a cross in the form of one square with four identical squares surrounding it
Cut it into the fewest number of pieces possible such that, when rearranged the pieces form two smaller crosses of identical size.
I have seen this dissection before. The two smaller crosses are similar to and exactly half the area of the original.
One of the five pieces is one of the two small crosses. That piece is centered in the large cross but rotated so that four of its corners touch the sides of the large cross.
The other four pieces are all congruent and make up the second small cross, with 90 degree rotational symmetry where the cuts meet.