I got a relatively muted response to this puzzle, but some very interesting answers among them. Specifically, I want to congratulate Mayuri+Mehul, as well as Tishyaa for very innovative answers. Jyotsana Dube also sent a correct answer, but not universally applicable to all cuts of the smaller rectangle.
The original intent of the question was a little different than the answers I received. I have recorded the answer on the following link – would strongly encourage parents to show the video to children, as it teaches the kids an important property of a rectangle.
Hope you enjoyed the puzzle!