The tangram puzzle has been a favorite of mine for many years. When I regularly taught a geometry course for teachers, I would use this puzzle as the opening activity for the course and would then come back to it periodically. Of all the things we did, this puzzle, and the ways it can be used to help students think about a variety of geometric shapes and relationships, was the thing students often said they found most immediately applicable to their classes. I would hear testimonies from teachers across the grades.
It was always my belief that students should have the opportunity to create the puzzle rather than simply be given the puzzle pieces. In the following video, which we did several years ago, I show both how to use paper folding as a way to create the puzzle pieces, and in the process how to see one way they can fit together to form a specified shape. In this case the pieces form a square, but there are many other familiar shapes that can be explored, as well as relationships between shapes (such as similarity of triangles).