Tangrams are almost certainly the best known and most widely available geometric puzzle found in US classrooms. They are inexpensive, available from many publishers, and there is a vast number of accessible tangram puzzles on the Web. On this site, I share tangram-based activities which prioritize geometric insight.
Tangram Puzzles is a freely downloadable book to use before, after, or alongside the below.
Tangrams in the Classroom
An activity you can do prior to handing out plastic tangram sets: Tearing Tangrams.
Hand out the sets so that neighbors have different-colored sets. This makes it more difficult for pieces to migrate to a different bag. It also makes it more obvious if a student "cheats" by using a piece from another set.
For students to familiarize themselves with the seven pieces you can ask them to make two-layer tangram figures, using at least three pieces. (In other words, create pairs of congruent figures. For example, cover the square with the two small triangles.)
In Geometry Labs (free download), Section 2 is all about tangrams, with activities involving measurement, vocabulary, symmetry, and convexity. See also Lab 10.6 on tangram similarity.
I commented and expanded on those activities in this blog post: Convex Tangram Polygons.
You can use my Virtual Tangrams if you need to project solutions on a screen, or create images for handouts.