Puzzles involving cutting shapes and forming them into different shapes helps reinforce the idea that area do not necessarily change with change in shape. It is also a good activity for developing visualisation skill and spatial ability.
The puzzle below is from one of the leaflets at the booth of Japan Society of Mathematical Education last ICME 12 in Seoul, Korea. The original puzzle is suited for Grade 4. The instruction was to cut the pentagon along the dotted lines and then form them into the shapes shown. The shapes shown in the leaflet is a parallelogram, a rectangle, an isosceles trapezoid, and a general trapezoid. I modified the puzzle for students in the higher level. I have indicated the measure of the two angles just in case you want your students to justify that the pieces really form into quadrilaterals. This is one way to assess your students knowledge of the properties of these parallelograms, trapezoid and trapezoids as they justify each shape formed.
Here are two solutions – rectangle and isosceles trapezoid. Form the other two shapes.