tiles (either the plastic kind sold as math manipulatives or bathroom tiles)
colored wooden cubes
any items collected in junk/treasure boxes
The children work in pairs. One child builds a design with the materials and the second child copies it using another group of the same materials. If they are things that the children can glue onto cardstock, such as the toothpicks, they can create permanent records of their discoveries. In the case of things that you don't want them to glue down, like pattern blocks, they can glue down paper shapes that match the pattern blocks. Either they can trace the blocks, or you can make up the shapes in advance for them. Some things such as wooden cubes can be drawn on a sheet of paper, or colored in on a sheet of 1/2 inch graph paper.
At another time they could explore single mirrors and two mirrors hinged together with magazines and books with colorful pictures to look at with mirrors. After they have done this awhile, provide them with 3" to 5" tagboard numerals and letters (capital and lower case) or a bowl of pattern blocks.