Abstract
The purpose of this article is to devise teaching materials using the lattice points in the space as a part of enrichment of the mathematical activity. The authors are considering that lessons using the lattice points have possibility to bear a cycle of the mathematics activity, which is typical to the plane lattice using the geoboards. In the case of the space lattice, there may be some obstacle to make and handle some good models for the space lattice, but the material is considered to be appropriate to promote the space recognition of students with enjoying the discrete thinking. In this article, we first show some characterization of the teaching materials relating to the lattice points. In particular, the materials using geoboards are discussed, and after Pick's theorem for the plane lattice figures is introduced, the impossibility of the direct generalization of the formula to the lattice polyhedra in the space is revealed. Although there are some formulas created by Reeve or Ehrhart, which can be regarded as generalizations of Pick's theorem, these formulas use some fractional lattices. We design several practical teaching materials relating to Pick's formulas using only lattice points for the lattice prisms or trees of lattice parallelepipeds. Through such teaching materials, we show some rich contents of the space lattice to enhance the space recognition of students.
