Lattice Points and Boundary Lattice Points

Bernard,

Lattice points are the points on the regular x,y coordinate plane that
have an integer value for both x and y. When you draw the shape by
drawing the edges, if the line contains one of these lattice points,
it is a boundary lattice point. If a lattice point is totally inside
the shapes edges, it is an interior lattice point. The number of
lattice points on the boundary and inside depend on where you put the
shape and how it is rotated. In some positions it may have more
boundary points and in others, more interior points.

If you draw the isosceles right triangle whose legs are 3 units long
with the right angle at the vertex (0,0) and the two legs along the x
and y axes, then there would be the following boundary lattice points:

(0,0), (0,1), (0,2), (0,3), (1,0), (2,0), (3,0), (2,1), and (1,2)

The only interior lattice point in this case would be (1,1). Plot
these and it will help.

But if you slide the triangle to the left 1/2 a unit and down 1/2 a
unit, suddenly there are no boundary points and 6 interior points.
Again, try plotting these. I think your teacher is working up to a
beautiful mathematical idea called Pick's theorem and a way to find
area.

I hope this helps. Good luck.