By reading the literature, we think that there is a method to calculate the polygon area by calculating the grid points, which can be used to estimate the area of curve graphics. This method originated from lattice polygons. The so-called lattice polygon means that the vertices of this polygon are all lattice points.

Let s be the area of Figure 2, L be the number of grid points on the boundary (the intersection of the horizontal and vertical lines forming the grid is just on the edge of the graph), and N be the number of internal grid points (the intersection is inside the graph), so it is easy to calculate the graph area as 1 1. If L=6 and N=9 are related to graphs, and the relationship of +N- 1 holds, is this method general?

2. Explain the rationality of the method of counting grid to estimate the area. Students should understand the method of estimating area by counting grid, but what is the basis of the method of estimating area by counting grid?

If the length and width of the rectangle are m and n, respectively, the area S=mn.

Then consider the number of boundary lattice points of this rectangle, L = 2 (m+1)+2 (n-1); Number of internal lattice points n = (m-1) (n-1); And+n-1= m+n+Mn-m-n+1= Mn, so the relation S=+N- 1 is valid for lattice rectangles.

It can be seen from the symmetry of the graph that the above relationship is also true for right-angled triangles with lattice points and general triangles.