jmc

Points with integer coordinates are called lattice points. The length of the rectangle is $5-(-5)=10$ units. There will be 9 lattice positions between the two vertical sides of the rectangle. The height of the rectangle is $4-(-4)=8$ units. There will be 7 lattice positions between the top and bottom of the rectangle. That's a total of $9\times 7=\boxed{63}$ lattice points.