Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
Polygons drawn on square dotty paper have dots on their perimeter
(p) and often internal (i) ones as well. Find a relationship
between p, i and the area of the polygons.
Pin those squares down! A geoboard is a piece of wood with pins
hammered onto it. The diagram shows a 5 by 5 board with 25 pins set
out in a square array. Squares are made by stretching rubber bands
round specific pins as shown. What is the total number of squares
that can be made on a 5 by 5 board? How many squares can be made on
an $N$ by $N$ board?
Ling Xiang Ning has illustrated his solution for $N = 2$, $3$ ,
$4$ and $5$ in a good clear diagram. There are 50 squares on a 5 by
5 board Can you find a general formula for an $N$ by $N$ board?
We count the squares on a 2 by 2 board, 3 by 3 board...