The first thought that pops into the mind is that 8*8 yields 64 squares. However, these only account for the 1-by-1 size squares. We quickly realize that squares can be seen in different sizes on a chessboard.
Let us consider how many 2-by-2 squares are possible.
I'd like to think that there are 7 horizontal positions for a 2-by-2 square since a square cannot move outside the board. Likewise, there are 7 vertical positions for squares of this size. 7 times 7 gives 49 squares here.
Similarly, for 3-by-3 squares, there are 6 horizontal positions and 6 vertical positions. =>36 squares.
Also, for 4-by-4 squares, there are 5 horizontal positions and 5 vertical positions. =>25 squares.
We see a pattern!
Taking it all the way, we get 1 square for 8-by-8 size.
The total number of squares is the sum of the squares: 64+49+36+25+16+9+4+1= 204 squares.
Ways we can extend this puzzle:
Q: How many 3-by-2 rectangles are in a chessboard? (The student must think about positioning these rectangles on the board.)
Q: How many triangles are there below?
Lovely! And great extensions to this problem -- I may ask you to share these with the class, Jessica.
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