First Assignment!

How many sqaures in a 8x8 chessboard?


My thought process behind this was to first identify which types of sqaures could possibly be in the chessboard.  Everything from a 1x1, 2x3, 3x3...... to an 8x8 sqaure. The next step was to figure out how many of each type there are. Once we do that we can simply add them up to get our answer!

For starters I looked at the 1x1 type squares.  It's pretty clear that there is a total of 64.  The reasoning is because there are 8 possible positions vertically and 8 possible positions horizontally.  8x8 = 64.


I applied this same method to all of the different types.  So for a 2x2 type we can see that there is 7 possible positions vertically and 7 possible positions horizontally.  We are limited to 7 because we would go "outside" the board otherwise.  so 7x7 = 49

After applying this method to all types, I ended up with

64 1x1's
49 2x2's
36 3x3's
25 4x4's
16 5x5's
9  6x6's
4  7x7's
1  8x8's

Add them all up and you get 204!

A few words of teacherly advice:
 For students who are stuck or having a difficult time:
               - Add a visual representation or visual aid, this helps a lot!
               - Show them that there is more then just 1x1 squares, 64 is not the answer!

Ways to expand this problem:
               - What happens if the 8x8 chessboard gets bigger or smaller? (ex. 7x7 or 9x9)
               - Other shapes(rectangles, triangles)

Hope you all enjoyed

Thanks :)