How Many Squares on a Checker Board?
(Use cutout models to count squares)

Use grid paper. Draw a picture of the 8 x 8 checker board on the grid paper. Now outline squares of different sizes, 8x8, 7x7, 6x6 and so on. Cut them out with scissors so that they can be placed on top of the picture of the checker board.


Start with
the 8x8 size
cutout.

    The 8x8 model can fit on the
checker board in only one place.
  Size 8x8
1 square

Now use the
7x7 size
cutout .

    The 7x7 model can fit on the
checker board in four places.
  Size 7x7
4 squares

Next use the
6x6 size
cutout.

    The 6x6 model can fit on the
checker board in nine places.
  Size 6x6
9 squares

This could continue until all the squares are counted, however using a math pattern will make it easier. Notice the numbers of squares are (1, 4, 9, __, __, ...). This follows the pattern of square numbers 12 = 1, 22 = 4, 32 = 9, 42 = 16, and so on.

Now this table can be easily completed using the math pattern, and the sum of all sizes of squares will be found.

8x8 size 12 = 1
7x7 size 22 = 4
6x6 size 32 = 9
5x5 size    
4x4 size    
3x3 size    
2x2 size    
1x1 size    
Sum =  

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