How many squares are in a checkerboard

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How many squares are in a checkerboard?

• By Gloria Gonzalez
• And Ieni Vargas

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How many squares are in a checkerboard?

• Let's see if we can get you started and then YOU
  can come up with the answer.
• This question implies "squares of any size." How
  about if we make a Table and collect our data we
  might find a pattern going on.
• If you look at the smallest size square on a
  checkerboard, we can call that a 1 unit by 1 unit
  square. There are 64 of that size.

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The next size square is a 2 unit by 2 units
square. It's okay to use These more than once in
different new squares. The top left-hand corner
Has 4 of the 1x1's which make a 2x2. See it?
Good, then move right One square and find a new
2x2 made with two of the 1x1's you just used And
two 1x1's right next to them. Likewise you can
shift down just one Row and use two that you've
already used. Count all of these 2x2's. You
should come up with 49.Our table would look like
this
Size number of squares
11
64 22 49 33
44 Ect..
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• You go ahead and complete this table. After
  you're done you still have To add up the right
  hand column to get the total number of squares on
  a Checkerboard.
• Let's get back to the question of a pattern. Is
  there anything you? Notice about the numbers in
  the right hand column? Is there a relation To
  the number of 1x1 squares on a side of the
  checkerboard? 2x2's? Etc.?
• Say you have an 8x8 checkerboard. There are many
  other little and Big squares inside of that so
  counting them all how many are there? (E.g.
  counting all the 1x1, 2x2, 3x3, 4x4, 5x5, 6x6,
  7x7, 8x8).

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Suppose you have a square checkerboard of some
other size - not 8x8 What is the easiest way to
find how many squares there are in it? Make sure
that when you are done you will be able to find
the number Of squares in any size checkerboard
within 2 minutes. I tried counting the squares
individually, but that wouldn't be quick Enough
for 12x12. Could you please give me some tips for
how to Approach this problem?
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THANKS FOR YOUR TIME.