Pow 1 - How Many Squares?

Ava Stills

POW #2 How Many Squares?

Process

In the problem “How Many Squares?” I was very confused at first. I thought that we had to draw out and count all of the squares to get the answer. Lauren told us to look for the pattern, and that is what really helped me. I wrote out all of the sizes (8x8, 7x7, 6x6, ect...) and began to figure out how many squares you could make out of those dimensions. I started with 8x8 and 1x1 because those were the fairly obvious dimensions. I tried with 2x2 to draw out all of the squares in different colors. It started out okay but eventually became too confusing. I moved to 7x7 and discovered that you can only fit four. One for each corner. I then drew an 8x8 square and started sectioning out the 6x6 squares in it. I got 9 6x6 squares. I again sketched out the 5x5 squares and got 16. I thought back to the pattern we were supposed to look for and tried to figure out what pattern had to do with the numbers. I thought about the square sizes and how many were there and I started to see the pattern. I saw the square numbers and continued down through the other square sizes having higher squarer numbers for the answer. At first i though it didn’t work out but once I double checked the numbers i realized that I had found the pattern. I added up all of the squares that each of the sizes made and I got 204 squares.

Solution

For the first problem I found the pattern of square numbers going from largest size square and smallest square number to the smallest size squares and largest square number. One you got the square numbers you added them all up to get the solution that an 8x8 grid has 204 squares in it. The way I had it all written to get the answer looked similar to this; 8x8=1 + 7x7=4 + 6x6=9 + 5x5=16 + 4x4=25 + 3x3=36 + 2x2=49 + 1x1=64 =204 squares .

For the second question asked I figured out that if you find