How Many Squares Are in the Compound Drawing?

We explored  the dimensions of our mathematical world. We should not take for granted that children understand the classifications of mathematical dimensions.

It is a challenge for most people to see all of the squares in a compound drawing. I taught the children a systematic way of first looking at the number of 1x1 squares first, then 2x2 squares, 3x3 squares and finally 4x4 squares. 

Although I provided the children with several square challenges, the most interesting one is looking for patterns in square grids. For example, in a 2x2 grid, there are four 1x1 squares and one 2x2 square. In a 3x3 grid, there are nine 1x1 squares, four 2x2 squares and one 3x3 square. In a 4x4 grid, there are sixteen 1x1 squares, nine 2x2 squares, four 3x3 squares and one 4x4 square. Yes, the pattern continues to find the sum of consecutive squares. 1 + 4 + 9 + 16 = 30. For a 5x5 square, since the pattern continues, we need to simply add 25 to the 4x4 solution of 30; so 30 + 25 = 55. For a 6x6 square, add 36 to 55 = 91; and so on.

I hope the children continue trying to solve each of these puzzles. See the attached pdf.