Chess board problem

We know the definition of a square is a 4 equal sided shape with interior angle of 90 degrees. So, in a chess board, there are different squares of dimension of 1x1, 2x2, 3x3, 4x4, 5x5, 6x6, 7x7, 8x8. If I was doing this with a class, I would put students into 8 different groups. Each group would be finding the number of possible squares of their dimension. At the end, we would put everyone’s results together, adding all the results for the total number of squares.  I think there can be a number of approaches. Students can draw the shapes out, cut the shapes, or use an overhead projector transparent sheets to trace with their shape. I hope the groups can find some kind of pattern and some efficient ways of counting. More importantly, students can practice break down a problem, team work and communication.

  I would start with the bigger dimensions, because the bigger the dimension, the less squares the chess board can fit. For example, the whole chess board is a 8x8 square. By drawing out a 7x7 square on the board we see that it leaves a strip of 1 cube on 2 edges, and by rotation that 7x7 square, we see that there are 4 possible positions. Then we are going to use the method called starting sides. For example, for the 6x6, we are going to draw a 6x6 square from the top left hand corner, and count how many possible starting sides of vertical edges there are from the left along the board moving horizontally. For the 6x6, we can start on the first 3 vertical edges but not the 4^th, because there are only 5 squares left to make the 4^th width. Same with the lengths, we are going to start from the top moving vertically down the board to find the number of possible starting widths to make a 6 units’ length. And we see that we can only make a length of 6 units with the top 3 horizontal edges. And 3 possible vertical edges times 3 possible horizontal edges give you 9 different positions for the 6x6. Repeat the process of possible starting edges for the rest of the dimensions and students should find 16 positions for the 5x5, 25 positions for the 4x4, 36 positions for the 3x3, 49 positions for the 2x2, and we know it is a 8x8 chess board for the 1x1squares making 64 squares. Add all the numbers and the answer is 204 possible squares.

        If students understand the logics of starting sides, students can solve the number of squares on any board. As a class we might try to come up with a function to represent the to number of squares. Students can see that there is a pattern of 1^2, 2^2, 3^3, 4^2…, and we can plot a graph of dimension vs. number of squares. This can be used in the chapter of parabolic functions, and to show how the function can be applied.