Square Grids Yield Square Number of Rectangles

Last week we explored the amazing pattern of squares withing square grids. The result is always a sum of square numbers such as 1 + 4 + 9 + 16 + 25 ... in a 5x5 grid.

This week we continued this geometric exploration looking at the more challenging problem of how many rectangles. The first challenge was listing the different rectangle dimensions in a systematic order such as 1x1, 1x2, 1x3, 1x4, 1x5, 2x2, 2x3, 2x4, 2x5, 3x3, 3x4, 3x5, 4x4, 4x5, and 5x5 for a 5x5 grid. We discovered several different methods of counting rectrangles of various dimensions. Each of the square rectangles follow the same pattern as last week's challenge so the 1x1 was 25 rectangles, the 2x2 was 16 rectangles, the 3x3 was 9 rectangles, the 4x4 was 4 rectangles, and the 5x5 was 1 rectangle. When counting the 1x2 rectangles, we can choose horizontal and count to 20; we simply double that number to address the vertical rectangles so we have a total of 40.

The number of rectangles in a 1x1 square grid was of course 1. 

The number of rectangles in a 2x2 square grid was 9.

The number of rectangles in a 3x3 square grid was 36.

Some of the children were able to make a conjecture (educated guess) about how many rectangles there would be in a 4x4 square grid. The children noticed that each solution was a square number of 1x1, 3x3, and 6x6. Most students guessed that the 4x4 grid would have 81 or 100 rectangles (the actual solution was 100 or 10x10), and then solved the 4x4 grid. Next, they made a conjecture about the number of rectangles in a 5x5 square grid (the actual solution was 225 or 15x15). The children should be able to find the solution for the 6x6 and so on as long as they follow the sequence. Some of the students found that the sequence of square numbers were based on triangular number squares (1, 3, 6, 10, 15, 21, 28, 36 and so on).