Fibonacci Squares and the Golden Spiral on a Grid

Grades 8-9

Question: Trace a Fibonacci sequence on a black-and-white checkerboard made of 6 x 10 black and white squares.

To represent a Fibonacci sequence on a checkerboard, we use a fascinating geometric construction where each number in the sequence corresponds to the side of a square. The sequence starts with

1, 1, 2, 3, 5, 8, \dots

In this diagram, we've drawn a

10 \times 6

checkerboard (black and white). On top of it, we've overlaid the Fibonacci squares:

The first two squares ( $1 \times 1$ ): Placed at the center of the board.
The $2 \times 2$ square: Attaches to the two previous squares.
The $3 \times 3$ square: Rests against the block formed by the 1 and 2 squares.
The $5 \times 5$ square: The largest square that can fit within a height of 6 units.

The Golden Spiral: The curved red line you see is the "Fibonacci spiral". It's made of circular arcs drawn inside each square. Each arc connects two opposite corners of a square, creating a continuous curve that illustrates the sequence's exponential growth. This shape appears throughout nature, from the arrangement of sunflower seeds to the shape of galaxies.

The squares are colored indigo to stand out clearly against the checkerboard background, and the sequence of numbers is labeled at each stage of the construction.