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Thought for the day: Guide students to notice and wonder throughout the problem-solving process: (1) creating or clarifying the problem (2) exploring the question (3) reasoning about their conjectures and strategies (4) reflecting on their conclusions and thinking of new questions to ask.

Concepts: Fibonacci numbers; the golden ratio; length and area; analyzing and extending geometric, numeric, and algebraic patterns; rational and irrational numbers; similarity


Examples of noticing and wondering

I notice that the squares get larger in a counterclockwise pattern.
I notice a pattern in the side lengths of the squares.
I notice that the next square in the pattern will be on the top, and it will have 13 squares on each side.
I notice rectangles that are almost, but not quite, similar.
I notice that the ratios of the side-lengths of these rectangles alternate up and down, changing by less each time: 2, 1.5, 1.66666..., 1.6, 1.625. (These ratios are both within and between the rectangles.)

I wonder if there are patterns in the areas of the squares and rectangles.
I wonder if the ratios of the side lengths are approaching some particular number.
I wonder if the areas also have ratios that approach a particular number.
I wonder if there is a simple relationship between the eventual ratios of the side lengths and areas.
I wonder how the patterns would change if I started with three 1-by-1 squares instead of two.



For a more structured investigation of this image, see Problem #7 in Exploration #8: Expanding and Contract from my book Advanced Common Core Math Explorations: Ratios, Proportions, and Similarity.

As you continue the pattern, the figure becomes closer to what is known as a golden rectangle—a rectangle whose longer side is "ø" ("phi", or about 1.61803) times as long as the shorter side.

If it were possible to begin with a perfect golden rectangle and to draw a figure like the one above from the "outside-in," the squares would continue spiraling inward forever with one square of each size! (There would not be two 1-by-1 squares.)

The number, ø, is known as the golden ratio. It is a famous irrational number that has many amazing properties, including the fact that you can (1) square it simply by adding 1, and (2) find its reciprocal by subtracting 1! The golden ratio is related to the equally famous Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, ... in which each number is the sum of the preceding two. This sequence appears as the side lengths of rectangles in the prompt! The Fibonacci sequence and the golden ratio are very popular topics for students to explore and read about!

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