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Which of these rectangles are the same shape? Can you find pairs of the same shape?
How can you be sure that they are exactly the same shape and not just nearly the same shape? The colours give a clue but there is a mathematical property here that you can use to test if two rectangles are the same shape.
Can you work out what that property is?
As in this diagram, draw two squares of unit area side by side on your squared paper, then a square of side 2 units to make a 3 by 2 rectangle, then a square of side 3 units to make a 5 by 3 rectangle, and continue drawing squares whose sides are given by the Fibonacci numbers until you fill your piece of paper.
OK, if you have explored the ratios using the spreadsheet you have some pretty convincing evidence that the ratio of successive terms of the Fibonacci sequence tends to the limit called the golden ratio which has a value $\phi \approx 1.618$.This is a fact but we have not proved it yet.
Now you might like to draw this spiral for yourself on the whirling squares diagram you have already drawn. Just draw the curve from corner to corner across each square.
I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?
I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?
The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.