1. Golden Spirals or ‘Whirling squares’ These are mathematical constructions which appear in many places.

2. Our spiral closely resembles that of the Nautilus shell. This is another genuine – and striking – occurrence of Fibonacci numbers cropping up in the natural world. Nautilus Shell

3. Square Longest side of rectangle 1 Starting from the square given; add a 1x1 square below to form a rectangle. add a 2x2 square to form a bigger rectangle add a 3x3 square to form an even bigger rectangle etc……… Whirling squares - Worksheet

4. SquareRectangles longest side 11 22 33 45 Whirling squares Starting from the square given; add a 1X1 square below to form a rectangle. add a 2X2 square to form a bigger rectangle add a 3x3 square to form an even bigger rectangle etc………

5. Starting from the square given; add a 1x1 square below to form a rectangle. add a 2x2 square to form a bigger rectangle add a 3x3 square to form an even bigger rectangle etc……… Each new rectangle has a side which is as long as the sum of the previous two square's sides. SquareRectangles longest side 11 2 3 4 5 6 7 8 Whirling squares 2 3 5 8 13 21 34

6. Use a compass to draw quarter- circles in each square to create an elegant Golden spiral. SquareRectangles longest side 11 22 33 45 58 613 721 834 Whirling squares Add a splash of colour to complete your drawing Golden Spiral

7. Each new rectangle has a side which is as long as the sum of the previous two square's sides. SquareRectangles longest side 11 22 33 45 58 613 721 834 Whirling squares Golden Spiral

8. 1 ÷ 1 = 1 2 ÷ 1 = 2 3 ÷ 2 = 1.5 5 ÷ 3 = 1.6666666666… 8 ÷ 5 = 1.6 13 ÷ 8 = 1.625 21 ÷ 13 = 1.61538461… What do you notice about 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584… Ф The Fibonacci ratio Ф these numbers?

9. The Golden ratio The number that these ratios tend towards = 1.618033988749… is called the; Golden ratio or ‘Phi’ The Golden ratio is not unique to just my Fibonacci numbers. Indeed, if you start with any 2 numbers you like, remarkably after a few terms ‘Phi’ will always appear Is Fibonacci correct? Try it for any 2 numbers as proof

10. My sequence ratio tends to TermMy sequenceSequence ratio 1 2 3 4 5 6 7 8 9 10 11 12 My sequence =,,,,, ………………. The Golden ratio is not unique to just my Fibonacci numbers. Indeed, if you start with any 2 numbers you like, remarkably after a few terms ‘Phi’ will always appear Is Fibonacci correct? Start with any 2 numbers Create your own sequence Find the ratio of your sequence Ф The ratio of ‘My’ sequence - worksheet Ф

11. My sequence ratio tends to TermMy sequenceSequence ratio 1 2 3 4 5 6 7 8 9 10 11 12 My sequence =,,,,,,,,,,,,………. The Golden ratio is not unique to just my Fibonacci numbers. Indeed, if you start with any 2 numbers you like, remarkably after a few terms ‘Phi’ will always appear Is Fibonacci correct? Start with any 2 numbers Create your own sequence Find the ratio of your sequence

12. Teacher notes; for Number Magic