3. Patterns in shapes

1.  Using the Fibonacci nos. to make a golden rectangle (chapter 7)

2.  Using the pentagon to get the golden mean (also similar triangles)

3.  The golden spiral from a pentagon (chapter 7)

4.  Start the binomial expansion using the area of a square, and volume of a cube

5.  Number of routes between points...taxicab geometry, people pieces, lead to Pascal's triangle (chapter 9)

6.  Following Archimedes to get Pi

8.  Find the slope of a ramp (or mountain,..)

10.  How lengths, squares and cubes grow (chapter 6). See also Genny doubles the size of a dog

11.  The surface area to volume ratios of  the Cuisenaire rods to find out why rodents are nocturnal animals (chapter 6)

12.
Fractals- finding the area and perimeter of the snowflake curve and see Emily's work on the same and the

13. How the Nautilus shell grows (chapter 6)- similar shapes within the Nautilus shell. See
the 'eye test' for similar shapes in Chapter 11.

14. Using a Pantograph (similar shapes) by Roxana and by Sheri.

15. Finding the area of rectangles with a constant perimeter (chapter 14).

16. Using squares and pieces of squares to find fractions which lead to infinite series

17. Finding the angle formed two chords in a circle

18. Using squares and rectangles to multiply fractions

19. Volume of a pyramid is 1/3 the Volume of a cube with the same base and height (Sheri's work) and see

20. See Abe's work on tessellations

21. Geometric transformations with matrices: Justin works on reflections with matrices . Also see Don's sample problems from his book "Changing Shapes With Matrices"

22.. Finding the sum of the angles of a polygon. See Paul's work

23. spirals

Study growth

a). in size of chamber, by finding ratios of radius vectors 360° apart and

b). in volume by pouring water in chambers

c). the angle between the radius vector and the tangent to the curve

Graph of  exCot(79.5*Pi/180) and the Nautilus shell

ii. Graph (1+i)n   See chapter 11

iii. Graph of ii i...  and the IES java applet inspired by Don's chapter 11

iv. Graph of  r = 2in chapter 11, the polar graph of a spiral and shows the ratios of the areas of 90° segments. See "On Size and Life" by McMahan & Bonner; Scientific American Library Series; W.H. Freeman and Co., NY, NY; pp 47-51

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