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
7. Area within shapes on a geoboard, leads to integral! (see also Geoboard Magic)
8. Find the slope of a ramp (or mountain,..)
9. # of images in the hinged mirrors vs the angle between the mirrors
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 IES java applet inspired by Don's chapter 11 .
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.
i. Nautilus shell
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
Graph of ii i... and the IES
java applet inspired by Don's chapter 11
iv. Graph of r = 2t in 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
24. Olivia makes a 3x3x3 cube with the Soma pieces.
25. Anna: " the sum of the angles of a triangle is not always equal to 180° "