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.
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 = 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' "