Increase students. familiarity with the Golden Ratio and its relationship to the Fibonacci sequence

Students will be able to construct a Golden Rectangle, compute using the Golden ratio and see Fibonacci connections.

- Knowing that Ø= √5/2 + 1/2, show that 1/Ø = √5/2 - 1/2

- Construct a Golden Rectangle. Start with a 2-unit square, mark a segment of length √5, strike an arc ending
to the level of the base and extend the base

- Sketch a graph of quotients of Fibonacci numbers, f
_{n+1}/f_{n}. Use the values from the table. Using graph paper, mark the horizontal scale from 1 to 14 in units of 1 and the vertical axis from 1 to 2 in units of 1/25 (0.04). Put a dot at each pair (n, f_{n+1}/f_{n}) and connect the dots. Lable Ø on the graph.

- Make a table of values for powers of Ø divided by √5 and find a pattern for the values of the nearest
integer.

n Ø ^{n}/√5Nearest Integer 1 Ø/√5 = 0.72360... 1 2 Ø ^{2}/√5 = 1.1782...1 3 Ø ^{3}/√5 = 1.89443...2 4 Ø ^{4}/√5 = 3.06525..... 5 Ø ^{5}/√5 = 4.95967..... 6 Ø ^{6}/√5 =.. 7 Ø ^{7}/√5 =.. 8 Ø ^{8}/√5 =..

- Fill in the table to find another pattern:

n (Ø ^{n}+ 1/Ø^{n})/√5Nearest Integer 1 (Ø + 1/Ø)/√5 1 2 (Ø ^{2}+ 1/Ø^{2})/√51 3 (Ø ^{3}+ 1/Ø^{3})/√52 4 (Ø ^{4}+ 1/Ø^{4})/√5.. 5 (Ø ^{5}+ 1/Ø^{5})/√5.. 6 (Ø ^{6}+ 1/Ø^{6})/√5.. 7 (Ø ^{7}+ 1/Ø^{7})/√5.. 8 (Ø ^{8}+ 1/Ø^{8})/√5.. - Finally, consider the optical illusion called Luis Carroll's paradox.
Does the area really increase from 8
^{2}= 64 to 5*13 = 65? Cut up an 8" square and reassemble to find out .