# LESSON PLAN IV

### Purpose

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

### Objectives

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

### Activities

1. Knowing that Ø= √5/2 + 1/2, show that 1/Ø = √5/2 - 1/2
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
3. Sketch a graph of quotients of Fibonacci numbers, fn+1/fn. 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, fn+1/fn) and connect the dots. Lable Ø on the graph.
4. Make a table of values for powers of Ø divided by √5 and find a pattern for the values of the nearest integer.
nØn/√5Nearest Integer
1Ø/√5 = 0.72360... 1
2Ø2/√5 = 1.1782...1
3Ø3/√5 = 1.89443...2
4Ø4/√5 = 3.06525.....
5Ø5/√5 = 4.95967.....
6Ø6/√5 = ..
7Ø7/√5 = ..
8Ø8/√5 = ..

5. Fill in the table to find another pattern:

 n Nearest Integer (Øn + 1/Øn)/√5 1 (Ø + 1/Ø)/√5 1 2 (Ø2 + 1/Ø2)/√5 1 3 (Ø3 + 1/Ø3)/√5 2 4 (Ø4 + 1/Ø4)/√5 .. 5 (Ø5 + 1/Ø5)/√5 .. 6 (Ø6 + 1/Ø6)/√5 .. 7 (Ø7 + 1/Ø7)/√5 .. 8 (Ø8 + 1/Ø8)/√5 ..
6. Finally, consider the optical illusion called Luis Carroll's paradox. Does the area really increase from 82 = 64 to 5*13 = 65? Cut up an 8" square and reassemble to find out .