TITLE: Do the Twist!

SUBJECT: Algebra II or Advanced Algebra

LESSON TOPIC: "Recursive Formulas for Sequences" from unit on "Patterns, Sequences and Series"

INTERACTIVITY: Following instructions on worksheet, and using the examples from nature, students remove Velcro-backed squares from the pocket and construct a "Fibonacci Spiral" by sticking the squares to the felt background on the work area of the board. Students next determine the dimensions of the squares by noting their positions relative to one another in the spiral. Using these dimensions and the order in which the squares were placed on the board, the students write the beginning of the "Fibonacci Sequence."

DIRECTIONS FOR USE:

- Make sure that seven (7) Velcro-backed squares are present either in the pocket at the bottom of the bulletin board, or on the felt-backed work area of the bulletin board.
- Each square has a quarter circle inscribed inside of it with black marker, except one of the two smallest squares, which has a semicircle inscribed inside.
- Begin by affixing the small square with the inscribed semicircle near the center of the felt work area of the bulletin board, oriented so that the semicircle looks like the letter ‘U’.
- Next, attach the other small square directly above and adjacent to the first square, oriented so that the arcs inscribed in the squares meet at their common edge.
- Then, affix the next larger square directly beside and adjacent to the rectangle formed by the first two squares, and oriented so that the arc beginning at the center appears continuous.
- Continue to create a larger and larger rectangle and extend the spiral outward by adding squares, each new square having a side which is as long as the sum of the previous two squares’ sides.
- Assuming the first two squares you placed on the board have sides of unit length, you can determine the sizes of all the squares by noting their positions relative to one another in the spiral.
- Write the sizes of the squares in the order that they were placed on the board. This is the beginning of the Fibonacci Sequence. Then see if you can determine the next three numbers in the sequence:

- Please move the squares around on the board or return them to the pocket so that the next student can give this a try!
- Complete the following recursive formula to define the Fibonacci Sequence:

- For extra credit, investigate Dr. Ron Knott's "Easier Fibonacci Puzzles" website (www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/ fibpuzzles.html) and write a one-page description and explanation for one of the puzzles contained there.

TIME: should take 15 minutes or less.

SPECIAL CONSTRUCTION: Squares of foam board with Velcro tape on back. Felt surface on work area of bulletin board.

CREDIT: Homework grade for completing worksheet. Extra credit if student investigates Dr. Ron Knott's "Easier Fibonacci Puzzles" website (www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/ fibpuzzles.html) and writes appropriate one-page description and explanation for one of the puzzles.