The Journal of Online Mathematics and Its Applications, Volume 8 (2008)

Modeling Spiral Growth in a GSP Environment, Dogan-Dunlap and Jordan

In this sample worksheet, we have omitted the white space in which students would ordinarily copy and paste the resuts from the GSP Program.

Use the file Spirals.gsp to enter the numbers.

- Define rational numbers.
- Define irrational numbers.
- Enter the following rational numbers (1/4, 1/5, 1/8, 2/10) :
- 1st Rational: 1/4. Copy and Paste the GSP result in the space provided below:
- 2nd Rational: 1/5. Copy and Paste the GSP result in the space provided below:
- 3rd Rational: 1/8. Copy and Paste the GSP result in the space provided below:
- 4th Rational: 2/10. Copy and Paste the GSP result in the space provided below:

- Enter the following irrational numbers (π,
`e`, √2 ).- 1st Irrational: π. Copy and Paste the GSP result in the space provided below:
- 2nd Irrational:
`e`. Copy and Paste the GSP result in the space provided below: - 3rd Irrational: √2. Copy and Paste the GSP result in the space provided below:

- Compare the sketches from questions 3 and 4, and describe the similarities and differences of each sketch focusing on rational and irrational number behaviors.
- What number (s) can you input to create a sketch that would produce ten rays?
- What number (s) would you use to create the sunflower seed placement appearance?
- Try the following number (1 + √5) / 2. This number is called the golden ratio.

Based upon your observations, make a conjecture as to why rational numbers spike out and eventually create rays while some irrational number approximations make more spirals.