GGBNA 2012

Steve Phelps

[math]Next = \frac{Now + 1}{Last}[/math]: From Elementary Mathematics to Elliptic Curves with GeoGebra Steve Phelps GeoGebra Institute of Ohio (USA) Madeira High School University of Cincinnati Email: steve.phelps@geogebra.org Twitter: @giohio Facebook: tinyurl.com/giohiofacebook This Talk: tinyurl.com/ggbna2012 GGBNA 2013: midwestggb.com/conference This is my presentation for the 2012 North American GeoGebra Conference, which chronicles my GeoGebra investigation into a simple little problem found in these two resources: [list] [*]Sawyer, W.W. (1961). Lyness’ Periodic Sequence. The Mathematical Gazette, Vol. 45, No. 353, p. 207. The Mathematical Association of America [*]Lyness, R.C. (1961). Cycles. The Mathematical Gazette, Vol. 45, No. 353, p. 207-209. The Mathematical Association of America [/list] The problem is so simple, it can be readily understood by an elementary school student, yet is deep and rich enough to be the focus of graduate level mathematics research.


  • 1. GGBNA 2012 #0
  • 2. GGBNA 2012 #1
  • 3. GGBNA 2012 #2
  • 4. GGBNA 2012 #3
  • 5. The Original Problem in 2D
  • 6. GGBNA 2012 #5
  • 7. GGBNA 2012 #6
  • 8. GGBNA 2012 #7
  • 9. What-If-Not: The Original Problem in 2D
  • 10. The Steps of the Sequence and Invariant Curve
  • 11. What-If-Not: Complex Number Lyness Sequence
  • 12. GGBNA 2012 #11 - Iterating Trig Functions

1.

GGBNA 2012 #0

This is a very simple problem on the surface, so simple that an elementary aged student could understand it, yet deep enough to be the focus of graduate level mathematics work. The problem is this: Pick two numbers to start with (Today and Yesterday, Now and Last, [math]a_2[/math] and [math]a_1[/math], preferably positive. Perform the calculation [math]\frac{Today + 1}{Yesterday}[/math] or [math]\frac{Now + 1}{Last}[/math] or [math]\frac{a_{n+1} + 1}{a_n}[/math] to get a new number. This new number will be referred to as Tomorrow or as Next or as [math]a_3[/math]. For younger students, it is one day later (Tomorrow is the new Today), or it is one step later (Next is the new Now). Repeat this calculation, continuing to repeat this process until you get to a nice number at which to stop.
– Steve Phelps

2.

–

3.

–

4.

–

5.

–

6.

–

7.

–

8.

–

9.

–

10.

–

11.

–

12.

–

Information