## Linear

Two packets:

**"Iterating Linear Functions: An Introduction to Dynamical Systems"**. These are activities which I co-authored with Jonathan Choate for*The Mathematics Teacher,*February 1997, special issue on Algebra. Use them to introduce these ideas.- Teacher's Guide (PDF)
- Student Sheets (PDF)

- Some lessons from
*Algebra: Themes, Tools, Concepts,*(1994) the book I co-authored with Anita Wah. You can use**"Instant Riches"**before the above packet, and the rest as a source of additional problems, homework, and assessments. Two PDF files: - Teacher's Guide (PDF)
- Student Sheets (PDF)

- Some lessons from

Note that there is a difference in terminology between the two packets: what is known as a "time series graph" in the first is called "linked function diagrams" in the second.

Both packets require technological support, such as a spreadsheet program, a graphing calculator, Fathom, Desmos, or GeoGebra.

**Fathom files**:- Arithmetic
- Linear
**GeoGebra:**- Iterating Linear Functions:with a table | with a 'cobweb' graph

- As a follow-up to these lessons, here is a worksheet I use in Infinity, a post-Algebra 2 elective course, to review the above material at a higher level, and to find an explicit formula for the `n^(th)` term:
- Explicit Formula(PDF)

## Non-Linear

- After this, we are ready for an exploration of iterating some non-linear functions:
- See Infinity, Unit 2: Chaos
- Excerpts: Iterating the Logistic Function | Assessment: Project

**GeoGebra:**- Iterating `f(x)=rx(1-x)` (in two representations)
- Bifurcation Graph for `f(x)=rx(1-x)`

- Finally, a number-theoretic extension:
- Iterating `f(x)=2|x|-2`