## 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

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.

**GeoGebra applets:**- with a table | with linked function diagrams | with a 'cobweb' graph
**Fathom files**:- Arithmetic
- Linear

Quick review/overview of key concepts, plus derivation of an explicit formula for the *n ^{th}* iterate. (For 11th/12th graders, or teachers.)

- I created this applet (Tri, Tri Again) to illustrate a wonderful (and unexpected?) application of these ideas to geometry, by Rachel Chou:
- Triangles and Iteration (PDF)

## Non-Linear

- After this, we are ready for a more formal approach to the above material, and an exploration of the iteration of some non-linear functions:
- See Infinity, Unit 2: Chaos

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