Mathematics Topics: The first task presents a classic non-routine problem that can be used in a variety of ways. The problem is then extended. The second task is a continued fraction, and the third concerns Fibonacci sequences and ratios. The fourth task concerns finding a fixed-point. All tasks can be solved with recursion and other methods.

Mathematical Thinking: Students can use a variety of heuristics to complete the collection of tasks. Examples of useful heuristics include: draw a diagram, solve a simpler problem, use algebraic equations, look for a pattern, use symmetry, use graphs. The tasks also ask students to generate recursive equations, compare recursive methods with other methods, conjecture about limits and prove conjectures.

Technology: Graphing calculators can be used to generate a table of values from recursive equations and plot the generated values.

Sample Screenshots: Below are screenshots illustrating graphing calculator-based solutions to some of the tasks.

Figure 2. Continued Fraction Screenshots