Fibonacci Paths
Discovering the Fibonacci sequence through counting paths
A hands-on, logic-based workshop that introduces the Fibonacci sequence through counting paths and recursive reasoning, rather than memorizing formulas.
Workshop Overview
This workshop introduces the Fibonacci sequence through a concrete counting problem. Students explore how the number of possible paths to a target changes under different step rules, leading naturally to a recursive pattern.
Conceptual Framework
Students build a simple linear path using marked positions arranged in a row. A starting point and a target position are defined.
The scenario is enacted through two puppet characters. One character moves under the rule of each episode to reach the second character, who remains at the destination.
The exploration unfolds across three episodes, each introducing a different movement rule.
In Episode 1, the character may move only one step at a time, from one position to the next.
In Episode 2, the character may move only two steps at a time, jumping directly to the second next position.
In Episode 3, the character may move either one step or two steps. Counting the possible paths under this rule leads naturally to the Fibonacci sequence.
In the final episode, students observe that the number of paths to a position equals the sum of the paths to the two previous positions.
Game Structure
- A linear path of positions is created using simple classroom materials.
- A character begins at the starting point.
- The target is placed at different positions along the path.
- Students count all valid paths using one-step and two-step moves.
Mathematical Mapping
- Positions represent steps along a number line.
- Paths represent different combinations of moves.
- The total number of paths forms a numerical sequence.
- The recurrence relation Fn = Fn−1 + Fn−2 emerges from the counting process.
Learning Outcomes
- Explain the Fibonacci sequence as a result of counting paths
- Understand recursive relationships through concrete examples
- Recognize how simple rules generate numerical patterns
- Connect physical models to abstract mathematical sequences
Target Group
Upper elementary and middle school students, with adaptable depth for different age groups.