Mathematics - Lesson 10
Fibonacci Sequence
The Fibonacci sequence begins with two simple numbers, then grows by adding the previous two terms. That tiny rule creates a pattern that shows up in algorithms, population models, plant growth, art, architecture, and the golden ratio.
The Rule
A sequence is an ordered list of numbers. In the Fibonacci sequence, each new term is the sum of the two terms immediately before it.
The first terms are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34.... Every number carries the memory of the two before it.
Build the Pattern One Term at a Time
Move the slider to reveal more terms. Watch the ratios between neighboring terms settle near the golden ratio.
Ratio Trail
Fibonacci Spiral
Why It Grows So Fast
The Fibonacci sequence is recursive: the next value depends on earlier values. Recursive rules can create surprisingly large growth from very simple instructions.
Practice
Use the rule. Each answer should be a whole number.
1. What comes next? 1, 1, 2, 3, 5, __
2. What comes next? 8, 13, 21, 34, __
3. If F(7) = 13 and F(8) = 21, what is F(9)?
Where Fibonacci Appears
Fibonacci-style patterns are useful because they model growth where each new stage depends on earlier stages. You can find related patterns in branching plants, pinecones, sunflower seed packing, music, art composition, and computer science.
Lesson Guide
Goal: recognize Fibonacci as a recursive sequence and connect it to ratios, spirals, and growth models.
- Start with the rule: add the previous two terms.
- Use the lab to compare terms and ratios.
- Ask why a simple rule can create complex-looking structure.
Teacher Prompts
- What information do you need before you can calculate the next term?
- How does the ratio change as the terms get larger?
- Where else do you see repeated growth patterns?