Golden Blueprint Worksheet
Golden Blueprint
Sequence Ratio Investigation
Name:
Date:
The Fibonacci Sequence
A recursive sequence where each term is the sum of the two preceding terms: \(F_n = F_{n-1} + F_{n-2}\).
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...
Calculator Tip
Find ratios by dividing a term by the one before it.
8 ÷ 5 = 1.6
Part 1: Video Observation
As you watch (2:00–5:00), record the decimal results for the ratios (\(F_n / F_{n-1}\)).
| Ratio | Decimal |
|---|
| 1 / 1 | 1.000 |
| 2 / 1 | 2.000 |
| 3 / 2 | 1.500 |
| 5 / 3 | |
| 8 / 5 | |
| Ratio | Decimal |
|---|
| 13 / 8 | |
| 21 / 13 | |
| 34 / 21 | |
| 55 / 34 | |
| 89 / 55 | |
Part 2: Lucas Sequence Challenge
Same rule (add previous two), different start (2, 1):
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123...
| Ratio (\(L_n / L_{n-1}\)) | Calculated Decimal (Round to 4 places) |
|---|
| 1 / 2 | |
| 3 / 1 | |
| 4 / 3 | |
| 7 / 4 | |
| 11 / 7 | |
| 18 / 11 | |
| 29 / 18 | |
| 47 / 29 | |
| 76 / 47 | |
| 123 / 76 | |
Part 3: Conclusions
1. Comparison: Look at the final ratios for both Fibonacci and Lucas. What do you notice about the number they are both "approaching"?
2. Defining the Limit: In math, a limit is a value that a sequence gets closer and closer to. What is the limit of these ratios (\(\phi\))?
\(\phi \approx\)
3. Theory: Why do you think sequences with different starting numbers end up at the same ratio?
4. Application: Use your value for \(\phi\) to predict the term after 500 in a sequence that follows this golden rule.
Golden Blueprint Slides
Golden Blueprint
Discovering Nature's Constant (\(\phi\))
The Hook
Petal Patterns
Shell Spirals
Galactic Arms
"What mathematical pattern connects these vastly different parts of our universe?"
Video: Fibonacci Ratios
Watch closely from 2:00 to 5:00.
Tracking Task
Record the decimal results in Part 1 of your worksheet.
Embedded media
The Lucas Sequence
The Lucas Sequence uses the exact same recursive rule as Fibonacci:
2, 1, 3, 4, 7, 11, 18...
If the starting numbers change, does the ratio limit change too?
Your Mission
- 1 Use Part 2 of your worksheet.
- 2 Calculate the next 5 ratios.
- 3 Find the convergence point.
The Result
Fibonacci Limit
≈ 1.618
=
Lucas Limit
≈ 1.618
Regardless of the starting numbers, any sequence with the rule \(F_n = F_{n-1} + F_{n-2}\) converges to the same magic ratio.
\(\phi\) Phi
The Golden Ratio
Mathematical Limit
The value a sequence "approaches" as the number of terms goes to infinity (\(n \to \infty\)).
The Exact Value
\[\phi = \frac{1 + \sqrt{5}}{2}\]
Golden Blueprint Teacher Guide
Golden Blueprint
Teacher Facilitation Guide
Geometry / Algebra II
Instructional Goal
Students will discover that the ratio of successive terms in any additive recursive sequence converges to the same mathematical constant: The Golden Ratio (\(\phi\)). This lesson introduces the concept of a limit in a concrete, numerical context.
Essential Questions
- How can a sequence of whole numbers reveal an irrational constant?
- What does it mean for a sequence of values to "converge" to a limit?
- Does the starting point of a sequence change its long-term behavior?
At a Glance
Duration
45–50 Minutes
Materials
Slide Deck, Worksheet, Scientific Calculators
Video Link
youtube.com/watch?v=mVO2dcuR7P0
Lesson Timeline
01
The Hook (5 min)
Use Slide 2 to display nature images. Ask students what math connects a sunflower and a galaxy. Don't reveal the answer; let them speculate about "spirals" or "proportions."
02
Video Observation (10 min)
Play the video from 2:00 to 5:00. Students must track the Fibonacci ratios in Part 1 of the worksheet. Pause at 2:54 to allow students to calculate 13/8 and 21/13 on their own before the video reveals them.
"Watch how the numbers jump above and below 1.618 before settling. This 'oscillation' is a key feature of the convergence."
03
The Lucas Challenge (20 min)
Students work individually or in pairs to calculate Lucas Sequence ratios. They may assume the limit will be different because the numbers are different (2, 1, 3...). Monitor their progress—the "Aha!" moment happens when they reach 123/76.
04
Discussion & Synthesis (10 min)
Define "Phi" (\(\phi\)) and the concept of a limit. Use Slide 6 to show the exact radical form. Discuss why this happens: the rule (\(F_n = F_{n-1} + F_{n-2}\)) dictates the ratio, not the starting values.
Answer Key & Data
Lucas Sequence Ratios (\(L_n / L_{n-1}\))
1 / 2 0.5
3 / 1 3.0
4 / 3 1.3333
7 / 4 1.7500
11 / 7 1.5714
18 / 11 1.6364
29 / 18 1.6111
47 / 29 1.6207
76 / 47 1.6170
123 / 76 1.6184
Synthesis Question 1: Comparisons
