Fibonacci Forest Quest

Overview

In this activity, students explore how math and science intersect in nature—spotting the Fibonacci sequence in pinecones, flowers, and trees as they work to protect the Fibonacci Forest from destruction.

Objectives

Upon completion of the activity, students will review the Fibonacci Sequence, be able to calculate/ identify the first 10 numbers of the sequence, connect the sequence to the spiral visualization, and navigate an immersive virtual environment to locate natural objects that align to the sequence.

Standards

NGSS Performance Expectations

• MS-LS1-4: Use argument based on empirical evidence and scientific reasoning to support an explanation for how characteristic animal behaviors and specialized plant structures affect the probability of successful reproduction of animals and plants respectively.
• MS-LS1-5: Construct a scientific explanation based on evidence for how environmental and genetic factors influence the growth of organisms
• MS-LS4-6: Use mathematical representations to support explanations of how natural selection may lead to increases and decreases of specific traits in populations over time.

CCSS Math Standards

• CCSS.Math.Content.4.OA.C.5: Generate a number or shape pattern that follows a given rule.
• CCSS.Math.Practice.MP7: Look for and make use of structure.

Activity Details

Duration of Activity: 15-20 minutes

Materials:

• Smartphone or tablet with the McGraw Hill AR Application installed
• Flat, non-patterned surface

Scientific and Engineering Practices:

• Analyzing and Interpreting Data: Students can analyze patterns in nature and interpret how they relate to the Fibonacci sequence.
• Using Mathematics and Computational Thinking: Applying mathematical concepts to understand natural phenomena aligns well with the Fibonacci sequence exploration.

Disciplinary Core Ideas

LS1.B: Growth and Development of Organisms - Understanding how plants and animals use various structures and behaviors to grow and reproduce can be linked to observing patterns like the Fibonacci sequence in nature.

Launch and Explore

Launch: Scanning

The device needs a variety of perspective information to understand the space.

• Slowly move the camera throughout the space.
• View surfaces at an angle.
• Aim the camera at multiple points throughout the space.

Exploration

• Move the phone closer in to increase the size of the objects in AR.
• Move the phone around the objects to view them from different angles.

Environment

Ideal spaces for AR should feature the following:

• A flat open space;
• A surface with non-patterned visual texture and contrast;
• A matte or minimally reflective surface;
• A static environment, where nothing in the space is in motion; and
• A well-lit space, where detail is visible in the darkest and brightest parts of the space.

During the Activity

Teacher Tips

• This is a spatial activity, which means that students' entire environment will become an enchanted forest. Make sure students have plenty of space to walk around without obstacles.
• Students will wander through the forest looking for the items on the list. Some may be hidden, so be on the lookout!
• The forest map, included at the end of this lesson plan, can be used to provide clues or can be distributed to students if needed.

Evaluate

Students will be presented with five randomly selected exercises from the following set.

1. What is the Fibonacci sequence?

   • A. A sequence of even numbers
   • B. A sequence where each number is the sum of the two preceding numbers (Correct Answer)
   • C. A sequence of prime numbers
   • D. A sequence of random numbers

2. What shape is formed when you graph the Fibonacci sequence?

   • A. A triangle
   • B. A circle
   • C. A spiral (Correct Answer)
   • D. A rectangle

3. What are some examples of Fibonacci numbers in plants? Select all that apply.

   • A. The number of petals on flowers (Correct Answer)
   • B. The arrangement of seeds in a sunflower (Correct Answer)
   • C. The height of a tree
   • D. The spacing of leaves on a stem (Correct Answer)

4. What are the first two numbers in the Fibonacci sequence?

   • A. 0 and 1 (Correct Answer)
   • B. 1 and 2
   • C. 1 and 1
   • D. 2 and 3

5. Why do plants use the Fibonacci sequence in their growth?

   • A. To grow faster
   • B. To optimize sunlight exposure (Correct Answer)
   • C. To produce more seeds
   • D. To grow taller

6. What is the 6th number in the Fibonacci sequence?

   • A. 5 (Correct Answer)
   • B. 8
   • C. 13
   • D. 21

7. Where can the Fibonacci sequence be observed in nature? (Select all that apply)

   • A. Hurricane patterns (Correct Answer)
   • B. Mountain peaks
   • C. Acorns
   • D. Nautilus shells (Correct Answer)

8. What number would come next in this part of the Fibonacci sequence? 5, 8, 13, ___

   • A. 15
   • B. 21 (Correct Answer)
   • C. 34
   • D. 55

9. Which of these is NOT a Fibonacci number?

   • A. 2
   • B. 3
   • C. 8
   • D. 9 (Correct Answer)

10. Choose the Fibonacci example that goes with the correct image.

    The images associated with this question are labeled "Plants", "Architecture", and "Space".

    • A. 5
    • B. 8
    • C. 13
    • D. 21

    Note: The correct answer choice for this question is not explicitly marked in the provided text. The images depict a sunflower head (Plants), a spiral staircase (Architecture), and a spiral galaxy (Space), all illustrating Fibonacci-like spirals.

Extension Activities

1. Fibonacci Sequence Calculation

The Fibonacci Sequence is a series of numbers that is calculated by adding the two previous numbers together and has no end. It starts with 0, 1, 1, and 2. Calculate the next 10 numbers in the sequence and list them below.

Student responses should be: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233.

2. Plant Benefits of Fibonacci Spiral

Many plants have petals, leaves, and seed heads arranged in the Fibonacci spiral. How does this benefit the plant?

Students should be able to state that the arrangement of petals and leaves in the Fibonacci spiral pattern ensure that the plant optimizes exposure to rain and sunlight. The spiral arrangement seen in seed heads ensures that the plant is optimizing the opportunities for reproduction by efficiently packing as many seeds as possible in the seed head.

3. Sunflower Spiral Count

The sunflower is a great example of how nature has used the spiral of the Fibonacci Sequence to maximize growing opportunities. The seeds in this flower head are arranged in spirals and the number of spirals will equal one of the Fibonacci numbers. Highlight and count the number of spirals. How many did you come up with?

Responses may vary, as there is more than one way to count the spirals. Each method, if properly counted, will provide a Fibonacci number. The typical methods will result in either 21, 34, or 55 spirals. Encourage students to compare their results and methods with each other.

Enrichment

The Connection Between Nature and Art

We've seen how the spiral design of the Fibonacci Sequence can be found all around us in the natural world. The design is not only practical, allowing plants to maximize exposure to sun and rain, but is also elegant and beautiful. Think about the curves in the sunflower head, the spirals in the snail shell, or the arrangement of the scales on the pinecone.

The spiral also provides visual balance and proportion, which has influenced architects and artists for generations. From ancient architecture to Renaissance paintings to photography, the beauty and balance of the Fibonacci spiral is still used today by artists and designers. Now it's your turn! Using the graph paper below, graph the first few numbers of the Fibonacci Sequence then turn it into art. Use whatever materials are available and make it into something beautiful! Use color, shapes, or even take some inspiration from nature to create your own masterpiece.

Student responses should all include the classic Fibonacci spiral but will vary depending upon the materials available and their personal artistic choices. Use an internet search for examples if students need inspiration. Consider displaying finished student work.

Visual Descriptions

Overhead Map of Fibonacci Forest

This is a colorful, stylized overhead map of a virtual forest environment used for the AR activity. It depicts various natural elements like trees (purple, green), flowers (yellow sunflowers, red roses), water features (blue streams/waterfalls), and rock formations. Some areas feature spiral patterns, and a distinct spiral galaxy graphic is visible in the upper right. The map shows the layout for the AR exploration activity.

Graph Paper

A blank grid of graph paper is provided for students to draw the Fibonacci sequence and create art.

Fibonacci Examples (Page 3, Question 10)

• Plants: An image of a sunflower head, showing the spiral arrangement of seeds.
• Architecture: An image of a spiral staircase in a building, demonstrating a Fibonacci-like spiral.
• Space: An image of a spiral galaxy, illustrating a large-scale Fibonacci-like spiral.