A Mathematical Nature Walk

[Catrin Muzquiz]

Beforeembarking on our mathematical nature walk, we gathered some essential supplies. You will need a notebook, some pencils, a ruler, a magnifying glass, and a camera. So, with our tools in hand and curiosity in our hearts, we set out to explore a nearby park. The park offered a variety of natural elements to observe and analyze.

Our first stop was a lush area filled with trees, bushes, and flowers. We were so amazed at the array of shapes and patterns in the leaves and petals. Therefore, we took out our magnifying glass to closely examined the intricate veins on a leaf and the symmetry in a blooming flower. Back at home, we turned our observations into a mini-lesson on geometry, discussing concepts like symmetry, polygons, and angles.

As we continued our walk, we encountered fallen leaves of various sizes scattered along the path. We decided to turn this into a counting and measurement activity. We gathered leaves and measured their lengths, then compared and sorted them based on size. This led to discussions about measurements, units, and organizing data.

One of the highlights of our walk was discovering a pinecone on the ground. We marveled at the way the spirals were arranged and realized it followed the Fibonacci sequence. We discussed the concept of the Fibonacci sequence and observed how it appears not only in pinecones but also in sunflowers and seashells. When we arrived home we read Leo + Tea and Growing Patterns: Fibonacci Numbers in Nature by Sara Campbell.

Near a tranquil pond, we noticed the reflection of the trees in the water. This sparked a conversation about symmetry and reflectional symmetry in nature. We used our notebooks to sketch the reflections and discussed how symmetry plays a role in art, architecture, and even animal markings.

Throughout our mathematical nature walk, we captured moments through photographs and notes in our notebooks. When we returned home, we compiled our findings and reflections into a nature-inspired math journal. This not only solidified the concepts we learned but also allowed us to create a beautiful keepsake of our outdoor adventure.

Little Lions Learn was founded in 2019 by Khara Sandoval, a certified teacher who left the public education system to homeschool her own children. Khara joined forces with Kristy Crandall, the author of "The Science of Homeschooling", to create a completely new educational experience for our nation's children. "We make homeschooling possible by offering an affordable online community, engaging classes, and essential resources that empower parents to confidently develop a homeschool plan that fits their unique family needs." Our goal is to help new and seasoned homeschool families experience a positive education journey. We know making the switch from public school to homeschooling is a huge adjustment, but we believe it is worth it even if you have to juggle work. We do it, and we know you can, too!

Get students to gather a collection of seeds from a wooded area. They can drag an old blanket or fuzzy cloth behind them on the ground. They can place a large, old, wool sock over hands or shoes to see what they can pick up. Or they can wear bracelets made of masking tape (sticky side out), to hold the seeds they find. Have children examine their seed collections and invent a system for sorting or classifying them. Explain that plants have developed many different methods of seed dispersal, to ensure the success of their species.

Local ponds, swamps, and dams are teeming with microorganisms and bugs. Why not take some dip nets and magnifying glasses with you on your nature walk? Stop at these ponds and allow children to do some dip netting to see what sort of critters live in the ponds. Students could also pick different environments during their nature walks and make observations on sunlight, soil moisture, temperature, wind, water flow, plants, and animals in each.

One of the best things to do on a nature walk is to simply observe what nature has to offer. With an app such as iNaturalist, students can record, catalog and share what they have found. The app also facilitates connection with thousands of scientists to help users identify the species they have observed. With other apps such as Leafsnap or the online tree identification field guide from the Arbor Day Foundation, students can also identify the trees and plant species in their area.

It was a little odd but one day I took a ruler with us on our walk in the nature reserve. Although I did get a few strange looks from passing walkers, we enjoyed a whole hour of learning and fun together. As soon as I put the ruler in my daughters hand (and explained about centimeters and millimeters) she was off measuring everything in sight.

After letting her get used to the instrument and giving her plenty of time to play with it, we sat down with a little more structured activity. We collected sticks and leaves and by measuring them we sorted them into longest to shortest. She was really focused in this activity too.

Penny Whitehouse is a mother of three, with tertiary qualifications in wildlife biology and early years education and 12 years' experience as an environmental education officer, she is a perfect storm of the passion, knowledge and skills required to get younger generations connecting with the natural world.

This is great! A good way to combine learning about nature and numbers! I would definitely want to do this with kids on a little hiking expedition. Also- if you measure with centimeters, they can convert it into meters, millimeters, etc.

Hi Anna, thanks so much for your lovely comment and for sharing at the After School linky. Miss Possum really loved this activity. Best ofall I can tell she really enjoyed it because I caught her measuring things around the house yesterday. I love that my daughter loves to learn!

British Wildlife is the leading natural history magazine in the UK, providing essential reading for both enthusiast and professional naturalists and wildlife conservationists. Published eight times a year, British Wildlife bridges the gap between popular writing and scientific literature through a combination of long-form articles, regular columns and reports, book reviews and letters.

Conservation Land Management (CLM) is a quarterly magazine that is widely regarded as essential reading for all who are involved in land management for nature conservation, across the British Isles. CLM includes long-form articles, events listings, publication reviews, new product information and updates, reports of conferences and letters.

Preface xv

Acknowledgments xix

Introduction 1

AT THE BEGINNING . . . 11

(General questions to challenge our powers of observation, estimation, and physical intuition)

Q.1-Q.6: Rainbows 11

Q.7: Shadows 11

Q.8-9: Clouds and cloud droplets 12

Q.10: Light 12

Q.11: Sound 12

Q.12-13: The rotation of the Earth 12

Q.14: The horizon 12

Q.15: The appearance of distant hills 12

IN THE "PLAYGROUND" 13

(just to get our feet wet. . .)

Q.16: Loch Ness--how long to empty it? 13

Q.17: The Grand Canyon--how long to fill it with sand? 14

Q.18: Just how large an area is a million acres? 15

Q.19: Twenty-five billion hamburgers--how many have you eaten? 16

Q.20: How many head of cattle would be required to satisfy the (1978) daily demand for meat in the United States? 16

Q.21: Why could King Kong never exist? 17

Q.22: Why do small bugs dislike taking showers? 18

Q.23: How fast is that raindrop falling? 18

Q.24: Why can haystacks explode if they're too big? 20

In the garden 24

Q.25: Why can I see the "whole universe" in my garden globe? 24

Q.26: How long is that bee going to collect nectar? 25

Q.27: Why are those drops on the spider's web so evenly spaced? 27

Q.28: What is the Fibonacci sequence? 31

Q.29: So what is the "golden angle"? 35

Q.30: Why are the angles between leaves "just so"? 36

IN THE NEIGHBORHOOD 43

Q.31: Can you infer fencepost (or bridge) "shapes"

just by walking past them? 43

Q.32: Can you weigh a pumpkin just by carefully looking at it? 48

Q.33: Can you determine the paths of low-flying ducks? 53

IN THE SHADOWS 58

Q.34: How high is that tree? (An estimate using elliptical light patches) 58

Q.35: Does my shadow accelerate? 59

Q.36: How long is the Earth's shadow? 61

Q.37: And Jupiter's? And Neptune's? 63

Q.38: How wide is the Moon's shadow? 63

IN THE SKY 64

Q.39: How far away is the horizon (neglecting refraction)? 64

Q.40: How far away is that cloud? 66

Q.41: How well is starlight reflected from a calm body of water? 67

Q.42: How heavy is that cloud? 71

Q.43: Why can we see farther in rain than in fog? 72

Q.44: How far away does that "road puddle" mirage appear to be? 73

Q.45: Why is the sky blue? 77

Q.46: So how much more is violet light scattered than red? 79

Q.47: What causes variation in colors of butterfly wings, bird plumage, and oil slicks? 80

Q.48: What causes the metallic colors in that cloud? 84

Q.49: How do rainbows form? And what are those fringes underneath the primary bow? 85

Q.50: What about the secondary rainbow? 92

Q.51: Are there higher-order rainbows? 93

Q.52: So what is that triple rainbow? 95

Q.53: Is there a "zeroth"-order rainbow? 98

Q.54: Can bubbles produce "rainbows"? 99

Q.55: What would "diamondbows" look like? 100

Q.56: What causes that ring around the Sun? 101

Q.57: What is that shaft of light above the setting Sun? 109

Q.58: What is that colored splotch of light beside the Sun? 111

Q.59: What's that "smiley face" in the sky? 113

Q.60: What are those colored rings around the shadow of my plane? 116

Q.61: Why does geometrical optics imply infinite intensity at the rainbow angle? 118

IN THE NEST 122

Q.62: How can you model the shape of birds' eggs? 122

Q.63: What is the sphericity index? 123

Q.64: Can the shape of an egg be modeled trigonometrically? 124

Q.65: Can the shape of an egg be modeled algebraically? 127

Q.66: Can the shape of an egg be modeled using calculus? 130

Q.67: Can the shape of an egg be modeled geometrically? 134

IN (OR ON) THE WATER 137

Q.68: What causes a glitter path? 137

Q.69: What is the path of wave intersections? 140

Q.70: How fast do waves move on the surface of water? 141

Q.71: How do moving ships produce that wave pattern? 148

Q.72: How do rocks in a flowing stream produce different patterns? 152

Q.73: Can waves be stopped by opposing streams? 154

Q.74: How far away is the storm? 157

Q.75: How fast is the calm region of that "puddle wave" expanding? 158

Q.76: How much energy do ocean waves have? 160

Q.77: Does a wave raise the average depth of the water? 162

Q.78: How can ship wakes prove the Earth is "round"? 164

In the forest 168

Q.79: How high can trees grow? 168

Q.80: How much shade does a layer of leaves provide for the layer below? 172

Q.81: What is the "murmur of the forest"? 174

Q.82: How opaque is a wood or forest? 176

Q.83: Why do some trees have "tumors"? 179

IN THE NATIONAL PARK 183

Q.84: What shapes are river meanders? 183

Q.85: Why are mountain shadows triangular? 189

Q.86: Why does Zion Arch appear circular? 191

IN THE NIGHT SKY 194

Q.87: How are star magnitudes measured? 194

Q.88: How can I stargaze with a flashlight? 196

Q.89: How can you model a star? 197

Q.90: How long would it take the Sun to collapse? 205

Q.91: What are those small rings around the Moon? 207

Q.92: How can you model an eclipse of the Sun? 210

AT THE END . . . 217

Q.93: How can you model walking? 217

Q.94: How "long" is that tree? 221

Q.95: What are those "rays" I sometimes see at or after sunset? 224

Q.96: How can twilight help determine the height of the atmosphere? 228

Appendix 1: A very short glossary of mathematical terms and functions 231

Appendix 2: Answers to questions 1-15 234

Appendix 3: Newton's law of cooling 238

Appendix 4: More mathematical patterns in nature 240

References 243

Index 247

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