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Free teacher notes and activity ideas available via the website

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About the Author:Julie Ellis is a very experienced writer in the field of education Julie has a number of works published by Learning Media, New Holland, Reed Education and others. About the Illustrator:Izzy Joy is a young but experienced illustrator. "My personal artwork is definitely about connecting people with each other and with nature ...

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Topics: numbers, measurement, geometry (radius, diameter, circumference, metres, ratios and scale)

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Measuring the Mighty: Maths Adventures in Kauri Education | by NZAEE

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Waikato schools learn key maths concepts through rich tasks related to their local forest and grounds, building skills and values. Gain access free today.

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Registered charity Number 1058911 Copyright © 2023 MEI

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Examples of how to incorporate maths learning with nature and the items we can easily find outdoors.

Topics: numbers, measurement, geometry

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Outdoor maths teaching inspiration for the summer term

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Outdoors primary maths teaching ideas for the summer term.

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This activity designed by Beyond Benign supports students calculate, interpret and identify ways to reduce their ecological footprint. The resource has a suggested footprint calculator, or there is also FutureFit, designed here in Aotearoa.

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To book a waste audit kit, visit the Waikato Regional Council: Resources For Teachers page.

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Topics: geometry

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Maeroa Intermediate Takes Maths Outdoors for a Real World Challenge

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Maeroa Intermediate Takes Maths Outdoors for a Real World Challenge

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When Maeroa Intermediate School received funding from the Waikato Regional Council for their gully restoration project, it wasn't just a chance to upgrade their native plant nursery - it was an opportunity to bring maths to life in a real-world context.

Since beginning their gully restoration in 2020, Maeroa students have grown thousands of eco-sourced native plants. Their nursery has been productive, but its location in an old set of abandoned cricket nets - with uneven pavers and remnants of ivy - meant it was far from ideal. To make the space safer and more functional, the school planned to pour a new concrete pad.

Lead teacher Julie Yeoman saw this as a perfect opportunity to teach measurement and geometry in a hands-on way. The site - a rectangular area with a small existing concrete slab and surrounding loose pavers - was ideal for a practical maths lesson.

Julie split her class, inviting students keen to get hands-on to come outside and calculate the materials needed. Armed with metre rulers, pencils, and paper, students set out to measure the area and work out the volume of concrete required.

Outside, the class discussed what a concrete pad is, how thick it needs to be, why soil needs to be removed beforehand, and the purpose of boxing. Before measuring, students made predictions about the dimensions and depth of the pad. They took turns carefully measuring the length and width of the space-learning how to read a metre ruler correctly as part of the process.

After calculating the total area, students faced another challenge: subtracting the area of the existing concrete cricket pitch. With Julie's guidance, they checked and corrected their calculations before moving on to the final step - calculating volume by multiplying the area by the depth of the concrete: 0.1 metres (or 10 cm).

To bring the lesson full circle, students began digging out the topsoil to a depth of 10 cm, trying to keep the surface as level as possible. The activity sparked enthusiasm.

When asked what they enjoyed, students said:

and even, "I like being outside"
"I like helping people"
"I like experiencing maths in new ways"
"Finding buried treasure when we dig!"
(A fair bit of 'treasure' has been unearthed in the gully over the years!)

"The maths practice books are cool, but doing it on paper, with a ruler, outside - that's real maths. It lets students use what they've learned in class and apply it in the real world." - Lead teacher Julie's reflection

The concrete truck is scheduled to arrive during a Saturday gully working bee, when students and their families will come together to pour the slab - bringing their learning to life, one wheelbarrow at a time.

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Maeroa Intermediate Takes Maths Outdoors for a Real-World Challenge

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When Maeroa Intermediate School received funding from the Waikato Regional Council for their gully restoration project, it wasn't just a chance to upgrade their native plant nursery - it was an opportunity to bring maths to life in a real-world context. Since beginning their gully...

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Topics: numbers, measurement, statistics

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Topics: numbers, measurement, geometry

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Water is essential for life, and the abundance or scarcity of water is a major influence on which life forms can exist in a place. The Enviroschools Water of Life Theme Area is filled with activities the support learning about this precious resource, and has many opportunities to incorporate maths.

Examples are:

How much is fresh? (pg 46) - numbers, measurement, ratios and scale

How much water is in us? (pg 42) - measurement, percentages

Water audit (pg 65) - numbers, measurement, statistics

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Measure and understand the electricity and fuel use by your school.

Topics: numbers, measurement, algebra, statistics.

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Measure and understand the transport choices of your school community. How do students and staff travel to school? Are there ways we can shift our community travel habits to low-carbon alternatives?

Topics: numbers, measurement, statistics

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You can do this any time of the year to find out which birds are at your school.

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25 June - 3 July 2022

Te Tatauranga o ngā Manu Māra o Aotearoa - The New Zealand Garden Bird Survey

Make our birds count

We need you to help make a difference

Healthy bird populations can indicate that the environment is healthy. We know lots about endangered native birds in the bush, but we don't always know what's happening in the populations of all the birds around us, particularly in urban and garden environments.

That's why we need you to help us count them.

What we learn from information about the birds in your garden collected by you and your fellow citizen scientists helps our researchers make decisions to protect our environment. It also tells us if what we are doing in areas such as predator control is working.

The 2022 survey starts in

Take part in the 2022 NZ Garden Bird Survey "

All you need is one hour, on one day, from 25th June to the 3rd July 2022.

Get your students engaged in science and nature for the day.

Thank you to our long-term supporters

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Te Tatauranga o ngā Manu Māra o Aotearoa - The New Zealand Garden Bird Survey Healthy bird populations can indicate that the environment is healthy. We know lots about endangered native birds in the bush, but we don't always know what's happening in the populations of all the birds around us, particularly in urban and garden environments.

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If your school is involved with predator trapping, Trap NZ helps you record and understand your predator control and biodiversity data. Many schools around the motu are using this app to record data.

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Nature's Geometry: Uncovering Glamorous Patterns and Shapes in the Wilderness

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Nature's Geometry: Uncovering Glamorous Patterns and Shapes in the Wilderness

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Topics: geometry

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Geometry is not solely confined to the chalk-dusted blackboards of classrooms or the abstract realm of mathematical theory. It is vividly alive and intricately woven into the very fabric of the natural world around us. In nature, geometry manifests through an array of patterns and shapes, from the fractal branching of trees to the hexagonal chambers of a honeycomb. These shapes are not random; they are the results of specific principles and processes that govern the physical and biological world. Observing nature through a geometric lens can reveal how mathematics structures life and our planet's inanimate elements.

Our planet is a mosaic of natural designs that can teach us a great deal about geometry. Spirals and rotational symmetries are widespread in the natural kingdom, exhibited in the growth patterns of seashells and the distributions of sunflower seeds. Phenomena such as the ripple of water waves demonstrate the fluid dynamics and geometric patterns that emerge from interactions between natural forces.

Exploring these patterns deepens our appreciation of the natural world's aesthetics and improves our understanding of the underlying scientific principles that shape our environment. By examining the shadows cast by trees or the structure of frost on a windowpane, we can begin to see how light, shadow, and temperature influence natural geometry.

The Essence of Geometry in Nature

Geometry is integral to the fabric of nature; it is the mathematics that structures the environment around us. The shapes and symmetry we observe in the natural world are manifestations of geometric principles. From the hexagonal chambers of a honeycomb to the spiral patterns of a seashell, nature displays an eclectic array of geometric designs.

We often see the order and regularity that geometry brings into nature, exhibiting a form of silent mathematics at work. For example, Fibonacci sequences appear in the arrangement of leaves, while fractals are evidenced in the intricate patterns of snowflakes and river networks. These patterns are aesthetically pleasing and serve specific scientific functions, underlying how natural selection favours certain shapes and structures for their efficiency.

Our understanding of this geometric language extends to various scientific fields. In biology, the study of morphogenesis explores how shapes form in living organisms. Meanwhile, in physics, the principles of geometry elucidate the symmetrical nature of fundamental particles and forces.

By exploring shapes in nature through the lens of math and science, we're able to gain greater insights into the world around us. Nature's geometry is more than just beauty; it's an essential part of how life is structured and how it functions. As we educate ourselves about these geometric properties, we learn to see the world in a richer, more interconnected way.

Mathematical Patterns in the Living World

In nature, we frequently observe intricate patterns that adhere to mathematical rules. From the tiniest cells to the most complex animals, these patterns can be found across various scales and forms in the living world.

Life's Building Blocks

Life at its most fundamental level is comprised of cells-self-contained units that establish the complexity of organisms. Delving into their core, we find DNA, life's blueprint that coils itself into a double helix. This structure is a prime example of the incredible organisation at the cellular level. It's not just about the construction, though. The sequences within DNA reflect a mathematical beauty, often showing repetitions and proportions consistent with the Fibonacci sequence, a series of numbers where each subsequent number is the sum of the previous two.

Floral Symmetry and Growth

Plants exhibit a fascinating array of symmetrical patterns, notably evident in their flowers. The layout of petals, the distribution of seeds, and even the growth of leaves often follow specific ratios such as the Fibonacci sequence. This sequence can be visualised in the way sunflower seeds are arranged in a head or how petals are spaced around a flower, allowing for optimal sunlight exposure and growth. It's a beautiful synergy between function and pattern, and we regularly encounter fractals in plants, self-repeating patterns that are similar at different scales, mirroring the endless complexity in nature.

Animal Body Plans

Examining the animal kingdom presents us with a diverse tapestry of body patterns and proportions. The Fibonacci sequence pops up again here, for example, in the spiral shells of certain molluscs. This mathematical principle is echoed in the growth and formation of animal body parts and their arrangements. Whether it's the way a zebra's stripes curve and angle, the spirals of a nautilus shell, or the segmentations of an insect's body, there's an underlying geometry to how these structures form.

In all these instances, animals and plants alike, nature's geometry underscores the elegance and functionality of these biological forms. Patterns such as the Fibonacci sequence ensure efficient growth, maximum exposure to environmental elements, and space optimisation-crucial factors that contribute to the survival and proliferation of species.

Mathematics, far from abstract, manifests vividly and vitally within the living tapestry of our natural world.

Geometric Shapes in Non-Living Nature

We're surrounded by geometry in the natural world, especially within the realm of non-living nature. From the intricate forms of minerals to the celestial bodies overhead, geometry is omnipresent.

Minerals and Crystals

Minerals and crystals intrigue us with their geometrical perfection. They are nature's masterpieces, showcasing a myriad of shapes, from cubic to trigonal systems. Take salt, for example, with its perfect cubes emerging due to its internal crystalline structure. The angles and planes seen in these crystalline shapes reflect the orderly arrangement of atoms within them, and this can be beautifully exemplified in the diverse formations of quartz.

Spheres and Circles in the Sky

Looking upwards, we're captivated by the spheres and circles that dance across the sky. The moon, a natural satellite of the Earth, demonstrates a near-perfect spherical shape due to the force of its own gravity. During a solar or lunar eclipse, we witness the circular shadows cast by these celestial bodies.

The sun, too, appears as a bright circle in the sky, its spherical shape holding together the solar system with its gravitational pull. Even the seemingly pinprick-sized stars that twinkle at night are mighty spheres of burning plasma dispersed throughout the vastness of space.

Phenomena and Patterns: From Snowflakes to Sand Dunes

Nature is a mesmerising gallery of geometric designs where patterns and shapes abound in every corner. We often marvel at the perfection of a snowflake, its delicate branches forming a precise hexagonal symmetry. This natural marvel showcases the inherent order in apparent chaos-each branch is a testament to the intricate patterns of freezing water molecules as they crystallise in clouds.

Similarly, shifting sand dunes are sculpted by the wind into soft, sinuous forms that, from afar, display a cadence comparable to the gentle undulations of ocean wave patterns. Their graceful contours tell a story of the relentless desert winds as their shapes are carved and reshaped by the elements.

But why hexagons in snowflakes? The answer lies in the efficient packing of water molecules as they transition from the fluid form to solid, with hexagons being the shape that allows for maximum efficiency in this context-a splendid convergence of physics and artistry. It is a recurring motif seen not just in snowflakes but also in the honeycombs of beehives and the packed cells of basalt columns.

Below is an overview of nature's propensity for geometric design:

Snowflakes: Hexagonal symmetry due to efficient water molecule packing.

Clouds: Ever-changing patterns influenced by atmospheric conditions.

Sand Dunes: Wavy lines formed by wind erosion and deposition.

Natural Patterns: Repetition of shapes like spirals in shells and branching trees.

In our effort to understand these natural phenomena, we're reminded of how mathematics and art fuse, manifesting in the environment that surrounds us. Our exploration into nature's geometry is enriching, revealing the rhythm of patterns that, although diverse, follow the universality of mathematical rules.

Botanical Geometry: Leaves and Trees

Nature is a vast tapestry woven with intricate patterns and shapes, where the geometry of leaves and trees plays a fundamental role. Let's examine the sophisticated design nature employs in the foliage and architecture of plants.

Leaf Arrangement and Function

Leaves are nature's brilliant invention, sophisticated in function and form. We notice that their arrangement on the stem-a pattern known as phyllotaxis -often follows distinct spiral or opposite patterns, each serving to maximise sunlight absorption and minimise shading of lower leaves. Interestingly, this arrangement can sometimes mirror the fascinating patterns of Romanesco broccoli, with each leaf positioned at a specific angle to its neighbours, conforming to a spiral pattern that's both efficient and aesthetically pleasing.

In terms of function, leaves are the powerhouses of the plant world. Their primary role is to perform photosynthesis, converting sunlight into energy stores, which are critical for growth and reproduction.

Trees' Branching Patterns

The overarching framework of trees showcases nature's aptitude for structural efficiency and resilience. Branching patterns of trees, which we call dendritic patterns, often display a fractal nature, meaning they repeat at different scales. A single glance at a tree's branches reveals a repeating pattern that starts from the trunk and continues all the way to the tips of the smallest twig. Each branch and twig is strategically placed to support the leaves in capturing sunlight, resembling how rivers branch into smaller streams.

Trees like the mighty oak or the slender birch stand as a testament to the strength of these natural designs. Each species follows its own unique geometric rules that dictate how it will grow and how its branches will support the leaves necessary for its survival. In essence, these patterns are not just visually remarkable but are the tree's solution to the problems of light, space, and gravity.

Spirals and Rotational Symmetry in Nature

Within the natural world, we find an astonishing array of forms and structures, all following geometric patterns that are both functional and aesthetically pleasing. Spirals and rotational symmetry play a fundamental role in the design and evolution of various natural entities, from the smallest shells on the seashore to the vast spiralling galaxies in the cosmos.

Snails and Shellfish

In observing the shells of snails and shellfish, we're seeing a classic example of logarithmic spirals in nature. These spirals allow for growth without changing the body's shape, which is essential for these creatures. The rotational symmetry within their shells ensures that each turn of the spiral is proportional to the next, creating a harmonious and recognisable pattern across different species.

Shells: Exhibiting both gorgeous designs and structural efficiency, they typically follow a logarithmic spiral. The growth follows a clear route, where each new section is a scaled-up copy of the previous, showcasing rotational symmetry at its finest.

Galactic Spirals

When we cast our eyes to the heavens and observe the structure of galaxies, we can't help but marvel at the grand spiral designs they often present. Those swirling arms extend from the centre and wrap around it in a sprawling dance of stars and cosmic dust.

Galaxies: Their shape is not merely for show but speaks volumes about the rotational movements within. The spiral arms are thought to be areas of high star formation, and the pattern they create can tell astronomers a great deal about the history and dynamics of the galaxy.

Our exploration of spirals and rotational symmetry in nature represents the incredible balance between mathematical order and the organic growth of living systems. From the cone-shaped patterns in pinecones and the spiralling seeds of sunflowers to the curves of our own ears, these designs tell the story of a universe consistently governed by a set of geometric rules. The beauty we witness in these natural formations is not accidental but is a testament to the elegant principles that guide our world's structure.

Light and Shadow: Nature's Impact on Architecture

We often see the brilliance of nature influencing design and architecture-how light plays with structures and how shadows carve out the form.

The Golden Ratio in Design

The golden ratio, a proportion deeply rooted in nature, is embraced in design for its aesthetically pleasing balance. In Greek architecture, this ratio was employed to achieve harmony and beauty, resonating with the natural world. Consider the Parthenon; the use of the golden ratio in its facade shows a thoughtful application of geometric proportions guided by natural principles.

Natural Light in Structures

Incorporating natural light into buildings is an art that balances the sun's angles with thoughtful architecture. Structures harnessed by Greek architects, for example, often optimised the interplay of light and shadow, creating dynamic spaces. It is about understanding the sun's path and designing windows and openings that maximise the use of daylight while also forming a connection between the interior and the natural world outside.

Water's Influence: From Bubbles to Waves

Our natural world is filled with patterns and structures that are largely influenced by water. From the delicate forms of bubbles to the rhythmic patterns of ocean waves, water shapes our environment in ways that are both profound and captivating.

Bubble Geometry

Water's surface tension gives life to bubbles' sphere-like shapes. When observing bubbles, we find that they embody a perfect balance, forming the most efficient and minimal-area structure possible. Fascinatingly, the geometry of bubbles can explain natural phenomena such as the hexagonal tessellations found in beehives, where bees instinctively follow this principle to build their honeycombs with the least amount of wax for the most space.

Patterns in Ocean and Air

The sea whispers the language of geometry through its waves. By their very nature, waves create a variety of patterns that can instil joy in those who observe their relentless advance and retreat. Meanwhile, high above, clouds form vast, idyllic formations on the canvas of the sky. These cloud patterns often resemble Lichtenberg figures, tree-like structures representing the branching paths of electrical discharges, which are a testament to the patterned chaos that governs our universe.

The Aesthetics of Nature's Geometry

When we observe the natural world, we're often struck by the geometric shapes that bring about a sense of balance and harmony. This aesthetic appeal isn't just a coincidence; it reflects the mathematical order that underpins our universe.

Stars twinkling in the night sky arrange themselves in patterns that have captivated humans since time immemorial. These celestial formations inspire us to look for meaning beyond our own existence, connecting us to the infinite universe through their geometric precision.

Design in nature, whether intentional by humans or inherently part of organic structures, adheres to geometrical principles. It resonates with us on a fundamental level, perhaps because the human body itself is a testament to nature's geometric design. The spirals of our DNA, our faces' symmetry, and our limbs' proportionate structure underscore this intrinsic relationship.

Nature's geometry provides more than just visual appeal; it evokes an innate joy within us. The recurring patterns and shapes in nature are not just pleasing to the eye; they evoke a profound appreciation for the complexities of life. From the Fibonacci sequence seen in the curl of a seashell to the hexagonal chambers of a beehive, these patterns are aesthetically appealing and universally recognised symbols of natural beauty.

When we create and interact with design inspired by nature's geometry, be it through art, architecture, or everyday objects, we often find these designs to be particularly pleasing.

This connection to nature can offer soothing, calming effects and even inspire creativity and innovation.

In our journey, we appreciate and integrate these natural elements into our lives, as they bring an undeniable aesthetic that speaks to the soul, a testament to the natural order that guides our entire cosmos.

The Fractals Around Us: Chaos and Structure

As we explore the natural world, it's truly fascinating to observe the prevalence of fractals - intricate patterns that repeat themselves at different scales. These fractals are found throughout nature in elements such as fern leaves, cloud formations, mountain ranges, and even coastlines. They are a perfect symphony of chaos and structure, displaying a complex yet predictable pattern.

Ferns are an excellent example of how fractal geometry manifests in living organisms. Each frond is a miniature replica of the whole, showcasing self-similarity, a distinct characteristic of fractal patterns. Similarly, when we gaze up at the clouds, we cannot help but notice their fluffy, irregular contours. Despite their seemingly random shapes, clouds follow fractal geometry, which allows for detailed weather models and forecasts.

The rugged terrain of mountains also embodies the essence of fractals. If you've ever marvelled at how mountain ranges seem to replicate their forms no matter the scale at which we observe them, you appreciate the fractal nature of their topography. This fractal dimension provides insights into geological processes and erosion patterns.

Moreover, coastlines display a vivid fractal geometry. Their intricate designs, weaving in and out around continents, are more than just scenic wonders; they are natural examples of fractals that challenge our ability to measure their true length due to their intricate detailing at every scale.

The concept of chaos in nature, often associated with unpredictability, is intimately connected with the order we find in fractals. The chaotic systems found in weather patterns and geological formations show an underlying fractal structure that, while seemingly disordered, abides by mathematical principles.

In essence, fractals are a testament to an organised complexity, showing us that within the apparent randomness of nature, there's an inherent order governed by mathematical rules. These patterns inspire us to appreciate the beauty of nature's geometry that is echoed throughout our natural environment.

Practical Applications and Relevance

The geometry we observe in nature is not merely an aesthetic wonder; it holds immense practical significance across various scientific disciplines. In chemistry, the understanding of molecular shapes is crucial for comprehending chemical reactions and properties. Similarly, in physics, the geometric arrangement of crystals affects the physical traits of materials.

The principles of natural geometry extend greatly into engineering as well. Geometric forms found in nature often inspire the design of structures and machines. For instance, the hexagonal shapes in honeycomb structures inform robust yet lightweight constructions. In technology, biomimicry utilises natural shapes to engineer more efficient and sustainable solutions, as well as to inspire innovations in robotics and artificial intelligence.

Mathematics is the backbone of understanding geometric patterns. Patterns such as spirals and fractals are integral to modelling complex phenomena, from weather systems to the spread of diseases. These patterns offer a foundation for applications in predictive modelling and analysis.

Our interventions through technology strive to imitate and harness the finesse of nature's designs. From the aerodynamics of birds influencing aviation design to the surface textures of lotus leaves inspiring self-cleaning materials, we see a harmony of nature's templates with man-made advancements.

In essence, the geometry seen in the natural world is foundational to advancing not only our understanding of the universe but also the development of pioneering technologies and enduring structures. Our commitment to detailed exploration allows us to unlock a plethora of applications that improve the quality, efficiency, and sustainability of human life.

Frequently Asked Questions

In this section, we'll answer some of the most common queries about the captivating ways that shapes and patterns of geometry occur naturally all around us.

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Nature's Geometry: Uncovering Glamorous Patterns and Shapes in the Wilderness - LearningMole

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Nature's Geometry: Geometry is not solely confined to the chalk-dusted blackboards of classrooms or the abstract realm of mathematical theory. It is vividly alive and intricately woven into the very fabric of the natural world around us. In nature, geometry manifests through an array of patterns and shapes, from the fractal branching of trees to the hexagonal chambers of a honeycomb.

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Geometry - Hut Building (Ecological Building Theme Area)

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Geometry - Hut Building (Ecological Building Theme Area)

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More information including learning objectives can be found on the MOE resources site: Tahurangi Hut building

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Picture books with measurement content can be found here Picture books with measurement content

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Making NZ Pest Free - maths & stats in real world context

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Making NZ Pest Free - maths & stats in real world context

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lesson plan from Zealandia - gathering, analysing and displaying data for real world outcomes.

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Mathematics and statistics in a real context

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Can we make New Zealand pest-free? Mathematics and statistics in a real context Urban ecosanctuary ZEALANDIA, with support from WWF New Zealand, has produced a comprehensive teaching resource supporti…

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Calculate how many trees are required to offset the carbon emitted by a flight.

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This Spotlight is part of a series highlighting environmental contexts for mathematics and statistics learning in Aotearoa primary and intermediate schools, using the new curriculum. Links to related mathematics stories and resources are included throughout and at the end.

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Maths Curriculum Series: Written by educator Annelies Judson this collection includes hands-on geometry learning in nature. Gain access free today.