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Explore the fundamentals of music via Ableton’s interactive website. Experiment with beats, melody, harmony, basslines, and song structure in your web browser.

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Free Technology For Teachers – A TeacherMeade site

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Engage every student in every lesson with Actively Learn's award-winning ELA, science, and social studies curriculum platform.

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This is a bite-sized taster of what ForSkills do. Our full bank of hundreds of expertly crafted video tutorials and resources are available, ad-free, within ...

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A brief history of numerical systems - Alessandra King

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There's a mathematical ratio commonly found in nature-the ratio of 1 to 1.618-that has many names. Most often we call it the Golden Section, Golden Ratio, or Golden Mean, but it's also occasionally referred to as the Golden Number, Divine Proportion, Golden Proportion, Fibonacci Number, and Phi.

You'll usually find the golden ratio depicted as a single large rectangle formed by a square and another rectangle. What's unique about this is that you can repeat the sequence infinitely and perfectly within each section.

If you take away the big square on the left, what remains is yet another golden rectangle. . . and so on.

The golden ratio in art and architecture

The appearance of this ratio in music, in patterns of human behavior, even in the proportion of the human body, all point to its universality as a principle of good structure and design.

Used in art, the golden ratio is the most mysterious of all compositional strategies. We know that by creating images based on this rectangle our art will be more likely to appeal to the human eye, but we don't know why.

Some scholars argue that the Egyptians applied the golden ratio when building the great pyramids, as far back as 3000 B.C.

In 300 B.C. Euclid described the golden section in his writing of Euclid's Elements, and before that, around 500 B.C., Pythagoras claimed that the golden ratio is the basis for the proportions of the human figure.

The ancient Greeks also used the golden ratio when building the Parthenon.

Artists throughout history, like Botticelli and Leonardo daVinci, have used the golden rectangle, or variations of it, as the basis for their compositions.

Here's da Vinci's painting, The Last Supper, with golden sections highlighted.

Golden rectangles are still the most visually pleasing rectangles known, and although they're based on a mathematical ratio, you won't need an iota of math to create one.

How to make a rectangle based on the golden ratio

If you want to use a golden rectangle in your own compositions, here's how you can make that happen without any special tools or mathematical formulas.

1. Begin with a square, which will be the length of the short side of the rectangle.

2. Then draw a line that divides it in half (forming two rectangles).

3. Draw a line going from corner to opposing corner of one of those halves.

4. Rotate the top point of that diagonal line downward until it extends your square.

5. Finish off the rectangle using that diagonal length as a guide for the long side of your golden rectangle. It's that simple.

Visual points of interest inside a golden rectangle

Any square or rectangle (but especially those based on the golden ratio) contain areas inside it that appeal to us visually as well. Here's how you find those points:

1. Draw a straight from each bottom corner to its opposite top corner on either side. They will cross in the exact center of the format.

2. From the center to each corner, locate the midway point to each opposing corner.

These points-represented by the green dots in the diagram above-are called the "eyes of the rectangle."

How to use the "eyes" of a golden rectangle

One strategy often used by artists is to locate focal points or areas of emphasis around and within these eyes, creating a strong visual path in their compositions.

Edward Hopper's composition, below, sets the sailboat right on the lower right eye (with the tip of the sails extending nearly to the upper right eye).

In this painting, Carolyn Anderson places her subject's hands around that spot too.

J.M.W. Turner uses the angle of his waves to create an arch that circles through the lower right and lower left eyes.

Notice that the focus of the scene is then captured within all four eyes, too.

How you use the golden ratio or these "eyes" to direct a viewers' visual path is limited only by your imagination. Try using the eyes of the rectangle in your next painting and see what difference it can make in the strength of your composition.
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Painters rely on an array of methods for finding the best placement of their subjects within a painting's edges. One of the easiest methods to use and understand is to place the subject within the rabatment of the rectangle.

What is the rabatment of a rectangle? Simply put, it's the perfect square found inside any rectangle (the rabatment's sides are all equal to the short. . . read more

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A Guide to the Golden Ratio (AKA Golden Section or Golden Mean) for Artists

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The appearance of a certain mathematical ratio - 1 to 1.618 - in nature, music, patterns of human behavior, even in the proportion of the human body, all point to its universality as a principle of good structure and design.

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Could be useful for practical tasks based on MSS

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Maps when you need them

Find all your maps in the Your Places menu on Google Maps.

Chart your favorite places or explore a new city.

Discover new destinations

Save maps you find on the web and use them on-the-go.

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Maps when you need them Find all your maps in the Your Places menu on Google Maps. Chart your favorite places or explore a new city. Discover new destinations Save maps you find on the web and use them on-the-go.

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This is Numberphile. We mainly post videos about mathematics and just numbers in general.

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how insects walk on water and how galaxies form...

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The Navier-Stokes equations explore how all fluids behave, from liquids and gases to weather systems and galaxies. For Dr Hannah Fry they are the perfect example of how powerful mathematics can be when it comes to unlocking the romance of nature

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Beautiful equations: how insects walk on water and galaxies form - video

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The Navier-Stokes equations explore how all fluids behave, from liquids and gases to weather systems and galaxies. For Dr Hannah Fry they are the perfect example of how powerful mathematics can be when it comes to unlocking the romance of nature

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beautiful equations

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Paul Dirac had an eye for beauty. In one essay, from May 1963, the British Nobel laureate referred to beauty nine times. It makes four appearances in four consecutive sentences. In the article he painted a picture of how physicists saw nature. But the word beauty never defined a sunset, nor a flower, or nature in any traditional sense. Dirac was talking quantum theory and gravity. The beauty lay in the mathematics.

What does it mean for maths to be beautiful? It is not about the appearance of the symbols on the page. That, at best, is secondary. Maths becomes beautiful through the power and elegance of its arguments and formulae; through the bridges it builds between previously unconnected worlds. When it surprises. For those who learn the language, maths has the same capacity for beauty as art, music, a full blanket of stars on the darkest night.

"The slow movement of the Mozart clarinet concerto is a really beautiful piece of music, but I don't print off a page of the score and put that on my wall. It's not about that. It's about the music and the ideas and the emotional response," says Vicky Neale, a mathematician at Oxford University. "It's the same with a piece of mathematics. It's not how it looks, it's about the underlying thought processes."

Brain scans of mathematicians show that gazing at formulae considered beautiful by the beholder elicits activity in the same emotional region as great art and music. The more beautiful the formula, the greater the activity in the medial orbito-frontal cortex. "So far as the brain is concerned, maths has beauty just like art. There is common neurophysiological ground," says Sir Michael Atiyah, an honorary professor of mathematics at Edinburgh University.

Ask mathematicians about the most beautiful equation and one crops up time and again. Written in the 18th century by the Swiss mathematician, Leonhard Euler, the relation is short and simple: e iπ+1 = 0. It is neat and compact even to the naive eye. But the beauty comes from a deeper understanding: here the five most important mathematical constants are brought together. Euler's formula marries the world of circles, imaginary numbers and exponentials.

The beauty of other formulae may be more obvious. With E=mc ², Albert Einstein built a bridge between energy and mass, two concepts that had previously seemed worlds apart. Maggie Aderin-Pocock, the space scientist, declared it the most beautiful equation and she is in good company. "Why is it so beautiful? Because it comes to life. Now energy will have mass and mass can be put into energy. These four symbols capture a complete world. It's difficult to imagine a shorter formula with more power," says Robbert Dijkgraaf, director of the Institute for Advanced Study in Princeton, where Einstein was one of the first faculty members.

"One of the reasons there's almost an objective beauty in mathematics is that we use the word beautiful also to indicate the raw power in an idea. The equations or results in mathematics that are seen to be beautiful are almost like poems.The power per variable is something that is part of the experience. Just seeing a huge part of mathematics or nature being described with just a few symbols gives a great sense of elegance or beauty," Dijkgraaf adds. "A second element is you feel its beauty is reflecting reality. It's reflecting a sense of order that's out there as part of the laws of nature."

The power of an equation to connect what seem like completely unrelated realms of mathematics comes up often. Marcus du Sautoy, a maths professor at Oxford, has more than a soft spot for Riemann's formula. Published by Bernhard Riemann in 1859 (the same year Charles Darwin stunned the world with On the Origin of Species), the formula reveals how many primes exist below a given number, where primes are whole numbers divisible only by themselves and one, such as 2, 3, 5, 7 and 11. While one side of the equation describes the primes, the other is controlled by zeros.

"This formula turns these these indivisible prime numbers, into something completely different," says du Sautoy. "On the one side, you've got these indivisible prime numbers and then Riemann takes you on this journey to somewhere completely unexpected, to these things which we now call the Riemann zeros. Each of these zeros gives rise to a note - and it's the combination of these notes together which tell us how the primes on the other side are distributed across all numbers."

More than 2,000 years ago, the ancient Greek mathematician, Euclid, solved a numerical puzzle so beautifully that it still makes Neale smile every time it comes to mind. "When I think about beauty in mathematics, my first thoughts are not about equations. For me it's much more about an argument, a line of thinking, or a particular proof," she says.

Euclid proved there are infinitely many prime numbers. How did he do it? He began by imagining a universe where the number of primes was not infinite. Given a big enough blackboard, one could chalk them all up.

He then asked what happened if all these primes were multiplied together: 2 x 3 x 5 and so on, all the way to the end of the list, and the result added to the number 1. This huge new number provides the answer. Either it is a prime number itself, and so the original list was incomplete, or it is divisible by a smaller prime. But divide Euclid's number by any prime on the list and always there is a 1 left over. The number is not divisible by any prime on the list. "It turns out you reach an absurdity, a contradiction," says Neale. The original assumption that the number of primes is finite must be wrong.

"The proof for me is really beautiful. It takes some thinking to get your head around it, but it doesn't involve learning lots of difficult concepts. It's surprising that you can prove something so difficult in such an elegant way," Neale adds.

Behind beautiful processes lie beautiful mathematics. Well, some of the time. Hannah Fry, a lecturer in the mathematics of cities at UCL spent years staring at the Navier-Stokes equations. "They're a single mathematical sentence that is capable of describing the miraculously beautiful and diverse behaviour of almost all of Earth's fluids," she says. With a grasp of the formulae, we can understand blood flow in the body, make boats glide through the water, and build awesome chocolate enrobers.

In his 1963 essay, Dirac elevated beauty from an aesthetic response to something far more profound: a route to the truth. "It is more important to have beauty in one's equations than to have them fit experiment," he wrote, continuing: "It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress." Shocking at first pass, Dirac captured what is now a common sentiment: when a beautiful equation seems at odds with nature, the fault may lie not with the maths, but in applying it to the wrong aspect of nature.

"Truth and beauty are closely related but not the same," says Atiyah. "You are never sure that you have the truth. All you are doing is striving towards better and better truths and the light that guides you is beauty. Beauty is the torch you hold up and follow in the belief that it will lead you to truth in the end."

Something approaching faith in mathematical beauty has led physicists to draw up two of the most compelling descriptions of reality: supersymmetry and string theory. In a supersymmetric universe, every known type of particle has a heavier, invisible twin. In string theory, reality has 10 dimensions, but six are curled up so tight they are hidden from us. The mathematics behind both theories are often described as beautiful, but it is not at all clear if either is true.

There is a danger here for mathematicians. Beauty is a fallible guide. "You can literally be seduced by something that is not correct. This is a risk," says Dijkgraaf, whose institute motto, "Truth and Beauty" features one naked and one dressed woman. "Sometimes I feel that physicists, like Odysseus, must tie themselves to the mast of the ship so they are not seduced by the sirens of mathematics."

It may be that mathematicians and scientists are the only groups that still use the word "beautiful" without hesitation. It is rarely employed by critics of literature, art or music, who perhaps fear it sounds superficial or kitschy.

"I'm very proud that in mathematics and science the concept of beauty is still there. I think it's an incredibly important concept in our lives," says Dijkgraaf. "The sense of beauty we experience in maths and science is a multidimensional sense of beauty. We don't feel it's in any conflict with being deep, or interesting, or powerful, or meaningful. For the mathematician it's all captured by that one word."
Topics

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Magic numbers: can maths equations be beautiful?

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The concept of beauty underpins how mathematicians solve quantum theory or describe gravity. From E=mc² to string theory, mathematical beauty has led physicists to draw up some of the most compelling descriptions of reality

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A-level mathematics, may be some useful ideas or resources for higher level GCSE.

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Overview

Our resources are organised into topics or stations on a map or tubemap showing the connections between them.

Watch this video or use the preceding menu to find out more.

The map

The mathematics has been organised along a system of tube lines, which reflect big themes that run through mathematics. The tubemap offers a way to navigate to resources for teaching a particular area of mathematics; resources can be accessed by clicking on a station name. Some stations are still in development and this is indicated by a construction symbol.

You can access the map at any time by clicking the Map button in the main menu.

Stations

Each station features an overarching question that could be used to stimulate curiosity. There are a number of key questions that summarise the ideas that arise in the resources. You can filter resources by selecting a key question. You can also filter resources by tube line, if you wish to choose a resource with a particular emphasis.

The resources that are available at each station may come under three headings. Resources that introduce new ideas relating to the station are in the "Introducing" section. "Developing" resources may build on these ideas or earlier work. "Review questions" are mostly drawn from past examination questions and have been chosen as ones that are interesting in nature and require non-routine thinking.

Resources

To help you find the resources you are looking for, we have classified them according to twelve resource types.

Resources are divided into sections. In some cases this may be a warm-up followed by a larger task, and others may have a problem followed by a suggestion of how to tackle it. Almost all resources contain a solution or a "Things you might have noticed" section. This may help you to think about questions you might ask students, to see areas where students could get stuck, or offer further insight into how you might use the resource in your classroom. Further support is offered in the teacher notes.

Pervasive ideas

By clicking on a pervasive idea you will find a description of how the idea arises within the resources. If pervasive ideas are highlighted in a resource they appear in the sidebar (or bottom-bar on a small screen) and in the list of resources at the station (on a wide screen).
What are pervasive ideas?
Search

Use the Search button in the top menu bar to open the search dialog.

You can search for resources by title, keywords or text within the resource. Review questions can be searched by question Ref. You can also search for pervasive ideas and mathematical processes such as visualising.

You can also browse all our material with filtering options by clicking the Browse button in the main menu.

Saving resources

You can save a resource for later by adding it to your resource collection. To do this click the star next to the resource title. Note you must log in or register to use this facility. To access your resource collection, click the User button in the top menu bar then Your resource collection.

Within your resource collection, you can organise your resources by creating ordered subcollections. To add a resource to a subcollection, click the add to subcollection icon in your resource collection table. To select a subcollection, use the drop-down list above the table. You can also generate a PDF of resources from a subcollection, which could be used to generate your own exam from our review questions, for example.

Bundles and more

The More menu in the top right contains links to the Underground hub, our About and contact page and the Your mathematical classroom section of the site.

Our About page contains a video explaining our philosophy.

The Your mathematical classroom area of the site is accessible from the More menu and has been coloured purple to distinguish it from the rest of the site, as well as having its own menu once inside this area. It contains support for using our resources in the classroom as well as professional development opportunities. This includes the Bundles page, which contains small collections of resources that have a particular focus or goal, our Resources in action page, which provides detailed teacher support for a handful of resources, our blog, upcoming and recorded webinars, a courses page, and information about our Champions programme.

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A level maths teaching resources | Underground Mathematics

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Rich resources for teaching A level maths, enabling all students to explore the connections that underpin mathematics. The resources are free for all users.

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A website to inspire creativity in maths