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What is measured in parts per million, parts per billion, or parts per trillion? Take a look at some concentration analogies here. Commonly, substances thought of as contaminants or toxins such a PCB's, lead, mercury, and pesticides, are measured in these small concentrations. Need a bit more about exactly what is meant by a ppm or ppb? This article will provide you with some more examples and explanations, as will what does ppm or ppb mean?

If you think parts per million is small, try imagining parts per billion. Parts per billion is a common measurement used for toxins in drinking water such as lead and mercury. Do you realize how minuscule of an amount that actually is? For lead, anything over 5 ppb is considered dangerous. That would be 5 parts of lead in a billion parts of water is cause for concern! This is how toxic Flint's water really is gives some great visual examples and comparisons to take a look at, based on the water crisis in Flint, Michigan. There's been a lot of news about lead in drinking water and the potential harmful effects. But, how much lead is too much? Watch and scroll through this Washington Post infographic and find out more about Flint, Michigan. The Center for Disease Control is another place to find out valuable information regarding lead in drinking water applicable to anywhere in the world.

Climate change is another topic where the term "parts per million" is discussed. This article: Atmospheric carbon dioxide levels are showing a startling increase is a great resource to learn more.

Have trouble understanding large numbers? Well, you are not alone. Humans have great difficulty conceptualizing large numbers. This video from the SciShow: Understanding the Most Extreme Numbers in the Universe suggests some ways to start to understand vast distances measured in the universe.

Another way to increase your understanding is to scale things down. How to Develop A Better Sense of Scale-Better Explained, describes a very effective method of communicating large numbers in understandable language.

Ginormous Numbers Could Create a Mental Black Hole also gives some insight into the difficulty of large numbers to the human brain. How could the fact that we don't really comprehend large numbers affect us? Read this article: Large numbers are dehumanising, so should big data worry us? to find out! Would a book help? Try Richard Evan Schwartz's book: Really Big Numbers. Click here to watch a trailer about it.

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How to visualize one part per million - Kim Preshoff + The TED-Ed Community

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"Parts per million" is a scientific unit of measurement that counts the number of units of one substance per one million units of another. But because it's hard to conceptualize really large numbers, it can be difficult to wrap our brains around what "one part per million" really means.

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How high can you count on your fingers? (Spoiler: much higher than 10) - James Tanton

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How high can you count on your fingers? It seems like a question with an obvious answer. After all, most of us have ten fingers -- or to be more precise, eight fingers and two thumbs. This gives us a total of ten digits on our two hands, which we use to count to ten.

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Let's Begin...

1, 2, 3, 4, 5, 6, 7, 8, 9... and 0. With just these ten symbols, we can write any rational number imaginable. But why these particular symbols? Why ten of them? And why do we arrange them the way we do? Alessandra King gives a brief history of numerical systems.

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About TED-Ed Animations

TED-Ed Animations feature the words and ideas of educators brought to life by professional animators. Are you an educator or animator interested in creating a TED-Ed Animation? Nominate yourself here "

Meet The Creators

Educator Alessandra King

Director Michael Kalopaidis

Script Editor Alex Gendler

Producer Zedem Media

Illustrator Jeanne Bornet

Animator Maria Savva

Sound Designer Andreas Trachonitis

Associate Producer Jessica Ruby

Content Producer Gerta Xhelo

Editorial Producer Alex Rosenthal

Narrator Julianna Zarzycki

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A brief history of numerical systems - Alessandra King

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1, 2, 3, 4, 5, 6, 7, 8, 9... and 0. With just these ten symbols, we can write any rational number imaginable. But why these particular symbols? Why ten of them? And why do we arrange them the way we do?

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One is one...or is it? - Christopher Danielson

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One bag of apples, one apple, one slice of apple -- which of these is one unit? Explore the basic unit of math (explained by a trip to the grocery store!) and discover the many meanings of one.

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To learn more about 10-digit autobiographical numbers, check out Martin Gardner's Mathematical Circus.

Interestingly, we can expand on autobiographical numbers. Consider this: if there is a notion of an autobiography of a number, then it would be logical to expect there to be a notion of a biography of a number. What would be the logical candidate for a biography of a number? Let us say that given a number N, its biography is another number M such that the first digit of M is the number of zeroes in N, the second digit of M is the number of ones in N and so on.

For example, it is possible to have two numbers that are biographies of each other: 130 and 1101. Can you find two 10-digit numbers that are biographies of each other?

If you want to find out more about the biographies of numbers, check out the educator's "A Story of Storytelling Numbers."

You can also explore the educator's math blog here.

There are many interesting sequences that describe themselves. For example, the Golomb sequence, a(n), is lexicographically the earliest non-decreasing sequence of positive integers such that a(n) is the number of times that n occurs in the sequence. It has to start with 1: a(1) = 1. That means we have 1 one. Therefore, a(2) can't be 1. We pick the next smallest number available, so a(2) =2. Now we have to have 2 twos, so we need one more two. Thus a(3) = 2. The last equation says that the sequence has to have 2 threes. As we already used all ones and twos, we have a(4) = a(5) = 3. The first few terms of the sequence are:

1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12.

This sequence is sequence A001462 in the Online Encyclopedia of Integer Sequences.

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Can you solve the Leonardo da Vinci riddle? - Tanya Khovanova

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You've found Leonardo da Vinci's secret vault, secured by a series of combination locks. Fortunately, your treasure map has three codes: 1210, 3211000, and... hmm. The last one appears to be missing. Can you figure out the last number and open the vault? Tanya Khovanova shows how.

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The Khan Academy also has an awesome educational lesson on this topic: Why dividing by zero is undefined! Mathforum.org also answers the same question!

What are some other problems associated with zeros? Learn more by watching this video from Numberphile: Problems with Zero.

Numbers can be fascinating! Look at this article: The Secret Magic of Dividing by Zero!

Love mathematics? TED-Ed has a number of amazing lessons that involve all types of math! Click here and find your next lesson!

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In the world of math, many strange results are possible when we change the rules. But there's one rule that most of us have been warned not to break: don't divide by zero. How can the simple combination of an everyday number and a basic operation cause such problems?

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Can you solve the rogue AI riddle? - Dan Finkel

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A hostile artificial intelligence called NIM has taken over the world's computers. You're the only person skilled enough to shut it down, and you'll only have one chance. Can you survive and shut off the artificial intelligence? Dan Finkel shows how. Lesson by Dan Finkel, directed by Artrake Studio.

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Read Conway's original paper here, or take a closer look at Conway's Constant and the visual graph of its roots. Or delve deeper into examining the elements and proofs regarding the limits of the sequence.

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Can you find the next number in this sequence? - Alex Gendler

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1, 11, 21, 1211, 111221. These are the first five elements of a number sequence. Can you figure out what comes next? Alex Gendler reveals the answer and explains how beyond just being a neat puzzle, this type of sequence has practical applications as well.

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There have been many efforts throughout the years to 'make America metric'. The biggest push in recent history came as a result of a three-year study in the late 1960's. This came to the conclusion that America was already on the road to 'metrification' (the official adoption and usage of the metric system) and that the trend should be officially encouraged and standardized.

YouTube features a window into this time period with an animated (if you can really call it that) mini-movie entitled 'Metric America', released in 1973. Whether you agree or not with the overall goal of pushing for the metrication of America, it offers a charming insight into the metric-supporters of that time. Watch this movie to see how they felt the following questions should be answered:

- What is the metric system?
- How will America adopt the metric system and how hard will it be?
- What advantages await America upon our adoption to the metric system?

There have also been several recent articles about this dilemma including this one from TIME: Why Won't America Go Metric? and Who's Afraid of the Metric System? From The Atlantic.

Time To Switch To The Metric System! is another quick video from DNews about why America should be using the metric system. Finally, watch this Seeker Network video: Why Don't Americans Use The Metric System? What do you think? Is it time to make the change?

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Why the metric system matters - Matt Anticole

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For the majority of recorded human history, units like the weight of a grain or the length of a hand weren't exact and varied from place to place. Now, consistent measurements are such an integral part of our daily lives that it's hard to appreciate what a major accomplishment for humanity they've been.

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This passcode riddle is an excellent way to introduce factors and multiples and shows the most effective way to understand organized factorization. A good way to understand this problem is by first understanding how a number can be written as all possible products of t natural numbers. Here is a more detailed explanation of how that can be done. Once we have understood this process of writing a number as all possible products of 2 natural numbers, we can move on to understanding how a number can be written as all possible products of 3 natural numbers. For instance, to find the products of 36, a good way to start off would be to write it as 1*1*36. Next, we can break 36 down further and write it using all possible products of 2 numbers. We should get 1*2*18; 1*3*12; 1*4*9; and 1*6*6. We continue this process to find all possible products of 3 natural numbers.

If you enjoyed this problem, here's another version of a problem with the number 72: "UNSOLVABLE" Logic Puzzle: What Are Their Ages?

You may also visit the educator's website here to understand factors and multiples in detail. Finally, if you wish to understand more math concepts (who wouldn't?), visit this link.

Love riddles? TED-Ed has plenty of them that will challenge you:

The famously difficult green-eyed logic puzzle - Alex Gendler
Can you solve the prisoner hat riddle? - Alex Gendler
Can you solve the control room riddle? - Dennis Shasha
Can you solve "Einstein's Riddle"? - Dan Van der Vieren Can you solve the temple riddle? - Dennis Shasha Can you solve the frog riddle? - Derek Abbott
Can you solve the locker riddle? - Lisa Winer

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Can you solve the passcode riddle? - Ganesh Pai

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In a dystopian world, your resistance group is humanity's last hope. Unfortunately, you've all been captured by the tyrannical rulers and brought to the ancient coliseum for their deadly entertainment. Will you be able to solve the passcode riddle and get everyone out safely? Ganesh Pai shows how.

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The binary system used in computers is a numeric system, like the decimal that we use in our day-to-day lives. The only difference between them is that the decimal uses ten symbols to represent numbers (0 to 9), while binary uses only two (0 and 1). This page has an interesting parallel between the two systems, including the representation of rational numbers. This lesson illustrates how the binary system is used, along with other numeric systems. In decimal systems, we know that any number can be represented-you just add digits as the represented number gets bigger. Hence, a system with "only" ten symbols is enough to represent virtually infinite possibilities. Binary works the exact same way: if you need to represent more information, you can add extra bits to your memory. However, using systems with less symbols is not easy: a number represented in binary requires around three times more digits to be written than its decimal representation.

Wouldn't it be better to use the widespread decimal system then? Actually, we use binary computers because binary devices are easier to implement than "decimal devices." Currently, a computer's central processing units (CPU) are made of electrical components called "transistors." Check out this lesson to see how transistors are made, how they operate, and how they have changed in the last century. Binary can also be implemented in multiple other ways. An optical fiber transmits data encoded in pulses of light, while hard disks store information using magnetic fields. You can check several examples of binary devices, even solutions that use more than one transistor to implement a single bit, here. The choice of which technology will be used depends specifically on each application and the most important parameters are cost, speed, size, and robustness.

Now that we know how to build devices that can represent numbers, we can expand their scope by mapping those numbers to a set of interest. A set of interest can be literally anything, such as letters in the alphabet or colors in a palette.

The effort to encode letters with industry standards began in the 1960s, when the American Standard Code for Information Interchange (ASCII) was developed. This encoding used 7 bits, which were mapped to the English letters and special characters. The values corresponding to each symbol are here. As computers became more popular worldwide, the need for tables containing more symbols emerged. Nowadays, the most used encoding is the UTF-8, which was created in 1993 and maps more than one million symbols to binary, including characters in all known alphabets and even emojis. You can navigate through the UTF-8 table here. In this encoding, each character can take one to four bytes to be represented, as shown in the first table here.

Colors are also mapped to binary sequences. Besides the RGB system, other systems were conceived to represent colors. The HSL system also uses three components to represent colors: hue, which varies from 0 to 360, with the color "red" being mapped to 0, "green" to 120 and "blue" to 240; saturation, which is the intensity of the colored component; and lightness, the "white level" of the color. The CMYK uses four components, corresponding to the levels of cyan, magenta, yellow, and black in the pixel. This system is called "subtractive," which means that, as a component gets larger, the pixel emits less light. It is very convenient for printers, where the ink acts as a "filter" for the white canvas of the paper. In this paragraph, the name of each color system has a link to interactive panels that allow you to see the colors corresponding to each possible encoding. If you want to see how those systems compare with each other, check this website.

It is possible to reduce the number of bits without losing information, optimizing data transmission or storage. Strategies that perform compression include the run-length encoding, the Lempel-Ziv algorithm, and the Huffman coding. The Lempel-Ziv algorithm replaces repeated patterns in the data with a token. Both the token and the pattern are added to the compressed file, so the decoder can accurately rebuild the original file. Although this post's discussion is not related to binary, it illustrates the Lempel-Ziv algorithm. The Huffman coding counts the number of occurrences of symbols in the file and creates a new binary encoding for each one of those symbols. Symbols that are more frequent receive a shorter binary sequence, reducing the size of the file. The ZIP files are created by using a combination of those algorithms.

Binary is a way to represent information like any other language. Engineers use international standards to attribute meaning to states of binary devices; they are able to represent letters, colors and even sounds. Just as a painting of pipe is not the pipe itself, the meaning of each bit is not embedded in itself, but in the program that reads it.

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How exactly does binary code work? - José Américo N L F Freitas

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Imagine trying to use words to describe every scene in a film, every note in a song, or every street in your town. Now imagine trying to do it using only the numbers 1 and 0.

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The mathematical secrets of Pascal’s triangle - Wajdi Mohamed Ratemi

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Pascal’s triangle, which at first may just look like a neatly arranged stack of numbers, is actually a mathematical treasure trove. But what about it has so intrigued mathematicians the world over? Wajdi Mohamed Ratemi shows how Pascal’s triangle is full of patterns and secrets.

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Can you solve the buried treasure riddle? - Daniel Griller

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After a massive storm tears through the Hex Archipelago, you find five grizzled survivors in the water. As an act of gratitude for saving them, they reveal a secret - the island they were just on holds some buried treasure. But when the sailors try to describe the island, something seems off.

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Let's Begin...

When they're used well, graphs can help us intuitively grasp complex data. But as visual software has enabled more usage of graphs throughout all media, it has also made them easier to use in a careless or dishonest way - and as it turns out, there are plenty of ways graphs can mislead and outright manipulate. Lea Gaslowitz shares some things to look out for.

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About TED-Ed Animations

TED-Ed Animations feature the words and ideas of educators brought to life by professional animators. Are you an educator or animator interested in creating a TED-Ed Animation? Nominate yourself here "

Meet The Creators

Educator Lea Gaslowitz

Director Mark Phillips

Script Editor Alex Gendler

Sound Designer Sam Bair

Associate Producer Jessica Ruby

Content Producer Gerta Xhelo

Editorial Producer Alex Rosenthal

Narrator Pen-Pen Chen

See more creators

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How to spot a misleading graph - Lea Gaslowitz

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When they're used well, graphs can help us intuitively grasp complex data. But as visual software has enabled more usage of graphs throughout all media, it has also made them easier to use in a careless or dishonest way - and as it turns out, there are plenty of ways graphs can mislead and outright manipulate.

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For more proofs of the Pythagorean theorem, including the one created by former U.S. President James Garfield, visit this site. Another resource, The Pythagorean Proposition, by Elisha Scott Loomis, contains an impressive collection of 367 proofs of the Pythagorean theorem.

The Pythagorean theorem can be extended in its breadth and usage in many ways. For example, the theorem can be extended to 3 dimensions: the squared distance between diagonal corners of a cube is equal to the squared distance of the length, width, and height of the cube. In the same way, though perhaps difficult to visualize, the theorem can be extended to any number of dimensions. As well, the theorem can be extended to apply to a trirectangular tetrahedron, as outlined in de Gua's theorem.

One of the implications that results from the Pythagorean theorem is the inevitability of irrational numbers, numbers that cannot be expressed as a ratio of two integers. For example, a triangle with side lengths of 1 would have a hypotenuse of √2. This TED Ed video explores more about the shattering impact of this discovery.

The Pythagorean theorem is based on the propositions of Euclidean geometry, the geometry of planes or flat surfaces. In fact, Pythagorean theorem is shown to be synonymous with the parallel postulate, the proposition that only one line can be drawn through a certain point so that it is parallel to a given line that does not contain the point. When the parallel postulate is altered, geometries of surfaces with positive and negative curvatures emerge, and the Pythagorean theorem no longer holds true. Modifications to the Pythagorean theorem are then needed for both elliptical and hyperbolic surfaces.

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How many ways are there to prove the Pythagorean theorem? - Betty Fei

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What do Euclid, 12-year-old Einstein, and American President James Garfield have in common? They all came up with elegant proofs for the famous Pythagorean theorem, one of the most fundamental rules of geometry and the basis for practical applications like constructing stable buildings and triangulating GPS coordinates.

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Can you solve the river crossing riddle? - Lisa Winer

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As a wildfire rages through the grasslands, three lions and three wildebeest flee for their lives. To escape the inferno, they must cross over to the left bank of a crocodile-infested river. Can you help them figure out how to get across on the one raft available without losing any lives? Lisa Winer…

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Can you solve the Mondrian squares riddle? - Gord Hamilton

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Dutch artist Piet Mondrian’s abstract, rectangular paintings inspired mathematicians to create a two-fold challenge. Can you solve the puzzle and get to the lowest score possible? Gordon Hamilton shows how.

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The Nrich Maths Project Cambridge,England. Mathematics resources for children,parents and teachers to enrich learning. Problems,children's solutions,interactivities,games,articles.

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This section of the nzmaths website has problem-solving lessons that you can use in your maths programme. The lessons provide coverage of Levels 1 to 6 of The New Zealand Curriculum. The lessons are organised by level and curriculum strand.
Accompanying each lesson is a copymaster of the problem in English and in Māori.

Choose a problem that involves your students in applying current learning.
Remember that the context of most problems can be adapted to suit your students and your current class inquiry.
Customise the problems for your class.

The site also includes Problem Solving Information. This provides you with practical information about how to implement problem solving in your maths programme as well as some of the philosophical ideas behind problem solving. We also have a collection of problems and solutions for students to use independently.

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This section of the nzmaths website has problem-solving lessons that you can use in your maths programme. The lessons provide coverage of Levels 1 to 6 of The New Zealand Curriculum. The lessons are organised by level and curriculum strand. Accompanying each lesson is a copymaster of the problem in…

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How to Learn Math is a free self-paced class for learners of all levels of mathematics. It combines really important information on the brain and learning with new evidence on the best ways to approach and learn math effectively. Many people have had negative experiences with math, and end up disliking math or failing. This class will give learners of math the information they need to become powerful math learners, it will correct any misconceptions they have about what math is, and it will teach them about their own potential to succeed and the strategies needed to approach math effectively. If you have had past negative experiences with math this will help change your relationship to one that is positive and powerful. The course will feature Jo and a team of undergraduates, as well as videos of math in action - in dance, juggling, snowflakes, soccer and many other applications. It is designed with a pedagogy of active engagement. The course is planned to remain available indefinitely.

Strategies needed to approach math effectively

Information about your own potential to succeed in math

Correct any misconceptions about what math is

Part 1: The Brain and Math Learning.
1. Knocking Down the Myths About Math.

Everyone can learn math well. There is no such thing as a "math person". This session give stunning new evidence on brain growth, and consider what it means for math learners.

2. Math and Mindset

When individuals change their mindset from fixed to growth their learning potential increases drastically. In this session participants will be encouraged to develop a growth mindset for math.

3. Mistakes and Speed

Recent brain evidence shows the value of students working on challenging work and even making mistakes. But many students are afraid of mistakes and think it means they are not a math person. This session will encourage students to think positively about mistakes. It will also help debunk myths about math and speed.
Part 2: Strategies for Success.
1. Number Flexibility, Mathematical Reasoning, and Connections

In this session participants will engage in a "number talk" and see different solutions of number problems to understand and learn ways to act on numbers flexibility. Number sense is critical to all levels of math and lack of number sense is the reason that many students fail courses in algebra and beyond. Participants will also learn about the value of talking, reasoning, and making connections in math.

2. Number Patterns and Representations

In this session participants will see that math is a subject that is made up of connected, big ideas. They will learn about the value of sense making, intuition, and mathematical drawing. A special section on fractions will help students learn the big ideas in fractions and the value of understanding big ideas in math more generally.

3. Math in Life, Nature and Work

In this session participants will see math as something valuable, exciting, and present throughout life. They will see mathematical patterns in nature and in different sports, exploring in depth the mathematics in dance and juggling. This session will review the key ideas from the course and help participants take the important strategies and ideas they have learned into their future.

Who is this course for?
This self-paced course is designed for any learner of math and anyone who wants to improve their relationship with math. The ideas should be understandable by students of all levels of mathematics.

Parents who have children under age 13 and who think their children would benefit from some of the course materials should register themselves (i.e., parent's name, email, username) for the course. The parent may then choose to share course materials with their child at their own discretion.
What is the course structure?
The course will consist of six short lessons, taking approximately 20 minutes each. The lessons will combine presentations from Dr. Boaler and a team of undergraduates, interviews with members of the public, cutting edge research ideas, interesting visuals and films, and explorations of math in nature, sport and design.
What is the pace of the course?
The course will be self-paced, and you can start and end the course at any time in the months it is open. It is recommended that you take no more than one session a week, to allow the ideas to be processed and understood.
How will I be assessed?
There will be no formal assessment. Participants will be asked to complete a pre-and post-survey. The course will include quizzes that combine opportunities to write, work on math and reflect. These will not be graded.
Does this course carry any kind of Stanford University credit?
No.

Interested in this course for your business or team?

Train your employees in the most in-demand topics, with edX For Business.

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Stanford University: How to Learn Math: For Students

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How to Learn Math is a free self-paced class for learners of all levels of mathematics. It combines important information on the brain with new evidence on the best ways to approach math effectively. Many people have had negative experiences with math.