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A beginner's guide to Native American art after 1600
Integrating our content in your curriculum (K-12 math)
Islamic Middle East
The term Islamic is used here to define the art and material culture of those lands where the dominant religion is Islam, the religion revealed to the prophet Muhammad in seventh century Arabia.
Later Zhao
Liao Dynasty (907–1125)
1.0—Welcome to Big History
Big History is an exciting course that tells the 13.8 billion year history of the Universe from the Big Bang to the present. Before we really dive into the core concepts of Big History, it’s important to get an overview of what Big History is and what you can expect from the course. Ready to challenge your idea of what studying history is all about?
Scale is incredibly important in understanding Big History. As you already know, this course looks at history at a larger scale than any other history course we’ve seen, and studying time and space over 13.8 billion years poses an interesting challenge. One way of dealing with this challenge is by using different scales so that each topic can be studied from the appropriate perspective. In this lesson, you’ll watch a video on scale, and then take part in an activity that will get you out of your chair. By the time you’ve seen the video and finished the activity, you’ll have a good sense of how scale can affect the way we view things.
1.2—Origin Stories
Origin stories are the emphasis of this lesson. Big History focuses on the modern, scientific origin story of how the world got to be the way it is. Big History is one origin story, and it’s important to recognize that many different types of origin stories exist. Some are thousands of years old and some are newer. They come from varying cultures and religions, but one theme runs through all of them: It seems that all humans are interested in understanding their origin to some degree.
1.3—What Are Disciplines?
Big History is an interdisciplinary course—in other words, it looks at the history of the Universe and universal change from a number of very distinctive perspectives. Each perspective represents one or more areas of study (we call these disciplines), and each area of study strives to answer a completely different set of questions about ourselves, our planet, and the Universe. No one discipline can know all there is to know about absolutely everything. It would be impossible to think of all the questions to ask, let alone have all the answers. Usually, if we look at any single object or event from the points of view of multiple areas of expertise, we can put together a rich understanding that goes far beyond a one-discipline approach.
1.4—My Big History
In this lesson, you’re going to spend a lot of time thinking about yourself in relation to the Big History narrative. What about your life is like Big History? It may seem like nothing is, but in fact, there are lots of things that have happened in your life that can be compared to the major events in Big History.
10.0—Looking Back
Big History tells the story of the Universe from the Big Bang to the present. It would be impossible for any history course to cover everything related to its subject matter, and this challenge is particularly great in Big History because of the scales of time and size involved. Big History deals with this challenge by focusing on the eight turning points, or thresholds, over the course of the 13.8 billion-year story. At each of these thresholds, the Universe became more complex, and things appeared with significant new emergent properties. In this lesson, you’ll review what you’ve learned about this story, which will put you in the perfect position to think about the future of Big History in subsequent lessons in this unit.
10.1—The Biosphere
As you learned in Unit 5, the biosphere is not static. The biosphere changes as a result of astronomical, geological, and biological influences. The dinosaurs, for example, became extinct as a result of an astronomical event—the effects of an asteroid impact on Earth. Every species impacts the biosphere, though the extent of that impact can vary dramatically. The impact of humans has changed over time. The impact of foragers was not dramatic, but these early humans did have the ability to destroy flora and fauna and cause fires in different parts of the Earth. Agriculture, and later the modern revolution, significantly increased the impact of humans on the biosphere. The acceleration of the last 100 years has seen an acceleration of these human impacts on the biosphere. What are the most significant of these impacts and what should humans be doing about them?
10.2—Looking Forward
Historians typically focus on the past, asking questions like: What happened in the past? Why did it happen? What lessons can be drawn from these events? Studying the past is possible because there is lots of interesting evidence left behind for scholars of many disciplines—not just historians—to look at. Physicists, for example, can look at the abundance of hydrogen and helium in the Universe today and draw conclusions about the early history of the Universe. Geologists can look at the distribution of plants and animals and rock formations on the Earth today and draw conclusions about the how the configuration of the Earth’s continents has changed over time. Each discipline has its own questions and evidence and is interested in investigating change over time. Historians don’t typically predict the future, but given the number of trends you’ve studied in this course, it seems appropriate to give some thought to what the future might be like.
16th to 18th century: Dynasty, revolution, and the Classical Age
It all starts here in the 16th and 17th centuries, an era of civil war, plague, a Glorious Revolution, and the birth of the United Kingdom. The following century sees Britain establish itself as a world power through economic growth and colonial ambitions. All the while, British art is flourishing in parallel to these dramatic historical events, shifting from images of family wealth to more personal portraits to scenes of classical drama and ongoing war.
1913 Centennial Celebration
1913 is a particularly important year within the history of modern art, marked by events and objects that would fundamentally change the way art was conceived and understood. In February of that year, the groundbreaking "Armory Show" introduced the American public to the work of Paul Cézanne, Pablo Picasso, Marcel Duchamp, and many other European artists exhibited alongside their American avant-garde counterparts. In this series of videos, curators from all areas of the Museum speak about their favorite works from 1913 in MoMA's collection.
19th century American sculpture, architecture and photography
19th century: Romanticism and the Victorian Era
The 19th century sees an outpouring of complex images and emotions from the Romantics, among them the great figures of JMW Turner and William Blake. While Queen Victoria reigns and new developments sweep across the country, artists bring realism together with an ongoing interest in mythical themes.
2-D Divergence theorem
Using Green's theorem (which you should already be familiar with) to establish that the "flux" through the boundary of a region is equal to the double integral of the divergence over the region. We'll also talk about why this makes conceptual sense.
2.0—How Did Our Understanding of the Universe Change?
Through the ages, astronomers have used the tools of their time to understand the origin and structure of the Universe. Their views built upon one another, leading to our modern view of the Universe.
2.1—The Big Bang
The Big Bang hasn’t always been the generally accepted explanation of how our Universe came to be. In fact, our views on the origin and structure of the Universe have changed drastically over the last thousand years. As new thinking and evidence have built upon one another, our understanding of the Universe has evolved. Over the ages, astronomers have used the tools of their time to understand the origin and structure of the Universe.
2.2—Claim Testing
The focus of this lesson is to begin to understand the process of claim testing. We use the term "claim testing" in Big History to mean the process that someone goes through when evaluating the truth of a statement that is made either in Big History or in other research that you might do. In general, when you encounter a claim, it’s important to ask why you should believe and trust in that claim. This is a core idea in Big History, and you should use claim testing on a regular basis throughout the course.
2003 AIME
2008 Bank bailout
In 2008, the entire financial system was at a potential breaking point because of a popping housing bubble. This tutorial breaks down how the government attempted to address this (historical note: Sal made these videos as the crisis was unfolding).
2013 AMC 10 A
Final five questions on the 2013 AMC 10 A. (Three of these problems are shared with the 2013 AMC 12 A.) Videos produced by Art of Problem Solving ( Problems from the MAA American Mathematics Competitions (
3.0—How Were Stars Formed?
In the years following the Big Bang, hydrogen atoms floated freely around the Universe. These atoms were slightly more packed together in some places than in others. In the more crowded areas, the hydrogen atoms were close enough to each other to let gravity do its work. In these little pockets of hydrogen, stars lit up across our Universe.
3.1—Creation of Complex Elements
With the birth of stars, new sources of light and energy emerged all over the Universe. They burned hydrogen to create helium. Helium was used to create carbon. Neon, oxygen, silicon, and iron were also created during the lives of stars. However, once these stars started running out of fuel is when things really got interesting. It’s in the massive explosions that resulted from certain stars running out of fuel that all of the elements of the periodic table were created. Without the death of stars, our world would not exist today.
3.2—Way of Knowing: Stars and Elements
Aging and dying stars get hotter than… well, they get hot. Hot enough to create new, heavier elements. What's so special about the heavier elements? Imagine life without metal.
3D Shapes
4.1—What Was Young Earth Like?
Scientists estimate that the Earth formed about 4.6 billion years ago. The Earth that we know today, the relatively peaceful home of myriad forms of life, didn’t appear overnight. Rather, it took billions of years to slowly evolve into its current form. The process of accretion that led to the formation of the Earth was a violent one, and it produced an Earth that was only a little less violent and hostile. For a few hundred million years, the early Earth was characterized by high temperatures, toxic gases, high levels of radiation, and ongoing high-impact collisions. Over time, these conditions improved and the Earth took on its distinctive structure with differentiated layers of core, mantle, crust, and atmosphere. This distinctive structure has important consequences: First, it helps explain why the surface of the Earth changes over time; and second, it helps explain why the Earth evolved into a suitable setting for living things.
4.2—Why Is Plate Tectonics Important?
In the last lesson, you learned that the interior of the Earth changed over time to give the planet the unique layering that it currently possesses. In this lesson, you will learn that the surface of the Earth changes as well. The high temperatures that caused the differentiation of the Earth’s layers continue even today. In a process known as plate tectonics, the heat and movement of the mantle floating under the Earth’s crust drive the movement of the Earth’s crust over its surface. The slow shifting of these tectonic plates explains the shape of our continents as well as mountain ranges and traumatic events like earthquakes.
4.3—Ways of Knowing: Our Solar System and Earth
Towering mountains and trembling earthquakes, the surface of our Earth is constantly in motion. Plate tectonics is responsible for the shape and position of our land.
5.0—What Is Life?
With the appearance of the first planets, the Universe became much more complex. Planets, which formed from clouds of chemicals created during the death of stars, are more chemically diverse than the stars from which they came. Planets also differ from stars because they don’t generate huge amounts of energy at their centers. As a result, their surface temperatures are much cooler. This combination of diverse chemical ingredients and less violent conditions enabled planets to become the setting for life. When life emerged on Earth, it had characteristics that made it very different from nonliving things. Many people will tell you they “know” the difference between living and nonliving things. However, biologists – the experts – have struggled to agree on a single definition of life, even though many of the proposed definitions share similarities. In this lesson, we’ll focus on understanding the characteristics all living things share.
5.1—How Did Life Begin and Change?
For centuries, scientists have searched for the answer to the question, how did life begin? Some have argued it began in a shallow pool of water. Others have argued that it began deep below the surface of the ocean. Still others believe that a meteor from some distant corner of the Universe brought to the Earth the ingredients for life. How life appeared and how it changed over time are some of the most interesting questions you’ll tackle in this course.
5.2—How Do Earth and Life Interact?
The biosphere is an amazing place, serving as home for humans and many other species. The atmosphere provides the air we need to breathe, but the atmosphere also helps to protect us from the Sun’s radiation. The water, plants, and animals of the biosphere provide humans with many of the resources we need for survival. But as amazing the biosphere is in creating a “comfort zone” for humans, it can also be quite fragile, and from time to time has been subject to significant threats from various astronomical, geological, and biological forces. Changes in these forces can result in both mild and extreme impacts on the biosphere.
5.3—Ways of Knowing: Life
Life first appeared on Earth around 3.8 billion years ago—not long after the Earth itself —and life forms have been changing and diversifying ever since. How is it possible for scientists to know about the history of life? What methods do they use to study life and how it has evolved over time? The methods scientists use to reconstruct the story of living things and how they change over time also change over time. New techniques and instruments are constantly being developed to look more deeply into the world of cells and DNA.
6.0—How Our Ancestors Evolved
How *did* life transform from a single-celled organism to our own species, one that can create, communicate, love, and hate? We’ll explore how bacteria, plants, and animals have transformed over time, leading to the arrival of Homo sapiens (fancy words for humans). We’ll then take a look at how this process of change led to the cultural evolution of humans and the development of our most powerful skill, collective learning.
6.1—Ways of Knowing: Early Humans
Considering we have never met a Homo habilis or an Australopithecus, how do we know about these early human species? Disciplines such as anthropology, archaeology, primatology, and history give us bits and pieces of evidence to help solve the puzzle of how humans have evolved over time. While we have learned a great deal about this topic in the last hundred years, there are still many unanswered questions and more evidence to be found!
6.2—Collective Learning
Considering we have never met a Homo habilis or an Australopithecus, how do we know about these early human species? Disciplines such as anthropology, archaeology, primatology, and history give us bits and pieces of evidence to help solve the puzzle of how humans have evolved over time. While we have learned a great deal about this topic in the last hundred years, there are still many unanswered questions and more evidence to be found!
6.3—How Did the First Humans Live?
We often teach dogs a few basic tricks, such as how to sit, stay, and fetch. However, have you ever seen dogs teach one another tricks? They might mimic each other’s behavior, but that’s really not the same thing. Humans, on the other hand, can deliberately pass information to each other and teach one another new skills. This makes it much easier for each generation to pick up where the previous one left off, and it’s why we are the only species able to dominate the biosphere. But why have humans evolved into such a powerful species? What makes us special? Here the story continues, as our ancestors evolve from early Homo sapiens to more intelligent beings who are able to learn collectively. Collective learning is the foundation on which modern society is built, and the reason our species continues to build upon what previous generations created.
7.0—The Rise of Agriculture
All living things need energy to survive. For a long time, humans met their energy needs almost exclusively by eating the food they foraged in their local environment. In other words, humans were completely dependent on the plants and animals that nature provided. The invention of farming about 12,000 years ago gave humans access to vast new food and energy resources, which helped to dramatically transform the way humans lived. Among other things, farming made possible dramatic population growth, and it allowed humans to settle in larger, denser communities than a foraging lifestyle could support. These larger and denser communities eventually led to the development of cities and civilizations, which accelerated collective learning and innovation.
7.1—The First Cities and States Appear
The invention of farming led to dramatic changes in the way humans got their food. By domesticating plants and animals, people could settle in one place. Because domestication gave people more control over the plants and animals in an area, human groups could grow bigger and denser. As these populations grew and turned into cities, states, and empires, greater opportunities for collective learning evolved but so did the types of challenges human communities had to face. Population growth, specialization of labor, and the emergence of rulers and social hierarchies all paved the way for increased collective learning, but they also created some complex problems and relationships that we are still struggling to resolve today.
7.2 Where and why did the first cities appear?
The mysterious pyramids of Ancient Egypt, the Great Wall of China, and the beauty of Machu Picchu (an old Incan city in Peru) are all impressive remnants left behind from some of the world’s ancient agrarian civilizations. But not every civilization has left behind such noticeable clues. In fact, many artifacts from agrarian civilizations fade away with time. But the clues that remain become important windows to the past. Researchers in a variety of disciplines such as history and archaeology use both written record and historical artifacts to pose, analyze, and answer questions about the past. As you will see, written record is of particular importance. Unlike other species, writing gives us the ability to preserve and pass on large amounts of information from one generation to the next. With this ability comes incredible power.
For most of the agrarian era, the world was divided into four separate and distinct world zones. Over time, these zones slowly became more connected as networks of communication and exchange expanded. While innovations did occur throughout this era, such as irrigation, iron plows, and fast-ripening rice, none of these innovations were able to sustain long-term population growth, which limited expansion. Each innovation led to immediate growth, but once populations had grown beyond a certain point, they fell. These cycles of rise and fall in population, called Malthusian cycles, characterized the agrarian era. Humans would not break out of these cycles until the world zones became more connected and rates of innovation were capable of sustaining growth over much longer periods of time.
8.1—Exploration & Interconnection
The rise of agriculture ushered in an era of increasing innovation in communication and transportation that led different parts of the world to connect in entirely new ways. The voyages of Christopher Columbus extended this exchange from Afro-Eurasia to the Americas, which saw a massive movement of ideas, people, diseases, plants, and animals between the two hemispheres. The results of these exchanges were dramatic. Potatoes and corn, first cultivated in the Americas, quickly became crucial in the diets of people across Eurasia. Horses and cattle, unknown in the New World in 1492, quickly took on crucial roles in many societies in the Americas. The linking of the different world zones in this period and the exchanges that this linking made possible, transformed the lifeways of the people and civilizations involved – and laid the foundation for modern exchange routes and the global balance of power.
8.2—Commerce & Collective Learning
Evading bandits through mountain passes, leading a caravan of yaks carrying silk and goods, sailing the trade winds off the Indian coastline – these are a few things you might have done as a trader in the age of agrarian civilizations. Systems of exchange and trade between large agrarian civilizations facilitated the transfer of goods from one civilization to the next, but they also helped share the world’s religions, ideas, innovations, diseases, and people. While each world zone had its own trade routes, none were as vast and intense as the Silk Road. This large system of exchange and trade, initially designed for commerce, dispersed goods and ideas throughout Afro-Eurasia, and paved the way for a substantial increase in both commerce and collective learning.
For most of the agrarian era, the four world zones operated independently of each other with little or no knowledge of what was going on in the other zones. The world, in effect, was divided into four unconnected regions, none of which was really interested in the others. With the improved transportation and communication technologies developed 500 years ago, humans acquired the means for connecting these formerly independent zones. After 1492, for example, the Americas and Afro-Eurasia were put in regular contact, and the Columbian Exchange saw the transfer of people, ideas, animals, plants, and diseases between these two once separate world zones. Exchanges like these fueled social, political, economic, and intellectual innovation. Within a few hundred years, this more fully connected world saw dramatic acceleration in innovation and population growth, which ushered in the Modern Revolution.
9.1—The Anthropocene
For most of the past 10,000 years or so, the biosphere has been a fairly stable and predictable place. Whether you look at temperature, types of vegetation, soils, or sea level, the basic characteristics of the biosphere have remained about the same, having shown only moderate variation at any point within most of that time frame. That type of consistency is what prompted geologists to label the last 10,000 years of geological history as the Holocene Epoch; an epoch that was ushered in at the end of the last ice age. But there are now a number of scientists who view the data from the last 250 years and conclude that the biosphere is showing fundamentally different characteristics from the previous 10,000 years. The rise of carbon dioxide levels, glacial melting, and the shrinking of tropical rainforests are just some of the factors that they cite as evidence that the biosphere has entered a new epoch. Because so much of the change they have identified seems to derive from human activity, these scientists propose that this new epoch be called the Anthropocene to reflect the tremendous impact that humans now exert in the biosphere.
9.2—Changing Economies
The Modern Revolution created the world we live in today. This world is very different from the world of 500 or 1,000 years ago, let alone 10,000 or 100,000 years ago. The connection of the four world zones allowed for the creation of a global network of exchange. Though this network was not built overnight, it emerged fairly quickly, and it increased the potential connections and diversity of connections for many members of the network. The result was an acceleration of both collective learning and innovation. Commerce was an important driver of change in this global network. Because commerce began to take on greater significance for many societies, a number of important thinkers began to ask questions about the nature of the exchange of goods, the nature of productivity and efficiency, and the interests of the individual and the state in business. All of this new inquiry gave birth to the discipline of economics. These economic thinkers, like the thinkers in any discipline, shared a set of concerns and questions but often came up with very different answers to those questions. The articulation of the ideas of capitalism and communism were the most influential economic ideas generated in the course of the Modern Revolution.
A Brief History of Women in Art
Despite being engaged with the art world in every way imaginable, many women artists have been invisible in the traditional narrative of art history. They have faced challenges due to gender biases, from finding difficulty in training to selling their work and gaining recognition. Follow the course of this contentious history, from the challenges facing women up through the 20th century, to the campaigns of the feminist art movement, to the continuing work of contemporary women artists.
A beginner's guide to 20th century art
If abstract art puzzles you, this is a great place to start.
A beginner's guide to Baroque art
Get a head start on the historical developments that shape Baroque art in the Catholic Italy, Spain, Flanders, France and in the Protestant north.
A beginner's guide to Buddhist art and culture
Among the founders of the world's major religions, the Buddha was the only teacher who did not claim to be other than an ordinary human being. The Buddha was simply a human being and he claimed no inspiration from any god or external power.
A beginner's guide to Byzantine art and culture
Learn about the Byzantine Iconoclasm and how it shapes the Byzantine art that remains today.
A beginner's guide to Imperial China
China was a highly literate society that greatly valued poetry and brush-written calligraphy, which along with painting, were called the Three Perfections, reflecting the esteemed position of the arts in Chinese life.
A beginner's guide to Renaissance Florence
The Renaissance really gets going in the early years of 15th century in Florence. In this period, which we call the Early Renaissance, Florence is not a city in the unified country of Italy, as it is now. Instead, Italy was divided into many city-states (Florence, Milan, Venice etc.), each with their own government (some were ruled by despots, and others were republics).
A beginner's guide to Southeast Asian art
A beginner's guide to ancient Egypt
Who were the ancient Egyptians? What did they believe? What did their art mean to them and what materials did they use to make it? We don't know everything about this culture, but we have learned quite a bit.
A beginner's guide to ancient Greece
Rio 2016, London 2012, Beijing 2008, Olympia 776 B.C.E. find out how the ancient pan-Hellenic games inspired our modern Olympiad. Then learn to identify the classical orders and you will never look at your city or town the same way.
A beginner's guide to ancient Rome
Roman art spans almost 1,000 years and three continents. The first Roman art can be dated back to 509 B.C.E., with the legendary founding of the Roman Republic, and lasted until 330 C.E. (or much longer, if you include Byzantine art). The city of Rome was a melting pot, and the Romans had no qualms about adapting artistic influences from the other Mediterranean cultures that surrounded and preceded them.
A beginner's guide to contemporary art
A beginner's guide to medieval Europe
The Middle Ages lasted 1000 years. Learn how manuscripts were made, how medieval calendars worked, and how Christ's life was represented.
A beginner's guide to the High Renaissance
As the Humanism of the Early Renaissance develops, a problem arises. How can an artist create figures that are at once an accurate naturalistic rendering of the human body in space while also describing the ideal beauty of the spiritual realm?
A beginner's guide to the Northern Renaissance
In the 15th century, the northern European countries we know today as Belgium, the Netherlands and Luxembourg were controlled by the enormously wealthy Dukes of Burgundy (Burgundy is a region in France). Today the 15th century in this region is often referred to today as the Burgundian Netherlands. The court of the Dukes of Burgundy were the most important patrons of the early Northern Renaissance, but newly wealthy private citizens also commissioned art as part of a growing interest in private meditation and prayer. Portraits were also commissioned in growing numbers.
A beginner's guide to the Northern Renaissance in the sixteenth century
A beginner's guide to the arts of the Islamic world
The Taj Mahal, a silk carpet, a Qur‘an; all of these are examples of Islamic art. But what exactly is Islamic art and architecture?
ATP and reaction coupling
You may have heard ATP described as the “energy currency” of a cell. But what exactly does that mean? Learn more about how ATP stores energy and how its breakdown (hydrolysis) powers energy-requiring reactions in a cell.
About pi and tau
When you want to make a circle, how is it done? Well you probably will start with the radius one.
About the SAT
Absolute and relative maxima and minima
Absolute value
Find the distance between any two numbers using absolute value.
Absolute value equations
You are absolutely tired of not knowing how to deal with equations that have absolute values in them. Well, this tutorial might help.
Abstract Expressionism
Action painting, drip painting, luminous fields of color so large that they envelop the viewer while exposing the process of their creation. Abstract Expressionism is not a single style and isn't even consistently abstract. Rather, it is intensely personal painting often on a very large scale that is more concerned with exposing the act of creative invention than in traditions of representation.
In a world full of unbalanced forces (which you learn more about when you study Newton's laws), you will have acceleration (which is the rate in change of velocity). Whether you're thinking about how fast a Porsche can get to 60mph or how long it takes for a passenger plane to get to the necessary speed for flight, this tutorial will help.
Access to college
Throw out your preconceived notions about who can go to college as admissions officers and students discuss the wide-variety of pathways to higher education.
Acid-base equilibria
In this tutorial, we will learn how the strength of an acid or base is related to the equilibrium reaction that is happening in our solution. We will also see what happens when we mix solutions of acids and bases in neutralization reactions.
Acid/base equilibria
Chill out! Characterizing solutions of acids and bases is easy when you use the simple ICE table technique. You will also learn how buffers regulate pH so well, a process essential to homeostasis.
Acids and bases
What makes a compound acidic or basic? We will learn about the different definitions for acids and bases, and how we measure how acidic or basic a substance is. We will be putting our chemical equilibrium knowledge to good use when we look at the reactivity of weak acids and bases.
Acids, bases, and pH
You may know that lemon juice is acidic, and that bleach is basic. But did you know that acid-base chemistry is happening all the time in your body? Learn what makes something an acid or base, what the pH scale really measures, and how buffers work in your blood!
Active transport
In active transport, the cell has to pay (often, in the form of ATP) to move molecules against their gradients. Learn more about the amazing protein machines that make this possible.
Want to test your knowledge of art? Try these activities first or come back to them after you finish the tutorial.
Acyanotic heart diseases
Adding and ones, tens, and hundreds
Learn how to solve addition problems by adding ones, tens, and hundreds.
Adding and subtracting decimals
You get the general idea of decimal is and what the digits in different places represent (place value). Now you're ready to do something with the decimals. Adding and subtracting is a good place to start. This will allow you to add your family's expenses to figure out if your little brother is laundering money (perhaps literally). Have fun!
Adding and subtracting decimals percentages and fractions
If you think you know how to convert between decimals, fractions and percentages and are familiar with negative numbers, this tutorial is a good place to test all of those skills at the same time.
Adding and subtracting decimals word problems
In this tutorial, we practice solving word problems by adding and subtracting decimal numbers.
Adding and subtracting fractions
You've already got 2 cups of sugar in the cupboard. Your grandmother's recipe for disgustingly-sweet-fudge-cake calls for 3 and 1/3 cups of sugar. How much sugar do you need to borrow from you robot neighbor? Adding and subtracting fractions is key. It might be a good idea to look at the equivalent fractions tutorial before tackling this one.
Adding and subtracting fractions with like denominators
Learn how to add and subtract fractions. This is an introductory tutorial, so the fractions you add and subtract have the same denominators.
Adding and subtracting fractions with unlike denominators
We've already had some good practice adding fractions with like denominators. We'll now add fractions with unlike denominators. This is a very big deal. After this tutorial, you'll be able to add, pretty much, any two (or three or four or... ) fractions!
Adding and subtracting fractions with unlike denominators word problems
Practice solving word problems by adding and subtracting fractions that have different denominators.
Adding and subtracting fractions: word problems
Learn how to solve word problems by adding and subtracting fractions. This is an introductory tutorial, so the fractions you add and subtract have the same denominators.
Adding and subtracting integers
Add and subtract negative numbers using a number line. It's the 7th grade mathematics shuffle: "Slide to the left for a negative value, and slide to the right for a positive value." Be careful, though. Which way do you move if you are subtracting a negative number? The answer awaits!
Adding and subtracting mixed number with unlike denominators
Learn how to add and subtract mixed numbers whose fractional parts have different denominators.
Adding and subtracting mixed numbers
Learn how to add and subtract mixed numbers. This is an introductory tutorial, so the fractions you add and subtract have the same denominators.
Adding and subtracting negative fractions
Learn how to add and subtract fractions, even when they're negative.
Adding and subtracting negative numbers
You understand that negative numbers represent how far we are "below zero". Now you are ready to add and subtract them! In this tutorial, we will explain, give examples, and give practice adding and subtracting negative numbers. This is a super-important concept for the rest of your mathematical career so, no pressure, learn it as well as you can!
Adding and subtracting negative numbers with variables
Feeling positive about all of this negative number stuff? Challenge yourself to solve some more advanced problems, including expressions with variables.
Adding and subtracting polynomials
Learn how to add and subtract two polynomials to obtain another polynomial. For instance, express (3x^3+2x-1)+(2x^4-x^3+7x) as a polynomial in standard form.
Adding and subtracting rational expressions
Learn how to add and subtract rational expressions. Like multiplication and division, this skill has a remarkable affinity to adding and subtracting fractions!
Adding and subtracting rational numbers
We're going to mix it up a little in this set of examples. Remember that a rational number is a real number that can be written as a simple fraction, or by extension a decimal or percentage. We have to be able to add and subtract numbers when they are in different formats, whether fraction, decimal or percentage. This gets to be fun as learn to move between these expressions of rational numbers, and start to understand they're all pointing to same thing! Common Core Standards: 7.NS.A.1d
Adding and subtracting with negatives on the number line
Learn how to use number lines to solve negative number addition and subtraction problems.
Adding and subtracting with unlike denominator word problems
You know what a fraction is and are now eager to apply this knowledge to real-world situations (especially ones where the denominators aren't equal)? Well, you're about to see that adding and subtracting fractions is far more powerful (and fun) then you've ever dreamed possible!
Adding decimals
Honestly, once you remember one very important rule, the rest of adding and subtracting decimals is a snap. That rule is to be sure and align your decimals first. In this tutorial, you'll get a chance to practice this with us and become fluent adding decimals. We'll tackle subtracting decimals in the next tutorial.
Adding fractions with denominators of 10 and 100
Practice working with fractions that have denominators of 10 and 100. This practice prepares us for the next tutorial on decimal numbers.
Adding multi-digit numbers
You know how to add multi-digit numbers from the 3rd grade. Now we will give you even more practice (and tackle even larger numbers)! We'll dive into something called "regrouping" or "carrying" - it's a nifty little trick that makes adding multi-digit numbers much easier! Common Core Standard: 4.NBT.B.4
Adding ones and tens
Learn how to solve addition problems like 23+4 and 23+40.
Adding with regrouping within 1000
Learn how to add numbers like 259 + 363. These addition problems are a harder than the ones we've done before because they require regrouping (also called carrying).
Addition and subtraction missing value problems
Learn how to solve problems like "___ - 237 = 448" where you don't know one of the values in an addition or subtraction equation.
Addition and subtraction within 10
Learn how to add and subtract small numbers. Numbers used are 10 or less.
Addition and subtraction word problems
Learn how to solve addition and subtraction word problems with small numbers. Numbers used are 10 or less.
Addition and subtraction word problems with negatives
See how adding and subtracting negative numbers can be used in different situations.
Addition with carrying
You're somewhat familar with adding, say, 17+12 or 21+32, but what happens for 13+19? Essentially, what happens when I max out the "ones place". In this tutorial, we'll introduce you to the powerful tool of carrying and why it works.
Addition within 100
Learn to add two-digit numbers like 37 and 49.
Addition within 20
Learn to add. Numbers used are 20 or less.
Additional topics in math
Admissions essays
Students often view the admissions essay as a chance to show off their "x-factor." Is that the best approach? Hear from admissions officers and current college students about what actually makes a great college application essay.
Admissions interviews
It pays to be prepared for your college interview. Learn how admissions officer view the interview and how current college students approached the process as they were applying to school.
Advanced sequences and series
You understand what sequences and series are and the mathematical notation for them. This tutorial takes things further by exploring ideas of convergence divergence and other, more challenging topics.
Advanced structure in expressions
This tutorial is all about *really* being able to interpret and see meaning in algebraic expressions--including those that involve rational expressions, exponentials, and polynomials. If you enjoy these ideas and problems, then you're really begun to develop your mathematical maturity.
Advanced trig examples
This tutorial is a catch-all for a bunch of things that we haven't been able (for lack of time or ability) to categorize into other tutorials :(
Because of Afghanistan's geographical position—on the edge of central Asia with India and China beyond to the east, and Iran, the Middle East and the numerous cultures of the Mediterranean and the rest of Europe to the west—it was criss-crossed by ancient trade routes. In many ways, then as now, it was a hub and meeting place for diverse cultures and neighbours, both near and distant, over thousands of years.
Discover pioneers of African modernism like Ibrahim El-Salahi alongside contemporary artists like Meschac Gaba, who brings art into the streets and questions the nature of contemporary African art. Learn about artists who bridge African and Western traditions in their work.
Africa: 1100-1980 C.E.
Human life, which is understood to have begun in Africa, developed over millions of years and radiated beyond the continent of Africa. The earliest African art dates to 77,000 years ago. While interpretation of this art is conjectural at best, the clarity and strength of design and expression in the work is obvious.
African art, an introduction
Find here a brief introduction to the peoples and cultures of Africa.
Aftermath of World War I
World war I (or the Great War) was a defining event for the 20th Century. It marked the end (or beginning of the end) of centuries-old empires and the dawn-of newly independent states based on ethnic and linguistic commonality. It didn't just change the face of Europe, it changed the face of the world. From the Paris Peace Conference and Treaty of Versailles, we'll see how the end of World War I may have been just the set up for even more conflict in Europe and the world.
Age of revolution (quiz)
Age word problems
In 72 years, Sal will be 3 times as old as he is today (although he might not be... um... capable of doing much). How old is Sal today? These classic questions have plagued philosophers through the ages. Actually, they haven't. But they have plagued algebra students! Even though few people ask questions like this in the real-world, these are strangely enjoyable problems.
Aggregate demand and aggregate supply
This tutorial looks at supply and demand in aggregate-from the perspective of the entire economy (not just the market for one good or service). Instead of thinking of quantity of one good, we think of total output (GDP). Very useful model for thinking through macroeconomic events.
Think Sargon and Narim-sin, the Akkadians ruled most of Mesopotamia for centuries a really long time ago.
Aksum was the name of a city and a kingdom which is essentially modern-day northern Ethiopia and Eritrea. Aksum was a major naval and trading power from the 1st to the 7th centuries C.E. As a civilization it had a profound impact upon the people of Egypt, southern Arabia, Europe and Asia, all of whom were visitors to its shores, and in some cases were residents.
Alcohol nomenclature and properties
It can clean a wound or kill your liver. Some religions ban it, others use it in their sacred rites. Some of the most stupid acts humanity every committed were done under its influence. It is even responsible for some of our births. In this tutorial, Sal and Jay name alcohols and discuss their properties.
Aldol condensations
In this tutorial, Sal and Jay show you the mechanism of the aldol condensation and how to predict the products of aldol reactions.
Alexander Hamilton
Hamilton was a strong believer is a central federal government, a debate that continues to this day.
Algebra II readiness warmup
Algebra readiness skill check
Algebraic equations basics
Explore what it means for equations with variables (algebraic equations) to be solved.
Algebraic expressions with fractions
Algebraic expressions can be composed of pretty much any operation, including fractions. Here we'll explore how to manipulate fractions algebraically. In some ways this is just a review of adding, subtracting, multiplying and dividing fractions, but now we are doing it with variables!
Algebraically determining segment length
In this tutorial, you'll flex both your algebra and geometry muscles at the same time. You'll do this by applying the right amount of spray tan (which is needed for properly flexing any muscle) and then solve problems about line segments using algebra!
Algeria is vast nation on the southern coast of the Mediterranean. It is largely desert and is known for its brilliant prehistoric, ancient and Islamic cultures.
Alkene nomenclature
In this tutorial, Jay names alkenes, discusses the stability of alkenes, and introduces the E/Z system.
Alkene reactions
In this tutorial, Sal shows several reactions of alkenes.
Alkyne reactions
In this tutorial, Jay shows the reactions of alkynes.
Alternate number bases
Most of us are use to using the digits 0-9 to represent numbers in the base-10 (decimal)number system. In this tutorial, we'll see that is just one of many (really infinite) number systems. In particular, we will focus on the binary (base-2) and hexadecimal (base-16) systems.
Ok. You knew triangles where cool, but you never imagined they were this cool! Well, this tutorial will take things even further. After perpendicular bisectors, angle bisector and medians, the only other thing (that I can think of) is a line that intersects a vertex and the opposite side (called an altitude). As we'll see, these are just as cool as the rest and, as you may have guessed, intersect at a unique point called the orthocenter (unbelievable!).
Amazonian peoples
The vast reaches of the Amazonian rainforest have been shaped by human hands for thousands of years. Long before Europeans arrived in the Americas this deep history of human interaction with river and forest gave rise to enduring cultural traditions and sophisticated art styles.
America: Civil War to the Gilded Age
Find here the art of Winslow Homer, the expat Mary Cassatt, and work by other American artists in the 2nd half of the 19th century.
American art to World War II
This tutorial includes many of the most iconic American images, American Gothic and Nighthawks for example. But also find here photographers that documented American poverty during the depression, the gritty cityscape, and the magic of looking up to a night sky through the canopy of a tree.
American civics
Videos about how government works in the United States.
American entry into the Great War (World War I)
Naval blockades in World War I to starve enemy nation of trade. Contrary to what many think, American entry into WWI was not due purely to the sinking of the Lusitania. Learn more about what caused the United States to play its first major direct role in a European conflict.
Amino acids and proteins
1A: We will come to an understanding of the central dogma of molecular biology: DNA makes RNA, and RNA makes protein. You will learn about how we classify the different amino acids and how they come together to form the building blocks of complex proteins.
Amino acids, peptides, proteins
An overview of blended learning
Get an overview of blended learning along with the definition of blended learning and an introduction to several different models of blended learning.
An overview of the teacher experience
Analysis of variance
You already know a good bit about hypothesis testing with one or two samples. Now we take things further by making inferences based on three or more samples. We'll use the very special F-distribution to do it (F stands for "fabulous").
Analyzing functions
You know a function when you see one, but are curious to start looking deeper at their properties. Some functions seem to be mirror images around the y-axis while others seems to be flipped mirror images while others are neither. How can we shift and reflect them? This tutorial addresses these questions by covering even and odd functions. It also covers how we can shift and reflect them. Enjoy!
Analyzing the solutions to systems of equations
Learn how to determine the number of solutions a system of equations has, without having to go through the entire solution process.
Ancestral Puebloan (formerly Anasazi)
Explore the incredible art and architecture of the ancient four corners region of the United States (where Colorado, New Mexico, Arizona, and Utah meet), including Mesa Verde.
Ancient Colombian chiefdoms
For centuries Europeans were dazzled by the legend of El Dorado, a lost city of gold in South America. El Dorado–literally “the golden one”–actually refers to the ritual that took place at Lake Guatavita, near modern Bogotá. The newly elected leader, covered in powdered gold, dived into the lake and emerged as the new chief of the Muisca people who lived in the central highlands of present-day Colombia's Eastern Range.
Ancient Egypt
Towards the end of the fourth millennium B.C.E. several independent city-states were unified to form a single state, marking the beginning of over 3,000 years of pharaonic civilization in the Nile Valley. Fertile earth left behind after the yearly Nile flood provided the basis for Egypt’s agricultural prosperity, a key factor in the longevity of the civilization.
Ancient Egypt
The art of dynastic Egypt embodies a sense of permanence. It was created for eternity in the service of a culture that focused on preserving a cycle of rebirth. By permission, © 2013 The College Board
Ancient Etruria
Ancient Greece
The British Museum collection includes objects from across the entire Greek world, ranging in date from the beginning of pre-history to early Christianity in the Byzantine era.
Ancient Greek vases
Mythic stories painted on an array of shapes and sizes—discover the artfully crafted Athenian vases of ancient Greece.
Ancient Near East
Between 6000 and 1550 B.C.E., Mesopotamia, the land between the Tigris and Euphrates rivers (now Iraq, north east Syria and part of south east Turkey) witnessed crucial advancements in the development of human civilization during the evolution from small agricultural settlements to large citie
Ancient Rome
Ancient Rome (quiz)
Ancient and Medieval
This tutorial includes the Ancient Near East, and Ancient Greece and Rome.
Ancient glassmaking
Created and refined by ancient Greeks, Romans, and Etruscans, these ancient glass objects were not only beautiful, they served many purposes.
Angle addition formula proofs
Let's see if we can prove the angle addition formulas for sine and cosine!
Angle addition formulas
We'll now see that we can express the sin(a+b) and the cos(a+b) in terms of sin a, sin b, cos a, and cos b. This will be handy in a whole set of applications.
Angle basics
What is an angle and how do we label, measure and construct them? Common Core Standards: 4.MD.C.5, 4.MD.C.5a, 4.MD.C.5b, 4.MD.C.6, 4.MD.C.7, 4.G.A.1
Angle basics and measurement
This tutorial will define what an angle is and help us think about how to measure them. If you're new to angles, this is a great place to start.
Angle bisectors
This tutorial experiments with lines that divide the angles of a triangle in two (angle bisectors). As we'll prove, all three angle bisectors actually intersect at one point called the incenter (amazing!). We'll also prove that this incenter is equidistant from the sides of the triangle (even more amazing!). This allows us to create a circle centered at the incenter that is tangent to the sides of the triangle (not surprisingly called the "incircle").
Learn three types of angle pairs: vertical angles, complementary angles, and supplementary angles.
Angles and polygons
Angles between intersecting and parallel lines
Welcome. I'd like to introduce you to Mr. Angle. Nice to meet you. So nice to meet you. This tutorial introduces us to angles. It includes how we measure them, how angles relate to each other and properties of angles created from various types of intersecting lines. Mr. Angle is actually far more fun than you might initially presume him to be.
Angles between intersecting lines
Welcome. I'd like to introduce you to Mr. Angle. Nice to meet you. So nice to meet you. This tutorial introduces us to angles. It includes how we measure them, how angles relate to each other and properties of angles created from various types of intersecting lines. Mr. Angle is actually far more fun than you might initially presume him to be.
Angular Movement
Animating parabolas
Discover how we create parabolic arcs by weaving together straight lines. This lesson touches on midpoints (grade 5) and takes 45-60min. Plus there's a hands-on activity at the end!
Animation basics
Learn how to animate your drawings.
Ant bot
Build an ant colony with simple behaviours
Antibiotics and antibiotic resistance
Antonín Dvořák. Symphony No. 9 "From the New World"
Antwerp and Bruges
Aortic dissection and aneurysm
When a patient with hypertension and chest pain radiating to the back presents to the hospital, one should suspect aortic dissection, which is a tear of the inner lining (tunica intima) of the aorta, with dissection of blood through its middle layer (tunica media). And when an elderly male presents with a pulsatile abdominal mass, one should suspect a rupturing abdominal aortic aneurysm. An aneurysm is simply a balloon-like dilation of a vessel wall. You will come to an understanding of the pathophysiology of these life-threatening diseases in addition to their surgical and medical management.
Applying differentiation in different fields
The idea of a derivative being the instantaneous rate of change is useful when studying or thinking about phenomena in a whole range of fields. In this tutorial, we begin to just scratch the surface as we apply derivatives in fields as disperse as biology and economics.
Approximating irrational numbers
The values of irrational numbers can't be written perfectly as decimals or fractions. However, we can approximate them, which is usually good enough. In this tutorial, we learn how to approximate and compare irrational numbers.
Arabia and Israel in the 20th Century
The Middle East is a center of cultures, religions, and, unfortunately, conflict in our modern world. This tutorial takes us from a declining Ottoman Empire to the modern Middle East which is still the center of many religions, cultures and conflicts.
Arc length
We'll now use integration to find the arc length of a curve. As we'll see, it is based on the same idea of summing up an infinite number of infinitely small line segments.
Arc length of polar graphs
You may already be familiar with finding arc length of graphs that are defined in terms of rectangular coordinates. We'll now extend our knowledge of arc length to include polar graphs!
Architecture and design
What about architecture? F.L. Wright’s Guggenheim and Mies’ Seagram Building each drew on classical precedents to create beauty in the modern city.
Area is how we thinking about how much space something takes in two dimensions such as comparing how much land one property takes up versus another. We'll take a conceptual look at how area is actually measured.
Area and circumference challenge problems
Problems that stretch your understanding of area and circumference of circles
Area and circumference of circles
A circle is at the foundation of geometry, and at the foundation of a circle is a special number called Pi. This number allows us to relate the radius, diameter, and circumference of a circle.
Area and perimeter
Rectangles are common shapes that you will often find the need to compare. Who has a bigger poster size? Who's yard has more grass to mow? We'll learn about length, width, area, and perimeter--about all you need to know to begin comparing!
Area and perimeter of polygons
Let's now extend our understanding of area to triangles and more interesting quadrilaterals! Common Core Standard: 6.G.A.1
Area basics
Area is how we thinking about how much space something takes in two dimensions such as comparing how much land one property takes up versus another. In this tutorial, we'll take a conceptual look at how area is actually measured (especially for rectangles).
Area between curves
Area defined by polar graphs
We'll now use the power of the integral to find areas defined by polar graphs!
Area models and multiplication
Most of us learn to multiply eventually, but only a select-few actually understand what the multiplication represents. This tutorial, with the help of area models, will allow you to be part of this elite group.
Area models to visualize multiplication
Most of us learn to multiply multi-digit numbers eventually, but only a select-few actually understand what the multiplication represents. This tutorial, with the help of grids and area models, will allow you to be part of this elite group. Common Core Standard: 4.NBT.B.5
Area of inscribed triangle
This more advanced (and very optional) tutorial is fun to look at for enrichment. It builds to figuring out the formula for the area of a triangle inscribed in a circle!
Area of shapes on grids
Area problems involving triangles and quadrilaterals on grids
Area of trapezoids and composite figures
Finding the area of more complex shapes like trapezoids and composite figures (shapes built from other shapes)
Area under a rate function as net change
Differential calculus was all about rates (that is, after all, what a derivative is). As we'll see, integral calculus is all about the idea of summing or "integrating" an infinitely many infinitely small small things to get a finite value (often the area under a curve). Despite not really having any calculus in it, this tutorial foreshadows the connection between rates and areas under curves. As we'll see, this is the foundation of the fundamental connections in all of calculus!
Areas of parallelograms
Understand the formula for the area of a parallelogram.
Areas of triangles
Understand the formula for the area of a triangle.
Arithmetic properties
2 + 3 = 3 + 2, 6 x 4 = 4 x 6. Adding zero to a number does not change the number. Likewise, multiplying a number by 1 does not change it. You may already know these things from working through other tutorials, but some people (not us) like to give these properties names that sound far more complicated than the property themselves. This tutorial (which we're not a fan of), is here just in case you're asked to identify the "Commutative Law of Multiplication". We believe the important thing isn't the fancy label, but the underlying idea (which isn't that fancy).
Arithmetic sequences
Learn how to find the explicit or recursive formula of an arithmetic sequence.
Arithmetic warmups
Arithmetic warmups
Aromatic stability
In this tutorial, Sal and Jay explain the concept of aromatic stabilization and show how to determine if a compound or an ion exhibits aromaticity. Knowledge of MO theory is assumed.
Store multiple values in your variables with arrays!
Arrhenius equation and reaction mechanisms
In this tutorial, we will learn about reaction mechanisms and how temperature and the activation energy affect the rate of a reaction.
Art 1010
Art 1010 is a short series of fun (and funny) animations created for the Utah System of Higher Education that introduce the history of Western art.
Art History: Cubism
The Spaniard Picasso changed the way we see the world. He could draw with academic perfection at a very young age but he gave it up in order to create a language of representation suited to the modern world. Together with the French artist George Braque, Picasso undertook an analysis of form and vision that would inspire radical new visual forms across Europe and in America. This tutorial explains the underlying principles of Cubism and the abstract experiments that followed including Italian Futurism, Russian Suprematism, and the Dutch movement, de Stijl.
Art History: Flanders
This tutorial focuses on the art of Peter Paul Rubens, whose work was in high demand by nearly every King, Queen and aristocrat in Catholic Europe (good thing he had a huge workshop!). Rubens was a master of color, dramatic compositions, and movement. Although he was from Northern Europe, he traveled to Italy and absorbed the art of the Renaissance, of classical antiquity, and of Caravaggio. He painted nearly every type of subject—landscapes, portraits, mythology, and history paintings.
Art History: France
In France, the LeNain Brothers painted scenes of every-day life (genre paintings), often depicting peasants. There was a renewal of interest in their art in the mid-Nineteenth Century, when the art critic, Champfluery wrote that the brothers “considered men in tatters more interesting than courtiers in embroidered garments.” At the same time, Poussin created a very different style—one that was highly intellectual and looked back to Renaissance, and ancient Greek and Roman art.
Art History: Holland
In the Protestant Dutch Republic of the 17th century there was an enormous demand for art from a wide cross-section of the public. This was a very good thing, since the institution that had been the main patron for art—the Church—was no longer in the business of commissioning art due to the Protestant Reformation. Dutch artists sought out new subjects of interest to their new clientele, scenes of everyday life (genre paintings), landscapes and still-lifes. There was also an enormous market for portraits. One of the greatest artist of this period, Rembrandt, made his name as a portrait painter, but was also a printmaker, and his work also includes moving interpretations of biblical subjects (though from a Protestant perspective).
Art History: Impressionism
Impressionism is both a style, and the name of a group of artists who did something radical—in 1874 they banded together and held their own independent exhibition. These artists described, in fleeting sensations of light, the new leisure pastimes of the city and its suburbs It’s hard to imagine, but at this time in France, the only place of consequence that artists could exhibit their work was the official government-sanctioned exhibitions (called salons), held just once a year, and controlled by a conservative jury. The Impressionists painted modern Paris and landscapes with a loose open brushstrokes, bright colors, and unconventional compositions—none of which was appreciated by the salon jury!
Art History: Late Victorian
British art saw a return to the classical after the 1860s, not just in terms of style, but also subject matter. Alma Tadema created sensual Victorian visions of the ancient Greeks and Romans, and Leighton too rendered classicizing figures and subjects. Both of these artists, together with Sargent, were influenced by the Aesthetic Movement, where the subject or narrative of a work of art was minimized in favor of a focus on issues of form (color harmonies, line, composition).
Art History: Neo-Classicism
Jacques Louis David invented a style reflecting Enlightenment ideas by looking back to ancient art. He became a revolutionary and 1st painter to Napoleon.
Art History: Neue Sachlichkeit
Germany was defeated and exhausted in 1918 at the end of WWI. The equally exhausted victors imposed harsh terms on Germany. It was forced to forfeit its overseas colonial possessions, to cede land to its neighbors, and to pay reparations. As demobilized troops returned, German cities filled with unemployed, often maimed veterans. The Socialists briefly seized power and by the early 1920s hyperinflation further destabilized the nation. Neue Sachlichkeit or the New Objectivity cast a cold sharp eye on Modern Germany’s hypocrisy, aggression, and destitution even as extremists on the political right consolidated power. The National Socialists or Nazi Party won the chancellorship in 1933 and quickly used art and architecture as a means build the myth of a pure German people shaped by the land and unsullied by modern industrial culture. This tutorial looks at the ways that competing political ideologies each used art for its own purposes.
Art History: Post-Impressionism
The work of van Gogh, Gauguin, Cézanne and Seurat together constitute Post-Impressionism and yet their work is so varied and unrelated, we might never otherwise think of these four artists as a group. Certainly van Gogh and Gauguin were friends and they briefly painted together, but each of these artists was concerned with solving particular issues that had to do with their own individual sensibility. Ironically, if anything ties these artists together it is this focus on subjectivity. This tutorial explores the sketchy multiperspectival views of Cézanne, Seurat’s systematized critiques of upper middle-class Paris, Gauguin’s fascination with the primitive and exotic, and van Gogh’s unerring ability to convey deeply human experiences.
Art History: Realism
In the mid-Nineteenth Century, great art was still defined as art that took it’s subjects from religion, history or mythology and its style from ancient Greece and Rome. Hardly what we would consider modern and appropriate for an industrial, commercial, urban culture! Courbet agreed, and so did his friend, the writer Charles Baudelaire who called for an art that would depict, as he called it, the beauty of modern life. Courbet painted the reality of life in the countryside—not the idealized peasants that were the usual fare at the exhibits in Paris. The revolution of 1848, in which both the working class and the middle class played a significant role, set the stage for Realism. Later, Manet and then Degas painted modern life in Paris, a city which was undergoing rapid modernization in the period after 1855 (the Second Empire).
Art History: Rococo
Overgrown gardens heavy with the scent of roses, palaces and luxury; the artists Watteau, Boucher, and Fragonard typify the aristocratic style known as Rococo.
Art History: Second Empire
Despite the brief dismantling of the Royal Academy during the French Revolution, art remained an extension of the power of the French State which regularly purchased art that it favored (often art that supported its political objectives). Through the Royal Academy (originally been founded by Louis XIV), the state extended its reach to the official exhibitions (salons) and to matters of style and subject matter through the École des Beaux Arts (School of Fine Arts). These were not just the official institutions of art, they were, in essence, the only institutions available for living artists to train and to make their work known. This tutorial looks at a crucial moment for painting, on the eve of the Revolution of 1848. We also examine one of the great State commissions of the Second Empire, the Opera House.
Art History: Spain
The main focus of this tutorial, and a leading artist at this time is the great Diego Velazquez, who spent most of his career as the court painter to the King of Spain painting official portraits. But in the hands of Velazquez, even mundane portraits became masterpieces of brushwork and color. His early work was influenced by the realism of Caravaggio. Get up close to the princess in his later masterpiece, Las Meninas (The Maids of Honor), and you’ll see broad brushstrokes of red, pink, black and white, but step back and they magically resolve to create a perfect illusion of the silk of her dress and the light moving across her face and hair. No other artist, except perhaps Titian and Rubens, revealed so honestly the alchemy of painting—how paint can be turned into reality.
Art History: Symbolism
By the 1880s artists such as Klimt, Khnopff, and Stuck turned from the academies and focused on the interior self by exploring dreams, myth and death.
Art and Conflict
Raging battlefields, victorious figures on horseback—and a wheelchair with knives for handles. This is the kind of art that has emerged from and in response to conflict. Read a short history of conflict in art, from painters who glorified scenes of war to photographers who shared images of its aftermath with the world, from performers tackling national disputes in the gallery to artists using irony and contradiction in their work to speak to bigger issues.
Art and Memory
How do artists engage with memory? Some believe the task of art is to catalogue and preserve images for the future, giving memory a visible form. Then there are artists who create objects meant to convey a personal memory or evoke shared cultural memories. Still others play with images and ideas from the past, reframing them in new ways to encourage us to think about the reliability of historical memory as well as our own. Can we ever trust memory to be objective, or even true? Can art help us explore the fickle nature of what we remember, and how?
Art and appropriation
Art isn't made in a vacuum. But where do you draw the line between inspiration and appropriation? Can a very close copy be a work of art in its own right? These artists (and forgers) share work that takes inspiration to its limit.
Art and the Great War
Here are several examples that show artists interpreting the rise of the machine immediately before and during WWI.
Art conservation
Go behind-the-scenes and learn about art conservation.
Art in Afghanistan
Archaeologists discovered treasures from Afghanistan's nomadic ancestors but then had to hide them to keep them safe. Learn more about this remarkable story of the survival and discovery of treasures from this ancient land.
Art of Mexico in the 19th century
The landscapes of José María Velasco and the first art school in the Americas, the Real Academia de San Carlos
Art of the Islamic World
Art terms in action
Watch these video demonstrations of key art terms relating to the Abstract Expressionists.
Arterial Stiffness
Believe it or not, the arteries are elastic and when they recoil they actually push blood along when the heart is relaxing (diastole). This is known as the windkessel effect and is the same basic principle used by some water guns. Unfortunately, with all the work that the circulatory system has to do, our arteries can become rigid with age. When the arteries get stiff like lead pipes, the problem is quite different then when the arteries actually get clogged up, but just as important.
Arthritis and rheumatoid arthritis
Artist interviews
Hear from contemporary artists as they share their art-making techniques and sources of inspiration.
Arts of the Islamic world (quiz)
Test your knowledge of the arts of the Islamic world!
Arts of the Islamic world: late period
Later Period (c. 1517 –1924 C.E.) The Ottomans, a small Turkic state in Anatolia, emerged as a major military and political force and conquered Constantinople, the Balkans, the Near East, and North Africa.
Ai WeiWei’s "Sunflower Seeds" takes us on a journey into the life of a Chinese village. Meanwhile, modern and contemporary artists from India and Pakistan, like Zarina Hashmi and Amrita Sher-Gil, explore the notions of home, immigration, and bridging cultures through their art.
Asia, Africa, Europe, Latin America and the Middle East
Meanwhile, European artists were also busy exploring issues of identity and the body. Gerhard Richter for example rejected the way artists branded by a particular style and creates a spectrum of work from the hyper realist to the purely abstract. In Britain, Bacon, Freud and Ofili find new power in representations of the human body far removed for the classical tradition.
Ask a Physicist
Hi everyone, I'm David and I love physics. Have a physics question/comment? I'll be checking the comments section every couple hours until Sunday August 2nd at 12:00 PM PST and replying as often as possible. Also, please feel free to answer other people's questions if you know the answer.
Ask a biologist!
Ask a chemist!
The Assyrian empire dominated Mesopotamia and all of the Near East for the first half of the first millennium, led by a series of highly ambitious and aggressive warrior kings. The culture of the Assyrians was brutal, the army seldom marching on the battlefield but rather terrorizing opponents into submission who, once conquered, were tortured, raped, beheaded, and flayed with their corpses publicly displayed. The Assyrians torched enemies' houses, salted their fields, and cut down their orchards.
Asymptotic Notation
Learn how to use asymptotic analysis to describe the efficiency of an algorithm, and how to use asymptotic notation (Big O, Big-Theta, and Big-Omega) to more precisely describe the efficiency.
Atomic nucleus
4E: An introduction to the nucleus including isotopes, mass defects, and radioactive decay.
6B: Have you ever tried to multi-task while frantically aiming to finish a large project? The phone goes off, you get a text message, all while you try to finish an essay or read a book. In an increasingly busy world, our attention is frequently distracted by so many inputs. Here you will explore the concept of selective and divided attention, as well as the role of language in cognition and development in our lives.
Attention and language
Have you ever tried to multi-task? Explore the concept of selective and divided attention, as well as the role of language in cognition and development.
Attributing behavior to persons or situations
Augusta Read Thomas: Of Paradise and Light
A leading composer of her generation, Augusta Read Thomas' works have been performed by major orchestras around the world. Here she discusses the compositional process that led to Of Paradise and Light. Music Director Gerard Schwarz joins in the discussion of this transcendent new work.
Average costs (ATC, MC) and marginal revenue (MR)
In this tutorial, Sal uses the example of an orange juice business to help us understand the ideas of average total cost (ATC), marginal cost (MC) and marginal revenue (MR). We then use this understanding to answer the age-old question, "how much orange juice should I produce?" Finally, we use these ideas to construct a long-run supply curve. A must watch if you're interested in making juice!
Average fixed, variable and marginal costs
Using a spreadsheet, Sal walks through an example of average costs per line of code as a firm hires more engineers. Really good primer to understand what average fixed costs, average variable costs, average total costs (ATC) and average marginal costs (MC) are (and how they are calculated).
Average rate of change
Learn what's the average rate of change of a function and how to find it over given intervals.
Average rate of change word problems
Solve word problems that concern the average rate of change of a functional relationship.
Average value of a function
We don't need calculus to figure out the average value of a linear function over an interval, but what about non-linear functions? Luckily, integral calculus comes to the rescue here. In this tutorial, we'll understand what "average value" of a function over an interval means. We'll also connect that notion to the Mean Value Theorem we first learned in differential calculus.
Aztec (Mexica)
The Aztec (or Mexica as they called themselves) were a small and obscure tribe when they settled in the Valley of Mexico and founded their capital, Tenochtitlan, in 1345. At the beginning of the sixteenth century Tenochtitlan (now Mexico City) was one of the largest cities in the world and the Mexica empire stretched from the Atlantic to the Pacific and into Guatemala and Nicaragua. Hernán Cortés and his small Spanish army arrived in 1519 and overthrew the Mexica ruler Moctezuma Xocoyotzin.
For two thousand years the myth of Babylon has haunted the European imagination. The Tower of Babel and the Hanging Gardens, Belshazzar’s Feast and the Fall of Babylon have inspired artists, writers, poets, philosophers and film makers. Learn here about the source of these myths, Babylonia itself.
Balanced and unbalanced forces
You will often hear physics professors be careful to say "net force" or "unbalanced force" rather than just "force". Why? This tutorial explains why and might give you more intuition about Newton's laws in the process.
Balancing chemical equations
We are now going to look at chemical reactions. But as we do, we need to make sure that atoms aren't magically appearing or disappearing. Put another way, we need to sure that we have the same number of each constituent atom in the product of the reaction as we do in the reactants (the molecules that react)!
Banking and Money
We all use money and most of us use banks. Despite this, the actual working of the banking system is a bit of a mystery to most (especially fractional reserve banking). This older tutorial (bad handwriting and resolution) starts from a basic society looking to do more than barter and incrementally builds to a modern society with fraction reserve banking. Through this process, you will hopefully gain a deep understanding of how money and banking works in our modern world.
Bar graphs
Learn how to read and make bar graphs.
Basic matrix operations
Keanu Reeves' virtual world in the The Matrix (I guess we can call all three movies "The Matrices") have more in common with this tutorial than you might suspect. Matrices are ways of organizing numbers. They are used extensively in computer graphics, simulations and information processing in general. The super-intelligent artificial intelligences that created The Matrix for Keanu must have used many matrices in the process. This tutorial introduces you to what a matrix is and how we define some basic operations on them.
Basic multiplication
If 3 kids each have two robot possums, how many total robot possums do we have? You liked addition, but now you're ready to go to the next level. Depending on how you view it, multiplication is about repeated addition or scaling a number or seeing what number you get when you have another number multiple times. If that last sentence made little sense, you might enjoy this tutorial.
Basic probability
Flip a quarter a hundred times. What's the probability that it will turn up heads? Tails? Even if we are unsure about whether something will happen, can we start to be mathematical about the "chances" of an event (essentially realizing that some things are more likely than others) occurring. This tutorial will introduce us to the tools that allow us to think about random events and the logic behind comparing, judging, and finding the probabilities of those events.
Basic scene construction
In this lesson you'll be given a scene from the Art Department to build. We'll explore how the operations of translation, rotation and scaling are used to position objects. It touches on concepts from Grades 7/8 and takes 50-60 minutes to complete.
Basic set operations
Whether you are learning computer science, logic, or probability (or a bunch of other things), it can be very, very useful to have this "set" of skills. From what a set is to how we can operate on them, this tutorial will have you familiar with the basics of sets!
Basic shapes
Learn the names of basic shapes
Basic trigonometric ratios
In this tutorial, you will learn all the trigonometry that you are likely to remember in ten years (assuming you are a lazy non-curious, non-lifelong learner). But even in that non-ideal world where you forgot everything else, you'll be able to do more than you might expect with the concentrated knowledge you are about to get.
Basic observations leading to homemade batteries
Becoming a better programmer
Now that you understand the basics of programming, learn techniques that will help you be more productive and write more beautiful code.
Beginner’s guide to Japanese Art
Discover the ancient temples of Nara, the process for making lacquer, the formats of Japanese painting, and the conservation of the Gan Ku Tiger Scroll in the British Museum.
Behavior and genetics
7A: Nature vs. nurture - it’s a dilemma scientists have aimed to answer for years. Do our surroundings or genetics have a greater impact on the individuals we eventually become? You will learn about the way our genes and experiences shape the ways we respond to our environment as we discuss experiments such as twin and adoption studies in the context of development.
Behind the scenes at MoMA
Come behind the scenes and watch the staff and artists at work.
Ben Milne - CEO of Dwolla
Ben Milne, CEO of Dwolla, discusses his motivation in founding his company and the excitement of starting something new. Ben advocates for the idea that failure, which can happen in big and small ways, does not have to be your legacy.
Until the late 19th century, one of the major powers in West Africa was the kingdom of Benin in what is now southwest Nigeria.
Benjamin Franklin
Franklin was a great inventor and statesman and had an invaluable role in the founding of America.
Bernard Rands: Adieu
Pulitzer-prize-winning composer and Harvard Professor of Music Bernard Rands discusses the art of composition. He and Music Director Gerard Schwarz look into the score of Rands' recent work "Adieu" ("Goodbye") as recorded by the All-Star Orchestra.
Bernoulli distributions and margin of error
Ever wondered what pollsters are talking about when they said that there is a 3% "margin of error" in their results. Well, this tutorial will not only explain what it means, but give you the tools and understanding to be a pollster yourself!
Best practices (K-12 math)
Best practices (out-of-school-time programs)
Beth Schmidt - Founder of
Beth Schmidt was a Teach for America Corps Member teaching 10th Grade English at Locke High School in South Central, Los Angeles, when she asked her class to write about their passion and an after-school and summer program they wanted to attend. Their essays prompted her to find a way to make these life-changing opportunities available to her students. She started Wishbone to send students to programs that they otherwise wouldn’t be able to afford.
Between perfect competition and monopoly
Most markets sit somewhere in-between perfect competition and monopolies. This tutorial explores some of those scenarios--from monopolistic competition to oligopolies and duopolies.
Bezier Curves
This lesson begins with linear interpolation and builds to Bezier curves using de Castiljau's algorithm. It's great to try after you've finished the Environment Modeling topic. It's appropriate for grade 8 through high school and takes 30-40 minutes.
Big bang and expansion of the universe
What does it mean for the universe to expand? Was the "big bang" an explosion of some sort or a rapid expansion of space-time (it was the latter)? If the universe was/is expanding, what is "outside" it? How do we know how far/old things are? This tutorial addresses some of the oldest questions known to man.
Binary search
Learn about binary search, a way to efficiently search an array of items by halving the search space each time.
Binomial distribution
Binomial theorem
You can keep taking the powers of a binomial by hand, but, as we'll see in this tutorial, there is a much more elegant way to do it using the binomial theorem and/or Pascal's Triangle.
Biodiversity Hotspots
Areas that have a high diversity of unique and threatened species are known as biodiversity hotspots.
Biodiversity analyses and unknowns
Collecting specimens in the field launches the lab work of analysis, documentation, and cataloging. Despite the ongoing work of scientists, only about 10% of Earth’s biodiversity has been documented.
Biodiversity and ecosystem function
A wealth and variety of species, or species richness, promote strong ecological networks and functions, making ecosystems more resilient to major disturbances and collapse.
Biodiversity and ecosystem services
Healthy ecosystems provide crucial direct, indirect, and aesthetic-ethical benefits to humans.
Biodiversity and the tree of life
Understanding the place of every species in the tree of life is crucial in our race to save biodiversity. See how the history of a species can be traced through evolutionary trees.
Biodiversity champions
Advocates for biodiversity conservation come in all forms, from single individuals to international agencies.
Biodiversity distribution patterns
Life is abundant on Earth, but is distributed unevenly, with species richness and population sizes greater in some areas than others. The physical environment, other organisms, evolutionary factors, and human actions all influence where species live.
Biodiversity fieldwork
For centuries, expeditions to discover biodiversity have taken scientists to the far corners of the planet. Today’s expeditions are multidisciplinary, incorporate new technologies and involve new ethics.
Biodiversity patterns of speciation and extinction
Over time, species richness has had significant ups and downs. Speciation increases the number of species. Extinction decreases richness.
Biological basis of behavior: Endocrine system
Consider any behavior, sleeping for instance, and think about all of the organs that have to work together to have it go smoothly. The heart and lungs need to slow down, the brain needs to stop taking in the cues from the environment, and the bladder needs to wait until morning to empty. This coordinated effort is achieved by a number of unique hormones acting on different tissues. Learn more about how this process works and why it’s so critical to our everyday lives!
Biological basis of behavior: The nervous system
3A: The very fact that you are able to understand this sentence means that neurons in your brain (85 billion in total) are talking to each other. Neurons are the living substance of the nervous system, which extends beyond the brain to the spinal cord and peripherally, allows you to think and process, make decisions, stand up straight, maintain your heart rate, rest and digest. You will come to appreciate the structure and function of the nervous system as we delve into its anatomy and physiology, from the gray and white matter to the cerebellum to the neurons.
Biological explanations of social behavior in animals
Nature vs. nurture: are animal behaviors taught or genetically passed down from generation to generation? Here we explore how social behaviors, like genuinely caring about the well being of those in your community, are more “nature” than “nurture.”
Biological sciences practice questions
These questions are available under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License (available at
3A: The human body is composed of about 100 trillion cells (this is not counting your bacterial buddies, who actually outnumber your cells 10 to 1!) Your cells must speak to each other to coordinate this massive symphony of life. In this tutorial, you will learn about the molecular basis of cellular signaling that makes this vast network speedy and efficient.
Bit-zee Bot
This project is a low cost robot made from every day items that are taken apart and described in the reverse engineering section. This project was created by Karl R. C. Wendt.
Learn about bitcoins and how they work. Videos by Zulfikar Ramzan. Zulfikar is a world-leading expert in computer security and cryptography and is currently the Chief Scientist at Sourcefire. He received his Ph.D. in computer science from MIT.
Black-Scholes Formula
Options have been bought and sold for ages, but finding a rational way to price them seemed beyond our mathematical know-how... until 1973 when Fischer Black and Myron Scholes showed up and gave us the Black-Scholes model. This work was later extended by Robert Merton and now underpins much of modern finance.
Bleeding and impaired hemostasis
Blended learning facilities and furniture
Blended learning hardware and infrastructure
Blended learning software
Blood Pressure
Using the stethoscope to check blood pressure is a technique that’s been used for >100 years! Blood pressure is one of the major vital signs frequently measured by health care workers, and it tells us a lot about our blood circulation. Learn what blood pressure is, how it relates to resistance in a tube, why it is necessary to get oxygen to your cells, and how it can change as you age. We’ll finally put it all together by relating pressure, flow, and resistance in one awesome equation!
Blood Pressure Control
The human body enjoys stability. For example, if your blood pressure changes, the body puts a couple of brilliant systems into motion in order to respond and bring your blood pressure back to normal. There are some quick responses using nerves and some slower responses using hormones. The system using hormones is sometimes called the renin-angiotensin-aldosterone-system (RAAS), which is the main system in the body for controlling blood pressure. When your blood pressure drops too low or gets too high, your kidneys, liver, and pituitary gland (part of your brain) talk to each other to solve the problem. They do this without you even noticing! Learn how the body knows when the blood pressure has changed, and how hormones like angiotensin 2, aldosterone, and ADH help return blood pressure to back to normal.
Blood Vessel Diseases
The ancient Greeks thought blood vessels actually carried air throughout the body. Although we know better today, many people are still often confused with the specifics! We now know that the vessels carry blood instead, and we are able to distinguish between two different types: arteries and veins. Learn about how arteries differ from veins and how vessels can get damaged over time.
Blood Vessels
Where does your blood go after it leaves the heart? Your body has a fantastic pipeline system that moves your blood around to drop off oxygen and food to those hungry cells, and removes cell waste. Learn how arteries carry blood away from the heart, how veins bring blood back to the heart, and about the different layers of cells that make up these blood vessels.
Blood pressure control
The human body enjoys stability. For example, if your blood pressure changes, the body puts a couple of brilliant systems into motion in order to respond and bring your blood pressure back to normal. There are some quick responses using nerves and some slower responses using hormones. The system using hormones is sometimes called the renin-angiotensin-aldosterone-system (RAAS), which is the main system in the body for controlling blood pressure. When your blood pressure drops too low or gets too high, your kidneys, liver, and pituitary gland (part of your brain) talk to each other to solve the problem. They do this without you even noticing! Learn how the body knows when the blood pressure has changed, and how hormones like angiotensin 2, aldosterone, and ADH help return blood pressure to back to normal.
Blood vessels
Where does your blood go after it leaves the heart? Your body has a fantastic pipeline system that moves your blood around to drop off oxygen and food to those hungry cells, and removes cell waste. Learn how arteries carry blood away from the heart, how veins bring blood back to the heart, and about the different layers of cells that make up these blood vessels.
Body and mind
Bohr's model of the hydrogen atom
Atomic theory is not Bohr-ing! While it doesn't work for atoms with more than one electron, the Bohr model successfully predicts the emission spectrum of hydrogen.
Bond-line structures
Call them Bonds. Covalent Bonds. Smart chemists need time to stir (and shake) their solutions. In this tutorial, we will explain how chemists use bond-line structures as a form of organic shorthand to skip time-consuming carbon and hydrogen atoms labeling. Watch this tutorial so you too can be in the Dr. Know.
Both corporations and governments can borrow money by selling bonds. This tutorial explains how this works and how bond prices relate to interest rates. In general, understanding this not only helps you with your own investing, but gives you a lens on the entire global economy.
Books, music, and literature
Box and whisker plots
Whether you're looking at scientific data or stock price charts, box-and-whisker plots can illuminate patterns in your life. This tutorial covers what they are, how to read them, and how to construct them.
Box-and-whisker plots
Whether you're looking at scientific data or stock price charts, box-and-whisker plots can show up in your life. This tutorial covers what they are, how to read them and how to construct them. We'd consider this tutorial very optional, but it is a good application of dealing with medians and ranges.
Brain teasers
Random logic puzzles and brain teasers. Fun to do and useful for many job interviews!
Each of the instruments in this family are made to sound by the vibrations of the player's lips combined with a steady stream of breath. Learn from the All-Star Orchestra players themselves about the special attributes of the trumpet, French horn, trombone, bass trombone and tuba.
Breadth-first search
Learn how to traverse a graph using breadth-first-search to find a particular node or to make sure you've visited all the notes, traversing one layer at a time.
Learn how a mother is able to nourish a baby through breastfeeding
Breathing Control
Luckily, we can breathe without thinking which means that we have autonomic control of breathing. If we couldn’t, we would risk dying if we went to sleep (look up Ondine’s curse)! There are times when the body wants more oxygen (like during heavy exercise), and when the body wants less (like when we’re resting). How does our body automatically seem to know when to inhale more, and when to inhale less? Also, if we do have autonomic control of breathing, how is it possible to also have conscious control of our breathing? These questions get to the fundamentals of breathing control.
Bringing it all together
This tutorial brings together all of the major ideas in this topic. First, it starts off with a light-weight review of the various ideas in the topic. It then goes into a heavy-weight proof of a truly, truly, truly amazing idea. It was amazing enough that orthocenters, circumcenters, and centroids exist , but we'll see in the videos on Euler lines that they sit on the same line themselves (incenters must be feeling lonely)!!!!!!!
Britain in the age of revolution
Buddhism has deeply influenced the character and evolution of Asian civilization over the past 2,500 years. It is based on the teachings of a historical figure, Siddhartha Gautama, who lived around the fifth century B.C.E. As it moved across Asia, Buddhism absorbed indigenous beliefs and incorporated a wide range of imagery, both local and foreign, into its art and religious practices. Buddhism continues to evolve as a religion in many parts of the world.
Buddhist art
Buddhism was established in the fifth century B.C.E. by the Buddha or "enlightened one." However it was not until the third century B.C.E. that Buddhism enjoyed royal patronage under the Mauryan kings—notably Ashoka—and spread to all parts of the subcontinent. Buddhism continued to flourish in subsequent centuries, reaching South East Asia in the fifth century C.E. and Tibet in the seventh.
Buffer solutions
Buffer solutions play an important role in many chemical and biological processes. In this tutorial we will learn how buffers resist changes in pH, and how to calculate the pH of a buffer solution.
Building crowds
Discover how combinations were used to design crowds of robots in WALL-E. This lesson introduces the counting principle using tree diagrams. It touches on concepts in grades 4 through 7 and takes 30-45 minutes.
Bulk transport
Macrophages (specialized immune cells) in your body can "eat" bacteria, pulling them into the cell and digesting them. Learn how they accomplish this feat, and how cells transport other large molecules across the plasma membrane.
Buoyant Force and Archimedes' Principle
Burnett Elementary
4th grade teacher Alison Elizondo shares her experience using Khan Academy at a Title I Public school (40% English language learners, 30% free/reduced lunch) in Milpitas, California.
Learn how to create clickable buttons for your programs that are easy to customize.
CAHSEE Example Problems
Sal working through the 53 problems from the practice test available at for the CAHSEE (California High School Exit Examination). Clearly useful if you're looking to take that exam. Probably still useful if you want to make sure you have a solid understanding of basic high school math.
CD spectroscope
Turn an old CD into a spectroscope to analyze light—you may be surprised by what you see. Try pointing your CD spectroscope at the fluorescent light in your room, sunlit clouds in the sky (but never the sun), even your friend’s colored shirt to reveal the wavelengths of light that mix together to create the color you see. It's your very own truth teller for light!
CSS Layout
Learn how to use the HTML span and div elements with CSS properties like position and float to change the layout of your webpages.
CSS text properties
Learn how to style your text, like font families, size, spacing, and alignment.
Calculus AB example questions
Many of you are planning on taking the Calculus AB advanced placement exam. These are example problems taken directly from previous years' exams. Even if you aren't taking the exam, these are very useful problem for making sure you understand your calculus (as always, best to pause the videos and try them yourself before Sal does).
Calculus BC sample questions
The Calculus BC AP exam is a super set of the AB exam. It covers everything in AB as well as some of the more advanced topics in integration, sequences and function approximation. This tutorial is great practice for anyone looking to test their calculus mettle!
Calculus readiness warmup
Calvin Carter - Founder of Bottle Rocket Apps
The day after Steve Jobs announced that Apple was opening its platform to third party developers Calvin Carter bought pencils and a pad of paper and began sketching applications. Today developers at Bottle Rocket Apps still use pencil and paper as they begin the process of producing custom, high-end mobil apps for some the world’s leading brands. Along the way they hold on to their passion and stay focused on their mission.
Cambodia and Myanmar
Cameroon is home to some 200 languages and numerous ethnic groups. It has a varied landscape of mountains, grasslands and coastline.
Campus visit
Once you've identified several target schools, a campus visit can be a great way to narrow down your options. Even if you can't visit target colleges that are located far away, taking time to explore your local college campus can give you a sense of what you want in a college.
4C: Capacitors are simply components which store electrostatic energy in a field. They are similar to batteries - however, capacitors only store new electricity rather than producing it through a chemical reaction like a battery does. You will walk through a mathematical description of how capacitors function and how they work within electrical circuits.
Capacity utilization and inflation
This tutorial starts with a very "micro" view of when firms decide to raise (or lower prices). It then jumps back to the macro view to discuss how capacity utilization can impact prices.
Carbohydrate Metabolism
1D: The glucose in the bread of the ham and cheese sandwich you just had for lunch goes on a productive journey within your cells after it is absorbed - the glucose in the bread is involved in several interlinked pathways. Your body has a decision to make - it can either break down the glucose for energy or store it for later. We will delve into the metabolic pathways of glucose - glycolysis, gluconeogenesis, and the pentose phosphate shunt.
Table sugar and corn syrup have the same chemical formula, so why do they taste different? The answer is in their stereochemistry. Learn how to assign chirality, interpret Fischer projections, and differentiate between different mono- and poly-saccharides in the following videos.
Carboxylic acid derivatives
5D: As the name would suggest, carboxylic acid derivatives are quite similar to carboxylic acids in their structure and function. If you’ve ever used soap to wash your hand, you have experienced firsthand (pun intended!) the lavatory effects of an ester, one of the many classes of carboxylic acid derivatives. In this tutorial, we will discuss the important nomenclature, properties, and reactions of carboxylic acids.
Carboxylic acids
5D: Have you ever used vinegar to make succulent barbecue wings? Then carboxylic acids are your friends. Vinegar, also known as acetic acid, is one of the simplest carboxylic acids. You will discover the important nomenclature, properties, and reactions of carboxylic acids in this tutorial.
Cardiac dysrhythmias and tachycardias
Your heart is an electrical organ, and it produces a rather rhythmic music - lub-dub, lub-dub. We are able to measure its music through the electrocardiogram (EKG), which is able to pick up pathological rhythms - supraventricular tachycardias, atrial fibrillation, and ventricular tachyardia for instance- through electrical leads placed on the chest. We will discover how to identify these dysrthymias as well as how these conditions are treated, such as through the use of anti-arrhythmic drugs and pacemakers.
Charlemagne, King of the Franks and later Holy Roman Emperor, instigated a cultural revival known as the Carolingian Renaissance that continues to impact the way European languages are written, the structure of modern law and the very notion of Europe itself.
Case Study: Earthquake Risk in Bangladesh
Only recently have scientists come to recognize that Bangladesh is positioned at the juncture of several active tectonic plate boundaries. Moreover, it sits atop the world’s largest river delta at close to sea level, facing both the risk posed by a quake and secondary risks of tsunamis and flooding in the quake’s aftermath. These teaching case materials show the fieldwork of an international team of scientists working to provide Bangladeshi leaders the tools they need to understand and minimize geologic risks. Using both seismic and sedimentary data, scientists are attempting to model the interactions between past earthquakes and river-course shifts – an invaluable aid in predicting future quakes and how they could impact this unique terrain.
Case studies (out-of-school-time programs)
Case studies: teaching in a blended learning environment
Cash versus accrual accounting
Just keeping track of cash that goes in and out of a business doesn't always reflect what's going on. This tutorial compares cash and accrual accounting. Very valuable if you ever plan on starting or investing in any type of business (you might also discover a nascent passion for accounting)!
Categorical data
Celiac disease
Cell membrane overview
2A: Not all shall pass! Yards have fences, and cells have membranes. You don’t want just anybody waltzing into your backyard. Similarly, a healthy cell doesn’t just let in any random molecule - this is the concept of selective permeability. Some molecules (e.g. steroids) more easily cross the barrier whereas others (e.g. charged ions) have a more difficult time getting inside the cell without a little help from transporters in the membrane.
Cell potentials
An introduction to calculating cell potentials at standard state.
Cell potentials under nonstandard conditions
Calculating the cell potential when operating at nonstandard conditions.
Cell theory
Let’s go on a journey that starts with Robert Hooke and Anton van Leeuwenhoek hunched over their microscopes looking at cork and bacteria, respectively. These videos will guide you through the events that led us to the knowledge we have today about cells.
Cell-cell interactions
2A: The human body is composed of about 100 trillion cells (this is not counting your bacterial buddies, who actually outnumber your cells 10 to 1!) Your cells must speak to each other to coordinate this massive symphony of life. In this tutorial, you will learn about the molecular basis of cellular signaling that makes this vast network speedy and efficient.
Cellular development
What is the difference between a heart cell and a liver cell? Not only do they look different, but they obviously have different roles in our bodies. Yet, both of these cells have the same 23 chromosomes in their nuclei, so shouldn’t they be similar? These videos explain why a heart cell becomes a heart cell and not a liver cell.
Cellular division
Central Italy
Beyond the city-states of Florence, Siena and Venice, artists like Piero della Francesca were busy creating among the most celebrated art of the Renaissance.
Central, inscribed and circumscribed angles
We'll now dig a bit deeper in our understanding of circles by looking at central, inscribed and circumscribed angles. This is fun and beautiful as is, but you'll also see that it shows up on a lot of math standardized tests. Why do people like to put geometry like this on standardized tests? Because it shows deductive reasoning skills which are super important in every walk of life!
Centripetal acceleration
Why do things move in circles? Seriously. Why does *anything* ever move in a circle (straight lines seem much more natural)? Is something moving in a circle at a constant speed accelerating? If so, in what direction? This tutorial will help you get your mind around this super-fun topic.
Cepheid variables
Stellar parallax can be used for "nearby" stars, but what if we want to measure further out? Well this tutorial will expose you to a class of stars that helps us do this. Cepheids are large, bright, variable stars that are visible in other galaxies. We know how bright they should be and can gauge how far they are by how bright they look to us.
Ceramics and glass
Chain rule
You can take the derivatives of f(x) and g(x), but what about f(g(x)) or g(f(x))? The chain rule gives us this ability. Because most complex and hairy functions can be thought of the composition of several simpler ones (ones that you can find derivatives of), you'll be able to take the derivative of almost any function after this tutorial. Just imagine.
Challenging complex number problem
This tutorial goes through a fancy problem from the IIT JEE exam in India (competitive exam for getting into their top engineering schools). Whether or not you live in India, this is a good example to test whether you are a complex number rock star.
Challenging existing assumptions
Challenging the State
With its capacity for commentary, art becomes the perfect tool for artists to create powerful visual and conceptual statements they might not be able make otherwise. Faced with hostile power structures at home or in exile, some artists rely upon art for a voice, while others see it as their duty to reveal the global power structures of industry and economics. Join these artists as they use their work to unearth, challenge, and speak out about issues around racism, nationalism, immigration, and oppression.
Change of basis
Finding a coordinate system boring. Even worse, does it make certain transformations difficult (especially transformations that you have to do over and over and over again)? Well, we have the tool for you: change your coordinate system to one that you like more. Sound strange? Watch this tutorial and it will be less so. Have fun!
Changing the PV Loop
Once you’ve learned about the PV loop, a natural question arises - Does it ever change shape? It turns out that there are precisely three things that can change the shape of the loop: 1. Preload, 2. Afterload, and 3. Contractility. That’s it! The tricky part comes when you try to change one and you realize that the body begins to change the other two as well as a natural consequence. In order to simplify, you’ll find that PV loops are sometimes even described as PV boxes. You’ll get to learn about PV loops, PV boxes, and even play around with them yourself in this tutorial!
Charge and electric force (Coulomb's law)
Chavín culture
The Chavín culture, named after the highland site of Chavín de Huántar, thrived from 900 B.C.E. to 200 B.C.E. The culture and site centered on a religious cult that had far-flung influence, with pilgrims coming from hundreds of kilometers away. Those same pilgrims often took Chavín-style artworks with them, extending the culture’s artistic influence throughout the Andean region.
Checking solutions of inequalities
Learn how to check possible solutions of two-variable linear inequalities. For example, is (2, 0) a solution to y>x+1? In this tutorial, we also check solutions to systems of inequalities.
Chemical bonds and reactions
Find out how atoms interact with each other to form bonds! We'll look at the different types of strong bonds, including nonpolar covalent, polar covalent, and ionic bonds, as well as weak bonds such as hydrogen bonds and London dispersion forces.
Chemistry Introduction
Chemistry and chemical reactions can be electric or explosive, sweet or sour, powerful or perplexing.
Chi-square probability distribution
You've gotten good at hypothesis testing when you can make assumptions about the underlying distributions. In this tutorial, we'll learn about a new distribution (the chi-square one) and how it can help you (yes, you) infer what an underlying distribution even is!
Ancient China includes the Neolithic period, defined as the age before the use of metal, when China witnessed a transition from a nomadic existence to one of settled farming. Imperial Chinese history is marked by the rise and fall of many dynasties and occasional periods of disunity, but overall the age was remarkably stable and marked by a sophisticated governing system that included the concept of a meritocracy.
Chinese currency and U.S. debt
This tutorial contains short videos that explain how China and the United States are intertwined through currency and debt. This is key for understanding the current global macro picture.
Are you right handed or sinister-handed? Have you ever thought that you might not be as attractive as you look in the mirror? Welcome to the world of chirality!
Chromosomal inheritance
Between 1856 and 1863, Gregor Mendel did a series of experiments with pea plants that established much of our fundamental understanding of heredity. In this section you will learn how traits are passed down from parents to their offspring and how genetic recombination can produce organisms with new gene combinations.
Chronic Bronchitis (COPD)
Chronic bronchitis (COPD)
Assess your understanding of the code breaking presented in the ancient cryptography lesson. This series of articles and exercises will prepare you for the upcoming challenge!
Circle arcs and sectors
This tutorial will review some of the basic of circles and then think about lengths of arcs and areas of sectors.
You've seen circles your entire life. You've even studied them a bit in math class. Now we go further, taking a deep look at the equations of circles.
Circuits with capacitors
Circulatory and pulmonary systems
As humans, we really like breathing oxygen. That's because the cells in our body will die if they don't get oxygen to function in a reasonable amount of time. This tutorials describes how we use the lungs to exchange gasses between our blood and the atmosphere and how the oxygen is then pumped through the body by way of blood and the circulatory system.
Circulatory system introduction
No organ quite symbolizes love like the heart. One reason may be that your heart helps you live, by moving ~5 liters (1.3 gallons) of blood through almost 100,000 kilometers (62,000 miles) of blood vessels every single minute! It has to do this all day, everyday, without ever taking a vacation! Now that is true love. Learn about how the heart works, how blood flows through the heart, where the blood goes after it leaves the heart, and what your heart is doing when it makes the sound “Lub Dub.”
Circumference and area of circles
Circles are everywhere. How can we measure how big they are? Well, we could think about the distance around the circle (circumference). Another option would be to think about how much space it takes up on our paper (area). Have fun!
Classic Veracruz culture
Learn about Classic Veracruz culture along what is now the gulf coast of Mexico
By around 500 B.C.E. ‘rule by the people,’ or democracy, had emerged in the city of Athens. Following the defeat of a Persian invasion in 480-479 B.C.E., mainland Greece and Athens in particular entered into a golden age. In drama and philosophy, literature, art and architecture Athens was second to none. The city’s empire stretched from the western Mediterranean to the Black Sea, creating enormous wealth. This paid for one of the biggest public building projects ever seen in Greece, which included the Parthenon.
Classifying quadrilaterals
Quadrilateral come in many shapes and sizes. We've got squares, rectangles, rhombi, parallelograms, kites, trapezoids, and more! Let's analyze the properties of these shapes and come up with a system for classifying them.
Classifying shapes
Some practice classifying geometric shapes.
Classifying triangles
Triangles can be classified in different ways. In this tutorial, we'll learn about scalene, isosceles, equilateral, acute, right, and obtuse triangles.
Claus Sluter and Claus de Werve
Covering some 70 percent of Earth's surface, clouds play a key role in our planet's well-being. But how do they form, why are there so many types, and what clues can they give us about the weather and climate to come?
Clovis culture
North America was one of the last continents in the world to be settled by humans. The first clear evidence of human activity in North America are spear heads called Clovis points.
Explore cognitive development and intelligence, as well as how our minds solve problems, make decisions, and represent knowledge.
Cohesion and adhesion
Water molecules like to stick together! Learn more about the cohesive (and adhesive) properties of water and why they're important for bugs, trees, and the tear ducts in your eyes!
Coin detector
Get to know your rotation sensor while building a 5 cent machine
With its synergetic and cooperative spirit, performance art can be a call for artists to work with one another. In this section, we'll look more closely at artists who work, think, and play together in order to make art that is greater than just the sum of its parts.
Collateralized debt obligations
College Application process
Start by addressing some of the basic questions of the college application process: when should you apply, how should you fill out the application forms, and when have you applied to enough schools?
College loans
College loans can be a scary proposition, because unlike scholarships and grants, they have to be paid back (with interest!) after you finish school. However, loans also represent an investment in your future and one that can absolutely be worth it if used in moderation. Learn what sort of loans are out there and how to prioritize between them.
College search: Other important choices
Now that you have a sense of the types of colleges that exist, it's time to start thinking about other important options - financial aid policies, campus size, location, etc. -that will have a major impact on your college experience.
College search: Type of college
Community college or 4-year program? Public or private? University or liberal arts college? There are countless exciting decisions to make as you consider your college options, and the first step is understanding the types of colleges out there!
Colonial period
Colored Shadows
Are shadows always black, or can they take on the colors of the rainbow? In this hands-on science snack, which was developed and demonstrated by Exploratorium Senior Scientist Paul Doherty, you’ll have fun with colored lightbulbs, experiment with additive color mixtures, and learn about human color perception. Watch the videos, build your own colored shadows, and see how exciting light and shadow can be.
We'll show you how to color and outline your shapes!
You are already familiar with calculating permutation ("How many ways could 7 different people sit in 4 different seats?"). But what if you didn't care about which seat they sat in? What if you just cared about which 4 people were in the car? Or put another way, you want to know how many combinations of 4 people can you stick in the car from a pool of 7 candidates. Or how many ways are there to choose 4 things from a pool of 7? Look no further than this tutorial to answer your questions.
Combining functions
Learn how to take two basic functions and combine them into a new function which is their sum, difference, product, or quotient.
Combining like terms
Combine the powers of Chuck Norris and polar bears to understand how to combine like terms.
Common Core conversations
Interested in how the Common Core standards cover key math concepts? In these videos, Sal Khan and Bill McCallum (one of the lead authors of the standards) explore how these concepts fit together.
Common denominators
Using our knowledge of equivalent fractions, let's rewrite fractions to have the same denominator.
Comparative advantage and gains from trade
Should you try to produce everything yourself or only what you are best at and trade for everything else? What if you're better than your trading partners at everything? This tutorial focuses on comparative advantage, specialization and gains from trade with a microeconomic lens.
Comparing and estimating length
Learn how to compare and estimate length.
Comparing and interpreting functions
In this tutorial, we'll dive deeper into actually thinking about what functions represent and how one function compares to another.
Comparing and sampling populations
When we are trying to make a judgement about a population, it is often impractical (or impossible) to observe every member of the population. Imagine trying to survey all 300+ million Americans to understand the likely outcome of the next presidential election! Because of this, much of statistics is collecting data from a representative and random sample. From the data collected from this random sample we can infer things about the greater population.
Comparing data displays
Learn how to interpret and compare dot plots, histograms, and box plots.
Comparing decimal place values
For a number like 34.135, learn how to compare the 3 in the tens place to the 3 in the hundredths place.
Comparing decimals
Learn to tell which of two decimal numbers is bigger by thinking about place value.
Comparing decimals visually
Learn how to compare decimals by drawing diagrams to represent each number.
Comparing exponential and polynomial functions
Learn how distinguish between linear and exponential growth, and learn the differences between the end behavior of polynomial and exponential functions.
Comparing fractions
In this tutorial, we'll practice understanding what quantities fractions actually represent and comparing those to each other.
Comparing fractions of different wholes
Comparing fractions with unlike denominators
Learn how to compare fractions with unlike denominators by drawing diagrams to represent each fraction.
Comparing linear functions
Learn to compare features of linear functions, such as slope and intercepts. Functions in this tutorial are given in different forms—tables, graphs, equations, and written descriptions of real-world situations.
Comparing multi-digit numbers
Comparing negative numbers
We all know that 6 is bigger than 4, but is -6 bigger than -4? This tutorial is designed to help you compare negative numbers.
Comparing small numbers
Learn to compare numbers between 0 and 10.
Comparing three-digit numbers
Learn how to compare three-digit numbers by thinking about place value (hundreds, tens, and ones).
Comparing two-digit numbers
Learn how to compare two-digit numbers by thinking about place value (tens and ones).
Comparing with multiplication
In this tutorial, we look at multiplication and division through the lens of comparison. For example, say that you are 9 and 3 times older than your cousin. How old would your cousin be? Multiplying a number times 3 gets you to your age, 9. Can you figure out the answer? We'll go through several exercises together so you get enough practice to feel confident multiplying. By the way, memorizing your multiplication tables helps a lot! Common Core Standards: 4.OA.A.1, 4.OA.A.2
Complementary and supplementary angles
In this tutorial we'll look at the most famous types of angle-pairs--complementary and supplementary angles. This aren't particularly deep concepts, but you'll find they do come in handy!
Completing the square
You're already familiar with factoring quadratics, but have begun to realize that it only is useful in certain cases. Well, this tutorial will introduce you to something far more powerful and general. Even better, it is the bridge to understanding and proving the famous quadratic formula. Welcome to the world of completing the square!
Completing the square and the quadratic formula
You're already familiar with factoring quadratics, but have begun to realize that it only is useful in certain cases. Well, this tutorial will introduce you to something far more powerful and general. Even better, it is the bridge to understanding and proving the famous quadratic formula. Welcome to the world of completing the square!
Complex and repeated roots of characteristic equation
Thinking about what happens when you have complex or repeated roots for your characteristic equation.
Complex numbers
Let's start constructing numbers that have both a real and imaginary part. We'll call them complex. We can even plot them on the complex plane and use them to find the roots of ANY quadratic equation. The fun must not stop!
Components of GDP
You already understand the circular nature of the economy and how GDP is defined from the last tutorial. Now let's think about how economists define the composition of GDP. In particular, we'll focus on consumption (C), investment (I), government spending (G) and net exports.
Composing functions
Learn what function composition is, how to to work with composite functions, and how to find the formulas of the composition of two functions.
Composing shapes
Practice making shapes from other shapes.
Compound and absolute value inequalities
You're starting to get comfortable with a world where everything isn't equal. In this tutorial, we'll add more constraints to think of at the same time. You may not realize it, but the ability to understand and manipulate compound and absolute value inequalities is key to many areas of science, engineering, and manufacturing (especially when tolerances are concerned)!
Compound events and sample spaces
Compound interest basics
Interest is the basis of modern capital markets. Depending on whether you are lending or borrowing, it can be viewed as a return on an asset (lending) or the cost of capital (borrowing). This tutorial gives an introduction to this fundamental concept, including what it means to compound. It also gives a rule of thumb that might make it easy to do some rough interest calculations in your head.
Compound, independent events
What is the probability of making three free throws in a row (LeBron literally asks this in this tutorial). In this tutorial, we'll explore compound events happening where the probability of one event is not dependent on the outcome of another (compound, independent, events).
Computing with numbers in scientific notation
Solve multiplication, division, addition, and subtraction problems involving numbers written in scientific notation.
Computing with scientific notation
You already understand what scientific notation is. Now you'll actually use it to compute values and solve real-world problems.
Concavity and inflection points
Concept of multiplication and division
Let's introduce ourselves to two of the most fundamental ideas in all of mathematics: multiplication and division!
Conceptual and Performance art
Does art need to be a physical thing, an object made of stone or canvas? Could art be an idea enacted, a process, or an environment inhabited and transformed?
Conceptualizing decimals and place notation
You've been using decimals all of your life. When you pay $0.75 at a vending machine, 0.75 is a decimal. When you see the ratings of gymnastics judges at the Olympics ("9.5, 9.4, 7.5 (booooo)"), those are decimals. This tutorial will help you understand this powerful tool all the better. Before you know it, you'll be representing numbers that are in-between whole numbers all the time!
Confident intervals
We all have confidence intervals ("I'm the king of the world!!!!") and non-confidence intervals ("No one loves me"). That is not what this tutorial is about. This tutorial takes what you already know about the central limit theorem, sampling distributions, and z-scores and uses these tools to dive into the world of inferential statistics. It may seem magical at first, but from our sample, we can now make inferences about the probability of our population mean actually being in an interval.
Do conformations float your boat? If so, pull up a chair and learn about the different conformations of cyclohexane.
Conformations of alkanes and cycloalkanes
Feeling unsure about conformations? If you need a little more perspective on alkanes, try looking at them from Newman’s point of view. The result might leave you feeling more stable.
Congruence and similarity
Conic section basics
What is a conic other than a jazz singer from New Orleans? Well, as you'll see in this tutorial, a conic section is formed when you intersect a plane with cones. You end up with some familiar shapes (like circles and ellipses) and some that are a bit unexpected (like hyperbolas). This tutorial gets you set up with the basics and is a good foundation for going deeper into the world of conic sections.
Conics from equations
You're familiar with the graphs and equations of all of the conic sections. Now you want practice identifying them given only their equations. You, my friend, are about to click on exactly the right tutorial.
Conics in the IIT JEE
Do you think that the math exams that you have to take are hard? Well, if you have the stomach, try the problem(s) in this tutorial. They are not only conceptually difficult, but they are also hairy. Don't worry if you have trouble with this. Most of us would. The IIT JEE is an exam administered to 200,000 students every year in India to select which 2000 go to the competitive IITs. They need to make sure that most of the students can't do most of the problems so that they can really whittle the applicants down.
Art conservation work includes treatment and preventive care, scholarly research on materials and techniques, and development of new conservation methods to address the changing needs of a growing museum. The Conservation Center at the Asian Art Museum shares the mandate of the museum to create a deeper level of understanding of Asian cultures by our visitors. Through cooperative exchanges, joint projects, and public outreach, art conservation can provide a unique window into shared traditions of art preservation, restoration and fabrication.
Conserving Art
Even the greatest works of art require care as they age—how do we preserve works of art, and how do we fix them if they have been damaged? In this tutorial, take a look behind the scenes at the work of the conservation team as they preserve works of art in unstable environments, restore great paintings that have been vandalised, and even create their own works of art in order to subject them to extensive experimentation. Learn about the art and science of conservation here.
Constantinople and the East
This tutorial focuses on Byzantine art made in the east, what is today Greece, Turkey, and the Middle East.
Constraining solutions of two-variable inequalities
Given a specific value for one of the variables in a two-variable linear inequality (or system of inequalities), learn how to find the corresponding range of values for the other variable. For example, given that x=3, what are the possible values for y in 2x+y<1 ?
Constructing a line tangent to a circle
Constructing basic exponential models
Learn how to construct exponential functions to model real-world situations.
Constructing bisectors of lines and angles
With just a compass and a straightedge (or virtual versions of them), you'll be amazed by how many geometric shapes you can construct perfectly. This tutorial gets you started with the building block of how to bisect angle and lines (and how to construct perpendicular bisectors of lines).
Constructing circumcircles and incircles
In our study of triangles, we spent a decent amount of time think about incenters (the intersections of the angle bisectors) and circumcenters (the intersections of the perpendicular bisectors). We'll now leverage this knowledge to actually construct circle inscribed and circumscribed about a triangle using only a compass and straightedge (actually virtual versions of them).
Constructing equations in slope-intercept form
You know a bit about slope and intercepts. Now we will develop that know-how even further to construct the equation of a line in slope-intercept form.
Constructing numeric expressions
Let's construct and interpret expressions from word problems. We can also think about what the effects of parentheses are.
Constructing regular polygons
Have you ever wondered how people would draw a square, equilateral triangle or even hexagon before there were computers? Well, now you're going to do just that (ironically, with a computer). Using our virtual compass and straightedge, you'll construct several regular shapes (by inscribing them inside circles).
Constructing triangles
Let's practice making triangles and start thinking about when it's possible or impossible to make a triangle.
Consumer and producer surplus
Many times, the equilibrium price is lower than the highest price some folks are willing to pay. For all consumers, this is called consumer surplus. Similarly, the price might be higher than the minimum price at which some are willing to produce. For all the producers, this is called producer surplus. This tutorial covers them both with an emphasis on the visual.
Consumer price index
$1 went a lot further in 1900 than today (you could probably buy a good meal for the family for $1 back then). Why? And how do we measure how much more expensive things have gotten (i.e., inflation)?
Consumption function
We are steadily building up the tools to understand the Keynesian Cross and the IS-LM model. In this tutorial, we begin to model consumption as a linear function of disposable income. Seems reasonable to me.
Contemporary Sculptors at the British Museum
The British Museum has one of the world’s most celebrated and diverse collections of sculpture, dating from prehistory to the present day. Since its foundation in 1753, the Museum has consistently engaged with the contemporary world, both in its collecting and its displays.
Continuity using limits
A function isn't continuous when there is a "break" in its graph. This tutorial uses limits to define this idea more formally and gives practice thinking about continuity (and discontinuity) in terms of limits.
Continuous compound interest and e
This is an older tutorial (notice the low-res, bad handwriting) about one of the coolest numbers in reality and how it falls out of our innate desire to compound interest continuously.
Continuous compounding and e
This tutorial introduces us to one of the derivations (from finance and continuously compounding interest) of the irrational number 'e' which is roughly 2.71...
Conversations with Sal: Talks and presentations
Converting between fractions and decimals
Both fractions and decimals are desperate to capture that little part of our heart that desires to express non-whole numbers. But must we commit? Can't we have business in the front and party in the back (younger people should look up the word "mullet" to see a hair-style worth considering for your next trip to the barber)? Can't it look like a pump, but feel like a sneaker? Well, if 18-wheelers can turn into self-righteous robots, then why can't decimals and fractions turn into each other?
Converting decimals to fractions
Learn how to rewrite decimals as fractions.
Converting fractions to decimals
This tutorial guides you through two very important skills: converting fractions to decimals and converting decimals to fractions.
Converting larger units to smaller units
Learn how convert from measurements in big units (like kilograms) to measurements in small units (like grams). For example, 2 kilograms is the same as 2000 grams.
Converting repeating decimals to fractions
You know that converting a fraction into a decimal can sometimes result in a repeating decimal. For example: 2/3 = 0.666666..., and 1/7 = 0.142857142857... But how do you convert a repeating decimal into a fraction? As we'll see in this tutorial, a little bit of algebra magic can do the trick!
Coordinate plane
We first explored the coordinate plane in the 5th grade, but that was only dealing with positive coordinates. Now we know all about negative numbers so why not have negative coordinates as well? Let's get cozy with the x and y axis, plotting ordered pairs, quadrants, and reflection points. We got it covered. Sit back, relax, and get ready to groove with us. Common Core Standards: 6.NS.C.6, 6.NS.C.6b, 6.NS.C.6c, 6.NS.C.8
Copy of Balancing chemical equations
We are now going to look at chemical reactions. But as we do, we need to make sure that atoms aren't magically appearing or disappearing. Put another way, we need to sure that we have the same number of each constituent atom in the product of the reaction as we do in the reactants (the molecules that react)!
Copy of Neurodevelopmental disorders
Copy of Slope
If you've ever struggled to tell someone just how steep something is, you'll find the answer here. In this tutorial, we cover the idea of the slope of a line. We also think about how slope relates to the equation of a line and how you can determine the slope or y-intercept given some clues. This tutorial is appropriate for someone who understands the basics of graphing equations and want to dig a bit deeper. After this tutorial, you will be prepared to start thinking deeper about the equation of a line.
Copy of Transformations, congruence, and similarity
Coronary artery disease
Coronary artery disease - clogging of the arteries supplying the heart- is the cause of about 30% of all deaths globally, making it the leading cause of death. Stroke is a similarly debilitating condition that results from lack of perfusion to the brain. Unfortunately, patients with heart disease are more likely to suffer from heart disease and vice versa. You will come to an understanding of the pathophysiology behind these common diseases and how they relate to one another.
Corporate bankruptcy
Anybody or anything (you can decide if a corporation is a person) can have trouble paying its debts. This tutorial walks through what happens to a corporation in these circumstances.
Corporate structure and taxation
In exchange for being treated as a person-like-legal entity (and the limited liability this gives for its owners), most corporations pay taxes. This tutorial focuses on what corporations are, "double taxation" and a few ways that multinationals might try to get out of paying taxes.
How many times do you need to cut a cake? How many fence posts do you need? These life altering decisions will be based on how well you count.
Counting crowds
In this lesson we construct snakes using permutations and develop a powerful counting tool: the binomial coefficient. It touches on concepts in grade 6 through high school and takes 45-60 minutes.
Counting electrons
Aspiring chemists need to be able to count electrons. Without this ability, your days as a chemist would be numbered. In this tutorial, you will learn how to count electrons to assign formal charges and oxidation states. Both forms of "electron bookkeeping" are important concepts to understand for organic reactions.
Counting objects
Learn to count up to 20 objects.
Course materials
These syllabi and sample discussion questions are for the use of instructors who are encouraged to build on and adapt them for their own classes. They can also be used by individual learners seeking a pathway through the art history content on Khan Academy akin to an introductory college-level survey course.
Cranach and Altdorfer
Crash Course Biology
Hank Green teaches you biology! Learn, study and understand the science of life. Topics covered range from: taxonomy, systems, biological molecules, photosynthesis, evolution, animals, plants, anatomy, and ecology.
Crash Course Chemistry
It's time to jump into Chemistry with Hank Green. He's been a life long lover of this subject and it's not just because of the rules, laws, bonds, and coefficients. But, also because of the men and women who shaped this science that deepened our understanding of our world so magnificently.
Crash Course Ecology
Hank Green teaches you ecology! Learn, study and understand how organisms relate to one another and to their surroundings. Start with the history of life on earth, then cover population ecology, community ecology, ecosystem ecology, and conservation and restoration ecology.
Creating Contradiction
Sometimes the conflict that art addresses is internal rather than external. Some artists deliberately employ an element of contradiction in their work in order to create tension or irony, highlighting certain qualities. Through unexpected juxtapositions, they challenge our expectations and perceptions of the everyday.
Credit Crisis
This tutorial talks about how the housing-bubble-induced credit crisis unfolded with a focus on the derivative securities that helped pump the bubble.
Credit default swaps
Critical analysis and reasoning skills (CARS) practice questions
Go ahead and practice some passage-based questions!
Critical points and graphing with calculus
Can calculus be used to figure out when a function takes on a local or global maximum value? Absolutely. Not only that, but derivatives and second derivatives can also help us understand the shape of the function (whether they are concave upward or downward). If you have a basic conceptual understanding of derivatives, then you can start applying that knowledge here to identify critical points, extrema, inflections points and even to graph functions.
Critical thinking
The critical thinking section will teach you the skills to think clearly and independently. It will help you identify valid arguments, detect inconsistencies in reasoning, understanding logical connections between ideas, and construct and evaluate arguments.
Cross sections of 3D objects
Cross topic arithmetic
You've probably been learning how to do arithmetic for some time and feel pretty good about it. This tutorial will make you feel even better once by showing you a bunch of examples of where it can be applied (using multiple skills at a time). Get through the exercises here and you really are an arithmetic rock star!
Why is it that in America it is acceptable to eat a quarter pound hamburger and guzzle down a large soft drink yet it is weird to eat a grasshopper? The answer is because of our culture. Culture varies across the world and provides rules for a society of people to live by. Here we explore what culture is and how it shapes our human experience.
Curiosity rover: discoveries
What did the curiosity rover find? Follow the mission timeline & findings here.
Curiosity rover: mission briefing
Why are we going to Mars today, where are we looking, what are we hoping to find?
Curl measures how much a vector field is "spinning". A bit of a pain to calculate, but could come in handy when we work with Stokes' and Greens' theorems later on.
This tutorial walks through how China's undervaluing of its currency impacts trade and prices (which also fuels cheap borrowing for the U.S.).
Currency reserves
This tutorial delves into how and why countries (usually their central banks) would want to keep other countries' currency in reserve. It then goes into why this sometime leaves the reserve-holding country open to a speculative attack (this is seriously high drama).
Current and capital accounts
In this tutorial we will see how trade and assets (including money) changing hands are fundamentally intertwined. Not only that, but we will see how this can be accounted for through the capital account (assets changing hands) and current account (trade).
Current and resistance
Always follow the path of least resistance. In these videos you will learn how to interpret circuit diagrams and the rules that govern the flow of electrons - or current - through wires and resistors.
Cyanotic heart diseases
Whenever we go online—to shop, chat with our friends, or do anything else—we put ourselves at risk of cybercrime. Computer viruses can corrupt our files, hackers can steal our data, and criminals can trick us into revealing sensitive information. But luckily there are simple steps we can take to protect our digital lives.
The art of the Cycladic Islands is best known for beautiful and enigmatic figures of varying sizes. Marbles once painted and with a strong abstract quality that attracted the attention of some of the leading artists of the 20th century.
Cystic fibrosis
The cytoskeleton provides an internal framework for the cell, much like our skeleton helps support our body. In these videos, you will learn about the different filaments that comprise the cytoskeleton, and how these filaments provide structural support, create roadways for intracellular transport, and generate cellular motion.
Cytoskeleton, junctions, and extracellular stru
We've spent a lot of time looking at what's inside a cell. But what about the stuff on the outside? Here, we'll look at the extracellular matrix of animal cells and the cell wall of plant cells, as well as the junctions that hold cells together.
1B: DNA makes RNA, and RNA makes protein - in a nutshell, this is the central dogma of molecular biology. Let’s delve into that simple notion here so we can come to a better understand of the flow of genetic information.
DNA technology
Thanks to advances in DNA technology, we can now clone genes, control gene expression, and sequence entire genomes. How is this possible? These videos will cover the techniques that revolutionized molecular biology and continue to be used almost daily in research labs around the world.
DOM access with jQuery
Learn how to access the DOM using jQuery and CSS selectors.
DOM animation & effects with jQuery
Learn how to use jQuery to animate elements with effects like fading and sliding.
DOM events with jQuery
Learn how to use jQuery to add event listeners to elements, so that you can respond when users do things like click, press keys, scroll, drag, and more.
DOM modification with jQuery
Learn how to modify your webpage DOM and create new DOM nodes with jQuery.
DVD Player
DVD Player
Dada was an anti-art movement that sought to subvert the function of the arts in an increasingly commercial and militaristic society. Dada develops in numerous cities including Zurich, Berlin, Paris, and New York in the context WWI.
Daedalic and Archaic
This tutorial traces the representation of the human body in monumental Greek sculpture from the earliest Egyptian influence to the increasing naturalism that lays the foundation for the Classical style.
Danny O'Neill - President of The Roasterie
Danny O’Neill, President of The Roasterie, describes the journey that led him to starting his own company as well as some of the key attributes of an entrepreneur.
Darwin and Evolution by Natural Selection
Happiest at home with his notebooks and his microscope, he shunned the public eye. Controversy made him ill. This brilliant observer of nature kept his most original and revolutionary idea under wraps for decades. Yet today, two centuries after Charles Darwin's birth, nearly everyone knows his name. What did Darwin do, and why does he still matter so much?
These tutorials are good for some basic understanding of how to read line graphs
Data management
This tutorial is less about statistics and more about interpreting data. Whether you're looking at scientific data or stock price: a table, pictograph, bar graph or line graph, box-and-whisker plots can show up in your life. This tutorial covers what they are, how to read them and how to construct them.
Dave Gilboa & Neil Blumenthal - Co-founders & Co-CEOs of Warby Parker
The visionaries who founded Warby Parker were friends and classmates at the Wharton School at the University of Pennsylvania who challenged convention, disrupted an industry and created an organization that did something good in the world. Dave Gilboa and Neil Blumenthal, along with Andrew Hunt and Jeffrey Raider, set ambitious targets for themselves and committed to have fun along the way.
Dave Smith - CEO & Founder of TekScape IT
When Dave Smith came to the harsh realization and he alone was in charge of his future, he took a resourceful route to become an expert in his field. Mixing the desire to make it with the imagination to fake it, he went to great lengths to connect with TekScape IT customers and make them believe that his tiny organization was big enough to solve their trickiest problems.
Deadweight loss
We can often lose economic efficiency because of things like price floors, ceilings and taxes. This loss in surplus (people who have more marginal benefit than marginal cost are not buying or people who have more marginal cost than benefit are buying) is called deadweight loss.
Debugging programs
How can you figure out what's wrong with your code? Learn tips and tricks for debugging your programs.
Decimal place value introduction
Learn the basics of decimal place value.
Decimals and fractions
If you already know a bit about both decimals and fractions, this tutorial will help build a bridge between the two. Through a bunch of examples and practice, you'll be able operate in both worlds. Have fun!
Decimals greater than one
In this tutorial, we use tenths and hundredths to describe numbers greater than 1.
Decimals in written form
Learn how to write decimals like 3.14 in words as "three and fourteen hundredths".
Decimals on a number line
Let's think about where decimals are on a number line. It will help us understand what decimals represent in general!
Decimals on the number line
See how decimals can be placed on the number line, just like fractions and whole numbers can.
Decimals vs. fractions
See how decimals and fractions are just two ways to write the same number.
Decimals, fractions and percentages
Let's review how to convert between fractions, decimals and percentages
Decomposing fractions
In this tutorial, we'll see that a fraction can be broken up (or decomposed) into a bunch of other fractions. You might see the world in a completely different way after this.
Decorative Arts
Discover the exquisitely crafted objects used in the daily life of European aristocracy. Also explore maiolica from Spain and Italy, and learn about the history and making of stained glass.
Deductive and inductive reasoning
You will hear the words "deductive reasoning" and "inductive reasoning" throughout your life. This very optional tutorial will give you context for what these mean.
Definite integrals
Until now, we have seen definite integrals as the area under a curve. We've approximated this area with reactangles using Riemann sums. We also realized that we could potentially find the exact area if we take the limit as we approach having an infinite, infinitely thin rectangles. But is there an easier way to evaluate an integral? Even more, does this somehow connect to everything we know about the derivative and differential calculus? Hold on to your seats, because everything is about to come together!
Prices don't always go up. They often go down. This might seem like a good thing, but it could be disastrous for a modern economy is it goes too far. This tutorial explains what deflation is, how it happens and what the effects of it might be.
Dementia, delirium, and Alzheimer's
Democratic Republic of the Congo
This large and populous nation is located in the center of the continent. It's climate is tropical and contains dense jungle, mountains, grasslands and the Congo River. More than 200 ethnic groups make the DRC their home including the Chokwe, Kongo, and Kuba peoples.
Density and Pressure
Dependent and independent variables
Sometimes one variable depends on another. For example, the amount of money you make might depend on how many hours you work.
Dependent events
What's the probability of picking two "e" from the bag in scrabble (assuming that I don't replace the tiles). Well, the probability of picking an 'e' on your second try depends on what happened in the first (if you picked an 'e' the first time around, then there is one less 'e' in the bag). This is just one of many, many type of scenarios involving dependent probability.
Dependent probability
What's the probability of picking two "e" from the bag in scrabble (assuming that I don't replace the tiles). Well, the probability of picking an 'e' on your second try depends on what happened in the first (if you picked an 'e' the first time around, then there is one less 'e' in the bag). This is just one of many, many type of scenarios involving dependent probability.
Depreciation and amortization
How do you account for things that get "used up" or a cost that should be spread over time. This tutorial has your answer. Depreciation and amortization might sound fancy, but you'll hopefully find them to be quite understandable.
Depression and related disorders
Derivative properties and intuition
Let's now get a better understanding of the different derivative-related notations and use them to better understand properties of derivatives.
Derivatives of common functions
We told you about the derivatives of many functions, but you might want proof that what we told you is actually true. That's what this tutorial tries to do!
Derivatives of inverse functions
In this tutorial we explore a common method to find the derivatives of inverse tangent (arctangent), inverse sine (arcsine), inverse cosine (arccosine) and the natural logarithm function.
Determining the domain of a function
Determine the domains of functions according to various considerations.
Determining whether a function is invertible
Learn about the conditions for a function to be invertible, and learn how to manipulate functions to make them invertible.
Diels-Alder reaction
What’s the big deal? Actually, the Diels-Alder reaction is a very big “diel” because it is commonly used in organic synthesis. Learn how to “diel” with this classic reaction in this tutorial.
Diffusion and osmosis
Ever wonder why plants wilt if you don't water them? Find out here, and learn more about the basic principles of probability that underlie movement of water and other molecules in biological systems.
Digital Camera
Digital Camera
Dilations or scaling around a point
In addition to the three rigid transformation, there are the dilations, which expand or shrink figures while keeping the same proportions. These are extremely important in the subject of Similarity! Gain experience by performing dilation, learn how to find the dilation that maps a given figure to another, and learn how to manually draw the image of a dilation.
When companies issue new shares, many people consider this a share "dilution"--implying that the value of each share has been "watered down" a bit. This tutorial walks through the mechanics and why--assuming management isn't doing something stupid--the shares might not be diluted at all.
Dinosaur extinction
The extinction of non-avian dinosaurs except birds at the end of the Cretaceous has intrigued paleontologists for more than a century. One theory is that an asteroid impact 65 million years ago off the coast of Mexico generated massive tsunamis, with impact debris cutting off sunlight for months, stopping photosynthesis and causing freezing temperatures. Chemical reactions in the atmosphere caused acid rain and long-term global warming, all of which extinguished non-avian dinosaurs. However, at the same time, massive lava flows erupted across what is now southwest India. The eruptions probably caused many of the same effects as the asteroid impact. Although most scientists believe that the impact was the final blow for non-avian dinosaurs, both events could well have played a role.
Dinosaur fossils
The American Museum of Natural History houses the largest and most spectacular collection of vertebrate fossils in the world. A fossil is any evidence of prehistoric life that is at least 10,000 years old. The most common fossils are bones and teeth, but footprints and skin impressions fossils as well. Fossils are excavated from ancient riverbeds and lakes, caves, volcanic ash falls, and tar pits.
Direct and inverse variation
Whether you are talking about how force relates to acceleration or how the cost of movie tickets relates to the number of people going, it is not uncommon in this universe for things to vary directly. Similarly, when you are, say, talking about how hunger might relate to seeing roadkill, things can vary inversely. This tutorial digs deeper into these ideas with a bunch of examples of direct and inverse variation.
Directing Effects
In this tutorial, Jay shows you the directing effects of substituents on a benzene ring. Knowledge of Electrophilic Aromatic Substitution reactions is assumed.
Discontinuities of rational functions
Learn about the ways in which rational functions behave when their denominator is equal to zero. This gets interesting when vertical asymptotes are involved! Learn how to determine this behavior for any kind of rational function.
Discovery Lab 2012
Discovery Lab 2013
Discovery of Magnetism
The discovery of magnetism. What can we do with this invisible force?
Discovery of magnetic fields
Let's find out more about this invisible force which guides the compass. How strong is it? What shape is it?
Why is it that some traits like skin color, religion, or cultural values have more meaning than others like eye color or personality? Here we explore how discrimination begins as a stereotype and can escalate into behaviors that can negatively influence whole communities of people.
Displacement, velocity and time
This tutorial is the backbone of your understanding of kinematics (i.e., the motion of objects). You might already know that distance = rate x time. This tutorial essentially reviews that idea with a vector lens (we introduce you to vectors here as well). So strap your belts (actually this might not be necessary since we don't plan on decelerating in this tutorial) and prepare for a gentle ride of foundational physics knowledge.
Distance, weight, and fluid volume: U.S. customary units
The International System of Units used today is based on the metric system. The United States, however, likes to dance to the beat of a different drummer and still uses the old British Imperial System (U.S. customary system) for most of its measuring. This tutorial introduces you U.S. customary units for measuring distance, volume, and weight.
Distances between points
We are now going to leverage our understanding of the coordinate plane to think about distances between points and ratios of lengths of segments between points.
Distribution warmup
Introduction to probability distributions, center, spread, and overall shape. In this warmup we will discover the binomial distribution!
Distributive property
Learn how to apply the distributive property to numerical expressions (without variables) and algebraic expressions (with variables). The distributive property is also sometimes called the distributive law.
Is a vector field "coming together" or "drawing apart" at a given point in space. The divergence is a vector operator that gives us a scalar value at any point in a vector field. If it is positive, then we are diverging. Otherwise, we are converging!
Divergence theorem (3D)
An earlier tutorial used Green's theorem to prove the divergence theorem in 2-D, this tutorial gives us the 3-D version (what most people are talking about when they refer to the "divergence theorem"). We will get an intuition for it (that the flux through a close surface--like a balloon--should be equal to the divergence across it's volume). We will use it in examples. We will prove it in another tutorial.
Divergence theorem proof
You know what the divergence theorem is, you can apply it and you conceptually understand it. This tutorial will actually prove it to you (references types of regions which are covered in the "types of regions in 3d" tutorial.
Dividing decimals
In this tutorial, we'll extend our division skills to include decimals! We get into some pretty fun stuff here including dividing decimals by whole numbers, and dividing whole numbers by decimals. Finally...dividing decimals by decimals. Common Core Standard 5.NBT.B.7
Dividing fractions and whole numbers word problems
Learn how to solve word problems by dividing fractions and whole numbers.
Dividing fractions by fractions
In this tutorial, we'll become fraction dividing experts! In particular, we'll understand what it means to divide a fraction by another fraction. Too much fun!!!
Dividing unit fractions by whole numbers
Learn how to divide unit fractions by whole numbers. Example: 1/5 ÷ 3 = 1/15.
Dividing whole numbers and fractions
This is one exciting tutorial. In it, we will understand that fractions can represent division (and the other way around). Then we will create fractions by dividing whole numbers and then start dividing the fractions themselves. We'll see that dividing by something is the exact same thing as multiplying by that thing's reciprocal!
Dividing whole numbers by unit fractions
Learn how to divide whole numbers by unit fractions. Example: 4 ÷ 1/3 = 12.
Divisibility and factors
In this tutorial, we'll begin to think about the numbers that "make up" the number. This will be useful throughout our study of math. Whether we are adding fractions, exploring mystical number patterns, or breaking computer codes, factoring numbers are key! Eye of the tiger!
Divisibility tests
Whether you are trying to impress your dog's friends (who are obsessed with figuring out number divisibility) or quickly factor a number, it can be very useful to tell whether a number is divisible by another. This tutorial walks through some of the more standard divisibility methods and describes why they work.
Learn long division on whole numbers. Solve problems like 9815 divided by 65.
Division and partial fraction expansion
When you're trying to integrate a rational expression, the techniques of partial fraction expansion and algebraic long division can be *very* useful.
Division facts
Learn your division facts
Division intro
Welcome to the concept of division.
Documenting Conflict
By its very nature, art is a kind of documentation—it reflects the society in which it was made, and the intention of the artist who created it. The following artists use art specifically as a way of documenting conflict and its effects all over the globe, creating records of conflict situations, issues, and outcomes. But is documentary art just a record, or does it have an inherent element of commentary? Can art ever be objective?
Domain and range
What values can you and can you not input into a function? What values can the function output? The domain is the set of values that the function is defined for (i.e., the values that you can input into a function). The range is the set of values that the function output can take on. This tutorial covers the ideas of domain and range through multiple worked examples. These are really important ideas as you study higher mathematics.
Doodling in Math
Let's say you're me and you're in math class…
Dot plots
A dot plot is like a bar chart, but it displays data using dots (not bars). And a frequency table is like a dot plot, but it displays data in a simple table (not a fancy diagram).
Dot structures
Going dotty trying to review? Don't worry about a single thing - or multiples for that matter! Get some points on simple organic dot structures in this tutorial.
Double integrals
A single definite integral can be used to find the area under a curve. with double integrals, we can start thinking about the volume under a surface!
Drawing basics
We'll show you the basics of programming and how to draw shapes.
Drawing polygons in the coordinate plane
Practice drawing shapes on the coordinate plane.
Learn about the art of drawing as revealed through the work of several masters.
Drug abuse and drug addictions
Drug dependence
6B: A drug is any chemical substance with a biological effect on an organism. Unfortunately, some people have addictions to drugs - 18.1% of all US adults smoke cigarettes, which have disastrous consequences on our health. You will come to appreciate the effects of drugs on our human physiology as we explore the mechanisms of psychoactive drugs such as depressants, opiates, stimulants, and hallucinogens.
E1 and E2 reactions
In this tutorial, Jay covers the E1 elimination mechanism, carbocation rearrangements, and the details of the E2 elimination reaction.
Early 20th century: War and dynamism
The first half of the 20th century is consumed by war–and the aftermath. While many British artists respond with images of pessimism and despair, others find a renewed sense of dynamism, a belief in the power of culture and in the artist’s ability to change the world for the better.
Early Christian
Persecutions of Christians ended in Rome with Emperor Constantine. 4th century churches, tombs and catacombs decorated with Christian imagery can still be visited.
Early Republic
Early Victorian
Modernity in the guise of history
Early classical
The Early Classical style describes the trends in Greek sculpture between c. 490 and 450 B.C.E. Artistically this stylistic phase represents a transition from the rather austere and static Archaic style of the sixth century B.C.E. to the more idealized Classical style.
Early empire
In 31 B.C.E. Octavian, the adopted son of Julius Caesar, defeated Cleopatra and Mark Anthony at Actium. This brought the last civil war of the republic to an end. Although it was hoped by many that the republic could be restored, it soon became clear that a new political system was forming: the emperor became the focus of the empire and its people. Although, in theory, Augustus (as Octavian became known) was only the first citizen and ruled by consent of the Senate, he was in fact the empire's supreme authority. As emperor he could pass his powers to the heir he decreed and was a king in all but name.
Early medieval
In the fifth century C.E., people from tribes called Angles, Saxons and Jutes left their homelands in northern Europe to look for a new home. They knew that the Romans had recently left the green land of Britain unguarded, so they sailed across the channel in small wooden boats. The Britons did not give in without a fight, but after many years the invaders managed to overcome them and were to rule for over 500 years.
Early period
Early Period (c. 640-900 C.E.) From the rightfully guided caliphs who succeeded Muhammad, through the Umayyads' Dome of the Rock and Great Mosque of Damascus, to the decline of Abbasids rule.
Early photography
The early photography of Niépce, Daguerre, Cameron posed questions about art, aesthetics, and technology we still try to answer today.
Earth & the Formation of Our Solar Systemm
Before 1995, most people believed that the only planets in the Universe were found in our Solar System. Since 1995, hundreds of “exoplanets,” or planets outside of our Solar System, have been discovered orbiting other stars. The Earth and our Solar System are not as unique as they were once thought to be. Planet formation is now considered to be very common in the Universe, and planets can form in the wake of the formation of any star. Star formation begins in giant gas clouds, and probably 99.9 percent of the material in these clouds goes into making the star. Only about 0.1 percent of the material in the original gas cloud is left for planet formation. This leftover material orbits the star and various forces cause the materials to begin crashing into one another. Over time, this process leads to the formation of very large objects—what we know as planets. Sometimes rocky like our Earth, sometimes gassy like Jupiter, these planets gather mass as other floating debris crashes onto their surfaces. In the early days of our Solar System, the Earth was constantly bombarded with floating debris. Over time, things settled down and the Earth cooled, making it the perfect place for life to form.
Earth's rotation and tilt
What causes the seasons? Even more, can Earth's climate change over long period just to "wobbles" in its orbit? This tutorial explains it all. You'll know more about orbits (and precession and Milankovitch cycles) than you ever thought possible. Have fun!
East Africa
East Africa is a huge region with Djibouti, Eritrea, Ethiopia and Somalia to the north, Tanzania, Kenya, Uganda, Rwanda and Burundi to the south and the Indian Ocean Islands of Comoros, Mauritius, Seychelles, Réunion and Mayotte.
Eastside Prep
Learn how 6th and 7th grade teachers Suney Park and Jen Johnson use KA at their school in East Palo Alto, California. All students at this independent school will be the first in their families to attend college.
Learn all about the Ebola outbreak occurring in Western Africa. Dr. Rishi Desai is a pediatric infectious disease physician and former epidemiologist with the Centers for Disease Control and Prevention (CDC)
Economic profit and opportunity cost
Economic profit and accounting profit are two different things (the difference being that economic profit takes into account opportunity cost). Confused? This tutorial lays it all out with the example of a restaurant.
Edo period (1615–1868)
Eigenvectors, eigenvalues, eigenspaces! We will not stop with the "eigens"! Seriously though, eigen-everythings have many applications including finding "good" bases for a transformation (yes, "good" is a technical term in this context).
Electric Motor
How can we turn electric current into rotary motion?
Electric field
Electric motors
Electric potential energy, electric potential, and voltage
Recall from chemistry that electrons are transferred between atoms during redox reactions. When redox reactions occur on a conductive surface, the moving electrons can be collected as a current, with the voltage being dependent on the species used in reaction. Find out how batteries are made and the chemistry that makes them function by watching these videos.
Electrochemistry, thermodynamics, and equilibrium
Relating the cell potential to the equilibrium constant K and Gibbs free energy.
Electrolytic cells and electrolysis
An introduction to electrolytic cells and electrolysis
Discoveries leading to the Right Hand rule
Electromagnetism Introduction
Explore the interactions between electricity and magnetism.
Electron configurations
In this tutorial, Sal and Jay show how to write electron configurations.
Electron shells and orbitals
Learn how electrons are organized in atoms! We will cover the Bohr model, electron shells, and orbitals, as well as periodic table trends in electron configurations and reactivity.
What is the most attractive concept in undergraduate organic chemistry? Currently most are polarized on the topic, but the pull of electronegativity is hard to resist. Within this tutorial, we will learn about electronegativity and see how it applies to polarity, intermolecular forces, and physical properties.
Electronic structure
1s, 2s, red s, blue s. Electrons spin around a the nucleus of an atom in various spherical and teardrop shaped orbitals. The following section will show how the periodic table is built around these orbitals, and how the electronic structure of atoms dictates how they interact with light and other atoms.
Electrophilic Aromatic Substitution
In this tutorial, Jay shows several electrophilic aromatic substitution reactions.
4C: Ouch! have you ever heard of people being struck by lightning? Have you ever seen a defibrillator used to shock a patient back from the jaws of death? These amazing phenomena are possible due to the action of electric charges. Like masses, electric charges can have an associated force and a potential energy within a field. In these tutorials, you will discover how electric charges interact with one another and will arrive at an understanding of the concepts of electric force, fields, and potential.
Elements and atoms
Learn what matter is made of! We'll cover elements, atoms, and subatomic particles (protons, neutrons, and electrons). We'll also check out isotopes, radioactivity, and carbon dating.
Elimination reactions
In this tutorial, Sal explains the difference between an E1 and an E2 elimination reaction.
What would you call a circle that isn't a circle? One that is is is taller or fatter rather than being perfectly round? An ellipse. (All circles are special cases of ellipses.) In this tutorial we go deep into the equations and graphs of ellipses.
Do you remember when you were just a sperm and an egg, or when you were just made up of 32 cells, or when you were a blastocyst attaching to your mom’s uterus? These videos should help jog your memory about the nine months before you came into this world as an human being.
Learn about the physiological, behavioral, and cognitive components of emotion. Appreciate how different areas of our brain play a role in emotion. Understand the basic theories of emotion.
Emphysema (COPD)
In this tutorial, we will explore enantiomers, compounds that have the same composition and bonding but are fundamentally different because they are mirror images of each other (kind of like Tomax and Xamot--the Crimson Guard Commanders from GI Joe).
End behavior and graphs of basic exponential functions
Learn how an exponential function behaves as the value of its input increases to positive infinity or decreases to negative infinity. Learn how to graph basic exponential functions.
End behavior of rational functions
Learn about the ways in which rational functions behave as x approaches positive or negative infinity. This gets interesting Learn how to determine this behavior for any kind of rational function.
Endocrine system
Endocrine system introduction
Glands are special organs that secrete chemical messages called hormones, which seep into the blood - it’s like putting a tea bag in hot water. As the heart pumps, this blood carries these chemical messages throughout the body, allowing the hormones to interact with specific target cells and organs. Endocrine glands help us to maintain our appetites, grow up, metabolize molecules, concentrate urine,- and oh, so much more! We will examine how these variegated hormones play a role in homeostasis as the body responds to a changing environment.
Endocrine system introduction
Glands are special organs that secrete chemical messages called hormones, which seep into the blood - it’s like putting a tea bag in hot water. As the heart pumps, this blood carries these chemical messages throughout the body, allowing the hormones to interact with specific target cells and organs. Endocrine glands help us to maintain our appetites, grow up, metabolize molecules, concentrate urine,- and oh, so much more! We will examine how these variegated hormones play a role in homeostasis as the body responds to a changing environment.
For something we use every day, energy is a pretty mysterious concept. Find out what energy is, how it can be converted into useful forms, and why some sources are running low.
Energy in metabolism
Can you feel the energy flowing through your body? The answer may be no (unless you’ve had something caffeinated recently), but it’s true: energy is constantly being transformed in your body. Learn about different types of energy and how they’re interconverted, and discover the metabolic “traffic” that takes place in each of your cells.
England (Blake and Fuseli)
As the industrial revolution transformed the British countryside, replacing fields with factories, painters turned to landscape. Constable painted his native suffolk, where he spent his childhood, and imbued it with a sense of affection for rural life. Turner, on the other hand, created dramatic and sublime landscapes with a sense of the heroic or even the tragic. What both of these artists have in common is a desire to make landscape painting—understood as a low subject by the Academy which dictated official views on art—carry serious meaning.
England (Constable, Turner, Martin and Nash)
England, France and Tyrol
Enlightenment and Revolution (4-1)
The Enlightenment set the stage for this era. Scientific inquiry and empirical evidence were promoted in order to reveal and understand the physical world. Belief in knowledge and progress led to revolutions and a new emphasis on human rights. Subsequently, Romanticism offered a critique of Enlightenment principles and industrialization. Philosophies of Marx and Darwin impacted worldviews, followed by the work of Freud and Einstein. Later, postmodern theory influenced art making and the study of art. In addition, artists were affected by exposure to diverse cultures, largely as a result of colonialism. The advent of mass production supplied artists with ready images, which they were quick to appropriate. By permission, © 2013 The College Board
An introduction to enthalpy and Hess's law
An introduction to the Carnot cycle and entropy
Enzyme kinetics
1A: You’ll come to understand how enzymes, biomolecular catalysts, speed up reactions in cells as well as interact with one another. With just a little algebra, we’ll come to a mathematical understanding of this fundamental process.
Enzyme regulation
Enzymes may control chemical reactions, but what controls enzymes? As it turns out, enzymes can be turned “on” or “off” by a wide range of molecules, some found normally in cells and others added from the outside. Learn about different types of enzyme regulators, which include nerve gas, HIV drugs, and the end products of metabolic pathways.
Enzyme structure and function
1A: The multitude of reactions within our cells are sped up by enzymes. Without these biomolecules, these biochemical pathways would be as slow as a turtle. For instance, without enzymes, your body may never be able to break down and absorb the hamburger you just had for lunch. The hamburger would simply sit there, a lump in your stomach, until reactions slowly started to happen on their own - enzymes speed that up!
Epsilon delta definition of limits
This tutorial introduces a "formal" definition of limits. So put on your ball gown and/or tuxedo to party with Mr. Epsilon Delta (no, this is not referring to a fraternity). This tends to be covered early in a traditional calculus class (right after basic limits), but we have mixed feelings about that. It is cool and rigorous, but also very "mathy" (as most rigorous things are). Don't fret if you have trouble with it the first time. If you have a basic conceptual understanding of what limits are (from the "Limits" tutorial), you're ready to start thinking about taking derivatives.
Equals sign
Learn the meaning of the equals sign.
Equation examples for beginners
Like the "Why of algebra" and "Super Yoga plans" tutorials, we'll introduce you to the most fundamental ideas of what equations mean and how to solve them. We'll then do a bunch of examples to make sure you're comfortable with things like 3x – 7 = 8. So relax, grab a cup of hot chocolate, and be on your way to becoming an algebra rockstar. And, by the way, in any of the "example" videos, try to solve the problem on your own before seeing how Sal does it. It makes the learning better!
Equation of a circle
You know that a circle can be viewed as the set of all points that whose distance from the center is equal to the radius. In this tutorial, we use this information and the Pythagorean Theorem to derive the equation of a circle.
Equation practice with vertical, complementary, and supplementary angles
Learn to solve for variables presented in geometry diagrams. These problems require understanding of vertical, complementary, and supplementary angles
Equations of normal and tangent lines
A derivative at a point in a curve can be viewed as the slope of the line tangent to that curve at that point. Given this, the natural next question is what the equation of that tangent line is. In this tutorial, we'll not only find equations of tangent lines, but normal ones as well!
Equations of parallel and perpendicular lines
Equations of proportional relationships
Learn how to write equations of proportional relationships in the form y=kx.
The bane of the pharmaceutical industry is that reactions don’t go from 100% reactants to 100% products. They end in equilibrium with each other. Learn how to calculate equilibrium constants and push reactions further towards products by watching these videos!
Equilibrium constant
In this tutorial, we will examine the equilibrium state and learn how to write the expression for the equilibrium constant.
Equivalent fractions
There are literally infinite ways to represent any fraction (or number for that matter). Don't believe us? Let's take 1/3. 2/6, 3/9, 4/12 ... 10001/30003 are all equivalent fractions (and we could keep going)! If you know the basics of what a fraction is, this is a great tutorial for recognizing when fractions are equivalent and then simplifying them as much as possible!
Equivalent fractions on the number line
Equivalent systems of equations and the elimination method
Learn about the elimination method, which is a way manipulate systems of equations in order to solve them algebraically. This is actually very similar to the way we manipulate single equations in order to solve them!
Estimating and rounding with decimals
Laziness is usually considered a bad thing. But sometimes, it is useful to be lazy in a smart way. Why do a big, hairy calculation if you just need a rough estimate? Why keep track of 2.345609 when you only need 2.35? This tutorial will get you comfortable with sometimes being a little rough with numbers. By being able to round and estimate them, it'll only add one more tool to your toolkit.
Estimating infinite series
We've spent a lot of time thinking about whether a series converges or diverges. But, even if we can determine that a series converges, how can we figure out what it converges to? This tutorial will show techniques of estimating what a series converges to and also determining how good our estimates are. This is super useful because most series can't be precisely evaluated (like we were able to do with infinite geometric series).
Estimating limits from graphs
In this tutorial, we will build our ability to visualize limits by estimating them based on graphs of functions. We will look at both one-sided and two-sided limits.
Estimating line of best fit
Lines are widely used to model relationships between two quantitative variables. In this tutorial, for scatter plots that suggest a linear association, we'll informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line Common Core Standard: 8.SP.A.2
Before Rome, the Etruscan civilization ruled much what is now Italy. The Etruscans left fine metalwork, elaborate tombs and a deep mark on ancient Roman culture.
Eukaryotic cells
Unlike prokaryotes, eukaryotic cells have a nucleus and other membrane-bound organelles that allow the cell to compartmentalize different functions. These videos will provide an overview of eukaryotic cells and will take you through the structures and functions of the various organelles found within the plasma membrane.
Euler's Method
Most differential equations cannot be solved by "hand" (through analytical means). However, this doesn't mean that we have to give up! As we'll see in this tutorial, we can use numerical methods to approximate a solution to a differential equation. In particular, we'll learn about Euler's Method which is a fairly intuitive way to do this!
Evaluating expressions
Wait, why are we using letters in math? How can an 'x' represent a number? What number is it? This tutorial is great if you're just beginning to delve into the world of variables and expressions.
Evaluating expressions with unknown variables
When solving equations, there is a natural hunger to figure out what an unknown is equal to. This is especially the case if we want to evaluate an expression that the unknown is part of. This tutorial exposes us to a class of solvable problems that challenges this hunger and forces us to be the thinking human beings that we are! In case you're curious, these types of problems are known to show up on standardized exams to see if you are really a thinking human (as opposed to a robot possum).
Evaluating expressions word problems
Learn how to plug in values to evaluate real-world expressions.
Evaluating functions
Learn how to find the value of a function for a given input value.
Evolution and human health (American Museum of Natural History)
Evolutionary knowledge about life on Earth affects our well-being every day, in ways perhaps less apparent but no less important than any of the other sciences. Phylogenetic sequence analysis, or understanding the relatedness of one species to others, can help scientists: identify viruses and develop effective vaccines, identify new species and thus inform biodiversity conservation efforts, or aid in the cultivation of pathogen-resistant crops. And identifying the evolutionary processes of disease agents that transfer between species may help scientists determine how to prevent and treat emerging diseases.
Evolution and natural selection
Evolution and population dynamics
1C: Charles Darwin inaugurated the field of evolutionary biology 150 years ago with the publication of “On the Origin of Species.” You will learn about the driving forces of evolution beyond natural selection and the relationship between populations and their environments. The story of Darwin’s finches will make a lot more sense.
Exact equations and integrating factors
A very special class of often non-linear differential equations. If you know a bit about partial derivatives, this tutorial will help you know how to 'exactly' solve these!
Expanded form for decimals
Learn how to write a decimal like 3.14 in expanded form (3 + 0.1 + 0.04).
Expected value
Now that we know what a random variable is, we can think about expected value. As we'll see, it can be viewed as a probability-weighted average of possible outcomes!
Calling great explainers of the world! Help us answer frequently asked biology questions by writing clear, deep, engaging explanations. Your explanation may even be selected for an article on Khan Academy (with full credit to you)!
Exponent properties
In this tutorial, you will learn about how to manipulate expressions with exponents in them. We'll give lots of example to make sure you see a lot of scenarios. For optimal learning (and fun), pause the video before Sal does an example.
Exponential form of complex numbers
Exponential growth and decay
From compound interest to population growth to half lives of radioactive materials, it all comes down to exponential growth and decay.
In 3rd grade, you learned that there is an easier way to write "5+5+5". You saw that 5+5+5=3x5. But is there an easier way to write repeated multiplication (like "5x5x5")? Absolutely! That's exactly what exponents are for! Common Core Standard: 6.EE.A.1
Exponents with negative bases
Evaluate exponential expressions with negative bases, use odd and even rules to determine the sign of an exponential expression, and learn about powers of 1 and 0.
Expressionism experiments in Germany were dominated by two groups of artists, Die Brücke (the Bridge) and Der Blaue Reiter. Here find Ernst Ludwig Kirchner, Emil Nolde, and the Russians, Wassily Kandinsky and Alexei von Jawlensky. Meanwhile in Vienna, Egon Schiele was exploring representations of the body with wild, restless energy.
Extracurricular and leadership activities
Whether you play sports, act in theatrical productions, or babysit younger siblings - extracurricular activities include everything you do when you're not in class. Admissions officers explain how they evaluate these activities in the context of your college application, and current college students tell their stories of extracurricular activities in high school.
Extravagant inventions
On what did the rulers of late eighteenth-century Europe spend their money? On wars, palaces, and the arts—but also extravagant mechanical furniture. Both the King of Prussia and the Empress of Russia entertained themselves and their guests with ingeniously concealed and automated drawers as well as hidden compartments built within some of the most elegant desks and tables ever devised.
FRENCH HORN: Interview and Demonstration with Principal John Cerminaro
Each of the instruments in this family are made to sound by the vibrations of the player's lips combined with a steady stream of breath. Learn from the All-Star Orchestra players themselves about the special attributes of the trumpet, French horn, trombone, bass trombone and tuba.
Factoring and roots of higher degree polynomials
Factoring quadratics are now second nature to you. Even when traditional factoring is difficult, you know about completing the square and the quadratic formula. Now you're ready for something more interesting. Well, as you'll see in this tutorial, factoring higher degree polynomials is definitely the challenge you're looking for!
Factoring polynomials 1 - Common factors
Learn how to take a common monomial factor out of a polynomial expression.
Factoring polynomials 2 - Quadratic forms
Learn how to factor quadratic polynomials of the form ax^2+bx+c as the product of two linear binomials. Learn how to identify these forms in more elaborate polynomials that aren't necessarily quadratic.
Factoring polynomials 3 - Special product forms
Learn how to factor quadratics that have the "perfect square" form and the "difference of squares" form. Learn how to identify these forms in more elaborate polynomials that aren't necessarily quadratic.
Factoring quadratic expressions
Not only is factoring quadratic expressions (essentially second-degree polynomials) fun, but it is good for you. It will allow you to analyze and solve a whole range of equations. It will allow you to impress people at parties and move up the career ladder. How exciting!
Factoring quadratics
Just saying the word "quadratic" will make you feel smart and powerful. Try it. Imagine how smart and powerful you would actually be if you know what a quadratic is. Even better, imagine being able to completely dominate these "quadratics" with new found powers of factorization. Well, dream no longer. This tutorial will be super fun. Just bring to it your equation solving skills, your ability to multiply binomials and a non-linear way of thinking!
Factoring simple expressions
You already know a bit about multiplying expressions. We'll now reverse course and look at how to think about an expression as the product of simpler ones (just like we did when we find the factors of numbers).
Factoring special products
You will encounter very factorable quadratics that don't always seem so. This tutorial will expand your arsenal by exposing you to special products like difference-of-squares and perfect square quadratics.
Factoring using imaginary numbers
Factors and multiples
In this tutorial, we'll begin to think about the numbers that "make up" the number. This will be useful throughout our study of math. Whether we are adding fractions, exploring mystical number patterns, or breaking computer codes, factoring numbers are key! Let's go, kids! Common Core Standard: 4.OA.B.4
Factors that affect chemical equilibrium
In this tutorial, we will learn about the different factors that affect a reaction mixture at equilibrium.
Fahrenheit and Celsius conversion
There are three major conventions for measuring temperature in the world, Fahrenheit, Celsius and Kelvin. If converting between the three gives you cold feet, then this tutorial might warm them up.
Family and community
Fancy multiplication and division word problems
In this tutorial, we'll start to challenge you with more sophisticated multiplication and division word problems. If you understand mult-digit multiplication and long division, you have all the tools you need to tackle these. May the force be with you!
Fashion meets art
Designers Jonathan Saunders, Preen, and Simone Rocha combine textures and materials in new and inventive ways after meeting their favourite modern works of art. Learn the terms and materials used in art and fashion, and test your knowledge with a quiz.
Fat and protein metabolism
1D: The ham and cheese sandwich you just enjoyed need to be processed by the cells of your body. In addition to the sweet glucose we happily consume, we also take in fat (great for storing energy compactly) and proteins (which can be metabolized to produce energy or used as building blocks for innumerable parts of your body). These tutorials will shed light on the key metabolic pathways governing the metabolism of fats and proteins.
Fauvism and Matisse
Les Fauves (the wild beasts) used color the way most artists use line, to define form in space. From these brilliant early experiments, Henri Matisse went on to create some of the most daring and satisfying art of the early 20th century avant-garde.
Features of quadratic functions
Federal reserve (central banks)
You know that the Federal Reserve (or central banks in general) controls the money supply and short-term interest rates. But how exactly do they do this. Even more, how is "quantitative easing" different than regular open market operations. This tutorial explains it all in the context of the Federal Reserves attempts to stave off deflation during the 2008-2012 recession.
Fetal Circulation
At one stage or another in development, every friend you know had gill slits and a tail. Pretty crazy thought, huh? Fetal development is incredible, and it’s important to understand exactly how it happens. The structure and function of the circulatory system is incredibly complex, and fetuses are no exception. Find out how the heart and circulatory system work in the fetus!
Figuration and abstraction in post-war Britain
In the years of recovery after WWII, British artists explored both abstraction and figural representation as we see with Henry Moore and Anthony Caro.
Film meets art
Three great filmmakers meet three great painters. Watch Chris Nolan, Mike Leigh, and Ken Loach be inspired by atmospheric scenes, depictions of society, and the personalities of the painters themselves. Learn the terms and materials used in art and film, and test your knowledge with a quiz.
Financial aid application process
Financial aid forms, including the FAFSA and CSS Profile, represent one of the first step in the financial aid process when you are applying to college.
Financial aid packages
Once a college accepts you, they will take the information that you provided on your financial aid forms to determine your financial aid package. Now comes one of the most important parts of the college admissions process: comparing offers to determine which of your top schools provided enough aid to make college affordable.
Finding angles in shapes
Do the angles in a triangle always add up to the same thing? Would I ask it if they didn't? What do we know about the angles of a triangle if two of the sides are congruent (an isosceles triangle) or all three are congruent (an equilateral)? This tutorial is the place to find out. Common Core Standard: 8.G.A.5
Finding inverse functions
Learn how to find the inverses of polynomials functions, radical functions, and rational functions.
Finding inverses of matrices
We've talked a lot about inverse transformations abstractly in the last tutorial. Now, we're ready to actually compute inverses. We start from "documenting" the row operations to get a matrix into reduced row echelon form and use this to come up with the formula for the inverse of a 2x2 matrix. After this we define a determinant for 2x2, 3x3 and nxn matrices.
Finding limits algebraically
We often attempt to find the limit at a point where the function itself is not defined. In this tutorial, we will use algebra to "simplify" functions into ones where it is defined. Given that the original function and the simplified one may be identical except for the limit point in question, this is a useful way of finding limits.
Finding mistakes in one-step equations
Practice spotting the mistake in someone else's work as they attempt to solve one-step equations.
Finite geometric series
Whether you are computing mortgage payments or calculating how many users your website will have after a few years, geometric series show up in life far more than you imagine. This tutorial will review all the important concepts and more!
First things first
New to art? This is a good place to start. Art gives us access to the way people at different moments in history have understood the world. Jump in and explore!
Florence, the Late Gothic
When Vasari wrote his enormously influential book, Lives of the Artists, in the 16th century, he credited Giotto, the 14th century Florentine artist with beginning "the great art of painting as we know it today, introducing the technique of drawing accurately from life, which had been neglected for more than two hundred years." For Vasari, Giotto was the first artist to leave behind the medieval practice of painting what one knows and believes, for what one sees. This tutorial looks at painting and sculpture in Florence to highlight some of the most influential art of the 14th century.
Fluid Dynamics
Fluids at rest
4B: Fluids can be fun even when they’re not moving. A still pond has more going on with it than meets the eye - why exactly do you even float when you decide to jump in on a scalding summer day? It’s because you displace a volume of water that provides a buoyant force upward, counteracting the downward pull of gravity. You will find out why the great Archimedes ran through the streets of Syracuse shouting "eureka" millennia ago in Ancient Greece as we also delve into the concepts of pressure, specific gravity, buoyancy, and Bernoulli's principle.
Fluids in motion
4B: In these videos you'll learn more about fluids. Bernoulli's equation will describe the behavior of fluids in motion. The Venturi effect, turbulence, surface tension and viscosity will be discussed as well as the difference between arteries and veins.
Flux in 3-D and constructing unit normal vectors to surface
Flux can be view as the rate at which "stuff" passes through a surface. Imagine a net placed in a river and imagine the water that is flowing directly across the net in a unit of time--this is flux (and it would depend on the orientation of the net, the shape of the net, and the speed and direction of the current). It is an important idea throughout physics and is key for understanding Stokes' theorem and the divergence theorem.
For kids
Watch videos suitable for kids.
Force of tension
4A: Let’s picture an adventurous rock-climber hanging at the edge of a cliff. Gravity is providing a downward force. The only reason he does not fall down to the valley below is due to the force of tension upward, provided by the muscles of his arm. You will work through some interesting real-world problems involving tensile forces in these tutorials.
Forces on inclined planes
4A: If you've ever moved from one town to another, you are likely familiar with inclined planes. When you push up that 50 pound box into the long flatbed truck , there are several forces at play: the weight of the box, the frictional force between the box and the ramp, and the force of your push. We will explain how these forces interact so next time the move won't be so back-breaking.
Form processing with jQuery
Learn how to use jQuery to process forms in the browser, like to make quizzes or games.
Formation of carboxylic acid derivatives
In this tutorial, Sal shows the formation of carboxylic acid derivatives including esters, acyl chlorides, and amides.
Formation of enolate anions
In this tutorial, Sal and Jay show you how to form enolate anions from aldehydes and ketones.
Forward and futures contracts
In many commodities markets, it is very helpful for buyers or sellers to lock-in future prices. This is what both forwards and futures allow for. This tutorial explains how they work and what the difference is between the two.
Four different blended learning models
Fractional reserve accounting
If you already know a bit of what fractional reserve banking involves, this tutorial will take you deeper by looking at the actual accounting of central banks and banks.
Fractional reserve banking
Most modern economies use a counter-intuitive model of banking called "fractional reserve banking." It is counter-intuitive (and some people would say wrong) because it allows banks to lend out money that it tells depositors is available at any time and essentially involves private banks in money creation. It also creates the possibility of mass instability through bank runs that tend to be mitigated through government regulation and insurance (some would say government subsidy of banks). This tutorial explains how fractional reserve lending works and outlines the good and bad. It also talks about the alternative of full reserve banking.
Let's help you relate improper fractions to mixed numbers!
Fractions and whole numbers
Fractions as division
**Spoiler Alert!** The fraction bar and the division symbol are really saying the same thing.
Fractions intro
Fractions of shapes
Learn how to break shapes into equal parts.
Fractions on the number line
France's many revolutions and republics
Unlike the American Revolution which fairly cleanly transitioned the United States from British rule to a republic, France's process of democratization was much longer and more painful. This tutorial gives a scaffold of that (and gives some context for the book/musical/movie "Les Miserables").
Free energy
Unsure whether a chemical reaction will be spontaneous (take place without added energy)? Gibbs free energy to the rescue! Learn about the Gibbs free energy change (ΔG) and how it can be used to predict reaction spontaneity.
Free radical reaction
In this tutorial, Sal introduces free radical reactions by showing the reaction of methane with chlorine.
French Revolution
"Let them eat cake!" "No, how about we cut your head off instead!" The French Revolution was ugly, bloody and idealistic. This tutorial covers the beginning of the end of the Bourbon rule (actually doesn't really go away for 60 years) and birth of France as a Republic (which will really take about 80 years).
Full-length SAT: Section 3
For more Reading practice, download a full-length SAT and do the questions in section 3. Then watch Sal think through the sentence completion questions from section 3, starting on pg. 48 of the downloadable SAT. Looking for the downloadable full-length SAT? Check out the Full-length SAT topic.
Full-length SAT: Section 5
For more Writing practice, download a full-length SAT and do the questions in section 5. Then watch Sal think through the identifying sentence errors and improving sentences questions from section 5, starting on pg. 54 of the downloadable SAT. Looking for the downloadable full-length SAT? Check out the Full-length SAT topic.
Function introduction
Relationships can be any association between sets of numbers while functions have only one output for a given input. This tutorial works through a bunch of examples of testing whether something is a valid function. As always, we really encourage you to pause the videos and try the problems before Sal does!
Functional groups
The chemistry of every organic compound is determined by the functional groups it contains. In this tutorial, we put the fun back into identifying functional groups!
Make your code more re-usable by grouping it into functions, and then make those functions accept parameters and return values.
Functions and equations
Understand the subtle differences and similarities between functions and equations. In this exercise, we will see how an equation can be turned into a function.
Functions and function notation
Functions and linear transformations
People have been telling you forever that linear algebra and matrices are useful for modeling, simulations and computer graphics, but it has been a little non-obvious. This tutorial will start to draw the lines by re-introducing you functions (a bit more rigor than you may remember from high school) and linear functions/transformations in particular.
Functions defined by integrals
Let's explore functions defined by definite integrals. It will hopefully give you a deeper appreciation of what a definite integral represents.
Fundamental Theorem of Algebra
This tutorial will better connect the world of complex numbers to roots of polynomials. It will show us that when we couldn't find roots, we just weren't looking hard enough. In particular, the Fundamental Theorem of Algebra tells us that every non-zero polynomial in one-variable of degree n has exactly n-roots (although they might not all be real!)
Fundamental Theorem of Calculus
You get the general idea that taking a definite integral of a function is related to evaluating the antiderivative, but where did this connection come from. This tutorial focuses on the fundamental theorem of calculus which ties the ideas of integration and differentiation together. We'll explain what it is, give a proof and then show examples of taking derivatives of integrals where the Fundamental Theorem is directly applicable.
Further learning
Ideas of how you could continue your learning journey in algorithms.
Further learning in SQL
Get ideas for how you can continue learning more about SQL and databases.
GDP and the circular flow of income and expenditures
Economics can some times get confusing because one person's expenditure is another person's income which can then be used for expenditure and on and on and on. Seems very circular. It is. This tutorial helps us grapple with this and introduces us to the primary tool economists use to measure a nations productivity/income/expenditure--GDP (gross domestic product).
GMAT Data Sufficiency
Sal works through the 155 data sufficiency questions in chapter 6 of the the 11th edition of the official GMAC GMAT Review (ISBN Number: 0-9765709-0-4 published in 2005)
GMAT: Problem Solving
Sal works through the 249 problem solving questions in chapter 5 of the the 11th edition of the official GMAC GMAT Review (ISBN Number: 0-9765709-0-4 published in 2005)
Gabon is an equatorial nation largely covered by rain forest. National parks comprise some 10 percent of the nation's territory. The Fang are likely the largest ethnic group.
Galaxies are the basic building blocks of the universe. Even in a tiny patch of sky, many hundreds can be seen. By surveying such regions, astronomers estimate that there may be 100 billion galaxies in the observable universe. These galaxies are not uniformly spread through space--they commonly occur in pairs, groups, clusters, and superclusters. The articles and videos in this tutorial depict the different types of galaxies in our Universe, and the interactions between them. The final video in this tutorial looks at how one of the world's largest telescopes is bringing distant galaxies into view.
Galvanic cells
Batteries power much of our lives (literally). In this tutorial, we'll use our knowledge of oxidation and reduction to understand how Galvanic/Voltaic cells actually work.
Gas Exchange
If you think of your lungs as a mini factory, you can think of the gases as goods that your body trades. Humans need oxygen for important metabolic activities. For example, when you exercise, your breathe more because your body needs more oxygen! These metabolic activities produce carbon dioxide, which is something your body needs to get rid of to avoid blood acidity. So, keeping with the example of your lungs as a factory, oxygen is an import good, and carbon dioxide is an export good! Learn more about the specific mechanisms of this “goods exchange” in the tiny air sacs of the lungs: the alveoli.
Gastrointestinal system
3B: You begin with mechanical digestion in your mouth, where starches are also broken down. The bolus of food travels down your esophagus into the stomach, where it sits and churns, further allowing for breakdown of biomolecules like fat and protein.The contents of the stomach then enter the small intestine, which is where the bulk of digestion and absorption of nutrients occurs through the work of shear mechanical (peristalsis) and chemical (digestive enzymes) force. Then our food commutes through the large intestine (AKA the colon), where most of the fluid absorption in the GI tract occurs. Phew! Eating sure is a lot of work! By Raja Narayan.
Gastrointestinal system introduction
Do you live to eat, or eat to live? Folks fall on both sides of this question, but who deny the powerful role that food and water play in our everyday lives. If we were cars, food and water would be the gasoline. Eating keeps us moving, laughing, playing, and learning. The energy from food is carefully extracted through a process of ingestion, digestion, and absorption, and requires one long (very long!) tube with a couple of key organs (liver, pancreas) sprouting off of it. Go ahead and grab a bite to eat before we get started…
Geithner Plan
The poop really started to hit the fan in the fall of 2008. When the new administration took office in early 2009, the poop was still there. This is tutorial explains an attempt--probably not a well thought out one--to clean the poop and slow the fan. Videos on the Geithner Plan to solve the continuing banking crisis in early 2009.
Gem carving in antiquity
Precious markers of culture and status, carved gemstones were prized by Greek and Roman elites. Discover how engraved gems were created, used, and admired.
Gene control
1B: Cells have many intricate mechanisms which regulate expression of genetic material - from transcription of RNA to translation of protein. At every point in this process, enzymes in your body can step in to modulate how much or how little RNA, or protein is produced from the genome. Sometimes, these genetic controls go awry, and so cells grow without inhibition - this is often how tumors develop, a pathogenic process we will also explore.
Genetic and ecosystem biodiversity
Diversity is important at the level of genes and ecosystems, as well as species.
Genetic mutations
1C: Mutations are not always a bad thing - they give rise to much of the spice and flavor of life. But sometimes they are a result of environmental injury and can give rise to malignant disease processes like cancer. We will look at the causes and types of genetics mutations in this series as we also examine their effect on biological systems.
Geometric sequences
What happens when the ratio between successive terms in a sequence is the same (or it has a "common ratio")? Well, then we'd be dealing with a geometric sequence (which comes up extremely frequently in mathematics).
Geometric series
Whether you are computing mortgage payments or calculating the distance traveled by a bouncing ball, geometric series show up in life far more than you imagine. This tutorial will review all the important concepts and more!
Geometric transformations with matrices
We'll now see one of the most powerful applications of matrix multiplication--geometric transformations. This is core of what videos games and computer-based animation uses to "transform" figures based on movement or perspective. You probably never thought matrices could be so much fun!
Geometry problems on the coordinate plane
You are familiar with the ideas of slope and distance on the coordinate plane. You also feel comfortable with congruence an similarity and many of the other core ideas in Euclidean geometry. In this, tutorial, Descartes and Euclid are forced to work together as we tackle geometry problems on the coordinate plane!
Geometry readiness warmup
George Washington
George Washington was the first President of the United States but before that he was the General of the Continental Army. Hear Pulitzer Prize winning historian Joe Ellis discuss George Washington with Aspen Institute's Walter Isaacson.
Get to know our site (higher education math)
Get to know our site (parents and tutors)
Getting a feel for equations and inequalities
The core underlying concepts in algebra are variables, expressions, equations and inequalities. You will see them throughout your math life (and even life after school). This tutorial won't give you all the tools that you'll later learn to analyze and interpret these ideas, but it'll get you started thinking about them.
Getting started
Getting started (K-12 math classrooms)
Getting started (out-of-school-time programs)
Republic of Ghana is a country in West Africa that is home to numerous peoples including the Asante
Gibbs free energy
An introduction to Gibbs free energy and how free energy relates to spontaneity and the equilibrium constant
Giles Shih - President and CEO of BioResource International
Giles Shih, President and CEO of BioResource International, describes his company and explains how producing "big green chickens" will help feed the world.
Global contemporary: 1980-present
Global contemporary art is characterized by a transcendence of traditional conceptions of art and is supported by technological developments and global awareness. Digital technology in particular provides increased access to imagery and contextual information about diverse artists and artworks throughout history and across the globe.
Global modernisms in the 21st century
Culture, like capitalism, is increasingly global. Artists work internationally and no single city is now the center of the art world. Istanbul, Beijing, Bogotá, Beirut, Lagos, and São Paulo all support thriving communities of artists.
Global threats to biodiversity
Local impacts can have global consequences. Humans are causing accelerating changes in the atmosphere, on land, and in the oceans. Life in these changing environments is tough for many species.
Review Big History vocabulary.
Review Big History vocabulary.
The name glycolysis means "sugar-splitting," and sure enough, this metabolic pathway splits glucose into two three-carbon molecules. Learn more about the steps of glycolysis and how it is used in both cellular respiration and fermentation.
Goryeo dynasty (918-1392)
No, we’re not talking about the dark subculture we know as Goth! We’re talking about the style of art and architecture In Europe from the 1100s to the beginnings of the Renaissance at about 1400. Hopefully by the end of this tutorial when someone says Gothic, you’ll think of enormous stained-glass windows in churches whose vaulted ceilings reach toward heaven and not black clothing and dark eyeliner!
Ever walk on hill (or any wacky surface) and wonder which way would be the fastest way up (or down). Now you can figure this out exactly with the gradient.
Grants and scholarships
Free money? It might sound too good to be true, but that's just what scholarships and grants - two terms used interchangeably - offer if you put in the time to apply!
Graph representation
Learn how to describe graphs, with their edges, vertices, and weights, and see different ways to store graph data, with edge lists, adjacency matrices, and adjacency lists.
Graphical representation of systems of equations
Learn how to solve a system of equations by considering its graph, which turns out to be a VERY effective method.
Graphing and analyzing proportional relationships
When you walk at a constant rate, the relationship between the distance you've walked and the time you've spent walking is proportional. In this tutorial, we learn about proportional relationships (constant rates) and their graphs..
Graphing linear equations in slope-intercept form
Math is beautiful because there are so many way to appreciate the same relationship. In this tutorial, we'll use our knowledge of slope to actually graph lines that have been expressed in slope-intercept form.
Graphing linear inequalities
In this tutorial we'll see how to graph linear inequalities on the coordinate plane. We'll also learn how to determine if a particular point is a solution of an inequality.
Graphing on the coordinate plane
How can we communicate exactly where something is in two dimensions? Who was this Descartes character? In this tutorial, we cover the basics of the coordinate plane. We then delve into graphing points and determining whether a point is a solution of an equation. This will be a great tutorial experience if you are just starting to ramp up your understanding of graphing or need some fundamental review.
Graphing two-variable inequalities
Learn how to graph two-variables linear inequalities and systems of two-variable linear inequalities.
Graphing with intercept
There are many ways to graph a line and this tutorial covers one of the simpler ones. Since you only need two points for a line, let's find what value an equation takes on when x = 0 (essentially the y-intercept) and what value it takes on when y = 0 (the x-intercept). Then we can graph the line by going through those two points.
Graphs of proportional relationships
Learn how to graph proportional relationships.
Graphs of rational functions
Combine your knowledge of intercepts, horizontal asymptotes, vertical asymptotes, and removable discontinuities, in order to analyze entire graphs of rational functions!
Graphs of trig functions
The unit circle definition allows us to define sine and cosine over all real numbers. Doesn't that make you curious what the graphs might look like? Well this tutorial will scratch that itch (and maybe a few others). Have fun.
Greater than and less than basics
Equality is usually a good thing, but the world is not a perfect place. No matter how hard we try, we can't help but compare one thing to another and realize how unequal they may be. This tutorial gives you the tools to do these comparisons in the mathematical world (which we call inequalities). You'll become familiar with the "greater than" and "less than symbols" and learn to use them.
Greatest common divisor
You know how to find factors of a number. But what about factors that are common to two numbers? Even better, imagine the largest factors that are common to two numbers. I know. Too exciting!
Greatest common factor
You know how to find factors of a number, but what about factors that are common to two numbers? Even better, imagine the largest factors that are common to two numbers. These are called the greatest common factors (GCF) or sometimes greatest common divisors (GCD). Yay, acronyms! Too exciting! Common Core Standard: 6.NS.B.4
Greek Debt Crisis
The Greek government incurred debt beyond its means but didn't have control over its own currency to inflate away its obligations. From austerity, to a bailout, to leaving the Eurozone, none of the options looked great. In this tutorial, Sal walks through the situation Greece was in and its options (these videos were made as the crisis was unfolding).
Greek art (quiz)
Green's theorem
It is sometimes easier to take a double integral (a particular double integral as we'll see) over a region and sometimes easier to take a line integral around the boundary. Green's theorem draws the connection between the two so we can go back and forth. This tutorial proves Green's theorem and then gives a few examples of using it. If you can take line integrals through vector fields, you're ready for Mr. Green.
Growth and Metabolism
Find out what helps you grow and how we can measure growth
HIV and
HTML/JS: DOM access methods
Learn how to use the DOM API from JavaScript to access elements in your webpages.
HTML/JS: DOM animation
Learn how to animate parts of your webpage, using three different techniques- window.setInterval, window.requestAnimationFrame, and CSS animations/transitions.
HTML/JS: DOM events
Learn how to use the DOM API from JavaScript to make your webpage react to user events, like clicking, scrolling, and entering fields in forms.
HTML/JS: DOM modification
Learn how to use the DOM API from JavaScript to modify aspects of your webpage elements like their styles and attributes, plus create new elements from scratch.
HTML/JS: Further learning
Now that you've learned how to manipulate your webpages with JavaScript, what else could you learn?
HTML/SS: Further learning
What can you do to keep learning HTML and CSS?
Hair Dryer
Hair Dryer
Haitian Revolution
Yes, you are right. Haiti is not in Europe. We put the tutorial here because it was a French colony and its own revolution is closely linked to that of France's. Possibly one of the saddest histories that a nation can have, this tutorial tries to give as much context as possible for the birth of Haiti.
Han dynasty (206 B.C.E–220 C.E.)
Hans Holbein
Hardware, software and facilities decisions overview
Harmonic motion
Ever watch a slinky gyrate back and forth. This is harmonic motion (a special class of oscillatory motion). In this tutorial we'll see how we can model and deal with this type of phenomena.
Harnessing science and education for biodiversity conservation
We are changing global environments and losing biodiversity at unprecedented rates. Our greatest tool for stemming that loss is the symbiosis between science and education.
Health Care System
The health care system in the United States is rapidly changing. To better understand these changes, we review the health care insurance, drug pricing, physician compensation, and much more! join us as we explore the basics about the Health Care system in the US, including a comparison with European healthcare.
Heart Depolarization
Your heart relies on the flow of electricity to maintain a steady, consistent beat - like an automatic pump that maintains a regular rate and rhythm throughout your life! There are specialized heart cells that allow positive current to travel quickly throughout the heart muscle. In these videos, we’ll check out the flow of this positive charge on a macroscopic and microscopic level.
Heart Muscle Contraction
Your heart is made of a special type of muscle, found nowhere else in the body! This unique muscle is specialized to perform the repetitive task of pumping your blood throughout your body, from the day you’re born to the day you die. We’ll take an in-depth look of how the heart accomplishes this on a cellular level, and learn about the proteins actin and myosin that are the workhorses that tug and pull on one another to create every single muscle contraction. You’ll appreciate the fact that your heart beat is a fairly sophisticated process!
Heart depolarization
Your heart relies on the flow of electricity to maintain a steady, consistent beat - like an automatic pump that maintains a regular rate and rhythm throughout your life! There are specialized heart cells that allow positive current to travel quickly throughout the heart muscle. In these videos, we’ll check out the flow of this positive charge on a macroscopic and microscopic level.
Heart failure
Heart of algebra
Heart valve diseases
Hedge funds
Hedge funds have absolutely nothing to do with shrubbery. Their name comes from the fact that early hedge funds (and some current ones) tried to "hedge" their exposure to the market (so they could, in theory, do well in an "up" or "down" market as long as they were good at picking the good companies). Today, hedge funds represent a huge class investment funds. They are far less regulated than, say, mutual funds. In exchange for this, they aren't allowed to market or take investments from "unsophisticated" investors. Some use their flexibility to mitigate risk, other use it to amplify it.
Heian period (794–1185)
Alexander the Great died in 323 B.C.E. leaving a vast empire to his generals, the Diadochi (successors). The Diadochi divided Alexander's empire amongst themselves—the Hellenistic dynasties of the Seleucids in the Near East, the Ptolemies in Egypt, the Antigonids in Macedonia, and the wealthy Attalid kings of Pergamon who ruled most of western Asia Minor. Greek culture flourished across an enormous area, but at the same time, these "Hellenized" peoples infused Greek art with a drama and breadth beyond anything the Greeks had previously produced.
Hematologic system introduction
Roughly 5 L of blood fill your arteries, veins, capillaries, and venules. What’s it good for you ask? It carries oxygen to help your cells carry out respiration in addition to a number of other substances like lipids and hormones throughout the body. In cases of blood loss, such as trauma situations, the physician must be wary of the different blood types. We will explore the intricacies of the hematologic system here.
Heredity and genetics
Heron's formula
Named after Heron of Alexandria, Heron's formula is a power (but often overlooked) method for finding the area of ANY triangle. In this tutorial we will explain how to use it and then prove it!
Since it's shaped like a hexagon and flex rhymes with hex, hexaflexagon it is!
Hieronymus Bosch
High Tech Middle School
8th grade teacher Bryan Harms shares how he used Khan Academy in his project-based classroom in Chula Vista, California.
High school classes
Take the challenging class or get the good grade? Learn more about what admissions looks for on your high school transcript and how current college students navigated the process when in high school.
Higher education case studies
Hindu art
Hinduism has very ancient roots but began to assume its mature form only in the fourth century C.E. The most characteristic features of mature Hinduism are the worship of divine images and the construction of temples to house these images. Hinduism has a vast pantheon of male and female deities but pre-eminent among them are Shiva and Vishnu.
Hindu art and culture, an introduction
Hinduism is one of the world’s oldest religions. It has complex roots, and involves a vast array of practices and a host of deities. Its plethora of forms and beliefs reflects the tremendous diversity of India, where most of its one billion followers reside. Hinduism is more than a religion. It is a culture, a way of life, and a code of behavior.
Hinduism has no historical founder, and no central authority. It includes enormously diverse beliefs and practices, which vary over time and among individuals, communities, and regional areas. Its authority—its beliefs and practices—rests on a large body of sacred texts that may date back more than 3,000 years.
Histograms are similar to dot plots and bar graphs, but they work a little bit differently. In this tutorial, we'll learn how histograms work and when to use them.
Historical circumstances explained by AD/AS
In the last tutorial, we claimed that the aggregate demand and aggregate supply model (AD-AS) would be useful for analyzing macroeconomic events. Well, in this tutorial, we'll do exactly that.
History of life on Earth
Earth is over 4.5 billion years old. How do we know this? When did life first emerge? From the dawn of Earth as a planet to the first primitive life forms to our "modern" species, this tutorial is an epic journey of the history of life on Earth.
History of philosophy
Are you interested in learning more about what philosophers have said in the past? Check out this section to learn more about the history of philosophical thought.
Home buying process
Buying a home involves a lot more than writing a check and getting keys. It could very well be the largest transaction that either the buyer or seller does in their life. Because that, it is an involved process that can sometimes be confusing. Whether we're talking title insurance, escrow, or contingencies, the goal of this tutorial is to demystify the process.
Home equity and personal balance sheets
This old and badly drawn tutorial covers a topic essential to anyone planning to not live in the woods -- your personal balance sheet. Since homes are usually the biggest part of these personal balance sheets, we cover that too.
Homogeneous equations
In this equations, all of the fat is fully mixed in so it doesn't collect at the top. No (that would be homogenized equations). Actually, the term "homogeneous" is way overused in differential equations. In this tutorial, we'll look at equations of the form y'=(F(y/x)).
Hormonal regulation
Hour of Code (for Teachers)
Hour of Code™
Hour of Databases
Hour of Drawing with Drag-and-Drop Code
Learn how to program drawings with JavaScript and ProcessingJS, by dragging and dropping blocks. Optimized for mobile tablets.
Hour of Webpages
Other household items you know
Housing price conundrum
Back before the 2008 credit crisis, Sal was perplexed by why housing prices were going up so fast and theorized that it was a bubble forming (he was right). These pre-2008 videos are fun from a historical point-of-view since they were made before all the poo poo hit the fan.
How 10 relates to place value
How do scientists study dinosaurs?
To learn about ancient life, scientists study fossils. Finding these traces of ancient life takes time and experience. Paleontologists search carefully for bits of exposed bone, then typically transport the large piece of rock that contains the fossil back to the lab. Trackways provide some of the best clues about dinosaur behavior. Studying living birds and other reptiles, which are related to dinosaurs, gives insight into behavior and biology. Paleontologists also turn to experts in other fields. For example, geochemists analyze fossil bones and teeth for clues about paleoclimate, while paleobotanists examine coprolites for the physical and chemical traces of ancient plants. Together, these scientists are filling in the picture of what these giant dinosaurs ate, how fast they grew, and how long they lived.
How the Constitution deals with civil liberties and privacy in an age of technological change
The Constitution is the supreme law of the United States of America. One of the most interesting question that American society and the U.S. Supreme Court is confronting today - how to translate an 18th century Constitution into the 21st century.
How to Prep for the SAT
How-to guides
Examine an exceptional sculpture from the ancient Huastec culture of Mexico.
Human Evolution: The Evidence
Humans have evolved just like all other species. We modern humans are the only remaining descendants of a once varied family of primates called hominids. All other hominids are now extinct. Fossils and DNA continue to reveal the details of our complex evolutionary history, which extends back millions of years and reveals that humans and other living primates share a common ancestor. New fossil finds continue to clarify what other hominids looked like, and how and when they lived.Technology to study DNA has emerged in the past few decades, adding to what fossils tell us.
Human development
Humanity on Earth
Where do we think humans come from? How and why have we developed as a species. This tutorial attempts to give an overview of these truly fundamental questions. From human evolution (which is covered in more depth in the biology playlist) to the development of agriculture, this tutorial will give you an appreciation of where we've been (and maybe where we're going).
Learn to see three-digit numbers as hundreds, tens, and ones.
How closely bonded are you to your idea of orbitals? In this tutorial, we will learn about hybridized orbitals, the ligers of organic chemistry.
Hybridization and hybrid orbitals
In this tutorial, we will learn about sp3, sp2, and sp hybridization.
Hydrocarbon structures and functional groups
Carbon-containing molecules come in many shapes, sizes and types. Here, get an overview of different hydrocarbon structures, learn about isomers (molecules with the same atoms arranged in different ways), and check out some of the key functional groups found in biological molecules.
Hydrogen bonding in water
Water is key to life, and hydrogen bonding is key to understanding water! Learn how the structure of water and the properties of O and H atoms allow water molecules to "stick" to each other via hydrogen bonds, as well as why water makes a great solvent (why you can dissolve stuff in it so well).
It is no hyperbole to say that hyperbolas are awesome. In this tutorial, we look closely at this wacky conic section. We pay special attention to its graph and equation.
Hypothesis testing
Hypothesis testing with one sample
This tutorial helps us answer one of the most important questions not only in statistics, but all of science: how confident are we that a result from a new drug or process is not due to random chance but due to an actual impact. If you are familiar with sampling distributions and confidence intervals, you're ready for this adventure!
Hypothesis testing with two samples
You're already familiar with hypothesis testing with one sample. In this tutorial, we'll go further by testing whether the difference between the means of two samples seems to be unlikely purely due to chance.
IIT JEE Questions
Questions from previous IIT JEEs
IS-LM Model
In this tutorial, we begin thinking about the impact of real interest rates on planned investment and output. We then use this to help us plot the IS curve. We then think about how, assuming a fixed money supply, as there is more economic activity, people are willing to pay more for money (helps us plot the LM curve). Finally, we use the IS-LM model to think about how fiscal policy can impact both GDP and real interest rates. You should watch the Keynesian Cross tutorial before this one.
Ideal gas equation
In this tutorial, Sal shows you how to use the ideal gas equation in calculations.
Ideas for parents and mentors
Identifying proportional relationships
Learn how to tell whether two quantities are proportional by thinking about equivalent ratios and drawing graphs.
Identity, the body, and the subversion of Modernism: United States
Here is a broad range of art from the United States including the Process art of Hesse, Benglis, and Windsor, the appropriation of Sherman, Colescott, and Levine and Carrie Mae Weems' confrontation with the brutality of slavery and racism.
Ife is today regarded as the spiritual heartland of the Yoruba people living in Nigeria, the Republic of Benin and their many descendants around the world. It is rightly regarded as the birthplace of some of the highest achievements of African art and culture, combining technical accomplishment with strong aesthetic appeal.
Igor Stravinsky: Suite from The Firebird
The magical world of Russian folklore comes to life as Music Director Gerard Schwarz tells the story of The Firebird ballet, and looks into the music of Igor Stravinsky's thrilling musical score that was commissioned by the legendary impresario Serge Diaghilev and his Ballets Russes.
Immune system
3B: Chances are , you’ve had a fever or a cough at least once in your life (unless you live in a bubble, in which case you should probably go out more!) Have you ever wondered why your body reacts this way? Your body has a deadly arsenal of weapons against microbial invaders, ranging from bacteria and viruses to protozoans and fungi. We have specialized cells that destroy foreign bodies through mechanisms such as consumption, expulsion, and degradation. You will become acquainted with the interplay of the numerous soldiers in your body’s army and how they keep you healthy!
Immunologic system introduction
Chances are, you’ve had a fever or a cough at least once in your life (unless you live in a bubble, in which case you should probably go out more!) Have you ever wondered why your body reacts this way? Your body has a deadly arsenal of weapons against microbial invaders, ranging from bacteria and viruses to protozoans and fungi. We have specialized cells that destroy foreign bodies through mechanisms such as consumption, expulsion, and degradation. You will become acquainted with the interplay of the numerous soldiers in your body’s army and how they keep you healthy!
Implicit differentiation
Like people, mathematical relations are not always explicit about their intentions. In this tutorial, we'll be able to take the derivative of one variable with respect to another even when they are implicitly defined (like "x^2 + y^2 = 1").
Importance of college
Find out just how much college can impact your life, both during your time in school and after graduation.
Improper integrals
Not everything (or everyone) should or could be proper all the time. Same is true for definite integrals. In this tutorial, we'll look at improper integrals--ones where one or both bounds are at infinity! Mind blowing!
From their capital, Cuzco, in the central Peruvian Andes, the Inca created a huge empire reaching over 2,400 miles along the length of the Andes. The supreme head of state was the king, considered a living god ruling by divine right and the royal family controlled important areas of government such as the army.
Inclined planes and friction
We've all slid down slides/snow-or-mud-covered-hills/railings at some point in our life (if not, you haven't really lived) and noticed that the smoother the surface the more we would accelerate (try to slide down a non-snow-or-mud-covered hill). This tutorial looks into this in some depth. We'll look at masses on inclined planes and think about static and kinetic friction.
Indefinite integral as anti-derivative
You are very familiar with taking the derivative of a function. Now we are going to go the other way around--if I give you a derivative of a function, can you come up with a possible original function. The symbol which we'll use to denote the anti-derivative will see strange at first, but it will all come together in a few tutorials when we see the connection between areas under curves, integrals and anti-derivatives.
Independent events warmup
Introduction to independent events and frequency analysis using histograms.
Indicating Electrolysis
Without much more than a 9-volt battery, you can break water into its two molecular ingredients, hydrogen and oxygen. The four video modules of this hands-on science snack show you how to build a simple electrolysis device and break up water. Using an acid-base indicator, you'll also observe groovy swirls of color in the water as it changes.
Indigenous Americas
Art of the Indigenous Americas is among the world’s oldest artistic traditions. While its roots lie in northern Asia, it developed independently between c. 10,000 B.C.E. and 1492 C.E., which marked the beginning of the European invasions. Regions and cultures are referred to as the Indigenous Americas to signal the priority of First Nations cultural traditions over those of the colonizing and migrant peoples that have progressively taken over the American continents for the last 500 years.
Proof by induction is a core tool. This tutorial walks you through the general idea that if 1) something is true for a base case (say when n=1) and 2) if it is true for n, then it is also true for n+1, then it must be true for all n! Amazing!
Algebraic inequalities (inequalities with variables) are useful for describing real-world situations. Learn how to write inequalities and graph them on the number line.
Infinite geometric series
You're already familiar with finite geometric series, but you don't want the summation to stop!! What happens if you keep adding? The terms are getting small fast! Can it be that the sum of an infinite number of rapidly shrinking terms can be finite! Yes, often times it can! Mind-blowing! Stupendous!
Infinity . . .
Understanding infinity (all different kinds!). Countable and uncountable infinities. Bigger and smaller infinities.
Inflammatory bowel disease
Inflation and unemployment
Economists have notices a correlation between unemployment and correlation (you may wan to guess what type of correlation). On some level, this tutorial is common sense, but it will give you fancy labels for this relation so that you can sound fancy at fancy parties.
Inflation scenarios
You know about inflation, but now want to look at how thing might play out in different scenarios. This tutorial focuses on when inflation is "acceptable" and when it isn't (and the causes and repercussions).
Inflationary and deflationary scenarios
This tutorial walks through various scenarios of moderate and extreme price changes. Very good way to understand how activity in the economy may impact price (and vice versa).
Most people have had the flu virus at least once in their lives, and it’s usually not a pleasant experience… Fight back with some good information! Learn about typical flu symptoms (and how tell it apart from the cold), and how the flu virus invades your cells to cause disease. Finally, learn how flu vaccines may help prevent you from getting sick, and how we can test and treat the flu just in case you get really ill. Stay healthy, my friends!
Infrared Spectroscopy
In this tutorial, Jay introduces the theory of IR spectroscopy and shows how to analyze simple IR spectra.
Infrared and Ultraviolet/Visible spectroscopy
They aren't the same as what you see in CSI:Miami, but molecules do have fingerprints. No two molecules have the same absorption spectra. We can use this fact to determine the composition of a material, its concentration, and measure it’s kinetics. Learn how to use this fantastic tool now!
Innova Schools
This ambitious educational private project in Latin America plans to become the largest school network in the region by 2018. Currently operating 11 schools in Lima, Peru, the company will have at least 70 world-class schools throughout Peru by 2016. Their vision is to provide affordable high-quality education to Peruvian children.
Inputs and outputs of a function
Extend your understanding of the relationship between the inputs of a function and the outputs of that function.
Insertion sort
Learn insertion sort, another simple but not very efficient way to sort an array of values.
Inspiration from the classical past
Artist and poet Jim Dine draws personal inspiration from antiquity for his contemporary installation at the Getty Villa in Malibu.
Integers Negative numbers
Inetegers can be positive or negative whole numbers. Did you notice any use of negative integers in our day-to-day life? You are right! Canadian winter temperatures, of course! Let's get a visual understanding of positive and negative integers
Integration by parts
When we wanted to take the derivative of f(x)g(x) in differential calculus, we used the product rule. In this tutorial, we use the product rule to derive a powerful way to take the anti-derivative of a class of functions--integration by parts.
Integration using trigonometric identities
You will occasionally encounter integrals in life that involve products of exponents of trig functions. In this tutorial, you will see examples of using trigonometric identities to get these types of integrals into a form that you can actually integrate (and you'll get some practice doing it as well)!
Integumentary system
3B: There is really more than meets the eye with skin. Yes, it does make us look nicer than a bag of bones, muscles, and organs. But it also serves other important purposes which range from guarding the body against infection to sensation to allowing for metabolism of vitamin D. We will explore the structure and function of skin from the macroscopic to the microscopic level in this tutorial.
Integumentary system introduction
There is really more than meets the eye with skin. Yes, it does make us look nicer than a bag of bones, muscles, and organs. But it also serves other important purposes which range from guarding the body against infection to sensation to allowing for metabolism of vitamin D. We will explore the structure and function of skin from the macroscopic to the microscopic level in this tutorial.
Interest as the price of money
Interest basics
This is a good introduction to the basic concept of interest. We will warn you that it is an older video so Sal's sound and handwriting weren't quite up to snuff then.
Interest on credit cards and loans
Most of us have borrowed to buy something. Credit cards, in particular, can be quite convenient (but dangerous if not used in moderation). This tutorial explains credit card interest, how credit card companies make money and a far more silly way of borrowing money called "payday" loans.
Interest rate swaps
Interference of light waves
Internal energy
An introduction to the first law of thermodynamics, internal energy, heat, work, and PV diagrams
Interpreting angles
Now that we know what angles are, let's dig a bit deeper and classify them and understand their properties a bit better. Common Core Standards: 4.MD.C.7, 4.G.A.1
Interpreting features of graphs
Interpret the graphs of functions in terms of the contexts that are modeled by the functions.
Interpreting formulas of basic exponential functions
Learn how to analyze the formulas of basic exponential functions in order to find their common ratio, initial value, and other parameters.
Interpreting function notation
Solve some word problems by interpreting expressions of modeling functions.
Interpreting graphs and tables of basic exponential functions
Learn how to analyze the graphs, or tables of values, of basic exponential functions in order to find their common ratio, initial value, and other parameters.
Interpreting linear expressions
Any expression (mathematical or otherwise) has meaning. Help us match the linear expression to the meaning options given. In some cases, more than one meaning may apply.
Interpreting linear functions and equations
Let's see how we can use linear models to learn about real-world situations
Interquartile range (IQR)
Interquartile range (IQR) is a tool for reasoning about the spread of data. Higher values of IQR tell you the data is more spread out. Lower values of IQR tell you the data is closer together.
Intervals where a function is positive, negative, increasing, or decreasing
Learn about features of functions that have distinct graphical representations: intervals where the function is always positive or always negative, and intervals where the function is always increasing or always decreasing.
Intro to Algorithms
What are algorithms and why should you care? We'll start with an overview of algorithms and then discuss two games that you could use an algorithm to solve more efficiently - the number guessing game and a route-finding game.
Intro to CSS
Learn how to write simple CSS rules, to select based on element, class, or ID, and change the colors of your page.
Intro to Games & Visualizations
A quick tour of the many components of games and visualizations, demonstrated by some of our favorite programs.
Intro to HTML
Learn what HTML is and how to make a webpage with marked up text and images.
Intro to Modern Cryptography
A new problem emerges in the 20th century. What happens if Alice and Bob can never meet to share a key in the first place?
Intro to Natural Simulations
Intro to adding negative numbers
Learn how to add negative numbers.
Intro to addition and subtraction
Adding and subtracting is the basis of all mathematics. This tutorial introduces you to one-digit addition and subtraction. You might become pretty familiar with the number line too!
Intro to addition with two-digit numbers
Learn how to add two-digit numbers. By the end of this tutorial, you'll be able to solve problems like 48+6 and 48+21. More advanced problems like 48+28 will be covered in a future tutorial.
Intro to differential equations
How is a differential equation different from a regular one? Well, the solution is a function (or a class of functions), not a number. How do you like me now (that is what the differential equation would say in response to your shock)!
Intro to hyperbolic functions
You know your regular trig functions that are defined with the help of the unit circle. We will now define a new class of functions constructed from exponentials that have an eery resemblance to those classic trig functions (but are still quite different).
Intro to making webpages interactive
Are you ready to learn how to make your webpages interactive with HTML, JavaScript, and the DOM API? You'll find out with our review quizzes in this tutorial. Get pumped!
Intro to percentages
Intro to programming
If you've never been here before, check out this introductory video first. Then get coding!
Intro to statistics
Statistics help us answer many questions, but not all questions are statistical. In this tutorial, we'll learn to tell the difference between a statistical question and a non-statistical question.
Intro to subtracting negative numbers
Learn how to subtract negative numbers.
Intro to subtraction with two-digit numbers
Learn how to subtract two-digit numbers. By the end of this tutorial, you'll be able to solve problems like 48-9 and 48-31. More advanced problems like 48-29 will be covered in a future tutorial.
Introducing photography
Discover the elements of photography and how to look closely at a photograph.
“Asia” is a term invented by the Greeks and Romans and developed by Western geographers. Culturally, no “Asia” exists, and the peoples who inhabit “Asia” often have little in common with each other. Recognizing the diversity of the huge area conventionally designated “Asia,” the Asian Art Museum has arranged its collections into seven general groupings.
Introduction to Africa
Humans first evolved in Africa, walking upright about five million years ago, and making the first tools about two and a half million years ago using the opposable thumb. The British Museum collection includes objects dating from this time, but also represents historic and contemporary societies across the continent.
Introduction to Euclidean geometry
Roughly 2400 years ago, Euclid of Alexandria wrote Elements which served as the world's geometry textbook until recently. Studied by Abraham Lincoln in order to sharpen his mind and truly appreciate mathematical deduction, it is still the basis of what we consider a first year course in geometry. This tutorial gives a bit of this background and then lays the conceptual foundation of points, lines, circles and planes that we will use as we journey through the world of Euclid.
Introduction to Philip Glass' "Harmonium Mountain"
Philip Glass is often called the most famous living composer today. His short work, Harmonium Mountain, was conceived in partnership with the visual artist Clifford Ross as the soundtrack for an animated film based on ever-changing images of a mountain in Colorado. This analysis explores connections between the visual and musical arts.
Introduction to animation
In this lesson you will use animation tools to bring a ball to life. Along the way you'll learn some basic principles of animation. It's appropriate for all ages and takes 45-60 minutes (beware: animation can be addicting).
Introduction to arithmetic squences
Learn about arithmetic sequences and their main features, the initial term and the common difference. Evaluate arithmetic sequences.
Introduction to cell division
Introduction to cells
Start your cellular journey the right way: with some history and some microscopy! Here, we'll learn more about how cells were discovered, how they can be observed, and how big they are in comparison to other very small things.
Introduction to cellular respiration
How does your body get usable energy from the snack you just ate? Learn the basics of how cells extract energy from fuel molecules, including what redox reactions are and why they are important in the breakdown of fuels.
Introduction to compounds
There are 118 elements on the periodic table, and often you will see them combine in specific ratios to form compounds. In this tutorial we will learn about the different ways that chemists represent, count, and categorize compounds.
Introduction to cultures and religions for the study of AP Art History
An introduction to Judaism, Hinduism, Christianity, Buddhism, and Islam
Introduction to decimals
What are decimal numbers? How are they like fractions?
Introduction to derivatives
Discover what magic we can derive when we take a derivative, which is the slope of the tangent line at any point on a curve.
Introduction to differential calculus
The topic that is now known as "calculus" was really called "the calculus of differentials" when first devised by Newton (and Leibniz) roughly four hundred years ago. To Newton, differentials were infinitely small "changes" in numbers that previous mathematics didn't know what to do with. Think this has no relevence to you? Well how would you figure out how fast something is going *right* at this moment (you'd have to figure out the very, very small change in distance over an infinitely small change in time)? This tutorial gives a gentle introduction to the world of Newton and Leibniz.
Introduction to economics
This tutorial (that only has one video) is an overview of what economics is. In particular it will tell you the difference between microeconomics (the subject you're in right now) and macroeconomics. Really good first watch to give you some context on the world of economics.
Introduction to enzymes
Every second of the day, tiny molecular machines called enzymes are working away busily inside your cells, catalyzing and directing the chemical reactions that keep you alive. Learn more about how enzymes speed up reactions and direct metabolic “traffic” in cells.
Introduction to equivalent algebraic expressions
Learn to tell whether or not two algebraic expressions are equivalent by combining like terms and using the distributive property.
Introduction to exponential growth and decay
Learn about exponential growth and decay, and specifically how it always grows (or diminishes) by equal factors.
Introduction to factorization
Learn what factorization is all about, and warm-up by factoring some monomials.
Introduction to geometric sequences
Learn about geometric sequences and their main features, the initial term and the common ratio. Evaluate geometric sequences.
Introduction to health care in the U.S.
This tutorial introduces the structure of the U.S. health care system, how money flows within it, and an overview of different types of public and private insurance. These videos and questions provide a clear explanation of what is and is not working within the health care system to help frame the health care reform discussion and inform clinicians and the public how to improve quality while decreasing health care spending. Narrated by Dr. Darshak Sanghavi, a pediatric cardiologist and fellow at Brookings Institution.
Introduction to hematologic system
Roughly 5 L of blood fill your arteries, veins, capillaries, and venules. What’s it good for you ask? It carries oxygen to help your cells carry out respiration in addition to a number of other substances like lipids and hormones throughout the body. In cases of blood loss, such as trauma situations, the physician must be wary of the different blood types. We will explore the intricacies of the hematologic system here.
Introduction to intermolecular forces
Introduction to inverses of functions
Learn what inverses of functions are, and gain some experience with them by finding the outputs of the inverse of a function given that function.
Introduction to light waves
Introduction to macromolecules
Learn about monomers, polymers, dehydration synthesis, and hydrolysis reactions! These will get you all set to learn more about the different types of macromolecules.
Introduction to percentages
Percent literally means "per cent" or "per hundred". Learn how to rewrite fractions as percents and vice versa.
Introduction to polynomial operations
Learn how to add and subtract polynomial expressions with one or two variables.
Introduction to polynomials
Learn about polynomial expressions: What are they? How are they constructed? What can we do with them?
Introduction to pulmonary diseases
Introduction to rates
Rates, such as kilometers per hour, help us solve all sorts of important and interesting problems.
Introduction to rational exponents and radicals
In this tutorial you will learn about radicals and about rational exponents and practice how to change back and forth from these two types of notation.
Introduction to ratios
Ratios show up everywhere in life—sports, finance, medicine, cooking, you name it! In this tutorial, we learn what ratios are and how to use them.
Introduction to rigid transformations
The three rigid transformations are translations, rotations, and reflections. Get to know them and gain some experience in performing them on different figures.
Introduction to sequences
Learn what sequences are, and about the explicit and recursive definitions of sequences. Evaluate sequences that are defined recursively.
Introduction to stocks
Many people own stocks, but, unfortunately, most of them don't really understand what they own. This tutorial will keep you from being one of those people (not keep you from owning stock, but keep you from being ignorant about your investments).
Introduction to systems of equations
Whether in the real world or a cliche fantasy one, systems of equations are key to solving super-important issues like "the make-up of change in a troll's pocket". This tutorial shows you how to solve systems of equations graphically.
Introduction to systems of linear equations
Learn what "systems of equations" are and what counts as a solution to such a system.
Introduction to the atom
In chemistry, we will often be thinking about the world on a much smaller scale than you can see with the naked eye. Here we will learn about atoms and elements, tiny particles that make up the world around us. What are atoms, and what kind of properties do they have? How do we weigh and count atoms? We will answer those questions in this section!
Introduction to the domain and range of a function
Learn what the domain and the range of a function are. Practice finding the domain and the range of a function given its graph.
Introduction to the periodic table
How are the elements organized in the periodic table? This tutorial will cover the classification of elements including groups, transition metals, and valence electrons.
Introduction: Applying to college
Hear from Sal and admissions officers on what it takes to put together a college application. Then, begin organizing your own materials with a timeline for the college application process.
Introduction: College admissions
Hear Sal's college admissions story and then start planning your own with a 4-year college admissions timeline.
Introduction: Exploring college options
Hear from Sal and several guidance counselors on what it takes to find great college matches. Then, begin planning your own search with a timeline for exploring college options.
Introduction: Making high school count
Hear from Sal and admissions officers on what it takes to make the most of high school. Then, begin planning your own high school-to-college story with a timeline for making high school count.
Introduction: Paying for college
Hear from Sal and various experts on what it takes to pay for college. Then, begin organizing your own financial aid plan with a timeline focused on paying for college.
Introduction: Wrapping up
Hear from Sal about how he wrapped up the admissions process, and then begin planning your own college transition.
Inverse functions and transformations
You can use a transformation/function to map from one set to another, but can you invert it? In other words, is there a function/transformation that given the output of the original mapping, can output the original input (this is much clearer with diagrams). This tutorial addresses this question in a linear algebra context. Since matrices can represent linear transformations, we're going to spend a lot of time thinking about matrices that represent the inverse transformation.
Inverse trig functions
Someone has taken the sine of an angle and got 0.85671 and they won't tell you what the angle is!!! You must know it! But how?!!! Inverse trig functions are here to save your day (they often go under the aliases arcsin, arccos, and arctan).
Inverting matrices
Multiplying by the inverse of a matrix is the closest thing we have to matrix division. Like multiplying a regular number by its reciprocal to get 1, multiplying a matrix by its inverse gives us the identity matrix (1 could be thought of as the "identity scalar"). This tutorial will walk you through this sometimes involved process which will become bizarrely fun once you get the hang of it.
Investment and consumption
When are you using capital to create more things (investment) vs. for consumption (we all need to consume a bit to be happy). When you do invest, how do you compare risk to return? Can capital include human abilities? This tutorial hodge-podge covers it all.
The following articles and videos explore seventeenth-century Iran through the reign and legacy of one of its most influential rulers, Shah 'Abbas I (reigned 1587–1629).
Iron deficiency anemia and anemia of chronic disease
Irrational numbers
Learn what irrational numbers are. Also learn how to classify numbers as whole, integer, rational, and irrational.
Islam has been an important cultural force in much of Asia for more than five hundred years, and in some parts for more than a thousand. Today, far more Muslims live in other parts of Asia than in the Arab areas of Asia such as Iraq, Saudi Arabia, and Syria.
This tutorial is about magic. Bernini turns stone into flesh and Caravaggio makes the distant stories of the Bible into immediate experiences that take place before our eyes. Here are glorious frescos that dissolve the ceilings of cathedrals and reach up to the infinity of heaven.
JS and the DOM
Learn how to use JavaScript to control the "DOM" (Document Object Model) of a webpage.
James Madison
James Madison is known as the “Father of the Constitution.” Learn more about this founding father with Lynne Cheney, author of “James Madison: A Life Reconsidered” in conversation with Walter Isaacson of the Aspen Institute.
Jason Christiansen - President & CEO of Rigid Industries
Jason Christiansen has heard all the familiar comparisons between running a business and being a team player, but as a former major league baseball player, he steps to the plate with a unique perspective. Christiansen talks about building Rigid Industries and how the company deals with imitation product lines and compares the pressure of standing on the mound to standing before his team of employees.
Johannes Brahms: Academic Festival Overture
Although Johannes Brahms never went to a university, he was offered a PhD from the University of Breslau, but he to receive this he had to compose a work for the school. This became the Academic Festival Overture, described by Brahms as a cheerful potpourri of student songs. The overture has the largest orchestration of any of Brahms’ works, and is among his most charming and optimistic pieces.
Joseon dynasty (1392–1910)
Journey into Cryptography
Explore how we have hidden secret messages through history.
Journey into Information Theory
Explore the history of communication from signal fires to the Information Age
Judaism and art
Judaism is an ancient monotheistic religion with a focus on sacred texts rather than sacred images, making its art an especially interesting area of study.
KIPP has charter schools all over the United States. Find out more about how they are using Khan Academy.
Kamakura period (1185-1333)
Key issues for the study of AP Art History
Here's where you'll find a list of resources for all 250 AP objects in one place. Also some helpful material about dates and maps and how to look at art.
Keynesian Cross
We now build on our consumption function models and start to explore ideas of planned expenditures as a function of output. When plotted with the actual output line, we get our Keynesian Cross which helps us think about whether the economy is operating at its potential.
Keynesian thinking
Whether you love him or hate him (or just consider him a friend that you respect but disagree with every-now-and-then), Keynes has helped define how many modern governments think about their economies. This tutorial explains how his thinking was a fundamental departure from classical economics.
Khan Academy in Idaho
Schools across the state of Idaho are using Khan Academy as part of this initiative. Learn more in this tutorial and at
Khan Academy living room chats
Kinematic formulas and projectile motion
We don't believe in memorizing formulas and neither should you (unless you want to live your life as a shadow of your true potential). This tutorial builds on what we know about displacement, velocity and acceleration to solve problems in kinematics (including projectile motion problems). Along the way, we derive (and re-derive) some of the classic formulas that you might see in your physics book.
Kinetic molecular theory of gases
4A: Though we may not be able to always visualize them, gases are comprised of atoms and molecules. We’ll explore the molecular behavior that determines the pressure and temperature of a gas. In addition, we will discuss the first law of thermodynamics and heat capacity.
Chemical reactions are like an assembly line. The right parts have to come together at the right time in order to make the right product. How fast this can occur depends on the slowest step of the process. These videos show how and why different reactions occur at different speeds, and how catalysts - like enzymes in the body - can make reactions happen faster.
Koch snowflake fractal
Named after Helge von Koch, the Koch snowflake is one of the first fractals to be discovered. It is created by adding smaller and smaller equilateral bumps to an existing equilateral triangle. Quite amazingly, it produces a figure of infinite perimeter and finite area!
Kofun period (300-552 C.E.)
Few people are aware that the name Korea is derived from the name of the Goryeo (previously transliterated as Koryo) dynasty. It was during this period (918–1392) that Korea became known to the world outside East Asia.
Korean art
Discover the Royal Palaces of Seoul, monastic practices that keep traditional painting alive, a Confucian scholar's house and the tradition of Korean ceramics.
Krebs (citric acid) cycle and oxidative phosphorylation
1D: You will learn about the latter steps in cellular respiration - the citric acid cycle and oxidative phosphorylation. It is through these elegant processes that your cells produce energy from sugars, fats, and proteins.
L'Hôpital's Rule
Limits have done their part helping to find derivatives. Now, under the guidance of l'Hôpital's rule, derivatives are looking to show their gratitude by helping to find limits. Ever try to evaluate a function at a point and get 0/0 or infinity/infinity? Well, that's a big clue that l'Hopital's rule can help you find the limit of the function at that point.
LaKeshia Grant - CEO & Founder of Virtual Enterprise Architects
LaKeshia Grant founded Virtual Enterprise Architects as a place where she would have a voice and create an environment where others could be heard. She discusses her industry and encourages would-be entrepreneurs to incorporate their core values in their business. Grant’s mother may not know exactly what the information technology business does, but she instilled a strong work ethic and the spirit of entrepreneurship in her daughter.
Lab Values and Concentrations
Ever wonder about your lab values and what they mean? Lab values measure amounts of electrolytes or cells in your blood and occasionally tell you about how hormones and enzymes are working! Dive deeper and get a good understanding of concentrations as well!
Labor and marginal product revenue
Constructing a demand curve for an individual firm by thinking about how much increment benefit they get from an incremental employee (marginal product of labor (MPL) and marginal product revenue (MPR). We later think about how we can add these "demand" curves to construct a "demand" curve for the market for labor in this industry.
6C: Were it not for the intricate structure of the English language, these letters on the webpage would be absolutely meaningless. As we grow up, we effortlessly pick up the syntax of our mother tongue, babbling as toddlers and maturing to write essays as teenagers. You will explore theories on the development of language and cognition and how our system of language may be disrupted by pathological neurological events like strokes.
Laplace transform
We now use one of the coolest techniques in mathematics to transform differential equations into algebraic ones. You'll also learn about transforms in general!
Laplace transform to solve a differential equation
You know a good bit about taking Laplace transform and useful properties of the transform. You are dying to actually apply these skills to an actual differential equation. Wait no longer!
Lara Morgan - Founder of Pacific Direct
Lara Morgan, Founder of Pacific Direct, shares her entrepreneurial story and describes her motivations in founding companies. Lara describes how understanding the mechanisms of money, along with a fearlessness of asking questionshelped her company grow.
Laryngeal conditions
Late Gothic art, an introduction
Italian art from the late 13th and 14th centuries was once known as primitive because it was seen largely as a transition from Medieval abstraction to the naturalism of the Renaissance (with a dose of Byzantine influence thrown in for good measure). Despite the Black Death, we now study the brilliant artist's of Florence and Siena in their own right.
Late Renaissance in Venice
Venetian painters pursued innovative compositional approaches and introduced new subjects, such as landscape and the female nude. In the Late Renaissance, Titian’s mastery was rivaled by Tintoretto and Veronese. Each attempted to out-paint the other with increasingly dynamic and sensual subjects for local churches and international patrons.
Late classical (4th century)
The late Classical style during the early 4th century was a time of experimentation and transition away from the strict canonical ideals of the high Classical moment.
Late empire
Financial pressures, urban decline, underpaid troops and consequently overstretched frontiers - all of these finally caused the collapse of the western empire under waves of barbarian incursions in the early fifth century C.E. The last western emperor, Romulus Augustus, was deposed in 476 C.E., though the empire in the east, centered on Byzantium (Constantinople), continued until the fifteenth century.
Latin America
Graciela Iturbide captures the spirit of Mexican life in her photographs, while Melanie Smith explores new territory and the colonial gaze. Latin American artists like Gabriel Orozco and Cildo Meireles confront modern life with humour and empathy. Learn more about the subversive and often playful art of Latin America here.
Latin American Modernism
Latin America produced many of the most important modernists of the 20th century. Here were artist who drew on their region's colonial and indigenous past and its political present to create entirely new forms of public and private art.
Lattice multiplication
Tired of "standard multiplication". In this tutorial we'll explore a different way. Not only is lattice multiplication interesting, it'll help us appreciate that there are many ways to do things. We'll also try to grasp why it works in the first place. Enjoy!
Law of cosines and law of sines
The primary tool that we've had to find the length of a side of a triangle given the other two sides has been the Pythagorean theorem, but that only applies to right triangles. In this tutorial, we'll extend this triangle-side-length toolkit with the law of cosines and the law of sines. Using these tool, given information about side lengths and angles, we can figure out things about even non-right triangles that you may have thought weren't even possible!
Laws of thermodynamics
LeBron asks
LeBron James asks questions about math and science, and we answer!
Leading change in blended learning
Learn Algorithms with Dartmouth College
In this collaboration with Dartmouth college professors Tom Cormen and Devin Balkcom, Khan Academy is offering an introduction to computer science algorithms, including searching, sorting, recursion, and graph theory.
Learn to paint like Turner
Have you ever wanted to paint a swirling seascape in oils? Capture the sunset with a palette of watercolours? Draw like a master? Learn the basics of Turner’s typical techniques in this series: tackle the use of line, tone, and colour, and draw copies of masterworks in the same way students would have done in the 19th century.
7C: This is the most “meta” of all modules, in which you will learn about learning! Your environment has a huge impact on your future behavior, and your behavior itself has consequences on the environment. You will come to appreciate the mechanisms of classical and operant conditioning (and how you can apply these concepts to training your dog!). We will apply these to contemporary issues like the issue of violence in the media. As you go through this module, you will gain an understanding of how your brain retains new information.
Least common multiple
Life is good, but it can always get better. Just imagine being able to find the smallest number that is a multiple of two other numbers! It's called the LCM or Least Common Multiple. Other than making your life more fulfilling, lcm will allow you to do incredible things like adding fractions. Common Core Standard: 6.NS.B.4
Length introduction
Learn how to compare things by length. No measuring required.
Length word problems
Learn how to solve length word problems.
Leonardo da Vinci
Letters of recommendation
Don't brag about yourself in the college essay; let others do it for you in their letters of recommendation! Experts explain who to ask for your recommendation letters and how to make sure they have the intended impact on your college application.
Leveraged buy-outs
Private equity firms often borrow money (use leverage) to buy companies. This tutorial explains how they do it and pay the debt.
Life and death of stars
Stars begin when material drifting in space condenses due to gravity to be dense enough for fusion to occur. Depending on the volume and make-up of this material, the star could then develop into very different things--from supernovae, to neutron stars, to black holes. This tutorial explores the life of stars and will have you appreciating the grand weirdness of our reality.
Life in the Universe
Are dolphins the only intelligent life in the universe? We don't know for sure, but this tutorial gives a framework for thinking about the problem.
Life insurance
It is a bit of a downer to think about, but we are all going to die. Do we care what happens to our loved ones (if they really are "loved" than the answer is obvious). This tutorial walks us through the options to insure our families against losing us. The reason why we stuck it in the "investment vehicles" topic is because it can also be an investment that we can use before we die.
Life of Benjamin Franklin
In many ways, Benjamin Franklin is the "Founding Father" of the United States of America that best represent many ideals of the country. In this series with Walter Isaacson, we go into the life and philosophy of Franklin.
Life of a company--from birth to death
This is an old set of videos, but if you put up with Sal's messy handwriting (it has since improved) and spotty sound, there is a lot to be learned here. In particular, this tutorial walks through starting, financing and taking public a company (and even talks about what happens if it has trouble paying its debts).
Light Guitar
Get to know your light sensor while building musical instruments
Light and color introduction
Seeing is believing—light is pretty amazing stuff.
Light and electromagnetic radiation
4D: Believe it or not, light has both wave-like and particle-like properties, as evidenced by the concepts of polarization, interference, and the photon model. In this modern age of medicine, we have seen a rise in the clinical use of the laser (which actually stands for “light amplification by stimulated emission of radiation”). As we discuss theories outlined by geniuses like Max Planck several decades ago, you’ll discover how light rays interfere in double slit, single slit, and diffraction gratings.
Light and fundamental forces
This tutorial gives an overview of light and the fundamental four forces. You won't have a degree in physics after this, but it'll give you some good context for understanding cosmology and the universe we are experiencing. It should be pretty understandable by someone with a very basic background in science.
Lights Puzzles
Try to light up all the tiles!
Limiting reagent stoichiometry
In a reaction, you often have extra of one molecule (or too little of the other) so all the reactant doesn't react. We'll explore how to identify which reactant is limiting which is helpful in a whole series of scenarios.
Limits are the core tool that we build upon for calculus. Many times, a function can be undefined at a point, but we can think about what the function "approaches" as it gets closer and closer to that point (this is the "limit"). Other times, the function may be defined at a point, but it may approach a different limit. There are many, many times where the function value is the same as the limit at a point. Either way, this is a powerful tool as we start thinking about slope of a tangent line to a curve. If you have a decent background in algebra (graphing and functions in particular), you'll hopefully enjoy this tutorial!
Limits and infinity
You have a basic understanding of what a limit is. Now, in this tutorial, we can explore situation where we take the limit as x approaches negative or positive infinity (and situations where the limit itself could be unbounded).
Linda Jeschofnig - Co-founder of Hands-On Labs
A passion for science education led Linda Jeschognig from her life in accounting to a second act as an entrepreneur. She talks about the inspiration behind Hands-on Labs and overcoming the obstacles with a company created to send kits containing hydrochloric acid, cobalt nitrate and other hazardous elements to college chemistry students. Along the way, Jeschofnig has gained support and reached out to guide other women on the entrepreneurial path.
Line integrals for scalar functions
With traditional integrals, our "path" was straight and linear (most of the time, we traversed the x-axis). Now we can explore taking integrals over any line or curve (called line integrals).
Line integrals in vector fields
You've done some work with line integral with scalar functions and you know something about parameterizing position-vector valued functions. In that case, welcome! You are now ready to explore a core tool math and physics: the line integral for vector fields. Need to know the work done as a mass is moved through a gravitational field. No sweat with line integrals.
Line of symmetry
A line of symmetry for a two-dimensional figure is a line across the figure such that the figure can be folded along the line into matching parts. In this tutorial, we'll identify line-symmetric figures and draw lines of symmetry. Common Core Standard: 4.G.A.3
Line plots
Learn how to read and make bar graphs.
Linear and nonlinear functions
Not every relationship in the universe can be represented by a line (in fact, most can't be). We call these "nonlinear". In this tutorial, you'll learn to tell the difference between a linear and nonlinear function! Have fun! Common Core Standare: 8.F.A.3
Linear combinations and spans
Given a set of vectors, what other vectors can you create by adding and/or subtracting scalar multiples of those vectors. The set of vectors that you can create through these linear combinations of the original set is called the "span" of the set.
Linear dependence and independence
If no vector in a set can be created from a linear combination of the other vectors in the set, then we say that the set is linearly independent. Linearly independent sets are great because there aren't any extra, unnecessary vectors lying around in the set. :)
Linear equation word problems
Now that we are reasonably familiar with what a linear equation is and how we can solve them, let's apply these skills to tackling real-world problems.
Linear equations in one variable
Extend your equation solving skills by tackling fancier equations that have variables on both sides.
Linear homogeneous equations
To make your life interesting, we'll now use the word "homogeneous" in a way that is not connected to the way we used the term when talking about first-order equations. As you'll see, second order linear homogeneous equations can be solved with a little bit of algebra (and a lot of love).
Linear inequalities
In this tutorial you'll discover that much of the logic you've used to solve equations can also be applied to think about inequalities!
Linear regression and correlation
Even when there might be a rough linear relationship between two variables, the data in the real-world is never as clean as you want it to be. This tutorial helps you think about how you can best fit a line to the relationship between two variables.
Linear systems of equations word problems
Now that we are able to solve any kind of system we are given, it's time to use that knowledge to solve some word problems.
Linear transformation examples
In this tutorial, we do several examples of actually constructing transformation matrices. Very useful if you've got some actual transforming to do (especially scaling, rotating and projecting) ;)
Lines, line segments and rays
Let's draw points, lines, line segments, and rays. We'll also think about perpendicular and parallel lines and identify these in two-dimensional figures. Common Core Standard: 4.G.A.1
Did you know that fats are beautiful little molecules with long, trailing hydrocarbon tails? Learn about the structure and function of fats and other lipids, including cholesterol and the phospholipids that make up cell membranes.
Literature meets art
George the Poet imagines what it's like to be in someone else's shoes as he looks at a photograph by Paul Graham. Children's writer and illustrator Maurice Sendak shares his admiration for fellow writer and illustrator William Blake. Here's where literature meets art. Learn the terms and materials used in art and literature, and test your knowledge with a quiz.
Local linearization
Let's see how we can local linearization can be used to approximate values of functions near values that we know.
Local threats to biodiversity
Human population increase and activities threaten biodiversity in almost every corner of our planet. Local threats to species richness include land-use changes, pollution, resource exploitation, and invasive species.
Logarithm basics
If you understand how to take an exponent and you're looking to take your mathematical game to a new level, then you've found the right tutorial. Put simply and confusingly, logarithms are inverse operators to exponents (just as subtraction to addition or division to multiplication). As you'll see, taking a logarithm of something tells you what exponent you need to raise a base to to get that number.
Logarithm properties
You want to go deeper in your understanding of logarithms. This tutorial does just that by exploring properties of logarithms that will help you manipulate them in entirely new ways (mostly falling out of exponent properties).
Logarithmic equations
Now that you are familiar with logarithms and logarithmic functions, let's think about how to solve equations involving logarithms.
Logarithmic functions
This tutorial shows you what a logarithmic function is. It will then go on to show the many times in nature and science that these type of functions are useful to describe what is happening.
Logarithmic scale and patterns
Logarithms show up in science and music far more than you might first imagine. This tutorial explores where these appearances occur!
Logic and if Statements
Teach your program to make decisions!
Logistic differential equation and function
Can population grow exponentially forever? Malthus would say "no". Well how do you model that mathematically. The logistic differential equation and logistic function are there to rescue us!
Long division of polynomials
Learn how to divide two polynomials using long division.
Long live Tau
Pi (3.14159...) seems to get all of the attention in mathematics. On some level this is warranted. The ratio of the circumference of a circle to the diameter. Seems pretty pure. But what about the ratio of the circumference to the radius (which is two times pi and referred to as "tau")? Now that you know a bit of trigonometry, you'll discover in videos made by Sal and Vi that "tau" may be much more deserving of the throne!
Loooong division!
You know your multiplication tables and are getting the hang of basic division. In this tutorial, we will journey into the world of loooong division (sometimes, referred to as "long division", but that's not as much fun to say). After this tutorial, you'll be able to divide any whole number by any other. The fun will not stop!
Repeating something over-and-over? Loops are here to help!
Los Altos School District
Los Altos School District was the first district to pilot Khan Academy. Based on the success of four initial teachers, the program is now districtwide and used in 5th-8th grades.
Ludwig van Beethoven: Symphony No. 5
Possibly the most iconic of all symphonies, "Beethoven's 5th" is instantly recognizable by its dramatic opening motive. Learn how this rhythmic and melodic idea permeates the entire work, holding the listener in thrall through all four movements to the blazing finale. Music Director Gerard Schwarz and noted expert Leon Botstein explore the many facets of Beethoven's masterpiece, including a demonstration by Maestro Schwarz of how to conduct the first movement.
Lung cancer
Lymphatic system
Lymphatic system introduction
Your heart pumps roughly 20 L of blood throughout the day to your tissues. The plasma component of blood (not containing blood cells) leaks out through capillaries (the tiniest of blood vessels) and is mostly reabsorbed. However, about 3L of the plasma is left behind in fluid surrounding tissues, and it is the job of the hard-working lymphatic system to bring back this fluid to the circulatory system. The lymphatic system moves fluid in one direction, but without the force of a pump like the heart.
Maclaurin series and Euler's identity
In this tutorial, we will learn to approximate differentiable functions with polynomials. Beyond just being super cool, this can be useful for approximating functions so that they are easier to calculate, differentiate or integrate. So whether you will have to write simulations or become a bond trader (bond traders use polynomial approximation to estimate changes in bond prices given interest rate changes and vice versa), this tutorial could be fun. If that isn't motivation enough, we also come up with one of the most epic and powerful conclusions in all of mathematics in this tutorial: Euler's identity.
Macromolecules skill check
Check your understanding of large biological molecules!
Magnetic field created by a current
Magnets - how do they work? Guaranteed way to win a nobel prize: find a magnetic monopole. If you don’t know what that is, these videos would be a good place to start. Notable achievements of magnets - they put current in our homes, the protect us from dangerous cosmic rays, and they make particles at CERN go really really fast so we can learn more about science!
Magnets and Magnetic Force
Making 10
Learn how to make the number 10 from two smaller numbers.
Making a Memory Game
Ever played the game where you flip over cards and try to find pairs? Learn how to program a digital version of it!
Making a Side Scroller: Hoppy Beaver
Learn how to make a simple side scroller, where you press a key to get your beaver to collect enough sticks for their den. You could easily extend this to make your favorite flappy game!
Making decisions with probability
Making small numbers
Learn to make numbers from other numbers by adding. Numbers used are 10 or less.
Making the medieval book
The production of a manuscript was a long, complex and expensive process. It involved making parchment from animal skin, pricking and ruling hundreds of pages, and writing down long texts by hand, one letter at the time. When the binding was finally added, an object was born that weighed several kilos and could cost as much as a car today.
Making, finding, and conserving
Go behind the scenes at The Metropolitan Museum of Art to explore how some of the world's finest art was created, how it is conserved, and even how it was discovered.
The Republic of Mali is part of West Africa and has a varied geography. In the south and west are verdant river valleys while the north and east jut into the Sahara Desert. It is a French speaking nation though dozens of languages are spoken there. Islam was introduced in the 11th century and is the dominant religion.
Manipulating expressions
Using the combined powers of Chuck Norris and polar bears (which are much less powerful than Mr. Norris) to better understand what expressions represent and how we can manipulate them. Great tutorial if you want to understand that expressions are just a way to express things!
The Mannerist style developed in the courts of the Italian elite. It built on Renaissance naturalism but distorts and deceives to delight its highly educated audience.
Learn about medieval and Renaissance manuscripts, and how these exquisite handmade objects were created and used.
Marble from antiquity to modern times
Learn how generations of marble sculptors have been influenced by the artistic achievements and skills of ancient Greek and Roman masters.
Marc Ecko - Founder of Ecko Unlimited
Marc Ecko, Founder of Ecko Unlimited, discusses his origins as an entrepreneur and the entrepreneurial culture of Hip Hop. Describing graffiti as the extreme sport of art, Marc talks about how this form of artistic expression was his gateway to entrepreneurship and offers advice to young people.
Marginal propensity to consume (MPC)
If you earn a $1, you might spend some fraction of it. This can then be income for someone else. This can keep going. In this tutorial, we'll explore how the incremental spend per incremental earnings (marginal propensity to consume) and the multiplier effect based on it can drive economic activity.
Marginal utility and budget lines
In this tutorial we look at the utility of getting one more of something and put numbers to it. We then use this to construct a budget line and think about indifference curves.
Market equilibrium
You understand demand and supply. This tutorial puts it all together by thinking about where the two curves intersect. This point represents the equilibrium price and quantity which is, in an ideal world, where the market would transact.
Mars: Ancient Observations
Where is Mars? How does it move? How far away is it? What are the conditions on the surface? This tutorial covers our initial observations of the Red Planet
Mars: Modern exploration
What are the conditions on the surface of Mars? Does it have life? Does it have water? Was the ancient environment habitable on Mars? This tutorial covers 20th century discoveries.
Martha Washington
Mass and volume
Estimate the volume and mass of items. Solve word problems involving mass and volume.
Watch fun, educational videos on all sorts of Materials, how they're created and what they can do.
Math patterns
Let's now use our mathematical toolkit to discover and make use of patterns! This is a seriously fun tutorial.
Mathematics of Rendering
Explore the most fundamental calculation a ray tracer does: line-plane intersection in 2D and 3D. This lesson involves solving systems of equations and is appropriate for the advanced high school level. It takes at least 60 minutes to complete.
Mathematics of parabolas 2
Discover how computers draw millions of parabolas really efficiently using DeCastlejau's algorithm. This lesson introduces weighted averages of two points leading to parametric curves. It touches on concepts in grade 7 through high school and takes 35-50min.
Mathematics of rotation
This lesson will push you to derive the rotation formulas we used in lesson 1. So roll up your sleeves! It's appropriate for the high school level and takes 35-45min.
Mathematics of subdivision
Explore how to calculate a weighted average of 3 points and connect this to the subdivision operation from lesson 1. This lesson touches on concepts in grades 7 through high school and takes 35-50 minutes.
Matrix equations
Matrix multiplication
You know what a matrix is, how to add them and multiply them by a scalar. Now we'll define multiplying one matrix by another matrix. The process may seem bizarre at first (and maybe even a little longer than that), but there is a certain naturalness to the process. When you study more advanced linear algebra and computer science, it has tons of applications (computer graphics, simulations, etc.)
Maurice Ravel: Suite No. 2 from Daphnis et Chloe
Analysis by Gerard Schwarz. A love story from ancient Greek mythology inspired the creation of an epic ballet in 1912. French composer Maurice Ravel expanded the possibilities of orchestral color and texture in his remarkable score. Music Director Gerard Schwarz explores the history and musical structure of this landmark work of musical Impressionism.
Maximum and minimum points
Learn about maximum and minimum points of functions.
The Maya civilization (beginning around 300 C.E.) was one of the most sophisticated in the Americas. It extended from southeastern Mexico across modern-day Guatemala, Belize and the western parts of Honduras and El Salvador. Maya cities usually had a dramatic stepped pyramid and the Maya developed a sophisticated writing system and used an elaborate calendar system. There are about six million Maya today.
Mean absolute deviation (MAD)
Mean absolute deviation (MAD) is a tool for reasoning about the spread of data, just like interquartile range (IQR). Higher values of MAD tell you the data is more spread out. Lower values of MAD tell you the data is closer together.
Mean and median
Mean and median are measures of "central tendency." That is, they help us find the center (or middle) of the data. In this tutorial, we'll learn how to compute mean and median.
Mean and median challenge problems
Practice solving more challenging problems using your knowledge of mean and median.
Mean value theorem
If over the last hour on the highway, you averaged 60 miles per hour, then you must have been going exactly 60 miles per hour at some point. This is the gist of the mean value theorem (which generalizes the idea for any continuous, differentiable function).
Everything in the universe can be measured.
Measures of central tendency
This is the foundational tutorial for the rest of statistics. We start thinking about how you can represent a set of numbers with one number that somehow represents the "center". We then talk about the differences between populations, samples, parameters and statistics.
Measuring age on Earth
Geologists and archaeologists will tell you how old things are or when they happened, but how do they know? This tutorial answers this question by covering some of the primary techniques of "dating" (not in the romantic sense).
Measuring cost of living --inflation and the consumer price index
We might generally sense that our cost of living is going up (inflation), but how can we measure it? This tutorial shows how it is done in the United States with the consumer price index (CPI).
Measuring length
Learn how to measure length using unit squares and rulers.
Measuring magnetic fields
How can we quantify the strength of magnetic fields? How strong is the Earth's field?
Measuring segments
Using a number line, let's learn how we measure segments.
Measuring the solar system
How can we apply geometry and trigonometry in order to measure the size of the earth, moon and sun?
Mechanical advantage
If you have ever used a tool of any kind (including the bones in your body), you have employed mechanical advantage. Whether you used an incline plane to drag something off of a pick-up truck, or the back of a hammer to remove a nail, the world of mechanical advantage surrounds us.
Mechanical waves
Medians and centroids
You've explored perpendicular bisectors and angle bisectors, but you're craving to study lines that intersect the vertices of a a triangle AND bisect the opposite sides. Well, you're luck because that (medians) is what we are going to study in this tutorial. We'll prove here that the medians intersect at a unique point (amazing!) called the centroid and divide the triangle into six mini triangles of equal area (even more amazing!). The centroid also always happens to divide all the medians in segments with lengths at a 1:2 ration (stupendous!).
Medieval Europe and the Islamic world (3-1 to 3-2)
Medieval artistic traditions include late antique, early Christian, Byzantine, Islamic, migratory, Carolingian*, Romanesque, and Gothic, named for their principal culture, religion, government, and/or artistic style. Continuities and exchanges between coexisting traditions in medieval Europe are evident in shared artistic forms, functions, and techniques. Contextual information comes primarily from literary, theological, and governmental (both secular and religious) records, which vary in quantity according to period and geographical region, and to a lesser extent from archaeological excavations. Elite religious and court cultures throughout the Middle Ages prioritized the study of theology, music, literary and poetic invention, and in the Islamic world, scientific and mathematical theory. Cultural and artistic exchanges were facilitated through trade and conquest. By permission, © 2013 The College Board
Medieval era
The centuries 300–1100 C.E. witnessed great change in Europe. The Roman Empire broke down in the west, but continued as the Byzantine Empire in the east. People, objects and ideas travelled across the continent, while Christianity and Islam emerged as major religions. By 1100, the precursors of several modern states had developed. Europe as we know it today was taking shape. Europe as we know it today was taking shape.
Medieval period
Medieval Period (c. 900-1517 C.E.) The Fatimids (909-1171) ruled north Africa, and parts of Syria and the Seljuqs contolled eastern Islamic lands and eventually Iran, Iraq and much of Anatolia.
Meet the Professional
What can you do with computer science and programming skills once you've learned them? We've invited people from all around the world and the industry to introduce themselves to you. Find out how diverse our field can be!
To the north and east of Australia lie the islands known as Melanesia. These islands form one of the most culturally complex regions of the entire world, with 1,293 languages spoken across the Solomon Islands, Vanuatu, New Caledonia and the island of New Guinea (politically divided into Indonesia’s West Papua Province and the nation of Papua New Guinea). It is also a region of great antiquity. New Guinea has been settled for around 45,000 years, the Solomon Islands for 35,000 years, and Vanuatu and New Caledonia for about 4,000.
Explore the structure of human memory; processes involved in normal encoding, retrieval, forgetting, and aging; and diseases affecting memory.
Mendelian genetics
1C: Why do some people have blue eyes and others brown? What determines your blood type? You will be able to answer questions like these as you have some fun with Punnett squares and discover the mechanisms of inheritance (and what all this has to do with a 19th-century German monk).
Merge sort
Learn merge sort, a more efficient sorting algorithm that relies heavily on the power of recursion to repeatedly sort and merge sub-arrays.
Mergers and acquisitions
Companies often buy or merge with other companies using shares (which is sometimes less intuitive than when they use cash). This tutorial walks through the mechanics of how this happens and details what is likely to happen in the public markets because of the transaction (including opportunities for arbitrage).
Mesoamerica and Central America
Explore the cultures of the Olmec, Maya and Mixtec people as well as the Mexica (Aztec) through objects in the collection of the British Museum.
Metaphysics and epistemology
Metaphysics is an area of philosophy concerned with what there is in the universe (ontology) and the nature of what exists. Epistemology is a related area interested in knowledge and how we know things about the universe.
Method of undetermined coefficients
Now we can apply some of our second order linear differential equations skills to nonhomogeneous equations. Yay!
Metric and U.S. customary units intro
An introduction to units for measuring distance, volume, and mass. The units used in this tutorial are metric units, not U.S. customary units.
Metric system
You might be surprised to realize that only two countries in the world use feet, miles, or yards or a bunch of other "English System" conventions (the irony being that the English don't even use the "English System" any more). Everyone else, including the English, now use the metric system which is actually much more logical. Learn about kilo and milli and centi and become a metric unit megastar!
The word Micronesia comes from the Greek and means "small" and "island." It contains the island groups north of Melanesia and east of the Philippines including the Mariana, Caroline, Marshall and Kiribati islands.
Mid-century to today: Modernism and its legacy
Emerging from the shadow of World War II, artists like Henry Moore grapple with a meaningful way forward for art. Through a period of social and political upheaval, British art sees groundbreaking genres, mediums, and relationships form. Art breaks with convention and defies categorization. And we find ourselves in the last room of the chronology–in the present.
Middle Kingdom and New Kingdom
Learn about the ruler Akhenaten, Nefertiti, and Queen Tiye as well as some of the most extraordinary paintings from all of Egyptian culture found in the tomb-chapel of Nebamun.
Middle empire
The imperial system of the Roman Empire depended heavily on the personality and standing of the emperor himself. The reigns of weak or unpopular emperors often ended in bloodshed at Rome and chaos throughout the empire as a whole. In the third century C.E. the very existence of the empire was threatened by a combination of economic crisis, weak and short-lived emperors and usurpers (and the violent civil wars between their rival supporting armies), and massive barbarian penetration into Roman territory.
Ming dynasty
Minimalism and Earthworks
Minimalism isn’t simple, although Judd, Smithson, Christo and others did use simple forms to convey complex issues about the nature of art.
Mirrors and lenses
Mixed number addition and subtraction
You know the basics of what mixed numbers are. You're now ready to add and subtract them. This tutorial gives you plenty of examples and practice in this core skill!
Mixed number multiplication and division
My recipe calls for a cup and a half of blueberries and serves 10 people. But I have 23 people coming over. How many cups of blueberries do I need? You know that mixed numbers and improper fractions are two sides of the same coin (and you can convert between the two). In this tutorial we'll learn to multiply and divide mixed numbers (mainly by converting them into improper fractions first).
Mixed numbers
Learn how to write improper fractions as mixed numbers (and mixed numbers as improper fractions).
Mixed numbers and improper fractions
Let's help you relate improper fractions to mixed numbers!
Explore a rare and important codex about the Mixtec region and the ruler Eight Deer-Jaguar Claw.
Mixtures and solutions
The history of modern art is not simply a linear progression of styles. Rather, artists respond to and participate in the intellectual, social, and cultural contexts of their time. MoMA has a long history of experimental approaches to engaging people with art, which is at the core of the museum's mission. Listen to MoMA educators discuss how they teach challenging works of art, hear tips for teaching, and learn about MoMA's programs for individuals with dementia.
Mobius strips
Playing mathematically with strips!
Modeling situations by combining and composing functions
Make use of knowledge in combining and composing functions, by modeling complex real-world situations.
Modeling with differential equations
Now that you know how to find solutions to separable differential equations, we can use this skill to model some real world phenomena like population growth and how things might cool down.
Modeling with exponential functions
As we'll see in this tutorial, anything from compound interest to radioactive decay can be modeled with exponential functions.
Modeling with linear functions and equations
Modeling with linear inequalities
Learn how to construct linear inequalities and systems of inequalities given a real-life context. Learn how to interpret such inequalities in terms of a real-life context, either algebraically or graphically.
Modeling with one-variable equations and inequalities
Now that you know how to solve linear, quadratic and exponential equations, we'll apply these incredible skills to a wide-range of real-world (and sometimes not-so-real-world) situations.
Modeling with periodic functions
By now, you are reasonably familiar with the graphs of sine and cosine and are beginning to appreciate that they can be used to model periodic phenomena. In this tutorial, you'll get experience doing just that--modeling with periodic functions!
Modeling with rational functions
See some examples of how rational functions and equations can come in handy when solving real-world word problems.
Modeling with sequences
Now that we've mastered arithmetic and geometric sequences, let's see how they can help us solve real-world word problems. In this tutorial you will find the formula for a sequence that models a real-world relationship.
Modeling with subdivision surfaces
Discover one of Pixar's key modeling technologies: subdivision surfaces. This lesson introduces the split and average operations used to make smooth surfaces. It's appropriate for all ages and takes 40-50 minutes.
Modelling the solar system
Astronomy begins when we look up and start asking questions. Where are we? How big is the earth? This lesson is for all ages, start here!
Modern Information Theory
Information Theory in the 20th Century
Diverse artists with a common dedication to innovation came to be discussed as the avant-garde. Subdivisions include Neoclassicism, Romanticism, Realism, Impressionism, Post-Impressionism, Symbolism, Expressionism, Cubism, Constructivism, Abstraction, Surrealism, Abstract Expressionism, Pop Art, performance art, and earth and environmental art. Many of these categories fall under the general heading of modernism. © 2013 The College Board
Modifying databases with SQL
Learn how to update data, delete data, and change table schemas with SQL commands UPDATE, DELETE, ALTER, and DROP.
Modular arithmetic
This is a system of arithmetic for integers. These lessons provide a foundation for the mathematics presented in the Modern Cryptography tutorial.
Molecular composition
We'll now explore two different ways of representing what elements are in a molecule: molecular and empirical formulas. Then we'll actually Molecular formulas actually represent the number of atoms in a molecule while empirical formulas show us the ratio of the constituents based on experiments. In order to help us connect these ideas, we'll also explore a quantity called the "mole". Just as a "dozen" represents 12 of something, a "mole" represents roughly 602,200,000,000,000,000,000,000 of something. This will help us think about mass composition of molecules.
Depending on your view of things, this may be the most violent of our tutorials. Things will crash and collide. We'll learn about momentum and how it is transferred. Whether you're playing pool (or "billiards") or deciding whether you want to get tackled by the 300lb. guy, this tutorial is of key importance.
Monetary and fiscal policy
Governments (and pseudo government entities like central banks) have two tools at their disposal to try to impact the business cycle --monetary and fiscal policy. This will help you understand what they are.
Learn how to count U.S. coins and dollars.
Money supply
This short tutorial explains how we measure how much "money" there is out there. As we'll see, this isn't as straightforward as counting dollars in people's pockets, especially because there are multiple types of money.
Money word problems
Learn how to solve a variety of problems involving money.
No, we aren't talking about the board game although the game does try to approximate what this tutorial is about--notice that you can charge more rent at either Boardwalk or Park Place if you own both (you have a "monopoly" in the navy blue market). The opposite of perfect competition is when you have only one firm operating. This tutorial explores what this firm would do to maximize economic profit.
Montessori Schools
More CSS selectors
Learn more complex selectors- using multiple classes, combing elements with classes, descendant selectors, grouped selectors, and dynamic pseudo-classes.
More HTML tags
Learn how to make links, tables, and comments.
More advanced SQL queries
Learn how to perform more advanced SQL queries using AND/OR, IN, LIKE, HAVING, and more.
More determinant depth
In the last tutorial on matrix inverses, we first defined what a determinant is and gave several examples of computing them. In this tutorial we go deeper. We will explore what happens to the determinant under several circumstances and conceptualize it in several ways.
More equation practice
This tutorial is for you if you already have the basics of solving equations and are looking to put your newfound powers to work in more examples.
More mathy functions
In this tutorial, we'll start to use and define functions in more "mathy" or formal ways.
Mortgage-backed securities
What started out as a creative way to spread risk ended up fueling a monster housing bubble. This tutorial explains what mortgage-backed securities are and how they work.
Most people buying a home need a mortgage to do so. This tutorial explains what a mortgage is and then actually does some math to figure out what your payments are (the last video is quite mathy so consider it optional).
Motion along a line
Derivatives can be used to calculate instantaneous rates of change. The rate of change of position with respect to time is velocity and the rate of change of velocity with respect to time is acceleration. Using these ideas, we'll be able to analyze one-dimensional particle movement given position as a function of time.
Motivation and attitudes
7A: What makes us do the things we do, or feel the way we feel in situations? Explore how the physiological and psycho-social theories, factors, and situations behind how motivation, attitudes, and behavior are inter-related.
Mound-building societies lived in the Ohio Valley from the last millennium B.C.E. to approximately 1500 C.E. and were influenced by the older Mississippian culture to the west.
Moving the decimal to multiply and divide by 10
In our decimal number system, as we move places to the left, the place values increase by a factor of 10 (likewise, they decrease by a factor of 10 as we move rightward). This idea gets direct application when we multiply or divide a decimal number by 10 because it has the effect of shifting every place value one to the right or left (sometime seen as moving the decimal point).
Multi-digit multiplication
You know your multiplication tables and are ready to learn how to multiply *any* number (actually, any whole number). Imagine the possibilities! This tutorial will make you unstoppable.
Multi-step equations with distribution
Apply the distributive property to solve more complex equations with variables on both sides.
Multi-step word problems
We are now going to solve real-world problems using multiple steps. Along the way, we'll be using letters to represent unknown quantities. Common Core Standard: 3.OA.D.8
Multiple sclerosis
You know your multiplication tables (and basic division) from the 3rd grade and are ready to learn how to multiply and divide multi-digit numbers. Imagine the possibilities! This tutorial will make you unstoppable. Common Core Standard: 4.NBT.B.5
Multiplication and decimals
Learn how to multiply multi-digit whole numbers like 345 times 892.
Multiplication and division
Let's handle slightly more complicated multiplications and divisions. Now you'll become super good at it with a bit more practice! After these tutorials, you'll be comfortable dividing two digit whole numbers and decimals, whether they are in the divisor or dividend position. Pretty exciting, huh? Let me a hear a "Yes!"
Multiplication and division word problems with negatives
Explore the meaning of negative numbers in different situations.
Multiplication as scaling
Learn to see multiplication as a way to grow (scale up) and shrink (scale down) other numbers.
Multiplication facts
Learn multiplication facts (sometimes called times tables).
Multiplication intro
Welcome to the concept of multiplication.
Multiplication table
Learn how to read a multiplication table. Can you see any patterns?
Multiply and divide negative fractions
We've already multiplied and divided fractions, and we've also multiplied and divided negative numbers. Now we combine those skills.
Multiplying a polynomial by a binomial
Learn how to multiply a polynomial expression by a binomial expression. Binomials are just polynomials with exactly two terms!
Multiplying a polynomial by a monomial
Learn how to multiply a polynomial expression by a monomial expression. Monomials are just polynomials with a single term!
Multiplying and dividing by 10, 100, and 1000
Learn how to, for example, multiply 34.5 by 100 to get 3450.
Multiplying and dividing complex numbers
Multiplying and dividing fractions 2
Solve more complex multiplication and division problems involving negative fractions.
Multiplying and dividing negative numbers
You already know how to multiply and divide whole numbers, decimals, and fractions. See what happens when we throw negative numbers into the mix. It's really not so different!
Multiplying and dividing rational expressions
Learn how to multiply and divide rational expressions. You will be surprised to see how similar it is to multiplying and dividing fractions!
Multiplying and dividing with powers of 10
Learn how to shift the decimal point when you multiply and divide by powers of ten.
Multiplying binomials
In this tutorial you'll learn that multiplying things like (4x-7)(-9x+5) just require the distributive property that you learned in elementary school. We'll touch on the FOIL method because it seems to be covered in a lot of schools, but we don't like it (we don't think it is good to memorize processes without knowing the why).
Multiplying by 10
As we'll see in this tutorial, multiplying a multiple of 10--like 10, 20, 30, 40, etc.--by a single digit number is not too much more complicated. Common Core Standard: 3.NBT.A.3
Multiplying decimals
The real world is seldom about whole numbers. If you precisely measure anything, you're likely to get a decimal. If you don't know how to multiply these decimals, then you won't be able to do all the powerful things that multiplication can do in the real world (figure out your commission as a robot possum salesperson, determining how much shag carpet you need for your secret lair, etc.).
Multiplying fractions
Learn how to multiply a fraction by a fraction.
Multiplying fractions and whole numbers
Learn how to multiply a fraction by a whole number.
Multiplying fractions and whole numbers word problems
Get some practice solving word problems by multiplying fractions by whole numbers.
Multiplying fractions word problems
Multiplying fractions is useful. Period. That's all we really have to say. Believe us don't believe us. You'll learn eventually. This tutorial will have you multiplying in real-world scenarios (which is almost as fun as completely artificial, fake scenarios).
Multiplying mixed numbers
Learn how to multiply mixed numbers.
Multiplying polynomials
Learn how to multiply two polynomials to obtain another polynomial. For instance, express (2x^2+5)*(x^3-2x+6) as a polynomial in standard form.
Multiplying unit fractions and whole numbers
Learn how to multiply unit fractions and whole numbers.
Multiplying whole numbers and fractions
Learn the art of multiplying whole numbers by fractions. See how 3 times 2/3 is the same as 6 times 1/3, which is the same as 6/3.
Multiplying whole numbers and fractions word problems
Learn how to solve word problems by multiplying whole numbers and fractions.
Multistep word problems
In this tutorial, we'll start to challenge you with more sophisticated multiplication and division word problems. If you understand mult-digit multiplication and long division, you have all the tools you need to tackle these. May the force be with you! Common Core Standard: 4.OA.A.3
Muromachi period (1392–1573)
Without muscles, we wouldn't be able to do much of anything. This tutorial begins to explore what muscle cells are and how they contract in order to move our bodies (or do things like breath and pump blood).
Muscular system
3B: Muscles never sleep (literally). If you have ever taken a breath, you have benefited from the work of the diaphragm, which contracts to create an area of low pressure within your thoracic cavity, allowing air in. How exactly are some weightlifters able to support 717 lbs without breaking anything more than a little sweat? Fun fact: the largest muscle in your body is the gluteus maximus (that’s your butt) while the smallest skeletal muscle is the stapedius (it stabilizes the smallest bone, the stapes, which is in your middle ear). By Raja Narayan.
Muscular system introduction
Muscles never sleep (literally). If you have ever taken a breath, you have benefited from the work of the diaphragm, which contracts to create an area of low pressure within your thoracic cavity, allowing air in. How exactly are some weightlifters able to support 717 lbs without breaking anything more than a little sweat? Fun fact: the largest muscle in your body is the gluteus maximus (that’s your butt) while the smallest skeletal muscle is the stapedius (it stabilizes the smallest bone, the stapes, which is in your middle ear).
Music Basics: Notes and Rhythm
The basic principles of music are explained in plain language with helpful graphics and live video demonstrations. If you have ever wondered "how does music work?" then you'll find answers here. Presented by the All-Star Orchestra music director Gerard Schwarz.
Music basics: notes and rhythm
The basic principles of music are explained in plain language with helpful graphics and live video demonstrations. If you have ever wondered "how does music work?" then you'll find answers here. Presented by the All-Star Orchestra music director Gerard Schwarz.
Mutual funds
If we're not in the mood to research and pick our own stocks, mutual funds and/or ETFs might be a good option. This tutorial explains what they are and how they are different.
The Mycenaean citadel on Mainland Greece stands high above the Aegean Sea surrounded by walls of stone so large they are called Cyclopean, as if only the mythic giant Cyclops could have moved them into place. The Bronze Age Mycenaeans were also renowned smiths, who's expertise evident in the gold masks and jewelry found in the shafts below their grave circles.
Myeloproliferative disorders
Myocarditis and pericarditis
Inflammation of the heart muscle and the fibrous sac surrounding the heart are called myocarditis and pericarditis, respectively. Each of these disorders present with specific signs and symptoms. For instance, pericarditis sometimes presents with a subtle finding when listening to the heart with the stethoscope, termed the friction rub. You will come to appreciate the clinical differences between these diseases as well as their therapeutic management.Hell
NCLEX-RN questions
A collection of questions from content covered on the NCLEX-RN. These questions are available under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License (available at
The Siq is a canyon leading to Petra, the greatest city of the Nabataeans, a people who occupied the area from Sinai to northern Arabia and southern Syria.
Naming alkanes
Methane, ethane, propane, butane . . . by the time you get to five carbons you might have some pent up frustration. By six carbons you might even believe that you are hexed! Don’t worry though, you’ll feel sane again once Sal shows you how to name alkanes in this tutorial.
Naming alkanes, cycloalkanes, and bicyclic compounds
Do you speak the language of organic chemistry? Nomenclature running rings around you? Feel like the victim of a vicious cycle? In this tutorial, we will learn how to be fluent in naming alkanes, cycloalkanes, and bicyclic compounds.
Naming alkenes
In this tutorial, Sal names alkenes and discusses the E-Z system.
Naming amines
In this tutorial, Sal shows how to name amines.
Naming and preparing alkynes
In this tutorial, Jay covers the nomenclature and preparation of alkynes, the acidity of terminal alkynes, and the alkylation of alkynes.
Naming benzene derivatives
Would a cyclohexatriene by any other name smell as sweet? In this tutorial, Sal and Jay explain how to name benzene derivatives, the sometimes sweet-smelling cyclic molecules that can be used in the synthesis of explosives and plastics.
Naming carboxylic acids
In this tutorial, Sal shows how to name carboxylic acids.
Napoleon Bonaparte
A man with such a huge "Napoleonic complex", that they named it after him. A military genius with a ginormous ego, some people consider him a hero or a tyrant or both. France has successfully overthrown Louis XVI in 1789. It has gone through a many-year period of bloodshed and instability. The monarch's of Europe are not happy about this "overthrow-your-king" business. A 5'6'' Corsican establishes himself as a strong military tactician during the wars with other European powers and soon comes to power in France. This tutorial covers the rise and fall of one of the most famous men in all of history: Napoleon Bonaparte (Napoleon I).
Nara period (710–794)
Nash equilibrium
If you haven't watched the movie "A Beautiful Mind", you should. It is about John Nash (played by Russell Crowe) who won the Nobel Prize in economics for his foundational contributions to game theory. This is what this tutorial is about. Nash put some structure around how players in a "game" can optimize their outcomes (if the movie is to be fully believed, this insight struck him when he realized that if all his friends hit on the most pretty girl, he should hit on the second-most pretty one). In this tutorial, we use the classic "prisoner's dilemma" to highlight this concept.
Nationalism, Imperialism & Globalization
Native North America
Celebrate the beautiful and complex material worlds of Native North America with an exploration of five regions across the continent.
Natural Science
What makes living (and nonliving) things tick?
Natural logarithms
e is a special number that shows up throughout nature (you will appreciate this more and more as you develop your mathematical understanding). Given this, logarithms with base e have a special name--natural logarithms. In this tutorial, we will learn to evaluate and graph this special function.
Negative and fractional exponents
It's normally a bad idea to hang around with negative people or do negative things, but we think it's OK to associate with negative exponents. And fractional exponents are even more fun. This idea will open up entirely new vistas to your mathematical life.
Negative decimals and fractions on the number line
Practice finding positive and negative decimals and fractions on the number line.
Negative exponents
It's normally a bad idea to hang around with negative people or do negative things, but we think it's OK to associate with negative exponents. As a matter of fact, we'll learn that positive and negative exponents have quite the opposite effect on their bases.
Negative number basics
What is a "negative number"? What happens when I add or subtract one of these? If you come to this tutorial armed with the basics of adding and subtracting numbers, you'll learn about what happens in the world below zero! You will learn what negative numbers are and how we can add and subtract them.
Negative numbers
Understanding and operating with negative numbers is key in algebra. This tutorial will make sure that you have the basics down!
Negative numbers introduction
What are negative numbers? When do we use them? Where do we find them on the number line? Let's learn what happens in the world below zero!
Neolithic art
The Neolithic Revolution was the most important development in human history. The way we live today—settled in cities, protected by laws, eating food from farms—dates from approximately 10,000 years ago.
Neolithic art in China
Nerve Regulation of the Heart
Although your heart can beat independently, your nervous system is important as an external regulator. Your brain can tell your heart to speed up or slow down, depending on the scenario. For example, when you’re falling asleep, your nervous system will cause your heart to slow down, and 8 hours later when your phone alarm goes off, your nervous system will speed up your heartbeat! So even though your heart muscle beats by itself, the nerves can ramp up or down the speed. Check out the videos to learn more about how the nerves help to regulate the heart.
Nervous system introduction
Neurons transmit information to one another through electrochemical signals. They make up the motor nerves that allow you to type an essay, the sensory nerves that let you feel a fluffy dog, and your brain, allowing to remember the content of this module. They have a number of helper cells, ranging from astrocytes, to microglia, to ependymal cells as well. You will come to appreciate the structure and function of neurons and the comrade cells which help to maintain the optimal function of the nervous system.
Neural calls
3A: Neurons transmit information to one another through electrochemical signals. They make up the motor nerves that allow you to type an essay, the sensory nerves that let you feel a fluffy dog, and your brain, allowing to remember the content of this module. They have a number of helper cells, ranging from astrocytes, to microglia, to ependymal cells as well. You will come to appreciate the structure and function of neurons and the comrade cells which help to maintain the optimal function of the nervous system.
Neural calls and neurotransmitters
Get an overview of the structure and function of neurons, and learn about the many other important cells needed to help our nervous system function optimally! By Matt Jensen.
Neural cells
3A: Neurons transmit information to one another through electrochemical signals. They make up the motor nerves that allow you to type an essay, the sensory nerves that let you feel a fluffy dog, and your brain, allowing to remember the content of this module. They have a number of helper cells, ranging from astrocytes, to microglia, to ependymal cells as well. You will come to appreciate the structure and function of neurons and the comrade cells which help to maintain the optimal function of the nervous system.
Neurodevelopmental disorders
Neuron membrane potentials
3A: As you visually process this webpage, countless action potentials are taking place across the neurons in the occipital lobe of your brain. An action potential is an extremely quick change in the membrane potential of the neuron that can propagate toward other neurons. It’s like one neuron is telling another neuron “Hey, wake up! We have work to do!” You will soon gain an appreciation for the mechanism of the action potential, which allows for communication between neurons of all flavors in your body- from those responsible for the maintenance of memory to those functioning in sensation.
Neuronal synapses
3A: Synapses are the means by which neurons, just like other cells of the body, communicate with each other as well as other cells, like muscles and glands. They are the equivalent of neural telephone lines allowing for the transmission of a neurotransmitter, that is, the message. Thus, you are able to tell a dog from a cat. and a ball from a rock. Without all this cross-talk, you might look at an object without truly seeing or hear a sound without actually listening.
New TopProto-Three Kingdoms (300 B.C.E.-300 C.E.)ic
New York School
The term New York School usually refers to both the younger Abstract Expressionists sometimes also known as 2nd generation Abstract Expressionists and artists directly influenced by this movement. The includes both the color field paintings championed by the critic Clement Greenberg who sought to advance formal aspects of AbEx and artists like Jasper Johns and Robert Rauschenberg who looked at the heroic nature of Abstract Expressionism with both irony and respect.
New and noteworthy
New operator definitions
Are you bored of the traditional operators of addition, subtraction, multiplication and division? Do even exponents seem a little run-of-the-mill? Well in this tutorial, we will--somewhat arbitrarily--define completely new operators and notation (which are essentially new function definitions without the function notation). Not only will this tutorial expand your mind, it could be the basis of a lot of fun at your next dinner party!
Newton's law of gravitation
Why are you sticking to your chair (ignoring the spilled glue)? Why does the earth orbit the sun (or does it)? How high could I throw my dog on the moon? Gravitation defines our everyday life and the structure of the universe. This tutorial will introduce it to you in the Newtonian sense.
Newton's laws and equilibrium
4A: Apples would not fall without a force to propel them downward. These three laws described by the famous Englishman centuries ago still inform our understanding of the physical world today. When you push a patient’s bed down the halls of the hospital, you are providing a force in a direction. If we know the magnitude of the force we provide as well as the mass of the patient and the hospital bed, we can calculate the acceleration of the patient and his bed using Newton’s Second law (F=ma).
Newton's laws of motion
This tutorial will expose you to the foundation of classical mechanics--Newton's laws. On one level they are intuitive, on another lever they are completely counter-intuitive. Challenge your take on reality and watch this tutorial. The world will look very different after you're done.
The Federal Republic of Nigeria is the most populous nation in Africa and is home to some 500 ethnic groups (and languages) including the Yoruba people. Nigeria has a large Christian population concentrated in the central and southern regions and an equally large Muslim population predominantly in the north and southwest.
Nomenclature and preparation of epoxides
In this tutorial, Sal and Jay name epoxides. Jay also shows the preparation of epoxides and includes the stereochemistry of the reaction.
Nomenclature and properties of ethers
In this tutorial, Sal and Jay name ethers and discuss the physical properties of ethers.
Nomenclature and reactions of carboxylic acid derivatives
In this tutorial, Jay shows you the nomenclature, properties, and reactions of carboxylic acid derivatives.
Nomenclature of aldehydes and ketones
In this tutorial, Sal and Jay show you the nomenclature, physical properties, and reactivity of aldehydes and ketones.
Non-linear systems of equations
Tired of linear systems? Well, we might just bring a little nonlinearity into your life, honey. (You might want to brush up on your solving quadratics before tackling the non-linear systems.) As always, try to pause the videos and do them before Sal does!
Normal distribution
The normal distribution (often referred to as the "bell curve" is at the core of most of inferential statistics. By assuming that most complex processes result in a normal distribution (we'll see why this is reasonable), we can gauge the probability of it happening by chance. To best enjoy this tutorial, it is good to come to it understanding what probability distributions and random variables are. You should also be very familiar with the notions of population and sample mean and standard deviation.
Normal force
A dog is balancing on one arm on my head. Is my head applying a force to the dog's hand? If it weren't, wouldn't there be nothing to offset the pull of gravity causing the acrobatic dog to fall? What would we call this force? Can we have a general term from the component of a contact force that acts perpendicular to the plane of contact? These are absolutely normal questions to ask.
Normal forces
4A: How do forces work inside an elevator or when you slide a box across the floor? Normal forces are forces which act perpendicularly to surfaces. When you see a patient in a hospital bed, the reason he does not fall through the bed is because the ground provides the patient with a normal force that directly opposes the force of the Earth’s gravity on your the body (this is why you aren’t sinking to the Earth’s core right now). We will walk through real-world physical examples like these to shed light on normal forces.
Normative and non-normative behavior
7B: Learn about how "normal" and "deviant" behavior is defined in today's society. This includes a discussion of the range of normal and abnormal behavior, common theories used to understand basic deviance, and discussion of some types of deviance that occur in groups.
Northern Qi dynasty (550 - 577 C.E.)
The Northern Qi dynasty existed during a period of instability known as the "Southern and Northern dynasties" and was one of several dynasties that ruled different regions of China simultaneously. The Northern Qi dynasty lasted from 550 until 577.
Northern Wei dynasty (386 - 534)
Northwest coast
Nose, sinus, and upper respiratory conditions
Nose, sinus, and upper respiratory track infections
Nucleic acids
Maybe you've heard DNA discussed as the "genetic material." But what exactly is DNA, when you get right down to it? Learn more about the structure and properties of DNA and its nucleic acid cousin, RNA.
Nucleic acids, lipids, and carbohydrates
The following videos outline the functions lipids and carbohydrates. Lipids are important in energy storage, signaling, and cellular structure. Carbohydrates are also important for energy storage, so we’ll talk about those too.
Nucleophilic Aromatic Substitution
In this tutorial, Jay shows the addition-elimination mechanism and the elimination-addition mechanism.
Nucleophilicity and basicity
In this tutorial, Sal discusses the difference between nucleophilicity and basicity.
Null space and column space
We will define matrix-vector multiplication and think about the set of vectors that satisfy Ax=0 for a given matrix A (this is the null space of A). We then proceed to think about the linear combinations of the columns of a matrix (column space). Both of these ideas help us think the possible solutions to the Matrix-vector equation Ax=b.
Number opposites
Opposite numbers are the same distance from 0 on opposite sides of the number line. A number opposite is sometimes called an additive inverse.
Number patterns
One of the fundamentals of algebra is learning to recognize patterns among numbers and being able to visualize those patterns and relationships. So...let's use some mathematical tools to visualize, interpret, and graph patterns involving the coordinate plane. Trust us, this sounds a lot more complicated than it really is. Just follow along and pause the videos as necssary. We're not going anywhere!
Number sets
The world of numbers can be split up into multiple "sets", many of which overlap with each other (integers, rational numbers, irrational numbers, etc.). This tutorial works through examples that expose you to the terminology of the various sets and how you can differentiate them.
Numbers 0 to 120
Learn the numbers from 0 to 120.
Oakland Unity
Oakland Unity, a charter school serving east Oakland, starting using Khan Academy with their 9th graders. By the end of the school year, additional grade levels were using the program based on the success they saw.
Object-Oriented Design
How to use object-oriented concepts in JavaScript to make more re-usable code.
Learn how to store complex data in objects.
Occupational lung diseases
Ocean Acidification in a Cup
Create a carbon dioxide-rich atmosphere in a cup and watch how it changes the water beneath it. This model of ocean–atmosphere interaction shows how carbon dioxide gas diffuses into water, causing the water to become more acidic. Ocean acidification is a change that can have big consequences.
Odds and evens
See how even numbers are multiples of 2.
Ohms law and circuits with resistors
Old limits tutorial
This tutorial covers much of the same material as the "Limits" tutorial, but does it with Sal's original "old school" videos. The sound, resolution or handwriting isn't as good, but some people find them more charming.
Old school equations with Sal
Some of Sal's oldest (and roughest) videos on algebra. Great tutorial if you want to see what Khan Academy was like around 2006. You might also like it if you feel like Sal has lost his magic now that he doesn't use the cheapest possible equipment to make the videos.
Old school similarity
These videos may look similar (pun-intended) to videos in another playlist but they are older, rougher and arguably more charming. These are some of the original videos that Sal made on similarity. They are less formal than those in the "other" similarity tutorial, but, who knows, you might like them more.
Old videos on projectile motion
This tutorial has some of the old videos that Sal first did around 2007. This content is covered elsewhere, but some folks like the raw (and masculine) simplicity of these first lessons (Sal added the bit about "masculine").
The Olmecs lived in the low-lying Gulf Coast of what is now Mexico about 1200-400 B.C.E. at sites such as San Lorenzo and La Venta. These and the other Olmec centers included many of the features that would be associated with later civilizations in Central America including the Mexica (Aztecs) and Maya. Alongside impressive architecture, there is evidence of a ceremonial ball game and complex astrological calendars.
Olney Charter High School,
TFA alum Tal Sztainer shares his experience teaching 12th grade math in Philadelphia and the impact of finding funding for 10 computers for his classroom.
One, zero, and infinite solutions
Not all linear equations in one variable have exactly one solution. Some have no solutions, others have exactly one, and then still others may have an infinite number of solutions.
One-digit division
Every time you split your avocado harvest with your 10 pet robot possums, you've been dividing. You don't farm avocados? You only have 8 robot possums? No worries. I'm sure you've divided as well. Multiplication is awesome, but you're ready for the next step. Division is the art of trying to split things into equal groups. Like subtraction undoes addition, division also undoes multiplication. After this tutorial, you'll have a basic understanding of all of the core operations in arithmetic!
One-step addition and subtraction equations
Solve equations by adding and subtracting numbers from both sides.
One-step inequalities
Learn how to solve (mathematical) inequalities. A key concept in this tutorial is flipping the inequality sign when multiplying or dividing both sides by a negative number.
One-step multiplication and division equations
Solve equations by multiplying and dividing numbers from both sides.
Operations with decimals
Optimal angle for a projectile
This tutorial tackles a fundamental question when trying to launch things as far as possible (key if you're looking to capture a fort with anything from water balloons to arrows). With a bit of calculus, we'll get to a fairly intuitive answer.
Optimization with calculus
Using calculus to solve optimization problems
Orbital mechanics 1
How do planets move? An introduction to orbital mechanics and the work of Johannes Kepler.
Order of operations
If you have the expression "3 + 4 x 5", do you add the 3 to the 4 first or multiply the 4 and 5 first? To clear up confusion here, the math world has defined which operation should get priority over others. This is super important. You won't really be able to do any involved math if you don't get this clear. But don't worry, this tutorial has your back.
Ordered-pair solutions to two-variable linear equations
Orders of magnitude
When people want to think about the general size of things but not worry about the exact number, they tend to think in terms of "orders of magnitude". This allows us to analyze and make comparisons between numbers very quickly, which allows us to make decisions about them quickly as well.
Organic acid-base chemistry
Do you remember the basics of acid-base chemistry? If your answer is neutral, then you might want to watch this tutorial and review these important concepts from general chemistry.
Orthogonal complements
We will know explore the set of vectors that is orthogonal to every vector in a second set (this is the second set's orthogonal complement).
Orthogonal projections
This is one of those tutorials that bring many ideas we've been building together into something applicable. Orthogonal projections (which can sometimes be conceptualized as a "vector's shadow" on a subspace if the light source is above it) can be used in fields varying from computer graphics and statistics! If you're familiar with orthogonal complements, then you're ready for this tutorial!
Orthonormal basis
As we'll see in this tutorial, it is hard not to love a basis where all the vectors are orthogonal to each other and each have length 1 (hey, this sounds pretty much like some coordinate systems you've known for a long time!). We explore these orthonormal bases in some depth and also give you a great tool for creating them: the Gram-Schmidt Process (which would also be a great name for a band).
Other K-12 case studies
See how Khan Academy is used in real life. These case studies cover various models in public, charter, and independent schools. We're excited about the way these organizations are using our resources and are eager to learn from more educators and students. For even more examples, check out our blog at
Other Materials
Access a range of other materials to support learning about this unit.
Other Reactions and Synthesis
In this tutorial, Jay covers a few more reactions of benzene derivatives and also shows how to approach the synthesis of substituted benzene rings.
Other cool stuff
Pythagoras, snakes, fractals, snowflakes...
Otto I (who became emperor in 962) lends his name to the “Ottonian” period. He forged an important alliance with the Pope, which allowed him to be crowned the first official Holy Roman Emperor since 924.
Download and practice with a real, full-length SAT provided by the College Board.
Overview (K-12 math classrooms)
Overview (out-of-school-time programs)
Overview and history of algebra
Did you realize that the word "algebra" comes from Arabic (just like "algorithm" and "al jazeera" and "Aladdin")? And what is so great about algebra anyway? This tutorial doesn't explore algebra so much as it introduces the history and ideas that underpin it.
Overview of Chinese history 1911-1949
The early 1900s marked the end of thousands of years of dynastic imperial rule in China. It also marked the beginning of a complex period of fragmentation, civil war and fending off Japanese imperial ambitions. This tutorial covers everything from the establishment of the Republic of China by Sun Yat-sen to the Warlord Era to the Chinese Civil War between the Chiang Kai-Shek led Kuomintang and the Communists led by Mao Zedong.
Overview of World War II
World War II was the bloodiest conflict in human history. Starting with the Second Sino-Japanese War (1937) and the Nazi invasion of Poland (1939) through the eventual Axis power surrenders, this tutorial walks through the major events, year by year, in both the European and Pacific Arenas.
Overview of metabolism
1D: You are breathing, your heart is beating, and you are reading this sentence. All these processes would be impossible were it not for the chemical energy produce within our cells. In this tutorial, we will integrate the biology and chemistry of metabolism as we walk you through the electron transport chain and the production of ATP, the ultimate energy currency in our bodies.
Overview: steps of cellular respiration
Before you jump into the nitty-gritty details, get the big picture of cellular respiration. Learn about the different stages of this process and how they fit together.
Oxidative phosphorylation
Oxidative phosphorylation produces most of the ATP made in cellular respiration. Learn more about how oxidative phosphorylation uses electron transport to drive ATP synthesis.
PIANO (as orchestral instrument): Interview and Demonstration with Kimberly Russ
This large family, including timpani, drums, cymbals, xylophones, gongs, bells, and rattles, is comprised of musical instruments played by striking with the hand or with a stick or beater, or by shaking or rubbing. The All-Star Orchestra percussion section demonstrates the remarkable variety of sounds that can be produced. Also included in this group are orchestral keyboard instruments like the piano and celesta.
Painting in Florence
In the 15th century, Florence was a proud republic where political power resided in the hands of wealthy merchant families (such as the Medici who would later seize control of Florence) and powerful guilds (organizations of merchants and craftsmen). Importantly for art history, all of these groups commissioned poetry, painting, sculpture and architecture—often as an expression of civic pride—making Florence the leading city-state in Italy during the cultural epoch we call the Renaissance. In the last half of the fifteenth century, Florentine artists were invited to Rome, to work for patrons like Pope Sixtus IV who had artists like Perugino, Ghirlandaio and Botticelli fresco the walls of the Sistine Chapel, in the Vatican.
Learn about paintings and portraits, and discover the techniques used by the artists who made them.
Paleolithic art
Humans make art. We do this for many reasons and with whatever technologies are available to us. But can we really know what such ancient images originally meant?
Paleolithic, Mesolithic and Neolithic
Periods of time before the written record are often defined in terms of geological eras or major shifts in climate and environment. The periods of global prehistory, known as lithic or stone ages, are Paleolithic (“old stone age”), Mesolithic (“middle stone age”), and Neolithic (“new stone age”). A glacial period produced European ice ages; Saharan agricultural grassland became desert; and tectonic shifts in southeast Asia created land bridges between the continent and the now-islands of the Pacific south of the equator. Human behavior and expression was influenced by the changing environments in which they lived. By permission, © 2013 The College Board
Palmyra, an ancient caravan city in the Syrian Desert, is renowned for its monumental architecture and colonnaded streets, as well as for its distinctive tower tombs. The ancient city and its material culture incorporated elements from Greek, Roman, and Near Eastern cultures and the city itself existed at a key point of connectivity where the spheres of the Mediterranean, Persian, and Near Eastern world overlapped.
You've seen parabolas already when you graphed quadratic functions. Now we will look at them from a conic perspective. In particular we will look at them as the set of all points equidistant from a point (focus) and a line (directrix). Have fun!
Paracas and Nasca cultures
The Paracas (c. 800 B.C.E. - 100 C.E.) culture immediately preceded and gave rise to the Nasca (c. 100 C.E. – 800 C.E.) on the southern coast of modern-day Peru. The dry conditions preserved a remarkable number of naturally-mummified burials, along with the textiles, ceramics, featherwork, jewelry, and other goods that honored the dead.
Parallel and perpendicular
Learn the definitions of "parallel" and "perpendicular".
Parametric equations
Here we will explore representing our x's and y's in terms of a third variable or parameter (often 't'). Not only can we describe new things, but it can be super useful for describing things like particle motion in physics.
Parametrizing a surface
You can parameterize a line with a position vector valued function and understand what a differential means in that context already. This tutorial will take things further by parametrizing surfaces (2 parameters baby!) and have us thinking about partial differentials.
Parkinson's disease
Partial derivatives
Let's jump out of that boring (okay, it wasn't THAT boring) 2-D world into the exciting 3-D world that we all live and breathe in. Instead of functions of x that can be visualized as lines, we can have functions of x and y that can be visualized as surfaces. But does the idea of a derivative still make sense? Of course it does! As long as you specify what direction you're going in. Welcome to the world of partial derivatives!
Partial fraction expansion
Partial quotient division
Feeling constrained by traditional long division? Want to impress your friends, family and even your enemies? Well, partial quotient division may be for you (or it might not). This very optional tutorial will show you that there are many ways to slice a walnut (just made up that colloquialism).
Performance art is more than a prompt for its audience to consider and question art–it's often an invitation for the audience to get involved. These artists push the boundary between life and art, creating works that rely on the participation of others. Learn more about what can happen when you join in.
Particle Systems
Passive transport
Sometimes it pays to go with the flow! Learn about passive transport mechanisms, in which molecules travel across the plasma membrane by moving down their concentration gradients (no energy expenditure required).
Passport to advanced mathematics
Patterns in arithmetic
Math is full of patterns that will keep emerging as you explore it (this is what makes math so beautiful). We begin to just scratch the surface in this tutorial! Common Core Standards: 3.OA.B.5, 3.OA.D.9
Patterns in data
Paulson Bailout
In the fall of 2008, it became clear that a cascade of bank failures was happening because of shoddy loans and exotic securities (both which fueled a now popping housing bubble). In an attempt to avoid a depression, the Treasury Secretary (Hank Paulson) wanted to pour $1 Trillion into the same banks that had created the mess. This tutorial walks us through the beginnings of the mess and possible solutions. Historical note: it was created as the crisis was unfolding.
Penny Battery
Can you light up a room with just five cents? A battery is a device that converts chemical energy into electrical energy. This activity, developed and demonstrated by Julie Yu, director of the Exploratorium Teacher Institute, shows you how to build a battery strong enough to power an LED for a whole day—using nothing more than a few pennies and a salty solution. Watch the videos to learn how to build a battery of your very own, and light the way for other chemistry lovers.
Percent problems
Learn how to solve percent problems like these: What is 80% of 25? Three is what percent of four?
Percent word problems
Learn how to solve a variety of word problems involving percentages.
Percent-decimal conversions
Decimals and percents are two ways of expressing the same number. For example, 0.5 and 50% mean the exact same thing. Learn how to convert decimals to percents and vice versa.
Perception, prejudice, and bias
Have you ever judged someone based on their appearance? Maybe they were tall and good looking and you assumed they were successful and intelligent. Here we will explore how and why we see the world the way we do as well as the implications of those perceptions.
This large family, including timpani, drums, cymbals, xylophones, gongs, bells, and rattles, is comprised of musical instruments played by striking with the hand or with a stick or beater, or by shaking or rubbing. The All-Star Orchestra percussion section demonstrates the remarkable variety of sounds that can be produced. Also included in this group are orchestral keyboard instruments like the piano and celesta.
Perfect competition
This tutorial looks at markets that are deemed to have "perfect competition." This means that there are many players with identical products, no barriers to entry, no advantage for existing players and good pricing information. Few to no real market completely matches this theoretical ideal, but many are close. Even the example we use in this tutorial (the airline industry) isn't quite perfect (you should think about why).
Performance art
Scripted or unscripted, spontaneous or carefully orchestrated, performance art challenges us to think outside the box through experiences, performances, and interventions. Learn more about the history of this innovative discipline and meet some of its standout figures, from Marina Abramović to Vito Acconci and more in between.
Performance art: Marina Abramović
Theatrical and staged elements have been a key feature of visual art throughout the 20th century. Movements like Futurism, Dada, and Bauhaus employed theater, dance, music, and poetry with live or broadcast performances to engage with audiences. In the 1960s and 1970s, performance gained renewed momentum when artists conceived of Happenings, Fluxus, "actions," experimental dance, and site-specific interventions. Throughout its history MoMA has been host to many artworks involving live and performative elements. While most of these activities previously took place at the periphery of MoMA's exhibition program, the 2008 addition of "and Performance Art" to what was then called the Department of Media introduced performance art as a central component in the Museum's programming.
Have you ever wondered how much fencing you need to surround a plot of land? No? Well, you should still go through this tutorial just in case. You'll learn all about how to think about and calculate perimeter--essentially the length of the boundary of a figure.
Perimeter and Area
Area and perimeter help us measure the size of 2D shapes. We'll take a conceptual look at how they are actually measured.
Perimeter and area
You first learned about perimeter and area when you were in grade school. In this tutorial, we will revisit those ideas with a more mathy lens. We will use our prior knowledge of congruence to really start to prove some neat (and useful) results (including some that will be useful in our study of similarity).
Perimeter and area of non-standard shapes
Not everything in the world is a rectangle, circle or triangle. In this tutorial, we give you practice at finding the perimeters and areas of these less-than-standard shapes!
Periodic table trends
Have you ever wondered why it is called the "periodic" table? The elements are organized based on their electronic structure, which results in some interesting trends in atomic properties as you move down a group or across a period. This tutorial will try to explain some of those trends!
You want to display your Chuck Norris dolls on your desk at school and there is only room for five of them. Unfortunately, you own 50. How many ways can you pick the dolls and arrange them on your desk?
Perpendicular bisectors
In this tutorial, we study lines that are perpendicular to the sides of a triangle and divide them in two (perpendicular bisectors). As we'll prove, they intersect at a unique point called the cicumcenter (which, quite amazingly, is equidistant to the vertices). We can then create a circle (circumcircle) centered at this point that goes through all the vertices. This tutorial is the extension of the core narrative of the Geometry "course". After this, you might want to look at the tutorial on angle bisectors.
Western histories often looked at the Persians only in relation to their confrontations with the ancient Greeks, but the Persian empire was long-lived, complex and sophisticated. The heart of ancient Persia is in what is now southwest Iran. In the second half of the 6th century B.C.E., the Persians created an enormous empire reaching from the Indus Valley to Northern Greece and from Central Asia to Egypt.
Personal bankruptcy
Back in the day (like medieval Europe), you would actually be thrown in jail if you couldn't pay your debts (debtor's prison). That seemed like a pretty awful thing to do (not to mention that lenders are much less likely to be paid by someone rotting in prison), so governments created an "out" called bankruptcy (which, as you'll see, is a pseudo-painful "reset" button on your finances).
Petrus c
Philip Rosedale - Founder and Chairman of Second Life
Philip Rosedale, Founder of Coffee and Power, discusses his recent venture and how a student’s education today can lead to a career tomorrow.
Learn how photographers and artists have explored innovative ways to use photography to represent and interpret the changing world.
Photographic processes
Explore the earliest photographic processes, techniques, and influences that led the way to modern photography.
Physical sciences practice questions
These questions are available under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License (available at
Watch fun, educational videos on all sorts of Physics questions.
Picture graphs
Learn how to read and make picture graphs.
Piecewise functions
In this tutorial, we will get practice looking at wackier functions that are defined interval by interval (or piece by piece)!
Piketty's Capital in the 21st Century
In this tutorial we will dig a bit into Thomas Piketty's popular "Capital in the 21st Century" that attempts to analyze the relationship(s) between economic growth, returns on capital and inequality. Our intent is to neither argue for nor against the ideas in the book, but rather use it as a tool for critical reasoning and discourse.
Pisa and Rome, the Late Gothic
Place value
You've been counting for a while now. It's second nature to go from "9" to "10" or "99" to "100", but what are you really doing when you add another digit? How do we represent so many numbers (really as many as we want) with only 10 number symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)? In this tutorial you'll learn about place value. This is key to better understanding what you're really doing when you count, carry, regroup, multiply and divide with mult-digit numbers. If you really think about it, it might change your worldview forever!
Place value and decimal operations
Here we will explore understanding decimals conceptually, using operations with decimals, rounding and estimating with decimals, and converting decimals. And why not go into the fun concepts of manipulating decimals through addition, subtraction, multiplication and division! It really is a breeze once you are done with these videos!
Place value and decimals
Here we will explore understanding decimals conceptually, using operations with decimals, rounding and estimating with decimals, and converting decimals. And why not go into the fun concepts of manipulating decimals through addition, subtraction, multiplication and division! It really is a breeze once you are done with these videos!
Place value introduction
The Sun, an unremarkable star, holds in orbit a system of planets. If most Sun-like stars have planets, then there should be billions of planets in our Milky Way Galaxy. Numerous other planetary systems have already been detected, many with configurations quite different than that of our solar system. The videos in this tutorial explore planetary systems, moving from those in our galactic neighborhood to those that orbit distant stars. These videos and articles survey the diverse planets in our solar system, from the forces that shape their surfaces to ring systems and moons to the search for the building blocks of life. Last is a documentary about how astrophysicists are using new tools to study the chemical characteristics of these faraway worlds.
Plasma cell dyscrasias
Plate Tectonics
Plate tectonics has emerged as one of the grand unifying theories of geology. It connects seemingly unrelated features and events of the planet — its continents and oceans, its mountains, its volcanoes and earthquakes — to a single global process. That process is the slow movement of plates on the Earth’s surface. These rigid plates are continually being formed, altered, and consumed. They move on the mantle below and carry the continents along with them.
Plate tectonics
Is it a coincidence that Africa and South America could fit like puzzle pieces? Why do earthquakes happen where they do? What about volcanoes and mountains? Are all of these ideas linked? Yes, they are. This tutorial on plate-tectonics explains how and why the continents have shifted over time. In the process, we also explore the structure of the Earth, all the way down to the core.
Pleural Effusion
Pleural effusion
Point-slope form
Poisson process
Polar coordinates
Feel that Cartesian coordinates are too "square". That they bias us towards lines and away from cool spirally things. Well polar coordinates be just what you need!
The Polynesian collection at the Museum date back to the earliest contact with European explorers and missionaries. Discover a massive stone Easter Island figure, an intricately carved Maori lintel from New Zealand and a feather cape from Hawaii.
Polynomial basics
"Polynomials" sound like a fancy word, but you just have to break down the root words. "Poly" means "many". So we're just talking about "many nomials" and everyone knows what a "nomial" is. Okay, most of us don't. Well, a polynomial has "many" terms. From understanding what a "term" is to basic simplification, addition and subtraction of polynomials, this tutorial will get you very familiar with the world of many "nomials." :)
Polynomial graphs and end behavior
In this tutorial, we will study the behavior of polynomials and their graphs. In particular, we'll look at which forms of a polynomial are best for determining various aspects of its graph.
Polynomial remainder theorem
The polynomial remainder theorem allows us to easily determine whether a linear expression is a factor of a given polynomial. Learn exactly what the theorem means, practice using it, and learn about its proof.
Wham! Warhol, Lichtenstein and Oldenburg crashed the gates of high culture asking what authentic art looks like in a society filled with tawdry commercial images.
Position vector functions and derivatives
In this tutorial, we will explore position vector functions and think about what it means to take a derivative of one. Very valuable for thinking about what it means to take a line integral along a path in a vector field (next tutorial).
Post college acceptance
Now comes the best part - it's time to make your pick! Select the college you will be attending, send in your deposit, and then begin the college transition . Life is about to change, and after you select your college, the decisions are only beginning. From housing and meal plans to orientation and course selection - it's going to be a very busy, very exciting summer!
Almost no ancient Greek wall painting survives, fortunately, their magnificent painted vases can tell us a great deal about lost wall painting as well as Greek culture and technology. This tutorial traces Greek pottery from the Geometric through to the Attic red figure style.
Power rule
Calculus is about to seem strangely straight forward. You've spent some time using the definition of a derivative to find the slope at a point. In this tutorial, we'll derive and apply the derivative for any term in a polynomial. By the end of this tutorial, you'll have the power to take the derivative of any polynomial like it's second nature!
Power series function representation using algebra
Now that we're familiar with the idea of an infinite series, we can now think about functions that are defined using infinite series. In particular, we'll begin to look at the power series representation of a function (and the special case of a geometric series). In later tutorials, we'll use calculus to find the power series of more types of functions.
Powers of 10
Learn how exponents are a simpler way to write something like 10x10x10.
Practice dividing polynomials with remainders
After learning about the different methods in which we can find the quotient and the remainder of two polynomials, gain some practice with actually performing polynomial division yourself.
Pre-Raphaelites and mid-Victorian art
In 1848, a small group of young artists banded together and formed “The Pre-Raphaelite Brotherhood,” a name which sounds intentionally backward-looking and medieval. The Pre-Raphaelites looked back to art before the time of Raphael (before about 1500 that is)—before the art of the Renaissance was reduced to formulas followed for centuries by artists associated with the art academies of Europe. Their idea was that the art before Raphael was more sincere, more true to nature and how we see, and therefore less formulaic. These artists also embraced a wide range of subjects, including modern life, biblical and literary subjects, and even history. By looking backwards, the Pre-Raphaelites led British art into the modern era.
Precalculus readiness warmup
Predynastic and Old Kingdom
While today we consider the Greco-Roman period to be in the distant past, it should be noted that Cleopatra VII's reign (which ended in 30 B.C.E.) is closer to our own time than it was to that of the construction of the pyramids of Giza. It took humans nearly 4000 years to build something--anything--taller than the Great Pyramids. Contrast that span to the modern era; we get excited when a sports record lasts longer than a decade.
Pregnancy and pregnancy complications
Preload and Afterload
After using your jeans for a while, you’ll begin to notice small tears and rips developing in the fabric. Why doesn’t this happen to your heart as well? Well, your heart manages to stay healthy despite all of the “wall stress” that pulls on the heart walls. During different parts of the heart cycle (afterload vs. preload) the mechanics of “wall stress” change dramatically. Learn exactly what preload and afterload mean, and how we can use pressure-volume loops to estimate their values.
Preload and Afterload
After using your jeans for a while, you’ll begin to notice small tears and rips developing in the fabric. Why doesn’t this happen to your heart as well? Well, your heart manages to stay healthy despite all of the “wall stress” that pulls on the heart walls. During different parts of the heart cycle (afterload vs. preload) the mechanics of “wall stress” change dramatically. Learn exactly what preload and afterload mean, and how we can use pressure-volume loops to estimate their values.
Preparing students for success
Present value
If you gladly pay for a hamburger on Tuesday for a hamburger today, is it equivalent to paying for it today? A reasonable argument can be made that most everything in finance really boils down to "present value". So pay attention to this tutorial.
Pressure Volume Loops
The pressure volume loop is one of the classic figures that helps us to conceptualize and understand the mechanics of the left ventricle of the heart. In addition to a filling up with blood and squeezing out blood there is a (very short) period of time when the heart muscle is contracting and relaxing with no volume change! As the left ventricle moves around the PV loop with each lub dub you get a sense for the amazing amount of work it does as pressures and volumes go up and down, all day, every day. This is a fascinating area where physics and biology meet to produce something miraculous.
Price discrimination
This short tutorial explores how a wine business can utilize first-degree price discrimination to maximize economic profit (it uses many of the ideas we've explored in the rest of this tutorial).
Price elasticity
You're familiar with supply and demand curves already. In this tutorial we'll explore what implications their steepness (or lack of) implies. Price elasticity is a measure of how sensitive something is to price.
Primality Test
Why do primes make some problems fundamentally hard? To find out we need to explore primality tests in more detail.
Prime and composite numbers
Prime numbers have been studied by mathematicians and mystics for ages (seriously). They are both basic and mysterious. The more you explore them, the more you will realize that the universe is a fascinating place. This tutorial will introduce you to the magical world of prime numbers as well as their composite number brethren. Common Core Standard: 4.OA.B.4
Prime factorization
You know what prime numbers are and how to identify them. In this tutorial, we'll see that *all* positive whole numbers can be broken down into products of prime numbers (In some way, prime numbers are the "atoms" of the number world that can be multiplied to create any other number). Besides being a fascinating idea, it is also extremely useful. Prime factorization can be used to decrypt encrypted information!
Prime numbers
Prime numbers have been studied by mathematicians and mystics for ages (seriously). They are both basic and mysterious. The more you explore them, the more you will realize that the universe is a fascinating place. This tutorial will introduce you to the magical world of prime numbers.
Prime numbers, factors and multiples
After these videos, understanding the concept of prime numbers and the ones for which you can finding factors and multiples, will be no brainers!
Principles of bioenergetics
1D: When you light a candle, energy in the form of heat is dissipated into the surroundings. Without energy transfer, frogs wouldn’t jump, and cheetahs wouldn’t run. We will discuss Gibbs free energy, enthalpy, and Le Chatelier’s principles, thermodynamic concepts governing energy transfer as we examine their relationship to metabolism. After this tutorial, you will understand what it really means to “burn calories” during exercise.
Artists have used printmaking to create some of their most profound and compelling works of art, yet the basic printmaking techniques remain a mystery to most people. These videos demonstrate three key printmaking processes—relief, intaglio, and lithography. They include prints from the Museum's collection to demonstrate the range of expressive effects associated with each technique.
Flip a quarter a hundred times. What's the probability that it will turn up heads? Tails? Explores probability models, sample spaces, predict the frequency of an outcome of a simple probability experiment or game and a whole lot more.
Flip a quarter a hundred times. What's the probability that it will turn up heads? Tails? Explores probability models, sample spaces, predict the frequency of an outcome of a simple probability experiment or game and a whole lot more.
Probability and statistics readiness warmup
Probability using combinatorics
This tutorial will apply the permutation and combination tools you learned in the last tutorial to problems of probability. You'll finally learn that there may be better "investments" than poring all your money into the Powerball Lottery.
Probability warmups
The 'problem of points' is a classic problem Fermat and Pascal famously debated in the 17th century. Their solution to this problem formed the basis of modern day probability theory. Now it's your turn to relive this challenge!
Problem solving and data analysis
Product and quotient rules
You can figure out the derivative of f(x). You're also good for g(x). But what about f(x) times g(x)? This is what the product rule is all about. This tutorial is all about the product rule. It also covers the quotient rule (which really is just a special case of the product rule).
Production possibilities frontier
This tutorial goes back to the basics. You are a hunter-gatherer with only so much time to hunt or gather. How do you allocate your time and energy to maximize your happiness? This is what we try to understand through our study of the production possibilities frontier and opportunity cost.
Programming Transformations
Learn how to use translate(), rotate(), and scale() for easier manipulation of the shapes in your programs.
Programming basics
Collection of programming basics using NXT-G
Projectile Launcher
When you think about prokaryotes, disease-causing bacteria might come to mind. However, most prokaryotes are not harmful. Prokaryotes live symbiotically with many plants and animals and are key players in the global ecosystem. They are incredibly diverse and are found in the most extreme of environments. These videos will take you through a general overview of these amazing organisms.
Prokaryotic and eukaryotic cells
Here's a question: what's the difference between you and a bacterium? Probably quite a lot of things! At the cellular level, though, some of the key differences include a cell nucleus and membrane-bound organelles, which you have (and the bacterium lacks). Here, we'll look in detail at the differences between prokaryotes and eukaryotes.
Proof of Stokes' theorem
You know what Stokes' theorem is and how to apply it, but are craving for some real proof that it is true. Well, you've found the right tutorial!
Proofs of theorems that use triangle congruence
Gain even more experience with using triangle congruence in proofs.
Properties and definitions of transformations
Let's continue our deep voyage through the world of transformations! Use your knowledge about the precise descriptions of the rigid transformations and their properties in various situations.
Properties of Numbers and Order of Operations
We'll hear really good explanations and practice rewriting expressions so that we can extend our understanding of the properties of numbers and of PEDMAS!
Properties of addition and subtraction
Apply the properties of addition and subtraction to negative numbers.
Properties of carbon
Learn why carbon is a fantastic building block for biological molecules! We'll look at carbon's atomic structure and ability to form bonds...and as a special bonus, we'll investigate the possibility of silicon-based life forms.
Properties of matrix multiplication
As we'll see, some of the properties of scalar multiplication (like the distributive and associative properties) have analogs in matrix multiplication while some don't (the commutative property).
Properties of multiplication
Learn about the commutative, associative, and distributive properties of multiplication. Solve multiplication problems by changing the order of factors (commutativity), the grouping of factors (associativity), and by decomposing the factors (distributive property).
Properties of multiplication and division
See how the properties of multiplication and division apply to negative numbers.
Properties of numbers
This group of tutorials will introduce us to some of the common properties of numbers, including the least common multiple (LCM), greatest common factor (GCF), and the distributive property. We'll hear really good explanations and practice rewriting expressions so that we can extend our understanding of the properties of numbers and of PEDMAS! All of these will be extremely useful going forward. Really, it's true!
Properties of the Laplace transform
You know how to use the definition of the Laplace transform. In this tutorial, we'll explore some of the properties of the transform that will start to make it clear why they are so useful for differential equations. This tutorial is paired well with the tutorial on using the "Laplace transform to solve differential equations". In fact you might come back to this tutorial over and over as you solve more and more problems.
Properties of the definite integral
You now know that the area under a curve can be denoted by a definite integral. In this tutorial, we'll extend that knowledge by exploring various properties of the definite integral (that will be really useful later on in life)!
Protecting student privacy on Khan Academy (for parents)
5D: DNA makes RNA, and RNA makes proteins,, which are one of the most important biomolecules of our bodies. They are comprised of even smaller amino acids, which are held together by peptide bonds to form proteins. You will discover the structure and functions of proteins at the cellular level in this tutorial.
Protestant reformation and Catholic counter-reformation (3-5)
Production of religious imagery declined in northern Europe, and nonreligious genres, such as landscape, still life, genre, history, mythology, and portraiture, developed and flourished. In the south, there was an increase in the production of political propaganda, religious imagery, and pageantry, with the elaboration of naturalism, dynamic compositions, bold color schemes, and the affective power of images and constructed spaces. Production of religious imagery declined in northern Europe, and nonreligious genres, such as landscape, still life, genre, history, mythology, and portraiture, developed and flourished. In the south, there was an increase in the production of political propaganda, religious imagery, and pageantry, with the elaboration of naturalism, dynamic compositions, bold color schemes, and the affective power of images and constructed spaces. By permission, © 2013 The College Board
Proton nuclear magnetic resonance
Let’s shift gears and take a peak at 1H-NMR. When you bring a magnet close to iron filings, the filings align with the applied field. The hydrogen atoms of molecules behave similarly under a strong magnetic field, but bounce back and radiate when the field is removed. in this section you will learn how to interpret the spectra created by proton NMR.
Proving the chain rule
We've already been using the chain rule, but let's take a moment to really convince ourselves that it'll always work. It's not necessary to watch these videos before moving on, but if you have the time, they'll help to deepen your understanding of derivatives, continuity, and the chain rule.
Psychological disorders
When someone breaks their leg they get a cast to fix the damage and when someone gets a papercut, they use a bandaid. But what happens when someone’s mind needs to be healed? Although many mental disorders are difficult to diagnose, these videos explore the symptoms and biological explanations of a few mental disorders that are extensively researched.
Psychotic disorders
Read the Rosetta stone, discover how mummies were prepared for the afterlife and how we care for them now.
Public goods and externalities
In many scenarios thinking only about producers' marginal cost or consumers' marginal benefit does not fully capture *all* of the costs or benefits from the production/use of a product. In this tutorial, we explore these externalities (negative and positive ones) to think a bit deeper about ways to maximize total surplus not just for producers and consumers, but for society as a whole.
Pulmonary Diseases
Pulmonary hypertension
Put and call options
Options allow investors and speculators to hedge downside (or upside). It allows them to trade on a belief that prices will change a lot--just not clear about direction. It allows them to benefit in any market (with leverage) if they speculate correctly. This tutorial walks through option basics and even goes into some fairly sophisticated option mechanics.
Put together, take apart
When we add, we put together. When we subtract, we take apart. Numbers used are 10 or less.
Puzzles and surprises
Pyruvate oxidation and the citric acid cycle
Glycolysis can extract a bit of the energy from a glucose molecule, but the citric acid cycle can squeeze out much more. Learn about the steps of the cycle, how it harvests energy, and how it's fueled by acetyl CoA (produced by oxidation of pyruvate from glycolysis).
Pythagorean Theorem Proofs
The Pythagorean theorem is one of the most famous ideas in all of mathematics. This tutorial proves it. Then proves it again... and again... and again. More than just satisfying any skepticism of whether the Pythagorean theorem is really true (only one proof would be sufficient for that), it will hopefully open your mind to new and beautiful ways to prove something very powerful. Common Core Standard: 8.G.B.6
Pythagorean identity
In this tutorial, we look at the relationship between the definitions of sine, cosine and tangent (both SOH CAH TOA and unit circle definitions) and the Pythagorean theorem to derive and apply the Pythagorean identity. This is the building block of much of the rest of the trigonometric identities and will be surprisingly useful the rest of your life!
Pythagorean theorem
Named after the Greek philosopher who lived nearly 2600 years ago, the Pythagorean theorem is as good as math theorems get (Pythagoras also started a religious movement). It's simple. It's beautiful. It's powerful. In this tutorial, we will cover what it is and how it can be used. We have another tutorial that gives you as many proofs of it as you might need.
Pythagorean theorem proofs
The Pythagorean theorem is one of the most famous ideas in all of mathematics. This tutorial proves it. Then proves it again... and again... and again. More than just satisfying any skepticism of whether the Pythagorean theorem is really true (only one proof would be sufficient for that), it will hopefully open your mind to new and beautiful ways to prove something very powerful.
Qin dynasty (221-206 B.C.E.)
Quadratic inequalities
You are familiar with factoring quadratic expressions and solving quadratic equations. Well, as you might guess, not everything in life has to be equal. In this short tutorial we will look at quadratic inequalities.
Quadratic odds and ends
This tutorial has a bunch of extra, but random, videos on quadratics. A completely optional tutorial that you may or may not want to look at. If you do, watch it last. There are some Sal oldies here and some random examples.
Not all things with four sides have to be squares or rectangles! We will now broaden our understanding of quadrilaterals.
Quadrilaterals on the coordinate plane
Math problems that test your understanding of Quadrilateral and points on the coordinate plane
Quantum numbers and orbitals
In this tutorial, we will see how the quantum numbers predict the orbitals available in different energy levels.
Quasars and galactive collisions
Quasars are the brightest objects in the universe. The gamma rays from them could sterilize a solar system (i.e. obliterate life). What do we think these objects are? Why don't we see any close by (which we should be thankful for)? Could they tell us what our own galaxy may have been like 1 billion or so years ago?
Quick sort
Learn quick sort, another efficient sorting algorithm that uses recursion to more quickly sort an array of values.
Quick start guides
Quiz: Ancient Near Eastern art
Quiz: Byzantine art
Quiz: materials and techniques
Quiz: prehistoric art
RNA: the wonder molecule
Nature’s best kept secret is a wonder molecule called RNA. Find out why it’s central to the origin of life, evolution, and the cellular machinery that keeps us alive––and how you can help scientists understand it better just by playing a game.
Most people know that you can measure angles with degrees, but only exceptionally worldly people know that radians can be an exciting alternative. As you'll see, degrees are somewhat arbitrary (if we lived on a planet that took 600 days to orbit its star, we'd probably have 600 degrees in a full revolution). Radians are pure. Seriously, they are measuring the angle in terms of how long the arc that subtends them is (measured in radiuseseses). If that makes no sense, imagine measuring a bridge with car lengths. If that still doesn't make sense, watch this tutorial!
Radical equations
In this tutorial you will solve radical equations.
Radioactive decay
Random sampling warmup
Introduction to random sampling (also known as the weak law of large numbers)
Random variables and probability distributions
Randomized Algorithms
Would access to coin flips speed up a primality test? How would this work?
Leonardo, Michelangelo, Raphael, and Bramante. The High Renaissance was short lived but it changed the way we see ourselves.
Rate problems with fractions
Learn how to compute rates associated with ratios of fractions or decimals. For example, if you run 6.5 meters in 1.5 seconds, what is your speed in meters per second?
How fast can a robot possum fly? What is the rate at which a hungry you can consume avocados? This tutorial helps you make sense of these fundamental questions in life.
Rates of change
Have you ever wondered how fast the area of a ripple of a pond is increasing based on how fast the ripple is? What about how fast a volcano's volume is increasing? This tutorial on related rates will satiate your curiosity and then some! Solving related rates problems using calculus
Ratio word problems
Get some practice solving ratio word problems.
Rational and irrational expressions
Determine whether various combinations of rational and irrational numbers are rational or irrational themselves.
Rational and irrational numbers
Learn about irrational numbers: what are they? How do we know they exist? How many of them there are? Also learn how to distinguish irrational numbers from rational numbers.
Rational exponents and the properties of exponents
In this tutorial you will rewrite variable expressions with rational exponents and radicals using the properties of exponents.
Ratios and percentages
A common application of numbers in real life (what is REAL life anyway, huh?) is through ratios, rates, and percentages. Whether following a recipe, changing the oil in a car, or determining the percentage earned on a test...these are skills you will use over and over. Ratios are simply another way to express fractions. Rates combine units of measurements to help us understand their relationship better. Percentages, like ratios, give us another way to express fractions and decimals. We'll explain each to you and then practice until you've mastered each concept.
Ratios and proportions
Would you rather go to a college with a high teacher-to-student ratio or a low one? What about the ratio of girls-to-boys? What is the ratio of eggs to butter in your favorite dessert? Ratios show up EVERYWHERE in life. This tutorial introduces you to what they (and proportions) are and how to make good use of them!
Ratios, rates and percentages
A common application of numbers in real life (what is REAL life anyway, huh?) is through ratios, rates, and percentages. Whether following a recipe, changing the oil in a car, or determining the percentage earned on a test...these are skills you will use over and over. Ratios are simply another way to express fractions. Rates combine units of measurements to help us understand their relationship better. Percentages, like ratios, give us another way to express fractions and decimals. We'll explain each to you and then practice until you've mastered each concept.
Reaction rates and rate laws
In this tutorial, we will examine the rates of chemical reactions and learn how to write rate laws.
Reactions of alcohols
In this tutorial, Jay assigns oxidation states to alcohols, shows an oxidation mechanism using the Jones reagent, shows the formation of nitrate esters from alcohols, and demonstrates how to make alkyl halides from alcohols. Biochemical redox reactions are also discussed.
Reactions of aldehydes and ketones
In this tutorial, Jay shows you how to predict the products of the reactions of aldehydes and ketones.
Reactions of benzene
In this tutorial, Sal shows the mechanism of Electrophilic Aromatic Substitution and the reactions of bromination and Friedel-Crafts Acylation.
Reading and interpreting data
This tutorial is less about statistics and more about interpreting data--whether it is presented as a table, pictograph, bar graph or line graph. Good for someone new to these ideas. For a student in high school or college looking to learn statistics, it might make sense to skip (although it might not hurt either).
Real and nominal GDP
The value of a currency is constantly changing (usually going down in terms of what you can buy). Given this, how can we compare GDP measured in dollars in one year to another year? This tutorial answers that question by introducing you to real GDP and GDP deflators.
Real and nominal return
If the value of money is constantly changing, can we compare investment return in the future or past to that earned in the present? This tutorial focuses on how to do this (another good tutorial to watch is the one on "present value").
Realism in the United States
Reciprocal trig functions
You're now familiar with sine, cosine and tangent. Now you'll see that mathematicians have also defined functions that are the reciprocal of those: cosecant, secant and cotangent.
Recognizing functions
Not all relationships are functions. In this tutorial, you'll learn which are!
Recognizing shapes
Learn how different shapes have different numbers of corners, different side lengths, and so on. Use these properties to help you recognize shapes.
Rectangle area and perimeter word problems
In this tutorial, you'll stretch your understanding of area and perimeter by applying it to word problems.
Recursive Algorithms
Learn the concept of recursion, a technique that is often used in algorithms. See how to use recursion to calculate factorial and powers of a number, plus to generate art.
Redox Reactions
4E: You’re the resident on call in the wards of the hospital and have been receiving calls all day long. When you finally plug in your phone for charging, a redox reaction takes place to refuel its battery. At one terminal of the battery, a reaction is producing free electrons, while the other end absorbs these electrons. We will delve into the mechanics of this elegant process - part of the reason we are able to live in an increasingly wireless world - in this tutorial.
Redox reactions
Oxidation and reduction are powerful ideas for thinking about how charge is transferred within a reaction. As we'll see, it is something of a hypothetical, but it is, nonetheless, very useful.
Reduced row echelon form
You've probably already appreciated that there are many ways to solve a system of equations. Well, we'll introduce you to another one in this tutorial. Reduced row echelon form has us performing operations on matrices to get them in a form that helps us solve the system.
Reflection and refraction
Learn how to find the reflection that maps a given figure to another, and how to manually draw the image of a reflection.
Regrouping decimal numbers
Let's tackle some more challenging problems that stretch our understanding of decimal place value. These problems involve regrouping place values in interesting ways.
Regrouping decimals
Let's explore how we can regroup and redistribute value among the various place values in a decimal number.
Regrouping whole numbers
Regrouping involves taking value from one place and giving it to another. It is a great way to make sure you understand place value. It is also super useful when subtracting multi-digit numbers (the process is often called "borrowing" even though you never really "pay back" the value taken from one place and given to another).
Relating multiplication and division
Let's see how multiplication and division are related.
Learn how to store related data in multiple tables and use joins to bring them together (inner joins, outer joins, and self joins).
Relationship between reaction concentrations and time
In this tutorial, we will convert rate laws into equations that allow us to determine the concentrations of reactants at any time during the course of a reaction.
Relationships between angles
Let's complement and supplement our knowledge of angles with some new geometry vocabulary. In this group of tutorials you'll learn about complementary, supplementary, vertical, adjacent, and straight angles. You'll quickly realize that the relationship bewten each of the angle types is quite logical and solving problems involving intersecting lines is a snap. Onward! Common Core Standards: 7.G.B.5
Reliquaries, manuscripts and tapestries
Some division problems don't work out perfectly. For example, we know that 6 divided by 3 is 2, and 9 divided by 3 is 3. But, what is 7 divided by 3? Or 8 divided by 3? In division problems like these, we're left with remainders.
Renaissance Art in Europe
The period dating from the late Middle Ages until the end of the eighteenth century was a time of great social change. Political revolution, religious upheaval and the discovery of new continents radically influenced European life.
Renal regulation of blood pressure
In addition to fluid and electrolyte balance, the kidneys play an essential role in regulating blood pressure. When the nephron, the functional unit of the kidney, senses that it is not receiving much blood flow, it knows something is up. So begins a cascade which allows the nephron to reabsorb more fluid from the urine, in attempt to fill up the perceived dip in blood volume. Here you will learn the mechanism of ADH and the renin-angiotensin-aldosterone system in sustaining this homeostasis.
Renal system
3B: If you want to learn more about the renal system, then urine the right place! (Pun aside, the kidneys are about more than just making urine). Every thirty minutes, your kidneys filter the entire blood supply in your body. Imagine a dirty pool filled with algae. Placing a filter in this pool will cause the algae to be flushed out, and after a time you’ll have a clean, crisp blue pool to enjoy. Just like the filter for a pool, our kidneys filter the blood and remove toxic wastes. These paired organs are key to maintaining electrolyte and water homeostasis in your body.
Renal system introduction
3B: If you want to learn more about the renal system, then urine the right place! (Pun aside, the kidneys are about more than just making urine). Every thirty minutes, your kidneys filter the entire blood supply in your body. Imagine a dirty pool filled with algae. Placing a filter in this pool will cause the algae to be flushed out, and after a time you’ll have a clean, crisp blue pool to enjoy. Just like the filter for a pool, our kidneys filter the blood and remove toxic wastes. These paired organs are key to maintaining electrolyte and water homeostasis in your body.
Renaud Laplanche, Founder of Lending Club
Renaud Laplanche was opening his mail when the idea for Lending Club came to him. He tells the story of seeing the opportunity and creating the online Lending Club to fill the gap in the financial industries market. Laplache’s competitive nature extends to one-man sailboat racing and he compares the risks and rewards of racing with setting the pace as an entrepreneur.
Rendering 101
Discover the basics behind the rendering equation. This lesson is designed for all ages and takes 35-40 minutes.
Renting vs. buying a home
Is it always better to buy than rent? What if home prices go up dramatically and rents don't? How can we compare home prices to rents to figure out what to do. This older tutorial (low-res, bad handwriting) walks us through this. It is about housing but similar thinking can be applied to any rent-vs-buy decision (spoiler alert, Sal did eventually buy a home).
Repeated addition
Adding the same number to itself over and over
Repeating decimals
Learn how to convert between repeating decimals and fractions.
Reproductive system
Starting with sexual reproduction and then the development of a baby, we’re going to take a journey that starts with our own microscopic beginning, and ends with a fully formed baby entering the world.
Reproductive system introduction
Our genes are survivors. Even though a person might die, his or her genes will go on to the next generation and then the next generation… Some genes are exactly the same as the ones in your great-great-great-grandmother! Yet, with the exception of identical twins, we are all genetically distinct from our family members. Starting with sexual reproduction and then the development of a baby, we’re going to take a journey that starts with our own microscopic beginning, and ends with a fully formed baby entering the world.
In legend Rome was founded in 753 B.C.E. by Romulus, its first king. In 509 B.C.E. Rome became a republic ruled by the Senate (wealthy landowners and elders) and the Roman people. During the 450 years of the republic Rome conquered the rest of Italy and then expanded into France, Spain, Turkey, North Africa and Greece. Rome became very Greek influenced or “Hellenized,” filled with Greek architecture, literature, statues, wall-paintings, mosaics, pottery and glass. But with Greek culture came Greek gold, and generals and senators fought over this new wealth. The Republic collapsed in civil war and the Roman empire began.
Resizing with variables
Learn how to use variable expressions to resize parts of your drawing relative to other parts. (This requires a bit more math, so if you don't feel you have a good grasp of fractions yet, you can skip over this.)
Resonance structures
Are bond-line drawings not resonating with you? Sometimes one drawing is not enough to accurately describe the distribution of electron density. In this tutorial, we will learn how two or more resonance structures are merged to give us a single representation (a resonance hybrid).
Download the teacher resources from the Big History Project classroom version of this course. Text transcripts for videos are also available.
Download the teacher resources from the Big History Project classroom version of this course. Text transcripts for videos are also available.
Respiratory system
Did you know that your right lung is larger than your left? That’s because the majority of your heart is on the left side of your body, and your left lung is slightly smaller to accommodate it. The lungs take in oxygen and help you breathe out carbon dioxide. Humans have an intricate respiratory system, with hundreds of millions of tiny air sacs called alveoli, where all of the magic happens. These videos will introduce you to the lungs, and show how they help you survive.
Respiratory system introduction
Did you know that your right lung is larger than your left? That’s because the majority of your heart is on the left side of your body, and your left lung is slightly smaller to accommodate it. The lungs take in oxygen and help you breathe out carbon dioxide. Humans have an intricate respiratory system, with hundreds of millions of tiny air sacs called alveoli, where all of the magic happens. These videos will introduce you to the lungs, and show how they help you survive.
Retirement accounts: IRAs and 401ks
The government apparently wants us to save for retirement (not always obvious because it also wants us to spend as much as possible to pump the economy going into the next election cycle). To encourage this, it has created some ways to save that avoid or defer taxes: IRAs and 401ks.
Reverse chain rule
The Chain Rule tells us that derivative of g(f(x)) = g'(f(x))f'(x). You already knew this. But what about going the other way around? What happens if you want to integrate g'(f(x))f'(x)? Well, that's what the "reverse chain rule" is for. As you can see, a lot of integrals you'll run into can be solved this way. It is also another way of doing u-substitution without having to substitute (so it is faster)!
Rewriting fractions as decimals
Learn how to convert fractions to decimals without performing long division. For example, learn how to write 21/60 as a decimal by reasoning about equivalent fractions.
Richard Branson - Chairman of the Virgin Group
Richard Branson, Chairman of the Virgin Group, shares his story as a successful entrepreneur with a diverse portfolio.
Riemann sums and definite integration
In this tutorial, we'll think about how we can find the area under a curve. We'll first approximate this with rectangles (and trapezoids)--generally called Riemann sums. We'll then think about find the exact area by having the number of rectangles approach infinity (they'll have infinitesimal widths) which we'll use the definite integral to denote.
Ring-opening reactions of epoxides
In this tutorial, Sal and Jay show the SN1 and SN2 ring opening reactions of epoxides.
Rise of Hitler and the Nazis
How did the National Socialists (Nazis) go from being a tiny, marginal party in the early 1920s to having full control of Germany and catalyzing World War II? Who was Hitler and what was his philosophy and how did he come to power?
Rise of Mussolini and Fascism
The word "Fascist" is now a pejorative term ("pejorative" means "negative" or "derogatory") to describe leaders or states that have absolute control and are aggressively nationalistic. The terms "fascism" and "fascist", however, were first embraced by Benito Mussolini in Italy in the 1920s and 1930s to describe their party and policies (that were absolutist and aggressively nationalistic). This tutorial described Mussolini and the Fascists' rapid rise to power and the influence it had on the rest of the world (including providing a model for Hitler in Germany).
Road trip! (conquest, trade, & more conquest)
Visogoths, Ostrogoths, and Vikings, oh my! Western Europe was not a peaceful place during the 600 years after the fall of the Roman Empire. Western Europe (what is now Italy, France, Spain, England, etc.) had been repeatedly invaded. The result was a fractured feudal society with little stability and less economic growth. It was only in the 11th century that everything began to change. Peace and prosperity allow for travel and for the widespread construction of large buildings. The faithful set out on pilgrimages in great numbers to visit holy relics in churches across Europe. This meant that ideas and styles also traveled, towns grew and churches were built and enlarged. These were, with rare exceptions, the first large structures to be built in the west since the fall of the Romans so many centuries before. We call the period Romanesque (Roman-like) because the masons of this period looked back to the architecture of ancient Rome.
Romanticism in France
Romanticism begins in France with the violent and exotic battle scenes of Gros and the famous shipwreck, the Raft of the Medusa, painted by Gericault. Soon after, two distinct trends emerge in French painting, one—represented by the artist Delacroix—was rebellious, and emphasized emotion, color and loose brushwork. The other—which can be seen in the art of Ingres—upheld tradition, and emphasized line and a highly finished surface. Of course, things were more complicated—but those were battle lines!
Romanticism in Germany
This tutorial focuses exclusively on the art of Caspar David Friedrich, whose work best exemplifies Romanticism’s interest in the big questions of man’s mortality and place in the universe. The world had changed dramatically since the time of Michelangelo, Bernini and Rembrandt, and as a result, Friedrich approached these big questions without the Christian narratives that dominated the art of the past. And like his English counterparts during this period, he imbues nature and the landscape with symbolic and often spiritual meaning.
Romanticism in Spain
The great artist Francisco Goya is the focus of this tutorial. Goya began his career designing tapestries for the royal residences, and eventually became court painter to the King of Spain. But after Napoleon’s army occupied Spain and deposed the King, Goya documented the horrors he witnessed. His work following the occupation, including the Third of May 1808, remains some of the most powerful anti-war images ever created. His later years were spent largely in a house outside Madrid which he painted with haunting scenes. Saturn Devouring his sons belongs to this late series, known as the “Black Paintings.”
Romanticism in the United States
Romanticism—An Introduction
Start here.
Learn how to find the rotation that maps a given figure to another, and how to manually draw the image of a rotation.
Rounding is useful when you are trying to roughly estimate numbers.
Rounding decimals
Learn how to round decimal numbers to specific place values. For example, round 3.14159 to the nearest hundredth as 3.14.
Rounding whole numbers
If you're looking to create an army of robot dogs, will it really make a difference if you have 10,300 dogs, 9,997 dogs or 10,005 dogs? Probably not. All you really care about is how many dogs you have to, say, the nearest thousand (10,000 dogs). In this tutorial, you'll learn about conventions for rounding whole numbers. Very useful when you might not need to (or cannot) be completely precise. Common Core Standard: 4.NBT.A.3
SAT Math: Level 1
In this tutorial, you'll find the SAT Math Practice: Level 1 exercise, featuring lots of previously-unreleased SAT math questions provided by College Board. Stuck on a problem? Check out videos where Sal solves each problem step by step. Let's do this!
SAT Math: Level 2
In this tutorial, you'll find the SAT Math Practice: Level 2 exercise, featuring lots of previously-unreleased SAT math questions provided by College Board. Stuck on a problem? Check out videos where Sal solves each problem step by step. Here we go!
SAT Math: Level 3
In this tutorial, you'll find the SAT Math Practice: Level 3 exercise, featuring lots of previously-unreleased SAT math questions provided by College Board. Stuck on a problem? Check out videos where Sal solves each problem step by step. Congrats on venturing into level 3 territory. You've got this!
SAT Math: Level 4
In this tutorial, you'll find the SAT Math Practice: Level 4 exercise, featuring lots of previously-unreleased SAT math questions provided by College Board. Stuck on a problem? Check out videos where Sal solves each problem step by step. You're tackling some of the harder SAT Math problems now. Rock it out!
SAT Math: Level 5
In this tutorial, you'll find the SAT Math Practice: Level 5 exercise, featuring lots of previously-unreleased SAT math questions provided by College Board. Stuck on a problem? Check out videos where Sal solves each problem step by step. These are the hardest math problems on the SAT. Eye of the tiger!
SAT Reading: Sentence completion
SAT Writing: Identifying sentence errors
SN1 and SN2
In this tutorial, Jay covers the definitions of nucleophile/electrophile, The Schwartz Rules (may the Schwartz be with you!), and the differences between SN1 and SN2 reactions.
SN1 vs SN2
In this tutorial, Sal analyzes the differences between SN1 and SN2 reactions.
In this tutorial, Sal compares the differences between SN1, SN2, E1, and E2 reactions.
SQL basics
We'll show you the basics of creating tables and selecting data in various different ways.
Sal chats with entrepreneurs
Sal's old Maclaurin and Taylor series tutorial
Everything in this tutorial is covered (with better resolution and handwriting) in the "other" Maclaurin and Taylor series tutorial, but this one has a bit of old-school charm so we are keeping it here for historical reasons.
Sal's old angle videos
These are some of the classic, original angle video that Sal had done way back when (like 2007). Other tutorials are more polished than this one, but this one has charm. Also not bad if you're looking for more examples of angles between intersected lines, transversals and parallel lines.
Sal's old statistics videos
This tutorial covers central tendency and dispersion. It is redundant with the other tutorials on this topic, but it has the benefit of messy handwriting and a cheap microphone. This is Sal circa 2007 so take it all with a grain of salt (or just skip it altogether).
Sampling distribution
In this tutorial, we experience one of the most exciting ideas in statistics--the central limit theorem. Without it, it would be a lot harder to make any inferences about population parameters given sample statistics. It tells us that, regardless of what the population distribution looks like, the distribution of the sample means (you'll learn what that is) can be normal. Good idea to understand a bit about normal distributions before diving into this tutorial.
Scale drawings
Now that you understand lines, you can put lines together to create drawings. A scale drawing depicts an actual object that has been reduced (or enlarged) by a certain amount. The "scale" shows the relationship between those two sizes. After this set of tutorials, you can draw your bedroom "to scale" and practice rearranging your furniture!
Scale of earth, sun, galaxy and universe
The Earth is huge, but it is tiny compared to the Sun (which is super huge). But the Sun is tiny compared to the solar system which is tiny compared to the distance to the next star. Oh, did we mention that there are over 100 billion stars in our galaxy (which is about 100,000 light years in diameter) which is one of hundreds of billions of galaxies in just the observable universe (which might be infinite for all we know). Don't feel small. We find it liberating. Your everyday human stresses are nothing compared to this enormity that we are a part of. Enjoy the fact that we get to be part of this vastness!
Scatter plots
Scene management
Learn how to change between multiple scenes in your program, even if they're animated or interactive.
School redesign overview
Schools using Khan Academy
Scientific notation
Scientists and engineers often have to deal with super huge (like 6,000,000,000,000,000,000,000) and super small numbers (like 0.0000000000532) . How can they do this without tiring their hands out? How can they look at a number and understand how large or small it is without counting the digits? The answer is to use scientific notation. If you come to this tutorial with a basic understanding of positive and negative exponents, it should leave you with a new appreciation for representing really huge and really small numbers!
Scientific notation intro
Scientists and engineers often deal with super huge numbers like 6,000,000,000,000,000,000,000 and super small numbers like 0.0000000000532. How can they do this without tiring their hands out? How can they look at a number and understand how large or small it is without counting the digits? The answer to both questions: scientific notation.
Scientific notation word problems
Solve word problems involving scientific notation.
A polychromed saint, a juggling man cast in bronze, and magnificent marble throughout the ages—explore the making and meaning of different types of sculpted treasures.
Sculpture and architecture in Florence
Sculpture and architecture are central to the cultural development of Renaissance Florence. Like chapels, palaces, and cathedrals, sculpture was used to express the wealth, power and piety of the city's leading patrons and guilds. Learn about the brilliant innovations of Brunelleschi, his friend Donatello and other leading artists that helped define the Renaissance.
Section 2
In section 2, you'll zero in on math practice with a bunch of math problems from a real SAT. Want even more math practice? Check out sections 6 and 8 and the SAT Math practice topic.
Section 3
In Section 3, let's switch gears and focus on Reading questions, starting on pg. 48 of the downloadable SAT. Want even more reading practice? Check out the SAT Reading and Writing practice topic.
Section 5
Have you ever seen a sentence in a newspaper or on a restaurant menu that just wasn't quite right? In this section, you'll have lots of opportunities to find errors and fix 'em. Section 5 starts on pg. 54 of the downloadable test. Want even more writing practice? Check out the SAT Reading and Writing practice topic.
Section 6
There are some really fun math problems in this section. Seriously. It all begins on pg. 60 in the downloadable SAT. Want even more math practice? Check out sections 2 and 8 and the SAT Math practice topic.
Section 8
More math practice coming your way in Section 8, starting on pg. 69 in the downloadable SAT. Want even more math practice? Check out sections 2 and 6 and the SAT Math practice topic.
Seismic waves and how we know Earth's structure
How do we know what the Earth is made up of? Has someone dug to the core? No, but we humans have been able to see how earthquake (seismic) waves have been bent and reflected through our planet to get a reasonable idea of what is down there.
Selection sort
Learn selection sort, a simple algorithm for sorting an array of values, and see why it isn't the most efficient algorithm.
Self-presentation and interacting with others
Sensory perception
Learn about how we perceive our various senses, including the theories, laws, and organizational principles that underly our ability to make sense of the world around us.
Separable equations
Arguably the 'easiest' class of differential equations. Here we use our powers of algebra to "separate" the y's from the x's on two different sides of the equation and then we just integrate!
Separations and purifications
5C: Did you know that digitalis, one of the oldest medicines used to increase cardiac contractility, is derived from the foxglove plant? When you are in the Amazon rainforest searching for a cure for cancer in a new exotic plant, your potential miracle drug of interest is not originally pure - it must be separated from the other contaminating components. Through these tutorials you will learn how to separate and purify chemical compounds using organic chemistry lab techniques such as extraction, distillation, chromatography, and gel electrophoresis.
Sequence convergence and divergence
Now that we understand what a sequence is, we're going to think about what happens to the terms of a sequence at infinity (do they approach 0, a finite value, or +- infinity?).
In this tutorial, we'll review what sequences are, associated notation and convergence/divergence of sequences.
Sequences and series
This sequence (pun intended) of videos and exercises will help us explore ordered lists of objects--even infinite ones--that often have some pattern to them. We will then explore constructing sequences where the nth term is the sum of the first n terms of another sequence (series). This is surprisingly useful in a whole series (pun intended) of applications from finance to drug dosage.
Serbian and Italian fronts in World War I
Contrary to what some history books and movies would have you believe, World War I was not just fought on the Western and/or Eastern fronts. Because of the empires involved, it was a truly global conflict. This tutorial will cover some of the campaigns that your history book might not (but are important to understanding the War).
You're familiar with sequences and have been eager to sum them up. Well wait no longer! In this tutorial, we'll see that series are just sums of sequences and familiarize ourselves with the notation.
Sexually transmitted diseases
Shang dynasty (c. 1600-1050 B.C.E.)
Shape of data distributions
Like people, no two data distributions look exactly the same. Well, maybe that's not always true... Anyway, the point is that each data distribution has it's own shape. In this tutorial, you'll learn new vocabulary that will have you discussing the shape of data distributions like a pro in no time!
Classify quadrilaterals, including rectangles, rhombuses, and squares. See what attributes they have in common and which ones make them unique.
Shapes on the coordinate plane
Let's think about shapes as collections of points on the coordinate plane. When you're done with this tutorial, you might be saying, "I have had it with these shapes on this plane!" But you'll be happy you went through it.
Shifting and reflecting functions
Shifting functions
Learn how to transform functions so their graph is shifted horizontally or vertically. Learn how to find the equation of a function that is a shift of another function.
Shock is a rather common clinical situation, especially in the emergency room. Quite simply, circulatory shock refers to poor perfusion of organs with blood. For example, shock may result from loss of blood (hemorrhage), a poorly functioning heart (heart failure), or dilated blood vessels (sepsis and anaphylactic shock). We will explore how to differentiate the many different causes of shock here.
Shorting stock
Can you sell something that you borrowed from someone else? Why, yes, you can and it is called "shorting". Why would you do this? Well, you can now make money if the price goes down. Is this bad? This tutorial has your answers.
Siena, the Late Gothic
When we think of the Renaissance, we tend to think of Florence (and Rome). But the city of Siena also deserves our attention. Today, the lovely walled city of Siena is one of the best preserved Medieval cities in Europe and it was chosen by the United Nations as a World Heritage Site. In the 14th century, Siena was a wealthy independent nation and often at war with its neighbor, Florence. Some of the most important art of the 14th century was commissioned for Siena’s Cathedral and town hall. Duccio and his students, the Lorenzetti brothers and Simone Martini produced large-scale painting with an intricacy and subtle coloration that is unique in the Renaissance.
Sierra Leone
This tutorial explores the Sowei masks of the female Sande society and the art of the Qisi and Vai peoples.
Sight (vision)
Photons are hitting your eye as you read this! Learn about our sense of sight, including the cells responsible for converting light into a neural impulse, the structure of the eye, and how we break down images to make sense of them.
Significant figures
There is a strong temptation in life to appear precise, even when you are aren't accurate. If you precisely measure one dimension of a carpet to be 3.256 meters and eyeball the other dimensional to be "roughly 2 meters", can you really claim that the area is 6.512 square meters (3.256 x 2)? Isn't that a little misleading? This tutorial gets us thinking about this conundrum and gives us the best practices that scientists and engineers use to not mislead each other.
Similarity and transformations
Two figures are similar if you can get from one to another through some combinations of translations, reflections, rotations AND DILATIONS (so you can scale up and down). This tutorial helps give us an intuition for this.
Simple Machines Explorations
Simplifying complicated equations
You feel good about your rapidly developing equation-solving ability. Now you're ready to fully flex your brain. In this tutorial, we'll explore equations that don't look so simple at first, but that, with a bit of skill, we can turn into equations that don't cause any stress! Have fun!
Simplifying numerical radical expressions
In this tutorial you will simplify numerical expressions with radicals by removing all factors that are perfect squares from inside the radical.
Simplifying rational expressions
Learn how to simplify rational expressions by canceling factors that are shared by the numerator and the denominator. Sometimes this calls for factoring the numerator and the denominator in various ways.
Sine, cosine and tangent trigonometric functions
In this tutorial, you will learn all the trigonometry that you are likely to remember in ten years (assuming you are a lazy non-curious, non-lifelong learner). But even in that non-ideal world where you forgot everything else, you'll be able to do more than you might expect with the concentrated knowledge you are about to get.
Singing (and noises)
The title says it all...
Skeletal system
3B: Were it not for your skeleton, you and I would be a mere sack of flesh. You will come to appreciate that the bones, together with muscles, are a scaffolding for your body. We will also explore their endocrine function, especially with regards to calcium and phosphate homeostasis. Fun fact: the bone most broken is the clavicle (AKA collar bone).
Skeletal system introduction
Skill check for atoms, molecules, an
Want to check your knowledge of atoms and molecules for the new school year? Try a skill check for some quick review and links to related videos!
Skill check for chemical reactions and stoichiometry
Want to check your knowledge of chemical reactions and stoichiometry for the new school year? Try a skill check for some quick review and links to related videos!
Skill check for fluids
This is a skill check that can help you determine what areas to review in fluids.
Skill check for one-dimensional motion
Test your knowledge about one-dimensional motion for the new school year. Try this skill check for a quick review and links to review videos.
Skill check: chemistry of life
Check your understanding of the basic chemistry that makes life possible!
Learn to count by fives, tens, and hundreds.
Slavery and the Civil War
The end of slavery. The beginning of a new era. The biggest conflict ever to happen on American soil. What caused the Civil War and how did it change the United States?
Sleep and consciousness
6B: One third of our lives is consumed by the mysterious process called “sleep.” There is a lot we don’t know about sleep, but we will discuss what we do understand in this module. You will find why napping for a twenty minutes in the afternoon may not be such a bad idea as you explore states of consciousness, circadian rhythms, sleep stages, and sleep disorders.
Slicing geometric shapes
When we slice (cut through) a 3D shape, we make a 2D shape called a cross section. In this tutorial, we think about the different cross sections we can make by slicing 3D shapes.
Slope-intercept form
Slow sock on Lubricon VI
This short tutorial will have you dealing with orbiting frozen socks in order to understand whether you understand Newton's Laws. We also quiz you a bit during the videos just to make sure that you aren't daydreaming about what you would do with a frozen sock.
Sneek peaks
So that’s where that comes from!
Soap Film Interference Model
Light reflecting off the surface of soapy water produces colors—why? What affects which colors you see? Try this activity and learn how to model what happens when light of different wavelengths encounters soap film of different thicknesses. This snack is a companion to the Soap Film on a Can tutorial.
Soap Film on a Can
Why do we see colors in oily water and soap bubbles? Sometimes we see red, sometimes blue, and sometimes it appears as though we see nothing at all. Experiment with soap film to observe the behavior and colorful appearance of different wavelengths of light.
Social behavior
8C: Who are we attracted to and why? Why do some people behave aggressively and others altruistically? These are questions we can begin to answer through an understanding of social behavior. This tutorial will help you answer these questions as well as give you insight into the basics of attachment and the relationships we have with our parents and those around us as we explore some of the aspects that govern our interactions with other human beings.
Social inequality
10A: The key here is to focus on a broad understanding of social class, including theories on how it's organized, how people move between classes, and what leads to poverty.
Social interactions
Here we will learn why people look at you funny when you do the worm in the middle of the hallway at school. Just kidding -- but you will learn about the biological influences and social theories which explain why we interact with others and the world around us the way we do.
Social sciences practice questions
These questions are available under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License (available at
Social structures
9A: Any group of people involved in interpersonal relationship is a society. How do we humans all live together in such highly populated cities? How does a society change? You will learn about some famous sociological theories which aim to address these important questions, from conflict theory and feminist theory to social constructionism.
Solderless Spout Bot
This project is based on a MAKE Beetle bot. The tutorial was created by Karl R. C. Wendt.
Solid of revolution
Using definite integration, we know how to find the area under a curve. But what about the volume of the 3-D shape generated by rotating a section of the curve about one of the axes (or any horizontal or vertical line for that matter). This in an older tutorial that is now covered in other tutorials. This tutorial will give you a powerful tool and stretch your powers of 3-D visualization!
Solids of revolution - disc method
You know how to use definite integrals to find areas under curves. We now take that idea for "spin" by thinking about the volumes of things created when you rotate functions around various lines. This tutorial focuses on the "disc method" and the "washer method" for these types of problems.
Solids of revolution - shell method
You want to rotate a function around a vertical line, but do all your integrating in terms of x and f(x), then the shell method is your new friend. It is similarly fantastic when you want to rotate around a horizontal line but integrate in terms of y.
Solubility equilibria
When fresh water runs over a rock a little bit of the rock dissolves. When this mineral rich water dries a little, the rock can be redeposited. Reactants and products are always in equilibrium with each other. Learn how to calculate equilibrium constants and by watching these videos!
Solutions and graphs of two-variable equations
In this tutorial, we'll work through examples that show how a line can be viewed as all of coordinates whose x and y values satisfy a linear equation. Likewise, a linear equation can be viewed as describing a relationship between the x and y values on a line.
Solutions to linear equations
Not every linear equation has exactly one solution. Some have no solutions and others might have an infinite number of solutions. This tutorial will give you the intuition on when these different situations arise.
Solving and graphing quadratics
Tired of lines? Not sure if a parabola is a disease of the gut or a new mode of transportation? Ever wondered what would happen to the graph of a function if you stuck an x² someplace? Well, look no further. In this tutorial, we will study the graphs of quadratic functions (parabolas), including their foci and whatever the plural of directrix is.
Solving any system of linear equations
Use everything you learned so far in order to solve any kind of system of equations!
Solving basic equations
Now we'll introduce you to the most fundamental ideas of what equations mean and how to solve them. We'll then do a bunch of examples to make sure you're comfortable with things like 3x – 7 = 8. So relax, grab a cup of hot chocolate, and be on your way to becoming an algebra rockstar. And, by the way, in any of the "example" videos, try to solve the problem on your own before seeing how Sal does it. It makes the learning better!
Solving basic exponential models
Learn how to construct exponential functions and then analyze them to model and solve real-world problems.
Solving equations examples and practice
You've been through "Equation examples for beginners" and are feeling good. Well, this tutorial continues that journey by addressing equations that are just a bit more fancy. By the end of this tutorial, you really will have some of the core algebraic tools in your toolkit!
Solving fancier linear equations
You've been through "Equations for beginners" and are feeling good. Well, this tutorial continues that journey by addressing equations that are just a bit more fancy. By the end of this tutorial, you really will have some of the core algebraic tools in your toolkit!
Solving for a variable
You feel comfortable solving for an unknown. But life is all about stepping outside of your comfort zone--it's the only way you can grow! This tutorial takes solving equations to another level by making things a little more abstract. You will now solve for a variable, but it will be in terms of other variables. Don't worry, we think you'll find it quite therapeutic once you get the hang of it.
Solving linear systems graphically
We already know that we can represent the set of all x-y pairs that satisfy a linear equation as a line. If there is a point where two of these lines intersect, then the x-y pair corresponding to that point must satisfy both equations. Common Core Standards: 8.EE.C.8, 8.EE.C.8a
Solving problems with similar and congruent triangles
We spend a lot of time in geometry proving that triangles are congruent or similar. We now apply this ability to some really interesting problems (seriously, these are fun)!
Solving quadratics by factoring
Just saying the word "quadratic" will make you feel smart and powerful. Try it. Imagine how smart and powerful you would actually be if you know what a quadratic is. Even better, imagine being able to completely dominate these "quadratics" with new found powers of factorization. Well, dream no longer. This tutorial will be super fun. Just bring to it your equation solving skills, your ability to multiply binomials and a non-linear way of thinking!
Solving quadratics by taking square root
Let's explore one of the most fundamental ways to solve a quadratic equation when it is already written in terms of a square of an expression in x--solve for the expression and take the square root. As we'll see, equations are not always in this form, and that is where completing the square--which puts equations in this form--is essential!
Solving rational equations
Learn how to solve equations that have a rational expression, or a few of those.
Solving systems by elimination
This tutorial is a bit of an excursion back to you Algebra II days when you first solved systems of equations (and possibly used matrices to do so). In this tutorial, we did a bit deeper than you may have then, with emphasis on valid row operations and getting a matrix into reduced row echelon form.
Solving systems of equations
Here we'll see that you may be able to actually solve for two unknowns if you have two different constraints or equations. You can view this solution as being represented by the intersection of the graphs of the two equations.
Solving systems of equations with substitution
Learn about the substitution method, which is another way to manipulate systems of equations in order to solve them.
Solving systems with elimination
You can solve a system of equations with either substitution or elimination. This tutorial focuses with a ton of examples on elimination. You'll learn best if you pause the videos and try to do the problem before Sal does. Once you get a hang for things, feel free to skip forward to the exercises. Common Core Standards: 8.EE.C.8, 8.EE.C.8b
Solving systems with substitution
This tutorial is focused on solving systems through substitution. It has more examples than you can shake a dog at. As always, pause the video and try to solve before Sal does. Once you get a hang for things, feel free to skip the rest of the videos and try the exercises. The best way to learn, after all, is to do rather than just listen! Common Core Standards: 8.EE.C.8, 8.EE.C.8b
6A: When you hold a glass of water, your brain perceives senses many bits of information about the glass - its temperature, its size, and its location in space. We perceive the environment through our bodily senses, including our sensation of pain, temperature, pressure, balance, and movement. In this module you will discover how our body gathers this information and processes it so that we can make sense of the colorfu lworld around us.
Song dynasty (960-1279)
The Song dynasty which is divided between the Northern Song and the later Southern Song when land north of the Yangtze River was lost to the Jin Dynasty. The Song Dynasty witnessed important historical developments such as the first use of gunpowder, paper currency, and the identification of true north. This was also a period that saw creative production at the highest level including monumental landscape painting. The Song dynasty existed from 960 until 1279.
Sound (Audition)
6A: Listen carefully - that’s the sound of fingers on a keyboard. Why does it sound different from the sound of nails on a chalkboard? And why might your grandfather have difficulty hearing that faint noise whereas you have difficulty zoning it out? In this module, you will learn about how we hear, including the structure of the outer, middle, and inner ear, as well as the basics of auditory processing and cochlear implants.
Sound Introduction
Activities to unlock the secrets of sound, hearing, and perception.
Sound intensity
South America
South America has witnessed the emergence of some of the most intriguing and diverse ancient cultures in the world. Explore the Nazca, Inca peoples and the early cultures of present-day Columbia through objects in the British Museum collection.
South Asia
What are often thought of as “Indian” art and culture spread not only throughout the modern nation of India but also through Pakistan and Bangladesh. This huge area was never politically unified except under British colonial rule (1858–1947). Earlier, various kingdoms and principalities controlled large or small areas, and occasionally a conqueror created a vast empire.
South Asia, an introduction
Special products of polynomials
Learn about the special types of products of polynomials: perfect squares and the difference of two squares. These will be very helpful once you tackle more advanced expressions in Algebra.
Special right triangles
We hate to pick favorites, but there really are certain right triangles that are more special than others. In this tutorial, we pick them out, show why they're special, and prove it! These include 30-60-90 and 45-45-90 triangles (the numbers refer to the measure of the angles in the triangle).
Special thanks to contributors
Specific heat and heat transfer
In the lab, it is useful to know how much of something you have or the concentration of a solute. In this tutorial, we'll light to do that!
There is much more to light than meets the eye. Introduction to the electromagnetic spectrum and the science of spectroscopy.
Speed and velocity
4A: With every beat, your heart pumps blood throughout the vessels of your body. Let’s look at the radial artery, one of the vessels in your arm. How fast is the blood going at a particular point in time, and in what direction? The speed is an absolute number without direction - 1.8 kilometers per hour in this case. Velocity expands on the speed, which is an absolute value, by adding direction - for instance, 1.8 kilometers per hour downward. It’s that easy! you already grasping the concept of speed and velocity, terms used to describe the rate of change in our distance over time.
Spherical mirrors
4D: Have you ever been to a house of mirrors at a carnival, or maybe seen one in the movies? Curved mirrors are fun, and so is a description of their physics. It’s actually possible to roughly predict what an image produced by a mirror will look like using just pen and paper. You will discover the difference between real and virtual images as we draw ray tracing diagrams to show how images are formed by spherical mirrors.
Spider Bot by Karl R. C. Wendt
Spider Bot is a low cost robot made of recycled components and designed by Karl R. C. Wendt
Spirals, Fibonacci and being a plant
You're feeling spirally today, and math class today is taking place in greenhouse #3...
Spout Bot with Solder
This project is based on a MAKE Beetle bot. The tutorial was created by Karl R.C. Wendt.
Springs and Hooke's Law
Weighing machines of all sorts employ springs that take a certain amount of force to keep compressed or stretched to a certain point. Hooke's law will give us all the tools to weigh in on the subject ourselves and spring into action (yes, the puns are annoying us too)!
Square roots and cube roots
A strong contender for coolest symbol in mathematics is the radical. What is it? How does it relate to exponents? How is the square root different than the cube root? Learn all about square roots and cube roots in this tutorial.
Squeeze theorem
If a function is always smaller than one function and always greater than another (i.e. it is always between them), then if the upper and lower function converge to a limit at a point, then so does the one in between. Not only is this useful for proving certain tricky limits (we use it to prove lim (x → 0) of (sin x)/x, but it is a useful metaphor to use in life (seriously). :) This tutorial is useful but optional. It is covered in most calculus courses, but it is not necessary to progress on to the "Introduction to derivatives" tutorial.
Standard form
Standardized tests
Nothing strikes fear into the hearts of students aspiring to college quite like the SAT/ACT, standardized tests that many schools require as part of their application package. Overcome that fear now by getting a primer on what these tests cover, how they are scored, and the impact they have on college admissions.
Standing waves
The human eye can see about 6,000 stars in the night sky. Photographs reveal millions more in every direction. All of these stars reside in our Milky Way galaxy. But they are just a tiny fraction of the several hundred billion stars in our galaxy alone. Counting all the stars in all the galaxies, there are perhaps a hundred billion billion stars in the observable universe. The videos in this tutorial show the tumultuous surface of our Sun, star systems in all their diversity, and the lifecycles of distant stars--from their birth in star clusters to the deaths of high-mass stars in spectacular explosions called supernovas.
Starter multiplication and division word problems
Math is super useful in the real world. In this tutorial, we'll get some practice figuring out things that would have been tough without the powerful tools of multiplication and division!
States of matter
Statistical questions
This is the foundational tutorial for the rest of statistics. Mean, median and mode are measures of "central tendency." That is, they help us find the center (or middle) of the data.
Stellar parallax
We've talked a lot about distances to stars, but how do we know? Stellar parallax--which looks at how much a star shifts in the sky when Earth is at various points in its orbit--is the oldest technique we have for measuring how far stars are. It is great for "nearby" stars even with precise instruments (i.e, in our part of our galaxy). To measure distance further, we have to start thinking about Cepheid variables (other tutorial).
Stem cells and cancer
5B: Even molecules with the same chemical formula can have different shapes even though they may be comprised of the same atoms. For instance, with one sheet of paper, you can make origami swans of so many different shapes - similarly molecules can come in different conformations. We will walk through the concepts of structural and conformational isomers as well as stereoisomers and diastereomers
Stereoisomeric relationships
Mirror, mirror on the wall . . . who is the fairest stereoisomer of all? In this tutorial, we will examine the relationships between different stereoisomers.
Now we are going to draw the connections between balancing equations and what happens in the lab (where you actually have a certain mass of a compound).
Stokes' theorem
Stokes' theorem relates the line integral around a surface to the curl on the surface. This tutorial explores the intuition behind Stokes' theorem, how it is an extension of Green's theorem to surfaces (as opposed to just regions) and gives some examples using it. We prove Stokes' theorem in another tutorial. Good to come to this tutorial having experienced the tutorial on "flux in 3D".
Strategies for adding two and three-digit numbers
Learn strategies for solving addition problems within 1000.
Strategies for adding within 100
Learn strategies for adding and subtracting two-digit numbers.
6C: Perhaps when you think about stress, you may think of that nerve-wracking sensation you may feel in your stomach before the final exam for organic chemistry you studied all month for or your shivering response to a chilly winter night. But stress can also be the microtraumas inflicted on yourself during exercise. Stress can be a good thing that forces our antifragile bodies to adapt to a changing external environment. Stress is the natural process by which we evaluate and respond to the challenges and threats of our environment. Jump into this playlist to learn about stressors, stress reactions, the effects of stress, and strategies to manage this wild process.
Stretching functions
Learn how to transform functions so their graph is stretched horizontally or vertically. Learn how to find the equation of a function that is a stretch of another function.
The largest group within the orchestra is comprised of instruments that produce sound from vibrating strings. The vibrations – resulting from the strings being played by a bow, or being plucked with the fingers – resonate within the body of each instrument. The violin, viola, cello and contrabass are all primary played with a bow. The harp is plucked and stroked with the hands. Principals from the All-Star Orchestra introduce the special qualities and histories of each instrument.
Structure in linear expressions
Algebra isn't just some voodoo concocted to keep you from running outside. It is a way of representing logic and manipulating ideas. This tutorial will have blood flowing to your brain in record quantities as you actually have to think about what the algebraic expressions and equations actually mean!
Student experience overview
Subspaces and the basis for a subspace
In this tutorial, we'll define what a "subspace" is --essentially a subset of vectors that has some special properties. We'll then think of a set of vectors that can most efficiently be use to construct a subspace which we will call a "basis".
Substitution and evaluating expressions
Learn how to substitute (or "plug in") values for variables and evaluate algebraic expressions.
Subtracting and ones, tens, and hundreds
Learn how to solve addition problems by adding ones, tens, and hundreds.
Subtracting decimals
Anything you can do with whole numbers, you can do with decimals. Subtraction is no exception. In this tutorial, you'll get some good practice subtracting decimals up the hundredths place.
Subtracting multi-digit numbers
In the 3rd grade, you learned to subtract multi-digit numbers. This includes subtraction through regrouping (sometimes known as "borrowing"). This tutorial will give you practice doing this with even larger numbers. Have fun! Common Core Standard: 4.NBT.B.4
Subtracting ones and tens
Learn how to solve addition problems like 24-3 and 45-30.
Subtraction with borrowing (regrouping)
You can subtract 21 from 45, but are a bit perplexed trying to subtract 26 from 45 (how do you subtract the 6 in 26 from the 5 in 45). This tutorial is your answer. You'll see that we can essentially "regroup" the value in a number from one place to another to solve your problem. This is also often called borrowing (although it is like "borrowing" sugar from your neighbor in that you never give it back).
Subtraction within 100
Learn to subtract two-digit numbers like 47 and 39.
Subtraction within 20
Learn to subtract. Numbers used are 20 or less.
Sui dynasty
We sometimes use the word "Ur" to speak of the origin of something (for example, "Adam spoke the ur-language"). In fact, Ur was an actual Sumerian city and we can go back there to learn about the origin of writing, cities, and even civilization. Ur really was the ur-Ur.
Summit Public Schools
This charter school network, also featured in Waiting for Superman, aimed to transform the learning experience for their 200 incoming 9th graders in their San Jose, California, schools.
Sums of consecutive integers problems
The sum of 5 consecutive integers is 15. What are the integers? In this tutorial, you'll learn to solve problems by thinking about the sums of consecutive integers.
Super Yoga Plans
Let's use our algebra tools to solve a problem of earth-shattering importance: which Super Yoga plan is the best value! In this word problem, you'll put to practice your knowledge of variables, substitution, and one-step equations.
Super Yoga plans
This tutorial is a survey of the major themes in basic algebra in five videos! From basic equations to graphing to systems, it has it all. Great for someone looking for a gentle, but broad understanding of the use of algebra. Also great for anyone unsure of which gym plan they should pick!
Surface area
Learn how to find the total area of the surfaces of 3D shapes. To do this, we wrap 3D shapes in 2D nets.
Surface integrals
Finding line integrals to be a bit boring? Well, this tutorial will add new dimension to your life by explore what surface integrals are and how we can calculate them.
Do we too readily accept the concrete rational world before us as all that is real? Could there be more? Could the dream be a doorway to a more primal creative experience no less real than our waking world? Influenced by ideas of psychoanalysis such as the unconscious artists built on the irrational art of Dada to explore the dark world of desire freed from rules created to protect us from inner ourselves.
Learn about different kinds of symmetries of two-dimensional shapes, and analyze various shapes according to their symmetries.
Symmetry and periodicity of trig functions
In this tutorial, we will explore the unit circle in more depth so that we can better appreciate how trig functions of an angle might relate to angles that are in some way symmetric within the unit circle. We'll also look at the periodicity of the functions themselves (why they repeat after a certain change in angle).
Synthesis and cleavage of ethers
In this tutorial, Jay shows how to synthesize ethers using the Williamson ether synthesis and how to cleave an ether linkage using acid.
Synthesis of alcohols
In this tutorial, Jay shows how to synthesize alcohols using sodium borohydride, lithium aluminum hydride, and grignard reagents.
Synthesis using alkynes
In this tutorial, Jay demonstrates how to use Dr. Schwartz's organic flowsheet to solve synthesis problems involving alkynes. Always remember, pain is temporary, orgo is forever!
Synthetic division of polynomials
Learn how to divide polynomials using long division.
Systems of equations word problems
Let's now use systems of equations to solve real-world problems!
Systems with three variables
Two equations with two unknowns not challenging enough for you? How about three equations with three unknowns? Visualizing lines in 2-D too easy? Well, now you're going to visualize intersecting planes in 3-D, baby. (Okay, we admit that it is weird for a website to call you "baby.")
Systems word problems
This tutorial doesn't involve talking parrots and greedy trolls, but it takes many of the ideas you might have learned in that tutorial and applies them to word problems. These include rate problems, mixture problems, and others. If you can pause and solve the example videos before Sal does, we'd say that you have a pretty good grasp of systems. Enjoy! Common Core Standards: 8.EE.C.8, 8.EE.C.8b, 8.EE.C.8c
T.A. McCann - Founder and CEO of Gist
T.A. McCann, Founder and CEO of Gist, talks about his entrepreneurial journey, including how he joined the America’s Cup sailing team. T.A. discusses how entrepreneurs need to show initiative and chart their own course, advising other founders to always ask questions and make progress.
Tailored tips for using KA with students
These are a few different models that some teachers have used with their students, depending on their students' needs and available resources.
Tang dynasty (618-907)
The Tang dynasty was long and prosperous and is known as China's golden age. It was a period when China was powerful and engaged in long distance trade across networks that reached western Asia and Europe. This was also a period of improved education, a strong civil bureaucracy, and some of the most revered poetry, painting and ceramics ever made. The Tang dynasty existed from 618 until 907.
Tangents to polar curves
Taste (gustation) and smell (olfaction)
Learn about our senses of smell (olfaction) and taste (gustation), including the anatomy and underlying molecular basis behind these important senses.
Benjamin Franklin (and several other writers/philosophers) tells us that "In this world nothing can be said to be certain, except death and taxes." He's right. This tutorial focus on personal income tax. Very important to watch if you ever plan on earning money (some of which the government will take for itself).
Taylor series approximations
As we've already seen, Maclaurin series are special cases of Taylor series centered at 0. We'll now focus on more generalized Taylor series.
Teaching Computer Programming
Want to teach computer programming in the classroom? Here are our guides and case studies.
Learn how teen numbers work.
Telling time
Learn to tell time with some fun exercises!
Temperature, kinetic theory, and the ideal gas law
Learn to see two-digit numbers as tens and ones.
Bad commute? Baby crying? Bills to pay? Looking to take a bath with some Calgon (do a search on YouTube for context) to ease your tension? This tutorial has nothing (actually little, not nothing) to do with that. So far, most of the forces we've been dealing with are forces of "pushing"--contact forces at the macro level because of atoms not wanting to get to close at the micro level. Now we'll deal with "pulling" force or tension (at a micro level this is the force of attraction between bonded atoms).
Tests for convergence and divergence
We will now deepen our convergence and divergence tool kits by exploring a series of "tests" we can apply to determine the behavior of some series.
Learn how to display text on the canvas, resize it, color it, and animate it.
Thanksgiving math
Mathed potatoes, Borromean onion rings, green bean matheroles and Turduckenen-duckenen (yes, you read that right)
The Ancient Near East, an introduction
Sumerians, Akkadians, Babylonians, Assyrians, and Persians - no wonder we need an introduction!
The Arctic
The Articles of Confederation
The Cold War
The cold war between the United States and the Soviet and their respective allies never involved direct conflict (which might have ended the world). Instead, it involved posturing, brinksmanship and proxy wars in far-flung regions of the world.
The Constitution of the United States
The Declaration of Independence
The Doppler effect
The Himalayas and the Tibetan Buddhist World
The Himalayas are the highest mountain ranges in the world, and from them flow the major rivers of Asia. The kingdoms of Nepal and Bhutan are located along the Himalayan ranges, and the Tibetan plateau lies to their north. Although the Himalayas are nearly impassible, many peoples have managed the crossing and left traces of their cultures.
The Middle East
Take a trip to Beirut, where one of Lebanon’s earliest abstract artists, Saloua Raouda Choucair, is having her collection of works prepared for a major international exhibition. Learn how artist Dia Al-Azzawi managed to learn more about his home in Iraq after coming to London, and watch as young artists from the Middle East grapple with exile and political unrest.
The Northern Renaissance in the 16th century
The Pacific (content area 9)
The arts of the Pacific vary by virtue of ecological situations, social structure, and impact of external influences, such as commerce, colonialism, and missionary activity. Created in a variety of media, Pacific arts are distinguished by the virtuosity with which materials are used and presented.
The Peredvizhniki (The Wanderers)
The Protestant Reformation
In 1517 a German theologian and monk, Martin Luther, challenged the authority of the Pope and sparked the Protestant Reformation. His ideas spread quickly, thanks in part to the printing press. By challenging the power of the Church, and asserting the authority of individual conscience (it was increasingly possible for people to read the bible in the language that they spoke), the Reformation laid the foundation for the value that modern culture places on the individual.
The Pythagorean theorem
Named after the Greek philosopher who lived nearly 2600 years ago, the Pythagorean theorem is as good as math theorems get (Pythagoras also started a religious movement). It's simple. It's beautiful. It's powerful. Common Core Standards: 8.G.B.7, 8.G.B.8
The Roman silversmith's craft
Learn about the opulent silver objects of the Roman Empire and see how one spectacular silver drinking cup was fashioned.
The Sun and solar storms
Despite its apparently steady glow, the Sun is a churning mass of superhot plasma that regularly produces powerful flares and storms that can knock out power and communication systems on Earth. Find out what makes the Sun so volatile and how you can get access to the same data, images, and tools that scientists use to predict solar storms.
The anatomy of expressions
Learn new words, such as "term" and "coefficient", which help us communicate about the different parts expressions.
The art of dress
The beginning of World War I
Called the Great War (before World War II came about), World War I was the bloody wake-up call that humanity was entering into a new stage of civilization. Really the defining conflict that took Europe from 19th Century Imperial states that saw heroism in war into a modern shape. Unfortunately, it had to go through World War II as well (that some would argue was due to imbalances created by World War I).
The business cycle
Economies never have a long steady march upwards. They constantly oscillate between growth and recession. This tutorial gives a little intuition for why that is.
The complex plane
You know what imaginary and complex numbers are, but want to start digging a bit deeper. In this tutorial, we will explore different ways of representing a complex number and mapping them on the complex plane.
The convolution integral
This tutorial won't be as convoluted as you might suspect. We'll see what multiplying transforms in the s-domain give us in the time domain.
The cube root
If you're familiar with the idea of a square root, we're about to take things one step (dimension?) further with the cube root. This generally refers to finding a number that ,when cubed, is equal to the number that you're trying to find the cube root of!
The demand curve
You've probably heard of supply and demand. Well, this tutorial focuses on the demand part. All else equal, do people want more or less of something if the price goes down (what would you do)? Not only will you get an intuition for the way we typically depict a demand curve, you'll get an understanding for what might shift it.
The distributive property
The distributive property is an idea that shows up over and over again in mathematics. It is the idea that 5 x (3 + 4) = (5 x 3) + (5 x 4). If that last statement made complete sense, no need to watch this tutorial. If it didn't or you don't know why it's true, then this tutorial might be a good way to pass the time :)
The distributive property and equivalent expressions
Learn how to factor and expand algebraic expressions (expressions with variables) by applying the distributive property.
The evolutionary causes of biodiversity
Environmental conditions play a critical role in determining if an individual will survive and contribute its genetic information to the next generation and how new species will evolve.
The founding mothers
The imaginary unit i
This is where math starts to get really cool. It may see strange to define a number whose square is negative one. Why do we do this? Because it fits a nice niche in the math ecosystem and can be used to solve problems in engineering and science (not to mention some of the coolest fractals are based on imaginary and complex numbers). The more you think about it, you might realize that all numbers, not just i, are very abstract.
The international avant-garde
This tutorial could be called the "the School of Paris" Here find the impact of Cubism and pre-war abstraction in de Chirico's Metaphysical painting, Brancusi's biomorphic sculpture, Mondrian's de Stijl grids and other leading members of the avant-garde.
The kidney and nephron
How do we get unwanted substances out of our blood? Through the kidney. This tutorial goes into some detail to describe just how this happens.
The learning environment in a station rotation, lab rotation and flex model
The modern U.S. Supreme Court on privacy and civil liberties
The moves of a blended learning teacher
The neuron and nervous system
Neurons are the primary way that our bodies transmit signals from one part to another quickly. In this tutorial, we'll explore the anatomy of a neuron and the mechanism by which a signal is actually transmitted through one.
The plasma membrane
Learn about the amazing and versatile barrier that surrounds all of your cells: the plasma membrane!
The possible number of solutions of systems of linear equations
Learn about the possible number of solutions of systems of linear equations. Spoiler: There can be a single solution, zero solutions, or infinite solutions. These different cases define the distinctions between "consistent" and "inconsistent" systems, and between "dependent" and "independent" systems.
The quadratic formula (quadratic equation)
Probably one of the most famous tools in mathematics, the quadratic formula (a.k.a. quadratic equation) helps you think about the roots of ANY quadratic (even ones that have no real roots)! As you'll see, it is just the by-product of completing the square, but understanding and applying the formula will take your algebra skills to new heights. In theory, one could apply the quadratic formula in a brainless way without understanding factoring or completing the square, but that's lame and uninteresting. We recommend coming to this tutorial with a solid background in both of those techniques. Have fun!
The reconstruction amendments
The square root
A strong contender for coolest symbol in mathematics is the radical. What is it? How does it relate to exponents? How is the square root different than the cube root? How can I simplify, multiply and add these things? This tutorial assumes you know the basics of exponents and exponent properties and takes you through the radical world for radicals (and gives you some good practice along the way)!
The supply curve
Now we'll focus on the "supply" part of supply and demand. Supply curves (as we typically depict them) come out of the idea that producers will make more if they get paid more.
The why of algebra
Algebra seems mysterious to me. I really don't "get" what an equation represents. Why do we do the same thing to both sides? This tutorial is a conceptual journey through the basics of algebra. It is made for someone just beginning their algebra adventure. But even folks who feel pretty good that they know how to manipulate equations might pick up a new intuition or two.
The world of exponents
Addition was nice. Multiplication was cooler. In the mood for a new operation that grows numbers even faster? Ever felt like expressing repeated multiplication with less writing? Ever wanted to describe how most things in the universe grow and shrink? Well, exponents are your answer! This tutorial covers everything from basic exponents to negative and fractional ones. It assumes you remember your multiplication, negative numbers and fractions.
Theorems concerning parallelogram properties
Triangle congruence is not only useful when working with triangles, it's also useful with any other kind of polygon! Prove a few theorems about the properties of parallelograms using triangle congruence.
Theorems concerning triangle properties
Now that we know how to show that two triangles are congruent, let's put that to use by proving some theorems about triangles.
Theoretical and experimental probability
In many situations, we don't know exact probabilities, so we estimate probability based on history of events.
Theories of attitude and behavior change
7C: Although people can learn new behaviors and change their attitudes, psychological, environmental, and biological factors influence whether those changes will be short-term or long-term. Understanding how people learn new behaviors, change their attitudes, and the conditions that affect learning helps us understand behavior and our interactions with others. The content in this category covers learning and theories of attitude and behavior change. This includes the elaboration likelihood model, theories of information processing, and social cognitive theory.
Theories of personality
7A: Curious about your personality? It’s a complex thing which is difficult to define in even a book. Nonetheless, throughout history, several notable psychologists and schools of thought have attempted to figure out how to organize and categorize human personalities. We will review these theories and see which one resonates the most with you! By Shreena Desai. Motivation and attitudes - What makes us dot he things we do, or feel the way we feel in various social situations? We will discuss how the physiological and psycho-social theories, factors, and situations behind motivation, attitudes, and behavior are interrelated.
Wouldn’t it be terrifying if our clothes, food, and houses spontaneously combusted? Molecular bonds store tremendous amounts of energy, but generally are very stable. In order to release the energy, you have to add more first - like applying a match to a firework. These videos will overview standard enthalpy, entropy, and how to find out if a reaction is spontaneous.
Thin lenses
4D: Without lenses, we would not be able to examine the layers of the skin or to observe the habits of the pink flamingo from a distance. If you have ever used a microscope, binoculars, or a magnifying glass, you have benefited from the workings of a thin lens, which refracts rays of light as it passes through the medium. In these tutorials, you’ll learn how to use ray tracings and the thin lens formula to predict the sizes and orientation of images created by thin lenses. We will apply these concepts to a discussion of the human eye.
Thinking about solutions to systems
You know how to solve systems of equations (for the most part). This tutorial will take things a bit deeper by exploring cases when you might have no solutions or an infinite number of them. Common Core Standards: 8.EE.C.8, 8.EE.C.8b
Thinking algebraically about inequalities
In this tutorial you'll discover that much of the logic you've used to solve equations can also be applied to think about inequalities!
Thiols and sulfides
In this tutorial, Jay shows how to prepare sulfides from thiols.
Three Kingdoms Period (57 B.C.E.-676 C.E.)
Three core financial statements
Corporations use three financial statements to report what's going on: balance sheets, cash flow statements and income statements. They can be derived from each other and each give a valuable lens on the operations and condition of a business. After you know the basics of accrual accounting (available in another tutorial), this tutorial will give you tools you need to responsibly understand any business.
Throat conditions
Learn how to tell time.
Time scale of the cosmos
Not only is the universe unimaginable large (possibly infinite), but it is also unimaginably old. If you were feeling small in space, wait until you realize that all of human history is but a tiny blip in the history of the universe.
Time word problems
Learn how to solve a variety of problems involving time.
Chemists use titrations to determine the concentration of an acid, base, or other substance in solution. We will cover the different types (strong acid/strong base, strong acid/weak base, weak acid/strong base, and redox), calculating the pH at different points of the titration, and choosing indicators.
Skrillex loved titrating solutions because he got to drop the base. Titrations are a technique that allow us to determine the concentration of an unknown by adding a known element to it until a specific change occurs. For instance, finding the acidity of a solution by adding standard base until the solution turns pink. Learn more about titrations and indicators by watching these examples.
Tlatilco was one village of several spread around the great lake that once formed the center of what is today Mexico City. Tlatilco figurines are wonderful small ceramic figures, often of women, found in Central Mexico from 1500-1200 B.C.E. Some of the motifs found on other Tlatilco ceramics, such as ducks and fish, would have come directly from their lakeside surroundings.
Toby Rush - CEO & Founder of EyeVerify
Toby Rush looks for hard problems to solve and ways to apply simple solutions for bringing products to market. He teamed with university scientist to develop EyeVerify Inc., which uses EyePrint ID, a highly accurate biometric technology that uses existing cameras on mobile devices to image and pattern match the blood vessels in the whites of the eye.
Tools for understanding art
Torque, moments, and angular momentum
Until this tutorial, we have been completely ignoring that things rotate. In this tutorial, we'll explore why they rotate and how they do it. It will leave your head spinning (no, it won't, but seemed like a fun thing to say given the subject matter).
Tour of a eukaryotic cell
Join us on a voyage through some of the specialized compartments and structures of eukaryotic cells. From the rolling folds of the endoplasmic reticulum to the majestic fibers of the microtubule cytoskeleton, it's a trip you won't regret!
Towers of Hanoi
Use the recursive technique to solve the Towers of Hanoi, a classic mathematical puzzle and one reportedly faced by monks in a temple.
Transformation Puzzles
Random puzzles involving rotating, reflecting, and dilating polygons.
It's time to learn about the most useful math concept for creating video game graphics: geometric transformations.
Transformations and congruence
Two figures are congruent if you can go from one to another through some combination of translations, reflections and rotations. In this tutorial, we'll really internalize this by working through the actual transformations.
Transformations and matrix multiplication
You probably remember how to multiply matrices from high school, but didn't know why or what it represented. This tutorial will address this. You'll see that multiplying two matrices can be view as the composition of linear transformations.
It's time to learn about the most useful math concept for creating video game graphics: geometric transformations. We will start with lines of symmetry and the order of rotational symmetry and then move on to reflecting, translating, and/or rotating a shape
Learn how to find the translation that maps a given figure to another, and how to manually draw the image of a translation.
Transport across a cell membrane
2A: Each cell in your body has a “membrane potential.” Think of it like rolling a ball to the top of a hill - once the ball is at the top, it is smooth sailing down. Similarly, this electric membrane potential allows ions to flow down a gradient of electrical energy (the inside of the cell is negative relative to the outside). We will discuss this concept as well as other mechanisms for movement of ions, water, and other molecules across cellular membranes.
Transpose of a matrix
We now explore what happens when you switch the rows and columns of a matrix!
Tree of life
Triangle angle properties
Do the angles in a triangle always add up to the same thing? Would I ask it if they didn't? What do we know about the angles of a triangle if two of the sides are congruent (an isosceles triangle) or all three are congruent (an equilateral)? This tutorial is the place to find out.
Triangle angles
Do the angles in a triangle always add up to the same thing? Would I ask it if they didn't? What do we know about the angles of a triangle if two of the sides are congruent (an isosceles triangle) or all three are congruent (an equilateral)? This tutorial is the place to find out. Common Core Standard: 8.G.A.5
Triangle congruence
Learn how the definition of congruence using rigid transformations can be simplified to simple criteria when studying congruent triangles.
Triangle inequality theorem
The triangle inequality theorem is, on some level, kind of simple. But, as you'll see as you go into high level mathematics, it is often used in fancy proofs. This tutorial introduces you to what it is and gives you some practice understanding the constraints on the dimensions of a triangle.
Triangle similarity
This tutorial explains a similar (but not congruent) idea to congruency (if that last sentence made sense, you might not need this tutorial). Seriously, we'll take a rigorous look at similarity and think of some reasonable postulates for it. We'll then use these to prove some results and solve some problems. The fun must not stop!
Triangles and Polygons
What do we know about a triangle if two of the sides are congruent (an isosceles triangle) or all three are congruent (an equilateral)? What abut their angles? Let's find out.
Trig functions of special angles
In this tutorial, we'll really digest how special triangles and angles that show up a lot in mathematics relate to each other and the various trig functions.
Trig identities
If you're starting to sense that there may be more to trig functions than meet the eye, you are sensing right. In this tutorial you'll discover exciting and beautiful and elegant and hilarious relationships between our favorite trig functions (and maybe a few that we don't particularly like). Warning: Many of these videos are the old, rougher Sal with the cheap equipment!
Trig problems on the unit circle
Trig ratio application problems
You are now familiar with the basic trig ratios. We'll now use them to solve a whole bunch of real-world problems. Seriously, trig shows up a lot in the real-world.
Trig ratios and similarity
In this tutorial, we will build on our understanding of similarity to get a deeper appreciation for the motivation behind trigonometric ratios and relationships.
Trig substitution
We will now do another substitution technique (the other was u-substitution) where we substitute variables with trig functions. This allows us to leverage some trigonometric identities to simplify the expression into one that it is easier to take the anti-derivative of.
Trigonometric ratios and similarity
In this tutorial, we will build on our understanding of similarity to get a deeper appreciation for the motivation behind trigonometric ratios and relationships.
Trigonometry readiness warmup
Triple integrals
This is about as many integrals we can use before our brains explode. Now we can sum variable quantities in three-dimensions (what is the mass of a 3-D wacky object that has variable density)!
Almost one third of the entire world’s population is infected with Mycobacterium tuberculosis, the type of bacteria that causes TB. Although only a fraction of these people will actually become sick with the disease, in 2012, the World Health Organization reported 1.3 million TB related deaths. The good news is that health care workers and public health officials around the world have done a great job of helping to detect and treat the disease early. The bad news is that TB is developing drug resistance. Learn more about this ancient disease, that still plagues us in the modern-day.
Turn light into sound
Can you listen to a beam of light? Audio signals can be transmitted along radio waves through space, and in electrical pulses through wires. Visible light and other forms of electromagnetic radiation can carry audio signals, too. In this activity, learn how to build a simple device in which the signal from a radio is transmitted along a beam of light traveling between a light-emitting diode (LED) and a solar cell. This science snack was developed by Exploratorium Teacher Institute staff Don Rathjen and demonstrated by Exploratorium Senior Scientist Paul Doherty.
Turner: The Artist and His Work
J.M.W. Turner was both an artist of his time and a radical “modernist,” a practitioner of traditional styles of painting and a precursor of those to come many decades later. He was a robust personality and a sensitive observer of the events that shaped the world in his lifetime. As you learn about Turner and look closely at his work, think about the relationship between biography and practice. Should art be separated from events in the artist’s life or seen objectively? Or do you think a creator’s personality will inevitably influence the work he or she makes?
Two digit addition and subtraction
In this tutorial, we'll start adding and subtracting numbers that have two (yes, two!) digits. We won't be doing any carrying or borrowing (you'll learn what those are shortly) so you can see that adding or subtracting two digit numbers is really just an extension of what you already know.
Two-dimensional projectile motion
Let's escape from the binds of one-dimension (where we were forced to launch things straight up) and start launching at angles. With a little bit of trig (might want to review sin and cos) we'll be figuring out just how long and far something can travel.
Two-step equation word problems
Learn how to set up and solve equations to find answers to real-world questions.
Two-step equations intro
Learn how to solve equations like 2x+7=13 and 3(x+4)=3, which can be solved in two steps.
Two-step equations with decimals and fractions
Get some practice solving two-step equations that involve decimals and fractions
Two-step inequalities
Learn how to solve more complex (mathematical) inequalities. Also learn to construct inequalities to model real-world situations.
Two-way tables
Types of chemical bonds
In this tutorial, we will learn about electronegativity and the different types of chemical bonds.
Types of chemical reactions
Types of regions in three dimensions
This tutorial classifies regions in three dimensions. Comes in useful for some types of double integrals and we use these ideas to prove the divergence theorem.
Types of statistical studies
US Declaration of Independence
In this tutorial Walter Isaacson walks Sal through the United States Declaration of Independence. In doing so, they discuss the philosophical underpinnings of the American Revolution and the United States in general. Walter Isaacson is the President and CEO of the Aspen Institute. He is the former CEO and Chairman of CNN and Managing Editor of Time Magazine. He has written best-selling biographies of Steve Jobs, Benjamin Franklin, Albert Einstein and Henry Kissinger.
UV/Vis Spectroscopy
In this tutorial, Jay introduces UV/Vis spectroscopy and color in organic compounds.
Undefined and Indeterminate
In second grade you may have raised your hand in class and asked what you get when you divide by zero. The answer was probably "it's not defined." In this tutorial we'll explore what that (and "indeterminate") means and why the math world has left this gap in arithmetic. (They could define something divided by 0 as 7 or 9 or 119.57 but have decided not to.)
Undefined quotients
Let's learn why dividing by zero is "undefined."
Understanding company statements and capital structure
If you understand what a stock is (also a good idea to look at the topic on accounting and financial statements), then you're ready to dig in a bit on a company's actual financials. This tutorial does this to help you understand what the price of a company really is.
Understanding fractions
If you don't understand fractions, you won't be even 1/3 educated. Glasses will seem half empty rather than half full. You'll be lucky to not be duped into some type of shady real-estate scheme or putting far too many eggs in your cake batter. Good thing this tutorial is here. You'll see that fractions allow us to view the world in entirely new ways. You'll see that everything doesn't have to be a whole. You'll be able to slice and dice and then put it all back together (and if you order now, we'll throw in a spatula warmer for no extra charge).
Understanding whole number representations
Whether with words or numbers, we'll try to understand multiple ways of representing a whole number quantity. We'll even play with place value a good bit to make sure that everything is clicking!
Unemployment is a key metric for judging the health of an economy (and even political stability). This tutorial is a primer on what it is and how it's measured (which you might find surprising).
Unit conversion
Wait, I'm in Europe and my car only tells my distance traveled in kilometers! But I'm used to a units of distance devised by the Romans to measure the average length of 1000 paces of a soldier (the "mile")! How do I operate? These tutorials are about the fundamental skill of unit conversion. Sal's cousin Nadia being a bit confused with this was actually the reason why he started tutoring her (which led to the creation of the Khan Academy). 
Unit sense
Learn which units of measurement you should use in different situations.
Unit word problems
Now that you have some experience converting between units, let's apply that skill in some real-world problems (okay, some of them are a bit concocted, but it's all about the learning)! We'll tackle word problems that deal with distance, weight, volume, time, and more! Common Core Standard: 5.MD.A.1
United States History
The universe is all the matter, energy, and space that exist. We can observe only a part of it--the observable universe. The entire, universe, including the part we cannot see, may be infinite. The observable universe contains as many as 100 billion galaxies and extends a billion light-years in every direction. The first five videos in this tutorial depict the different types of galaxies in our Universe, including spiral, giant and dwarf ellipticals. The interactions between these galaxy types are the subject of the five time-lapse simulations that follow. The last two videos describe two galactic phenomena: the Phoenix galaxy, which produces stars faster than any other known galaxy; and the formation of an elliptical galaxy 11 billion years ago, when two massive spiral galaxies merged.
Unknown angle algebra problems
Practice finding missing angles in figures using your knowledge of vertical, supplementary, and complementary angles.
Urinary system introduction
If you want to learn more about the renal system, then urine the right place! (Pun aside, the kidneys are about more than just making urine). Every thirty minutes, your kidneys filter the entire blood supply in your body. Imagine a dirty pool filled with algae. Placing a filter in this pool will cause the algae to be flushed out, and after a time you’ll have a clean, crisp blue pool to enjoy. Just like the filter for a pool, our kidneys filter the blood and remove toxic wastes. These paired organs are key to maintaining electrolyte and water homeostasis in your body.
Using JS libraries in your webpage
Learn how to include JS libraries on your webpage, so that you can use functionality built by other web developers.
Using KA in developmental math
Using KA in homeschool
Using KA to tutor
Using regrouping to subtract within 1000
Learn how to subtract numbers like 363 - 259. These subtraction problems are a harder than the ones we've done before because they require regrouping (also called borrowing).
Using secant line slopes to approximate tangent slope
The idea of slope is fairly straightforward-- (change in vertical) over (change in horizontal). But how do we measure this if the (change in horizontal) is zero (which would be the case when finding the slope of the tangent line. In this tutorial, we'll approximate this by finding the slopes of secant lines.
Using the medieval book
This tutorial discusses how books were used in medieval times. After a manuscript was produced it came into circulation in a monastery, became part of a private library, or ended up in the hands of a student. Readers’ interactions with books left physical traces, such as wear-and-tear, bookmarks, corrections and marginal notes. They reflect how the book was handled, what was deemed important information, and how that information was used.
Using the site
Valuation and Investing
Life is full of people who will try to convince you that something is a good or bad idea by spouting technical jargon. Most of them have no idea what they are talking about. Don't be one of those people or their victims when it comes to stocks. From P/E rations to EV/EBITDA, we've got your back!
Value theory
Value theory is an evaluative area of philosophy that includes ethics, aesthetics, social and political philosophy, feminist philosophy, and other areas.
Van Eyck
Van der Goes
Van der Weyden
We'll cover how to use variables to hold values, animate your drawings, and more.
Variables and equations
What is "preservation of equality"? Wait, why are we using letters in math? How can an 'x' represent a number? What number is it? I must figure this out!!! Yes, you must. This tutorial is great if you're just beginning to delve into the world of algebraic variables and expressions.
Variables and expressions
Wait, why are we using letters in math? How can an 'x' represent a number? What number is it? I must figure this out!!! Yes, you must. This tutorial is great if you're just beginning to delve into the world of algebraic variables and expressions.
Variance and standard deviation
We have tools (like the arithmetic mean) to measure central tendency and are now curious about representing how much the data in a set varies from that central tendency. In this tutorial we introduce the variance and standard deviation (which is just the square root of the variance) as two commonly used tools for doing this.
Variations on cellular respiration
There's more than one way to extract energy from a fuel molecule! Learn how cells break down fuels without oxygen, how non-glucose molecules can enter cellular respiration, and how cellular respiration is regulated.
Inflammation of the blood vessel wall is termed “vasculitis,” though the etiology of these diseases are rather nebulous. They present with nonspecific symptoms like fever, fatigue, weight loss. Large, medium, and small vessels can all be involved. We will explore the specific differences between the various vasculitides, which range from Takayasu arteritis to microscopic polyangitis.
Vector basics
Vector dot and cross products
In this tutorial, we define two ways to "multiply" vectors-- the dot product and the cross product. As we progress, we'll get an intuitive feel for their meaning, how they can used and how the two vector products relate to each other.
Vectors and scalars
4A: It’s gym day. Today you are going to run, swim, and lift weights. In physics terms, we can describe these actions in terms of vectors and scalars. For instance, if you throw a discus across a field with a 20-pound force westward, that is an example of a vector: an entity with both magnitude and direction. However, when you show off about how many laps you can swim in an hour, you are referring to speed, a scalar, which does not specify direction. An understanding of vectors and scalar will provide a foundation to many of the other basic concepts in physics.
Vectors in magnitude and direction form
Vectors in rectangular form
Venice and Ravenna
Learn here about the impact of the Byzantine Empire on the Italian cities of Venice and Ravenna and discover some of the most glorious mosaics ever made.
Venn diagrams and adding probabilities
What is the probability of getting a diamond or an ace from a deck of cards? Well I could get a diamond that is not an ace, an ace that is not a diamond, or the ace of diamonds. This tutorial helps us think these types of situations through a bit better (especially with the help of our good friend, the Venn diagram).
Venn diagrams and the addition rule
What is the probability of getting a diamond or an ace from a deck of cards? Well I could get a diamond that is not an ace, an ace that is not a diamond, or the ace of diamonds. This tutorial helps us think these types of situations through a bit better (especially with the help of our good friend, the Venn diagram).
Verifying that functions are inverses
Learn how to check whether a pair of functions are the inverses of one another.
Viceroyalty of New Spain
Viceroyalty of Peru
Videos for scenarios from math teacher training
Viruses can cause sickness and disease, however they are not living entities. If they are not alive, then what are they? In this section, you will learn about the different types of viruses and how they survive, replicate, and spread inside a host organism.
Visit Turner's Gallery
In 1822, Turner opened a gallery on Queen Anne Street in London in order show his work to its best advantage and promote it to potential buyers, having designed everything from heating and lighting to the arrangement of the paintings himself. Step inside a virtual recreation of Turner’s gallery, where you can see how the artist exhibited a lifetime of work in a single space, plus get a sneak peek into how the gallery was accurately recreated in real life for Mike Leigh’s feature film "Mr. Turner."
Visualizing derivatives and antiderivatives
You understand that a derivative can be viewed as the slope of the tangent line at a point or the instantaneous rate of change of a function with respect to x. This tutorial will deepen your ability to visualize and conceptualize derivatives through videos and exercises. We think you'll find this tutorial incredibly fun and satisfying (seriously).
Visualizing equivalent fractions
Do you want 2/3 or 4/6 of this pizza? Doesn't matter because they are both the same fraction. This tutorial will help us explore this idea by really visualizing what equivalent fractions represent.
Visually adding and subtracting fractions with unlike denominators
An introduction to adding and subtracting fractions with unlike denominators that uses visual fraction models
Vocal Visualizer
Croon a tune, croak like a frog, hum, hoot, or holler—you don’t need to be Pavarotti to transform laser light into dazzling shapes and patterns with the power of your voice. By humming, singing, or talking into the Vocal Visualizer—also known as a simple sound oscilloscope—you’ll be able to see sound as vibration, and create a variety of cool laser light patterns that are reflected onto a wall or any other flat surface.
Volume measures how much 3-dimensional "space" an object takes up. We'll see in this tutorial that it is an extension of length (1-D) or area (2-D) to three dimensions!
Volume formula intuition
The volume formula for boxes (rectangular prisms) makes sense if we take time to think about it.
Volume of a box or rectangular prism
Volume measures how much 3-dimensional "space" an object takes up. We'll see in this tutorial that it is an extension of length (1-D) or area (2-D) to three dimensions!
Volume of cones, cylinders, and spheres
Volume of solids with known cross sections
We will now leverage the definite integral to find volumes of figures where we know what the cross sections look like. It is surprisingly fun.
Volume problems
Volume with fractions
Finding the volume of rectangular prisms that have fractional side lengths
Volume with unit cubes
Welcome to the 3rd dimension! [cue spooky music] This tutorial explains how to find volume by counting unit cubes.
Wall painting
Paintings from antiquity rarely survive—paint, after all, is a much less durable medium than stone or bronze sculpture. But thanks to the ancient Roman city of Pompeii we can trace the history of Roman wall painting. The entire city was buried in volcanic ash in 79 C.E. when the volcano at Mount Vesuvius erupted—preserving the rich colors in the paintings in the houses and monuments there for thousands of years until their rediscovery.
War and conquest
Water as a solid, liquid, and gas
Ever wonder why sweating cools you off? Or why ice cubes float in a glass? If so, you're in the right place! Learn about what happens to water when it heats up, cools down, and changes state (becoming a solid or a gas). We'll see why water is great at resisting changes in temperature, and why ice is less dense than liquid water.
Ways to write numbers (standard, written, and expanded forms)
This video is a short introduction to the resources developed by NASA and Khan Academy.
Welcome to 3rd grade
We're about to start our third grade journey! Before we start, let's brush up on core concepts from early math.
Welcome to 4th grade
We're about to start our fourth grade journey! Before we start, let's brush up on core math concepts from third grade.
Welcome to 500 years of British art
Your journey through British art begins here. Director at Tate Britain Penelope Curtis introduces the 500-year circuit that traces the history of British art at Tate and takes you on a walk through time.
Welcome to 5th grade
We're about to start our fifth grade journey! Before we start, let's brush up on core math concepts from fourth grade.
Welcome to 6th grade
We're about to start our sixth grade journey! Before we start, let's brush up on core math concepts from fifth grade.
Welcome to 7th grade
We're about to start our seventh grade journey! Before we start, let's brush up on core math concepts from sixth grade.
Welcome to 8th grade
We're about to start our eighth grade journey! Before we start, let's brush up on core math concepts from seventh grade.
Welcome to Tate on Khan Academy
What is Tate on Khan Academy? What can I learn about here? I have a question - who can I ask? Find out more about our tutorials and begin your journey through art here.
Welcome to jQuery
Get an overview of the jQuery library, meet the inventor, and try using jQuery in a webpage.
West and central Asia: 500 B.C.E.-1980 C.E.
The arts of West and Central Asia play a key role in the history of world art, giving form to the vast cultural interchanges that have occurred in these lands that link the European and Asian peoples.
Western and Eastern fronts of World War I
This tutorial goes into some detail to describe the tactics and battles of the two major fronts of World War I--the Western Front and the Eastern Front.
What drives oil prices
This tutorial tries to address a very important question in the real world--what drives oil prices? And we will do it using the tools of the supply and demand curves.
What expressions express
Using the combined powers of Chuck Norris and polar bears (which are much less powerful than Mr. Norris) to better understand what expressions represent and how we can manipulate them. Great tutorial if you want to understand that expressions are just a way to express things!
What fractions mean
What happens when you stay put
What is a dinosaur?
It's not what they ate or when they lived that distinguishes dinosaurs from other reptiles. It's the hole in their hip socket, which is associated with their upright gait.
What is addition? What is subtraction?
Learn what it means to add. Learn what it means to subtract.
What is an archive?
So what exactly is an archive? In this series of videos, you will come across everything from jukeboxes to sketchbooks, from careful documentation to experiments in chance. Take a look at how an archive can become a creative laboratory, not only tracing and documenting an artist’s work, but also inspiring others in the process. How can sketchbooks bring an artist’s art and practice to life? How can we preserve something as time-based and ephemeral as performance art? How can old archives inspire us to make art in new ways?
What’s God got to do with it?
When people do great and really terrible things
Whole number division with decimal quotients
Sometimes when we divide our answer (the quotient) is a decimal number.
Why parties in a cartel will cheat
You know what Nash equilibrium is (from the other tutorial). Now we apply it to a scenario that is fairly realistic--parties to a cartel cheating. A cartel is a group of actors that agree (sometimes illegally) to coordinate their production/pricing to maximize their collective economic profit. What we will see, however, is that this is not a "Pareto optimal" state and they will soon start producing more than agreed on.
Why we do the same thing to both sides of an equation
A high-level, conceptual walk-through of what an equation represents and why we do the same thing to both sides of it
Women and the expansion of American
Women as Artists
While women take roles in every aspect of the art world, it is the women artists who have created, crafted, and forged their way forward with their work. In this tutorial, meet some of the female pioneers in painting, sculpture, and ideas of modernism who set the tone for art to come, alongside contemporary women painters, photographers, and professionals whose practices continue to be rich and varied.
Women's Issues in Art
Many women artists have found ways to engage with the art world despite facing challenge and marginalisation. With these new voices, some artists have used their work to speak about issues that are unique to them as women, artists, and creative practitioners. Take a look at women artists who create platforms for thinking about gender, sexuality, equality, political rights, and more.
This family of instruments is situated in the middle of the orchestra, and is comprised of three groups: the flutes (flutes, piccolo), single reeds (clarinets, bass-clarinet), and double reeds (oboe, English horn, bassoons, contra-bassoon) Watch and listen as principal players of the All-Star Orchestra demonstrate their instruments and tell about their musical lives.
Word problems with "more" and "fewer"
Learn how to solve word problems that use the words "more" and "fewer". Numbers used are 20 or less.
Word problems with "more" and "fewer" (within 100)
Learn how to solve word problems that use the words "more" and "fewer". Numbers used are 100 or less.
Word problems within 100
Learn how to solve addition and subtraction word problems. Numbers used 100 or less.
Word problems within 20
Learn how to solve addition and subtraction word problems. Numbers used are 20 or less.
Work and energy
You're doing a lot more work than you realize (most of which goes unpaid). This tutorial will have you seeing the world in terms of potentials and energy and work (which is more fun than you can possibly imagine).
Work study
Work-study, one component of a financial aid package, offers you the chance to earn extra funds through on-campus jobs while still going to college full-time. Hear from admissions on how work-study operates and why it just might be the extra money you need to make college affordable.
Worked examples: geometry
Sal does the 80 problems from the released questions from the California Standards Test for Geometry. Basic understanding of Algebra I necessary.
Working with powers of 10
This tutorial leads in to scientific notation. It features multiplication and division problems involving powers of 10.
Working with triangles
Learn how to solve different geometric problems using triangle congruence.
Working with units algebraically
You already know some basic algebra and you've been exposed to units for some time now. In this tutorial, we meld these two ideas to think about units within the context of algebraic expressions.
World War I Quiz
Test your comprehension of the causes, dynamics and aftermath of World War I (as covered in the tutorials in this topic) by taking this quiz.
Writing algebraic expressions introduction
Learn the basics of writing expressions with variables.
Writing and interpreting expressions
All the symbols you write in math are just a language or short-hand to represent real-world ideas. In this tutorial, we'll get experience writing algebraic expressions to elegantly represent real-life ideas.
Writing and solving proportions
Learn how to write proportions and how to solve proportions.
Writing basic algebraic expressions word problems
Learn how to write algebraic expressions (expressions with variables) to model real-world situations.
Writing clean code
Learn how to write JavaScript code that is easy to read and understand, with good use of indentation, spacing, naming, and comments.
Writing expressions
In this tutorial, we will learn to describe situations using mathematical expressions, and consider the effect of using parentheses in the outcome of those expressions.
Writing: Grammar
Writing: Passages
You ain’t the boss of me!
Yuan dynasty (1271-1368)
The Yuan dynasty was established by the Mongol invader Kublai Khan and was a period of unified rule, trade and important developments in ceramics, painting, poetry, and theater. The Yuan dynasty lasted from 1271 until 1368.
Zach Kaplan - CEO of Inventables
Zach Kaplan, CEO of Inventables, discusses how the revolution of digital manufacturing and desktop publishing can impact a new generation of entrepreneurs.
Zero and identity matrices
In arithmetic, we learned than a number times 1 is still that number and that anything times 0 is 0. In this tutorial, we attempt to extend these ideas to the world of matrices!
Zhou dynasty (c. 1046 B.C.E.-256 B.C.E.)
alcohols and phenols
5D: If you’ve ever walked through the wards of a hospital, you’ve probably noticed dozens of alcohol-based hand sanitizer dispensers, which quickly clean the hands of healthcare workers between seeing patients. Alcohols serve many other functions. You will gain a strong understanding of the nomenclature, properties and reactions of alcohols and phenols, along with the criteria for determining aromaticity in heterocyles. By the way, if you’re wondering where the term “aromaticity” comes from, many of the earliest aromatic compounds, like benzene and toluene, were noted to have pleasant odors, and the name for this structural class has stuck ever since.
aldehydes and ketones
5D: Have you ever dissected a preserved cadaver in anatomy class? That stench you remember is the smell of formaldehyde used to preserve it. Formaldehyde is an aldehyde a class of molecules we will discuss along with its cousin the ketone. We will discuss their formation as well as how they interact in various chemical reactions as we walk through some real-world examples.
This topic explains how Spout works! It was created by Karl R. C. Wendt.
alpha-carbon chemistry
5D: Aldol condensations are one of the most important, frequently seen reactions in biochemistry. In fact, the very first reaction of the Krebs (TCA) cycle is an aldol condensation in which acetyl CoA condenses with oxaloacetate, forming citrate. You will learn the mechanism of these reactions formed as we extend this concept to predict the products of aldol condensations.
biological basis of behavior: Nervous system
7A: The very fact that you are able to understand this sentence means that neurons in your brain (85 billion in total) are talking to each other. Neurons are the living substance of the nervous system, which extends beyond the brain to the spinal cord and peripherally, allows you to think and process, make decisions, stand up straight, maintain your heart rate, rest and digest. You will come to appreciate the structure and function of the nervous system as we delve into its anatomy and physiology, from the gray and white matter to the cerebellum to the neurons.
circulatory system
3B: With every beat, the heart pumps blood throughout the body through an intricate system of blood vessels to provide oxygenation to tissues - from about 4 weeks after conception until the day you die. Over your lifetime, it will pump about 175 million liters of blood (that’s only the amount of water that falls over the Niagara in a few minutes). This little pump in the middle of your chest is only the size of a clenched fist (if you’re an adult), and yet it does so much.
covalent bonds
5B: Eating popcorn alone at a movie theater can be quite lonely - but sharing with someone special can feel...well, special! Sharing is caring is caring with atoms too! In a covalent bond, two atoms share electron pairs in their orbitals. We will discuss the mechanism of this bonding as well as the idea of electron orbital hybridization in this tutorial.
crypto challenge
Ready to try your hand at real-world code breaking? This adventure contains a beginner, intermediate and super-advanced level. See how far you can go!
Prices don't always go up. Sometimes they go down (we call this deflation). This tutorial explains how this happens.
9B: There are many different ways of looking at a population. You can separate the population into different groups to view statistics. For instance, one may want to compare the efficacy of a drug in treating diabetics with congestive heart failure compared to diabetics without heart failure - these each form two separate populations of patients. In this module, you will come to see how different populations interact and change over time.
Discovering static electricity & electrostatic force. What is it? How can it be created, detected, and measured?
5D - The multitude of reactions within our cells are sped up by enzymes. Without these biomolecules, these biochemical pathways would be as slow as a turtle. For instance, without enzymes, your body may never be able to break down and absorb the hamburger you just had for lunch. The hamburger would simply sit there, a lump in your stomach, until reactions slowly started to happen on their own - enzymes speed that up!
gas phase
4B: Imagine taking a balloon and heating it up by putting it in a sauna. What would happen? Gases, though you may not be able to see them, are perpetually in motion. You will encounter them if you ever frequent an operating room, as the anesthesiologist holds in his arsenal several gases with doze-inducing properties. We will walk through the history and application of the ideal gas law to real-world problems as you also come to appreciate the meaning of partial pressures and STP in these tutorials.
hematologic system
3B: Roughly 5 L of blood fill your arteries, veins, capillaries, and venules. What’s it good for you ask? It carries oxygen to help your cells carry out respiration in addition to a number of other substances like lipids and hormones throughout the body. In cases of blood loss, such as trauma situations, the physician must be wary of the different blood types. We will explore the intricacies of the hematologic system here.
jQuery techniques
Learn a few common techniques used by jQuery developers, like chaining function calls and looping through collections.
neuron membrane potentials
Learn the causes and functions of neuron membrane potentials, including resting, graded, and action potentials.
nomenclature and reactions of carboxylic acids
In this tutorial, Jay shows you how to name carboxylic acids and the products of different reactions of carboxylic acids.
Wait, why are we using letters in math? How can an 'x' represent a number? What number is it? I must figure this out!!! Yes, you must. This tutorial is great if you're just beginning to delve into the world of algebraic variables and expressions.
periodic table
4E: A little more than a century ago, the chemist Dmitri Mendeleev published an early form of the periodic table, which organizes the known elements of our world by ionization energy and electron affinity. His method of classifying the elements was so useful that we still use it even today. We will learn to apply this elegant table to an understanding of atoms and molecules in this tutorial. Hydrogen, helium, lithium, beryllium, boron, carbon…
proton NMR
In this tutorial, Jay introduces the theory of proton NMR and shows how to analyze NMR spectra.
Exploring materials which cause a decrease in deflection when added in series with our meter.
8A: Who exactly are we? How do we develop our morals and patterns of learning? What influences our behaviors in social situations? Physicians must have a keen understanding of their patients’ definitions of themselves in order to treat holistically. Over the past few centuries, several notable psychologists - from Freud to Erikson to Vygotsky - have attempted to answer these grand question. In this module you will explore some of their ideas surrounding the concept of self-identity as we delve into the different phases of our life as we transform from infants to teenagers to full-fledged adults.
sensory perception
6A: Each day, we encounter various sights, sounds, smells, and tastes. Without an integrative centers, these many inputs would mean nothing to us. We will learn about how we perceive our various senses, including the theories, laws, and organizational principles that underly our ability to make sense of the world around us.
skeletal system introduction
Were it not for your skeleton, you and I would be a mere sack of flesh. You will come to appreciate that the bones, together with muscles, are a scaffolding for your body. We will also explore their endocrine function, especially with regards to calcium and phosphate homeostasis. Fun fact: the bone most broken is the clavicle (AKA collar bone).
Birds of a feather flock together. How do we explain these and other observations in the way we interact with fellow human beings? We will dive into the fascinating and relevant world of social psychology as we discuss concepts which you may have noticed in real life. You will come to know the names of such phenomena as well as the specific factors that motivate people to behave the way they do in groups.
U-substitution is a must-have tool for any integrating arsenal (tools aren't normally put in arsenals, but that sounds better than toolkit). It is essentially the reverse chain rule. U-substitution is very useful for any integral where an expression is of the form g(f(x))f'(x)(and a few other cases). Over time, you'll be able to do these in your head without necessarily even explicitly substituting. Why the letter "u"? Well, it could have been anything, but this is the convention. I guess why not the letter "u" :)
work and energy
4A: Work doesn’t always have to break your back. In physics, it has a different meaning. When a force is applied to an object and results in displacement, work has been done. When an apple falls from a tree, gravity has done work on the object as it descends earthward. And as work is done by gravity on the object, its gravitational potential energy is converted into kinetic energy (which is also why it hurts as it lands on your head, hopefully resulting in a brilliant Newtonian idea). Let’s get moving!
x-intercepts and y-intercepts