## Schoology Public Resources

"The One World Schoolhouse" book

1.0 Welcome to the Big History Project

Embark on a 13.8 billion year journey through the history of the Universe and humans, an adventure that will challenge you to think about school, life, and the Universe like never before.

1.1 What Is Big History

What is Big History? How does it differ from other courses and what are of the major themes it emphasizes?

1.2 Complexity and Thresholds

Thresholds of increasing complexity result in the creation of something fragile, diverse, precise, and entirely new in the Universe.

1.3 Scale

A mile, a year, a foot: all our everyday measures relate to a familiar scale. Big history is so big that we need to use entirely different measurements on an entirely different scale.

1.4 Origin Stories

An origin story is a culture’s reflections about the beginning of the Universe, the Earth, and life, stitched together into a narrative.

1.5 Ways of Knowing: What We Believe

What should we believe? These days we receive information from countless sources. Claim testing empowers us to assess the trustworthiness of a concept, assertion, or story.

10.0 What does the future hold?

The study of big history inevitably leads to the future. So what’s next? Join some of the world’s great thinkers to try to predict the next threshold.

10.1 Big history's most important unanswered question

Even experts don’t have all the answers. Sometimes they don’t even know which questions to ask. What do you think the most important unanswered question might be?

1400s Quiz

This quiz features is a painting that does not appear in any tutorial in this section, though it does date to this century (1400-1500). It is an “unknown” for you to test your skill and knowledge about Renaissance art. Some of the questions may require knowledge beyond the videos in this set of tutorials.

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.

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 The Big Bang

The Big Bang is where big history begins. Everything that's ever existed, including you, traces back to this unimaginably profound event.

2.1 How Did Our View 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.

2.2 What Emerged from the Big Bang?

Out of the chaos of the Big Bang came the first atoms, the building blocks of every single thing.

2.3 Ways of Knowing: How Do We Know About the Big Bang?

Do the laws of physics and mathematics apply to everything in the Universe? Janna Levin explains astrophysics.

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 (www.aops.com). Problems from the MAA American Mathematics Competitions (amc.maa.org)

3.0 Stars and big history

With the birth of the first stars, a fantastic chain of events occurred that made possible a diversity of elements and chemistry. Anything was possible.

3.1 How were stars formed?

Ever spill jelly beans on the floor? In the new Universe, some atoms – like the jelly beans – wound up together in pockets. Unlike the jelly bean scenario, gravity did its thing. Long story short…stars.

3.2 What did stars give us?

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.

3.3 Ways of knowing: Stars and elements

All those new elements. What exactly were they and how did they bind to or repel each other? The science of chemistry was born as early scientists studied the properties and structure of chemical elements and compounds.

4.0 Earth and the solar system

Earth formed from the leftovers of the Sun accumulating over time. Despite its violent and unstable beginning, Earth slowly (very, very slowly) became the world we know today.

4.1 How our solar system formed

Chemically rich clouds of tasty leftover matter orbited our newly formed Sun. Chunks of debris collided and smashed into each other, eventually creating our Earth and Solar System.

4.2 What was the young earth like?

As giant hunks of rock, metal, and ice slammed into the Earth’s surface, it became a planet with three layers. The interplay between the layers resulted in the Earth as we know it.

4.3 Why is plate tectonics important?

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.

4.4 Ways of knowing: Our solar system

The history of our planet – along with clues about its future – is written in rock record. Rock detectives (geologists) study these clues and often observe Earth's changes firsthand.

5.0 Life and big history

The young Earth was a dangerous and unfriendly place. Deep in the oceans, though, conditions were perfect for the emergence of life.

5.1 What is life?

Among living things, there are strong similarities and remarkable differences, many explained by the amount of shared DNA. But what does that really mean? What is life?

5.2 How did life begin and change

The appearance of life marked a profound beginning. Over time, simple life forms transformed, evolving into complex organisms.

5.3 How do earth and life interact?

Life can be fragile. The continued existence of a species can depend on even the slightest changes to astronomical, geological, and biological conditions.

5.4 Ways of knowing: Life

The history of our planet is written in rock record, along with clues about the future. Rock detectives – geologists – study these clues and often observe Earth’s changes first hand.

6.0 Collective learning and big history

Our talent for preserving and sharing information, passing it from one generation to the next, has made us the most powerful species on the planet.

6.1 How did our ancestors evolve

We might share a lot with our primate cousins, but our bigger brains, our ability to walk upright, and other physical “improvements,” are all adaptations that make humans unique.

6.2 What makes humans different?

You may think you can talk to the animals, but our ability to use language separates us from other species. Without it, we wouldn’t have the talent for collective learning that allows us to dominate the biosphere.

6.3 How did the first humans live?

Our Paleolithic ancestors were foraging nomads who eventually migrated across six continents. These early humans made tools, used fire, and sustained themselves in diverse environmental conditions.

6.4 Ways of knowing early humans

We’re obsessed with understanding the roots of who we are as a species. Anthropologists, archaeologists, and primatologists are the most obsessed, as they work to paint a picture of early human life.

7.0 Agriculture

Foraging was a way of life for thousands of years, but a very difficult one. Once humans discovered agriculture, they were able to stay in one place and feed many more people than ever before.

7.1 Why was agriculture so important?

Planting crops and raising livestock meant that people didn't have to move around to follow their food. They could stay in one place, build things to last, and for some, not worry about food every day.

7.2 Where and why did the first cities appear?

Larger and larger communities of people living together…sounds like you might need some kind of organization.

7.3 Ways of knowing: How do we know about agriculture and civilization?

Like detectives solving a murder mystery, historians use artifacts and written records of past civilizations as clues to understand our history as a species.

8.1 How did the world become interconnected?

Crossing dangerous deserts and deep ocean waters to connect the four world zones wasn't easy. After 1400, innovation and collective learning took a giant leap forward.

8.2 Why was commerce so important?

As systems of exchange and trade made the world a smaller place, the Afro-Eurasian world zone gained power.

8.3 Ways of knowing: Expansion and interconnection

Travelers and explorers came back from the corners of the earth with fantastic stories and new ideas and technologies. Their adventures, and they were true adventures, offer insight into the newly connected world.

9.0 Acceleration

In the last 500 years, our world has undergone a transformation. Connecting the four world zones fostered astounding innovation and pushed our species into the modern era.

9.1 Why did change accelerate

Communication, transportation, and greater connection: the pace of innovation continued to accelerate, stimulating a tremendous appetite for energy.

9.2 How was the modern world created?

For the first time, a single species can effect major change on the entire biosphere. The Industrial Revolution has led us into the modern world. Are we on the brink of a new threshold?

9.3 Ways of knowing: The modern revolution

Smith, Marx, and Keynes – acceleration gave rise to three dynamic thinkers who had great influence on the ideas of commerce, labor, and the global economy.

A Beginner's Guide to the Baroque

A Beginner's Guide to the Medieval Period

An overview of Europe in the middle ages.

A beginner's guide to the Renaissance

About pi and tau

When you want to make a circle, how is it done? Well you probably will start with the radius one.

Absolute and relative maxima and minima

Absolute value

You'll find absolute value absolutely straightforward--it is just the "distance from zero." If you have a positive number, it is its own absolute value. If you have a negative number, just make it positive to get the absolute value.
As you'll see as you develop mathematically, this idea will eventually extend to more contexts and dimensions, so it's super important that you understand this core concept now.
Common Core Standards: 7.NS.A.1a, 7.NS.A.1b

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

More than sixty years have passed since the critic Robert Coates, writing in "The New Yorker" in 1946, first used the term “Abstract Expressionism” to describe the richly colored canvases of Hans Hofmann. Over the years the name has come to designate the paintings and sculptures of artists as different as Jackson Pollock and Barnett Newman, Willem de Kooning and Mark Rothko, Lee Krasner and David Smith. Watch these short videos to explore some of the most important abstract art of the 20th century and the artists' experiments with techniques.

Acceleration

4A: Learn how to calculate acceleration, and how distance, velocity, and acceleration can be related to one another using velocity graphs. By Sal.

Acid/base

Do you remember the basics of acid/base chemistry? In this tutorial, Jay reminds you of a few definitions, shows you how the stability of the conjugate base affects the acidity of the molecule, and demonstrates the importance of pKa values.

Acids and bases

Adding and multiplying polynomials

"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 polynomials 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." :)

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 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 with unlike denominators is key. It might be a good idea to look at the equivalent fractions tutorial before tackling this one.
Common Core Standard: 5.NF.A.1

Adding and subtracting fractions with common denominators

You've already got 1/4 cups of sugar in the cupboard. Your grandmother's recipe for disgustingly-sweet-fudge-cake calls for 3/4 cups of sugar. How much sugar do you need to borrow from you robot neighbor? Adding and subtracting fractions is key.
Common Core Standards 4.NF.B.3, 4.NF.B.3a

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 word problems

You know what a fraction is and are now eager to apply this knowledge to real-world situations? Well, you're about to see that adding and subtracting fractions is far more powerful (and fun) then you've ever dreamed possible!

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!
Common Core Standards: 7.NS.A.1, 7.NS.A.1b, 7.NS.A.1c, 7.NS.A.1d

Adding and subtracting rational expressions

Well, rational expressions are just algebraic expressions formed by dividing one expression by another. In this tutorial, we'll see that, even though they may look hairy, adding and subtracting rational expressions involves most of what we know about adding and subtracting numeric fractions.

Adding and subtracting rational numbers

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.
Common Core Standards: 7.NS.A.1d

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

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!
Common Core Standard: 5.NBT.B.7

Adding fractions with unlike denominators

We've already had some good practice adding fractions with like denominators. We'll now begin to explore adding fractions with unlike denominators. In particular, we'll think about adding fractions with denominators of 10 and 100. Later on, in 5th, grade we'll extend this to adding fractions of any denominator to fractions of any denominator.

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)!
Common Core Standard: 4.NBT.B.4

Adding with regrouping within 1000

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 regrouping and why it works.
Common Core Standard: 3.NBT.A.2

Addition and subtraction within 10

Addition and subtraction within 100

Addition and subtraction within 1000

Addition and subtraction within 20

Addition reactions of conjugated dienes

In this tutorial, Jay shows the possible products for an addition to a conjugated diene and how the end product can be controlled by changing the reaction conditions.

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.

Advanced ratios and proportions

In this tutorial, we will explore more advanced examples involving ratios and proportions.

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 :(

Aegean Art

Excavations led by teams of archaeologists in the nineteenth century hoped to find evidence for places and people in Homer's epic poems The Iliad and The Odyssey. They may not have discovered what they sought, but along the way, they made remarkable discoveries. The tutorial covers the art of the Cycladic Islands, and the Island of Crete (Minoan), and of Mainland Greece (Mycenaean).

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 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.

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.

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!

Alkene nomenclature

In this tutorial, Jay names alkenes, discusses the stability of alkenes, and introduces the E/Z system.

Alkene reactions

In this tutorial, Jay explains the addition reactions of alkenes.

Alkyne reactions

In this tutorial, Jay shows the reactions of alkynes.

Altitudes

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!).

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.

American-Modernism

Art had never been especially important in America. Before the Civil War, many of America’s best artists went to Europe and stayed. Even after the war, American artists found little enthusiasm for their work unless it was directly informed by European precedents. By the first years of the 20th Century, a small group of American artists began to paint the gritty streets of New York and were called the Ashcan School for their portrayal of life in the tenements. In 1913 however, the Armory Show exhibited advanced American and European art and helped to create a market for the work of Georgia O’Keeffe and other members of modern galleries like Alfred Steiglitz’s 291 and Peggy Guggenheim’s Art of This Century. During the Great Depression artists such as Grant Wood portrayed rural life in the south and midwest and became known as regionalists while other realists such as Edward Hopper rendered the alienation of the modern city. Meanwhile, Surrealist ideas infused a younger generation of artists’ work in Mexico and the US which would result, by the end of WWII, in the first internationally important American art movement, Abstract Expressionism.

Amino acids and proteins

1A: What is the central dogma of biology? Learn the fundamental relationships that tie together DNA, RNA, and proteins. Then go on to discover the different types of amino acids, and how they come together to form complex proteins. By Tracy Kim Kovach.

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").

Analytic geometry

Use what you know about the coordinate plane to tackle these word problems!

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 linear functions

Linear functions show up throughout life (even though you might not realize it). This tutorial will have you thinking much deeper about what a linear function means and various ways to interpret one. Like always, pause the video and try the problem before Sal does. Then test your understanding by practicing the problems at the end of the tutorial.
Common Core Standards: 8.F.A.2, 8.F.A.4, 8.F.A.5

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 art of Ancient Greece and Rome is grounded in civic ideals and polytheism. Etruscan and Roman artists and architects accumulated and creatively adapted Greek objects and forms to create buildings and artworks that appealed to their tastes for eclecticism and historicism. Contextual information for ancient Greek and Roman art can be derived from contemporary literary, political, legal, and economic records as well as from archaeological excavations conducted from the mid-18th century onward. Etruscan art, by contrast, is illuminated primarily by modern archaeological record and by descriptions of contemporary external observers.
By permission, © 2013 The College Board

Ancient Near East

Religion plays a significant role in the art and architecture of the ancient Near East, with cosmology guiding representation of deities and kings who themselves assume divine attributes.
By permission, © 2013 The College Board

Ancient Rome

Ancient and Medieval

This tutorial includes the Ancient Near East, and Ancient Greece and Rome.

Angela Ahrendts - CEO of Burberry

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 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").

Angles

Let's complement and supplement our knowledge of angles with some new vocabulary. Onward!
Common Core Standards: 7.G.B.5

Angles and intersecting lines

Work through these problems to solidify your understanding of how angles can relate to each other.

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.

Animation basics

Learn how to animate your drawings.

Ant bot

Build an ant colony with simple behaviours

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.

Applying linear equations

Did you think that we were playing with equations just for fun? Nope. They are actually useful for solving real problems.
Common Core Standards: 7.EE.B.4, 7.EE.B.4a

Applying mathematical reasoning

This is a big one. But look you're over half way through this brush up on Arithmetic! This section is all about understanding how to work with data (like balancing a checkbook and reading an article).

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.

Area

Let's now extend our understanding of area to triangles and more interesting quadrilaterals!
Common Core Standard: 6.G.A.1

Area and circumference of circles

You already know about area and perimeter of lots of shapes. Now we'll round out those concepts by applying them to circles. Mathematicians call the distance around a circle its circumference and the space inside a circle its area.
In this tutorial, we'll learn that there's another type of pi in the math world, and it's even more awesome than apple pie. We'll use pi to find the circumference and area of any circle in the world, no matter how big or how small!
Common Core Standards: 7.G.B.4

Area and perimeter

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 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, volume, and surface area

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 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.

Arrays

Store multiple values in your variables with arrays!

Art History Introduction

Art History: Ancient Egypt

Woody Allen famously said, “I don't want to achieve immortality through my work. I want to achieve it through not dying.” The ancient Egyptians on the other hand, confronted death head-on. In fact, the art of the ancient Egyptians was (for the most part) never meant to be seen by the eyes of the living—it was meant to benefit the dead in the afterlife. Throughout human history, art has been recognized for its ability to outlive us and has been used as a receptacle for our fears and hopes about our own mortality.

Art History: Ancient Greece

The ancient Greeks kept busy. They produced painting and sculpture that was copied by the ancient Romans, by Renaissance and Baroque masters, and by the royal academies up until the 19th century. We still copy ancient Greek architecture, refer to their philosophy, use their geometry, perform their theatre, hold olympic games, and redefine their democracy.

Art History: Ancient Near East

Was writing invented to record poetry or great literature? Nope. Writing was invented to help keep track of beer and other goods and services! The ancient Sumerians were nothing if not practical. Ancient cultures established the first cities and large scale architecture. From the Gates of the City of Ancient Babylon, to the ancient code of laws instituted by King Hammurabi, the Ancient Near East is strangely distant but also remarkably familiar.

Art History: Ancient Rome

The Romans weren’t very original (they borrowed the Greek’s and Egyptian’s gods, architecture, etc.), but they sure knew the political value of art and they were brilliant engineers and administrators. This tutorial traces Roman art and architecture from the Republic through the collapse of the Empire and the rise Christianity. Fly over a reconstruction of the ancient city of Rome. Understand how the Colosseum was built to appease a population angry at the excesses of the former Emperor Nero, and uncover the secret initiation rites buried by the ash of Mount Vesuvius.

Art History: Architecture

Towers of glass and steel from the mid-20th Century suggest, for many people, the rationalization of urban space that dehumanized our cities with empty plazas, rigorous geometries and uniformity. But International style architecture was born of the utopian idea that innovative design could improve the lives millions and its forms recall the clarity and harmony of ancient Greek architecture. This tutorial treats the late work of Ludwig Mies van der Rohe, Gordon Bunshaft, and Frank Lloyd Wright.

Art History: Art of the Americas

If you are an art historian with expertise in this area and would like to contribute content, please contact Beth or Steven.

Art History: Buddhist Art

Buddhism began when Siddhārtha renounced his princely life and succeeded in attaining enlightenment. As the philosophy and practice of Buddhism traveled, it was shaped and reshaped first across Northern India, Pakistan and Afghanistan and later in China, Japan and South East Asia. Learn how we trace the history of Buddhism through the representation of the historical Buddha and the many other buddhas that are still widely venerated today.

Art History: Byzantine

Today, we know the city of Constantinople as Istanbul (in fact there’s a song about that!). But even before it was Constantinople, it was the ancient city of Byzantium, and it was renamed Constantinople (city of Constantine) by none other than the Emperor Constantine himself (it’s good to be the emperor!). From there a succession of emperors ruled the Byzantine empire as the Roman empire dissolved. For two centuries, the Byzantine empire even included the Italian city of Ravenna, where many churches decorated with astoundingly beautiful mosaics can still be found. In Byzantine art we see a departure from the naturalism of the ancient Greek and Roman world. Figures float in ethereal gold heavenly spaces in mosaics, and we find intricate carvings made from ivory, a luxurious material imported from Africa.

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: Early Christian

The first Christians were often persecuted in the Ancient Roman empire, but when—thanks to Constantine—Christianity became legal in the 4th century, Christians could worship openly and the first churches were built in Rome and Christians could be buried in tombs sculpted with Christian imagery. This tutorial takes you on a tour of early Christian churches, and takes a close look at the sarcophagus of a Roman senator who died in 359.

Art History: Early Medieval Art

Christian art, which was initially influenced by the illusionary quality of classical art, started to move away from naturalistic representation and instead pushed toward abstraction. Artists began to abandon classical artistic conventions like shading, modeling and perspective—conventions that make the image appear more real. They no longer observed details in nature to record them in paint, bronze, marble, or mosaic.

Art History: English Portraiture

Britain and America in the Age of Revolutions (Reynolds, Copley, Peale)
It was hard to be an artist in America during the colonial period, and for decades after too. There were no real art schools, no grand tradition of art, and no wealthy aristocratic patrons to commission heroic subjects. Americans were practical, and they wanted portraits—and not paintings of classical mythology (which didn’t always make American artists, at least those with wider ambitions, happy). As you’ll learn in this tutorial, Copley was the greatest American portrait painter of the period, and Peale, who studied with Copley, painted portraits of American heroes such as George Washington, Thomas Jefferson and Benjamin Franklin, and founded what became the first real American art museum. American artists looked to England for support and inspiration, often to the older Sir Joshua Reynolds, who was the painter to the King, and first President of the Royal Academy in London.

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: Florence

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." In other words, for Vasari, Giotto was the first artist to leave behind the medieval practice of painting what one knows and believes, for the practice of painting what one sees. This tutorial looks at painting and sculpture in both Pisa and Florence to highlight some of the most forward-thinking art of this century.

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: Gothic

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!

Art History: High Renaissance

We don’t call this the “High” Renaissance for nothing! This period sees hugely ambitious projects, from Michelangelo’s painting of the Sistine Chapel ceiling and the rebuilding of St. Peter’s in Rome, to Raphael’s frescoes in the papal palace. We use this term to refer to the art of the Italian Renaissance, beginning with Leonardo, whose great masterpiece the Last Supper, actually dates to the last decades of the 15th century (history is never neat!). In the painting and sculpture of this period, ideally beautiful figures who move gracefully, often in complex, multi-figure compositions conveying the sense that human beings are an echo of the perfection of God.

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: Islamic Art

More than 1 billion people call themselves muslim and this monotheistic religion is now estimated to be the second largest in the world. Islamic culture was among the most advanced and tolerant during the Medieval era. Cities from Isfahan in the East to Grenada in the West became important centers of art and learning. This tutorial looks at the sculpture, tilework, costume and interior spaces of this brilliant culture.

Art History: Italy

Baroque painting, sculpture and architecture in Italy seems to miraculously unite the heavenly and the earthly to deepen the faith of believers. We’ll look at the utterly convincing illusions of heaven on the painted ceilings of Il Gesu and St. Ignazio, and the way Caravaggio painted biblical scenes with a gritty realism that makes them look as though they are taking place on the streets of Rome. And of course we explore the great genius of the period, Bernini, who used every means at his disposal—painting, sculpture and architecture—in works like The Ecstasy of St. Theresa and the Cathedra Petri to bring the viewer closer to the divine.

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: Mannerism

You could say that High Renaissance painters had achieved it all—ideally beautiful, graceful figures, rational spaces, and unified compositions with dozens of figures. If you were a young artist in the early decades of the 1500s you might have felt that there was nothing left to accomplish! Renaissance art was always based on the visual world—on representing things as we see them, but Mannerist art was more artificial, it looked to other art rather than to nature, and Mannerist artists purposely looked for complexity and difficulty to showcase their skills. Figures are elongated, the illusion of space that was so important for the Renaissance no longer makes sense, and the human body is often impossibly twisted.

Art History: Media

Watching a soccer game is a lot more interesting if you’ve actually played the game yourself. Similarly, art makes a lot more sense if you understand how it was made. Different materials can have a profound effect on how a work of art looks and how we respond to it. Learn about how artist’s made their own supplies before there were art supply stores! Understand the differences between oil and tempera and how marble is quarried and carved.

Art History: Neo-Classicism

Jacques Louis David, an active supporter of the Revolution of 1789, is the star of this tutorial. David served in the revolutionary government, used his art in the service of its cause—and voted to behead King Louis XVI. He captured the patriotism of the revolution’s early phase and later, memorialized its dead heroes. And when the revolution failed, and Napoleon came to power, David used his great talents to present a heroic image of that military general-turned emperor. David invents a new style for the democratic values of the Enlightenment—one that is the very opposite of the luxuriousness of the Rococo—and that looks back to Renaissance and ancient Greek art, hence the name—Neo-Classicism (new classicism).

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: Northern Renaissance

The Renaissance in the North continues, but now with the impact of the Protestant Reformation, where there was growing concern that images in the church violated the commandment against making likenesses, as part of the prohibition against worshipping idols. The Reformation had a direct impact on some of the greatest painters of this period, including the German artists Durer, who converted, and Cranach, who was a close friend of Luther. There was increasing exchange during this period between artists in Italy and those in Northern Europe in terms of both methods and style, though the two styles remain distinct. Here we see some of the most complex painting in the work of Holbein, some of the most playful in the work of Bruegel and some of the most terrifying in the work of Bosch.

Art History: Pop Art

When people walk into an art museum they often expect to see treasures of their cultural history—beautifully crafted precious objects that express profound truths—images of God, nature, man’s heroism, but Campbell soup cans hawked on TV? Pop art sought to upend our comfortable understanding of what art is and it did just that. Warhol, Lichtenstein, Oldenburg, and others confronted the visual reality of our commercial consumer culture by focusing on the mechanics of representation and the subject matter of daily life in the middle of the 20th Century.

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: Postmodernism

If you’ve done the tutorials on Nineteenth and Twentieth Century art then you likely have an idea that by “Modernism,” art historians are referring to a set of ideas that characterize art and culture after about 1848. Key to understanding Modern art are the ideas of a heroic “avant-garde” that challenges authority and the expression of the individual (one that is invariably white and male). The term “Post-Modernism,” was initially used in the art world in 1979 for architecture that arbitrarily borrowed historical styles with little regard for original meaning or context. The term was quickly and broadly adopted, and came to refer to a strategy to undermine Modernism’s utopian and heroizing tendencies by using multiple yet simultaneous critical perspectives. Post-Modernism was not anti-Modernism, it was instead, an effort to destabilize Modernist narratives with deeply skeptical critical strategies that emphasized the plurality of gender, race, nationality, politics, and economic inequality.

Art History: Pre-Raphaelites

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.

Art History: Prehistoric

You’ve seen the drawing of human evolution showing a procession of monkey, ape, primitive and modern man? Well, somewhere along that line early man could be shown holding a paint brush and a chisel. Mankind has been making art for at least 100,000 years. Why was the earliest art made? What might it have once meant? This tutorial focuses on what we do and do not know about one of the earliest representations of the human body, The Venus of Willendorf.

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

It’s hard not to like Rococo art. After all, it’s subjects are often about luxury and pleasure, which makes sense since its patrons were the extremely wealthy French aristocracy. This tutorial features two romantic liaisons—Fragonard’s The Swing and The Meeting, portraits and a mythological subject, “Venus Consoling Love.” You get the idea.

Art History: Romanesque

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 little economic growth. Charlemagne and the Ottonians had partially and briefly unified the West, and of course the Church was a stabilizing institution, but it was only in the 11th Century that everything changed. Now there was finally enough peace and prosperity to allow for travel and for the widespread construction of large buildings. 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. The relative calm of the Romanesque period also meant it was possible to travel, and 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.

Art History: Romanticism in England

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.

Art History: 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!

Art History: 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.

Art History: 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.”

Art History: Romanticism in the United States

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, as well as The Dance, a sculpture that adorned its façade.

Art History: Siena

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 an 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.

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

The 1880s saw a shift away from the modern-life focus of Impressionism, as artists turned toward the interior self, to dreams, and myth. There was a sense that Impressionism had been too tied up with the materialism of middle-class culture. In some ways, van Gogh and Gauguin can also be seen as Symbolists. Many Symbolist belonged to groups of artists who broke away (or seceded) from the art establishment in their respective countries, to hold their own exhibitions. For example, Klimt belonged to the Vienna Secession (he was its first president), Khnopff to a similar group in Belgium called Lex XX (The Twenty), and Stuck co-founded the Munich Secession.

Art History: The Basics

If you’ve ever found yourself saying “How can that be art?,” then this tutorial is for you! We’ve tried to answer the real questions that come up during a typical visit to a modern art museum. Learn how describing can help you slow down and better understand what a work of art is communicating. Learn how important historical context is and how meaning can change over time. This tutorial asks: Why is art important? How can a snow shovel be art? How can we find meaning in abstract art?

Art History: Tyrol

Currently this tutorial contains the work of only one artist, Michael Pacher. It would have been so much easier if we could have included him in Venice, since Pacher was influenced by the work of the great Venetian painter Mantegna. But Pacher spoke German, and the specific area he was from (today in the north-east of Italy), was known as Tyrol (though there is a state of Tyrol today in Austria). Confusing, we know. In this tutorial, we take a look at Pacher’s amazing St. Wolfgang Altarpiece which is still in the church it was made for—a church on a lake surrounded by mountains in Austria, still visited by religious pilgrims. Most altarpieces in the Renaissance were made of many interconnected panels that were later sold and ended up in different collections, and this means that to see one altarpiece you usually have to travel to many museums. But this is not true of the St. Wolfgang altarpiece. Seeing a Renaissance work of art in the space it was made for helps us to travel back in time to the late 15th Century.

Art History: Venice

Renaissance Venice was a city of merchants that traded with the Byzantine and Islamic Empires to the East, with the Germanic nations North of the Alps, and with Kingdoms to the West. In the 15th Century, Venice was at the height of its power with colonies and a well equipped navy to protect its merchant fleet abroad. Its fabulous wealth financed the construction of sumptuous churches and palaces—and art to fill them. The city’s salty, humid air meant that frescos faded quickly and so in the late 15th century, artists adopted oil, a medium they had seen used on panels from the north. By the late 15th century, they had also adopted canvas as their support of choice and artists like Giovanni Bellini created some of the most subtle and engaging art ever made.

Art conservation

Go behind-the-scenes and learn about art conservation in the lab and in the field.

Art history: Burgundy

No, not the color and not the wine! In the 15th century, the duchy of Burgundy was one of the most powerful regions in Europe, and stretched from what is today central France up to what is today Belgium and Holland. The home of the Dukes of Burgundy was Dijon, and Duke Philip the Bold commissioned some of the century's greatest works there, including a Carthusian monastery just outside the city walls, where he hoped to be buried so the monks could pray for his soul for eternity. He hired some of the most brilliant artists in Europe to work for him there, including Claus Sluter. Very little of the monastery survives today, but thankfully Sluter’s great work, The Well of Moses, can still be seen there.

Art history: Dada

Do we know who we really are? What parts of our mind do we know and what parts are hidden from us? Should art only focus on the rational, the conscious, or should we also pay attention to the irrational, the uncanny, the powerful impulses that remain unarticulated and just beyond the reach of our awareness. Dada was born during WWI when poets, artists, and actors, sickened by the violence around them, chose to celebrate the irrational. They created an anti-art that challenged the cultural assumptions that they felt supported the ruling elite that had, in turn, caused the war. In the years after the war, Dada gave way to Surrealism which reinstituted traditional forms of art-making but focused on Freud’s theories of the unconscious.

Art history: Expressionism

Wild Beasts! Les Fauve (wild beasts) is what one critic called the brilliant expressive canvases of Matisse and other artists who exhibited together in 1905. This tutorial traces the work of Henri Matisse from his early Fauvist work with its jarringly bright colors to the stricter geometries he introduced during the First World War. It also tracks Expressionist developments in Germany and Austria with videos on Kirchner, Kandinsky and Jawlensky, artists who adopted a rough, “primitive” style, and on Egon Schiele’s taut, sexually charged paintings from Vienna.

Art history: Minimalism

“Primary Structures,” “ABC Art,” and “Minimalism” are terms that attempted to categorize the work of Donald Judd, Dan Flavin, Sol Lewitt and other artists who produced hard edged, often geometric and seemingly machined objects in the later 1960s and early 1970s. These stark, often cold and cerebral forms were the very antithesis of the deeply emotive gestural art of the Abstract Expressionists and their followers who had dominated the New York art scene since the 1940s. Here was an art that renounced the authentic “hand’ of the artist and sought instead to create forms without reference to the world beyond the object’s own logic except perhaps as Platonic expressions of a pure ideal.

Art history: Performance art

Can art be an enactment? Can the “art” be relocated from the object crafted by an artist to the more ephemeral reaction of the viewer? Must an artist actually “make” a work of art at all? Conceptual artists recognized that when the 19th Century avant-garde broke with the academies and their emphasis on technical execution (the blending color or compositional clarity for example), they were freer to focus on more conceptual issues such as modern urban life, subjectivity, or pictorial language. Fast forward to the second half of the Twentieth Century when intellectual content became the defining characteristic of art and concept fully eclipsed craft. In the 1960 and 70s the art of Beuys, Haacke and others became increasingly conceptual. These artists used found objects, performance, and installation while de-emphasizing the act of “object making.”

Art history: Process art

By the second half of the 20th Century, the avant-garde had an avant-garde of its own. Even as Pop, Minimal and Concept artists renounced the handmade work of art, a small group of women recognized the subversive value of handicraft in an age of industrial manufacturing and in an art world dominated by male artists and critics who sought theoretical purity. During her short career, Eva Hesse resituated the ancient question of how to meaningfully represent the human body and unleashed decades of experimentation that continues to this day. Judy Chicago, Linda Benglis, Jackie Windsor, Faith Ringgold, and others used the lowly status of craft and its historical association with female artisans to contrast with “high art.” By focusing of the act of making, on process and craft, these women began the process of fracturing Modernism’s reductive and largely male narratives.

Art history: The postwar figure

Artists first represented the human body more than 30,000 years ago and haven’t stopped since. Figurative art has been a continuous tradition through human history. Even in societies where the biblical law forbidding the graven image is most strictly interpreted (Judaism and Islam for example), there have always been instances of figural art. The same is true for the era of modernist abstraction when artists found new ways to portray the body on canvas or with the lens of a camera that could profoundly describe the human condition in , and abstraction the post-holocaust era.

Art of the Islamic World

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.

Artist interviews

Hear from contemporary artists as they share their art-making techniques and sources of inspiration.

Artist's films

Asthma

Learn how asthma causes breathing difficulties in adults and children

Asymptotes and graphing rational functions

Attention and language

6B: 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. By Carole Yue.

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

Even when a function is nonlinear, we can calculate the average rate of change over an interval (we'll need calculus to calculate the rate of change at a particular value of the independent variable). This tutorial will give you practice doing just that.

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.

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 charts

Bar charts are everywhere. They make it easy for us to compare data and see trends.
Common Core Standard: 6.SP.B.4

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

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 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 trigonometry

Even more fun than Geometry...let's play with triangles! A review and expansion on understanding and calculating missing information in triangles.

Batteries

Basic observations leading to homemade batteries

Battery basics

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.

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.

Behavior and genetics

7A: Learn about the way our genes and experiences shape the way we respond to our environment. By Ryan Patton.

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.

Benjamin Franklin

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 classrooms)

Best practices (higher education math)

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.

Big History staging

Big Questions

Why is that art? Can art be an idea? Does art have to represent the world we see?

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.

Big picture talks and media

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 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 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.

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!

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.

Bitcoin

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.

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.

Bond-line structures

Call them Bonds. Covalent Bonds. Smart chemists need time to stir (and shake) their solutions. In this tutorial, Jay explains 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.

Bonds

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.

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.
Common Core Standard: 6.SP.B.4

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!

Breastfeeding

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)!!!!!!!

Brush up plan

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.

CAHSEE Example Problems

Sal working through the 53 problems from the practice test available at http://www.cde.ca.gov/ta/tg/hs/documents/mathpractest.pdf 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.

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!

Capacitors

4C: Learn how a capacitor works, and the mathematical formulas that explain their behavior. By David Santo Pietro.

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.

Capstone exercises

Carbohydrate Metabolism

1D: What are the key metabolic pathways that are intimately involved in breaking down or building up glucose? Learn about glycolysis, gluconeogenesis, and the pentose phosphate pathway and how they fit into cellular respiration as a whole. By Jasmine Rana.

Carboxylic acid derivatives

5D: The nomenclature, properties, and reactions of carboxylic acid derivatives

Carboxylic acids

5D: This tutorial will cover the important nomenclature, properties, and reactions of carboxylic acids

Carolingian

Charlemagne, King of the Franks and later Holy Roman Emperor, instigated a cultural revival known as the Carolingian Renaissance.

Case studies (higher education math)

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

Cell division

Cell membrane overview

2A: Learn about the basics behind the cell membrane. What makes up the cell membrane and how does it work? How does our cell membrane help sustain life? By William Tsai.

Cellular respiration

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 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.

Cervical Spine

Your cervical spine is the uppermost part of your spine, the part that makes up your neck. Take a look at the 7 vertebra that compose the cervical spine, and see different views of a real person’s spine! Join Sal and Dr. Mahadevan as they inspect these X-rays and discuss its alignment and protection in airway management.

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

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!

Chemical reactions (stoichiometry)

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!

China

Many products and technologies that were first developed in China—silk, porcelain, gunpowder, tea, paper, and woodblock printing—were much sought after by cultures far beyond its borders. In exchange the Chinese sought exotic goods, horses, and jade, as well as access to the sources of Buddhism.

Chinese art from the Yuan dynasty

Art from the Yuan dynasty

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.

Chirality and absolute configuration

Mirror, mirror on the wall . . . who is the fairest stereoisomer of all? In this tutorial, Jay explains chirality and how to determine the absolute configuration at a chirality center.

Chirality and the R,S system

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.
In this tutorial, Sal explores molecules 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).

Ciphers

Learn about algorithms for performing encryption & decryption. Then practice making and breaking codes!

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.

Circles

What do you get when you take the sun and divide its circumference by its diameter?…Pi in the sky! How to play with circles using pi!

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.

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 function videos

These oldie-but-maybe-goodies are the original function videos that Sal made years ago for his cousins. Despite the messy handwriting, some people claim that they like these better than the new ones (they claim that there is a certain charm to them). We'll let you decide.

Classifying shapes

In this tutorial, we'll classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. We'll also learn about special triangles called "right triangles".
Common Core Standard: 4.G.A.2

Coaching in a K-12 environment

Explore recommended practices from the many educators who are using Khan Academy

Cognition

Explore cognitive development and intelligence, as well as how our minds solve problems, make decisions, and represent knowledge.

Coin detector

Get to know your rotation sensor while building a 5 cent machine

Collateralized debt obligations

College Brush Up how to

Colon Disease

The colon, otherwise known as the “large intestine,” is a tube that’s about 5 feet long (1.5 meters). This is where the majority of fluid reabsorption occurs in your GI tract (the tract extending from your mouth to your anal sphincter). The colon is susceptible to multiple diseases, including hyperplasia, dysplasia, and cancer. Learn more about healthy colon tissue and these three diseases in the following videos. Join Sal and Dr. Andy Connolly as they (and you) take a microscopic look at colon tissue!

Colonial Americas

Art and culture from the European invasion of the Americas to the end of the colonial era.

Color in organic molecules

In this tutorial, Jay explains basic color theory and shows how conjugation determines the color of organic molecules. Basic knowledge of MO theory is assumed.

Colored shadows

Have fun with colored light bulbs and experiment with different properties of light! Developed and demonstrated by Exploratorium Senior Scientist Paul Doherty.

Coloring

We'll show you how to color and outline your shapes!

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 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. For example, imaging trying to survey all 300+ million Americans to understand the likely outcome of the next presidential election! Because of this, much of statistics is making estimates about a population based on a random sample. This tutorial introduces you to the idea of what makes a good random sample. I'm sure all of you would enjoy it (but I haven't had the time to ask all of you).

Comparing decimals

Let's test our understanding of decimals by comparing them to one another!
Common Core Standard: 5.NBT.A.3b

Comparing fractions

In this tutorial, we'll practice understanding what quantities fractions actually represent and comparing those to each other.
Common Core Standard 4.NF.A.2

Comparing numbers through 10

Comparing with multiplication

In this tutorial, we look at multiplication and division through the lens of comparison. For example, say that are 9 and 3 times older than your cousin. How old would your cousin be?
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

Composing 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

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).
Common Core Standards: 7.SP.C.8, 7.SP.C.8a, 7.SP.C.8b

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 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!

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!

Conceptualizing decimals and place value

In this tutorial, we'll really think about what the different digits represent in a decimal (and how they relate to each other).
Common Core Standards: 5.NBT.A.1 , 5.NBT.A.3, 5.NBT.A.3a

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.

Conformations

In this tutorial, Sal draws Newman projections and also explains chair and boat conformations for cyclohexane.

Conformations of alkanes and cycloalkanes

In this tutorial, Jay shows the different conformations of straight chain alkanes and cyclohexane.

Congruence

Review how to demonstrate if two angles are identical or different.

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.

Conic sections

Review of how to identify and graph circles, ellipses, parabolas, and hyperbolas.

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.

Conservation

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.

Constructing a line tangent to a circle

Constructing and slicing geometric shapes

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 proportions

Now that we know what a proportional relationship is, let's construct some to solve real problems!
Common Core Standards: 7.RP.A.2, 7.RP.A.3

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).

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.

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...

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 fractions to decimals

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!
Common Core Standards: 7.NS.A.2d

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

How can we communicate exactly where something is in two dimensions? In this tutorial, we cover the basics of the coordinate plane.
Common Core Standard: 5.G.A.1, 5.G.A.2

Coordinate planes

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.

Counting

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 objects

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 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.

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 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.

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!

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

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.

Currency

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).

DNA

1B: Learn about gene expression and how information flows between DNA and protein. By Tracy Kim Kovach.

DVD Player

DVD Player

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

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?

Data

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.

Davis Guggenheim - Filmmaker

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.

Decimals

This set of exercises works on different types of real world examples of working with decimals. 100% of pure math fun!

Decimals and fractions

Decimals and fractions are two different ways of representing the same number. In this tutorial, we'll explore converting between the two and thinking about what exactly decimals represent.
Common Core Standards: 4.NF.C.5, 4.NF.C.6, 4.NF.C.7

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!

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.

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 been viewing integrals as anti-derivatives. Now we explore them as the area under a curve between two boundaries (we will now construct definite integrals by defining the boundaries). This is the real meat of integral calculus!

Deflation

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.

Dependent and independent variables

In some relationships, one variable (the independent one) is thought to drive the behavior of the other one (the dependent one). This tutorial explores this idea further.
Common Core Standards: 6.EE.C.9

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.

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!

Describing ratios

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!
Common Core Standards: 6.RP.A.1, 6.RP.A.2

Diastereomers and meso compounds

In this tutorial, Sal and Jay define stereoisomers, diastereomers, and meso compounds.

Diels-alder reaction

In this tutorial, Jay shows the mechanism, stereochemistry, and regiochemistry for the classic Diels-Alder reaction.

Differential calculus

Digital Camera

Digital Camera

Dilations or scaling around a point

We understand the idea of scaling/dilation from everyday life (hey, let's make it bigger or smaller keeping the same proportions!). In this tutorial, you'll understand this type of transformation in a much, much deeper way.

Dilution

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.

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.

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.

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?

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.

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

You've already seen the distributive property in action multiple times. Now we'll use it again!

Divergence

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!
Common Core Standard 5.NBT.B.7

Dividing 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!
Common Core Standards: 5.NF.B.3, 5.NF.B.7, 5.NF.B.7a, 5.NF.B.7b, 5.NF.B.7c

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 polynomials

You know what polynomials are. You know how to add, subtract, and multiply them. Unless you are completely incurious, you must be wondering how to divide them!
In this tutorial we'll explore how we divide polynomials--both through algebraic long division and synthetic division. (We like classic algebraic long division more since you can actually understand what you're doing.)

Dividing whole numbers and fractions

Let's explore how we can think about dividing a fraction by a whole number and a whole number by a fraction. This is more useful than you might think!

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.

Division

After this tutorial, you know how to divide any positive number by any other. Pretty exciting.. eh?!
Common Core Standards: 6.NS.B.2, 6.NS.B.3

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 structures

5B: Learn how to draw Lewis dot structures, assign formal charges, draw resonance structures, and analyze the geometry of molecules and ions using VSEPR. By Jay.

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.

Drawings

Learn about drawings in the Getty Museum's collection.

E1 and E2 reactions

In this tutorial, Jay covers the E1 elimination mechanism, carbocation rearrangements, and the details of the E2 elimination reaction.

Early Humans

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!

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.

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.

Eigen-everything

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 rotational motion?

Electricity and magnetism

Electromagnet

Discoveries leading to the Right Hand rule

Electronegativity

What is the most important concept to understand in undergraduate organic chemistry? Of course, it's electronegativity! In this tutorial, Jay explains the concept of electronegativity and shows how it applies to polarity, intermolecular forces, and physical properties.

Electrophilic Aromatic Substitution

In this tutorial, Jay shows several electrophilic aromatic substitution reactions.

Elementary content

Ellipses

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.

Elon Musk - CEO of Tesla Motors and SpaceX

Emotion

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.

Endocrine system

3A: Dive into the endocrine system! See how the body uses special organs (called glands) that secrete chemical messages (called hormones) in order to properly respond to it's changing environment. By Ryan Patton.

Endocrinology and Diabetes

In this section, we’ll revisit the endocrine system. After a review, we’ll explore how our hormones can cause different kinds of symptoms and behaviors, including normal childhood growth and precocious puberty (puberty kicking in at an earlier age than normal). After that, we’ll take a closer look at diabetes, which is a growing endemic in the world as we see a greater availability of cheap, low quality foods. This will include a focus on glucose concentration and other blood sugar levels, and what your body (and modern medicine) can do to maintain a healthy balance in your body.

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

Enzyme kinetics

1A: Come learn about enzyme kinetics. How can we look at enzyme activity from a mathematical point of view? By Ross Firestone.

Enzyme structure and function

1A: Learn the basics of enzyme structure and function. Enzymes are able to catalyze lots of different biochemical reactions, and make them occur at much faster rates. By Ross Firestone.

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.

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.

Equations 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!
Common Core Standard: 6.EE.B.7

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!

Equivalent 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!
Common Core Standards: 6.EE.A.3, 6.EE.A.4

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!

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 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

Estimating probability

If you know all of the possible outcomes of a trial (and the associated probabilities of each of them), you can find the exact probability. In many situations, however, we don't know this and instead, we estimate the probability based on history of events. That's what we're going to do in this tutorial.

Etruscan art

Before the small village of Rome became “Rome” with a capital R (to paraphrase D.H. Lawrence), a brilliant civilization once controlled almost the entire peninsula we now call Italy. This was the Etruscan civilization, a vanished culture whose achievements set the stage not only for the development of ancient Roman art and culture but for the Italian Renaissance as well.

Evaluating 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.
Common Core Standards: 6.EE.A.1, 6.EE.A.2b, 6.EE.A.2c

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 function expressions

This is a super fun tutorial where we'll evaluate expressions that involve functions. We'll add, subtract, multiply and divide them. We'll also do composite functions which involves taking the output of one function to be the input of another one!
As always, pause the video and try the problem before Sal does!

Evolution and natural selection

Evolution and population dynamics

1C: How do organisms evolve from one form to another? Learn about all of the different driving forces of evolution and how there might be a few more than just natural selection. How can populations evolve and adapt to changes in their environment? Do groups of organisms behave differently than individuals? By Ross Firestone.

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!

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!

Exploring rigid transformations

Let's use some pretty cool software widgets to see how we can transform a shape through translations, rotations, dilations and reflections.

Exponent expressions and equations

More practice solving equations, but now they've got exponents & radicals. Radical!!

Exponent properties

Tired of hairy exponent expressions? Feel compelled to clean them up? Well, this tutorial might just give you the tools you need.
If you know a bit about exponents, you'll learn a ton more in this tutorial as you learn about the rules for simplifying exponents.

Exponent properties examples with variables

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 and logarithmic functions

An exciting look at how to use logarithmic functions, their properties, and graphs. Logarithms are used everywhere, from measuring sound to earthquakes.

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.

Exponents

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

Exponents can be viewed as repeated multiplication. Now that we know how to multiply negative numbers, we can multiply them repeatedly with exponents!

Exponents, radicals and scientific notation

Practice these exercises to brush up on what getting squared is all about.

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 by grouping

Factoring by grouping is probably the one thing that most people never really learn well. Your fate doesn't have to be the same. In this tutorial, you'll see how factoring by grouping can be used to factor a quadratic expression where the coefficient on the x^2 term is something other than 1?

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 quadratics in two variables

We'll now extend the application of our quadratic-factoring toolkit, by factoring expressions with two variables. As we'll see, this is really just an extension of what you probably already know (or at least will know after working through this tutorial). Onward!

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! Eye of the tiger!
Common Core Standard: 4.OA.B.4

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!

Farenheit and Celsius unit conversion

There are three major conventions for measuring temperature in the world, Farenheit, Celsius and Kelvin. If converting between the three gives you cold feet, then this tutorial might warm them up.

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!

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.

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!

Fischer projections

In this tutorial, Jay shows how to draw a fischer projection and how to assign an absolute configuration to a chirality center in a fischer projection.

Fluids

Fluids at rest

4B: Learn the physics behind fluids at rest. Some of the ideas explored are density, pressure, Pascal's principle, Archimedes' principle, buoyant forces, and specific gravity. By Sal.

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 next 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: Discover the force of tension and how we can use it to solve very interesting problems! By Sal.

Forces on inclined planes

4A: Learn about the forces that an object experiences on a ramp or inclined plane. Examples include inclines with and without friction. By Sal.

Formal charge and resonance

Positive and negative charges are everywhere in orgo! In this tutorial, Jay shows you how to assign formal charges to molecules and how to draw resonance structures.

Formal understanding of congruency

We begin to seriously channel Euclid in this tutorial to really, really (no, really) prove things--in particular, that triangles are congruents. You'll appreciate (and love) what rigorous proofs are. It will sharpen your mind and make you a better friend, relative and citizen (and make you more popular in general). Don't have too much fun.

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 exponents

If you're familiar with taking square roots and cube roots (and other roots), then you're ready to see that we can also express these as exponents in ways that are consistent with the exponent properties you know and love!

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.

Fractions

Just 4 practice sets covering everything you'll need to do with fractions. They may be challenging, but if you can do these then you're ready to divide and conquer!

France

Art in France excluding Burgundy

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 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 expressions

We'll now see what it means to add, subtract, multiply and divide functions!

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!

Function inverses

Functions associate a set of inputs with a set of outputs (in fancy language, they "map" one set to another). But can we go the other way around? Are there functions that can start with the outputs as inputs and produce the original inputs as outputs? Yes, there are! They are called function inverses!
This tutorial works through a bunch of examples to get you familiar with the world of function inverses.

Function notation

f(x), g(x), etc. What does this mean? Well they are ways of referring to "functions of x". This is an idea that will show up throughout more advanced mathematics and computer science so it is a good idea to understand them now!

Functional groups

In this tutorial, Jay puts the "fun" back into recognizing functional groups.

Functions

Make your code more re-usable by grouping it into functions, and then make those functions accept parameters and return values.

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 and their graphs

Here we will revisit some of the most important pieces to understanding functions!

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.

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)

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.

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.

Gene control

1B: Regulation of genes occurs at every step of the pathway for gene expression. Gene control is critical to the health of the cell, especially with regards to oncogenes and tumor suppressor genes. By Tracy Kim Kovach.

Geometric models

Let's use what we know about geometry to answer some really, really interesting questions.

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

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!

Get to know our site (K-12 math classrooms)

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.

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.

Glassmaking: An Introduction to its History and Technique

Glass is magical stuff. When you look at a glass object, you might never guess that it was once a hot, flowing liquid that could be inflated or molded or swirled into almost any shape.

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.

Glossary

Review big history vocabulary and take the glossary challenge.

Gradient

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.

Graphing and analyzing linear equations

Use the power of algebra to understand and interpret points and lines (something we typically do in geometry). Pretty cool!

Graphing and analyzing proportional relationships

In proportional relationships, the ratio between one variable and the other is always constant. In the context of rate problems, this constant ratio can also be considered a rate of change. This tutorial allows you dig deeper into this idea.
Common Core Standard: 8.EE.B.5

Graphing functions

You've already graphed functions when you graphed lines and curves in other topics so this really isn't anything new. Now we'll do a few more examples in this tutorial, but we'll use the function notation to make things a bit more explicit.

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 rational functions

Rational functions are often not defined at certain points and have very interesting behavior has the input variable becomes very large in magnitude. This tutorial explores how to graph these functions, paying attention to these special features. We'll talk a lot about vertical and horizontal asymptotes.

Graphing solutions to 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.

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. I know. 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).

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

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.

Hardware, software and facilities decisions overview

Harmonic motion

Every 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.

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.

Healthy Lifestyle

If you looked at our “Circulatory system diseases” section, you already know that diseases related to an unhealthy lifestyle are on a critical rise. We hope that the following set of videos will allow you to develop a healthier lifestyle, and help you improve the lives of others as well. This is important for parents, children, students, and anyone who wants to take better care of their body. Learn some of the fundamentals behind staying healthy: Reducing your salt, keeping your weight in a healthy range, and exercising regularly.

Hear from leaders in blended learning

Silicon Schools Fund and the Clayton Christensen Institute provide insight and guidance on delivering high-quality blended learning.

Heart Disease and Stroke

Heart disease is the number one cause of death globally (~30% of all deaths!). It affects different parts of the world unequally, with the majority of deaths occurring in low-income or middle-income countries. Find out what heart disease and stroke are, what can cause them, and how the two are related!

Heart 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”.

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!

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.

Hematologic system

It takes between 30 seconds to a minute for your blood to travel from your heart, to your body, and back to the heart again - perhaps a bit longer if the trip is out to your big toe! Our blood is incredibly important for transporting oxygen throughout the body. Hemoglobin, the protein that fills our blood cells, has wonderful mechanisms to allow it to bind to both oxygen and carbon dioxide. This is important for effective and quick transport of the gases around our body. Our blood is about 45% cells and 55% plasma, so the old adage “blood is thicker than water” quite literally holds true in scientific terms! Learn more about how this amazing system works in the following videos.

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!

Hexaflexagons

Since it's shaped like a hexagon and flex rhymes with hex, hexaflexagon it is!

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 content

Hinduism

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.

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.

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.

Homeless stats exercises

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)).

Hot off the press

Hour of Code (for Teachers)

Hour of Code™

Household

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 to use this Brush Up with your Learning Dashboard

How to use this Brush Up with your Learning dashboard

How-to guides

Human Origins

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).

Hundreds

Hybrid orbitals

In this tutorial, Jay goes over sp3, sp2, and sp hybridization.

Hybridization

It's the liger of orbitals and essential to organic chemistry! In this tutorial, Sal discusses the hybridization of orbitals.

Hyperbolas

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.

Hypertension

Nearly one billion people in the world have high blood pressure. That’s 1 in every 7 people! With the amount of unhealthy foods becoming increasingly available to everyone, it makes sense that this number is climbing. This set of videos will explore high blood pressure, also known as hypertension. Learn more about it, what it does to different parts of the body, symptoms of hypertension, and what you can do with your everyday life to manage it!

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 laws

Imaginary and complex numbers

Stop being rational and review how to understand and solve equations with imaginary and complex numbers!

Immunology

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").

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!

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. In other words, we'll be taking the anti-derivative!

Independent events warmup

Introduction to independent events and frequency analysis using histograms.

Induction

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!

Inequalities

Not all expressions are created equal, but we can still use some algebra to compare them in interesting ways. This tutorial will start you on that journey!
Common Core Standard: 6.EE.B.8

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!

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).

Influenza

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!

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.

Integral calculus

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.

Intercepts of linear functions

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.

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

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 linear expressions

Algebra is the language that aliens will use to communicate with us (that or Esperanto). In this tutorial, we'll learn to express and understand this language (Algebra, not Esperanto) a bit better.
Common Core Standard: 7.EE.A.2

Interpreting the Balance Sheet

Interpreting the Income Statement

Intro algebra

Intro to Games & Visualizations

A quick tour of the many components of games and visualizations, demonstrated by some of our favorite programs.

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 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 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 percentages

At least 50% of the math that you end up doing in your real life will involve percentages. We're not really sure about that figure, but it sounds authoritative. Anyway, unless you've watched this tutorial, you're really in no position to argue otherwise.
As you'll see "percent" literally means "per cent" or "per hundred". It's a pseudo-decimally thing that our society likes to use even though decimals or fractions alone would have done the trick. Either way, we're 100% sure you'll find this useful.

Intro to programming

If you've never been here before, check out this introductory video first. Then get coding!

Introduction

Introduction to the Lego NXT environment and what it is capable of. We begin with a few mini projects.

Introduction

“Asia” is a term invented by the Greeks and Romans, and developed by Western geographers to indicate the land mass east of the Ural Mountains and Ural River, together with offshore islands such as Japan and Java. 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: South Asia, the Persian World and West Asia, Southeast Asia, the Himalayas and the Tibetan Buddhist World, China, Korea, and Japan.

Introduction to Cultures and Religions for the study of AP Art History

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 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 very short tutorial gives us the big picture of what economics is all about and, in particular, compares macroeconomics (where you are now) to microeconomics.

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 hormones

You might hear the word “hormone” thrown around a lot. But what exactly is a hormone? And, more importantly, why should you care what a hormone is? Hormones are built from single amino acids, chains of amino acids (peptides), or cholesterol, and they are a form of chemical communication between cells. Sometimes the signal is being sent a long distance, for example from the brain down to the bones in the toe, and other times the signal is being sent to cells that are just a few millimeters away. Either way, hormones allow important conversations to take place within the body.

Introduction to muscles

How do our muscles work? When we decide to kick a ball or shake a leg, how do we get our bodies to do that? Which muscles do we control? Which muscles control us? Learn how our muscles work at the smallest, most cellular level. Then see how nature scales up those microscopic processes into a kick or a dance move. Finally, learn how our brain tells muscle to contract and how that helps us respond to changes in temperature or even a lion chasing us.

Introduction to statistics

This tutorial will get you started in your statistics journey. In particular, we'll think about what types of questions can be answered with statistics and come up with some basic measures of central tendency!
Common Core Standards: 6.SP.A.1, 6.SP.A.2, 6.SP.A.3

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 the Atom

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.

Isosceles and equilateral triangles

This tutorial uses our understanding of congruence postulates to prove some neat properties of isosceles and equilateral triangles.

Japan

Part of a long archipelago off the eastern rim of the Asian continent, the island country of Japan has four main islands: Hokkaido, Honshu, Shikoku, and Kyushu. Numerous smaller islands lie on either end to form a sweeping arc formation that extends northeast to southwest. Another distinctive feature of the Japanese landscape is its volcanic, mountainous terrain. More than two thirds of the land is adorned with low to steep mountains traversed by swift-flowing rivers. These dramatic geographic features have shaped Japan's political history and artistic culture.

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.

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

KIPP

KIPP has charter schools all over the United States. Find out more about how they are using Khan Academy.

Key issues for the study of AP Art History

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 at home

Khan Academy in other learning environments

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.

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!

Korea

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.

Krebs (citric acid) cycle and oxidative phosphorylation

1D: Learn about two of the most vital energy-producing pathways that take place in the mitochondria: The Krebs/TCA cycle and oxidative phosphorylation in the electron transport chain. Learn about the products and reactants of these pathways and how they are regulated. By Jasmine Rana.

Kuna Middle School

Shelby Harris' 7th grade classroom at a public school in Kuna, Idaho, was featured in Davis Guggenheim's documentary TEACH. Shelby faced the challenge of starting to use Khan Academy in the spring for 3 months before state testing. She transformed her teaching as she learned about using technology as a tool, not a replacement for teaching.

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

LaKeskia 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.

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.

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!

LeBron asks

LeBron James asks questions about math and science, and we answer!

Leading change in blended learning

Learning

7C: Learn about learning! How does our environment influence our behavior? What are the consequences of our behavior? Understand the basics of classical and operant conditioning. Appreciate how we learn through observation, and how it relates to the issue of violence in the media. By Jeffrey Walsh.

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! Other than making your life more fulfilling, it will allow you to do incredible things like adding fractions.
Common Core Standard: 6.NS.B.4

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 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.

Limits

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

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 in 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

These exercises review the most fundamental ideas of what equations mean and how to solve them.

Linear equations in one variable

You started first solving equations in sixth and seventh grade. You'll now extend this skill by tackling fancier equations that have variables on both sides.
Common Core Standards: 8.EE.C.7, 8.EE.C.7b

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

Not everything in the world is equal, and that is true for expressions. We can still come to some useful conclusions though!
Common Core Standard: 7.EE.B.4b

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 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

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!

Logarithms

Put simply and confusingly, logarithms are inverse operators to exponents (just as subtraction to addition or division to multiplication). This section will help you evaluate and apply more complex math.

Logic and if Statements

Teach your program to make decisions!

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!

Looping

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.

Lung 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 sing and survive!

Lymphatic system

3B: Welcome to the lymphatic system! Learn about how it is a critical part of the circulatory system. Find out how it comes to the rescue of the cardiovascular system and the immune system. Also discover how it moves fluid in one direction, like blood, but without a heart! By Patrick van Nieuwenhuizen.

Lymphatics

Welcome to the lymphatic system! Learn about how it is a critical part of the circulatory system. Find out how it comes to the rescue of the cardiovascular system and the immune system. Also discover how it moves fluid in one direction, like blood, but without a heart!

MCAT Competition

Make great content for our MCAT Competition! We are looking for talented video-based educators and fantastic question and article writers, that will join us in helping students learn the 2015 MCAT Foundational concepts.
Send in your submission by June 13, 2014!

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.

Making Art

Carved carnelians, painted parchment, finessed flesh tones... explore art techniques in these videos.

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

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!
Common Core Standard: 7.EE.A.1

Manuscripts

Learn about medieval manuscripts. Find out how these handmade objects were created, how the medieval calendars within them work, and see the book considered by many to be the finest illuminated manuscript ever made.

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.

Mass and volume

Materials

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 parabolas

This is simply collection of related KA content (this is NOT a finished tutorial)

Matrices

More than just 0's and 1's, matrices are ways of organizing numbers. They are used extensively in computer graphics, simulations and information processing in general.

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.)

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).

Measurement

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 segments

Most of what we call "lines" in everyday life are really line segments from a mathematical point of view. This exercise makes you a bit more familiar with line segments by giving you some practice measuring and comparing them.
Have fun!

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.

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

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

Meet 500 years of British art

The BP Walk through British Art offers a circuit of Tate Britain’s unparalleled collection from its beginnings to its end. This ‘walk through time’ has been arranged to ensure that the collection’s full historical range, from 1545 to the present, is always on show. There are no designated themes or movements; instead, you can see a range of art made at any one moment in an open conversational manner.
Walk through time with Tate's curators as they introduce the new displays at Tate Britain.

Meet Tate Britain

On 19 November, 2013, Tate Britain opened its doors to the newly refurbished building. See how some of Britain's top creative minds have made personal connections with works which have inspired them.

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!

Memory

Explore the structure of human memory; processes involved in normal encoding, retrieval, forgetting, and aging; and diseases affecting memory.

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).

Metacognition

How to learn better

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

The International System of Units used today is based on the metric system. The United States, however, likes to dance to a different drummer and still uses the old British Imperial System (U.S. customary system) for most of its measuring. This tutorial will introduce you to both for measuring distance, volume, weight and time.
Common Core Standard: 4.MD.A.1

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!

Middle school content

Miscellaneous

Enjoy!

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

We can often have fractions whose numerators are not less than the denominators (like 23/4 or 3/2 or even 6/6). These top-heavy friends are called improper fractions. Since they represent a whole or more (in absolute terms), they can also be expressed as a combination of a whole number and a "proper fraction" (one where the numerator is less than the denominator) which is called a "mixed number." They are both awesome ways of representing a number and getting acquainted with both (as this tutorial does) is super useful in life!
Common Core Standard: 4.NF.B.3c

Mixed numbers and improper fractions

We can often have fractions whose numerators are not less than the denominators (like 23/4 or 3/2 or even 6/6). These top-heavy friends are called improper fractions. Since they represent a whole or more (in absolute terms), they can also be expressed as a combination of a whole number and a "proper fraction" (one where the numerator is less than the denominator) which is called a "mixed number." They are both awesome ways of representing a number and getting acquainted with both (as this tutorial does) is super useful in life!

MoMA-learning

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 constraints

In this tutorial, we'll use what we know about equations, inequalities and systems to answer some very practical real-world problems (and a few fake, impractical ones as well just for fun).

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 graphs

Much of the reason why math is interesting is that it can be used to model real-world phenomena. In this tutorial, we'll do just this with graphs.

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!

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

Modernism (4-2 to 4-3)

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

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 orbital theory

In this tutorial, Jay introduces molecular orbital (MO) theory and shows how MO theory explains the experimental observations of the Diels-Alder reaction.

Momentum

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.

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 type of money.

Monopolies

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.

More analytic geometry

You're familiar with graphing lines, slope and y-intercepts. Now we are going to go further into analytic geometry by thinking about distances between points, midpoints, parallel lines and perpendicular ones. Enjoy!

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.

Mortgages

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. By Shreena Desai.

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 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

Multiplication

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

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.).
Common Core Standards: 5.NBT.B.5, 5.NBT.B.7

Multiplication and division

Practice a couple of word problems with multiplication and division. Take your time, write down the question and go step by step. Your dashboard can take you through some easier practice exercises if you need some review.

Multiplication by powers of 10

This tutorial will be your first exposure to exponents which we will build on in later grades. In particular, we're going to think about what happens when you multiply by 10 multiple times (and think about how the number of zeroes relates to the number of times we multiply by 10). Later on, we'll do the same thing with other numbers.
Common Core Standard: 5.NBT.A.2

Multiplying and dividing complex numbers

Multiplying and dividing fractions

What is 2/3 of 2/3? If 4/7 of the class are boys, how many boys are there? Multiplying fractions is not only super-useful, but super-fun as well.
Did you also know that fractions can represent division (and the other way around). We can create fractions by dividing whole numbers and then even divide the fractions themselves. We'll see that dividing by something is the exact same thing as multiplying by that thing's reciprocal!
Common Core Standards: 7.NS.A.2a, 7.NS.A.2b

Multiplying and dividing negative numbers

It is starting to dawn on you that negative numbers are incredibly awesome. So awesome that you are feeling embarrassed to think how excited you are about them!
Well, the journey is just beginning. In this tutorial we will think about multiplying and dividing numbers throughout the number line!
Common Core Standards: 7.NS.A.2a, 7.NS.A.2b

Multiplying and dividing rational expressions

Let's extend what we know about multiplying and dividing fractions to rational expressions. It may look complicated, but it really is about applying some core principles of what fractions represent.

Multiplying and factoring expressions

This reviews ways to multiply and factor any expression. Refresh these skills to solve a broad array of problems in algebra.

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

What is 2/3 of 2/3? If 4/7 of the class are boys, how many boys are there? Multiplying fractions is not only super-useful, but super-fun as well.
Common Core Standards: 5.NF.B.4, 5.NF.B.4a, 5.NF.B.4b, 5.NF.B.5, 5.NF.B.5a, 5.NF.B.5b

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).
Common Core Standard: 5.NF.B.6

Multiplying monomials

"Monomials" sounds like a fancy word, but it just refers to single terms like "4x" or "8xy" or "17x^2z". In this tutorial, we'll learn to multiply and divide them using ideas you're already familiar with (like exponent properties and greatest common factor).

Multiplying monomials, binomials and polynomials in general

You'll see in this tutorial that multiplying polynomials is just an extension of the same distributive property that you've already learned to multiply simpler expression (that's why we think FOIL is lame--it doesn't generalize and it is more memorization than real understanding).

Multiplying whole numbers and fractions

If I cut a pizza into eight equal slices, we already know that each slice is 1/8 of the pizza. But what fraction of the pizza have I eaten if I eat 3 slices? Well, what is 3 x 1/8? This tutorial will explain all!
Common Core Standards: 4.NF.B.4, 4.NF.B.4a, 4.NF.B.4b , 4.NF.B.4c

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

Muscles

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: How do our muscles work? When we decide to kick a ball or shake a leg, how do we get our bodies to do that? Which muscles do we control? Which muscles control us? Learn how our muscles work at the smallest, most cellular level. Then see how nature scales up those microscopic processes into a kick or a dance move. Finally, learn how our brain tells muscle to contract and how that helps us respond to changes in temperature or even a lion chasing us. By Raja Narayan.

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.

My Benefits

Naming alkanes

In this tutorial, Sal shows how to name alkanes.

Naming alkanes, cycloalkanes, and bicyclic compounds

Do you speak the language of organic chemistry? In this tutorial, Jay shows you 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).

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 realize 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

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 number basics

What is a "negative number"? In this tutorial, you'll learn about what happens in the world below zero!
Common Core Standards: 6.NS.C.6, 6.NS.C.6a, 6.NS.C.6c

Negative numbers and absolute value

Try out some more word problems to practice with negative numbers & absolute value.

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.

Neural calls

3A: 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.

Neuronal synapses

3A: An overview of how neurons communicate with their target cells at synapses, and the roles of neurotransmitters and their receptors. By Matt Jensen.

New HS exercises

New Topic

How can we quantify the strength of magnetic fields? How strong is the Earth's field?

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: Go back in time and rediscover Newton's three laws! These powerful laws help explain inertia, equal and opposite forces, and calculation of acceleration based upon forces. By Sal.

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.

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 of aldehydes and ketones

In this tutorial, Sal names 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: Learn about the normal force, and try some examples that demonstrate how the normal force is affected by the motion of an elevator, as well as how the normal force affects friction. By Sal.

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. By Jeff Walsh.

Nucleophilic Aromatic Substitution

In this tutorial, Jay shows the addition-elimination mechanism and the elimination-addition mechanism.

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 Theory warmups

Warmups related to number theory

Number patterns

Let's use some mathematical tools to visualize and interpret some patterns involving the coordinate plane.

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.

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.

Objects

Learn how to store complex data in objects.

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 probability

Sal's old videos on probability. Covered better in other tutorials but here because some people actually like these better.

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").

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-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!

Optical activity

In this tutorial, Jay explains the concept of optical activity and demonstrates how to calculate specific rotation and enantiomeric excess.

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.

Orbitals and Electrons

Order of operations

Mathematics wouldn't be so useful if, interpreted in different ways, the same expression could be viewed to represent different values. To combat this issue, the mathematical community defined "orders of operations" to remove an ambiguity there might be when evaluating an expression!

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.

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 schools.khanacademy.org.

Other Materials

[note: move web links into this section]
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...

Other features

Other materials

Other reference materials

Ottonian

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.

Our Species

Our vision

Overview

Download and practice with a real, full-length SAT provided by the College Board.

Overview Quiz on Medieval Art

These questions focus on works of art from across the different periods of the middle ages.

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 SAT prep on Khan Academy

Overview of metabolism

1D: What is metabolism? What molecules are involved? How is useful chemical energy produced? Apply principles from general chemistry and biology to learn the common themes of metabolism. By Jasmine Rana.

Oxidation and reduction

LEO the Lion goes GERRRR!!!! If that is all that you remember about redox from general chemistry, then this tutorial is for you! Jay shows you how to assign oxidation states in organic molecules.

Paintings

Learn about some of the great paintings in the Getty Museum's collection and discover the techniques used by the artists that made them.

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

Parabolas

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!

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.

Paritial fraction expansion

If you add several rational expressions with lower degree denominator, you are likely to get a sum with a higher degree denominator (which is the least-common multiple of the lower-degree ones). This tutorial lets us think about going the other way--start with a rational expression with a higher degree denominator and break it up as the sum of simpler rational expressions.

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 breath 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

If you add several rational expressions with lower degree denominator, you are likely to get a sum with a higher degree denominator (which is the least-common multiple of the lower-degree ones). This tutorial lets us think about going the other way--start with a rational expression with a higher degree denominator and break it up as the sum of simpler rational expressions.
This has many uses throughout mathematics. In particular, it is key when taking inverse Laplace transforms in differential equations (which you'll take, and rock, after calculus).

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).

Patterns

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

Build a battery strong enough to power an LED using the change in your pocket! Developed and demonstrated by Julie Yu, director of the Exploratorium Teacher Institute.

Percent word problems

Whether you're calculating a tip at your favorite restaurant or figuring out how many decades you'll be paying your student debt because of the interest, percents will show up again and again and again in your life. This tutorial will expose you to some of these problems before they show up in your actual life so you can handle them with ease (kind of like a vaccine for the brain). Enjoy.

Percentages

At least 50% of the math that you end up doing in your real life will involve percentages. We're not really sure about that figure, but it sounds authoritative. Anyway, unless you've watched this tutorial, you're really in no position to argue otherwise. As you'll see "percent" literally means "per cent" or "per hundred". It's a pseudo-decimally thing that our society likes to use even though decimals or fractions alone would have done the trick. Either way, we're 100% sure you'll find this useful.
Common Core Standard: 6.RP.A.3c

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: 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.

Perimeter

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

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 lense. 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!

Perimeter, area and volume

A review of how to calculate quantities of shapes.

Periodic table, trends, and bonding

Permutations and combinations

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? What if you don't care what order they are in or how they are posed (okay, of course you care about their awesome poses)?

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.

Perseus basics

Perseus one beta

First exercises made from scratch in Perseus one.

Persistence is key

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).

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.

Photography after 1970

Photography before 1970

Photosynthesis

Physics

Watch fun, educational videos on all sorts of Physics questions.

Pictographs and line graphs

Line graphs are very common ways to see trends in data. Pictographs are less common, but they can be cute to look at (this is why newspapers like to use them).
Common Core Standards: 6.SP.B.4, 6.SP.B.5

Piecewise functions

In this tutorial, we will get practice looking at wackier functions that are defined interval by interval (or piece by piece)!

Place value

In this tutorial we'll really dissect numbers to think about what they represent.
Common Core Standards: 4.NBT.A.1 , 4.NBT.A.2

Plan and teach with Khan Academy (K-12 math classrooms)

Plan and teach with Khan Academy (higher education math)

Planning your implementation

Think through the essential parts of you Khan Academy implementation, including figuring out the purpose it will serve, how to integrate it into your curriculum, and how to hold students accountable

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.

Point-slope form and standard form

You know the slope of a line and you know that it contains a certain point. Well, in this tutorial, you'll see that you can quickly take this information (and that knowledge the definition of what slope is) to construct the equation of this line in point-slope form! You'll also manipulate between point-slope, slope-intercept and standard 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!

Polygons in the coordinate plane

We've learned about polygons and have explored all four quadrants of the coordinate plane.
Now, let's combine those skills to draw, measure, and solve problems with polygons in the coordinate plane.
Common Core Standards: CC.6.G.A.3

Polynomial and rational functions

Explore quadratics and higher degree polynomials with these exponentially awesome functions!

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 polynomials 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.

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).

Positive and 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.
Common Core Standards: 8.EE.A.1

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.

Practice and feedback

Test your knowledge and skills! Investigate how language makes humans different, and then take on the quiz and glossary challenge.

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? Build machines to perform primality tests!

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.
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.

Principles of bioenergetics

1D: What are the thermodynamic principles that are fundamental to understanding metabolism? Learn about Gibbs Free Energy, Enthalpy, and Le Chatelier's Principle. By Jasmine Rana.

Printmaking

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.

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!

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 you happiness? This is what we try to understand through our study of the production possibilities frontier and opportunity cost.

Programming basics

Collection of programming basics using NXT-G

Projectile Launcher

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!

Properties and patterns in arithmetic

Math is full of properties and 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

Properties of integer exponents

In this tutorial, we're going to significantly deepen our understanding of the world of exponents. In particular, we'll look at what it means to raise something to a negative exponent and also think about various exponent properties for rewriting expressions.

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 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.

Proportional relationships

In this tutorial we'll think deeper about how one variable changes with respect to another. Pay attention because you'll find that these ideas will keep popping up in your life!

Proteins

5D: What are amino acids? How do they come together to form proteins? Learn about how amino acids synthesized, and then how they come together to form proteins. Discover how proteins are structured and what functions they perform on the cellular level. By Tracy Kim Kovach.

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

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.

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.

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.

Quadratic equations

Just saying "quadratic" will make you feel smart. Try these review exercises and feel your brain tingle as you get even smarter!

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.

Quadrilaterals

Not all things with four sides have to be squares or rectangles! We will now broaden our understanding of quadrilaterals!

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 start guides

Radians

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

You're enjoying algebra and equations, but you miss radicals. Wouldn't it be unbelievably awesome if you could solve equations with radicals in them. Well, your dreams can come true.
In this tutorial, we work through a bunch of examples to help you understand how to solve radical equations. As always, pause the videos and try to solve the example before Sal does.

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?

Rates

How fast can a robot possum fly? What is the rate at which a hungry person can consume avocados? This tutorial helps you make sense of these fundamental questions in life.
Common Core Standard: 6.RP.A.3b

Rates for proportional relationships

In proportional relationships, the ratio between one variable and the other is always constant. In the context of rate problems, this constant ratio can also be considered a rate of change. This tutorial allows you dig deeper into this idea.

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

Let's apply our knowledge of ratios in real-world settings!
Common Core Standards: 6.RP.A.3, 6.RP.A.3a

Rational and irrational numbers

In this tutorial, we'll use the powers of algebra to actually prove what we've always suspected to be true of rational and irrational numbers!

Rational expressions

Learn how to interpret and manipulate rational expressions.

Rational functions

Have you ever wondered what would happen if you divide one polynomial by another? What if you set that equal to something else? Would it be as unbelievably epic as you suspect it would be?

Rational number word problems

You know that fractions, decimals and percentages are all ways to represent rational numbers. In this tutorial, you'll take things to the next level by using these representations together to solve problems.
Common Core Standards 7.NS.A.3

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, proportions, units and rates

What a mouthful! Say that 10x fast.. and then do it in half the time. Wow, you're already a rate and ratio expert.

Reaction rates

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.

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").

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

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 and explicit functions

In this tutorial, we'll see that we can often define a function in terms of itself! This may seem circular and illogical at first, but, as we'll see, it is actually quite reasonable and useful!

Redox Reactions

4E: Discover how to assign oxidation states/numbers in redox reactions, then go on to balance redox reactions! By Jay.

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.

Regrouping decimal numbers

Let's explore how we can regroup and redistribute value among the various place values in a decimal number.
Common Core Standard: 5.NBT.A.1

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).

Renaissance Art in Europe

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.

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).

Resources

Download the teacher resources from the Big History Project classroom version of this course. Text captions for videos are also available.

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.

Rewriting fractions as decimals

We already know that the same quantity can be represented as a decimal or a fraction. In this tutorial, we'll begin to see how a fraction can be rewritten as a decimal.

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.

Right triangles and trigonometry

Review how to find unknown parts of triangles.

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)

Rounding

Rounding is useful when you are trying to roughly estimate numbers.
Common Core Standard: 3.NBT.A.1

Rounding decimals

One robot hamster rabbit weighs 1.51 kg while another weights 1.49 kg. Is this a big different or a little one? Say you want a robot hamster that roughly weighs 1.5kg. Will either do?
Let's get some practice rounding with decimals!
Common Core Standard: 5.NBT.A.4

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

SEO Landing Pages Articles To Test

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.

SN1/SN2/E1/E2

In this tutorial, Jay discusses the strength of a nucleophile and the differences between SN1, SN2, E1, and E2 reactions.

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

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!

Scale of the small and large

We humans have trouble comprehending something larger than, say, our planet (and even that isn't easy to conceptualize) and smaller than, say, a cell (once again, still not easy to think about). This tutorial explores the scales of the universe well beyond that of normal human comprehension, but does so in a way that makes them at least a little more understandable.
How does a bacteria compare to an atom? What about a galaxy to a star? Turn on your inertial dampeners. You're in store for quite a ride!

Scatter plots

School pilots

School redesign overview

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!
Common Core Standard: 8.EE.A.4

Scott Cook - Founder and Chairman of the Executive Committee, Intuit

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.

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: Learn how to separate and purify chemical compounds using organic chemistry laboratory techniques, including extraction, distillation, chromatography, and gel electrophoresis. By Angela Guerrero.

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?).

Sequences

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).

Series

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.

Set goals and create a plan

Set goals and create a study plan

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.

Shifting and reflecting functions

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.

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

How do we know something is similar? Congruent?

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 radical expressions

You already know what square roots and cube roots are, now you will apply that knowledge to simplify variable expressions that involve radicals.

Simplifying rational expressions

You get a rational expression when you divide one polynomial by another. If you have a good understanding of factoring quadratics, you'll be able to apply this skill here to help realize where a rational expression may not be defined and how we can go about simplifying it.

Simulating grass

In this tutorial we will design a model for a blade of grass and use it to create an entire field!

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...

Skip-counting

Sleep and consciousness

6B: Explore states of consciousness, circadian rhythms, sleep stages, and sleep disorders. By Carole Yue.

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.

Slope of a line

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.
Common Core Standard: 8.F.B.4

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.

So that’s where that comes from!

Social structures

9A: What is a society? How do we humans all live together in such highly populated cities? How does a society change? Learn about some sociologic theories that have been proposed that try to address some of these important questions! By Sydney Brown.

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!

Solutions to linear equations

No all equations in one variable have exactly one solution. Some have no solutions and some are true for any value of the unknown. In this tutorial, we'll learn to tell the difference (and understand why this is).
Common Core Standard: 8.EE.C.7a

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 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.
Common Core Standards: 6.EE.B.5, 6.EE.B.6, 6.EE.B.7

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 equations with distribution

In this tutorial, we'll look at slightly more complicated equations that just having variables on both sides. If you can solve these, you're well on your way to mastering equations!
Common Core Standards: 8.EE.C.7, 8.EE.C.7b

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

The equations you are about to see are some of the hairiest in all of algebra. The key is to keep calm and don't let the rational equation be the boss of you.

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

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" or "how can order the right amount of potato chips for a King's party." Join us as we cover (and practice with examples and exercises) all of the major ways of solving a system: graphically, elimination, and substitution. This tutorial will also help you think about when system might have no solution or an infinite number of solutions. Very, very exciting stuff!

Solving systems through examples

This tutorial focuses on solving systems graphically. This is covered in several other tutorials, but this one gives you more examples than you can shake a chicken at. Pause the videos and try to do them before Sal does.

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

Somatosensation

6A: We perceive the environment through our bodily senses, including our sensation of pain, temperature, pressure, balance, and movement. Discover how our body gathers this information and processes it so that we can make sense of the world. By Ronald Sahyouni.

Sound (Audition)

Learn about how we hear, including the structure of the outer, middle, and inner ear, as well as the basics of auditory processing & cochlear implants.

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.

South, East, and Southeast Asia

The arts of South, East, and Southeast Asia represent some of the world’s oldest, most diverse, and most sophisticated visual traditions.

Southeast Asia

Only in the past sixty years has “Southeast Asia” been used to refer to the region comprising modern-day Burma (Myanmar), Thailand, Laos, Cambodia, Vietnam, Malaysia, Singapore, Indonesia, Brunei, and the Philippines. These ten countries cover an area more than three times that of Great Britain, France, and Germany combined, and they have a population about twice as great.

Special properties and parts of triangles

Who doesn't like triangles? They're useful. They show up a lot. What you'll see in this topic is that they are far more magical and mystical than you ever imagined!

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).

Spectroscopy

There is much more to light than meets the eye. Introduction to the electromagnetic spectrum and the science of spectroscopy.

Speed and velocity

4A: Think to calculate speed and velocity, and then go on to learn about average and instantaneous speed and velocity. By Sal and David SantoPietro.

Spherical mirrors

4D: These videos go over the images formed by curved mirrors. It will be explained what a real and virtual image is and how to draw ray tracings to determine the location and size of the image formed. By David Santo Pietro.

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?
Common Core Standards: 8.EE.A.2

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.

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

Statistics

You made it! Last section.. you're so close to finishing your brush up. You often use arithmetic to apply these basic statistics concepts.

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).

Stereochemistry

5B: Get an overview of isomers and stereochemistry including structural isomers, stereoisomers, diastereomers, and conformational isomers. By Jay.

Stoichiometry

4E: Think about the concept of the "mole" and how we can use it to help us balance chemical reactions. By Sal.

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".

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".

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!

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

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).

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.

Super Yoga Plans

Let's use our algebra tools to solve problems of earth-shattering importance (like which gym plan you should choose)!
Common Core Standard: 6.EE.B.6

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!

Super fast systems of equations

Have no time for trolls, kings and parrots and just want to get to the essence of system. This might be a good tutorial for you. As you can see, this stuff is so important that we're covering it in several tutorials!

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.

Symmetry

Let's get an intuitive understanding for symmetry of two dimensional shapes.

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

In this tutorial, we'll learn a technique for dividing one polynomial by another--synthetic division. As always, we'll also explore why it works!

System for solving the King's problems

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" or "how can order the right amount of potato chips for a King's party." Join us as we cover (and practice with examples and exercises) all of the major ways of solving a system: graphically, elimination, and substitution. This tutorial will also help you think about when system might have no solution or an infinite number of solutions. Very, very exciting stuff!
If you want more examples, feel free to look at the other tutorials in this topic.
Common Core Standards: 8.EE.C.8, 8.EE.C.8a, 8.EE.C.8b, 8.EE.C.8c

Systems of equations and inequalities

Solving a system of equations or inequalities in two variables by elimination, substitution, and graphing. These are powerful ways to help you solve all kinds of real world problems.

Systems of equations 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!

Systems of inequalities

You feel comfortable with systems of equations, but you begin to realize that the world is not always fair. Not everything is equal! In this short tutorial, we will explore systems of inequalities. We'll graph them. We'll think about whether a point satisfies them. We'll even give you as much practice as you need. All for 3 easy installments of... just kidding, it's free (although the knowledge obtained in priceless). A good deal if we say so ourselves!

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.") 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!

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.

Taste (gustation) and smell (olfaction)

6A: Learn about our senses of smell (olfaction) and taste (gustation), including the anatomy and underlying molecular basis behind these important senses. By Ronald Sahyouni.

Taxes

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.

Technology considerations

Telling time

Tens

Tension

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).

Text

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 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 Crusades

Crusading took many different forms, and attempting to precisely define crusading has engaged historians in intense debates for more than 150 years.

The Declaration of Independence

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 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. Unforunately, 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 conservation of works of art

See conservators in action and learn about the work a conservator does in different media.

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 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 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 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 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 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.

Theories of personality

7A: Curious about your personality? Throughout history, famous psychologists and schools of thought have tried to figure out how to organize and categorize personalities. Review these theories and see which one resonates the most with you! By Shreena Desai.

Thermodynamics

These topics have moved!

Things Justin couldn't get to work

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 about solving equations

Much of algebra seems obsessed with "doing the same thing to both sides". Why is this? How can we develop an intuition for which algebraic operations are valid and which ones aren't? This tutorial takes a high-level, conceptual walk-through of what an equation represents and why we do the same thing to both sides of it.
Common Core Standard: 6.EE.B.7

Thinking about the number line

In this tutorial, we'll use what we know about negative numbers, absolute value, and the number line to think just a little bit deeper!

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!

Thinking critically about multiplication and division

Let's now apply your new found understanding of multiplication and division!
Common Core Standards: 3.OA.A.3, 3.OA.A.4, 3.OA.B.6

Thiols and sulfides

In this tutorial, Jay shows how to prepare sulfides from thiols.

Thomas Friedman - Author

Thoughts on education

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.

Time and money

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.

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).

Transformations

Practice thinking more visually by rotating, scaling, and shifting polygons.

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.

Transformations on the coordinate plane

Let's continue our deep voyage through the world of transformations by thinking about how points map to each other after a transformation.

Transpose of a matrix

We now explore what happens when you switch the rows and columns of a matrix!

Tree of life

Trenceto

An overview of this transitional period in Europe.

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 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!

Triangle similarity and constant slope

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. We'll connect this idea to the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b (cc.8.ee.6).

Triangle similarity and slope

In this tutorial, we use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. We'll connect this idea to the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
Common Core Standard: 8.EE.B.6

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 identities and examples

More complex identities...How to work with ANY triangle!

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.

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 mass)!

Tuberculosis

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

Listen to a beam of light! Learn how to build a simple device in which the signal from a radio is transmitted on a beam of light traveling between a light-emitting diode (LED) and a solar cell. Developed by Exploratorium Teacher Institute staff Don Rathjen and demonstrated by Exploratorium Senior Scientist Paul Doherty.

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 equations

You saw very basic equations in 6th grade. Now we'll step things up a bit (literally), by tackling 2-step equations. Yee-Hah!
Common Core Standard: 7.EE.B.4

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

U-substitution

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 reverise 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" :)

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.

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.)

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

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

When you first started measuring things in units in 2nd grade, you saw that you might want to you different units depending on the scale, application or culture that you are in. Well, you could imagine that you'd want to convert from one unit to another as well. For example, say you walked 2 kilometers. How many meters would that be?
Common Core Standard: 4.MD.A.1

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)!
Common Core Standard: 5.MD.A.1

United States History

Using data

Using playlists to integrate Khan Academy into your curriculum

Using regrouping to subtract within 1000

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).
Common Core Standard: 3.NBT.A.2

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.

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!

Variables

We'll cover how to use variables to hold values, animate your drawings, and more.

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.

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

We will begin our journey through linear algebra by defining and conceptualizing what a vector is (rather than starting with matrices and matrix operations like in a more basic algebra course) and defining some basic operations (like addition, subtraction and scalar multiplication).

Vectors and scalars

4A: Find out what makes vectors different from scalars! Find out how to break vectors into components, and go over the different ways to represent vectors. By Sal.

Vectors in magnitude and direction form

Vectors in rectangular form

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).

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.

Volume

Tired of perimeter and area and now want to measure 3-D space-take-upness. Well you've found the right tutorial. Enjoy!
Common Core Standards:
5.MD.C.3, 5.MD.C.3a, 5.MD.C.3b, 5.MD.C.4, 5.MD.C.5, 5.MD.C.5a, 5.MD.C.5b , 5.MD.C.5c

Volume and surface area

Let's explore the volume and surface area of 3D shapes.
Common Core Standards: 6.G.A.2, 6.G.A.4

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!

Waves and optics

Ways to use Khan Academy

Discover the most common ways Khan Academy is being used in a variety of learning environments

Welcome

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

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).
Common Core Standards: 3.NF.A.1, 3.NF.A.2, 3.NF.A.2a, 3.NF.A.2b

What happens when you stay put

What’s God got to do with it?

When people do great and really terrible things

Why Khan Academy? (K-12 math classrooms)

Why Khan Academy? (higher education math)

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.

Word problems with units

Let's solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit.
Common Core Standard: 4.MD.A.2

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).

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 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 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 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.
Common Core Standard: 6.EE.A.2, 6.EE.A.2b

Yoga plans

This tutorial is a survey of all the core ideas in a traditional first-year algebra course. It is by no means comprehensive (that's what the other 600+ videos are for), but it will hopefully whet your appetite for more algebra!

You ain’t the boss of me!

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!

alcohols and phenols

5D: Get a strong understanding of the nomenclature, properties, and reactions of alcohols and phenols, along with the criteria for determining aromaticity in benzene and aromatic heterocycles. By Jay.

aldehydes and ketones

5D: Learn how to name aldehydes and ketones, and then dive deeper and explore how they are formed and interact in chemical reactions. By Jay.

all-about-spout

This topic explains how Spout works! It was created by Karl R. C. Wendt.

alpha-carbon chemistry

5D: How to form enolate anions and how to use them to predict the products of aldol condensations. By Jay.

covalent bonds

5B: Learn about covalent bonds and SP hybridization. By Jay.

crypto challenge

The first challenge in the Cryptography series

deflation

Prices don't always go up. Sometimes they go down (we call this deflation). This tutorial explains how this happens.

electrostatics

Discovering static electricity & electrostatic force. What is it? How can it be created, detected, and measured?

enzymes

Come learn about enzymes and the basic principles behind enzyme mechanisms catalysis. By Ross Firestone.

evolution-today

gas phase

4B: Let's learn about gas phase chemistry! Discover the history and application of the ideal gas law. We'll walk through topics like temperature, STP, partial pressure, and the difference between real gases and their ideal counterparts. By Ryan Patton.

neuron membrane potentials

Learn the causes and functions of neuron membrane potentials, including resting, graded, and action potentials.

periodic table

4E: Learn about how the periodic table is organized and then explore ionization energy and electron affinity. By Jay.

resistance

Exploring materials which cause a decrease in deflection when added in series with our meter.

self-identity

8A: Explore the ideas surrounding the concept of Self-Identity. Who are we? How do we develop our morals and patterns of learning? What influences our behaviors in social situations? Discover the importance of different phases of our life in our transformation into adulthood and old age.

sensory perception

6A: 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. By Ron Sahyouni

social-psychology

7B: Dive into the fascinating and relevant world of social psychology! You may have noticed many of these concepts in your everyday life, but perhaps you did not know the proper terminology or the specific factors that motivate people to behave the way they do in groups. In this tutorial, you will learn some important aspects of social psychology. By Jeffrey Walsh.

work and energy

4A: Learn about the concepts of work, mechanical advantage, and power. After understanding the principle of conservation of energy, you can apply it to different types of energy including kinetic energy, gravitational potential energy, and spring potential energy. Finally, learn about the conservative force as well as how to find mechanical advantage. By Sal and David SantoPietro.

x-intercepts and y-intercepts of linear functions

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.