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Mathematics

Within Mathematics

Man in suit with eyes bulging and lips pursed raises tensed hands in the air.

People feeding dried hay to goats. Caption: How can we figure out how much energy is stored in this hay?

Señora Alicia rests her hands on a blue counter while she talks. She's next to Susana the Iguana puppet, and they are surrounded by green plants.

Teal background with 6 boxes arranged in 2 rows on it. The 2 left boxes in the top row are yellow, and the other is blue. The middle box in the bottom row is yellow, the others are blue, and the right box has a question mark on it.

Señora Alicia and Susana the Iguana puppet look toward the camera. Señora Alicia rests her arms on a blue counter.

Señora Alicia looks down at 2 yellow rubber ducks and 2 white teddy bears with red ribbons on the wooden table in front of her. She has her hands on some papers. Colorful plants and wall decorations are behind her.

Señora Alicia is smiling and wearing a yellow shirt. She is next to a green iguana puppet named Susana and some green plants. Her elbows rest on a blue counter with her palms together and facing up.

Storybook cover with an orange background and a drawing of a baby wearing a diaper splashing in a puddle. The Spanish title reads, "Los Charcos Son Pequeños."

Person standing in a colorful room with cacti in pots all around the walls. Spanish captions.

Señora Alicia and a dark-haired man holding a small red guitar look at each other as they sing. There's a mountain and potted plant painted on the wall behind them.

Green background. The number 2 in orange with a yellow outline is on the left. On the right is the number 4 in cheetah print, the number 5 is a yellow and red gradient, and the number 6 in a brown pattern.

Señora Alicia sits on a green metal chair on a colorful tile floor. There are colorful plants, drawings, and wall decorations behind her.

Señora Alicia wears a yellow shirt and moves her hands in front of her shoulders. There's a red and blue striped background behind her.

Closeup of a hand holding a yellow pencil, pointing at a Panamanian mola of a bird, made of black and brightly colored cloth.

Aerial closeup of 2 hands, the right hand using scissors to cut an outline of a cartoon fish out of pink paper.

Close up of a woman's hands over a white paper with dashed horizontal lines. Her right hand moves around circles of various colors and sizes.

Storybook page on an orange background. The page has illustrations of a kite, a yellow shirt, and multiple diamond shapes. The Spanish text on the page reads, "Rombos. Dulces. Papalote. Estampado."

Señor Enrique uses a paintbrush up to paint on a canvas sitting on a wooden easel. His back is to us, and there's a small table with water, more paintbrushes, and paints on it.

Señora Alicia stands in front of a green chair on a colorful tile floor, partially extends her arms in front of her. There are colorful plants, drawings, and wall decorations behind her.

Señora Alicia holds her hands up next to her face, palms facing outward. Next to her, Susana the Iguana looks at her from a blue counter.

Señora Alicia and Susana the Iguana stand at a blue counter and look over at Goldilocks, who is wearing a pink shirt and bows in her hair. Spanish Caption. "¿Como estas, Ricitos de Oro?"

Señora Alicia holds up a painter's palette with red, green, and blue paint on it. The tubs of paint sit on the blue counter in front of her, each with paintbrushes resting on top. Next to her is Susana the Iguana.

Señora Alicia stands behind a wood table, looking down into a clay pot resting on top of the table. She rubs the handle of a wooden spoon in the pot between her hands. There's pitcher of milk and 2 cups on the table.

Person speaking. Caption: between one and one hundred nanometers.

Person speaking and standing next to a poster board with "Control of matter at the nanoscale" surrounded by logos from different organizations. Caption: of matter at the atomic and molecular level.

Cartoon clouds against a blue sky with an animated banner that's gradually revealing text. Partial text reads, "Skills and."

Floating frame over an animated night sky. Frame shows an aerial shot of pink all star sneakers jumping on hopscotch squares. Text on image reads, "Look for Numbers!"

Floating frame over an animated night sky with colorful floating numbers. The frame has overlapping geometric shapes. Text on image reads, "Numbers."

Floating frame over a cloudy sky with a ship passing on the sea below. Frame shoes parents and 2 children playing in cardboard moving boxes. Text on image reads, "Directional Words."

Floating frame over an animated night sky shows a kindergarten-aged child reading a book with a large number 10 on the front. Text nearby reads, "Books that teach math directly."

Closeup aerial view of a board with flippers and raised attachments creating lanes and obstacles. A person off camera gestures toward board with 1 hand and holds a ball in the other.

Infographic shows stacks of paper above text labels. The first is a tall white stack of paper labeled, “Endorse, Human-caused global Warming is Real.” The second is a yellow stack of paper falls down and is labeled, “Excluded, No Position.” The third is a very short stack of red paper is labeled, “Reject, Human-caused global warming is not real.”

Digital DNA strands with two orange arrows measure the width and length of the DNA. Text reads, “34A, 21A.”

Cartoon sports scoreboard with a red team that has two hundred thirty points and a yellow whale team that has two hundred thirty-one points. A dodgeball is thrown in front of the board as a crowd of people behind the board cheer.

Cartoon scoreboard labeled, “Jugadores Restantes,” with a red E.M. team that has scored two hundred thirty points versus a Yellow Beluga whale team that has scored two hundred thirty-one points. Red dodgeballs fly through the air as a crowd of people cheer behind the board.

Photos of a smiling woman and a serious looking man are cut out and placed on cartoon bodies. They sit at a table with a vase of flowers in between them. Behind them is a chart with a graph showing that the woman spoke thirty-five words while the man only spoke six words.

Photo of a woman’s head cut out and placed on top of a cartoon body. She holds a phone up to her ear and has her mouth open like she’s talking. Over her head is the number thirty six.

Large square leans forward in front of a fence next to an oval that's next to a pole and flowers. There’s a ruler measurement under the image that’s labeled, “four, fifteen over three, six.”

Curved cartoon house with a fence around it. A person in front of the house stands next to two large eggs and a mailbox. Below the image is a ruler measurement labeled, “four, fifteen over three, and six.”

Cartoon characters that look like queens on playing cards are dressed in red and blue gowns. They look at each other with confusion and question marks over their heads.

Cartoon person that looks like the king on playing cards has an angry expression on their face. They point in front of them and say, “Pie de la Reina.”

Cartoon girl dressed in all black with a skull hair clip sits in a pink room as she holds a video game controller in her hands. An orange cat hangs from the ceiling and points at her. The girl looks angry and annoyed.

Cartoon girl dressed in all black with a skull hair clip sits in a pink room as she holds a video game controller in her hands. An orange cat hangs from the ceiling and points at her. The girl looks angry and annoyed.

Cartoon woman dressed in a black jumpsuit stands in a power stance and holds her arm out. Out of her hand is a green ray of powers. Text reads, “Outta Sight.”

Cartoon green tire labeled “Tamaño Normal" with lines that create a box around it labeled twenty-one inches tall and twenty-one inches at the base. There’s a question mark for the width measurement. Over the tire is a smaller version of the tire labeled “Tamaño reducido en escala" that only has the width labeled one inch. Text, “twenty one times que numero equals question mark.”

Still image from: One Big Pair of Underwear!

White ball with craft eyes sits on top of a red, plastic pedestal. 1 side of the ball has its outer layer torn away and exposing the ball's inner core. It appears to have been damaged by some type of impact.

Dianna wears a navy shirt with red, yellow, and pink stripes on the wrists and shoulders. She sits behind a large piece of paper and holds a black marker. A colorful background with science items on a shelf is behind her, and equations appear on screen.

Overhead view of a person's hands demonstrating how to solve a physics problem related to rotation on a large blue sheet of paper.

A C, G, I graphic of a spaceship made in the shape of a planetary ring rotates around a brown planet. The side closest to the planet looks like oceans with clouds and green continents, and the other side looks like a spaceship with glowing lights.

Aerial closeup of 2 arms. The left has a pink sleeve, the right has a red sleeve and uses a black marker to write on a large sheet of paper. The paper shows an X, Y axis with Y axis labeled Force in Newtons and X axis labeled X in meters. Next to this, a drawing that looks like a spring coil.

Host Dianna Cowern, a blonde woman in a red sweater and black scarf, smiles as she holds an old, large book up. Text on screen reads, "Newton's Principia, Original Manuscript," and an arrow points at the book.

In front of a blurred-out shot of this video's set, text reads, "Very Useful Trick. When any two objects collide elastically and energy is conserved, then the difference in their two velocities before the collision equals the negative of the difference between the two velocities after the collision."

Video host looks at us while sitting at a table with physics equations written on a large sheet of paper. Behind her, the set is decorated with a modular shelf holding scientific models, floating equations, and hanging gears.

Aerial closeup of 2 arms wearing denim sleeves with white stars. The right hand uses a large black marker to write on a large sheet of paper. The paper reads, "Energy and work" and has various physics equations.

Man is suspended in the air by a clump of large balloons over a crowd of people. There are multiple desert towers and rock features in the distance behind them. Text on screen reads, "David Blaine," with an arrow pointing to the man.

Aerial closeup of 2 arms in light blue sleeves over a paper. The right hand writes on a large sheet of paper with a thick black marker. The paper reads, "Equation of motion. Y equals Y sub 0 plus V sub 0, T plus 1/2, A, T sup to the power of 2."

Man wearing a gray winter hat, a dark gray hoodie, black snow pants, and red gloves snowboards down a snowy ski slope on a foggy day. A yellow snow making machine and a little building with a red roof are nearby.

Physics equation related to orbital motion is written on a large blue sheet of paper. The host's hand points at a line that represents the Earth's radius.

Aerial closeup of 2 arms. The right hand uses a thick black marker to write various physics equations on a large sheet of paper.

Video host speaks to us with her hands clasped together while sitting at a table with physics equations written on paper. Behind her, the set is decorated with a modular shelf holding scientific models, floating equations, and hanging gears.

Overhead view of the host's hands as she demonstrates how to work out an equation dealing with the relationships among centrifugal and centripetal force and the Earth's rotation.

Overhead view of the host demonstrating how to work out equations of motion involving acceleration, derivatives, sines, and cosines.

Aerial closeup of 2 arms in black sleeves. The right hand writes on a large sheet of paper with a thick black marker. The paper reads, "tangent equals opposite over adjacent (tan). S, O, H, C, A, H, T, O, A. 1." There's a diagram of a 37.5 degree angle and an directional axis labeled N and E.

Video host speaks to us, holding a marker while sitting at a table with physics equations written on paper. Behind, the set is decorated with a modular shelf holding scientific models, floating equations, and hanging gears.

Video host speaks to us, glances to the side and gestures with her hands while sitting at a table with physics equations written on paper. Behind, the set is decorated with a modular shelf holding scientific models, floating equations, and hanging gears.

A screenshot of 2008 physics exam problem 4, labeled sticky disks.

Simple line drawing of a house on a hill with a roof and 2 windows.

School of fish swimming in a tank. Caption: Sometimes they sink like these rocks.

Medium view of a man speaking while sitting before a backdrop of the U, S flag and an overlay of the Georgia state flag. Title card below the man reads, "Randy Howe. Author. Flags of the 50 States."

Still image from: Ten Dragon Eggs: A Book About Counting Down

Logarithmic scale shows a horizontal line with equal-sized notched increments labeled 1, 10, 100, and 1,000. Notches between each point are spaced out differently to represent the scale difference. Between 100 and 1,000, the notches representing smaller numbers are labeled "far," while the notches representing the larger numbers are labeled "close."

Digital art. Zoomed-in superscript number 2, representing a number squared, to the left of the quotation-mark notation for "inches."

Person holding a twisted strip of material. Caption: This shape is called a Möbius strip.

Trace wears scarfs around his shoulders and talks. Caption: We are here, today, to lay to rest the kilogram.

Woman with glasses in a blue shirt demonstrates sign language in front of a black curtain. White text in a red box at the bottom of the screen reads, "Exponent slash power."

Split screen. The left is a white background with a polynomial math equation written and simplified in blue. To the right, a woman works through the problem using a screen in front of her. Caption. Now I need to distribute that negative 1 to that negative 6.

Aerial side view of a computer screen displaying a white screen with a math word problem written and worked out in long form in blue. The text at the top reads, "Example 3. What is the slope of the line that contains the given points?" Caption. The slope of the line that contains these two points.

Split screen. The left is a white background with a math word problem written in blue. Text at the top reads, "Example 3. Determine if the situation presents a direct variation. Below this, the problem is in typed text and the answer is in handwritten text. To the right, a woman works through the problem using a screen in front of her.

Split screen. The left is a white background with a math word problem written in blue. The prompt reads, "Determine if the situation represents an inverse variation." To the right, a woman works through the problem using a screen in front of her. Caption. We want to determine if this is an inverse variation situation.

Aerial side view of a left hand touching a computer screen displaying a partially visible math problem written in blue. Caption. After you've factored your numerator, it's smooth sailing.

Split screen. The left is a white background with blue text that reads, "Example 2. Evaluate 3 minus 10, X, when X equals 2 over 5." Handwriting below reads, 3 minus 10 times 2 over 5. To the right, a woman works through the problem using a screen in front of her.

Aerial side view of a hand touching a computer screen displaying a math problem written and worked out in long form in blue. The problem at the top reads, "How did I get that?" Caption. And I know there's going to be a plus sign in between."

Split screen. The left is a white background with blue text that reads, "Example 4, factor completely. 4x squared minus X minus 3." To the right, a woman works through the problem using a screen in front of her.

Split screen. The left is a white background with blue text that reads, "Example 5, factor completely," above a handwritten polynomial equation in the midst being solved. a woman works through the problem using a screen in front of her.

Split screen. The left is a white background with an algebra equation being worked out in long form in blue. Given F, X equals 6, X minus 4, Find F, 1. To the right, a woman works through the problem using a screen in front of her.

Aerial side view of a computer screen displaying a math problem written and worked out in long form in blue. The problem at the top reads, "How did I get that?" Caption. When negative 2 is my input, 12 is my output.

Split screen. The left is a white background with an algebra equation is being worked out in long form in blue. To the right, a woman works through the problem using a screen in front of her.

Aerial side view of a computer screen displaying a math problem written and worked out in long form in blue. The problem at the top reads, "Example 1. Find the zeros of the quadratic function." Caption. I'll set both parts equal to 0, individually.

Split screen. The left is a white background with blue text that reads, "How did I get that?" followed by the algebra equation, 9, X, sup 5, Y minus 4, X sup 3, plus X squared. To the right, a woman works through the problem using a screen in front of her.

Split screen. The left is a white background with blue text that reads, "Example 3. The bill at the restaurant was split evenly 4 ways." Handwriting below reads, B is to Bill. To the right, a woman works through the problem using a screen in front of her.

Split screen. The left is a white background with a solved math equation with division and exponents. To the right, a woman works through the problem using a screen in front of her. Caption. I could represent this as X to the 3 minus 5.

White background with blue text that reads, "Example 1, simplify. 5 sup 3 to the power of 10. Multiply exponents." Caption. So 3 times 10, that's 30.

Blue handwriting working out a math problem in long form on a white background. The equation reads, open parentheses 5, B cubed times C closed parentheses, squared.

$White background with blue writing showing how to solve the math equation 7 over 2 to the power of 3 using fractions. Caption. That means that 7 halves to the third power.$

Split screen. The left is a white background with blue text that reads, "Simplify the following expressions," followed by 2 math equations. X sup 11 times X sup 5. And, 5 sup 8 times 5 sup 20. Answer buttons are next to each equation. To the right, a woman works through the problem using a screen in front of her. Caption. Work through these problems.

Aerial side view of 2 hands touching a computer screen displaying an algebra problem written in blue. Text above the equation reads, "Find the Product" and the letters, F, O, I, L. Caption. Then the last term came from distributing the 3 to the 7.

Split screen. The left is a white background with blue text that reads, "Properties of Real Numbers. Let R, S, and T be real numbers." Beneath is a table with title fields labeled Property and Explanation. To the right, a woman works through the problem using a screen in front of her. Caption. Still keeps the equation equal and balanced.

Closeup of a large screen on top of a podium with a hand holding a pen writing numbers. Title on the screen reads, "Multiplication Property of Inequality. A equals 1, B equals 2, C equals 3." An if and then table appears below and reads, "If A is less than B and C is positive, then A, C is less than B, C."

White background with blue text that reads, "Properties of Real Numbers. Let R, S, and T be real numbers." Beneath is a table with title fields labeled, "Property, Addition, Multiplication." Caption. Now for multiplication.

$White background with blue text that reads, "Example 4, Simplify," above a math division problem with fractions and exponents written and worked out in long form. Caption. Just subtract my exponents.$

Split screen. The left is a white background with a partially visible math word problem written and worked out in blue handwriting below. To the right, a woman works through the problem using a screen in front of her. Caption. You really just need to pick one, one ordered pair.

Woman with dark hair in a bun wearing a green shirt stands at a podium in front of an illuminated blue and purple background. She holds a pen in her right hand while looking down at a screen in front of her.

Aerial side view of a hand using a stylus to work out a math problem on a computer screen. Text on screen reads, "Example 2. Write the numeric expression in its simplest radical form."

Split screen. The left is a white background with blue text that reads, "Solve for x. 3 times 2 over 3, X equals 4." To the right, a woman works through the problem using a screen in front of her. Caption. Put my multiplication symbol and a 3.

Aerial side view of 2 hands touching a computer screen displaying a white screen with a long form algebra problem written in blue. Caption. That's how we represent that solution.

Woman with dark hair wearing a gray shirt stand at a podium in front of an illuminated blue and purple background. She holds a pen in her right hand while looking down at a screen in front of her.

Aerial side view of 2 hands touching a computer screen displaying a math word problem involving tickets sold for a high school sporting event. Some parts of the problem are highlighted yellow.

Split screen. The left is a white background with a solved polynomial math equation in blue. To the right, a woman works through the problem using a screen in front of her. Caption. Let's look at this last one.

Split screen. The left is a white background with a chart with 2 rows and 3 columns labeled, "Verbal, Explanation, Example." The boxes are filled in with a combination of typed and handwritten text. To the right, a woman works through the problem using a screen in front of her.

White background with an X,Y axis with gray grid lines and a blue line sloping downward from left to write. Blue text in the top right reads, "Write the equation of the line." Caption. The Y intercept and count off that slope.

Computer generated image of a sphere composed of twisted interlocking tubes of various colors. Caption: Fractal research is changing fields such as biology.

Woman signs in front of a storybook cover of the number 0 wearing a mask and cape and flying. Text on book cover reads, "Zero the Hero."