Critical thinking is the foundation of thoughtful writing. It involves getting beneath the surface of events and ideas, exploration of more than one perspective, developing informed opinions, and trying new approaches to situations.

An inquisitive thinker asks probing questions that get beneath the surface of events and ideas. It is not enough to know what happened and when it happened, but also why and how it happened. It’s more than “just the facts”, but the situations and reasoning behind those facts.

An open-minded thinker sees other aspects to a situation, besides the one that is the most comfortable. Other perspectives must be evaluated carefully before arriving at an informed conclusion. It is not enough to consider just a single dimension. Analytic thought about a number of perspectives, and then synthesizing common elements is also part of the process.

A knowledgeable thinker uses evidence to support their position. Objective evidence is factual and observable rather than subjective and unsupported. For example, a patient may say that they are in pain. The pain they are feeling may come from a source shown on a MRI, such as torn ligaments in their right knee. Those torn ligaments are observable to those who read the MRI, and then the source of their pain can be treated effectively. If there is not enough objective evidence, a critical thinker will admit the need for more and seek it before forming an opinion. It might involve more research in specific areas, and then evaluating the type of evidence. Suppose a product is being sold by infomercial, and the only evidence presented was testimonials from celebrities. No objective evidence existed to show how the product works, or even why it works. When you think critically about the claims, the evidence, and the state of your finances, will you rush out to buy it?

Critical thinking about a situation often demands a new approach, outside the tried and true. The traditional ways to express ideas may not been as effective as they were in the past. Clichés and conventional expressions were once new and fresh, but now they read as hackneyed and stale. Effective communication demands creative thinking.

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]]>Clouds on Earth form in the atmosphere from condensed water vapor. They are classified both by their location in the troposphere and by their appearance. Some clouds are low-level, with bases closest to the surface; others are mid-level; and others are at the highest levels of the troposphere. There are over 90 different types and subtypes of clouds.

The troposphere is the part of the atmosphere where weather takes place. On Earth, it is the lowest level of the atmosphere. It extends roughly from 0 to 12 km above the surface, and contains almost all of the water vapor. Temperature differences between the highest levels of the troposphere and the surface create convection currents. Those currents produce wind.

The lowest-level clouds are from the surface to about 2000 meters (6500 ft.). Surface clouds are known as fog or mist. Stratus clouds are flat clouds that make up most blanketing cloud cover, cumulus clouds at the lowest level are fluffy and detached, and stratocumulus clouds carry the most precipitation.

Mid-level clouds are usually between 2000 meters (6500 feet) and 7000 meters (23,000 feet). These include altostratus and altocumulus clouds. The thickest clouds are those that produce the most precipitation. If the clouds are not quite as thick and heavy, most of the precipitation remains in the sky and does not reach the ground. Mid-level clouds are hard to tell apart by satellite images alone.

The highest-level clouds form up to the highest levels of the troposphere, and occasionally into the tropopause. They are types of cirrus clouds, often wispy and made of ice crystals. Cirrocumulus clouds look like flakes or ripples, and cirrostratus clouds are a very thin veil. Mixed clouds can have low bases and billow upwards through different levels. For example, nimbostratus clouds carry heavy rain and extend across a wide area horizontally and vertically. Towering cumulus and cumulonimbus clouds are formed in convection areas and can carry thunderstorms, downbursts, and cause flash floods. On a weather map, they are associated with the most severe disturbances.

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]]>Penguins are flightless birds that spend most of their time in the ocean. They are native to many locations in the Southern Hemisphere, including Antarctica, New Zealand, southern Australia, Chile, and Argentina. They have unique coloration and markings, and most species are threatened or endangered.

There are 18 different species of penguins. They are unable to fly, as they do not have wings. Instead, they are skilled swimmers, using flippers that are specially adapted for effective movement through the water, similar to birds that fly through the air. They have an insulating layer of air underneath their plumage, as well as thick feathers, to help them conserve heat in the water, and to help them live in cold climates. Penguins have a distinctive dark and white pattern of feathers called countershading. The patterns make them nearly invisible to predators in the water. Although most penguins are black, white, and shades of grey, some kinds have yellow or red bills, and some have yellow or blue feathers. The largest penguin species is the emperor penguin, at 4 feet tall, between 49 and 99 lb. The smallest are the little blue or fairy penguins, who are only about 13-16 inches high and weigh about 2 lb. They eat types of fish, krill, and squid, caught while they are underwater.

Penguins live in the Southern Hemisphere. Many varieties live in Antarctica, the islands around Antarctica, southern New Zealand, southern Australia, Chile, and Argentina. Most live in very cold climates, where they have few predators and safe places to lay their eggs. They live in large colonies to bring up their young. The Galapagos penguin is the only species of penguin to live partially in the Northern Hemisphere, because Isabela Island touches the equator. They live in a temperate area, but rely on cold currents to bring them food. Like other penguins, they feed from the ocean.

Fossils exist for many species of penguins, including a large prehistoric penguin that was nearly 6 feet tall. It lived in temperate regions, and was found in New Zealand. Penguins flourished during periods of cooling, and lived close together on islands of Antarctica, which were much closer together in prehistoric times. They have heavier bones than other birds, so they can be distinguished from them.

The rarest penguins are the Galapagos penguins, with only about 1000 breeding pairs. They are an endangered species, with many enemies, from cats, dogs, and rats what attack them on land, to sharks, fur seals, and sea lions. Yellow-eyed penguins, African penguins, and erect-crested penguins are also endangered. Other species are of threatened status. Most penguins are affected by global warming, because their habitat and food sources are impacted.

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]]>Adding rational expressions with unlike denominators is similar to adding fractions with unlike denominators. The first step is to find the least common multiple (LCM) of the denominators. This can then form the least common denominator. Then multiply each rational expression so that it has the least common denominator. Next, add the numerators now that the denominators are alike. They can be factored and simplified. Subtraction is the process of adding the inverse.

To find the LCM of two or more algebraic expressions, first factor each expression. Then, multiply all of the factors. This is similar to finding the common factors of ordinary numbers without variables. For example, the number 6 can be factored as 2·3, and the number 10 can be factored as 2·5. Their least common multiple is 2·3·5 or 30. Compare that to just multiplying 6·10, or 60. The process is similar with expressions containing variables. Suppose the expressions are 8x^{2}y^{2} and 12xy^{3}. The coefficient 8 can be factored as 2·2·2 and the coefficient 12 can be factored as 2·2·3. The LCM of 8 and 12 can be factored as 2·2·2·3 or 24. Similarly, the LCM of x^{2} and x is x^{2} and the LCM of y^{2} and y^{3} is y^{3}. Putting them together, the LCM of 8x^{2}y^{2} and 12xy^{3} is 24x^{2}y^{3}.

In order to find the LCM of polynomial expressions, it is also necessary to factor them first, if they can be factored. Suppose one expression is y^{2} +5y +4 and the other is y^{2} +2y +1. The first polynomial can be factored as (y +4) (y +1), and the second one can be factored as (y +1) (y +1). The LCM is (y +4) (y+1) (y +1), or (y +1)^{2}(y +4). Expressions such as (t^{2}+16) and (t-2) have no factors in common, so their LCM is (t^{2}+16) (t-2).

In order to add expressions with unlike denominators, first find the LCM of the denominators. This can be used to form their LCD. Then write each expression as an equivalent expression by multiplying it by a form of 1 using the LCD. The numerators can then be added, and the rational expression can be simplified. Suppose the expressions are (7x^{2})/6 and (3x)/16. First, find the LCM of 6 and 16. The number 6 factors as 2·3 and the number 16 factors as 2·2·2·2. Their LCM will be 2·2·2·2·3 or 48. The expression is [(7x^{2})/6] ·8/8, or (56x^{2})/48 + [(3x)/16] ·3/3 or (9x)/48. Adding expressions over like denominators, the solution is then (56x^{2} +9x)/48.

Subtraction with unlike denominators is similar to addition with unlike denominators, with an additional step. First, find the LCM of the unlike denominators to form the LCD. Then write each expression as an equivalent expression by multiplying it by a form of 1 using the LCD. Then, subtract the numerators and write the difference over the LCD. The resulting rational expression can then be simplified. For example, the expression is [(y +2)/(y-4)]-[(y +1)/(y +4)]. First, multiply [(y +2)/(y-4)] by [(y+4)/(y+4)]. The numerator is then (y +2) (y +4). Then multiply – [(y +1)/(y +4)] by [(y-4)/(y-4). The numerator is then – [(y+1) (y-4)]. Both are multiplied by the LCM, so the new numerator is {[(y +2) (y +4)] – [(y+1)(y-4)]}. Expanding the numerator, (y +2 )(y +4 equals y^{2} +6y +8, and -[(y+1)(y-4)] equals –(y^{2}-3y-4) or -y^{2} +3y +4, or y^{2}-y^{2} +6y +3y+8 +4 or (9y +12)/[(y +4)(y-4)] or 3(3y+4)/[(y +4)(y-4)].

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Adding and subtracting rational expressions with like denominators is similar to adding and subtracting fractions with like denominators. It is a three-step process: first add the numerators, then simplify the numerator, then simplify the result by factoring. Since subtraction is the inverse of addition, subtraction is a matter of adding the inverse of the numerator.

Suppose the problem is 4x/5 + 6x/5. Since the numerators have a variable in common, 4x +6x can be added as 10x. Then (10x)/5 has the common factor of 5 in both the numerator and denominator. The simplified result is (2x)/1, or 2x. Suppose there were no variable in the expression. It would then be a fraction, so that 4/5 +6/5 would equal 10/5 or 2.

Suppose the problem is (8y^{2})/(y +3) + (2y^{2})/(y+3). The numerators have variables, and so do the denominators. In this case, the denominator in each expression is the same, so the numerators can be added as 8y^{2} +2y^{2} =10y^{2}. The expression is then (10y^{2}/(y +3). There are no factors in common, so it is already in simplest terms.

Suppose the problem is (x^{2}-4x-10)/(x-7) +(x-18)/(x-7). The denominator is the same for both rational expressions, so the numerators can be added. Combining like terms, (x^{2}-4x+x -10 -18), the x^{2} is the only exponent. Then, -4x +x equals -3x, for the variables. Then, (-10 -18) equals -28, so the numerator is now (x^{2}-3x-28), which can be factored as (x +4)(x-7). Using FOIL to check, x·x equals x^{2}, 4x-7x equals -3x, and (-7·4) equals -28. The numerator [(x +4)(x-7)] can now be placed over the common denominator (x-7). The (x-7) in the numerator and denominator cancel out, so the answer in simplest terms is (x +4).

As subtraction is adding the inverse, the process is very similar to addition of rational expressions with like denominators. The numerator is subtracted, then the rational expression is simplified. For example, the first step to solve[(4m +5)/(m-1)] – [(2m-1)/(m-1)] is to combine the numerators as [(4m +5) –(2m -1)] or 4m-2m +5 +1 or 2m +6. The expression 2m +6 is not in simplest terms, as it can be factored as 2(m +3). Then using the common denominator, the entire solution is [2(m +3)]/(m-1).

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]]>Shakespeare’s plays have been performed all over the world, translated into many other languages, and sets have been adapted beyond the traditional stage settings. In addition to movies made from the plays, there have been movies based on Shakespearean themes, operas and other musical works set to Shakespeare, as well as books, dance, and other forms of visual arts.

Shakespeare’s plays were staged elaborately when they were originally performed, with elaborate scenery, fencing, dance, music, sound effects, and even fireworks. The Globe Theatre itself burned in 1613 when a cannonball caught fire. *Hamlet* has been reset to Haiti, *Macbeth* to World War I, and *Richard III* to Nazi Germany. *The Taming of the Shrew* was reset as the Broadway musical *Kiss Me Kate*. *Romeo and Juliet *was reset to rival gangs in New York City in the Broadway musical *West Side Story*, later to be made into a movie.

There have been more than 400 movie and TV adaptations of Shakespeare plots. Many of them have been based on *Romeo and Juliet*, besides the film adaptations such as the 1968 film *Romeo and Juliet* directed by Zeffirelli and Baz Luhrmann’s 1996* Romeo + Juliet*. The movie *Shakespeare in* *Love* is a retelling of Shakespeare writing the play, and* Romeo Must Die* (2000) uses mixed martial arts to tell the story. There have also been many adaptations of other plays or Shakespearean concepts. The movie *10 Things I Hate About You* is based on The *Taming of the Shrew*. *Hamlet* has been adapted to *The Lion King*, *Let the Devil Wear Black*, and *Rosencrantz and Guildenstern Are Dead*. Science fiction based on Shakespeare includes the* Star Trek* episode “The Conscience of the King”, a *Doctor Who* episode, “The Shakespeare Code”, and the android love story *Romie-0* *and Julie-8.*

*A Thousand Acres* by Jane Smiley and *The Serpent’s Tooth* by Diana L. Paxson are among the many books based on* King Lear*. *The Daughter of* *Time* by Josephine Tey is connected to *Richard III*, *Brave New World* by Aldous Huxley to *The Tempest*, and *The Talented Mr. Ripley* by Patricia Highsmith to *Macbeth*. Every year brings new books and stories, as Shakespeare’s themes are universal.

Since music and dance were such a part of Shakespeare’s plays, many ballets have been staged to tell the stories, such as *Romeo and Juliet* with music by Prokofiev and by Tchaikovsky, *A Midsummer Night’s Dream* by Mendelssohn, and a one-act version, The Dream by Ashton, along with many others. Verdi scored the dramatic operas *Otello* and *Macbeth*, as well as the comedy *Falstaff*, based on *The Merry Wives of Windsor*. Vaughn Williams composed the opera *Sir John in Love*, also based on *The Merry Wives of Windso*r. Many paintings were inspired by scenes and characters from Shakespeare, including work by William Blake, Millais, and others.

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]]>William Shakespeare (1564-1616) is known as the greatest writer in the English language and dramatist to the world. He wrote over 35 plays, including histories, comedies, and tragedies. His sonnets are a collection of 154, published in 1609. They contain some of the most beautiful love poetry ever written.

Shakespeare wrote historical dramas that appeared in the First Folio collection. Most of these plays dealt with English history, such as *King John* and *Henry VIII*, and a complete series of plays about the Wars of the Roses: *Richard II*, *Richard III*, *Henry IV* parts I and II, *Henry V*, and *Henry* *VI*, parts I, II, and III. As Shakespeare was living during the reign of Elizabeth I, the daughter of Henry VIII and Anne Boleyn, he wrote to advance the Tudor cause and the Tudor dynasty. Richard III is seen as a monster, and the just end of the York line, *Henry VIII* ends with the birth of Elizabeth I. Many of his plays that he called comedies and some of his plays that he called tragedies are also historical in nature.

In Shakespeare’s time, comedies were plays with happy endings, sometimes with happy marriages, and they were more lighthearted in tone than the histories or tragedies. Some of those comedies include *A Midsummer Night’s Dream*, *As You Like It*, *The Merry Wives of Windsor*, *The Two* *Gentlemen of Verona*, *Much Ado about Nothing,* and *Twelfth Night.* Some of his later plays, such as *The Tempest*,* Cymbeline*, and *A Winter’s Tale* came to be known as romances. Plays such as *All’s Well that Ends Well* and *Measure for Measure* were also known as problem plays, as they mixed humor and tragedy in an unusual way.

Tragedies present characters in plots that blend death, madness, and murder. Some of Shakespeare’s great tragedies were based on historical Roman figures, such as *Antony and Cleopatra*, *Julius Caesar*, and *Coriolanus*. *Romeo and Juliet* and *Othello* used Italian narratives as sources, and *King Lear* was derived from tales about ancient kings. Hamlet was a prince in Denmark, and *Macbeth* was believed by many to carry a curse, as murder, intrigue, and deception are throughout the play.

The Sonnets are a collection of 154 poems, published in 1609. The first 126 sonnets were addressed to a young man (The Fair Youth), and the last 28 were addressed to a woman (The Dark Lady). From the time that they were first printed, readers wondered to whom they were dedicated. Were the Fair Youth, the Dark Lady, and the Rival Poet real people, characterizations, or aspects of Shakespeare’s culture? Critics have specified many different opinions on their nature.

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]]>During solar eclipses, more is revealed about the features in the sun’s corona. The corona is not normally visible to the naked eye, so scientists began studying it long before satellites were launched into space. In addition, observations during a solar eclipse brought important early proof of Einstein’s theory of relativity.

People have viewed solar eclipses throughout history. A megalithic monument in Ireland shows spiral carvings on rocks, called petroglyphs, which code the positions of the sun at various times. One of those may correspond to a total solar eclipse in 3340 BCE. In Ancient China, solar eclipses were thought to be predictions of doom to the Emperor who ruled at the time, and Chinese astrologers kept careful records of them. They thought that the sun was devoured by a huge green dragon that lived somewhere in the heavens. To scare away the dragon, the Chinese people beat pots and drums to make noise and scare away the dragon. In 2134 BCE, they described an eclipse in a phrase that translated as “the sun and the moon did not meet harmoniously.”

In Babylon, astronomers kept very careful records of the sun and moon’s position, so that they eventually used a mathematical formula to predict eclipses. Some of the eclipses they observed included one in 1063 BCE that “turned day into night”, and a famous one in 763 BCE, recorded by Assyrians in the ancient city of Nineveh. The ancient Greeks recorded eclipses as well as developing geometric predictions of when and where both lunar and solar eclipses would occur. By 200 CE, eclipses were a predictable scientific fact rather than an unpredictable sign of cosmic doom for much of the world.

**Early Observations of the Solar Corona**

The earliest observations of solar prominences were in 334 CE, and the corona was described 600 years later. By the 1700s, the corona was named as a crown around the sun, as part of the sun that could only be seen during the eclipse. The first wet plate photograph of a solar eclipse was taken in 1860, and many other discoveries followed.

A new line was found in the sun’s spectrum during an eclipse in 1868, and the chemist named it “helium” after the ancient Greek word for the sun, Helios. The element helium was not formally discovered on earth until 30 years later. On May 29, 1919, Sir Arthur Eddington carried out an experiment during the solar eclipse that was an early proof of Einstein’s theory of relativity. The positions of stars during the solar eclipse were carefully measured and compared with measurements of those same stars in their normal positions (not during a solar eclipse). Those careful measurements showed that space-time was warped by the gravity of the sun, exactly as predicted by Einstein’s equations. Eddington’s findings have been replicated in experiments elsewhere in the world, and scientists will also use the solar eclipse of August 21, 2017 as another opportunity to simulate his famous experiment, 98 years later.

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]]>On Monday, August 21, 2017, many people in North America will be able to see either a partial solar eclipse or a total solar eclipse, depending on where they live. Parts of South America, Africa, and Europe will also see a partial solar eclipse.

During a solar eclipse, the moon passes between the earth and the sun. It blocks either all of the light from the sun or part of the light from the sun during the time it takes for the moon to travel across the face of the sun. The darkest portion of the moon’s shadow, the umbra, falls along a narrow path on the earth, for a total solar eclipse. The penumbra, or half of the moon’s shadow, falls on either side of the umbra. The sun is much larger than the earth or the moon, so it produces both umbra and penumbra.

The path of totality is a 70 mile wide region that stretches from Lincoln Beach, OR through Oregon, Idaho, Wyoming, Montana, Nebraska, Iowa, Kansas, Missouri, Illinois, Kentucky, Tennessee, Georgia, and North and South Carolina. Other regions north and south of the path of totality will see a partial eclipse. Although the sun appears to be covered for about two minutes, the entire eclipse from start to finish takes about two hours. Many different websites on the Internet show whether the eclipse will be total or partial based on the location of the city.

A partial eclipse begins when the sun appears to be partially blocked by the moon. That phase can last over an hour, and the sun looks like a partially-obscured crescent. The next phase before totality is often called the “diamond ring” because a single brilliant point of light shows at one place, while the corona of the sun forms a ring. The phase just before totality is referred to as “Baily’s beads” which are shining points of sunlight that are formed from the hills and valleys of the moon as it almost obscures the sun. During totality, the moon covers the entire disk of the sun and only the corona shows. In the final stages, the growing crescent sun shows as the moon moves away from blocking it. If the eclipse is a partial eclipse, none of the phases just before totality show.

Do not look at the sun directly without proper eye protection. Regular sunglasses do not provide enough protection, so special “eclipse glasses” are readily available. There are also many ways to view the eclipse indirectly, such as by using a sun funnel to reflect light coming from a telescope, projecting the image of the eclipse onto a screen, or use a pinhole projection.

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]]>The process of multiplying and dividing rational expressions is similar to multiplying and dividing rational numbers. In order to multiply, first multiply the numerators, then the denominators, and then simplify the expression. Division is the inverse of multiplication, so multiply by the reciprocal (or inverse) of the divisor, and then simplify the expression.

Rational numbers are also known as fractions. The fraction ¾ is a rational number, as is the fraction 3/5. The numerator 3 is in a ratio to the denominator 4. To multiply ¾ by 3/5, multiply the numerators 3·3, and then multiply the denominators 4·5, to result in the new fraction or ratio of 9/20. The fraction 9/20 is in the simplest form, because there are no factors common to both 9 and 20 except for 1. Suppose one ratio were a/9 and another were 7/10. The process of multiplication would be similar, so that a/9 times 7/10 would equal 7a/90.

Suppose the problem were (5a^{3})/4 times 2/ (5a). The numerator is 5a^{3}·2 or 10a^{3}, and the denominator is 4 ·5a or 20a. The new expression, (10a^{3})/20a can be simplified to ½ ·a^{2} or a^{2}/2. If the problem were 4/ (5x^{2}) ·(x-2)/ (2x^{3}), the new numerator would be 4(x-2) and the new denominator would be (5x^{2}) · (2x^{3}) or 10x^{5}. The new expression is then [4(x-2)]/ (10x^{5}), which is not in simplest form. The fraction 4/10 can be simplified to 2/5, so the expression in simplest form is [2(x-2)]/ (5x^{5})].

Remember that dividing rational numbers is the same as multiplying by the reciprocal of the divisor, so that 4/5 ÷2/3 is the same as 4/5 ·3/2, so that 4·3 equals 12 and 5·2 equals 10. Since 2 is a common factor in both the numerator and denominator, the fraction in simplest terms is 6/5. There are no common factors to both 6 and 5 except for 1 so the fraction is in simplest terms.

Similar to dividing rational numbers, when dividing rational expressions, also multiply by the reciprocal of the divisor. Therefore (8n^{5})/3 ÷ (2n^{2})/9 becomes (8n^{5})/3 ·9/ (2n^{2}). Multiply [(8n^{5}) · 9], the numerator and [3· (2n^{2})]. However, the resulting expression (72n^{5})/ (6n^{2}) is not in simplest terms. First, factor out the common coefficients, so that 12 is left. (The number 72 divided by 6 equals 12.) Next factor out the common variables, so that n^{5}/n^{2} equals n^{3}. The quotient in simplest form is 12n^{3}. Likewise, suppose the problem were [(4m-8)/5] ÷ [(m-2)/10]. The new expression would then be [(4m-8)/5] · [10/ (m-2)]. The new numerator can be factored as [4(m-2)]10, and the new denominator can be factored as 5(m-2). Since (m-2) is a common factor for both the numerator and the denominator, (m-2)/(m-2) equals 1 and cancels out, leaving 4(10)/5, or 8.

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