Varying the types of sentences used within paragraphs can lead to effective writing. Some types of sentences include the simple sentence, the compound sentence, the complex sentence, and the compound-complex sentence.

The simple sentence contains one independent clause. It can be as brief as a subject and verb, such as “Birds fly.” Even if modifiers are added to the subject and the verb, such as “Many different migratory birds fly long distances in the winter”, it remains a simple sentence. The sentence can have more than one subject, such as “Birds and insects fly” or more than one verb, such as “Birds walk and fly, “ but it is still a simple sentence, because it constructed with only one independent clause.

The compound sentence contains more than one independent clause, such as “Birds fly, and fish swim.” As long as there are no dependent clauses in either part of the sentence, it can be modified many different ways. For example, “Migratory birds fly long distances in the winter, and some fish swim long distances to spawn.” It is very important to make sure that both parts of the compound sentence are joined by either a coordinate conjunction such as **and**, **but**,** or**, **so**, or a semicolon, in order to avoid run-on sentences.

The complex sentence contains only one independent clause and one or more dependent clauses. For example, “Although many types of birds nest in a small territorial area, migratory birds fly long distances in the winter.” The dependent clause is not a separate sentence, so it cannot stand alone. One common writing error is to separate the dependent clause from the rest of the sentence, ending it with a period. In order to correct this error, either recast it as an independent clause by removing the subordinating conjunction **although**, or connect it with the independent clause.

The compound-complex sentence contains more than one independent clause, as well as dependent clauses that modify them. For example, “Migratory birds fly long distances in the winter rather than remain under adverse conditions; but some fish swim long distances in order to spawn.” Some words that signify dependent clauses are subordinating conjunctions such as **before**, **after**, **rather than**, **if**, **than**, **unless**, **until**, **whether**, **when**, and **while**.

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]]>Data in real-world situations doesn’t always fall into neat linear relationships. The actual values may be plotted in a scatterplot. Trend lines approximate the direction and size of the relationship between data points, as do median-fit lines.

Actual data values can be plotted by ordered pairs on a graph called a scatterplot. The domain or independent value is on the x axis and the range or dependent value is on the y axis. The more data values there are available, the more can be plotted, even if they don’t fall on a line. Scatterplots are useful because they show the range and direction of many different data points.

Scatterplots are also useful because they can show the relationships between variables, such as height and weight, average temperature and power usage, or hours of study and test scores. The relationship between two different types of data is called correlation. If the values for both types of variables increase, the correlation is positive. Suppose a seed is planted and the height of the plant is measured every day. There is a positive correlation between the height of the plant and the number of days since it was planted, because the plant grows. If the values along one axis increase while the values along the other axis decrease, the correlation is negative. For example, as it is darker outside, the more light is needed inside. The darkness outside increases, while the light inside increases. Some variables have no relationship, called a zero correlation. For example, there is probably no correlation between the number of sunny days in Seattle, Washington with the number of cars sold in Denver, Colorado. The scatterplot for those two variables will not show a clear relationship.

Once data points have been plotted on a scatterplot and the direction of a correlation has been observed, a line can be drawn estimating that relationship, called a trend line. It may or may not go through specific data points, and usually has as many points showing above the line as below the line. If data are plotted on a graphing calculator or a computer spreadsheet, the trend line is often calculated automatically.

Median-fit lines are specific types of trend lines that use the values found in the data set. First, all values in the data set can be plotted on a graph as a scatterplot to see the direction of the correlation. Divide the data into three groups with equal numbers of data points in each group. If there are not three groups of equal size, make sure the first and third groups are equal. Next, find the median value for the x values and y values for each group. Plot the median values on the same graph and draw a line connecting the median points of the first group and the third group. The median-fit line will be parallel to that line.

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]]>A function is a special relationship between variables in an equation such that for every domain value, there is only one range value. Functions can be expressed using a function rule that defines those values. Once enough values have been derived, the function can be graphed.

In order to begin thinking about writing a function, first determine how the variables are related. Does the value of one variable depend upon the value of another? Suppose a plumber charges $75.00 per hour plus a labor charge of $50.00. The cost of the repair depends upon the number of hours it takes. Therefore, if the repair takes 1 hour, the cost of the repair is $125.00, and if it takes 2 hours, the cost is $200.00, and so on. Another way to say that is that the cost of the repair is the dependent variable, and the number of hours is the independent variable. Suppose that a car gets 30 miles to the gallon. The number of gallons used depends upon the number of miles driven. If the car is driven 30 miles, it uses one gallon; 60 miles, 2 gallons; and so on.

After the independent and dependent variables have been identified, the next step is to write an equation. In the plumbing example, y, the cost of the repair, equals $75.00x + $50.00, where x is the number of hours. Since $75.00x + $50 defines the function, it is the function rule. Similarly, the number of gallons used, y, equals the number of miles driven, x, divided by 30. The function rule of x/30 defines the function. Function notation is substituted for the dependent variable y, so that f(x) = $75.00x + $50.00, or f(x) =x/30. Functions are usually represented with the letters f, g, or h, so they may be expressed as f(x), g(x), or h(x).

The first step to graphing a function is to organize a table with values of x and y. It is better to use many values to show a pattern. Suppose that the function is a simple one, such as f(x) = 10x. Then if x =1, f(x) = 10; if x=2, f(x) = 20, x =3, f(x) =30, and so on. If the domain of a function is given in a problem as specific values of x, only use those values, but if it is not specified, the domain is all real numbers.

After the table with values of x and y is generated, points can be plotted using Cartesian coordinates. Once the pattern is established, a line or smooth curve can be drawn through them. If the domain of x is all real numbers, the endpoints of the line can be drawn as arrowheads.

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]]>The sport of soccer is popular worldwide. Like other sports, the ball and the players follow laws of physics as games are played, especially Newton’s Laws of Motion, friction, momentum, and aerodynamics.

Newton’s First Law

According to Newton’s First Law of Motion, a soccer ball will stay at rest unless a force of some sort moves it, and it will stay in motion unless a different force stops it. The force that usually moves the soccer ball is the player’s kick. Another player blocks its motion. The game would be very boring if the eleven players of one team and the eleven players of the opposing team stood on opposite sides of the field and the ball stayed at rest.

Newton’s Second Law

According to Newton’s Second Law, the force behind the soccer ball equals its mass times acceleration, in the equation F =ma. A hard kick will move the soccer ball farther and faster than a soft kick. The acceleration of the ball depends upon how much force behind the kick.

Newton’s Third Law

According to Newton’s Third Law, for every action there is an equal and opposite reaction. The player kicks the soccer ball, and the ball “kicks” back, but the player doesn’t feel the reaction because the player’s leg has more mass and force than the soccer ball.

The soccer ball doesn’t continue to move indefinitely because of friction between the ball and the grass as it bounces across the field. Players use friction to slow the motion of the ball, as a way to control it and keep it from going out of bounds. When the ball is kicked off-center, the friction between the air and the ball creates curvature of motion.

The players are constantly in motion, and transfer momentum to the ball by the motion of their feet, so that the ball goes in the direction they want. They can kick the ball in such a way to spin it, making it harder for opposing players to stop its motion. Players can use the momentum of the ball to control its motion.

A soccer ball is a sphere that has less resistance and drag from its motion through the air. Its center of gravity is in the center of the ball. The official World Cup soccer ball looks somewhat different than the typical polygon-tiled soccer ball. The panels are joined with deeper seams and covered with tiny bumps to maximize the airflow, so it can go faster and straighter.

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]]>Students read and study different types of text, including expository text, persuasive text, expressive text, and literary text. Different types of text have different purposes, and understanding the author’s purpose in writing helps in active reading.

Most of the textbooks and reference books students are required to read are expository text, written to explain a concept or give information about a topic. Usually the purpose is stated in clear objectives. Paragraphs may follow a chronological order in a clear time sequence, or they may follow a series of steps, describing a process. When headings and subheadings are used in textbooks, use them as a study guide to question and summarize what is learned. For example, a chapter in a biology textbook on the science of biology might include headings about a definition of science, how scientists work, characteristics of living things, and tools and procedures used in experiments. Subheadings under how scientists work include designing experiments, publishing and repeating investigations, and how a theory develops.

Persuasive text is written in order to sway the reader’s opinion to the author’s point of view. It may be as brief as an advertisement or as involved as a political document. The goal of persuasive text is a call to action. Editorials may be slanted towards a particular opinion, and present reasons why the reader might support or oppose a candidate or activity. Often, writers will use comparison and contrast to support their position. Persuasive text can be straightforward, or it can be distorted to the point of propaganda.

Expressive text is written in order to convey emotional content from the author’s own point of view. Usually, expressive text is intended to be private, such as letters, journals, and diaries. However, when those letters are exchanged between famous people, such as the letters between John and Abigail Adams, or the journals and diaries are written during a particular time in history or by famous people, they are no longer private.

Literary texts are writings such as novels, short stories, drama, poetry, and literary biographies and autobiographies. They often convey moods through description, keep the reader engaged through a fast-moving plot and memorable characters, and emphasize the finished product. Readers may be entertained, enthralled, and encouraged to find meaning within themes and characters.

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On July 14, 2015, the New Horizons probe flew past Pluto in its closest approach, over 7700 miles or 12,500 km from the dwarf planet. From the time it was launched in January 2006, it has flown an incredible 3 billion miles from Earth. The distance is so far away that pictures and messages from the spacecraft take more than 4 hours to reach scientists. Much of the data from the probe will take months to process. Telemetry was received from the probe at about 9 PM EDT, showing that it was not damaged in the flyby.

Photographs taken of Pluto and its moons Charon, Nix, Hydra, Styx, and Kerberos show an incredible amount of detail, and those are just the ones that have been seen during New Horizons’ approach to Pluto. During the time immediately after the flyby at 7:49 EDT, all cameras and other devices on the probe were so busy that antennas were not turned in the right direction to send a message back. In order to capture all the features of the planetary system, cameras on the probe had to twist and turn many different ways.

The New Horizons probe is about the size of a baby grand piano, traveling through space at more than 13 km/second. It is the fastest spacecraft ever launched, and got an extra boost from the planet Jupiter as it flew by the giant planet. It has seven different types of instruments. Three of them are optical: a long-range camera LORRI, an ultraviolet spectrometer ALICE, and a telescope RALPH. Plasma sensors SWAP (Solar Wind at Pluto) and PEPSSI (Pluto Energetic Particle Spectrometer Science Investigation) are joined by the SDC (Student Dust Counter), built by students at the University of Colorado at Boulder, and REX, the Radio Science Experiment. A very detailed itinerary of exactly which instruments are operating can be found on the NASA New Horizons website.

The size of Pluto has been revised from information sent back. At 2370 km, it is larger than the next largest Kuiper Belt Object (KBO), Eris. It has a polar ice cap made of methane ice, different than the ices at the equator. Pluto also has a thin atmosphere of nitrogen that is larger than scientists thought it would be. As the probe traveled closer to Pluto, features such as a heart-shaped patch probably made of methane, nitrogen, and carbon monoxide snow could be clearly seen. Pluto’s largest moon Charon is over half its size, at 1,208 km. It has huge chasms and craters. Pluto and Charon orbit around a common center of gravity, and the other moons orbit Pluto and Charon.

After the Pluto flyby, NASA scientists hope to extend the mission to other objects within the Kuiper Belt. There are two nearby objects that it will take until 2019 to reach. The radioactive power source, powered by plutonium, is projected to last until the 2030s, when the battery will have decayed so much it will no longer be able to transmit data.

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]]>It is very important to have a good vocabulary in order to do the amount and type of reading required in high school and college. Some strategies for building vocabulary include understanding root words, focusing on specialized vocabulary, and using a variety of reading skills to determine meaning.

The English language is a hodgepodge of words borrowed from other languages, coined words, acronyms, and idioms. Sometimes, the same word will have different meanings depending upon how it is used within a sentence or phrase. Compare the difference of the word **run** between the sentences “He will run in a race”; “There was a run on the bank during the Great Depression”; and “She had a run in her stocking”. Many words have changed meaning over time. Hundreds of years ago, the word awful meant awe-inspiring, not horrible, and terrible had the connotation of fearsomeness rather than rottenness. Similarly, to call someone **silly** in Shakespeare’s time meant they were blessed, and to call them **stupid **meant they were amazed.

Root words give clues to the meanings of words, as do prefixes and suffixes. For example, the root –spect is from a Latin word meaning “to look”. Other related words include spectacle, inspect, spectator, and introspection. The prefix in + spect leads to a starting definition of looking into something, while the suffix –tor leads to a starting definition of one who looks. An actor is someone who acts, with a similar suffix.

Each content area has words in common with others, but also has a specialized vocabulary. The words **gravity**, **work**, and **force** have a general meaning, as well as specialized definitions in physics. The word **affect **is a noun in psychology specifically referring to emotions. Elements are specific substances in chemistry. Reading and understanding texts in areas as diverse as English, mathematics, history, and the sciences depends upon adding those specialized word meanings and definitions to the vocabulary. As people move from novice to expert within a particular field, they become more comfortable with the many different instances and nuances of its specialized vocabulary.

Students in high school and college use many of the same reading skills to develop vocabulary as younger children, just at a more automatic level. A reader might first approach an unfamiliar word by guessing its meaning by the context of the passage around it. If the meaning isn’t immediately clear, the reader might break down the structure of the word into roots, prefixes, and suffixes, or sound it out syllable by syllable. The reader might check their understanding by using a dictionary or thesaurus as a starting point.

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]]>Relationships between data points can often be represented on graphs. They can also be expressed in ordered pairs as relations or functions.

One of the many ways that relationships between the values of variables can be represented is on a graph. Traditionally, the x axis is the horizontal axis (also known as the independent variable), and the y axis is the vertical axis (also known as the dependent variable). The different relationships between variables can be described by key words. For example, a horizontal line stays at the same value, such as a temperature staying constant. If a line slants upward, the value is rising, such as the temperature rising during the day to reach its highest value. If a line slants downward, the value is falling, such as the temperature falling after sundown towards its lowest point. Graphs are often read from left to right, because the x axis is measured from 0 to the highest value.

Another way that relationships can be measured on a graph is whether the values are measured and plotted continuously or in a discrete sequence of individual items. Suppose a neurologist is studying a patient’s EEG pattern. Typically, the graph produced by the EEG is a continuous reading of brain waves, showing many peaks and valleys. In contrast, suppose that same neurologist needs to determine if a patient is taking medication on time. She may order blood tests measuring the level of medication every three hours. The level of medication in the patient’s blood can be plotted as separate values on a graph, with discrete measurements in a sequence showing when the patient takes the medication, at least during the time the blood tests are being taken.

Although relationships between the values of variables can be plotted on graphs, they can also be represented as ordered pairs in brackets, tables, or diagrams. They are presented with the values of x first and the values of y second, such as {(1, 2) (2, 4) (3, 6)}. The values of x {1, 2, 3} are known as the domain, and the values of y {2, 4, 6}, the range.

Functions are a special type of relations in which every domain value can be paired with one range value. The set {(1, 2) (2, 4) (3, 6)} is a function. The set {(1,2) (2,2) (3,2) } is also a function, although it isn’t as interesting. The set {(1,2) (1,3) (2,2)} is not a function because the domain value 1 has 2 range values 2 and 3. To determine if a relation is a function, examine the values for the x variable. If a particular value of x is paired with more than one value of y, it is not a function. Therefore, if more than one x has the same value, the relation is not a function.

Function equations express the relationship between variables. Suppose a person makes $10.00 an hour. The first hour they work, they make $10.00, the second hour, $20.00; the third, $30.00 and so on. One way to express the relationship in an equation is that the time worked in hours (x) times the amount per hour ($10.00) equals the amount earned (y), or $10.00x =y.

Function notation is a special way that functions are represented in math language. It is a shorthand to describe the relationship between variables. The equation $10.00x =y represents a function where the amount of money that is earned depends on the amount of time worked, or f(x) = 10x.

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]]>Percents are based on the decimal system and have many useful applications in business, sales, and finance. Other applications of percents include percent increase and decrease.

Many of the most useful applications of percents involve estimating and calculating percentages for sales tax and tips at a restaurant. Suppose the sales tax is 8.7% in a particular town. That means that for any taxable item, the cost to buy it is the cost C plus .087C, calculated automatically at the cash register. It is always better to estimate high to make sure there is enough money for a purchase, and it is easier to use a value such as 10%. Then, the price of that item is the cost C plus .10C. If the cost of a CD is $15.99, estimate $16.00, and add 10% of $16.00 or $1.60, so the estimated cost will be a little less than $16.00 + $1.60 or $17.60. If the cost is higher, the estimated tax will also be higher. Suppose a DVD is $24.00. Its estimated price with tax will be $24.00 + $2.40 or about $26.40.

Tips

Estimating tips works on the same principle, except the percentage of tip to leave is up to the diner, and depends on the quality of service and the type of service. Suppose the waitperson brings the check and it is $25.36, including tax. The quality of service is good, and the diners decide to leave a 20% tip. The cost of the meal will then be the price P plus 0.20C. It is easier to estimate the tip by adding 0.10C + 0.10C, or $2.54 + 2.54, so the cost of the meal is $25.36 + $5.08 or $30.44.

Commissions are typically a percentage of total sales and are either added to a base salary or paid as commission only. Suppose that Lori works in a salon, and her employer pays her $9.50 an hour plus a 1.5% commission on all products she sells. Her gross wages after a 40-hour work week are $9.50 times 40, or $380.00. Suppose she sells $125.00 in products during that week. Her commission will be $125.00 times 0.015, or $1.88. (When using percents in a calculation, change the percent to a decimal by multiplying by 1/100 and using decimal points.) $380.00 +$1.88 = $381.88.

Simple interest follows a formula of I=Prt, where I is the amount of interest, P is the principal amount of money invested or loaned, r is the interest rate, given as a percentage, and t is the amount of time in years. It is only paid on the principal. In order to find the amount of simple interest paid on a loan of $1000.00 at a rate of 15% over 2 years, first substitute the known values in the formula such that I = $1000.00 (0.15)(2). Again, the 15% is changed to a decimal 0.15. Solving the equation, I = $300.00.

Percent change refers to the amount of increase or decrease given as a percentage. Percent increase is often referred to as a markup, and percent decrease is a discount. Suppose the regular price of an item is $12.00 and it is being sold at 30% off. In this case, the sale price is equal to the regular price P minus the discount of 30% P, or the sale price is equal to $12.00 – (0.30) P or $12.00 – 3.60 = $8.40.

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]]>Rainbows, glories, fogbows, and glassbows are all related phenomena caused by the nature of light. Each color is created by light hitting raindrops or beads of glass. They have fascinated people for millennia, including Aristotle and Sir Isaac Newton.

Rainbows are phenomena that are created whenever the conditions are right for them, according to physical laws of optics. The light source must be behind the observer, water droplets must be in the air in front of the observer, and there must be a 42 degree angle between the observer and the water droplets. The 42-degree angle produces the wavelength for red, and a 40-degree angle produces violet, but blue is the easiest to see for most. Light refracts through the water droplets as it enters, is bent into a spectrum of colors, reflects, and refracts again as it exits. Different droplets reflect the light at different angles, so some reflect red while others reflect the other colors. Although the eye separates the different colors into bands, colors are actually in a continuous gradation.

How Rainbows Are Made

Rainbows can be produced by any source of water droplets – raindrops, the arc of water from a garden hose, the mist from a waterfall, or spray from the wake of a water skier, speedboat, or pounding surf. They appear circular rather than straight because they are part of the circumference of a circle with the center on a line with the observer’s head, their shadow, and a point below the horizon called the antisolar point. Half the rainbow is below the horizon and invisible to the observer. The size of the rainbow depends on the amount of raindrops producing it.

Types of Rainbows

Most rainbows consist of a primary rainbow, with red at the outside and violet on the inside, and a secondary rainbow, with the colors reversed so that violet is on the outside and red is on the inside. This is because the light is reflected twice by some raindrops. The secondary rainbow appears outside the primary rainbow and is much fainter, because it is produced by fewer raindrops. The area between the primary and secondary rainbows appears darker than the rest of the sky, because of the way light is reflected from raindrops at a greater angle than 42 degrees.

Circular rainbows can be seen if the observer is in an airplane or at a high enough elevation that they can see the entire circle. Some observers consider the circular rainbow that can be seen from an airplane window as a glory. In that case, the plane is flying low enough above cloud cover that its shadow is visible. It can also be observed from high mountain peaks, and the ancient Chinese believed that a glory around someone’s head in shadow was a sign of enlightenment.

Rainbows that appear from fog, such as the bows that are created by car headlights, appear white rather than brightly-colored. That is because the droplets that make up fog are smaller than colored wavelengths. Sometimes faint reddish or bluish colors can be seen.

Rainbows can also be made by refraction and reflection from tiny transparent glass beads, such as those used in sandblasting or art projects. The transparent spherical beads take the place of raindrops, and are glued close together on a piece of black poster board 12 inches square. It is best to use a spray adhesive to adhere the beads to the poster board or black foam core board, and work outside to avoid the fumes and most of the mess. Students can then use strong sunlight outside or a spotlight inside to cast the bow from the transparent beads.

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