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Our education staff publish regular articles, tips and tutorials to help students with their homeworkThu, 12 Jan 2017 18:04:36 +0000en-UShourly1https://wordpress.org/?v=4.6.175589453Math Review of Solving Systems by Substitution
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Wed, 11 Jan 2017 16:12:39 +0000http://schooltutoring.com/help/?p=9076Overview
One of the ways to solve systems of equations is by graphing the equations. However, graphing the equations is not always the most accurate method to solve them. If one variable in a system is represented in terms of the other variable in the system, the systems can be solved by substitution.

Using Substitution

Suppose one of the equations in the system is x + y = 5 and the other equation is x = y +1. The expression y +1 can be substituted for x, so that y +1 +y =5. Then, there is just one variable so that 2y +1 =5, 2y +1 -1 = 5-1, or 2y = 4, or y =2. In order to check, substitute the value of y to solve for x, such that x +2 = 5, or x +2-2 = 5-2, or x = 3. Check the second equation also, so that 3 =2 +1. That is the way to use substitution to solve a system of equations.

Isolating the Variables

Sometimes, the variables cannot be isolated as easily in a system of equations, but the system of substitution can still be used. Suppose the equations were x-2y = 8 and 2x +y = 8. The first equation can be rearranged such that x = 8 +2y. Using substitution, the second equation then becomes 2(8 +2y) +y =8, or 16 +4y +y =8. As before, there is only one variable, such that 5y = 8-16 or 5y=-8, or y = -8/5. Again, check the value of x, so that x – (2)(-8/5) =8, or x +16/5 =8. (Notice how the sign changes when two negative values are multiplied.) Then multiply both sides by 5, so that 5x +16 = 40, or 5x =24 or x = 24/5. To check the first equation, 24/5 – 2[-8/5] equals 24/5 +16/5 = 40/5, or 8. To check the second equation 2 (24/5) – (8/5) = 48/5 – 8/5) = 40/5 = 8.

Understanding the Problem and Developing a Plan

Math problems that are written in words can often be translated into systems of equations, then solved by using substitution. Suppose the statement were “The sum of two numbers is 82. One number is 12 more than the other. What is the larger number?” The first sentence can be represented by the equation x +y = 82. The second sentence can be represented by the equation x=12 +y.

Problem-Solving: Solving the Problem and Checking the Answer

To solve the problem, take the system of equations and use substitution, so that 12 +y +y = 82, then 2y = 82-12, or 2y = 70, then y = 70/2, or 35. Using the second equation to solve for x, 12 +35 = 47, and using the first equation, 47 +35 = 82.
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Thu, 27 Oct 2016 23:43:51 +0000http://schooltutoring.com/help/?p=8239Overview
The SAT consists mostly of multiple-choice questions, except for the essay and some mathematics questions that require an answer grid. Each multiple-choice question has 5 alternative answers. Four of them are distractors, and one alternative is the correct answer.

Following Directions

It is very important to follow directions exactly when taking the SAT or any other standardized test. Study the directions for each type of test as part of preparation before taking the SAT. Bring several Number 2 pencils and erasers on the day of the test, and remember to fill in the bubbles on the answer sheet completely, so that the machine will read the answers that are chosen. Use the approved calculator for the math portion, and practice with it before the test so that it is familiar. The essay portion must be written legibly in pencil, so that it can be scanned.

Multiple-Choice Format

Most of the questions are in multiple-choice format, with 5 possible answers for each question. All the math questions in multiple-choice format range from easy to hard. The multiple-choice questions on the critical reading portions refer to that reading portion alone. The advantage for students using the multiple-choice format is that the correct answer is already one of the alternatives.

Eliminating Alternatives

Raw scores on the SAT are calculated so that each correct answer receives a full point, each incorrect answer subtracts ¼ of a point, and each unanswered question receives no points. That formula is supposed to discourage students from random guessing. However, students can narrow the field by eliminating distractors that are clearly incorrect. Suppose a student can confidently state that 2 of the 5 possible alternatives on a question are incorrect. The probability of getting any one of the three alternatives correct is 33%, which is already higher than the 25% incorrect-answer penalty. If 3 alternatives can be eliminated, the probability is 50%. Skillful preparation for the SAT ensures students can eliminate alternatives, and thus raise their SAT scores.

Choosing the Best Answer

Students can use their answer books as scratch paper during the test, and are encouraged to do so. That way, they can draw diagrams for geometry problems, cross out incorrect alternatives, underline key words in sentences; and do anything to help choose the best answer for every question. Students can also circle questions in the answer book that stump them, as a reminder to go back and try again if there is extra time. During SAT preparation, individual students can learn the methods and strategies to help them achieve the best scores possible on the SAT, and get into the colleges of their choice.
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Tue, 18 Oct 2016 01:17:53 +0000http://schooltutoring.com/help/?p=8204Overview
Curling and bobsled are winter sports that rely on friction against an icy track or surface. In curling, a special stone is moved by a combination of momentum and friction against a sheet of ice. Team players use a variety of strategies to determine which of their stones will score the highest. In bobsledding and related sports, two and 4 person teams gain maximum velocity at the start, steering their sleds to minimize the effects of gravity, wind resistance, and drag.

Curling

The sport of curling probably originated in Scotland in medieval times. Weavers used their large granite stones from warp beams to skim across the ice. They built special shallow ponds and used frozen rivers. The object of the game is to move the stone down the pond to a target, called “home.” Once the team member pushes off the block to give momentum to the stone, it is not kicked or thrown. The team captain, or “skip”, guides the stone with one broom, while two other players sweep the ice back and forth in front of the stone with special brooms, generating enough heat from friction to melt a thin film of water for the stone to glide upon. If stones collide, they exchange momentum.

Equipment for Curling

The stones weigh around 45 pounds, and are made from a type of granite that resists water, so any melting ice becomes the glide path. Absorbed water would slow the stone’s movement. At one time, curling brooms were made of corn husks, but curling brooms used today are made from synthetic materials to stand up to the rapid sweeping action across the ice.

Bobsledding

The winter sport of bobsledding calls for two or four person teams. They push an aerodynamically-designed sled down a 50 meter start course, jump into the sled, keeping it steady and straight, and careen down the course against the force of gravity. The force of gravity can reach as much as 5G, similar to the forces on fighter pilots.

Minimizing Drag and Air Resistance

After the bobsled team pushes off the sled, no further acceleration is possible, as the vehicles are not motorized. The smooth design of the bobsled, as well as the rubber surface of the suits competitors wear, are designed to minimize the amount of drag and air resistance that would slow them down.
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Mon, 15 Aug 2016 21:55:49 +0000http://schooltutoring.com/help/?p=8183Overview
Three-dimensional solid figures can be represented by the two-dimensional pattern of polygons that create them. The pattern, called a net, is a visual representation that illustrates the formula for the surface area of the three-dimensional figure. If the net were folded, it would produce that figure.

Representation of Cylinders

The net for a cylinder consists of two circles adjoining a rectangle. This relates to the formula for the surface area of a cylinder; 2 πr^{2}+ 2πrh. The 2 πr^{2} is the formula for the area of the two circles. In order for the rectangle to fit around the circumference of the circle, the width of the rectangle is 2πr. The height of the rectangle is the height of the cylinder. Since the formula for the area of a rectangle is length multiplied by width, the area of the rectangle will have the measurement 2πrh.
Representation of Rectangular Prisms
The net for a cube consists of the six square faces that make up the cube. A square has the same length, width, and height. It is a special type of a rectangular prism, also known as a rectangular cuboid. A rectangular prism also has 6 faces, in three parallel pairs that meet at right angles. The top and bottom faces are congruent, as are the two other pairs of opposite sides. The formula for the surface area of a rectangular prism is 2(lw +wh +lh), which the net illustrates perfectly.

Representations of Pyramids

Pyramids are solid figures with triangular faces that meet at a single point called an apex, and a polygon base. A tetrahedron is a special type of pyramid with 4 triangular faces, and a regular tetrahedron, with all triangles equilateral and congruent, is a Platonic solid. Another type of pyramid has a square or rectangular base and three triangular sides. The net that illustrates the pyramids has the base bounded by triangles on each side. A regular tetrahedron has a net with all four triangles inside a larger triangle.

Representations of Other Figures

Many other solid figures can be represented by their nets. For example, a cone with a circular base is represented by the circular base adjoined by a quarter circle. Solid figures have been extrapolated into more than three dimensions. A tesseract is a four-dimensional figure with three-dimensional faces. It has been used in surreal art, science fiction, music, and popular culture.
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Sun, 24 Jul 2016 21:03:33 +0000http://schooltutoring.com/help/?p=9048Overview
Writers use basic structures to put their ideas in order and present them. Some of those basic structures include chronological, sequential, comparative, causal (or cause-and-effect), categorical, and evaluative.
Chronological and Sequential
The simplest way to order ideas is to present them as they happen in time, with a beginning, middle, and end. Most stories and novels follow this sequence, with early events followed by later events. For example, an essay on the growth of industry may discuss developments in the 1700s, followed by events in the 1800s, and events in the 1900s. Another related structure is sequential, where items are discussed from step to step (or sometimes in reverse). Suppose a writer is discussing how to play Pokemon Go as a beginner to the game. First, he or she might talk about signing up for the game; then, choosing and customizing the avatar; next, entering the animated map; then traveling to stops in the real world in order to capture each Pokemon. Writers will give transition clues in this structure with words such as first, second, third, next, and finally.

Comparative

Comparative structure is used to compare and contrast ideas, and often the most relevant ideas are discussed first, with the others discussed afterwards. This structure may be used when answering a “compare and contrast” essay. Suppose the writer were asked to compare and contrast igneous, sedimentary, and metamorphic rocks. He or she might discuss the ways they are similar in composition and structure, before describing differences in the way each type of rock is formed. Writers also use comparative structure when comparing and contrasting ideas for a debate, to explain why their side of the argument is stronger than that of the opposing side.

Casual

Writers develop this structure to describe causes and effects. For example, a writer may develop an essay on air pollution by discussing the different chemicals that cause air pollution, as well as how those chemicals are produced, before discussing the health and environmental effects of air pollution. Causes are discussed before effects, and writers may also discuss the solutions to the problem.

Categorical and Evaluative

In categorical writing, the order of ideas is less important than in the other types. For example, a writer is writing about different types of trucks. They may choose to write about GMC trucks, then Ford trucks, or Ford trucks, then GMC trucks. Similarly, in evaluative writing, writers may choose to present arguments for a particular position, against a particular position, and neutral to that position, or in any other order.

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Sun, 17 Jul 2016 19:44:31 +0000http://schooltutoring.com/help/?p=9043Overview
One of the ways to help the reader understand the points made or the direction the argument is going is by the use of transitions. Transition words and phrases signal illustration, contrast, continuation, or conclusion.

Illustration

One of the ways to develop an idea is by illustrating examples. They add more information about a thesis, reinforcing it, agreeing with material that has gone before. For example, many people say that summer is their favorite season. First, the weather is pleasant, so that people can get outdoors and enjoy favorite activities. Second, some people have vacations during the summer, so they can travel to other places like the beach. In addition, people can dress more casually and comfortably. Words and phrases such as first, second, for instance, and for example, signal to the reader that the illustrations continue the previous idea.

Contrast

However, another way to develop an idea is by using contrasting examples. Contrast shows that there is another way of looking at an idea by pointing out alternatives, changing direction. On the other hand, other people prefer winter, because they can take part in winter sports like skiing, snowshoeing, or ice skating. In contrast, winter vacations can be less expensive and destinations less crowded. Unlike the heat of summer, the cool of winter brings the beauty of blanketing snow. Words and phrases such as on the other hand, in contrast, unlike, otherwise, and however signal contrast to the reader.

Continuation

Writers can also continue with earlier points made in the essay. Transition words such as especially, furthermore, and moreover allow the reader to stop and consider further points. Some people especially like summer because of the long, sunny days, allowing them more time to spend outdoors. The warm weather allows many crops, such as corn, peas, strawberries, and raspberries to grow and ripen.

Conclusion

During the conclusion, the writer summarizes and restates the points made in the essay. Transition words and phrases such as in conclusion, finally, as aresult, and after all signal the reader that the argument is coming to a close. In conclusion, people prefer a particular season of the year because that season contains their preferred weather and activities.
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Tue, 12 Jul 2016 03:10:44 +0000http://schooltutoring.com/help/?p=9039Overview
Metalloids are elements that have some properties of metals and some of nonmetals. They are on the periodic table along the dividing line between metals and nonmetals. The most commonly recognized metalloids include the elements boron (B), silicon (Si), germanium (Ge), arsenic (As), antimony (Sb), and tellurium (Te).

Metalloids

Elements are commonly classified as either metals, nonmetals, or metalloids. Most metalloids are brittle (a non-metallic property), act as semiconductors of electricity, and have a metallic luster (also a metallic property). They are solid at room temperature. In chemical reactions, they often act more like nonmetals, but they form alloys like metals. Whether elements are classified as metalloids or not depend upon the chemist’s decision. For example, polonium (Po), and astatine (At) are sometimes included in the list of metalloids, because of their chemical properties and their location on the periodic table.

Semiconductor Properties

Metalloids are good semiconductors, which mean that they are between the electrical conductivity of metals and materials used for insulation. Semiconductors can conduct electricity under some conditions, so electrical current can be controlled. Semiconductor chips, transistors, and other electronic parts form integrated circuits for everything from computers to cell phones. The metalloids, especially silicon, boron, germanium, and compounds of arsenic and antimony, are natural semiconductors. Silicon and germanium revolutionized the electronics and computer industries.

Alloys

The metalloids are often too brittle to be used as pure substances, but form many useful alloys. For example, boron is used in alloys with steel and with nickel for welding components. Germanium is alloyed with silver to make tarnish-resistant sterling silver. Pewter is an alloy of tin and antimony.

Other Uses of Metalloids

Many compounds of metalloids are highly toxic, such as those containing arsenic and antimony. However, other compounds can be used as disinfectants and antiviral agents. Compounds of boron are used as catalysts in many chemical reactions. Many compounds are used to form glassware, especially in chemical and industrial uses, such as optical fibers. Silicon and boron compounds are also used in fireworks, as they are less toxic than some other compounds.

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]]>9039Science Review of the Juno Mission
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Sun, 10 Jul 2016 00:34:53 +0000http://schooltutoring.com/help/?p=9034Overview
The Juno probe entered orbit around the planet Jupiter on July 4, 2016. Jupiter is the largest planet in the Solar System, and the Juno probe will study the magnetic fields of the planet, as well as clues to its origin and composition.

The Juno Probe

The Juno probe launched in August 2011. Its three giant solar panels extend to 9 meters, about the size of a basketball court. They are needed to power the spacecraft, since Jupiter is about five times further from the Sun than Earth and gets about 25 times less sunlight. All the scientific instruments are within a thick vault to protect them from Jupiter’s intense magnetic field.

The Planet Jupiter

Jupiter is the largest planet in the solar system. It is so large that within the Solar System, everything in the solar system except for the sun could fit inside it. It has an ocean of liquid hydrogen rather than water. Although the planet is so huge, its day is only about 10 hours long, as it rotates very quickly. Bands of clouds and spots are formed from ammonia. It has around 60 moons, including the four large moons first discovered by Galileo – Io, Europa, Ganymede, and Callisto. Ganymede is the largest moon in the Solar System, even larger than Mercury and Pluto.

The Juno Mission

The Juno probe will orbit Jupiter in a highly elliptical orbit around its poles. The probe itself spins, while the scientific instruments are fixed, so that the most area can be covered by each instrument. Detailed measurements will be made of Jupiter’s strong magnetic field, the clouds, and what lies beneath them. Study of its gravity will provide clues to its structure, as will study of its chemical composition. The spacecraft will orbit Jupiter until February 2018, when it will burn up in its atmosphere.

Mission Goals

Some of the goals for the Juno mission include how the planet was formed, if it has a solid core, and how its magnetic field was generated. If Jupiter has a solid, rocky core, it would have formed later in the history of the solar system than if its core is not solid. The amount of water and other elements also contain clues to its formation. Scientists theorize that Jupiter was the first planet to form, so its composition is closer to the early solar system than the other planets. The Juno mission is the first mission to see beneath the clouds of the planet.

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Mon, 04 Jul 2016 18:27:05 +0000http://schooltutoring.com/help/?p=9029Overview
One of the ways to solve systems of equations is by graphing the equations on the same coordinate plane. By graphing the equations, it is possible to tell whether they have no solutions in common, one solution in common, or an infinite number of solutions in common.

No Solutions in Common

These linear equations are also known as parallel lines, those with the same slope and different y-intercepts. Another way to describe them is that the solutions of that particular system are inconsistent. For example, suppose that the equations are y = 3x – 1 and 2y = 6x +4. Solving the second equation, y = 3x +2. Both equations have the same slope, but their y-intercepts are different, so they are parallel.

One Solution in Common

Some of these linear equations are perpendicular lines, where the product of their slopes is equal to -1, but lines can also meet at other angles and still have one solution in common. A system of equations that has at least one solution in common is consistent. Both equations have one point in common, although it is the only solution of the system.

Identifying Solutions

One way to identify if a particular point is a solution of both equations in a system is to see if its coordinates solve both equations. For example, check to see if a point with the coordinates (1, 2) is a solution of the system y= x +1 and 2x +y = 4. The point is a solution of the equation y = x +1, because 2 = 1 +1, and it is also a solution of the equation 2x +y = 4, because 2 +2 = 4. It is a solution of that system of equations. A point with the coordinates (5, 6) is a solution of the equation y=x +1, but is it a solution of the equation 2x +y = 4? 10 +6 is equal to 16, which is not equal to 4. The point (5, 6) is not a solution of that system of equations.

All Solutions in Common

Some systems of equations have all solutions in common, so that any solution of one equation is also a solution of the other equation. The lines coincide along the same graph. They are both consistent and dependent. Suppose the system of equations is x +y =9 and 3x +3y =27. The simplest form of 3x +3y = 27 is x +y =9, just by dividing every member of the equation 3x +3y =27 by 3.
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Mon, 04 Jul 2016 02:33:10 +0000http://schooltutoring.com/help/?p=9025Overview
Parallel lines never intersect when they are graphed on the same plane, while perpendicular lines are lines that intersect at one point at right angles to each other. Their linear equations have special relationships.

Parallel Lines

Parallel lines are lines in the same plane that have no points in common. Suppose that one line has the equation y = 2x. In slope-intercept form, its slope would be 2 and the y-intercept would be 0. Suppose that another line in the same plane has the equation y = 2x +4. In that case, its slope is still 2 but the y-intercept is 4. Those lines would have no points in common, because there isn’t any point that would be a solution of both equations. Therefore the lines would not intersect, and they are parallel.

Solving Equations for Parallel Lines

In the example of y=2x and y=2x +4, both lines have the same slope, 2, and the y-intercepts are different. Both equations are already solved for y. Given pairs of equations, they can both be put in slope-intercept form and solved for y to determine the slope and the y-intercept. If the slopes of the lines are equal and the y-intercepts are not the same, the lines are parallel. Suppose the equations for the lines are y = -3x +4 and 6x +2y = -10. Are those lines parallel? The slope of the line y = -3x +4 is already -3 and the y intercept is +4. Solving the second equation for y takes place in 2 steps, because 2y = -6x -10, moving the 6x, so y equals (-6/2) x – (10/2), or -3x -5. The slope of both lines is -3 but the y-intercepts are different, so they are parallel.

Perpendicular Lines

Perpendicular lines are lines that are in the same plane that intersect at one point, forming a 90° angle (a right angle). Slopes that have a product of -1 are perpendicular. Suppose a line has the equation y = 2x -3 and another line has the equation y = ( -1/2) x -4. The product of the slopes, 2(-1/2) is -1, so they are perpendicular.

Solving Equations for Perpendicular Lines

In order to determine of two equations are for perpendicular lines, solve for y and determine the product of the slopes. Suppose the equations are 3y = 9x +3 and 6y +2x =6 are perpendicular. Solving for y, 3y=9x +3 can be simplified to y = 3x +1 by dividing both sides by 3. Solving for y, 6y = -2x +6, or y = (-1/3) x +1. The product of the slopes, 3 (-1/3) = -1.

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